Numbers represented by a cubic form

EDIT, April 11, 2013: See answer at https://mathoverflow.net/questions/127160/numbers-integrally-represented-by-a-ternary-cubic-form/127295#127295

This is part 2 ( of 25 discriminants of class number exactly three) of primes represented integrally by a homogeneous cubic form I think two will be enough.

Given integers $a,b,c,$ and cubic form
$$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b – a b^2 + 3 a^2 c – a c^2 + b^2 c – b c^2 – 4 a b c $$
$$ f(a,b,c) =
\det \left( \begin{array}{ccc}
a & b & c \\
c & a + c & b + c \\
b + c & b + 2 c & a + b + 2 c
\end{array}
\right) . $$
what primes $p$ can be integrally represented as
$$ p = f(a,b,c)? $$

I think it is all primes $(p| 11) = -1 ,$ and all $p = u^2 + 11 v^2$ in integers, but not any $q = 3 u^2 + 2 u v + 4 v^2.$
Note that, if $-p$ is represented, so is $p.$

I also suspect that if prime $q = 3 u^2 + 2 u v + 4 v^2$ and $f(a,b,c) \equiv 0 \pmod q,$ then all three $a,b,c \equiv 0 \pmod q,$ and $f(a,b,c) \equiv 0 \pmod {q^3}.$

Note that if $f$ integrally represents both $m,n$ then it represents $mn.$ That is because $f(a,b,c) = \det(aI + b X + c X^2),$ where
$$ X = \left( \begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1
\end{array}
\right) $$ Then $X^3 = X^2 + X + I$ and $X^4 = 2 X^2 + 2 X + I.$

If all suspicions are correct, we can correctly describe all numbers integrally represented by this polynomial: positive or negative are unimportant, most prime factors are unimportant, all that matters is that every exponent of a prime factor $q = 3 u^2 + 2 u v + 4 v^2$ must be divisible by 3.

I should have done this last time: most of the class field part has already been done, by Hudson and Williams (1991), Theorem 1 and Table 1 on page 134. You get my version of the polynomial by negating their variable $x.$ Note the connection to the “tribonacci” numbers, with precisely the formalism I described at Fibonacci Sequence in $\mathbb Z_n$.

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           p           a           b           c 
................................................
           2           0           1           1
           7           0         -11           6
          11           0          -3           2
          13           0          -1           2
          17          -1           0           2
          19           1           2           4
          29           0          -7           4
          41           0           3           2
          43           0           4          -1
          47           0           5          -2
          53           0           1           4
          61           0          46         -25
          73           2         -36          19
          79           0           3           4
          83           0          24         -13
         101          -1          12          -6
         103           0          15          -8
         107           1          -9           5
         109           1           2           6
         127           1          -2           4
         131           1           7          -3
         139           1          -6           4
         149          -1           4           2
         151           0         -20          11
         163           0           5           2
         167          -1           1           5
         173           0           6          -1
         193           1         -52          28
         197           0           9          -4
         199          -1           5           1
         211           0         -12           7
         227          -2           0           5
         233           0         -16           9
         239           0          -6           5
         241           0          -4           5
         257           0          -1           6
         263           2           4           9
         269          -1           0           6
         271           2           8          -3
         277           1          -7           5
         281           0           2           7
         283          -1           2           6
         293          -1          -8           6
         307           2          -1           6
         311           0           5           6
         337          -2           5           2
         347           1           7           5
         349           0          19         -10
         359          -1           9          -3
         373           2           5          10
         397           1          -1           7
         401           0         -68          37
         409           3         -77          41
         419           0          -7           6
         421           0           7           2
         431           1         -14           8
         439           0           8          -1
         457           0           1           8
         461           0          -2           7
         479           1          -8           6
         491           0           7           4
         499           0          13          -6
         503          -1         -36          20
         523           0           9          -2
         541           2         -12           7
         547           1         -11           7
         557          -1          25         -13
         563          -2         -11           8
         569           0           8           1
         571           1          -3           7
         587           0         -29          16
         593           3         -25          13
         599          -1           0           8
         601           0           7           6
         607           0          11          -4
         613           0           4           9
         617           2          -1           8
         659           0           8           3
         673           0          -6           7
         677           0         -17          10
         683          -1           4           8
         701           2          13          -6
         733           1          10          -2
         739          -1          14          -6
         743          -2           1           8
         757           0          81         -44
         761          -1           8           2
         769           0         -25          14
         773          -1           7           5
         787           2           5          12
         809          -1         -10           8
         811          -4           0           7
         821          -1           3           9
         827           2          10           7
         853           0         -11           8
         857          -2           3           8
         863           0           9           2
         877          -2         -15          10
         883           0         -14           9
         887           2          -3           8
         907           0          -5           8
         911           0           8           7
         919           0          -2           9
         929           1           7          11
         937           3           8          14
         941           3          -1           9
         953          -1           6           8
         967           1          13          -5
         991           1         -35          19
         997          -3           7           3

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