# Properties

 Label 1008.2.t.g Level $1008$ Weight $2$ Character orbit 1008.t Analytic conductor $8.049$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(193,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.t (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - 1) q^{3} + (\beta_{3} + \beta_{2} + 2) q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10})$$ q + (b5 - 1) * q^3 + (b3 + b2 + 2) * q^5 + (b5 - b4 + b3) * q^7 + (-b4 - 2*b3 - b2 - 2) * q^9 $$q + (\beta_{5} - 1) q^{3} + (\beta_{3} + \beta_{2} + 2) q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9} + (2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{13} + (3 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{15} + (4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2) q^{17} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{19} + (2 \beta_{5} + \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 3) q^{21} + ( - \beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{23} + (\beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_1 + 1) q^{25} + ( - 2 \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_1 + 2) q^{27} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 1) q^{29} + ( - \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{31} + ( - \beta_{5} - 7 \beta_{4} + 4 \beta_{3} + \beta_{2} - 4 \beta_1) q^{33} + (4 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{35} + ( - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{37} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{39} + (5 \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{41} + ( - 3 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 5) q^{43} + (\beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 7) q^{45} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{47} + (5 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{49} + ( - 8 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 10) q^{51} + ( - 3 \beta_{5} - \beta_{4} - \beta_{3} + 5 \beta_1 - 1) q^{53} + (3 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 4) q^{55} + ( - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1) q^{57} + ( - 4 \beta_{5} + 5 \beta_{3} + 4 \beta_{2} + 5 \beta_1) q^{59} + (2 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} - 8 \beta_1 + 3) q^{61} + ( - 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{63} + ( - 3 \beta_{5} + 4 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{65} + (4 \beta_{5} - 3 \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1 - 3) q^{67} + ( - 4 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{69} + (3 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 4) q^{71} + ( - 2 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} - 4 \beta_1 + 3) q^{73} + (5 \beta_{5} - 7 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 2) q^{75} + ( - 2 \beta_{5} - 6 \beta_{4} + \beta_{3} + 5 \beta_{2} + \beta_1 - 9) q^{77} + (7 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + \beta_1 - 4) q^{79} + ( - \beta_{5} + 8 \beta_{4} - 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 6) q^{81} + (\beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{83} + (10 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 2 \beta_1 - 6) q^{85} + (3 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{87} + (3 \beta_{5} - 4 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 4) q^{89} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{91} + (2 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + 5 \beta_{2} + 4 \beta_1 - 4) q^{93} + ( - 2 \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{95} + (2 \beta_{5} - 10 \beta_{4} - 2 \beta_{2} - 10) q^{97} + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 7 \beta_{2} + 7) q^{99}+O(q^{100})$$ q + (b5 - 1) * q^3 + (b3 + b2 + 2) * q^5 + (b5 - b4 + b3) * q^7 + (-b4 - 2*b3 - b2 - 2) * q^9 + (2*b4 - 3*b3 + b2 + 2*b1 - 2) * q^11 + (-b5 + b4 + b1) * q^13 + (3*b5 - b4 - 2*b3 - b2 - 2) * q^15 + (4*b5 + 2*b4 - 2*b3 - 2) * q^17 + (b4 - b3 - b1 + 1) * q^19 + (2*b5 + b4 - 4*b3 - 2*b2 - 3) * q^21 + (-b4 - 2*b2 - b1 - 2) * q^23 + (b4 + 2*b3 + 4*b2 + b1 + 1) * q^25 + (-2*b5 - b4 + 4*b3 - 2*b1 + 2) * q^27 + (-2*b5 - b4 + 3*b3 + 2*b2 + 3*b1 - 1) * q^29 + (-b5 + 5*b4 + 2*b3 + b2 + 2*b1 + 5) * q^31 + (-b5 - 7*b4 + 4*b3 + b2 - 4*b1) * q^33 + (4*b5 - 2*b4 - b3 - b2 - b1 + 1) * q^35 + (-3*b4 + 2*b3 + 2*b1 - 3) * q^37 + (-b5 + 2*b4 + b3 + 2*b2 + 2) * q^39 + (5*b4 - b3 + 2*b1 - 1) * q^41 + (-3*b5 - 5*b4 + 2*b3 + 3*b2 + 2*b1 - 5) * q^43 + (b5 - 4*b4 - 2*b3 - 3*b2 - 2*b1 - 7) * q^45 + (-3*b5 + b4 + b3 + b1 + 1) * q^47 + (5*b5 + 2*b4 - 4*b3 - 4*b2 - 2*b1 - 1) * q^49 + (-8*b4 - 4*b3 - 2*b2 - 10) * q^51 + (-3*b5 - b4 - b3 + 5*b1 - 1) * q^53 + (3*b4 - 2*b3 + 4*b2 + 3*b1 - 4) * q^55 + (-b5 - 3*b4 + 3*b3 + b2 + b1) * q^57 + (-4*b5 + 5*b3 + 4*b2 + 5*b1) * q^59 + (2*b5 - 5*b4 + 3*b3 - 8*b1 + 3) * q^61 + (-4*b5 - 4*b4 + 4*b3 + b2 - b1 - 3) * q^63 + (-3*b5 + 4*b4 + b3 + b1 + 1) * q^65 + (4*b5 - 3*b4 - b3 - 4*b2 - b1 - 3) * q^67 + (-4*b5 + 5*b4 + b3 + b2 + 2*b1 + 3) * q^69 + (3*b4 - 2*b3 + 4*b2 + 3*b1 - 4) * q^71 + (-2*b5 + 6*b4 + 3*b3 - 4*b1 + 3) * q^73 + (5*b5 - 7*b4 - 5*b3 - 3*b2 - 2*b1 - 2) * q^75 + (-2*b5 - 6*b4 + b3 + 5*b2 + b1 - 9) * q^77 + (7*b5 + 4*b4 - 4*b3 + b1 - 4) * q^79 + (-b5 + 8*b4 - 2*b3 + b2 + 5*b1 + 6) * q^81 + (b5 - 3*b4 + b3 - b2 + b1 - 3) * q^83 + (10*b5 - 2*b4 - 6*b3 + 2*b1 - 6) * q^85 + (3*b5 + 5*b4 - 5*b3 - b2 - 3*b1 + 4) * q^87 + (3*b5 - 4*b4 - b3 - 3*b2 - b1 - 4) * q^89 + (-2*b5 + 4*b4 + 2*b3 + 3*b2 + b1 + 2) * q^91 + (2*b5 + 4*b4 - 4*b3 + 5*b2 + 4*b1 - 4) * q^93 + (-2*b5 + b3 + 2*b2 + b1) * q^95 + (2*b5 - 10*b4 - 2*b2 - 10) * q^97 + (3*b5 + 2*b4 - 2*b3 - 7*b2 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} + 10 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10})$$ 6 * q - 4 * q^3 + 10 * q^5 + 2 * q^7 - 4 * q^9 $$6 q - 4 q^{3} + 10 q^{5} + 2 q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} + 3 q^{19} - 7 q^{21} - 14 q^{23} + 4 q^{25} - 7 q^{27} - 5 q^{29} + 14 q^{31} - 4 q^{33} + 19 q^{35} - 9 q^{37} + 3 q^{39} - 12 q^{41} - 18 q^{43} - 31 q^{45} - 3 q^{47} - 26 q^{51} + 9 q^{53} - 14 q^{55} + 2 q^{57} - 4 q^{59} + 4 q^{61} - 28 q^{63} - 12 q^{65} - 5 q^{67} - q^{69} - 14 q^{71} - 25 q^{73} + 25 q^{75} - 35 q^{77} - 7 q^{79} + 32 q^{81} - 8 q^{83} + 14 q^{85} + 20 q^{87} - 9 q^{89} - 4 q^{91} - 3 q^{93} - 2 q^{95} - 28 q^{97} + 41 q^{99}+O(q^{100})$$ 6 * q - 4 * q^3 + 10 * q^5 + 2 * q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^17 + 3 * q^19 - 7 * q^21 - 14 * q^23 + 4 * q^25 - 7 * q^27 - 5 * q^29 + 14 * q^31 - 4 * q^33 + 19 * q^35 - 9 * q^37 + 3 * q^39 - 12 * q^41 - 18 * q^43 - 31 * q^45 - 3 * q^47 - 26 * q^51 + 9 * q^53 - 14 * q^55 + 2 * q^57 - 4 * q^59 + 4 * q^61 - 28 * q^63 - 12 * q^65 - 5 * q^67 - q^69 - 14 * q^71 - 25 * q^73 + 25 * q^75 - 35 * q^77 - 7 * q^79 + 32 * q^81 - 8 * q^83 + 14 * q^85 + 20 * q^87 - 9 * q^89 - 4 * q^91 - 3 * q^93 - 2 * q^95 - 28 * q^97 + 41 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3$$ (v^5 - v^4 + 5*v^3 + v^2 + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3$$ (-v^5 + v^4 - 5*v^3 + 2*v^2 - 3*v) / 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 6) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 33*v - 9) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 - 2$$ b3 + b2 + b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1$$ b5 + b4 + b3 + b2 - 3*b1 - 1 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6$$ 2*b5 + 3*b4 - 5*b3 - 3*b2 - 6*b1 + 6 $$\nu^{5}$$ $$=$$ $$-3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7$$ -3*b5 - 2*b4 - 11*b3 - 6*b2 + 8*b1 + 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{4}$$ $$1$$ $$\beta_{4}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i
0 −1.73025 + 0.0789082i 0 −0.460505 0 −2.25729 1.38008i 0 2.98755 0.273062i 0
193.2 0 −0.619562 + 1.61745i 0 1.76088 0 1.85185 + 1.88962i 0 −2.23229 2.00422i 0
193.3 0 0.349814 1.69636i 0 3.69963 0 1.40545 2.24159i 0 −2.75526 1.18682i 0
961.1 0 −1.73025 0.0789082i 0 −0.460505 0 −2.25729 + 1.38008i 0 2.98755 + 0.273062i 0
961.2 0 −0.619562 1.61745i 0 1.76088 0 1.85185 1.88962i 0 −2.23229 + 2.00422i 0
961.3 0 0.349814 + 1.69636i 0 3.69963 0 1.40545 + 2.24159i 0 −2.75526 + 1.18682i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.t.g 6
3.b odd 2 1 3024.2.t.g 6
4.b odd 2 1 126.2.h.c yes 6
7.c even 3 1 1008.2.q.h 6
9.c even 3 1 1008.2.q.h 6
9.d odd 6 1 3024.2.q.h 6
12.b even 2 1 378.2.h.d 6
21.h odd 6 1 3024.2.q.h 6
28.d even 2 1 882.2.h.o 6
28.f even 6 1 882.2.e.p 6
28.f even 6 1 882.2.f.m 6
28.g odd 6 1 126.2.e.d 6
28.g odd 6 1 882.2.f.l 6
36.f odd 6 1 126.2.e.d 6
36.f odd 6 1 1134.2.g.k 6
36.h even 6 1 378.2.e.c 6
36.h even 6 1 1134.2.g.n 6
63.g even 3 1 inner 1008.2.t.g 6
63.n odd 6 1 3024.2.t.g 6
84.h odd 2 1 2646.2.h.p 6
84.j odd 6 1 2646.2.e.o 6
84.j odd 6 1 2646.2.f.n 6
84.n even 6 1 378.2.e.c 6
84.n even 6 1 2646.2.f.o 6
252.n even 6 1 882.2.h.o 6
252.n even 6 1 7938.2.a.by 3
252.o even 6 1 378.2.h.d 6
252.o even 6 1 7938.2.a.bu 3
252.r odd 6 1 2646.2.f.n 6
252.s odd 6 1 2646.2.e.o 6
252.u odd 6 1 882.2.f.l 6
252.u odd 6 1 1134.2.g.k 6
252.bb even 6 1 1134.2.g.n 6
252.bb even 6 1 2646.2.f.o 6
252.bi even 6 1 882.2.e.p 6
252.bj even 6 1 882.2.f.m 6
252.bl odd 6 1 126.2.h.c yes 6
252.bl odd 6 1 7938.2.a.cb 3
252.bn odd 6 1 2646.2.h.p 6
252.bn odd 6 1 7938.2.a.bx 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 28.g odd 6 1
126.2.e.d 6 36.f odd 6 1
126.2.h.c yes 6 4.b odd 2 1
126.2.h.c yes 6 252.bl odd 6 1
378.2.e.c 6 36.h even 6 1
378.2.e.c 6 84.n even 6 1
378.2.h.d 6 12.b even 2 1
378.2.h.d 6 252.o even 6 1
882.2.e.p 6 28.f even 6 1
882.2.e.p 6 252.bi even 6 1
882.2.f.l 6 28.g odd 6 1
882.2.f.l 6 252.u odd 6 1
882.2.f.m 6 28.f even 6 1
882.2.f.m 6 252.bj even 6 1
882.2.h.o 6 28.d even 2 1
882.2.h.o 6 252.n even 6 1
1008.2.q.h 6 7.c even 3 1
1008.2.q.h 6 9.c even 3 1
1008.2.t.g 6 1.a even 1 1 trivial
1008.2.t.g 6 63.g even 3 1 inner
1134.2.g.k 6 36.f odd 6 1
1134.2.g.k 6 252.u odd 6 1
1134.2.g.n 6 36.h even 6 1
1134.2.g.n 6 252.bb even 6 1
2646.2.e.o 6 84.j odd 6 1
2646.2.e.o 6 252.s odd 6 1
2646.2.f.n 6 84.j odd 6 1
2646.2.f.n 6 252.r odd 6 1
2646.2.f.o 6 84.n even 6 1
2646.2.f.o 6 252.bb even 6 1
2646.2.h.p 6 84.h odd 2 1
2646.2.h.p 6 252.bn odd 6 1
3024.2.q.h 6 9.d odd 6 1
3024.2.q.h 6 21.h odd 6 1
3024.2.t.g 6 3.b odd 2 1
3024.2.t.g 6 63.n odd 6 1
7938.2.a.bu 3 252.o even 6 1
7938.2.a.bx 3 252.bn odd 6 1
7938.2.a.by 3 252.n even 6 1
7938.2.a.cb 3 252.bl odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{3} - 5T_{5}^{2} + 4T_{5} + 3$$ T5^3 - 5*T5^2 + 4*T5 + 3 $$T_{11}^{3} + T_{11}^{2} - 26T_{11} + 33$$ T11^3 + T11^2 - 26*T11 + 33

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 4 T^{5} + 10 T^{4} + 21 T^{3} + \cdots + 27$$
$5$ $$(T^{3} - 5 T^{2} + 4 T + 3)^{2}$$
$7$ $$T^{6} - 2 T^{5} + 2 T^{4} + 19 T^{3} + \cdots + 343$$
$11$ $$(T^{3} + T^{2} - 26 T + 33)^{2}$$
$13$ $$T^{6} + 2 T^{5} + 7 T^{4} + 15 T^{2} + \cdots + 9$$
$17$ $$T^{6} + 4 T^{5} + 60 T^{4} + \cdots + 28224$$
$19$ $$T^{6} - 3 T^{5} + 15 T^{4} + 4 T^{3} + \cdots + 49$$
$23$ $$(T^{3} + 7 T^{2} + 4 T - 3)^{2}$$
$29$ $$T^{6} + 5 T^{5} + 57 T^{4} + \cdots + 1089$$
$31$ $$T^{6} - 14 T^{5} + 151 T^{4} + \cdots + 729$$
$37$ $$T^{6} + 9 T^{5} + 90 T^{4} + \cdots + 5329$$
$41$ $$T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 729$$
$43$ $$T^{6} + 18 T^{5} + 243 T^{4} + 1456 T^{3} + \cdots + 1$$
$47$ $$T^{6} + 3 T^{5} + 33 T^{4} - 126 T^{3} + \cdots + 729$$
$53$ $$T^{6} - 9 T^{5} + 123 T^{4} + 396 T^{3} + \cdots + 81$$
$59$ $$T^{6} + 4 T^{5} + 117 T^{4} + \cdots + 31329$$
$61$ $$T^{6} - 4 T^{5} + 151 T^{4} + \cdots + 514089$$
$67$ $$T^{6} + 5 T^{5} + 83 T^{4} + \cdots + 22201$$
$71$ $$(T^{3} + 7 T^{2} - 50 T - 99)^{2}$$
$73$ $$T^{6} + 25 T^{5} + 473 T^{4} + \cdots + 2401$$
$79$ $$T^{6} + 7 T^{5} + 193 T^{4} + \cdots + 594441$$
$83$ $$T^{6} + 8 T^{5} + 69 T^{4} + \cdots + 8649$$
$89$ $$T^{6} + 9 T^{5} + 87 T^{4} + \cdots + 3969$$
$97$ $$T^{6} + 28 T^{5} + 548 T^{4} + \cdots + 287296$$