# Properties

 Label 378.2.h.c Level $378$ Weight $2$ Character orbit 378.h Analytic conductor $3.018$ 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] = [378,2,Mod(289,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("378.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.h (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: $$3.01834519640$$ 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: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - 1) q^{2} - \beta_{4} q^{4} - \beta_{3} q^{5} + (\beta_{5} + \beta_{2} - \beta_1 - 1) q^{7} + q^{8}+O(q^{10})$$ q + (b4 - 1) * q^2 - b4 * q^4 - b3 * q^5 + (b5 + b2 - b1 - 1) * q^7 + q^8 $$q + (\beta_{4} - 1) q^{2} - \beta_{4} q^{4} - \beta_{3} q^{5} + (\beta_{5} + \beta_{2} - \beta_1 - 1) q^{7} + q^{8} - \beta_{2} q^{10} + \beta_{3} q^{11} + (2 \beta_{5} - 3 \beta_{4} + \beta_{2} + 3) q^{13} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{14} + (\beta_{4} - 1) q^{16} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{17} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{19} + (\beta_{3} + \beta_{2}) q^{20} + \beta_{2} q^{22} + (\beta_1 - 2) q^{23} + ( - 2 \beta_{3} + \beta_1 - 1) q^{25} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{26} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{28} + (3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{29} + (\beta_{5} + 6 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{31} - \beta_{4} q^{32} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{34} + ( - 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{35} + \beta_{4} q^{37} + (3 \beta_1 + 2) q^{38} - \beta_{3} q^{40} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 1) q^{41} + ( - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{43} + ( - \beta_{3} - \beta_{2}) q^{44} + ( - \beta_{5} - 2 \beta_{4} + 2) q^{46} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} + 3) q^{47} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{49} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{50} + (\beta_{3} - 2 \beta_1 - 3) q^{52} + (\beta_{5} + 5 \beta_{4} + \beta_{2} - 5) q^{53} + (2 \beta_{3} - \beta_1 - 4) q^{55} + (\beta_{5} + \beta_{2} - \beta_1 - 1) q^{56} + (2 \beta_{3} + 3 \beta_1) q^{58} + (\beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{59} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} + 3) q^{61} + (\beta_{3} + \beta_1 - 6) q^{62} + q^{64} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{2} + 2) q^{65} + ( - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + \beta_1) q^{67} + ( - 2 \beta_1 - 2) q^{68} + ( - \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{70} + (2 \beta_{3} - 7 \beta_1 - 4) q^{71} + (\beta_{5} - 8 \beta_{4} - 4 \beta_{2} + 8) q^{73} - q^{74} + ( - 3 \beta_{5} + 2 \beta_{4} - 2) q^{76} + (3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{77} + ( - 4 \beta_{5} + \beta_{2}) q^{79} - \beta_{2} q^{80} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{82} + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{83} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2) q^{85} + (3 \beta_{3} - 3 \beta_1 + 2) q^{86} + \beta_{3} q^{88} + (4 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{89} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 8) q^{91} + (\beta_{5} + 2 \beta_{4} - \beta_1) q^{92} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{94} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{95} + (2 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{97} + ( - 3 \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{98}+O(q^{100})$$ q + (b4 - 1) * q^2 - b4 * q^4 - b3 * q^5 + (b5 + b2 - b1 - 1) * q^7 + q^8 - b2 * q^10 + b3 * q^11 + (2*b5 - 3*b4 + b2 + 3) * q^13 + (-b4 - b3 - b2 + b1 + 1) * q^14 + (b4 - 1) * q^16 + (2*b5 - 2*b4 + 2) * q^17 + (3*b5 - 2*b4 - 3*b1) * q^19 + (b3 + b2) * q^20 + b2 * q^22 + (b1 - 2) * q^23 + (-2*b3 + b1 - 1) * q^25 + (-2*b5 + 3*b4 - b3 - b2 + 2*b1) * q^26 + (-b5 + b4 + b3) * q^28 + (3*b5 - 2*b3 - 2*b2 - 3*b1) * q^29 + (b5 + 6*b4 - b3 - b2 - b1) * q^31 - b4 * q^32 + (-2*b5 + 2*b4 + 2*b1) * q^34 + (-3*b4 + b3 + 2*b2 + b1 + 4) * q^35 + b4 * q^37 + (3*b1 + 2) * q^38 - b3 * q^40 + (-b5 + b4 + 2*b2 - 1) * q^41 + (-3*b5 - 2*b4 - 3*b3 - 3*b2 + 3*b1) * q^43 + (-b3 - b2) * q^44 + (-b5 - 2*b4 + 2) * q^46 + (-3*b5 - 3*b4 - 3*b2 + 3) * q^47 + (-b4 + 2*b3 + b2 + 3*b1) * q^49 + (-b5 - b4 - 2*b2 + 1) * q^50 + (b3 - 2*b1 - 3) * q^52 + (b5 + 5*b4 + b2 - 5) * q^53 + (2*b3 - b1 - 4) * q^55 + (b5 + b2 - b1 - 1) * q^56 + (2*b3 + 3*b1) * q^58 + (b5 + 5*b4 + 2*b3 + 2*b2 - b1) * q^59 + (-b5 - 3*b4 - 2*b2 + 3) * q^61 + (b3 + b1 - 6) * q^62 + q^64 + (-b5 - 2*b4 + 5*b2 + 2) * q^65 + (-b5 + 2*b4 + 4*b3 + 4*b2 + b1) * q^67 + (-2*b1 - 2) * q^68 + (-b5 + 4*b4 - 2*b3 - b2 - 1) * q^70 + (2*b3 - 7*b1 - 4) * q^71 + (b5 - 8*b4 - 4*b2 + 8) * q^73 - q^74 + (-3*b5 + 2*b4 - 2) * q^76 + (3*b4 - b3 - 2*b2 - b1 - 4) * q^77 + (-4*b5 + b2) * q^79 - b2 * q^80 + (b5 - b4 - 2*b3 - 2*b2 - b1) * q^82 + (-2*b5 - b4 - 3*b3 - 3*b2 + 2*b1) * q^83 + (-2*b5 + 2*b4 + 2*b2 - 2) * q^85 + (3*b3 - 3*b1 + 2) * q^86 + b3 * q^88 + (4*b5 + 2*b4 + b3 + b2 - 4*b1) * q^89 + (-3*b5 + b4 + 3*b3 + 4*b2 + b1 - 8) * q^91 + (b5 + 2*b4 - b1) * q^92 + (3*b5 + 3*b4 + 3*b3 + 3*b2 - 3*b1) * q^94 + (-3*b5 + 3*b4 + 2*b3 + 2*b2 + 3*b1) * q^95 + (2*b5 + 8*b4 - 2*b3 - 2*b2 - 2*b1) * q^97 + (-3*b5 - b3 + b2 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} - 3 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10})$$ 6 * q - 3 * q^2 - 3 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8 $$6 q - 3 q^{2} - 3 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8} - q^{10} - 2 q^{11} + 8 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} - q^{20} + q^{22} - 14 q^{23} - 4 q^{25} + 8 q^{26} + 2 q^{28} + 5 q^{29} + 20 q^{31} - 3 q^{32} + 4 q^{34} + 13 q^{35} + 3 q^{37} + 6 q^{38} + 2 q^{40} - 6 q^{43} + q^{44} + 7 q^{46} + 9 q^{47} - 12 q^{49} + 2 q^{50} - 16 q^{52} - 15 q^{53} - 26 q^{55} - 4 q^{56} - 10 q^{58} + 14 q^{59} + 8 q^{61} - 40 q^{62} + 6 q^{64} + 12 q^{65} + q^{67} - 8 q^{68} + 10 q^{70} - 14 q^{71} + 19 q^{73} - 6 q^{74} - 3 q^{76} - 13 q^{77} + 5 q^{79} - q^{80} - 2 q^{83} - 2 q^{85} + 12 q^{86} - 2 q^{88} + 9 q^{89} - 46 q^{91} + 7 q^{92} + 9 q^{94} + 4 q^{95} + 28 q^{97} + 12 q^{98}+O(q^{100})$$ 6 * q - 3 * q^2 - 3 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8 - q^10 - 2 * q^11 + 8 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^19 - q^20 + q^22 - 14 * q^23 - 4 * q^25 + 8 * q^26 + 2 * q^28 + 5 * q^29 + 20 * q^31 - 3 * q^32 + 4 * q^34 + 13 * q^35 + 3 * q^37 + 6 * q^38 + 2 * q^40 - 6 * q^43 + q^44 + 7 * q^46 + 9 * q^47 - 12 * q^49 + 2 * q^50 - 16 * q^52 - 15 * q^53 - 26 * q^55 - 4 * q^56 - 10 * q^58 + 14 * q^59 + 8 * q^61 - 40 * q^62 + 6 * q^64 + 12 * q^65 + q^67 - 8 * q^68 + 10 * q^70 - 14 * q^71 + 19 * q^73 - 6 * q^74 - 3 * q^76 - 13 * q^77 + 5 * q^79 - q^80 - 2 * q^83 - 2 * q^85 + 12 * q^86 - 2 * q^88 + 9 * q^89 - 46 * q^91 + 7 * q^92 + 9 * q^94 + 4 * q^95 + 28 * q^97 + 12 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ v^2 - v + 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3$$ (-v^5 + v^4 - 8*v^3 + 5*v^2 - 18*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3$$ v^4 - 2*v^3 + 6*v^2 - 5*v + 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 9) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 30*v - 9) / 3
 $$\nu$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + 4*b1 - 4) / 3 $$\nu^{3}$$ $$=$$ $$( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3$$ (7*b5 + 5*b4 + 2*b3 + 4*b2 + b1 - 10) / 3 $$\nu^{4}$$ $$=$$ $$( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3$$ (16*b5 + 11*b4 + 8*b3 + 10*b2 - 17*b1 + 5) / 3 $$\nu^{5}$$ $$=$$ $$( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3$$ (-14*b5 - 16*b4 + 5*b3 - 5*b2 - 23*b1 + 47) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 + 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 − 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.58836 0 −2.64400 + 0.0963576i 1.00000 0 0.794182 + 1.37556i
289.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.593579 0 −0.0665372 2.64491i 1.00000 0 0.296790 + 0.514055i
289.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 3.18194 0 0.710533 + 2.54856i 1.00000 0 −1.59097 2.75564i
361.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.58836 0 −2.64400 0.0963576i 1.00000 0 0.794182 1.37556i
361.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.593579 0 −0.0665372 + 2.64491i 1.00000 0 0.296790 0.514055i
361.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 3.18194 0 0.710533 2.54856i 1.00000 0 −1.59097 + 2.75564i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.3 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 378.2.h.c 6
3.b odd 2 1 126.2.h.d yes 6
4.b odd 2 1 3024.2.t.h 6
7.b odd 2 1 2646.2.h.o 6
7.c even 3 1 378.2.e.d 6
7.c even 3 1 2646.2.f.l 6
7.d odd 6 1 2646.2.e.p 6
7.d odd 6 1 2646.2.f.m 6
9.c even 3 1 378.2.e.d 6
9.c even 3 1 1134.2.g.l 6
9.d odd 6 1 126.2.e.c 6
9.d odd 6 1 1134.2.g.m 6
12.b even 2 1 1008.2.t.h 6
21.c even 2 1 882.2.h.p 6
21.g even 6 1 882.2.e.o 6
21.g even 6 1 882.2.f.o 6
21.h odd 6 1 126.2.e.c 6
21.h odd 6 1 882.2.f.n 6
28.g odd 6 1 3024.2.q.g 6
36.f odd 6 1 3024.2.q.g 6
36.h even 6 1 1008.2.q.g 6
63.g even 3 1 inner 378.2.h.c 6
63.g even 3 1 7938.2.a.ca 3
63.h even 3 1 1134.2.g.l 6
63.h even 3 1 2646.2.f.l 6
63.i even 6 1 882.2.f.o 6
63.j odd 6 1 882.2.f.n 6
63.j odd 6 1 1134.2.g.m 6
63.k odd 6 1 2646.2.h.o 6
63.k odd 6 1 7938.2.a.bz 3
63.l odd 6 1 2646.2.e.p 6
63.n odd 6 1 126.2.h.d yes 6
63.n odd 6 1 7938.2.a.bv 3
63.o even 6 1 882.2.e.o 6
63.s even 6 1 882.2.h.p 6
63.s even 6 1 7938.2.a.bw 3
63.t odd 6 1 2646.2.f.m 6
84.n even 6 1 1008.2.q.g 6
252.o even 6 1 1008.2.t.h 6
252.bl odd 6 1 3024.2.t.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 9.d odd 6 1
126.2.e.c 6 21.h odd 6 1
126.2.h.d yes 6 3.b odd 2 1
126.2.h.d yes 6 63.n odd 6 1
378.2.e.d 6 7.c even 3 1
378.2.e.d 6 9.c even 3 1
378.2.h.c 6 1.a even 1 1 trivial
378.2.h.c 6 63.g even 3 1 inner
882.2.e.o 6 21.g even 6 1
882.2.e.o 6 63.o even 6 1
882.2.f.n 6 21.h odd 6 1
882.2.f.n 6 63.j odd 6 1
882.2.f.o 6 21.g even 6 1
882.2.f.o 6 63.i even 6 1
882.2.h.p 6 21.c even 2 1
882.2.h.p 6 63.s even 6 1
1008.2.q.g 6 36.h even 6 1
1008.2.q.g 6 84.n even 6 1
1008.2.t.h 6 12.b even 2 1
1008.2.t.h 6 252.o even 6 1
1134.2.g.l 6 9.c even 3 1
1134.2.g.l 6 63.h even 3 1
1134.2.g.m 6 9.d odd 6 1
1134.2.g.m 6 63.j odd 6 1
2646.2.e.p 6 7.d odd 6 1
2646.2.e.p 6 63.l odd 6 1
2646.2.f.l 6 7.c even 3 1
2646.2.f.l 6 63.h even 3 1
2646.2.f.m 6 7.d odd 6 1
2646.2.f.m 6 63.t odd 6 1
2646.2.h.o 6 7.b odd 2 1
2646.2.h.o 6 63.k odd 6 1
3024.2.q.g 6 28.g odd 6 1
3024.2.q.g 6 36.f odd 6 1
3024.2.t.h 6 4.b odd 2 1
3024.2.t.h 6 252.bl odd 6 1
7938.2.a.bv 3 63.n odd 6 1
7938.2.a.bw 3 63.s even 6 1
7938.2.a.bz 3 63.k odd 6 1
7938.2.a.ca 3 63.g even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - T_{5}^{2} - 6T_{5} - 3$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$(T^{3} - T^{2} - 6 T - 3)^{2}$$
$7$ $$T^{6} + 4 T^{5} + 14 T^{4} + 55 T^{3} + \cdots + 343$$
$11$ $$(T^{3} + T^{2} - 6 T + 3)^{2}$$
$13$ $$T^{6} - 8 T^{5} + 63 T^{4} + \cdots + 4761$$
$17$ $$T^{6} - 4 T^{5} + 28 T^{4} + 240 T^{2} + \cdots + 576$$
$19$ $$T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 2401$$
$23$ $$(T^{3} + 7 T^{2} + 12 T + 3)^{2}$$
$29$ $$T^{6} - 5 T^{5} + 91 T^{4} + \cdots + 131769$$
$31$ $$T^{6} - 20 T^{5} + 279 T^{4} + \cdots + 40401$$
$37$ $$(T^{2} - T + 1)^{3}$$
$41$ $$T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81$$
$43$ $$T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 16129$$
$47$ $$T^{6} - 9 T^{5} + 135 T^{4} + \cdots + 35721$$
$53$ $$T^{6} + 15 T^{5} + 159 T^{4} + \cdots + 6561$$
$59$ $$T^{6} - 14 T^{5} + 157 T^{4} + \cdots + 3969$$
$61$ $$T^{6} - 8 T^{5} + 69 T^{4} + \cdots + 8649$$
$67$ $$T^{6} - T^{5} + 113 T^{4} + \cdots + 44521$$
$71$ $$(T^{3} + 7 T^{2} - 198 T - 1593)^{2}$$
$73$ $$T^{6} - 19 T^{5} + 353 T^{4} + \cdots + 398161$$
$79$ $$T^{6} - 5 T^{5} + 99 T^{4} + \cdots + 103041$$
$83$ $$T^{6} + 2 T^{5} + 67 T^{4} + \cdots + 21609$$
$89$ $$T^{6} - 9 T^{5} + 123 T^{4} + 396 T^{3} + \cdots + 81$$
$97$ $$T^{6} - 28 T^{5} + 572 T^{4} + \cdots + 61504$$