# Properties

 Label 819.2.o.d Level $819$ Weight $2$ Character orbit 819.o Analytic conductor $6.540$ 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] = [819,2,Mod(568,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("819.568");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.o (of order $$3$$, degree $$2$$, 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: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.771147.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 5x^{4} + 6x^{3} + 15x^{2} + 4x + 1$$ x^6 - x^5 + 5*x^4 + 6*x^3 + 15*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{2} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{4} + (2 \beta_{3} + \beta_{2} - 1) q^{5} + (\beta_{4} + 1) q^{7} + (\beta_{3} - \beta_{2} + 1) q^{8}+O(q^{10})$$ q + (b5 + b4 - b3 - b2 + 1) * q^2 + (-b5 - 2*b4 + b2 - b1 - 2) * q^4 + (2*b3 + b2 - 1) * q^5 + (b4 + 1) * q^7 + (b3 - b2 + 1) * q^8 $$q + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{2} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{4} + (2 \beta_{3} + \beta_{2} - 1) q^{5} + (\beta_{4} + 1) q^{7} + (\beta_{3} - \beta_{2} + 1) q^{8} + (2 \beta_{5} + 6 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 2) q^{10} + (3 \beta_{4} + \beta_1) q^{11} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{13} + ( - \beta_{3} - \beta_{2}) q^{14} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{16} + ( - \beta_{5} + \beta_{4} + 1) q^{17} + (3 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 3) q^{19} + ( - 5 \beta_{5} - 7 \beta_{4} + 3 \beta_{2} - 3 \beta_1 - 7) q^{20} + ( - 4 \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{22} + (\beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + (3 \beta_{3} - \beta_{2} + 3) q^{25} + (3 \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 3 \beta_1 + 7) q^{26} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{28} + (3 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{29} + (4 \beta_{3} + \beta_{2} - 4) q^{31} + (2 \beta_{5} - \beta_{2} + \beta_1) q^{32} + ( - \beta_{3} - 2 \beta_{2} + 3) q^{34} + (2 \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 + 1) q^{35} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{37} + ( - 5 \beta_{3} - 3 \beta_{2} + 4) q^{38} + (6 \beta_{3} + 4 \beta_{2} + 1) q^{40} + ( - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4) q^{41} + (\beta_{5} + 7 \beta_{4} - \beta_{2} + \beta_1 + 7) q^{43} + (3 \beta_{3} + 4) q^{44} + (4 \beta_{5} - \beta_{4} - 1) q^{46} + (2 \beta_{3} - 3 \beta_{2} + 6) q^{47} + \beta_{4} q^{49} + (10 \beta_{5} + 11 \beta_{4} - 10 \beta_{3} - 10 \beta_{2} + 7 \beta_1 + 10) q^{50} + (2 \beta_{5} + \beta_{4} - 5 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{52} + ( - 2 \beta_{3} - 2 \beta_{2} - 3) q^{53} + (7 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} - 7 \beta_{2} + 2 \beta_1 + 7) q^{55} + (\beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 2) q^{56} + ( - 6 \beta_{5} - 11 \beta_{4} + 5 \beta_{2} - 5 \beta_1 - 11) q^{58} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{59} + ( - 2 \beta_{5} - 5 \beta_{4} - 5) q^{61} + (3 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 7 \beta_1 + 3) q^{62} + ( - 5 \beta_{3} + 2) q^{64} + ( - 3 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} - 4 \beta_1 - 9) q^{65} + (3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{67} + (\beta_{5} - \beta_{3} - \beta_{2} + 1) q^{68} + ( - 2 \beta_{3} + \beta_{2} - 4) q^{70} + ( - \beta_{5} + 4 \beta_{4} - 5 \beta_{2} + 5 \beta_1 + 4) q^{71} + (3 \beta_{3} - 2 \beta_{2} - 1) q^{73} + ( - 2 \beta_{5} - 6 \beta_{4} + 3 \beta_{2} - 3 \beta_1 - 6) q^{74} + ( - 3 \beta_{5} - 8 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{76} + (\beta_{2} - 3) q^{77} + (2 \beta_{3} + 9 \beta_{2} - 4) q^{79} + ( - \beta_{5} + 9 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{80} + (2 \beta_{5} + 14 \beta_{4} - 4 \beta_{2} + 4 \beta_1 + 14) q^{82} + ( - 6 \beta_{3} + \beta_{2} + 1) q^{83} + (2 \beta_{5} - 4 \beta_{4} + \beta_{2} - \beta_1 - 4) q^{85} + ( - 8 \beta_{3} - 6 \beta_{2} - 1) q^{86} + (2 \beta_{5} + 9 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 2) q^{88} + ( - 6 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} - 4 \beta_1 - 6) q^{89} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 - 1) q^{91} + ( - \beta_{3} + \beta_{2} - 4) q^{92} + (13 \beta_{5} + 9 \beta_{4} - 13 \beta_{3} - 13 \beta_{2} + 7 \beta_1 + 13) q^{94} + (8 \beta_{5} + 3 \beta_{4} - \beta_{2} + \beta_1 + 3) q^{95} + (2 \beta_{5} - 8 \beta_{4} - \beta_{2} + \beta_1 - 8) q^{97} + ( - \beta_{5} - \beta_{4} - 1) q^{98}+O(q^{100})$$ q + (b5 + b4 - b3 - b2 + 1) * q^2 + (-b5 - 2*b4 + b2 - b1 - 2) * q^4 + (2*b3 + b2 - 1) * q^5 + (b4 + 1) * q^7 + (b3 - b2 + 1) * q^8 + (2*b5 + 6*b4 - 2*b3 - 2*b2 + 3*b1 + 2) * q^10 + (3*b4 + b1) * q^11 + (-2*b5 - b4 + b3 - b1 - 1) * q^13 + (-b3 - b2) * q^14 + (2*b5 - b4 - 2*b3 - 2*b2 + b1 + 2) * q^16 + (-b5 + b4 + 1) * q^17 + (3*b4 - 2*b2 + 2*b1 + 3) * q^19 + (-5*b5 - 7*b4 + 3*b2 - 3*b1 - 7) * q^20 + (-4*b5 - 2*b4 + b2 - b1 - 2) * q^22 + (b5 - 3*b4 - b3 - b2 - b1 + 1) * q^23 + (3*b3 - b2 + 3) * q^25 + (3*b5 + 2*b4 - b3 - 4*b2 + 3*b1 + 7) * q^26 + (-b5 - 2*b4 + b3 + b2 - b1 - 1) * q^28 + (3*b5 + 4*b4 - 3*b3 - 3*b2 + 2*b1 + 3) * q^29 + (4*b3 + b2 - 4) * q^31 + (2*b5 - b2 + b1) * q^32 + (-b3 - 2*b2 + 3) * q^34 + (2*b5 + b4 - b2 + b1 + 1) * q^35 + (2*b5 + b4 - 2*b3 - 2*b2 + b1 + 2) * q^37 + (-5*b3 - 3*b2 + 4) * q^38 + (6*b3 + 4*b2 + 1) * q^40 + (-4*b5 - 2*b4 + 4*b3 + 4*b2 - 4) * q^41 + (b5 + 7*b4 - b2 + b1 + 7) * q^43 + (3*b3 + 4) * q^44 + (4*b5 - b4 - 1) * q^46 + (2*b3 - 3*b2 + 6) * q^47 + b4 * q^49 + (10*b5 + 11*b4 - 10*b3 - 10*b2 + 7*b1 + 10) * q^50 + (2*b5 + b4 - 5*b3 - 3*b2 + b1 - 1) * q^52 + (-2*b3 - 2*b2 - 3) * q^53 + (7*b5 + 3*b4 - 7*b3 - 7*b2 + 2*b1 + 7) * q^55 + (b5 + 2*b4 - 2*b2 + 2*b1 + 2) * q^56 + (-6*b5 - 11*b4 + 5*b2 - 5*b1 - 11) * q^58 + (-b5 - b4 - b2 + b1 - 1) * q^59 + (-2*b5 - 5*b4 - 5) * q^61 + (3*b5 + 9*b4 - 3*b3 - 3*b2 + 7*b1 + 3) * q^62 + (-5*b3 + 2) * q^64 + (-3*b5 - 11*b4 + 4*b3 + 7*b2 - 4*b1 - 9) * q^65 + (3*b5 + 2*b4 - 3*b3 - 3*b2 - 2*b1 + 3) * q^67 + (b5 - b3 - b2 + 1) * q^68 + (-2*b3 + b2 - 4) * q^70 + (-b5 + 4*b4 - 5*b2 + 5*b1 + 4) * q^71 + (3*b3 - 2*b2 - 1) * q^73 + (-2*b5 - 6*b4 + 3*b2 - 3*b1 - 6) * q^74 + (-3*b5 - 8*b4 + 3*b3 + 3*b2 - 3*b1 - 3) * q^76 + (b2 - 3) * q^77 + (2*b3 + 9*b2 - 4) * q^79 + (-b5 + 9*b4 + b3 + b2 + 2*b1 - 1) * q^80 + (2*b5 + 14*b4 - 4*b2 + 4*b1 + 14) * q^82 + (-6*b3 + b2 + 1) * q^83 + (2*b5 - 4*b4 + b2 - b1 - 4) * q^85 + (-8*b3 - 6*b2 - 1) * q^86 + (2*b5 + 9*b4 - 2*b3 - 2*b2 + 4*b1 + 2) * q^88 + (-6*b5 + 3*b4 + 6*b3 + 6*b2 - 4*b1 - 6) * q^89 + (-b5 + 2*b3 + b1 - 1) * q^91 + (-b3 + b2 - 4) * q^92 + (13*b5 + 9*b4 - 13*b3 - 13*b2 + 7*b1 + 13) * q^94 + (8*b5 + 3*b4 - b2 + b1 + 3) * q^95 + (2*b5 - 8*b4 - b2 + b1 - 8) * q^97 + (-b5 - b4 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} - 4 q^{4} + 3 q^{7} + 6 q^{8}+O(q^{10})$$ 6 * q - 2 * q^2 - 4 * q^4 + 3 * q^7 + 6 * q^8 $$6 q - 2 q^{2} - 4 q^{4} + 3 q^{7} + 6 q^{8} - 13 q^{10} - 8 q^{11} - 4 q^{14} + 6 q^{16} + 4 q^{17} + 7 q^{19} - 13 q^{20} - q^{22} + 9 q^{23} + 22 q^{25} + 26 q^{26} + 4 q^{28} - 7 q^{29} - 14 q^{31} - 3 q^{32} + 12 q^{34} + 8 q^{38} + 26 q^{40} + 2 q^{41} + 19 q^{43} + 30 q^{44} - 7 q^{46} + 34 q^{47} - 3 q^{49} - 16 q^{50} - 26 q^{52} - 26 q^{53} + 3 q^{56} - 22 q^{58} - 3 q^{59} - 13 q^{61} - 17 q^{62} + 2 q^{64} - 5 q^{67} + q^{68} - 26 q^{70} + 8 q^{71} - 4 q^{73} - 13 q^{74} + 18 q^{76} - 16 q^{77} - 2 q^{79} - 26 q^{80} + 36 q^{82} - 4 q^{83} - 13 q^{85} - 34 q^{86} - 21 q^{88} - 19 q^{89} - 24 q^{92} - 7 q^{94} - 27 q^{97} - 2 q^{98}+O(q^{100})$$ 6 * q - 2 * q^2 - 4 * q^4 + 3 * q^7 + 6 * q^8 - 13 * q^10 - 8 * q^11 - 4 * q^14 + 6 * q^16 + 4 * q^17 + 7 * q^19 - 13 * q^20 - q^22 + 9 * q^23 + 22 * q^25 + 26 * q^26 + 4 * q^28 - 7 * q^29 - 14 * q^31 - 3 * q^32 + 12 * q^34 + 8 * q^38 + 26 * q^40 + 2 * q^41 + 19 * q^43 + 30 * q^44 - 7 * q^46 + 34 * q^47 - 3 * q^49 - 16 * q^50 - 26 * q^52 - 26 * q^53 + 3 * q^56 - 22 * q^58 - 3 * q^59 - 13 * q^61 - 17 * q^62 + 2 * q^64 - 5 * q^67 + q^68 - 26 * q^70 + 8 * q^71 - 4 * q^73 - 13 * q^74 + 18 * q^76 - 16 * q^77 - 2 * q^79 - 26 * q^80 + 36 * q^82 - 4 * q^83 - 13 * q^85 - 34 * q^86 - 21 * q^88 - 19 * q^89 - 24 * q^92 - 7 * q^94 - 27 * q^97 - 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5x^{4} + 6x^{3} + 15x^{2} + 4x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 5\nu^{4} - 25\nu^{3} + 15\nu^{2} + 4\nu - 20 ) / 79$$ (-v^5 + 5*v^4 - 25*v^3 + 15*v^2 + 4*v - 20) / 79 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} + 25\nu^{4} - 46\nu^{3} + 75\nu^{2} + 20\nu + 216 ) / 79$$ (-5*v^5 + 25*v^4 - 46*v^3 + 75*v^2 + 20*v + 216) / 79 $$\beta_{4}$$ $$=$$ $$( 20\nu^{5} - 21\nu^{4} + 105\nu^{3} + 95\nu^{2} + 315\nu + 5 ) / 79$$ (20*v^5 - 21*v^4 + 105*v^3 + 95*v^2 + 315*v + 5) / 79 $$\beta_{5}$$ $$=$$ $$( 38\nu^{5} - 32\nu^{4} + 160\nu^{3} + 299\nu^{2} + 480\nu + 128 ) / 79$$ (38*v^5 - 32*v^4 + 160*v^3 + 299*v^2 + 480*v + 128) / 79
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2$$ b5 - 2*b4 - 2*b2 + 2*b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 5\beta_{2} - 4$$ b3 - 5*b2 - 4 $$\nu^{4}$$ $$=$$ $$-5\beta_{5} + 11\beta_{4} + 5\beta_{3} + 5\beta_{2} - 15\beta _1 - 5$$ -5*b5 + 11*b4 + 5*b3 + 5*b2 - 15*b1 - 5 $$\nu^{5}$$ $$=$$ $$-10\beta_{5} + 25\beta_{4} + 41\beta_{2} - 41\beta _1 + 25$$ -10*b5 + 25*b4 + 41*b2 - 41*b1 + 25

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
568.1
 −0.136945 + 0.237196i 1.32555 − 2.29591i −0.688601 + 1.19269i −0.136945 − 0.237196i 1.32555 + 2.29591i −0.688601 − 1.19269i
−1.18860 + 2.05872i 0 −1.82555 3.16194i 4.02830 0 0.500000 + 0.866025i 3.92498 0 −4.78804 + 8.29313i
568.2 −0.636945 + 1.10322i 0 0.188601 + 0.326667i −1.10331 0 0.500000 + 0.866025i −3.02830 0 0.702750 1.21720i
568.3 0.825547 1.42989i 0 −0.363055 0.628829i −2.92498 0 0.500000 + 0.866025i 2.10331 0 −2.41471 + 4.18240i
757.1 −1.18860 2.05872i 0 −1.82555 + 3.16194i 4.02830 0 0.500000 0.866025i 3.92498 0 −4.78804 8.29313i
757.2 −0.636945 1.10322i 0 0.188601 0.326667i −1.10331 0 0.500000 0.866025i −3.02830 0 0.702750 + 1.21720i
757.3 0.825547 + 1.42989i 0 −0.363055 + 0.628829i −2.92498 0 0.500000 0.866025i 2.10331 0 −2.41471 4.18240i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 568.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.o.d 6
3.b odd 2 1 273.2.k.d 6
13.c even 3 1 inner 819.2.o.d 6
39.h odd 6 1 3549.2.a.s 3
39.i odd 6 1 273.2.k.d 6
39.i odd 6 1 3549.2.a.h 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.d 6 3.b odd 2 1
273.2.k.d 6 39.i odd 6 1
819.2.o.d 6 1.a even 1 1 trivial
819.2.o.d 6 13.c even 3 1 inner
3549.2.a.h 3 39.i odd 6 1
3549.2.a.s 3 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}^{6} + 2T_{2}^{5} + 7T_{2}^{4} + 4T_{2}^{3} + 19T_{2}^{2} + 15T_{2} + 25$$ T2^6 + 2*T2^5 + 7*T2^4 + 4*T2^3 + 19*T2^2 + 15*T2 + 25 $$T_{11}^{6} + 8T_{11}^{5} + 47T_{11}^{4} + 126T_{11}^{3} + 249T_{11}^{2} + 85T_{11} + 25$$ T11^6 + 8*T11^5 + 47*T11^4 + 126*T11^3 + 249*T11^2 + 85*T11 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 2 T^{5} + 7 T^{4} + 4 T^{3} + \cdots + 25$$
$3$ $$T^{6}$$
$5$ $$(T^{3} - 13 T - 13)^{2}$$
$7$ $$(T^{2} - T + 1)^{3}$$
$11$ $$T^{6} + 8 T^{5} + 47 T^{4} + 126 T^{3} + \cdots + 25$$
$13$ $$T^{6} + 65T^{3} + 2197$$
$17$ $$T^{6} - 4 T^{5} + 15 T^{4} - 6 T^{3} + \cdots + 1$$
$19$ $$T^{6} - 7 T^{5} + 50 T^{4} + \cdots + 2209$$
$23$ $$T^{6} - 9 T^{5} + 67 T^{4} - 124 T^{3} + \cdots + 1$$
$29$ $$T^{6} + 7 T^{5} + 63 T^{4} - 108 T^{3} + \cdots + 25$$
$31$ $$(T^{3} + 7 T^{2} - 40 T - 281)^{2}$$
$37$ $$T^{6} + 13 T^{4} - 26 T^{3} + \cdots + 169$$
$41$ $$T^{6} - 2 T^{5} + 72 T^{4} + \cdots + 40000$$
$43$ $$T^{6} - 19 T^{5} + 245 T^{4} + \cdots + 52441$$
$47$ $$(T^{3} - 17 T^{2} + 14 T + 547)^{2}$$
$53$ $$(T^{3} + 13 T^{2} + 39 T - 13)^{2}$$
$59$ $$T^{6} + 3 T^{5} + 19 T^{4} + 20 T^{3} + \cdots + 625$$
$61$ $$T^{6} + 13 T^{5} + 130 T^{4} + \cdots + 169$$
$67$ $$T^{6} + 5 T^{5} + 99 T^{4} + \cdots + 156025$$
$71$ $$T^{6} - 8 T^{5} + 177 T^{4} + \cdots + 265225$$
$73$ $$(T^{3} + 2 T^{2} - 81 T + 73)^{2}$$
$79$ $$(T^{3} + T^{2} - 290 T - 337)^{2}$$
$83$ $$(T^{3} + 2 T^{2} - 185 T + 229)^{2}$$
$89$ $$T^{6} + 19 T^{5} + 362 T^{4} + \cdots + 1038361$$
$97$ $$T^{6} + 27 T^{5} + 499 T^{4} + \cdots + 358801$$