# Properties

 Label 405.4.e.s Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ 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] = [405,4,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (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: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.148347072.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 29x^{4} + 223x^{2} + 243$$ x^6 + 29*x^4 + 223*x^2 + 243 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{2} + ( - \beta_{5} - 2 \beta_{2} - \beta_1 - 2) q^{4} + (5 \beta_{2} + 5) q^{5} + (3 \beta_{4} + 5 \beta_{3} - 10 \beta_{2} - 5 \beta_1) q^{7} + (\beta_{5} - \beta_{4} + 3 \beta_{3} - 8) q^{8}+O(q^{10})$$ q + (b3 - b1) * q^2 + (-b5 - 2*b2 - b1 - 2) * q^4 + (5*b2 + 5) * q^5 + (3*b4 + 5*b3 - 10*b2 - 5*b1) * q^7 + (b5 - b4 + 3*b3 - 8) * q^8 $$q + (\beta_{3} - \beta_1) q^{2} + ( - \beta_{5} - 2 \beta_{2} - \beta_1 - 2) q^{4} + (5 \beta_{2} + 5) q^{5} + (3 \beta_{4} + 5 \beta_{3} - 10 \beta_{2} - 5 \beta_1) q^{7} + (\beta_{5} - \beta_{4} + 3 \beta_{3} - 8) q^{8} + 5 \beta_{3} q^{10} + (5 \beta_{4} + 7 \beta_{3} + 17 \beta_{2} - 7 \beta_1) q^{11} + (13 \beta_{5} + 20 \beta_{2} + 13 \beta_1 + 20) q^{13} + ( - 11 \beta_{5} - 56 \beta_{2} - 9 \beta_1 - 56) q^{14} + ( - 9 \beta_{4} + 5 \beta_{3} - 44 \beta_{2} - 5 \beta_1) q^{16} + (8 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 14) q^{17} + (14 \beta_{5} - 14 \beta_{4} - 10 \beta_{3} - 5) q^{19} + ( - 5 \beta_{4} + 5 \beta_{3} - 10 \beta_{2} - 5 \beta_1) q^{20} + ( - 17 \beta_{5} - 80 \beta_{2} + 20 \beta_1 - 80) q^{22} + (19 \beta_{5} + 22 \beta_{2} + 15 \beta_1 + 22) q^{23} + 25 \beta_{2} q^{25} + ( - 13 \beta_{5} + 13 \beta_{4} + 59 \beta_{3} + 104) q^{26} + ( - 11 \beta_{5} + 11 \beta_{4} - 47 \beta_{3} + 12) q^{28} + ( - 13 \beta_{4} + 65 \beta_{3} + 95 \beta_{2} - 65 \beta_1) q^{29} + (\beta_{5} - 211 \beta_{2} + 5 \beta_1 - 211) q^{31} + (21 \beta_{5} - 96 \beta_{2} - 43 \beta_1 - 96) q^{32} + (8 \beta_{4} + 38 \beta_{3} - 64 \beta_{2} - 38 \beta_1) q^{34} + ( - 15 \beta_{5} + 15 \beta_{4} + 25 \beta_{3} + 50) q^{35} + ( - 2 \beta_{5} + 2 \beta_{4} - 54 \beta_{3} - 156) q^{37} + (38 \beta_{4} + 13 \beta_{3} + 128 \beta_{2} - 13 \beta_1) q^{38} + (5 \beta_{5} - 40 \beta_{2} + 15 \beta_1 - 40) q^{40} + ( - 62 \beta_{5} - 59 \beta_{2} + 2 \beta_1 - 59) q^{41} + ( - 8 \beta_{4} + 28 \beta_{3} - 288 \beta_{2} - 28 \beta_1) q^{43} + (14 \beta_{5} - 14 \beta_{4} - 38 \beta_{3} + 98) q^{44} + ( - 23 \beta_{5} + 23 \beta_{4} + 75 \beta_{3} + 112) q^{46} + ( - 31 \beta_{4} + 67 \beta_{3} + 56 \beta_{2} - 67 \beta_1) q^{47} + ( - 7 \beta_{5} - 301 \beta_{2} - 11 \beta_1 - 301) q^{49} + 25 \beta_1 q^{50} + (19 \beta_{4} + 33 \beta_{3} - 456 \beta_{2} - 33 \beta_1) q^{52} + (21 \beta_{5} - 21 \beta_{4} + 53 \beta_{3} + 186) q^{53} + ( - 25 \beta_{5} + 25 \beta_{4} + 35 \beta_{3} - 85) q^{55} + ( - 63 \beta_{4} + 15 \beta_{3} - 15 \beta_1) q^{56} + ( - 39 \beta_{5} - 624 \beta_{2} + 4 \beta_1 - 624) q^{58} + (90 \beta_{5} - 159 \beta_{2} + 58 \beta_1 - 159) q^{59} + (58 \beta_{4} - 122 \beta_{3} + 6 \beta_{2} + 122 \beta_1) q^{61} + (3 \beta_{5} - 3 \beta_{4} - 204 \beta_{3} + 48) q^{62} + ( - 13 \beta_{5} + 13 \beta_{4} - 57 \beta_{3} - 120) q^{64} + (65 \beta_{4} - 65 \beta_{3} + 100 \beta_{2} + 65 \beta_1) q^{65} + (18 \beta_{5} - 38 \beta_{2} - 154 \beta_1 - 38) q^{67} + (10 \beta_{5} - 284 \beta_{2} - 22 \beta_1 - 284) q^{68} + ( - 55 \beta_{4} + 45 \beta_{3} - 280 \beta_{2} - 45 \beta_1) q^{70} + ( - 69 \beta_{5} + 69 \beta_{4} + 79 \beta_{3} + 177) q^{71} + ( - 12 \beta_{5} + 12 \beta_{4} + 32 \beta_{3} - 226) q^{73} + (50 \beta_{4} - 214 \beta_{3} + 536 \beta_{2} + 214 \beta_1) q^{74} + (23 \beta_{5} - 246 \beta_{2} + 111 \beta_1 - 246) q^{76} + ( - 98 \beta_{5} - 662 \beta_{2} + 162 \beta_1 - 662) q^{77} + ( - 54 \beta_{4} - 82 \beta_{3} - 184 \beta_{2} + 82 \beta_1) q^{79} + (45 \beta_{5} - 45 \beta_{4} + 25 \beta_{3} + 220) q^{80} + (126 \beta_{5} - 126 \beta_{4} - 181 \beta_{3} + 144) q^{82} + ( - 30 \beta_{4} - 210 \beta_{3} + 648 \beta_{2} + 210 \beta_1) q^{83} + (40 \beta_{5} + 70 \beta_{2} + 40 \beta_1 + 70) q^{85} + ( - 12 \beta_{5} - 264 \beta_{2} - 332 \beta_1 - 264) q^{86} + ( - 70 \beta_{4} - 72 \beta_{3} - 232 \beta_{2} + 72 \beta_1) q^{88} + ( - 3 \beta_{5} + 3 \beta_{4} - 39 \beta_{3} - 297) q^{89} + (161 \beta_{5} - 161 \beta_{4} + 581 \beta_{3} - 216) q^{91} + (31 \beta_{4} + 21 \beta_{3} - 620 \beta_{2} - 21 \beta_1) q^{92} + ( - 5 \beta_{5} - 608 \beta_{2} - 73 \beta_1 - 608) q^{94} + (70 \beta_{5} - 25 \beta_{2} - 50 \beta_1 - 25) q^{95} + (32 \beta_{4} - 336 \beta_{3} + 112 \beta_{2} + 336 \beta_1) q^{97} + (3 \beta_{5} - 3 \beta_{4} - 326 \beta_{3} - 96) q^{98}+O(q^{100})$$ q + (b3 - b1) * q^2 + (-b5 - 2*b2 - b1 - 2) * q^4 + (5*b2 + 5) * q^5 + (3*b4 + 5*b3 - 10*b2 - 5*b1) * q^7 + (b5 - b4 + 3*b3 - 8) * q^8 + 5*b3 * q^10 + (5*b4 + 7*b3 + 17*b2 - 7*b1) * q^11 + (13*b5 + 20*b2 + 13*b1 + 20) * q^13 + (-11*b5 - 56*b2 - 9*b1 - 56) * q^14 + (-9*b4 + 5*b3 - 44*b2 - 5*b1) * q^16 + (8*b5 - 8*b4 + 8*b3 + 14) * q^17 + (14*b5 - 14*b4 - 10*b3 - 5) * q^19 + (-5*b4 + 5*b3 - 10*b2 - 5*b1) * q^20 + (-17*b5 - 80*b2 + 20*b1 - 80) * q^22 + (19*b5 + 22*b2 + 15*b1 + 22) * q^23 + 25*b2 * q^25 + (-13*b5 + 13*b4 + 59*b3 + 104) * q^26 + (-11*b5 + 11*b4 - 47*b3 + 12) * q^28 + (-13*b4 + 65*b3 + 95*b2 - 65*b1) * q^29 + (b5 - 211*b2 + 5*b1 - 211) * q^31 + (21*b5 - 96*b2 - 43*b1 - 96) * q^32 + (8*b4 + 38*b3 - 64*b2 - 38*b1) * q^34 + (-15*b5 + 15*b4 + 25*b3 + 50) * q^35 + (-2*b5 + 2*b4 - 54*b3 - 156) * q^37 + (38*b4 + 13*b3 + 128*b2 - 13*b1) * q^38 + (5*b5 - 40*b2 + 15*b1 - 40) * q^40 + (-62*b5 - 59*b2 + 2*b1 - 59) * q^41 + (-8*b4 + 28*b3 - 288*b2 - 28*b1) * q^43 + (14*b5 - 14*b4 - 38*b3 + 98) * q^44 + (-23*b5 + 23*b4 + 75*b3 + 112) * q^46 + (-31*b4 + 67*b3 + 56*b2 - 67*b1) * q^47 + (-7*b5 - 301*b2 - 11*b1 - 301) * q^49 + 25*b1 * q^50 + (19*b4 + 33*b3 - 456*b2 - 33*b1) * q^52 + (21*b5 - 21*b4 + 53*b3 + 186) * q^53 + (-25*b5 + 25*b4 + 35*b3 - 85) * q^55 + (-63*b4 + 15*b3 - 15*b1) * q^56 + (-39*b5 - 624*b2 + 4*b1 - 624) * q^58 + (90*b5 - 159*b2 + 58*b1 - 159) * q^59 + (58*b4 - 122*b3 + 6*b2 + 122*b1) * q^61 + (3*b5 - 3*b4 - 204*b3 + 48) * q^62 + (-13*b5 + 13*b4 - 57*b3 - 120) * q^64 + (65*b4 - 65*b3 + 100*b2 + 65*b1) * q^65 + (18*b5 - 38*b2 - 154*b1 - 38) * q^67 + (10*b5 - 284*b2 - 22*b1 - 284) * q^68 + (-55*b4 + 45*b3 - 280*b2 - 45*b1) * q^70 + (-69*b5 + 69*b4 + 79*b3 + 177) * q^71 + (-12*b5 + 12*b4 + 32*b3 - 226) * q^73 + (50*b4 - 214*b3 + 536*b2 + 214*b1) * q^74 + (23*b5 - 246*b2 + 111*b1 - 246) * q^76 + (-98*b5 - 662*b2 + 162*b1 - 662) * q^77 + (-54*b4 - 82*b3 - 184*b2 + 82*b1) * q^79 + (45*b5 - 45*b4 + 25*b3 + 220) * q^80 + (126*b5 - 126*b4 - 181*b3 + 144) * q^82 + (-30*b4 - 210*b3 + 648*b2 + 210*b1) * q^83 + (40*b5 + 70*b2 + 40*b1 + 70) * q^85 + (-12*b5 - 264*b2 - 332*b1 - 264) * q^86 + (-70*b4 - 72*b3 - 232*b2 + 72*b1) * q^88 + (-3*b5 + 3*b4 - 39*b3 - 297) * q^89 + (161*b5 - 161*b4 + 581*b3 - 216) * q^91 + (31*b4 + 21*b3 - 620*b2 - 21*b1) * q^92 + (-5*b5 - 608*b2 - 73*b1 - 608) * q^94 + (70*b5 - 25*b2 - 50*b1 - 25) * q^95 + (32*b4 - 336*b3 + 112*b2 + 336*b1) * q^97 + (3*b5 - 3*b4 - 326*b3 - 96) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8}+O(q^{10})$$ 6 * q - q^2 - 5 * q^4 + 15 * q^5 + 25 * q^7 - 54 * q^8 $$6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8} - 10 q^{10} - 58 q^{11} + 47 q^{13} - 159 q^{14} + 127 q^{16} + 68 q^{17} - 10 q^{19} + 25 q^{20} - 260 q^{22} + 51 q^{23} - 75 q^{25} + 506 q^{26} + 166 q^{28} - 350 q^{29} - 638 q^{31} - 245 q^{32} + 154 q^{34} + 250 q^{35} - 828 q^{37} - 397 q^{38} - 135 q^{40} - 179 q^{41} + 836 q^{43} + 664 q^{44} + 522 q^{46} - 235 q^{47} - 892 q^{49} - 25 q^{50} + 1335 q^{52} + 1010 q^{53} - 580 q^{55} - 15 q^{56} - 1876 q^{58} - 535 q^{59} + 104 q^{61} + 696 q^{62} - 606 q^{64} - 235 q^{65} + 40 q^{67} - 830 q^{68} + 795 q^{70} + 904 q^{71} - 1420 q^{73} - 1394 q^{74} - 849 q^{76} - 2148 q^{77} + 634 q^{79} + 1270 q^{80} + 1226 q^{82} - 1734 q^{83} + 170 q^{85} - 460 q^{86} + 768 q^{88} - 1704 q^{89} - 2458 q^{91} + 1839 q^{92} - 1751 q^{94} - 25 q^{95} + 76 q^{98}+O(q^{100})$$ 6 * q - q^2 - 5 * q^4 + 15 * q^5 + 25 * q^7 - 54 * q^8 - 10 * q^10 - 58 * q^11 + 47 * q^13 - 159 * q^14 + 127 * q^16 + 68 * q^17 - 10 * q^19 + 25 * q^20 - 260 * q^22 + 51 * q^23 - 75 * q^25 + 506 * q^26 + 166 * q^28 - 350 * q^29 - 638 * q^31 - 245 * q^32 + 154 * q^34 + 250 * q^35 - 828 * q^37 - 397 * q^38 - 135 * q^40 - 179 * q^41 + 836 * q^43 + 664 * q^44 + 522 * q^46 - 235 * q^47 - 892 * q^49 - 25 * q^50 + 1335 * q^52 + 1010 * q^53 - 580 * q^55 - 15 * q^56 - 1876 * q^58 - 535 * q^59 + 104 * q^61 + 696 * q^62 - 606 * q^64 - 235 * q^65 + 40 * q^67 - 830 * q^68 + 795 * q^70 + 904 * q^71 - 1420 * q^73 - 1394 * q^74 - 849 * q^76 - 2148 * q^77 + 634 * q^79 + 1270 * q^80 + 1226 * q^82 - 1734 * q^83 + 170 * q^85 - 460 * q^86 + 768 * q^88 - 1704 * q^89 - 2458 * q^91 + 1839 * q^92 - 1751 * q^94 - 25 * q^95 + 76 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 29x^{4} + 223x^{2} + 243$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu^{2} + 13\nu + 9 ) / 4$$ (v^3 + v^2 + 13*v + 9) / 4 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 20\nu^{3} + 79\nu - 36 ) / 72$$ (v^5 + 20*v^3 + 79*v - 36) / 72 $$\beta_{3}$$ $$=$$ $$( \nu^{2} + 9 ) / 2$$ (v^2 + 9) / 2 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} - 9\nu^{4} - 20\nu^{3} - 144\nu^{2} + 29\nu - 207 ) / 72$$ (-v^5 - 9*v^4 - 20*v^3 - 144*v^2 + 29*v - 207) / 72 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 9\nu^{4} - 20\nu^{3} + 144\nu^{2} + 29\nu + 207 ) / 72$$ (-v^5 + 9*v^4 - 20*v^3 + 144*v^2 + 29*v + 207) / 72
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{2} + 1 ) / 3$$ (b5 + b4 + 2*b2 + 1) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_{3} - 9$$ 2*b3 - 9 $$\nu^{3}$$ $$=$$ $$( -13\beta_{5} - 13\beta_{4} - 6\beta_{3} - 26\beta_{2} + 12\beta _1 - 13 ) / 3$$ (-13*b5 - 13*b4 - 6*b3 - 26*b2 + 12*b1 - 13) / 3 $$\nu^{4}$$ $$=$$ $$4\beta_{5} - 4\beta_{4} - 32\beta_{3} + 121$$ 4*b5 - 4*b4 - 32*b3 + 121 $$\nu^{5}$$ $$=$$ $$( 181\beta_{5} + 181\beta_{4} + 120\beta_{3} + 578\beta_{2} - 240\beta _1 + 289 ) / 3$$ (181*b5 + 181*b4 + 120*b3 + 578*b2 - 240*b1 + 289) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 4.00586i − 3.41374i 1.13993i − 4.00586i 3.41374i − 1.13993i
−1.76174 + 3.05142i 0 −2.20744 3.82340i 2.50000 + 4.33013i 0 −12.7162 + 22.0252i −12.6321 0 −17.6174
136.2 −0.663404 + 1.14905i 0 3.11979 + 5.40363i 2.50000 + 4.33013i 0 12.0521 20.8749i −18.8932 0 −6.63404
136.3 1.92514 3.33444i 0 −3.41235 5.91036i 2.50000 + 4.33013i 0 13.1641 22.8009i 4.52526 0 19.2514
271.1 −1.76174 3.05142i 0 −2.20744 + 3.82340i 2.50000 4.33013i 0 −12.7162 22.0252i −12.6321 0 −17.6174
271.2 −0.663404 1.14905i 0 3.11979 5.40363i 2.50000 4.33013i 0 12.0521 + 20.8749i −18.8932 0 −6.63404
271.3 1.92514 + 3.33444i 0 −3.41235 + 5.91036i 2.50000 4.33013i 0 13.1641 + 22.8009i 4.52526 0 19.2514
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.e.s 6
3.b odd 2 1 405.4.e.u 6
9.c even 3 1 405.4.a.i yes 3
9.c even 3 1 inner 405.4.e.s 6
9.d odd 6 1 405.4.a.g 3
9.d odd 6 1 405.4.e.u 6
45.h odd 6 1 2025.4.a.r 3
45.j even 6 1 2025.4.a.p 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.g 3 9.d odd 6 1
405.4.a.i yes 3 9.c even 3 1
405.4.e.s 6 1.a even 1 1 trivial
405.4.e.s 6 9.c even 3 1 inner
405.4.e.u 6 3.b odd 2 1
405.4.e.u 6 9.d odd 6 1
2025.4.a.p 3 45.j even 6 1
2025.4.a.r 3 45.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 $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{6} + T_{2}^{5} + 15T_{2}^{4} + 22T_{2}^{3} + 214T_{2}^{2} + 252T_{2} + 324$$ T2^6 + T2^5 + 15*T2^4 + 22*T2^3 + 214*T2^2 + 252*T2 + 324 $$T_{7}^{6} - 25T_{7}^{5} + 1273T_{7}^{4} - 16080T_{7}^{3} + 823404T_{7}^{2} - 10458720T_{7} + 260499600$$ T7^6 - 25*T7^5 + 1273*T7^4 - 16080*T7^3 + 823404*T7^2 - 10458720*T7 + 260499600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 15 T^{4} + 22 T^{3} + \cdots + 324$$
$3$ $$T^{6}$$
$5$ $$(T^{2} - 5 T + 25)^{3}$$
$7$ $$T^{6} - 25 T^{5} + \cdots + 260499600$$
$11$ $$T^{6} + 58 T^{5} + 4275 T^{4} + \cdots + 9000000$$
$13$ $$T^{6} - 47 T^{5} + \cdots + 137160603904$$
$17$ $$(T^{3} - 34 T^{2} - 2708 T + 90984)^{2}$$
$19$ $$(T^{3} + 5 T^{2} - 10777 T - 299645)^{2}$$
$23$ $$T^{6} - 51 T^{5} + \cdots + 1084005816336$$
$29$ $$T^{6} + \cdots + 126287249817600$$
$31$ $$T^{6} + 638 T^{5} + \cdots + 90993741996096$$
$37$ $$(T^{3} + 414 T^{2} + 16032 T - 577760)^{2}$$
$41$ $$T^{6} + \cdots + 316826721339129$$
$43$ $$T^{6} + \cdots + 349427949912064$$
$47$ $$T^{6} + 235 T^{5} + \cdots + 81096796901376$$
$53$ $$(T^{3} - 505 T^{2} + 35128 T + 1500684)^{2}$$
$59$ $$T^{6} + \cdots + 498077813969649$$
$61$ $$T^{6} + \cdots + 554264938580224$$
$67$ $$T^{6} + \cdots + 246999193899264$$
$71$ $$(T^{3} - 452 T^{2} - 264923 T + 116183454)^{2}$$
$73$ $$(T^{3} + 710 T^{2} + 144236 T + 8707528)^{2}$$
$79$ $$T^{6} - 634 T^{5} + \cdots + 25533374939136$$
$83$ $$T^{6} + 1734 T^{5} + \cdots + 49\!\cdots\!04$$
$89$ $$(T^{3} + 852 T^{2} + 220725 T + 17926434)^{2}$$
$97$ $$T^{6} + 1575168 T^{4} + \cdots + 49\!\cdots\!64$$