# Properties

 Label 63.6.c.b Level $63$ Weight $6$ Character orbit 63.c Analytic conductor $10.104$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(62,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("63.62");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 63.c (of order $$2$$, degree $$1$$, 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: $$10.1041806482$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 456x^{6} - 612x^{5} + 66744x^{4} + 32004x^{3} + 3729464x^{2} + 2503804x + 66026577$$ x^8 - 4*x^7 + 456*x^6 - 612*x^5 + 66744*x^4 + 32004*x^3 + 3729464*x^2 + 2503804*x + 66026577 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_{2}) q^{2} - 2 \beta_1 q^{4} + \beta_{6} q^{5} + (\beta_{5} + 2 \beta_1 - 14) q^{7} + (4 \beta_{4} + 4 \beta_{2}) q^{8}+O(q^{10})$$ q + (b4 - b2) * q^2 - 2*b1 * q^4 + b6 * q^5 + (b5 + 2*b1 - 14) * q^7 + (4*b4 + 4*b2) * q^8 $$q + (\beta_{4} - \beta_{2}) q^{2} - 2 \beta_1 q^{4} + \beta_{6} q^{5} + (\beta_{5} + 2 \beta_1 - 14) q^{7} + (4 \beta_{4} + 4 \beta_{2}) q^{8} + (2 \beta_{5} + \beta_{3} - \beta_1) q^{10} + (4 \beta_{4} + 53 \beta_{2}) q^{11} + (4 \beta_{5} + \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{7} - 8 \beta_{6} + 21 \beta_{4} - 31 \beta_{2}) q^{14} + ( - 64 \beta_1 - 16) q^{16} + ( - 2 \beta_{7} + 13 \beta_{6}) q^{17} + (6 \beta_{5} - 6 \beta_{3} - 3 \beta_1) q^{19} + ( - 4 \beta_{7} - 4 \beta_{6}) q^{20} + (49 \beta_1 + 670) q^{22} + (158 \beta_{4} + 845 \beta_{2}) q^{23} + (20 \beta_1 + 1171) q^{25} + ( - 6 \beta_{7} - 52 \beta_{6}) q^{26} + ( - 14 \beta_{3} + 28 \beta_1 - 1260) q^{28} + (143 \beta_{4} - 850 \beta_{2}) q^{29} + (4 \beta_{5} + 24 \beta_{3} - 2 \beta_1) q^{31} + ( - 784 \beta_{4} + 1296 \beta_{2}) q^{32} + ( - 6 \beta_{5} + \beta_{3} + 3 \beta_1) q^{34} + (5 \beta_{7} - 9 \beta_{6} + 140 \beta_{4} - 2148 \beta_{2}) q^{35} + ( - 526 \beta_1 - 4288) q^{37} + (6 \beta_{7} + 72 \beta_{6}) q^{38} + ( - 8 \beta_{5} + 4 \beta_{3} + 4 \beta_1) q^{40} + (26 \beta_{7} + 111 \beta_{6}) q^{41} + (613 \beta_1 + 4052) q^{43} + (1484 \beta_{4} + 144 \beta_{2}) q^{44} + (687 \beta_1 + 8986) q^{46} + (38 \beta_{7} + 82 \beta_{6}) q^{47} + ( - 28 \beta_{5} + 35 \beta_{3} - 126 \beta_1 - 13265) q^{49} + (1451 \beta_{4} - 1531 \beta_{2}) q^{50} + ( - 72 \beta_{5} - 56 \beta_{3} + 36 \beta_1) q^{52} + ( - 6091 \beta_{4} + 326 \beta_{2}) q^{53} + ( - 106 \beta_{5} + 4 \beta_{3} + 53 \beta_1) q^{55} + ( - 4 \beta_{7} + 24 \beta_{6} - 196 \beta_{4} - 236 \beta_{2}) q^{56} + ( - 993 \beta_1 - 14474) q^{58} + (28 \beta_{7} - 328 \beta_{6}) q^{59} + ( - 172 \beta_{5} - 2 \beta_{3} + 86 \beta_1) q^{61} + ( - 52 \beta_{7} - 512 \beta_{6}) q^{62} + (32 \beta_1 + 31744) q^{64} + (4856 \beta_{4} - 8952 \beta_{2}) q^{65} + (2548 \beta_1 + 19076) q^{67} + ( - 60 \beta_{7} + 444 \beta_{6}) q^{68} + (62 \beta_{5} + 21 \beta_{3} - 2319 \beta_1 - 32592) q^{70} + (7634 \beta_{4} + 11765 \beta_{2}) q^{71} + (36 \beta_{5} - 151 \beta_{3} - 18 \beta_1) q^{73} + ( - 11652 \beta_{4} + 13756 \beta_{2}) q^{74} + (432 \beta_{5} - 84 \beta_{3} - 216 \beta_1) q^{76} + ( - 4 \beta_{7} + 367 \beta_{6} - 1911 \beta_{4} - 922 \beta_{2}) q^{77} + (1708 \beta_1 - 11824) q^{79} + ( - 128 \beta_{7} - 144 \beta_{6}) q^{80} + (638 \beta_{5} + 267 \beta_{3} - 319 \beta_1) q^{82} + (22 \beta_{7} + 798 \beta_{6}) q^{83} + ( - 3996 \beta_1 + 59400) q^{85} + (12634 \beta_{4} - 15086 \beta_{2}) q^{86} + (228 \beta_1 - 3256) q^{88} + ( - 54 \beta_{7} - 1155 \beta_{6}) q^{89} + (34 \beta_{5} + 56 \beta_{3} - 2445 \beta_1 - 62664) q^{91} + (23660 \beta_{4} + 5688 \beta_{2}) q^{92} + (772 \beta_{5} + 310 \beta_{3} - 386 \beta_1) q^{94} + ( - 24936 \beta_{4} - 10728 \beta_{2}) q^{95} + ( - 484 \beta_{5} + 163 \beta_{3} + 242 \beta_1) q^{97} + ( - 42 \beta_{7} - 476 \beta_{6} - 15225 \beta_{4} + \cdots + 15785 \beta_{2}) q^{98}+O(q^{100})$$ q + (b4 - b2) * q^2 - 2*b1 * q^4 + b6 * q^5 + (b5 + 2*b1 - 14) * q^7 + (4*b4 + 4*b2) * q^8 + (2*b5 + b3 - b1) * q^10 + (4*b4 + 53*b2) * q^11 + (4*b5 + b3 - 2*b1) * q^13 + (-b7 - 8*b6 + 21*b4 - 31*b2) * q^14 + (-64*b1 - 16) * q^16 + (-2*b7 + 13*b6) * q^17 + (6*b5 - 6*b3 - 3*b1) * q^19 + (-4*b7 - 4*b6) * q^20 + (49*b1 + 670) * q^22 + (158*b4 + 845*b2) * q^23 + (20*b1 + 1171) * q^25 + (-6*b7 - 52*b6) * q^26 + (-14*b3 + 28*b1 - 1260) * q^28 + (143*b4 - 850*b2) * q^29 + (4*b5 + 24*b3 - 2*b1) * q^31 + (-784*b4 + 1296*b2) * q^32 + (-6*b5 + b3 + 3*b1) * q^34 + (5*b7 - 9*b6 + 140*b4 - 2148*b2) * q^35 + (-526*b1 - 4288) * q^37 + (6*b7 + 72*b6) * q^38 + (-8*b5 + 4*b3 + 4*b1) * q^40 + (26*b7 + 111*b6) * q^41 + (613*b1 + 4052) * q^43 + (1484*b4 + 144*b2) * q^44 + (687*b1 + 8986) * q^46 + (38*b7 + 82*b6) * q^47 + (-28*b5 + 35*b3 - 126*b1 - 13265) * q^49 + (1451*b4 - 1531*b2) * q^50 + (-72*b5 - 56*b3 + 36*b1) * q^52 + (-6091*b4 + 326*b2) * q^53 + (-106*b5 + 4*b3 + 53*b1) * q^55 + (-4*b7 + 24*b6 - 196*b4 - 236*b2) * q^56 + (-993*b1 - 14474) * q^58 + (28*b7 - 328*b6) * q^59 + (-172*b5 - 2*b3 + 86*b1) * q^61 + (-52*b7 - 512*b6) * q^62 + (32*b1 + 31744) * q^64 + (4856*b4 - 8952*b2) * q^65 + (2548*b1 + 19076) * q^67 + (-60*b7 + 444*b6) * q^68 + (62*b5 + 21*b3 - 2319*b1 - 32592) * q^70 + (7634*b4 + 11765*b2) * q^71 + (36*b5 - 151*b3 - 18*b1) * q^73 + (-11652*b4 + 13756*b2) * q^74 + (432*b5 - 84*b3 - 216*b1) * q^76 + (-4*b7 + 367*b6 - 1911*b4 - 922*b2) * q^77 + (1708*b1 - 11824) * q^79 + (-128*b7 - 144*b6) * q^80 + (638*b5 + 267*b3 - 319*b1) * q^82 + (22*b7 + 798*b6) * q^83 + (-3996*b1 + 59400) * q^85 + (12634*b4 - 15086*b2) * q^86 + (228*b1 - 3256) * q^88 + (-54*b7 - 1155*b6) * q^89 + (34*b5 + 56*b3 - 2445*b1 - 62664) * q^91 + (23660*b4 + 5688*b2) * q^92 + (772*b5 + 310*b3 - 386*b1) * q^94 + (-24936*b4 - 10728*b2) * q^95 + (-484*b5 + 163*b3 + 242*b1) * q^97 + (-42*b7 - 476*b6 - 15225*b4 + 15785*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 112 q^{7}+O(q^{10})$$ 8 * q - 112 * q^7 $$8 q - 112 q^{7} - 128 q^{16} + 5360 q^{22} + 9368 q^{25} - 10080 q^{28} - 34304 q^{37} + 32416 q^{43} + 71888 q^{46} - 106120 q^{49} - 115792 q^{58} + 253952 q^{64} + 152608 q^{67} - 260736 q^{70} - 94592 q^{79} + 475200 q^{85} - 26048 q^{88} - 501312 q^{91}+O(q^{100})$$ 8 * q - 112 * q^7 - 128 * q^16 + 5360 * q^22 + 9368 * q^25 - 10080 * q^28 - 34304 * q^37 + 32416 * q^43 + 71888 * q^46 - 106120 * q^49 - 115792 * q^58 + 253952 * q^64 + 152608 * q^67 - 260736 * q^70 - 94592 * q^79 + 475200 * q^85 - 26048 * q^88 - 501312 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 456x^{6} - 612x^{5} + 66744x^{4} + 32004x^{3} + 3729464x^{2} + 2503804x + 66026577$$ :

 $$\beta_{1}$$ $$=$$ $$( - 7276 \nu^{7} - 6016 \nu^{6} - 2355636 \nu^{5} - 9413506 \nu^{4} - 185550156 \nu^{3} - 611944144 \nu^{2} - 3939559436 \nu + 62846012472 ) / 4466767135$$ (-7276*v^7 - 6016*v^6 - 2355636*v^5 - 9413506*v^4 - 185550156*v^3 - 611944144*v^2 - 3939559436*v + 62846012472) / 4466767135 $$\beta_{2}$$ $$=$$ $$( 106 \nu^{7} - 597 \nu^{6} + 49014 \nu^{5} - 159684 \nu^{4} + 6374124 \nu^{3} - 8686185 \nu^{2} + 195744008 \nu - 314036202 ) / 51360255$$ (106*v^7 - 597*v^6 + 49014*v^5 - 159684*v^4 + 6374124*v^3 - 8686185*v^2 + 195744008*v - 314036202) / 51360255 $$\beta_{3}$$ $$=$$ $$( 136314924 \nu^{7} + 13885773036 \nu^{6} + 99207113184 \nu^{5} + 4638877777212 \nu^{4} + 40666053647532 \nu^{3} + \cdots + 87\!\cdots\!64 ) / 46771518670585$$ (136314924*v^7 + 13885773036*v^6 + 99207113184*v^5 + 4638877777212*v^4 + 40666053647532*v^3 + 388661146589760*v^2 + 3722129576222472*v + 8741415421523664) / 46771518670585 $$\beta_{4}$$ $$=$$ $$( - 4026892 \nu^{7} + 41326635 \nu^{6} - 1675985952 \nu^{5} + 10635929286 \nu^{4} - 189591871494 \nu^{3} + 683245379841 \nu^{2} + \cdots + 17714021093796 ) / 1287289504695$$ (-4026892*v^7 + 41326635*v^6 - 1675985952*v^5 + 10635929286*v^4 - 189591871494*v^3 + 683245379841*v^2 - 5417930459474*v + 17714021093796) / 1287289504695 $$\beta_{5}$$ $$=$$ $$( 603178436 \nu^{7} - 2349790824 \nu^{6} + 303972103626 \nu^{5} - 442753391709 \nu^{4} + 52921024233816 \nu^{3} + \cdots + 26\!\cdots\!78 ) / 140314556011755$$ (603178436*v^7 - 2349790824*v^6 + 303972103626*v^5 - 442753391709*v^4 + 52921024233816*v^3 + 16548578396334*v^2 + 4540318229573026*v + 2628436460918178) / 140314556011755 $$\beta_{6}$$ $$=$$ $$( - 174176 \nu^{7} - 1612878 \nu^{6} - 68934144 \nu^{5} - 1127164206 \nu^{4} - 7806037224 \nu^{3} - 184380040050 \nu^{2} + \cdots - 5955704489118 ) / 13400301405$$ (-174176*v^7 - 1612878*v^6 - 68934144*v^5 - 1127164206*v^4 - 7806037224*v^3 - 184380040050*v^2 - 378525797788*v - 5955704489118) / 13400301405 $$\beta_{7}$$ $$=$$ $$( - 3252328 \nu^{7} - 23469492 \nu^{6} - 1033111860 \nu^{5} - 12230717712 \nu^{4} - 98965955544 \nu^{3} - 1566104155968 \nu^{2} + \cdots - 45975274320660 ) / 13400301405$$ (-3252328*v^7 - 23469492*v^6 - 1033111860*v^5 - 12230717712*v^4 - 98965955544*v^3 - 1566104155968*v^2 - 3648413057660*v - 45975274320660) / 13400301405
 $$\nu$$ $$=$$ $$( 6\beta_{5} + 12\beta_{4} - \beta_{3} - 9\beta _1 + 36 ) / 72$$ (6*b5 + 12*b4 - b3 - 9*b1 + 36) / 72 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} + 2\beta_{6} + 6\beta_{5} + 6\beta_{4} - 4\beta_{3} + 18\beta_{2} + 153\beta _1 - 4032 ) / 36$$ (-b7 + 2*b6 + 6*b5 + 6*b4 - 4*b3 + 18*b2 + 153*b1 - 4032) / 36 $$\nu^{3}$$ $$=$$ $$( -3\beta_{7} + 33\beta_{6} - 402\beta_{5} - 2010\beta_{4} + 142\beta_{3} - 1404\beta_{2} + 1458\beta _1 - 16074 ) / 36$$ (-3*b7 + 33*b6 - 402*b5 - 2010*b4 + 142*b3 - 1404*b2 + 1458*b1 - 16074) / 36 $$\nu^{4}$$ $$=$$ $$( 133 \beta_{7} - 716 \beta_{6} - 1302 \beta_{5} - 5352 \beta_{4} + 634 \beta_{3} - 7524 \beta_{2} - 15660 \beta _1 + 291960 ) / 18$$ (133*b7 - 716*b6 - 1302*b5 - 5352*b4 + 634*b3 - 7524*b2 - 15660*b1 + 291960) / 18 $$\nu^{5}$$ $$=$$ $$( 4320 \beta_{7} - 33480 \beta_{6} + 121326 \beta_{5} + 959808 \beta_{4} - 50539 \beta_{3} + 969300 \beta_{2} - 761877 \beta _1 + 10552716 ) / 72$$ (4320*b7 - 33480*b6 + 121326*b5 + 959808*b4 - 50539*b3 + 969300*b2 - 761877*b1 + 10552716) / 72 $$\nu^{6}$$ $$=$$ $$( - 59335 \beta_{7} + 387032 \beta_{6} + 727620 \beta_{5} + 5222850 \beta_{4} - 328975 \beta_{3} + 6235722 \beta_{2} + 4954770 \beta _1 - 87159168 ) / 36$$ (-59335*b7 + 387032*b6 + 727620*b5 + 5222850*b4 - 328975*b3 + 6235722*b2 + 4954770*b1 - 87159168) / 36 $$\nu^{7}$$ $$=$$ $$( - 833784 \beta_{7} + 5942559 \beta_{6} - 8749824 \beta_{5} - 98335740 \beta_{4} + 3798511 \beta_{3} - 108304182 \beta_{2} + 90041283 \beta _1 - 1341413874 ) / 36$$ (-833784*b7 + 5942559*b6 - 8749824*b5 - 98335740*b4 + 3798511*b3 - 108304182*b2 + 90041283*b1 - 1341413874) / 36

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 −0.822876 + 5.87982i −0.822876 − 7.29403i 1.82288 − 14.2575i 1.82288 + 12.8433i 1.82288 + 14.2575i 1.82288 − 12.8433i −0.822876 − 5.87982i −0.822876 + 7.29403i
7.98430i 0 −31.7490 −67.9227 0 25.6863 + 127.072i 2.00393i 0 542.315i
62.2 7.98430i 0 −31.7490 67.9227 0 25.6863 127.072i 2.00393i 0 542.315i
62.3 0.500983i 0 31.7490 −63.0754 0 −53.6863 118.003i 31.9372i 0 31.5997i
62.4 0.500983i 0 31.7490 63.0754 0 −53.6863 + 118.003i 31.9372i 0 31.5997i
62.5 0.500983i 0 31.7490 −63.0754 0 −53.6863 + 118.003i 31.9372i 0 31.5997i
62.6 0.500983i 0 31.7490 63.0754 0 −53.6863 118.003i 31.9372i 0 31.5997i
62.7 7.98430i 0 −31.7490 −67.9227 0 25.6863 127.072i 2.00393i 0 542.315i
62.8 7.98430i 0 −31.7490 67.9227 0 25.6863 + 127.072i 2.00393i 0 542.315i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 62.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.c.b 8
3.b odd 2 1 inner 63.6.c.b 8
4.b odd 2 1 1008.6.k.b 8
7.b odd 2 1 inner 63.6.c.b 8
12.b even 2 1 1008.6.k.b 8
21.c even 2 1 inner 63.6.c.b 8
28.d even 2 1 1008.6.k.b 8
84.h odd 2 1 1008.6.k.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.c.b 8 1.a even 1 1 trivial
63.6.c.b 8 3.b odd 2 1 inner
63.6.c.b 8 7.b odd 2 1 inner
63.6.c.b 8 21.c even 2 1 inner
1008.6.k.b 8 4.b odd 2 1
1008.6.k.b 8 12.b even 2 1
1008.6.k.b 8 28.d even 2 1
1008.6.k.b 8 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 64T_{2}^{2} + 16$$ acting on $$S_{6}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 64 T^{2} + 16)^{2}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 8592 T^{2} + 18354816)^{2}$$
$7$ $$(T^{4} + 56 T^{3} + 28098 T^{2} + \cdots + 282475249)^{2}$$
$11$ $$(T^{4} + 79228 T^{2} + \cdots + 1523965444)^{2}$$
$13$ $$(T^{4} + 676128 T^{2} + \cdots + 26504354304)^{2}$$
$17$ $$(T^{4} - 3795984 T^{2} + \cdots + 13380660864)^{2}$$
$19$ $$(T^{4} + 6287328 T^{2} + \cdots + 5001393682944)^{2}$$
$23$ $$(T^{4} + 20891404 T^{2} + \cdots + 91145170812004)^{2}$$
$29$ $$(T^{4} + 20966164 T^{2} + \cdots + 95002410498724)^{2}$$
$31$ $$(T^{4} + 90530688 T^{2} + \cdots + 19\!\cdots\!04)^{2}$$
$37$ $$(T^{2} + 8576 T - 51335408)^{4}$$
$41$ $$(T^{4} - 450270096 T^{2} + \cdots + 22\!\cdots\!04)^{2}$$
$43$ $$(T^{2} - 8104 T - 78275084)^{4}$$
$47$ $$(T^{4} - 815119296 T^{2} + \cdots + 14\!\cdots\!36)^{2}$$
$53$ $$(T^{4} + 1338585844 T^{2} + \cdots + 44\!\cdots\!36)^{2}$$
$59$ $$(T^{4} - 1412814336 T^{2} + \cdots + 10\!\cdots\!76)^{2}$$
$61$ $$(T^{4} + 893736192 T^{2} + \cdots + 19\!\cdots\!44)^{2}$$
$67$ $$(T^{2} - 38152 T - 1272166832)^{4}$$
$71$ $$(T^{4} + 5973632716 T^{2} + \cdots + 78\!\cdots\!64)^{2}$$
$73$ $$(T^{4} + 3510489888 T^{2} + \cdots + 30\!\cdots\!84)^{2}$$
$79$ $$(T^{2} + 23648 T - 595343552)^{4}$$
$83$ $$(T^{4} - 5607969216 T^{2} + \cdots + 59\!\cdots\!96)^{2}$$
$89$ $$(T^{4} - 12592946448 T^{2} + \cdots + 19\!\cdots\!84)^{2}$$
$97$ $$(T^{4} + 10358370336 T^{2} + \cdots + 21\!\cdots\!56)^{2}$$