# Properties

 Label 441.4.e.y Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ 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] = [441,4,Mod(226,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.5922408960000.19 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 54x^{6} + 176x^{5} + 1307x^{4} - 2912x^{3} - 15314x^{2} + 16800x + 86044$$ x^8 - 4*x^7 - 54*x^6 + 176*x^5 + 1307*x^4 - 2912*x^3 - 15314*x^2 + 16800*x + 86044 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$7^{2}$$ Twist minimal: no (minimal twist has level 49) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_{2} + 1) q^{2} + (\beta_{6} - 8 \beta_1 - 8) q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{2} - 17) q^{8}+O(q^{10})$$ q + (b6 - b2 + 1) * q^2 + (b6 - 8*b1 - 8) * q^4 + (b7 + b4) * q^5 + (b2 - 17) * q^8 $$q + (\beta_{6} - \beta_{2} + 1) q^{2} + (\beta_{6} - 8 \beta_1 - 8) q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{2} - 17) q^{8} + (4 \beta_{4} + 2 \beta_{3}) q^{10} + (6 \beta_{6} + 28 \beta_1 + 28) q^{11} + ( - \beta_{7} - 6 \beta_{5} + 6 \beta_{3}) q^{13} + ( - 9 \beta_{6} + 9 \beta_{2} - 48 \beta_1 - 9) q^{16} + (9 \beta_{4} - \beta_{3}) q^{17} + ( - 4 \beta_{7} - 11 \beta_{5} - 4 \beta_{4}) q^{19} + ( - 12 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{20} + ( - 22 \beta_{2} - 74) q^{22} + (8 \beta_{6} - 8 \beta_{2} - 84 \beta_1 + 8) q^{23} + (22 \beta_{6} + 43 \beta_1 + 43) q^{25} + ( - 16 \beta_{7} + 16 \beta_{5} - 16 \beta_{4}) q^{26} + ( - 14 \beta_{2} - 58) q^{29} + ( - 4 \beta_{4} + 38 \beta_{3}) q^{31} + (47 \beta_{6} + 16 \beta_1 + 16) q^{32} + ( - 34 \beta_{7} - 21 \beta_{5} + 21 \beta_{3}) q^{34} + (6 \beta_{6} - 6 \beta_{2} + 56 \beta_1 + 6) q^{37} + ( - 38 \beta_{4} + 25 \beta_{3}) q^{38} + ( - 20 \beta_{7} - 2 \beta_{5} - 20 \beta_{4}) q^{40} + (3 \beta_{7} - 31 \beta_{5} + 31 \beta_{3}) q^{41} + ( - 70 \beta_{2} + 170) q^{43} + ( - 26 \beta_{6} + 26 \beta_{2} - 128 \beta_1 - 26) q^{44} + (92 \beta_{6} - 128 \beta_1 - 128) q^{46} + ( - 32 \beta_{7} + 26 \beta_{5} - 32 \beta_{4}) q^{47} + ( - 21 \beta_{2} - 331) q^{50} + ( - 24 \beta_{4} - 32 \beta_{3}) q^{52} + ( - 36 \beta_{6} - 14 \beta_1 - 14) q^{53} + (4 \beta_{7} - 12 \beta_{5} + 12 \beta_{3}) q^{55} + ( - 58 \beta_{6} + 58 \beta_{2} - 224 \beta_1 - 58) q^{58} + ( - 20 \beta_{4} + 49 \beta_{3}) q^{59} + ( - 27 \beta_{7} + 68 \beta_{5} - 27 \beta_{4}) q^{61} + ( - 60 \beta_{7} + 122 \beta_{5} - 122 \beta_{3}) q^{62} + (103 \beta_{2} - 471) q^{64} + ( - 70 \beta_{6} + 70 \beta_{2} + 154 \beta_1 - 70) q^{65} + ( - 76 \beta_{6} + 448 \beta_1 + 448) q^{67} + ( - 106 \beta_{7} - 13 \beta_{5} - 106 \beta_{4}) q^{68} + ( - 28 \beta_{2} - 548) q^{71} + (25 \beta_{4} + 37 \beta_{3}) q^{73} + ( - 50 \beta_{6} - 96 \beta_1 - 96) q^{74} + (70 \beta_{7} + 63 \beta_{5} - 63 \beta_{3}) q^{76} + (188 \beta_{6} - 188 \beta_{2} - 168 \beta_1 + 188) q^{79} + (12 \beta_{4} - 18 \beta_{3}) q^{80} + ( - 50 \beta_{7} + 99 \beta_{5} - 50 \beta_{4}) q^{82} + ( - 8 \beta_{7} + \beta_{5} - \beta_{3}) q^{83} + (190 \beta_{2} - 916) q^{85} + (170 \beta_{6} - 170 \beta_{2} - 1120 \beta_1 + 170) q^{86} + ( - 74 \beta_{6} - 352 \beta_1 - 352) q^{88} + (75 \beta_{7} + 157 \beta_{5} + 75 \beta_{4}) q^{89} + (156 \beta_{2} - 956) q^{92} + ( - 76 \beta_{4} - 142 \beta_{3}) q^{94} + ( - 176 \beta_{6} + 460 \beta_1 + 460) q^{95} + ( - 91 \beta_{7} + 189 \beta_{5} - 189 \beta_{3}) q^{97}+O(q^{100})$$ q + (b6 - b2 + 1) * q^2 + (b6 - 8*b1 - 8) * q^4 + (b7 + b4) * q^5 + (b2 - 17) * q^8 + (4*b4 + 2*b3) * q^10 + (6*b6 + 28*b1 + 28) * q^11 + (-b7 - 6*b5 + 6*b3) * q^13 + (-9*b6 + 9*b2 - 48*b1 - 9) * q^16 + (9*b4 - b3) * q^17 + (-4*b7 - 11*b5 - 4*b4) * q^19 + (-12*b7 - 2*b5 + 2*b3) * q^20 + (-22*b2 - 74) * q^22 + (8*b6 - 8*b2 - 84*b1 + 8) * q^23 + (22*b6 + 43*b1 + 43) * q^25 + (-16*b7 + 16*b5 - 16*b4) * q^26 + (-14*b2 - 58) * q^29 + (-4*b4 + 38*b3) * q^31 + (47*b6 + 16*b1 + 16) * q^32 + (-34*b7 - 21*b5 + 21*b3) * q^34 + (6*b6 - 6*b2 + 56*b1 + 6) * q^37 + (-38*b4 + 25*b3) * q^38 + (-20*b7 - 2*b5 - 20*b4) * q^40 + (3*b7 - 31*b5 + 31*b3) * q^41 + (-70*b2 + 170) * q^43 + (-26*b6 + 26*b2 - 128*b1 - 26) * q^44 + (92*b6 - 128*b1 - 128) * q^46 + (-32*b7 + 26*b5 - 32*b4) * q^47 + (-21*b2 - 331) * q^50 + (-24*b4 - 32*b3) * q^52 + (-36*b6 - 14*b1 - 14) * q^53 + (4*b7 - 12*b5 + 12*b3) * q^55 + (-58*b6 + 58*b2 - 224*b1 - 58) * q^58 + (-20*b4 + 49*b3) * q^59 + (-27*b7 + 68*b5 - 27*b4) * q^61 + (-60*b7 + 122*b5 - 122*b3) * q^62 + (103*b2 - 471) * q^64 + (-70*b6 + 70*b2 + 154*b1 - 70) * q^65 + (-76*b6 + 448*b1 + 448) * q^67 + (-106*b7 - 13*b5 - 106*b4) * q^68 + (-28*b2 - 548) * q^71 + (25*b4 + 37*b3) * q^73 + (-50*b6 - 96*b1 - 96) * q^74 + (70*b7 + 63*b5 - 63*b3) * q^76 + (188*b6 - 188*b2 - 168*b1 + 188) * q^79 + (12*b4 - 18*b3) * q^80 + (-50*b7 + 99*b5 - 50*b4) * q^82 + (-8*b7 + b5 - b3) * q^83 + (190*b2 - 916) * q^85 + (170*b6 - 170*b2 - 1120*b1 + 170) * q^86 + (-74*b6 - 352*b1 - 352) * q^88 + (75*b7 + 157*b5 + 75*b4) * q^89 + (156*b2 - 956) * q^92 + (-76*b4 - 142*b3) * q^94 + (-176*b6 + 460*b1 + 460) * q^95 + (-91*b7 + 189*b5 - 189*b3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{2} - 34 q^{4} - 132 q^{8}+O(q^{10})$$ 8 * q + 2 * q^2 - 34 * q^4 - 132 * q^8 $$8 q + 2 q^{2} - 34 q^{4} - 132 q^{8} + 100 q^{11} + 174 q^{16} - 680 q^{22} + 352 q^{23} + 128 q^{25} - 520 q^{29} - 30 q^{32} - 212 q^{37} + 1080 q^{43} + 460 q^{44} - 696 q^{46} - 2732 q^{50} + 16 q^{53} + 780 q^{58} - 3356 q^{64} - 756 q^{65} + 1944 q^{67} - 4496 q^{71} - 284 q^{74} + 1048 q^{79} - 6568 q^{85} + 4820 q^{86} - 1260 q^{88} - 7024 q^{92} + 2192 q^{95}+O(q^{100})$$ 8 * q + 2 * q^2 - 34 * q^4 - 132 * q^8 + 100 * q^11 + 174 * q^16 - 680 * q^22 + 352 * q^23 + 128 * q^25 - 520 * q^29 - 30 * q^32 - 212 * q^37 + 1080 * q^43 + 460 * q^44 - 696 * q^46 - 2732 * q^50 + 16 * q^53 + 780 * q^58 - 3356 * q^64 - 756 * q^65 + 1944 * q^67 - 4496 * q^71 - 284 * q^74 + 1048 * q^79 - 6568 * q^85 + 4820 * q^86 - 1260 * q^88 - 7024 * q^92 + 2192 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 54x^{6} + 176x^{5} + 1307x^{4} - 2912x^{3} - 15314x^{2} + 16800x + 86044$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{6} + 6\nu^{5} + 149\nu^{4} - 308\nu^{3} - 3293\nu^{2} + 3448\nu + 22372 ) / 8946$$ (-2*v^6 + 6*v^5 + 149*v^4 - 308*v^3 - 3293*v^2 + 3448*v + 22372) / 8946 $$\beta_{2}$$ $$=$$ $$( 188 \nu^{7} - 658 \nu^{6} - 11168 \nu^{5} + 29565 \nu^{4} + 311680 \nu^{3} - 497414 \nu^{2} - 5795360 \nu + 4518059 ) / 3072951$$ (188*v^7 - 658*v^6 - 11168*v^5 + 29565*v^4 + 311680*v^3 - 497414*v^2 - 5795360*v + 4518059) / 3072951 $$\beta_{3}$$ $$=$$ $$( - 94 \nu^{7} + 558 \nu^{6} + 4897 \nu^{5} - 31843 \nu^{4} - 120574 \nu^{3} + 1137914 \nu^{2} + 454250 \nu - 11478684 ) / 1024317$$ (-94*v^7 + 558*v^6 + 4897*v^5 - 31843*v^4 - 120574*v^3 + 1137914*v^2 + 454250*v - 11478684) / 1024317 $$\beta_{4}$$ $$=$$ $$( 1034 \nu^{7} - 2932 \nu^{6} - 63485 \nu^{5} + 111426 \nu^{4} + 1820038 \nu^{3} - 68156 \nu^{2} - 17694113 \nu - 24320296 ) / 3072951$$ (1034*v^7 - 2932*v^6 - 63485*v^5 + 111426*v^4 + 1820038*v^3 - 68156*v^2 - 17694113*v - 24320296) / 3072951 $$\beta_{5}$$ $$=$$ $$( - 685 \nu^{7} + 10069 \nu^{6} + 3667 \nu^{5} - 388145 \nu^{4} + 210780 \nu^{3} + 6478590 \nu^{2} - 3760032 \nu - 36898988 ) / 2048634$$ (-685*v^7 + 10069*v^6 + 3667*v^5 - 388145*v^4 + 210780*v^3 + 6478590*v^2 - 3760032*v - 36898988) / 2048634 $$\beta_{6}$$ $$=$$ $$( 2744 \nu^{7} - 8917 \nu^{6} - 155726 \nu^{5} + 356991 \nu^{4} + 3179236 \nu^{3} - 3891986 \nu^{2} - 26106296 \nu + 2554244 ) / 6145902$$ (2744*v^7 - 8917*v^6 - 155726*v^5 + 356991*v^4 + 3179236*v^3 - 3891986*v^2 - 26106296*v + 2554244) / 6145902 $$\beta_{7}$$ $$=$$ $$( 5467 \nu^{7} + 2506 \nu^{6} - 235570 \nu^{5} - 369522 \nu^{4} + 3410879 \nu^{3} + 8909612 \nu^{2} - 8027680 \nu - 44176664 ) / 6145902$$ (5467*v^7 + 2506*v^6 - 235570*v^5 - 369522*v^4 + 3410879*v^3 + 8909612*v^2 - 8027680*v - 44176664) / 6145902
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{3} - 7\beta_{2} + 7 ) / 7$$ (b4 - b3 - 7*b2 + 7) / 7 $$\nu^{2}$$ $$=$$ $$( 4\beta_{4} + 10\beta_{3} - 7\beta_{2} + 14\beta _1 + 119 ) / 7$$ (4*b4 + 10*b3 - 7*b2 + 14*b1 + 119) / 7 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 42\beta_{6} - 2\beta_{5} + 54\beta_{4} - 31\beta_{3} - 77\beta_{2} + 189 ) / 7$$ (2*b7 - 42*b6 - 2*b5 + 54*b4 - 31*b3 - 77*b2 + 189) / 7 $$\nu^{4}$$ $$=$$ $$( 16\beta_{7} - 84\beta_{6} + 40\beta_{5} + 200\beta_{4} + 236\beta_{3} - 147\beta_{2} + 1316\beta _1 + 2023 ) / 7$$ (16*b7 - 84*b6 + 40*b5 + 200*b4 + 236*b3 - 147*b2 + 1316*b1 + 2023) / 7 $$\nu^{5}$$ $$=$$ $$( 356\beta_{7} - 2240\beta_{6} - 216\beta_{5} + 1686\beta_{4} - 315\beta_{3} + 245\beta_{2} + 2240\beta _1 + 3451 ) / 7$$ (356*b7 - 2240*b6 - 216*b5 + 1686*b4 - 315*b3 + 245*b2 + 2240*b1 + 3451) / 7 $$\nu^{6}$$ $$=$$ $$( 1952 \beta_{7} - 6510 \beta_{6} + 2640 \beta_{5} + 6780 \beta_{4} + 3222 \beta_{3} + 1099 \beta_{2} + 50400 \beta _1 + 26397 ) / 7$$ (1952*b7 - 6510*b6 + 2640*b5 + 6780*b4 + 3222*b3 + 1099*b2 + 50400*b1 + 26397) / 7 $$\nu^{7}$$ $$=$$ $$( 22148 \beta_{7} - 73010 \beta_{6} - 6566 \beta_{5} + 44318 \beta_{4} + 2477 \beta_{3} + 49287 \beta_{2} + 139552 \beta _1 + 28329 ) / 7$$ (22148*b7 - 73010*b6 - 6566*b5 + 44318*b4 + 2477*b3 + 49287*b2 + 139552*b1 + 28329) / 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −2.82402 + 1.22474i −4.23824 − 1.22474i 5.23824 + 1.22474i 3.82402 − 1.22474i −2.82402 − 1.22474i −4.23824 + 1.22474i 5.23824 − 1.22474i 3.82402 + 1.22474i
−1.76556 + 3.05805i 0 −2.23444 3.87016i −1.03865 + 1.79899i 0 0 −12.4689 0 −3.66760 6.35247i
226.2 −1.76556 + 3.05805i 0 −2.23444 3.87016i 1.03865 1.79899i 0 0 −12.4689 0 3.66760 + 6.35247i
226.3 2.26556 3.92407i 0 −6.26556 10.8523i −6.73953 + 11.6732i 0 0 −20.5311 0 30.5377 + 52.8928i
226.4 2.26556 3.92407i 0 −6.26556 10.8523i 6.73953 11.6732i 0 0 −20.5311 0 −30.5377 52.8928i
361.1 −1.76556 3.05805i 0 −2.23444 + 3.87016i −1.03865 1.79899i 0 0 −12.4689 0 −3.66760 + 6.35247i
361.2 −1.76556 3.05805i 0 −2.23444 + 3.87016i 1.03865 + 1.79899i 0 0 −12.4689 0 3.66760 6.35247i
361.3 2.26556 + 3.92407i 0 −6.26556 + 10.8523i −6.73953 11.6732i 0 0 −20.5311 0 30.5377 52.8928i
361.4 2.26556 + 3.92407i 0 −6.26556 + 10.8523i 6.73953 + 11.6732i 0 0 −20.5311 0 −30.5377 + 52.8928i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.y 8
3.b odd 2 1 49.4.c.e 8
7.b odd 2 1 inner 441.4.e.y 8
7.c even 3 1 441.4.a.u 4
7.c even 3 1 inner 441.4.e.y 8
7.d odd 6 1 441.4.a.u 4
7.d odd 6 1 inner 441.4.e.y 8
21.c even 2 1 49.4.c.e 8
21.g even 6 1 49.4.a.e 4
21.g even 6 1 49.4.c.e 8
21.h odd 6 1 49.4.a.e 4
21.h odd 6 1 49.4.c.e 8
84.j odd 6 1 784.4.a.bf 4
84.n even 6 1 784.4.a.bf 4
105.o odd 6 1 1225.4.a.bb 4
105.p even 6 1 1225.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 21.g even 6 1
49.4.a.e 4 21.h odd 6 1
49.4.c.e 8 3.b odd 2 1
49.4.c.e 8 21.c even 2 1
49.4.c.e 8 21.g even 6 1
49.4.c.e 8 21.h odd 6 1
441.4.a.u 4 7.c even 3 1
441.4.a.u 4 7.d odd 6 1
441.4.e.y 8 1.a even 1 1 trivial
441.4.e.y 8 7.b odd 2 1 inner
441.4.e.y 8 7.c even 3 1 inner
441.4.e.y 8 7.d odd 6 1 inner
784.4.a.bf 4 84.j odd 6 1
784.4.a.bf 4 84.n even 6 1
1225.4.a.bb 4 105.o odd 6 1
1225.4.a.bb 4 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 17T_{2}^{2} + 16T_{2} + 256$$ T2^4 - T2^3 + 17*T2^2 + 16*T2 + 256 $$T_{5}^{8} + 186T_{5}^{6} + 33812T_{5}^{4} + 145824T_{5}^{2} + 614656$$ T5^8 + 186*T5^6 + 33812*T5^4 + 145824*T5^2 + 614656 $$T_{13}^{4} - 3234T_{13}^{2} + 2458624$$ T13^4 - 3234*T13^2 + 2458624

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{3} + 17 T^{2} + 16 T + 256)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 186 T^{6} + 33812 T^{4} + \cdots + 614656$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - 50 T^{3} + 2460 T^{2} + \cdots + 1600)^{2}$$
$13$ $$(T^{4} - 3234 T^{2} + 2458624)^{2}$$
$17$ $$T^{8} + 14564 T^{6} + \cdots + 95001747709456$$
$19$ $$T^{8} + 14746 T^{6} + \cdots + 27\!\cdots\!16$$
$23$ $$(T^{4} - 176 T^{3} + 24272 T^{2} + \cdots + 44943616)^{2}$$
$29$ $$(T^{2} + 130 T + 1040)^{4}$$
$31$ $$T^{8} + 100104 T^{6} + \cdots + 39\!\cdots\!36$$
$37$ $$(T^{4} + 106 T^{3} + 9012 T^{2} + \cdots + 4946176)^{2}$$
$41$ $$(T^{4} - 66836 T^{2} + \cdots + 307721764)^{2}$$
$43$ $$(T^{2} - 270 T - 61400)^{4}$$
$47$ $$T^{8} + 187240 T^{6} + \cdots + 69\!\cdots\!00$$
$53$ $$(T^{4} - 8 T^{3} + 21108 T^{2} + \cdots + 442849936)^{2}$$
$59$ $$T^{8} + 189354 T^{6} + \cdots + 25\!\cdots\!36$$
$61$ $$T^{8} + 360266 T^{6} + \cdots + 25\!\cdots\!76$$
$67$ $$(T^{4} - 972 T^{3} + \cdots + 20259536896)^{2}$$
$71$ $$(T^{2} + 1124 T + 303104)^{4}$$
$73$ $$T^{8} + 276756 T^{6} + \cdots + 15\!\cdots\!76$$
$79$ $$(T^{4} - 524 T^{3} + \cdots + 255728444416)^{2}$$
$83$ $$(T^{4} - 11466 T^{2} + 6492304)^{2}$$
$89$ $$T^{8} + 3623876 T^{6} + \cdots + 80\!\cdots\!56$$
$97$ $$(T^{4} - 3082884 T^{2} + \cdots + 841222821124)^{2}$$