# Properties

 Label 320.6.c.k Level 320 Weight 6 Character orbit 320.c Analytic conductor 51.323 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{44}\cdot 5^{4}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + ( 5 + \beta_{5} + \beta_{7} ) q^{5} + \beta_{9} q^{7} + ( -142 + 7 \beta_{1} + 2 \beta_{7} - 2 \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + ( 5 + \beta_{5} + \beta_{7} ) q^{5} + \beta_{9} q^{7} + ( -142 + 7 \beta_{1} + 2 \beta_{7} - 2 \beta_{8} ) q^{9} -\beta_{3} q^{11} + ( -3 \beta_{5} + 11 \beta_{6} ) q^{13} + ( \beta_{3} - 13 \beta_{4} + \beta_{11} ) q^{15} + ( -2 \beta_{5} + \beta_{6} + 7 \beta_{7} + 7 \beta_{8} ) q^{17} + ( \beta_{2} - 3 \beta_{3} - \beta_{10} - \beta_{11} ) q^{19} + ( -80 + 68 \beta_{1} - 3 \beta_{7} + 3 \beta_{8} ) q^{21} + ( -36 \beta_{4} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{23} + ( 150 - 75 \beta_{1} - 10 \beta_{5} + 25 \beta_{6} + 15 \beta_{7} + 25 \beta_{8} ) q^{25} + ( -48 \beta_{4} - 14 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{27} + ( 1740 + 106 \beta_{1} - 42 \beta_{7} + 42 \beta_{8} ) q^{29} + ( 4 \beta_{2} - 4 \beta_{3} - 7 \beta_{10} - 7 \beta_{11} ) q^{31} + ( 88 \beta_{5} - 191 \beta_{6} + 69 \beta_{7} + 69 \beta_{8} ) q^{33} + ( -5 \beta_{2} + 7 \beta_{3} + 9 \beta_{4} + 20 \beta_{9} - 5 \beta_{10} - 3 \beta_{11} ) q^{35} + ( 98 \beta_{5} + 118 \beta_{6} - 53 \beta_{7} - 53 \beta_{8} ) q^{37} + ( -14 \beta_{2} - 22 \beta_{3} ) q^{39} + ( -9105 - 103 \beta_{1} + 80 \beta_{7} - 80 \beta_{8} ) q^{41} + ( -261 \beta_{4} + 26 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} ) q^{43} + ( 6265 - 10 \beta_{1} + 28 \beta_{5} + 155 \beta_{6} - 167 \beta_{7} + 120 \beta_{8} ) q^{45} + ( -42 \beta_{4} - 101 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{47} + ( -2418 - 833 \beta_{1} - 74 \beta_{7} + 74 \beta_{8} ) q^{49} + ( -17 \beta_{2} + 12 \beta_{3} + 7 \beta_{10} + 7 \beta_{11} ) q^{51} + ( 317 \beta_{5} + 65 \beta_{6} - 68 \beta_{7} - 68 \beta_{8} ) q^{53} + ( 30 \beta_{2} + 13 \beta_{3} + 381 \beta_{4} + 105 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} ) q^{55} + ( 194 \beta_{5} - 365 \beta_{6} + 367 \beta_{7} + 367 \beta_{8} ) q^{57} + ( 63 \beta_{2} - 41 \beta_{3} - 3 \beta_{10} - 3 \beta_{11} ) q^{59} + ( -5820 + 1322 \beta_{1} - 221 \beta_{7} + 221 \beta_{8} ) q^{61} + ( -720 \beta_{4} + 107 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{63} + ( 185 - 595 \beta_{1} + 602 \beta_{5} - 390 \beta_{6} + 12 \beta_{7} + 390 \beta_{8} ) q^{65} + ( 867 \beta_{4} - 210 \beta_{9} + 16 \beta_{10} - 16 \beta_{11} ) q^{67} + ( 13870 - 22 \beta_{1} - 553 \beta_{7} + 553 \beta_{8} ) q^{69} + ( 26 \beta_{2} + 74 \beta_{3} + 43 \beta_{10} + 43 \beta_{11} ) q^{71} + ( 672 \beta_{5} + 453 \beta_{6} + 565 \beta_{7} + 565 \beta_{8} ) q^{73} + ( -75 \beta_{2} - 10 \beta_{3} + 1155 \beta_{4} + 150 \beta_{9} + 25 \beta_{10} + 15 \beta_{11} ) q^{75} + ( 1541 \beta_{5} + 311 \beta_{6} + 603 \beta_{7} + 603 \beta_{8} ) q^{77} + ( -130 \beta_{2} + 64 \beta_{3} + \beta_{10} + \beta_{11} ) q^{79} + ( -14726 + 737 \beta_{1} - 524 \beta_{7} + 524 \beta_{8} ) q^{81} + ( -1509 \beta_{4} + 78 \beta_{9} + 32 \beta_{10} - 32 \beta_{11} ) q^{83} + ( -12630 + 50 \beta_{1} - 31 \beta_{5} - 175 \beta_{6} + 169 \beta_{7} + 525 \beta_{8} ) q^{85} + ( 2086 \beta_{4} - 212 \beta_{9} + 42 \beta_{10} - 42 \beta_{11} ) q^{87} + ( -1040 + 2190 \beta_{1} - 622 \beta_{7} + 622 \beta_{8} ) q^{89} + ( -47 \beta_{2} + 22 \beta_{3} - 43 \beta_{10} - 43 \beta_{11} ) q^{91} + ( 834 \beta_{5} + 284 \beta_{6} + 1372 \beta_{7} + 1372 \beta_{8} ) q^{93} + ( 110 \beta_{2} - 41 \beta_{3} + 1983 \beta_{4} + 35 \beta_{9} - 40 \beta_{10} - 21 \beta_{11} ) q^{95} + ( 100 \beta_{5} + 965 \beta_{6} + 487 \beta_{7} + 487 \beta_{8} ) q^{97} + ( 141 \beta_{2} + 277 \beta_{3} + 69 \beta_{10} + 69 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 60q^{5} - 1676q^{9} + O(q^{10})$$ $$12q + 60q^{5} - 1676q^{9} - 688q^{21} + 1500q^{25} + 21304q^{29} - 109672q^{41} + 75140q^{45} - 32348q^{49} - 64552q^{61} - 160q^{65} + 166352q^{69} - 173764q^{81} - 151360q^{85} - 3720q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 165 x^{10} + 9528 x^{8} + 254984 x^{6} + 3245664 x^{4} + 15975501 x^{2} + 588289$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-170020 \nu^{10} - 24454244 \nu^{8} - 1102611568 \nu^{6} - 20045402992 \nu^{4} - 128060923628 \nu^{2} + 2177766131$$$$)/ 200642913$$ $$\beta_{2}$$ $$=$$ $$($$$$-411520 \nu^{10} - 60010112 \nu^{8} - 2780210368 \nu^{6} - 52857554368 \nu^{4} - 362353494656 \nu^{2} - 34380714880$$$$)/66880971$$ $$\beta_{3}$$ $$=$$ $$($$$$-5323964 \nu^{10} - 786315016 \nu^{8} - 37088343272 \nu^{6} - 711962887784 \nu^{4} - 4837404881512 \nu^{2} - 96708608876$$$$)/ 200642913$$ $$\beta_{4}$$ $$=$$ $$($$$$-1310906 \nu^{11} - 195210058 \nu^{9} - 9277834112 \nu^{7} - 175028468600 \nu^{5} - 1019260920568 \nu^{3} + 2576633734216 \nu$$$$)/ 51297704757$$ $$\beta_{5}$$ $$=$$ $$($$$$1957360 \nu^{11} + 286234304 \nu^{9} + 13567743616 \nu^{7} + 284453317984 \nu^{5} + 2688495819008 \nu^{3} + 8621360914288 \nu$$$$)/ 51297704757$$ $$\beta_{6}$$ $$=$$ $$($$$$-67875062 \nu^{11} - 10019691676 \nu^{9} - 471366569900 \nu^{7} - 8947051389500 \nu^{5} - 57637006550236 \nu^{3} + 38667851889478 \nu$$$$)/ 153893114271$$ $$\beta_{7}$$ $$=$$ $$($$$$85161215 \nu^{11} + 484184090 \nu^{10} + 12477922798 \nu^{9} + 70984599790 \nu^{8} + 581043147182 \nu^{7} + 3308494583420 \nu^{6} + 11012560403798 \nu^{5} + 62703306419780 \nu^{4} + 74871373684486 \nu^{3} + 421179959346250 \nu^{2} + 19752862444721 \nu + 17382114505340$$$$)/ 153893114271$$ $$\beta_{8}$$ $$=$$ $$($$$$85161215 \nu^{11} - 484184090 \nu^{10} + 12477922798 \nu^{9} - 70984599790 \nu^{8} + 581043147182 \nu^{7} - 3308494583420 \nu^{6} + 11012560403798 \nu^{5} - 62703306419780 \nu^{4} + 74871373684486 \nu^{3} - 421179959346250 \nu^{2} + 19752862444721 \nu - 17382114505340$$$$)/ 153893114271$$ $$\beta_{9}$$ $$=$$ $$($$$$175955038 \nu^{11} + 25169463050 \nu^{9} + 1119493454860 \nu^{7} + 19636262498968 \nu^{5} + 109318479348956 \nu^{3} - 173990789836964 \nu$$$$)/ 153893114271$$ $$\beta_{10}$$ $$=$$ $$($$$$-2541138896 \nu^{11} - 3284674432 \nu^{10} - 372595783468 \nu^{9} - 475746157328 \nu^{8} - 17374777187732 \nu^{7} - 21714641655616 \nu^{6} - 329909822608880 \nu^{5} - 401095958765632 \nu^{4} - 2234157186356428 \nu^{3} - 2626945209511376 \nu^{2} - 317433643570484 \nu - 240122345851648$$$$)/ 153893114271$$ $$\beta_{11}$$ $$=$$ $$($$$$2541138896 \nu^{11} - 3284674432 \nu^{10} + 372595783468 \nu^{9} - 475746157328 \nu^{8} + 17374777187732 \nu^{7} - 21714641655616 \nu^{6} + 329909822608880 \nu^{5} - 401095958765632 \nu^{4} + 2234157186356428 \nu^{3} - 2626945209511376 \nu^{2} + 317433643570484 \nu - 240122345851648$$$$)/ 153893114271$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{8} - \beta_{7} - 5 \beta_{6} - 2 \beta_{5} + 40 \beta_{4}$$$$)/160$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} - 48 \beta_{8} + 48 \beta_{7} + 8 \beta_{3} + 5 \beta_{2} + 120 \beta_{1} - 8840$$$$)/320$$ $$\nu^{3}$$ $$=$$ $$($$$$16 \beta_{11} - 16 \beta_{10} - 40 \beta_{9} - 130 \beta_{8} - 130 \beta_{7} + 910 \beta_{6} - 755 \beta_{5} - 3576 \beta_{4}$$$$)/320$$ $$\nu^{4}$$ $$=$$ $$($$$$-80 \beta_{11} - 80 \beta_{10} + 9816 \beta_{8} - 9816 \beta_{7} - 1520 \beta_{3} - 995 \beta_{2} - 14040 \beta_{1} + 876040$$$$)/640$$ $$\nu^{5}$$ $$=$$ $$($$$$-835 \beta_{11} + 835 \beta_{10} + 1930 \beta_{9} + 9416 \beta_{8} + 9416 \beta_{7} - 36920 \beta_{6} + 40567 \beta_{5} + 112850 \beta_{4}$$$$)/160$$ $$\nu^{6}$$ $$=$$ $$($$$$12696 \beta_{11} + 12696 \beta_{10} - 839160 \beta_{8} + 839160 \beta_{7} + 128112 \beta_{3} + 79845 \beta_{2} + 1000440 \beta_{1} - 58009960$$$$)/640$$ $$\nu^{7}$$ $$=$$ $$($$$$142620 \beta_{11} - 142620 \beta_{10} - 319200 \beta_{9} - 1729042 \beta_{8} - 1729042 \beta_{7} + 5885470 \beta_{6} - 6833759 \beta_{5} - 16637360 \beta_{4}$$$$)/320$$ $$\nu^{8}$$ $$=$$ $$($$$$-586649 \beta_{11} - 586649 \beta_{10} + 34087152 \beta_{8} - 34087152 \beta_{7} - 5190248 \beta_{3} - 3135725 \beta_{2} - 38276760 \beta_{1} + 2171360360$$$$)/320$$ $$\nu^{9}$$ $$=$$ $$($$$$-5783906 \beta_{11} + 5783906 \beta_{10} + 12748460 \beta_{9} + 71743835 \beta_{8} + 71743835 \beta_{7} - 233642345 \beta_{6} + 274570960 \beta_{5} + 645384276 \beta_{4}$$$$)/160$$ $$\nu^{10}$$ $$=$$ $$($$$$24339950 \beta_{11} + 24339950 \beta_{10} - 1362127056 \beta_{8} + 1362127056 \beta_{7} + 207342440 \beta_{3} + 123500345 \beta_{2} + 1499141520 \beta_{1} - 84593817040$$$$)/160$$ $$\nu^{11}$$ $$=$$ $$($$$$461870614 \beta_{11} - 461870614 \beta_{10} - 1010984020 \beta_{9} - 5773585979 \beta_{8} - 5773585979 \beta_{7} + 18531101945 \beta_{6} - 21831110698 \beta_{5} - 50835708524 \beta_{4}$$$$)/160$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 8.90339i − 4.23250i − 4.32684i − 5.17887i − 0.192622i − 4.71554i 4.71554i 0.192622i 5.17887i 4.32684i 4.23250i 8.90339i
0 26.2718i 0 −48.8634 27.1545i 0 23.9306i 0 −447.206 0
129.2 0 26.2718i 0 −48.8634 + 27.1545i 0 23.9306i 0 −447.206 0
129.3 0 19.0114i 0 46.9000 30.4203i 0 91.0286i 0 −118.434 0
129.4 0 19.0114i 0 46.9000 + 30.4203i 0 91.0286i 0 −118.434 0
129.5 0 9.81633i 0 16.9634 53.2658i 0 222.821i 0 146.640 0
129.6 0 9.81633i 0 16.9634 + 53.2658i 0 222.821i 0 146.640 0
129.7 0 9.81633i 0 16.9634 53.2658i 0 222.821i 0 146.640 0
129.8 0 9.81633i 0 16.9634 + 53.2658i 0 222.821i 0 146.640 0
129.9 0 19.0114i 0 46.9000 30.4203i 0 91.0286i 0 −118.434 0
129.10 0 19.0114i 0 46.9000 + 30.4203i 0 91.0286i 0 −118.434 0
129.11 0 26.2718i 0 −48.8634 27.1545i 0 23.9306i 0 −447.206 0
129.12 0 26.2718i 0 −48.8634 + 27.1545i 0 23.9306i 0 −447.206 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.c.k 12
4.b odd 2 1 inner 320.6.c.k 12
5.b even 2 1 inner 320.6.c.k 12
8.b even 2 1 160.6.c.c 12
8.d odd 2 1 160.6.c.c 12
20.d odd 2 1 inner 320.6.c.k 12
40.e odd 2 1 160.6.c.c 12
40.f even 2 1 160.6.c.c 12
40.i odd 4 1 800.6.a.z 6
40.i odd 4 1 800.6.a.ba 6
40.k even 4 1 800.6.a.z 6
40.k even 4 1 800.6.a.ba 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.c 12 8.b even 2 1
160.6.c.c 12 8.d odd 2 1
160.6.c.c 12 40.e odd 2 1
160.6.c.c 12 40.f even 2 1
320.6.c.k 12 1.a even 1 1 trivial
320.6.c.k 12 4.b odd 2 1 inner
320.6.c.k 12 5.b even 2 1 inner
320.6.c.k 12 20.d odd 2 1 inner
800.6.a.z 6 40.i odd 4 1
800.6.a.z 6 40.k even 4 1
800.6.a.ba 6 40.i odd 4 1
800.6.a.ba 6 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(320, [\chi])$$:

 $$T_{3}^{6} + 1148 T_{3}^{4} + 350800 T_{3}^{2} + 24038400$$ $$T_{11}^{6} - 746112 T_{11}^{4} + 138141388800 T_{11}^{2} -$$$$69\!\cdots\!00$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 310 T^{2} + 120679 T^{4} - 26699028 T^{6} + 7125974271 T^{8} - 1080903164310 T^{10} + 205891132094649 T^{12} )^{2}$$
$5$ $$( 1 - 30 T + 75 T^{2} + 123500 T^{3} + 234375 T^{4} - 292968750 T^{5} + 30517578125 T^{6} )^{2}$$
$7$ $$( 1 - 42334 T^{2} + 748334111 T^{4} - 10497411086308 T^{6} + 211385864339918639 T^{8} -$$$$33\!\cdots\!34$$$$T^{10} +$$$$22\!\cdots\!49$$$$T^{12} )^{2}$$
$11$ $$( 1 + 220194 T^{2} + 46554422967 T^{4} + 4986677846505148 T^{6} +$$$$12\!\cdots\!67$$$$T^{8} +$$$$14\!\cdots\!94$$$$T^{10} +$$$$17\!\cdots\!01$$$$T^{12} )^{2}$$
$13$ $$( 1 - 921822 T^{2} + 378030377607 T^{4} - 125995996576493956 T^{6} +$$$$52\!\cdots\!43$$$$T^{8} -$$$$17\!\cdots\!22$$$$T^{10} +$$$$26\!\cdots\!49$$$$T^{12} )^{2}$$
$17$ $$( 1 - 7508390 T^{2} + 24616526523247 T^{4} - 45351621021996027220 T^{6} +$$$$49\!\cdots\!03$$$$T^{8} -$$$$30\!\cdots\!90$$$$T^{10} +$$$$81\!\cdots\!49$$$$T^{12} )^{2}$$
$19$ $$( 1 + 8862994 T^{2} + 40251921150215 T^{4} +$$$$12\!\cdots\!80$$$$T^{6} +$$$$24\!\cdots\!15$$$$T^{8} +$$$$33\!\cdots\!94$$$$T^{10} +$$$$23\!\cdots\!01$$$$T^{12} )^{2}$$
$23$ $$( 1 - 27906430 T^{2} + 367961637958719 T^{4} -$$$$29\!\cdots\!08$$$$T^{6} +$$$$15\!\cdots\!31$$$$T^{8} -$$$$47\!\cdots\!30$$$$T^{10} +$$$$71\!\cdots\!49$$$$T^{12} )^{2}$$
$29$ $$( 1 - 5326 T + 46243827 T^{2} - 133362268948 T^{3} + 948514025927223 T^{4} - 2240686724556870526 T^{5} +$$$$86\!\cdots\!49$$$$T^{6} )^{4}$$
$31$ $$( 1 + 10594234 T^{2} - 153174449706673 T^{4} +$$$$19\!\cdots\!88$$$$T^{6} -$$$$12\!\cdots\!73$$$$T^{8} +$$$$71\!\cdots\!34$$$$T^{10} +$$$$55\!\cdots\!01$$$$T^{12} )^{2}$$
$37$ $$( 1 - 234974254 T^{2} + 32080895329334807 T^{4} -$$$$26\!\cdots\!92$$$$T^{6} +$$$$15\!\cdots\!43$$$$T^{8} -$$$$54\!\cdots\!54$$$$T^{10} +$$$$11\!\cdots\!49$$$$T^{12} )^{2}$$
$41$ $$( 1 + 27418 T + 531273543 T^{2} + 6402679071436 T^{3} + 61551334383790143 T^{4} +$$$$36\!\cdots\!18$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6} )^{4}$$
$43$ $$( 1 - 613900486 T^{2} + 182904754783784951 T^{4} -$$$$33\!\cdots\!72$$$$T^{6} +$$$$39\!\cdots\!99$$$$T^{8} -$$$$28\!\cdots\!86$$$$T^{10} +$$$$10\!\cdots\!49$$$$T^{12} )^{2}$$
$47$ $$( 1 - 579920494 T^{2} + 165136673035562991 T^{4} -$$$$38\!\cdots\!48$$$$T^{6} +$$$$86\!\cdots\!59$$$$T^{8} -$$$$16\!\cdots\!94$$$$T^{10} +$$$$14\!\cdots\!49$$$$T^{12} )^{2}$$
$53$ $$( 1 - 2026325390 T^{2} + 1866514545072595447 T^{4} -$$$$99\!\cdots\!20$$$$T^{6} +$$$$32\!\cdots\!03$$$$T^{8} -$$$$61\!\cdots\!90$$$$T^{10} +$$$$53\!\cdots\!49$$$$T^{12} )^{2}$$
$59$ $$( 1 + 55950786 T^{2} + 253972763776609047 T^{4} -$$$$45\!\cdots\!68$$$$T^{6} +$$$$12\!\cdots\!47$$$$T^{8} +$$$$14\!\cdots\!86$$$$T^{10} +$$$$13\!\cdots\!01$$$$T^{12} )^{2}$$
$61$ $$( 1 + 16138 T + 744681843 T^{2} + 29097452426876 T^{3} + 628955530019662743 T^{4} +$$$$11\!\cdots\!38$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6} )^{4}$$
$67$ $$( 1 - 2173336374 T^{2} + 6882868324909910631 T^{4} -$$$$82\!\cdots\!68$$$$T^{6} +$$$$12\!\cdots\!19$$$$T^{8} -$$$$72\!\cdots\!74$$$$T^{10} +$$$$60\!\cdots\!49$$$$T^{12} )^{2}$$
$71$ $$( 1 + 942628714 T^{2} + 4622069102042239647 T^{4} +$$$$48\!\cdots\!68$$$$T^{6} +$$$$15\!\cdots\!47$$$$T^{8} +$$$$99\!\cdots\!14$$$$T^{10} +$$$$34\!\cdots\!01$$$$T^{12} )^{2}$$
$73$ $$( 1 - 4736672630 T^{2} + 15959781190221205247 T^{4} -$$$$37\!\cdots\!40$$$$T^{6} +$$$$68\!\cdots\!03$$$$T^{8} -$$$$87\!\cdots\!30$$$$T^{10} +$$$$79\!\cdots\!49$$$$T^{12} )^{2}$$
$79$ $$( 1 + 454164826 T^{2} + 18487132928554429487 T^{4} -$$$$50\!\cdots\!28$$$$T^{6} +$$$$17\!\cdots\!87$$$$T^{8} +$$$$40\!\cdots\!26$$$$T^{10} +$$$$84\!\cdots\!01$$$$T^{12} )^{2}$$
$83$ $$( 1 - 11599602390 T^{2} + 72197091204656028039 T^{4} -$$$$31\!\cdots\!48$$$$T^{6} +$$$$11\!\cdots\!11$$$$T^{8} -$$$$27\!\cdots\!90$$$$T^{10} +$$$$37\!\cdots\!49$$$$T^{12} )^{2}$$
$89$ $$( 1 + 930 T + 9446079447 T^{2} + 231471892494140 T^{3} + 52747469192025044703 T^{4} +$$$$28\!\cdots\!30$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6} )^{4}$$
$97$ $$( 1 - 35609466630 T^{2} +$$$$64\!\cdots\!47$$$$T^{4} -$$$$68\!\cdots\!40$$$$T^{6} +$$$$47\!\cdots\!03$$$$T^{8} -$$$$19\!\cdots\!30$$$$T^{10} +$$$$40\!\cdots\!49$$$$T^{12} )^{2}$$