# Properties

 Label 320.6.c.i Level 320 Weight 6 Character orbit 320.c Analytic conductor 51.323 Analytic rank 0 Dimension 8 CM no Inner twists 2

# 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{31}\cdot 5^{3}$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + ( -125 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + ( -125 + \beta_{2} - \beta_{3} ) q^{9} + ( -92 - \beta_{2} + \beta_{4} ) q^{11} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{13} + ( 124 + 10 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{15} + ( -40 \beta_{1} - 9 \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{17} + ( 172 + 8 \beta_{1} - 19 \beta_{2} - \beta_{4} - 4 \beta_{7} ) q^{19} + ( -248 + 14 \beta_{1} - 27 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 7 \beta_{7} ) q^{21} + ( 22 \beta_{1} - \beta_{2} + 10 \beta_{5} + 9 \beta_{6} + 4 \beta_{7} ) q^{23} + ( -267 + 80 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} + 9 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{25} + ( 212 \beta_{1} + 42 \beta_{2} + 8 \beta_{5} - 6 \beta_{6} - 8 \beta_{7} ) q^{27} + ( -734 + 12 \beta_{1} - 28 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} - 6 \beta_{7} ) q^{29} + ( -528 + 12 \beta_{2} - 12 \beta_{3} ) q^{31} + ( 188 \beta_{1} - 87 \beta_{2} - 43 \beta_{5} - 24 \beta_{6} + 4 \beta_{7} ) q^{33} + ( 2404 - 215 \beta_{1} + 19 \beta_{2} + 12 \beta_{3} + \beta_{4} + 34 \beta_{5} - 18 \beta_{6} - 8 \beta_{7} ) q^{35} + ( 410 \beta_{1} - 5 \beta_{2} - 26 \beta_{5} + 16 \beta_{6} - \beta_{7} ) q^{37} + ( 376 + 16 \beta_{1} - 46 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 8 \beta_{7} ) q^{39} + ( 2950 + 28 \beta_{1} - 39 \beta_{2} - 15 \beta_{3} - 16 \beta_{4} - 14 \beta_{7} ) q^{41} + ( -243 \beta_{1} - 54 \beta_{2} - 108 \beta_{5} + 54 \beta_{6} ) q^{43} + ( 3541 - 430 \beta_{1} - 151 \beta_{2} - 10 \beta_{3} - 20 \beta_{4} - 75 \beta_{5} - 60 \beta_{6} - 5 \beta_{7} ) q^{45} + ( -532 \beta_{1} + 81 \beta_{2} - 50 \beta_{5} - 9 \beta_{6} - 28 \beta_{7} ) q^{47} + ( -5625 + 24 \beta_{1} - 75 \beta_{2} - 9 \beta_{3} + 24 \beta_{4} - 12 \beta_{7} ) q^{49} + ( -15600 + 56 \beta_{1} - 52 \beta_{2} - 72 \beta_{3} - 16 \beta_{4} - 28 \beta_{7} ) q^{51} + ( 134 \beta_{1} + 70 \beta_{2} + 191 \beta_{5} - 36 \beta_{6} + 17 \beta_{7} ) q^{53} + ( -1876 + 560 \beta_{1} - 194 \beta_{2} + 30 \beta_{3} - 15 \beta_{4} - 125 \beta_{5} - 20 \beta_{6} - 70 \beta_{7} ) q^{55} + ( -756 \beta_{1} + 327 \beta_{2} + 83 \beta_{5} + 144 \beta_{6} - 20 \beta_{7} ) q^{57} + ( 11460 - 8 \beta_{1} - 41 \beta_{2} + 48 \beta_{3} + 13 \beta_{4} + 4 \beta_{7} ) q^{59} + ( -15482 - 118 \beta_{1} + 269 \beta_{2} + 18 \beta_{3} + 8 \beta_{4} + 59 \beta_{7} ) q^{61} + ( 330 \beta_{1} + 693 \beta_{2} + 274 \beta_{5} + 39 \beta_{6} - 76 \beta_{7} ) q^{63} + ( -9008 - 340 \beta_{1} + 57 \beta_{2} + 21 \beta_{3} - 12 \beta_{4} + 182 \beta_{5} - 144 \beta_{6} - 34 \beta_{7} ) q^{65} + ( -115 \beta_{1} + 214 \beta_{2} + 124 \beta_{5} + 10 \beta_{6} - 16 \beta_{7} ) q^{67} + ( 9592 - 78 \beta_{1} + 133 \beta_{2} + 26 \beta_{3} + 36 \beta_{4} + 39 \beta_{7} ) q^{69} + ( 15704 + 240 \beta_{1} - 494 \beta_{2} - 108 \beta_{3} + 2 \beta_{4} - 120 \beta_{7} ) q^{71} + ( 1124 \beta_{1} - 103 \beta_{2} - 311 \beta_{5} + 8 \beta_{6} - 40 \beta_{7} ) q^{73} + ( 27848 + 995 \beta_{1} + 180 \beta_{2} + 52 \beta_{3} - 14 \beta_{4} + 194 \beta_{5} + 72 \beta_{6} - 88 \beta_{7} ) q^{75} + ( -2384 \beta_{1} + 217 \beta_{2} - 111 \beta_{5} + 468 \beta_{6} + 28 \beta_{7} ) q^{77} + ( -5408 + 208 \beta_{1} - 608 \beta_{2} + 60 \beta_{3} + 28 \beta_{4} - 104 \beta_{7} ) q^{79} + ( 51209 - 236 \beta_{1} + 381 \beta_{2} + 121 \beta_{3} + 88 \beta_{4} + 118 \beta_{7} ) q^{81} + ( 1669 \beta_{1} + 302 \beta_{2} - 236 \beta_{5} - 342 \beta_{6} - 176 \beta_{7} ) q^{83} + ( 36720 - 320 \beta_{1} - 55 \beta_{2} + 90 \beta_{3} + 80 \beta_{4} - 125 \beta_{5} - 60 \beta_{6} + 40 \beta_{7} ) q^{85} + ( 2366 \beta_{1} + 264 \beta_{2} - 64 \beta_{5} + 48 \beta_{6} - 56 \beta_{7} ) q^{87} + ( -5238 - 156 \beta_{1} + 312 \beta_{2} + 126 \beta_{3} - 48 \beta_{4} + 78 \beta_{7} ) q^{89} + ( -60952 + 200 \beta_{1} - 730 \beta_{2} + 120 \beta_{3} + 110 \beta_{4} - 100 \beta_{7} ) q^{91} + ( 4488 \beta_{1} + 504 \beta_{2} + 96 \beta_{5} - 72 \beta_{6} - 96 \beta_{7} ) q^{93} + ( -55324 - 2160 \beta_{1} + 234 \beta_{2} + 30 \beta_{3} + 95 \beta_{4} - 55 \beta_{5} + 180 \beta_{6} + 30 \beta_{7} ) q^{95} + ( 1544 \beta_{1} + 107 \beta_{2} + 49 \beta_{5} + 488 \beta_{6} + 86 \beta_{7} ) q^{97} + ( 33356 + 808 \beta_{1} - 1803 \beta_{2} - 152 \beta_{3} - 65 \beta_{4} - 404 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} - 1000q^{9} + O(q^{10})$$ $$8q - 8q^{5} - 1000q^{9} - 736q^{11} + 992q^{15} + 1376q^{19} - 1984q^{21} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} + 3008q^{39} + 23600q^{41} + 28328q^{45} - 45000q^{49} - 124800q^{51} - 15008q^{55} + 91680q^{59} - 123856q^{61} - 72064q^{65} + 76736q^{69} + 125632q^{71} + 222784q^{75} - 43264q^{79} + 409672q^{81} + 293760q^{85} - 41904q^{89} - 487616q^{91} - 442592q^{95} + 266848q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-182 \nu^{7} - 7420 \nu^{5} - 82106 \nu^{3} - 160458 \nu$$$$)/639$$ $$\beta_{2}$$ $$=$$ $$($$$$176 \nu^{7} - 66 \nu^{6} + 7372 \nu^{5} - 2232 \nu^{4} + 86042 \nu^{3} - 19752 \nu^{2} + 203676 \nu - 29412$$$$)/639$$ $$\beta_{3}$$ $$=$$ $$($$$$176 \nu^{7} - 438 \nu^{6} + 7372 \nu^{5} - 15432 \nu^{4} + 86042 \nu^{3} - 113112 \nu^{2} + 203676 \nu + 106884$$$$)/639$$ $$\beta_{4}$$ $$=$$ $$($$$$176 \nu^{7} - 282 \nu^{6} + 7372 \nu^{5} - 17592 \nu^{4} + 86042 \nu^{3} - 259752 \nu^{2} + 203676 \nu - 390564$$$$)/639$$ $$\beta_{5}$$ $$=$$ $$($$$$656 \nu^{7} + 66 \nu^{6} + 26548 \nu^{5} + 2232 \nu^{4} + 295142 \nu^{3} + 19752 \nu^{2} + 646692 \nu + 29412$$$$)/639$$ $$\beta_{6}$$ $$=$$ $$($$$$974 \nu^{7} + 66 \nu^{6} + 40168 \nu^{5} + 2232 \nu^{4} + 454172 \nu^{3} + 19752 \nu^{2} + 991374 \nu + 29412$$$$)/639$$ $$\beta_{7}$$ $$=$$ $$($$$$-1244 \nu^{7} - 330 \nu^{6} - 51700 \nu^{5} - 11160 \nu^{4} - 594422 \nu^{3} - 98760 \nu^{2} - 1339296 \nu - 147060$$$$)/639$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$8 \beta_{7} + 24 \beta_{6} + 3 \beta_{5} - 13 \beta_{2} + 72 \beta_{1}$$$$)/1280$$ $$\nu^{2}$$ $$=$$ $$($$$$-7 \beta_{7} - \beta_{4} + 13 \beta_{3} - 47 \beta_{2} + 14 \beta_{1} - 6560$$$$)/640$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} - 6 \beta_{6} + \beta_{5} + 5 \beta_{2} - 10 \beta_{1}$$$$)/16$$ $$\nu^{4}$$ $$=$$ $$($$$$169 \beta_{7} - 29 \beta_{4} - 283 \beta_{3} + 1157 \beta_{2} - 338 \beta_{1} + 121760$$$$)/640$$ $$\nu^{5}$$ $$=$$ $$($$$$3016 \beta_{7} + 10296 \beta_{6} - 3443 \beta_{5} - 8227 \beta_{2} + 14120 \beta_{1}$$$$)/1280$$ $$\nu^{6}$$ $$=$$ $$($$$$-53 \beta_{7} + 16 \beta_{4} + 71 \beta_{3} - 352 \beta_{2} + 106 \beta_{1} - 30496$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$-57832 \beta_{7} - 224376 \beta_{6} + 101633 \beta_{5} + 166417 \beta_{2} - 282728 \beta_{1}$$$$)/1280$$

## 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
 1.64654i − 0.0965878i 4.73066i 3.98753i − 3.98753i − 4.73066i 0.0965878i − 1.64654i
0 28.9338i 0 13.1588 + 54.3309i 0 146.828i 0 −594.165 0
129.2 0 24.1383i 0 −46.7401 30.6653i 0 179.876i 0 −339.657 0
129.3 0 5.49000i 0 53.0051 17.7613i 0 188.968i 0 212.860 0
129.4 0 4.69449i 0 −23.4238 50.7575i 0 10.2635i 0 220.962 0
129.5 0 4.69449i 0 −23.4238 + 50.7575i 0 10.2635i 0 220.962 0
129.6 0 5.49000i 0 53.0051 + 17.7613i 0 188.968i 0 212.860 0
129.7 0 24.1383i 0 −46.7401 + 30.6653i 0 179.876i 0 −339.657 0
129.8 0 28.9338i 0 13.1588 54.3309i 0 146.828i 0 −594.165 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 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.i 8
4.b odd 2 1 320.6.c.j 8
5.b even 2 1 inner 320.6.c.i 8
8.b even 2 1 80.6.c.d 8
8.d odd 2 1 40.6.c.a 8
20.d odd 2 1 320.6.c.j 8
24.f even 2 1 360.6.f.b 8
24.h odd 2 1 720.6.f.n 8
40.e odd 2 1 40.6.c.a 8
40.f even 2 1 80.6.c.d 8
40.i odd 4 1 400.6.a.z 4
40.i odd 4 1 400.6.a.ba 4
40.k even 4 1 200.6.a.j 4
40.k even 4 1 200.6.a.k 4
120.i odd 2 1 720.6.f.n 8
120.m even 2 1 360.6.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 8.d odd 2 1
40.6.c.a 8 40.e odd 2 1
80.6.c.d 8 8.b even 2 1
80.6.c.d 8 40.f even 2 1
200.6.a.j 4 40.k even 4 1
200.6.a.k 4 40.k even 4 1
320.6.c.i 8 1.a even 1 1 trivial
320.6.c.i 8 5.b even 2 1 inner
320.6.c.j 8 4.b odd 2 1
320.6.c.j 8 20.d odd 2 1
360.6.f.b 8 24.f even 2 1
360.6.f.b 8 120.m even 2 1
400.6.a.z 4 40.i odd 4 1
400.6.a.ba 4 40.i odd 4 1
720.6.f.n 8 24.h odd 2 1
720.6.f.n 8 120.i odd 2 1

## Hecke kernels

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

 $$T_{3}^{8} + 1472 T_{3}^{6} + 562528 T_{3}^{4} + 26394624 T_{3}^{2} + 324000000$$ $$T_{11}^{4} + 368 T_{11}^{3} - 468000 T_{11}^{2} - 126641408 T_{11} + 37397137664$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 472 T^{2} + 69724 T^{4} - 20119464 T^{6} + 8439593958 T^{8} - 1188034229736 T^{10} + 243112555575324 T^{12} - 97180614348674328 T^{14} + 12157665459056928801 T^{16}$$
$5$ $$1 + 8 T + 1100 T^{2} - 113000 T^{3} - 438250 T^{4} - 353125000 T^{5} + 10742187500 T^{6} + 244140625000 T^{7} + 95367431640625 T^{8}$$
$7$ $$1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} -$$$$10\!\cdots\!84$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{12} -$$$$10\!\cdots\!72$$$$T^{14} +$$$$63\!\cdots\!01$$$$T^{16}$$
$11$ $$( 1 + 368 T + 176204 T^{2} + 51158896 T^{3} + 42277949270 T^{4} + 8239191359696 T^{5} + 4570277964394604 T^{6} + 1537227326344959568 T^{7} +$$$$67\!\cdots\!01$$$$T^{8} )^{2}$$
$13$ $$1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} +$$$$15\!\cdots\!86$$$$T^{8} -$$$$60\!\cdots\!16$$$$T^{10} +$$$$20\!\cdots\!56$$$$T^{12} -$$$$41\!\cdots\!28$$$$T^{14} +$$$$36\!\cdots\!01$$$$T^{16}$$
$17$ $$1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} +$$$$45\!\cdots\!30$$$$T^{8} -$$$$63\!\cdots\!32$$$$T^{10} +$$$$71\!\cdots\!44$$$$T^{12} -$$$$51\!\cdots\!28$$$$T^{14} +$$$$16\!\cdots\!01$$$$T^{16}$$
$19$ $$( 1 - 688 T + 5308396 T^{2} - 6058136368 T^{3} + 15069081422710 T^{4} - 15000545402668432 T^{5} + 32546127598645797196 T^{6} -$$$$10\!\cdots\!12$$$$T^{7} +$$$$37\!\cdots\!01$$$$T^{8} )^{2}$$
$23$ $$1 - 34675896 T^{2} + 569897415616828 T^{4} -$$$$59\!\cdots\!20$$$$T^{6} +$$$$44\!\cdots\!82$$$$T^{8} -$$$$24\!\cdots\!80$$$$T^{10} +$$$$97\!\cdots\!28$$$$T^{12} -$$$$24\!\cdots\!04$$$$T^{14} +$$$$29\!\cdots\!01$$$$T^{16}$$
$29$ $$( 1 + 2936 T + 58625996 T^{2} + 77951973928 T^{3} + 1466411094282230 T^{4} + 1598884552081323272 T^{5} +$$$$24\!\cdots\!96$$$$T^{6} +$$$$25\!\cdots\!64$$$$T^{7} +$$$$17\!\cdots\!01$$$$T^{8} )^{2}$$
$31$ $$( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} +$$$$65\!\cdots\!04$$$$T^{6} +$$$$49\!\cdots\!12$$$$T^{7} +$$$$67\!\cdots\!01$$$$T^{8} )^{2}$$
$37$ $$1 - 251774632 T^{2} + 38631208311838780 T^{4} -$$$$41\!\cdots\!52$$$$T^{6} +$$$$32\!\cdots\!34$$$$T^{8} -$$$$19\!\cdots\!48$$$$T^{10} +$$$$89\!\cdots\!80$$$$T^{12} -$$$$27\!\cdots\!68$$$$T^{14} +$$$$53\!\cdots\!01$$$$T^{16}$$
$41$ $$( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} -$$$$34\!\cdots\!76$$$$T^{5} +$$$$45\!\cdots\!40$$$$T^{6} -$$$$18\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{2}$$
$43$ $$1 - 283211672 T^{2} + 48021567531024796 T^{4} -$$$$54\!\cdots\!84$$$$T^{6} +$$$$43\!\cdots\!06$$$$T^{8} -$$$$11\!\cdots\!16$$$$T^{10} +$$$$22\!\cdots\!96$$$$T^{12} -$$$$28\!\cdots\!28$$$$T^{14} +$$$$21\!\cdots\!01$$$$T^{16}$$
$47$ $$1 - 963352312 T^{2} + 473832864723586300 T^{4} -$$$$15\!\cdots\!72$$$$T^{6} +$$$$40\!\cdots\!54$$$$T^{8} -$$$$83\!\cdots\!28$$$$T^{10} +$$$$13\!\cdots\!00$$$$T^{12} -$$$$14\!\cdots\!88$$$$T^{14} +$$$$76\!\cdots\!01$$$$T^{16}$$
$53$ $$1 - 1183385640 T^{2} + 874785099161623996 T^{4} -$$$$49\!\cdots\!80$$$$T^{6} +$$$$23\!\cdots\!06$$$$T^{8} -$$$$86\!\cdots\!20$$$$T^{10} +$$$$26\!\cdots\!96$$$$T^{12} -$$$$63\!\cdots\!60$$$$T^{14} +$$$$93\!\cdots\!01$$$$T^{16}$$
$59$ $$( 1 - 45840 T + 3064286732 T^{2} - 94721285480976 T^{3} + 3348109683185502486 T^{4} -$$$$67\!\cdots\!24$$$$T^{5} +$$$$15\!\cdots\!32$$$$T^{6} -$$$$16\!\cdots\!60$$$$T^{7} +$$$$26\!\cdots\!01$$$$T^{8} )^{2}$$
$61$ $$( 1 + 61928 T + 3903014764 T^{2} + 145287706763384 T^{3} + 5198153942066716726 T^{4} +$$$$12\!\cdots\!84$$$$T^{5} +$$$$27\!\cdots\!64$$$$T^{6} +$$$$37\!\cdots\!28$$$$T^{7} +$$$$50\!\cdots\!01$$$$T^{8} )^{2}$$
$67$ $$1 - 9281919064 T^{2} + 39492482666681482588 T^{4} -$$$$10\!\cdots\!40$$$$T^{6} +$$$$16\!\cdots\!62$$$$T^{8} -$$$$18\!\cdots\!60$$$$T^{10} +$$$$13\!\cdots\!88$$$$T^{12} -$$$$56\!\cdots\!36$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16}$$
$71$ $$( 1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} -$$$$33\!\cdots\!96$$$$T^{5} +$$$$11\!\cdots\!56$$$$T^{6} -$$$$36\!\cdots\!16$$$$T^{7} +$$$$10\!\cdots\!01$$$$T^{8} )^{2}$$
$73$ $$1 - 9140679496 T^{2} + 46078306824990298588 T^{4} -$$$$15\!\cdots\!60$$$$T^{6} +$$$$37\!\cdots\!02$$$$T^{8} -$$$$66\!\cdots\!40$$$$T^{10} +$$$$85\!\cdots\!88$$$$T^{12} -$$$$72\!\cdots\!04$$$$T^{14} +$$$$34\!\cdots\!01$$$$T^{16}$$
$79$ $$( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} +$$$$10\!\cdots\!12$$$$T^{5} +$$$$67\!\cdots\!04$$$$T^{6} +$$$$63\!\cdots\!68$$$$T^{7} +$$$$89\!\cdots\!01$$$$T^{8} )^{2}$$
$83$ $$1 - 6759897816 T^{2} + 40943120759345365468 T^{4} -$$$$86\!\cdots\!80$$$$T^{6} +$$$$39\!\cdots\!42$$$$T^{8} -$$$$13\!\cdots\!20$$$$T^{10} +$$$$98\!\cdots\!68$$$$T^{12} -$$$$25\!\cdots\!84$$$$T^{14} +$$$$57\!\cdots\!01$$$$T^{16}$$
$89$ $$( 1 + 20952 T + 16118164796 T^{2} + 497915996461992 T^{3} +$$$$11\!\cdots\!30$$$$T^{4} +$$$$27\!\cdots\!08$$$$T^{5} +$$$$50\!\cdots\!96$$$$T^{6} +$$$$36\!\cdots\!48$$$$T^{7} +$$$$97\!\cdots\!01$$$$T^{8} )^{2}$$
$97$ $$1 - 45263915272 T^{2} +$$$$10\!\cdots\!40$$$$T^{4} -$$$$14\!\cdots\!12$$$$T^{6} +$$$$15\!\cdots\!94$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$55\!\cdots\!40$$$$T^{12} -$$$$18\!\cdots\!28$$$$T^{14} +$$$$29\!\cdots\!01$$$$T^{16}$$