# Properties

 Label 1008.4.bt.a Level $1008$ Weight $4$ Character orbit 1008.bt Analytic conductor $59.474$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1008.bt (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$59.4739252858$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{13} q^{5} + ( -3 + \beta_{4} ) q^{7} +O(q^{10})$$ $$q -\beta_{13} q^{5} + ( -3 + \beta_{4} ) q^{7} + ( -\beta_{1} - 3 \beta_{6} + \beta_{14} ) q^{11} + ( 9 - 18 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + \beta_{9} - \beta_{10} ) q^{13} + ( 2 \beta_{6} - 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{17} + ( 28 + 26 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + 4 \beta_{9} + 4 \beta_{10} ) q^{19} + ( 4 \beta_{1} - 11 \beta_{6} + 4 \beta_{8} - 17 \beta_{11} + 6 \beta_{12} - 3 \beta_{13} + \beta_{15} ) q^{23} + ( -\beta_{2} - 5 \beta_{3} - 5 \beta_{5} - \beta_{7} - \beta_{9} ) q^{25} + ( -4 \beta_{6} + 10 \beta_{8} - 2 \beta_{11} - 4 \beta_{12} + 8 \beta_{13} + \beta_{14} + \beta_{15} ) q^{29} + ( -96 + 46 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} - 3 \beta_{9} + 4 \beta_{10} ) q^{31} + ( -7 \beta_{1} - 36 \beta_{6} - 7 \beta_{8} - 5 \beta_{11} + 7 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{35} + ( -148 + 149 \beta_{2} + \beta_{4} + 6 \beta_{5} - 3 \beta_{7} + 4 \beta_{9} - 4 \beta_{10} ) q^{37} + ( -14 \beta_{1} - 25 \beta_{6} - 7 \beta_{8} - 10 \beta_{11} - 5 \beta_{12} - 3 \beta_{14} + 3 \beta_{15} ) q^{41} + ( -10 - 17 \beta_{3} + 10 \beta_{4} - 10 \beta_{7} + 4 \beta_{9} - 6 \beta_{10} ) q^{43} + ( 25 \beta_{6} - 25 \beta_{11} + 2 \beta_{13} + 8 \beta_{14} + 4 \beta_{15} ) q^{47} + ( -63 + 189 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} - 7 \beta_{7} - 14 \beta_{9} - 21 \beta_{10} ) q^{49} + ( 5 \beta_{1} - 34 \beta_{6} - 20 \beta_{12} - 20 \beta_{13} + 3 \beta_{14} ) q^{53} + ( -14 - 32 \beta_{2} + 6 \beta_{3} - 30 \beta_{4} + 12 \beta_{5} - 30 \beta_{7} - 17 \beta_{9} - 13 \beta_{10} ) q^{55} + ( -14 \beta_{1} + 46 \beta_{6} - 28 \beta_{8} + 126 \beta_{11} - 17 \beta_{12} + 17 \beta_{13} ) q^{59} + ( -48 - 56 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} - 5 \beta_{5} + 19 \beta_{7} + 27 \beta_{9} + 27 \beta_{10} ) q^{61} + ( 7 \beta_{1} + 73 \beta_{6} + 7 \beta_{8} + 109 \beta_{11} - 36 \beta_{12} + 18 \beta_{13} - 4 \beta_{15} ) q^{65} + ( 3 - 40 \beta_{2} + 16 \beta_{3} + 3 \beta_{4} + 16 \beta_{5} + 8 \beta_{7} + 8 \beta_{9} + 3 \beta_{10} ) q^{67} + ( -12 \beta_{6} + 12 \beta_{8} - \beta_{11} - 12 \beta_{12} + 24 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{71} + ( 320 - 157 \beta_{2} + 19 \beta_{3} + 6 \beta_{4} - 19 \beta_{5} - 9 \beta_{7} + 9 \beta_{9} - 6 \beta_{10} ) q^{73} + ( -40 \beta_{1} + 99 \beta_{6} - 43 \beta_{8} + 12 \beta_{11} - 11 \beta_{12} + 26 \beta_{13} + 3 \beta_{14} - 4 \beta_{15} ) q^{77} + ( 267 - 248 \beta_{2} + 19 \beta_{4} - 4 \beta_{5} - 6 \beta_{7} + 25 \beta_{9} - 25 \beta_{10} ) q^{79} + ( 132 \beta_{6} + 79 \beta_{11} - 26 \beta_{12} - 5 \beta_{14} + 5 \beta_{15} ) q^{83} + ( -289 - 15 \beta_{3} + 23 \beta_{4} - 23 \beta_{7} - 17 \beta_{9} - 40 \beta_{10} ) q^{85} + ( -14 \beta_{1} - 54 \beta_{6} + 14 \beta_{8} + 54 \beta_{11} + 72 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{89} + ( -3 - 114 \beta_{2} + 7 \beta_{3} + 15 \beta_{4} - 63 \beta_{5} + 21 \beta_{7} + 3 \beta_{9} + 14 \beta_{10} ) q^{91} + ( -42 \beta_{1} - 53 \beta_{6} - 44 \beta_{12} - 44 \beta_{13} + 12 \beta_{14} ) q^{95} + ( 365 - 816 \beta_{2} + 11 \beta_{3} - 43 \beta_{4} + 22 \beta_{5} - 43 \beta_{7} - 36 \beta_{9} - 7 \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 56q^{7} + O(q^{10})$$ $$16q - 56q^{7} + 612q^{19} - 20q^{25} - 1128q^{31} - 1196q^{37} - 328q^{43} + 784q^{49} - 1632q^{61} - 308q^{67} + 4068q^{73} + 2176q^{79} - 4608q^{85} - 924q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2582250337 \nu^{14} + 122400945426 \nu^{12} - 4183683881139 \nu^{10} + 68992918331690 \nu^{8} - 834373374448305 \nu^{6} + 2691078814742472 \nu^{4} - 8010514076168368 \nu^{2} + 4353170949761400$$$$)/ 3653393210006400$$ $$\beta_{3}$$ $$=$$ $$($$$$-50455191 \nu^{14} + 2307248098 \nu^{12} - 75951189685 \nu^{10} + 1149235164582 \nu^{8} - 12455835189187 \nu^{6} + 11817118117336 \nu^{4} - 6433658902800 \nu^{2} + 266811223623624$$$$)/ 36533932100064$$ $$\beta_{4}$$ $$=$$ $$($$$$3065808613 \nu^{14} - 119299855274 \nu^{12} + 3755758223311 \nu^{10} - 39894438005410 \nu^{8} + 304761942701045 \nu^{6} + 5181178973387672 \nu^{4} - 13365801003441168 \nu^{2} + 37862868003762600$$$$)/ 1565739947145600$$ $$\beta_{5}$$ $$=$$ $$($$$$-14645418817 \nu^{14} + 682142342066 \nu^{12} - 23315734296099 \nu^{10} + 380321573724090 \nu^{8} - 4649975584933505 \nu^{6} + 14997423418452552 \nu^{4} - 55639302718135088 \nu^{2} + 3899872052955000$$$$)/ 3653393210006400$$ $$\beta_{6}$$ $$=$$ $$($$$$-937699527 \nu^{15} + 42257710318 \nu^{13} - 1411537509445 \nu^{11} + 21358303256454 \nu^{9} - 240064275625255 \nu^{7} + 219618751797592 \nu^{5} - 119568250371600 \nu^{3} - 4112177675462712 \nu$$$$)/ 730678642001280$$ $$\beta_{7}$$ $$=$$ $$($$$$128271858533 \nu^{14} - 6298544893834 \nu^{12} + 217538780633951 \nu^{10} - 3754243087666610 \nu^{8} + 46494742729915045 \nu^{6} - 193106893664041448 \nu^{4} + 505510895322272112 \nu^{2} - 316783969979117400$$$$)/ 10960179630019200$$ $$\beta_{8}$$ $$=$$ $$($$$$-2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + 68992918331690 \nu^{9} - 834373374448305 \nu^{7} + 2691078814742472 \nu^{5} - 8010514076168368 \nu^{3} + 699777739755000 \nu$$$$)/ 913348302501600$$ $$\beta_{9}$$ $$=$$ $$($$$$101634687971 \nu^{14} - 4897105890358 \nu^{12} + 168224430945137 \nu^{10} - 2836021484362070 \nu^{8} + 34614407806423915 \nu^{6} - 125983315626057176 \nu^{4} + 305685554606760144 \nu^{2} - 145127159268013800$$$$)/ 5480089815009600$$ $$\beta_{10}$$ $$=$$ $$($$$$-113844178871 \nu^{14} + 5451040376758 \nu^{12} - 186603617326637 \nu^{10} + 3114121239923870 \nu^{8} - 37705694334280615 \nu^{6} + 128842902385283576 \nu^{4} - 307242415224480144 \nu^{2} + 54421465755227400$$$$)/ 5480089815009600$$ $$\beta_{11}$$ $$=$$ $$($$$$253797 \nu^{15} - 12085406 \nu^{13} + 413081109 \nu^{11} - 6837155440 \nu^{9} + 82382868455 \nu^{7} - 265706935032 \nu^{5} + 646432035208 \nu^{3} - 69093405000 \nu$$$$)/ 49371246000$$ $$\beta_{12}$$ $$=$$ $$($$$$360099673999 \nu^{15} - 17075216598302 \nu^{13} + 583205171768653 \nu^{11} - 9610560627221830 \nu^{9} + 115843575769268735 \nu^{7} - 369284393770126744 \nu^{5} + 1022975623455773136 \nu^{3} - 702499753463272200 \nu$$$$)/ 54800898150096000$$ $$\beta_{13}$$ $$=$$ $$($$$$-383305735249 \nu^{15} + 18114452956802 \nu^{13} - 618137711412403 \nu^{11} + 10139133031594330 \nu^{9} - 121958132016661985 \nu^{7} + 374719488978256744 \nu^{5} - 1025934682367273136 \nu^{3} - 362143867944097800 \nu$$$$)/ 54800898150096000$$ $$\beta_{14}$$ $$=$$ $$($$$$-1280411313 \nu^{15} + 57618445232 \nu^{13} - 1927428289955 \nu^{11} + 29164366973226 \nu^{9} - 328956825464915 \nu^{7} + 299885332403048 \nu^{5} - 163268228300400 \nu^{3} - 7474638384004248 \nu$$$$)/ 91334830250160$$ $$\beta_{15}$$ $$=$$ $$($$$$-64647369298 \nu^{15} + 3114318637179 \nu^{13} - 106972651093931 \nu^{11} + 1802487525141385 \nu^{9} - 22027100507637720 \nu^{7} + 80556943300247288 \nu^{5} - 212250439965289272 \nu^{3} + 115355458949385600 \nu$$$$)/ 1141685378127000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} - \beta_{4} - 25 \beta_{2} + 24$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{13} - \beta_{12} + \beta_{11} - 5 \beta_{8} - \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$-13 \beta_{10} - 12 \beta_{9} - 12 \beta_{7} + 12 \beta_{5} - 13 \beta_{4} + 12 \beta_{3} - 259 \beta_{2} - 13$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{15} + 27 \beta_{13} - 54 \beta_{12} + 13 \beta_{11} - 110 \beta_{8} - 41 \beta_{6} - 110 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-283 \beta_{10} - 314 \beta_{9} + 31 \beta_{7} - 31 \beta_{4} + 259 \beta_{3} - 5751$$ $$\nu^{7}$$ $$=$$ $$86 \beta_{14} - 659 \beta_{13} - 659 \beta_{12} - 722 \beta_{6} - 2457 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$831 \beta_{10} - 831 \beta_{9} + 7510 \beta_{7} - 5527 \beta_{5} + 6679 \beta_{4} + 127972 \beta_{2} - 121293$$ $$\nu^{9}$$ $$=$$ $$2814 \beta_{15} + 2814 \beta_{14} - 31702 \beta_{13} + 15851 \beta_{12} + 9607 \beta_{11} + 55231 \beta_{8} + 15851 \beta_{6}$$ $$\nu^{10}$$ $$=$$ $$179186 \beta_{10} + 157707 \beta_{9} + 157707 \beta_{7} - 118179 \beta_{5} + 179186 \beta_{4} - 118179 \beta_{3} + 2880488 \beta_{2} + 179186$$ $$\nu^{11}$$ $$=$$ $$82486 \beta_{15} - 379851 \beta_{13} + 759702 \beta_{12} + 395191 \beta_{11} + 1247719 \beta_{8} + 1154893 \beta_{6} + 1247719 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$3725147 \beta_{10} + 4269970 \beta_{9} - 544823 \beta_{7} + 544823 \beta_{4} - 2537675 \beta_{3} + 65697303$$ $$\nu^{13}$$ $$=$$ $$-2277118 \beta_{14} + 9084763 \beta_{13} + 9084763 \beta_{12} + 21877186 \beta_{6} + 28315551 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-13638999 \beta_{10} + 13638999 \beta_{9} - 101655338 \beta_{7} + 54753563 \beta_{5} - 88016339 \beta_{4} - 1480259600 \beta_{2} + 1392243261$$ $$\nu^{15}$$ $$=$$ $$-60540774 \beta_{15} - 60540774 \beta_{14} + 433899350 \beta_{13} - 216949675 \beta_{12} - 373149623 \beta_{11} - 645326063 \beta_{8} - 216949675 \beta_{6}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −1.57646 − 0.910170i 4.21355 + 2.43270i −0.648633 − 0.374489i −3.91663 − 2.26127i 3.91663 + 2.26127i 0.648633 + 0.374489i −4.21355 − 2.43270i 1.57646 + 0.910170i −1.57646 + 0.910170i 4.21355 − 2.43270i −0.648633 + 0.374489i −3.91663 + 2.26127i 3.91663 − 2.26127i 0.648633 − 0.374489i −4.21355 + 2.43270i 1.57646 − 0.910170i
0 0 0 −7.54372 13.0661i 0 −16.2919 + 8.80760i 0 0 0
17.2 0 0 0 −6.38217 11.0542i 0 −2.53897 18.3454i 0 0 0
17.3 0 0 0 −5.42768 9.40102i 0 18.2341 3.24321i 0 0 0
17.4 0 0 0 −0.632851 1.09613i 0 −13.4032 + 12.7810i 0 0 0
17.5 0 0 0 0.632851 + 1.09613i 0 −13.4032 + 12.7810i 0 0 0
17.6 0 0 0 5.42768 + 9.40102i 0 18.2341 3.24321i 0 0 0
17.7 0 0 0 6.38217 + 11.0542i 0 −2.53897 18.3454i 0 0 0
17.8 0 0 0 7.54372 + 13.0661i 0 −16.2919 + 8.80760i 0 0 0
593.1 0 0 0 −7.54372 + 13.0661i 0 −16.2919 8.80760i 0 0 0
593.2 0 0 0 −6.38217 + 11.0542i 0 −2.53897 + 18.3454i 0 0 0
593.3 0 0 0 −5.42768 + 9.40102i 0 18.2341 + 3.24321i 0 0 0
593.4 0 0 0 −0.632851 + 1.09613i 0 −13.4032 12.7810i 0 0 0
593.5 0 0 0 0.632851 1.09613i 0 −13.4032 12.7810i 0 0 0
593.6 0 0 0 5.42768 9.40102i 0 18.2341 + 3.24321i 0 0 0
593.7 0 0 0 6.38217 11.0542i 0 −2.53897 + 18.3454i 0 0 0
593.8 0 0 0 7.54372 13.0661i 0 −16.2919 8.80760i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 593.8 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.bt.a 16
3.b odd 2 1 inner 1008.4.bt.a 16
4.b odd 2 1 63.4.p.a 16
7.d odd 6 1 inner 1008.4.bt.a 16
12.b even 2 1 63.4.p.a 16
21.g even 6 1 inner 1008.4.bt.a 16
28.d even 2 1 441.4.p.c 16
28.f even 6 1 63.4.p.a 16
28.f even 6 1 441.4.c.a 16
28.g odd 6 1 441.4.c.a 16
28.g odd 6 1 441.4.p.c 16
84.h odd 2 1 441.4.p.c 16
84.j odd 6 1 63.4.p.a 16
84.j odd 6 1 441.4.c.a 16
84.n even 6 1 441.4.c.a 16
84.n even 6 1 441.4.p.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.p.a 16 4.b odd 2 1
63.4.p.a 16 12.b even 2 1
63.4.p.a 16 28.f even 6 1
63.4.p.a 16 84.j odd 6 1
441.4.c.a 16 28.f even 6 1
441.4.c.a 16 28.g odd 6 1
441.4.c.a 16 84.j odd 6 1
441.4.c.a 16 84.n even 6 1
441.4.p.c 16 28.d even 2 1
441.4.p.c 16 28.g odd 6 1
441.4.p.c 16 84.h odd 2 1
441.4.p.c 16 84.n even 6 1
1008.4.bt.a 16 1.a even 1 1 trivial
1008.4.bt.a 16 3.b odd 2 1 inner
1008.4.bt.a 16 7.d odd 6 1 inner
1008.4.bt.a 16 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$19\!\cdots\!64$$$$T_{5}^{4} +$$$$31\!\cdots\!72$$$$T_{5}^{2} +$$$$49\!\cdots\!56$$">$$T_{5}^{16} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(1008, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$49018425731856 + 31530370595472 T^{2} + 19693854748764 T^{4} + 370814223780 T^{6} + 4739623389 T^{8} + 33794766 T^{10} + 176175 T^{12} + 510 T^{14} + T^{16}$$
$7$ $$( 13841287201 + 1129900996 T + 23059204 T^{2} - 3294172 T^{3} - 262591 T^{4} - 9604 T^{5} + 196 T^{6} + 28 T^{7} + T^{8} )^{2}$$
$11$ $$17\!\cdots\!16$$$$-$$$$20\!\cdots\!60$$$$T^{2} +$$$$21\!\cdots\!20$$$$T^{4} - 261274857054043068 T^{6} + 219830806138269 T^{8} - 105678094310 T^{10} + 37127511 T^{12} - 7446 T^{14} + T^{16}$$
$13$ $$( 2752420357764 + 38728522980 T^{2} + 47288277 T^{4} + 13074 T^{6} + T^{8} )^{2}$$
$17$ $$88\!\cdots\!56$$$$+$$$$64\!\cdots\!28$$$$T^{2} +$$$$29\!\cdots\!52$$$$T^{4} + 85531884162965662464 T^{6} + 18421656867340560 T^{8} + 2793093124176 T^{10} + 313601004 T^{12} + 22356 T^{14} + T^{16}$$
$19$ $$( 567106596 - 20223801360 T + 240424077474 T^{2} - 756672840 T^{3} - 85804785 T^{4} + 272646 T^{5} + 30321 T^{6} - 306 T^{7} + T^{8} )^{2}$$
$23$ $$10\!\cdots\!76$$$$-$$$$42\!\cdots\!28$$$$T^{2} +$$$$11\!\cdots\!40$$$$T^{4} -$$$$18\!\cdots\!88$$$$T^{6} + 225387671576342544 T^{8} - 18841701469712 T^{10} + 1152260940 T^{12} - 42372 T^{14} + T^{16}$$
$29$ $$( 253130837240241216 + 47051469656336 T^{2} + 3179040801 T^{4} + 93078 T^{6} + T^{8} )^{2}$$
$31$ $$( 20564942686042041 + 464084387243064 T + 882141790320 T^{2} - 58872659328 T^{3} - 134048907 T^{4} + 10260288 T^{5} + 124224 T^{6} + 564 T^{7} + T^{8} )^{2}$$
$37$ $$( 36871673487280324 + 1475505352165768 T + 38424543760414 T^{2} + 595549862012 T^{3} + 6745700747 T^{4} + 48851570 T^{5} + 250213 T^{6} + 598 T^{7} + T^{8} )^{2}$$
$41$ $$( 32324337336074547456 - 2566451699751744 T^{2} + 53591867172 T^{4} - 402972 T^{6} + T^{8} )^{2}$$
$43$ $$( 144774182 + 27889316 T - 195789 T^{2} + 82 T^{3} + T^{4} )^{4}$$
$47$ $$16\!\cdots\!36$$$$+$$$$61\!\cdots\!20$$$$T^{2} +$$$$19\!\cdots\!40$$$$T^{4} +$$$$13\!\cdots\!36$$$$T^{6} +$$$$68\!\cdots\!04$$$$T^{8} + 13314700787201280 T^{10} + 192810529224 T^{12} + 472272 T^{14} + T^{16}$$
$53$ $$24\!\cdots\!16$$$$-$$$$74\!\cdots\!80$$$$T^{2} +$$$$15\!\cdots\!04$$$$T^{4} -$$$$17\!\cdots\!44$$$$T^{6} +$$$$15\!\cdots\!57$$$$T^{8} - 79972866291229718 T^{10} + 306776980515 T^{12} - 684342 T^{14} + T^{16}$$
$59$ $$20\!\cdots\!36$$$$+$$$$66\!\cdots\!48$$$$T^{2} +$$$$15\!\cdots\!72$$$$T^{4} +$$$$16\!\cdots\!36$$$$T^{6} +$$$$11\!\cdots\!25$$$$T^{8} + 430452358387636086 T^{10} + 1125567820107 T^{12} + 1243590 T^{14} + T^{16}$$
$61$ $$( 345703894859082816 - 41677416905608512 T + 1502930502375120 T^{2} + 20726360884656 T^{3} + 65628156924 T^{4} - 238596768 T^{5} - 70446 T^{6} + 816 T^{7} + T^{8} )^{2}$$
$67$ $$( 20749646356594032784 - 38600086175507024 T + 846328311384796 T^{2} + 37829515276 T^{3} + 25660342357 T^{4} - 9236990 T^{5} + 193747 T^{6} + 154 T^{7} + T^{8} )^{2}$$
$71$ $$( 2630939135312864016 + 579610290699200 T^{2} + 28173681528 T^{4} + 317232 T^{6} + T^{8} )^{2}$$
$73$ $$($$$$91\!\cdots\!00$$$$-$$$$52\!\cdots\!00$$$$T + 1158009552606105108 T^{2} - 844955676736164 T^{3} - 643525312275 T^{4} + 986638482 T^{5} + 893979 T^{6} - 2034 T^{7} + T^{8} )^{2}$$
$79$ $$($$$$84\!\cdots\!49$$$$+ 24278780400738095116 T + 117506333264689432 T^{2} + 62832634078240 T^{3} + 465213348493 T^{4} + 36829816 T^{5} + 1703128 T^{6} - 1088 T^{7} + T^{8} )^{2}$$
$83$ $$($$$$75\!\cdots\!64$$$$- 194675614013208132 T^{2} + 1143650214837 T^{4} - 1941810 T^{6} + T^{8} )^{2}$$
$89$ $$47\!\cdots\!16$$$$+$$$$12\!\cdots\!12$$$$T^{2} +$$$$25\!\cdots\!88$$$$T^{4} +$$$$21\!\cdots\!68$$$$T^{6} +$$$$14\!\cdots\!08$$$$T^{8} + 3681748657198486656 T^{10} + 8361242100720 T^{12} + 3090888 T^{14} + T^{16}$$
$97$ $$($$$$64\!\cdots\!56$$$$+ 3466733814045027288 T^{2} + 6247433928969 T^{4} + 4322814 T^{6} + T^{8} )^{2}$$