# Properties

 Label 38.6.c.a Level 38 Weight 6 Character orbit 38.c Analytic conductor 6.095 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.09458515289$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 133 x^{4} - 60 x^{3} + 17689 x^{2} - 3990 x + 900$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 - 4 \beta_{3} ) q^{2} + ( -5 + \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{3} -16 \beta_{3} q^{4} + ( 5 - \beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{5} ) q^{5} + ( 4 \beta_{1} + 20 \beta_{3} ) q^{6} + ( -104 + 4 \beta_{2} ) q^{7} -64 q^{8} + ( -10 \beta_{1} - 138 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( 4 - 4 \beta_{3} ) q^{2} + ( -5 + \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{3} -16 \beta_{3} q^{4} + ( 5 - \beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{5} ) q^{5} + ( 4 \beta_{1} + 20 \beta_{3} ) q^{6} + ( -104 + 4 \beta_{2} ) q^{7} -64 q^{8} + ( -10 \beta_{1} - 138 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{9} + ( -4 \beta_{1} - 20 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{10} + ( -156 + \beta_{2} - \beta_{4} ) q^{11} + ( 80 - 16 \beta_{2} ) q^{12} + ( -15 \beta_{1} - 247 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} ) q^{13} + ( -416 + 16 \beta_{1} + 16 \beta_{2} + 416 \beta_{3} ) q^{14} + ( 54 \beta_{1} + 321 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{15} + ( -256 + 256 \beta_{3} ) q^{16} + ( -335 + 45 \beta_{1} + 45 \beta_{2} + 335 \beta_{3} + 5 \beta_{5} ) q^{17} + ( -552 + 40 \beta_{2} + 16 \beta_{4} ) q^{18} + ( 893 + 19 \beta_{1} + 114 \beta_{3} - 19 \beta_{5} ) q^{19} + ( -80 + 16 \beta_{2} + 16 \beta_{4} ) q^{20} + ( 1944 - 124 \beta_{1} - 124 \beta_{2} - 1944 \beta_{3} - 16 \beta_{5} ) q^{21} + ( -624 + 4 \beta_{1} + 4 \beta_{2} + 624 \beta_{3} - 4 \beta_{5} ) q^{22} + ( -14 \beta_{1} - 711 \beta_{3} - 35 \beta_{4} + 35 \beta_{5} ) q^{23} + ( 320 - 64 \beta_{1} - 64 \beta_{2} - 320 \beta_{3} ) q^{24} + ( -83 \beta_{1} - 1052 \beta_{3} - 31 \beta_{4} + 31 \beta_{5} ) q^{25} + ( -988 + 60 \beta_{2} + 20 \beta_{4} ) q^{26} + ( 2795 - 121 \beta_{2} - 60 \beta_{4} ) q^{27} + ( 64 \beta_{1} + 1664 \beta_{3} ) q^{28} + ( -61 \beta_{1} + 505 \beta_{3} - 29 \beta_{4} + 29 \beta_{5} ) q^{29} + ( 1284 - 216 \beta_{2} - 36 \beta_{4} ) q^{30} + ( 48 + 356 \beta_{2} + 54 \beta_{4} ) q^{31} + 1024 \beta_{3} q^{32} + ( 1196 - 117 \beta_{1} - 117 \beta_{2} - 1196 \beta_{3} + \beta_{5} ) q^{33} + ( 180 \beta_{1} + 1340 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} ) q^{34} + ( -1704 + 300 \beta_{1} + 300 \beta_{2} + 1704 \beta_{3} + 120 \beta_{5} ) q^{35} + ( -2208 + 160 \beta_{1} + 160 \beta_{2} + 2208 \beta_{3} + 64 \beta_{5} ) q^{36} + ( 3326 - 94 \beta_{2} + 52 \beta_{4} ) q^{37} + ( 4028 - 76 \beta_{2} - 3572 \beta_{3} + 76 \beta_{4} - 76 \beta_{5} ) q^{38} + ( 6275 - 542 \beta_{2} - 85 \beta_{4} ) q^{39} + ( -320 + 64 \beta_{1} + 64 \beta_{2} + 320 \beta_{3} + 64 \beta_{5} ) q^{40} + ( -6463 - 212 \beta_{1} - 212 \beta_{2} + 6463 \beta_{3} - 162 \beta_{5} ) q^{41} + ( -496 \beta_{1} - 7776 \beta_{3} + 64 \beta_{4} - 64 \beta_{5} ) q^{42} + ( -1701 + 202 \beta_{1} + 202 \beta_{2} + 1701 \beta_{3} + 45 \beta_{5} ) q^{43} + ( 16 \beta_{1} + 2496 \beta_{3} + 16 \beta_{4} - 16 \beta_{5} ) q^{44} + ( -19074 + 744 \beta_{2} + 18 \beta_{4} ) q^{45} + ( -2844 + 56 \beta_{2} - 140 \beta_{4} ) q^{46} + ( 166 \beta_{1} + 2409 \beta_{3} + 199 \beta_{4} - 199 \beta_{5} ) q^{47} + ( -256 \beta_{1} - 1280 \beta_{3} ) q^{48} + ( -295 - 832 \beta_{2} - 64 \beta_{4} ) q^{49} + ( -4208 + 332 \beta_{2} - 124 \beta_{4} ) q^{50} + ( -780 \beta_{1} - 17395 \beta_{3} + 205 \beta_{4} - 205 \beta_{5} ) q^{51} + ( -3952 + 240 \beta_{1} + 240 \beta_{2} + 3952 \beta_{3} + 80 \beta_{5} ) q^{52} + ( 319 \beta_{1} - 1199 \beta_{3} + 441 \beta_{4} - 441 \beta_{5} ) q^{53} + ( 11180 - 484 \beta_{1} - 484 \beta_{2} - 11180 \beta_{3} - 240 \beta_{5} ) q^{54} + ( 2780 + 176 \beta_{1} + 176 \beta_{2} - 2780 \beta_{3} + 200 \beta_{5} ) q^{55} + ( 6656 - 256 \beta_{2} ) q^{56} + ( -11799 + 1729 \beta_{1} + 1102 \beta_{2} + 3325 \beta_{3} - 19 \beta_{4} + 95 \beta_{5} ) q^{57} + ( 2020 + 244 \beta_{2} - 116 \beta_{4} ) q^{58} + ( -125 - 303 \beta_{1} - 303 \beta_{2} + 125 \beta_{3} + 50 \beta_{5} ) q^{59} + ( 5136 - 864 \beta_{1} - 864 \beta_{2} - 5136 \beta_{3} - 144 \beta_{5} ) q^{60} + ( -243 \beta_{1} + 6481 \beta_{3} + 231 \beta_{4} - 231 \beta_{5} ) q^{61} + ( 192 + 1424 \beta_{1} + 1424 \beta_{2} - 192 \beta_{3} + 216 \beta_{5} ) q^{62} + ( 2296 \beta_{1} + 27632 \beta_{3} - 576 \beta_{4} + 576 \beta_{5} ) q^{63} + 4096 q^{64} + ( -24955 + 1127 \beta_{2} + 107 \beta_{4} ) q^{65} + ( -468 \beta_{1} - 4784 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{66} + ( -1781 \beta_{1} + 20665 \beta_{3} - 158 \beta_{4} + 158 \beta_{5} ) q^{67} + ( 5360 - 720 \beta_{2} - 80 \beta_{4} ) q^{68} + ( 10639 + 759 \beta_{2} + 119 \beta_{4} ) q^{69} + ( 1200 \beta_{1} + 6816 \beta_{3} - 480 \beta_{4} + 480 \beta_{5} ) q^{70} + ( 1951 - 2818 \beta_{1} - 2818 \beta_{2} - 1951 \beta_{3} - 93 \beta_{5} ) q^{71} + ( 640 \beta_{1} + 8832 \beta_{3} - 256 \beta_{4} + 256 \beta_{5} ) q^{72} + ( 6137 - 1132 \beta_{1} - 1132 \beta_{2} - 6137 \beta_{3} + 336 \beta_{5} ) q^{73} + ( 13304 - 376 \beta_{1} - 376 \beta_{2} - 13304 \beta_{3} + 208 \beta_{5} ) q^{74} + ( 36668 - 103 \beta_{2} - 177 \beta_{4} ) q^{75} + ( 1824 - 304 \beta_{1} - 304 \beta_{2} - 16112 \beta_{3} + 304 \beta_{4} ) q^{76} + ( 17888 - 552 \beta_{2} + 88 \beta_{4} ) q^{77} + ( 25100 - 2168 \beta_{1} - 2168 \beta_{2} - 25100 \beta_{3} - 340 \beta_{5} ) q^{78} + ( 26487 - 3960 \beta_{1} - 3960 \beta_{2} - 26487 \beta_{3} + 697 \beta_{5} ) q^{79} + ( 256 \beta_{1} + 1280 \beta_{3} - 256 \beta_{4} + 256 \beta_{5} ) q^{80} + ( -19917 + 3610 \beta_{1} + 3610 \beta_{2} + 19917 \beta_{3} - 188 \beta_{5} ) q^{81} + ( -848 \beta_{1} + 25852 \beta_{3} + 648 \beta_{4} - 648 \beta_{5} ) q^{82} + ( -55748 + 3447 \beta_{2} - 457 \beta_{4} ) q^{83} + ( -31104 + 1984 \beta_{2} + 256 \beta_{4} ) q^{84} + ( 2685 \beta_{1} + 34275 \beta_{3} - 315 \beta_{4} + 315 \beta_{5} ) q^{85} + ( 808 \beta_{1} + 6804 \beta_{3} - 180 \beta_{4} + 180 \beta_{5} ) q^{86} + ( 20931 + 1476 \beta_{2} - 99 \beta_{4} ) q^{87} + ( 9984 - 64 \beta_{2} + 64 \beta_{4} ) q^{88} + ( 1455 \beta_{1} - 55439 \beta_{3} + 89 \beta_{4} - 89 \beta_{5} ) q^{89} + ( -76296 + 2976 \beta_{1} + 2976 \beta_{2} + 76296 \beta_{3} + 72 \beta_{5} ) q^{90} + ( 3428 \beta_{1} + 45848 \beta_{3} - 760 \beta_{4} + 760 \beta_{5} ) q^{91} + ( -11376 + 224 \beta_{1} + 224 \beta_{2} + 11376 \beta_{3} - 560 \beta_{5} ) q^{92} + ( 123256 - 4108 \beta_{1} - 4108 \beta_{2} - 123256 \beta_{3} - 1694 \beta_{5} ) q^{93} + ( 9636 - 664 \beta_{2} + 796 \beta_{4} ) q^{94} + ( 10659 - 1444 \beta_{1} - 1938 \beta_{2} - 77729 \beta_{3} - 950 \beta_{4} - 133 \beta_{5} ) q^{95} + ( -5120 + 1024 \beta_{2} ) q^{96} + ( -29047 + 3472 \beta_{1} + 3472 \beta_{2} + 29047 \beta_{3} + 526 \beta_{5} ) q^{97} + ( -1180 - 3328 \beta_{1} - 3328 \beta_{2} + 1180 \beta_{3} - 256 \beta_{5} ) q^{98} + ( 1494 \beta_{1} + 9784 \beta_{3} - 706 \beta_{4} + 706 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 12q^{2} - 15q^{3} - 48q^{4} + 14q^{5} + 60q^{6} - 624q^{7} - 384q^{8} - 410q^{9} + O(q^{10})$$ $$6q + 12q^{2} - 15q^{3} - 48q^{4} + 14q^{5} + 60q^{6} - 624q^{7} - 384q^{8} - 410q^{9} - 56q^{10} - 938q^{11} + 480q^{12} - 736q^{13} - 1248q^{14} + 954q^{15} - 768q^{16} - 1000q^{17} - 3280q^{18} + 5681q^{19} - 448q^{20} + 5816q^{21} - 1876q^{22} - 2168q^{23} + 960q^{24} - 3187q^{25} - 5888q^{26} + 16650q^{27} + 4992q^{28} + 1486q^{29} + 7632q^{30} + 396q^{31} + 3072q^{32} + 3589q^{33} + 4000q^{34} - 4992q^{35} - 6560q^{36} + 20060q^{37} + 13528q^{38} + 37480q^{39} - 896q^{40} - 19551q^{41} - 23264q^{42} - 5058q^{43} + 7504q^{44} - 114408q^{45} - 17344q^{46} + 7426q^{47} - 3840q^{48} - 1898q^{49} - 25496q^{50} - 51980q^{51} - 11776q^{52} - 3156q^{53} + 33300q^{54} + 8540q^{55} + 39936q^{56} - 60762q^{57} + 11888q^{58} - 325q^{59} + 15264q^{60} + 19674q^{61} + 792q^{62} + 82320q^{63} + 24576q^{64} - 149516q^{65} - 14356q^{66} + 61837q^{67} + 32000q^{68} + 64072q^{69} + 19968q^{70} + 5760q^{71} + 26240q^{72} + 18747q^{73} + 40120q^{74} + 219654q^{75} - 36784q^{76} + 107504q^{77} + 74960q^{78} + 80158q^{79} + 3584q^{80} - 59939q^{81} + 78204q^{82} - 335402q^{83} - 186112q^{84} + 102510q^{85} + 20232q^{86} + 125388q^{87} + 60032q^{88} - 166228q^{89} - 228816q^{90} + 136784q^{91} - 34688q^{92} + 368074q^{93} + 59408q^{94} - 171266q^{95} - 30720q^{96} - 86615q^{97} - 3796q^{98} + 28646q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 133 x^{4} - 60 x^{3} + 17689 x^{2} - 3990 x + 900$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} - 60$$$$)/133$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 133 \nu^{3} - 30 \nu^{2} + 17689 \nu$$$$)/3990$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 133 \nu^{2} - 30 \nu + 11837$$$$)/133$$ $$\beta_{5}$$ $$=$$ $$($$$$-89 \nu^{5} - 11837 \nu^{3} + 6660 \nu^{2} - 1574321 \nu + 355110$$$$)/3990$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 89 \beta_{3} - 89$$ $$\nu^{3}$$ $$=$$ $$($$$$133 \beta_{2} + 60$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-133 \beta_{5} + 133 \beta_{4} - 11837 \beta_{3} + 15 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$60 \beta_{5} + 13320 \beta_{3} - 17689 \beta_{2} - 17689 \beta_{1} - 13320$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 5.70904 − 9.88835i 0.112825 − 0.195419i −5.82187 + 10.0838i 5.70904 + 9.88835i 0.112825 + 0.195419i −5.82187 − 10.0838i
2.00000 + 3.46410i −13.9181 24.1068i −8.00000 + 13.8564i 34.6044 + 59.9365i 55.6723 96.4273i −195.345 −64.0000 −265.926 + 460.597i −138.418 + 239.746i
7.2 2.00000 + 3.46410i −2.72565 4.72096i −8.00000 + 13.8564i −41.7489 72.3112i 10.9026 18.8839i −105.805 −64.0000 106.642 184.709i 166.996 289.245i
7.3 2.00000 + 3.46410i 9.14373 + 15.8374i −8.00000 + 13.8564i 14.1445 + 24.4990i −36.5749 + 63.3496i −10.8501 −64.0000 −45.7157 + 79.1819i −56.5781 + 97.9961i
11.1 2.00000 3.46410i −13.9181 + 24.1068i −8.00000 13.8564i 34.6044 59.9365i 55.6723 + 96.4273i −195.345 −64.0000 −265.926 460.597i −138.418 239.746i
11.2 2.00000 3.46410i −2.72565 + 4.72096i −8.00000 13.8564i −41.7489 + 72.3112i 10.9026 + 18.8839i −105.805 −64.0000 106.642 + 184.709i 166.996 + 289.245i
11.3 2.00000 3.46410i 9.14373 15.8374i −8.00000 13.8564i 14.1445 24.4990i −36.5749 63.3496i −10.8501 −64.0000 −45.7157 79.1819i −56.5781 97.9961i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.6.c.a 6
3.b odd 2 1 342.6.g.a 6
4.b odd 2 1 304.6.i.a 6
19.c even 3 1 inner 38.6.c.a 6
19.c even 3 1 722.6.a.e 3
19.d odd 6 1 722.6.a.f 3
57.h odd 6 1 342.6.g.a 6
76.g odd 6 1 304.6.i.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.c.a 6 1.a even 1 1 trivial
38.6.c.a 6 19.c even 3 1 inner
304.6.i.a 6 4.b odd 2 1
304.6.i.a 6 76.g odd 6 1
342.6.g.a 6 3.b odd 2 1
342.6.g.a 6 57.h odd 6 1
722.6.a.e 3 19.c even 3 1
722.6.a.f 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 15 T_{3}^{5} + 682 T_{3}^{4} - 1305 T_{3}^{3} + 250474 T_{3}^{2} + 1268175 T_{3} + 7700625$$ acting on $$S_{6}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 4 T + 16 T^{2} )^{3}$$
$3$ $$1 + 15 T - 47 T^{2} - 4950 T^{3} - 59837 T^{4} + 130935 T^{5} + 17818902 T^{6} + 31817205 T^{7} - 3533315013 T^{8} - 71027089650 T^{9} - 163878866847 T^{10} + 12709329141645 T^{11} + 205891132094649 T^{12}$$
$5$ $$1 - 14 T - 2996 T^{2} - 196640 T^{3} + 1277528 T^{4} + 385096642 T^{5} + 33053246326 T^{6} + 1203427006250 T^{7} + 12475859375000 T^{8} - 6000976562500000 T^{9} - 285720825195312500 T^{10} - 4172325134277343750 T^{11} +$$$$93\!\cdots\!25$$$$T^{12}$$
$7$ $$( 1 + 312 T + 74357 T^{2} + 10711824 T^{3} + 1249718099 T^{4} + 88132277688 T^{5} + 4747561509943 T^{6} )^{2}$$
$11$ $$( 1 + 469 T + 549865 T^{2} + 153971518 T^{3} + 88556308115 T^{4} + 12164652137869 T^{5} + 4177248169415651 T^{6} )^{2}$$
$13$ $$1 + 736 T - 499140 T^{2} - 287313252 T^{3} + 309244832284 T^{4} + 80793864978472 T^{5} - 97145685360579842 T^{6} + 29998196509451804296 T^{7} +$$$$42\!\cdots\!16$$$$T^{8} -$$$$14\!\cdots\!64$$$$T^{9} -$$$$94\!\cdots\!40$$$$T^{10} +$$$$51\!\cdots\!48$$$$T^{11} +$$$$26\!\cdots\!49$$$$T^{12}$$
$17$ $$1 + 1000 T - 2408696 T^{2} - 2405502500 T^{3} + 3872248293944 T^{4} + 2245689732583000 T^{5} - 4337536158265406426 T^{6} +$$$$31\!\cdots\!00$$$$T^{7} +$$$$78\!\cdots\!56$$$$T^{8} -$$$$68\!\cdots\!00$$$$T^{9} -$$$$97\!\cdots\!96$$$$T^{10} +$$$$57\!\cdots\!00$$$$T^{11} +$$$$81\!\cdots\!49$$$$T^{12}$$
$19$ $$1 - 5681 T + 15930569 T^{2} - 29730651014 T^{3} + 39445665970331 T^{4} - 34830587410567481 T^{5} + 15181127029874798299 T^{6}$$
$23$ $$1 + 2168 T - 8760066 T^{2} - 30615252044 T^{3} + 29724241296454 T^{4} + 116606691007999892 T^{5} + 66348890302406753462 T^{6} +$$$$75\!\cdots\!56$$$$T^{7} +$$$$12\!\cdots\!46$$$$T^{8} -$$$$81\!\cdots\!08$$$$T^{9} -$$$$15\!\cdots\!66$$$$T^{10} +$$$$23\!\cdots\!24$$$$T^{11} +$$$$71\!\cdots\!49$$$$T^{12}$$
$29$ $$1 - 1486 T - 52804508 T^{2} + 35874759344 T^{3} + 1810629470101136 T^{4} - 507041284560567550 T^{5} -$$$$42\!\cdots\!46$$$$T^{6} -$$$$10\!\cdots\!50$$$$T^{7} +$$$$76\!\cdots\!36$$$$T^{8} +$$$$30\!\cdots\!56$$$$T^{9} -$$$$93\!\cdots\!08$$$$T^{10} -$$$$53\!\cdots\!14$$$$T^{11} +$$$$74\!\cdots\!01$$$$T^{12}$$
$31$ $$( 1 - 198 T + 4743581 T^{2} + 270431123436 T^{3} + 135804696729731 T^{4} - 162286400822198598 T^{5} +$$$$23\!\cdots\!51$$$$T^{6} )^{2}$$
$37$ $$( 1 - 10030 T + 220041227 T^{2} - 1356924175348 T^{3} + 15258529383315239 T^{4} - 48230101255351025470 T^{5} +$$$$33\!\cdots\!93$$$$T^{6} )^{2}$$
$41$ $$1 + 19551 T + 79730791 T^{2} - 2332252585848 T^{3} - 24150006361237239 T^{4} +$$$$13\!\cdots\!21$$$$T^{5} +$$$$43\!\cdots\!42$$$$T^{6} +$$$$16\!\cdots\!21$$$$T^{7} -$$$$32\!\cdots\!39$$$$T^{8} -$$$$36\!\cdots\!48$$$$T^{9} +$$$$14\!\cdots\!91$$$$T^{10} +$$$$40\!\cdots\!51$$$$T^{11} +$$$$24\!\cdots\!01$$$$T^{12}$$
$43$ $$1 + 5058 T - 391958150 T^{2} - 895138250460 T^{3} + 105353925148109606 T^{4} +$$$$11\!\cdots\!14$$$$T^{5} -$$$$17\!\cdots\!54$$$$T^{6} +$$$$17\!\cdots\!02$$$$T^{7} +$$$$22\!\cdots\!94$$$$T^{8} -$$$$28\!\cdots\!20$$$$T^{9} -$$$$18\!\cdots\!50$$$$T^{10} +$$$$34\!\cdots\!94$$$$T^{11} +$$$$10\!\cdots\!49$$$$T^{12}$$
$47$ $$1 - 7426 T - 397164762 T^{2} + 4334007544900 T^{3} + 72285854998029634 T^{4} -$$$$58\!\cdots\!14$$$$T^{5} -$$$$10\!\cdots\!10$$$$T^{6} -$$$$13\!\cdots\!98$$$$T^{7} +$$$$38\!\cdots\!66$$$$T^{8} +$$$$52\!\cdots\!00$$$$T^{9} -$$$$10\!\cdots\!62$$$$T^{10} -$$$$47\!\cdots\!82$$$$T^{11} +$$$$14\!\cdots\!49$$$$T^{12}$$
$53$ $$1 + 3156 T - 21762380 T^{2} + 21897373038084 T^{3} + 22136104052497932 T^{4} +$$$$12\!\cdots\!84$$$$T^{5} +$$$$26\!\cdots\!78$$$$T^{6} +$$$$51\!\cdots\!12$$$$T^{7} +$$$$38\!\cdots\!68$$$$T^{8} +$$$$16\!\cdots\!88$$$$T^{9} -$$$$66\!\cdots\!80$$$$T^{10} +$$$$40\!\cdots\!08$$$$T^{11} +$$$$53\!\cdots\!49$$$$T^{12}$$
$59$ $$1 + 325 T - 2078392259 T^{2} - 654694327310 T^{3} + 2833968652603107515 T^{4} +$$$$58\!\cdots\!45$$$$T^{5} -$$$$23\!\cdots\!46$$$$T^{6} +$$$$41\!\cdots\!55$$$$T^{7} +$$$$14\!\cdots\!15$$$$T^{8} -$$$$23\!\cdots\!90$$$$T^{9} -$$$$54\!\cdots\!59$$$$T^{10} +$$$$60\!\cdots\!75$$$$T^{11} +$$$$13\!\cdots\!01$$$$T^{12}$$
$61$ $$1 - 19674 T - 1939800348 T^{2} + 20046443458280 T^{3} + 2801522678585113080 T^{4} -$$$$13\!\cdots\!94$$$$T^{5} -$$$$25\!\cdots\!66$$$$T^{6} -$$$$11\!\cdots\!94$$$$T^{7} +$$$$19\!\cdots\!80$$$$T^{8} +$$$$12\!\cdots\!80$$$$T^{9} -$$$$98\!\cdots\!48$$$$T^{10} -$$$$84\!\cdots\!74$$$$T^{11} +$$$$36\!\cdots\!01$$$$T^{12}$$
$67$ $$1 - 61837 T + 384165349 T^{2} + 110380695724918 T^{3} - 2801865955161978413 T^{4} -$$$$94\!\cdots\!33$$$$T^{5} +$$$$80\!\cdots\!46$$$$T^{6} -$$$$12\!\cdots\!31$$$$T^{7} -$$$$51\!\cdots\!37$$$$T^{8} +$$$$27\!\cdots\!74$$$$T^{9} +$$$$12\!\cdots\!49$$$$T^{10} -$$$$27\!\cdots\!59$$$$T^{11} +$$$$60\!\cdots\!49$$$$T^{12}$$
$71$ $$1 - 5760 T - 1162067618 T^{2} + 76725553844268 T^{3} - 968682651749976234 T^{4} -$$$$43\!\cdots\!32$$$$T^{5} +$$$$10\!\cdots\!54$$$$T^{6} -$$$$79\!\cdots\!32$$$$T^{7} -$$$$31\!\cdots\!34$$$$T^{8} +$$$$45\!\cdots\!68$$$$T^{9} -$$$$12\!\cdots\!18$$$$T^{10} -$$$$11\!\cdots\!60$$$$T^{11} +$$$$34\!\cdots\!01$$$$T^{12}$$
$73$ $$1 - 18747 T - 4569061397 T^{2} + 15800845049124 T^{3} + 12998093967735391937 T^{4} +$$$$36\!\cdots\!31$$$$T^{5} -$$$$30\!\cdots\!66$$$$T^{6} +$$$$74\!\cdots\!83$$$$T^{7} +$$$$55\!\cdots\!13$$$$T^{8} +$$$$14\!\cdots\!68$$$$T^{9} -$$$$84\!\cdots\!97$$$$T^{10} -$$$$71\!\cdots\!71$$$$T^{11} +$$$$79\!\cdots\!49$$$$T^{12}$$
$79$ $$1 - 80158 T + 6756292046 T^{2} - 548417554298436 T^{3} + 24171391996329566410 T^{4} -$$$$10\!\cdots\!18$$$$T^{5} +$$$$79\!\cdots\!74$$$$T^{6} -$$$$31\!\cdots\!82$$$$T^{7} +$$$$22\!\cdots\!10$$$$T^{8} -$$$$15\!\cdots\!64$$$$T^{9} +$$$$60\!\cdots\!46$$$$T^{10} -$$$$22\!\cdots\!42$$$$T^{11} +$$$$84\!\cdots\!01$$$$T^{12}$$
$83$ $$( 1 + 167701 T + 13355547801 T^{2} + 802637191479838 T^{3} + 52608045597668276043 T^{4} +$$$$26\!\cdots\!49$$$$T^{5} +$$$$61\!\cdots\!07$$$$T^{6} )^{2}$$
$89$ $$1 + 166228 T + 2865258232 T^{2} + 177116904254068 T^{3} +$$$$14\!\cdots\!24$$$$T^{4} +$$$$79\!\cdots\!28$$$$T^{5} -$$$$58\!\cdots\!46$$$$T^{6} +$$$$44\!\cdots\!72$$$$T^{7} +$$$$46\!\cdots\!24$$$$T^{8} +$$$$30\!\cdots\!32$$$$T^{9} +$$$$27\!\cdots\!32$$$$T^{10} +$$$$90\!\cdots\!72$$$$T^{11} +$$$$30\!\cdots\!01$$$$T^{12}$$
$97$ $$1 + 86615 T - 13044791905 T^{2} - 1310411799333648 T^{3} +$$$$11\!\cdots\!05$$$$T^{4} +$$$$75\!\cdots\!05$$$$T^{5} -$$$$52\!\cdots\!38$$$$T^{6} +$$$$64\!\cdots\!85$$$$T^{7} +$$$$82\!\cdots\!45$$$$T^{8} -$$$$82\!\cdots\!64$$$$T^{9} -$$$$70\!\cdots\!05$$$$T^{10} +$$$$40\!\cdots\!55$$$$T^{11} +$$$$40\!\cdots\!49$$$$T^{12}$$