# Properties

 Label 38.4.c.c Level 38 Weight 4 Character orbit 38.c Analytic conductor 2.242 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} + 64 x^{4} + 33 x^{3} + 3984 x^{2} - 945 x + 225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \beta_{4} ) q^{2} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{3} -4 \beta_{4} q^{4} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( -2 \beta_{1} + 4 \beta_{4} ) q^{6} + ( 9 + \beta_{2} ) q^{7} -8 q^{8} + ( 4 \beta_{1} - 2 \beta_{3} - 20 \beta_{4} + 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \beta_{4} ) q^{2} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{3} -4 \beta_{4} q^{4} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( -2 \beta_{1} + 4 \beta_{4} ) q^{6} + ( 9 + \beta_{2} ) q^{7} -8 q^{8} + ( 4 \beta_{1} - 2 \beta_{3} - 20 \beta_{4} + 2 \beta_{5} ) q^{9} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{10} + ( 2 - \beta_{2} + 3 \beta_{3} ) q^{11} + ( 8 + 4 \beta_{2} ) q^{12} + ( -4 \beta_{1} - \beta_{3} + 44 \beta_{4} + \beta_{5} ) q^{13} + ( 18 + 2 \beta_{1} + 2 \beta_{2} - 18 \beta_{4} ) q^{14} + ( 13 \beta_{1} - 3 \beta_{3} - 31 \beta_{4} + 3 \beta_{5} ) q^{15} + ( -16 + 16 \beta_{4} ) q^{16} + ( -18 + 18 \beta_{4} - 3 \beta_{5} ) q^{17} + ( -40 - 8 \beta_{2} - 4 \beta_{3} ) q^{18} + ( -6 + 13 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} + 26 \beta_{4} + 3 \beta_{5} ) q^{19} + ( 4 + 4 \beta_{2} + 4 \beta_{3} ) q^{20} + ( -61 - 11 \beta_{1} - 11 \beta_{2} + 61 \beta_{4} - 2 \beta_{5} ) q^{21} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + 6 \beta_{5} ) q^{22} + ( -5 \beta_{1} - \beta_{3} + 17 \beta_{4} + \beta_{5} ) q^{23} + ( 16 + 8 \beta_{1} + 8 \beta_{2} - 16 \beta_{4} ) q^{24} + ( 10 \beta_{1} + 9 \beta_{3} - 113 \beta_{4} - 9 \beta_{5} ) q^{25} + ( 88 + 8 \beta_{2} - 2 \beta_{3} ) q^{26} + ( 130 + 21 \beta_{2} + 10 \beta_{3} ) q^{27} + ( 4 \beta_{1} - 36 \beta_{4} ) q^{28} + ( -25 \beta_{1} - 5 \beta_{3} - 35 \beta_{4} + 5 \beta_{5} ) q^{29} + ( -62 - 26 \beta_{2} - 6 \beta_{3} ) q^{30} + ( -11 + 23 \beta_{2} - 6 \beta_{3} ) q^{31} + 32 \beta_{4} q^{32} + ( 81 - 30 \beta_{1} - 30 \beta_{2} - 81 \beta_{4} - \beta_{5} ) q^{33} + ( 6 \beta_{3} + 36 \beta_{4} - 6 \beta_{5} ) q^{34} + ( -38 - 20 \beta_{1} - 20 \beta_{2} + 38 \beta_{4} - 10 \beta_{5} ) q^{35} + ( -80 - 16 \beta_{1} - 16 \beta_{2} + 80 \beta_{4} - 8 \beta_{5} ) q^{36} + ( -59 + 5 \beta_{2} + 6 \beta_{3} ) q^{37} + ( 40 + 16 \beta_{1} - 10 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} + 2 \beta_{5} ) q^{38} + ( -274 - 42 \beta_{2} - 7 \beta_{3} ) q^{39} + ( 8 + 8 \beta_{1} + 8 \beta_{2} - 8 \beta_{4} + 8 \beta_{5} ) q^{40} + ( 147 - 30 \beta_{1} - 30 \beta_{2} - 147 \beta_{4} - 4 \beta_{5} ) q^{41} + ( -22 \beta_{1} + 4 \beta_{3} + 122 \beta_{4} - 4 \beta_{5} ) q^{42} + ( -8 + 42 \beta_{1} + 42 \beta_{2} + 8 \beta_{4} + 7 \beta_{5} ) q^{43} + ( -4 \beta_{1} - 12 \beta_{3} - 8 \beta_{4} + 12 \beta_{5} ) q^{44} + ( 552 + 60 \beta_{2} + 2 \beta_{3} ) q^{45} + ( 34 + 10 \beta_{2} - 2 \beta_{3} ) q^{46} + ( 19 \beta_{1} + 5 \beta_{3} - 85 \beta_{4} - 5 \beta_{5} ) q^{47} + ( 16 \beta_{1} - 32 \beta_{4} ) q^{48} + ( -219 + 18 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -226 - 20 \beta_{2} + 18 \beta_{3} ) q^{50} + ( 48 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} ) q^{51} + ( 176 + 16 \beta_{1} + 16 \beta_{2} - 176 \beta_{4} - 4 \beta_{5} ) q^{52} + ( 24 \beta_{1} - 5 \beta_{3} + 5 \beta_{5} ) q^{53} + ( 260 + 42 \beta_{1} + 42 \beta_{2} - 260 \beta_{4} + 20 \beta_{5} ) q^{54} + ( -597 + 15 \beta_{1} + 15 \beta_{2} + 597 \beta_{4} + 32 \beta_{5} ) q^{55} + ( -72 - 8 \beta_{2} ) q^{56} + ( 147 - 20 \beta_{1} + 10 \beta_{2} + 29 \beta_{3} + 318 \beta_{4} - 17 \beta_{5} ) q^{57} + ( -70 + 50 \beta_{2} - 10 \beta_{3} ) q^{58} + ( -358 - \beta_{1} - \beta_{2} + 358 \beta_{4} - 8 \beta_{5} ) q^{59} + ( -124 - 52 \beta_{1} - 52 \beta_{2} + 124 \beta_{4} - 12 \beta_{5} ) q^{60} + ( 29 \beta_{1} - \beta_{3} - 337 \beta_{4} + \beta_{5} ) q^{61} + ( -22 + 46 \beta_{1} + 46 \beta_{2} + 22 \beta_{4} - 12 \beta_{5} ) q^{62} + ( 76 \beta_{1} - 24 \beta_{3} - 324 \beta_{4} + 24 \beta_{5} ) q^{63} + 64 q^{64} + ( 48 - 90 \beta_{2} - 59 \beta_{3} ) q^{65} + ( -60 \beta_{1} + 2 \beta_{3} - 162 \beta_{4} - 2 \beta_{5} ) q^{66} + ( -67 \beta_{1} + 16 \beta_{3} + 320 \beta_{4} - 16 \beta_{5} ) q^{67} + ( 72 + 12 \beta_{3} ) q^{68} + ( -263 - 17 \beta_{2} - 9 \beta_{3} ) q^{69} + ( -40 \beta_{1} + 20 \beta_{3} + 76 \beta_{4} - 20 \beta_{5} ) q^{70} + ( 710 - 22 \beta_{1} - 22 \beta_{2} - 710 \beta_{4} + 17 \beta_{5} ) q^{71} + ( -32 \beta_{1} + 16 \beta_{3} + 160 \beta_{4} - 16 \beta_{5} ) q^{72} + ( 221 - 14 \beta_{1} - 14 \beta_{2} - 221 \beta_{4} + 10 \beta_{5} ) q^{73} + ( -118 + 10 \beta_{1} + 10 \beta_{2} + 118 \beta_{4} + 12 \beta_{5} ) q^{74} + ( 782 + 43 \beta_{2} + 11 \beta_{3} ) q^{75} + ( 104 - 20 \beta_{1} - 52 \beta_{2} - 4 \beta_{3} - 80 \beta_{4} - 8 \beta_{5} ) q^{76} + ( -67 + 23 \beta_{2} + 22 \beta_{3} ) q^{77} + ( -548 - 84 \beta_{1} - 84 \beta_{2} + 548 \beta_{4} - 14 \beta_{5} ) q^{78} + ( 570 + 68 \beta_{1} + 68 \beta_{2} - 570 \beta_{4} - 29 \beta_{5} ) q^{79} + ( 16 \beta_{1} - 16 \beta_{3} - 16 \beta_{4} + 16 \beta_{5} ) q^{80} + ( -483 - 164 \beta_{1} - 164 \beta_{2} + 483 \beta_{4} + 2 \beta_{5} ) q^{81} + ( -60 \beta_{1} + 8 \beta_{3} - 294 \beta_{4} - 8 \beta_{5} ) q^{82} + ( 168 - 105 \beta_{2} + 21 \beta_{3} ) q^{83} + ( 244 + 44 \beta_{2} + 8 \beta_{3} ) q^{84} + ( 12 \beta_{1} + 15 \beta_{3} - 642 \beta_{4} - 15 \beta_{5} ) q^{85} + ( 84 \beta_{1} - 14 \beta_{3} + 16 \beta_{4} + 14 \beta_{5} ) q^{86} + ( -1075 + 35 \beta_{2} - 45 \beta_{3} ) q^{87} + ( -16 + 8 \beta_{2} - 24 \beta_{3} ) q^{88} + ( -88 \beta_{1} + 5 \beta_{3} + 262 \beta_{4} - 5 \beta_{5} ) q^{89} + ( 1104 + 120 \beta_{1} + 120 \beta_{2} - 1104 \beta_{4} + 4 \beta_{5} ) q^{90} + ( -70 \beta_{1} + 582 \beta_{4} ) q^{91} + ( 68 + 20 \beta_{1} + 20 \beta_{2} - 68 \beta_{4} - 4 \beta_{5} ) q^{92} + ( -1051 + 25 \beta_{1} + 25 \beta_{2} + 1051 \beta_{4} - 40 \beta_{5} ) q^{93} + ( -170 - 38 \beta_{2} + 10 \beta_{3} ) q^{94} + ( 541 - 80 \beta_{1} + 31 \beta_{2} - 46 \beta_{3} + 434 \beta_{4} + 9 \beta_{5} ) q^{95} + ( -64 - 32 \beta_{2} ) q^{96} + ( -247 + 202 \beta_{1} + 202 \beta_{2} + 247 \beta_{4} + 42 \beta_{5} ) q^{97} + ( -438 + 36 \beta_{1} + 36 \beta_{2} + 438 \beta_{4} + 4 \beta_{5} ) q^{98} + ( 16 \beta_{1} + 20 \beta_{3} - 1060 \beta_{4} - 20 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{2} - 5q^{3} - 12q^{4} - q^{5} + 10q^{6} + 52q^{7} - 48q^{8} - 54q^{9} + O(q^{10})$$ $$6q + 6q^{2} - 5q^{3} - 12q^{4} - q^{5} + 10q^{6} + 52q^{7} - 48q^{8} - 54q^{9} + 2q^{10} + 8q^{11} + 40q^{12} + 129q^{13} + 52q^{14} - 77q^{15} - 48q^{16} - 51q^{17} - 216q^{18} + 40q^{19} + 8q^{20} - 170q^{21} + 8q^{22} + 47q^{23} + 40q^{24} - 338q^{25} + 516q^{26} + 718q^{27} - 104q^{28} - 125q^{29} - 308q^{30} - 100q^{31} + 96q^{32} + 274q^{33} + 102q^{34} - 84q^{35} - 216q^{36} - 376q^{37} + 322q^{38} - 1546q^{39} + 8q^{40} + 475q^{41} + 340q^{42} - 73q^{43} - 16q^{44} + 3188q^{45} + 188q^{46} - 241q^{47} - 80q^{48} - 1354q^{49} - 1352q^{50} + 69q^{51} + 516q^{52} + 29q^{53} + 718q^{54} - 1838q^{55} - 416q^{56} + 1755q^{57} - 500q^{58} - 1065q^{59} - 308q^{60} - 981q^{61} - 100q^{62} - 872q^{63} + 384q^{64} + 586q^{65} - 548q^{66} + 877q^{67} + 408q^{68} - 1526q^{69} + 168q^{70} + 2135q^{71} + 432q^{72} + 667q^{73} - 376q^{74} + 4584q^{75} + 484q^{76} - 492q^{77} - 1546q^{78} + 1671q^{79} - 16q^{80} - 1287q^{81} - 950q^{82} + 1176q^{83} + 1360q^{84} - 1929q^{85} + 146q^{86} - 6430q^{87} - 64q^{88} + 693q^{89} + 3188q^{90} + 1676q^{91} + 188q^{92} - 3138q^{93} - 964q^{94} + 4489q^{95} - 320q^{96} - 985q^{97} - 1354q^{98} - 3184q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 64 x^{4} + 33 x^{3} + 3984 x^{2} - 945 x + 225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 64 \nu^{4} + 4096 \nu^{3} - 3984 \nu^{2} + 945 \nu - 60480$$$$)/254031$$ $$\beta_{3}$$ $$=$$ $$($$$$21 \nu^{5} - 1344 \nu^{4} + 1339 \nu^{3} - 83664 \nu^{2} + 19845 \nu - 3641036$$$$)/169354$$ $$\beta_{4}$$ $$=$$ $$($$$$1344 \nu^{5} - 1339 \nu^{4} + 85696 \nu^{3} + 64832 \nu^{2} + 5334576 \nu + 4800$$$$)/1270155$$ $$\beta_{5}$$ $$=$$ $$($$$$57792 \nu^{5} - 57577 \nu^{4} + 3684928 \nu^{3} + 1517621 \nu^{2} + 229386768 \nu - 54410265$$$$)/2540310$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-2 \beta_{5} + 43 \beta_{4} - 43$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 63 \beta_{2} - 28$$ $$\nu^{4}$$ $$=$$ $$128 \beta_{5} - 2737 \beta_{4} - 128 \beta_{3} - 48 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$224 \beta_{5} - 3856 \beta_{4} - 4017 \beta_{2} - 4017 \beta_{1} + 3856$$

## Character values

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

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{4}$$

## 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
 −3.78825 + 6.56144i 0.118706 − 0.205606i 4.16954 − 7.22186i −3.78825 − 6.56144i 0.118706 + 0.205606i 4.16954 + 7.22186i
1.00000 + 1.73205i −4.78825 8.29349i −2.00000 + 3.46410i −7.88908 13.6643i 9.57650 16.5870i 16.5765 −8.00000 −32.3546 + 56.0399i 15.7782 27.3286i
7.2 1.00000 + 1.73205i −0.881294 1.52645i −2.00000 + 3.46410i 10.3546 + 17.9347i 1.76259 3.05289i 8.76259 −8.00000 11.9466 20.6922i −20.7092 + 35.8694i
7.3 1.00000 + 1.73205i 3.16954 + 5.48981i −2.00000 + 3.46410i −2.96554 5.13646i −6.33908 + 10.9796i 0.660916 −8.00000 −6.59199 + 11.4177i 5.93108 10.2729i
11.1 1.00000 1.73205i −4.78825 + 8.29349i −2.00000 3.46410i −7.88908 + 13.6643i 9.57650 + 16.5870i 16.5765 −8.00000 −32.3546 56.0399i 15.7782 + 27.3286i
11.2 1.00000 1.73205i −0.881294 + 1.52645i −2.00000 3.46410i 10.3546 17.9347i 1.76259 + 3.05289i 8.76259 −8.00000 11.9466 + 20.6922i −20.7092 35.8694i
11.3 1.00000 1.73205i 3.16954 5.48981i −2.00000 3.46410i −2.96554 + 5.13646i −6.33908 10.9796i 0.660916 −8.00000 −6.59199 11.4177i 5.93108 + 10.2729i
 $$n$$: 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 38.4.c.c 6
3.b odd 2 1 342.4.g.f 6
4.b odd 2 1 304.4.i.e 6
19.c even 3 1 inner 38.4.c.c 6
19.c even 3 1 722.4.a.j 3
19.d odd 6 1 722.4.a.k 3
57.h odd 6 1 342.4.g.f 6
76.g odd 6 1 304.4.i.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.c.c 6 1.a even 1 1 trivial
38.4.c.c 6 19.c even 3 1 inner
304.4.i.e 6 4.b odd 2 1
304.4.i.e 6 76.g odd 6 1
342.4.g.f 6 3.b odd 2 1
342.4.g.f 6 57.h odd 6 1
722.4.a.j 3 19.c even 3 1
722.4.a.k 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 5 T_{3}^{5} + 80 T_{3}^{4} - 61 T_{3}^{3} + 3560 T_{3}^{2} + 5885 T_{3} + 11449$$ acting on $$S_{4}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 4 T^{2} )^{3}$$
$3$ $$1 + 5 T - T^{2} - 196 T^{3} - 841 T^{4} + 863 T^{5} + 29458 T^{6} + 23301 T^{7} - 613089 T^{8} - 3857868 T^{9} - 531441 T^{10} + 71744535 T^{11} + 387420489 T^{12}$$
$5$ $$1 + T - 18 T^{2} + 3395 T^{3} - 326 T^{4} - 21197 T^{5} + 6694844 T^{6} - 2649625 T^{7} - 5093750 T^{8} + 6630859375 T^{9} - 4394531250 T^{10} + 30517578125 T^{11} + 3814697265625 T^{12}$$
$7$ $$( 1 - 26 T + 1191 T^{2} - 17932 T^{3} + 408513 T^{4} - 3058874 T^{5} + 40353607 T^{6} )^{2}$$
$11$ $$( 1 - 4 T + 682 T^{2} + 39332 T^{3} + 907742 T^{4} - 7086244 T^{5} + 2357947691 T^{6} )^{2}$$
$13$ $$1 - 129 T + 6066 T^{2} - 253267 T^{3} + 16943784 T^{4} - 502399461 T^{5} + 3723333168 T^{6} - 1103771615817 T^{7} + 81784409105256 T^{8} - 2685769742701591 T^{9} + 141326184352969746 T^{10} - 6602980198817707653 T^{11} +$$$$11\!\cdots\!29$$$$T^{12}$$
$17$ $$1 + 51 T - 9906 T^{2} - 151851 T^{3} + 74840604 T^{4} - 98127885 T^{5} - 438212648396 T^{6} - 482102299005 T^{7} + 1806470243051676 T^{8} - 18007687633945947 T^{9} - 5771455881998012466 T^{10} +$$$$14\!\cdots\!43$$$$T^{11} +$$$$14\!\cdots\!09$$$$T^{12}$$
$19$ $$1 - 40 T + 20042 T^{2} - 523792 T^{3} + 137468078 T^{4} - 1881835240 T^{5} + 322687697779 T^{6}$$
$23$ $$1 - 47 T - 32844 T^{2} + 504917 T^{3} + 751641370 T^{4} - 4591819601 T^{5} - 10516239653362 T^{6} - 55868669085367 T^{7} + 111269898417127930 T^{8} + 909432598367913571 T^{9} -$$$$71\!\cdots\!24$$$$T^{10} -$$$$12\!\cdots\!29$$$$T^{11} +$$$$32\!\cdots\!69$$$$T^{12}$$
$29$ $$1 + 125 T - 8142 T^{2} - 14097125 T^{3} - 1081533074 T^{4} + 132267152375 T^{5} + 78772712082692 T^{6} + 3225863579273875 T^{7} - 643321094848018754 T^{8} -$$$$20\!\cdots\!25$$$$T^{9} -$$$$28\!\cdots\!22$$$$T^{10} +$$$$10\!\cdots\!25$$$$T^{11} +$$$$21\!\cdots\!61$$$$T^{12}$$
$31$ $$( 1 + 50 T + 37223 T^{2} + 6788948 T^{3} + 1108910393 T^{4} + 44375184050 T^{5} + 26439622160671 T^{6} )^{2}$$
$37$ $$( 1 + 188 T + 151301 T^{2} + 18957524 T^{3} + 7663849553 T^{4} + 482356564892 T^{5} + 129961739795077 T^{6} )^{2}$$
$41$ $$1 - 475 T + 3 T^{2} + 5759068 T^{3} + 8534432497 T^{4} - 223402200673 T^{5} - 737152668991786 T^{6} - 15397103072583833 T^{7} + 40539443998527919777 T^{8} +$$$$18\!\cdots\!48$$$$T^{9} + 67690470901098558243 T^{10} -$$$$73\!\cdots\!75$$$$T^{11} +$$$$10\!\cdots\!21$$$$T^{12}$$
$43$ $$1 + 73 T - 121468 T^{2} + 18245049 T^{3} + 6324323812 T^{4} - 1580283610855 T^{5} - 286573522304042 T^{6} - 125643609048248485 T^{7} + 39978346855087622788 T^{8} +$$$$91\!\cdots\!07$$$$T^{9} -$$$$48\!\cdots\!68$$$$T^{10} +$$$$23\!\cdots\!11$$$$T^{11} +$$$$25\!\cdots\!49$$$$T^{12}$$
$47$ $$1 + 241 T - 236400 T^{2} - 18981991 T^{3} + 45305953102 T^{4} + 1105735177159 T^{5} - 5368402941241282 T^{6} + 114800743298178857 T^{7} +$$$$48\!\cdots\!58$$$$T^{8} -$$$$21\!\cdots\!97$$$$T^{9} -$$$$27\!\cdots\!00$$$$T^{10} +$$$$29\!\cdots\!63$$$$T^{11} +$$$$12\!\cdots\!89$$$$T^{12}$$
$53$ $$1 - 29 T - 407142 T^{2} - 1151815 T^{3} + 105555987664 T^{4} + 701545671919 T^{5} - 18171194838082960 T^{6} + 104444014998284963 T^{7} +$$$$23\!\cdots\!56$$$$T^{8} -$$$$38\!\cdots\!95$$$$T^{9} -$$$$20\!\cdots\!22$$$$T^{10} -$$$$21\!\cdots\!53$$$$T^{11} +$$$$10\!\cdots\!89$$$$T^{12}$$
$59$ $$1 + 1065 T + 161703 T^{2} + 89067468 T^{3} + 241962488631 T^{4} + 79813402193307 T^{5} - 718191030855422 T^{6} + 16391996729059198353 T^{7} +$$$$10\!\cdots\!71$$$$T^{8} +$$$$77\!\cdots\!52$$$$T^{9} +$$$$28\!\cdots\!43$$$$T^{10} +$$$$38\!\cdots\!35$$$$T^{11} +$$$$75\!\cdots\!21$$$$T^{12}$$
$61$ $$1 + 981 T + 12750 T^{2} + 3148103 T^{3} + 230827245666 T^{4} + 68463896386983 T^{5} - 15384474202149204 T^{6} + 15540003665813788323 T^{7} +$$$$11\!\cdots\!26$$$$T^{8} +$$$$36\!\cdots\!23$$$$T^{9} +$$$$33\!\cdots\!50$$$$T^{10} +$$$$59\!\cdots\!81$$$$T^{11} +$$$$13\!\cdots\!81$$$$T^{12}$$
$67$ $$1 - 877 T - 72113 T^{2} + 319271284 T^{3} - 8838632225 T^{4} - 80158064632807 T^{5} + 51326171155299842 T^{6} - 24108579993156931741 T^{7} -$$$$79\!\cdots\!25$$$$T^{8} +$$$$86\!\cdots\!48$$$$T^{9} -$$$$59\!\cdots\!93$$$$T^{10} -$$$$21\!\cdots\!11$$$$T^{11} +$$$$74\!\cdots\!09$$$$T^{12}$$
$71$ $$1 - 2135 T + 2114448 T^{2} - 1639099963 T^{3} + 1277487858472 T^{4} - 911080403250227 T^{5} + 571498227106030874 T^{6} -$$$$32\!\cdots\!97$$$$T^{7} +$$$$16\!\cdots\!12$$$$T^{8} -$$$$75\!\cdots\!53$$$$T^{9} +$$$$34\!\cdots\!68$$$$T^{10} -$$$$12\!\cdots\!85$$$$T^{11} +$$$$21\!\cdots\!61$$$$T^{12}$$
$73$ $$1 - 667 T - 816425 T^{2} + 198290512 T^{3} + 753538170433 T^{4} - 102005483218021 T^{5} - 298290707617331602 T^{6} - 39681867065024875357 T^{7} +$$$$11\!\cdots\!37$$$$T^{8} +$$$$11\!\cdots\!56$$$$T^{9} -$$$$18\!\cdots\!25$$$$T^{10} -$$$$59\!\cdots\!19$$$$T^{11} +$$$$34\!\cdots\!69$$$$T^{12}$$
$79$ $$1 - 1671 T + 1066044 T^{2} - 376513711 T^{3} + 33271826928 T^{4} + 273500597277609 T^{5} - 335723835963217830 T^{6} +$$$$13\!\cdots\!51$$$$T^{7} +$$$$80\!\cdots\!88$$$$T^{8} -$$$$45\!\cdots\!09$$$$T^{9} +$$$$62\!\cdots\!04$$$$T^{10} -$$$$48\!\cdots\!29$$$$T^{11} +$$$$14\!\cdots\!61$$$$T^{12}$$
$83$ $$( 1 - 588 T + 867318 T^{2} - 834896496 T^{3} + 495921157266 T^{4} - 192240939540972 T^{5} + 186940255267540403 T^{6} )^{2}$$
$89$ $$1 - 693 T - 1318266 T^{2} + 603160677 T^{3} + 1325896194492 T^{4} - 250307485289253 T^{5} - 963102551972824100 T^{6} -$$$$17\!\cdots\!57$$$$T^{7} +$$$$65\!\cdots\!12$$$$T^{8} +$$$$21\!\cdots\!93$$$$T^{9} -$$$$32\!\cdots\!86$$$$T^{10} -$$$$12\!\cdots\!57$$$$T^{11} +$$$$12\!\cdots\!81$$$$T^{12}$$
$97$ $$1 + 985 T + 664943 T^{2} + 1921009440 T^{3} + 612575495713 T^{4} + 24265596031175 T^{5} + 1283581803624645982 T^{6} + 22146554326560580775 T^{7} +$$$$51\!\cdots\!77$$$$T^{8} +$$$$14\!\cdots\!80$$$$T^{9} +$$$$46\!\cdots\!63$$$$T^{10} +$$$$62\!\cdots\!05$$$$T^{11} +$$$$57\!\cdots\!89$$$$T^{12}$$