# Properties

 Label 2100.2.bi.k Level 2100 Weight 2 Character orbit 2100.bi Analytic conductor 16.769 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.bi (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: 10.0.29471584693248.1 Defining polynomial: $$x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 13 x^{6} - 36 x^{5} + 39 x^{4} - 36 x^{3} + 54 x^{2} - 162 x + 243$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{7} + ( \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{7} + ( \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} + ( -1 + \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{11} + ( -1 - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{13} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{17} + ( \beta_{6} - \beta_{8} ) q^{19} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{21} + ( -3 + 2 \beta_{2} - \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{27} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{29} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} ) q^{31} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{33} + ( \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{37} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{39} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{41} + ( 3 + 2 \beta_{2} + \beta_{3} - \beta_{9} ) q^{43} + ( -2 - 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{9} ) q^{47} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{49} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{51} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -4 + \beta_{2} + \beta_{3} - 4 \beta_{7} + \beta_{9} ) q^{57} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{59} + ( 3 - \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{61} + ( -1 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{63} + ( -\beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{67} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{69} + ( -1 + 3 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{71} + ( 1 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} + 4 \beta_{8} ) q^{73} + ( 1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - 4 \beta_{7} + \beta_{8} ) q^{77} + ( -4 \beta_{3} + 4 \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{79} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{81} + ( -1 + 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{5} - 4 \beta_{6} ) q^{83} + ( 3 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{87} + ( 6 + \beta_{1} - 3 \beta_{4} + \beta_{5} + 6 \beta_{7} ) q^{89} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{91} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} - 4 \beta_{7} - \beta_{8} + 4 \beta_{9} ) q^{93} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{97} + ( 5 - 2 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + q^{3} + 5q^{7} - 3q^{9} + O(q^{10})$$ $$10q + q^{3} + 5q^{7} - 3q^{9} - 6q^{11} - 6q^{17} + 3q^{19} + 10q^{21} - 24q^{23} - 8q^{27} + 15q^{31} + 20q^{33} + q^{37} + 15q^{39} - 8q^{41} + 26q^{43} - 14q^{47} - 13q^{49} - 44q^{51} + 24q^{53} - 18q^{57} + 42q^{61} + q^{63} - 7q^{67} - 14q^{69} + 3q^{73} + 26q^{77} + q^{79} + 41q^{81} + 8q^{83} + 26q^{87} + 28q^{89} - 11q^{91} + 47q^{93} + 36q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 13 x^{6} - 36 x^{5} + 39 x^{4} - 36 x^{3} + 54 x^{2} - 162 x + 243$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - 2 \nu^{5} + 4 \nu^{4} - 4 \nu^{3} + 18 \nu^{2} - 21 \nu + 18$$$$)/18$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{9} - \nu^{8} - 9 \nu^{7} + 7 \nu^{6} - 18 \nu^{5} + 22 \nu^{4} - 51 \nu^{3} + 21 \nu^{2} - 171 \nu + 135$$$$)/216$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{9} - 4 \nu^{8} + 7 \nu^{7} + 28 \nu^{6} + 32 \nu^{5} - 30 \nu^{4} + 123 \nu^{3} - 216 \nu^{2} - 243 \nu - 486$$$$)/648$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{9} + 2 \nu^{8} - \nu^{7} + 4 \nu^{6} + 16 \nu^{5} - 58 \nu^{4} + 9 \nu^{3} + 6 \nu^{2} + 9 \nu - 486$$$$)/216$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{9} - \nu^{7} + 6 \nu^{6} - 20 \nu^{5} + 22 \nu^{4} - 15 \nu^{3} + 48 \nu^{2} - 63 \nu + 216$$$$)/72$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{9} - 7 \nu^{8} + 16 \nu^{7} - 23 \nu^{6} + 50 \nu^{5} - 108 \nu^{4} + 114 \nu^{3} - 153 \nu^{2} - 729$$$$)/648$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{9} - 2 \nu^{8} + 2 \nu^{7} - 4 \nu^{6} + 13 \nu^{5} - 36 \nu^{4} + 39 \nu^{3} - 36 \nu^{2} + 54 \nu - 162$$$$)/81$$ $$\beta_{9}$$ $$=$$ $$($$$$-13 \nu^{9} + 5 \nu^{8} + 25 \nu^{7} - 35 \nu^{6} - 94 \nu^{5} + 186 \nu^{4} + 231 \nu^{3} - 297 \nu^{2} + 243 \nu + 1701$$$$)/648$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 2 \beta_{1} - 3$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 6$$ $$\nu^{6}$$ $$=$$ $$-4 \beta_{9} - 8 \beta_{7} + 4 \beta_{6} + 12 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 3$$ $$\nu^{7}$$ $$=$$ $$-8 \beta_{9} - 10 \beta_{8} + 18 \beta_{7} + 12 \beta_{6} - 4 \beta_{5} + 6 \beta_{4} - 18 \beta_{3} - 8 \beta_{2} - \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$16 \beta_{9} - 16 \beta_{8} - 7 \beta_{7} - 10 \beta_{6} + 38 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - 8 \beta_{1} + 30$$ $$\nu^{9}$$ $$=$$ $$16 \beta_{9} - 29 \beta_{8} + 88 \beta_{7} - 9 \beta_{6} + 25 \beta_{5} + 26 \beta_{3} + 18 \beta_{2} + 46 \beta_{1} + 48$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1 + \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −1.08831 + 1.34743i 1.15038 + 1.29484i −1.31611 − 1.12599i 1.72689 + 0.133595i 0.527154 − 1.64988i −1.08831 − 1.34743i 1.15038 − 1.29484i −1.31611 + 1.12599i 1.72689 − 0.133595i 0.527154 + 1.64988i
0 −1.71107 + 0.268793i 0 0 0 −2.57325 + 0.615143i 0 2.85550 0.919845i 0
101.2 0 −0.546177 1.64368i 0 0 0 1.08214 2.41433i 0 −2.40338 + 1.79548i 0
101.3 0 0.317079 + 1.70278i 0 0 0 1.73439 + 1.99797i 0 −2.79892 + 1.07983i 0
101.4 0 0.747749 1.56233i 0 0 0 0.456468 + 2.60608i 0 −1.88174 2.33646i 0
101.5 0 1.69242 + 0.368412i 0 0 0 1.80025 1.93884i 0 2.72854 + 1.24701i 0
1601.1 0 −1.71107 0.268793i 0 0 0 −2.57325 0.615143i 0 2.85550 + 0.919845i 0
1601.2 0 −0.546177 + 1.64368i 0 0 0 1.08214 + 2.41433i 0 −2.40338 1.79548i 0
1601.3 0 0.317079 1.70278i 0 0 0 1.73439 1.99797i 0 −2.79892 1.07983i 0
1601.4 0 0.747749 + 1.56233i 0 0 0 0.456468 2.60608i 0 −1.88174 + 2.33646i 0
1601.5 0 1.69242 0.368412i 0 0 0 1.80025 + 1.93884i 0 2.72854 1.24701i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 2100.2.bi.k 10
3.b odd 2 1 2100.2.bi.j 10
5.b even 2 1 420.2.bh.a 10
5.c odd 4 2 2100.2.bo.h 20
7.d odd 6 1 2100.2.bi.j 10
15.d odd 2 1 420.2.bh.b yes 10
15.e even 4 2 2100.2.bo.g 20
21.g even 6 1 inner 2100.2.bi.k 10
35.i odd 6 1 420.2.bh.b yes 10
35.i odd 6 1 2940.2.d.a 10
35.j even 6 1 2940.2.d.b 10
35.k even 12 2 2100.2.bo.g 20
105.o odd 6 1 2940.2.d.a 10
105.p even 6 1 420.2.bh.a 10
105.p even 6 1 2940.2.d.b 10
105.w odd 12 2 2100.2.bo.h 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a 10 5.b even 2 1
420.2.bh.a 10 105.p even 6 1
420.2.bh.b yes 10 15.d odd 2 1
420.2.bh.b yes 10 35.i odd 6 1
2100.2.bi.j 10 3.b odd 2 1
2100.2.bi.j 10 7.d odd 6 1
2100.2.bi.k 10 1.a even 1 1 trivial
2100.2.bi.k 10 21.g even 6 1 inner
2100.2.bo.g 20 15.e even 4 2
2100.2.bo.g 20 35.k even 12 2
2100.2.bo.h 20 5.c odd 4 2
2100.2.bo.h 20 105.w odd 12 2
2940.2.d.a 10 35.i odd 6 1
2940.2.d.a 10 105.o odd 6 1
2940.2.d.b 10 35.j even 6 1
2940.2.d.b 10 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{10} + \cdots$$ $$T_{13}^{10} + 81 T_{13}^{8} + 2183 T_{13}^{6} + 22251 T_{13}^{4} + 65704 T_{13}^{2} + 3888$$ $$T_{19}^{10} - \cdots$$ $$T_{37}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + 2 T^{2} + T^{3} - 11 T^{4} + 12 T^{5} - 33 T^{6} + 9 T^{7} + 54 T^{8} - 81 T^{9} + 243 T^{10}$$
$5$ 1
$7$ $$1 - 5 T + 19 T^{2} - 6 T^{3} - 91 T^{4} + 533 T^{5} - 637 T^{6} - 294 T^{7} + 6517 T^{8} - 12005 T^{9} + 16807 T^{10}$$
$11$ $$1 + 6 T + 51 T^{2} + 234 T^{3} + 1195 T^{4} + 3666 T^{5} + 12622 T^{6} + 19434 T^{7} + 25645 T^{8} - 184704 T^{9} - 509783 T^{10} - 2031744 T^{11} + 3103045 T^{12} + 25866654 T^{13} + 184798702 T^{14} + 590412966 T^{15} + 2117015395 T^{16} + 4559998014 T^{17} + 10932302931 T^{18} + 14147686146 T^{19} + 25937424601 T^{20}$$
$13$ $$1 - 49 T^{2} + 1364 T^{4} - 28371 T^{6} + 474775 T^{8} - 6687888 T^{10} + 80236975 T^{12} - 810304131 T^{14} + 6583767476 T^{16} - 39970805329 T^{18} + 137858491849 T^{20}$$
$17$ $$1 + 6 T - 15 T^{2} - 138 T^{3} + 193 T^{4} + 2754 T^{5} + 4928 T^{6} - 30306 T^{7} - 220151 T^{8} + 325680 T^{9} + 6075881 T^{10} + 5536560 T^{11} - 63623639 T^{12} - 148893378 T^{13} + 411591488 T^{14} + 3910286178 T^{15} + 4658550817 T^{16} - 56626736874 T^{17} - 104636361615 T^{18} + 711527258982 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 - 3 T + 83 T^{2} - 240 T^{3} + 3800 T^{4} - 10380 T^{5} + 124845 T^{6} - 315315 T^{7} + 3213415 T^{8} - 7312140 T^{9} + 67225536 T^{10} - 138930660 T^{11} + 1160042815 T^{12} - 2162745585 T^{13} + 16269925245 T^{14} - 25701907620 T^{15} + 178774347800 T^{16} - 214529217360 T^{17} + 1409635732403 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20}$$
$23$ $$1 + 24 T + 338 T^{2} + 3504 T^{3} + 29201 T^{4} + 203310 T^{5} + 1216412 T^{6} + 6430392 T^{7} + 31045441 T^{8} + 143820006 T^{9} + 675418266 T^{10} + 3307860138 T^{11} + 16423038289 T^{12} + 78238579464 T^{13} + 340401950492 T^{14} + 1308572895330 T^{15} + 4322795994689 T^{16} + 11930508366288 T^{17} + 26469113024978 T^{18} + 43227663875112 T^{19} + 41426511213649 T^{20}$$
$29$ $$1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 21444875137 T^{12} - 438227063914 T^{14} + 6803589145598 T^{16} - 74036469118228 T^{18} + 420707233300201 T^{20}$$
$31$ $$1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 2125123656 T^{11} + 11687054467 T^{12} - 63439189725 T^{13} + 335324010453 T^{14} - 1757944388004 T^{15} + 8580385587908 T^{16} - 37967407473180 T^{17} + 142432803252647 T^{18} - 396594332410065 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - T - 117 T^{2} + 176 T^{3} + 6668 T^{4} - 11380 T^{5} - 299439 T^{6} + 438173 T^{7} + 12712375 T^{8} - 7307172 T^{9} - 492022188 T^{10} - 270365364 T^{11} + 17403241375 T^{12} + 22194776969 T^{13} - 561196895679 T^{14} - 789134230660 T^{15} + 17108263695212 T^{16} + 16708010375408 T^{17} - 410960096108757 T^{18} - 129961739795077 T^{19} + 4808584372417849 T^{20}$$
$41$ $$( 1 + 4 T + 90 T^{2} + 592 T^{3} + 3749 T^{4} + 36434 T^{5} + 153709 T^{6} + 995152 T^{7} + 6202890 T^{8} + 11303044 T^{9} + 115856201 T^{10} )^{2}$$
$43$ $$( 1 - 13 T + 173 T^{2} - 1668 T^{3} + 14545 T^{4} - 96933 T^{5} + 625435 T^{6} - 3084132 T^{7} + 13754711 T^{8} - 44444413 T^{9} + 147008443 T^{10} )^{2}$$
$47$ $$1 + 14 T + 49 T^{2} - 70 T^{3} - 1449 T^{4} - 16780 T^{5} - 75362 T^{6} + 133028 T^{7} + 1885261 T^{8} + 57961722 T^{9} + 732803503 T^{10} + 2724200934 T^{11} + 4164541549 T^{12} + 13811366044 T^{13} - 367742519522 T^{14} - 3848409217460 T^{15} - 15619083011721 T^{16} - 35463618432410 T^{17} + 1166753046426289 T^{18} + 15667826623438738 T^{19} + 52599132235830049 T^{20}$$
$53$ $$1 - 24 T + 337 T^{2} - 3480 T^{3} + 28655 T^{4} - 224976 T^{5} + 1724958 T^{6} - 14318448 T^{7} + 120990445 T^{8} - 964554888 T^{9} + 7393360287 T^{10} - 51121409064 T^{11} + 339862160005 T^{12} - 2131687582896 T^{13} + 13610748324798 T^{14} - 94083949233168 T^{15} + 635119768151495 T^{16} - 4087994766632760 T^{17} + 20981515668628657 T^{18} - 79194326203251192 T^{19} + 174887470365513049 T^{20}$$
$59$ $$1 - 135 T^{2} - 756 T^{3} + 7351 T^{4} + 84450 T^{5} - 41974 T^{6} - 3597210 T^{7} - 8726459 T^{8} + 56020614 T^{9} + 414268763 T^{10} + 3305216226 T^{11} - 30376803779 T^{12} - 738791392590 T^{13} - 508614110614 T^{14} + 60375357050550 T^{15} + 310069102794991 T^{16} - 1881420522523164 T^{17} - 19822109076583335 T^{18} + 511116753300641401 T^{20}$$
$61$ $$1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 3740861826432 T^{11} + 25382030036233 T^{12} - 155990965832514 T^{13} + 865105397597886 T^{14} - 4275515394922200 T^{15} + 18602616131649353 T^{16} - 69165484335150168 T^{17} + 213178532052976472 T^{18} - 491154135899033922 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 + 7 T - 94 T^{2} - 1731 T^{3} + 26 T^{4} + 170241 T^{5} + 879508 T^{6} - 10247793 T^{7} - 116682655 T^{8} + 286659444 T^{9} + 9801469996 T^{10} + 19206182748 T^{11} - 523788438295 T^{12} - 3082156966059 T^{13} + 17723072128468 T^{14} + 229846648340787 T^{15} + 2351917936394 T^{16} - 10491091788814113 T^{17} - 38170361690324254 T^{18} + 190445740774064629 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 7205364893890 T^{12} - 340770947150172 T^{14} + 11797139447136653 T^{16} - 272507990185711142 T^{18} + 3255243551009881201 T^{20}$$
$73$ $$1 - 3 T + 115 T^{2} - 336 T^{3} + 8912 T^{4} - 58260 T^{5} + 290497 T^{6} - 1508853 T^{7} - 7279937 T^{8} + 55999116 T^{9} - 744209196 T^{10} + 4087935468 T^{11} - 38794784273 T^{12} - 586969467501 T^{13} + 8249603815777 T^{14} - 120777151008180 T^{15} + 1348690624687568 T^{16} - 3711925902416592 T^{17} + 92742910567819315 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 - T - 201 T^{2} + 740 T^{3} + 18128 T^{4} - 100900 T^{5} - 805347 T^{6} + 5115611 T^{7} + 21370651 T^{8} - 77003688 T^{9} - 699661200 T^{10} - 6083291352 T^{11} + 133374232891 T^{12} + 2522195731829 T^{13} - 31368330883107 T^{14} - 310474990659100 T^{15} + 4406689393684688 T^{16} + 14210892649757660 T^{17} - 304938870791218761 T^{18} - 119851595982618319 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$( 1 - 4 T + 36 T^{2} - 304 T^{3} + 3029 T^{4} + 33442 T^{5} + 251407 T^{6} - 2094256 T^{7} + 20584332 T^{8} - 189833284 T^{9} + 3939040643 T^{10} )^{2}$$
$89$ $$1 - 28 T + 118 T^{2} + 2240 T^{3} + 12999 T^{4} - 635638 T^{5} + 2053984 T^{6} + 15180164 T^{7} + 276922069 T^{8} - 3387791550 T^{9} + 10662822610 T^{10} - 301513447950 T^{11} + 2193499708549 T^{12} + 10701545034916 T^{13} + 128871559138144 T^{14} - 3549440380043462 T^{15} + 6460259801202039 T^{16} + 99078190165984960 T^{17} + 464517479072845558 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 - 758 T^{2} + 269933 T^{4} - 59888792 T^{6} + 9235543570 T^{8} - 1040392525668 T^{10} + 86897229450130 T^{12} - 5301911695718552 T^{14} + 224846632206499757 T^{16} - 5940774664537736438 T^{18} + 73742412689492826049 T^{20}$$