# Properties

 Label 430.2.e.g Level 430 Weight 2 Character orbit 430.e Analytic conductor 3.434 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + ( 1 - \beta_{5} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{4} + \beta_{5} + \beta_{7} ) q^{7} - q^{8} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + ( 1 - \beta_{5} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{4} + \beta_{5} + \beta_{7} ) q^{7} - q^{8} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} ) q^{9} + ( -1 + \beta_{5} ) q^{10} + ( 1 - \beta_{2} + \beta_{6} ) q^{11} + \beta_{1} q^{12} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{13} + ( \beta_{4} - \beta_{5} - \beta_{7} ) q^{14} + ( \beta_{1} - \beta_{2} ) q^{15} + q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{17} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{18} + ( 1 + \beta_{1} - \beta_{5} - \beta_{9} ) q^{19} + ( 1 - \beta_{5} ) q^{20} + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{6} ) q^{21} + ( -1 + \beta_{2} - \beta_{6} ) q^{22} + ( 1 - \beta_{5} - \beta_{7} + \beta_{9} ) q^{23} -\beta_{1} q^{24} -\beta_{5} q^{25} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{26} + ( -1 - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{27} + ( -\beta_{4} + \beta_{5} + \beta_{7} ) q^{28} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{29} + ( -\beta_{1} + \beta_{2} ) q^{30} + ( -2 + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{31} - q^{32} + ( -2 + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} ) q^{33} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{34} + ( 1 - \beta_{4} ) q^{35} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} ) q^{36} + ( 3 - 3 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{37} + ( -1 - \beta_{1} + \beta_{5} + \beta_{9} ) q^{38} + ( 2 - 4 \beta_{2} - \beta_{3} ) q^{39} + ( -1 + \beta_{5} ) q^{40} + ( -1 + 2 \beta_{2} ) q^{41} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{42} + ( 2 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{8} ) q^{43} + ( 1 - \beta_{2} + \beta_{6} ) q^{44} + ( -1 - \beta_{2} - \beta_{3} ) q^{45} + ( -1 + \beta_{5} + \beta_{7} - \beta_{9} ) q^{46} + ( -3 + \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{6} ) q^{47} + \beta_{1} q^{48} + ( -1 + 2 \beta_{1} + \beta_{5} + 2 \beta_{7} ) q^{49} + \beta_{5} q^{50} + ( 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{51} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{52} + ( -4 \beta_{1} - 2 \beta_{9} ) q^{53} + ( 1 + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{54} + ( 1 - \beta_{1} - \beta_{5} - \beta_{9} ) q^{55} + ( \beta_{4} - \beta_{5} - \beta_{7} ) q^{56} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{57} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{58} + ( \beta_{2} + \beta_{3} - \beta_{4} ) q^{59} + ( \beta_{1} - \beta_{2} ) q^{60} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{61} + ( 2 - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{62} + ( 4 - 4 \beta_{5} + 2 \beta_{8} - \beta_{9} ) q^{63} + q^{64} + ( -1 - \beta_{3} - \beta_{4} ) q^{65} + ( 2 - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{66} + ( 4 - 4 \beta_{5} - 3 \beta_{7} - \beta_{8} - \beta_{9} ) q^{67} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{68} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{69} + ( -1 + \beta_{4} ) q^{70} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 8 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{71} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{72} + ( \beta_{1} - \beta_{2} + \beta_{4} + 6 \beta_{5} - \beta_{7} ) q^{73} + ( -3 + 3 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{74} -\beta_{2} q^{75} + ( 1 + \beta_{1} - \beta_{5} - \beta_{9} ) q^{76} + ( 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{77} + ( -2 + 4 \beta_{2} + \beta_{3} ) q^{78} + ( -4 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{79} + ( 1 - \beta_{5} ) q^{80} + ( -2 - 8 \beta_{1} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{81} + ( 1 - 2 \beta_{2} ) q^{82} + ( -7 + 3 \beta_{1} + 7 \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{83} + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{6} ) q^{84} + ( \beta_{2} + \beta_{3} - \beta_{4} ) q^{85} + ( -2 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} ) q^{86} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{87} + ( -1 + \beta_{2} - \beta_{6} ) q^{88} + ( 3 - 2 \beta_{1} - 3 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} ) q^{89} + ( 1 + \beta_{2} + \beta_{3} ) q^{90} + ( -2 + 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{91} + ( 1 - \beta_{5} - \beta_{7} + \beta_{9} ) q^{92} + ( -\beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{93} + ( 3 - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{6} ) q^{94} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{95} -\beta_{1} q^{96} + ( 3 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{97} + ( 1 - 2 \beta_{1} - \beta_{5} - 2 \beta_{7} ) q^{98} + ( -5 \beta_{1} + 5 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} - \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 10q^{2} + q^{3} + 10q^{4} + 5q^{5} - q^{6} + 6q^{7} - 10q^{8} - 8q^{9} + O(q^{10})$$ $$10q - 10q^{2} + q^{3} + 10q^{4} + 5q^{5} - q^{6} + 6q^{7} - 10q^{8} - 8q^{9} - 5q^{10} + 4q^{11} + q^{12} - 6q^{13} - 6q^{14} - q^{15} + 10q^{16} + 4q^{17} + 8q^{18} + 4q^{19} + 5q^{20} - 4q^{21} - 4q^{22} + 8q^{23} - q^{24} - 5q^{25} + 6q^{26} - 26q^{27} + 6q^{28} - q^{29} + q^{30} - 8q^{31} - 10q^{32} - 14q^{33} - 4q^{34} + 12q^{35} - 8q^{36} + 14q^{37} - 4q^{38} + 8q^{39} - 5q^{40} - 6q^{41} + 4q^{42} + 15q^{43} + 4q^{44} - 16q^{45} - 8q^{46} - 22q^{47} + q^{48} - 5q^{49} + 5q^{50} + 24q^{51} - 6q^{52} - 8q^{53} + 26q^{54} + 2q^{55} - 6q^{56} - 32q^{57} + q^{58} + 8q^{59} - q^{60} - 14q^{61} + 8q^{62} + 22q^{63} + 10q^{64} - 12q^{65} + 14q^{66} + 19q^{67} + 4q^{68} + 10q^{69} - 12q^{70} + 36q^{71} + 8q^{72} + 28q^{73} - 14q^{74} - 2q^{75} + 4q^{76} + 8q^{77} - 8q^{78} - 10q^{79} + 5q^{80} - 25q^{81} + 6q^{82} - 25q^{83} - 4q^{84} + 8q^{85} - 15q^{86} + 22q^{87} - 4q^{88} + 19q^{89} + 16q^{90} - 10q^{91} + 8q^{92} + 12q^{93} + 22q^{94} - 4q^{95} - q^{96} + 20q^{97} + 5q^{98} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 12 x^{8} + x^{7} + 106 x^{6} - 27 x^{5} + 233 x^{4} - 164 x^{3} + 460 x^{2} - 240 x + 144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$4541 \nu^{9} + 13275 \nu^{8} - 36108 \nu^{7} + 250973 \nu^{6} - 194346 \nu^{5} + 1203777 \nu^{4} - 7157555 \nu^{3} + 2737836 \nu^{2} - 1478304 \nu + 8999376$$$$)/15118180$$ $$\beta_{3}$$ $$=$$ $$($$$$2655 \nu^{9} - 20775 \nu^{8} + 56508 \nu^{7} - 185333 \nu^{6} + 304146 \nu^{5} - 1883877 \nu^{4} + 2128023 \nu^{3} - 4284636 \nu^{2} + 2313504 \nu - 14025200$$$$)/3023636$$ $$\beta_{4}$$ $$=$$ $$($$$$115659 \nu^{9} - 648775 \nu^{8} + 1764668 \nu^{7} - 6351673 \nu^{6} + 9498066 \nu^{5} - 58830917 \nu^{4} + 13339695 \nu^{3} - 133803356 \nu^{2} + 72247584 \nu - 125349056$$$$)/30236360$$ $$\beta_{5}$$ $$=$$ $$($$$$-187487 \nu^{9} + 201110 \nu^{8} - 2210019 \nu^{7} - 295811 \nu^{6} - 19120703 \nu^{5} + 4479111 \nu^{4} - 40073140 \nu^{3} + 9275203 \nu^{2} - 78030512 \nu + 40561968$$$$)/45354540$$ $$\beta_{6}$$ $$=$$ $$($$$$250497 \nu^{9} - 825325 \nu^{8} + 2244884 \nu^{7} - 5540819 \nu^{6} + 12082758 \nu^{5} - 74840471 \nu^{4} - 42699475 \nu^{3} - 170215028 \nu^{2} + 91908192 \nu - 213627928$$$$)/30236360$$ $$\beta_{7}$$ $$=$$ $$($$$$-309304 \nu^{9} - 184850 \nu^{8} - 3276753 \nu^{7} - 5574127 \nu^{6} - 35089246 \nu^{5} - 35659923 \nu^{4} - 68461640 \nu^{3} - 4107559 \nu^{2} - 96580999 \nu - 15923124$$$$)/22677270$$ $$\beta_{8}$$ $$=$$ $$($$$$-34825 \nu^{9} + 26632 \nu^{8} - 405039 \nu^{7} - 160516 \nu^{6} - 3625183 \nu^{5} - 336519 \nu^{4} - 7492244 \nu^{3} + 1319966 \nu^{2} - 16360447 \nu - 1056600$$$$)/2267727$$ $$\beta_{9}$$ $$=$$ $$($$$$145979 \nu^{9} - 36050 \nu^{8} + 1678593 \nu^{7} + 1416977 \nu^{6} + 15714671 \nu^{5} + 7932183 \nu^{4} + 31749940 \nu^{3} - 2693341 \nu^{2} + 33838574 \nu + 5601924$$$$)/4123140$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - 4 \beta_{5} - \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{4} - 2 \beta_{3} - 9 \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} - 11 \beta_{8} + \beta_{7} + 29 \beta_{5} - 17 \beta_{1} - 29$$ $$\nu^{5}$$ $$=$$ $$-12 \beta_{9} - 28 \beta_{8} - 10 \beta_{7} - 12 \beta_{6} + 32 \beta_{5} + 10 \beta_{4} + 28 \beta_{3} + 91 \beta_{2} - 91 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-18 \beta_{6} - 6 \beta_{4} + 119 \beta_{3} + 225 \beta_{2} + 266$$ $$\nu^{7}$$ $$=$$ $$125 \beta_{9} + 344 \beta_{8} + 89 \beta_{7} - 501 \beta_{5} + 973 \beta_{1} + 501$$ $$\nu^{8}$$ $$=$$ $$255 \beta_{9} + 1317 \beta_{8} + 5 \beta_{7} + 255 \beta_{6} - 2699 \beta_{5} - 5 \beta_{4} - 1317 \beta_{3} - 2761 \beta_{2} + 2761 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$1312 \beta_{6} - 802 \beta_{4} - 4078 \beta_{3} - 10723 \beta_{2} - 6588$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/430\mathbb{Z}\right)^\times$$.

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 −1.28976 + 2.23393i −0.850932 + 1.47386i 0.323824 − 0.560879i 0.622887 − 1.07887i 1.69398 − 2.93406i −1.28976 − 2.23393i −0.850932 − 1.47386i 0.323824 + 0.560879i 0.622887 + 1.07887i 1.69398 + 2.93406i
−1.00000 −1.28976 + 2.23393i 1.00000 0.500000 0.866025i 1.28976 2.23393i 2.24349 + 3.88584i −1.00000 −1.82697 3.16441i −0.500000 + 0.866025i
221.2 −1.00000 −0.850932 + 1.47386i 1.00000 0.500000 0.866025i 0.850932 1.47386i −1.11274 1.92732i −1.00000 0.0518303 + 0.0897728i −0.500000 + 0.866025i
221.3 −1.00000 0.323824 0.560879i 1.00000 0.500000 0.866025i −0.323824 + 0.560879i 1.69423 + 2.93449i −1.00000 1.29028 + 2.23482i −0.500000 + 0.866025i
221.4 −1.00000 0.622887 1.07887i 1.00000 0.500000 0.866025i −0.622887 + 1.07887i −0.561656 0.972816i −1.00000 0.724024 + 1.25405i −0.500000 + 0.866025i
221.5 −1.00000 1.69398 2.93406i 1.00000 0.500000 0.866025i −1.69398 + 2.93406i 0.736679 + 1.27596i −1.00000 −4.23916 7.34244i −0.500000 + 0.866025i
251.1 −1.00000 −1.28976 2.23393i 1.00000 0.500000 + 0.866025i 1.28976 + 2.23393i 2.24349 3.88584i −1.00000 −1.82697 + 3.16441i −0.500000 0.866025i
251.2 −1.00000 −0.850932 1.47386i 1.00000 0.500000 + 0.866025i 0.850932 + 1.47386i −1.11274 + 1.92732i −1.00000 0.0518303 0.0897728i −0.500000 0.866025i
251.3 −1.00000 0.323824 + 0.560879i 1.00000 0.500000 + 0.866025i −0.323824 0.560879i 1.69423 2.93449i −1.00000 1.29028 2.23482i −0.500000 0.866025i
251.4 −1.00000 0.622887 + 1.07887i 1.00000 0.500000 + 0.866025i −0.622887 1.07887i −0.561656 + 0.972816i −1.00000 0.724024 1.25405i −0.500000 0.866025i
251.5 −1.00000 1.69398 + 2.93406i 1.00000 0.500000 + 0.866025i −1.69398 2.93406i 0.736679 1.27596i −1.00000 −4.23916 + 7.34244i −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.e.g 10
43.c even 3 1 inner 430.2.e.g 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.g 10 1.a even 1 1 trivial
430.2.e.g 10 43.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{10}$$
$3$ $$1 - T - 3 T^{2} + 10 T^{3} - 2 T^{4} - 42 T^{5} + 35 T^{6} + 115 T^{7} - 245 T^{8} - 132 T^{9} + 1032 T^{10} - 396 T^{11} - 2205 T^{12} + 3105 T^{13} + 2835 T^{14} - 10206 T^{15} - 1458 T^{16} + 21870 T^{17} - 19683 T^{18} - 19683 T^{19} + 59049 T^{20}$$
$5$ $$( 1 - T + T^{2} )^{5}$$
$7$ $$1 - 6 T + 3 T^{2} + 50 T^{3} - 95 T^{4} - 16 T^{5} - 208 T^{6} + 348 T^{7} + 4257 T^{8} - 5530 T^{9} - 15197 T^{10} - 38710 T^{11} + 208593 T^{12} + 119364 T^{13} - 499408 T^{14} - 268912 T^{15} - 11176655 T^{16} + 41177150 T^{17} + 17294403 T^{18} - 242121642 T^{19} + 282475249 T^{20}$$
$11$ $$( 1 - 2 T + 18 T^{2} - 76 T^{3} + 371 T^{4} - 702 T^{5} + 4081 T^{6} - 9196 T^{7} + 23958 T^{8} - 29282 T^{9} + 161051 T^{10} )^{2}$$
$13$ $$1 + 6 T - 5 T^{2} + 18 T^{3} + 467 T^{4} - 554 T^{5} - 1746 T^{6} + 28202 T^{7} + 10477 T^{8} + 16676 T^{9} + 1420225 T^{10} + 216788 T^{11} + 1770613 T^{12} + 61959794 T^{13} - 49867506 T^{14} - 205696322 T^{15} + 2254119803 T^{16} + 1129473306 T^{17} - 4078653605 T^{18} + 63626996238 T^{19} + 137858491849 T^{20}$$
$17$ $$1 - 4 T - 6 T^{2} + 280 T^{3} - 1081 T^{4} - 208 T^{5} + 32636 T^{6} - 139236 T^{7} + 180789 T^{8} + 1937608 T^{9} - 11672626 T^{10} + 32939336 T^{11} + 52248021 T^{12} - 684066468 T^{13} + 2725791356 T^{14} - 295330256 T^{15} - 26092712089 T^{16} + 114894828440 T^{17} - 41854544646 T^{18} - 474351505988 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 - 4 T - 30 T^{2} + 180 T^{3} + 97 T^{4} - 2250 T^{5} + 5972 T^{6} - 11828 T^{7} - 17043 T^{8} + 256830 T^{9} - 1468214 T^{10} + 4879770 T^{11} - 6152523 T^{12} - 81128252 T^{13} + 778277012 T^{14} - 5571222750 T^{15} + 4563450457 T^{16} + 160896913020 T^{17} - 509506891230 T^{18} - 1290750791116 T^{19} + 6131066257801 T^{20}$$
$23$ $$1 - 8 T - 27 T^{2} + 192 T^{3} + 1575 T^{4} - 5756 T^{5} - 23066 T^{6} + 48700 T^{7} - 54019 T^{8} + 249988 T^{9} + 6209083 T^{10} + 5749724 T^{11} - 28576051 T^{12} + 592532900 T^{13} - 6454812506 T^{14} - 37047590308 T^{15} + 233156525175 T^{16} + 653726485824 T^{17} - 2114396602587 T^{18} - 14409221291704 T^{19} + 41426511213649 T^{20}$$
$29$ $$1 + T - 95 T^{2} - 82 T^{3} + 4736 T^{4} + 3458 T^{5} - 162633 T^{6} - 116645 T^{7} + 4535747 T^{8} + 1675080 T^{9} - 123827076 T^{10} + 48577320 T^{11} + 3814563227 T^{12} - 2844854905 T^{13} - 115027230873 T^{14} + 70927553242 T^{15} + 2817083248256 T^{16} - 1414489857338 T^{17} - 47523409231295 T^{18} + 14507145975869 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 + 8 T - 13 T^{2} + 388 T^{3} + 4345 T^{4} - 8230 T^{5} + 52964 T^{6} + 1316242 T^{7} - 980339 T^{8} + 3056850 T^{9} + 278176459 T^{10} + 94762350 T^{11} - 942105779 T^{12} + 39212165422 T^{13} + 48913366244 T^{14} - 235617912730 T^{15} + 3856203493945 T^{16} + 10674894275068 T^{17} - 11087583486733 T^{18} + 211516977285368 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 - 14 T + 49 T^{2} + 226 T^{3} - 2811 T^{4} + 16774 T^{5} - 31644 T^{6} - 434806 T^{7} + 2812513 T^{8} - 2824952 T^{9} - 16711059 T^{10} - 104523224 T^{11} + 3850330297 T^{12} - 22024228318 T^{13} - 59305950684 T^{14} + 1163175534718 T^{15} - 7212256935699 T^{16} + 21454604232058 T^{17} + 172111493242129 T^{18} - 1819464357131078 T^{19} + 4808584372417849 T^{20}$$
$41$ $$( 1 + 3 T + 163 T^{2} + 398 T^{3} + 11909 T^{4} + 22481 T^{5} + 488269 T^{6} + 669038 T^{7} + 11234123 T^{8} + 8477283 T^{9} + 115856201 T^{10} )^{2}$$
$43$ $$1 - 15 T + 13 T^{2} + 604 T^{3} + 2365 T^{4} - 59039 T^{5} + 101695 T^{6} + 1116796 T^{7} + 1033591 T^{8} - 51282015 T^{9} + 147008443 T^{10}$$
$47$ $$( 1 + 11 T + 106 T^{2} + 41 T^{3} - 3215 T^{4} - 52196 T^{5} - 151105 T^{6} + 90569 T^{7} + 11005238 T^{8} + 53676491 T^{9} + 229345007 T^{10} )^{2}$$
$53$ $$1 + 8 T + 31 T^{2} + 520 T^{3} + 3871 T^{4} - 2352 T^{5} + 98098 T^{6} + 852976 T^{7} - 7861235 T^{8} - 76503880 T^{9} - 119377183 T^{10} - 4054705640 T^{11} - 22082209115 T^{12} + 126988507952 T^{13} + 774040405138 T^{14} - 983595799536 T^{15} + 85798241930359 T^{16} + 610849792715240 T^{17} + 1930050402752191 T^{18} + 26398108734417064 T^{19} + 174887470365513049 T^{20}$$
$59$ $$( 1 - 4 T + 232 T^{2} - 856 T^{3} + 24811 T^{4} - 72008 T^{5} + 1463849 T^{6} - 2979736 T^{7} + 47647928 T^{8} - 48469444 T^{9} + 714924299 T^{10} )^{2}$$
$61$ $$1 + 14 T + 45 T^{2} - 806 T^{3} - 8787 T^{4} - 17696 T^{5} + 772476 T^{6} + 8884756 T^{7} + 31411545 T^{8} - 248377590 T^{9} - 3246005175 T^{10} - 15151032990 T^{11} + 116882358945 T^{12} + 2016670801636 T^{13} + 10695579872316 T^{14} - 14945976142496 T^{15} - 452709529510107 T^{16} - 2533050725832926 T^{17} + 8626829084877645 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 - 19 T + 54 T^{2} - 221 T^{3} + 14098 T^{4} - 32577 T^{5} - 397044 T^{6} - 4826483 T^{7} + 18752721 T^{8} + 110914612 T^{9} + 927211540 T^{10} + 7431279004 T^{11} + 84180964569 T^{12} - 1451627506529 T^{13} - 8000881686324 T^{14} - 43983025610739 T^{15} + 1275282271818562 T^{16} - 1339417264776383 T^{17} + 21927654588058614 T^{18} - 516924153529603993 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$1 - 36 T + 567 T^{2} - 5968 T^{3} + 54917 T^{4} - 373762 T^{5} + 748760 T^{6} + 12952414 T^{7} - 202612107 T^{8} + 2525412690 T^{9} - 25391593021 T^{10} + 179304300990 T^{11} - 1021367631387 T^{12} + 4635811447154 T^{13} + 19027250265560 T^{14} - 674352370688462 T^{15} + 7034883292089557 T^{16} - 54279677105277488 T^{17} + 366142252216346487 T^{18} - 1650546025864165116 T^{19} + 3255243551009881201 T^{20}$$
$73$ $$1 - 28 T + 130 T^{2} + 788 T^{3} + 40731 T^{4} - 647242 T^{5} - 299136 T^{6} - 781424 T^{7} + 698093137 T^{8} - 3419792278 T^{9} - 15252313914 T^{10} - 249644836294 T^{11} + 3720138327073 T^{12} - 303987220208 T^{13} - 8494936219776 T^{14} - 1341779003996506 T^{15} + 6163994370977259 T^{16} + 8705350033048436 T^{17} + 104839811946230530 T^{18} - 1648404427831501564 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + 10 T + 3 T^{2} - 790 T^{3} - 11175 T^{4} + 7900 T^{5} - 88976 T^{6} - 5523320 T^{7} - 28654919 T^{8} + 457995090 T^{9} + 11536331579 T^{10} + 36181612110 T^{11} - 178835349479 T^{12} - 2723212169480 T^{13} - 3465622407056 T^{14} + 24308745552100 T^{15} - 2716502315447175 T^{16} - 15171088099065610 T^{17} + 4551326429719683 T^{18} + 1198515959826183190 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 + 25 T + 191 T^{2} + 240 T^{3} + 2762 T^{4} + 155288 T^{5} + 1413133 T^{6} - 7730059 T^{7} - 182939905 T^{8} - 397129048 T^{9} + 6416763512 T^{10} - 32961710984 T^{11} - 1260273005545 T^{12} - 4419947245433 T^{13} + 67064919529693 T^{14} + 611685743370184 T^{15} + 903009311245178 T^{16} + 6512652237510480 T^{17} + 430187816338556831 T^{18} + 4673506381688510075 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 - 19 T + 62 T^{2} - 165 T^{3} + 5234 T^{4} + 112771 T^{5} - 1360484 T^{6} - 1913717 T^{7} + 39369557 T^{8} - 1124166758 T^{9} + 20476807868 T^{10} - 100050841462 T^{11} + 311846260997 T^{12} - 1349111159773 T^{13} - 85359815004644 T^{14} + 629719968123179 T^{15} + 2601200076889874 T^{16} - 7298170257762285 T^{17} + 244068505953529022 T^{18} - 6656771670442218971 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$( 1 - 10 T + 318 T^{2} - 2434 T^{3} + 47465 T^{4} - 293384 T^{5} + 4604105 T^{6} - 22901506 T^{7} + 290230014 T^{8} - 885292810 T^{9} + 8587340257 T^{10} )^{2}$$