Properties

Label 4730.2.a.bd
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 3 x^{10} - 25 x^{9} + 72 x^{8} + 216 x^{7} - 572 x^{6} - 767 x^{5} + 1665 x^{4} + 993 x^{3} - 1256 x^{2} + 96 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -103 \nu^{10} + \nu^{9} + 3107 \nu^{8} + 860 \nu^{7} - 32880 \nu^{6} - 18212 \nu^{5} + 143569 \nu^{4} + 104157 \nu^{3} - 229595 \nu^{2} - 108836 \nu + 73488 \)\()/20016\)
\(\beta_{3}\)\(=\)\((\)\( -119 \nu^{10} - 123 \nu^{9} + 3147 \nu^{8} + 3752 \nu^{7} - 25124 \nu^{6} - 35632 \nu^{5} + 40717 \nu^{4} + 91781 \nu^{3} + 155513 \nu^{2} + 115108 \nu - 54064 \)\()/20016\)
\(\beta_{4}\)\(=\)\((\)\( -261 \nu^{10} + 931 \nu^{9} + 6977 \nu^{8} - 22220 \nu^{7} - 67976 \nu^{6} + 172764 \nu^{5} + 295019 \nu^{4} - 453265 \nu^{3} - 504705 \nu^{2} + 113660 \nu - 12928 \)\()/20016\)
\(\beta_{5}\)\(=\)\((\)\( -261 \nu^{10} + 931 \nu^{9} + 6977 \nu^{8} - 22220 \nu^{7} - 67976 \nu^{6} + 172764 \nu^{5} + 295019 \nu^{4} - 473281 \nu^{3} - 504705 \nu^{2} + 273788 \nu + 7088 \)\()/20016\)
\(\beta_{6}\)\(=\)\((\)\( -55 \nu^{10} + 95 \nu^{9} + 1319 \nu^{8} - 1978 \nu^{7} - 10278 \nu^{6} + 12642 \nu^{5} + 27619 \nu^{4} - 26071 \nu^{3} - 11877 \nu^{2} + 16080 \nu - 7560 \)\()/3336\)
\(\beta_{7}\)\(=\)\((\)\( 93 \nu^{10} - 565 \nu^{9} - 2387 \nu^{8} + 14222 \nu^{7} + 23480 \nu^{6} - 117984 \nu^{5} - 116459 \nu^{4} + 339163 \nu^{3} + 266583 \nu^{2} - 136646 \nu - 10316 \)\()/5004\)
\(\beta_{8}\)\(=\)\((\)\( 591 \nu^{10} - 1501 \nu^{9} - 14891 \nu^{8} + 34088 \nu^{7} + 129644 \nu^{6} - 248616 \nu^{5} - 460733 \nu^{4} + 629707 \nu^{3} + 595983 \nu^{2} - 390284 \nu - 61808 \)\()/20016\)
\(\beta_{9}\)\(=\)\((\)\( -1175 \nu^{10} + 2257 \nu^{9} + 30251 \nu^{8} - 51684 \nu^{7} - 269944 \nu^{6} + 380900 \nu^{5} + 978105 \nu^{4} - 967451 \nu^{3} - 1227115 \nu^{2} + 528372 \nu + 72112 \)\()/20016\)
\(\beta_{10}\)\(=\)\((\)\( -1845 \nu^{10} + 3667 \nu^{9} + 47633 \nu^{8} - 84524 \nu^{7} - 429368 \nu^{6} + 631596 \nu^{5} + 1606859 \nu^{4} - 1648465 \nu^{3} - 2222529 \nu^{2} + 923708 \nu + 209696 \)\()/20016\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{6} + \beta_{5} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} - 2 \beta_{9} + 10 \beta_{8} - \beta_{7} + 11 \beta_{6} + 9 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 9 \beta_{1} + 42\)
\(\nu^{5}\)\(=\)\(3 \beta_{10} - 5 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 13 \beta_{5} + 15 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 71 \beta_{1} + 14\)
\(\nu^{6}\)\(=\)\(15 \beta_{10} - 33 \beta_{9} + 98 \beta_{8} - 20 \beta_{7} + 117 \beta_{6} + 76 \beta_{5} - 14 \beta_{4} + 25 \beta_{3} + 37 \beta_{2} + 77 \beta_{1} + 384\)
\(\nu^{7}\)\(=\)\(56 \beta_{10} - 98 \beta_{9} - 23 \beta_{8} + 23 \beta_{7} - 13 \beta_{6} - 144 \beta_{5} + 192 \beta_{4} + 32 \beta_{3} - 51 \beta_{2} + 663 \beta_{1} + 162\)
\(\nu^{8}\)\(=\)\(187 \beta_{10} - 440 \beta_{9} + 971 \beta_{8} - 286 \beta_{7} + 1227 \beta_{6} + 643 \beta_{5} - 139 \beta_{4} + 412 \beta_{3} + 524 \beta_{2} + 668 \beta_{1} + 3687\)
\(\nu^{9}\)\(=\)\(771 \beta_{10} - 1424 \beta_{9} - 393 \beta_{8} + 349 \beta_{7} - 123 \beta_{6} - 1543 \beta_{5} + 2326 \beta_{4} + 428 \beta_{3} - 645 \beta_{2} + 6399 \beta_{1} + 1853\)
\(\nu^{10}\)\(=\)\(2191 \beta_{10} - 5474 \beta_{9} + 9697 \beta_{8} - 3618 \beta_{7} + 12834 \beta_{6} + 5521 \beta_{5} - 1133 \beta_{4} + 5759 \beta_{3} + 6687 \beta_{2} + 5963 \beta_{1} + 36654\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.26154
−2.79443
−1.98664
−1.33938
−0.183468
0.384923
0.533016
2.54495
2.66316
3.07012
3.36929
1.00000 −3.26154 1.00000 1.00000 −3.26154 2.49864 1.00000 7.63767 1.00000
1.2 1.00000 −2.79443 1.00000 1.00000 −2.79443 −2.04908 1.00000 4.80882 1.00000
1.3 1.00000 −1.98664 1.00000 1.00000 −1.98664 2.71702 1.00000 0.946742 1.00000
1.4 1.00000 −1.33938 1.00000 1.00000 −1.33938 −0.356366 1.00000 −1.20607 1.00000
1.5 1.00000 −0.183468 1.00000 1.00000 −0.183468 −3.52029 1.00000 −2.96634 1.00000
1.6 1.00000 0.384923 1.00000 1.00000 0.384923 0.951091 1.00000 −2.85183 1.00000
1.7 1.00000 0.533016 1.00000 1.00000 0.533016 3.32805 1.00000 −2.71589 1.00000
1.8 1.00000 2.54495 1.00000 1.00000 2.54495 3.09459 1.00000 3.47677 1.00000
1.9 1.00000 2.66316 1.00000 1.00000 2.66316 −1.80847 1.00000 4.09240 1.00000
1.10 1.00000 3.07012 1.00000 1.00000 3.07012 −3.37856 1.00000 6.42564 1.00000
1.11 1.00000 3.36929 1.00000 1.00000 3.36929 3.52337 1.00000 8.35210 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4730.2.a.bd 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bd 11 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\(T_{3}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)
\(T_{13}^{11} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{11} \)
$3$ \( 1 - 3 T + 8 T^{2} - 18 T^{3} + 36 T^{4} - 59 T^{5} + 124 T^{6} - 207 T^{7} + 342 T^{8} - 662 T^{9} + 1239 T^{10} - 1910 T^{11} + 3717 T^{12} - 5958 T^{13} + 9234 T^{14} - 16767 T^{15} + 30132 T^{16} - 43011 T^{17} + 78732 T^{18} - 118098 T^{19} + 157464 T^{20} - 177147 T^{21} + 177147 T^{22} \)
$5$ \( ( 1 - T )^{11} \)
$7$ \( 1 - 5 T + 50 T^{2} - 190 T^{3} + 1196 T^{4} - 3839 T^{5} + 18826 T^{6} - 52789 T^{7} + 217656 T^{8} - 541254 T^{9} + 1940685 T^{10} - 4289798 T^{11} + 13584795 T^{12} - 26521446 T^{13} + 74656008 T^{14} - 126746389 T^{15} + 316408582 T^{16} - 451654511 T^{17} + 984957428 T^{18} - 1095312190 T^{19} + 2017680350 T^{20} - 1412376245 T^{21} + 1977326743 T^{22} \)
$11$ \( ( 1 - T )^{11} \)
$13$ \( 1 - T + 81 T^{2} - 78 T^{3} + 3405 T^{4} - 3133 T^{5} + 96573 T^{6} - 84232 T^{7} + 2036230 T^{8} - 1653738 T^{9} + 33403870 T^{10} - 24588436 T^{11} + 434250310 T^{12} - 279481722 T^{13} + 4473597310 T^{14} - 2405750152 T^{15} + 35856878889 T^{16} - 15122392597 T^{17} + 213658700385 T^{18} - 63626996238 T^{19} + 858964449213 T^{20} - 137858491849 T^{21} + 1792160394037 T^{22} \)
$17$ \( 1 - 8 T + 102 T^{2} - 673 T^{3} + 5299 T^{4} - 29681 T^{5} + 185052 T^{6} - 912175 T^{7} + 4851713 T^{8} - 21430231 T^{9} + 101099041 T^{10} - 403343552 T^{11} + 1718683697 T^{12} - 6193336759 T^{13} + 23836465969 T^{14} - 76185768175 T^{15} + 262747377564 T^{16} - 716427185489 T^{17} + 2174384628227 T^{18} - 4694684757793 T^{19} + 12095963402694 T^{20} - 16127951203592 T^{21} + 34271896307633 T^{22} \)
$19$ \( 1 - T + 102 T^{2} - 46 T^{3} + 5342 T^{4} - 1261 T^{5} + 194504 T^{6} - 48101 T^{7} + 5474534 T^{8} - 1617136 T^{9} + 124760351 T^{10} - 37040174 T^{11} + 2370446669 T^{12} - 583786096 T^{13} + 37549828706 T^{14} - 6268570421 T^{15} + 481611159896 T^{16} - 59324855941 T^{17} + 4775062829738 T^{18} - 781243899886 T^{19} + 32914145173458 T^{20} - 6131066257801 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 - 20 T + 325 T^{2} - 3740 T^{3} + 37775 T^{4} - 320356 T^{5} + 2467139 T^{6} - 16860976 T^{7} + 106678186 T^{8} - 613971656 T^{9} + 3305956354 T^{10} - 16363048296 T^{11} + 76036996142 T^{12} - 324791006024 T^{13} + 1297953489062 T^{14} - 4718392384816 T^{15} + 15879352832677 T^{16} - 47424185256484 T^{17} + 128617281260425 T^{18} - 292883084950940 T^{19} + 585374614975475 T^{20} - 828530224272980 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 - 9 T + 162 T^{2} - 1053 T^{3} + 12166 T^{4} - 63513 T^{5} + 608249 T^{6} - 2695188 T^{7} + 23466090 T^{8} - 91017590 T^{9} + 758047824 T^{10} - 2714096862 T^{11} + 21983386896 T^{12} - 76545793190 T^{13} + 572314469010 T^{14} - 1906255263828 T^{15} + 12475885868101 T^{16} - 37779013586673 T^{17} + 209861995175294 T^{18} - 526759472847933 T^{19} + 2350157648090778 T^{20} - 3786365099701809 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 - 12 T + 164 T^{2} - 1556 T^{3} + 15616 T^{4} - 124012 T^{5} + 1013895 T^{6} - 7040784 T^{7} + 49839566 T^{8} - 309293704 T^{9} + 1931687428 T^{10} - 10680001976 T^{11} + 59882310268 T^{12} - 297231249544 T^{13} + 1484770510706 T^{14} - 6502311880464 T^{15} + 29026953053145 T^{16} - 110061106488172 T^{17} + 429636981957376 T^{18} - 1327098454258196 T^{19} + 4336098034350044 T^{20} - 9835539443769612 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 - 19 T + 299 T^{2} - 3026 T^{3} + 29855 T^{4} - 232367 T^{5} + 1893333 T^{6} - 13007240 T^{7} + 97247462 T^{8} - 622220358 T^{9} + 4293909026 T^{10} - 25192306700 T^{11} + 158874633962 T^{12} - 851819670102 T^{13} + 4925875692686 T^{14} - 24377661925640 T^{15} + 131291202138681 T^{16} - 596190148480103 T^{17} + 2834191191805715 T^{18} - 10628762827564946 T^{19} + 38858560198728023 T^{20} - 91363103075939131 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 - 9 T + 263 T^{2} - 1710 T^{3} + 29847 T^{4} - 133349 T^{5} + 1922353 T^{6} - 4465016 T^{7} + 77896422 T^{8} + 25594334 T^{9} + 2429925018 T^{10} + 6297341132 T^{11} + 99626925738 T^{12} + 43024075454 T^{13} + 5368699300662 T^{14} - 12617068077176 T^{15} + 222716515560953 T^{16} - 633421650433109 T^{17} + 5812830812526207 T^{18} - 13654222141796910 T^{19} + 86101448745611743 T^{20} - 120803933791371609 T^{21} + 550329031716248441 T^{22} \)
$43$ \( ( 1 + T )^{11} \)
$47$ \( 1 - 15 T + 255 T^{2} - 2455 T^{3} + 27185 T^{4} - 217894 T^{5} + 1846022 T^{6} - 13249734 T^{7} + 98662127 T^{8} - 718191795 T^{9} + 4886685036 T^{10} - 34934321878 T^{11} + 229674196692 T^{12} - 1586485675155 T^{13} + 10243398011521 T^{14} - 64654475254854 T^{15} + 423375928512154 T^{16} - 2348726344897126 T^{17} + 13772549529786655 T^{18} - 58456708754623255 T^{19} + 285378270641205585 T^{20} - 788986983537450735 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 - 35 T + 810 T^{2} - 14004 T^{3} + 207380 T^{4} - 2657861 T^{5} + 30622062 T^{6} - 317995297 T^{7} + 3039322164 T^{8} - 26684452634 T^{9} + 217536830827 T^{10} - 1640482517442 T^{11} + 11529452033831 T^{12} - 74956627448906 T^{13} + 452485165809828 T^{14} - 2509135849067857 T^{15} + 12806008314766566 T^{16} - 58909791034685069 T^{17} + 243611596179397060 T^{18} - 871884704520699444 T^{19} + 2672808509359727730 T^{20} - 6121061462792956715 T^{21} + 9269035929372191597 T^{22} \)
$59$ \( 1 - 13 T + 355 T^{2} - 3917 T^{3} + 59667 T^{4} - 587600 T^{5} + 6624242 T^{6} - 60601636 T^{7} + 563071603 T^{8} - 4858682619 T^{9} + 39304507990 T^{10} - 316414839214 T^{11} + 2318965971410 T^{12} - 16913074196739 T^{13} + 115643082752537 T^{14} - 734331900602596 T^{15} + 4735831568256358 T^{16} - 24785281567451600 T^{17} + 148490368144695273 T^{18} - 575134824096125357 T^{19} + 3075363515622503345 T^{20} - 6644517792908338213 T^{21} + 30155888444737842659 T^{22} \)
$61$ \( 1 - 9 T + 422 T^{2} - 3261 T^{3} + 84106 T^{4} - 585093 T^{5} + 10888153 T^{6} - 70680988 T^{7} + 1048482778 T^{8} - 6384957130 T^{9} + 79640272008 T^{10} - 443464733902 T^{11} + 4858056592488 T^{12} - 23758425480730 T^{13} + 237985669433218 T^{14} - 978637721570908 T^{15} + 9196093748522053 T^{16} - 30144210396000573 T^{17} + 264323528966382226 T^{18} - 625157547684133341 T^{19} + 4934929651176007502 T^{20} - 6420086204965943409 T^{21} + 43513917611435838661 T^{22} \)
$67$ \( 1 + T + 333 T^{2} + 638 T^{3} + 59387 T^{4} + 183285 T^{5} + 7347167 T^{6} + 30560872 T^{7} + 700322818 T^{8} + 3400020626 T^{9} + 55047335898 T^{10} + 268569881332 T^{11} + 3688171505166 T^{12} + 15262692590114 T^{13} + 210631191710134 T^{14} + 615835829537512 T^{15} + 9919594632021869 T^{16} + 16579664575845165 T^{17} + 359927480105317001 T^{18} + 259071178281136958 T^{19} + 9059775953966217351 T^{20} + 1822837804551761449 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 - 14 T + 460 T^{2} - 4985 T^{3} + 97579 T^{4} - 889779 T^{5} + 13375484 T^{6} - 107292585 T^{7} + 1371734479 T^{8} - 10002402779 T^{9} + 114707718403 T^{10} - 772344883748 T^{11} + 8144248006613 T^{12} - 50422112408939 T^{13} + 490958859113369 T^{14} - 2726484943685385 T^{15} + 24132440816630884 T^{16} - 113980942526943459 T^{17} + 887492729935635389 T^{18} - 3219081353260118585 T^{19} + 21090310330486554260 T^{20} - 45573409714138336814 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 + 20 T + 434 T^{2} + 6864 T^{3} + 101726 T^{4} + 1297304 T^{5} + 15856793 T^{6} + 173960008 T^{7} + 1848478870 T^{8} + 17939764980 T^{9} + 168530649416 T^{10} + 1469464726928 T^{11} + 12302737407368 T^{12} + 95601007578420 T^{13} + 719089704570790 T^{14} + 4940158231545928 T^{15} + 32872267124381249 T^{16} + 196326497101624856 T^{17} + 1123807661753661422 T^{18} + 5535542070760971984 T^{19} + 25550268631388274242 T^{20} + 85952516594071152980 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 + 23 T + 736 T^{2} + 12240 T^{3} + 228094 T^{4} + 3009647 T^{5} + 42087186 T^{6} + 467303733 T^{7} + 5429720888 T^{8} + 52768209540 T^{9} + 536578013057 T^{10} + 4660295820994 T^{11} + 42389663031503 T^{12} + 329326395739140 T^{13} + 2677064156898632 T^{14} + 18201518251952373 T^{15} + 129504644997203214 T^{16} + 731607431246411087 T^{17} + 4380296416288950946 T^{18} + 18569411833256306640 T^{19} + 88210774643207082784 T^{20} + \)\(21\!\cdots\!23\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 - 12 T + 592 T^{2} - 6193 T^{3} + 164761 T^{4} - 1477529 T^{5} + 28827088 T^{6} - 221017883 T^{7} + 3620521637 T^{8} - 24209524583 T^{9} + 358500020515 T^{10} - 2169978265488 T^{11} + 29755501702745 T^{12} - 166779414852287 T^{13} + 2070167205255319 T^{14} - 10489137638154443 T^{15} + 113551071251337584 T^{16} - 483063882923525201 T^{17} + 4470962897101934147 T^{18} - 13948445793637080913 T^{19} + \)\(11\!\cdots\!76\)\( T^{20} - \)\(18\!\cdots\!88\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 + T + 673 T^{2} + 1442 T^{3} + 221665 T^{4} + 632093 T^{5} + 47406969 T^{6} + 149040168 T^{7} + 7309766046 T^{8} + 22635956562 T^{9} + 849378499526 T^{10} + 2389655595756 T^{11} + 75594686457814 T^{12} + 179299411927602 T^{13} + 5153158459682574 T^{14} + 9351114139336488 T^{15} + 264723333192900081 T^{16} + 314138395147411373 T^{17} + 9804538849617435785 T^{18} + 5676561057822400802 T^{19} + \)\(23\!\cdots\!57\)\( T^{20} + 31181719929966183601 T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 - 17 T + 733 T^{2} - 8806 T^{3} + 226517 T^{4} - 2124109 T^{5} + 43873857 T^{6} - 352763480 T^{7} + 6494674638 T^{8} - 47525276130 T^{9} + 783419884094 T^{10} - 5192510072708 T^{11} + 75991728757118 T^{12} - 447165323107170 T^{13} + 5927514185887374 T^{14} - 31229897247457880 T^{15} + 376759738445961249 T^{16} - 1769323332417733261 T^{17} + 18302185005128722421 T^{18} - 69016440232083518566 T^{19} + \)\(55\!\cdots\!61\)\( T^{20} - \)\(12\!\cdots\!33\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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