Properties

Label 4730.2.a.bb
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 22 x^{9} + 21 x^{8} + 165 x^{7} - 130 x^{6} - 535 x^{5} + 323 x^{4} + 710 x^{3} - 343 x^{2} - 261 x + 128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 179 \nu^{10} - 98 \nu^{9} - 3948 \nu^{8} + 1857 \nu^{7} + 29718 \nu^{6} - 7762 \nu^{5} - 95629 \nu^{4} + 4348 \nu^{3} + 121284 \nu^{2} + 10075 \nu - 37066 \)\()/106\)
\(\beta_{4}\)\(=\)\((\)\( 147 \nu^{10} - 79 \nu^{9} - 3228 \nu^{8} + 1504 \nu^{7} + 24177 \nu^{6} - 6282 \nu^{5} - 77497 \nu^{4} + 3571 \nu^{3} + 98197 \nu^{2} + 8106 \nu - 30135 \)\()/53\)
\(\beta_{5}\)\(=\)\((\)\( 337 \nu^{10} - 162 \nu^{9} - 7450 \nu^{8} + 3113 \nu^{7} + 56338 \nu^{6} - 12764 \nu^{5} - 182201 \nu^{4} + 5074 \nu^{3} + 232742 \nu^{2} + 20953 \nu - 72106 \)\()/106\)
\(\beta_{6}\)\(=\)\((\)\( 177 \nu^{10} - 67 \nu^{9} - 3956 \nu^{8} + 1295 \nu^{7} + 30382 \nu^{6} - 4834 \nu^{5} - 99703 \nu^{4} - 1090 \nu^{3} + 128967 \nu^{2} + 12963 \nu - 40313 \)\()/53\)
\(\beta_{7}\)\(=\)\((\)\( -421 \nu^{10} + 192 \nu^{9} + 9340 \nu^{8} - 3715 \nu^{7} - 70956 \nu^{6} + 15218 \nu^{5} + 230195 \nu^{4} - 5464 \nu^{3} - 294094 \nu^{2} - 26115 \nu + 90598 \)\()/106\)
\(\beta_{8}\)\(=\)\((\)\( -325 \nu^{10} + 135 \nu^{9} + 7233 \nu^{8} - 2603 \nu^{7} - 55234 \nu^{6} + 10089 \nu^{5} + 180357 \nu^{4} - 271 \nu^{3} - 232306 \nu^{2} - 23017 \nu + 72296 \)\()/53\)
\(\beta_{9}\)\(=\)\((\)\( 325 \nu^{10} - 135 \nu^{9} - 7233 \nu^{8} + 2603 \nu^{7} + 55234 \nu^{6} - 10089 \nu^{5} - 180357 \nu^{4} + 324 \nu^{3} + 232306 \nu^{2} + 22646 \nu - 72296 \)\()/53\)
\(\beta_{10}\)\(=\)\((\)\( -474 \nu^{10} + 192 \nu^{9} + 10559 \nu^{8} - 3715 \nu^{7} - 80708 \nu^{6} + 14370 \nu^{5} + 263479 \nu^{4} + 48 \nu^{3} - 338879 \nu^{2} - 33429 \nu + 105438 \)\()/53\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 3 \beta_{5} + \beta_{4} + 10 \beta_{2} - 3 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(-\beta_{10} + 13 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 4 \beta_{2} + 60 \beta_{1} - 4\)
\(\nu^{6}\)\(=\)\(2 \beta_{10} - 16 \beta_{9} - 32 \beta_{8} - 15 \beta_{7} - 13 \beta_{6} - 42 \beta_{5} + 14 \beta_{4} - \beta_{3} + 98 \beta_{2} - 59 \beta_{1} + 211\)
\(\nu^{7}\)\(=\)\(-14 \beta_{10} + 145 \beta_{9} + 149 \beta_{8} + 31 \beta_{7} - 26 \beta_{6} + 58 \beta_{5} - 35 \beta_{4} + 13 \beta_{3} - 84 \beta_{2} + 571 \beta_{1} - 113\)
\(\nu^{8}\)\(=\)\(35 \beta_{10} - 216 \beta_{9} - 405 \beta_{8} - 176 \beta_{7} - 135 \beta_{6} - 475 \beta_{5} + 164 \beta_{4} - 23 \beta_{3} + 979 \beta_{2} - 861 \beta_{1} + 1929\)
\(\nu^{9}\)\(=\)\(-164 \beta_{10} + 1554 \beta_{9} + 1658 \beta_{8} + 392 \beta_{7} - 253 \beta_{6} + 820 \beta_{5} - 452 \beta_{4} + 130 \beta_{3} - 1281 \beta_{2} + 5744 \beta_{1} - 1947\)
\(\nu^{10}\)\(=\)\(452 \beta_{10} - 2756 \beta_{9} - 4787 \beta_{8} - 1946 \beta_{7} - 1309 \beta_{6} - 5129 \beta_{5} + 1856 \beta_{4} - 361 \beta_{3} + 9984 \beta_{2} - 11201 \beta_{1} + 18835\)

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
2.87555
2.78036
2.17792
1.48284
0.580450
0.577714
−0.797340
−1.56687
−1.80931
−1.95364
−3.34767
−1.00000 −2.87555 1.00000 −1.00000 2.87555 −0.638121 −1.00000 5.26880 1.00000
1.2 −1.00000 −2.78036 1.00000 −1.00000 2.78036 −2.23633 −1.00000 4.73038 1.00000
1.3 −1.00000 −2.17792 1.00000 −1.00000 2.17792 4.63803 −1.00000 1.74332 1.00000
1.4 −1.00000 −1.48284 1.00000 −1.00000 1.48284 4.82678 −1.00000 −0.801199 1.00000
1.5 −1.00000 −0.580450 1.00000 −1.00000 0.580450 1.64698 −1.00000 −2.66308 1.00000
1.6 −1.00000 −0.577714 1.00000 −1.00000 0.577714 −2.69986 −1.00000 −2.66625 1.00000
1.7 −1.00000 0.797340 1.00000 −1.00000 −0.797340 −1.54185 −1.00000 −2.36425 1.00000
1.8 −1.00000 1.56687 1.00000 −1.00000 −1.56687 −4.55535 −1.00000 −0.544925 1.00000
1.9 −1.00000 1.80931 1.00000 −1.00000 −1.80931 4.53794 −1.00000 0.273608 1.00000
1.10 −1.00000 1.95364 1.00000 −1.00000 −1.95364 0.00701571 −1.00000 0.816706 1.00000
1.11 −1.00000 3.34767 1.00000 −1.00000 −3.34767 2.01478 −1.00000 8.20687 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.bb 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bb 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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{11} \)
$3$ \( 1 + T + 11 T^{2} + 9 T^{3} + 66 T^{4} + 31 T^{5} + 257 T^{6} - 35 T^{7} + 704 T^{8} - 725 T^{9} + 1638 T^{10} - 3146 T^{11} + 4914 T^{12} - 6525 T^{13} + 19008 T^{14} - 2835 T^{15} + 62451 T^{16} + 22599 T^{17} + 144342 T^{18} + 59049 T^{19} + 216513 T^{20} + 59049 T^{21} + 177147 T^{22} \)
$5$ \( ( 1 + T )^{11} \)
$7$ \( 1 - 6 T + 41 T^{2} - 195 T^{3} + 853 T^{4} - 3192 T^{5} + 11332 T^{6} - 35308 T^{7} + 109189 T^{8} - 304757 T^{9} + 854564 T^{10} - 2261356 T^{11} + 5981948 T^{12} - 14933093 T^{13} + 37451827 T^{14} - 84774508 T^{15} + 190456924 T^{16} - 375535608 T^{17} + 702482179 T^{18} - 1124136195 T^{19} + 1654497887 T^{20} - 1694851494 T^{21} + 1977326743 T^{22} \)
$11$ \( ( 1 + T )^{11} \)
$13$ \( 1 - 4 T + 69 T^{2} - 214 T^{3} + 2317 T^{4} - 6388 T^{5} + 55721 T^{6} - 144536 T^{7} + 1064202 T^{8} - 2539080 T^{9} + 16477738 T^{10} - 35978532 T^{11} + 214210594 T^{12} - 429104520 T^{13} + 2338051794 T^{14} - 4128092696 T^{15} + 20688817253 T^{16} - 30833655892 T^{17} + 145388313889 T^{18} - 174566374294 T^{19} + 731710456737 T^{20} - 551433967396 T^{21} + 1792160394037 T^{22} \)
$17$ \( 1 + 10 T + 113 T^{2} + 787 T^{3} + 5635 T^{4} + 31710 T^{5} + 178382 T^{6} + 861634 T^{7} + 4228065 T^{8} + 18442633 T^{9} + 82974330 T^{10} + 335085016 T^{11} + 1410563610 T^{12} + 5329920937 T^{13} + 20772483345 T^{14} + 71964533314 T^{15} + 253276931374 T^{16} + 765402312990 T^{17} + 2312258422355 T^{18} + 5489921106067 T^{19} + 13400430044161 T^{20} + 20159939004490 T^{21} + 34271896307633 T^{22} \)
$19$ \( 1 - 14 T + 139 T^{2} - 1081 T^{3} + 7803 T^{4} - 51898 T^{5} + 318510 T^{6} - 1798606 T^{7} + 9535247 T^{8} - 47724665 T^{9} + 226979392 T^{10} - 1016995472 T^{11} + 4312608448 T^{12} - 17228604065 T^{13} + 65402259173 T^{14} - 234396132526 T^{15} + 788662292490 T^{16} - 2441587132138 T^{17} + 6974881179417 T^{18} - 18359231647321 T^{19} + 44853589991281 T^{20} - 85834927609214 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 + 18 T + 319 T^{2} + 3714 T^{3} + 40117 T^{4} + 351722 T^{5} + 2862803 T^{6} + 20204856 T^{7} + 133116154 T^{8} + 781361156 T^{9} + 4297024446 T^{10} + 21277738828 T^{11} + 98831562258 T^{12} + 413340051524 T^{13} + 1619624245718 T^{14} + 5654147107896 T^{15} + 18425982049429 T^{16} + 52067478950858 T^{17} + 136591382457299 T^{18} + 290846999333634 T^{19} + 574567699006697 T^{20} + 745677201845682 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 + 6 T + 185 T^{2} + 1098 T^{3} + 17276 T^{4} + 100744 T^{5} + 1083130 T^{6} + 6059792 T^{7} + 50753057 T^{8} + 264521298 T^{9} + 1857219847 T^{10} + 8754314412 T^{11} + 53859375563 T^{12} + 222462411618 T^{13} + 1237816307173 T^{14} + 4285975745552 T^{15} + 22216240816370 T^{16} + 59924880650824 T^{17} + 298008863114284 T^{18} + 549270561431178 T^{19} + 2683822005535765 T^{20} + 2524243399801206 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 - 13 T + 316 T^{2} - 3389 T^{3} + 45996 T^{4} - 414137 T^{5} + 4077083 T^{6} - 31182876 T^{7} + 244722226 T^{8} - 1600202938 T^{9} + 10449201896 T^{10} - 58438954190 T^{11} + 323925258776 T^{12} - 1537795023418 T^{13} + 7290519834766 T^{14} - 28798040826396 T^{15} + 116723424846533 T^{16} - 367548111938297 T^{17} + 1265470198649556 T^{18} - 2890447725887549 T^{19} + 8354920602772036 T^{20} - 10655167730750413 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 - T + 186 T^{2} + 69 T^{3} + 18790 T^{4} + 19123 T^{5} + 1371913 T^{6} + 1719148 T^{7} + 77809346 T^{8} + 102179878 T^{9} + 3522899448 T^{10} + 4450627678 T^{11} + 130347279576 T^{12} + 139884252982 T^{13} + 3941276802938 T^{14} + 3221960134828 T^{15} + 95133876079741 T^{16} + 49064386119307 T^{17} + 1783769971329070 T^{18} + 242361082320549 T^{19} + 24172883601884322 T^{20} - 4808584372417849 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 - 12 T + 281 T^{2} - 2910 T^{3} + 39921 T^{4} - 367844 T^{5} + 3783993 T^{6} - 31007864 T^{7} + 261909310 T^{8} - 1908124592 T^{9} + 13837625822 T^{10} - 89265217428 T^{11} + 567342658702 T^{12} - 3207557439152 T^{13} + 18051051554510 T^{14} - 87620812784504 T^{15} + 438399053590593 T^{16} - 1747297344426404 T^{17} + 7774785367603401 T^{18} - 23236132416742110 T^{19} + 91994323564703041 T^{20} - 161071911721828812 T^{21} + 550329031716248441 T^{22} \)
$43$ \( ( 1 - T )^{11} \)
$47$ \( 1 + 5 T + 211 T^{2} + 1191 T^{3} + 26133 T^{4} + 146594 T^{5} + 2359788 T^{6} + 12684950 T^{7} + 166644597 T^{8} + 834900889 T^{9} + 9555736092 T^{10} + 43627217926 T^{11} + 449119596324 T^{12} + 1844296063801 T^{13} + 17301541994331 T^{14} + 61898509500950 T^{15} + 541205595378516 T^{16} + 1580168291939426 T^{17} + 13239582007059579 T^{18} + 28359242414157351 T^{19} + 236136529824683837 T^{20} + 262995661179150245 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 + 27 T + 683 T^{2} + 11403 T^{3} + 176440 T^{4} + 2209291 T^{5} + 26016523 T^{6} + 265446321 T^{7} + 2573710310 T^{8} + 22256645035 T^{9} + 183789838266 T^{10} + 1368384439916 T^{11} + 9740861428098 T^{12} + 62518915903315 T^{13} + 383166269821870 T^{14} + 2094499152370401 T^{15} + 10879992662130839 T^{16} + 48967523563049539 T^{17} + 207266033512840280 T^{18} + 709947249760749483 T^{19} + 2253738533200856839 T^{20} + 4721961699868852323 T^{21} + 9269035929372191597 T^{22} \)
$59$ \( 1 - 11 T + 467 T^{2} - 3647 T^{3} + 89683 T^{4} - 438686 T^{5} + 9084162 T^{6} - 13845946 T^{7} + 519801447 T^{8} + 1822460995 T^{9} + 19762126954 T^{10} + 203907662094 T^{11} + 1165965490286 T^{12} + 6343986723595 T^{13} + 106756301383413 T^{14} - 167776326068506 T^{15} + 6494488149852438 T^{16} - 18504009580835726 T^{17} + 223189731113022377 T^{18} - 535490605942958687 T^{19} + 4045619047311856513 T^{20} - 5622284286307055411 T^{21} + 30155888444737842659 T^{22} \)
$61$ \( 1 - 36 T + 839 T^{2} - 14612 T^{3} + 218860 T^{4} - 2882022 T^{5} + 34433520 T^{6} - 373718240 T^{7} + 3744303379 T^{8} - 34749872166 T^{9} + 301335794361 T^{10} - 2434628504696 T^{11} + 18381483456021 T^{12} - 129304274329686 T^{13} + 849885725268799 T^{14} - 5174443329839840 T^{15} + 29082423622409520 T^{16} - 148482852356637942 T^{17} + 687820697091556060 T^{18} - 2801227257516269972 T^{19} + 9811388571887844299 T^{20} - 25680344819863773636 T^{21} + 43513917611435838661 T^{22} \)
$67$ \( 1 - 18 T + 557 T^{2} - 7280 T^{3} + 133099 T^{4} - 1446074 T^{5} + 20250607 T^{6} - 194050496 T^{7} + 2253964178 T^{8} - 19229019076 T^{9} + 192042617706 T^{10} - 1459223924640 T^{11} + 12866855386302 T^{12} - 86319066632164 T^{13} + 677909028067814 T^{14} - 3910335025006016 T^{15} + 27340852942689949 T^{16} - 130809514536654506 T^{17} + 806674653956885977 T^{18} - 2956172692612346480 T^{19} + 15154039658736285479 T^{20} - 32811080481931706082 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 + 14 T + 437 T^{2} + 4223 T^{3} + 71293 T^{4} + 481786 T^{5} + 6090758 T^{6} + 32381282 T^{7} + 417026685 T^{8} + 2611708381 T^{9} + 33828315852 T^{10} + 225169195076 T^{11} + 2401810425492 T^{12} + 13165621948621 T^{13} + 149258437855035 T^{14} + 822862808555042 T^{15} + 10989124353438058 T^{16} + 61716923389162906 T^{17} + 648418401452169563 T^{18} + 2727017162450848703 T^{19} + 20035794813962226547 T^{20} + 45573409714138336814 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 + 11 T + 419 T^{2} + 4678 T^{3} + 89099 T^{4} + 977295 T^{5} + 12973961 T^{6} + 132506040 T^{7} + 1444417742 T^{8} + 13285903126 T^{9} + 129124939882 T^{10} + 1067800677380 T^{11} + 9426120611386 T^{12} + 70800577758454 T^{13} + 561903056739614 T^{14} + 3762938457875640 T^{15} + 26895949997789873 T^{16} + 147898182681108255 T^{17} + 984312160653023603 T^{18} + 3772620309880510918 T^{19} + 24667194830764255547 T^{20} + 47273884126739134139 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 - 28 T + 932 T^{2} - 18511 T^{3} + 366453 T^{4} - 5675049 T^{5} + 84285456 T^{6} - 1068914179 T^{7} + 12909518055 T^{8} - 137778481749 T^{9} + 1401316366039 T^{10} - 12753157348064 T^{11} + 110703992917081 T^{12} - 859875504595509 T^{13} + 6364895872319145 T^{14} - 41634293854098499 T^{15} + 259351101727432944 T^{16} - 1379533221366995529 T^{17} + 7037330059704924027 T^{18} - 28083201180180350671 T^{19} + \)\(11\!\cdots\!08\)\( T^{20} - \)\(26\!\cdots\!28\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 + 4 T + 376 T^{2} + 1039 T^{3} + 71899 T^{4} + 183149 T^{5} + 10363006 T^{6} + 29383751 T^{7} + 1227190655 T^{8} + 3388895417 T^{9} + 119250890355 T^{10} + 299932335704 T^{11} + 9897823899465 T^{12} + 23346100527713 T^{13} + 701691663050485 T^{14} + 1394503487142071 T^{15} + 40820301817652858 T^{16} + 59878802442158981 T^{17} + 1951054930103191673 T^{18} + 2340131629192463599 T^{19} + 70289535980595191528 T^{20} + 62064164748823413796 T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 + 7 T + 582 T^{2} + 4287 T^{3} + 168480 T^{4} + 1270715 T^{5} + 32203935 T^{6} + 241593516 T^{7} + 4553415280 T^{8} + 32830868478 T^{9} + 504268929186 T^{10} + 3348280634826 T^{11} + 44879934697554 T^{12} + 260053309214238 T^{13} + 3210016616526320 T^{14} + 15158118604909356 T^{15} + 179828687531731815 T^{16} + 631521581143507115 T^{17} + 7452095303198725920 T^{18} + 16876156210044821247 T^{19} + \)\(20\!\cdots\!38\)\( T^{20} + \)\(21\!\cdots\!07\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 + T + 387 T^{2} + 1342 T^{3} + 82555 T^{4} + 470349 T^{5} + 13051513 T^{6} + 93397112 T^{7} + 1738013254 T^{8} + 13081357618 T^{9} + 198916145922 T^{10} + 1415625834516 T^{11} + 19294866154434 T^{12} + 123082493827762 T^{13} + 1586237770567942 T^{14} + 8268379172836472 T^{15} + 112077782999658841 T^{16} + 391787549546350221 T^{17} + 6670302375090618715 T^{18} + 10517835883653881662 T^{19} + \)\(29\!\cdots\!79\)\( T^{20} + 73742412689492826049 T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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