Properties

Label 4730.2.a.bf
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 32 x^{11} - 5 x^{10} + 376 x^{9} + 100 x^{8} - 1985 x^{7} - 576 x^{6} + 4708 x^{5} + 889 x^{4} - 4774 x^{3} - 296 x^{2} + 1648 x - 128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(-1209723 \nu^{12} + 95374748 \nu^{11} - 33549200 \nu^{10} - 2743522569 \nu^{9} + 925347580 \nu^{8} + 27643360772 \nu^{7} - 4575966069 \nu^{6} - 113746877852 \nu^{5} - 29379276 \nu^{4} + 169384509677 \nu^{3} + 4304466878 \nu^{2} - 83293017552 \nu + 1796287264\)\()/ 2054583056 \)
\(\beta_{4}\)\(=\)\((\)\(-5481053 \nu^{12} + 43657412 \nu^{11} + 142029872 \nu^{10} - 1180471903 \nu^{9} - 1458988540 \nu^{8} + 10768673468 \nu^{7} + 8502787741 \nu^{6} - 36350024564 \nu^{5} - 29481549172 \nu^{4} + 27932078459 \nu^{3} + 39779836946 \nu^{2} + 9035066960 \nu - 10321684480\)\()/ 2054583056 \)
\(\beta_{5}\)\(=\)\((\)\(-1552491 \nu^{12} + 12856529 \nu^{11} + 28305288 \nu^{10} - 353658669 \nu^{9} - 66962761 \nu^{8} + 3432214148 \nu^{7} - 1065584245 \nu^{6} - 14002567981 \nu^{5} + 5222853248 \nu^{4} + 22264625561 \nu^{3} - 4442731421 \nu^{2} - 10640613778 \nu - 126650044\)\()/ 513645764 \)
\(\beta_{6}\)\(=\)\((\)\(-7498623 \nu^{12} + 119840596 \nu^{11} + 152682496 \nu^{10} - 3489939941 \nu^{9} - 1110548172 \nu^{8} + 36121754996 \nu^{7} + 5494069471 \nu^{6} - 157358169460 \nu^{5} - 21132901100 \nu^{4} + 263223860793 \nu^{3} + 12404294110 \nu^{2} - 133095724096 \nu + 9257057216\)\()/ 2054583056 \)
\(\beta_{7}\)\(=\)\((\)\(727017 \nu^{12} + 1618382 \nu^{11} - 22069016 \nu^{10} - 54545229 \nu^{9} + 235343334 \nu^{8} + 638419143 \nu^{7} - 1043909357 \nu^{6} - 3006216222 \nu^{5} + 1923637015 \nu^{4} + 4831785188 \nu^{3} - 2184380390 \nu^{2} - 1579243760 \nu + 617584666\)\()/ 128411441 \)
\(\beta_{8}\)\(=\)\((\)\(37859597 \nu^{12} + 18959624 \nu^{11} - 1105975856 \nu^{10} - 870485553 \nu^{9} + 11145550800 \nu^{8} + 12205431316 \nu^{7} - 43669040893 \nu^{6} - 63677149080 \nu^{5} + 50088512132 \nu^{4} + 107832991333 \nu^{3} - 7926495062 \nu^{2} - 53582557768 \nu + 1147504160\)\()/ 2054583056 \)
\(\beta_{9}\)\(=\)\((\)\(-21709793 \nu^{12} - 1131240 \nu^{11} + 660508136 \nu^{10} + 179115477 \nu^{9} - 7186292024 \nu^{8} - 3343610108 \nu^{7} + 33252100577 \nu^{6} + 19863052416 \nu^{5} - 60782498060 \nu^{4} - 36532740473 \nu^{3} + 36788542462 \nu^{2} + 18443062896 \nu - 2778200784\)\()/ 1027291528 \)
\(\beta_{10}\)\(=\)\((\)\(-57521037 \nu^{12} + 2964296 \nu^{11} + 1781628496 \nu^{10} + 152007089 \nu^{9} - 19772954064 \nu^{8} - 3198910436 \nu^{7} + 93374155469 \nu^{6} + 10580600648 \nu^{5} - 172412437988 \nu^{4} + 31477383355 \nu^{3} + 96187853574 \nu^{2} - 52892942840 \nu + 4872284112\)\()/ 2054583056 \)
\(\beta_{11}\)\(=\)\((\)\(-58907007 \nu^{12} - 134757996 \nu^{11} + 1894620016 \nu^{10} + 4238230907 \nu^{9} - 21372141628 \nu^{8} - 46129163084 \nu^{7} + 98233765135 \nu^{6} + 198773038460 \nu^{5} - 161245957452 \nu^{4} - 281557192999 \nu^{3} + 89669389822 \nu^{2} + 115425178880 \nu - 18968508576\)\()/ 2054583056 \)
\(\beta_{12}\)\(=\)\((\)\(-34294587 \nu^{12} - 28183806 \nu^{11} + 1057712496 \nu^{10} + 1016626975 \nu^{9} - 11529953354 \nu^{8} - 12220311388 \nu^{7} + 52055942691 \nu^{6} + 55935151726 \nu^{5} - 87424696364 \nu^{4} - 78411099859 \nu^{3} + 49091138148 \nu^{2} + 28811682716 \nu - 4427223552\)\()/ 1027291528 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 9 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{3} + 12 \beta_{2} + 2 \beta_{1} + 42\)
\(\nu^{5}\)\(=\)\(\beta_{11} - 2 \beta_{10} + 16 \beta_{9} + 15 \beta_{8} + \beta_{7} - 12 \beta_{6} + 15 \beta_{5} - 14 \beta_{4} + 12 \beta_{3} + 90 \beta_{1} + 21\)
\(\nu^{6}\)\(=\)\(-14 \beta_{12} - 35 \beta_{11} + 34 \beta_{10} + 18 \beta_{9} - 11 \beta_{8} - \beta_{7} - 20 \beta_{6} + 7 \beta_{5} + \beta_{4} - 35 \beta_{3} + 136 \beta_{2} + 38 \beta_{1} + 403\)
\(\nu^{7}\)\(=\)\(-11 \beta_{12} + 18 \beta_{11} - 30 \beta_{10} + 214 \beta_{9} + 188 \beta_{8} + 15 \beta_{7} - 134 \beta_{6} + 195 \beta_{5} - 171 \beta_{4} + 125 \beta_{3} + 6 \beta_{2} + 949 \beta_{1} + 305\)
\(\nu^{8}\)\(=\)\(-173 \beta_{12} - 479 \beta_{11} + 467 \beta_{10} + 268 \beta_{9} - 88 \beta_{8} - 4 \beta_{7} - 320 \beta_{6} + 178 \beta_{5} + 12 \beta_{4} - 468 \beta_{3} + 1524 \beta_{2} + 561 \beta_{1} + 4093\)
\(\nu^{9}\)\(=\)\(-319 \beta_{12} + 234 \beta_{11} - 302 \beta_{10} + 2714 \beta_{9} + 2231 \beta_{8} + 177 \beta_{7} - 1542 \beta_{6} + 2442 \beta_{5} - 2003 \beta_{4} + 1262 \beta_{3} + 165 \beta_{2} + 10329 \beta_{1} + 3953\)
\(\nu^{10}\)\(=\)\(-2171 \beta_{12} - 5994 \beta_{11} + 5991 \beta_{10} + 3754 \beta_{9} - 466 \beta_{8} + 142 \beta_{7} - 4631 \beta_{6} + 3166 \beta_{5} + 69 \beta_{4} - 5715 \beta_{3} + 17032 \beta_{2} + 7657 \beta_{1} + 42977\)
\(\nu^{11}\)\(=\)\(-6101 \beta_{12} + 2654 \beta_{11} - 2223 \beta_{10} + 33580 \beta_{9} + 25975 \beta_{8} + 2029 \beta_{7} - 18340 \beta_{6} + 30181 \beta_{5} - 23026 \beta_{4} + 12628 \beta_{3} + 3111 \beta_{2} + 114645 \beta_{1} + 48941\)
\(\nu^{12}\)\(=\)\(-28049 \beta_{12} - 71881 \beta_{11} + 74580 \beta_{10} + 50863 \beta_{9} + 1043 \beta_{8} + 4072 \beta_{7} - 63229 \beta_{6} + 48708 \beta_{5} - 457 \beta_{4} - 67170 \beta_{3} + 190399 \beta_{2} + 100998 \beta_{1} + 461337\)

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
3.45880
3.26258
2.61361
1.30046
1.16663
0.743787
0.0802958
−0.921464
−1.13397
−2.15443
−2.24942
−2.86226
−3.30461
1.00000 −3.45880 1.00000 −1.00000 −3.45880 −3.54413 1.00000 8.96327 −1.00000
1.2 1.00000 −3.26258 1.00000 −1.00000 −3.26258 2.73152 1.00000 7.64446 −1.00000
1.3 1.00000 −2.61361 1.00000 −1.00000 −2.61361 0.881450 1.00000 3.83093 −1.00000
1.4 1.00000 −1.30046 1.00000 −1.00000 −1.30046 3.42310 1.00000 −1.30880 −1.00000
1.5 1.00000 −1.16663 1.00000 −1.00000 −1.16663 −3.85859 1.00000 −1.63898 −1.00000
1.6 1.00000 −0.743787 1.00000 −1.00000 −0.743787 2.37023 1.00000 −2.44678 −1.00000
1.7 1.00000 −0.0802958 1.00000 −1.00000 −0.0802958 2.53237 1.00000 −2.99355 −1.00000
1.8 1.00000 0.921464 1.00000 −1.00000 0.921464 −3.84192 1.00000 −2.15090 −1.00000
1.9 1.00000 1.13397 1.00000 −1.00000 1.13397 −0.848165 1.00000 −1.71412 −1.00000
1.10 1.00000 2.15443 1.00000 −1.00000 2.15443 5.20049 1.00000 1.64159 −1.00000
1.11 1.00000 2.24942 1.00000 −1.00000 2.24942 −1.43432 1.00000 2.05989 −1.00000
1.12 1.00000 2.86226 1.00000 −1.00000 2.86226 4.06491 1.00000 5.19253 −1.00000
1.13 1.00000 3.30461 1.00000 −1.00000 3.30461 −0.676945 1.00000 7.92047 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.bf 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bf 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{13} \)
$3$ \( 1 + 7 T^{2} + 5 T^{3} + 22 T^{4} + 50 T^{5} + 49 T^{6} + 201 T^{7} + 202 T^{8} + 479 T^{9} + 830 T^{10} + 1238 T^{11} + 2485 T^{12} + 4118 T^{13} + 7455 T^{14} + 11142 T^{15} + 22410 T^{16} + 38799 T^{17} + 49086 T^{18} + 146529 T^{19} + 107163 T^{20} + 328050 T^{21} + 433026 T^{22} + 295245 T^{23} + 1240029 T^{24} + 1594323 T^{26} \)
$5$ \( ( 1 + T )^{13} \)
$7$ \( 1 - 7 T + 55 T^{2} - 267 T^{3} + 1347 T^{4} - 5442 T^{5} + 22070 T^{6} - 79322 T^{7} + 278950 T^{8} - 908818 T^{9} + 2841829 T^{10} - 8427511 T^{11} + 23807104 T^{12} - 64576498 T^{13} + 166649728 T^{14} - 412948039 T^{15} + 974747347 T^{16} - 2182072018 T^{17} + 4688312650 T^{18} - 9332153978 T^{19} + 18175594010 T^{20} - 31372047042 T^{21} + 54356308629 T^{22} - 75420891483 T^{23} + 108752970865 T^{24} - 96889010407 T^{25} + 96889010407 T^{26} \)
$11$ \( ( 1 - T )^{13} \)
$13$ \( 1 - 11 T + 75 T^{2} - 382 T^{3} + 1978 T^{4} - 9930 T^{5} + 46646 T^{6} - 195198 T^{7} + 796185 T^{8} - 3305893 T^{9} + 13747971 T^{10} - 54137268 T^{11} + 200442712 T^{12} - 724655468 T^{13} + 2605755256 T^{14} - 9149198292 T^{15} + 30204292287 T^{16} - 94419609973 T^{17} + 295617917205 T^{18} - 942183463182 T^{19} + 2926967323982 T^{20} - 8100206059530 T^{21} + 20975699759794 T^{22} - 52661943886318 T^{23} + 134412029552775 T^{24} - 256278936347291 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 + 2 T + 67 T^{2} + 163 T^{3} + 2718 T^{4} + 6056 T^{5} + 82177 T^{6} + 154121 T^{7} + 2009116 T^{8} + 3235927 T^{9} + 42051082 T^{10} + 59922182 T^{11} + 791261731 T^{12} + 1033688050 T^{13} + 13451449427 T^{14} + 17317510598 T^{15} + 206596965866 T^{16} + 270267858967 T^{17} + 2852657416412 T^{18} + 3720106271849 T^{19} + 33720401131121 T^{20} + 42245187062696 T^{21} + 322321848318846 T^{22} + 328607005773187 T^{23} + 2296217052611411 T^{24} + 1165244474459522 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 - 7 T + 133 T^{2} - 733 T^{3} + 8113 T^{4} - 36066 T^{5} + 308714 T^{6} - 1152722 T^{7} + 8733404 T^{8} - 29094256 T^{9} + 209356519 T^{10} - 655415059 T^{11} + 4489611260 T^{12} - 13285488074 T^{13} + 85302613940 T^{14} - 236604836299 T^{15} + 1435976363821 T^{16} - 3791592536176 T^{17} + 21624772910996 T^{18} - 54230822038082 T^{19} + 275950720033646 T^{20} - 612529184636706 T^{21} + 2617965292081027 T^{22} - 4494071566968133 T^{23} + 15493204433463127 T^{24} - 15493204433463127 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 - 12 T + 235 T^{2} - 2110 T^{3} + 24018 T^{4} - 178048 T^{5} + 1515766 T^{6} - 9755958 T^{7} + 68250451 T^{8} - 391837716 T^{9} + 2366333961 T^{10} - 12302060204 T^{11} + 66093027700 T^{12} - 312696182080 T^{13} + 1520139637100 T^{14} - 6507789847916 T^{15} + 28791185303487 T^{16} - 109652258283156 T^{17} + 439283312540693 T^{18} - 1444231915576662 T^{19} + 5160918648497402 T^{20} - 13943114307311488 T^{21} + 43260084623018334 T^{22} - 87409938660799390 T^{23} + 223910293109772845 T^{24} - 262975493184243852 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 - 14 T + 263 T^{2} - 2504 T^{3} + 29073 T^{4} - 223736 T^{5} + 2057591 T^{6} - 13777228 T^{7} + 109005347 T^{8} - 654890530 T^{9} + 4613166005 T^{10} - 25182646956 T^{11} + 160523919836 T^{12} - 800258776688 T^{13} + 4655193675244 T^{14} - 21178606089996 T^{15} + 112510505695945 T^{16} - 463191628948930 T^{17} + 2235824914113703 T^{18} - 8195016513134188 T^{19} + 35493190244511619 T^{20} - 111923131450242296 T^{21} + 421766254956439437 T^{22} - 1053450912183703304 T^{23} + 3208734068380633027 T^{24} - 4953406964876566574 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 - 20 T + 345 T^{2} - 4284 T^{3} + 49213 T^{4} - 475096 T^{5} + 4320803 T^{6} - 35098012 T^{7} + 271794529 T^{8} - 1926651660 T^{9} + 13114415697 T^{10} - 82829278120 T^{11} + 503646686670 T^{12} - 2855576892432 T^{13} + 15613047286770 T^{14} - 79598936273320 T^{15} + 390691558029327 T^{16} - 1779303267694860 T^{17} + 7781246611714879 T^{18} - 31149614845782172 T^{19} + 118876585588651133 T^{20} - 405205120324069336 T^{21} + 1301173125393101923 T^{22} - 3511287581425751484 T^{23} + 8765924529259666695 T^{24} - 15753255675770995220 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 - 17 T + 338 T^{2} - 4461 T^{3} + 53723 T^{4} - 563092 T^{5} + 5297663 T^{6} - 45734857 T^{7} + 365946568 T^{8} - 2720371175 T^{9} + 19249323905 T^{10} - 128806105154 T^{11} + 828541603074 T^{12} - 5137279773064 T^{13} + 30656039313738 T^{14} - 176335557955826 T^{15} + 975036003759965 T^{16} - 5098413561709175 T^{17} + 25376183075689576 T^{18} - 117343130416738513 T^{19} + 502917093008040179 T^{20} - 1977849080667283732 T^{21} + 6981934547010921671 T^{22} - 21451094885356024389 T^{23} + 60136156161457619594 T^{24} - \)\(11\!\cdots\!77\)\( T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 9 T + 319 T^{2} - 2412 T^{3} + 50174 T^{4} - 334930 T^{5} + 5251398 T^{6} - 31645460 T^{7} + 409958581 T^{8} - 2254156951 T^{9} + 25246017899 T^{10} - 126927673728 T^{11} + 1261064550492 T^{12} - 5772191105500 T^{13} + 51703646570172 T^{14} - 213365419536768 T^{15} + 1739980799616979 T^{16} - 6369708800014711 T^{17} + 47496243762010781 T^{18} - 150319233754395860 T^{19} + 1022732204350135638 T^{20} - 2674391006989496530 T^{21} + 16426061176282599214 T^{22} - 32375454256087591212 T^{23} + \)\(17\!\cdots\!79\)\( T^{24} - \)\(20\!\cdots\!29\)\( T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( ( 1 - T )^{13} \)
$47$ \( 1 + 9 T + 209 T^{2} + 1615 T^{3} + 22878 T^{4} + 139949 T^{5} + 1554899 T^{6} + 6643799 T^{7} + 65457736 T^{8} + 83709231 T^{9} + 1300502392 T^{10} - 11429352470 T^{11} - 17174985611 T^{12} - 887798750826 T^{13} - 807224323717 T^{14} - 25247439606230 T^{15} + 135022059844616 T^{16} + 408474344035311 T^{17} + 15012404921124152 T^{18} + 71614940023594871 T^{19} + 787747783384798237 T^{20} + 3332365757026790189 T^{21} + 25603466963645103426 T^{22} + 84947598560865529135 T^{23} + \)\(51\!\cdots\!27\)\( T^{24} + \)\(10\!\cdots\!69\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 - 22 T + 581 T^{2} - 8749 T^{3} + 139630 T^{4} - 1655570 T^{5} + 20285399 T^{6} - 203571811 T^{7} + 2098784666 T^{8} - 18574923189 T^{9} + 168407676946 T^{10} - 1341109666016 T^{11} + 10908213683249 T^{12} - 78633585686470 T^{13} + 578135325212197 T^{14} - 3767177051838944 T^{15} + 25072029720689642 T^{16} - 146565078499263909 T^{17} + 877702288098710338 T^{18} - 4512039134688534619 T^{19} + 23829484181338339963 T^{20} - \)\(10\!\cdots\!70\)\( T^{21} + \)\(46\!\cdots\!90\)\( T^{22} - \)\(15\!\cdots\!01\)\( T^{23} + \)\(53\!\cdots\!57\)\( T^{24} - \)\(10\!\cdots\!02\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 - 19 T + 537 T^{2} - 6619 T^{3} + 106978 T^{4} - 934941 T^{5} + 11300703 T^{6} - 70764537 T^{7} + 753279690 T^{8} - 3098541459 T^{9} + 36702064482 T^{10} - 70332779484 T^{11} + 1641598773685 T^{12} - 998829502442 T^{13} + 96854327647415 T^{14} - 244828405383804 T^{15} + 7537833301248678 T^{16} - 37546145432169699 T^{17} + 538537954324187310 T^{18} - 2984885933518289217 T^{19} + 28123511300448527757 T^{20} - \)\(13\!\cdots\!61\)\( T^{21} + \)\(92\!\cdots\!42\)\( T^{22} - \)\(33\!\cdots\!19\)\( T^{23} + \)\(16\!\cdots\!83\)\( T^{24} - \)\(33\!\cdots\!39\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 - 2 T + 315 T^{2} - 412 T^{3} + 55409 T^{4} - 9840 T^{5} + 6674971 T^{6} + 7460960 T^{7} + 615781307 T^{8} + 1516526530 T^{9} + 46748738921 T^{10} + 165053609212 T^{11} + 3113898277640 T^{12} + 12045027655904 T^{13} + 189947794936040 T^{14} + 614164479877852 T^{15} + 10611075509027501 T^{16} + 20997585206661730 T^{17} + 520086614117145407 T^{18} + 384391452292446560 T^{19} + 20977717290897930391 T^{20} - 1886399959893245040 T^{21} + \)\(64\!\cdots\!69\)\( T^{22} - \)\(29\!\cdots\!12\)\( T^{23} + \)\(13\!\cdots\!15\)\( T^{24} - \)\(53\!\cdots\!42\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 - 9 T + 441 T^{2} - 3782 T^{3} + 108252 T^{4} - 853310 T^{5} + 18230676 T^{6} - 132834654 T^{7} + 2326362913 T^{8} - 15576421799 T^{9} + 234801332769 T^{10} - 1441780778508 T^{11} + 19214354289480 T^{12} - 107290355193796 T^{13} + 1287361737395160 T^{14} - 6472153914722412 T^{15} + 70619553247602747 T^{16} - 313882360418686679 T^{17} + 3140880976834956691 T^{18} - 12016007896818884526 T^{19} + \)\(11\!\cdots\!48\)\( T^{20} - \)\(34\!\cdots\!10\)\( T^{21} + \)\(29\!\cdots\!44\)\( T^{22} - \)\(68\!\cdots\!18\)\( T^{23} + \)\(53\!\cdots\!03\)\( T^{24} - \)\(73\!\cdots\!49\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 6 T + 501 T^{2} - 3749 T^{3} + 130316 T^{4} - 1104060 T^{5} + 23219737 T^{6} - 206533023 T^{7} + 3142906526 T^{8} - 27675354965 T^{9} + 337836323572 T^{10} - 2821257371214 T^{11} + 29403904461635 T^{12} - 225311662741582 T^{13} + 2087677216776085 T^{14} - 14221958408289774 T^{15} + 120915336405978092 T^{16} - 703277291932346165 T^{17} + 5670524201658644626 T^{18} - 26456938885362423183 T^{19} + \)\(21\!\cdots\!67\)\( T^{20} - \)\(71\!\cdots\!60\)\( T^{21} + \)\(59\!\cdots\!96\)\( T^{22} - \)\(12\!\cdots\!49\)\( T^{23} + \)\(11\!\cdots\!71\)\( T^{24} - \)\(98\!\cdots\!46\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 7 T + 399 T^{2} - 3012 T^{3} + 89944 T^{4} - 668670 T^{5} + 14571168 T^{6} - 103530660 T^{7} + 1842824813 T^{8} - 12439646537 T^{9} + 190400204787 T^{10} - 1206715535064 T^{11} + 16465182480168 T^{12} - 96590643124004 T^{13} + 1201958321052264 T^{14} - 6430587086356056 T^{15} + 74068916465624379 T^{16} - 353264080312541417 T^{17} + 3820307770705837109 T^{18} - 15667732328289520740 T^{19} + \)\(16\!\cdots\!96\)\( T^{20} - \)\(53\!\cdots\!70\)\( T^{21} + \)\(52\!\cdots\!72\)\( T^{22} - \)\(12\!\cdots\!88\)\( T^{23} + \)\(12\!\cdots\!23\)\( T^{24} - \)\(16\!\cdots\!47\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 - 50 T + 1658 T^{2} - 39081 T^{3} + 764788 T^{4} - 12664163 T^{5} + 188574694 T^{6} - 2535590314 T^{7} + 31718108822 T^{8} - 367378810578 T^{9} + 4008883879703 T^{10} - 40939732372457 T^{11} + 396495909847004 T^{12} - 3614086581463274 T^{13} + 31323176877913316 T^{14} - 255504869736504137 T^{15} + 1976536099164887417 T^{16} - 14309434429696756818 T^{17} + 97598409714913451978 T^{18} - \)\(61\!\cdots\!94\)\( T^{19} + \)\(36\!\cdots\!46\)\( T^{20} - \)\(19\!\cdots\!43\)\( T^{21} + \)\(91\!\cdots\!72\)\( T^{22} - \)\(37\!\cdots\!81\)\( T^{23} + \)\(12\!\cdots\!82\)\( T^{24} - \)\(29\!\cdots\!50\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - T + 529 T^{2} - 1775 T^{3} + 148198 T^{4} - 707625 T^{5} + 29150747 T^{6} - 159852999 T^{7} + 4425883026 T^{8} - 25095323897 T^{9} + 540940826210 T^{10} - 2977236851834 T^{11} + 54281517087965 T^{12} - 277223885418538 T^{13} + 4505365918301095 T^{14} - 20510184672284426 T^{15} + 309302932196137270 T^{16} - 1190981937102796937 T^{17} + 17433733120577825718 T^{18} - 52262399177214383631 T^{19} + \)\(79\!\cdots\!69\)\( T^{20} - \)\(15\!\cdots\!25\)\( T^{21} + \)\(27\!\cdots\!94\)\( T^{22} - \)\(27\!\cdots\!75\)\( T^{23} + \)\(68\!\cdots\!43\)\( T^{24} - \)\(10\!\cdots\!61\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 5 T + 558 T^{2} - 2547 T^{3} + 155717 T^{4} - 584188 T^{5} + 29267765 T^{6} - 83723951 T^{7} + 4235160724 T^{8} - 9150731107 T^{9} + 508578230213 T^{10} - 887989578286 T^{11} + 52399791106270 T^{12} - 81332559882616 T^{13} + 4663581408458030 T^{14} - 7033765449603406 T^{15} + 358531886375028397 T^{16} - 574137376441590787 T^{17} + 23649389258885881076 T^{18} - 41609237252335506911 T^{19} + \)\(12\!\cdots\!85\)\( T^{20} - \)\(22\!\cdots\!28\)\( T^{21} + \)\(54\!\cdots\!53\)\( T^{22} - \)\(79\!\cdots\!47\)\( T^{23} + \)\(15\!\cdots\!62\)\( T^{24} - \)\(12\!\cdots\!05\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 - 20 T + 785 T^{2} - 11420 T^{3} + 271778 T^{4} - 3258592 T^{5} + 60901010 T^{6} - 637222508 T^{7} + 10178636847 T^{8} - 95182889548 T^{9} + 1360029722975 T^{10} - 11619142030840 T^{11} + 152968772535020 T^{12} - 1212016541443456 T^{13} + 14837970935896940 T^{14} - 109324507368173560 T^{15} + 1241262407356762175 T^{16} - 8426472775186854988 T^{17} + 87407417957626649679 T^{18} - \)\(53\!\cdots\!32\)\( T^{19} + \)\(49\!\cdots\!30\)\( T^{20} - \)\(25\!\cdots\!12\)\( T^{21} + \)\(20\!\cdots\!26\)\( T^{22} - \)\(84\!\cdots\!80\)\( T^{23} + \)\(56\!\cdots\!05\)\( T^{24} - \)\(13\!\cdots\!20\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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