Properties

Label 6042.2.a.bh
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 3 x^{12} - 40 x^{11} + 123 x^{10} + 537 x^{9} - 1707 x^{8} - 2914 x^{7} + 9639 x^{6} + 6938 x^{5} - 22200 x^{4} - 9466 x^{3} + 16812 x^{2} + 9304 x + 1200\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 3 x^{12} - 40 x^{11} + 123 x^{10} + 537 x^{9} - 1707 x^{8} - 2914 x^{7} + 9639 x^{6} + 6938 x^{5} - 22200 x^{4} - 9466 x^{3} + 16812 x^{2} + 9304 x + 1200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(1104465971 \nu^{12} + 6679376015 \nu^{11} - 55180929014 \nu^{10} - 276650415923 \nu^{9} + 1022243116203 \nu^{8} + 3980299416517 \nu^{7} - 8515787410070 \nu^{6} - 24476327772653 \nu^{5} + 30037232426544 \nu^{4} + 66746578762844 \nu^{3} - 30721425263802 \nu^{2} - 69505164946224 \nu - 15131253915640\)\()/ 887834031664 \)
\(\beta_{3}\)\(=\)\((\)\(-2632284209 \nu^{12} + 18882316439 \nu^{11} + 89540587722 \nu^{10} - 780424603231 \nu^{9} - 786204074125 \nu^{8} + 11062731147833 \nu^{7} - 443211211426 \nu^{6} - 65484111774973 \nu^{5} + 20272416631968 \nu^{4} + 165465205024724 \nu^{3} - 30222105954378 \nu^{2} - 150515869605488 \nu - 33713612784760\)\()/ 887834031664 \)
\(\beta_{4}\)\(=\)\((\)\(-2727750661 \nu^{12} + 14646261136 \nu^{11} + 102820445846 \nu^{10} - 605287792477 \nu^{9} - 1209915311616 \nu^{8} + 8558511877653 \nu^{7} + 4585058152073 \nu^{6} - 50218247672710 \nu^{5} - 2416172669298 \nu^{4} + 123875804125670 \nu^{3} - 60768123000 \nu^{2} - 106411914676176 \nu - 25057438289328\)\()/ 443917015832 \)
\(\beta_{5}\)\(=\)\((\)\(3825967735 \nu^{12} - 5504342876 \nu^{11} - 156363041234 \nu^{10} + 219870360311 \nu^{9} + 2181905208796 \nu^{8} - 2859488827735 \nu^{7} - 12585440057303 \nu^{6} + 13755460366038 \nu^{5} + 30424461859070 \nu^{4} - 20469284564146 \nu^{3} - 25651497176616 \nu^{2} - 1427999257888 \nu - 955171318320\)\()/ 443917015832 \)
\(\beta_{6}\)\(=\)\((\)\(4180312254 \nu^{12} + 7962412967 \nu^{11} - 186403032696 \nu^{10} - 333593208456 \nu^{9} + 2999393383979 \nu^{8} + 4949970794130 \nu^{7} - 21545221318909 \nu^{6} - 32224399272003 \nu^{5} + 68240550554438 \nu^{4} + 94219150878618 \nu^{3} - 69773601238538 \nu^{2} - 102911563743592 \nu - 20324471216384\)\()/ 443917015832 \)
\(\beta_{7}\)\(=\)\((\)\(-9076089647 \nu^{12} + 14744154019 \nu^{11} + 376726997738 \nu^{10} - 592303202345 \nu^{9} - 5407907514181 \nu^{8} + 7818704648743 \nu^{7} + 33013088468736 \nu^{6} - 39050414261665 \nu^{5} - 89720193617028 \nu^{4} + 64874222641024 \nu^{3} + 99166364858438 \nu^{2} - 8967190274096 \nu - 15375912065432\)\()/ 887834031664 \)
\(\beta_{8}\)\(=\)\((\)\(10002084743 \nu^{12} + 15569453759 \nu^{11} - 450595476454 \nu^{10} - 664510529295 \nu^{9} + 7352334071027 \nu^{8} + 10188017922425 \nu^{7} - 53794158630402 \nu^{6} - 69934638097797 \nu^{5} + 174198902752088 \nu^{4} + 219914719422692 \nu^{3} - 182388318974730 \nu^{2} - 261526193845840 \nu - 57489181360808\)\()/ 887834031664 \)
\(\beta_{9}\)\(=\)\((\)\(2761861453 \nu^{12} - 2173491833 \nu^{11} - 115007316091 \nu^{10} + 84482827509 \nu^{9} + 1657562870917 \nu^{8} - 1017129760174 \nu^{7} - 10090594369310 \nu^{6} + 3874256295364 \nu^{5} + 26156616453175 \nu^{4} - 909737621208 \nu^{3} - 23030561263496 \nu^{2} - 9936848006134 \nu - 1802645531584\)\()/ 221958507916 \)
\(\beta_{10}\)\(=\)\((\)\(15738407933 \nu^{12} - 24588080919 \nu^{11} - 643851470174 \nu^{10} + 983188381227 \nu^{9} + 8996584023733 \nu^{8} - 12861301367465 \nu^{7} - 52033807512746 \nu^{6} + 63229291380089 \nu^{5} + 127116995436076 \nu^{4} - 104200446673436 \nu^{3} - 114796805478126 \nu^{2} + 23466620266904 \nu + 10900451890120\)\()/ 887834031664 \)
\(\beta_{11}\)\(=\)\((\)\(4706616569 \nu^{12} - 8016452396 \nu^{11} - 190884176433 \nu^{10} + 324090176126 \nu^{9} + 2627378123161 \nu^{8} - 4339680041921 \nu^{7} - 14800691726179 \nu^{6} + 22580897844015 \nu^{5} + 34875401841134 \nu^{4} - 43741985100670 \nu^{3} - 31658973603658 \nu^{2} + 22658914077896 \nu + 7501450230024\)\()/ 221958507916 \)
\(\beta_{12}\)\(=\)\((\)\(20467926511 \nu^{12} - 32421181759 \nu^{11} - 841326116794 \nu^{10} + 1299036592121 \nu^{9} + 11863059795713 \nu^{8} - 17070643833519 \nu^{7} - 69906482897300 \nu^{6} + 84837751822269 \nu^{5} + 177219409007748 \nu^{4} - 143491522737224 \nu^{3} - 170364844943950 \nu^{2} + 34044755921264 \nu + 19380400213704\)\()/ 887834031664 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{12} + \beta_{11} + \beta_{8} - \beta_{6} - \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{3} + 4 \beta_{2} + 10 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(-18 \beta_{12} + 17 \beta_{11} - \beta_{10} + 3 \beta_{9} + 15 \beta_{8} - 3 \beta_{7} - 17 \beta_{6} + 2 \beta_{3} + \beta_{2} - 20 \beta_{1} + 89\)
\(\nu^{5}\)\(=\)\(44 \beta_{12} + 23 \beta_{11} - 39 \beta_{10} - \beta_{9} - 24 \beta_{8} + 27 \beta_{7} - 20 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 55 \beta_{3} + 83 \beta_{2} + 128 \beta_{1} - 57\)
\(\nu^{6}\)\(=\)\(-313 \beta_{12} + 279 \beta_{11} - 39 \beta_{10} + 91 \beta_{9} + 237 \beta_{8} - 78 \beta_{7} - 283 \beta_{6} + 21 \beta_{5} - 5 \beta_{4} + 52 \beta_{3} - \beta_{2} - 351 \beta_{1} + 1380\)
\(\nu^{7}\)\(=\)\(830 \beta_{12} + 422 \beta_{11} - 650 \beta_{10} - 42 \beta_{9} - 512 \beta_{8} + 568 \beta_{7} - 323 \beta_{6} - 89 \beta_{5} - 178 \beta_{4} + 1220 \beta_{3} + 1502 \beta_{2} + 1891 \beta_{1} - 1159\)
\(\nu^{8}\)\(=\)\(-5483 \beta_{12} + 4669 \beta_{11} - 1013 \beta_{10} + 2055 \beta_{9} + 3938 \beta_{8} - 1660 \beta_{7} - 4780 \beta_{6} + 729 \beta_{5} - 148 \beta_{4} + 1051 \beta_{3} - 493 \beta_{2} - 6057 \beta_{1} + 23136\)
\(\nu^{9}\)\(=\)\(15171 \beta_{12} + 7406 \beta_{11} - 10670 \beta_{10} - 1190 \beta_{9} - 10321 \beta_{8} + 11105 \beta_{7} - 4992 \beta_{6} - 1516 \beta_{5} - 4464 \beta_{4} + 24829 \beta_{3} + 26700 \beta_{2} + 30208 \beta_{1} - 22032\)
\(\nu^{10}\)\(=\)\(-97176 \beta_{12} + 79790 \beta_{11} - 22368 \beta_{10} + 41914 \beta_{9} + 67591 \beta_{8} - 33120 \beta_{7} - 82132 \beta_{6} + 18057 \beta_{5} - 3145 \beta_{4} + 19465 \beta_{3} - 16520 \beta_{2} - 105479 \beta_{1} + 401275\)
\(\nu^{11}\)\(=\)\(276242 \beta_{12} + 128935 \beta_{11} - 177279 \beta_{10} - 28567 \beta_{9} - 201325 \beta_{8} + 210979 \beta_{7} - 76793 \beta_{6} - 24672 \beta_{5} - 97122 \beta_{4} + 483242 \beta_{3} + 476005 \beta_{2} + 505032 \beta_{1} - 414057\)
\(\nu^{12}\)\(=\)\(-1739338 \beta_{12} + 1385481 \beta_{11} - 454943 \beta_{10} + 816015 \beta_{9} + 1185360 \beta_{8} - 641477 \beta_{7} - 1431558 \beta_{6} + 390914 \beta_{5} - 58827 \beta_{4} + 344423 \beta_{3} - 409247 \beta_{2} - 1861262 \beta_{1} + 7082839\)

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
−4.29179
−3.54278
−2.34301
−1.64331
−0.942914
−0.312107
−0.229612
1.76686
2.08556
2.23588
2.48443
3.50660
4.22619
1.00000 1.00000 1.00000 −4.29179 1.00000 4.11570 1.00000 1.00000 −4.29179
1.2 1.00000 1.00000 1.00000 −3.54278 1.00000 −0.903867 1.00000 1.00000 −3.54278
1.3 1.00000 1.00000 1.00000 −2.34301 1.00000 4.92290 1.00000 1.00000 −2.34301
1.4 1.00000 1.00000 1.00000 −1.64331 1.00000 −4.90896 1.00000 1.00000 −1.64331
1.5 1.00000 1.00000 1.00000 −0.942914 1.00000 −0.429327 1.00000 1.00000 −0.942914
1.6 1.00000 1.00000 1.00000 −0.312107 1.00000 −1.12710 1.00000 1.00000 −0.312107
1.7 1.00000 1.00000 1.00000 −0.229612 1.00000 2.69176 1.00000 1.00000 −0.229612
1.8 1.00000 1.00000 1.00000 1.76686 1.00000 1.80214 1.00000 1.00000 1.76686
1.9 1.00000 1.00000 1.00000 2.08556 1.00000 1.83609 1.00000 1.00000 2.08556
1.10 1.00000 1.00000 1.00000 2.23588 1.00000 4.85906 1.00000 1.00000 2.23588
1.11 1.00000 1.00000 1.00000 2.48443 1.00000 3.12498 1.00000 1.00000 2.48443
1.12 1.00000 1.00000 1.00000 3.50660 1.00000 −4.14603 1.00000 1.00000 3.50660
1.13 1.00000 1.00000 1.00000 4.22619 1.00000 0.162672 1.00000 1.00000 4.22619
\(n\): 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 6042.2.a.bh 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bh 13 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(-1\)
\(53\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} - \cdots\)
\(T_{11}^{13} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{13} \)
$3$ \( ( 1 - T )^{13} \)
$5$ \( 1 - 3 T + 25 T^{2} - 57 T^{3} + 287 T^{4} - 507 T^{5} + 2001 T^{6} - 2766 T^{7} + 10123 T^{8} - 11055 T^{9} + 44509 T^{10} - 42813 T^{11} + 204314 T^{12} - 194430 T^{13} + 1021570 T^{14} - 1070325 T^{15} + 5563625 T^{16} - 6909375 T^{17} + 31634375 T^{18} - 43218750 T^{19} + 156328125 T^{20} - 198046875 T^{21} + 560546875 T^{22} - 556640625 T^{23} + 1220703125 T^{24} - 732421875 T^{25} + 1220703125 T^{26} \)
$7$ \( 1 - 12 T + 97 T^{2} - 562 T^{3} + 2650 T^{4} - 10334 T^{5} + 34930 T^{6} - 104063 T^{7} + 288341 T^{8} - 774238 T^{9} + 2144461 T^{10} - 6089343 T^{11} + 17381680 T^{12} - 47202352 T^{13} + 121671760 T^{14} - 298377807 T^{15} + 735550123 T^{16} - 1858945438 T^{17} + 4846147187 T^{18} - 12242907887 T^{19} + 28766356990 T^{20} - 59573453534 T^{21} + 106937058550 T^{22} - 158751089938 T^{23} + 191800694071 T^{24} - 166095446412 T^{25} + 96889010407 T^{26} \)
$11$ \( 1 - 4 T + 68 T^{2} - 163 T^{3} + 2044 T^{4} - 2586 T^{5} + 40824 T^{6} - 16517 T^{7} + 662738 T^{8} + 127192 T^{9} + 9314419 T^{10} + 5165200 T^{11} + 114475028 T^{12} + 76332556 T^{13} + 1259225308 T^{14} + 624989200 T^{15} + 12397491689 T^{16} + 1862218072 T^{17} + 106734617638 T^{18} - 29260873037 T^{19} + 795544268904 T^{20} - 554332066266 T^{21} + 4819645080404 T^{22} - 4227800209963 T^{23} + 19401193601548 T^{24} - 12553713506884 T^{25} + 34522712143931 T^{26} \)
$13$ \( 1 - 18 T + 229 T^{2} - 2230 T^{3} + 18426 T^{4} - 132180 T^{5} + 849664 T^{6} - 4944275 T^{7} + 26403235 T^{8} - 130094222 T^{9} + 595492543 T^{10} - 2538749167 T^{11} + 10115848214 T^{12} - 37692765128 T^{13} + 131506026782 T^{14} - 429048609223 T^{15} + 1308297116971 T^{16} - 3715621074542 T^{17} + 9803336332855 T^{18} - 23865071068475 T^{19} + 53315155948288 T^{20} - 107823286701780 T^{21} + 195398505446898 T^{22} - 307424436823270 T^{23} + 410404730234473 T^{24} - 419365532204658 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 - 3 T + 123 T^{2} - 466 T^{3} + 8032 T^{4} - 32730 T^{5} + 361778 T^{6} - 1461553 T^{7} + 12226232 T^{8} - 47067853 T^{9} + 322825250 T^{10} - 1154302277 T^{11} + 6808471240 T^{12} - 22113367388 T^{13} + 115744011080 T^{14} - 333593358053 T^{15} + 1586040453250 T^{16} - 3931154150413 T^{17} + 17359501088824 T^{18} - 35278336384657 T^{19} + 148451504440594 T^{20} - 228316541043930 T^{21} + 952497824023904 T^{22} - 939453157609234 T^{23} + 4215443245838859 T^{24} - 1747866711689283 T^{25} + 9904578032905937 T^{26} \)
$19$ \( ( 1 - T )^{13} \)
$23$ \( 1 - 8 T + 150 T^{2} - 995 T^{3} + 11176 T^{4} - 64492 T^{5} + 564008 T^{6} - 2919675 T^{7} + 21809092 T^{8} - 103215072 T^{9} + 688580607 T^{10} - 3011745618 T^{11} + 18401255144 T^{12} - 74730405848 T^{13} + 423228868312 T^{14} - 1593213431922 T^{15} + 8377960245369 T^{16} - 28883808963552 T^{17} + 140370796630556 T^{18} - 432216684216075 T^{19} + 1920348790711576 T^{20} - 5050432062742252 T^{21} + 20129682144510488 T^{22} - 41219378657580755 T^{23} + 142921463687089050 T^{24} - 175316995456162568 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 - 11 T + 293 T^{2} - 2706 T^{3} + 40193 T^{4} - 321951 T^{5} + 3481856 T^{6} - 24653939 T^{7} + 214943476 T^{8} - 1359010431 T^{9} + 10051369346 T^{10} - 56946727427 T^{11} + 367587386791 T^{12} - 1862125179534 T^{13} + 10660034216939 T^{14} - 47892197766107 T^{15} + 245142846979594 T^{16} - 961202256648111 T^{17} + 4408737662813924 T^{18} - 14664737871711419 T^{19} + 60061585325749504 T^{20} - 161054832899206911 T^{21} + 583085718208102717 T^{22} - 1138433773310343906 T^{23} + 3574749361351807897 T^{24} - 3891962615260159451 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 - 2 T + 199 T^{2} - 327 T^{3} + 17769 T^{4} - 23857 T^{5} + 969035 T^{6} - 1132915 T^{7} + 38927187 T^{8} - 47515578 T^{9} + 1371322541 T^{10} - 1972588838 T^{11} + 46011033826 T^{12} - 69587930342 T^{13} + 1426342048606 T^{14} - 1895657873318 T^{15} + 40853069818931 T^{16} - 43881634110138 T^{17} + 1114452314628237 T^{18} - 1005466232760115 T^{19} + 26660686015052885 T^{20} - 20347421480229937 T^{21} + 469805646172962999 T^{22} - 268018449842721927 T^{23} + 5056286902384561369 T^{24} - 1575325567577099522 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 - 26 T + 529 T^{2} - 7904 T^{3} + 103262 T^{4} - 1169056 T^{5} + 12019094 T^{6} - 112358959 T^{7} + 973023469 T^{8} - 7821880362 T^{9} + 58927065033 T^{10} - 415931966177 T^{11} + 2767794504844 T^{12} - 17322216197496 T^{13} + 102408396679228 T^{14} - 569410861696313 T^{15} + 2984832625116549 T^{16} - 14659463121126282 T^{17} + 67473297594326833 T^{18} - 288282348394048231 T^{19} + 1140995154857977502 T^{20} - 4106285180483068576 T^{21} + 13420109174719241174 T^{22} - 38007050879590678496 T^{23} + 94118421921334558477 T^{24} - \)\(17\!\cdots\!06\)\( T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 3 T + 257 T^{2} - 1467 T^{3} + 34740 T^{4} - 253508 T^{5} + 3491069 T^{6} - 25669713 T^{7} + 278109082 T^{8} - 1883021097 T^{9} + 17430580176 T^{10} - 108869351172 T^{11} + 872814080755 T^{12} - 5013746853344 T^{13} + 35785377310955 T^{14} - 183009379320132 T^{15} + 1201333016310096 T^{16} - 5320967578079817 T^{17} + 32220661704117482 T^{18} - 121933812586552833 T^{19} + 679900608163468789 T^{20} - 2024242424984006468 T^{21} + 11373248400846205140 T^{22} - 19691041207993572267 T^{23} + \)\(14\!\cdots\!37\)\( T^{24} - 67690470901098558243 T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( 1 - 24 T + 530 T^{2} - 7595 T^{3} + 104532 T^{4} - 1156702 T^{5} + 12558348 T^{6} - 117673007 T^{7} + 1088298600 T^{8} - 8950722176 T^{9} + 72757627443 T^{10} - 533590726990 T^{11} + 3871341804992 T^{12} - 25513634843988 T^{13} + 166467697614656 T^{14} - 986609254204510 T^{15} + 5784740685110601 T^{16} - 30600737926030976 T^{17} + 159989082705079800 T^{18} - 743853798314518343 T^{19} + 3413592711158371236 T^{20} - 13519764637501631902 T^{21} + 52537010910982072476 T^{22} - \)\(16\!\cdots\!55\)\( T^{23} + \)\(49\!\cdots\!10\)\( T^{24} - \)\(95\!\cdots\!24\)\( T^{25} + \)\(17\!\cdots\!43\)\( T^{26} \)
$47$ \( 1 - 4 T + 268 T^{2} - 1033 T^{3} + 36836 T^{4} - 122200 T^{5} + 3395347 T^{6} - 8423020 T^{7} + 234170305 T^{8} - 353238168 T^{9} + 13083450776 T^{10} - 8602214943 T^{11} + 645830400529 T^{12} - 191333851936 T^{13} + 30354028824863 T^{14} - 19002292809087 T^{15} + 1358363109916648 T^{16} - 1723689576864408 T^{17} + 53705790239417135 T^{18} - 90793546300473580 T^{19} + 1720161292194685661 T^{20} - 2909739230067194200 T^{21} + 41224290107213525212 T^{22} - 54334903599612440617 T^{23} + \)\(66\!\cdots\!04\)\( T^{24} - \)\(46\!\cdots\!64\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( ( 1 + T )^{13} \)
$59$ \( 1 - 8 T + 488 T^{2} - 4056 T^{3} + 116702 T^{4} - 955302 T^{5} + 18233248 T^{6} - 141093644 T^{7} + 2084160979 T^{8} - 14904118150 T^{9} + 185090156460 T^{10} - 1213436893580 T^{11} + 13261524523706 T^{12} - 79351524183736 T^{13} + 782429946898654 T^{14} - 4223973826551980 T^{15} + 38013631243598340 T^{16} - 180598580010202150 T^{17} + 1490017326914728721 T^{18} - 5951405197273277804 T^{19} + 45376199708273062112 T^{20} - \)\(14\!\cdots\!42\)\( T^{21} + \)\(10\!\cdots\!78\)\( T^{22} - \)\(20\!\cdots\!56\)\( T^{23} + \)\(14\!\cdots\!92\)\( T^{24} - \)\(14\!\cdots\!48\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 - 29 T + 719 T^{2} - 12106 T^{3} + 182257 T^{4} - 2246365 T^{5} + 25885890 T^{6} - 263725067 T^{7} + 2623379820 T^{8} - 24187988835 T^{9} + 222738683596 T^{10} - 1921212995771 T^{11} + 16380793423473 T^{12} - 129356675228598 T^{13} + 999228398831853 T^{14} - 7148833557263891 T^{15} + 50557449141303676 T^{16} - 334903047519185235 T^{17} + 2215696892090045820 T^{18} - 13587214180219807187 T^{19} + 81352695351527643690 T^{20} - \)\(43\!\cdots\!65\)\( T^{21} + \)\(21\!\cdots\!37\)\( T^{22} - \)\(86\!\cdots\!06\)\( T^{23} + \)\(31\!\cdots\!59\)\( T^{24} - \)\(76\!\cdots\!09\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 + 5 T + 479 T^{2} + 2461 T^{3} + 113540 T^{4} + 565244 T^{5} + 17806940 T^{6} + 82160097 T^{7} + 2083232523 T^{8} + 8672645107 T^{9} + 194767833389 T^{10} + 729094736874 T^{11} + 15225633883928 T^{12} + 52217255381240 T^{13} + 1020117470223176 T^{14} + 3272906273827386 T^{15} + 58578957873575807 T^{16} + 174763520941214947 T^{17} + 2812624533021254961 T^{18} + 7432069453468110393 T^{19} + \)\(10\!\cdots\!20\)\( T^{20} + \)\(22\!\cdots\!04\)\( T^{21} + \)\(30\!\cdots\!80\)\( T^{22} + \)\(44\!\cdots\!89\)\( T^{23} + \)\(58\!\cdots\!57\)\( T^{24} + \)\(40\!\cdots\!05\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 29 T + 720 T^{2} - 12572 T^{3} + 193566 T^{4} - 2524156 T^{5} + 30049566 T^{6} - 321700200 T^{7} + 3227480510 T^{8} - 30090521795 T^{9} + 270116549823 T^{10} - 2324136264700 T^{11} + 19845964258776 T^{12} - 166590249768584 T^{13} + 1409063462373096 T^{14} - 11715970910352700 T^{15} + 96677684463699753 T^{16} - 764650740978087395 T^{17} + 5823115065922449010 T^{18} - 41209886957442484200 T^{19} + \)\(27\!\cdots\!06\)\( T^{20} - \)\(16\!\cdots\!16\)\( T^{21} + \)\(88\!\cdots\!46\)\( T^{22} - \)\(40\!\cdots\!72\)\( T^{23} + \)\(16\!\cdots\!20\)\( T^{24} - \)\(47\!\cdots\!89\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 15 T + 557 T^{2} - 8027 T^{3} + 167520 T^{4} - 2162404 T^{5} + 33852096 T^{6} - 389448255 T^{7} + 5033721195 T^{8} - 51826456265 T^{9} + 578222783383 T^{10} - 5345218355286 T^{11} + 52641732870928 T^{12} - 436897445699016 T^{13} + 3842846499577744 T^{14} - 28484668615319094 T^{15} + 224938492523304511 T^{16} - 1471780195189429865 T^{17} + 10435264416436513635 T^{18} - 58936850350026175695 T^{19} + \)\(37\!\cdots\!12\)\( T^{20} - \)\(17\!\cdots\!24\)\( T^{21} + \)\(98\!\cdots\!60\)\( T^{22} - \)\(34\!\cdots\!23\)\( T^{23} + \)\(17\!\cdots\!89\)\( T^{24} - \)\(34\!\cdots\!15\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 - 15 T + 655 T^{2} - 7729 T^{3} + 200993 T^{4} - 1936471 T^{5} + 38812327 T^{6} - 310349774 T^{7} + 5363658003 T^{8} - 36115904519 T^{9} + 577889673817 T^{10} - 3372883398881 T^{11} + 52084492714874 T^{12} - 277652174402022 T^{13} + 4114674924475046 T^{14} - 21050165292416321 T^{15} + 284922146889059863 T^{16} - 1406717406403316039 T^{17} + 16504278180178711197 T^{18} - 75442136883177402254 T^{19} + \)\(74\!\cdots\!93\)\( T^{20} - \)\(29\!\cdots\!31\)\( T^{21} + \)\(24\!\cdots\!67\)\( T^{22} - \)\(73\!\cdots\!29\)\( T^{23} + \)\(48\!\cdots\!45\)\( T^{24} - \)\(88\!\cdots\!15\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - 4 T + 473 T^{2} - 1122 T^{3} + 107130 T^{4} - 71316 T^{5} + 15994894 T^{6} + 17009403 T^{7} + 1826527295 T^{8} + 4796438420 T^{9} + 176988246491 T^{10} + 654794730599 T^{11} + 15654692771376 T^{12} + 62609920158744 T^{13} + 1299339500024208 T^{14} + 4510880899096511 T^{15} + 101199578496349417 T^{16} + 227630914193092820 T^{17} + 7194765250553850685 T^{18} + 5561060567603788707 T^{19} + \)\(43\!\cdots\!38\)\( T^{20} - \)\(16\!\cdots\!56\)\( T^{21} + \)\(20\!\cdots\!90\)\( T^{22} - \)\(17\!\cdots\!78\)\( T^{23} + \)\(60\!\cdots\!91\)\( T^{24} - \)\(42\!\cdots\!44\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 3 T + 446 T^{2} - 1828 T^{3} + 98490 T^{4} - 515170 T^{5} + 14334284 T^{6} - 99999528 T^{7} + 1555706438 T^{8} - 15158087141 T^{9} + 138708660035 T^{10} - 1848037464820 T^{11} + 11570893576074 T^{12} - 182365098822956 T^{13} + 1029809528270586 T^{14} - 14638304758839220 T^{15} + 97785305356213915 T^{16} - 951052356499622981 T^{17} + 8687157234984032662 T^{18} - 49697894520930666408 T^{19} + \)\(63\!\cdots\!36\)\( T^{20} - \)\(20\!\cdots\!70\)\( T^{21} + \)\(34\!\cdots\!10\)\( T^{22} - \)\(57\!\cdots\!28\)\( T^{23} + \)\(12\!\cdots\!94\)\( T^{24} - \)\(74\!\cdots\!63\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 - 50 T + 1882 T^{2} - 50535 T^{3} + 1142892 T^{4} - 21649886 T^{5} + 362096140 T^{6} - 5357588155 T^{7} + 72052590140 T^{8} - 885465616422 T^{9} + 10155880272401 T^{10} - 109630009836830 T^{11} + 1135304366473216 T^{12} - 11336309613424532 T^{13} + 110124523547901952 T^{14} - 1031508762554733470 T^{15} + 9268997715853037873 T^{16} - 78389634372061452582 T^{17} + \)\(61\!\cdots\!80\)\( T^{18} - \)\(44\!\cdots\!95\)\( T^{19} + \)\(29\!\cdots\!20\)\( T^{20} - \)\(16\!\cdots\!46\)\( T^{21} + \)\(86\!\cdots\!64\)\( T^{22} - \)\(37\!\cdots\!15\)\( T^{23} + \)\(13\!\cdots\!46\)\( T^{24} - \)\(34\!\cdots\!50\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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