Properties

Label 8034.2.a.x
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - x^{12} - 40 x^{11} + 34 x^{10} + 604 x^{9} - 381 x^{8} - 4352 x^{7} + 1474 x^{6} + 15809 x^{5} - 368 x^{4} - 27369 x^{3} - 7621 x^{2} + 17300 x + 8832\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - x^{12} - 40 x^{11} + 34 x^{10} + 604 x^{9} - 381 x^{8} - 4352 x^{7} + 1474 x^{6} + 15809 x^{5} - 368 x^{4} - 27369 x^{3} - 7621 x^{2} + 17300 x + 8832\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(82571179 \nu^{12} - 4674469543 \nu^{11} + 4045229380 \nu^{10} + 153227692402 \nu^{9} - 190916406966 \nu^{8} - 1693362162947 \nu^{7} + 2186335491570 \nu^{6} + 7311077583168 \nu^{5} - 7974918999565 \nu^{4} - 12395566278704 \nu^{3} + 7496600830547 \nu^{2} + 7178855402017 \nu - 94191942330\)\()/ 316533270094 \)
\(\beta_{3}\)\(=\)\((\)\(-821635533 \nu^{12} - 14688528363 \nu^{11} + 23628366040 \nu^{10} + 567415004562 \nu^{9} - 132389839580 \nu^{8} - 8016782345595 \nu^{7} - 1886872567880 \nu^{6} + 50022172152038 \nu^{5} + 24369898063691 \nu^{4} - 130289551131540 \nu^{3} - 81913590142811 \nu^{2} + 104592523119481 \nu + 74380301896924\)\()/ 633066540188 \)
\(\beta_{4}\)\(=\)\((\)\(464768053 \nu^{12} - 3020129323 \nu^{11} - 17668924332 \nu^{10} + 110747092838 \nu^{9} + 253508520438 \nu^{8} - 1447775318393 \nu^{7} - 1820042919050 \nu^{6} + 8096109949528 \nu^{5} + 7507442330311 \nu^{4} - 18589696680318 \nu^{3} - 15726961057905 \nu^{2} + 13031570002261 \nu + 10932434824650\)\()/ 316533270094 \)
\(\beta_{5}\)\(=\)\((\)\(1800405167 \nu^{12} - 9821990403 \nu^{11} - 66844233676 \nu^{10} + 372531445822 \nu^{9} + 923116143424 \nu^{8} - 5111459259651 \nu^{7} - 6174254472800 \nu^{6} + 30716279416750 \nu^{5} + 22995984560079 \nu^{4} - 78136989988656 \nu^{3} - 46139543962307 \nu^{2} + 63313812889133 \nu + 37558928545592\)\()/ 633066540188 \)
\(\beta_{6}\)\(=\)\((\)\(459562957 \nu^{12} + 518271920 \nu^{11} - 16642314467 \nu^{10} - 17632719959 \nu^{9} + 219058129306 \nu^{8} + 220995431293 \nu^{7} - 1289495659648 \nu^{6} - 1257189213960 \nu^{5} + 3402241451437 \nu^{4} + 3081177289065 \nu^{3} - 3076261637442 \nu^{2} - 2594200123208 \nu - 261207897518\)\()/ 158266635047 \)
\(\beta_{7}\)\(=\)\((\)\(-1001697093 \nu^{12} + 3637925703 \nu^{11} + 29239399554 \nu^{10} - 117962252484 \nu^{9} - 247199851646 \nu^{8} + 1251371300361 \nu^{7} + 392655827726 \nu^{6} - 4796699155248 \nu^{5} + 1170436096691 \nu^{4} + 6233211700574 \nu^{3} - 1027544285569 \nu^{2} - 1990455155601 \nu - 1599125153292\)\()/ 316533270094 \)
\(\beta_{8}\)\(=\)\((\)\(-1301185969 \nu^{12} - 3095768311 \nu^{11} + 48687375005 \nu^{10} + 117305988087 \nu^{9} - 652269947584 \nu^{8} - 1636981333955 \nu^{7} + 3626438951305 \nu^{6} + 10185117990171 \nu^{5} - 6405537841610 \nu^{4} - 26666364418514 \nu^{3} - 4445159036711 \nu^{2} + 21974554085695 \nu + 11198108957142\)\()/ 158266635047 \)
\(\beta_{9}\)\(=\)\((\)\(-5709491399 \nu^{12} + 1884380107 \nu^{11} + 220983218852 \nu^{10} - 60804918910 \nu^{9} - 3140305882384 \nu^{8} + 596077416591 \nu^{7} + 20057540387488 \nu^{6} - 1615844172274 \nu^{5} - 57139879804783 \nu^{4} - 114622375696 \nu^{3} + 63720036244919 \nu^{2} + 3045455679311 \nu - 22176589418488\)\()/ 633066540188 \)
\(\beta_{10}\)\(=\)\((\)\(-1572124490 \nu^{12} + 1168195578 \nu^{11} + 59893192506 \nu^{10} - 46050449196 \nu^{9} - 844559885012 \nu^{8} + 638669608161 \nu^{7} + 5500753705133 \nu^{6} - 3769562907595 \nu^{5} - 17246842712312 \nu^{4} + 9535362819479 \nu^{3} + 24610988707373 \nu^{2} - 8054888236900 \nu - 13075070948772\)\()/ 158266635047 \)
\(\beta_{11}\)\(=\)\((\)\(4905042985 \nu^{12} - 8698615733 \nu^{11} - 187812103994 \nu^{10} + 306055612136 \nu^{9} + 2649761260912 \nu^{8} - 3752712058411 \nu^{7} - 17188892321224 \nu^{6} + 19215174884610 \nu^{5} + 53780502305993 \nu^{4} - 41767633025424 \nu^{3} - 76738490139713 \nu^{2} + 30387722952273 \nu + 40291809896428\)\()/ 316533270094 \)
\(\beta_{12}\)\(=\)\((\)\(5235281189 \nu^{12} - 6247478525 \nu^{11} - 196255326952 \nu^{10} + 205900657400 \nu^{9} + 2669456439700 \nu^{8} - 2236009627109 \nu^{7} - 16177418069660 \nu^{6} + 9001746461066 \nu^{5} + 44509961821057 \nu^{4} - 13527828592988 \nu^{3} - 50559136604119 \nu^{2} + 6830665264495 \nu + 19623283268508\)\()/ 316533270094 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{10} - 2 \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 9 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-\beta_{11} - 2 \beta_{10} - \beta_{8} + 15 \beta_{7} + 12 \beta_{6} - \beta_{5} + \beta_{3} + 16 \beta_{2} - 4 \beta_{1} + 75\)
\(\nu^{5}\)\(=\)\(2 \beta_{12} + 2 \beta_{11} + 16 \beta_{10} + 7 \beta_{9} - 33 \beta_{8} + 16 \beta_{6} - 16 \beta_{5} - 37 \beta_{4} + 49 \beta_{3} + 14 \beta_{2} + 100 \beta_{1} + 24\)
\(\nu^{6}\)\(=\)\(-4 \beta_{12} - 12 \beta_{11} - 36 \beta_{10} + 7 \beta_{9} - 22 \beta_{8} + 198 \beta_{7} + 139 \beta_{6} - 14 \beta_{5} + 2 \beta_{4} + 15 \beta_{3} + 215 \beta_{2} - 86 \beta_{1} + 896\)
\(\nu^{7}\)\(=\)\(52 \beta_{12} + 42 \beta_{11} + 216 \beta_{10} + 178 \beta_{9} - 477 \beta_{8} - 9 \beta_{7} + 201 \beta_{6} - 229 \beta_{5} - 535 \beta_{4} + 693 \beta_{3} + 163 \beta_{2} + 1187 \beta_{1} + 237\)
\(\nu^{8}\)\(=\)\(-136 \beta_{12} - 70 \beta_{11} - 520 \beta_{10} + 190 \beta_{9} - 367 \beta_{8} + 2549 \beta_{7} + 1644 \beta_{6} - 154 \beta_{5} + 57 \beta_{4} + 166 \beta_{3} + 2789 \beta_{2} - 1416 \beta_{1} + 11090\)
\(\nu^{9}\)\(=\)\(1011 \beta_{12} + 650 \beta_{11} + 2812 \beta_{10} + 3318 \beta_{9} - 6682 \beta_{8} - 241 \beta_{7} + 2403 \beta_{6} - 3190 \beta_{5} - 7268 \beta_{4} + 9510 \beta_{3} + 1815 \beta_{2} + 14464 \beta_{1} + 2129\)
\(\nu^{10}\)\(=\)\(-3017 \beta_{12} + 419 \beta_{11} - 7116 \beta_{10} + 3536 \beta_{9} - 5573 \beta_{8} + 32694 \beta_{7} + 19796 \beta_{6} - 1499 \beta_{5} + 1131 \beta_{4} + 1557 \beta_{3} + 36022 \beta_{2} - 21290 \beta_{1} + 139500\)
\(\nu^{11}\)\(=\)\(17512 \beta_{12} + 8931 \beta_{11} + 36263 \beta_{10} + 54796 \beta_{9} - 92393 \beta_{8} - 4410 \beta_{7} + 28728 \beta_{6} - 44052 \beta_{5} - 97071 \beta_{4} + 129454 \beta_{3} + 20001 \beta_{2} + 178798 \beta_{1} + 17574\)
\(\nu^{12}\)\(=\)\(-56146 \beta_{12} + 22414 \beta_{11} - 95843 \beta_{10} + 56188 \beta_{9} - 80875 \beta_{8} + 420209 \beta_{7} + 241672 \beta_{6} - 12416 \beta_{5} + 19472 \beta_{4} + 11516 \beta_{3} + 466863 \beta_{2} - 306338 \beta_{1} + 1771965\)

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
−3.67523
−3.54523
−2.08649
−2.01149
−1.22001
−0.975286
−0.757993
1.20657
1.95504
1.98373
3.21537
3.25869
3.65233
−1.00000 1.00000 1.00000 −3.67523 −1.00000 −3.11602 −1.00000 1.00000 3.67523
1.2 −1.00000 1.00000 1.00000 −3.54523 −1.00000 0.166701 −1.00000 1.00000 3.54523
1.3 −1.00000 1.00000 1.00000 −2.08649 −1.00000 −3.62308 −1.00000 1.00000 2.08649
1.4 −1.00000 1.00000 1.00000 −2.01149 −1.00000 0.821182 −1.00000 1.00000 2.01149
1.5 −1.00000 1.00000 1.00000 −1.22001 −1.00000 3.96620 −1.00000 1.00000 1.22001
1.6 −1.00000 1.00000 1.00000 −0.975286 −1.00000 −3.46524 −1.00000 1.00000 0.975286
1.7 −1.00000 1.00000 1.00000 −0.757993 −1.00000 1.64575 −1.00000 1.00000 0.757993
1.8 −1.00000 1.00000 1.00000 1.20657 −1.00000 −1.77104 −1.00000 1.00000 −1.20657
1.9 −1.00000 1.00000 1.00000 1.95504 −1.00000 −2.89499 −1.00000 1.00000 −1.95504
1.10 −1.00000 1.00000 1.00000 1.98373 −1.00000 3.61896 −1.00000 1.00000 −1.98373
1.11 −1.00000 1.00000 1.00000 3.21537 −1.00000 −0.928130 −1.00000 1.00000 −3.21537
1.12 −1.00000 1.00000 1.00000 3.25869 −1.00000 3.01501 −1.00000 1.00000 −3.25869
1.13 −1.00000 1.00000 1.00000 3.65233 −1.00000 1.56471 −1.00000 1.00000 −3.65233
\(n\): 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 8034.2.a.x 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.x 13 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{13} \)
$3$ \( ( 1 - T )^{13} \)
$5$ \( 1 - T + 25 T^{2} - 26 T^{3} + 354 T^{4} - 331 T^{5} + 3578 T^{6} - 3016 T^{7} + 28964 T^{8} - 22223 T^{9} + 196931 T^{10} - 141731 T^{11} + 1146515 T^{12} - 768828 T^{13} + 5732575 T^{14} - 3543275 T^{15} + 24616375 T^{16} - 13889375 T^{17} + 90512500 T^{18} - 47125000 T^{19} + 279531250 T^{20} - 129296875 T^{21} + 691406250 T^{22} - 253906250 T^{23} + 1220703125 T^{24} - 244140625 T^{25} + 1220703125 T^{26} \)
$7$ \( 1 + T + 46 T^{2} + 40 T^{3} + 1119 T^{4} + 823 T^{5} + 18792 T^{6} + 11823 T^{7} + 241082 T^{8} + 132193 T^{9} + 2478482 T^{10} + 1206362 T^{11} + 20915377 T^{12} + 9199108 T^{13} + 146407639 T^{14} + 59111738 T^{15} + 850119326 T^{16} + 317395393 T^{17} + 4051865174 T^{18} + 1390964127 T^{19} + 15476020056 T^{20} + 4744431223 T^{21} + 45155686233 T^{22} + 11299009960 T^{23} + 90957030178 T^{24} + 13841287201 T^{25} + 96889010407 T^{26} \)
$11$ \( 1 - 6 T + 95 T^{2} - 445 T^{3} + 4078 T^{4} - 15973 T^{5} + 111449 T^{6} - 383000 T^{7} + 2252090 T^{8} - 6966559 T^{9} + 36075172 T^{10} - 101525851 T^{11} + 474251051 T^{12} - 1221561420 T^{13} + 5216761561 T^{14} - 12284627971 T^{15} + 48016053932 T^{16} - 101997390319 T^{17} + 362701346590 T^{18} - 678507863000 T^{19} + 2171825720779 T^{20} - 3423954406213 T^{21} + 9615710683898 T^{22} - 11542153947445 T^{23} + 27104608708045 T^{24} - 18830570260326 T^{25} + 34522712143931 T^{26} \)
$13$ \( ( 1 - T )^{13} \)
$17$ \( 1 - 18 T + 271 T^{2} - 2833 T^{3} + 26652 T^{4} - 209033 T^{5} + 1518293 T^{6} - 9758968 T^{7} + 58951093 T^{8} - 323718808 T^{9} + 1684297661 T^{10} - 8072157719 T^{11} + 36809878173 T^{12} - 155489187802 T^{13} + 625767928941 T^{14} - 2332853580791 T^{15} + 8274954408493 T^{16} - 27037318562968 T^{17} + 83702122053701 T^{18} - 235557763468792 T^{19} + 623014334845189 T^{20} - 1458163505164553 T^{21} + 3160604084398044 T^{22} - 5711310719972017 T^{23} + 9287683899368543 T^{24} - 10487200270135698 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 + T + 89 T^{2} + 118 T^{3} + 3551 T^{4} + 7307 T^{5} + 79541 T^{6} + 254589 T^{7} + 946180 T^{8} + 4707191 T^{9} + 979270 T^{10} + 33002165 T^{11} - 181609952 T^{12} - 63805734 T^{13} - 3450589088 T^{14} + 11913781565 T^{15} + 6716812930 T^{16} + 613445838311 T^{17} + 2342835351820 T^{18} + 11977363797909 T^{19} + 71099451991799 T^{20} + 124098895140587 T^{21} + 1145864014813229 T^{22} + 723465818420518 T^{23} + 10367633041941491 T^{24} + 2213314919066161 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 - 20 T + 352 T^{2} - 4101 T^{3} + 43585 T^{4} - 372518 T^{5} + 2979330 T^{6} - 20485645 T^{7} + 134510618 T^{8} - 786186384 T^{9} + 4479260759 T^{10} - 23276946973 T^{11} + 120274516911 T^{12} - 575825457774 T^{13} + 2766313888953 T^{14} - 12313504948717 T^{15} + 54499165654753 T^{16} - 220007183884944 T^{17} + 865756474589974 T^{18} - 3032610669313405 T^{19} + 10144098599010510 T^{20} - 29172251614907558 T^{21} + 78503238749864855 T^{22} - 169890122487174549 T^{23} + 335389034785702304 T^{24} - 438292488640406420 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 - 25 T + 514 T^{2} - 7329 T^{3} + 92318 T^{4} - 968258 T^{5} + 9295705 T^{6} - 79030818 T^{7} + 626852495 T^{8} - 4527137118 T^{9} + 30817131221 T^{10} - 193620860489 T^{11} + 1152727712381 T^{12} - 6371576750030 T^{13} + 33429103659049 T^{14} - 162835143671249 T^{15} + 751599013348969 T^{16} - 3201958067956158 T^{17} + 12857464925966755 T^{18} - 47009373624106578 T^{19} + 160349761454952845 T^{20} - 484367591320791938 T^{21} + 1339270702200274342 T^{22} - 3083363312857173129 T^{23} + 6271062019572796106 T^{24} - 8845369580136726025 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 + 5 T + 229 T^{2} + 1662 T^{3} + 27643 T^{4} + 228773 T^{5} + 2386285 T^{6} + 19044655 T^{7} + 157334264 T^{8} + 1121682255 T^{9} + 8005905466 T^{10} + 50136749907 T^{11} + 317045808232 T^{12} + 1751049030286 T^{13} + 9828420055192 T^{14} + 48181416660627 T^{15} + 238503929737606 T^{16} + 1035897117819855 T^{17} + 4504346401529864 T^{18} + 16902201415875055 T^{19} + 65652938363867635 T^{20} + 195118441308489893 T^{21} + 730870475387428453 T^{22} + 1362222212962091262 T^{23} + 5818541209276706299 T^{24} + 3938313918942748805 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 + 6 T + 229 T^{2} + 1563 T^{3} + 29319 T^{4} + 205714 T^{5} + 2622655 T^{6} + 18188219 T^{7} + 179633626 T^{8} + 1195429316 T^{9} + 9840351898 T^{10} + 61459463482 T^{11} + 440891702264 T^{12} + 2531076009400 T^{13} + 16312992983768 T^{14} + 84138005506858 T^{15} + 498443344689394 T^{16} + 2240427002303876 T^{17} + 12456506437098082 T^{18} + 46665993820975571 T^{19} + 248973562222248115 T^{20} + 722566198383904594 T^{21} + 3810348249051862563 T^{22} + 7515817374089097987 T^{23} + 40743135387496434577 T^{24} + 39497712035040211686 T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 13 T + 329 T^{2} - 4008 T^{3} + 57462 T^{4} - 611726 T^{5} + 6720724 T^{6} - 62188861 T^{7} + 572898395 T^{8} - 4677423501 T^{9} + 37417333446 T^{10} - 272037711237 T^{11} + 1925600799031 T^{12} - 12501862012612 T^{13} + 78949632760271 T^{14} - 457295392589397 T^{15} + 2578840038431766 T^{16} - 13217280909609261 T^{17} + 66373831603697395 T^{18} - 295403572379059501 T^{19} + 1308889722574609844 T^{20} - 4884586370709272846 T^{21} + 18812020714145786982 T^{22} - 53798018515090823208 T^{23} + \)\(18\!\cdots\!89\)\( T^{24} - \)\(29\!\cdots\!53\)\( T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( 1 + 2 T + 227 T^{2} + 94 T^{3} + 29608 T^{4} - 10088 T^{5} + 2824117 T^{6} - 2157767 T^{7} + 211815474 T^{8} - 212412520 T^{9} + 12970645628 T^{10} - 14056618991 T^{11} + 663393251065 T^{12} - 691184010500 T^{13} + 28525909795795 T^{14} - 25990688514359 T^{15} + 1031257121945396 T^{16} - 726196135788520 T^{17} + 31138663036046982 T^{18} - 13640028582151583 T^{19} + 767647560543667519 T^{20} - 117910564400438888 T^{21} + 14880762054226047544 T^{22} + 2031479337448719406 T^{23} + \)\(21\!\cdots\!89\)\( T^{24} + 79919261594525152802 T^{25} + \)\(17\!\cdots\!43\)\( T^{26} \)
$47$ \( 1 - 5 T + 286 T^{2} - 1338 T^{3} + 39347 T^{4} - 178291 T^{5} + 3533298 T^{6} - 15709377 T^{7} + 237763181 T^{8} - 1033499488 T^{9} + 13212651748 T^{10} - 55950858606 T^{11} + 658244739205 T^{12} - 2721293956630 T^{13} + 30937502742635 T^{14} - 123595446660654 T^{15} + 1371777142432604 T^{16} - 5043147815103328 T^{17} + 54529798410787267 T^{18} - 169334757367440033 T^{19} + 1790050458285676974 T^{20} - 4245338110212030451 T^{21} + 44034426725174573149 T^{22} - 70377638931540605562 T^{23} + \)\(70\!\cdots\!58\)\( T^{24} - \)\(58\!\cdots\!05\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 - 21 T + 589 T^{2} - 8383 T^{3} + 136294 T^{4} - 1463302 T^{5} + 17488039 T^{6} - 148771630 T^{7} + 1441224347 T^{8} - 10024221099 T^{9} + 85365669693 T^{10} - 512392979211 T^{11} + 4298837598977 T^{12} - 25283227199172 T^{13} + 227838392745781 T^{14} - 1439311878603699 T^{15} + 12708984806884761 T^{16} - 79095926121458619 T^{17} + 602713526317268071 T^{18} - 3297428133069970270 T^{19} + 20543394227203909643 T^{20} - 91104729498325374022 T^{21} + \)\(44\!\cdots\!02\)\( T^{22} - \)\(14\!\cdots\!67\)\( T^{23} + \)\(54\!\cdots\!33\)\( T^{24} - \)\(10\!\cdots\!61\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 - 22 T + 600 T^{2} - 9425 T^{3} + 151882 T^{4} - 1899182 T^{5} + 23244267 T^{6} - 248040679 T^{7} + 2554423688 T^{8} - 24346566537 T^{9} + 221972413919 T^{10} - 1927189113375 T^{11} + 15853218736929 T^{12} - 125386473045648 T^{13} + 935339905478811 T^{14} - 6708545303658375 T^{15} + 45588472398270301 T^{16} - 295016135839348857 T^{17} + 1826219564492394712 T^{18} - 10462488204895982239 T^{19} + 57846879583079282673 T^{20} - \)\(27\!\cdots\!22\)\( T^{21} + \)\(13\!\cdots\!98\)\( T^{22} - \)\(48\!\cdots\!25\)\( T^{23} + \)\(18\!\cdots\!00\)\( T^{24} - \)\(39\!\cdots\!82\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 - 2 T + 295 T^{2} - 490 T^{3} + 48484 T^{4} - 81010 T^{5} + 5923599 T^{6} - 9592871 T^{7} + 583396486 T^{8} - 895865936 T^{9} + 48288553634 T^{10} - 70761375911 T^{11} + 3416338637165 T^{12} - 4697682281336 T^{13} + 208396656867065 T^{14} - 263303079764831 T^{15} + 10960584192398954 T^{16} - 12404017307172176 T^{17} + 492734514091998286 T^{18} - 494228305116780431 T^{19} + 18616348320711159579 T^{20} - 15530209425909733810 T^{21} + \)\(56\!\cdots\!44\)\( T^{22} - \)\(34\!\cdots\!90\)\( T^{23} + \)\(12\!\cdots\!95\)\( T^{24} - \)\(53\!\cdots\!42\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 + T + 325 T^{2} + 1023 T^{3} + 55755 T^{4} + 241106 T^{5} + 7121201 T^{6} + 32604747 T^{7} + 752311908 T^{8} + 3317852515 T^{9} + 67250045058 T^{10} + 286352503860 T^{11} + 5145893316345 T^{12} + 21040737104174 T^{13} + 344774852195115 T^{14} + 1285436389827540 T^{15} + 20226325301779254 T^{16} + 66858447489919315 T^{17} + 1015715195285874156 T^{18} + 2949372664649556243 T^{19} + 43159545544537752923 T^{20} + 97905353464971484946 T^{21} + \)\(15\!\cdots\!85\)\( T^{22} + \)\(18\!\cdots\!27\)\( T^{23} + \)\(39\!\cdots\!75\)\( T^{24} + \)\(81\!\cdots\!61\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 32 T + 978 T^{2} - 20218 T^{3} + 383291 T^{4} - 6037364 T^{5} + 88023846 T^{6} - 1138407353 T^{7} + 13794584265 T^{8} - 152690668115 T^{9} + 1598671561773 T^{10} - 15523677790207 T^{11} + 143439830649220 T^{12} - 1237750583694574 T^{13} + 10184227976094620 T^{14} - 78254859740433487 T^{15} + 572182137345736203 T^{16} - 3880126549815251315 T^{17} + 24888593815755762015 T^{18} - \)\(14\!\cdots\!13\)\( T^{19} + \)\(80\!\cdots\!86\)\( T^{20} - \)\(38\!\cdots\!04\)\( T^{21} + \)\(17\!\cdots\!21\)\( T^{22} - \)\(65\!\cdots\!18\)\( T^{23} + \)\(22\!\cdots\!38\)\( T^{24} - \)\(52\!\cdots\!12\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 + 13 T + 618 T^{2} + 7904 T^{3} + 195357 T^{4} + 2321074 T^{5} + 40912383 T^{6} + 440852563 T^{7} + 6245448502 T^{8} + 60624484621 T^{9} + 729440872702 T^{10} + 6366369654909 T^{11} + 66944930027678 T^{12} + 523353492943988 T^{13} + 4886979892020494 T^{14} + 33926383891010061 T^{15} + 283764899975913934 T^{16} + 1721628724767951661 T^{17} + 12947261875040603686 T^{18} + 66716081529127628707 T^{19} + \)\(45\!\cdots\!51\)\( T^{20} + \)\(18\!\cdots\!94\)\( T^{21} + \)\(11\!\cdots\!41\)\( T^{22} + \)\(33\!\cdots\!96\)\( T^{23} + \)\(19\!\cdots\!86\)\( T^{24} + \)\(29\!\cdots\!73\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 + 7 T + 666 T^{2} + 3966 T^{3} + 216212 T^{4} + 1126763 T^{5} + 45845413 T^{6} + 213787293 T^{7} + 7134900695 T^{8} + 30164548415 T^{9} + 863876465774 T^{10} + 3324058355493 T^{11} + 83908955488337 T^{12} + 292728058117086 T^{13} + 6628807483578623 T^{14} + 20745448196631813 T^{15} + 425924788808747186 T^{16} + 1174911604092671615 T^{17} + 21954491839779297305 T^{18} + 51969009078092494653 T^{19} + \)\(88\!\cdots\!67\)\( T^{20} + \)\(17\!\cdots\!43\)\( T^{21} + \)\(25\!\cdots\!28\)\( T^{22} + \)\(37\!\cdots\!66\)\( T^{23} + \)\(49\!\cdots\!14\)\( T^{24} + \)\(41\!\cdots\!87\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - 17 T + 747 T^{2} - 11982 T^{3} + 282330 T^{4} - 4121306 T^{5} + 70097218 T^{6} - 915823506 T^{7} + 12584642041 T^{8} - 146432891558 T^{9} + 1711928377951 T^{10} - 17730424063177 T^{11} + 180891835710606 T^{12} - 1665108893172028 T^{13} + 15014022363980298 T^{14} - 122144891371226353 T^{15} + 978858391443468437 T^{16} - 6949459172517754118 T^{17} + 49571416477105472363 T^{18} - \)\(29\!\cdots\!14\)\( T^{19} + \)\(19\!\cdots\!86\)\( T^{20} - \)\(92\!\cdots\!46\)\( T^{21} + \)\(52\!\cdots\!90\)\( T^{22} - \)\(18\!\cdots\!18\)\( T^{23} + \)\(96\!\cdots\!49\)\( T^{24} - \)\(18\!\cdots\!37\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 + 12 T + 531 T^{2} + 6016 T^{3} + 150579 T^{4} + 1541597 T^{5} + 29147971 T^{6} + 265730610 T^{7} + 4248135242 T^{8} + 34812908136 T^{9} + 500190450476 T^{10} + 3751238541798 T^{11} + 50320475683808 T^{12} + 352861386205326 T^{13} + 4478522335858912 T^{14} + 29713560489581958 T^{15} + 352618761681615244 T^{16} + 2184239872179772776 T^{17} + 23721839738720001658 T^{18} + \)\(13\!\cdots\!10\)\( T^{19} + \)\(12\!\cdots\!59\)\( T^{20} + \)\(60\!\cdots\!57\)\( T^{21} + \)\(52\!\cdots\!11\)\( T^{22} + \)\(18\!\cdots\!16\)\( T^{23} + \)\(14\!\cdots\!59\)\( T^{24} + \)\(29\!\cdots\!52\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 + 42 T + 1908 T^{2} + 52434 T^{3} + 1409752 T^{4} + 29235853 T^{5} + 581942751 T^{6} + 9713128072 T^{7} + 154994147480 T^{8} + 2153837439916 T^{9} + 28605686577873 T^{10} + 337031300350094 T^{11} + 3796582535716027 T^{12} + 38237088196994506 T^{13} + 368268505964454619 T^{14} + 3171127504994034446 T^{15} + 26107637786087084529 T^{16} + \)\(19\!\cdots\!96\)\( T^{17} + \)\(13\!\cdots\!60\)\( T^{18} + \)\(80\!\cdots\!88\)\( T^{19} + \)\(47\!\cdots\!63\)\( T^{20} + \)\(22\!\cdots\!33\)\( T^{21} + \)\(10\!\cdots\!84\)\( T^{22} + \)\(38\!\cdots\!66\)\( T^{23} + \)\(13\!\cdots\!24\)\( T^{24} + \)\(29\!\cdots\!22\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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