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:

Atkin-Lehner signs

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

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

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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{13} \)
$3$ \( ( -1 + T )^{13} \)
$5$ \( 8832 + 17300 T - 7621 T^{2} - 27369 T^{3} - 368 T^{4} + 15809 T^{5} + 1474 T^{6} - 4352 T^{7} - 381 T^{8} + 604 T^{9} + 34 T^{10} - 40 T^{11} - T^{12} + T^{13} \)
$7$ \( -2840 + 18319 T - 1775 T^{2} - 37493 T^{3} + 8272 T^{4} + 22787 T^{5} - 4081 T^{6} - 6037 T^{7} + 669 T^{8} + 762 T^{9} - 44 T^{10} - 45 T^{11} + T^{12} + T^{13} \)
$11$ \( 1296 + 472 T - 17356 T^{2} + 13872 T^{3} + 47701 T^{4} - 78260 T^{5} + 32481 T^{6} + 5871 T^{7} - 6227 T^{8} + 448 T^{9} + 347 T^{10} - 48 T^{11} - 6 T^{12} + T^{13} \)
$13$ \( ( -1 + T )^{13} \)
$17$ \( 4990080 - 5377808 T - 5995968 T^{2} + 3903472 T^{3} + 1556948 T^{4} - 1078967 T^{5} - 81667 T^{6} + 120145 T^{7} - 8331 T^{8} - 5240 T^{9} + 839 T^{10} + 50 T^{11} - 18 T^{12} + T^{13} \)
$19$ \( 2495488 + 3848896 T - 4176712 T^{2} - 5000600 T^{3} + 1656190 T^{4} + 1692519 T^{5} - 133353 T^{6} - 184008 T^{7} + 4381 T^{8} + 8415 T^{9} - 110 T^{10} - 158 T^{11} + T^{12} + T^{13} \)
$23$ \( 4424592 + 14328706 T - 17662959 T^{2} - 1488634 T^{3} + 6769436 T^{4} - 1480263 T^{5} - 618268 T^{6} + 252335 T^{7} - 608 T^{8} - 11086 T^{9} + 1419 T^{10} + 53 T^{11} - 20 T^{12} + T^{13} \)
$29$ \( 4049256 - 23034137 T + 48766424 T^{2} - 46841633 T^{3} + 18862079 T^{4} - 453494 T^{5} - 1834877 T^{6} + 416079 T^{7} + 21802 T^{8} - 16983 T^{9} + 1371 T^{10} + 137 T^{11} - 25 T^{12} + T^{13} \)
$31$ \( -9725696 + 39758880 T - 61601336 T^{2} + 45288800 T^{3} - 14390090 T^{4} - 460541 T^{5} + 1527361 T^{6} - 290472 T^{7} - 26977 T^{8} + 12019 T^{9} - 198 T^{10} - 174 T^{11} + 5 T^{12} + T^{13} \)
$37$ \( -1312715840 + 1501158416 T + 167017504 T^{2} - 384164684 T^{3} + 12283498 T^{4} + 32017131 T^{5} - 1850796 T^{6} - 1248396 T^{7} + 70960 T^{8} + 25101 T^{9} - 1101 T^{10} - 252 T^{11} + 6 T^{12} + T^{13} \)
$41$ \( -12873600 - 80665528 T - 143802938 T^{2} - 82242969 T^{3} + 9412820 T^{4} + 20391210 T^{5} + 2995563 T^{6} - 900274 T^{7} - 148508 T^{8} + 18348 T^{9} + 2388 T^{10} - 204 T^{11} - 13 T^{12} + T^{13} \)
$43$ \( 3969021440 + 2766907488 T - 3146425888 T^{2} - 779924128 T^{3} + 315502436 T^{4} + 72395370 T^{5} - 10413853 T^{6} - 2569459 T^{7} + 149184 T^{8} + 42422 T^{9} - 938 T^{10} - 332 T^{11} + 2 T^{12} + T^{13} \)
$47$ \( 3985992 + 109671122 T + 256996699 T^{2} - 57642776 T^{3} - 96425379 T^{4} + 23793096 T^{5} + 6021449 T^{6} - 1508815 T^{7} - 145861 T^{8} + 35070 T^{9} + 1482 T^{10} - 325 T^{11} - 5 T^{12} + T^{13} \)
$53$ \( -48406272 + 199648240 T - 213257456 T^{2} + 926444 T^{3} + 80134826 T^{4} - 25843319 T^{5} - 2347523 T^{6} + 2049033 T^{7} - 205718 T^{8} - 24508 T^{9} + 4973 T^{10} - 100 T^{11} - 21 T^{12} + T^{13} \)
$59$ \( -28914768 + 183352106 T - 365384203 T^{2} + 217137388 T^{3} + 32532829 T^{4} - 53675147 T^{5} + 6132206 T^{6} + 2454201 T^{7} - 473860 T^{8} - 11253 T^{9} + 6151 T^{10} - 167 T^{11} - 22 T^{12} + T^{13} \)
$61$ \( 1289907712 - 7420362816 T + 11977329936 T^{2} - 3421305576 T^{3} - 760231460 T^{4} + 289131388 T^{5} + 16968603 T^{6} - 7804573 T^{7} - 183978 T^{8} + 92404 T^{9} + 974 T^{10} - 498 T^{11} - 2 T^{12} + T^{13} \)
$67$ \( -19748422250 - 17287926927 T + 20940607062 T^{2} - 2966774888 T^{3} - 1410918844 T^{4} + 311051851 T^{5} + 30415120 T^{6} - 9225392 T^{7} - 201898 T^{8} + 108015 T^{9} + 219 T^{10} - 546 T^{11} + T^{12} + T^{13} \)
$71$ \( -361413120 - 1612896592 T - 369407238 T^{2} + 976552321 T^{3} + 12807523 T^{4} - 112598884 T^{5} + 6405593 T^{6} + 4191093 T^{7} - 393432 T^{8} - 52862 T^{9} + 7046 T^{10} + 55 T^{11} - 32 T^{12} + T^{13} \)
$73$ \( -6590780212 + 2827515029 T + 2048927568 T^{2} - 1147709873 T^{3} - 64790488 T^{4} + 104488920 T^{5} - 6848845 T^{6} - 3217650 T^{7} + 292112 T^{8} + 45488 T^{9} - 3484 T^{10} - 331 T^{11} + 13 T^{12} + T^{13} \)
$79$ \( 818214400 + 669187872 T - 1847808640 T^{2} - 11306608 T^{3} + 330605028 T^{4} + 20511534 T^{5} - 18556289 T^{6} - 1909139 T^{7} + 352721 T^{8} + 43123 T^{9} - 2670 T^{10} - 361 T^{11} + 7 T^{12} + T^{13} \)
$83$ \( -257420472 + 8216584726 T + 2022657691 T^{2} - 1708073364 T^{3} - 341837535 T^{4} + 121183342 T^{5} + 20366196 T^{6} - 3574412 T^{7} - 500348 T^{8} + 48104 T^{9} + 4950 T^{10} - 332 T^{11} - 17 T^{12} + T^{13} \)
$89$ \( 169980344064 + 51350599568 T - 62817461784 T^{2} - 18262709124 T^{3} + 4133107596 T^{4} + 887848661 T^{5} - 111519030 T^{6} - 16374728 T^{7} + 1320165 T^{8} + 145595 T^{9} - 6800 T^{10} - 626 T^{11} + 12 T^{12} + T^{13} \)
$97$ \( -58931008 - 127647408 T - 57183120 T^{2} + 53040172 T^{3} + 59044662 T^{4} + 18709561 T^{5} + 195062 T^{6} - 1243365 T^{7} - 285515 T^{8} - 14499 T^{9} + 3546 T^{10} + 647 T^{11} + 42 T^{12} + T^{13} \)
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