Properties

Label 8034.2.a.y
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: \(x^{13} - 5 x^{12} - 25 x^{11} + 134 x^{10} + 196 x^{9} - 1151 x^{8} - 801 x^{7} + 4263 x^{6} + 2205 x^{5} - 6840 x^{4} - 3579 x^{3} + 3559 x^{2} + 1839 x - 180\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 5 x^{12} - 25 x^{11} + 134 x^{10} + 196 x^{9} - 1151 x^{8} - 801 x^{7} + 4263 x^{6} + 2205 x^{5} - 6840 x^{4} - 3579 x^{3} + 3559 x^{2} + 1839 x - 180\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(29990952597 \nu^{12} - 43375192180 \nu^{11} - 1136475186639 \nu^{10} + 1007708200373 \nu^{9} + 15167079562047 \nu^{8} - 5134111470190 \nu^{7} - 86013366708565 \nu^{6} - 14172806973364 \nu^{5} + 192287295025937 \nu^{4} + 102799778649133 \nu^{3} - 112700868033808 \nu^{2} - 76422756434293 \nu + 4629967598166\)\()/ 1742391215594 \)
\(\beta_{3}\)\(=\)\((\)\(65603934565 \nu^{12} - 181553890487 \nu^{11} - 2150123291131 \nu^{10} + 4562233057208 \nu^{9} + 25073320454392 \nu^{8} - 32931751958387 \nu^{7} - 132998549746986 \nu^{6} + 69118693945305 \nu^{5} + 295215629097864 \nu^{4} + 15139766651310 \nu^{3} - 186156246848460 \nu^{2} - 51907864728215 \nu + 10324372473579\)\()/ 2613586823391 \)
\(\beta_{4}\)\(=\)\((\)\(162857412779 \nu^{12} - 677670898672 \nu^{11} - 4278067082381 \nu^{10} + 16667349248383 \nu^{9} + 36727857551753 \nu^{8} - 118345177553344 \nu^{7} - 156262962783561 \nu^{6} + 294974331366942 \nu^{5} + 316924790401641 \nu^{4} - 194513299987389 \nu^{3} - 188494938439110 \nu^{2} + 13543461373865 \nu + 13092136637310\)\()/ 5227173646782 \)
\(\beta_{5}\)\(=\)\((\)\(102100965329 \nu^{12} - 304209647056 \nu^{11} - 3169887944810 \nu^{10} + 7262339101606 \nu^{9} + 34848573902027 \nu^{8} - 46654184977558 \nu^{7} - 178805884240980 \nu^{6} + 68608084577061 \nu^{5} + 379650761799534 \nu^{4} + 101715657395343 \nu^{3} - 187075360189239 \nu^{2} - 98831180192488 \nu - 14347124251584\)\()/ 2613586823391 \)
\(\beta_{6}\)\(=\)\((\)\(-217714208045 \nu^{12} + 1005449553016 \nu^{11} + 5411101570403 \nu^{10} - 25448761107733 \nu^{9} - 41088867689801 \nu^{8} + 193780513946266 \nu^{7} + 145599003207645 \nu^{6} - 575826243150648 \nu^{5} - 243229487220105 \nu^{4} + 638472984040929 \nu^{3} + 104596802213862 \nu^{2} - 206073950699651 \nu + 13131093814746\)\()/ 5227173646782 \)
\(\beta_{7}\)\(=\)\((\)\(-255095470579 \nu^{12} + 661810602824 \nu^{11} + 8285169875425 \nu^{10} - 15232827751271 \nu^{9} - 96662482099681 \nu^{8} + 86623598556140 \nu^{7} + 524052689971935 \nu^{6} - 27010984297314 \nu^{5} - 1182427264108857 \nu^{4} - 557297580549693 \nu^{3} + 699141028801284 \nu^{2} + 417558588046067 \nu - 13275752967450\)\()/ 5227173646782 \)
\(\beta_{8}\)\(=\)\((\)\(-87548046093 \nu^{12} + 264959999010 \nu^{11} + 2727333681067 \nu^{10} - 6425835183533 \nu^{9} - 30213793968393 \nu^{8} + 43083002164404 \nu^{7} + 157724847260109 \nu^{6} - 76859446109462 \nu^{5} - 352937772776757 \nu^{4} - 51039017330433 \nu^{3} + 224469599916190 \nu^{2} + 76919571886465 \nu - 15396019776588\)\()/ 1742391215594 \)
\(\beta_{9}\)\(=\)\((\)\(-137156022856 \nu^{12} + 292035064997 \nu^{11} + 4686634469818 \nu^{10} - 6331690706375 \nu^{9} - 57883623377098 \nu^{8} + 27934282538669 \nu^{7} + 327020126294178 \nu^{6} + 73722367174863 \nu^{5} - 764557275936216 \nu^{4} - 483649010501916 \nu^{3} + 497688206439438 \nu^{2} + 341001851855144 \nu - 33755880545082\)\()/ 2613586823391 \)
\(\beta_{10}\)\(=\)\((\)\(48721679165 \nu^{12} - 146572031608 \nu^{11} - 1525594606540 \nu^{10} + 3583008630877 \nu^{9} + 16951906999904 \nu^{8} - 24498849631117 \nu^{7} - 87888224114164 \nu^{6} + 46835853163763 \nu^{5} + 192397099265341 \nu^{4} + 17354205928509 \nu^{3} - 113951919651177 \nu^{2} - 30844919826139 \nu + 4959261111660\)\()/ 871195607797 \)
\(\beta_{11}\)\(=\)\((\)\(-389665509547 \nu^{12} + 1140098743058 \nu^{11} + 12323418899743 \nu^{10} - 27777758676353 \nu^{9} - 138454266189829 \nu^{8} + 187683228295316 \nu^{7} + 721098516997287 \nu^{6} - 334211798029062 \nu^{5} - 1567675103232729 \nu^{4} - 233571225428787 \nu^{3} + 874832876252430 \nu^{2} + 315223219608983 \nu + 1172172013944\)\()/ 5227173646782 \)
\(\beta_{12}\)\(=\)\((\)\(-114093379497 \nu^{12} + 401458140693 \nu^{11} + 3322146417204 \nu^{10} - 9875157476924 \nu^{9} - 33571232899839 \nu^{8} + 69389648399099 \nu^{7} + 162730885578322 \nu^{6} - 157270125105526 \nu^{5} - 343557388191574 \nu^{4} + 46987436209824 \nu^{3} + 192897249331282 \nu^{2} + 35315463008526 \nu - 5083097540960\)\()/ 871195607797 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12} - \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{11} - 3 \beta_{10} + \beta_{9} - 6 \beta_{8} - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 16 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(12 \beta_{12} + 9 \beta_{11} - \beta_{10} + 4 \beta_{9} - 22 \beta_{8} - 5 \beta_{7} - 14 \beta_{6} + 18 \beta_{5} - 6 \beta_{4} + \beta_{3} + 2 \beta_{2} + 36 \beta_{1} + 58\)
\(\nu^{5}\)\(=\)\(\beta_{12} + 37 \beta_{11} - 62 \beta_{10} + 34 \beta_{9} - 144 \beta_{8} - 13 \beta_{7} - 16 \beta_{6} + 54 \beta_{5} - 63 \beta_{4} - 36 \beta_{3} + 21 \beta_{2} + 273 \beta_{1} + 65\)
\(\nu^{6}\)\(=\)\(161 \beta_{12} + 243 \beta_{11} - 67 \beta_{10} + 127 \beta_{9} - 479 \beta_{8} - 145 \beta_{7} - 206 \beta_{6} + 347 \beta_{5} - 170 \beta_{4} + 34 \beta_{3} + 45 \beta_{2} + 696 \beta_{1} + 900\)
\(\nu^{7}\)\(=\)\(35 \beta_{12} + 951 \beta_{11} - 1180 \beta_{10} + 801 \beta_{9} - 2920 \beta_{8} - 447 \beta_{7} - 267 \beta_{6} + 1218 \beta_{5} - 1212 \beta_{4} - 495 \beta_{3} + 375 \beta_{2} + 4847 \beta_{1} + 1468\)
\(\nu^{8}\)\(=\)\(2393 \beta_{12} + 5389 \beta_{11} - 2073 \beta_{10} + 3071 \beta_{9} - 10213 \beta_{8} - 3254 \beta_{7} - 3203 \beta_{6} + 6899 \beta_{5} - 3815 \beta_{4} + 848 \beta_{3} + 866 \beta_{2} + 14054 \beta_{1} + 15532\)
\(\nu^{9}\)\(=\)\(1150 \beta_{12} + 21643 \beta_{11} - 22293 \beta_{10} + 17056 \beta_{9} - 57599 \beta_{8} - 11144 \beta_{7} - 4950 \beta_{6} + 26262 \beta_{5} - 23245 \beta_{4} - 6042 \beta_{3} + 6485 \beta_{2} + 89148 \beta_{1} + 33244\)
\(\nu^{10}\)\(=\)\(38079 \beta_{12} + 113170 \beta_{11} - 52003 \beta_{10} + 67753 \beta_{9} - 214027 \beta_{8} - 67665 \beta_{7} - 52315 \beta_{6} + 138605 \beta_{5} - 80678 \beta_{4} + 18531 \beta_{3} + 16603 \beta_{2} + 288396 \beta_{1} + 281896\)
\(\nu^{11}\)\(=\)\(33272 \beta_{12} + 467279 \beta_{11} - 424267 \beta_{10} + 350742 \beta_{9} - 1135805 \beta_{8} - 249258 \beta_{7} - 98641 \beta_{6} + 554534 \beta_{5} - 449856 \beta_{4} - 64501 \beta_{3} + 113521 \beta_{2} + 1685330 \beta_{1} + 737045\)
\(\nu^{12}\)\(=\)\(637279 \beta_{12} + 2327795 \beta_{11} - 1193552 \beta_{10} + 1436886 \beta_{9} - 4436546 \beta_{8} - 1370991 \beta_{7} - 893575 \beta_{6} + 2793619 \beta_{5} - 1673509 \beta_{4} + 379292 \beta_{3} + 324468 \beta_{2} + 5942369 \beta_{1} + 5275921\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.14079
0.0852190
4.50145
−1.72356
1.94320
−1.46489
−0.723662
−3.83598
−0.892751
3.49310
2.89547
1.94844
0.914750
1.00000 −1.00000 1.00000 −4.27728 −1.00000 −2.14079 1.00000 1.00000 −4.27728
1.2 1.00000 −1.00000 1.00000 −4.16745 −1.00000 0.0852190 1.00000 1.00000 −4.16745
1.3 1.00000 −1.00000 1.00000 −1.68854 −1.00000 4.50145 1.00000 1.00000 −1.68854
1.4 1.00000 −1.00000 1.00000 −1.45246 −1.00000 −1.72356 1.00000 1.00000 −1.45246
1.5 1.00000 −1.00000 1.00000 −0.714571 −1.00000 1.94320 1.00000 1.00000 −0.714571
1.6 1.00000 −1.00000 1.00000 −0.556182 −1.00000 −1.46489 1.00000 1.00000 −0.556182
1.7 1.00000 −1.00000 1.00000 0.370604 −1.00000 −0.723662 1.00000 1.00000 0.370604
1.8 1.00000 −1.00000 1.00000 1.02710 −1.00000 −3.83598 1.00000 1.00000 1.02710
1.9 1.00000 −1.00000 1.00000 1.04935 −1.00000 −0.892751 1.00000 1.00000 1.04935
1.10 1.00000 −1.00000 1.00000 2.90785 −1.00000 3.49310 1.00000 1.00000 2.90785
1.11 1.00000 −1.00000 1.00000 3.35434 −1.00000 2.89547 1.00000 1.00000 3.35434
1.12 1.00000 −1.00000 1.00000 3.48931 −1.00000 1.94844 1.00000 1.00000 3.48931
1.13 1.00000 −1.00000 1.00000 3.65794 −1.00000 0.914750 1.00000 1.00000 3.65794
\(n\): 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.y 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.y 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

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{13} \)
$3$ \( ( 1 + T )^{13} \)
$5$ \( -864 + 756 T + 6421 T^{2} - 2231 T^{3} - 11958 T^{4} + 3399 T^{5} + 7614 T^{6} - 1960 T^{7} - 1749 T^{8} + 496 T^{9} + 130 T^{10} - 40 T^{11} - 3 T^{12} + T^{13} \)
$7$ \( -180 + 1839 T + 3559 T^{2} - 3579 T^{3} - 6840 T^{4} + 2205 T^{5} + 4263 T^{6} - 801 T^{7} - 1151 T^{8} + 196 T^{9} + 134 T^{10} - 25 T^{11} - 5 T^{12} + T^{13} \)
$11$ \( -587920 - 583128 T + 864508 T^{2} + 559752 T^{3} - 534603 T^{4} - 140156 T^{5} + 137699 T^{6} + 7359 T^{7} - 14727 T^{8} + 750 T^{9} + 615 T^{10} - 62 T^{11} - 8 T^{12} + T^{13} \)
$13$ \( ( -1 + T )^{13} \)
$17$ \( 5617408 + 47592544 T + 6225680 T^{2} - 21806448 T^{3} - 1072696 T^{4} + 3383950 T^{5} + 37789 T^{6} - 241094 T^{7} + 1281 T^{8} + 8590 T^{9} - 82 T^{10} - 149 T^{11} + T^{12} + T^{13} \)
$19$ \( -5142528 - 15421216 T + 20086936 T^{2} + 407912 T^{3} - 5844438 T^{4} + 730611 T^{5} + 647227 T^{6} - 109074 T^{7} - 32175 T^{8} + 5957 T^{9} + 682 T^{10} - 132 T^{11} - 5 T^{12} + T^{13} \)
$23$ \( 305262088 - 44915989 T - 141845391 T^{2} + 8598634 T^{3} + 25235225 T^{4} + 119119 T^{5} - 2139303 T^{6} - 110075 T^{7} + 87728 T^{8} + 7267 T^{9} - 1556 T^{10} - 157 T^{11} + 9 T^{12} + T^{13} \)
$29$ \( -21563136 + 70542597 T - 37959732 T^{2} - 28997537 T^{3} + 13335527 T^{4} + 5702442 T^{5} - 1173265 T^{6} - 466583 T^{7} + 33114 T^{8} + 15301 T^{9} - 339 T^{10} - 211 T^{11} + T^{12} + T^{13} \)
$31$ \( -5142528 - 15421216 T + 20086936 T^{2} + 407912 T^{3} - 5844438 T^{4} + 730611 T^{5} + 647227 T^{6} - 109074 T^{7} - 32175 T^{8} + 5957 T^{9} + 682 T^{10} - 132 T^{11} - 5 T^{12} + T^{13} \)
$37$ \( -1583312 + 25133712 T - 18479412 T^{2} - 21233950 T^{3} + 11739265 T^{4} + 3179259 T^{5} - 2185434 T^{6} + 30606 T^{7} + 133423 T^{8} - 21206 T^{9} - 547 T^{10} + 398 T^{11} - 35 T^{12} + T^{13} \)
$41$ \( -10405627608 - 3746688362 T + 3246515303 T^{2} + 669026065 T^{3} - 331706040 T^{4} - 41852707 T^{5} + 15690505 T^{6} + 1100936 T^{7} - 382550 T^{8} - 9368 T^{9} + 4758 T^{10} - 61 T^{11} - 24 T^{12} + T^{13} \)
$43$ \( 2236128 - 13372160 T + 14565328 T^{2} + 13686316 T^{3} - 11836822 T^{4} - 1219953 T^{5} + 1947516 T^{6} - 120715 T^{7} - 92026 T^{8} + 9130 T^{9} + 1626 T^{10} - 178 T^{11} - 9 T^{12} + T^{13} \)
$47$ \( 3379806 - 23185599 T + 13948491 T^{2} + 35566033 T^{3} - 1340319 T^{4} - 13389979 T^{5} - 4097310 T^{6} + 106724 T^{7} + 178563 T^{8} + 12338 T^{9} - 2313 T^{10} - 236 T^{11} + 8 T^{12} + T^{13} \)
$53$ \( 2231257728 - 16662880288 T - 1013300080 T^{2} + 4199793384 T^{3} + 204890896 T^{4} - 298260900 T^{5} + 3596319 T^{6} + 8764050 T^{7} - 617847 T^{8} - 76599 T^{9} + 9159 T^{10} + 14 T^{11} - 32 T^{12} + T^{13} \)
$59$ \( -2461440 + 14214308 T + 116632787 T^{2} - 147411406 T^{3} - 24155309 T^{4} + 28384917 T^{5} + 3561080 T^{6} - 1574281 T^{7} - 184034 T^{8} + 30459 T^{9} + 2677 T^{10} - 275 T^{11} - 12 T^{12} + T^{13} \)
$61$ \( 11383480832 + 8381122912 T - 5582269304 T^{2} - 6236778120 T^{3} - 1211682182 T^{4} + 217951571 T^{5} + 60948240 T^{6} - 4385083 T^{7} - 1015324 T^{8} + 57966 T^{9} + 7030 T^{10} - 398 T^{11} - 17 T^{12} + T^{13} \)
$67$ \( 6772775080 - 12509511986 T + 2987533597 T^{2} + 1660242089 T^{3} - 552924391 T^{4} - 65758713 T^{5} + 30353484 T^{6} + 639952 T^{7} - 699880 T^{8} + 13320 T^{9} + 6715 T^{10} - 270 T^{11} - 20 T^{12} + T^{13} \)
$71$ \( 552276992 - 164613568 T - 529789552 T^{2} + 256000483 T^{3} + 102868019 T^{4} - 85147214 T^{5} + 13097649 T^{6} + 1627089 T^{7} - 514048 T^{8} + 7974 T^{9} + 5464 T^{10} - 273 T^{11} - 16 T^{12} + T^{13} \)
$73$ \( -445541544 + 1569500022 T - 2095868381 T^{2} + 1419685585 T^{3} - 520748672 T^{4} + 90007760 T^{5} + 2250182 T^{6} - 4124667 T^{7} + 747191 T^{8} - 41751 T^{9} - 4009 T^{10} + 771 T^{11} - 46 T^{12} + T^{13} \)
$79$ \( -1034356945408 - 88696398528 T + 189036375216 T^{2} + 7331942776 T^{3} - 10621386292 T^{4} + 86731656 T^{5} + 231413307 T^{6} - 7357357 T^{7} - 2294399 T^{8} + 103479 T^{9} + 10362 T^{10} - 553 T^{11} - 17 T^{12} + T^{13} \)
$83$ \( -27181773312 - 35026282472 T + 16846671837 T^{2} + 5220800392 T^{3} - 2365179907 T^{4} - 150319584 T^{5} + 111437464 T^{6} - 3978300 T^{7} - 1514030 T^{8} + 92944 T^{9} + 7738 T^{10} - 554 T^{11} - 13 T^{12} + T^{13} \)
$89$ \( -1536000 + 12993120 T - 19222424 T^{2} - 21379000 T^{3} + 58890258 T^{4} - 32605469 T^{5} + 3996832 T^{6} + 1284588 T^{7} - 322165 T^{8} - 1271 T^{9} + 4976 T^{10} - 264 T^{11} - 14 T^{12} + T^{13} \)
$97$ \( 4262466688 - 11959082224 T + 11232447552 T^{2} - 3855385620 T^{3} - 48020034 T^{4} + 340716217 T^{5} - 72584134 T^{6} + 1163781 T^{7} + 1437309 T^{8} - 187175 T^{9} + 5434 T^{10} + 553 T^{11} - 46 T^{12} + T^{13} \)
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