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

Related objects

Downloads

Learn more about

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: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
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:

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

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

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{13} \)
$3$ \( ( 1 + T )^{13} \)
$5$ \( 1 - 3 T + 25 T^{2} - 50 T^{3} + 246 T^{4} - 199 T^{5} + 1110 T^{6} + 1404 T^{7} + 3074 T^{8} + 14037 T^{9} + 33619 T^{10} + 17011 T^{11} + 364541 T^{12} - 151604 T^{13} + 1822705 T^{14} + 425275 T^{15} + 4202375 T^{16} + 8773125 T^{17} + 9606250 T^{18} + 21937500 T^{19} + 86718750 T^{20} - 77734375 T^{21} + 480468750 T^{22} - 488281250 T^{23} + 1220703125 T^{24} - 732421875 T^{25} + 1220703125 T^{26} \)
$7$ \( 1 - 5 T + 66 T^{2} - 286 T^{3} + 2093 T^{4} - 7941 T^{5} + 42270 T^{6} - 142023 T^{7} + 610540 T^{8} - 1834001 T^{9} + 6718878 T^{10} - 18154644 T^{11} + 58451055 T^{12} - 142165308 T^{13} + 409157385 T^{14} - 889577556 T^{15} + 2304575154 T^{16} - 4403436401 T^{17} + 10261345780 T^{18} - 16708863927 T^{19} + 34811162610 T^{20} - 45778284741 T^{21} + 84460099451 T^{22} - 80787921214 T^{23} + 130503565038 T^{24} - 69206436005 T^{25} + 96889010407 T^{26} \)
$11$ \( 1 - 8 T + 81 T^{2} - 441 T^{3} + 2686 T^{4} - 10965 T^{5} + 49665 T^{6} - 152162 T^{7} + 545672 T^{8} - 1092105 T^{9} + 3121168 T^{10} + 40003 T^{11} + 1547275 T^{12} + 66901436 T^{13} + 17020025 T^{14} + 4840363 T^{15} + 4154274608 T^{16} - 15989509305 T^{17} + 87881021272 T^{18} - 269564264882 T^{19} + 967830347715 T^{20} - 2350445130165 T^{21} + 6333447498026 T^{22} - 11438404249041 T^{23} + 23110245319491 T^{24} - 25107427013768 T^{25} + 34522712143931 T^{26} \)
$13$ \( ( 1 - T )^{13} \)
$17$ \( 1 + T + 72 T^{2} + 122 T^{3} + 3269 T^{4} + 6415 T^{5} + 109939 T^{6} + 226455 T^{7} + 2995534 T^{8} + 6146609 T^{9} + 68282485 T^{10} + 135831759 T^{11} + 1335831804 T^{12} + 2522899582 T^{13} + 22709140668 T^{14} + 39255378351 T^{15} + 335471848805 T^{16} + 513370930289 T^{17} + 4253229918638 T^{18} + 5466073187895 T^{19} + 45112223370947 T^{20} + 44749483984015 T^{21} + 387663768268693 T^{22} + 245951255854778 T^{23} + 2467576534149576 T^{24} + 582622237229761 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 - 5 T + 115 T^{2} - 458 T^{3} + 6527 T^{4} - 21725 T^{5} + 250387 T^{6} - 709183 T^{7} + 7431436 T^{8} - 18489375 T^{9} + 185043908 T^{10} - 418680767 T^{11} + 4005392870 T^{12} - 8429455742 T^{13} + 76102464530 T^{14} - 151143756887 T^{15} + 1269216164972 T^{16} - 2409553839375 T^{17} + 18400971248164 T^{18} - 33364139025223 T^{19} + 223813863112993 T^{20} - 368967907065725 T^{21} + 2106182603403533 T^{22} - 2808028346072858 T^{23} + 13396379773295185 T^{24} - 11066574595330805 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 + 9 T + 142 T^{2} + 928 T^{3} + 8808 T^{4} + 44074 T^{5} + 306041 T^{6} + 1052729 T^{7} + 5689972 T^{8} + 4307962 T^{9} + 11562851 T^{10} - 584517996 T^{11} - 2286303824 T^{12} - 20909499900 T^{13} - 52584987952 T^{14} - 309210019884 T^{15} + 140685208117 T^{16} + 1205544394042 T^{17} + 36622611452396 T^{18} + 155841673391081 T^{19} + 1042016184625327 T^{20} + 3451478365274794 T^{21} + 15864552642166104 T^{22} + 38443802406266272 T^{23} + 135298985623777634 T^{24} + 197231619888182889 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 + T + 166 T^{2} + 9 T^{3} + 13590 T^{4} - 9690 T^{5} + 742427 T^{6} - 954692 T^{7} + 30842049 T^{8} - 53084536 T^{9} + 1054198399 T^{10} - 2171709581 T^{11} + 32057467777 T^{12} - 70373835838 T^{13} + 929666565533 T^{14} - 1826407757621 T^{15} + 25710844753211 T^{16} - 37545683706616 T^{17} + 632605862504301 T^{18} - 567873065972132 T^{19} + 12806773918461943 T^{20} - 4847387741592090 T^{21} + 197152113812059710 T^{22} + 3786365099701809 T^{23} + 2025284621107167614 T^{24} + 353814783205469041 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 - 5 T + 271 T^{2} - 1178 T^{3} + 35903 T^{4} - 137885 T^{5} + 3096295 T^{6} - 10609183 T^{7} + 194619460 T^{8} - 598844151 T^{9} + 9436631708 T^{10} - 26157030311 T^{11} + 363528273854 T^{12} - 906774761870 T^{13} + 11269376489474 T^{14} - 25136906128871 T^{15} + 281126695213028 T^{16} - 553045149175671 T^{17} + 5571789907878460 T^{18} - 9415688964902623 T^{19} + 85187169508818745 T^{20} - 117600880697552285 T^{21} + 949261754434570913 T^{22} - 965522122063383578 T^{23} + 6885697238925709201 T^{24} - 3938313918942748805 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 - 35 T + 879 T^{2} - 16087 T^{3} + 247562 T^{4} - 3231357 T^{5} + 37423176 T^{6} - 386418261 T^{7} + 3632397334 T^{8} - 31153758892 T^{9} + 246616407746 T^{10} - 1802194905228 T^{11} + 12241252447362 T^{12} - 77149226163688 T^{13} + 452926340552394 T^{14} - 2467204825257132 T^{15} + 12491860901558138 T^{16} - 58387159918789612 T^{17} + 251884804535810638 T^{18} - 991443537167554749 T^{19} + 3552652345958634408 T^{20} - 11350075070783800797 T^{21} + 32173588227148852274 T^{22} - 77355696799085936863 T^{23} + \)\(15\!\cdots\!27\)\( T^{24} - \)\(23\!\cdots\!35\)\( T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 24 T + 472 T^{2} - 7050 T^{3} + 94239 T^{4} - 1094474 T^{5} + 11715795 T^{6} - 113769865 T^{7} + 1033931287 T^{8} - 8696611730 T^{9} + 68942962671 T^{10} - 510748783110 T^{11} + 3581397425849 T^{12} - 23556186478802 T^{13} + 146837294459809 T^{14} - 858568704407910 T^{15} + 4751617930247991 T^{16} - 24574546258776530 T^{17} + 119787351006860687 T^{18} - 540418718234497465 T^{19} + 2281701148163650395 T^{20} - 8739293055216977354 T^{21} + 30852146115352490679 T^{22} - 94629748136574427050 T^{23} + \)\(25\!\cdots\!52\)\( T^{24} - \)\(54\!\cdots\!44\)\( T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( 1 - 9 T + 381 T^{2} - 3018 T^{3} + 69158 T^{4} - 491152 T^{5} + 8049887 T^{6} - 51841958 T^{7} + 679496277 T^{8} - 3991100381 T^{9} + 44419157527 T^{10} - 238235230232 T^{11} + 2334269136875 T^{12} - 11387991993860 T^{13} + 100373572885625 T^{14} - 440496940698968 T^{15} + 3531633957499189 T^{16} - 13644777973663181 T^{17} + 99891689706066711 T^{18} - 327711837689009942 T^{19} + 2188109103908294909 T^{20} - 5740682942744286352 T^{21} + 34758299856328188194 T^{22} - 65223453621491863482 T^{23} + \)\(35\!\cdots\!67\)\( T^{24} - \)\(35\!\cdots\!09\)\( T^{25} + \)\(17\!\cdots\!43\)\( T^{26} \)
$47$ \( 1 + 8 T + 375 T^{2} + 2199 T^{3} + 62628 T^{4} + 257805 T^{5} + 6346256 T^{6} + 15847093 T^{7} + 448993624 T^{8} + 394121817 T^{9} + 24578534949 T^{10} - 14898890565 T^{11} + 1174103421532 T^{12} - 1530957472620 T^{13} + 55182860812004 T^{14} - 32911649258085 T^{15} + 2551817234010027 T^{16} + 1923188742100377 T^{17} + 102974445839235368 T^{18} + 170819227785688597 T^{19} + 3215160017977036528 T^{20} + 6138668757835294605 T^{21} + 70088903269480091676 T^{22} + \)\(11\!\cdots\!51\)\( T^{23} + \)\(92\!\cdots\!25\)\( T^{24} + \)\(92\!\cdots\!28\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 - 32 T + 703 T^{2} - 11193 T^{3} + 150665 T^{4} - 1696185 T^{5} + 16968079 T^{6} - 148721494 T^{7} + 1192804159 T^{8} - 8603867786 T^{9} + 58893158913 T^{10} - 379379490929 T^{11} + 2523907028612 T^{12} - 17333659447546 T^{13} + 133767072516436 T^{14} - 1065676990019561 T^{15} + 8767836819490701 T^{16} - 67888655291945066 T^{17} + 498825323325455387 T^{18} - 3296316900660406726 T^{19} + 19932591422934263123 T^{20} - \)\(10\!\cdots\!85\)\( T^{21} + \)\(49\!\cdots\!45\)\( T^{22} - \)\(19\!\cdots\!57\)\( T^{23} + \)\(65\!\cdots\!91\)\( T^{24} - \)\(15\!\cdots\!12\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 - 12 T + 492 T^{2} - 5819 T^{3} + 123502 T^{4} - 1361556 T^{5} + 20687717 T^{6} - 206164863 T^{7} + 2540047898 T^{8} - 22702133281 T^{9} + 239075358515 T^{10} - 1918882189879 T^{11} + 17700735919831 T^{12} - 127370740797940 T^{13} + 1044343419270029 T^{14} - 6679628902968799 T^{15} + 49101058056452185 T^{16} - 275089944435991441 T^{17} + 1815941962904073502 T^{18} - 8696143939363656183 T^{19} + 51484517629565268223 T^{20} - \)\(19\!\cdots\!76\)\( T^{21} + \)\(10\!\cdots\!78\)\( T^{22} - \)\(29\!\cdots\!19\)\( T^{23} + \)\(14\!\cdots\!28\)\( T^{24} - \)\(21\!\cdots\!72\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 - 17 T + 395 T^{2} - 5414 T^{3} + 81146 T^{4} - 901986 T^{5} + 10902127 T^{6} - 106300462 T^{7} + 1104348671 T^{8} - 9720783669 T^{9} + 91269537043 T^{10} - 736154617740 T^{11} + 6374523735013 T^{12} - 48129864687696 T^{13} + 388845947835793 T^{14} - 2739231332610540 T^{15} + 20716450787557183 T^{16} - 134592425076370629 T^{17} + 932728802540865971 T^{18} - 5476639596987254782 T^{19} + 34262581526641116667 T^{20} - \)\(17\!\cdots\!66\)\( T^{21} + \)\(94\!\cdots\!86\)\( T^{22} - \)\(38\!\cdots\!14\)\( T^{23} + \)\(17\!\cdots\!95\)\( T^{24} - \)\(45\!\cdots\!57\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 - 20 T + 601 T^{2} - 9365 T^{3} + 164472 T^{4} - 2126310 T^{5} + 28028480 T^{6} - 311675821 T^{7} + 3396003920 T^{8} - 33461413283 T^{9} + 318622545371 T^{10} - 2859111247421 T^{11} + 24737883791037 T^{12} - 206153123607736 T^{13} + 1657438213999479 T^{14} - 12834550389672869 T^{15} + 95829872613418073 T^{16} - 674284987896740243 T^{17} + 4585030155862419440 T^{18} - 28193690528854835749 T^{19} + \)\(16\!\cdots\!40\)\( T^{20} - \)\(86\!\cdots\!10\)\( T^{21} + \)\(44\!\cdots\!84\)\( T^{22} - \)\(17\!\cdots\!85\)\( T^{23} + \)\(73\!\cdots\!83\)\( T^{24} - \)\(16\!\cdots\!20\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 16 T + 650 T^{2} - 8168 T^{3} + 187959 T^{4} - 1957904 T^{5} + 33394406 T^{6} - 299247255 T^{7} + 4217856563 T^{8} - 33459809651 T^{9} + 414713355965 T^{10} - 2988823921021 T^{11} + 33936031084590 T^{12} - 227145830014434 T^{13} + 2409458207005890 T^{14} - 15066661385866861 T^{15} + 148430471946789115 T^{16} - 850270009171933331 T^{17} + 7609980609272580613 T^{18} - 38333658328079886855 T^{19} + \)\(30\!\cdots\!46\)\( T^{20} - \)\(12\!\cdots\!44\)\( T^{21} + \)\(86\!\cdots\!29\)\( T^{22} - \)\(26\!\cdots\!68\)\( T^{23} + \)\(15\!\cdots\!50\)\( T^{24} - \)\(26\!\cdots\!56\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 46 T + 1720 T^{2} - 44305 T^{3} + 993024 T^{4} - 18358223 T^{5} + 305680033 T^{6} - 4459620559 T^{7} + 59766164049 T^{8} - 721821551394 T^{9} + 8101777460075 T^{10} - 83131105504269 T^{11} + 797678453714516 T^{12} - 7040793819274228 T^{13} + 58230527121159668 T^{14} - 443005661232249501 T^{15} + 3151729162185996275 T^{16} - 20498462375480697954 T^{17} + \)\(12\!\cdots\!57\)\( T^{18} - \)\(67\!\cdots\!51\)\( T^{19} + \)\(33\!\cdots\!01\)\( T^{20} - \)\(14\!\cdots\!63\)\( T^{21} + \)\(58\!\cdots\!12\)\( T^{22} - \)\(19\!\cdots\!45\)\( T^{23} + \)\(53\!\cdots\!40\)\( T^{24} - \)\(10\!\cdots\!66\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 - 17 T + 474 T^{2} - 5754 T^{3} + 109720 T^{4} - 1110821 T^{5} + 17405351 T^{6} - 152496831 T^{7} + 2129325399 T^{8} - 16573632681 T^{9} + 215111030146 T^{10} - 1525675616183 T^{11} + 18938845788835 T^{12} - 126205446995874 T^{13} + 1496168817317965 T^{14} - 9521741520598103 T^{15} + 106058127192153694 T^{16} - 645544335389197161 T^{17} + 6552054344546178201 T^{18} - 37070066622805953951 T^{19} + \)\(33\!\cdots\!09\)\( T^{20} - \)\(16\!\cdots\!81\)\( T^{21} + \)\(13\!\cdots\!80\)\( T^{22} - \)\(54\!\cdots\!54\)\( T^{23} + \)\(35\!\cdots\!46\)\( T^{24} - \)\(10\!\cdots\!97\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - 13 T + 525 T^{2} - 5210 T^{3} + 124484 T^{4} - 1002252 T^{5} + 19074120 T^{6} - 130370586 T^{7} + 2254421737 T^{8} - 13369356510 T^{9} + 224626910605 T^{10} - 1169892326755 T^{11} + 19915374871540 T^{12} - 96528686048908 T^{13} + 1652976114337820 T^{14} - 8059388239015195 T^{15} + 128438747334101135 T^{16} - 634487212815019710 T^{17} + 8880258848505656891 T^{18} - 42623408063175324234 T^{19} + \)\(51\!\cdots\!40\)\( T^{20} - \)\(22\!\cdots\!32\)\( T^{21} + \)\(23\!\cdots\!52\)\( T^{22} - \)\(80\!\cdots\!90\)\( T^{23} + \)\(67\!\cdots\!75\)\( T^{24} - \)\(13\!\cdots\!93\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 14 T + 893 T^{2} - 9976 T^{3} + 358111 T^{4} - 3212529 T^{5} + 86874731 T^{6} - 623018848 T^{7} + 14557512254 T^{8} - 83111773324 T^{9} + 1844464004430 T^{10} - 8575604633208 T^{11} + 190802854208908 T^{12} - 782594002264202 T^{13} + 16981454024592812 T^{14} - 67927364299640568 T^{15} + 1300289944739012670 T^{16} - 5214618911831779084 T^{17} + 81290013855881988046 T^{18} - \)\(30\!\cdots\!28\)\( T^{19} + \)\(38\!\cdots\!99\)\( T^{20} - \)\(12\!\cdots\!49\)\( T^{21} + \)\(12\!\cdots\!99\)\( T^{22} - \)\(31\!\cdots\!76\)\( T^{23} + \)\(24\!\cdots\!77\)\( T^{24} - \)\(34\!\cdots\!94\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 - 46 T + 1814 T^{2} - 48110 T^{3} + 1136778 T^{4} - 21857435 T^{5} + 384959219 T^{6} - 5892700340 T^{7} + 84305542616 T^{8} - 1084305973884 T^{9} + 13249273926983 T^{10} - 148620823050822 T^{11} + 1603545598731357 T^{12} - 16063777032627910 T^{13} + 155543923076941629 T^{14} - 1398373324085184198 T^{15} + 12092254582761355559 T^{16} - 95992828251955297404 T^{17} + \)\(72\!\cdots\!12\)\( T^{18} - \)\(49\!\cdots\!60\)\( T^{19} + \)\(31\!\cdots\!47\)\( T^{20} - \)\(17\!\cdots\!35\)\( T^{21} + \)\(86\!\cdots\!26\)\( T^{22} - \)\(35\!\cdots\!90\)\( T^{23} + \)\(12\!\cdots\!42\)\( T^{24} - \)\(31\!\cdots\!86\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
show more
show less