Properties

Label 8030.2.a.bf
Level 8030
Weight 2
Character orbit 8030.a
Self dual Yes
Analytic conductor 64.120
Analytic rank 1
Dimension 15
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(4\) \(x^{14}\mathstrut -\mathstrut \) \(23\) \(x^{13}\mathstrut +\mathstrut \) \(99\) \(x^{12}\mathstrut +\mathstrut \) \(191\) \(x^{11}\mathstrut -\mathstrut \) \(922\) \(x^{10}\mathstrut -\mathstrut \) \(702\) \(x^{9}\mathstrut +\mathstrut \) \(4108\) \(x^{8}\mathstrut +\mathstrut \) \(957\) \(x^{7}\mathstrut -\mathstrut \) \(8875\) \(x^{6}\mathstrut +\mathstrut \) \(479\) \(x^{5}\mathstrut +\mathstrut \) \(7698\) \(x^{4}\mathstrut -\mathstrut \) \(1731\) \(x^{3}\mathstrut -\mathstrut \) \(626\) \(x^{2}\mathstrut +\mathstrut \) \(46\) \(x\mathstrut +\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(366035\) \(\nu^{14}\mathstrut -\mathstrut \) \(1741279\) \(\nu^{13}\mathstrut -\mathstrut \) \(10239084\) \(\nu^{12}\mathstrut +\mathstrut \) \(56953737\) \(\nu^{11}\mathstrut +\mathstrut \) \(90005878\) \(\nu^{10}\mathstrut -\mathstrut \) \(692885868\) \(\nu^{9}\mathstrut -\mathstrut \) \(160464288\) \(\nu^{8}\mathstrut +\mathstrut \) \(3898870430\) \(\nu^{7}\mathstrut -\mathstrut \) \(1431593699\) \(\nu^{6}\mathstrut -\mathstrut \) \(10202941704\) \(\nu^{5}\mathstrut +\mathstrut \) \(6932586289\) \(\nu^{4}\mathstrut +\mathstrut \) \(9832804595\) \(\nu^{3}\mathstrut -\mathstrut \) \(8728032188\) \(\nu^{2}\mathstrut +\mathstrut \) \(521031646\) \(\nu\mathstrut +\mathstrut \) \(261843832\)\()/85835268\)
\(\beta_{4}\)\(=\)\((\)\(221319\) \(\nu^{14}\mathstrut -\mathstrut \) \(1050335\) \(\nu^{13}\mathstrut -\mathstrut \) \(1195170\) \(\nu^{12}\mathstrut +\mathstrut \) \(13572861\) \(\nu^{11}\mathstrut -\mathstrut \) \(36046464\) \(\nu^{10}\mathstrut +\mathstrut \) \(17357120\) \(\nu^{9}\mathstrut +\mathstrut \) \(359392846\) \(\nu^{8}\mathstrut -\mathstrut \) \(782126404\) \(\nu^{7}\mathstrut -\mathstrut \) \(979688875\) \(\nu^{6}\mathstrut +\mathstrut \) \(3235047750\) \(\nu^{5}\mathstrut +\mathstrut \) \(29989029\) \(\nu^{4}\mathstrut -\mathstrut \) \(3986272579\) \(\nu^{3}\mathstrut +\mathstrut \) \(2073107780\) \(\nu^{2}\mathstrut +\mathstrut \) \(49446534\) \(\nu\mathstrut -\mathstrut \) \(70626556\)\()/28611756\)
\(\beta_{5}\)\(=\)\((\)\(960341\) \(\nu^{14}\mathstrut -\mathstrut \) \(2079073\) \(\nu^{13}\mathstrut -\mathstrut \) \(24085422\) \(\nu^{12}\mathstrut +\mathstrut \) \(48433869\) \(\nu^{11}\mathstrut +\mathstrut \) \(218289340\) \(\nu^{10}\mathstrut -\mathstrut \) \(426649374\) \(\nu^{9}\mathstrut -\mathstrut \) \(810364518\) \(\nu^{8}\mathstrut +\mathstrut \) \(1857649106\) \(\nu^{7}\mathstrut +\mathstrut \) \(448142761\) \(\nu^{6}\mathstrut -\mathstrut \) \(4071404154\) \(\nu^{5}\mathstrut +\mathstrut \) \(4203159601\) \(\nu^{4}\mathstrut +\mathstrut \) \(3189119675\) \(\nu^{3}\mathstrut -\mathstrut \) \(7486177226\) \(\nu^{2}\mathstrut +\mathstrut \) \(1214106778\) \(\nu\mathstrut +\mathstrut \) \(498906736\)\()/85835268\)
\(\beta_{6}\)\(=\)\((\)\(1401323\) \(\nu^{14}\mathstrut -\mathstrut \) \(3706999\) \(\nu^{13}\mathstrut -\mathstrut \) \(32508126\) \(\nu^{12}\mathstrut +\mathstrut \) \(80618787\) \(\nu^{11}\mathstrut +\mathstrut \) \(269290000\) \(\nu^{10}\mathstrut -\mathstrut \) \(618823506\) \(\nu^{9}\mathstrut -\mathstrut \) \(949617366\) \(\nu^{8}\mathstrut +\mathstrut \) \(2084627546\) \(\nu^{7}\mathstrut +\mathstrut \) \(1045234987\) \(\nu^{6}\mathstrut -\mathstrut \) \(2989722630\) \(\nu^{5}\mathstrut +\mathstrut \) \(1208212627\) \(\nu^{4}\mathstrut +\mathstrut \) \(1426926773\) \(\nu^{3}\mathstrut -\mathstrut \) \(2405458214\) \(\nu^{2}\mathstrut -\mathstrut \) \(487849418\) \(\nu\mathstrut +\mathstrut \) \(246176608\)\()/85835268\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(3305617\) \(\nu^{14}\mathstrut +\mathstrut \) \(10474127\) \(\nu^{13}\mathstrut +\mathstrut \) \(80134722\) \(\nu^{12}\mathstrut -\mathstrut \) \(252534357\) \(\nu^{11}\mathstrut -\mathstrut \) \(718017836\) \(\nu^{10}\mathstrut +\mathstrut \) \(2266987038\) \(\nu^{9}\mathstrut +\mathstrut \) \(2954517636\) \(\nu^{8}\mathstrut -\mathstrut \) \(9607202860\) \(\nu^{7}\mathstrut -\mathstrut \) \(5141897405\) \(\nu^{6}\mathstrut +\mathstrut \) \(19320093438\) \(\nu^{5}\mathstrut +\mathstrut \) \(943404775\) \(\nu^{4}\mathstrut -\mathstrut \) \(14598872695\) \(\nu^{3}\mathstrut +\mathstrut \) \(4690178716\) \(\nu^{2}\mathstrut -\mathstrut \) \(529393910\) \(\nu\mathstrut +\mathstrut \) \(60835336\)\()/85835268\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1280809\) \(\nu^{14}\mathstrut +\mathstrut \) \(4426957\) \(\nu^{13}\mathstrut +\mathstrut \) \(33773790\) \(\nu^{12}\mathstrut -\mathstrut \) \(119493219\) \(\nu^{11}\mathstrut -\mathstrut \) \(333339020\) \(\nu^{10}\mathstrut +\mathstrut \) \(1227637336\) \(\nu^{9}\mathstrut +\mathstrut \) \(1518438146\) \(\nu^{8}\mathstrut -\mathstrut \) \(6030088596\) \(\nu^{7}\mathstrut -\mathstrut \) \(2987617231\) \(\nu^{6}\mathstrut +\mathstrut \) \(14236309866\) \(\nu^{5}\mathstrut +\mathstrut \) \(1004705713\) \(\nu^{4}\mathstrut -\mathstrut \) \(13138428591\) \(\nu^{3}\mathstrut +\mathstrut \) \(2615461640\) \(\nu^{2}\mathstrut +\mathstrut \) \(577080874\) \(\nu\mathstrut -\mathstrut \) \(18158464\)\()/28611756\)
\(\beta_{9}\)\(=\)\((\)\(2319073\) \(\nu^{14}\mathstrut -\mathstrut \) \(9327080\) \(\nu^{13}\mathstrut -\mathstrut \) \(53426214\) \(\nu^{12}\mathstrut +\mathstrut \) \(228472740\) \(\nu^{11}\mathstrut +\mathstrut \) \(453199208\) \(\nu^{10}\mathstrut -\mathstrut \) \(2100319149\) \(\nu^{9}\mathstrut -\mathstrut \) \(1804697133\) \(\nu^{8}\mathstrut +\mathstrut \) \(9216665638\) \(\nu^{7}\mathstrut +\mathstrut \) \(3306920312\) \(\nu^{6}\mathstrut -\mathstrut \) \(19523419710\) \(\nu^{5}\mathstrut -\mathstrut \) \(1610019898\) \(\nu^{4}\mathstrut +\mathstrut \) \(16363093441\) \(\nu^{3}\mathstrut -\mathstrut \) \(1640072710\) \(\nu^{2}\mathstrut -\mathstrut \) \(831757504\) \(\nu\mathstrut -\mathstrut \) \(66381772\)\()/42917634\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(5483185\) \(\nu^{14}\mathstrut +\mathstrut \) \(25183331\) \(\nu^{13}\mathstrut +\mathstrut \) \(106053138\) \(\nu^{12}\mathstrut -\mathstrut \) \(580526073\) \(\nu^{11}\mathstrut -\mathstrut \) \(625755272\) \(\nu^{10}\mathstrut +\mathstrut \) \(4911689490\) \(\nu^{9}\mathstrut +\mathstrut \) \(780653028\) \(\nu^{8}\mathstrut -\mathstrut \) \(19433892424\) \(\nu^{7}\mathstrut +\mathstrut \) \(4280176927\) \(\nu^{6}\mathstrut +\mathstrut \) \(36275272134\) \(\nu^{5}\mathstrut -\mathstrut \) \(14021936813\) \(\nu^{4}\mathstrut -\mathstrut \) \(25646611771\) \(\nu^{3}\mathstrut +\mathstrut \) \(11525790424\) \(\nu^{2}\mathstrut -\mathstrut \) \(141037466\) \(\nu\mathstrut -\mathstrut \) \(214919840\)\()/85835268\)
\(\beta_{11}\)\(=\)\((\)\(1064697\) \(\nu^{14}\mathstrut -\mathstrut \) \(3945448\) \(\nu^{13}\mathstrut -\mathstrut \) \(25941384\) \(\nu^{12}\mathstrut +\mathstrut \) \(100332648\) \(\nu^{11}\mathstrut +\mathstrut \) \(233803794\) \(\nu^{10}\mathstrut -\mathstrut \) \(966740297\) \(\nu^{9}\mathstrut -\mathstrut \) \(965567359\) \(\nu^{8}\mathstrut +\mathstrut \) \(4476683578\) \(\nu^{7}\mathstrut +\mathstrut \) \(1656705742\) \(\nu^{6}\mathstrut -\mathstrut \) \(10082589522\) \(\nu^{5}\mathstrut -\mathstrut \) \(83605584\) \(\nu^{4}\mathstrut +\mathstrut \) \(9118578187\) \(\nu^{3}\mathstrut -\mathstrut \) \(2014418084\) \(\nu^{2}\mathstrut -\mathstrut \) \(730702446\) \(\nu\mathstrut +\mathstrut \) \(23240302\)\()/14305878\)
\(\beta_{12}\)\(=\)\((\)\(7527395\) \(\nu^{14}\mathstrut -\mathstrut \) \(29036557\) \(\nu^{13}\mathstrut -\mathstrut \) \(178480998\) \(\nu^{12}\mathstrut +\mathstrut \) \(733439763\) \(\nu^{11}\mathstrut +\mathstrut \) \(1531273792\) \(\nu^{10}\mathstrut -\mathstrut \) \(6991802706\) \(\nu^{9}\mathstrut -\mathstrut \) \(5708071680\) \(\nu^{8}\mathstrut +\mathstrut \) \(31839705608\) \(\nu^{7}\mathstrut +\mathstrut \) \(6904612423\) \(\nu^{6}\mathstrut -\mathstrut \) \(69771551034\) \(\nu^{5}\mathstrut +\mathstrut \) \(9165309703\) \(\nu^{4}\mathstrut +\mathstrut \) \(59543491481\) \(\nu^{3}\mathstrut -\mathstrut \) \(22043262632\) \(\nu^{2}\mathstrut -\mathstrut \) \(1450048838\) \(\nu\mathstrut +\mathstrut \) \(771923284\)\()/85835268\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(438707\) \(\nu^{14}\mathstrut +\mathstrut \) \(1765566\) \(\nu^{13}\mathstrut +\mathstrut \) \(10104105\) \(\nu^{12}\mathstrut -\mathstrut \) \(43697817\) \(\nu^{11}\mathstrut -\mathstrut \) \(84343819\) \(\nu^{10}\mathstrut +\mathstrut \) \(406761742\) \(\nu^{9}\mathstrut +\mathstrut \) \(315281564\) \(\nu^{8}\mathstrut -\mathstrut \) \(1809931594\) \(\nu^{7}\mathstrut -\mathstrut \) \(460844871\) \(\nu^{6}\mathstrut +\mathstrut \) \(3901113249\) \(\nu^{5}\mathstrut -\mathstrut \) \(112742041\) \(\nu^{4}\mathstrut -\mathstrut \) \(3368581534\) \(\nu^{3}\mathstrut +\mathstrut \) \(677370717\) \(\nu^{2}\mathstrut +\mathstrut \) \(249964550\) \(\nu\mathstrut -\mathstrut \) \(15724668\)\()/4768626\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(3990373\) \(\nu^{14}\mathstrut +\mathstrut \) \(16003502\) \(\nu^{13}\mathstrut +\mathstrut \) \(91665171\) \(\nu^{12}\mathstrut -\mathstrut \) \(395775153\) \(\nu^{11}\mathstrut -\mathstrut \) \(759666443\) \(\nu^{10}\mathstrut +\mathstrut \) \(3679695978\) \(\nu^{9}\mathstrut +\mathstrut \) \(2782401546\) \(\nu^{8}\mathstrut -\mathstrut \) \(16337319754\) \(\nu^{7}\mathstrut -\mathstrut \) \(3770851553\) \(\nu^{6}\mathstrut +\mathstrut \) \(35037808089\) \(\nu^{5}\mathstrut -\mathstrut \) \(1839177515\) \(\nu^{4}\mathstrut -\mathstrut \) \(29893392208\) \(\nu^{3}\mathstrut +\mathstrut \) \(6483494065\) \(\nu^{2}\mathstrut +\mathstrut \) \(2067898252\) \(\nu\mathstrut -\mathstrut \) \(1774460\)\()/42917634\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)
\(\nu^{5}\)\(=\)\(-\)\(14\) \(\beta_{14}\mathstrut -\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut -\mathstrut \) \(4\) \(\beta_{11}\mathstrut +\mathstrut \) \(13\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(28\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut -\mathstrut \) \(17\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(44\) \(\beta_{1}\mathstrut +\mathstrut \) \(30\)
\(\nu^{6}\)\(=\)\(-\)\(7\) \(\beta_{14}\mathstrut -\mathstrut \) \(5\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut -\mathstrut \) \(23\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{9}\mathstrut -\mathstrut \) \(20\) \(\beta_{8}\mathstrut +\mathstrut \) \(31\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut +\mathstrut \) \(24\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(40\) \(\beta_{3}\mathstrut +\mathstrut \) \(109\) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(240\)
\(\nu^{7}\)\(=\)\(-\)\(163\) \(\beta_{14}\mathstrut -\mathstrut \) \(60\) \(\beta_{13}\mathstrut +\mathstrut \) \(141\) \(\beta_{12}\mathstrut -\mathstrut \) \(79\) \(\beta_{11}\mathstrut +\mathstrut \) \(137\) \(\beta_{10}\mathstrut -\mathstrut \) \(208\) \(\beta_{9}\mathstrut +\mathstrut \) \(79\) \(\beta_{8}\mathstrut +\mathstrut \) \(26\) \(\beta_{7}\mathstrut -\mathstrut \) \(315\) \(\beta_{6}\mathstrut +\mathstrut \) \(222\) \(\beta_{5}\mathstrut +\mathstrut \) \(133\) \(\beta_{4}\mathstrut -\mathstrut \) \(235\) \(\beta_{3}\mathstrut +\mathstrut \) \(248\) \(\beta_{2}\mathstrut +\mathstrut \) \(373\) \(\beta_{1}\mathstrut +\mathstrut \) \(359\)
\(\nu^{8}\)\(=\)\(-\)\(147\) \(\beta_{14}\mathstrut -\mathstrut \) \(110\) \(\beta_{13}\mathstrut +\mathstrut \) \(151\) \(\beta_{12}\mathstrut -\mathstrut \) \(348\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(213\) \(\beta_{9}\mathstrut -\mathstrut \) \(286\) \(\beta_{8}\mathstrut +\mathstrut \) \(385\) \(\beta_{7}\mathstrut +\mathstrut \) \(57\) \(\beta_{6}\mathstrut +\mathstrut \) \(382\) \(\beta_{5}\mathstrut -\mathstrut \) \(44\) \(\beta_{4}\mathstrut -\mathstrut \) \(607\) \(\beta_{3}\mathstrut +\mathstrut \) \(1224\) \(\beta_{2}\mathstrut +\mathstrut \) \(318\) \(\beta_{1}\mathstrut +\mathstrut \) \(2305\)
\(\nu^{9}\)\(=\)\(-\)\(1806\) \(\beta_{14}\mathstrut -\mathstrut \) \(849\) \(\beta_{13}\mathstrut +\mathstrut \) \(1479\) \(\beta_{12}\mathstrut -\mathstrut \) \(1140\) \(\beta_{11}\mathstrut +\mathstrut \) \(1356\) \(\beta_{10}\mathstrut -\mathstrut \) \(2511\) \(\beta_{9}\mathstrut +\mathstrut \) \(527\) \(\beta_{8}\mathstrut +\mathstrut \) \(443\) \(\beta_{7}\mathstrut -\mathstrut \) \(3345\) \(\beta_{6}\mathstrut +\mathstrut \) \(2652\) \(\beta_{5}\mathstrut +\mathstrut \) \(1267\) \(\beta_{4}\mathstrut -\mathstrut \) \(2999\) \(\beta_{3}\mathstrut +\mathstrut \) \(3110\) \(\beta_{2}\mathstrut +\mathstrut \) \(3438\) \(\beta_{1}\mathstrut +\mathstrut \) \(4104\)
\(\nu^{10}\)\(=\)\(-\)\(2195\) \(\beta_{14}\mathstrut -\mathstrut \) \(1700\) \(\beta_{13}\mathstrut +\mathstrut \) \(1761\) \(\beta_{12}\mathstrut -\mathstrut \) \(4531\) \(\beta_{11}\mathstrut -\mathstrut \) \(57\) \(\beta_{10}\mathstrut -\mathstrut \) \(3498\) \(\beta_{9}\mathstrut -\mathstrut \) \(3640\) \(\beta_{8}\mathstrut +\mathstrut \) \(4476\) \(\beta_{7}\mathstrut -\mathstrut \) \(56\) \(\beta_{6}\mathstrut +\mathstrut \) \(5203\) \(\beta_{5}\mathstrut -\mathstrut \) \(664\) \(\beta_{4}\mathstrut -\mathstrut \) \(8221\) \(\beta_{3}\mathstrut +\mathstrut \) \(13832\) \(\beta_{2}\mathstrut +\mathstrut \) \(3466\) \(\beta_{1}\mathstrut +\mathstrut \) \(23478\)
\(\nu^{11}\)\(=\)\(-\)\(19682\) \(\beta_{14}\mathstrut -\mathstrut \) \(10588\) \(\beta_{13}\mathstrut +\mathstrut \) \(15478\) \(\beta_{12}\mathstrut -\mathstrut \) \(14625\) \(\beta_{11}\mathstrut +\mathstrut \) \(13103\) \(\beta_{10}\mathstrut -\mathstrut \) \(29304\) \(\beta_{9}\mathstrut +\mathstrut \) \(2483\) \(\beta_{8}\mathstrut +\mathstrut \) \(6410\) \(\beta_{7}\mathstrut -\mathstrut \) \(34927\) \(\beta_{6}\mathstrut +\mathstrut \) \(30511\) \(\beta_{5}\mathstrut +\mathstrut \) \(11740\) \(\beta_{4}\mathstrut -\mathstrut \) \(36688\) \(\beta_{3}\mathstrut +\mathstrut \) \(37382\) \(\beta_{2}\mathstrut +\mathstrut \) \(33217\) \(\beta_{1}\mathstrut +\mathstrut \) \(46607\)
\(\nu^{12}\)\(=\)\(-\)\(28912\) \(\beta_{14}\mathstrut -\mathstrut \) \(22917\) \(\beta_{13}\mathstrut +\mathstrut \) \(20741\) \(\beta_{12}\mathstrut -\mathstrut \) \(55160\) \(\beta_{11}\mathstrut -\mathstrut \) \(1004\) \(\beta_{10}\mathstrut -\mathstrut \) \(49590\) \(\beta_{9}\mathstrut -\mathstrut \) \(43920\) \(\beta_{8}\mathstrut +\mathstrut \) \(50750\) \(\beta_{7}\mathstrut -\mathstrut \) \(7934\) \(\beta_{6}\mathstrut +\mathstrut \) \(65839\) \(\beta_{5}\mathstrut -\mathstrut \) \(8623\) \(\beta_{4}\mathstrut -\mathstrut \) \(104759\) \(\beta_{3}\mathstrut +\mathstrut \) \(156247\) \(\beta_{2}\mathstrut +\mathstrut \) \(37881\) \(\beta_{1}\mathstrut +\mathstrut \) \(246608\)
\(\nu^{13}\)\(=\)\(-\)\(213402\) \(\beta_{14}\mathstrut -\mathstrut \) \(124866\) \(\beta_{13}\mathstrut +\mathstrut \) \(162850\) \(\beta_{12}\mathstrut -\mathstrut \) \(177555\) \(\beta_{11}\mathstrut +\mathstrut \) \(125423\) \(\beta_{10}\mathstrut -\mathstrut \) \(336065\) \(\beta_{9}\mathstrut -\mathstrut \) \(3879\) \(\beta_{8}\mathstrut +\mathstrut \) \(85351\) \(\beta_{7}\mathstrut -\mathstrut \) \(363634\) \(\beta_{6}\mathstrut +\mathstrut \) \(345042\) \(\beta_{5}\mathstrut +\mathstrut \) \(107373\) \(\beta_{4}\mathstrut -\mathstrut \) \(437482\) \(\beta_{3}\mathstrut +\mathstrut \) \(439621\) \(\beta_{2}\mathstrut +\mathstrut \) \(330209\) \(\beta_{1}\mathstrut +\mathstrut \) \(530114\)
\(\nu^{14}\)\(=\)\(-\)\(359398\) \(\beta_{14}\mathstrut -\mathstrut \) \(289003\) \(\beta_{13}\mathstrut +\mathstrut \) \(244941\) \(\beta_{12}\mathstrut -\mathstrut \) \(649452\) \(\beta_{11}\mathstrut -\mathstrut \) \(14003\) \(\beta_{10}\mathstrut -\mathstrut \) \(651509\) \(\beta_{9}\mathstrut -\mathstrut \) \(514992\) \(\beta_{8}\mathstrut +\mathstrut \) \(569310\) \(\beta_{7}\mathstrut -\mathstrut \) \(161754\) \(\beta_{6}\mathstrut +\mathstrut \) \(801163\) \(\beta_{5}\mathstrut -\mathstrut \) \(103760\) \(\beta_{4}\mathstrut -\mathstrut \) \(1286815\) \(\beta_{3}\mathstrut +\mathstrut \) \(1761997\) \(\beta_{2}\mathstrut +\mathstrut \) \(419582\) \(\beta_{1}\mathstrut +\mathstrut \) \(2635917\)

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
3.35557
2.96469
2.44568
2.39402
1.82644
1.49088
0.440451
0.142263
−0.0958822
−0.205398
−1.67023
−1.82374
−1.99698
−2.16715
−3.10061
1.00000 −3.35557 1.00000 −1.00000 −3.35557 3.99638 1.00000 8.25982 −1.00000
1.2 1.00000 −2.96469 1.00000 −1.00000 −2.96469 −3.63894 1.00000 5.78937 −1.00000
1.3 1.00000 −2.44568 1.00000 −1.00000 −2.44568 −0.745785 1.00000 2.98137 −1.00000
1.4 1.00000 −2.39402 1.00000 −1.00000 −2.39402 −0.390362 1.00000 2.73133 −1.00000
1.5 1.00000 −1.82644 1.00000 −1.00000 −1.82644 3.00453 1.00000 0.335869 −1.00000
1.6 1.00000 −1.49088 1.00000 −1.00000 −1.49088 −3.98101 1.00000 −0.777287 −1.00000
1.7 1.00000 −0.440451 1.00000 −1.00000 −0.440451 3.12770 1.00000 −2.80600 −1.00000
1.8 1.00000 −0.142263 1.00000 −1.00000 −0.142263 −2.21205 1.00000 −2.97976 −1.00000
1.9 1.00000 0.0958822 1.00000 −1.00000 0.0958822 −1.44050 1.00000 −2.99081 −1.00000
1.10 1.00000 0.205398 1.00000 −1.00000 0.205398 3.29352 1.00000 −2.95781 −1.00000
1.11 1.00000 1.67023 1.00000 −1.00000 1.67023 1.44867 1.00000 −0.210343 −1.00000
1.12 1.00000 1.82374 1.00000 −1.00000 1.82374 −3.76421 1.00000 0.326027 −1.00000
1.13 1.00000 1.99698 1.00000 −1.00000 1.99698 0.970755 1.00000 0.987937 −1.00000
1.14 1.00000 2.16715 1.00000 −1.00000 2.16715 −3.23897 1.00000 1.69652 −1.00000
1.15 1.00000 3.10061 1.00000 −1.00000 3.10061 −2.42975 1.00000 6.61377 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)
\(73\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\(T_{3}^{15} + \cdots\)
\(T_{7}^{15} + \cdots\)