Properties

Label 8034.2.a.bb
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 1
Dimension 14
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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(6\) \(x^{13}\mathstrut -\mathstrut \) \(29\) \(x^{12}\mathstrut +\mathstrut \) \(207\) \(x^{11}\mathstrut +\mathstrut \) \(269\) \(x^{10}\mathstrut -\mathstrut \) \(2601\) \(x^{9}\mathstrut -\mathstrut \) \(847\) \(x^{8}\mathstrut +\mathstrut \) \(14851\) \(x^{7}\mathstrut +\mathstrut \) \(678\) \(x^{6}\mathstrut -\mathstrut \) \(39390\) \(x^{5}\mathstrut -\mathstrut \) \(3280\) \(x^{4}\mathstrut +\mathstrut \) \(42456\) \(x^{3}\mathstrut +\mathstrut \) \(10816\) \(x^{2}\mathstrut -\mathstrut \) \(7296\) \(x\mathstrut -\mathstrut \) \(2048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(17211124339\) \(\nu^{13}\mathstrut +\mathstrut \) \(48838930211\) \(\nu^{12}\mathstrut -\mathstrut \) \(1083388884247\) \(\nu^{11}\mathstrut -\mathstrut \) \(1278700159484\) \(\nu^{10}\mathstrut +\mathstrut \) \(21896289725388\) \(\nu^{9}\mathstrut +\mathstrut \) \(8353374678764\) \(\nu^{8}\mathstrut -\mathstrut \) \(175798632679800\) \(\nu^{7}\mathstrut +\mathstrut \) \(21533806125636\) \(\nu^{6}\mathstrut +\mathstrut \) \(467406199485803\) \(\nu^{5}\mathstrut -\mathstrut \) \(351416224121502\) \(\nu^{4}\mathstrut +\mathstrut \) \(243900222298282\) \(\nu^{3}\mathstrut +\mathstrut \) \(769318123450260\) \(\nu^{2}\mathstrut -\mathstrut \) \(1166460932605168\) \(\nu\mathstrut +\mathstrut \) \(118321024199296\)\()/\)\(165587393151104\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(88374380939\) \(\nu^{13}\mathstrut +\mathstrut \) \(341475795883\) \(\nu^{12}\mathstrut +\mathstrut \) \(3738713297463\) \(\nu^{11}\mathstrut -\mathstrut \) \(13680291423778\) \(\nu^{10}\mathstrut -\mathstrut \) \(60950374089966\) \(\nu^{9}\mathstrut +\mathstrut \) \(203902612897310\) \(\nu^{8}\mathstrut +\mathstrut \) \(480284475172790\) \(\nu^{7}\mathstrut -\mathstrut \) \(1384162777559138\) \(\nu^{6}\mathstrut -\mathstrut \) \(1866871026854413\) \(\nu^{5}\mathstrut +\mathstrut \) \(4166130863706398\) \(\nu^{4}\mathstrut +\mathstrut \) \(3191603596475186\) \(\nu^{3}\mathstrut -\mathstrut \) \(4474034344329852\) \(\nu^{2}\mathstrut -\mathstrut \) \(1493240738430736\) \(\nu\mathstrut +\mathstrut \) \(823088103896320\)\()/\)\(165587393151104\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(406907271083\) \(\nu^{13}\mathstrut +\mathstrut \) \(2304692170158\) \(\nu^{12}\mathstrut +\mathstrut \) \(11004973389543\) \(\nu^{11}\mathstrut -\mathstrut \) \(75016193619169\) \(\nu^{10}\mathstrut -\mathstrut \) \(82526311620203\) \(\nu^{9}\mathstrut +\mathstrut \) \(861495599563087\) \(\nu^{8}\mathstrut +\mathstrut \) \(21324319463753\) \(\nu^{7}\mathstrut -\mathstrut \) \(4294335680262885\) \(\nu^{6}\mathstrut +\mathstrut \) \(1434806545518450\) \(\nu^{5}\mathstrut +\mathstrut \) \(9639491681032898\) \(\nu^{4}\mathstrut -\mathstrut \) \(3273111940991512\) \(\nu^{3}\mathstrut -\mathstrut \) \(10523459905050216\) \(\nu^{2}\mathstrut +\mathstrut \) \(2930798928626144\) \(\nu\mathstrut +\mathstrut \) \(3910063812049920\)\()/\)\(662349572604416\)
\(\beta_{5}\)\(=\)\((\)\(113401884803\) \(\nu^{13}\mathstrut -\mathstrut \) \(883549115785\) \(\nu^{12}\mathstrut -\mathstrut \) \(2323893522087\) \(\nu^{11}\mathstrut +\mathstrut \) \(29836393456848\) \(\nu^{10}\mathstrut -\mathstrut \) \(852019922224\) \(\nu^{9}\mathstrut -\mathstrut \) \(364296872907848\) \(\nu^{8}\mathstrut +\mathstrut \) \(263847579785132\) \(\nu^{7}\mathstrut +\mathstrut \) \(2001792019065248\) \(\nu^{6}\mathstrut -\mathstrut \) \(1727833814125777\) \(\nu^{5}\mathstrut -\mathstrut \) \(5028642484757934\) \(\nu^{4}\mathstrut +\mathstrut \) \(3738614769502962\) \(\nu^{3}\mathstrut +\mathstrut \) \(5009595167008196\) \(\nu^{2}\mathstrut -\mathstrut \) \(2681914257270768\) \(\nu\mathstrut -\mathstrut \) \(1049016746216704\)\()/\)\(165587393151104\)
\(\beta_{6}\)\(=\)\((\)\(736531555801\) \(\nu^{13}\mathstrut -\mathstrut \) \(3946026327910\) \(\nu^{12}\mathstrut -\mathstrut \) \(23999796454509\) \(\nu^{11}\mathstrut +\mathstrut \) \(138185242548231\) \(\nu^{10}\mathstrut +\mathstrut \) \(283131019369069\) \(\nu^{9}\mathstrut -\mathstrut \) \(1765360355157129\) \(\nu^{8}\mathstrut -\mathstrut \) \(1550184318259343\) \(\nu^{7}\mathstrut +\mathstrut \) \(10222081364164867\) \(\nu^{6}\mathstrut +\mathstrut \) \(4328705466068094\) \(\nu^{5}\mathstrut -\mathstrut \) \(27089707111959846\) \(\nu^{4}\mathstrut -\mathstrut \) \(6353469675911824\) \(\nu^{3}\mathstrut +\mathstrut \) \(27368166174349864\) \(\nu^{2}\mathstrut +\mathstrut \) \(3859293469715616\) \(\nu\mathstrut -\mathstrut \) \(3240111054661120\)\()/\)\(662349572604416\)
\(\beta_{7}\)\(=\)\((\)\(1012895827421\) \(\nu^{13}\mathstrut -\mathstrut \) \(5401357076630\) \(\nu^{12}\mathstrut -\mathstrut \) \(30269645318433\) \(\nu^{11}\mathstrut +\mathstrut \) \(182412406997691\) \(\nu^{10}\mathstrut +\mathstrut \) \(294059953736009\) \(\nu^{9}\mathstrut -\mathstrut \) \(2216963035494389\) \(\nu^{8}\mathstrut -\mathstrut \) \(971532240127523\) \(\nu^{7}\mathstrut +\mathstrut \) \(11978958293225447\) \(\nu^{6}\mathstrut +\mathstrut \) \(149793013139294\) \(\nu^{5}\mathstrut -\mathstrut \) \(28653365696975870\) \(\nu^{4}\mathstrut +\mathstrut \) \(2303549139588352\) \(\nu^{3}\mathstrut +\mathstrut \) \(24319897537607016\) \(\nu^{2}\mathstrut -\mathstrut \) \(796238968943584\) \(\nu\mathstrut -\mathstrut \) \(804113198053376\)\()/\)\(662349572604416\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(161946406606\) \(\nu^{13}\mathstrut +\mathstrut \) \(993100465781\) \(\nu^{12}\mathstrut +\mathstrut \) \(4568699998010\) \(\nu^{11}\mathstrut -\mathstrut \) \(33836085473159\) \(\nu^{10}\mathstrut -\mathstrut \) \(39248816007673\) \(\nu^{9}\mathstrut +\mathstrut \) \(415409487608205\) \(\nu^{8}\mathstrut +\mathstrut \) \(80934994695967\) \(\nu^{7}\mathstrut -\mathstrut \) \(2274108863937751\) \(\nu^{6}\mathstrut +\mathstrut \) \(260702734356941\) \(\nu^{5}\mathstrut +\mathstrut \) \(5633078350696404\) \(\nu^{4}\mathstrut -\mathstrut \) \(672606967463486\) \(\nu^{3}\mathstrut -\mathstrut \) \(5548224775659980\) \(\nu^{2}\mathstrut -\mathstrut \) \(339551833683136\) \(\nu\mathstrut +\mathstrut \) \(821129125068864\)\()/\)\(82793696575552\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(339042867519\) \(\nu^{13}\mathstrut +\mathstrut \) \(2193728871065\) \(\nu^{12}\mathstrut +\mathstrut \) \(8513285647275\) \(\nu^{11}\mathstrut -\mathstrut \) \(73385354065404\) \(\nu^{10}\mathstrut -\mathstrut \) \(47754006020300\) \(\nu^{9}\mathstrut +\mathstrut \) \(882287181466036\) \(\nu^{8}\mathstrut -\mathstrut \) \(215439634992760\) \(\nu^{7}\mathstrut -\mathstrut \) \(4744161357889636\) \(\nu^{6}\mathstrut +\mathstrut \) \(2237814174755737\) \(\nu^{5}\mathstrut +\mathstrut \) \(11755366222327694\) \(\nu^{4}\mathstrut -\mathstrut \) \(3969078366908066\) \(\nu^{3}\mathstrut -\mathstrut \) \(12224135566363796\) \(\nu^{2}\mathstrut +\mathstrut \) \(525183514160720\) \(\nu\mathstrut +\mathstrut \) \(2100534074540032\)\()/\)\(165587393151104\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(2206595272859\) \(\nu^{13}\mathstrut +\mathstrut \) \(14412109193718\) \(\nu^{12}\mathstrut +\mathstrut \) \(54849889967511\) \(\nu^{11}\mathstrut -\mathstrut \) \(474790047743417\) \(\nu^{10}\mathstrut -\mathstrut \) \(307896247879923\) \(\nu^{9}\mathstrut +\mathstrut \) \(5544743002953591\) \(\nu^{8}\mathstrut -\mathstrut \) \(1212936140282751\) \(\nu^{7}\mathstrut -\mathstrut \) \(28137851717749085\) \(\nu^{6}\mathstrut +\mathstrut \) \(12427081549409178\) \(\nu^{5}\mathstrut +\mathstrut \) \(62275142320830146\) \(\nu^{4}\mathstrut -\mathstrut \) \(19724650983886152\) \(\nu^{3}\mathstrut -\mathstrut \) \(53935123111369352\) \(\nu^{2}\mathstrut +\mathstrut \) \(943866195323488\) \(\nu\mathstrut +\mathstrut \) \(6787275004993536\)\()/\)\(662349572604416\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(2312493819663\) \(\nu^{13}\mathstrut +\mathstrut \) \(15946662074178\) \(\nu^{12}\mathstrut +\mathstrut \) \(56053557441147\) \(\nu^{11}\mathstrut -\mathstrut \) \(536833570831273\) \(\nu^{10}\mathstrut -\mathstrut \) \(265156769412035\) \(\nu^{9}\mathstrut +\mathstrut \) \(6499156388583719\) \(\nu^{8}\mathstrut -\mathstrut \) \(2119780149955167\) \(\nu^{7}\mathstrut -\mathstrut \) \(35077355093355085\) \(\nu^{6}\mathstrut +\mathstrut \) \(18558566709942790\) \(\nu^{5}\mathstrut +\mathstrut \) \(85869291262886602\) \(\nu^{4}\mathstrut -\mathstrut \) \(35303846107753504\) \(\nu^{3}\mathstrut -\mathstrut \) \(83761116115370808\) \(\nu^{2}\mathstrut +\mathstrut \) \(9764388104189088\) \(\nu\mathstrut +\mathstrut \) \(10136073926334976\)\()/\)\(662349572604416\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(155979666211\) \(\nu^{13}\mathstrut +\mathstrut \) \(1092105641867\) \(\nu^{12}\mathstrut +\mathstrut \) \(3542117432817\) \(\nu^{11}\mathstrut -\mathstrut \) \(36139437565114\) \(\nu^{10}\mathstrut -\mathstrut \) \(10207803685344\) \(\nu^{9}\mathstrut +\mathstrut \) \(425935692445960\) \(\nu^{8}\mathstrut -\mathstrut \) \(227141704110620\) \(\nu^{7}\mathstrut -\mathstrut \) \(2204586185386584\) \(\nu^{6}\mathstrut +\mathstrut \) \(1611592294003569\) \(\nu^{5}\mathstrut +\mathstrut \) \(5085371693596568\) \(\nu^{4}\mathstrut -\mathstrut \) \(2867839136841206\) \(\nu^{3}\mathstrut -\mathstrut \) \(4698153606819416\) \(\nu^{2}\mathstrut +\mathstrut \) \(774956085111656\) \(\nu\mathstrut +\mathstrut \) \(564375014073344\)\()/\)\(41396848287776\)
\(\beta_{13}\)\(=\)\((\)\(2734835840775\) \(\nu^{13}\mathstrut -\mathstrut \) \(17117209536866\) \(\nu^{12}\mathstrut -\mathstrut \) \(75341966167987\) \(\nu^{11}\mathstrut +\mathstrut \) \(586439935888449\) \(\nu^{10}\mathstrut +\mathstrut \) \(596543424673451\) \(\nu^{9}\mathstrut -\mathstrut \) \(7281353341457903\) \(\nu^{8}\mathstrut -\mathstrut \) \(504552281455193\) \(\nu^{7}\mathstrut +\mathstrut \) \(40700141428094181\) \(\nu^{6}\mathstrut -\mathstrut \) \(9221457804581926\) \(\nu^{5}\mathstrut -\mathstrut \) \(103939479614198202\) \(\nu^{4}\mathstrut +\mathstrut \) \(23513032611045888\) \(\nu^{3}\mathstrut +\mathstrut \) \(105396875559095672\) \(\nu^{2}\mathstrut -\mathstrut \) \(9119618453491232\) \(\nu\mathstrut -\mathstrut \) \(17591589253353984\)\()/\)\(662349572604416\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{4}\)\(=\)\(13\) \(\beta_{13}\mathstrut -\mathstrut \) \(4\) \(\beta_{12}\mathstrut +\mathstrut \) \(19\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(88\)
\(\nu^{5}\)\(=\)\(26\) \(\beta_{13}\mathstrut -\mathstrut \) \(37\) \(\beta_{12}\mathstrut +\mathstrut \) \(65\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(18\) \(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(33\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(31\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(148\)
\(\nu^{6}\)\(=\)\(167\) \(\beta_{13}\mathstrut -\mathstrut \) \(103\) \(\beta_{12}\mathstrut +\mathstrut \) \(335\) \(\beta_{11}\mathstrut +\mathstrut \) \(18\) \(\beta_{10}\mathstrut -\mathstrut \) \(41\) \(\beta_{9}\mathstrut -\mathstrut \) \(37\) \(\beta_{8}\mathstrut -\mathstrut \) \(17\) \(\beta_{7}\mathstrut +\mathstrut \) \(111\) \(\beta_{6}\mathstrut +\mathstrut \) \(113\) \(\beta_{5}\mathstrut +\mathstrut \) \(108\) \(\beta_{4}\mathstrut +\mathstrut \) \(177\) \(\beta_{3}\mathstrut -\mathstrut \) \(177\) \(\beta_{2}\mathstrut +\mathstrut \) \(276\) \(\beta_{1}\mathstrut +\mathstrut \) \(1263\)
\(\nu^{7}\)\(=\)\(479\) \(\beta_{13}\mathstrut -\mathstrut \) \(611\) \(\beta_{12}\mathstrut +\mathstrut \) \(1200\) \(\beta_{11}\mathstrut -\mathstrut \) \(93\) \(\beta_{10}\mathstrut -\mathstrut \) \(162\) \(\beta_{9}\mathstrut +\mathstrut \) \(242\) \(\beta_{8}\mathstrut -\mathstrut \) \(350\) \(\beta_{7}\mathstrut +\mathstrut \) \(483\) \(\beta_{6}\mathstrut +\mathstrut \) \(315\) \(\beta_{5}\mathstrut +\mathstrut \) \(679\) \(\beta_{4}\mathstrut +\mathstrut \) \(107\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(1682\) \(\beta_{1}\mathstrut +\mathstrut \) \(2702\)
\(\nu^{8}\)\(=\)\(2222\) \(\beta_{13}\mathstrut -\mathstrut \) \(2060\) \(\beta_{12}\mathstrut +\mathstrut \) \(5775\) \(\beta_{11}\mathstrut +\mathstrut \) \(203\) \(\beta_{10}\mathstrut -\mathstrut \) \(1006\) \(\beta_{9}\mathstrut -\mathstrut \) \(885\) \(\beta_{8}\mathstrut -\mathstrut \) \(542\) \(\beta_{7}\mathstrut +\mathstrut \) \(1884\) \(\beta_{6}\mathstrut +\mathstrut \) \(2415\) \(\beta_{5}\mathstrut +\mathstrut \) \(2350\) \(\beta_{4}\mathstrut +\mathstrut \) \(2137\) \(\beta_{3}\mathstrut -\mathstrut \) \(2159\) \(\beta_{2}\mathstrut +\mathstrut \) \(4443\) \(\beta_{1}\mathstrut +\mathstrut \) \(19087\)
\(\nu^{9}\)\(=\)\(7842\) \(\beta_{13}\mathstrut -\mathstrut \) \(10023\) \(\beta_{12}\mathstrut +\mathstrut \) \(21287\) \(\beta_{11}\mathstrut -\mathstrut \) \(2124\) \(\beta_{10}\mathstrut -\mathstrut \) \(3977\) \(\beta_{9}\mathstrut +\mathstrut \) \(2676\) \(\beta_{8}\mathstrut -\mathstrut \) \(5991\) \(\beta_{7}\mathstrut +\mathstrut \) \(6985\) \(\beta_{6}\mathstrut +\mathstrut \) \(6668\) \(\beta_{5}\mathstrut +\mathstrut \) \(13401\) \(\beta_{4}\mathstrut +\mathstrut \) \(1459\) \(\beta_{3}\mathstrut +\mathstrut \) \(285\) \(\beta_{2}\mathstrut +\mathstrut \) \(24370\) \(\beta_{1}\mathstrut +\mathstrut \) \(47413\)
\(\nu^{10}\)\(=\)\(30616\) \(\beta_{13}\mathstrut -\mathstrut \) \(37773\) \(\beta_{12}\mathstrut +\mathstrut \) \(98893\) \(\beta_{11}\mathstrut +\mathstrut \) \(986\) \(\beta_{10}\mathstrut -\mathstrut \) \(21437\) \(\beta_{9}\mathstrut -\mathstrut \) \(18052\) \(\beta_{8}\mathstrut -\mathstrut \) \(12588\) \(\beta_{7}\mathstrut +\mathstrut \) \(29338\) \(\beta_{6}\mathstrut +\mathstrut \) \(46972\) \(\beta_{5}\mathstrut +\mathstrut \) \(47547\) \(\beta_{4}\mathstrut +\mathstrut \) \(24544\) \(\beta_{3}\mathstrut -\mathstrut \) \(25202\) \(\beta_{2}\mathstrut +\mathstrut \) \(71435\) \(\beta_{1}\mathstrut +\mathstrut \) \(296729\)
\(\nu^{11}\)\(=\)\(122139\) \(\beta_{13}\mathstrut -\mathstrut \) \(166504\) \(\beta_{12}\mathstrut +\mathstrut \) \(373041\) \(\beta_{11}\mathstrut -\mathstrut \) \(43026\) \(\beta_{10}\mathstrut -\mathstrut \) \(86704\) \(\beta_{9}\mathstrut +\mathstrut \) \(20835\) \(\beta_{8}\mathstrut -\mathstrut \) \(102258\) \(\beta_{7}\mathstrut +\mathstrut \) \(101217\) \(\beta_{6}\mathstrut +\mathstrut \) \(138595\) \(\beta_{5}\mathstrut +\mathstrut \) \(252965\) \(\beta_{4}\mathstrut +\mathstrut \) \(12729\) \(\beta_{3}\mathstrut +\mathstrut \) \(16621\) \(\beta_{2}\mathstrut +\mathstrut \) \(365621\) \(\beta_{1}\mathstrut +\mathstrut \) \(819867\)
\(\nu^{12}\)\(=\)\(434104\) \(\beta_{13}\mathstrut -\mathstrut \) \(667453\) \(\beta_{12}\mathstrut +\mathstrut \) \(1693145\) \(\beta_{11}\mathstrut -\mathstrut \) \(27689\) \(\beta_{10}\mathstrut -\mathstrut \) \(430192\) \(\beta_{9}\mathstrut -\mathstrut \) \(343448\) \(\beta_{8}\mathstrut -\mathstrut \) \(259336\) \(\beta_{7}\mathstrut +\mathstrut \) \(440956\) \(\beta_{6}\mathstrut +\mathstrut \) \(876463\) \(\beta_{5}\mathstrut +\mathstrut \) \(925564\) \(\beta_{4}\mathstrut +\mathstrut \) \(258691\) \(\beta_{3}\mathstrut -\mathstrut \) \(270723\) \(\beta_{2}\mathstrut +\mathstrut \) \(1151467\) \(\beta_{1}\mathstrut +\mathstrut \) \(4704078\)
\(\nu^{13}\)\(=\)\(1860057\) \(\beta_{13}\mathstrut -\mathstrut \) \(2802582\) \(\beta_{12}\mathstrut +\mathstrut \) \(6513435\) \(\beta_{11}\mathstrut -\mathstrut \) \(823424\) \(\beta_{10}\mathstrut -\mathstrut \) \(1769865\) \(\beta_{9}\mathstrut -\mathstrut \) \(35291\) \(\beta_{8}\mathstrut -\mathstrut \) \(1747648\) \(\beta_{7}\mathstrut +\mathstrut \) \(1471030\) \(\beta_{6}\mathstrut +\mathstrut \) \(2780674\) \(\beta_{5}\mathstrut +\mathstrut \) \(4665904\) \(\beta_{4}\mathstrut -\mathstrut \) \(25839\) \(\beta_{3}\mathstrut +\mathstrut \) \(487738\) \(\beta_{2}\mathstrut +\mathstrut \) \(5617405\) \(\beta_{1}\mathstrut +\mathstrut \) \(14098951\)

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
4.19123
3.81560
3.75223
2.57230
2.22430
1.73527
0.453790
−0.290663
−0.480706
−1.34813
−1.50270
−2.36221
−3.07028
−3.69005
−1.00000 1.00000 1.00000 −4.19123 −1.00000 −2.48485 −1.00000 1.00000 4.19123
1.2 −1.00000 1.00000 1.00000 −3.81560 −1.00000 4.94203 −1.00000 1.00000 3.81560
1.3 −1.00000 1.00000 1.00000 −3.75223 −1.00000 1.91373 −1.00000 1.00000 3.75223
1.4 −1.00000 1.00000 1.00000 −2.57230 −1.00000 −4.81484 −1.00000 1.00000 2.57230
1.5 −1.00000 1.00000 1.00000 −2.22430 −1.00000 1.37131 −1.00000 1.00000 2.22430
1.6 −1.00000 1.00000 1.00000 −1.73527 −1.00000 −3.07502 −1.00000 1.00000 1.73527
1.7 −1.00000 1.00000 1.00000 −0.453790 −1.00000 3.87720 −1.00000 1.00000 0.453790
1.8 −1.00000 1.00000 1.00000 0.290663 −1.00000 −3.15490 −1.00000 1.00000 −0.290663
1.9 −1.00000 1.00000 1.00000 0.480706 −1.00000 0.765483 −1.00000 1.00000 −0.480706
1.10 −1.00000 1.00000 1.00000 1.34813 −1.00000 2.71387 −1.00000 1.00000 −1.34813
1.11 −1.00000 1.00000 1.00000 1.50270 −1.00000 0.210711 −1.00000 1.00000 −1.50270
1.12 −1.00000 1.00000 1.00000 2.36221 −1.00000 −4.26027 −1.00000 1.00000 −2.36221
1.13 −1.00000 1.00000 1.00000 3.07028 −1.00000 −1.20257 −1.00000 1.00000 −3.07028
1.14 −1.00000 1.00000 1.00000 3.69005 −1.00000 −0.801879 −1.00000 1.00000 −3.69005
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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

\(T_{5}^{14} + \cdots\)
\(T_{7}^{14} + \cdots\)