Properties

Label 8030.2.a.bd
Level 8030
Weight 2
Character orbit 8030.a
Self dual Yes
Analytic conductor 64.120
Analytic rank 0
Dimension 14
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(6\) \(x^{13}\mathstrut -\mathstrut \) \(15\) \(x^{12}\mathstrut +\mathstrut \) \(143\) \(x^{11}\mathstrut -\mathstrut \) \(13\) \(x^{10}\mathstrut -\mathstrut \) \(1176\) \(x^{9}\mathstrut +\mathstrut \) \(1018\) \(x^{8}\mathstrut +\mathstrut \) \(4076\) \(x^{7}\mathstrut -\mathstrut \) \(4865\) \(x^{6}\mathstrut -\mathstrut \) \(5483\) \(x^{5}\mathstrut +\mathstrut \) \(7607\) \(x^{4}\mathstrut +\mathstrut \) \(1210\) \(x^{3}\mathstrut -\mathstrut \) \(3153\) \(x^{2}\mathstrut +\mathstrut \) \(878\) \(x\mathstrut -\mathstrut \) \(54\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(2914868715\) \(\nu^{13}\mathstrut -\mathstrut \) \(1481677836\) \(\nu^{12}\mathstrut -\mathstrut \) \(92329276273\) \(\nu^{11}\mathstrut +\mathstrut \) \(36500239287\) \(\nu^{10}\mathstrut +\mathstrut \) \(1083304971166\) \(\nu^{9}\mathstrut -\mathstrut \) \(283103779133\) \(\nu^{8}\mathstrut -\mathstrut \) \(5700905875877\) \(\nu^{7}\mathstrut +\mathstrut \) \(558270197709\) \(\nu^{6}\mathstrut +\mathstrut \) \(12398302271535\) \(\nu^{5}\mathstrut +\mathstrut \) \(1389266066603\) \(\nu^{4}\mathstrut -\mathstrut \) \(5703730355632\) \(\nu^{3}\mathstrut -\mathstrut \) \(3296864096055\) \(\nu^{2}\mathstrut -\mathstrut \) \(3341457726265\) \(\nu\mathstrut +\mathstrut \) \(1286475808555\)\()/\)\(482819744623\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(3013739076\) \(\nu^{13}\mathstrut +\mathstrut \) \(25971820874\) \(\nu^{12}\mathstrut +\mathstrut \) \(24044059112\) \(\nu^{11}\mathstrut -\mathstrut \) \(634643609480\) \(\nu^{10}\mathstrut +\mathstrut \) \(538710938358\) \(\nu^{9}\mathstrut +\mathstrut \) \(5487295766313\) \(\nu^{8}\mathstrut -\mathstrut \) \(7077616441041\) \(\nu^{7}\mathstrut -\mathstrut \) \(21124488583388\) \(\nu^{6}\mathstrut +\mathstrut \) \(27735357910256\) \(\nu^{5}\mathstrut +\mathstrut \) \(35990246964364\) \(\nu^{4}\mathstrut -\mathstrut \) \(38216521144563\) \(\nu^{3}\mathstrut -\mathstrut \) \(19205530692950\) \(\nu^{2}\mathstrut +\mathstrut \) \(13262432789978\) \(\nu\mathstrut -\mathstrut \) \(1183349774126\)\()/\)\(482819744623\)
\(\beta_{4}\)\(=\)\((\)\(5956196641\) \(\nu^{13}\mathstrut -\mathstrut \) \(30375734869\) \(\nu^{12}\mathstrut -\mathstrut \) \(122300275334\) \(\nu^{11}\mathstrut +\mathstrut \) \(768938904475\) \(\nu^{10}\mathstrut +\mathstrut \) \(717920990999\) \(\nu^{9}\mathstrut -\mathstrut \) \(6986634081528\) \(\nu^{8}\mathstrut -\mathstrut \) \(642289677611\) \(\nu^{7}\mathstrut +\mathstrut \) \(28584504146557\) \(\nu^{6}\mathstrut -\mathstrut \) \(4480093356336\) \(\nu^{5}\mathstrut -\mathstrut \) \(52213239513774\) \(\nu^{4}\mathstrut +\mathstrut \) \(7729035386177\) \(\nu^{3}\mathstrut +\mathstrut \) \(33030321873865\) \(\nu^{2}\mathstrut -\mathstrut \) \(1431003798141\) \(\nu\mathstrut -\mathstrut \) \(2680171248537\)\()/\)\(482819744623\)
\(\beta_{5}\)\(=\)\((\)\(6000025764\) \(\nu^{13}\mathstrut -\mathstrut \) \(45403692907\) \(\nu^{12}\mathstrut -\mathstrut \) \(71158156672\) \(\nu^{11}\mathstrut +\mathstrut \) \(1109907562806\) \(\nu^{10}\mathstrut -\mathstrut \) \(509729037226\) \(\nu^{9}\mathstrut -\mathstrut \) \(9545555571287\) \(\nu^{8}\mathstrut +\mathstrut \) \(9385150318324\) \(\nu^{7}\mathstrut +\mathstrut \) \(35851778259498\) \(\nu^{6}\mathstrut -\mathstrut \) \(38688550611021\) \(\nu^{5}\mathstrut -\mathstrut \) \(56311055503292\) \(\nu^{4}\mathstrut +\mathstrut \) \(53847674213760\) \(\nu^{3}\mathstrut +\mathstrut \) \(22154233001447\) \(\nu^{2}\mathstrut -\mathstrut \) \(20395203586154\) \(\nu\mathstrut +\mathstrut \) \(4161812783791\)\()/\)\(482819744623\)
\(\beta_{6}\)\(=\)\((\)\(6388173845\) \(\nu^{13}\mathstrut -\mathstrut \) \(29903795688\) \(\nu^{12}\mathstrut -\mathstrut \) \(123641888828\) \(\nu^{11}\mathstrut +\mathstrut \) \(713909753046\) \(\nu^{10}\mathstrut +\mathstrut \) \(589523550970\) \(\nu^{9}\mathstrut -\mathstrut \) \(5893337239748\) \(\nu^{8}\mathstrut +\mathstrut \) \(810711181527\) \(\nu^{7}\mathstrut +\mathstrut \) \(20675574287462\) \(\nu^{6}\mathstrut -\mathstrut \) \(10044358890977\) \(\nu^{5}\mathstrut -\mathstrut \) \(29630682458330\) \(\nu^{4}\mathstrut +\mathstrut \) \(14730787829598\) \(\nu^{3}\mathstrut +\mathstrut \) \(13083260308723\) \(\nu^{2}\mathstrut -\mathstrut \) \(1126946044631\) \(\nu\mathstrut -\mathstrut \) \(961873041978\)\()/\)\(482819744623\)
\(\beta_{7}\)\(=\)\((\)\(8203270317\) \(\nu^{13}\mathstrut -\mathstrut \) \(53522196132\) \(\nu^{12}\mathstrut -\mathstrut \) \(116123739560\) \(\nu^{11}\mathstrut +\mathstrut \) \(1309912732812\) \(\nu^{10}\mathstrut -\mathstrut \) \(283231450723\) \(\nu^{9}\mathstrut -\mathstrut \) \(11317154526930\) \(\nu^{8}\mathstrut +\mathstrut \) \(9881701876806\) \(\nu^{7}\mathstrut +\mathstrut \) \(43187691623126\) \(\nu^{6}\mathstrut -\mathstrut \) \(44998247127849\) \(\nu^{5}\mathstrut -\mathstrut \) \(71895862985458\) \(\nu^{4}\mathstrut +\mathstrut \) \(67109040703585\) \(\nu^{3}\mathstrut +\mathstrut \) \(38135302615113\) \(\nu^{2}\mathstrut -\mathstrut \) \(26335183439660\) \(\nu\mathstrut +\mathstrut \) \(944421094086\)\()/\)\(482819744623\)
\(\beta_{8}\)\(=\)\((\)\(13083447036\) \(\nu^{13}\mathstrut -\mathstrut \) \(60182654009\) \(\nu^{12}\mathstrut -\mathstrut \) \(270235819099\) \(\nu^{11}\mathstrut +\mathstrut \) \(1478723382956\) \(\nu^{10}\mathstrut +\mathstrut \) \(1596559087142\) \(\nu^{9}\mathstrut -\mathstrut \) \(12820512106105\) \(\nu^{8}\mathstrut -\mathstrut \) \(1258953366685\) \(\nu^{7}\mathstrut +\mathstrut \) \(48879634175378\) \(\nu^{6}\mathstrut -\mathstrut \) \(12598618668441\) \(\nu^{5}\mathstrut -\mathstrut \) \(80539286848771\) \(\nu^{4}\mathstrut +\mathstrut \) \(26562914190307\) \(\nu^{3}\mathstrut +\mathstrut \) \(43092476857320\) \(\nu^{2}\mathstrut -\mathstrut \) \(9951647666317\) \(\nu\mathstrut -\mathstrut \) \(1416109772417\)\()/\)\(482819744623\)
\(\beta_{9}\)\(=\)\((\)\(14031049713\) \(\nu^{13}\mathstrut -\mathstrut \) \(93711317903\) \(\nu^{12}\mathstrut -\mathstrut \) \(204783187150\) \(\nu^{11}\mathstrut +\mathstrut \) \(2305886641289\) \(\nu^{10}\mathstrut -\mathstrut \) \(295470716745\) \(\nu^{9}\mathstrut -\mathstrut \) \(20075729637319\) \(\nu^{8}\mathstrut +\mathstrut \) \(14835344892468\) \(\nu^{7}\mathstrut +\mathstrut \) \(77325077506102\) \(\nu^{6}\mathstrut -\mathstrut \) \(67341316580738\) \(\nu^{5}\mathstrut -\mathstrut \) \(129680110818591\) \(\nu^{4}\mathstrut +\mathstrut \) \(96943799182377\) \(\nu^{3}\mathstrut +\mathstrut \) \(69223163681453\) \(\nu^{2}\mathstrut -\mathstrut \) \(34905959431298\) \(\nu\mathstrut +\mathstrut \) \(402909869499\)\()/\)\(482819744623\)
\(\beta_{10}\)\(=\)\((\)\(28180525363\) \(\nu^{13}\mathstrut -\mathstrut \) \(154398148196\) \(\nu^{12}\mathstrut -\mathstrut \) \(490244344188\) \(\nu^{11}\mathstrut +\mathstrut \) \(3740214959036\) \(\nu^{10}\mathstrut +\mathstrut \) \(1247167738284\) \(\nu^{9}\mathstrut -\mathstrut \) \(31661779330967\) \(\nu^{8}\mathstrut +\mathstrut \) \(15330046758242\) \(\nu^{7}\mathstrut +\mathstrut \) \(115907842949630\) \(\nu^{6}\mathstrut -\mathstrut \) \(89835693002166\) \(\nu^{5}\mathstrut -\mathstrut \) \(177025480819054\) \(\nu^{4}\mathstrut +\mathstrut \) \(145303515049335\) \(\nu^{3}\mathstrut +\mathstrut \) \(76771336312339\) \(\nu^{2}\mathstrut -\mathstrut \) \(58697430879687\) \(\nu\mathstrut +\mathstrut \) \(6525355140337\)\()/\)\(482819744623\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(35738278950\) \(\nu^{13}\mathstrut +\mathstrut \) \(192127692799\) \(\nu^{12}\mathstrut +\mathstrut \) \(631908968296\) \(\nu^{11}\mathstrut -\mathstrut \) \(4643945048558\) \(\nu^{10}\mathstrut -\mathstrut \) \(1847048138062\) \(\nu^{9}\mathstrut +\mathstrut \) \(39185629367682\) \(\nu^{8}\mathstrut -\mathstrut \) \(16990732017358\) \(\nu^{7}\mathstrut -\mathstrut \) \(142875275809638\) \(\nu^{6}\mathstrut +\mathstrut \) \(104228827943786\) \(\nu^{5}\mathstrut +\mathstrut \) \(217765174871063\) \(\nu^{4}\mathstrut -\mathstrut \) \(169142536868314\) \(\nu^{3}\mathstrut -\mathstrut \) \(97347766663359\) \(\nu^{2}\mathstrut +\mathstrut \) \(68876752884511\) \(\nu\mathstrut -\mathstrut \) \(4851763055327\)\()/\)\(482819744623\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(36555709416\) \(\nu^{13}\mathstrut +\mathstrut \) \(213327294789\) \(\nu^{12}\mathstrut +\mathstrut \) \(597085618488\) \(\nu^{11}\mathstrut -\mathstrut \) \(5170210115848\) \(\nu^{10}\mathstrut -\mathstrut \) \(726309968214\) \(\nu^{9}\mathstrut +\mathstrut \) \(43854772955429\) \(\nu^{8}\mathstrut -\mathstrut \) \(26714709837124\) \(\nu^{7}\mathstrut -\mathstrut \) \(161529134835147\) \(\nu^{6}\mathstrut +\mathstrut \) \(137126464243593\) \(\nu^{5}\mathstrut +\mathstrut \) \(250595480245328\) \(\nu^{4}\mathstrut -\mathstrut \) \(209343360132153\) \(\nu^{3}\mathstrut -\mathstrut \) \(113325751216221\) \(\nu^{2}\mathstrut +\mathstrut \) \(80352395313623\) \(\nu\mathstrut -\mathstrut \) \(7547519788454\)\()/\)\(482819744623\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(41694475591\) \(\nu^{13}\mathstrut +\mathstrut \) \(222503427668\) \(\nu^{12}\mathstrut +\mathstrut \) \(754209243630\) \(\nu^{11}\mathstrut -\mathstrut \) \(5412883953033\) \(\nu^{10}\mathstrut -\mathstrut \) \(2564969129061\) \(\nu^{9}\mathstrut +\mathstrut \) \(46172263449210\) \(\nu^{8}\mathstrut -\mathstrut \) \(16348442339747\) \(\nu^{7}\mathstrut -\mathstrut \) \(171459779956195\) \(\nu^{6}\mathstrut +\mathstrut \) \(108708921300122\) \(\nu^{5}\mathstrut +\mathstrut \) \(269978414384837\) \(\nu^{4}\mathstrut -\mathstrut \) \(176871572254491\) \(\nu^{3}\mathstrut -\mathstrut \) \(129895268792601\) \(\nu^{2}\mathstrut +\mathstrut \) \(70307756682652\) \(\nu\mathstrut -\mathstrut \) \(4102870785282\)\()/\)\(482819744623\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(10\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(8\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\)
\(\nu^{5}\)\(=\)\(-\)\(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(4\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(18\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(20\) \(\beta_{4}\mathstrut +\mathstrut \) \(31\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(76\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{6}\)\(=\)\(98\) \(\beta_{13}\mathstrut -\mathstrut \) \(17\) \(\beta_{12}\mathstrut -\mathstrut \) \(62\) \(\beta_{11}\mathstrut +\mathstrut \) \(38\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(35\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(37\) \(\beta_{6}\mathstrut -\mathstrut \) \(25\) \(\beta_{5}\mathstrut +\mathstrut \) \(77\) \(\beta_{4}\mathstrut +\mathstrut \) \(26\) \(\beta_{3}\mathstrut -\mathstrut \) \(39\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\mathstrut +\mathstrut \) \(278\)
\(\nu^{7}\)\(=\)\(-\)\(45\) \(\beta_{12}\mathstrut +\mathstrut \) \(79\) \(\beta_{11}\mathstrut +\mathstrut \) \(19\) \(\beta_{10}\mathstrut +\mathstrut \) \(77\) \(\beta_{9}\mathstrut +\mathstrut \) \(243\) \(\beta_{8}\mathstrut +\mathstrut \) \(125\) \(\beta_{7}\mathstrut -\mathstrut \) \(158\) \(\beta_{6}\mathstrut -\mathstrut \) \(46\) \(\beta_{5}\mathstrut -\mathstrut \) \(282\) \(\beta_{4}\mathstrut +\mathstrut \) \(395\) \(\beta_{3}\mathstrut -\mathstrut \) \(167\) \(\beta_{2}\mathstrut +\mathstrut \) \(769\) \(\beta_{1}\mathstrut +\mathstrut \) \(167\)
\(\nu^{8}\)\(=\)\(988\) \(\beta_{13}\mathstrut -\mathstrut \) \(241\) \(\beta_{12}\mathstrut -\mathstrut \) \(481\) \(\beta_{11}\mathstrut +\mathstrut \) \(533\) \(\beta_{10}\mathstrut -\mathstrut \) \(115\) \(\beta_{9}\mathstrut +\mathstrut \) \(474\) \(\beta_{8}\mathstrut +\mathstrut \) \(18\) \(\beta_{7}\mathstrut -\mathstrut \) \(513\) \(\beta_{6}\mathstrut -\mathstrut \) \(392\) \(\beta_{5}\mathstrut +\mathstrut \) \(671\) \(\beta_{4}\mathstrut +\mathstrut \) \(440\) \(\beta_{3}\mathstrut -\mathstrut \) \(550\) \(\beta_{2}\mathstrut +\mathstrut \) \(429\) \(\beta_{1}\mathstrut +\mathstrut \) \(2668\)
\(\nu^{9}\)\(=\)\(-\)\(4\) \(\beta_{13}\mathstrut -\mathstrut \) \(731\) \(\beta_{12}\mathstrut +\mathstrut \) \(1167\) \(\beta_{11}\mathstrut +\mathstrut \) \(271\) \(\beta_{10}\mathstrut +\mathstrut \) \(1073\) \(\beta_{9}\mathstrut +\mathstrut \) \(2976\) \(\beta_{8}\mathstrut +\mathstrut \) \(1281\) \(\beta_{7}\mathstrut -\mathstrut \) \(1884\) \(\beta_{6}\mathstrut -\mathstrut \) \(738\) \(\beta_{5}\mathstrut -\mathstrut \) \(3521\) \(\beta_{4}\mathstrut +\mathstrut \) \(4737\) \(\beta_{3}\mathstrut -\mathstrut \) \(1924\) \(\beta_{2}\mathstrut +\mathstrut \) \(8048\) \(\beta_{1}\mathstrut +\mathstrut \) \(1730\)
\(\nu^{10}\)\(=\)\(10225\) \(\beta_{13}\mathstrut -\mathstrut \) \(3209\) \(\beta_{12}\mathstrut -\mathstrut \) \(3721\) \(\beta_{11}\mathstrut +\mathstrut \) \(6640\) \(\beta_{10}\mathstrut -\mathstrut \) \(1026\) \(\beta_{9}\mathstrut +\mathstrut \) \(5884\) \(\beta_{8}\mathstrut +\mathstrut \) \(247\) \(\beta_{7}\mathstrut -\mathstrut \) \(6428\) \(\beta_{6}\mathstrut -\mathstrut \) \(5246\) \(\beta_{5}\mathstrut +\mathstrut \) \(6010\) \(\beta_{4}\mathstrut +\mathstrut \) \(6265\) \(\beta_{3}\mathstrut -\mathstrut \) \(6886\) \(\beta_{2}\mathstrut +\mathstrut \) \(5192\) \(\beta_{1}\mathstrut +\mathstrut \) \(26703\)
\(\nu^{11}\)\(=\)\(-\)\(43\) \(\beta_{13}\mathstrut -\mathstrut \) \(10370\) \(\beta_{12}\mathstrut +\mathstrut \) \(15404\) \(\beta_{11}\mathstrut +\mathstrut \) \(3503\) \(\beta_{10}\mathstrut +\mathstrut \) \(13282\) \(\beta_{9}\mathstrut +\mathstrut \) \(34905\) \(\beta_{8}\mathstrut +\mathstrut \) \(13300\) \(\beta_{7}\mathstrut -\mathstrut \) \(22127\) \(\beta_{6}\mathstrut -\mathstrut \) \(10262\) \(\beta_{5}\mathstrut -\mathstrut \) \(41591\) \(\beta_{4}\mathstrut +\mathstrut \) \(55324\) \(\beta_{3}\mathstrut -\mathstrut \) \(21902\) \(\beta_{2}\mathstrut +\mathstrut \) \(85942\) \(\beta_{1}\mathstrut +\mathstrut \) \(17707\)
\(\nu^{12}\)\(=\)\(107883\) \(\beta_{13}\mathstrut -\mathstrut \) \(41167\) \(\beta_{12}\mathstrut -\mathstrut \) \(28344\) \(\beta_{11}\mathstrut +\mathstrut \) \(78038\) \(\beta_{10}\mathstrut -\mathstrut \) \(8955\) \(\beta_{9}\mathstrut +\mathstrut \) \(70250\) \(\beta_{8}\mathstrut +\mathstrut \) \(3208\) \(\beta_{7}\mathstrut -\mathstrut \) \(76886\) \(\beta_{6}\mathstrut -\mathstrut \) \(65289\) \(\beta_{5}\mathstrut +\mathstrut \) \(55190\) \(\beta_{4}\mathstrut +\mathstrut \) \(81823\) \(\beta_{3}\mathstrut -\mathstrut \) \(81614\) \(\beta_{2}\mathstrut +\mathstrut \) \(61303\) \(\beta_{1}\mathstrut +\mathstrut \) \(274930\)
\(\nu^{13}\)\(=\)\(479\) \(\beta_{13}\mathstrut -\mathstrut \) \(136719\) \(\beta_{12}\mathstrut +\mathstrut \) \(191734\) \(\beta_{11}\mathstrut +\mathstrut \) \(43206\) \(\beta_{10}\mathstrut +\mathstrut \) \(155417\) \(\beta_{9}\mathstrut +\mathstrut \) \(400665\) \(\beta_{8}\mathstrut +\mathstrut \) \(140685\) \(\beta_{7}\mathstrut -\mathstrut \) \(256748\) \(\beta_{6}\mathstrut -\mathstrut \) \(132540\) \(\beta_{5}\mathstrut -\mathstrut \) \(477197\) \(\beta_{4}\mathstrut +\mathstrut \) \(636924\) \(\beta_{3}\mathstrut -\mathstrut \) \(247817\) \(\beta_{2}\mathstrut +\mathstrut \) \(929302\) \(\beta_{1}\mathstrut +\mathstrut \) \(183262\)

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
−3.28632
−2.47616
−1.98092
−1.51960
−0.856641
0.0877386
0.331408
0.454872
1.13475
2.25086
2.65302
2.86338
2.99130
3.35230
1.00000 −3.28632 1.00000 −1.00000 −3.28632 −0.411624 1.00000 7.79988 −1.00000
1.2 1.00000 −2.47616 1.00000 −1.00000 −2.47616 −1.89245 1.00000 3.13135 −1.00000
1.3 1.00000 −1.98092 1.00000 −1.00000 −1.98092 1.99444 1.00000 0.924028 −1.00000
1.4 1.00000 −1.51960 1.00000 −1.00000 −1.51960 −1.36556 1.00000 −0.690822 −1.00000
1.5 1.00000 −0.856641 1.00000 −1.00000 −0.856641 1.42002 1.00000 −2.26617 −1.00000
1.6 1.00000 0.0877386 1.00000 −1.00000 0.0877386 −3.09806 1.00000 −2.99230 −1.00000
1.7 1.00000 0.331408 1.00000 −1.00000 0.331408 4.90237 1.00000 −2.89017 −1.00000
1.8 1.00000 0.454872 1.00000 −1.00000 0.454872 −4.57770 1.00000 −2.79309 −1.00000
1.9 1.00000 1.13475 1.00000 −1.00000 1.13475 −2.24922 1.00000 −1.71235 −1.00000
1.10 1.00000 2.25086 1.00000 −1.00000 2.25086 4.10088 1.00000 2.06636 −1.00000
1.11 1.00000 2.65302 1.00000 −1.00000 2.65302 −2.26267 1.00000 4.03849 −1.00000
1.12 1.00000 2.86338 1.00000 −1.00000 2.86338 2.76153 1.00000 5.19896 −1.00000
1.13 1.00000 2.99130 1.00000 −1.00000 2.99130 4.10532 1.00000 5.94789 −1.00000
1.14 1.00000 3.35230 1.00000 −1.00000 3.35230 0.572732 1.00000 8.23794 −1.00000
\(n\): 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
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)
\(73\) \(-1\)

Hecke kernels

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

\(T_{3}^{14} - \cdots\)
\(T_{7}^{14} - \cdots\)