Properties

Label 8030.2.a.be
Level 8030
Weight 2
Character orbit 8030.a
Self dual Yes
Analytic conductor 64.120
Analytic rank 0
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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(3\) \(x^{14}\mathstrut -\mathstrut \) \(24\) \(x^{13}\mathstrut +\mathstrut \) \(64\) \(x^{12}\mathstrut +\mathstrut \) \(237\) \(x^{11}\mathstrut -\mathstrut \) \(524\) \(x^{10}\mathstrut -\mathstrut \) \(1225\) \(x^{9}\mathstrut +\mathstrut \) \(2074\) \(x^{8}\mathstrut +\mathstrut \) \(3463\) \(x^{7}\mathstrut -\mathstrut \) \(4142\) \(x^{6}\mathstrut -\mathstrut \) \(5157\) \(x^{5}\mathstrut +\mathstrut \) \(3892\) \(x^{4}\mathstrut +\mathstrut \) \(3622\) \(x^{3}\mathstrut -\mathstrut \) \(1239\) \(x^{2}\mathstrut -\mathstrut \) \(907\) \(x\mathstrut -\mathstrut \) \(52\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(6135455\) \(\nu^{14}\mathstrut +\mathstrut \) \(34883556\) \(\nu^{13}\mathstrut +\mathstrut \) \(73715727\) \(\nu^{12}\mathstrut -\mathstrut \) \(679696493\) \(\nu^{11}\mathstrut +\mathstrut \) \(67269559\) \(\nu^{10}\mathstrut +\mathstrut \) \(4744001082\) \(\nu^{9}\mathstrut -\mathstrut \) \(4217825796\) \(\nu^{8}\mathstrut -\mathstrut \) \(13747836174\) \(\nu^{7}\mathstrut +\mathstrut \) \(20746207779\) \(\nu^{6}\mathstrut +\mathstrut \) \(12690525605\) \(\nu^{5}\mathstrut -\mathstrut \) \(39315304221\) \(\nu^{4}\mathstrut +\mathstrut \) \(2956713916\) \(\nu^{3}\mathstrut +\mathstrut \) \(27532331027\) \(\nu^{2}\mathstrut +\mathstrut \) \(3839031984\) \(\nu\mathstrut -\mathstrut \) \(5186399506\)\()/\)\(2177153354\)
\(\beta_{4}\)\(=\)\((\)\(7950717\) \(\nu^{14}\mathstrut +\mathstrut \) \(11327530\) \(\nu^{13}\mathstrut -\mathstrut \) \(241656913\) \(\nu^{12}\mathstrut -\mathstrut \) \(452867377\) \(\nu^{11}\mathstrut +\mathstrut \) \(2818120875\) \(\nu^{10}\mathstrut +\mathstrut \) \(6117736632\) \(\nu^{9}\mathstrut -\mathstrut \) \(15127942180\) \(\nu^{8}\mathstrut -\mathstrut \) \(36349354126\) \(\nu^{7}\mathstrut +\mathstrut \) \(35107273271\) \(\nu^{6}\mathstrut +\mathstrut \) \(95890554923\) \(\nu^{5}\mathstrut -\mathstrut \) \(22387758233\) \(\nu^{4}\mathstrut -\mathstrut \) \(97161579378\) \(\nu^{3}\mathstrut -\mathstrut \) \(19134815857\) \(\nu^{2}\mathstrut +\mathstrut \) \(24406922512\) \(\nu\mathstrut +\mathstrut \) \(16257697244\)\()/\)\(2177153354\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(24837945\) \(\nu^{14}\mathstrut +\mathstrut \) \(12188068\) \(\nu^{13}\mathstrut +\mathstrut \) \(748401619\) \(\nu^{12}\mathstrut -\mathstrut \) \(92125045\) \(\nu^{11}\mathstrut -\mathstrut \) \(8687620119\) \(\nu^{10}\mathstrut -\mathstrut \) \(1601443234\) \(\nu^{9}\mathstrut +\mathstrut \) \(47598412884\) \(\nu^{8}\mathstrut +\mathstrut \) \(20671511970\) \(\nu^{7}\mathstrut -\mathstrut \) \(120798745963\) \(\nu^{6}\mathstrut -\mathstrut \) \(76879409387\) \(\nu^{5}\mathstrut +\mathstrut \) \(115626370497\) \(\nu^{4}\mathstrut +\mathstrut \) \(100643975784\) \(\nu^{3}\mathstrut -\mathstrut \) \(8311173255\) \(\nu^{2}\mathstrut -\mathstrut \) \(36912682832\) \(\nu\mathstrut -\mathstrut \) \(12450340512\)\()/\)\(2177153354\)
\(\beta_{6}\)\(=\)\((\)\(41839165\) \(\nu^{14}\mathstrut -\mathstrut \) \(97335759\) \(\nu^{13}\mathstrut -\mathstrut \) \(950444152\) \(\nu^{12}\mathstrut +\mathstrut \) \(1659753197\) \(\nu^{11}\mathstrut +\mathstrut \) \(8530051604\) \(\nu^{10}\mathstrut -\mathstrut \) \(8753931606\) \(\nu^{9}\mathstrut -\mathstrut \) \(36848460158\) \(\nu^{8}\mathstrut +\mathstrut \) \(7755983846\) \(\nu^{7}\mathstrut +\mathstrut \) \(72071650321\) \(\nu^{6}\mathstrut +\mathstrut \) \(55228228592\) \(\nu^{5}\mathstrut -\mathstrut \) \(41339133295\) \(\nu^{4}\mathstrut -\mathstrut \) \(131259374745\) \(\nu^{3}\mathstrut -\mathstrut \) \(21481263848\) \(\nu^{2}\mathstrut +\mathstrut \) \(75474187418\) \(\nu\mathstrut +\mathstrut \) \(17646606370\)\()/\)\(2177153354\)
\(\beta_{7}\)\(=\)\((\)\(56714341\) \(\nu^{14}\mathstrut -\mathstrut \) \(129019881\) \(\nu^{13}\mathstrut -\mathstrut \) \(1338802868\) \(\nu^{12}\mathstrut +\mathstrut \) \(2167586419\) \(\nu^{11}\mathstrut +\mathstrut \) \(12997204796\) \(\nu^{10}\mathstrut -\mathstrut \) \(11049545846\) \(\nu^{9}\mathstrut -\mathstrut \) \(64630380466\) \(\nu^{8}\mathstrut +\mathstrut \) \(7471313626\) \(\nu^{7}\mathstrut +\mathstrut \) \(164499818719\) \(\nu^{6}\mathstrut +\mathstrut \) \(74458102334\) \(\nu^{5}\mathstrut -\mathstrut \) \(189177229519\) \(\nu^{4}\mathstrut -\mathstrut \) \(156888344263\) \(\nu^{3}\mathstrut +\mathstrut \) \(67012434018\) \(\nu^{2}\mathstrut +\mathstrut \) \(77244383618\) \(\nu\mathstrut +\mathstrut \) \(8151943214\)\()/\)\(2177153354\)
\(\beta_{8}\)\(=\)\((\)\(61936144\) \(\nu^{14}\mathstrut -\mathstrut \) \(225208367\) \(\nu^{13}\mathstrut -\mathstrut \) \(1288366493\) \(\nu^{12}\mathstrut +\mathstrut \) \(4591312948\) \(\nu^{11}\mathstrut +\mathstrut \) \(10745591943\) \(\nu^{10}\mathstrut -\mathstrut \) \(35752487220\) \(\nu^{9}\mathstrut -\mathstrut \) \(46132733754\) \(\nu^{8}\mathstrut +\mathstrut \) \(133503050370\) \(\nu^{7}\mathstrut +\mathstrut \) \(107882694636\) \(\nu^{6}\mathstrut -\mathstrut \) \(247774605843\) \(\nu^{5}\mathstrut -\mathstrut \) \(135509066570\) \(\nu^{4}\mathstrut +\mathstrut \) \(212376010591\) \(\nu^{3}\mathstrut +\mathstrut \) \(86714808271\) \(\nu^{2}\mathstrut -\mathstrut \) \(65954539994\) \(\nu\mathstrut -\mathstrut \) \(18624471416\)\()/\)\(2177153354\)
\(\beta_{9}\)\(=\)\((\)\(65242380\) \(\nu^{14}\mathstrut -\mathstrut \) \(179003743\) \(\nu^{13}\mathstrut -\mathstrut \) \(1562396749\) \(\nu^{12}\mathstrut +\mathstrut \) \(3633751796\) \(\nu^{11}\mathstrut +\mathstrut \) \(15268939361\) \(\nu^{10}\mathstrut -\mathstrut \) \(27574646724\) \(\nu^{9}\mathstrut -\mathstrut \) \(76493019332\) \(\nu^{8}\mathstrut +\mathstrut \) \(97447776890\) \(\nu^{7}\mathstrut +\mathstrut \) \(199912233004\) \(\nu^{6}\mathstrut -\mathstrut \) \(167355932817\) \(\nu^{5}\mathstrut -\mathstrut \) \(248989059686\) \(\nu^{4}\mathstrut +\mathstrut \) \(134816092571\) \(\nu^{3}\mathstrut +\mathstrut \) \(118920622125\) \(\nu^{2}\mathstrut -\mathstrut \) \(37744064900\) \(\nu\mathstrut -\mathstrut \) \(11100190102\)\()/\)\(2177153354\)
\(\beta_{10}\)\(=\)\((\)\(71064649\) \(\nu^{14}\mathstrut -\mathstrut \) \(368116072\) \(\nu^{13}\mathstrut -\mathstrut \) \(1126751873\) \(\nu^{12}\mathstrut +\mathstrut \) \(7646764207\) \(\nu^{11}\mathstrut +\mathstrut \) \(5344844255\) \(\nu^{10}\mathstrut -\mathstrut \) \(61363519446\) \(\nu^{9}\mathstrut -\mathstrut \) \(1942348058\) \(\nu^{8}\mathstrut +\mathstrut \) \(238753967290\) \(\nu^{7}\mathstrut -\mathstrut \) \(42825791741\) \(\nu^{6}\mathstrut -\mathstrut \) \(460051369589\) \(\nu^{5}\mathstrut +\mathstrut \) \(84647326467\) \(\nu^{4}\mathstrut +\mathstrut \) \(391395922546\) \(\nu^{3}\mathstrut -\mathstrut \) \(29263842569\) \(\nu^{2}\mathstrut -\mathstrut \) \(104754666984\) \(\nu\mathstrut -\mathstrut \) \(7735840732\)\()/\)\(2177153354\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(76022316\) \(\nu^{14}\mathstrut +\mathstrut \) \(248764393\) \(\nu^{13}\mathstrut +\mathstrut \) \(1603739133\) \(\nu^{12}\mathstrut -\mathstrut \) \(4818142064\) \(\nu^{11}\mathstrut -\mathstrut \) \(13496443259\) \(\nu^{10}\mathstrut +\mathstrut \) \(34378751670\) \(\nu^{9}\mathstrut +\mathstrut \) \(57042850138\) \(\nu^{8}\mathstrut -\mathstrut \) \(110901532418\) \(\nu^{7}\mathstrut -\mathstrut \) \(122243760128\) \(\nu^{6}\mathstrut +\mathstrut \) \(164574576525\) \(\nu^{5}\mathstrut +\mathstrut \) \(118581520582\) \(\nu^{4}\mathstrut -\mathstrut \) \(110080563943\) \(\nu^{3}\mathstrut -\mathstrut \) \(44401968095\) \(\nu^{2}\mathstrut +\mathstrut \) \(34500882696\) \(\nu\mathstrut +\mathstrut \) \(8066141436\)\()/\)\(2177153354\)
\(\beta_{12}\)\(=\)\((\)\(91585352\) \(\nu^{14}\mathstrut -\mathstrut \) \(318996982\) \(\nu^{13}\mathstrut -\mathstrut \) \(1872097191\) \(\nu^{12}\mathstrut +\mathstrut \) \(6110728165\) \(\nu^{11}\mathstrut +\mathstrut \) \(15534144122\) \(\nu^{10}\mathstrut -\mathstrut \) \(43241172145\) \(\nu^{9}\mathstrut -\mathstrut \) \(67657466690\) \(\nu^{8}\mathstrut +\mathstrut \) \(138751040662\) \(\nu^{7}\mathstrut +\mathstrut \) \(161480829438\) \(\nu^{6}\mathstrut -\mathstrut \) \(202517077432\) \(\nu^{5}\mathstrut -\mathstrut \) \(193099646255\) \(\nu^{4}\mathstrut +\mathstrut \) \(117986128256\) \(\nu^{3}\mathstrut +\mathstrut \) \(89043617457\) \(\nu^{2}\mathstrut -\mathstrut \) \(14632506249\) \(\nu\mathstrut -\mathstrut \) \(6182535442\)\()/\)\(2177153354\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(112912991\) \(\nu^{14}\mathstrut +\mathstrut \) \(678967023\) \(\nu^{13}\mathstrut +\mathstrut \) \(1226139356\) \(\nu^{12}\mathstrut -\mathstrut \) \(13031879625\) \(\nu^{11}\mathstrut +\mathstrut \) \(1743712172\) \(\nu^{10}\mathstrut +\mathstrut \) \(94117009160\) \(\nu^{9}\mathstrut -\mathstrut \) \(65335095990\) \(\nu^{8}\mathstrut -\mathstrut \) \(318204584738\) \(\nu^{7}\mathstrut +\mathstrut \) \(275813864357\) \(\nu^{6}\mathstrut +\mathstrut \) \(510133874052\) \(\nu^{5}\mathstrut -\mathstrut \) \(420061369229\) \(\nu^{4}\mathstrut -\mathstrut \) \(341590452637\) \(\nu^{3}\mathstrut +\mathstrut \) \(192306033104\) \(\nu^{2}\mathstrut +\mathstrut \) \(72667782626\) \(\nu\mathstrut -\mathstrut \) \(5408793724\)\()/\)\(2177153354\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(129594252\) \(\nu^{14}\mathstrut +\mathstrut \) \(450924867\) \(\nu^{13}\mathstrut +\mathstrut \) \(2633495927\) \(\nu^{12}\mathstrut -\mathstrut \) \(8681786756\) \(\nu^{11}\mathstrut -\mathstrut \) \(21227439485\) \(\nu^{10}\mathstrut +\mathstrut \) \(61551327578\) \(\nu^{9}\mathstrut +\mathstrut \) \(85918496500\) \(\nu^{8}\mathstrut -\mathstrut \) \(196128090616\) \(\nu^{7}\mathstrut -\mathstrut \) \(177527508028\) \(\nu^{6}\mathstrut +\mathstrut \) \(277012186727\) \(\nu^{5}\mathstrut +\mathstrut \) \(168410118860\) \(\nu^{4}\mathstrut -\mathstrut \) \(144657019529\) \(\nu^{3}\mathstrut -\mathstrut \) \(63563333451\) \(\nu^{2}\mathstrut +\mathstrut \) \(18858709056\) \(\nu\mathstrut +\mathstrut \) \(10284133320\)\()/\)\(2177153354\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(14\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(59\) \(\beta_{1}\mathstrut +\mathstrut \) \(43\)
\(\nu^{6}\)\(=\)\(18\) \(\beta_{14}\mathstrut -\mathstrut \) \(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(6\) \(\beta_{12}\mathstrut +\mathstrut \) \(33\) \(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(6\) \(\beta_{9}\mathstrut +\mathstrut \) \(37\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(45\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(134\) \(\beta_{2}\mathstrut +\mathstrut \) \(148\) \(\beta_{1}\mathstrut +\mathstrut \) \(239\)
\(\nu^{7}\)\(=\)\(52\) \(\beta_{14}\mathstrut -\mathstrut \) \(21\) \(\beta_{13}\mathstrut -\mathstrut \) \(38\) \(\beta_{12}\mathstrut +\mathstrut \) \(162\) \(\beta_{11}\mathstrut +\mathstrut \) \(19\) \(\beta_{10}\mathstrut +\mathstrut \) \(55\) \(\beta_{9}\mathstrut +\mathstrut \) \(162\) \(\beta_{8}\mathstrut +\mathstrut \) \(43\) \(\beta_{7}\mathstrut +\mathstrut \) \(19\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(172\) \(\beta_{4}\mathstrut -\mathstrut \) \(80\) \(\beta_{3}\mathstrut +\mathstrut \) \(402\) \(\beta_{2}\mathstrut +\mathstrut \) \(566\) \(\beta_{1}\mathstrut +\mathstrut \) \(523\)
\(\nu^{8}\)\(=\)\(270\) \(\beta_{14}\mathstrut -\mathstrut \) \(69\) \(\beta_{13}\mathstrut -\mathstrut \) \(122\) \(\beta_{12}\mathstrut +\mathstrut \) \(443\) \(\beta_{11}\mathstrut +\mathstrut \) \(124\) \(\beta_{10}\mathstrut +\mathstrut \) \(141\) \(\beta_{9}\mathstrut +\mathstrut \) \(528\) \(\beta_{8}\mathstrut +\mathstrut \) \(226\) \(\beta_{7}\mathstrut +\mathstrut \) \(53\) \(\beta_{6}\mathstrut -\mathstrut \) \(30\) \(\beta_{5}\mathstrut +\mathstrut \) \(567\) \(\beta_{4}\mathstrut -\mathstrut \) \(171\) \(\beta_{3}\mathstrut +\mathstrut \) \(1535\) \(\beta_{2}\mathstrut +\mathstrut \) \(1646\) \(\beta_{1}\mathstrut +\mathstrut \) \(2293\)
\(\nu^{9}\)\(=\)\(881\) \(\beta_{14}\mathstrut -\mathstrut \) \(323\) \(\beta_{13}\mathstrut -\mathstrut \) \(544\) \(\beta_{12}\mathstrut +\mathstrut \) \(1836\) \(\beta_{11}\mathstrut +\mathstrut \) \(266\) \(\beta_{10}\mathstrut +\mathstrut \) \(771\) \(\beta_{9}\mathstrut +\mathstrut \) \(2015\) \(\beta_{8}\mathstrut +\mathstrut \) \(663\) \(\beta_{7}\mathstrut +\mathstrut \) \(285\) \(\beta_{6}\mathstrut -\mathstrut \) \(103\) \(\beta_{5}\mathstrut +\mathstrut \) \(2043\) \(\beta_{4}\mathstrut -\mathstrut \) \(777\) \(\beta_{3}\mathstrut +\mathstrut \) \(4944\) \(\beta_{2}\mathstrut +\mathstrut \) \(5868\) \(\beta_{1}\mathstrut +\mathstrut \) \(6090\)
\(\nu^{10}\)\(=\)\(3716\) \(\beta_{14}\mathstrut -\mathstrut \) \(1102\) \(\beta_{13}\mathstrut -\mathstrut \) \(1804\) \(\beta_{12}\mathstrut +\mathstrut \) \(5543\) \(\beta_{11}\mathstrut +\mathstrut \) \(1285\) \(\beta_{10}\mathstrut +\mathstrut \) \(2269\) \(\beta_{9}\mathstrut +\mathstrut \) \(6882\) \(\beta_{8}\mathstrut +\mathstrut \) \(2822\) \(\beta_{7}\mathstrut +\mathstrut \) \(912\) \(\beta_{6}\mathstrut -\mathstrut \) \(526\) \(\beta_{5}\mathstrut +\mathstrut \) \(6843\) \(\beta_{4}\mathstrut -\mathstrut \) \(2034\) \(\beta_{3}\mathstrut +\mathstrut \) \(17881\) \(\beta_{2}\mathstrut +\mathstrut \) \(18375\) \(\beta_{1}\mathstrut +\mathstrut \) \(23669\)
\(\nu^{11}\)\(=\)\(12635\) \(\beta_{14}\mathstrut -\mathstrut \) \(4418\) \(\beta_{13}\mathstrut -\mathstrut \) \(7062\) \(\beta_{12}\mathstrut +\mathstrut \) \(20997\) \(\beta_{11}\mathstrut +\mathstrut \) \(3346\) \(\beta_{10}\mathstrut +\mathstrut \) \(9943\) \(\beta_{9}\mathstrut +\mathstrut \) \(24828\) \(\beta_{8}\mathstrut +\mathstrut \) \(8963\) \(\beta_{7}\mathstrut +\mathstrut \) \(3922\) \(\beta_{6}\mathstrut -\mathstrut \) \(1762\) \(\beta_{5}\mathstrut +\mathstrut \) \(24057\) \(\beta_{4}\mathstrut -\mathstrut \) \(8132\) \(\beta_{3}\mathstrut +\mathstrut \) \(59523\) \(\beta_{2}\mathstrut +\mathstrut \) \(63659\) \(\beta_{1}\mathstrut +\mathstrut \) \(70120\)
\(\nu^{12}\)\(=\)\(48467\) \(\beta_{14}\mathstrut -\mathstrut \) \(15279\) \(\beta_{13}\mathstrut -\mathstrut \) \(23800\) \(\beta_{12}\mathstrut +\mathstrut \) \(67189\) \(\beta_{11}\mathstrut +\mathstrut \) \(13729\) \(\beta_{10}\mathstrut +\mathstrut \) \(31522\) \(\beta_{9}\mathstrut +\mathstrut \) \(85945\) \(\beta_{8}\mathstrut +\mathstrut \) \(34397\) \(\beta_{7}\mathstrut +\mathstrut \) \(13259\) \(\beta_{6}\mathstrut -\mathstrut \) \(7577\) \(\beta_{5}\mathstrut +\mathstrut \) \(81403\) \(\beta_{4}\mathstrut -\mathstrut \) \(23986\) \(\beta_{3}\mathstrut +\mathstrut \) \(209822\) \(\beta_{2}\mathstrut +\mathstrut \) \(207182\) \(\beta_{1}\mathstrut +\mathstrut \) \(255986\)
\(\nu^{13}\)\(=\)\(167007\) \(\beta_{14}\mathstrut -\mathstrut \) \(56988\) \(\beta_{13}\mathstrut -\mathstrut \) \(87682\) \(\beta_{12}\mathstrut +\mathstrut \) \(242649\) \(\beta_{11}\mathstrut +\mathstrut \) \(40201\) \(\beta_{10}\mathstrut +\mathstrut \) \(123684\) \(\beta_{9}\mathstrut +\mathstrut \) \(302425\) \(\beta_{8}\mathstrut +\mathstrut \) \(113697\) \(\beta_{7}\mathstrut +\mathstrut \) \(51426\) \(\beta_{6}\mathstrut -\mathstrut \) \(25439\) \(\beta_{5}\mathstrut +\mathstrut \) \(282789\) \(\beta_{4}\mathstrut -\mathstrut \) \(89340\) \(\beta_{3}\mathstrut +\mathstrut \) \(709373\) \(\beta_{2}\mathstrut +\mathstrut \) \(709870\) \(\beta_{1}\mathstrut +\mathstrut \) \(807039\)
\(\nu^{14}\)\(=\)\(610525\) \(\beta_{14}\mathstrut -\mathstrut \) \(197701\) \(\beta_{13}\mathstrut -\mathstrut \) \(297946\) \(\beta_{12}\mathstrut +\mathstrut \) \(801830\) \(\beta_{11}\mathstrut +\mathstrut \) \(151022\) \(\beta_{10}\mathstrut +\mathstrut \) \(407785\) \(\beta_{9}\mathstrut +\mathstrut \) \(1048743\) \(\beta_{8}\mathstrut +\mathstrut \) \(413842\) \(\beta_{7}\mathstrut +\mathstrut \) \(177390\) \(\beta_{6}\mathstrut -\mathstrut \) \(99676\) \(\beta_{5}\mathstrut +\mathstrut \) \(962659\) \(\beta_{4}\mathstrut -\mathstrut \) \(281914\) \(\beta_{3}\mathstrut +\mathstrut \) \(2468654\) \(\beta_{2}\mathstrut +\mathstrut \) \(2359527\) \(\beta_{1}\mathstrut +\mathstrut \) \(2852415\)

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
−2.53445
−2.37460
−1.85209
−1.72375
−1.51118
−0.907771
−0.424375
−0.0638672
0.823119
1.24368
1.38748
1.90080
2.77411
2.83620
3.42670
−1.00000 −2.53445 1.00000 1.00000 2.53445 −3.43636 −1.00000 3.42346 −1.00000
1.2 −1.00000 −2.37460 1.00000 1.00000 2.37460 −1.30659 −1.00000 2.63872 −1.00000
1.3 −1.00000 −1.85209 1.00000 1.00000 1.85209 1.81131 −1.00000 0.430244 −1.00000
1.4 −1.00000 −1.72375 1.00000 1.00000 1.72375 4.05342 −1.00000 −0.0286805 −1.00000
1.5 −1.00000 −1.51118 1.00000 1.00000 1.51118 −0.990203 −1.00000 −0.716341 −1.00000
1.6 −1.00000 −0.907771 1.00000 1.00000 0.907771 −0.750305 −1.00000 −2.17595 −1.00000
1.7 −1.00000 −0.424375 1.00000 1.00000 0.424375 −2.28477 −1.00000 −2.81991 −1.00000
1.8 −1.00000 −0.0638672 1.00000 1.00000 0.0638672 2.62362 −1.00000 −2.99592 −1.00000
1.9 −1.00000 0.823119 1.00000 1.00000 −0.823119 4.98726 −1.00000 −2.32248 −1.00000
1.10 −1.00000 1.24368 1.00000 1.00000 −1.24368 2.03579 −1.00000 −1.45326 −1.00000
1.11 −1.00000 1.38748 1.00000 1.00000 −1.38748 −5.08575 −1.00000 −1.07490 −1.00000
1.12 −1.00000 1.90080 1.00000 1.00000 −1.90080 −1.33242 −1.00000 0.613049 −1.00000
1.13 −1.00000 2.77411 1.00000 1.00000 −2.77411 0.220732 −1.00000 4.69567 −1.00000
1.14 −1.00000 2.83620 1.00000 1.00000 −2.83620 3.77087 −1.00000 5.04400 −1.00000
1.15 −1.00000 3.42670 1.00000 1.00000 −3.42670 2.68339 −1.00000 8.74228 −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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

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