Properties

Label 4025.2.a.bb
Level 4025
Weight 2
Character orbit 4025.a
Self dual Yes
Analytic conductor 32.140
Analytic rank 0
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(x^{13}\mathstrut -\mathstrut \) \(22\) \(x^{12}\mathstrut +\mathstrut \) \(18\) \(x^{11}\mathstrut +\mathstrut \) \(187\) \(x^{10}\mathstrut -\mathstrut \) \(118\) \(x^{9}\mathstrut -\mathstrut \) \(772\) \(x^{8}\mathstrut +\mathstrut \) \(346\) \(x^{7}\mathstrut +\mathstrut \) \(1581\) \(x^{6}\mathstrut -\mathstrut \) \(443\) \(x^{5}\mathstrut -\mathstrut \) \(1429\) \(x^{4}\mathstrut +\mathstrut \) \(193\) \(x^{3}\mathstrut +\mathstrut \) \(386\) \(x^{2}\mathstrut -\mathstrut \) \(3\) \(x\mathstrut -\mathstrut \) \(5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(134\) \(\nu^{13}\mathstrut +\mathstrut \) \(37911\) \(\nu^{12}\mathstrut +\mathstrut \) \(9431\) \(\nu^{11}\mathstrut -\mathstrut \) \(833971\) \(\nu^{10}\mathstrut -\mathstrut \) \(178425\) \(\nu^{9}\mathstrut +\mathstrut \) \(6807185\) \(\nu^{8}\mathstrut +\mathstrut \) \(1395439\) \(\nu^{7}\mathstrut -\mathstrut \) \(25174921\) \(\nu^{6}\mathstrut -\mathstrut \) \(5051683\) \(\nu^{5}\mathstrut +\mathstrut \) \(40706850\) \(\nu^{4}\mathstrut +\mathstrut \) \(8180960\) \(\nu^{3}\mathstrut -\mathstrut \) \(22002987\) \(\nu^{2}\mathstrut -\mathstrut \) \(4823059\) \(\nu\mathstrut +\mathstrut \) \(1719962\)\()/1242873\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(8617\) \(\nu^{13}\mathstrut -\mathstrut \) \(22078\) \(\nu^{12}\mathstrut +\mathstrut \) \(273594\) \(\nu^{11}\mathstrut +\mathstrut \) \(349093\) \(\nu^{10}\mathstrut -\mathstrut \) \(3046784\) \(\nu^{9}\mathstrut -\mathstrut \) \(1722211\) \(\nu^{8}\mathstrut +\mathstrut \) \(15312112\) \(\nu^{7}\mathstrut +\mathstrut \) \(2054981\) \(\nu^{6}\mathstrut -\mathstrut \) \(35751429\) \(\nu^{5}\mathstrut +\mathstrut \) \(4401504\) \(\nu^{4}\mathstrut +\mathstrut \) \(34327336\) \(\nu^{3}\mathstrut -\mathstrut \) \(7553195\) \(\nu^{2}\mathstrut -\mathstrut \) \(8567221\) \(\nu\mathstrut +\mathstrut \) \(208671\)\()/414291\)
\(\beta_{6}\)\(=\)\((\)\(3427\) \(\nu^{13}\mathstrut -\mathstrut \) \(14217\) \(\nu^{12}\mathstrut -\mathstrut \) \(50538\) \(\nu^{11}\mathstrut +\mathstrut \) \(246032\) \(\nu^{10}\mathstrut +\mathstrut \) \(208975\) \(\nu^{9}\mathstrut -\mathstrut \) \(1485425\) \(\nu^{8}\mathstrut +\mathstrut \) \(41160\) \(\nu^{7}\mathstrut +\mathstrut \) \(3684384\) \(\nu^{6}\mathstrut -\mathstrut \) \(1876665\) \(\nu^{5}\mathstrut -\mathstrut \) \(3337746\) \(\nu^{4}\mathstrut +\mathstrut \) \(3436136\) \(\nu^{3}\mathstrut +\mathstrut \) \(746869\) \(\nu^{2}\mathstrut -\mathstrut \) \(1764856\) \(\nu\mathstrut -\mathstrut \) \(136436\)\()/138097\)
\(\beta_{7}\)\(=\)\((\)\(37777\) \(\nu^{13}\mathstrut +\mathstrut \) \(6483\) \(\nu^{12}\mathstrut -\mathstrut \) \(831559\) \(\nu^{11}\mathstrut -\mathstrut \) \(153367\) \(\nu^{10}\mathstrut +\mathstrut \) \(6791373\) \(\nu^{9}\mathstrut +\mathstrut \) \(1291991\) \(\nu^{8}\mathstrut -\mathstrut \) \(25128557\) \(\nu^{7}\mathstrut -\mathstrut \) \(4839829\) \(\nu^{6}\mathstrut +\mathstrut \) \(40647488\) \(\nu^{5}\mathstrut +\mathstrut \) \(7989474\) \(\nu^{4}\mathstrut -\mathstrut \) \(21977125\) \(\nu^{3}\mathstrut -\mathstrut \) \(4771335\) \(\nu^{2}\mathstrut +\mathstrut \) \(476687\) \(\nu\mathstrut -\mathstrut \) \(670\)\()/1242873\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(17266\) \(\nu^{13}\mathstrut +\mathstrut \) \(28793\) \(\nu^{12}\mathstrut +\mathstrut \) \(353631\) \(\nu^{11}\mathstrut -\mathstrut \) \(477956\) \(\nu^{10}\mathstrut -\mathstrut \) \(2836277\) \(\nu^{9}\mathstrut +\mathstrut \) \(2649746\) \(\nu^{8}\mathstrut +\mathstrut \) \(11524948\) \(\nu^{7}\mathstrut -\mathstrut \) \(5168881\) \(\nu^{6}\mathstrut -\mathstrut \) \(25230657\) \(\nu^{5}\mathstrut +\mathstrut \) \(449829\) \(\nu^{4}\mathstrut +\mathstrut \) \(27355117\) \(\nu^{3}\mathstrut +\mathstrut \) \(5168560\) \(\nu^{2}\mathstrut -\mathstrut \) \(9402664\) \(\nu\mathstrut -\mathstrut \) \(1259865\)\()/414291\)
\(\beta_{9}\)\(=\)\((\)\(74561\) \(\nu^{13}\mathstrut -\mathstrut \) \(92562\) \(\nu^{12}\mathstrut -\mathstrut \) \(1571588\) \(\nu^{11}\mathstrut +\mathstrut \) \(1647493\) \(\nu^{10}\mathstrut +\mathstrut \) \(12628452\) \(\nu^{9}\mathstrut -\mathstrut \) \(10583948\) \(\nu^{8}\mathstrut -\mathstrut \) \(48372151\) \(\nu^{7}\mathstrut +\mathstrut \) \(29999800\) \(\nu^{6}\mathstrut +\mathstrut \) \(89700733\) \(\nu^{5}\mathstrut -\mathstrut \) \(36175764\) \(\nu^{4}\mathstrut -\mathstrut \) \(72909281\) \(\nu^{3}\mathstrut +\mathstrut \) \(12237873\) \(\nu^{2}\mathstrut +\mathstrut \) \(22678219\) \(\nu\mathstrut +\mathstrut \) \(3221839\)\()/1242873\)
\(\beta_{10}\)\(=\)\((\)\(92428\) \(\nu^{13}\mathstrut -\mathstrut \) \(123405\) \(\nu^{12}\mathstrut -\mathstrut \) \(1867552\) \(\nu^{11}\mathstrut +\mathstrut \) \(2127977\) \(\nu^{10}\mathstrut +\mathstrut \) \(14235777\) \(\nu^{9}\mathstrut -\mathstrut \) \(12965704\) \(\nu^{8}\mathstrut -\mathstrut \) \(51178406\) \(\nu^{7}\mathstrut +\mathstrut \) \(33005087\) \(\nu^{6}\mathstrut +\mathstrut \) \(87794291\) \(\nu^{5}\mathstrut -\mathstrut \) \(29107302\) \(\nu^{4}\mathstrut -\mathstrut \) \(61333048\) \(\nu^{3}\mathstrut -\mathstrut \) \(3662787\) \(\nu^{2}\mathstrut +\mathstrut \) \(6952400\) \(\nu\mathstrut +\mathstrut \) \(4568996\)\()/1242873\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(114541\) \(\nu^{13}\mathstrut +\mathstrut \) \(106461\) \(\nu^{12}\mathstrut +\mathstrut \) \(2641672\) \(\nu^{11}\mathstrut -\mathstrut \) \(2117591\) \(\nu^{10}\mathstrut -\mathstrut \) \(23419830\) \(\nu^{9}\mathstrut +\mathstrut \) \(15754684\) \(\nu^{8}\mathstrut +\mathstrut \) \(99159767\) \(\nu^{7}\mathstrut -\mathstrut \) \(53798933\) \(\nu^{6}\mathstrut -\mathstrut \) \(199755620\) \(\nu^{5}\mathstrut +\mathstrut \) \(81640410\) \(\nu^{4}\mathstrut +\mathstrut \) \(158892526\) \(\nu^{3}\mathstrut -\mathstrut \) \(42632694\) \(\nu^{2}\mathstrut -\mathstrut \) \(23191094\) \(\nu\mathstrut +\mathstrut \) \(1997443\)\()/1242873\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(64550\) \(\nu^{13}\mathstrut +\mathstrut \) \(87137\) \(\nu^{12}\mathstrut +\mathstrut \) \(1331797\) \(\nu^{11}\mathstrut -\mathstrut \) \(1563334\) \(\nu^{10}\mathstrut -\mathstrut \) \(10454477\) \(\nu^{9}\mathstrut +\mathstrut \) \(10248993\) \(\nu^{8}\mathstrut +\mathstrut \) \(39148617\) \(\nu^{7}\mathstrut -\mathstrut \) \(30423831\) \(\nu^{6}\mathstrut -\mathstrut \) \(70995548\) \(\nu^{5}\mathstrut +\mathstrut \) \(41000631\) \(\nu^{4}\mathstrut +\mathstrut \) \(53938016\) \(\nu^{3}\mathstrut -\mathstrut \) \(21439823\) \(\nu^{2}\mathstrut -\mathstrut \) \(9671628\) \(\nu\mathstrut +\mathstrut \) \(1835824\)\()/414291\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(203639\) \(\nu^{13}\mathstrut +\mathstrut \) \(277101\) \(\nu^{12}\mathstrut +\mathstrut \) \(4169738\) \(\nu^{11}\mathstrut -\mathstrut \) \(4742359\) \(\nu^{10}\mathstrut -\mathstrut \) \(32522883\) \(\nu^{9}\mathstrut +\mathstrut \) \(28618406\) \(\nu^{8}\mathstrut +\mathstrut \) \(121398712\) \(\nu^{7}\mathstrut -\mathstrut \) \(72669595\) \(\nu^{6}\mathstrut -\mathstrut \) \(221496586\) \(\nu^{5}\mathstrut +\mathstrut \) \(69984327\) \(\nu^{4}\mathstrut +\mathstrut \) \(175750652\) \(\nu^{3}\mathstrut -\mathstrut \) \(12281418\) \(\nu^{2}\mathstrut -\mathstrut \) \(40794727\) \(\nu\mathstrut -\mathstrut \) \(2790322\)\()/1242873\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(27\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(12\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(9\) \(\beta_{10}\mathstrut +\mathstrut \) \(23\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(48\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(76\)
\(\nu^{7}\)\(=\)\(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(29\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(83\) \(\beta_{3}\mathstrut +\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(155\) \(\beta_{1}\mathstrut +\mathstrut \) \(78\)
\(\nu^{8}\)\(=\)\(111\) \(\beta_{13}\mathstrut -\mathstrut \) \(14\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(67\) \(\beta_{10}\mathstrut +\mathstrut \) \(205\) \(\beta_{9}\mathstrut -\mathstrut \) \(17\) \(\beta_{8}\mathstrut -\mathstrut \) \(78\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(109\) \(\beta_{4}\mathstrut +\mathstrut \) \(110\) \(\beta_{3}\mathstrut +\mathstrut \) \(333\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(453\)
\(\nu^{9}\)\(=\)\(57\) \(\beta_{13}\mathstrut -\mathstrut \) \(21\) \(\beta_{12}\mathstrut -\mathstrut \) \(33\) \(\beta_{11}\mathstrut -\mathstrut \) \(107\) \(\beta_{10}\mathstrut +\mathstrut \) \(307\) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(37\) \(\beta_{7}\mathstrut -\mathstrut \) \(129\) \(\beta_{6}\mathstrut +\mathstrut \) \(161\) \(\beta_{5}\mathstrut +\mathstrut \) \(88\) \(\beta_{4}\mathstrut +\mathstrut \) \(654\) \(\beta_{3}\mathstrut +\mathstrut \) \(130\) \(\beta_{2}\mathstrut +\mathstrut \) \(932\) \(\beta_{1}\mathstrut +\mathstrut \) \(567\)
\(\nu^{10}\)\(=\)\(945\) \(\beta_{13}\mathstrut -\mathstrut \) \(149\) \(\beta_{12}\mathstrut -\mathstrut \) \(38\) \(\beta_{11}\mathstrut +\mathstrut \) \(476\) \(\beta_{10}\mathstrut +\mathstrut \) \(1692\) \(\beta_{9}\mathstrut -\mathstrut \) \(194\) \(\beta_{8}\mathstrut -\mathstrut \) \(571\) \(\beta_{7}\mathstrut +\mathstrut \) \(93\) \(\beta_{6}\mathstrut +\mathstrut \) \(83\) \(\beta_{5}\mathstrut +\mathstrut \) \(906\) \(\beta_{4}\mathstrut +\mathstrut \) \(931\) \(\beta_{3}\mathstrut +\mathstrut \) \(2344\) \(\beta_{2}\mathstrut +\mathstrut \) \(186\) \(\beta_{1}\mathstrut +\mathstrut \) \(2876\)
\(\nu^{11}\)\(=\)\(720\) \(\beta_{13}\mathstrut -\mathstrut \) \(281\) \(\beta_{12}\mathstrut -\mathstrut \) \(370\) \(\beta_{11}\mathstrut -\mathstrut \) \(854\) \(\beta_{10}\mathstrut +\mathstrut \) \(2872\) \(\beta_{9}\mathstrut -\mathstrut \) \(121\) \(\beta_{8}\mathstrut -\mathstrut \) \(444\) \(\beta_{7}\mathstrut -\mathstrut \) \(1155\) \(\beta_{6}\mathstrut +\mathstrut \) \(1513\) \(\beta_{5}\mathstrut +\mathstrut \) \(1043\) \(\beta_{4}\mathstrut +\mathstrut \) \(5055\) \(\beta_{3}\mathstrut +\mathstrut \) \(1186\) \(\beta_{2}\mathstrut +\mathstrut \) \(5815\) \(\beta_{1}\mathstrut +\mathstrut \) \(4049\)
\(\nu^{12}\)\(=\)\(7758\) \(\beta_{13}\mathstrut -\mathstrut \) \(1431\) \(\beta_{12}\mathstrut -\mathstrut \) \(479\) \(\beta_{11}\mathstrut +\mathstrut \) \(3338\) \(\beta_{10}\mathstrut +\mathstrut \) \(13556\) \(\beta_{9}\mathstrut -\mathstrut \) \(1883\) \(\beta_{8}\mathstrut -\mathstrut \) \(4128\) \(\beta_{7}\mathstrut +\mathstrut \) \(546\) \(\beta_{6}\mathstrut +\mathstrut \) \(1117\) \(\beta_{5}\mathstrut +\mathstrut \) \(7294\) \(\beta_{4}\mathstrut +\mathstrut \) \(7643\) \(\beta_{3}\mathstrut +\mathstrut \) \(16718\) \(\beta_{2}\mathstrut +\mathstrut \) \(1716\) \(\beta_{1}\mathstrut +\mathstrut \) \(19068\)
\(\nu^{13}\)\(=\)\(7632\) \(\beta_{13}\mathstrut -\mathstrut \) \(3084\) \(\beta_{12}\mathstrut -\mathstrut \) \(3546\) \(\beta_{11}\mathstrut -\mathstrut \) \(6459\) \(\beta_{10}\mathstrut +\mathstrut \) \(25222\) \(\beta_{9}\mathstrut -\mathstrut \) \(1596\) \(\beta_{8}\mathstrut -\mathstrut \) \(4431\) \(\beta_{7}\mathstrut -\mathstrut \) \(9803\) \(\beta_{6}\mathstrut +\mathstrut \) \(13271\) \(\beta_{5}\mathstrut +\mathstrut \) \(10489\) \(\beta_{4}\mathstrut +\mathstrut \) \(38763\) \(\beta_{3}\mathstrut +\mathstrut \) \(10361\) \(\beta_{2}\mathstrut +\mathstrut \) \(37406\) \(\beta_{1}\mathstrut +\mathstrut \) \(28967\)

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
−2.64833
−2.25329
−1.83943
−1.76659
−1.25067
−0.582976
−0.116065
0.117364
0.704319
1.31493
1.82692
2.17410
2.52898
2.79074
−2.64833 0.547903 5.01366 0 −1.45103 1.00000 −7.98118 −2.69980 0
1.2 −2.25329 2.73097 3.07734 0 −6.15367 1.00000 −2.42755 4.45818 0
1.3 −1.83943 −1.64563 1.38349 0 3.02702 1.00000 1.13403 −0.291898 0
1.4 −1.76659 −0.991358 1.12083 0 1.75132 1.00000 1.55314 −2.01721 0
1.5 −1.25067 3.28603 −0.435826 0 −4.10974 1.00000 3.04641 7.79801 0
1.6 −0.582976 −0.367598 −1.66014 0 0.214301 1.00000 2.13377 −2.86487 0
1.7 −0.116065 1.59146 −1.98653 0 −0.184713 1.00000 0.462695 −0.467244 0
1.8 0.117364 0.701283 −1.98623 0 0.0823052 1.00000 −0.467839 −2.50820 0
1.9 0.704319 −2.54639 −1.50393 0 −1.79347 1.00000 −2.46789 3.48411 0
1.10 1.31493 3.36299 −0.270967 0 4.42209 1.00000 −2.98616 8.30972 0
1.11 1.82692 −2.52976 1.33763 0 −4.62167 1.00000 −1.21009 3.39970 0
1.12 2.17410 1.37249 2.72672 0 2.98394 1.00000 1.57997 −1.11626 0
1.13 2.52898 −1.80521 4.39572 0 −4.56532 1.00000 6.05871 0.258773 0
1.14 2.79074 2.29281 5.78823 0 6.39865 1.00000 10.5720 2.25700 0
\(n\): 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
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{14} - \cdots\)
\(T_{3}^{14} - \cdots\)
\(T_{11}^{14} + \cdots\)