Properties

Label 4025.2.a.bc
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{7} q^{3} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{1} - \beta_{11} ) q^{6} \) \(+ q^{7}\) \( + ( \beta_{1} + \beta_{3} ) q^{8} \) \( + ( 1 + \beta_{5} - \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{7} q^{3} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{1} - \beta_{11} ) q^{6} \) \(+ q^{7}\) \( + ( \beta_{1} + \beta_{3} ) q^{8} \) \( + ( 1 + \beta_{5} - \beta_{7} ) q^{9} \) \( + \beta_{8} q^{11} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{12} \) \( + ( 1 + \beta_{6} ) q^{13} \) \( + \beta_{1} q^{14} \) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{16} \) \( + ( 1 - \beta_{3} - \beta_{4} - \beta_{12} ) q^{17} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{18} \) \( + ( \beta_{1} - \beta_{2} - \beta_{7} + \beta_{9} ) q^{19} \) \( -\beta_{7} q^{21} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{10} ) q^{22} \) \(+ q^{23}\) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} ) q^{24} \) \( + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{26} \) \( + ( 1 + \beta_{2} + \beta_{5} - \beta_{7} + \beta_{12} ) q^{27} \) \( + ( 1 + \beta_{2} ) q^{28} \) \( + ( 1 + \beta_{1} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{29} \) \( + ( 1 - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{13} ) q^{31} \) \( + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{32} \) \( + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{33} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{34} \) \( + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{36} \) \( + ( 2 + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{37} \) \( + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{38} \) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{39} \) \( + ( -1 - \beta_{2} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{41} \) \( + ( -1 + \beta_{1} - \beta_{11} ) q^{42} \) \( + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{43} \) \( + ( 2 + 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{13} ) q^{44} \) \( + \beta_{1} q^{46} \) \( + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{47} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{48} \) \(+ q^{49}\) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{51} \) \( + ( 2 + \beta_{2} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{52} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{53} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{54} \) \( + ( \beta_{1} + \beta_{3} ) q^{56} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{57} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{58} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{59} \) \( + ( \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{61} \) \( + ( -2 + \beta_{1} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{62} \) \( + ( 1 + \beta_{5} - \beta_{7} ) q^{63} \) \( + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 4 \beta_{11} + 3 \beta_{12} ) q^{64} \) \( + ( -2 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{66} \) \( + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} - 2 \beta_{12} - \beta_{13} ) q^{67} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{68} \) \( -\beta_{7} q^{69} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{71} \) \( + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{11} ) q^{72} \) \( + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} ) q^{73} \) \( + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{74} \) \( + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{13} ) q^{76} \) \( + \beta_{8} q^{77} \) \( + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} ) q^{78} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{79} \) \( + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{81} \) \( + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{82} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} + 2 \beta_{13} ) q^{83} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{84} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 4 \beta_{7} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{86} \) \( + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{12} ) q^{87} \) \( + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{88} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{89} \) \( + ( 1 + \beta_{6} ) q^{91} \) \( + ( 1 + \beta_{2} ) q^{92} \) \( + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{93} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{13} ) q^{94} \) \( + ( -9 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{11} - \beta_{12} ) q^{96} \) \( + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{97} \) \( + \beta_{1} q^{98} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 17q^{12} \) \(\mathstrut +\mathstrut 11q^{13} \) \(\mathstrut +\mathstrut 3q^{14} \) \(\mathstrut +\mathstrut 23q^{16} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 15q^{18} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 23q^{22} \) \(\mathstrut +\mathstrut 14q^{23} \) \(\mathstrut -\mathstrut 12q^{24} \) \(\mathstrut -\mathstrut 9q^{26} \) \(\mathstrut +\mathstrut 25q^{27} \) \(\mathstrut +\mathstrut 17q^{28} \) \(\mathstrut +\mathstrut 7q^{29} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut +\mathstrut 24q^{32} \) \(\mathstrut +\mathstrut 6q^{33} \) \(\mathstrut -\mathstrut 14q^{34} \) \(\mathstrut +\mathstrut 13q^{36} \) \(\mathstrut +\mathstrut 22q^{37} \) \(\mathstrut +\mathstrut 20q^{38} \) \(\mathstrut -\mathstrut 10q^{39} \) \(\mathstrut -\mathstrut 17q^{41} \) \(\mathstrut -\mathstrut 4q^{42} \) \(\mathstrut +\mathstrut 18q^{43} \) \(\mathstrut +\mathstrut 28q^{44} \) \(\mathstrut +\mathstrut 3q^{46} \) \(\mathstrut +\mathstrut 30q^{47} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut +\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 8q^{52} \) \(\mathstrut +\mathstrut 11q^{53} \) \(\mathstrut +\mathstrut 20q^{54} \) \(\mathstrut +\mathstrut 3q^{56} \) \(\mathstrut +\mathstrut 18q^{57} \) \(\mathstrut +\mathstrut 38q^{58} \) \(\mathstrut -\mathstrut 22q^{59} \) \(\mathstrut -\mathstrut 8q^{61} \) \(\mathstrut -\mathstrut 22q^{62} \) \(\mathstrut +\mathstrut 18q^{63} \) \(\mathstrut +\mathstrut 29q^{64} \) \(\mathstrut -\mathstrut 9q^{66} \) \(\mathstrut +\mathstrut 39q^{67} \) \(\mathstrut +\mathstrut q^{68} \) \(\mathstrut +\mathstrut 4q^{69} \) \(\mathstrut -\mathstrut 5q^{71} \) \(\mathstrut -\mathstrut 24q^{72} \) \(\mathstrut +\mathstrut 18q^{73} \) \(\mathstrut +\mathstrut 35q^{74} \) \(\mathstrut -\mathstrut 41q^{76} \) \(\mathstrut +\mathstrut 5q^{77} \) \(\mathstrut -\mathstrut 22q^{78} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 8q^{82} \) \(\mathstrut +\mathstrut 24q^{83} \) \(\mathstrut +\mathstrut 17q^{84} \) \(\mathstrut -\mathstrut 26q^{86} \) \(\mathstrut +\mathstrut 5q^{87} \) \(\mathstrut +\mathstrut 58q^{88} \) \(\mathstrut +\mathstrut 25q^{89} \) \(\mathstrut +\mathstrut 11q^{91} \) \(\mathstrut +\mathstrut 17q^{92} \) \(\mathstrut +\mathstrut 47q^{93} \) \(\mathstrut -\mathstrut 2q^{94} \) \(\mathstrut -\mathstrut 117q^{96} \) \(\mathstrut +\mathstrut 43q^{97} \) \(\mathstrut +\mathstrut 3q^{98} \) \(\mathstrut +\mathstrut 55q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(3\) \(x^{13}\mathstrut -\mathstrut \) \(18\) \(x^{12}\mathstrut +\mathstrut \) \(58\) \(x^{11}\mathstrut +\mathstrut \) \(111\) \(x^{10}\mathstrut -\mathstrut \) \(414\) \(x^{9}\mathstrut -\mathstrut \) \(244\) \(x^{8}\mathstrut +\mathstrut \) \(1330\) \(x^{7}\mathstrut -\mathstrut \) \(27\) \(x^{6}\mathstrut -\mathstrut \) \(1853\) \(x^{5}\mathstrut +\mathstrut \) \(539\) \(x^{4}\mathstrut +\mathstrut \) \(891\) \(x^{3}\mathstrut -\mathstrut \) \(218\) \(x^{2}\mathstrut -\mathstrut \) \(133\) \(x\mathstrut -\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(4055\) \(\nu^{13}\mathstrut +\mathstrut \) \(7975\) \(\nu^{12}\mathstrut +\mathstrut \) \(81881\) \(\nu^{11}\mathstrut -\mathstrut \) \(152987\) \(\nu^{10}\mathstrut -\mathstrut \) \(615369\) \(\nu^{9}\mathstrut +\mathstrut \) \(1080275\) \(\nu^{8}\mathstrut +\mathstrut \) \(2103680\) \(\nu^{7}\mathstrut -\mathstrut \) \(3407257\) \(\nu^{6}\mathstrut -\mathstrut \) \(3155676\) \(\nu^{5}\mathstrut +\mathstrut \) \(4556819\) \(\nu^{4}\mathstrut +\mathstrut \) \(1597821\) \(\nu^{3}\mathstrut -\mathstrut \) \(1900192\) \(\nu^{2}\mathstrut -\mathstrut \) \(75581\) \(\nu\mathstrut +\mathstrut \) \(142730\)\()/22937\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(21776\) \(\nu^{13}\mathstrut +\mathstrut \) \(33409\) \(\nu^{12}\mathstrut +\mathstrut \) \(446796\) \(\nu^{11}\mathstrut -\mathstrut \) \(619227\) \(\nu^{10}\mathstrut -\mathstrut \) \(3443457\) \(\nu^{9}\mathstrut +\mathstrut \) \(4180444\) \(\nu^{8}\mathstrut +\mathstrut \) \(12330084\) \(\nu^{7}\mathstrut -\mathstrut \) \(12373995\) \(\nu^{6}\mathstrut -\mathstrut \) \(20539065\) \(\nu^{5}\mathstrut +\mathstrut \) \(14830170\) \(\nu^{4}\mathstrut +\mathstrut \) \(14277936\) \(\nu^{3}\mathstrut -\mathstrut \) \(4309323\) \(\nu^{2}\mathstrut -\mathstrut \) \(3505754\) \(\nu\mathstrut -\mathstrut \) \(316562\)\()/22937\)
\(\beta_{6}\)\(=\)\((\)\(25090\) \(\nu^{13}\mathstrut -\mathstrut \) \(41963\) \(\nu^{12}\mathstrut -\mathstrut \) \(504992\) \(\nu^{11}\mathstrut +\mathstrut \) \(782303\) \(\nu^{10}\mathstrut +\mathstrut \) \(3774769\) \(\nu^{9}\mathstrut -\mathstrut \) \(5321389\) \(\nu^{8}\mathstrut -\mathstrut \) \(12816147\) \(\nu^{7}\mathstrut +\mathstrut \) \(15921004\) \(\nu^{6}\mathstrut +\mathstrut \) \(19231422\) \(\nu^{5}\mathstrut -\mathstrut \) \(19453226\) \(\nu^{4}\mathstrut -\mathstrut \) \(10653101\) \(\nu^{3}\mathstrut +\mathstrut \) \(6121945\) \(\nu^{2}\mathstrut +\mathstrut \) \(2002366\) \(\nu\mathstrut +\mathstrut \) \(89726\)\()/22937\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(33730\) \(\nu^{13}\mathstrut +\mathstrut \) \(54713\) \(\nu^{12}\mathstrut +\mathstrut \) \(686159\) \(\nu^{11}\mathstrut -\mathstrut \) \(1017543\) \(\nu^{10}\mathstrut -\mathstrut \) \(5217903\) \(\nu^{9}\mathstrut +\mathstrut \) \(6900067\) \(\nu^{8}\mathstrut +\mathstrut \) \(18260121\) \(\nu^{7}\mathstrut -\mathstrut \) \(20566123\) \(\nu^{6}\mathstrut -\mathstrut \) \(29115247\) \(\nu^{5}\mathstrut +\mathstrut \) \(25007434\) \(\nu^{4}\mathstrut +\mathstrut \) \(18540140\) \(\nu^{3}\mathstrut -\mathstrut \) \(7714479\) \(\nu^{2}\mathstrut -\mathstrut \) \(4174085\) \(\nu\mathstrut -\mathstrut \) \(320996\)\()/22937\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(34523\) \(\nu^{13}\mathstrut +\mathstrut \) \(57715\) \(\nu^{12}\mathstrut +\mathstrut \) \(699434\) \(\nu^{11}\mathstrut -\mathstrut \) \(1075557\) \(\nu^{10}\mathstrut -\mathstrut \) \(5289707\) \(\nu^{9}\mathstrut +\mathstrut \) \(7310378\) \(\nu^{8}\mathstrut +\mathstrut \) \(18364740\) \(\nu^{7}\mathstrut -\mathstrut \) \(21833829\) \(\nu^{6}\mathstrut -\mathstrut \) \(28918074\) \(\nu^{5}\mathstrut +\mathstrut \) \(26520409\) \(\nu^{4}\mathstrut +\mathstrut \) \(18099684\) \(\nu^{3}\mathstrut -\mathstrut \) \(7961510\) \(\nu^{2}\mathstrut -\mathstrut \) \(4230503\) \(\nu\mathstrut -\mathstrut \) \(436758\)\()/22937\)
\(\beta_{9}\)\(=\)\((\)\(42027\) \(\nu^{13}\mathstrut -\mathstrut \) \(75160\) \(\nu^{12}\mathstrut -\mathstrut \) \(839833\) \(\nu^{11}\mathstrut +\mathstrut \) \(1403897\) \(\nu^{10}\mathstrut +\mathstrut \) \(6219514\) \(\nu^{9}\mathstrut -\mathstrut \) \(9565583\) \(\nu^{8}\mathstrut -\mathstrut \) \(20836696\) \(\nu^{7}\mathstrut +\mathstrut \) \(28645999\) \(\nu^{6}\mathstrut +\mathstrut \) \(30567527\) \(\nu^{5}\mathstrut -\mathstrut \) \(34949868\) \(\nu^{4}\mathstrut -\mathstrut \) \(16164725\) \(\nu^{3}\mathstrut +\mathstrut \) \(10894262\) \(\nu^{2}\mathstrut +\mathstrut \) \(3009015\) \(\nu\mathstrut +\mathstrut \) \(131732\)\()/22937\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(45854\) \(\nu^{13}\mathstrut +\mathstrut \) \(78020\) \(\nu^{12}\mathstrut +\mathstrut \) \(926777\) \(\nu^{11}\mathstrut -\mathstrut \) \(1457654\) \(\nu^{10}\mathstrut -\mathstrut \) \(6982144\) \(\nu^{9}\mathstrut +\mathstrut \) \(9941128\) \(\nu^{8}\mathstrut +\mathstrut \) \(24081761\) \(\nu^{7}\mathstrut -\mathstrut \) \(29850195\) \(\nu^{6}\mathstrut -\mathstrut \) \(37450710\) \(\nu^{5}\mathstrut +\mathstrut \) \(36707581\) \(\nu^{4}\mathstrut +\mathstrut \) \(22798483\) \(\nu^{3}\mathstrut -\mathstrut \) \(11779454\) \(\nu^{2}\mathstrut -\mathstrut \) \(5051254\) \(\nu\mathstrut -\mathstrut \) \(264833\)\()/22937\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(46477\) \(\nu^{13}\mathstrut +\mathstrut \) \(79019\) \(\nu^{12}\mathstrut +\mathstrut \) \(938797\) \(\nu^{11}\mathstrut -\mathstrut \) \(1473873\) \(\nu^{10}\mathstrut -\mathstrut \) \(7064153\) \(\nu^{9}\mathstrut +\mathstrut \) \(10030001\) \(\nu^{8}\mathstrut +\mathstrut \) \(24294777\) \(\nu^{7}\mathstrut -\mathstrut \) \(30025957\) \(\nu^{6}\mathstrut -\mathstrut \) \(37494256\) \(\nu^{5}\mathstrut +\mathstrut \) \(36720610\) \(\nu^{4}\mathstrut +\mathstrut \) \(22338951\) \(\nu^{3}\mathstrut -\mathstrut \) \(11527225\) \(\nu^{2}\mathstrut -\mathstrut \) \(4784149\) \(\nu\mathstrut -\mathstrut \) \(326507\)\()/22937\)
\(\beta_{12}\)\(=\)\((\)\(52513\) \(\nu^{13}\mathstrut -\mathstrut \) \(88882\) \(\nu^{12}\mathstrut -\mathstrut \) \(1064311\) \(\nu^{11}\mathstrut +\mathstrut \) \(1659877\) \(\nu^{10}\mathstrut +\mathstrut \) \(8052400\) \(\nu^{9}\mathstrut -\mathstrut \) \(11315041\) \(\nu^{8}\mathstrut -\mathstrut \) \(27962096\) \(\nu^{7}\mathstrut +\mathstrut \) \(33964264\) \(\nu^{6}\mathstrut +\mathstrut \) \(44003702\) \(\nu^{5}\mathstrut -\mathstrut \) \(41769609\) \(\nu^{4}\mathstrut -\mathstrut \) \(27425832\) \(\nu^{3}\mathstrut +\mathstrut \) \(13388234\) \(\nu^{2}\mathstrut +\mathstrut \) \(6282190\) \(\nu\mathstrut +\mathstrut \) \(362877\)\()/22937\)
\(\beta_{13}\)\(=\)\((\)\(82827\) \(\nu^{13}\mathstrut -\mathstrut \) \(143014\) \(\nu^{12}\mathstrut -\mathstrut \) \(1668584\) \(\nu^{11}\mathstrut +\mathstrut \) \(2672763\) \(\nu^{10}\mathstrut +\mathstrut \) \(12509311\) \(\nu^{9}\mathstrut -\mathstrut \) \(18243758\) \(\nu^{8}\mathstrut -\mathstrut \) \(42782683\) \(\nu^{7}\mathstrut +\mathstrut \) \(54897262\) \(\nu^{6}\mathstrut +\mathstrut \) \(65368699\) \(\nu^{5}\mathstrut -\mathstrut \) \(67878225\) \(\nu^{4}\mathstrut -\mathstrut \) \(38037463\) \(\nu^{3}\mathstrut +\mathstrut \) \(22247589\) \(\nu^{2}\mathstrut +\mathstrut \) \(7829560\) \(\nu\mathstrut +\mathstrut \) \(291069\)\()/22937\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(14\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(46\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(98\)
\(\nu^{7}\)\(=\)\(28\) \(\beta_{12}\mathstrut +\mathstrut \) \(18\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(83\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(182\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)
\(\nu^{8}\)\(=\)\(-\)\(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(47\) \(\beta_{12}\mathstrut +\mathstrut \) \(143\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(17\) \(\beta_{9}\mathstrut -\mathstrut \) \(68\) \(\beta_{8}\mathstrut -\mathstrut \) \(110\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(95\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(114\) \(\beta_{3}\mathstrut +\mathstrut \) \(303\) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(643\)
\(\nu^{9}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(285\) \(\beta_{12}\mathstrut +\mathstrut \) \(215\) \(\beta_{11}\mathstrut +\mathstrut \) \(173\) \(\beta_{10}\mathstrut +\mathstrut \) \(34\) \(\beta_{9}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut +\mathstrut \) \(58\) \(\beta_{7}\mathstrut +\mathstrut \) \(141\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(95\) \(\beta_{4}\mathstrut +\mathstrut \) \(655\) \(\beta_{3}\mathstrut +\mathstrut \) \(114\) \(\beta_{2}\mathstrut +\mathstrut \) \(1195\) \(\beta_{1}\mathstrut +\mathstrut \) \(244\)
\(\nu^{10}\)\(=\)\(-\)\(32\) \(\beta_{13}\mathstrut +\mathstrut \) \(517\) \(\beta_{12}\mathstrut +\mathstrut \) \(1293\) \(\beta_{11}\mathstrut +\mathstrut \) \(181\) \(\beta_{10}\mathstrut +\mathstrut \) \(190\) \(\beta_{9}\mathstrut -\mathstrut \) \(500\) \(\beta_{8}\mathstrut -\mathstrut \) \(907\) \(\beta_{7}\mathstrut +\mathstrut \) \(201\) \(\beta_{6}\mathstrut +\mathstrut \) \(757\) \(\beta_{5}\mathstrut +\mathstrut \) \(124\) \(\beta_{4}\mathstrut +\mathstrut \) \(1001\) \(\beta_{3}\mathstrut +\mathstrut \) \(2030\) \(\beta_{2}\mathstrut +\mathstrut \) \(241\) \(\beta_{1}\mathstrut +\mathstrut \) \(4397\)
\(\nu^{11}\)\(=\)\(20\) \(\beta_{13}\mathstrut +\mathstrut \) \(2567\) \(\beta_{12}\mathstrut +\mathstrut \) \(2171\) \(\beta_{11}\mathstrut +\mathstrut \) \(1602\) \(\beta_{10}\mathstrut +\mathstrut \) \(393\) \(\beta_{9}\mathstrut -\mathstrut \) \(221\) \(\beta_{8}\mathstrut +\mathstrut \) \(304\) \(\beta_{7}\mathstrut +\mathstrut \) \(1259\) \(\beta_{6}\mathstrut +\mathstrut \) \(205\) \(\beta_{5}\mathstrut +\mathstrut \) \(768\) \(\beta_{4}\mathstrut +\mathstrut \) \(5081\) \(\beta_{3}\mathstrut +\mathstrut \) \(1013\) \(\beta_{2}\mathstrut +\mathstrut \) \(8084\) \(\beta_{1}\mathstrut +\mathstrut \) \(2489\)
\(\nu^{12}\)\(=\)\(-\)\(343\) \(\beta_{13}\mathstrut +\mathstrut \) \(4943\) \(\beta_{12}\mathstrut +\mathstrut \) \(11007\) \(\beta_{11}\mathstrut +\mathstrut \) \(1792\) \(\beta_{10}\mathstrut +\mathstrut \) \(1799\) \(\beta_{9}\mathstrut -\mathstrut \) \(3687\) \(\beta_{8}\mathstrut -\mathstrut \) \(7100\) \(\beta_{7}\mathstrut +\mathstrut \) \(2044\) \(\beta_{6}\mathstrut +\mathstrut \) \(5820\) \(\beta_{5}\mathstrut +\mathstrut \) \(1071\) \(\beta_{4}\mathstrut +\mathstrut \) \(8470\) \(\beta_{3}\mathstrut +\mathstrut \) \(13876\) \(\beta_{2}\mathstrut +\mathstrut \) \(2596\) \(\beta_{1}\mathstrut +\mathstrut \) \(30917\)
\(\nu^{13}\)\(=\)\(252\) \(\beta_{13}\mathstrut +\mathstrut \) \(21770\) \(\beta_{12}\mathstrut +\mathstrut \) \(20092\) \(\beta_{11}\mathstrut +\mathstrut \) \(13735\) \(\beta_{10}\mathstrut +\mathstrut \) \(3872\) \(\beta_{9}\mathstrut -\mathstrut \) \(2314\) \(\beta_{8}\mathstrut +\mathstrut \) \(1138\) \(\beta_{7}\mathstrut +\mathstrut \) \(10635\) \(\beta_{6}\mathstrut +\mathstrut \) \(2154\) \(\beta_{5}\mathstrut +\mathstrut \) \(6065\) \(\beta_{4}\mathstrut +\mathstrut \) \(39173\) \(\beta_{3}\mathstrut +\mathstrut \) \(8771\) \(\beta_{2}\mathstrut +\mathstrut \) \(55926\) \(\beta_{1}\mathstrut +\mathstrut \) \(23306\)

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
−2.60812
−2.33696
−2.04050
−1.51853
−0.689960
−0.256730
−0.0822464
0.744662
1.29908
1.31920
1.63325
2.30520
2.42992
2.80172
−2.60812 2.93022 4.80228 0 −7.64237 1.00000 −7.30867 5.58621 0
1.2 −2.33696 0.00757499 3.46136 0 −0.0177024 1.00000 −3.41514 −2.99994 0
1.3 −2.04050 −1.31355 2.16365 0 2.68030 1.00000 −0.333918 −1.27458 0
1.4 −1.51853 0.630051 0.305932 0 −0.956751 1.00000 2.57249 −2.60304 0
1.5 −0.689960 2.51200 −1.52395 0 −1.73318 1.00000 2.43139 3.31014 0
1.6 −0.256730 −2.72326 −1.93409 0 0.699143 1.00000 1.01000 4.41613 0
1.7 −0.0822464 1.69790 −1.99324 0 −0.139646 1.00000 0.328429 −0.117136 0
1.8 0.744662 −1.85918 −1.44548 0 −1.38446 1.00000 −2.56572 0.456544 0
1.9 1.29908 −2.64602 −0.312383 0 −3.43741 1.00000 −3.00398 4.00145 0
1.10 1.31920 2.92477 −0.259709 0 3.85835 1.00000 −2.98101 5.55425 0
1.11 1.63325 −0.456743 0.667506 0 −0.745975 1.00000 −2.17630 −2.79139 0
1.12 2.30520 0.930494 3.31395 0 2.14497 1.00000 3.02891 −2.13418 0
1.13 2.42992 3.09771 3.90452 0 7.52721 1.00000 4.62785 6.59584 0
1.14 2.80172 −1.73197 5.84965 0 −4.85249 1.00000 10.7857 −0.000289796 0 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

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