Properties

Label 4030.2.a.p
Level 4030
Weight 2
Character orbit 4030.a
Self dual Yes
Analytic conductor 32.180
Analytic rank 0
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1797120146\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(3\) \(x^{8}\mathstrut -\mathstrut \) \(16\) \(x^{7}\mathstrut +\mathstrut \) \(46\) \(x^{6}\mathstrut +\mathstrut \) \(80\) \(x^{5}\mathstrut -\mathstrut \) \(212\) \(x^{4}\mathstrut -\mathstrut \) \(133\) \(x^{3}\mathstrut +\mathstrut \) \(294\) \(x^{2}\mathstrut +\mathstrut \) \(52\) \(x\mathstrut -\mathstrut \) \(112\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -95 \nu^{8} + 187 \nu^{7} + 1698 \nu^{6} - 2678 \nu^{5} - 9980 \nu^{4} + 10528 \nu^{3} + 21287 \nu^{2} - 7824 \nu - 11056 \)\()/236\)
\(\beta_{3}\)\(=\)\((\)\( -56 \nu^{8} + 127 \nu^{7} + 1009 \nu^{6} - 1832 \nu^{5} - 6150 \nu^{4} + 7186 \nu^{3} + 14220 \nu^{2} - 5071 \nu - 7850 \)\()/118\)
\(\beta_{4}\)\(=\)\((\)\( 71 \nu^{8} - 141 \nu^{7} - 1274 \nu^{6} + 1994 \nu^{5} + 7614 \nu^{4} - 7718 \nu^{3} - 16811 \nu^{2} + 5676 \nu + 8838 \)\()/118\)
\(\beta_{5}\)\(=\)\((\)\( -161 \nu^{8} + 343 \nu^{7} + 2864 \nu^{6} - 4854 \nu^{5} - 16988 \nu^{4} + 18580 \nu^{3} + 37313 \nu^{2} - 12374 \nu - 19368 \)\()/236\)
\(\beta_{6}\)\(=\)\((\)\( -207 \nu^{8} + 441 \nu^{7} + 3716 \nu^{6} - 6342 \nu^{5} - 22280 \nu^{4} + 24900 \nu^{3} + 49491 \nu^{2} - 17730 \nu - 25812 \)\()/236\)
\(\beta_{7}\)\(=\)\((\)\( 247 \nu^{8} - 557 \nu^{7} - 4344 \nu^{6} + 7954 \nu^{5} + 25476 \nu^{4} - 30960 \nu^{3} - 55535 \nu^{2} + 22018 \nu + 28604 \)\()/236\)
\(\beta_{8}\)\(=\)\((\)\( 198 \nu^{8} - 409 \nu^{7} - 3557 \nu^{6} + 5820 \nu^{5} + 21378 \nu^{4} - 22504 \nu^{3} - 47842 \nu^{2} + 15479 \nu + 25526 \)\()/118\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{5}\)\(=\)\(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{5}\mathstrut -\mathstrut \) \(23\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(66\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(35\) \(\beta_{8}\mathstrut +\mathstrut \) \(19\) \(\beta_{7}\mathstrut -\mathstrut \) \(58\) \(\beta_{6}\mathstrut +\mathstrut \) \(51\) \(\beta_{5}\mathstrut -\mathstrut \) \(22\) \(\beta_{4}\mathstrut +\mathstrut \) \(97\) \(\beta_{3}\mathstrut +\mathstrut \) \(88\) \(\beta_{2}\mathstrut +\mathstrut \) \(97\) \(\beta_{1}\mathstrut +\mathstrut \) \(191\)
\(\nu^{7}\)\(=\)\(153\) \(\beta_{8}\mathstrut +\mathstrut \) \(37\) \(\beta_{7}\mathstrut -\mathstrut \) \(30\) \(\beta_{6}\mathstrut +\mathstrut \) \(129\) \(\beta_{5}\mathstrut -\mathstrut \) \(236\) \(\beta_{4}\mathstrut +\mathstrut \) \(173\) \(\beta_{3}\mathstrut +\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(570\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\)
\(\nu^{8}\)\(=\)\(461\) \(\beta_{8}\mathstrut +\mathstrut \) \(251\) \(\beta_{7}\mathstrut -\mathstrut \) \(423\) \(\beta_{6}\mathstrut +\mathstrut \) \(653\) \(\beta_{5}\mathstrut -\mathstrut \) \(326\) \(\beta_{4}\mathstrut +\mathstrut \) \(964\) \(\beta_{3}\mathstrut +\mathstrut \) \(763\) \(\beta_{2}\mathstrut +\mathstrut \) \(973\) \(\beta_{1}\mathstrut +\mathstrut \) \(1471\)

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
3.25457
2.72026
2.42233
0.905402
0.887962
−0.777449
−1.27856
−2.33128
−2.80323
−1.00000 −3.25457 1.00000 −1.00000 3.25457 −1.51784 −1.00000 7.59221 1.00000
1.2 −1.00000 −2.72026 1.00000 −1.00000 2.72026 4.31759 −1.00000 4.39981 1.00000
1.3 −1.00000 −2.42233 1.00000 −1.00000 2.42233 −2.05608 −1.00000 2.86766 1.00000
1.4 −1.00000 −0.905402 1.00000 −1.00000 0.905402 −0.957463 −1.00000 −2.18025 1.00000
1.5 −1.00000 −0.887962 1.00000 −1.00000 0.887962 −2.77086 −1.00000 −2.21152 1.00000
1.6 −1.00000 0.777449 1.00000 −1.00000 −0.777449 1.65571 −1.00000 −2.39557 1.00000
1.7 −1.00000 1.27856 1.00000 −1.00000 −1.27856 −0.354220 −1.00000 −1.36528 1.00000
1.8 −1.00000 2.33128 1.00000 −1.00000 −2.33128 −4.98848 −1.00000 2.43487 1.00000
1.9 −1.00000 2.80323 1.00000 −1.00000 −2.80323 3.67164 −1.00000 4.85808 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(13\) \(1\)
\(31\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{9} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).