Properties

Label 4030.2.a.l
Level 4030
Weight 2
Character orbit 4030.a
Self dual Yes
Analytic conductor 32.180
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(14\) \(x^{6}\mathstrut -\mathstrut \) \(6\) \(x^{5}\mathstrut +\mathstrut \) \(54\) \(x^{4}\mathstrut +\mathstrut \) \(46\) \(x^{3}\mathstrut -\mathstrut \) \(32\) \(x^{2}\mathstrut -\mathstrut \) \(43\) \(x\mathstrut -\mathstrut \) \(11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{7} + \nu^{6} + 13 \nu^{5} - 7 \nu^{4} - 47 \nu^{3} + 2 \nu^{2} + 30 \nu + 7 \)
\(\beta_{3}\)\(=\)\( \nu^{7} - \nu^{6} - 13 \nu^{5} + 7 \nu^{4} + 47 \nu^{3} - \nu^{2} - 31 \nu - 11 \)
\(\beta_{4}\)\(=\)\((\)\( 9 \nu^{7} - 7 \nu^{6} - 120 \nu^{5} + 41 \nu^{4} + 448 \nu^{3} + 50 \nu^{2} - 318 \nu - 118 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{7} + 4 \nu^{6} + 110 \nu^{5} - 7 \nu^{4} - 426 \nu^{3} - 155 \nu^{2} + 316 \nu + 176 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( 13 \nu^{7} - 9 \nu^{6} - 175 \nu^{5} + 42 \nu^{4} + 666 \nu^{3} + 145 \nu^{2} - 501 \nu - 221 \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( -24 \nu^{7} + 17 \nu^{6} + 325 \nu^{5} - 86 \nu^{4} - 1243 \nu^{3} - 230 \nu^{2} + 933 \nu + 388 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(53\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\)
\(\nu^{6}\)\(=\)\(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(29\) \(\beta_{4}\mathstrut +\mathstrut \) \(46\) \(\beta_{3}\mathstrut +\mathstrut \) \(75\) \(\beta_{2}\mathstrut +\mathstrut \) \(103\) \(\beta_{1}\mathstrut +\mathstrut \) \(187\)
\(\nu^{7}\)\(=\)\(29\) \(\beta_{7}\mathstrut +\mathstrut \) \(100\) \(\beta_{6}\mathstrut +\mathstrut \) \(75\) \(\beta_{5}\mathstrut +\mathstrut \) \(54\) \(\beta_{4}\mathstrut +\mathstrut \) \(25\) \(\beta_{3}\mathstrut +\mathstrut \) \(122\) \(\beta_{2}\mathstrut +\mathstrut \) \(418\) \(\beta_{1}\mathstrut +\mathstrut \) \(225\)

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
−1.94234
−0.717467
2.92608
2.57048
1.03393
−0.839784
−2.55848
−0.472415
−1.00000 −3.08723 1.00000 1.00000 3.08723 4.56306 −1.00000 6.53101 −1.00000
1.2 −1.00000 −1.80420 1.00000 1.00000 1.80420 1.26189 −1.00000 0.255155 −1.00000
1.3 −1.00000 −1.43624 1.00000 1.00000 1.43624 −2.74822 −1.00000 −0.937215 −1.00000
1.4 −1.00000 0.229462 1.00000 1.00000 −0.229462 2.13623 −1.00000 −2.94735 −1.00000
1.5 −1.00000 0.244211 1.00000 1.00000 −0.244211 −1.97655 −1.00000 −2.94036 −1.00000
1.6 −1.00000 1.08982 1.00000 1.00000 −1.08982 4.96118 −1.00000 −1.81229 −1.00000
1.7 −1.00000 2.53701 1.00000 1.00000 −2.53701 1.50878 −1.00000 3.43644 −1.00000
1.8 −1.00000 3.22717 1.00000 1.00000 −3.22717 1.29362 −1.00000 7.41461 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).