Properties

Label 4030.2.a.k
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: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(18\) \(x^{6}\mathstrut +\mathstrut \) \(12\) \(x^{5}\mathstrut +\mathstrut \) \(98\) \(x^{4}\mathstrut -\mathstrut \) \(18\) \(x^{3}\mathstrut -\mathstrut \) \(173\) \(x^{2}\mathstrut -\mathstrut \) \(48\) \(x\mathstrut +\mathstrut \) \(32\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} + 19 \nu^{5} + 31 \nu^{4} - 96 \nu^{3} - 109 \nu^{2} + 70 \nu + 20 \)\()/21\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{7} - 3 \nu^{6} - 31 \nu^{5} + 36 \nu^{4} + 129 \nu^{3} - 69 \nu^{2} - 147 \nu - 19 \)\()/21\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{7} + \nu^{6} + 50 \nu^{5} - 26 \nu^{4} - 246 \nu^{3} + 170 \nu^{2} + 385 \nu - 150 \)\()/42\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} - 11 \nu^{6} - 74 \nu^{5} + 160 \nu^{4} + 270 \nu^{3} - 526 \nu^{2} - 245 \nu + 236 \)\()/42\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{7} + 11 \nu^{6} + 74 \nu^{5} - 160 \nu^{4} - 270 \nu^{3} + 568 \nu^{2} + 203 \nu - 404 \)\()/42\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{6} + 50 \nu^{5} - 110 \nu^{4} - 239 \nu^{3} + 331 \nu^{2} + 364 \nu - 101 \)\()/21\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(10\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(29\)
\(\nu^{5}\)\(=\)\(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(56\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)
\(\nu^{6}\)\(=\)\(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(97\) \(\beta_{6}\mathstrut +\mathstrut \) \(110\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut -\mathstrut \) \(25\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(105\) \(\beta_{1}\mathstrut +\mathstrut \) \(247\)
\(\nu^{7}\)\(=\)\(51\) \(\beta_{7}\mathstrut -\mathstrut \) \(31\) \(\beta_{6}\mathstrut -\mathstrut \) \(120\) \(\beta_{5}\mathstrut -\mathstrut \) \(129\) \(\beta_{4}\mathstrut +\mathstrut \) \(177\) \(\beta_{3}\mathstrut +\mathstrut \) \(151\) \(\beta_{2}\mathstrut +\mathstrut \) \(484\) \(\beta_{1}\mathstrut +\mathstrut \) \(177\)

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
3.14543
2.76768
1.89749
0.315905
−0.815721
−1.21997
−1.98236
−3.10845
−1.00000 −3.14543 1.00000 −1.00000 3.14543 −3.35025 −1.00000 6.89376 1.00000
1.2 −1.00000 −2.76768 1.00000 −1.00000 2.76768 −3.85986 −1.00000 4.66003 1.00000
1.3 −1.00000 −1.89749 1.00000 −1.00000 1.89749 0.730984 −1.00000 0.600463 1.00000
1.4 −1.00000 −0.315905 1.00000 −1.00000 0.315905 −4.55393 −1.00000 −2.90020 1.00000
1.5 −1.00000 0.815721 1.00000 −1.00000 −0.815721 0.327652 −1.00000 −2.33460 1.00000
1.6 −1.00000 1.21997 1.00000 −1.00000 −1.21997 −1.25636 −1.00000 −1.51167 1.00000
1.7 −1.00000 1.98236 1.00000 −1.00000 −1.98236 4.72598 −1.00000 0.929744 1.00000
1.8 −1.00000 3.10845 1.00000 −1.00000 −3.10845 −0.764219 −1.00000 6.66247 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).