Properties

Label 4030.2.a.j
Level 4030
Weight 2
Character orbit 4030.a
Self dual Yes
Analytic conductor 32.180
Analytic rank 1
Dimension 7
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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(6\) \(x^{5}\mathstrut +\mathstrut \) \(20\) \(x^{4}\mathstrut +\mathstrut \) \(9\) \(x^{3}\mathstrut -\mathstrut \) \(37\) \(x^{2}\mathstrut -\mathstrut \) \(3\) \(x\mathstrut +\mathstrut \) \(20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 7 \nu^{4} + 11 \nu^{3} + 14 \nu^{2} - 12 \nu - 8 \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 2 \nu^{5} + 8 \nu^{4} - 11 \nu^{3} - 21 \nu^{2} + 11 \nu + 17 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + 2 \nu^{5} + 8 \nu^{4} - 12 \nu^{3} - 20 \nu^{2} + 15 \nu + 15 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - \nu^{5} - 10 \nu^{4} + 6 \nu^{3} + 30 \nu^{2} - 7 \nu - 24 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)
\(\nu^{5}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{6}\)\(=\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(44\) \(\beta_{2}\mathstrut +\mathstrut \) \(55\) \(\beta_{1}\mathstrut +\mathstrut \) \(57\)

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
−0.937425
2.64004
1.30511
−1.28870
1.13299
−1.98668
2.13467
1.00000 −3.21109 1.00000 −1.00000 −3.21109 3.80132 1.00000 7.31111 −1.00000
1.2 1.00000 −2.33934 1.00000 −1.00000 −2.33934 −3.11075 1.00000 2.47253 −1.00000
1.3 1.00000 −1.69801 1.00000 −1.00000 −1.69801 4.41789 1.00000 −0.116762 −1.00000
1.4 1.00000 0.444821 1.00000 −1.00000 0.444821 −0.0224858 1.00000 −2.80213 −1.00000
1.5 1.00000 0.779589 1.00000 −1.00000 0.779589 −2.22130 1.00000 −2.39224 −1.00000
1.6 1.00000 0.820963 1.00000 −1.00000 0.820963 2.40836 1.00000 −2.32602 −1.00000
1.7 1.00000 2.20307 1.00000 −1.00000 2.20307 −1.27303 1.00000 1.85352 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(T_{3}^{7} \) \(\mathstrut +\mathstrut 3 T_{3}^{6} \) \(\mathstrut -\mathstrut 8 T_{3}^{5} \) \(\mathstrut -\mathstrut 20 T_{3}^{4} \) \(\mathstrut +\mathstrut 22 T_{3}^{3} \) \(\mathstrut +\mathstrut 24 T_{3}^{2} \) \(\mathstrut -\mathstrut 31 T_{3} \) \(\mathstrut +\mathstrut 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).