Properties

Label 8030.2.a.u
Level 8030
Weight 2
Character orbit 8030.a
Self dual Yes
Analytic conductor 64.120
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3981.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(4\) \(x^{2}\mathstrut +\mathstrut \) \(2\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)

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
−1.75080
0.785261
2.28400
−0.318459
1.00000 −2.17963 1.00000 1.00000 −2.17963 −3.00000 1.00000 1.75080 1.00000
1.2 1.00000 −1.48820 1.00000 1.00000 −1.48820 −3.00000 1.00000 −0.785261 1.00000
1.3 1.00000 0.846169 1.00000 1.00000 0.846169 −3.00000 1.00000 −2.28400 1.00000
1.4 1.00000 1.82166 1.00000 1.00000 1.82166 −3.00000 1.00000 0.318459 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)
\(73\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\(T_{3}^{4} \) \(\mathstrut +\mathstrut T_{3}^{3} \) \(\mathstrut -\mathstrut 5 T_{3}^{2} \) \(\mathstrut -\mathstrut 3 T_{3} \) \(\mathstrut +\mathstrut 5 \)
\(T_{7} \) \(\mathstrut +\mathstrut 3 \)