Properties

Label 4030.2.a.n
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 16 x^{6} + 18 x^{5} + 64 x^{4} - 84 x^{3} - 19 x^{2} + 22 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{7} + 7 \nu^{6} - 44 \nu^{5} - 112 \nu^{4} + 186 \nu^{3} + 426 \nu^{2} - 319 \nu - 108 \)\()/58\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{7} + 5 \nu^{6} + 39 \nu^{5} - 80 \nu^{4} - 211 \nu^{3} + 325 \nu^{2} + 203 \nu - 102 \)\()/29\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{7} - 3 \nu^{6} - 122 \nu^{5} + 48 \nu^{4} + 608 \nu^{3} - 166 \nu^{2} - 725 \nu - 136 \)\()/58\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{7} + 20 \nu^{6} + 127 \nu^{5} - 320 \nu^{4} - 467 \nu^{3} + 1271 \nu^{2} - 203 \nu - 205 \)\()/29\)
\(\beta_{6}\)\(=\)\((\)\( -25 \nu^{7} + 19 \nu^{6} + 386 \nu^{5} - 362 \nu^{4} - 1434 \nu^{3} + 1786 \nu^{2} + 145 \nu - 318 \)\()/58\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + 16 \nu^{5} - 18 \nu^{4} - 62 \nu^{3} + 82 \nu^{2} + 3 \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} - \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + 8 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} - 2 \beta_{6} + \beta_{5} + 10 \beta_{4} + 9 \beta_{3} - 13 \beta_{2} + \beta_{1} + 31\)
\(\nu^{5}\)\(=\)\(13 \beta_{7} - 13 \beta_{6} - \beta_{5} + 12 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} + 69 \beta_{1} - 10\)
\(\nu^{6}\)\(=\)\(12 \beta_{7} - 28 \beta_{6} + 17 \beta_{5} + 94 \beta_{4} + 81 \beta_{3} - 138 \beta_{2} + 20 \beta_{1} + 263\)
\(\nu^{7}\)\(=\)\(138 \beta_{7} - 138 \beta_{6} - 17 \beta_{5} + 126 \beta_{4} + 49 \beta_{3} - 116 \beta_{2} + 613 \beta_{1} - 73\)

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
3.08271
2.04806
1.54913
0.635022
−0.178838
−0.447568
−2.63528
−3.05324
1.00000 −3.08271 1.00000 −1.00000 −3.08271 −2.62527 1.00000 6.50312 −1.00000
1.2 1.00000 −2.04806 1.00000 −1.00000 −2.04806 3.38208 1.00000 1.19454 −1.00000
1.3 1.00000 −1.54913 1.00000 −1.00000 −1.54913 −2.44745 1.00000 −0.600186 −1.00000
1.4 1.00000 −0.635022 1.00000 −1.00000 −0.635022 1.23180 1.00000 −2.59675 −1.00000
1.5 1.00000 0.178838 1.00000 −1.00000 0.178838 −4.63233 1.00000 −2.96802 −1.00000
1.6 1.00000 0.447568 1.00000 −1.00000 0.447568 1.86515 1.00000 −2.79968 −1.00000
1.7 1.00000 2.63528 1.00000 −1.00000 2.63528 0.203142 1.00000 3.94469 −1.00000
1.8 1.00000 3.05324 1.00000 −1.00000 3.05324 4.02288 1.00000 6.32229 −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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).