Properties

Label 4002.2.a.bi
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(24\) \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut +\mathstrut \) \(194\) \(x^{4}\mathstrut +\mathstrut \) \(39\) \(x^{3}\mathstrut -\mathstrut \) \(607\) \(x^{2}\mathstrut -\mathstrut \) \(104\) \(x\mathstrut +\mathstrut \) \(600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -11 \nu^{7} - 79 \nu^{6} + 173 \nu^{5} + 1466 \nu^{4} - 180 \nu^{3} - 8105 \nu^{2} - 2752 \nu + 12248 \)\()/1048\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{7} + 17 \nu^{6} - 213 \nu^{5} - 342 \nu^{4} + 1362 \nu^{3} + 1701 \nu^{2} - 1726 \nu - 732 \)\()/524\)
\(\beta_{3}\)\(=\)\((\)\( -21 \nu^{7} + 135 \nu^{6} + 235 \nu^{5} - 2346 \nu^{4} + 228 \nu^{3} + 11489 \nu^{2} - 3920 \nu - 15584 \)\()/1048\)
\(\beta_{4}\)\(=\)\((\)\( 57 \nu^{7} - 67 \nu^{6} - 1087 \nu^{5} + 978 \nu^{4} + 6268 \nu^{3} - 3637 \nu^{2} - 11368 \nu + 4272 \)\()/1048\)
\(\beta_{5}\)\(=\)\((\)\( 57 \nu^{7} - 67 \nu^{6} - 1087 \nu^{5} + 978 \nu^{4} + 6268 \nu^{3} - 3637 \nu^{2} - 9272 \nu + 4272 \)\()/1048\)
\(\beta_{6}\)\(=\)\((\)\( -20 \nu^{7} + 35 \nu^{6} + 386 \nu^{5} - 681 \nu^{4} - 2197 \nu^{3} + 3949 \nu^{2} + 3690 \nu - 6408 \)\()/262\)
\(\beta_{7}\)\(=\)\((\)\( 103 \nu^{7} - 213 \nu^{6} - 2001 \nu^{5} + 3422 \nu^{4} + 11308 \nu^{3} - 14331 \nu^{2} - 17104 \nu + 15552 \)\()/1048\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\) \(\beta_{2}\mathstrut -\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(109\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(77\) \(\beta_{5}\mathstrut -\mathstrut \) \(27\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(44\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(99\) \(\beta_{7}\mathstrut -\mathstrut \) \(72\) \(\beta_{6}\mathstrut +\mathstrut \) \(23\) \(\beta_{5}\mathstrut -\mathstrut \) \(188\) \(\beta_{4}\mathstrut +\mathstrut \) \(147\) \(\beta_{3}\mathstrut -\mathstrut \) \(286\) \(\beta_{2}\mathstrut -\mathstrut \) \(153\) \(\beta_{1}\mathstrut +\mathstrut \) \(1017\)\()/2\)
\(\nu^{7}\)\(=\)\(-\)\(140\) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(399\) \(\beta_{5}\mathstrut -\mathstrut \) \(69\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut -\mathstrut \) \(204\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(257\)

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
−2.45069
−1.57534
−2.15995
1.98746
3.34107
2.85533
−3.18819
1.19031
−1.00000 1.00000 1.00000 −4.40973 −1.00000 −0.551242 −1.00000 1.00000 4.40973
1.2 −1.00000 1.00000 1.00000 −2.69359 −1.00000 −3.87906 −1.00000 1.00000 2.69359
1.3 −1.00000 1.00000 1.00000 −1.69873 −1.00000 3.41134 −1.00000 1.00000 1.69873
1.4 −1.00000 1.00000 1.00000 −0.881455 −1.00000 0.253486 −1.00000 1.00000 0.881455
1.5 −1.00000 1.00000 1.00000 −0.464023 −1.00000 −2.35800 −1.00000 1.00000 0.464023
1.6 −1.00000 1.00000 1.00000 1.27098 −1.00000 3.55617 −1.00000 1.00000 −1.27098
1.7 −1.00000 1.00000 1.00000 1.60812 −1.00000 −3.37648 −1.00000 1.00000 −1.60812
1.8 −1.00000 1.00000 1.00000 4.26843 −1.00000 −3.05622 −1.00000 1.00000 −4.26843
\(n\): 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
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)
\(T_{11}^{8} + \cdots\)