Properties

Label 4002.2.a.bg
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 1
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(x^{6}\mathstrut -\mathstrut \) \(13\) \(x^{5}\mathstrut +\mathstrut \) \(16\) \(x^{4}\mathstrut +\mathstrut \) \(19\) \(x^{3}\mathstrut -\mathstrut \) \(8\) \(x^{2}\mathstrut -\mathstrut \) \(10\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu^{6} - 7 \nu^{5} - 47 \nu^{4} + 100 \nu^{3} + 4 \nu^{2} - 44 \nu - 6 \)
\(\beta_{2}\)\(=\)\( -5 \nu^{6} + 8 \nu^{5} + 60 \nu^{4} - 116 \nu^{3} - 23 \nu^{2} + 54 \nu + 14 \)
\(\beta_{3}\)\(=\)\( 7 \nu^{6} - 10 \nu^{5} - 86 \nu^{4} + 149 \nu^{3} + 61 \nu^{2} - 79 \nu - 29 \)
\(\beta_{4}\)\(=\)\( -7 \nu^{6} + 11 \nu^{5} + 85 \nu^{4} - 161 \nu^{3} - 45 \nu^{2} + 87 \nu + 26 \)
\(\beta_{5}\)\(=\)\( 8 \nu^{6} - 12 \nu^{5} - 98 \nu^{4} + 177 \nu^{3} + 64 \nu^{2} - 95 \nu - 35 \)
\(\beta_{6}\)\(=\)\( -11 \nu^{6} + 17 \nu^{5} + 134 \nu^{4} - 249 \nu^{3} - 76 \nu^{2} + 130 \nu + 41 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(24\) \(\beta_{6}\mathstrut +\mathstrut \) \(26\) \(\beta_{5}\mathstrut -\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(69\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(30\) \(\beta_{5}\mathstrut +\mathstrut \) \(47\) \(\beta_{4}\mathstrut +\mathstrut \) \(33\) \(\beta_{3}\mathstrut +\mathstrut \) \(122\) \(\beta_{2}\mathstrut +\mathstrut \) \(95\) \(\beta_{1}\mathstrut -\mathstrut \) \(25\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(266\) \(\beta_{6}\mathstrut +\mathstrut \) \(241\) \(\beta_{5}\mathstrut -\mathstrut \) \(114\) \(\beta_{4}\mathstrut -\mathstrut \) \(97\) \(\beta_{3}\mathstrut -\mathstrut \) \(261\) \(\beta_{2}\mathstrut -\mathstrut \) \(106\) \(\beta_{1}\mathstrut +\mathstrut \) \(698\)\()/2\)

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
2.99277
−3.51756
2.02256
−0.518095
0.932053
−0.571135
−0.340585
−1.00000 −1.00000 1.00000 −3.83000 1.00000 −4.91072 −1.00000 1.00000 3.83000
1.2 −1.00000 −1.00000 1.00000 −0.694170 1.00000 1.42319 −1.00000 1.00000 0.694170
1.3 −1.00000 −1.00000 1.00000 −0.662522 1.00000 −0.537195 −1.00000 1.00000 0.662522
1.4 −1.00000 −1.00000 1.00000 −0.324686 1.00000 2.40824 −1.00000 1.00000 0.324686
1.5 −1.00000 −1.00000 1.00000 1.78524 1.00000 0.541489 −1.00000 1.00000 −1.78524
1.6 −1.00000 −1.00000 1.00000 2.26164 1.00000 −5.18548 −1.00000 1.00000 −2.26164
1.7 −1.00000 −1.00000 1.00000 3.46450 1.00000 1.26047 −1.00000 1.00000 −3.46450
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

Hecke kernels

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

\(T_{5}^{7} \) \(\mathstrut -\mathstrut 2 T_{5}^{6} \) \(\mathstrut -\mathstrut 16 T_{5}^{5} \) \(\mathstrut +\mathstrut 34 T_{5}^{4} \) \(\mathstrut +\mathstrut 29 T_{5}^{3} \) \(\mathstrut -\mathstrut 42 T_{5}^{2} \) \(\mathstrut -\mathstrut 40 T_{5} \) \(\mathstrut -\mathstrut 8 \)
\(T_{7}^{7} \) \(\mathstrut +\mathstrut 5 T_{7}^{6} \) \(\mathstrut -\mathstrut 18 T_{7}^{5} \) \(\mathstrut -\mathstrut 52 T_{7}^{4} \) \(\mathstrut +\mathstrut 172 T_{7}^{3} \) \(\mathstrut -\mathstrut 96 T_{7}^{2} \) \(\mathstrut -\mathstrut 48 T_{7} \) \(\mathstrut +\mathstrut 32 \)
\(T_{11}^{7} \) \(\mathstrut +\mathstrut T_{11}^{6} \) \(\mathstrut -\mathstrut 37 T_{11}^{5} \) \(\mathstrut -\mathstrut 47 T_{11}^{4} \) \(\mathstrut +\mathstrut 248 T_{11}^{3} \) \(\mathstrut +\mathstrut 204 T_{11}^{2} \) \(\mathstrut -\mathstrut 456 T_{11} \) \(\mathstrut -\mathstrut 176 \)