Properties

Label 4002.2.a.be
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 1
Dimension 5
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: \(5\)
Coefficient field: 5.5.2389280.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(11\) \(x^{3}\mathstrut +\mathstrut \) \(7\) \(x^{2}\mathstrut +\mathstrut \) \(26\) \(x\mathstrut -\mathstrut \) \(18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 8 \nu - 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 7 \nu^{2} + 8 \nu + 4 \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 6 \nu + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(24\)

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
1.61972
−2.11043
0.711151
−2.35912
3.13869
−1.00000 −1.00000 1.00000 −2.88781 1.00000 −3.71597 −1.00000 1.00000 2.88781
1.2 −1.00000 −1.00000 1.00000 −2.25808 1.00000 1.35501 −1.00000 1.00000 2.25808
1.3 −1.00000 −1.00000 1.00000 −0.651506 1.00000 2.08128 −1.00000 1.00000 0.651506
1.4 −1.00000 −1.00000 1.00000 3.26670 1.00000 −3.10583 −1.00000 1.00000 −3.26670
1.5 −1.00000 −1.00000 1.00000 3.53070 1.00000 −0.614480 −1.00000 1.00000 −3.53070
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{5} \) \(\mathstrut -\mathstrut T_{5}^{4} \) \(\mathstrut -\mathstrut 18 T_{5}^{3} \) \(\mathstrut +\mathstrut 4 T_{5}^{2} \) \(\mathstrut +\mathstrut 85 T_{5} \) \(\mathstrut +\mathstrut 49 \)
\(T_{7}^{5} \) \(\mathstrut +\mathstrut 4 T_{7}^{4} \) \(\mathstrut -\mathstrut 7 T_{7}^{3} \) \(\mathstrut -\mathstrut 26 T_{7}^{2} \) \(\mathstrut +\mathstrut 20 T_{7} \) \(\mathstrut +\mathstrut 20 \)
\(T_{11}^{5} \) \(\mathstrut -\mathstrut 5 T_{11}^{4} \) \(\mathstrut -\mathstrut 17 T_{11}^{3} \) \(\mathstrut +\mathstrut 99 T_{11}^{2} \) \(\mathstrut -\mathstrut 122 T_{11} \) \(\mathstrut +\mathstrut 42 \)