Properties

Label 4002.2.a.bf
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 0
Dimension 6
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.61157024.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut -\mathstrut \) \(8\) \(x^{4}\mathstrut +\mathstrut \) \(8\) \(x^{3}\mathstrut +\mathstrut \) \(17\) \(x^{2}\mathstrut -\mathstrut \) \(4\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} - 3 \nu^{3} - 3 \nu^{2} + 9 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 3 \nu^{3} + 5 \nu^{2} - 9 \nu - 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 6 \nu^{3} + 4 \nu^{2} + 7 \nu + 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 5 \nu^{3} + 11 \nu^{2} + 8 \nu - 2 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 5 \nu^{3} + 13 \nu^{2} + 4 \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)\()/2\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(21\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(35\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(28\) \(\beta_{3}\mathstrut +\mathstrut \) \(87\) \(\beta_{2}\mathstrut +\mathstrut \) \(71\) \(\beta_{1}\mathstrut +\mathstrut \) \(104\)\()/2\)

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
−1.82560
3.25648
−0.430873
−1.44200
1.85969
0.582305
−1.00000 1.00000 1.00000 −1.56155 −1.00000 −3.98403 −1.00000 1.00000 1.56155
1.2 −1.00000 1.00000 1.00000 −1.56155 −1.00000 −1.09168 −1.00000 1.00000 1.56155
1.3 −1.00000 1.00000 1.00000 −1.56155 −1.00000 1.95260 −1.00000 1.00000 1.56155
1.4 −1.00000 1.00000 1.00000 2.56155 −1.00000 −1.96335 −1.00000 1.00000 −2.56155
1.5 −1.00000 1.00000 1.00000 2.56155 −1.00000 3.26093 −1.00000 1.00000 −2.56155
1.6 −1.00000 1.00000 1.00000 2.56155 −1.00000 3.82553 −1.00000 1.00000 −2.56155
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{2} \) \(\mathstrut -\mathstrut T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{7}^{6} \) \(\mathstrut -\mathstrut 2 T_{7}^{5} \) \(\mathstrut -\mathstrut 23 T_{7}^{4} \) \(\mathstrut +\mathstrut 40 T_{7}^{3} \) \(\mathstrut +\mathstrut 128 T_{7}^{2} \) \(\mathstrut -\mathstrut 124 T_{7} \) \(\mathstrut -\mathstrut 208 \)
\(T_{11}^{6} \) \(\mathstrut -\mathstrut 7 T_{11}^{5} \) \(\mathstrut +\mathstrut 6 T_{11}^{4} \) \(\mathstrut +\mathstrut 38 T_{11}^{3} \) \(\mathstrut -\mathstrut 48 T_{11}^{2} \) \(\mathstrut -\mathstrut 48 T_{11} \) \(\mathstrut +\mathstrut 32 \)