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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4002.2.a.bf 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bf 6 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(-1\)

Hecke kernels

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

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