Properties

Label 4002.2.a.r
Level 4002
Weight 2
Character orbit 4002.a
Self dual yes
Analytic conductor 31.956
Analytic rank 0
Dimension 1
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
1.00000 1.00000 1.00000 3.00000 1.00000 1.00000 1.00000 1.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
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.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.r 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5} - 3 \)
\( T_{7} - 1 \)
\( T_{11} \)