Properties

Label 4002.2.a.e
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

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

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 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
\(n\): 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.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.e 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 - T \)
$5$ \( 1 + T + 5 T^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( 1 - 5 T + 11 T^{2} \)
$13$ \( 1 - 3 T + 13 T^{2} \)
$17$ \( 1 - 4 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + T \)
$29$ \( 1 + T \)
$31$ \( 1 - 5 T + 31 T^{2} \)
$37$ \( 1 + 3 T + 37 T^{2} \)
$41$ \( 1 - 9 T + 41 T^{2} \)
$43$ \( 1 + 6 T + 43 T^{2} \)
$47$ \( 1 - 10 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 7 T + 59 T^{2} \)
$61$ \( 1 - T + 61 T^{2} \)
$67$ \( 1 + 13 T + 67 T^{2} \)
$71$ \( 1 + 5 T + 71 T^{2} \)
$73$ \( 1 + 4 T + 73 T^{2} \)
$79$ \( 1 + 2 T + 79 T^{2} \)
$83$ \( 1 + 6 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 - 18 T + 97 T^{2} \)
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