Properties

Label 4002.2.a.k
Level 4002
Weight 2
Character orbit 4002.a
Self dual yes
Analytic conductor 31.956
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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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: \(1\)
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

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

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 1.00000 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.k 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5} - 1 \)
\( T_{7} - 1 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + T \)
$5$ \( 1 - T + 5 T^{2} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 - T + 19 T^{2} \)
$23$ \( 1 - T \)
$29$ \( 1 + T \)
$31$ \( 1 + 2 T + 31 T^{2} \)
$37$ \( 1 + T + 37 T^{2} \)
$41$ \( 1 + 9 T + 41 T^{2} \)
$43$ \( 1 + 5 T + 43 T^{2} \)
$47$ \( 1 - 7 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 9 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 8 T + 67 T^{2} \)
$71$ \( 1 + 2 T + 71 T^{2} \)
$73$ \( 1 - 14 T + 73 T^{2} \)
$79$ \( 1 + 4 T + 79 T^{2} \)
$83$ \( 1 + 8 T + 83 T^{2} \)
$89$ \( 1 - 18 T + 89 T^{2} \)
$97$ \( 1 + 8 T + 97 T^{2} \)
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