Properties

Label 4030.2.a.a
Level 4030
Weight 2
Character orbit 4030.a
Self dual yes
Analytic conductor 32.180
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1797120146\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} + q^{5} - q^{8} - 3q^{9} - q^{10} - 2q^{11} + q^{13} + q^{16} + 4q^{17} + 3q^{18} + 4q^{19} + q^{20} + 2q^{22} - 2q^{23} + q^{25} - q^{26} + q^{31} - q^{32} - 4q^{34} - 3q^{36} + 2q^{37} - 4q^{38} - q^{40} + 6q^{41} - 8q^{43} - 2q^{44} - 3q^{45} + 2q^{46} - 12q^{47} - 7q^{49} - q^{50} + q^{52} + 10q^{53} - 2q^{55} + 12q^{59} - 4q^{61} - q^{62} + q^{64} + q^{65} + 4q^{67} + 4q^{68} + 3q^{72} - 2q^{74} + 4q^{76} - 4q^{79} + q^{80} + 9q^{81} - 6q^{82} + 4q^{83} + 4q^{85} + 8q^{86} + 2q^{88} + 6q^{89} + 3q^{90} - 2q^{92} + 12q^{94} + 4q^{95} + 2q^{97} + 7q^{98} + 6q^{99} + O(q^{100}) \)

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 0 1.00000 1.00000 0 0 −1.00000 −3.00000 −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 4030.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4030.2.a.a 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(13\) \(-1\)
\(31\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).