Properties

Label 4730.2.a.f
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
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

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

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 3.00000 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 4730.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.f 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3} \)
\( T_{7} - 3 \)
\( T_{13} - 1 \)