Properties

Label 58.2.a.a
Level 58
Weight 2
Character orbit 58.a
Self dual yes
Analytic conductor 0.463
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 58 = 2 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 58.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.463132331723\)
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

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

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 −3.00000 1.00000 −3.00000 3.00000 −2.00000 −1.00000 6.00000 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 58.2.a.a 1
3.b odd 2 1 522.2.a.k 1
4.b odd 2 1 464.2.a.f 1
5.b even 2 1 1450.2.a.i 1
5.c odd 4 2 1450.2.b.f 2
7.b odd 2 1 2842.2.a.d 1
8.b even 2 1 1856.2.a.p 1
8.d odd 2 1 1856.2.a.b 1
11.b odd 2 1 7018.2.a.c 1
12.b even 2 1 4176.2.a.bh 1
13.b even 2 1 9802.2.a.d 1
29.b even 2 1 1682.2.a.j 1
29.c odd 4 2 1682.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.a 1 1.a even 1 1 trivial
464.2.a.f 1 4.b odd 2 1
522.2.a.k 1 3.b odd 2 1
1450.2.a.i 1 5.b even 2 1
1450.2.b.f 2 5.c odd 4 2
1682.2.a.j 1 29.b even 2 1
1682.2.b.e 2 29.c odd 4 2
1856.2.a.b 1 8.d odd 2 1
1856.2.a.p 1 8.b even 2 1
2842.2.a.d 1 7.b odd 2 1
4176.2.a.bh 1 12.b even 2 1
7018.2.a.c 1 11.b odd 2 1
9802.2.a.d 1 13.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(58))\).