Properties

Label 58.2.a.b
Level $58$
Weight $2$
Character orbit 58.a
Self dual yes
Analytic conductor $0.463$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
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

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

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 −2.00000 1.00000 −2.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(58))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 8 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T + 3 \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 11 \) Copy content Toggle raw display
$47$ \( T - 13 \) Copy content Toggle raw display
$53$ \( T + 11 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 2 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T - 15 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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