Properties

Label 130.2.a.c
Level $130$
Weight $2$
Character orbit 130.a
Self dual yes
Analytic conductor $1.038$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.03805522628\)
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

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

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 2.00000 1.00000 −1.00000 2.00000 −4.00000 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 130.2.a.c 1
3.b odd 2 1 1170.2.a.d 1
4.b odd 2 1 1040.2.a.b 1
5.b even 2 1 650.2.a.c 1
5.c odd 4 2 650.2.b.g 2
7.b odd 2 1 6370.2.a.l 1
8.b even 2 1 4160.2.a.c 1
8.d odd 2 1 4160.2.a.t 1
12.b even 2 1 9360.2.a.by 1
13.b even 2 1 1690.2.a.e 1
13.c even 3 2 1690.2.e.a 2
13.d odd 4 2 1690.2.d.e 2
13.e even 6 2 1690.2.e.g 2
13.f odd 12 4 1690.2.l.a 4
15.d odd 2 1 5850.2.a.cb 1
15.e even 4 2 5850.2.e.u 2
20.d odd 2 1 5200.2.a.bd 1
65.d even 2 1 8450.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.a.c 1 1.a even 1 1 trivial
650.2.a.c 1 5.b even 2 1
650.2.b.g 2 5.c odd 4 2
1040.2.a.b 1 4.b odd 2 1
1170.2.a.d 1 3.b odd 2 1
1690.2.a.e 1 13.b even 2 1
1690.2.d.e 2 13.d odd 4 2
1690.2.e.a 2 13.c even 3 2
1690.2.e.g 2 13.e even 6 2
1690.2.l.a 4 13.f odd 12 4
4160.2.a.c 1 8.b even 2 1
4160.2.a.t 1 8.d odd 2 1
5200.2.a.bd 1 20.d odd 2 1
5850.2.a.cb 1 15.d odd 2 1
5850.2.e.u 2 15.e even 4 2
6370.2.a.l 1 7.b odd 2 1
8450.2.a.n 1 65.d even 2 1
9360.2.a.by 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(130))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -2 + T \)
$5$ \( 1 + T \)
$7$ \( 4 + T \)
$11$ \( 2 + T \)
$13$ \( 1 + T \)
$17$ \( -2 + T \)
$19$ \( -6 + T \)
$23$ \( -6 + T \)
$29$ \( -2 + T \)
$31$ \( 6 + T \)
$37$ \( 2 + T \)
$41$ \( -10 + T \)
$43$ \( 10 + T \)
$47$ \( 12 + T \)
$53$ \( -2 + T \)
$59$ \( -10 + T \)
$61$ \( -2 + T \)
$67$ \( 12 + T \)
$71$ \( -10 + T \)
$73$ \( -10 + T \)
$79$ \( 4 + T \)
$83$ \( T \)
$89$ \( 14 + T \)
$97$ \( -14 + T \)
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