Properties

Label 930.2.a.j
Level $930$
Weight $2$
Character orbit 930.a
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} - q^{10} + 2q^{11} + q^{12} + 2q^{13} - 4q^{14} + q^{15} + q^{16} - q^{18} + q^{20} + 4q^{21} - 2q^{22} - 6q^{23} - q^{24} + q^{25} - 2q^{26} + q^{27} + 4q^{28} - q^{30} + q^{31} - q^{32} + 2q^{33} + 4q^{35} + q^{36} - 2q^{37} + 2q^{39} - q^{40} - 10q^{41} - 4q^{42} - 4q^{43} + 2q^{44} + q^{45} + 6q^{46} + 4q^{47} + q^{48} + 9q^{49} - q^{50} + 2q^{52} + 6q^{53} - q^{54} + 2q^{55} - 4q^{56} - 4q^{59} + q^{60} - q^{62} + 4q^{63} + q^{64} + 2q^{65} - 2q^{66} + 4q^{67} - 6q^{69} - 4q^{70} - 16q^{71} - q^{72} + 4q^{73} + 2q^{74} + q^{75} + 8q^{77} - 2q^{78} + 4q^{79} + q^{80} + q^{81} + 10q^{82} + 8q^{83} + 4q^{84} + 4q^{86} - 2q^{88} + 6q^{89} - q^{90} + 8q^{91} - 6q^{92} + q^{93} - 4q^{94} - q^{96} + 14q^{97} - 9q^{98} + 2q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(31\) \(-1\)

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 930.2.a.j 1
3.b odd 2 1 2790.2.a.v 1
4.b odd 2 1 7440.2.a.g 1
5.b even 2 1 4650.2.a.x 1
5.c odd 4 2 4650.2.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.j 1 1.a even 1 1 trivial
2790.2.a.v 1 3.b odd 2 1
4650.2.a.x 1 5.b even 2 1
4650.2.d.h 2 5.c odd 4 2
7440.2.a.g 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(930))\):

\( T_{7} - 4 \)
\( T_{11} - 2 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( -4 + T \)
$11$ \( -2 + T \)
$13$ \( -2 + T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( -1 + T \)
$37$ \( 2 + T \)
$41$ \( 10 + T \)
$43$ \( 4 + T \)
$47$ \( -4 + T \)
$53$ \( -6 + T \)
$59$ \( 4 + T \)
$61$ \( T \)
$67$ \( -4 + T \)
$71$ \( 16 + T \)
$73$ \( -4 + T \)
$79$ \( -4 + T \)
$83$ \( -8 + T \)
$89$ \( -6 + T \)
$97$ \( -14 + T \)
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