Properties

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

Related objects

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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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + 5q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - 4q^{17} - q^{18} + q^{19} - q^{20} + q^{21} - 5q^{22} - 5q^{23} - q^{24} + q^{25} - 2q^{26} + q^{27} + q^{28} + 4q^{29} + q^{30} - q^{31} - q^{32} + 5q^{33} + 4q^{34} - q^{35} + q^{36} + 12q^{37} - q^{38} + 2q^{39} + q^{40} + 4q^{41} - q^{42} + 11q^{43} + 5q^{44} - q^{45} + 5q^{46} - 10q^{47} + q^{48} - 6q^{49} - q^{50} - 4q^{51} + 2q^{52} + 9q^{53} - q^{54} - 5q^{55} - q^{56} + q^{57} - 4q^{58} - 10q^{59} - q^{60} + 10q^{61} + q^{62} + q^{63} + q^{64} - 2q^{65} - 5q^{66} + 6q^{67} - 4q^{68} - 5q^{69} + q^{70} + 15q^{71} - q^{72} - 13q^{73} - 12q^{74} + q^{75} + q^{76} + 5q^{77} - 2q^{78} + 13q^{79} - q^{80} + q^{81} - 4q^{82} - 8q^{83} + q^{84} + 4q^{85} - 11q^{86} + 4q^{87} - 5q^{88} + 3q^{89} + q^{90} + 2q^{91} - 5q^{92} - q^{93} + 10q^{94} - q^{95} - q^{96} - 18q^{97} + 6q^{98} + 5q^{99} + O(q^{100}) \)

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 1.00000 −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.f 1
3.b odd 2 1 2790.2.a.bb 1
4.b odd 2 1 7440.2.a.c 1
5.b even 2 1 4650.2.a.bb 1
5.c odd 4 2 4650.2.d.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.f 1 1.a even 1 1 trivial
2790.2.a.bb 1 3.b odd 2 1
4650.2.a.bb 1 5.b even 2 1
4650.2.d.l 2 5.c odd 4 2
7440.2.a.c 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(930))\):

\( T_{7} - 1 \)
\( T_{11} - 5 \)
\( T_{19} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( -1 + T \)
$11$ \( -5 + T \)
$13$ \( -2 + T \)
$17$ \( 4 + T \)
$19$ \( -1 + T \)
$23$ \( 5 + T \)
$29$ \( -4 + T \)
$31$ \( 1 + T \)
$37$ \( -12 + T \)
$41$ \( -4 + T \)
$43$ \( -11 + T \)
$47$ \( 10 + T \)
$53$ \( -9 + T \)
$59$ \( 10 + T \)
$61$ \( -10 + T \)
$67$ \( -6 + T \)
$71$ \( -15 + T \)
$73$ \( 13 + T \)
$79$ \( -13 + T \)
$83$ \( 8 + T \)
$89$ \( -3 + T \)
$97$ \( 18 + T \)
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