Properties

Label 2310.2.a.k
Level $2310$
Weight $2$
Character orbit 2310.a
Self dual yes
Analytic conductor $18.445$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.4454428669\)
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

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

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
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 2310.2.a.k 1
3.b odd 2 1 6930.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.k 1 1.a even 1 1 trivial
6930.2.a.w 1 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2310))\):

\( T_{13} - 2 \)
\( T_{17} + 2 \)
\( T_{19} + 8 \)
\( T_{23} + 8 \)
\( T_{29} + 6 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( 1 + T \)
$11$ \( 1 + T \)
$13$ \( -2 + T \)
$17$ \( 2 + T \)
$19$ \( 8 + T \)
$23$ \( 8 + T \)
$29$ \( 6 + T \)
$31$ \( T \)
$37$ \( 6 + T \)
$41$ \( 2 + T \)
$43$ \( -4 + T \)
$47$ \( -12 + T \)
$53$ \( 14 + T \)
$59$ \( -4 + T \)
$61$ \( 10 + T \)
$67$ \( -8 + T \)
$71$ \( T \)
$73$ \( 10 + T \)
$79$ \( -4 + T \)
$83$ \( 4 + T \)
$89$ \( 14 + T \)
$97$ \( -14 + T \)
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