Properties

Label 2310.2.a.r
Level $2310$
Weight $2$
Character orbit 2310.a
Self dual yes
Analytic conductor $18.445$
Analytic rank $0$
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: \(0\)
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

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

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

\( T_{13} + 2 \)
\( T_{17} - 6 \)
\( T_{19} - 8 \)
\( T_{23} \)
\( T_{29} + 2 \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( -1 + T \)
$13$ \( 2 + T \)
$17$ \( -6 + T \)
$19$ \( -8 + T \)
$23$ \( T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( 2 + T \)
$41$ \( 2 + T \)
$43$ \( -4 + T \)
$47$ \( 4 + T \)
$53$ \( -14 + T \)
$59$ \( -4 + T \)
$61$ \( -14 + T \)
$67$ \( -12 + T \)
$71$ \( 8 + T \)
$73$ \( 2 + T \)
$79$ \( 8 + T \)
$83$ \( -16 + T \)
$89$ \( -2 + T \)
$97$ \( -2 + T \)
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