Properties

Label 2352.2.a.j
Level $2352$
Weight $2$
Character orbit 2352.a
Self dual yes
Analytic conductor $18.781$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.7808145554\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 588)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 2q^{5} + q^{9} + O(q^{10}) \) \( q - q^{3} + 2q^{5} + q^{9} - 2q^{11} - 4q^{13} - 2q^{15} + 6q^{17} - 8q^{19} + 6q^{23} - q^{25} - q^{27} - 10q^{29} - 4q^{31} + 2q^{33} + 6q^{37} + 4q^{39} - 6q^{41} - 4q^{43} + 2q^{45} - 8q^{47} - 6q^{51} + 2q^{53} - 4q^{55} + 8q^{57} + 4q^{59} - 8q^{61} - 8q^{65} + 8q^{67} - 6q^{69} + 10q^{71} + 4q^{73} + q^{75} - 4q^{79} + q^{81} - 12q^{83} + 12q^{85} + 10q^{87} - 14q^{89} + 4q^{93} - 16q^{95} + 4q^{97} - 2q^{99} + O(q^{100}) \)

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
0 −1.00000 0 2.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

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 2352.2.a.j 1
3.b odd 2 1 7056.2.a.n 1
4.b odd 2 1 588.2.a.e yes 1
7.b odd 2 1 2352.2.a.p 1
7.c even 3 2 2352.2.q.p 2
7.d odd 6 2 2352.2.q.k 2
8.b even 2 1 9408.2.a.ca 1
8.d odd 2 1 9408.2.a.l 1
12.b even 2 1 1764.2.a.b 1
21.c even 2 1 7056.2.a.bu 1
28.d even 2 1 588.2.a.b 1
28.f even 6 2 588.2.i.g 2
28.g odd 6 2 588.2.i.a 2
56.e even 2 1 9408.2.a.cu 1
56.h odd 2 1 9408.2.a.bf 1
84.h odd 2 1 1764.2.a.i 1
84.j odd 6 2 1764.2.k.c 2
84.n even 6 2 1764.2.k.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.a.b 1 28.d even 2 1
588.2.a.e yes 1 4.b odd 2 1
588.2.i.a 2 28.g odd 6 2
588.2.i.g 2 28.f even 6 2
1764.2.a.b 1 12.b even 2 1
1764.2.a.i 1 84.h odd 2 1
1764.2.k.c 2 84.j odd 6 2
1764.2.k.i 2 84.n even 6 2
2352.2.a.j 1 1.a even 1 1 trivial
2352.2.a.p 1 7.b odd 2 1
2352.2.q.k 2 7.d odd 6 2
2352.2.q.p 2 7.c even 3 2
7056.2.a.n 1 3.b odd 2 1
7056.2.a.bu 1 21.c even 2 1
9408.2.a.l 1 8.d odd 2 1
9408.2.a.bf 1 56.h odd 2 1
9408.2.a.ca 1 8.b even 2 1
9408.2.a.cu 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} - 2 \)
\( T_{11} + 2 \)
\( T_{13} + 4 \)
\( T_{17} - 6 \)