Properties

Label 4704.2.a.g
Level $4704$
Weight $2$
Character orbit 4704.a
Self dual yes
Analytic conductor $37.562$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5616291108\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{9} - 2q^{11} + q^{13} - 2q^{17} + 5q^{19} + 6q^{23} - 5q^{25} - q^{27} - 8q^{29} - 3q^{31} + 2q^{33} - 9q^{37} - q^{39} + 2q^{41} + q^{43} + 8q^{47} + 2q^{51} + 6q^{53} - 5q^{57} + 6q^{59} - 2q^{61} - 5q^{67} - 6q^{69} + 4q^{71} - 11q^{73} + 5q^{75} - 5q^{79} + q^{81} + 8q^{87} + 12q^{89} + 3q^{93} + 18q^{97} - 2q^{99} + O(q^{100}) \)

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 0 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 4704.2.a.g 1
4.b odd 2 1 4704.2.a.ba 1
7.b odd 2 1 4704.2.a.y 1
7.c even 3 2 672.2.q.h yes 2
8.b even 2 1 9408.2.a.cn 1
8.d odd 2 1 9408.2.a.t 1
21.h odd 6 2 2016.2.s.e 2
28.d even 2 1 4704.2.a.j 1
28.g odd 6 2 672.2.q.d 2
56.e even 2 1 9408.2.a.ci 1
56.h odd 2 1 9408.2.a.x 1
56.k odd 6 2 1344.2.q.q 2
56.p even 6 2 1344.2.q.e 2
84.n even 6 2 2016.2.s.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.d 2 28.g odd 6 2
672.2.q.h yes 2 7.c even 3 2
1344.2.q.e 2 56.p even 6 2
1344.2.q.q 2 56.k odd 6 2
2016.2.s.e 2 21.h odd 6 2
2016.2.s.h 2 84.n even 6 2
4704.2.a.g 1 1.a even 1 1 trivial
4704.2.a.j 1 28.d even 2 1
4704.2.a.y 1 7.b odd 2 1
4704.2.a.ba 1 4.b odd 2 1
9408.2.a.t 1 8.d odd 2 1
9408.2.a.x 1 56.h odd 2 1
9408.2.a.ci 1 56.e even 2 1
9408.2.a.cn 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4704))\):

\( T_{5} \)
\( T_{11} + 2 \)
\( T_{13} - 1 \)
\( T_{19} - 5 \)
\( T_{31} + 3 \)