Properties

Label 4704.2.a.bb
Level $4704$
Weight $2$
Character orbit 4704.a
Self dual yes
Analytic conductor $37.562$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{3} + q^{9} + O(q^{10}) \) \( q + q^{3} + q^{9} + 2q^{11} + 5q^{13} - 2q^{17} + 3q^{19} + 2q^{23} - 5q^{25} + q^{27} + 8q^{29} - q^{31} + 2q^{33} - 5q^{37} + 5q^{39} + 2q^{41} + 7q^{43} - 8q^{47} - 2q^{51} - 2q^{53} + 3q^{57} + 10q^{59} - 2q^{61} - 11q^{67} + 2q^{69} + 12q^{71} - 3q^{73} - 5q^{75} + 17q^{79} + q^{81} - 16q^{83} + 8q^{87} + 12q^{89} - q^{93} - 14q^{97} + 2q^{99} + O(q^{100}) \)

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

Hecke kernels

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

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