Properties

Label 2352.4.a.bh
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 10q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 10q^{5} + 9q^{9} + 52q^{11} + 10q^{13} + 30q^{15} + 54q^{17} - 52q^{19} - 48q^{23} - 25q^{25} + 27q^{27} - 186q^{29} + 224q^{31} + 156q^{33} + 94q^{37} + 30q^{39} + 478q^{41} + 316q^{43} + 90q^{45} + 256q^{47} + 162q^{51} - 66q^{53} + 520q^{55} - 156q^{57} + 420q^{59} - 342q^{61} + 100q^{65} - 668q^{67} - 144q^{69} + 272q^{71} + 86q^{73} - 75q^{75} - 1360q^{79} + 81q^{81} + 188q^{83} + 540q^{85} - 558q^{87} + 366q^{89} + 672q^{93} - 520q^{95} - 1554q^{97} + 468q^{99} + O(q^{100}) \)

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 3.00000 0 10.0000 0 0 0 9.00000 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 2352.4.a.bh 1
4.b odd 2 1 1176.4.a.g 1
7.b odd 2 1 336.4.a.b 1
21.c even 2 1 1008.4.a.q 1
28.d even 2 1 168.4.a.e 1
56.e even 2 1 1344.4.a.k 1
56.h odd 2 1 1344.4.a.x 1
84.h odd 2 1 504.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.e 1 28.d even 2 1
336.4.a.b 1 7.b odd 2 1
504.4.a.e 1 84.h odd 2 1
1008.4.a.q 1 21.c even 2 1
1176.4.a.g 1 4.b odd 2 1
1344.4.a.k 1 56.e even 2 1
1344.4.a.x 1 56.h odd 2 1
2352.4.a.bh 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} - 10 \)
\( T_{11} - 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -10 + T \)
$7$ \( T \)
$11$ \( -52 + T \)
$13$ \( -10 + T \)
$17$ \( -54 + T \)
$19$ \( 52 + T \)
$23$ \( 48 + T \)
$29$ \( 186 + T \)
$31$ \( -224 + T \)
$37$ \( -94 + T \)
$41$ \( -478 + T \)
$43$ \( -316 + T \)
$47$ \( -256 + T \)
$53$ \( 66 + T \)
$59$ \( -420 + T \)
$61$ \( 342 + T \)
$67$ \( 668 + T \)
$71$ \( -272 + T \)
$73$ \( -86 + T \)
$79$ \( 1360 + T \)
$83$ \( -188 + T \)
$89$ \( -366 + T \)
$97$ \( 1554 + T \)
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