Properties

Label 2352.4.a.b
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 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 12q^{5} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 12q^{5} + 9q^{9} - 20q^{11} + 84q^{13} + 36q^{15} + 96q^{17} + 12q^{19} + 176q^{23} + 19q^{25} - 27q^{27} + 58q^{29} - 264q^{31} + 60q^{33} + 258q^{37} - 252q^{39} - 156q^{43} - 108q^{45} - 408q^{47} - 288q^{51} - 722q^{53} + 240q^{55} - 36q^{57} + 492q^{59} + 492q^{61} - 1008q^{65} - 412q^{67} - 528q^{69} - 296q^{71} - 240q^{73} - 57q^{75} - 776q^{79} + 81q^{81} + 924q^{83} - 1152q^{85} - 174q^{87} + 744q^{89} + 792q^{93} - 144q^{95} + 168q^{97} - 180q^{99} + O(q^{100}) \)

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 −12.0000 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.b 1
4.b odd 2 1 147.4.a.e yes 1
7.b odd 2 1 2352.4.a.bi 1
12.b even 2 1 441.4.a.h 1
28.d even 2 1 147.4.a.d 1
28.f even 6 2 147.4.e.f 2
28.g odd 6 2 147.4.e.e 2
84.h odd 2 1 441.4.a.g 1
84.j odd 6 2 441.4.e.g 2
84.n even 6 2 441.4.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 28.d even 2 1
147.4.a.e yes 1 4.b odd 2 1
147.4.e.e 2 28.g odd 6 2
147.4.e.f 2 28.f even 6 2
441.4.a.g 1 84.h odd 2 1
441.4.a.h 1 12.b even 2 1
441.4.e.f 2 84.n even 6 2
441.4.e.g 2 84.j odd 6 2
2352.4.a.b 1 1.a even 1 1 trivial
2352.4.a.bi 1 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 12 \)
\( T_{11} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 12 + T \)
$7$ \( T \)
$11$ \( 20 + T \)
$13$ \( -84 + T \)
$17$ \( -96 + T \)
$19$ \( -12 + T \)
$23$ \( -176 + T \)
$29$ \( -58 + T \)
$31$ \( 264 + T \)
$37$ \( -258 + T \)
$41$ \( T \)
$43$ \( 156 + T \)
$47$ \( 408 + T \)
$53$ \( 722 + T \)
$59$ \( -492 + T \)
$61$ \( -492 + T \)
$67$ \( 412 + T \)
$71$ \( 296 + T \)
$73$ \( 240 + T \)
$79$ \( 776 + T \)
$83$ \( -924 + T \)
$89$ \( -744 + T \)
$97$ \( -168 + T \)
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