Properties

Label 2352.4.a.x
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 1176)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 12q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 12q^{5} + 9q^{9} + 60q^{11} - 44q^{13} - 36q^{15} + 128q^{17} + 52q^{19} + 160q^{23} + 19q^{25} + 27q^{27} - 230q^{29} + 136q^{31} + 180q^{33} - 318q^{37} - 132q^{39} + 192q^{41} - 220q^{43} - 108q^{45} + 184q^{47} + 384q^{51} - 498q^{53} - 720q^{55} + 156q^{57} - 492q^{59} - 20q^{61} + 528q^{65} - 380q^{67} + 480q^{69} + 264q^{71} + 560q^{73} + 57q^{75} - 104q^{79} + 81q^{81} + 1508q^{83} - 1536q^{85} - 690q^{87} - 1144q^{89} + 408q^{93} - 624q^{95} + 904q^{97} + 540q^{99} + O(q^{100}) \)

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.x 1
4.b odd 2 1 1176.4.a.b 1
7.b odd 2 1 2352.4.a.p 1
28.d even 2 1 1176.4.a.o yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.4.a.b 1 4.b odd 2 1
1176.4.a.o yes 1 28.d even 2 1
2352.4.a.p 1 7.b odd 2 1
2352.4.a.x 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 12 \)
\( T_{11} - 60 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 12 + T \)
$7$ \( T \)
$11$ \( -60 + T \)
$13$ \( 44 + T \)
$17$ \( -128 + T \)
$19$ \( -52 + T \)
$23$ \( -160 + T \)
$29$ \( 230 + T \)
$31$ \( -136 + T \)
$37$ \( 318 + T \)
$41$ \( -192 + T \)
$43$ \( 220 + T \)
$47$ \( -184 + T \)
$53$ \( 498 + T \)
$59$ \( 492 + T \)
$61$ \( 20 + T \)
$67$ \( 380 + T \)
$71$ \( -264 + T \)
$73$ \( -560 + T \)
$79$ \( 104 + T \)
$83$ \( -1508 + T \)
$89$ \( 1144 + T \)
$97$ \( -904 + T \)
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