Properties

Label 2352.4.a.c
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 3q^{3} - 7q^{5} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 7q^{5} + 9q^{9} - 7q^{11} + 52q^{13} + 21q^{15} - 72q^{17} + 20q^{19} + 48q^{23} - 76q^{25} - 27q^{27} - 243q^{29} + 95q^{31} + 21q^{33} + 352q^{37} - 156q^{39} + 296q^{41} - 158q^{43} - 63q^{45} - 142q^{47} + 216q^{51} - 375q^{53} + 49q^{55} - 60q^{57} + 279q^{59} - 246q^{61} - 364q^{65} + 730q^{67} - 144q^{69} - 338q^{71} + 542q^{73} + 228q^{75} + 305q^{79} + 81q^{81} + 1123q^{83} + 504q^{85} + 729q^{87} + 426q^{89} - 285q^{93} - 140q^{95} + 369q^{97} - 63q^{99} + O(q^{100}) \)

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 −7.00000 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.c 1
4.b odd 2 1 1176.4.a.i 1
7.b odd 2 1 2352.4.a.bg 1
7.d odd 6 2 336.4.q.a 2
28.d even 2 1 1176.4.a.f 1
28.f even 6 2 168.4.q.a 2
84.j odd 6 2 504.4.s.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.a 2 28.f even 6 2
336.4.q.a 2 7.d odd 6 2
504.4.s.e 2 84.j odd 6 2
1176.4.a.f 1 28.d even 2 1
1176.4.a.i 1 4.b odd 2 1
2352.4.a.c 1 1.a even 1 1 trivial
2352.4.a.bg 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 7 \)
\( T_{11} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 7 + T \)
$7$ \( T \)
$11$ \( 7 + T \)
$13$ \( -52 + T \)
$17$ \( 72 + T \)
$19$ \( -20 + T \)
$23$ \( -48 + T \)
$29$ \( 243 + T \)
$31$ \( -95 + T \)
$37$ \( -352 + T \)
$41$ \( -296 + T \)
$43$ \( 158 + T \)
$47$ \( 142 + T \)
$53$ \( 375 + T \)
$59$ \( -279 + T \)
$61$ \( 246 + T \)
$67$ \( -730 + T \)
$71$ \( 338 + T \)
$73$ \( -542 + T \)
$79$ \( -305 + T \)
$83$ \( -1123 + T \)
$89$ \( -426 + T \)
$97$ \( -369 + T \)
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