Properties

Label 2352.4.a.bb
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

\(f(q)\) \(=\) \( q + 3q^{3} + 2q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 2q^{5} + 9q^{9} - 52q^{11} - 86q^{13} + 6q^{15} + 30q^{17} - 4q^{19} - 120q^{23} - 121q^{25} + 27q^{27} + 246q^{29} + 80q^{31} - 156q^{33} - 290q^{37} - 258q^{39} + 374q^{41} - 164q^{43} + 18q^{45} + 464q^{47} + 90q^{51} - 162q^{53} - 104q^{55} - 12q^{57} + 180q^{59} + 666q^{61} - 172q^{65} + 628q^{67} - 360q^{69} - 296q^{71} + 518q^{73} - 363q^{75} + 1184q^{79} + 81q^{81} + 220q^{83} + 60q^{85} + 738q^{87} + 774q^{89} + 240q^{93} - 8q^{95} + 1086q^{97} - 468q^{99} + O(q^{100}) \)

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 2.00000 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.bb 1
4.b odd 2 1 1176.4.a.e 1
7.b odd 2 1 336.4.a.c 1
21.c even 2 1 1008.4.a.l 1
28.d even 2 1 168.4.a.f 1
56.e even 2 1 1344.4.a.g 1
56.h odd 2 1 1344.4.a.v 1
84.h odd 2 1 504.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.f 1 28.d even 2 1
336.4.a.c 1 7.b odd 2 1
504.4.a.c 1 84.h odd 2 1
1008.4.a.l 1 21.c even 2 1
1176.4.a.e 1 4.b odd 2 1
1344.4.a.g 1 56.e even 2 1
1344.4.a.v 1 56.h odd 2 1
2352.4.a.bb 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} - 2 \)
\( T_{11} + 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( 52 + T \)
$13$ \( 86 + T \)
$17$ \( -30 + T \)
$19$ \( 4 + T \)
$23$ \( 120 + T \)
$29$ \( -246 + T \)
$31$ \( -80 + T \)
$37$ \( 290 + T \)
$41$ \( -374 + T \)
$43$ \( 164 + T \)
$47$ \( -464 + T \)
$53$ \( 162 + T \)
$59$ \( -180 + T \)
$61$ \( -666 + T \)
$67$ \( -628 + T \)
$71$ \( 296 + T \)
$73$ \( -518 + T \)
$79$ \( -1184 + T \)
$83$ \( -220 + T \)
$89$ \( -774 + T \)
$97$ \( -1086 + T \)
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