Properties

Label 2352.4.a.w
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 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 14q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 14q^{5} + 9q^{9} + 28q^{11} + 74q^{13} - 42q^{15} - 82q^{17} + 92q^{19} - 8q^{23} + 71q^{25} + 27q^{27} - 138q^{29} + 80q^{31} + 84q^{33} + 30q^{37} + 222q^{39} - 282q^{41} - 4q^{43} - 126q^{45} + 240q^{47} - 246q^{51} - 130q^{53} - 392q^{55} + 276q^{57} + 596q^{59} + 218q^{61} - 1036q^{65} + 436q^{67} - 24q^{69} - 856q^{71} + 998q^{73} + 213q^{75} + 32q^{79} + 81q^{81} - 1508q^{83} + 1148q^{85} - 414q^{87} + 246q^{89} + 240q^{93} - 1288q^{95} - 866q^{97} + 252q^{99} + O(q^{100}) \)

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 −14.0000 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.w 1
4.b odd 2 1 1176.4.a.a 1
7.b odd 2 1 48.4.a.b 1
21.c even 2 1 144.4.a.b 1
28.d even 2 1 24.4.a.a 1
35.c odd 2 1 1200.4.a.u 1
35.f even 4 2 1200.4.f.p 2
56.e even 2 1 192.4.a.a 1
56.h odd 2 1 192.4.a.g 1
84.h odd 2 1 72.4.a.b 1
112.j even 4 2 768.4.d.o 2
112.l odd 4 2 768.4.d.b 2
140.c even 2 1 600.4.a.h 1
140.j odd 4 2 600.4.f.b 2
168.e odd 2 1 576.4.a.u 1
168.i even 2 1 576.4.a.v 1
252.s odd 6 2 648.4.i.k 2
252.bi even 6 2 648.4.i.b 2
420.o odd 2 1 1800.4.a.bg 1
420.w even 4 2 1800.4.f.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.a.a 1 28.d even 2 1
48.4.a.b 1 7.b odd 2 1
72.4.a.b 1 84.h odd 2 1
144.4.a.b 1 21.c even 2 1
192.4.a.a 1 56.e even 2 1
192.4.a.g 1 56.h odd 2 1
576.4.a.u 1 168.e odd 2 1
576.4.a.v 1 168.i even 2 1
600.4.a.h 1 140.c even 2 1
600.4.f.b 2 140.j odd 4 2
648.4.i.b 2 252.bi even 6 2
648.4.i.k 2 252.s odd 6 2
768.4.d.b 2 112.l odd 4 2
768.4.d.o 2 112.j even 4 2
1176.4.a.a 1 4.b odd 2 1
1200.4.a.u 1 35.c odd 2 1
1200.4.f.p 2 35.f even 4 2
1800.4.a.bg 1 420.o odd 2 1
1800.4.f.q 2 420.w even 4 2
2352.4.a.w 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 14 \)
\( T_{11} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 14 + T \)
$7$ \( T \)
$11$ \( -28 + T \)
$13$ \( -74 + T \)
$17$ \( 82 + T \)
$19$ \( -92 + T \)
$23$ \( 8 + T \)
$29$ \( 138 + T \)
$31$ \( -80 + T \)
$37$ \( -30 + T \)
$41$ \( 282 + T \)
$43$ \( 4 + T \)
$47$ \( -240 + T \)
$53$ \( 130 + T \)
$59$ \( -596 + T \)
$61$ \( -218 + T \)
$67$ \( -436 + T \)
$71$ \( 856 + T \)
$73$ \( -998 + T \)
$79$ \( -32 + T \)
$83$ \( 1508 + T \)
$89$ \( -246 + T \)
$97$ \( 866 + T \)
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