Properties

Label 2352.4.a.bk
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 12)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 18q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 18q^{5} + 9q^{9} - 36q^{11} + 10q^{13} + 54q^{15} - 18q^{17} - 100q^{19} - 72q^{23} + 199q^{25} + 27q^{27} - 234q^{29} - 16q^{31} - 108q^{33} - 226q^{37} + 30q^{39} - 90q^{41} - 452q^{43} + 162q^{45} + 432q^{47} - 54q^{51} + 414q^{53} - 648q^{55} - 300q^{57} - 684q^{59} - 422q^{61} + 180q^{65} - 332q^{67} - 216q^{69} + 360q^{71} - 26q^{73} + 597q^{75} - 512q^{79} + 81q^{81} - 1188q^{83} - 324q^{85} - 702q^{87} + 630q^{89} - 48q^{93} - 1800q^{95} + 1054q^{97} - 324q^{99} + O(q^{100}) \)

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 18.0000 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.bk 1
4.b odd 2 1 588.4.a.c 1
7.b odd 2 1 48.4.a.a 1
12.b even 2 1 1764.4.a.b 1
21.c even 2 1 144.4.a.g 1
28.d even 2 1 12.4.a.a 1
28.f even 6 2 588.4.i.d 2
28.g odd 6 2 588.4.i.e 2
35.c odd 2 1 1200.4.a.be 1
35.f even 4 2 1200.4.f.d 2
56.e even 2 1 192.4.a.f 1
56.h odd 2 1 192.4.a.l 1
84.h odd 2 1 36.4.a.a 1
84.j odd 6 2 1764.4.k.b 2
84.n even 6 2 1764.4.k.o 2
112.j even 4 2 768.4.d.g 2
112.l odd 4 2 768.4.d.j 2
140.c even 2 1 300.4.a.b 1
140.j odd 4 2 300.4.d.e 2
168.e odd 2 1 576.4.a.b 1
168.i even 2 1 576.4.a.a 1
252.s odd 6 2 324.4.e.a 2
252.bi even 6 2 324.4.e.h 2
308.g odd 2 1 1452.4.a.d 1
364.h even 2 1 2028.4.a.c 1
364.p odd 4 2 2028.4.b.c 2
420.o odd 2 1 900.4.a.g 1
420.w even 4 2 900.4.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 28.d even 2 1
36.4.a.a 1 84.h odd 2 1
48.4.a.a 1 7.b odd 2 1
144.4.a.g 1 21.c even 2 1
192.4.a.f 1 56.e even 2 1
192.4.a.l 1 56.h odd 2 1
300.4.a.b 1 140.c even 2 1
300.4.d.e 2 140.j odd 4 2
324.4.e.a 2 252.s odd 6 2
324.4.e.h 2 252.bi even 6 2
576.4.a.a 1 168.i even 2 1
576.4.a.b 1 168.e odd 2 1
588.4.a.c 1 4.b odd 2 1
588.4.i.d 2 28.f even 6 2
588.4.i.e 2 28.g odd 6 2
768.4.d.g 2 112.j even 4 2
768.4.d.j 2 112.l odd 4 2
900.4.a.g 1 420.o odd 2 1
900.4.d.c 2 420.w even 4 2
1200.4.a.be 1 35.c odd 2 1
1200.4.f.d 2 35.f even 4 2
1452.4.a.d 1 308.g odd 2 1
1764.4.a.b 1 12.b even 2 1
1764.4.k.b 2 84.j odd 6 2
1764.4.k.o 2 84.n even 6 2
2028.4.a.c 1 364.h even 2 1
2028.4.b.c 2 364.p odd 4 2
2352.4.a.bk 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} - 18 \)
\( T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -18 + T \)
$7$ \( T \)
$11$ \( 36 + T \)
$13$ \( -10 + T \)
$17$ \( 18 + T \)
$19$ \( 100 + T \)
$23$ \( 72 + T \)
$29$ \( 234 + T \)
$31$ \( 16 + T \)
$37$ \( 226 + T \)
$41$ \( 90 + T \)
$43$ \( 452 + T \)
$47$ \( -432 + T \)
$53$ \( -414 + T \)
$59$ \( 684 + T \)
$61$ \( 422 + T \)
$67$ \( 332 + T \)
$71$ \( -360 + T \)
$73$ \( 26 + T \)
$79$ \( 512 + T \)
$83$ \( 1188 + T \)
$89$ \( -630 + T \)
$97$ \( -1054 + T \)
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