Properties

Label 2352.4.a.e
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 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 6q^{5} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 6q^{5} + 9q^{9} - 12q^{11} - 38q^{13} + 18q^{15} + 126q^{17} + 20q^{19} - 168q^{23} - 89q^{25} - 27q^{27} + 30q^{29} - 88q^{31} + 36q^{33} + 254q^{37} + 114q^{39} - 42q^{41} + 52q^{43} - 54q^{45} - 96q^{47} - 378q^{51} + 198q^{53} + 72q^{55} - 60q^{57} - 660q^{59} + 538q^{61} + 228q^{65} - 884q^{67} + 504q^{69} - 792q^{71} - 218q^{73} + 267q^{75} + 520q^{79} + 81q^{81} - 492q^{83} - 756q^{85} - 90q^{87} - 810q^{89} + 264q^{93} - 120q^{95} - 1154q^{97} - 108q^{99} + O(q^{100}) \)

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 −6.00000 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.e 1
4.b odd 2 1 294.4.a.e 1
7.b odd 2 1 48.4.a.c 1
12.b even 2 1 882.4.a.n 1
21.c even 2 1 144.4.a.c 1
28.d even 2 1 6.4.a.a 1
28.f even 6 2 294.4.e.h 2
28.g odd 6 2 294.4.e.g 2
35.c odd 2 1 1200.4.a.b 1
35.f even 4 2 1200.4.f.j 2
56.e even 2 1 192.4.a.i 1
56.h odd 2 1 192.4.a.c 1
84.h odd 2 1 18.4.a.a 1
84.j odd 6 2 882.4.g.i 2
84.n even 6 2 882.4.g.f 2
112.j even 4 2 768.4.d.n 2
112.l odd 4 2 768.4.d.c 2
140.c even 2 1 150.4.a.i 1
140.j odd 4 2 150.4.c.d 2
168.e odd 2 1 576.4.a.q 1
168.i even 2 1 576.4.a.r 1
252.s odd 6 2 162.4.c.c 2
252.bi even 6 2 162.4.c.f 2
308.g odd 2 1 726.4.a.f 1
364.h even 2 1 1014.4.a.g 1
364.p odd 4 2 1014.4.b.d 2
420.o odd 2 1 450.4.a.h 1
420.w even 4 2 450.4.c.e 2
476.e even 2 1 1734.4.a.d 1
532.b odd 2 1 2166.4.a.i 1
924.n even 2 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 28.d even 2 1
18.4.a.a 1 84.h odd 2 1
48.4.a.c 1 7.b odd 2 1
144.4.a.c 1 21.c even 2 1
150.4.a.i 1 140.c even 2 1
150.4.c.d 2 140.j odd 4 2
162.4.c.c 2 252.s odd 6 2
162.4.c.f 2 252.bi even 6 2
192.4.a.c 1 56.h odd 2 1
192.4.a.i 1 56.e even 2 1
294.4.a.e 1 4.b odd 2 1
294.4.e.g 2 28.g odd 6 2
294.4.e.h 2 28.f even 6 2
450.4.a.h 1 420.o odd 2 1
450.4.c.e 2 420.w even 4 2
576.4.a.q 1 168.e odd 2 1
576.4.a.r 1 168.i even 2 1
726.4.a.f 1 308.g odd 2 1
768.4.d.c 2 112.l odd 4 2
768.4.d.n 2 112.j even 4 2
882.4.a.n 1 12.b even 2 1
882.4.g.f 2 84.n even 6 2
882.4.g.i 2 84.j odd 6 2
1014.4.a.g 1 364.h even 2 1
1014.4.b.d 2 364.p odd 4 2
1200.4.a.b 1 35.c odd 2 1
1200.4.f.j 2 35.f even 4 2
1734.4.a.d 1 476.e even 2 1
2166.4.a.i 1 532.b odd 2 1
2178.4.a.e 1 924.n even 2 1
2352.4.a.e 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5} + 6 \)
\( T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T \)
$5$ \( 1 + 6 T + 125 T^{2} \)
$7$ 1
$11$ \( 1 + 12 T + 1331 T^{2} \)
$13$ \( 1 + 38 T + 2197 T^{2} \)
$17$ \( 1 - 126 T + 4913 T^{2} \)
$19$ \( 1 - 20 T + 6859 T^{2} \)
$23$ \( 1 + 168 T + 12167 T^{2} \)
$29$ \( 1 - 30 T + 24389 T^{2} \)
$31$ \( 1 + 88 T + 29791 T^{2} \)
$37$ \( 1 - 254 T + 50653 T^{2} \)
$41$ \( 1 + 42 T + 68921 T^{2} \)
$43$ \( 1 - 52 T + 79507 T^{2} \)
$47$ \( 1 + 96 T + 103823 T^{2} \)
$53$ \( 1 - 198 T + 148877 T^{2} \)
$59$ \( 1 + 660 T + 205379 T^{2} \)
$61$ \( 1 - 538 T + 226981 T^{2} \)
$67$ \( 1 + 884 T + 300763 T^{2} \)
$71$ \( 1 + 792 T + 357911 T^{2} \)
$73$ \( 1 + 218 T + 389017 T^{2} \)
$79$ \( 1 - 520 T + 493039 T^{2} \)
$83$ \( 1 + 492 T + 571787 T^{2} \)
$89$ \( 1 + 810 T + 704969 T^{2} \)
$97$ \( 1 + 1154 T + 912673 T^{2} \)
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