Properties

Label 392.4.a.e
Level $392$
Weight $4$
Character orbit 392.a
Self dual yes
Analytic conductor $23.129$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1287487223\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{3} + 2q^{5} - 11q^{9} + O(q^{10}) \) \( q + 4q^{3} + 2q^{5} - 11q^{9} - 44q^{11} - 22q^{13} + 8q^{15} - 50q^{17} - 44q^{19} - 56q^{23} - 121q^{25} - 152q^{27} + 198q^{29} + 160q^{31} - 176q^{33} - 162q^{37} - 88q^{39} + 198q^{41} + 52q^{43} - 22q^{45} - 528q^{47} - 200q^{51} - 242q^{53} - 88q^{55} - 176q^{57} + 668q^{59} - 550q^{61} - 44q^{65} + 188q^{67} - 224q^{69} + 728q^{71} - 154q^{73} - 484q^{75} - 656q^{79} - 311q^{81} - 236q^{83} - 100q^{85} + 792q^{87} - 714q^{89} + 640q^{93} - 88q^{95} + 478q^{97} + 484q^{99} + O(q^{100}) \)

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 4.00000 0 2.00000 0 0 0 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 392.4.a.e 1
4.b odd 2 1 784.4.a.e 1
7.b odd 2 1 8.4.a.a 1
7.c even 3 2 392.4.i.b 2
7.d odd 6 2 392.4.i.g 2
21.c even 2 1 72.4.a.c 1
28.d even 2 1 16.4.a.a 1
35.c odd 2 1 200.4.a.g 1
35.f even 4 2 200.4.c.e 2
56.e even 2 1 64.4.a.b 1
56.h odd 2 1 64.4.a.d 1
63.l odd 6 2 648.4.i.h 2
63.o even 6 2 648.4.i.e 2
77.b even 2 1 968.4.a.a 1
84.h odd 2 1 144.4.a.e 1
91.b odd 2 1 1352.4.a.a 1
105.g even 2 1 1800.4.a.d 1
105.k odd 4 2 1800.4.f.u 2
112.j even 4 2 256.4.b.g 2
112.l odd 4 2 256.4.b.a 2
119.d odd 2 1 2312.4.a.a 1
140.c even 2 1 400.4.a.g 1
140.j odd 4 2 400.4.c.i 2
168.e odd 2 1 576.4.a.j 1
168.i even 2 1 576.4.a.k 1
280.c odd 2 1 1600.4.a.o 1
280.n even 2 1 1600.4.a.bm 1
308.g odd 2 1 1936.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 7.b odd 2 1
16.4.a.a 1 28.d even 2 1
64.4.a.b 1 56.e even 2 1
64.4.a.d 1 56.h odd 2 1
72.4.a.c 1 21.c even 2 1
144.4.a.e 1 84.h odd 2 1
200.4.a.g 1 35.c odd 2 1
200.4.c.e 2 35.f even 4 2
256.4.b.a 2 112.l odd 4 2
256.4.b.g 2 112.j even 4 2
392.4.a.e 1 1.a even 1 1 trivial
392.4.i.b 2 7.c even 3 2
392.4.i.g 2 7.d odd 6 2
400.4.a.g 1 140.c even 2 1
400.4.c.i 2 140.j odd 4 2
576.4.a.j 1 168.e odd 2 1
576.4.a.k 1 168.i even 2 1
648.4.i.e 2 63.o even 6 2
648.4.i.h 2 63.l odd 6 2
784.4.a.e 1 4.b odd 2 1
968.4.a.a 1 77.b even 2 1
1352.4.a.a 1 91.b odd 2 1
1600.4.a.o 1 280.c odd 2 1
1600.4.a.bm 1 280.n even 2 1
1800.4.a.d 1 105.g even 2 1
1800.4.f.u 2 105.k odd 4 2
1936.4.a.l 1 308.g odd 2 1
2312.4.a.a 1 119.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(392))\):

\( T_{3} - 4 \)
\( T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 27 T^{2} \)
$5$ \( 1 - 2 T + 125 T^{2} \)
$7$ 1
$11$ \( 1 + 44 T + 1331 T^{2} \)
$13$ \( 1 + 22 T + 2197 T^{2} \)
$17$ \( 1 + 50 T + 4913 T^{2} \)
$19$ \( 1 + 44 T + 6859 T^{2} \)
$23$ \( 1 + 56 T + 12167 T^{2} \)
$29$ \( 1 - 198 T + 24389 T^{2} \)
$31$ \( 1 - 160 T + 29791 T^{2} \)
$37$ \( 1 + 162 T + 50653 T^{2} \)
$41$ \( 1 - 198 T + 68921 T^{2} \)
$43$ \( 1 - 52 T + 79507 T^{2} \)
$47$ \( 1 + 528 T + 103823 T^{2} \)
$53$ \( 1 + 242 T + 148877 T^{2} \)
$59$ \( 1 - 668 T + 205379 T^{2} \)
$61$ \( 1 + 550 T + 226981 T^{2} \)
$67$ \( 1 - 188 T + 300763 T^{2} \)
$71$ \( 1 - 728 T + 357911 T^{2} \)
$73$ \( 1 + 154 T + 389017 T^{2} \)
$79$ \( 1 + 656 T + 493039 T^{2} \)
$83$ \( 1 + 236 T + 571787 T^{2} \)
$89$ \( 1 + 714 T + 704969 T^{2} \)
$97$ \( 1 - 478 T + 912673 T^{2} \)
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