Properties

Label 392.2.a.f
Level $392$
Weight $2$
Character orbit 392.a
Self dual yes
Analytic conductor $3.130$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - q^{5} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} - q^{5} + 6q^{9} - q^{11} + 2q^{13} - 3q^{15} + 3q^{17} + 5q^{19} - 3q^{23} - 4q^{25} + 9q^{27} - 6q^{29} - q^{31} - 3q^{33} - 5q^{37} + 6q^{39} - 10q^{41} - 4q^{43} - 6q^{45} + q^{47} + 9q^{51} - 9q^{53} + q^{55} + 15q^{57} + 3q^{59} + 3q^{61} - 2q^{65} + 11q^{67} - 9q^{69} + 16q^{71} + 7q^{73} - 12q^{75} - 11q^{79} + 9q^{81} - 4q^{83} - 3q^{85} - 18q^{87} - 9q^{89} - 3q^{93} - 5q^{95} + 6q^{97} - 6q^{99} + O(q^{100}) \)

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 −1.00000 0 0 0 6.00000 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 392.2.a.f 1
3.b odd 2 1 3528.2.a.r 1
4.b odd 2 1 784.2.a.a 1
5.b even 2 1 9800.2.a.b 1
7.b odd 2 1 392.2.a.a 1
7.c even 3 2 56.2.i.a 2
7.d odd 6 2 392.2.i.f 2
8.b even 2 1 3136.2.a.b 1
8.d odd 2 1 3136.2.a.bc 1
12.b even 2 1 7056.2.a.bi 1
21.c even 2 1 3528.2.a.k 1
21.g even 6 2 3528.2.s.o 2
21.h odd 6 2 504.2.s.e 2
28.d even 2 1 784.2.a.j 1
28.f even 6 2 784.2.i.a 2
28.g odd 6 2 112.2.i.c 2
35.c odd 2 1 9800.2.a.bp 1
35.j even 6 2 1400.2.q.g 2
35.l odd 12 4 1400.2.bh.f 4
56.e even 2 1 3136.2.a.a 1
56.h odd 2 1 3136.2.a.bb 1
56.k odd 6 2 448.2.i.a 2
56.p even 6 2 448.2.i.f 2
84.h odd 2 1 7056.2.a.s 1
84.n even 6 2 1008.2.s.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 7.c even 3 2
112.2.i.c 2 28.g odd 6 2
392.2.a.a 1 7.b odd 2 1
392.2.a.f 1 1.a even 1 1 trivial
392.2.i.f 2 7.d odd 6 2
448.2.i.a 2 56.k odd 6 2
448.2.i.f 2 56.p even 6 2
504.2.s.e 2 21.h odd 6 2
784.2.a.a 1 4.b odd 2 1
784.2.a.j 1 28.d even 2 1
784.2.i.a 2 28.f even 6 2
1008.2.s.e 2 84.n even 6 2
1400.2.q.g 2 35.j even 6 2
1400.2.bh.f 4 35.l odd 12 4
3136.2.a.a 1 56.e even 2 1
3136.2.a.b 1 8.b even 2 1
3136.2.a.bb 1 56.h odd 2 1
3136.2.a.bc 1 8.d odd 2 1
3528.2.a.k 1 21.c even 2 1
3528.2.a.r 1 3.b odd 2 1
3528.2.s.o 2 21.g even 6 2
7056.2.a.s 1 84.h odd 2 1
7056.2.a.bi 1 12.b even 2 1
9800.2.a.b 1 5.b even 2 1
9800.2.a.bp 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( -2 + T \)
$17$ \( -3 + T \)
$19$ \( -5 + T \)
$23$ \( 3 + T \)
$29$ \( 6 + T \)
$31$ \( 1 + T \)
$37$ \( 5 + T \)
$41$ \( 10 + T \)
$43$ \( 4 + T \)
$47$ \( -1 + T \)
$53$ \( 9 + T \)
$59$ \( -3 + T \)
$61$ \( -3 + T \)
$67$ \( -11 + T \)
$71$ \( -16 + T \)
$73$ \( -7 + T \)
$79$ \( 11 + T \)
$83$ \( 4 + T \)
$89$ \( 9 + T \)
$97$ \( -6 + T \)
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