Properties

Label 392.2.a.e
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

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

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 1.00000 0 1.00000 0 0 0 −2.00000 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.e 1
3.b odd 2 1 3528.2.a.j 1
4.b odd 2 1 784.2.a.c 1
5.b even 2 1 9800.2.a.s 1
7.b odd 2 1 392.2.a.c 1
7.c even 3 2 392.2.i.b 2
7.d odd 6 2 56.2.i.b 2
8.b even 2 1 3136.2.a.i 1
8.d odd 2 1 3136.2.a.t 1
12.b even 2 1 7056.2.a.u 1
21.c even 2 1 3528.2.a.p 1
21.g even 6 2 504.2.s.c 2
21.h odd 6 2 3528.2.s.q 2
28.d even 2 1 784.2.a.h 1
28.f even 6 2 112.2.i.a 2
28.g odd 6 2 784.2.i.h 2
35.c odd 2 1 9800.2.a.be 1
35.i odd 6 2 1400.2.q.d 2
35.k even 12 4 1400.2.bh.a 4
56.e even 2 1 3136.2.a.j 1
56.h odd 2 1 3136.2.a.u 1
56.j odd 6 2 448.2.i.b 2
56.m even 6 2 448.2.i.d 2
84.h odd 2 1 7056.2.a.bj 1
84.j odd 6 2 1008.2.s.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 7.d odd 6 2
112.2.i.a 2 28.f even 6 2
392.2.a.c 1 7.b odd 2 1
392.2.a.e 1 1.a even 1 1 trivial
392.2.i.b 2 7.c even 3 2
448.2.i.b 2 56.j odd 6 2
448.2.i.d 2 56.m even 6 2
504.2.s.c 2 21.g even 6 2
784.2.a.c 1 4.b odd 2 1
784.2.a.h 1 28.d even 2 1
784.2.i.h 2 28.g odd 6 2
1008.2.s.g 2 84.j odd 6 2
1400.2.q.d 2 35.i odd 6 2
1400.2.bh.a 4 35.k even 12 4
3136.2.a.i 1 8.b even 2 1
3136.2.a.j 1 56.e even 2 1
3136.2.a.t 1 8.d odd 2 1
3136.2.a.u 1 56.h odd 2 1
3528.2.a.j 1 3.b odd 2 1
3528.2.a.p 1 21.c even 2 1
3528.2.s.q 2 21.h odd 6 2
7056.2.a.u 1 12.b even 2 1
7056.2.a.bj 1 84.h odd 2 1
9800.2.a.s 1 5.b even 2 1
9800.2.a.be 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + 3 T^{2} \)
$5$ \( 1 - T + 5 T^{2} \)
$7$ 1
$11$ \( 1 - 3 T + 11 T^{2} \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 - 5 T + 17 T^{2} \)
$19$ \( 1 + T + 19 T^{2} \)
$23$ \( 1 + 7 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 - 5 T + 31 T^{2} \)
$37$ \( 1 - 3 T + 37 T^{2} \)
$41$ \( 1 - 2 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 5 T + 47 T^{2} \)
$53$ \( 1 + T + 53 T^{2} \)
$59$ \( 1 + 15 T + 59 T^{2} \)
$61$ \( 1 - 5 T + 61 T^{2} \)
$67$ \( 1 + 9 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 7 T + 73 T^{2} \)
$79$ \( 1 - T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 7 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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