Properties

Label 392.1.g.b
Level $392$
Weight $1$
Character orbit 392.g
Self dual yes
Analytic conductor $0.196$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 392.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.195633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.2744.1
Artin image $D_8$
Artin field Galois closure of 8.0.421654016.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} - q^{8} + q^{9} -\beta q^{12} + q^{16} + \beta q^{17} - q^{18} + \beta q^{19} + \beta q^{24} + q^{25} - q^{32} -\beta q^{34} + q^{36} -\beta q^{38} -\beta q^{41} -\beta q^{48} - q^{50} -2 q^{51} -2 q^{57} + \beta q^{59} + q^{64} -2 q^{67} + \beta q^{68} - q^{72} -\beta q^{73} -\beta q^{75} + \beta q^{76} - q^{81} + \beta q^{82} + \beta q^{83} -\beta q^{89} + \beta q^{96} + \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + 2q^{9} + 2q^{16} - 2q^{18} + 2q^{25} - 2q^{32} + 2q^{36} - 2q^{50} - 4q^{51} - 4q^{57} + 2q^{64} - 4q^{67} - 2q^{72} - 2q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.41421
−1.41421
−1.00000 −1.41421 1.00000 0 1.41421 0 −1.00000 1.00000 0
99.2 −1.00000 1.41421 1.00000 0 −1.41421 0 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
7.b odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.1.g.b 2
3.b odd 2 1 3528.1.g.c 2
4.b odd 2 1 1568.1.g.b 2
7.b odd 2 1 inner 392.1.g.b 2
7.c even 3 2 392.1.k.b 4
7.d odd 6 2 392.1.k.b 4
8.b even 2 1 1568.1.g.b 2
8.d odd 2 1 CM 392.1.g.b 2
21.c even 2 1 3528.1.g.c 2
21.g even 6 2 3528.1.bx.b 4
21.h odd 6 2 3528.1.bx.b 4
24.f even 2 1 3528.1.g.c 2
28.d even 2 1 1568.1.g.b 2
28.f even 6 2 1568.1.o.b 4
28.g odd 6 2 1568.1.o.b 4
56.e even 2 1 inner 392.1.g.b 2
56.h odd 2 1 1568.1.g.b 2
56.j odd 6 2 1568.1.o.b 4
56.k odd 6 2 392.1.k.b 4
56.m even 6 2 392.1.k.b 4
56.p even 6 2 1568.1.o.b 4
168.e odd 2 1 3528.1.g.c 2
168.v even 6 2 3528.1.bx.b 4
168.be odd 6 2 3528.1.bx.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.1.g.b 2 1.a even 1 1 trivial
392.1.g.b 2 7.b odd 2 1 inner
392.1.g.b 2 8.d odd 2 1 CM
392.1.g.b 2 56.e even 2 1 inner
392.1.k.b 4 7.c even 3 2
392.1.k.b 4 7.d odd 6 2
392.1.k.b 4 56.k odd 6 2
392.1.k.b 4 56.m even 6 2
1568.1.g.b 2 4.b odd 2 1
1568.1.g.b 2 8.b even 2 1
1568.1.g.b 2 28.d even 2 1
1568.1.g.b 2 56.h odd 2 1
1568.1.o.b 4 28.f even 6 2
1568.1.o.b 4 28.g odd 6 2
1568.1.o.b 4 56.j odd 6 2
1568.1.o.b 4 56.p even 6 2
3528.1.g.c 2 3.b odd 2 1
3528.1.g.c 2 21.c even 2 1
3528.1.g.c 2 24.f even 2 1
3528.1.g.c 2 168.e odd 2 1
3528.1.bx.b 4 21.g even 6 2
3528.1.bx.b 4 21.h odd 6 2
3528.1.bx.b 4 168.v even 6 2
3528.1.bx.b 4 168.be odd 6 2

Hecke kernels

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

Hecke characteristic polynomials

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