Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 392.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.195633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
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 $C_3\times D_8$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{3} q^{6} - q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{3} q^{6} - q^{8} + \beta_{2} q^{9} + ( \beta_{1} + \beta_{3} ) q^{12} + \beta_{2} q^{16} + \beta_{1} q^{17} + ( 1 + \beta_{2} ) q^{18} + ( -\beta_{1} - \beta_{3} ) q^{19} + \beta_{1} q^{24} + ( -1 - \beta_{2} ) q^{25} + ( 1 + \beta_{2} ) q^{32} -\beta_{3} q^{34} + q^{36} -\beta_{1} q^{38} -\beta_{3} q^{41} -\beta_{3} q^{48} - q^{50} -2 \beta_{2} q^{51} -2 q^{57} + \beta_{1} q^{59} + q^{64} + ( 2 + 2 \beta_{2} ) q^{67} + ( -\beta_{1} - \beta_{3} ) q^{68} -\beta_{2} q^{72} -\beta_{1} q^{73} + ( \beta_{1} + \beta_{3} ) q^{75} + \beta_{3} q^{76} + ( 1 + \beta_{2} ) q^{81} + ( -\beta_{1} - \beta_{3} ) q^{82} + \beta_{3} q^{83} + ( \beta_{1} + \beta_{3} ) q^{89} + ( -\beta_{1} - \beta_{3} ) q^{96} + \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} - 4q^{8} - 2q^{9} - 2q^{16} + 2q^{18} - 2q^{25} + 2q^{32} + 4q^{36} - 4q^{50} + 4q^{51} - 8q^{57} + 4q^{64} + 4q^{67} + 2q^{72} + 2q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
0.500000 + 0.866025i −0.707107 + 1.22474i −0.500000 + 0.866025i 0 −1.41421 0 −1.00000 −0.500000 0.866025i 0
67.2 0.500000 + 0.866025i 0.707107 1.22474i −0.500000 + 0.866025i 0 1.41421 0 −1.00000 −0.500000 0.866025i 0
275.1 0.500000 0.866025i −0.707107 1.22474i −0.500000 0.866025i 0 −1.41421 0 −1.00000 −0.500000 + 0.866025i 0
275.2 0.500000 0.866025i 0.707107 + 1.22474i −0.500000 0.866025i 0 1.41421 0 −1.00000 −0.500000 + 0.866025i 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
7.c even 3 1 inner
7.d odd 6 1 inner
56.e even 2 1 inner
56.k odd 6 1 inner
56.m even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2 T_{3}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(392, [\chi])\).

Hecke characteristic polynomials

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