# 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 $8$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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