# 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 $4$

# Related objects

## 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: 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
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

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