# Properties

 Label 1568.1.g.b Level $1568$ Weight $1$ Character orbit 1568.g Self dual yes Analytic conductor $0.783$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1568.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.782533939809$$ 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: no (minimal twist has level 392) Projective image $$D_{4}$$ Projective field Galois closure of 4.0.2744.1 Artin image $D_8$ Artin field Galois closure of 8.0.6746464256.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + q^{9} +O(q^{10})$$ $$q -\beta q^{3} + q^{9} -\beta q^{17} + \beta q^{19} + q^{25} + \beta q^{41} + 2 q^{51} -2 q^{57} + \beta q^{59} + 2 q^{67} + \beta q^{73} -\beta q^{75} - q^{81} + \beta q^{83} + \beta q^{89} -\beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{9} + O(q^{10})$$ $$2q + 2q^{9} + 2q^{25} + 4q^{51} - 4q^{57} + 4q^{67} - 2q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
687.1
 1.41421 −1.41421
0 −1.41421 0 0 0 0 0 1.00000 0
687.2 0 1.41421 0 0 0 0 0 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 1568.1.g.b 2
4.b odd 2 1 392.1.g.b 2
7.b odd 2 1 inner 1568.1.g.b 2
7.c even 3 2 1568.1.o.b 4
7.d odd 6 2 1568.1.o.b 4
8.b even 2 1 392.1.g.b 2
8.d odd 2 1 CM 1568.1.g.b 2
12.b even 2 1 3528.1.g.c 2
24.h odd 2 1 3528.1.g.c 2
28.d even 2 1 392.1.g.b 2
28.f even 6 2 392.1.k.b 4
28.g odd 6 2 392.1.k.b 4
56.e even 2 1 inner 1568.1.g.b 2
56.h odd 2 1 392.1.g.b 2
56.j odd 6 2 392.1.k.b 4
56.k odd 6 2 1568.1.o.b 4
56.m even 6 2 1568.1.o.b 4
56.p even 6 2 392.1.k.b 4
84.h odd 2 1 3528.1.g.c 2
84.j odd 6 2 3528.1.bx.b 4
84.n even 6 2 3528.1.bx.b 4
168.i even 2 1 3528.1.g.c 2
168.s odd 6 2 3528.1.bx.b 4
168.ba even 6 2 3528.1.bx.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.1.g.b 2 4.b odd 2 1
392.1.g.b 2 8.b even 2 1
392.1.g.b 2 28.d even 2 1
392.1.g.b 2 56.h odd 2 1
392.1.k.b 4 28.f even 6 2
392.1.k.b 4 28.g odd 6 2
392.1.k.b 4 56.j odd 6 2
392.1.k.b 4 56.p even 6 2
1568.1.g.b 2 1.a even 1 1 trivial
1568.1.g.b 2 7.b odd 2 1 inner
1568.1.g.b 2 8.d odd 2 1 CM
1568.1.g.b 2 56.e even 2 1 inner
1568.1.o.b 4 7.c even 3 2
1568.1.o.b 4 7.d odd 6 2
1568.1.o.b 4 56.k odd 6 2
1568.1.o.b 4 56.m even 6 2
3528.1.g.c 2 12.b even 2 1
3528.1.g.c 2 24.h odd 2 1
3528.1.g.c 2 84.h odd 2 1
3528.1.g.c 2 168.i even 2 1
3528.1.bx.b 4 84.j odd 6 2
3528.1.bx.b 4 84.n even 6 2
3528.1.bx.b 4 168.s odd 6 2
3528.1.bx.b 4 168.ba even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2$$ acting on $$S_{1}^{\mathrm{new}}(1568, [\chi])$$.

## Hecke characteristic polynomials

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