# Properties

 Label 768.1.e.b Level $768$ Weight $1$ Character orbit 768.e Self dual yes Analytic conductor $0.383$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -8, 24 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,1,Mod(257,768)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("768.257");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 768.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.383281929702$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 192) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-2}, \sqrt{-3})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.6144.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9}+O(q^{10})$$ q + q^3 + q^9 $$q + q^{3} + q^{9} - 2 q^{19} + q^{25} + q^{27} - 2 q^{43} - q^{49} - 2 q^{57} + 2 q^{67} - 2 q^{73} + q^{75} + q^{81} - 2 q^{97}+O(q^{100})$$ q + q^3 + q^9 - 2 * q^19 + q^25 + q^27 - 2 * q^43 - q^49 - 2 * q^57 + 2 * q^67 - 2 * q^73 + q^75 + q^81 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$0$$ $$0$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0
0 1.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
24.f even 2 1 RM by $$\Q(\sqrt{6})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.1.e.b 1
3.b odd 2 1 CM 768.1.e.b 1
4.b odd 2 1 768.1.e.a 1
8.b even 2 1 768.1.e.a 1
8.d odd 2 1 CM 768.1.e.b 1
12.b even 2 1 768.1.e.a 1
16.e even 4 2 192.1.h.a 2
16.f odd 4 2 192.1.h.a 2
24.f even 2 1 RM 768.1.e.b 1
24.h odd 2 1 768.1.e.a 1
32.g even 8 4 3072.1.i.g 4
32.h odd 8 4 3072.1.i.g 4
48.i odd 4 2 192.1.h.a 2
48.k even 4 2 192.1.h.a 2
96.o even 8 4 3072.1.i.g 4
96.p odd 8 4 3072.1.i.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.1.h.a 2 16.e even 4 2
192.1.h.a 2 16.f odd 4 2
192.1.h.a 2 48.i odd 4 2
192.1.h.a 2 48.k even 4 2
768.1.e.a 1 4.b odd 2 1
768.1.e.a 1 8.b even 2 1
768.1.e.a 1 12.b even 2 1
768.1.e.a 1 24.h odd 2 1
768.1.e.b 1 1.a even 1 1 trivial
768.1.e.b 1 3.b odd 2 1 CM
768.1.e.b 1 8.d odd 2 1 CM
768.1.e.b 1 24.f even 2 1 RM
3072.1.i.g 4 32.g even 8 4
3072.1.i.g 4 32.h odd 8 4
3072.1.i.g 4 96.o even 8 4
3072.1.i.g 4 96.p odd 8 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{11}$$ T11 $$T_{19} + 2$$ T19 + 2

## Hecke characteristic polynomials

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