# Properties

 Label 192.1.h.a Level $192$ Weight $1$ Character orbit 192.h Analytic conductor $0.096$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -8, 24 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,1,Mod(161,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.161");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 192.h (of order $$2$$, degree $$1$$, 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: no Analytic conductor: $$0.0958204824255$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-2}, \sqrt{-3})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.5308416.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{3} - q^{9} +O(q^{10})$$ q - z * q^3 - q^9 $$q - i q^{3} - q^{9} + 2 i q^{19} - q^{25} + i q^{27} - 2 i q^{43} - q^{49} + 2 q^{57} - 2 i q^{67} + 2 q^{73} + i q^{75} + q^{81} - 2 q^{97} +O(q^{100})$$ q - z * q^3 - q^9 + 2*z * q^19 - q^25 + z * q^27 - 2*z * q^43 - q^49 + 2 * q^57 - 2*z * q^67 + 2 * q^73 + z * q^75 + q^81 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 2 q^{25} - 2 q^{49} + 4 q^{57} + 4 q^{73} + 2 q^{81} - 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^9 - 2 * q^25 - 2 * q^49 + 4 * q^57 + 4 * q^73 + 2 * q^81 - 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
161.2 0 1.00000i 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 $$\Q(\sqrt{6})$$
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.1.h.a 2
3.b odd 2 1 CM 192.1.h.a 2
4.b odd 2 1 inner 192.1.h.a 2
8.b even 2 1 inner 192.1.h.a 2
8.d odd 2 1 CM 192.1.h.a 2
12.b even 2 1 inner 192.1.h.a 2
16.e even 4 1 768.1.e.a 1
16.e even 4 1 768.1.e.b 1
16.f odd 4 1 768.1.e.a 1
16.f odd 4 1 768.1.e.b 1
24.f even 2 1 RM 192.1.h.a 2
24.h odd 2 1 inner 192.1.h.a 2
32.g even 8 4 3072.1.i.g 4
32.h odd 8 4 3072.1.i.g 4
48.i odd 4 1 768.1.e.a 1
48.i odd 4 1 768.1.e.b 1
48.k even 4 1 768.1.e.a 1
48.k even 4 1 768.1.e.b 1
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 1.a even 1 1 trivial
192.1.h.a 2 3.b odd 2 1 CM
192.1.h.a 2 4.b odd 2 1 inner
192.1.h.a 2 8.b even 2 1 inner
192.1.h.a 2 8.d odd 2 1 CM
192.1.h.a 2 12.b even 2 1 inner
192.1.h.a 2 24.f even 2 1 RM
192.1.h.a 2 24.h odd 2 1 inner
768.1.e.a 1 16.e even 4 1
768.1.e.a 1 16.f odd 4 1
768.1.e.a 1 48.i odd 4 1
768.1.e.a 1 48.k even 4 1
768.1.e.b 1 16.e even 4 1
768.1.e.b 1 16.f odd 4 1
768.1.e.b 1 48.i odd 4 1
768.1.e.b 1 48.k even 4 1
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 is the entire newspace $$S_{1}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

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