# Properties

 Label 3072.1.i.g Level $3072$ Weight $1$ Character orbit 3072.i Analytic conductor $1.533$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -3, -8, 24 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,1,Mod(257,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("3072.257");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3072.i (of order $$4$$, degree $$2$$, 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: no Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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: $\OD_{16}:C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ q - z * q^3 + z^2 * q^9 $$q - \zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} - \zeta_{8} q^{19} - \zeta_{8}^{2} q^{25} - \zeta_{8}^{3} q^{27} - \zeta_{8}^{3} q^{43} + q^{49} + 2 \zeta_{8}^{2} q^{57} - \zeta_{8} q^{67} - \zeta_{8}^{2} q^{73} + \zeta_{8}^{3} q^{75} - q^{81} - q^{97} +O(q^{100})$$ q - z * q^3 + z^2 * q^9 - z * q^19 - z^2 * q^25 - z^3 * q^27 - z^3 * q^43 + q^49 + 2*z^2 * q^57 - z * q^67 - z^2 * q^73 + z^3 * q^75 - q^81 - q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{49} - 4 q^{81} - 8 q^{97}+O(q^{100})$$ 4 * q + 4 * q^49 - 4 * q^81 - 8 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$$.

 $$n$$ $$1025$$ $$2047$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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.

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
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 −0.707107 0.707107i 0 0 0 0 0 1.00000i 0
257.2 0 0.707107 + 0.707107i 0 0 0 0 0 1.00000i 0
1793.1 0 −0.707107 + 0.707107i 0 0 0 0 0 1.00000i 0
1793.2 0 0.707107 0.707107i 0 0 0 0 0 1.00000i 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
16.e even 4 2 inner
16.f odd 4 2 inner
24.h odd 2 1 inner
48.i odd 4 2 inner
48.k even 4 2 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3072, [\chi])$$:

 $$T_{5}$$ T5 $$T_{11}$$ T11 $$T_{19}^{4} + 16$$ T19^4 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4} + 16$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 16$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4} + 16$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$(T + 2)^{4}$$