# Properties

 Label 3267.1.c.a Level $3267$ Weight $1$ Character orbit 3267.c Analytic conductor $1.630$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3267,1,Mod(2782,3267)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3267, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3267.2782");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3267.c (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: $$1.63044539627$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} + \cdots)$$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{4} - \beta_1 q^{7}+O(q^{10})$$ q + q^4 - b1 * q^7 $$q + q^{4} - \beta_1 q^{7} - \beta_{3} q^{13} + q^{16} + ( - \beta_{3} + \beta_1) q^{19} - q^{25} - \beta_1 q^{28} - \beta_{2} q^{31} + (\beta_{3} - \beta_1) q^{43} + (\beta_{2} - 1) q^{49} - \beta_{3} q^{52} + (\beta_{3} - \beta_1) q^{61} + q^{64} + \beta_{2} q^{67} - \beta_{3} q^{73} + ( - \beta_{3} + \beta_1) q^{76} + \beta_{3} q^{79} - q^{91} + q^{97}+O(q^{100})$$ q + q^4 - b1 * q^7 - b3 * q^13 + q^16 + (-b3 + b1) * q^19 - q^25 - b1 * q^28 - b2 * q^31 + (b3 - b1) * q^43 + (b2 - 1) * q^49 - b3 * q^52 + (b3 - b1) * q^61 + q^64 + b2 * q^67 - b3 * q^73 + (-b3 + b1) * q^76 + b3 * q^79 - q^91 + q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4}+O(q^{10})$$ 4 * q + 4 * q^4 $$4 q + 4 q^{4} + 4 q^{16} - 4 q^{25} - 4 q^{49} + 4 q^{64} - 4 q^{91} + 4 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 + 4 * q^16 - 4 * q^25 - 4 * q^49 + 4 * q^64 - 4 * q^91 + 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$3026$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2782.1
 1.93185i 0.517638i − 0.517638i − 1.93185i
0 0 1.00000 0 0 1.93185i 0 0 0
2782.2 0 0 1.00000 0 0 0.517638i 0 0 0
2782.3 0 0 1.00000 0 0 0.517638i 0 0 0
2782.4 0 0 1.00000 0 0 1.93185i 0 0 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 $$\Q(\sqrt{-3})$$
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.c.a 4
3.b odd 2 1 CM 3267.1.c.a 4
11.b odd 2 1 inner 3267.1.c.a 4
11.c even 5 4 3267.1.l.a 16
11.d odd 10 4 3267.1.l.a 16
33.d even 2 1 inner 3267.1.c.a 4
33.f even 10 4 3267.1.l.a 16
33.h odd 10 4 3267.1.l.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.c.a 4 1.a even 1 1 trivial
3267.1.c.a 4 3.b odd 2 1 CM
3267.1.c.a 4 11.b odd 2 1 inner
3267.1.c.a 4 33.d even 2 1 inner
3267.1.l.a 16 11.c even 5 4
3267.1.l.a 16 11.d odd 10 4
3267.1.l.a 16 33.f even 10 4
3267.1.l.a 16 33.h odd 10 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3267, [\chi])$$.

## Hecke characteristic polynomials

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