# Properties

 Label 441.1.m.a Level $441$ Weight $1$ Character orbit 441.m Analytic conductor $0.220$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -7, 21 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,1,Mod(19,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("441.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 441.m (of order $$6$$, 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: $$0.220087670571$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-7})$$ Artin image: $C_3\times D_4$ Artin field: Galois closure of 12.0.794280046581.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{4} +O(q^{10})$$ q - z^2 * q^4 $$q - \zeta_{6}^{2} q^{4} - \zeta_{6} q^{16} + \zeta_{6}^{2} q^{25} + \zeta_{6} q^{37} - q^{43} - q^{64} + \zeta_{6}^{2} q^{67} - \zeta_{6} q^{79} +O(q^{100})$$ q - z^2 * q^4 - z * q^16 + z^2 * q^25 + z * q^37 - q^43 - q^64 + z^2 * q^67 - z * q^79 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{4}+O(q^{10})$$ 2 * q + q^4 $$2 q + q^{4} - q^{16} - q^{25} + 2 q^{37} - 4 q^{43} - 2 q^{64} - 2 q^{67} - 2 q^{79}+O(q^{100})$$ 2 * q + q^4 - q^16 - q^25 + 2 * q^37 - 4 * q^43 - 2 * q^64 - 2 * q^67 - 2 * q^79

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0.500000 + 0.866025i 0 0 0 0 0 0
325.1 0 0 0.500000 0.866025i 0 0 0 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 $$\Q(\sqrt{-3})$$
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
21.c even 2 1 RM by $$\Q(\sqrt{21})$$
7.c even 3 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.1.m.a 2
3.b odd 2 1 CM 441.1.m.a 2
7.b odd 2 1 CM 441.1.m.a 2
7.c even 3 1 63.1.d.a 1
7.c even 3 1 inner 441.1.m.a 2
7.d odd 6 1 63.1.d.a 1
7.d odd 6 1 inner 441.1.m.a 2
9.c even 3 1 3969.1.k.b 2
9.c even 3 1 3969.1.t.c 2
9.d odd 6 1 3969.1.k.b 2
9.d odd 6 1 3969.1.t.c 2
21.c even 2 1 RM 441.1.m.a 2
21.g even 6 1 63.1.d.a 1
21.g even 6 1 inner 441.1.m.a 2
21.h odd 6 1 63.1.d.a 1
21.h odd 6 1 inner 441.1.m.a 2
28.f even 6 1 1008.1.f.a 1
28.g odd 6 1 1008.1.f.a 1
35.i odd 6 1 1575.1.h.b 1
35.j even 6 1 1575.1.h.b 1
35.k even 12 2 1575.1.e.b 2
35.l odd 12 2 1575.1.e.b 2
63.g even 3 1 567.1.l.b 2
63.g even 3 1 3969.1.t.c 2
63.h even 3 1 567.1.l.b 2
63.h even 3 1 3969.1.k.b 2
63.i even 6 1 567.1.l.b 2
63.i even 6 1 3969.1.k.b 2
63.j odd 6 1 567.1.l.b 2
63.j odd 6 1 3969.1.k.b 2
63.k odd 6 1 567.1.l.b 2
63.k odd 6 1 3969.1.t.c 2
63.l odd 6 1 3969.1.k.b 2
63.l odd 6 1 3969.1.t.c 2
63.n odd 6 1 567.1.l.b 2
63.n odd 6 1 3969.1.t.c 2
63.o even 6 1 3969.1.k.b 2
63.o even 6 1 3969.1.t.c 2
63.s even 6 1 567.1.l.b 2
63.s even 6 1 3969.1.t.c 2
63.t odd 6 1 567.1.l.b 2
63.t odd 6 1 3969.1.k.b 2
84.j odd 6 1 1008.1.f.a 1
84.n even 6 1 1008.1.f.a 1
105.o odd 6 1 1575.1.h.b 1
105.p even 6 1 1575.1.h.b 1
105.w odd 12 2 1575.1.e.b 2
105.x even 12 2 1575.1.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.1.d.a 1 7.c even 3 1
63.1.d.a 1 7.d odd 6 1
63.1.d.a 1 21.g even 6 1
63.1.d.a 1 21.h odd 6 1
441.1.m.a 2 1.a even 1 1 trivial
441.1.m.a 2 3.b odd 2 1 CM
441.1.m.a 2 7.b odd 2 1 CM
441.1.m.a 2 7.c even 3 1 inner
441.1.m.a 2 7.d odd 6 1 inner
441.1.m.a 2 21.c even 2 1 RM
441.1.m.a 2 21.g even 6 1 inner
441.1.m.a 2 21.h odd 6 1 inner
567.1.l.b 2 63.g even 3 1
567.1.l.b 2 63.h even 3 1
567.1.l.b 2 63.i even 6 1
567.1.l.b 2 63.j odd 6 1
567.1.l.b 2 63.k odd 6 1
567.1.l.b 2 63.n odd 6 1
567.1.l.b 2 63.s even 6 1
567.1.l.b 2 63.t odd 6 1
1008.1.f.a 1 28.f even 6 1
1008.1.f.a 1 28.g odd 6 1
1008.1.f.a 1 84.j odd 6 1
1008.1.f.a 1 84.n even 6 1
1575.1.e.b 2 35.k even 12 2
1575.1.e.b 2 35.l odd 12 2
1575.1.e.b 2 105.w odd 12 2
1575.1.e.b 2 105.x even 12 2
1575.1.h.b 1 35.i odd 6 1
1575.1.h.b 1 35.j even 6 1
1575.1.h.b 1 105.o odd 6 1
1575.1.h.b 1 105.p even 6 1
3969.1.k.b 2 9.c even 3 1
3969.1.k.b 2 9.d odd 6 1
3969.1.k.b 2 63.h even 3 1
3969.1.k.b 2 63.i even 6 1
3969.1.k.b 2 63.j odd 6 1
3969.1.k.b 2 63.l odd 6 1
3969.1.k.b 2 63.o even 6 1
3969.1.k.b 2 63.t odd 6 1
3969.1.t.c 2 9.c even 3 1
3969.1.t.c 2 9.d odd 6 1
3969.1.t.c 2 63.g even 3 1
3969.1.t.c 2 63.k odd 6 1
3969.1.t.c 2 63.l odd 6 1
3969.1.t.c 2 63.n odd 6 1
3969.1.t.c 2 63.o even 6 1
3969.1.t.c 2 63.s even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

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