# Properties

 Label 441.2.e.c Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(226,441)]

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, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, 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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{4}+O(q^{10})$$ q + (-2*z + 2) * q^4 $$q + ( - 2 \zeta_{6} + 2) q^{4} + 7 q^{13} - 4 \zeta_{6} q^{16} - 7 \zeta_{6} q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + (7 \zeta_{6} - 7) q^{31} + \zeta_{6} q^{37} + 5 q^{43} + ( - 14 \zeta_{6} + 14) q^{52} + 14 \zeta_{6} q^{61} - 8 q^{64} + (11 \zeta_{6} - 11) q^{67} + (7 \zeta_{6} - 7) q^{73} - 14 q^{76} + 13 \zeta_{6} q^{79} - 14 q^{97} +O(q^{100})$$ q + (-2*z + 2) * q^4 + 7 * q^13 - 4*z * q^16 - 7*z * q^19 + (-5*z + 5) * q^25 + (7*z - 7) * q^31 + z * q^37 + 5 * q^43 + (-14*z + 14) * q^52 + 14*z * q^61 - 8 * q^64 + (11*z - 11) * q^67 + (7*z - 7) * q^73 - 14 * q^76 + 13*z * q^79 - 14 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} + 14 q^{13} - 4 q^{16} - 7 q^{19} + 5 q^{25} - 7 q^{31} + q^{37} + 10 q^{43} + 14 q^{52} + 14 q^{61} - 16 q^{64} - 11 q^{67} - 7 q^{73} - 28 q^{76} + 13 q^{79} - 28 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + 14 * q^13 - 4 * q^16 - 7 * q^19 + 5 * q^25 - 7 * q^31 + q^37 + 10 * q^43 + 14 * q^52 + 14 * q^61 - 16 * q^64 - 11 * q^67 - 7 * q^73 - 28 * q^76 + 13 * q^79 - 28 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 1.00000 + 1.73205i 0 0 0 0 0 0
361.1 0 0 1.00000 1.73205i 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.c even 3 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 441.2.e.c 2
3.b odd 2 1 CM 441.2.e.c 2
7.b odd 2 1 63.2.e.a 2
7.c even 3 1 441.2.a.e 1
7.c even 3 1 inner 441.2.e.c 2
7.d odd 6 1 63.2.e.a 2
7.d odd 6 1 441.2.a.d 1
21.c even 2 1 63.2.e.a 2
21.g even 6 1 63.2.e.a 2
21.g even 6 1 441.2.a.d 1
21.h odd 6 1 441.2.a.e 1
21.h odd 6 1 inner 441.2.e.c 2
28.d even 2 1 1008.2.s.j 2
28.f even 6 1 1008.2.s.j 2
28.f even 6 1 7056.2.a.y 1
28.g odd 6 1 7056.2.a.bf 1
63.i even 6 1 567.2.g.d 2
63.k odd 6 1 567.2.h.c 2
63.l odd 6 1 567.2.g.d 2
63.l odd 6 1 567.2.h.c 2
63.o even 6 1 567.2.g.d 2
63.o even 6 1 567.2.h.c 2
63.s even 6 1 567.2.h.c 2
63.t odd 6 1 567.2.g.d 2
84.h odd 2 1 1008.2.s.j 2
84.j odd 6 1 1008.2.s.j 2
84.j odd 6 1 7056.2.a.y 1
84.n even 6 1 7056.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 7.b odd 2 1
63.2.e.a 2 7.d odd 6 1
63.2.e.a 2 21.c even 2 1
63.2.e.a 2 21.g even 6 1
441.2.a.d 1 7.d odd 6 1
441.2.a.d 1 21.g even 6 1
441.2.a.e 1 7.c even 3 1
441.2.a.e 1 21.h odd 6 1
441.2.e.c 2 1.a even 1 1 trivial
441.2.e.c 2 3.b odd 2 1 CM
441.2.e.c 2 7.c even 3 1 inner
441.2.e.c 2 21.h odd 6 1 inner
567.2.g.d 2 63.i even 6 1
567.2.g.d 2 63.l odd 6 1
567.2.g.d 2 63.o even 6 1
567.2.g.d 2 63.t odd 6 1
567.2.h.c 2 63.k odd 6 1
567.2.h.c 2 63.l odd 6 1
567.2.h.c 2 63.o even 6 1
567.2.h.c 2 63.s even 6 1
1008.2.s.j 2 28.d even 2 1
1008.2.s.j 2 28.f even 6 1
1008.2.s.j 2 84.h odd 2 1
1008.2.s.j 2 84.j odd 6 1
7056.2.a.y 1 28.f even 6 1
7056.2.a.y 1 84.j odd 6 1
7056.2.a.bf 1 28.g odd 6 1
7056.2.a.bf 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{13} - 7$$ T13 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T - 7)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 7T + 49$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2}$$
$43$ $$(T - 5)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 7T + 49$$
$79$ $$T^{2} - 13T + 169$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 14)^{2}$$