# Properties

 Label 441.2.e.d Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM discriminant -7 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 3 q^{8}+O(q^{10})$$ q + z * q^2 + (-z + 1) * q^4 + 3 * q^8 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 3 q^{8} + ( - 4 \zeta_{6} + 4) q^{11} + \zeta_{6} q^{16} + 4 q^{22} + 8 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - 2 q^{29} + ( - 5 \zeta_{6} + 5) q^{32} + 6 \zeta_{6} q^{37} - 12 q^{43} - 4 \zeta_{6} q^{44} + (8 \zeta_{6} - 8) q^{46} + 5 q^{50} + (10 \zeta_{6} - 10) q^{53} - 2 \zeta_{6} q^{58} + 7 q^{64} + (4 \zeta_{6} - 4) q^{67} - 16 q^{71} + (6 \zeta_{6} - 6) q^{74} - 8 \zeta_{6} q^{79} - 12 \zeta_{6} q^{86} + ( - 12 \zeta_{6} + 12) q^{88} + 8 q^{92} +O(q^{100})$$ q + z * q^2 + (-z + 1) * q^4 + 3 * q^8 + (-4*z + 4) * q^11 + z * q^16 + 4 * q^22 + 8*z * q^23 + (-5*z + 5) * q^25 - 2 * q^29 + (-5*z + 5) * q^32 + 6*z * q^37 - 12 * q^43 - 4*z * q^44 + (8*z - 8) * q^46 + 5 * q^50 + (10*z - 10) * q^53 - 2*z * q^58 + 7 * q^64 + (4*z - 4) * q^67 - 16 * q^71 + (6*z - 6) * q^74 - 8*z * q^79 - 12*z * q^86 + (-12*z + 12) * q^88 + 8 * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 + 6 * q^8 $$2 q + q^{2} + q^{4} + 6 q^{8} + 4 q^{11} + q^{16} + 8 q^{22} + 8 q^{23} + 5 q^{25} - 4 q^{29} + 5 q^{32} + 6 q^{37} - 24 q^{43} - 4 q^{44} - 8 q^{46} + 10 q^{50} - 10 q^{53} - 2 q^{58} + 14 q^{64} - 4 q^{67} - 32 q^{71} - 6 q^{74} - 8 q^{79} - 12 q^{86} + 12 q^{88} + 16 q^{92}+O(q^{100})$$ 2 * q + q^2 + q^4 + 6 * q^8 + 4 * q^11 + q^16 + 8 * q^22 + 8 * q^23 + 5 * q^25 - 4 * q^29 + 5 * q^32 + 6 * q^37 - 24 * q^43 - 4 * q^44 - 8 * q^46 + 10 * q^50 - 10 * q^53 - 2 * q^58 + 14 * q^64 - 4 * q^67 - 32 * q^71 - 6 * q^74 - 8 * q^79 - 12 * q^86 + 12 * q^88 + 16 * q^92

## 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
0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 0 3.00000 0 0
361.1 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0 3.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d 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.d 2
3.b odd 2 1 49.2.c.a 2
7.b odd 2 1 CM 441.2.e.d 2
7.c even 3 1 441.2.a.c 1
7.c even 3 1 inner 441.2.e.d 2
7.d odd 6 1 441.2.a.c 1
7.d odd 6 1 inner 441.2.e.d 2
12.b even 2 1 784.2.i.f 2
21.c even 2 1 49.2.c.a 2
21.g even 6 1 49.2.a.a 1
21.g even 6 1 49.2.c.a 2
21.h odd 6 1 49.2.a.a 1
21.h odd 6 1 49.2.c.a 2
28.f even 6 1 7056.2.a.bg 1
28.g odd 6 1 7056.2.a.bg 1
84.h odd 2 1 784.2.i.f 2
84.j odd 6 1 784.2.a.f 1
84.j odd 6 1 784.2.i.f 2
84.n even 6 1 784.2.a.f 1
84.n even 6 1 784.2.i.f 2
105.o odd 6 1 1225.2.a.c 1
105.p even 6 1 1225.2.a.c 1
105.w odd 12 2 1225.2.b.c 2
105.x even 12 2 1225.2.b.c 2
168.s odd 6 1 3136.2.a.n 1
168.v even 6 1 3136.2.a.o 1
168.ba even 6 1 3136.2.a.n 1
168.be odd 6 1 3136.2.a.o 1
231.k odd 6 1 5929.2.a.c 1
231.l even 6 1 5929.2.a.c 1
273.w odd 6 1 8281.2.a.d 1
273.ba even 6 1 8281.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 21.g even 6 1
49.2.a.a 1 21.h odd 6 1
49.2.c.a 2 3.b odd 2 1
49.2.c.a 2 21.c even 2 1
49.2.c.a 2 21.g even 6 1
49.2.c.a 2 21.h odd 6 1
441.2.a.c 1 7.c even 3 1
441.2.a.c 1 7.d odd 6 1
441.2.e.d 2 1.a even 1 1 trivial
441.2.e.d 2 7.b odd 2 1 CM
441.2.e.d 2 7.c even 3 1 inner
441.2.e.d 2 7.d odd 6 1 inner
784.2.a.f 1 84.j odd 6 1
784.2.a.f 1 84.n even 6 1
784.2.i.f 2 12.b even 2 1
784.2.i.f 2 84.h odd 2 1
784.2.i.f 2 84.j odd 6 1
784.2.i.f 2 84.n even 6 1
1225.2.a.c 1 105.o odd 6 1
1225.2.a.c 1 105.p even 6 1
1225.2.b.c 2 105.w odd 12 2
1225.2.b.c 2 105.x even 12 2
3136.2.a.n 1 168.s odd 6 1
3136.2.a.n 1 168.ba even 6 1
3136.2.a.o 1 168.v even 6 1
3136.2.a.o 1 168.be odd 6 1
5929.2.a.c 1 231.k odd 6 1
5929.2.a.c 1 231.l even 6 1
7056.2.a.bg 1 28.f even 6 1
7056.2.a.bg 1 28.g odd 6 1
8281.2.a.d 1 273.w odd 6 1
8281.2.a.d 1 273.ba even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{5}$$ T5 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$T^{2}$$
$43$ $$(T + 12)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 10T + 100$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T + 16)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$