# Properties

 Label 441.2.e.b Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} - 3 q^{8} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^4 + 2*z * q^5 - 3 * q^8 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} - 3 q^{8} + ( - 2 \zeta_{6} + 2) q^{10} + ( - 4 \zeta_{6} + 4) q^{11} + 2 q^{13} + \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + 4 \zeta_{6} q^{19} + 2 q^{20} - 4 q^{22} + ( - \zeta_{6} + 1) q^{25} - 2 \zeta_{6} q^{26} + 2 q^{29} + (5 \zeta_{6} - 5) q^{32} - 6 q^{34} - 6 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} - 6 \zeta_{6} q^{40} + 2 q^{41} - 4 q^{43} - 4 \zeta_{6} q^{44} - q^{50} + ( - 2 \zeta_{6} + 2) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} + 8 q^{55} - 2 \zeta_{6} q^{58} + (12 \zeta_{6} - 12) q^{59} - 2 \zeta_{6} q^{61} + 7 q^{64} + 4 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} - 6 \zeta_{6} q^{68} + (6 \zeta_{6} - 6) q^{73} + (6 \zeta_{6} - 6) q^{74} + 4 q^{76} + 16 \zeta_{6} q^{79} + (2 \zeta_{6} - 2) q^{80} - 2 \zeta_{6} q^{82} - 12 q^{83} + 12 q^{85} + 4 \zeta_{6} q^{86} + (12 \zeta_{6} - 12) q^{88} + 14 \zeta_{6} q^{89} + (8 \zeta_{6} - 8) q^{95} - 18 q^{97} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^4 + 2*z * q^5 - 3 * q^8 + (-2*z + 2) * q^10 + (-4*z + 4) * q^11 + 2 * q^13 + z * q^16 + (-6*z + 6) * q^17 + 4*z * q^19 + 2 * q^20 - 4 * q^22 + (-z + 1) * q^25 - 2*z * q^26 + 2 * q^29 + (5*z - 5) * q^32 - 6 * q^34 - 6*z * q^37 + (-4*z + 4) * q^38 - 6*z * q^40 + 2 * q^41 - 4 * q^43 - 4*z * q^44 - q^50 + (-2*z + 2) * q^52 + (-6*z + 6) * q^53 + 8 * q^55 - 2*z * q^58 + (12*z - 12) * q^59 - 2*z * q^61 + 7 * q^64 + 4*z * q^65 + (4*z - 4) * q^67 - 6*z * q^68 + (6*z - 6) * q^73 + (6*z - 6) * q^74 + 4 * q^76 + 16*z * q^79 + (2*z - 2) * q^80 - 2*z * q^82 - 12 * q^83 + 12 * q^85 + 4*z * q^86 + (12*z - 12) * q^88 + 14*z * q^89 + (8*z - 8) * q^95 - 18 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 + 2 * q^5 - 6 * q^8 $$2 q - q^{2} + q^{4} + 2 q^{5} - 6 q^{8} + 2 q^{10} + 4 q^{11} + 4 q^{13} + q^{16} + 6 q^{17} + 4 q^{19} + 4 q^{20} - 8 q^{22} + q^{25} - 2 q^{26} + 4 q^{29} - 5 q^{32} - 12 q^{34} - 6 q^{37} + 4 q^{38} - 6 q^{40} + 4 q^{41} - 8 q^{43} - 4 q^{44} - 2 q^{50} + 2 q^{52} + 6 q^{53} + 16 q^{55} - 2 q^{58} - 12 q^{59} - 2 q^{61} + 14 q^{64} + 4 q^{65} - 4 q^{67} - 6 q^{68} - 6 q^{73} - 6 q^{74} + 8 q^{76} + 16 q^{79} - 2 q^{80} - 2 q^{82} - 24 q^{83} + 24 q^{85} + 4 q^{86} - 12 q^{88} + 14 q^{89} - 8 q^{95} - 36 q^{97}+O(q^{100})$$ 2 * q - q^2 + q^4 + 2 * q^5 - 6 * q^8 + 2 * q^10 + 4 * q^11 + 4 * q^13 + q^16 + 6 * q^17 + 4 * q^19 + 4 * q^20 - 8 * q^22 + q^25 - 2 * q^26 + 4 * q^29 - 5 * q^32 - 12 * q^34 - 6 * q^37 + 4 * q^38 - 6 * q^40 + 4 * q^41 - 8 * q^43 - 4 * q^44 - 2 * q^50 + 2 * q^52 + 6 * q^53 + 16 * q^55 - 2 * q^58 - 12 * q^59 - 2 * q^61 + 14 * q^64 + 4 * q^65 - 4 * q^67 - 6 * q^68 - 6 * q^73 - 6 * q^74 + 8 * q^76 + 16 * q^79 - 2 * q^80 - 2 * q^82 - 24 * q^83 + 24 * q^85 + 4 * q^86 - 12 * q^88 + 14 * q^89 - 8 * q^95 - 36 * 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
−0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 1.73205i 0 0 −3.00000 0 1.00000 + 1.73205i
361.1 −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 + 1.73205i 0 0 −3.00000 0 1.00000 1.73205i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.b 2
3.b odd 2 1 147.2.e.c 2
7.b odd 2 1 441.2.e.a 2
7.c even 3 1 441.2.a.f 1
7.c even 3 1 inner 441.2.e.b 2
7.d odd 6 1 63.2.a.a 1
7.d odd 6 1 441.2.e.a 2
12.b even 2 1 2352.2.q.e 2
21.c even 2 1 147.2.e.b 2
21.g even 6 1 21.2.a.a 1
21.g even 6 1 147.2.e.b 2
21.h odd 6 1 147.2.a.a 1
21.h odd 6 1 147.2.e.c 2
28.f even 6 1 1008.2.a.l 1
28.g odd 6 1 7056.2.a.p 1
35.i odd 6 1 1575.2.a.c 1
35.k even 12 2 1575.2.d.a 2
56.j odd 6 1 4032.2.a.h 1
56.m even 6 1 4032.2.a.k 1
63.i even 6 1 567.2.f.g 2
63.k odd 6 1 567.2.f.b 2
63.s even 6 1 567.2.f.g 2
63.t odd 6 1 567.2.f.b 2
77.i even 6 1 7623.2.a.g 1
84.h odd 2 1 2352.2.q.x 2
84.j odd 6 1 336.2.a.a 1
84.j odd 6 1 2352.2.q.x 2
84.n even 6 1 2352.2.a.v 1
84.n even 6 1 2352.2.q.e 2
105.o odd 6 1 3675.2.a.n 1
105.p even 6 1 525.2.a.d 1
105.w odd 12 2 525.2.d.a 2
168.s odd 6 1 9408.2.a.bv 1
168.v even 6 1 9408.2.a.m 1
168.ba even 6 1 1344.2.a.g 1
168.be odd 6 1 1344.2.a.s 1
231.k odd 6 1 2541.2.a.j 1
273.ba even 6 1 3549.2.a.c 1
336.bo even 12 2 5376.2.c.r 2
336.br odd 12 2 5376.2.c.l 2
357.s even 6 1 6069.2.a.b 1
399.s odd 6 1 7581.2.a.d 1
420.be odd 6 1 8400.2.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 21.g even 6 1
63.2.a.a 1 7.d odd 6 1
147.2.a.a 1 21.h odd 6 1
147.2.e.b 2 21.c even 2 1
147.2.e.b 2 21.g even 6 1
147.2.e.c 2 3.b odd 2 1
147.2.e.c 2 21.h odd 6 1
336.2.a.a 1 84.j odd 6 1
441.2.a.f 1 7.c even 3 1
441.2.e.a 2 7.b odd 2 1
441.2.e.a 2 7.d odd 6 1
441.2.e.b 2 1.a even 1 1 trivial
441.2.e.b 2 7.c even 3 1 inner
525.2.a.d 1 105.p even 6 1
525.2.d.a 2 105.w odd 12 2
567.2.f.b 2 63.k odd 6 1
567.2.f.b 2 63.t odd 6 1
567.2.f.g 2 63.i even 6 1
567.2.f.g 2 63.s even 6 1
1008.2.a.l 1 28.f even 6 1
1344.2.a.g 1 168.ba even 6 1
1344.2.a.s 1 168.be odd 6 1
1575.2.a.c 1 35.i odd 6 1
1575.2.d.a 2 35.k even 12 2
2352.2.a.v 1 84.n even 6 1
2352.2.q.e 2 12.b even 2 1
2352.2.q.e 2 84.n even 6 1
2352.2.q.x 2 84.h odd 2 1
2352.2.q.x 2 84.j odd 6 1
2541.2.a.j 1 231.k odd 6 1
3549.2.a.c 1 273.ba even 6 1
3675.2.a.n 1 105.o odd 6 1
4032.2.a.h 1 56.j odd 6 1
4032.2.a.k 1 56.m even 6 1
5376.2.c.l 2 336.br odd 12 2
5376.2.c.r 2 336.bo even 12 2
6069.2.a.b 1 357.s even 6 1
7056.2.a.p 1 28.g odd 6 1
7581.2.a.d 1 399.s odd 6 1
7623.2.a.g 1 77.i even 6 1
8400.2.a.bn 1 420.be odd 6 1
9408.2.a.m 1 168.v even 6 1
9408.2.a.bv 1 168.s odd 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 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 6T + 36$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 6T + 36$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 14T + 196$$
$97$ $$(T + 18)^{2}$$