# Properties

 Label 441.2.e.j Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 63) 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + 2 \beta_{2} q^{5} - \beta_{3} q^{8}+O(q^{10})$$ q - b2 * q^2 + (b1 - 1) * q^4 + 2*b2 * q^5 - b3 * q^8 $$q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + 2 \beta_{2} q^{5} - \beta_{3} q^{8} + ( - 6 \beta_1 + 6) q^{10} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{11} + 2 q^{13} + 5 \beta_1 q^{16} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{17} + 4 \beta_1 q^{19} - 2 \beta_{3} q^{20} + 6 q^{22} + 2 \beta_{2} q^{23} + (7 \beta_1 - 7) q^{25} - 2 \beta_{2} q^{26} + ( - 4 \beta_1 + 4) q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} + 6 q^{34} - 2 \beta_1 q^{37} + (4 \beta_{3} - 4 \beta_{2}) q^{38} - 6 \beta_1 q^{40} + 6 \beta_{3} q^{41} - 4 q^{43} - 2 \beta_{2} q^{44} + ( - 6 \beta_1 + 6) q^{46} - 4 \beta_{2} q^{47} + 7 \beta_{3} q^{50} + (2 \beta_1 - 2) q^{52} + (4 \beta_{3} - 4 \beta_{2}) q^{53} - 12 q^{55} + (4 \beta_{3} - 4 \beta_{2}) q^{59} + 10 \beta_1 q^{61} - 4 \beta_{3} q^{62} + q^{64} + 4 \beta_{2} q^{65} + ( - 4 \beta_1 + 4) q^{67} - 2 \beta_{2} q^{68} - 6 \beta_{3} q^{71} + (14 \beta_1 - 14) q^{73} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{74} - 4 q^{76} - 8 \beta_1 q^{79} + ( - 10 \beta_{3} + 10 \beta_{2}) q^{80} - 18 \beta_1 q^{82} - 12 q^{85} + 4 \beta_{2} q^{86} + ( - 6 \beta_1 + 6) q^{88} + 2 \beta_{2} q^{89} - 2 \beta_{3} q^{92} + (12 \beta_1 - 12) q^{94} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{95} + 14 q^{97}+O(q^{100})$$ q - b2 * q^2 + (b1 - 1) * q^4 + 2*b2 * q^5 - b3 * q^8 + (-6*b1 + 6) * q^10 + (-2*b3 + 2*b2) * q^11 + 2 * q^13 + 5*b1 * q^16 + (-2*b3 + 2*b2) * q^17 + 4*b1 * q^19 - 2*b3 * q^20 + 6 * q^22 + 2*b2 * q^23 + (7*b1 - 7) * q^25 - 2*b2 * q^26 + (-4*b1 + 4) * q^31 + (3*b3 - 3*b2) * q^32 + 6 * q^34 - 2*b1 * q^37 + (4*b3 - 4*b2) * q^38 - 6*b1 * q^40 + 6*b3 * q^41 - 4 * q^43 - 2*b2 * q^44 + (-6*b1 + 6) * q^46 - 4*b2 * q^47 + 7*b3 * q^50 + (2*b1 - 2) * q^52 + (4*b3 - 4*b2) * q^53 - 12 * q^55 + (4*b3 - 4*b2) * q^59 + 10*b1 * q^61 - 4*b3 * q^62 + q^64 + 4*b2 * q^65 + (-4*b1 + 4) * q^67 - 2*b2 * q^68 - 6*b3 * q^71 + (14*b1 - 14) * q^73 + (-2*b3 + 2*b2) * q^74 - 4 * q^76 - 8*b1 * q^79 + (-10*b3 + 10*b2) * q^80 - 18*b1 * q^82 - 12 * q^85 + 4*b2 * q^86 + (-6*b1 + 6) * q^88 + 2*b2 * q^89 - 2*b3 * q^92 + (12*b1 - 12) * q^94 + (-8*b3 + 8*b2) * q^95 + 14 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4}+O(q^{10})$$ 4 * q - 2 * q^4 $$4 q - 2 q^{4} + 12 q^{10} + 8 q^{13} + 10 q^{16} + 8 q^{19} + 24 q^{22} - 14 q^{25} + 8 q^{31} + 24 q^{34} - 4 q^{37} - 12 q^{40} - 16 q^{43} + 12 q^{46} - 4 q^{52} - 48 q^{55} + 20 q^{61} + 4 q^{64} + 8 q^{67} - 28 q^{73} - 16 q^{76} - 16 q^{79} - 36 q^{82} - 48 q^{85} + 12 q^{88} - 24 q^{94} + 56 q^{97}+O(q^{100})$$ 4 * q - 2 * q^4 + 12 * q^10 + 8 * q^13 + 10 * q^16 + 8 * q^19 + 24 * q^22 - 14 * q^25 + 8 * q^31 + 24 * q^34 - 4 * q^37 - 12 * q^40 - 16 * q^43 + 12 * q^46 - 4 * q^52 - 48 * q^55 + 20 * q^61 + 4 * q^64 + 8 * q^67 - 28 * q^73 - 16 * q^76 - 16 * q^79 - 36 * q^82 - 48 * q^85 + 12 * q^88 - 24 * q^94 + 56 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 1.50000i 0 −0.500000 0.866025i 1.73205 3.00000i 0 0 −1.73205 0 3.00000 + 5.19615i
226.2 0.866025 1.50000i 0 −0.500000 0.866025i −1.73205 + 3.00000i 0 0 1.73205 0 3.00000 + 5.19615i
361.1 −0.866025 1.50000i 0 −0.500000 + 0.866025i 1.73205 + 3.00000i 0 0 −1.73205 0 3.00000 5.19615i
361.2 0.866025 + 1.50000i 0 −0.500000 + 0.866025i −1.73205 3.00000i 0 0 1.73205 0 3.00000 5.19615i
 $$n$$: 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 inner
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.j 4
3.b odd 2 1 inner 441.2.e.j 4
7.b odd 2 1 441.2.e.i 4
7.c even 3 1 63.2.a.b 2
7.c even 3 1 inner 441.2.e.j 4
7.d odd 6 1 441.2.a.g 2
7.d odd 6 1 441.2.e.i 4
21.c even 2 1 441.2.e.i 4
21.g even 6 1 441.2.a.g 2
21.g even 6 1 441.2.e.i 4
21.h odd 6 1 63.2.a.b 2
21.h odd 6 1 inner 441.2.e.j 4
28.f even 6 1 7056.2.a.cm 2
28.g odd 6 1 1008.2.a.n 2
35.j even 6 1 1575.2.a.q 2
35.l odd 12 2 1575.2.d.i 4
56.k odd 6 1 4032.2.a.bq 2
56.p even 6 1 4032.2.a.bt 2
63.g even 3 1 567.2.f.j 4
63.h even 3 1 567.2.f.j 4
63.j odd 6 1 567.2.f.j 4
63.n odd 6 1 567.2.f.j 4
77.h odd 6 1 7623.2.a.bi 2
84.j odd 6 1 7056.2.a.cm 2
84.n even 6 1 1008.2.a.n 2
105.o odd 6 1 1575.2.a.q 2
105.x even 12 2 1575.2.d.i 4
168.s odd 6 1 4032.2.a.bt 2
168.v even 6 1 4032.2.a.bq 2
231.l even 6 1 7623.2.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 7.c even 3 1
63.2.a.b 2 21.h odd 6 1
441.2.a.g 2 7.d odd 6 1
441.2.a.g 2 21.g even 6 1
441.2.e.i 4 7.b odd 2 1
441.2.e.i 4 7.d odd 6 1
441.2.e.i 4 21.c even 2 1
441.2.e.i 4 21.g even 6 1
441.2.e.j 4 1.a even 1 1 trivial
441.2.e.j 4 3.b odd 2 1 inner
441.2.e.j 4 7.c even 3 1 inner
441.2.e.j 4 21.h odd 6 1 inner
567.2.f.j 4 63.g even 3 1
567.2.f.j 4 63.h even 3 1
567.2.f.j 4 63.j odd 6 1
567.2.f.j 4 63.n odd 6 1
1008.2.a.n 2 28.g odd 6 1
1008.2.a.n 2 84.n even 6 1
1575.2.a.q 2 35.j even 6 1
1575.2.a.q 2 105.o odd 6 1
1575.2.d.i 4 35.l odd 12 2
1575.2.d.i 4 105.x even 12 2
4032.2.a.bq 2 56.k odd 6 1
4032.2.a.bq 2 168.v even 6 1
4032.2.a.bt 2 56.p even 6 1
4032.2.a.bt 2 168.s odd 6 1
7056.2.a.cm 2 28.f even 6 1
7056.2.a.cm 2 84.j odd 6 1
7623.2.a.bi 2 77.h odd 6 1
7623.2.a.bi 2 231.l even 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 3T_{2}^{2} + 9$$ T2^4 + 3*T2^2 + 9 $$T_{5}^{4} + 12T_{5}^{2} + 144$$ T5^4 + 12*T5^2 + 144 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 12T^{2} + 144$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 12T^{2} + 144$$
$13$ $$(T - 2)^{4}$$
$17$ $$T^{4} + 12T^{2} + 144$$
$19$ $$(T^{2} - 4 T + 16)^{2}$$
$23$ $$T^{4} + 12T^{2} + 144$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$(T^{2} + 2 T + 4)^{2}$$
$41$ $$(T^{2} - 108)^{2}$$
$43$ $$(T + 4)^{4}$$
$47$ $$T^{4} + 48T^{2} + 2304$$
$53$ $$T^{4} + 48T^{2} + 2304$$
$59$ $$T^{4} + 48T^{2} + 2304$$
$61$ $$(T^{2} - 10 T + 100)^{2}$$
$67$ $$(T^{2} - 4 T + 16)^{2}$$
$71$ $$(T^{2} - 108)^{2}$$
$73$ $$(T^{2} + 14 T + 196)^{2}$$
$79$ $$(T^{2} + 8 T + 64)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 12T^{2} + 144$$
$97$ $$(T - 14)^{4}$$
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