# Properties

 Label 441.4.e.i Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ 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] = [441,4,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, 4, names="a")

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$26.0198423125$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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.

 $$f(q)$$ $$=$$ $$q + ( - 8 \zeta_{6} + 8) q^{4}+O(q^{10})$$ q + (-8*z + 8) * q^4 $$q + ( - 8 \zeta_{6} + 8) q^{4} - 70 q^{13} - 64 \zeta_{6} q^{16} - 56 \zeta_{6} q^{19} + ( - 125 \zeta_{6} + 125) q^{25} + (308 \zeta_{6} - 308) q^{31} - 110 \zeta_{6} q^{37} - 520 q^{43} + (560 \zeta_{6} - 560) q^{52} - 182 \zeta_{6} q^{61} - 512 q^{64} + ( - 880 \zeta_{6} + 880) q^{67} + (1190 \zeta_{6} - 1190) q^{73} - 448 q^{76} - 884 \zeta_{6} q^{79} - 1330 q^{97} +O(q^{100})$$ q + (-8*z + 8) * q^4 - 70 * q^13 - 64*z * q^16 - 56*z * q^19 + (-125*z + 125) * q^25 + (308*z - 308) * q^31 - 110*z * q^37 - 520 * q^43 + (560*z - 560) * q^52 - 182*z * q^61 - 512 * q^64 + (-880*z + 880) * q^67 + (1190*z - 1190) * q^73 - 448 * q^76 - 884*z * q^79 - 1330 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4}+O(q^{10})$$ 2 * q + 8 * q^4 $$2 q + 8 q^{4} - 140 q^{13} - 64 q^{16} - 56 q^{19} + 125 q^{25} - 308 q^{31} - 110 q^{37} - 1040 q^{43} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} - 1190 q^{73} - 896 q^{76} - 884 q^{79} - 2660 q^{97}+O(q^{100})$$ 2 * q + 8 * q^4 - 140 * q^13 - 64 * q^16 - 56 * q^19 + 125 * q^25 - 308 * q^31 - 110 * q^37 - 1040 * q^43 - 560 * q^52 - 182 * q^61 - 1024 * q^64 + 880 * q^67 - 1190 * q^73 - 896 * q^76 - 884 * q^79 - 2660 * 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 4.00000 + 6.92820i 0 0 0 0 0 0
361.1 0 0 4.00000 6.92820i 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 441.4.e.i 2
3.b odd 2 1 CM 441.4.e.i 2
7.b odd 2 1 441.4.e.j 2
7.c even 3 1 9.4.a.a 1
7.c even 3 1 inner 441.4.e.i 2
7.d odd 6 1 441.4.a.f 1
7.d odd 6 1 441.4.e.j 2
21.c even 2 1 441.4.e.j 2
21.g even 6 1 441.4.a.f 1
21.g even 6 1 441.4.e.j 2
21.h odd 6 1 9.4.a.a 1
21.h odd 6 1 inner 441.4.e.i 2
28.g odd 6 1 144.4.a.d 1
35.j even 6 1 225.4.a.d 1
35.l odd 12 2 225.4.b.g 2
56.k odd 6 1 576.4.a.l 1
56.p even 6 1 576.4.a.m 1
63.g even 3 1 81.4.c.b 2
63.h even 3 1 81.4.c.b 2
63.j odd 6 1 81.4.c.b 2
63.n odd 6 1 81.4.c.b 2
77.h odd 6 1 1089.4.a.g 1
84.n even 6 1 144.4.a.d 1
91.r even 6 1 1521.4.a.g 1
105.o odd 6 1 225.4.a.d 1
105.x even 12 2 225.4.b.g 2
168.s odd 6 1 576.4.a.m 1
168.v even 6 1 576.4.a.l 1
231.l even 6 1 1089.4.a.g 1
273.w odd 6 1 1521.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 7.c even 3 1
9.4.a.a 1 21.h odd 6 1
81.4.c.b 2 63.g even 3 1
81.4.c.b 2 63.h even 3 1
81.4.c.b 2 63.j odd 6 1
81.4.c.b 2 63.n odd 6 1
144.4.a.d 1 28.g odd 6 1
144.4.a.d 1 84.n even 6 1
225.4.a.d 1 35.j even 6 1
225.4.a.d 1 105.o odd 6 1
225.4.b.g 2 35.l odd 12 2
225.4.b.g 2 105.x even 12 2
441.4.a.f 1 7.d odd 6 1
441.4.a.f 1 21.g even 6 1
441.4.e.i 2 1.a even 1 1 trivial
441.4.e.i 2 3.b odd 2 1 CM
441.4.e.i 2 7.c even 3 1 inner
441.4.e.i 2 21.h odd 6 1 inner
441.4.e.j 2 7.b odd 2 1
441.4.e.j 2 7.d odd 6 1
441.4.e.j 2 21.c even 2 1
441.4.e.j 2 21.g even 6 1
576.4.a.l 1 56.k odd 6 1
576.4.a.l 1 168.v even 6 1
576.4.a.m 1 56.p even 6 1
576.4.a.m 1 168.s odd 6 1
1089.4.a.g 1 77.h odd 6 1
1089.4.a.g 1 231.l even 6 1
1521.4.a.g 1 91.r even 6 1
1521.4.a.g 1 273.w odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{13} + 70$$ T13 + 70

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 70)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 56T + 3136$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 308T + 94864$$
$37$ $$T^{2} + 110T + 12100$$
$41$ $$T^{2}$$
$43$ $$(T + 520)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 182T + 33124$$
$67$ $$T^{2} - 880T + 774400$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 1190 T + 1416100$$
$79$ $$T^{2} + 884T + 781456$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 1330)^{2}$$