# Properties

 Label 441.4.e.a Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ 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] = [441,4,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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 5 \zeta_{6} q^{2} + (17 \zeta_{6} - 17) q^{4} + 45 q^{8} +O(q^{10})$$ q - 5*z * q^2 + (17*z - 17) * q^4 + 45 * q^8 $$q - 5 \zeta_{6} q^{2} + (17 \zeta_{6} - 17) q^{4} + 45 q^{8} + (68 \zeta_{6} - 68) q^{11} - 89 \zeta_{6} q^{16} + 340 q^{22} - 40 \zeta_{6} q^{23} + ( - 125 \zeta_{6} + 125) q^{25} + 166 q^{29} + (85 \zeta_{6} - 85) q^{32} - 450 \zeta_{6} q^{37} - 180 q^{43} - 1156 \zeta_{6} q^{44} + (200 \zeta_{6} - 200) q^{46} - 625 q^{50} + ( - 590 \zeta_{6} + 590) q^{53} - 830 \zeta_{6} q^{58} - 287 q^{64} + ( - 740 \zeta_{6} + 740) q^{67} - 688 q^{71} + (2250 \zeta_{6} - 2250) q^{74} + 1384 \zeta_{6} q^{79} + 900 \zeta_{6} q^{86} + (3060 \zeta_{6} - 3060) q^{88} + 680 q^{92} +O(q^{100})$$ q - 5*z * q^2 + (17*z - 17) * q^4 + 45 * q^8 + (68*z - 68) * q^11 - 89*z * q^16 + 340 * q^22 - 40*z * q^23 + (-125*z + 125) * q^25 + 166 * q^29 + (85*z - 85) * q^32 - 450*z * q^37 - 180 * q^43 - 1156*z * q^44 + (200*z - 200) * q^46 - 625 * q^50 + (-590*z + 590) * q^53 - 830*z * q^58 - 287 * q^64 + (-740*z + 740) * q^67 - 688 * q^71 + (2250*z - 2250) * q^74 + 1384*z * q^79 + 900*z * q^86 + (3060*z - 3060) * q^88 + 680 * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{2} - 17 q^{4} + 90 q^{8}+O(q^{10})$$ 2 * q - 5 * q^2 - 17 * q^4 + 90 * q^8 $$2 q - 5 q^{2} - 17 q^{4} + 90 q^{8} - 68 q^{11} - 89 q^{16} + 680 q^{22} - 40 q^{23} + 125 q^{25} + 332 q^{29} - 85 q^{32} - 450 q^{37} - 360 q^{43} - 1156 q^{44} - 200 q^{46} - 1250 q^{50} + 590 q^{53} - 830 q^{58} - 574 q^{64} + 740 q^{67} - 1376 q^{71} - 2250 q^{74} + 1384 q^{79} + 900 q^{86} - 3060 q^{88} + 1360 q^{92}+O(q^{100})$$ 2 * q - 5 * q^2 - 17 * q^4 + 90 * q^8 - 68 * q^11 - 89 * q^16 + 680 * q^22 - 40 * q^23 + 125 * q^25 + 332 * q^29 - 85 * q^32 - 450 * q^37 - 360 * q^43 - 1156 * q^44 - 200 * q^46 - 1250 * q^50 + 590 * q^53 - 830 * q^58 - 574 * q^64 + 740 * q^67 - 1376 * q^71 - 2250 * q^74 + 1384 * q^79 + 900 * q^86 - 3060 * q^88 + 1360 * 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
−2.50000 + 4.33013i 0 −8.50000 14.7224i 0 0 0 45.0000 0 0
361.1 −2.50000 4.33013i 0 −8.50000 + 14.7224i 0 0 0 45.0000 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 441.4.e.a 2
3.b odd 2 1 49.4.c.d 2
7.b odd 2 1 CM 441.4.e.a 2
7.c even 3 1 441.4.a.m 1
7.c even 3 1 inner 441.4.e.a 2
7.d odd 6 1 441.4.a.m 1
7.d odd 6 1 inner 441.4.e.a 2
21.c even 2 1 49.4.c.d 2
21.g even 6 1 49.4.a.a 1
21.g even 6 1 49.4.c.d 2
21.h odd 6 1 49.4.a.a 1
21.h odd 6 1 49.4.c.d 2
84.j odd 6 1 784.4.a.k 1
84.n even 6 1 784.4.a.k 1
105.o odd 6 1 1225.4.a.l 1
105.p even 6 1 1225.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 21.g even 6 1
49.4.a.a 1 21.h odd 6 1
49.4.c.d 2 3.b odd 2 1
49.4.c.d 2 21.c even 2 1
49.4.c.d 2 21.g even 6 1
49.4.c.d 2 21.h odd 6 1
441.4.a.m 1 7.c even 3 1
441.4.a.m 1 7.d odd 6 1
441.4.e.a 2 1.a even 1 1 trivial
441.4.e.a 2 7.b odd 2 1 CM
441.4.e.a 2 7.c even 3 1 inner
441.4.e.a 2 7.d odd 6 1 inner
784.4.a.k 1 84.j odd 6 1
784.4.a.k 1 84.n even 6 1
1225.4.a.l 1 105.o odd 6 1
1225.4.a.l 1 105.p even 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 5T_{2} + 25$$ T2^2 + 5*T2 + 25 $$T_{5}$$ T5 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5T + 25$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 68T + 4624$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 40T + 1600$$
$29$ $$(T - 166)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 450T + 202500$$
$41$ $$T^{2}$$
$43$ $$(T + 180)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 590T + 348100$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 740T + 547600$$
$71$ $$(T + 688)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} - 1384 T + 1915456$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$