# Properties

 Label 441.2.e.h Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $8$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 5*b2 * q^4 + 3*b3 * q^8 $$q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8} + (2 \beta_{3} + 2 \beta_1) q^{11} + ( - 11 \beta_{2} - 11) q^{16} - 14 q^{22} + 2 \beta_1 q^{23} - 5 \beta_{2} q^{25} - 4 \beta_{3} q^{29} + ( - 5 \beta_{3} - 5 \beta_1) q^{32} + ( - 6 \beta_{2} - 6) q^{37} + 12 q^{43} - 10 \beta_1 q^{44} + 14 \beta_{2} q^{46} - 5 \beta_{3} q^{50} + (4 \beta_{3} + 4 \beta_1) q^{53} + (28 \beta_{2} + 28) q^{58} + 13 q^{64} + 4 \beta_{2} q^{67} - 2 \beta_{3} q^{71} + ( - 6 \beta_{3} - 6 \beta_1) q^{74} + ( - 8 \beta_{2} - 8) q^{79} + 12 \beta_1 q^{86} - 42 \beta_{2} q^{88} + 10 \beta_{3} q^{92}+O(q^{100})$$ q + b1 * q^2 + 5*b2 * q^4 + 3*b3 * q^8 + (2*b3 + 2*b1) * q^11 + (-11*b2 - 11) * q^16 - 14 * q^22 + 2*b1 * q^23 - 5*b2 * q^25 - 4*b3 * q^29 + (-5*b3 - 5*b1) * q^32 + (-6*b2 - 6) * q^37 + 12 * q^43 - 10*b1 * q^44 + 14*b2 * q^46 - 5*b3 * q^50 + (4*b3 + 4*b1) * q^53 + (28*b2 + 28) * q^58 + 13 * q^64 + 4*b2 * q^67 - 2*b3 * q^71 + (-6*b3 - 6*b1) * q^74 + (-8*b2 - 8) * q^79 + 12*b1 * q^86 - 42*b2 * q^88 + 10*b3 * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{4}+O(q^{10})$$ 4 * q - 10 * q^4 $$4 q - 10 q^{4} - 22 q^{16} - 56 q^{22} + 10 q^{25} - 12 q^{37} + 48 q^{43} - 28 q^{46} + 56 q^{58} + 52 q^{64} - 8 q^{67} - 16 q^{79} + 84 q^{88}+O(q^{100})$$ 4 * q - 10 * q^4 - 22 * q^16 - 56 * q^22 + 10 * q^25 - 12 * q^37 + 48 * q^43 - 28 * q^46 + 56 * q^58 + 52 * q^64 - 8 * q^67 - 16 * q^79 + 84 * q^88

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
−1.32288 + 2.29129i 0 −2.50000 4.33013i 0 0 0 7.93725 0 0
226.2 1.32288 2.29129i 0 −2.50000 4.33013i 0 0 0 −7.93725 0 0
361.1 −1.32288 2.29129i 0 −2.50000 + 4.33013i 0 0 0 7.93725 0 0
361.2 1.32288 + 2.29129i 0 −2.50000 + 4.33013i 0 0 0 −7.93725 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 $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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.h 4
3.b odd 2 1 inner 441.2.e.h 4
7.b odd 2 1 CM 441.2.e.h 4
7.c even 3 1 441.2.a.h 2
7.c even 3 1 inner 441.2.e.h 4
7.d odd 6 1 441.2.a.h 2
7.d odd 6 1 inner 441.2.e.h 4
21.c even 2 1 inner 441.2.e.h 4
21.g even 6 1 441.2.a.h 2
21.g even 6 1 inner 441.2.e.h 4
21.h odd 6 1 441.2.a.h 2
21.h odd 6 1 inner 441.2.e.h 4
28.f even 6 1 7056.2.a.co 2
28.g odd 6 1 7056.2.a.co 2
84.j odd 6 1 7056.2.a.co 2
84.n even 6 1 7056.2.a.co 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 7.c even 3 1
441.2.a.h 2 7.d odd 6 1
441.2.a.h 2 21.g even 6 1
441.2.a.h 2 21.h odd 6 1
441.2.e.h 4 1.a even 1 1 trivial
441.2.e.h 4 3.b odd 2 1 inner
441.2.e.h 4 7.b odd 2 1 CM
441.2.e.h 4 7.c even 3 1 inner
441.2.e.h 4 7.d odd 6 1 inner
441.2.e.h 4 21.c even 2 1 inner
441.2.e.h 4 21.g even 6 1 inner
441.2.e.h 4 21.h odd 6 1 inner
7056.2.a.co 2 28.f even 6 1
7056.2.a.co 2 28.g odd 6 1
7056.2.a.co 2 84.j odd 6 1
7056.2.a.co 2 84.n even 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 7T_{2}^{2} + 49$$ T2^4 + 7*T2^2 + 49 $$T_{5}$$ T5 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 49$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 28T^{2} + 784$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 28T^{2} + 784$$
$29$ $$(T^{2} - 112)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} + 6 T + 36)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T - 12)^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 112 T^{2} + 12544$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 4 T + 16)^{2}$$
$71$ $$(T^{2} - 28)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 8 T + 64)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$