# Properties

 Label 441.2.a.h Level $441$ Weight $2$ Character orbit 441.a Self dual yes Analytic conductor $3.521$ 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] = [441,2,Mod(1,441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## 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: yes Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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 $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 5 q^{4} + 3 \beta q^{8}+O(q^{10})$$ q + b * q^2 + 5 * q^4 + 3*b * q^8 $$q + \beta q^{2} + 5 q^{4} + 3 \beta q^{8} - 2 \beta q^{11} + 11 q^{16} - 14 q^{22} + 2 \beta q^{23} - 5 q^{25} - 4 \beta q^{29} + 5 \beta q^{32} + 6 q^{37} + 12 q^{43} - 10 \beta q^{44} + 14 q^{46} - 5 \beta q^{50} - 4 \beta q^{53} - 28 q^{58} + 13 q^{64} + 4 q^{67} - 2 \beta q^{71} + 6 \beta q^{74} + 8 q^{79} + 12 \beta q^{86} - 42 q^{88} + 10 \beta q^{92} +O(q^{100})$$ q + b * q^2 + 5 * q^4 + 3*b * q^8 - 2*b * q^11 + 11 * q^16 - 14 * q^22 + 2*b * q^23 - 5 * q^25 - 4*b * q^29 + 5*b * q^32 + 6 * q^37 + 12 * q^43 - 10*b * q^44 + 14 * q^46 - 5*b * q^50 - 4*b * q^53 - 28 * q^58 + 13 * q^64 + 4 * q^67 - 2*b * q^71 + 6*b * q^74 + 8 * q^79 + 12*b * q^86 - 42 * q^88 + 10*b * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{4}+O(q^{10})$$ 2 * q + 10 * q^4 $$2 q + 10 q^{4} + 22 q^{16} - 28 q^{22} - 10 q^{25} + 12 q^{37} + 24 q^{43} + 28 q^{46} - 56 q^{58} + 26 q^{64} + 8 q^{67} + 16 q^{79} - 84 q^{88}+O(q^{100})$$ 2 * q + 10 * q^4 + 22 * q^16 - 28 * q^22 - 10 * q^25 + 12 * q^37 + 24 * q^43 + 28 * q^46 - 56 * q^58 + 26 * q^64 + 8 * q^67 + 16 * q^79 - 84 * q^88

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64575 2.64575
−2.64575 0 5.00000 0 0 0 −7.93725 0 0
1.2 2.64575 0 5.00000 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

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## 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
21.c even 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 1.a even 1 1 trivial
441.2.a.h 2 3.b odd 2 1 inner
441.2.a.h 2 7.b odd 2 1 CM
441.2.a.h 2 21.c even 2 1 inner
441.2.e.h 4 7.c even 3 2
441.2.e.h 4 7.d odd 6 2
441.2.e.h 4 21.g even 6 2
441.2.e.h 4 21.h odd 6 2
7056.2.a.co 2 4.b odd 2 1
7056.2.a.co 2 12.b even 2 1
7056.2.a.co 2 28.d even 2 1
7056.2.a.co 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} - 7$$ T2^2 - 7 $$T_{5}$$ T5 $$T_{13}$$ T13

## Hecke characteristic polynomials

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