# Properties

 Label 441.6.a.f Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,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, 6, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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)

 $$f(q)$$ $$=$$ $$q - 32 q^{4}+O(q^{10})$$ q - 32 * q^4 $$q - 32 q^{4} + 427 q^{13} + 1024 q^{16} - 3143 q^{19} - 3125 q^{25} - 2723 q^{31} - 6661 q^{37} + 22475 q^{43} - 13664 q^{52} + 38626 q^{61} - 32768 q^{64} - 37939 q^{67} + 78127 q^{73} + 100576 q^{76} + 90857 q^{79} + 134386 q^{97}+O(q^{100})$$ q - 32 * q^4 + 427 * q^13 + 1024 * q^16 - 3143 * q^19 - 3125 * q^25 - 2723 * q^31 - 6661 * q^37 + 22475 * q^43 - 13664 * q^52 + 38626 * q^61 - 32768 * q^64 - 37939 * q^67 + 78127 * q^73 + 100576 * q^76 + 90857 * q^79 + 134386 * q^97

## 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
 0
0 0 −32.0000 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

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.f 1
3.b odd 2 1 CM 441.6.a.f 1
7.b odd 2 1 441.6.a.e 1
7.d odd 6 2 63.6.e.b 2
21.c even 2 1 441.6.a.e 1
21.g even 6 2 63.6.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.e.b 2 7.d odd 6 2
63.6.e.b 2 21.g even 6 2
441.6.a.e 1 7.b odd 2 1
441.6.a.e 1 21.c even 2 1
441.6.a.f 1 1.a even 1 1 trivial
441.6.a.f 1 3.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{13} - 427$$ T13 - 427

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 427$$
$17$ $$T$$
$19$ $$T + 3143$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 2723$$
$37$ $$T + 6661$$
$41$ $$T$$
$43$ $$T - 22475$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 38626$$
$67$ $$T + 37939$$
$71$ $$T$$
$73$ $$T - 78127$$
$79$ $$T - 90857$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 134386$$
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