# Properties

 Label 44.1.d.a Level $44$ Weight $1$ Character orbit 44.d Self dual yes Analytic conductor $0.022$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -11 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [44,1,Mod(21,44)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(44, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("44.21");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$44 = 2^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 44.d (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$0.0219588605559$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.44.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.44.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5}+O(q^{10})$$ q - q^3 - q^5 $$q - q^{3} - q^{5} + q^{11} + q^{15} - q^{23} + q^{27} - q^{31} - q^{33} - q^{37} + 2 q^{47} + q^{49} + 2 q^{53} - q^{55} - q^{59} - q^{67} + q^{69} - q^{71} - q^{81} - q^{89} + q^{93} - q^{97}+O(q^{100})$$ q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2 * q^47 + q^49 + 2 * q^53 - q^55 - q^59 - q^67 + q^69 - q^71 - q^81 - q^89 + q^93 - q^97

## Expression as an eta quotient

$$f(z) = \eta(2z)\eta(22z)=q\prod_{n=1}^\infty(1 - q^{2n})^{}(1 - q^{22n})^{}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/44\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$1$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1
 0
0 −1.00000 0 −1.00000 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.1.d.a 1
3.b odd 2 1 396.1.f.a 1
4.b odd 2 1 176.1.h.a 1
5.b even 2 1 1100.1.f.a 1
5.c odd 4 2 1100.1.e.a 2
7.b odd 2 1 2156.1.h.a 1
7.c even 3 2 2156.1.k.b 2
7.d odd 6 2 2156.1.k.a 2
8.b even 2 1 704.1.h.b 1
8.d odd 2 1 704.1.h.a 1
9.c even 3 2 3564.1.m.b 2
9.d odd 6 2 3564.1.m.a 2
11.b odd 2 1 CM 44.1.d.a 1
11.c even 5 4 484.1.f.a 4
11.d odd 10 4 484.1.f.a 4
12.b even 2 1 1584.1.j.a 1
16.e even 4 2 2816.1.b.b 2
16.f odd 4 2 2816.1.b.a 2
33.d even 2 1 396.1.f.a 1
44.c even 2 1 176.1.h.a 1
44.g even 10 4 1936.1.n.a 4
44.h odd 10 4 1936.1.n.a 4
55.d odd 2 1 1100.1.f.a 1
55.e even 4 2 1100.1.e.a 2
77.b even 2 1 2156.1.h.a 1
77.h odd 6 2 2156.1.k.b 2
77.i even 6 2 2156.1.k.a 2
88.b odd 2 1 704.1.h.b 1
88.g even 2 1 704.1.h.a 1
99.g even 6 2 3564.1.m.a 2
99.h odd 6 2 3564.1.m.b 2
132.d odd 2 1 1584.1.j.a 1
176.i even 4 2 2816.1.b.a 2
176.l odd 4 2 2816.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.1.d.a 1 1.a even 1 1 trivial
44.1.d.a 1 11.b odd 2 1 CM
176.1.h.a 1 4.b odd 2 1
176.1.h.a 1 44.c even 2 1
396.1.f.a 1 3.b odd 2 1
396.1.f.a 1 33.d even 2 1
484.1.f.a 4 11.c even 5 4
484.1.f.a 4 11.d odd 10 4
704.1.h.a 1 8.d odd 2 1
704.1.h.a 1 88.g even 2 1
704.1.h.b 1 8.b even 2 1
704.1.h.b 1 88.b odd 2 1
1100.1.e.a 2 5.c odd 4 2
1100.1.e.a 2 55.e even 4 2
1100.1.f.a 1 5.b even 2 1
1100.1.f.a 1 55.d odd 2 1
1584.1.j.a 1 12.b even 2 1
1584.1.j.a 1 132.d odd 2 1
1936.1.n.a 4 44.g even 10 4
1936.1.n.a 4 44.h odd 10 4
2156.1.h.a 1 7.b odd 2 1
2156.1.h.a 1 77.b even 2 1
2156.1.k.a 2 7.d odd 6 2
2156.1.k.a 2 77.i even 6 2
2156.1.k.b 2 7.c even 3 2
2156.1.k.b 2 77.h odd 6 2
2816.1.b.a 2 16.f odd 4 2
2816.1.b.a 2 176.i even 4 2
2816.1.b.b 2 16.e even 4 2
2816.1.b.b 2 176.l odd 4 2
3564.1.m.a 2 9.d odd 6 2
3564.1.m.a 2 99.g even 6 2
3564.1.m.b 2 9.c even 3 2
3564.1.m.b 2 99.h odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(44, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T$$
$31$ $$T + 1$$
$37$ $$T + 1$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T - 2$$
$53$ $$T - 2$$
$59$ $$T + 1$$
$61$ $$T$$
$67$ $$T + 1$$
$71$ $$T + 1$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 1$$
$97$ $$T + 1$$
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