# Properties

 Label 55.1.d.a Level $55$ Weight $1$ Character orbit 55.d Self dual yes Analytic conductor $0.027$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -11, -55, 5 Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,1,Mod(54,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 55.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: $$0.0274485756948$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-11})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{4} - q^{5} + q^{9}+O(q^{10})$$ q - q^4 - q^5 + q^9 $$q - q^{4} - q^{5} + q^{9} - q^{11} + q^{16} + q^{20} + q^{25} - 2 q^{31} - q^{36} + q^{44} - q^{45} - q^{49} + q^{55} + 2 q^{59} - q^{64} + 2 q^{71} - q^{80} + q^{81} - 2 q^{89} - q^{99}+O(q^{100})$$ q - q^4 - q^5 + q^9 - q^11 + q^16 + q^20 + q^25 - 2 * q^31 - q^36 + q^44 - q^45 - q^49 + q^55 + 2 * q^59 - q^64 + 2 * q^71 - q^80 + q^81 - 2 * q^89 - q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
54.1
 0
0 0 −1.00000 −1.00000 0 0 0 1.00000 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
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
55.d odd 2 1 CM by $$\Q(\sqrt{-55})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.1.d.a 1
3.b odd 2 1 495.1.h.a 1
4.b odd 2 1 880.1.i.a 1
5.b even 2 1 RM 55.1.d.a 1
5.c odd 4 2 275.1.c.a 1
7.b odd 2 1 2695.1.g.c 1
7.c even 3 2 2695.1.q.c 2
7.d odd 6 2 2695.1.q.b 2
8.b even 2 1 3520.1.i.b 1
8.d odd 2 1 3520.1.i.a 1
11.b odd 2 1 CM 55.1.d.a 1
11.c even 5 4 605.1.h.a 4
11.d odd 10 4 605.1.h.a 4
15.d odd 2 1 495.1.h.a 1
15.e even 4 2 2475.1.b.a 1
20.d odd 2 1 880.1.i.a 1
33.d even 2 1 495.1.h.a 1
35.c odd 2 1 2695.1.g.c 1
35.i odd 6 2 2695.1.q.b 2
35.j even 6 2 2695.1.q.c 2
40.e odd 2 1 3520.1.i.a 1
40.f even 2 1 3520.1.i.b 1
44.c even 2 1 880.1.i.a 1
55.d odd 2 1 CM 55.1.d.a 1
55.e even 4 2 275.1.c.a 1
55.h odd 10 4 605.1.h.a 4
55.j even 10 4 605.1.h.a 4
55.k odd 20 8 3025.1.x.a 4
55.l even 20 8 3025.1.x.a 4
77.b even 2 1 2695.1.g.c 1
77.h odd 6 2 2695.1.q.c 2
77.i even 6 2 2695.1.q.b 2
88.b odd 2 1 3520.1.i.b 1
88.g even 2 1 3520.1.i.a 1
165.d even 2 1 495.1.h.a 1
165.l odd 4 2 2475.1.b.a 1
220.g even 2 1 880.1.i.a 1
385.h even 2 1 2695.1.g.c 1
385.o even 6 2 2695.1.q.b 2
385.q odd 6 2 2695.1.q.c 2
440.c even 2 1 3520.1.i.a 1
440.o odd 2 1 3520.1.i.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 1.a even 1 1 trivial
55.1.d.a 1 5.b even 2 1 RM
55.1.d.a 1 11.b odd 2 1 CM
55.1.d.a 1 55.d odd 2 1 CM
275.1.c.a 1 5.c odd 4 2
275.1.c.a 1 55.e even 4 2
495.1.h.a 1 3.b odd 2 1
495.1.h.a 1 15.d odd 2 1
495.1.h.a 1 33.d even 2 1
495.1.h.a 1 165.d even 2 1
605.1.h.a 4 11.c even 5 4
605.1.h.a 4 11.d odd 10 4
605.1.h.a 4 55.h odd 10 4
605.1.h.a 4 55.j even 10 4
880.1.i.a 1 4.b odd 2 1
880.1.i.a 1 20.d odd 2 1
880.1.i.a 1 44.c even 2 1
880.1.i.a 1 220.g even 2 1
2475.1.b.a 1 15.e even 4 2
2475.1.b.a 1 165.l odd 4 2
2695.1.g.c 1 7.b odd 2 1
2695.1.g.c 1 35.c odd 2 1
2695.1.g.c 1 77.b even 2 1
2695.1.g.c 1 385.h even 2 1
2695.1.q.b 2 7.d odd 6 2
2695.1.q.b 2 35.i odd 6 2
2695.1.q.b 2 77.i even 6 2
2695.1.q.b 2 385.o even 6 2
2695.1.q.c 2 7.c even 3 2
2695.1.q.c 2 35.j even 6 2
2695.1.q.c 2 77.h odd 6 2
2695.1.q.c 2 385.q odd 6 2
3025.1.x.a 4 55.k odd 20 8
3025.1.x.a 4 55.l even 20 8
3520.1.i.a 1 8.d odd 2 1
3520.1.i.a 1 40.e odd 2 1
3520.1.i.a 1 88.g even 2 1
3520.1.i.a 1 440.c even 2 1
3520.1.i.b 1 8.b even 2 1
3520.1.i.b 1 40.f even 2 1
3520.1.i.b 1 88.b odd 2 1
3520.1.i.b 1 440.o odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

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