LMFDB - Newform orbit 253.1.k.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [253,1,Mod(32,253)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(253, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 10]))

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

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

chi := DirichletCharacter("253.32");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$253 = 11 \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	253.k (of order $22$ , degree $10$ , 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:	$0.126263448196$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} - \cdots)$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q + (\zeta_{22}^{10} + \zeta_{22}^{8}) q^{3} - \zeta_{22}^{5} q^{4} + ( - \zeta_{22}^{7} - \zeta_{22}) q^{5} + ( - \zeta_{22}^{9} + \cdots - \zeta_{22}^{5}) q^{9} - \zeta_{22}^{3} q^{11} + (\zeta_{22}^{4} + \zeta_{22}^{2}) q^{12} + \cdots + (\zeta_{22}^{10} + \zeta_{22}^{8} - \zeta_{22}) q^{99} +O(q^{100})$ q + (z^10 + z^8) * q^3 - z^5 * q^4 + (-z^7 - z) * q^5 + (-z^9 - z^7 - z^5) * q^9 - z^3 * q^11 + (z^4 + z^2) * q^12 + (-z^9 + z^6 + z^4 + 1) * q^15 + z^10 * q^16 + (z^6 - z) * q^20 - z^3 * q^23 + (z^8 - z^3 + z^2) * q^25 + (z^8 + z^6 + z^4 + z^2) * q^27 + (-z^9 + z^6) * q^31 + (z^2 + 1) * q^33 + (z^10 - z^3 - z) * q^36 + (z^2 - z) * q^37 + z^8 * q^44 + (z^10 + z^8 + z^6 - z^5 - z^3 - z) * q^45 + (-z^7 + z^4) * q^47 + (-z^9 - z^7) * q^48 - z^9 * q^49 + (-z^5 + z^4) * q^53 + (z^10 + z^4) * q^55 + (-z^7 + z^6) * q^59 + (-z^9 - z^5 - z^3 + 1) * q^60 + z^4 * q^64 + (-z^9 - z^7) * q^67 + (z^2 + 1) * q^69 + (-z^5 + 1) * q^71 + (z^10 - z^7 - z^5 + z^2 - z + 1) * q^75 + (z^6 + 1) * q^80 + (z^10 - z^7 - z^5 - z^3 - z) * q^81 + (z^10 + z^8) * q^89 + z^8 * q^92 + (z^8 + z^6 - z^5 - z^3) * q^93 + (z^8 + 1) * q^97 + (z^10 + z^8 - z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{9} - q^{11} - 2 q^{12} + 7 q^{15} - q^{16} - 2 q^{20} - q^{23} - 3 q^{25} - 4 q^{27} - 2 q^{31} + 9 q^{33} - 3 q^{36} - 2 q^{37} - q^{44} - 6 q^{45} - 2 q^{47}+ \cdots - 3 q^{99}+O(q^{100})$ 10 * q - 2 * q^3 - q^4 - 2 * q^5 - 3 * q^9 - q^11 - 2 * q^12 + 7 * q^15 - q^16 - 2 * q^20 - q^23 - 3 * q^25 - 4 * q^27 - 2 * q^31 + 9 * q^33 - 3 * q^36 - 2 * q^37 - q^44 - 6 * q^45 - 2 * q^47 - 2 * q^48 - q^49 - 2 * q^53 - 2 * q^55 - 2 * q^59 + 7 * q^60 - q^64 - 2 * q^67 + 9 * q^69 + 9 * q^71 + 5 * q^75 + 9 * q^80 - 5 * q^81 - 2 * q^89 - q^92 - 4 * q^93 + 9 * q^97 - 3 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/253\mathbb{Z}\right)^\times$ .

$n$	$24$	$166$
$\chi(n)$	$-1$	$\zeta_{22}^{8}$

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}

32.1

−0.415415	−	0.909632i
−0.841254	+	0.540641i
−0.415415	+	0.909632i
0.142315	+	0.989821i
0.959493	+	0.281733i
0.142315	−	0.989821i
−0.841254	−	0.540641i
0.654861	+	0.755750i
0.959493	−	0.281733i
0.654861	−	0.755750i

−0.544078

−

0.627899i

0.841254

−

0.540641i

0.273100

1.89945i

0.0440780

−

0.306569i

54.1

0.698939

1.53046i

−0.959493

−

0.281733i

0.186393

0.215109i

−1.19894

1.38365i

87.1

−0.544078

0.627899i

0.841254

0.540641i

0.273100

−

1.89945i

0.0440780

0.306569i

98.1

0.273100

0.0801894i

−0.654861

−

0.755750i

0.698939

−

0.449181i

−0.773100

−

0.496841i

131.1

−1.61435

1.03748i

−0.142315

−

0.989821i

−0.544078

−

1.19136i

1.11435

−

2.44009i

142.1

0.273100

−

0.0801894i

−0.654861

0.755750i

0.698939

0.449181i

−0.773100

0.496841i

164.1

0.698939

−

1.53046i

−0.959493

0.281733i

0.186393

−

0.215109i

−1.19894

−

1.38365i

186.1

0.186393

1.29639i

0.415415

0.909632i

−1.61435

−

0.474017i

−0.686393

0.201543i

197.1

−1.61435

−

1.03748i

−0.142315

0.989821i

−0.544078

1.19136i

1.11435

2.44009i

219.1

0.186393

−

1.29639i

0.415415

−

0.909632i

−1.61435

0.474017i

−0.686393

−

0.201543i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11})$
23.c	even	11	1	inner
253.k	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	253.1.k.a	✓	10
3.b	odd	2	1		2277.1.be.a		10
11.b	odd	2	1	CM	253.1.k.a	✓	10
11.c	even	5	4		2783.1.dh.a		40
11.d	odd	10	4		2783.1.dh.a		40
23.c	even	11	1	inner	253.1.k.a	✓	10
33.d	even	2	1		2277.1.be.a		10
69.h	odd	22	1		2277.1.be.a		10
253.k	odd	22	1	inner	253.1.k.a	✓	10
253.m	even	55	4		2783.1.dh.a		40
253.o	odd	110	4		2783.1.dh.a		40
759.t	even	22	1		2277.1.be.a		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
253.1.k.a	✓	10	1.a	even	1	1	trivial
253.1.k.a	✓	10	11.b	odd	2	1	CM
253.1.k.a	✓	10	23.c	even	11	1	inner
253.1.k.a	✓	10	253.k	odd	22	1	inner
2277.1.be.a		10	3.b	odd	2	1
2277.1.be.a		10	33.d	even	2	1
2277.1.be.a		10	69.h	odd	22	1
2277.1.be.a		10	759.t	even	22	1
2783.1.dh.a		40	11.c	even	5	4
2783.1.dh.a		40	11.d	odd	10	4
2783.1.dh.a		40	253.m	even	55	4
2783.1.dh.a		40	253.o	odd	110	4

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(253, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10}$ T^10
$3$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$5$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1
$7$	$T^{10}$ T^10
$11$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$13$	$T^{10}$ T^10
$17$	$T^{10}$ T^10
$19$	$T^{10}$ T^10
$23$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$29$	$T^{10}$ T^10
$31$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1
$37$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1
$41$	$T^{10}$ T^10
$43$	$T^{10}$ T^10
$47$	$(T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2}$ (T^5 + T^4 - 4T^3 - 3T^2 + 3*T + 1)^2
$53$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$59$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$61$	$T^{10}$ T^10
$67$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1
$71$	$T^{10} - 9 T^{9} + \cdots + 1$ T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$73$	$T^{10}$ T^10
$79$	$T^{10}$ T^10
$83$	$T^{10}$ T^10
$89$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$97$	$T^{10} - 9 T^{9} + \cdots + 1$ T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 32.1
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	253.1.k.a

Level	$253$
Weight	$1$
Character orbit	253.k
Analytic conductor	$0.126$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{11}$
CM discriminant	-11
Inner twists	$4$