LMFDB - Newform orbit 345.1.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [345,1,Mod(29,345)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(345, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 18]))

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

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

chi := DirichletCharacter("345.29");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$345 = 3 \cdot 5 \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	345.p (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:	$0.172177429358$
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:	$\Z[a_1, a_2, a_3]$
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.

$f(q)$	$=$	$q + ( - \zeta_{22}^{5} - \zeta_{22}^{3}) q^{2} + \zeta_{22}^{10} q^{3} + (\zeta_{22}^{10} + \cdots + \zeta_{22}^{6}) q^{4} + \zeta_{22}^{2} q^{5} + (\zeta_{22}^{4} + \zeta_{22}^{2}) q^{6} + ( - \zeta_{22}^{9} + \zeta_{22}^{4} + \cdots + 1) q^{8}+ \cdots + (\zeta_{22}^{4} + \zeta_{22}^{2}) q^{98}+O(q^{100})$ q + (-z^5 - z^3) * q^2 + z^10 * q^3 + (z^10 + z^8 + z^6) * q^4 + z^2 * q^5 + (z^4 + z^2) * q^6 + (-z^9 + z^4 + z^2 + 1) * q^8 - z^9 * q^9 + (-z^7 - z^5) * q^10 + (-z^9 - z^7 - z^5) * q^12 - z * q^15 + (-z^9 - z^7 - z^5 - z^3 - z) * q^16 + (z^6 + 1) * q^17 + (-z^3 - z) * q^18 + (-z^3 + z^2) * q^19 + (z^10 + z^8 - z) * q^20 - z^7 * q^23 + (z^10 + z^8 - z^3 - z) * q^24 + z^4 * q^25 + z^8 * q^27 + (z^6 + z^4) * q^30 + (-z^7 + z^6) * q^31 + (z^10 + z^8 + z^6 + z^4 - z^3 - z) * q^32 + (-z^9 - z^5 - z^3 + 1) * q^34 + (z^8 + z^6 + z^4) * q^36 + (z^8 - z^7 + z^6 - z^5) * q^38 + (z^6 + z^4 + z^2 + 1) * q^40 + q^45 + (z^10 - z) * q^46 + (-z^7 + z^4) * q^47 + (z^8 + z^6 + z^4 + z^2 + 1) * q^48 + z^10 * q^49 + (-z^9 - z^7) * q^50 + (z^10 - z^5) * q^51 + (-z^9 - z) * q^53 + (z^2 + 1) * q^54 + (z^2 - z) * q^57 + (-z^9 - z^7 + 1) * q^60 + (z^8 - z^5) * q^61 + (z^10 - z^9 - z + 1) * q^62 + (-z^9 + z^8 - z^7 + z^6 + z^4 + z^2 + 1) * q^64 + (z^10 + z^8 + z^6 - z^5 - z^3 - z) * q^68 + z^6 * q^69 + (-z^9 - z^7 + z^2 + 1) * q^72 - z^3 * q^75 + (z^10 - z^9 + z^8 + z^2 - z + 1) * q^76 + (z^8 + z^4) * q^79 + (-z^9 - z^7 - z^5 - z^3 + 1) * q^80 - z^7 * q^81 + (z^10 + z^8) * q^83 + (z^8 + z^2) * q^85 + (-z^5 - z^3) * q^90 + (z^6 + z^4 + z^2) * q^92 + (z^6 - z^5) * q^93 + (z^10 - z^9 - z^7 - z) * q^94 + (-z^5 + z^4) * q^95 + (-z^9 - z^7 - z^5 - z^3 + z^2 + 1) * q^96 + (z^4 + z^2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 2 q^{2} - q^{3} - 3 q^{4} - q^{5} - 2 q^{6} + 7 q^{8} - q^{9} - 2 q^{10} - 3 q^{12} - q^{15} - 5 q^{16} + 9 q^{17} - 2 q^{18} - 2 q^{19} - 3 q^{20} - q^{23} - 4 q^{24} - q^{25} - q^{27} - 2 q^{30}+ \cdots - 2 q^{98}+O(q^{100})$ 10 * q - 2 * q^2 - q^3 - 3 * q^4 - q^5 - 2 * q^6 + 7 * q^8 - q^9 - 2 * q^10 - 3 * q^12 - q^15 - 5 * q^16 + 9 * q^17 - 2 * q^18 - 2 * q^19 - 3 * q^20 - q^23 - 4 * q^24 - q^25 - q^27 - 2 * q^30 - 2 * q^31 - 6 * q^32 + 7 * q^34 - 3 * q^36 - 4 * q^38 + 7 * q^40 + 10 * q^45 - 2 * q^46 - 2 * q^47 + 6 * q^48 - q^49 - 2 * q^50 - 2 * q^51 - 2 * q^53 + 9 * q^54 - 2 * q^57 + 8 * q^60 - 2 * q^61 + 7 * q^62 + 4 * q^64 - 6 * q^68 - q^69 + 7 * q^72 - q^75 + 5 * q^76 - 2 * q^79 + 6 * q^80 - q^81 - 2 * q^83 - 2 * q^85 - 2 * q^90 - 3 * q^92 - 2 * q^93 - 4 * q^94 - 2 * q^95 + 5 * q^96 - 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/345\mathbb{Z}\right)^\times$ .

$n$	$116$	$166$	$277$
$\chi(n)$	$-1$	$\zeta_{22}^{4}$	$-1$

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}

29.1

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

−0.797176

1.74557i

−0.959493

−

0.281733i

−1.75667

−

2.02730i

0.841254

−

0.540641i

1.25667

−

1.45027i

3.09792

−

0.909632i

0.841254

0.540641i

0.273100

1.89945i

59.1

−1.10181

−

1.27155i

0.841254

0.540641i

−0.260554

1.81219i

0.415415

−

0.909632i

−0.239446

−

1.66538i

1.17597

−

0.755750i

0.415415

0.909632i

−1.61435

0.474017i

104.1

−0.239446

0.153882i

−0.142315

0.989821i

−0.381761

0.835939i

−0.959493

0.281733i

−0.118239

−

0.258908i

−0.0777324

−

0.540641i

−0.959493

−

0.281733i

0.186393

−

0.215109i

119.1

−0.797176

−

1.74557i

−0.959493

0.281733i

−1.75667

2.02730i

0.841254

0.540641i

1.25667

1.45027i

3.09792

0.909632i

0.841254

−

0.540641i

0.273100

−

1.89945i

164.1

−0.118239

−

0.822373i

0.415415

−

0.909632i

0.297176

−

0.0872586i

−0.654861

0.755750i

−0.797176

−

0.234072i

−0.452036

−

0.989821i

−0.654861

−

0.755750i

0.698939

0.449181i

179.1

1.25667

0.368991i

−0.654861

0.755750i

0.601808

0.386758i

−0.142315

0.989821i

−1.10181

0.708089i

−0.244123

−

0.281733i

−0.142315

−

0.989821i

−0.544078

1.19136i

209.1

−0.239446

−

0.153882i

−0.142315

−

0.989821i

−0.381761

−

0.835939i

−0.959493

−

0.281733i

−0.118239

0.258908i

−0.0777324

0.540641i

−0.959493

0.281733i

0.186393

0.215109i

239.1

1.25667

−

0.368991i

−0.654861

−

0.755750i

0.601808

−

0.386758i

−0.142315

−

0.989821i

−1.10181

−

0.708089i

−0.244123

0.281733i

−0.142315

0.989821i

−0.544078

−

1.19136i

269.1

−1.10181

1.27155i

0.841254

−

0.540641i

−0.260554

−

1.81219i

0.415415

0.909632i

−0.239446

1.66538i

1.17597

0.755750i

0.415415

−

0.909632i

−1.61435

−

0.474017i

284.1

−0.118239

0.822373i

0.415415

0.909632i

0.297176

0.0872586i

−0.654861

−

0.755750i

−0.797176

0.234072i

−0.452036

0.989821i

−0.654861

0.755750i

0.698939

−

0.449181i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
23.c	even	11	1	inner
345.p	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	345.1.p.a	✓	10
3.b	odd	2	1		345.1.p.b	yes	10
5.b	even	2	1		345.1.p.b	yes	10
5.c	odd	4	2		1725.1.bc.a		20
15.d	odd	2	1	CM	345.1.p.a	✓	10
15.e	even	4	2		1725.1.bc.a		20
23.c	even	11	1	inner	345.1.p.a	✓	10
69.h	odd	22	1		345.1.p.b	yes	10
115.j	even	22	1		345.1.p.b	yes	10
115.k	odd	44	2		1725.1.bc.a		20
345.p	odd	22	1	inner	345.1.p.a	✓	10
345.x	even	44	2		1725.1.bc.a		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
345.1.p.a	✓	10	1.a	even	1	1	trivial
345.1.p.a	✓	10	15.d	odd	2	1	CM
345.1.p.a	✓	10	23.c	even	11	1	inner
345.1.p.a	✓	10	345.p	odd	22	1	inner
345.1.p.b	yes	10	3.b	odd	2	1
345.1.p.b	yes	10	5.b	even	2	1
345.1.p.b	yes	10	69.h	odd	22	1
345.1.p.b	yes	10	115.j	even	22	1
1725.1.bc.a		20	5.c	odd	4	2
1725.1.bc.a		20	15.e	even	4	2
1725.1.bc.a		20	115.k	odd	44	2
1725.1.bc.a		20	345.x	even	44	2

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{10} + 2T_{2}^{9} + 4T_{2}^{8} - 3T_{2}^{7} - 6T_{2}^{6} - 12T_{2}^{5} + 9T_{2}^{4} + 7T_{2}^{3} + 14T_{2}^{2} + 6T_{2} + 1$ T2^10 + 2*T2^9 + 4*T2^8 - 3*T2^7 - 6*T2^6 - 12*T2^5 + 9*T2^4 + 7*T2^3 + 14*T2^2 + 6*T2 + 1 acting on $S_{1}^{\mathrm{new}}(345, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 - 3T^7 - 6T^6 - 12T^5 + 9T^4 + 7T^3 + 14T^2 + 6*T + 1
$3$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$7$	$T^{10}$ T^10
$11$	$T^{10}$ T^10
$13$	$T^{10}$ T^10
$17$	$T^{10} - 9 T^{9} + \cdots + 1$ T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$19$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1
$23$	$T^{10} + T^{9} + \cdots + 1$ 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 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$37$	$T^{10}$ T^10
$41$	$T^{10}$ 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 - 3T^7 - 6T^6 - 12T^5 + 9T^4 + 7T^3 + 14T^2 + 6*T + 1
$59$	$T^{10}$ T^10
$61$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1
$67$	$T^{10}$ T^10
$71$	$T^{10}$ T^10
$73$	$T^{10}$ T^10
$79$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1
$83$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$89$	$T^{10}$ T^10
$97$	$T^{10}$ T^10
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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 29.1
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	345.1.p.a

Level	$345$
Weight	$1$
Character orbit	345.p
Analytic conductor	$0.172$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{11}$
CM discriminant	-15
Inner twists	$4$