LMFDB - Newform orbit 960.1.cl.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,1,Mod(29,960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 11, 8, 8])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("960.29"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	960.cl (of order $16$ , degree $8$ , minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.479102412128$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{16})$
Coefficient field:	$\Q(\zeta_{32})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 1$ x^16 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

$q$ -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.

$f(q)$	$=$	$q - \zeta_{32}^{13} q^{2} + \zeta_{32} q^{3} - \zeta_{32}^{10} q^{4} + \zeta_{32}^{11} q^{5} - \zeta_{32}^{14} q^{6} - \zeta_{32}^{7} q^{8} + \zeta_{32}^{2} q^{9} + \zeta_{32}^{8} q^{10} - \zeta_{32}^{11} q^{12} + \cdots - \zeta_{32}^{9} q^{98} +O(q^{100})$ q - z^13 * q^2 + z * q^3 - z^10 * q^4 + z^11 * q^5 - z^14 * q^6 - z^7 * q^8 + z^2 * q^9 + z^8 * q^10 - z^11 * q^12 + z^12 * q^15 - z^4 * q^16 + (z^15 - z^9) * q^17 - z^15 * q^18 + (z^6 - z^4) * q^19 + z^5 * q^20 + (z^13 + z^7) * q^23 - z^8 * q^24 - z^6 * q^25 + z^3 * q^27 + z^9 * q^30 + (-z^14 - z^2) * q^31 - z * q^32 + (z^12 - z^6) * q^34 - z^12 * q^36 + (z^3 - z) * q^38 + z^2 * q^40 + z^13 * q^45 + (z^10 + z^4) * q^46 + (-z^5 - z^3) * q^47 - z^5 * q^48 - z^12 * q^49 - z^3 * q^50 + (-z^10 - 1) * q^51 + (z^9 + z^5) * q^53 + q^54 + (z^7 - z^5) * q^57 + z^6 * q^60 + (z^10 - z^8) * q^61 + (z^15 - z^11) * q^62 + z^14 * q^64 + (z^9 - z^3) * q^68 + (z^14 + z^8) * q^69 - z^9 * q^72 - z^7 * q^75 + (z^14 + 1) * q^76 + (z^8 - 1) * q^79 - z^15 * q^80 + z^4 * q^81 + (z^15 + z^11) * q^83 + (-z^10 + z^4) * q^85 + z^10 * q^90 + (z^7 + z) * q^92 + (-z^15 - z^3) * q^93 + (-z^2 - 1) * q^94 + (-z^15 - z) * q^95 - z^2 * q^96 - z^9 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 16 q^{51} + 16 q^{54} + 16 q^{76} - 16 q^{79} - 16 q^{94}+O(q^{100})$ 16 * q - 16 * q^51 + 16 * q^54 + 16 * q^76 - 16 * q^79 - 16 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$ .

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$-\zeta_{32}^{6}$

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

−0.980785	−	0.195090i
0.980785	+	0.195090i
0.195090	+	0.980785i
−0.195090	−	0.980785i
0.831470	−	0.555570i
−0.831470	+	0.555570i
0.831470	+	0.555570i
−0.831470	−	0.555570i
0.195090	−	0.980785i
−0.195090	+	0.980785i
−0.980785	+	0.195090i
0.980785	−	0.195090i
0.555570	+	0.831470i
−0.555570	−	0.831470i
0.555570	−	0.831470i
−0.555570	+	0.831470i

−0.831470

0.555570i

−0.980785

−

0.195090i

0.382683

−

0.923880i

0.555570

−

0.831470i

0.923880

−

0.382683i

0.195090

0.980785i

0.923880

0.382683i

1.00000i

29.2

0.831470

−

0.555570i

0.980785

0.195090i

0.382683

−

0.923880i

−0.555570

0.831470i

0.923880

−

0.382683i

−0.195090

−

0.980785i

0.923880

0.382683i

1.00000i

149.1

−0.555570

0.831470i

0.195090

0.980785i

−0.382683

−

0.923880i

−0.831470

0.555570i

−0.923880

−

0.382683i

0.980785

0.195090i

−0.923880

0.382683i

−

1.00000i

149.2

0.555570

−

0.831470i

−0.195090

−

0.980785i

−0.382683

−

0.923880i

0.831470

−

0.555570i

−0.923880

−

0.382683i

−0.980785

−

0.195090i

−0.923880

0.382683i

−

1.00000i

269.1

−0.195090

0.980785i

0.831470

−

0.555570i

−0.923880

−

0.382683i

0.980785

−

0.195090i

0.382683

0.923880i

0.555570

−

0.831470i

0.382683

−

0.923880i

1.00000i

269.2

0.195090

−

0.980785i

−0.831470

0.555570i

−0.923880

−

0.382683i

−0.980785

0.195090i

0.382683

0.923880i

−0.555570

0.831470i

0.382683

−

0.923880i

1.00000i

389.1

−0.195090

−

0.980785i

0.831470

0.555570i

−0.923880

0.382683i

0.980785

0.195090i

0.382683

−

0.923880i

0.555570

0.831470i

0.382683

0.923880i

−

1.00000i

389.2

0.195090

0.980785i

−0.831470

−

0.555570i

−0.923880

0.382683i

−0.980785

−

0.195090i

0.382683

−

0.923880i

−0.555570

−

0.831470i

0.382683

0.923880i

−

1.00000i

509.1

−0.555570

−

0.831470i

0.195090

−

0.980785i

−0.382683

0.923880i

−0.831470

−

0.555570i

−0.923880

0.382683i

0.980785

−

0.195090i

−0.923880

−

0.382683i

1.00000i

509.2

0.555570

0.831470i

−0.195090

0.980785i

−0.382683

0.923880i

0.831470

0.555570i

−0.923880

0.382683i

−0.980785

0.195090i

−0.923880

−

0.382683i

1.00000i

629.1

−0.831470

−

0.555570i

−0.980785

0.195090i

0.382683

0.923880i

0.555570

0.831470i

0.923880

0.382683i

0.195090

−

0.980785i

0.923880

−

0.382683i

−

1.00000i

629.2

0.831470

0.555570i

0.980785

−

0.195090i

0.382683

0.923880i

−0.555570

−

0.831470i

0.923880

0.382683i

−0.195090

0.980785i

0.923880

−

0.382683i

−

1.00000i

749.1

−0.980785

−

0.195090i

0.555570

0.831470i

0.923880

0.382683i

−0.195090

−

0.980785i

−0.382683

−

0.923880i

−0.831470

−

0.555570i

−0.382683

0.923880i

1.00000i

749.2

0.980785

0.195090i

−0.555570

−

0.831470i

0.923880

0.382683i

0.195090

0.980785i

−0.382683

−

0.923880i

0.831470

0.555570i

−0.382683

0.923880i

1.00000i

869.1

−0.980785

0.195090i

0.555570

−

0.831470i

0.923880

−

0.382683i

−0.195090

0.980785i

−0.382683

0.923880i

−0.831470

0.555570i

−0.382683

−

0.923880i

−

1.00000i

869.2

0.980785

−

0.195090i

−0.555570

0.831470i

0.923880

−

0.382683i

0.195090

−

0.980785i

−0.382683

0.923880i

0.831470

−

0.555570i

−0.382683

−

0.923880i

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
3.b	odd	2	1	inner
5.b	even	2	1	inner
64.i	even	16	1	inner
192.q	odd	16	1	inner
320.bf	even	16	1	inner
960.cl	odd	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.1.cl.a	✓	16
3.b	odd	2	1	inner	960.1.cl.a	✓	16
4.b	odd	2	1		3840.1.cl.a		16
5.b	even	2	1	inner	960.1.cl.a	✓	16
12.b	even	2	1		3840.1.cl.a		16
15.d	odd	2	1	CM	960.1.cl.a	✓	16
20.d	odd	2	1		3840.1.cl.a		16
60.h	even	2	1		3840.1.cl.a		16
64.i	even	16	1	inner	960.1.cl.a	✓	16
64.j	odd	16	1		3840.1.cl.a		16
192.q	odd	16	1	inner	960.1.cl.a	✓	16
192.s	even	16	1		3840.1.cl.a		16
320.bf	even	16	1	inner	960.1.cl.a	✓	16
320.bh	odd	16	1		3840.1.cl.a		16
960.cl	odd	16	1	inner	960.1.cl.a	✓	16
960.cp	even	16	1		3840.1.cl.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.1.cl.a	✓	16	1.a	even	1	1	trivial
960.1.cl.a	✓	16	3.b	odd	2	1	inner
960.1.cl.a	✓	16	5.b	even	2	1	inner
960.1.cl.a	✓	16	15.d	odd	2	1	CM
960.1.cl.a	✓	16	64.i	even	16	1	inner
960.1.cl.a	✓	16	192.q	odd	16	1	inner
960.1.cl.a	✓	16	320.bf	even	16	1	inner
960.1.cl.a	✓	16	960.cl	odd	16	1	inner
3840.1.cl.a		16	4.b	odd	2	1
3840.1.cl.a		16	12.b	even	2	1
3840.1.cl.a		16	20.d	odd	2	1
3840.1.cl.a		16	60.h	even	2	1
3840.1.cl.a		16	64.j	odd	16	1
3840.1.cl.a		16	192.s	even	16	1
3840.1.cl.a		16	320.bh	odd	16	1
3840.1.cl.a		16	960.cp	even	16	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(960, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16} + 1$ T^16 + 1
$3$	$T^{16} + 1$ T^16 + 1
$5$	$T^{16} + 1$ T^16 + 1
$7$	$T^{16}$ T^16
$11$	$T^{16}$ T^16
$13$	$T^{16}$ T^16
$17$	$T^{16} + 24 T^{12} + \cdots + 4$ T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$19$	$(T^{8} - 8 T^{5} + 2 T^{4} + \cdots + 2)^{2}$ (T^8 - 8T^5 + 2T^4 + 12T^2 + 8T + 2)^2
$23$	$T^{16} + 16 T^{10} + \cdots + 4$ T^16 + 16T^10 + 140T^8 + 192T^6 + 128T^4 - 32*T^2 + 4
$29$	$T^{16}$ T^16
$31$	$(T^{4} + 4 T^{2} + 2)^{4}$ (T^4 + 4*T^2 + 2)^4
$37$	$T^{16}$ T^16
$41$	$T^{16}$ T^16
$43$	$T^{16}$ T^16
$47$	$T^{16} + 24 T^{12} + \cdots + 4$ T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$53$	$T^{16} - 16 T^{12} + \cdots + 16$ T^16 - 16T^12 + 128T^8 + 64*T^4 + 16
$59$	$T^{16}$ T^16
$61$	$(T^{8} + 4 T^{6} + 6 T^{4} + \cdots + 2)^{2}$ (T^8 + 4T^6 + 6T^4 - 8T^3 + 4T^2 + 8*T + 2)^2
$67$	$T^{16}$ T^16
$71$	$T^{16}$ T^16
$73$	$T^{16}$ T^16
$79$	$(T^{2} + 2 T + 2)^{8}$ (T^2 + 2*T + 2)^8
$83$	$T^{16} + 16 T^{12} + \cdots + 16$ T^16 + 16T^12 + 128T^8 - 64*T^4 + 16
$89$	$T^{16}$ T^16
$97$	$T^{16}$ T^16
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 29.2
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	960.1.cl.a

Level	$960$
Weight	$1$
Character orbit	960.cl
Analytic conductor	$0.479$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{16}$
CM discriminant	-15
Inner twists	$8$