LMFDB - Newform orbit 1248.1.cm.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,1,Mod(77,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 4, 4])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.77"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1248.cm (of order $8$, degree $4$, 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:	$0.622833135766$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
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}^{3} q^{2} + \zeta_{32}^{10} q^{3} + \zeta_{32}^{6} q^{4} + (\zeta_{32}^{11} - \zeta_{32}) q^{5} - \zeta_{32}^{13} q^{6} - \zeta_{32}^{9} q^{8} - \zeta_{32}^{4} q^{9} + ( - \zeta_{32}^{14} + \zeta_{32}^{4}) q^{10} + \cdots + (\zeta_{32}^{3} - \zeta_{32}) q^{99} +O(q^{100}) $ q - z^3 * q^2 + z^10 * q^3 + z^6 * q^4 + (z^11 - z) * q^5 - z^13 * q^6 - z^9 * q^8 - z^4 * q^9 + (-z^14 + z^4) * q^10 + (z^15 - z^13) * q^11 - q^12 - z^2 * q^13 + (-z^11 - z^5) * q^15 + z^12 * q^16 + z^7 * q^18 + (-z^7 - z) * q^20 + (z^2 - 1) * q^22 + z^3 * q^24 + (-z^12 - z^6 + z^2) * q^25 + z^5 * q^26 - z^14 * q^27 + (z^14 + z^8) * q^30 - z^15 * q^32 + (-z^9 + z^7) * q^33 - z^10 * q^36 - z^12 * q^39 + (z^10 + z^4) * q^40 + (-z^15 + z^9) * q^41 + (-z^8 - z^4) * q^43 + (-z^5 + z^3) * q^44 + (-z^15 + z^5) * q^45 + (-z^9 - z^7) * q^47 - z^6 * q^48 - z^8 * q^49 + (z^15 + z^9 - z^5) * q^50 - z^8 * q^52 - z * q^54 + (z^14 - z^10 + z^8 + 1) * q^55 + (-z^11 + z) * q^59 + (-z^11 + z) * q^60 + (z^14 + z^6) * q^61 - z^2 * q^64 + (-z^13 + z^3) * q^65 + (z^12 - z^10) * q^66 + (z^5 - z^3) * q^71 + z^13 * q^72 + (z^12 + z^6 + 1) * q^75 + z^15 * q^78 + (-z^10 - z^6) * q^79 + (-z^13 - z^7) * q^80 + z^8 * q^81 + (-z^12 - z^2) * q^82 + (z^13 - z^7) * q^83 + (z^11 + z^7) * q^86 + (z^8 - z^6) * q^88 + (z^13 + z^11) * q^89 + (-z^8 - z^2) * q^90 + (z^12 + z^10) * q^94 + z^9 * q^96 + z^11 * q^98 + (z^3 - z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{12} - 16 q^{22} + 16 q^{55} + 16 q^{75}+O(q^{100}) $ 16 * q - 16 * q^12 - 16 * q^22 + 16 * q^55 + 16 * q^75

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$-1$	$-\zeta_{32}^{12}$

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} $

77.1

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

−0.831470

−

0.555570i

−0.382683

0.923880i

0.382683

0.923880i

−1.53636

0.636379i

0.831470

−

0.555570i

0.195090

−

0.980785i

−0.707107

−

0.707107i

1.63099

0.324423i

77.2

−0.555570

0.831470i

0.382683

−

0.923880i

−0.382683

−

0.923880i

1.02656

−

0.425215i

0.555570

0.831470i

0.980785

0.195090i

−0.707107

−

0.707107i

−0.216773

1.08979i

77.3

0.555570

−

0.831470i

0.382683

−

0.923880i

−0.382683

−

0.923880i

−1.02656

0.425215i

−0.555570

−

0.831470i

−0.980785

−

0.195090i

−0.707107

−

0.707107i

−0.216773

1.08979i

77.4

0.831470

0.555570i

−0.382683

0.923880i

0.382683

0.923880i

1.53636

−

0.636379i

−0.831470

0.555570i

−0.195090

0.980785i

−0.707107

−

0.707107i

1.63099

0.324423i

389.1

−0.831470

0.555570i

−0.382683

−

0.923880i

0.382683

−

0.923880i

−1.53636

−

0.636379i

0.831470

0.555570i

0.195090

0.980785i

−0.707107

0.707107i

1.63099

−

0.324423i

389.2

−0.555570

−

0.831470i

0.382683

0.923880i

−0.382683

0.923880i

1.02656

0.425215i

0.555570

−

0.831470i

0.980785

−

0.195090i

−0.707107

0.707107i

−0.216773

−

1.08979i

389.3

0.555570

0.831470i

0.382683

0.923880i

−0.382683

0.923880i

−1.02656

−

0.425215i

−0.555570

0.831470i

−0.980785

0.195090i

−0.707107

0.707107i

−0.216773

−

1.08979i

389.4

0.831470

−

0.555570i

−0.382683

−

0.923880i

0.382683

−

0.923880i

1.53636

0.636379i

−0.831470

−

0.555570i

−0.195090

−

0.980785i

−0.707107

0.707107i

1.63099

−

0.324423i

701.1

−0.980785

0.195090i

−0.923880

−

0.382683i

0.923880

−

0.382683i

0.750661

1.81225i

0.980785

0.195090i

−0.831470

0.555570i

0.707107

0.707107i

−1.08979

−

1.63099i

701.2

−0.195090

−

0.980785i

0.923880

0.382683i

−0.923880

0.382683i

−0.149316

−

0.360480i

0.195090

−

0.980785i

0.555570

0.831470i

0.707107

0.707107i

−0.324423

0.216773i

701.3

0.195090

0.980785i

0.923880

0.382683i

−0.923880

0.382683i

0.149316

0.360480i

−0.195090

0.980785i

−0.555570

−

0.831470i

0.707107

0.707107i

−0.324423

0.216773i

701.4

0.980785

−

0.195090i

−0.923880

−

0.382683i

0.923880

−

0.382683i

−0.750661

−

1.81225i

−0.980785

−

0.195090i

0.831470

−

0.555570i

0.707107

0.707107i

−1.08979

−

1.63099i

1013.1

−0.980785

−

0.195090i

−0.923880

0.382683i

0.923880

0.382683i

0.750661

−

1.81225i

0.980785

−

0.195090i

−0.831470

−

0.555570i

0.707107

−

0.707107i

−1.08979

1.63099i

1013.2

−0.195090

0.980785i

0.923880

−

0.382683i

−0.923880

−

0.382683i

−0.149316

0.360480i

0.195090

0.980785i

0.555570

−

0.831470i

0.707107

−

0.707107i

−0.324423

−

0.216773i

1013.3

0.195090

−

0.980785i

0.923880

−

0.382683i

−0.923880

−

0.382683i

0.149316

−

0.360480i

−0.195090

−

0.980785i

−0.555570

0.831470i

0.707107

−

0.707107i

−0.324423

−

0.216773i

1013.4

0.980785

0.195090i

−0.923880

0.382683i

0.923880

0.382683i

−0.750661

1.81225i

−0.980785

0.195090i

0.831470

0.555570i

0.707107

−

0.707107i

−1.08979

1.63099i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.d	odd	2	1	CM by $\Q(\sqrt{-39}) $
3.b	odd	2	1	inner
13.b	even	2	1	inner
32.g	even	8	1	inner
96.p	odd	8	1	inner
416.bg	even	8	1	inner
1248.cm	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1248.1.cm.a	✓	16
3.b	odd	2	1	inner	1248.1.cm.a	✓	16
13.b	even	2	1	inner	1248.1.cm.a	✓	16
32.g	even	8	1	inner	1248.1.cm.a	✓	16
39.d	odd	2	1	CM	1248.1.cm.a	✓	16
96.p	odd	8	1	inner	1248.1.cm.a	✓	16
416.bg	even	8	1	inner	1248.1.cm.a	✓	16
1248.cm	odd	8	1	inner	1248.1.cm.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1248.1.cm.a	✓	16	1.a	even	1	1	trivial
1248.1.cm.a	✓	16	3.b	odd	2	1	inner
1248.1.cm.a	✓	16	13.b	even	2	1	inner
1248.1.cm.a	✓	16	32.g	even	8	1	inner
1248.1.cm.a	✓	16	39.d	odd	2	1	CM
1248.1.cm.a	✓	16	96.p	odd	8	1	inner
1248.1.cm.a	✓	16	416.bg	even	8	1	inner
1248.1.cm.a	✓	16	1248.cm	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1248, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 1 $ T^16 + 1
$3$	$ (T^{8} + 1)^{2} $ (T^8 + 1)^2
$5$	$ T^{16} - 16 T^{10} + \cdots + 4 $ T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$7$	$ T^{16} $ T^16
$11$	$ T^{16} - 16 T^{10} + \cdots + 4 $ T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$13$	$ (T^{8} + 1)^{2} $ (T^8 + 1)^2
$17$	$ T^{16} $ T^16
$19$	$ T^{16} $ T^16
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ T^{16} + 24 T^{12} + \cdots + 4 $ T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$43$	$ (T^{4} + 2 T^{2} - 4 T + 2)^{4} $ (T^4 + 2T^2 - 4T + 2)^4
$47$	$ (T^{8} + 8 T^{6} + 20 T^{4} + \cdots + 2)^{2} $ (T^8 + 8T^6 + 20T^4 + 16*T^2 + 2)^2
$53$	$ T^{16} $ T^16
$59$	$ T^{16} - 16 T^{10} + \cdots + 4 $ T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$61$	$ (T^{8} + 16)^{2} $ (T^8 + 16)^2
$67$	$ T^{16} $ T^16
$71$	$ T^{16} + 24 T^{12} + \cdots + 4 $ T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$73$	$ T^{16} $ T^16
$79$	$ (T^{4} + 4 T^{2} + 2)^{4} $ (T^4 + 4*T^2 + 2)^4
$83$	$ T^{16} - 16 T^{10} + \cdots + 4 $ T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$89$	$ T^{16} + 24 T^{12} + \cdots + 4 $ T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$97$	$ T^{16} $ T^16
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1248.cm (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{32}^{3} q^{2} + \zeta_{32}^{10} q^{3} + \zeta_{32}^{6} q^{4} + (\zeta_{32}^{11} - \zeta_{32}) q^{5} - \zeta_{32}^{13} q^{6} - \zeta_{32}^{9} q^{8} - \zeta_{32}^{4} q^{9} + ( - \zeta_{32}^{14} + \zeta_{32}^{4}) q^{10} + \cdots + (\zeta_{32}^{3} - \zeta_{32}) q^{99} +O(q^{100}) \) q - z^3 * q^2 + z^10 * q^3 + z^6 * q^4 + (z^11 - z) * q^5 - z^13 * q^6 - z^9 * q^8 - z^4 * q^9 + (-z^14 + z^4) * q^10 + (z^15 - z^13) * q^11 - q^12 - z^2 * q^13 + (-z^11 - z^5) * q^15 + z^12 * q^16 + z^7 * q^18 + (-z^7 - z) * q^20 + (z^2 - 1) * q^22 + z^3 * q^24 + (-z^12 - z^6 + z^2) * q^25 + z^5 * q^26 - z^14 * q^27 + (z^14 + z^8) * q^30 - z^15 * q^32 + (-z^9 + z^7) * q^33 - z^10 * q^36 - z^12 * q^39 + (z^10 + z^4) * q^40 + (-z^15 + z^9) * q^41 + (-z^8 - z^4) * q^43 + (-z^5 + z^3) * q^44 + (-z^15 + z^5) * q^45 + (-z^9 - z^7) * q^47 - z^6 * q^48 - z^8 * q^49 + (z^15 + z^9 - z^5) * q^50 - z^8 * q^52 - z * q^54 + (z^14 - z^10 + z^8 + 1) * q^55 + (-z^11 + z) * q^59 + (-z^11 + z) * q^60 + (z^14 + z^6) * q^61 - z^2 * q^64 + (-z^13 + z^3) * q^65 + (z^12 - z^10) * q^66 + (z^5 - z^3) * q^71 + z^13 * q^72 + (z^12 + z^6 + 1) * q^75 + z^15 * q^78 + (-z^10 - z^6) * q^79 + (-z^13 - z^7) * q^80 + z^8 * q^81 + (-z^12 - z^2) * q^82 + (z^13 - z^7) * q^83 + (z^11 + z^7) * q^86 + (z^8 - z^6) * q^88 + (z^13 + z^11) * q^89 + (-z^8 - z^2) * q^90 + (z^12 + z^10) * q^94 + z^9 * q^96 + z^11 * q^98 + (z^3 - z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{12} - 16 q^{22} + 16 q^{55} + 16 q^{75}+O(q^{100}) \) 16 * q - 16 * q^12 - 16 * q^22 + 16 * q^55 + 16 * q^75

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	1248.1.cm.a

Level	$1248$
Weight	$1$
Character orbit	1248.cm
Analytic conductor	$0.623$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{16}$
CM discriminant	-39
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.622833135766\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{8})\)
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 \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

\(n\)	\(703\)	\(769\)	\(833\)	\(1093\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-\zeta_{32}^{12}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 77.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 1 \) T^16 + 1
$3$	\( (T^{8} + 1)^{2} \) (T^8 + 1)^2
$5$	\( T^{16} - 16 T^{10} + \cdots + 4 \) T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$7$	\( T^{16} \) T^16
$11$	\( T^{16} - 16 T^{10} + \cdots + 4 \) T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$13$	\( (T^{8} + 1)^{2} \) (T^8 + 1)^2
$17$	\( T^{16} \) T^16
$19$	\( T^{16} \) T^16
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( T^{16} + 24 T^{12} + \cdots + 4 \) T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$43$	\( (T^{4} + 2 T^{2} - 4 T + 2)^{4} \) (T^4 + 2T^2 - 4T + 2)^4
$47$	\( (T^{8} + 8 T^{6} + 20 T^{4} + \cdots + 2)^{2} \) (T^8 + 8T^6 + 20T^4 + 16*T^2 + 2)^2
$53$	\( T^{16} \) T^16
$59$	\( T^{16} - 16 T^{10} + \cdots + 4 \) T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$61$	\( (T^{8} + 16)^{2} \) (T^8 + 16)^2
$67$	\( T^{16} \) T^16
$71$	\( T^{16} + 24 T^{12} + \cdots + 4 \) T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$73$	\( T^{16} \) T^16
$79$	\( (T^{4} + 4 T^{2} + 2)^{4} \) (T^4 + 4*T^2 + 2)^4
$83$	\( T^{16} - 16 T^{10} + \cdots + 4 \) T^16 - 16T^10 + 140T^8 - 192T^6 + 128T^4 + 32*T^2 + 4
$89$	\( T^{16} + 24 T^{12} + \cdots + 4 \) T^16 + 24T^12 + 148T^8 + 176*T^4 + 4
$97$	\( T^{16} \) T^16
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