LMFDB - Newform orbit 1920.1.db.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,1,Mod(29,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(32))

chi = DirichletCharacter(H, H._module([0, 27, 16, 16]))

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

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

chi := DirichletCharacter("1920.29");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1920.db (of order $32$, degree $16$, 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:	$0.958204824255$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$2$ over $\Q(\zeta_{32})$
Coefficient field:	$\Q(\zeta_{64})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + 1 $ x^32 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{32}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{32} + \cdots)$

$q$-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_{64}^{27} q^{2} - \zeta_{64}^{23} q^{3} - \zeta_{64}^{22} q^{4} + \zeta_{64}^{29} q^{5} - \zeta_{64}^{18} q^{6} - \zeta_{64}^{17} q^{8} - \zeta_{64}^{14} q^{9} + \zeta_{64}^{24} q^{10} - \zeta_{64}^{13} q^{12} + \cdots - \zeta_{64}^{31} q^{98} +O(q^{100}) $ q - z^27 * q^2 - z^23 * q^3 - z^22 * q^4 + z^29 * q^5 - z^18 * q^6 - z^17 * q^8 - z^14 * q^9 + z^24 * q^10 - z^13 * q^12 + z^20 * q^15 - z^12 * q^16 + (z^31 - z^25) * q^17 - z^9 * q^18 + (z^12 - z^10) * q^19 + z^19 * q^20 + (-z^11 + z) * q^23 - z^8 * q^24 - z^26 * q^25 - z^5 * q^27 + z^15 * q^30 + (-z^30 + z^18) * q^31 - z^7 * q^32 + (z^26 - z^20) * q^34 - z^4 * q^36 + (z^7 - z^5) * q^38 + z^14 * q^40 + z^11 * q^45 + (-z^28 - z^6) * q^46 + (z^21 - z^3) * q^47 - z^3 * q^48 + z^4 * q^49 - z^21 * q^50 + (z^22 - z^16) * q^51 + (z^19 + z^15) * q^53 - q^54 + (z^3 - z) * q^57 + z^10 * q^60 + (z^8 + z^6) * q^61 + (-z^25 + z^13) * q^62 - z^2 * q^64 + (z^21 - z^15) * q^68 + (-z^24 - z^2) * q^69 + z^31 * q^72 - z^17 * q^75 + (z^2 - 1) * q^76 + (-z^24 + z^16) * q^79 + z^9 * q^80 + z^28 * q^81 + (-z^29 + z^25) * q^83 + (-z^28 + z^22) * q^85 + z^6 * q^90 + (-z^23 - z) * q^92 + (-z^21 + z^9) * q^93 + (z^30 + z^16) * q^94 + (-z^9 + z^7) * q^95 + z^30 * q^96 - z^31 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{54} - 32 q^{76}+O(q^{100}) $ 32 * q - 32 * q^54 - 32 * q^76

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$1$	$-1$	$-\zeta_{64}^{26}$	$-1$

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

0.634393	−	0.773010i
−0.634393	+	0.773010i
−0.0980171	+	0.995185i
0.0980171	−	0.995185i
0.881921	−	0.471397i
−0.881921	+	0.471397i
−0.471397	+	0.881921i
0.471397	−	0.881921i
−0.995185	+	0.0980171i
0.995185	−	0.0980171i
0.773010	−	0.634393i
−0.773010	+	0.634393i
−0.956940	−	0.290285i
0.956940	+	0.290285i
−0.956940	+	0.290285i
0.956940	−	0.290285i
0.773010	+	0.634393i
−0.773010	−	0.634393i
−0.995185	−	0.0980171i
0.995185	+	0.0980171i

−0.290285

−

0.956940i

−0.0980171

0.995185i

−0.831470

0.555570i

0.881921

−

0.471397i

0.980785

−

0.195090i

0.773010

0.634393i

−0.980785

−

0.195090i

−0.707107

−

0.707107i

29.2

0.290285

0.956940i

0.0980171

−

0.995185i

−0.831470

0.555570i

−0.881921

0.471397i

0.980785

−

0.195090i

−0.773010

−

0.634393i

−0.980785

−

0.195090i

−0.707107

−

0.707107i

149.1

−0.471397

−

0.881921i

−0.773010

−

0.634393i

−0.555570

0.831470i

−0.290285

−

0.956940i

−0.195090

0.980785i

0.995185

0.0980171i

0.195090

0.980785i

−0.707107

0.707107i

149.2

0.471397

0.881921i

0.773010

0.634393i

−0.555570

0.831470i

0.290285

0.956940i

−0.195090

0.980785i

−0.995185

−

0.0980171i

0.195090

0.980785i

−0.707107

0.707107i

269.1

−0.773010

0.634393i

−0.290285

−

0.956940i

0.195090

−

0.980785i

−0.0980171

−

0.995185i

0.831470

0.555570i

0.471397

0.881921i

−0.831470

0.555570i

0.707107

0.707107i

269.2

0.773010

−

0.634393i

0.290285

0.956940i

0.195090

−

0.980785i

0.0980171

0.995185i

0.831470

0.555570i

−0.471397

−

0.881921i

−0.831470

0.555570i

0.707107

0.707107i

389.1

−0.634393

0.773010i

0.956940

0.290285i

−0.195090

−

0.980785i

−0.995185

−

0.0980171i

−0.831470

0.555570i

0.881921

0.471397i

0.831470

0.555570i

0.707107

−

0.707107i

389.2

0.634393

−

0.773010i

−0.956940

−

0.290285i

−0.195090

−

0.980785i

0.995185

0.0980171i

−0.831470

0.555570i

−0.881921

−

0.471397i

0.831470

0.555570i

0.707107

−

0.707107i

509.1

−0.881921

−

0.471397i

−0.634393

−

0.773010i

0.555570

0.831470i

0.956940

0.290285i

0.195090

0.980785i

−0.0980171

−

0.995185i

−0.195090

0.980785i

−0.707107

−

0.707107i

509.2

0.881921

0.471397i

0.634393

0.773010i

0.555570

0.831470i

−0.956940

−

0.290285i

0.195090

0.980785i

0.0980171

0.995185i

−0.195090

0.980785i

−0.707107

−

0.707107i

629.1

−0.956940

−

0.290285i

0.995185

−

0.0980171i

0.831470

0.555570i

0.471397

−

0.881921i

−0.980785

−

0.195090i

−0.634393

−

0.773010i

0.980785

−

0.195090i

−0.707107

0.707107i

629.2

0.956940

0.290285i

−0.995185

0.0980171i

0.831470

0.555570i

−0.471397

0.881921i

−0.980785

−

0.195090i

0.634393

0.773010i

0.980785

−

0.195090i

−0.707107

0.707107i

749.1

−0.0980171

0.995185i

0.881921

0.471397i

−0.980785

−

0.195090i

0.634393

−

0.773010i

−0.555570

0.831470i

0.290285

−

0.956940i

0.555570

0.831470i

0.707107

0.707107i

749.2

0.0980171

−

0.995185i

−0.881921

−

0.471397i

−0.980785

−

0.195090i

−0.634393

0.773010i

−0.555570

0.831470i

−0.290285

0.956940i

0.555570

0.831470i

0.707107

0.707107i

869.1

−0.0980171

−

0.995185i

0.881921

−

0.471397i

−0.980785

0.195090i

0.634393

0.773010i

−0.555570

−

0.831470i

0.290285

0.956940i

0.555570

−

0.831470i

0.707107

−

0.707107i

869.2

0.0980171

0.995185i

−0.881921

0.471397i

−0.980785

0.195090i

−0.634393

−

0.773010i

−0.555570

−

0.831470i

−0.290285

−

0.956940i

0.555570

−

0.831470i

0.707107

−

0.707107i

989.1

−0.956940

0.290285i

0.995185

0.0980171i

0.831470

−

0.555570i

0.471397

0.881921i

−0.980785

0.195090i

−0.634393

0.773010i

0.980785

0.195090i

−0.707107

−

0.707107i

989.2

0.956940

−

0.290285i

−0.995185

−

0.0980171i

0.831470

−

0.555570i

−0.471397

−

0.881921i

−0.980785

0.195090i

0.634393

−

0.773010i

0.980785

0.195090i

−0.707107

−

0.707107i

1109.1

−0.881921

0.471397i

−0.634393

0.773010i

0.555570

−

0.831470i

0.956940

−

0.290285i

0.195090

−

0.980785i

−0.0980171

0.995185i

−0.195090

−

0.980785i

−0.707107

0.707107i

1109.2

0.881921

−

0.471397i

0.634393

−

0.773010i

0.555570

−

0.831470i

−0.956940

0.290285i

0.195090

−

0.980785i

0.0980171

−

0.995185i

−0.195090

−

0.980785i

−0.707107

0.707107i

See all 32 embeddings

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
128.k	even	32	1	inner
384.x	odd	32	1	inner
640.bo	even	32	1	inner
1920.db	odd	32	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1920.1.db.a	✓	32
3.b	odd	2	1	inner	1920.1.db.a	✓	32
5.b	even	2	1	inner	1920.1.db.a	✓	32
15.d	odd	2	1	CM	1920.1.db.a	✓	32
128.k	even	32	1	inner	1920.1.db.a	✓	32
384.x	odd	32	1	inner	1920.1.db.a	✓	32
640.bo	even	32	1	inner	1920.1.db.a	✓	32
1920.db	odd	32	1	inner	1920.1.db.a	✓	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1920.1.db.a	✓	32	1.a	even	1	1	trivial
1920.1.db.a	✓	32	3.b	odd	2	1	inner
1920.1.db.a	✓	32	5.b	even	2	1	inner
1920.1.db.a	✓	32	15.d	odd	2	1	CM
1920.1.db.a	✓	32	128.k	even	32	1	inner
1920.1.db.a	✓	32	384.x	odd	32	1	inner
1920.1.db.a	✓	32	640.bo	even	32	1	inner
1920.1.db.a	✓	32	1920.db	odd	32	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1920, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{32} + 1 $ T^32 + 1
$3$	$ T^{32} + 1 $ T^32 + 1
$5$	$ T^{32} + 1 $ T^32 + 1
$7$	$ T^{32} $ T^32
$11$	$ T^{32} $ T^32
$13$	$ T^{32} $ T^32
$17$	$ T^{32} + 280 T^{24} + \cdots + 4 $ T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4
$19$	$ (T^{16} + 16 T^{13} + \cdots + 2)^{2} $ (T^16 + 16T^13 + 88T^10 + 2T^8 + 192T^7 - 48T^5 + 140T^4 + 40T^2 + 16T + 2)^2
$23$	$ T^{32} + 176 T^{20} + \cdots + 4 $ T^32 + 176T^20 + 4576T^18 + 25740T^16 + 56576T^14 + 62016T^12 + 38272T^10 + 15488T^8 - 1024T^6 + 2720T^4 - 192T^2 + 4
$29$	$ T^{32} $ T^32
$31$	$ (T^{16} + 24 T^{12} + \cdots + 4)^{2} $ (T^16 + 24T^12 + 148T^8 + 176*T^4 + 4)^2
$37$	$ T^{32} $ T^32
$41$	$ T^{32} $ T^32
$43$	$ T^{32} $ T^32
$47$	$ T^{32} + 280 T^{24} + \cdots + 4 $ T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4
$53$	$ T^{32} - 32 T^{28} + \cdots + 16 $ T^32 - 32T^28 + 400T^24 - 2432T^20 + 10368T^16 + 8704T^12 + 30144T^8 + 1280*T^4 + 16
$59$	$ T^{32} $ T^32
$61$	$ (T^{16} + 4 T^{12} + \cdots + 2)^{2} $ (T^16 + 4T^12 + 16T^11 + 6T^8 - 160T^7 + 56T^6 + 4T^4 + 80T^3 + 72T^2 + 16*T + 2)^2
$67$	$ T^{32} $ T^32
$71$	$ T^{32} $ T^32
$73$	$ T^{32} $ T^32
$79$	$ (T^{4} + 2 T^{2} - 4 T + 2)^{8} $ (T^4 + 2T^2 - 4T + 2)^8
$83$	$ T^{32} + 112 T^{24} + \cdots + 16 $ T^32 + 112T^24 + 2112T^20 + 6272T^16 - 30336T^12 + 29248T^8 + 512T^4 + 16
$89$	$ T^{32} $ T^32
$97$	$ T^{32} $ T^32
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1920.db (of order \(32\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{64}^{27} q^{2} - \zeta_{64}^{23} q^{3} - \zeta_{64}^{22} q^{4} + \zeta_{64}^{29} q^{5} - \zeta_{64}^{18} q^{6} - \zeta_{64}^{17} q^{8} - \zeta_{64}^{14} q^{9} + \zeta_{64}^{24} q^{10} - \zeta_{64}^{13} q^{12} + \cdots - \zeta_{64}^{31} q^{98} +O(q^{100}) \) q - z^27 * q^2 - z^23 * q^3 - z^22 * q^4 + z^29 * q^5 - z^18 * q^6 - z^17 * q^8 - z^14 * q^9 + z^24 * q^10 - z^13 * q^12 + z^20 * q^15 - z^12 * q^16 + (z^31 - z^25) * q^17 - z^9 * q^18 + (z^12 - z^10) * q^19 + z^19 * q^20 + (-z^11 + z) * q^23 - z^8 * q^24 - z^26 * q^25 - z^5 * q^27 + z^15 * q^30 + (-z^30 + z^18) * q^31 - z^7 * q^32 + (z^26 - z^20) * q^34 - z^4 * q^36 + (z^7 - z^5) * q^38 + z^14 * q^40 + z^11 * q^45 + (-z^28 - z^6) * q^46 + (z^21 - z^3) * q^47 - z^3 * q^48 + z^4 * q^49 - z^21 * q^50 + (z^22 - z^16) * q^51 + (z^19 + z^15) * q^53 - q^54 + (z^3 - z) * q^57 + z^10 * q^60 + (z^8 + z^6) * q^61 + (-z^25 + z^13) * q^62 - z^2 * q^64 + (z^21 - z^15) * q^68 + (-z^24 - z^2) * q^69 + z^31 * q^72 - z^17 * q^75 + (z^2 - 1) * q^76 + (-z^24 + z^16) * q^79 + z^9 * q^80 + z^28 * q^81 + (-z^29 + z^25) * q^83 + (-z^28 + z^22) * q^85 + z^6 * q^90 + (-z^23 - z) * q^92 + (-z^21 + z^9) * q^93 + (z^30 + z^16) * q^94 + (-z^9 + z^7) * q^95 + z^30 * q^96 - z^31 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{54} - 32 q^{76}+O(q^{100}) \) 32 * q - 32 * q^54 - 32 * q^76

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	1920.1.db.a

Level	$1920$
Weight	$1$
Character orbit	1920.db
Analytic conductor	$0.958$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{32}$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.958204824255\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(2\) over \(\Q(\zeta_{32})\)
Coefficient field:	\(\Q(\zeta_{64})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{32} + 1 \) x^32 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{32}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{32} + \cdots)\)

\(n\)	\(511\)	\(641\)	\(901\)	\(1537\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-\zeta_{64}^{26}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 29.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{32} + 1 \) T^32 + 1
$3$	\( T^{32} + 1 \) T^32 + 1
$5$	\( T^{32} + 1 \) T^32 + 1
$7$	\( T^{32} \) T^32
$11$	\( T^{32} \) T^32
$13$	\( T^{32} \) T^32
$17$	\( T^{32} + 280 T^{24} + \cdots + 4 \) T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4
$19$	\( (T^{16} + 16 T^{13} + \cdots + 2)^{2} \) (T^16 + 16T^13 + 88T^10 + 2T^8 + 192T^7 - 48T^5 + 140T^4 + 40T^2 + 16T + 2)^2
$23$	\( T^{32} + 176 T^{20} + \cdots + 4 \) T^32 + 176T^20 + 4576T^18 + 25740T^16 + 56576T^14 + 62016T^12 + 38272T^10 + 15488T^8 - 1024T^6 + 2720T^4 - 192T^2 + 4
$29$	\( T^{32} \) T^32
$31$	\( (T^{16} + 24 T^{12} + \cdots + 4)^{2} \) (T^16 + 24T^12 + 148T^8 + 176*T^4 + 4)^2
$37$	\( T^{32} \) T^32
$41$	\( T^{32} \) T^32
$43$	\( T^{32} \) T^32
$47$	\( T^{32} + 280 T^{24} + \cdots + 4 \) T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4
$53$	\( T^{32} - 32 T^{28} + \cdots + 16 \) T^32 - 32T^28 + 400T^24 - 2432T^20 + 10368T^16 + 8704T^12 + 30144T^8 + 1280*T^4 + 16
$59$	\( T^{32} \) T^32
$61$	\( (T^{16} + 4 T^{12} + \cdots + 2)^{2} \) (T^16 + 4T^12 + 16T^11 + 6T^8 - 160T^7 + 56T^6 + 4T^4 + 80T^3 + 72T^2 + 16*T + 2)^2
$67$	\( T^{32} \) T^32
$71$	\( T^{32} \) T^32
$73$	\( T^{32} \) T^32
$79$	\( (T^{4} + 2 T^{2} - 4 T + 2)^{8} \) (T^4 + 2T^2 - 4T + 2)^8
$83$	\( T^{32} + 112 T^{24} + \cdots + 16 \) T^32 + 112T^24 + 2112T^20 + 6272T^16 - 30336T^12 + 29248T^8 + 512T^4 + 16
$89$	\( T^{32} \) T^32
$97$	\( T^{32} \) T^32
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