LMFDB - Newform orbit 3808.1.cu.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,1,Mod(237,3808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7, 4, 4]))

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

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

chi := DirichletCharacter("3808.237");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3808.cu (of order $8$, degree $4$, 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:	$1.90043956811$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} + \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3808\mathbb{Z}\right)^\times$.

$n$	$2143$	$2689$	$3265$	$3333$
$\chi(n)$	$1$	$-1$	$-1$	$-\zeta_{40}^{15}$

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} $

237.1

0.987688	+	0.156434i
0.453990	−	0.891007i
0.156434	+	0.987688i
−0.891007	+	0.453990i
0.987688	−	0.156434i
0.453990	+	0.891007i
0.156434	−	0.987688i
−0.891007	−	0.453990i
0.891007	−	0.453990i
−0.156434	−	0.987688i
−0.453990	+	0.891007i
−0.987688	−	0.156434i
0.891007	+	0.453990i
−0.156434	+	0.987688i
−0.453990	−	0.891007i
−0.987688	+	0.156434i

−0.987688

−

0.156434i

0.399903

−

0.965451i

0.951057

0.309017i

1.40505

−

0.581990i

−0.546010

0.891007i

−0.707107

0.707107i

−0.891007

−

0.453990i

−0.0650673

−

0.0650673i

−1.47879

0.355026i

237.2

−0.453990

0.891007i

−0.497066

1.20002i

−0.587785

−

0.809017i

−0.431351

0.178671i

−0.843566

−

0.987688i

−0.707107

0.707107i

0.987688

−

0.156434i

−0.485875

−

0.485875i

0.0366318

−

0.465451i

237.3

−0.156434

−

0.987688i

0.744220

−

1.79671i

−0.951057

0.309017i

−1.84206

0.763007i

−1.89101

−

0.453990i

−0.707107

0.707107i

0.453990

0.891007i

−1.96718

−

1.96718i

1.04178

1.70002i

237.4

0.891007

−

0.453990i

0.0600500

−

0.144974i

0.587785

−

0.809017i

1.57547

−

0.652583i

−0.0123117

−

0.156434i

−0.707107

0.707107i

0.156434

−

0.987688i

0.689695

0.689695i

1.10749

−

1.29671i

1189.1

−0.987688

0.156434i

0.399903

0.965451i

0.951057

−

0.309017i

1.40505

0.581990i

−0.546010

−

0.891007i

−0.707107

−

0.707107i

−0.891007

0.453990i

−0.0650673

0.0650673i

−1.47879

−

0.355026i

1189.2

−0.453990

−

0.891007i

−0.497066

−

1.20002i

−0.587785

0.809017i

−0.431351

−

0.178671i

−0.843566

0.987688i

−0.707107

−

0.707107i

0.987688

0.156434i

−0.485875

0.485875i

0.0366318

0.465451i

1189.3

−0.156434

0.987688i

0.744220

1.79671i

−0.951057

−

0.309017i

−1.84206

−

0.763007i

−1.89101

0.453990i

−0.707107

−

0.707107i

0.453990

−

0.891007i

−1.96718

1.96718i

1.04178

−

1.70002i

1189.4

0.891007

0.453990i

0.0600500

0.144974i

0.587785

0.809017i

1.57547

0.652583i

−0.0123117

0.156434i

−0.707107

−

0.707107i

0.156434

0.987688i

0.689695

−

0.689695i

1.10749

1.29671i

2141.1

−0.891007

0.453990i

1.84206

0.763007i

0.587785

−

0.809017i

−0.399903

−

0.965451i

−1.98769

0.156434i

0.707107

−

0.707107i

−0.156434

0.987688i

2.10391

2.10391i

0.794622

0.678671i

2141.2

0.156434

0.987688i

0.431351

0.178671i

−0.951057

0.309017i

−0.0600500

−

0.144974i

−0.108993

0.453990i

0.707107

−

0.707107i

−0.453990

−

0.891007i

−0.552967

−

0.552967i

0.133795

−

0.0819895i

2141.3

0.453990

−

0.891007i

−1.40505

−

0.581990i

−0.587785

−

0.809017i

−0.744220

−

1.79671i

−1.15643

0.987688i

0.707107

−

0.707107i

−0.987688

0.156434i

0.928339

0.928339i

−1.93874

−

0.152583i

2141.4

0.987688

0.156434i

−1.57547

−

0.652583i

0.951057

0.309017i

0.497066

1.20002i

−1.45399

−

0.891007i

0.707107

−

0.707107i

0.891007

0.453990i

1.34915

1.34915i

0.303221

1.26301i

3093.1

−0.891007

−

0.453990i

1.84206

−

0.763007i

0.587785

0.809017i

−0.399903

0.965451i

−1.98769

−

0.156434i

0.707107

0.707107i

−0.156434

−

0.987688i

2.10391

−

2.10391i

0.794622

−

0.678671i

3093.2

0.156434

−

0.987688i

0.431351

−

0.178671i

−0.951057

−

0.309017i

−0.0600500

0.144974i

−0.108993

−

0.453990i

0.707107

0.707107i

−0.453990

0.891007i

−0.552967

0.552967i

0.133795

0.0819895i

3093.3

0.453990

0.891007i

−1.40505

0.581990i

−0.587785

0.809017i

−0.744220

1.79671i

−1.15643

−

0.987688i

0.707107

0.707107i

−0.987688

−

0.156434i

0.928339

−

0.928339i

−1.93874

0.152583i

3093.4

0.987688

−

0.156434i

−1.57547

0.652583i

0.951057

−

0.309017i

0.497066

−

1.20002i

−1.45399

0.891007i

0.707107

0.707107i

0.891007

−

0.453990i

1.34915

−

1.34915i

0.303221

−

1.26301i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
119.d	odd	2	1	CM by $\Q(\sqrt{-119}) $
32.g	even	8	1	inner
3808.cu	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3808.1.cu.c	✓	16
7.b	odd	2	1		3808.1.cu.d	yes	16
17.b	even	2	1		3808.1.cu.d	yes	16
32.g	even	8	1	inner	3808.1.cu.c	✓	16
119.d	odd	2	1	CM	3808.1.cu.c	✓	16
224.v	odd	8	1		3808.1.cu.d	yes	16
544.bc	even	8	1		3808.1.cu.d	yes	16
3808.cu	odd	8	1	inner	3808.1.cu.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3808.1.cu.c	✓	16	1.a	even	1	1	trivial
3808.1.cu.c	✓	16	32.g	even	8	1	inner
3808.1.cu.c	✓	16	119.d	odd	2	1	CM
3808.1.cu.c	✓	16	3808.cu	odd	8	1	inner
3808.1.cu.d	yes	16	7.b	odd	2	1
3808.1.cu.d	yes	16	17.b	even	2	1
3808.1.cu.d	yes	16	224.v	odd	8	1
3808.1.cu.d	yes	16	544.bc	even	8	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{16} - 2 T_{3}^{14} + 4 T_{3}^{13} + 2 T_{3}^{12} - 16 T_{3}^{11} + 16 T_{3}^{10} + 52 T_{3}^{9} + \cdots + 1 $ T3^16 - 2*T3^14 + 4*T3^13 + 2*T3^12 - 16*T3^11 + 16*T3^10 + 52*T3^9 + 75*T3^8 + 72*T3^7 + 28*T3^6 + 52*T3^5 + 82*T3^4 - 136*T3^3 + 58*T3^2 - 8*T3 + 1 acting on $S_{1}^{\mathrm{new}}(3808, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - T^{12} + \cdots + 1 $ T^16 - T^12 + T^8 - T^4 + 1
$3$	$ T^{16} - 2 T^{14} + \cdots + 1 $ T^16 - 2T^14 + 4T^13 + 2T^12 - 16T^11 + 16T^10 + 52T^9 + 75T^8 + 72T^7 + 28T^6 + 52T^5 + 82T^4 - 136T^3 + 58T^2 - 8T + 1
$5$	$ T^{16} - 2 T^{14} + \cdots + 1 $ T^16 - 2T^14 - 4T^13 + 2T^12 + 16T^11 + 16T^10 - 52T^9 + 75T^8 - 72T^7 + 28T^6 - 52T^5 + 82T^4 + 136T^3 + 58T^2 + 8T + 1
$7$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$19$	$ T^{16} $ T^16
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ (T^{2} + T - 1)^{8} $ (T^2 + T - 1)^8
$37$	$ T^{16} $ T^16
$41$	$ (T^{8} + 7 T^{4} + 1)^{2} $ (T^8 + 7*T^4 + 1)^2
$43$	$ T^{16} - 2 T^{14} + \cdots + 1 $ T^16 - 2T^14 - 4T^13 + 2T^12 + 16T^11 + 16T^10 - 52T^9 + 75T^8 - 72T^7 + 28T^6 - 52T^5 + 82T^4 + 136T^3 + 58T^2 + 8T + 1
$47$	$ T^{16} $ T^16
$53$	$ T^{16} - 2 T^{14} + \cdots + 1 $ T^16 - 2T^14 - 4T^13 + 2T^12 + 16T^11 + 16T^10 - 52T^9 + 75T^8 - 72T^7 + 28T^6 - 52T^5 + 82T^4 + 136T^3 + 58T^2 + 8T + 1
$59$	$ T^{16} $ T^16
$61$	$ T^{16} - 4 T^{15} + \cdots + 1 $ T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 32T^10 + 40T^9 - 69T^8 - 4T^7 + 212T^6 - 328T^5 + 194T^4 - 48T^3 + 46T^2 - 12*T + 1
$67$	$ T^{16} + 4 T^{15} + \cdots + 1 $ T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 44T^11 + 32T^10 - 40T^9 - 69T^8 + 4T^7 + 212T^6 + 328T^5 + 194T^4 + 48T^3 + 46T^2 + 12*T + 1
$71$	$ T^{16} $ T^16
$73$	$ (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} $ (T^8 - 2T^7 + 2T^6 + 11T^4 - 20T^3 + 18T^2 - 6T + 1)^2
$79$	$ T^{16} $ T^16
$83$	$ T^{16} $ T^16
$89$	$ T^{16} $ T^16
$97$	$ (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} $ (T^8 - 8T^6 + 19T^4 - 12*T^2 + 1)^2
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3808.cu (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{40} q^{2} + ( - \zeta_{40}^{19} - \zeta_{40}^{6}) q^{3} + \zeta_{40}^{2} q^{4} + ( - \zeta_{40}^{13} + \zeta_{40}^{2}) q^{5} + (\zeta_{40}^{7} - 1) q^{6} + \zeta_{40}^{15} q^{7} - \zeta_{40}^{3} q^{8} + ( - \zeta_{40}^{18} + \cdots - \zeta_{40}^{5}) q^{9} +O(q^{10}) \) q - z * q^2 + (-z^19 - z^6) * q^3 + z^2 * q^4 + (-z^13 + z^2) * q^5 + (z^7 - 1) * q^6 + z^15 * q^7 - z^3 * q^8 + (-z^18 + z^12 - z^5) * q^9	\( q - \zeta_{40} q^{2} + ( - \zeta_{40}^{19} - \zeta_{40}^{6}) q^{3} + \zeta_{40}^{2} q^{4} + ( - \zeta_{40}^{13} + \zeta_{40}^{2}) q^{5} + (\zeta_{40}^{7} - 1) q^{6} + \zeta_{40}^{15} q^{7} - \zeta_{40}^{3} q^{8} + ( - \zeta_{40}^{18} + \cdots - \zeta_{40}^{5}) q^{9} + \cdots + \zeta_{40}^{11} q^{98} +O(q^{100}) \) q - z * q^2 + (-z^19 - z^6) * q^3 + z^2 * q^4 + (-z^13 + z^2) * q^5 + (z^7 - 1) * q^6 + z^15 * q^7 - z^3 * q^8 + (-z^18 + z^12 - z^5) * q^9 + (z^14 - z^3) * q^10 + (-z^8 + z) * q^12 - z^16 * q^14 + (z^19 - z^12 - z^8 + z) * q^15 + z^4 * q^16 + z^10 * q^17 + (z^19 - z^13 + z^6) * q^18 + (-z^15 + z^4) * q^20 + (z^14 + z) * q^21 + (z^9 - z^2) * q^24 + (-z^15 - z^6 + z^4) * q^25 + (-z^18 - z^17 + z^11 - z^4) * q^27 + z^17 * q^28 + (z^13 + z^9 - z^2 + 1) * q^30 + (-z^12 + z^8) * q^31 - z^5 * q^32 - z^11 * q^34 + (z^17 + z^8) * q^35 + (z^14 - z^7 + 1) * q^36 + (z^16 - z^5) * q^40 + (z^9 + z) * q^41 + (-z^15 - z^2) * q^42 + (-z^9 - z^6) * q^43 + (z^18 + z^14 - z^11 - z^7 + z^5 + 1) * q^45 + (-z^10 + z^3) * q^48 - z^10 * q^49 + (z^16 + z^7 - z^5) * q^50 + (-z^16 + z^9) * q^51 + (z^18 - z^17) * q^53 + (z^19 + z^18 - z^12 + z^5) * q^54 - z^18 * q^56 + (-z^14 - z^10 + z^3 - z) * q^60 + (-z^16 - z^9) * q^61 + (z^13 - z^9) * q^62 + (z^13 - z^7 + 1) * q^63 + z^6 * q^64 + (-z^13 - z^12) * q^67 + z^12 * q^68 + (-z^18 - z^9) * q^70 + (-z^15 + z^8 - z) * q^72 + (z^6 + z^4) * q^73 + (-z^14 + z^12 - z^10 - z^5 + z^3 - z) * q^75 + (-z^17 + z^6) * q^80 + (-z^17 - z^16 + z^10 - z^4 - z^3) * q^81 + (-z^10 - z^2) * q^82 + (z^16 + z^3) * q^84 + (z^12 + z^3) * q^85 + (z^10 + z^7) * q^86 + (-z^19 - z^15 + z^12 + z^8 - z^6 - z) * q^90 + (z^18 - z^14 - z^11 + z^7) * q^93 + (z^11 - z^4) * q^96 + (z^17 - z^3) * q^97 + z^11 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{6} + 4 q^{9}+O(q^{10}) \) 16 * q - 16 * q^6 + 4 * q^9	\( 16 q - 16 q^{6} + 4 q^{9} + 4 q^{12} + 4 q^{14} + 4 q^{16} + 4 q^{20} + 4 q^{25} - 4 q^{27} + 16 q^{30} - 8 q^{31} - 4 q^{35} + 16 q^{36} - 4 q^{40} + 16 q^{45} - 4 q^{50} + 4 q^{51} - 4 q^{54} + 4 q^{61} + 16 q^{63} - 4 q^{67} + 4 q^{68} - 4 q^{72} + 4 q^{73} + 4 q^{75} - 4 q^{84} + 4 q^{85} - 4 q^{96}+O(q^{100}) \) 16 * q - 16 * q^6 + 4 * q^9 + 4 * q^12 + 4 * q^14 + 4 * q^16 + 4 * q^20 + 4 * q^25 - 4 * q^27 + 16 * q^30 - 8 * q^31 - 4 * q^35 + 16 * q^36 - 4 * q^40 + 16 * q^45 - 4 * q^50 + 4 * q^51 - 4 * q^54 + 4 * q^61 + 16 * q^63 - 4 * q^67 + 4 * q^68 - 4 * q^72 + 4 * q^73 + 4 * q^75 - 4 * q^84 + 4 * q^85 - 4 * q^96

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	3808.1.cu.c

Level	$3808$
Weight	$1$
Character orbit	3808.cu
Analytic conductor	$1.900$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{40}$
CM discriminant	-119
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.90043956811\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{8})\)
Coefficient field:	\(\Q(\zeta_{40})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{40}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

\(n\)	\(2143\)	\(2689\)	\(3265\)	\(3333\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-\zeta_{40}^{15}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 237.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - T^{12} + \cdots + 1 \) T^16 - T^12 + T^8 - T^4 + 1
$3$	\( T^{16} - 2 T^{14} + \cdots + 1 \) T^16 - 2T^14 + 4T^13 + 2T^12 - 16T^11 + 16T^10 + 52T^9 + 75T^8 + 72T^7 + 28T^6 + 52T^5 + 82T^4 - 136T^3 + 58T^2 - 8T + 1
$5$	\( T^{16} - 2 T^{14} + \cdots + 1 \) T^16 - 2T^14 - 4T^13 + 2T^12 + 16T^11 + 16T^10 - 52T^9 + 75T^8 - 72T^7 + 28T^6 - 52T^5 + 82T^4 + 136T^3 + 58T^2 + 8T + 1
$7$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$19$	\( T^{16} \) T^16
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( (T^{2} + T - 1)^{8} \) (T^2 + T - 1)^8
$37$	\( T^{16} \) T^16
$41$	\( (T^{8} + 7 T^{4} + 1)^{2} \) (T^8 + 7*T^4 + 1)^2
$43$	\( T^{16} - 2 T^{14} + \cdots + 1 \) T^16 - 2T^14 - 4T^13 + 2T^12 + 16T^11 + 16T^10 - 52T^9 + 75T^8 - 72T^7 + 28T^6 - 52T^5 + 82T^4 + 136T^3 + 58T^2 + 8T + 1
$47$	\( T^{16} \) T^16
$53$	\( T^{16} - 2 T^{14} + \cdots + 1 \) T^16 - 2T^14 - 4T^13 + 2T^12 + 16T^11 + 16T^10 - 52T^9 + 75T^8 - 72T^7 + 28T^6 - 52T^5 + 82T^4 + 136T^3 + 58T^2 + 8T + 1
$59$	\( T^{16} \) T^16
$61$	\( T^{16} - 4 T^{15} + \cdots + 1 \) T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 32T^10 + 40T^9 - 69T^8 - 4T^7 + 212T^6 - 328T^5 + 194T^4 - 48T^3 + 46T^2 - 12*T + 1
$67$	\( T^{16} + 4 T^{15} + \cdots + 1 \) T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 44T^11 + 32T^10 - 40T^9 - 69T^8 + 4T^7 + 212T^6 + 328T^5 + 194T^4 + 48T^3 + 46T^2 + 12*T + 1
$71$	\( T^{16} \) T^16
$73$	\( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) (T^8 - 2T^7 + 2T^6 + 11T^4 - 20T^3 + 18T^2 - 6T + 1)^2
$79$	\( T^{16} \) T^16
$83$	\( T^{16} \) T^16
$89$	\( T^{16} \) T^16
$97$	\( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) (T^8 - 8T^6 + 19T^4 - 12*T^2 + 1)^2
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