LMFDB - Newform orbit 3920.1.di.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,1,Mod(239,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 0, 7, 2]))

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

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

chi := DirichletCharacter("3920.239");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3920.di (of order $14$, degree $6$, 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:	$1.95633484952$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{14})$
Coefficient field:	$\Q(\zeta_{28})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{14}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{14} + \cdots)$

$q$-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_{28}^{11} - \zeta_{28}^{7}) q^{3} + \zeta_{28}^{2} q^{5} + \zeta_{28}^{13} q^{7} + ( - \zeta_{28}^{8} + \zeta_{28}^{4} - 1) q^{9} + (\zeta_{28}^{13} - \zeta_{28}^{9}) q^{15} + ( - \zeta_{28}^{10} + \zeta_{28}^{6}) q^{21}+ \cdots + (\zeta_{28}^{12} - \zeta_{28}^{10}) q^{89}+O(q^{100}) $ q + (z^11 - z^7) * q^3 + z^2 * q^5 + z^13 * q^7 + (-z^8 + z^4 - 1) * q^9 + (z^13 - z^9) * q^15 + (-z^10 + z^6) * q^21 + (z^11 + z) * q^23 + z^4 * q^25 + (-z^11 + z^7 + z^5 - z) * q^27 + (-z^2 + 1) * q^29 - z * q^35 + (-z^12 + z^6) * q^41 + (-z^9 + z) * q^43 + (-z^10 + z^6 - z^2) * q^45 + (-z^5 + z) * q^47 - z^12 * q^49 + (z^10 + z^6) * q^61 + (-z^13 + z^7 - z^3) * q^63 + (z^12 - 2z^8 + z^4) q^69 + (-z^11 - z) * q^75 + (-z^12 + z^8 - z^4 - z^2 + 1) * q^81 + (z^5 + z^3) * q^83 + (-z^13 + z^11 + z^9 - z^7) * q^87 + (z^12 - z^10) * q^89
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{5} - 12 q^{9} - 2 q^{25} + 10 q^{29} + 4 q^{41} - 2 q^{45} + 2 q^{49} + 4 q^{61} + 12 q^{81} - 4 q^{89}+O(q^{100}) $ 12 * q + 2 * q^5 - 12 * q^9 - 2 * q^25 + 10 * q^29 + 4 * q^41 - 2 * q^45 + 2 * q^49 + 4 * q^61 + 12 * q^81 - 4 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$.

$n$	$981$	$1471$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$\zeta_{28}^{4}$	$-1$

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} $

239.1

0.974928	+	0.222521i
−0.974928	−	0.222521i
−0.433884	−	0.900969i
0.433884	+	0.900969i
−0.433884	+	0.900969i
0.433884	−	0.900969i
0.974928	−	0.222521i
−0.974928	+	0.222521i
−0.781831	−	0.623490i
0.781831	+	0.623490i
−0.781831	+	0.623490i
0.781831	−	0.623490i

−0.781831

−

0.376510i

0.900969

0.433884i

−0.974928

0.222521i

−0.153989

−

0.193096i

239.2

0.781831

0.376510i

0.900969

0.433884i

0.974928

−

0.222521i

−0.153989

−

0.193096i

799.1

−0.974928

1.22252i

−0.623490

0.781831i

0.433884

−

0.900969i

−0.321552

−

1.40881i

799.2

0.974928

−

1.22252i

−0.623490

0.781831i

−0.433884

0.900969i

−0.321552

−

1.40881i

1359.1

−0.974928

−

1.22252i

−0.623490

−

0.781831i

0.433884

0.900969i

−0.321552

1.40881i

1359.2

0.974928

1.22252i

−0.623490

−

0.781831i

−0.433884

−

0.900969i

−0.321552

1.40881i

1919.1

−0.781831

0.376510i

0.900969

−

0.433884i

−0.974928

−

0.222521i

−0.153989

0.193096i

1919.2

0.781831

−

0.376510i

0.900969

−

0.433884i

0.974928

0.222521i

−0.153989

0.193096i

2479.1

−0.433884

−

1.90097i

0.222521

0.974928i

0.781831

−

0.623490i

−2.52446

1.21572i

2479.2

0.433884

1.90097i

0.222521

0.974928i

−0.781831

0.623490i

−2.52446

1.21572i

3599.1

−0.433884

1.90097i

0.222521

−

0.974928i

0.781831

0.623490i

−2.52446

−

1.21572i

3599.2

0.433884

−

1.90097i

0.222521

−

0.974928i

−0.781831

−

0.623490i

−2.52446

−

1.21572i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
4.b	odd	2	1	inner
5.b	even	2	1	inner
49.e	even	7	1	inner
196.k	odd	14	1	inner
245.p	even	14	1	inner
980.ba	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3920.1.di.a	✓	12
4.b	odd	2	1	inner	3920.1.di.a	✓	12
5.b	even	2	1	inner	3920.1.di.a	✓	12
20.d	odd	2	1	CM	3920.1.di.a	✓	12
49.e	even	7	1	inner	3920.1.di.a	✓	12
196.k	odd	14	1	inner	3920.1.di.a	✓	12
245.p	even	14	1	inner	3920.1.di.a	✓	12
980.ba	odd	14	1	inner	3920.1.di.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3920.1.di.a	✓	12	1.a	even	1	1	trivial
3920.1.di.a	✓	12	4.b	odd	2	1	inner
3920.1.di.a	✓	12	5.b	even	2	1	inner
3920.1.di.a	✓	12	20.d	odd	2	1	CM
3920.1.di.a	✓	12	49.e	even	7	1	inner
3920.1.di.a	✓	12	196.k	odd	14	1	inner
3920.1.di.a	✓	12	245.p	even	14	1	inner
3920.1.di.a	✓	12	980.ba	odd	14	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3920, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 7 T^{10} + \cdots + 49 $ T^12 + 7T^10 + 21T^8 + 35T^6 + 49T^4 - 49*T^2 + 49
$5$	$ (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$7$	$ T^{12} - T^{10} + \cdots + 1 $ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$11$	$ T^{12} $ T^12
$13$	$ T^{12} $ T^12
$17$	$ T^{12} $ T^12
$19$	$ T^{12} $ T^12
$23$	$ T^{12} + 14 T^{8} + \cdots + 49 $ T^12 + 14T^8 - 14T^6 + 49T^4 + 98T^2 + 49
$29$	$ (T^{6} - 5 T^{5} + 11 T^{4} + \cdots + 1)^{2} $ (T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1)^2
$31$	$ T^{12} $ T^12
$37$	$ T^{12} $ T^12
$41$	$ (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} $ (T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1)^2
$43$	$ T^{12} + 14 T^{8} + \cdots + 49 $ T^12 + 14T^8 - 14T^6 + 49T^4 + 98T^2 + 49
$47$	$ T^{12} + 35 T^{6} + \cdots + 49 $ T^12 + 35T^6 + 98T^4 + 49*T^2 + 49
$53$	$ T^{12} $ T^12
$59$	$ T^{12} $ T^12
$61$	$ (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} $ (T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1)^2
$67$	$ T^{12} $ T^12
$71$	$ T^{12} $ T^12
$73$	$ T^{12} $ T^12
$79$	$ T^{12} $ T^12
$83$	$ T^{12} + 14 T^{8} + \cdots + 49 $ T^12 + 14T^8 - 14T^6 + 49T^4 + 98T^2 + 49
$89$	$ (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} $ (T^6 + 2T^5 + 4T^4 + 8T^3 + 9T^2 + 4*T + 1)^2
$97$	$ T^{12} $ T^12
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3920.di (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{28}^{11} - \zeta_{28}^{7}) q^{3} + \zeta_{28}^{2} q^{5} + \zeta_{28}^{13} q^{7} + ( - \zeta_{28}^{8} + \zeta_{28}^{4} - 1) q^{9} + (\zeta_{28}^{13} - \zeta_{28}^{9}) q^{15} + ( - \zeta_{28}^{10} + \zeta_{28}^{6}) q^{21}+ \cdots + (\zeta_{28}^{12} - \zeta_{28}^{10}) q^{89}+O(q^{100}) \) q + (z^11 - z^7) * q^3 + z^2 * q^5 + z^13 * q^7 + (-z^8 + z^4 - 1) * q^9 + (z^13 - z^9) * q^15 + (-z^10 + z^6) * q^21 + (z^11 + z) * q^23 + z^4 * q^25 + (-z^11 + z^7 + z^5 - z) * q^27 + (-z^2 + 1) * q^29 - z * q^35 + (-z^12 + z^6) * q^41 + (-z^9 + z) * q^43 + (-z^10 + z^6 - z^2) * q^45 + (-z^5 + z) * q^47 - z^12 * q^49 + (z^10 + z^6) * q^61 + (-z^13 + z^7 - z^3) * q^63 + (z^12 - 2z^8 + z^4) q^69 + (-z^11 - z) * q^75 + (-z^12 + z^8 - z^4 - z^2 + 1) * q^81 + (z^5 + z^3) * q^83 + (-z^13 + z^11 + z^9 - z^7) * q^87 + (z^12 - z^10) * q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{5} - 12 q^{9} - 2 q^{25} + 10 q^{29} + 4 q^{41} - 2 q^{45} + 2 q^{49} + 4 q^{61} + 12 q^{81} - 4 q^{89}+O(q^{100}) \) 12 * q + 2 * q^5 - 12 * q^9 - 2 * q^25 + 10 * q^29 + 4 * q^41 - 2 * q^45 + 2 * q^49 + 4 * q^61 + 12 * q^81 - 4 * q^89

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	3920.1.di.a

Level	$3920$
Weight	$1$
Character orbit	3920.di
Analytic conductor	$1.956$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{14}$
CM discriminant	-20
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{14})\)
Coefficient field:	\(\Q(\zeta_{28})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{14}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{14} + \cdots)\)

\(n\)	\(981\)	\(1471\)	\(3041\)	\(3137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\zeta_{28}^{4}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 239.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 7 T^{10} + \cdots + 49 \) T^12 + 7T^10 + 21T^8 + 35T^6 + 49T^4 - 49*T^2 + 49
$5$	\( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$7$	\( T^{12} - T^{10} + \cdots + 1 \) T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$11$	\( T^{12} \) T^12
$13$	\( T^{12} \) T^12
$17$	\( T^{12} \) T^12
$19$	\( T^{12} \) T^12
$23$	\( T^{12} + 14 T^{8} + \cdots + 49 \) T^12 + 14T^8 - 14T^6 + 49T^4 + 98T^2 + 49
$29$	\( (T^{6} - 5 T^{5} + 11 T^{4} + \cdots + 1)^{2} \) (T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1)^2
$31$	\( T^{12} \) T^12
$37$	\( T^{12} \) T^12
$41$	\( (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) (T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1)^2
$43$	\( T^{12} + 14 T^{8} + \cdots + 49 \) T^12 + 14T^8 - 14T^6 + 49T^4 + 98T^2 + 49
$47$	\( T^{12} + 35 T^{6} + \cdots + 49 \) T^12 + 35T^6 + 98T^4 + 49*T^2 + 49
$53$	\( T^{12} \) T^12
$59$	\( T^{12} \) T^12
$61$	\( (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) (T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1)^2
$67$	\( T^{12} \) T^12
$71$	\( T^{12} \) T^12
$73$	\( T^{12} \) T^12
$79$	\( T^{12} \) T^12
$83$	\( T^{12} + 14 T^{8} + \cdots + 49 \) T^12 + 14T^8 - 14T^6 + 49T^4 + 98T^2 + 49
$89$	\( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) (T^6 + 2T^5 + 4T^4 + 8T^3 + 9T^2 + 4*T + 1)^2
$97$	\( T^{12} \) T^12
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