LMFDB - Newform orbit 2816.1.r.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,1,Mod(513,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("2816.513");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2816.r (of order $10$, degree $4$, not 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:	$1.40536707557$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 704)
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.0.9658153742336.1

$q$-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_{10}^{2} - \zeta_{10}) q^{3} + (\zeta_{10}^{4} + \cdots + \zeta_{10}^{2}) q^{9} - \zeta_{10}^{4} q^{11} + ( - \zeta_{10}^{3} + \zeta_{10}) q^{17} + ( - \zeta_{10}^{3} - 1) q^{19} - \zeta_{10}^{4} q^{25} + \cdots + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{99} +O(q^{100}) $ q + (z^2 - z) * q^3 + (z^4 - z^3 + z^2) * q^9 - z^4 * q^11 + (-z^3 + z) * q^17 + (-z^3 - 1) * q^19 - z^4 * q^25 + (z^4 - z^3 - z + 1) * q^27 + (z - 1) * q^33 + (z^2 + z) * q^41 + (z^3 + z^2) * q^43 + z^2 * q^49 + (z^4 + z^3 - z^2 + 1) * q^51 + (z^4 - z^2 + z + 1) * q^57 + (-z^2 - 1) * q^59 + (-z^4 + z) * q^67 + (-z^4 - z^3) * q^73 + (z - 1) * q^75 + (z^4 - z^3 + z^2 - z + 1) * q^81 + (-z^4 + 1) * q^83 + (z^3 - z^2) * q^89 + (-z + 1) * q^97 + (z^3 - z^2 + z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 3 q^{9} + q^{11} - 5 q^{19} + q^{25} + q^{27} - 3 q^{33} - q^{49} + 5 q^{51} + 5 q^{57} - 3 q^{59} + 2 q^{67} - 3 q^{75} + 5 q^{83} + 2 q^{89} + 3 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 3 * q^9 + q^11 - 5 * q^19 + q^25 + q^27 - 3 * q^33 - q^49 + 5 * q^51 + 5 * q^57 - 3 * q^59 + 2 * q^67 - 3 * q^75 + 5 * q^83 + 2 * q^89 + 3 * q^97 + 3 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$\zeta_{10}$	$1$	$1$

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} $

513.1

−0.309017	−	0.951057i
0.809017	+	0.587785i
−0.309017	+	0.951057i
0.809017	−	0.587785i

−0.500000

1.53884i

−1.30902

−

0.951057i

1025.1

−0.500000

0.363271i

−0.190983

0.587785i

1537.1

−0.500000

−

1.53884i

−1.30902

0.951057i

2305.1

−0.500000

−

0.363271i

−0.190983

−

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
11.d	odd	10	1	inner
88.k	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2816.1.r.a		4
4.b	odd	2	1		2816.1.r.b		4
8.b	even	2	1		2816.1.r.b		4
8.d	odd	2	1	CM	2816.1.r.a		4
11.d	odd	10	1	inner	2816.1.r.a		4
16.e	even	4	2		704.1.x.a	✓	8
16.f	odd	4	2		704.1.x.a	✓	8
44.g	even	10	1		2816.1.r.b		4
88.k	even	10	1	inner	2816.1.r.a		4
88.p	odd	10	1		2816.1.r.b		4
176.u	odd	20	2		704.1.x.a	✓	8
176.x	even	20	2		704.1.x.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
704.1.x.a	✓	8	16.e	even	4	2
704.1.x.a	✓	8	16.f	odd	4	2
704.1.x.a	✓	8	176.u	odd	20	2
704.1.x.a	✓	8	176.x	even	20	2
2816.1.r.a		4	1.a	even	1	1	trivial
2816.1.r.a		4	8.d	odd	2	1	CM
2816.1.r.a		4	11.d	odd	10	1	inner
2816.1.r.a		4	88.k	even	10	1	inner
2816.1.r.b		4	4.b	odd	2	1
2816.1.r.b		4	8.b	even	2	1
2816.1.r.b		4	44.g	even	10	1
2816.1.r.b		4	88.p	odd	10	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 1 $ T3^4 + 2*T3^3 + 4*T3^2 + 3*T3 + 1 acting on $S_{1}^{\mathrm{new}}(2816, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$5$	$ T^{4} $ T^4
$7$	$ T^{4} $ T^4
$11$	$ T^{4} - T^{3} + T^{2} + \cdots + 1 $ T^4 - T^3 + T^2 - T + 1
$13$	$ T^{4} $ T^4
$17$	$ T^{4} - 5T + 5 $ T^4 - 5*T + 5
$19$	$ T^{4} + 5 T^{3} + \cdots + 5 $ T^4 + 5T^3 + 10T^2 + 10*T + 5
$23$	$ T^{4} $ T^4
$29$	$ T^{4} $ T^4
$31$	$ T^{4} $ T^4
$37$	$ T^{4} $ T^4
$41$	$ T^{4} + 5T + 5 $ T^4 + 5*T + 5
$43$	$ T^{4} + 5T^{2} + 5 $ T^4 + 5*T^2 + 5
$47$	$ T^{4} $ T^4
$53$	$ T^{4} $ T^4
$59$	$ T^{4} + 3 T^{3} + \cdots + 1 $ T^4 + 3T^3 + 4T^2 + 2*T + 1
$61$	$ T^{4} $ T^4
$67$	$ (T^{2} - T - 1)^{2} $ (T^2 - T - 1)^2
$71$	$ T^{4} $ T^4
$73$	$ T^{4} + 5T + 5 $ T^4 + 5*T + 5
$79$	$ T^{4} $ T^4
$83$	$ T^{4} - 5 T^{3} + \cdots + 5 $ T^4 - 5T^3 + 10T^2 - 10*T + 5
$89$	$ (T^{2} - T - 1)^{2} $ (T^2 - T - 1)^2
$97$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
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Level:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2816.r (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{10}^{2} - \zeta_{10}) q^{3} + (\zeta_{10}^{4} + \cdots + \zeta_{10}^{2}) q^{9} - \zeta_{10}^{4} q^{11} + ( - \zeta_{10}^{3} + \zeta_{10}) q^{17} + ( - \zeta_{10}^{3} - 1) q^{19} - \zeta_{10}^{4} q^{25} + \cdots + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{99} +O(q^{100}) \) q + (z^2 - z) * q^3 + (z^4 - z^3 + z^2) * q^9 - z^4 * q^11 + (-z^3 + z) * q^17 + (-z^3 - 1) * q^19 - z^4 * q^25 + (z^4 - z^3 - z + 1) * q^27 + (z - 1) * q^33 + (z^2 + z) * q^41 + (z^3 + z^2) * q^43 + z^2 * q^49 + (z^4 + z^3 - z^2 + 1) * q^51 + (z^4 - z^2 + z + 1) * q^57 + (-z^2 - 1) * q^59 + (-z^4 + z) * q^67 + (-z^4 - z^3) * q^73 + (z - 1) * q^75 + (z^4 - z^3 + z^2 - z + 1) * q^81 + (-z^4 + 1) * q^83 + (z^3 - z^2) * q^89 + (-z + 1) * q^97 + (z^3 - z^2 + z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 3 q^{9} + q^{11} - 5 q^{19} + q^{25} + q^{27} - 3 q^{33} - q^{49} + 5 q^{51} + 5 q^{57} - 3 q^{59} + 2 q^{67} - 3 q^{75} + 5 q^{83} + 2 q^{89} + 3 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 3 * q^9 + q^11 - 5 * q^19 + q^25 + q^27 - 3 * q^33 - q^49 + 5 * q^51 + 5 * q^57 - 3 * q^59 + 2 * q^67 - 3 * q^75 + 5 * q^83 + 2 * q^89 + 3 * q^97 + 3 * q^99

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	2816.1.r.a

Level	$2816$
Weight	$1$
Character orbit	2816.r
Analytic conductor	$1.405$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{10}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.40536707557\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \) x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 704)
Projective image:	\(D_{10}\)
Projective field:	Galois closure of 10.0.9658153742336.1

\(n\)	\(1025\)	\(1541\)	\(2047\)
\(\chi(n)\)	\(\zeta_{10}\)	\(1\)	\(1\)

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
11.d	odd	10	1	inner
88.k	even	10	1	inner

$p$	$F_p(T)$
$2$	\( T^{4} \) T^4
$3$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$5$	\( T^{4} \) T^4
$7$	\( T^{4} \) T^4
$11$	\( T^{4} - T^{3} + T^{2} + \cdots + 1 \) T^4 - T^3 + T^2 - T + 1
$13$	\( T^{4} \) T^4
$17$	\( T^{4} - 5T + 5 \) T^4 - 5*T + 5
$19$	\( T^{4} + 5 T^{3} + \cdots + 5 \) T^4 + 5T^3 + 10T^2 + 10*T + 5
$23$	\( T^{4} \) T^4
$29$	\( T^{4} \) T^4
$31$	\( T^{4} \) T^4
$37$	\( T^{4} \) T^4
$41$	\( T^{4} + 5T + 5 \) T^4 + 5*T + 5
$43$	\( T^{4} + 5T^{2} + 5 \) T^4 + 5*T^2 + 5
$47$	\( T^{4} \) T^4
$53$	\( T^{4} \) T^4
$59$	\( T^{4} + 3 T^{3} + \cdots + 1 \) T^4 + 3T^3 + 4T^2 + 2*T + 1
$61$	\( T^{4} \) T^4
$67$	\( (T^{2} - T - 1)^{2} \) (T^2 - T - 1)^2
$71$	\( T^{4} \) T^4
$73$	\( T^{4} + 5T + 5 \) T^4 + 5*T + 5
$79$	\( T^{4} \) T^4
$83$	\( T^{4} - 5 T^{3} + \cdots + 5 \) T^4 - 5T^3 + 10T^2 - 10*T + 5
$89$	\( (T^{2} - T - 1)^{2} \) (T^2 - T - 1)^2
$97$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
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\(n\):		e.g. 2-40 or 990-1000
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