LMFDB - Newform orbit 3113.1.l.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3113,1,Mod(565,3113)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3113, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([2, 5]))

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

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

chi := DirichletCharacter("3113.565");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3113 = 11 \cdot 283 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3113.l (of order $10$, 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:	$1.55358938433$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.1172583049.1

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3113\mathbb{Z}\right)^\times$.

$n$	$1982$	$2267$
$\chi(n)$	$-\zeta_{10}$	$-1$

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} $

565.1

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

0.309017

0.951057i

0.618034

1.90211i

−0.809017

−

0.587785i

1131.1

−0.809017

−

0.587785i

−1.61803

−

1.17557i

0.309017

−

0.951057i

1697.1

0.309017

−

0.951057i

0.618034

−

1.90211i

−0.809017

0.587785i

2546.1

−0.809017

0.587785i

−1.61803

1.17557i

0.309017

0.951057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
283.b	odd	2	1	CM by $\Q(\sqrt{-283}) $
11.c	even	5	1	inner
3113.l	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3113.1.l.a	✓	4
11.c	even	5	1	inner	3113.1.l.a	✓	4
283.b	odd	2	1	CM	3113.1.l.a	✓	4
3113.l	odd	10	1	inner	3113.1.l.a	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3113.1.l.a	✓	4	1.a	even	1	1	trivial
3113.1.l.a	✓	4	11.c	even	5	1	inner
3113.1.l.a	✓	4	283.b	odd	2	1	CM
3113.1.l.a	✓	4	3113.l	odd	10	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16 $ T7^4 + 2*T7^3 + 4*T7^2 + 8*T7 + 16 acting on $S_{1}^{\mathrm{new}}(3113, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ T^{4} + 2 T^{3} + \cdots + 16 $ T^4 + 2T^3 + 4T^2 + 8*T + 16
$11$	$ T^{4} + T^{3} + T^{2} + \cdots + 1 $ T^4 + T^3 + T^2 + T + 1
$13$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ (T^{2} + T - 1)^{2} $ (T^2 + T - 1)^2
$29$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$31$	$ T^{4} $ T^4
$37$	$ T^{4} $ T^4
$41$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$43$	$ T^{4} $ T^4
$47$	$ T^{4} $ T^4
$53$	$ T^{4} $ T^4
$59$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$61$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$67$	$ T^{4} $ T^4
$71$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$73$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$79$	$ T^{4} $ T^4
$83$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$89$	$ (T^{2} + T - 1)^{2} $ (T^2 + T - 1)^2
$97$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
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Overview	Random
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3113 = 11 \cdot 283 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3113.l (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{10} q^{4} - \zeta_{10} q^{7} - \zeta_{10}^{3} q^{9} +O(q^{10}) \) q - z * q^4 - z * q^7 - z^3 * q^9	\( q - \zeta_{10} q^{4} - \zeta_{10} q^{7} - \zeta_{10}^{3} q^{9} - \zeta_{10}^{3} q^{11} + ( - \zeta_{10} + 1) q^{13} + \zeta_{10}^{2} q^{16} + (\zeta_{10}^{4} - \zeta_{10}) q^{23} + \zeta_{10}^{4} q^{25} + 2 \zeta_{10}^{2} q^{28} + (\zeta_{10}^{2} + 1) q^{29} + \zeta_{10}^{4} q^{36} + (\zeta_{10}^{2} - \zeta_{10}) q^{41} + \zeta_{10}^{4} q^{44} + 3 \zeta_{10}^{2} q^{49} + (\zeta_{10}^{2} - \zeta_{10}) q^{52} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{59} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{61} + 2 \zeta_{10}^{4} q^{63} - \zeta_{10}^{3} q^{64} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{71} + (\zeta_{10}^{2} + 1) q^{73} + 2 \zeta_{10}^{4} q^{77} - \zeta_{10} q^{81} + (\zeta_{10}^{4} + 1) q^{83} + (\zeta_{10}^{4} - \zeta_{10}) q^{89} + (2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{91} + (\zeta_{10}^{2} + 1) q^{92} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{97} - \zeta_{10} q^{99} +O(q^{100}) \) q - z * q^4 - z * q^7 - z^3 * q^9 - z^3 * q^11 + (-z + 1) * q^13 + z^2 * q^16 + (z^4 - z) * q^23 + z^4 * q^25 + 2z^2 q^28 + (z^2 + 1) * q^29 + z^4 * q^36 + (z^2 - z) * q^41 + z^4 * q^44 + 3z^2 q^49 + (z^2 - z) * q^52 + (z^4 - z^3) * q^59 + (-z^3 - z) * q^61 + 2z^4 q^63 - z^3 * q^64 + (-z^3 - z) * q^71 + (z^2 + 1) * q^73 + 2z^4 q^77 - z * q^81 + (z^4 + 1) * q^83 + (z^4 - z) * q^89 + (2z^2 - 2z) * q^91 + (z^2 + 1) * q^92 + (z^4 + z^2) * q^97 - z * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) \) 4 * q - q^4 - 2 * q^7 - q^9	\( 4 q - q^{4} - 2 q^{7} - q^{9} - q^{11} + 3 q^{13} - q^{16} - 2 q^{23} - q^{25} - 2 q^{28} + 3 q^{29} - q^{36} - 2 q^{41} - q^{44} - 3 q^{49} - 2 q^{52} - 2 q^{59} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{71} + 3 q^{73} - 2 q^{77} - q^{81} + 3 q^{83} - 2 q^{89} - 4 q^{91} + 3 q^{92} - 2 q^{97} - q^{99}+O(q^{100}) \) 4 * q - q^4 - 2 * q^7 - q^9 - q^11 + 3 * q^13 - q^16 - 2 * q^23 - q^25 - 2 * q^28 + 3 * q^29 - q^36 - 2 * q^41 - q^44 - 3 * q^49 - 2 * q^52 - 2 * q^59 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^71 + 3 * q^73 - 2 * q^77 - q^81 + 3 * q^83 - 2 * q^89 - 4 * q^91 + 3 * q^92 - 2 * q^97 - q^99

Introduction

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Database

Properties

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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	3113.1.l.a

Level	$3113$
Weight	$1$
Character orbit	3113.l
Analytic conductor	$1.554$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{5}$
CM discriminant	-283
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.55358938433\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.1172583049.1

$p$	$F_p(T)$
$2$	\( T^{4} \) T^4
$3$	\( T^{4} \) T^4
$5$	\( T^{4} \) T^4
$7$	\( T^{4} + 2 T^{3} + \cdots + 16 \) T^4 + 2T^3 + 4T^2 + 8*T + 16
$11$	\( T^{4} + T^{3} + T^{2} + \cdots + 1 \) T^4 + T^3 + T^2 + T + 1
$13$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$17$	\( T^{4} \) T^4
$19$	\( T^{4} \) T^4
$23$	\( (T^{2} + T - 1)^{2} \) (T^2 + T - 1)^2
$29$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$31$	\( T^{4} \) T^4
$37$	\( T^{4} \) T^4
$41$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$43$	\( T^{4} \) T^4
$47$	\( T^{4} \) T^4
$53$	\( T^{4} \) T^4
$59$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$61$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$67$	\( T^{4} \) T^4
$71$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$73$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$79$	\( T^{4} \) T^4
$83$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$89$	\( (T^{2} + T - 1)^{2} \) (T^2 + T - 1)^2
$97$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
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\(n\)	\(1982\)	\(2267\)
\(\chi(n)\)	\(-\zeta_{10}\)	\(-1\)

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