LMFDB - Newform orbit 3328.1.bl.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,1,Mod(513,3328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3328, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("3328.513");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3328.bl (of order $12$, 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:	$1.66088836204$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1664)
Projective image:	$D_{12}$
Projective field:	Galois closure of 12.2.7516865509350965248.6

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$1$	$-\zeta_{12}^{5}$	$1$

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

−0.866025	−	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i
0.866025	+	0.500000i

−0.366025

0.366025i

0.500000

0.866025i

769.1

1.36603

−

1.36603i

0.500000

−

0.866025i

1025.1

−0.366025

−

0.366025i

0.500000

−

0.866025i

1281.1

1.36603

1.36603i

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
13.f	odd	12	1	inner
52.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3328.1.bl.b		4
4.b	odd	2	1	CM	3328.1.bl.b		4
8.b	even	2	1		3328.1.bl.a		4
8.d	odd	2	1		3328.1.bl.a		4
13.f	odd	12	1	inner	3328.1.bl.b		4
16.e	even	4	1		1664.1.bv.a	✓	4
16.e	even	4	1		1664.1.bv.b	yes	4
16.f	odd	4	1		1664.1.bv.a	✓	4
16.f	odd	4	1		1664.1.bv.b	yes	4
52.l	even	12	1	inner	3328.1.bl.b		4
104.u	even	12	1		3328.1.bl.a		4
104.x	odd	12	1		3328.1.bl.a		4
208.be	odd	12	1		1664.1.bv.a	✓	4
208.bf	even	12	1		1664.1.bv.a	✓	4
208.bk	even	12	1		1664.1.bv.b	yes	4
208.bl	odd	12	1		1664.1.bv.b	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1664.1.bv.a	✓	4	16.e	even	4	1
1664.1.bv.a	✓	4	16.f	odd	4	1
1664.1.bv.a	✓	4	208.be	odd	12	1
1664.1.bv.a	✓	4	208.bf	even	12	1
1664.1.bv.b	yes	4	16.e	even	4	1
1664.1.bv.b	yes	4	16.f	odd	4	1
1664.1.bv.b	yes	4	208.bk	even	12	1
1664.1.bv.b	yes	4	208.bl	odd	12	1
3328.1.bl.a		4	8.b	even	2	1
3328.1.bl.a		4	8.d	odd	2	1
3328.1.bl.a		4	104.u	even	12	1
3328.1.bl.a		4	104.x	odd	12	1
3328.1.bl.b		4	1.a	even	1	1	trivial
3328.1.bl.b		4	4.b	odd	2	1	CM
3328.1.bl.b		4	13.f	odd	12	1	inner
3328.1.bl.b		4	52.l	even	12	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 2T_{5}^{3} + 2T_{5}^{2} + 2T_{5} + 1 $ T5^4 - 2*T5^3 + 2*T5^2 + 2*T5 + 1 acting on $S_{1}^{\mathrm{new}}(3328, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} - 2 T^{3} + \cdots + 1 $ T^4 - 2T^3 + 2T^2 + 2*T + 1
$7$	$ T^{4} $ T^4
$11$	$ T^{4} $ T^4
$13$	$ (T^{2} - T + 1)^{2} $ (T^2 - T + 1)^2
$17$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$19$	$ T^{4} $ T^4
$23$	$ T^{4} $ T^4
$29$	$ (T^{2} - T + 1)^{2} $ (T^2 - T + 1)^2
$31$	$ T^{4} $ T^4
$37$	$ T^{4} + 4 T^{3} + \cdots + 1 $ T^4 + 4T^3 + 5T^2 + 2*T + 1
$41$	$ T^{4} - 4 T^{3} + \cdots + 1 $ T^4 - 4T^3 + 5T^2 - 2*T + 1
$43$	$ T^{4} $ T^4
$47$	$ T^{4} $ T^4
$53$	$ (T^{2} - 3)^{2} $ (T^2 - 3)^2
$59$	$ T^{4} $ T^4
$61$	$ T^{4} + 3T^{2} + 9 $ T^4 + 3*T^2 + 9
$67$	$ T^{4} $ T^4
$71$	$ T^{4} $ T^4
$73$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 2T^2 - 2*T + 1
$79$	$ T^{4} $ T^4
$83$	$ T^{4} $ T^4
$89$	$ T^{4} - 2 T^{3} + \cdots + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$97$	$ T^{4} - 2 T^{3} + \cdots + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
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Classical	Maass
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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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3328.bl (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{12}^{2} + \zeta_{12}) q^{5} + \zeta_{12}^{2} q^{9}+O(q^{10}) \) q + (z^2 + z) * q^5 + z^2 * q^9	\( q + (\zeta_{12}^{2} + \zeta_{12}) q^{5} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{4} q^{13} + \zeta_{12}^{5} q^{17} + (\zeta_{12}^{4} + \cdots + \zeta_{12}^{2}) q^{25} + \cdots + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{97} +O(q^{100}) \) q + (z^2 + z) * q^5 + z^2 * q^9 - z^4 * q^13 + z^5 * q^17 + (z^4 + z^3 + z^2) * q^25 - z^4 * q^29 + (-z^5 - 1) * q^37 + (-z^5 + 1) * q^41 + (z^4 + z^3) * q^45 - z * q^49 + (z^5 - z) * q^53 + (-z^3 - z) * q^61 + (-z^5 + 1) * q^65 + (z^5 + z^4) * q^73 + z^4 * q^81 + (-z - 1) * q^85 + (-z^4 - z) * q^89 + (z^5 + z^2) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^5 + 2 * q^9	\( 4 q + 2 q^{5} + 2 q^{9} + 2 q^{13} + 2 q^{29} - 4 q^{37} + 4 q^{41} - 2 q^{45} + 4 q^{65} - 2 q^{73} - 2 q^{81} - 4 q^{85} + 2 q^{89} + 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 2 * q^9 + 2 * q^13 + 2 * q^29 - 4 * q^37 + 4 * q^41 - 2 * q^45 + 4 * q^65 - 2 * q^73 - 2 * q^81 - 4 * q^85 + 2 * q^89 + 2 * q^97

Introduction

L-functions

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	3328.1.bl.b

Level	$3328$
Weight	$1$
Character orbit	3328.bl
Analytic conductor	$1.661$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{12}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.66088836204\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{2} + 1 \) x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 1664)
Projective image:	\(D_{12}\)
Projective field:	Galois closure of 12.2.7516865509350965248.6

\(n\)	\(261\)	\(769\)	\(1535\)
\(\chi(n)\)	\(1\)	\(-\zeta_{12}^{5}\)	\(1\)

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
13.f	odd	12	1	inner
52.l	even	12	1	inner

$p$	$F_p(T)$
$2$	\( T^{4} \) T^4
$3$	\( T^{4} \) T^4
$5$	\( T^{4} - 2 T^{3} + \cdots + 1 \) T^4 - 2T^3 + 2T^2 + 2*T + 1
$7$	\( T^{4} \) T^4
$11$	\( T^{4} \) T^4
$13$	\( (T^{2} - T + 1)^{2} \) (T^2 - T + 1)^2
$17$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$19$	\( T^{4} \) T^4
$23$	\( T^{4} \) T^4
$29$	\( (T^{2} - T + 1)^{2} \) (T^2 - T + 1)^2
$31$	\( T^{4} \) T^4
$37$	\( T^{4} + 4 T^{3} + \cdots + 1 \) T^4 + 4T^3 + 5T^2 + 2*T + 1
$41$	\( T^{4} - 4 T^{3} + \cdots + 1 \) T^4 - 4T^3 + 5T^2 - 2*T + 1
$43$	\( T^{4} \) T^4
$47$	\( T^{4} \) T^4
$53$	\( (T^{2} - 3)^{2} \) (T^2 - 3)^2
$59$	\( T^{4} \) T^4
$61$	\( T^{4} + 3T^{2} + 9 \) T^4 + 3*T^2 + 9
$67$	\( T^{4} \) T^4
$71$	\( T^{4} \) T^4
$73$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 2T^2 - 2*T + 1
$79$	\( T^{4} \) T^4
$83$	\( T^{4} \) T^4
$89$	\( T^{4} - 2 T^{3} + \cdots + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$97$	\( T^{4} - 2 T^{3} + \cdots + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
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\(n\):		e.g. 2-40 or 990-1000
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