LMFDB - Newform orbit 1107.1.g.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1107,1,Mod(296,1107)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1107, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("1107.296");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1107 = 3^{3} \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1107.g (of order $4$, degree $2$, 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:	$0.552464968985$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.1860867.2

$q$-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_{8}^{3} + \zeta_{8}) q^{2} + q^{4} + (\zeta_{8}^{2} + 1) q^{7} + \zeta_{8}^{3} q^{11} + 2 \zeta_{8} q^{14} - q^{16} + ( - \zeta_{8}^{2} + 1) q^{19} + (\zeta_{8}^{2} - 1) q^{22} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{23}+ \cdots + (\zeta_{8}^{3} + \zeta_{8}) q^{98}+O(q^{100}) $ q + (-z^3 + z) * q^2 + q^4 + (z^2 + 1) * q^7 + z^3 * q^11 + 2z q^14 - q^16 + (-z^2 + 1) * q^19 + (z^2 - 1) * q^22 + (-z^3 - z) * q^23 - q^25 + (z^2 + 1) * q^28 - z^3 * q^29 - q^31 + (z^3 - z) * q^32 + q^37 - 2z^3 q^38 + z^3 * q^41 - z^2 * q^43 + z^3 * q^44 - 2z^2 q^46 - z * q^47 + z^2 * q^49 + (z^3 - z) * q^50 + z^3 * q^53 + (-z^2 + 1) * q^58 + z^2 * q^61 + (z^3 - z) * q^62 - q^64 + z^2 * q^73 + (-z^3 + z) * q^74 + (-z^2 + 1) * q^76 + (z^3 - z) * q^77 + (z^2 - 1) * q^82 + (z^3 + z) * q^83 + (-z^3 - z) * q^86 - z^3 * q^89 + (-z^3 - z) * q^92 + (-z^2 - 1) * q^94 + (-z^2 + 1) * q^97 + (z^3 + z) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 4 q^{7} - 4 q^{16} + 4 q^{19} - 4 q^{22} - 4 q^{25} + 4 q^{28} - 4 q^{31} + 4 q^{37} + 4 q^{58} - 4 q^{64} + 4 q^{76} - 4 q^{82} - 4 q^{94} + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 4 * q^7 - 4 * q^16 + 4 * q^19 - 4 * q^22 - 4 * q^25 + 4 * q^28 - 4 * q^31 + 4 * q^37 + 4 * q^58 - 4 * q^64 + 4 * q^76 - 4 * q^82 - 4 * q^94 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1107\mathbb{Z}\right)^\times$.

$n$	$83$	$703$
$\chi(n)$	$-1$	$-\zeta_{8}^{2}$

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} $

296.1

−0.707107	−	0.707107i
0.707107	+	0.707107i
−0.707107	+	0.707107i
0.707107	−	0.707107i

−1.41421

1.00000

1.00000i

296.2

1.41421

1.00000

1.00000i

647.1

−1.41421

1.00000

−

1.00000i

647.2

1.41421

1.00000

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
41.c	even	4	1	inner
123.f	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1107.1.g.b	✓	4
3.b	odd	2	1	inner	1107.1.g.b	✓	4
9.c	even	3	2		3321.1.r.a		8
9.d	odd	6	2		3321.1.r.a		8
41.c	even	4	1	inner	1107.1.g.b	✓	4
123.f	odd	4	1	inner	1107.1.g.b	✓	4
369.q	odd	12	2		3321.1.r.a		8
369.r	even	12	2		3321.1.r.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1107.1.g.b	✓	4	1.a	even	1	1	trivial
1107.1.g.b	✓	4	3.b	odd	2	1	inner
1107.1.g.b	✓	4	41.c	even	4	1	inner
1107.1.g.b	✓	4	123.f	odd	4	1	inner
3321.1.r.a		8	9.c	even	3	2
3321.1.r.a		8	9.d	odd	6	2
3321.1.r.a		8	369.q	odd	12	2
3321.1.r.a		8	369.r	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} - 2 $ T2^2 - 2 acting on $S_{1}^{\mathrm{new}}(1107, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ (T^{2} - 2 T + 2)^{2} $ (T^2 - 2*T + 2)^2
$11$	$ T^{4} + 1 $ T^4 + 1
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ (T^{2} - 2 T + 2)^{2} $ (T^2 - 2*T + 2)^2
$23$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$29$	$ T^{4} + 1 $ T^4 + 1
$31$	$ (T + 1)^{4} $ (T + 1)^4
$37$	$ (T - 1)^{4} $ (T - 1)^4
$41$	$ T^{4} + 1 $ T^4 + 1
$43$	$ (T^{2} + 1)^{2} $ (T^2 + 1)^2
$47$	$ T^{4} + 1 $ T^4 + 1
$53$	$ T^{4} + 1 $ T^4 + 1
$59$	$ T^{4} $ T^4
$61$	$ (T^{2} + 1)^{2} $ (T^2 + 1)^2
$67$	$ T^{4} $ T^4
$71$	$ T^{4} $ T^4
$73$	$ (T^{2} + 1)^{2} $ (T^2 + 1)^2
$79$	$ T^{4} $ T^4
$83$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$89$	$ T^{4} + 1 $ T^4 + 1
$97$	$ (T^{2} - 2 T + 2)^{2} $ (T^2 - 2*T + 2)^2
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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:	\( N \)	\(=\)	\( 1107 = 3^{3} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1107.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{2} + q^{4} + (\zeta_{8}^{2} + 1) q^{7} + \zeta_{8}^{3} q^{11} + 2 \zeta_{8} q^{14} - q^{16} + ( - \zeta_{8}^{2} + 1) q^{19} + (\zeta_{8}^{2} - 1) q^{22} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{23}+ \cdots + (\zeta_{8}^{3} + \zeta_{8}) q^{98}+O(q^{100}) \) q + (-z^3 + z) * q^2 + q^4 + (z^2 + 1) * q^7 + z^3 * q^11 + 2z q^14 - q^16 + (-z^2 + 1) * q^19 + (z^2 - 1) * q^22 + (-z^3 - z) * q^23 - q^25 + (z^2 + 1) * q^28 - z^3 * q^29 - q^31 + (z^3 - z) * q^32 + q^37 - 2z^3 q^38 + z^3 * q^41 - z^2 * q^43 + z^3 * q^44 - 2z^2 q^46 - z * q^47 + z^2 * q^49 + (z^3 - z) * q^50 + z^3 * q^53 + (-z^2 + 1) * q^58 + z^2 * q^61 + (z^3 - z) * q^62 - q^64 + z^2 * q^73 + (-z^3 + z) * q^74 + (-z^2 + 1) * q^76 + (z^3 - z) * q^77 + (z^2 - 1) * q^82 + (z^3 + z) * q^83 + (-z^3 - z) * q^86 - z^3 * q^89 + (-z^3 - z) * q^92 + (-z^2 - 1) * q^94 + (-z^2 + 1) * q^97 + (z^3 + z) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 4 q^{7} - 4 q^{16} + 4 q^{19} - 4 q^{22} - 4 q^{25} + 4 q^{28} - 4 q^{31} + 4 q^{37} + 4 q^{58} - 4 q^{64} + 4 q^{76} - 4 q^{82} - 4 q^{94} + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 4 * q^7 - 4 * q^16 + 4 * q^19 - 4 * q^22 - 4 * q^25 + 4 * q^28 - 4 * q^31 + 4 * q^37 + 4 * q^58 - 4 * q^64 + 4 * q^76 - 4 * q^82 - 4 * q^94 + 4 * q^97

Introduction

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Database

Properties

Related objects

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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	1107.1.g.b

Level	$1107$
Weight	$1$
Character orbit	1107.g
Analytic conductor	$0.552$
Analytic rank	$0$
Dimension	$4$
Projective image	$S_{4}$
CM/RM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.552464968985\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{8})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 1 \) x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.2.1860867.2

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$3$	\( T^{4} \) T^4
$5$	\( T^{4} \) T^4
$7$	\( (T^{2} - 2 T + 2)^{2} \) (T^2 - 2*T + 2)^2
$11$	\( T^{4} + 1 \) T^4 + 1
$13$	\( T^{4} \) T^4
$17$	\( T^{4} \) T^4
$19$	\( (T^{2} - 2 T + 2)^{2} \) (T^2 - 2*T + 2)^2
$23$	\( (T^{2} + 2)^{2} \) (T^2 + 2)^2
$29$	\( T^{4} + 1 \) T^4 + 1
$31$	\( (T + 1)^{4} \) (T + 1)^4
$37$	\( (T - 1)^{4} \) (T - 1)^4
$41$	\( T^{4} + 1 \) T^4 + 1
$43$	\( (T^{2} + 1)^{2} \) (T^2 + 1)^2
$47$	\( T^{4} + 1 \) T^4 + 1
$53$	\( T^{4} + 1 \) T^4 + 1
$59$	\( T^{4} \) T^4
$61$	\( (T^{2} + 1)^{2} \) (T^2 + 1)^2
$67$	\( T^{4} \) T^4
$71$	\( T^{4} \) T^4
$73$	\( (T^{2} + 1)^{2} \) (T^2 + 1)^2
$79$	\( T^{4} \) T^4
$83$	\( (T^{2} + 2)^{2} \) (T^2 + 2)^2
$89$	\( T^{4} + 1 \) T^4 + 1
$97$	\( (T^{2} - 2 T + 2)^{2} \) (T^2 - 2*T + 2)^2
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\(n\)	\(83\)	\(703\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{8}^{2}\)

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