LMFDB - Newform orbit 752.1.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [752,1,Mod(657,752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(752, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("752.657");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 752 = 2^{4} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	752.g (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$0.375296889500$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 47)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.2209.1
Artin image:	$D_{10}$
Artin field:	Galois closure of 10.0.4996793344.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{5})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta + 1) q^{3} + \beta q^{7} + ( - \beta + 1) q^{9}+O(q^{10}) $ q + (-b + 1) * q^3 + b * q^7 + (-b + 1) * q^9	$ q + ( - \beta + 1) q^{3} + \beta q^{7} + ( - \beta + 1) q^{9} + (\beta - 1) q^{17} - q^{21} + q^{25} + q^{27} + (\beta - 1) q^{37} - q^{47} + \beta q^{49} + (\beta - 2) q^{51} - \beta q^{53} + \beta q^{59} - \beta q^{61} - q^{63} + ( - \beta + 1) q^{71} + ( - \beta + 1) q^{75} + ( - \beta + 1) q^{79} - 2 q^{83} - \beta q^{89} - \beta q^{97} +O(q^{100}) $ q + (-b + 1) * q^3 + b * q^7 + (-b + 1) * q^9 + (b - 1) * q^17 - q^21 + q^25 + q^27 + (b - 1) * q^37 - q^47 + b * q^49 + (b - 2) * q^51 - b * q^53 + b * q^59 - b * q^61 - q^63 + (-b + 1) * q^71 + (-b + 1) * q^75 + (-b + 1) * q^79 - 2 * q^83 - b * q^89 - b * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{7} + q^{9}+O(q^{10}) $ 2 * q + q^3 + q^7 + q^9	$ 2 q + q^{3} + q^{7} + q^{9} - q^{17} - 2 q^{21} + 2 q^{25} + 2 q^{27} - q^{37} - 2 q^{47} + q^{49} - 3 q^{51} - q^{53} + q^{59} - q^{61} - 2 q^{63} + q^{71} + q^{75} + q^{79} - 4 q^{83} - q^{89} - q^{97}+O(q^{100}) $ 2 * q + q^3 + q^7 + q^9 - q^17 - 2 * q^21 + 2 * q^25 + 2 * q^27 - q^37 - 2 * q^47 + q^49 - 3 * q^51 - q^53 + q^59 - q^61 - 2 * q^63 + q^71 + q^75 + q^79 - 4 * q^83 - q^89 - q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/752\mathbb{Z}\right)^\times$.

$n$	$95$	$193$	$565$
$\chi(n)$	$1$	$-1$	$1$

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} $

657.1

1.61803
−0.618034

−0.618034

1.61803

−0.618034

657.2

1.61803

−0.618034

1.61803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
47.b	odd	2	1	CM by $\Q(\sqrt{-47}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	752.1.g.a		2
4.b	odd	2	1		47.1.b.a	✓	2
8.b	even	2	1		3008.1.g.a		2
8.d	odd	2	1		3008.1.g.b		2
12.b	even	2	1		423.1.d.a		2
20.d	odd	2	1		1175.1.d.c		2
20.e	even	4	2		1175.1.b.b		4
28.d	even	2	1		2303.1.d.c		2
28.f	even	6	2		2303.1.f.b		4
28.g	odd	6	2		2303.1.f.c		4
36.f	odd	6	2		3807.1.f.b		4
36.h	even	6	2		3807.1.f.a		4
47.b	odd	2	1	CM	752.1.g.a		2
188.b	even	2	1		47.1.b.a	✓	2
188.g	odd	46	22		2209.1.d.a		44
188.h	even	46	22		2209.1.d.a		44
376.e	odd	2	1		3008.1.g.a		2
376.h	even	2	1		3008.1.g.b		2
564.f	odd	2	1		423.1.d.a		2
940.h	even	2	1		1175.1.d.c		2
940.l	odd	4	2		1175.1.b.b		4
1316.h	odd	2	1		2303.1.d.c		2
1316.j	odd	6	2		2303.1.f.b		4
1316.o	even	6	2		2303.1.f.c		4
1692.l	odd	6	2		3807.1.f.a		4
1692.p	even	6	2		3807.1.f.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
47.1.b.a	✓	2	4.b	odd	2	1
47.1.b.a	✓	2	188.b	even	2	1
423.1.d.a		2	12.b	even	2	1
423.1.d.a		2	564.f	odd	2	1
752.1.g.a		2	1.a	even	1	1	trivial
752.1.g.a		2	47.b	odd	2	1	CM
1175.1.b.b		4	20.e	even	4	2
1175.1.b.b		4	940.l	odd	4	2
1175.1.d.c		2	20.d	odd	2	1
1175.1.d.c		2	940.h	even	2	1
2209.1.d.a		44	188.g	odd	46	22
2209.1.d.a		44	188.h	even	46	22
2303.1.d.c		2	28.d	even	2	1
2303.1.d.c		2	1316.h	odd	2	1
2303.1.f.b		4	28.f	even	6	2
2303.1.f.b		4	1316.j	odd	6	2
2303.1.f.c		4	28.g	odd	6	2
2303.1.f.c		4	1316.o	even	6	2
3008.1.g.a		2	8.b	even	2	1
3008.1.g.a		2	376.e	odd	2	1
3008.1.g.b		2	8.d	odd	2	1
3008.1.g.b		2	376.h	even	2	1
3807.1.f.a		4	36.h	even	6	2
3807.1.f.a		4	1692.l	odd	6	2
3807.1.f.b		4	36.f	odd	6	2
3807.1.f.b		4	1692.p	even	6	2

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(752, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} - T - 1 $ T^2 - T - 1
$5$	$ T^{2} $ T^2
$7$	$ T^{2} - T - 1 $ T^2 - T - 1
$11$	$ T^{2} $ T^2
$13$	$ T^{2} $ T^2
$17$	$ T^{2} + T - 1 $ T^2 + T - 1
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} $ T^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + T - 1 $ T^2 + T - 1
$41$	$ T^{2} $ T^2
$43$	$ T^{2} $ T^2
$47$	$ (T + 1)^{2} $ (T + 1)^2
$53$	$ T^{2} + T - 1 $ T^2 + T - 1
$59$	$ T^{2} - T - 1 $ T^2 - T - 1
$61$	$ T^{2} + T - 1 $ T^2 + T - 1
$67$	$ T^{2} $ T^2
$71$	$ T^{2} - T - 1 $ T^2 - T - 1
$73$	$ T^{2} $ T^2
$79$	$ T^{2} - T - 1 $ T^2 - T - 1
$83$	$ (T + 2)^{2} $ (T + 2)^2
$89$	$ T^{2} + T - 1 $ T^2 + T - 1
$97$	$ T^{2} + T - 1 $ T^2 + T - 1
show more
show less

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:	\( N \)	\(=\)	\( 752 = 2^{4} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	752.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta + 1) q^{3} + \beta q^{7} + ( - \beta + 1) q^{9}+O(q^{10}) \) q + (-b + 1) * q^3 + b * q^7 + (-b + 1) * q^9	\( q + ( - \beta + 1) q^{3} + \beta q^{7} + ( - \beta + 1) q^{9} + (\beta - 1) q^{17} - q^{21} + q^{25} + q^{27} + (\beta - 1) q^{37} - q^{47} + \beta q^{49} + (\beta - 2) q^{51} - \beta q^{53} + \beta q^{59} - \beta q^{61} - q^{63} + ( - \beta + 1) q^{71} + ( - \beta + 1) q^{75} + ( - \beta + 1) q^{79} - 2 q^{83} - \beta q^{89} - \beta q^{97} +O(q^{100}) \) q + (-b + 1) * q^3 + b * q^7 + (-b + 1) * q^9 + (b - 1) * q^17 - q^21 + q^25 + q^27 + (b - 1) * q^37 - q^47 + b * q^49 + (b - 2) * q^51 - b * q^53 + b * q^59 - b * q^61 - q^63 + (-b + 1) * q^71 + (-b + 1) * q^75 + (-b + 1) * q^79 - 2 * q^83 - b * q^89 - b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{7} + q^{9}+O(q^{10}) \) 2 * q + q^3 + q^7 + q^9	\( 2 q + q^{3} + q^{7} + q^{9} - q^{17} - 2 q^{21} + 2 q^{25} + 2 q^{27} - q^{37} - 2 q^{47} + q^{49} - 3 q^{51} - q^{53} + q^{59} - q^{61} - 2 q^{63} + q^{71} + q^{75} + q^{79} - 4 q^{83} - q^{89} - q^{97}+O(q^{100}) \) 2 * q + q^3 + q^7 + q^9 - q^17 - 2 * q^21 + 2 * q^25 + 2 * q^27 - q^37 - 2 * q^47 + q^49 - 3 * q^51 - q^53 + q^59 - q^61 - 2 * q^63 + q^71 + q^75 + q^79 - 4 * q^83 - q^89 - q^97

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	752.1.g.a

Level	$752$
Weight	$1$
Character orbit	752.g
Self dual	yes
Analytic conductor	$0.375$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{5}$
CM discriminant	-47
Inner twists	$2$

Self dual:	yes
Analytic conductor:	\(0.375296889500\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{2} - x - 1 \) x^2 - x - 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 47)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.2209.1
Artin image:	$D_{10}$
Artin field:	Galois closure of 10.0.4996793344.3

$p$	$F_p(T)$
$2$	\( T^{2} \) T^2
$3$	\( T^{2} - T - 1 \) T^2 - T - 1
$5$	\( T^{2} \) T^2
$7$	\( T^{2} - T - 1 \) T^2 - T - 1
$11$	\( T^{2} \) T^2
$13$	\( T^{2} \) T^2
$17$	\( T^{2} + T - 1 \) T^2 + T - 1
$19$	\( T^{2} \) T^2
$23$	\( T^{2} \) T^2
$29$	\( T^{2} \) T^2
$31$	\( T^{2} \) T^2
$37$	\( T^{2} + T - 1 \) T^2 + T - 1
$41$	\( T^{2} \) T^2
$43$	\( T^{2} \) T^2
$47$	\( (T + 1)^{2} \) (T + 1)^2
$53$	\( T^{2} + T - 1 \) T^2 + T - 1
$59$	\( T^{2} - T - 1 \) T^2 - T - 1
$61$	\( T^{2} + T - 1 \) T^2 + T - 1
$67$	\( T^{2} \) T^2
$71$	\( T^{2} - T - 1 \) T^2 - T - 1
$73$	\( T^{2} \) T^2
$79$	\( T^{2} - T - 1 \) T^2 - T - 1
$83$	\( (T + 2)^{2} \) (T + 2)^2
$89$	\( T^{2} + T - 1 \) T^2 + T - 1
$97$	\( T^{2} + T - 1 \) T^2 + T - 1
show more
show less

\(n\)	\(95\)	\(193\)	\(565\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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