LMFDB - Newform orbit 119.1.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [119,1,Mod(118,119)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(119, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("119.118");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 119 = 7 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	119.d (of order $2$, degree $1$, 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:	$0.0593887365033$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.14161.1
Artin image:	$D_5$
Artin field:	Galois closure of 5.1.14161.1

$q$-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^{2} + (\beta - 1) q^{3} + ( - \beta + 1) q^{4} - \beta q^{5} + ( - \beta + 2) q^{6} + q^{7} - q^{8} + ( - \beta + 1) q^{9} - q^{10} + (\beta - 2) q^{12} + (\beta - 1) q^{14} - q^{15} + \cdots + (\beta - 1) q^{98} +O(q^{100}) $ q + (b - 1) * q^2 + (b - 1) * q^3 + (-b + 1) * q^4 - b * q^5 + (-b + 2) * q^6 + q^7 - q^8 + (-b + 1) * q^9 - q^10 + (b - 2) * q^12 + (b - 1) * q^14 - q^15 + q^17 + (b - 2) * q^18 + q^20 + (b - 1) * q^21 + (-b + 1) * q^24 + b * q^25 - q^27 + (-b + 1) * q^28 + (-b + 1) * q^30 - b * q^31 + q^32 + (b - 1) * q^34 - b * q^35 + (-b + 2) * q^36 + b * q^40 + (b - 1) * q^41 + (-b + 2) * q^42 + (b - 1) * q^43 + q^45 + q^49 + q^50 + (b - 1) * q^51 - b * q^53 + (-b + 1) * q^54 - q^56 + (b - 1) * q^60 + (b - 1) * q^61 - q^62 + (-b + 1) * q^63 + (b - 1) * q^64 - b * q^67 + (-b + 1) * q^68 - q^70 + (b - 1) * q^72 + (b - 1) * q^73 + q^75 + (-b + 2) * q^82 + (b - 2) * q^84 - b * q^85 + (-b + 2) * q^86 + (b - 1) * q^90 - q^93 + (b - 1) * q^96 - b * q^97 + (b - 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 2 q^{7} - 2 q^{8} + q^{9} - 2 q^{10} - 3 q^{12} - q^{14} - 2 q^{15} + 2 q^{17} - 3 q^{18} + 2 q^{20} - q^{21} + q^{24} + q^{25} - 2 q^{27} + q^{28}+ \cdots - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 2 * q^7 - 2 * q^8 + q^9 - 2 * q^10 - 3 * q^12 - q^14 - 2 * q^15 + 2 * q^17 - 3 * q^18 + 2 * q^20 - q^21 + q^24 + q^25 - 2 * q^27 + q^28 + q^30 - q^31 + 2 * q^32 - q^34 - q^35 + 3 * q^36 + q^40 - q^41 + 3 * q^42 - q^43 + 2 * q^45 + 2 * q^49 + 2 * q^50 - q^51 - q^53 + q^54 - 2 * q^56 - q^60 - q^61 - 2 * q^62 + q^63 - q^64 - q^67 + q^68 - 2 * q^70 - q^72 - q^73 + 2 * q^75 + 3 * q^82 - 3 * q^84 - q^85 + 3 * q^86 - q^90 - 2 * q^93 - q^96 - q^97 - q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/119\mathbb{Z}\right)^\times$.

$n$	$52$	$71$
$\chi(n)$	$-1$	$-1$

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} $

118.1

−0.618034
1.61803

−1.61803

1.61803

0.618034

2.61803

1.00000

−1.00000

1.61803

−1.00000

118.2

0.618034

−0.618034

−1.61803

0.381966

1.00000

−1.00000

−0.618034

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
119.d	odd	2	1	CM by $\Q(\sqrt{-119}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	119.1.d.a	✓	2
3.b	odd	2	1		1071.1.h.b		2
4.b	odd	2	1		1904.1.n.b		2
5.b	even	2	1		2975.1.h.d		2
5.c	odd	4	2		2975.1.b.b		4
7.b	odd	2	1		119.1.d.b	yes	2
7.c	even	3	2		833.1.h.b		4
7.d	odd	6	2		833.1.h.a		4
17.b	even	2	1		119.1.d.b	yes	2
17.c	even	4	2		2023.1.c.e		4
17.d	even	8	4		2023.1.f.b		8
17.e	odd	16	8		2023.1.l.b		16
21.c	even	2	1		1071.1.h.a		2
28.d	even	2	1		1904.1.n.a		2
35.c	odd	2	1		2975.1.h.c		2
35.f	even	4	2		2975.1.b.a		4
51.c	odd	2	1		1071.1.h.a		2
68.d	odd	2	1		1904.1.n.a		2
85.c	even	2	1		2975.1.h.c		2
85.g	odd	4	2		2975.1.b.a		4
119.d	odd	2	1	CM	119.1.d.a	✓	2
119.f	odd	4	2		2023.1.c.e		4
119.h	odd	6	2		833.1.h.b		4
119.j	even	6	2		833.1.h.a		4
119.l	odd	8	4		2023.1.f.b		8
119.p	even	16	8		2023.1.l.b		16
357.c	even	2	1		1071.1.h.b		2
476.e	even	2	1		1904.1.n.b		2
595.b	odd	2	1		2975.1.h.d		2
595.p	even	4	2		2975.1.b.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
119.1.d.a	✓	2	1.a	even	1	1	trivial
119.1.d.a	✓	2	119.d	odd	2	1	CM
119.1.d.b	yes	2	7.b	odd	2	1
119.1.d.b	yes	2	17.b	even	2	1
833.1.h.a		4	7.d	odd	6	2
833.1.h.a		4	119.j	even	6	2
833.1.h.b		4	7.c	even	3	2
833.1.h.b		4	119.h	odd	6	2
1071.1.h.a		2	21.c	even	2	1
1071.1.h.a		2	51.c	odd	2	1
1071.1.h.b		2	3.b	odd	2	1
1071.1.h.b		2	357.c	even	2	1
1904.1.n.a		2	28.d	even	2	1
1904.1.n.a		2	68.d	odd	2	1
1904.1.n.b		2	4.b	odd	2	1
1904.1.n.b		2	476.e	even	2	1
2023.1.c.e		4	17.c	even	4	2
2023.1.c.e		4	119.f	odd	4	2
2023.1.f.b		8	17.d	even	8	4
2023.1.f.b		8	119.l	odd	8	4
2023.1.l.b		16	17.e	odd	16	8
2023.1.l.b		16	119.p	even	16	8
2975.1.b.a		4	35.f	even	4	2
2975.1.b.a		4	85.g	odd	4	2
2975.1.b.b		4	5.c	odd	4	2
2975.1.b.b		4	595.p	even	4	2
2975.1.h.c		2	35.c	odd	2	1
2975.1.h.c		2	85.c	even	2	1
2975.1.h.d		2	5.b	even	2	1
2975.1.h.d		2	595.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} + T_{3} - 1 $ T3^2 + T3 - 1 acting on $S_{1}^{\mathrm{new}}(119, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + T - 1 $ T^2 + T - 1
$3$	$ T^{2} + T - 1 $ T^2 + T - 1
$5$	$ T^{2} + T - 1 $ T^2 + T - 1
$7$	$ (T - 1)^{2} $ (T - 1)^2
$11$	$ T^{2} $ T^2
$13$	$ T^{2} $ T^2
$17$	$ (T - 1)^{2} $ (T - 1)^2
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} $ T^2
$31$	$ T^{2} + T - 1 $ T^2 + T - 1
$37$	$ T^{2} $ T^2
$41$	$ T^{2} + T - 1 $ T^2 + T - 1
$43$	$ T^{2} + T - 1 $ T^2 + T - 1
$47$	$ T^{2} $ T^2
$53$	$ T^{2} + T - 1 $ T^2 + T - 1
$59$	$ T^{2} $ T^2
$61$	$ T^{2} + T - 1 $ T^2 + T - 1
$67$	$ T^{2} + T - 1 $ T^2 + T - 1
$71$	$ T^{2} $ T^2
$73$	$ T^{2} + T - 1 $ T^2 + T - 1
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ T^{2} + T - 1 $ T^2 + T - 1
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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:	\( N \)	\(=\)	\( 119 = 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	119.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{2} + (\beta - 1) q^{3} + ( - \beta + 1) q^{4} - \beta q^{5} + ( - \beta + 2) q^{6} + q^{7} - q^{8} + ( - \beta + 1) q^{9} - q^{10} + (\beta - 2) q^{12} + (\beta - 1) q^{14} - q^{15} + \cdots + (\beta - 1) q^{98} +O(q^{100}) \) q + (b - 1) * q^2 + (b - 1) * q^3 + (-b + 1) * q^4 - b * q^5 + (-b + 2) * q^6 + q^7 - q^8 + (-b + 1) * q^9 - q^10 + (b - 2) * q^12 + (b - 1) * q^14 - q^15 + q^17 + (b - 2) * q^18 + q^20 + (b - 1) * q^21 + (-b + 1) * q^24 + b * q^25 - q^27 + (-b + 1) * q^28 + (-b + 1) * q^30 - b * q^31 + q^32 + (b - 1) * q^34 - b * q^35 + (-b + 2) * q^36 + b * q^40 + (b - 1) * q^41 + (-b + 2) * q^42 + (b - 1) * q^43 + q^45 + q^49 + q^50 + (b - 1) * q^51 - b * q^53 + (-b + 1) * q^54 - q^56 + (b - 1) * q^60 + (b - 1) * q^61 - q^62 + (-b + 1) * q^63 + (b - 1) * q^64 - b * q^67 + (-b + 1) * q^68 - q^70 + (b - 1) * q^72 + (b - 1) * q^73 + q^75 + (-b + 2) * q^82 + (b - 2) * q^84 - b * q^85 + (-b + 2) * q^86 + (b - 1) * q^90 - q^93 + (b - 1) * q^96 - b * q^97 + (b - 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 2 q^{7} - 2 q^{8} + q^{9} - 2 q^{10} - 3 q^{12} - q^{14} - 2 q^{15} + 2 q^{17} - 3 q^{18} + 2 q^{20} - q^{21} + q^{24} + q^{25} - 2 q^{27} + q^{28}+ \cdots - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 2 * q^7 - 2 * q^8 + q^9 - 2 * q^10 - 3 * q^12 - q^14 - 2 * q^15 + 2 * q^17 - 3 * q^18 + 2 * q^20 - q^21 + q^24 + q^25 - 2 * q^27 + q^28 + q^30 - q^31 + 2 * q^32 - q^34 - q^35 + 3 * q^36 + q^40 - q^41 + 3 * q^42 - q^43 + 2 * q^45 + 2 * q^49 + 2 * q^50 - q^51 - q^53 + q^54 - 2 * q^56 - q^60 - q^61 - 2 * q^62 + q^63 - q^64 - q^67 + q^68 - 2 * q^70 - q^72 - q^73 + 2 * q^75 + 3 * q^82 - 3 * q^84 - q^85 + 3 * q^86 - q^90 - 2 * q^93 - q^96 - q^97 - q^98

Introduction

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Varieties

Fields

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Groups

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	119.1.d.a

Level	$119$
Weight	$1$
Character orbit	119.d
Self dual	yes
Analytic conductor	$0.059$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{5}$
CM discriminant	-119
Inner twists	$2$

Self dual:	yes
Analytic conductor:	\(0.0593887365033\)
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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.14161.1
Artin image:	$D_5$
Artin field:	Galois closure of 5.1.14161.1

$p$	$F_p(T)$
$2$	\( T^{2} + T - 1 \) T^2 + T - 1
$3$	\( T^{2} + T - 1 \) T^2 + T - 1
$5$	\( T^{2} + T - 1 \) T^2 + T - 1
$7$	\( (T - 1)^{2} \) (T - 1)^2
$11$	\( T^{2} \) T^2
$13$	\( T^{2} \) T^2
$17$	\( (T - 1)^{2} \) (T - 1)^2
$19$	\( T^{2} \) T^2
$23$	\( T^{2} \) T^2
$29$	\( T^{2} \) T^2
$31$	\( T^{2} + T - 1 \) T^2 + T - 1
$37$	\( T^{2} \) T^2
$41$	\( T^{2} + T - 1 \) T^2 + T - 1
$43$	\( T^{2} + T - 1 \) T^2 + T - 1
$47$	\( T^{2} \) T^2
$53$	\( T^{2} + T - 1 \) T^2 + T - 1
$59$	\( T^{2} \) T^2
$61$	\( T^{2} + T - 1 \) T^2 + T - 1
$67$	\( T^{2} + T - 1 \) T^2 + T - 1
$71$	\( T^{2} \) T^2
$73$	\( T^{2} + T - 1 \) T^2 + T - 1
$79$	\( T^{2} \) T^2
$83$	\( T^{2} \) T^2
$89$	\( T^{2} \) T^2
$97$	\( T^{2} + T - 1 \) T^2 + T - 1
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\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(-1\)	\(-1\)

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