LMFDB - Newform orbit 784.2.f.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,2,Mod(783,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.783"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$784 = 2^{4} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	784.f (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.26027151847$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $\beta = \sqrt{-3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - q^{3} + \beta q^{5} - 2 q^{9} - \beta q^{11} - \beta q^{15} + 3 \beta q^{17} - 7 q^{19} - 5 \beta q^{23} + 2 q^{25} + 5 q^{27} - 6 q^{29} - 5 q^{31} + \beta q^{33} - 5 q^{37} - 4 \beta q^{41} - 2 \beta q^{43} + \cdots + 2 \beta q^{99} +O(q^{100})$ q - q^3 + b * q^5 - 2 * q^9 - b * q^11 - b * q^15 + 3b q^17 - 7 * q^19 - 5b q^23 + 2 * q^25 + 5 * q^27 - 6 * q^29 - 5 * q^31 + b * q^33 - 5 * q^37 - 4b q^41 - 2b q^43 - 2b q^45 - 3 * q^47 - 3b q^51 - 9 * q^53 + 3 * q^55 + 7 * q^57 - 9 * q^59 + 5b q^61 + 3b q^67 + 5b q^69 + 2b q^71 - b * q^73 - 2 * q^75 + 3b q^79 + q^81 - 12 * q^83 - 9 * q^85 + 6 * q^87 - 7b q^89 + 5 * q^93 - 7b q^95 - 4b q^97 + 2b q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{3} - 4 q^{9} - 14 q^{19} + 4 q^{25} + 10 q^{27} - 12 q^{29} - 10 q^{31} - 10 q^{37} - 6 q^{47} - 18 q^{53} + 6 q^{55} + 14 q^{57} - 18 q^{59} - 4 q^{75} + 2 q^{81} - 24 q^{83} - 18 q^{85} + 12 q^{87}+ \cdots + 10 q^{93}+O(q^{100})$ 2 * q - 2 * q^3 - 4 * q^9 - 14 * q^19 + 4 * q^25 + 10 * q^27 - 12 * q^29 - 10 * q^31 - 10 * q^37 - 6 * q^47 - 18 * q^53 + 6 * q^55 + 14 * q^57 - 18 * q^59 - 4 * q^75 + 2 * q^81 - 24 * q^83 - 18 * q^85 + 12 * q^87 + 10 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$ .

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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}

783.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

−2.00000

783.2

−1.00000

1.73205i

−2.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.2.f.a		2
3.b	odd	2	1		7056.2.b.b		2
4.b	odd	2	1		784.2.f.b		2
7.b	odd	2	1		784.2.f.b		2
7.c	even	3	1		112.2.p.b	yes	2
7.c	even	3	1		784.2.p.d		2
7.d	odd	6	1		112.2.p.a	✓	2
7.d	odd	6	1		784.2.p.c		2
8.b	even	2	1		3136.2.f.b		2
8.d	odd	2	1		3136.2.f.a		2
12.b	even	2	1		7056.2.b.m		2
21.c	even	2	1		7056.2.b.m		2
21.g	even	6	1		1008.2.cs.f		2
21.h	odd	6	1		1008.2.cs.c		2
28.d	even	2	1	inner	784.2.f.a		2
28.f	even	6	1		112.2.p.b	yes	2
28.f	even	6	1		784.2.p.d		2
28.g	odd	6	1		112.2.p.a	✓	2
28.g	odd	6	1		784.2.p.c		2
56.e	even	2	1		3136.2.f.b		2
56.h	odd	2	1		3136.2.f.a		2
56.j	odd	6	1		448.2.p.b		2
56.k	odd	6	1		448.2.p.b		2
56.m	even	6	1		448.2.p.a		2
56.p	even	6	1		448.2.p.a		2
84.h	odd	2	1		7056.2.b.b		2
84.j	odd	6	1		1008.2.cs.c		2
84.n	even	6	1		1008.2.cs.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.p.a	✓	2	7.d	odd	6	1
112.2.p.a	✓	2	28.g	odd	6	1
112.2.p.b	yes	2	7.c	even	3	1
112.2.p.b	yes	2	28.f	even	6	1
448.2.p.a		2	56.m	even	6	1
448.2.p.a		2	56.p	even	6	1
448.2.p.b		2	56.j	odd	6	1
448.2.p.b		2	56.k	odd	6	1
784.2.f.a		2	1.a	even	1	1	trivial
784.2.f.a		2	28.d	even	2	1	inner
784.2.f.b		2	4.b	odd	2	1
784.2.f.b		2	7.b	odd	2	1
784.2.p.c		2	7.d	odd	6	1
784.2.p.c		2	28.g	odd	6	1
784.2.p.d		2	7.c	even	3	1
784.2.p.d		2	28.f	even	6	1
1008.2.cs.c		2	21.h	odd	6	1
1008.2.cs.c		2	84.j	odd	6	1
1008.2.cs.f		2	21.g	even	6	1
1008.2.cs.f		2	84.n	even	6	1
3136.2.f.a		2	8.d	odd	2	1
3136.2.f.a		2	56.h	odd	2	1
3136.2.f.b		2	8.b	even	2	1
3136.2.f.b		2	56.e	even	2	1
7056.2.b.b		2	3.b	odd	2	1
7056.2.b.b		2	84.h	odd	2	1
7056.2.b.m		2	12.b	even	2	1
7056.2.b.m		2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 1$ T3 + 1 acting on $S_{2}^{\mathrm{new}}(784, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T + 1)^{2}$ (T + 1)^2
$5$	$T^{2} + 3$ T^2 + 3
$7$	$T^{2}$ T^2
$11$	$T^{2} + 3$ T^2 + 3
$13$	$T^{2}$ T^2
$17$	$T^{2} + 27$ T^2 + 27
$19$	$(T + 7)^{2}$ (T + 7)^2
$23$	$T^{2} + 75$ T^2 + 75
$29$	$(T + 6)^{2}$ (T + 6)^2
$31$	$(T + 5)^{2}$ (T + 5)^2
$37$	$(T + 5)^{2}$ (T + 5)^2
$41$	$T^{2} + 48$ T^2 + 48
$43$	$T^{2} + 12$ T^2 + 12
$47$	$(T + 3)^{2}$ (T + 3)^2
$53$	$(T + 9)^{2}$ (T + 9)^2
$59$	$(T + 9)^{2}$ (T + 9)^2
$61$	$T^{2} + 75$ T^2 + 75
$67$	$T^{2} + 27$ T^2 + 27
$71$	$T^{2} + 12$ T^2 + 12
$73$	$T^{2} + 3$ T^2 + 3
$79$	$T^{2} + 27$ T^2 + 27
$83$	$(T + 12)^{2}$ (T + 12)^2
$89$	$T^{2} + 147$ T^2 + 147
$97$	$T^{2} + 48$ T^2 + 48
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 $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more