LMFDB - Newform orbit 387.3.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,3,Mod(343,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("387.343");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	387.b (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:	$10.5449862307$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + 4 q^{4}+O(q^{10}) $

$ q + 4 q^{4} + 21 q^{11} - 17 q^{13} + 16 q^{16} + 9 q^{17} - 3 q^{23} + 25 q^{25} + 19 q^{31} - 39 q^{41} - 43 q^{43} + 84 q^{44} + 78 q^{47} + 49 q^{49} - 68 q^{52} - 63 q^{53} + 54 q^{59} + 64 q^{64} + 91 q^{67} + 36 q^{68} - 14 q^{79} - 123 q^{83} - 12 q^{92} - 193 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$.

$n$	$46$	$173$
$\chi(n)$	$-1$	$1$

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} $

343.1

4.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
43.b	odd	2	1	CM by $\Q(\sqrt{-43}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	387.3.b.a		1
3.b	odd	2	1		43.3.b.a	✓	1
12.b	even	2	1		688.3.b.a		1
43.b	odd	2	1	CM	387.3.b.a		1
129.d	even	2	1		43.3.b.a	✓	1
516.h	odd	2	1		688.3.b.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
43.3.b.a	✓	1	3.b	odd	2	1
43.3.b.a	✓	1	129.d	even	2	1
387.3.b.a		1	1.a	even	1	1	trivial
387.3.b.a		1	43.b	odd	2	1	CM
688.3.b.a		1	12.b	even	2	1
688.3.b.a		1	516.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(387, [\chi])$:

$ T_{2} $

$ T_{11} - 21 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 21 $ T - 21
$13$	$ T + 17 $ T + 17
$17$	$ T - 9 $ T - 9
$19$	$ T $ T
$23$	$ T + 3 $ T + 3
$29$	$ T $ T
$31$	$ T - 19 $ T - 19
$37$	$ T $ T
$41$	$ T + 39 $ T + 39
$43$	$ T + 43 $ T + 43
$47$	$ T - 78 $ T - 78
$53$	$ T + 63 $ T + 63
$59$	$ T - 54 $ T - 54
$61$	$ T $ T
$67$	$ T - 91 $ T - 91
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T + 14 $ T + 14
$83$	$ T + 123 $ T + 123
$89$	$ T $ T
$97$	$ T + 193 $ T + 193
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Overview	Random
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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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	387.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(46\)	\(173\)
\(\chi(n)\)	\(-1\)	\(1\)

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