LMFDB - Newform orbit 399.1.h.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [399,1,Mod(398,399)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("399.398"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	399.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$0.199126940041$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-3}, \sqrt{133})$
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.1197.1
Stark unit:	Root of $x^{4} - 9x^{3} - 11x^{2} - 9x + 1$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

398.1

−1.00000

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
133.c	even	2	1	RM by $\Q(\sqrt{133}) $
399.h	odd	2	1	CM by $\Q(\sqrt{-399}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	399.1.h.a	✓	1
3.b	odd	2	1	CM	399.1.h.a	✓	1
7.b	odd	2	1		399.1.h.b	yes	1
7.c	even	3	2		2793.1.s.b		2
7.d	odd	6	2		2793.1.s.a		2
19.b	odd	2	1		399.1.h.b	yes	1
21.c	even	2	1		399.1.h.b	yes	1
21.g	even	6	2		2793.1.s.a		2
21.h	odd	6	2		2793.1.s.b		2
57.d	even	2	1		399.1.h.b	yes	1
133.c	even	2	1	RM	399.1.h.a	✓	1
133.o	even	6	2		2793.1.s.b		2
133.r	odd	6	2		2793.1.s.a		2
399.h	odd	2	1	CM	399.1.h.a	✓	1
399.s	odd	6	2		2793.1.s.b		2
399.w	even	6	2		2793.1.s.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
399.1.h.a	✓	1	1.a	even	1	1	trivial
399.1.h.a	✓	1	3.b	odd	2	1	CM
399.1.h.a	✓	1	133.c	even	2	1	RM
399.1.h.a	✓	1	399.h	odd	2	1	CM
399.1.h.b	yes	1	7.b	odd	2	1
399.1.h.b	yes	1	19.b	odd	2	1
399.1.h.b	yes	1	21.c	even	2	1
399.1.h.b	yes	1	57.d	even	2	1
2793.1.s.a		2	7.d	odd	6	2
2793.1.s.a		2	21.g	even	6	2
2793.1.s.a		2	133.r	odd	6	2
2793.1.s.a		2	399.w	even	6	2
2793.1.s.b		2	7.c	even	3	2
2793.1.s.b		2	21.h	odd	6	2
2793.1.s.b		2	133.o	even	6	2
2793.1.s.b		2	399.s	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(399, [\chi])$:

$ T_{2} $

T2

$ T_{13} - 2 $

T13 - 2

$ T_{97} + 2 $

T97 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T - 1 $ T - 1
$11$	$ T $ T
$13$	$ T - 2 $ T - 2
$17$	$ T $ T
$19$	$ T + 1 $ T + 1
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T - 2 $ T - 2
$37$	$ T $ T
$41$	$ T $ T
$43$	$ T + 2 $ T + 2
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T + 2 $ T + 2
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Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	399.h (of order \(2\), degree \(1\), minimal)

\(n\)	\(115\)	\(134\)	\(211\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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