LMFDB - Newform orbit 336.6.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.a (trivial)

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:	$53.8889634572$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - 9 q^{3} + 26 q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10}) $

$ q - 9 q^{3} + 26 q^{5} + 49 q^{7} + 81 q^{9} - 664 q^{11} + 318 q^{13} - 234 q^{15} + 1582 q^{17} - 236 q^{19} - 441 q^{21} - 2212 q^{23} - 2449 q^{25} - 729 q^{27} - 4954 q^{29} + 7128 q^{31} + 5976 q^{33} + 1274 q^{35} + 4358 q^{37} - 2862 q^{39} + 10542 q^{41} + 8452 q^{43} + 2106 q^{45} - 5352 q^{47} + 2401 q^{49} - 14238 q^{51} - 33354 q^{53} - 17264 q^{55} + 2124 q^{57} + 15436 q^{59} - 36762 q^{61} + 3969 q^{63} + 8268 q^{65} - 40972 q^{67} + 19908 q^{69} + 9092 q^{71} - 73454 q^{73} + 22041 q^{75} - 32536 q^{77} - 89400 q^{79} + 6561 q^{81} + 6428 q^{83} + 41132 q^{85} + 44586 q^{87} - 122658 q^{89} + 15582 q^{91} - 64152 q^{93} - 6136 q^{95} + 21370 q^{97} - 53784 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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} $

1.1

−9.00000

26.0000

49.0000

81.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$7$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.6.a.h		1
3.b	odd	2	1		1008.6.a.j		1
4.b	odd	2	1		42.6.a.d	✓	1
12.b	even	2	1		126.6.a.i		1
20.d	odd	2	1		1050.6.a.k		1
20.e	even	4	2		1050.6.g.i		2
28.d	even	2	1		294.6.a.b		1
28.f	even	6	2		294.6.e.p		2
28.g	odd	6	2		294.6.e.i		2
84.h	odd	2	1		882.6.a.s		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.d	✓	1	4.b	odd	2	1
126.6.a.i		1	12.b	even	2	1
294.6.a.b		1	28.d	even	2	1
294.6.e.i		2	28.g	odd	6	2
294.6.e.p		2	28.f	even	6	2
336.6.a.h		1	1.a	even	1	1	trivial
882.6.a.s		1	84.h	odd	2	1
1008.6.a.j		1	3.b	odd	2	1
1050.6.a.k		1	20.d	odd	2	1
1050.6.g.i		2	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(336))$:

$ T_{5} - 26 $

$ T_{11} + 664 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 9 $ T + 9
$5$	$ T - 26 $ T - 26
$7$	$ T - 49 $ T - 49
$11$	$ T + 664 $ T + 664
$13$	$ T - 318 $ T - 318
$17$	$ T - 1582 $ T - 1582
$19$	$ T + 236 $ T + 236
$23$	$ T + 2212 $ T + 2212
$29$	$ T + 4954 $ T + 4954
$31$	$ T - 7128 $ T - 7128
$37$	$ T - 4358 $ T - 4358
$41$	$ T - 10542 $ T - 10542
$43$	$ T - 8452 $ T - 8452
$47$	$ T + 5352 $ T + 5352
$53$	$ T + 33354 $ T + 33354
$59$	$ T - 15436 $ T - 15436
$61$	$ T + 36762 $ T + 36762
$67$	$ T + 40972 $ T + 40972
$71$	$ T - 9092 $ T - 9092
$73$	$ T + 73454 $ T + 73454
$79$	$ T + 89400 $ T + 89400
$83$	$ T - 6428 $ T - 6428
$89$	$ T + 122658 $ T + 122658
$97$	$ T - 21370 $ T - 21370
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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

\(n\):		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