LMFDB - Newform orbit 756.1.h.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,1,Mod(755,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("756.755");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	756.h (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:	$0.377293149551$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.756.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.1714608.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8}+O(q^{10}) $

$ q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + q^{11} - q^{14} + q^{16} - 2 q^{17} - q^{19} + q^{20} - q^{22} + q^{23} + q^{28} - q^{31} - q^{32} + 2 q^{34} + q^{35} - q^{37} + q^{38} - q^{40} + q^{41} + q^{44} - q^{46} + q^{49} + q^{55} - q^{56} + q^{62} + q^{64} - 2 q^{68} - q^{70} + q^{71} + q^{74} - q^{76} + q^{77} + q^{80} - q^{82} - 2 q^{85} - q^{88} + q^{89} + q^{92} - q^{95} - q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

755.1

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
84.h	odd	2	1	CM by $\Q(\sqrt{-21}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.1.h.b	yes	1
3.b	odd	2	1		756.1.h.c	yes	1
4.b	odd	2	1		756.1.h.d	yes	1
7.b	odd	2	1		756.1.h.a	✓	1
9.c	even	3	2		2268.1.s.c		2
9.d	odd	6	2		2268.1.s.b		2
12.b	even	2	1		756.1.h.a	✓	1
21.c	even	2	1		756.1.h.d	yes	1
28.d	even	2	1		756.1.h.c	yes	1
36.f	odd	6	2		2268.1.s.a		2
36.h	even	6	2		2268.1.s.d		2
63.l	odd	6	2		2268.1.s.d		2
63.o	even	6	2		2268.1.s.a		2
84.h	odd	2	1	CM	756.1.h.b	yes	1
252.s	odd	6	2		2268.1.s.c		2
252.bi	even	6	2		2268.1.s.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.1.h.a	✓	1	7.b	odd	2	1
756.1.h.a	✓	1	12.b	even	2	1
756.1.h.b	yes	1	1.a	even	1	1	trivial
756.1.h.b	yes	1	84.h	odd	2	1	CM
756.1.h.c	yes	1	3.b	odd	2	1
756.1.h.c	yes	1	28.d	even	2	1
756.1.h.d	yes	1	4.b	odd	2	1
756.1.h.d	yes	1	21.c	even	2	1
2268.1.s.a		2	36.f	odd	6	2
2268.1.s.a		2	63.o	even	6	2
2268.1.s.b		2	9.d	odd	6	2
2268.1.s.b		2	252.bi	even	6	2
2268.1.s.c		2	9.c	even	3	2
2268.1.s.c		2	252.s	odd	6	2
2268.1.s.d		2	36.h	even	6	2
2268.1.s.d		2	63.l	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(756, [\chi])$:

$ T_{5} - 1 $

$ T_{11} - 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T $ T
$5$	$ T - 1 $ T - 1
$7$	$ T - 1 $ T - 1
$11$	$ T - 1 $ T - 1
$13$	$ T $ T
$17$	$ T + 2 $ T + 2
$19$	$ T + 1 $ T + 1
$23$	$ T - 1 $ T - 1
$29$	$ T $ T
$31$	$ T + 1 $ T + 1
$37$	$ T + 1 $ T + 1
$41$	$ T - 1 $ T - 1
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T - 1 $ T - 1
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T - 1 $ T - 1
$97$	$ T $ T
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	756.h (of order \(2\), degree \(1\), minimal)

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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