LMFDB - Newform orbit 756.2.f.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,2,Mod(377,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 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("756.377"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.03669039281$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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)

Coefficients of the $q$ -expansion are expressed in terms of a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \zeta_{6} - 2) q^{7} + ( - 2 \zeta_{6} + 1) q^{13} + ( - 10 \zeta_{6} + 5) q^{19} - 5 q^{25} + ( - 12 \zeta_{6} + 6) q^{31} + q^{37} - 8 q^{43} + (5 \zeta_{6} + 3) q^{49} + ( - 10 \zeta_{6} + 5) q^{61}+ \cdots + (22 \zeta_{6} - 11) q^{97}+O(q^{100})$ q + (-z - 2) * q^7 + (-2z + 1) q^13 + (-10z + 5) q^19 - 5 * q^25 + (-12z + 6) q^31 + q^37 - 8 * q^43 + (5z + 3) q^49 + (-10z + 5) q^61 + 11 * q^67 + (2z - 1) q^73 - 13 * q^79 + (5z - 4) q^91 + (22z - 11) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 5 q^{7} - 10 q^{25} + 2 q^{37} - 16 q^{43} + 11 q^{49} + 22 q^{67} - 26 q^{79} - 3 q^{91}+O(q^{100})$ 2 * q - 5 * q^7 - 10 * q^25 + 2 * q^37 - 16 * q^43 + 11 * q^49 + 22 * q^67 - 26 * q^79 - 3 * q^91

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}

377.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−2.50000

−

0.866025i

377.2

−2.50000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.2.f.b	✓	2
3.b	odd	2	1	CM	756.2.f.b	✓	2
4.b	odd	2	1		3024.2.k.c		2
7.b	odd	2	1	inner	756.2.f.b	✓	2
9.c	even	3	1		2268.2.x.d		2
9.c	even	3	1		2268.2.x.f		2
9.d	odd	6	1		2268.2.x.d		2
9.d	odd	6	1		2268.2.x.f		2
12.b	even	2	1		3024.2.k.c		2
21.c	even	2	1	inner	756.2.f.b	✓	2
28.d	even	2	1		3024.2.k.c		2
63.l	odd	6	1		2268.2.x.d		2
63.l	odd	6	1		2268.2.x.f		2
63.o	even	6	1		2268.2.x.d		2
63.o	even	6	1		2268.2.x.f		2
84.h	odd	2	1		3024.2.k.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.f.b	✓	2	1.a	even	1	1	trivial
756.2.f.b	✓	2	3.b	odd	2	1	CM
756.2.f.b	✓	2	7.b	odd	2	1	inner
756.2.f.b	✓	2	21.c	even	2	1	inner
2268.2.x.d		2	9.c	even	3	1
2268.2.x.d		2	9.d	odd	6	1
2268.2.x.d		2	63.l	odd	6	1
2268.2.x.d		2	63.o	even	6	1
2268.2.x.f		2	9.c	even	3	1
2268.2.x.f		2	9.d	odd	6	1
2268.2.x.f		2	63.l	odd	6	1
2268.2.x.f		2	63.o	even	6	1
3024.2.k.c		2	4.b	odd	2	1
3024.2.k.c		2	12.b	even	2	1
3024.2.k.c		2	28.d	even	2	1
3024.2.k.c		2	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}$ T5 acting on $S_{2}^{\mathrm{new}}(756, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 5T + 7$ T^2 + 5*T + 7
$11$	$T^{2}$ T^2
$13$	$T^{2} + 3$ T^2 + 3
$17$	$T^{2}$ T^2
$19$	$T^{2} + 75$ T^2 + 75
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} + 108$ T^2 + 108
$37$	$(T - 1)^{2}$ (T - 1)^2
$41$	$T^{2}$ T^2
$43$	$(T + 8)^{2}$ (T + 8)^2
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} + 75$ T^2 + 75
$67$	$(T - 11)^{2}$ (T - 11)^2
$71$	$T^{2}$ T^2
$73$	$T^{2} + 3$ T^2 + 3
$79$	$(T + 13)^{2}$ (T + 13)^2
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2} + 363$ T^2 + 363
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