LMFDB - Newform orbit 756.2.f.a

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,-6] 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:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $\beta = \sqrt{-3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 3 q^{5} + ( - \beta + 2) q^{7} + 3 \beta q^{11} - 2 \beta q^{13} + 6 q^{17} - \beta q^{19} - 3 \beta q^{23} + 4 q^{25} - 6 \beta q^{29} - 3 \beta q^{31} + (3 \beta - 6) q^{35} + q^{37} + 3 q^{41} + \cdots + 4 \beta q^{97} +O(q^{100})$ q - 3 * q^5 + (-b + 2) * q^7 + 3b q^11 - 2b q^13 + 6 * q^17 - b * q^19 - 3b q^23 + 4 * q^25 - 6b q^29 - 3b q^31 + (3b - 6) q^35 + q^37 + 3 * q^41 + 10 * q^43 + 6 * q^47 + (-4b + 1) q^49 - 9b q^55 - 6 * q^59 + 8b q^61 + 6b q^65 + 2 * q^67 - 3b q^71 + 2b q^73 + (6b + 9) q^77 + 14 * q^79 - 6 * q^83 - 18 * q^85 - 9 * q^89 + (-4b - 6) q^91 + 3b q^95 + 4b q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{5} + 4 q^{7} + 12 q^{17} + 8 q^{25} - 12 q^{35} + 2 q^{37} + 6 q^{41} + 20 q^{43} + 12 q^{47} + 2 q^{49} - 12 q^{59} + 4 q^{67} + 18 q^{77} + 28 q^{79} - 12 q^{83} - 36 q^{85} - 18 q^{89} - 12 q^{91}+O(q^{100})$ 2 * q - 6 * q^5 + 4 * q^7 + 12 * q^17 + 8 * q^25 - 12 * q^35 + 2 * q^37 + 6 * q^41 + 20 * q^43 + 12 * q^47 + 2 * q^49 - 12 * q^59 + 4 * q^67 + 18 * q^77 + 28 * q^79 - 12 * q^83 - 36 * q^85 - 18 * q^89 - 12 * 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

−3.00000

2.00000

−

1.73205i

377.2

−3.00000

2.00000

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
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.a	✓	2
3.b	odd	2	1		756.2.f.c	yes	2
4.b	odd	2	1		3024.2.k.a		2
7.b	odd	2	1		756.2.f.c	yes	2
9.c	even	3	1		2268.2.x.g		2
9.c	even	3	1		2268.2.x.h		2
9.d	odd	6	1		2268.2.x.a		2
9.d	odd	6	1		2268.2.x.b		2
12.b	even	2	1		3024.2.k.d		2
21.c	even	2	1	inner	756.2.f.a	✓	2
28.d	even	2	1		3024.2.k.d		2
63.l	odd	6	1		2268.2.x.a		2
63.l	odd	6	1		2268.2.x.b		2
63.o	even	6	1		2268.2.x.g		2
63.o	even	6	1		2268.2.x.h		2
84.h	odd	2	1		3024.2.k.a		2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} + 3$ T5 + 3 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
$5$	$(T + 3)^{2}$ (T + 3)^2
$7$	$T^{2} - 4T + 7$ T^2 - 4*T + 7
$11$	$T^{2} + 27$ T^2 + 27
$13$	$T^{2} + 12$ T^2 + 12
$17$	$(T - 6)^{2}$ (T - 6)^2
$19$	$T^{2} + 3$ T^2 + 3
$23$	$T^{2} + 27$ T^2 + 27
$29$	$T^{2} + 108$ T^2 + 108
$31$	$T^{2} + 27$ T^2 + 27
$37$	$(T - 1)^{2}$ (T - 1)^2
$41$	$(T - 3)^{2}$ (T - 3)^2
$43$	$(T - 10)^{2}$ (T - 10)^2
$47$	$(T - 6)^{2}$ (T - 6)^2
$53$	$T^{2}$ T^2
$59$	$(T + 6)^{2}$ (T + 6)^2
$61$	$T^{2} + 192$ T^2 + 192
$67$	$(T - 2)^{2}$ (T - 2)^2
$71$	$T^{2} + 27$ T^2 + 27
$73$	$T^{2} + 12$ T^2 + 12
$79$	$(T - 14)^{2}$ (T - 14)^2
$83$	$(T + 6)^{2}$ (T + 6)^2
$89$	$(T + 9)^{2}$ (T + 9)^2
$97$	$T^{2} + 48$ T^2 + 48
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