LMFDB - Newform orbit 3528.2.s.bd

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$ , degree $2$ , not 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:	no
Analytic conductor:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 2x^{2} + 4$ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{5} + (2 \beta_{2} - 2 \beta_1 + 2) q^{11} - \beta_{3} q^{13} + (2 \beta_{2} - 3 \beta_1 + 2) q^{17} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{19} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{23}+ \cdots + (3 \beta_{3} + 12) q^{97}+O(q^{100})$ q + (-b3 + 2b2 - b1) q^5 + (2b2 - 2b1 + 2) * q^11 - b3 * q^13 + (2b2 - 3b1 + 2) * q^17 + (-2b3 - 4b2 - 2b1) q^19 + (-2b3 - 2b2 - 2b1) q^23 + (-b2 + 4b1 - 1) q^25 + 6b3 q^29 + (8b2 - 2b1 + 8) * q^31 + (4b3 - 4b2 + 4b1) q^37 + (-b3 - 2) * q^41 - 8 * q^43 + (2b3 + 4b2 + 2b1) q^47 + (-2b2 - 8b1 - 2) * q^53 + (-6b3 - 8) q^55 + (-8b2 + 2b1 - 8) * q^59 + (-7b3 - 4b2 - 7b1) q^61 + (2b3 - 2b2 + 2b1) q^65 + (8b2 + 8) q^67 + (-2b3 - 2) q^71 + (-4b2 - 5b1 - 4) * q^73 + (4b3 - 8b2 + 4b1) q^79 + (-8b3 + 4) q^83 + (-8b3 - 10) q^85 + (-9b3 + 2b2 - 9b1) q^89 + (4b2 + 4) q^95 + (3b3 + 12) q^97
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{5} + 4 q^{11} + 4 q^{17} + 8 q^{19} + 4 q^{23} - 2 q^{25} + 16 q^{31} + 8 q^{37} - 8 q^{41} - 32 q^{43} - 8 q^{47} - 4 q^{53} - 32 q^{55} - 16 q^{59} + 8 q^{61} + 4 q^{65} + 16 q^{67} - 8 q^{71}+ \cdots + 48 q^{97}+O(q^{100})$ 4 * q - 4 * q^5 + 4 * q^11 + 4 * q^17 + 8 * q^19 + 4 * q^23 - 2 * q^25 + 16 * q^31 + 8 * q^37 - 8 * q^41 - 32 * q^43 - 8 * q^47 - 4 * q^53 - 32 * q^55 - 16 * q^59 + 8 * q^61 + 4 * q^65 + 16 * q^67 - 8 * q^71 - 8 * q^73 + 16 * q^79 + 16 * q^83 - 40 * q^85 - 4 * q^89 + 8 * q^95 + 48 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} + 2x^{2} + 4$ x^4 + 2*x^2 + 4 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{2} ) / 2$ (v^2) / 2
$\beta_{3}$	$=$	$( \nu^{3} ) / 2$ (v^3) / 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$2\beta_{2}$ 2*b2
$\nu^{3}$	$=$	$2\beta_{3}$ 2*b3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$\beta_{2}$	$1$	$1$

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}

361.1

−0.707107	+	1.22474i
0.707107	−	1.22474i
−0.707107	−	1.22474i
0.707107	+	1.22474i

−1.70711

−

2.95680i

361.2

−0.292893

−

0.507306i

3313.1

−1.70711

2.95680i

3313.2

−0.292893

0.507306i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3528.2.s.bd		4
3.b	odd	2	1		1176.2.q.o		4
7.b	odd	2	1		3528.2.s.bm		4
7.c	even	3	1		3528.2.a.bl		2
7.c	even	3	1	inner	3528.2.s.bd		4
7.d	odd	6	1		3528.2.a.bb		2
7.d	odd	6	1		3528.2.s.bm		4
12.b	even	2	1		2352.2.q.bc		4
21.c	even	2	1		1176.2.q.k		4
21.g	even	6	1		1176.2.a.o	yes	2
21.g	even	6	1		1176.2.q.k		4
21.h	odd	6	1		1176.2.a.j	✓	2
21.h	odd	6	1		1176.2.q.o		4
28.f	even	6	1		7056.2.a.cg		2
28.g	odd	6	1		7056.2.a.cx		2
84.h	odd	2	1		2352.2.q.be		4
84.j	odd	6	1		2352.2.a.bb		2
84.j	odd	6	1		2352.2.q.be		4
84.n	even	6	1		2352.2.a.bd		2
84.n	even	6	1		2352.2.q.bc		4
168.s	odd	6	1		9408.2.a.ee		2
168.v	even	6	1		9408.2.a.ds		2
168.ba	even	6	1		9408.2.a.dg		2
168.be	odd	6	1		9408.2.a.du		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.2.a.j	✓	2	21.h	odd	6	1
1176.2.a.o	yes	2	21.g	even	6	1
1176.2.q.k		4	21.c	even	2	1
1176.2.q.k		4	21.g	even	6	1
1176.2.q.o		4	3.b	odd	2	1
1176.2.q.o		4	21.h	odd	6	1
2352.2.a.bb		2	84.j	odd	6	1
2352.2.a.bd		2	84.n	even	6	1
2352.2.q.bc		4	12.b	even	2	1
2352.2.q.bc		4	84.n	even	6	1
2352.2.q.be		4	84.h	odd	2	1
2352.2.q.be		4	84.j	odd	6	1
3528.2.a.bb		2	7.d	odd	6	1
3528.2.a.bl		2	7.c	even	3	1
3528.2.s.bd		4	1.a	even	1	1	trivial
3528.2.s.bd		4	7.c	even	3	1	inner
3528.2.s.bm		4	7.b	odd	2	1
3528.2.s.bm		4	7.d	odd	6	1
7056.2.a.cg		2	28.f	even	6	1
7056.2.a.cx		2	28.g	odd	6	1
9408.2.a.dg		2	168.ba	even	6	1
9408.2.a.ds		2	168.v	even	6	1
9408.2.a.du		2	168.be	odd	6	1
9408.2.a.ee		2	168.s	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3528, [\chi])$ :

T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 8T_{5} + 4

T_{11}^{4} - 4T_{11}^{3} + 20T_{11}^{2} + 16T_{11} + 16

T_{13}^{2} - 2

T_{23}^{4} - 4T_{23}^{3} + 20T_{23}^{2} + 16T_{23} + 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4}$ T^4
$5$	$T^{4} + 4 T^{3} + \cdots + 4$ T^4 + 4T^3 + 14T^2 + 8*T + 4
$7$	$T^{4}$ T^4
$11$	$T^{4} - 4 T^{3} + \cdots + 16$ T^4 - 4T^3 + 20T^2 + 16*T + 16
$13$	$(T^{2} - 2)^{2}$ (T^2 - 2)^2
$17$	$T^{4} - 4 T^{3} + \cdots + 196$ T^4 - 4T^3 + 30T^2 + 56*T + 196
$19$	$T^{4} - 8 T^{3} + \cdots + 64$ T^4 - 8T^3 + 56T^2 - 64*T + 64
$23$	$T^{4} - 4 T^{3} + \cdots + 16$ T^4 - 4T^3 + 20T^2 + 16*T + 16
$29$	$(T^{2} - 72)^{2}$ (T^2 - 72)^2
$31$	$T^{4} - 16 T^{3} + \cdots + 3136$ T^4 - 16T^3 + 200T^2 - 896*T + 3136
$37$	$T^{4} - 8 T^{3} + \cdots + 256$ T^4 - 8T^3 + 80T^2 + 128*T + 256
$41$	$(T^{2} + 4 T + 2)^{2}$ (T^2 + 4*T + 2)^2
$43$	$(T + 8)^{4}$ (T + 8)^4
$47$	$T^{4} + 8 T^{3} + \cdots + 64$ T^4 + 8T^3 + 56T^2 + 64*T + 64
$53$	$T^{4} + 4 T^{3} + \cdots + 15376$ T^4 + 4T^3 + 140T^2 - 496*T + 15376
$59$	$T^{4} + 16 T^{3} + \cdots + 3136$ T^4 + 16T^3 + 200T^2 + 896*T + 3136
$61$	$T^{4} - 8 T^{3} + \cdots + 6724$ T^4 - 8T^3 + 146T^2 + 656*T + 6724
$67$	$(T^{2} - 8 T + 64)^{2}$ (T^2 - 8*T + 64)^2
$71$	$(T^{2} + 4 T - 4)^{2}$ (T^2 + 4*T - 4)^2
$73$	$T^{4} + 8 T^{3} + \cdots + 1156$ T^4 + 8T^3 + 98T^2 - 272*T + 1156
$79$	$T^{4} - 16 T^{3} + \cdots + 1024$ T^4 - 16T^3 + 224T^2 - 512*T + 1024
$83$	$(T^{2} - 8 T - 112)^{2}$ (T^2 - 8*T - 112)^2
$89$	$T^{4} + 4 T^{3} + \cdots + 24964$ T^4 + 4T^3 + 174T^2 - 632*T + 24964
$97$	$(T^{2} - 24 T + 126)^{2}$ (T^2 - 24*T + 126)^2
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