Newform orbit 3528.2.s.y

• Label: 3528.2.s.y
• Level: $3528$
• Weight: $2$
• Character orbit: 3528.s
• Analytic conductor: $28.171$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Self dual:	no
Analytic conductor:	\(28.1712218331\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\)	\(=\)	\( q + 2 \zeta_{6} q^{5}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \)

Character values

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

\(n\)	\(785\)	\(1081\)	\(1765\)	\(2647\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

1.00000

+

1.73205i

0

0

0

0

0

3313.1

0

0

0

1.00000

−

1.73205i

0

0

0

0

0

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.y		2
3.b	odd	2	1		1176.2.q.a		2
7.b	odd	2	1		3528.2.s.j		2
7.c	even	3	1		3528.2.a.d		1
7.c	even	3	1	inner	3528.2.s.y		2
7.d	odd	6	1		72.2.a.a		1
7.d	odd	6	1		3528.2.s.j		2
12.b	even	2	1		2352.2.q.r		2
21.c	even	2	1		1176.2.q.i		2
21.g	even	6	1		24.2.a.a	✓	1
21.g	even	6	1		1176.2.q.i		2
21.h	odd	6	1		1176.2.a.i		1
21.h	odd	6	1		1176.2.q.a		2
28.f	even	6	1		144.2.a.b		1
28.g	odd	6	1		7056.2.a.q		1
35.i	odd	6	1		1800.2.a.m		1
35.k	even	12	2		1800.2.f.c		2
56.j	odd	6	1		576.2.a.d		1
56.m	even	6	1		576.2.a.b		1
63.i	even	6	1		648.2.i.g		2
63.k	odd	6	1		648.2.i.b		2
63.s	even	6	1		648.2.i.g		2
63.t	odd	6	1		648.2.i.b		2
77.i	even	6	1		8712.2.a.u		1
84.h	odd	2	1		2352.2.q.l		2
84.j	odd	6	1		48.2.a.a		1
84.j	odd	6	1		2352.2.q.l		2
84.n	even	6	1		2352.2.a.i		1
84.n	even	6	1		2352.2.q.r		2
105.p	even	6	1		600.2.a.h		1
105.w	odd	12	2		600.2.f.e		2
112.v	even	12	2		2304.2.d.k		2
112.x	odd	12	2		2304.2.d.i		2
140.s	even	6	1		3600.2.a.v		1
140.x	odd	12	2		3600.2.f.r		2
168.s	odd	6	1		9408.2.a.h		1
168.v	even	6	1		9408.2.a.cc		1
168.ba	even	6	1		192.2.a.d		1
168.be	odd	6	1		192.2.a.b		1
231.k	odd	6	1		2904.2.a.c		1
252.n	even	6	1		1296.2.i.e		2
252.r	odd	6	1		1296.2.i.m		2
252.bj	even	6	1		1296.2.i.e		2
252.bn	odd	6	1		1296.2.i.m		2
273.ba	even	6	1		4056.2.a.i		1
273.cb	odd	12	2		4056.2.c.e		2
336.bo	even	12	2		768.2.d.e		2
336.br	odd	12	2		768.2.d.d		2
357.s	even	6	1		6936.2.a.p		1
399.s	odd	6	1		8664.2.a.j		1
420.be	odd	6	1		1200.2.a.d		1
420.br	even	12	2		1200.2.f.b		2
840.cb	even	6	1		4800.2.a.q		1
840.ct	odd	6	1		4800.2.a.cc		1
840.dh	odd	12	2		4800.2.f.d		2
840.dk	even	12	2		4800.2.f.bg		2
924.y	even	6	1		5808.2.a.s		1
1092.ct	odd	6	1		8112.2.a.be		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.2.a.a	✓	1	21.g	even	6	1
48.2.a.a		1	84.j	odd	6	1
72.2.a.a		1	7.d	odd	6	1
144.2.a.b		1	28.f	even	6	1
192.2.a.b		1	168.be	odd	6	1
192.2.a.d		1	168.ba	even	6	1
576.2.a.b		1	56.m	even	6	1
576.2.a.d		1	56.j	odd	6	1
600.2.a.h		1	105.p	even	6	1
600.2.f.e		2	105.w	odd	12	2
648.2.i.b		2	63.k	odd	6	1
648.2.i.b		2	63.t	odd	6	1
648.2.i.g		2	63.i	even	6	1
648.2.i.g		2	63.s	even	6	1
768.2.d.d		2	336.br	odd	12	2
768.2.d.e		2	336.bo	even	12	2
1176.2.a.i		1	21.h	odd	6	1
1176.2.q.a		2	3.b	odd	2	1
1176.2.q.a		2	21.h	odd	6	1
1176.2.q.i		2	21.c	even	2	1
1176.2.q.i		2	21.g	even	6	1
1200.2.a.d		1	420.be	odd	6	1
1200.2.f.b		2	420.br	even	12	2
1296.2.i.e		2	252.n	even	6	1
1296.2.i.e		2	252.bj	even	6	1
1296.2.i.m		2	252.r	odd	6	1
1296.2.i.m		2	252.bn	odd	6	1
1800.2.a.m		1	35.i	odd	6	1
1800.2.f.c		2	35.k	even	12	2
2304.2.d.i		2	112.x	odd	12	2
2304.2.d.k		2	112.v	even	12	2
2352.2.a.i		1	84.n	even	6	1
2352.2.q.l		2	84.h	odd	2	1
2352.2.q.l		2	84.j	odd	6	1
2352.2.q.r		2	12.b	even	2	1
2352.2.q.r		2	84.n	even	6	1
2904.2.a.c		1	231.k	odd	6	1
3528.2.a.d		1	7.c	even	3	1
3528.2.s.j		2	7.b	odd	2	1
3528.2.s.j		2	7.d	odd	6	1
3528.2.s.y		2	1.a	even	1	1	trivial
3528.2.s.y		2	7.c	even	3	1	inner
3600.2.a.v		1	140.s	even	6	1
3600.2.f.r		2	140.x	odd	12	2
4056.2.a.i		1	273.ba	even	6	1
4056.2.c.e		2	273.cb	odd	12	2
4800.2.a.q		1	840.cb	even	6	1
4800.2.a.cc		1	840.ct	odd	6	1
4800.2.f.d		2	840.dh	odd	12	2
4800.2.f.bg		2	840.dk	even	12	2
5808.2.a.s		1	924.y	even	6	1
6936.2.a.p		1	357.s	even	6	1
7056.2.a.q		1	28.g	odd	6	1
8112.2.a.be		1	1092.ct	odd	6	1
8664.2.a.j		1	399.s	odd	6	1
8712.2.a.u		1	77.i	even	6	1
9408.2.a.h		1	168.s	odd	6	1
9408.2.a.cc		1	168.v	even	6	1

Hecke kernels

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

\( T_{5}^{2} - 2T_{5} + 4 \)

\( T_{11}^{2} - 4T_{11} + 16 \)

\( T_{13} - 2 \)

\( T_{23}^{2} + 8T_{23} + 64 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 2T + 4 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 4T + 16 \)