Newform orbit 528.2.b.a

• Label: 528.2.b.a
• Level: $528$
• Weight: $2$
• Character orbit: 528.b
• Analytic conductor: $4.216$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	528.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.21610122672\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-11}) \)
Defining polynomial:	\( x^{2} - x + 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{3} + ( - 2 \beta + 1) q^{5} + (\beta - 3) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\)	\(133\)	\(145\)	\(353\)	\(463\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

65.1

0.500000	+	1.65831i
0.500000	−	1.65831i

0

−0.500000

−

1.65831i

0

−

3.31662i

0

0

0

−2.50000

+

1.65831i

0

65.2

0

−0.500000

+

1.65831i

0

3.31662i

0

0

0

−2.50000

−

1.65831i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)
3.b	odd	2	1	inner
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	528.2.b.a		2
3.b	odd	2	1	inner	528.2.b.a		2
4.b	odd	2	1		33.2.d.a	✓	2
8.b	even	2	1		2112.2.b.f		2
8.d	odd	2	1		2112.2.b.e		2
11.b	odd	2	1	CM	528.2.b.a		2
12.b	even	2	1		33.2.d.a	✓	2
20.d	odd	2	1		825.2.f.a		2
20.e	even	4	2		825.2.d.a		4
24.f	even	2	1		2112.2.b.e		2
24.h	odd	2	1		2112.2.b.f		2
33.d	even	2	1	inner	528.2.b.a		2
36.f	odd	6	2		891.2.g.a		4
36.h	even	6	2		891.2.g.a		4
44.c	even	2	1		33.2.d.a	✓	2
44.g	even	10	4		363.2.f.c		8
44.h	odd	10	4		363.2.f.c		8
60.h	even	2	1		825.2.f.a		2
60.l	odd	4	2		825.2.d.a		4
88.b	odd	2	1		2112.2.b.f		2
88.g	even	2	1		2112.2.b.e		2
132.d	odd	2	1		33.2.d.a	✓	2
132.n	odd	10	4		363.2.f.c		8
132.o	even	10	4		363.2.f.c		8
220.g	even	2	1		825.2.f.a		2
220.i	odd	4	2		825.2.d.a		4
264.m	even	2	1		2112.2.b.f		2
264.p	odd	2	1		2112.2.b.e		2
396.k	even	6	2		891.2.g.a		4
396.o	odd	6	2		891.2.g.a		4
660.g	odd	2	1		825.2.f.a		2
660.q	even	4	2		825.2.d.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.d.a	✓	2	4.b	odd	2	1
33.2.d.a	✓	2	12.b	even	2	1
33.2.d.a	✓	2	44.c	even	2	1
33.2.d.a	✓	2	132.d	odd	2	1
363.2.f.c		8	44.g	even	10	4
363.2.f.c		8	44.h	odd	10	4
363.2.f.c		8	132.n	odd	10	4
363.2.f.c		8	132.o	even	10	4
528.2.b.a		2	1.a	even	1	1	trivial
528.2.b.a		2	3.b	odd	2	1	inner
528.2.b.a		2	11.b	odd	2	1	CM
528.2.b.a		2	33.d	even	2	1	inner
825.2.d.a		4	20.e	even	4	2
825.2.d.a		4	60.l	odd	4	2
825.2.d.a		4	220.i	odd	4	2
825.2.d.a		4	660.q	even	4	2
825.2.f.a		2	20.d	odd	2	1
825.2.f.a		2	60.h	even	2	1
825.2.f.a		2	220.g	even	2	1
825.2.f.a		2	660.g	odd	2	1
891.2.g.a		4	36.f	odd	6	2
891.2.g.a		4	36.h	even	6	2
891.2.g.a		4	396.k	even	6	2
891.2.g.a		4	396.o	odd	6	2
2112.2.b.e		2	8.d	odd	2	1
2112.2.b.e		2	24.f	even	2	1
2112.2.b.e		2	88.g	even	2	1
2112.2.b.e		2	264.p	odd	2	1
2112.2.b.f		2	8.b	even	2	1
2112.2.b.f		2	24.h	odd	2	1
2112.2.b.f		2	88.b	odd	2	1
2112.2.b.f		2	264.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\):

\( T_{5}^{2} + 11 \)

\( T_{17} \)

\( T_{29} \)

Hecke characteristic polynomials

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