Newform orbit 66.2.b.b

• Label: 66.2.b.b
• Level: $66$
• Weight: $2$
• Character orbit: 66.b
• Analytic conductor: $0.527$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 66 = 2 \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	66.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.527012653340\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + q^{2} + (\beta - 1) q^{3} + q^{4} + \beta q^{5} + (\beta - 1) q^{6} - 3 \beta q^{7} + q^{8} + ( - 2 \beta - 1) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(-1\)	\(-1\)

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

	−	1.41421i
		1.41421i

1.00000

−1.00000

−

1.41421i

1.00000

−

1.41421i

−1.00000

−

1.41421i

4.24264i

1.00000

−1.00000

+

2.82843i

−

1.41421i

65.2

1.00000

−1.00000

+

1.41421i

1.00000

1.41421i

−1.00000

+

1.41421i

−

4.24264i

1.00000

−1.00000

−

2.82843i

1.41421i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
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	66.2.b.b	yes	2
3.b	odd	2	1		66.2.b.a	✓	2
4.b	odd	2	1		528.2.b.c		2
5.b	even	2	1		1650.2.d.a		2
5.c	odd	4	2		1650.2.f.a		4
8.b	even	2	1		2112.2.b.i		2
8.d	odd	2	1		2112.2.b.b		2
9.c	even	3	2		1782.2.i.c		4
9.d	odd	6	2		1782.2.i.f		4
11.b	odd	2	1		66.2.b.a	✓	2
11.c	even	5	4		726.2.h.e		8
11.d	odd	10	4		726.2.h.i		8
12.b	even	2	1		528.2.b.b		2
15.d	odd	2	1		1650.2.d.b		2
15.e	even	4	2		1650.2.f.b		4
24.f	even	2	1		2112.2.b.d		2
24.h	odd	2	1		2112.2.b.g		2
33.d	even	2	1	inner	66.2.b.b	yes	2
33.f	even	10	4		726.2.h.e		8
33.h	odd	10	4		726.2.h.i		8
44.c	even	2	1		528.2.b.b		2
55.d	odd	2	1		1650.2.d.b		2
55.e	even	4	2		1650.2.f.b		4
88.b	odd	2	1		2112.2.b.g		2
88.g	even	2	1		2112.2.b.d		2
99.g	even	6	2		1782.2.i.c		4
99.h	odd	6	2		1782.2.i.f		4
132.d	odd	2	1		528.2.b.c		2
165.d	even	2	1		1650.2.d.a		2
165.l	odd	4	2		1650.2.f.a		4
264.m	even	2	1		2112.2.b.i		2
264.p	odd	2	1		2112.2.b.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.2.b.a	✓	2	3.b	odd	2	1
66.2.b.a	✓	2	11.b	odd	2	1
66.2.b.b	yes	2	1.a	even	1	1	trivial
66.2.b.b	yes	2	33.d	even	2	1	inner
528.2.b.b		2	12.b	even	2	1
528.2.b.b		2	44.c	even	2	1
528.2.b.c		2	4.b	odd	2	1
528.2.b.c		2	132.d	odd	2	1
726.2.h.e		8	11.c	even	5	4
726.2.h.e		8	33.f	even	10	4
726.2.h.i		8	11.d	odd	10	4
726.2.h.i		8	33.h	odd	10	4
1650.2.d.a		2	5.b	even	2	1
1650.2.d.a		2	165.d	even	2	1
1650.2.d.b		2	15.d	odd	2	1
1650.2.d.b		2	55.d	odd	2	1
1650.2.f.a		4	5.c	odd	4	2
1650.2.f.a		4	165.l	odd	4	2
1650.2.f.b		4	15.e	even	4	2
1650.2.f.b		4	55.e	even	4	2
1782.2.i.c		4	9.c	even	3	2
1782.2.i.c		4	99.g	even	6	2
1782.2.i.f		4	9.d	odd	6	2
1782.2.i.f		4	99.h	odd	6	2
2112.2.b.b		2	8.d	odd	2	1
2112.2.b.b		2	264.p	odd	2	1
2112.2.b.d		2	24.f	even	2	1
2112.2.b.d		2	88.g	even	2	1
2112.2.b.g		2	24.h	odd	2	1
2112.2.b.g		2	88.b	odd	2	1
2112.2.b.i		2	8.b	even	2	1
2112.2.b.i		2	264.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{29} + 6 \) acting on \(S_{2}^{\mathrm{new}}(66, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( T^{2} + 2T + 3 \)
$5$	\( T^{2} + 2 \)
$7$	\( T^{2} + 18 \)
$11$	\( T^{2} + 6T + 11 \)