Newform orbit 33.2.d.a

• Label: 33.2.d.a
• Level: $33$
• Weight: $2$
• Character orbit: 33.d
• Analytic conductor: $0.264$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.263506326670\)
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:	yes
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

32.1

0.500000	−	1.65831i
0.500000	+	1.65831i

0

0.500000

−

1.65831i

−2.00000

3.31662i

0

0

0

−2.50000

−

1.65831i

0

32.2

0

0.500000

+

1.65831i

−2.00000

−

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	33.2.d.a	✓	2
3.b	odd	2	1	inner	33.2.d.a	✓	2
4.b	odd	2	1		528.2.b.a		2
5.b	even	2	1		825.2.f.a		2
5.c	odd	4	2		825.2.d.a		4
8.b	even	2	1		2112.2.b.e		2
8.d	odd	2	1		2112.2.b.f		2
9.c	even	3	2		891.2.g.a		4
9.d	odd	6	2		891.2.g.a		4
11.b	odd	2	1	CM	33.2.d.a	✓	2
11.c	even	5	4		363.2.f.c		8
11.d	odd	10	4		363.2.f.c		8
12.b	even	2	1		528.2.b.a		2
15.d	odd	2	1		825.2.f.a		2
15.e	even	4	2		825.2.d.a		4
24.f	even	2	1		2112.2.b.f		2
24.h	odd	2	1		2112.2.b.e		2
33.d	even	2	1	inner	33.2.d.a	✓	2
33.f	even	10	4		363.2.f.c		8
33.h	odd	10	4		363.2.f.c		8
44.c	even	2	1		528.2.b.a		2
55.d	odd	2	1		825.2.f.a		2
55.e	even	4	2		825.2.d.a		4
88.b	odd	2	1		2112.2.b.e		2
88.g	even	2	1		2112.2.b.f		2
99.g	even	6	2		891.2.g.a		4
99.h	odd	6	2		891.2.g.a		4
132.d	odd	2	1		528.2.b.a		2
165.d	even	2	1		825.2.f.a		2
165.l	odd	4	2		825.2.d.a		4
264.m	even	2	1		2112.2.b.e		2
264.p	odd	2	1		2112.2.b.f		2

    

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

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

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