Newform orbit 2112.2.b.f

• Label: 2112.2.b.f
• Level: $2112$
• Weight: $2$
• Character orbit: 2112.b
• Analytic conductor: $16.864$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.8644049069\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times\).

\(n\)	\(133\)	\(1409\)	\(1729\)	\(2047\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(1\)

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	2112.2.b.f		2
3.b	odd	2	1	inner	2112.2.b.f		2
4.b	odd	2	1		2112.2.b.e		2
8.b	even	2	1		528.2.b.a		2
8.d	odd	2	1		33.2.d.a	✓	2
11.b	odd	2	1	CM	2112.2.b.f		2
12.b	even	2	1		2112.2.b.e		2
24.f	even	2	1		33.2.d.a	✓	2
24.h	odd	2	1		528.2.b.a		2
33.d	even	2	1	inner	2112.2.b.f		2
40.e	odd	2	1		825.2.f.a		2
40.k	even	4	2		825.2.d.a		4
44.c	even	2	1		2112.2.b.e		2
72.l	even	6	2		891.2.g.a		4
72.p	odd	6	2		891.2.g.a		4
88.b	odd	2	1		528.2.b.a		2
88.g	even	2	1		33.2.d.a	✓	2
88.k	even	10	4		363.2.f.c		8
88.l	odd	10	4		363.2.f.c		8
120.m	even	2	1		825.2.f.a		2
120.q	odd	4	2		825.2.d.a		4
132.d	odd	2	1		2112.2.b.e		2
264.m	even	2	1		528.2.b.a		2
264.p	odd	2	1		33.2.d.a	✓	2
264.r	odd	10	4		363.2.f.c		8
264.w	even	10	4		363.2.f.c		8
440.c	even	2	1		825.2.f.a		2
440.w	odd	4	2		825.2.d.a		4
792.s	odd	6	2		891.2.g.a		4
792.z	even	6	2		891.2.g.a		4
1320.b	odd	2	1		825.2.f.a		2
1320.bt	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	8.d	odd	2	1
33.2.d.a	✓	2	24.f	even	2	1
33.2.d.a	✓	2	88.g	even	2	1
33.2.d.a	✓	2	264.p	odd	2	1
363.2.f.c		8	88.k	even	10	4
363.2.f.c		8	88.l	odd	10	4
363.2.f.c		8	264.r	odd	10	4
363.2.f.c		8	264.w	even	10	4
528.2.b.a		2	8.b	even	2	1
528.2.b.a		2	24.h	odd	2	1
528.2.b.a		2	88.b	odd	2	1
528.2.b.a		2	264.m	even	2	1
825.2.d.a		4	40.k	even	4	2
825.2.d.a		4	120.q	odd	4	2
825.2.d.a		4	440.w	odd	4	2
825.2.d.a		4	1320.bt	even	4	2
825.2.f.a		2	40.e	odd	2	1
825.2.f.a		2	120.m	even	2	1
825.2.f.a		2	440.c	even	2	1
825.2.f.a		2	1320.b	odd	2	1
891.2.g.a		4	72.l	even	6	2
891.2.g.a		4	72.p	odd	6	2
891.2.g.a		4	792.s	odd	6	2
891.2.g.a		4	792.z	even	6	2
2112.2.b.e		2	4.b	odd	2	1
2112.2.b.e		2	12.b	even	2	1
2112.2.b.e		2	44.c	even	2	1
2112.2.b.e		2	132.d	odd	2	1
2112.2.b.f		2	1.a	even	1	1	trivial
2112.2.b.f		2	3.b	odd	2	1	inner
2112.2.b.f		2	11.b	odd	2	1	CM
2112.2.b.f		2	33.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2112, [\chi])\):

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

\( T_{7} \)

\( T_{17} \)

\( T_{29} \)

\( T_{31} + 5 \)

\( T_{83} \)

Hecke characteristic polynomials

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