Newform orbit 11.5.b.b

• Label: 11.5.b.b
• Level: $11$
• Weight: $5$
• Character orbit: 11.b
• Analytic conductor: $1.137$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	11.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.13706959392\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-30}) \)
Defining polynomial:	\( x^{2} + 30 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{-30}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 3 q^{3} - 14 q^{4} + 31 q^{5} - 3 \beta q^{6} - 10 \beta q^{7} + 2 \beta q^{8} - 72 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 28 q^{4} + 62 q^{5} - 144 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

10.1

	−	5.47723i
		5.47723i

−

5.47723i

−3.00000

−14.0000

31.0000

16.4317i

54.7723i

−

10.9545i

−72.0000

−

169.794i

10.2

5.47723i

−3.00000

−14.0000

31.0000

−

16.4317i

−

54.7723i

10.9545i

−72.0000

169.794i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	11.5.b.b	✓	2
3.b	odd	2	1		99.5.c.b		2
4.b	odd	2	1		176.5.h.c		2
5.b	even	2	1		275.5.c.e		2
5.c	odd	4	2		275.5.d.b		4
8.b	even	2	1		704.5.h.f		2
8.d	odd	2	1		704.5.h.d		2
11.b	odd	2	1	inner	11.5.b.b	✓	2
11.c	even	5	4		121.5.d.b		8
11.d	odd	10	4		121.5.d.b		8
33.d	even	2	1		99.5.c.b		2
44.c	even	2	1		176.5.h.c		2
55.d	odd	2	1		275.5.c.e		2
55.e	even	4	2		275.5.d.b		4
88.b	odd	2	1		704.5.h.f		2
88.g	even	2	1		704.5.h.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.5.b.b	✓	2	1.a	even	1	1	trivial
11.5.b.b	✓	2	11.b	odd	2	1	inner
99.5.c.b		2	3.b	odd	2	1
99.5.c.b		2	33.d	even	2	1
121.5.d.b		8	11.c	even	5	4
121.5.d.b		8	11.d	odd	10	4
176.5.h.c		2	4.b	odd	2	1
176.5.h.c		2	44.c	even	2	1
275.5.c.e		2	5.b	even	2	1
275.5.c.e		2	55.d	odd	2	1
275.5.d.b		4	5.c	odd	4	2
275.5.d.b		4	55.e	even	4	2
704.5.h.d		2	8.d	odd	2	1
704.5.h.d		2	88.g	even	2	1
704.5.h.f		2	8.b	even	2	1
704.5.h.f		2	88.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 30 \) acting on \(S_{5}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 30 \)
$3$	\( (T + 3)^{2} \)
$5$	\( (T - 31)^{2} \)
$7$	\( T^{2} + 3000 \)
$11$	\( T^{2} - 22T + 14641 \)