Newform orbit 1089.2.a.l

• Label: 1089.2.a.l
• Level: $1089$
• Weight: $2$
• Character orbit: 1089.a
• Self dual: yes
• Analytic conductor: $8.696$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.69570878012\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

1.61803
−0.618034

−2.61803

0

4.85410

0.618034

0

1.00000

−7.47214

0

−1.61803

1.2

−0.381966

0

−1.85410

−1.61803

0

1.00000

1.47214

0

0.618034

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(11\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1089.2.a.l		2
3.b	odd	2	1		363.2.a.i		2
11.b	odd	2	1		1089.2.a.t		2
11.d	odd	10	2		99.2.f.a		4
12.b	even	2	1		5808.2.a.ci		2
15.d	odd	2	1		9075.2.a.u		2
33.d	even	2	1		363.2.a.d		2
33.f	even	10	2		33.2.e.b	✓	4
33.f	even	10	2		363.2.e.k		4
33.h	odd	10	2		363.2.e.b		4
33.h	odd	10	2		363.2.e.f		4
99.o	odd	30	4		891.2.n.b		8
99.p	even	30	4		891.2.n.c		8
132.d	odd	2	1		5808.2.a.cj		2
132.n	odd	10	2		528.2.y.b		4
165.d	even	2	1		9075.2.a.cb		2
165.r	even	10	2		825.2.n.c		4
165.u	odd	20	4		825.2.bx.d		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.e.b	✓	4	33.f	even	10	2
99.2.f.a		4	11.d	odd	10	2
363.2.a.d		2	33.d	even	2	1
363.2.a.i		2	3.b	odd	2	1
363.2.e.b		4	33.h	odd	10	2
363.2.e.f		4	33.h	odd	10	2
363.2.e.k		4	33.f	even	10	2
528.2.y.b		4	132.n	odd	10	2
825.2.n.c		4	165.r	even	10	2
825.2.bx.d		8	165.u	odd	20	4
891.2.n.b		8	99.o	odd	30	4
891.2.n.c		8	99.p	even	30	4
1089.2.a.l		2	1.a	even	1	1	trivial
1089.2.a.t		2	11.b	odd	2	1
5808.2.a.ci		2	12.b	even	2	1
5808.2.a.cj		2	132.d	odd	2	1
9075.2.a.u		2	15.d	odd	2	1
9075.2.a.cb		2	165.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{2} + 3T_{2} + 1 \)

\( T_{5}^{2} + T_{5} - 1 \)

\( T_{7} - 1 \)

Hecke characteristic polynomials

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