Newform orbit 550.2.a.n

• Label: 550.2.a.n
• Level: $550$
• Weight: $2$
• Character orbit: 550.a
• Self dual: yes
• Analytic conductor: $4.392$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	550.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.39177211117\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{33}) \)
Defining polynomial:	\( x^{2} - x - 8 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + (\beta + 5) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} - q^{7} + 2 q^{8} + 11 q^{9}+O(q^{10}) \)

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

−2.37228
3.37228

1.00000

−2.37228

1.00000

0

−2.37228

2.37228

1.00000

2.62772

0

1.2

1.00000

3.37228

1.00000

0

3.37228

−3.37228

1.00000

8.37228

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(1\)
\(11\)	\(1\)

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	550.2.a.n		2
3.b	odd	2	1		4950.2.a.bw		2
4.b	odd	2	1		4400.2.a.bl		2
5.b	even	2	1		110.2.a.d	✓	2
5.c	odd	4	2		550.2.b.f		4
11.b	odd	2	1		6050.2.a.cb		2
15.d	odd	2	1		990.2.a.m		2
15.e	even	4	2		4950.2.c.bc		4
20.d	odd	2	1		880.2.a.n		2
20.e	even	4	2		4400.2.b.p		4
35.c	odd	2	1		5390.2.a.bp		2
40.e	odd	2	1		3520.2.a.bj		2
40.f	even	2	1		3520.2.a.bq		2
55.d	odd	2	1		1210.2.a.r		2
60.h	even	2	1		7920.2.a.bq		2
220.g	even	2	1		9680.2.a.bt		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
110.2.a.d	✓	2	5.b	even	2	1
550.2.a.n		2	1.a	even	1	1	trivial
550.2.b.f		4	5.c	odd	4	2
880.2.a.n		2	20.d	odd	2	1
990.2.a.m		2	15.d	odd	2	1
1210.2.a.r		2	55.d	odd	2	1
3520.2.a.bj		2	40.e	odd	2	1
3520.2.a.bq		2	40.f	even	2	1
4400.2.a.bl		2	4.b	odd	2	1
4400.2.b.p		4	20.e	even	4	2
4950.2.a.bw		2	3.b	odd	2	1
4950.2.c.bc		4	15.e	even	4	2
5390.2.a.bp		2	35.c	odd	2	1
6050.2.a.cb		2	11.b	odd	2	1
7920.2.a.bq		2	60.h	even	2	1
9680.2.a.bt		2	220.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(550))\):

\( T_{3}^{2} - T_{3} - 8 \)

\( T_{7}^{2} + T_{7} - 8 \)

\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( T^{2} - T - 8 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + T - 8 \)
$11$	\( (T + 1)^{2} \)