Newform orbit 363.2.a.e

• Label: 363.2.a.e
• Level: $363$
• Weight: $2$
• Character orbit: 363.a
• Self dual: yes
• Analytic conductor: $2.899$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.89856959337\)
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.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} + (\beta - 2) q^{5} - \beta q^{6} - 3 q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \)

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

1.61803
−0.618034

−1.61803

1.00000

0.618034

−0.381966

−1.61803

−3.00000

2.23607

1.00000

0.618034

1.2

0.618034

1.00000

−1.61803

−2.61803

0.618034

−3.00000

−2.23607

1.00000

−1.61803

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	363.2.a.e		2
3.b	odd	2	1		1089.2.a.s		2
4.b	odd	2	1		5808.2.a.bm		2
5.b	even	2	1		9075.2.a.bv		2
11.b	odd	2	1		363.2.a.h		2
11.c	even	5	2		363.2.e.c		4
11.c	even	5	2		363.2.e.j		4
11.d	odd	10	2		33.2.e.a	✓	4
11.d	odd	10	2		363.2.e.h		4
33.d	even	2	1		1089.2.a.m		2
33.f	even	10	2		99.2.f.b		4
44.c	even	2	1		5808.2.a.bl		2
44.g	even	10	2		528.2.y.f		4
55.d	odd	2	1		9075.2.a.x		2
55.h	odd	10	2		825.2.n.f		4
55.l	even	20	4		825.2.bx.b		8
99.o	odd	30	4		891.2.n.d		8
99.p	even	30	4		891.2.n.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.e.a	✓	4	11.d	odd	10	2
99.2.f.b		4	33.f	even	10	2
363.2.a.e		2	1.a	even	1	1	trivial
363.2.a.h		2	11.b	odd	2	1
363.2.e.c		4	11.c	even	5	2
363.2.e.h		4	11.d	odd	10	2
363.2.e.j		4	11.c	even	5	2
528.2.y.f		4	44.g	even	10	2
825.2.n.f		4	55.h	odd	10	2
825.2.bx.b		8	55.l	even	20	4
891.2.n.a		8	99.p	even	30	4
891.2.n.d		8	99.o	odd	30	4
1089.2.a.m		2	33.d	even	2	1
1089.2.a.s		2	3.b	odd	2	1
5808.2.a.bl		2	44.c	even	2	1
5808.2.a.bm		2	4.b	odd	2	1
9075.2.a.x		2	55.d	odd	2	1
9075.2.a.bv		2	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\).

Hecke characteristic polynomials

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