Newform orbit 363.2.a.f

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

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:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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

−1.73205
1.73205

−1.73205

−1.00000

1.00000

−3.00000

1.73205

3.46410

1.73205

1.00000

5.19615

1.2

1.73205

−1.00000

1.00000

−3.00000

−1.73205

−3.46410

−1.73205

1.00000

−5.19615

Atkin-Lehner signs

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

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	363.2.a.f	✓	2
3.b	odd	2	1		1089.2.a.o		2
4.b	odd	2	1		5808.2.a.ca		2
5.b	even	2	1		9075.2.a.bo		2
11.b	odd	2	1	inner	363.2.a.f	✓	2
11.c	even	5	4		363.2.e.m		8
11.d	odd	10	4		363.2.e.m		8
33.d	even	2	1		1089.2.a.o		2
44.c	even	2	1		5808.2.a.ca		2
55.d	odd	2	1		9075.2.a.bo		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.2.a.f	✓	2	1.a	even	1	1	trivial
363.2.a.f	✓	2	11.b	odd	2	1	inner
363.2.e.m		8	11.c	even	5	4
363.2.e.m		8	11.d	odd	10	4
1089.2.a.o		2	3.b	odd	2	1
1089.2.a.o		2	33.d	even	2	1
5808.2.a.ca		2	4.b	odd	2	1
5808.2.a.ca		2	44.c	even	2	1
9075.2.a.bo		2	5.b	even	2	1
9075.2.a.bo		2	55.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3 \)
$3$	\( (T + 1)^{2} \)
$5$	\( (T + 3)^{2} \)
$7$	\( T^{2} - 12 \)
$11$	\( T^{2} \)