Newform orbit 1386.2.a.m

• Label: 1386.2.a.m
• Level: $1386$
• Weight: $2$
• Character orbit: 1386.a
• Self dual: yes
• Analytic conductor: $11.067$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.0672657201\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

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

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

−1.00000

0

1.00000

−3.23607

0

1.00000

−1.00000

0

3.23607

1.2

−1.00000

0

1.00000

1.23607

0

1.00000

−1.00000

0

−1.23607

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(-1\)
\(7\)	\(-1\)
\(11\)	\(1\)

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	1386.2.a.m		2
3.b	odd	2	1		154.2.a.d	✓	2
7.b	odd	2	1		9702.2.a.cu		2
12.b	even	2	1		1232.2.a.p		2
15.d	odd	2	1		3850.2.a.bj		2
15.e	even	4	2		3850.2.c.q		4
21.c	even	2	1		1078.2.a.w		2
21.g	even	6	2		1078.2.e.n		4
21.h	odd	6	2		1078.2.e.q		4
24.f	even	2	1		4928.2.a.bk		2
24.h	odd	2	1		4928.2.a.bt		2
33.d	even	2	1		1694.2.a.l		2
84.h	odd	2	1		8624.2.a.bf		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
154.2.a.d	✓	2	3.b	odd	2	1
1078.2.a.w		2	21.c	even	2	1
1078.2.e.n		4	21.g	even	6	2
1078.2.e.q		4	21.h	odd	6	2
1232.2.a.p		2	12.b	even	2	1
1386.2.a.m		2	1.a	even	1	1	trivial
1694.2.a.l		2	33.d	even	2	1
3850.2.a.bj		2	15.d	odd	2	1
3850.2.c.q		4	15.e	even	4	2
4928.2.a.bk		2	24.f	even	2	1
4928.2.a.bt		2	24.h	odd	2	1
8624.2.a.bf		2	84.h	odd	2	1
9702.2.a.cu		2	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5}^{2} + 2T_{5} - 4 \)

\( T_{13}^{2} + 2T_{13} - 4 \)

\( T_{17}^{2} - 4T_{17} - 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T - 4 \)
$7$	\( (T - 1)^{2} \)
$11$	\( (T + 1)^{2} \)