Newform orbit 3887.2.a.i

• Label: 3887.2.a.i
• Level: $3887$
• Weight: $2$
• Character orbit: 3887.a
• Self dual: yes
• Analytic conductor: $31.038$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3887 = 13^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3887.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(31.0378512657\)
Analytic rank:	\(0\)
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 23)
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.

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

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

−0.618034
1.61803

−0.618034

−2.23607

−1.61803

−1.23607

1.38197

−3.23607

2.23607

2.00000

0.763932

1.2

1.61803

2.23607

0.618034

3.23607

3.61803

1.23607

−2.23607

2.00000

5.23607

Atkin-Lehner signs

\( p \)	Sign
\(13\)	\(1\)
\(23\)	\(-1\)

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	3887.2.a.i		2
13.b	even	2	1		23.2.a.a	✓	2
39.d	odd	2	1		207.2.a.d		2
52.b	odd	2	1		368.2.a.h		2
65.d	even	2	1		575.2.a.f		2
65.h	odd	4	2		575.2.b.d		4
91.b	odd	2	1		1127.2.a.c		2
104.e	even	2	1		1472.2.a.t		2
104.h	odd	2	1		1472.2.a.s		2
143.d	odd	2	1		2783.2.a.c		2
156.h	even	2	1		3312.2.a.ba		2
195.e	odd	2	1		5175.2.a.be		2
221.b	even	2	1		6647.2.a.b		2
247.d	odd	2	1		8303.2.a.e		2
260.g	odd	2	1		9200.2.a.bt		2
299.c	odd	2	1		529.2.a.a		2
299.o	odd	22	10		529.2.c.n		20
299.p	even	22	10		529.2.c.o		20
897.g	even	2	1		4761.2.a.w		2
1196.d	even	2	1		8464.2.a.bb		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
23.2.a.a	✓	2	13.b	even	2	1
207.2.a.d		2	39.d	odd	2	1
368.2.a.h		2	52.b	odd	2	1
529.2.a.a		2	299.c	odd	2	1
529.2.c.n		20	299.o	odd	22	10
529.2.c.o		20	299.p	even	22	10
575.2.a.f		2	65.d	even	2	1
575.2.b.d		4	65.h	odd	4	2
1127.2.a.c		2	91.b	odd	2	1
1472.2.a.s		2	104.h	odd	2	1
1472.2.a.t		2	104.e	even	2	1
2783.2.a.c		2	143.d	odd	2	1
3312.2.a.ba		2	156.h	even	2	1
3887.2.a.i		2	1.a	even	1	1	trivial
4761.2.a.w		2	897.g	even	2	1
5175.2.a.be		2	195.e	odd	2	1
6647.2.a.b		2	221.b	even	2	1
8303.2.a.e		2	247.d	odd	2	1
8464.2.a.bb		2	1196.d	even	2	1
9200.2.a.bt		2	260.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3887))\):

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

\( T_{3}^{2} - 5 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 1 \)
$3$	\( T^{2} - 5 \)
$5$	\( T^{2} - 2T - 4 \)
$7$	\( T^{2} + 2T - 4 \)
$11$	\( T^{2} - 6T + 4 \)