Newform orbit 3920.2.a.bt

• Label: 3920.2.a.bt
• Level: $3920$
• Weight: $2$
• Character orbit: 3920.a
• Self dual: yes
• Analytic conductor: $31.301$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(31.3013575923\)
Analytic rank:	\(1\)
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 280)
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.

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

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

3.37228
−2.37228

0

−3.37228

0

1.00000

0

0

0

8.37228

0

1.2

0

2.37228

0

1.00000

0

0

0

2.62772

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(5\)	\(-1\)
\(7\)	\(-1\)

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	3920.2.a.bt		2
4.b	odd	2	1		1960.2.a.s		2
7.b	odd	2	1		560.2.a.h		2
20.d	odd	2	1		9800.2.a.bu		2
21.c	even	2	1		5040.2.a.by		2
28.d	even	2	1		280.2.a.c	✓	2
28.f	even	6	2		1960.2.q.t		4
28.g	odd	6	2		1960.2.q.r		4
35.c	odd	2	1		2800.2.a.bk		2
35.f	even	4	2		2800.2.g.r		4
56.e	even	2	1		2240.2.a.bk		2
56.h	odd	2	1		2240.2.a.bg		2
84.h	odd	2	1		2520.2.a.x		2
140.c	even	2	1		1400.2.a.r		2
140.j	odd	4	2		1400.2.g.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.2.a.c	✓	2	28.d	even	2	1
560.2.a.h		2	7.b	odd	2	1
1400.2.a.r		2	140.c	even	2	1
1400.2.g.i		4	140.j	odd	4	2
1960.2.a.s		2	4.b	odd	2	1
1960.2.q.r		4	28.g	odd	6	2
1960.2.q.t		4	28.f	even	6	2
2240.2.a.bg		2	56.h	odd	2	1
2240.2.a.bk		2	56.e	even	2	1
2520.2.a.x		2	84.h	odd	2	1
2800.2.a.bk		2	35.c	odd	2	1
2800.2.g.r		4	35.f	even	4	2
3920.2.a.bt		2	1.a	even	1	1	trivial
5040.2.a.by		2	21.c	even	2	1
9800.2.a.bu		2	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3920))\):

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

\( T_{11}^{2} + 7T_{11} + 4 \)

\( T_{13}^{2} + 3T_{13} - 6 \)

\( T_{17}^{2} + 5T_{17} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T - 8 \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 7T + 4 \)