Newform orbit 2016.2.s.a

• Label: 2016.2.s.a
• Level: $2016$
• Weight: $2$
• Character orbit: 2016.s
• Analytic conductor: $16.098$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.0978410475\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(577\)	\(1765\)	\(1793\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

289.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

−2.00000

+

3.46410i

0

−2.50000

+

0.866025i

0

0

0

865.1

0

0

0

−2.00000

−

3.46410i

0

−2.50000

−

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2016.2.s.a		2
3.b	odd	2	1		672.2.q.j	yes	2
4.b	odd	2	1		2016.2.s.b		2
7.c	even	3	1	inner	2016.2.s.a		2
12.b	even	2	1		672.2.q.e	✓	2
21.g	even	6	1		4704.2.a.bh		1
21.h	odd	6	1		672.2.q.j	yes	2
21.h	odd	6	1		4704.2.a.a		1
24.f	even	2	1		1344.2.q.l		2
24.h	odd	2	1		1344.2.q.a		2
28.g	odd	6	1		2016.2.s.b		2
84.j	odd	6	1		4704.2.a.p		1
84.n	even	6	1		672.2.q.e	✓	2
84.n	even	6	1		4704.2.a.r		1
168.s	odd	6	1		1344.2.q.a		2
168.s	odd	6	1		9408.2.a.dd		1
168.v	even	6	1		1344.2.q.l		2
168.v	even	6	1		9408.2.a.bp		1
168.ba	even	6	1		9408.2.a.a		1
168.be	odd	6	1		9408.2.a.bs		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.2.q.e	✓	2	12.b	even	2	1
672.2.q.e	✓	2	84.n	even	6	1
672.2.q.j	yes	2	3.b	odd	2	1
672.2.q.j	yes	2	21.h	odd	6	1
1344.2.q.a		2	24.h	odd	2	1
1344.2.q.a		2	168.s	odd	6	1
1344.2.q.l		2	24.f	even	2	1
1344.2.q.l		2	168.v	even	6	1
2016.2.s.a		2	1.a	even	1	1	trivial
2016.2.s.a		2	7.c	even	3	1	inner
2016.2.s.b		2	4.b	odd	2	1
2016.2.s.b		2	28.g	odd	6	1
4704.2.a.a		1	21.h	odd	6	1
4704.2.a.p		1	84.j	odd	6	1
4704.2.a.r		1	84.n	even	6	1
4704.2.a.bh		1	21.g	even	6	1
9408.2.a.a		1	168.ba	even	6	1
9408.2.a.bp		1	168.v	even	6	1
9408.2.a.bs		1	168.be	odd	6	1
9408.2.a.dd		1	168.s	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\):

\( T_{5}^{2} + 4T_{5} + 16 \)

\( T_{11}^{2} - 6T_{11} + 36 \)

\( T_{13} - 5 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4T + 16 \)
$7$	\( T^{2} + 5T + 7 \)
$11$	\( T^{2} - 6T + 36 \)