Newform orbit 1638.2.r.e

• Label: 1638.2.r.e
• Level: $1638$
• Weight: $2$
• Character orbit: 1638.r
• Analytic conductor: $13.079$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0794958511\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 182)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\)	\(379\)	\(703\)	\(911\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

757.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−1.00000

0

0.500000

−

0.866025i

1.00000

0

0.500000

+

0.866025i

1387.1

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−1.00000

0

0.500000

+

0.866025i

1.00000

0

0.500000

−

0.866025i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1638.2.r.e		2
3.b	odd	2	1		182.2.g.d	✓	2
12.b	even	2	1		1456.2.s.b		2
13.c	even	3	1	inner	1638.2.r.e		2
21.c	even	2	1		1274.2.g.b		2
21.g	even	6	1		1274.2.e.c		2
21.g	even	6	1		1274.2.h.m		2
21.h	odd	6	1		1274.2.e.j		2
21.h	odd	6	1		1274.2.h.d		2
39.h	odd	6	1		2366.2.a.i		1
39.i	odd	6	1		182.2.g.d	✓	2
39.i	odd	6	1		2366.2.a.b		1
39.k	even	12	2		2366.2.d.c		2
156.p	even	6	1		1456.2.s.b		2
273.r	even	6	1		1274.2.h.m		2
273.s	odd	6	1		1274.2.h.d		2
273.bf	even	6	1		1274.2.e.c		2
273.bm	odd	6	1		1274.2.e.j		2
273.bn	even	6	1		1274.2.g.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
182.2.g.d	✓	2	3.b	odd	2	1
182.2.g.d	✓	2	39.i	odd	6	1
1274.2.e.c		2	21.g	even	6	1
1274.2.e.c		2	273.bf	even	6	1
1274.2.e.j		2	21.h	odd	6	1
1274.2.e.j		2	273.bm	odd	6	1
1274.2.g.b		2	21.c	even	2	1
1274.2.g.b		2	273.bn	even	6	1
1274.2.h.d		2	21.h	odd	6	1
1274.2.h.d		2	273.s	odd	6	1
1274.2.h.m		2	21.g	even	6	1
1274.2.h.m		2	273.r	even	6	1
1456.2.s.b		2	12.b	even	2	1
1456.2.s.b		2	156.p	even	6	1
1638.2.r.e		2	1.a	even	1	1	trivial
1638.2.r.e		2	13.c	even	3	1	inner
2366.2.a.b		1	39.i	odd	6	1
2366.2.a.i		1	39.h	odd	6	1
2366.2.d.c		2	39.k	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5} + 1 \)

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

\( T_{17}^{2} - T_{17} + 1 \)

Hecke characteristic polynomials

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