Newform orbit 1782.2.e.e

• Label: 1782.2.e.e
• Level: $1782$
• Weight: $2$
• Character orbit: 1782.e
• Analytic conductor: $14.229$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1782 = 2 \cdot 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1782.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2293416402\)
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 66)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times\).

\(n\)	\(1135\)	\(1541\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

595.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−1.00000

+

1.73205i

0

2.00000

+

3.46410i

1.00000

0

2.00000

1189.1

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−1.00000

−

1.73205i

0

2.00000

−

3.46410i

1.00000

0

2.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1782.2.e.e		2
3.b	odd	2	1		1782.2.e.v		2
9.c	even	3	1		66.2.a.b	✓	1
9.c	even	3	1	inner	1782.2.e.e		2
9.d	odd	6	1		198.2.a.a		1
9.d	odd	6	1		1782.2.e.v		2
36.f	odd	6	1		528.2.a.j		1
36.h	even	6	1		1584.2.a.f		1
45.h	odd	6	1		4950.2.a.bu		1
45.j	even	6	1		1650.2.a.k		1
45.k	odd	12	2		1650.2.c.e		2
45.l	even	12	2		4950.2.c.p		2
63.l	odd	6	1		3234.2.a.t		1
63.o	even	6	1		9702.2.a.x		1
72.j	odd	6	1		6336.2.a.bw		1
72.l	even	6	1		6336.2.a.cj		1
72.n	even	6	1		2112.2.a.r		1
72.p	odd	6	1		2112.2.a.e		1
99.g	even	6	1		2178.2.a.g		1
99.h	odd	6	1		726.2.a.c		1
99.m	even	15	4		726.2.e.g		4
99.o	odd	30	4		726.2.e.o		4
396.k	even	6	1		5808.2.a.bc		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.2.a.b	✓	1	9.c	even	3	1
198.2.a.a		1	9.d	odd	6	1
528.2.a.j		1	36.f	odd	6	1
726.2.a.c		1	99.h	odd	6	1
726.2.e.g		4	99.m	even	15	4
726.2.e.o		4	99.o	odd	30	4
1584.2.a.f		1	36.h	even	6	1
1650.2.a.k		1	45.j	even	6	1
1650.2.c.e		2	45.k	odd	12	2
1782.2.e.e		2	1.a	even	1	1	trivial
1782.2.e.e		2	9.c	even	3	1	inner
1782.2.e.v		2	3.b	odd	2	1
1782.2.e.v		2	9.d	odd	6	1
2112.2.a.e		1	72.p	odd	6	1
2112.2.a.r		1	72.n	even	6	1
2178.2.a.g		1	99.g	even	6	1
3234.2.a.t		1	63.l	odd	6	1
4950.2.a.bu		1	45.h	odd	6	1
4950.2.c.p		2	45.l	even	12	2
5808.2.a.bc		1	396.k	even	6	1
6336.2.a.bw		1	72.j	odd	6	1
6336.2.a.cj		1	72.l	even	6	1
9702.2.a.x		1	63.o	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1782, [\chi])\):

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

\( T_{7}^{2} - 4T_{7} + 16 \)

\( T_{13}^{2} - 6T_{13} + 36 \)

\( T_{17} - 2 \)

\( T_{23}^{2} + 4T_{23} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T + 4 \)
$7$	\( T^{2} - 4T + 16 \)
$11$	\( T^{2} - T + 1 \)