Newform orbit 784.2.p.e

• Label: 784.2.p.e
• Level: $784$
• Weight: $2$
• Character orbit: 784.p
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(687\)	\(689\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

31.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

1.00000

−

1.73205i

0

−3.00000

+

1.73205i

0

0

0

−0.500000

−

0.866025i

0

607.1

0

1.00000

+

1.73205i

0

−3.00000

−

1.73205i

0

0

0

−0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.2.p.e		2
4.b	odd	2	1		784.2.p.a		2
7.b	odd	2	1		784.2.p.b		2
7.c	even	3	1		112.2.f.a	✓	2
7.c	even	3	1		784.2.p.f		2
7.d	odd	6	1		112.2.f.b	yes	2
7.d	odd	6	1		784.2.p.a		2
21.g	even	6	1		1008.2.b.b		2
21.h	odd	6	1		1008.2.b.g		2
28.d	even	2	1		784.2.p.f		2
28.f	even	6	1		112.2.f.a	✓	2
28.f	even	6	1	inner	784.2.p.e		2
28.g	odd	6	1		112.2.f.b	yes	2
28.g	odd	6	1		784.2.p.b		2
35.i	odd	6	1		2800.2.k.b		2
35.j	even	6	1		2800.2.k.e		2
35.k	even	12	2		2800.2.e.b		4
35.l	odd	12	2		2800.2.e.c		4
56.j	odd	6	1		448.2.f.a		2
56.k	odd	6	1		448.2.f.a		2
56.m	even	6	1		448.2.f.c		2
56.p	even	6	1		448.2.f.c		2
84.j	odd	6	1		1008.2.b.g		2
84.n	even	6	1		1008.2.b.b		2
112.u	odd	12	2		1792.2.e.c		4
112.v	even	12	2		1792.2.e.a		4
112.w	even	12	2		1792.2.e.a		4
112.x	odd	12	2		1792.2.e.c		4
140.p	odd	6	1		2800.2.k.b		2
140.s	even	6	1		2800.2.k.e		2
140.w	even	12	2		2800.2.e.b		4
140.x	odd	12	2		2800.2.e.c		4
168.s	odd	6	1		4032.2.b.h		2
168.v	even	6	1		4032.2.b.b		2
168.ba	even	6	1		4032.2.b.b		2
168.be	odd	6	1		4032.2.b.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.f.a	✓	2	7.c	even	3	1
112.2.f.a	✓	2	28.f	even	6	1
112.2.f.b	yes	2	7.d	odd	6	1
112.2.f.b	yes	2	28.g	odd	6	1
448.2.f.a		2	56.j	odd	6	1
448.2.f.a		2	56.k	odd	6	1
448.2.f.c		2	56.m	even	6	1
448.2.f.c		2	56.p	even	6	1
784.2.p.a		2	4.b	odd	2	1
784.2.p.a		2	7.d	odd	6	1
784.2.p.b		2	7.b	odd	2	1
784.2.p.b		2	28.g	odd	6	1
784.2.p.e		2	1.a	even	1	1	trivial
784.2.p.e		2	28.f	even	6	1	inner
784.2.p.f		2	7.c	even	3	1
784.2.p.f		2	28.d	even	2	1
1008.2.b.b		2	21.g	even	6	1
1008.2.b.b		2	84.n	even	6	1
1008.2.b.g		2	21.h	odd	6	1
1008.2.b.g		2	84.j	odd	6	1
1792.2.e.a		4	112.v	even	12	2
1792.2.e.a		4	112.w	even	12	2
1792.2.e.c		4	112.u	odd	12	2
1792.2.e.c		4	112.x	odd	12	2
2800.2.e.b		4	35.k	even	12	2
2800.2.e.b		4	140.w	even	12	2
2800.2.e.c		4	35.l	odd	12	2
2800.2.e.c		4	140.x	odd	12	2
2800.2.k.b		2	35.i	odd	6	1
2800.2.k.b		2	140.p	odd	6	1
2800.2.k.e		2	35.j	even	6	1
2800.2.k.e		2	140.s	even	6	1
4032.2.b.b		2	168.v	even	6	1
4032.2.b.b		2	168.ba	even	6	1
4032.2.b.h		2	168.s	odd	6	1
4032.2.b.h		2	168.be	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 4 \)

\( T_{5}^{2} + 6T_{5} + 12 \)

\( T_{11}^{2} - 6T_{11} + 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T + 4 \)
$5$	\( T^{2} + 6T + 12 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 6T + 12 \)