Newform orbit 784.3.s.a

• Label: 784.3.s.a
• Level: $784$
• Weight: $3$
• Character orbit: 784.s
• Analytic conductor: $21.362$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -7
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.3624527258\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

129.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

0

0

0

0

−4.50000

−

7.79423i

0

705.1

0

0

0

0

0

0

0

−4.50000

+

7.79423i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
7.c	even	3	1	inner
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.3.s.a		2
4.b	odd	2	1		49.3.d.a		2
7.b	odd	2	1	CM	784.3.s.a		2
7.c	even	3	1		112.3.c.a		1
7.c	even	3	1	inner	784.3.s.a		2
7.d	odd	6	1		112.3.c.a		1
7.d	odd	6	1	inner	784.3.s.a		2
12.b	even	2	1		441.3.m.a		2
21.g	even	6	1		1008.3.f.a		1
21.h	odd	6	1		1008.3.f.a		1
28.d	even	2	1		49.3.d.a		2
28.f	even	6	1		7.3.b.a	✓	1
28.f	even	6	1		49.3.d.a		2
28.g	odd	6	1		7.3.b.a	✓	1
28.g	odd	6	1		49.3.d.a		2
56.j	odd	6	1		448.3.c.b		1
56.k	odd	6	1		448.3.c.a		1
56.m	even	6	1		448.3.c.a		1
56.p	even	6	1		448.3.c.b		1
84.h	odd	2	1		441.3.m.a		2
84.j	odd	6	1		63.3.d.a		1
84.j	odd	6	1		441.3.m.a		2
84.n	even	6	1		63.3.d.a		1
84.n	even	6	1		441.3.m.a		2
140.p	odd	6	1		175.3.d.a		1
140.s	even	6	1		175.3.d.a		1
140.w	even	12	2		175.3.c.a		2
140.x	odd	12	2		175.3.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.3.b.a	✓	1	28.f	even	6	1
7.3.b.a	✓	1	28.g	odd	6	1
49.3.d.a		2	4.b	odd	2	1
49.3.d.a		2	28.d	even	2	1
49.3.d.a		2	28.f	even	6	1
49.3.d.a		2	28.g	odd	6	1
63.3.d.a		1	84.j	odd	6	1
63.3.d.a		1	84.n	even	6	1
112.3.c.a		1	7.c	even	3	1
112.3.c.a		1	7.d	odd	6	1
175.3.c.a		2	140.w	even	12	2
175.3.c.a		2	140.x	odd	12	2
175.3.d.a		1	140.p	odd	6	1
175.3.d.a		1	140.s	even	6	1
441.3.m.a		2	12.b	even	2	1
441.3.m.a		2	84.h	odd	2	1
441.3.m.a		2	84.j	odd	6	1
441.3.m.a		2	84.n	even	6	1
448.3.c.a		1	56.k	odd	6	1
448.3.c.a		1	56.m	even	6	1
448.3.c.b		1	56.j	odd	6	1
448.3.c.b		1	56.p	even	6	1
784.3.s.a		2	1.a	even	1	1	trivial
784.3.s.a		2	7.b	odd	2	1	CM
784.3.s.a		2	7.c	even	3	1	inner
784.3.s.a		2	7.d	odd	6	1	inner
1008.3.f.a		1	21.g	even	6	1
1008.3.f.a		1	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{3}^{\mathrm{new}}(784, [\chi])\).

Hecke characteristic polynomials

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