Newform orbit 1568.3.g.c

• Label: 1568.3.g.c
• Level: $1568$
• Weight: $3$
• Character orbit: 1568.g
• Analytic conductor: $42.725$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(42.7249054517\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{3} - 3 \beta q^{5} - 8 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(1471\)	\(1473\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

687.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−1.00000

0

−

5.19615i

0

0

0

−8.00000

0

687.2

0

−1.00000

0

5.19615i

0

0

0

−8.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1568.3.g.c		2
4.b	odd	2	1		392.3.g.e		2
7.b	odd	2	1		1568.3.g.f		2
7.c	even	3	1		224.3.o.a		2
7.c	even	3	1		224.3.o.b		2
8.b	even	2	1		392.3.g.e		2
8.d	odd	2	1	inner	1568.3.g.c		2
28.d	even	2	1		392.3.g.d		2
28.f	even	6	1		392.3.k.a		2
28.f	even	6	1		392.3.k.c		2
28.g	odd	6	1		56.3.k.a	✓	2
28.g	odd	6	1		56.3.k.b	yes	2
56.e	even	2	1		1568.3.g.f		2
56.h	odd	2	1		392.3.g.d		2
56.j	odd	6	1		392.3.k.a		2
56.j	odd	6	1		392.3.k.c		2
56.k	odd	6	1		224.3.o.a		2
56.k	odd	6	1		224.3.o.b		2
56.p	even	6	1		56.3.k.a	✓	2
56.p	even	6	1		56.3.k.b	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.3.k.a	✓	2	28.g	odd	6	1
56.3.k.a	✓	2	56.p	even	6	1
56.3.k.b	yes	2	28.g	odd	6	1
56.3.k.b	yes	2	56.p	even	6	1
224.3.o.a		2	7.c	even	3	1
224.3.o.a		2	56.k	odd	6	1
224.3.o.b		2	7.c	even	3	1
224.3.o.b		2	56.k	odd	6	1
392.3.g.d		2	28.d	even	2	1
392.3.g.d		2	56.h	odd	2	1
392.3.g.e		2	4.b	odd	2	1
392.3.g.e		2	8.b	even	2	1
392.3.k.a		2	28.f	even	6	1
392.3.k.a		2	56.j	odd	6	1
392.3.k.c		2	28.f	even	6	1
392.3.k.c		2	56.j	odd	6	1
1568.3.g.c		2	1.a	even	1	1	trivial
1568.3.g.c		2	8.d	odd	2	1	inner
1568.3.g.f		2	7.b	odd	2	1
1568.3.g.f		2	56.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 1)^{2} \)
$5$	\( T^{2} + 27 \)
$7$	\( T^{2} \)
$11$	\( (T + 17)^{2} \)