Newform orbit 1568.2.i.m

• Label: 1568.2.i.m
• Level: $1568$
• Weight: $2$
• Character orbit: 1568.i
• Analytic conductor: $12.521$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.5205430369\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial:	\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_{2} - \beta_1 - 1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{5} - 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} + 2 \)

Character values

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

\(n\)	\(197\)	\(1471\)	\(1473\)
\(\chi(n)\)	\(1\)	\(1\)	\(\beta_{1}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

961.1

0.809017	+	1.40126i
−0.309017	−	0.535233i
0.809017	−	1.40126i
−0.309017	+	0.535233i

0

−1.61803

−

2.80252i

0

0.618034

−

1.07047i

0

0

0

−3.73607

+

6.47106i

0

961.2

0

0.618034

+

1.07047i

0

−1.61803

+

2.80252i

0

0

0

0.736068

−

1.27491i

0

1537.1

0

−1.61803

+

2.80252i

0

0.618034

+

1.07047i

0

0

0

−3.73607

−

6.47106i

0

1537.2

0

0.618034

−

1.07047i

0

−1.61803

−

2.80252i

0

0

0

0.736068

+

1.27491i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1568.2.i.m		4
4.b	odd	2	1		1568.2.i.v		4
7.b	odd	2	1		1568.2.i.w		4
7.c	even	3	1		224.2.a.d	yes	2
7.c	even	3	1	inner	1568.2.i.m		4
7.d	odd	6	1		1568.2.a.k		2
7.d	odd	6	1		1568.2.i.w		4
21.h	odd	6	1		2016.2.a.o		2
28.d	even	2	1		1568.2.i.n		4
28.f	even	6	1		1568.2.a.v		2
28.f	even	6	1		1568.2.i.n		4
28.g	odd	6	1		224.2.a.c	✓	2
28.g	odd	6	1		1568.2.i.v		4
35.j	even	6	1		5600.2.a.z		2
56.j	odd	6	1		3136.2.a.by		2
56.k	odd	6	1		448.2.a.j		2
56.m	even	6	1		3136.2.a.bf		2
56.p	even	6	1		448.2.a.i		2
84.n	even	6	1		2016.2.a.r		2
112.u	odd	12	2		1792.2.b.k		4
112.w	even	12	2		1792.2.b.m		4
140.p	odd	6	1		5600.2.a.bk		2
168.s	odd	6	1		4032.2.a.bv		2
168.v	even	6	1		4032.2.a.bw		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.a.c	✓	2	28.g	odd	6	1
224.2.a.d	yes	2	7.c	even	3	1
448.2.a.i		2	56.p	even	6	1
448.2.a.j		2	56.k	odd	6	1
1568.2.a.k		2	7.d	odd	6	1
1568.2.a.v		2	28.f	even	6	1
1568.2.i.m		4	1.a	even	1	1	trivial
1568.2.i.m		4	7.c	even	3	1	inner
1568.2.i.n		4	28.d	even	2	1
1568.2.i.n		4	28.f	even	6	1
1568.2.i.v		4	4.b	odd	2	1
1568.2.i.v		4	28.g	odd	6	1
1568.2.i.w		4	7.b	odd	2	1
1568.2.i.w		4	7.d	odd	6	1
1792.2.b.k		4	112.u	odd	12	2
1792.2.b.m		4	112.w	even	12	2
2016.2.a.o		2	21.h	odd	6	1
2016.2.a.r		2	84.n	even	6	1
3136.2.a.bf		2	56.m	even	6	1
3136.2.a.by		2	56.j	odd	6	1
4032.2.a.bv		2	168.s	odd	6	1
4032.2.a.bw		2	168.v	even	6	1
5600.2.a.z		2	35.j	even	6	1
5600.2.a.bk		2	140.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 16 \)

\( T_{5}^{4} + 2T_{5}^{3} + 8T_{5}^{2} - 8T_{5} + 16 \)

\( T_{11}^{4} + 4T_{11}^{3} + 32T_{11}^{2} - 64T_{11} + 256 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	T^4 + 2*T^3 + 8*T^2 - 8*T + 16
$5$	T^4 + 2*T^3 + 8*T^2 - 8*T + 16
$7$	\( T^{4} \)
$11$	T^4 + 4*T^3 + 32*T^2 - 64*T + 256