Newform orbit 1728.2.s.f

• Label: 1728.2.s.f
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.s
• Analytic conductor: $13.798$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.170772624.1
Defining polynomial:	\( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_{2} - 1) q^{5} + ( - \beta_{6} + \beta_1) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + \nu^{5} + 2\nu^{3} + 4\nu^{2} + 4\nu - 8 ) / 8 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{6} + \nu^{5} - \nu^{4} + 2\nu^{3} - 4\nu^{2} + 4\nu - 4 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} - 3\nu^{5} + 4\nu^{4} - 2\nu^{3} + 12\nu^{2} - 12\nu + 16 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 32 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 4\nu^{3} + 10\nu^{2} - 16\nu + 20 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{7} - 3\nu^{6} + 9\nu^{5} - 8\nu^{4} + 10\nu^{3} - 24\nu^{2} + 20\nu - 40 ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( 3\nu^{7} - 5\nu^{6} + 11\nu^{5} - 6\nu^{4} + 10\nu^{3} - 28\nu^{2} + 36\nu - 48 ) / 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(703\)	\(1217\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-\beta_{4}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

575.1

−1.02187	−	0.977642i
0.335728	+	1.37379i
0.774115	−	1.18353i
1.41203	−	0.0786378i
−1.02187	+	0.977642i
0.335728	−	1.37379i
0.774115	+	1.18353i
1.41203	+	0.0786378i

0

0

0

−2.18614

+

1.26217i

0

−1.10489

−

0.637910i

0

0

0

575.2

0

0

0

−2.18614

+

1.26217i

0

1.10489

+

0.637910i

0

0

0

575.3

0

0

0

0.686141

−

0.396143i

0

−2.35143

−

1.35760i

0

0

0

575.4

0

0

0

0.686141

−

0.396143i

0

2.35143

+

1.35760i

0

0

0

1151.1

0

0

0

−2.18614

−

1.26217i

0

−1.10489

+

0.637910i

0

0

0

1151.2

0

0

0

−2.18614

−

1.26217i

0

1.10489

−

0.637910i

0

0

0

1151.3

0

0

0

0.686141

+

0.396143i

0

−2.35143

+

1.35760i

0

0

0

1151.4

0

0

0

0.686141

+

0.396143i

0

2.35143

−

1.35760i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
36.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1728.2.s.f		8
3.b	odd	2	1		576.2.s.f		8
4.b	odd	2	1	inner	1728.2.s.f		8
8.b	even	2	1		108.2.h.a		8
8.d	odd	2	1		108.2.h.a		8
9.c	even	3	1		576.2.s.f		8
9.c	even	3	1		5184.2.c.j		8
9.d	odd	6	1	inner	1728.2.s.f		8
9.d	odd	6	1		5184.2.c.j		8
12.b	even	2	1		576.2.s.f		8
24.f	even	2	1		36.2.h.a	✓	8
24.h	odd	2	1		36.2.h.a	✓	8
36.f	odd	6	1		576.2.s.f		8
36.f	odd	6	1		5184.2.c.j		8
36.h	even	6	1	inner	1728.2.s.f		8
36.h	even	6	1		5184.2.c.j		8
72.j	odd	6	1		108.2.h.a		8
72.j	odd	6	1		324.2.b.b		8
72.l	even	6	1		108.2.h.a		8
72.l	even	6	1		324.2.b.b		8
72.n	even	6	1		36.2.h.a	✓	8
72.n	even	6	1		324.2.b.b		8
72.p	odd	6	1		36.2.h.a	✓	8
72.p	odd	6	1		324.2.b.b		8
120.i	odd	2	1		900.2.r.c		8
120.m	even	2	1		900.2.r.c		8
120.q	odd	4	2		900.2.o.a		16
120.w	even	4	2		900.2.o.a		16
360.z	odd	6	1		900.2.r.c		8
360.bk	even	6	1		900.2.r.c		8
360.bo	even	12	2		900.2.o.a		16
360.bu	odd	12	2		900.2.o.a		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.h.a	✓	8	24.f	even	2	1
36.2.h.a	✓	8	24.h	odd	2	1
36.2.h.a	✓	8	72.n	even	6	1
36.2.h.a	✓	8	72.p	odd	6	1
108.2.h.a		8	8.b	even	2	1
108.2.h.a		8	8.d	odd	2	1
108.2.h.a		8	72.j	odd	6	1
108.2.h.a		8	72.l	even	6	1
324.2.b.b		8	72.j	odd	6	1
324.2.b.b		8	72.l	even	6	1
324.2.b.b		8	72.n	even	6	1
324.2.b.b		8	72.p	odd	6	1
576.2.s.f		8	3.b	odd	2	1
576.2.s.f		8	9.c	even	3	1
576.2.s.f		8	12.b	even	2	1
576.2.s.f		8	36.f	odd	6	1
900.2.o.a		16	120.q	odd	4	2
900.2.o.a		16	120.w	even	4	2
900.2.o.a		16	360.bo	even	12	2
900.2.o.a		16	360.bu	odd	12	2
900.2.r.c		8	120.i	odd	2	1
900.2.r.c		8	120.m	even	2	1
900.2.r.c		8	360.z	odd	6	1
900.2.r.c		8	360.bk	even	6	1
1728.2.s.f		8	1.a	even	1	1	trivial
1728.2.s.f		8	4.b	odd	2	1	inner
1728.2.s.f		8	9.d	odd	6	1	inner
1728.2.s.f		8	36.h	even	6	1	inner
5184.2.c.j		8	9.c	even	3	1
5184.2.c.j		8	9.d	odd	6	1
5184.2.c.j		8	36.f	odd	6	1
5184.2.c.j		8	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} + 3T_{5}^{3} + T_{5}^{2} - 6T_{5} + 4 \)

\( T_{7}^{8} - 9T_{7}^{6} + 69T_{7}^{4} - 108T_{7}^{2} + 144 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	\( T^{8} \)
$5$	(T^4 + 3*T^3 + T^2 - 6*T + 4)^2
$7$	T^8 - 9*T^6 + 69*T^4 - 108*T^2 + 144
$11$	T^8 + 12*T^6 + 141*T^4 + 36*T^2 + 9