Newform orbit 1728.2.s.d

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

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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 144)
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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 2) q^{5} + (\zeta_{6} + 1) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} + 3 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(703\)	\(1217\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

575.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

1.50000

−

0.866025i

0

1.50000

+

0.866025i

0

0

0

1151.1

0

0

0

1.50000

+

0.866025i

0

1.50000

−

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
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.d		2
3.b	odd	2	1		576.2.s.c		2
4.b	odd	2	1		1728.2.s.c		2
8.b	even	2	1		432.2.s.b		2
8.d	odd	2	1		432.2.s.a		2
9.c	even	3	1		576.2.s.b		2
9.c	even	3	1		5184.2.c.d		2
9.d	odd	6	1		1728.2.s.c		2
9.d	odd	6	1		5184.2.c.b		2
12.b	even	2	1		576.2.s.b		2
24.f	even	2	1		144.2.s.b	✓	2
24.h	odd	2	1		144.2.s.c	yes	2
36.f	odd	6	1		576.2.s.c		2
36.f	odd	6	1		5184.2.c.b		2
36.h	even	6	1	inner	1728.2.s.d		2
36.h	even	6	1		5184.2.c.d		2
72.j	odd	6	1		432.2.s.a		2
72.j	odd	6	1		1296.2.c.c		2
72.l	even	6	1		432.2.s.b		2
72.l	even	6	1		1296.2.c.a		2
72.n	even	6	1		144.2.s.b	✓	2
72.n	even	6	1		1296.2.c.a		2
72.p	odd	6	1		144.2.s.c	yes	2
72.p	odd	6	1		1296.2.c.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
144.2.s.b	✓	2	24.f	even	2	1
144.2.s.b	✓	2	72.n	even	6	1
144.2.s.c	yes	2	24.h	odd	2	1
144.2.s.c	yes	2	72.p	odd	6	1
432.2.s.a		2	8.d	odd	2	1
432.2.s.a		2	72.j	odd	6	1
432.2.s.b		2	8.b	even	2	1
432.2.s.b		2	72.l	even	6	1
576.2.s.b		2	9.c	even	3	1
576.2.s.b		2	12.b	even	2	1
576.2.s.c		2	3.b	odd	2	1
576.2.s.c		2	36.f	odd	6	1
1296.2.c.a		2	72.l	even	6	1
1296.2.c.a		2	72.n	even	6	1
1296.2.c.c		2	72.j	odd	6	1
1296.2.c.c		2	72.p	odd	6	1
1728.2.s.c		2	4.b	odd	2	1
1728.2.s.c		2	9.d	odd	6	1
1728.2.s.d		2	1.a	even	1	1	trivial
1728.2.s.d		2	36.h	even	6	1	inner
5184.2.c.b		2	9.d	odd	6	1
5184.2.c.b		2	36.f	odd	6	1
5184.2.c.d		2	9.c	even	3	1
5184.2.c.d		2	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 \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

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

\( T_{7}^{2} - 3T_{7} + 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 3T + 3 \)
$7$	\( T^{2} - 3T + 3 \)
$11$	\( T^{2} + 3T + 9 \)