Newform orbit 1728.3.q.b

• Label: 1728.3.q.b
• Level: $1728$
• Weight: $3$
• Character orbit: 1728.q
• Analytic conductor: $47.085$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(47.0845896815\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 9)
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.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 4) q^{5} + ( - 2 \zeta_{6} + 2) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 2 q^{7}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

449.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

3.00000

+

1.73205i

0

1.00000

+

1.73205i

0

0

0

1601.1

0

0

0

3.00000

−

1.73205i

0

1.00000

−

1.73205i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1728.3.q.b		2
3.b	odd	2	1		576.3.q.a		2
4.b	odd	2	1		1728.3.q.a		2
8.b	even	2	1		432.3.q.a		2
8.d	odd	2	1		27.3.d.a		2
9.c	even	3	1		576.3.q.a		2
9.d	odd	6	1	inner	1728.3.q.b		2
12.b	even	2	1		576.3.q.b		2
24.f	even	2	1		9.3.d.a	✓	2
24.h	odd	2	1		144.3.q.a		2
36.f	odd	6	1		576.3.q.b		2
36.h	even	6	1		1728.3.q.a		2
40.e	odd	2	1		675.3.j.a		2
40.k	even	4	2		675.3.i.a		4
72.j	odd	6	1		432.3.q.a		2
72.j	odd	6	1		1296.3.e.a		2
72.l	even	6	1		27.3.d.a		2
72.l	even	6	1		81.3.b.a		2
72.n	even	6	1		144.3.q.a		2
72.n	even	6	1		1296.3.e.a		2
72.p	odd	6	1		9.3.d.a	✓	2
72.p	odd	6	1		81.3.b.a		2
120.m	even	2	1		225.3.j.a		2
120.q	odd	4	2		225.3.i.a		4
168.e	odd	2	1		441.3.r.a		2
168.v	even	6	1		441.3.j.a		2
168.v	even	6	1		441.3.n.b		2
168.be	odd	6	1		441.3.j.b		2
168.be	odd	6	1		441.3.n.a		2
360.z	odd	6	1		225.3.j.a		2
360.bd	even	6	1		675.3.j.a		2
360.bo	even	12	2		225.3.i.a		4
360.bt	odd	12	2		675.3.i.a		4
504.ba	odd	6	1		441.3.j.a		2
504.be	even	6	1		441.3.r.a		2
504.bf	even	6	1		441.3.n.a		2
504.ce	odd	6	1		441.3.n.b		2
504.cz	even	6	1		441.3.j.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.3.d.a	✓	2	24.f	even	2	1
9.3.d.a	✓	2	72.p	odd	6	1
27.3.d.a		2	8.d	odd	2	1
27.3.d.a		2	72.l	even	6	1
81.3.b.a		2	72.l	even	6	1
81.3.b.a		2	72.p	odd	6	1
144.3.q.a		2	24.h	odd	2	1
144.3.q.a		2	72.n	even	6	1
225.3.i.a		4	120.q	odd	4	2
225.3.i.a		4	360.bo	even	12	2
225.3.j.a		2	120.m	even	2	1
225.3.j.a		2	360.z	odd	6	1
432.3.q.a		2	8.b	even	2	1
432.3.q.a		2	72.j	odd	6	1
441.3.j.a		2	168.v	even	6	1
441.3.j.a		2	504.ba	odd	6	1
441.3.j.b		2	168.be	odd	6	1
441.3.j.b		2	504.cz	even	6	1
441.3.n.a		2	168.be	odd	6	1
441.3.n.a		2	504.bf	even	6	1
441.3.n.b		2	168.v	even	6	1
441.3.n.b		2	504.ce	odd	6	1
441.3.r.a		2	168.e	odd	2	1
441.3.r.a		2	504.be	even	6	1
576.3.q.a		2	3.b	odd	2	1
576.3.q.a		2	9.c	even	3	1
576.3.q.b		2	12.b	even	2	1
576.3.q.b		2	36.f	odd	6	1
675.3.i.a		4	40.k	even	4	2
675.3.i.a		4	360.bt	odd	12	2
675.3.j.a		2	40.e	odd	2	1
675.3.j.a		2	360.bd	even	6	1
1296.3.e.a		2	72.j	odd	6	1
1296.3.e.a		2	72.n	even	6	1
1728.3.q.a		2	4.b	odd	2	1
1728.3.q.a		2	36.h	even	6	1
1728.3.q.b		2	1.a	even	1	1	trivial
1728.3.q.b		2	9.d	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} - 6T_{5} + 12 \)

\( T_{7}^{2} - 2T_{7} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 6T + 12 \)
$7$	\( T^{2} - 2T + 4 \)
$11$	\( T^{2} - 3T + 3 \)