Newform orbit 1764.2.i.g

• Label: 1764.2.i.g
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.i
• Analytic conductor: $14.086$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.309123.1
Defining polynomial:	\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b5 - b3 - b2) * q^3 + (-b5 + b3 + b2 + b1) * q^5 + (2*b4 - 2*b3 - b2 + b1 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - q^{5} - 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \)
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \)
\(\beta_{5}\)	\(=\)	\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(\beta_{4}\)	\(1\)	\(\beta_{4}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

373.1

0.500000	−	0.224437i
0.500000	+	1.41036i
0.500000	−	2.05195i
0.500000	+	0.224437i
0.500000	−	1.41036i
0.500000	+	2.05195i

0

−0.349814

−

1.69636i

0

0.849814

+

1.47192i

0

0

0

−2.75526

+

1.18682i

0

373.2

0

0.619562

+

1.61745i

0

−0.119562

−

0.207087i

0

0

0

−2.23229

+

2.00422i

0

373.3

0

1.73025

+

0.0789082i

0

−1.23025

−

2.13086i

0

0

0

2.98755

+

0.273062i

0

1537.1

0

−0.349814

+

1.69636i

0

0.849814

−

1.47192i

0

0

0

−2.75526

−

1.18682i

0

1537.2

0

0.619562

−

1.61745i

0

−0.119562

+

0.207087i

0

0

0

−2.23229

−

2.00422i

0

1537.3

0

1.73025

−

0.0789082i

0

−1.23025

+

2.13086i

0

0

0

2.98755

−

0.273062i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.2.i.g		6
3.b	odd	2	1		5292.2.i.f		6
7.b	odd	2	1		1764.2.i.d		6
7.c	even	3	1		252.2.j.a	✓	6
7.c	even	3	1		1764.2.l.e		6
7.d	odd	6	1		1764.2.j.e		6
7.d	odd	6	1		1764.2.l.f		6
9.c	even	3	1		1764.2.l.e		6
9.d	odd	6	1		5292.2.l.e		6
21.c	even	2	1		5292.2.i.e		6
21.g	even	6	1		5292.2.j.d		6
21.g	even	6	1		5292.2.l.f		6
21.h	odd	6	1		756.2.j.b		6
21.h	odd	6	1		5292.2.l.e		6
28.g	odd	6	1		1008.2.r.j		6
63.g	even	3	1		252.2.j.a	✓	6
63.h	even	3	1	inner	1764.2.i.g		6
63.h	even	3	1		2268.2.a.i		3
63.i	even	6	1		5292.2.i.e		6
63.j	odd	6	1		2268.2.a.h		3
63.j	odd	6	1		5292.2.i.f		6
63.k	odd	6	1		1764.2.j.e		6
63.l	odd	6	1		1764.2.l.f		6
63.n	odd	6	1		756.2.j.b		6
63.o	even	6	1		5292.2.l.f		6
63.s	even	6	1		5292.2.j.d		6
63.t	odd	6	1		1764.2.i.d		6
84.n	even	6	1		3024.2.r.j		6
252.o	even	6	1		3024.2.r.j		6
252.u	odd	6	1		9072.2.a.by		3
252.bb	even	6	1		9072.2.a.bv		3
252.bl	odd	6	1		1008.2.r.j		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.j.a	✓	6	7.c	even	3	1
252.2.j.a	✓	6	63.g	even	3	1
756.2.j.b		6	21.h	odd	6	1
756.2.j.b		6	63.n	odd	6	1
1008.2.r.j		6	28.g	odd	6	1
1008.2.r.j		6	252.bl	odd	6	1
1764.2.i.d		6	7.b	odd	2	1
1764.2.i.d		6	63.t	odd	6	1
1764.2.i.g		6	1.a	even	1	1	trivial
1764.2.i.g		6	63.h	even	3	1	inner
1764.2.j.e		6	7.d	odd	6	1
1764.2.j.e		6	63.k	odd	6	1
1764.2.l.e		6	7.c	even	3	1
1764.2.l.e		6	9.c	even	3	1
1764.2.l.f		6	7.d	odd	6	1
1764.2.l.f		6	63.l	odd	6	1
2268.2.a.h		3	63.j	odd	6	1
2268.2.a.i		3	63.h	even	3	1
3024.2.r.j		6	84.n	even	6	1
3024.2.r.j		6	252.o	even	6	1
5292.2.i.e		6	21.c	even	2	1
5292.2.i.e		6	63.i	even	6	1
5292.2.i.f		6	3.b	odd	2	1
5292.2.i.f		6	63.j	odd	6	1
5292.2.j.d		6	21.g	even	6	1
5292.2.j.d		6	63.s	even	6	1
5292.2.l.e		6	9.d	odd	6	1
5292.2.l.e		6	21.h	odd	6	1
5292.2.l.f		6	21.g	even	6	1
5292.2.l.f		6	63.o	even	6	1
9072.2.a.bv		3	252.bb	even	6	1
9072.2.a.by		3	252.u	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + T_{5}^{5} + 5T_{5}^{4} - 2T_{5}^{3} + 17T_{5}^{2} + 4T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{6} \)
$3$	T^6 - 4*T^5 + 10*T^4 - 21*T^3 + 30*T^2 - 36*T + 27
$5$	T^6 + T^5 + 5*T^4 - 2*T^3 + 17*T^2 + 4*T + 1
$7$	\( T^{6} \)
$11$	T^6 + 2*T^5 + 29*T^4 + 68*T^3 + 743*T^2 + 1475*T + 3481