Newform orbit 1764.2.l.d

• Label: 1764.2.l.d
• Level: $1764$
• Weight: $2$
• Character orbit: 1764.l
• 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.l (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:	\( 3 \)
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.

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

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

\(\beta_{1}\)	\(=\)	\( \nu^{2} - 2\nu + 3 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} - 4\nu^{4} + 11\nu^{3} - 20\nu^{2} + 18\nu - 12 ) / 3 \)
\(\beta_{3}\)	\(=\)	\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \)
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \)
\(\beta_{5}\)	\(=\)	\( -\nu^{5} + 2\nu^{4} - 8\nu^{3} + 9\nu^{2} - 13\nu + 6 \)

Character values

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

\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(-\beta_{3}\)	\(1\)	\(-1 + \beta_{3}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

949.1

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

0

−1.73025

−

0.0789082i

0

2.05408

0

0

0

2.98755

+

0.273062i

0

949.2

0

−0.619562

−

1.61745i

0

−3.94282

0

0

0

−2.23229

+

2.00422i

0

949.3

0

0.349814

+

1.69636i

0

−1.11126

0

0

0

−2.75526

+

1.18682i

0

961.1

0

−1.73025

+

0.0789082i

0

2.05408

0

0

0

2.98755

−

0.273062i

0

961.2

0

−0.619562

+

1.61745i

0

−3.94282

0

0

0

−2.23229

−

2.00422i

0

961.3

0

0.349814

−

1.69636i

0

−1.11126

0

0

0

−2.75526

−

1.18682i

0

Inner twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 3T_{5}^{2} - 6T_{5} - 9 \) 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
$5$	\( (T^{3} + 3 T^{2} - 6 T - 9)^{2} \)
$7$	\( T^{6} \)
$11$	\( (T^{3} + 6 T^{2} + 3 T - 9)^{2} \)