Newform orbit 1764.3.z.d

• Label: 1764.3.z.d
• Level: $1764$
• Weight: $3$
• Character orbit: 1764.z
• Analytic conductor: $48.066$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(48.0655186332\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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 + (16*z - 8) * q^13 + (-16*z + 32) * q^19 + (25*z - 25) * q^25 + (-24*z - 24) * q^31 + 26*z * q^37 + 22 * q^43 + (56*z - 112) * q^61 + (122*z - 122) * q^67 + (80*z + 80) * q^73 + 142*z * q^79 + (-224*z + 112) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 48 q^{19} - 25 q^{25} - 72 q^{31} + 26 q^{37} + 44 q^{43} - 168 q^{61} - 122 q^{67} + 240 q^{73} + 142 q^{79}+O(q^{100}) \)

Character values

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

\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

325.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

0

0

0

0

0

0

901.1

0

0

0

0

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.3.z.d		2
3.b	odd	2	1	CM	1764.3.z.d		2
7.b	odd	2	1		1764.3.z.b		2
7.c	even	3	1		252.3.d.b	✓	2
7.c	even	3	1		1764.3.z.b		2
7.d	odd	6	1		252.3.d.b	✓	2
7.d	odd	6	1	inner	1764.3.z.d		2
21.c	even	2	1		1764.3.z.b		2
21.g	even	6	1		252.3.d.b	✓	2
21.g	even	6	1	inner	1764.3.z.d		2
21.h	odd	6	1		252.3.d.b	✓	2
21.h	odd	6	1		1764.3.z.b		2
28.f	even	6	1		1008.3.f.e		2
28.g	odd	6	1		1008.3.f.e		2
84.j	odd	6	1		1008.3.f.e		2
84.n	even	6	1		1008.3.f.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.3.d.b	✓	2	7.c	even	3	1
252.3.d.b	✓	2	7.d	odd	6	1
252.3.d.b	✓	2	21.g	even	6	1
252.3.d.b	✓	2	21.h	odd	6	1
1008.3.f.e		2	28.f	even	6	1
1008.3.f.e		2	28.g	odd	6	1
1008.3.f.e		2	84.j	odd	6	1
1008.3.f.e		2	84.n	even	6	1
1764.3.z.b		2	7.b	odd	2	1
1764.3.z.b		2	7.c	even	3	1
1764.3.z.b		2	21.c	even	2	1
1764.3.z.b		2	21.h	odd	6	1
1764.3.z.d		2	1.a	even	1	1	trivial
1764.3.z.d		2	3.b	odd	2	1	CM
1764.3.z.d		2	7.d	odd	6	1	inner
1764.3.z.d		2	21.g	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5} \)

\( T_{11} \)

\( T_{13}^{2} + 192 \)

\( T_{19}^{2} - 48T_{19} + 768 \)

Hecke characteristic polynomials

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