Newform orbit 648.2.n.p

• Label: 648.2.n.p
• Level: $648$
• Weight: $2$
• Character orbit: 648.n
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $8$
• CM discriminant: -24
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{2} + 2 \beta_{2} q^{4} + (\beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{2} + 1) q^{7} + (2 \beta_{5} - 2 \beta_{4}) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 4 q^{7}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( 3\zeta_{24}^{2} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{24}^{4} \)
\(\beta_{3}\)	\(=\)	\( 3\zeta_{24}^{6} \)
\(\beta_{4}\)	\(=\)	\( -\zeta_{24}^{7} + \zeta_{24} \)
\(\beta_{5}\)	\(=\)	\( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \)
\(\beta_{6}\)	\(=\)	\( 3\zeta_{24}^{7} + 3\zeta_{24} \)
\(\beta_{7}\)	\(=\)	\( -3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 3\zeta_{24} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

109.1

−0.258819	−	0.965926i
−0.965926	+	0.258819i
0.258819	+	0.965926i
0.965926	−	0.258819i
−0.258819	+	0.965926i
−0.965926	−	0.258819i
0.258819	−	0.965926i
0.965926	+	0.258819i

−1.22474

+

0.707107i

0

1.00000

−

1.73205i

−3.82282

−

2.20711i

0

−1.62132

−

2.80821i

2.82843i

0

6.24264

109.2

−1.22474

+

0.707107i

0

1.00000

−

1.73205i

1.37333

+

0.792893i

0

2.62132

+

4.54026i

2.82843i

0

−2.24264

109.3

1.22474

−

0.707107i

0

1.00000

−

1.73205i

−1.37333

−

0.792893i

0

2.62132

+

4.54026i

−

2.82843i

0

−2.24264

109.4

1.22474

−

0.707107i

0

1.00000

−

1.73205i

3.82282

+

2.20711i

0

−1.62132

−

2.80821i

−

2.82843i

0

6.24264

541.1

−1.22474

−

0.707107i

0

1.00000

+

1.73205i

−3.82282

+

2.20711i

0

−1.62132

+

2.80821i

−

2.82843i

0

6.24264

541.2

−1.22474

−

0.707107i

0

1.00000

+

1.73205i

1.37333

−

0.792893i

0

2.62132

−

4.54026i

−

2.82843i

0

−2.24264

541.3

1.22474

+

0.707107i

0

1.00000

+

1.73205i

−1.37333

+

0.792893i

0

2.62132

−

4.54026i

2.82843i

0

−2.24264

541.4

1.22474

+

0.707107i

0

1.00000

+

1.73205i

3.82282

−

2.20711i

0

−1.62132

+

2.80821i

2.82843i

0

6.24264

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by \(\Q(\sqrt{-6}) \)
3.b	odd	2	1	inner
8.b	even	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
72.j	odd	6	1	inner
72.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.2.n.p		8
3.b	odd	2	1	inner	648.2.n.p		8
4.b	odd	2	1		2592.2.r.o		8
8.b	even	2	1	inner	648.2.n.p		8
8.d	odd	2	1		2592.2.r.o		8
9.c	even	3	1		216.2.d.a	✓	4
9.c	even	3	1	inner	648.2.n.p		8
9.d	odd	6	1		216.2.d.a	✓	4
9.d	odd	6	1	inner	648.2.n.p		8
12.b	even	2	1		2592.2.r.o		8
24.f	even	2	1		2592.2.r.o		8
24.h	odd	2	1	CM	648.2.n.p		8
36.f	odd	6	1		864.2.d.b		4
36.f	odd	6	1		2592.2.r.o		8
36.h	even	6	1		864.2.d.b		4
36.h	even	6	1		2592.2.r.o		8
72.j	odd	6	1		216.2.d.a	✓	4
72.j	odd	6	1	inner	648.2.n.p		8
72.l	even	6	1		864.2.d.b		4
72.l	even	6	1		2592.2.r.o		8
72.n	even	6	1		216.2.d.a	✓	4
72.n	even	6	1	inner	648.2.n.p		8
72.p	odd	6	1		864.2.d.b		4
72.p	odd	6	1		2592.2.r.o		8
144.u	even	12	1		6912.2.a.y		2
144.u	even	12	1		6912.2.a.by		2
144.v	odd	12	1		6912.2.a.y		2
144.v	odd	12	1		6912.2.a.by		2
144.w	odd	12	1		6912.2.a.z		2
144.w	odd	12	1		6912.2.a.bz		2
144.x	even	12	1		6912.2.a.z		2
144.x	even	12	1		6912.2.a.bz		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
216.2.d.a	✓	4	9.c	even	3	1
216.2.d.a	✓	4	9.d	odd	6	1
216.2.d.a	✓	4	72.j	odd	6	1
216.2.d.a	✓	4	72.n	even	6	1
648.2.n.p		8	1.a	even	1	1	trivial
648.2.n.p		8	3.b	odd	2	1	inner
648.2.n.p		8	8.b	even	2	1	inner
648.2.n.p		8	9.c	even	3	1	inner
648.2.n.p		8	9.d	odd	6	1	inner
648.2.n.p		8	24.h	odd	2	1	CM
648.2.n.p		8	72.j	odd	6	1	inner
648.2.n.p		8	72.n	even	6	1	inner
864.2.d.b		4	36.f	odd	6	1
864.2.d.b		4	36.h	even	6	1
864.2.d.b		4	72.l	even	6	1
864.2.d.b		4	72.p	odd	6	1
2592.2.r.o		8	4.b	odd	2	1
2592.2.r.o		8	8.d	odd	2	1
2592.2.r.o		8	12.b	even	2	1
2592.2.r.o		8	24.f	even	2	1
2592.2.r.o		8	36.f	odd	6	1
2592.2.r.o		8	36.h	even	6	1
2592.2.r.o		8	72.l	even	6	1
2592.2.r.o		8	72.p	odd	6	1
6912.2.a.y		2	144.u	even	12	1
6912.2.a.y		2	144.v	odd	12	1
6912.2.a.z		2	144.w	odd	12	1
6912.2.a.z		2	144.x	even	12	1
6912.2.a.by		2	144.u	even	12	1
6912.2.a.by		2	144.v	odd	12	1
6912.2.a.bz		2	144.w	odd	12	1
6912.2.a.bz		2	144.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\):

\( T_{5}^{8} - 22T_{5}^{6} + 435T_{5}^{4} - 1078T_{5}^{2} + 2401 \)

\( T_{13} \)

\( T_{17} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - 2 T^{2} + 4)^{2} \)
$3$	\( T^{8} \)
$5$	T^8 - 22*T^6 + 435*T^4 - 1078*T^2 + 2401
$7$	(T^4 - 2*T^3 + 21*T^2 + 34*T + 289)^2
$11$	T^8 - 34*T^6 + 1155*T^4 - 34*T^2 + 1