Newform orbit 81.8.c.a

• Label: 81.8.c.a
• Level: $81$
• Weight: $8$
• Character orbit: 81.c
• Analytic conductor: $25.303$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(25.3031870642\)
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 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (6 \zeta_{6} - 6) q^{2} + 92 \zeta_{6} q^{4} - 390 \zeta_{6} q^{5} + ( - 64 \zeta_{6} + 64) q^{7} - 1320 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 92 q^{4} - 390 q^{5} + 64 q^{7} - 2640 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

28.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−3.00000

−

5.19615i

0

46.0000

−

79.6743i

−195.000

+

337.750i

0

32.0000

+

55.4256i

−1320.00

0

2340.00

55.1

−3.00000

+

5.19615i

0

46.0000

+

79.6743i

−195.000

−

337.750i

0

32.0000

−

55.4256i

−1320.00

0

2340.00

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.8.c.a		2
3.b	odd	2	1		81.8.c.c		2
9.c	even	3	1		3.8.a.a	✓	1
9.c	even	3	1	inner	81.8.c.a		2
9.d	odd	6	1		9.8.a.a		1
9.d	odd	6	1		81.8.c.c		2
36.f	odd	6	1		48.8.a.g		1
36.h	even	6	1		144.8.a.b		1
45.h	odd	6	1		225.8.a.i		1
45.j	even	6	1		75.8.a.a		1
45.k	odd	12	2		75.8.b.c		2
45.l	even	12	2		225.8.b.f		2
63.g	even	3	1		147.8.e.b		2
63.h	even	3	1		147.8.e.b		2
63.k	odd	6	1		147.8.e.a		2
63.l	odd	6	1		147.8.a.b		1
63.o	even	6	1		441.8.a.a		1
63.t	odd	6	1		147.8.e.a		2
72.j	odd	6	1		576.8.a.w		1
72.l	even	6	1		576.8.a.x		1
72.n	even	6	1		192.8.a.i		1
72.p	odd	6	1		192.8.a.a		1
99.h	odd	6	1		363.8.a.b		1
117.t	even	6	1		507.8.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.8.a.a	✓	1	9.c	even	3	1
9.8.a.a		1	9.d	odd	6	1
48.8.a.g		1	36.f	odd	6	1
75.8.a.a		1	45.j	even	6	1
75.8.b.c		2	45.k	odd	12	2
81.8.c.a		2	1.a	even	1	1	trivial
81.8.c.a		2	9.c	even	3	1	inner
81.8.c.c		2	3.b	odd	2	1
81.8.c.c		2	9.d	odd	6	1
144.8.a.b		1	36.h	even	6	1
147.8.a.b		1	63.l	odd	6	1
147.8.e.a		2	63.k	odd	6	1
147.8.e.a		2	63.t	odd	6	1
147.8.e.b		2	63.g	even	3	1
147.8.e.b		2	63.h	even	3	1
192.8.a.a		1	72.p	odd	6	1
192.8.a.i		1	72.n	even	6	1
225.8.a.i		1	45.h	odd	6	1
225.8.b.f		2	45.l	even	12	2
363.8.a.b		1	99.h	odd	6	1
441.8.a.a		1	63.o	even	6	1
507.8.a.a		1	117.t	even	6	1
576.8.a.w		1	72.j	odd	6	1
576.8.a.x		1	72.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 6T_{2} + 36 \) acting on \(S_{8}^{\mathrm{new}}(81, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 6T + 36 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 390T + 152100 \)
$7$	\( T^{2} - 64T + 4096 \)
$11$	\( T^{2} - 948T + 898704 \)