Newform orbit 81.3.b.a

• Label: 81.3.b.a
• Level: $81$
• Weight: $3$
• Character orbit: 81.b
• Analytic conductor: $2.207$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	81.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.20709014132\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + q^{4} - 2 \beta q^{5} + 2 q^{7} - 5 \beta q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

80.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−

1.73205i

0

1.00000

−

3.46410i

0

2.00000

−

8.66025i

0

−6.00000

80.2

1.73205i

0

1.00000

3.46410i

0

2.00000

8.66025i

0

−6.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.3.b.a		2
3.b	odd	2	1	inner	81.3.b.a		2
4.b	odd	2	1		1296.3.e.a		2
9.c	even	3	1		9.3.d.a	✓	2
9.c	even	3	1		27.3.d.a		2
9.d	odd	6	1		9.3.d.a	✓	2
9.d	odd	6	1		27.3.d.a		2
12.b	even	2	1		1296.3.e.a		2
36.f	odd	6	1		144.3.q.a		2
36.f	odd	6	1		432.3.q.a		2
36.h	even	6	1		144.3.q.a		2
36.h	even	6	1		432.3.q.a		2
45.h	odd	6	1		225.3.j.a		2
45.h	odd	6	1		675.3.j.a		2
45.j	even	6	1		225.3.j.a		2
45.j	even	6	1		675.3.j.a		2
45.k	odd	12	2		225.3.i.a		4
45.k	odd	12	2		675.3.i.a		4
45.l	even	12	2		225.3.i.a		4
45.l	even	12	2		675.3.i.a		4
63.g	even	3	1		441.3.n.b		2
63.h	even	3	1		441.3.j.a		2
63.i	even	6	1		441.3.j.b		2
63.j	odd	6	1		441.3.j.a		2
63.k	odd	6	1		441.3.n.a		2
63.l	odd	6	1		441.3.r.a		2
63.n	odd	6	1		441.3.n.b		2
63.o	even	6	1		441.3.r.a		2
63.s	even	6	1		441.3.n.a		2
63.t	odd	6	1		441.3.j.b		2
72.j	odd	6	1		576.3.q.b		2
72.j	odd	6	1		1728.3.q.a		2
72.l	even	6	1		576.3.q.a		2
72.l	even	6	1		1728.3.q.b		2
72.n	even	6	1		576.3.q.b		2
72.n	even	6	1		1728.3.q.a		2
72.p	odd	6	1		576.3.q.a		2
72.p	odd	6	1		1728.3.q.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.3.d.a	✓	2	9.c	even	3	1
9.3.d.a	✓	2	9.d	odd	6	1
27.3.d.a		2	9.c	even	3	1
27.3.d.a		2	9.d	odd	6	1
81.3.b.a		2	1.a	even	1	1	trivial
81.3.b.a		2	3.b	odd	2	1	inner
144.3.q.a		2	36.f	odd	6	1
144.3.q.a		2	36.h	even	6	1
225.3.i.a		4	45.k	odd	12	2
225.3.i.a		4	45.l	even	12	2
225.3.j.a		2	45.h	odd	6	1
225.3.j.a		2	45.j	even	6	1
432.3.q.a		2	36.f	odd	6	1
432.3.q.a		2	36.h	even	6	1
441.3.j.a		2	63.h	even	3	1
441.3.j.a		2	63.j	odd	6	1
441.3.j.b		2	63.i	even	6	1
441.3.j.b		2	63.t	odd	6	1
441.3.n.a		2	63.k	odd	6	1
441.3.n.a		2	63.s	even	6	1
441.3.n.b		2	63.g	even	3	1
441.3.n.b		2	63.n	odd	6	1
441.3.r.a		2	63.l	odd	6	1
441.3.r.a		2	63.o	even	6	1
576.3.q.a		2	72.l	even	6	1
576.3.q.a		2	72.p	odd	6	1
576.3.q.b		2	72.j	odd	6	1
576.3.q.b		2	72.n	even	6	1
675.3.i.a		4	45.k	odd	12	2
675.3.i.a		4	45.l	even	12	2
675.3.j.a		2	45.h	odd	6	1
675.3.j.a		2	45.j	even	6	1
1296.3.e.a		2	4.b	odd	2	1
1296.3.e.a		2	12.b	even	2	1
1728.3.q.a		2	72.j	odd	6	1
1728.3.q.a		2	72.n	even	6	1
1728.3.q.b		2	72.l	even	6	1
1728.3.q.b		2	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{3}^{\mathrm{new}}(81, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 12 \)
$7$	\( (T - 2)^{2} \)
$11$	\( T^{2} + 3 \)