Newform orbit 81.4.c.b

• Label: 81.4.c.b
• Level: $81$
• Weight: $4$
• Character orbit: 81.c
• Analytic conductor: $4.779$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.77915471046\)
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 9)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + 8 \zeta_{6} q^{4} + (20 \zeta_{6} - 20) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4} - 20 q^{7}+O(q^{10}) \)

Character values

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

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

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

0

0

4.00000

−

6.92820i

0

0

−10.0000

−

17.3205i

0

0

0

55.1

0

0

4.00000

+

6.92820i

0

0

−10.0000

+

17.3205i

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.4.c.b		2
3.b	odd	2	1	CM	81.4.c.b		2
9.c	even	3	1		9.4.a.a	✓	1
9.c	even	3	1	inner	81.4.c.b		2
9.d	odd	6	1		9.4.a.a	✓	1
9.d	odd	6	1	inner	81.4.c.b		2
36.f	odd	6	1		144.4.a.d		1
36.h	even	6	1		144.4.a.d		1
45.h	odd	6	1		225.4.a.d		1
45.j	even	6	1		225.4.a.d		1
45.k	odd	12	2		225.4.b.g		2
45.l	even	12	2		225.4.b.g		2
63.g	even	3	1		441.4.e.i		2
63.h	even	3	1		441.4.e.i		2
63.i	even	6	1		441.4.e.j		2
63.j	odd	6	1		441.4.e.i		2
63.k	odd	6	1		441.4.e.j		2
63.l	odd	6	1		441.4.a.f		1
63.n	odd	6	1		441.4.e.i		2
63.o	even	6	1		441.4.a.f		1
63.s	even	6	1		441.4.e.j		2
63.t	odd	6	1		441.4.e.j		2
72.j	odd	6	1		576.4.a.m		1
72.l	even	6	1		576.4.a.l		1
72.n	even	6	1		576.4.a.m		1
72.p	odd	6	1		576.4.a.l		1
99.g	even	6	1		1089.4.a.g		1
99.h	odd	6	1		1089.4.a.g		1
117.n	odd	6	1		1521.4.a.g		1
117.t	even	6	1		1521.4.a.g		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.4.a.a	✓	1	9.c	even	3	1
9.4.a.a	✓	1	9.d	odd	6	1
81.4.c.b		2	1.a	even	1	1	trivial
81.4.c.b		2	3.b	odd	2	1	CM
81.4.c.b		2	9.c	even	3	1	inner
81.4.c.b		2	9.d	odd	6	1	inner
144.4.a.d		1	36.f	odd	6	1
144.4.a.d		1	36.h	even	6	1
225.4.a.d		1	45.h	odd	6	1
225.4.a.d		1	45.j	even	6	1
225.4.b.g		2	45.k	odd	12	2
225.4.b.g		2	45.l	even	12	2
441.4.a.f		1	63.l	odd	6	1
441.4.a.f		1	63.o	even	6	1
441.4.e.i		2	63.g	even	3	1
441.4.e.i		2	63.h	even	3	1
441.4.e.i		2	63.j	odd	6	1
441.4.e.i		2	63.n	odd	6	1
441.4.e.j		2	63.i	even	6	1
441.4.e.j		2	63.k	odd	6	1
441.4.e.j		2	63.s	even	6	1
441.4.e.j		2	63.t	odd	6	1
576.4.a.l		1	72.l	even	6	1
576.4.a.l		1	72.p	odd	6	1
576.4.a.m		1	72.j	odd	6	1
576.4.a.m		1	72.n	even	6	1
1089.4.a.g		1	99.g	even	6	1
1089.4.a.g		1	99.h	odd	6	1
1521.4.a.g		1	117.n	odd	6	1
1521.4.a.g		1	117.t	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(81, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 20T + 400 \)
$11$	\( T^{2} \)