Newform orbit 81.4.c.c

• Label: 81.4.c.c
• Level: $81$
• Weight: $4$
• Character orbit: 81.c
• Analytic conductor: $4.779$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 27)
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.

\(f(q)\)	\(=\)	\( q + ( - 3 \zeta_{6} + 3) q^{2} - \zeta_{6} q^{4} + 15 \zeta_{6} q^{5} + ( - 25 \zeta_{6} + 25) q^{7} + 21 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - q^{4} + 15 q^{5} + 25 q^{7} + 42 q^{8}+O(q^{10}) \)

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

1.50000

+

2.59808i

0

−0.500000

+

0.866025i

7.50000

−

12.9904i

0

12.5000

+

21.6506i

21.0000

0

45.0000

55.1

1.50000

−

2.59808i

0

−0.500000

−

0.866025i

7.50000

+

12.9904i

0

12.5000

−

21.6506i

21.0000

0

45.0000

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	81.4.c.c		2
3.b	odd	2	1		81.4.c.a		2
9.c	even	3	1		27.4.a.a	✓	1
9.c	even	3	1	inner	81.4.c.c		2
9.d	odd	6	1		27.4.a.b	yes	1
9.d	odd	6	1		81.4.c.a		2
36.f	odd	6	1		432.4.a.a		1
36.h	even	6	1		432.4.a.n		1
45.h	odd	6	1		675.4.a.a		1
45.j	even	6	1		675.4.a.j		1
45.k	odd	12	2		675.4.b.b		2
45.l	even	12	2		675.4.b.a		2
63.l	odd	6	1		1323.4.a.d		1
63.o	even	6	1		1323.4.a.k		1
72.j	odd	6	1		1728.4.a.c		1
72.l	even	6	1		1728.4.a.d		1
72.n	even	6	1		1728.4.a.bc		1
72.p	odd	6	1		1728.4.a.bd		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.4.a.a	✓	1	9.c	even	3	1
27.4.a.b	yes	1	9.d	odd	6	1
81.4.c.a		2	3.b	odd	2	1
81.4.c.a		2	9.d	odd	6	1
81.4.c.c		2	1.a	even	1	1	trivial
81.4.c.c		2	9.c	even	3	1	inner
432.4.a.a		1	36.f	odd	6	1
432.4.a.n		1	36.h	even	6	1
675.4.a.a		1	45.h	odd	6	1
675.4.a.j		1	45.j	even	6	1
675.4.b.a		2	45.l	even	12	2
675.4.b.b		2	45.k	odd	12	2
1323.4.a.d		1	63.l	odd	6	1
1323.4.a.k		1	63.o	even	6	1
1728.4.a.c		1	72.j	odd	6	1
1728.4.a.d		1	72.l	even	6	1
1728.4.a.bc		1	72.n	even	6	1
1728.4.a.bd		1	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(81, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3T + 9 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 15T + 225 \)
$7$	\( T^{2} - 25T + 625 \)
$11$	\( T^{2} + 15T + 225 \)