Newform orbit 81.5.d.b

• Label: 81.5.d.b
• Level: $81$
• Weight: $5$
• Character orbit: 81.d
• Analytic conductor: $8.373$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.37296700979\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} - 7 \beta_{2} q^{4} + (11 \beta_{3} - 11 \beta_1) q^{5} + ( - 19 \beta_{2} + 19) q^{7} - 23 \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{4} + 38 q^{7}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( 3\zeta_{12} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{12}^{2} \)
\(\beta_{3}\)	\(=\)	\( 3\zeta_{12}^{3} \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

26.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

−2.59808

−

1.50000i

0

−3.50000

−

6.06218i

28.5788

−

16.5000i

0

9.50000

−

16.4545i

69.0000i

0

−99.0000

26.2

2.59808

+

1.50000i

0

−3.50000

−

6.06218i

−28.5788

+

16.5000i

0

9.50000

−

16.4545i

−

69.0000i

0

−99.0000

53.1

−2.59808

+

1.50000i

0

−3.50000

+

6.06218i

28.5788

+

16.5000i

0

9.50000

+

16.4545i

−

69.0000i

0

−99.0000

53.2

2.59808

−

1.50000i

0

−3.50000

+

6.06218i

−28.5788

−

16.5000i

0

9.50000

+

16.4545i

69.0000i

0

−99.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
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	81.5.d.b		4
3.b	odd	2	1	inner	81.5.d.b		4
9.c	even	3	1		27.5.b.c	✓	2
9.c	even	3	1	inner	81.5.d.b		4
9.d	odd	6	1		27.5.b.c	✓	2
9.d	odd	6	1	inner	81.5.d.b		4
36.f	odd	6	1		432.5.e.e		2
36.h	even	6	1		432.5.e.e		2
45.h	odd	6	1		675.5.c.h		2
45.j	even	6	1		675.5.c.h		2
45.k	odd	12	1		675.5.d.a		2
45.k	odd	12	1		675.5.d.d		2
45.l	even	12	1		675.5.d.a		2
45.l	even	12	1		675.5.d.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.5.b.c	✓	2	9.c	even	3	1
27.5.b.c	✓	2	9.d	odd	6	1
81.5.d.b		4	1.a	even	1	1	trivial
81.5.d.b		4	3.b	odd	2	1	inner
81.5.d.b		4	9.c	even	3	1	inner
81.5.d.b		4	9.d	odd	6	1	inner
432.5.e.e		2	36.f	odd	6	1
432.5.e.e		2	36.h	even	6	1
675.5.c.h		2	45.h	odd	6	1
675.5.c.h		2	45.j	even	6	1
675.5.d.a		2	45.k	odd	12	1
675.5.d.a		2	45.l	even	12	1
675.5.d.d		2	45.k	odd	12	1
675.5.d.d		2	45.l	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 9T_{2}^{2} + 81 \) acting on \(S_{5}^{\mathrm{new}}(81, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - 9T^{2} + 81 \)
$3$	\( T^{4} \)
$5$	\( T^{4} - 1089 T^{2} + 1185921 \)
$7$	\( (T^{2} - 19 T + 361)^{2} \)
$11$	\( T^{4} - 15129 T^{2} + 228886641 \)