Newform orbit 81.4.a.a

• Label: 81.4.a.a
• Level: $81$
• Weight: $4$
• Character orbit: 81.a
• Self dual: yes
• Analytic conductor: $4.779$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.77915471046\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{33}) \)
Defining polynomial:	\( x^{2} - x - 8 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta - 1) q^{2} + (3 \beta + 1) q^{4} + (\beta - 8) q^{5} + (3 \beta + 2) q^{7} + (\beta - 17) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 5 q^{4} - 15 q^{5} + 7 q^{7} - 33 q^{8}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

3.37228
−2.37228

−4.37228

0

11.1168

−4.62772

0

12.1168

−13.6277

0

20.2337

1.2

1.37228

0

−6.11684

−10.3723

0

−5.11684

−19.3723

0

−14.2337

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.4.a.a		2
3.b	odd	2	1		81.4.a.d		2
4.b	odd	2	1		1296.4.a.i		2
5.b	even	2	1		2025.4.a.n		2
9.c	even	3	2		27.4.c.a		4
9.d	odd	6	2		9.4.c.a	✓	4
12.b	even	2	1		1296.4.a.u		2
15.d	odd	2	1		2025.4.a.g		2
36.f	odd	6	2		432.4.i.c		4
36.h	even	6	2		144.4.i.c		4
45.h	odd	6	2		225.4.e.b		4
45.l	even	12	4		225.4.k.b		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.4.c.a	✓	4	9.d	odd	6	2
27.4.c.a		4	9.c	even	3	2
81.4.a.a		2	1.a	even	1	1	trivial
81.4.a.d		2	3.b	odd	2	1
144.4.i.c		4	36.h	even	6	2
225.4.e.b		4	45.h	odd	6	2
225.4.k.b		8	45.l	even	12	4
432.4.i.c		4	36.f	odd	6	2
1296.4.a.i		2	4.b	odd	2	1
1296.4.a.u		2	12.b	even	2	1
2025.4.a.g		2	15.d	odd	2	1
2025.4.a.n		2	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} - 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(81))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T - 6 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 15T + 48 \)
$7$	\( T^{2} - 7T - 62 \)
$11$	\( T^{2} + 66T + 561 \)