Newform orbit 1296.4.a.u

• Label: 1296.4.a.u
• Level: $1296$
• Weight: $4$
• Character orbit: 1296.a
• Self dual: yes
• Analytic conductor: $76.466$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(76.4664753674\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{33}) \)
Defining polynomial:	\( x^{2} - x - 8 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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.

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

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

0

0

0

4.62772

0

−12.1168

0

0

0

1.2

0

0

0

10.3723

0

5.11684

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)

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	1296.4.a.u		2
3.b	odd	2	1		1296.4.a.i		2
4.b	odd	2	1		81.4.a.d		2
9.c	even	3	2		144.4.i.c		4
9.d	odd	6	2		432.4.i.c		4
12.b	even	2	1		81.4.a.a		2
20.d	odd	2	1		2025.4.a.g		2
36.f	odd	6	2		9.4.c.a	✓	4
36.h	even	6	2		27.4.c.a		4
60.h	even	2	1		2025.4.a.n		2
180.p	odd	6	2		225.4.e.b		4
180.x	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	36.f	odd	6	2
27.4.c.a		4	36.h	even	6	2
81.4.a.a		2	12.b	even	2	1
81.4.a.d		2	4.b	odd	2	1
144.4.i.c		4	9.c	even	3	2
225.4.e.b		4	180.p	odd	6	2
225.4.k.b		8	180.x	even	12	4
432.4.i.c		4	9.d	odd	6	2
1296.4.a.i		2	3.b	odd	2	1
1296.4.a.u		2	1.a	even	1	1	trivial
2025.4.a.g		2	20.d	odd	2	1
2025.4.a.n		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 15T_{5} + 48 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\).

Hecke characteristic polynomials

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