Newform orbit 1296.2.i.d

• Label: 1296.2.i.d
• Level: $1296$
• Weight: $2$
• Character orbit: 1296.i
• Analytic conductor: $10.349$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3486121020\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 324)
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.

\(f(q)\)	\(=\)	\( q - 3 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(1135\)	\(1217\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

433.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

−1.50000

+

2.59808i

0

1.00000

+

1.73205i

0

0

0

865.1

0

0

0

−1.50000

−

2.59808i

0

1.00000

−

1.73205i

0

0

0

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	1296.2.i.d		2
3.b	odd	2	1		1296.2.i.p		2
4.b	odd	2	1		324.2.e.a		2
9.c	even	3	1		1296.2.a.j		1
9.c	even	3	1	inner	1296.2.i.d		2
9.d	odd	6	1		1296.2.a.a		1
9.d	odd	6	1		1296.2.i.p		2
12.b	even	2	1		324.2.e.d		2
36.f	odd	6	1		324.2.a.d	yes	1
36.f	odd	6	1		324.2.e.a		2
36.h	even	6	1		324.2.a.b	✓	1
36.h	even	6	1		324.2.e.d		2
72.j	odd	6	1		5184.2.a.z		1
72.l	even	6	1		5184.2.a.bc		1
72.n	even	6	1		5184.2.a.d		1
72.p	odd	6	1		5184.2.a.g		1
180.n	even	6	1		8100.2.a.f		1
180.p	odd	6	1		8100.2.a.a		1
180.v	odd	12	2		8100.2.d.j		2
180.x	even	12	2		8100.2.d.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
324.2.a.b	✓	1	36.h	even	6	1
324.2.a.d	yes	1	36.f	odd	6	1
324.2.e.a		2	4.b	odd	2	1
324.2.e.a		2	36.f	odd	6	1
324.2.e.d		2	12.b	even	2	1
324.2.e.d		2	36.h	even	6	1
1296.2.a.a		1	9.d	odd	6	1
1296.2.a.j		1	9.c	even	3	1
1296.2.i.d		2	1.a	even	1	1	trivial
1296.2.i.d		2	9.c	even	3	1	inner
1296.2.i.p		2	3.b	odd	2	1
1296.2.i.p		2	9.d	odd	6	1
5184.2.a.d		1	72.n	even	6	1
5184.2.a.g		1	72.p	odd	6	1
5184.2.a.z		1	72.j	odd	6	1
5184.2.a.bc		1	72.l	even	6	1
8100.2.a.a		1	180.p	odd	6	1
8100.2.a.f		1	180.n	even	6	1
8100.2.d.a		2	180.x	even	12	2
8100.2.d.j		2	180.v	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \)

\( T_{7}^{2} - 2T_{7} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} - 2T + 4 \)
$11$	\( T^{2} + 6T + 36 \)