Newform orbit 1296.2.i.h

• Label: 1296.2.i.h
• Level: $1296$
• Weight: $2$
• Character orbit: 1296.i
• Analytic conductor: $10.349$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

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:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (4 \zeta_{6} - 4) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7}+O(q^{10}) \)

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

0

0

−2.00000

−

3.46410i

0

0

0

865.1

0

0

0

0

0

−2.00000

+

3.46410i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
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	1296.2.i.h		2
3.b	odd	2	1	CM	1296.2.i.h		2
4.b	odd	2	1		324.2.e.c		2
9.c	even	3	1		144.2.a.a		1
9.c	even	3	1	inner	1296.2.i.h		2
9.d	odd	6	1		144.2.a.a		1
9.d	odd	6	1	inner	1296.2.i.h		2
12.b	even	2	1		324.2.e.c		2
36.f	odd	6	1		36.2.a.a	✓	1
36.f	odd	6	1		324.2.e.c		2
36.h	even	6	1		36.2.a.a	✓	1
36.h	even	6	1		324.2.e.c		2
45.h	odd	6	1		3600.2.a.e		1
45.j	even	6	1		3600.2.a.e		1
45.k	odd	12	2		3600.2.f.m		2
45.l	even	12	2		3600.2.f.m		2
63.l	odd	6	1		7056.2.a.bb		1
63.o	even	6	1		7056.2.a.bb		1
72.j	odd	6	1		576.2.a.f		1
72.l	even	6	1		576.2.a.e		1
72.n	even	6	1		576.2.a.f		1
72.p	odd	6	1		576.2.a.e		1
144.u	even	12	2		2304.2.d.q		2
144.v	odd	12	2		2304.2.d.q		2
144.w	odd	12	2		2304.2.d.a		2
144.x	even	12	2		2304.2.d.a		2
180.n	even	6	1		900.2.a.g		1
180.p	odd	6	1		900.2.a.g		1
180.v	odd	12	2		900.2.d.b		2
180.x	even	12	2		900.2.d.b		2
252.n	even	6	1		1764.2.k.g		2
252.o	even	6	1		1764.2.k.h		2
252.r	odd	6	1		1764.2.k.g		2
252.s	odd	6	1		1764.2.a.e		1
252.u	odd	6	1		1764.2.k.h		2
252.bb	even	6	1		1764.2.k.h		2
252.bi	even	6	1		1764.2.a.e		1
252.bj	even	6	1		1764.2.k.g		2
252.bl	odd	6	1		1764.2.k.h		2
252.bn	odd	6	1		1764.2.k.g		2
396.k	even	6	1		4356.2.a.g		1
396.o	odd	6	1		4356.2.a.g		1
468.x	even	6	1		6084.2.a.i		1
468.bg	odd	6	1		6084.2.a.i		1
468.bs	even	12	2		6084.2.b.f		2
468.ch	odd	12	2		6084.2.b.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.a.a	✓	1	36.f	odd	6	1
36.2.a.a	✓	1	36.h	even	6	1
144.2.a.a		1	9.c	even	3	1
144.2.a.a		1	9.d	odd	6	1
324.2.e.c		2	4.b	odd	2	1
324.2.e.c		2	12.b	even	2	1
324.2.e.c		2	36.f	odd	6	1
324.2.e.c		2	36.h	even	6	1
576.2.a.e		1	72.l	even	6	1
576.2.a.e		1	72.p	odd	6	1
576.2.a.f		1	72.j	odd	6	1
576.2.a.f		1	72.n	even	6	1
900.2.a.g		1	180.n	even	6	1
900.2.a.g		1	180.p	odd	6	1
900.2.d.b		2	180.v	odd	12	2
900.2.d.b		2	180.x	even	12	2
1296.2.i.h		2	1.a	even	1	1	trivial
1296.2.i.h		2	3.b	odd	2	1	CM
1296.2.i.h		2	9.c	even	3	1	inner
1296.2.i.h		2	9.d	odd	6	1	inner
1764.2.a.e		1	252.s	odd	6	1
1764.2.a.e		1	252.bi	even	6	1
1764.2.k.g		2	252.n	even	6	1
1764.2.k.g		2	252.r	odd	6	1
1764.2.k.g		2	252.bj	even	6	1
1764.2.k.g		2	252.bn	odd	6	1
1764.2.k.h		2	252.o	even	6	1
1764.2.k.h		2	252.u	odd	6	1
1764.2.k.h		2	252.bb	even	6	1
1764.2.k.h		2	252.bl	odd	6	1
2304.2.d.a		2	144.w	odd	12	2
2304.2.d.a		2	144.x	even	12	2
2304.2.d.q		2	144.u	even	12	2
2304.2.d.q		2	144.v	odd	12	2
3600.2.a.e		1	45.h	odd	6	1
3600.2.a.e		1	45.j	even	6	1
3600.2.f.m		2	45.k	odd	12	2
3600.2.f.m		2	45.l	even	12	2
4356.2.a.g		1	396.k	even	6	1
4356.2.a.g		1	396.o	odd	6	1
6084.2.a.i		1	468.x	even	6	1
6084.2.a.i		1	468.bg	odd	6	1
6084.2.b.f		2	468.bs	even	12	2
6084.2.b.f		2	468.ch	odd	12	2
7056.2.a.bb		1	63.l	odd	6	1
7056.2.a.bb		1	63.o	even	6	1

Hecke kernels

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

\( T_{5} \)

\( T_{7}^{2} + 4T_{7} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 4T + 16 \)
$11$	\( T^{2} \)