Newform orbit 324.6.e.d

• Label: 324.6.e.d
• Level: $324$
• Weight: $6$
• Character orbit: 324.e
• Analytic conductor: $51.964$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(51.9643576194\)
Analytic rank:	\(0\)
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 4)
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.

\(f(q)\)	\(=\)	\( q + 54 \zeta_{6} q^{5} + ( - 88 \zeta_{6} + 88) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{5} + 88 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

109.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

27.0000

−

46.7654i

0

44.0000

+

76.2102i

0

0

0

217.1

0

0

0

27.0000

+

46.7654i

0

44.0000

−

76.2102i

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	324.6.e.d		2
3.b	odd	2	1		324.6.e.a		2
9.c	even	3	1		36.6.a.a		1
9.c	even	3	1	inner	324.6.e.d		2
9.d	odd	6	1		4.6.a.a	✓	1
9.d	odd	6	1		324.6.e.a		2
36.f	odd	6	1		144.6.a.c		1
36.h	even	6	1		16.6.a.b		1
45.h	odd	6	1		100.6.a.b		1
45.j	even	6	1		900.6.a.h		1
45.k	odd	12	2		900.6.d.a		2
45.l	even	12	2		100.6.c.b		2
63.i	even	6	1		196.6.e.d		2
63.j	odd	6	1		196.6.e.g		2
63.n	odd	6	1		196.6.e.g		2
63.o	even	6	1		196.6.a.e		1
63.s	even	6	1		196.6.e.d		2
72.j	odd	6	1		64.6.a.f		1
72.l	even	6	1		64.6.a.b		1
72.n	even	6	1		576.6.a.bc		1
72.p	odd	6	1		576.6.a.bd		1
99.g	even	6	1		484.6.a.a		1
117.n	odd	6	1		676.6.a.a		1
117.z	even	12	2		676.6.d.a		2
144.u	even	12	2		256.6.b.c		2
144.w	odd	12	2		256.6.b.g		2
180.n	even	6	1		400.6.a.d		1
180.v	odd	12	2		400.6.c.f		2
252.s	odd	6	1		784.6.a.d		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.6.a.a	✓	1	9.d	odd	6	1
16.6.a.b		1	36.h	even	6	1
36.6.a.a		1	9.c	even	3	1
64.6.a.b		1	72.l	even	6	1
64.6.a.f		1	72.j	odd	6	1
100.6.a.b		1	45.h	odd	6	1
100.6.c.b		2	45.l	even	12	2
144.6.a.c		1	36.f	odd	6	1
196.6.a.e		1	63.o	even	6	1
196.6.e.d		2	63.i	even	6	1
196.6.e.d		2	63.s	even	6	1
196.6.e.g		2	63.j	odd	6	1
196.6.e.g		2	63.n	odd	6	1
256.6.b.c		2	144.u	even	12	2
256.6.b.g		2	144.w	odd	12	2
324.6.e.a		2	3.b	odd	2	1
324.6.e.a		2	9.d	odd	6	1
324.6.e.d		2	1.a	even	1	1	trivial
324.6.e.d		2	9.c	even	3	1	inner
400.6.a.d		1	180.n	even	6	1
400.6.c.f		2	180.v	odd	12	2
484.6.a.a		1	99.g	even	6	1
576.6.a.bc		1	72.n	even	6	1
576.6.a.bd		1	72.p	odd	6	1
676.6.a.a		1	117.n	odd	6	1
676.6.d.a		2	117.z	even	12	2
784.6.a.d		1	252.s	odd	6	1
900.6.a.h		1	45.j	even	6	1
900.6.d.a		2	45.k	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} - 54T_{5} + 2916 \)

\( T_{7}^{2} - 88T_{7} + 7744 \)

\( T_{11}^{2} - 540T_{11} + 291600 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 54T + 2916 \)
$7$	\( T^{2} - 88T + 7744 \)
$11$	\( T^{2} - 540T + 291600 \)