Newform orbit 324.3.f.a

• Label: 324.3.f.a
• Level: $324$
• Weight: $3$
• Character orbit: 324.f
• Analytic conductor: $8.828$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.82836056527\)
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 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - 2 q^{2} + 4 q^{4} + (2 \zeta_{6} - 2) q^{5} + ( - 4 \zeta_{6} + 8) q^{7} - 8 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 2 q^{5} + 12 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

55.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−2.00000

0

4.00000

−1.00000

−

1.73205i

0

6.00000

+

3.46410i

−8.00000

0

2.00000

+

3.46410i

271.1

−2.00000

0

4.00000

−1.00000

+

1.73205i

0

6.00000

−

3.46410i

−8.00000

0

2.00000

−

3.46410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
36.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.3.f.a		2
3.b	odd	2	1		324.3.f.j		2
4.b	odd	2	1		324.3.f.g		2
9.c	even	3	1		36.3.d.c		2
9.c	even	3	1		324.3.f.g		2
9.d	odd	6	1		12.3.d.a	✓	2
9.d	odd	6	1		324.3.f.d		2
12.b	even	2	1		324.3.f.d		2
36.f	odd	6	1		36.3.d.c		2
36.f	odd	6	1	inner	324.3.f.a		2
36.h	even	6	1		12.3.d.a	✓	2
36.h	even	6	1		324.3.f.j		2
45.h	odd	6	1		300.3.c.b		2
45.j	even	6	1		900.3.c.e		2
45.k	odd	12	2		900.3.f.c		4
45.l	even	12	2		300.3.f.a		4
63.o	even	6	1		588.3.g.b		2
72.j	odd	6	1		192.3.g.b		2
72.l	even	6	1		192.3.g.b		2
72.n	even	6	1		576.3.g.e		2
72.p	odd	6	1		576.3.g.e		2
144.u	even	12	2		768.3.b.c		4
144.v	odd	12	2		2304.3.b.l		4
144.w	odd	12	2		768.3.b.c		4
144.x	even	12	2		2304.3.b.l		4
180.n	even	6	1		300.3.c.b		2
180.p	odd	6	1		900.3.c.e		2
180.v	odd	12	2		300.3.f.a		4
180.x	even	12	2		900.3.f.c		4
252.s	odd	6	1		588.3.g.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.3.d.a	✓	2	9.d	odd	6	1
12.3.d.a	✓	2	36.h	even	6	1
36.3.d.c		2	9.c	even	3	1
36.3.d.c		2	36.f	odd	6	1
192.3.g.b		2	72.j	odd	6	1
192.3.g.b		2	72.l	even	6	1
300.3.c.b		2	45.h	odd	6	1
300.3.c.b		2	180.n	even	6	1
300.3.f.a		4	45.l	even	12	2
300.3.f.a		4	180.v	odd	12	2
324.3.f.a		2	1.a	even	1	1	trivial
324.3.f.a		2	36.f	odd	6	1	inner
324.3.f.d		2	9.d	odd	6	1
324.3.f.d		2	12.b	even	2	1
324.3.f.g		2	4.b	odd	2	1
324.3.f.g		2	9.c	even	3	1
324.3.f.j		2	3.b	odd	2	1
324.3.f.j		2	36.h	even	6	1
576.3.g.e		2	72.n	even	6	1
576.3.g.e		2	72.p	odd	6	1
588.3.g.b		2	63.o	even	6	1
588.3.g.b		2	252.s	odd	6	1
768.3.b.c		4	144.u	even	12	2
768.3.b.c		4	144.w	odd	12	2
900.3.c.e		2	45.j	even	6	1
900.3.c.e		2	180.p	odd	6	1
900.3.f.c		4	45.k	odd	12	2
900.3.f.c		4	180.x	even	12	2
2304.3.b.l		4	144.v	odd	12	2
2304.3.b.l		4	144.x	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 4 \)

\( T_{7}^{2} - 12T_{7} + 48 \)

\( T_{11}^{2} - 12T_{11} + 48 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T + 4 \)
$7$	\( T^{2} - 12T + 48 \)
$11$	\( T^{2} - 12T + 48 \)