Newform orbit 324.3.g.a

• Label: 324.3.g.a
• Level: $324$
• Weight: $3$
• Character orbit: 324.g
• Analytic conductor: $8.828$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (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:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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 + (11*z - 11) * q^7 - 23*z * q^13 - 37 * q^19 + (25*z - 25) * q^25 + 46*z * q^31 - 73 * q^37 + (-22*z + 22) * q^43 - 72*z * q^49 + (47*z - 47) * q^61 + 13*z * q^67 + 143 * q^73 + (11*z - 11) * q^79 + 253 * q^91 + (-169*z + 169) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 11 q^{7} - 23 q^{13} - 74 q^{19} - 25 q^{25} + 46 q^{31} - 146 q^{37} + 22 q^{43} - 72 q^{49} - 47 q^{61} + 13 q^{67} + 286 q^{73} - 11 q^{79} + 506 q^{91} + 169 q^{97}+O(q^{100}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

53.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

0

0

−5.50000

−

9.52628i

0

0

0

269.1

0

0

0

0

0

−5.50000

+

9.52628i

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	324.3.g.a		2
3.b	odd	2	1	CM	324.3.g.a		2
4.b	odd	2	1		1296.3.q.c		2
9.c	even	3	1		108.3.c.a	✓	1
9.c	even	3	1	inner	324.3.g.a		2
9.d	odd	6	1		108.3.c.a	✓	1
9.d	odd	6	1	inner	324.3.g.a		2
12.b	even	2	1		1296.3.q.c		2
36.f	odd	6	1		432.3.e.a		1
36.f	odd	6	1		1296.3.q.c		2
36.h	even	6	1		432.3.e.a		1
36.h	even	6	1		1296.3.q.c		2
45.h	odd	6	1		2700.3.g.b		1
45.j	even	6	1		2700.3.g.b		1
45.k	odd	12	2		2700.3.b.d		2
45.l	even	12	2		2700.3.b.d		2
72.j	odd	6	1		1728.3.e.c		1
72.l	even	6	1		1728.3.e.b		1
72.n	even	6	1		1728.3.e.c		1
72.p	odd	6	1		1728.3.e.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.3.c.a	✓	1	9.c	even	3	1
108.3.c.a	✓	1	9.d	odd	6	1
324.3.g.a		2	1.a	even	1	1	trivial
324.3.g.a		2	3.b	odd	2	1	CM
324.3.g.a		2	9.c	even	3	1	inner
324.3.g.a		2	9.d	odd	6	1	inner
432.3.e.a		1	36.f	odd	6	1
432.3.e.a		1	36.h	even	6	1
1296.3.q.c		2	4.b	odd	2	1
1296.3.q.c		2	12.b	even	2	1
1296.3.q.c		2	36.f	odd	6	1
1296.3.q.c		2	36.h	even	6	1
1728.3.e.b		1	72.l	even	6	1
1728.3.e.b		1	72.p	odd	6	1
1728.3.e.c		1	72.j	odd	6	1
1728.3.e.c		1	72.n	even	6	1
2700.3.b.d		2	45.k	odd	12	2
2700.3.b.d		2	45.l	even	12	2
2700.3.g.b		1	45.h	odd	6	1
2700.3.g.b		1	45.j	even	6	1

Hecke kernels

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

\( T_{5} \)

\( T_{7}^{2} + 11T_{7} + 121 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 11T + 121 \)
$11$	\( T^{2} \)