Newform orbit 324.10.e.b

• Label: 324.10.e.b
• Level: $324$
• Weight: $10$
• Character orbit: 324.e
• Analytic conductor: $166.872$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(166.871610917\)
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.

\(f(q)\)	\(=\)	\( q - 666 \zeta_{6} q^{5} + ( - 6328 \zeta_{6} + 6328) q^{7} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 666 q^{5} + 6328 q^{7}+O(q^{10}) \)

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

−333.000

+

576.773i

0

3164.00

+

5480.21i

0

0

0

217.1

0

0

0

−333.000

−

576.773i

0

3164.00

−

5480.21i

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	324.10.e.b		2
3.b	odd	2	1		324.10.e.e		2
9.c	even	3	1		36.10.a.b		1
9.c	even	3	1	inner	324.10.e.b		2
9.d	odd	6	1		4.10.a.a	✓	1
9.d	odd	6	1		324.10.e.e		2
36.f	odd	6	1		144.10.a.j		1
36.h	even	6	1		16.10.a.a		1
45.h	odd	6	1		100.10.a.a		1
45.l	even	12	2		100.10.c.a		2
63.i	even	6	1		196.10.e.b		2
63.j	odd	6	1		196.10.e.a		2
63.n	odd	6	1		196.10.e.a		2
63.o	even	6	1		196.10.a.a		1
63.s	even	6	1		196.10.e.b		2
72.j	odd	6	1		64.10.a.a		1
72.l	even	6	1		64.10.a.i		1
144.u	even	12	2		256.10.b.b		2
144.w	odd	12	2		256.10.b.j		2
180.n	even	6	1		400.10.a.k		1
180.v	odd	12	2		400.10.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.10.a.a	✓	1	9.d	odd	6	1
16.10.a.a		1	36.h	even	6	1
36.10.a.b		1	9.c	even	3	1
64.10.a.a		1	72.j	odd	6	1
64.10.a.i		1	72.l	even	6	1
100.10.a.a		1	45.h	odd	6	1
100.10.c.a		2	45.l	even	12	2
144.10.a.j		1	36.f	odd	6	1
196.10.a.a		1	63.o	even	6	1
196.10.e.a		2	63.j	odd	6	1
196.10.e.a		2	63.n	odd	6	1
196.10.e.b		2	63.i	even	6	1
196.10.e.b		2	63.s	even	6	1
256.10.b.b		2	144.u	even	12	2
256.10.b.j		2	144.w	odd	12	2
324.10.e.b		2	1.a	even	1	1	trivial
324.10.e.b		2	9.c	even	3	1	inner
324.10.e.e		2	3.b	odd	2	1
324.10.e.e		2	9.d	odd	6	1
400.10.a.k		1	180.n	even	6	1
400.10.c.a		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 \(S_{10}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} + 666T_{5} + 443556 \)

\( T_{7}^{2} - 6328T_{7} + 40043584 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 666T + 443556 \)
$7$	\( T^{2} - 6328 T + 40043584 \)
$11$	\( T^{2} + 30420 T + 925376400 \)