Newform orbit 1323.2.h.b

• Label: 1323.2.h.b
• Level: $1323$
• Weight: $2$
• Character orbit: 1323.h
• Analytic conductor: $10.564$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5642081874\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b4 + b3 - 1) * q^2 + (-2*b4 - b3 + 1) * q^4 + (b5 - b4 - b3 + b2 - b1) * q^5 + (2*b4 - b3 - 2) * q^8
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 3 q^{5} - 12 q^{8}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{18}^{3} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{18}^{5} + \zeta_{18} \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \)
\(\beta_{4}\)	\(=\)	\( -\zeta_{18}^{5} + \zeta_{18}^{4} \)
\(\beta_{5}\)	\(=\)	\( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(-1 + \beta_{1}\)	\(-1 + \beta_{1}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

226.1

−0.766044	+	0.642788i
−0.173648	−	0.984808i
0.939693	+	0.342020i
−0.766044	−	0.642788i
−0.173648	+	0.984808i
0.939693	−	0.342020i

−2.53209

0

4.41147

0.439693

+

0.761570i

0

0

−6.10607

0

−1.11334

−

1.92836i

226.2

−1.34730

0

−0.184793

−1.26604

−

2.19285i

0

0

2.94356

0

1.70574

+

2.95442i

226.3

0.879385

0

−1.22668

−0.673648

−

1.16679i

0

0

−2.83750

0

−0.592396

−

1.02606i

802.1

−2.53209

0

4.41147

0.439693

−

0.761570i

0

0

−6.10607

0

−1.11334

+

1.92836i

802.2

−1.34730

0

−0.184793

−1.26604

+

2.19285i

0

0

2.94356

0

1.70574

−

2.95442i

802.3

0.879385

0

−1.22668

−0.673648

+

1.16679i

0

0

−2.83750

0

−0.592396

+

1.02606i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1323.2.h.b		6
3.b	odd	2	1		441.2.h.e		6
7.b	odd	2	1		1323.2.h.c		6
7.c	even	3	1		1323.2.f.d		6
7.c	even	3	1		1323.2.g.e		6
7.d	odd	6	1		189.2.f.b		6
7.d	odd	6	1		1323.2.g.d		6
9.c	even	3	1		1323.2.g.e		6
9.d	odd	6	1		441.2.g.b		6
21.c	even	2	1		441.2.h.d		6
21.g	even	6	1		63.2.f.a	✓	6
21.g	even	6	1		441.2.g.c		6
21.h	odd	6	1		441.2.f.c		6
21.h	odd	6	1		441.2.g.b		6
28.f	even	6	1		3024.2.r.k		6
63.g	even	3	1		1323.2.f.d		6
63.h	even	3	1	inner	1323.2.h.b		6
63.h	even	3	1		3969.2.a.l		3
63.i	even	6	1		441.2.h.d		6
63.i	even	6	1		567.2.a.h		3
63.j	odd	6	1		441.2.h.e		6
63.j	odd	6	1		3969.2.a.q		3
63.k	odd	6	1		189.2.f.b		6
63.l	odd	6	1		1323.2.g.d		6
63.n	odd	6	1		441.2.f.c		6
63.o	even	6	1		441.2.g.c		6
63.s	even	6	1		63.2.f.a	✓	6
63.t	odd	6	1		567.2.a.c		3
63.t	odd	6	1		1323.2.h.c		6
84.j	odd	6	1		1008.2.r.h		6
252.n	even	6	1		3024.2.r.k		6
252.r	odd	6	1		9072.2.a.ca		3
252.bj	even	6	1		9072.2.a.bs		3
252.bn	odd	6	1		1008.2.r.h		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.f.a	✓	6	21.g	even	6	1
63.2.f.a	✓	6	63.s	even	6	1
189.2.f.b		6	7.d	odd	6	1
189.2.f.b		6	63.k	odd	6	1
441.2.f.c		6	21.h	odd	6	1
441.2.f.c		6	63.n	odd	6	1
441.2.g.b		6	9.d	odd	6	1
441.2.g.b		6	21.h	odd	6	1
441.2.g.c		6	21.g	even	6	1
441.2.g.c		6	63.o	even	6	1
441.2.h.d		6	21.c	even	2	1
441.2.h.d		6	63.i	even	6	1
441.2.h.e		6	3.b	odd	2	1
441.2.h.e		6	63.j	odd	6	1
567.2.a.c		3	63.t	odd	6	1
567.2.a.h		3	63.i	even	6	1
1008.2.r.h		6	84.j	odd	6	1
1008.2.r.h		6	252.bn	odd	6	1
1323.2.f.d		6	7.c	even	3	1
1323.2.f.d		6	63.g	even	3	1
1323.2.g.d		6	7.d	odd	6	1
1323.2.g.d		6	63.l	odd	6	1
1323.2.g.e		6	7.c	even	3	1
1323.2.g.e		6	9.c	even	3	1
1323.2.h.b		6	1.a	even	1	1	trivial
1323.2.h.b		6	63.h	even	3	1	inner
1323.2.h.c		6	7.b	odd	2	1
1323.2.h.c		6	63.t	odd	6	1
3024.2.r.k		6	28.f	even	6	1
3024.2.r.k		6	252.n	even	6	1
3969.2.a.l		3	63.h	even	3	1
3969.2.a.q		3	63.j	odd	6	1
9072.2.a.bs		3	252.bj	even	6	1
9072.2.a.ca		3	252.r	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\( T_{2}^{3} + 3T_{2}^{2} - 3 \)

\( T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{3} + 3 T^{2} - 3)^{2} \)
$3$	\( T^{6} \)
$5$	T^6 + 3*T^5 + 9*T^4 + 6*T^3 + 9*T^2 + 9
$7$	\( T^{6} \)
$11$	T^6 - 6*T^5 + 27*T^4 - 48*T^3 + 63*T^2 - 27*T + 9