Newform orbit 1323.2.s.d

• Label: 1323.2.s.d
• Level: $1323$
• Weight: $2$
• Character orbit: 1323.s
• Analytic conductor: $10.564$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5642081874\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 441)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
656.1	−2.34591	+	1.35441i		0		2.66888	−	4.62263i	1.20293				0			0				9.04141i		0		−2.82197	+	1.62926i
656.2	−2.34591	+	1.35441i		0		2.66888	−	4.62263i	−1.20293				0			0				9.04141i		0		2.82197	−	1.62926i
656.3	−2.05485	+	1.18637i		0		1.81495	−	3.14358i	−3.43548				0			0				3.86732i		0		7.05942	−	4.07576i
656.4	−2.05485	+	1.18637i		0		1.81495	−	3.14358i	3.43548				0			0				3.86732i		0		−7.05942	+	4.07576i
656.5	−1.28562	+	0.742253i		0		0.101880	−	0.176462i	0.308431				0			0			−	2.66653i		0		−0.396525	+	0.228934i
656.6	−1.28562	+	0.742253i		0		0.101880	−	0.176462i	−0.308431				0			0			−	2.66653i		0		0.396525	−	0.228934i
656.7	−1.02035	+	0.589100i		0		−0.305921	+	0.529871i	−4.33202				0			0			−	3.07728i		0		4.42019	−	2.55200i
656.8	−1.02035	+	0.589100i		0		−0.305921	+	0.529871i	4.33202				0			0			−	3.07728i		0		−4.42019	+	2.55200i
656.9	−0.850109	+	0.490811i		0		−0.518210	+	0.897565i	−1.88120				0			0			−	2.98061i		0		1.59922	−	0.923312i
656.10	−0.850109	+	0.490811i		0		−0.518210	+	0.897565i	1.88120				0			0			−	2.98061i		0		−1.59922	+	0.923312i
656.11	−0.367369	+	0.212101i		0		−0.910027	+	1.57621i	3.60763				0			0			−	1.62047i		0		−1.32533	+	0.765180i
656.12	−0.367369	+	0.212101i		0		−0.910027	+	1.57621i	−3.60763				0			0			−	1.62047i		0		1.32533	−	0.765180i
656.13	0.105953	−	0.0611722i		0		−0.992516	+	1.71909i	0.529430				0			0				0.487547i		0		0.0560949	−	0.0323864i
656.14	0.105953	−	0.0611722i		0		−0.992516	+	1.71909i	−0.529430				0			0				0.487547i		0		−0.0560949	+	0.0323864i
656.15	0.575298	−	0.332148i		0		−0.779355	+	1.34988i	0.0283039				0			0				2.36404i		0		0.0162832	−	0.00940110i
656.16	0.575298	−	0.332148i		0		−0.779355	+	1.34988i	−0.0283039				0			0				2.36404i		0		−0.0162832	+	0.00940110i
656.17	1.58658	−	0.916012i		0		0.678156	−	1.17460i	−0.645568				0			0				1.17925i		0		−1.02425	+	0.591348i
656.18	1.58658	−	0.916012i		0		0.678156	−	1.17460i	0.645568				0			0				1.17925i		0		1.02425	−	0.591348i
656.19	1.61855	−	0.934468i		0		0.746462	−	1.29291i	−2.50573				0			0				0.947692i		0		−4.05565	+	2.34153i
656.20	1.61855	−	0.934468i		0		0.746462	−	1.29291i	2.50573				0			0				0.947692i		0		4.05565	−	2.34153i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
63.n	odd	6	1	inner
63.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1323.2.s.d		48
3.b	odd	2	1		441.2.s.d		48
7.b	odd	2	1	inner	1323.2.s.d		48
7.c	even	3	1		1323.2.i.d		48
7.c	even	3	1		1323.2.o.e		48
7.d	odd	6	1		1323.2.i.d		48
7.d	odd	6	1		1323.2.o.e		48
9.c	even	3	1		441.2.i.d		48
9.d	odd	6	1		1323.2.i.d		48
21.c	even	2	1		441.2.s.d		48
21.g	even	6	1		441.2.i.d		48
21.g	even	6	1		441.2.o.e	✓	48
21.h	odd	6	1		441.2.i.d		48
21.h	odd	6	1		441.2.o.e	✓	48
63.g	even	3	1		441.2.s.d		48
63.h	even	3	1		441.2.o.e	✓	48
63.i	even	6	1		1323.2.o.e		48
63.j	odd	6	1		1323.2.o.e		48
63.k	odd	6	1		441.2.s.d		48
63.l	odd	6	1		441.2.i.d		48
63.n	odd	6	1	inner	1323.2.s.d		48
63.o	even	6	1		1323.2.i.d		48
63.s	even	6	1	inner	1323.2.s.d		48
63.t	odd	6	1		441.2.o.e	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.2.i.d		48	9.c	even	3	1
441.2.i.d		48	21.g	even	6	1
441.2.i.d		48	21.h	odd	6	1
441.2.i.d		48	63.l	odd	6	1
441.2.o.e	✓	48	21.g	even	6	1
441.2.o.e	✓	48	21.h	odd	6	1
441.2.o.e	✓	48	63.h	even	3	1
441.2.o.e	✓	48	63.t	odd	6	1
441.2.s.d		48	3.b	odd	2	1
441.2.s.d		48	21.c	even	2	1
441.2.s.d		48	63.g	even	3	1
441.2.s.d		48	63.k	odd	6	1
1323.2.i.d		48	7.c	even	3	1
1323.2.i.d		48	7.d	odd	6	1
1323.2.i.d		48	9.d	odd	6	1
1323.2.i.d		48	63.o	even	6	1
1323.2.o.e		48	7.c	even	3	1
1323.2.o.e		48	7.d	odd	6	1
1323.2.o.e		48	63.i	even	6	1
1323.2.o.e		48	63.j	odd	6	1
1323.2.s.d		48	1.a	even	1	1	trivial
1323.2.s.d		48	7.b	odd	2	1	inner
1323.2.s.d		48	63.n	odd	6	1	inner
1323.2.s.d		48	63.s	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^24 - 18*T2^22 + 207*T2^20 - 12*T2^19 - 1434*T2^18 + 108*T2^17 + 7227*T2^16 - 540*T2^15 - 24510*T2^14 + 61340*T2^12 + 9612*T2^11 - 101154*T2^10 - 34488*T2^9 + 112545*T2^8 + 70116*T2^7 - 42660*T2^6 - 33444*T2^5 + 14130*T2^4 + 10800*T2^3 + 678*T2^2 - 504*T2 + 49