Newform orbit 648.2.q.a

• Label: 648.2.q.a
• Level: $648$
• Weight: $2$
• Character orbit: 648.q
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.q (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
73.1		0			0			0		−2.75433	+	2.31115i		0		−1.28512	+	0.467744i		0			0			0
73.2		0			0			0		−0.407020	+	0.341530i		0		0.507439	−	0.184693i		0			0			0
73.3		0			0			0		1.80911	−	1.51802i		0		−3.12406	+	1.13706i		0			0			0
73.4		0			0			0		2.11828	−	1.77745i		0		4.34143	−	1.58015i		0			0			0
145.1		0			0			0		−0.444259	+	2.51952i		0		−0.612199	+	0.513696i		0			0			0
145.2		0			0			0		−0.198034	+	1.12311i		0		0.914338	−	0.767221i		0			0			0
145.3		0			0			0		0.0770674	−	0.437071i		0		0.935232	−	0.784753i		0			0			0
145.4		0			0			0		0.738874	−	4.19036i		0		−2.50342	+	2.10062i		0			0			0
289.1		0			0			0		−2.42978	−	0.884366i		0		−0.245784	+	1.39391i		0			0			0
289.2		0			0			0		−0.307563	−	0.111944i		0		0.551939	−	3.13020i		0			0			0
289.3		0			0			0		0.00848388	+	0.00308788i		0		−0.356397	+	2.02123i		0			0			0
289.4		0			0			0		1.78916	+	0.651202i		0		−0.623407	+	3.53552i		0			0			0
361.1		0			0			0		−2.42978	+	0.884366i		0		−0.245784	−	1.39391i		0			0			0
361.2		0			0			0		−0.307563	+	0.111944i		0		0.551939	+	3.13020i		0			0			0
361.3		0			0			0		0.00848388	−	0.00308788i		0		−0.356397	−	2.02123i		0			0			0
361.4		0			0			0		1.78916	−	0.651202i		0		−0.623407	−	3.53552i		0			0			0
505.1		0			0			0		−0.444259	−	2.51952i		0		−0.612199	−	0.513696i		0			0			0
505.2		0			0			0		−0.198034	−	1.12311i		0		0.914338	+	0.767221i		0			0			0
505.3		0			0			0		0.0770674	+	0.437071i		0		0.935232	+	0.784753i		0			0			0
505.4		0			0			0		0.738874	+	4.19036i		0		−2.50342	−	2.10062i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.2.q.a		24
3.b	odd	2	1		216.2.q.a	✓	24
12.b	even	2	1		432.2.u.e		24
27.e	even	9	1	inner	648.2.q.a		24
27.e	even	9	1		5832.2.a.i		12
27.f	odd	18	1		216.2.q.a	✓	24
27.f	odd	18	1		5832.2.a.h		12
108.l	even	18	1		432.2.u.e		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
216.2.q.a	✓	24	3.b	odd	2	1
216.2.q.a	✓	24	27.f	odd	18	1
432.2.u.e		24	12.b	even	2	1
432.2.u.e		24	108.l	even	18	1
648.2.q.a		24	1.a	even	1	1	trivial
648.2.q.a		24	27.e	even	9	1	inner
5832.2.a.h		12	27.f	odd	18	1
5832.2.a.i		12	27.e	even	9	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^24 + 12*T5^22 + 19*T5^21 + 45*T5^20 - 309*T5^19 + 1702*T5^18 + 1152*T5^17 - 486*T5^16 + 4858*T5^15 - 23877*T5^14 + 6159*T5^13 + 454615*T5^12 - 763902*T5^11 + 260358*T5^10 - 389045*T5^9 + 1168164*T5^8 + 1826226*T5^7 + 1385830*T5^6 + 644256*T5^5 + 273360*T5^4 + 83125*T5^3 + 10800*T5^2 - 201*T5 + 1