Newform orbit 540.2.bj.a

• Label: 540.2.bj.a
• Level: $540$
• Weight: $2$
• Character orbit: 540.bj
• Analytic conductor: $4.312$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	540.bj (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.31192170915\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(18\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
77.1		0		−1.72265	−	0.180256i		0		−1.88508	+	1.20270i		0		1.78959	+	0.156569i		0		2.93502	+	0.621035i		0
77.2		0		−1.46545	−	0.923290i		0		1.95679	+	1.08211i		0		−4.76532	−	0.416911i		0		1.29507	+	2.70606i		0
77.3		0		−1.45611	−	0.937947i		0		2.00716	−	0.985553i		0		3.74089	+	0.327285i		0		1.24051	+	2.73151i		0
77.4		0		−1.43153	+	0.975044i		0		1.93809	−	1.11526i		0		−2.72135	−	0.238087i		0		1.09858	−	2.79162i		0
77.5		0		−1.36940	+	1.06055i		0		0.570085	+	2.16218i		0		2.28167	+	0.199620i		0		0.750488	−	2.90461i		0
77.6		0		−1.25051	+	1.19843i		0		−2.17068	−	0.536787i		0		−2.10186	−	0.183889i		0		0.127526	−	2.99729i		0
77.7		0		−0.581369	+	1.63157i		0		−0.300886	−	2.21573i		0		4.97000	+	0.434819i		0		−2.32402	−	1.89708i		0
77.8		0		−0.470819	−	1.66683i		0		−0.769420	+	2.09952i		0		−0.335573	−	0.0293589i		0		−2.55666	+	1.56955i		0
77.9		0		−0.324859	+	1.70131i		0		0.420978	+	2.19608i		0		−3.01994	−	0.264210i		0		−2.78893	−	1.10537i		0
77.10		0		0.0656921	−	1.73080i		0		0.696960	−	2.12468i		0		−1.42486	−	0.124659i		0		−2.99137	−	0.227400i		0
77.11		0		0.230415	−	1.71666i		0		−1.83635	−	1.27586i		0		3.92499	+	0.343393i		0		−2.89382	−	0.791087i		0
77.12		0		0.709167	+	1.58022i		0		−1.49644	−	1.66153i		0		−1.61202	−	0.141034i		0		−1.99417	+	2.24127i		0
77.13		0		0.839467	+	1.51502i		0		2.21884	−	0.277019i		0		0.702012	+	0.0614181i		0		−1.59059	+	2.54362i		0
77.14		0		1.22933	−	1.22014i		0		1.94303	+	1.10664i		0		0.707351	+	0.0618852i		0		0.0225061	−	2.99992i		0
77.15		0		1.39727	−	1.02355i		0		−1.92320	+	1.14075i		0		−5.04582	−	0.441452i		0		0.904709	−	2.86033i		0
77.16		0		1.57719	+	0.715876i		0		0.315012	+	2.21377i		0		2.77962	+	0.243185i		0		1.97504	+	2.25814i		0
77.17		0		1.69522	+	0.355293i		0		−2.18535	+	0.473534i		0		2.18731	+	0.191365i		0		2.74753	+	1.20460i		0
77.18		0		1.73127	−	0.0521224i		0		0.277223	−	2.21882i		0		−2.05670	−	0.179938i		0		2.99457	−	0.180476i		0
113.1		0		−1.73205	+	0.00335186i		0		0.0855458	+	2.23443i		0		0.869779	+	0.609026i		0		2.99998	−	0.0116112i		0
113.2		0		−1.64097	−	0.554265i		0		1.70842	−	1.44267i		0		−2.84188	−	1.98991i		0		2.38558	+	1.81907i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
27.f	odd	18	1	inner
135.q	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	540.2.bj.a	✓	216
5.c	odd	4	1	inner	540.2.bj.a	✓	216
27.f	odd	18	1	inner	540.2.bj.a	✓	216
135.q	even	36	1	inner	540.2.bj.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
540.2.bj.a	✓	216	1.a	even	1	1	trivial
540.2.bj.a	✓	216	5.c	odd	4	1	inner
540.2.bj.a	✓	216	27.f	odd	18	1	inner
540.2.bj.a	✓	216	135.q	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(540, [\chi])\).