Newform orbit 540.2.bb.a

• Label: 540.2.bb.a
• Level: $540$
• Weight: $2$
• Character orbit: 540.bb
• Analytic conductor: $4.312$
• Analytic rank: $0$
• Dimension: $24$
• CM discriminant: -20
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	540.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.31192170915\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−0.909039	−	1.08335i	−1.24394	+	1.20524i	−0.347296	+	1.96962i	−0.764780	−	2.10122i	2.43649	+	0.252009i	0.767983	+	4.35545i	2.44949	−	1.41421i	0.0947721	−	2.99850i	−1.58114	+	2.73861i
59.2	−0.909039	−	1.08335i	1.72763	+	0.123682i	−0.347296	+	1.96962i	0.764780	+	2.10122i	−1.43649	−	1.98406i	−0.264032	−	1.49740i	2.44949	−	1.41421i	2.96941	+	0.427352i	1.58114	−	2.73861i
59.3	0.909039	+	1.08335i	−1.72763	−	0.123682i	−0.347296	+	1.96962i	0.764780	+	2.10122i	−1.43649	−	1.98406i	0.264032	+	1.49740i	−2.44949	+	1.41421i	2.96941	+	0.427352i	−1.58114	+	2.73861i
59.4	0.909039	+	1.08335i	1.24394	−	1.20524i	−0.347296	+	1.96962i	−0.764780	−	2.10122i	2.43649	+	0.252009i	−0.767983	−	4.35545i	−2.44949	+	1.41421i	0.0947721	−	2.99850i	1.58114	−	2.73861i
119.1	−0.909039	+	1.08335i	−1.24394	−	1.20524i	−0.347296	−	1.96962i	−0.764780	+	2.10122i	2.43649	−	0.252009i	0.767983	−	4.35545i	2.44949	+	1.41421i	0.0947721	+	2.99850i	−1.58114	−	2.73861i
119.2	−0.909039	+	1.08335i	1.72763	−	0.123682i	−0.347296	−	1.96962i	0.764780	−	2.10122i	−1.43649	+	1.98406i	−0.264032	+	1.49740i	2.44949	+	1.41421i	2.96941	−	0.427352i	1.58114	+	2.73861i
119.3	0.909039	−	1.08335i	−1.72763	+	0.123682i	−0.347296	−	1.96962i	0.764780	−	2.10122i	−1.43649	+	1.98406i	0.264032	−	1.49740i	−2.44949	−	1.41421i	2.96941	−	0.427352i	−1.58114	−	2.73861i
119.4	0.909039	−	1.08335i	1.24394	+	1.20524i	−0.347296	−	1.96962i	−0.764780	+	2.10122i	2.43649	−	0.252009i	−0.767983	+	4.35545i	−2.44949	−	1.41421i	0.0947721	+	2.99850i	1.58114	+	2.73861i
239.1	−0.483690	+	1.32893i	−0.970926	+	1.43433i	−1.53209	−	1.28558i	−2.20210	−	0.388289i	−1.43649	−	1.98406i	−2.78002	+	2.33272i	2.44949	−	1.41421i	−1.11460	−	2.78526i	1.58114	−	2.73861i
239.2	−0.483690	+	1.32893i	−0.421802	−	1.67991i	−1.53209	−	1.28558i	2.20210	+	0.388289i	2.43649	+	0.252009i	−3.62133	+	3.03866i	2.44949	−	1.41421i	−2.64417	+	1.41718i	−1.58114	+	2.73861i
239.3	0.483690	−	1.32893i	0.421802	+	1.67991i	−1.53209	−	1.28558i	2.20210	+	0.388289i	2.43649	+	0.252009i	3.62133	−	3.03866i	−2.44949	+	1.41421i	−2.64417	+	1.41718i	1.58114	−	2.73861i
239.4	0.483690	−	1.32893i	0.970926	−	1.43433i	−1.53209	−	1.28558i	−2.20210	−	0.388289i	−1.43649	−	1.98406i	2.78002	−	2.33272i	−2.44949	+	1.41421i	−1.11460	−	2.78526i	−1.58114	+	2.73861i
299.1	−1.39273	−	0.245576i	−1.66574	+	0.474661i	1.87939	+	0.684040i	−1.43732	−	1.71293i	2.43649	−	0.252009i	0.286298	−	0.104204i	−2.44949	−	1.41421i	2.54939	−	1.58133i	1.58114	+	2.73861i
299.2	−1.39273	−	0.245576i	0.756703	−	1.55801i	1.87939	+	0.684040i	1.43732	+	1.71293i	−1.43649	+	1.98406i	4.83901	−	1.76125i	−2.44949	−	1.41421i	−1.85480	−	2.35790i	−1.58114	−	2.73861i
299.3	1.39273	+	0.245576i	−0.756703	+	1.55801i	1.87939	+	0.684040i	1.43732	+	1.71293i	−1.43649	+	1.98406i	−4.83901	+	1.76125i	2.44949	+	1.41421i	−1.85480	−	2.35790i	1.58114	+	2.73861i
299.4	1.39273	+	0.245576i	1.66574	−	0.474661i	1.87939	+	0.684040i	−1.43732	−	1.71293i	2.43649	−	0.252009i	−0.286298	+	0.104204i	2.44949	+	1.41421i	2.54939	−	1.58133i	−1.58114	−	2.73861i
419.1	−1.39273	+	0.245576i	−1.66574	−	0.474661i	1.87939	−	0.684040i	−1.43732	+	1.71293i	2.43649	+	0.252009i	0.286298	+	0.104204i	−2.44949	+	1.41421i	2.54939	+	1.58133i	1.58114	−	2.73861i
419.2	−1.39273	+	0.245576i	0.756703	+	1.55801i	1.87939	−	0.684040i	1.43732	−	1.71293i	−1.43649	−	1.98406i	4.83901	+	1.76125i	−2.44949	+	1.41421i	−1.85480	+	2.35790i	−1.58114	+	2.73861i
419.3	1.39273	−	0.245576i	−0.756703	−	1.55801i	1.87939	−	0.684040i	1.43732	−	1.71293i	−1.43649	−	1.98406i	−4.83901	−	1.76125i	2.44949	−	1.41421i	−1.85480	+	2.35790i	1.58114	−	2.73861i
419.4	1.39273	−	0.245576i	1.66574	+	0.474661i	1.87939	−	0.684040i	−1.43732	+	1.71293i	2.43649	+	0.252009i	−0.286298	−	0.104204i	2.44949	−	1.41421i	2.54939	+	1.58133i	−1.58114	+	2.73861i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)
4.b	odd	2	1	inner
5.b	even	2	1	inner
27.f	odd	18	1	inner
108.l	even	18	1	inner
135.n	odd	18	1	inner
540.bb	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	540.2.bb.a	✓	24
4.b	odd	2	1	inner	540.2.bb.a	✓	24
5.b	even	2	1	inner	540.2.bb.a	✓	24
20.d	odd	2	1	CM	540.2.bb.a	✓	24
27.f	odd	18	1	inner	540.2.bb.a	✓	24
108.l	even	18	1	inner	540.2.bb.a	✓	24
135.n	odd	18	1	inner	540.2.bb.a	✓	24
540.bb	even	18	1	inner	540.2.bb.a	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
540.2.bb.a	✓	24	1.a	even	1	1	trivial
540.2.bb.a	✓	24	4.b	odd	2	1	inner
540.2.bb.a	✓	24	5.b	even	2	1	inner
540.2.bb.a	✓	24	20.d	odd	2	1	CM
540.2.bb.a	✓	24	27.f	odd	18	1	inner
540.2.bb.a	✓	24	108.l	even	18	1	inner
540.2.bb.a	✓	24	135.n	odd	18	1	inner
540.2.bb.a	✓	24	540.bb	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^24 - 12*T7^22 + 285*T7^20 + 10352*T7^18 - 6894*T7^16 + 4975092*T7^14 + 139161864*T7^12 + 435625308*T7^10 + 23506073469*T7^8 + 97461196328*T7^6 + 110428591899*T7^4 - 16846099332*T7^2 + 1073283121