Newform orbit 270.3.n.a

• Label: 270.3.n.a
• Level: $270$
• Weight: $3$
• Character orbit: 270.n
• Analytic conductor: $7.357$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	270.n (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.35696713773\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(36\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 18 q^{5} + 12 q^{6} + 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−0.245576	+	1.39273i	−2.95765	−	0.502313i	−1.87939	−	0.684040i	−4.92358	+	0.870849i	1.42591	−	3.99584i	−1.65791	−	4.55508i	1.41421	−	2.44949i	8.49536	+	2.97133i	−0.00374550	−	7.07107i
29.2	−0.245576	+	1.39273i	−2.93139	+	0.637912i	−1.87939	−	0.684040i	0.772686	+	4.93993i	−0.168560	−	4.23929i	3.98607	+	10.9516i	1.41421	−	2.44949i	8.18614	−	3.73994i	−7.06974	−	0.136986i
29.3	−0.245576	+	1.39273i	−2.77364	−	1.14319i	−1.87939	−	0.684040i	3.94044	−	3.07781i	2.27330	−	3.58219i	−1.03960	−	2.85628i	1.41421	−	2.44949i	6.38621	+	6.34163i	3.31888	+	6.24380i
29.4	−0.245576	+	1.39273i	−2.44467	+	1.73884i	−1.87939	−	0.684040i	1.14986	−	4.86599i	−1.82139	−	3.83178i	1.77882	+	4.88727i	1.41421	−	2.44949i	2.95284	−	8.50181i	6.49462	+	2.79641i
29.5	−0.245576	+	1.39273i	−2.13533	+	2.10722i	−1.87939	−	0.684040i	4.31675	+	2.52302i	−2.41039	−	3.49142i	−3.64772	−	10.0220i	1.41421	−	2.44949i	0.119286	−	8.99921i	−4.57398	+	5.39247i
29.6	−0.245576	+	1.39273i	−1.81716	−	2.38704i	−1.87939	−	0.684040i	1.90906	+	4.62120i	3.77074	−	1.94461i	−2.55043	−	7.00726i	1.41421	−	2.44949i	−2.39588	+	8.67524i	−6.90490	+	1.52395i
29.7	−0.245576	+	1.39273i	−0.936605	+	2.85005i	−1.87939	−	0.684040i	−4.96948	+	0.551583i	−3.73934	−	2.00434i	1.06409	+	2.92356i	1.41421	−	2.44949i	−7.24554	−	5.33874i	0.452178	−	7.05660i
29.8	−0.245576	+	1.39273i	−0.642397	−	2.93041i	−1.87939	−	0.684040i	−3.29699	−	3.75897i	4.23903	−	0.175046i	−0.265944	−	0.730676i	1.41421	−	2.44949i	−8.17465	+	3.76498i	6.04489	−	3.66870i
29.9	−0.245576	+	1.39273i	−0.221554	−	2.99181i	−1.87939	−	0.684040i	4.65626	−	1.82189i	4.22118	+	0.426150i	4.28972	+	11.7859i	1.41421	−	2.44949i	−8.90183	+	1.32569i	1.39393	+	6.93231i
29.10	−0.245576	+	1.39273i	0.0244460	−	2.99990i	−1.87939	−	0.684040i	−1.99837	+	4.58329i	4.17204	+	0.770749i	0.118587	+	0.325814i	1.41421	−	2.44949i	−8.99880	−	0.146671i	−5.89253	−	3.90873i
29.11	−0.245576	+	1.39273i	0.324527	+	2.98240i	−1.87939	−	0.684040i	4.96091	+	0.623972i	−4.23336	−	0.280425i	0.158266	+	0.434833i	1.41421	−	2.44949i	−8.78936	+	1.93574i	−2.08730	+	6.75597i
29.12	−0.245576	+	1.39273i	1.53299	+	2.57875i	−1.87939	−	0.684040i	−0.871603	−	4.92344i	−3.96796	+	1.50176i	−3.41693	−	9.38793i	1.41421	−	2.44949i	−4.29990	+	7.90638i	7.07107	−	0.00482815i
29.13	−0.245576	+	1.39273i	2.40493	−	1.79340i	−1.87939	−	0.684040i	−0.604237	−	4.96336i	1.90713	+	3.78983i	−0.969685	−	2.66419i	1.41421	−	2.44949i	2.56740	−	8.62603i	7.06099	+	0.377342i
29.14	−0.245576	+	1.39273i	2.51712	+	1.63221i	−1.87939	−	0.684040i	−2.75970	+	4.16942i	−2.89136	+	3.10484i	−4.37787	−	12.0281i	1.41421	−	2.44949i	3.67181	+	8.21692i	−5.12915	−	4.86742i
29.15	−0.245576	+	1.39273i	2.69147	−	1.32513i	−1.87939	−	0.684040i	−4.68006	+	1.75983i	1.18458	+	4.07391i	−1.40747	−	3.86699i	1.41421	−	2.44949i	5.48807	−	7.13310i	−1.30165	−	6.95023i
29.16	−0.245576	+	1.39273i	2.72301	+	1.25905i	−1.87939	−	0.684040i	−4.13198	−	2.81545i	−2.42222	+	3.48322i	3.91947	+	10.7686i	1.41421	−	2.44949i	5.82957	+	6.85683i	4.93587	−	5.06332i
29.17	−0.245576	+	1.39273i	2.77167	−	1.14797i	−1.87939	−	0.684040i	2.92999	+	4.05156i	0.918153	+	4.14210i	2.42688	+	6.66780i	1.41421	−	2.44949i	6.36434	−	6.36358i	−6.36226	+	3.08572i
29.18	−0.245576	+	1.39273i	2.95358	+	0.525705i	−1.87939	−	0.684040i	4.91911	−	0.895728i	−1.45749	+	3.98443i	−0.273799	−	0.752258i	1.41421	−	2.44949i	8.44727	+	3.10542i	0.0394919	+	7.07096i
29.19	0.245576	−	1.39273i	−2.95358	−	0.525705i	−1.87939	−	0.684040i	−1.73631	+	4.68884i	−1.45749	+	3.98443i	0.273799	+	0.752258i	−1.41421	+	2.44949i	8.44727	+	3.10542i	6.10388	+	3.56968i
29.20	0.245576	−	1.39273i	−2.77167	+	1.14797i	−1.87939	−	0.684040i	3.48122	+	3.58903i	0.918153	+	4.14210i	−2.42688	−	6.66780i	−1.41421	+	2.44949i	6.36434	−	6.36358i	5.85344	−	3.96701i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
27.f	odd	18	1	inner
135.n	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	270.3.n.a	✓	216
5.b	even	2	1	inner	270.3.n.a	✓	216
27.f	odd	18	1	inner	270.3.n.a	✓	216
135.n	odd	18	1	inner	270.3.n.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.3.n.a	✓	216	1.a	even	1	1	trivial
270.3.n.a	✓	216	5.b	even	2	1	inner
270.3.n.a	✓	216	27.f	odd	18	1	inner
270.3.n.a	✓	216	135.n	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(270, [\chi])\).