Newform orbit 540.2.be.a

• Label: 540.2.be.a
• Level: $540$
• Weight: $2$
• Character orbit: 540.be
• Analytic conductor: $4.312$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.31192170915\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 432 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41370	−	0.0380122i	0.0948588	+	1.72945i	1.99711	+	0.107476i	−0.342020	+	0.939693i	−0.0683620	−	2.44854i	0.592569	+	0.104486i	−2.81923	−	0.227853i	−2.98200	+	0.328108i	0.519235	−	1.31544i
11.2	−1.41024	−	0.105903i	1.59941	−	0.664741i	1.97757	+	0.298697i	0.342020	−	0.939693i	−2.32596	+	0.768065i	2.87648	+	0.507202i	−2.75722	−	0.630665i	2.11624	−	2.12639i	−0.581847	+	1.28897i
11.3	−1.40514	+	0.159958i	−0.0604462	+	1.73100i	1.94883	−	0.449526i	0.342020	−	0.939693i	−0.191951	−	2.44196i	−4.54659	−	0.801686i	−2.66647	+	0.943376i	−2.99269	−	0.209264i	−0.330274	+	1.37511i
11.4	−1.40297	+	0.177991i	−1.72248	−	0.181830i	1.93664	−	0.499431i	0.342020	−	0.939693i	2.44895	−	0.0514835i	−1.67026	−	0.294513i	−2.62815	+	1.04539i	2.93388	+	0.626399i	−0.312587	+	1.37924i
11.5	−1.39864	+	0.209324i	0.570476	−	1.63541i	1.91237	−	0.585538i	−0.342020	+	0.939693i	−0.455558	+	2.40675i	1.79764	+	0.316972i	−2.55214	+	1.21926i	−2.34911	−	1.86592i	0.281661	−	1.38588i
11.6	−1.38456	−	0.288063i	0.103186	−	1.72897i	1.83404	+	0.797684i	0.342020	−	0.939693i	−0.640921	+	2.36415i	−2.16019	−	0.380900i	−2.30956	−	1.63276i	−2.97871	−	0.356810i	−0.744240	+	1.20254i
11.7	−1.34150	+	0.447643i	1.63820	−	0.562416i	1.59923	−	1.20102i	−0.342020	+	0.939693i	−1.94587	+	1.48781i	−3.58594	−	0.632299i	−1.60774	+	2.32706i	2.36738	−	1.84270i	0.0381725	−	1.41370i
11.8	−1.32876	−	0.484134i	−1.24536	−	1.20377i	1.53123	+	1.28660i	−0.342020	+	0.939693i	1.07201	+	2.20245i	4.81853	+	0.849637i	−1.41175	−	2.45091i	0.101864	+	2.99827i	0.909402	−	1.08305i
11.9	−1.30501	+	0.544932i	−1.34721	+	1.08859i	1.40610	−	1.42228i	−0.342020	+	0.939693i	1.16492	−	2.15476i	3.23494	+	0.570407i	−1.05993	+	2.62232i	0.629953	−	2.93311i	−0.0657288	−	1.41269i
11.10	−1.29418	−	0.570165i	−0.816567	−	1.52749i	1.34982	+	1.47580i	−0.342020	+	0.939693i	0.185867	+	2.44243i	−4.81738	−	0.849433i	−0.905471	−	2.67958i	−1.66644	+	2.49459i	0.978417	−	1.02113i
11.11	−1.28988	+	0.579828i	−1.53475	−	0.802841i	1.32760	−	1.49582i	−0.342020	+	0.939693i	2.44515	+	0.145681i	−2.53856	−	0.447617i	−0.845126	+	2.69922i	1.71089	+	2.46431i	−0.103694	−	1.41041i
11.12	−1.23687	−	0.685681i	−0.641893	+	1.60872i	1.05968	+	1.69619i	0.342020	−	0.939693i	1.89700	−	1.54964i	0.0767354	+	0.0135305i	−0.147641	−	2.82457i	−2.17595	−	2.06525i	−1.06736	+	0.927759i
11.13	−1.20831	−	0.734831i	−1.50938	+	0.849577i	0.920046	+	1.77581i	−0.342020	+	0.939693i	2.44810	+	0.0825815i	−2.61755	−	0.461544i	0.193219	−	2.82182i	1.55644	−	2.56466i	1.10378	−	0.884117i
11.14	−1.19467	+	0.756808i	−1.12619	−	1.31594i	0.854484	−	1.80827i	0.342020	−	0.939693i	2.34134	+	0.719816i	4.64650	+	0.819302i	0.347688	+	2.80698i	−0.463415	+	2.96399i	0.302565	+	1.38147i
11.15	−1.14304	−	0.832741i	0.932759	+	1.45944i	0.613085	+	1.90371i	0.342020	−	0.939693i	0.149152	−	2.44494i	3.63963	+	0.641765i	0.884519	−	2.68656i	−1.25992	+	2.72261i	−1.17346	+	0.789293i
11.16	−1.13219	+	0.847440i	0.315979	−	1.70298i	0.563689	−	1.91892i	0.342020	−	0.939693i	1.08543	+	2.19587i	−2.38535	−	0.420602i	0.987970	+	2.65027i	−2.80032	−	1.07621i	0.409103	+	1.35375i
11.17	−1.11807	+	0.865978i	1.71259	−	0.258927i	0.500164	−	1.93645i	0.342020	−	0.939693i	−1.69057	+	1.77256i	−1.15631	−	0.203889i	1.11770	+	2.59822i	2.86591	−	0.886872i	0.431350	+	1.34682i
11.18	−1.05484	+	0.941973i	1.54262	+	0.787603i	0.225375	−	1.98726i	−0.342020	+	0.939693i	−2.36912	+	0.622312i	−0.308184	−	0.0543412i	1.63421	+	2.30854i	1.75936	+	2.42995i	−0.524388	−	1.31340i
11.19	−0.992140	−	1.00780i	1.73090	−	0.0631559i	−0.0313148	+	1.99975i	−0.342020	+	0.939693i	−1.78094	−	1.68174i	3.02663	+	0.533676i	2.04642	−	1.95248i	2.99202	−	0.218633i	1.28635	−	0.587620i
11.20	−0.977486	−	1.02202i	−1.42519	−	0.984291i	−0.0890420	+	1.99802i	0.342020	−	0.939693i	0.387142	+	2.41870i	0.242074	+	0.0426842i	2.12905	−	1.86203i	1.06234	+	2.80561i	−1.29470	+	0.568986i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
27.f	odd	18	1	inner
108.l	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.be.a	✓	432
4.b	odd	2	1	inner	540.2.be.a	✓	432
27.f	odd	18	1	inner	540.2.be.a	✓	432
108.l	even	18	1	inner	540.2.be.a	✓	432

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
540.2.be.a	✓	432	1.a	even	1	1	trivial
540.2.be.a	✓	432	4.b	odd	2	1	inner
540.2.be.a	✓	432	27.f	odd	18	1	inner
540.2.be.a	✓	432	108.l	even	18	1	inner

Hecke kernels

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