Newform orbit 585.2.ca.a

• Label: 585.2.ca.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.ca
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.ca (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.67124851824\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(80\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 2 q^{3} - 304 q^{4} - 2 q^{5} - 8 q^{6} + 2 q^{7} + 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
58.1		−	2.74212i	−0.788234	−	1.54230i	−5.51923			1.63311	+	1.52740i	−4.22917	+	2.16143i	−0.999815	+	1.73173i			9.65015i	−1.75738	+	2.43139i	4.18831	−	4.47819i
58.2		−	2.66053i	1.12828	−	1.31415i	−5.07843			0.0728242	−	2.23488i	−3.49634	−	3.00182i	−1.86279	+	3.22644i			8.19025i	−0.453983	−	2.96545i	−5.94597	−	0.193751i
58.3		−	2.65574i	−1.68340	+	0.407622i	−5.05295			1.82773	−	1.28817i	1.08254	+	4.47068i	2.01891	−	3.49685i			8.10783i	2.66769	−	1.37238i	−3.42106	−	4.85398i
58.4		−	2.60486i	1.61477	+	0.626514i	−4.78531			1.82078	+	1.29798i	1.63198	−	4.20625i	0.106425	−	0.184334i			7.25535i	2.21496	+	2.02335i	3.38106	−	4.74289i
58.5		−	2.57181i	−1.35707	+	1.07628i	−4.61419			−1.80330	−	1.32216i	2.76798	+	3.49011i	−2.21895	+	3.84333i			6.72319i	0.683251	−	2.92116i	−3.40034	+	4.63774i
58.6		−	2.54907i	−1.36646	−	1.06432i	−4.49778			−2.19362	−	0.433645i	−2.71304	+	3.48321i	0.727014	−	1.25923i			6.36702i	0.734437	+	2.90871i	−1.10539	+	5.59169i
58.7		−	2.45868i	1.16934	−	1.27775i	−4.04511			−0.887561	+	2.05237i	−3.14158	−	2.87503i	1.60020	−	2.77163i			5.02826i	−0.265294	−	2.98825i	5.04613	+	2.18223i
58.8		−	2.37704i	1.71493	−	0.242904i	−3.65031			−1.07099	−	1.96290i	−0.577391	−	4.07646i	1.74707	−	3.02601i			3.92284i	2.88200	−	0.833127i	−4.66589	+	2.54578i
58.9		−	2.30512i	1.67336	+	0.447082i	−3.31357			−1.85931	+	1.24217i	1.03058	−	3.85728i	−1.86470	+	3.22975i			3.02793i	2.60024	+	1.49625i	2.86334	+	4.28592i
58.10		−	2.27576i	−0.685267	+	1.59073i	−3.17907			2.10624	+	0.750837i	3.62011	+	1.55950i	−0.618123	+	1.07062i			2.68327i	−2.06082	−	2.18014i	1.70872	−	4.79329i
58.11		−	2.20453i	0.106040	+	1.72880i	−2.85995			−0.758370	+	2.10354i	3.81119	−	0.233769i	−0.828210	+	1.43450i			1.89577i	−2.97751	+	0.366645i	4.63731	+	1.67185i
58.12		−	2.10402i	−1.71160	−	0.265389i	−2.42689			−0.257164	+	2.22123i	−0.558384	+	3.60123i	0.344866	−	0.597325i			0.898186i	2.85914	+	0.908480i	4.67351	+	0.541078i
58.13		−	2.09364i	−0.395593	−	1.68627i	−2.38334			1.23936	−	1.86118i	−3.53045	+	0.828230i	1.93359	−	3.34908i			0.802581i	−2.68701	+	1.33415i	−3.89665	−	2.59477i
58.14		−	1.99427i	−0.506102	+	1.65646i	−1.97712			−0.297436	−	2.21620i	3.30343	+	1.00931i	0.719291	−	1.24585i		−	0.0456243i	−2.48772	−	1.67668i	−4.41970	+	0.593167i
58.15		−	1.94398i	0.815198	−	1.52822i	−1.77906			2.18642	−	0.468590i	−2.97083	−	1.58473i	−0.636252	+	1.10202i		−	0.429495i	−1.67090	−	2.49160i	−0.910930	−	4.25036i
58.16		−	1.77500i	−1.09543	−	1.34165i	−1.15063			−0.688545	−	2.12742i	−2.38144	+	1.94439i	−0.808718	+	1.40074i		−	1.50763i	−0.600064	+	2.93937i	−3.77617	+	1.22217i
58.17		−	1.76612i	−1.73076	−	0.0669573i	−1.11918			2.23595	−	0.0225610i	−0.118255	+	3.05672i	−1.78019	+	3.08337i		−	1.55563i	2.99103	+	0.231774i	−0.0398455	−	3.94896i
58.18		−	1.75231i	−1.56638	+	0.739226i	−1.07060			−2.16402	+	0.563061i	1.29536	+	2.74479i	0.692379	−	1.19924i		−	1.62860i	1.90709	−	2.31582i	0.986658	+	3.79203i
58.19		−	1.74919i	1.23086	+	1.21860i	−1.05968			−1.74457	−	1.39874i	2.13157	−	2.15301i	0.948991	−	1.64370i		−	1.64481i	0.0300184	+	2.99985i	−2.44667	+	3.05159i
58.20		−	1.61264i	0.866170	+	1.49992i	−0.600607			1.84546	+	1.26264i	2.41883	−	1.39682i	2.40872	−	4.17203i		−	2.25672i	−1.49950	+	2.59836i	2.03619	−	2.97607i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
585.ca	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	585.2.ca.a	✓	320
5.c	odd	4	1		585.2.dv.a	yes	320
9.c	even	3	1		585.2.dt.a	yes	320
13.f	odd	12	1		585.2.cc.a	yes	320
45.k	odd	12	1		585.2.cc.a	yes	320
65.t	even	12	1		585.2.dt.a	yes	320
117.w	odd	12	1		585.2.dv.a	yes	320
585.ca	even	12	1	inner	585.2.ca.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.ca.a	✓	320	1.a	even	1	1	trivial
585.2.ca.a	✓	320	585.ca	even	12	1	inner
585.2.cc.a	yes	320	13.f	odd	12	1
585.2.cc.a	yes	320	45.k	odd	12	1
585.2.dt.a	yes	320	9.c	even	3	1
585.2.dt.a	yes	320	65.t	even	12	1
585.2.dv.a	yes	320	5.c	odd	4	1
585.2.dv.a	yes	320	117.w	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).