Newform orbit 585.2.cg.a

• Label: 585.2.cg.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.cg
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.cg (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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 4 q^{2} - 8 q^{3} - 152 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
187.1	−1.40853	+	2.43965i	0.220459	+	1.71796i	−2.96792	−	5.14059i	1.83454	+	1.27846i	−4.50175	−	1.88196i	1.94956	+	1.12558i	11.0875			−2.90280	+	0.757482i	−5.70300	+	2.67488i
187.2	−1.34555	+	2.33056i	1.48852	+	0.885604i	−2.62099	−	4.53969i	−2.22639	+	0.207818i	−4.06683	+	2.27747i	−1.73355	−	1.00086i	8.72449			1.43141	+	2.63649i	2.51138	−	5.46835i
187.3	−1.33671	+	2.31526i	−1.56099	+	0.750545i	−2.57361	−	4.45762i	−0.671518	−	2.13285i	0.348891	−	4.61735i	−2.81530	−	1.62541i	8.41387			1.87337	−	2.34318i	5.83573	+	1.29628i
187.4	−1.28035	+	2.21763i	−1.36042	−	1.07204i	−2.27857	−	3.94661i	−2.08412	+	0.810200i	4.11918	−	1.64431i	1.87822	+	1.08439i	6.54808			0.701467	+	2.91684i	0.871682	−	5.65914i
187.5	−1.24938	+	2.16398i	−1.54228	−	0.788262i	−2.12188	−	3.67521i	1.41315	+	1.73292i	3.63268	−	2.35264i	−3.61058	−	2.08457i	5.60662			1.75729	+	2.43145i	−5.51557	+	0.892963i
187.6	−1.24193	+	2.15108i	0.632622	−	1.61239i	−2.08476	−	3.61091i	−1.73545	−	1.41004i	2.68270	+	3.36328i	0.772595	+	0.446058i	5.38875			−2.19958	−	2.04006i	5.18841	−	1.98193i
187.7	−1.23057	+	2.13140i	−0.834401	−	1.51782i	−2.02858	−	3.51361i	1.42252	−	1.72524i	4.26187	+	0.0893324i	1.30481	+	0.753333i	5.06296			−1.60755	+	2.53294i	1.92667	+	5.15498i
187.8	−1.19124	+	2.06329i	1.65319	−	0.516695i	−1.83811	−	3.18370i	0.767180	+	2.10034i	−0.903251	+	4.02651i	2.85067	+	1.64583i	3.99355			2.46605	−	1.70839i	−5.24751	−	0.919098i
187.9	−1.18788	+	2.05748i	1.12780	−	1.31456i	−1.82214	−	3.15604i	−1.07548	+	1.96044i	1.36498	+	3.88197i	−2.54867	−	1.47148i	3.90642			−0.456137	−	2.96512i	−2.75601	−	4.54156i
187.10	−1.14526	+	1.98365i	1.67020	−	0.458738i	−1.62326	−	2.81156i	1.51540	−	1.64425i	−1.00284	+	3.83847i	1.97486	+	1.14018i	2.85517			2.57912	−	1.53237i	1.52609	+	4.88913i
187.11	−1.13215	+	1.96095i	1.35748	+	1.07575i	−1.56354	−	2.70814i	0.774839	−	2.09753i	−3.64637	+	1.44404i	−0.926644	−	0.534998i	2.55207			0.685519	+	2.92063i	3.23591	+	3.89414i
187.12	−1.08216	+	1.87436i	−0.780957	+	1.54600i	−1.34214	−	2.32465i	−1.56410	−	1.59799i	−2.05263	−	3.13681i	3.72350	+	2.14976i	1.48099			−1.78021	−	2.41471i	4.68782	−	1.20240i
187.13	−1.06895	+	1.85148i	−0.635947	+	1.61108i	−1.28531	−	2.22623i	2.06095	−	0.867461i	−2.30308	−	2.89960i	−1.58340	−	0.914175i	1.21994			−2.19114	−	2.04912i	−0.596969	+	4.74307i
187.14	−1.06008	+	1.83611i	−1.57465	+	0.721433i	−1.24753	−	2.16078i	−0.0981709	+	2.23391i	0.344627	−	3.65601i	1.88589	+	1.08882i	1.04960			1.95907	−	2.27201i	−3.99763	−	2.54837i
187.15	−1.04339	+	1.80721i	−0.112469	+	1.72840i	−1.17733	−	2.03920i	−1.02394	+	1.98785i	−3.00622	−	2.00665i	−1.64282	−	0.948481i	0.740109			−2.97470	−	0.388781i	−2.52409	−	3.92457i
187.16	−0.864716	+	1.49773i	1.45246	+	0.943596i	−0.495466	−	0.858172i	−2.11363	+	0.729768i	−2.66921	+	1.35945i	3.20124	+	1.84824i	−1.74511			1.21925	+	2.74106i	0.734693	−	3.79669i
187.17	−0.863065	+	1.49487i	0.109682	−	1.72857i	−0.489761	−	0.848291i	2.03112	+	0.935172i	2.48933	+	1.65583i	−1.49343	−	0.862234i	−1.76148			−2.97594	−	0.379187i	−3.15095	+	2.22915i
187.18	−0.847006	+	1.46706i	−1.64683	+	0.536618i	−0.434837	−	0.753160i	2.14353	+	0.636611i	0.607623	−	2.87051i	0.580662	+	0.335245i	−1.91478			2.42408	−	1.76743i	−2.74953	+	2.60547i
187.19	−0.829047	+	1.43595i	1.57564	−	0.719282i	−0.374638	−	0.648893i	−1.40935	−	1.73601i	−0.273424	+	2.85886i	−2.06808	−	1.19401i	−2.07382			1.96527	−	2.26665i	3.66125	−	0.584522i
187.20	−0.811328	+	1.40526i	−1.66380	−	0.481412i	−0.316506	−	0.548205i	−1.93631	−	1.11834i	2.02640	−	1.94750i	1.33109	+	0.768508i	−2.21815			2.53649	+	1.60195i	3.14255	−	1.81368i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
65.k	even	4	1	inner
585.cg	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.cg.a	✓	320
5.c	odd	4	1		585.2.dq.a	yes	320
9.c	even	3	1	inner	585.2.cg.a	✓	320
13.d	odd	4	1		585.2.dq.a	yes	320
45.k	odd	12	1		585.2.dq.a	yes	320
65.k	even	4	1	inner	585.2.cg.a	✓	320
117.y	odd	12	1		585.2.dq.a	yes	320
585.cg	even	12	1	inner	585.2.cg.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.cg.a	✓	320	1.a	even	1	1	trivial
585.2.cg.a	✓	320	9.c	even	3	1	inner
585.2.cg.a	✓	320	65.k	even	4	1	inner
585.2.cg.a	✓	320	585.cg	even	12	1	inner
585.2.dq.a	yes	320	5.c	odd	4	1
585.2.dq.a	yes	320	13.d	odd	4	1
585.2.dq.a	yes	320	45.k	odd	12	1
585.2.dq.a	yes	320	117.y	odd	12	1

Hecke kernels

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