Newform orbit 585.2.dc.a

• Label: 585.2.dc.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.dc
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $8$

Newspace parameters

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 6 q^{5} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
164.1	−2.65715	+	0.711982i	1.15525	−	1.29050i	4.82149	−	2.78369i	2.09210	−	0.789376i	−2.15088	+	4.25157i	−1.94401	+	0.520895i	−6.93917	+	6.93917i	−0.330775	−	2.98171i	−4.99701	+	3.58703i
164.2	−2.59389	+	0.695032i	0.0985268	+	1.72925i	4.51317	−	2.60568i	2.21166	+	0.329494i	−1.45745	−	4.41700i	4.44855	−	1.19199i	−6.09793	+	6.09793i	−2.98058	+	0.340754i	−5.96582	+	0.682500i
164.3	−2.59100	+	0.694255i	0.0569524	−	1.73111i	4.49922	−	2.59762i	−1.56218	+	1.59987i	1.05427	+	4.52485i	1.12544	−	0.301561i	−6.06056	+	6.06056i	−2.99351	−	0.197182i	2.93690	−	5.22981i
164.4	−2.48492	+	0.665831i	1.61898	+	0.615552i	3.99943	−	2.30907i	−1.66241	+	1.49546i	−4.43288	−	0.451627i	−0.432814	+	0.115972i	−4.76263	+	4.76263i	2.24219	+	1.99313i	3.13522	−	4.82298i
164.5	−2.48424	+	0.665651i	−1.55851	+	0.755686i	3.99632	−	2.30727i	−1.88020	−	1.21030i	3.36868	−	2.91473i	−0.669803	+	0.179473i	−4.75480	+	4.75480i	1.85788	−	2.35548i	5.47652	+	1.75512i
164.6	−2.47472	+	0.663100i	−1.27504	−	1.17230i	3.95250	−	2.28198i	0.191249	−	2.22787i	3.93271	+	2.05563i	1.25633	−	0.336634i	−4.64492	+	4.64492i	0.251440	+	2.98944i	1.00402	+	5.64019i
164.7	−2.30772	+	0.618351i	0.461745	+	1.66937i	3.21115	−	1.85396i	0.372905	−	2.20475i	−2.09783	−	3.56691i	−4.52558	+	1.21263i	−2.88529	+	2.88529i	−2.57358	+	1.54164i	0.502753	+	5.31853i
164.8	−2.28388	+	0.611965i	−0.287814	+	1.70797i	3.10957	−	1.79531i	−2.17778	+	0.507230i	−0.387885	−	4.07694i	0.678622	−	0.181836i	−2.65940	+	2.65940i	−2.83433	−	0.983154i	4.66338	−	2.49118i
164.9	−2.27743	+	0.610235i	−1.72455	−	0.160992i	3.08224	−	1.77953i	0.385486	+	2.20259i	4.02579	−	0.685736i	3.94614	−	1.05736i	−2.59927	+	2.59927i	2.94816	+	0.555277i	−2.22201	−	4.78100i
164.10	−2.16015	+	0.578809i	1.58432	+	0.699949i	2.59916	−	1.50062i	1.77996	+	1.35342i	−3.82750	−	0.594973i	−1.16452	+	0.312032i	−1.58331	+	1.58331i	2.02014	+	2.21789i	−4.62834	−	1.89334i
164.11	−2.13111	+	0.571030i	1.35367	−	1.08054i	2.48352	−	1.43386i	−1.73129	−	1.41515i	−2.26781	+	3.07575i	4.08026	−	1.09330i	−1.35372	+	1.35372i	0.664858	−	2.92540i	4.49766	+	2.02722i
164.12	−2.07313	+	0.555494i	1.73181	−	0.0286733i	2.25725	−	1.30323i	1.36037	−	1.77465i	−3.57435	+	1.02146i	1.35683	−	0.363563i	−0.920373	+	0.920373i	2.99836	−	0.0993136i	−1.83443	+	4.43476i
164.13	−1.98498	+	0.531874i	−1.32883	−	1.11095i	1.92521	−	1.11152i	−1.17646	+	1.90156i	3.22859	+	1.49845i	−4.43313	+	1.18785i	−0.324104	+	0.324104i	0.531565	+	2.95253i	1.32387	−	4.40029i
164.14	−1.88049	+	0.503875i	−1.27426	+	1.17314i	1.55029	−	0.895062i	2.23491	+	0.0719709i	1.80512	−	2.84814i	−1.72694	+	0.462732i	0.288918	−	0.288918i	0.247484	−	2.98977i	−4.23898	+	0.990775i
164.15	−1.86715	+	0.500300i	−0.305679	−	1.70486i	1.50389	−	0.868268i	−0.913957	−	2.04076i	1.42369	+	3.03030i	−2.65203	+	0.710610i	0.360114	−	0.360114i	−2.81312	+	1.04228i	2.72748	+	3.35314i
164.16	−1.84968	+	0.495619i	1.05401	−	1.37443i	1.44361	−	0.833470i	0.719605	+	2.11711i	−1.26838	+	3.06464i	−2.83413	+	0.759403i	0.450982	−	0.450982i	−0.778129	−	2.89733i	−2.38032	−	3.55932i
164.17	−1.79970	+	0.482228i	1.03443	+	1.38923i	1.27432	−	0.735731i	−1.26424	−	1.84437i	−2.53159	−	2.00136i	1.48564	−	0.398076i	0.696333	−	0.696333i	−0.859899	+	2.87412i	3.16466	+	2.70966i
164.18	−1.65763	+	0.444161i	−1.73065	−	0.0696861i	0.818415	−	0.472512i	−2.23607	+	0.00194582i	2.89973	−	0.653173i	−1.68932	+	0.452653i	1.28018	−	1.28018i	2.99029	+	0.241204i	3.70571	−	0.996400i
164.19	−1.64428	+	0.440584i	−0.676410	+	1.59451i	0.777498	−	0.448889i	0.163465	+	2.23009i	0.409693	−	2.91984i	−1.08555	+	0.290873i	1.32674	−	1.32674i	−2.08494	−	2.15709i	−1.25132	−	3.59487i
164.20	−1.60998	+	0.431393i	−1.72789	+	0.120044i	0.673887	−	0.389069i	1.75763	−	1.38230i	2.73008	−	0.938666i	1.42948	−	0.383028i	1.44007	−	1.44007i	2.97118	−	0.414843i	−2.23343	+	2.98370i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.d	odd	6	1	inner
13.d	odd	4	1	inner
45.h	odd	6	1	inner
65.g	odd	4	1	inner
117.z	even	12	1	inner
585.dc	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.dc.a	✓	320
5.b	even	2	1	inner	585.2.dc.a	✓	320
9.d	odd	6	1	inner	585.2.dc.a	✓	320
13.d	odd	4	1	inner	585.2.dc.a	✓	320
45.h	odd	6	1	inner	585.2.dc.a	✓	320
65.g	odd	4	1	inner	585.2.dc.a	✓	320
117.z	even	12	1	inner	585.2.dc.a	✓	320
585.dc	even	12	1	inner	585.2.dc.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.dc.a	✓	320	1.a	even	1	1	trivial
585.2.dc.a	✓	320	5.b	even	2	1	inner
585.2.dc.a	✓	320	9.d	odd	6	1	inner
585.2.dc.a	✓	320	13.d	odd	4	1	inner
585.2.dc.a	✓	320	45.h	odd	6	1	inner
585.2.dc.a	✓	320	65.g	odd	4	1	inner
585.2.dc.a	✓	320	117.z	even	12	1	inner
585.2.dc.a	✓	320	585.dc	even	12	1	inner

Hecke kernels

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