Newform orbit 585.2.cs.a

• Label: 585.2.cs.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.cs
• 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.cs (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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 8 q^{3}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
38.1	−0.709028	−	2.64613i	1.38003	+	1.04667i	−4.76722	+	2.75236i	2.17472	−	0.520187i	1.79114	−	4.39385i	0.965003	−	0.258572i	6.78898	+	6.78898i	0.808963	+	2.88887i	−2.91842	−	5.38576i
38.2	−0.704212	−	2.62815i	−1.70492	−	0.305385i	−4.67923	+	2.70155i	−1.82265	−	1.29536i	0.398024	+	4.69584i	3.98974	−	1.06905i	6.54738	+	6.54738i	2.81348	+	1.04131i	−2.12088	+	5.70240i
38.3	−0.688133	−	2.56815i	−1.66314	+	0.483716i	−4.38979	+	2.53445i	1.02976	+	1.98484i	2.38671	+	3.93831i	−3.87320	+	1.03782i	5.76957	+	5.76957i	2.53204	−	1.60897i	4.38876	−	4.01040i
38.4	−0.664833	−	2.48119i	−0.163652	−	1.72430i	−3.98226	+	2.29916i	−1.85653	+	1.24631i	−4.16952	+	1.55243i	−0.996020	+	0.266883i	4.71947	+	4.71947i	−2.94644	+	0.564372i	4.32662	+	3.77782i
38.5	−0.650930	−	2.42930i	0.953959	−	1.44567i	−3.74576	+	2.16261i	1.55067	+	1.61103i	−4.13293	−	1.37643i	2.31407	−	0.620054i	4.13513	+	4.13513i	−1.17992	−	2.75822i	2.90431	−	4.81572i
38.6	−0.650340	−	2.42710i	−0.734088	+	1.56879i	−3.73583	+	2.15688i	0.0163040	−	2.23601i	4.28503	+	0.761457i	−0.186040	+	0.0498492i	4.11101	+	4.11101i	−1.92223	−	2.30327i	−5.43762	+	1.41459i
38.7	−0.644279	−	2.40448i	−0.960455	−	1.44136i	−3.63439	+	2.09832i	0.996091	−	2.00195i	−2.84693	+	3.23804i	−4.27757	+	1.14617i	3.86653	+	3.86653i	−1.15505	+	2.76873i	−5.45542	−	1.10527i
38.8	−0.625431	−	2.33414i	1.68528	−	0.399803i	−3.32500	+	1.91969i	−1.79269	−	1.33652i	−1.98722	−	3.68363i	1.39674	−	0.374256i	3.14297	+	3.14297i	2.68031	−	1.34756i	−1.99842	+	5.02028i
38.9	−0.618802	−	2.30940i	1.72509	−	0.155166i	−3.21836	+	1.85812i	−1.06539	+	1.96595i	−1.42583	−	3.88790i	−1.84407	+	0.494117i	2.90147	+	2.90147i	2.95185	−	0.535351i	5.19942	+	1.24388i
38.10	−0.579664	−	2.16333i	−0.182499	−	1.72241i	−2.61195	+	1.50801i	1.75722	−	1.38282i	−3.62036	+	1.39322i	3.33203	−	0.892815i	1.60905	+	1.60905i	−2.93339	+	0.628676i	−4.01009	−	2.99988i
38.11	−0.548947	−	2.04870i	0.857420	+	1.50494i	−2.16377	+	1.24926i	−0.402933	−	2.19946i	2.61248	−	2.58273i	−2.06970	+	0.554575i	0.747642	+	0.747642i	−1.52966	+	2.58072i	−4.28485	+	2.03288i
38.12	−0.525469	−	1.96108i	−0.349854	+	1.69635i	−1.83766	+	1.06097i	2.16160	+	0.572253i	3.51051	−	0.205289i	0.762735	−	0.204374i	0.175061	+	0.175061i	−2.75520	−	1.18695i	−0.0136227	−	4.53977i
38.13	−0.520814	−	1.94370i	1.01119	+	1.40623i	−1.77468	+	1.02461i	−2.19568	+	0.423047i	2.20666	−	2.69784i	3.97745	−	1.06575i	0.0700465	+	0.0700465i	−0.954987	+	2.84394i	1.96582	+	4.04743i
38.14	−0.519146	−	1.93748i	−1.30408	+	1.13990i	−1.75227	+	1.01167i	−2.22437	+	0.228463i	2.88554	+	1.93486i	−2.16199	+	0.579304i	0.0331069	+	0.0331069i	0.401265	−	2.97304i	1.59741	+	4.19106i
38.15	−0.517136	−	1.92998i	−1.34798	−	1.08763i	−1.72533	+	0.996119i	1.76074	+	1.37834i	−1.40202	+	3.16402i	0.0304589	−	0.00816144i	−0.0109672	−	0.0109672i	0.634100	+	2.93222i	1.74962	−	4.11096i
38.16	−0.515506	−	1.92390i	−1.60554	−	0.649798i	−1.70358	+	0.983561i	−0.578506	+	2.15994i	−0.422477	+	3.42387i	2.94359	−	0.788734i	−0.0463048	−	0.0463048i	2.15553	+	2.08655i	4.45372		0.000476924i
38.17	−0.498144	−	1.85910i	1.73183	+	0.0273740i	−1.47606	+	0.852201i	2.14340	−	0.637051i	−0.811813	−	3.23329i	−4.05526	+	1.08660i	−0.402296	−	0.402296i	2.99850	+	0.0948144i	−2.25206	−	3.66745i
38.18	−0.474122	−	1.76945i	0.723274	+	1.57381i	−1.17410	+	0.677866i	0.0271834	+	2.23590i	2.44185	−	2.02597i	−3.05427	+	0.818389i	−0.834537	−	0.834537i	−1.95375	+	2.27659i	3.94342	−	1.10819i
38.19	−0.442672	−	1.65207i	0.818635	−	1.52638i	−0.801338	+	0.462653i	−1.92064	−	1.14505i	−2.88408	−	0.676759i	−2.93765	+	0.787141i	−1.29974	−	1.29974i	−1.65967	−	2.49910i	−1.04150	+	3.67992i
38.20	−0.413818	−	1.54439i	1.13899	−	1.30488i	−0.481842	+	0.278191i	1.05003	−	1.97419i	−2.48657	−	1.21906i	0.357007	−	0.0956596i	−1.63211	−	1.63211i	−0.405418	−	2.97248i	−3.48344	−	0.804707i

Inner twists

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

    

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

Hecke kernels

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