Newform orbit 592.2.cd.a

• Label: 592.2.cd.a
• Level: $592$
• Weight: $2$
• Character orbit: 592.cd
• Analytic conductor: $4.727$
• Analytic rank: $0$
• Dimension: $888$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.cd (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.72714379966\)
Analytic rank:	\(0\)
Dimension:	\(888\)
Relative dimension:	\(74\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 888 q - 18 q^{2} - 12 q^{3} - 6 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41400	−	0.0243566i	−0.549472	+	1.17835i	1.99881	+	0.0688807i	0.996072	+	0.175634i	0.805657	−	1.65280i	−0.212008	+	1.20236i	−2.82465	−	0.146082i	0.841780	+	1.00319i	−1.40417	−	0.272609i
59.2	−1.41399	+	0.0251076i	0.838078	−	1.79726i	1.99874	−	0.0710039i	3.36748	+	0.593778i	−1.13991	+	2.56236i	0.0998909	−	0.566509i	−2.82442	+	0.150583i	−0.599418	−	0.714359i	−4.77650	−	0.755047i
59.3	−1.41325	+	0.0521943i	0.251641	−	0.539646i	1.99455	−	0.147527i	−1.05989	−	0.186887i	−0.327465	+	0.775789i	0.880221	−	4.99198i	−2.81110	+	0.312597i	1.70047	+	2.02654i	1.50764	+	0.208797i
59.4	−1.39240	−	0.247459i	0.674429	−	1.44632i	1.87753	+	0.689122i	−0.437583	−	0.0771577i	−1.29698	+	1.84695i	−0.183695	+	1.04179i	−2.44373	−	1.42414i	0.291385	+	0.347259i	0.590195	+	0.215718i
59.5	−1.36455	+	0.371496i	1.29956	−	2.78691i	1.72398	−	1.01385i	−0.633268	−	0.111662i	−0.737981	+	4.28565i	−0.0242083	+	0.137292i	−1.97581	+	2.02390i	−4.14964	−	4.94535i	0.905607	−	0.0828882i
59.6	−1.36183	+	0.381338i	−0.801159	+	1.71809i	1.70916	−	1.03863i	3.60992	+	0.636527i	0.435870	−	2.64526i	0.330638	−	1.87514i	−1.93152	+	2.06621i	−0.381619	−	0.454796i	−5.15883	+	0.509758i
59.7	−1.35855	+	0.392854i	0.736418	−	1.57925i	1.69133	−	1.06743i	−4.35992	−	0.768771i	−0.380047	+	2.43480i	−0.742838	+	4.21285i	−1.87842	+	2.11460i	−0.0233676	−	0.0278484i	6.22519	−	0.668395i
59.8	−1.33521	−	0.466069i	−1.09670	+	2.35188i	1.56556	+	1.24460i	−3.51490	−	0.619772i	2.56046	−	2.62911i	0.128700	−	0.729896i	−1.51028	−	2.39145i	−2.40023	−	2.86048i	4.40427	+	2.46571i
59.9	−1.32722	+	0.488359i	−1.32075	+	2.83235i	1.52301	−	1.29632i	0.141936	+	0.0250272i	0.369715	−	4.40415i	−0.482455	+	2.73614i	−1.38830	+	2.46427i	−4.34948	−	5.18351i	−0.200603	+	0.0360993i
59.10	−1.32684	−	0.489371i	−0.924017	+	1.98156i	1.52103	+	1.29864i	0.814764	+	0.143665i	2.19575	−	2.17704i	0.655667	−	3.71847i	−1.38266	−	2.46744i	−1.14441	−	1.36386i	−1.01076	−	0.589343i
59.11	−1.31273	+	0.526050i	−0.131337	+	0.281653i	1.44654	−	1.38113i	−2.27097	−	0.400434i	0.0242470	−	0.438825i	0.325998	−	1.84883i	−1.17239	+	2.57401i	1.86628	+	2.22415i	3.19184	−	0.668982i
59.12	−1.26367	−	0.634934i	−0.116467	+	0.249765i	1.19372	+	1.60469i	3.46200	+	0.610444i	0.305760	−	0.241671i	−0.338062	+	1.91724i	−0.489592	−	2.78573i	1.87954	+	2.23995i	−3.98723	−	2.96954i
59.13	−1.18866	+	0.766217i	0.202916	−	0.435155i	0.825823	−	1.82154i	2.36349	+	0.416747i	0.0922249	+	0.672728i	−0.724850	+	4.11083i	0.414075	+	2.79795i	1.78018	+	2.12153i	−3.12871	+	1.31558i
59.14	−1.16242	−	0.805464i	0.792880	−	1.70034i	0.702456	+	1.87258i	−0.737626	−	0.130063i	−2.29122	+	1.33787i	−0.00398011	+	0.0225723i	0.691745	−	2.74253i	−0.334124	−	0.398194i	0.752672	+	0.745320i
59.15	−1.13850	+	0.838935i	1.20643	−	2.58719i	0.592376	−	1.91026i	1.69225	+	0.298389i	0.796967	+	3.95764i	0.0620693	−	0.352013i	0.928162	+	2.67180i	−3.30974	−	3.94440i	−2.17696	+	1.07997i
59.16	−1.12365	−	0.858727i	0.280797	−	0.602170i	0.525175	+	1.92982i	−3.85529	−	0.679792i	−0.832617	+	0.435501i	0.293824	−	1.66636i	1.06707	−	2.61942i	1.64460	+	1.95996i	3.74824	+	4.07449i
59.17	−1.03989	+	0.958450i	−0.104246	+	0.223556i	0.162747	−	1.99337i	−0.783481	−	0.138149i	−0.105863	−	0.332388i	0.00439145	−	0.0249051i	1.74130	+	2.22887i	1.88925	+	2.25152i	0.947144	−	0.607268i
59.18	−1.01084	−	0.989039i	1.31506	−	2.82015i	0.0436037	+	1.99952i	2.94925	+	0.520033i	−4.11855	+	1.55008i	0.867821	−	4.92166i	1.93353	−	2.06433i	−4.29551	−	5.11919i	−2.46690	−	3.44260i
59.19	−0.983499	−	1.01623i	−1.20153	+	2.57669i	−0.0654585	+	1.99893i	−0.550404	−	0.0970511i	3.80022	−	1.31314i	−0.461321	+	2.61628i	2.09576	−	1.89942i	−3.26730	−	3.89381i	0.442695	+	0.654788i
59.20	−0.954813	+	1.04323i	−0.493287	+	1.05786i	−0.176665	−	1.99218i	−1.89308	−	0.333802i	−0.632594	−	1.52467i	−0.144383	+	0.818838i	2.24699	+	1.71786i	1.05263	+	1.25448i	2.15577	−	1.65621i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
592.cd	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	592.2.cd.a	yes	888
16.f	odd	4	1		592.2.ca.a	✓	888
37.i	odd	36	1		592.2.ca.a	✓	888
592.cd	even	36	1	inner	592.2.cd.a	yes	888

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
592.2.ca.a	✓	888	16.f	odd	4	1
592.2.ca.a	✓	888	37.i	odd	36	1
592.2.cd.a	yes	888	1.a	even	1	1	trivial
592.2.cd.a	yes	888	592.cd	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).