Newform orbit 819.2.dt.a

• Label: 819.2.dt.a
• Level: $819$
• Weight: $2$
• Character orbit: 819.dt
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.dt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(84\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 84 q^{4} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
337.1	−2.41651	+	1.39517i	−0.424462	+	1.67924i	2.89301	−	5.01083i	−2.68905	−	1.55252i	−1.31711	−	4.65008i	−0.866025	+	0.500000i			10.5643i	−2.63966	−	1.42554i	8.66414
337.2	−2.37650	+	1.37207i	−1.56357	−	0.745148i	2.76516	−	4.78940i	3.62031	+	2.09019i	4.73822	−	0.374488i	−0.866025	+	0.500000i			9.68772i	1.88951	+	2.33018i	−11.4716
337.3	−2.33754	+	1.34958i	1.54698	+	0.778999i	2.64273	−	4.57734i	0.804286	+	0.464355i	−4.66746	+	0.266836i	−0.866025	+	0.500000i			8.86798i	1.78632	+	2.41020i	−2.50673
337.4	−2.30480	+	1.33068i	0.148040	−	1.72571i	2.54140	−	4.40184i	−0.291608	−	0.168360i	1.95516	+	4.17442i	0.866025	−	0.500000i			8.20443i	−2.95617	−	0.510948i	0.896131
337.5	−2.24889	+	1.29840i	−1.32426	+	1.11640i	2.37166	−	4.10784i	0.816362	+	0.471327i	1.52859	−	4.23006i	0.866025	−	0.500000i			7.12384i	0.507324	−	2.95679i	−2.44787
337.6	−2.21181	+	1.27699i	1.71517	+	0.241244i	2.26141	−	3.91688i	0.995531	+	0.574770i	−4.10170	+	1.65667i	0.866025	−	0.500000i			6.44325i	2.88360	+	0.827549i	−2.93590
337.7	−2.16695	+	1.25109i	−0.959189	−	1.44221i	2.13045	−	3.69005i	−2.59086	−	1.49584i	3.88285	+	1.92516i	−0.866025	+	0.500000i			5.65720i	−1.15991	+	2.76670i	7.48570
337.8	−2.06384	+	1.19156i	1.49926	−	0.867312i	1.83963	−	3.18634i	−3.32879	−	1.92188i	−2.06078	+	3.57645i	0.866025	−	0.500000i			4.00190i	1.49554	−	2.60065i	9.16013
337.9	−2.03052	+	1.17232i	1.22264	−	1.22685i	1.74867	−	3.02878i	1.75889	+	1.01550i	−1.04433	+	3.92446i	−0.866025	+	0.500000i			3.51071i	−0.0103136	−	2.99998i	−4.76194
337.10	−1.95078	+	1.12628i	0.472299	+	1.66641i	1.53703	−	2.66221i	−2.16475	−	1.24982i	−2.79820	−	2.71886i	0.866025	−	0.500000i			2.41938i	−2.55387	+	1.57409i	5.63061
337.11	−1.94920	+	1.12537i	0.0645279	−	1.73085i	1.53293	−	2.65511i	1.74401	+	1.00691i	1.82207	+	3.44639i	−0.866025	+	0.500000i			2.39896i	−2.99167	−	0.223376i	−4.53257
337.12	−1.84819	+	1.06705i	−1.53241	−	0.807287i	1.27721	−	2.21219i	−2.40783	−	1.39016i	3.69361	−	0.143147i	0.866025	−	0.500000i			1.18319i	1.69658	+	2.47419i	5.93352
337.13	−1.80299	+	1.04096i	−0.863909	+	1.50122i	1.16718	−	2.02161i	2.68582	+	1.55066i	−0.00508612	−	3.60597i	−0.866025	+	0.500000i			0.696099i	−1.50732	−	2.59383i	−6.45668
337.14	−1.74716	+	1.00872i	1.15382	+	1.29178i	1.03505	−	1.79275i	−1.27957	−	0.738758i	−3.31896	−	1.09306i	−0.866025	+	0.500000i			0.141404i	−0.337385	+	2.98097i	2.98081
337.15	−1.73298	+	1.00053i	1.57453	−	0.721697i	1.00214	−	1.73575i	−0.636607	−	0.367545i	−2.00654	+	2.82606i	−0.866025	+	0.500000i			0.00855189i	1.95831	−	2.27267i	1.47097
337.16	−1.71093	+	0.987807i	−0.608882	−	1.62150i	0.951527	−	1.64809i	2.28326	+	1.31824i	2.64349	+	2.17282i	0.866025	−	0.500000i		−	0.191527i	−2.25853	+	1.97460i	−5.20867
337.17	−1.69479	+	0.978485i	−0.566986	+	1.63662i	0.914865	−	1.58459i	0.936100	+	0.540458i	−0.640489	−	3.32851i	−0.866025	+	0.500000i		−	0.333212i	−2.35705	−	1.85588i	−2.11532
337.18	−1.63987	+	0.946779i	−1.67703	−	0.433111i	0.792780	−	1.37313i	1.42878	+	0.824906i	3.16016	−	0.877527i	0.866025	−	0.500000i		−	0.784766i	2.62483	+	1.45268i	−3.12401
337.19	−1.56120	+	0.901358i	−1.24185	+	1.20740i	0.624892	−	1.08235i	−2.25541	−	1.30216i	0.850471	−	3.00434i	0.866025	−	0.500000i		−	1.35243i	0.0843734	−	2.99881i	4.69485
337.20	−1.34614	+	0.777193i	−1.61190	+	0.633864i	0.208059	−	0.360369i	3.42864	+	1.97952i	1.67720	−	2.10603i	0.866025	−	0.500000i		−	2.46197i	2.19643	−	2.04345i	−6.15389

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
13.b	even	2	1	inner
117.t	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	819.2.dt.a	✓	168
9.c	even	3	1	inner	819.2.dt.a	✓	168
13.b	even	2	1	inner	819.2.dt.a	✓	168
117.t	even	6	1	inner	819.2.dt.a	✓	168

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
819.2.dt.a	✓	168	1.a	even	1	1	trivial
819.2.dt.a	✓	168	9.c	even	3	1	inner
819.2.dt.a	✓	168	13.b	even	2	1	inner
819.2.dt.a	✓	168	117.t	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).