Newform orbit 504.2.w.a

• Label: 504.2.w.a
• Level: $504$
• Weight: $2$
• Character orbit: 504.w
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.w (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 184 q + q^{2} + q^{4} - 2 q^{6} - 2 q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
205.1	−1.41403	−	0.0227401i	1.48240	+	0.895821i	1.99897	+	0.0643103i			3.06618i	−2.07579	−	1.30043i	−1.96423	−	1.77251i	−2.82514	−	0.136393i	1.39501	+	2.65593i	0.0697253	−	4.33568i
205.2	−1.41269	−	0.0655984i	−0.165594	+	1.72412i	1.99139	+	0.185341i			1.74289i	0.347032	−	2.42478i	1.38436	+	2.25467i	−2.80107	−	0.392461i	−2.94516	−	0.571007i	0.114331	−	2.46217i
205.3	−1.40591	−	0.153043i	1.60884	−	0.641583i	1.95316	+	0.430329i			0.532179i	−2.36007	+	0.655785i	0.0251237	+	2.64563i	−2.68010	−	0.903920i	2.17674	−	2.06441i	0.0814462	−	0.748194i
205.4	−1.40394	+	0.170165i	0.436895	−	1.67604i	1.94209	−	0.477802i			2.31532i	−0.328169	+	2.42741i	1.54699	−	2.14635i	−2.64527	+	1.00128i	−2.61825	−	1.46451i	−0.393985	−	3.25056i
205.5	−1.40100	−	0.192870i	−1.38115	+	1.04519i	1.92560	+	0.540421i		−	2.37580i	2.13658	−	1.19793i	0.371350	−	2.61956i	−2.59354	−	1.12852i	0.815145	−	2.88713i	−0.458220	+	3.32850i
205.6	−1.40065	+	0.195431i	−1.32655	−	1.11367i	1.92361	−	0.547458i		−	1.05207i	2.07567	+	1.30061i	2.60895	−	0.439732i	−2.58731	+	1.14273i	0.519470	+	2.95468i	0.205606	+	1.47357i
205.7	−1.40063	−	0.195508i	1.23756	+	1.21180i	1.92355	+	0.547670i		−	3.52202i	−1.49645	−	1.93924i	2.62220	−	0.352244i	−2.58712	−	1.14316i	0.0630863	+	2.99934i	−0.688583	+	4.93306i
205.8	−1.37849	+	0.315851i	−0.628421	−	1.61403i	1.80048	−	0.870795i		−	3.22312i	1.37607	+	2.02644i	−2.64436	+	0.0858396i	−2.20690	+	1.76906i	−2.21017	+	2.02858i	1.01802	+	4.44304i
205.9	−1.37435	+	0.333425i	1.66932	−	0.461909i	1.77765	−	0.916484i		−	3.28102i	−2.14022	+	1.19142i	−2.10198	−	1.60676i	−2.13753	+	1.85228i	2.57328	−	1.54215i	1.09397	+	4.50925i
205.10	−1.31761	+	0.513706i	−1.35345	+	1.08082i	1.47221	−	1.35373i			0.334033i	1.22810	−	2.11938i	−2.60627	−	0.455379i	−1.24439	+	2.53998i	0.663641	−	2.92568i	−0.171595	−	0.440127i
205.11	−1.31691	−	0.515516i	−0.229486	−	1.71678i	1.46849	+	1.35777i			1.20405i	−0.582816	+	2.37914i	−2.62096	+	0.361324i	−1.23391	−	2.54509i	−2.89467	+	0.787953i	0.620709	−	1.58563i
205.12	−1.31047	+	0.531673i	−1.72905	+	0.101939i	1.43465	−	1.39348i			0.970877i	2.21166	−	1.05288i	1.53976	+	2.15154i	−1.13918	+	2.58887i	2.97922	−	0.352514i	−0.516189	−	1.27230i
205.13	−1.28717	−	0.585837i	−1.65782	+	0.501625i	1.31359	+	1.50814i			4.18515i	2.42776	+	0.325538i	1.88927	−	1.85221i	−0.807285	−	2.71077i	2.49674	−	1.66321i	2.45182	−	5.38698i
205.14	−1.28512	−	0.590314i	−1.73078	−	0.0662226i	1.30306	+	1.51725i		−	2.85273i	2.18517	+	1.10681i	−0.733697	+	2.54199i	−0.778934	−	2.71906i	2.99123	+	0.229234i	−1.68401	+	3.66610i
205.15	−1.22663	−	0.703825i	−0.0860069	+	1.72991i	1.00926	+	1.72667i			0.633242i	1.32306	−	2.06144i	−2.18459	−	1.49250i	−0.0227161	−	2.82834i	−2.98521	−	0.297569i	0.445692	−	0.776756i
205.16	−1.19869	−	0.750436i	−0.854748	−	1.50645i	0.873692	+	1.79907i			2.34627i	−0.105924	+	2.44720i	1.58767	+	2.11644i	0.302808	−	2.81217i	−1.53881	+	2.57528i	1.76073	−	2.81244i
205.17	−1.18039	+	0.778903i	0.433261	+	1.67699i	0.786619	−	1.83881i		−	0.823850i	−1.81763	−	1.64202i	0.822054	−	2.51480i	0.503744	+	2.78321i	−2.62457	+	1.45315i	0.641700	+	0.972460i
205.18	−1.16839	+	0.796778i	1.04913	−	1.37816i	0.730290	−	1.86190i			2.16527i	−0.127710	+	2.44616i	−1.74348	+	1.99004i	0.630255	+	2.75731i	−0.798651	−	2.89174i	−1.72524	−	2.52989i
205.19	−1.12540	−	0.856430i	1.26962	+	1.17816i	0.533054	+	1.92765i		−	2.29636i	−0.419826	−	2.41324i	−2.07761	+	1.63815i	1.05100	−	2.62591i	0.223889	+	2.99163i	−1.96667	+	2.58432i
205.20	−1.12401	−	0.858252i	1.72775	−	0.121975i	0.526807	+	1.92937i			0.697625i	−2.04670	−	1.34574i	2.42006	−	1.06926i	1.06375	−	2.62077i	2.97024	−	0.421485i	0.598738	−	0.784139i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
63.g	even	3	1	inner
504.w	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.2.w.a	✓	184
7.c	even	3	1		504.2.cq.a	yes	184
8.b	even	2	1	inner	504.2.w.a	✓	184
9.c	even	3	1		504.2.cq.a	yes	184
56.p	even	6	1		504.2.cq.a	yes	184
63.g	even	3	1	inner	504.2.w.a	✓	184
72.n	even	6	1		504.2.cq.a	yes	184
504.w	even	6	1	inner	504.2.w.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.w.a	✓	184	1.a	even	1	1	trivial
504.2.w.a	✓	184	8.b	even	2	1	inner
504.2.w.a	✓	184	63.g	even	3	1	inner
504.2.w.a	✓	184	504.w	even	6	1	inner
504.2.cq.a	yes	184	7.c	even	3	1
504.2.cq.a	yes	184	9.c	even	3	1
504.2.cq.a	yes	184	56.p	even	6	1
504.2.cq.a	yes	184	72.n	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).