Newform orbit 504.2.y.a

• Label: 504.2.y.a
• Level: $504$
• Weight: $2$
• Character orbit: 504.y
• 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.y (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.

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 3 q^{2} + q^{4} + 6 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
173.1	−1.41306	+	0.0570616i	0.0165996	+	1.73197i	1.99349	−	0.161263i			0.186906i	−0.122285	−	2.44644i	−0.0220103	+	2.64566i	−2.80772	+	0.341626i	−2.99945	+	0.0574999i	−0.0106652	−	0.264110i
173.2	−1.41306	−	0.0571030i	0.216493	−	1.71847i	1.99348	+	0.161380i			1.61593i	−0.404047	+	2.41594i	2.40829	−	1.09550i	−2.80769	−	0.341873i	−2.90626	−	0.744072i	0.0922743	−	2.28340i
173.3	−1.40941	−	0.116469i	1.69912	+	0.336136i	1.97287	+	0.328305i		−	3.14427i	−2.35561	−	0.671648i	0.623342	+	2.57127i	−2.74234	−	0.692495i	2.77403	+	1.14227i	−0.366210	+	4.43156i
173.4	−1.40767	−	0.135848i	−1.56804	+	0.735705i	1.96309	+	0.382459i			3.12398i	2.30723	−	0.822619i	2.22515	+	1.43133i	−2.71143	−	0.805059i	1.91747	−	2.30723i	0.424387	−	4.39755i
173.5	−1.40645	−	0.147961i	1.68230	−	0.412165i	1.95621	+	0.416201i			0.157459i	−2.42705	+	0.330775i	−2.21308	−	1.44992i	−2.68974	−	0.874811i	2.66024	−	1.38677i	0.0232979	−	0.221459i
173.6	−1.39680	−	0.221244i	−1.40670	+	1.01054i	1.90210	+	0.618068i		−	4.36497i	2.18846	−	1.10030i	1.72321	−	2.00762i	−2.52011	−	1.28415i	0.957621	−	2.84306i	−0.965725	+	6.09699i
173.7	−1.39532	+	0.230384i	−0.267989	+	1.71119i	1.89385	−	0.642919i		−	0.170174i	−0.0203002	−	2.44941i	−1.51202	−	2.17113i	−2.49441	+	1.33339i	−2.85636	−	0.917161i	0.0392054	+	0.237448i
173.8	−1.38113	+	0.304086i	1.31500	+	1.12729i	1.81506	−	0.839967i			3.74037i	−2.15898	−	1.15707i	0.706907	−	2.54957i	−2.25142	+	1.71204i	0.458435	+	2.96477i	−1.13739	−	5.16595i
173.9	−1.34205	+	0.445987i	−1.28072	−	1.16609i	1.60219	−	1.19707i		−	0.636978i	2.23885	+	0.993762i	2.14318	+	1.55138i	−1.61634	+	2.32109i	0.280479	+	2.98686i	0.284084	+	0.854856i
173.10	−1.33822	−	0.457359i	0.789770	−	1.54151i	1.58165	+	1.22409i			2.01312i	−1.76191	+	1.70167i	−1.48443	+	2.19008i	−1.55673	−	2.36148i	−1.75253	−	2.43488i	0.920719	−	2.69399i
173.11	−1.33533	−	0.465720i	−1.66375	−	0.481605i	1.56621	+	1.24378i		−	1.79202i	1.99736	+	1.41794i	−1.87653	+	1.86511i	−1.51215	−	2.39027i	2.53611	+	1.60254i	−0.834578	+	2.39293i
173.12	−1.31489	+	0.520628i	1.30232	−	1.14191i	1.45789	−	1.36914i		−	1.77147i	−1.11790	+	2.17952i	1.98439	−	1.74991i	−1.20416	+	2.55930i	0.392079	−	2.97427i	0.922278	+	2.32930i
173.13	−1.30575	+	0.543166i	−0.944647	−	1.45177i	1.40994	−	1.41847i			3.97106i	2.02202	+	1.38254i	−2.57827	−	0.593742i	−1.07056	+	2.61800i	−1.21528	+	2.74282i	−2.15695	−	5.18519i
173.14	−1.26838	+	0.625475i	−1.71321	+	0.254787i	1.21756	−	1.58668i		−	1.07899i	2.01363	−	1.39473i	−2.47543	+	0.933935i	−0.551904	+	2.77406i	2.87017	−	0.873007i	0.674878	+	1.36856i
173.15	−1.26678	−	0.628698i	1.22962	+	1.21985i	1.20948	+	1.59285i		−	1.38607i	−0.790746	−	2.31834i	1.97507	−	1.76042i	−0.530723	−	2.77819i	0.0239358	+	2.99990i	−0.871418	+	1.75585i
173.16	−1.23015	−	0.697666i	−1.59170	−	0.683004i	1.02652	+	1.71646i			2.06988i	1.48151	+	1.95067i	0.440031	−	2.60890i	−0.0652554	−	2.82767i	2.06701	+	2.17427i	1.44408	−	2.54625i
173.17	−1.17587	+	0.785710i	1.71321	−	0.254787i	0.765320	−	1.84778i			1.07899i	−1.81431	+	1.64568i	−2.47543	+	0.933935i	0.551904	+	2.77406i	2.87017	−	0.873007i	−0.847769	−	1.26874i
173.18	−1.14306	−	0.832711i	1.10317	+	1.33530i	0.613184	+	1.90368i			4.07380i	−0.149078	−	2.44495i	−1.75571	+	1.97926i	0.884310	−	2.68663i	−0.566029	+	2.94612i	3.39230	−	4.65661i
173.19	−1.14122	−	0.835238i	−0.987509	+	1.42296i	0.604756	+	1.90638i			0.709981i	2.31548	−	0.799106i	−2.47476	−	0.935707i	0.902119	−	2.68071i	−1.04965	−	2.81038i	0.593003	−	0.810243i
173.20	−1.12327	+	0.859225i	0.944647	+	1.45177i	0.523464	−	1.93028i		−	3.97106i	−2.30849	−	0.819065i	−2.57827	−	0.593742i	1.07056	+	2.61800i	−1.21528	+	2.74282i	3.41204	+	4.46057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
63.s	even	6	1	inner
504.y	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.y.a	✓	184
7.d	odd	6	1		504.2.ca.a	yes	184
8.b	even	2	1	inner	504.2.y.a	✓	184
9.d	odd	6	1		504.2.ca.a	yes	184
56.j	odd	6	1		504.2.ca.a	yes	184
63.s	even	6	1	inner	504.2.y.a	✓	184
72.j	odd	6	1		504.2.ca.a	yes	184
504.y	even	6	1	inner	504.2.y.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.y.a	✓	184	1.a	even	1	1	trivial
504.2.y.a	✓	184	8.b	even	2	1	inner
504.2.y.a	✓	184	63.s	even	6	1	inner
504.2.y.a	✓	184	504.y	even	6	1	inner
504.2.ca.a	yes	184	7.d	odd	6	1
504.2.ca.a	yes	184	9.d	odd	6	1
504.2.ca.a	yes	184	56.j	odd	6	1
504.2.ca.a	yes	184	72.j	odd	6	1

Hecke kernels

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