Newform orbit 501.2.g.a

• Label: 501.2.g.a
• Level: $501$
• Weight: $2$
• Character orbit: 501.g
• Analytic conductor: $4.001$
• Analytic rank: $0$
• Dimension: $4428$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 501 = 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	501.g (of order \(166\), degree \(82\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00050514127\)
Analytic rank:	\(0\)
Dimension:	\(4428\)
Relative dimension:	\(54\) over \(\Q(\zeta_{166})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{166}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 4428 q - 81 q^{3} - 106 q^{4} - 87 q^{6} - 162 q^{7} - 81 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.88312	+	1.99319i	−1.64341	+	0.546991i	−0.313173	−	5.51003i	−0.0409340	+	2.16267i	2.00448	−	4.30567i	−2.10916	+	2.70428i	7.37860	+	6.21786i	2.40160	−	1.79786i	−4.23352	−	4.15415i
5.2	−1.84577	+	1.95365i	0.327186	+	1.70087i	−0.296415	−	5.21519i	−0.00550829	+	0.291020i	−3.92681	−	2.50019i	1.20363	−	1.54325i	6.62530	+	5.58307i	−2.78590	+	1.11300i	−0.558385	−	0.547916i
5.3	−1.78590	+	1.89028i	1.09246	−	1.34407i	−0.270255	−	4.75493i	0.0203932	−	1.07744i	0.589650	+	4.46544i	1.17764	−	1.50993i	5.49366	+	4.62945i	−0.613051	−	2.93669i	2.00025	+	1.96274i
5.4	−1.74419	+	1.84614i	−1.57791	−	0.714286i	−0.252541	−	4.44326i	0.0823462	−	4.35061i	4.07084	−	1.66719i	0.408584	−	0.523871i	4.75908	+	4.01043i	1.97959	+	2.25416i	7.88820	+	7.74031i
5.5	−1.66319	+	1.76041i	−0.776220	−	1.54838i	−0.219334	−	3.85901i	−0.0182397	+	0.963660i	4.01678	+	1.20879i	−1.37376	+	1.76138i	3.45433	+	2.91092i	−1.79497	+	2.40377i	−1.66610	−	1.63486i
5.6	−1.63742	+	1.73313i	1.66998	+	0.459533i	−0.209099	−	3.67893i	−0.0634161	+	3.35047i	−3.53088	+	2.14184i	1.16863	−	1.49838i	3.07194	+	2.58869i	2.57766	+	1.53482i	−5.70295	−	5.59603i
5.7	−1.53382	+	1.62347i	0.368509	+	1.69240i	−0.169568	−	2.98342i	0.0331235	−	1.75002i	−3.31277	−	1.99756i	−2.46502	+	3.16056i	1.68780	+	1.42229i	−2.72840	+	1.24732i	2.79030	+	2.73798i
5.8	−1.41212	+	1.49466i	−1.05582	+	1.37304i	−0.126435	−	2.22453i	0.0402183	−	2.12486i	−0.561277	−	3.51700i	1.15944	−	1.48659i	0.358687	+	0.302262i	−0.770472	−	2.89937i	3.11916	+	3.06068i
5.9	−1.38920	+	1.47040i	−0.0758177	−	1.73039i	−0.118708	−	2.08858i	−0.00471370	+	0.249040i	2.64969	+	2.29238i	0.481586	−	0.617472i	0.142237	+	0.119862i	−2.98850	+	0.262388i	−0.359639	−	0.352897i
5.10	−1.35209	+	1.43112i	−1.38310	+	1.04261i	−0.106469	−	1.87323i	−0.0524556	+	2.77140i	0.377974	−	3.38909i	1.71132	−	2.19419i	−0.186302	−	0.156995i	0.825929	−	2.88407i	−3.89528	−	3.82225i
5.11	−1.18027	+	1.24926i	−1.70445	+	0.307994i	−0.0541168	−	0.952143i	0.0125081	−	0.660841i	1.62694	−	2.49281i	−1.29745	+	1.66354i	−1.37509	−	1.15878i	2.81028	−	1.04992i	0.810797	+	0.795596i
5.12	−1.17399	+	1.24262i	1.72690	+	0.133489i	−0.0523393	−	0.920870i	0.0311985	−	1.64831i	−2.19325	+	1.98916i	−0.123013	+	0.157723i	−1.40873	−	1.18712i	2.96436	+	0.461046i	2.01159	+	1.97388i
5.13	−1.16384	+	1.23187i	1.23896	+	1.21036i	−0.0494826	−	0.870608i	−0.0281112	+	1.48520i	−2.93295	+	0.117573i	−1.50437	+	1.92885i	−1.46178	−	1.23183i	0.0700597	+	2.99918i	−1.79685	−	1.76316i
5.14	−1.10029	+	1.16461i	0.619849	−	1.61734i	−0.0321721	−	0.566043i	0.0823063	−	4.34850i	1.20155	+	2.50143i	−3.01020	+	3.85957i	−1.75572	−	1.47952i	−2.23158	−	2.00501i	4.97373	+	4.88048i
5.15	−0.997374	+	1.05567i	1.57180	−	0.727637i	−0.00619636	−	0.109020i	0.0269453	−	1.42361i	−0.799524	+	2.38503i	1.63514	−	2.09652i	−2.09986	−	1.76953i	1.94109	−	2.28739i	1.47598	+	1.44831i
5.16	−0.964236	+	1.02060i	−1.41173	−	1.00350i	0.00162479	+	0.0285869i	−0.0746725	+	3.94518i	2.38541	−	0.473193i	−0.786791	+	1.00880i	−2.17808	−	1.83544i	0.985963	+	2.83335i	−3.95444	−	3.88030i
5.17	−0.845251	+	0.894656i	0.844668	−	1.51213i	0.0275295	+	0.484360i	−0.0637482	+	3.36802i	0.638880	+	2.03382i	2.58489	−	3.31425i	−2.33896	−	1.97102i	−1.57307	−	2.55450i	−2.95934	−	2.90385i
5.18	−0.723493	+	0.765782i	0.288713	+	1.70782i	0.0505110	+	0.888702i	−0.0392781	+	2.07519i	−1.51670	−	1.01450i	1.97989	−	2.53854i	−2.32830	−	1.96203i	−2.83329	+	0.986140i	−1.56072	−	1.53146i
5.19	−0.721629	+	0.763809i	−0.726640	−	1.57226i	0.0508351	+	0.894404i	0.0559940	−	2.95834i	1.72527	+	0.579573i	2.65134	−	3.39945i	−2.32689	−	1.96085i	−1.94399	+	2.28493i	2.21920	+	2.17759i
5.20	−0.611342	+	0.647075i	0.254573	+	1.71324i	0.0685230	+	1.20561i	0.0751782	−	3.97190i	−1.26423	−	0.882648i	0.904026	−	1.15911i	−2.18346	−	1.83998i	−2.87038	+	0.872291i	2.52416	+	2.47684i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
167.d	odd	166	1	inner
501.g	even	166	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	501.2.g.a	✓	4428
3.b	odd	2	1	inner	501.2.g.a	✓	4428
167.d	odd	166	1	inner	501.2.g.a	✓	4428
501.g	even	166	1	inner	501.2.g.a	✓	4428

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
501.2.g.a	✓	4428	1.a	even	1	1	trivial
501.2.g.a	✓	4428	3.b	odd	2	1	inner
501.2.g.a	✓	4428	167.d	odd	166	1	inner
501.2.g.a	✓	4428	501.g	even	166	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(501, [\chi])\).