Newform orbit 164.3.j.a

• Label: 164.3.j.a
• Level: $164$
• Weight: $3$
• Character orbit: 164.j
• Analytic conductor: $4.469$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	164.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.46867633551\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 3 q^{2} - 3 q^{4} - 10 q^{5} + 3 q^{6} + 30 q^{8} - 440 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
51.1	−1.99963	−	0.0384436i			1.51702i	3.99704	+	0.153746i	2.19856	+	6.76648i	0.0583199	−	3.03349i	0.981428	+	1.35082i	−7.98670	−	0.461096i	6.69864			−4.13619	−	13.6150i
51.2	−1.99907	−	0.0609218i			4.62833i	3.99258	+	0.243574i	−2.53676	−	7.80734i	0.281966	−	9.25237i	1.15544	+	1.59033i	−7.96661	−	0.730157i	−12.4215			4.59552	+	15.7620i
51.3	−1.98934	+	0.206214i		−	3.47556i	3.91495	−	0.820460i	−1.98281	−	6.10245i	0.716709	+	6.91407i	−4.28268	−	5.89460i	−7.61898	+	2.43949i	−3.07951			5.20289	+	11.7310i
51.4	−1.96656	−	0.364219i		−	1.41424i	3.73469	+	1.43252i	−0.621293	−	1.91214i	−0.515095	+	2.78119i	3.11238	+	4.28382i	−6.82273	−	4.17737i	6.99991			0.525369	+	3.98663i
51.5	−1.83024	+	0.806360i		−	5.33091i	2.69957	−	2.95167i	2.10067	+	6.46520i	4.29863	+	9.75686i	5.42005	+	7.46006i	−2.56075	+	7.57909i	−19.4186			−9.05802	−	10.1390i
51.6	−1.72991	−	1.00370i			2.24404i	1.98516	+	3.47263i	−0.159083	−	0.489606i	2.25235	−	3.88199i	−7.65530	−	10.5366i	0.0513562	−	7.99984i	3.96427			−0.216222	+	1.00665i
51.7	−1.69584	+	1.06024i			2.07018i	1.75176	−	3.59601i	−0.968289	−	2.98009i	−2.19489	−	3.51069i	1.30405	+	1.79488i	0.841937	+	7.95557i	4.71437			4.80169	+	4.02714i
51.8	−1.68618	−	1.07555i		−	4.98179i	1.68639	+	3.62713i	0.982375	+	3.02344i	−5.35816	+	8.40018i	−3.59000	−	4.94121i	1.05761	−	7.92978i	−15.8182			1.59540	−	6.15464i
51.9	−1.59001	+	1.21321i		−	1.40742i	1.05624	−	3.85803i	1.19593	+	3.68069i	1.70750	+	2.23781i	−6.09806	−	8.39326i	3.00117	+	7.41572i	7.01916			−6.36699	−	4.40141i
51.10	−1.54397	−	1.27129i			4.98179i	0.767660	+	3.92565i	0.982375	+	3.02344i	6.33328	−	7.69171i	3.59000	+	4.94121i	3.80538	−	7.03698i	−15.8182			2.32690	−	5.91696i
51.11	−1.48915	−	1.33508i		−	2.24404i	0.435135	+	3.97626i	−0.159083	−	0.489606i	−2.99597	+	3.34172i	7.65530	+	10.5366i	4.66064	−	6.50219i	3.96427			−0.416765	+	0.941485i
51.12	−1.28954	+	1.52875i		−	3.83178i	−0.674166	−	3.94278i	−2.23545	−	6.88001i	5.85783	+	4.94123i	5.28857	+	7.27909i	6.89689	+	4.05374i	−5.68250			13.4005	+	5.45461i
51.13	−1.19251	+	1.60559i			4.60760i	−1.15586	−	3.82936i	1.84256	+	5.67083i	−7.39793	−	5.49460i	5.74265	+	7.90408i	7.52676	+	2.71070i	−12.2300			−11.3023	−	3.80409i
51.14	−0.954093	−	1.75776i			1.41424i	−2.17941	+	3.35412i	−0.621293	−	1.91214i	2.48590	−	1.34932i	−3.11238	−	4.28382i	7.97510	+	0.630736i	6.99991			−2.76831	+	2.91644i
51.15	−0.693375	+	1.87596i			4.17983i	−3.03846	−	2.60149i	−0.910607	−	2.80256i	−7.84119	−	2.89819i	−5.61198	−	7.72423i	6.98708	−	3.89624i	−8.47095			5.88889	+	0.234960i
51.16	−0.675687	−	1.88240i		−	4.62833i	−3.08689	+	2.54383i	−2.53676	−	7.80734i	−8.71239	+	3.12730i	−1.15544	−	1.59033i	6.87430	+	4.09195i	−12.4215			−12.9825	+	10.0505i
51.17	−0.654482	−	1.88988i		−	1.51702i	−3.14331	+	2.47379i	2.19856	+	6.76648i	−2.86699	+	0.992864i	−0.981428	−	1.35082i	6.73240	+	4.32143i	6.69864			11.3489	−	8.58356i
51.18	−0.612182	+	1.90400i		−	2.35755i	−3.25047	−	2.33120i	1.74583	+	5.37312i	4.48878	+	1.44325i	0.418361	+	0.575825i	6.42849	−	4.76178i	3.44197			−11.2992	+	0.0347430i
51.19	−0.418619	−	1.95570i			3.47556i	−3.64952	+	1.63739i	−1.98281	−	6.10245i	6.79715	−	1.45493i	4.28268	+	5.89460i	4.72999	+	6.45191i	−3.07951			−11.1045	+	6.43238i
51.20	−0.331829	+	1.97228i		−	0.324143i	−3.77978	−	1.30892i	−1.37920	−	4.24474i	0.639301	+	0.107560i	0.790707	+	1.08831i	3.83580	−	7.02045i	8.89493			8.82947	−	1.31164i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
41.d	even	5	1	inner
164.j	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	164.3.j.a	✓	160
4.b	odd	2	1	inner	164.3.j.a	✓	160
41.d	even	5	1	inner	164.3.j.a	✓	160
164.j	odd	10	1	inner	164.3.j.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
164.3.j.a	✓	160	1.a	even	1	1	trivial
164.3.j.a	✓	160	4.b	odd	2	1	inner
164.3.j.a	✓	160	41.d	even	5	1	inner
164.3.j.a	✓	160	164.j	odd	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(164, [\chi])\).