Newform orbit 164.3.l.a

• Label: 164.3.l.a
• Level: $164$
• Weight: $3$
• Character orbit: 164.l
• 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.l (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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 3 q^{2} - 3 q^{4} - 2 q^{5} - 25 q^{6} - 36 q^{8} + 424 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.99597	−	0.126903i	2.39245			3.96779	+	0.506589i	−2.73540	+	1.98738i	−4.77525	−	0.303609i	−2.96576	+	9.12767i	−7.85530	−	1.51466i	−3.27620			5.71198	−	3.61963i
23.2	−1.98941	−	0.205567i	5.80392			3.91548	+	0.817912i	−0.392502	+	0.285169i	−11.5464	−	1.19309i	3.50661	−	10.7922i	−7.62136	−	2.43205i	24.6855			0.839467	−	0.486633i
23.3	−1.94462	+	0.467387i	−3.71133			3.56310	−	1.81778i	−4.22575	+	3.07019i	7.21714	−	1.73463i	−1.55229	+	4.77745i	−6.07927	+	5.20024i	4.77400			6.78252	−	7.94541i
23.4	−1.94108	+	0.481888i	0.0526689			3.53557	−	1.87076i	6.01711	−	4.37168i	−0.102234	+	0.0253805i	1.71992	−	5.29336i	−5.96132	+	5.33504i	−8.99723			−9.57301	+	11.3853i
23.5	−1.88686	−	0.663140i	−5.42935			3.12049	+	2.50251i	2.11565	−	1.53711i	10.2444	+	3.60042i	2.69024	−	8.27972i	−4.22842	−	6.79121i	20.4779			−5.01127	+	1.49734i
23.6	−1.87660	−	0.691635i	−1.80399			3.04328	+	2.59585i	1.98179	−	1.43986i	3.38537	+	1.24770i	−1.53649	+	4.72884i	−3.91565	−	6.97622i	−5.74562			−4.71489	+	1.33136i
23.7	−1.82820	−	0.810985i	0.174784			2.68461	+	2.96528i	−5.60132	+	4.06960i	−0.319540	−	0.141747i	3.43849	−	10.5826i	−2.50319	−	7.59829i	−8.96945			13.5407	−	2.89744i
23.8	−1.59233	+	1.21015i	4.23702			1.07105	−	3.85394i	2.10237	−	1.52746i	−6.74676	+	5.12745i	−2.22561	+	6.84972i	2.95838	+	7.43290i	8.95238			−1.49921	+	4.97643i
23.9	−1.56964	−	1.23944i	3.24329			0.927554	+	3.89097i	7.04687	−	5.11985i	−5.09081	−	4.01988i	−0.790554	+	2.43307i	3.36671	−	7.25708i	1.51893			−17.4068	−	0.697870i
23.10	−1.49088	+	1.33315i	−3.04044			0.445422	−	3.97512i	0.357384	−	0.259655i	4.53292	−	4.05337i	1.22026	−	3.75559i	4.63537	+	6.52023i	0.244298			−0.186657	+	0.863560i
23.11	−1.47996	+	1.34526i	2.13105			0.380577	−	3.98185i	−7.01448	+	5.09632i	−3.15388	+	2.86681i	1.33584	−	4.11128i	4.79337	+	6.40497i	−4.45862			3.52532	−	16.9786i
23.12	−1.21580	−	1.58803i	0.677342			−1.04367	+	3.86144i	−1.99860	+	1.45207i	−0.823511	−	1.07564i	0.934712	−	2.87675i	7.40098	−	3.03736i	−8.54121			4.73583	+	1.40842i
23.13	−0.995129	−	1.73485i	−3.65070			−2.01944	+	3.45281i	2.46264	−	1.78921i	3.63292	+	6.33344i	−1.94656	+	5.99089i	7.99972	+	0.0674398i	4.32763			−5.55467	−	2.49182i
23.14	−0.981376	−	1.74267i	5.08911			−2.07380	+	3.42043i	−6.37097	+	4.62878i	−4.99432	−	8.86863i	−2.93081	+	9.02010i	7.99586	+	0.257229i	16.8990			14.3188	+	6.55993i
23.15	−0.888883	+	1.79162i	−1.25447			−2.41978	−	3.18507i	4.13475	−	3.00407i	1.11507	−	2.24752i	−3.84198	+	11.8244i	7.85732	−	1.50415i	−7.42631			1.70683	+	10.0781i
23.16	−0.568924	+	1.91737i	4.43662			−3.35265	−	2.18168i	1.63991	−	1.19147i	−2.52410	+	8.50667i	2.22047	−	6.83389i	6.09050	−	5.18708i	10.6836			1.35150	+	3.82218i
23.17	−0.491389	+	1.93869i	0.0745071			−3.51707	−	1.90530i	1.56299	−	1.13558i	−0.0366119	+	0.144447i	2.91894	−	8.98359i	5.42205	−	5.88229i	−8.99445			1.43351	+	3.58818i
23.18	−0.288297	+	1.97911i	−4.83617			−3.83377	−	1.14114i	−5.05367	+	3.67170i	1.39425	−	9.57132i	−1.02466	+	3.15358i	3.36371	−	7.25847i	14.3885			−5.80976	−	11.0603i
23.19	−0.230366	−	1.98669i	−5.08911			−3.89386	+	0.915333i	−6.37097	+	4.62878i	1.17236	+	10.1105i	2.93081	−	9.02010i	2.71550	+	7.52503i	16.8990			10.6636	+	11.5908i
23.20	−0.214645	−	1.98845i	3.65070			−3.90785	+	0.853622i	2.46264	−	1.78921i	−0.783606	−	7.25923i	1.94656	−	5.99089i	2.53619	+	7.58734i	4.32763			−4.08635	−	4.51279i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
41.f	even	10	1	inner
164.l	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.l.a	✓	160
4.b	odd	2	1	inner	164.3.l.a	✓	160
41.f	even	10	1	inner	164.3.l.a	✓	160
164.l	odd	10	1	inner	164.3.l.a	✓	160

    

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

Hecke kernels

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