Newform orbit 444.2.z.a

• Label: 444.2.z.a
• Level: $444$
• Weight: $2$
• Character orbit: 444.z
• Analytic conductor: $3.545$
• Analytic rank: $0$
• Dimension: $432$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	444.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54535784974\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(72\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 432 * q - 12 * q^4 - 12 * q^6 - 12 * q^9 - 6 * q^10 + 9 * q^12 - 36 * q^16 - 27 * q^18 - 12 * q^21 - 24 * q^22 - 33 * q^24 - 24 * q^25 + 12 * q^28 + 60 * q^30 + 6 * q^33 - 12 * q^34 - 12 * q^36 - 12 * q^37 - 72 * q^40 + 39 * q^42 - 6 * q^45 - 78 * q^46 - 3 * q^48 + 12 * q^49 + 18 * q^52 - 39 * q^54 - 12 * q^57 + 12 * q^58 - 24 * q^60 - 72 * q^61 - 6 * q^64 - 111 * q^66 + 6 * q^69 + 18 * q^70 - 153 * q^72 - 48 * q^73 - 6 * q^76 - 117 * q^78 - 12 * q^81 + 24 * q^82 - 30 * q^84 - 36 * q^85 + 90 * q^88 - 84 * q^90 - 12 * q^93 + 36 * q^94 - 45 * q^96 - 12 * q^97

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
71.1	−1.41159	+	0.0861638i	0.637647	−	1.61041i	1.98515	−	0.243255i	0.603572	−	1.65830i	−0.761336	+	2.32817i	1.57611	−	4.33033i	−2.78125	+	0.514424i	−2.18681	−	2.05374i	−0.709108	+	2.39284i
71.2	−1.40829	+	0.129273i	1.37047	+	1.05916i	1.96658	−	0.364110i	0.193108	−	0.530561i	−2.06694	−	1.31444i	0.279484	−	0.767876i	−2.72245	+	0.766999i	0.756358	+	2.90309i	−0.203366	+	0.772148i
71.3	−1.40302	−	0.177605i	−1.70139	+	0.324449i	1.93691	+	0.498366i	−0.520600	+	1.43034i	2.44470	−	0.153031i	0.268735	−	0.738344i	−2.62901	−	1.04322i	2.78947	−	1.10403i	0.984446	−	1.91433i
71.4	−1.40222	+	0.183808i	−0.566361	−	1.63684i	1.93243	−	0.515479i	−0.996154	+	2.73691i	1.09503	+	2.19110i	−0.104414	+	0.286876i	−2.61494	+	1.07801i	−2.35847	+	1.85408i	0.893758	−	4.02085i
71.5	−1.40147	+	0.189448i	−1.06273	−	1.36770i	1.92822	−	0.531011i	1.14924	−	3.15751i	1.74849	+	1.71545i	−1.54293	+	4.23917i	−2.60174	+	1.10949i	−0.741192	+	2.90700i	−1.01244	+	4.64287i
71.6	−1.37804	−	0.317829i	1.58204	−	0.705094i	1.79797	+	0.875961i	0.777340	−	2.13572i	−2.40420	+	0.468827i	−0.807737	+	2.21924i	−2.19926	−	1.77855i	2.00569	−	2.23097i	−1.75000	+	2.69604i
71.7	−1.32568	−	0.492506i	1.57766	+	0.714842i	1.51488	+	1.30581i	−1.31357	+	3.60900i	−1.73941	−	1.72466i	−0.825175	+	2.26715i	−1.36513	−	2.47718i	1.97800	+	2.25555i	3.51883	−	4.13746i
71.8	−1.32140	+	0.503894i	0.246064	+	1.71448i	1.49218	−	1.33169i	1.20445	−	3.30919i	−1.18907	−	2.14152i	−0.0363915	+	0.0999847i	−1.30073	+	2.51159i	−2.87891	+	0.843744i	0.0759287	+	4.97967i
71.9	−1.30499	+	0.544971i	−1.57958	+	0.710573i	1.40601	−	1.42237i	0.157602	−	0.433007i	1.67410	−	1.78812i	−0.511998	+	1.40670i	−1.05969	+	2.62242i	1.99017	−	2.24482i	0.0303073	+	0.650959i
71.10	−1.26731	−	0.627630i	−1.34607	+	1.09000i	1.21216	+	1.59081i	1.40803	−	3.86854i	2.39001	−	0.536531i	0.479593	−	1.31767i	−0.537752	−	2.77684i	0.623817	−	2.93443i	−4.21242	+	4.01892i
71.11	−1.24496	+	0.670881i	1.34558	−	1.09060i	1.09984	−	1.67044i	−0.666581	+	1.83142i	−0.943528	+	2.26048i	−1.01841	+	2.79807i	−0.248585	+	2.81748i	0.621181	−	2.93498i	−0.398798	−	2.72723i
71.12	−1.23808	−	0.683498i	−1.41271	−	1.00212i	1.06566	+	1.69244i	−0.294884	+	0.810188i	1.06409	+	2.20629i	0.810777	−	2.22759i	−0.162588	−	2.82375i	0.991502	+	2.83142i	0.918851	−	0.801521i
71.13	−1.18818	−	0.766964i	0.135442	+	1.72675i	0.823531	+	1.82258i	−0.368617	+	1.01277i	1.16342	−	2.15556i	1.37915	−	3.78917i	0.419352	−	2.79717i	−2.96331	+	0.467747i	1.21474	−	0.920631i
71.14	−1.16292	+	0.804748i	1.72676	+	0.135283i	0.704762	−	1.87171i	−0.762821	+	2.09583i	−2.11695	+	1.23228i	1.30360	−	3.58160i	0.686676	+	2.74381i	2.96340	+	0.467202i	−0.799518	−	3.05116i
71.15	−1.15409	+	0.817357i	0.0325093	+	1.73175i	0.663855	−	1.88661i	−1.15886	+	3.18395i	−1.45297	−	1.97202i	−0.966628	+	2.65579i	0.775885	+	2.71993i	−2.99789	+	0.112596i	−1.26499	−	4.62178i
71.16	−1.15347	−	0.818236i	0.391784	−	1.68716i	0.660979	+	1.88762i	−0.567490	+	1.55917i	−1.83241	+	1.62551i	−0.810356	+	2.22644i	0.782100	−	2.71815i	−2.69301	−	1.32201i	1.93035	−	1.33411i
71.17	−1.08328	+	0.909122i	−1.52091	−	0.828755i	0.346993	−	1.96967i	0.537386	−	1.47646i	2.40101	−	0.484919i	1.29384	−	3.55479i	1.41478	+	2.44916i	1.62633	+	2.52092i	0.760140	+	2.08797i
71.18	−1.00610	−	0.993859i	1.72956	−	0.0928602i	0.0244869	+	1.99985i	0.567490	−	1.55917i	−1.83241	−	1.62551i	0.810356	−	2.22644i	1.96293	−	2.03639i	2.98275	−	0.321215i	−2.12055	+	1.00468i
71.19	−0.969319	+	1.02977i	0.832954	−	1.51861i	−0.120840	−	1.99635i	0.607091	−	1.66797i	0.756419	+	2.32977i	−0.498794	+	1.37042i	2.17290	+	1.81066i	−1.61237	−	2.52987i	1.12915	+	2.24196i
71.20	−0.961637	−	1.03694i	−1.67699	−	0.433231i	−0.150507	+	1.99433i	0.368617	−	1.01277i	1.16342	+	2.15556i	−1.37915	+	3.78917i	2.21274	−	1.76175i	2.62462	+	1.45305i	−1.40466	+	0.591679i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
37.f	even	9	1	inner
111.p	odd	18	1	inner
148.p	odd	18	1	inner
444.z	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	444.2.z.a	✓	432
3.b	odd	2	1	inner	444.2.z.a	✓	432
4.b	odd	2	1	inner	444.2.z.a	✓	432
12.b	even	2	1	inner	444.2.z.a	✓	432
37.f	even	9	1	inner	444.2.z.a	✓	432
111.p	odd	18	1	inner	444.2.z.a	✓	432
148.p	odd	18	1	inner	444.2.z.a	✓	432
444.z	even	18	1	inner	444.2.z.a	✓	432

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
444.2.z.a	✓	432	1.a	even	1	1	trivial
444.2.z.a	✓	432	3.b	odd	2	1	inner
444.2.z.a	✓	432	4.b	odd	2	1	inner
444.2.z.a	✓	432	12.b	even	2	1	inner
444.2.z.a	✓	432	37.f	even	9	1	inner
444.2.z.a	✓	432	111.p	odd	18	1	inner
444.2.z.a	✓	432	148.p	odd	18	1	inner
444.2.z.a	✓	432	444.z	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(444, [\chi])\).