Newform orbit 333.3.n.a

• Label: 333.3.n.a
• Level: $333$
• Weight: $3$
• Character orbit: 333.n
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $148$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(148\)
Relative dimension:	\(74\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 148 q - 6 q^{3} - 146 q^{4} - 14 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
110.1	−1.97208	+	3.41575i	0.912910	−	2.85773i	−5.77823	−	10.0082i	−2.24263	−	3.88436i	7.96094	+	8.75395i	2.86017	−	4.95397i	29.8039			−7.33319	−	5.21769i	17.6906
110.2	−1.90242	+	3.29509i	0.0481523	+	2.99961i	−5.23841	−	9.07319i	0.976991	+	1.69220i	−9.97560	−	5.54786i	−4.73015	+	8.19286i	24.6433			−8.99536	+	0.288876i	−7.43459
110.3	−1.87397	+	3.24581i	−2.75452	−	1.18855i	−5.02350	−	8.70096i	3.55614	+	6.15941i	9.01966	−	6.71333i	−0.0383313	+	0.0663917i	22.6638			6.17472	+	6.54774i	−26.6563
110.4	−1.85833	+	3.21872i	2.56675	+	1.55300i	−4.90678	−	8.49879i	2.76762	+	4.79366i	−9.76853	+	5.37567i	4.38697	−	7.59845i	21.6070			4.17641	+	7.97230i	−20.5726
110.5	−1.84270	+	3.19165i	−2.89994	+	0.768344i	−4.79108	−	8.29839i	−3.27112	−	5.66574i	2.89143	−	10.6714i	−3.96379	+	6.86548i	20.5725			7.81929	−	4.45630i	24.1107
110.6	−1.69407	+	2.93422i	−1.20316	+	2.74816i	−3.73977	−	6.47748i	−1.22746	−	2.12602i	−6.02549	−	8.18593i	4.27756	−	7.40894i	11.7892			−6.10482	−	6.61296i	8.31761
110.7	−1.67032	+	2.89308i	2.96393	+	0.463800i	−3.57993	−	6.20062i	−4.26153	−	7.38119i	−6.29252	+	7.80019i	−2.88339	+	4.99418i	10.5559			8.56978	+	2.74935i	28.4724
110.8	−1.64003	+	2.84061i	−0.892672	−	2.86411i	−3.37936	−	5.85323i	−0.285652	−	0.494764i	9.59982	+	2.16148i	−3.14706	+	5.45087i	9.04877			−7.40627	+	5.11343i	1.87391
110.9	−1.63980	+	2.84022i	2.79695	−	1.08494i	−3.37789	−	5.85067i	−0.286787	−	0.496729i	−1.50496	+	9.72302i	0.940265	−	1.62859i	9.03783			6.64581	−	6.06904i	1.88109
110.10	−1.62323	+	2.81152i	1.74800	−	2.43813i	−3.26975	−	5.66338i	3.68770	+	6.38729i	4.01745	+	8.87219i	−4.37225	+	7.57295i	8.24441			−2.88899	−	8.52372i	−23.9440
110.11	−1.54962	+	2.68403i	−2.86411	−	0.892678i	−2.80267	−	4.85436i	−3.91166	−	6.77519i	6.83426	−	6.30403i	5.93243	−	10.2753i	4.97532			7.40625	+	5.11346i	24.2464
110.12	−1.54008	+	2.66750i	−2.74016	+	1.22129i	−2.74370	−	4.75223i	2.40249	+	4.16123i	0.962288	−	9.19024i	1.86033	−	3.22219i	4.58142			6.01693	−	6.69303i	−14.8001
110.13	−1.47315	+	2.55158i	1.83443	+	2.37379i	−2.34037	−	4.05363i	−1.42559	−	2.46920i	−8.75930	+	1.18375i	0.889782	−	1.54115i	2.00564			−2.26971	+	8.70910i	8.40046
110.14	−1.33317	+	2.30912i	−0.706891	−	2.91553i	−1.55469	−	2.69280i	2.82419	+	4.89164i	7.67471	+	2.25460i	5.96669	−	10.3346i	−2.37470			−8.00061	+	4.12192i	−15.0605
110.15	−1.28545	+	2.22646i	1.78103	+	2.41411i	−1.30475	−	2.25989i	0.943859	+	1.63481i	−7.66434	+	0.862186i	−6.07806	+	10.5275i	−3.57484			−2.65585	+	8.59921i	−4.85312
110.16	−1.24444	+	2.15544i	−2.27524	−	1.95532i	−1.09728	−	1.90054i	−1.82791	−	3.16603i	7.04598	−	2.47086i	−4.50975	+	7.81112i	−4.49356			1.35345	+	8.89765i	9.09891
110.17	−1.21885	+	2.11111i	2.94107	+	0.591713i	−0.971192	−	1.68215i	4.22130	+	7.31151i	−4.83389	+	5.48771i	−0.385107	+	0.667025i	−5.01585			8.29975	+	3.48053i	−20.5805
110.18	−1.17556	+	2.03613i	−2.54227	+	1.59275i	−0.763889	−	1.32309i	1.61998	+	2.80589i	−0.254445	−	7.04878i	−3.47522	+	6.01925i	−5.81250			3.92631	−	8.09840i	−7.61754
110.19	−1.11519	+	1.93157i	2.21265	−	2.02587i	−0.487296	−	0.844021i	−0.743796	−	1.28829i	1.44558	+	6.53312i	1.57738	−	2.73211i	−6.74781			0.791665	−	8.96511i	3.31789
110.20	−1.10869	+	1.92032i	−0.0807228	−	2.99891i	−0.458409	−	0.793987i	−4.43892	−	7.68843i	5.84836	+	3.16987i	1.06556	−	1.84560i	−6.83662			−8.98697	+	0.484161i	19.6856

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
37.b	even	2	1	inner
333.n	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	333.3.n.a	✓	148
9.d	odd	6	1	inner	333.3.n.a	✓	148
37.b	even	2	1	inner	333.3.n.a	✓	148
333.n	odd	6	1	inner	333.3.n.a	✓	148

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
333.3.n.a	✓	148	1.a	even	1	1	trivial
333.3.n.a	✓	148	9.d	odd	6	1	inner
333.3.n.a	✓	148	37.b	even	2	1	inner
333.3.n.a	✓	148	333.n	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(333, [\chi])\).