Newform orbit 381.2.n.b

• Label: 381.2.n.b
• Level: $381$
• Weight: $2$
• Character orbit: 381.n
• Analytic conductor: $3.042$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 381 = 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	381.n (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.04230031701\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(40\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 6 q^{3} - 276 q^{4} + 15 q^{6} - 6 q^{7} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1		−	2.68533i	−1.67535	−	0.439540i	−5.21100			−1.43350	+	2.48289i	−1.18031	+	4.49887i	0.788957	−	2.16764i			8.62259i	2.61361	+	1.47277i	6.66738	+	3.84941i
59.2		−	2.66203i	−1.18742	+	1.26097i	−5.08641			1.43409	−	2.48392i	3.35673	+	3.16094i	−1.67256	+	4.59532i			8.21611i	−0.180079	−	2.99459i	−6.61226	−	3.81759i
59.3		−	2.57426i	1.72555	−	0.149889i	−4.62680			1.07649	−	1.86453i	−0.385853	−	4.44202i	0.166087	−	0.456321i			6.76205i	2.95507	−	0.517284i	−4.79978	−	2.77116i
59.4		−	2.50061i	0.558132	+	1.63966i	−4.25306			0.367272	−	0.636134i	4.10016	−	1.39567i	1.27659	−	3.50740i			5.63404i	−2.37698	+	1.83029i	−1.59072	−	0.918405i
59.5		−	2.47307i	1.66896	+	0.463225i	−4.11606			−2.02471	+	3.50690i	1.14559	−	4.12744i	−1.10001	+	3.02224i			5.23314i	2.57085	+	1.54621i	8.67279	+	5.00724i
59.6		−	2.44577i	−0.639114	−	1.60982i	−3.98177			1.35689	−	2.35020i	−3.93725	+	1.56312i	0.497312	−	1.36635i			4.84694i	−2.18307	+	2.05772i	−5.74803	−	3.31863i
59.7		−	2.01206i	−1.21328	+	1.23610i	−2.04840			−0.860805	+	1.49096i	2.48711	+	2.44120i	−0.199157	+	0.547180i			0.0973849i	−0.0558931	−	2.99948i	2.99990	+	1.73199i
59.8		−	1.91489i	0.912228	−	1.47236i	−1.66681			−1.61325	+	2.79423i	−2.81941	−	1.74682i	1.59386	−	4.37910i		−	0.638018i	−1.33568	−	2.68625i	5.35066	+	3.08920i
59.9		−	1.85316i	1.33711	−	1.10097i	−1.43419			0.679362	−	1.17669i	−2.04027	−	2.47787i	−0.528416	+	1.45181i		−	1.04853i	0.575726	−	2.94424i	−2.18059	−	1.25897i
59.10		−	1.77580i	−0.904604	−	1.47706i	−1.15348			−0.879556	+	1.52344i	−2.62296	+	1.60640i	−1.15906	+	3.18449i		−	1.50326i	−1.36338	+	2.67230i	2.70532	+	1.56192i
59.11		−	1.60698i	−1.70470	−	0.306599i	−0.582373			1.34147	−	2.32350i	−0.492697	+	2.73941i	−0.00862836	+	0.0237062i		−	2.27809i	2.81199	+	1.04532i	−3.73381	−	2.15571i
59.12		−	1.28869i	1.04188	+	1.38365i	0.339280			0.645116	−	1.11737i	1.78309	−	1.34266i	−1.80449	+	4.95781i		−	3.01460i	−0.828968	+	2.88319i	−1.43995	−	0.831354i
59.13		−	1.20224i	1.59649	+	0.671736i	0.554615			−0.198583	+	0.343956i	0.807589	−	1.91936i	0.817292	−	2.24549i		−	3.07126i	2.09754	+	2.14484i	0.413518	+	0.238745i
59.14		−	1.19529i	−0.159636	+	1.72468i	0.571294			1.84219	−	3.19077i	2.06148	+	0.190810i	0.605935	−	1.66479i		−	3.07343i	−2.94903	−	0.550641i	−3.81388	−	2.20195i
59.15		−	0.918043i	−0.490902	+	1.66103i	1.15720			−1.36181	+	2.35872i	1.52490	+	0.450669i	−0.655442	+	1.80081i		−	2.89844i	−2.51803	−	1.63080i	2.16541	+	1.25020i
59.16		−	0.909262i	−1.58565	−	0.696931i	1.17324			−1.62652	+	2.81721i	−0.633693	+	1.44177i	0.0343421	−	0.0943541i		−	2.88531i	2.02857	+	2.21018i	2.56159	+	1.47893i
59.17		−	0.618769i	−0.0975313	−	1.72930i	1.61713			−0.592317	+	1.02592i	−1.07004	+	0.0603493i	−0.127538	+	0.350408i		−	2.23816i	−2.98098	+	0.337322i	0.634809	+	0.366507i
59.18		−	0.433500i	0.223520	−	1.71757i	1.81208			1.33347	−	2.30965i	−0.744566	−	0.0968959i	0.296480	−	0.814572i		−	1.65254i	−2.90008	−	0.767821i	−1.00123	−	0.578062i
59.19		−	0.249569i	1.61448	−	0.627256i	1.93772			−1.84375	+	3.19347i	−0.156544	−	0.402925i	−1.19290	+	3.27746i		−	0.982733i	2.21310	−	2.02539i	0.796991	+	0.460143i
59.20		−	0.0469092i	−1.13722	+	1.30642i	1.99780			−0.545153	+	0.944233i	0.0612830	+	0.0533461i	1.69769	−	4.66437i		−	0.187534i	−0.413458	−	2.97137i	0.0442932	+	0.0255727i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
127.h	odd	18	1	inner
381.n	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	381.2.n.b	✓	240
3.b	odd	2	1	inner	381.2.n.b	✓	240
127.h	odd	18	1	inner	381.2.n.b	✓	240
381.n	even	18	1	inner	381.2.n.b	✓	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
381.2.n.b	✓	240	1.a	even	1	1	trivial
381.2.n.b	✓	240	3.b	odd	2	1	inner
381.2.n.b	✓	240	127.h	odd	18	1	inner
381.2.n.b	✓	240	381.n	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^120 + 189*T2^118 + 17382*T2^116 + 1036644*T2^114 + 45078882*T2^112 + 1523771070*T2^110 + 41682148929*T2^108 + 948508757733*T2^106 + 18318127871637*T2^104 + 304811357239426*T2^102 + 4421788419714333*T2^100 + 56447680999818618*T2^98 + 638950420365970454*T2^96 + 6453004441159889136*T2^94 + 58447839042549546627*T2^92 + 476819377237469580014*T2^90 + 3516275596214602944099*T2^88 + 23510872005984448265901*T2^86 + 142893490282491494589666*T2^84 + 791103436140523592413536*T2^82 + 3996616137285613328554878*T2^80 + 18450660619108153696885219*T2^78 + 77927248220838998472529824*T2^76 + 301376779245888182912033088*T2^74 + 1067960474326324302636119832*T2^72 + 3469058418813994257229578678*T2^70 + 10331777179266697607353378971*T2^68 + 28213402335990949403243296031*T2^66 + 70628505167136185872433681322*T2^64 + 162029748514007854076379235341*T2^62 + 340459593961434955116154338768*T2^60 + 654753843152031813015276162960*T2^58 + 1151427016976362298028116258532*T2^56 + 1849530441961414472043584202703*T2^54 + 2710114728828149069359646847411*T2^52 + 3617086515371269917962936079576*T2^50 + 4389582872133945358517973136791*T2^48 + 4834220229333745694931794487714*T2^46 + 4820626791141413308638091852308*T2^44 + 4341790999480002965044463672274*T2^42 + 3522119805634165797467987447394*T2^40 + 2565326217997172439852024803697*T2^38 + 1671672412133445030946782610599*T2^36 + 970748951000891397878791468407*T2^34 + 500112946687945432356574626486*T2^32 + 227424377041408769337215219717*T2^30 + 90763702827057022766476680876*T2^28 + 31580965097582628771259554186*T2^26 + 9507299421846966503759441971*T2^24 + 2454266678697759760897587396*T2^22 + 537534231199322355558603648*T2^20 + 98614267417597323914684176*T2^18 + 14916094404074716028987514*T2^16 + 1823347759638329585110038*T2^14 + 175492166787579108323113*T2^12 + 12836709792429684682287*T2^10 + 678417979080939467373*T2^8 + 23968831551472706586*T2^6 + 495488736512368542*T2^4 + 4546785152751759*T2^2 + 7845999027129