Newform orbit 387.3.s.a

• Label: 387.3.s.a
• Level: $387$
• Weight: $3$
• Character orbit: 387.s
• Analytic conductor: $10.545$
• Analytic rank: $0$
• Dimension: $172$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	387.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5449862307\)
Analytic rank:	\(0\)
Dimension:	\(172\)
Relative dimension:	\(86\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 172 q + 166 q^{4} - 36 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
85.1	−3.40634	+	1.96665i	2.98601	−	0.289359i	5.73543	−	9.93405i	5.90568	+	3.40964i	−9.60230	+	6.85810i	6.88271	−	3.97373i			29.3851i	8.83254	−	1.72806i	−26.8223
85.2	−3.36645	+	1.94362i	2.16723	+	2.07439i	5.55531	−	9.62208i	−7.55242	−	4.36039i	−11.3277	−	2.77105i	1.95488	−	1.12865i			27.6407i	0.393808	+	8.99138i	33.8998
85.3	−3.28525	+	1.89674i	1.02188	−	2.82060i	5.19526	−	8.99845i	−1.26253	−	0.728921i	1.99283	+	11.2046i	−4.78002	+	2.75975i			24.2423i	−6.91154	−	5.76460i	5.53030
85.4	−3.19623	+	1.84535i	−2.97325	+	0.399754i	4.81060	−	8.33220i	−0.159562	−	0.0921232i	8.76550	−	6.76437i	4.17790	−	2.41211i			20.7461i	8.68039	−	2.37713i	0.679997
85.5	−3.11356	+	1.79761i	−0.431319	+	2.96883i	4.46283	−	7.72985i	−0.324020	−	0.187073i	−3.99388	−	10.0190i	−10.2245	+	5.90313i			17.7089i	−8.62793	−	2.56103i	1.34514
85.6	−3.04273	+	1.75672i	−2.35061	−	1.86404i	4.17213	−	7.22634i	7.69910	+	4.44508i	10.4269	+	1.54239i	−6.84815	+	3.95378i			15.2633i	2.05074	+	8.76324i	−31.2350
85.7	−3.03061	+	1.74972i	0.161604	+	2.99564i	4.12305	−	7.14134i	2.57556	+	1.48700i	−5.73130	−	8.79586i	4.78153	−	2.76062i			14.8590i	−8.94777	+	0.968216i	−10.4074
85.8	−3.00891	+	1.73720i	−1.89239	−	2.32785i	4.03571	−	6.99005i	−4.77681	−	2.75789i	9.73796	+	3.71684i	−5.27007	+	3.04268i			14.1457i	−1.83774	+	8.81037i	19.1640
85.9	−2.88071	+	1.66318i	−1.45467	−	2.62373i	3.53231	−	6.11815i	−5.34559	−	3.08628i	8.55419	+	5.13882i	11.3002	−	6.52416i			10.1940i	−4.76789	+	7.63330i	20.5321
85.10	−2.81609	+	1.62587i	2.40103	+	1.79862i	3.28692	−	5.69310i	4.73991	+	2.73659i	−9.68586	−	1.16131i	−8.85766	+	5.11397i			8.36944i	2.52992	+	8.63710i	−17.7973
85.11	−2.73858	+	1.58112i	2.82092	−	1.02100i	2.99987	−	5.19592i	−1.03697	−	0.598695i	−6.11098	+	7.25628i	−6.65545	+	3.84252i			6.32363i	6.91513	−	5.76029i	3.78643
85.12	−2.67387	+	1.54376i	−2.09965	+	2.14278i	2.76638	−	4.79152i	6.78082	+	3.91491i	2.30625	−	8.97085i	0.957215	−	0.552649i			4.73244i	−0.182974	−	8.99814i	−24.1747
85.13	−2.61200	+	1.50804i	1.31387	−	2.69699i	2.54835	−	4.41388i	3.60296	+	2.08017i	0.635324	+	9.02589i	4.83108	−	2.78923i			3.30774i	−5.54748	−	7.08700i	−12.5479
85.14	−2.56126	+	1.47875i	2.62632	−	1.44999i	2.37338	−	4.11082i	−5.56021	−	3.21019i	−4.58252	+	7.59746i	8.71849	−	5.03362i			2.20855i	4.79507	−	7.61625i	18.9882
85.15	−2.54524	+	1.46949i	−1.80187	+	2.39860i	2.31882	−	4.01631i	−6.19712	−	3.57791i	1.06145	−	8.75283i	0.635505	−	0.366909i			1.87401i	−2.50656	−	8.64391i	21.0309
85.16	−2.54465	+	1.46916i	2.82743	+	1.00280i	2.31684	−	4.01288i	−4.04953	−	2.33799i	−8.66811	+	1.60215i	−3.93144	+	2.26982i			1.86194i	6.98877	+	5.67073i	13.7395
85.17	−2.39791	+	1.38443i	0.276113	−	2.98727i	1.83330	−	3.17537i	4.36934	+	2.52264i	3.47357	+	7.54545i	1.75168	−	1.01133i		−	0.923121i	−8.84752	−	1.64965i	−13.9697
85.18	−2.17345	+	1.25484i	2.81938	+	1.02524i	1.14927	−	1.99059i	3.74018	+	2.15939i	−7.41430	+	1.30956i	4.16320	−	2.40362i		−	4.27015i	6.89775	+	5.78109i	−10.8388
85.19	−2.13572	+	1.23306i	−2.86900	−	0.876822i	1.04087	−	1.80283i	−1.42813	−	0.824532i	7.20856	−	1.66500i	−2.65259	+	1.53147i		−	4.73067i	7.46237	+	5.03121i	4.06678
85.20	−2.08803	+	1.20552i	1.02373	+	2.81992i	0.906577	−	1.57024i	−0.584192	−	0.337283i	−5.53707	−	4.65394i	10.2934	−	5.94288i		−	5.27259i	−6.90393	+	5.77371i	1.62641

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
43.b	odd	2	1	inner
387.s	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	387.3.s.a	✓	172
9.c	even	3	1	inner	387.3.s.a	✓	172
43.b	odd	2	1	inner	387.3.s.a	✓	172
387.s	odd	6	1	inner	387.3.s.a	✓	172

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
387.3.s.a	✓	172	1.a	even	1	1	trivial
387.3.s.a	✓	172	9.c	even	3	1	inner
387.3.s.a	✓	172	43.b	odd	2	1	inner
387.3.s.a	✓	172	387.s	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(387, [\chi])\).