Newform orbit 363.3.n.a

• Label: 363.3.n.a
• Level: $363$
• Weight: $3$
• Character orbit: 363.n
• Analytic conductor: $9.891$
• Analytic rank: $0$
• Dimension: $3440$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.n (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89103359628\)
Analytic rank:	\(0\)
Dimension:	\(3440\)
Relative dimension:	\(86\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3440 q - 26 q^{3} - 250 q^{4} - 65 q^{6} - 90 q^{7} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−3.31754	−	2.00055i	−1.64773	+	2.50699i	5.13721	+	9.73609i	−2.69199	−	3.29216i	10.4818	−	5.02066i	−0.0790614	+	0.102529i	1.54999	−	27.1062i	−3.56996	−	8.26168i	2.34464	+	16.3073i
5.2	−3.24568	−	1.95722i	2.91003	−	0.729177i	4.83707	+	9.16727i	1.26675	+	1.54918i	−10.8722	−	3.32889i	1.36366	−	1.76843i	1.37725	−	24.0854i	7.93660	−	4.24386i	−1.07941	−	7.50744i
5.3	−3.23086	−	1.94828i	−2.66376	−	1.38000i	4.77598	+	9.05149i	4.90463	+	5.99811i	5.91759	+	9.64833i	5.06993	−	6.57481i	1.34277	−	23.4823i	5.19119	+	7.35197i	−4.16017	−	28.9346i
5.4	−3.22389	−	1.94408i	0.202670	−	2.99315i	4.74737	+	8.99727i	−0.0726671	−	0.0888681i	−6.47230	+	9.25557i	−1.21016	+	1.56936i	1.32670	−	23.2013i	−8.91785	−	1.21324i	0.0615043	+	0.427772i
5.5	−3.20546	−	1.93296i	1.85620	+	2.35681i	4.67196	+	8.85435i	3.62795	+	4.43680i	−1.39434	−	11.1426i	−7.30778	+	9.47689i	1.28458	−	22.4647i	−2.10907	+	8.74939i	−3.05309	−	21.2347i
5.6	−2.97826	−	1.79596i	−2.97143	−	0.413034i	3.77792	+	7.15995i	−3.34568	−	4.09159i	8.10791	+	6.56669i	−1.59918	+	2.07385i	0.813153	−	14.2204i	8.65881	+	2.45460i	2.61598	+	18.1945i
5.7	−2.94331	−	1.77488i	−0.0418778	+	2.99971i	3.64620	+	6.91032i	2.45609	+	3.00367i	5.44739	−	8.75474i	5.00337	−	6.48849i	0.748230	−	13.0850i	−8.99649	−	0.251243i	−1.89787	−	13.2000i
5.8	−2.90486	−	1.75170i	2.92495	−	0.666820i	3.50312	+	6.63916i	−3.55972	−	4.35336i	−9.66466	−	3.18661i	5.18573	−	6.72498i	0.679083	−	11.8758i	8.11070	−	3.90083i	2.71475	+	18.8815i
5.9	−2.75966	−	1.66414i	1.64205	−	2.51071i	2.97970	+	5.64716i	−5.25349	−	6.42475i	−8.70967	+	4.19610i	−6.49355	+	8.42099i	0.438790	−	7.67356i	−3.60732	−	8.24544i	3.80619	+	26.4726i
5.10	−2.75604	−	1.66195i	−2.46031	−	1.71665i	2.96701	+	5.62311i	1.97088	+	2.41029i	3.92772	+	8.82008i	−7.11417	+	9.22582i	0.433217	−	7.57610i	3.10623	+	8.44697i	−1.42605	−	9.91837i
5.11	−2.67332	−	1.61207i	−1.49131	−	2.60307i	2.68121	+	5.08146i	−5.41605	−	6.62355i	−0.209589	+	9.36297i	7.10555	−	9.21465i	0.311071	−	5.44001i	−4.55198	+	7.76398i	3.80121	+	26.4380i
5.12	−2.66681	−	1.60815i	1.91909	+	2.30588i	2.65908	+	5.03951i	−3.15324	−	3.85625i	−1.40967	−	9.23552i	2.35051	−	3.04820i	0.301879	−	5.27926i	−1.63415	+	8.85040i	2.20768	+	15.3548i
5.13	−2.64782	−	1.59669i	2.57721	+	1.53557i	2.59483	+	4.91776i	−0.201699	−	0.246667i	−4.37214	−	8.18093i	−1.44294	+	1.87124i	0.275430	−	4.81671i	4.28402	+	7.91500i	0.140210	+	0.975180i
5.14	−2.52840	−	1.52468i	0.133319	−	2.99704i	2.20149	+	4.17228i	2.22813	+	2.72489i	−4.90661	+	7.37444i	5.37934	−	6.97605i	0.120934	−	2.11489i	−8.96445	−	0.799126i	−1.47902	−	10.2868i
5.15	−2.47416	−	1.49198i	−2.58205	+	1.52742i	2.02883	+	3.84506i	4.19426	+	5.12937i	8.66729	+	0.0732824i	0.570809	−	0.740239i	0.0573139	−	1.00231i	4.33400	−	7.88774i	−2.72440	−	18.9486i
5.16	−2.43018	−	1.46546i	2.61810	−	1.46477i	1.89157	+	3.58492i	6.28815	+	7.69009i	−8.50902	−	0.277042i	−3.69888	+	4.79680i	0.00864436	−	0.151173i	4.70889	−	7.66983i	−4.01190	−	27.9033i
5.17	−2.40500	−	1.45027i	−2.54171	+	1.59364i	1.81408	+	3.43806i	−0.992099	−	1.21329i	8.42403	−	0.146535i	0.415775	−	0.539187i	−0.0180677	+	0.315968i	3.92063	−	8.10115i	0.626408	+	4.35677i
5.18	−2.33613	−	1.40874i	−0.472288	+	2.96259i	1.60628	+	3.04424i	−3.79498	−	4.64107i	5.27683	−	6.25566i	−7.97665	+	10.3443i	−0.0868981	+	1.51967i	−8.55389	−	2.79839i	2.32752	+	16.1882i
5.19	−2.15083	−	1.29700i	−0.283974	−	2.98653i	1.07719	+	2.04151i	2.98915	+	3.65557i	−3.26274	+	6.79183i	−4.32135	+	5.60403i	−0.242570	+	4.24207i	−8.83872	+	1.69620i	−1.68788	−	11.7394i
5.20	−2.03049	−	1.22443i	2.78410	+	1.11748i	0.756991	+	1.43466i	5.33715	+	6.52706i	−4.28482	−	5.67798i	7.27334	−	9.43223i	−0.321878	+	5.62900i	6.50247	+	6.22237i	−2.84510	−	19.7881i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
121.g	even	55	1	inner
363.n	odd	110	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.3.n.a	✓	3440
3.b	odd	2	1	inner	363.3.n.a	✓	3440
121.g	even	55	1	inner	363.3.n.a	✓	3440
363.n	odd	110	1	inner	363.3.n.a	✓	3440

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.3.n.a	✓	3440	1.a	even	1	1	trivial
363.3.n.a	✓	3440	3.b	odd	2	1	inner
363.3.n.a	✓	3440	121.g	even	55	1	inner
363.3.n.a	✓	3440	363.n	odd	110	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(363, [\chi])\).