Newform orbit 147.4.o.a

• Label: 147.4.o.a
• Level: $147$
• Weight: $4$
• Character orbit: 147.o
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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	−0.818852	−	5.43273i	−4.35577	+	2.83324i	−21.1994	+	6.53916i	−0.438859	+	5.85617i	18.9589	+	21.3437i	15.6934	−	9.83451i	33.8143	+	70.2162i	10.9455	−	24.6819i	32.1744	−	2.41113i
5.2	−0.784114	−	5.20226i	−3.88768	−	3.44760i	−18.8041	+	5.80029i	1.21887	−	16.2647i	−14.8869	+	22.9280i	−13.2361	−	12.9540i	26.6578	+	55.3555i	3.22814	+	26.8063i	−85.5687	+	6.41249i
5.3	−0.770524	−	5.11209i	1.54388	+	4.96150i	−17.8952	+	5.51994i	1.06354	−	14.1919i	24.1740	−	11.7154i	−7.96540	+	16.7198i	24.0622	+	49.9657i	−22.2329	+	15.3199i	−73.3698	+	5.49831i
5.4	−0.767839	−	5.09428i	1.51895	−	4.96918i	−17.7175	+	5.46513i	−1.39020	+	18.5509i	−26.4807	−	3.92241i	−16.9769	−	7.40157i	23.5628	+	48.9286i	−22.3856	−	15.0959i	95.5709	−	7.16205i
5.5	−0.759454	−	5.03865i	5.03986	+	1.26482i	−17.1666	+	5.29521i	−0.854479	+	11.4022i	2.54544	−	26.3547i	17.3640	+	6.44148i	22.0309	+	45.7477i	23.8005	+	12.7490i	58.1008	−	4.35405i
5.6	−0.689180	−	4.57241i	5.15093	−	0.684017i	−12.7874	+	3.94438i	1.04338	−	13.9229i	−6.67752	−	23.0808i	0.682921	−	18.5077i	10.7977	+	22.4216i	26.0642	−	7.04665i	−64.3802	+	4.82463i
5.7	−0.662136	−	4.39299i	−3.87216	−	3.46502i	−11.2153	+	3.45947i	−0.269294	+	3.59347i	−12.6579	+	19.3047i	5.98839	+	17.5254i	7.20285	+	14.9569i	2.98725	+	26.8342i	15.9644	−	1.19637i
5.8	−0.649258	−	4.30754i	4.35143	−	2.83990i	−10.4888	+	3.23537i	0.356756	−	4.76057i	−15.0582	−	16.9002i	−10.5546	+	15.2184i	5.62577	+	11.6820i	10.8699	−	24.7153i	−20.7380	+	1.55410i
5.9	−0.636041	−	4.21986i	1.34771	+	5.01834i	−9.75808	+	3.00997i	−0.672851	+	8.97857i	20.3195	−	8.87899i	−5.80514	−	17.5869i	4.09531	+	8.50399i	−23.3674	+	13.5265i	38.3163	−	2.87141i
5.10	−0.624126	−	4.14080i	−3.21654	+	4.08091i	−9.11213	+	2.81072i	−0.364644	+	4.86583i	18.9058	+	10.7721i	−15.9938	+	9.33802i	2.79039	+	5.79431i	−6.30768	−	26.2529i	20.3760	−	1.52697i
5.11	−0.561990	−	3.72856i	0.548275	−	5.16715i	−5.94175	+	1.83279i	0.187339	−	2.49987i	−19.5741	+	0.859607i	15.6805	−	9.85511i	−2.91542	−	6.05394i	−26.3988	−	5.66604i	−9.42620	+	0.706396i
5.12	−0.530428	−	3.51916i	−4.46523	+	2.65738i	−4.45854	+	1.37528i	1.28581	−	17.1580i	11.7202	+	14.3043i	17.2607	+	6.71337i	−5.14847	−	10.6909i	12.8766	−	23.7317i	−61.0638	+	4.57610i
5.13	−0.457551	−	3.03565i	−5.18186	+	0.385177i	−1.36123	+	0.419884i	0.320431	−	4.27585i	3.54022	+	15.5541i	−14.3135	−	11.7526i	−8.75852	−	18.1872i	26.7033	−	3.99187i	−13.1266	+	0.983703i
5.14	−0.438503	−	2.90928i	−4.93977	−	1.61204i	−0.627036	+	0.193415i	−1.63665	+	21.8396i	−2.52378	+	15.0781i	10.2249	−	15.4418i	−9.37472	−	19.4668i	21.8026	+	15.9263i	64.2550	−	4.81525i
5.15	−0.431465	−	2.86258i	4.61685	+	2.38427i	−0.363637	+	0.112167i	−1.19719	+	15.9753i	4.83316	−	14.2448i	−14.9527	+	10.9278i	−9.57047	−	19.8733i	15.6305	+	22.0156i	46.2473	−	3.46575i
5.16	−0.427111	−	2.83369i	2.80575	+	4.37353i	−0.202821	+	0.0625619i	0.635279	−	8.47721i	11.1949	−	9.81863i	18.5201	−	0.0759273i	−9.68315	−	20.1073i	−11.2555	+	24.5421i	−24.2932	+	1.82052i
5.17	−0.355651	−	2.35959i	0.637236	−	5.15693i	2.20341	−	0.679661i	1.51744	−	20.2488i	−12.3949	+	0.330452i	−12.0884	+	14.0311i	−10.6702	−	22.1569i	−26.1879	−	6.57236i	−48.3185	+	3.62097i
5.18	−0.352491	−	2.33863i	−2.06206	+	4.76948i	2.29965	−	0.709349i	−1.10419	+	14.7344i	11.8809	+	3.14118i	10.5236	+	15.2399i	−10.6787	−	22.1746i	−18.4959	−	19.6699i	34.8476	−	2.61146i
5.19	−0.316876	−	2.10234i	4.92925	+	1.64391i	3.32518	−	1.02568i	0.531903	−	7.09775i	1.89409	−	10.8839i	−14.7008	−	11.2644i	−10.5898	−	21.9899i	21.5951	+	16.2065i	−15.0904	+	1.13087i
5.20	−0.313610	−	2.08066i	4.28367	−	2.94111i	3.41377	−	1.05301i	−0.646097	+	8.62157i	−7.46286	−	7.99053i	12.6820	+	13.4969i	−10.5653	−	21.9390i	9.69974	−	25.1975i	18.1412	−	1.35950i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
49.h	odd	42	1	inner
147.o	even	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.4.o.a	✓	648
3.b	odd	2	1	inner	147.4.o.a	✓	648
49.h	odd	42	1	inner	147.4.o.a	✓	648
147.o	even	42	1	inner	147.4.o.a	✓	648

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
147.4.o.a	✓	648	1.a	even	1	1	trivial
147.4.o.a	✓	648	3.b	odd	2	1	inner
147.4.o.a	✓	648	49.h	odd	42	1	inner
147.4.o.a	✓	648	147.o	even	42	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(147, [\chi])\).