Newform orbit 387.3.bh.a

• Label: 387.3.bh.a
• Level: $387$
• Weight: $3$
• Character orbit: 387.bh
• Analytic conductor: $10.545$
• Analytic rank: $0$
• Dimension: $1032$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	387.bh (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5449862307\)
Analytic rank:	\(0\)
Dimension:	\(1032\)
Relative dimension:	\(86\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1032 q - 7 q^{2} - 11 q^{3} - 173 q^{4} - 7 q^{5} - 10 q^{6} - 28 q^{8} + 39 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
106.1	−0.580137	−	3.84895i	1.11991	+	2.78313i	−10.6556	+	3.28682i	0.820883	+	1.70458i	10.0624	−	5.92507i		−	6.51449i	12.0771	+	25.0783i	−6.49161	+	6.23370i	6.08463	−	4.14843i
106.2	−0.579825	−	3.84688i	−1.28465	−	2.71103i	−10.6400	+	3.28202i	0.134751	+	0.279814i	−9.68415	+	6.51381i			7.70557i	12.0431	+	25.0077i	−5.69936	+	6.96544i	0.998279	−	0.680615i
106.3	−0.570079	−	3.78223i	−2.98557	+	0.293920i	−10.1580	+	3.13331i	−2.11377	−	4.38929i	2.81368	+	11.1245i			1.50584i	11.0034	+	22.8488i	8.82722	−	1.75504i	−15.3963	+	10.4970i
106.4	−0.539599	−	3.58000i	2.50108	−	1.65669i	−8.70297	+	2.68451i	2.70872	+	5.62472i	−7.28052	−	8.05992i			5.14566i	8.02327	+	16.6605i	3.51079	−	8.28700i	18.6749	−	12.7323i
106.5	−0.526085	−	3.49035i	−2.97864	+	0.357348i	−8.08346	+	2.49342i	4.06848	+	8.44829i	2.81429	+	10.2085i		−	5.62075i	6.82944	+	14.1815i	8.74461	−	2.12882i	27.3471	−	18.6449i
106.6	−0.523995	−	3.47648i	2.87601	+	0.853549i	−7.98903	+	2.46429i	0.887031	+	1.84194i	1.46033	−	10.4457i			5.36575i	6.65156	+	13.8121i	7.54291	+	4.90964i	5.93866	−	4.04891i
106.7	−0.510253	−	3.38530i	−1.81037	+	2.39219i	−7.37764	+	2.27570i	−1.59280	−	3.30748i	9.02203	+	4.90803i		−	5.89070i	5.52672	+	11.4764i	−2.44513	−	8.66149i	−10.3841	+	7.07975i
106.8	−0.509405	−	3.37968i	1.19773	+	2.75054i	−7.34046	+	2.26423i	−3.81518	−	7.92229i	8.68581	−	5.44907i			10.6677i	5.45984	+	11.3375i	−6.13090	+	6.58878i	−24.8314	+	16.9297i
106.9	−0.507459	−	3.36677i	2.99929	−	0.0650992i	−7.25533	+	2.23797i	−2.28834	−	4.75178i	−1.74119	−	10.0649i		−	8.57736i	5.30738	+	11.0209i	8.99152	−	0.390503i	−14.8369	+	10.1156i
106.10	−0.501151	−	3.32492i	−1.51651	−	2.58847i	−6.98164	+	2.15355i	1.70088	+	3.53191i	−7.84645	+	6.33950i		−	2.76779i	4.82354	+	10.0162i	−4.40037	+	7.85091i	10.8909	−	7.42531i
106.11	−0.499296	−	3.31261i	1.43674	−	2.63359i	−6.90181	+	2.12893i	−2.70957	−	5.62647i	−9.44141	−	3.44441i			3.02755i	4.68426	+	9.72697i	−4.87158	−	7.56755i	−17.2854	+	11.7850i
106.12	−0.471062	−	3.12529i	−2.09092	+	2.15129i	−5.72326	+	1.76539i	0.482000	+	1.00088i	7.70836	+	5.52135i			13.9443i	2.72806	+	5.66487i	−0.256093	−	8.99636i	2.90100	−	1.97787i
106.13	−0.462008	−	3.06522i	−2.21110	−	2.02757i	−5.35984	+	1.65329i	−2.32877	−	4.83574i	−5.19340	+	7.71426i		−	12.1908i	2.16410	+	4.49380i	0.777932	+	8.96632i	−13.7467	+	9.37235i
106.14	−0.433716	−	2.87752i	−1.36289	+	2.67255i	−4.26970	+	1.31703i	1.08939	+	2.26215i	8.28141	+	2.76262i		−	4.67552i	0.591163	+	1.22756i	−5.28505	−	7.28479i	6.03689	−	4.11588i
106.15	−0.428081	−	2.84013i	1.01441	+	2.82329i	−4.06080	+	1.25259i	3.08528	+	6.40665i	7.58427	−	4.08965i			5.06434i	0.311044	+	0.645889i	−6.94195	+	5.72794i	16.8750	−	11.5052i
106.16	−0.426805	−	2.83166i	0.307668	−	2.98418i	−4.01386	+	1.23811i	2.19272	+	4.55323i	−8.58151	+	0.402451i		−	5.90323i	0.249093	+	0.517247i	−8.81068	−	1.83627i	11.9574	−	8.15239i
106.17	−0.406086	−	2.69420i	−2.89455	−	0.788405i	−3.27153	+	1.00913i	0.408409	+	0.848071i	−0.948687	+	8.11866i			7.41981i	−0.681365	−	1.41487i	7.75684	+	4.56415i	2.11903	−	1.44473i
106.18	−0.353486	−	2.34522i	2.76847	+	1.15566i	−1.55283	+	0.478985i	3.10191	+	6.44118i	1.73167	−	6.90120i		−	11.7491i	−2.44396	−	5.07494i	6.32890	+	6.39883i	14.0095	−	9.55153i
106.19	−0.337018	−	2.23597i	2.53103	−	1.61056i	−1.06368	+	0.328102i	0.735317	+	1.52690i	−4.45415	−	5.11651i		−	1.34380i	−2.83233	−	5.88139i	3.81222	−	8.15273i	3.16629	−	2.15874i
106.20	−0.332339	−	2.20493i	2.66937	+	1.36912i	−0.928959	+	0.286546i	−1.54176	−	3.20150i	2.13166	−	6.34077i			4.69051i	−2.92941	−	6.08298i	5.25105	+	7.30934i	−6.54667	+	4.46345i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
387.bh	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	387.3.bh.a	yes	1032
9.c	even	3	1		387.3.bg.a	✓	1032
43.h	odd	42	1		387.3.bg.a	✓	1032
387.bh	odd	42	1	inner	387.3.bh.a	yes	1032

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
387.3.bg.a	✓	1032	9.c	even	3	1
387.3.bg.a	✓	1032	43.h	odd	42	1
387.3.bh.a	yes	1032	1.a	even	1	1	trivial
387.3.bh.a	yes	1032	387.bh	odd	42	1	inner

Hecke kernels

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