Newform orbit 74.4.h.a

• Label: 74.4.h.a
• Level: $74$
• Weight: $4$
• Character orbit: 74.h
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.28558	+	1.53209i	−7.78411	+	6.53164i	−0.694593	−	3.93923i	−6.20210	+	17.0401i		−	20.3229i	7.25695	+	2.64131i	6.92820	+	4.00000i	13.2415	−	75.0961i	−18.1337	−	31.4085i
3.2	−1.28558	+	1.53209i	−3.90885	+	3.27991i	−0.694593	−	3.93923i	3.66320	−	10.0646i		−	10.2053i	−15.5322	−	5.65327i	6.92820	+	4.00000i	−0.167232	+	0.948422i	10.7105	+	18.5511i
3.3	−1.28558	+	1.53209i	−1.02557	+	0.860552i	−0.694593	−	3.93923i	2.03799	−	5.59934i		−	2.67756i	27.8721	+	10.1446i	6.92820	+	4.00000i	−4.37727	+	24.8247i	5.95870	+	10.3208i
3.4	−1.28558	+	1.53209i	0.613106	−	0.514457i	−0.694593	−	3.93923i	−3.29690	+	9.05815i			1.60071i	−17.1096	−	6.22737i	6.92820	+	4.00000i	−4.57727	+	25.9590i	−9.63948	−	16.6961i
3.5	−1.28558	+	1.53209i	6.92790	−	5.81319i	−0.694593	−	3.93923i	−5.26900	+	14.4765i			18.0875i	22.8859	+	8.32979i	6.92820	+	4.00000i	9.51400	−	53.9566i	−15.4055	−	26.6831i
3.6	1.28558	−	1.53209i	−6.88175	+	5.77448i	−0.694593	−	3.93923i	5.49624	−	15.1008i			17.9670i	27.7081	+	10.0849i	−6.92820	−	4.00000i	9.32545	−	52.8873i	−16.0699	−	27.8339i
3.7	1.28558	−	1.53209i	−4.42123	+	3.70986i	−0.694593	−	3.93923i	−1.12359	+	3.08705i			11.5430i	−33.1337	−	12.0597i	−6.92820	−	4.00000i	1.09578	−	6.21447i	3.28516	+	5.69007i
3.8	1.28558	−	1.53209i	−3.00702	+	2.52319i	−0.694593	−	3.93923i	−4.82880	+	13.2670i			7.85076i	21.8021	+	7.93531i	−6.92820	−	4.00000i	−2.01282	+	11.4153i	14.1185	+	24.4539i
3.9	1.28558	−	1.53209i	2.61104	−	2.19092i	−0.694593	−	3.93923i	3.30841	−	9.08979i		−	6.81693i	−7.17041	−	2.60982i	−6.92820	−	4.00000i	−2.67112	+	15.1487i	−9.67315	−	16.7544i
3.10	1.28558	−	1.53209i	6.52145	−	5.47215i	−0.694593	−	3.93923i	−1.16279	+	3.19474i		−	17.0263i	12.9120	+	4.69957i	−6.92820	−	4.00000i	7.89642	−	44.7828i	3.39977	+	5.88857i
21.1	−0.684040	+	1.87939i	−8.01099	+	2.91576i	−3.06418	−	2.57115i	−1.36441	−	0.240582i		−	17.0502i	0.234093	−	1.32761i	6.92820	−	4.00000i	34.9911	−	29.3610i	1.38546	−	2.39968i
21.2	−0.684040	+	1.87939i	−1.59789	+	0.581585i	−3.06418	−	2.57115i	19.8439	+	3.49901i		−	3.40088i	1.06400	−	6.03424i	6.92820	−	4.00000i	−18.4682	+	15.4966i	−20.1500	+	34.9008i
21.3	−0.684040	+	1.87939i	1.13456	−	0.412945i	−3.06418	−	2.57115i	−13.1250	−	2.31430i			2.41474i	5.34436	−	30.3094i	6.92820	−	4.00000i	−19.5665	+	16.4182i	13.3275	−	23.0839i
21.4	−0.684040	+	1.87939i	2.51768	−	0.916361i	−3.06418	−	2.57115i	−10.4603	−	1.84444i			5.35852i	−6.29270	+	35.6877i	6.92820	−	4.00000i	−15.1842	+	12.7411i	10.6217	−	18.3974i
21.5	−0.684040	+	1.87939i	8.12302	−	2.95654i	−3.06418	−	2.57115i	5.91804	+	1.04351i			17.2887i	0.991911	−	5.62541i	6.92820	−	4.00000i	36.5591	−	30.6767i	−6.00933	+	10.4085i
21.6	0.684040	−	1.87939i	−5.71416	+	2.07978i	−3.06418	−	2.57115i	5.14822	+	0.907770i			12.1618i	−3.41363	+	19.3596i	−6.92820	+	4.00000i	7.64290	−	6.41315i	5.22764	−	9.05454i
21.7	0.684040	−	1.87939i	−4.20598	+	1.53085i	−3.06418	−	2.57115i	16.4608	+	2.90248i			8.95182i	5.69922	−	32.3219i	−6.92820	+	4.00000i	−5.33644	+	4.47780i	16.7147	−	28.9507i
21.8	0.684040	−	1.87939i	−1.20040	+	0.436908i	−3.06418	−	2.57115i	−14.3629	−	2.53256i			2.55487i	−1.05161	+	5.96397i	−6.92820	+	4.00000i	−19.4331	+	16.3063i	−14.5844	+	25.2610i
21.9	0.684040	−	1.87939i	6.26608	−	2.28067i	−3.06418	−	2.57115i	17.1326	+	3.02094i		−	13.3365i	−3.13680	+	17.7897i	−6.92820	+	4.00000i	13.3792	−	11.2264i	17.3969	−	30.1323i
21.10	0.684040	−	1.87939i	7.02082	−	2.55537i	−3.06418	−	2.57115i	−7.50303	−	1.32299i		−	14.9428i	3.84603	−	21.8119i	−6.92820	+	4.00000i	22.0789	−	18.5264i	−7.61878	+	13.1961i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.h	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	74.4.h.a	✓	60
37.h	even	18	1	inner	74.4.h.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
74.4.h.a	✓	60	1.a	even	1	1	trivial
74.4.h.a	✓	60	37.h	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(74, [\chi])\).