Embedded newform 74.4.h.a.3.1

• Label: 74.4.h.a.3.1
• Level: $74$
• Weight: $4$
• Character: 74.3
• 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:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	74.3
Dual form			74.4.h.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28558 + 1.53209i) q^{2} +(-7.78411 + 6.53164i) q^{3} +(-0.694593 - 3.93923i) q^{4} +(-6.20210 + 17.0401i) q^{5} -20.3229i q^{6} +(7.25695 + 2.64131i) q^{7} +(6.92820 + 4.00000i) q^{8} +(13.2415 - 75.0961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28558	+	1.53209i	−0.454519	+	0.541675i
							\(3\)	−7.78411	+	6.53164i	−1.49805	+	1.25701i	−0.614287	+	0.789082i	\(0.710556\pi\)
−0.883764	+	0.467932i	\(0.844999\pi\)
\(5\)	−6.20210	+	17.0401i	−0.554733	+	1.52412i	0.272443	+	0.962172i	\(0.412168\pi\)
							−0.827176	+	0.561944i	\(0.810054\pi\)
\(7\)	7.25695	+	2.64131i	0.391838	+	0.142617i	0.530423	−	0.847733i	\(-0.322033\pi\)
							−0.138585	+	0.990351i	\(0.544255\pi\)
\(11\)	−10.8734	+	18.8332i	−0.298041	+	0.516222i	−0.975688	−	0.219166i	\(-0.929666\pi\)
							0.677647	+	0.735387i	\(0.263000\pi\)
\(13\)	−8.25908	+	1.45630i	−0.176204	+	0.0310696i	−0.261054	−	0.965324i	\(-0.584070\pi\)
							0.0848498	+	0.996394i	\(0.472959\pi\)
\(17\)	108.954	+	19.2115i	1.55442	+	0.274087i	0.883854	−	0.467762i	\(-0.154940\pi\)
							0.670567	+	0.741849i	\(0.266051\pi\)
\(19\)	−27.4662	−	32.7329i	−0.331641	−	0.395234i	0.574295	−	0.818648i	\(-0.305276\pi\)
							−0.905936	+	0.423414i	\(0.860832\pi\)
\(23\)	−156.398	+	90.2965i	−1.41788	+	0.818614i	−0.996113	−	0.0880890i	\(-0.971924\pi\)
							−0.421769	+	0.906703i	\(0.638591\pi\)
\(29\)	−37.7412	−	21.7899i	−0.241668	−	0.139527i	0.374275	−	0.927318i	\(-0.377892\pi\)
							−0.615943	+	0.787791i	\(0.711225\pi\)
\(31\)			29.9041i			0.173256i	0.996241	+	0.0866281i	\(0.0276092\pi\)
							−0.996241	+	0.0866281i	\(0.972391\pi\)
\(37\)	32.1848	−	222.749i	0.143004	−	0.989722i
							\(41\)	−5.88485	−	33.3746i	−0.0224161	−	0.127128i	0.971546	−	0.236850i	\(-0.0761150\pi\)
−0.993962	+	0.109722i	\(0.965004\pi\)
\(43\)			305.014i			1.08173i	0.841111	+	0.540863i	\(0.181902\pi\)
							−0.841111	+	0.540863i	\(0.818098\pi\)
\(47\)	−186.606	−	323.211i	−0.579134	−	1.00309i	−0.995579	−	0.0939280i	\(-0.970058\pi\)
							0.416445	−	0.909161i	\(-0.363276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.3.1	✓	60
37.25	even	18	inner	74.4.h.a.25.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.3.1	✓	60	1.1	even	1	trivial
74.4.h.a.25.1	yes	60	37.25	even	18	inner