Embedded newform 74.4.h.a.3.4

• Label: 74.4.h.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	74.3
Dual form			74.4.h.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28558 + 1.53209i) q^{2} +(0.613106 - 0.514457i) q^{3} +(-0.694593 - 3.93923i) q^{4} +(-3.29690 + 9.05815i) q^{5} +1.60071i q^{6} +(-17.1096 - 6.22737i) q^{7} +(6.92820 + 4.00000i) q^{8} +(-4.57727 + 25.9590i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

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
							\(3\)	0.613106	−	0.514457i	0.117992	−	0.0990074i	−0.581882	−	0.813273i	\(-0.697684\pi\)
0.699875	+	0.714266i	\(0.253239\pi\)
\(5\)	−3.29690	+	9.05815i	−0.294883	+	0.810186i	0.700451	+	0.713701i	\(0.252982\pi\)
							−0.995334	+	0.0964851i	\(0.969240\pi\)
\(7\)	−17.1096	−	6.22737i	−0.923829	−	0.336246i	−0.164068	−	0.986449i	\(-0.552462\pi\)
							−0.759761	+	0.650203i	\(0.774684\pi\)
\(11\)	−2.04871	+	3.54847i	−0.0561555	+	0.0972641i	−0.892737	−	0.450579i	\(-0.851218\pi\)
							0.836581	+	0.547843i	\(0.184551\pi\)
\(13\)	−69.3910	+	12.2355i	−1.48043	+	0.261040i	−0.854750	−	0.519040i	\(-0.826290\pi\)
							−0.625680	+	0.780079i	\(0.715179\pi\)
\(17\)	−59.8733	−	10.5573i	−0.854201	−	0.150619i	−0.270633	−	0.962683i	\(-0.587233\pi\)
							−0.583568	+	0.812064i	\(0.698344\pi\)
\(19\)	−12.1480	−	14.4775i	−0.146681	−	0.174808i	0.687701	−	0.725994i	\(-0.258620\pi\)
							−0.834383	+	0.551186i	\(0.814176\pi\)
\(23\)	60.3507	−	34.8435i	0.547130	−	0.315886i	−0.200833	−	0.979625i	\(-0.564365\pi\)
							0.747964	+	0.663740i	\(0.231032\pi\)
\(29\)	164.804	+	95.1495i	1.05529	+	0.609270i	0.924124	−	0.382092i	\(-0.124796\pi\)
							0.131161	+	0.991361i	\(0.458129\pi\)
\(31\)			281.152i			1.62891i	0.580225	+	0.814457i	\(0.302965\pi\)
							−0.580225	+	0.814457i	\(0.697035\pi\)
\(37\)	223.106	+	29.6096i	0.991308	+	0.131562i
							\(41\)	−57.6585	−	326.998i	−0.219628	−	1.24557i	−0.872692	−	0.488270i	\(-0.837628\pi\)
0.653064	−	0.757302i	\(-0.273483\pi\)
\(43\)		−	187.225i		−	0.663988i	−0.943281	−	0.331994i	\(-0.892279\pi\)
							0.943281	−	0.331994i	\(-0.107721\pi\)
\(47\)	39.4134	+	68.2660i	0.122320	+	0.211864i	0.920682	−	0.390313i	\(-0.127633\pi\)
							−0.798362	+	0.602178i	\(0.794300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.3.4	✓	60
37.25	even	18	inner	74.4.h.a.25.4	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.3.4	✓	60	1.1	even	1	trivial
74.4.h.a.25.4	yes	60	37.25	even	18	inner