Embedded newform 74.4.h.a.3.10

• Label: 74.4.h.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	74.3
Dual form			74.4.h.a.25.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28558 - 1.53209i) q^{2} +(6.52145 - 5.47215i) q^{3} +(-0.694593 - 3.93923i) q^{4} +(-1.16279 + 3.19474i) q^{5} -17.0263i q^{6} +(12.9120 + 4.69957i) q^{7} +(-6.92820 - 4.00000i) q^{8} +(7.89642 - 44.7828i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

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
							\(3\)	6.52145	−	5.47215i	1.25505	−	1.05312i	0.258863	−	0.965914i	\(-0.416652\pi\)
0.996191	−	0.0872009i	\(-0.0277922\pi\)
\(5\)	−1.16279	+	3.19474i	−0.104003	+	0.285746i	−0.980769	−	0.195171i	\(-0.937474\pi\)
							0.876766	+	0.480917i	\(0.159696\pi\)
\(7\)	12.9120	+	4.69957i	0.697180	+	0.253753i	0.666206	−	0.745767i	\(-0.267917\pi\)
							0.0309738	+	0.999520i	\(0.490139\pi\)
\(11\)	−24.3183	+	42.1205i	−0.666567	+	1.15453i	0.312291	+	0.949987i	\(0.398904\pi\)
							−0.978858	+	0.204542i	\(0.934430\pi\)
\(13\)	−46.4376	+	8.18820i	−0.990729	+	0.174692i	−0.645446	−	0.763806i	\(-0.723328\pi\)
							−0.345283	+	0.938499i	\(0.612217\pi\)
\(17\)	106.871	+	18.8443i	1.52471	+	0.268848i	0.872283	−	0.489002i	\(-0.162639\pi\)
							0.652429	+	0.757850i	\(0.273750\pi\)
\(19\)	−71.6189	−	85.3521i	−0.864763	−	1.03058i	−0.999213	−	0.0396642i	\(-0.987371\pi\)
							0.134450	−	0.990920i	\(-0.457073\pi\)
\(23\)	15.6931	−	9.06041i	0.142271	−	0.0821403i	−0.427175	−	0.904169i	\(-0.640491\pi\)
							0.569446	+	0.822029i	\(0.307158\pi\)
\(29\)	−33.6987	−	19.4559i	−0.215782	−	0.124582i	0.388214	−	0.921569i	\(-0.373092\pi\)
							−0.603996	+	0.796987i	\(0.706426\pi\)
\(31\)			152.428i			0.883126i	0.897230	+	0.441563i	\(0.145576\pi\)
							−0.897230	+	0.441563i	\(0.854424\pi\)
\(37\)	31.7123	+	222.817i	0.140905	+	0.990023i
							\(41\)	−57.9651	−	328.736i	−0.220796	−	1.25220i	−0.870561	−	0.492060i	\(-0.836244\pi\)
0.649765	−	0.760135i	\(-0.274867\pi\)
\(43\)		−	85.7425i		−	0.304084i	−0.988374	−	0.152042i	\(-0.951415\pi\)
							0.988374	−	0.152042i	\(-0.0485849\pi\)
\(47\)	106.338	+	184.182i	0.330020	+	0.571611i	0.982515	−	0.186181i	\(-0.0596112\pi\)
							−0.652495	+	0.757793i	\(0.726278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.3.10	✓	60
37.25	even	18	inner	74.4.h.a.25.10	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.3.10	✓	60	1.1	even	1	trivial
74.4.h.a.25.10	yes	60	37.25	even	18	inner