Embedded newform 74.4.h.a.3.9

• Label: 74.4.h.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	74.3
Dual form			74.4.h.a.25.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28558 - 1.53209i) q^{2} +(2.61104 - 2.19092i) q^{3} +(-0.694593 - 3.93923i) q^{4} +(3.30841 - 9.08979i) q^{5} -6.81693i q^{6} +(-7.17041 - 2.60982i) q^{7} +(-6.92820 - 4.00000i) q^{8} +(-2.67112 + 15.1487i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

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
							\(3\)	2.61104	−	2.19092i	0.502494	−	0.421643i	−0.355985	−	0.934492i	\(-0.615854\pi\)
0.858479	+	0.512849i	\(0.171410\pi\)
\(5\)	3.30841	−	9.08979i	0.295913	−	0.813016i	−0.699258	−	0.714869i	\(-0.746486\pi\)
							0.995172	−	0.0981467i	\(-0.0312914\pi\)
\(7\)	−7.17041	−	2.60982i	−0.387166	−	0.140917i	0.141101	−	0.989995i	\(-0.454936\pi\)
							−0.528267	+	0.849078i	\(0.677158\pi\)
\(11\)	14.0887	−	24.4023i	0.386173	−	0.668871i	−0.605758	−	0.795649i	\(-0.707130\pi\)
							0.991931	+	0.126778i	\(0.0404635\pi\)
\(13\)	41.6256	−	7.33972i	0.888066	−	0.156590i	0.289038	−	0.957318i	\(-0.406665\pi\)
							0.599029	+	0.800728i	\(0.295554\pi\)
\(17\)	−56.1736	−	9.90492i	−0.801417	−	0.141311i	−0.242085	−	0.970255i	\(-0.577831\pi\)
							−0.559332	+	0.828944i	\(0.688942\pi\)
\(19\)	51.2793	+	61.1122i	0.619172	+	0.737901i	0.980928	−	0.194372i	\(-0.0622670\pi\)
							−0.361756	+	0.932273i	\(0.617823\pi\)
\(23\)	13.4358	−	7.75715i	0.121807	−	0.0703251i	−0.437859	−	0.899044i	\(-0.644263\pi\)
							0.559665	+	0.828719i	\(0.310930\pi\)
\(29\)	126.535	+	73.0551i	0.810241	+	0.467793i	0.847040	−	0.531530i	\(-0.178383\pi\)
							−0.0367986	+	0.999323i	\(0.511716\pi\)
\(31\)			18.6218i			0.107890i	0.998544	+	0.0539448i	\(0.0171795\pi\)
							−0.998544	+	0.0539448i	\(0.982820\pi\)
\(37\)	−15.3150	+	224.541i	−0.0680481	+	0.997682i
							\(41\)	43.5279	+	246.859i	0.165803	+	0.940315i	0.948233	+	0.317576i	\(0.102869\pi\)
−0.782430	+	0.622739i	\(0.786020\pi\)
\(43\)			43.4798i			0.154200i	0.997023	+	0.0771001i	\(0.0245661\pi\)
							−0.997023	+	0.0771001i	\(0.975434\pi\)
\(47\)	−156.149	−	270.458i	−0.484609	−	0.839368i	0.515234	−	0.857049i	\(-0.327705\pi\)
							−0.999844	+	0.0176814i	\(0.994372\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.3.9	✓	60
37.25	even	18	inner	74.4.h.a.25.9	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.3.9	✓	60	1.1	even	1	trivial
74.4.h.a.25.9	yes	60	37.25	even	18	inner