Embedded newform 74.4.h.a.3.3

• Label: 74.4.h.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	74.3
Dual form			74.4.h.a.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28558 + 1.53209i) q^{2} +(-1.02557 + 0.860552i) q^{3} +(-0.694593 - 3.93923i) q^{4} +(2.03799 - 5.59934i) q^{5} -2.67756i q^{6} +(27.8721 + 10.1446i) q^{7} +(6.92820 + 4.00000i) q^{8} +(-4.37727 + 24.8247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.02557	+	0.860552i	−0.197370	+	0.165613i	−0.736117	−	0.676855i	\(-0.763343\pi\)
0.538746	+	0.842468i	\(0.318898\pi\)
\(5\)	2.03799	−	5.59934i	0.182284	−	0.500821i	−0.814572	−	0.580063i	\(-0.803028\pi\)
							0.996855	+	0.0792425i	\(0.0252501\pi\)
\(7\)	27.8721	+	10.1446i	1.50495	+	0.547758i	0.957337	−	0.288973i	\(-0.0933139\pi\)
							0.547614	+	0.836731i	\(0.315536\pi\)
\(11\)	−14.5368	+	25.1784i	−0.398455	+	0.690144i	−0.993535	−	0.113522i	\(-0.963787\pi\)
							0.595081	+	0.803666i	\(0.297120\pi\)
\(13\)	3.46536	−	0.611037i	0.0739322	−	0.0130362i	−0.136560	−	0.990632i	\(-0.543605\pi\)
							0.210492	+	0.977596i	\(0.432493\pi\)
\(17\)	50.1772	+	8.84759i	0.715868	+	0.126227i	0.519707	−	0.854345i	\(-0.326041\pi\)
							0.196161	+	0.980572i	\(0.437152\pi\)
\(19\)	63.9715	+	76.2382i	0.772424	+	0.920539i	0.998565	−	0.0535554i	\(-0.0170554\pi\)
							−0.226141	+	0.974095i	\(0.572611\pi\)
\(23\)	67.7553	−	39.1186i	0.614259	−	0.354643i	−0.160371	−	0.987057i	\(-0.551269\pi\)
							0.774631	+	0.632414i	\(0.217936\pi\)
\(29\)	−132.713	−	76.6219i	−0.849800	−	0.490632i	0.0107836	−	0.999942i	\(-0.496567\pi\)
							−0.860583	+	0.509310i	\(0.829901\pi\)
\(31\)		−	259.250i		−	1.50202i	−0.660288	−	0.751012i	\(-0.729566\pi\)
							0.660288	−	0.751012i	\(-0.270434\pi\)
\(37\)	−70.5592	+	213.716i	−0.313510	+	0.949585i
							\(41\)	−11.8581	−	67.2508i	−0.0451690	−	0.256166i	0.953859	−	0.300256i	\(-0.0970722\pi\)
−0.999028	+	0.0440902i	\(0.985961\pi\)
\(43\)		−	29.7365i		−	0.105460i	−0.998609	−	0.0527299i	\(-0.983208\pi\)
							0.998609	−	0.0527299i	\(-0.0167923\pi\)
\(47\)	−300.811	−	521.019i	−0.933569	−	1.61699i	−0.777166	−	0.629295i	\(-0.783344\pi\)
							−0.156403	−	0.987693i	\(-0.549990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.3.3	✓	60
37.25	even	18	inner	74.4.h.a.25.3	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.3.3	✓	60	1.1	even	1	trivial
74.4.h.a.25.3	yes	60	37.25	even	18	inner