Embedded newform 74.4.f.b.9.3

• Label: 74.4.f.b.9.3
• Level: $74$
• Weight: $4$
• Character: 74.9
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.3
Character	\(\chi\)	\(=\)	74.9
Dual form			74.4.f.b.33.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347296 + 1.96962i) q^{2} +(-0.267400 + 1.51650i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(-13.2707 - 11.1355i) q^{5} -3.07979 q^{6} +(-24.3073 - 20.3962i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(23.1434 + 8.42352i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 6 q^{5} - 36 q^{6} - 75 q^{7} - 120 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.347296	+	1.96962i	0.122788	+	0.696364i
							\(3\)	−0.267400	+	1.51650i	−0.0514611	+	0.291850i	−0.999667	−	0.0258060i	\(-0.991785\pi\)
0.948206	+	0.317656i	\(0.102896\pi\)
\(5\)	−13.2707	−	11.1355i	−1.18697	−	0.995988i	−0.999907	−	0.0136400i	\(-0.995658\pi\)
							−0.187065	−	0.982348i	\(-0.559897\pi\)
\(7\)	−24.3073	−	20.3962i	−1.31247	−	1.10129i	−0.987844	−	0.155451i	\(-0.950317\pi\)
							−0.324626	−	0.945842i	\(-0.605238\pi\)
\(11\)	13.5164	+	23.4111i	0.370487	+	0.641702i	0.989641	−	0.143568i	\(-0.0458575\pi\)
							−0.619154	+	0.785270i	\(0.712524\pi\)
\(13\)	−45.8158	+	16.6756i	−0.977462	+	0.355767i	−0.780853	−	0.624715i	\(-0.785215\pi\)
							−0.196609	+	0.980482i	\(0.562993\pi\)
\(17\)	−78.3235	−	28.5074i	−1.11743	−	0.406710i	−0.283714	−	0.958909i	\(-0.591567\pi\)
							−0.833712	+	0.552199i	\(0.813789\pi\)
\(19\)	8.75074	−	49.6279i	0.105661	−	0.599233i	−0.885293	−	0.465033i	\(-0.846042\pi\)
							0.990954	−	0.134200i	\(-0.0428464\pi\)
\(23\)	−26.8765	+	46.5514i	−0.243658	+	0.422028i	−0.961753	−	0.273917i	\(-0.911681\pi\)
							0.718096	+	0.695944i	\(0.245014\pi\)
\(29\)	−98.9409	−	171.371i	−0.633547	−	1.09734i	−0.986821	−	0.161816i	\(-0.948265\pi\)
							0.353274	−	0.935520i	\(-0.385068\pi\)
\(31\)	30.3258			0.175699			0.0878495	−	0.996134i	\(-0.472001\pi\)
							0.0878495	+	0.996134i	\(0.472001\pi\)
\(37\)	5.93493	+	224.984i	0.0263702	+	0.999652i
							\(41\)	179.248	−	65.2409i	0.682777	−	0.248510i	0.0227373	−	0.999741i	\(-0.492762\pi\)
0.660039	+	0.751231i	\(0.270540\pi\)
\(43\)	−125.026			−0.443401			−0.221700	−	0.975115i	\(-0.571161\pi\)
							−0.221700	+	0.975115i	\(0.571161\pi\)
\(47\)	297.126	−	514.637i	0.922133	−	1.59718i	0.126024	−	0.992027i	\(-0.459778\pi\)
							0.796108	−	0.605154i	\(-0.206888\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.b.9.3	✓	30
37.33	even	9	inner	74.4.f.b.33.3	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.9.3	✓	30	1.1	even	1	trivial
74.4.f.b.33.3	yes	30	37.33	even	9	inner