Embedded newform 74.4.c.b.47.3

• Label: 74.4.c.b.47.3
• Level: $74$
• Weight: $4$
• Character: 74.47
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial:	x^10 + 84*x^8 - 140*x^7 + 6309*x^6 - 5214*x^5 + 67648*x^4 + 164178*x^3 + 511389*x^2 + 497502*x + 443556
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			47.3
Root			\(-0.632114 - 1.09485i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.47
Dual form			74.4.c.b.63.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-1.13211 - 1.96088i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(2.10613 + 3.64793i) q^{5} +4.52846 q^{6} +(-10.8960 - 18.8724i) q^{7} +8.00000 q^{8} +(10.9366 - 18.9428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} - 5 q^{3} - 20 q^{4} - q^{5} + 20 q^{6} - q^{7} + 80 q^{8} - 38 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.73205i	−0.353553	+	0.612372i
							\(3\)	−1.13211	−	1.96088i	−0.217875	−	0.377371i	0.736283	−	0.676674i	\(-0.236579\pi\)
−0.954158	+	0.299303i	\(0.903246\pi\)
\(5\)	2.10613	+	3.64793i	0.188378	+	0.326281i	0.944710	−	0.327908i	\(-0.106344\pi\)
							−0.756331	+	0.654189i	\(0.773010\pi\)
\(7\)	−10.8960	−	18.8724i	−0.588327	−	1.01901i	−0.994452	−	0.105194i	\(-0.966454\pi\)
							0.406125	−	0.913818i	\(-0.366880\pi\)
\(11\)	40.2373			1.10291			0.551455	−	0.834205i	\(-0.314073\pi\)
							0.551455	+	0.834205i	\(0.314073\pi\)
\(13\)	−36.3412	−	62.9448i	−0.775325	−	1.34290i	−0.934611	−	0.355670i	\(-0.884253\pi\)
							0.159286	−	0.987232i	\(-0.449081\pi\)
\(17\)	47.5554	−	82.3684i	0.678463	−	1.17513i	−0.296980	−	0.954884i	\(-0.595980\pi\)
							0.975444	−	0.220249i	\(-0.0706871\pi\)
\(19\)	65.0623	+	112.691i	0.785595	+	1.36069i	0.928643	+	0.370975i	\(0.120977\pi\)
							−0.143047	+	0.989716i	\(0.545690\pi\)
\(23\)	−88.1512			−0.799165			−0.399583	−	0.916697i	\(-0.630845\pi\)
							−0.399583	+	0.916697i	\(0.630845\pi\)
\(29\)	−237.554			−1.52112			−0.760562	−	0.649266i	\(-0.775076\pi\)
							−0.760562	+	0.649266i	\(0.775076\pi\)
\(31\)	−138.733			−0.803778			−0.401889	−	0.915688i	\(-0.631646\pi\)
							−0.401889	+	0.915688i	\(0.631646\pi\)
\(37\)	106.472	+	198.285i	0.473077	+	0.881021i
							\(41\)	136.057	+	235.658i	0.518258	+	0.897649i	0.999775	+	0.0212125i	\(0.00675265\pi\)
−0.481517	+	0.876437i	\(0.659914\pi\)
\(43\)	−70.8560			−0.251289			−0.125645	−	0.992075i	\(-0.540100\pi\)
							−0.125645	+	0.992075i	\(0.540100\pi\)
\(47\)	420.442			1.30485			0.652423	−	0.757855i	\(-0.273753\pi\)
							0.652423	+	0.757855i	\(0.273753\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.c.b.47.3	✓	10
3.2	odd	2		666.4.f.d.343.3		10
37.26	even	3	inner	74.4.c.b.63.3	yes	10
111.26	odd	6		666.4.f.d.433.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.c.b.47.3	✓	10	1.1	even	1	trivial
74.4.c.b.63.3	yes	10	37.26	even	3	inner
666.4.f.d.343.3		10	3.2	odd	2
666.4.f.d.433.3		10	111.26	odd	6