Embedded newform 74.4.c.b.47.2

• Label: 74.4.c.b.47.2
• 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			47.2
Root			\(-0.858393 - 1.48678i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.47
Dual form			74.4.c.b.63.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-1.35839 - 2.35281i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(0.388081 + 0.672176i) q^{5} +5.43357 q^{6} +(11.9492 + 20.6965i) q^{7} +8.00000 q^{8} +(9.80954 - 16.9906i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} - 5 q^{3} - 20 q^{4} - q^{5} + 20 q^{6} - q^{7} + 80 q^{8} - 38 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.35839	−	2.35281i	−0.261423	−	0.452798i	0.705197	−	0.709011i	\(-0.250858\pi\)
−0.966620	+	0.256213i	\(0.917525\pi\)
\(5\)	0.388081	+	0.672176i	0.0347110	+	0.0601213i	0.882859	−	0.469638i	\(-0.155616\pi\)
							−0.848148	+	0.529759i	\(0.822282\pi\)
\(7\)	11.9492	+	20.6965i	0.645194	+	1.11751i	0.984257	+	0.176745i	\(0.0565567\pi\)
							−0.339063	+	0.940764i	\(0.610110\pi\)
\(11\)	52.4685			1.43817			0.719084	−	0.694923i	\(-0.244561\pi\)
							0.719084	+	0.694923i	\(0.244561\pi\)
\(13\)	34.3127	+	59.4314i	0.732049	+	1.26795i	0.956006	+	0.293347i	\(0.0947692\pi\)
							−0.223957	+	0.974599i	\(0.571897\pi\)
\(17\)	−44.7302	+	77.4750i	−0.638157	+	1.10532i	0.347680	+	0.937613i	\(0.386969\pi\)
							−0.985837	+	0.167707i	\(0.946364\pi\)
\(19\)	−47.2533	−	81.8450i	−0.570560	−	0.988239i	−0.996508	−	0.0834915i	\(-0.973393\pi\)
							0.425948	−	0.904747i	\(-0.359940\pi\)
\(23\)	135.523			1.22863			0.614317	−	0.789059i	\(-0.289432\pi\)
							0.614317	+	0.789059i	\(0.289432\pi\)
\(29\)	172.177			1.10250			0.551248	−	0.834341i	\(-0.314152\pi\)
							0.551248	+	0.834341i	\(0.314152\pi\)
\(31\)	−286.679			−1.66094			−0.830468	−	0.557066i	\(-0.811927\pi\)
							−0.830468	+	0.557066i	\(0.811927\pi\)
\(37\)	184.044	−	129.541i	0.817748	−	0.575577i
							\(41\)	80.2235	+	138.951i	0.305581	+	0.529281i	0.977390	−	0.211443i	\(-0.0678161\pi\)
−0.671810	+	0.740724i	\(0.734483\pi\)
\(43\)	−459.761			−1.63053			−0.815266	−	0.579087i	\(-0.803409\pi\)
							−0.815266	+	0.579087i	\(0.803409\pi\)
\(47\)	−57.4596			−0.178326			−0.0891632	−	0.996017i	\(-0.528419\pi\)
							−0.0891632	+	0.996017i	\(0.528419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.c.b.47.2	✓	10
3.2	odd	2		666.4.f.d.343.4		10
37.26	even	3	inner	74.4.c.b.63.2	yes	10
111.26	odd	6		666.4.f.d.433.4		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.c.b.47.2	✓	10	1.1	even	1	trivial
74.4.c.b.63.2	yes	10	37.26	even	3	inner
666.4.f.d.343.4		10	3.2	odd	2
666.4.f.d.433.4		10	111.26	odd	6