Embedded newform 74.4.c.b.47.1

• Label: 74.4.c.b.47.1
• 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			47.1
Root			\(-4.57142 - 7.91792i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.47
Dual form			74.4.c.b.63.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-5.07142 - 8.78395i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(3.82762 + 6.62963i) q^{5} +20.2857 q^{6} +(-2.78651 - 4.82638i) q^{7} +8.00000 q^{8} +(-37.9385 + 65.7114i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} - 5 q^{3} - 20 q^{4} - q^{5} + 20 q^{6} - q^{7} + 80 q^{8} - 38 q^{9}+O(q^{10}) \)

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
							\(3\)	−5.07142	−	8.78395i	−0.975994	−	1.69047i	−0.676614	−	0.736338i	\(-0.736553\pi\)
−0.299381	−	0.954134i	\(-0.596780\pi\)
\(5\)	3.82762	+	6.62963i	0.342353	+	0.592973i	0.984869	−	0.173299i	\(-0.0554428\pi\)
							−0.642516	+	0.766272i	\(0.722110\pi\)
\(7\)	−2.78651	−	4.82638i	−0.150457	−	0.260600i	0.780938	−	0.624608i	\(-0.214741\pi\)
							−0.931396	+	0.364008i	\(0.881408\pi\)
\(11\)	−66.0355			−1.81004			−0.905020	−	0.425369i	\(-0.860144\pi\)
							−0.905020	+	0.425369i	\(0.860144\pi\)
\(13\)	28.6948	+	49.7008i	0.612192	+	1.06035i	0.990870	+	0.134819i	\(0.0430454\pi\)
							−0.378678	+	0.925528i	\(0.623621\pi\)
\(17\)	−0.639621	+	1.10786i	−0.00912535	+	0.0158056i	−0.870552	−	0.492076i	\(-0.836238\pi\)
							0.861427	+	0.507882i	\(0.169571\pi\)
\(19\)	16.1071	+	27.8984i	0.194486	+	0.336859i	0.946732	−	0.322023i	\(-0.104363\pi\)
							−0.752246	+	0.658882i	\(0.771030\pi\)
\(23\)	−143.964			−1.30516			−0.652578	−	0.757721i	\(-0.726313\pi\)
							−0.652578	+	0.757721i	\(0.726313\pi\)
\(29\)	−102.082			−0.653662			−0.326831	−	0.945083i	\(-0.605981\pi\)
							−0.326831	+	0.945083i	\(0.605981\pi\)
\(31\)	−87.0826			−0.504532			−0.252266	−	0.967658i	\(-0.581176\pi\)
							−0.252266	+	0.967658i	\(0.581176\pi\)
\(37\)	−171.144	−	146.160i	−0.760431	−	0.649418i
							\(41\)	−31.8414	−	55.1509i	−0.121288	−	0.210076i	0.798988	−	0.601347i	\(-0.205369\pi\)
−0.920276	+	0.391271i	\(0.872036\pi\)
\(43\)	−310.394			−1.10080			−0.550402	−	0.834900i	\(-0.685526\pi\)
							−0.550402	+	0.834900i	\(0.685526\pi\)
\(47\)	−52.7493			−0.163708			−0.0818540	−	0.996644i	\(-0.526084\pi\)
							−0.0818540	+	0.996644i	\(0.526084\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.c.b.47.1	✓	10
3.2	odd	2		666.4.f.d.343.1		10
37.26	even	3	inner	74.4.c.b.63.1	yes	10
111.26	odd	6		666.4.f.d.433.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.c.b.47.1	✓	10	1.1	even	1	trivial
74.4.c.b.63.1	yes	10	37.26	even	3	inner
666.4.f.d.343.1		10	3.2	odd	2
666.4.f.d.433.1		10	111.26	odd	6