Embedded newform 74.4.c.b.47.5

• Label: 74.4.c.b.47.5
• 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			47.5
Root			\(3.92323 + 6.79524i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.47
Dual form			74.4.c.b.63.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(3.42323 + 5.92921i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(2.94299 + 5.09740i) q^{5} -13.6929 q^{6} +(8.40192 + 14.5526i) q^{7} +8.00000 q^{8} +(-9.93705 + 17.2115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} - 5 q^{3} - 20 q^{4} - q^{5} + 20 q^{6} - q^{7} + 80 q^{8} - 38 q^{9}+O(q^{10}) \)

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
							\(3\)	3.42323	+	5.92921i	0.658801	+	1.14108i	0.980926	+	0.194380i	\(0.0622696\pi\)
−0.322125	+	0.946697i	\(0.604397\pi\)
\(5\)	2.94299	+	5.09740i	0.263229	+	0.455925i	0.967098	−	0.254404i	\(-0.0818793\pi\)
							−0.703869	+	0.710329i	\(0.748546\pi\)
\(7\)	8.40192	+	14.5526i	0.453661	+	0.785764i	0.998610	−	0.0527052i	\(-0.0167844\pi\)
							−0.544949	+	0.838469i	\(0.683451\pi\)
\(11\)	−26.2337			−0.719069			−0.359534	−	0.933132i	\(-0.617064\pi\)
							−0.359534	+	0.933132i	\(0.617064\pi\)
\(13\)	−19.5419	−	33.8476i	−0.416920	−	0.722127i	0.578708	−	0.815535i	\(-0.303557\pi\)
							−0.995628	+	0.0934082i	\(0.970224\pi\)
\(17\)	−11.4147	+	19.7709i	−0.162852	+	0.282068i	−0.935890	−	0.352291i	\(-0.885403\pi\)
							0.773038	+	0.634359i	\(0.218736\pi\)
\(19\)	38.0743	+	65.9466i	0.459728	+	0.796273i	0.998946	−	0.0458936i	\(-0.0146135\pi\)
							−0.539218	+	0.842166i	\(0.681280\pi\)
\(23\)	−40.7683			−0.369599			−0.184799	−	0.982776i	\(-0.559164\pi\)
							−0.184799	+	0.982776i	\(0.559164\pi\)
\(29\)	145.866			0.934022			0.467011	−	0.884252i	\(-0.345331\pi\)
							0.467011	+	0.884252i	\(0.345331\pi\)
\(31\)	242.321			1.40394			0.701969	−	0.712207i	\(-0.252304\pi\)
							0.701969	+	0.712207i	\(0.252304\pi\)
\(37\)	−131.380	−	182.735i	−0.583751	−	0.811933i
							\(41\)	108.622	+	188.140i	0.413755	+	0.716645i	0.995297	−	0.0968713i	\(-0.0308835\pi\)
−0.581542	+	0.813517i	\(0.697550\pi\)
\(43\)	372.125			1.31973			0.659867	−	0.751382i	\(-0.270612\pi\)
							0.659867	+	0.751382i	\(0.270612\pi\)
\(47\)	−83.4851			−0.259097			−0.129548	−	0.991573i	\(-0.541353\pi\)
							−0.129548	+	0.991573i	\(0.541353\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.c.b.47.5	✓	10
3.2	odd	2		666.4.f.d.343.2		10
37.26	even	3	inner	74.4.c.b.63.5	yes	10
111.26	odd	6		666.4.f.d.433.2		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.c.b.47.5	✓	10	1.1	even	1	trivial
74.4.c.b.63.5	yes	10	37.26	even	3	inner
666.4.f.d.343.2		10	3.2	odd	2
666.4.f.d.433.2		10	111.26	odd	6