Embedded newform 150.3.i.a.29.14

• Label: 150.3.i.a.29.14
• Level: $150$
• Weight: $3$
• Character: 150.29
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.i (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.08720396540\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(20\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			29.14
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.437016 - 1.34500i) q^{2} +(-1.51513 + 2.58928i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(-2.92277 + 4.05678i) q^{5} +(2.82044 + 3.16940i) q^{6} -10.3258i q^{7} +(-2.28825 + 1.66251i) q^{8} +(-4.40877 - 7.84619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.437016	−	1.34500i	0.218508	−	0.672499i
							\(3\)	−1.51513	+	2.58928i	−0.505043	+	0.863094i
\(5\)	−2.92277	+	4.05678i	−0.584554	+	0.811355i
							\(7\)		−	10.3258i		−	1.47511i	−0.675285	−	0.737557i	\(-0.735979\pi\)
0.675285	−	0.737557i	\(-0.264021\pi\)
\(11\)	−15.3878	−	4.99979i	−1.39889	−	0.454526i	−0.490057	−	0.871690i	\(-0.663024\pi\)
							−0.908831	+	0.417164i	\(0.863024\pi\)
\(13\)	−12.4481	+	4.04463i	−0.957545	+	0.311125i	−0.745778	−	0.666195i	\(-0.767922\pi\)
							−0.211767	+	0.977320i	\(0.567922\pi\)
\(17\)	−1.44783	+	1.05191i	−0.0851666	+	0.0618772i	−0.629554	−	0.776957i	\(-0.716762\pi\)
							0.544387	+	0.838834i	\(0.316762\pi\)
\(19\)	7.67604	−	5.57697i	0.404002	−	0.293525i	−0.367167	−	0.930155i	\(-0.619672\pi\)
							0.771169	+	0.636630i	\(0.219672\pi\)
\(23\)	−10.8632	+	33.4335i	−0.472313	+	1.45363i	0.377236	+	0.926117i	\(0.376875\pi\)
							−0.849548	+	0.527511i	\(0.823125\pi\)
\(29\)	3.88473	−	5.34688i	0.133956	−	0.184375i	−0.736769	−	0.676144i	\(-0.763650\pi\)
							0.870726	+	0.491769i	\(0.163650\pi\)
\(31\)	32.4553	−	23.5802i	1.04695	−	0.760651i	0.0753170	−	0.997160i	\(-0.476003\pi\)
							0.971629	+	0.236509i	\(0.0760031\pi\)
\(37\)	−52.7078	+	17.1258i	−1.42454	+	0.462860i	−0.917040	−	0.398795i	\(-0.869429\pi\)
							−0.507495	+	0.861655i	\(0.669429\pi\)
\(41\)	−43.7419	+	14.2126i	−1.06688	+	0.346649i	−0.789270	−	0.614046i	\(-0.789541\pi\)
							−0.277605	+	0.960695i	\(0.589541\pi\)
\(43\)			8.12640i			0.188986i	0.995526	+	0.0944930i	\(0.0301230\pi\)
							−0.995526	+	0.0944930i	\(0.969877\pi\)
\(47\)	−17.8973	−	13.0031i	−0.380793	−	0.276663i	0.380879	−	0.924625i	\(-0.375621\pi\)
							−0.761672	+	0.647962i	\(0.775621\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.i.a.29.14	yes	80
3.2	odd	2	inner	150.3.i.a.29.4	✓	80
25.19	even	10	inner	150.3.i.a.119.4	yes	80
75.44	odd	10	inner	150.3.i.a.119.14	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.i.a.29.4	✓	80	3.2	odd	2	inner
150.3.i.a.29.14	yes	80	1.1	even	1	trivial
150.3.i.a.119.4	yes	80	25.19	even	10	inner
150.3.i.a.119.14	yes	80	75.44	odd	10	inner