Embedded newform 150.6.g.d.61.5

• Label: 150.6.g.d.61.5
• Level: $150$
• Weight: $6$
• Character: 150.61
• Analytic conductor: $24.058$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.0575729719\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			61.5
Character	\(\chi\)	\(=\)	150.61
Dual form			150.6.g.d.91.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 + 2.35114i) q^{2} +(-2.78115 + 8.55951i) q^{3} +(4.94427 - 15.2169i) q^{4} +(29.0425 + 47.7654i) q^{5} +(-11.1246 - 34.2380i) q^{6} -12.1985 q^{7} +(19.7771 + 60.8676i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{2} + 63 q^{3} - 112 q^{4} + 11 q^{5} + 252 q^{6} - 62 q^{7} - 448 q^{8} - 567 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.23607	+	2.35114i	−0.572061	+	0.415627i
							\(3\)	−2.78115	+	8.55951i	−0.178411	+	0.549093i
\(5\)	29.0425	+	47.7654i	0.519529	+	0.854453i
							\(7\)	−12.1985			−0.0940939			−0.0470469	−	0.998893i	\(-0.514981\pi\)
−0.0470469	+	0.998893i	\(0.514981\pi\)
\(11\)	−206.610	+	150.111i	−0.514838	+	0.374051i	−0.814656	−	0.579945i	\(-0.803074\pi\)
							0.299818	+	0.953996i	\(0.403074\pi\)
\(13\)	80.3839	+	58.4023i	0.131920	+	0.0958455i	0.651788	−	0.758401i	\(-0.274019\pi\)
							−0.519868	+	0.854246i	\(0.674019\pi\)
\(17\)	505.963	+	1557.19i	0.424616	+	1.30683i	0.903362	+	0.428880i	\(0.141092\pi\)
							−0.478746	+	0.877954i	\(0.658908\pi\)
\(19\)	−8.81470	−	27.1288i	−0.00560175	−	0.0172404i	0.948217	−	0.317625i	\(-0.102885\pi\)
							−0.953818	+	0.300384i	\(0.902885\pi\)
\(23\)	−570.255	+	414.314i	−0.224776	+	0.163309i	−0.694474	−	0.719518i	\(-0.744363\pi\)
							0.469698	+	0.882827i	\(0.344363\pi\)
\(29\)	−1059.46	+	3260.68i	−0.233932	+	0.719969i	0.763329	+	0.646010i	\(0.223563\pi\)
							−0.997261	+	0.0739590i	\(0.976437\pi\)
\(31\)	−1270.06	−	3908.84i	−0.237367	−	0.730539i	−0.996799	−	0.0799526i	\(-0.974523\pi\)
							0.759432	−	0.650587i	\(-0.225477\pi\)
\(37\)	−6998.70	−	5084.85i	−0.840452	−	0.610624i	0.0820450	−	0.996629i	\(-0.473855\pi\)
							−0.922497	+	0.386005i	\(0.873855\pi\)
\(41\)	−5046.51	−	3666.50i	−0.468847	−	0.340637i	0.328145	−	0.944627i	\(-0.393577\pi\)
							−0.796992	+	0.603990i	\(0.793577\pi\)
\(43\)	−4457.32			−0.367623			−0.183812	−	0.982961i	\(-0.558844\pi\)
							−0.183812	+	0.982961i	\(0.558844\pi\)
\(47\)	5384.78	−	16572.6i	0.355569	−	1.09433i	−0.600111	−	0.799917i	\(-0.704877\pi\)
							0.955679	−	0.294410i	\(-0.0951232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.d.61.5	✓	28
25.16	even	5	inner	150.6.g.d.91.5	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.d.61.5	✓	28	1.1	even	1	trivial
150.6.g.d.91.5	yes	28	25.16	even	5	inner