Embedded newform 150.3.i.a.29.1

• Label: 150.3.i.a.29.1
• 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			29.1
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.437016 + 1.34500i) q^{2} +(-2.95317 + 0.527988i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(4.87810 + 1.09731i) q^{5} +(0.580441 - 4.20275i) q^{6} +10.6357i q^{7} +(2.28825 - 1.66251i) q^{8} +(8.44246 - 3.11848i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.95317	+	0.527988i	−0.984391	+	0.175996i
\(5\)	4.87810	+	1.09731i	0.975621	+	0.219463i
							\(7\)			10.6357i			1.51939i	0.650282	+	0.759693i	\(0.274651\pi\)
−0.650282	+	0.759693i	\(0.725349\pi\)
\(11\)	−20.0520	−	6.51529i	−1.82291	−	0.592299i	−0.999698	−	0.0245621i	\(-0.992181\pi\)
							−0.823210	−	0.567737i	\(-0.807819\pi\)
\(13\)	−16.6239	+	5.40145i	−1.27876	+	0.415496i	−0.868145	−	0.496311i	\(-0.834688\pi\)
							−0.410620	+	0.911807i	\(0.634688\pi\)
\(17\)	5.13214	−	3.72872i	0.301891	−	0.219337i	−0.426518	−	0.904479i	\(-0.640260\pi\)
							0.728409	+	0.685142i	\(0.240260\pi\)
\(19\)	−12.8322	+	9.32315i	−0.675380	+	0.490692i	−0.871822	−	0.489823i	\(-0.837061\pi\)
							0.196442	+	0.980515i	\(0.437061\pi\)
\(23\)	−0.812484	+	2.50057i	−0.0353254	+	0.108720i	−0.967164	−	0.254152i	\(-0.918204\pi\)
							0.931839	+	0.362872i	\(0.118204\pi\)
\(29\)	−22.9279	+	31.5575i	−0.790617	+	1.08819i	0.203414	+	0.979093i	\(0.434796\pi\)
							−0.994031	+	0.109098i	\(0.965204\pi\)
\(31\)	17.4027	−	12.6438i	0.561378	−	0.407865i	−0.270585	−	0.962696i	\(-0.587217\pi\)
							0.831963	+	0.554831i	\(0.187217\pi\)
\(37\)	18.2067	−	5.91573i	0.492074	−	0.159885i	−0.0524605	−	0.998623i	\(-0.516706\pi\)
							0.544534	+	0.838738i	\(0.316706\pi\)
\(41\)	−5.07341	+	1.64845i	−0.123742	+	0.0402061i	−0.370233	−	0.928939i	\(-0.620722\pi\)
							0.246492	+	0.969145i	\(0.420722\pi\)
\(43\)			48.0577i			1.11762i	0.829295	+	0.558811i	\(0.188742\pi\)
							−0.829295	+	0.558811i	\(0.811258\pi\)
\(47\)	−5.01957	−	3.64693i	−0.106799	−	0.0775943i	0.533104	−	0.846050i	\(-0.321026\pi\)
							−0.639903	+	0.768456i	\(0.721026\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.i.a.29.1	✓	80
3.2	odd	2	inner	150.3.i.a.29.16	yes	80
25.19	even	10	inner	150.3.i.a.119.16	yes	80
75.44	odd	10	inner	150.3.i.a.119.1	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.i.a.29.1	✓	80	1.1	even	1	trivial
150.3.i.a.29.16	yes	80	3.2	odd	2	inner
150.3.i.a.119.1	yes	80	75.44	odd	10	inner
150.3.i.a.119.16	yes	80	25.19	even	10	inner