Embedded newform 150.3.i.a.29.16

• Label: 150.3.i.a.29.16
• 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			29.16
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.437016 - 1.34500i) q^{2} +(0.410434 + 2.97179i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(-4.87810 - 1.09731i) q^{5} +(4.17642 + 0.746688i) q^{6} +10.6357i q^{7} +(-2.28825 + 1.66251i) q^{8} +(-8.66309 + 2.43945i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	0.410434	+	2.97179i	0.136811	+	0.990597i
\(5\)	−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	\(0.00781913\pi\)
							0.823210	+	0.567737i	\(0.192181\pi\)
\(13\)	−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	−0.728409	−	0.685142i	\(-0.759740\pi\)
							0.426518	+	0.904479i	\(0.359740\pi\)
\(19\)	−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	−0.931839	−	0.362872i	\(-0.881796\pi\)
							0.967164	+	0.254152i	\(0.0817962\pi\)
\(29\)	22.9279	−	31.5575i	0.790617	−	1.08819i	−0.203414	−	0.979093i	\(-0.565204\pi\)
							0.994031	−	0.109098i	\(-0.0347961\pi\)
\(31\)	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	−0.246492	−	0.969145i	\(-0.579278\pi\)
							0.370233	+	0.928939i	\(0.379278\pi\)
\(43\)			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	0.639903	−	0.768456i	\(-0.278974\pi\)
							−0.533104	+	0.846050i	\(0.678974\pi\)

(See \(a_n\) instead)

Twists

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

    

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