Embedded newform 150.3.i.a.29.15

• Label: 150.3.i.a.29.15
• 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			29.15
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.437016 - 1.34500i) q^{2} +(-0.277644 - 2.98712i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(4.08750 + 2.87964i) q^{5} +(-4.13901 - 0.931990i) q^{6} -12.3067i q^{7} +(-2.28825 + 1.66251i) q^{8} +(-8.84583 + 1.65872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.277644	−	2.98712i	−0.0925482	−	0.995708i
\(5\)	4.08750	+	2.87964i	0.817500	+	0.575928i
							\(7\)		−	12.3067i		−	1.75810i	−0.476732	−	0.879049i	\(-0.658179\pi\)
0.476732	−	0.879049i	\(-0.341821\pi\)
\(11\)	−0.160129	−	0.0520290i	−0.0145572	−	0.00472991i	0.301729	−	0.953394i	\(-0.402436\pi\)
							−0.316287	+	0.948664i	\(0.602436\pi\)
\(13\)	3.28958	−	1.06885i	0.253044	−	0.0822191i	−0.179748	−	0.983713i	\(-0.557528\pi\)
							0.432792	+	0.901494i	\(0.357528\pi\)
\(17\)	−5.09181	+	3.69941i	−0.299518	+	0.217613i	−0.727386	−	0.686229i	\(-0.759265\pi\)
							0.427868	+	0.903841i	\(0.359265\pi\)
\(19\)	−10.0004	+	7.26573i	−0.526338	+	0.382407i	−0.818986	−	0.573813i	\(-0.805464\pi\)
							0.292648	+	0.956220i	\(0.405464\pi\)
\(23\)	8.44200	−	25.9818i	0.367044	−	1.12964i	−0.581648	−	0.813441i	\(-0.697592\pi\)
							0.948691	−	0.316203i	\(-0.102408\pi\)
\(29\)	−12.7605	+	17.5633i	−0.440018	+	0.605632i	−0.970216	−	0.242241i	\(-0.922117\pi\)
							0.530199	+	0.847874i	\(0.322117\pi\)
\(31\)	35.5750	−	25.8467i	1.14758	−	0.833765i	0.159422	−	0.987210i	\(-0.449037\pi\)
							0.988157	+	0.153445i	\(0.0490369\pi\)
\(37\)	61.6178	−	20.0208i	1.66535	−	0.541104i	0.683364	−	0.730078i	\(-0.260516\pi\)
							0.981982	+	0.188974i	\(0.0605163\pi\)
\(41\)	68.9685	−	22.4092i	1.68216	−	0.546566i	0.696829	−	0.717237i	\(-0.254593\pi\)
							0.985328	+	0.170671i	\(0.0545934\pi\)
\(43\)			41.4339i			0.963580i	0.876287	+	0.481790i	\(0.160013\pi\)
							−0.876287	+	0.481790i	\(0.839987\pi\)
\(47\)	28.7978	+	20.9228i	0.612719	+	0.445167i	0.850371	−	0.526184i	\(-0.176378\pi\)
							−0.237652	+	0.971350i	\(0.576378\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.i.a.29.15	yes	80
3.2	odd	2	inner	150.3.i.a.29.10	✓	80
25.19	even	10	inner	150.3.i.a.119.10	yes	80
75.44	odd	10	inner	150.3.i.a.119.15	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.i.a.29.10	✓	80	3.2	odd	2	inner
150.3.i.a.29.15	yes	80	1.1	even	1	trivial
150.3.i.a.119.10	yes	80	25.19	even	10	inner
150.3.i.a.119.15	yes	80	75.44	odd	10	inner