Embedded newform 150.3.j.a.11.18

• Label: 150.3.j.a.11.18
• Level: $150$
• Weight: $3$
• Character: 150.11
• 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.j (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			11.18
Character	\(\chi\)	\(=\)	150.11
Dual form			150.3.j.a.41.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.831254 + 1.14412i) q^{2} +(1.89250 + 2.32775i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(-4.91294 + 0.928983i) q^{5} +(-1.09009 + 4.10021i) q^{6} -5.64982 q^{7} +(-2.68999 + 0.874032i) q^{8} +(-1.83688 + 8.81055i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.831254	+	1.14412i	0.415627	+	0.572061i
							\(3\)	1.89250	+	2.32775i	0.630834	+	0.775918i
\(5\)	−4.91294	+	0.928983i	−0.982588	+	0.185797i
							\(7\)	−5.64982			−0.807118			−0.403559	−	0.914954i	\(-0.632227\pi\)
−0.403559	+	0.914954i	\(0.632227\pi\)
\(11\)	9.35064	+	12.8700i	0.850058	+	1.17000i	0.983850	+	0.178995i	\(0.0572845\pi\)
							−0.133792	+	0.991009i	\(0.542715\pi\)
\(13\)	−1.90792	−	1.38619i	−0.146763	−	0.106630i	0.511981	−	0.858997i	\(-0.328912\pi\)
							−0.658744	+	0.752367i	\(0.728912\pi\)
\(17\)	27.0787	−	8.79841i	1.59287	−	0.517554i	0.627537	−	0.778587i	\(-0.284063\pi\)
							0.965330	+	0.261033i	\(0.0840631\pi\)
\(19\)	−4.36382	−	13.4305i	−0.229675	−	0.706866i	−0.997783	−	0.0665467i	\(-0.978802\pi\)
							0.768108	−	0.640320i	\(-0.221198\pi\)
\(23\)	26.2815	+	36.1733i	1.14267	+	1.57275i	0.761389	+	0.648295i	\(0.224518\pi\)
							0.381283	+	0.924458i	\(0.375482\pi\)
\(29\)	−21.0526	−	6.84041i	−0.725953	−	0.235876i	−0.0773505	−	0.997004i	\(-0.524646\pi\)
							−0.648602	+	0.761128i	\(0.724646\pi\)
\(31\)	−2.10324	−	6.47312i	−0.0678466	−	0.208810i	0.911385	−	0.411554i	\(-0.135014\pi\)
							−0.979232	+	0.202744i	\(0.935014\pi\)
\(37\)	−12.0598	−	8.76194i	−0.325940	−	0.236809i	0.412766	−	0.910837i	\(-0.364563\pi\)
							−0.738706	+	0.674028i	\(0.764563\pi\)
\(41\)	−12.0302	+	16.5581i	−0.293419	+	0.403857i	−0.930121	−	0.367253i	\(-0.880298\pi\)
							0.636702	+	0.771110i	\(0.280298\pi\)
\(43\)	76.3844			1.77638			0.888191	−	0.459474i	\(-0.151962\pi\)
							0.888191	+	0.459474i	\(0.151962\pi\)
\(47\)	−25.6756	−	8.34249i	−0.546288	−	0.177500i	0.0228540	−	0.999739i	\(-0.492725\pi\)
							−0.569142	+	0.822239i	\(0.692725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.j.a.11.18	yes	80
3.2	odd	2	inner	150.3.j.a.11.1	✓	80
25.16	even	5	inner	150.3.j.a.41.1	yes	80
75.41	odd	10	inner	150.3.j.a.41.18	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.j.a.11.1	✓	80	3.2	odd	2	inner
150.3.j.a.11.18	yes	80	1.1	even	1	trivial
150.3.j.a.41.1	yes	80	25.16	even	5	inner
150.3.j.a.41.18	yes	80	75.41	odd	10	inner