Embedded newform 150.3.k.a.13.3

• Label: 150.3.k.a.13.3
• Level: $150$
• Weight: $3$
• Character: 150.13
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.k (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.08720396540\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	150.13
Dual form			150.3.k.a.127.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39680 - 0.221232i) q^{2} +(0.786335 - 1.54327i) q^{3} +(1.90211 - 0.618034i) q^{4} +(-3.06706 - 3.94881i) q^{5} +(0.756934 - 2.32960i) q^{6} +(2.82371 - 2.82371i) q^{7} +(2.52015 - 1.28408i) q^{8} +(-1.76336 - 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{2} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39680	−	0.221232i	0.698401	−	0.110616i
							\(3\)	0.786335	−	1.54327i	0.262112	−	0.514423i
\(5\)	−3.06706	−	3.94881i	−0.613413	−	0.789762i
							\(7\)	2.82371	−	2.82371i	0.403387	−	0.403387i	−0.476038	−	0.879425i	\(-0.657927\pi\)
0.879425	+	0.476038i	\(0.157927\pi\)
\(11\)	−3.47973	−	2.52817i	−0.316339	−	0.229834i	0.418273	−	0.908322i	\(-0.362636\pi\)
							−0.734612	+	0.678488i	\(0.762636\pi\)
\(13\)	8.74279	+	1.38472i	0.672522	+	0.106517i	0.483350	−	0.875427i	\(-0.339420\pi\)
							0.189172	+	0.981944i	\(0.439420\pi\)
\(17\)	−1.04634	−	2.05355i	−0.0615493	−	0.120797i	0.858182	−	0.513345i	\(-0.171594\pi\)
							−0.919732	+	0.392548i	\(0.871594\pi\)
\(19\)	13.1187	+	4.26252i	0.690457	+	0.224343i	0.633168	−	0.774015i	\(-0.281754\pi\)
							0.0572890	+	0.998358i	\(0.481754\pi\)
\(23\)	4.16922	+	26.3234i	0.181270	+	1.14450i	0.895657	+	0.444745i	\(0.146706\pi\)
							−0.714387	+	0.699751i	\(0.753294\pi\)
\(29\)	10.5664	−	3.43323i	0.364359	−	0.118387i	−0.121115	−	0.992638i	\(-0.538647\pi\)
							0.485474	+	0.874251i	\(0.338647\pi\)
\(31\)	−1.07525	+	3.30929i	−0.0346856	+	0.106751i	−0.966900	−	0.255154i	\(-0.917874\pi\)
							0.932215	+	0.361906i	\(0.117874\pi\)
\(37\)	−0.317055	+	2.00181i	−0.00856905	+	0.0541028i	−0.991602	−	0.129324i	\(-0.958719\pi\)
							0.983033	+	0.183427i	\(0.0587192\pi\)
\(41\)	6.99693	−	5.08357i	0.170657	−	0.123989i	−0.499178	−	0.866499i	\(-0.666365\pi\)
							0.669835	+	0.742510i	\(0.266365\pi\)
\(43\)	42.8426	+	42.8426i	0.996340	+	0.996340i	0.999993	−	0.00365363i	\(-0.00116299\pi\)
							−0.00365363	+	0.999993i	\(0.501163\pi\)
\(47\)	−40.5741	−	20.6735i	−0.863279	−	0.439863i	−0.0344777	−	0.999405i	\(-0.510977\pi\)
							−0.828801	+	0.559543i	\(0.810977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.a.13.3	✓	32
25.2	odd	20	inner	150.3.k.a.127.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.a.13.3	✓	32	1.1	even	1	trivial
150.3.k.a.127.3	yes	32	25.2	odd	20	inner