Embedded newform 150.3.k.b.37.3

• Label: 150.3.k.b.37.3
• Level: $150$
• Weight: $3$
• Character: 150.37
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $48$
• 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:	\(48\)
Relative dimension:	\(6\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			37.3
Character	\(\chi\)	\(=\)	150.37
Dual form			150.3.k.b.73.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.221232 - 1.39680i) q^{2} +(-1.54327 - 0.786335i) q^{3} +(-1.90211 + 0.618034i) q^{4} +(4.33062 + 2.49915i) q^{5} +(-0.756934 + 2.32960i) q^{6} +(5.58271 + 5.58271i) q^{7} +(1.28408 + 2.52015i) q^{8} +(1.76336 + 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{2} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.221232	−	1.39680i	−0.110616	−	0.698401i
							\(3\)	−1.54327	−	0.786335i	−0.514423	−	0.262112i
\(5\)	4.33062	+	2.49915i	0.866123	+	0.499831i
							\(7\)	5.58271	+	5.58271i	0.797530	+	0.797530i	0.982705	−	0.185176i	\(-0.0592854\pi\)
−0.185176	+	0.982705i	\(0.559285\pi\)
\(11\)	5.31002	+	3.85796i	0.482729	+	0.350723i	0.802381	−	0.596812i	\(-0.203566\pi\)
							−0.319652	+	0.947535i	\(0.603566\pi\)
\(13\)	3.64180	−	22.9934i	0.280139	−	1.76873i	−0.299738	−	0.954022i	\(-0.596899\pi\)
							0.579876	−	0.814705i	\(-0.303101\pi\)
\(17\)	−8.72868	+	4.44749i	−0.513452	+	0.261617i	−0.691463	−	0.722412i	\(-0.743034\pi\)
							0.178011	+	0.984028i	\(0.443034\pi\)
\(19\)	17.6112	+	5.72221i	0.926903	+	0.301169i	0.733296	−	0.679910i	\(-0.237981\pi\)
							0.193607	+	0.981079i	\(0.437981\pi\)
\(23\)	34.0240	−	5.38887i	1.47930	−	0.234299i	0.635981	−	0.771705i	\(-0.280596\pi\)
							0.843324	+	0.537406i	\(0.180596\pi\)
\(29\)	−44.0406	+	14.3097i	−1.51864	+	0.493437i	−0.945390	−	0.325942i	\(-0.894319\pi\)
							−0.573253	+	0.819379i	\(0.694319\pi\)
\(31\)	2.01376	−	6.19770i	0.0649599	−	0.199926i	−0.913309	−	0.407268i	\(-0.866481\pi\)
							0.978268	+	0.207342i	\(0.0664814\pi\)
\(37\)	40.3413	+	6.38944i	1.09031	+	0.172688i	0.675600	−	0.737268i	\(-0.263884\pi\)
							0.414706	+	0.909956i	\(0.363884\pi\)
\(41\)	−58.1838	+	42.2730i	−1.41912	+	1.03105i	−0.427200	+	0.904157i	\(0.640500\pi\)
							−0.991917	+	0.126892i	\(0.959500\pi\)
\(43\)	36.5705	−	36.5705i	0.850477	−	0.850477i	−0.139715	−	0.990192i	\(-0.544619\pi\)
							0.990192	+	0.139715i	\(0.0446186\pi\)
\(47\)	−7.64878	+	15.0116i	−0.162740	+	0.319395i	−0.957948	−	0.286941i	\(-0.907362\pi\)
							0.795208	+	0.606336i	\(0.207362\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.b.37.3	✓	48
25.23	odd	20	inner	150.3.k.b.73.3	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.b.37.3	✓	48	1.1	even	1	trivial
150.3.k.b.73.3	yes	48	25.23	odd	20	inner