Embedded newform 150.3.k.b.13.5

• Label: 150.3.k.b.13.5
• Level: $150$
• Weight: $3$
• Character: 150.13
• 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			13.5
Character	\(\chi\)	\(=\)	150.13
Dual form			150.3.k.b.127.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39680 + 0.221232i) q^{2} +(0.786335 - 1.54327i) q^{3} +(1.90211 - 0.618034i) q^{4} +(2.03179 + 4.56857i) q^{5} +(-0.756934 + 2.32960i) q^{6} +(-7.48537 + 7.48537i) q^{7} +(-2.52015 + 1.28408i) q^{8} +(-1.76336 - 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{2} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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
\(5\)	2.03179	+	4.56857i	0.406357	+	0.913714i
							\(7\)	−7.48537	+	7.48537i	−1.06934	+	1.06934i	−0.0719282	+	0.997410i	\(0.522915\pi\)
−0.997410	+	0.0719282i	\(0.977085\pi\)
\(11\)	6.98510	+	5.07497i	0.635009	+	0.461361i	0.858132	−	0.513429i	\(-0.171625\pi\)
							−0.223123	+	0.974790i	\(0.571625\pi\)
\(13\)	−2.37886	−	0.376775i	−0.182989	−	0.0289827i	0.0642671	−	0.997933i	\(-0.479529\pi\)
							−0.247257	+	0.968950i	\(0.579529\pi\)
\(17\)	14.2550	+	27.9770i	0.838529	+	1.64570i	0.761034	+	0.648712i	\(0.224692\pi\)
							0.0774946	+	0.996993i	\(0.475308\pi\)
\(19\)	−6.61901	−	2.15065i	−0.348369	−	0.113192i	0.129605	−	0.991566i	\(-0.458629\pi\)
							−0.477975	+	0.878374i	\(0.658629\pi\)
\(23\)	1.07547	+	6.79023i	0.0467594	+	0.295227i	0.999975	−	0.00706380i	\(-0.00224850\pi\)
							−0.953216	+	0.302291i	\(0.902248\pi\)
\(29\)	27.9332	−	9.07604i	0.963213	−	0.312967i	0.215140	−	0.976583i	\(-0.430979\pi\)
							0.748073	+	0.663616i	\(0.230979\pi\)
\(31\)	−4.28751	+	13.1956i	−0.138307	+	0.425664i	−0.996090	−	0.0883478i	\(-0.971841\pi\)
							0.857783	+	0.514012i	\(0.171841\pi\)
\(37\)	3.51753	−	22.2088i	0.0950685	−	0.600239i	−0.893453	−	0.449157i	\(-0.851724\pi\)
							0.988521	−	0.151082i	\(-0.0482756\pi\)
\(41\)	50.3267	−	36.5645i	1.22748	−	0.891817i	0.230782	−	0.973006i	\(-0.425872\pi\)
							0.996699	+	0.0811890i	\(0.0258717\pi\)
\(43\)	6.29869	+	6.29869i	0.146481	+	0.146481i	0.776544	−	0.630063i	\(-0.216971\pi\)
							−0.630063	+	0.776544i	\(0.716971\pi\)
\(47\)	−33.0107	−	16.8198i	−0.702356	−	0.357868i	0.0660628	−	0.997815i	\(-0.478956\pi\)
							−0.768419	+	0.639947i	\(0.778956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.b.13.5	✓	48
25.2	odd	20	inner	150.3.k.b.127.5	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.b.13.5	✓	48	1.1	even	1	trivial
150.3.k.b.127.5	yes	48	25.2	odd	20	inner