Embedded newform 150.3.k.b.13.6

• Label: 150.3.k.b.13.6
• 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			13.6
Character	\(\chi\)	\(=\)	150.13
Dual form			150.3.k.b.127.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39680 + 0.221232i) q^{2} +(0.786335 - 1.54327i) q^{3} +(1.90211 - 0.618034i) q^{4} +(3.79961 - 3.25008i) q^{5} +(-0.756934 + 2.32960i) q^{6} +(0.904842 - 0.904842i) q^{7} +(-2.52015 + 1.28408i) q^{8} +(-1.76336 - 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{2} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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
\(5\)	3.79961	−	3.25008i	0.759921	−	0.650015i
							\(7\)	0.904842	−	0.904842i	0.129263	−	0.129263i	−0.639515	−	0.768778i	\(-0.720865\pi\)
0.768778	+	0.639515i	\(0.220865\pi\)
\(11\)	−3.26652	−	2.37326i	−0.296956	−	0.215751i	0.429323	−	0.903151i	\(-0.358752\pi\)
							−0.726279	+	0.687400i	\(0.758752\pi\)
\(13\)	4.03564	+	0.639183i	0.310434	+	0.0491679i	0.309708	−	0.950832i	\(-0.399769\pi\)
							0.000726244		1.00000i	\(0.499769\pi\)
\(17\)	−12.9208	−	25.3584i	−0.760045	−	1.49167i	−0.867487	−	0.497460i	\(-0.834266\pi\)
							0.107442	−	0.994211i	\(-0.465734\pi\)
\(19\)	26.0204	+	8.45455i	1.36950	+	0.444976i	0.899202	−	0.437534i	\(-0.144148\pi\)
							0.470294	+	0.882510i	\(0.344148\pi\)
\(23\)	−3.54825	−	22.4028i	−0.154272	−	0.974033i	−0.936405	−	0.350922i	\(-0.885868\pi\)
							0.782133	−	0.623111i	\(-0.214132\pi\)
\(29\)	27.9962	−	9.09653i	0.965388	−	0.313673i	0.216435	−	0.976297i	\(-0.430557\pi\)
							0.748953	+	0.662624i	\(0.230557\pi\)
\(31\)	−12.2998	+	37.8548i	−0.396767	+	1.22112i	0.530810	+	0.847491i	\(0.321888\pi\)
							−0.927577	+	0.373632i	\(0.878112\pi\)
\(37\)	1.53504	−	9.69188i	0.0414877	−	0.261943i	−0.958222	−	0.286024i	\(-0.907666\pi\)
							0.999710	+	0.0240816i	\(0.00766616\pi\)
\(41\)	−37.0427	+	26.9131i	−0.903481	+	0.656417i	−0.939358	−	0.342939i	\(-0.888578\pi\)
							0.0358769	+	0.999356i	\(0.488578\pi\)
\(43\)	9.68415	+	9.68415i	0.225213	+	0.225213i	0.810689	−	0.585477i	\(-0.199093\pi\)
							−0.585477	+	0.810689i	\(0.699093\pi\)
\(47\)	26.7625	+	13.6362i	0.569415	+	0.290131i	0.714897	−	0.699230i	\(-0.246473\pi\)
							−0.145483	+	0.989361i	\(0.546473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.b.13.6	✓	48
25.2	odd	20	inner	150.3.k.b.127.6	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.b.13.6	✓	48	1.1	even	1	trivial
150.3.k.b.127.6	yes	48	25.2	odd	20	inner