Embedded newform 150.3.k.a.67.3

• Label: 150.3.k.a.67.3
• Level: $150$
• Weight: $3$
• Character: 150.67
• 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			67.3
Character	\(\chi\)	\(=\)	150.67
Dual form			150.3.k.a.103.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642040 - 1.26007i) q^{2} +(0.270952 - 1.71073i) q^{3} +(-1.17557 - 1.61803i) q^{4} +(-3.90603 + 3.12137i) q^{5} +(-1.98168 - 1.43977i) q^{6} +(-6.00700 - 6.00700i) q^{7} +(-2.79360 + 0.442463i) q^{8} +(-2.85317 - 0.927051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{2} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.642040	−	1.26007i	0.321020	−	0.630037i
							\(3\)	0.270952	−	1.71073i	0.0903175	−	0.570242i
\(5\)	−3.90603	+	3.12137i	−0.781206	+	0.624274i
							\(7\)	−6.00700	−	6.00700i	−0.858143	−	0.858143i	0.132976	−	0.991119i	\(-0.457547\pi\)
−0.991119	+	0.132976i	\(0.957547\pi\)
\(11\)	−5.41078	−	16.6527i	−0.491889	−	1.51388i	−0.821749	−	0.569849i	\(-0.807002\pi\)
							0.329860	−	0.944030i	\(-0.392998\pi\)
\(13\)	−0.892883	−	1.75238i	−0.0686833	−	0.134799i	0.854112	−	0.520089i	\(-0.174101\pi\)
							−0.922795	+	0.385291i	\(0.874101\pi\)
\(17\)	2.71208	+	17.1234i	0.159534	+	1.00726i	0.929406	+	0.369060i	\(0.120320\pi\)
							−0.769872	+	0.638199i	\(0.779680\pi\)
\(19\)	9.77088	−	13.4485i	0.514257	−	0.707814i	−0.470373	−	0.882468i	\(-0.655881\pi\)
							0.984630	+	0.174654i	\(0.0558806\pi\)
\(23\)	22.5711	+	11.5006i	0.981353	+	0.500024i	0.869624	−	0.493714i	\(-0.164361\pi\)
							0.111729	+	0.993739i	\(0.464361\pi\)
\(29\)	11.6592	+	16.0475i	0.402040	+	0.553361i	0.961255	−	0.275662i	\(-0.0888971\pi\)
							−0.559214	+	0.829023i	\(0.688897\pi\)
\(31\)	−45.5739	−	33.1114i	−1.47013	−	1.06811i	−0.980577	−	0.196134i	\(-0.937161\pi\)
							−0.489549	−	0.871976i	\(-0.662839\pi\)
\(37\)	39.3601	−	20.0550i	1.06379	−	0.542026i	0.167668	−	0.985843i	\(-0.446376\pi\)
							0.896117	+	0.443818i	\(0.146376\pi\)
\(41\)	15.1510	−	46.6298i	0.369535	−	1.13731i	−0.577556	−	0.816351i	\(-0.695994\pi\)
							0.947092	−	0.320962i	\(-0.104006\pi\)
\(43\)	16.8137	−	16.8137i	0.391017	−	0.391017i	−0.484033	−	0.875050i	\(-0.660828\pi\)
							0.875050	+	0.484033i	\(0.160828\pi\)
\(47\)	−38.7013	−	6.12969i	−0.823432	−	0.130419i	−0.269521	−	0.962995i	\(-0.586865\pi\)
							−0.553911	+	0.832576i	\(0.686865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.a.67.3	✓	32
25.3	odd	20	inner	150.3.k.a.103.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.a.67.3	✓	32	1.1	even	1	trivial
150.3.k.a.103.3	yes	32	25.3	odd	20	inner