Embedded newform 150.3.k.b.13.2

• Label: 150.3.k.b.13.2
• 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			13.2
Character	\(\chi\)	\(=\)	150.13
Dual form			150.3.k.b.127.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39680 + 0.221232i) q^{2} +(-0.786335 + 1.54327i) q^{3} +(1.90211 - 0.618034i) q^{4} +(0.283257 - 4.99197i) q^{5} +(0.756934 - 2.32960i) q^{6} +(-6.64753 + 6.64753i) q^{7} +(-2.52015 + 1.28408i) q^{8} +(-1.76336 - 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{2} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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
\(5\)	0.283257	−	4.99197i	0.0566514	−	0.998394i
							\(7\)	−6.64753	+	6.64753i	−0.949648	+	0.949648i	−0.998792	−	0.0491440i	\(-0.984351\pi\)
0.0491440	+	0.998792i	\(0.484351\pi\)
\(11\)	4.00165	+	2.90737i	0.363787	+	0.264307i	0.754630	−	0.656151i	\(-0.227816\pi\)
							−0.390843	+	0.920457i	\(0.627816\pi\)
\(13\)	−20.2108	−	3.20107i	−1.55467	−	0.246236i	−0.680830	−	0.732441i	\(-0.738381\pi\)
							−0.873845	+	0.486205i	\(0.838381\pi\)
\(17\)	−8.04607	−	15.7913i	−0.473298	−	0.928900i	−0.997030	−	0.0770170i	\(-0.975460\pi\)
							0.523731	−	0.851883i	\(-0.324540\pi\)
\(19\)	−23.0111	−	7.47675i	−1.21111	−	0.393513i	−0.367272	−	0.930114i	\(-0.619708\pi\)
							−0.843836	+	0.536601i	\(0.819708\pi\)
\(23\)	0.111675	+	0.705088i	0.00485543	+	0.0306560i	0.989997	−	0.141088i	\(-0.0450599\pi\)
							−0.985142	+	0.171744i	\(0.945060\pi\)
\(29\)	−19.2498	+	6.25465i	−0.663787	+	0.215678i	−0.621484	−	0.783427i	\(-0.713470\pi\)
							−0.0423037	+	0.999105i	\(0.513470\pi\)
\(31\)	3.15087	−	9.69738i	0.101641	−	0.312819i	−0.887286	−	0.461219i	\(-0.847412\pi\)
							0.988927	+	0.148400i	\(0.0474123\pi\)
\(37\)	−5.87948	+	37.1216i	−0.158905	+	1.00329i	0.771360	+	0.636398i	\(0.219577\pi\)
							−0.930265	+	0.366887i	\(0.880423\pi\)
\(41\)	25.2600	−	18.3525i	0.616099	−	0.447622i	−0.235458	−	0.971885i	\(-0.575659\pi\)
							0.851557	+	0.524263i	\(0.175659\pi\)
\(43\)	53.1919	+	53.1919i	1.23702	+	1.23702i	0.961213	+	0.275809i	\(0.0889456\pi\)
							0.275809	+	0.961213i	\(0.411054\pi\)
\(47\)	−35.6998	−	18.1900i	−0.759570	−	0.387020i	0.0308988	−	0.999523i	\(-0.490163\pi\)
							−0.790469	+	0.612502i	\(0.790163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.b.13.2	✓	48
25.2	odd	20	inner	150.3.k.b.127.2	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.b.13.2	✓	48	1.1	even	1	trivial
150.3.k.b.127.2	yes	48	25.2	odd	20	inner