Embedded newform 150.3.k.b.37.2

• Label: 150.3.k.b.37.2
• 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			37.2
Character	\(\chi\)	\(=\)	150.37
Dual form			150.3.k.b.73.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.221232 - 1.39680i) q^{2} +(-1.54327 - 0.786335i) q^{3} +(-1.90211 + 0.618034i) q^{4} +(1.63241 - 4.72602i) q^{5} +(-0.756934 + 2.32960i) q^{6} +(-1.66389 - 1.66389i) q^{7} +(1.28408 + 2.52015i) q^{8} +(1.76336 + 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{2} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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
\(5\)	1.63241	−	4.72602i	0.326481	−	0.945204i
							\(7\)	−1.66389	−	1.66389i	−0.237698	−	0.237698i	0.578198	−	0.815896i	\(-0.303756\pi\)
−0.815896	+	0.578198i	\(0.803756\pi\)
\(11\)	−15.0189	−	10.9119i	−1.36535	−	0.991988i	−0.998084	−	0.0618744i	\(-0.980292\pi\)
							−0.367271	−	0.930114i	\(-0.619708\pi\)
\(13\)	−2.19795	+	13.8773i	−0.169073	+	1.06748i	0.746516	+	0.665367i	\(0.231725\pi\)
							−0.915589	+	0.402116i	\(0.868275\pi\)
\(17\)	−9.92156	+	5.05529i	−0.583621	+	0.297370i	−0.720763	−	0.693182i	\(-0.756208\pi\)
							0.137142	+	0.990551i	\(0.456208\pi\)
\(19\)	−33.6799	−	10.9433i	−1.77263	−	0.575961i	−0.774248	−	0.632882i	\(-0.781872\pi\)
							−0.998379	+	0.0569208i	\(0.981872\pi\)
\(23\)	44.7573	−	7.08886i	1.94597	−	0.308211i	0.946092	−	0.323899i	\(-0.104994\pi\)
							0.999877	+	0.0156881i	\(0.00499387\pi\)
\(29\)	4.44644	−	1.44473i	0.153325	−	0.0498184i	−0.231349	−	0.972871i	\(-0.574314\pi\)
							0.384674	+	0.923052i	\(0.374314\pi\)
\(31\)	3.88987	−	11.9718i	0.125480	−	0.386187i	−0.868508	−	0.495674i	\(-0.834921\pi\)
							0.993988	+	0.109487i	\(0.0349209\pi\)
\(37\)	−11.5187	−	1.82438i	−0.311315	−	0.0493075i	−0.00117796	−	0.999999i	\(-0.500375\pi\)
							−0.310137	+	0.950692i	\(0.600375\pi\)
\(41\)	25.6503	−	18.6360i	0.625616	−	0.454537i	−0.229263	−	0.973365i	\(-0.573631\pi\)
							0.854879	+	0.518828i	\(0.173631\pi\)
\(43\)	22.2502	−	22.2502i	0.517448	−	0.517448i	−0.399351	−	0.916798i	\(-0.630764\pi\)
							0.916798	+	0.399351i	\(0.130764\pi\)
\(47\)	26.8973	−	52.7889i	0.572283	−	1.12317i	−0.405607	−	0.914048i	\(-0.632940\pi\)
							0.977890	−	0.209121i	\(-0.0670602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.b.37.2	✓	48
25.23	odd	20	inner	150.3.k.b.73.2	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.b.37.2	✓	48	1.1	even	1	trivial
150.3.k.b.73.2	yes	48	25.23	odd	20	inner