Embedded newform 150.3.k.a.73.1

• Label: 150.3.k.a.73.1
• Level: $150$
• Weight: $3$
• Character: 150.73
• 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			73.1
Character	\(\chi\)	\(=\)	150.73
Dual form			150.3.k.a.37.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.221232 - 1.39680i) q^{2} +(-1.54327 + 0.786335i) q^{3} +(-1.90211 - 0.618034i) q^{4} +(-1.48914 + 4.77310i) q^{5} +(0.756934 + 2.32960i) q^{6} +(9.35986 - 9.35986i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(1.76336 - 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{2} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{20}\right)\)

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
\(5\)	−1.48914	+	4.77310i	−0.297827	+	0.954620i
							\(7\)	9.35986	−	9.35986i	1.33712	−	1.33712i	0.438287	−	0.898835i	\(-0.355585\pi\)
0.898835	−	0.438287i	\(-0.144415\pi\)
\(11\)	13.3667	−	9.71148i	1.21515	−	0.882861i	0.219466	−	0.975620i	\(-0.429569\pi\)
							0.995689	+	0.0927588i	\(0.0295685\pi\)
\(13\)	−1.86256	−	11.7598i	−0.143274	−	0.904597i	−0.949677	−	0.313230i	\(-0.898589\pi\)
							0.806403	−	0.591366i	\(-0.201411\pi\)
\(17\)	17.3995	+	8.86551i	1.02350	+	0.521500i	0.883392	−	0.468636i	\(-0.155254\pi\)
							0.140110	+	0.990136i	\(0.455254\pi\)
\(19\)	−16.4119	+	5.33255i	−0.863785	+	0.280661i	−0.707209	−	0.707005i	\(-0.750046\pi\)
							−0.156577	+	0.987666i	\(0.550046\pi\)
\(23\)	17.1062	+	2.70936i	0.743749	+	0.117798i	0.516798	−	0.856108i	\(-0.327124\pi\)
							0.226952	+	0.973906i	\(0.427124\pi\)
\(29\)	7.90234	+	2.56763i	0.272494	+	0.0885388i	0.442077	−	0.896977i	\(-0.354242\pi\)
							−0.169582	+	0.985516i	\(0.554242\pi\)
\(31\)	−0.527352	−	1.62302i	−0.0170114	−	0.0523556i	0.942190	−	0.335078i	\(-0.108763\pi\)
							−0.959202	+	0.282722i	\(0.908763\pi\)
\(37\)	26.7701	−	4.23997i	0.723516	−	0.114594i	0.216194	−	0.976350i	\(-0.430636\pi\)
							0.507322	+	0.861757i	\(0.330636\pi\)
\(41\)	−41.3813	−	30.0653i	−1.00930	−	0.733300i	−0.0452386	−	0.998976i	\(-0.514405\pi\)
							−0.964062	+	0.265676i	\(0.914405\pi\)
\(43\)	16.8295	+	16.8295i	0.391385	+	0.391385i	0.875181	−	0.483796i	\(-0.160742\pi\)
							−0.483796	+	0.875181i	\(0.660742\pi\)
\(47\)	9.38971	+	18.4284i	0.199781	+	0.392093i	0.969062	−	0.246819i	\(-0.0793852\pi\)
							−0.769281	+	0.638911i	\(0.779385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.a.73.1	yes	32
25.12	odd	20	inner	150.3.k.a.37.1	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.a.37.1	✓	32	25.12	odd	20	inner
150.3.k.a.73.1	yes	32	1.1	even	1	trivial