Embedded newform 150.6.g.b.61.4

• Label: 150.6.g.b.61.4
• Level: $150$
• Weight: $6$
• Character: 150.61
• Analytic conductor: $24.058$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.0575729719\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			61.4
Character	\(\chi\)	\(=\)	150.61
Dual form			150.6.g.b.91.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 + 2.35114i) q^{2} +(2.78115 - 8.55951i) q^{3} +(4.94427 - 15.2169i) q^{4} +(10.3231 - 54.9403i) q^{5} +(11.1246 + 34.2380i) q^{6} +175.692 q^{7} +(19.7771 + 60.8676i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 54 q^{3} - 96 q^{4} + 30 q^{5} - 216 q^{6} - 212 q^{7} - 384 q^{8} - 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.23607	+	2.35114i	−0.572061	+	0.415627i
							\(3\)	2.78115	−	8.55951i	0.178411	−	0.549093i
\(5\)	10.3231	−	54.9403i	0.184665	−	0.982802i
							\(7\)	175.692			1.35521			0.677606	−	0.735425i	\(-0.263017\pi\)
0.677606	+	0.735425i	\(0.263017\pi\)
\(11\)	201.117	−	146.120i	0.501149	−	0.364106i	−0.308307	−	0.951287i	\(-0.599762\pi\)
							0.809456	+	0.587181i	\(0.199762\pi\)
\(13\)	779.331	+	566.217i	1.27898	+	0.929233i	0.999522	−	0.0309252i	\(-0.00984537\pi\)
							0.279458	+	0.960158i	\(0.409845\pi\)
\(17\)	61.6746	+	189.815i	0.0517587	+	0.159297i	0.973595	−	0.228283i	\(-0.0733111\pi\)
							−0.921836	+	0.387580i	\(0.873311\pi\)
\(19\)	151.706	+	466.903i	0.0964092	+	0.296717i	0.987618	−	0.156875i	\(-0.0501421\pi\)
							−0.891209	+	0.453593i	\(0.850142\pi\)
\(23\)	990.820	−	719.873i	0.390549	−	0.283750i	−0.375132	−	0.926972i	\(-0.622402\pi\)
							0.765680	+	0.643221i	\(0.222402\pi\)
\(29\)	632.817	−	1947.61i	0.139728	−	0.430038i	−0.856567	−	0.516035i	\(-0.827407\pi\)
							0.996295	+	0.0859966i	\(0.0274074\pi\)
\(31\)	−455.299	−	1401.27i	−0.0850928	−	0.261889i	0.899453	−	0.437018i	\(-0.143966\pi\)
							−0.984545	+	0.175130i	\(0.943966\pi\)
\(37\)	5427.89	+	3943.60i	0.651819	+	0.473574i	0.863890	−	0.503680i	\(-0.168021\pi\)
							−0.212071	+	0.977254i	\(0.568021\pi\)
\(41\)	6081.55	+	4418.50i	0.565008	+	0.410502i	0.833288	−	0.552839i	\(-0.186455\pi\)
							−0.268280	+	0.963341i	\(0.586455\pi\)
\(43\)	−5367.13			−0.442661			−0.221330	−	0.975199i	\(-0.571040\pi\)
							−0.221330	+	0.975199i	\(0.571040\pi\)
\(47\)	691.203	−	2127.30i	0.0456416	−	0.140470i	−0.925639	−	0.378408i	\(-0.876472\pi\)
							0.971280	+	0.237938i	\(0.0764715\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.b.61.4	✓	24
25.16	even	5	inner	150.6.g.b.91.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.b.61.4	✓	24	1.1	even	1	trivial
150.6.g.b.91.4	yes	24	25.16	even	5	inner