Embedded newform 150.6.g.b.31.6

• Label: 150.6.g.b.31.6
• Level: $150$
• Weight: $6$
• Character: 150.31
• 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			31.6
Character	\(\chi\)	\(=\)	150.31
Dual form			150.6.g.b.121.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23607 + 3.80423i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(55.5770 + 6.01600i) q^{5} +(-29.1246 - 21.1603i) q^{6} +98.1755 q^{7} +(-51.7771 - 37.6183i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 54 q^{3} - 96 q^{4} + 30 q^{5} - 216 q^{6} - 212 q^{7} - 384 q^{8} - 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.23607	+	3.80423i	0.218508	+	0.672499i
							\(3\)	−7.28115	+	5.29007i	−0.467086	+	0.339358i
\(5\)	55.5770	+	6.01600i	0.994192	+	0.107618i
							\(7\)	98.1755			0.757283			0.378641	−	0.925543i	\(-0.376391\pi\)
0.378641	+	0.925543i	\(0.376391\pi\)
\(11\)	−177.243	−	545.498i	−0.441660	−	1.35929i	−0.886106	−	0.463483i	\(-0.846599\pi\)
							0.444446	−	0.895806i	\(-0.353401\pi\)
\(13\)	262.512	−	807.930i	0.430815	−	1.32591i	−0.466499	−	0.884522i	\(-0.654485\pi\)
							0.897315	−	0.441392i	\(-0.145515\pi\)
\(17\)	1287.73	+	935.592i	1.08069	+	0.785171i	0.977804	−	0.209522i	\(-0.0671908\pi\)
							0.102891	+	0.994693i	\(0.467191\pi\)
\(19\)	−2346.15	−	1704.58i	−1.49098	−	1.08326i	−0.973806	−	0.227381i	\(-0.926984\pi\)
							−0.517175	−	0.855880i	\(-0.673016\pi\)
\(23\)	−831.614	−	2559.45i	−0.327795	−	1.00885i	−0.970163	−	0.242453i	\(-0.922048\pi\)
							0.642368	−	0.766396i	\(-0.277952\pi\)
\(29\)	4248.13	−	3086.45i	0.937999	−	0.681496i	−0.00993884	−	0.999951i	\(-0.503164\pi\)
							0.947938	+	0.318454i	\(0.103164\pi\)
\(31\)	3881.52	+	2820.09i	0.725434	+	0.527059i	0.888116	−	0.459620i	\(-0.152014\pi\)
							−0.162681	+	0.986679i	\(0.552014\pi\)
\(37\)	−3032.69	+	9333.65i	−0.364186	+	1.12085i	0.586303	+	0.810092i	\(0.300583\pi\)
							−0.950489	+	0.310757i	\(0.899417\pi\)
\(41\)	1714.04	−	5275.28i	0.159244	−	0.490101i	−0.839323	−	0.543634i	\(-0.817048\pi\)
							0.998566	+	0.0535324i	\(0.0170480\pi\)
\(43\)	19989.9			1.64869			0.824347	−	0.566085i	\(-0.191542\pi\)
							0.824347	+	0.566085i	\(0.191542\pi\)
\(47\)	8792.08	−	6387.82i	0.580560	−	0.421801i	−0.258366	−	0.966047i	\(-0.583184\pi\)
							0.838926	+	0.544246i	\(0.183184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.b.31.6	✓	24
25.21	even	5	inner	150.6.g.b.121.6	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.b.31.6	✓	24	1.1	even	1	trivial
150.6.g.b.121.6	yes	24	25.21	even	5	inner