Embedded newform 150.6.g.c.31.5

• Label: 150.6.g.c.31.5
• 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			31.5
Character	\(\chi\)	\(=\)	150.31
Dual form			150.6.g.c.121.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23607 - 3.80423i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(43.0312 - 35.6835i) q^{5} +(29.1246 + 21.1603i) q^{6} +32.5493 q^{7} +(51.7771 + 37.6183i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} - 54 q^{3} - 96 q^{4} + 80 q^{5} + 216 q^{6} - 454 q^{7} + 384 q^{8} - 486 q^{9}+O(q^{10}) \)

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
\(5\)	43.0312	−	35.6835i	0.769766	−	0.638326i
							\(7\)	32.5493			0.251071			0.125535	−	0.992089i	\(-0.459935\pi\)
0.125535	+	0.992089i	\(0.459935\pi\)
\(11\)	−95.0205	−	292.443i	−0.236775	−	0.728718i	−0.996881	−	0.0789192i	\(-0.974853\pi\)
							0.760106	−	0.649799i	\(-0.225147\pi\)
\(13\)	−262.489	+	807.859i	−0.430778	+	1.32580i	0.466574	+	0.884482i	\(0.345488\pi\)
							−0.897352	+	0.441315i	\(0.854512\pi\)
\(17\)	870.731	+	632.623i	0.730738	+	0.530912i	0.889797	−	0.456357i	\(-0.150846\pi\)
							−0.159059	+	0.987269i	\(0.550846\pi\)
\(19\)	2244.16	+	1630.48i	1.42617	+	1.03617i	0.990714	+	0.135959i	\(0.0434116\pi\)
							0.435452	+	0.900212i	\(0.356588\pi\)
\(23\)	−1426.79	−	4391.21i	−0.562394	−	1.73087i	−0.675570	−	0.737296i	\(-0.736102\pi\)
							0.113176	−	0.993575i	\(-0.463898\pi\)
\(29\)	2748.44	−	1996.86i	0.606863	−	0.440912i	−0.241445	−	0.970414i	\(-0.577621\pi\)
							0.848308	+	0.529503i	\(0.177621\pi\)
\(31\)	−3658.54	−	2658.08i	−0.683760	−	0.496781i	0.190843	−	0.981621i	\(-0.438878\pi\)
							−0.874603	+	0.484840i	\(0.838878\pi\)
\(37\)	1120.42	−	3448.29i	0.134547	−	0.414094i	−0.860972	−	0.508652i	\(-0.830144\pi\)
							0.995519	+	0.0945582i	\(0.0301439\pi\)
\(41\)	2022.72	−	6225.30i	0.187922	−	0.578363i	−0.812065	−	0.583567i	\(-0.801656\pi\)
							0.999986	+	0.00520397i	\(0.00165648\pi\)
\(43\)	14896.8			1.22864			0.614318	−	0.789059i	\(-0.289431\pi\)
							0.614318	+	0.789059i	\(0.289431\pi\)
\(47\)	20027.7	−	14551.0i	1.32247	−	0.960833i	0.322575	−	0.946544i	\(-0.395452\pi\)
							0.999898	−	0.0142885i	\(-0.00454831\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.c.31.5	✓	24
25.21	even	5	inner	150.6.g.c.121.5	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.c.31.5	✓	24	1.1	even	1	trivial
150.6.g.c.121.5	yes	24	25.21	even	5	inner