Embedded newform 150.6.g.b.31.1

• Label: 150.6.g.b.31.1
• 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			31.1
Character	\(\chi\)	\(=\)	150.31
Dual form			150.6.g.b.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23607 + 3.80423i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(-55.5966 - 5.83204i) q^{5} +(-29.1246 - 21.1603i) q^{6} -184.364 q^{7} +(-51.7771 - 37.6183i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 54 q^{3} - 96 q^{4} + 30 q^{5} - 216 q^{6} - 212 q^{7} - 384 q^{8} - 486 q^{9}+O(q^{10}) \)

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
\(5\)	−55.5966	−	5.83204i	−0.994543	−	0.104327i
							\(7\)	−184.364			−1.42210			−0.711050	−	0.703142i	\(-0.751780\pi\)
−0.711050	+	0.703142i	\(0.751780\pi\)
\(11\)	−0.761061	−	2.34231i	−0.00189643	−	0.00583663i	0.950104	−	0.311934i	\(-0.100977\pi\)
							−0.952000	+	0.306097i	\(0.900977\pi\)
\(13\)	−50.6926	+	156.016i	−0.0831929	+	0.256041i	−0.983997	−	0.178184i	\(-0.942978\pi\)
							0.900804	+	0.434225i	\(0.142978\pi\)
\(17\)	983.698	+	714.698i	0.825542	+	0.599792i	0.918295	−	0.395897i	\(-0.129566\pi\)
							−0.0927522	+	0.995689i	\(0.529566\pi\)
\(19\)	−57.1750	−	41.5401i	−0.0363347	−	0.0263987i	0.569470	−	0.822012i	\(-0.307149\pi\)
							−0.605804	+	0.795614i	\(0.707149\pi\)
\(23\)	−197.717	−	608.509i	−0.0779334	−	0.239854i	0.904498	−	0.426477i	\(-0.140246\pi\)
							−0.982432	+	0.186623i	\(0.940246\pi\)
\(29\)	−580.413	+	421.695i	−0.128157	+	0.0931114i	−0.650017	−	0.759920i	\(-0.725238\pi\)
							0.521860	+	0.853031i	\(0.325238\pi\)
\(31\)	−3425.10	−	2488.48i	−0.640131	−	0.465083i	0.219764	−	0.975553i	\(-0.429471\pi\)
							−0.859895	+	0.510470i	\(0.829471\pi\)
\(37\)	328.461	−	1010.90i	0.0394439	−	0.121396i	−0.929396	−	0.369085i	\(-0.879671\pi\)
							0.968840	+	0.247689i	\(0.0796712\pi\)
\(41\)	806.764	−	2482.97i	0.0749527	−	0.230681i	−0.906560	−	0.422076i	\(-0.861301\pi\)
							0.981513	+	0.191396i	\(0.0613014\pi\)
\(43\)	21189.4			1.74762			0.873811	−	0.486265i	\(-0.161641\pi\)
							0.873811	+	0.486265i	\(0.161641\pi\)
\(47\)	−8436.44	+	6129.43i	−0.557076	+	0.404739i	−0.830388	−	0.557186i	\(-0.811881\pi\)
							0.273312	+	0.961926i	\(0.411881\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.b.31.1	✓	24
25.21	even	5	inner	150.6.g.b.121.1	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.b.31.1	✓	24	1.1	even	1	trivial
150.6.g.b.121.1	yes	24	25.21	even	5	inner