Embedded newform 150.6.g.b.31.4

• Label: 150.6.g.b.31.4
• 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			31.4
Character	\(\chi\)	\(=\)	150.31
Dual form			150.6.g.b.121.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23607 + 3.80423i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(37.7606 + 41.2206i) q^{5} +(-29.1246 - 21.1603i) q^{6} -165.301 q^{7} +(-51.7771 - 37.6183i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 54 q^{3} - 96 q^{4} + 30 q^{5} - 216 q^{6} - 212 q^{7} - 384 q^{8} - 486 q^{9}+O(q^{10}) \)

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
\(5\)	37.7606	+	41.2206i	0.675482	+	0.737377i
							\(7\)	−165.301			−1.27506			−0.637531	−	0.770425i	\(-0.720044\pi\)
−0.637531	+	0.770425i	\(0.720044\pi\)
\(11\)	20.3700	+	62.6924i	0.0507585	+	0.156219i	0.973223	−	0.229864i	\(-0.0738280\pi\)
							−0.922464	+	0.386082i	\(0.873828\pi\)
\(13\)	−74.3515	+	228.830i	−0.122020	+	0.375539i	−0.993346	−	0.115165i	\(-0.963260\pi\)
							0.871326	+	0.490704i	\(0.163260\pi\)
\(17\)	−1329.69	−	966.075i	−1.11591	−	0.810753i	−0.132323	−	0.991207i	\(-0.542244\pi\)
							−0.983584	+	0.180453i	\(0.942244\pi\)
\(19\)	436.138	+	316.873i	0.277166	+	0.201373i	0.717680	−	0.696373i	\(-0.245204\pi\)
							−0.440514	+	0.897746i	\(0.645204\pi\)
\(23\)	−504.570	−	1552.91i	−0.198885	−	0.612104i	−0.999909	−	0.0134727i	\(-0.995711\pi\)
							0.801025	−	0.598632i	\(-0.204289\pi\)
\(29\)	1210.64	−	879.585i	0.267314	−	0.194215i	−0.446051	−	0.895007i	\(-0.647170\pi\)
							0.713365	+	0.700792i	\(0.247170\pi\)
\(31\)	1313.22	+	954.111i	0.245433	+	0.178318i	0.703701	−	0.710497i	\(-0.251530\pi\)
							−0.458267	+	0.888815i	\(0.651530\pi\)
\(37\)	4636.30	−	14269.1i	0.556760	−	1.71353i	−0.134491	−	0.990915i	\(-0.542940\pi\)
							0.691251	−	0.722615i	\(-0.257060\pi\)
\(41\)	4306.67	−	13254.6i	0.400113	−	1.23142i	−0.524795	−	0.851229i	\(-0.675858\pi\)
							0.924908	−	0.380192i	\(-0.124142\pi\)
\(43\)	−12253.3			−1.01060			−0.505302	−	0.862942i	\(-0.668619\pi\)
							−0.505302	+	0.862942i	\(0.668619\pi\)
\(47\)	4533.10	−	3293.49i	0.299330	−	0.217476i	−0.427975	−	0.903791i	\(-0.640773\pi\)
							0.727305	+	0.686315i	\(0.240773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.b.31.4	✓	24
25.21	even	5	inner	150.6.g.b.121.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.b.31.4	✓	24	1.1	even	1	trivial
150.6.g.b.121.4	yes	24	25.21	even	5	inner