Embedded newform 150.6.g.d.61.7

• Label: 150.6.g.d.61.7
• Level: $150$
• Weight: $6$
• Character: 150.61
• Analytic conductor: $24.058$
• Analytic rank: $0$
• Dimension: $28$
• 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:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			61.7
Character	\(\chi\)	\(=\)	150.61
Dual form			150.6.g.d.91.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 + 2.35114i) q^{2} +(-2.78115 + 8.55951i) q^{3} +(4.94427 - 15.2169i) q^{4} +(55.4808 - 6.84686i) q^{5} +(-11.1246 - 34.2380i) q^{6} +85.6043 q^{7} +(19.7771 + 60.8676i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{2} + 63 q^{3} - 112 q^{4} + 11 q^{5} + 252 q^{6} - 62 q^{7} - 448 q^{8} - 567 q^{9}+O(q^{10}) \)

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
\(5\)	55.4808	−	6.84686i	0.992471	−	0.122480i
							\(7\)	85.6043			0.660314			0.330157	−	0.943926i	\(-0.392898\pi\)
0.330157	+	0.943926i	\(0.392898\pi\)
\(11\)	337.570	−	245.259i	0.841168	−	0.611144i	−0.0815291	−	0.996671i	\(-0.525980\pi\)
							0.922697	+	0.385527i	\(0.125980\pi\)
\(13\)	15.9222	+	11.5681i	0.0261303	+	0.0189848i	0.600774	−	0.799419i	\(-0.294859\pi\)
							−0.574643	+	0.818404i	\(0.694859\pi\)
\(17\)	−653.470	−	2011.17i	−0.548407	−	1.68782i	−0.712748	−	0.701420i	\(-0.752550\pi\)
							0.164340	−	0.986404i	\(-0.447450\pi\)
\(19\)	−538.640	−	1657.76i	−0.342306	−	1.05351i	−0.963010	−	0.269465i	\(-0.913153\pi\)
							0.620704	−	0.784045i	\(-0.286847\pi\)
\(23\)	−3018.98	+	2193.42i	−1.18998	+	0.864572i	−0.993262	−	0.115888i	\(-0.963029\pi\)
							−0.196719	+	0.980460i	\(0.563029\pi\)
\(29\)	878.815	−	2704.71i	0.194045	−	0.597209i	−0.805941	−	0.591995i	\(-0.798340\pi\)
							0.999986	−	0.00521372i	\(-0.00165959\pi\)
\(31\)	−2987.49	−	9194.56i	−0.558345	−	1.71841i	−0.686941	−	0.726713i	\(-0.741047\pi\)
							0.128596	−	0.991697i	\(-0.458953\pi\)
\(37\)	8602.07	+	6249.77i	1.03300	+	0.750516i	0.968906	−	0.247428i	\(-0.0795855\pi\)
							0.0640901	+	0.997944i	\(0.479585\pi\)
\(41\)	7799.82	+	5666.90i	0.724644	+	0.526485i	0.887865	−	0.460105i	\(-0.152188\pi\)
							−0.163220	+	0.986590i	\(0.552188\pi\)
\(43\)	14970.3			1.23470			0.617348	−	0.786690i	\(-0.288207\pi\)
							0.617348	+	0.786690i	\(0.288207\pi\)
\(47\)	−2555.74	+	7865.75i	−0.168761	+	0.519393i	−0.999294	−	0.0375771i	\(-0.988036\pi\)
							0.830533	+	0.556970i	\(0.188036\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.d.61.7	✓	28
25.16	even	5	inner	150.6.g.d.91.7	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.d.61.7	✓	28	1.1	even	1	trivial
150.6.g.d.91.7	yes	28	25.16	even	5	inner