Embedded newform 150.6.g.d.91.6

• Label: 150.6.g.d.91.6
• Level: $150$
• Weight: $6$
• Character: 150.91
• 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			91.6
Character	\(\chi\)	\(=\)	150.91
Dual form			150.6.g.d.61.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 - 2.35114i) q^{2} +(-2.78115 - 8.55951i) q^{3} +(4.94427 + 15.2169i) q^{4} +(41.0391 + 37.9577i) q^{5} +(-11.1246 + 34.2380i) q^{6} -217.896 q^{7} +(19.7771 - 60.8676i) q^{8} +(-65.5304 + 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{2} + 63 q^{3} - 112 q^{4} + 11 q^{5} + 252 q^{6} - 62 q^{7} - 448 q^{8} - 567 q^{9}+O(q^{10}) \)

Character values

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

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

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
\(5\)	41.0391	+	37.9577i	0.734130	+	0.679009i
							\(7\)	−217.896			−1.68075			−0.840376	−	0.542005i	\(-0.817666\pi\)
−0.840376	+	0.542005i	\(0.817666\pi\)
\(11\)	4.90062	+	3.56051i	0.0122115	+	0.00887218i	0.593874	−	0.804558i	\(-0.297598\pi\)
							−0.581663	+	0.813430i	\(0.697598\pi\)
\(13\)	49.9815	−	36.3137i	0.0820259	−	0.0595953i	−0.546017	−	0.837774i	\(-0.683856\pi\)
							0.628042	+	0.778179i	\(0.283856\pi\)
\(17\)	472.147	−	1453.12i	0.396237	−	1.21949i	−0.531758	−	0.846896i	\(-0.678468\pi\)
							0.927994	−	0.372594i	\(-0.121532\pi\)
\(19\)	−339.610	+	1045.21i	−0.215822	+	0.664232i	0.783272	+	0.621679i	\(0.213549\pi\)
							−0.999094	+	0.0425531i	\(0.986451\pi\)
\(23\)	3027.50	+	2199.61i	1.19334	+	0.867014i	0.993613	−	0.112837i	\(-0.0359939\pi\)
							0.199729	+	0.979851i	\(0.435994\pi\)
\(29\)	813.022	+	2502.22i	0.179518	+	0.552498i	0.999811	−	0.0194447i	\(-0.00618982\pi\)
							−0.820293	+	0.571943i	\(0.806190\pi\)
\(31\)	2747.68	−	8456.49i	0.513526	−	1.58047i	−0.272423	−	0.962178i	\(-0.587825\pi\)
							0.785949	−	0.618292i	\(-0.212175\pi\)
\(37\)	11585.8	−	8417.60i	1.39131	−	1.01084i	0.395586	−	0.918429i	\(-0.370541\pi\)
							0.995721	−	0.0924149i	\(-0.0294586\pi\)
\(41\)	4664.37	−	3388.86i	0.433344	−	0.314843i	−0.349641	−	0.936884i	\(-0.613696\pi\)
							0.782985	+	0.622041i	\(0.213696\pi\)
\(43\)	5419.24			0.446959			0.223479	−	0.974709i	\(-0.428258\pi\)
							0.223479	+	0.974709i	\(0.428258\pi\)
\(47\)	−5776.83	−	17779.3i	−0.381456	−	1.17400i	−0.939018	−	0.343867i	\(-0.888263\pi\)
							0.557562	−	0.830135i	\(-0.311737\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.d.91.6	yes	28
25.11	even	5	inner	150.6.g.d.61.6	✓	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.d.61.6	✓	28	25.11	even	5	inner
150.6.g.d.91.6	yes	28	1.1	even	1	trivial