Embedded newform 151.2.h.a.72.11

• Label: 151.2.h.a.72.11
• Level: $151$
• Weight: $2$
• Character: 151.72
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.h (of order \(25\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{25})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{25}]$

Embedding invariants

Embedding label			72.11
Character	\(\chi\)	\(=\)	151.72
Dual form			151.2.h.a.86.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.759381 - 2.33713i) q^{2} +(0.877834 - 0.347559i) q^{3} +(-3.26750 - 2.37398i) q^{4} +(0.146769 - 0.137825i) q^{5} +(-0.145682 - 2.31554i) q^{6} +(0.635506 + 0.349372i) q^{7} +(-4.05342 + 2.94498i) q^{8} +(-1.53711 + 1.44344i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 15 q^{2} - 20 q^{3} - 55 q^{4} - 20 q^{5} - 20 q^{7} + 5 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/151\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{12}{25}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.759381	−	2.33713i	0.536963	−	1.65260i	−0.202405	−	0.979302i	\(-0.564876\pi\)
							0.739368	−	0.673301i	\(-0.235124\pi\)
\(3\)	0.877834	−	0.347559i	0.506817	−	0.200663i	−0.100779	−	0.994909i	\(-0.532134\pi\)
							0.607597	+	0.794246i	\(0.292134\pi\)
\(5\)	0.146769	−	0.137825i	0.0656371	−	0.0616373i	−0.651034	−	0.759048i	\(-0.725665\pi\)
							0.716672	+	0.697411i	\(0.245665\pi\)
\(7\)	0.635506	+	0.349372i	0.240199	+	0.132050i	0.597307	−	0.802013i	\(-0.296237\pi\)
							−0.357109	+	0.934063i	\(0.616237\pi\)
\(11\)	2.30369	+	0.912097i	0.694590	+	0.275007i	0.688775	−	0.724976i	\(-0.258149\pi\)
							0.00581509	+	0.999983i	\(0.498149\pi\)
\(13\)	−0.628521	+	0.990391i	−0.174320	+	0.274685i	−0.920243	−	0.391348i	\(-0.872009\pi\)
							0.745922	+	0.666033i	\(0.232009\pi\)
\(17\)	1.58260	−	0.870044i	0.383838	−	0.211017i	−0.278139	−	0.960541i	\(-0.589718\pi\)
							0.661977	+	0.749524i	\(0.269718\pi\)
\(19\)	−0.974683	−	0.708149i	−0.223608	−	0.162460i	0.470341	−	0.882484i	\(-0.344131\pi\)
							−0.693949	+	0.720024i	\(0.744131\pi\)
\(23\)	2.05625	+	1.49395i	0.428757	+	0.311510i	0.781152	−	0.624341i	\(-0.214633\pi\)
							−0.352395	+	0.935851i	\(0.614633\pi\)
\(29\)	−0.922789	+	1.11546i	−0.171358	+	0.207136i	−0.849200	−	0.528072i	\(-0.822915\pi\)
							0.677842	+	0.735207i	\(0.262915\pi\)
\(31\)	−1.14060	−	5.97925i	−0.204858	−	1.07390i	−0.924968	−	0.380045i	\(-0.875909\pi\)
							0.720110	−	0.693860i	\(-0.244091\pi\)
\(37\)	6.30287	−	9.93174i	1.03619	−	1.63277i	0.302671	−	0.953095i	\(-0.402122\pi\)
							0.733515	−	0.679673i	\(-0.237878\pi\)
\(41\)	−0.0786136	−	1.24953i	−0.0122774	−	0.195143i	−0.999384	−	0.0350820i	\(-0.988831\pi\)
							0.987107	−	0.160061i	\(-0.0511692\pi\)
\(43\)	−4.69958	+	2.58362i	−0.716680	+	0.393998i	−0.797974	−	0.602692i	\(-0.794095\pi\)
							0.0812939	+	0.996690i	\(0.474095\pi\)
\(47\)	0.753792	+	11.9812i	0.109952	+	1.74764i	0.537158	+	0.843481i	\(0.319498\pi\)
							−0.427206	+	0.904154i	\(0.640502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.h.a.72.11	✓	220
151.86	even	25	inner	151.2.h.a.86.11	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.h.a.72.11	✓	220	1.1	even	1	trivial
151.2.h.a.86.11	yes	220	151.86	even	25	inner