Embedded newform 151.2.k.a.90.12

• Label: 151.2.k.a.90.12
• Level: $151$
• Weight: $2$
• Character: 151.90
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.k (of order \(75\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(12\) over \(\Q(\zeta_{75})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{75}]$

Embedding invariants

Embedding label			90.12
Character	\(\chi\)	\(=\)	151.90
Dual form			151.2.k.a.99.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.287949 - 2.73965i) q^{2} +(0.548911 - 0.515462i) q^{3} +(-5.46646 - 1.16193i) q^{4} +(-0.949437 + 0.471305i) q^{5} +(-1.25413 - 1.65225i) q^{6} +(1.63742 - 3.13210i) q^{7} +(-3.05482 + 9.40178i) q^{8} +(-0.152769 + 2.42819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 45 q^{2} - 40 q^{3} + 15 q^{4} - 40 q^{5} - 60 q^{6} - 40 q^{7} - 50 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{22}{75}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.287949	−	2.73965i	0.203610	−	1.93722i	−0.123727	−	0.992316i	\(-0.539485\pi\)
							0.327338	−	0.944907i	\(-0.393849\pi\)
\(3\)	0.548911	−	0.515462i	0.316914	−	0.297602i	−0.509722	−	0.860339i	\(-0.670252\pi\)
							0.826636	+	0.562737i	\(0.190252\pi\)
\(5\)	−0.949437	+	0.471305i	−0.424601	+	0.210774i	−0.644266	−	0.764802i	\(-0.722837\pi\)
							0.219665	+	0.975575i	\(0.429504\pi\)
\(7\)	1.63742	−	3.13210i	0.618887	−	1.18382i	−0.350100	−	0.936712i	\(-0.613852\pi\)
							0.968987	−	0.247110i	\(-0.0794809\pi\)
\(11\)	0.863081	−	3.67977i	0.260229	−	1.10949i	−0.669956	−	0.742401i	\(-0.733687\pi\)
							0.930185	−	0.367091i	\(-0.119646\pi\)
\(13\)	−1.19632	−	0.100458i	−0.331799	−	0.0278619i	−0.0833893	−	0.996517i	\(-0.526574\pi\)
							−0.248409	+	0.968655i	\(0.579908\pi\)
\(17\)	6.40353	+	0.268388i	1.55308	+	0.0650935i	0.802892	−	0.596125i	\(-0.203294\pi\)
							0.750193	+	0.661219i	\(0.229960\pi\)
\(19\)	−0.616282	−	1.89672i	−0.141385	−	0.435138i	0.855144	−	0.518391i	\(-0.173469\pi\)
							−0.996528	+	0.0832534i	\(0.973469\pi\)
\(23\)	2.17254	−	2.41285i	0.453007	−	0.503115i	−0.472770	−	0.881186i	\(-0.656746\pi\)
							0.925777	+	0.378071i	\(0.123413\pi\)
\(29\)	−0.230070	−	1.20607i	−0.0427230	−	0.223962i	0.954522	−	0.298141i	\(-0.0963664\pi\)
							−0.997245	+	0.0741789i	\(0.976366\pi\)
\(31\)	−1.09938	−	0.871357i	−0.197454	−	0.156500i	0.517472	−	0.855700i	\(-0.326873\pi\)
							−0.714926	+	0.699200i	\(0.753540\pi\)
\(37\)	−5.99571	+	8.62670i	−0.985689	+	1.41822i	−0.0787809	+	0.996892i	\(0.525103\pi\)
							−0.906908	+	0.421329i	\(0.861564\pi\)
\(41\)	10.3828	−	1.31165i	1.62151	−	0.204845i	0.738521	−	0.674230i	\(-0.235525\pi\)
							0.882993	+	0.469386i	\(0.155525\pi\)
\(43\)	−0.771276	−	1.47531i	−0.117618	−	0.224983i	0.820226	−	0.572040i	\(-0.193848\pi\)
							−0.937844	+	0.347057i	\(0.887181\pi\)
\(47\)	−2.82638	+	6.72370i	−0.412270	+	0.980753i	0.574482	+	0.818517i	\(0.305204\pi\)
							−0.986752	+	0.162235i	\(0.948130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.k.a.90.12	✓	480
151.99	even	75	inner	151.2.k.a.99.12	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.k.a.90.12	✓	480	1.1	even	1	trivial
151.2.k.a.99.12	yes	480	151.99	even	75	inner