Embedded newform 151.2.h.a.9.11

• Label: 151.2.h.a.9.11
• Level: $151$
• Weight: $2$
• Character: 151.9
• 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			9.11
Character	\(\chi\)	\(=\)	151.9
Dual form			151.2.h.a.84.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.849962 - 2.61591i) q^{2} +(0.0476841 + 0.757917i) q^{3} +(-4.50253 - 3.27128i) q^{4} +(1.44306 - 0.182301i) q^{5} +(2.02318 + 0.519463i) q^{6} +(-1.39396 - 2.96231i) q^{7} +(-7.93392 + 5.76433i) q^{8} +(2.40418 - 0.303718i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 15 q^{2} - 20 q^{3} - 55 q^{4} - 20 q^{5} - 20 q^{7} + 5 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{2}{25}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.849962	−	2.61591i	0.601014	−	1.84973i	0.0788472	−	0.996887i	\(-0.474876\pi\)
							0.522167	−	0.852843i	\(-0.325124\pi\)
\(3\)	0.0476841	+	0.757917i	0.0275304	+	0.437584i	0.987514	+	0.157529i	\(0.0503527\pi\)
							−0.959984	+	0.280055i	\(0.909647\pi\)
\(5\)	1.44306	−	0.182301i	0.645356	−	0.0815275i	0.204159	−	0.978938i	\(-0.434554\pi\)
							0.441197	+	0.897410i	\(0.354554\pi\)
\(7\)	−1.39396	−	2.96231i	−0.526866	−	1.11965i	−0.974284	−	0.225323i	\(-0.927656\pi\)
							0.447418	−	0.894325i	\(-0.352344\pi\)
\(11\)	−0.289014	+	4.59375i	−0.0871411	+	1.38507i	0.675077	+	0.737748i	\(0.264111\pi\)
							−0.762218	+	0.647321i	\(0.775889\pi\)
\(13\)	−0.448952	−	0.542689i	−0.124517	−	0.150515i	0.704571	−	0.709634i	\(-0.251140\pi\)
							−0.829088	+	0.559119i	\(0.811140\pi\)
\(17\)	2.16701	−	4.60514i	0.525578	−	1.11691i	−0.449138	−	0.893462i	\(-0.648269\pi\)
							0.974716	−	0.223448i	\(-0.0717312\pi\)
\(19\)	4.59021	+	3.33498i	1.05307	+	0.765097i	0.972793	−	0.231676i	\(-0.0744208\pi\)
							0.0802728	+	0.996773i	\(0.474421\pi\)
\(23\)	2.09019	+	1.51861i	0.435835	+	0.316653i	0.783978	−	0.620789i	\(-0.213188\pi\)
							−0.348143	+	0.937441i	\(0.613188\pi\)
\(29\)	3.50602	+	1.38813i	0.651051	+	0.257769i	0.670351	−	0.742044i	\(-0.266144\pi\)
							−0.0192995	+	0.999814i	\(0.506144\pi\)
\(31\)	−0.950032	−	0.892139i	−0.170631	−	0.160233i	0.594527	−	0.804075i	\(-0.297339\pi\)
							−0.765158	+	0.643842i	\(0.777339\pi\)
\(37\)	−3.74778	−	4.53029i	−0.616131	−	0.744775i	0.366485	−	0.930424i	\(-0.380561\pi\)
							−0.982616	+	0.185649i	\(0.940561\pi\)
\(41\)	−3.44747	−	0.885160i	−0.538405	−	0.138239i	−0.0303108	−	0.999541i	\(-0.509650\pi\)
							−0.508094	+	0.861302i	\(0.669650\pi\)
\(43\)	−2.95715	+	6.28425i	−0.450960	+	0.958340i	0.542054	+	0.840344i	\(0.317647\pi\)
							−0.993014	+	0.117996i	\(0.962353\pi\)
\(47\)	−7.93146	−	2.03645i	−1.15692	−	0.297047i	−0.378993	−	0.925399i	\(-0.623730\pi\)
							−0.777929	+	0.628352i	\(0.783730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.h.a.9.11	✓	220
151.84	even	25	inner	151.2.h.a.84.11	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.h.a.9.11	✓	220	1.1	even	1	trivial
151.2.h.a.84.11	yes	220	151.84	even	25	inner