Embedded newform 151.2.g.a.76.4

• Label: 151.2.g.a.76.4
• Level: $151$
• Weight: $2$
• Character: 151.76
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $96$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(12\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			76.4
Character	\(\chi\)	\(=\)	151.76
Dual form			151.2.g.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.579506 - 1.00373i) q^{2} +(2.23683 + 1.62515i) q^{3} +(0.328346 - 0.568712i) q^{4} +(2.59512 - 2.88217i) q^{5} +(0.334964 - 3.18697i) q^{6} +(-3.20705 + 3.56179i) q^{7} -3.07914 q^{8} +(1.43524 + 4.41720i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 4 q^{3} - 49 q^{4} - 9 q^{5} + 7 q^{6} - 7 q^{7} + 12 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{8}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.579506	−	1.00373i	−0.409772	−	0.709747i	0.585092	−	0.810967i	\(-0.301059\pi\)
							−0.994864	+	0.101220i	\(0.967725\pi\)
\(3\)	2.23683	+	1.62515i	1.29143	+	0.938282i	0.999833	−	0.0182585i	\(-0.00581218\pi\)
							0.291601	+	0.956540i	\(0.405812\pi\)
\(5\)	2.59512	−	2.88217i	1.16057	−	1.28895i	0.210263	−	0.977645i	\(-0.432568\pi\)
							0.950311	−	0.311303i	\(-0.100765\pi\)
\(7\)	−3.20705	+	3.56179i	−1.21215	+	1.34623i	−0.291144	+	0.956679i	\(0.594036\pi\)
							−0.921006	+	0.389549i	\(0.872631\pi\)
\(11\)	−2.36314	−	1.05214i	−0.712515	−	0.317232i	0.0182764	−	0.999833i	\(-0.494182\pi\)
							−0.730791	+	0.682601i	\(0.760849\pi\)
\(13\)	−0.237336	+	0.105669i	−0.0658252	+	0.0293073i	−0.439385	−	0.898299i	\(-0.644804\pi\)
							0.373560	+	0.927606i	\(0.378137\pi\)
\(17\)	−0.923528	+	0.196302i	−0.223988	+	0.0476102i	−0.318539	−	0.947910i	\(-0.603192\pi\)
							0.0945502	+	0.995520i	\(0.469859\pi\)
\(19\)	−1.00249			−0.229987			−0.114994	−	0.993366i	\(-0.536685\pi\)
							−0.114994	+	0.993366i	\(0.536685\pi\)
\(23\)	3.07236	+	5.32148i	0.640630	+	1.10960i	0.985292	+	0.170877i	\(0.0546602\pi\)
							−0.344662	+	0.938727i	\(0.612006\pi\)
\(29\)	6.26582	−	4.55238i	1.16353	−	0.845357i	0.173313	−	0.984867i	\(-0.444553\pi\)
							0.990221	+	0.139510i	\(0.0445528\pi\)
\(31\)	−4.24631	+	0.902581i	−0.762660	+	0.162108i	−0.572791	−	0.819701i	\(-0.694139\pi\)
							−0.189868	+	0.981810i	\(0.560806\pi\)
\(37\)	−0.549079	+	5.22414i	−0.0902680	+	0.858843i	0.851901	+	0.523704i	\(0.175450\pi\)
							−0.942169	+	0.335139i	\(0.891217\pi\)
\(41\)	−1.12044	−	0.814048i	−0.174983	−	0.127133i	0.496846	−	0.867839i	\(-0.334491\pi\)
							−0.671830	+	0.740706i	\(0.734491\pi\)
\(43\)	−3.09752	−	3.44014i	−0.472367	−	0.524617i	0.459128	−	0.888370i	\(-0.348162\pi\)
							−0.931495	+	0.363753i	\(0.881495\pi\)
\(47\)	2.26562	−	1.00872i	0.330474	−	0.147136i	−0.234791	−	0.972046i	\(-0.575441\pi\)
							0.565265	+	0.824909i	\(0.308774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.g.a.76.4	yes	96
151.2	even	15	inner	151.2.g.a.2.4	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.g.a.2.4	✓	96	151.2	even	15	inner
151.2.g.a.76.4	yes	96	1.1	even	1	trivial