Embedded newform 91.2.r.a.25.4

• Label: 91.2.r.a.25.4
• Level: $91$
• Weight: $2$
• Character: 91.25
• Analytic conductor: $0.727$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	91.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.726638658394\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			25.4
Root			\(0.287846 + 0.166188i\) of defining polynomial
Character	\(\chi\)	\(=\)	91.25
Dual form			91.2.r.a.51.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.287846 + 0.166188i) q^{2} +(-0.729919 + 1.26426i) q^{3} +(-0.944763 + 1.63638i) q^{4} +(-1.25195 + 0.722811i) q^{5} -0.485214i q^{6} +(-2.26391 - 1.36920i) q^{7} -1.29278i q^{8} +(0.434437 + 0.752468i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} + 6 q^{4} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.287846	+	0.166188i	−0.203538	+	0.117512i	−0.598304	−	0.801269i	\(-0.704159\pi\)
							0.394767	+	0.918781i	\(0.370825\pi\)
\(3\)	−0.729919	+	1.26426i	−0.421419	+	0.729919i	−0.996079	−	0.0884737i	\(-0.971801\pi\)
							0.574660	+	0.818392i	\(0.305134\pi\)
\(5\)	−1.25195	+	0.722811i	−0.559887	+	0.323251i	−0.753100	−	0.657906i	\(-0.771442\pi\)
							0.193213	+	0.981157i	\(0.438109\pi\)
\(7\)	−2.26391	−	1.36920i	−0.855677	−	0.517510i
							\(11\)	5.15732	+	2.97758i	1.55499	+	0.897774i	0.997723	+	0.0674405i	\(0.0214833\pi\)
0.557267	+	0.830333i	\(0.311850\pi\)
\(13\)	1.88953	+	3.07078i	0.524060	+	0.851681i
							\(17\)	2.16436	−	3.74877i	0.524933	−	0.909211i	−0.474645	−	0.880177i	\(-0.657424\pi\)
0.999578	−	0.0290341i	\(-0.00924314\pi\)
\(19\)	−1.69527	+	0.978767i	−0.388923	+	0.224545i	−0.681693	−	0.731638i	\(-0.738756\pi\)
							0.292771	+	0.956183i	\(0.405423\pi\)
\(23\)	−0.270081	−	0.467795i	−0.0563158	−	0.0975419i	0.836493	−	0.547977i	\(-0.184602\pi\)
							−0.892809	+	0.450436i	\(0.851269\pi\)
\(29\)	7.15857			1.32931			0.664656	−	0.747149i	\(-0.268578\pi\)
							0.664656	+	0.747149i	\(0.268578\pi\)
\(31\)	−5.28968	−	3.05400i	−0.950055	−	0.548514i	−0.0569568	−	0.998377i	\(-0.518140\pi\)
							−0.893098	+	0.449862i	\(0.851473\pi\)
\(37\)	6.95316	−	4.01441i	1.14309	−	0.659964i	0.195897	−	0.980624i	\(-0.437238\pi\)
							0.947194	+	0.320660i	\(0.103905\pi\)
\(41\)			7.55362i			1.17968i	0.807521	+	0.589839i	\(0.200809\pi\)
							−0.807521	+	0.589839i	\(0.799191\pi\)
\(43\)	−4.24839			−0.647873			−0.323936	−	0.946079i	\(-0.605006\pi\)
							−0.323936	+	0.946079i	\(0.605006\pi\)
\(47\)	5.42204	−	3.13042i	0.790886	−	0.456618i	−0.0493882	−	0.998780i	\(-0.515727\pi\)
							0.840274	+	0.542161i	\(0.182394\pi\)