Embedded newform 91.10.h.a.16.1

• Label: 91.10.h.a.16.1
• Level: $91$
• Weight: $10$
• Character: 91.16
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.1
Character	\(\chi\)	\(=\)	91.16
Dual form			91.10.h.a.74.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-44.9267 q^{2} +(-48.7021 + 84.3546i) q^{3} +1506.41 q^{4} +(-218.727 + 378.846i) q^{5} +(2188.03 - 3789.78i) q^{6} +(-3419.91 + 5353.30i) q^{7} -44675.7 q^{8} +(5097.70 + 8829.48i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 2 q^{2} + 163 q^{3} + 40958 q^{4} - 2 q^{5} - 1026 q^{6} - 913 q^{7} - 1032 q^{8} - 512605 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−44.9267			−1.98550			−0.992750	−	0.120198i	\(-0.961647\pi\)
							−0.992750	+	0.120198i	\(0.961647\pi\)
\(3\)	−48.7021	+	84.3546i	−0.347138	+	0.601261i	−0.985740	−	0.168276i	\(-0.946180\pi\)
							0.638602	+	0.769537i	\(0.279513\pi\)
\(5\)	−218.727	+	378.846i	−0.156508	+	0.271080i	−0.933607	−	0.358298i	\(-0.883357\pi\)
							0.777099	+	0.629378i	\(0.216690\pi\)
\(7\)	−3419.91	+	5353.30i	−0.538361	+	0.842714i
							\(11\)	11764.3	−	20376.4i	0.242270	−	0.419624i	−0.719091	−	0.694916i	\(-0.755441\pi\)
0.961360	+	0.275293i	\(0.0887748\pi\)
\(13\)	102767.	+	6596.89i	0.997946	+	0.0640611i
							\(17\)	91913.3			0.266906			0.133453	−	0.991055i	\(-0.457393\pi\)
0.133453	+	0.991055i	\(0.457393\pi\)
\(19\)	−279773.	−	484580.i	−0.492509	−	0.853050i	0.507454	−	0.861679i	\(-0.330587\pi\)
							−0.999963	+	0.00862859i	\(0.997253\pi\)
\(23\)	540351.			0.402625			0.201312	−	0.979527i	\(-0.435479\pi\)
							0.201312	+	0.979527i	\(0.435479\pi\)
\(29\)	−1.03720e6	−	1.79649e6i	−0.272316	−	0.471665i	0.697139	−	0.716936i	\(-0.254456\pi\)
							−0.969454	+	0.245272i	\(0.921123\pi\)
\(31\)	−3.50895e6	−	6.07767e6i	−0.682416	−	1.18198i	−0.974241	−	0.225507i	\(-0.927596\pi\)
							0.291826	−	0.956471i	\(-0.405737\pi\)
\(37\)	1.04842e7			0.919657			0.459828	−	0.888008i	\(-0.347911\pi\)
							0.459828	+	0.888008i	\(0.347911\pi\)
\(41\)	−1.43294e7	−	2.48193e7i	−0.791956	−	1.37171i	−0.924754	−	0.380566i	\(-0.875729\pi\)
							0.132798	−	0.991143i	\(-0.457604\pi\)
\(43\)	−9.79536e6	+	1.69661e7i	−0.436931	+	0.756786i	−0.997451	−	0.0713546i	\(-0.977268\pi\)
							0.560520	+	0.828141i	\(0.310601\pi\)
\(47\)	−1.42021e7	+	2.45988e7i	−0.424534	+	0.735314i	−0.996377	−	0.0850491i	\(-0.972895\pi\)
							0.571843	+	0.820363i	\(0.306229\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.h.a.16.1	yes	164
7.4	even	3		91.10.g.a.81.82	yes	164
13.9	even	3		91.10.g.a.9.82	✓	164
91.74	even	3	inner	91.10.h.a.74.1	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.g.a.9.82	✓	164	13.9	even	3
91.10.g.a.81.82	yes	164	7.4	even	3
91.10.h.a.16.1	yes	164	1.1	even	1	trivial
91.10.h.a.74.1	yes	164	91.74	even	3	inner