Embedded newform 91.10.h.a.16.5

• Label: 91.10.h.a.16.5
• 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			16.5
Character	\(\chi\)	\(=\)	91.16
Dual form			91.10.h.a.74.5

$q$-expansion

\(f(q)\)	\(=\)	\(q-41.2552 q^{2} +(-8.92868 + 15.4649i) q^{3} +1189.99 q^{4} +(-842.207 + 1458.75i) q^{5} +(368.355 - 638.009i) q^{6} +(6320.66 + 634.750i) q^{7} -27970.8 q^{8} +(9682.06 + 16769.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 2 q^{2} + 163 q^{3} + 40958 q^{4} - 2 q^{5} - 1026 q^{6} - 913 q^{7} - 1032 q^{8} - 512605 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−41.2552			−1.82324			−0.911620	−	0.411034i	\(-0.865168\pi\)
							−0.911620	+	0.411034i	\(0.865168\pi\)
\(3\)	−8.92868	+	15.4649i	−0.0636417	+	0.110231i	−0.896091	−	0.443871i	\(-0.853605\pi\)
							0.832449	+	0.554102i	\(0.186938\pi\)
\(5\)	−842.207	+	1458.75i	−0.602634	+	1.04379i	0.389786	+	0.920905i	\(0.372549\pi\)
							−0.992421	+	0.122888i	\(0.960784\pi\)
\(7\)	6320.66	+	634.750i	0.994995	+	0.0999221i
							\(11\)	−40305.0	+	69810.3i	−0.830026	+	1.43765i	0.0679909	+	0.997686i	\(0.478341\pi\)
−0.898017	+	0.439961i	\(0.854992\pi\)
\(13\)	−63241.9	−	81270.9i	−0.614129	−	0.789206i
							\(17\)	−265639.			−0.771385			−0.385692	−	0.922627i	\(-0.626037\pi\)
−0.385692	+	0.922627i	\(0.626037\pi\)
\(19\)	108947.	+	188702.i	0.191790	+	0.332190i	0.945843	−	0.324623i	\(-0.105237\pi\)
							−0.754054	+	0.656813i	\(0.771904\pi\)
\(23\)	−2.08241e6			−1.55164			−0.775821	−	0.630952i	\(-0.782664\pi\)
							−0.775821	+	0.630952i	\(0.782664\pi\)
\(29\)	2.30623e6	+	3.99450e6i	0.605496	+	1.04875i	0.991973	+	0.126451i	\(0.0403586\pi\)
							−0.386477	+	0.922299i	\(0.626308\pi\)
\(31\)	4.58835e6	+	7.94726e6i	0.892337	+	1.54557i	0.837066	+	0.547102i	\(0.184269\pi\)
							0.0552709	+	0.998471i	\(0.482398\pi\)
\(37\)	3.89785e6			0.341915			0.170957	−	0.985278i	\(-0.445314\pi\)
							0.170957	+	0.985278i	\(0.445314\pi\)
\(41\)	−516496.	−	894598.i	−0.0285456	−	0.0494425i	0.851400	−	0.524517i	\(-0.175754\pi\)
							−0.879945	+	0.475075i	\(0.842421\pi\)
\(43\)	−1.38778e7	+	2.40371e7i	−0.619033	+	1.07220i	0.370629	+	0.928781i	\(0.379142\pi\)
							−0.989662	+	0.143416i	\(0.954191\pi\)
\(47\)	4.78590e6	−	8.28942e6i	0.143062	−	0.247790i	−0.785586	−	0.618752i	\(-0.787639\pi\)
							0.928648	+	0.370962i	\(0.120972\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.h.a.16.5	yes	164
7.4	even	3		91.10.g.a.81.78	yes	164
13.9	even	3		91.10.g.a.9.78	✓	164
91.74	even	3	inner	91.10.h.a.74.5	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.g.a.9.78	✓	164	13.9	even	3
91.10.g.a.81.78	yes	164	7.4	even	3
91.10.h.a.16.5	yes	164	1.1	even	1	trivial
91.10.h.a.74.5	yes	164	91.74	even	3	inner