Embedded newform 91.10.h.a.16.19

• Label: 91.10.h.a.16.19
• 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			16.19
Character	\(\chi\)	\(=\)	91.16
Dual form			91.10.h.a.74.19

$q$-expansion

\(f(q)\)	\(=\)	\(q-26.7467 q^{2} +(112.136 - 194.225i) q^{3} +203.387 q^{4} +(1156.03 - 2002.30i) q^{5} +(-2999.27 + 5194.89i) q^{6} +(-2358.40 - 5898.44i) q^{7} +8254.38 q^{8} +(-15307.5 - 26513.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 2 q^{2} + 163 q^{3} + 40958 q^{4} - 2 q^{5} - 1026 q^{6} - 913 q^{7} - 1032 q^{8} - 512605 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.7467			−1.18205			−0.591025	−	0.806653i	\(-0.701276\pi\)
							−0.591025	+	0.806653i	\(0.701276\pi\)
\(3\)	112.136	−	194.225i	0.799281	−	1.38439i	−0.120804	−	0.992676i	\(-0.538547\pi\)
							0.920085	−	0.391719i	\(-0.128119\pi\)
\(5\)	1156.03	−	2002.30i	0.827187	−	1.43273i	−0.0730498	−	0.997328i	\(-0.523273\pi\)
							0.900237	−	0.435401i	\(-0.143393\pi\)
\(7\)	−2358.40	−	5898.44i	−0.371259	−	0.928530i
							\(11\)	28156.1	−	48767.9i	0.579837	−	1.00431i	−0.415660	−	0.909520i	\(-0.636449\pi\)
0.995498	−	0.0947877i	\(-0.0302172\pi\)
\(13\)	−15905.8	+	101742.i	−0.154458	+	0.987999i
							\(17\)	−414960.			−1.20500			−0.602498	−	0.798120i	\(-0.705828\pi\)
−0.602498	+	0.798120i	\(0.705828\pi\)
\(19\)	160375.	+	277778.i	0.282323	+	0.488998i	0.971957	−	0.235161i	\(-0.0755616\pi\)
							−0.689633	+	0.724159i	\(0.742228\pi\)
\(23\)	−482282.			−0.359357			−0.179679	−	0.983725i	\(-0.557506\pi\)
							−0.179679	+	0.983725i	\(0.557506\pi\)
\(29\)	−2.09819e6	−	3.63418e6i	−0.550877	−	0.954146i	−0.998212	−	0.0597798i	\(-0.980960\pi\)
							0.447335	−	0.894366i	\(-0.352373\pi\)
\(31\)	2.28769e6	+	3.96240e6i	0.444908	+	0.770603i	0.998046	−	0.0624865i	\(-0.0199030\pi\)
							−0.553138	+	0.833090i	\(0.686570\pi\)
\(37\)	2.19020e7			1.92121			0.960606	−	0.277913i	\(-0.0896427\pi\)
							0.960606	+	0.277913i	\(0.0896427\pi\)
\(41\)	−1.05183e7	−	1.82182e7i	−0.581324	−	1.00688i	−0.995323	−	0.0966057i	\(-0.969201\pi\)
							0.413998	−	0.910278i	\(-0.364132\pi\)
\(43\)	−985772.	+	1.70741e6i	−0.0439712	+	0.0761604i	−0.887173	−	0.461436i	\(-0.847334\pi\)
							0.843202	+	0.537597i	\(0.180668\pi\)
\(47\)	2.53106e7	−	4.38393e7i	0.756593	−	1.31046i	−0.187985	−	0.982172i	\(-0.560196\pi\)
							0.944578	−	0.328286i	\(-0.106471\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.h.a.16.19	yes	164
7.4	even	3		91.10.g.a.81.64	yes	164
13.9	even	3		91.10.g.a.9.64	✓	164
91.74	even	3	inner	91.10.h.a.74.19	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.g.a.9.64	✓	164	13.9	even	3
91.10.g.a.81.64	yes	164	7.4	even	3
91.10.h.a.16.19	yes	164	1.1	even	1	trivial
91.10.h.a.74.19	yes	164	91.74	even	3	inner