Embedded newform 91.10.e.a.53.13

• Label: 91.10.e.a.53.13
• Level: $91$
• Weight: $10$
• Character: 91.53
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			53.13
Character	\(\chi\)	\(=\)	91.53
Dual form			91.10.e.a.79.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.79810 - 13.5067i) q^{2} +(70.3015 - 121.766i) q^{3} +(134.379 - 232.752i) q^{4} +(740.523 + 1282.62i) q^{5} -2192.87 q^{6} +(-6071.29 - 1868.97i) q^{7} -12176.9 q^{8} +(-43.0895 - 74.6332i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q + 65 q^{2} - 8533 q^{4} - 2118 q^{5} + 17358 q^{6} - 8764 q^{7} - 106980 q^{8} - 215597 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.79810	−	13.5067i	−0.344631	−	0.596918i	0.640656	−	0.767828i	\(-0.278663\pi\)
							−0.985287	+	0.170910i	\(0.945329\pi\)
\(3\)	70.3015	−	121.766i	0.501093	−	0.867919i	−0.498906	−	0.866656i	\(-0.666265\pi\)
							0.999999	−	0.00126300i	\(-0.000402025\pi\)
\(5\)	740.523	+	1282.62i	0.529875	+	0.917770i	0.999393	+	0.0348474i	\(0.0110945\pi\)
							−0.469518	+	0.882923i	\(0.655572\pi\)
\(7\)	−6071.29	−	1868.97i	−0.955740	−	0.294213i
							\(11\)	−22926.3	+	39709.5i	−0.472136	+	0.817764i	−0.999492	−	0.0318811i	\(-0.989850\pi\)
0.527356	+	0.849645i	\(0.323184\pi\)
\(13\)	−28561.0			−0.277350
							\(17\)	139383.	−	241418.i	0.404752	−	0.701052i	−0.589540	−	0.807739i	\(-0.700691\pi\)
0.994293	+	0.106687i	\(0.0340244\pi\)
\(19\)	243036.	+	420951.i	0.427838	+	0.741037i	0.996681	−	0.0814092i	\(-0.0259421\pi\)
							−0.568843	+	0.822446i	\(0.692609\pi\)
\(23\)	1.01611e6	+	1.75996e6i	0.757125	+	1.31138i	0.944311	+	0.329054i	\(0.106730\pi\)
							−0.187186	+	0.982324i	\(0.559937\pi\)
\(29\)	−6.15582e6			−1.61620			−0.808099	−	0.589046i	\(-0.799504\pi\)
							−0.808099	+	0.589046i	\(0.799504\pi\)
\(31\)	−1.84025e6	+	3.18740e6i	−0.357889	+	0.619882i	−0.987608	−	0.156941i	\(-0.949837\pi\)
							0.629719	+	0.776823i	\(0.283170\pi\)
\(37\)	4.58581e6	+	7.94286e6i	0.402261	+	0.696737i	0.993998	−	0.109394i	\(-0.0348910\pi\)
							−0.591737	+	0.806131i	\(0.701558\pi\)
\(41\)	3.44538e7			1.90419			0.952094	−	0.305804i	\(-0.0989254\pi\)
							0.952094	+	0.305804i	\(0.0989254\pi\)
\(43\)	−1.66095e7			−0.740880			−0.370440	−	0.928856i	\(-0.620793\pi\)
							−0.370440	+	0.928856i	\(0.620793\pi\)
\(47\)	−1.17340e7	−	2.03238e7i	−0.350756	−	0.607527i	0.635626	−	0.771997i	\(-0.280742\pi\)
							−0.986382	+	0.164470i	\(0.947409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.e.a.53.13	✓	70
7.2	even	3	inner	91.10.e.a.79.13	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.e.a.53.13	✓	70	1.1	even	1	trivial
91.10.e.a.79.13	yes	70	7.2	even	3	inner