Embedded newform 91.10.u.a.30.13

• Label: 91.10.u.a.30.13
• Level: $91$
• Weight: $10$
• Character: 91.30
• 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.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			30.13
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-28.4816 + 16.4439i) q^{2} +94.5061 q^{3} +(284.803 - 493.293i) q^{4} +(1116.08 + 644.368i) q^{5} +(-2691.69 + 1554.05i) q^{6} +(-1759.05 + 6104.04i) q^{7} +1894.52i q^{8} -10751.6 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−28.4816	+	16.4439i	−1.25872	+	0.726724i	−0.972826	−	0.231537i	\(-0.925625\pi\)
							−0.285897	+	0.958260i	\(0.592291\pi\)
\(3\)	94.5061			0.673619			0.336809	−	0.941573i	\(-0.390652\pi\)
							0.336809	+	0.941573i	\(0.390652\pi\)
\(5\)	1116.08	+	644.368i	0.798600	+	0.461072i	0.842981	−	0.537943i	\(-0.180798\pi\)
							−0.0443814	+	0.999015i	\(0.514132\pi\)
\(7\)	−1759.05	+	6104.04i	−0.276909	+	0.960896i
							\(11\)		−	83600.6i		−	1.72164i	−0.508910	−	0.860820i	\(-0.669951\pi\)
0.508910	−	0.860820i	\(-0.330049\pi\)
\(13\)	60912.8	+	83030.9i	0.591512	+	0.806296i
							\(17\)	100352.	−	173815.i	0.291411	−	0.504738i	−0.682733	−	0.730668i	\(-0.739209\pi\)
0.974144	+	0.225930i	\(0.0725421\pi\)
\(19\)			318456.i			0.560607i	0.959911	+	0.280304i	\(0.0904352\pi\)
							−0.959911	+	0.280304i	\(0.909565\pi\)
\(23\)	−993496.	−	1.72079e6i	−0.740271	−	1.28219i	−0.952372	−	0.304940i	\(-0.901364\pi\)
							0.212100	−	0.977248i	\(-0.431970\pi\)
\(29\)	2.81272e6	−	4.87177e6i	0.738473	−	1.27907i	−0.214709	−	0.976678i	\(-0.568880\pi\)
							0.953183	−	0.302395i	\(-0.0977863\pi\)
\(31\)	−1.06713e6	+	616106.i	−0.207534	+	0.119820i	−0.600165	−	0.799876i	\(-0.704898\pi\)
							0.392631	+	0.919696i	\(0.371565\pi\)
\(37\)	−1.20741e7	+	6.97098e6i	−1.05912	+	0.611485i	−0.925190	−	0.379505i	\(-0.876094\pi\)
							−0.133934	+	0.990990i	\(0.542761\pi\)
\(41\)	−1.35080e7	−	7.79883e6i	−0.746557	−	0.431025i	0.0778916	−	0.996962i	\(-0.475181\pi\)
							−0.824448	+	0.565937i	\(0.808515\pi\)
\(43\)	−1.22137e6	−	2.11548e6i	−0.0544803	−	0.0943627i	0.837499	−	0.546439i	\(-0.184017\pi\)
							−0.891979	+	0.452076i	\(0.850684\pi\)
\(47\)	3.24789e7	+	1.87517e7i	0.970869	+	0.560532i	0.899501	−	0.436918i	\(-0.143930\pi\)
							0.0713682	+	0.997450i	\(0.477263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.13	yes	164
7.4	even	3		91.10.k.a.4.13	✓	164
13.10	even	6		91.10.k.a.23.70	yes	164
91.88	even	6	inner	91.10.u.a.88.13	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.13	✓	164	7.4	even	3
91.10.k.a.23.70	yes	164	13.10	even	6
91.10.u.a.30.13	yes	164	1.1	even	1	trivial
91.10.u.a.88.13	yes	164	91.88	even	6	inner