Embedded newform 91.10.u.a.30.7

• Label: 91.10.u.a.30.7
• 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			30.7
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-34.8194 + 20.1030i) q^{2} +195.742 q^{3} +(552.260 - 956.543i) q^{4} +(-1750.16 - 1010.46i) q^{5} +(-6815.62 + 3935.00i) q^{6} +(-6019.15 - 2030.62i) q^{7} +23822.8i q^{8} +18631.9 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−34.8194	+	20.1030i	−1.53881	+	0.888435i	−0.539906	+	0.841725i	\(0.681540\pi\)
							−0.998909	+	0.0467094i	\(0.985127\pi\)
\(3\)	195.742			1.39521			0.697603	−	0.716484i	\(-0.254250\pi\)
							0.697603	+	0.716484i	\(0.254250\pi\)
\(5\)	−1750.16	−	1010.46i	−1.25231	−	0.723024i	−0.280746	−	0.959782i	\(-0.590582\pi\)
							−0.971569	+	0.236758i	\(0.923915\pi\)
\(7\)	−6019.15	−	2030.62i	−0.947533	−	0.319659i
							\(11\)		−	58104.8i		−	1.19659i	−0.801277	−	0.598294i	\(-0.795845\pi\)
0.801277	−	0.598294i	\(-0.204155\pi\)
\(13\)	−99058.2	+	28142.1i	−0.961934	+	0.273282i
							\(17\)	−196191.	+	339812.i	−0.569715	+	0.986776i	0.426879	+	0.904309i	\(0.359613\pi\)
−0.996594	+	0.0824668i	\(0.973720\pi\)
\(19\)		−	722361.i		−	1.27164i	−0.771839	−	0.635818i	\(-0.780663\pi\)
							0.771839	−	0.635818i	\(-0.219337\pi\)
\(23\)	−754860.	−	1.30746e6i	−0.562459	−	0.974208i	−0.997281	−	0.0736914i	\(-0.976522\pi\)
							0.434822	−	0.900516i	\(-0.356811\pi\)
\(29\)	−475941.	+	824353.i	−0.124957	+	0.216433i	−0.921716	−	0.387865i	\(-0.873213\pi\)
							0.796759	+	0.604297i	\(0.206546\pi\)
\(31\)	7.07500e6	−	4.08475e6i	1.37594	−	0.794398i	0.384270	−	0.923221i	\(-0.374453\pi\)
							0.991668	+	0.128823i	\(0.0411198\pi\)
\(37\)	−8.29948e6	+	4.79171e6i	−0.728020	+	0.420322i	−0.817697	−	0.575648i	\(-0.804750\pi\)
							0.0896775	+	0.995971i	\(0.471416\pi\)
\(41\)	3.27421e6	+	1.89037e6i	0.180959	+	0.104477i	0.587743	−	0.809048i	\(-0.300017\pi\)
							−0.406784	+	0.913524i	\(0.633350\pi\)
\(43\)	1.86429e7	+	3.22904e7i	0.831582	+	1.44034i	0.896783	+	0.442470i	\(0.145898\pi\)
							−0.0652011	+	0.997872i	\(0.520769\pi\)
\(47\)	4.04133e7	+	2.33326e7i	1.20805	+	0.697467i	0.962333	−	0.271875i	\(-0.0876438\pi\)
							0.245715	+	0.969342i	\(0.420977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.7	yes	164
7.4	even	3		91.10.k.a.4.7	✓	164
13.10	even	6		91.10.k.a.23.76	yes	164
91.88	even	6	inner	91.10.u.a.88.7	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.7	✓	164	7.4	even	3
91.10.k.a.23.76	yes	164	13.10	even	6
91.10.u.a.30.7	yes	164	1.1	even	1	trivial
91.10.u.a.88.7	yes	164	91.88	even	6	inner