Embedded newform 91.10.u.a.30.4

• Label: 91.10.u.a.30.4
• 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			30.4
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-35.5005 + 20.4962i) q^{2} +132.067 q^{3} +(584.192 - 1011.85i) q^{4} +(1239.03 + 715.356i) q^{5} +(-4688.46 + 2706.88i) q^{6} +(-3482.51 + 5312.79i) q^{7} +26906.8i q^{8} -2241.26 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−35.5005	+	20.4962i	−1.56892	+	0.905815i	−0.572621	+	0.819820i	\(0.694073\pi\)
							−0.996296	+	0.0859946i	\(0.972593\pi\)
\(3\)	132.067			0.941346			0.470673	−	0.882308i	\(-0.344011\pi\)
							0.470673	+	0.882308i	\(0.344011\pi\)
\(5\)	1239.03	+	715.356i	0.886579	+	0.511867i	0.872822	−	0.488039i	\(-0.162287\pi\)
							0.0137572	+	0.999905i	\(0.495621\pi\)
\(7\)	−3482.51	+	5312.79i	−0.548216	+	0.836337i
							\(11\)			86843.6i			1.78842i	0.447643	+	0.894212i	\(0.352264\pi\)
−0.447643	+	0.894212i	\(0.647736\pi\)
\(13\)	−65762.6	+	79245.0i	−0.638608	+	0.769532i
							\(17\)	44361.8	−	76837.0i	0.128822	−	0.223126i	−0.794399	−	0.607397i	\(-0.792214\pi\)
0.923220	+	0.384271i	\(0.125547\pi\)
\(19\)		−	864429.i		−	1.52173i	−0.648909	−	0.760866i	\(-0.724774\pi\)
							0.648909	−	0.760866i	\(-0.275226\pi\)
\(23\)	971797.	+	1.68320e6i	0.724103	+	1.25418i	0.959342	+	0.282246i	\(0.0910794\pi\)
							−0.235239	+	0.971938i	\(0.575587\pi\)
\(29\)	−1.40287e6	+	2.42984e6i	−0.368320	+	0.637949i	−0.989303	−	0.145875i	\(-0.953400\pi\)
							0.620983	+	0.783824i	\(0.286734\pi\)
\(31\)	1.34625e6	−	777258.i	0.261817	−	0.151160i	−0.363346	−	0.931654i	\(-0.618366\pi\)
							0.625163	+	0.780494i	\(0.285032\pi\)
\(37\)	407519.	−	235281.i	0.0357471	−	0.0206386i	−0.482020	−	0.876160i	\(-0.660097\pi\)
							0.517767	+	0.855522i	\(0.326763\pi\)
\(41\)	1.79624e7	+	1.03706e7i	0.992741	+	0.573159i	0.906093	−	0.423080i	\(-0.139051\pi\)
							0.0866485	+	0.996239i	\(0.472384\pi\)
\(43\)	−1.63156e7	−	2.82595e7i	−0.727772	−	1.26054i	−0.957823	−	0.287360i	\(-0.907222\pi\)
							0.230050	−	0.973179i	\(-0.426111\pi\)
\(47\)	−1.65713e7	−	9.56746e6i	−0.495355	−	0.285994i	0.231438	−	0.972850i	\(-0.425657\pi\)
							−0.726793	+	0.686856i	\(0.758990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.4	yes	164
7.4	even	3		91.10.k.a.4.4	✓	164
13.10	even	6		91.10.k.a.23.79	yes	164
91.88	even	6	inner	91.10.u.a.88.4	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.4	✓	164	7.4	even	3
91.10.k.a.23.79	yes	164	13.10	even	6
91.10.u.a.30.4	yes	164	1.1	even	1	trivial
91.10.u.a.88.4	yes	164	91.88	even	6	inner