Embedded newform 91.10.q.a.36.13

• Label: 91.10.q.a.36.13
• Level: $91$
• Weight: $10$
• Character: 91.36
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $124$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(124\)
Relative dimension:	\(62\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			36.13
Character	\(\chi\)	\(=\)	91.36
Dual form			91.10.q.a.43.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-25.2553 + 14.5812i) q^{2} +(-95.5625 - 165.519i) q^{3} +(169.221 - 293.099i) q^{4} -1073.46i q^{5} +(4826.92 + 2786.83i) q^{6} +(2079.33 + 1200.50i) q^{7} -5061.36i q^{8} +(-8422.88 + 14588.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q + 15360 q^{4} + 6570 q^{6} - 381386 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−25.2553	+	14.5812i	−1.11614	+	0.644403i	−0.940412	−	0.340036i	\(-0.889561\pi\)
							−0.175726	+	0.984439i	\(0.556227\pi\)
\(3\)	−95.5625	−	165.519i	−0.681149	−	1.17978i	−0.974631	−	0.223819i	\(-0.928147\pi\)
							0.293482	−	0.955965i	\(-0.405186\pi\)
\(5\)		−	1073.46i		−	0.768109i	−0.923311	−	0.384054i	\(-0.874528\pi\)
							0.923311	−	0.384054i	\(-0.125472\pi\)
\(7\)	2079.33	+	1200.50i	0.327327	+	0.188982i
							\(11\)	−53158.6	+	30691.1i	−1.09473	+	0.632042i	−0.934831	−	0.355092i	\(-0.884449\pi\)
−0.159897	+	0.987134i	\(0.551116\pi\)
\(13\)	−55492.3	−	86747.4i	−0.538875	−	0.842386i
							\(17\)	295411.	−	511667.i	0.857841	−	1.48582i	−0.0161438	−	0.999870i	\(-0.505139\pi\)
0.873984	−	0.485954i	\(-0.161528\pi\)
\(19\)	−725202.	−	418696.i	−1.27664	−	0.737068i	−0.300410	−	0.953810i	\(-0.597123\pi\)
							−0.976229	+	0.216743i	\(0.930457\pi\)
\(23\)	−31168.5	−	53985.3i	−0.0232242	−	0.0402254i	0.854180	−	0.519978i	\(-0.174060\pi\)
							−0.877404	+	0.479752i	\(0.840726\pi\)
\(29\)	−2.60922e6	−	4.51930e6i	−0.685046	−	1.18653i	−0.973422	−	0.229018i	\(-0.926449\pi\)
							0.288376	−	0.957517i	\(-0.406885\pi\)
\(31\)		−	7.23269e6i		−	1.40661i	−0.710890	−	0.703303i	\(-0.751708\pi\)
							0.710890	−	0.703303i	\(-0.248292\pi\)
\(37\)	7.90737e6	−	4.56532e6i	0.693624	−	0.400464i	−0.111344	−	0.993782i	\(-0.535516\pi\)
							0.804968	+	0.593318i	\(0.202182\pi\)
\(41\)	−7.29285e6	+	4.21053e6i	−0.403060	+	0.232707i	−0.687804	−	0.725897i	\(-0.741425\pi\)
							0.284743	+	0.958604i	\(0.408092\pi\)
\(43\)	1.26267e7	−	2.18700e7i	0.563223	−	0.975531i	−0.433990	−	0.900918i	\(-0.642895\pi\)
							0.997213	−	0.0746129i	\(-0.0237721\pi\)
\(47\)		−	1.53917e7i		−	0.460093i	−0.973180	−	0.230046i	\(-0.926112\pi\)
							0.973180	−	0.230046i	\(-0.0738878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.q.a.36.13	✓	124
13.4	even	6	inner	91.10.q.a.43.13	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.q.a.36.13	✓	124	1.1	even	1	trivial
91.10.q.a.43.13	yes	124	13.4	even	6	inner