Embedded newform 91.8.h.a.16.16

• Label: 91.8.h.a.16.16
• Level: $91$
• Weight: $8$
• Character: 91.16
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(63\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.16
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.16

$q$-expansion

\(f(q)\)	\(=\)	\(q-13.3993 q^{2} +(9.37130 - 16.2316i) q^{3} +51.5416 q^{4} +(-257.461 + 445.935i) q^{5} +(-125.569 + 217.492i) q^{6} +(237.569 - 875.845i) q^{7} +1024.49 q^{8} +(917.858 + 1589.78i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.3993			−1.18434			−0.592172	−	0.805812i	\(-0.701729\pi\)
							−0.592172	+	0.805812i	\(0.701729\pi\)
\(3\)	9.37130	−	16.2316i	0.200390	−	0.347085i	−0.748264	−	0.663401i	\(-0.769113\pi\)
							0.948654	+	0.316316i	\(0.102446\pi\)
\(5\)	−257.461	+	445.935i	−0.921120	+	1.59543i	−0.123436	+	0.992353i	\(0.539391\pi\)
							−0.797685	+	0.603075i	\(0.793942\pi\)
\(7\)	237.569	−	875.845i	0.261786	−	0.965126i
							\(11\)	539.896	−	935.127i	0.122303	−	0.211834i	−0.798373	−	0.602164i	\(-0.794306\pi\)
0.920675	+	0.390329i	\(0.127639\pi\)
\(13\)	271.572	−	7916.74i	0.0342833	−	0.999412i
							\(17\)	−5747.64			−0.283739			−0.141869	−	0.989885i	\(-0.545311\pi\)
−0.141869	+	0.989885i	\(0.545311\pi\)
\(19\)	2508.15	+	4344.25i	0.0838911	+	0.145304i	0.904918	−	0.425586i	\(-0.139932\pi\)
							−0.821027	+	0.570889i	\(0.806599\pi\)
\(23\)	57678.0			0.988469			0.494234	−	0.869329i	\(-0.335448\pi\)
							0.494234	+	0.869329i	\(0.335448\pi\)
\(29\)	−23253.6	−	40276.4i	−0.177050	−	0.306660i	0.763819	−	0.645431i	\(-0.223322\pi\)
							−0.940869	+	0.338771i	\(0.889989\pi\)
\(31\)	1345.26	+	2330.06i	0.00811037	+	0.0140476i	0.870052	−	0.492960i	\(-0.164085\pi\)
							−0.861942	+	0.507007i	\(0.830752\pi\)
\(37\)	305913.			0.992868			0.496434	−	0.868074i	\(-0.334642\pi\)
							0.496434	+	0.868074i	\(0.334642\pi\)
\(41\)	234460.	+	406097.i	0.531283	+	0.920208i	0.999333	+	0.0365069i	\(0.0116231\pi\)
							−0.468051	+	0.883702i	\(0.655044\pi\)
\(43\)	−367278.	+	636143.i	−0.704457	+	1.22016i	0.262430	+	0.964951i	\(0.415476\pi\)
							−0.966887	+	0.255205i	\(0.917857\pi\)
\(47\)	−343600.	+	595132.i	−0.482736	+	0.836124i	−0.999804	−	0.0198208i	\(-0.993690\pi\)
							0.517067	+	0.855945i	\(0.327024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.16	yes	126
7.4	even	3		91.8.g.a.81.48	yes	126
13.9	even	3		91.8.g.a.9.48	✓	126
91.74	even	3	inner	91.8.h.a.74.16	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.48	✓	126	13.9	even	3
91.8.g.a.81.48	yes	126	7.4	even	3
91.8.h.a.16.16	yes	126	1.1	even	1	trivial
91.8.h.a.74.16	yes	126	91.74	even	3	inner