Embedded newform 91.8.h.a.16.6

• Label: 91.8.h.a.16.6
• 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			16.6
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.6

$q$-expansion

\(f(q)\)	\(=\)	\(q-18.3767 q^{2} +(-24.0823 + 41.7118i) q^{3} +209.703 q^{4} +(-251.617 + 435.813i) q^{5} +(442.554 - 766.525i) q^{6} +(-798.848 + 430.564i) q^{7} -1501.43 q^{8} +(-66.4166 - 115.037i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.3767			−1.62429			−0.812143	−	0.583458i	\(-0.801699\pi\)
							−0.812143	+	0.583458i	\(0.801699\pi\)
\(3\)	−24.0823	+	41.7118i	−0.514961	+	0.891938i	0.484889	+	0.874576i	\(0.338860\pi\)
							−0.999849	+	0.0173620i	\(0.994473\pi\)
\(5\)	−251.617	+	435.813i	−0.900212	+	1.55921i	−0.0729938	+	0.997332i	\(0.523255\pi\)
							−0.827218	+	0.561881i	\(0.810078\pi\)
\(7\)	−798.848	+	430.564i	−0.880280	+	0.474455i
							\(11\)	2299.31	−	3982.53i	0.520864	−	0.902162i	−0.478842	−	0.877901i	\(-0.658943\pi\)
0.999706	−	0.0242612i	\(-0.00772334\pi\)
\(13\)	−5461.13	+	5737.99i	−0.689415	+	0.724366i
							\(17\)	6885.37			0.339904			0.169952	−	0.985452i	\(-0.445639\pi\)
0.169952	+	0.985452i	\(0.445639\pi\)
\(19\)	−11392.2	−	19731.9i	−0.381040	−	0.659980i	0.610171	−	0.792269i	\(-0.291101\pi\)
							−0.991211	+	0.132289i	\(0.957767\pi\)
\(23\)	−112325.			−1.92499			−0.962496	−	0.271295i	\(-0.912548\pi\)
							−0.962496	+	0.271295i	\(0.912548\pi\)
\(29\)	−26273.2	−	45506.5i	−0.200041	−	0.346482i	0.748500	−	0.663135i	\(-0.230774\pi\)
							−0.948541	+	0.316653i	\(0.897441\pi\)
\(31\)	115439.	+	199947.i	0.695966	+	1.20545i	0.969854	+	0.243687i	\(0.0783571\pi\)
							−0.273887	+	0.961762i	\(0.588310\pi\)
\(37\)	313873.			1.01870			0.509352	−	0.860558i	\(-0.329885\pi\)
							0.509352	+	0.860558i	\(0.329885\pi\)
\(41\)	−174630.	−	302468.i	−0.395709	−	0.685388i	0.597483	−	0.801882i	\(-0.296168\pi\)
							−0.993191	+	0.116494i	\(0.962834\pi\)
\(43\)	−95945.6	+	166183.i	−0.184029	+	0.318747i	−0.943249	−	0.332087i	\(-0.892247\pi\)
							0.759220	+	0.650834i	\(0.225581\pi\)
\(47\)	23849.6	−	41308.8i	0.0335073	−	0.0580363i	−0.848785	−	0.528737i	\(-0.822666\pi\)
							0.882293	+	0.470701i	\(0.155999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.6	yes	126
7.4	even	3		91.8.g.a.81.58	yes	126
13.9	even	3		91.8.g.a.9.58	✓	126
91.74	even	3	inner	91.8.h.a.74.6	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.58	✓	126	13.9	even	3
91.8.g.a.81.58	yes	126	7.4	even	3
91.8.h.a.16.6	yes	126	1.1	even	1	trivial
91.8.h.a.74.6	yes	126	91.74	even	3	inner