Embedded newform 91.8.h.a.16.2

• Label: 91.8.h.a.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-21.6162 q^{2} +(-8.26223 + 14.3106i) q^{3} +339.261 q^{4} +(269.902 - 467.485i) q^{5} +(178.598 - 309.341i) q^{6} +(280.582 + 863.028i) q^{7} -4566.66 q^{8} +(956.971 + 1657.52i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−21.6162			−1.91062			−0.955311	−	0.295603i	\(-0.904480\pi\)
							−0.955311	+	0.295603i	\(0.904480\pi\)
\(3\)	−8.26223	+	14.3106i	−0.176674	+	0.306008i	−0.940739	−	0.339130i	\(-0.889867\pi\)
							0.764065	+	0.645139i	\(0.223201\pi\)
\(5\)	269.902	−	467.485i	0.965632	−	1.67252i	0.257726	−	0.966218i	\(-0.417027\pi\)
							0.707906	−	0.706307i	\(-0.249640\pi\)
\(7\)	280.582	+	863.028i	0.309184	+	0.951002i
							\(11\)	820.415	−	1421.00i	0.185849	−	0.321899i	−0.758014	−	0.652239i	\(-0.773830\pi\)
0.943862	+	0.330340i	\(0.107163\pi\)
\(13\)	−3783.93	+	6959.20i	−0.477684	+	0.878531i
							\(17\)	−9094.08			−0.448939			−0.224470	−	0.974481i	\(-0.572065\pi\)
−0.224470	+	0.974481i	\(0.572065\pi\)
\(19\)	21515.7	+	37266.3i	0.719645	+	1.24646i	0.961140	+	0.276060i	\(0.0890288\pi\)
							−0.241495	+	0.970402i	\(0.577638\pi\)
\(23\)	−35295.8			−0.604889			−0.302444	−	0.953167i	\(-0.597803\pi\)
							−0.302444	+	0.953167i	\(0.597803\pi\)
\(29\)	7981.72	+	13824.8i	0.0607720	+	0.105260i	0.894811	−	0.446446i	\(-0.147310\pi\)
							−0.834039	+	0.551706i	\(0.813977\pi\)
\(31\)	108251.	+	187497.i	0.652630	+	1.13039i	0.982482	+	0.186356i	\(0.0596679\pi\)
							−0.329852	+	0.944033i	\(0.606999\pi\)
\(37\)	−271277.			−0.880456			−0.440228	−	0.897886i	\(-0.645102\pi\)
							−0.440228	+	0.897886i	\(0.645102\pi\)
\(41\)	−12959.9	−	22447.1i	−0.0293668	−	0.0508648i	0.850968	−	0.525217i	\(-0.176016\pi\)
							−0.880335	+	0.474352i	\(0.842682\pi\)
\(43\)	121297.	−	210093.i	0.232655	−	0.402970i	−0.725934	−	0.687765i	\(-0.758592\pi\)
							0.958589	+	0.284795i	\(0.0919255\pi\)
\(47\)	−429565.	+	744029.i	−0.603513	+	1.04532i	0.388771	+	0.921334i	\(0.372900\pi\)
							−0.992285	+	0.123981i	\(0.960434\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.2	yes	126
7.4	even	3		91.8.g.a.81.62	yes	126
13.9	even	3		91.8.g.a.9.62	✓	126
91.74	even	3	inner	91.8.h.a.74.2	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.62	✓	126	13.9	even	3
91.8.g.a.81.62	yes	126	7.4	even	3
91.8.h.a.16.2	yes	126	1.1	even	1	trivial
91.8.h.a.74.2	yes	126	91.74	even	3	inner