Embedded newform 91.8.h.a.16.14

• Label: 91.8.h.a.16.14
• 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			16.14
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.14

$q$-expansion

\(f(q)\)	\(=\)	\(q-14.2960 q^{2} +(-8.02646 + 13.9022i) q^{3} +76.3766 q^{4} +(107.975 - 187.018i) q^{5} +(114.747 - 198.747i) q^{6} +(868.950 + 261.665i) q^{7} +738.010 q^{8} +(964.652 + 1670.83i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.2960			−1.26360			−0.631801	−	0.775130i	\(-0.717684\pi\)
							−0.631801	+	0.775130i	\(0.717684\pi\)
\(3\)	−8.02646	+	13.9022i	−0.171633	+	0.297276i	−0.938991	−	0.343942i	\(-0.888238\pi\)
							0.767358	+	0.641219i	\(0.221571\pi\)
\(5\)	107.975	−	187.018i	0.386302	−	0.669094i	−0.605647	−	0.795733i	\(-0.707086\pi\)
							0.991949	+	0.126639i	\(0.0404190\pi\)
\(7\)	868.950	+	261.665i	0.957529	+	0.288338i
							\(11\)	−823.307	+	1426.01i	−0.186504	+	0.323034i	−0.944082	−	0.329710i	\(-0.893049\pi\)
0.757578	+	0.652744i	\(0.226382\pi\)
\(13\)	−882.496	−	7872.08i	−0.111407	−	0.993775i
							\(17\)	31348.7			1.54756			0.773782	−	0.633453i	\(-0.218363\pi\)
0.773782	+	0.633453i	\(0.218363\pi\)
\(19\)	−9417.34	−	16311.3i	−0.314986	−	0.545571i	0.664449	−	0.747334i	\(-0.268666\pi\)
							−0.979434	+	0.201763i	\(0.935333\pi\)
\(23\)	−93974.5			−1.61051			−0.805254	−	0.592931i	\(-0.797971\pi\)
							−0.805254	+	0.592931i	\(0.797971\pi\)
\(29\)	−48830.0	−	84576.0i	−0.371786	−	0.643953i	0.618054	−	0.786136i	\(-0.287921\pi\)
							−0.989840	+	0.142183i	\(0.954588\pi\)
\(31\)	34776.4	+	60234.5i	0.209662	+	0.363145i	0.951608	−	0.307315i	\(-0.0994304\pi\)
							−0.741946	+	0.670459i	\(0.766097\pi\)
\(37\)	435473.			1.41337			0.706684	−	0.707529i	\(-0.250190\pi\)
							0.706684	+	0.707529i	\(0.250190\pi\)
\(41\)	368516.	+	638288.i	0.835051	+	1.44635i	0.893989	+	0.448089i	\(0.147895\pi\)
							−0.0589386	+	0.998262i	\(0.518772\pi\)
\(43\)	147423.	−	255345.i	0.282766	−	0.489764i	−0.689299	−	0.724477i	\(-0.742081\pi\)
							0.972065	+	0.234712i	\(0.0754148\pi\)
\(47\)	147086.	−	254760.i	0.206647	−	0.357923i	−0.744009	−	0.668169i	\(-0.767078\pi\)
							0.950656	+	0.310247i	\(0.100412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.14	yes	126
7.4	even	3		91.8.g.a.81.50	yes	126
13.9	even	3		91.8.g.a.9.50	✓	126
91.74	even	3	inner	91.8.h.a.74.14	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.50	✓	126	13.9	even	3
91.8.g.a.81.50	yes	126	7.4	even	3
91.8.h.a.16.14	yes	126	1.1	even	1	trivial
91.8.h.a.74.14	yes	126	91.74	even	3	inner