Embedded newform 91.8.h.a.16.17

• Label: 91.8.h.a.16.17
• 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			16.17
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.17

$q$-expansion

\(f(q)\)	\(=\)	\(q-13.0238 q^{2} +(-26.2436 + 45.4552i) q^{3} +41.6200 q^{4} +(-63.3839 + 109.784i) q^{5} +(341.792 - 592.001i) q^{6} +(185.431 - 888.346i) q^{7} +1125.00 q^{8} +(-283.950 - 491.816i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.0238			−1.15115			−0.575577	−	0.817747i	\(-0.695223\pi\)
							−0.575577	+	0.817747i	\(0.695223\pi\)
\(3\)	−26.2436	+	45.4552i	−0.561175	+	0.971984i	0.436219	+	0.899841i	\(0.356317\pi\)
							−0.997394	+	0.0721436i	\(0.977016\pi\)
\(5\)	−63.3839	+	109.784i	−0.226769	+	0.392776i	−0.956849	−	0.290586i	\(-0.906150\pi\)
							0.730080	+	0.683362i	\(0.239483\pi\)
\(7\)	185.431	−	888.346i	0.204334	−	0.978901i
							\(11\)	−3912.36	+	6776.40i	−0.886266	+	1.53506i	−0.0420102	+	0.999117i	\(0.513376\pi\)
−0.844256	+	0.535941i	\(0.819957\pi\)
\(13\)	−3112.50	+	7284.29i	−0.392923	+	0.919571i
							\(17\)	−11611.1			−0.573193			−0.286596	−	0.958051i	\(-0.592524\pi\)
−0.286596	+	0.958051i	\(0.592524\pi\)
\(19\)	24035.9	+	41631.4i	0.803938	+	1.39246i	0.917006	+	0.398874i	\(0.130599\pi\)
							−0.113068	+	0.993587i	\(0.536068\pi\)
\(23\)	−45292.3			−0.776207			−0.388103	−	0.921616i	\(-0.626870\pi\)
							−0.388103	+	0.921616i	\(0.626870\pi\)
\(29\)	15696.9	+	27187.9i	0.119515	+	0.207006i	0.919575	−	0.392913i	\(-0.128533\pi\)
							−0.800061	+	0.599919i	\(0.795199\pi\)
\(31\)	81657.7	+	141435.i	0.492301	+	0.852691i	0.999961	−	0.00886702i	\(-0.00282250\pi\)
							−0.507659	+	0.861558i	\(0.669489\pi\)
\(37\)	158638.			0.514875			0.257438	−	0.966295i	\(-0.417122\pi\)
							0.257438	+	0.966295i	\(0.417122\pi\)
\(41\)	−165700.	−	287000.i	−0.375472	−	0.650337i	0.614925	−	0.788585i	\(-0.289186\pi\)
							−0.990398	+	0.138248i	\(0.955853\pi\)
\(43\)	53024.9	−	91841.9i	0.101705	−	0.176157i	−0.810682	−	0.585486i	\(-0.800904\pi\)
							0.912387	+	0.409329i	\(0.134237\pi\)
\(47\)	104408.	−	180840.i	0.146687	−	0.254069i	−0.783314	−	0.621626i	\(-0.786472\pi\)
							0.930001	+	0.367557i	\(0.119806\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.17	yes	126
7.4	even	3		91.8.g.a.81.47	yes	126
13.9	even	3		91.8.g.a.9.47	✓	126
91.74	even	3	inner	91.8.h.a.74.17	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.47	✓	126	13.9	even	3
91.8.g.a.81.47	yes	126	7.4	even	3
91.8.h.a.16.17	yes	126	1.1	even	1	trivial
91.8.h.a.74.17	yes	126	91.74	even	3	inner