Embedded newform 91.8.h.a.16.7

• Label: 91.8.h.a.16.7
• 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			16.7
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.7

$q$-expansion

\(f(q)\)	\(=\)	\(q-18.3303 q^{2} +(-41.3019 + 71.5369i) q^{3} +208.000 q^{4} +(68.8190 - 119.198i) q^{5} +(757.075 - 1311.29i) q^{6} +(-737.556 + 528.729i) q^{7} -1466.42 q^{8} +(-2318.19 - 4015.22i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.3303			−1.62018			−0.810092	−	0.586303i	\(-0.800583\pi\)
							−0.810092	+	0.586303i	\(0.800583\pi\)
\(3\)	−41.3019	+	71.5369i	−0.883172	+	1.52970i	−0.0353777	+	0.999374i	\(0.511263\pi\)
							−0.847794	+	0.530325i	\(0.822070\pi\)
\(5\)	68.8190	−	119.198i	0.246214	−	0.426456i	−0.716258	−	0.697836i	\(-0.754147\pi\)
							0.962472	+	0.271380i	\(0.0874799\pi\)
\(7\)	−737.556	+	528.729i	−0.812740	+	0.582627i
							\(11\)	−1892.45	+	3277.83i	−0.428697	+	0.742526i	−0.996758	−	0.0804613i	\(-0.974361\pi\)
0.568060	+	0.822987i	\(0.307694\pi\)
\(13\)	7283.69	−	3113.90i	0.919496	−	0.393100i
							\(17\)	26170.6			1.29194			0.645971	−	0.763362i	\(-0.276453\pi\)
0.645971	+	0.763362i	\(0.276453\pi\)
\(19\)	23704.3	+	41057.0i	0.792846	+	1.37325i	0.924198	+	0.381914i	\(0.124735\pi\)
							−0.131352	+	0.991336i	\(0.541932\pi\)
\(23\)	16291.1			0.279192			0.139596	−	0.990209i	\(-0.455420\pi\)
							0.139596	+	0.990209i	\(0.455420\pi\)
\(29\)	54675.3	+	94700.4i	0.416292	+	0.721039i	0.995563	−	0.0940959i	\(-0.0299960\pi\)
							−0.579271	+	0.815135i	\(0.696663\pi\)
\(31\)	−28713.0	−	49732.4i	−0.173106	−	0.299829i	0.766398	−	0.642366i	\(-0.222047\pi\)
							−0.939504	+	0.342537i	\(0.888714\pi\)
\(37\)	329397.			1.06909			0.534544	−	0.845140i	\(-0.320483\pi\)
							0.534544	+	0.845140i	\(0.320483\pi\)
\(41\)	183883.	+	318495.i	0.416676	+	0.721704i	0.995603	−	0.0936756i	\(-0.0298616\pi\)
							−0.578927	+	0.815380i	\(0.696528\pi\)
\(43\)	−317194.	+	549396.i	−0.608395	+	1.05377i	0.383110	+	0.923703i	\(0.374853\pi\)
							−0.991505	+	0.130068i	\(0.958480\pi\)
\(47\)	−53546.3	+	92744.8i	−0.0752292	+	0.130301i	−0.901186	−	0.433433i	\(-0.857302\pi\)
							0.825957	+	0.563734i	\(0.190636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.7	yes	126
7.4	even	3		91.8.g.a.81.57	yes	126
13.9	even	3		91.8.g.a.9.57	✓	126
91.74	even	3	inner	91.8.h.a.74.7	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.57	✓	126	13.9	even	3
91.8.g.a.81.57	yes	126	7.4	even	3
91.8.h.a.16.7	yes	126	1.1	even	1	trivial
91.8.h.a.74.7	yes	126	91.74	even	3	inner