Embedded newform 91.8.h.a.16.19

• Label: 91.8.h.a.16.19
• 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			16.19
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.19

$q$-expansion

\(f(q)\)	\(=\)	\(q-11.4327 q^{2} +(-25.3886 + 43.9743i) q^{3} +2.70595 q^{4} +(116.445 - 201.688i) q^{5} +(290.259 - 502.743i) q^{6} +(-337.374 + 842.450i) q^{7} +1432.45 q^{8} +(-195.658 - 338.890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.4327			−1.01051			−0.505257	−	0.862969i	\(-0.668602\pi\)
							−0.505257	+	0.862969i	\(0.668602\pi\)
\(3\)	−25.3886	+	43.9743i	−0.542892	+	0.940317i	0.455844	+	0.890060i	\(0.349338\pi\)
							−0.998736	+	0.0502574i	\(0.983996\pi\)
\(5\)	116.445	−	201.688i	0.416605	−	0.721581i	−0.578990	−	0.815334i	\(-0.696553\pi\)
							0.995595	+	0.0937532i	\(0.0298865\pi\)
\(7\)	−337.374	+	842.450i	−0.371765	+	0.928327i
							\(11\)	−587.943	+	1018.35i	−0.133187	+	0.230686i	−0.924903	−	0.380202i	\(-0.875854\pi\)
0.791717	+	0.610889i	\(0.209188\pi\)
\(13\)	−7433.63	−	2736.72i	−0.938424	−	0.345484i
							\(17\)	−34638.0			−1.70995			−0.854973	−	0.518673i	\(-0.826426\pi\)
−0.854973	+	0.518673i	\(0.826426\pi\)
\(19\)	−20952.4	−	36290.7i	−0.700804	−	1.21383i	−0.968185	−	0.250237i	\(-0.919491\pi\)
							0.267381	−	0.963591i	\(-0.413842\pi\)
\(23\)	40946.1			0.701722			0.350861	−	0.936427i	\(-0.385889\pi\)
							0.350861	+	0.936427i	\(0.385889\pi\)
\(29\)	102045.	+	176748.i	0.776963	+	1.34574i	0.933684	+	0.358097i	\(0.116574\pi\)
							−0.156721	+	0.987643i	\(0.550092\pi\)
\(31\)	−10098.1	−	17490.5i	−0.0608801	−	0.105447i	0.833979	−	0.551796i	\(-0.186057\pi\)
							−0.894859	+	0.446349i	\(0.852724\pi\)
\(37\)	113873.			0.369586			0.184793	−	0.982777i	\(-0.440839\pi\)
							0.184793	+	0.982777i	\(0.440839\pi\)
\(41\)	−251032.	−	434800.i	−0.568834	−	0.985249i	−0.996682	−	0.0813985i	\(-0.974061\pi\)
							0.427848	−	0.903851i	\(-0.359272\pi\)
\(43\)	249578.	−	432281.i	0.478703	−	0.829138i	−0.520999	−	0.853557i	\(-0.674440\pi\)
							0.999702	+	0.0244195i	\(0.00777375\pi\)
\(47\)	173211.	−	300010.i	0.243351	−	0.421496i	−0.718316	−	0.695717i	\(-0.755087\pi\)
							0.961667	+	0.274221i	\(0.0884201\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.19	yes	126
7.4	even	3		91.8.g.a.81.45	yes	126
13.9	even	3		91.8.g.a.9.45	✓	126
91.74	even	3	inner	91.8.h.a.74.19	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.45	✓	126	13.9	even	3
91.8.g.a.81.45	yes	126	7.4	even	3
91.8.h.a.16.19	yes	126	1.1	even	1	trivial
91.8.h.a.74.19	yes	126	91.74	even	3	inner