Embedded newform 91.8.u.a.30.13

• Label: 91.8.u.a.30.13
• Level: $91$
• Weight: $8$
• Character: 91.30
• 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.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(63\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			30.13
Character	\(\chi\)	\(=\)	91.30
Dual form			91.8.u.a.88.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.3706 + 7.14217i) q^{2} -51.8106 q^{3} +(38.0211 - 65.8544i) q^{4} +(189.087 + 109.169i) q^{5} +(640.928 - 370.040i) q^{6} +(746.771 - 515.632i) q^{7} -742.183i q^{8} +497.341 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 3 q^{2} - 164 q^{3} + 3841 q^{4} - 6 q^{6} + 754 q^{7} + 85710 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.3706	+	7.14217i	−1.09342	+	0.631284i	−0.934484	−	0.356005i	\(-0.884139\pi\)
							−0.158932	+	0.987289i	\(0.550805\pi\)
\(3\)	−51.8106			−1.10788			−0.553942	−	0.832555i	\(-0.686877\pi\)
							−0.553942	+	0.832555i	\(0.686877\pi\)
\(5\)	189.087	+	109.169i	0.676498	+	0.390576i	0.798534	−	0.601949i	\(-0.205609\pi\)
							−0.122036	+	0.992526i	\(0.538942\pi\)
\(7\)	746.771	−	515.632i	0.822895	−	0.568194i
							\(11\)		−	3987.67i		−	0.903328i	−0.892188	−	0.451664i	\(-0.850831\pi\)
0.892188	−	0.451664i	\(-0.149169\pi\)
\(13\)	3950.81	+	6865.83i	0.498751	+	0.866745i
							\(17\)	−15270.3	+	26448.9i	−0.753834	+	1.30568i	0.192119	+	0.981372i	\(0.438464\pi\)
−0.945952	+	0.324306i	\(0.894869\pi\)
\(19\)			19113.7i			0.639304i	0.947535	+	0.319652i	\(0.103566\pi\)
							−0.947535	+	0.319652i	\(0.896434\pi\)
\(23\)	−39654.3	−	68683.2i	−0.679583	−	1.17707i	−0.975107	−	0.221737i	\(-0.928827\pi\)
							0.295523	−	0.955336i	\(-0.404506\pi\)
\(29\)	59005.9	−	102201.i	0.449265	−	0.778150i	−0.549073	−	0.835774i	\(-0.685019\pi\)
							0.998338	+	0.0576244i	\(0.0183526\pi\)
\(31\)	−236578.	+	136589.i	−1.42629	+	0.823471i	−0.996826	−	0.0796111i	\(-0.974632\pi\)
							−0.429468	+	0.903082i	\(0.641299\pi\)
\(37\)	−64440.5	+	37204.7i	−0.209148	+	0.120751i	−0.600915	−	0.799313i	\(-0.705197\pi\)
							0.391768	+	0.920064i	\(0.371864\pi\)
\(41\)	583714.	+	337007.i	1.32268	+	0.763652i	0.984156	−	0.177304i	\(-0.0567376\pi\)
							0.338528	+	0.940956i	\(0.390071\pi\)
\(43\)	−239126.	−	414178.i	−0.458656	−	0.794415i	0.540234	−	0.841515i	\(-0.318336\pi\)
							−0.998890	+	0.0470992i	\(0.985002\pi\)
\(47\)	203985.	+	117771.i	0.286587	+	0.165461i	0.636401	−	0.771358i	\(-0.280422\pi\)
							−0.349815	+	0.936819i	\(0.613756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.u.a.30.13	yes	126
7.4	even	3		91.8.k.a.4.13	✓	126
13.10	even	6		91.8.k.a.23.51	yes	126
91.88	even	6	inner	91.8.u.a.88.13	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.k.a.4.13	✓	126	7.4	even	3
91.8.k.a.23.51	yes	126	13.10	even	6
91.8.u.a.30.13	yes	126	1.1	even	1	trivial
91.8.u.a.88.13	yes	126	91.88	even	6	inner