Embedded newform 91.8.u.a.30.3

• Label: 91.8.u.a.30.3
• 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			30.3
Character	\(\chi\)	\(=\)	91.30
Dual form			91.8.u.a.88.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.4889 + 10.6746i) q^{2} +37.9468 q^{3} +(163.893 - 283.870i) q^{4} +(411.268 + 237.445i) q^{5} +(-701.593 + 405.065i) q^{6} +(874.632 + 241.997i) q^{7} +4265.24i q^{8} -747.044 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 3 q^{2} - 164 q^{3} + 3841 q^{4} - 6 q^{6} + 754 q^{7} + 85710 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.4889	+	10.6746i	−1.63420	+	0.943507i	−0.651425	+	0.758713i	\(0.725828\pi\)
							−0.982777	+	0.184794i	\(0.940838\pi\)
\(3\)	37.9468			0.811428			0.405714	−	0.914000i	\(-0.367023\pi\)
							0.405714	+	0.914000i	\(0.367023\pi\)
\(5\)	411.268	+	237.445i	1.47140	+	0.849511i	0.999484	−	0.0321350i	\(-0.0102306\pi\)
							0.471912	+	0.881646i	\(0.343564\pi\)
\(7\)	874.632	+	241.997i	0.963789	+	0.266665i
							\(11\)		−	2997.59i		−	0.679045i	−0.940598	−	0.339522i	\(-0.889735\pi\)
0.940598	−	0.339522i	\(-0.110265\pi\)
\(13\)	−6827.70	+	4016.34i	−0.861932	+	0.507024i
							\(17\)	−6254.63	+	10833.3i	−0.308767	+	0.534800i	−0.978093	−	0.208169i	\(-0.933250\pi\)
0.669326	+	0.742969i	\(0.266583\pi\)
\(19\)			55198.7i			1.84625i	0.384498	+	0.923126i	\(0.374375\pi\)
							−0.384498	+	0.923126i	\(0.625625\pi\)
\(23\)	−6713.17	−	11627.6i	−0.115048	−	0.199270i	0.802751	−	0.596315i	\(-0.203369\pi\)
							−0.917799	+	0.397045i	\(0.870036\pi\)
\(29\)	−117.991	+	204.367i	−0.000898372	+	0.00155603i	−0.866474	−	0.499222i	\(-0.833619\pi\)
							0.865576	+	0.500778i	\(0.166953\pi\)
\(31\)	179053.	−	103376.i	1.07948	−	0.623241i	0.148727	−	0.988878i	\(-0.452482\pi\)
							0.930757	+	0.365638i	\(0.119149\pi\)
\(37\)	−145033.	+	83734.8i	−0.470718	+	0.271769i	−0.716540	−	0.697546i	\(-0.754275\pi\)
							0.245822	+	0.969315i	\(0.420942\pi\)
\(41\)	55154.6	+	31843.5i	0.124979	+	0.0721569i	0.561186	−	0.827689i	\(-0.310345\pi\)
							−0.436207	+	0.899846i	\(0.643678\pi\)
\(43\)	438678.	+	759813.i	0.841408	+	1.45736i	0.888705	+	0.458480i	\(0.151606\pi\)
							−0.0472966	+	0.998881i	\(0.515061\pi\)
\(47\)	−704971.	−	407015.i	−0.990442	−	0.571832i	−0.0850354	−	0.996378i	\(-0.527100\pi\)
							−0.905406	+	0.424546i	\(0.860434\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.u.a.30.3	yes	126
7.4	even	3		91.8.k.a.4.3	✓	126
13.10	even	6		91.8.k.a.23.61	yes	126
91.88	even	6	inner	91.8.u.a.88.3	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.k.a.4.3	✓	126	7.4	even	3
91.8.k.a.23.61	yes	126	13.10	even	6
91.8.u.a.30.3	yes	126	1.1	even	1	trivial
91.8.u.a.88.3	yes	126	91.88	even	6	inner