Embedded newform 99.6.e.a.34.10

• Label: 99.6.e.a.34.10
• Level: $99$
• Weight: $6$
• Character: 99.34
• Analytic conductor: $15.878$
• Analytic rank: $0$
• Dimension: $46$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	99.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8779981615\)
Analytic rank:	\(0\)
Dimension:	\(46\)
Relative dimension:	\(23\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			34.10
Character	\(\chi\)	\(=\)	99.34
Dual form			99.6.e.a.67.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.909131 + 1.57466i) q^{2} +(-7.01311 - 13.9218i) q^{3} +(14.3470 + 24.8497i) q^{4} +(48.3226 + 83.6972i) q^{5} +(28.2979 + 1.61345i) q^{6} +(3.10482 - 5.37770i) q^{7} -110.357 q^{8} +(-144.632 + 195.270i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 15 q^{3} - 320 q^{4} - 36 q^{5} + 273 q^{6} + 167 q^{7} + 426 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.909131	+	1.57466i	−0.160713	+	0.278363i	−0.935125	−	0.354319i	\(-0.884713\pi\)
							0.774411	+	0.632682i	\(0.218046\pi\)
\(3\)	−7.01311	−	13.9218i	−0.449892	−	0.893083i
							\(5\)	48.3226	+	83.6972i	0.864421	+	1.49722i	0.867621	+	0.497227i	\(0.165648\pi\)
−0.00319938	+	0.999995i	\(0.501018\pi\)
\(7\)	3.10482	−	5.37770i	0.0239492	−	0.0414812i	−0.853802	−	0.520597i	\(-0.825709\pi\)
							0.877752	+	0.479116i	\(0.159043\pi\)
\(11\)	60.5000	−	104.789i	0.150756	−	0.261116i
							\(13\)	−97.0804	−	168.148i	−0.159321	−	0.275952i	0.775303	−	0.631589i	\(-0.217597\pi\)
−0.934624	+	0.355637i	\(0.884264\pi\)
\(17\)	88.6272			0.0743781			0.0371890	−	0.999308i	\(-0.488160\pi\)
							0.0371890	+	0.999308i	\(0.488160\pi\)
\(19\)	−1673.27			−1.06336			−0.531681	−	0.846945i	\(-0.678440\pi\)
							−0.531681	+	0.846945i	\(0.678440\pi\)
\(23\)	273.048	+	472.933i	0.107627	+	0.186415i	0.914808	−	0.403888i	\(-0.132342\pi\)
							−0.807182	+	0.590303i	\(0.799008\pi\)
\(29\)	−3659.70	+	6338.78i	−0.808072	+	1.39962i	0.106126	+	0.994353i	\(0.466155\pi\)
							−0.914198	+	0.405269i	\(0.867178\pi\)
\(31\)	1987.05	+	3441.67i	0.371368	+	0.643229i	0.989776	−	0.142628i	\(-0.0455553\pi\)
							−0.618408	+	0.785857i	\(0.712222\pi\)
\(37\)	−1000.21			−0.120112			−0.0600562	−	0.998195i	\(-0.519128\pi\)
							−0.0600562	+	0.998195i	\(0.519128\pi\)
\(41\)	−7801.02	−	13511.8i	−0.724756	−	1.25531i	−0.959074	−	0.283154i	\(-0.908619\pi\)
							0.234318	−	0.972160i	\(-0.424714\pi\)
\(43\)	−6036.17	+	10455.0i	−0.497841	+	0.862285i	−0.999997	−	0.00249178i	\(-0.999207\pi\)
							0.502156	+	0.864777i	\(0.332540\pi\)
\(47\)	4488.95	−	7775.08i	0.296415	−	0.513405i	−0.678898	−	0.734232i	\(-0.737542\pi\)
							0.975313	+	0.220827i	\(0.0708756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.6.e.a.34.10	✓	46
3.2	odd	2		297.6.e.a.100.14		46
9.2	odd	6		891.6.a.e.1.10		23
9.4	even	3	inner	99.6.e.a.67.10	yes	46
9.5	odd	6		297.6.e.a.199.14		46
9.7	even	3		891.6.a.f.1.14		23

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.10	✓	46	1.1	even	1	trivial
99.6.e.a.67.10	yes	46	9.4	even	3	inner
297.6.e.a.100.14		46	3.2	odd	2
297.6.e.a.199.14		46	9.5	odd	6
891.6.a.e.1.10		23	9.2	odd	6
891.6.a.f.1.14		23	9.7	even	3