Embedded newform 99.6.e.a.34.8

• Label: 99.6.e.a.34.8
• 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			34.8
Character	\(\chi\)	\(=\)	99.34
Dual form			99.6.e.a.67.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.16807 + 3.75521i) q^{2} +(11.8040 + 10.1817i) q^{3} +(6.59895 + 11.4297i) q^{4} +(29.6963 + 51.4354i) q^{5} +(-63.8260 + 22.2518i) q^{6} +(-30.4063 + 52.6652i) q^{7} -195.984 q^{8} +(35.6678 + 240.368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 15 q^{3} - 320 q^{4} - 36 q^{5} + 273 q^{6} + 167 q^{7} + 426 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.16807	+	3.75521i	−0.383264	+	0.663833i	−0.991527	−	0.129903i	\(-0.958533\pi\)
							0.608263	+	0.793736i	\(0.291867\pi\)
\(3\)	11.8040	+	10.1817i	0.757226	+	0.653153i
							\(5\)	29.6963	+	51.4354i	0.531223	+	0.920105i	0.999336	+	0.0364364i	\(0.0116006\pi\)
−0.468113	+	0.883669i	\(0.655066\pi\)
\(7\)	−30.4063	+	52.6652i	−0.234541	+	0.406236i	−0.959139	−	0.282935i	\(-0.908692\pi\)
							0.724598	+	0.689171i	\(0.242025\pi\)
\(11\)	60.5000	−	104.789i	0.150756	−	0.261116i
							\(13\)	11.6657	+	20.2057i	0.0191450	+	0.0331600i	0.875439	−	0.483328i	\(-0.160572\pi\)
−0.856294	+	0.516488i	\(0.827239\pi\)
\(17\)	520.434			0.436761			0.218380	−	0.975864i	\(-0.429923\pi\)
							0.218380	+	0.975864i	\(0.429923\pi\)
\(19\)	2153.91			1.36881			0.684405	−	0.729102i	\(-0.260062\pi\)
							0.684405	+	0.729102i	\(0.260062\pi\)
\(23\)	−825.429	−	1429.68i	−0.325357	−	0.563535i	0.656228	−	0.754563i	\(-0.272151\pi\)
							−0.981585	+	0.191028i	\(0.938818\pi\)
\(29\)	−942.110	+	1631.78i	−0.208021	+	0.360302i	−0.951091	−	0.308911i	\(-0.900035\pi\)
							0.743070	+	0.669213i	\(0.233369\pi\)
\(31\)	−796.291	−	1379.22i	−0.148822	−	0.257767i	0.781970	−	0.623316i	\(-0.214215\pi\)
							−0.930792	+	0.365548i	\(0.880881\pi\)
\(37\)	−1682.52			−0.202049			−0.101024	−	0.994884i	\(-0.532212\pi\)
							−0.101024	+	0.994884i	\(0.532212\pi\)
\(41\)	−6324.28	−	10954.0i	−0.587559	−	1.01768i	−0.994551	−	0.104250i	\(-0.966756\pi\)
							0.406992	−	0.913432i	\(-0.366578\pi\)
\(43\)	7078.79	−	12260.8i	0.583832	−	1.01123i	−0.411188	−	0.911551i	\(-0.634886\pi\)
							0.995020	−	0.0996765i	\(-0.0317808\pi\)
\(47\)	−12830.5	+	22223.0i	−0.847222	+	1.46743i	0.0364550	+	0.999335i	\(0.488393\pi\)
							−0.883677	+	0.468097i	\(0.844940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.6.e.a.34.8	✓	46
3.2	odd	2		297.6.e.a.100.16		46
9.2	odd	6		891.6.a.e.1.8		23
9.4	even	3	inner	99.6.e.a.67.8	yes	46
9.5	odd	6		297.6.e.a.199.16		46
9.7	even	3		891.6.a.f.1.16		23

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.8	✓	46	1.1	even	1	trivial
99.6.e.a.67.8	yes	46	9.4	even	3	inner
297.6.e.a.100.16		46	3.2	odd	2
297.6.e.a.199.16		46	9.5	odd	6
891.6.a.e.1.8		23	9.2	odd	6
891.6.a.f.1.16		23	9.7	even	3