Embedded newform 99.6.e.a.67.9

• Label: 99.6.e.a.67.9
• Level: $99$
• Weight: $6$
• Character: 99.67
• 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			67.9
Character	\(\chi\)	\(=\)	99.67
Dual form			99.6.e.a.34.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69570 - 2.93704i) q^{2} +(-10.7528 - 11.2862i) q^{3} +(10.2492 - 17.7521i) q^{4} +(12.5191 - 21.6836i) q^{5} +(-14.9145 + 50.7193i) q^{6} +(124.918 + 216.365i) q^{7} -178.043 q^{8} +(-11.7558 + 242.715i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 15 q^{3} - 320 q^{4} - 36 q^{5} + 273 q^{6} + 167 q^{7} + 426 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69570	−	2.93704i	−0.299760	−	0.519200i	0.676321	−	0.736607i	\(-0.263573\pi\)
							−0.976081	+	0.217407i	\(0.930240\pi\)
\(3\)	−10.7528	−	11.2862i	−0.689791	−	0.724009i
							\(5\)	12.5191	−	21.6836i	0.223948	−	0.387889i	−0.732056	−	0.681245i	\(-0.761439\pi\)
0.956003	+	0.293356i	\(0.0947722\pi\)
\(7\)	124.918	+	216.365i	0.963566	+	1.66895i	0.713418	+	0.700738i	\(0.247146\pi\)
							0.250148	+	0.968208i	\(0.419521\pi\)
\(11\)	60.5000	+	104.789i	0.150756	+	0.261116i
							\(13\)	−492.811	+	853.573i	−0.808764	+	1.40082i	0.104957	+	0.994477i	\(0.466530\pi\)
−0.913721	+	0.406343i	\(0.866804\pi\)
\(17\)	1349.42			1.13246			0.566231	−	0.824247i	\(-0.308401\pi\)
							0.566231	+	0.824247i	\(0.308401\pi\)
\(19\)	63.6150			0.0404274			0.0202137	−	0.999796i	\(-0.493565\pi\)
							0.0202137	+	0.999796i	\(0.493565\pi\)
\(23\)	−1660.38	+	2875.87i	−0.654468	+	1.13357i	0.327559	+	0.944831i	\(0.393774\pi\)
							−0.982027	+	0.188741i	\(0.939559\pi\)
\(29\)	−584.254	−	1011.96i	−0.129005	−	0.223443i	0.794286	−	0.607544i	\(-0.207845\pi\)
							−0.923291	+	0.384100i	\(0.874512\pi\)
\(31\)	−1293.10	+	2239.71i	−0.241672	+	0.418589i	−0.961191	−	0.275885i	\(-0.911029\pi\)
							0.719518	+	0.694473i	\(0.244363\pi\)
\(37\)	−1418.91			−0.170393			−0.0851964	−	0.996364i	\(-0.527152\pi\)
							−0.0851964	+	0.996364i	\(0.527152\pi\)
\(41\)	8321.03	−	14412.5i	0.773068	−	1.33899i	−0.162806	−	0.986658i	\(-0.552054\pi\)
							0.935874	−	0.352335i	\(-0.114612\pi\)
\(43\)	−8382.22	−	14518.4i	−0.691334	−	1.19743i	−0.971401	−	0.237445i	\(-0.923690\pi\)
							0.280067	−	0.959980i	\(-0.409643\pi\)
\(47\)	4016.68	+	6957.09i	0.265230	+	0.459391i	0.967624	−	0.252397i	\(-0.0812189\pi\)
							−0.702394	+	0.711788i	\(0.747886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.6.e.a.67.9	yes	46
3.2	odd	2		297.6.e.a.199.15		46
9.2	odd	6		297.6.e.a.100.15		46
9.4	even	3		891.6.a.f.1.15		23
9.5	odd	6		891.6.a.e.1.9		23
9.7	even	3	inner	99.6.e.a.34.9	✓	46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.9	✓	46	9.7	even	3	inner
99.6.e.a.67.9	yes	46	1.1	even	1	trivial
297.6.e.a.100.15		46	9.2	odd	6
297.6.e.a.199.15		46	3.2	odd	2
891.6.a.e.1.9		23	9.5	odd	6
891.6.a.f.1.15		23	9.4	even	3