Embedded newform 99.6.g.b.32.20

• Label: 99.6.g.b.32.20
• Level: $99$
• Weight: $6$
• Character: 99.32
• Analytic conductor: $15.878$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8779981615\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			32.20
Character	\(\chi\)	\(=\)	99.32
Dual form			99.6.g.b.65.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.08560 - 3.61236i) q^{2} +(11.7874 - 10.2009i) q^{3} +(7.30056 - 12.6449i) q^{4} +(-69.0112 - 39.8436i) q^{5} +(-61.4329 - 21.3054i) q^{6} +(165.542 - 95.5759i) q^{7} -194.382 q^{8} +(34.8852 - 240.483i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 48 q^{3} - 962 q^{4} - 90 q^{5} + 876 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.08560	−	3.61236i	−0.368685	−	0.638581i	0.620675	−	0.784068i	\(-0.286859\pi\)
							−0.989360	+	0.145487i	\(0.953525\pi\)
\(3\)	11.7874	−	10.2009i	0.756162	−	0.654385i
							\(5\)	−69.0112	−	39.8436i	−1.23451	−	0.712744i	−0.266543	−	0.963823i	\(-0.585881\pi\)
−0.967967	+	0.251079i	\(0.919215\pi\)
\(7\)	165.542	−	95.5759i	1.27692	−	0.737231i	0.300640	−	0.953738i	\(-0.402800\pi\)
							0.976281	+	0.216507i	\(0.0694665\pi\)
\(11\)	−400.858	+	19.0833i	−0.998869	+	0.0475523i
							\(13\)	463.157	+	267.404i	0.760098	+	0.438843i	0.829331	−	0.558758i	\(-0.188722\pi\)
−0.0692327	+	0.997601i	\(0.522055\pi\)
\(17\)	799.547			0.670999			0.335499	−	0.942040i	\(-0.391095\pi\)
							0.335499	+	0.942040i	\(0.391095\pi\)
\(19\)		−	783.444i		−	0.497879i	−0.968519	−	0.248940i	\(-0.919918\pi\)
							0.968519	−	0.248940i	\(-0.0800821\pi\)
\(23\)	539.160	+	311.284i	0.212519	+	0.122698i	0.602482	−	0.798133i	\(-0.294179\pi\)
							−0.389962	+	0.920831i	\(0.627512\pi\)
\(29\)	3168.74	+	5488.41i	0.699666	+	1.21186i	0.968582	+	0.248694i	\(0.0800013\pi\)
							−0.268916	+	0.963164i	\(0.586665\pi\)
\(31\)	782.974	−	1356.15i	0.146333	−	0.253457i	−0.783536	−	0.621346i	\(-0.786586\pi\)
							0.929870	+	0.367889i	\(0.119919\pi\)
\(37\)	−12757.3			−1.53199			−0.765995	−	0.642847i	\(-0.777753\pi\)
							−0.765995	+	0.642847i	\(0.777753\pi\)
\(41\)	−3263.89	+	5653.22i	−0.303232	+	0.525214i	−0.976866	−	0.213851i	\(-0.931399\pi\)
							0.673634	+	0.739065i	\(0.264732\pi\)
\(43\)	−15785.4	+	9113.69i	−1.30192	+	0.751663i	−0.980733	−	0.195354i	\(-0.937415\pi\)
							−0.321185	+	0.947016i	\(0.604081\pi\)
\(47\)	17819.0	−	10287.8i	1.17663	−	0.679326i	0.221395	−	0.975184i	\(-0.428939\pi\)
							0.955232	+	0.295858i	\(0.0956056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.6.g.b.32.20	✓	112
9.2	odd	6	inner	99.6.g.b.65.37	yes	112
11.10	odd	2	inner	99.6.g.b.32.37	yes	112
99.65	even	6	inner	99.6.g.b.65.20	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.g.b.32.20	✓	112	1.1	even	1	trivial
99.6.g.b.32.37	yes	112	11.10	odd	2	inner
99.6.g.b.65.20	yes	112	99.65	even	6	inner
99.6.g.b.65.37	yes	112	9.2	odd	6	inner