Embedded newform 99.6.e.b.34.14

• Label: 99.6.e.b.34.14
• Level: $99$
• Weight: $6$
• Character: 99.34
• Analytic conductor: $15.878$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• 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:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			34.14
Character	\(\chi\)	\(=\)	99.34
Dual form			99.6.e.b.67.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.201317 - 0.348691i) q^{2} +(15.5707 + 0.744389i) q^{3} +(15.9189 + 27.5724i) q^{4} +(-39.4247 - 68.2856i) q^{5} +(3.39420 - 5.27950i) q^{6} +(-99.3099 + 172.010i) q^{7} +25.7033 q^{8} +(241.892 + 23.1813i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 16 q^{3} - 480 q^{4} - 93 q^{5} - 547 q^{6} - 225 q^{7} + 426 q^{8} + 412 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.201317	−	0.348691i	0.0355882	−	0.0616405i	−0.847683	−	0.530504i	\(-0.822003\pi\)
							0.883271	+	0.468863i	\(0.155336\pi\)
\(3\)	15.5707	+	0.744389i	0.998859	+	0.0477525i
							\(5\)	−39.4247	−	68.2856i	−0.705251	−	1.22153i	−0.966601	−	0.256286i	\(-0.917501\pi\)
0.261350	−	0.965244i	\(-0.415832\pi\)
\(7\)	−99.3099	+	172.010i	−0.766033	+	1.32681i	0.173666	+	0.984805i	\(0.444439\pi\)
							−0.939699	+	0.342003i	\(0.888895\pi\)
\(11\)	−60.5000	+	104.789i	−0.150756	+	0.261116i
							\(13\)	607.102	+	1051.53i	0.996330	+	1.72569i	0.572295	+	0.820048i	\(0.306053\pi\)
0.424034	+	0.905646i	\(0.360614\pi\)
\(17\)	207.720			0.174323			0.0871616	−	0.996194i	\(-0.472220\pi\)
							0.0871616	+	0.996194i	\(0.472220\pi\)
\(19\)	801.598			0.509416			0.254708	−	0.967018i	\(-0.418021\pi\)
							0.254708	+	0.967018i	\(0.418021\pi\)
\(23\)	−1112.27	−	1926.51i	−0.438420	−	0.759366i	0.559148	−	0.829068i	\(-0.311128\pi\)
							−0.997568	+	0.0697023i	\(0.977795\pi\)
\(29\)	−2172.45	+	3762.80i	−0.479684	+	0.830837i	−0.999728	−	0.0233024i	\(-0.992582\pi\)
							0.520045	+	0.854139i	\(0.325915\pi\)
\(31\)	865.211	+	1498.59i	0.161703	+	0.280078i	0.935480	−	0.353381i	\(-0.114968\pi\)
							−0.773777	+	0.633459i	\(0.781635\pi\)
\(37\)	6524.73			0.783534			0.391767	−	0.920064i	\(-0.371864\pi\)
							0.391767	+	0.920064i	\(0.371864\pi\)
\(41\)	1158.67	+	2006.87i	0.107646	+	0.186449i	0.914816	−	0.403870i	\(-0.132335\pi\)
							−0.807170	+	0.590319i	\(0.799002\pi\)
\(43\)	7662.81	−	13272.4i	0.632000	−	1.09466i	−0.355142	−	0.934812i	\(-0.615568\pi\)
							0.987142	−	0.159844i	\(-0.0510991\pi\)
\(47\)	−13112.6	+	22711.7i	−0.865852	+	1.49970i	0.000346239		1.00000i	\(0.499890\pi\)
							−0.866198	+	0.499700i	\(0.833444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.6.e.b.34.14	✓	54
3.2	odd	2		297.6.e.b.100.14		54
9.2	odd	6		891.6.a.i.1.14		27
9.4	even	3	inner	99.6.e.b.67.14	yes	54
9.5	odd	6		297.6.e.b.199.14		54
9.7	even	3		891.6.a.j.1.14		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.b.34.14	✓	54	1.1	even	1	trivial
99.6.e.b.67.14	yes	54	9.4	even	3	inner
297.6.e.b.100.14		54	3.2	odd	2
297.6.e.b.199.14		54	9.5	odd	6
891.6.a.i.1.14		27	9.2	odd	6
891.6.a.j.1.14		27	9.7	even	3