Embedded newform 108.4.l.a.11.19

• Label: 108.4.l.a.11.19
• Level: $108$
• Weight: $4$
• Character: 108.11
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37220628062\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(52\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			11.19
Character	\(\chi\)	\(=\)	108.11
Dual form			108.4.l.a.59.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17164 + 2.57435i) q^{2} +(4.92086 + 1.66888i) q^{3} +(-5.25451 - 6.03242i) q^{4} +(3.67143 - 10.0872i) q^{5} +(-10.0618 + 10.7127i) q^{6} +(6.04634 + 1.06613i) q^{7} +(21.6859 - 6.45911i) q^{8} +(21.4296 + 16.4247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17164	+	2.57435i	−0.414238	+	0.910169i
							\(3\)	4.92086	+	1.66888i	0.947019	+	0.321177i
\(5\)	3.67143	−	10.0872i	0.328382	−	0.902223i								−0.660139	−	0.751143i	\(-0.729503\pi\)
							0.988522	−	0.151080i	\(-0.0482751\pi\)
\(7\)	6.04634	+	1.06613i	0.326472	+	0.0575658i	0.334482	−	0.942402i	\(-0.391439\pi\)
							−0.00801032	+	0.999968i	\(0.502550\pi\)
\(11\)	31.0921	−	11.3166i	0.852239	−	0.310190i	0.121286	−	0.992618i	\(-0.461298\pi\)
							0.730953	+	0.682428i	\(0.239076\pi\)
\(13\)	33.3096	−	27.9501i	0.710649	−	0.596305i	−0.214132	−	0.976805i	\(-0.568692\pi\)
							0.924781	+	0.380500i	\(0.124248\pi\)
\(17\)	−97.6451	+	56.3755i	−1.39308	+	0.804297i	−0.993655	−	0.112467i	\(-0.964125\pi\)
							−0.399428	+	0.916764i	\(0.630791\pi\)
\(19\)	87.7369	+	50.6549i	1.05938	+	0.611633i	0.925261	−	0.379332i	\(-0.123846\pi\)
							0.134119	+	0.990965i	\(0.457179\pi\)
\(23\)	−28.7938	−	163.298i	−0.261040	−	1.48043i	−0.780080	−	0.625680i	\(-0.784821\pi\)
							0.519040	−	0.854750i	\(-0.326290\pi\)
\(29\)	−127.814	+	152.323i	−0.818432	+	0.975369i	−0.999968	−	0.00801543i	\(-0.997449\pi\)
							0.181536	+	0.983384i	\(0.441893\pi\)
\(31\)	66.6999	−	11.7610i	0.386440	−	0.0681398i	0.0229470	−	0.999737i	\(-0.492695\pi\)
							0.363493	+	0.931597i	\(0.381584\pi\)
\(37\)	98.0242	+	169.783i	0.435543	+	0.754382i	0.997340	−	0.0728930i	\(-0.0232232\pi\)
							−0.561797	+	0.827275i	\(0.689890\pi\)
\(41\)	−133.306	−	158.868i	−0.507779	−	0.605147i	0.449867	−	0.893095i	\(-0.351471\pi\)
							−0.957646	+	0.287948i	\(0.907027\pi\)
\(43\)	−98.0684	−	269.441i	−0.347798	−	0.955566i	−0.983062	−	0.183272i	\(-0.941331\pi\)
							0.635265	−	0.772294i	\(-0.280891\pi\)
\(47\)	10.0291	−	56.8776i	0.0311253	−	0.176520i	−0.965282	−	0.261210i	\(-0.915879\pi\)
							0.996407	+	0.0846893i	\(0.0269898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.11.19	yes	312
4.3	odd	2	inner	108.4.l.a.11.10	✓	312
27.5	odd	18	inner	108.4.l.a.59.10	yes	312
108.59	even	18	inner	108.4.l.a.59.19	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.10	✓	312	4.3	odd	2	inner
108.4.l.a.11.19	yes	312	1.1	even	1	trivial
108.4.l.a.59.10	yes	312	27.5	odd	18	inner
108.4.l.a.59.19	yes	312	108.59	even	18	inner