Embedded newform 108.4.l.a.59.19

• Label: 108.4.l.a.59.19
• Level: $108$
• Weight: $4$
• Character: 108.59
• 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			59.19
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.19

$q$-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 \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.10	✓	312	27.11	odd	18	inner
108.4.l.a.11.19	yes	312	108.11	even	18	inner
108.4.l.a.59.10	yes	312	4.3	odd	2	inner
108.4.l.a.59.19	yes	312	1.1	even	1	trivial