Embedded newform 108.4.l.a.11.17

• Label: 108.4.l.a.11.17
• 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			11.17
Character	\(\chi\)	\(=\)	108.11
Dual form			108.4.l.a.59.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74755 + 2.22397i) q^{2} +(-2.88986 + 4.31841i) q^{3} +(-1.89211 - 7.77303i) q^{4} +(3.28651 - 9.02962i) q^{5} +(-4.55384 - 13.9736i) q^{6} +(18.0956 + 3.19074i) q^{7} +(20.5936 + 9.37578i) q^{8} +(-10.2974 - 24.9592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74755	+	2.22397i	−0.617854	+	0.786293i
							\(3\)	−2.88986	+	4.31841i	−0.556154	+	0.831079i
\(5\)	3.28651	−	9.02962i	0.293955	−	0.807634i								−0.701524	−	0.712646i	\(-0.747497\pi\)
							0.995478	−	0.0949876i	\(-0.0302812\pi\)
\(7\)	18.0956	+	3.19074i	0.977070	+	0.172284i	0.639311	−	0.768949i	\(-0.279220\pi\)
							0.337760	+	0.941232i	\(0.390331\pi\)
\(11\)	49.0304	−	17.8456i	1.34393	−	0.489150i	0.432882	−	0.901451i	\(-0.357497\pi\)
							0.911048	+	0.412300i	\(0.135275\pi\)
\(13\)	−45.2507	+	37.9699i	−0.965407	+	0.810073i	−0.981824	−	0.189792i	\(-0.939219\pi\)
							0.0164169	+	0.999865i	\(0.494774\pi\)
\(17\)	48.9268	−	28.2479i	0.698028	−	0.403007i	−0.108584	−	0.994087i	\(-0.534632\pi\)
							0.806613	+	0.591080i	\(0.201298\pi\)
\(19\)	−11.8846	−	6.86160i	−0.143501	−	0.0828505i	0.426530	−	0.904473i	\(-0.359736\pi\)
							−0.570032	+	0.821623i	\(0.693069\pi\)
\(23\)	32.1761	+	182.479i	0.291703	+	1.65433i	0.680308	+	0.732926i	\(0.261846\pi\)
							−0.388605	+	0.921404i	\(0.627043\pi\)
\(29\)	199.157	−	237.347i	1.27526	−	1.51980i	0.540796	−	0.841154i	\(-0.318123\pi\)
							0.734467	−	0.678645i	\(-0.237432\pi\)
\(31\)	123.956	−	21.8568i	0.718167	−	0.126632i	0.197390	−	0.980325i	\(-0.436753\pi\)
							0.520777	+	0.853693i	\(0.325642\pi\)
\(37\)	41.7796	+	72.3645i	0.185636	+	0.321531i	0.943791	−	0.330544i	\(-0.107232\pi\)
							−0.758155	+	0.652075i	\(0.773899\pi\)
\(41\)	29.6614	+	35.3491i	0.112984	+	0.134649i	0.819572	−	0.572976i	\(-0.194211\pi\)
							−0.706588	+	0.707625i	\(0.749767\pi\)
\(43\)	117.701	+	323.380i	0.417422	+	1.14686i	0.953158	+	0.302473i	\(0.0978121\pi\)
							−0.535736	+	0.844386i	\(0.679966\pi\)
\(47\)	40.4260	−	229.267i	0.125463	−	0.711533i	−0.855570	−	0.517688i	\(-0.826793\pi\)
							0.981032	−	0.193846i	\(-0.0620961\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.11.17	yes	312
4.3	odd	2	inner	108.4.l.a.11.14	✓	312
27.5	odd	18	inner	108.4.l.a.59.14	yes	312
108.59	even	18	inner	108.4.l.a.59.17	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.14	✓	312	4.3	odd	2	inner
108.4.l.a.11.17	yes	312	1.1	even	1	trivial
108.4.l.a.59.14	yes	312	27.5	odd	18	inner
108.4.l.a.59.17	yes	312	108.59	even	18	inner