Embedded newform 108.5.j.a.7.8

• Label: 108.5.j.a.7.8
• Level: $108$
• Weight: $5$
• Character: 108.7
• Analytic conductor: $11.164$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1639560131\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			7.8
Character	\(\chi\)	\(=\)	108.7
Dual form			108.5.j.a.31.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.76323 - 1.35575i) q^{2} +(4.63322 + 7.71578i) q^{3} +(12.3239 + 10.2040i) q^{4} +(-5.82341 - 33.0262i) q^{5} +(-6.97519 - 35.3178i) q^{6} +(-25.3950 + 30.2645i) q^{7} +(-32.5435 - 55.1083i) q^{8} +(-38.0666 + 71.4978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.76323	−	1.35575i	−0.940809	−	0.338938i
							\(3\)	4.63322	+	7.71578i	0.514802	+	0.857309i
\(5\)	−5.82341	−	33.0262i	−0.232936	−	1.32105i								−0.846916	−	0.531726i	\(-0.821544\pi\)
							0.613980	−	0.789322i	\(-0.289568\pi\)
\(7\)	−25.3950	+	30.2645i	−0.518264	+	0.617643i	−0.960169	−	0.279418i	\(-0.909858\pi\)
							0.441905	+	0.897062i	\(0.354303\pi\)
\(11\)	48.1603	+	8.49196i	0.398019	+	0.0701815i	0.369075	−	0.929400i	\(-0.379674\pi\)
							0.0289439	+	0.999581i	\(0.490786\pi\)
\(13\)	−119.837	−	43.6172i	−0.709096	−	0.258090i	−0.0378065	−	0.999285i	\(-0.512037\pi\)
							−0.671290	+	0.741195i	\(0.734259\pi\)
\(17\)	−107.004	−	185.337i	−0.370257	−	0.641305i	0.619348	−	0.785117i	\(-0.287397\pi\)
							−0.989605	+	0.143812i	\(0.954064\pi\)
\(19\)	−304.406	−	175.749i	−0.843231	−	0.486840i	0.0151303	−	0.999886i	\(-0.495184\pi\)
							−0.858361	+	0.513046i	\(0.828517\pi\)
\(23\)	64.3179	+	76.6511i	0.121584	+	0.144898i	0.823403	−	0.567457i	\(-0.192073\pi\)
							−0.701819	+	0.712356i	\(0.747628\pi\)
\(29\)	−1217.57	+	443.158i	−1.44776	+	0.526942i	−0.941965	−	0.335712i	\(-0.891023\pi\)
							−0.505795	+	0.862654i	\(0.668801\pi\)
\(31\)	−762.924	−	909.217i	−0.793886	−	0.946116i	0.205585	−	0.978639i	\(-0.434090\pi\)
							−0.999471	+	0.0325232i	\(0.989646\pi\)
\(37\)	−403.723	−	699.268i	−0.294903	−	0.510787i	0.680059	−	0.733157i	\(-0.261954\pi\)
							−0.974962	+	0.222370i	\(0.928621\pi\)
\(41\)	−2627.10	−	956.186i	−1.56282	−	0.568820i	−0.591439	−	0.806350i	\(-0.701440\pi\)
							−0.971380	+	0.237530i	\(0.923662\pi\)
\(43\)	−357.633	−	63.0603i	−0.193419	−	0.0341051i	0.0760991	−	0.997100i	\(-0.475753\pi\)
							−0.269519	+	0.962995i	\(0.586865\pi\)
\(47\)	2633.91	−	3138.97i	1.19235	−	1.42099i	0.309787	−	0.950806i	\(-0.399742\pi\)
							0.882567	−	0.470186i	\(-0.155813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.j.a.7.8	✓	420
4.3	odd	2	inner	108.5.j.a.7.25	yes	420
27.4	even	9	inner	108.5.j.a.31.25	yes	420
108.31	odd	18	inner	108.5.j.a.31.8	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.j.a.7.8	✓	420	1.1	even	1	trivial
108.5.j.a.7.25	yes	420	4.3	odd	2	inner
108.5.j.a.31.8	yes	420	108.31	odd	18	inner
108.5.j.a.31.25	yes	420	27.4	even	9	inner