Embedded newform 108.5.j.a.7.1

• Label: 108.5.j.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	108.7
Dual form			108.5.j.a.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.99762 - 0.137857i) q^{2} +(5.81196 - 6.87176i) q^{3} +(15.9620 + 1.10220i) q^{4} +(-8.59701 - 48.7561i) q^{5} +(-24.1814 + 26.6695i) q^{6} +(32.6195 - 38.8744i) q^{7} +(-63.6581 - 6.60665i) q^{8} +(-13.4422 - 79.8768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.99762	−	0.137857i	−0.999406	−	0.0344642i
							\(3\)	5.81196	−	6.87176i	0.645774	−	0.763529i
\(5\)	−8.59701	−	48.7561i	−0.343881	−	1.95024i								−0.309723	−	0.950827i	\(-0.600236\pi\)
							−0.0341580	−	0.999416i	\(-0.510875\pi\)
\(7\)	32.6195	−	38.8744i	0.665704	−	0.793355i	−0.322488	−	0.946574i	\(-0.604519\pi\)
							0.988192	+	0.153218i	\(0.0489637\pi\)
\(11\)	−36.7617	−	6.48208i	−0.303816	−	0.0535709i	0.0196621	−	0.999807i	\(-0.493741\pi\)
							−0.323478	+	0.946236i	\(0.604852\pi\)
\(13\)	62.6299	+	22.7954i	0.370591	+	0.134884i	0.520600	−	0.853800i	\(-0.325708\pi\)
							−0.150009	+	0.988685i	\(0.547930\pi\)
\(17\)	192.604	+	333.601i	0.666451	+	1.15433i	0.978890	+	0.204389i	\(0.0655208\pi\)
							−0.312439	+	0.949938i	\(0.601146\pi\)
\(19\)	421.117	+	243.132i	1.16653	+	0.673496i	0.952860	−	0.303411i	\(-0.0981254\pi\)
							0.213669	+	0.976906i	\(0.431459\pi\)
\(23\)	−330.594	−	393.986i	−0.624941	−	0.744775i	0.356971	−	0.934116i	\(-0.383810\pi\)
							−0.981912	+	0.189340i	\(0.939365\pi\)
\(29\)	170.909	−	62.2057i	0.203221	−	0.0739664i	−0.238404	−	0.971166i	\(-0.576624\pi\)
							0.441625	+	0.897200i	\(0.354402\pi\)
\(31\)	−290.749	−	346.501i	−0.302548	−	0.360563i	0.593254	−	0.805015i	\(-0.297843\pi\)
							−0.895803	+	0.444452i	\(0.853398\pi\)
\(37\)	167.651	+	290.380i	0.122463	+	0.212111i	0.920738	−	0.390181i	\(-0.127588\pi\)
							−0.798276	+	0.602292i	\(0.794254\pi\)
\(41\)	1844.58	+	671.373i	1.09731	+	0.399389i	0.826325	−	0.563194i	\(-0.190428\pi\)
							0.270988	+	0.962583i	\(0.412650\pi\)
\(43\)	−899.006	−	158.519i	−0.486212	−	0.0857323i	−0.0748320	−	0.997196i	\(-0.523842\pi\)
							−0.411380	+	0.911464i	\(0.634953\pi\)
\(47\)	2077.74	−	2476.15i	0.940577	−	1.12094i	−0.0519175	−	0.998651i	\(-0.516533\pi\)
							0.992495	−	0.122285i	\(-0.0390223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.j.a.7.1	✓	420
4.3	odd	2	inner	108.5.j.a.7.16	yes	420
27.4	even	9	inner	108.5.j.a.31.16	yes	420
108.31	odd	18	inner	108.5.j.a.31.1	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.j.a.7.1	✓	420	1.1	even	1	trivial
108.5.j.a.7.16	yes	420	4.3	odd	2	inner
108.5.j.a.31.1	yes	420	108.31	odd	18	inner
108.5.j.a.31.16	yes	420	27.4	even	9	inner