Embedded newform 108.4.l.a.59.8

• Label: 108.4.l.a.59.8
• 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			59.8
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.61271 - 1.08339i) q^{2} +(2.79011 - 4.38352i) q^{3} +(5.65252 + 5.66118i) q^{4} +(-1.04436 - 2.86937i) q^{5} +(-12.0388 + 8.43010i) q^{6} +(-28.5377 + 5.03197i) q^{7} +(-8.63512 - 20.9149i) q^{8} +(-11.4306 - 24.4610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.61271	−	1.08339i	−0.923733	−	0.383037i
							\(3\)	2.79011	−	4.38352i	0.536957	−	0.843610i
\(5\)	−1.04436	−	2.86937i	−0.0934108	−	0.256644i								0.884184	−	0.467138i	\(-0.154715\pi\)
							−0.977595	+	0.210494i	\(0.932493\pi\)
\(7\)	−28.5377	+	5.03197i	−1.54089	+	0.271701i	−0.878604	−	0.477550i	\(-0.841525\pi\)
							−0.662286	+	0.749251i	\(0.730414\pi\)
\(11\)	41.0637	+	14.9460i	1.12556	+	0.409671i	0.836679	−	0.547693i	\(-0.184494\pi\)
							0.288883	+	0.957364i	\(0.406716\pi\)
\(13\)	−49.8954	−	41.8672i	−1.06450	−	0.893222i	−0.0699573	−	0.997550i	\(-0.522286\pi\)
							−0.994543	+	0.104328i	\(0.966731\pi\)
\(17\)	−54.6110	−	31.5297i	−0.779125	−	0.449828i	0.0569954	−	0.998374i	\(-0.481848\pi\)
							−0.836120	+	0.548547i	\(0.815181\pi\)
\(19\)	−112.171	+	64.7622i	−1.35441	+	0.781971i	−0.988864	−	0.148820i	\(-0.952452\pi\)
							−0.365550	+	0.930792i	\(0.619119\pi\)
\(23\)	−16.6095	+	94.1969i	−0.150579	+	0.853975i	0.812138	+	0.583465i	\(0.198303\pi\)
							−0.962717	+	0.270510i	\(0.912808\pi\)
\(29\)	−50.6360	−	60.3456i	−0.324237	−	0.386410i	0.579162	−	0.815213i	\(-0.303380\pi\)
							−0.903398	+	0.428803i	\(0.858936\pi\)
\(31\)	31.3820	+	5.53349i	0.181818	+	0.0320595i	0.263816	−	0.964573i	\(-0.415019\pi\)
							−0.0819975	+	0.996633i	\(0.526130\pi\)
\(37\)	−178.935	+	309.925i	−0.795048	+	1.37706i	0.127761	+	0.991805i	\(0.459221\pi\)
							−0.922809	+	0.385258i	\(0.874112\pi\)
\(41\)	165.509	−	197.245i	0.630441	−	0.751330i	−0.352387	−	0.935854i	\(-0.614630\pi\)
							0.982828	+	0.184524i	\(0.0590743\pi\)
\(43\)	89.4918	−	245.877i	0.317381	−	0.871997i	−0.673732	−	0.738976i	\(-0.735310\pi\)
							0.991113	−	0.133021i	\(-0.0424679\pi\)
\(47\)	−43.0897	−	244.374i	−0.133729	−	0.758416i	−0.975736	−	0.218950i	\(-0.929737\pi\)
							0.842007	−	0.539467i	\(-0.181374\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.8	yes	312
4.3	odd	2	inner	108.4.l.a.59.22	yes	312
27.11	odd	18	inner	108.4.l.a.11.22	yes	312
108.11	even	18	inner	108.4.l.a.11.8	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.8	✓	312	108.11	even	18	inner
108.4.l.a.11.22	yes	312	27.11	odd	18	inner
108.4.l.a.59.8	yes	312	1.1	even	1	trivial
108.4.l.a.59.22	yes	312	4.3	odd	2	inner