Embedded newform 108.4.l.a.59.3

• Label: 108.4.l.a.59.3
• 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			59.3
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.76593 - 0.591311i) q^{2} +(-5.08011 - 1.09199i) q^{3} +(7.30070 + 3.27105i) q^{4} +(-6.87889 - 18.8996i) q^{5} +(13.4055 + 6.02429i) q^{6} +(22.1047 - 3.89766i) q^{7} +(-18.2590 - 13.3645i) q^{8} +(24.6151 + 11.0949i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.76593	−	0.591311i	−0.977903	−	0.209060i
							\(3\)	−5.08011	−	1.09199i	−0.977668	−	0.210154i
\(5\)	−6.87889	−	18.8996i	−0.615266	−	1.69043i								−0.718283	−	0.695751i	\(-0.755072\pi\)
							0.103016	−	0.994680i	\(-0.467151\pi\)
\(7\)	22.1047	−	3.89766i	1.19354	−	0.210454i	0.458637	−	0.888624i	\(-0.348338\pi\)
							0.734905	+	0.678170i	\(0.237227\pi\)
\(11\)	−8.65672	−	3.15079i	−0.237282	−	0.0863635i	0.220642	−	0.975355i	\(-0.429185\pi\)
							−0.457924	+	0.888991i	\(0.651407\pi\)
\(13\)	−50.5865	−	42.4471i	−1.07924	−	0.905594i	−0.0833869	−	0.996517i	\(-0.526574\pi\)
							−0.995858	+	0.0909234i	\(0.971018\pi\)
\(17\)	−28.8132	−	16.6353i	−0.411072	−	0.237333i	0.280178	−	0.959948i	\(-0.409607\pi\)
							−0.691250	+	0.722615i	\(0.742940\pi\)
\(19\)	−47.2000	+	27.2509i	−0.569917	+	0.329042i	−0.757116	−	0.653280i	\(-0.773392\pi\)
							0.187199	+	0.982322i	\(0.440059\pi\)
\(23\)	−24.3591	+	138.147i	−0.220836	+	1.25242i	0.649651	+	0.760233i	\(0.274915\pi\)
							−0.870487	+	0.492191i	\(0.836196\pi\)
\(29\)	60.0544	+	71.5700i	0.384546	+	0.458284i	0.923243	−	0.384215i	\(-0.125528\pi\)
							−0.538698	+	0.842499i	\(0.681084\pi\)
\(31\)	−13.8120	−	2.43543i	−0.0800231	−	0.0141102i	0.133493	−	0.991050i	\(-0.457381\pi\)
							−0.213516	+	0.976939i	\(0.568492\pi\)
\(37\)	171.170	−	296.475i	0.760545	−	1.31730i	−0.182025	−	0.983294i	\(-0.558265\pi\)
							0.942570	−	0.334009i	\(-0.108402\pi\)
\(41\)	39.0468	−	46.5342i	0.148734	−	0.177254i	−0.686533	−	0.727098i	\(-0.740868\pi\)
							0.835267	+	0.549844i	\(0.185313\pi\)
\(43\)	−123.873	+	340.337i	−0.439312	+	1.20700i	0.500629	+	0.865662i	\(0.333102\pi\)
							−0.939941	+	0.341337i	\(0.889120\pi\)
\(47\)	−38.5295	−	218.511i	−0.119577	−	0.678152i	−0.984382	−	0.176045i	\(-0.943669\pi\)
							0.864805	−	0.502107i	\(-0.167442\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.3	yes	312
4.3	odd	2	inner	108.4.l.a.59.26	yes	312
27.11	odd	18	inner	108.4.l.a.11.26	yes	312
108.11	even	18	inner	108.4.l.a.11.3	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.3	✓	312	108.11	even	18	inner
108.4.l.a.11.26	yes	312	27.11	odd	18	inner
108.4.l.a.59.3	yes	312	1.1	even	1	trivial
108.4.l.a.59.26	yes	312	4.3	odd	2	inner