Embedded newform 108.4.l.a.59.30

• Label: 108.4.l.a.59.30
• 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			59.30
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.274863 - 2.81504i) q^{2} +(2.56535 + 4.51874i) q^{3} +(-7.84890 - 1.54750i) q^{4} +(2.35563 + 6.47204i) q^{5} +(13.4255 - 5.97952i) q^{6} +(6.19890 - 1.09303i) q^{7} +(-6.51364 + 21.6696i) q^{8} +(-13.8380 + 23.1843i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.274863	−	2.81504i	0.0971786	−	0.995267i
							\(3\)	2.56535	+	4.51874i	0.493701	+	0.869632i
\(5\)	2.35563	+	6.47204i	0.210694	+	0.578877i								0.999353	−	0.0359534i	\(-0.0114468\pi\)
							−0.788660	+	0.614830i	\(0.789225\pi\)
\(7\)	6.19890	−	1.09303i	0.334709	−	0.0590183i	−0.00376784	−	0.999993i	\(-0.501199\pi\)
							0.338477	+	0.940975i	\(0.390088\pi\)
\(11\)	20.4109	+	7.42895i	0.559465	+	0.203628i	0.606247	−	0.795277i	\(-0.292674\pi\)
							−0.0467821	+	0.998905i	\(0.514897\pi\)
\(13\)	59.5758	+	49.9900i	1.27103	+	1.06652i	0.994415	+	0.105537i	\(0.0336562\pi\)
							0.276612	+	0.960982i	\(0.410788\pi\)
\(17\)	34.1099	+	19.6934i	0.486640	+	0.280962i	0.723179	−	0.690660i	\(-0.242680\pi\)
							−0.236540	+	0.971622i	\(0.576013\pi\)
\(19\)	−102.949	+	59.4377i	−1.24306	+	0.717682i	−0.969716	−	0.244234i	\(-0.921463\pi\)
							−0.273345	+	0.961916i	\(0.588130\pi\)
\(23\)	29.8149	−	169.088i	0.270297	−	1.53293i	−0.483219	−	0.875500i	\(-0.660532\pi\)
							0.753516	−	0.657430i	\(-0.228356\pi\)
\(29\)	13.5934	+	16.1999i	0.0870422	+	0.103733i	0.807808	−	0.589446i	\(-0.200654\pi\)
							−0.720765	+	0.693179i	\(0.756210\pi\)
\(31\)	−123.473	−	21.7717i	−0.715370	−	0.126139i	−0.195895	−	0.980625i	\(-0.562761\pi\)
							−0.519475	+	0.854486i	\(0.673872\pi\)
\(37\)	−100.091	+	173.363i	−0.444727	+	0.770290i	−0.998033	−	0.0626882i	\(-0.980033\pi\)
							0.553306	+	0.832978i	\(0.313366\pi\)
\(41\)	154.750	−	184.424i	0.589461	−	0.702492i	−0.386041	−	0.922481i	\(-0.626158\pi\)
							0.975502	+	0.219989i	\(0.0706023\pi\)
\(43\)	20.3028	−	55.7814i	0.0720034	−	0.197828i	−0.898470	−	0.439034i	\(-0.855321\pi\)
							0.970474	+	0.241206i	\(0.0775430\pi\)
\(47\)	−45.4402	−	257.704i	−0.141024	−	0.799788i	−0.970474	−	0.241207i	\(-0.922457\pi\)
							0.829450	−	0.558582i	\(-0.188654\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.30	yes	312
4.3	odd	2	inner	108.4.l.a.59.1	yes	312
27.11	odd	18	inner	108.4.l.a.11.1	✓	312
108.11	even	18	inner	108.4.l.a.11.30	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.1	✓	312	27.11	odd	18	inner
108.4.l.a.11.30	yes	312	108.11	even	18	inner
108.4.l.a.59.1	yes	312	4.3	odd	2	inner
108.4.l.a.59.30	yes	312	1.1	even	1	trivial