Embedded newform 108.4.l.a.11.1

• Label: 108.4.l.a.11.1
• Level: $108$
• Weight: $4$
• Character: 108.11
• 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			11.1
Character	\(\chi\)	\(=\)	108.11
Dual form			108.4.l.a.59.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.82000 + 0.218140i) q^{2} +(-2.56535 + 4.51874i) q^{3} +(7.90483 - 1.23031i) q^{4} +(2.35563 - 6.47204i) q^{5} +(6.24857 - 13.3025i) q^{6} +(-6.19890 - 1.09303i) q^{7} +(-22.0233 + 5.19383i) q^{8} +(-13.8380 - 23.1843i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82000	+	0.218140i	−0.997022	+	0.0771241i
							\(3\)	−2.56535	+	4.51874i	−0.493701	+	0.869632i
\(5\)	2.35563	−	6.47204i	0.210694	−	0.578877i								−0.788660	−	0.614830i	\(-0.789225\pi\)
							0.999353	+	0.0359534i	\(0.0114468\pi\)
\(7\)	−6.19890	−	1.09303i	−0.334709	−	0.0590183i	0.00376784	−	0.999993i	\(-0.498801\pi\)
							−0.338477	+	0.940975i	\(0.609912\pi\)
\(11\)	−20.4109	+	7.42895i	−0.559465	+	0.203628i	−0.606247	−	0.795277i	\(-0.707326\pi\)
							0.0467821	+	0.998905i	\(0.485103\pi\)
\(13\)	59.5758	−	49.9900i	1.27103	−	1.06652i	0.276612	−	0.960982i	\(-0.410788\pi\)
							0.994415	−	0.105537i	\(-0.0336562\pi\)
\(17\)	34.1099	−	19.6934i	0.486640	−	0.280962i	−0.236540	−	0.971622i	\(-0.576013\pi\)
							0.723179	+	0.690660i	\(0.242680\pi\)
\(19\)	102.949	+	59.4377i	1.24306	+	0.717682i	0.969716	−	0.244234i	\(-0.0785366\pi\)
							0.273345	+	0.961916i	\(0.411870\pi\)
\(23\)	−29.8149	−	169.088i	−0.270297	−	1.53293i	−0.753516	−	0.657430i	\(-0.771644\pi\)
							0.483219	−	0.875500i	\(-0.339468\pi\)
\(29\)	13.5934	−	16.1999i	0.0870422	−	0.103733i	−0.720765	−	0.693179i	\(-0.756210\pi\)
							0.807808	+	0.589446i	\(0.200654\pi\)
\(31\)	123.473	−	21.7717i	0.715370	−	0.126139i	0.195895	−	0.980625i	\(-0.437239\pi\)
							0.519475	+	0.854486i	\(0.326128\pi\)
\(37\)	−100.091	−	173.363i	−0.444727	−	0.770290i	0.553306	−	0.832978i	\(-0.313366\pi\)
							−0.998033	+	0.0626882i	\(0.980033\pi\)
\(41\)	154.750	+	184.424i	0.589461	+	0.702492i	0.975502	−	0.219989i	\(-0.0706023\pi\)
							−0.386041	+	0.922481i	\(0.626158\pi\)
\(43\)	−20.3028	−	55.7814i	−0.0720034	−	0.197828i	0.898470	−	0.439034i	\(-0.144679\pi\)
							−0.970474	+	0.241206i	\(0.922457\pi\)
\(47\)	45.4402	−	257.704i	0.141024	−	0.799788i	−0.829450	−	0.558582i	\(-0.811346\pi\)
							0.970474	−	0.241207i	\(-0.0775431\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.1	✓	312	1.1	even	1	trivial
108.4.l.a.11.30	yes	312	4.3	odd	2	inner
108.4.l.a.59.1	yes	312	108.59	even	18	inner
108.4.l.a.59.30	yes	312	27.5	odd	18	inner