Embedded newform 108.4.l.a.11.9

• Label: 108.4.l.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	108.11
Dual form			108.4.l.a.59.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60460 - 1.10274i) q^{2} +(2.16828 + 4.72213i) q^{3} +(5.56791 + 5.74442i) q^{4} +(-0.836765 + 2.29899i) q^{5} +(-0.440216 - 14.6903i) q^{6} +(22.6981 + 4.00229i) q^{7} +(-8.16758 - 21.1019i) q^{8} +(-17.5971 + 20.4779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60460	−	1.10274i	−0.920866	−	0.389879i
							\(3\)	2.16828	+	4.72213i	0.417286	+	0.908775i
\(5\)	−0.836765	+	2.29899i	−0.0748425	+	0.205628i								−0.971472	−	0.237153i	\(-0.923786\pi\)
							0.896630	+	0.442781i	\(0.146008\pi\)
\(7\)	22.6981	+	4.00229i	1.22558	+	0.216103i	0.748727	−	0.662879i	\(-0.230666\pi\)
							0.476855	+	0.878982i	\(0.341777\pi\)
\(11\)	2.30985	−	0.840718i	0.0633134	−	0.0230442i	−0.310169	−	0.950681i	\(-0.600386\pi\)
							0.373483	+	0.927637i	\(0.378164\pi\)
\(13\)	−22.0818	+	18.5288i	−0.471106	+	0.395305i	−0.847198	−	0.531277i	\(-0.821712\pi\)
							0.376092	+	0.926583i	\(0.377268\pi\)
\(17\)	−42.9739	+	24.8110i	−0.613100	+	0.353973i	−0.774178	−	0.632968i	\(-0.781836\pi\)
							0.161078	+	0.986942i	\(0.448503\pi\)
\(19\)	53.4555	+	30.8625i	0.645449	+	0.372650i	0.786710	−	0.617322i	\(-0.211783\pi\)
							−0.141262	+	0.989972i	\(0.545116\pi\)
\(23\)	18.5100	+	104.975i	0.167809	+	0.951690i	0.946121	+	0.323813i	\(0.104965\pi\)
							−0.778313	+	0.627877i	\(0.783924\pi\)
\(29\)	−13.7345	+	16.3682i	−0.0879462	+	0.104810i	−0.808223	−	0.588877i	\(-0.799570\pi\)
							0.720277	+	0.693687i	\(0.244015\pi\)
\(31\)	−119.672	+	21.1014i	−0.693347	+	0.122256i	−0.509207	−	0.860644i	\(-0.670061\pi\)
							−0.184140	+	0.982900i	\(0.558950\pi\)
\(37\)	146.742	+	254.165i	0.652008	+	1.12931i	0.982635	+	0.185549i	\(0.0594064\pi\)
							−0.330627	+	0.943761i	\(0.607260\pi\)
\(41\)	−230.454	−	274.644i	−0.877824	−	1.04615i	−0.998570	−	0.0534666i	\(-0.982973\pi\)
							0.120745	−	0.992684i	\(-0.461472\pi\)
\(43\)	−101.863	−	279.866i	−0.361255	−	0.992539i	−0.978586	−	0.205836i	\(-0.934009\pi\)
							0.617332	−	0.786703i	\(-0.288214\pi\)
\(47\)	85.6590	−	485.796i	0.265844	−	1.50767i	−0.500779	−	0.865575i	\(-0.666953\pi\)
							0.766622	−	0.642098i	\(-0.221936\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.11.9	✓	312
4.3	odd	2	inner	108.4.l.a.11.38	yes	312
27.5	odd	18	inner	108.4.l.a.59.38	yes	312
108.59	even	18	inner	108.4.l.a.59.9	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.9	✓	312	1.1	even	1	trivial
108.4.l.a.11.38	yes	312	4.3	odd	2	inner
108.4.l.a.59.9	yes	312	108.59	even	18	inner
108.4.l.a.59.38	yes	312	27.5	odd	18	inner