Embedded newform 108.4.l.a.59.20

• Label: 108.4.l.a.59.20
• 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			59.20
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11352 + 2.60001i) q^{2} +(-0.629751 + 5.15785i) q^{3} +(-5.52015 - 5.79033i) q^{4} +(4.68740 + 12.8785i) q^{5} +(-12.7092 - 7.38072i) q^{6} +(-21.3342 + 3.76180i) q^{7} +(21.2017 - 7.90483i) q^{8} +(-26.2068 - 6.49632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11352	+	2.60001i	−0.393688	+	0.919244i
							\(3\)	−0.629751	+	5.15785i	−0.121196	+	0.992629i
\(5\)	4.68740	+	12.8785i	0.419253	+	1.15189i								0.952129	+	0.305696i	\(0.0988891\pi\)
							−0.532876	+	0.846193i	\(0.678889\pi\)
\(7\)	−21.3342	+	3.76180i	−1.15194	+	0.203118i	−0.716823	−	0.697255i	\(-0.754404\pi\)
							−0.435118	+	0.900374i	\(0.643293\pi\)
\(11\)	21.5280	+	7.83555i	0.590085	+	0.214773i	0.619767	−	0.784786i	\(-0.287227\pi\)
							−0.0296817	+	0.999559i	\(0.509449\pi\)
\(13\)	15.3981	+	12.9206i	0.328514	+	0.275656i	0.792094	−	0.610399i	\(-0.208991\pi\)
							−0.463580	+	0.886055i	\(0.653435\pi\)
\(17\)	−99.6316	−	57.5223i	−1.42142	−	0.820660i	−0.425003	−	0.905192i	\(-0.639727\pi\)
							−0.996421	+	0.0845322i	\(0.973060\pi\)
\(19\)	50.4854	−	29.1478i	0.609587	−	0.351945i	−0.163217	−	0.986590i	\(-0.552187\pi\)
							0.772804	+	0.634645i	\(0.218854\pi\)
\(23\)	11.3991	−	64.6474i	0.103342	−	0.586083i	−0.888527	−	0.458824i	\(-0.848271\pi\)
							0.991870	−	0.127259i	\(-0.0406180\pi\)
\(29\)	133.108	+	158.632i	0.852327	+	1.01576i	0.999644	+	0.0266747i	\(0.00849183\pi\)
							−0.147317	+	0.989089i	\(0.547064\pi\)
\(31\)	−169.958	−	29.9682i	−0.984689	−	0.173627i	−0.341955	−	0.939716i	\(-0.611089\pi\)
							−0.642734	+	0.766089i	\(0.722200\pi\)
\(37\)	−182.946	+	316.872i	−0.812870	+	1.40793i	0.0979765	+	0.995189i	\(0.468763\pi\)
							−0.910847	+	0.412744i	\(0.864570\pi\)
\(41\)	−234.229	+	279.143i	−0.892206	+	1.06329i	0.105421	+	0.994428i	\(0.466381\pi\)
							−0.997626	+	0.0688616i	\(0.978063\pi\)
\(43\)	−141.773	+	389.519i	−0.502797	+	1.38142i	0.385736	+	0.922609i	\(0.373947\pi\)
							−0.888533	+	0.458813i	\(0.848275\pi\)
\(47\)	41.1721	+	233.498i	0.127778	+	0.724665i	0.979619	+	0.200864i	\(0.0643750\pi\)
							−0.851841	+	0.523800i	\(0.824514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.20	yes	312
4.3	odd	2	inner	108.4.l.a.59.48	yes	312
27.11	odd	18	inner	108.4.l.a.11.48	yes	312
108.11	even	18	inner	108.4.l.a.11.20	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.20	✓	312	108.11	even	18	inner
108.4.l.a.11.48	yes	312	27.11	odd	18	inner
108.4.l.a.59.20	yes	312	1.1	even	1	trivial
108.4.l.a.59.48	yes	312	4.3	odd	2	inner