Embedded newform 108.2.l.a.11.8

• Label: 108.2.l.a.11.8
• Level: $108$
• Weight: $2$
• Character: 108.11
• Analytic conductor: $0.862$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	108.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.862384341830\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			11.8
Character	\(\chi\)	\(=\)	108.11
Dual form			108.2.l.a.59.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.376137 + 1.36328i) q^{2} +(-0.971698 - 1.43381i) q^{3} +(-1.71704 - 1.02556i) q^{4} +(1.29726 - 3.56419i) q^{5} +(2.32017 - 0.785385i) q^{6} +(-2.58045 - 0.455003i) q^{7} +(2.04396 - 1.95505i) q^{8} +(-1.11160 + 2.78646i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.376137	+	1.36328i	−0.265969	+	0.963982i
							\(3\)	−0.971698	−	1.43381i	−0.561010	−	0.827809i
\(5\)	1.29726	−	3.56419i	0.580152	−	1.59395i								−0.207769	−	0.978178i	\(-0.566620\pi\)
							0.787921	−	0.615776i	\(-0.211157\pi\)
\(7\)	−2.58045	−	0.455003i	−0.975319	−	0.171975i	−0.336796	−	0.941578i	\(-0.609343\pi\)
							−0.638523	+	0.769603i	\(0.720454\pi\)
\(11\)	3.39295	−	1.23493i	1.02301	−	0.372347i	0.224597	−	0.974452i	\(-0.427893\pi\)
							0.798417	+	0.602105i	\(0.205671\pi\)
\(13\)	−0.819980	+	0.688045i	−0.227421	+	0.190829i	−0.749377	−	0.662143i	\(-0.769647\pi\)
							0.521956	+	0.852973i	\(0.325203\pi\)
\(17\)	0.980503	−	0.566093i	0.237807	−	0.137298i	−0.376361	−	0.926473i	\(-0.622825\pi\)
							0.614168	+	0.789175i	\(0.289492\pi\)
\(19\)	0.0627092	+	0.0362052i	0.0143865	+	0.00830603i	0.507176	−	0.861843i	\(-0.330689\pi\)
							−0.492790	+	0.870149i	\(0.664023\pi\)
\(23\)	−0.0731435	−	0.414817i	−0.0152515	−	0.0864954i	0.976232	−	0.216728i	\(-0.0695386\pi\)
							−0.991483	+	0.130233i	\(0.958427\pi\)
\(29\)	−0.192281	+	0.229152i	−0.0357057	+	0.0425524i	−0.783602	−	0.621263i	\(-0.786620\pi\)
							0.747896	+	0.663816i	\(0.231064\pi\)
\(31\)	6.88084	−	1.21328i	1.23583	−	0.217911i	0.482705	−	0.875783i	\(-0.339654\pi\)
							0.753130	+	0.657872i	\(0.228543\pi\)
\(37\)	0.377668	+	0.654141i	0.0620883	+	0.107540i	0.895399	−	0.445265i	\(-0.146891\pi\)
							−0.833310	+	0.552805i	\(0.813557\pi\)
\(41\)	6.36586	+	7.58654i	0.994180	+	1.18482i	0.982761	+	0.184883i	\(0.0591906\pi\)
							0.0114196	+	0.999935i	\(0.496365\pi\)
\(43\)	3.18308	+	8.74545i	0.485416	+	1.33367i	0.904791	+	0.425856i	\(0.140027\pi\)
							−0.419375	+	0.907813i	\(0.637751\pi\)
\(47\)	0.0443203	−	0.251353i	0.00646479	−	0.0366636i	−0.981405	−	0.191950i	\(-0.938519\pi\)
							0.987870	+	0.155286i	\(0.0496300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.2.l.a.11.8	yes	96
3.2	odd	2		324.2.l.a.35.9		96
4.3	odd	2	inner	108.2.l.a.11.3	✓	96
9.2	odd	6		972.2.l.a.755.2		96
9.4	even	3		972.2.l.c.431.4		96
9.5	odd	6		972.2.l.b.431.13		96
9.7	even	3		972.2.l.d.755.15		96
12.11	even	2		324.2.l.a.35.14		96
27.4	even	9		972.2.l.a.215.4		96
27.5	odd	18	inner	108.2.l.a.59.3	yes	96
27.13	even	9		972.2.l.b.539.7		96
27.14	odd	18		972.2.l.c.539.10		96
27.22	even	9		324.2.l.a.287.14		96
27.23	odd	18		972.2.l.d.215.13		96
36.7	odd	6		972.2.l.d.755.13		96
36.11	even	6		972.2.l.a.755.4		96
36.23	even	6		972.2.l.b.431.7		96
36.31	odd	6		972.2.l.c.431.10		96
108.23	even	18		972.2.l.d.215.15		96
108.31	odd	18		972.2.l.a.215.2		96
108.59	even	18	inner	108.2.l.a.59.8	yes	96
108.67	odd	18		972.2.l.b.539.13		96
108.95	even	18		972.2.l.c.539.4		96
108.103	odd	18		324.2.l.a.287.9		96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.2.l.a.11.3	✓	96	4.3	odd	2	inner
108.2.l.a.11.8	yes	96	1.1	even	1	trivial
108.2.l.a.59.3	yes	96	27.5	odd	18	inner
108.2.l.a.59.8	yes	96	108.59	even	18	inner
324.2.l.a.35.9		96	3.2	odd	2
324.2.l.a.35.14		96	12.11	even	2
324.2.l.a.287.9		96	108.103	odd	18
324.2.l.a.287.14		96	27.22	even	9
972.2.l.a.215.2		96	108.31	odd	18
972.2.l.a.215.4		96	27.4	even	9
972.2.l.a.755.2		96	9.2	odd	6
972.2.l.a.755.4		96	36.11	even	6
972.2.l.b.431.7		96	36.23	even	6
972.2.l.b.431.13		96	9.5	odd	6
972.2.l.b.539.7		96	27.13	even	9
972.2.l.b.539.13		96	108.67	odd	18
972.2.l.c.431.4		96	9.4	even	3
972.2.l.c.431.10		96	36.31	odd	6
972.2.l.c.539.4		96	108.95	even	18
972.2.l.c.539.10		96	27.14	odd	18
972.2.l.d.215.13		96	27.23	odd	18
972.2.l.d.215.15		96	108.23	even	18
972.2.l.d.755.13		96	36.7	odd	6
972.2.l.d.755.15		96	9.7	even	3