Embedded newform 81.3.h.a.2.9

• Label: 81.3.h.a.2.9
• Level: $81$
• Weight: $3$
• Character: 81.2
• Analytic conductor: $2.207$
• Analytic rank: $0$
• Dimension: $306$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	81.h (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.20709014132\)
Analytic rank:	\(0\)
Dimension:	\(306\)
Relative dimension:	\(17\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	81.2
Dual form			81.3.h.a.41.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.272030 - 0.0158440i) q^{2} +(2.02197 + 2.21622i) q^{3} +(-3.89920 - 0.455752i) q^{4} +(0.613475 + 2.58845i) q^{5} +(-0.514925 - 0.634914i) q^{6} +(5.77313 + 7.75466i) q^{7} +(2.12689 + 0.375028i) q^{8} +(-0.823240 + 8.96227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.272030	−	0.0158440i	−0.136015	−	0.00792198i	−0.00999782	−	0.999950i	\(-0.503182\pi\)
							−0.126017	+	0.992028i	\(0.540219\pi\)
\(3\)	2.02197	+	2.21622i	0.673991	+	0.738739i
							\(5\)	0.613475	+	2.58845i	0.122695	+	0.517691i	0.999332	+	0.0365388i	\(0.0116332\pi\)
−0.876637	+	0.481152i	\(0.840219\pi\)
\(7\)	5.77313	+	7.75466i	0.824733	+	1.10781i	0.992532	+	0.121988i	\(0.0389268\pi\)
							−0.167798	+	0.985821i	\(0.553666\pi\)
\(11\)	−5.84163	−	5.51130i	−0.531058	−	0.501027i	0.373353	−	0.927689i	\(-0.378208\pi\)
							−0.904411	+	0.426662i	\(0.859689\pi\)
\(13\)	0.939216	+	0.471692i	0.0722474	+	0.0362840i	0.484556	−	0.874760i	\(-0.338981\pi\)
							−0.412309	+	0.911044i	\(0.635277\pi\)
\(17\)	0.554194	−	1.52264i	0.0325997	−	0.0895668i	−0.922325	−	0.386415i	\(-0.873713\pi\)
							0.954925	+	0.296848i	\(0.0959355\pi\)
\(19\)	24.0582	−	8.75646i	1.26622	−	0.460866i	0.380369	−	0.924835i	\(-0.375797\pi\)
							0.885852	+	0.463968i	\(0.153575\pi\)
\(23\)	−34.0875	−	25.3772i	−1.48206	−	1.10336i	−0.969614	−	0.244638i	\(-0.921331\pi\)
							−0.512449	−	0.858717i	\(-0.671262\pi\)
\(29\)	24.4153	−	37.1217i	0.841908	−	1.28006i	−0.116078	−	0.993240i	\(-0.537032\pi\)
							0.957985	−	0.286818i	\(-0.0925975\pi\)
\(31\)	2.95290	+	6.84558i	0.0952547	+	0.220825i	0.959181	−	0.282793i	\(-0.0912610\pi\)
							−0.863926	+	0.503618i	\(0.832002\pi\)
\(37\)	−29.6352	+	24.8669i	−0.800950	+	0.672077i	−0.948430	−	0.316988i	\(-0.897328\pi\)
							0.147479	+	0.989065i	\(0.452884\pi\)
\(41\)	26.4135	−	1.53841i	0.644231	−	0.0375222i	0.267081	−	0.963674i	\(-0.413941\pi\)
							0.377151	+	0.926152i	\(0.376904\pi\)
\(43\)	−1.97512	−	6.59737i	−0.0459331	−	0.153427i	0.931946	−	0.362596i	\(-0.118109\pi\)
							−0.977880	+	0.209169i	\(0.932924\pi\)
\(47\)	69.5827	+	30.0150i	1.48048	+	0.638618i	0.974594	−	0.223981i	\(-0.0719052\pi\)
							0.505889	+	0.862598i	\(0.331164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.h.a.2.9	✓	306
3.2	odd	2		243.3.h.a.8.9		306
81.40	even	27		243.3.h.a.152.9		306
81.41	odd	54	inner	81.3.h.a.41.9	yes	306

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.3.h.a.2.9	✓	306	1.1	even	1	trivial
81.3.h.a.41.9	yes	306	81.41	odd	54	inner
243.3.h.a.8.9		306	3.2	odd	2
243.3.h.a.152.9		306	81.40	even	27