Embedded newform 81.3.h.a.2.15

• Label: 81.3.h.a.2.15
• 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			2.15
Character	\(\chi\)	\(=\)	81.2
Dual form			81.3.h.a.41.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.79319 + 0.162685i) q^{2} +(1.52144 + 2.58558i) q^{3} +(3.80251 + 0.444450i) q^{4} +(-0.618939 - 2.61151i) q^{5} +(3.82905 + 7.46953i) q^{6} +(-3.52360 - 4.73302i) q^{7} +(-0.472829 - 0.0833725i) q^{8} +(-4.37041 + 7.86762i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.79319	+	0.162685i	1.39660	+	0.0813425i	0.739852	−	0.672770i	\(-0.234896\pi\)
							0.656745	+	0.754112i	\(0.271933\pi\)
\(3\)	1.52144	+	2.58558i	0.507148	+	0.861859i
							\(5\)	−0.618939	−	2.61151i	−0.123788	−	0.522302i	−0.999239	−	0.0389960i	\(-0.987584\pi\)
0.875452	−	0.483306i	\(-0.160564\pi\)
\(7\)	−3.52360	−	4.73302i	−0.503372	−	0.676146i	0.475712	−	0.879601i	\(-0.342190\pi\)
							−0.979084	+	0.203455i	\(0.934783\pi\)
\(11\)	1.92260	+	1.81388i	0.174782	+	0.164898i	0.768662	−	0.639655i	\(-0.220923\pi\)
							−0.593880	+	0.804554i	\(0.702404\pi\)
\(13\)	4.16369	+	2.09108i	0.320284	+	0.160853i	0.601676	−	0.798740i	\(-0.294500\pi\)
							−0.281392	+	0.959593i	\(0.590796\pi\)
\(17\)	3.19119	−	8.76773i	0.187717	−	0.515749i	−0.809758	−	0.586764i	\(-0.800402\pi\)
							0.997475	+	0.0710152i	\(0.0226239\pi\)
\(19\)	5.04651	−	1.83678i	0.265606	−	0.0966725i	−0.205785	−	0.978597i	\(-0.565975\pi\)
							0.471390	+	0.881925i	\(0.343752\pi\)
\(23\)	−6.19117	−	4.60916i	−0.269181	−	0.200398i	0.454101	−	0.890950i	\(-0.349960\pi\)
							−0.723283	+	0.690552i	\(0.757368\pi\)
\(29\)	−23.2482	+	35.3471i	−0.801661	+	1.21887i	0.170686	+	0.985325i	\(0.445402\pi\)
							−0.972347	+	0.233541i	\(0.924969\pi\)
\(31\)	21.3325	+	49.4543i	0.688146	+	1.59530i	0.798907	+	0.601454i	\(0.205412\pi\)
							−0.110761	+	0.993847i	\(0.535329\pi\)
\(37\)	51.9840	−	43.6198i	1.40497	−	1.17891i	0.446133	−	0.894967i	\(-0.352801\pi\)
							0.958840	−	0.283946i	\(-0.0916437\pi\)
\(41\)	−67.0181	+	3.90336i	−1.63459	+	0.0952038i	−0.850683	−	0.525679i	\(-0.823811\pi\)
							−0.783904	+	0.620883i	\(0.786774\pi\)
\(43\)	13.0248	+	43.5058i	0.302902	+	1.01176i	0.964979	+	0.262326i	\(0.0844897\pi\)
							−0.662078	+	0.749435i	\(0.730325\pi\)
\(47\)	−19.7479	−	8.51840i	−0.420168	−	0.181243i	0.175490	−	0.984481i	\(-0.443849\pi\)
							−0.595657	+	0.803239i	\(0.703108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.h.a.2.15	✓	306
3.2	odd	2		243.3.h.a.8.3		306
81.40	even	27		243.3.h.a.152.3		306
81.41	odd	54	inner	81.3.h.a.41.15	yes	306

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.3.h.a.2.15	✓	306	1.1	even	1	trivial
81.3.h.a.41.15	yes	306	81.41	odd	54	inner
243.3.h.a.8.3		306	3.2	odd	2
243.3.h.a.152.3		306	81.40	even	27