Embedded newform 81.3.h.a.2.14

• Label: 81.3.h.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	81.2
Dual form			81.3.h.a.41.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.62917 + 0.153131i) q^{2} +(-0.579386 - 2.94352i) q^{3} +(2.91611 + 0.340844i) q^{4} +(-1.61217 - 6.80228i) q^{5} +(-1.07256 - 7.82773i) q^{6} +(6.82619 + 9.16917i) q^{7} +(-2.75970 - 0.486609i) q^{8} +(-8.32862 + 3.41087i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.62917	+	0.153131i	1.31458	+	0.0765657i	0.701055	−	0.713107i	\(-0.252713\pi\)
							0.613528	+	0.789673i	\(0.289750\pi\)
\(3\)	−0.579386	−	2.94352i	−0.193129	−	0.981173i
							\(5\)	−1.61217	−	6.80228i	−0.322434	−	1.36046i	−0.856995	−	0.515325i	\(-0.827671\pi\)
0.534561	−	0.845130i	\(-0.320477\pi\)
\(7\)	6.82619	+	9.16917i	0.975170	+	1.30988i	0.950678	+	0.310181i	\(0.100390\pi\)
							0.0244928	+	0.999700i	\(0.492203\pi\)
\(11\)	12.3255	+	11.6285i	1.12050	+	1.05713i	0.997906	+	0.0646765i	\(0.0206015\pi\)
							0.122590	+	0.992457i	\(0.460880\pi\)
\(13\)	9.14384	+	4.59221i	0.703372	+	0.353247i	0.764269	−	0.644897i	\(-0.223100\pi\)
							−0.0608971	+	0.998144i	\(0.519396\pi\)
\(17\)	1.66535	−	4.57552i	0.0979619	−	0.269148i	−0.881026	−	0.473069i	\(-0.843146\pi\)
							0.978987	+	0.203921i	\(0.0653684\pi\)
\(19\)	−6.46644	+	2.35359i	−0.340339	+	0.123873i	−0.506535	−	0.862220i	\(-0.669074\pi\)
							0.166196	+	0.986093i	\(0.446852\pi\)
\(23\)	−24.3526	−	18.1298i	−1.05881	−	0.788254i	−0.0807814	−	0.996732i	\(-0.525742\pi\)
							−0.978027	+	0.208478i	\(0.933149\pi\)
\(29\)	7.90060	−	12.0123i	0.272434	−	0.414217i	−0.673194	−	0.739466i	\(-0.735078\pi\)
							0.945628	+	0.325250i	\(0.105448\pi\)
\(31\)	−6.74915	−	15.6463i	−0.217714	−	0.504719i	0.773999	−	0.633186i	\(-0.218253\pi\)
							−0.991714	+	0.128468i	\(0.958994\pi\)
\(37\)	−48.5369	+	40.7273i	−1.31181	+	1.10074i	−0.323833	+	0.946114i	\(0.604972\pi\)
							−0.987974	+	0.154622i	\(0.950584\pi\)
\(41\)	−39.9167	+	2.32488i	−0.973579	+	0.0567045i	−0.537553	−	0.843230i	\(-0.680651\pi\)
							−0.436026	+	0.899934i	\(0.643614\pi\)
\(43\)	1.03672	+	3.46290i	0.0241099	+	0.0805326i	0.969198	−	0.246283i	\(-0.0792094\pi\)
							−0.945088	+	0.326816i	\(0.894024\pi\)
\(47\)	53.0224	+	22.8716i	1.12814	+	0.486630i	0.876601	−	0.481218i	\(-0.159805\pi\)
							0.251535	+	0.967848i	\(0.419065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.h.a.2.14	✓	306
3.2	odd	2		243.3.h.a.8.4		306
81.40	even	27		243.3.h.a.152.4		306
81.41	odd	54	inner	81.3.h.a.41.14	yes	306

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.3.h.a.2.14	✓	306	1.1	even	1	trivial
81.3.h.a.41.14	yes	306	81.41	odd	54	inner
243.3.h.a.8.4		306	3.2	odd	2
243.3.h.a.152.4		306	81.40	even	27