Embedded newform 81.3.h.a.2.10

• Label: 81.3.h.a.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	81.2
Dual form			81.3.h.a.41.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.218910 - 0.0127500i) q^{2} +(0.445650 - 2.96671i) q^{3} +(-3.92519 - 0.458789i) q^{4} +(-0.196368 - 0.828540i) q^{5} +(-0.135383 + 0.643760i) q^{6} +(-2.81710 - 3.78402i) q^{7} +(1.71721 + 0.302791i) q^{8} +(-8.60279 - 2.64424i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.218910	−	0.0127500i	−0.109455	−	0.00637502i	0.00332998	−	0.999994i	\(-0.498940\pi\)
							−0.112785	+	0.993619i	\(0.535977\pi\)
\(3\)	0.445650	−	2.96671i	0.148550	−	0.988905i
							\(5\)	−0.196368	−	0.828540i	−0.0392735	−	0.165708i	0.949874	−	0.312632i	\(-0.101211\pi\)
−0.989148	+	0.146924i	\(0.953063\pi\)
\(7\)	−2.81710	−	3.78402i	−0.402443	−	0.540574i	0.553992	−	0.832522i	\(-0.313104\pi\)
							−0.956434	+	0.291948i	\(0.905697\pi\)
\(11\)	−13.2477	−	12.4986i	−1.20434	−	1.13624i	−0.987098	−	0.160120i	\(-0.948812\pi\)
							−0.217243	−	0.976117i	\(-0.569706\pi\)
\(13\)	21.5238	+	10.8097i	1.65568	+	0.831513i	0.996858	+	0.0792127i	\(0.0252406\pi\)
							0.658821	+	0.752300i	\(0.271056\pi\)
\(17\)	9.73273	−	26.7405i	0.572514	−	1.57297i	−0.228005	−	0.973660i	\(-0.573220\pi\)
							0.800518	−	0.599308i	\(-0.204558\pi\)
\(19\)	−16.4605	+	5.99113i	−0.866341	+	0.315323i	−0.736685	−	0.676236i	\(-0.763610\pi\)
							−0.129657	+	0.991559i	\(0.541388\pi\)
\(23\)	17.0591	+	12.7000i	0.741699	+	0.552174i	0.900353	−	0.435160i	\(-0.143308\pi\)
							−0.158655	+	0.987334i	\(0.550716\pi\)
\(29\)	8.64601	−	13.1456i	0.298138	−	0.453297i	−0.655089	−	0.755552i	\(-0.727369\pi\)
							0.953227	+	0.302254i	\(0.0977392\pi\)
\(31\)	10.5139	+	24.3739i	0.339157	+	0.786255i	0.999381	+	0.0351859i	\(0.0112023\pi\)
							−0.660223	+	0.751069i	\(0.729538\pi\)
\(37\)	−0.715692	+	0.600537i	−0.0193430	+	0.0162307i	−0.652408	−	0.757868i	\(-0.726241\pi\)
							0.633065	+	0.774099i	\(0.281797\pi\)
\(41\)	−33.9480	+	1.97725i	−0.828001	+	0.0482256i	−0.466910	−	0.884305i	\(-0.654633\pi\)
							−0.361091	+	0.932531i	\(0.617596\pi\)
\(43\)	−9.09025	−	30.3636i	−0.211401	−	0.706129i	−0.996256	−	0.0864498i	\(-0.972448\pi\)
							0.784855	−	0.619680i	\(-0.212737\pi\)
\(47\)	−51.2127	−	22.0910i	−1.08963	−	0.470022i	−0.225992	−	0.974129i	\(-0.572562\pi\)
							−0.863641	+	0.504107i	\(0.831822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.h.a.2.10	✓	306
3.2	odd	2		243.3.h.a.8.8		306
81.40	even	27		243.3.h.a.152.8		306
81.41	odd	54	inner	81.3.h.a.41.10	yes	306

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.3.h.a.2.10	✓	306	1.1	even	1	trivial
81.3.h.a.41.10	yes	306	81.41	odd	54	inner
243.3.h.a.8.8		306	3.2	odd	2
243.3.h.a.152.8		306	81.40	even	27