Embedded newform 81.2.g.a.13.1

• Label: 81.2.g.a.13.1
• Level: $81$
• Weight: $2$
• Character: 81.13
• Analytic conductor: $0.647$
• Analytic rank: $0$
• Dimension: $144$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	81.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.646788256372\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(8\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	81.13
Dual form			81.2.g.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38576 - 1.19817i) q^{2} +(1.71560 - 0.238151i) q^{3} +(3.06192 + 4.11287i) q^{4} +(0.727126 + 2.42877i) q^{5} +(-4.37836 - 1.48742i) q^{6} +(0.202369 + 0.469143i) q^{7} +(-1.44988 - 8.22270i) q^{8} +(2.88657 - 0.817145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38576	−	1.19817i	−1.68699	−	0.847237i	−0.991710	−	0.128497i	\(-0.958985\pi\)
							−0.695278	−	0.718741i	\(-0.744719\pi\)
\(3\)	1.71560	−	0.238151i	0.990502	−	0.137497i
							\(5\)	0.727126	+	2.42877i	0.325181	+	1.08618i	0.951766	+	0.306826i	\(0.0992670\pi\)
−0.626585	+	0.779353i	\(0.715548\pi\)
\(7\)	0.202369	+	0.469143i	0.0764882	+	0.177320i	0.952170	−	0.305569i	\(-0.0988468\pi\)
							−0.875682	+	0.482889i	\(0.839588\pi\)
\(11\)	1.49541	−	0.354419i	0.450883	−	0.106861i	0.00110057	−	0.999999i	\(-0.499650\pi\)
							0.449782	+	0.893138i	\(0.351502\pi\)
\(13\)	−4.71325	−	3.09996i	−1.30722	−	0.859773i	−0.311327	−	0.950303i	\(-0.600774\pi\)
							−0.995894	+	0.0905298i	\(0.971144\pi\)
\(17\)	−1.33086	+	0.484393i	−0.322781	+	0.117483i	−0.498328	−	0.866988i	\(-0.666053\pi\)
							0.175548	+	0.984471i	\(0.443830\pi\)
\(19\)	0.986977	+	0.359230i	0.226428	+	0.0824131i	0.452743	−	0.891641i	\(-0.350446\pi\)
							−0.226315	+	0.974054i	\(0.572668\pi\)
\(23\)	0.103125	−	0.239071i	0.0215031	−	0.0498497i	−0.907126	−	0.420859i	\(-0.861729\pi\)
							0.928629	+	0.371009i	\(0.120988\pi\)
\(29\)	−0.103412	−	1.77552i	−0.0192031	−	0.329705i	−0.994204	−	0.107506i	\(-0.965713\pi\)
							0.975001	−	0.222199i	\(-0.0713235\pi\)
\(31\)	−5.13181	+	0.599822i	−0.921700	+	0.107731i	−0.563690	−	0.825986i	\(-0.690619\pi\)
							−0.358010	+	0.933718i	\(0.616545\pi\)
\(37\)	−5.04999	−	4.23745i	−0.830214	−	0.696632i	0.125126	−	0.992141i	\(-0.460066\pi\)
							−0.955340	+	0.295509i	\(0.904511\pi\)
\(41\)	−6.03866	+	3.03273i	−0.943080	+	0.473633i	−0.852773	−	0.522281i	\(-0.825081\pi\)
							−0.0903071	+	0.995914i	\(0.528785\pi\)
\(43\)	4.19282	+	4.44412i	0.639398	+	0.677723i	0.963318	−	0.268362i	\(-0.0864823\pi\)
							−0.323920	+	0.946085i	\(0.605001\pi\)
\(47\)	−7.26206	−	0.848813i	−1.05928	−	0.123812i	−0.431430	−	0.902146i	\(-0.641991\pi\)
							−0.627850	+	0.778334i	\(0.716065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.2.g.a.13.1	✓	144
3.2	odd	2		243.2.g.a.10.8		144
9.2	odd	6		729.2.g.a.514.1		144
9.4	even	3		729.2.g.c.28.8		144
9.5	odd	6		729.2.g.b.28.1		144
9.7	even	3		729.2.g.d.514.8		144
81.2	odd	54		729.2.g.b.703.1		144
81.5	odd	54		6561.2.a.d.1.71		72
81.25	even	27	inner	81.2.g.a.25.1	yes	144
81.29	odd	54		729.2.g.a.217.1		144
81.52	even	27		729.2.g.d.217.8		144
81.56	odd	54		243.2.g.a.73.8		144
81.76	even	27		6561.2.a.c.1.2		72
81.79	even	27		729.2.g.c.703.8		144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.1	✓	144	1.1	even	1	trivial
81.2.g.a.25.1	yes	144	81.25	even	27	inner
243.2.g.a.10.8		144	3.2	odd	2
243.2.g.a.73.8		144	81.56	odd	54
729.2.g.a.217.1		144	81.29	odd	54
729.2.g.a.514.1		144	9.2	odd	6
729.2.g.b.28.1		144	9.5	odd	6
729.2.g.b.703.1		144	81.2	odd	54
729.2.g.c.28.8		144	9.4	even	3
729.2.g.c.703.8		144	81.79	even	27
729.2.g.d.217.8		144	81.52	even	27
729.2.g.d.514.8		144	9.7	even	3
6561.2.a.c.1.2		72	81.76	even	27
6561.2.a.d.1.71		72	81.5	odd	54