Embedded newform 81.3.h.a.2.8

• Label: 81.3.h.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	81.2
Dual form			81.3.h.a.41.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.614022 - 0.0357627i) q^{2} +(2.99545 - 0.165205i) q^{3} +(-3.59721 - 0.420453i) q^{4} +(-2.20305 - 9.29539i) q^{5} +(-1.84518 - 0.00568553i) q^{6} +(-3.03111 - 4.07148i) q^{7} +(4.61661 + 0.814032i) q^{8} +(8.94541 - 0.989729i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.614022	−	0.0357627i	−0.307011	−	0.0178814i	−0.0960523	−	0.995376i	\(-0.530622\pi\)
							−0.210959	+	0.977495i	\(0.567659\pi\)
\(3\)	2.99545	−	0.165205i	0.998483	−	0.0550685i
							\(5\)	−2.20305	−	9.29539i	−0.440610	−	1.85908i	−0.508008	−	0.861352i	\(-0.669618\pi\)
0.0673981	−	0.997726i	\(-0.478530\pi\)
\(7\)	−3.03111	−	4.07148i	−0.433015	−	0.581641i	0.531038	−	0.847348i	\(-0.321802\pi\)
							−0.964054	+	0.265707i	\(0.914395\pi\)
\(11\)	13.0974	+	12.3567i	1.19067	+	1.12334i	0.989613	+	0.143758i	\(0.0459187\pi\)
							0.201056	+	0.979580i	\(0.435563\pi\)
\(13\)	−10.6768	−	5.36209i	−0.821292	−	0.412468i	−0.0120380	−	0.999928i	\(-0.503832\pi\)
							−0.809254	+	0.587459i	\(0.800128\pi\)
\(17\)	2.20772	−	6.06566i	0.129866	−	0.356804i	−0.857669	−	0.514202i	\(-0.828088\pi\)
							0.987535	+	0.157398i	\(0.0503105\pi\)
\(19\)	0.235479	−	0.0857072i	0.0123936	−	0.00451091i	−0.335816	−	0.941928i	\(-0.609012\pi\)
							0.348210	+	0.937417i	\(0.386790\pi\)
\(23\)	11.7015	+	8.71143i	0.508760	+	0.378758i	0.820714	−	0.571339i	\(-0.193576\pi\)
							−0.311954	+	0.950097i	\(0.600983\pi\)
\(29\)	16.4818	−	25.0593i	0.568337	−	0.864114i	−0.430930	−	0.902385i	\(-0.641815\pi\)
							0.999267	+	0.0382708i	\(0.0121849\pi\)
\(31\)	12.0661	+	27.9722i	0.389227	+	0.902331i	0.994345	+	0.106195i	\(0.0338667\pi\)
							−0.605118	+	0.796136i	\(0.706874\pi\)
\(37\)	1.02484	−	0.859943i	0.0276984	−	0.0232417i	−0.628833	−	0.777540i	\(-0.716467\pi\)
							0.656532	+	0.754298i	\(0.272023\pi\)
\(41\)	49.7323	−	2.89658i	1.21298	−	0.0706482i	0.560307	−	0.828285i	\(-0.310683\pi\)
							0.652676	+	0.757637i	\(0.273646\pi\)
\(43\)	10.2543	+	34.2516i	0.238471	+	0.796549i	0.990511	+	0.137432i	\(0.0438848\pi\)
							−0.752040	+	0.659117i	\(0.770930\pi\)
\(47\)	−14.4473	−	6.23196i	−0.307390	−	0.132595i	0.236797	−	0.971559i	\(-0.423902\pi\)
							−0.544187	+	0.838964i	\(0.683162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.h.a.2.8	✓	306
3.2	odd	2		243.3.h.a.8.10		306
81.40	even	27		243.3.h.a.152.10		306
81.41	odd	54	inner	81.3.h.a.41.8	yes	306

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.3.h.a.2.8	✓	306	1.1	even	1	trivial
81.3.h.a.41.8	yes	306	81.41	odd	54	inner
243.3.h.a.8.10		306	3.2	odd	2
243.3.h.a.152.10		306	81.40	even	27