Embedded newform 81.3.h.a.41.8

• Label: 81.3.h.a.41.8
• Level: $81$
• Weight: $3$
• Character: 81.41
• 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			41.8
Character	\(\chi\)	\(=\)	81.41
Dual form			81.3.h.a.2.8

$q$-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 \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{53}{54}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.3.h.a.2.8	✓	306	81.2	odd	54	inner
81.3.h.a.41.8	yes	306	1.1	even	1	trivial
243.3.h.a.8.10		306	81.79	even	27
243.3.h.a.152.10		306	3.2	odd	2