Embedded newform 81.3.h.a.2.7

• Label: 81.3.h.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	81.2
Dual form			81.3.h.a.41.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04279 - 0.0607357i) q^{2} +(-2.04391 - 2.19600i) q^{3} +(-2.88923 - 0.337702i) q^{4} +(1.21177 + 5.11287i) q^{5} +(1.99800 + 2.41411i) q^{6} +(4.42126 + 5.93878i) q^{7} +(7.10711 + 1.25318i) q^{8} +(-0.644854 + 8.97687i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04279	−	0.0607357i	−0.521396	−	0.0303678i	−0.204571	−	0.978852i	\(-0.565580\pi\)
							−0.316825	+	0.948484i	\(0.602617\pi\)
\(3\)	−2.04391	−	2.19600i	−0.681304	−	0.732001i
							\(5\)	1.21177	+	5.11287i	0.242354	+	1.02257i	0.949067	+	0.315073i	\(0.102029\pi\)
−0.706713	+	0.707501i	\(0.749823\pi\)
\(7\)	4.42126	+	5.93878i	0.631608	+	0.848397i	0.996581	−	0.0826214i	\(-0.0263292\pi\)
							−0.364973	+	0.931018i	\(0.618922\pi\)
\(11\)	7.09307	+	6.69197i	0.644825	+	0.608361i	0.937556	−	0.347834i	\(-0.113083\pi\)
							−0.292731	+	0.956195i	\(0.594564\pi\)
\(13\)	−17.3263	−	8.70161i	−1.33279	−	0.669354i	−0.367379	−	0.930071i	\(-0.619745\pi\)
							−0.965415	+	0.260717i	\(0.916041\pi\)
\(17\)	−6.80049	+	18.6842i	−0.400029	+	1.09907i	0.562241	+	0.826973i	\(0.309939\pi\)
							−0.962270	+	0.272096i	\(0.912283\pi\)
\(19\)	−31.2030	+	11.3570i	−1.64227	+	0.597736i	−0.987433	−	0.158038i	\(-0.949483\pi\)
							−0.654833	+	0.755774i	\(0.727261\pi\)
\(23\)	23.2164	+	17.2839i	1.00941	+	0.751475i	0.968836	−	0.247701i	\(-0.0796751\pi\)
							0.0405702	+	0.999177i	\(0.487083\pi\)
\(29\)	8.81850	−	13.4079i	0.304086	−	0.462341i	−0.650827	−	0.759226i	\(-0.725578\pi\)
							0.954913	+	0.296885i	\(0.0959479\pi\)
\(31\)	7.12701	+	16.5223i	0.229904	+	0.532977i	0.993647	−	0.112544i	\(-0.0358998\pi\)
							−0.763743	+	0.645520i	\(0.776641\pi\)
\(37\)	38.7151	−	32.4858i	1.04635	−	0.877996i	0.0536492	−	0.998560i	\(-0.482915\pi\)
							0.992706	+	0.120564i	\(0.0384703\pi\)
\(41\)	−59.2114	+	3.44867i	−1.44418	+	0.0841139i	−0.762240	−	0.647294i	\(-0.775900\pi\)
							−0.681940	+	0.731408i	\(0.738863\pi\)
\(43\)	7.32261	+	24.4592i	0.170293	+	0.568819i	0.999944	+	0.0105371i	\(0.00335414\pi\)
							−0.829651	+	0.558282i	\(0.811461\pi\)
\(47\)	26.7112	+	11.5221i	0.568323	+	0.245151i	0.660797	−	0.750564i	\(-0.270218\pi\)
							−0.0924741	+	0.995715i	\(0.529478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.h.a.2.7	✓	306
3.2	odd	2		243.3.h.a.8.11		306
81.40	even	27		243.3.h.a.152.11		306
81.41	odd	54	inner	81.3.h.a.41.7	yes	306

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.3.h.a.2.7	✓	306	1.1	even	1	trivial
81.3.h.a.41.7	yes	306	81.41	odd	54	inner
243.3.h.a.8.11		306	3.2	odd	2
243.3.h.a.152.11		306	81.40	even	27