Embedded newform 54.3.f.a.47.3

• Label: 54.3.f.a.47.3
• Level: $54$
• Weight: $3$
• Character: 54.47
• Analytic conductor: $1.471$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	54.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47139342755\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			47.3
Character	\(\chi\)	\(=\)	54.47
Dual form			54.3.f.a.23.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.483690 - 1.32893i) q^{2} +(-0.391734 + 2.97431i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(7.52350 - 1.32660i) q^{5} +(4.14212 - 0.918059i) q^{6} +(4.40811 + 3.69885i) q^{7} +(2.44949 + 1.41421i) q^{8} +(-8.69309 - 2.33028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 18 q^{5} + 12 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.483690	−	1.32893i	−0.241845	−	0.664463i
							\(3\)	−0.391734	+	2.97431i	−0.130578	+	0.991438i
\(5\)	7.52350	−	1.32660i	1.50470	−	0.265319i								0.640301	−	0.768124i	\(-0.278810\pi\)
							0.864400	+	0.502805i	\(0.167699\pi\)
\(7\)	4.40811	+	3.69885i	0.629731	+	0.528407i	0.900845	−	0.434140i	\(-0.142948\pi\)
							−0.271115	+	0.962547i	\(0.587392\pi\)
\(11\)	8.02644	+	1.41528i	0.729677	+	0.128662i	0.526131	−	0.850404i	\(-0.323642\pi\)
							0.203546	+	0.979065i	\(0.434753\pi\)
\(13\)	−22.0464	−	8.02423i	−1.69588	−	0.617248i	−0.700531	−	0.713622i	\(-0.747053\pi\)
							−0.995345	+	0.0963739i	\(0.969276\pi\)
\(17\)	6.39327	−	3.69115i	0.376075	−	0.217127i	−0.300034	−	0.953928i	\(-0.596998\pi\)
							0.676109	+	0.736802i	\(0.263665\pi\)
\(19\)	−7.80130	+	13.5122i	−0.410595	+	0.711171i	−0.994955	−	0.100324i	\(-0.968012\pi\)
							0.584360	+	0.811494i	\(0.301346\pi\)
\(23\)	−19.9755	−	23.8059i	−0.868500	−	1.03504i	−0.999049	−	0.0435964i	\(-0.986118\pi\)
							0.130549	−	0.991442i	\(-0.458326\pi\)
\(29\)	−9.68546	−	26.6106i	−0.333981	−	0.917606i	−0.987065	−	0.160320i	\(-0.948747\pi\)
							0.653084	−	0.757286i	\(-0.273475\pi\)
\(31\)	12.2601	−	10.2874i	0.395487	−	0.331853i	−0.423259	−	0.906009i	\(-0.639114\pi\)
							0.818746	+	0.574156i	\(0.194670\pi\)
\(37\)	5.99046	+	10.3758i	0.161904	+	0.280427i	0.935552	−	0.353190i	\(-0.114903\pi\)
							−0.773647	+	0.633617i	\(0.781570\pi\)
\(41\)	2.82625	−	7.76505i	0.0689329	−	0.189391i	−0.900443	−	0.434975i	\(-0.856757\pi\)
							0.969375	+	0.245583i	\(0.0789795\pi\)
\(43\)	−7.82854	+	44.3978i	−0.182059	+	1.03251i	0.747617	+	0.664130i	\(0.231198\pi\)
							−0.929676	+	0.368378i	\(0.879913\pi\)
\(47\)	−43.4592	+	51.7927i	−0.924664	+	1.10197i	0.0698695	+	0.997556i	\(0.477742\pi\)
							−0.994534	+	0.104416i	\(0.966703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.3.f.a.47.3	yes	36
3.2	odd	2		162.3.f.a.143.4		36
4.3	odd	2		432.3.bc.c.209.3		36
27.2	odd	18		1458.3.b.c.1457.16		36
27.4	even	9		162.3.f.a.17.4		36
27.23	odd	18	inner	54.3.f.a.23.3	✓	36
27.25	even	9		1458.3.b.c.1457.21		36
108.23	even	18		432.3.bc.c.401.3		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.23.3	✓	36	27.23	odd	18	inner
54.3.f.a.47.3	yes	36	1.1	even	1	trivial
162.3.f.a.17.4		36	27.4	even	9
162.3.f.a.143.4		36	3.2	odd	2
432.3.bc.c.209.3		36	4.3	odd	2
432.3.bc.c.401.3		36	108.23	even	18
1458.3.b.c.1457.16		36	27.2	odd	18
1458.3.b.c.1457.21		36	27.25	even	9