Embedded newform 27.5.f.a.5.2

• Label: 27.5.f.a.5.2
• Level: $27$
• Weight: $5$
• Character: 27.5
• Analytic conductor: $2.791$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79098900326\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	27.5
Dual form			27.5.f.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.69447 - 4.40289i) q^{2} +(-2.23464 - 8.71817i) q^{3} +(-2.95802 + 16.7758i) q^{4} +(-4.30375 - 11.8244i) q^{5} +(-30.1294 + 42.0478i) q^{6} +(8.03917 + 45.5924i) q^{7} +(5.14963 - 2.97314i) q^{8} +(-71.0128 + 38.9638i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.69447	−	4.40289i	−0.923617	−	1.10072i	−0.994655	−	0.103250i	\(-0.967076\pi\)
							0.0710386	−	0.997474i	\(-0.477369\pi\)
\(3\)	−2.23464	−	8.71817i	−0.248293	−	0.968685i
							\(5\)	−4.30375	−	11.8244i	−0.172150	−	0.472978i	0.823373	−	0.567501i	\(-0.192090\pi\)
−0.995523	+	0.0945230i	\(0.969867\pi\)
\(7\)	8.03917	+	45.5924i	0.164065	+	0.930457i	0.950024	+	0.312178i	\(0.101059\pi\)
							−0.785959	+	0.618279i	\(0.787830\pi\)
\(11\)	60.7190	−	166.824i	0.501810	−	1.37871i	−0.387697	−	0.921787i	\(-0.626729\pi\)
							0.889506	−	0.456923i	\(-0.151049\pi\)
\(13\)	−143.455	−	120.373i	−0.848846	−	0.712266i	0.110689	−	0.993855i	\(-0.464694\pi\)
							−0.959535	+	0.281589i	\(0.909139\pi\)
\(17\)	−55.9875	−	32.3244i	−0.193728	−	0.111849i	0.399998	−	0.916516i	\(-0.369011\pi\)
							−0.593727	+	0.804667i	\(0.702344\pi\)
\(19\)	−103.533	−	179.324i	−0.286794	−	0.496742i	0.686248	−	0.727367i	\(-0.259256\pi\)
							−0.973043	+	0.230625i	\(0.925923\pi\)
\(23\)	−851.182	−	150.086i	−1.60904	−	0.283717i	−0.704371	−	0.709832i	\(-0.748771\pi\)
							−0.904669	+	0.426114i	\(0.859882\pi\)
\(29\)	402.512	+	479.695i	0.478611	+	0.570386i	0.950283	−	0.311388i	\(-0.100794\pi\)
							−0.471672	+	0.881774i	\(0.656349\pi\)
\(31\)	0.246143	−	1.39594i	0.000256132	−	0.00145259i	−0.984679	−	0.174374i	\(-0.944210\pi\)
							0.984936	+	0.172922i	\(0.0553208\pi\)
\(37\)	1020.83	−	1768.13i	0.745677	−	1.29155i	−0.204201	−	0.978929i	\(-0.565460\pi\)
							0.949878	−	0.312621i	\(-0.101207\pi\)
\(41\)	1231.33	−	1467.44i	0.732497	−	0.872956i	−0.263284	−	0.964718i	\(-0.584806\pi\)
							0.995781	+	0.0917623i	\(0.0292500\pi\)
\(43\)	2275.15	+	828.085i	1.23047	+	0.447856i	0.873760	−	0.486358i	\(-0.161675\pi\)
							0.356714	+	0.934214i	\(0.383897\pi\)
\(47\)	−2361.56	+	416.406i	−1.06906	+	0.188504i	−0.680374	−	0.732865i	\(-0.738183\pi\)
							−0.388687	+	0.921370i	\(0.627071\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.5.2	✓	66
3.2	odd	2		81.5.f.a.44.10		66
27.11	odd	18	inner	27.5.f.a.11.2	yes	66
27.16	even	9		81.5.f.a.35.10		66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.5.2	✓	66	1.1	even	1	trivial
27.5.f.a.11.2	yes	66	27.11	odd	18	inner
81.5.f.a.35.10		66	27.16	even	9
81.5.f.a.44.10		66	3.2	odd	2