Embedded newform 27.7.f.a.5.2

• Label: 27.7.f.a.5.2
• Level: $27$
• Weight: $7$
• Character: 27.5
• Analytic conductor: $6.211$
• Analytic rank: $0$
• Dimension: $102$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.21146025774\)
Analytic rank:	\(0\)
Dimension:	\(102\)
Relative dimension:	\(17\) 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			27.7.f.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.78575 - 10.4705i) q^{2} +(21.2323 - 16.6790i) q^{3} +(-21.3274 + 120.954i) q^{4} +(-83.1319 - 228.403i) q^{5} +(-361.178 - 75.7744i) q^{6} +(3.85901 + 21.8856i) q^{7} +(696.250 - 401.980i) q^{8} +(172.622 - 708.267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.78575	−	10.4705i	−1.09822	−	1.30881i	−0.947331	−	0.320255i	\(-0.896231\pi\)
							−0.150888	−	0.988551i	\(-0.548213\pi\)
\(3\)	21.2323	−	16.6790i	0.786382	−	0.617740i
							\(5\)	−83.1319	−	228.403i	−0.665055	−	1.82722i	−0.552379	−	0.833593i	\(-0.686280\pi\)
−0.112676	−	0.993632i	\(-0.535942\pi\)
\(7\)	3.85901	+	21.8856i	0.0112508	+	0.0638063i	0.989916	−	0.141654i	\(-0.0452420\pi\)
							−0.978665	+	0.205460i	\(0.934131\pi\)
\(11\)	−470.609	+	1292.99i	−0.353576	+	0.971441i	0.627636	+	0.778507i	\(0.284023\pi\)
							−0.981212	+	0.192934i	\(0.938200\pi\)
\(13\)	1442.04	+	1210.02i	0.656368	+	0.550758i	0.908996	−	0.416805i	\(-0.136850\pi\)
							−0.252628	+	0.967564i	\(0.581295\pi\)
\(17\)	739.593	+	427.004i	0.150538	+	0.0869131i	0.573377	−	0.819292i	\(-0.305633\pi\)
							−0.422839	+	0.906205i	\(0.638966\pi\)
\(19\)	−4651.50	−	8056.63i	−0.678160	−	1.17461i	−0.975535	−	0.219846i	\(-0.929444\pi\)
							0.297375	−	0.954761i	\(-0.403889\pi\)
\(23\)	−845.594	−	149.101i	−0.0694990	−	0.0122546i	0.138791	−	0.990322i	\(-0.455679\pi\)
							−0.208290	+	0.978067i	\(0.566790\pi\)
\(29\)	−6312.41	−	7522.84i	−0.258822	−	0.308452i	0.620948	−	0.783852i	\(-0.286748\pi\)
							−0.879770	+	0.475400i	\(0.842303\pi\)
\(31\)	5062.00	−	28708.0i	0.169917	−	0.963648i	−0.773931	−	0.633269i	\(-0.781713\pi\)
							0.943849	−	0.330378i	\(-0.107176\pi\)
\(37\)	18624.3	−	32258.2i	0.367683	−	0.636846i	−0.621520	−	0.783399i	\(-0.713484\pi\)
							0.989203	+	0.146553i	\(0.0468177\pi\)
\(41\)	31749.6	−	37837.7i	0.460666	−	0.549001i	−0.484841	−	0.874602i	\(-0.661122\pi\)
							0.945507	+	0.325602i	\(0.105567\pi\)
\(43\)	10987.9	+	3999.26i	0.138200	+	0.0503007i	0.410194	−	0.911998i	\(-0.365461\pi\)
							−0.271994	+	0.962299i	\(0.587683\pi\)
\(47\)	−123131.	+	21711.4i	−1.18597	+	0.209119i	−0.731627	−	0.681705i	\(-0.761239\pi\)
							−0.454348	+	0.890824i	\(0.650128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.5.2	✓	102
3.2	odd	2		81.7.f.a.44.16		102
27.11	odd	18	inner	27.7.f.a.11.2	yes	102
27.16	even	9		81.7.f.a.35.16		102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.5.2	✓	102	1.1	even	1	trivial
27.7.f.a.11.2	yes	102	27.11	odd	18	inner
81.7.f.a.35.16		102	27.16	even	9
81.7.f.a.44.16		102	3.2	odd	2