Embedded newform 27.7.f.a.2.2

• Label: 27.7.f.a.2.2
• Level: $27$
• Weight: $7$
• Character: 27.2
• 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			2.2
Character	\(\chi\)	\(=\)	27.2
Dual form			27.7.f.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.9309 - 2.28007i) q^{2} +(19.5181 + 18.6559i) q^{3} +(101.870 + 37.0775i) q^{4} +(-78.6006 - 93.6725i) q^{5} +(-209.851 - 285.740i) q^{6} +(149.932 - 54.5707i) q^{7} +(-504.967 - 291.543i) q^{8} +(32.9160 + 728.257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.9309	−	2.28007i	−1.61636	−	0.285009i	−0.708954	−	0.705254i	\(-0.750833\pi\)
							−0.907410	+	0.420246i	\(0.861944\pi\)
\(3\)	19.5181	+	18.6559i	0.722894	+	0.690959i
							\(5\)	−78.6006	−	93.6725i	−0.628805	−	0.749380i	0.353753	−	0.935339i	\(-0.384905\pi\)
−0.982558	+	0.185959i	\(0.940461\pi\)
\(7\)	149.932	−	54.5707i	0.437119	−	0.159098i	−0.114081	−	0.993471i	\(-0.536392\pi\)
							0.551200	+	0.834373i	\(0.314170\pi\)
\(11\)	−771.586	+	919.541i	−0.579704	+	0.690865i	−0.973592	−	0.228294i	\(-0.926685\pi\)
							0.393888	+	0.919158i	\(0.371130\pi\)
\(13\)	595.843	+	3379.19i	0.271208	+	1.53809i	0.750756	+	0.660580i	\(0.229689\pi\)
							−0.479548	+	0.877515i	\(0.659200\pi\)
\(17\)	194.413	−	112.244i	0.0395711	−	0.0228464i	−0.480084	−	0.877223i	\(-0.659394\pi\)
							0.519655	+	0.854376i	\(0.326060\pi\)
\(19\)	−4509.62	+	7810.89i	−0.657474	+	1.13878i	0.323793	+	0.946128i	\(0.395042\pi\)
							−0.981267	+	0.192651i	\(0.938291\pi\)
\(23\)	−4908.20	+	13485.2i	−0.403403	+	1.10834i	0.557191	+	0.830384i	\(0.311879\pi\)
							−0.960594	+	0.277956i	\(0.910343\pi\)
\(29\)	−8347.98	−	1471.97i	−0.342285	−	0.0603540i	−0.000136405	−	1.00000i	\(-0.500043\pi\)
							−0.342148	+	0.939646i	\(0.611155\pi\)
\(31\)	53068.2	+	19315.3i	1.78135	+	0.648359i	0.999696	+	0.0246482i	\(0.00784656\pi\)
							0.781655	+	0.623711i	\(0.214376\pi\)
\(37\)	−44133.5	−	76441.5i	−0.871291	−	1.50912i	−0.860662	−	0.509176i	\(-0.829950\pi\)
							−0.0106286	−	0.999944i	\(-0.503383\pi\)
\(41\)	93692.0	−	16520.4i	1.35941	−	0.239701i	0.554049	−	0.832484i	\(-0.313082\pi\)
							0.805362	+	0.592783i	\(0.201971\pi\)
\(43\)	−54840.2	−	46016.4i	−0.689754	−	0.578772i	0.229085	−	0.973407i	\(-0.426427\pi\)
							−0.918838	+	0.394635i	\(0.870871\pi\)
\(47\)	31800.7	+	87371.7i	0.306297	+	0.841545i	0.993370	+	0.114957i	\(0.0366730\pi\)
							−0.687073	+	0.726588i	\(0.741105\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.2.2	✓	102
3.2	odd	2		81.7.f.a.8.16		102
27.13	even	9		81.7.f.a.71.16		102
27.14	odd	18	inner	27.7.f.a.14.2	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.2.2	✓	102	1.1	even	1	trivial
27.7.f.a.14.2	yes	102	27.14	odd	18	inner
81.7.f.a.8.16		102	3.2	odd	2
81.7.f.a.71.16		102	27.13	even	9