Embedded newform 27.7.f.a.2.4

• Label: 27.7.f.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	27.2
Dual form			27.7.f.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.97059 - 1.75808i) q^{2} +(24.1318 - 12.1101i) q^{3} +(36.1814 + 13.1690i) q^{4} +(60.9980 + 72.6946i) q^{5} +(-261.899 + 78.3191i) q^{6} +(-133.791 + 48.6960i) q^{7} +(223.553 + 129.069i) q^{8} +(435.691 - 584.478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.97059	−	1.75808i	−1.24632	−	0.219760i	−0.488700	−	0.872452i	\(-0.662529\pi\)
							−0.757624	+	0.652692i	\(0.773640\pi\)
\(3\)	24.1318	−	12.1101i	0.893772	−	0.448522i
							\(5\)	60.9980	+	72.6946i	0.487984	+	0.581556i	0.952703	−	0.303901i	\(-0.0982893\pi\)
−0.464720	+	0.885458i	\(0.653845\pi\)
\(7\)	−133.791	+	48.6960i	−0.390062	+	0.141971i	−0.529604	−	0.848245i	\(-0.677659\pi\)
							0.139542	+	0.990216i	\(0.455437\pi\)
\(11\)	1624.00	−	1935.41i	1.22014	−	1.45410i	0.368782	−	0.929516i	\(-0.379775\pi\)
							0.851356	−	0.524588i	\(-0.175781\pi\)
\(13\)	−126.490	−	717.362i	−0.0575741	−	0.326519i	0.942394	−	0.334505i	\(-0.108569\pi\)
							−0.999968	+	0.00798588i	\(0.997458\pi\)
\(17\)	4210.29	−	2430.81i	0.856968	−	0.494771i	−0.00602755	−	0.999982i	\(-0.501919\pi\)
							0.862996	+	0.505211i	\(0.168585\pi\)
\(19\)	62.0141	−	107.412i	0.00904127	−	0.0156599i	−0.861469	−	0.507810i	\(-0.830455\pi\)
							0.870511	+	0.492150i	\(0.163789\pi\)
\(23\)	−4137.25	+	11367.0i	−0.340039	+	0.934249i	0.645344	+	0.763892i	\(0.276714\pi\)
							−0.985382	+	0.170357i	\(0.945508\pi\)
\(29\)	22191.9	+	3913.03i	0.909915	+	0.160443i	0.608965	−	0.793197i	\(-0.291585\pi\)
							0.300950	+	0.953640i	\(0.402696\pi\)
\(31\)	14120.1	+	5139.28i	0.473971	+	0.172511i	0.567950	−	0.823063i	\(-0.307737\pi\)
							−0.0939796	+	0.995574i	\(0.529959\pi\)
\(37\)	17270.1	+	29912.7i	0.340950	+	0.590542i	0.984609	−	0.174770i	\(-0.0559180\pi\)
							−0.643660	+	0.765312i	\(0.722585\pi\)
\(41\)	−87091.6	+	15356.6i	−1.26364	+	0.222815i	−0.765022	−	0.644004i	\(-0.777272\pi\)
							−0.498623	+	0.866819i	\(0.666161\pi\)
\(43\)	−4953.54	−	4156.52i	−0.0623032	−	0.0522786i	0.611104	−	0.791550i	\(-0.290726\pi\)
							−0.673408	+	0.739271i	\(0.735170\pi\)
\(47\)	59675.6	+	163957.i	0.574782	+	1.57920i	0.796855	+	0.604171i	\(0.206496\pi\)
							−0.222072	+	0.975030i	\(0.571282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.2.4	✓	102
3.2	odd	2		81.7.f.a.8.14		102
27.13	even	9		81.7.f.a.71.14		102
27.14	odd	18	inner	27.7.f.a.14.4	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.2.4	✓	102	1.1	even	1	trivial
27.7.f.a.14.4	yes	102	27.14	odd	18	inner
81.7.f.a.8.14		102	3.2	odd	2
81.7.f.a.71.14		102	27.13	even	9