Embedded newform 27.8.e.a.13.2

• Label: 27.8.e.a.13.2
• Level: $27$
• Weight: $8$
• Character: 27.13
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	27.13
Dual form			27.8.e.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.52519 - 19.9924i) q^{2} +(40.7783 + 22.8939i) q^{3} +(-266.987 + 97.1752i) q^{4} +(333.918 + 280.190i) q^{5} +(313.953 - 895.959i) q^{6} +(999.181 + 363.672i) q^{7} +(1584.69 + 2744.76i) q^{8} +(1138.73 + 1867.15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.52519	−	19.9924i	−0.311586	−	1.76709i	−0.590757	−	0.806850i	\(-0.701171\pi\)
							0.279171	−	0.960241i	\(-0.409940\pi\)
\(3\)	40.7783	+	22.8939i	0.871976	+	0.489549i
							\(5\)	333.918	+	280.190i	1.19466	+	1.00244i	0.999766	+	0.0216158i	\(0.00688105\pi\)
0.194895	+	0.980824i	\(0.437563\pi\)
\(7\)	999.181	+	363.672i	1.10103	+	0.400744i	0.827697	−	0.561175i	\(-0.189650\pi\)
							0.273337	+	0.961918i	\(0.411873\pi\)
\(11\)	−4388.30	+	3682.22i	−0.994081	+	0.834133i	−0.986153	−	0.165835i	\(-0.946968\pi\)
							−0.00792791	+	0.999969i	\(0.502524\pi\)
\(13\)	370.974	−	2103.90i	0.0468319	−	0.265597i	−0.952397	−	0.304861i	\(-0.901390\pi\)
							0.999229	+	0.0392636i	\(0.0125012\pi\)
\(17\)	8553.32	−	14814.8i	0.422244	−	0.731348i	−0.573915	−	0.818915i	\(-0.694576\pi\)
							0.996159	+	0.0875673i	\(0.0279093\pi\)
\(19\)	−10704.0	−	18539.9i	−0.358022	−	0.620113i	0.629608	−	0.776913i	\(-0.283215\pi\)
							−0.987630	+	0.156800i	\(0.949882\pi\)
\(23\)	−16062.8	+	5846.39i	−0.275280	+	0.100194i	−0.475971	−	0.879461i	\(-0.657903\pi\)
							0.200691	+	0.979654i	\(0.435681\pi\)
\(29\)	−18811.9	−	106687.i	−0.143232	−	0.812307i	−0.968770	−	0.247961i	\(-0.920239\pi\)
							0.825538	−	0.564346i	\(-0.190872\pi\)
\(31\)	294235.	−	107093.i	1.77390	−	0.645645i	0.773973	−	0.633219i	\(-0.218267\pi\)
							0.999923	−	0.0124260i	\(-0.00395542\pi\)
\(37\)	138567.	−	240006.i	0.449733	−	0.778961i	−0.548635	−	0.836062i	\(-0.684852\pi\)
							0.998368	+	0.0571010i	\(0.0181857\pi\)
\(41\)	−16752.0	+	95005.1i	−0.0379596	+	0.215280i	−0.997887	−	0.0649679i	\(-0.979305\pi\)
							0.959928	+	0.280248i	\(0.0904166\pi\)
\(43\)	−71877.2	+	60312.2i	−0.137864	+	0.115682i	−0.709112	−	0.705096i	\(-0.750904\pi\)
							0.571248	+	0.820778i	\(0.306460\pi\)
\(47\)	−1.07339e6	−	390680.i	−1.50804	−	0.548882i	−0.549913	−	0.835222i	\(-0.685339\pi\)
							−0.958128	+	0.286340i	\(0.907561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.13.2	✓	120
3.2	odd	2		81.8.e.a.10.19		120
27.2	odd	18		81.8.e.a.73.19		120
27.25	even	9	inner	27.8.e.a.25.2	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.13.2	✓	120	1.1	even	1	trivial
27.8.e.a.25.2	yes	120	27.25	even	9	inner
81.8.e.a.10.19		120	3.2	odd	2
81.8.e.a.73.19		120	27.2	odd	18