Embedded newform 27.10.e.a.4.5

• Label: 27.10.e.a.4.5
• Level: $27$
• Weight: $10$
• Character: 27.4
• Analytic conductor: $13.906$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.9059675764\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.1224 - 10.5997i) q^{2} +(-114.519 - 81.0463i) q^{3} +(343.545 + 288.269i) q^{4} +(182.505 - 1035.04i) q^{5} +(2475.99 + 3574.12i) q^{6} +(-6317.76 + 5301.23i) q^{7} +(984.496 + 1705.20i) q^{8} +(6546.00 + 18562.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.1224	−	10.5997i	−1.28704	−	0.468444i	−0.394285	−	0.918988i	\(-0.629008\pi\)
							−0.892754	+	0.450544i	\(0.851230\pi\)
\(3\)	−114.519	−	81.0463i	−0.816263	−	0.577680i
							\(5\)	182.505	−	1035.04i	0.130590	−	0.740613i	−0.847240	−	0.531211i	\(-0.821737\pi\)
0.977830	−	0.209402i	\(-0.0671517\pi\)
\(7\)	−6317.76	+	5301.23i	−0.994539	+	0.834517i	−0.986219	−	0.165448i	\(-0.947093\pi\)
							−0.00832059	+	0.999965i	\(0.502649\pi\)
\(11\)	5041.59	+	28592.3i	0.103825	+	0.588819i	0.991683	+	0.128702i	\(0.0410812\pi\)
							−0.887859	+	0.460116i	\(0.847808\pi\)
\(13\)	11755.8	−	4278.76i	0.114158	−	0.0415501i	−0.284310	−	0.958733i	\(-0.591764\pi\)
							0.398468	+	0.917182i	\(0.369542\pi\)
\(17\)	−40572.7	+	70273.9i	−0.117818	+	0.204068i	−0.918903	−	0.394484i	\(-0.870923\pi\)
							0.801084	+	0.598551i	\(0.204257\pi\)
\(19\)	−355751.	−	616178.i	−0.626260	−	1.08471i	−0.988296	−	0.152550i	\(-0.951252\pi\)
							0.362036	−	0.932164i	\(-0.382082\pi\)
\(23\)	830877.	+	697189.i	0.619101	+	0.519488i	0.897521	−	0.440972i	\(-0.145366\pi\)
							−0.278420	+	0.960460i	\(0.589811\pi\)
\(29\)	−1.22765e6	−	446827.i	−0.322316	−	0.117314i	0.175795	−	0.984427i	\(-0.443751\pi\)
							−0.498111	+	0.867113i	\(0.665973\pi\)
\(31\)	−3.77049e6	−	3.16382e6i	−0.733281	−	0.615296i	0.197743	−	0.980254i	\(-0.436639\pi\)
							−0.931024	+	0.364958i	\(0.881083\pi\)
\(37\)	−1.09573e7	+	1.89786e7i	−0.961160	+	1.66478i	−0.241563	+	0.970385i	\(0.577660\pi\)
							−0.719597	+	0.694392i	\(0.755673\pi\)
\(41\)	2.43653e7	−	8.86823e6i	1.34662	−	0.490128i	0.434726	−	0.900563i	\(-0.356845\pi\)
							0.911890	+	0.410434i	\(0.134623\pi\)
\(43\)	−5.16626e6	−	2.92993e7i	−0.230446	−	1.30692i	−0.851996	−	0.523548i	\(-0.824608\pi\)
							0.621551	−	0.783374i	\(-0.286503\pi\)
\(47\)	1.72175e6	−	1.44472e6i	0.0514671	−	0.0431860i	−0.616691	−	0.787205i	\(-0.711527\pi\)
							0.668158	+	0.744019i	\(0.267083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.5	✓	156
3.2	odd	2		81.10.e.a.64.22		156
27.7	even	9	inner	27.10.e.a.7.5	yes	156
27.20	odd	18		81.10.e.a.19.22		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.5	✓	156	1.1	even	1	trivial
27.10.e.a.7.5	yes	156	27.7	even	9	inner
81.10.e.a.19.22		156	27.20	odd	18
81.10.e.a.64.22		156	3.2	odd	2