Embedded newform 27.8.e.a.7.8

• Label: 27.8.e.a.7.8
• Level: $27$
• Weight: $8$
• Character: 27.7
• 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			7.8
Character	\(\chi\)	\(=\)	27.7
Dual form			27.8.e.a.4.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.04432 + 3.29186i) q^{2} +(-20.3312 - 42.1146i) q^{3} +(-27.0904 + 22.7315i) q^{4} +(82.6399 + 468.674i) q^{5} +(322.518 + 313.970i) q^{6} +(-558.283 - 468.455i) q^{7} +(786.170 - 1361.69i) q^{8} +(-1360.28 + 1712.48i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.04432	+	3.29186i	−0.799412	+	0.290962i	−0.709243	−	0.704964i	\(-0.750963\pi\)
							−0.0901693	+	0.995926i	\(0.528741\pi\)
\(3\)	−20.3312	−	42.1146i	−0.434750	−	0.900551i
							\(5\)	82.6399	+	468.674i	0.295661	+	1.67678i	0.664502	+	0.747286i	\(0.268644\pi\)
−0.368841	+	0.929493i	\(0.620245\pi\)
\(7\)	−558.283	−	468.455i	−0.615192	−	0.516208i	0.281096	−	0.959680i	\(-0.409302\pi\)
							−0.896288	+	0.443472i	\(0.853747\pi\)
\(11\)	1148.19	−	6511.69i	0.260099	−	1.47509i	−0.522539	−	0.852615i	\(-0.675015\pi\)
							0.782638	−	0.622477i	\(-0.213874\pi\)
\(13\)	7649.25	+	2784.10i	0.965644	+	0.351466i	0.776243	−	0.630434i	\(-0.217123\pi\)
							0.189401	+	0.981900i	\(0.439345\pi\)
\(17\)	−9782.26	−	16943.4i	−0.482912	−	0.836428i	0.516896	−	0.856048i	\(-0.327088\pi\)
							−0.999807	+	0.0196205i	\(0.993754\pi\)
\(19\)	8437.34	−	14613.9i	0.282207	−	0.488797i	−0.689721	−	0.724075i	\(-0.742267\pi\)
							0.971928	+	0.235278i	\(0.0756001\pi\)
\(23\)	49888.7	−	41861.6i	0.854978	−	0.717411i	−0.105902	−	0.994377i	\(-0.533773\pi\)
							0.960880	+	0.276965i	\(0.0893287\pi\)
\(29\)	170386.	−	62015.3i	1.29730	−	0.472178i	0.401183	−	0.915998i	\(-0.368599\pi\)
							0.896116	+	0.443820i	\(0.146377\pi\)
\(31\)	−55988.0	+	46979.5i	−0.337543	+	0.283232i	−0.795765	−	0.605606i	\(-0.792931\pi\)
							0.458222	+	0.888838i	\(0.348486\pi\)
\(37\)	51934.0	+	89952.3i	0.168556	+	0.291948i	0.937913	−	0.346872i	\(-0.112756\pi\)
							−0.769356	+	0.638820i	\(0.779423\pi\)
\(41\)	−427049.	−	155433.i	−0.967686	−	0.352209i	−0.190645	−	0.981659i	\(-0.561058\pi\)
							−0.777041	+	0.629450i	\(0.783280\pi\)
\(43\)	40840.2	−	231616.i	0.0783336	−	0.444252i	−0.920263	−	0.391299i	\(-0.872026\pi\)
							0.998597	−	0.0529524i	\(-0.0168632\pi\)
\(47\)	2850.34	+	2391.72i	0.00400455	+	0.00336022i	0.644788	−	0.764362i	\(-0.276946\pi\)
							−0.640783	+	0.767722i	\(0.721390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.7.8	yes	120
3.2	odd	2		81.8.e.a.19.13		120
27.4	even	9	inner	27.8.e.a.4.8	✓	120
27.23	odd	18		81.8.e.a.64.13		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.8	✓	120	27.4	even	9	inner
27.8.e.a.7.8	yes	120	1.1	even	1	trivial
81.8.e.a.19.13		120	3.2	odd	2
81.8.e.a.64.13		120	27.23	odd	18