Embedded newform 27.6.e.a.7.4

• Label: 27.6.e.a.7.4
• Level: $27$
• Weight: $6$
• Character: 27.7
• Analytic conductor: $4.330$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.33036313495\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	27.7
Dual form			27.6.e.a.4.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.55790 + 2.02291i) q^{2} +(-15.5209 + 1.44989i) q^{3} +(2.28464 - 1.91704i) q^{4} +(-3.90711 - 22.1583i) q^{5} +(83.3305 - 39.4557i) q^{6} +(69.5798 + 58.3844i) q^{7} +(85.8137 - 148.634i) q^{8} +(238.796 - 45.0070i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.55790	+	2.02291i	−0.982507	+	0.357603i	−0.782814	−	0.622255i	\(-0.786217\pi\)
							−0.199692	+	0.979859i	\(0.563994\pi\)
\(3\)	−15.5209	+	1.44989i	−0.995665	+	0.0930102i
							\(5\)	−3.90711	−	22.1583i	−0.0698925	−	0.396380i	−0.999605	−	0.0280948i	\(-0.991056\pi\)
0.929713	−	0.368286i	\(-0.120055\pi\)
\(7\)	69.5798	+	58.3844i	0.536708	+	0.450351i	0.870410	−	0.492327i	\(-0.163854\pi\)
							−0.333702	+	0.942678i	\(0.608298\pi\)
\(11\)	47.2935	−	268.215i	0.117847	−	0.668345i	−0.867454	−	0.497518i	\(-0.834245\pi\)
							0.985301	−	0.170827i	\(-0.0546440\pi\)
\(13\)	263.041	+	95.7391i	0.431683	+	0.157120i	0.548717	−	0.836008i	\(-0.315116\pi\)
							−0.117034	+	0.993128i	\(0.537339\pi\)
\(17\)	−736.555	−	1275.75i	−0.618135	−	1.07064i	−0.989826	−	0.142284i	\(-0.954555\pi\)
							0.371691	−	0.928356i	\(-0.378778\pi\)
\(19\)	121.121	−	209.787i	0.0769723	−	0.133320i	−0.824970	−	0.565177i	\(-0.808808\pi\)
							0.901942	+	0.431857i	\(0.142141\pi\)
\(23\)	−101.784	+	85.4070i	−0.0401200	+	0.0336646i	−0.662627	−	0.748949i	\(-0.730559\pi\)
							0.622507	+	0.782614i	\(0.286114\pi\)
\(29\)	7780.54	−	2831.88i	1.71797	−	0.625289i	0.720306	−	0.693657i	\(-0.244001\pi\)
							0.997660	+	0.0683681i	\(0.0217792\pi\)
\(31\)	5831.61	−	4893.30i	1.08989	−	0.914530i	0.0931898	−	0.995648i	\(-0.470294\pi\)
							0.996704	+	0.0811185i	\(0.0258492\pi\)
\(37\)	−5732.04	−	9928.18i	−0.688343	−	1.19224i	−0.972374	−	0.233429i	\(-0.925005\pi\)
							0.284031	−	0.958815i	\(-0.408328\pi\)
\(41\)	−5445.05	−	1981.84i	−0.505874	−	0.184123i	0.0764603	−	0.997073i	\(-0.475638\pi\)
							−0.582334	+	0.812950i	\(0.697860\pi\)
\(43\)	−3076.71	+	17448.9i	−0.253755	+	1.43912i	0.545492	+	0.838116i	\(0.316343\pi\)
							−0.799247	+	0.601003i	\(0.794768\pi\)
\(47\)	9830.35	+	8248.65i	0.649119	+	0.544676i	0.906804	−	0.421553i	\(-0.138515\pi\)
							−0.257684	+	0.966229i	\(0.582959\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.7.4	yes	84
3.2	odd	2		81.6.e.a.19.11		84
27.2	odd	18		729.6.a.e.1.11		42
27.4	even	9	inner	27.6.e.a.4.4	✓	84
27.23	odd	18		81.6.e.a.64.11		84
27.25	even	9		729.6.a.c.1.32		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.4	✓	84	27.4	even	9	inner
27.6.e.a.7.4	yes	84	1.1	even	1	trivial
81.6.e.a.19.11		84	3.2	odd	2
81.6.e.a.64.11		84	27.23	odd	18
729.6.a.c.1.32		42	27.25	even	9
729.6.a.e.1.11		42	27.2	odd	18