Embedded newform 27.6.e.a.4.12

• Label: 27.6.e.a.4.12
• Level: $27$
• Weight: $6$
• Character: 27.4
• 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			4.12
Character	\(\chi\)	\(=\)	27.4
Dual form			27.6.e.a.7.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.01235 + 2.91626i) q^{2} +(10.1819 - 11.8038i) q^{3} +(31.1798 + 26.1630i) q^{4} +(-3.30454 + 18.7410i) q^{5} +(116.004 - 64.8829i) q^{6} +(-6.74337 + 5.65836i) q^{7} +(37.1007 + 64.2603i) q^{8} +(-35.6577 - 240.370i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.01235	+	2.91626i	1.41640	+	0.515526i	0.933001	−	0.359875i	\(-0.117181\pi\)
							0.483397	+	0.875401i	\(0.339403\pi\)
\(3\)	10.1819	−	11.8038i	0.653169	−	0.757212i
							\(5\)	−3.30454	+	18.7410i	−0.0591135	+	0.335249i	−0.999994	−	0.00348971i	\(-0.998889\pi\)
0.940880	+	0.338739i	\(0.110000\pi\)
\(7\)	−6.74337	+	5.65836i	−0.0520154	+	0.0436461i	−0.668425	−	0.743780i	\(-0.733031\pi\)
							0.616409	+	0.787426i	\(0.288587\pi\)
\(11\)	81.7910	+	463.860i	0.203809	+	1.15586i	0.899303	+	0.437327i	\(0.144075\pi\)
							−0.695493	+	0.718533i	\(0.744814\pi\)
\(13\)	−883.772	+	321.667i	−1.45038	+	0.527895i	−0.942698	−	0.333648i	\(-0.891720\pi\)
							−0.507683	+	0.861544i	\(0.669498\pi\)
\(17\)	380.791	−	659.550i	0.319569	−	0.553510i	−0.660829	−	0.750536i	\(-0.729795\pi\)
							0.980398	+	0.197027i	\(0.0631286\pi\)
\(19\)	40.7246	+	70.5370i	0.0258805	+	0.0448263i	0.878676	−	0.477419i	\(-0.158428\pi\)
							−0.852795	+	0.522246i	\(0.825094\pi\)
\(23\)	−2601.59	−	2182.99i	−1.02546	−	0.860463i	−0.0351563	−	0.999382i	\(-0.511193\pi\)
							−0.990304	+	0.138919i	\(0.955637\pi\)
\(29\)	558.459	+	203.263i	0.123310	+	0.0448810i	0.402938	−	0.915227i	\(-0.367989\pi\)
							−0.279628	+	0.960108i	\(0.590211\pi\)
\(31\)	7078.78	+	5939.80i	1.32298	+	1.11011i	0.985663	+	0.168725i	\(0.0539649\pi\)
							0.337320	+	0.941390i	\(0.390480\pi\)
\(37\)	392.306	−	679.493i	0.0471108	−	0.0815983i	−0.841508	−	0.540244i	\(-0.818332\pi\)
							0.888619	+	0.458646i	\(0.151665\pi\)
\(41\)	17791.0	−	6475.38i	1.65288	−	0.601597i	0.663656	−	0.748038i	\(-0.269004\pi\)
							0.989219	+	0.146441i	\(0.0467817\pi\)
\(43\)	1368.62	+	7761.85i	0.112879	+	0.640168i	0.987779	+	0.155864i	\(0.0498162\pi\)
							−0.874900	+	0.484304i	\(0.839073\pi\)
\(47\)	2902.57	−	2435.54i	0.191663	−	0.160824i	−0.541907	−	0.840439i	\(-0.682297\pi\)
							0.733569	+	0.679615i	\(0.237853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.4.12	✓	84
3.2	odd	2		81.6.e.a.64.3		84
27.7	even	9	inner	27.6.e.a.7.12	yes	84
27.13	even	9		729.6.a.c.1.7		42
27.14	odd	18		729.6.a.e.1.36		42
27.20	odd	18		81.6.e.a.19.3		84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.12	✓	84	1.1	even	1	trivial
27.6.e.a.7.12	yes	84	27.7	even	9	inner
81.6.e.a.19.3		84	27.20	odd	18
81.6.e.a.64.3		84	3.2	odd	2
729.6.a.c.1.7		42	27.13	even	9
729.6.a.e.1.36		42	27.14	odd	18