Embedded newform 27.6.e.a.7.14

• Label: 27.6.e.a.7.14
• 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			7.14
Character	\(\chi\)	\(=\)	27.7
Dual form			27.6.e.a.4.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.0391 - 3.65394i) q^{2} +(-15.3240 - 2.85912i) q^{3} +(62.9193 - 52.7956i) q^{4} +(-7.04748 - 39.9683i) q^{5} +(-164.287 + 27.2900i) q^{6} +(72.7973 + 61.0842i) q^{7} +(267.808 - 463.857i) q^{8} +(226.651 + 87.6263i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.0391	−	3.65394i	1.77468	−	0.645932i	0.774777	−	0.632235i	\(-0.217862\pi\)
							0.999906	−	0.0136973i	\(-0.00436011\pi\)
\(3\)	−15.3240	−	2.85912i	−0.983036	−	0.183412i
							\(5\)	−7.04748	−	39.9683i	−0.126069	−	0.714974i	−0.980667	−	0.195684i	\(-0.937307\pi\)
0.854598	−	0.519290i	\(-0.173804\pi\)
\(7\)	72.7973	+	61.0842i	0.561526	+	0.471176i	0.878822	−	0.477150i	\(-0.158330\pi\)
							−0.317295	+	0.948327i	\(0.602775\pi\)
\(11\)	−133.797	+	758.801i	−0.333400	+	1.89080i	0.109093	+	0.994032i	\(0.465205\pi\)
							−0.442493	+	0.896772i	\(0.645906\pi\)
\(13\)	−233.784	−	85.0905i	−0.383669	−	0.139644i	0.142983	−	0.989725i	\(-0.454331\pi\)
							−0.526652	+	0.850081i	\(0.676553\pi\)
\(17\)	73.8617	+	127.932i	0.0619865	+	0.107364i	0.895353	−	0.445357i	\(-0.146923\pi\)
							−0.833367	+	0.552720i	\(0.813590\pi\)
\(19\)	63.9386	−	110.745i	0.0406330	−	0.0703785i	−0.844994	−	0.534776i	\(-0.820396\pi\)
							0.885627	+	0.464398i	\(0.153729\pi\)
\(23\)	1263.19	−	1059.94i	0.497909	−	0.417795i	−0.358942	−	0.933360i	\(-0.616862\pi\)
							0.856851	+	0.515565i	\(0.172418\pi\)
\(29\)	−6454.19	+	2349.13i	−1.42511	+	0.518696i	−0.935524	−	0.353262i	\(-0.885072\pi\)
							−0.489581	+	0.871958i	\(0.662850\pi\)
\(31\)	395.907	−	332.206i	0.0739928	−	0.0620873i	−0.605041	−	0.796194i	\(-0.706843\pi\)
							0.679034	+	0.734107i	\(0.262399\pi\)
\(37\)	2892.03	+	5009.15i	0.347295	+	0.601533i	0.985768	−	0.168111i	\(-0.0537669\pi\)
							−0.638473	+	0.769644i	\(0.720434\pi\)
\(41\)	−16796.5	−	6113.44i	−1.56049	−	0.567971i	−0.589639	−	0.807667i	\(-0.700730\pi\)
							−0.970848	+	0.239696i	\(0.922952\pi\)
\(43\)	649.157	−	3681.55i	0.0535400	−	0.303640i	−0.946265	−	0.323393i	\(-0.895177\pi\)
							0.999805	+	0.0197521i	\(0.00628770\pi\)
\(47\)	1917.07	+	1608.61i	0.126588	+	0.106220i	0.703884	−	0.710315i	\(-0.251448\pi\)
							−0.577296	+	0.816535i	\(0.695892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.7.14	yes	84
3.2	odd	2		81.6.e.a.19.1		84
27.2	odd	18		729.6.a.e.1.41		42
27.4	even	9	inner	27.6.e.a.4.14	✓	84
27.23	odd	18		81.6.e.a.64.1		84
27.25	even	9		729.6.a.c.1.2		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.14	✓	84	27.4	even	9	inner
27.6.e.a.7.14	yes	84	1.1	even	1	trivial
81.6.e.a.19.1		84	3.2	odd	2
81.6.e.a.64.1		84	27.23	odd	18
729.6.a.c.1.2		42	27.25	even	9
729.6.a.e.1.41		42	27.2	odd	18