Embedded newform 27.10.e.a.4.17

• Label: 27.10.e.a.4.17
• 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			4.17
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.5174 + 4.91993i) q^{2} +(-10.0180 - 139.938i) q^{3} +(-233.700 - 196.098i) q^{4} +(-142.171 + 806.289i) q^{5} +(553.068 - 1940.89i) q^{6} +(-461.968 + 387.637i) q^{7} +(-5876.77 - 10178.9i) q^{8} +(-19482.3 + 2803.80i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.5174	+	4.91993i	0.597390	+	0.217432i	0.622977	−	0.782240i	\(-0.285923\pi\)
							−0.0255864	+	0.999673i	\(0.508145\pi\)
\(3\)	−10.0180	−	139.938i	−0.0714062	−	0.997447i
							\(5\)	−142.171	+	806.289i	−0.101729	+	0.576934i	0.890748	+	0.454498i	\(0.150181\pi\)
−0.992477	+	0.122435i	\(0.960930\pi\)
\(7\)	−461.968	+	387.637i	−0.0727228	+	0.0610217i	−0.678424	−	0.734670i	\(-0.737337\pi\)
							0.605702	+	0.795692i	\(0.292892\pi\)
\(11\)	12504.2	+	70915.0i	0.257507	+	1.46040i	0.789554	+	0.613681i	\(0.210312\pi\)
							−0.532047	+	0.846715i	\(0.678577\pi\)
\(13\)	−37507.5	+	13651.6i	−0.364228	+	0.132568i	−0.517650	−	0.855593i	\(-0.673193\pi\)
							0.153422	+	0.988161i	\(0.450971\pi\)
\(17\)	−295337.	+	511538.i	−0.857624	+	1.48545i	0.0165642	+	0.999863i	\(0.494727\pi\)
							−0.874189	+	0.485586i	\(0.838606\pi\)
\(19\)	−424711.	−	735622.i	−0.747657	−	1.29498i	−0.948943	−	0.315448i	\(-0.897845\pi\)
							0.201285	−	0.979533i	\(-0.435488\pi\)
\(23\)	−1.08911e6	−	913870.i	−0.811513	−	0.680940i	0.139455	−	0.990228i	\(-0.455465\pi\)
							−0.950968	+	0.309288i	\(0.899909\pi\)
\(29\)	−1.20737e6	−	439449.i	−0.316994	−	0.115376i	0.178623	−	0.983918i	\(-0.442836\pi\)
							−0.495617	+	0.868541i	\(0.665058\pi\)
\(31\)	−5.64303e6	−	4.73507e6i	−1.09745	−	0.920870i	−0.100199	−	0.994967i	\(-0.531948\pi\)
							−0.997251	+	0.0740971i	\(0.976393\pi\)
\(37\)	3.87543e6	−	6.71245e6i	0.339948	−	0.588807i	−0.644475	−	0.764626i	\(-0.722924\pi\)
							0.984423	+	0.175819i	\(0.0562572\pi\)
\(41\)	9.60570e6	−	3.49619e6i	0.530886	−	0.193227i	−0.0626477	−	0.998036i	\(-0.519954\pi\)
							0.593534	+	0.804809i	\(0.297732\pi\)
\(43\)	−2.25267e6	−	1.27756e7i	−0.100483	−	0.569865i	−0.992929	−	0.118711i	\(-0.962124\pi\)
							0.892446	−	0.451154i	\(-0.148987\pi\)
\(47\)	2.02868e7	−	1.70226e7i	0.606420	−	0.508846i	−0.287082	−	0.957906i	\(-0.592685\pi\)
							0.893502	+	0.449059i	\(0.148241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.17	✓	156
3.2	odd	2		81.10.e.a.64.10		156
27.7	even	9	inner	27.10.e.a.7.17	yes	156
27.20	odd	18		81.10.e.a.19.10		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.17	✓	156	1.1	even	1	trivial
27.10.e.a.7.17	yes	156	27.7	even	9	inner
81.10.e.a.19.10		156	27.20	odd	18
81.10.e.a.64.10		156	3.2	odd	2