Embedded newform 27.10.e.a.4.11

• Label: 27.10.e.a.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.47952 - 2.35835i) q^{2} +(83.5586 + 112.699i) q^{3} +(-355.792 - 298.545i) q^{4} +(-63.8763 + 362.261i) q^{5} +(-275.637 - 927.294i) q^{6} +(6593.75 - 5532.82i) q^{7} +(3366.50 + 5830.96i) q^{8} +(-5718.93 + 18833.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.47952	−	2.35835i	−0.286357	−	0.104226i	0.194848	−	0.980833i	\(-0.437579\pi\)
							−0.481205	+	0.876608i	\(0.659801\pi\)
\(3\)	83.5586	+	112.699i	0.595587	+	0.803291i
							\(5\)	−63.8763	+	362.261i	−0.0457062	+	0.259213i	−0.999095	−	0.0425315i	\(-0.986458\pi\)
0.953389	+	0.301744i	\(0.0975688\pi\)
\(7\)	6593.75	−	5532.82i	1.03799	−	0.870974i	0.0462067	−	0.998932i	\(-0.485287\pi\)
							0.991780	+	0.127958i	\(0.0408423\pi\)
\(11\)	4585.32	+	26004.6i	0.0944283	+	0.535530i	0.994921	+	0.100659i	\(0.0320950\pi\)
							−0.900493	+	0.434871i	\(0.856794\pi\)
\(13\)	12128.3	−	4414.34i	0.117775	−	0.0428668i	−0.282460	−	0.959279i	\(-0.591150\pi\)
							0.400236	+	0.916412i	\(0.368928\pi\)
\(17\)	−171522.	+	297085.i	−0.498081	+	0.862701i	−0.999998	−	0.00221490i	\(-0.999295\pi\)
							0.501917	+	0.864916i	\(0.332628\pi\)
\(19\)	355097.	+	615046.i	0.625109	+	1.08272i	0.988520	+	0.151091i	\(0.0482787\pi\)
							−0.363411	+	0.931629i	\(0.618388\pi\)
\(23\)	1.53237e6	+	1.28581e6i	1.14180	+	0.958082i	0.999496	−	0.0317417i	\(-0.0101054\pi\)
							0.142301	+	0.989823i	\(0.454550\pi\)
\(29\)	2.22946e6	+	811457.i	0.585341	+	0.213047i	0.617679	−	0.786430i	\(-0.288073\pi\)
							−0.0323381	+	0.999477i	\(0.510295\pi\)
\(31\)	−5.47937e6	−	4.59773e6i	−1.06562	−	0.894162i	−0.0709721	−	0.997478i	\(-0.522610\pi\)
							−0.994649	+	0.103316i	\(0.967055\pi\)
\(37\)	−2.84853e6	+	4.93380e6i	−0.249869	+	0.432786i	−0.963489	−	0.267747i	\(-0.913721\pi\)
							0.713620	+	0.700533i	\(0.247054\pi\)
\(41\)	1.26360e7	−	4.59911e6i	0.698362	−	0.254183i	0.0316509	−	0.999499i	\(-0.489924\pi\)
							0.666712	+	0.745316i	\(0.267701\pi\)
\(43\)	−4.20355e6	−	2.38395e7i	−0.187503	−	1.06338i	−0.922697	−	0.385526i	\(-0.874020\pi\)
							0.735194	−	0.677857i	\(-0.237091\pi\)
\(47\)	−3.30774e7	+	2.77552e7i	−0.988761	+	0.829669i	−0.985388	−	0.170326i	\(-0.945518\pi\)
							−0.00337286	+	0.999994i	\(0.501074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.11	✓	156
3.2	odd	2		81.10.e.a.64.16		156
27.7	even	9	inner	27.10.e.a.7.11	yes	156
27.20	odd	18		81.10.e.a.19.16		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.11	✓	156	1.1	even	1	trivial
27.10.e.a.7.11	yes	156	27.7	even	9	inner
81.10.e.a.19.16		156	27.20	odd	18
81.10.e.a.64.16		156	3.2	odd	2