Embedded newform 54.5.f.a.11.4

• Label: 54.5.f.a.11.4
• Level: $54$
• Weight: $5$
• Character: 54.11
• Analytic conductor: $5.582$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	54.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58197800653\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			11.4
Character	\(\chi\)	\(=\)	54.11
Dual form			54.5.f.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81808 + 2.16670i) q^{2} +(-0.346638 + 8.99332i) q^{3} +(-1.38919 - 7.87846i) q^{4} +(-2.28016 + 6.26470i) q^{5} +(-18.8556 - 17.1016i) q^{6} +(-10.8277 + 61.4069i) q^{7} +(19.5959 + 11.3137i) q^{8} +(-80.7597 - 6.23486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 18 q^{5} - 96 q^{6} + 204 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81808	+	2.16670i	−0.454519	+	0.541675i
							\(3\)	−0.346638	+	8.99332i	−0.0385154	+	0.999258i
\(5\)	−2.28016	+	6.26470i	−0.0912065	+	0.250588i								−0.976905	−	0.213672i	\(-0.931458\pi\)
							0.885699	+	0.464260i	\(0.153680\pi\)
\(7\)	−10.8277	+	61.4069i	−0.220973	+	1.25320i	0.649262	+	0.760565i	\(0.275078\pi\)
							−0.870235	+	0.492637i	\(0.836033\pi\)
\(11\)	−78.0056	−	214.319i	−0.644674	−	1.77123i	−0.636518	−	0.771262i	\(-0.719626\pi\)
							−0.00815634	−	0.999967i	\(-0.502596\pi\)
\(13\)	16.0775	−	13.4906i	0.0951330	−	0.0798260i	−0.593981	−	0.804479i	\(-0.702444\pi\)
							0.689114	+	0.724653i	\(0.258000\pi\)
\(17\)	−356.069	+	205.577i	−1.23207	+	0.711338i	−0.967462	−	0.253017i	\(-0.918577\pi\)
							−0.264612	+	0.964355i	\(0.585244\pi\)
\(19\)	−93.7450	+	162.371i	−0.259681	+	0.449781i	−0.966157	−	0.257956i	\(-0.916951\pi\)
							0.706475	+	0.707738i	\(0.250284\pi\)
\(23\)	436.761	−	77.0128i	0.825635	−	0.145582i	0.255162	−	0.966898i	\(-0.417871\pi\)
							0.570473	+	0.821316i	\(0.306760\pi\)
\(29\)	−764.478	+	911.069i	−0.909010	+	1.08332i	0.0871870	+	0.996192i	\(0.472212\pi\)
							−0.996197	+	0.0871245i	\(0.972232\pi\)
\(31\)	136.363	+	773.355i	0.141897	+	0.804740i	0.969806	+	0.243877i	\(0.0784194\pi\)
							−0.827909	+	0.560863i	\(0.810469\pi\)
\(37\)	969.274	+	1678.83i	0.708016	+	1.22632i	0.965592	+	0.260062i	\(0.0837429\pi\)
							−0.257576	+	0.966258i	\(0.582924\pi\)
\(41\)	−441.321	−	525.946i	−0.262535	−	0.312877i	0.618633	−	0.785680i	\(-0.287687\pi\)
							−0.881168	+	0.472803i	\(0.843242\pi\)
\(43\)	−1300.41	+	473.311i	−0.703306	+	0.255982i	−0.668822	−	0.743423i	\(-0.733201\pi\)
							−0.0344841	+	0.999405i	\(0.510979\pi\)
\(47\)	580.588	+	102.373i	0.262828	+	0.0463437i	0.303509	−	0.952828i	\(-0.401842\pi\)
							−0.0406810	+	0.999172i	\(0.512953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.5.f.a.11.4	yes	72
3.2	odd	2		162.5.f.a.35.8		72
27.5	odd	18	inner	54.5.f.a.5.4	✓	72
27.22	even	9		162.5.f.a.125.8		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.5.f.a.5.4	✓	72	27.5	odd	18	inner
54.5.f.a.11.4	yes	72	1.1	even	1	trivial
162.5.f.a.35.8		72	3.2	odd	2
162.5.f.a.125.8		72	27.22	even	9