Embedded newform 229.4.i.a.3.20

• Label: 229.4.i.a.3.20
• Level: $229$
• Weight: $4$
• Character: 229.3
• Analytic conductor: $13.511$
• Analytic rank: $0$
• Dimension: $2052$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	229.i (of order \(57\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5114373913\)
Analytic rank:	\(0\)
Dimension:	\(2052\)
Relative dimension:	\(57\) over \(\Q(\zeta_{57})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{57}]$

Embedding invariants

Embedding label			3.20
Character	\(\chi\)	\(=\)	229.3
Dual form			229.4.i.a.153.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73238 - 0.937518i) q^{2} +(2.72537 - 2.65128i) q^{3} +(-2.25338 - 3.44905i) q^{4} +(5.31963 + 17.0078i) q^{5} +(-7.20702 + 2.03795i) q^{6} +(7.16729 - 19.1430i) q^{7} +(1.97148 + 23.7922i) q^{8} +(-0.345615 + 12.5383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2052 q - 30 q^{2} - 33 q^{3} - 514 q^{4} - 99 q^{5} - 33 q^{6} - 43 q^{7} + 46 q^{8} + 464 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/229\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{52}{57}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73238	−	0.937518i	−0.612490	−	0.331463i	0.139915	−	0.990163i	\(-0.455317\pi\)
							−0.752405	+	0.658701i	\(0.771106\pi\)
\(3\)	2.72537	−	2.65128i	0.524498	−	0.510240i	−0.386701	−	0.922205i	\(-0.626386\pi\)
							0.911200	+	0.411965i	\(0.135157\pi\)
\(5\)	5.31963	+	17.0078i	0.475802	+	1.52122i	0.814406	+	0.580296i	\(0.197063\pi\)
							−0.338603	+	0.940929i	\(0.609955\pi\)
\(7\)	7.16729	−	19.1430i	0.386997	−	1.03362i	−0.587500	−	0.809224i	\(-0.699888\pi\)
							0.974497	−	0.224399i	\(-0.0720420\pi\)
\(11\)	5.90602	−	4.59684i	0.161885	−	0.126000i	−0.530031	−	0.847978i	\(-0.677820\pi\)
							0.691916	+	0.721978i	\(0.256767\pi\)
\(13\)	−31.3309	+	34.0344i	−0.668433	+	0.726112i	−0.974123	−	0.226019i	\(-0.927429\pi\)
							0.305690	+	0.952131i	\(0.401113\pi\)
\(17\)	−51.1173	+	55.5282i	−0.729280	+	0.792209i	−0.984633	−	0.174639i	\(-0.944124\pi\)
							0.255353	+	0.966848i	\(0.417808\pi\)
\(19\)	−34.9734	+	7.83773i	−0.422287	+	0.0946368i	−0.424644	−	0.905360i	\(-0.639601\pi\)
							0.00235671	+	0.999997i	\(0.499250\pi\)
\(23\)	36.4155	+	186.427i	0.330137	+	1.69012i	0.664191	+	0.747563i	\(0.268776\pi\)
							−0.334054	+	0.942554i	\(0.608417\pi\)
\(29\)	80.9580	+	98.3019i	0.518398	+	0.629455i	0.964040	−	0.265755i	\(-0.0856214\pi\)
							−0.445643	+	0.895211i	\(0.647025\pi\)
\(31\)	−36.1654	−	260.806i	−0.209532	−	1.51104i	−0.745333	−	0.666693i	\(-0.767709\pi\)
							0.535800	−	0.844345i	\(-0.320010\pi\)
\(37\)	190.621	−	259.349i	0.846970	−	1.15234i	−0.139793	−	0.990181i	\(-0.544644\pi\)
							0.986764	−	0.162164i	\(-0.0518474\pi\)
\(41\)	11.0544	−	6.79509i	0.0421073	−	0.0258833i	−0.502459	−	0.864601i	\(-0.667571\pi\)
							0.544567	+	0.838718i	\(0.316694\pi\)
\(43\)	162.020	+	369.368i	0.574599	+	1.30995i	0.927335	+	0.374232i	\(0.122094\pi\)
							−0.352736	+	0.935723i	\(0.614749\pi\)
\(47\)	−3.97765	+	144.302i	−0.0123447	+	0.447842i	0.968045	+	0.250775i	\(0.0806855\pi\)
							−0.980390	+	0.197067i	\(0.936858\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.4.i.a.3.20	✓	2052
229.153	even	57	inner	229.4.i.a.153.20	yes	2052

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.4.i.a.3.20	✓	2052	1.1	even	1	trivial
229.4.i.a.153.20	yes	2052	229.153	even	57	inner