Embedded newform 229.4.g.a.43.3

• Label: 229.4.g.a.43.3
• Level: $229$
• Weight: $4$
• Character: 229.43
• Analytic conductor: $13.511$
• Analytic rank: $0$
• Dimension: $1026$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	229.g (of order \(19\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5114373913\)
Analytic rank:	\(0\)
Dimension:	\(1026\)
Relative dimension:	\(57\) over \(\Q(\zeta_{19})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{19}]$

Embedding invariants

Embedding label			43.3
Character	\(\chi\)	\(=\)	229.43
Dual form			229.4.g.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.54340 - 3.84916i) q^{2} +(-0.642373 - 1.46446i) q^{3} +(-1.59970 + 19.3055i) q^{4} +(-15.3330 + 5.26382i) q^{5} +(-3.36076 + 7.66177i) q^{6} +(-0.615535 + 2.43069i) q^{7} +(46.9494 - 36.5422i) q^{8} +(16.5546 - 17.9831i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1026 q - 15 q^{2} - 21 q^{3} - 257 q^{4} + 57 q^{5} + 3 q^{6} + 7 q^{7} + 23 q^{8} - 470 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{12}{19}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.54340	−	3.84916i	−1.25278	−	1.36088i	−0.906193	−	0.422865i	\(-0.861024\pi\)
							−0.346588	−	0.938017i	\(-0.612660\pi\)
\(3\)	−0.642373	−	1.46446i	−0.123625	−	0.281836i	0.843054	−	0.537829i	\(-0.180755\pi\)
							−0.966679	+	0.255993i	\(0.917598\pi\)
\(5\)	−15.3330	+	5.26382i	−1.37142	+	0.470811i	−0.909351	−	0.416031i	\(-0.863421\pi\)
							−0.462074	+	0.886841i	\(0.652895\pi\)
\(7\)	−0.615535	+	2.43069i	−0.0332358	+	0.131245i	−0.983794	−	0.179300i	\(-0.942617\pi\)
							0.950559	+	0.310545i	\(0.100512\pi\)
\(11\)	20.0169	+	30.6382i	0.548666	+	0.839796i	0.998580	−	0.0532763i	\(-0.0169664\pi\)
							−0.449914	+	0.893072i	\(0.648545\pi\)
\(13\)	20.1090	−	6.90342i	0.429018	−	0.147282i	−0.101730	−	0.994812i	\(-0.532438\pi\)
							0.530747	+	0.847530i	\(0.321911\pi\)
\(17\)	−23.2211	+	7.97182i	−0.331291	+	0.113732i	−0.485327	−	0.874333i	\(-0.661299\pi\)
							0.154035	+	0.988065i	\(0.450773\pi\)
\(19\)	39.1602	+	13.4437i	0.472841	+	0.162326i	0.550812	−	0.834629i	\(-0.314318\pi\)
							−0.0779717	+	0.996956i	\(0.524844\pi\)
\(23\)	−115.494	+	62.5021i	−1.04705	+	0.566634i	−0.905960	−	0.423363i	\(-0.860850\pi\)
							−0.141088	+	0.989997i	\(0.545060\pi\)
\(29\)	52.6731	−	208.001i	0.337281	−	1.33189i	−0.535812	−	0.844337i	\(-0.679994\pi\)
							0.873093	−	0.487554i	\(-0.162111\pi\)
\(31\)	78.5324	+	120.203i	0.454995	+	0.696421i	0.988810	−	0.149180i	\(-0.0476633\pi\)
							−0.533815	+	0.845601i	\(0.679242\pi\)
\(37\)	−185.463	+	30.9483i	−0.824053	+	0.137510i	−0.561573	−	0.827427i	\(-0.689804\pi\)
							−0.262480	+	0.964937i	\(0.584540\pi\)
\(41\)	−267.473	−	290.553i	−1.01883	−	1.10675i	−0.994254	−	0.107051i	\(-0.965859\pi\)
							−0.0245806	−	0.999698i	\(-0.507825\pi\)
\(43\)	−123.794	+	20.6576i	−0.439032	+	0.0732616i	−0.379983	−	0.924994i	\(-0.624070\pi\)
							−0.0590495	+	0.998255i	\(0.518807\pi\)
\(47\)	138.500	−	150.451i	0.429836	−	0.466927i	−0.482781	−	0.875741i	\(-0.660373\pi\)
							0.912618	+	0.408814i	\(0.134058\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.4.g.a.43.3	yes	1026
229.16	even	19	inner	229.4.g.a.16.3	✓	1026

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.4.g.a.16.3	✓	1026	229.16	even	19	inner
229.4.g.a.43.3	yes	1026	1.1	even	1	trivial