Embedded newform 229.4.g.a.16.19

• Label: 229.4.g.a.16.19
• Level: $229$
• Weight: $4$
• Character: 229.16
• Analytic conductor: $13.511$
• Analytic rank: $0$
• Dimension: $1026$
• CM: no
• 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			16.19
Character	\(\chi\)	\(=\)	229.16
Dual form			229.4.g.a.43.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45929 + 1.58521i) q^{2} +(-3.13463 + 7.14624i) q^{3} +(0.277268 + 3.34612i) q^{4} +(16.9279 + 5.81135i) q^{5} +(-6.75396 - 15.3975i) q^{6} +(0.683761 + 2.70011i) q^{7} +(-19.3113 - 15.0306i) q^{8} +(-22.9562 - 24.9371i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1026 q - 15 q^{2} - 21 q^{3} - 257 q^{4} + 57 q^{5} + 3 q^{6} + 7 q^{7} + 23 q^{8} - 470 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{7}{19}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45929	+	1.58521i	−0.515936	+	0.560456i	−0.938234	−	0.346001i	\(-0.887539\pi\)
							0.422298	+	0.906457i	\(0.361224\pi\)
\(3\)	−3.13463	+	7.14624i	−0.603260	+	1.37529i	0.303169	+	0.952937i	\(0.401955\pi\)
							−0.906429	+	0.422358i	\(0.861203\pi\)
\(5\)	16.9279	+	5.81135i	1.51408	+	0.519783i	0.951682	−	0.307085i	\(-0.0993538\pi\)
							0.562395	+	0.826869i	\(0.309880\pi\)
\(7\)	0.683761	+	2.70011i	0.0369196	+	0.145792i	0.985115	−	0.171894i	\(-0.0549887\pi\)
							−0.948196	+	0.317687i	\(0.897094\pi\)
\(11\)	−27.2194	+	41.6624i	−0.746086	+	1.14197i	0.239185	+	0.970974i	\(0.423120\pi\)
							−0.985272	+	0.170996i	\(0.945301\pi\)
\(13\)	23.9582	+	8.22486i	0.511139	+	0.175474i	0.568191	−	0.822897i	\(-0.307644\pi\)
							−0.0570514	+	0.998371i	\(0.518170\pi\)
\(17\)	6.50838	+	2.23433i	0.0928537	+	0.0318767i	0.370735	−	0.928739i	\(-0.379106\pi\)
							−0.277881	+	0.960615i	\(0.589632\pi\)
\(19\)	−60.5649	+	20.7920i	−0.731292	+	0.251053i	−0.665100	−	0.746754i	\(-0.731611\pi\)
							−0.0661917	+	0.997807i	\(0.521085\pi\)
\(23\)	44.6953	+	24.1879i	0.405201	+	0.219284i	0.665747	−	0.746178i	\(-0.268113\pi\)
							−0.260546	+	0.965461i	\(0.583903\pi\)
\(29\)	−35.3974	−	139.781i	−0.226660	−	0.895058i	−0.973245	−	0.229769i	\(-0.926203\pi\)
							0.746586	−	0.665289i	\(-0.231692\pi\)
\(31\)	−1.64028	+	2.51064i	−0.00950332	+	0.0145459i	−0.841887	−	0.539655i	\(-0.818555\pi\)
							0.832383	+	0.554200i	\(0.186976\pi\)
\(37\)	−314.276	−	52.4434i	−1.39640	−	0.233017i	−0.581935	−	0.813235i	\(-0.697704\pi\)
							−0.814461	+	0.580218i	\(0.802967\pi\)
\(41\)	13.5141	−	14.6802i	0.0514768	−	0.0559187i	−0.709454	−	0.704752i	\(-0.751058\pi\)
							0.760931	+	0.648833i	\(0.224743\pi\)
\(43\)	507.165	+	84.6310i	1.79865	+	0.300142i	0.966928	−	0.255049i	\(-0.0820915\pi\)
							0.831723	+	0.555190i	\(0.187355\pi\)
\(47\)	160.681	+	174.547i	0.498676	+	0.541707i	0.933391	−	0.358861i	\(-0.116835\pi\)
							−0.434714	+	0.900568i	\(0.643151\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.4.g.a.16.19	✓	1026
229.43	even	19	inner	229.4.g.a.43.19	yes	1026

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.4.g.a.16.19	✓	1026	1.1	even	1	trivial
229.4.g.a.43.19	yes	1026	229.43	even	19	inner