Embedded newform 229.3.j.a.2.4

• Label: 229.3.j.a.2.4
• Level: $229$
• Weight: $3$
• Character: 229.2
• Analytic conductor: $6.240$
• Analytic rank: $0$
• Dimension: $1368$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	229.j (of order \(76\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.23979805385\)
Analytic rank:	\(0\)
Dimension:	\(1368\)
Relative dimension:	\(38\) over \(\Q(\zeta_{76})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{76}]$

Embedding invariants

Embedding label			2.4
Character	\(\chi\)	\(=\)	229.2
Dual form			229.3.j.a.115.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.53314 + 0.740823i) q^{2} +(2.71340 + 1.46842i) q^{3} +(8.27120 - 3.62808i) q^{4} +(7.69168 + 0.637351i) q^{5} +(-10.6747 - 3.17798i) q^{6} +(-11.5224 + 5.63294i) q^{7} +(-14.7837 + 10.5553i) q^{8} +(0.283741 + 0.434297i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1368 q - 38 q^{2} - 34 q^{3} - 38 q^{4} - 38 q^{5} - 32 q^{6} - 20 q^{7} - 32 q^{8} - 242 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{7}{76}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.53314	+	0.740823i	−1.76657	+	0.370411i	−0.971673	−	0.236330i	\(-0.924055\pi\)
							−0.794899	+	0.606741i	\(0.792476\pi\)
\(3\)	2.71340	+	1.46842i	0.904466	+	0.489472i	0.860436	−	0.509558i	\(-0.170191\pi\)
							0.0440293	+	0.999030i	\(0.485981\pi\)
\(5\)	7.69168	+	0.637351i	1.53834	+	0.127470i	0.821677	−	0.569954i	\(-0.193039\pi\)
							0.716659	+	0.697424i	\(0.245670\pi\)
\(7\)	−11.5224	+	5.63294i	−1.64605	+	0.804706i	−0.646942	+	0.762539i	\(0.723952\pi\)
							−0.999111	+	0.0421668i	\(0.986574\pi\)
\(11\)	−16.6816	+	4.22436i	−1.51651	+	0.384033i	−0.911202	−	0.411959i	\(-0.864845\pi\)
							−0.605308	+	0.795991i	\(0.706950\pi\)
\(13\)	−0.0907182	+	0.0768344i	−0.00697832	+	0.00591034i	−0.649782	−	0.760121i	\(-0.725140\pi\)
							0.642803	+	0.766031i	\(0.277771\pi\)
\(17\)	−1.49393	+	18.0291i	−0.0878783	+	1.06053i	0.799860	+	0.600186i	\(0.204907\pi\)
							−0.887739	+	0.460348i	\(0.847725\pi\)
\(19\)	1.74097	+	21.0104i	0.0916302	+	1.10581i	0.874957	+	0.484201i	\(0.160890\pi\)
							−0.783327	+	0.621610i	\(0.786479\pi\)
\(23\)	−10.2856	−	1.28210i	−0.447201	−	0.0557436i	−0.103089	−	0.994672i	\(-0.532873\pi\)
							−0.344112	+	0.938929i	\(0.611820\pi\)
\(29\)	−3.24312	−	6.63391i	−0.111832	−	0.228756i	0.835164	−	0.550000i	\(-0.185372\pi\)
							−0.946996	+	0.321245i	\(0.895899\pi\)
\(31\)	−14.4099	+	24.1829i	−0.464836	+	0.780095i	−0.997818	−	0.0660318i	\(-0.978966\pi\)
							0.532982	+	0.846127i	\(0.321071\pi\)
\(37\)	49.1909	+	53.4355i	1.32948	+	1.44420i	0.805663	+	0.592375i	\(0.201809\pi\)
							0.523820	+	0.851829i	\(0.324506\pi\)
\(41\)	−8.36454	−	39.8923i	−0.204013	−	0.972983i	−0.951249	−	0.308424i	\(-0.900198\pi\)
							0.747236	−	0.664559i	\(-0.231381\pi\)
\(43\)	55.4361	+	60.2197i	1.28921	+	1.40046i	0.870370	+	0.492399i	\(0.163880\pi\)
							0.418843	+	0.908059i	\(0.362436\pi\)
\(47\)	26.6567	+	5.58933i	0.567165	+	0.118922i	0.479994	−	0.877272i	\(-0.340639\pi\)
							0.0871702	+	0.996193i	\(0.472218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.3.j.a.2.4	✓	1368
229.115	odd	76	inner	229.3.j.a.115.4	yes	1368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.3.j.a.2.4	✓	1368	1.1	even	1	trivial
229.3.j.a.115.4	yes	1368	229.115	odd	76	inner