Embedded newform 229.3.j.a.2.6

• Label: 229.3.j.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	229.2
Dual form			229.3.j.a.115.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.99165 + 0.627283i) q^{2} +(-4.16513 - 2.25406i) q^{3} +(4.89338 - 2.14644i) q^{4} +(-1.25098 - 0.103659i) q^{5} +(13.8745 + 4.13063i) q^{6} +(-9.36336 + 4.57747i) q^{7} +(-3.34208 + 2.38620i) q^{8} +(7.34501 + 11.2424i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1368 q - 38 q^{2} - 34 q^{3} - 38 q^{4} - 38 q^{5} - 32 q^{6} - 20 q^{7} - 32 q^{8} - 242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.99165	+	0.627283i	−1.49582	+	0.313641i	−0.880278	−	0.474459i	\(-0.842644\pi\)
							−0.615547	+	0.788100i	\(0.711065\pi\)
\(3\)	−4.16513	−	2.25406i	−1.38838	−	0.751352i	−0.401967	−	0.915654i	\(-0.631673\pi\)
							−0.986410	+	0.164302i	\(0.947463\pi\)
\(5\)	−1.25098	−	0.103659i	−0.250195	−	0.0207318i	−0.0431715	−	0.999068i	\(-0.513746\pi\)
							−0.207024	+	0.978336i	\(0.566378\pi\)
\(7\)	−9.36336	+	4.57747i	−1.33762	+	0.653924i	−0.962051	−	0.272869i	\(-0.912027\pi\)
							−0.375572	+	0.926793i	\(0.622554\pi\)
\(11\)	−13.0897	+	3.31476i	−1.18997	+	0.301342i	−0.788794	−	0.614658i	\(-0.789294\pi\)
							−0.401178	+	0.916000i	\(0.631399\pi\)
\(13\)	−13.9914	+	11.8501i	−1.07626	+	0.911549i	−0.996359	−	0.0852534i	\(-0.972830\pi\)
							−0.0799047	+	0.996803i	\(0.525462\pi\)
\(17\)	−0.856904	+	10.3413i	−0.0504061	+	0.608311i	0.923833	+	0.382795i	\(0.125038\pi\)
							−0.974239	+	0.225516i	\(0.927593\pi\)
\(19\)	−2.42240	−	29.2340i	−0.127494	−	1.53863i	−0.697258	−	0.716821i	\(-0.745597\pi\)
							0.569763	−	0.821809i	\(-0.307035\pi\)
\(23\)	7.83000	+	0.976009i	0.340435	+	0.0424352i	0.292077	−	0.956395i	\(-0.405654\pi\)
							0.0483582	+	0.998830i	\(0.484601\pi\)
\(29\)	20.3028	+	41.5301i	0.700098	+	1.43207i	0.892634	+	0.450783i	\(0.148855\pi\)
							−0.192536	+	0.981290i	\(0.561671\pi\)
\(31\)	29.6294	−	49.7245i	0.955786	−	1.60402i	0.170066	−	0.985433i	\(-0.445602\pi\)
							0.785720	−	0.618583i	\(-0.212293\pi\)
\(37\)	−29.7948	−	32.3658i	−0.805265	−	0.874751i	0.188969	−	0.981983i	\(-0.439485\pi\)
							−0.994234	+	0.107232i	\(0.965801\pi\)
\(41\)	3.58518	+	17.0985i	0.0874433	+	0.417036i	0.999968	+	0.00801448i	\(0.00255112\pi\)
							−0.912525	+	0.409022i	\(0.865870\pi\)
\(43\)	5.27206	+	5.72698i	0.122606	+	0.133186i	0.794007	−	0.607909i	\(-0.207991\pi\)
							−0.671401	+	0.741094i	\(0.734307\pi\)
\(47\)	−20.6554	−	4.33099i	−0.439478	−	0.0921488i	−0.0197626	−	0.999805i	\(-0.506291\pi\)
							−0.419715	+	0.907656i	\(0.637870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.3.j.a.2.6	✓	1368
229.115	odd	76	inner	229.3.j.a.115.6	yes	1368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.3.j.a.2.6	✓	1368	1.1	even	1	trivial
229.3.j.a.115.6	yes	1368	229.115	odd	76	inner