Embedded newform 229.3.j.a.2.16

• Label: 229.3.j.a.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	229.2
Dual form			229.3.j.a.115.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.723194 + 0.151638i) q^{2} +(1.12055 + 0.606409i) q^{3} +(-3.16308 + 1.38745i) q^{4} +(1.80041 + 0.149186i) q^{5} +(-0.902326 - 0.268634i) q^{6} +(-4.91157 + 2.40112i) q^{7} +(4.48261 - 3.20052i) q^{8} +(-4.03464 - 6.17548i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1368 q - 38 q^{2} - 34 q^{3} - 38 q^{4} - 38 q^{5} - 32 q^{6} - 20 q^{7} - 32 q^{8} - 242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.723194	+	0.151638i	−0.361597	+	0.0758189i	−0.382482	−	0.923963i	\(-0.624931\pi\)
							0.0208850	+	0.999782i	\(0.493352\pi\)
\(3\)	1.12055	+	0.606409i	0.373515	+	0.202136i	0.651850	−	0.758348i	\(-0.273993\pi\)
							−0.278335	+	0.960484i	\(0.589783\pi\)
\(5\)	1.80041	+	0.149186i	0.360083	+	0.0298373i	0.261262	−	0.965268i	\(-0.415861\pi\)
							0.0988207	+	0.995105i	\(0.468493\pi\)
\(7\)	−4.91157	+	2.40112i	−0.701653	+	0.343017i	−0.755151	−	0.655551i	\(-0.772437\pi\)
							0.0534982	+	0.998568i	\(0.482963\pi\)
\(11\)	−10.6110	+	2.68707i	−0.964636	+	0.244279i	−0.695262	−	0.718757i	\(-0.744712\pi\)
							−0.269374	+	0.963036i	\(0.586817\pi\)
\(13\)	0.284592	−	0.241037i	0.0218917	−	0.0185413i	−0.635287	−	0.772276i	\(-0.719118\pi\)
							0.657179	+	0.753735i	\(0.271750\pi\)
\(17\)	0.0898027	−	1.08376i	0.00528251	−	0.0637505i	−0.993433	−	0.114412i	\(-0.963501\pi\)
							0.998716	+	0.0506619i	\(0.0161331\pi\)
\(19\)	−1.73012	−	20.8795i	−0.0910592	−	1.09892i	−0.876958	−	0.480566i	\(-0.840431\pi\)
							0.785899	−	0.618355i	\(-0.212200\pi\)
\(23\)	−31.2689	−	3.89767i	−1.35952	−	0.169464i	−0.589647	−	0.807661i	\(-0.700733\pi\)
							−0.769873	+	0.638197i	\(0.779680\pi\)
\(29\)	1.38930	+	2.84186i	0.0479070	+	0.0979953i	0.921007	−	0.389545i	\(-0.127368\pi\)
							−0.873100	+	0.487541i	\(0.837894\pi\)
\(31\)	−2.36763	+	3.97339i	−0.0763750	+	0.128174i	−0.894848	−	0.446372i	\(-0.852716\pi\)
							0.818473	+	0.574546i	\(0.194821\pi\)
\(37\)	12.3926	+	13.4619i	0.334934	+	0.363835i	0.880345	−	0.474333i	\(-0.157311\pi\)
							−0.545411	+	0.838168i	\(0.683627\pi\)
\(41\)	−0.708162	−	3.37738i	−0.0172722	−	0.0823751i	0.969214	−	0.246221i	\(-0.0791889\pi\)
							−0.986486	+	0.163846i	\(0.947610\pi\)
\(43\)	−18.0859	−	19.6466i	−0.420603	−	0.456897i	0.489056	−	0.872253i	\(-0.337341\pi\)
							−0.909659	+	0.415355i	\(0.863657\pi\)
\(47\)	−81.9664	−	17.1865i	−1.74397	−	0.365671i	−0.778792	−	0.627282i	\(-0.784167\pi\)
							−0.965174	+	0.261610i	\(0.915746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.3.j.a.2.16	✓	1368
229.115	odd	76	inner	229.3.j.a.115.16	yes	1368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.3.j.a.2.16	✓	1368	1.1	even	1	trivial
229.3.j.a.115.16	yes	1368	229.115	odd	76	inner