Embedded newform 429.2.y.b.116.18

• Label: 429.2.y.b.116.18
• Level: $429$
• Weight: $2$
• Character: 429.116
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			116.18
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.687043 + 0.223234i) q^{2} +(-1.03953 - 1.38542i) q^{3} +(-1.19584 + 0.868828i) q^{4} +(-0.0885211 + 0.272440i) q^{5} +(1.02347 + 0.719784i) q^{6} +(1.47481 - 1.07151i) q^{7} +(1.47687 - 2.03274i) q^{8} +(-0.838768 + 2.88036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.687043	+	0.223234i	−0.485813	+	0.157850i	−0.541674	−	0.840588i	\(-0.682209\pi\)
							0.0558617	+	0.998439i	\(0.482209\pi\)
\(3\)	−1.03953	−	1.38542i	−0.600171	−	0.799872i
							\(5\)	−0.0885211	+	0.272440i	−0.0395879	+	0.121839i	−0.968897	−	0.247463i	\(-0.920403\pi\)
0.929309	+	0.369302i	\(0.120403\pi\)
\(7\)	1.47481	−	1.07151i	0.557426	−	0.404994i	−0.273090	−	0.961989i	\(-0.588046\pi\)
							0.830516	+	0.556995i	\(0.188046\pi\)
\(11\)	−3.25098	+	0.656614i	−0.980207	+	0.197977i
							\(13\)	−1.68516	−	3.18751i	−0.467378	−	0.884057i
\(17\)	−1.63781	+	5.04066i	−0.397227	+	1.22254i								0.529987	+	0.848006i	\(0.322197\pi\)
							−0.927214	+	0.374533i	\(0.877803\pi\)
\(19\)	3.22058	+	2.33989i	0.738852	+	0.536808i	0.892351	−	0.451341i	\(-0.149054\pi\)
							−0.153499	+	0.988149i	\(0.549054\pi\)
\(23\)			7.26056i			1.51393i	0.653454	+	0.756966i	\(0.273319\pi\)
							−0.653454	+	0.756966i	\(0.726681\pi\)
\(29\)	−5.45777	+	3.96531i	−1.01348	+	0.736339i	−0.964937	−	0.262482i	\(-0.915459\pi\)
							−0.0485464	+	0.998821i	\(0.515459\pi\)
\(31\)	−4.20750	+	1.36710i	−0.755689	+	0.245538i	−0.661427	−	0.750009i	\(-0.730049\pi\)
							−0.0942615	+	0.995547i	\(0.530049\pi\)
\(37\)	5.70924	+	7.85809i	0.938593	+	1.29186i	0.956412	+	0.292021i	\(0.0943276\pi\)
							−0.0178195	+	0.999841i	\(0.505672\pi\)
\(41\)	0.974403	−	1.34115i	0.152176	−	0.209453i	−0.726122	−	0.687566i	\(-0.758679\pi\)
							0.878298	+	0.478113i	\(0.158679\pi\)
\(43\)		−	1.58274i		−	0.241366i	−0.992691	−	0.120683i	\(-0.961492\pi\)
							0.992691	−	0.120683i	\(-0.0385084\pi\)
\(47\)	−2.77942	−	2.01936i	−0.405420	−	0.294555i	0.366325	−	0.930487i	\(-0.380616\pi\)
							−0.771745	+	0.635932i	\(0.780616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.18	yes	176
3.2	odd	2	inner	429.2.y.b.116.27	yes	176
11.2	odd	10	inner	429.2.y.b.233.17	yes	176
13.12	even	2	inner	429.2.y.b.116.28	yes	176
33.2	even	10	inner	429.2.y.b.233.28	yes	176
39.38	odd	2	inner	429.2.y.b.116.17	✓	176
143.90	odd	10	inner	429.2.y.b.233.27	yes	176
429.233	even	10	inner	429.2.y.b.233.18	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.17	✓	176	39.38	odd	2	inner
429.2.y.b.116.18	yes	176	1.1	even	1	trivial
429.2.y.b.116.27	yes	176	3.2	odd	2	inner
429.2.y.b.116.28	yes	176	13.12	even	2	inner
429.2.y.b.233.17	yes	176	11.2	odd	10	inner
429.2.y.b.233.18	yes	176	429.233	even	10	inner
429.2.y.b.233.27	yes	176	143.90	odd	10	inner
429.2.y.b.233.28	yes	176	33.2	even	10	inner