Embedded newform 429.2.y.b.116.5

• Label: 429.2.y.b.116.5
• 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			116.5
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.06588 + 0.671246i) q^{2} +(-1.48049 + 0.898973i) q^{3} +(2.19926 - 1.59786i) q^{4} +(0.803773 - 2.47376i) q^{5} +(2.45508 - 2.85094i) q^{6} +(3.30516 - 2.40134i) q^{7} +(-0.917289 + 1.26254i) q^{8} +(1.38370 - 2.66184i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.06588	+	0.671246i	−1.46080	+	0.474642i	−0.928314	−	0.371797i	\(-0.878742\pi\)
							−0.532485	+	0.846440i	\(0.678742\pi\)
\(3\)	−1.48049	+	0.898973i	−0.854761	+	0.519022i
							\(5\)	0.803773	−	2.47376i	0.359458	−	1.10630i	−0.593921	−	0.804524i	\(-0.702421\pi\)
0.953379	−	0.301775i	\(-0.0975792\pi\)
\(7\)	3.30516	−	2.40134i	1.24923	−	0.907622i	0.251058	−	0.967972i	\(-0.419222\pi\)
							0.998177	+	0.0603499i	\(0.0192217\pi\)
\(11\)	−3.29472	+	0.380585i	−0.993394	+	0.114751i
							\(13\)	−3.57897	+	0.436980i	−0.992629	+	0.121196i
\(17\)	−0.644745	+	1.98432i	−0.156374	+	0.481268i								−0.998297	−	0.0583279i	\(-0.981423\pi\)
							0.841924	+	0.539596i	\(0.181423\pi\)
\(19\)	−4.06762	−	2.95530i	−0.933175	−	0.677991i	0.0135931	−	0.999908i	\(-0.495673\pi\)
							−0.946768	+	0.321916i	\(0.895673\pi\)
\(23\)		−	6.35469i		−	1.32504i	−0.749042	−	0.662522i	\(-0.769486\pi\)
							0.749042	−	0.662522i	\(-0.230514\pi\)
\(29\)	−0.0346794	+	0.0251961i	−0.00643980	+	0.00467879i	−0.591000	−	0.806671i	\(-0.701267\pi\)
							0.584561	+	0.811350i	\(0.301267\pi\)
\(31\)	−5.58864	+	1.81586i	−1.00375	+	0.326138i	−0.764363	−	0.644786i	\(-0.776946\pi\)
							−0.239386	+	0.970924i	\(0.576946\pi\)
\(37\)	3.36857	+	4.63644i	0.553789	+	0.762226i	0.990520	−	0.137366i	\(-0.0438638\pi\)
							−0.436731	+	0.899592i	\(0.643864\pi\)
\(41\)	−3.60400	+	4.96048i	−0.562850	+	0.774697i	−0.991685	−	0.128686i	\(-0.958924\pi\)
							0.428835	+	0.903383i	\(0.358924\pi\)
\(43\)			3.06838i			0.467924i	0.972246	+	0.233962i	\(0.0751691\pi\)
							−0.972246	+	0.233962i	\(0.924831\pi\)
\(47\)	−6.28659	−	4.56748i	−0.916994	−	0.666235i	0.0257801	−	0.999668i	\(-0.491793\pi\)
							−0.942774	+	0.333433i	\(0.891793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.5	✓	176
3.2	odd	2	inner	429.2.y.b.116.40	yes	176
11.2	odd	10	inner	429.2.y.b.233.6	yes	176
13.12	even	2	inner	429.2.y.b.116.39	yes	176
33.2	even	10	inner	429.2.y.b.233.39	yes	176
39.38	odd	2	inner	429.2.y.b.116.6	yes	176
143.90	odd	10	inner	429.2.y.b.233.40	yes	176
429.233	even	10	inner	429.2.y.b.233.5	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.5	✓	176	1.1	even	1	trivial
429.2.y.b.116.6	yes	176	39.38	odd	2	inner
429.2.y.b.116.39	yes	176	13.12	even	2	inner
429.2.y.b.116.40	yes	176	3.2	odd	2	inner
429.2.y.b.233.5	yes	176	429.233	even	10	inner
429.2.y.b.233.6	yes	176	11.2	odd	10	inner
429.2.y.b.233.39	yes	176	33.2	even	10	inner
429.2.y.b.233.40	yes	176	143.90	odd	10	inner