Embedded newform 429.2.y.b.116.13

• Label: 429.2.y.b.116.13
• 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			116.13
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15602 + 0.375612i) q^{2} +(-1.73161 - 0.0391064i) q^{3} +(-0.422745 + 0.307142i) q^{4} +(-1.20121 + 3.69693i) q^{5} +(2.01646 - 0.605206i) q^{6} +(3.46009 - 2.51391i) q^{7} +(1.80225 - 2.48058i) q^{8} +(2.99694 + 0.135434i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15602	+	0.375612i	−0.817427	+	0.265598i	−0.687740	−	0.725957i	\(-0.741397\pi\)
							−0.129687	+	0.991555i	\(0.541397\pi\)
\(3\)	−1.73161	−	0.0391064i	−0.999745	−	0.0225781i
							\(5\)	−1.20121	+	3.69693i	−0.537196	+	1.65332i	0.201660	+	0.979456i	\(0.435367\pi\)
−0.738856	+	0.673864i	\(0.764633\pi\)
\(7\)	3.46009	−	2.51391i	1.30779	−	0.950167i	0.307793	−	0.951453i	\(-0.400409\pi\)
							0.999999	+	0.00128638i	\(0.000409468\pi\)
\(11\)	3.22684	−	0.766509i	0.972927	−	0.231111i
							\(13\)	0.200392	+	3.59998i	0.0555788	+	0.998454i
\(17\)	−1.91814	+	5.90343i	−0.465217	+	1.43179i								0.393492	+	0.919328i	\(0.371267\pi\)
							−0.858709	+	0.512463i	\(0.828733\pi\)
\(19\)	0.399101	+	0.289964i	0.0915600	+	0.0665222i	0.632623	−	0.774460i	\(-0.281978\pi\)
							−0.541063	+	0.840982i	\(0.681978\pi\)
\(23\)		−	1.11016i		−	0.231484i	−0.993279	−	0.115742i	\(-0.963075\pi\)
							0.993279	−	0.115742i	\(-0.0369246\pi\)
\(29\)	−0.695089	+	0.505011i	−0.129075	+	0.0937783i	−0.650449	−	0.759550i	\(-0.725419\pi\)
							0.521375	+	0.853328i	\(0.325419\pi\)
\(31\)	2.18882	−	0.711190i	0.393123	−	0.127733i	−0.105783	−	0.994389i	\(-0.533735\pi\)
							0.498907	+	0.866656i	\(0.333735\pi\)
\(37\)	3.61562	+	4.97648i	0.594405	+	0.818128i	0.995182	−	0.0980477i	\(-0.0312598\pi\)
							−0.400777	+	0.916176i	\(0.631260\pi\)
\(41\)	−0.314312	+	0.432614i	−0.0490873	+	0.0675629i	−0.832855	−	0.553491i	\(-0.813295\pi\)
							0.783768	+	0.621054i	\(0.213295\pi\)
\(43\)		−	1.87618i		−	0.286115i	−0.989714	−	0.143057i	\(-0.954307\pi\)
							0.989714	−	0.143057i	\(-0.0456933\pi\)
\(47\)	−0.710613	−	0.516291i	−0.103654	−	0.0753088i	0.534751	−	0.845010i	\(-0.320405\pi\)
							−0.638405	+	0.769701i	\(0.720405\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.13	✓	176
3.2	odd	2	inner	429.2.y.b.116.32	yes	176
11.2	odd	10	inner	429.2.y.b.233.14	yes	176
13.12	even	2	inner	429.2.y.b.116.31	yes	176
33.2	even	10	inner	429.2.y.b.233.31	yes	176
39.38	odd	2	inner	429.2.y.b.116.14	yes	176
143.90	odd	10	inner	429.2.y.b.233.32	yes	176
429.233	even	10	inner	429.2.y.b.233.13	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.13	✓	176	1.1	even	1	trivial
429.2.y.b.116.14	yes	176	39.38	odd	2	inner
429.2.y.b.116.31	yes	176	13.12	even	2	inner
429.2.y.b.116.32	yes	176	3.2	odd	2	inner
429.2.y.b.233.13	yes	176	429.233	even	10	inner
429.2.y.b.233.14	yes	176	11.2	odd	10	inner
429.2.y.b.233.31	yes	176	33.2	even	10	inner
429.2.y.b.233.32	yes	176	143.90	odd	10	inner