Embedded newform 429.2.bg.a.16.14

• Label: 429.2.bg.a.16.14
• Level: $429$
• Weight: $2$
• Character: 429.16
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.14
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.a.295.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.71487 - 1.90455i) q^{2} +(0.104528 - 0.994522i) q^{3} +(-0.477496 - 4.54307i) q^{4} +(1.12068 - 3.44910i) q^{5} +(-1.71487 - 1.90455i) q^{6} +(0.446007 + 4.24348i) q^{7} +(-5.32461 - 3.86856i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.71487	−	1.90455i	1.21259	−	1.34672i	0.291895	−	0.956450i	\(-0.405714\pi\)
							0.920700	−	0.390272i	\(-0.127619\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	1.12068	−	3.44910i	0.501184	−	1.54249i	−0.305909	−	0.952061i	\(-0.598960\pi\)
0.807093	−	0.590424i	\(-0.201040\pi\)
\(7\)	0.446007	+	4.24348i	0.168575	+	1.60388i	0.672474	+	0.740121i	\(0.265232\pi\)
							−0.503899	+	0.863763i	\(0.668102\pi\)
\(11\)	2.52717	+	2.14788i	0.761972	+	0.647611i
							\(13\)	1.70089	+	3.17915i	0.471741	+	0.881737i
\(17\)	1.01471	+	1.12695i	0.246104	+	0.273326i								0.853522	−	0.521056i	\(-0.174462\pi\)
							−0.607419	+	0.794382i	\(0.707795\pi\)
\(19\)	−4.48205	+	1.99554i	−1.02825	+	0.457808i	−0.850337	−	0.526238i	\(-0.823602\pi\)
							−0.177915	+	0.984046i	\(0.556935\pi\)
\(23\)	1.10822	+	1.91949i	0.231079	+	0.400241i	0.958126	−	0.286347i	\(-0.0924410\pi\)
							−0.727047	+	0.686588i	\(0.759108\pi\)
\(29\)	−2.59287	−	1.15442i	−0.481484	−	0.214370i	0.151618	−	0.988439i	\(-0.451552\pi\)
							−0.633102	+	0.774069i	\(0.718218\pi\)
\(31\)	1.50664	+	4.63696i	0.270601	+	0.832823i	0.990350	+	0.138589i	\(0.0442568\pi\)
							−0.719749	+	0.694234i	\(0.755743\pi\)
\(37\)	−7.88320	−	3.50983i	−1.29599	−	0.577012i	−0.361292	−	0.932453i	\(-0.617664\pi\)
							−0.934699	+	0.355441i	\(0.884331\pi\)
\(41\)	0.860741	−	8.18941i	0.134425	−	1.27897i	−0.694452	−	0.719539i	\(-0.744353\pi\)
							0.828877	−	0.559431i	\(-0.188980\pi\)
\(43\)	−1.65623	+	2.86868i	−0.252573	+	0.437470i	−0.964234	−	0.265054i	\(-0.914610\pi\)
							0.711660	+	0.702524i	\(0.247944\pi\)
\(47\)	−10.6769	−	7.75719i	−1.55738	−	1.13150i	−0.938118	−	0.346317i	\(-0.887432\pi\)
							−0.619261	−	0.785185i	\(-0.712568\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.16.14	✓	112
11.9	even	5	inner	429.2.bg.a.328.1	yes	112
13.9	even	3	inner	429.2.bg.a.412.1	yes	112
143.9	even	15	inner	429.2.bg.a.295.14	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.16.14	✓	112	1.1	even	1	trivial
429.2.bg.a.295.14	yes	112	143.9	even	15	inner
429.2.bg.a.328.1	yes	112	11.9	even	5	inner
429.2.bg.a.412.1	yes	112	13.9	even	3	inner