Embedded newform 429.2.bg.a.334.1

• Label: 429.2.bg.a.334.1
• Level: $429$
• Weight: $2$
• Character: 429.334
• 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			334.1
Character	\(\chi\)	\(=\)	429.334
Dual form			429.2.bg.a.289.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10664 + 0.937937i) q^{2} +(-0.669131 - 0.743145i) q^{3} +(2.21995 - 2.46551i) q^{4} +(1.88471 - 1.36932i) q^{5} +(2.10664 + 0.937937i) q^{6} +(-1.97834 + 2.19717i) q^{7} +(-0.938963 + 2.88983i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10664	+	0.937937i	−1.48962	+	0.663222i	−0.980329	−	0.197372i	\(-0.936759\pi\)
							−0.509292	+	0.860594i	\(0.670093\pi\)
\(3\)	−0.669131	−	0.743145i	−0.386323	−	0.429055i
							\(5\)	1.88471	−	1.36932i	0.842869	−	0.612380i	−0.0803014	−	0.996771i	\(-0.525588\pi\)
0.923171	+	0.384390i	\(0.125588\pi\)
\(7\)	−1.97834	+	2.19717i	−0.747742	+	0.830451i	−0.990192	−	0.139712i	\(-0.955382\pi\)
							0.242450	+	0.970164i	\(0.422049\pi\)
\(11\)	3.31643	+	0.0356820i	0.999942	+	0.0107585i
							\(13\)	−1.80461	+	3.12144i	−0.500508	+	0.865732i
\(17\)	−4.40590	−	1.96163i	−1.06859	−	0.475766i								−0.204376	−	0.978893i	\(-0.565516\pi\)
							−0.864213	+	0.503126i	\(0.832183\pi\)
\(19\)	6.89060	−	1.46464i	1.58081	−	0.336012i	0.667926	−	0.744228i	\(-0.267182\pi\)
							0.912885	+	0.408216i	\(0.133849\pi\)
\(23\)	2.81828	−	4.88140i	0.587651	−	1.01784i	−0.406888	−	0.913478i	\(-0.633386\pi\)
							0.994539	−	0.104364i	\(-0.0332806\pi\)
\(29\)	7.02228	+	1.49263i	1.30400	+	0.277175i	0.806989	−	0.590567i	\(-0.201096\pi\)
							0.497015	+	0.867742i	\(0.334429\pi\)
\(31\)	−4.32544	−	3.14262i	−0.776873	−	0.564431i	0.127166	−	0.991881i	\(-0.459412\pi\)
							−0.904039	+	0.427450i	\(0.859412\pi\)
\(37\)	8.14804	+	1.73192i	1.33953	+	0.284726i	0.821290	−	0.570510i	\(-0.193255\pi\)
							0.518239	+	0.855236i	\(0.326588\pi\)
\(41\)	2.99050	+	3.32129i	0.467038	+	0.518698i	0.929940	−	0.367711i	\(-0.119858\pi\)
							−0.462902	+	0.886409i	\(0.653192\pi\)
\(43\)	4.41523	+	7.64740i	0.673316	+	1.16622i	0.976958	+	0.213431i	\(0.0684639\pi\)
							−0.303642	+	0.952786i	\(0.598203\pi\)
\(47\)	1.23699	−	3.80706i	0.180433	−	0.555316i	−0.819407	−	0.573213i	\(-0.805697\pi\)
							0.999840	+	0.0178964i	\(0.00569689\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.334.1	yes	112
11.3	even	5	inner	429.2.bg.a.256.14	✓	112
13.3	even	3	inner	429.2.bg.a.367.14	yes	112
143.3	even	15	inner	429.2.bg.a.289.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.256.14	✓	112	11.3	even	5	inner
429.2.bg.a.289.1	yes	112	143.3	even	15	inner
429.2.bg.a.334.1	yes	112	1.1	even	1	trivial
429.2.bg.a.367.14	yes	112	13.3	even	3	inner