Embedded newform 429.2.bg.b.16.10

• Label: 429.2.bg.b.16.10
• 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			16.10
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.b.295.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.682919 - 0.758458i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(0.100176 + 0.953113i) q^{4} +(0.0526100 - 0.161917i) q^{5} +(0.682919 + 0.758458i) q^{6} +(-0.171511 - 1.63181i) q^{7} +(2.44268 + 1.77471i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.682919	−	0.758458i	0.482896	−	0.536311i	−0.451630	−	0.892206i	\(-0.649157\pi\)
							0.934526	+	0.355895i	\(0.115824\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	0.0526100	−	0.161917i	0.0235279	−	0.0724115i	−0.938603	−	0.344999i	\(-0.887879\pi\)
0.962131	+	0.272587i	\(0.0878793\pi\)
\(7\)	−0.171511	−	1.63181i	−0.0648249	−	0.616768i	−0.977914	−	0.209008i	\(-0.932977\pi\)
							0.913089	−	0.407760i	\(-0.133690\pi\)
\(11\)	2.19467	+	2.48665i	0.661717	+	0.749754i
							\(13\)	2.54632	+	2.55270i	0.706221	+	0.707991i
\(17\)	−1.38264	−	1.53557i	−0.335338	−	0.372431i								0.551767	−	0.833999i	\(-0.313954\pi\)
							−0.887105	+	0.461567i	\(0.847287\pi\)
\(19\)	−2.99913	+	1.33530i	−0.688048	+	0.306339i	−0.720818	−	0.693124i	\(-0.756234\pi\)
							0.0327702	+	0.999463i	\(0.489567\pi\)
\(23\)	2.82551	+	4.89393i	0.589160	+	1.02045i	0.994343	+	0.106219i	\(0.0338745\pi\)
							−0.405183	+	0.914236i	\(0.632792\pi\)
\(29\)	3.98072	+	1.77233i	0.739202	+	0.329114i	0.741563	−	0.670883i	\(-0.234085\pi\)
							−0.00236123	+	0.999997i	\(0.500752\pi\)
\(31\)	−3.12553	−	9.61940i	−0.561362	−	1.72769i	−0.678521	−	0.734581i	\(-0.737379\pi\)
							0.117159	−	0.993113i	\(-0.462621\pi\)
\(37\)	−8.15267	−	3.62980i	−1.34029	−	0.596736i	−0.393719	−	0.919231i	\(-0.628812\pi\)
							−0.946571	+	0.322495i	\(0.895478\pi\)
\(41\)	−0.683216	+	6.50037i	−0.106700	+	1.01519i	0.801882	+	0.597482i	\(0.203832\pi\)
							−0.908583	+	0.417705i	\(0.862835\pi\)
\(43\)	5.65130	−	9.78834i	0.861815	−	1.49271i	−0.00835944	−	0.999965i	\(-0.502661\pi\)
							0.870175	−	0.492743i	\(-0.164006\pi\)
\(47\)	−2.48324	−	1.80418i	−0.362218	−	0.263167i	0.391758	−	0.920068i	\(-0.371867\pi\)
							−0.753977	+	0.656901i	\(0.771867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.b.16.10	✓	112
11.9	even	5	inner	429.2.bg.b.328.5	yes	112
13.9	even	3	inner	429.2.bg.b.412.5	yes	112
143.9	even	15	inner	429.2.bg.b.295.10	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.b.16.10	✓	112	1.1	even	1	trivial
429.2.bg.b.295.10	yes	112	143.9	even	15	inner
429.2.bg.b.328.5	yes	112	11.9	even	5	inner
429.2.bg.b.412.5	yes	112	13.9	even	3	inner