Embedded newform 429.2.bn.b.4.12

• Label: 429.2.bn.b.4.12
• Level: $429$
• Weight: $2$
• Character: 429.4
• 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.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.12
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.b.322.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.772619 + 1.73533i) q^{2} +(0.669131 + 0.743145i) q^{3} +(-1.07617 + 1.19521i) q^{4} +(1.38712 + 1.90920i) q^{5} +(-0.772619 + 1.73533i) q^{6} +(-2.05909 - 1.85401i) q^{7} +(0.707612 + 0.229917i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.772619	+	1.73533i	0.546324	+	1.22706i	0.950025	+	0.312173i	\(0.101057\pi\)
							−0.403701	+	0.914891i	\(0.632276\pi\)
\(3\)	0.669131	+	0.743145i	0.386323	+	0.429055i
							\(5\)	1.38712	+	1.90920i	0.620338	+	0.853822i	0.997377	−	0.0723753i	\(-0.0230579\pi\)
−0.377040	+	0.926197i	\(0.623058\pi\)
\(7\)	−2.05909	−	1.85401i	−0.778263	−	0.700751i	0.180932	−	0.983496i	\(-0.442088\pi\)
							−0.959195	+	0.282745i	\(0.908755\pi\)
\(11\)	3.31484	−	0.108680i	0.999463	−	0.0327683i
							\(13\)	−1.58260	+	3.23966i	−0.438933	+	0.898520i
\(17\)	−5.13566	−	2.28654i	−1.24558	−	0.554568i								−0.325219	−	0.945639i	\(-0.605438\pi\)
							−0.920361	+	0.391071i	\(0.872105\pi\)
\(19\)	−0.136703	−	0.643137i	−0.0313618	−	0.147546i	0.959681	−	0.281092i	\(-0.0906967\pi\)
							−0.991043	+	0.133547i	\(0.957363\pi\)
\(23\)	2.26056	−	3.91541i	0.471360	−	0.816419i	−0.528104	−	0.849180i	\(-0.677097\pi\)
							0.999463	+	0.0327611i	\(0.0104300\pi\)
\(29\)	0.787013	+	0.167285i	0.146145	+	0.0310640i	0.280403	−	0.959882i	\(-0.409532\pi\)
							−0.134258	+	0.990946i	\(0.542865\pi\)
\(31\)	1.33516	−	1.83768i	0.239801	−	0.330058i	−0.672106	−	0.740455i	\(-0.734610\pi\)
							0.911907	+	0.410397i	\(0.134610\pi\)
\(37\)	1.58527	−	7.45811i	0.260617	−	1.22611i	−0.631885	−	0.775062i	\(-0.717718\pi\)
							0.892502	−	0.451044i	\(-0.148948\pi\)
\(41\)	0.0788457	−	0.0709930i	0.0123136	−	0.0110872i	−0.662951	−	0.748663i	\(-0.730696\pi\)
							0.675264	+	0.737576i	\(0.264029\pi\)
\(43\)	1.30720	+	2.26414i	0.199346	+	0.345278i	0.948317	−	0.317326i	\(-0.102785\pi\)
							−0.748970	+	0.662603i	\(0.769452\pi\)
\(47\)	−0.365591	−	0.118788i	−0.0533269	−	0.0173270i	0.282232	−	0.959346i	\(-0.408925\pi\)
							−0.335559	+	0.942019i	\(0.608925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.b.4.12	✓	112
11.3	even	5	inner	429.2.bn.b.355.12	yes	112
13.10	even	6	inner	429.2.bn.b.400.12	yes	112
143.36	even	30	inner	429.2.bn.b.322.12	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.4.12	✓	112	1.1	even	1	trivial
429.2.bn.b.322.12	yes	112	143.36	even	30	inner
429.2.bn.b.355.12	yes	112	11.3	even	5	inner
429.2.bn.b.400.12	yes	112	13.10	even	6	inner