Embedded newform 429.2.bn.b.49.1

• Label: 429.2.bn.b.49.1
• Level: $429$
• Weight: $2$
• Character: 429.49
• 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			49.1
Character	\(\chi\)	\(=\)	429.49
Dual form			429.2.bn.b.394.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91017 - 1.71993i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(0.481555 + 4.58169i) q^{4} +(3.19249 + 1.03730i) q^{5} +(1.91017 - 1.71993i) q^{6} +(-2.98254 + 0.313478i) q^{7} +(3.93864 - 5.42108i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91017	−	1.71993i	−1.35070	−	1.21617i	−0.954573	−	0.297977i	\(-0.903688\pi\)
							−0.396124	−	0.918197i	\(-0.629645\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	3.19249	+	1.03730i	1.42772	+	0.463895i	0.918047	−	0.396471i	\(-0.129765\pi\)
0.509676	+	0.860367i	\(0.329765\pi\)
\(7\)	−2.98254	+	0.313478i	−1.12730	+	0.118484i	−0.649768	−	0.760133i	\(-0.725134\pi\)
							−0.477528	+	0.878616i	\(0.658467\pi\)
\(11\)	−0.488116	−	3.28051i	−0.147172	−	0.989111i
							\(13\)	2.02198	−	2.98523i	0.560796	−	0.827954i
\(17\)	3.81428	+	4.23618i	0.925098	+	1.02743i								0.999544	+	0.0302003i	\(0.00961452\pi\)
							−0.0744459	+	0.997225i	\(0.523719\pi\)
\(19\)	0.570409	+	1.28116i	0.130861	+	0.293918i	0.967078	−	0.254480i	\(-0.0819044\pi\)
							−0.836217	+	0.548398i	\(0.815238\pi\)
\(23\)	4.21370	+	7.29834i	0.878617	+	1.52181i	0.852859	+	0.522141i	\(0.174867\pi\)
							0.0257577	+	0.999668i	\(0.491800\pi\)
\(29\)	4.35228	+	1.93776i	0.808198	+	0.359833i	0.768880	−	0.639394i	\(-0.220815\pi\)
							0.0393188	+	0.999227i	\(0.487481\pi\)
\(31\)	2.62865	−	0.854099i	0.472119	−	0.153401i	−0.0632874	−	0.997995i	\(-0.520158\pi\)
							0.535406	+	0.844595i	\(0.320158\pi\)
\(37\)	−2.69758	+	6.05886i	−0.443479	+	0.996071i	0.544110	+	0.839014i	\(0.316867\pi\)
							−0.987590	+	0.157057i	\(0.949799\pi\)
\(41\)	7.74209	+	0.813727i	1.20911	+	0.127083i	0.687554	−	0.726133i	\(-0.258684\pi\)
							0.521558	+	0.853216i	\(0.325351\pi\)
\(43\)	−5.86676	+	10.1615i	−0.894673	+	1.54962i	−0.0604636	+	0.998170i	\(0.519258\pi\)
							−0.834209	+	0.551448i	\(0.814075\pi\)
\(47\)	−1.25888	+	1.73271i	−0.183627	+	0.252741i	−0.890900	−	0.454200i	\(-0.849925\pi\)
							0.707273	+	0.706941i	\(0.249925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.b.49.1	✓	112
11.9	even	5	inner	429.2.bn.b.361.14	yes	112
13.4	even	6	inner	429.2.bn.b.82.14	yes	112
143.108	even	30	inner	429.2.bn.b.394.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.49.1	✓	112	1.1	even	1	trivial
429.2.bn.b.82.14	yes	112	13.4	even	6	inner
429.2.bn.b.361.14	yes	112	11.9	even	5	inner
429.2.bn.b.394.1	yes	112	143.108	even	30	inner