Embedded newform 429.2.bg.b.334.2

• Label: 429.2.bg.b.334.2
• 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			334.2
Character	\(\chi\)	\(=\)	429.334
Dual form			429.2.bg.b.289.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19603 + 0.977734i) q^{2} +(0.669131 + 0.743145i) q^{3} +(2.52831 - 2.80797i) q^{4} +(-1.82283 + 1.32436i) q^{5} +(-2.19603 - 0.977734i) q^{6} +(1.64888 - 1.83127i) q^{7} +(-1.32112 + 4.06599i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19603	+	0.977734i	−1.55283	+	0.691362i	−0.990747	−	0.135722i	\(-0.956665\pi\)
							−0.562078	+	0.827084i	\(0.689998\pi\)
\(3\)	0.669131	+	0.743145i	0.386323	+	0.429055i
							\(5\)	−1.82283	+	1.32436i	−0.815192	+	0.592272i	−0.915331	−	0.402702i	\(-0.868071\pi\)
0.100139	+	0.994973i	\(0.468071\pi\)
\(7\)	1.64888	−	1.83127i	0.623220	−	0.692156i	−0.346033	−	0.938222i	\(-0.612472\pi\)
							0.969253	+	0.246067i	\(0.0791382\pi\)
\(11\)	−0.0651161	−	3.31599i	−0.0196332	−	0.999807i
							\(13\)	3.60289	+	0.138547i	0.999261	+	0.0384259i
\(17\)	1.22811	+	0.546791i	0.297861	+	0.132616i								0.550226	−	0.835016i	\(-0.314542\pi\)
							−0.252365	+	0.967632i	\(0.581208\pi\)
\(19\)	0.130521	−	0.0277431i	0.0299435	−	0.00636469i	−0.192916	−	0.981215i	\(-0.561794\pi\)
							0.222859	+	0.974851i	\(0.428461\pi\)
\(23\)	4.08305	−	7.07205i	0.851375	−	1.47462i	−0.0285924	−	0.999591i	\(-0.509102\pi\)
							0.879968	−	0.475034i	\(-0.157564\pi\)
\(29\)	8.18130	+	1.73899i	1.51923	+	0.322922i	0.890601	−	0.454785i	\(-0.150284\pi\)
							0.628627	+	0.777707i	\(0.283617\pi\)
\(31\)	−2.23949	−	1.62709i	−0.402225	−	0.292233i	0.368222	−	0.929738i	\(-0.379967\pi\)
							−0.770446	+	0.637505i	\(0.779967\pi\)
\(37\)	−3.04532	−	0.647303i	−0.500648	−	0.106416i	−0.0493359	−	0.998782i	\(-0.515710\pi\)
							−0.451312	+	0.892366i	\(0.649044\pi\)
\(41\)	7.70209	+	8.55404i	1.20286	+	1.33592i	0.927156	+	0.374676i	\(0.122246\pi\)
							0.275709	+	0.961241i	\(0.411087\pi\)
\(43\)	0.273887	+	0.474386i	0.0417674	+	0.0723432i	0.886153	−	0.463392i	\(-0.153368\pi\)
							−0.844386	+	0.535735i	\(0.820034\pi\)
\(47\)	−2.82886	+	8.70633i	−0.412631	+	1.26995i	0.501721	+	0.865029i	\(0.332700\pi\)
							−0.914353	+	0.404919i	\(0.867300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.b.334.2	yes	112
11.3	even	5	inner	429.2.bg.b.256.13	✓	112
13.3	even	3	inner	429.2.bg.b.367.13	yes	112
143.3	even	15	inner	429.2.bg.b.289.2	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.b.256.13	✓	112	11.3	even	5	inner
429.2.bg.b.289.2	yes	112	143.3	even	15	inner
429.2.bg.b.334.2	yes	112	1.1	even	1	trivial
429.2.bg.b.367.13	yes	112	13.3	even	3	inner