Embedded newform 429.2.x.a.248.6

• Label: 429.2.x.a.248.6
• Level: $429$
• Weight: $2$
• Character: 429.248
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.x (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			248.6
Character	\(\chi\)	\(=\)	429.248
Dual form			429.2.x.a.365.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.742222 - 2.28432i) q^{2} +(-0.941406 - 1.45388i) q^{3} +(-3.04921 + 2.21538i) q^{4} +(-1.40487 - 0.456471i) q^{5} +(-2.62239 + 3.22957i) q^{6} +(2.72019 + 3.74402i) q^{7} +(3.43751 + 2.49750i) q^{8} +(-1.22751 + 2.73737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.742222	−	2.28432i	−0.524830	−	1.61526i	−0.764652	−	0.644444i	\(-0.777089\pi\)
							0.239821	−	0.970817i	\(-0.422911\pi\)
\(3\)	−0.941406	−	1.45388i	−0.543521	−	0.839396i
							\(5\)	−1.40487	−	0.456471i	−0.628279	−	0.204140i	−0.0224664	−	0.999748i	\(-0.507152\pi\)
−0.605813	+	0.795607i	\(0.707152\pi\)
\(7\)	2.72019	+	3.74402i	1.02813	+	1.41510i	0.906340	+	0.422549i	\(0.138865\pi\)
							0.121794	+	0.992555i	\(0.461135\pi\)
\(11\)	3.17939	+	0.944167i	0.958624	+	0.284677i
							\(13\)	−0.951057	+	0.309017i	−0.263776	+	0.0857059i
\(17\)	−0.822831	+	2.53241i	−0.199566	+	0.614201i								0.800327	+	0.599564i	\(0.204659\pi\)
							−0.999893	+	0.0146369i	\(0.995341\pi\)
\(19\)	−2.82297	+	3.88549i	−0.647635	+	0.891393i	−0.998994	−	0.0448480i	\(-0.985720\pi\)
							0.351359	+	0.936241i	\(0.385720\pi\)
\(23\)		−	0.350790i		−	0.0731448i	−0.999331	−	0.0365724i	\(-0.988356\pi\)
							0.999331	−	0.0365724i	\(-0.0116440\pi\)
\(29\)	−6.30827	+	4.58323i	−1.17142	+	0.851084i	−0.991178	−	0.132538i	\(-0.957687\pi\)
							−0.180239	+	0.983623i	\(0.557687\pi\)
\(31\)	1.14538	+	3.52513i	0.205717	+	0.633132i	0.999683	+	0.0251699i	\(0.00801266\pi\)
							−0.793966	+	0.607962i	\(0.791987\pi\)
\(37\)	1.37479	−	0.998842i	0.226014	−	0.164209i	−0.469016	−	0.883190i	\(-0.655391\pi\)
							0.695030	+	0.718981i	\(0.255391\pi\)
\(41\)	−0.557850	−	0.405302i	−0.0871215	−	0.0632975i	0.543372	−	0.839492i	\(-0.317147\pi\)
							−0.630493	+	0.776195i	\(0.717147\pi\)
\(43\)			2.92382i			0.445878i	0.974832	+	0.222939i	\(0.0715650\pi\)
							−0.974832	+	0.222939i	\(0.928435\pi\)
\(47\)	−2.21933	+	3.05464i	−0.323722	+	0.445565i	−0.939599	−	0.342277i	\(-0.888802\pi\)
							0.615877	+	0.787842i	\(0.288802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.x.a.248.6	✓	192
3.2	odd	2	inner	429.2.x.a.248.43	yes	192
11.2	odd	10	inner	429.2.x.a.365.43	yes	192
33.2	even	10	inner	429.2.x.a.365.6	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.x.a.248.6	✓	192	1.1	even	1	trivial
429.2.x.a.248.43	yes	192	3.2	odd	2	inner
429.2.x.a.365.6	yes	192	33.2	even	10	inner
429.2.x.a.365.43	yes	192	11.2	odd	10	inner