Embedded newform 429.2.x.a.248.19

• Label: 429.2.x.a.248.19
• 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			248.19
Character	\(\chi\)	\(=\)	429.248
Dual form			429.2.x.a.365.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.214393 - 0.659835i) q^{2} +(-0.480417 + 1.66409i) q^{3} +(1.22862 - 0.892642i) q^{4} +(1.23103 + 0.399986i) q^{5} +(1.20102 - 0.0397745i) q^{6} +(2.53769 + 3.49283i) q^{7} +(-1.97498 - 1.43491i) q^{8} +(-2.53840 - 1.59891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.214393	−	0.659835i	−0.151599	−	0.466574i	0.846201	−	0.532863i	\(-0.178884\pi\)
							−0.997800	+	0.0662894i	\(0.978884\pi\)
\(3\)	−0.480417	+	1.66409i	−0.277369	+	0.960764i
							\(5\)	1.23103	+	0.399986i	0.550534	+	0.178879i	0.571057	−	0.820910i	\(-0.306533\pi\)
−0.0205236	+	0.999789i	\(0.506533\pi\)
\(7\)	2.53769	+	3.49283i	0.959157	+	1.32017i	0.947338	+	0.320237i	\(0.103762\pi\)
							0.0118198	+	0.999930i	\(0.496238\pi\)
\(11\)	0.900317	+	3.19209i	0.271456	+	0.962451i
							\(13\)	−0.951057	+	0.309017i	−0.263776	+	0.0857059i
\(17\)	1.39778	−	4.30192i	0.339011	−	1.04337i								−0.625701	−	0.780063i	\(-0.715187\pi\)
							0.964712	−	0.263307i	\(-0.0848131\pi\)
\(19\)	−1.95623	+	2.69252i	−0.448790	+	0.617707i	−0.972137	−	0.234413i	\(-0.924683\pi\)
							0.523347	+	0.852120i	\(0.324683\pi\)
\(23\)			4.61164i			0.961594i	0.876832	+	0.480797i	\(0.159653\pi\)
							−0.876832	+	0.480797i	\(0.840347\pi\)
\(29\)	1.01839	−	0.739904i	0.189110	−	0.137397i	−0.489202	−	0.872171i	\(-0.662712\pi\)
							0.678312	+	0.734774i	\(0.262712\pi\)
\(31\)	1.83683	+	5.65318i	0.329904	+	1.01534i	0.969178	+	0.246362i	\(0.0792351\pi\)
							−0.639274	+	0.768979i	\(0.720765\pi\)
\(37\)	9.03773	−	6.56630i	1.48579	−	1.07949i	0.510162	−	0.860078i	\(-0.329585\pi\)
							0.975632	−	0.219414i	\(-0.0704145\pi\)
\(41\)	6.68499	+	4.85693i	1.04402	+	0.758525i	0.971066	−	0.238810i	\(-0.0767574\pi\)
							0.0729539	+	0.997335i	\(0.476757\pi\)
\(43\)			1.64596i			0.251006i	0.992093	+	0.125503i	\(0.0400544\pi\)
							−0.992093	+	0.125503i	\(0.959946\pi\)
\(47\)	4.38464	−	6.03494i	0.639565	−	0.880286i	−0.359027	−	0.933327i	\(-0.616891\pi\)
							0.998592	+	0.0530413i	\(0.0168915\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.x.a.248.19	✓	192
3.2	odd	2	inner	429.2.x.a.248.30	yes	192
11.2	odd	10	inner	429.2.x.a.365.30	yes	192
33.2	even	10	inner	429.2.x.a.365.19	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.x.a.248.19	✓	192	1.1	even	1	trivial
429.2.x.a.248.30	yes	192	3.2	odd	2	inner
429.2.x.a.365.19	yes	192	33.2	even	10	inner
429.2.x.a.365.30	yes	192	11.2	odd	10	inner