Embedded newform 429.2.x.a.248.18

• Label: 429.2.x.a.248.18
• 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			248.18
Character	\(\chi\)	\(=\)	429.248
Dual form			429.2.x.a.365.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.268430 - 0.826144i) q^{2} +(-1.73136 + 0.0488003i) q^{3} +(1.00758 - 0.732046i) q^{4} +(-1.29465 - 0.420659i) q^{5} +(0.505067 + 1.41726i) q^{6} +(2.39963 + 3.30281i) q^{7} +(-2.28076 - 1.65707i) q^{8} +(2.99524 - 0.168982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.268430	−	0.826144i	−0.189809	−	0.584172i	0.810189	−	0.586169i	\(-0.199364\pi\)
							−0.999998	+	0.00199674i	\(0.999364\pi\)
\(3\)	−1.73136	+	0.0488003i	−0.999603	+	0.0281748i
							\(5\)	−1.29465	−	0.420659i	−0.578987	−	0.188124i	0.00485967	−	0.999988i	\(-0.498453\pi\)
−0.583847	+	0.811864i	\(0.698453\pi\)
\(7\)	2.39963	+	3.30281i	0.906975	+	1.24834i	0.968189	+	0.250219i	\(0.0805028\pi\)
							−0.0612141	+	0.998125i	\(0.519497\pi\)
\(11\)	1.96934	−	2.66865i	0.593780	−	0.804628i
							\(13\)	0.951057	−	0.309017i	0.263776	−	0.0857059i
\(17\)	−0.259950	+	0.800045i	−0.0630472	+	0.194039i								−0.977619	−	0.210385i	\(-0.932528\pi\)
							0.914571	+	0.404425i	\(0.132528\pi\)
\(19\)	3.00117	−	4.13075i	0.688515	−	0.947659i	−0.311482	−	0.950252i	\(-0.600825\pi\)
							0.999997	+	0.00259286i	\(0.000825334\pi\)
\(23\)		−	0.778991i		−	0.162431i	−0.996697	−	0.0812154i	\(-0.974120\pi\)
							0.996697	−	0.0812154i	\(-0.0258802\pi\)
\(29\)	3.44661	−	2.50411i	0.640019	−	0.465001i	−0.219838	−	0.975536i	\(-0.570553\pi\)
							0.859857	+	0.510535i	\(0.170553\pi\)
\(31\)	−1.21620	−	3.74307i	−0.218436	−	0.672276i	−0.998892	−	0.0470652i	\(-0.985013\pi\)
							0.780456	−	0.625210i	\(-0.214987\pi\)
\(37\)	−0.186314	+	0.135365i	−0.0306298	+	0.0222539i	−0.602995	−	0.797745i	\(-0.706026\pi\)
							0.572365	+	0.819999i	\(0.306026\pi\)
\(41\)	5.04747	+	3.66720i	0.788283	+	0.572721i	0.907453	−	0.420153i	\(-0.138023\pi\)
							−0.119170	+	0.992874i	\(0.538023\pi\)
\(43\)		−	11.4105i		−	1.74009i	−0.492975	−	0.870044i	\(-0.664090\pi\)
							0.492975	−	0.870044i	\(-0.335910\pi\)
\(47\)	1.99429	−	2.74491i	0.290897	−	0.400386i	−0.638408	−	0.769698i	\(-0.720407\pi\)
							0.929305	+	0.369312i	\(0.120407\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.x.a.248.18	✓	192
3.2	odd	2	inner	429.2.x.a.248.31	yes	192
11.2	odd	10	inner	429.2.x.a.365.31	yes	192
33.2	even	10	inner	429.2.x.a.365.18	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.x.a.248.18	✓	192	1.1	even	1	trivial
429.2.x.a.248.31	yes	192	3.2	odd	2	inner
429.2.x.a.365.18	yes	192	33.2	even	10	inner
429.2.x.a.365.31	yes	192	11.2	odd	10	inner