Embedded newform 430.2.r.a.27.16

• Label: 430.2.r.a.27.16
• Level: $430$
• Weight: $2$
• Character: 430.27
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			27.16
Character	\(\chi\)	\(=\)	430.27
Dual form			430.2.r.a.223.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.846724 + 0.532032i) q^{2} +(-0.527328 + 0.331342i) q^{3} +(0.433884 + 0.900969i) q^{4} +(1.70699 + 1.44436i) q^{5} -0.622786 q^{6} +(-1.95690 + 1.95690i) q^{7} +(-0.111964 + 0.993712i) q^{8} +(-1.13336 + 2.35345i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.846724	+	0.532032i	0.598724	+	0.376203i
							\(3\)	−0.527328	+	0.331342i	−0.304453	+	0.191301i	−0.675590	−	0.737278i	\(-0.736111\pi\)
0.371137	+	0.928578i	\(0.378968\pi\)
\(5\)	1.70699	+	1.44436i	0.763389	+	0.645939i
							\(7\)	−1.95690	+	1.95690i	−0.739639	+	0.739639i	−0.972508	−	0.232869i	\(-0.925189\pi\)
0.232869	+	0.972508i	\(0.425189\pi\)
\(11\)	−3.93608	−	1.89552i	−1.18677	−	0.571519i	−0.266895	−	0.963726i	\(-0.585998\pi\)
							−0.919877	+	0.392206i	\(0.871712\pi\)
\(13\)	−0.393993	−	0.0443923i	−0.109274	−	0.0123122i	0.0571581	−	0.998365i	\(-0.481796\pi\)
							−0.166432	+	0.986053i	\(0.553225\pi\)
\(17\)	5.62781	−	0.634101i	1.36494	−	0.153792i	0.601090	−	0.799182i	\(-0.294733\pi\)
							0.763854	+	0.645389i	\(0.223305\pi\)
\(19\)	1.05443	−	0.507785i	0.241902	−	0.116494i	−0.309005	−	0.951060i	\(-0.599996\pi\)
							0.550907	+	0.834566i	\(0.314282\pi\)
\(23\)	2.16972	−	0.759217i	0.452417	−	0.158308i	−0.0944453	−	0.995530i	\(-0.530108\pi\)
							0.546863	+	0.837222i	\(0.315822\pi\)
\(29\)	0.737904	+	3.23297i	0.137025	+	0.600347i	0.996080	+	0.0884625i	\(0.0281953\pi\)
							−0.859054	+	0.511885i	\(0.828948\pi\)
\(31\)	1.07387	+	4.70494i	0.192873	+	0.845032i	0.975052	+	0.221978i	\(0.0712513\pi\)
							−0.782179	+	0.623054i	\(0.785892\pi\)
\(37\)	1.23264	−	1.23264i	0.202644	−	0.202644i	−0.598488	−	0.801132i	\(-0.704231\pi\)
							0.801132	+	0.598488i	\(0.204231\pi\)
\(41\)	0.850031	+	3.72423i	0.132753	+	0.581627i	0.996920	+	0.0784230i	\(0.0249885\pi\)
							−0.864168	+	0.503204i	\(0.832154\pi\)
\(43\)	−3.47361	−	5.56184i	−0.529721	−	0.848172i
							\(47\)	9.20973	+	3.22262i	1.34338	+	0.470068i	0.903730	−	0.428103i	\(-0.140818\pi\)
0.439647	+	0.898171i	\(0.355104\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.27.16	✓	264
5.3	odd	4	inner	430.2.r.a.113.16	yes	264
43.8	odd	14	inner	430.2.r.a.137.16	yes	264
215.8	even	28	inner	430.2.r.a.223.16	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.16	✓	264	1.1	even	1	trivial
430.2.r.a.113.16	yes	264	5.3	odd	4	inner
430.2.r.a.137.16	yes	264	43.8	odd	14	inner
430.2.r.a.223.16	yes	264	215.8	even	28	inner