Embedded newform 450.2.w.a.317.16

• Label: 450.2.w.a.317.16
• Level: $450$
• Weight: $2$
• Character: 450.317
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(30\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			317.16
Character	\(\chi\)	\(=\)	450.317
Dual form			450.2.w.a.203.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.838671 - 0.544639i) q^{2} +(-1.68209 + 0.412985i) q^{3} +(0.406737 - 0.913545i) q^{4} +(1.37401 + 1.76412i) q^{5} +(-1.18580 + 1.26249i) q^{6} +(-4.81799 + 1.29098i) q^{7} +(-0.156434 - 0.987688i) q^{8} +(2.65889 - 1.38936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.838671	−	0.544639i	0.593030	−	0.385118i
							\(3\)	−1.68209	+	0.412985i	−0.971158	+	0.238437i
\(5\)	1.37401	+	1.76412i	0.614474	+	0.788937i
							\(7\)	−4.81799	+	1.29098i	−1.82103	+	0.487943i	−0.996916	−	0.0784803i	\(-0.974993\pi\)
−0.824114	+	0.566424i	\(0.808327\pi\)
\(11\)	−0.151624	+	0.713334i	−0.0457163	+	0.215078i	−0.995076	−	0.0991122i	\(-0.968400\pi\)
							0.949360	+	0.314191i	\(0.101733\pi\)
\(13\)	−2.26885	+	3.49373i	−0.629267	+	0.968986i	0.369935	+	0.929057i	\(0.379380\pi\)
							−0.999203	+	0.0399288i	\(0.987287\pi\)
\(17\)	−4.18301	+	0.662524i	−1.01453	+	0.160686i	−0.641491	−	0.767130i	\(-0.721684\pi\)
							−0.373038	+	0.927816i	\(0.621684\pi\)
\(19\)	−1.62597	+	2.23795i	−0.373023	+	0.513422i	−0.953719	−	0.300698i	\(-0.902780\pi\)
							0.580696	+	0.814120i	\(0.302780\pi\)
\(23\)	1.15144	−	0.0603447i	0.240093	−	0.0125827i	0.0680900	−	0.997679i	\(-0.478310\pi\)
							0.172003	+	0.985096i	\(0.444976\pi\)
\(29\)	0.733326	+	6.97713i	0.136175	+	1.29562i	0.822684	+	0.568499i	\(0.192476\pi\)
							−0.686509	+	0.727121i	\(0.740858\pi\)
\(31\)	0.418872	−	3.98530i	0.0752317	−	0.715782i	−0.890278	−	0.455417i	\(-0.849490\pi\)
							0.965510	−	0.260365i	\(-0.0838430\pi\)
\(37\)	7.34692	−	3.74344i	1.20783	−	0.615418i	0.270113	−	0.962829i	\(-0.412939\pi\)
							0.937713	+	0.347410i	\(0.112939\pi\)
\(41\)	0.220821	+	1.03888i	0.0344864	+	0.162246i	0.992021	−	0.126073i	\(-0.0402372\pi\)
							−0.957535	+	0.288318i	\(0.906904\pi\)
\(43\)	2.27470	+	8.48928i	0.346888	+	1.29460i	0.890391	+	0.455196i	\(0.150431\pi\)
							−0.543503	+	0.839407i	\(0.682902\pi\)
\(47\)	−6.96206	−	5.63776i	−1.01552	−	0.822352i	−0.0313376	−	0.999509i	\(-0.509977\pi\)
							−0.984183	+	0.177157i	\(0.943310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.317.16	yes	480
9.5	odd	6	inner	450.2.w.a.167.12	✓	480
25.3	odd	20	inner	450.2.w.a.353.12	yes	480
225.203	even	60	inner	450.2.w.a.203.16	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.167.12	✓	480	9.5	odd	6	inner
450.2.w.a.203.16	yes	480	225.203	even	60	inner
450.2.w.a.317.16	yes	480	1.1	even	1	trivial
450.2.w.a.353.12	yes	480	25.3	odd	20	inner