Embedded newform 450.2.w.a.83.7

• Label: 450.2.w.a.83.7
• Level: $450$
• Weight: $2$
• Character: 450.83
• 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			83.7
Character	\(\chi\)	\(=\)	450.83
Dual form			450.2.w.a.347.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.544639 - 0.838671i) q^{2} +(-0.115750 + 1.72818i) q^{3} +(-0.406737 + 0.913545i) q^{4} +(-2.23198 - 0.135137i) q^{5} +(1.51241 - 0.844158i) q^{6} +(-0.326112 - 1.21707i) q^{7} +(0.987688 - 0.156434i) q^{8} +(-2.97320 - 0.400072i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.544639	−	0.838671i	−0.385118	−	0.593030i
							\(3\)	−0.115750	+	1.72818i	−0.0668281	+	0.997765i
\(5\)	−2.23198	−	0.135137i	−0.998172	−	0.0604353i
							\(7\)	−0.326112	−	1.21707i	−0.123259	−	0.460009i	0.876513	−	0.481379i	\(-0.159864\pi\)
−0.999772	+	0.0213702i	\(0.993197\pi\)
\(11\)	0.696688	−	3.27766i	0.210059	−	0.988252i	−0.739134	−	0.673559i	\(-0.764765\pi\)
							0.949193	−	0.314693i	\(-0.101902\pi\)
\(13\)	1.40422	+	0.911909i	0.389460	+	0.252918i	0.724484	−	0.689291i	\(-0.242078\pi\)
							−0.335025	+	0.942209i	\(0.608745\pi\)
\(17\)	−0.399239	−	2.52069i	−0.0968296	−	0.611358i	−0.987611	−	0.156925i	\(-0.949842\pi\)
							0.890781	−	0.454433i	\(-0.150158\pi\)
\(19\)	1.14834	−	1.58056i	0.263448	−	0.362605i	−0.656716	−	0.754138i	\(-0.728055\pi\)
							0.920164	+	0.391533i	\(0.128055\pi\)
\(23\)	−0.0806965	−	1.53978i	−0.0168264	−	0.321066i	−0.994055	−	0.108878i	\(-0.965274\pi\)
							0.977229	−	0.212188i	\(-0.0680591\pi\)
\(29\)	−0.848012	−	8.06829i	−0.157472	−	1.49824i	−0.732869	−	0.680370i	\(-0.761819\pi\)
							0.575397	−	0.817874i	\(-0.304848\pi\)
\(31\)	0.364489	−	3.46788i	0.0654641	−	0.622850i	−0.911772	−	0.410696i	\(-0.865286\pi\)
							0.977236	−	0.212153i	\(-0.0680477\pi\)
\(37\)	1.36256	+	2.67418i	0.224004	+	0.439632i	0.975468	−	0.220142i	\(-0.0706521\pi\)
							−0.751464	+	0.659774i	\(0.770652\pi\)
\(41\)	−1.76875	−	8.32130i	−0.276232	−	1.29957i	−0.869247	−	0.494377i	\(-0.835396\pi\)
							0.593016	−	0.805191i	\(-0.297937\pi\)
\(43\)	−0.737838	+	0.197703i	−0.112519	+	0.0301494i	−0.314639	−	0.949211i	\(-0.601884\pi\)
							0.202120	+	0.979361i	\(0.435217\pi\)
\(47\)	−2.64790	+	3.26989i	−0.386236	+	0.476962i	−0.932768	−	0.360476i	\(-0.882614\pi\)
							0.546532	+	0.837438i	\(0.315948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.83.7	yes	480
9.5	odd	6	inner	450.2.w.a.383.9	yes	480
25.22	odd	20	inner	450.2.w.a.47.9	✓	480
225.122	even	60	inner	450.2.w.a.347.7	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.47.9	✓	480	25.22	odd	20	inner
450.2.w.a.83.7	yes	480	1.1	even	1	trivial
450.2.w.a.347.7	yes	480	225.122	even	60	inner
450.2.w.a.383.9	yes	480	9.5	odd	6	inner