Embedded newform 450.2.w.a.263.1

• Label: 450.2.w.a.263.1
• Level: $450$
• Weight: $2$
• Character: 450.263
• 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			263.1
Character	\(\chi\)	\(=\)	450.263
Dual form			450.2.w.a.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.933580 - 0.358368i) q^{2} +(-1.70815 - 0.286738i) q^{3} +(0.743145 + 0.669131i) q^{4} +(-0.417533 - 2.19674i) q^{5} +(1.49194 + 0.879840i) q^{6} +(-0.520141 - 1.94119i) q^{7} +(-0.453990 - 0.891007i) q^{8} +(2.83556 + 0.979584i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.933580	−	0.358368i	−0.660141	−	0.253404i
							\(3\)	−1.70815	−	0.286738i	−0.986202	−	0.165548i
\(5\)	−0.417533	−	2.19674i	−0.186726	−	0.982412i
							\(7\)	−0.520141	−	1.94119i	−0.196595	−	0.733702i	−0.991848	−	0.127425i	\(-0.959329\pi\)
0.795253	−	0.606277i	\(-0.207338\pi\)
\(11\)	−0.844414	−	1.89658i	−0.254600	−	0.571842i	0.740349	−	0.672223i	\(-0.234660\pi\)
							−0.994949	+	0.100381i	\(0.967994\pi\)
\(13\)	0.114300	+	0.297761i	0.0317010	+	0.0825840i	0.948517	−	0.316725i	\(-0.102583\pi\)
							−0.916816	+	0.399309i	\(0.869250\pi\)
\(17\)	−1.68277	+	0.857415i	−0.408132	+	0.207954i	−0.645989	−	0.763346i	\(-0.723555\pi\)
							0.237857	+	0.971300i	\(0.423555\pi\)
\(19\)	3.79093	+	1.23175i	0.869700	+	0.282583i	0.709674	−	0.704530i	\(-0.248842\pi\)
							0.160026	+	0.987113i	\(0.448842\pi\)
\(23\)	−3.87778	−	4.78866i	−0.808572	−	0.998504i	−0.999856	−	0.0169689i	\(-0.994598\pi\)
							0.191284	−	0.981535i	\(-0.438735\pi\)
\(29\)	−3.27143	−	0.695363i	−0.607489	−	0.129126i	−0.106111	−	0.994354i	\(-0.533840\pi\)
							−0.501378	+	0.865229i	\(0.667173\pi\)
\(31\)	−7.45109	+	1.58378i	−1.33825	+	0.284455i	−0.820780	−	0.571244i	\(-0.806461\pi\)
							−0.517474	+	0.855699i	\(0.673128\pi\)
\(37\)	−1.48257	+	9.36058i	−0.243733	+	1.53887i	0.497406	+	0.867518i	\(0.334286\pi\)
							−0.741139	+	0.671352i	\(0.765714\pi\)
\(41\)	−2.41980	+	5.43496i	−0.377909	+	0.848798i	0.620027	+	0.784581i	\(0.287122\pi\)
							−0.997936	+	0.0642174i	\(0.979545\pi\)
\(43\)	−3.97819	+	1.06595i	−0.606668	+	0.162556i	−0.549060	−	0.835783i	\(-0.685014\pi\)
							−0.0576081	+	0.998339i	\(0.518347\pi\)
\(47\)	−6.72970	−	10.3628i	−0.981628	−	1.51157i	−0.854344	−	0.519708i	\(-0.826041\pi\)
							−0.127283	−	0.991866i	\(-0.540626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.263.1	yes	480
9.5	odd	6	inner	450.2.w.a.113.12	yes	480
25.2	odd	20	inner	450.2.w.a.227.12	yes	480
225.77	even	60	inner	450.2.w.a.77.1	✓	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.77.1	✓	480	225.77	even	60	inner
450.2.w.a.113.12	yes	480	9.5	odd	6	inner
450.2.w.a.227.12	yes	480	25.2	odd	20	inner
450.2.w.a.263.1	yes	480	1.1	even	1	trivial