Embedded newform 450.2.w.a.23.9

• Label: 450.2.w.a.23.9
• Level: $450$
• Weight: $2$
• Character: 450.23
• 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			23.9
Character	\(\chi\)	\(=\)	450.23
Dual form			450.2.w.a.137.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.358368 - 0.933580i) q^{2} +(0.169956 + 1.72369i) q^{3} +(-0.743145 + 0.669131i) q^{4} +(-1.60127 + 1.56075i) q^{5} +(1.54830 - 0.776384i) q^{6} +(-2.59992 - 0.696648i) q^{7} +(0.891007 + 0.453990i) q^{8} +(-2.94223 + 0.585904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.358368	−	0.933580i	−0.253404	−	0.660141i
							\(3\)	0.169956	+	1.72369i	0.0981242	+	0.995174i
\(5\)	−1.60127	+	1.56075i	−0.716108	+	0.697990i
							\(7\)	−2.59992	−	0.696648i	−0.982679	−	0.263308i	−0.268507	−	0.963278i	\(-0.586530\pi\)
−0.714173	+	0.699970i	\(0.753197\pi\)
\(11\)	1.86287	−	4.18408i	0.561677	−	1.26155i	−0.379986	−	0.924992i	\(-0.624071\pi\)
							0.941663	−	0.336556i	\(-0.109262\pi\)
\(13\)	−3.55536	−	1.36477i	−0.986079	−	0.378520i	−0.188724	−	0.982030i	\(-0.560435\pi\)
							−0.797356	+	0.603510i	\(0.793768\pi\)
\(17\)	1.21745	−	2.38938i	0.295275	−	0.579510i	−0.694938	−	0.719069i	\(-0.744568\pi\)
							0.990214	+	0.139559i	\(0.0445684\pi\)
\(19\)	1.31955	−	0.428747i	0.302725	−	0.0983613i	−0.153716	−	0.988115i	\(-0.549124\pi\)
							0.456441	+	0.889754i	\(0.349124\pi\)
\(23\)	−0.687072	−	0.556380i	−0.143264	−	0.116013i	0.555009	−	0.831844i	\(-0.312715\pi\)
							−0.698274	+	0.715831i	\(0.746048\pi\)
\(29\)	0.833388	−	0.177142i	0.154756	−	0.0328945i	−0.129882	−	0.991529i	\(-0.541460\pi\)
							0.284638	+	0.958635i	\(0.408127\pi\)
\(31\)	−5.53194	−	1.17585i	−0.993565	−	0.211189i	−0.317682	−	0.948197i	\(-0.602904\pi\)
							−0.675883	+	0.737009i	\(0.736238\pi\)
\(37\)	−10.3567	+	1.64034i	−1.70263	+	0.269670i	−0.930632	−	0.365956i	\(-0.880742\pi\)
							−0.771997	+	0.635626i	\(0.780742\pi\)
\(41\)	2.07844	+	4.66825i	0.324598	+	0.729058i	0.999965	−	0.00842369i	\(-0.00268138\pi\)
							−0.675367	+	0.737482i	\(0.736015\pi\)
\(43\)	−2.60247	+	9.71254i	−0.396873	+	1.48115i	0.421695	+	0.906738i	\(0.361435\pi\)
							−0.818567	+	0.574411i	\(0.805231\pi\)
\(47\)	3.72705	+	2.42037i	0.543646	+	0.353048i	0.787062	−	0.616874i	\(-0.211601\pi\)
							−0.243416	+	0.969922i	\(0.578268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.23.9	✓	480
9.2	odd	6	inner	450.2.w.a.173.19	yes	480
25.12	odd	20	inner	450.2.w.a.437.19	yes	480
225.137	even	60	inner	450.2.w.a.137.9	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.23.9	✓	480	1.1	even	1	trivial
450.2.w.a.137.9	yes	480	225.137	even	60	inner
450.2.w.a.173.19	yes	480	9.2	odd	6	inner
450.2.w.a.437.19	yes	480	25.12	odd	20	inner