Embedded newform 450.2.w.a.23.12

• Label: 450.2.w.a.23.12
• 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			23.12
Character	\(\chi\)	\(=\)	450.23
Dual form			450.2.w.a.137.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.358368 - 0.933580i) q^{2} +(1.34709 - 1.08874i) q^{3} +(-0.743145 + 0.669131i) q^{4} +(1.80264 + 1.32305i) q^{5} +(-1.49918 - 0.867446i) q^{6} +(0.216796 + 0.0580902i) q^{7} +(0.891007 + 0.453990i) q^{8} +(0.629292 - 2.93326i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

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
							\(3\)	1.34709	−	1.08874i	0.777742	−	0.628584i
\(5\)	1.80264	+	1.32305i	0.806167	+	0.591688i
							\(7\)	0.216796	+	0.0580902i	0.0819410	+	0.0219560i	0.299557	−	0.954079i	\(-0.403161\pi\)
−0.217616	+	0.976035i	\(0.569828\pi\)
\(11\)	−0.521954	+	1.17233i	−0.157375	+	0.353470i	−0.974974	−	0.222318i	\(-0.928638\pi\)
							0.817599	+	0.575788i	\(0.195305\pi\)
\(13\)	2.23713	+	0.858755i	0.620469	+	0.238176i	0.648237	−	0.761439i	\(-0.275507\pi\)
							−0.0277679	+	0.999614i	\(0.508840\pi\)
\(17\)	3.12471	−	6.13259i	0.757853	−	1.48737i	−0.111812	−	0.993729i	\(-0.535666\pi\)
							0.869666	−	0.493641i	\(-0.164334\pi\)
\(19\)	−1.72628	+	0.560901i	−0.396035	+	0.128680i	−0.500262	−	0.865874i	\(-0.666763\pi\)
							0.104227	+	0.994554i	\(0.466763\pi\)
\(23\)	2.24140	+	1.81505i	0.467365	+	0.378465i	0.833879	−	0.551948i	\(-0.186115\pi\)
							−0.366513	+	0.930413i	\(0.619449\pi\)
\(29\)	2.29667	−	0.488173i	0.426482	−	0.0906515i	0.0103299	−	0.999947i	\(-0.496712\pi\)
							0.416152	+	0.909295i	\(0.363378\pi\)
\(31\)	1.05017	+	0.223220i	0.188616	+	0.0400915i	0.301251	−	0.953545i	\(-0.402596\pi\)
							−0.112635	+	0.993636i	\(0.535929\pi\)
\(37\)	−7.25170	+	1.14856i	−1.19217	+	0.188822i	−0.720819	−	0.693123i	\(-0.756234\pi\)
							−0.471353	+	0.881945i	\(0.656234\pi\)
\(41\)	−4.46455	−	10.0275i	−0.697246	−	1.56604i	−0.819203	−	0.573504i	\(-0.805584\pi\)
							0.121957	−	0.992535i	\(-0.461083\pi\)
\(43\)	−3.15536	+	11.7760i	−0.481188	+	1.79582i	0.115456	+	0.993313i	\(0.463167\pi\)
							−0.596644	+	0.802506i	\(0.703499\pi\)
\(47\)	1.24627	+	0.809337i	0.181787	+	0.118054i	0.632325	−	0.774704i	\(-0.282101\pi\)
							−0.450537	+	0.892758i	\(0.648768\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.23.12	✓	480
9.2	odd	6	inner	450.2.w.a.173.29	yes	480
25.12	odd	20	inner	450.2.w.a.437.29	yes	480
225.137	even	60	inner	450.2.w.a.137.12	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.23.12	✓	480	1.1	even	1	trivial
450.2.w.a.137.12	yes	480	225.137	even	60	inner
450.2.w.a.173.29	yes	480	9.2	odd	6	inner
450.2.w.a.437.29	yes	480	25.12	odd	20	inner