Embedded newform 450.2.w.a.23.13

• Label: 450.2.w.a.23.13
• 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			23.13
Character	\(\chi\)	\(=\)	450.23
Dual form			450.2.w.a.137.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.358368 - 0.933580i) q^{2} +(1.39329 - 1.02895i) q^{3} +(-0.743145 + 0.669131i) q^{4} +(-1.12618 - 1.93176i) q^{5} +(-1.45992 - 0.932005i) q^{6} +(3.27132 + 0.876546i) q^{7} +(0.891007 + 0.453990i) q^{8} +(0.882516 - 2.86726i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

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
							\(3\)	1.39329	−	1.02895i	0.804417	−	0.594066i
\(5\)	−1.12618	−	1.93176i	−0.503644	−	0.863911i
							\(7\)	3.27132	+	0.876546i	1.23644	+	0.331303i	0.817084	−	0.576518i	\(-0.195589\pi\)
0.419357	+	0.907821i	\(0.362256\pi\)
\(11\)	0.780328	−	1.75265i	0.235278	−	0.528443i	−0.756862	−	0.653574i	\(-0.773269\pi\)
							0.992140	+	0.125132i	\(0.0399353\pi\)
\(13\)	0.854670	+	0.328077i	0.237043	+	0.0909922i	0.473990	−	0.880530i	\(-0.342813\pi\)
							−0.236947	+	0.971523i	\(0.576147\pi\)
\(17\)	−0.905589	+	1.77732i	−0.219638	+	0.431063i	−0.974364	−	0.224976i	\(-0.927769\pi\)
							0.754727	+	0.656039i	\(0.227769\pi\)
\(19\)	1.31698	−	0.427914i	0.302137	−	0.0981702i	−0.154025	−	0.988067i	\(-0.549224\pi\)
							0.456162	+	0.889897i	\(0.349224\pi\)
\(23\)	−6.25570	−	5.06577i	−1.30440	−	1.05629i	−0.994414	−	0.105546i	\(-0.966341\pi\)
							−0.309990	−	0.950740i	\(-0.600326\pi\)
\(29\)	0.247869	−	0.0526863i	0.0460282	−	0.00978359i	−0.184840	−	0.982769i	\(-0.559177\pi\)
							0.230868	+	0.972985i	\(0.425843\pi\)
\(31\)	−4.03688	−	0.858065i	−0.725045	−	0.154113i	−0.169417	−	0.985544i	\(-0.554189\pi\)
							−0.555627	+	0.831431i	\(0.687522\pi\)
\(37\)	2.98233	−	0.472355i	0.490293	−	0.0776547i	0.0936070	−	0.995609i	\(-0.470160\pi\)
							0.396686	+	0.917954i	\(0.370160\pi\)
\(41\)	4.68321	+	10.5187i	0.731395	+	1.64274i	0.765593	+	0.643326i	\(0.222446\pi\)
							−0.0341975	+	0.999415i	\(0.510888\pi\)
\(43\)	−0.690729	+	2.57784i	−0.105335	+	0.393116i	−0.998383	−	0.0568456i	\(-0.981896\pi\)
							0.893048	+	0.449962i	\(0.148562\pi\)
\(47\)	9.99776	+	6.49262i	1.45832	+	0.947046i	0.998248	+	0.0591662i	\(0.0188442\pi\)
							0.460075	+	0.887880i	\(0.347822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.23.13	✓	480
9.2	odd	6	inner	450.2.w.a.173.28	yes	480
25.12	odd	20	inner	450.2.w.a.437.28	yes	480
225.137	even	60	inner	450.2.w.a.137.13	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.23.13	✓	480	1.1	even	1	trivial
450.2.w.a.137.13	yes	480	225.137	even	60	inner
450.2.w.a.173.28	yes	480	9.2	odd	6	inner
450.2.w.a.437.28	yes	480	25.12	odd	20	inner