Embedded newform 450.2.w.a.23.16

• Label: 450.2.w.a.23.16
• 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			23.16
Character	\(\chi\)	\(=\)	450.23
Dual form			450.2.w.a.137.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.358368 + 0.933580i) q^{2} +(-1.73185 - 0.0261705i) q^{3} +(-0.743145 + 0.669131i) q^{4} +(-1.81212 + 1.31005i) q^{5} +(-0.596208 - 1.62620i) q^{6} +(4.79708 + 1.28537i) q^{7} +(-0.891007 - 0.453990i) q^{8} +(2.99863 + 0.0906470i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

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
							\(3\)	−1.73185	−	0.0261705i	−0.999886	−	0.0151096i
\(5\)	−1.81212	+	1.31005i	−0.810402	+	0.585874i
							\(7\)	4.79708	+	1.28537i	1.81312	+	0.485825i	0.995897	−	0.0904900i	\(-0.0288433\pi\)
0.817227	+	0.576315i	\(0.195510\pi\)
\(11\)	−1.12735	+	2.53207i	−0.339910	+	0.763449i	0.660016	+	0.751251i	\(0.270549\pi\)
							−0.999926	+	0.0121978i	\(0.996117\pi\)
\(13\)	−5.50832	−	2.11445i	−1.52773	−	0.586442i	−0.557835	−	0.829952i	\(-0.688368\pi\)
							−0.969898	+	0.243510i	\(0.921701\pi\)
\(17\)	−2.56541	+	5.03490i	−0.622203	+	1.22114i	0.337818	+	0.941212i	\(0.390311\pi\)
							−0.960020	+	0.279930i	\(0.909689\pi\)
\(19\)	−2.85228	+	0.926763i	−0.654359	+	0.212614i	−0.617335	−	0.786700i	\(-0.711788\pi\)
							−0.0370236	+	0.999314i	\(0.511788\pi\)
\(23\)	−3.42455	−	2.77315i	−0.714069	−	0.578242i	0.201941	−	0.979398i	\(-0.435275\pi\)
							−0.916010	+	0.401156i	\(0.868608\pi\)
\(29\)	0.113668	−	0.0241609i	0.0211076	−	0.00448656i	−0.197346	−	0.980334i	\(-0.563232\pi\)
							0.218453	+	0.975847i	\(0.429899\pi\)
\(31\)	−1.39293	−	0.296077i	−0.250178	−	0.0531770i	0.0811154	−	0.996705i	\(-0.474152\pi\)
							−0.331294	+	0.943528i	\(0.607485\pi\)
\(37\)	1.56264	−	0.247498i	0.256897	−	0.0406885i	−0.0266574	−	0.999645i	\(-0.508486\pi\)
							0.283554	+	0.958956i	\(0.408486\pi\)
\(41\)	2.72921	+	6.12992i	0.426232	+	0.957332i	0.991219	+	0.132232i	\(0.0422145\pi\)
							−0.564987	+	0.825100i	\(0.691119\pi\)
\(43\)	−1.92566	+	7.18667i	−0.293661	+	1.09596i	0.648614	+	0.761117i	\(0.275349\pi\)
							−0.942275	+	0.334840i	\(0.891318\pi\)
\(47\)	−8.48970	−	5.51328i	−1.23835	−	0.804194i	−0.251821	−	0.967774i	\(-0.581029\pi\)
							−0.986530	+	0.163580i	\(0.947696\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.23.16	✓	480
9.2	odd	6	inner	450.2.w.a.173.6	yes	480
25.12	odd	20	inner	450.2.w.a.437.6	yes	480
225.137	even	60	inner	450.2.w.a.137.16	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.23.16	✓	480	1.1	even	1	trivial
450.2.w.a.137.16	yes	480	225.137	even	60	inner
450.2.w.a.173.6	yes	480	9.2	odd	6	inner
450.2.w.a.437.6	yes	480	25.12	odd	20	inner