Embedded newform 450.2.w.a.263.5

• Label: 450.2.w.a.263.5
• 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			263.5
Character	\(\chi\)	\(=\)	450.263
Dual form			450.2.w.a.77.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.933580 - 0.358368i) q^{2} +(-0.621432 + 1.61673i) q^{3} +(0.743145 + 0.669131i) q^{4} +(0.120697 - 2.23281i) q^{5} +(1.15954 - 1.28665i) q^{6} +(0.461093 + 1.72082i) q^{7} +(-0.453990 - 0.891007i) q^{8} +(-2.22764 - 2.00938i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

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
							\(3\)	−0.621432	+	1.61673i	−0.358784	+	0.933421i
\(5\)	0.120697	−	2.23281i	0.0539772	−	0.998542i
							\(7\)	0.461093	+	1.72082i	0.174277	+	0.650409i	0.996674	+	0.0814955i	\(0.0259696\pi\)
−0.822397	+	0.568914i	\(0.807364\pi\)
\(11\)	−1.80640	−	4.05725i	−0.544651	−	1.22331i	−0.950887	−	0.309538i	\(-0.899826\pi\)
							0.406236	−	0.913768i	\(-0.366841\pi\)
\(13\)	−0.577917	−	1.50552i	−0.160285	−	0.417557i	0.829794	−	0.558070i	\(-0.188458\pi\)
							−0.990079	+	0.140513i	\(0.955125\pi\)
\(17\)	5.47158	−	2.78791i	1.32705	−	0.676168i	0.360531	−	0.932747i	\(-0.382596\pi\)
							0.966523	+	0.256580i	\(0.0825956\pi\)
\(19\)	−3.58379	−	1.16445i	−0.822179	−	0.267142i	−0.132431	−	0.991192i	\(-0.542278\pi\)
							−0.689747	+	0.724050i	\(0.742278\pi\)
\(23\)	2.89516	+	3.57522i	0.603682	+	0.745485i	0.983764	−	0.179468i	\(-0.0574376\pi\)
							−0.380082	+	0.924953i	\(0.624104\pi\)
\(29\)	−5.60388	−	1.19114i	−1.04061	−	0.221189i	−0.344250	−	0.938878i	\(-0.611867\pi\)
							−0.696364	+	0.717689i	\(0.745200\pi\)
\(31\)	7.04324	−	1.49709i	1.26500	−	0.268885i	0.473907	−	0.880575i	\(-0.342843\pi\)
							0.791097	+	0.611690i	\(0.209510\pi\)
\(37\)	1.49361	−	9.43029i	0.245548	−	1.55033i	−0.489311	−	0.872110i	\(-0.662752\pi\)
							0.734859	−	0.678220i	\(-0.237248\pi\)
\(41\)	4.19374	−	9.41930i	0.654953	−	1.47105i	−0.214344	−	0.976758i	\(-0.568761\pi\)
							0.869297	−	0.494290i	\(-0.164572\pi\)
\(43\)	4.67836	−	1.25356i	0.713443	−	0.191166i	0.116199	−	0.993226i	\(-0.462929\pi\)
							0.597244	+	0.802060i	\(0.296262\pi\)
\(47\)	−2.66386	−	4.10199i	−0.388564	−	0.598337i	0.589214	−	0.807977i	\(-0.299437\pi\)
							−0.977779	+	0.209640i	\(0.932771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.263.5	yes	480
9.5	odd	6	inner	450.2.w.a.113.5	yes	480
25.2	odd	20	inner	450.2.w.a.227.5	yes	480
225.77	even	60	inner	450.2.w.a.77.5	✓	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.77.5	✓	480	225.77	even	60	inner
450.2.w.a.113.5	yes	480	9.5	odd	6	inner
450.2.w.a.227.5	yes	480	25.2	odd	20	inner
450.2.w.a.263.5	yes	480	1.1	even	1	trivial