Embedded newform 450.2.w.a.83.13

• Label: 450.2.w.a.83.13
• Level: $450$
• Weight: $2$
• Character: 450.83
• 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			83.13
Character	\(\chi\)	\(=\)	450.83
Dual form			450.2.w.a.347.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.544639 - 0.838671i) q^{2} +(1.53738 + 0.797779i) q^{3} +(-0.406737 + 0.913545i) q^{4} +(2.04607 + 0.901991i) q^{5} +(-0.168245 - 1.72386i) q^{6} +(-0.474925 - 1.77244i) q^{7} +(0.987688 - 0.156434i) q^{8} +(1.72710 + 2.45299i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.544639	−	0.838671i	−0.385118	−	0.593030i
							\(3\)	1.53738	+	0.797779i	0.887609	+	0.460598i
\(5\)	2.04607	+	0.901991i	0.915031	+	0.403383i
							\(7\)	−0.474925	−	1.77244i	−0.179505	−	0.669921i	−0.995740	−	0.0922019i	\(-0.970609\pi\)
0.816235	−	0.577719i	\(-0.196057\pi\)
\(11\)	0.0321943	−	0.151462i	0.00970696	−	0.0456676i	−0.973028	−	0.230689i	\(-0.925902\pi\)
							0.982735	+	0.185021i	\(0.0592354\pi\)
\(13\)	2.53008	+	1.64305i	0.701719	+	0.455701i	0.845547	−	0.533901i	\(-0.179275\pi\)
							−0.143828	+	0.989603i	\(0.545941\pi\)
\(17\)	−0.809474	−	5.11082i	−0.196326	−	1.23956i	−0.867191	−	0.497976i	\(-0.834077\pi\)
							0.670864	−	0.741580i	\(-0.265923\pi\)
\(19\)	−1.01046	+	1.39077i	−0.231815	+	0.319065i	−0.909039	−	0.416711i	\(-0.863183\pi\)
							0.677224	+	0.735777i	\(0.263183\pi\)
\(23\)	0.117593	+	2.24381i	0.0245199	+	0.467867i	0.983103	+	0.183054i	\(0.0585982\pi\)
							−0.958583	+	0.284813i	\(0.908068\pi\)
\(29\)	−0.147269	−	1.40117i	−0.0273471	−	0.260191i	−0.999650	−	0.0264650i	\(-0.991575\pi\)
							0.972303	−	0.233726i	\(-0.0750917\pi\)
\(31\)	−0.604030	+	5.74696i	−0.108487	+	1.03219i	0.795887	+	0.605445i	\(0.207005\pi\)
							−0.904374	+	0.426740i	\(0.859662\pi\)
\(37\)	1.38915	+	2.72636i	0.228375	+	0.448211i	0.976550	−	0.215290i	\(-0.0690698\pi\)
							−0.748175	+	0.663501i	\(0.769070\pi\)
\(41\)	−1.42493	−	6.70376i	−0.222536	−	1.04695i	−0.937553	−	0.347844i	\(-0.886914\pi\)
							0.715016	−	0.699108i	\(-0.246419\pi\)
\(43\)	−2.74174	+	0.734648i	−0.418112	+	0.112033i	−0.461741	−	0.887015i	\(-0.652775\pi\)
							0.0436285	+	0.999048i	\(0.486108\pi\)
\(47\)	2.80469	−	3.46351i	0.409106	−	0.505204i	−0.530394	−	0.847751i	\(-0.677956\pi\)
							0.939501	+	0.342547i	\(0.111289\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.83.13	yes	480
9.5	odd	6	inner	450.2.w.a.383.15	yes	480
25.22	odd	20	inner	450.2.w.a.47.15	✓	480
225.122	even	60	inner	450.2.w.a.347.13	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.47.15	✓	480	25.22	odd	20	inner
450.2.w.a.83.13	yes	480	1.1	even	1	trivial
450.2.w.a.347.13	yes	480	225.122	even	60	inner
450.2.w.a.383.15	yes	480	9.5	odd	6	inner