Embedded newform 450.2.w.a.317.29

• Label: 450.2.w.a.317.29
• Level: $450$
• Weight: $2$
• Character: 450.317
• 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			317.29
Character	\(\chi\)	\(=\)	450.317
Dual form			450.2.w.a.203.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.838671 - 0.544639i) q^{2} +(1.69655 + 0.348891i) q^{3} +(0.406737 - 0.913545i) q^{4} +(0.632384 - 2.14478i) q^{5} +(1.61286 - 0.631402i) q^{6} +(-0.183128 + 0.0490689i) q^{7} +(-0.156434 - 0.987688i) q^{8} +(2.75655 + 1.18382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.838671	−	0.544639i	0.593030	−	0.385118i
							\(3\)	1.69655	+	0.348891i	0.979502	+	0.201432i
\(5\)	0.632384	−	2.14478i	0.282811	−	0.959176i
							\(7\)	−0.183128	+	0.0490689i	−0.0692158	+	0.0185463i	−0.293261	−	0.956032i	\(-0.594740\pi\)
0.224045	+	0.974579i	\(0.428074\pi\)
\(11\)	−0.580223	+	2.72973i	−0.174944	+	0.823045i	0.799896	+	0.600139i	\(0.204888\pi\)
							−0.974840	+	0.222907i	\(0.928445\pi\)
\(13\)	−0.0685798	+	0.105604i	−0.0190206	+	0.0292892i	−0.848053	−	0.529911i	\(-0.822225\pi\)
							0.829032	+	0.559201i	\(0.188892\pi\)
\(17\)	−2.12250	+	0.336171i	−0.514781	+	0.0815334i	−0.408420	−	0.912794i	\(-0.633920\pi\)
							−0.106361	+	0.994328i	\(0.533920\pi\)
\(19\)	2.08643	−	2.87173i	0.478660	−	0.658819i	−0.499587	−	0.866264i	\(-0.666515\pi\)
							0.978247	+	0.207445i	\(0.0665148\pi\)
\(23\)	−6.19726	+	0.324785i	−1.29222	+	0.0677223i	−0.686015	−	0.727588i	\(-0.740642\pi\)
							−0.606203	+	0.795310i	\(0.707308\pi\)
\(29\)	−0.141316	−	1.34453i	−0.0262417	−	0.249673i	−0.999777	−	0.0211348i	\(-0.993272\pi\)
							0.973535	−	0.228538i	\(-0.0733946\pi\)
\(31\)	−0.620723	+	5.90579i	−0.111485	+	1.06071i	0.785564	+	0.618780i	\(0.212373\pi\)
							−0.897049	+	0.441931i	\(0.854294\pi\)
\(37\)	9.62313	−	4.90323i	1.58203	−	0.806086i	0.582058	−	0.813147i	\(-0.302248\pi\)
							0.999975	+	0.00706138i	\(0.00224773\pi\)
\(41\)	0.824806	+	3.88041i	0.128813	+	0.606018i	0.994438	+	0.105325i	\(0.0335882\pi\)
							−0.865625	+	0.500693i	\(0.833078\pi\)
\(43\)	2.95661	+	11.0342i	0.450879	+	1.68270i	0.699928	+	0.714214i	\(0.253216\pi\)
							−0.249048	+	0.968491i	\(0.580118\pi\)
\(47\)	−1.96306	−	1.58965i	−0.286341	−	0.231875i	0.475378	−	0.879782i	\(-0.342311\pi\)
							−0.761720	+	0.647907i	\(0.775645\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.317.29	yes	480
9.5	odd	6	inner	450.2.w.a.167.1	✓	480
25.3	odd	20	inner	450.2.w.a.353.1	yes	480
225.203	even	60	inner	450.2.w.a.203.29	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.167.1	✓	480	9.5	odd	6	inner
450.2.w.a.203.29	yes	480	225.203	even	60	inner
450.2.w.a.317.29	yes	480	1.1	even	1	trivial
450.2.w.a.353.1	yes	480	25.3	odd	20	inner