Embedded newform 450.2.w.a.83.25

• Label: 450.2.w.a.83.25
• 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			83.25
Character	\(\chi\)	\(=\)	450.83
Dual form			450.2.w.a.347.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.544639 + 0.838671i) q^{2} +(0.883752 - 1.48962i) q^{3} +(-0.406737 + 0.913545i) q^{4} +(2.23112 - 0.148619i) q^{5} +(1.73063 - 0.0701307i) q^{6} +(1.34501 + 5.01966i) q^{7} +(-0.987688 + 0.156434i) q^{8} +(-1.43796 - 2.63292i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

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
							\(3\)	0.883752	−	1.48962i	0.510235	−	0.860035i
\(5\)	2.23112	−	0.148619i	0.997789	−	0.0664645i
							\(7\)	1.34501	+	5.01966i	0.508367	+	1.89725i	0.436173	+	0.899863i	\(0.356333\pi\)
0.0721941	+	0.997391i	\(0.477000\pi\)
\(11\)	−1.17614	+	5.53331i	−0.354620	+	1.66836i	0.333494	+	0.942752i	\(0.391772\pi\)
							−0.688114	+	0.725603i	\(0.741561\pi\)
\(13\)	−1.82215	−	1.18332i	−0.505374	−	0.328194i	0.266643	−	0.963795i	\(-0.414086\pi\)
							−0.772017	+	0.635601i	\(0.780752\pi\)
\(17\)	−0.305351	−	1.92791i	−0.0740584	−	0.467587i	−0.996648	−	0.0818045i	\(-0.973932\pi\)
							0.922590	−	0.385782i	\(-0.126068\pi\)
\(19\)	3.15513	−	4.34266i	0.723836	−	0.996275i	−0.275552	−	0.961286i	\(-0.588861\pi\)
							0.999388	−	0.0349883i	\(-0.0111394\pi\)
\(23\)	−0.215391	−	4.10991i	−0.0449122	−	0.856976i	−0.924487	−	0.381215i	\(-0.875506\pi\)
							0.879574	−	0.475761i	\(-0.157827\pi\)
\(29\)	−0.566575	−	5.39060i	−0.105210	−	1.00101i	−0.912005	−	0.410178i	\(-0.865467\pi\)
							0.806795	−	0.590831i	\(-0.201200\pi\)
\(31\)	−0.152138	+	1.44749i	−0.0273247	+	0.259977i	0.972328	+	0.233620i	\(0.0750573\pi\)
							−0.999653	+	0.0263568i	\(0.991609\pi\)
\(37\)	1.44678	+	2.83947i	0.237850	+	0.466806i	0.978818	−	0.204734i	\(-0.0656329\pi\)
							−0.740968	+	0.671540i	\(0.765633\pi\)
\(41\)	−1.55693	−	7.32479i	−0.243152	−	1.14394i	−0.915064	−	0.403309i	\(-0.867860\pi\)
							0.671912	−	0.740631i	\(-0.265473\pi\)
\(43\)	−5.27583	+	1.41366i	−0.804557	+	0.215580i	−0.637584	−	0.770381i	\(-0.720066\pi\)
							−0.166973	+	0.985961i	\(0.553399\pi\)
\(47\)	6.92944	−	8.55715i	1.01076	−	1.24819i	0.0422810	−	0.999106i	\(-0.486538\pi\)
							0.968482	−	0.249083i	\(-0.0801291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.83.25	yes	480
9.5	odd	6	inner	450.2.w.a.383.24	yes	480
25.22	odd	20	inner	450.2.w.a.47.24	✓	480
225.122	even	60	inner	450.2.w.a.347.25	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.47.24	✓	480	25.22	odd	20	inner
450.2.w.a.83.25	yes	480	1.1	even	1	trivial
450.2.w.a.347.25	yes	480	225.122	even	60	inner
450.2.w.a.383.24	yes	480	9.5	odd	6	inner