Embedded newform 450.2.w.a.23.18

• Label: 450.2.w.a.23.18
• 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			23.18
Character	\(\chi\)	\(=\)	450.23
Dual form			450.2.w.a.137.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.358368 + 0.933580i) q^{2} +(-1.63302 - 0.577277i) q^{3} +(-0.743145 + 0.669131i) q^{4} +(0.309636 + 2.21453i) q^{5} +(-0.0462871 - 1.73143i) q^{6} +(-4.14621 - 1.11097i) q^{7} +(-0.891007 - 0.453990i) q^{8} +(2.33350 + 1.88541i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

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
							\(3\)	−1.63302	−	0.577277i	−0.942824	−	0.333291i
\(5\)	0.309636	+	2.21453i	0.138473	+	0.990366i
							\(7\)	−4.14621	−	1.11097i	−1.56712	−	0.419909i	−0.632212	−	0.774796i	\(-0.717853\pi\)
−0.934909	+	0.354887i	\(0.884519\pi\)
\(11\)	1.69937	−	3.81685i	0.512379	−	1.15082i	−0.453365	−	0.891325i	\(-0.649776\pi\)
							0.965744	−	0.259497i	\(-0.0835569\pi\)
\(13\)	−2.83743	−	1.08919i	−0.786961	−	0.302086i	−0.0684982	−	0.997651i	\(-0.521821\pi\)
							−0.718463	+	0.695565i	\(0.755154\pi\)
\(17\)	1.43021	−	2.80695i	0.346877	−	0.680784i	−0.649985	−	0.759947i	\(-0.725225\pi\)
							0.996862	+	0.0791629i	\(0.0252247\pi\)
\(19\)	3.41582	−	1.10987i	0.783643	−	0.254621i	0.110248	−	0.993904i	\(-0.464835\pi\)
							0.673395	+	0.739283i	\(0.264835\pi\)
\(23\)	−4.93765	−	3.99843i	−1.02957	−	0.833731i	−0.0433336	−	0.999061i	\(-0.513798\pi\)
							−0.986238	+	0.165330i	\(0.947131\pi\)
\(29\)	−8.43271	+	1.79243i	−1.56591	+	0.332846i	−0.907580	−	0.419879i	\(-0.862072\pi\)
							−0.658335	+	0.752725i	\(0.728739\pi\)
\(31\)	−0.898095	−	0.190896i	−0.161303	−	0.0342859i	0.126552	−	0.991960i	\(-0.459609\pi\)
							−0.287855	+	0.957674i	\(0.592942\pi\)
\(37\)	4.58563	−	0.726293i	0.753873	−	0.119402i	0.232342	−	0.972634i	\(-0.425361\pi\)
							0.521531	+	0.853232i	\(0.325361\pi\)
\(41\)	−4.24929	−	9.54406i	−0.663628	−	1.49053i	−0.860154	−	0.510034i	\(-0.829633\pi\)
							0.196527	−	0.980498i	\(-0.437034\pi\)
\(43\)	0.702617	−	2.62220i	0.107148	−	0.399882i	−0.891432	−	0.453155i	\(-0.850299\pi\)
							0.998580	+	0.0532726i	\(0.0169652\pi\)
\(47\)	2.13435	+	1.38606i	0.311326	+	0.202178i	0.690839	−	0.723009i	\(-0.257242\pi\)
							−0.379512	+	0.925187i	\(0.623908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.23.18	✓	480
9.2	odd	6	inner	450.2.w.a.173.7	yes	480
25.12	odd	20	inner	450.2.w.a.437.7	yes	480
225.137	even	60	inner	450.2.w.a.137.18	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.23.18	✓	480	1.1	even	1	trivial
450.2.w.a.137.18	yes	480	225.137	even	60	inner
450.2.w.a.173.7	yes	480	9.2	odd	6	inner
450.2.w.a.437.7	yes	480	25.12	odd	20	inner