Embedded newform 450.2.w.a.23.17

• Label: 450.2.w.a.23.17
• 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			23.17
Character	\(\chi\)	\(=\)	450.23
Dual form			450.2.w.a.137.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.358368 + 0.933580i) q^{2} +(-1.69592 + 0.351926i) q^{3} +(-0.743145 + 0.669131i) q^{4} +(2.22290 + 0.242291i) q^{5} +(-0.936315 - 1.45716i) q^{6} +(-0.250623 - 0.0671543i) q^{7} +(-0.891007 - 0.453990i) q^{8} +(2.75230 - 1.19368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

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
							\(3\)	−1.69592	+	0.351926i	−0.979140	+	0.203184i
\(5\)	2.22290	+	0.242291i	0.994112	+	0.108356i
							\(7\)	−0.250623	−	0.0671543i	−0.0947267	−	0.0253819i	0.211144	−	0.977455i	\(-0.432281\pi\)
−0.305871	+	0.952073i	\(0.598948\pi\)
\(11\)	−1.89571	+	4.25783i	−0.571578	+	1.28378i	0.364241	+	0.931305i	\(0.381328\pi\)
							−0.935819	+	0.352480i	\(0.885338\pi\)
\(13\)	5.06141	+	1.94289i	1.40378	+	0.538862i	0.938185	−	0.346134i	\(-0.112506\pi\)
							0.465598	+	0.884996i	\(0.345839\pi\)
\(17\)	0.318397	−	0.624889i	0.0772225	−	0.151558i	−0.849167	−	0.528124i	\(-0.822895\pi\)
							0.926389	+	0.376567i	\(0.122895\pi\)
\(19\)	−7.33096	+	2.38197i	−1.68184	+	0.546462i	−0.985267	−	0.171026i	\(-0.945292\pi\)
							−0.696571	+	0.717488i	\(0.745292\pi\)
\(23\)	3.83023	+	3.10166i	0.798658	+	0.646741i	0.939214	−	0.343332i	\(-0.111556\pi\)
							−0.140556	+	0.990073i	\(0.544889\pi\)
\(29\)	−8.99583	+	1.91212i	−1.67048	+	0.355072i	−0.943446	−	0.331525i	\(-0.892437\pi\)
							−0.727038	+	0.686598i	\(0.759103\pi\)
\(31\)	2.23693	+	0.475474i	0.401765	+	0.0853977i	0.404362	−	0.914599i	\(-0.367493\pi\)
							−0.00259754	+	0.999997i	\(0.500827\pi\)
\(37\)	−3.15328	+	0.499431i	−0.518397	+	0.0821060i	−0.410150	−	0.912018i	\(-0.634524\pi\)
							−0.108247	+	0.994124i	\(0.534524\pi\)
\(41\)	1.50777	+	3.38650i	0.235474	+	0.528883i	0.992172	−	0.124882i	\(-0.0398553\pi\)
							−0.756698	+	0.653765i	\(0.773189\pi\)
\(43\)	1.56211	−	5.82987i	0.238220	−	0.889047i	−0.738452	−	0.674307i	\(-0.764443\pi\)
							0.976671	−	0.214741i	\(-0.0688907\pi\)
\(47\)	7.31410	+	4.74983i	1.06687	+	0.692834i	0.953706	−	0.300742i	\(-0.0972342\pi\)
							0.113166	+	0.993576i	\(0.463901\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.23.17	✓	480
9.2	odd	6	inner	450.2.w.a.173.3	yes	480
25.12	odd	20	inner	450.2.w.a.437.3	yes	480
225.137	even	60	inner	450.2.w.a.137.17	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.23.17	✓	480	1.1	even	1	trivial
450.2.w.a.137.17	yes	480	225.137	even	60	inner
450.2.w.a.173.3	yes	480	9.2	odd	6	inner
450.2.w.a.437.3	yes	480	25.12	odd	20	inner