Embedded newform 450.2.v.a.79.3

• Label: 450.2.v.a.79.3
• Level: $450$
• Weight: $2$
• Character: 450.79
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.v (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			79.3
Character	\(\chi\)	\(=\)	450.79
Dual form			450.2.v.a.319.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 + 0.669131i) q^{2} +(-1.44495 + 0.955053i) q^{3} +(0.104528 - 0.994522i) q^{4} +(-1.82705 - 1.28914i) q^{5} +(0.434750 - 1.67660i) q^{6} +(2.39808 + 1.38453i) q^{7} +(0.587785 + 0.809017i) q^{8} +(1.17575 - 2.76000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.743145	+	0.669131i	−0.525483	+	0.473147i
							\(3\)	−1.44495	+	0.955053i	−0.834241	+	0.551400i
\(5\)	−1.82705	−	1.28914i	−0.817083	−	0.576520i
							\(7\)	2.39808	+	1.38453i	0.906388	+	0.523303i	0.879267	−	0.476329i	\(-0.158033\pi\)
0.0271204	+	0.999632i	\(0.491366\pi\)
\(11\)	0.299362	+	0.332475i	0.0902609	+	0.100245i	0.786583	−	0.617484i	\(-0.211848\pi\)
							−0.696322	+	0.717729i	\(0.745181\pi\)
\(13\)	3.86357	+	3.47877i	1.07156	+	0.964837i	0.999470	−	0.0325607i	\(-0.0103662\pi\)
							0.0720907	+	0.997398i	\(0.477033\pi\)
\(17\)	−3.03981	−	4.18393i	−0.737261	−	1.01475i	−0.998772	−	0.0495521i	\(-0.984221\pi\)
							0.261511	−	0.965201i	\(-0.415779\pi\)
\(19\)	−6.94401	+	5.04512i	−1.59307	+	1.15743i	−0.693672	+	0.720291i	\(0.744008\pi\)
							−0.899394	+	0.437140i	\(0.855992\pi\)
\(23\)	1.40074	+	6.58994i	0.292074	+	1.37410i	0.842267	+	0.539061i	\(0.181221\pi\)
							−0.550193	+	0.835037i	\(0.685446\pi\)
\(29\)	2.33116	−	1.03790i	0.432886	−	0.192733i	−0.178713	−	0.983901i	\(-0.557193\pi\)
							0.611599	+	0.791168i	\(0.290527\pi\)
\(31\)	−4.08915	−	1.82061i	−0.734433	−	0.326990i	0.00521359	−	0.999986i	\(-0.498340\pi\)
							−0.739646	+	0.672996i	\(0.765007\pi\)
\(37\)	−3.98764	+	1.29566i	−0.655564	+	0.213006i	−0.617866	−	0.786283i	\(-0.712003\pi\)
							−0.0376983	+	0.999289i	\(0.512003\pi\)
\(41\)	−3.87550	+	4.30417i	−0.605251	+	0.672199i	−0.965423	−	0.260687i	\(-0.916051\pi\)
							0.360173	+	0.932886i	\(0.382718\pi\)
\(43\)	6.59650	+	3.80849i	1.00596	+	0.580789i	0.910005	−	0.414597i	\(-0.136077\pi\)
							0.0959511	+	0.995386i	\(0.469411\pi\)
\(47\)	1.29248	+	2.90296i	0.188528	+	0.423441i	0.982937	−	0.183942i	\(-0.0588860\pi\)
							−0.794409	+	0.607383i	\(0.792219\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.79.3	✓	240
9.4	even	3	inner	450.2.v.a.229.7	yes	240
25.19	even	10	inner	450.2.v.a.169.7	yes	240
225.94	even	30	inner	450.2.v.a.319.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.3	✓	240	1.1	even	1	trivial
450.2.v.a.169.7	yes	240	25.19	even	10	inner
450.2.v.a.229.7	yes	240	9.4	even	3	inner
450.2.v.a.319.3	yes	240	225.94	even	30	inner