Embedded newform 450.2.v.a.79.16

• Label: 450.2.v.a.79.16
• 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			79.16
Character	\(\chi\)	\(=\)	450.79
Dual form			450.2.v.a.319.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.743145 - 0.669131i) q^{2} +(-1.71340 + 0.253483i) q^{3} +(0.104528 - 0.994522i) q^{4} +(-2.04514 + 0.904112i) q^{5} +(-1.10369 + 1.33486i) q^{6} +(0.528158 + 0.304932i) q^{7} +(-0.587785 - 0.809017i) q^{8} +(2.87149 - 0.868637i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.71340	+	0.253483i	−0.989233	+	0.146348i
\(5\)	−2.04514	+	0.904112i	−0.914613	+	0.404331i
							\(7\)	0.528158	+	0.304932i	0.199625	+	0.115254i	0.596481	−	0.802628i	\(-0.296565\pi\)
−0.396856	+	0.917881i	\(0.629899\pi\)
\(11\)	2.74684	+	3.05068i	0.828204	+	0.919813i	0.997840	−	0.0656934i	\(-0.0209259\pi\)
							−0.169636	+	0.985507i	\(0.554259\pi\)
\(13\)	0.173707	+	0.156407i	0.0481778	+	0.0433794i	0.692868	−	0.721065i	\(-0.256347\pi\)
							−0.644690	+	0.764444i	\(0.723014\pi\)
\(17\)	3.38589	+	4.66028i	0.821200	+	1.13028i	0.989498	+	0.144548i	\(0.0461728\pi\)
							−0.168298	+	0.985736i	\(0.553827\pi\)
\(19\)	1.51314	−	1.09936i	0.347139	−	0.252211i	−0.400529	−	0.916284i	\(-0.631174\pi\)
							0.747668	+	0.664073i	\(0.231174\pi\)
\(23\)	1.12860	+	5.30967i	0.235330	+	1.10714i	0.924097	+	0.382159i	\(0.124819\pi\)
							−0.688767	+	0.724983i	\(0.741847\pi\)
\(29\)	5.20982	−	2.31956i	0.967439	−	0.430732i	0.138680	−	0.990337i	\(-0.455714\pi\)
							0.828759	+	0.559606i	\(0.189047\pi\)
\(31\)	5.98291	+	2.66376i	1.07456	+	0.478426i	0.866237	−	0.499634i	\(-0.166532\pi\)
							0.208325	+	0.978060i	\(0.433199\pi\)
\(37\)	−8.15384	+	2.64934i	−1.34048	+	0.435549i	−0.889481	−	0.456973i	\(-0.848934\pi\)
							−0.451003	+	0.892523i	\(0.648934\pi\)
\(41\)	−4.27148	+	4.74395i	−0.667092	+	0.740881i	−0.977781	−	0.209631i	\(-0.932774\pi\)
							0.310688	+	0.950512i	\(0.399441\pi\)
\(43\)	−1.51980	−	0.877459i	−0.231768	−	0.133811i	0.379619	−	0.925143i	\(-0.376055\pi\)
							−0.611387	+	0.791331i	\(0.709388\pi\)
\(47\)	0.853487	+	1.91696i	0.124494	+	0.279618i	0.965031	−	0.262136i	\(-0.0844270\pi\)
							−0.840537	+	0.541754i	\(0.817760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.79.16	✓	240
9.4	even	3	inner	450.2.v.a.229.24	yes	240
25.19	even	10	inner	450.2.v.a.169.24	yes	240
225.94	even	30	inner	450.2.v.a.319.16	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.16	✓	240	1.1	even	1	trivial
450.2.v.a.169.24	yes	240	25.19	even	10	inner
450.2.v.a.229.24	yes	240	9.4	even	3	inner
450.2.v.a.319.16	yes	240	225.94	even	30	inner