Embedded newform 450.2.v.a.319.5

• Label: 450.2.v.a.319.5
• Level: $450$
• Weight: $2$
• Character: 450.319
• 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			319.5
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 - 0.669131i) q^{2} +(-0.818331 + 1.52654i) q^{3} +(0.104528 + 0.994522i) q^{4} +(-2.19550 + 0.423984i) q^{5} +(1.62960 - 0.586873i) q^{6} +(4.25551 - 2.45692i) q^{7} +(0.587785 - 0.809017i) q^{8} +(-1.66067 - 2.49844i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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
							\(3\)	−0.818331	+	1.52654i	−0.472464	+	0.881350i
\(5\)	−2.19550	+	0.423984i	−0.981859	+	0.189611i
							\(7\)	4.25551	−	2.45692i	1.60843	−	0.928628i	0.618708	−	0.785621i	\(-0.287656\pi\)
0.989722	−	0.143007i	\(-0.0456771\pi\)
\(11\)	1.37120	−	1.52287i	0.413432	−	0.459163i	−0.500077	−	0.865981i	\(-0.666695\pi\)
							0.913509	+	0.406818i	\(0.133362\pi\)
\(13\)	−3.01385	+	2.71368i	−0.835892	+	0.752640i	−0.971226	−	0.238160i	\(-0.923456\pi\)
							0.135334	+	0.990800i	\(0.456789\pi\)
\(17\)	0.396865	−	0.546238i	0.0962539	−	0.132482i	−0.758174	−	0.652053i	\(-0.773908\pi\)
							0.854428	+	0.519571i	\(0.173908\pi\)
\(19\)	5.52782	+	4.01619i	1.26817	+	0.921378i	0.999128	−	0.0417512i	\(-0.0132937\pi\)
							0.269040	+	0.963129i	\(0.413294\pi\)
\(23\)	−0.0790176	+	0.371749i	−0.0164763	+	0.0775150i	−0.985618	−	0.168991i	\(-0.945949\pi\)
							0.969141	+	0.246506i	\(0.0792825\pi\)
\(29\)	6.51915	+	2.90251i	1.21058	+	0.538983i	0.909934	−	0.414754i	\(-0.136132\pi\)
							0.300642	+	0.953737i	\(0.402799\pi\)
\(31\)	7.78636	−	3.46671i	1.39847	−	0.622639i	0.437482	−	0.899227i	\(-0.355870\pi\)
							0.960989	+	0.276588i	\(0.0892036\pi\)
\(37\)	3.50250	+	1.13803i	0.575808	+	0.187091i	0.582422	−	0.812887i	\(-0.302105\pi\)
							−0.00661391	+	0.999978i	\(0.502105\pi\)
\(41\)	2.15988	+	2.39879i	0.337317	+	0.374628i	0.887809	−	0.460212i	\(-0.152227\pi\)
							−0.550492	+	0.834840i	\(0.685560\pi\)
\(43\)	−9.07493	+	5.23941i	−1.38391	+	0.799003i	−0.992620	−	0.121263i	\(-0.961306\pi\)
							−0.391294	+	0.920266i	\(0.627972\pi\)
\(47\)	−3.43543	+	7.71611i	−0.501109	+	1.12551i	0.469079	+	0.883156i	\(0.344586\pi\)
							−0.970188	+	0.242353i	\(0.922081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.5	yes	240
9.7	even	3	inner	450.2.v.a.169.14	yes	240
25.4	even	10	inner	450.2.v.a.229.14	yes	240
225.79	even	30	inner	450.2.v.a.79.5	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.5	✓	240	225.79	even	30	inner
450.2.v.a.169.14	yes	240	9.7	even	3	inner
450.2.v.a.229.14	yes	240	25.4	even	10	inner
450.2.v.a.319.5	yes	240	1.1	even	1	trivial