Embedded newform 450.2.v.a.79.9

• Label: 450.2.v.a.79.9
• 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			79.9
Character	\(\chi\)	\(=\)	450.79
Dual form			450.2.v.a.319.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 + 0.669131i) q^{2} +(0.547614 + 1.64320i) q^{3} +(0.104528 - 0.994522i) q^{4} +(-0.449052 + 2.19051i) q^{5} +(-1.50647 - 0.854713i) q^{6} +(2.70306 + 1.56061i) q^{7} +(0.587785 + 0.809017i) q^{8} +(-2.40024 + 1.79968i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	0.547614	+	1.64320i	0.316165	+	0.948704i
\(5\)	−0.449052	+	2.19051i	−0.200822	+	0.979628i
							\(7\)	2.70306	+	1.56061i	1.02166	+	0.589856i	0.914585	−	0.404394i	\(-0.132518\pi\)
0.107076	+	0.994251i	\(0.465851\pi\)
\(11\)	−1.59867	−	1.77550i	−0.482016	−	0.535333i	0.452259	−	0.891887i	\(-0.350618\pi\)
							−0.934275	+	0.356554i	\(0.883952\pi\)
\(13\)	3.97014	+	3.57473i	1.10112	+	0.991452i	0.999994	−	0.00342162i	\(-0.00108914\pi\)
							0.101125	+	0.994874i	\(0.467756\pi\)
\(17\)	−1.15545	−	1.59034i	−0.280238	−	0.385715i	0.645574	−	0.763697i	\(-0.276618\pi\)
							−0.925813	+	0.377982i	\(0.876618\pi\)
\(19\)	0.947812	−	0.688626i	0.217443	−	0.157982i	−0.473732	−	0.880669i	\(-0.657093\pi\)
							0.691175	+	0.722688i	\(0.257093\pi\)
\(23\)	−0.199277	−	0.937524i	−0.0415521	−	0.195487i	0.952476	−	0.304615i	\(-0.0985277\pi\)
							−0.994028	+	0.109127i	\(0.965194\pi\)
\(29\)	−4.42714	+	1.97109i	−0.822100	+	0.366023i	−0.774288	−	0.632834i	\(-0.781892\pi\)
							−0.0478124	+	0.998856i	\(0.515225\pi\)
\(31\)	6.48092	+	2.88549i	1.16401	+	0.518250i	0.895515	−	0.445032i	\(-0.146808\pi\)
							0.268493	+	0.963282i	\(0.413474\pi\)
\(37\)	−2.31523	+	0.752265i	−0.380622	+	0.123672i	−0.493078	−	0.869985i	\(-0.664128\pi\)
							0.112456	+	0.993657i	\(0.464128\pi\)
\(41\)	4.00861	−	4.45201i	0.626039	−	0.695287i	−0.343797	−	0.939044i	\(-0.611713\pi\)
							0.969836	+	0.243757i	\(0.0783799\pi\)
\(43\)	−10.7429	−	6.20243i	−1.63828	−	0.945861i	−0.981426	−	0.191843i	\(-0.938553\pi\)
							−0.656854	−	0.754018i	\(-0.728113\pi\)
\(47\)	2.99627	+	6.72973i	0.437051	+	0.981632i	0.989022	+	0.147769i	\(0.0472091\pi\)
							−0.551971	+	0.833863i	\(0.686124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.79.9	✓	240
9.4	even	3	inner	450.2.v.a.229.1	yes	240
25.19	even	10	inner	450.2.v.a.169.1	yes	240
225.94	even	30	inner	450.2.v.a.319.9	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.9	✓	240	1.1	even	1	trivial
450.2.v.a.169.1	yes	240	25.19	even	10	inner
450.2.v.a.229.1	yes	240	9.4	even	3	inner
450.2.v.a.319.9	yes	240	225.94	even	30	inner