Embedded newform 450.2.v.a.319.9

• Label: 450.2.v.a.319.9
• 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			319.9
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.9

$q$-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 \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.9	✓	240	225.79	even	30	inner
450.2.v.a.169.1	yes	240	9.7	even	3	inner
450.2.v.a.229.1	yes	240	25.4	even	10	inner
450.2.v.a.319.9	yes	240	1.1	even	1	trivial