Embedded newform 450.2.q.c.331.12

• Label: 450.2.q.c.331.12
• Level: $450$
• Weight: $2$
• Character: 450.331
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			331.12
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.c.121.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(1.25401 - 1.19477i) q^{3} +(0.913545 + 0.406737i) q^{4} +(0.968741 + 2.01533i) q^{5} +(1.47501 - 0.907935i) q^{6} +(-1.10262 + 1.90980i) q^{7} +(0.809017 + 0.587785i) q^{8} +(0.145066 - 2.99649i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.978148	+	0.207912i	0.691655	+	0.147016i
							\(3\)	1.25401	−	1.19477i	0.724001	−	0.689799i
\(5\)	0.968741	+	2.01533i	0.433234	+	0.901281i
							\(7\)	−1.10262	+	1.90980i	−0.416753	+	0.721837i	−0.995611	−	0.0935917i	\(-0.970165\pi\)
0.578858	+	0.815428i	\(0.303499\pi\)
\(11\)	1.53568	+	0.326418i	0.463024	+	0.0984188i	0.433516	−	0.901146i	\(-0.357273\pi\)
							0.0295078	+	0.999565i	\(0.490606\pi\)
\(13\)	0.900186	−	0.191340i	0.249667	−	0.0530683i	−0.0813782	−	0.996683i	\(-0.525932\pi\)
							0.331045	+	0.943615i	\(0.392599\pi\)
\(17\)	−0.915421	−	0.665093i	−0.222022	−	0.161309i	0.471214	−	0.882019i	\(-0.343816\pi\)
							−0.693237	+	0.720710i	\(0.743816\pi\)
\(19\)	−1.51724	−	1.10234i	−0.348079	−	0.252894i	0.399984	−	0.916522i	\(-0.369016\pi\)
							−0.748063	+	0.663628i	\(0.769016\pi\)
\(23\)	−0.697809	+	0.774995i	−0.145503	+	0.161598i	−0.811491	−	0.584365i	\(-0.801344\pi\)
							0.665988	+	0.745963i	\(0.268010\pi\)
\(29\)	−0.875522	−	8.33004i	−0.162580	−	1.54685i	−0.706501	−	0.707712i	\(-0.749727\pi\)
							0.543921	−	0.839137i	\(-0.316939\pi\)
\(31\)	0.485504	−	4.61926i	0.0871991	−	0.829644i	−0.860279	−	0.509824i	\(-0.829711\pi\)
							0.947478	−	0.319821i	\(-0.103623\pi\)
\(37\)	−0.828913	+	2.55113i	−0.136272	+	0.419403i	−0.995786	−	0.0917100i	\(-0.970767\pi\)
							0.859513	+	0.511113i	\(0.170767\pi\)
\(41\)	−3.61015	+	0.767361i	−0.563811	+	0.119842i	−0.480995	−	0.876723i	\(-0.659725\pi\)
							−0.0828156	+	0.996565i	\(0.526391\pi\)
\(43\)	−1.39116	+	2.40956i	−0.212150	+	0.367454i	−0.952387	−	0.304892i	\(-0.901380\pi\)
							0.740237	+	0.672345i	\(0.234713\pi\)
\(47\)	−0.883690	−	8.40775i	−0.128899	−	1.22640i	−0.847432	−	0.530903i	\(-0.821853\pi\)
							0.718533	−	0.695493i	\(-0.244814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.331.12	yes	120
9.4	even	3	inner	450.2.q.c.31.13	✓	120
25.21	even	5	inner	450.2.q.c.421.13	yes	120
225.121	even	15	inner	450.2.q.c.121.12	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.13	✓	120	9.4	even	3	inner
450.2.q.c.121.12	yes	120	225.121	even	15	inner
450.2.q.c.331.12	yes	120	1.1	even	1	trivial
450.2.q.c.421.13	yes	120	25.21	even	5	inner