Embedded newform 450.2.q.c.331.9

• Label: 450.2.q.c.331.9
• 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			331.9
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.c.121.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(0.232836 - 1.71633i) q^{3} +(0.913545 + 0.406737i) q^{4} +(-1.75460 + 1.38614i) q^{5} +(0.584593 - 1.63041i) q^{6} +(2.29291 - 3.97143i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-2.89158 - 0.799246i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

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
							\(3\)	0.232836	−	1.71633i	0.134428	−	0.990923i
\(5\)	−1.75460	+	1.38614i	−0.784681	+	0.619900i
							\(7\)	2.29291	−	3.97143i	0.866637	−	1.50106i	0.00122405	−	0.999999i	\(-0.499610\pi\)
0.865413	−	0.501060i	\(-0.167056\pi\)
\(11\)	−0.858490	−	0.182478i	−0.258845	−	0.0550191i	0.0766614	−	0.997057i	\(-0.475574\pi\)
							−0.335506	+	0.942038i	\(0.608907\pi\)
\(13\)	6.02900	−	1.28150i	1.67214	−	0.355425i	0.728158	−	0.685409i	\(-0.240377\pi\)
							0.943986	+	0.329984i	\(0.107043\pi\)
\(17\)	−5.31050	−	3.85831i	−1.28799	−	0.935777i	−0.288224	−	0.957563i	\(-0.593065\pi\)
							−0.999763	+	0.0217860i	\(0.993065\pi\)
\(19\)	1.65462	+	1.20215i	0.379596	+	0.275792i	0.761179	−	0.648542i	\(-0.224621\pi\)
							−0.381583	+	0.924335i	\(0.624621\pi\)
\(23\)	3.52151	−	3.91103i	0.734285	−	0.815506i	−0.254148	−	0.967165i	\(-0.581795\pi\)
							0.988433	+	0.151659i	\(0.0484617\pi\)
\(29\)	0.426332	+	4.05627i	0.0791678	+	0.753231i	0.960037	+	0.279872i	\(0.0902921\pi\)
							−0.880869	+	0.473359i	\(0.843041\pi\)
\(31\)	−0.871688	+	8.29355i	−0.156560	+	1.48957i	0.580789	+	0.814054i	\(0.302744\pi\)
							−0.737349	+	0.675512i	\(0.763923\pi\)
\(37\)	−1.37787	+	4.24064i	−0.226520	+	0.697156i	0.771614	+	0.636091i	\(0.219450\pi\)
							−0.998134	+	0.0610651i	\(0.980550\pi\)
\(41\)	−2.49626	+	0.530596i	−0.389850	+	0.0828651i	−0.398666	−	0.917096i	\(-0.630527\pi\)
							0.00881666	+	0.999961i	\(0.497194\pi\)
\(43\)	−0.217476	+	0.376679i	−0.0331647	+	0.0574430i	−0.882131	−	0.471003i	\(-0.843892\pi\)
							0.848967	+	0.528446i	\(0.177225\pi\)
\(47\)	0.638273	+	6.07277i	0.0931017	+	0.885804i	0.937007	+	0.349311i	\(0.113584\pi\)
							−0.843905	+	0.536493i	\(0.819749\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.331.9	yes	120
9.4	even	3	inner	450.2.q.c.31.15	✓	120
25.21	even	5	inner	450.2.q.c.421.15	yes	120
225.121	even	15	inner	450.2.q.c.121.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.15	✓	120	9.4	even	3	inner
450.2.q.c.121.9	yes	120	225.121	even	15	inner
450.2.q.c.331.9	yes	120	1.1	even	1	trivial
450.2.q.c.421.15	yes	120	25.21	even	5	inner