Embedded newform 450.2.q.b.331.10

• Label: 450.2.q.b.331.10
• Level: $450$
• Weight: $2$
• Character: 450.331
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $112$
• 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:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			331.10
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.b.121.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978148 - 0.207912i) q^{2} +(1.14723 - 1.29764i) q^{3} +(0.913545 + 0.406737i) q^{4} +(-2.11027 + 0.739425i) q^{5} +(-1.39195 + 1.03076i) q^{6} +(-2.09520 + 3.62899i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(-0.367749 - 2.97737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{2} + 2 q^{3} + 14 q^{4} + 9 q^{5} - 7 q^{6} - 8 q^{7} - 28 q^{8} + 8 q^{9}+O(q^{10}) \)

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
							\(3\)	1.14723	−	1.29764i	0.662351	−	0.749194i
\(5\)	−2.11027	+	0.739425i	−0.943743	+	0.330681i
							\(7\)	−2.09520	+	3.62899i	−0.791910	+	1.37163i	0.132873	+	0.991133i	\(0.457580\pi\)
−0.924783	+	0.380495i	\(0.875754\pi\)
\(11\)	2.20282	+	0.468225i	0.664176	+	0.141175i	0.527651	−	0.849462i	\(-0.323073\pi\)
							0.136526	+	0.990637i	\(0.456406\pi\)
\(13\)	5.45074	−	1.15859i	1.51176	−	0.321335i	0.623923	−	0.781486i	\(-0.285538\pi\)
							0.887841	+	0.460151i	\(0.152205\pi\)
\(17\)	2.94043	+	2.13635i	0.713158	+	0.518140i	0.884191	−	0.467125i	\(-0.154710\pi\)
							−0.171033	+	0.985265i	\(0.554710\pi\)
\(19\)	6.87589	+	4.99563i	1.57744	+	1.14608i	0.919559	+	0.392952i	\(0.128546\pi\)
							0.657879	+	0.753124i	\(0.271454\pi\)
\(23\)	−0.275933	+	0.306455i	−0.0575361	+	0.0639003i	−0.771226	−	0.636562i	\(-0.780356\pi\)
							0.713690	+	0.700462i	\(0.247023\pi\)
\(29\)	−0.639838	−	6.08766i	−0.118815	−	1.13045i	−0.877695	−	0.479219i	\(-0.840920\pi\)
							0.758880	−	0.651230i	\(-0.225747\pi\)
\(31\)	0.0164066	−	0.156099i	0.00294672	−	0.0280362i	−0.992950	−	0.118536i	\(-0.962180\pi\)
							0.995896	+	0.0905000i	\(0.0288465\pi\)
\(37\)	−3.36649	+	10.3610i	−0.553448	+	1.70334i	0.146559	+	0.989202i	\(0.453180\pi\)
							−0.700007	+	0.714136i	\(0.746820\pi\)
\(41\)	−0.114671	+	0.0243741i	−0.0179086	+	0.00380659i	−0.216857	−	0.976203i	\(-0.569581\pi\)
							0.198949	+	0.980010i	\(0.436247\pi\)
\(43\)	−2.45795	+	4.25730i	−0.374834	+	0.649232i	−0.990302	−	0.138930i	\(-0.955634\pi\)
							0.615468	+	0.788162i	\(0.288967\pi\)
\(47\)	0.521484	+	4.96159i	0.0760663	+	0.723723i	0.964386	+	0.264498i	\(0.0852062\pi\)
							−0.888320	+	0.459225i	\(0.848127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.b.331.10	yes	112
9.4	even	3	inner	450.2.q.b.31.13	✓	112
25.21	even	5	inner	450.2.q.b.421.13	yes	112
225.121	even	15	inner	450.2.q.b.121.10	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.b.31.13	✓	112	9.4	even	3	inner
450.2.q.b.121.10	yes	112	225.121	even	15	inner
450.2.q.b.331.10	yes	112	1.1	even	1	trivial
450.2.q.b.421.13	yes	112	25.21	even	5	inner