Embedded newform 450.2.q.c.331.14

• Label: 450.2.q.c.331.14
• 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			331.14
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.c.121.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(1.61999 + 0.612891i) q^{3} +(0.913545 + 0.406737i) q^{4} +(2.21715 - 0.290246i) q^{5} +(1.45716 + 0.936312i) q^{6} +(0.132744 - 0.229920i) q^{7} +(0.809017 + 0.587785i) q^{8} +(2.24873 + 1.98575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

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
							\(3\)	1.61999	+	0.612891i	0.935301	+	0.353853i
\(5\)	2.21715	−	0.290246i	0.991540	−	0.129802i
							\(7\)	0.132744	−	0.229920i	0.0501727	−	0.0869017i	−0.839848	−	0.542821i	\(-0.817356\pi\)
0.890021	+	0.455919i	\(0.150689\pi\)
\(11\)	−3.86720	−	0.822000i	−1.16601	−	0.247842i	−0.416057	−	0.909338i	\(-0.636588\pi\)
							−0.749949	+	0.661496i	\(0.769922\pi\)
\(13\)	−3.18491	+	0.676973i	−0.883334	+	0.187758i	−0.627177	−	0.778877i	\(-0.715790\pi\)
							−0.256157	+	0.966635i	\(0.582456\pi\)
\(17\)	−3.78818	−	2.75227i	−0.918768	−	0.667524i	0.0244492	−	0.999701i	\(-0.492217\pi\)
							−0.943217	+	0.332177i	\(0.892217\pi\)
\(19\)	−4.92923	−	3.58129i	−1.13084	−	0.821605i	−0.145026	−	0.989428i	\(-0.546326\pi\)
							−0.985817	+	0.167823i	\(0.946326\pi\)
\(23\)	2.53657	−	2.81715i	0.528911	−	0.587415i	−0.418186	−	0.908361i	\(-0.637334\pi\)
							0.947098	+	0.320946i	\(0.104001\pi\)
\(29\)	0.628719	+	5.98186i	0.116750	+	1.11080i	0.883361	+	0.468693i	\(0.155275\pi\)
							−0.766611	+	0.642112i	\(0.778059\pi\)
\(31\)	0.253042	−	2.40753i	0.0454477	−	0.432406i	−0.948013	−	0.318232i	\(-0.896911\pi\)
							0.993461	−	0.114174i	\(-0.0364223\pi\)
\(37\)	−2.82125	+	8.68290i	−0.463810	+	1.42746i	0.396663	+	0.917964i	\(0.370168\pi\)
							−0.860473	+	0.509496i	\(0.829832\pi\)
\(41\)	3.58064	−	0.761089i	0.559203	−	0.118862i	0.0803647	−	0.996766i	\(-0.474391\pi\)
							0.478838	+	0.877903i	\(0.341058\pi\)
\(43\)	0.506130	−	0.876643i	0.0771841	−	0.133687i	−0.824850	−	0.565352i	\(-0.808740\pi\)
							0.902034	+	0.431665i	\(0.142074\pi\)
\(47\)	−0.0870389	−	0.828120i	−0.0126959	−	0.120794i	0.986338	−	0.164732i	\(-0.0526761\pi\)
							−0.999034	+	0.0439387i	\(0.986009\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.331.14	yes	120
9.4	even	3	inner	450.2.q.c.31.8	✓	120
25.21	even	5	inner	450.2.q.c.421.8	yes	120
225.121	even	15	inner	450.2.q.c.121.14	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.8	✓	120	9.4	even	3	inner
450.2.q.c.121.14	yes	120	225.121	even	15	inner
450.2.q.c.331.14	yes	120	1.1	even	1	trivial
450.2.q.c.421.8	yes	120	25.21	even	5	inner