Embedded newform 450.2.q.c.331.10

• Label: 450.2.q.c.331.10
• 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			331.10
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.c.121.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(0.714022 + 1.57803i) q^{3} +(0.913545 + 0.406737i) q^{4} +(0.496594 - 2.18023i) q^{5} +(0.370328 + 1.69200i) q^{6} +(-2.25745 + 3.91002i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-1.98035 + 2.25349i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

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
							\(3\)	0.714022	+	1.57803i	0.412241	+	0.911075i
\(5\)	0.496594	−	2.18023i	0.222083	−	0.975028i
							\(7\)	−2.25745	+	3.91002i	−0.853237	+	1.47785i	0.0250340	+	0.999687i	\(0.492031\pi\)
−0.878271	+	0.478163i	\(0.841303\pi\)
\(11\)	1.82847	+	0.388653i	0.551304	+	0.117183i	0.475137	−	0.879912i	\(-0.342398\pi\)
							0.0761670	+	0.997095i	\(0.475732\pi\)
\(13\)	−1.12744	+	0.239646i	−0.312697	+	0.0664658i	−0.361587	−	0.932338i	\(-0.617765\pi\)
							0.0488901	+	0.998804i	\(0.484432\pi\)
\(17\)	4.68957	+	3.40717i	1.13739	+	0.826360i	0.986753	−	0.162230i	\(-0.0518686\pi\)
							0.150634	+	0.988590i	\(0.451869\pi\)
\(19\)	3.10643	+	2.25696i	0.712665	+	0.517781i	0.884032	−	0.467426i	\(-0.154819\pi\)
							−0.171367	+	0.985207i	\(0.554819\pi\)
\(23\)	3.21709	−	3.57294i	0.670810	−	0.745010i	−0.307638	−	0.951503i	\(-0.599539\pi\)
							0.978448	+	0.206494i	\(0.0662053\pi\)
\(29\)	−0.778646	−	7.40832i	−0.144591	−	1.37569i	−0.790587	−	0.612350i	\(-0.790224\pi\)
							0.645996	−	0.763341i	\(-0.276442\pi\)
\(31\)	1.05138	−	10.0032i	0.188833	−	1.79662i	−0.332329	−	0.943164i	\(-0.607834\pi\)
							0.521161	−	0.853458i	\(-0.325499\pi\)
\(37\)	0.200386	−	0.616724i	0.0329432	−	0.101389i	−0.933233	−	0.359272i	\(-0.883025\pi\)
							0.966176	+	0.257883i	\(0.0830250\pi\)
\(41\)	−7.88767	+	1.67658i	−1.23185	+	0.261837i	−0.777440	−	0.628957i	\(-0.783482\pi\)
							−0.454407	+	0.890794i	\(0.650149\pi\)
\(43\)	3.87371	−	6.70947i	0.590736	−	1.02318i	−0.403398	−	0.915025i	\(-0.632171\pi\)
							0.994134	−	0.108159i	\(-0.0344957\pi\)
\(47\)	0.159235	+	1.51502i	0.0232269	+	0.220989i	0.999978	+	0.00661314i	\(0.00210504\pi\)
							−0.976751	+	0.214376i	\(0.931228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.331.10	yes	120
9.4	even	3	inner	450.2.q.c.31.4	✓	120
25.21	even	5	inner	450.2.q.c.421.4	yes	120
225.121	even	15	inner	450.2.q.c.121.10	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.4	✓	120	9.4	even	3	inner
450.2.q.c.121.10	yes	120	225.121	even	15	inner
450.2.q.c.331.10	yes	120	1.1	even	1	trivial
450.2.q.c.421.4	yes	120	25.21	even	5	inner