Embedded newform 450.2.q.b.331.6

• Label: 450.2.q.b.331.6
• 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			331.6
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.b.121.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978148 - 0.207912i) q^{2} +(-0.155760 + 1.72503i) q^{3} +(0.913545 + 0.406737i) q^{4} +(-0.0873632 - 2.23436i) q^{5} +(0.511011 - 1.65495i) q^{6} +(1.26481 - 2.19072i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(-2.95148 - 0.537382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{2} + 2 q^{3} + 14 q^{4} + 9 q^{5} - 7 q^{6} - 8 q^{7} - 28 q^{8} + 8 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.155760	+	1.72503i	−0.0899280	+	0.995948i
\(5\)	−0.0873632	−	2.23436i	−0.0390700	−	0.999236i
							\(7\)	1.26481	−	2.19072i	0.478055	−	0.828015i	−0.521629	−	0.853173i	\(-0.674675\pi\)
0.999683	+	0.0251576i	\(0.00800876\pi\)
\(11\)	−3.51018	−	0.746111i	−1.05836	−	0.224961i	−0.354321	−	0.935124i	\(-0.615288\pi\)
							−0.704037	+	0.710163i	\(0.748621\pi\)
\(13\)	−4.89605	+	1.04069i	−1.35792	+	0.288635i	−0.828625	−	0.559804i	\(-0.810876\pi\)
							−0.529294	+	0.848439i	\(0.677543\pi\)
\(17\)	−3.54324	−	2.57432i	−0.859363	−	0.624363i	0.0683489	−	0.997661i	\(-0.478227\pi\)
							−0.927711	+	0.373298i	\(0.878227\pi\)
\(19\)	2.12573	+	1.54443i	0.487675	+	0.354317i	0.804290	−	0.594238i	\(-0.202546\pi\)
							−0.316614	+	0.948554i	\(0.602546\pi\)
\(23\)	−5.56561	+	6.18124i	−1.16051	+	1.28888i	−0.210169	+	0.977665i	\(0.567402\pi\)
							−0.950341	+	0.311212i	\(0.899265\pi\)
\(29\)	−0.969782	−	9.22686i	−0.180084	−	1.71338i	−0.595168	−	0.803601i	\(-0.702914\pi\)
							0.415084	−	0.909783i	\(-0.363752\pi\)
\(31\)	0.355197	−	3.37947i	0.0637952	−	0.606971i	−0.915188	−	0.403028i	\(-0.867958\pi\)
							0.978983	−	0.203943i	\(-0.0653757\pi\)
\(37\)	−1.69327	+	5.21136i	−0.278373	+	0.856743i	0.709935	+	0.704268i	\(0.248724\pi\)
							−0.988307	+	0.152475i	\(0.951276\pi\)
\(41\)	9.70817	−	2.06354i	1.51616	−	0.322270i	0.626693	−	0.779266i	\(-0.284408\pi\)
							0.889469	+	0.456996i	\(0.151075\pi\)
\(43\)	4.13303	−	7.15862i	0.630282	−	1.09168i	−0.357212	−	0.934023i	\(-0.616273\pi\)
							0.987494	−	0.157657i	\(-0.0503939\pi\)
\(47\)	−0.625996	−	5.95595i	−0.0913109	−	0.868765i	−0.940298	−	0.340352i	\(-0.889454\pi\)
							0.848987	−	0.528413i	\(-0.177213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.b.331.6	yes	112
9.4	even	3	inner	450.2.q.b.31.1	✓	112
25.21	even	5	inner	450.2.q.b.421.1	yes	112
225.121	even	15	inner	450.2.q.b.121.6	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.b.31.1	✓	112	9.4	even	3	inner
450.2.q.b.121.6	yes	112	225.121	even	15	inner
450.2.q.b.331.6	yes	112	1.1	even	1	trivial
450.2.q.b.421.1	yes	112	25.21	even	5	inner