Embedded newform 450.2.q.b.331.4

• Label: 450.2.q.b.331.4
• 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			331.4
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.b.121.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978148 - 0.207912i) q^{2} +(-1.14937 + 1.29574i) q^{3} +(0.913545 + 0.406737i) q^{4} +(-1.90142 - 1.17669i) q^{5} +(1.39365 - 1.02846i) q^{6} +(-2.23964 + 3.87917i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(-0.357894 - 2.97858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{2} + 2 q^{3} + 14 q^{4} + 9 q^{5} - 7 q^{6} - 8 q^{7} - 28 q^{8} + 8 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.14937	+	1.29574i	−0.663589	+	0.748097i
\(5\)	−1.90142	−	1.17669i	−0.850342	−	0.526231i
							\(7\)	−2.23964	+	3.87917i	−0.846503	+	1.46619i	0.0378056	+	0.999285i	\(0.487963\pi\)
−0.884309	+	0.466902i	\(0.845370\pi\)
\(11\)	4.32847	+	0.920046i	1.30508	+	0.277404i	0.807426	−	0.589969i	\(-0.200860\pi\)
							0.497658	+	0.867373i	\(0.334193\pi\)
\(13\)	−1.76935	+	0.376088i	−0.490731	+	0.104308i	−0.446629	−	0.894719i	\(-0.647376\pi\)
							−0.0441013	+	0.999027i	\(0.514042\pi\)
\(17\)	−4.08538	−	2.96821i	−0.990851	−	0.719896i	−0.0307440	−	0.999527i	\(-0.509788\pi\)
							−0.960107	+	0.279632i	\(0.909788\pi\)
\(19\)	−3.86308	−	2.80669i	−0.886252	−	0.643900i	0.0486463	−	0.998816i	\(-0.484509\pi\)
							−0.934898	+	0.354916i	\(0.884509\pi\)
\(23\)	5.23895	−	5.81845i	1.09240	−	1.21323i	0.116917	−	0.993142i	\(-0.462699\pi\)
							0.975480	−	0.220089i	\(-0.0706346\pi\)
\(29\)	−0.130243	−	1.23918i	−0.0241855	−	0.230109i	−0.999937	−	0.0112280i	\(-0.996426\pi\)
							0.975752	−	0.218881i	\(-0.0702407\pi\)
\(31\)	0.309573	−	2.94539i	0.0556009	−	0.529007i	−0.930903	−	0.365267i	\(-0.880978\pi\)
							0.986504	−	0.163740i	\(-0.0523557\pi\)
\(37\)	3.22341	−	9.92063i	0.529925	−	1.63094i	−0.224441	−	0.974488i	\(-0.572056\pi\)
							0.754366	−	0.656454i	\(-0.227944\pi\)
\(41\)	−3.40749	+	0.724285i	−0.532161	+	0.113114i	−0.466152	−	0.884705i	\(-0.654360\pi\)
							−0.0660091	+	0.997819i	\(0.521027\pi\)
\(43\)	−0.851893	+	1.47552i	−0.129912	+	0.225015i	−0.923642	−	0.383255i	\(-0.874803\pi\)
							0.793730	+	0.608270i	\(0.208136\pi\)
\(47\)	−1.09179	−	10.3877i	−0.159254	−	1.51520i	−0.723923	−	0.689881i	\(-0.757663\pi\)
							0.564669	−	0.825317i	\(-0.309004\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.b.331.4	yes	112
9.4	even	3	inner	450.2.q.b.31.2	✓	112
25.21	even	5	inner	450.2.q.b.421.2	yes	112
225.121	even	15	inner	450.2.q.b.121.4	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.b.31.2	✓	112	9.4	even	3	inner
450.2.q.b.121.4	yes	112	225.121	even	15	inner
450.2.q.b.331.4	yes	112	1.1	even	1	trivial
450.2.q.b.421.2	yes	112	25.21	even	5	inner