Embedded newform 450.2.w.a.23.6

• Label: 450.2.w.a.23.6
• Level: $450$
• Weight: $2$
• Character: 450.23
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(30\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			23.6
Character	\(\chi\)	\(=\)	450.23
Dual form			450.2.w.a.137.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.358368 - 0.933580i) q^{2} +(-0.466852 - 1.66795i) q^{3} +(-0.743145 + 0.669131i) q^{4} +(-0.954188 + 2.02226i) q^{5} +(-1.38986 + 1.03358i) q^{6} +(3.63027 + 0.972729i) q^{7} +(0.891007 + 0.453990i) q^{8} +(-2.56410 + 1.55737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{3}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.358368	−	0.933580i	−0.253404	−	0.660141i
							\(3\)	−0.466852	−	1.66795i	−0.269537	−	0.962990i
\(5\)	−0.954188	+	2.02226i	−0.426726	+	0.904381i
							\(7\)	3.63027	+	0.972729i	1.37211	+	0.367657i	0.868251	−	0.496126i	\(-0.165245\pi\)
0.503864	+	0.863783i	\(0.331911\pi\)
\(11\)	2.50642	−	5.62951i	0.755713	−	1.69736i	0.0397624	−	0.999209i	\(-0.487340\pi\)
							0.715951	−	0.698151i	\(-0.245993\pi\)
\(13\)	0.733072	+	0.281400i	0.203318	+	0.0780463i	0.457896	−	0.889006i	\(-0.348603\pi\)
							−0.254578	+	0.967052i	\(0.581937\pi\)
\(17\)	0.704086	−	1.38185i	0.170766	−	0.335147i	−0.789724	−	0.613463i	\(-0.789776\pi\)
							0.960490	+	0.278315i	\(0.0897761\pi\)
\(19\)	2.89871	−	0.941848i	0.665009	−	0.216075i	0.0429888	−	0.999076i	\(-0.486312\pi\)
							0.622021	+	0.783001i	\(0.286312\pi\)
\(23\)	−0.883449	−	0.715403i	−0.184212	−	0.149172i	0.532779	−	0.846254i	\(-0.321148\pi\)
							−0.716991	+	0.697083i	\(0.754481\pi\)
\(29\)	1.69676	−	0.360658i	0.315081	−	0.0669725i	−0.0476570	−	0.998864i	\(-0.515175\pi\)
							0.362738	+	0.931891i	\(0.381842\pi\)
\(31\)	6.11188	+	1.29912i	1.09773	+	0.233329i	0.720958	−	0.692979i	\(-0.243702\pi\)
							0.376768	+	0.926308i	\(0.377036\pi\)
\(37\)	−1.06111	+	0.168064i	−0.174446	+	0.0276295i	−0.243046	−	0.970015i	\(-0.578147\pi\)
							0.0686004	+	0.997644i	\(0.478147\pi\)
\(41\)	−3.73805	−	8.39579i	−0.583785	−	1.31120i	−0.928094	−	0.372346i	\(-0.878553\pi\)
							0.344309	−	0.938856i	\(-0.388113\pi\)
\(43\)	0.0549914	−	0.205231i	0.00838610	−	0.0312974i	−0.961606	−	0.274434i	\(-0.911510\pi\)
							0.969992	+	0.243136i	\(0.0781763\pi\)
\(47\)	−9.31996	−	6.05245i	−1.35946	−	0.882841i	−0.360698	−	0.932683i	\(-0.617462\pi\)
							−0.998757	+	0.0498418i	\(0.984128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.w.a.23.6	✓	480
9.2	odd	6	inner	450.2.w.a.173.26	yes	480
25.12	odd	20	inner	450.2.w.a.437.26	yes	480
225.137	even	60	inner	450.2.w.a.137.6	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.w.a.23.6	✓	480	1.1	even	1	trivial
450.2.w.a.137.6	yes	480	225.137	even	60	inner
450.2.w.a.173.26	yes	480	9.2	odd	6	inner
450.2.w.a.437.26	yes	480	25.12	odd	20	inner