Embedded newform 720.2.ce.a.229.8

• Label: 720.2.ce.a.229.8
• Level: $720$
• Weight: $2$
• Character: 720.229
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.ce (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(140\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			229.8
Character	\(\chi\)	\(=\)	720.229
Dual form			720.2.ce.a.349.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38957 + 0.262858i) q^{2} +(1.68223 + 0.412450i) q^{3} +(1.86181 - 0.730521i) q^{4} +(0.976053 + 2.01180i) q^{5} +(-2.44599 - 0.130941i) q^{6} +(1.36034 + 2.35618i) q^{7} +(-2.39509 + 1.50450i) q^{8} +(2.65977 + 1.38767i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 4 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\)	\(181\)	\(271\)	\(577\)	\(641\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38957	+	0.262858i	−0.982575	+	0.185869i
							\(3\)	1.68223	+	0.412450i	0.971234	+	0.238128i
\(5\)	0.976053	+	2.01180i	0.436504	+	0.899702i
							\(7\)	1.36034	+	2.35618i	0.514160	+	0.890552i	0.999865	+	0.0164289i	\(0.00522972\pi\)
−0.485705	+	0.874123i	\(0.661437\pi\)
\(11\)	0.749134	−	2.79581i	0.225873	−	0.842968i	−0.756181	−	0.654363i	\(-0.772937\pi\)
							0.982053	−	0.188605i	\(-0.0603965\pi\)
\(13\)	5.97695	−	1.60152i	1.65771	−	0.444182i	0.695952	−	0.718088i	\(-0.254982\pi\)
							0.961756	+	0.273906i	\(0.0883158\pi\)
\(17\)		−	5.74843i		−	1.39420i	−0.716974	−	0.697100i	\(-0.754473\pi\)
							0.716974	−	0.697100i	\(-0.245527\pi\)
\(19\)	−2.54181	+	2.54181i	−0.583130	+	0.583130i	−0.935762	−	0.352632i	\(-0.885287\pi\)
							0.352632	+	0.935762i	\(0.385287\pi\)
\(23\)	−1.99200	+	3.45024i	−0.415361	+	0.719426i	−0.995466	−	0.0951157i	\(-0.969678\pi\)
							0.580106	+	0.814541i	\(0.303011\pi\)
\(29\)	−4.90315	−	1.31380i	−0.910493	−	0.243966i	−0.226976	−	0.973900i	\(-0.572884\pi\)
							−0.683517	+	0.729935i	\(0.739550\pi\)
\(31\)	5.14941	−	8.91904i	0.924861	−	1.60191i	0.133077	−	0.991106i	\(-0.457514\pi\)
							0.791784	−	0.610801i	\(-0.209152\pi\)
\(37\)	−5.57420	+	5.57420i	−0.916393	+	0.916393i	−0.996765	−	0.0803722i	\(-0.974389\pi\)
							0.0803722	+	0.996765i	\(0.474389\pi\)
\(41\)	−2.67237	−	1.54289i	−0.417354	−	0.240960i	0.276590	−	0.960988i	\(-0.410795\pi\)
							−0.693945	+	0.720028i	\(0.744129\pi\)
\(43\)	−2.60764	−	0.698716i	−0.397662	−	0.106553i	0.0544453	−	0.998517i	\(-0.482661\pi\)
							−0.452107	+	0.891964i	\(0.649328\pi\)
\(47\)	−5.60502	+	3.23606i	−0.817577	+	0.472028i	−0.849580	−	0.527460i	\(-0.823145\pi\)
							0.0320033	+	0.999488i	\(0.489811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.ce.a.229.8	✓	560
5.4	even	2	inner	720.2.ce.a.229.133	yes	560
9.7	even	3	inner	720.2.ce.a.709.102	yes	560
16.13	even	4	inner	720.2.ce.a.589.39	yes	560
45.34	even	6	inner	720.2.ce.a.709.39	yes	560
80.29	even	4	inner	720.2.ce.a.589.102	yes	560
144.61	even	12	inner	720.2.ce.a.349.133	yes	560
720.349	even	12	inner	720.2.ce.a.349.8	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.ce.a.229.8	✓	560	1.1	even	1	trivial
720.2.ce.a.229.133	yes	560	5.4	even	2	inner
720.2.ce.a.349.8	yes	560	720.349	even	12	inner
720.2.ce.a.349.133	yes	560	144.61	even	12	inner
720.2.ce.a.589.39	yes	560	16.13	even	4	inner
720.2.ce.a.589.102	yes	560	80.29	even	4	inner
720.2.ce.a.709.39	yes	560	45.34	even	6	inner
720.2.ce.a.709.102	yes	560	9.7	even	3	inner