Embedded newform 712.2.t.a.189.58

• Label: 712.2.t.a.189.58
• Level: $712$
• Weight: $2$
• Character: 712.189
• Analytic conductor: $5.685$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	712.t (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.68534862392\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(88\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			189.58
Character	\(\chi\)	\(=\)	712.189
Dual form			712.2.t.a.437.58

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.664009 + 1.24864i) q^{2} +(-0.463401 - 3.22303i) q^{3} +(-1.11818 + 1.65821i) q^{4} +(0.304395 + 0.473647i) q^{5} +(3.71669 - 2.71874i) q^{6} +(2.18697 + 3.40299i) q^{7} +(-2.81299 - 0.295138i) q^{8} +(-7.29468 + 2.14191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q - 9 q^{2} - 9 q^{4} - 11 q^{6} - 22 q^{7} - 9 q^{8} - 102 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/712\mathbb{Z}\right)^\times\).

\(n\)	\(357\)	\(535\)	\(537\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.664009	+	1.24864i	0.469525	+	0.882919i
							\(3\)	−0.463401	−	3.22303i	−0.267545	−	1.86082i	−0.471541	−	0.881844i	\(-0.656302\pi\)
0.203997	−	0.978972i	\(-0.434607\pi\)
\(5\)	0.304395	+	0.473647i	0.136129	+	0.211821i	0.902625	−	0.430428i	\(-0.141638\pi\)
							−0.766495	+	0.642250i	\(0.778001\pi\)
\(7\)	2.18697	+	3.40299i	0.826598	+	1.28621i	0.955630	+	0.294571i	\(0.0951766\pi\)
							−0.129032	+	0.991640i	\(0.541187\pi\)
\(11\)	−1.48300	+	2.30760i	−0.447142	+	0.695767i	−0.989525	−	0.144365i	\(-0.953886\pi\)
							0.542382	+	0.840132i	\(0.317522\pi\)
\(13\)	0.728718	+	5.06834i	0.202110	+	1.40571i	0.798009	+	0.602646i	\(0.205887\pi\)
							−0.595899	+	0.803059i	\(0.703204\pi\)
\(17\)	3.01006	−	3.47379i	0.730046	−	0.842518i	−0.262431	−	0.964951i	\(-0.584524\pi\)
							0.992477	+	0.122433i	\(0.0390697\pi\)
\(19\)	−7.00838	+	2.05785i	−1.60783	+	0.472102i	−0.957712	−	0.287728i	\(-0.907100\pi\)
							−0.650121	+	0.759831i	\(0.725282\pi\)
\(23\)	1.25079	+	4.25979i	0.260807	+	0.888227i	0.980927	+	0.194377i	\(0.0622686\pi\)
							−0.720120	+	0.693850i	\(0.755913\pi\)
\(29\)	−2.93155	+	1.88399i	−0.544375	+	0.349848i	−0.783748	−	0.621079i	\(-0.786694\pi\)
							0.239373	+	0.970928i	\(0.423058\pi\)
\(31\)	−1.52252	+	5.18522i	−0.273452	+	0.931293i	0.702202	+	0.711978i	\(0.252200\pi\)
							−0.975654	+	0.219315i	\(0.929618\pi\)
\(37\)	10.1529			1.66912			0.834562	−	0.550915i	\(-0.185721\pi\)
							0.834562	+	0.550915i	\(0.185721\pi\)
\(41\)	2.25335	+	0.323982i	0.351913	+	0.0505975i	0.316005	−	0.948758i	\(-0.397658\pi\)
							0.0359085	+	0.999355i	\(0.488568\pi\)
\(43\)	4.19568	+	2.69640i	0.639836	+	0.411198i	0.819940	−	0.572450i	\(-0.194007\pi\)
							−0.180104	+	0.983648i	\(0.557643\pi\)
\(47\)	0.118213	−	0.822190i	0.0172431	−	0.119929i	−0.979382	−	0.202018i	\(-0.935250\pi\)
							0.996625	+	0.0820891i	\(0.0261592\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.t.a.189.58	✓	880
8.5	even	2	inner	712.2.t.a.189.72	yes	880
89.81	even	22	inner	712.2.t.a.437.72	yes	880
712.437	even	22	inner	712.2.t.a.437.58	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.t.a.189.58	✓	880	1.1	even	1	trivial
712.2.t.a.189.72	yes	880	8.5	even	2	inner
712.2.t.a.437.58	yes	880	712.437	even	22	inner
712.2.t.a.437.72	yes	880	89.81	even	22	inner