Embedded newform 712.2.t.a.189.70

• Label: 712.2.t.a.189.70
• 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			189.70
Character	\(\chi\)	\(=\)	712.189
Dual form			712.2.t.a.437.70

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.10192 - 0.886432i) q^{2} +(0.0174622 + 0.121452i) q^{3} +(0.428476 - 1.95356i) q^{4} +(-1.89750 - 2.95256i) q^{5} +(0.126901 + 0.118352i) q^{6} +(-0.950948 - 1.47970i) q^{7} +(-1.25955 - 2.53249i) q^{8} +(2.86403 - 0.840956i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q - 9 q^{2} - 9 q^{4} - 11 q^{6} - 22 q^{7} - 9 q^{8} - 102 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.10192	−	0.886432i	0.779178	−	0.626802i
							\(3\)	0.0174622	+	0.121452i	0.0100818	+	0.0701206i	0.994241	−	0.107165i	\(-0.0341774\pi\)
−0.984159	+	0.177286i	\(0.943268\pi\)
\(5\)	−1.89750	−	2.95256i	−0.848586	−	1.32043i	−0.945663	−	0.325148i	\(-0.894586\pi\)
							0.0970773	−	0.995277i	\(-0.469051\pi\)
\(7\)	−0.950948	−	1.47970i	−0.359425	−	0.559276i	0.613704	−	0.789536i	\(-0.289679\pi\)
							−0.973129	+	0.230260i	\(0.926042\pi\)
\(11\)	−2.42777	+	3.77768i	−0.731999	+	1.13901i	0.253169	+	0.967422i	\(0.418527\pi\)
							−0.985168	+	0.171591i	\(0.945109\pi\)
\(13\)	0.228427	+	1.58874i	0.0633541	+	0.440638i	0.996667	+	0.0815742i	\(0.0259947\pi\)
							−0.933313	+	0.359063i	\(0.883096\pi\)
\(17\)	0.759989	−	0.877074i	0.184324	−	0.212722i	−0.656066	−	0.754704i	\(-0.727781\pi\)
							0.840390	+	0.541982i	\(0.182326\pi\)
\(19\)	−2.75176	+	0.807989i	−0.631296	+	0.185365i	−0.581699	−	0.813404i	\(-0.697612\pi\)
							−0.0495971	+	0.998769i	\(0.515794\pi\)
\(23\)	−2.08459	−	7.09946i	−0.434667	−	1.48034i	−0.827887	−	0.560895i	\(-0.810457\pi\)
							0.393220	−	0.919444i	\(-0.371361\pi\)
\(29\)	2.19621	−	1.41142i	0.407825	−	0.262094i	−0.320605	−	0.947213i	\(-0.603886\pi\)
							0.728431	+	0.685119i	\(0.240250\pi\)
\(31\)	−0.765357	+	2.60657i	−0.137462	+	0.468153i	−0.999234	−	0.0391435i	\(-0.987537\pi\)
							0.861771	+	0.507297i	\(0.169355\pi\)
\(37\)	3.52650			0.579752			0.289876	−	0.957064i	\(-0.406386\pi\)
							0.289876	+	0.957064i	\(0.406386\pi\)
\(41\)	8.81940	+	1.26804i	1.37736	+	0.198034i	0.790900	−	0.611945i	\(-0.209613\pi\)
							0.586458	+	0.809979i	\(0.300522\pi\)
\(43\)	−3.12646	−	2.00925i	−0.476780	−	0.306408i	0.280089	−	0.959974i	\(-0.409636\pi\)
							−0.756870	+	0.653566i	\(0.773272\pi\)
\(47\)	0.717556	−	4.99071i	0.104666	−	0.727970i	−0.868135	−	0.496328i	\(-0.834681\pi\)
							0.972801	−	0.231642i	\(-0.0744097\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.t.a.189.70	yes	880
8.5	even	2	inner	712.2.t.a.189.23	✓	880
89.81	even	22	inner	712.2.t.a.437.23	yes	880
712.437	even	22	inner	712.2.t.a.437.70	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.t.a.189.23	✓	880	8.5	even	2	inner
712.2.t.a.189.70	yes	880	1.1	even	1	trivial
712.2.t.a.437.23	yes	880	89.81	even	22	inner
712.2.t.a.437.70	yes	880	712.437	even	22	inner