Embedded newform 712.2.bf.c.171.6

• Label: 712.2.bf.c.171.6
• Level: $712$
• Weight: $2$
• Character: 712.171
• Analytic conductor: $5.685$
• Analytic rank: $0$
• Dimension: $3440$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	712.bf (of order \(88\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.68534862392\)
Analytic rank:	\(0\)
Dimension:	\(3440\)
Relative dimension:	\(86\) over \(\Q(\zeta_{88})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{88}]$

Embedding invariants

Embedding label			171.6
Character	\(\chi\)	\(=\)	712.171
Dual form			712.2.bf.c.483.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38774 - 0.272381i) q^{2} +(-0.703764 - 2.75732i) q^{3} +(1.85162 + 0.755984i) q^{4} +(-3.43459 + 0.747149i) q^{5} +(0.225598 + 4.01812i) q^{6} +(-0.125289 + 0.694431i) q^{7} +(-2.36364 - 1.55345i) q^{8} +(-4.47450 + 2.44326i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3440 q - 36 q^{2} - 88 q^{3} - 20 q^{4} - 32 q^{6} - 36 q^{8} - 88 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(357\)	\(535\)	\(537\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{37}{88}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38774	−	0.272381i	−0.981277	−	0.192602i
							\(3\)	−0.703764	−	2.75732i	−0.406319	−	1.59194i	−0.755673	−	0.654949i	\(-0.772690\pi\)
0.349354	−	0.936991i	\(-0.386401\pi\)
\(5\)	−3.43459	+	0.747149i	−1.53599	+	0.334135i	−0.899422	−	0.437081i	\(-0.856012\pi\)
							−0.636573	+	0.771217i	\(0.719649\pi\)
\(7\)	−0.125289	+	0.694431i	−0.0473549	+	0.262470i	−0.998999	−	0.0447322i	\(-0.985757\pi\)
							0.951644	+	0.307203i	\(0.0993929\pi\)
\(11\)	1.78943	+	2.78440i	0.539533	+	0.839529i	0.998810	−	0.0487634i	\(-0.0155280\pi\)
							−0.459278	+	0.888293i	\(0.651892\pi\)
\(13\)	−4.92665	−	2.92312i	−1.36641	−	0.810726i	−0.373274	−	0.927721i	\(-0.621765\pi\)
							−0.993133	+	0.116995i	\(0.962674\pi\)
\(17\)	−0.135618	+	1.89619i	−0.0328922	+	0.459893i	0.954139	+	0.299364i	\(0.0967745\pi\)
							−0.987031	+	0.160529i	\(0.948680\pi\)
\(19\)	3.37999	−	4.19431i	0.775423	−	0.962241i	−0.224502	−	0.974474i	\(-0.572076\pi\)
							0.999925	+	0.0122324i	\(0.00389378\pi\)
\(23\)	0.523747	+	4.87156i	0.109209	+	1.01579i	0.909365	+	0.415999i	\(0.136568\pi\)
							−0.800156	+	0.599791i	\(0.795250\pi\)
\(29\)	3.75162	+	5.40337i	0.696659	+	1.00338i	0.998744	+	0.0500991i	\(0.0159537\pi\)
							−0.302085	+	0.953281i	\(0.597683\pi\)
\(31\)	−0.00905238	+	0.0841994i	−0.00162585	+	0.0151227i	−0.995054	−	0.0993307i	\(-0.968330\pi\)
							0.993429	+	0.114453i	\(0.0365116\pi\)
\(37\)	5.94107	+	2.46087i	0.976705	+	0.404565i	0.813204	−	0.581979i	\(-0.197721\pi\)
							0.163501	+	0.986543i	\(0.447721\pi\)
\(41\)	2.26735	−	1.34528i	0.354101	−	0.210098i	−0.322289	−	0.946641i	\(-0.604452\pi\)
							0.676390	+	0.736543i	\(0.263543\pi\)
\(43\)	−7.99983	−	5.55438i	−1.21996	−	0.847035i	−0.228012	−	0.973658i	\(-0.573223\pi\)
							−0.991951	+	0.126623i	\(0.959586\pi\)
\(47\)	−2.35810	−	1.76525i	−0.343965	−	0.257489i	0.413303	−	0.910594i	\(-0.364375\pi\)
							−0.757267	+	0.653105i	\(0.773466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.bf.c.171.6	✓	3440
8.3	odd	2	inner	712.2.bf.c.171.53	yes	3440
89.38	odd	88	inner	712.2.bf.c.483.53	yes	3440
712.483	even	88	inner	712.2.bf.c.483.6	yes	3440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.bf.c.171.6	✓	3440	1.1	even	1	trivial
712.2.bf.c.171.53	yes	3440	8.3	odd	2	inner
712.2.bf.c.483.6	yes	3440	712.483	even	88	inner
712.2.bf.c.483.53	yes	3440	89.38	odd	88	inner