Embedded newform 712.2.bf.c.171.41

• Label: 712.2.bf.c.171.41
• 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			171.41
Character	\(\chi\)	\(=\)	712.171
Dual form			712.2.bf.c.483.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.126330 + 1.40856i) q^{2} +(-0.474177 - 1.85781i) q^{3} +(-1.96808 - 0.355887i) q^{4} +(1.80462 - 0.392571i) q^{5} +(2.67674 - 0.433210i) q^{6} +(0.264921 - 1.46836i) q^{7} +(0.749917 - 2.72720i) q^{8} +(-0.593566 + 0.324112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3440 q - 36 q^{2} - 88 q^{3} - 20 q^{4} - 32 q^{6} - 36 q^{8} - 88 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.126330	+	1.40856i	−0.0893289	+	0.996002i
							\(3\)	−0.474177	−	1.85781i	−0.273766	−	1.07261i	−0.943864	−	0.330333i	\(-0.892839\pi\)
0.670098	−	0.742272i	\(-0.266252\pi\)
\(5\)	1.80462	−	0.392571i	0.807050	−	0.175563i	0.209931	−	0.977716i	\(-0.432676\pi\)
							0.597118	+	0.802153i	\(0.296312\pi\)
\(7\)	0.264921	−	1.46836i	0.100131	−	0.554987i	−0.894113	−	0.447842i	\(-0.852193\pi\)
							0.994243	−	0.107145i	\(-0.0341709\pi\)
\(11\)	−1.47888	−	2.30119i	−0.445900	−	0.693834i	0.543442	−	0.839447i	\(-0.317121\pi\)
							−0.989342	+	0.145613i	\(0.953485\pi\)
\(13\)	−2.08866	−	1.23926i	−0.579289	−	0.343708i	0.190815	−	0.981626i	\(-0.438887\pi\)
							−0.770104	+	0.637918i	\(0.779796\pi\)
\(17\)	0.00811775	−	0.113501i	0.00196884	−	0.0275280i	−0.996373	−	0.0850964i	\(-0.972880\pi\)
							0.998342	+	0.0575684i	\(0.0183347\pi\)
\(19\)	−4.56140	+	5.66035i	−1.04646	+	1.29857i	−0.0934498	+	0.995624i	\(0.529789\pi\)
							−0.953007	+	0.302949i	\(0.902029\pi\)
\(23\)	−0.431979	−	4.01799i	−0.0900738	−	0.837809i	−0.946753	−	0.321960i	\(-0.895658\pi\)
							0.856679	−	0.515849i	\(-0.172523\pi\)
\(29\)	3.55740	+	5.12363i	0.660592	+	0.951434i	0.999942	+	0.0107735i	\(0.00342936\pi\)
							−0.339350	+	0.940660i	\(0.610207\pi\)
\(31\)	0.886631	−	8.24688i	0.159244	−	1.48118i	−0.583776	−	0.811914i	\(-0.698426\pi\)
							0.743020	−	0.669269i	\(-0.233393\pi\)
\(37\)	−8.77022	−	3.63275i	−1.44182	−	0.597220i	−0.481578	−	0.876403i	\(-0.659936\pi\)
							−0.960238	+	0.279183i	\(0.909936\pi\)
\(41\)	7.85402	−	4.66001i	1.22659	−	0.727771i	0.255577	−	0.966789i	\(-0.417735\pi\)
							0.971015	+	0.239018i	\(0.0768254\pi\)
\(43\)	5.62737	+	3.90716i	0.858167	+	0.595836i	0.915409	−	0.402525i	\(-0.131867\pi\)
							−0.0572426	+	0.998360i	\(0.518231\pi\)
\(47\)	−7.90040	−	5.91416i	−1.15239	−	0.862670i	−0.160162	−	0.987091i	\(-0.551202\pi\)
							−0.992230	+	0.124421i	\(0.960293\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.bf.c.171.41	yes	3440
8.3	odd	2	inner	712.2.bf.c.171.7	✓	3440
89.38	odd	88	inner	712.2.bf.c.483.7	yes	3440
712.483	even	88	inner	712.2.bf.c.483.41	yes	3440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.bf.c.171.7	✓	3440	8.3	odd	2	inner
712.2.bf.c.171.41	yes	3440	1.1	even	1	trivial
712.2.bf.c.483.7	yes	3440	89.38	odd	88	inner
712.2.bf.c.483.41	yes	3440	712.483	even	88	inner