Embedded newform 712.2.bf.c.171.24

• Label: 712.2.bf.c.171.24
• 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			171.24
Character	\(\chi\)	\(=\)	712.171
Dual form			712.2.bf.c.483.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.922094 - 1.07226i) q^{2} +(0.574898 + 2.25243i) q^{3} +(-0.299485 + 1.97745i) q^{4} +(-0.512138 + 0.111409i) q^{5} +(1.88508 - 2.69339i) q^{6} +(0.421436 - 2.33586i) q^{7} +(2.39650 - 1.50227i) q^{8} +(-2.10989 + 1.15209i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3440 q - 36 q^{2} - 88 q^{3} - 20 q^{4} - 32 q^{6} - 36 q^{8} - 88 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.922094	−	1.07226i	−0.652019	−	0.758203i
							\(3\)	0.574898	+	2.25243i	0.331918	+	1.30044i	0.884327	+	0.466868i	\(0.154618\pi\)
−0.552409	+	0.833573i	\(0.686292\pi\)
\(5\)	−0.512138	+	0.111409i	−0.229035	+	0.0498235i	−0.325618	−	0.945501i	\(-0.605572\pi\)
							0.0965829	+	0.995325i	\(0.469209\pi\)
\(7\)	0.421436	−	2.33586i	0.159288	−	0.882873i	−0.799907	−	0.600125i	\(-0.795118\pi\)
							0.959194	−	0.282748i	\(-0.0912460\pi\)
\(11\)	−1.66284	−	2.58743i	−0.501364	−	0.780138i	0.494672	−	0.869079i	\(-0.335288\pi\)
							−0.996037	+	0.0889412i	\(0.971652\pi\)
\(13\)	−3.68648	−	2.18729i	−1.02245	−	0.606645i	−0.100892	−	0.994897i	\(-0.532170\pi\)
							−0.921553	+	0.388252i	\(0.873079\pi\)
\(17\)	0.503004	−	7.03292i	0.121996	−	1.70573i	−0.456234	−	0.889860i	\(-0.650802\pi\)
							0.578230	−	0.815874i	\(-0.303744\pi\)
\(19\)	2.28527	−	2.83585i	0.524277	−	0.650589i	−0.445298	−	0.895383i	\(-0.646902\pi\)
							0.969575	+	0.244794i	\(0.0787203\pi\)
\(23\)	0.00844482	+	0.0785483i	0.00176087	+	0.0163785i	0.995117	−	0.0987023i	\(-0.0314691\pi\)
							−0.993356	+	0.115081i	\(0.963287\pi\)
\(29\)	3.33577	+	4.80442i	0.619436	+	0.892158i	0.999461	−	0.0328181i	\(-0.0104482\pi\)
							−0.380025	+	0.924976i	\(0.624085\pi\)
\(31\)	0.466232	−	4.33659i	0.0837377	−	0.778875i	−0.872960	−	0.487792i	\(-0.837803\pi\)
							0.956698	−	0.291083i	\(-0.0940157\pi\)
\(37\)	−4.90428	−	2.03142i	−0.806259	−	0.333963i	−0.0587987	−	0.998270i	\(-0.518727\pi\)
							−0.747460	+	0.664306i	\(0.768727\pi\)
\(41\)	8.62698	−	5.11862i	1.34731	−	0.799395i	0.356452	−	0.934314i	\(-0.383986\pi\)
							0.990857	+	0.134919i	\(0.0430774\pi\)
\(43\)	−1.80519	−	1.25337i	−0.275289	−	0.191137i	0.424614	−	0.905374i	\(-0.360410\pi\)
							−0.699903	+	0.714238i	\(0.746773\pi\)
\(47\)	−3.88654	−	2.90943i	−0.566910	−	0.424384i	0.276999	−	0.960870i	\(-0.410660\pi\)
							−0.843909	+	0.536487i	\(0.819751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.bf.c.171.24	✓	3440
8.3	odd	2	inner	712.2.bf.c.171.70	yes	3440
89.38	odd	88	inner	712.2.bf.c.483.70	yes	3440
712.483	even	88	inner	712.2.bf.c.483.24	yes	3440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.bf.c.171.24	✓	3440	1.1	even	1	trivial
712.2.bf.c.171.70	yes	3440	8.3	odd	2	inner
712.2.bf.c.483.24	yes	3440	712.483	even	88	inner
712.2.bf.c.483.70	yes	3440	89.38	odd	88	inner