Embedded newform 712.2.bf.c.171.75

• Label: 712.2.bf.c.171.75
• 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			171.75
Character	\(\chi\)	\(=\)	712.171
Dual form			712.2.bf.c.483.75

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.27047 + 0.621223i) q^{2} +(0.772771 + 3.02769i) q^{3} +(1.22817 + 1.57848i) q^{4} +(3.58128 - 0.779059i) q^{5} +(-0.899089 + 4.32663i) q^{6} +(0.806285 - 4.46894i) q^{7} +(0.579752 + 2.76837i) q^{8} +(-5.93668 + 3.24167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3440 q - 36 q^{2} - 88 q^{3} - 20 q^{4} - 32 q^{6} - 36 q^{8} - 88 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.27047	+	0.621223i	0.898355	+	0.439271i
							\(3\)	0.772771	+	3.02769i	0.446160	+	1.74804i	0.641329	+	0.767266i	\(0.278384\pi\)
−0.195169	+	0.980770i	\(0.562526\pi\)
\(5\)	3.58128	−	0.779059i	1.60160	−	0.348406i	0.678990	−	0.734148i	\(-0.262418\pi\)
							0.922607	+	0.385742i	\(0.126055\pi\)
\(7\)	0.806285	−	4.46894i	0.304747	−	1.68910i	−0.352811	−	0.935695i	\(-0.614774\pi\)
							0.657558	−	0.753404i	\(-0.271590\pi\)
\(11\)	−0.0801664	−	0.124741i	−0.0241711	−	0.0376109i	0.828958	−	0.559311i	\(-0.188934\pi\)
							−0.853129	+	0.521700i	\(0.825298\pi\)
\(13\)	−4.90951	−	2.91295i	−1.36165	−	0.807906i	−0.369062	−	0.929405i	\(-0.620321\pi\)
							−0.992592	+	0.121498i	\(0.961230\pi\)
\(17\)	−0.0373710	+	0.522515i	−0.00906381	+	0.126729i	−0.999969	−	0.00783078i	\(-0.997507\pi\)
							0.990906	+	0.134559i	\(0.0429619\pi\)
\(19\)	−0.723402	+	0.897687i	−0.165960	+	0.205943i	−0.854783	−	0.518986i	\(-0.826310\pi\)
							0.688823	+	0.724930i	\(0.258128\pi\)
\(23\)	0.576972	+	5.36663i	0.120307	+	1.11902i	0.882055	+	0.471146i	\(0.156160\pi\)
							−0.761748	+	0.647873i	\(0.775659\pi\)
\(29\)	−3.72537	−	5.36555i	−0.691784	−	0.996358i	−0.999000	−	0.0447183i	\(-0.985761\pi\)
							0.307216	−	0.951640i	\(-0.400603\pi\)
\(31\)	0.0309154	−	0.287555i	0.00555257	−	0.0516465i	−0.991159	−	0.132682i	\(-0.957641\pi\)
							0.996711	+	0.0810358i	\(0.0258228\pi\)
\(37\)	−8.62723	−	3.57352i	−1.41831	−	0.587482i	−0.463873	−	0.885902i	\(-0.653541\pi\)
							−0.954435	+	0.298419i	\(0.903541\pi\)
\(41\)	1.95930	−	1.16251i	0.305991	−	0.181553i	−0.349135	−	0.937072i	\(-0.613525\pi\)
							0.655126	+	0.755520i	\(0.272615\pi\)
\(43\)	−3.71853	−	2.58182i	−0.567071	−	0.393725i	0.251734	−	0.967797i	\(-0.418999\pi\)
							−0.818805	+	0.574072i	\(0.805363\pi\)
\(47\)	7.33223	+	5.48884i	1.06952	+	0.800630i	0.980720	−	0.195420i	\(-0.0626070\pi\)
							0.0887965	+	0.996050i	\(0.471698\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.bf.c.171.75	yes	3440
8.3	odd	2	inner	712.2.bf.c.171.28	✓	3440
89.38	odd	88	inner	712.2.bf.c.483.28	yes	3440
712.483	even	88	inner	712.2.bf.c.483.75	yes	3440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.bf.c.171.28	✓	3440	8.3	odd	2	inner
712.2.bf.c.171.75	yes	3440	1.1	even	1	trivial
712.2.bf.c.483.28	yes	3440	89.38	odd	88	inner
712.2.bf.c.483.75	yes	3440	712.483	even	88	inner