Embedded newform 935.2.q.a.208.3

• Label: 935.2.q.a.208.3
• Level: $935$
• Weight: $2$
• Character: 935.208
• Analytic conductor: $7.466$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 935 = 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	935.q (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.46601258899\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(104\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			208.3
Character	\(\chi\)	\(=\)	935.208
Dual form			935.2.q.a.472.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88825 + 1.88825i) q^{2} -0.0115176 q^{3} -5.13101i q^{4} +(2.06686 + 0.853278i) q^{5} +(0.0217481 - 0.0217481i) q^{6} +3.91069i q^{7} +(5.91215 + 5.91215i) q^{8} -2.99987 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 8 q^{3} - 8 q^{5} + 176 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/935\mathbb{Z}\right)^\times\).

\(n\)	\(496\)	\(562\)	\(596\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88825	+	1.88825i	−1.33520	+	1.33520i	−0.434550	+	0.900648i	\(0.643093\pi\)
							−0.900648	+	0.434550i	\(0.856907\pi\)
\(3\)	−0.0115176			−0.00664967			−0.00332484	−	0.999994i	\(-0.501058\pi\)
							−0.00332484	+	0.999994i	\(0.501058\pi\)
\(5\)	2.06686	+	0.853278i	0.924329	+	0.381597i
							\(7\)			3.91069i			1.47810i	0.673650	+	0.739051i	\(0.264726\pi\)
−0.673650	+	0.739051i	\(0.735274\pi\)
\(11\)	2.05267	−	2.60510i	0.618904	−	0.785467i
							\(13\)	−0.524550	+	0.524550i	−0.145484	+	0.145484i	−0.776097	−	0.630613i	\(-0.782803\pi\)
0.630613	+	0.776097i	\(0.282803\pi\)
\(17\)	−4.09903	−	0.444903i	−0.994161	−	0.107905i
							\(19\)			1.10028i			0.252421i	0.992003	+	0.126211i	\(0.0402816\pi\)
−0.992003	+	0.126211i	\(0.959718\pi\)
\(23\)			4.74988i			0.990418i	0.868774	+	0.495209i	\(0.164909\pi\)
							−0.868774	+	0.495209i	\(0.835091\pi\)
\(29\)	−2.56524	+	2.56524i	−0.476353	+	0.476353i	−0.903963	−	0.427610i	\(-0.859356\pi\)
							0.427610	+	0.903963i	\(0.359356\pi\)
\(31\)	4.02604	−	4.02604i	0.723098	−	0.723098i	−0.246137	−	0.969235i	\(-0.579161\pi\)
							0.969235	+	0.246137i	\(0.0791613\pi\)
\(37\)			1.06402i			0.174923i	0.996168	+	0.0874617i	\(0.0278755\pi\)
							−0.996168	+	0.0874617i	\(0.972124\pi\)
\(41\)	−6.38368	+	6.38368i	−0.996964	+	0.996964i	−0.999995	−	0.00303136i	\(-0.999035\pi\)
							0.00303136	+	0.999995i	\(0.499035\pi\)
\(43\)	−0.274132	−	0.274132i	−0.0418048	−	0.0418048i	0.685895	−	0.727700i	\(-0.259411\pi\)
							−0.727700	+	0.685895i	\(0.759411\pi\)
\(47\)	−7.52711	+	7.52711i	−1.09794	+	1.09794i	−0.103291	+	0.994651i	\(0.532937\pi\)
							−0.994651	+	0.103291i	\(0.967063\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	935.2.q.a.208.3	yes	208
5.2	odd	4		935.2.l.a.582.3	yes	208
11.10	odd	2	inner	935.2.q.a.208.102	yes	208
17.13	even	4		935.2.l.a.98.102	yes	208
55.32	even	4		935.2.l.a.582.102	yes	208
85.47	odd	4	inner	935.2.q.a.472.102	yes	208
187.98	odd	4		935.2.l.a.98.3	✓	208
935.472	even	4	inner	935.2.q.a.472.3	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
935.2.l.a.98.3	✓	208	187.98	odd	4
935.2.l.a.98.102	yes	208	17.13	even	4
935.2.l.a.582.3	yes	208	5.2	odd	4
935.2.l.a.582.102	yes	208	55.32	even	4
935.2.q.a.208.3	yes	208	1.1	even	1	trivial
935.2.q.a.208.102	yes	208	11.10	odd	2	inner
935.2.q.a.472.3	yes	208	935.472	even	4	inner
935.2.q.a.472.102	yes	208	85.47	odd	4	inner