Embedded newform 403.2.v.a.36.3

• Label: 403.2.v.a.36.3
• Level: $403$
• Weight: $2$
• Character: 403.36
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			36.3
Character	\(\chi\)	\(=\)	403.36
Dual form			403.2.v.a.56.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.12624 + 1.22758i) q^{2} -0.797547 q^{3} +(2.01392 - 3.48821i) q^{4} +(3.14114 - 1.81354i) q^{5} +(1.69577 - 0.979055i) q^{6} +(-4.27757 + 2.46966i) q^{7} +4.97868i q^{8} -2.36392 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.12624	+	1.22758i	−1.50348	+	0.868032i	−0.503484	+	0.864005i	\(0.667949\pi\)
							−0.999992	+	0.00402754i	\(0.998718\pi\)
\(3\)	−0.797547			−0.460464			−0.230232	−	0.973136i	\(-0.573948\pi\)
							−0.230232	+	0.973136i	\(0.573948\pi\)
\(5\)	3.14114	−	1.81354i	1.40476	−	0.811039i	0.409885	−	0.912137i	\(-0.365569\pi\)
							0.994876	+	0.101098i	\(0.0322356\pi\)
\(7\)	−4.27757	+	2.46966i	−1.61677	+	0.933442i	−0.629021	+	0.777389i	\(0.716544\pi\)
							−0.987749	+	0.156053i	\(0.950123\pi\)
\(11\)	−1.32911	−	0.767364i	−0.400743	−	0.231369i	0.286062	−	0.958211i	\(-0.407654\pi\)
							−0.686804	+	0.726842i	\(0.740987\pi\)
\(13\)	3.58407	−	0.392984i	0.994042	−	0.108994i
							\(17\)	2.94372	+	5.09867i	0.713956	+	1.23661i	0.963361	+	0.268209i	\(0.0864317\pi\)
−0.249405	+	0.968399i	\(0.580235\pi\)
\(19\)	3.77622	−	2.18020i	0.866325	−	0.500173i	0.000199869	−	1.00000i	\(-0.499936\pi\)
							0.866125	+	0.499827i	\(0.166603\pi\)
\(23\)	1.48885	+	2.57877i	0.310447	+	0.537710i	0.978459	−	0.206440i	\(-0.0661879\pi\)
							−0.668012	+	0.744150i	\(0.732855\pi\)
\(29\)	0.886250	+	1.53503i	0.164572	+	0.285048i	0.936503	−	0.350659i	\(-0.114042\pi\)
							−0.771931	+	0.635706i	\(0.780709\pi\)
\(31\)	5.54265	+	0.528277i	0.995489	+	0.0948814i
							\(37\)		−	6.93903i		−	1.14077i	−0.821378	−	0.570385i	\(-0.806794\pi\)
0.821378	−	0.570385i	\(-0.193206\pi\)
\(41\)	6.61074	+	3.81671i	1.03242	+	0.596071i	0.917678	−	0.397324i	\(-0.130061\pi\)
							0.114746	+	0.993395i	\(0.463395\pi\)
\(43\)	4.52300	+	7.83407i	0.689752	+	1.19469i	0.971918	+	0.235320i	\(0.0756137\pi\)
							−0.282166	+	0.959365i	\(0.591053\pi\)
\(47\)			4.38154i			0.639114i	0.947567	+	0.319557i	\(0.103534\pi\)
							−0.947567	+	0.319557i	\(0.896466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.36.3	yes	70
13.4	even	6		403.2.s.a.160.3	✓	70
31.25	even	3		403.2.s.a.335.3	yes	70
403.56	even	6	inner	403.2.v.a.56.3	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.3	✓	70	13.4	even	6
403.2.s.a.335.3	yes	70	31.25	even	3
403.2.v.a.36.3	yes	70	1.1	even	1	trivial
403.2.v.a.56.3	yes	70	403.56	even	6	inner