Embedded newform 403.2.s.a.160.9

• Label: 403.2.s.a.160.9
• Level: $403$
• Weight: $2$
• Character: 403.160
• 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.s (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			160.9
Character	\(\chi\)	\(=\)	403.160
Dual form			403.2.s.a.335.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.42222 - 0.821121i) q^{2} +(-0.972983 - 1.68526i) q^{3} +(0.348479 + 0.603584i) q^{4} +(-3.04464 + 1.75782i) q^{5} +3.19575i q^{6} -0.764751i q^{7} +2.13991i q^{8} +(-0.393391 + 0.681372i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.42222	−	0.821121i	−1.00566	−	0.580620i	−0.0957446	−	0.995406i	\(-0.530523\pi\)
							−0.909919	+	0.414786i	\(0.863857\pi\)
\(3\)	−0.972983	−	1.68526i	−0.561752	−	0.972983i	−0.997344	−	0.0728384i	\(-0.976794\pi\)
							0.435592	−	0.900144i	\(-0.356539\pi\)
\(5\)	−3.04464	+	1.75782i	−1.36161	+	0.786123i	−0.989838	−	0.142203i	\(-0.954582\pi\)
							−0.371768	+	0.928326i	\(0.621248\pi\)
\(7\)		−	0.764751i		−	0.289049i	−0.989501	−	0.144524i	\(-0.953835\pi\)
							0.989501	−	0.144524i	\(-0.0461652\pi\)
\(11\)			2.18698i			0.659398i	0.944086	+	0.329699i	\(0.106947\pi\)
							−0.944086	+	0.329699i	\(0.893053\pi\)
\(13\)	−3.50738	−	0.835617i	−0.972773	−	0.231758i
							\(17\)	5.07658			1.23125			0.615625	−	0.788039i	\(-0.288903\pi\)
0.615625	+	0.788039i	\(0.288903\pi\)
\(19\)			2.93746i			0.673899i	0.941523	+	0.336949i	\(0.109395\pi\)
							−0.941523	+	0.336949i	\(0.890605\pi\)
\(23\)	−1.69696	+	2.93922i	−0.353840	+	0.612870i	−0.986919	−	0.161218i	\(-0.948458\pi\)
							0.633078	+	0.774088i	\(0.281791\pi\)
\(29\)	3.16447	−	5.48102i	0.587627	−	1.01780i	−0.406915	−	0.913466i	\(-0.633396\pi\)
							0.994542	−	0.104334i	\(-0.0332711\pi\)
\(31\)	3.72649	−	4.13682i	0.669297	−	0.742995i
							\(37\)	−0.560160	+	0.323409i	−0.0920898	+	0.0531681i	−0.545338	−	0.838216i	\(-0.683599\pi\)
0.453248	+	0.891385i	\(0.350265\pi\)
\(41\)			11.1278i			1.73788i	0.494920	+	0.868939i	\(0.335198\pi\)
							−0.494920	+	0.868939i	\(0.664802\pi\)
\(43\)	1.42314			0.217027			0.108514	−	0.994095i	\(-0.465391\pi\)
							0.108514	+	0.994095i	\(0.465391\pi\)
\(47\)		−	9.64513i		−	1.40689i	−0.710751	−	0.703443i	\(-0.751645\pi\)
							0.710751	−	0.703443i	\(-0.248355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.s.a.160.9	✓	70
13.10	even	6		403.2.v.a.36.9	yes	70
31.25	even	3		403.2.v.a.56.9	yes	70
403.335	even	6	inner	403.2.s.a.335.9	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.9	✓	70	1.1	even	1	trivial
403.2.s.a.335.9	yes	70	403.335	even	6	inner
403.2.v.a.36.9	yes	70	13.10	even	6
403.2.v.a.56.9	yes	70	31.25	even	3