Embedded newform 403.2.s.a.160.16

• Label: 403.2.s.a.160.16
• 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			160.16
Character	\(\chi\)	\(=\)	403.160
Dual form			403.2.s.a.335.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.605874 - 0.349801i) q^{2} +(1.45071 + 2.51270i) q^{3} +(-0.755278 - 1.30818i) q^{4} +(-1.79550 + 1.03663i) q^{5} -2.02984i q^{6} +0.152727i q^{7} +2.45599i q^{8} +(-2.70911 + 4.69232i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.605874	−	0.349801i	−0.428417	−	0.247347i	0.270255	−	0.962789i	\(-0.412892\pi\)
							−0.698672	+	0.715442i	\(0.746225\pi\)
\(3\)	1.45071	+	2.51270i	0.837567	+	1.45071i	0.891923	+	0.452187i	\(0.149356\pi\)
							−0.0543561	+	0.998522i	\(0.517311\pi\)
\(5\)	−1.79550	+	1.03663i	−0.802974	+	0.463597i	−0.844510	−	0.535540i	\(-0.820108\pi\)
							0.0415363	+	0.999137i	\(0.486775\pi\)
\(7\)			0.152727i			0.0577256i	0.999583	+	0.0288628i	\(0.00918859\pi\)
							−0.999583	+	0.0288628i	\(0.990811\pi\)
\(11\)		−	1.26156i		−	0.380374i	−0.981748	−	0.190187i	\(-0.939091\pi\)
							0.981748	−	0.190187i	\(-0.0609094\pi\)
\(13\)	−3.25633	+	1.54801i	−0.903143	+	0.429340i
							\(17\)	−3.89596			−0.944909			−0.472455	−	0.881355i	\(-0.656632\pi\)
−0.472455	+	0.881355i	\(0.656632\pi\)
\(19\)			7.92815i			1.81884i	0.415877	+	0.909421i	\(0.363475\pi\)
							−0.415877	+	0.909421i	\(0.636525\pi\)
\(23\)	4.11312	−	7.12413i	0.857644	−	1.48548i	−0.0165260	−	0.999863i	\(-0.505261\pi\)
							0.874170	−	0.485620i	\(-0.161406\pi\)
\(29\)	−0.981201	+	1.69949i	−0.182204	+	0.315587i	−0.942631	−	0.333837i	\(-0.891657\pi\)
							0.760426	+	0.649424i	\(0.224990\pi\)
\(31\)	−5.51441	−	0.768913i	−0.990418	−	0.138101i
							\(37\)	−8.26694	+	4.77292i	−1.35908	+	0.784663i	−0.989499	−	0.144537i	\(-0.953831\pi\)
−0.369577	+	0.929200i	\(0.620497\pi\)
\(41\)		−	2.23620i		−	0.349235i	−0.984636	−	0.174618i	\(-0.944131\pi\)
							0.984636	−	0.174618i	\(-0.0558689\pi\)
\(43\)	9.26643			1.41312			0.706558	−	0.707655i	\(-0.250247\pi\)
							0.706558	+	0.707655i	\(0.250247\pi\)
\(47\)		−	4.61814i		−	0.673625i	−0.941572	−	0.336812i	\(-0.890651\pi\)
							0.941572	−	0.336812i	\(-0.109349\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.s.a.160.16	✓	70
13.10	even	6		403.2.v.a.36.16	yes	70
31.25	even	3		403.2.v.a.56.16	yes	70
403.335	even	6	inner	403.2.s.a.335.16	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.16	✓	70	1.1	even	1	trivial
403.2.s.a.335.16	yes	70	403.335	even	6	inner
403.2.v.a.36.16	yes	70	13.10	even	6
403.2.v.a.56.16	yes	70	31.25	even	3