Embedded newform 403.2.r.a.218.18

• Label: 403.2.r.a.218.18
• Level: $403$
• Weight: $2$
• Character: 403.218
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			218.18
Character	\(\chi\)	\(=\)	403.218
Dual form			403.2.r.a.342.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.117721 + 0.0679661i) q^{2} +(0.0351399 + 0.0608641i) q^{3} +(-0.990761 + 1.71605i) q^{4} -1.52124i q^{5} +(-0.00827339 - 0.00477665i) q^{6} +(-2.01496 - 1.16334i) q^{7} -0.541217i q^{8} +(1.49753 - 2.59380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.117721	+	0.0679661i	−0.0832412	+	0.0480593i	−0.541043	−	0.840995i	\(-0.681970\pi\)
							0.457802	+	0.889054i	\(0.348637\pi\)
\(3\)	0.0351399	+	0.0608641i	0.0202880	+	0.0351399i	0.875991	−	0.482327i	\(-0.160208\pi\)
							−0.855703	+	0.517467i	\(0.826875\pi\)
\(5\)		−	1.52124i		−	0.680319i	−0.940368	−	0.340159i	\(-0.889519\pi\)
							0.940368	−	0.340159i	\(-0.110481\pi\)
\(7\)	−2.01496	−	1.16334i	−0.761584	−	0.439701i	0.0682804	−	0.997666i	\(-0.478249\pi\)
							−0.829864	+	0.557966i	\(0.811582\pi\)
\(11\)	3.90006	−	2.25170i	1.17591	−	0.678912i	0.220846	−	0.975309i	\(-0.429118\pi\)
							0.955065	+	0.296396i	\(0.0957849\pi\)
\(13\)	−0.579711	+	3.55864i	−0.160783	+	0.986990i
							\(17\)	3.84500	−	6.65973i	0.932549	−	1.61522i	0.153603	−	0.988133i	\(-0.450912\pi\)
0.778947	−	0.627090i	\(-0.215754\pi\)
\(19\)	2.14174	+	1.23653i	0.491349	+	0.283680i	0.725134	−	0.688608i	\(-0.241778\pi\)
							−0.233785	+	0.972288i	\(0.575111\pi\)
\(23\)	2.33127	+	4.03787i	0.486103	+	0.841955i	0.999872	−	0.0159734i	\(-0.00508470\pi\)
							−0.513770	+	0.857928i	\(0.671751\pi\)
\(29\)	−2.95605	−	5.12003i	−0.548925	−	0.950767i	−0.998349	−	0.0574476i	\(-0.981704\pi\)
							0.449423	−	0.893319i	\(-0.351630\pi\)
\(31\)			1.00000i			0.179605i
							\(37\)	−9.11472	+	5.26239i	−1.49845	+	0.865131i	−0.999998	−	0.00178569i	\(-0.999432\pi\)
−0.498453	+	0.866917i	\(0.666098\pi\)
\(41\)	2.07918	−	1.20041i	0.324713	−	0.187473i	−0.328778	−	0.944407i	\(-0.606637\pi\)
							0.653491	+	0.756934i	\(0.273304\pi\)
\(43\)	5.71438	−	9.89759i	0.871434	−	1.50937i	0.0109212	−	0.999940i	\(-0.496524\pi\)
							0.860513	−	0.509428i	\(-0.170143\pi\)
\(47\)		−	4.21200i		−	0.614383i	−0.951648	−	0.307191i	\(-0.900611\pi\)
							0.951648	−	0.307191i	\(-0.0993892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.218.18	✓	68
13.2	odd	12		5239.2.a.q.1.19		34
13.4	even	6	inner	403.2.r.a.342.18	yes	68
13.11	odd	12		5239.2.a.r.1.16		34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.18	✓	68	1.1	even	1	trivial
403.2.r.a.342.18	yes	68	13.4	even	6	inner
5239.2.a.q.1.19		34	13.2	odd	12
5239.2.a.r.1.16		34	13.11	odd	12