Embedded newform 403.2.r.a.218.17

• Label: 403.2.r.a.218.17
• 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			218.17
Character	\(\chi\)	\(=\)	403.218
Dual form			403.2.r.a.342.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.244810 + 0.141341i) q^{2} +(1.61186 + 2.79182i) q^{3} +(-0.960045 + 1.66285i) q^{4} -0.0582096i q^{5} +(-0.789198 - 0.455643i) q^{6} +(2.44759 + 1.41312i) q^{7} -1.10814i q^{8} +(-3.69618 + 6.40198i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.244810	+	0.141341i	−0.173107	+	0.0999431i	−0.584050	−	0.811718i	\(-0.698533\pi\)
							0.410943	+	0.911661i	\(0.365199\pi\)
\(3\)	1.61186	+	2.79182i	0.930608	+	1.61186i	0.782285	+	0.622921i	\(0.214054\pi\)
							0.148322	+	0.988939i	\(0.452613\pi\)
\(5\)		−	0.0582096i		−	0.0260321i	−0.999915	−	0.0130161i	\(-0.995857\pi\)
							0.999915	−	0.0130161i	\(-0.00414326\pi\)
\(7\)	2.44759	+	1.41312i	0.925104	+	0.534109i	0.885260	−	0.465097i	\(-0.153981\pi\)
							0.0398439	+	0.999206i	\(0.487314\pi\)
\(11\)	−1.55743	+	0.899182i	−0.469583	+	0.271114i	−0.716065	−	0.698034i	\(-0.754059\pi\)
							0.246482	+	0.969147i	\(0.420725\pi\)
\(13\)	0.590093	−	3.55694i	0.163662	−	0.986516i
							\(17\)	3.07966	−	5.33413i	0.746927	−	1.29372i	−0.202362	−	0.979311i	\(-0.564862\pi\)
0.949289	−	0.314405i	\(-0.101805\pi\)
\(19\)	4.61694	+	2.66559i	1.05920	+	0.611528i	0.925210	−	0.379455i	\(-0.123888\pi\)
							0.133988	+	0.990983i	\(0.457222\pi\)
\(23\)	−2.85580	−	4.94640i	−0.595476	−	1.03140i	−0.993479	−	0.114011i	\(-0.963630\pi\)
							0.398003	−	0.917384i	\(-0.369703\pi\)
\(29\)	1.14591	+	1.98478i	0.212790	+	0.368564i	0.952587	−	0.304267i	\(-0.0984115\pi\)
							−0.739796	+	0.672831i	\(0.765078\pi\)
\(31\)			1.00000i			0.179605i
							\(37\)	−5.60501	+	3.23605i	−0.921458	+	0.532004i	−0.884100	−	0.467298i	\(-0.845228\pi\)
−0.0373581	+	0.999302i	\(0.511894\pi\)
\(41\)	−7.73446	+	4.46549i	−1.20792	+	0.697393i	−0.962305	−	0.271974i	\(-0.912323\pi\)
							−0.245616	+	0.969367i	\(0.578990\pi\)
\(43\)	1.27522	−	2.20875i	0.194470	−	0.336831i	−0.752257	−	0.658870i	\(-0.771035\pi\)
							0.946727	+	0.322039i	\(0.104368\pi\)
\(47\)		−	1.27517i		−	0.186002i	−0.995666	−	0.0930012i	\(-0.970354\pi\)
							0.995666	−	0.0930012i	\(-0.0296460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.218.17	✓	68
13.2	odd	12		5239.2.a.q.1.18		34
13.4	even	6	inner	403.2.r.a.342.17	yes	68
13.11	odd	12		5239.2.a.r.1.17		34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.17	✓	68	1.1	even	1	trivial
403.2.r.a.342.17	yes	68	13.4	even	6	inner
5239.2.a.q.1.18		34	13.2	odd	12
5239.2.a.r.1.17		34	13.11	odd	12