Embedded newform 403.2.r.a.342.5

• Label: 403.2.r.a.342.5
• Level: $403$
• Weight: $2$
• Character: 403.342
• 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			342.5
Character	\(\chi\)	\(=\)	403.342
Dual form			403.2.r.a.218.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98176 - 1.14417i) q^{2} +(0.827863 - 1.43390i) q^{3} +(1.61825 + 2.80289i) q^{4} +1.16564i q^{5} +(-3.28125 + 1.89443i) q^{6} +(-2.58921 + 1.49488i) q^{7} -2.82952i q^{8} +(0.129286 + 0.223931i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98176	−	1.14417i	−1.40132	−	0.809050i	−0.406788	−	0.913523i	\(-0.633351\pi\)
							−0.994528	+	0.104473i	\(0.966685\pi\)
\(3\)	0.827863	−	1.43390i	0.477967	−	0.827863i	−0.521714	−	0.853120i	\(-0.674707\pi\)
							0.999681	+	0.0252576i	\(0.00804059\pi\)
\(5\)			1.16564i			0.521288i	0.965435	+	0.260644i	\(0.0839349\pi\)
							−0.965435	+	0.260644i	\(0.916065\pi\)
\(7\)	−2.58921	+	1.49488i	−0.978631	+	0.565013i	−0.901857	−	0.432036i	\(-0.857796\pi\)
							−0.0767746	+	0.997048i	\(0.524462\pi\)
\(11\)	−1.16460	−	0.672380i	−0.351139	−	0.202730i	0.314048	−	0.949407i	\(-0.398315\pi\)
							−0.665187	+	0.746677i	\(0.731648\pi\)
\(13\)	3.15059	−	1.75322i	0.873816	−	0.486256i
							\(17\)	3.33708	+	5.77999i	0.809361	+	1.40185i	0.913307	+	0.407272i	\(0.133520\pi\)
−0.103946	+	0.994583i	\(0.533147\pi\)
\(19\)	2.43890	−	1.40810i	0.559523	−	0.323041i	−0.193431	−	0.981114i	\(-0.561962\pi\)
							0.752954	+	0.658073i	\(0.228628\pi\)
\(23\)	−1.71975	+	2.97870i	−0.358593	+	0.621102i	−0.987726	−	0.156196i	\(-0.950077\pi\)
							0.629133	+	0.777298i	\(0.283410\pi\)
\(29\)	0.110886	−	0.192060i	0.0205910	−	0.0356647i	−0.855546	−	0.517726i	\(-0.826779\pi\)
							0.876137	+	0.482062i	\(0.160112\pi\)
\(31\)		−	1.00000i		−	0.179605i
							\(37\)	9.86577	+	5.69600i	1.62192	+	0.936417i	0.986405	+	0.164330i	\(0.0525462\pi\)
0.635517	+	0.772087i	\(0.280787\pi\)
\(41\)	−4.17402	−	2.40987i	−0.651873	−	0.376359i	0.137301	−	0.990529i	\(-0.456157\pi\)
							−0.789173	+	0.614170i	\(0.789491\pi\)
\(43\)	2.63099	+	4.55700i	0.401222	+	0.694936i	0.993874	−	0.110522i	\(-0.0352524\pi\)
							−0.592652	+	0.805459i	\(0.701919\pi\)
\(47\)		−	6.66306i		−	0.971908i	−0.873984	−	0.485954i	\(-0.838472\pi\)
							0.873984	−	0.485954i	\(-0.161528\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.342.5	yes	68
13.6	odd	12		5239.2.a.r.1.29		34
13.7	odd	12		5239.2.a.q.1.6		34
13.10	even	6	inner	403.2.r.a.218.5	✓	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.5	✓	68	13.10	even	6	inner
403.2.r.a.342.5	yes	68	1.1	even	1	trivial
5239.2.a.q.1.6		34	13.7	odd	12
5239.2.a.r.1.29		34	13.6	odd	12