Embedded newform 403.2.v.a.56.16

• Label: 403.2.v.a.56.16
• Level: $403$
• Weight: $2$
• Character: 403.56
• 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.v (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			56.16
Character	\(\chi\)	\(=\)	403.56
Dual form			403.2.v.a.36.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.605874 - 0.349801i) q^{2} -2.90142 q^{3} +(-0.755278 - 1.30818i) q^{4} +(1.79550 + 1.03663i) q^{5} +(1.75789 + 1.01492i) q^{6} +(-0.132266 - 0.0763637i) q^{7} +2.45599i q^{8} +5.41822 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

Character values

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

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

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	\(0.746225\pi\)
\(3\)	−2.90142			−1.67513			−0.837567	−	0.546335i	\(-0.816023\pi\)
							−0.837567	+	0.546335i	\(0.816023\pi\)
\(5\)	1.79550	+	1.03663i	0.802974	+	0.463597i	0.844510	−	0.535540i	\(-0.179892\pi\)
							−0.0415363	+	0.999137i	\(0.513225\pi\)
\(7\)	−0.132266	−	0.0763637i	−0.0499918	−	0.0288628i	0.474796	−	0.880096i	\(-0.342522\pi\)
							−0.524788	+	0.851233i	\(0.675855\pi\)
\(11\)	−1.09254	+	0.630779i	−0.329413	+	0.190187i	−0.655581	−	0.755125i	\(-0.727576\pi\)
							0.326167	+	0.945312i	\(0.394243\pi\)
\(13\)	2.96878	+	2.04606i	0.823391	+	0.567475i
							\(17\)	1.94798	−	3.37400i	0.472455	−	0.818316i	−0.527048	−	0.849835i	\(-0.676701\pi\)
0.999503	+	0.0315197i	\(0.0100347\pi\)
\(19\)	−6.86598	−	3.96407i	−1.57516	−	0.909421i	−0.995520	−	0.0945498i	\(-0.969859\pi\)
							−0.579643	−	0.814871i	\(-0.696808\pi\)
\(23\)	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
							\(37\)		−	9.54584i		−	1.56933i	−0.619922	−	0.784663i	\(-0.712836\pi\)
0.619922	−	0.784663i	\(-0.287164\pi\)
\(41\)	−1.93660	+	1.11810i	−0.302446	+	0.174618i	−0.643541	−	0.765411i	\(-0.722536\pi\)
							0.341095	+	0.940029i	\(0.389202\pi\)
\(43\)	−4.63321	+	8.02496i	−0.706558	+	1.22380i	0.259568	+	0.965725i	\(0.416420\pi\)
							−0.966126	+	0.258070i	\(0.916913\pi\)
\(47\)		−	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.v.a.56.16	yes	70
13.10	even	6		403.2.s.a.335.16	yes	70
31.5	even	3		403.2.s.a.160.16	✓	70
403.36	even	6	inner	403.2.v.a.36.16	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.16	✓	70	31.5	even	3
403.2.s.a.335.16	yes	70	13.10	even	6
403.2.v.a.36.16	yes	70	403.36	even	6	inner
403.2.v.a.56.16	yes	70	1.1	even	1	trivial