Embedded newform 403.2.v.a.56.5

• Label: 403.2.v.a.56.5
• 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			56.5
Character	\(\chi\)	\(=\)	403.56
Dual form			403.2.v.a.36.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90799 - 1.10158i) q^{2} -1.41212 q^{3} +(1.42696 + 2.47156i) q^{4} +(1.04549 + 0.603615i) q^{5} +(2.69431 + 1.55556i) q^{6} +(0.354004 + 0.204384i) q^{7} -1.88131i q^{8} -1.00592 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90799	−	1.10158i	−1.34915	−	0.778935i	−0.361025	−	0.932556i	\(-0.617573\pi\)
							−0.988130	+	0.153622i	\(0.950906\pi\)
\(3\)	−1.41212			−0.815287			−0.407643	−	0.913141i	\(-0.633649\pi\)
							−0.407643	+	0.913141i	\(0.633649\pi\)
\(5\)	1.04549	+	0.603615i	0.467558	+	0.269945i	0.715217	−	0.698902i	\(-0.246328\pi\)
							−0.247659	+	0.968847i	\(0.579661\pi\)
\(7\)	0.354004	+	0.204384i	0.133801	+	0.0772501i	0.565406	−	0.824812i	\(-0.308719\pi\)
							−0.431605	+	0.902063i	\(0.642053\pi\)
\(11\)	3.29939	−	1.90490i	0.994802	−	0.574349i	0.0880959	−	0.996112i	\(-0.471922\pi\)
							0.906706	+	0.421763i	\(0.138588\pi\)
\(13\)	−3.33193	+	1.37776i	−0.924112	+	0.382122i
							\(17\)	−0.765109	+	1.32521i	−0.185566	+	0.321410i	−0.943767	−	0.330611i	\(-0.892745\pi\)
0.758201	+	0.652021i	\(0.226079\pi\)
\(19\)	−6.57574	−	3.79650i	−1.50858	−	0.870978i	−0.999950	−	0.00999115i	\(-0.996820\pi\)
							−0.508628	−	0.860987i	\(-0.669847\pi\)
\(23\)	−2.09503	+	3.62870i	−0.436844	+	0.756636i	−0.997444	−	0.0714508i	\(-0.977237\pi\)
							0.560600	+	0.828087i	\(0.310570\pi\)
\(29\)	−3.41689	+	5.91823i	−0.634501	+	1.09899i	0.352120	+	0.935955i	\(0.385461\pi\)
							−0.986621	+	0.163033i	\(0.947872\pi\)
\(31\)	4.92093	+	2.60469i	0.883825	+	0.467817i
							\(37\)			2.62490i			0.431531i	0.976445	+	0.215765i	\(0.0692246\pi\)
−0.976445	+	0.215765i	\(0.930775\pi\)
\(41\)	−4.05831	+	2.34306i	−0.633801	+	0.365925i	−0.782223	−	0.622999i	\(-0.785914\pi\)
							0.148422	+	0.988924i	\(0.452581\pi\)
\(43\)	−4.53890	+	7.86160i	−0.692175	+	1.19888i	0.278948	+	0.960306i	\(0.410014\pi\)
							−0.971124	+	0.238577i	\(0.923319\pi\)
\(47\)			13.0806i			1.90801i	0.299793	+	0.954004i	\(0.403082\pi\)
							−0.299793	+	0.954004i	\(0.596918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.56.5	yes	70
13.10	even	6		403.2.s.a.335.5	yes	70
31.5	even	3		403.2.s.a.160.5	✓	70
403.36	even	6	inner	403.2.v.a.36.5	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.5	✓	70	31.5	even	3
403.2.s.a.335.5	yes	70	13.10	even	6
403.2.v.a.36.5	yes	70	403.36	even	6	inner
403.2.v.a.56.5	yes	70	1.1	even	1	trivial