Embedded newform 403.2.v.a.36.20

• Label: 403.2.v.a.36.20
• Level: $403$
• Weight: $2$
• Character: 403.36
• 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			36.20
Character	\(\chi\)	\(=\)	403.36
Dual form			403.2.v.a.56.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.388678 - 0.224403i) q^{2} +0.0944788 q^{3} +(-0.899286 + 1.55761i) q^{4} +(-0.0856400 + 0.0494443i) q^{5} +(0.0367218 - 0.0212014i) q^{6} +(-2.25811 + 1.30372i) q^{7} +1.70483i q^{8} -2.99107 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.388678	−	0.224403i	0.274837	−	0.158677i	−0.356247	−	0.934392i	\(-0.615944\pi\)
							0.631084	+	0.775715i	\(0.282610\pi\)
\(3\)	0.0944788			0.0545474			0.0272737	−	0.999628i	\(-0.491317\pi\)
							0.0272737	+	0.999628i	\(0.491317\pi\)
\(5\)	−0.0856400	+	0.0494443i	−0.0382994	+	0.0221121i	−0.519027	−	0.854758i	\(-0.673706\pi\)
							0.480728	+	0.876870i	\(0.340372\pi\)
\(7\)	−2.25811	+	1.30372i	−0.853484	+	0.492759i	−0.861825	−	0.507206i	\(-0.830678\pi\)
							0.00834117	+	0.999965i	\(0.497345\pi\)
\(11\)	−1.93419	−	1.11671i	−0.583180	−	0.336699i	0.179216	−	0.983810i	\(-0.442644\pi\)
							−0.762396	+	0.647110i	\(0.775977\pi\)
\(13\)	−3.54379	−	0.664498i	−0.982870	−	0.184299i
							\(17\)	−0.885285	−	1.53336i	−0.214713	−	0.371894i	0.738471	−	0.674286i	\(-0.235548\pi\)
−0.953184	+	0.302391i	\(0.902215\pi\)
\(19\)	3.86573	−	2.23188i	0.886859	−	0.512028i	0.0139449	−	0.999903i	\(-0.495561\pi\)
							0.872914	+	0.487875i	\(0.162228\pi\)
\(23\)	3.39507	+	5.88044i	0.707921	+	1.22616i	0.965627	+	0.259932i	\(0.0837003\pi\)
							−0.257705	+	0.966224i	\(0.582966\pi\)
\(29\)	−0.184885	−	0.320230i	−0.0343323	−	0.0594653i	0.848349	−	0.529438i	\(-0.177597\pi\)
							−0.882681	+	0.469973i	\(0.844264\pi\)
\(31\)	0.745833	+	5.51758i	0.133956	+	0.990987i
							\(37\)			7.11573i			1.16982i	0.811098	+	0.584910i	\(0.198870\pi\)
−0.811098	+	0.584910i	\(0.801130\pi\)
\(41\)	5.23921	+	3.02486i	0.818226	+	0.472403i	0.849804	−	0.527098i	\(-0.176720\pi\)
							−0.0315781	+	0.999501i	\(0.510053\pi\)
\(43\)	−5.09543	−	8.82554i	−0.777046	−	1.34588i	−0.933637	−	0.358220i	\(-0.883384\pi\)
							0.156591	−	0.987663i	\(-0.449949\pi\)
\(47\)			2.87313i			0.419090i	0.977799	+	0.209545i	\(0.0671982\pi\)
							−0.977799	+	0.209545i	\(0.932802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.36.20	yes	70
13.4	even	6		403.2.s.a.160.20	✓	70
31.25	even	3		403.2.s.a.335.20	yes	70
403.56	even	6	inner	403.2.v.a.56.20	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.20	✓	70	13.4	even	6
403.2.s.a.335.20	yes	70	31.25	even	3
403.2.v.a.36.20	yes	70	1.1	even	1	trivial
403.2.v.a.56.20	yes	70	403.56	even	6	inner