Embedded newform 403.3.m.a.181.1

• Label: 403.3.m.a.181.1
• Level: $403$
• Weight: $3$
• Character: 403.181
• Analytic conductor: $10.981$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	403.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9809546537\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(72\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			181.1
Character	\(\chi\)	\(=\)	403.181
Dual form			403.3.m.a.285.72

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.89295i q^{2} +(-3.19746 - 1.84605i) q^{3} -11.1551 q^{4} +(-5.73793 + 3.31280i) q^{5} +(-7.18660 + 12.4475i) q^{6} +(0.262115 + 0.151332i) q^{7} +27.8544i q^{8} +(2.31582 + 4.01112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{3} - 280 q^{4} + 196 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	3.89295i		−	1.94648i	−0.229800	−	0.973238i	\(-0.573807\pi\)
							0.229800	−	0.973238i	\(-0.426193\pi\)
\(3\)	−3.19746	−	1.84605i	−1.06582	−	0.615351i	−0.138783	−	0.990323i	\(-0.544319\pi\)
							−0.927036	+	0.374972i	\(0.877652\pi\)
\(5\)	−5.73793	+	3.31280i	−1.14759	+	0.662559i	−0.948298	−	0.317382i	\(-0.897196\pi\)
							−0.199288	+	0.979941i	\(0.563863\pi\)
\(7\)	0.262115	+	0.151332i	0.0374449	+	0.0216188i	0.518606	−	0.855014i	\(-0.326451\pi\)
							−0.481161	+	0.876632i	\(0.659785\pi\)
\(11\)	−8.24209	−	14.2757i	−0.749281	−	1.29779i	−0.948168	−	0.317770i	\(-0.897066\pi\)
							0.198887	−	0.980022i	\(-0.436267\pi\)
\(13\)	−12.5711	−	3.31154i	−0.967011	−	0.254734i
							\(17\)	−14.9223	−	8.61541i	−0.877784	−	0.506789i	−0.00785671	−	0.999969i	\(-0.502501\pi\)
−0.869927	+	0.493180i	\(0.835834\pi\)
\(19\)	5.38844	+	3.11101i	0.283602	+	0.163738i	0.635053	−	0.772469i	\(-0.280978\pi\)
							−0.351451	+	0.936206i	\(0.614312\pi\)
\(23\)		−	22.3456i		−	0.971546i	−0.874085	−	0.485773i	\(-0.838538\pi\)
							0.874085	−	0.485773i	\(-0.161462\pi\)
\(29\)			35.8074i			1.23474i	0.786673	+	0.617370i	\(0.211802\pi\)
							−0.786673	+	0.617370i	\(0.788198\pi\)
\(31\)	−22.2206	−	21.6159i	−0.716793	−	0.697286i
							\(37\)	17.3443	−	30.0412i	0.468765	−	0.811925i	−0.530597	−	0.847624i	\(-0.678032\pi\)
0.999363	+	0.0356987i	\(0.0113657\pi\)
\(41\)	28.1870	−	16.2737i	0.687487	−	0.396921i	−0.115183	−	0.993344i	\(-0.536745\pi\)
							0.802670	+	0.596424i	\(0.203412\pi\)
\(43\)	44.6310	+	25.7677i	1.03793	+	0.599249i	0.919246	−	0.393683i	\(-0.128799\pi\)
							0.118683	+	0.992932i	\(0.462133\pi\)
\(47\)		−	25.5796i		−	0.544246i	−0.962262	−	0.272123i	\(-0.912274\pi\)
							0.962262	−	0.272123i	\(-0.0877258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.3.m.a.181.1	✓	144
13.12	even	2	inner	403.3.m.a.181.72	yes	144
31.6	odd	6	inner	403.3.m.a.285.1	yes	144
403.285	odd	6	inner	403.3.m.a.285.72	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.3.m.a.181.1	✓	144	1.1	even	1	trivial
403.3.m.a.181.72	yes	144	13.12	even	2	inner
403.3.m.a.285.1	yes	144	31.6	odd	6	inner
403.3.m.a.285.72	yes	144	403.285	odd	6	inner