Embedded newform 403.3.m.a.285.1

• Label: 403.3.m.a.285.1
• Level: $403$
• Weight: $3$
• Character: 403.285
• Analytic conductor: $10.981$
• Analytic rank: $0$
• Dimension: $144$
• 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			285.1
Character	\(\chi\)	\(=\)	403.285
Dual form			403.3.m.a.181.72

$q$-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 \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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	−0.927036	−	0.374972i	\(-0.877652\pi\)
							−0.138783	+	0.990323i	\(0.544319\pi\)
\(5\)	5.73793	+	3.31280i	1.14759	+	0.662559i	0.948298	−	0.317382i	\(-0.102804\pi\)
							0.199288	+	0.979941i	\(0.436137\pi\)
\(7\)	−0.262115	+	0.151332i	−0.0374449	+	0.0216188i	−0.518606	−	0.855014i	\(-0.673549\pi\)
							0.481161	+	0.876632i	\(0.340215\pi\)
\(11\)	8.24209	−	14.2757i	0.749281	−	1.29779i	−0.198887	−	0.980022i	\(-0.563733\pi\)
							0.948168	−	0.317770i	\(-0.102934\pi\)
\(13\)	−3.41770	−	12.5427i	−0.262900	−	0.964823i
							\(17\)	−14.9223	+	8.61541i	−0.877784	+	0.506789i	−0.869927	−	0.493180i	\(-0.835834\pi\)
−0.00785671	+	0.999969i	\(0.502501\pi\)
\(19\)	−5.38844	+	3.11101i	−0.283602	+	0.163738i	−0.635053	−	0.772469i	\(-0.719022\pi\)
							0.351451	+	0.936206i	\(0.385688\pi\)
\(23\)			22.3456i			0.971546i	0.874085	+	0.485773i	\(0.161462\pi\)
							−0.874085	+	0.485773i	\(0.838538\pi\)
\(29\)		−	35.8074i		−	1.23474i	−0.786673	−	0.617370i	\(-0.788198\pi\)
							0.786673	−	0.617370i	\(-0.211802\pi\)
\(31\)	22.2206	−	21.6159i	0.716793	−	0.697286i
							\(37\)	−17.3443	−	30.0412i	−0.468765	−	0.811925i	0.530597	−	0.847624i	\(-0.321968\pi\)
−0.999363	+	0.0356987i	\(0.988634\pi\)
\(41\)	−28.1870	−	16.2737i	−0.687487	−	0.396921i	0.115183	−	0.993344i	\(-0.463255\pi\)
							−0.802670	+	0.596424i	\(0.796588\pi\)
\(43\)	44.6310	−	25.7677i	1.03793	−	0.599249i	0.118683	−	0.992932i	\(-0.462133\pi\)
							0.919246	+	0.393683i	\(0.128799\pi\)
\(47\)		−	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.285.1	yes	144
13.12	even	2	inner	403.3.m.a.285.72	yes	144
31.26	odd	6	inner	403.3.m.a.181.1	✓	144
403.181	odd	6	inner	403.3.m.a.181.72	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.3.m.a.181.1	✓	144	31.26	odd	6	inner
403.3.m.a.181.72	yes	144	403.181	odd	6	inner
403.3.m.a.285.1	yes	144	1.1	even	1	trivial
403.3.m.a.285.72	yes	144	13.12	even	2	inner