Embedded newform 403.2.bi.a.14.15

• Label: 403.2.bi.a.14.15
• Level: $403$
• Weight: $2$
• Character: 403.14
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bi (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			14.15
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.a.144.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.773640 - 2.38102i) q^{2} +(1.00840 - 1.11994i) q^{3} +(-3.45270 - 2.50854i) q^{4} +(1.04075 - 1.80263i) q^{5} +(-1.88646 - 3.26744i) q^{6} +(-0.0586254 + 0.557783i) q^{7} +(-4.59319 + 3.33715i) q^{8} +(0.0761884 + 0.724884i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.773640	−	2.38102i	0.547046	−	1.68364i	−0.169028	−	0.985611i	\(-0.554063\pi\)
							0.716074	−	0.698024i	\(-0.245937\pi\)
\(3\)	1.00840	−	1.11994i	0.582198	−	0.646596i	−0.378037	−	0.925791i	\(-0.623401\pi\)
							0.960235	+	0.279194i	\(0.0900674\pi\)
\(5\)	1.04075	−	1.80263i	0.465438	−	0.806163i	−0.533783	−	0.845622i	\(-0.679230\pi\)
							0.999221	+	0.0394588i	\(0.0125634\pi\)
\(7\)	−0.0586254	+	0.557783i	−0.0221583	+	0.210822i	0.977841	+	0.209351i	\(0.0671350\pi\)
							−0.999999	+	0.00147144i	\(0.999532\pi\)
\(11\)	−1.11063	+	0.494483i	−0.334867	+	0.149092i	−0.567280	−	0.823525i	\(-0.692004\pi\)
							0.232414	+	0.972617i	\(0.425338\pi\)
\(13\)	0.978148	+	0.207912i	0.271289	+	0.0576643i
							\(17\)	−0.761216	−	0.338915i	−0.184622	−	0.0821990i	0.312344	−	0.949969i	\(-0.398886\pi\)
−0.496966	+	0.867770i	\(0.665553\pi\)
\(19\)	2.37161	−	0.504102i	0.544085	−	0.115649i	0.0723337	−	0.997380i	\(-0.476955\pi\)
							0.471751	+	0.881732i	\(0.343622\pi\)
\(23\)	3.05064	−	2.21642i	0.636102	−	0.462155i	−0.222407	−	0.974954i	\(-0.571391\pi\)
							0.858509	+	0.512799i	\(0.171391\pi\)
\(29\)	−0.343696	+	1.05779i	−0.0638227	+	0.196426i	−0.977883	−	0.209152i	\(-0.932930\pi\)
							0.914060	+	0.405578i	\(0.132930\pi\)
\(31\)	1.68920	+	5.30534i	0.303389	+	0.952867i
							\(37\)	−0.910261	−	1.57662i	−0.149646	−	0.259194i	0.781451	−	0.623967i	\(-0.214480\pi\)
−0.931097	+	0.364773i	\(0.881147\pi\)
\(41\)	−6.74158	−	7.48729i	−1.05286	−	1.16932i	−0.985165	−	0.171611i	\(-0.945103\pi\)
							−0.0676936	−	0.997706i	\(-0.521564\pi\)
\(43\)	−6.06732	+	1.28965i	−0.925258	+	0.196670i	−0.645816	−	0.763493i	\(-0.723483\pi\)
							−0.279442	+	0.960163i	\(0.590149\pi\)
\(47\)	0.969631	+	2.98422i	0.141435	+	0.435293i	0.996535	−	0.0831699i	\(-0.0265044\pi\)
							−0.855100	+	0.518463i	\(0.826504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.14.15	✓	120
31.20	even	15	inner	403.2.bi.a.144.15	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.14.15	✓	120	1.1	even	1	trivial
403.2.bi.a.144.15	yes	120	31.20	even	15	inner