Embedded newform 403.2.bi.a.14.14

• Label: 403.2.bi.a.14.14
• 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			14.14
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.a.144.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.635252 - 1.95510i) q^{2} +(-1.26940 + 1.40981i) q^{3} +(-1.80085 - 1.30840i) q^{4} +(0.148342 - 0.256935i) q^{5} +(1.94994 + 3.37739i) q^{6} +(-0.266274 + 2.53343i) q^{7} +(-0.375822 + 0.273051i) q^{8} +(-0.0626061 - 0.595658i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.635252	−	1.95510i	0.449191	−	1.38247i	−0.428631	−	0.903480i	\(-0.641004\pi\)
							0.877822	−	0.478987i	\(-0.158996\pi\)
\(3\)	−1.26940	+	1.40981i	−0.732888	+	0.813954i	−0.988243	−	0.152893i	\(-0.951141\pi\)
							0.255355	+	0.966847i	\(0.417808\pi\)
\(5\)	0.148342	−	0.256935i	0.0663404	−	0.114905i	−0.830947	−	0.556351i	\(-0.812201\pi\)
							0.897288	+	0.441446i	\(0.145534\pi\)
\(7\)	−0.266274	+	2.53343i	−0.100642	+	0.957547i	0.821373	+	0.570392i	\(0.193209\pi\)
							−0.922015	+	0.387155i	\(0.873458\pi\)
\(11\)	2.92449	−	1.30207i	0.881767	−	0.392588i	0.0846488	−	0.996411i	\(-0.473023\pi\)
							0.797119	+	0.603823i	\(0.206357\pi\)
\(13\)	0.978148	+	0.207912i	0.271289	+	0.0576643i
							\(17\)	4.95118	+	2.20441i	1.20084	+	0.534647i	0.906968	−	0.421199i	\(-0.138391\pi\)
0.293869	+	0.955846i	\(0.405057\pi\)
\(19\)	6.19407	−	1.31659i	1.42102	−	0.302046i	0.567613	−	0.823296i	\(-0.307867\pi\)
							0.853404	+	0.521249i	\(0.174534\pi\)
\(23\)	−1.49148	+	1.08362i	−0.310995	+	0.225951i	−0.732323	−	0.680957i	\(-0.761564\pi\)
							0.421328	+	0.906908i	\(0.361564\pi\)
\(29\)	−3.12311	+	9.61196i	−0.579948	+	1.78490i	0.0387293	+	0.999250i	\(0.487669\pi\)
							−0.618677	+	0.785646i	\(0.712331\pi\)
\(31\)	1.28506	−	5.41744i	0.230804	−	0.973000i
							\(37\)	−5.90207	−	10.2227i	−0.970294	−	1.68060i	−0.694664	−	0.719335i	\(-0.744447\pi\)
−0.275630	−	0.961264i	\(-0.588887\pi\)
\(41\)	5.04474	+	5.60275i	0.787856	+	0.875002i	0.994642	−	0.103384i	\(-0.0329670\pi\)
							−0.206786	+	0.978386i	\(0.566300\pi\)
\(43\)	1.13643	−	0.241555i	0.173303	−	0.0368368i	−0.120443	−	0.992720i	\(-0.538431\pi\)
							0.293746	+	0.955884i	\(0.405098\pi\)
\(47\)	−2.90974	−	8.95526i	−0.424429	−	1.30626i	−0.903540	−	0.428505i	\(-0.859041\pi\)
							0.479110	−	0.877755i	\(-0.340959\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.14.14	✓	120
31.20	even	15	inner	403.2.bi.a.144.14	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.14.14	✓	120	1.1	even	1	trivial
403.2.bi.a.144.14	yes	120	31.20	even	15	inner