Embedded newform 403.2.bi.b.14.8

• Label: 403.2.bi.b.14.8
• Level: $403$
• Weight: $2$
• Character: 403.14
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• 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:	\(136\)
Relative dimension:	\(17\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			14.8
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.b.144.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0129642 - 0.0398998i) q^{2} +(1.94913 - 2.16473i) q^{3} +(1.61661 + 1.17454i) q^{4} +(-1.81601 + 3.14542i) q^{5} +(-0.0611034 - 0.105834i) q^{6} +(-0.401482 + 3.81985i) q^{7} +(0.135703 - 0.0985943i) q^{8} +(-0.573359 - 5.45514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 6 q^{2} + 2 q^{3} - 34 q^{4} - 15 q^{5} - 10 q^{6} - 8 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0129642	−	0.0398998i	0.00916709	−	0.0282134i	−0.946368	−	0.323090i	\(-0.895278\pi\)
							0.955535	+	0.294876i	\(0.0952784\pi\)
\(3\)	1.94913	−	2.16473i	1.12533	−	1.24981i	0.160475	−	0.987040i	\(-0.448698\pi\)
							0.964859	−	0.262769i	\(-0.0846358\pi\)
\(5\)	−1.81601	+	3.14542i	−0.812143	+	1.40667i	0.0992179	+	0.995066i	\(0.468366\pi\)
							−0.911361	+	0.411608i	\(0.864967\pi\)
\(7\)	−0.401482	+	3.81985i	−0.151746	+	1.44377i	0.608201	+	0.793783i	\(0.291891\pi\)
							−0.759947	+	0.649985i	\(0.774775\pi\)
\(11\)	3.78193	−	1.68382i	1.14029	−	0.507692i	0.252347	−	0.967637i	\(-0.418798\pi\)
							0.887947	+	0.459945i	\(0.152131\pi\)
\(13\)	−0.978148	−	0.207912i	−0.271289	−	0.0576643i
							\(17\)	−3.64606	−	1.62333i	−0.884300	−	0.393716i	−0.0862265	−	0.996276i	\(-0.527481\pi\)
−0.798074	+	0.602560i	\(0.794148\pi\)
\(19\)	−0.942682	+	0.200373i	−0.216266	+	0.0459688i	−0.314770	−	0.949168i	\(-0.601928\pi\)
							0.0985043	+	0.995137i	\(0.468594\pi\)
\(23\)	4.66135	−	3.38667i	0.971959	−	0.706169i	0.0160616	−	0.999871i	\(-0.494887\pi\)
							0.955897	+	0.293702i	\(0.0948872\pi\)
\(29\)	0.829217	−	2.55207i	0.153982	−	0.473907i	−0.844075	−	0.536226i	\(-0.819850\pi\)
							0.998056	+	0.0623188i	\(0.0198496\pi\)
\(31\)	3.64334	+	4.21023i	0.654364	+	0.756180i
							\(37\)	−5.85814	−	10.1466i	−0.963072	−	1.66809i	−0.714708	−	0.699423i	\(-0.753440\pi\)
−0.248364	−	0.968667i	\(-0.579893\pi\)
\(41\)	−5.63096	−	6.25382i	−0.879408	−	0.976682i	0.120463	−	0.992718i	\(-0.461562\pi\)
							−0.999871	+	0.0160360i	\(0.994895\pi\)
\(43\)	5.26597	−	1.11932i	0.803053	−	0.170694i	0.211940	−	0.977283i	\(-0.432022\pi\)
							0.591113	+	0.806588i	\(0.298689\pi\)
\(47\)	−1.97116	−	6.06659i	−0.287523	−	0.884904i	−0.985631	−	0.168912i	\(-0.945975\pi\)
							0.698109	−	0.715992i	\(-0.254025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.b.14.8	✓	136
31.20	even	15	inner	403.2.bi.b.144.8	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.b.14.8	✓	136	1.1	even	1	trivial
403.2.bi.b.144.8	yes	136	31.20	even	15	inner