Embedded newform 403.2.bt.a.82.9

• Label: 403.2.bt.a.82.9
• Level: $403$
• Weight: $2$
• Character: 403.82
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bt (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			82.9
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.309669 + 1.45688i) q^{2} +(2.00373 - 0.425907i) q^{3} +(-0.199502 - 0.0888240i) q^{4} +(3.03693 - 1.75337i) q^{5} +3.05108i q^{6} +(0.544135 + 0.748938i) q^{7} +(-1.55974 + 2.14679i) q^{8} +(1.09292 - 0.486598i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.309669	+	1.45688i	−0.218969	+	1.03017i	0.722057	+	0.691833i	\(0.243197\pi\)
							−0.941026	+	0.338334i	\(0.890137\pi\)
\(3\)	2.00373	−	0.425907i	1.15686	−	0.245897i	0.410765	−	0.911741i	\(-0.365262\pi\)
							0.746091	+	0.665844i	\(0.231928\pi\)
\(5\)	3.03693	−	1.75337i	1.35816	−	0.784133i	0.368782	−	0.929516i	\(-0.379775\pi\)
							0.989375	+	0.145383i	\(0.0464414\pi\)
\(7\)	0.544135	+	0.748938i	0.205664	+	0.283072i	0.899372	−	0.437184i	\(-0.144024\pi\)
							−0.693708	+	0.720256i	\(0.744024\pi\)
\(11\)	−1.17914	−	1.62295i	−0.355525	−	0.489339i	0.593370	−	0.804930i	\(-0.297797\pi\)
							−0.948895	+	0.315591i	\(0.897797\pi\)
\(13\)	−3.60553	+	0.0109024i	−0.999995	+	0.00302379i
							\(17\)	0.837176	+	0.608244i	0.203045	+	0.147521i	0.684661	−	0.728861i	\(-0.259950\pi\)
−0.481616	+	0.876382i	\(0.659950\pi\)
\(19\)	0.0507013	−	0.0164739i	0.0116317	−	0.00377936i	−0.303195	−	0.952928i	\(-0.598053\pi\)
							0.314827	+	0.949149i	\(0.398053\pi\)
\(23\)	1.95813	−	0.871814i	0.408297	−	0.181786i	−0.192302	−	0.981336i	\(-0.561595\pi\)
							0.600599	+	0.799550i	\(0.294929\pi\)
\(29\)	2.39288	+	0.508623i	0.444347	+	0.0944489i	0.424651	−	0.905357i	\(-0.360397\pi\)
							0.0196960	+	0.999806i	\(0.493730\pi\)
\(31\)	−1.21259	−	5.43412i	−0.217787	−	0.975996i
							\(37\)	−3.41501	+	1.97165i	−0.561423	+	0.324138i	−0.753717	−	0.657200i	\(-0.771741\pi\)
0.192293	+	0.981338i	\(0.438408\pi\)
\(41\)	−5.38745	+	1.75049i	−0.841378	+	0.273380i	−0.697830	−	0.716263i	\(-0.745851\pi\)
							−0.143548	+	0.989643i	\(0.545851\pi\)
\(43\)	2.28677	+	7.03794i	0.348728	+	1.07328i	0.959557	+	0.281513i	\(0.0908363\pi\)
							−0.610829	+	0.791763i	\(0.709164\pi\)
\(47\)	−9.92877	−	3.22605i	−1.44826	−	0.470568i	−0.523798	−	0.851843i	\(-0.675485\pi\)
							−0.924462	+	0.381275i	\(0.875485\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.9	yes	280
13.10	even	6		403.2.bp.a.361.27	yes	280
31.14	even	15		403.2.bp.a.355.27	✓	280
403.231	even	30	inner	403.2.bt.a.231.9	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.27	✓	280	31.14	even	15
403.2.bp.a.361.27	yes	280	13.10	even	6
403.2.bt.a.82.9	yes	280	1.1	even	1	trivial
403.2.bt.a.231.9	yes	280	403.231	even	30	inner