Embedded newform 403.2.bp.a.10.19

• Label: 403.2.bp.a.10.19
• Level: $403$
• Weight: $2$
• Character: 403.10
• 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.bp (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			10.19
Character	\(\chi\)	\(=\)	403.10
Dual form			403.2.bp.a.121.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0782606 + 0.00822552i) q^{2} +(-1.50523 + 1.09361i) q^{3} +(-1.95024 - 0.414536i) q^{4} +(-0.722573 + 0.417178i) q^{5} +(-0.126795 + 0.0732053i) q^{6} +(-0.267898 - 0.241217i) q^{7} +(-0.298897 - 0.0971177i) q^{8} +(0.142670 - 0.439093i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0782606	+	0.00822552i	0.0553386	+	0.00581632i	0.132157	−	0.991229i	\(-0.457810\pi\)
							−0.0768187	+	0.997045i	\(0.524476\pi\)
\(3\)	−1.50523	+	1.09361i	−0.869043	+	0.631396i	−0.930330	−	0.366724i	\(-0.880479\pi\)
							0.0612873	+	0.998120i	\(0.480479\pi\)
\(5\)	−0.722573	+	0.417178i	−0.323145	+	0.186568i	−0.652793	−	0.757536i	\(-0.726403\pi\)
							0.329649	+	0.944104i	\(0.393070\pi\)
\(7\)	−0.267898	−	0.241217i	−0.101256	−	0.0911713i	0.616948	−	0.787004i	\(-0.288369\pi\)
							−0.718204	+	0.695833i	\(0.755036\pi\)
\(11\)	0.512072	−	2.40911i	0.154395	−	0.726373i	−0.831025	−	0.556235i	\(-0.812245\pi\)
							0.985420	−	0.170138i	\(-0.0544214\pi\)
\(13\)	0.708341	−	3.53529i	0.196459	−	0.980512i
							\(17\)	2.88272	−	0.612740i	0.699161	−	0.148611i	0.155400	−	0.987852i	\(-0.450333\pi\)
0.543761	+	0.839240i	\(0.317000\pi\)
\(19\)	4.00576	+	0.421022i	0.918983	+	0.0965890i	0.552195	−	0.833715i	\(-0.313791\pi\)
							0.366789	+	0.930304i	\(0.380457\pi\)
\(23\)	1.43045	−	0.304052i	0.298270	−	0.0633992i	−0.0563459	−	0.998411i	\(-0.517945\pi\)
							0.354616	+	0.935012i	\(0.384612\pi\)
\(29\)	0.736560	−	7.00790i	0.136776	−	1.30133i	−0.683747	−	0.729719i	\(-0.739651\pi\)
							0.820522	−	0.571614i	\(-0.193683\pi\)
\(31\)	−0.956861	−	5.48493i	−0.171857	−	0.985122i
							\(37\)		−	9.99105i		−	1.64252i	−0.570556	−	0.821259i	\(-0.693272\pi\)
0.570556	−	0.821259i	\(-0.306728\pi\)
\(41\)	3.02815	+	6.80134i	0.472918	+	1.06219i	0.979764	+	0.200155i	\(0.0641444\pi\)
							−0.506847	+	0.862036i	\(0.669189\pi\)
\(43\)	−1.11778	+	10.6350i	−0.170460	+	1.62182i	0.490527	+	0.871426i	\(0.336804\pi\)
							−0.660987	+	0.750397i	\(0.729862\pi\)
\(47\)	−7.05188	−	9.70608i	−1.02862	−	1.41578i	−0.905977	−	0.423327i	\(-0.860862\pi\)
							−0.122645	−	0.992451i	\(-0.539138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.10.19	✓	280
13.4	even	6		403.2.bt.a.134.19	yes	280
31.28	even	15		403.2.bt.a.400.19	yes	280
403.121	even	30	inner	403.2.bp.a.121.19	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.10.19	✓	280	1.1	even	1	trivial
403.2.bp.a.121.19	yes	280	403.121	even	30	inner
403.2.bt.a.134.19	yes	280	13.4	even	6
403.2.bt.a.400.19	yes	280	31.28	even	15