Embedded newform 403.2.bt.a.82.3

• Label: 403.2.bt.a.82.3
• 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			82.3
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.526303 + 2.47606i) q^{2} +(1.87677 - 0.398919i) q^{3} +(-4.02678 - 1.79284i) q^{4} +(-3.24110 + 1.87125i) q^{5} +4.85693i q^{6} +(-1.04985 - 1.44500i) q^{7} +(3.58267 - 4.93112i) q^{8} +(0.622476 - 0.277144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.526303	+	2.47606i	−0.372152	+	1.75084i	0.250289	+	0.968171i	\(0.419474\pi\)
							−0.622441	+	0.782667i	\(0.713859\pi\)
\(3\)	1.87677	−	0.398919i	1.08355	−	0.230316i	0.368669	−	0.929561i	\(-0.379814\pi\)
							0.714882	+	0.699245i	\(0.246480\pi\)
\(5\)	−3.24110	+	1.87125i	−1.44946	+	0.836848i	−0.998449	−	0.0556684i	\(-0.982271\pi\)
							−0.451014	+	0.892517i	\(0.648938\pi\)
\(7\)	−1.04985	−	1.44500i	−0.396806	−	0.546157i	0.563133	−	0.826367i	\(-0.309596\pi\)
							−0.959939	+	0.280210i	\(0.909596\pi\)
\(11\)	−1.90978	−	2.62859i	−0.575821	−	0.792550i	0.417408	−	0.908719i	\(-0.362938\pi\)
							−0.993229	+	0.116169i	\(0.962938\pi\)
\(13\)	−2.92397	−	2.10959i	−0.810964	−	0.585096i
							\(17\)	6.26294	+	4.55029i	1.51899	+	1.10361i	0.961987	+	0.273095i	\(0.0880473\pi\)
0.556999	+	0.830513i	\(0.311953\pi\)
\(19\)	−5.62661	+	1.82820i	−1.29083	+	0.419417i	−0.872383	−	0.488822i	\(-0.837427\pi\)
							−0.418450	+	0.908240i	\(0.637427\pi\)
\(23\)	−3.04879	+	1.35741i	−0.635717	+	0.283040i	−0.699181	−	0.714945i	\(-0.746452\pi\)
							0.0634639	+	0.997984i	\(0.479785\pi\)
\(29\)	−3.18267	−	0.676498i	−0.591008	−	0.125623i	−0.0973070	−	0.995254i	\(-0.531023\pi\)
							−0.493701	+	0.869632i	\(0.664356\pi\)
\(31\)	5.55591	+	0.363143i	0.997871	+	0.0652224i
							\(37\)	−8.87414	+	5.12348i	−1.45890	+	0.842296i	−0.998957	−	0.0456534i	\(-0.985463\pi\)
−0.459942	+	0.887949i	\(0.652130\pi\)
\(41\)	−1.72004	+	0.558873i	−0.268624	+	0.0872813i	−0.440232	−	0.897884i	\(-0.645104\pi\)
							0.171608	+	0.985165i	\(0.445104\pi\)
\(43\)	0.119979	+	0.369257i	0.0182966	+	0.0563112i	0.959788	−	0.280726i	\(-0.0905752\pi\)
							−0.941491	+	0.337037i	\(0.890575\pi\)
\(47\)	0.415821	+	0.135108i	0.0606537	+	0.0197076i	0.339187	−	0.940719i	\(-0.389848\pi\)
							−0.278533	+	0.960427i	\(0.589848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.3	yes	280
13.10	even	6		403.2.bp.a.361.33	yes	280
31.14	even	15		403.2.bp.a.355.33	✓	280
403.231	even	30	inner	403.2.bt.a.231.3	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.33	✓	280	31.14	even	15
403.2.bp.a.361.33	yes	280	13.10	even	6
403.2.bt.a.82.3	yes	280	1.1	even	1	trivial
403.2.bt.a.231.3	yes	280	403.231	even	30	inner