Embedded newform 403.2.bn.a.60.1

• Label: 403.2.bn.a.60.1
• Level: $403$
• Weight: $2$
• Character: 403.60
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $272$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bn (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(34\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			60.1
Character	\(\chi\)	\(=\)	403.60
Dual form			403.2.bn.a.356.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.399110 - 2.51988i) q^{2} +(0.464849 + 0.639810i) q^{3} +(-4.28839 + 1.39338i) q^{4} +(-1.89475 - 1.89475i) q^{5} +(1.42672 - 1.42672i) q^{6} +(-1.39495 - 0.710764i) q^{7} +(2.90617 + 5.70368i) q^{8} +(0.733779 - 2.25834i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 6 q^{2} - 20 q^{3} - 16 q^{5} - 18 q^{7} + 4 q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.399110	−	2.51988i	−0.282213	−	1.78182i	−0.567492	−	0.823379i	\(-0.692086\pi\)
							0.285279	−	0.958445i	\(-0.407914\pi\)
\(3\)	0.464849	+	0.639810i	0.268381	+	0.369395i	0.921842	−	0.387565i	\(-0.126684\pi\)
							−0.653461	+	0.756960i	\(0.726684\pi\)
\(5\)	−1.89475	−	1.89475i	−0.847358	−	0.847358i	0.142445	−	0.989803i	\(-0.454504\pi\)
							−0.989803	+	0.142445i	\(0.954504\pi\)
\(7\)	−1.39495	−	0.710764i	−0.527243	−	0.268643i	0.170043	−	0.985437i	\(-0.445609\pi\)
							−0.697286	+	0.716793i	\(0.745609\pi\)
\(11\)	−0.267129	+	0.524270i	−0.0805424	+	0.158073i	−0.927765	−	0.373164i	\(-0.878273\pi\)
							0.847223	+	0.531237i	\(0.178273\pi\)
\(13\)	−2.40560	+	2.68572i	−0.667192	+	0.744886i
							\(17\)	−0.755452	+	2.32504i	−0.183224	+	0.563905i	−0.999913	−	0.0131724i	\(-0.995807\pi\)
0.816689	+	0.577078i	\(0.195807\pi\)
\(19\)	−0.117316	−	0.740705i	−0.0269142	−	0.169929i	0.970570	−	0.240819i	\(-0.0774161\pi\)
							−0.997484	+	0.0708899i	\(0.977416\pi\)
\(23\)	−2.02881	+	6.24404i	−0.423037	+	1.30197i	0.481825	+	0.876267i	\(0.339974\pi\)
							−0.904862	+	0.425706i	\(0.860026\pi\)
\(29\)	−2.86029	+	3.93685i	−0.531142	+	0.731055i	−0.987304	−	0.158842i	\(-0.949224\pi\)
							0.456162	+	0.889897i	\(0.349224\pi\)
\(31\)	−4.57845	−	3.16825i	−0.822314	−	0.569034i
							\(37\)	−7.52362	−	7.52362i	−1.23688	−	1.23688i	−0.961271	−	0.275604i	\(-0.911122\pi\)
−0.275604	−	0.961271i	\(-0.588878\pi\)
\(41\)	0.517744	+	3.26891i	0.0808580	+	0.510518i	0.994563	+	0.104139i	\(0.0332086\pi\)
							−0.913705	+	0.406379i	\(0.866791\pi\)
\(43\)	−4.01334	−	2.91586i	−0.612029	−	0.444665i	0.238099	−	0.971241i	\(-0.423476\pi\)
							−0.850128	+	0.526576i	\(0.823476\pi\)
\(47\)	1.71129	−	10.8047i	0.249618	−	1.57603i	−0.470644	−	0.882323i	\(-0.655978\pi\)
							0.720262	−	0.693702i	\(-0.244022\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bn.a.60.1	✓	272
13.5	odd	4	inner	403.2.bn.a.122.1	yes	272
31.15	odd	10	inner	403.2.bn.a.294.1	yes	272
403.356	even	20	inner	403.2.bn.a.356.1	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bn.a.60.1	✓	272	1.1	even	1	trivial
403.2.bn.a.122.1	yes	272	13.5	odd	4	inner
403.2.bn.a.294.1	yes	272	31.15	odd	10	inner
403.2.bn.a.356.1	yes	272	403.356	even	20	inner