Embedded newform 403.2.bn.a.60.9

• Label: 403.2.bn.a.60.9
• 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			60.9
Character	\(\chi\)	\(=\)	403.60
Dual form			403.2.bn.a.356.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.241622 - 1.52554i) q^{2} +(-1.55970 - 2.14675i) q^{3} +(-0.366780 + 0.119174i) q^{4} +(2.50940 + 2.50940i) q^{5} +(-2.89809 + 2.89809i) q^{6} +(3.71488 + 1.89282i) q^{7} +(-1.13200 - 2.22168i) q^{8} +(-1.24880 + 3.84340i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 6 q^{2} - 20 q^{3} - 16 q^{5} - 18 q^{7} + 4 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.241622	−	1.52554i	−0.170852	−	1.07872i	−0.912843	−	0.408310i	\(-0.866118\pi\)
							0.741991	−	0.670410i	\(-0.233882\pi\)
\(3\)	−1.55970	−	2.14675i	−0.900495	−	1.23942i	−0.970310	−	0.241865i	\(-0.922241\pi\)
							0.0698154	−	0.997560i	\(-0.477759\pi\)
\(5\)	2.50940	+	2.50940i	1.12224	+	1.12224i	0.991404	+	0.130833i	\(0.0417651\pi\)
							0.130833	+	0.991404i	\(0.458235\pi\)
\(7\)	3.71488	+	1.89282i	1.40409	+	0.715420i	0.981600	−	0.190947i	\(-0.0611558\pi\)
							0.422491	+	0.906367i	\(0.361156\pi\)
\(11\)	2.75969	−	5.41620i	0.832079	−	1.63305i	0.0594136	−	0.998233i	\(-0.481077\pi\)
							0.772665	−	0.634813i	\(-0.218923\pi\)
\(13\)	2.96615	+	2.04986i	0.822663	+	0.568530i
							\(17\)	−0.713860	+	2.19704i	−0.173137	+	0.532859i	−0.999543	−	0.0302139i	\(-0.990381\pi\)
0.826407	+	0.563073i	\(0.190381\pi\)
\(19\)	0.356451	+	2.25054i	0.0817754	+	0.516310i	0.994242	+	0.107158i	\(0.0341752\pi\)
							−0.912467	+	0.409151i	\(0.865825\pi\)
\(23\)	0.0451955	−	0.139098i	0.00942392	−	0.0290039i	−0.946234	−	0.323483i	\(-0.895146\pi\)
							0.955658	+	0.294479i	\(0.0951462\pi\)
\(29\)	−0.222225	+	0.305867i	−0.0412662	+	0.0567980i	−0.829151	−	0.559024i	\(-0.811176\pi\)
							0.787885	+	0.615822i	\(0.211176\pi\)
\(31\)	−4.96105	+	2.52745i	−0.891031	+	0.453943i
							\(37\)	−5.83310	−	5.83310i	−0.958956	−	0.958956i	0.0402344	−	0.999190i	\(-0.487190\pi\)
−0.999190	+	0.0402344i	\(0.987190\pi\)
\(41\)	0.868531	+	5.48369i	0.135642	+	0.856408i	0.957860	+	0.287236i	\(0.0927363\pi\)
							−0.822218	+	0.569172i	\(0.807264\pi\)
\(43\)	−2.95285	−	2.14537i	−0.450305	−	0.327166i	0.339411	−	0.940638i	\(-0.389772\pi\)
							−0.789716	+	0.613472i	\(0.789772\pi\)
\(47\)	0.733417	−	4.63061i	0.106980	−	0.675444i	−0.874665	−	0.484727i	\(-0.838919\pi\)
							0.981645	−	0.190717i	\(-0.0610812\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bn.a.60.9	✓	272
13.5	odd	4	inner	403.2.bn.a.122.9	yes	272
31.15	odd	10	inner	403.2.bn.a.294.9	yes	272
403.356	even	20	inner	403.2.bn.a.356.9	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bn.a.60.9	✓	272	1.1	even	1	trivial
403.2.bn.a.122.9	yes	272	13.5	odd	4	inner
403.2.bn.a.294.9	yes	272	31.15	odd	10	inner
403.2.bn.a.356.9	yes	272	403.356	even	20	inner