Embedded newform 403.2.bk.a.9.1

• Label: 403.2.bk.a.9.1
• Level: $403$
• Weight: $2$
• Character: 403.9
• 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.bk (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			9.1
Character	\(\chi\)	\(=\)	403.9
Dual form			403.2.bk.a.224.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83126 - 2.03382i) q^{2} +(-0.500677 + 0.556058i) q^{3} +(-0.573858 + 5.45990i) q^{4} +(-0.330308 - 0.572109i) q^{5} +2.04780 q^{6} +(-3.67674 - 2.67131i) q^{7} +(7.72715 - 5.61410i) q^{8} +(0.255062 + 2.42676i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83126	−	2.03382i	−1.29490	−	1.43813i	−0.835090	−	0.550113i	\(-0.814584\pi\)
							−0.459809	−	0.888018i	\(-0.652082\pi\)
\(3\)	−0.500677	+	0.556058i	−0.289066	+	0.321040i	−0.870134	−	0.492814i	\(-0.835968\pi\)
							0.581069	+	0.813855i	\(0.302635\pi\)
\(5\)	−0.330308	−	0.572109i	−0.147718	−	0.255855i	0.782666	−	0.622442i	\(-0.213859\pi\)
							−0.930384	+	0.366587i	\(0.880526\pi\)
\(7\)	−3.67674	−	2.67131i	−1.38968	−	1.00966i	−0.995901	−	0.0904534i	\(-0.971168\pi\)
							−0.393777	−	0.919206i	\(-0.628832\pi\)
\(11\)	2.97929	+	2.16458i	0.898289	+	0.652645i	0.938026	−	0.346565i	\(-0.112652\pi\)
							−0.0397369	+	0.999210i	\(0.512652\pi\)
\(13\)	0.517230	+	3.56826i	0.143454	+	0.989657i
							\(17\)	1.40380	−	1.01992i	0.340473	−	0.247368i	−0.404389	−	0.914587i	\(-0.632516\pi\)
0.744861	+	0.667219i	\(0.232516\pi\)
\(19\)	1.91178	−	5.88385i	0.438592	−	1.34985i	−0.450769	−	0.892641i	\(-0.648850\pi\)
							0.889361	−	0.457206i	\(-0.151150\pi\)
\(23\)	−0.139782	−	1.32994i	−0.0291467	−	0.277312i	−0.999382	−	0.0351484i	\(-0.988810\pi\)
							0.970235	−	0.242164i	\(-0.0778570\pi\)
\(29\)	0.304318	+	0.337979i	0.0565104	+	0.0627611i	0.770737	−	0.637153i	\(-0.219888\pi\)
							−0.714227	+	0.699914i	\(0.753221\pi\)
\(31\)	5.04660	+	2.35199i	0.906395	+	0.422430i
							\(37\)	−2.35856	−	4.08515i	−0.387745	−	0.671595i	0.604401	−	0.796681i	\(-0.293413\pi\)
−0.992146	+	0.125086i	\(0.960079\pi\)
\(41\)	1.93575	−	5.95762i	0.302313	−	0.930423i	−0.678353	−	0.734736i	\(-0.737306\pi\)
							0.980666	−	0.195688i	\(-0.0626939\pi\)
\(43\)	3.71866	−	11.4449i	0.567091	−	1.74533i	−0.0945659	−	0.995519i	\(-0.530146\pi\)
							0.661657	−	0.749807i	\(-0.269854\pi\)
\(47\)	0.133132	+	0.409738i	0.0194193	+	0.0597665i	0.960297	−	0.278981i	\(-0.0899967\pi\)
							−0.940877	+	0.338748i	\(0.889997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bk.a.9.1	yes	280
13.3	even	3		403.2.bj.a.133.35	yes	280
31.7	even	15		403.2.bj.a.100.35	✓	280
403.224	even	15	inner	403.2.bk.a.224.1	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.35	✓	280	31.7	even	15
403.2.bj.a.133.35	yes	280	13.3	even	3
403.2.bk.a.9.1	yes	280	1.1	even	1	trivial
403.2.bk.a.224.1	yes	280	403.224	even	15	inner