Embedded newform 403.2.bk.a.9.17

• Label: 403.2.bk.a.9.17
• 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			9.17
Character	\(\chi\)	\(=\)	403.9
Dual form			403.2.bk.a.224.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.133023 - 0.147737i) q^{2} +(0.0833490 - 0.0925684i) q^{3} +(0.204926 - 1.94974i) q^{4} +(-0.998333 - 1.72916i) q^{5} -0.0247631 q^{6} +(-1.37062 - 0.995817i) q^{7} +(-0.636973 + 0.462788i) q^{8} +(0.311964 + 2.96813i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.133023	−	0.147737i	−0.0940614	−	0.104466i	0.694276	−	0.719709i	\(-0.255725\pi\)
							−0.788338	+	0.615243i	\(0.789058\pi\)
\(3\)	0.0833490	−	0.0925684i	0.0481216	−	0.0534444i	−0.718603	−	0.695420i	\(-0.755218\pi\)
							0.766725	+	0.641976i	\(0.221885\pi\)
\(5\)	−0.998333	−	1.72916i	−0.446468	−	0.773305i	0.551685	−	0.834052i	\(-0.313985\pi\)
							−0.998153	+	0.0607471i	\(0.980652\pi\)
\(7\)	−1.37062	−	0.995817i	−0.518047	−	0.376383i	0.297820	−	0.954622i	\(-0.403740\pi\)
							−0.815868	+	0.578238i	\(0.803740\pi\)
\(11\)	−2.57458	−	1.87054i	−0.776266	−	0.563990i	0.127590	−	0.991827i	\(-0.459276\pi\)
							−0.903856	+	0.427837i	\(0.859276\pi\)
\(13\)	1.82100	−	3.11191i	0.505055	−	0.863087i
							\(17\)	−3.51884	+	2.55659i	−0.853445	+	0.620064i	−0.926094	−	0.377294i	\(-0.876855\pi\)
0.0726491	+	0.997358i	\(0.476855\pi\)
\(19\)	−0.323292	+	0.994989i	−0.0741682	+	0.228266i	−0.981267	−	0.192651i	\(-0.938292\pi\)
							0.907099	+	0.420917i	\(0.138292\pi\)
\(23\)	0.109463	+	1.04147i	0.0228246	+	0.217162i	0.999989	+	0.00467723i	\(0.00148881\pi\)
							−0.977164	+	0.212484i	\(0.931845\pi\)
\(29\)	0.167526	+	0.186057i	0.0311089	+	0.0345499i	0.758498	−	0.651675i	\(-0.225933\pi\)
							−0.727390	+	0.686225i	\(0.759267\pi\)
\(31\)	−4.17661	−	3.68184i	−0.750141	−	0.661278i
							\(37\)	0.583993	+	1.01150i	0.0960078	+	0.166290i	0.910029	−	0.414545i	\(-0.136059\pi\)
−0.814021	+	0.580835i	\(0.802726\pi\)
\(41\)	2.97207	−	9.14710i	0.464160	−	1.42854i	−0.395876	−	0.918304i	\(-0.629559\pi\)
							0.860036	−	0.510233i	\(-0.170441\pi\)
\(43\)	3.30752	−	10.1795i	0.504392	−	1.55236i	−0.297399	−	0.954753i	\(-0.596119\pi\)
							0.801791	−	0.597605i	\(-0.203881\pi\)
\(47\)	−1.31294	−	4.04082i	−0.191512	−	0.589414i	−1.00000		0.000900727i	\(-0.999713\pi\)
							0.808487	−	0.588514i	\(-0.200287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bk.a.9.17	yes	280
13.3	even	3		403.2.bj.a.133.19	yes	280
31.7	even	15		403.2.bj.a.100.19	✓	280
403.224	even	15	inner	403.2.bk.a.224.17	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.19	✓	280	31.7	even	15
403.2.bj.a.133.19	yes	280	13.3	even	3
403.2.bk.a.9.17	yes	280	1.1	even	1	trivial
403.2.bk.a.224.17	yes	280	403.224	even	15	inner