Embedded newform 403.2.bs.a.4.3

• Label: 403.2.bs.a.4.3
• Level: $403$
• Weight: $2$
• Character: 403.4
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bs (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.518746 + 2.44051i) q^{2} +(-1.86014 - 2.06590i) q^{3} +(-3.85989 - 1.71854i) q^{4} +2.00004i q^{5} +(6.00678 - 3.46802i) q^{6} +(-1.59632 + 3.58539i) q^{7} +(3.26332 - 4.49158i) q^{8} +(-0.494216 + 4.70215i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.518746	+	2.44051i	−0.366809	+	1.72570i	0.277315	+	0.960779i	\(0.410555\pi\)
							−0.644124	+	0.764921i	\(0.722778\pi\)
\(3\)	−1.86014	−	2.06590i	−1.07395	−	1.19275i	−0.980377	−	0.197134i	\(-0.936837\pi\)
							−0.0935766	−	0.995612i	\(-0.529830\pi\)
\(5\)			2.00004i			0.894446i	0.894422	+	0.447223i	\(0.147587\pi\)
							−0.894422	+	0.447223i	\(0.852413\pi\)
\(7\)	−1.59632	+	3.58539i	−0.603351	+	1.35515i	0.311071	+	0.950387i	\(0.399312\pi\)
							−0.914422	+	0.404762i	\(0.867354\pi\)
\(11\)	−1.84978	+	0.194419i	−0.557729	+	0.0586197i	−0.379200	−	0.925315i	\(-0.623801\pi\)
							−0.178529	+	0.983935i	\(0.557134\pi\)
\(13\)	0.726405	−	3.53162i	0.201468	−	0.979495i
							\(17\)	0.706683	−	6.72364i	0.171396	−	1.63072i	−0.483741	−	0.875211i	\(-0.660722\pi\)
0.655136	−	0.755511i	\(-0.272611\pi\)
\(19\)	0.410670	+	0.369769i	0.0942141	+	0.0848307i	0.714892	−	0.699235i	\(-0.246476\pi\)
							−0.620678	+	0.784066i	\(0.713143\pi\)
\(23\)	0.0161603	−	0.00719504i	0.00336966	−	0.00150027i	−0.405051	−	0.914294i	\(-0.632746\pi\)
							0.408421	+	0.912794i	\(0.366080\pi\)
\(29\)	−8.78316	−	1.86692i	−1.63099	−	0.346678i	−0.700688	−	0.713468i	\(-0.747123\pi\)
							−0.930304	+	0.366790i	\(0.880457\pi\)
\(31\)	5.34893	−	1.54562i	0.960696	−	0.277602i
							\(37\)	−0.381900	−	0.220490i	−0.0627839	−	0.0362483i	0.468279	−	0.883580i	\(-0.344874\pi\)
−0.531063	+	0.847332i	\(0.678207\pi\)
\(41\)	2.16483	−	10.1847i	0.338090	−	1.59059i	−0.400412	−	0.916335i	\(-0.631133\pi\)
							0.738501	−	0.674252i	\(-0.235534\pi\)
\(43\)	−0.304383	+	0.338052i	−0.0464180	+	0.0515524i	−0.765907	−	0.642952i	\(-0.777710\pi\)
							0.719489	+	0.694504i	\(0.244376\pi\)
\(47\)	−11.7978	−	3.83334i	−1.72089	−	0.559151i	−0.728804	−	0.684722i	\(-0.759924\pi\)
							−0.992085	+	0.125572i	\(0.959924\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.3	✓	288
13.10	even	6	inner	403.2.bs.a.283.34	yes	288
31.8	even	5	inner	403.2.bs.a.225.34	yes	288
403.101	even	30	inner	403.2.bs.a.101.3	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.3	✓	288	1.1	even	1	trivial
403.2.bs.a.101.3	yes	288	403.101	even	30	inner
403.2.bs.a.225.34	yes	288	31.8	even	5	inner
403.2.bs.a.283.34	yes	288	13.10	even	6	inner