Embedded newform 403.2.bs.a.4.17

• Label: 403.2.bs.a.4.17
• 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			4.17
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0525799 + 0.247369i) q^{2} +(0.690263 + 0.766614i) q^{3} +(1.76866 + 0.787460i) q^{4} +3.00880i q^{5} +(-0.225931 + 0.130441i) q^{6} +(-0.278018 + 0.624439i) q^{7} +(-0.585086 + 0.805302i) q^{8} +(0.202350 - 1.92523i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0525799	+	0.247369i	−0.0371796	+	0.174916i	−0.992820	−	0.119621i	\(-0.961832\pi\)
							0.955640	+	0.294537i	\(0.0951655\pi\)
\(3\)	0.690263	+	0.766614i	0.398523	+	0.442605i	0.908691	−	0.417470i	\(-0.137083\pi\)
							−0.510167	+	0.860075i	\(0.670416\pi\)
\(5\)			3.00880i			1.34558i	0.739835	+	0.672789i	\(0.234904\pi\)
							−0.739835	+	0.672789i	\(0.765096\pi\)
\(7\)	−0.278018	+	0.624439i	−0.105081	+	0.236016i	−0.958432	−	0.285321i	\(-0.907900\pi\)
							0.853351	+	0.521337i	\(0.174566\pi\)
\(11\)	−0.385366	+	0.0405036i	−0.116192	+	0.0122123i	−0.162446	−	0.986717i	\(-0.551938\pi\)
							0.0462539	+	0.998930i	\(0.485272\pi\)
\(13\)	−0.696679	+	3.53760i	−0.193224	+	0.981155i
							\(17\)	0.685186	−	6.51911i	0.166182	−	1.58112i	−0.520309	−	0.853978i	\(-0.674183\pi\)
0.686491	−	0.727138i	\(-0.259150\pi\)
\(19\)	−4.45601	−	4.01221i	−1.02228	−	0.920463i	−0.0254203	−	0.999677i	\(-0.508092\pi\)
							−0.996858	+	0.0792137i	\(0.974759\pi\)
\(23\)	2.54744	−	1.13419i	0.531178	−	0.236496i	−0.123579	−	0.992335i	\(-0.539437\pi\)
							0.654758	+	0.755839i	\(0.272771\pi\)
\(29\)	−7.49118	−	1.59230i	−1.39108	−	0.295683i	−0.549354	−	0.835590i	\(-0.685126\pi\)
							−0.841725	+	0.539907i	\(0.818459\pi\)
\(31\)	4.78673	+	2.84381i	0.859721	+	0.510764i
							\(37\)	9.45331	+	5.45787i	1.55411	+	0.897268i	0.997800	+	0.0662962i	\(0.0211182\pi\)
0.556314	+	0.830972i	\(0.312215\pi\)
\(41\)	−0.546642	+	2.57175i	−0.0853711	+	0.401640i	−0.999996	−	0.00266294i	\(-0.999152\pi\)
							0.914625	+	0.404302i	\(0.132486\pi\)
\(43\)	2.40896	−	2.67543i	0.367364	−	0.407999i	−0.530915	−	0.847425i	\(-0.678152\pi\)
							0.898279	+	0.439426i	\(0.144818\pi\)
\(47\)	6.78350	+	2.20409i	0.989475	+	0.321500i	0.758652	−	0.651496i	\(-0.225858\pi\)
							0.230823	+	0.972996i	\(0.425858\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.17	✓	288
13.10	even	6	inner	403.2.bs.a.283.20	yes	288
31.8	even	5	inner	403.2.bs.a.225.20	yes	288
403.101	even	30	inner	403.2.bs.a.101.17	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.17	✓	288	1.1	even	1	trivial
403.2.bs.a.101.17	yes	288	403.101	even	30	inner
403.2.bs.a.225.20	yes	288	31.8	even	5	inner
403.2.bs.a.283.20	yes	288	13.10	even	6	inner