Embedded newform 403.2.bk.a.9.6

• Label: 403.2.bk.a.9.6
• 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			9.6
Character	\(\chi\)	\(=\)	403.9
Dual form			403.2.bk.a.224.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44960 - 1.60995i) q^{2} +(0.384158 - 0.426651i) q^{3} +(-0.281525 + 2.67853i) q^{4} +(-1.83417 - 3.17687i) q^{5} -1.24376 q^{6} +(0.633388 + 0.460183i) q^{7} +(1.21509 - 0.882816i) q^{8} +(0.279132 + 2.65576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44960	−	1.60995i	−1.02502	−	1.13841i	−0.990291	−	0.139009i	\(-0.955608\pi\)
							−0.0347338	−	0.999397i	\(-0.511058\pi\)
\(3\)	0.384158	−	0.426651i	0.221794	−	0.246327i	−0.621971	−	0.783040i	\(-0.713668\pi\)
							0.843764	+	0.536714i	\(0.180334\pi\)
\(5\)	−1.83417	−	3.17687i	−0.820264	−	1.42074i	−0.905485	−	0.424377i	\(-0.860493\pi\)
							0.0852212	−	0.996362i	\(-0.472840\pi\)
\(7\)	0.633388	+	0.460183i	0.239398	+	0.173933i	0.701015	−	0.713146i	\(-0.252731\pi\)
							−0.461617	+	0.887079i	\(0.652731\pi\)
\(11\)	−2.98693	−	2.17013i	−0.900593	−	0.654319i	0.0380250	−	0.999277i	\(-0.487893\pi\)
							−0.938618	+	0.344957i	\(0.887893\pi\)
\(13\)	−3.50438	+	0.848113i	−0.971941	+	0.235224i
							\(17\)	−2.21307	+	1.60789i	−0.536748	+	0.389970i	−0.822876	−	0.568221i	\(-0.807632\pi\)
0.286128	+	0.958191i	\(0.407632\pi\)
\(19\)	1.02674	−	3.15999i	0.235551	−	0.724951i	−0.761497	−	0.648168i	\(-0.775535\pi\)
							0.997048	−	0.0767826i	\(-0.0244647\pi\)
\(23\)	0.120254	+	1.14414i	0.0250747	+	0.238570i	0.999880	+	0.0155111i	\(0.00493753\pi\)
							−0.974805	+	0.223059i	\(0.928396\pi\)
\(29\)	−3.17329	−	3.52429i	−0.589265	−	0.654445i	0.372593	−	0.927995i	\(-0.378469\pi\)
							−0.961858	+	0.273550i	\(0.911802\pi\)
\(31\)	−2.68177	+	4.87936i	−0.481659	+	0.876359i
							\(37\)	2.33561	+	4.04540i	0.383972	+	0.665059i	0.991626	−	0.129143i	\(-0.0412225\pi\)
−0.607654	+	0.794202i	\(0.707889\pi\)
\(41\)	−0.933334	+	2.87251i	−0.145762	+	0.448610i	−0.997108	−	0.0759945i	\(-0.975787\pi\)
							0.851346	+	0.524605i	\(0.175787\pi\)
\(43\)	0.102716	−	0.316128i	0.0156641	−	0.0482091i	−0.942919	−	0.333022i	\(-0.891932\pi\)
							0.958583	+	0.284813i	\(0.0919316\pi\)
\(47\)	1.72441	+	5.30717i	0.251530	+	0.774131i	0.994493	+	0.104799i	\(0.0334198\pi\)
							−0.742963	+	0.669332i	\(0.766580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bk.a.9.6	yes	280
13.3	even	3		403.2.bj.a.133.30	yes	280
31.7	even	15		403.2.bj.a.100.30	✓	280
403.224	even	15	inner	403.2.bk.a.224.6	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.30	✓	280	31.7	even	15
403.2.bj.a.133.30	yes	280	13.3	even	3
403.2.bk.a.9.6	yes	280	1.1	even	1	trivial
403.2.bk.a.224.6	yes	280	403.224	even	15	inner