Embedded newform 403.2.bj.a.100.2

• Label: 403.2.bj.a.100.2
• Level: $403$
• Weight: $2$
• Character: 403.100
• 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.bj (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			100.2
Character	\(\chi\)	\(=\)	403.100
Dual form			403.2.bj.a.133.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57899 - 0.548181i) q^{2} +(-0.805625 + 2.47946i) q^{3} +(4.52359 + 2.01403i) q^{4} +(-1.73519 + 3.00543i) q^{5} +(3.43689 - 5.95286i) q^{6} +(-0.217284 - 2.06732i) q^{7} +(-6.29611 - 4.57439i) q^{8} +(-3.07163 - 2.23167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57899	−	0.548181i	−1.82362	−	0.387622i	−0.836547	−	0.547895i	\(-0.815429\pi\)
							−0.987073	+	0.160272i	\(0.948763\pi\)
\(3\)	−0.805625	+	2.47946i	−0.465128	+	1.43152i	0.393694	+	0.919241i	\(0.371197\pi\)
							−0.858822	+	0.512274i	\(0.828803\pi\)
\(5\)	−1.73519	+	3.00543i	−0.775999	+	1.34407i	0.158233	+	0.987402i	\(0.449420\pi\)
							−0.934232	+	0.356667i	\(0.883913\pi\)
\(7\)	−0.217284	−	2.06732i	−0.0821258	−	0.781375i	−0.955632	−	0.294562i	\(-0.904826\pi\)
							0.873507	−	0.486812i	\(-0.161840\pi\)
\(11\)	−3.58309	−	1.59530i	−1.08034	−	0.481000i	−0.212156	−	0.977236i	\(-0.568048\pi\)
							−0.868188	+	0.496236i	\(0.834715\pi\)
\(13\)	−1.82969	−	3.10680i	−0.507466	−	0.861672i
							\(17\)	4.39324	−	1.95600i	1.06552	−	0.474399i	0.202348	−	0.979314i	\(-0.435143\pi\)
0.863169	+	0.504915i	\(0.168476\pi\)
\(19\)	2.44661	+	0.520043i	0.561291	+	0.119306i	0.479816	−	0.877369i	\(-0.340704\pi\)
							0.0814751	+	0.996675i	\(0.474037\pi\)
\(23\)	−4.03972	+	1.79860i	−0.842339	+	0.375034i	−0.782102	−	0.623151i	\(-0.785852\pi\)
							−0.0602372	+	0.998184i	\(0.519186\pi\)
\(29\)	4.24194	+	0.901652i	0.787709	+	0.167433i	0.584167	−	0.811634i	\(-0.301421\pi\)
							0.203542	+	0.979066i	\(0.434755\pi\)
\(31\)	5.24451	+	1.86952i	0.941942	+	0.335776i
							\(37\)	−9.49556			−1.56106			−0.780530	−	0.625118i	\(-0.785051\pi\)
−0.780530	+	0.625118i	\(0.785051\pi\)
\(41\)	−0.847329	+	0.941054i	−0.132331	+	0.146968i	−0.805668	−	0.592367i	\(-0.798193\pi\)
							0.673337	+	0.739335i	\(0.264860\pi\)
\(43\)	−8.84813	−	1.88073i	−1.34933	−	0.286808i	−0.524123	−	0.851643i	\(-0.675607\pi\)
							−0.825204	+	0.564835i	\(0.808940\pi\)
\(47\)	2.83344	−	8.72042i	0.413299	−	1.27200i	−0.500464	−	0.865757i	\(-0.666837\pi\)
							0.913763	−	0.406247i	\(-0.133163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bj.a.100.2	✓	280
13.3	even	3		403.2.bk.a.224.34	yes	280
31.9	even	15		403.2.bk.a.9.34	yes	280
403.133	even	15	inner	403.2.bj.a.133.2	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.2	✓	280	1.1	even	1	trivial
403.2.bj.a.133.2	yes	280	403.133	even	15	inner
403.2.bk.a.9.34	yes	280	31.9	even	15
403.2.bk.a.224.34	yes	280	13.3	even	3