Embedded newform 403.2.bj.a.100.11

• Label: 403.2.bj.a.100.11
• 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			100.11
Character	\(\chi\)	\(=\)	403.100
Dual form			403.2.bj.a.133.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34661 - 0.286231i) q^{2} +(0.272316 - 0.838103i) q^{3} +(-0.0956586 - 0.0425900i) q^{4} +(-1.29583 + 2.24445i) q^{5} +(-0.606595 + 1.05065i) q^{6} +(-0.453578 - 4.31551i) q^{7} +(2.34416 + 1.70313i) q^{8} +(1.79879 + 1.30690i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34661	−	0.286231i	−0.952198	−	0.202396i	−0.294478	−	0.955658i	\(-0.595146\pi\)
							−0.657720	+	0.753263i	\(0.728479\pi\)
\(3\)	0.272316	−	0.838103i	0.157222	−	0.483879i	−0.841157	−	0.540790i	\(-0.818125\pi\)
							0.998379	+	0.0569114i	\(0.0181253\pi\)
\(5\)	−1.29583	+	2.24445i	−0.579514	+	1.00375i	0.416021	+	0.909355i	\(0.363424\pi\)
							−0.995535	+	0.0943930i	\(0.969909\pi\)
\(7\)	−0.453578	−	4.31551i	−0.171436	−	1.63111i	−0.654880	−	0.755733i	\(-0.727281\pi\)
							0.483443	−	0.875376i	\(-0.339386\pi\)
\(11\)	−1.06014	−	0.472005i	−0.319644	−	0.142315i	0.240642	−	0.970614i	\(-0.422642\pi\)
							−0.560286	+	0.828299i	\(0.689309\pi\)
\(13\)	0.385101	−	3.58493i	0.106808	−	0.994280i
							\(17\)	−4.41501	+	1.96569i	−1.07080	+	0.476750i	−0.864962	−	0.501837i	\(-0.832658\pi\)
−0.205835	+	0.978587i	\(0.565991\pi\)
\(19\)	−6.82304	−	1.45028i	−1.56531	−	0.332718i	−0.657949	−	0.753062i	\(-0.728576\pi\)
							−0.907365	+	0.420345i	\(0.861909\pi\)
\(23\)	−2.68239	+	1.19428i	−0.559317	+	0.249024i	−0.666869	−	0.745175i	\(-0.732366\pi\)
							0.107551	+	0.994200i	\(0.465699\pi\)
\(29\)	−5.13190	−	1.09082i	−0.952969	−	0.202560i	−0.294909	−	0.955525i	\(-0.595289\pi\)
							−0.658060	+	0.752965i	\(0.728623\pi\)
\(31\)	−1.90385	+	5.23215i	−0.341942	+	0.939721i
							\(37\)	−8.78065			−1.44353			−0.721765	−	0.692138i	\(-0.756669\pi\)
−0.721765	+	0.692138i	\(0.756669\pi\)
\(41\)	−0.518398	+	0.575739i	−0.0809602	+	0.0899154i	−0.782264	−	0.622947i	\(-0.785935\pi\)
							0.701304	+	0.712863i	\(0.252602\pi\)
\(43\)	7.75731	+	1.64887i	1.18298	+	0.251450i	0.757081	−	0.653321i	\(-0.226625\pi\)
							0.425898	+	0.904771i	\(0.359958\pi\)
\(47\)	2.58592	−	7.95864i	0.377195	−	1.16089i	−0.564791	−	0.825234i	\(-0.691043\pi\)
							0.941986	−	0.335653i	\(-0.108957\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bj.a.100.11	✓	280
13.3	even	3		403.2.bk.a.224.25	yes	280
31.9	even	15		403.2.bk.a.9.25	yes	280
403.133	even	15	inner	403.2.bj.a.133.11	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.11	✓	280	1.1	even	1	trivial
403.2.bj.a.133.11	yes	280	403.133	even	15	inner
403.2.bk.a.9.25	yes	280	31.9	even	15
403.2.bk.a.224.25	yes	280	13.3	even	3