Embedded newform 403.2.bj.a.100.18

• Label: 403.2.bj.a.100.18
• 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			100.18
Character	\(\chi\)	\(=\)	403.100
Dual form			403.2.bj.a.133.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0768429 + 0.0163335i) q^{2} +(-0.788714 + 2.42741i) q^{3} +(-1.82145 - 0.810963i) q^{4} +(1.96469 - 3.40295i) q^{5} +(-0.100255 + 0.173647i) q^{6} +(0.299883 + 2.85320i) q^{7} +(-0.253832 - 0.184420i) q^{8} +(-2.84321 - 2.06571i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0768429	+	0.0163335i	0.0543361	+	0.0115495i	0.235000	−	0.971995i	\(-0.424491\pi\)
							−0.180663	+	0.983545i	\(0.557824\pi\)
\(3\)	−0.788714	+	2.42741i	−0.455364	+	1.40147i	0.415343	+	0.909665i	\(0.363662\pi\)
							−0.870707	+	0.491802i	\(0.836338\pi\)
\(5\)	1.96469	−	3.40295i	0.878638	−	1.52185i	0.0258021	−	0.999667i	\(-0.491786\pi\)
							0.852836	−	0.522179i	\(-0.174881\pi\)
\(7\)	0.299883	+	2.85320i	0.113345	+	1.07841i	0.892336	+	0.451372i	\(0.149065\pi\)
							−0.778990	+	0.627036i	\(0.784268\pi\)
\(11\)	2.31607	+	1.03118i	0.698321	+	0.310912i	0.725017	−	0.688731i	\(-0.241832\pi\)
							−0.0266961	+	0.999644i	\(0.508499\pi\)
\(13\)	1.42220	+	3.31321i	0.394448	+	0.918918i
							\(17\)	−1.59150	+	0.708580i	−0.385994	+	0.171856i	−0.590554	−	0.806998i	\(-0.701091\pi\)
0.204559	+	0.978854i	\(0.434424\pi\)
\(19\)	7.80300	+	1.65858i	1.79013	+	0.380504i	0.978919	−	0.204247i	\(-0.0654747\pi\)
							0.811212	+	0.584752i	\(0.198808\pi\)
\(23\)	−6.48430	+	2.88699i	−1.35207	+	0.601980i	−0.949603	−	0.313455i	\(-0.898514\pi\)
							−0.402466	+	0.915435i	\(0.631847\pi\)
\(29\)	−3.43454	−	0.730034i	−0.637778	−	0.135564i	−0.122337	−	0.992489i	\(-0.539039\pi\)
							−0.515441	+	0.856925i	\(0.672372\pi\)
\(31\)	2.64362	+	4.90013i	0.474809	+	0.880089i
							\(37\)	12.0378			1.97900			0.989500	−	0.144534i	\(-0.0461682\pi\)
0.989500	+	0.144534i	\(0.0461682\pi\)
\(41\)	−2.84890	+	3.16403i	−0.444924	+	0.494138i	−0.923334	−	0.383999i	\(-0.874547\pi\)
							0.478410	+	0.878137i	\(0.341213\pi\)
\(43\)	1.71809	+	0.365192i	0.262007	+	0.0556913i	0.337042	−	0.941490i	\(-0.390574\pi\)
							−0.0750351	+	0.997181i	\(0.523907\pi\)
\(47\)	1.55963	−	4.80004i	0.227495	−	0.700158i	−0.770534	−	0.637399i	\(-0.780010\pi\)
							0.998029	−	0.0627586i	\(-0.0199898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bj.a.100.18	✓	280
13.3	even	3		403.2.bk.a.224.18	yes	280
31.9	even	15		403.2.bk.a.9.18	yes	280
403.133	even	15	inner	403.2.bj.a.133.18	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.18	✓	280	1.1	even	1	trivial
403.2.bj.a.133.18	yes	280	403.133	even	15	inner
403.2.bk.a.9.18	yes	280	31.9	even	15
403.2.bk.a.224.18	yes	280	13.3	even	3