Embedded newform 403.2.bl.b.126.18

• Label: 403.2.bl.b.126.18
• Level: $403$
• Weight: $2$
• Character: 403.126
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bl (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			126.18
Character	\(\chi\)	\(=\)	403.126
Dual form			403.2.bl.b.16.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0464992 - 0.0207028i) q^{2} +(0.316586 + 3.01212i) q^{3} +(-1.33653 + 1.48436i) q^{4} +3.26010 q^{5} +(0.0770802 + 0.133507i) q^{6} +(-0.734123 + 0.815327i) q^{7} +(-0.0628748 + 0.193509i) q^{8} +(-6.03818 + 1.28346i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0464992	−	0.0207028i	0.0328799	−	0.0146391i	−0.390231	−	0.920717i	\(-0.627605\pi\)
							0.423111	+	0.906078i	\(0.360938\pi\)
\(3\)	0.316586	+	3.01212i	0.182781	+	1.73905i	0.574094	+	0.818789i	\(0.305354\pi\)
							−0.391313	+	0.920258i	\(0.627979\pi\)
\(5\)	3.26010			1.45796			0.728981	−	0.684534i	\(-0.239994\pi\)
							0.728981	+	0.684534i	\(0.239994\pi\)
\(7\)	−0.734123	+	0.815327i	−0.277473	+	0.308165i	−0.865732	−	0.500508i	\(-0.833147\pi\)
							0.588260	+	0.808672i	\(0.299813\pi\)
\(11\)	5.61283	+	1.19304i	1.69233	+	0.359716i	0.950465	−	0.310832i	\(-0.100608\pi\)
							0.741866	+	0.670548i	\(0.233941\pi\)
\(13\)	3.60499	+	0.0633711i	0.999846	+	0.0175760i
							\(17\)	−4.12661	+	0.877138i	−1.00085	+	0.212737i	−0.679067	−	0.734076i	\(-0.737615\pi\)
−0.321783	+	0.946814i	\(0.604282\pi\)
\(19\)	0.436794	−	4.15581i	0.100207	−	0.953409i	−0.822723	−	0.568442i	\(-0.807546\pi\)
							0.922930	−	0.384967i	\(-0.125787\pi\)
\(23\)	−5.65852	−	6.28442i	−1.17988	−	1.31039i	−0.940629	−	0.339436i	\(-0.889764\pi\)
							−0.239254	−	0.970957i	\(-0.576903\pi\)
\(29\)	−3.07076	+	1.36719i	−0.570227	+	0.253881i	−0.671534	−	0.740974i	\(-0.734364\pi\)
							0.101307	+	0.994855i	\(0.467698\pi\)
\(31\)	2.53302	−	4.95820i	0.454944	−	0.890520i
							\(37\)	−1.85882	+	3.21958i	−0.305589	+	0.529295i	−0.977392	−	0.211434i	\(-0.932187\pi\)
0.671803	+	0.740729i	\(0.265520\pi\)
\(41\)	1.76947	−	0.787821i	0.276346	−	0.123037i	−0.263885	−	0.964554i	\(-0.585004\pi\)
							0.540230	+	0.841517i	\(0.318337\pi\)
\(43\)	0.732469	−	6.96898i	0.111701	−	1.06276i	−0.784810	−	0.619737i	\(-0.787239\pi\)
							0.896510	−	0.443023i	\(-0.146094\pi\)
\(47\)	7.25820	−	5.27339i	1.05872	−	0.769203i	0.0848669	−	0.996392i	\(-0.472954\pi\)
							0.973851	+	0.227189i	\(0.0729535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.126.18	yes	280
13.3	even	3	inner	403.2.bl.b.250.18	yes	280
31.16	even	5	inner	403.2.bl.b.295.18	yes	280
403.16	even	15	inner	403.2.bl.b.16.18	✓	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.18	✓	280	403.16	even	15	inner
403.2.bl.b.126.18	yes	280	1.1	even	1	trivial
403.2.bl.b.250.18	yes	280	13.3	even	3	inner
403.2.bl.b.295.18	yes	280	31.16	even	5	inner