Embedded newform 403.2.bl.b.126.17

• Label: 403.2.bl.b.126.17
• 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			126.17
Character	\(\chi\)	\(=\)	403.126
Dual form			403.2.bl.b.16.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0396375 + 0.0176478i) q^{2} +(-0.210379 - 2.00162i) q^{3} +(-1.33700 + 1.48489i) q^{4} +0.511564 q^{5} +(0.0436630 + 0.0756265i) q^{6} +(1.64562 - 1.82764i) q^{7} +(0.0536061 - 0.164983i) q^{8} +(-1.02778 + 0.218461i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0396375	+	0.0176478i	−0.0280279	+	0.0124788i	−0.420703	−	0.907199i	\(-0.638216\pi\)
							0.392675	+	0.919677i	\(0.371550\pi\)
\(3\)	−0.210379	−	2.00162i	−0.121462	−	1.15564i	−0.870176	−	0.492742i	\(-0.835995\pi\)
							0.748713	−	0.662894i	\(-0.230672\pi\)
\(5\)	0.511564			0.228778			0.114389	−	0.993436i	\(-0.463509\pi\)
							0.114389	+	0.993436i	\(0.463509\pi\)
\(7\)	1.64562	−	1.82764i	0.621985	−	0.690784i	−0.347011	−	0.937861i	\(-0.612803\pi\)
							0.968996	+	0.247077i	\(0.0794700\pi\)
\(11\)	−2.25522	−	0.479362i	−0.679975	−	0.144533i	−0.145039	−	0.989426i	\(-0.546331\pi\)
							−0.534936	+	0.844893i	\(0.679664\pi\)
\(13\)	2.26084	−	2.80867i	0.627043	−	0.778985i
							\(17\)	−2.98552	+	0.634591i	−0.724094	+	0.153911i	−0.555193	−	0.831722i	\(-0.687355\pi\)
−0.168902	+	0.985633i	\(0.554022\pi\)
\(19\)	0.527832	−	5.02198i	0.121093	−	1.15212i	−0.750149	−	0.661269i	\(-0.770018\pi\)
							0.871242	−	0.490854i	\(-0.163315\pi\)
\(23\)	1.74544	+	1.93851i	0.363950	+	0.404208i	0.897110	−	0.441807i	\(-0.145662\pi\)
							−0.533160	+	0.846014i	\(0.678996\pi\)
\(29\)	8.91076	−	3.96733i	1.65469	−	0.736714i	0.654869	−	0.755743i	\(-0.272724\pi\)
							0.999819	+	0.0190285i	\(0.00605733\pi\)
\(31\)	−4.35284	−	3.47171i	−0.781793	−	0.623538i
							\(37\)	−0.852503	+	1.47658i	−0.140151	+	0.242748i	−0.927553	−	0.373691i	\(-0.878092\pi\)
0.787403	+	0.616439i	\(0.211425\pi\)
\(41\)	−0.466435	+	0.207670i	−0.0728449	+	0.0324327i	−0.442836	−	0.896603i	\(-0.646028\pi\)
							0.369991	+	0.929035i	\(0.379361\pi\)
\(43\)	0.0869595	−	0.827365i	0.0132612	−	0.126172i	−0.985888	−	0.167404i	\(-0.946462\pi\)
							0.999150	+	0.0412320i	\(0.0131283\pi\)
\(47\)	0.0827363	−	0.0601114i	0.0120683	−	0.00876815i	−0.581735	−	0.813379i	\(-0.697626\pi\)
							0.593803	+	0.804610i	\(0.297626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.126.17	yes	280
13.3	even	3	inner	403.2.bl.b.250.19	yes	280
31.16	even	5	inner	403.2.bl.b.295.19	yes	280
403.16	even	15	inner	403.2.bl.b.16.17	✓	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.17	✓	280	403.16	even	15	inner
403.2.bl.b.126.17	yes	280	1.1	even	1	trivial
403.2.bl.b.250.19	yes	280	13.3	even	3	inner
403.2.bl.b.295.19	yes	280	31.16	even	5	inner