Embedded newform 403.2.bl.b.16.5

• Label: 403.2.bl.b.16.5
• Level: $403$
• Weight: $2$
• Character: 403.16
• 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			16.5
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.04851 - 0.912054i) q^{2} +(-0.0396616 + 0.377355i) q^{3} +(2.02628 + 2.25041i) q^{4} +0.301769 q^{5} +(0.425415 - 0.736841i) q^{6} +(2.74290 + 3.04630i) q^{7} +(-0.712488 - 2.19281i) q^{8} +(2.79362 + 0.593802i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.04851	−	0.912054i	−1.44851	−	0.644920i	−0.476357	−	0.879252i	\(-0.658043\pi\)
							−0.972157	+	0.234332i	\(0.924710\pi\)
\(3\)	−0.0396616	+	0.377355i	−0.0228986	+	0.217866i	0.977089	+	0.212833i	\(0.0682689\pi\)
							−0.999987	+	0.00503350i	\(0.998398\pi\)
\(5\)	0.301769			0.134955			0.0674777	−	0.997721i	\(-0.478505\pi\)
							0.0674777	+	0.997721i	\(0.478505\pi\)
\(7\)	2.74290	+	3.04630i	1.03672	+	1.15139i	0.988293	+	0.152566i	\(0.0487535\pi\)
							0.0484250	+	0.998827i	\(0.484580\pi\)
\(11\)	−5.66025	+	1.20312i	−1.70663	+	0.362755i	−0.954949	−	0.296771i	\(-0.904090\pi\)
							−0.751682	+	0.659526i	\(0.770757\pi\)
\(13\)	−3.59489	+	0.277095i	−0.997042	+	0.0768523i
							\(17\)	−0.851389	−	0.180968i	−0.206492	−	0.0438913i	0.103504	−	0.994629i	\(-0.466994\pi\)
−0.309996	+	0.950738i	\(0.600328\pi\)
\(19\)	0.349274	+	3.32312i	0.0801289	+	0.762375i	0.958634	+	0.284642i	\(0.0918748\pi\)
							−0.878505	+	0.477733i	\(0.841459\pi\)
\(23\)	−0.161078	+	0.178895i	−0.0335870	+	0.0373021i	−0.759704	−	0.650269i	\(-0.774656\pi\)
							0.726117	+	0.687571i	\(0.241323\pi\)
\(29\)	−5.06011	−	2.25291i	−0.939640	−	0.418354i	−0.120994	−	0.992653i	\(-0.538608\pi\)
							−0.818646	+	0.574299i	\(0.805275\pi\)
\(31\)	4.27927	+	3.56200i	0.768579	+	0.639754i
							\(37\)	2.95160	+	5.11232i	0.485240	+	0.840460i	0.999856	−	0.0169607i	\(-0.00539902\pi\)
−0.514616	+	0.857420i	\(0.672066\pi\)
\(41\)	−1.88108	−	0.837513i	−0.293776	−	0.130798i	0.254556	−	0.967058i	\(-0.418071\pi\)
							−0.548333	+	0.836260i	\(0.684737\pi\)
\(43\)	−0.344608	−	3.27872i	−0.0525522	−	0.500001i	−0.988863	−	0.148828i	\(-0.952450\pi\)
							0.936311	−	0.351172i	\(-0.114217\pi\)
\(47\)	10.3253	+	7.50178i	1.50610	+	1.09425i	0.967871	+	0.251449i	\(0.0809070\pi\)
							0.538231	+	0.842798i	\(0.319093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.5	✓	280
13.9	even	3	inner	403.2.bl.b.295.31	yes	280
31.2	even	5	inner	403.2.bl.b.250.31	yes	280
403.126	even	15	inner	403.2.bl.b.126.5	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.5	✓	280	1.1	even	1	trivial
403.2.bl.b.126.5	yes	280	403.126	even	15	inner
403.2.bl.b.250.31	yes	280	31.2	even	5	inner
403.2.bl.b.295.31	yes	280	13.9	even	3	inner