Embedded newform 403.2.bk.a.9.7

• Label: 403.2.bk.a.9.7
• Level: $403$
• Weight: $2$
• Character: 403.9
• 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.bk (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			9.7
Character	\(\chi\)	\(=\)	403.9
Dual form			403.2.bk.a.224.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32661 - 1.47335i) q^{2} +(0.650439 - 0.722386i) q^{3} +(-0.201810 + 1.92010i) q^{4} +(1.17248 + 2.03079i) q^{5} -1.92721 q^{6} +(0.184244 + 0.133861i) q^{7} +(-0.111198 + 0.0807903i) q^{8} +(0.214815 + 2.04383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32661	−	1.47335i	−0.938057	−	1.04182i	−0.999046	−	0.0436796i	\(-0.986092\pi\)
							0.0609884	−	0.998138i	\(-0.480575\pi\)
\(3\)	0.650439	−	0.722386i	0.375531	−	0.417070i	−0.525521	−	0.850781i	\(-0.676129\pi\)
							0.901052	+	0.433711i	\(0.142796\pi\)
\(5\)	1.17248	+	2.03079i	0.524348	+	0.908198i	0.999598	+	0.0283469i	\(0.00902431\pi\)
							−0.475250	+	0.879851i	\(0.657642\pi\)
\(7\)	0.184244	+	0.133861i	0.0696377	+	0.0505948i	0.622059	−	0.782970i	\(-0.286296\pi\)
							−0.552422	+	0.833565i	\(0.686296\pi\)
\(11\)	−0.0157269	−	0.0114262i	−0.00474183	−	0.00344514i	0.585412	−	0.810736i	\(-0.300933\pi\)
							−0.590154	+	0.807291i	\(0.700933\pi\)
\(13\)	0.796867	+	3.51639i	0.221011	+	0.975271i
							\(17\)	1.64172	−	1.19278i	0.398176	−	0.289292i	−0.370621	−	0.928784i	\(-0.620855\pi\)
0.768798	+	0.639492i	\(0.220855\pi\)
\(19\)	−1.92622	+	5.92830i	−0.441906	+	1.36005i	0.443936	+	0.896059i	\(0.353582\pi\)
							−0.885842	+	0.463988i	\(0.846418\pi\)
\(23\)	0.538492	+	5.12341i	0.112283	+	1.06831i	0.895043	+	0.445981i	\(0.147145\pi\)
							−0.782759	+	0.622325i	\(0.786188\pi\)
\(29\)	−4.45166	−	4.94407i	−0.826652	−	0.918090i	0.171089	−	0.985255i	\(-0.445271\pi\)
							−0.997742	+	0.0671650i	\(0.978605\pi\)
\(31\)	3.27177	−	4.50505i	0.587628	−	0.809131i
							\(37\)	1.92029	+	3.32604i	0.315694	+	0.546797i	0.979585	−	0.201032i	\(-0.0644295\pi\)
−0.663891	+	0.747829i	\(0.731096\pi\)
\(41\)	−2.96721	+	9.13215i	−0.463401	+	1.42620i	0.397581	+	0.917567i	\(0.369850\pi\)
							−0.860982	+	0.508635i	\(0.830150\pi\)
\(43\)	3.46561	−	10.6660i	0.528500	−	1.62656i	−0.228788	−	0.973476i	\(-0.573476\pi\)
							0.757289	−	0.653080i	\(-0.226524\pi\)
\(47\)	−2.96858	−	9.13634i	−0.433012	−	1.33267i	−0.895110	−	0.445844i	\(-0.852903\pi\)
							0.462099	−	0.886828i	\(-0.347097\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bk.a.9.7	yes	280
13.3	even	3		403.2.bj.a.133.29	yes	280
31.7	even	15		403.2.bj.a.100.29	✓	280
403.224	even	15	inner	403.2.bk.a.224.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.29	✓	280	31.7	even	15
403.2.bj.a.133.29	yes	280	13.3	even	3
403.2.bk.a.9.7	yes	280	1.1	even	1	trivial
403.2.bk.a.224.7	yes	280	403.224	even	15	inner