Embedded newform 403.2.bl.b.16.16

• Label: 403.2.bl.b.16.16
• 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			16.16
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.187032 - 0.0832719i) q^{2} +(-0.0175618 + 0.167089i) q^{3} +(-1.31021 - 1.45514i) q^{4} -4.19195 q^{5} +(0.0171984 - 0.0297885i) q^{6} +(1.71571 + 1.90549i) q^{7} +(0.250410 + 0.770684i) q^{8} +(2.90683 + 0.617866i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.187032	−	0.0832719i	−0.132251	−	0.0588821i	0.339544	−	0.940590i	\(-0.389727\pi\)
							−0.471795	+	0.881708i	\(0.656394\pi\)
\(3\)	−0.0175618	+	0.167089i	−0.0101393	+	0.0964688i	−0.998421	−	0.0561710i	\(-0.982111\pi\)
							0.988282	+	0.152640i	\(0.0487775\pi\)
\(5\)	−4.19195			−1.87470			−0.937349	−	0.348393i	\(-0.886727\pi\)
							−0.937349	+	0.348393i	\(0.886727\pi\)
\(7\)	1.71571	+	1.90549i	0.648477	+	0.720207i	0.974308	−	0.225221i	\(-0.0723106\pi\)
							−0.325831	+	0.945428i	\(0.605644\pi\)
\(11\)	2.54449	−	0.540847i	0.767192	−	0.163072i	0.192339	−	0.981329i	\(-0.438393\pi\)
							0.574853	+	0.818257i	\(0.305059\pi\)
\(13\)	−3.60467	−	0.0795667i	−0.999756	−	0.0220678i
							\(17\)	5.68212	+	1.20777i	1.37812	+	0.292928i	0.836624	−	0.547778i	\(-0.184526\pi\)
0.541493	+	0.840706i	\(0.317859\pi\)
\(19\)	0.420086	+	3.99685i	0.0963742	+	0.916940i	0.930728	+	0.365711i	\(0.119174\pi\)
							−0.834354	+	0.551229i	\(0.814159\pi\)
\(23\)	−0.257030	+	0.285461i	−0.0535945	+	0.0595228i	−0.769346	−	0.638832i	\(-0.779418\pi\)
							0.715751	+	0.698355i	\(0.246084\pi\)
\(29\)	1.02128	+	0.454701i	0.189646	+	0.0844359i	0.499363	−	0.866393i	\(-0.333568\pi\)
							−0.309717	+	0.950829i	\(0.600234\pi\)
\(31\)	2.11649	+	5.14980i	0.380133	+	0.924932i
							\(37\)	2.51832	+	4.36186i	0.414010	+	0.717086i	0.995324	−	0.0965935i	\(-0.0307947\pi\)
−0.581314	+	0.813679i	\(0.697461\pi\)
\(41\)	−8.25122	−	3.67368i	−1.28862	−	0.573733i	−0.355968	−	0.934498i	\(-0.615849\pi\)
							−0.932657	+	0.360766i	\(0.882515\pi\)
\(43\)	1.14486	+	10.8926i	0.174589	+	1.66110i	0.634340	+	0.773054i	\(0.281272\pi\)
							−0.459751	+	0.888048i	\(0.652061\pi\)
\(47\)	−2.84593	−	2.06769i	−0.415122	−	0.301604i	0.360550	−	0.932740i	\(-0.382589\pi\)
							−0.775672	+	0.631136i	\(0.782589\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.16	✓	280
13.9	even	3	inner	403.2.bl.b.295.20	yes	280
31.2	even	5	inner	403.2.bl.b.250.20	yes	280
403.126	even	15	inner	403.2.bl.b.126.16	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.16	✓	280	1.1	even	1	trivial
403.2.bl.b.126.16	yes	280	403.126	even	15	inner
403.2.bl.b.250.20	yes	280	31.2	even	5	inner
403.2.bl.b.295.20	yes	280	13.9	even	3	inner