Embedded newform 403.2.bp.a.10.12

• Label: 403.2.bp.a.10.12
• Level: $403$
• Weight: $2$
• Character: 403.10
• 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.bp (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.12
Character	\(\chi\)	\(=\)	403.10
Dual form			403.2.bp.a.121.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17237 - 0.123221i) q^{2} +(-1.03818 + 0.754283i) q^{3} +(-0.597037 - 0.126904i) q^{4} +(-2.09944 + 1.21211i) q^{5} +(1.31007 - 0.756370i) q^{6} +(2.47362 + 2.22725i) q^{7} +(2.92656 + 0.950898i) q^{8} +(-0.418173 + 1.28701i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17237	−	0.123221i	−0.828988	−	0.0871301i	−0.319477	−	0.947594i	\(-0.603507\pi\)
							−0.509511	+	0.860464i	\(0.670174\pi\)
\(3\)	−1.03818	+	0.754283i	−0.599394	+	0.435485i	−0.845664	−	0.533716i	\(-0.820795\pi\)
							0.246270	+	0.969201i	\(0.420795\pi\)
\(5\)	−2.09944	+	1.21211i	−0.938898	+	0.542073i	−0.889615	−	0.456712i	\(-0.849027\pi\)
							−0.0492835	+	0.998785i	\(0.515694\pi\)
\(7\)	2.47362	+	2.22725i	0.934939	+	0.841823i	0.987632	−	0.156788i	\(-0.0501138\pi\)
							−0.0526930	+	0.998611i	\(0.516780\pi\)
\(11\)	−0.168094	+	0.790821i	−0.0506823	+	0.238441i	−0.996196	−	0.0871409i	\(-0.972227\pi\)
							0.945514	+	0.325582i	\(0.105560\pi\)
\(13\)	1.69431	−	3.18266i	0.469916	−	0.882711i
							\(17\)	−1.74857	+	0.371669i	−0.424089	+	0.0901430i	−0.415012	−	0.909816i	\(-0.636223\pi\)
−0.00907714	+	0.999959i	\(0.502889\pi\)
\(19\)	−6.79736	−	0.714431i	−1.55942	−	0.163902i	−0.714824	−	0.699304i	\(-0.753493\pi\)
							−0.844597	+	0.535403i	\(0.820160\pi\)
\(23\)	−5.88991	+	1.25194i	−1.22813	+	0.261047i	−0.775901	−	0.630855i	\(-0.782704\pi\)
							−0.452229	+	0.891902i	\(0.649371\pi\)
\(29\)	−0.275523	+	2.62143i	−0.0511633	+	0.486787i	0.938697	+	0.344743i	\(0.112034\pi\)
							−0.989860	+	0.142044i	\(0.954633\pi\)
\(31\)	5.51887	−	0.736258i	0.991218	−	0.132236i
							\(37\)		−	1.24630i		−	0.204890i	−0.994739	−	0.102445i	\(-0.967333\pi\)
0.994739	−	0.102445i	\(-0.0326665\pi\)
\(41\)	−2.88135	−	6.47162i	−0.449992	−	1.01070i	−0.986040	−	0.166510i	\(-0.946750\pi\)
							0.536048	−	0.844187i	\(-0.319917\pi\)
\(43\)	0.821110	−	7.81234i	0.125218	−	1.19137i	−0.733778	−	0.679389i	\(-0.762245\pi\)
							0.858996	−	0.511982i	\(-0.171088\pi\)
\(47\)	4.21270	+	5.79829i	0.614486	+	0.845768i	0.996937	−	0.0782082i	\(-0.0249199\pi\)
							−0.382451	+	0.923976i	\(0.624920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.10.12	✓	280
13.4	even	6		403.2.bt.a.134.12	yes	280
31.28	even	15		403.2.bt.a.400.12	yes	280
403.121	even	30	inner	403.2.bp.a.121.12	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.10.12	✓	280	1.1	even	1	trivial
403.2.bp.a.121.12	yes	280	403.121	even	30	inner
403.2.bt.a.134.12	yes	280	13.4	even	6
403.2.bt.a.400.12	yes	280	31.28	even	15