Embedded newform 403.2.bi.a.40.5

• Label: 403.2.bi.a.40.5
• Level: $403$
• Weight: $2$
• Character: 403.40
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bi (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			40.5
Character	\(\chi\)	\(=\)	403.40
Dual form			403.2.bi.a.131.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.395989 + 1.21873i) q^{2} +(2.04438 + 0.434547i) q^{3} +(0.289543 + 0.210365i) q^{4} +(-0.128714 - 0.222939i) q^{5} +(-1.33915 + 2.31947i) q^{6} +(1.64432 - 0.732098i) q^{7} +(-2.44446 + 1.77600i) q^{8} +(1.25003 + 0.556550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.395989	+	1.21873i	−0.280006	+	0.861771i	0.707845	+	0.706368i	\(0.249668\pi\)
							−0.987851	+	0.155403i	\(0.950332\pi\)
\(3\)	2.04438	+	0.434547i	1.18032	+	0.250886i	0.755968	−	0.654609i	\(-0.227167\pi\)
							0.424357	+	0.905495i	\(0.360500\pi\)
\(5\)	−0.128714	−	0.222939i	−0.0575627	−	0.0997015i	0.835808	−	0.549022i	\(-0.185000\pi\)
							−0.893371	+	0.449320i	\(0.851666\pi\)
\(7\)	1.64432	−	0.732098i	0.621494	−	0.276707i	−0.0717356	−	0.997424i	\(-0.522854\pi\)
							0.693230	+	0.720717i	\(0.256187\pi\)
\(11\)	−0.246626	+	2.34649i	−0.0743606	+	0.707494i	0.892302	+	0.451439i	\(0.149089\pi\)
							−0.966662	+	0.256054i	\(0.917577\pi\)
\(13\)	−0.669131	+	0.743145i	−0.185583	+	0.206111i
							\(17\)	−0.246692	−	2.34712i	−0.0598317	−	0.569261i	−0.982837	−	0.184474i	\(-0.940942\pi\)
0.923006	−	0.384786i	\(-0.125725\pi\)
\(19\)	5.39249	+	5.98897i	1.23712	+	1.37396i	0.901970	+	0.431800i	\(0.142121\pi\)
							0.335153	+	0.942164i	\(0.391212\pi\)
\(23\)	−2.86710	+	2.08307i	−0.597831	+	0.434349i	−0.845108	−	0.534595i	\(-0.820464\pi\)
							0.247278	+	0.968945i	\(0.420464\pi\)
\(29\)	2.50944	−	7.72326i	0.465991	−	1.43417i	−0.391740	−	0.920076i	\(-0.628127\pi\)
							0.857731	−	0.514098i	\(-0.171873\pi\)
\(31\)	4.56006	+	3.19466i	0.819012	+	0.573777i
							\(37\)	1.99198	−	3.45021i	0.327480	−	0.567212i	−0.654531	−	0.756035i	\(-0.727134\pi\)
0.982011	+	0.188823i	\(0.0604673\pi\)
\(41\)	−2.05150	+	0.436059i	−0.320390	+	0.0681010i	−0.365299	−	0.930890i	\(-0.619033\pi\)
							0.0449095	+	0.998991i	\(0.485700\pi\)
\(43\)	−6.74564	−	7.49179i	−1.02870	−	1.14249i	−0.989687	−	0.143248i	\(-0.954245\pi\)
							−0.0390132	−	0.999239i	\(-0.512421\pi\)
\(47\)	−2.35519	−	7.24854i	−0.343540	−	1.05731i	−0.962361	−	0.271776i	\(-0.912389\pi\)
							0.618820	−	0.785533i	\(-0.287611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.40.5	✓	120
31.7	even	15	inner	403.2.bi.a.131.5	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.40.5	✓	120	1.1	even	1	trivial
403.2.bi.a.131.5	yes	120	31.7	even	15	inner