Embedded newform 403.2.bt.a.82.11

• Label: 403.2.bt.a.82.11
• Level: $403$
• Weight: $2$
• Character: 403.82
• 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.bt (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			82.11
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.226015 + 1.06331i) q^{2} +(-0.562471 + 0.119557i) q^{3} +(0.747535 + 0.332824i) q^{4} +(1.49545 - 0.863396i) q^{5} -0.625106i q^{6} +(-0.503379 - 0.692842i) q^{7} +(-1.80078 + 2.47856i) q^{8} +(-2.43856 + 1.08572i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.226015	+	1.06331i	−0.159816	+	0.751877i	0.823111	+	0.567881i	\(0.192237\pi\)
							−0.982927	+	0.183996i	\(0.941097\pi\)
\(3\)	−0.562471	+	0.119557i	−0.324743	+	0.0690262i	−0.367399	−	0.930064i	\(-0.619752\pi\)
							0.0426556	+	0.999090i	\(0.486418\pi\)
\(5\)	1.49545	−	0.863396i	0.668784	−	0.386122i	−0.126832	−	0.991924i	\(-0.540481\pi\)
							0.795616	+	0.605802i	\(0.207148\pi\)
\(7\)	−0.503379	−	0.692842i	−0.190259	−	0.261870i	0.703221	−	0.710971i	\(-0.251744\pi\)
							−0.893481	+	0.449101i	\(0.851744\pi\)
\(11\)	1.20910	+	1.66418i	0.364556	+	0.501769i	0.951411	−	0.307923i	\(-0.0996341\pi\)
							−0.586855	+	0.809692i	\(0.699634\pi\)
\(13\)	2.30407	+	2.77331i	0.639035	+	0.769178i
							\(17\)	4.07661	+	2.96183i	0.988722	+	0.718349i	0.959641	−	0.281228i	\(-0.0907418\pi\)
0.0290815	+	0.999577i	\(0.490742\pi\)
\(19\)	3.75160	−	1.21897i	0.860677	−	0.279651i	0.154766	−	0.987951i	\(-0.450538\pi\)
							0.705911	+	0.708300i	\(0.250538\pi\)
\(23\)	−1.35001	+	0.601064i	−0.281497	+	0.125331i	−0.542629	−	0.839973i	\(-0.682571\pi\)
							0.261132	+	0.965303i	\(0.415904\pi\)
\(29\)	−2.62823	−	0.558648i	−0.488050	−	0.103738i	−0.0426874	−	0.999088i	\(-0.513592\pi\)
							−0.445363	+	0.895350i	\(0.646925\pi\)
\(31\)	5.37560	−	1.45016i	0.965485	−	0.260457i
							\(37\)	−4.54827	+	2.62595i	−0.747731	+	0.431703i	−0.824874	−	0.565317i	\(-0.808754\pi\)
0.0771422	+	0.997020i	\(0.475420\pi\)
\(41\)	−6.93235	+	2.25246i	−1.08265	+	0.351775i	−0.795403	−	0.606081i	\(-0.792741\pi\)
							−0.287249	+	0.957856i	\(0.592741\pi\)
\(43\)	−1.50284	−	4.62526i	−0.229181	−	0.705346i	−0.997840	−	0.0656878i	\(-0.979076\pi\)
							0.768659	−	0.639658i	\(-0.220924\pi\)
\(47\)	7.90258	+	2.56770i	1.15271	+	0.374538i	0.822163	−	0.569252i	\(-0.192767\pi\)
							0.330546	+	0.943790i	\(0.392767\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.11	yes	280
13.10	even	6		403.2.bp.a.361.25	yes	280
31.14	even	15		403.2.bp.a.355.25	✓	280
403.231	even	30	inner	403.2.bt.a.231.11	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.25	✓	280	31.14	even	15
403.2.bp.a.361.25	yes	280	13.10	even	6
403.2.bt.a.82.11	yes	280	1.1	even	1	trivial
403.2.bt.a.231.11	yes	280	403.231	even	30	inner