Embedded newform 403.2.bt.a.82.20

• Label: 403.2.bt.a.82.20
• 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			82.20
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0900306 - 0.423561i) q^{2} +(-1.34476 + 0.285838i) q^{3} +(1.65579 + 0.737206i) q^{4} +(1.67453 - 0.966790i) q^{5} +0.595322i q^{6} +(3.08797 + 4.25023i) q^{7} +(0.970373 - 1.33560i) q^{8} +(-1.01396 + 0.451444i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0900306	−	0.423561i	0.0636612	−	0.299503i	−0.934786	−	0.355212i	\(-0.884409\pi\)
							0.998447	+	0.0557094i	\(0.0177420\pi\)
\(3\)	−1.34476	+	0.285838i	−0.776398	+	0.165028i	−0.579036	−	0.815302i	\(-0.696571\pi\)
							−0.197362	+	0.980331i	\(0.563237\pi\)
\(5\)	1.67453	−	0.966790i	0.748873	−	0.432362i	−0.0764138	−	0.997076i	\(-0.524347\pi\)
							0.825286	+	0.564714i	\(0.191014\pi\)
\(7\)	3.08797	+	4.25023i	1.16714	+	1.60644i	0.680260	+	0.732971i	\(0.261867\pi\)
							0.486885	+	0.873466i	\(0.338133\pi\)
\(11\)	−1.76166	−	2.42472i	−0.531161	−	0.731081i	0.456146	−	0.889905i	\(-0.349230\pi\)
							−0.987307	+	0.158824i	\(0.949230\pi\)
\(13\)	−3.08778	+	1.86162i	−0.856396	+	0.516320i
							\(17\)	2.55110	+	1.85348i	0.618733	+	0.449536i	0.852479	−	0.522762i	\(-0.175098\pi\)
−0.233746	+	0.972298i	\(0.575098\pi\)
\(19\)	−1.52155	+	0.494383i	−0.349068	+	0.113419i	−0.478303	−	0.878195i	\(-0.658748\pi\)
							0.129235	+	0.991614i	\(0.458748\pi\)
\(23\)	3.63037	−	1.61634i	0.756984	−	0.337031i	0.00830670	−	0.999965i	\(-0.497356\pi\)
							0.748677	+	0.662934i	\(0.230689\pi\)
\(29\)	−2.16104	−	0.459342i	−0.401294	−	0.0852977i	0.00284329	−	0.999996i	\(-0.499095\pi\)
							−0.404138	+	0.914698i	\(0.632428\pi\)
\(31\)	4.13830	−	3.72485i	0.743260	−	0.669003i
							\(37\)	1.48863	−	0.859460i	0.244729	−	0.141294i	−0.372619	−	0.927984i	\(-0.621540\pi\)
0.617348	+	0.786690i	\(0.288207\pi\)
\(41\)	6.40656	−	2.08162i	1.00054	−	0.325094i	0.237457	−	0.971398i	\(-0.423686\pi\)
							0.763080	+	0.646304i	\(0.223686\pi\)
\(43\)	−1.39261	−	4.28600i	−0.212370	−	0.653609i	−0.999330	−	0.0366040i	\(-0.988346\pi\)
							0.786960	−	0.617005i	\(-0.211654\pi\)
\(47\)	−4.89592	−	1.59078i	−0.714143	−	0.232039i	−0.0706610	−	0.997500i	\(-0.522511\pi\)
							−0.643482	+	0.765461i	\(0.722511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.20	yes	280
13.10	even	6		403.2.bp.a.361.16	yes	280
31.14	even	15		403.2.bp.a.355.16	✓	280
403.231	even	30	inner	403.2.bt.a.231.20	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.16	✓	280	31.14	even	15
403.2.bp.a.361.16	yes	280	13.10	even	6
403.2.bt.a.82.20	yes	280	1.1	even	1	trivial
403.2.bt.a.231.20	yes	280	403.231	even	30	inner