Embedded newform 403.2.bp.a.361.24

• Label: 403.2.bp.a.361.24
• Level: $403$
• Weight: $2$
• Character: 403.361
• 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			361.24
Character	\(\chi\)	\(=\)	403.361
Dual form			403.2.bp.a.355.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.784997 + 0.706815i) q^{2} +(-0.662695 + 2.03956i) q^{3} +(-0.0924232 - 0.879348i) q^{4} +(3.00000 - 1.73205i) q^{5} +(-1.96181 + 1.13265i) q^{6} +(0.0587641 - 0.131986i) q^{7} +(1.79076 - 2.46477i) q^{8} +(-1.29361 - 0.939860i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{4}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.784997	+	0.706815i	0.555077	+	0.499794i	0.898254	−	0.439477i	\(-0.144836\pi\)
							−0.343177	+	0.939271i	\(0.611503\pi\)
\(3\)	−0.662695	+	2.03956i	−0.382607	+	1.17754i	0.555595	+	0.831453i	\(0.312491\pi\)
							−0.938201	+	0.346090i	\(0.887509\pi\)
\(5\)	3.00000	−	1.73205i	1.34164	−	0.774596i	0.354592	−	0.935021i	\(-0.384620\pi\)
							0.987048	+	0.160425i	\(0.0512865\pi\)
\(7\)	0.0587641	−	0.131986i	0.0222107	−	0.0498861i	−0.902100	−	0.431528i	\(-0.857975\pi\)
							0.924310	+	0.381642i	\(0.124641\pi\)
\(11\)	1.81467	−	0.190729i	0.547142	−	0.0575070i	0.173075	−	0.984909i	\(-0.444630\pi\)
							0.374067	+	0.927402i	\(0.377963\pi\)
\(13\)	−0.390310	−	3.58436i	−0.108252	−	0.994123i
							\(17\)	−0.174537	+	1.66061i	−0.0423314	+	0.402756i	0.952755	+	0.303741i	\(0.0982359\pi\)
−0.995086	+	0.0990151i	\(0.968431\pi\)
\(19\)	−0.407203	−	0.366647i	−0.0934187	−	0.0841146i	0.621098	−	0.783733i	\(-0.286687\pi\)
							−0.714517	+	0.699618i	\(0.753353\pi\)
\(23\)	−0.764128	+	7.27020i	−0.159332	+	1.51594i	0.564193	+	0.825643i	\(0.309187\pi\)
							−0.723525	+	0.690298i	\(0.757479\pi\)
\(29\)	−2.74888	+	3.05294i	−0.510454	+	0.566917i	−0.942188	−	0.335084i	\(-0.891235\pi\)
							0.431734	+	0.902001i	\(0.357902\pi\)
\(31\)	−3.75098	+	4.11462i	−0.673697	+	0.739008i
							\(37\)		−	11.2148i		−	1.84371i	−0.387538	−	0.921854i	\(-0.626675\pi\)
0.387538	−	0.921854i	\(-0.373325\pi\)
\(41\)	−0.261538	+	1.23044i	−0.0408454	+	0.192163i	−0.993840	−	0.110825i	\(-0.964651\pi\)
							0.952995	+	0.302988i	\(0.0979841\pi\)
\(43\)	−3.74943	+	4.16416i	−0.571783	+	0.635029i	−0.957791	−	0.287466i	\(-0.907187\pi\)
							0.386008	+	0.922495i	\(0.373854\pi\)
\(47\)	5.37460	+	1.74631i	0.783967	+	0.254726i	0.673533	−	0.739157i	\(-0.264776\pi\)
							0.110434	+	0.993883i	\(0.464776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.361.24	yes	280
13.4	even	6		403.2.bt.a.82.12	yes	280
31.14	even	15		403.2.bt.a.231.12	yes	280
403.355	even	30	inner	403.2.bp.a.355.24	✓	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.24	✓	280	403.355	even	30	inner
403.2.bp.a.361.24	yes	280	1.1	even	1	trivial
403.2.bt.a.82.12	yes	280	13.4	even	6
403.2.bt.a.231.12	yes	280	31.14	even	15