Embedded newform 403.2.bp.a.10.4

• Label: 403.2.bp.a.10.4
• 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			10.4
Character	\(\chi\)	\(=\)	403.10
Dual form			403.2.bp.a.121.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19585 - 0.230793i) q^{2} +(-0.847098 + 0.615453i) q^{3} +(2.81219 + 0.597750i) q^{4} +(2.76986 - 1.59918i) q^{5} +(2.00214 - 1.15594i) q^{6} +(3.22981 + 2.90814i) q^{7} +(-1.83743 - 0.597018i) q^{8} +(-0.588258 + 1.81047i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19585	−	0.230793i	−1.55270	−	0.163195i	−0.711020	−	0.703172i	\(-0.751766\pi\)
							−0.841680	+	0.539977i	\(0.818433\pi\)
\(3\)	−0.847098	+	0.615453i	−0.489073	+	0.355332i	−0.804828	−	0.593509i	\(-0.797742\pi\)
							0.315755	+	0.948841i	\(0.397742\pi\)
\(5\)	2.76986	−	1.59918i	1.23872	−	0.715175i	0.269886	−	0.962892i	\(-0.413014\pi\)
							0.968832	+	0.247718i	\(0.0796805\pi\)
\(7\)	3.22981	+	2.90814i	1.22075	+	1.09917i	0.992072	+	0.125671i	\(0.0401083\pi\)
							0.228682	+	0.973501i	\(0.426558\pi\)
\(11\)	0.834175	−	3.92449i	0.251513	−	1.18328i	−0.653176	−	0.757206i	\(-0.726564\pi\)
							0.904690	−	0.426071i	\(-0.140103\pi\)
\(13\)	3.44937	+	1.04967i	0.956684	+	0.291127i
							\(17\)	−5.50599	+	1.17033i	−1.33540	+	0.283848i	−0.819636	−	0.572884i	\(-0.805825\pi\)
−0.515762	+	0.856732i	\(0.672491\pi\)
\(19\)	−1.47471	−	0.154998i	−0.338322	−	0.0355591i	−0.0661558	−	0.997809i	\(-0.521073\pi\)
							−0.272166	+	0.962250i	\(0.587740\pi\)
\(23\)	6.62826	−	1.40888i	1.38209	−	0.293772i	0.543898	−	0.839151i	\(-0.316948\pi\)
							0.838189	+	0.545380i	\(0.183614\pi\)
\(29\)	−0.346120	+	3.29312i	−0.0642729	+	0.611516i	0.914217	+	0.405224i	\(0.132806\pi\)
							−0.978490	+	0.206292i	\(0.933860\pi\)
\(31\)	−0.883804	+	5.49717i	−0.158736	+	0.987321i
							\(37\)		−	3.14650i		−	0.517282i	−0.965974	−	0.258641i	\(-0.916725\pi\)
0.965974	−	0.258641i	\(-0.0832746\pi\)
\(41\)	3.98646	+	8.95373i	0.622580	+	1.39834i	0.899277	+	0.437380i	\(0.144094\pi\)
							−0.276697	+	0.960957i	\(0.589240\pi\)
\(43\)	−0.0228746	+	0.217637i	−0.00348834	+	0.0331894i	−0.996128	−	0.0879192i	\(-0.971978\pi\)
							0.992639	+	0.121109i	\(0.0386449\pi\)
\(47\)	−5.52412	−	7.60330i	−0.805776	−	1.10906i	−0.991961	−	0.126541i	\(-0.959613\pi\)
							0.186186	−	0.982515i	\(-0.440387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.10.4	✓	280
13.4	even	6		403.2.bt.a.134.4	yes	280
31.28	even	15		403.2.bt.a.400.4	yes	280
403.121	even	30	inner	403.2.bp.a.121.4	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.10.4	✓	280	1.1	even	1	trivial
403.2.bp.a.121.4	yes	280	403.121	even	30	inner
403.2.bt.a.134.4	yes	280	13.4	even	6
403.2.bt.a.400.4	yes	280	31.28	even	15