Embedded newform 403.2.bt.a.82.7

• Label: 403.2.bt.a.82.7
• 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			82.7
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.387542 + 1.82324i) q^{2} +(-0.271363 + 0.0576799i) q^{3} +(-1.34692 - 0.599689i) q^{4} +(0.212568 - 0.122726i) q^{5} -0.517113i q^{6} +(-2.14262 - 2.94906i) q^{7} +(-0.575865 + 0.792611i) q^{8} +(-2.67033 + 1.18891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.387542	+	1.82324i	−0.274033	+	1.28923i	0.598679	+	0.800989i	\(0.295693\pi\)
							−0.872712	+	0.488236i	\(0.837641\pi\)
\(3\)	−0.271363	+	0.0576799i	−0.156671	+	0.0333015i	−0.285580	−	0.958355i	\(-0.592186\pi\)
							0.128908	+	0.991657i	\(0.458853\pi\)
\(5\)	0.212568	−	0.122726i	0.0950631	−	0.0548847i	−0.451715	−	0.892162i	\(-0.649188\pi\)
							0.546778	+	0.837278i	\(0.315854\pi\)
\(7\)	−2.14262	−	2.94906i	−0.809834	−	1.11464i	−0.991349	−	0.131253i	\(-0.958100\pi\)
							0.181515	−	0.983388i	\(-0.441900\pi\)
\(11\)	1.34828	+	1.85574i	0.406520	+	0.559527i	0.962365	−	0.271759i	\(-0.0876052\pi\)
							−0.555845	+	0.831286i	\(0.687605\pi\)
\(13\)	−3.60301	−	0.135441i	−0.999294	−	0.0375646i
							\(17\)	−4.33579	−	3.15013i	−1.05158	−	0.764019i	−0.0790698	−	0.996869i	\(-0.525195\pi\)
−0.972513	+	0.232850i	\(0.925195\pi\)
\(19\)	−4.21106	+	1.36826i	−0.966084	+	0.313900i	−0.749234	−	0.662305i	\(-0.769578\pi\)
							−0.216850	+	0.976205i	\(0.569578\pi\)
\(23\)	0.319226	−	0.142129i	0.0665633	−	0.0296359i	−0.373185	−	0.927757i	\(-0.621734\pi\)
							0.439748	+	0.898121i	\(0.355068\pi\)
\(29\)	3.91319	+	0.831775i	0.726661	+	0.154457i	0.556367	−	0.830937i	\(-0.312195\pi\)
							0.170294	+	0.985393i	\(0.445528\pi\)
\(31\)	2.51726	+	4.96623i	0.452113	+	0.891961i
							\(37\)	3.28455	−	1.89634i	0.539977	−	0.311756i	−0.205093	−	0.978743i	\(-0.565750\pi\)
0.745069	+	0.666987i	\(0.232416\pi\)
\(41\)	1.94697	−	0.632610i	0.304066	−	0.0987971i	−0.153010	−	0.988225i	\(-0.548897\pi\)
							0.457076	+	0.889428i	\(0.348897\pi\)
\(43\)	−0.506993	−	1.56036i	−0.0773157	−	0.237953i	0.904927	−	0.425566i	\(-0.139925\pi\)
							−0.982243	+	0.187613i	\(0.939925\pi\)
\(47\)	−0.292057	−	0.0948949i	−0.0426008	−	0.0138418i	0.287639	−	0.957739i	\(-0.407130\pi\)
							−0.330240	+	0.943897i	\(0.607130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.7	yes	280
13.10	even	6		403.2.bp.a.361.29	yes	280
31.14	even	15		403.2.bp.a.355.29	✓	280
403.231	even	30	inner	403.2.bt.a.231.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.29	✓	280	31.14	even	15
403.2.bp.a.361.29	yes	280	13.10	even	6
403.2.bt.a.82.7	yes	280	1.1	even	1	trivial
403.2.bt.a.231.7	yes	280	403.231	even	30	inner