Embedded newform 403.2.bp.a.10.7

• Label: 403.2.bp.a.10.7
• 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			10.7
Character	\(\chi\)	\(=\)	403.10
Dual form			403.2.bp.a.121.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.02543 - 0.212881i) q^{2} +(1.58944 - 1.15479i) q^{3} +(2.10075 + 0.446527i) q^{4} +(1.71112 - 0.987913i) q^{5} +(-3.46513 + 2.00059i) q^{6} +(-1.42691 - 1.28479i) q^{7} +(-0.286037 - 0.0929391i) q^{8} +(0.265713 - 0.817780i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.02543	−	0.212881i	−1.43219	−	0.150530i	−0.643561	−	0.765394i	\(-0.722544\pi\)
							−0.788633	+	0.614865i	\(0.789211\pi\)
\(3\)	1.58944	−	1.15479i	0.917663	−	0.666721i	−0.0252784	−	0.999680i	\(-0.508047\pi\)
							0.942941	+	0.332959i	\(0.108047\pi\)
\(5\)	1.71112	−	0.987913i	0.765234	−	0.441808i	−0.0659377	−	0.997824i	\(-0.521004\pi\)
							0.831172	+	0.556016i	\(0.187671\pi\)
\(7\)	−1.42691	−	1.28479i	−0.539321	−	0.485607i	0.353864	−	0.935297i	\(-0.384868\pi\)
							−0.893185	+	0.449690i	\(0.851534\pi\)
\(11\)	0.529255	−	2.48995i	0.159576	−	0.750748i	−0.823465	−	0.567367i	\(-0.807962\pi\)
							0.983042	−	0.183382i	\(-0.0587044\pi\)
\(13\)	1.89152	−	3.06955i	0.524613	−	0.851341i
							\(17\)	−2.88666	+	0.613578i	−0.700118	+	0.148815i	−0.544200	−	0.838955i	\(-0.683167\pi\)
−0.155918	+	0.987770i	\(0.549833\pi\)
\(19\)	6.59901	+	0.693583i	1.51392	+	0.159119i	0.824755	−	0.565490i	\(-0.191313\pi\)
							0.689161	+	0.724609i	\(0.257979\pi\)
\(23\)	−3.56208	+	0.757143i	−0.742745	+	0.157875i	−0.563716	−	0.825969i	\(-0.690629\pi\)
							−0.179029	+	0.983844i	\(0.557295\pi\)
\(29\)	0.167731	−	1.59585i	0.0311469	−	0.296343i	−0.967849	−	0.251533i	\(-0.919065\pi\)
							0.998996	−	0.0448094i	\(-0.0142681\pi\)
\(31\)	2.37483	−	5.03589i	0.426531	−	0.904473i
							\(37\)			1.47828i			0.243028i	0.992590	+	0.121514i	\(0.0387749\pi\)
−0.992590	+	0.121514i	\(0.961225\pi\)
\(41\)	−4.03853	−	9.07069i	−0.630713	−	1.41660i	−0.892308	−	0.451427i	\(-0.850915\pi\)
							0.261595	−	0.965178i	\(-0.415751\pi\)
\(43\)	−0.223864	+	2.12993i	−0.0341390	+	0.324811i	0.964102	+	0.265531i	\(0.0855472\pi\)
							−0.998241	+	0.0592801i	\(0.981119\pi\)
\(47\)	−1.99781	−	2.74976i	−0.291411	−	0.401093i	0.638061	−	0.769986i	\(-0.279737\pi\)
							−0.929472	+	0.368893i	\(0.879737\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.10.7	✓	280
13.4	even	6		403.2.bt.a.134.7	yes	280
31.28	even	15		403.2.bt.a.400.7	yes	280
403.121	even	30	inner	403.2.bp.a.121.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.10.7	✓	280	1.1	even	1	trivial
403.2.bp.a.121.7	yes	280	403.121	even	30	inner
403.2.bt.a.134.7	yes	280	13.4	even	6
403.2.bt.a.400.7	yes	280	31.28	even	15