Embedded newform 403.2.bp.a.10.5

• Label: 403.2.bp.a.10.5
• 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			10.5
Character	\(\chi\)	\(=\)	403.10
Dual form			403.2.bp.a.121.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.18417 - 0.229566i) q^{2} +(0.117173 - 0.0851309i) q^{3} +(2.76161 + 0.586998i) q^{4} +(0.0491735 - 0.0283903i) q^{5} +(-0.275468 + 0.159042i) q^{6} +(-2.29210 - 2.06382i) q^{7} +(-1.71965 - 0.558748i) q^{8} +(-0.920569 + 2.83322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.18417	−	0.229566i	−1.54444	−	0.162327i	−0.706353	−	0.707860i	\(-0.749661\pi\)
							−0.838090	+	0.545532i	\(0.816327\pi\)
\(3\)	0.117173	−	0.0851309i	0.0676496	−	0.0491503i	−0.553446	−	0.832885i	\(-0.686688\pi\)
							0.621096	+	0.783734i	\(0.286688\pi\)
\(5\)	0.0491735	−	0.0283903i	0.0219911	−	0.0126965i	−0.488964	−	0.872304i	\(-0.662625\pi\)
							0.510955	+	0.859607i	\(0.329292\pi\)
\(7\)	−2.29210	−	2.06382i	−0.866334	−	0.780051i	0.110539	−	0.993872i	\(-0.464742\pi\)
							−0.976873	+	0.213821i	\(0.931409\pi\)
\(11\)	0.0543592	−	0.255740i	0.0163899	−	0.0771085i	−0.969193	−	0.246304i	\(-0.920784\pi\)
							0.985582	+	0.169196i	\(0.0541170\pi\)
\(13\)	0.935641	+	3.48204i	0.259500	+	0.965743i
							\(17\)	5.94850	−	1.26439i	1.44272	−	0.306660i	0.580944	−	0.813944i	\(-0.302684\pi\)
0.861779	+	0.507283i	\(0.169350\pi\)
\(19\)	−6.05524	−	0.636432i	−1.38917	−	0.146007i	−0.619773	−	0.784781i	\(-0.712775\pi\)
							−0.769395	+	0.638774i	\(0.779442\pi\)
\(23\)	−0.909714	+	0.193366i	−0.189688	+	0.0403195i	−0.301776	−	0.953379i	\(-0.597580\pi\)
							0.112088	+	0.993698i	\(0.464246\pi\)
\(29\)	−0.648275	+	6.16792i	−0.120382	+	1.14535i	0.752899	+	0.658136i	\(0.228655\pi\)
							−0.873280	+	0.487218i	\(0.838012\pi\)
\(31\)	5.39373	−	1.38119i	0.968742	−	0.248070i
							\(37\)			3.57092i			0.587055i	0.955951	+	0.293528i	\(0.0948293\pi\)
−0.955951	+	0.293528i	\(0.905171\pi\)
\(41\)	2.81007	+	6.31152i	0.438859	+	0.985694i	0.988629	+	0.150377i	\(0.0480488\pi\)
							−0.549770	+	0.835316i	\(0.685284\pi\)
\(43\)	−0.976364	+	9.28949i	−0.148894	+	1.41663i	0.623666	+	0.781691i	\(0.285643\pi\)
							−0.772560	+	0.634942i	\(0.781024\pi\)
\(47\)	3.95116	+	5.43831i	0.576336	+	0.793258i	0.993288	−	0.115670i	\(-0.0369016\pi\)
							−0.416952	+	0.908929i	\(0.636902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.10.5	✓	280
13.4	even	6		403.2.bt.a.134.5	yes	280
31.28	even	15		403.2.bt.a.400.5	yes	280
403.121	even	30	inner	403.2.bp.a.121.5	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.10.5	✓	280	1.1	even	1	trivial
403.2.bp.a.121.5	yes	280	403.121	even	30	inner
403.2.bt.a.134.5	yes	280	13.4	even	6
403.2.bt.a.400.5	yes	280	31.28	even	15