Embedded newform 403.2.bl.b.16.4

• Label: 403.2.bl.b.16.4
• Level: $403$
• Weight: $2$
• Character: 403.16
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bl (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.4
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.07437 - 0.923570i) q^{2} +(0.0811184 - 0.771790i) q^{3} +(2.11178 + 2.34536i) q^{4} +2.89709 q^{5} +(-0.881072 + 1.52606i) q^{6} +(-1.22242 - 1.35763i) q^{7} +(-0.811140 - 2.49643i) q^{8} +(2.34536 + 0.498522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.07437	−	0.923570i	−1.46680	−	0.653062i	−0.490889	−	0.871222i	\(-0.663328\pi\)
							−0.975913	+	0.218160i	\(0.929995\pi\)
\(3\)	0.0811184	−	0.771790i	0.0468337	−	0.445593i	−0.945828	−	0.324668i	\(-0.894747\pi\)
							0.992662	−	0.120925i	\(-0.0385860\pi\)
\(5\)	2.89709			1.29562			0.647810	−	0.761802i	\(-0.275685\pi\)
							0.647810	+	0.761802i	\(0.275685\pi\)
\(7\)	−1.22242	−	1.35763i	−0.462030	−	0.513137i	0.466435	−	0.884555i	\(-0.345538\pi\)
							−0.928465	+	0.371419i	\(0.878871\pi\)
\(11\)	−0.592020	+	0.125838i	−0.178501	+	0.0379415i	−0.296295	−	0.955097i	\(-0.595751\pi\)
							0.117794	+	0.993038i	\(0.462418\pi\)
\(13\)	2.79216	+	2.28119i	0.774406	+	0.632689i
							\(17\)	3.62886	+	0.771338i	0.880128	+	0.187077i	0.625746	−	0.780027i	\(-0.284795\pi\)
0.254382	+	0.967104i	\(0.418128\pi\)
\(19\)	0.537735	+	5.11621i	0.123365	+	1.17374i	0.864588	+	0.502481i	\(0.167579\pi\)
							−0.741224	+	0.671258i	\(0.765754\pi\)
\(23\)	−4.24104	+	4.71015i	−0.884318	+	0.982135i	−0.999938	−	0.0111585i	\(-0.996448\pi\)
							0.115619	+	0.993294i	\(0.463115\pi\)
\(29\)	8.09087	+	3.60229i	1.50244	+	0.668928i	0.982667	−	0.185378i	\(-0.0593508\pi\)
							0.519770	+	0.854306i	\(0.326017\pi\)
\(31\)	3.60608	−	4.24219i	0.647671	−	0.761920i
							\(37\)	−5.91946	−	10.2528i	−0.973153	−	1.68555i	−0.685900	−	0.727696i	\(-0.740591\pi\)
−0.287253	−	0.957855i	\(-0.592742\pi\)
\(41\)	−0.274572	−	0.122247i	−0.0428809	−	0.0190918i	0.385184	−	0.922840i	\(-0.374138\pi\)
							−0.428065	+	0.903748i	\(0.640805\pi\)
\(43\)	−0.976975	−	9.29529i	−0.148987	−	1.41752i	−0.772157	−	0.635432i	\(-0.780822\pi\)
							0.623170	−	0.782087i	\(-0.285845\pi\)
\(47\)	−2.48042	−	1.80213i	−0.361806	−	0.262868i	0.391999	−	0.919966i	\(-0.371784\pi\)
							−0.753805	+	0.657098i	\(0.771784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.4	✓	280
13.9	even	3	inner	403.2.bl.b.295.32	yes	280
31.2	even	5	inner	403.2.bl.b.250.32	yes	280
403.126	even	15	inner	403.2.bl.b.126.4	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.4	✓	280	1.1	even	1	trivial
403.2.bl.b.126.4	yes	280	403.126	even	15	inner
403.2.bl.b.250.32	yes	280	31.2	even	5	inner
403.2.bl.b.295.32	yes	280	13.9	even	3	inner