Embedded newform 403.3.n.a.347.11

• Label: 403.3.n.a.347.11
• Level: $403$
• Weight: $3$
• Character: 403.347
• Analytic conductor: $10.981$
• Analytic rank: $0$
• Dimension: $146$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	403.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9809546537\)
Analytic rank:	\(0\)
Dimension:	\(146\)
Relative dimension:	\(73\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			347.11
Character	\(\chi\)	\(=\)	403.347
Dual form			403.3.n.a.367.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47699 + 2.55822i) q^{2} +1.72252i q^{3} +(-2.36298 - 4.09281i) q^{4} +(0.496131 - 0.859324i) q^{5} +(-4.40657 - 2.54414i) q^{6} +(5.78997 - 10.0285i) q^{7} +2.14448 q^{8} +6.03294 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 146 q - 2 q^{2} - 146 q^{4} - 2 q^{5} + 12 q^{6} + 16 q^{7} - 10 q^{8} - 422 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47699	+	2.55822i	−0.738494	+	1.27911i	0.214680	+	0.976684i	\(0.431129\pi\)
							−0.953174	+	0.302424i	\(0.902204\pi\)
\(3\)			1.72252i			0.574172i	0.957905	+	0.287086i	\(0.0926866\pi\)
							−0.957905	+	0.287086i	\(0.907313\pi\)
\(5\)	0.496131	−	0.859324i	0.0992262	−	0.171865i	−0.812138	−	0.583465i	\(-0.801697\pi\)
							0.911365	+	0.411600i	\(0.135030\pi\)
\(7\)	5.78997	−	10.0285i	0.827139	−	1.43265i	−0.0731347	−	0.997322i	\(-0.523300\pi\)
							0.900274	−	0.435324i	\(-0.143366\pi\)
\(11\)	−16.5636	+	9.56299i	−1.50578	+	0.869363i	−0.505803	+	0.862649i	\(0.668804\pi\)
							−0.999977	+	0.00671323i	\(0.997863\pi\)
\(13\)	−2.52871	−	12.7517i	−0.194516	−	0.980899i
							\(17\)	25.7390	+	14.8604i	1.51406	+	0.874142i	0.999864	+	0.0164683i	\(0.00524227\pi\)
0.514194	+	0.857674i	\(0.328091\pi\)
\(19\)	−14.5691	+	25.2345i	−0.766797	+	1.32813i	0.172494	+	0.985011i	\(0.444818\pi\)
							−0.939291	+	0.343121i	\(0.888516\pi\)
\(23\)	28.0640	+	16.2028i	1.22018	+	0.704469i	0.964955	−	0.262414i	\(-0.0845186\pi\)
							0.255220	+	0.966883i	\(0.417852\pi\)
\(29\)	−7.29989	−	4.21460i	−0.251720	−	0.145331i	0.368831	−	0.929496i	\(-0.379758\pi\)
							−0.620552	+	0.784165i	\(0.713091\pi\)
\(31\)	1.42008	+	30.9675i	0.0458090	+	0.998950i
							\(37\)		−	24.1506i		−	0.652719i	−0.945246	−	0.326360i	\(-0.894178\pi\)
0.945246	−	0.326360i	\(-0.105822\pi\)
\(41\)	12.3411	+	21.3755i	0.301003	+	0.521353i	0.976363	−	0.216135i	\(-0.0693452\pi\)
							−0.675360	+	0.737488i	\(0.736012\pi\)
\(43\)	3.73923	+	2.15885i	0.0869589	+	0.0502057i	0.542849	−	0.839830i	\(-0.317346\pi\)
							−0.455890	+	0.890036i	\(0.650679\pi\)
\(47\)	78.9873			1.68058			0.840291	−	0.542136i	\(-0.182384\pi\)
							0.840291	+	0.542136i	\(0.182384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.3.n.a.347.11	✓	146
13.3	even	3		403.3.o.a.68.11	yes	146
31.26	odd	6		403.3.o.a.243.11	yes	146
403.367	odd	6	inner	403.3.n.a.367.11	yes	146

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.3.n.a.347.11	✓	146	1.1	even	1	trivial
403.3.n.a.367.11	yes	146	403.367	odd	6	inner
403.3.o.a.68.11	yes	146	13.3	even	3
403.3.o.a.243.11	yes	146	31.26	odd	6