Embedded newform 403.2.bn.a.60.6

• Label: 403.2.bn.a.60.6
• Level: $403$
• Weight: $2$
• Character: 403.60
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $272$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bn (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(34\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			60.6
Character	\(\chi\)	\(=\)	403.60
Dual form			403.2.bn.a.356.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327123 - 2.06537i) q^{2} +(-1.61302 - 2.22013i) q^{3} +(-2.25663 + 0.733225i) q^{4} +(0.152237 + 0.152237i) q^{5} +(-4.05774 + 4.05774i) q^{6} +(0.794879 + 0.405011i) q^{7} +(0.353884 + 0.694537i) q^{8} +(-1.40010 + 4.30907i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 6 q^{2} - 20 q^{3} - 16 q^{5} - 18 q^{7} + 4 q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.327123	−	2.06537i	−0.231311	−	1.46044i	−0.780718	−	0.624883i	\(-0.785147\pi\)
							0.549408	−	0.835554i	\(-0.314853\pi\)
\(3\)	−1.61302	−	2.22013i	−0.931278	−	1.28179i	−0.959359	−	0.282188i	\(-0.908940\pi\)
							0.0280812	−	0.999606i	\(-0.491060\pi\)
\(5\)	0.152237	+	0.152237i	0.0680825	+	0.0680825i	0.740328	−	0.672246i	\(-0.234670\pi\)
							−0.672246	+	0.740328i	\(0.734670\pi\)
\(7\)	0.794879	+	0.405011i	0.300436	+	0.153080i	0.597710	−	0.801712i	\(-0.296077\pi\)
							−0.297274	+	0.954792i	\(0.596077\pi\)
\(11\)	−2.88882	+	5.66964i	−0.871013	+	1.70946i	−0.183853	+	0.982954i	\(0.558857\pi\)
							−0.687160	+	0.726506i	\(0.741143\pi\)
\(13\)	−3.45346	−	1.03616i	−0.957817	−	0.287379i
							\(17\)	−0.572489	+	1.76194i	−0.138849	+	0.427333i	−0.996169	−	0.0874509i	\(-0.972128\pi\)
0.857320	+	0.514784i	\(0.172128\pi\)
\(19\)	0.0926623	+	0.585047i	0.0212582	+	0.134219i	0.996035	−	0.0889635i	\(-0.0283554\pi\)
							−0.974777	+	0.223182i	\(0.928355\pi\)
\(23\)	1.45249	−	4.47030i	0.302865	−	0.932122i	−0.677601	−	0.735430i	\(-0.736980\pi\)
							0.980465	−	0.196692i	\(-0.0630198\pi\)
\(29\)	0.223981	−	0.308284i	0.0415923	−	0.0572469i	−0.787714	−	0.616041i	\(-0.788736\pi\)
							0.829307	+	0.558794i	\(0.188736\pi\)
\(31\)	−5.47812	−	0.995111i	−0.983899	−	0.178727i
							\(37\)	−5.03724	−	5.03724i	−0.828117	−	0.828117i	0.159140	−	0.987256i	\(-0.449128\pi\)
−0.987256	+	0.159140i	\(0.949128\pi\)
\(41\)	−0.812201	−	5.12804i	−0.126844	−	0.800865i	−0.966297	−	0.257431i	\(-0.917124\pi\)
							0.839452	−	0.543434i	\(-0.182876\pi\)
\(43\)	−6.88452	−	5.00190i	−1.04988	−	0.762782i	−0.0776903	−	0.996978i	\(-0.524755\pi\)
							−0.972190	+	0.234195i	\(0.924755\pi\)
\(47\)	−0.899204	+	5.67735i	−0.131162	+	0.828127i	0.831124	+	0.556088i	\(0.187698\pi\)
							−0.962286	+	0.272039i	\(0.912302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bn.a.60.6	✓	272
13.5	odd	4	inner	403.2.bn.a.122.6	yes	272
31.15	odd	10	inner	403.2.bn.a.294.6	yes	272
403.356	even	20	inner	403.2.bn.a.356.6	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bn.a.60.6	✓	272	1.1	even	1	trivial
403.2.bn.a.122.6	yes	272	13.5	odd	4	inner
403.2.bn.a.294.6	yes	272	31.15	odd	10	inner
403.2.bn.a.356.6	yes	272	403.356	even	20	inner