Embedded newform 403.2.r.a.218.25

• Label: 403.2.r.a.218.25
• Level: $403$
• Weight: $2$
• Character: 403.218
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			218.25
Character	\(\chi\)	\(=\)	403.218
Dual form			403.2.r.a.342.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14747 - 0.662491i) q^{2} +(0.774529 + 1.34152i) q^{3} +(-0.122211 + 0.211676i) q^{4} +1.40687i q^{5} +(1.77750 + 1.02624i) q^{6} +(-0.464032 - 0.267909i) q^{7} +2.97382i q^{8} +(0.300209 - 0.519977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14747	−	0.662491i	0.811383	−	0.468452i	−0.0360532	−	0.999350i	\(-0.511479\pi\)
							0.847436	+	0.530898i	\(0.178145\pi\)
\(3\)	0.774529	+	1.34152i	0.447175	+	0.774529i	0.998201	−	0.0599587i	\(-0.0190969\pi\)
							−0.551026	+	0.834488i	\(0.685764\pi\)
\(5\)			1.40687i			0.629170i	0.949229	+	0.314585i	\(0.101865\pi\)
							−0.949229	+	0.314585i	\(0.898135\pi\)
\(7\)	−0.464032	−	0.267909i	−0.175388	−	0.101260i	0.409736	−	0.912204i	\(-0.365621\pi\)
							−0.585124	+	0.810944i	\(0.698954\pi\)
\(11\)	−3.45340	+	1.99382i	−1.04124	+	0.601159i	−0.920184	−	0.391486i	\(-0.871961\pi\)
							−0.121055	+	0.992646i	\(0.538628\pi\)
\(13\)	3.60549	+	0.0203369i	0.999984	+	0.00564045i
							\(17\)	0.143863	−	0.249177i	0.0348918	−	0.0604344i	−0.848052	−	0.529913i	\(-0.822225\pi\)
0.882944	+	0.469478i	\(0.155558\pi\)
\(19\)	0.116063	+	0.0670089i	0.0266267	+	0.0153729i	0.513254	−	0.858237i	\(-0.328440\pi\)
							−0.486628	+	0.873610i	\(0.661773\pi\)
\(23\)	−1.26198	−	2.18581i	−0.263141	−	0.455773i	0.703934	−	0.710265i	\(-0.251425\pi\)
							−0.967075	+	0.254492i	\(0.918092\pi\)
\(29\)	−1.26949	−	2.19882i	−0.235738	−	0.408310i	0.723749	−	0.690064i	\(-0.242417\pi\)
							−0.959487	+	0.281753i	\(0.909084\pi\)
\(31\)			1.00000i			0.179605i
							\(37\)	1.36843	−	0.790065i	0.224969	−	0.129886i	−0.383280	−	0.923632i	\(-0.625206\pi\)
0.608249	+	0.793746i	\(0.291872\pi\)
\(41\)	8.41706	−	4.85959i	1.31452	−	0.758941i	0.331682	−	0.943391i	\(-0.392384\pi\)
							0.982842	+	0.184450i	\(0.0590504\pi\)
\(43\)	−0.202925	+	0.351477i	−0.0309458	+	0.0535997i	−0.881084	−	0.472961i	\(-0.843185\pi\)
							0.850138	+	0.526560i	\(0.176519\pi\)
\(47\)			0.0603089i			0.00879696i	0.999990	+	0.00439848i	\(0.00140008\pi\)
							−0.999990	+	0.00439848i	\(0.998600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.218.25	✓	68
13.2	odd	12		5239.2.a.q.1.26		34
13.4	even	6	inner	403.2.r.a.342.25	yes	68
13.11	odd	12		5239.2.a.r.1.9		34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.25	✓	68	1.1	even	1	trivial
403.2.r.a.342.25	yes	68	13.4	even	6	inner
5239.2.a.q.1.26		34	13.2	odd	12
5239.2.a.r.1.9		34	13.11	odd	12