Embedded newform 403.2.r.a.342.25

• Label: 403.2.r.a.342.25
• Level: $403$
• Weight: $2$
• Character: 403.342
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• 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			342.25
Character	\(\chi\)	\(=\)	403.342
Dual form			403.2.r.a.218.25

$q$-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 \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.25	✓	68	13.10	even	6	inner
403.2.r.a.342.25	yes	68	1.1	even	1	trivial
5239.2.a.q.1.26		34	13.7	odd	12
5239.2.a.r.1.9		34	13.6	odd	12