Embedded newform 501.2.e.a.4.4

• Label: 501.2.e.a.4.4
• Level: $501$
• Weight: $2$
• Character: 501.4
• Analytic conductor: $4.001$
• Analytic rank: $0$
• Dimension: $1148$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 501 = 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	501.e (of order \(83\), degree \(82\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00050514127\)
Analytic rank:	\(0\)
Dimension:	\(1148\)
Relative dimension:	\(14\) over \(\Q(\zeta_{83})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{83}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	501.4
Dual form			501.2.e.a.376.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23206 - 1.46206i) q^{2} +(-0.584719 - 0.811236i) q^{3} +(-0.280634 + 1.63165i) q^{4} +(0.0557298 + 0.980522i) q^{5} +(-0.465666 + 1.85439i) q^{6} +(-4.08270 + 1.80541i) q^{7} +(-0.568155 + 0.332821i) q^{8} +(-0.316208 + 0.948690i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1148 q - 2 q^{2} - 14 q^{3} - 20 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/501\mathbb{Z}\right)^\times\).

\(n\)	\(172\)	\(335\)
\(\chi(n)\)	\(e\left(\frac{40}{83}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23206	−	1.46206i	−0.871201	−	1.03383i	−0.999116	−	0.0420492i	\(-0.986611\pi\)
							0.127914	−	0.991785i	\(-0.459172\pi\)
\(3\)	−0.584719	−	0.811236i	−0.337588	−	0.468367i
							\(5\)	0.0557298	+	0.980522i	0.0249231	+	0.438503i	0.986478	+	0.163891i	\(0.0524046\pi\)
−0.961555	+	0.274611i	\(0.911451\pi\)
\(7\)	−4.08270	+	1.80541i	−1.54312	+	0.682379i	−0.988707	−	0.149864i	\(-0.952116\pi\)
							−0.554410	+	0.832244i	\(0.687056\pi\)
\(11\)	0.0368948	+	1.94927i	0.0111242	+	0.587726i	0.961222	+	0.275775i	\(0.0889343\pi\)
							−0.950098	+	0.311951i	\(0.899017\pi\)
\(13\)	2.83344	−	6.09304i	0.785854	−	1.68991i	0.0639097	−	0.997956i	\(-0.479643\pi\)
							0.721945	−	0.691951i	\(-0.243249\pi\)
\(17\)	0.0835166	−	0.626732i	0.0202558	−	0.152005i	−0.978192	−	0.207703i	\(-0.933401\pi\)
							0.998448	+	0.0556984i	\(0.0177385\pi\)
\(19\)	4.62008	−	0.704884i	1.05992	−	0.161711i	0.402640	−	0.915359i	\(-0.368093\pi\)
							0.657279	+	0.753647i	\(0.271707\pi\)
\(23\)	−1.76234	+	2.25960i	−0.367473	+	0.471160i	−0.936252	−	0.351330i	\(-0.885729\pi\)
							0.568779	+	0.822491i	\(0.307416\pi\)
\(29\)	4.76102	+	6.10440i	0.884098	+	1.13356i	0.990263	+	0.139210i	\(0.0444563\pi\)
							−0.106164	+	0.994349i	\(0.533857\pi\)
\(31\)	0.754709	+	1.79792i	0.135550	+	0.322916i	0.975587	−	0.219612i	\(-0.0704792\pi\)
							−0.840038	+	0.542528i	\(0.817467\pi\)
\(37\)	2.34729	+	7.04236i	0.385891	+	1.15776i	0.944553	+	0.328360i	\(0.106496\pi\)
							−0.558661	+	0.829396i	\(0.688685\pi\)
\(41\)	0.421857	−	0.429918i	0.0658831	−	0.0671419i	−0.680035	−	0.733180i	\(-0.738035\pi\)
							0.745918	+	0.666038i	\(0.232011\pi\)
\(43\)	3.57042	−	0.825088i	0.544485	−	0.125825i	0.0560543	−	0.998428i	\(-0.482148\pi\)
							0.488430	+	0.872603i	\(0.337570\pi\)
\(47\)	2.14555	−	5.70742i	0.312961	−	0.832512i	−0.681917	−	0.731429i	\(-0.738854\pi\)
							0.994878	−	0.101083i	\(-0.0322308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	501.2.e.a.4.4	✓	1148
167.42	even	83	inner	501.2.e.a.376.4	yes	1148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.e.a.4.4	✓	1148	1.1	even	1	trivial
501.2.e.a.376.4	yes	1148	167.42	even	83	inner