Embedded newform 731.2.y.a.4.10

• Label: 731.2.y.a.4.10
• Level: $731$
• Weight: $2$
• Character: 731.4
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.y (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			4.10
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59420 - 1.27133i) q^{2} +(3.17551 + 0.357794i) q^{3} +(0.480150 + 2.10368i) q^{4} +(3.26776 - 1.14344i) q^{5} +(-4.60753 - 4.60753i) q^{6} +(2.24837 + 2.24837i) q^{7} +(0.139587 - 0.289855i) q^{8} +(7.03107 + 1.60480i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59420	−	1.27133i	−1.12727	−	0.898969i	−0.131544	−	0.991310i	\(-0.541994\pi\)
							−0.995727	+	0.0923409i	\(0.970565\pi\)
\(3\)	3.17551	+	0.357794i	1.83338	+	0.206573i	0.959911	−	0.280303i	\(-0.0904351\pi\)
							0.873471	+	0.486876i	\(0.161864\pi\)
\(5\)	3.26776	−	1.14344i	1.46139	−	0.511362i	0.521621	−	0.853177i	\(-0.325327\pi\)
							0.939767	+	0.341815i	\(0.111042\pi\)
\(7\)	2.24837	+	2.24837i	0.849803	+	0.849803i	0.990108	−	0.140305i	\(-0.0448085\pi\)
							−0.140305	+	0.990108i	\(0.544808\pi\)
\(11\)	−1.86731	+	1.17331i	−0.563014	+	0.353765i	−0.783275	−	0.621676i	\(-0.786452\pi\)
							0.220260	+	0.975441i	\(0.429309\pi\)
\(13\)	−6.19240	−	2.98210i	−1.71746	−	0.827087i	−0.990021	−	0.140917i	\(-0.954995\pi\)
							−0.727442	−	0.686169i	\(-0.759291\pi\)
\(17\)	2.71905	−	3.09948i	0.659466	−	0.751735i
							\(19\)	−3.46350	+	0.790521i	−0.794581	+	0.181358i	−0.600497	−	0.799627i	\(-0.705031\pi\)
−0.194084	+	0.980985i	\(0.562173\pi\)
\(23\)	−1.28533	+	0.807628i	−0.268010	+	0.168402i	−0.659332	−	0.751852i	\(-0.729161\pi\)
							0.391322	+	0.920254i	\(0.372018\pi\)
\(29\)	−0.326792	−	2.90036i	−0.0606837	−	0.538583i	−0.986870	−	0.161517i	\(-0.948361\pi\)
							0.926186	−	0.377066i	\(-0.123067\pi\)
\(31\)	−0.480252	−	4.26235i	−0.0862558	−	0.765541i	−0.960162	−	0.279446i	\(-0.909849\pi\)
							0.873906	−	0.486095i	\(-0.161579\pi\)
\(37\)	−7.22495	+	7.22495i	−1.18777	+	1.18777i	−0.210093	+	0.977681i	\(0.567377\pi\)
							−0.977681	+	0.210093i	\(0.932623\pi\)
\(41\)	−3.30815	+	0.372739i	−0.516646	+	0.0582121i	−0.366438	−	0.930442i	\(-0.619423\pi\)
							−0.150208	+	0.988654i	\(0.547994\pi\)
\(43\)	6.53542	+	0.536903i	0.996642	+	0.0818769i
							\(47\)	2.03020	+	8.89487i	0.296135	+	1.29745i	0.875831	+	0.482618i	\(0.160314\pi\)
−0.579697	+	0.814832i	\(0.696829\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.10	✓	768
17.13	even	4	inner	731.2.y.a.47.55	yes	768
43.11	even	7	inner	731.2.y.a.140.55	yes	768
731.183	even	28	inner	731.2.y.a.183.10	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.10	✓	768	1.1	even	1	trivial
731.2.y.a.47.55	yes	768	17.13	even	4	inner
731.2.y.a.140.55	yes	768	43.11	even	7	inner
731.2.y.a.183.10	yes	768	731.183	even	28	inner