Embedded newform 731.2.y.a.4.6

• Label: 731.2.y.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98792 - 1.58532i) q^{2} +(1.88514 + 0.212405i) q^{3} +(0.993571 + 4.35312i) q^{4} +(0.785952 - 0.275017i) q^{5} +(-3.41079 - 3.41079i) q^{6} +(-0.526144 - 0.526144i) q^{7} +(2.71950 - 5.64710i) q^{8} +(0.583864 + 0.133263i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98792	−	1.58532i	−1.40567	−	1.12099i	−0.975942	−	0.218032i	\(-0.930036\pi\)
							−0.429733	−	0.902956i	\(-0.641392\pi\)
\(3\)	1.88514	+	0.212405i	1.08839	+	0.122632i	0.637876	−	0.770139i	\(-0.279813\pi\)
							0.450511	+	0.892771i	\(0.351242\pi\)
\(5\)	0.785952	−	0.275017i	0.351489	−	0.122991i	−0.148759	−	0.988873i	\(-0.547528\pi\)
							0.500248	+	0.865882i	\(0.333242\pi\)
\(7\)	−0.526144	−	0.526144i	−0.198864	−	0.198864i	0.600649	−	0.799513i	\(-0.294909\pi\)
							−0.799513	+	0.600649i	\(0.794909\pi\)
\(11\)	3.47956	−	2.18635i	1.04913	−	0.659210i	0.106929	−	0.994267i	\(-0.465898\pi\)
							0.942198	+	0.335057i	\(0.108756\pi\)
\(13\)	1.17996	+	0.568241i	0.327263	+	0.157602i	0.590300	−	0.807184i	\(-0.299010\pi\)
							−0.263036	+	0.964786i	\(0.584724\pi\)
\(17\)	3.40106	−	2.33083i	0.824879	−	0.565310i
							\(19\)	−1.30387	+	0.297601i	−0.299129	+	0.0682743i	−0.369451	−	0.929250i	\(-0.620454\pi\)
0.0703222	+	0.997524i	\(0.477597\pi\)
\(23\)	0.942906	−	0.592467i	0.196609	−	0.123538i	−0.430129	−	0.902767i	\(-0.641532\pi\)
							0.626739	+	0.779229i	\(0.284389\pi\)
\(29\)	0.908884	+	8.06657i	0.168775	+	1.49792i	0.737473	+	0.675377i	\(0.236019\pi\)
							−0.568697	+	0.822547i	\(0.692552\pi\)
\(31\)	−0.285439	−	2.53334i	−0.0512663	−	0.455001i	−0.992953	−	0.118510i	\(-0.962188\pi\)
							0.941687	−	0.336491i	\(-0.109240\pi\)
\(37\)	7.40059	−	7.40059i	1.21665	−	1.21665i	0.247851	−	0.968798i	\(-0.420276\pi\)
							0.968798	−	0.247851i	\(-0.0797245\pi\)
\(41\)	10.9249	−	1.23094i	1.70618	−	0.192240i	0.795671	−	0.605728i	\(-0.207118\pi\)
							0.910510	+	0.413488i	\(0.135690\pi\)
\(43\)	5.87534	+	2.91211i	0.895981	+	0.444093i
							\(47\)	−0.516540	−	2.26311i	−0.0753451	−	0.330108i	0.923182	−	0.384363i	\(-0.125579\pi\)
−0.998527	+	0.0542546i	\(0.982722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.6	✓	768
17.13	even	4	inner	731.2.y.a.47.59	yes	768
43.11	even	7	inner	731.2.y.a.140.59	yes	768
731.183	even	28	inner	731.2.y.a.183.6	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.6	✓	768	1.1	even	1	trivial
731.2.y.a.47.59	yes	768	17.13	even	4	inner
731.2.y.a.140.59	yes	768	43.11	even	7	inner
731.2.y.a.183.6	yes	768	731.183	even	28	inner