Embedded newform 731.2.bd.a.7.12

• Label: 731.2.bd.a.7.12
• Level: $731$
• Weight: $2$
• Character: 731.7
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bd (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(64\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			7.12
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80829 - 0.749020i) q^{2} +(0.223690 - 3.41284i) q^{3} +(1.29469 + 1.29469i) q^{4} +(0.118178 + 0.134757i) q^{5} +(-2.96079 + 6.00388i) q^{6} +(-1.90537 + 2.17266i) q^{7} +(0.126613 + 0.305671i) q^{8} +(-8.62313 - 1.13526i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80829	−	0.749020i	−1.27866	−	0.529637i	−0.363072	−	0.931761i	\(-0.618272\pi\)
							−0.915585	+	0.402124i	\(0.868272\pi\)
\(3\)	0.223690	−	3.41284i	0.129147	−	1.97041i	−0.0938483	−	0.995587i	\(-0.529917\pi\)
							0.222996	−	0.974819i	\(-0.428416\pi\)
\(5\)	0.118178	+	0.134757i	0.0528510	+	0.0602650i	0.777661	−	0.628684i	\(-0.216406\pi\)
							−0.724810	+	0.688949i	\(0.758073\pi\)
\(7\)	−1.90537	+	2.17266i	−0.720164	+	0.821190i	−0.989903	−	0.141744i	\(-0.954729\pi\)
							0.269739	+	0.962933i	\(0.413062\pi\)
\(11\)	−3.05902	+	2.04397i	−0.922329	+	0.616281i	−0.923450	−	0.383719i	\(-0.874643\pi\)
							0.00112089	+	0.999999i	\(0.499643\pi\)
\(13\)	0.994386	+	3.71110i	0.275793	+	1.02927i	0.955308	+	0.295612i	\(0.0955236\pi\)
							−0.679515	+	0.733662i	\(0.737810\pi\)
\(17\)	4.11557	+	0.249154i	0.998173	+	0.0604287i
							\(19\)	0.100345	+	0.762193i	0.0230206	+	0.174859i	0.999093	−	0.0425881i	\(-0.0135603\pi\)
−0.976072	+	0.217447i	\(0.930227\pi\)
\(23\)	3.75831	−	7.62110i	0.783661	−	1.58911i	−0.0241794	−	0.999708i	\(-0.507697\pi\)
							0.807841	−	0.589401i	\(-0.200636\pi\)
\(29\)	−1.04299	−	0.354046i	−0.193678	−	0.0657447i	0.222915	−	0.974838i	\(-0.428443\pi\)
							−0.416593	+	0.909093i	\(0.636776\pi\)
\(31\)	−0.341604	+	5.21188i	−0.0613540	+	0.936081i	0.850607	+	0.525803i	\(0.176235\pi\)
							−0.911961	+	0.410278i	\(0.865432\pi\)
\(37\)	−0.00586791	+	0.0895270i	−0.000964678	+	0.0147181i	−0.998314	−	0.0580423i	\(-0.981514\pi\)
							0.997349	+	0.0727604i	\(0.0231808\pi\)
\(41\)	−0.553992	+	2.78510i	−0.0865190	+	0.434960i	0.913110	+	0.407714i	\(0.133674\pi\)
							−0.999629	+	0.0272464i	\(0.991326\pi\)
\(43\)	−2.73726	+	5.95881i	−0.417428	+	0.908710i
							\(47\)	−2.00792	+	2.00792i	−0.292886	+	0.292886i	−0.838219	−	0.545333i	\(-0.816403\pi\)
0.545333	+	0.838219i	\(0.316403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.12	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.53	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.53	yes	1024
731.209	even	48	inner	731.2.bd.a.209.12	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.12	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.12	yes	1024	731.209	even	48	inner
731.2.bd.a.381.53	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.53	yes	1024	17.5	odd	16	inner