Embedded newform 731.2.bd.a.7.19

• Label: 731.2.bd.a.7.19
• 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			7.19
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29704 - 0.537254i) q^{2} +(0.119073 - 1.81670i) q^{3} +(-0.0205300 - 0.0205300i) q^{4} +(-2.41704 - 2.75611i) q^{5} +(-1.13047 + 2.29237i) q^{6} +(-3.01254 + 3.43514i) q^{7} +(1.09011 + 2.63175i) q^{8} +(-0.311883 - 0.0410601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29704	−	0.537254i	−0.917149	−	0.379896i	−0.126360	−	0.991984i	\(-0.540329\pi\)
							−0.790789	+	0.612089i	\(0.790329\pi\)
\(3\)	0.119073	−	1.81670i	0.0687467	−	1.04887i	−0.814561	−	0.580078i	\(-0.803022\pi\)
							0.883308	−	0.468794i	\(-0.155311\pi\)
\(5\)	−2.41704	−	2.75611i	−1.08094	−	1.23257i	−0.971113	−	0.238619i	\(-0.923305\pi\)
							−0.109822	−	0.993951i	\(-0.535028\pi\)
\(7\)	−3.01254	+	3.43514i	−1.13863	+	1.29836i	−0.190071	+	0.981770i	\(0.560872\pi\)
							−0.948562	+	0.316592i	\(0.897461\pi\)
\(11\)	1.77625	−	1.18685i	0.535560	−	0.357850i	−0.258189	−	0.966094i	\(-0.583126\pi\)
							0.793750	+	0.608244i	\(0.208126\pi\)
\(13\)	−0.0570798	−	0.213025i	−0.0158311	−	0.0590824i	0.957558	−	0.288239i	\(-0.0930698\pi\)
							−0.973390	+	0.229157i	\(0.926403\pi\)
\(17\)	−4.05775	−	0.731233i	−0.984148	−	0.177350i
							\(19\)	0.902880	+	6.85805i	0.207135	+	1.57335i	0.706982	+	0.707232i	\(0.250056\pi\)
−0.499847	+	0.866114i	\(0.666610\pi\)
\(23\)	0.672045	−	1.36277i	0.140131	−	0.284158i	−0.815482	−	0.578783i	\(-0.803528\pi\)
							0.955613	+	0.294625i	\(0.0951947\pi\)
\(29\)	−2.83354	−	0.961857i	−0.526175	−	0.178612i	0.0456967	−	0.998955i	\(-0.485449\pi\)
							−0.571872	+	0.820343i	\(0.693783\pi\)
\(31\)	−0.415456	+	6.33863i	−0.0746181	+	1.13845i	0.782267	+	0.622944i	\(0.214064\pi\)
							−0.856885	+	0.515508i	\(0.827603\pi\)
\(37\)	0.128084	−	1.95418i	0.0210569	−	0.321266i	−0.974316	−	0.225183i	\(-0.927702\pi\)
							0.995373	−	0.0960828i	\(-0.0306314\pi\)
\(41\)	−2.39453	+	12.0381i	−0.373962	+	1.88003i	0.0928537	+	0.995680i	\(0.470401\pi\)
							−0.466816	+	0.884355i	\(0.654599\pi\)
\(43\)	−0.905748	−	6.49458i	−0.138125	−	0.990415i
							\(47\)	5.57202	−	5.57202i	0.812763	−	0.812763i	−0.172284	−	0.985047i	\(-0.555115\pi\)
0.985047	+	0.172284i	\(0.0551148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.19	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.46	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.46	yes	1024
731.209	even	48	inner	731.2.bd.a.209.19	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.19	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.19	yes	1024	731.209	even	48	inner
731.2.bd.a.381.46	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.46	yes	1024	17.5	odd	16	inner