Embedded newform 731.2.z.a.67.2

• Label: 731.2.z.a.67.2
• Level: $731$
• Weight: $2$
• Character: 731.67
• 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.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			67.2
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69621 + 2.12698i) q^{2} +(0.339828 + 2.25461i) q^{3} +(-1.20187 - 5.26573i) q^{4} +(-0.0789863 - 0.115852i) q^{5} +(-5.37192 - 3.10148i) q^{6} +(0.309020 - 0.178413i) q^{7} +(8.33653 + 4.01466i) q^{8} +(-2.10106 + 0.648091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69621	+	2.12698i	−1.19940	+	1.50400i	−0.385792	+	0.922586i	\(0.626072\pi\)
							−0.813608	+	0.581414i	\(0.802500\pi\)
\(3\)	0.339828	+	2.25461i	0.196200	+	1.30170i	0.842533	+	0.538644i	\(0.181063\pi\)
							−0.646334	+	0.763055i	\(0.723699\pi\)
\(5\)	−0.0789863	−	0.115852i	−0.0353238	−	0.0518104i	0.808171	−	0.588948i	\(-0.200458\pi\)
							−0.843494	+	0.537138i	\(0.819505\pi\)
\(7\)	0.309020	−	0.178413i	0.116799	−	0.0674338i	−0.440463	−	0.897771i	\(-0.645186\pi\)
							0.557261	+	0.830337i	\(0.311852\pi\)
\(11\)	3.60912	+	0.823758i	1.08819	+	0.248372i	0.728739	−	0.684792i	\(-0.240107\pi\)
							0.359451	+	0.933164i	\(0.382964\pi\)
\(13\)	0.376466	+	5.02359i	0.104413	+	1.39329i	0.765718	+	0.643176i	\(0.222384\pi\)
							−0.661305	+	0.750117i	\(0.729997\pi\)
\(17\)	3.50997	−	2.16336i	0.851292	−	0.524692i
							\(19\)	6.96983	+	2.14991i	1.59899	+	0.493223i	0.960936	−	0.276770i	\(-0.0892639\pi\)
0.638053	+	0.769992i	\(0.279740\pi\)
\(23\)	4.15616	−	4.47928i	0.866620	−	0.933994i	−0.131738	−	0.991285i	\(-0.542056\pi\)
							0.998358	+	0.0572906i	\(0.0182462\pi\)
\(29\)	−0.915033	+	6.07085i	−0.169917	+	1.12733i	0.727488	+	0.686121i	\(0.240688\pi\)
							−0.897405	+	0.441207i	\(0.854550\pi\)
\(31\)	−8.32384	+	3.26687i	−1.49501	+	0.586746i	−0.965232	−	0.261393i	\(-0.915818\pi\)
							−0.529773	+	0.848139i	\(0.677723\pi\)
\(37\)	−6.18125	−	3.56874i	−1.01619	−	0.586698i	−0.103192	−	0.994661i	\(-0.532906\pi\)
							−0.912998	+	0.407963i	\(0.866239\pi\)
\(41\)	−3.91499	−	3.12210i	−0.611419	−	0.487590i	0.268140	−	0.963380i	\(-0.413591\pi\)
							−0.879559	+	0.475790i	\(0.842162\pi\)
\(43\)	6.39212	+	1.46313i	0.974790	+	0.223125i
							\(47\)	−1.53957	−	6.74531i	−0.224570	−	0.983904i	−0.953990	−	0.299838i	\(-0.903067\pi\)
0.729420	−	0.684066i	\(-0.239790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.67.2	yes	768
17.16	even	2	inner	731.2.z.a.67.1	✓	768
43.9	even	21	inner	731.2.z.a.611.1	yes	768
731.611	even	42	inner	731.2.z.a.611.2	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.1	✓	768	17.16	even	2	inner
731.2.z.a.67.2	yes	768	1.1	even	1	trivial
731.2.z.a.611.1	yes	768	43.9	even	21	inner
731.2.z.a.611.2	yes	768	731.611	even	42	inner