Embedded newform 731.2.z.a.67.19

• Label: 731.2.z.a.67.19
• 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			67.19
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.755649 + 0.947554i) q^{2} +(-0.215818 - 1.43186i) q^{3} +(0.118189 + 0.517820i) q^{4} +(-1.58922 - 2.33095i) q^{5} +(1.51984 + 0.877483i) q^{6} +(1.01277 - 0.584720i) q^{7} +(-2.76386 - 1.33100i) q^{8} +(0.863080 - 0.266225i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.755649	+	0.947554i	−0.534325	+	0.670022i	−0.973582	−	0.228340i	\(-0.926670\pi\)
							0.439257	+	0.898361i	\(0.355242\pi\)
\(3\)	−0.215818	−	1.43186i	−0.124602	−	0.826683i	−0.960597	−	0.277945i	\(-0.910347\pi\)
							0.835994	−	0.548738i	\(-0.184892\pi\)
\(5\)	−1.58922	−	2.33095i	−0.710719	−	1.04243i	−0.996455	−	0.0841255i	\(-0.973190\pi\)
							0.285736	−	0.958308i	\(-0.407762\pi\)
\(7\)	1.01277	−	0.584720i	0.382789	−	0.221003i	−0.296242	−	0.955113i	\(-0.595733\pi\)
							0.679031	+	0.734109i	\(0.262400\pi\)
\(11\)	4.33416	+	0.989244i	1.30680	+	0.298268i	0.818550	−	0.574436i	\(-0.194779\pi\)
							0.488249	+	0.872704i	\(0.337636\pi\)
\(13\)	0.132431	+	1.76717i	0.0367299	+	0.490126i	0.984988	+	0.172625i	\(0.0552247\pi\)
							−0.948258	+	0.317501i	\(0.897156\pi\)
\(17\)	−4.04247	−	0.811437i	−0.980443	−	0.196802i
							\(19\)	−3.67852	−	1.13467i	−0.843911	−	0.260312i	−0.157494	−	0.987520i	\(-0.550341\pi\)
−0.686417	+	0.727208i	\(0.740818\pi\)
\(23\)	−1.03335	+	1.11368i	−0.215467	+	0.232219i	−0.831532	−	0.555477i	\(-0.812536\pi\)
							0.616064	+	0.787696i	\(0.288726\pi\)
\(29\)	1.31114	−	8.69885i	0.243473	−	1.61534i	−0.448754	−	0.893655i	\(-0.648132\pi\)
							0.692227	−	0.721680i	\(-0.256630\pi\)
\(31\)	5.10967	−	2.00540i	0.917723	−	0.360180i	0.140991	−	0.990011i	\(-0.454971\pi\)
							0.776732	+	0.629831i	\(0.216876\pi\)
\(37\)	−8.49656	−	4.90549i	−1.39683	−	0.806457i	−0.402767	−	0.915303i	\(-0.631951\pi\)
							−0.994059	+	0.108845i	\(0.965285\pi\)
\(41\)	−3.60772	−	2.87706i	−0.563431	−	0.449322i	0.299892	−	0.953973i	\(-0.403049\pi\)
							−0.863323	+	0.504652i	\(0.831621\pi\)
\(43\)	3.21446	+	5.71553i	0.490200	+	0.871610i
							\(47\)	−2.29884	−	10.0719i	−0.335320	−	1.46913i	−0.808671	−	0.588261i	\(-0.799813\pi\)
0.473351	−	0.880874i	\(-0.343044\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.67.19	✓	768
17.16	even	2	inner	731.2.z.a.67.20	yes	768
43.9	even	21	inner	731.2.z.a.611.20	yes	768
731.611	even	42	inner	731.2.z.a.611.19	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.19	✓	768	1.1	even	1	trivial
731.2.z.a.67.20	yes	768	17.16	even	2	inner
731.2.z.a.611.19	yes	768	731.611	even	42	inner
731.2.z.a.611.20	yes	768	43.9	even	21	inner