Embedded newform 731.2.z.a.67.20

• Label: 731.2.z.a.67.20
• 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			67.20
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.20

$q$-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	\(0.0896534\pi\)
							−0.835994	+	0.548738i	\(0.815108\pi\)
\(5\)	1.58922	+	2.33095i	0.710719	+	1.04243i	0.996455	+	0.0841255i	\(0.0268096\pi\)
							−0.285736	+	0.958308i	\(0.592238\pi\)
\(7\)	−1.01277	+	0.584720i	−0.382789	+	0.221003i	−0.679031	−	0.734109i	\(-0.737600\pi\)
							0.296242	+	0.955113i	\(0.404267\pi\)
\(11\)	−4.33416	−	0.989244i	−1.30680	−	0.298268i	−0.488249	−	0.872704i	\(-0.662364\pi\)
							−0.818550	+	0.574436i	\(0.805221\pi\)
\(13\)	0.132431	+	1.76717i	0.0367299	+	0.490126i	0.984988	+	0.172625i	\(0.0552247\pi\)
							−0.948258	+	0.317501i	\(0.897156\pi\)
\(17\)	−2.23223	−	3.46658i	−0.541394	−	0.840769i
							\(19\)	−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	−0.616064	−	0.787696i	\(-0.711274\pi\)
							0.831532	+	0.555477i	\(0.187464\pi\)
\(29\)	−1.31114	+	8.69885i	−0.243473	+	1.61534i	0.448754	+	0.893655i	\(0.351868\pi\)
							−0.692227	+	0.721680i	\(0.743370\pi\)
\(31\)	−5.10967	+	2.00540i	−0.917723	+	0.360180i	−0.776732	−	0.629831i	\(-0.783124\pi\)
							−0.140991	+	0.990011i	\(0.545029\pi\)
\(37\)	8.49656	+	4.90549i	1.39683	+	0.806457i	0.994059	−	0.108845i	\(-0.0347153\pi\)
							0.402767	+	0.915303i	\(0.368049\pi\)
\(41\)	3.60772	+	2.87706i	0.563431	+	0.449322i	0.863323	−	0.504652i	\(-0.168379\pi\)
							−0.299892	+	0.953973i	\(0.596951\pi\)
\(43\)	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.67.20	yes	768
17.16	even	2	inner	731.2.z.a.67.19	✓	768
43.9	even	21	inner	731.2.z.a.611.19	yes	768
731.611	even	42	inner	731.2.z.a.611.20	yes	768

    

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