Embedded newform 731.2.z.a.67.4

• Label: 731.2.z.a.67.4
• 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			67.4
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67449 + 2.09974i) q^{2} +(0.149191 + 0.989818i) q^{3} +(-1.15996 - 5.08212i) q^{4} +(2.44528 + 3.58657i) q^{5} +(-2.32818 - 1.34418i) q^{6} +(-0.298243 + 0.172191i) q^{7} +(7.77406 + 3.74379i) q^{8} +(1.90924 - 0.588921i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67449	+	2.09974i	−1.18404	+	1.48474i	−0.346772	+	0.937949i	\(0.612722\pi\)
							−0.837269	+	0.546791i	\(0.815849\pi\)
\(3\)	0.149191	+	0.989818i	0.0861355	+	0.571472i	0.989715	+	0.143052i	\(0.0456915\pi\)
							−0.903580	+	0.428420i	\(0.859070\pi\)
\(5\)	2.44528	+	3.58657i	1.09356	+	1.60396i	0.745516	+	0.666488i	\(0.232203\pi\)
							0.348049	+	0.937476i	\(0.386844\pi\)
\(7\)	−0.298243	+	0.172191i	−0.112725	+	0.0650819i	−0.555303	−	0.831648i	\(-0.687398\pi\)
							0.442577	+	0.896730i	\(0.354064\pi\)
\(11\)	−3.18945	−	0.727972i	−0.961656	−	0.219492i	−0.287253	−	0.957855i	\(-0.592742\pi\)
							−0.674403	+	0.738363i	\(0.735599\pi\)
\(13\)	−0.285303	−	3.80710i	−0.0791287	−	1.05590i	−0.885546	−	0.464552i	\(-0.846215\pi\)
							0.806417	−	0.591347i	\(-0.201404\pi\)
\(17\)	2.15873	+	3.51281i	0.523570	+	0.851983i
							\(19\)	−3.96623	−	1.22342i	−0.909916	−	0.280672i	−0.195754	−	0.980653i	\(-0.562716\pi\)
−0.714161	+	0.699981i	\(0.753192\pi\)
\(23\)	−4.78248	+	5.15428i	−0.997215	+	1.07474i	−1.24088e−5		1.00000i	\(0.500004\pi\)
							−0.997203	+	0.0747425i	\(0.976187\pi\)
\(29\)	−0.840643	+	5.57730i	−0.156104	+	1.03568i	0.764343	+	0.644810i	\(0.223064\pi\)
							−0.920446	+	0.390869i	\(0.872175\pi\)
\(31\)	−3.71087	+	1.45641i	−0.666492	+	0.261579i	−0.674374	−	0.738390i	\(-0.735586\pi\)
							0.00788183	+	0.999969i	\(0.497491\pi\)
\(37\)	−1.26753	−	0.731808i	−0.208380	−	0.120309i	0.392178	−	0.919889i	\(-0.371722\pi\)
							−0.600558	+	0.799581i	\(0.705055\pi\)
\(41\)	−2.16785	−	1.72880i	−0.338561	−	0.269994i	0.439419	−	0.898282i	\(-0.355184\pi\)
							−0.777980	+	0.628289i	\(0.783756\pi\)
\(43\)	−3.70677	−	5.40924i	−0.565277	−	0.824901i
							\(47\)	1.12220	+	4.91667i	0.163689	+	0.717170i	0.988432	+	0.151662i	\(0.0484626\pi\)
−0.824743	+	0.565508i	\(0.808680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.67.4	yes	768
17.16	even	2	inner	731.2.z.a.67.3	✓	768
43.9	even	21	inner	731.2.z.a.611.3	yes	768
731.611	even	42	inner	731.2.z.a.611.4	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.3	✓	768	17.16	even	2	inner
731.2.z.a.67.4	yes	768	1.1	even	1	trivial
731.2.z.a.611.3	yes	768	43.9	even	21	inner
731.2.z.a.611.4	yes	768	731.611	even	42	inner