Embedded newform 731.2.z.a.611.4

• Label: 731.2.z.a.611.4
• Level: $731$
• Weight: $2$
• Character: 731.611
• 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			611.4
Character	\(\chi\)	\(=\)	731.611
Dual form			731.2.z.a.67.4

$q$-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 \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.3	✓	768	43.24	even	21	inner
731.2.z.a.67.4	yes	768	731.67	even	42	inner
731.2.z.a.611.3	yes	768	17.16	even	2	inner
731.2.z.a.611.4	yes	768	1.1	even	1	trivial