Embedded newform 731.2.z.a.611.16

• Label: 731.2.z.a.611.16
• 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			611.16
Character	\(\chi\)	\(=\)	731.611
Dual form			731.2.z.a.67.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11327 - 1.39600i) q^{2} +(0.311016 - 2.06346i) q^{3} +(-0.264394 + 1.15839i) q^{4} +(-1.52042 + 2.23004i) q^{5} +(-3.22682 + 1.86301i) q^{6} +(2.35848 + 1.36167i) q^{7} +(-1.30600 + 0.628935i) q^{8} +(-1.29440 - 0.399269i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11327	−	1.39600i	−0.787201	−	0.987119i	−0.999950	−	0.0100119i	\(-0.996813\pi\)
							0.212749	−	0.977107i	\(-0.431758\pi\)
\(3\)	0.311016	−	2.06346i	0.179565	−	1.19134i	−0.699499	−	0.714633i	\(-0.746594\pi\)
							0.879064	−	0.476703i	\(-0.158168\pi\)
\(5\)	−1.52042	+	2.23004i	−0.679951	+	0.997305i	0.318823	+	0.947814i	\(0.396712\pi\)
							−0.998775	+	0.0494912i	\(0.984240\pi\)
\(7\)	2.35848	+	1.36167i	0.891421	+	0.514662i	0.874407	−	0.485193i	\(-0.161251\pi\)
							0.0170138	+	0.999855i	\(0.494584\pi\)
\(11\)	4.82473	−	1.10121i	1.45471	−	0.332028i	0.579183	−	0.815197i	\(-0.303371\pi\)
							0.875527	+	0.483169i	\(0.160514\pi\)
\(13\)	−0.0244481	+	0.326237i	−0.00678068	+	0.0904819i	−0.999567	−	0.0294122i	\(-0.990636\pi\)
							0.992787	+	0.119894i	\(0.0382555\pi\)
\(17\)	−1.71004	−	3.75177i	−0.414747	−	0.909937i
							\(19\)	−0.176855	+	0.0545526i	−0.0405733	+	0.0125152i	−0.314975	−	0.949100i	\(-0.601996\pi\)
0.274402	+	0.961615i	\(0.411520\pi\)
\(23\)	0.368639	+	0.397298i	0.0768665	+	0.0828424i	0.770314	−	0.637665i	\(-0.220100\pi\)
							−0.693447	+	0.720507i	\(0.743909\pi\)
\(29\)	−1.42106	−	9.42809i	−0.263883	−	1.75075i	−0.591837	−	0.806058i	\(-0.701597\pi\)
							0.327954	−	0.944694i	\(-0.393641\pi\)
\(31\)	5.08228	+	1.99465i	0.912804	+	0.358249i	0.774817	−	0.632185i	\(-0.217842\pi\)
							0.137986	+	0.990434i	\(0.455937\pi\)
\(37\)	8.80100	−	5.08126i	1.44688	−	0.835354i	0.448582	−	0.893742i	\(-0.351929\pi\)
							0.998294	+	0.0583872i	\(0.0185958\pi\)
\(41\)	−7.50110	+	5.98192i	−1.17147	+	0.934220i	−0.998712	−	0.0507439i	\(-0.983841\pi\)
							−0.172763	+	0.984963i	\(0.555269\pi\)
\(43\)	2.01284	+	6.24087i	0.306956	+	0.951724i
							\(47\)	2.03298	−	8.90708i	0.296541	−	1.29923i	−0.578699	−	0.815541i	\(-0.696439\pi\)
0.875240	−	0.483689i	\(-0.160704\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.611.16	yes	768
17.16	even	2	inner	731.2.z.a.611.15	yes	768
43.24	even	21	inner	731.2.z.a.67.15	✓	768
731.67	even	42	inner	731.2.z.a.67.16	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.15	✓	768	43.24	even	21	inner
731.2.z.a.67.16	yes	768	731.67	even	42	inner
731.2.z.a.611.15	yes	768	17.16	even	2	inner
731.2.z.a.611.16	yes	768	1.1	even	1	trivial