Embedded newform 731.2.z.a.67.15

• Label: 731.2.z.a.67.15
• 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			67.15
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.15

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11327	+	1.39600i	−0.787201	+	0.987119i	0.212749	+	0.977107i	\(0.431758\pi\)
							−0.999950	+	0.0100119i	\(0.996813\pi\)
\(3\)	−0.311016	−	2.06346i	−0.179565	−	1.19134i	−0.879064	−	0.476703i	\(-0.841832\pi\)
							0.699499	−	0.714633i	\(-0.253406\pi\)
\(5\)	1.52042	+	2.23004i	0.679951	+	0.997305i	0.998775	+	0.0494912i	\(0.0157600\pi\)
							−0.318823	+	0.947814i	\(0.603288\pi\)
\(7\)	−2.35848	+	1.36167i	−0.891421	+	0.514662i	−0.874407	−	0.485193i	\(-0.838749\pi\)
							−0.0170138	+	0.999855i	\(0.505416\pi\)
\(11\)	−4.82473	−	1.10121i	−1.45471	−	0.332028i	−0.579183	−	0.815197i	\(-0.696629\pi\)
							−0.875527	+	0.483169i	\(0.839486\pi\)
\(13\)	−0.0244481	−	0.326237i	−0.00678068	−	0.0904819i	0.992787	−	0.119894i	\(-0.0382555\pi\)
							−0.999567	+	0.0294122i	\(0.990636\pi\)
\(17\)	2.86767	−	2.96251i	0.695512	−	0.718514i
							\(19\)	−0.176855	−	0.0545526i	−0.0405733	−	0.0125152i	0.274402	−	0.961615i	\(-0.411520\pi\)
−0.314975	+	0.949100i	\(0.601996\pi\)
\(23\)	−0.368639	+	0.397298i	−0.0768665	+	0.0828424i	−0.770314	−	0.637665i	\(-0.779900\pi\)
							0.693447	+	0.720507i	\(0.256091\pi\)
\(29\)	1.42106	−	9.42809i	0.263883	−	1.75075i	−0.327954	−	0.944694i	\(-0.606359\pi\)
							0.591837	−	0.806058i	\(-0.298403\pi\)
\(31\)	−5.08228	+	1.99465i	−0.912804	+	0.358249i	−0.774817	−	0.632185i	\(-0.782158\pi\)
							−0.137986	+	0.990434i	\(0.544063\pi\)
\(37\)	−8.80100	−	5.08126i	−1.44688	−	0.835354i	−0.448582	−	0.893742i	\(-0.648071\pi\)
							−0.998294	+	0.0583872i	\(0.981404\pi\)
\(41\)	7.50110	+	5.98192i	1.17147	+	0.934220i	0.998712	−	0.0507439i	\(-0.0161592\pi\)
							0.172763	+	0.984963i	\(0.444731\pi\)
\(43\)	2.01284	−	6.24087i	0.306956	−	0.951724i
							\(47\)	2.03298	+	8.90708i	0.296541	+	1.29923i	0.875240	+	0.483689i	\(0.160704\pi\)
−0.578699	+	0.815541i	\(0.696439\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.15	✓	768	1.1	even	1	trivial
731.2.z.a.67.16	yes	768	17.16	even	2	inner
731.2.z.a.611.15	yes	768	731.611	even	42	inner
731.2.z.a.611.16	yes	768	43.9	even	21	inner