Embedded newform 731.2.s.b.214.11

• Label: 731.2.s.b.214.11
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			214.11
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93285 - 0.800613i) q^{2} +(0.211222 - 0.316117i) q^{3} +(1.68072 + 1.68072i) q^{4} +(-0.167026 - 0.839698i) q^{5} +(-0.661349 + 0.441899i) q^{6} +(0.314174 - 1.57946i) q^{7} +(-0.301744 - 0.728474i) q^{8} +(1.09274 + 2.63810i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93285	−	0.800613i	−1.36673	−	0.566119i	−0.425831	−	0.904803i	\(-0.640018\pi\)
							−0.940900	+	0.338684i	\(0.890018\pi\)
\(3\)	0.211222	−	0.316117i	0.121949	−	0.182510i	−0.765472	−	0.643469i	\(-0.777494\pi\)
							0.887421	+	0.460959i	\(0.152494\pi\)
\(5\)	−0.167026	−	0.839698i	−0.0746964	−	0.375524i	0.925297	−	0.379244i	\(-0.123816\pi\)
							−0.999993	+	0.00371987i	\(0.998816\pi\)
\(7\)	0.314174	−	1.57946i	0.118747	−	0.596980i	−0.874888	−	0.484326i	\(-0.839065\pi\)
							0.993634	−	0.112654i	\(-0.0359352\pi\)
\(11\)	−0.258209	−	0.386438i	−0.0778531	−	0.116515i	0.790500	−	0.612463i	\(-0.209821\pi\)
							−0.868353	+	0.495947i	\(0.834821\pi\)
\(13\)	−2.79426	+	2.79426i	−0.774989	+	0.774989i	−0.978974	−	0.203985i	\(-0.934611\pi\)
							0.203985	+	0.978974i	\(0.434611\pi\)
\(17\)	−3.49231	+	2.19176i	−0.847008	+	0.531579i
							\(19\)	−6.15758	−	2.55055i	−1.41264	−	0.585137i	−0.459644	−	0.888103i	\(-0.652023\pi\)
−0.953001	+	0.302967i	\(0.902023\pi\)
\(23\)	0.721254	−	0.481927i	0.150392	−	0.100489i	−0.478098	−	0.878307i	\(-0.658673\pi\)
							0.628490	+	0.777818i	\(0.283673\pi\)
\(29\)	−7.60421	+	1.51257i	−1.41207	+	0.280877i	−0.841448	−	0.540338i	\(-0.818296\pi\)
							−0.570618	+	0.821216i	\(0.693296\pi\)
\(31\)	0.265714	−	0.397668i	0.0477236	−	0.0714234i	−0.806841	−	0.590769i	\(-0.798824\pi\)
							0.854564	+	0.519346i	\(0.173824\pi\)
\(37\)	−1.13859	+	1.70402i	−0.187183	+	0.280139i	−0.913178	−	0.407561i	\(-0.866380\pi\)
							0.725995	+	0.687700i	\(0.241380\pi\)
\(41\)	−10.7767	−	2.14362i	−1.68304	−	0.334777i	−0.741312	−	0.671161i	\(-0.765796\pi\)
							−0.941725	+	0.336384i	\(0.890796\pi\)
\(43\)	−2.81313	−	5.92337i	−0.428998	−	0.903306i
							\(47\)	−1.39398	+	1.39398i	−0.203333	+	0.203333i	−0.801427	−	0.598093i	\(-0.795925\pi\)
0.598093	+	0.801427i	\(0.295925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.11	✓	496
17.12	odd	16	inner	731.2.s.b.386.52	yes	496
43.42	odd	2	inner	731.2.s.b.214.52	yes	496
731.386	even	16	inner	731.2.s.b.386.11	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.11	✓	496	1.1	even	1	trivial
731.2.s.b.214.52	yes	496	43.42	odd	2	inner
731.2.s.b.386.11	yes	496	731.386	even	16	inner
731.2.s.b.386.52	yes	496	17.12	odd	16	inner