Embedded newform 731.2.n.a.208.6

• Label: 731.2.n.a.208.6
• Level: $731$
• Weight: $2$
• Character: 731.208
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			208.6
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.59

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.49716i q^{2} +(0.186349 + 0.0499322i) q^{3} -4.23579 q^{4} +(0.0534817 + 0.0143304i) q^{5} +(0.124688 - 0.465344i) q^{6} +(4.60333 - 1.23346i) q^{7} +5.58312i q^{8} +(-2.56584 - 1.48139i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.49716i		−	1.76576i	−0.469602	−	0.882878i	\(-0.655603\pi\)
							0.469602	−	0.882878i	\(-0.344397\pi\)
\(3\)	0.186349	+	0.0499322i	0.107589	+	0.0288284i	0.312212	−	0.950013i	\(-0.398930\pi\)
							−0.204623	+	0.978841i	\(0.565597\pi\)
\(5\)	0.0534817	+	0.0143304i	0.0239177	+	0.00640874i	0.270758	−	0.962647i	\(-0.412726\pi\)
							−0.246840	+	0.969056i	\(0.579392\pi\)
\(7\)	4.60333	−	1.23346i	1.73989	−	0.466203i	0.757470	−	0.652870i	\(-0.226435\pi\)
							0.982423	+	0.186667i	\(0.0597684\pi\)
\(11\)	0.593469	+	0.593469i	0.178938	+	0.178938i	0.790893	−	0.611955i	\(-0.209617\pi\)
							−0.611955	+	0.790893i	\(0.709617\pi\)
\(13\)	0.665594	−	1.15284i	0.184603	−	0.319741i	−0.758840	−	0.651277i	\(-0.774234\pi\)
							0.943443	+	0.331536i	\(0.107567\pi\)
\(17\)	1.20948	−	3.94172i	0.293341	−	0.956008i
							\(19\)	−2.62489	+	1.51548i	−0.602191	+	0.347675i	−0.769903	−	0.638161i	\(-0.779695\pi\)
0.167712	+	0.985836i	\(0.446362\pi\)
\(23\)	1.48450	−	5.54022i	0.309539	−	1.15522i	−0.619428	−	0.785054i	\(-0.712635\pi\)
							0.928967	−	0.370162i	\(-0.120698\pi\)
\(29\)	−1.51395	−	5.65012i	−0.281133	−	1.04920i	−0.951619	−	0.307280i	\(-0.900581\pi\)
							0.670486	−	0.741922i	\(-0.266085\pi\)
\(31\)	0.300035	+	0.0803943i	0.0538880	+	0.0144392i	0.285662	−	0.958330i	\(-0.407786\pi\)
							−0.231774	+	0.972770i	\(0.574453\pi\)
\(37\)	8.01553	+	2.14776i	1.31775	+	0.353089i	0.848133	−	0.529784i	\(-0.177727\pi\)
							0.469613	+	0.882872i	\(0.344394\pi\)
\(41\)	5.77234	+	5.77234i	0.901488	+	0.901488i	0.995565	−	0.0940767i	\(-0.0299899\pi\)
							−0.0940767	+	0.995565i	\(0.529990\pi\)
\(43\)	−4.78898	+	4.47947i	−0.730313	+	0.683113i
							\(47\)	−12.9440			−1.88807			−0.944035	−	0.329846i	\(-0.893003\pi\)
−0.944035	+	0.329846i	\(0.893003\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.6	✓	256
17.13	even	4	inner	731.2.n.a.251.59	yes	256
43.6	even	3	inner	731.2.n.a.565.6	yes	256
731.608	even	12	inner	731.2.n.a.608.59	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.6	✓	256	1.1	even	1	trivial
731.2.n.a.251.59	yes	256	17.13	even	4	inner
731.2.n.a.565.6	yes	256	43.6	even	3	inner
731.2.n.a.608.59	yes	256	731.608	even	12	inner