Embedded newform 637.2.bi.a.256.27

• Label: 637.2.bi.a.256.27
• Level: $637$
• Weight: $2$
• Character: 637.256
• Analytic conductor: $5.086$
• Analytic rank: $0$
• Dimension: $756$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.bi (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.08647060876\)
Analytic rank:	\(0\)
Dimension:	\(756\)
Relative dimension:	\(63\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			256.27
Character	\(\chi\)	\(=\)	637.256
Dual form			637.2.bi.a.107.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.109709 - 0.480666i) q^{2} +(0.181260 - 0.461843i) q^{3} +(1.58293 - 0.762301i) q^{4} +(-0.911649 + 2.32284i) q^{5} +(-0.241878 - 0.0364572i) q^{6} +(-1.18792 + 2.36407i) q^{7} +(-1.15487 - 1.44816i) q^{8} +(2.01871 + 1.87309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 5 q^{2} - 11 q^{3} - 125 q^{4} - 26 q^{5} - q^{6} - 8 q^{7} - 16 q^{8} + 50 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(248\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.109709	−	0.480666i	−0.0775758	−	0.339882i	0.921214	−	0.389055i	\(-0.127198\pi\)
							−0.998790	+	0.0491733i	\(0.984341\pi\)
\(3\)	0.181260	−	0.461843i	0.104650	−	0.266645i	−0.868952	−	0.494897i	\(-0.835206\pi\)
							0.973602	+	0.228252i	\(0.0733010\pi\)
\(5\)	−0.911649	+	2.32284i	−0.407702	+	1.03881i	0.568643	+	0.822585i	\(0.307469\pi\)
							−0.976344	+	0.216222i	\(0.930626\pi\)
\(7\)	−1.18792	+	2.36407i	−0.448992	+	0.893536i
							\(11\)	−1.01029	+	0.937411i	−0.304613	+	0.282640i	−0.817636	−	0.575735i	\(-0.804716\pi\)
0.513023	+	0.858375i	\(0.328526\pi\)
\(13\)	−1.17643	+	3.40823i	−0.326284	+	0.945272i
							\(17\)	−2.94402	−	1.41776i	−0.714029	−	0.343858i	0.0413516	−	0.999145i	\(-0.486834\pi\)
−0.755380	+	0.655286i	\(0.772548\pi\)
\(19\)	−4.07629	+	7.06034i	−0.935165	+	1.61975i	−0.160827	+	0.986983i	\(0.551416\pi\)
							−0.774339	+	0.632771i	\(0.781917\pi\)
\(23\)	4.25183	−	2.04757i	0.886568	−	0.426949i	0.0655487	−	0.997849i	\(-0.479120\pi\)
							0.821019	+	0.570901i	\(0.193406\pi\)
\(29\)	−2.11511	+	1.44206i	−0.392766	+	0.267783i	−0.743560	−	0.668669i	\(-0.766864\pi\)
							0.350794	+	0.936453i	\(0.385912\pi\)
\(31\)	−0.961088	+	1.66465i	−0.172616	+	0.298981i	−0.939334	−	0.343004i	\(-0.888555\pi\)
							0.766717	+	0.641985i	\(0.221889\pi\)
\(37\)	0.590478	+	0.284359i	0.0970739	+	0.0467483i	0.481791	−	0.876286i	\(-0.339987\pi\)
							−0.384717	+	0.923035i	\(0.625701\pi\)
\(41\)	5.91677	−	0.891810i	0.924044	−	0.139277i	0.330249	−	0.943894i	\(-0.392867\pi\)
							0.593795	+	0.804616i	\(0.297629\pi\)
\(43\)	3.01835	+	0.454944i	0.460295	+	0.0693782i	0.375096	−	0.926986i	\(-0.377610\pi\)
							0.0851981	+	0.996364i	\(0.472848\pi\)
\(47\)	6.62826	−	6.15013i	0.966831	−	0.897088i	−0.0278995	−	0.999611i	\(-0.508882\pi\)
							0.994731	+	0.102522i	\(0.0326914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.bi.a.256.27	yes	756
13.3	even	3		637.2.bj.a.354.37	yes	756
49.9	even	21		637.2.bj.a.9.37	yes	756
637.107	even	21	inner	637.2.bi.a.107.27	✓	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.bi.a.107.27	✓	756	637.107	even	21	inner
637.2.bi.a.256.27	yes	756	1.1	even	1	trivial
637.2.bj.a.9.37	yes	756	49.9	even	21
637.2.bj.a.354.37	yes	756	13.3	even	3