Embedded newform 861.2.l.a.419.15

• Label: 861.2.l.a.419.15
• Level: $861$
• Weight: $2$
• Character: 861.419
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(108\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			419.15
Character	\(\chi\)	\(=\)	861.419
Dual form			861.2.l.a.524.15

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.09764 q^{2} +(-0.839690 + 1.51490i) q^{3} +2.40008 q^{4} +2.02468i q^{5} +(1.76136 - 3.17771i) q^{6} +(1.67336 + 2.04936i) q^{7} -0.839219 q^{8} +(-1.58984 - 2.54409i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 192 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/861\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(493\)	\(575\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.09764			−1.48325			−0.741626	−	0.670813i	\(-0.765945\pi\)
							−0.741626	+	0.670813i	\(0.765945\pi\)
\(3\)	−0.839690	+	1.51490i	−0.484795	+	0.874628i
							\(5\)			2.02468i			0.905465i	0.891646	+	0.452733i	\(0.149551\pi\)
−0.891646	+	0.452733i	\(0.850449\pi\)
\(7\)	1.67336	+	2.04936i	0.632472	+	0.774584i
							\(11\)	−1.97738	+	1.97738i	−0.596203	+	0.596203i	−0.939300	−	0.343097i	\(-0.888524\pi\)
0.343097	+	0.939300i	\(0.388524\pi\)
\(13\)	0.746924	−	0.746924i	0.207160	−	0.207160i	−0.595900	−	0.803059i	\(-0.703204\pi\)
							0.803059	+	0.595900i	\(0.203204\pi\)
\(17\)	−4.42368	+	4.42368i	−1.07290	+	1.07290i	−0.0757755	+	0.997125i	\(0.524143\pi\)
							−0.997125	+	0.0757755i	\(0.975857\pi\)
\(19\)	3.72167	+	3.72167i	0.853810	+	0.853810i	0.990600	−	0.136790i	\(-0.0436785\pi\)
							−0.136790	+	0.990600i	\(0.543678\pi\)
\(23\)		−	1.99753i		−	0.416514i	−0.978074	−	0.208257i	\(-0.933221\pi\)
							0.978074	−	0.208257i	\(-0.0667791\pi\)
\(29\)	−2.41447	+	2.41447i	−0.448355	+	0.448355i	−0.894807	−	0.446452i	\(-0.852687\pi\)
							0.446452	+	0.894807i	\(0.352687\pi\)
\(31\)			6.69591i			1.20262i	0.799016	+	0.601310i	\(0.205354\pi\)
							−0.799016	+	0.601310i	\(0.794646\pi\)
\(37\)	−0.270291			−0.0444356			−0.0222178	−	0.999753i	\(-0.507073\pi\)
							−0.0222178	+	0.999753i	\(0.507073\pi\)
\(41\)	−0.494936	+	6.38397i	−0.0772961	+	0.997008i
							\(43\)		−	9.03578i		−	1.37794i	−0.724788	−	0.688972i	\(-0.758062\pi\)
0.724788	−	0.688972i	\(-0.241938\pi\)
\(47\)	3.64140	−	3.64140i	0.531153	−	0.531153i	−0.389762	−	0.920916i	\(-0.627443\pi\)
							0.920916	+	0.389762i	\(0.127443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	861.2.l.a.419.15	✓	216
3.2	odd	2	inner	861.2.l.a.419.93	yes	216
7.6	odd	2	inner	861.2.l.a.419.16	yes	216
21.20	even	2	inner	861.2.l.a.419.94	yes	216
41.32	even	4	inner	861.2.l.a.524.94	yes	216
123.32	odd	4	inner	861.2.l.a.524.16	yes	216
287.237	odd	4	inner	861.2.l.a.524.93	yes	216
861.524	even	4	inner	861.2.l.a.524.15	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.l.a.419.15	✓	216	1.1	even	1	trivial
861.2.l.a.419.16	yes	216	7.6	odd	2	inner
861.2.l.a.419.93	yes	216	3.2	odd	2	inner
861.2.l.a.419.94	yes	216	21.20	even	2	inner
861.2.l.a.524.15	yes	216	861.524	even	4	inner
861.2.l.a.524.16	yes	216	123.32	odd	4	inner
861.2.l.a.524.93	yes	216	287.237	odd	4	inner
861.2.l.a.524.94	yes	216	41.32	even	4	inner