Embedded newform 861.2.bz.a.4.5

• Label: 861.2.bz.a.4.5
• Level: $861$
• Weight: $2$
• Character: 861.4
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $448$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.bz (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(448\)
Relative dimension:	\(56\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	861.4
Dual form			861.2.bz.a.646.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22722 - 0.991620i) q^{2} +(-0.866025 - 0.500000i) q^{3} +(2.63892 + 2.93081i) q^{4} +(-2.27705 + 0.484002i) q^{5} +(1.43302 + 1.97238i) q^{6} +(-2.08770 + 1.62528i) q^{7} +(-1.46442 - 4.50703i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 448 q + 56 q^{4} - 24 q^{8} + 224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(211\)	\(493\)	\(575\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22722	−	0.991620i	−1.57488	−	0.701181i	−0.581234	−	0.813737i	\(-0.697430\pi\)
							−0.993645	+	0.112555i	\(0.964096\pi\)
\(3\)	−0.866025	−	0.500000i	−0.500000	−	0.288675i
							\(5\)	−2.27705	+	0.484002i	−1.01833	+	0.216452i	−0.686687	−	0.726953i	\(-0.740936\pi\)
−0.331641	+	0.943406i	\(0.607602\pi\)
\(7\)	−2.08770	+	1.62528i	−0.789075	+	0.614297i
							\(11\)	1.03299	−	4.85985i	0.311459	−	1.46530i	−0.492342	−	0.870402i	\(-0.663859\pi\)
0.803801	−	0.594898i	\(-0.202807\pi\)
\(13\)	−1.41369	−	1.94578i	−0.392088	−	0.539663i	0.566648	−	0.823960i	\(-0.308240\pi\)
							−0.958736	+	0.284297i	\(0.908240\pi\)
\(17\)	1.26254	−	5.93979i	0.306211	−	1.44061i	−0.508660	−	0.860968i	\(-0.669859\pi\)
							0.814871	−	0.579643i	\(-0.196808\pi\)
\(19\)	5.31076	+	0.558183i	1.21837	+	0.128056i	0.691809	−	0.722080i	\(-0.256814\pi\)
							0.526563	+	0.850136i	\(0.323481\pi\)
\(23\)	0.870138	+	0.387410i	0.181436	+	0.0807806i	0.495444	−	0.868640i	\(-0.335005\pi\)
							−0.314008	+	0.949420i	\(0.601672\pi\)
\(29\)	−0.853436	−	0.277298i	−0.158479	−	0.0514930i	0.228703	−	0.973496i	\(-0.426552\pi\)
							−0.387182	+	0.922003i	\(0.626552\pi\)
\(31\)	1.81822	+	0.386475i	0.326562	+	0.0694129i	0.368275	−	0.929717i	\(-0.379948\pi\)
							−0.0417135	+	0.999130i	\(0.513282\pi\)
\(37\)	−1.57798	+	0.335409i	−0.259418	+	0.0551409i	−0.335784	−	0.941939i	\(-0.609001\pi\)
							0.0763668	+	0.997080i	\(0.475668\pi\)
\(41\)	−2.14461	+	6.03330i	−0.334932	+	0.942242i
							\(43\)	−4.15383	+	3.01793i	−0.633453	+	0.460231i	−0.857595	−	0.514326i	\(-0.828042\pi\)
0.224142	+	0.974557i	\(0.428042\pi\)
\(47\)	1.26769	−	2.84728i	0.184912	−	0.415318i	−0.797175	−	0.603748i	\(-0.793673\pi\)
							0.982087	+	0.188430i	\(0.0603399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	861.2.bz.a.4.5	✓	448
7.2	even	3	inner	861.2.bz.a.373.52	yes	448
41.31	even	10	inner	861.2.bz.a.277.52	yes	448
287.72	even	30	inner	861.2.bz.a.646.5	yes	448

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.bz.a.4.5	✓	448	1.1	even	1	trivial
861.2.bz.a.277.52	yes	448	41.31	even	10	inner
861.2.bz.a.373.52	yes	448	7.2	even	3	inner
861.2.bz.a.646.5	yes	448	287.72	even	30	inner