Embedded newform 861.2.bz.a.4.10

• Label: 861.2.bz.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	861.4
Dual form			861.2.bz.a.646.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92977 - 0.859188i) q^{2} +(-0.866025 - 0.500000i) q^{3} +(1.64754 + 1.82978i) q^{4} +(1.61753 - 0.343817i) q^{5} +(1.24163 + 1.70896i) q^{6} +(1.98413 - 1.75020i) q^{7} +(-0.301713 - 0.928578i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 448 q + 56 q^{4} - 24 q^{8} + 224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92977	−	0.859188i	−1.36455	−	0.607538i	−0.411796	−	0.911276i	\(-0.635099\pi\)
							−0.952756	+	0.303738i	\(0.901765\pi\)
\(3\)	−0.866025	−	0.500000i	−0.500000	−	0.288675i
							\(5\)	1.61753	−	0.343817i	0.723383	−	0.153760i	0.168516	−	0.985699i	\(-0.446103\pi\)
0.554867	+	0.831939i	\(0.312769\pi\)
\(7\)	1.98413	−	1.75020i	0.749932	−	0.661515i
							\(11\)	−0.756098	+	3.55716i	−0.227972	+	1.07253i	0.704057	+	0.710143i	\(0.251370\pi\)
−0.932030	+	0.362382i	\(0.881964\pi\)
\(13\)	−2.61951	−	3.60545i	−0.726522	−	0.999972i	−0.999282	−	0.0378895i	\(-0.987936\pi\)
							0.272760	−	0.962082i	\(-0.412064\pi\)
\(17\)	0.966162	−	4.54544i	0.234329	−	1.10243i	−0.690879	−	0.722970i	\(-0.742776\pi\)
							0.925208	−	0.379460i	\(-0.123890\pi\)
\(19\)	0.669547	+	0.0703722i	0.153605	+	0.0161445i	0.181019	−	0.983480i	\(-0.442061\pi\)
							−0.0274140	+	0.999624i	\(0.508727\pi\)
\(23\)	7.97667	+	3.55144i	1.66325	+	0.740527i	0.999969	−	0.00784350i	\(-0.00249669\pi\)
							0.663281	+	0.748370i	\(0.269163\pi\)
\(29\)	7.76285	+	2.52230i	1.44153	+	0.468380i	0.922373	−	0.386301i	\(-0.126248\pi\)
							0.519153	+	0.854681i	\(0.326248\pi\)
\(31\)	−2.40021	−	0.510180i	−0.431090	−	0.0916310i	−0.0127441	−	0.999919i	\(-0.504057\pi\)
							−0.418346	+	0.908288i	\(0.637390\pi\)
\(37\)	3.14662	−	0.668835i	0.517301	−	0.109956i	0.0581396	−	0.998308i	\(-0.481483\pi\)
							0.459162	+	0.888353i	\(0.348150\pi\)
\(41\)	−3.45984	+	5.38791i	−0.540336	+	0.841450i
							\(43\)	4.87450	−	3.54153i	0.743355	−	0.540079i	−0.150405	−	0.988624i	\(-0.548058\pi\)
0.893760	+	0.448546i	\(0.148058\pi\)
\(47\)	0.476740	−	1.07078i	0.0695396	−	0.156189i	−0.875432	−	0.483342i	\(-0.839423\pi\)
							0.944971	+	0.327153i	\(0.106089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	861.2.bz.a.4.10	✓	448
7.2	even	3	inner	861.2.bz.a.373.47	yes	448
41.31	even	10	inner	861.2.bz.a.277.47	yes	448
287.72	even	30	inner	861.2.bz.a.646.10	yes	448

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.bz.a.4.10	✓	448	1.1	even	1	trivial
861.2.bz.a.277.47	yes	448	41.31	even	10	inner
861.2.bz.a.373.47	yes	448	7.2	even	3	inner
861.2.bz.a.646.10	yes	448	287.72	even	30	inner