Embedded newform 570.2.v.c.83.7

• Label: 570.2.v.c.83.7
• Level: $570$
• Weight: $2$
• Character: 570.83
• Analytic conductor: $4.551$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.55147291521\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			83.7
Character	\(\chi\)	\(=\)	570.83
Dual form			570.2.v.c.467.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.258819 + 0.965926i) q^{2} +(-0.822951 - 1.52406i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(-0.462693 + 2.18767i) q^{5} +(1.68512 - 0.400455i) q^{6} +(2.22340 - 2.22340i) q^{7} +(0.707107 - 0.707107i) q^{8} +(-1.64550 + 2.50845i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 4 q^{3} + 8 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(211\)	\(457\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.258819	+	0.965926i	−0.183013	+	0.683013i
							\(3\)	−0.822951	−	1.52406i	−0.475131	−	0.879915i
\(5\)	−0.462693	+	2.18767i	−0.206922	+	0.978357i
							\(7\)	2.22340	−	2.22340i	0.840367	−	0.840367i	−0.148539	−	0.988907i	\(-0.547457\pi\)
0.988907	+	0.148539i	\(0.0474571\pi\)
\(11\)		−	2.55653i		−	0.770824i	−0.922745	−	0.385412i	\(-0.874059\pi\)
							0.922745	−	0.385412i	\(-0.125941\pi\)
\(13\)	−2.14434	+	0.574574i	−0.594733	+	0.159358i	−0.543615	−	0.839334i	\(-0.682945\pi\)
							−0.0511175	+	0.998693i	\(0.516278\pi\)
\(17\)	0.895783	−	3.34311i	0.217259	−	0.810823i	−0.768100	−	0.640330i	\(-0.778797\pi\)
							0.985359	−	0.170492i	\(-0.0545359\pi\)
\(19\)	3.91389	+	1.91871i	0.897908	+	0.440183i
							\(23\)	−0.436569	−	1.62930i	−0.0910310	−	0.339732i	0.905357	−	0.424652i	\(-0.139603\pi\)
−0.996388	+	0.0849194i	\(0.972937\pi\)
\(29\)	5.25111	−	9.09519i	0.975106	−	1.68893i	0.295523	−	0.955336i	\(-0.404506\pi\)
							0.679583	−	0.733598i	\(-0.262161\pi\)
\(31\)	0.288720			0.0518557			0.0259279	−	0.999664i	\(-0.491746\pi\)
							0.0259279	+	0.999664i	\(0.491746\pi\)
\(37\)	5.51813	−	5.51813i	0.907174	−	0.907174i	−0.0888690	−	0.996043i	\(-0.528325\pi\)
							0.996043	+	0.0888690i	\(0.0283252\pi\)
\(41\)	−3.95805	+	2.28518i	−0.618143	+	0.356885i	−0.776146	−	0.630554i	\(-0.782828\pi\)
							0.158003	+	0.987439i	\(0.449495\pi\)
\(43\)	3.23130	−	12.0594i	0.492768	−	1.83904i	−0.0494183	−	0.998778i	\(-0.515737\pi\)
							0.542187	−	0.840258i	\(-0.317597\pi\)
\(47\)	12.2898	−	3.29305i	1.79265	−	0.480340i	0.799862	−	0.600184i	\(-0.204906\pi\)
							0.992793	+	0.119844i	\(0.0382394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	570.2.v.c.83.7	✓	144
3.2	odd	2	inner	570.2.v.c.83.34	yes	144
5.2	odd	4	inner	570.2.v.c.197.3	yes	144
15.2	even	4	inner	570.2.v.c.197.25	yes	144
19.11	even	3	inner	570.2.v.c.353.25	yes	144
57.11	odd	6	inner	570.2.v.c.353.3	yes	144
95.87	odd	12	inner	570.2.v.c.467.34	yes	144
285.182	even	12	inner	570.2.v.c.467.7	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.v.c.83.7	✓	144	1.1	even	1	trivial
570.2.v.c.83.34	yes	144	3.2	odd	2	inner
570.2.v.c.197.3	yes	144	5.2	odd	4	inner
570.2.v.c.197.25	yes	144	15.2	even	4	inner
570.2.v.c.353.3	yes	144	57.11	odd	6	inner
570.2.v.c.353.25	yes	144	19.11	even	3	inner
570.2.v.c.467.7	yes	144	285.182	even	12	inner
570.2.v.c.467.34	yes	144	95.87	odd	12	inner