Embedded newform 585.2.bt.b.376.3

• Label: 585.2.bt.b.376.3
• Level: $585$
• Weight: $2$
• Character: 585.376
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.67124851824\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			376.3
Character	\(\chi\)	\(=\)	585.376
Dual form			585.2.bt.b.571.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.26372 - 1.30696i) q^{2} +(-0.538321 + 1.64627i) q^{3} +(2.41629 + 4.18514i) q^{4} +(0.866025 - 0.500000i) q^{5} +(3.37022 - 3.02314i) q^{6} +(3.14827 + 1.81766i) q^{7} -7.40414i q^{8} +(-2.42042 - 1.77245i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{3} + 52 q^{4} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\)	\(326\)	\(352\)	\(496\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.26372	−	1.30696i	−1.60069	−	0.924160i	−0.991349	−	0.131255i	\(-0.958099\pi\)
							−0.609344	−	0.792906i	\(-0.708567\pi\)
\(3\)	−0.538321	+	1.64627i	−0.310800	+	0.950475i
							\(5\)	0.866025	−	0.500000i	0.387298	−	0.223607i
\(7\)	3.14827	+	1.81766i	1.18993	+	0.687009i								0.958292	−	0.285792i	\(-0.0922566\pi\)
							0.231643	+	0.972801i	\(0.425590\pi\)
\(11\)	−5.33710	−	3.08138i	−1.60920	−	0.929070i	−0.989550	−	0.144188i	\(-0.953943\pi\)
							−0.619645	−	0.784882i	\(-0.712724\pi\)
\(13\)	0.233392	−	3.59799i	0.0647312	−	0.997903i
							\(17\)	2.59963			0.630502			0.315251	−	0.949008i	\(-0.397911\pi\)
0.315251	+	0.949008i	\(0.397911\pi\)
\(19\)		−	6.95618i		−	1.59586i	−0.602752	−	0.797929i	\(-0.705929\pi\)
							0.602752	−	0.797929i	\(-0.294071\pi\)
\(23\)	1.37773	+	2.38631i	0.287277	+	0.497579i	0.973159	−	0.230134i	\(-0.0739165\pi\)
							−0.685881	+	0.727713i	\(0.740583\pi\)
\(29\)	2.36365	−	4.09396i	0.438919	−	0.760229i	−0.558688	−	0.829378i	\(-0.688695\pi\)
							0.997606	+	0.0691488i	\(0.0220283\pi\)
\(31\)	−4.56322	+	2.63457i	−0.819578	+	0.473184i	−0.850271	−	0.526345i	\(-0.823562\pi\)
							0.0306930	+	0.999529i	\(0.490229\pi\)
\(37\)		−	2.85385i		−	0.469170i	−0.972096	−	0.234585i	\(-0.924627\pi\)
							0.972096	−	0.234585i	\(-0.0753731\pi\)
\(41\)	0.632395	−	0.365113i	0.0987635	−	0.0570211i	−0.449805	−	0.893127i	\(-0.648506\pi\)
							0.548568	+	0.836106i	\(0.315173\pi\)
\(43\)	2.01702	−	3.49359i	0.307593	−	0.532767i	−0.670242	−	0.742142i	\(-0.733810\pi\)
							0.977835	+	0.209376i	\(0.0671431\pi\)
\(47\)	−0.0766416	−	0.0442490i	−0.0111793	−	0.00645439i	0.494400	−	0.869235i	\(-0.335388\pi\)
							−0.505579	+	0.862780i	\(0.668721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.bt.b.376.3	✓	108
9.4	even	3	inner	585.2.bt.b.571.52	yes	108
13.12	even	2	inner	585.2.bt.b.376.52	yes	108
117.103	even	6	inner	585.2.bt.b.571.3	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.bt.b.376.3	✓	108	1.1	even	1	trivial
585.2.bt.b.376.52	yes	108	13.12	even	2	inner
585.2.bt.b.571.3	yes	108	117.103	even	6	inner
585.2.bt.b.571.52	yes	108	9.4	even	3	inner