Embedded newform 1008.2.s.o.289.1

• Label: 1008.2.s.o.289.1
• Level: $1008$
• Weight: $2$
• Character: 1008.289
• Analytic conductor: $8.049$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$1008 = 2^{4} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1008.s (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.04892052375$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1008.289
Dual form			1008.2.s.o.865.1

$q$-expansion

$f(q)$	$=$	$q+(1.50000 - 2.59808i) q^{5} +(0.500000 - 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{11} +2.00000 q^{13} +(3.00000 + 5.19615i) q^{17} +(1.00000 - 1.73205i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} -9.00000 q^{29} +(-3.50000 - 6.06218i) q^{31} +(-6.00000 - 5.19615i) q^{35} +(5.00000 - 8.66025i) q^{37} +4.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-1.50000 - 2.59808i) q^{53} +9.00000 q^{55} +(1.50000 + 2.59808i) q^{59} +(2.00000 - 3.46410i) q^{61} +(3.00000 - 5.19615i) q^{65} +(1.00000 + 1.73205i) q^{67} +(-1.00000 - 1.73205i) q^{73} +(7.50000 - 2.59808i) q^{77} +(2.50000 - 4.33013i) q^{79} +9.00000 q^{83} +18.0000 q^{85} +(-3.00000 + 5.19615i) q^{89} +(1.00000 - 5.19615i) q^{91} +(-3.00000 - 5.19615i) q^{95} -13.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 3 * q^5 + q^7 + 3 * q^11 + 4 * q^13 + 6 * q^17 + 2 * q^19 + 6 * q^23 - 4 * q^25 - 18 * q^29 - 7 * q^31 - 12 * q^35 + 10 * q^37 + 8 * q^43 - 12 * q^47 - 13 * q^49 - 3 * q^53 + 18 * q^55 + 3 * q^59 + 4 * q^61 + 6 * q^65 + 2 * q^67 - 2 * q^73 + 15 * q^77 + 5 * q^79 + 18 * q^83 + 36 * q^85 - 6 * q^89 + 2 * q^91 - 6 * q^95 - 26 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$		0			0
							$3$		0			0
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i								−0.306851	−	0.951757i	$-0.599275\pi$
							0.977672	−	0.210138i	$-0.0673912\pi$
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
							$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	0.998526	−	0.0542666i	$-0.0172821\pi$
−0.546259	+	0.837616i	$0.683949\pi$
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
							0.277350	+	0.960769i	$0.410544\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
							−0.230285	+	0.973123i	$0.573966\pi$
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	−0.728219	−	0.685344i	$-0.759652\pi$
							0.957635	+	0.287984i	$0.0929851\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
							0.988436	−	0.151642i	$-0.0484560\pi$
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
							−0.835629	+	0.549294i	$0.814897\pi$
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	−0.987829	−	0.155543i	$-0.950287\pi$
							0.359211	−	0.933257i	$-0.383046\pi$
$37$	5.00000	−	8.66025i	0.821995	−	1.42374i	−0.0821995	−	0.996616i	$-0.526194\pi$
							0.904194	−	0.427121i	$-0.140472\pi$
$41$		0			0			−	1.00000i	$-0.5\pi$
									1.00000i	$0.5\pi$
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
							0.304997	+	0.952353i	$0.401344\pi$
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	$0.505930\pi$
							−0.856560	+	0.516047i	$0.827403\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.s.o.289.1		2
3.2	odd	2		1008.2.s.b.289.1		2
4.3	odd	2		126.2.g.d.37.1	yes	2
7.2	even	3		7056.2.a.e.1.1		1
7.4	even	3	inner	1008.2.s.o.865.1		2
7.5	odd	6		7056.2.a.bx.1.1		1
12.11	even	2		126.2.g.a.37.1	✓	2
21.2	odd	6		7056.2.a.by.1.1		1
21.5	even	6		7056.2.a.h.1.1		1
21.11	odd	6		1008.2.s.b.865.1		2
28.3	even	6		882.2.g.g.361.1		2
28.11	odd	6		126.2.g.d.109.1	yes	2
28.19	even	6		882.2.a.e.1.1		1
28.23	odd	6		882.2.a.a.1.1		1
28.27	even	2		882.2.g.g.667.1		2
36.7	odd	6		1134.2.e.g.919.1		2
36.11	even	6		1134.2.e.k.919.1		2
36.23	even	6		1134.2.h.f.541.1		2
36.31	odd	6		1134.2.h.j.541.1		2
84.11	even	6		126.2.g.a.109.1	yes	2
84.23	even	6		882.2.a.j.1.1		1
84.47	odd	6		882.2.a.h.1.1		1
84.59	odd	6		882.2.g.e.361.1		2
84.83	odd	2		882.2.g.e.667.1		2
252.11	even	6		1134.2.h.f.109.1		2
252.67	odd	6		1134.2.e.g.865.1		2
252.95	even	6		1134.2.e.k.865.1		2
252.151	odd	6		1134.2.h.j.109.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.g.a.37.1	✓	2	12.11	even	2
126.2.g.a.109.1	yes	2	84.11	even	6
126.2.g.d.37.1	yes	2	4.3	odd	2
126.2.g.d.109.1	yes	2	28.11	odd	6
882.2.a.a.1.1		1	28.23	odd	6
882.2.a.e.1.1		1	28.19	even	6
882.2.a.h.1.1		1	84.47	odd	6
882.2.a.j.1.1		1	84.23	even	6
882.2.g.e.361.1		2	84.59	odd	6
882.2.g.e.667.1		2	84.83	odd	2
882.2.g.g.361.1		2	28.3	even	6
882.2.g.g.667.1		2	28.27	even	2
1008.2.s.b.289.1		2	3.2	odd	2
1008.2.s.b.865.1		2	21.11	odd	6
1008.2.s.o.289.1		2	1.1	even	1	trivial
1008.2.s.o.865.1		2	7.4	even	3	inner
1134.2.e.g.865.1		2	252.67	odd	6
1134.2.e.g.919.1		2	36.7	odd	6
1134.2.e.k.865.1		2	252.95	even	6
1134.2.e.k.919.1		2	36.11	even	6
1134.2.h.f.109.1		2	252.11	even	6
1134.2.h.f.541.1		2	36.23	even	6
1134.2.h.j.109.1		2	252.151	odd	6
1134.2.h.j.541.1		2	36.31	odd	6
7056.2.a.e.1.1		1	7.2	even	3
7056.2.a.h.1.1		1	21.5	even	6
7056.2.a.bx.1.1		1	7.5	odd	6
7056.2.a.by.1.1		1	21.2	odd	6