Embedded newform 1728.2.bc.e.1585.14

• Label: 1728.2.bc.e.1585.14
• Level: $1728$
• Weight: $2$
• Character: 1728.1585
• Analytic conductor: $13.798$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$1728 = 2^{6} \cdot 3^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1728.bc (of order $12$, degree $4$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$18$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			1585.14
Character	$\chi$	$=$	1728.1585
Dual form			1728.2.bc.e.145.14

$q$-expansion

$f(q)$	$=$	$q+(0.733432 - 2.73721i) q^{5} +(-1.14487 - 0.660988i) q^{7} +(1.28367 - 0.343957i) q^{11} +(3.36696 + 0.902174i) q^{13} +7.60772 q^{17} +(-4.32297 + 4.32297i) q^{19} +(3.46087 - 1.99814i) q^{23} +(-2.62424 - 1.51511i) q^{25} +(-0.950303 - 3.54658i) q^{29} +(-0.569129 - 0.985760i) q^{31} +(-2.64894 + 2.64894i) q^{35} +(2.26014 + 2.26014i) q^{37} +(-1.42311 + 0.821634i) q^{41} +(6.17424 - 1.65438i) q^{43} +(4.58731 - 7.94546i) q^{47} +(-2.62619 - 4.54869i) q^{49} +(-7.72215 - 7.72215i) q^{53} -3.76593i q^{55} +(1.28879 - 4.80982i) q^{59} +(-2.66068 - 9.92979i) q^{61} +(4.93887 - 8.55438i) q^{65} +(-13.9276 - 3.73189i) q^{67} +7.87498i q^{71} +0.577222i q^{73} +(-1.69698 - 0.454704i) q^{77} +(-0.716890 + 1.24169i) q^{79} +(-0.885341 - 3.30414i) q^{83} +(5.57975 - 20.8239i) q^{85} +16.2114i q^{89} +(-3.25839 - 3.25839i) q^{91} +(8.66225 + 15.0035i) q^{95} +(-0.648931 + 1.12398i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	72 * q - 4 * q^5 - 2 * q^11 - 16 * q^13 + 16 * q^17 - 28 * q^19 - 4 * q^29 - 28 * q^31 - 16 * q^35 + 16 * q^37 + 10 * q^43 - 56 * q^47 + 4 * q^49 + 8 * q^53 - 14 * q^59 - 32 * q^61 + 64 * q^65 + 18 * q^67 + 36 * q^77 - 44 * q^79 + 20 * q^83 - 8 * q^85 + 80 * q^91 + 48 * q^95 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$e\left(\frac{2}{3}\right)$

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			0
							$3$		0			0
$5$	0.733432	−	2.73721i	0.328001	−	1.22412i								−0.583260	−	0.812286i	$-0.698223\pi$
							0.911261	−	0.411830i	$-0.135110\pi$
$7$	−1.14487	−	0.660988i	−0.432718	−	0.249830i	0.267786	−	0.963479i	$-0.413708\pi$
							−0.700504	+	0.713648i	$0.747041\pi$
$11$	1.28367	−	0.343957i	0.387040	−	0.103707i	−0.0600515	−	0.998195i	$-0.519126\pi$
							0.447092	+	0.894488i	$0.352460\pi$
$13$	3.36696	+	0.902174i	0.933827	+	0.250218i	0.693486	−	0.720470i	$-0.256074\pi$
							0.240341	+	0.970689i	$0.422741\pi$
$17$	7.60772			1.84514			0.922572	−	0.385825i	$-0.126083\pi$
							0.922572	+	0.385825i	$0.126083\pi$
$19$	−4.32297	+	4.32297i	−0.991757	+	0.991757i	−0.999966	−	0.00820954i	$-0.997387\pi$
							0.00820954	+	0.999966i	$0.497387\pi$
$23$	3.46087	−	1.99814i	0.721642	−	0.416640i	−0.0937148	−	0.995599i	$-0.529874\pi$
							0.815357	+	0.578959i	$0.196541\pi$
$29$	−0.950303	−	3.54658i	−0.176467	−	0.658583i	−0.996297	−	0.0859765i	$-0.972599\pi$
							0.819830	−	0.572606i	$-0.194068\pi$
$31$	−0.569129	−	0.985760i	−0.102219	−	0.177048i	0.810380	−	0.585905i	$-0.199261\pi$
							−0.912598	+	0.408857i	$0.865927\pi$
$37$	2.26014	+	2.26014i	0.371565	+	0.371565i	0.868047	−	0.496482i	$-0.165375\pi$
							−0.496482	+	0.868047i	$0.665375\pi$
$41$	−1.42311	+	0.821634i	−0.222253	+	0.128318i	−0.606993	−	0.794707i	$-0.707624\pi$
							0.384740	+	0.923025i	$0.374291\pi$
$43$	6.17424	−	1.65438i	0.941563	−	0.252291i	0.244784	−	0.969578i	$-0.421283\pi$
							0.696778	+	0.717287i	$0.254616\pi$
$47$	4.58731	−	7.94546i	0.669129	−	1.15896i	−0.309020	−	0.951056i	$-0.600001\pi$
							0.978148	−	0.207909i	$-0.0666658\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.bc.e.1585.14		72
3.2	odd	2		576.2.bb.e.241.11		72
4.3	odd	2		432.2.y.e.181.12		72
9.4	even	3	inner	1728.2.bc.e.1009.5		72
9.5	odd	6		576.2.bb.e.49.18		72
12.11	even	2		144.2.x.e.133.7	yes	72
16.3	odd	4		432.2.y.e.397.2		72
16.13	even	4	inner	1728.2.bc.e.721.5		72
36.23	even	6		144.2.x.e.85.17	yes	72
36.31	odd	6		432.2.y.e.37.2		72
48.29	odd	4		576.2.bb.e.529.18		72
48.35	even	4		144.2.x.e.61.17	yes	72
144.13	even	12	inner	1728.2.bc.e.145.14		72
144.67	odd	12		432.2.y.e.253.12		72
144.77	odd	12		576.2.bb.e.337.11		72
144.131	even	12		144.2.x.e.13.7	✓	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.7	✓	72	144.131	even	12
144.2.x.e.61.17	yes	72	48.35	even	4
144.2.x.e.85.17	yes	72	36.23	even	6
144.2.x.e.133.7	yes	72	12.11	even	2
432.2.y.e.37.2		72	36.31	odd	6
432.2.y.e.181.12		72	4.3	odd	2
432.2.y.e.253.12		72	144.67	odd	12
432.2.y.e.397.2		72	16.3	odd	4
576.2.bb.e.49.18		72	9.5	odd	6
576.2.bb.e.241.11		72	3.2	odd	2
576.2.bb.e.337.11		72	144.77	odd	12
576.2.bb.e.529.18		72	48.29	odd	4
1728.2.bc.e.145.14		72	144.13	even	12	inner
1728.2.bc.e.721.5		72	16.13	even	4	inner
1728.2.bc.e.1009.5		72	9.4	even	3	inner
1728.2.bc.e.1585.14		72	1.1	even	1	trivial