LMFDB - Embedded newform 1728.3.g.m.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.2
Root			$0.500000 - 1.19293i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.m.703.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.59655 q^{5} +4.56106i q^{7} +O(q^{10})$	$q-6.59655 q^{5} +4.56106i q^{7} -1.16066i q^{11} -13.5580 q^{13} -5.32636 q^{17} -25.1900i q^{19} +10.4309i q^{23} +18.5144 q^{25} -47.0288 q^{29} -22.3862i q^{31} -30.0872i q^{35} +60.7288 q^{37} -8.81696 q^{41} +29.1160i q^{43} +78.2321i q^{47} +28.1968 q^{49} -62.7623 q^{53} +7.65637i q^{55} -109.116i q^{59} +66.4997 q^{61} +89.4360 q^{65} +81.9685i q^{67} -40.7276i q^{71} -4.29030 q^{73} +5.29386 q^{77} -5.20371i q^{79} +115.114i q^{83} +35.1356 q^{85} +141.504 q^{89} -61.8388i q^{91} +166.167i q^{95} +136.769 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} - 8 q^{13} - 24 q^{17} + 24 q^{25} - 128 q^{29} - 24 q^{37} - 160 q^{41} - 144 q^{49} - 48 q^{53} + 136 q^{61} + 280 q^{65} + 72 q^{73} + 520 q^{77} - 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 - 8 * q^13 - 24 * q^17 + 24 * q^25 - 128 * q^29 - 24 * q^37 - 160 * q^41 - 144 * q^49 - 48 * q^53 + 136 * q^61 + 280 * q^65 + 72 * q^73 + 520 * q^77 - 96 * q^85 + 168 * q^89 + 104 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−6.59655	−1.31931	−0.659655	−	0.751569i	$-0.729297\pi$
$5$	−6.59655	−1.31931	−0.659655	+	0.751569i	$0.729297\pi$
$6$	0	0
$7$	4.56106i	0.651580i	0.945442	+	0.325790i	$0.105630\pi$
$7$	4.56106i	0.651580i	−0.945442	+	0.325790i	$0.894370\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.16066i	− 0.105515i	−0.998607	−	0.0527575i	$-0.983199\pi$
$11$	− 1.16066i	− 0.105515i	0.998607	−	0.0527575i	$-0.0168010\pi$
$12$	0	0
$13$	−13.5580	−1.04292	−0.521461	−	0.853275i	$-0.674613\pi$
$13$	−13.5580	−1.04292	−0.521461	+	0.853275i	$0.674613\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.32636	−0.313315	−0.156658	−	0.987653i	$-0.550072\pi$
$17$	−5.32636	−0.313315	−0.156658	+	0.987653i	$0.550072\pi$
$18$	0	0
$19$	− 25.1900i	− 1.32579i	−0.748712	−	0.662896i	$-0.769327\pi$
$19$	− 25.1900i	− 1.32579i	0.748712	−	0.662896i	$-0.230673\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	10.4309i	0.453515i	0.973951	+	0.226758i	$0.0728125\pi$
$23$	10.4309i	0.453515i	−0.973951	+	0.226758i	$0.927187\pi$
$24$	0	0
$25$	18.5144	0.740577
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−47.0288	−1.62168	−0.810842	−	0.585265i	$-0.800990\pi$
$29$	−47.0288	−1.62168	−0.810842	+	0.585265i	$0.800990\pi$
$30$	0	0
$31$	− 22.3862i	− 0.722135i	−0.932540	−	0.361067i	$-0.882412\pi$
$31$	− 22.3862i	− 0.722135i	0.932540	−	0.361067i	$-0.117588\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 30.0872i	− 0.859635i
$36$	0	0
$37$	60.7288	1.64132	0.820659	−	0.571418i	$-0.193606\pi$
$37$	60.7288	1.64132	0.820659	+	0.571418i	$0.193606\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.81696	−0.215048	−0.107524	−	0.994202i	$-0.534292\pi$
$41$	−8.81696	−0.215048	−0.107524	+	0.994202i	$0.534292\pi$
$42$	0	0
$43$	29.1160i	0.677116i	0.940945	+	0.338558i	$0.109939\pi$
$43$	29.1160i	0.677116i	−0.940945	+	0.338558i	$0.890061\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	78.2321i	1.66451i	0.554392	+	0.832256i	$0.312951\pi$
$47$	78.2321i	1.66451i	−0.554392	+	0.832256i	$0.687049\pi$
$48$	0	0
$49$	28.1968	0.575444
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−62.7623	−1.18419	−0.592097	−	0.805866i	$-0.701700\pi$
$53$	−62.7623	−1.18419	−0.592097	+	0.805866i	$0.701700\pi$
$54$	0	0
$55$	7.65637i	0.139207i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 109.116i	− 1.84942i	−0.380666	−	0.924712i	$-0.624305\pi$
$59$	− 109.116i	− 1.84942i	0.380666	−	0.924712i	$-0.375695\pi$
$60$	0	0
$61$	66.4997	1.09016	0.545079	−	0.838384i	$-0.316500\pi$
$61$	66.4997	1.09016	0.545079	+	0.838384i	$0.316500\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	89.4360	1.37594
$66$	0	0
$67$	81.9685i	1.22341i	0.791086	+	0.611705i	$0.209516\pi$
$67$	81.9685i	1.22341i	−0.791086	+	0.611705i	$0.790484\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 40.7276i	− 0.573629i	−0.957986	−	0.286814i	$-0.907404\pi$
$71$	− 40.7276i	− 0.573629i	0.957986	−	0.286814i	$-0.0925963\pi$
$72$	0	0
$73$	−4.29030	−0.0587713	−0.0293856	−	0.999568i	$-0.509355\pi$
$73$	−4.29030	−0.0587713	−0.0293856	+	0.999568i	$0.509355\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.29386	0.0687514
$78$	0	0
$79$	− 5.20371i	− 0.0658698i	−0.999457	−	0.0329349i	$-0.989515\pi$
$79$	− 5.20371i	− 0.0658698i	0.999457	−	0.0329349i	$-0.0104854\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	115.114i	1.38691i	0.720498	+	0.693457i	$0.243913\pi$
$83$	115.114i	1.38691i	−0.720498	+	0.693457i	$0.756087\pi$
$84$	0	0
$85$	35.1356	0.413359
$86$	0	0
$87$	0	0
$88$	0	0
$89$	141.504	1.58993	0.794965	−	0.606656i	$-0.207489\pi$
$89$	141.504	1.58993	0.794965	+	0.606656i	$0.207489\pi$
$90$	0	0
$91$	− 61.8388i	− 0.679547i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	166.167i	1.74913i
$96$	0	0
$97$	136.769	1.40999	0.704994	−	0.709213i	$-0.250950\pi$
$97$	136.769	1.40999	0.704994	+	0.709213i	$0.250950\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−45.2350	−0.447871	−0.223935	−	0.974604i	$-0.571890\pi$
$101$	−45.2350	−0.447871	−0.223935	+	0.974604i	$0.571890\pi$
$102$	0	0
$103$	114.604i	1.11266i	0.830963	+	0.556328i	$0.187790\pi$
$103$	114.604i	1.11266i	−0.830963	+	0.556328i	$0.812210\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 194.930i	− 1.82178i	−0.412652	−	0.910889i	$-0.635397\pi$
$107$	− 194.930i	− 1.82178i	0.412652	−	0.910889i	$-0.364603\pi$
$108$	0	0
$109$	−6.35150	−0.0582706	−0.0291353	−	0.999575i	$-0.509275\pi$
$109$	−6.35150	−0.0582706	−0.0291353	+	0.999575i	$0.509275\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	48.7068	0.431034	0.215517	−	0.976500i	$-0.430856\pi$
$113$	48.7068	0.431034	0.215517	+	0.976500i	$0.430856\pi$
$114$	0	0
$115$	− 68.8076i	− 0.598327i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 24.2938i	− 0.204150i
$120$	0	0
$121$	119.653	0.988867
$122$	0	0
$123$	0	0
$124$	0	0
$125$	42.7824	0.342259
$126$	0	0
$127$	− 86.3509i	− 0.679929i	−0.940438	−	0.339964i	$-0.889585\pi$
$127$	− 86.3509i	− 0.679929i	0.940438	−	0.339964i	$-0.110415\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 90.6946i	− 0.692325i	−0.938175	−	0.346162i	$-0.887485\pi$
$131$	− 90.6946i	− 0.692325i	0.938175	−	0.346162i	$-0.112515\pi$
$132$	0	0
$133$	114.893	0.863859
$134$	0	0
$135$	0	0
$136$	0	0
$137$	148.355	1.08288	0.541442	−	0.840738i	$-0.317878\pi$
$137$	148.355	1.08288	0.541442	+	0.840738i	$0.317878\pi$
$138$	0	0
$139$	− 101.426i	− 0.729684i	−0.931069	−	0.364842i	$-0.881123\pi$
$139$	− 101.426i	− 0.729684i	0.931069	−	0.364842i	$-0.118877\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.7363i	0.110044i
$144$	0	0
$145$	310.228	2.13950
$146$	0	0
$147$	0	0
$148$	0	0
$149$	219.221	1.47128	0.735639	−	0.677373i	$-0.236882\pi$
$149$	219.221	1.47128	0.735639	+	0.677373i	$0.236882\pi$
$150$	0	0
$151$	26.1638i	0.173270i	0.996240	+	0.0866350i	$0.0276114\pi$
$151$	26.1638i	0.173270i	−0.996240	+	0.0866350i	$0.972389\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	147.671i	0.952719i
$156$	0	0
$157$	257.715	1.64150	0.820748	−	0.571290i	$-0.193557\pi$
$157$	257.715	1.64150	0.820748	+	0.571290i	$0.193557\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−47.5757	−0.295501
$162$	0	0
$163$	140.310i	0.860796i	0.902639	+	0.430398i	$0.141627\pi$
$163$	140.310i	0.860796i	−0.902639	+	0.430398i	$0.858373\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	28.2435i	0.169123i	0.996418	+	0.0845613i	$0.0269489\pi$
$167$	28.2435i	0.169123i	−0.996418	+	0.0845613i	$0.973051\pi$
$168$	0	0
$169$	14.8193	0.0876881
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−127.532	−0.737177	−0.368589	−	0.929593i	$-0.620159\pi$
$173$	−127.532	−0.737177	−0.368589	+	0.929593i	$0.620159\pi$
$174$	0	0
$175$	84.4453i	0.482545i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	167.254i	0.934378i	0.884157	+	0.467189i	$0.154733\pi$
$179$	167.254i	0.934378i	−0.884157	+	0.467189i	$0.845267\pi$
$180$	0	0
$181$	166.841	0.921775	0.460887	−	0.887459i	$-0.347531\pi$
$181$	166.841	0.921775	0.460887	+	0.887459i	$0.347531\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−400.600	−2.16541
$186$	0	0
$187$	6.18211i	0.0330594i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 163.153i	− 0.854203i	−0.904204	−	0.427102i	$-0.859535\pi$
$191$	− 163.153i	− 0.854203i	0.904204	−	0.427102i	$-0.140465\pi$
$192$	0	0
$193$	−137.611	−0.713010	−0.356505	−	0.934293i	$-0.616032\pi$
$193$	−137.611	−0.713010	−0.356505	+	0.934293i	$0.616032\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−132.978	−0.675017	−0.337509	−	0.941322i	$-0.609584\pi$
$197$	−132.978	−0.675017	−0.337509	+	0.941322i	$0.609584\pi$
$198$	0	0
$199$	− 371.097i	− 1.86481i	−0.361418	−	0.932404i	$-0.617707\pi$
$199$	− 371.097i	− 1.86481i	0.361418	−	0.932404i	$-0.382293\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 214.501i	− 1.05666i
$204$	0	0
$205$	58.1615	0.283715
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−29.2372	−0.139891
$210$	0	0
$211$	239.571i	1.13541i	0.823233	+	0.567704i	$0.192168\pi$
$211$	239.571i	1.13541i	−0.823233	+	0.567704i	$0.807832\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 192.065i	− 0.893326i
$216$	0	0
$217$	102.105	0.470528
$218$	0	0
$219$	0	0
$220$	0	0
$221$	72.2147	0.326763
$222$	0	0
$223$	− 121.555i	− 0.545089i	−0.962143	−	0.272545i	$-0.912135\pi$
$223$	− 121.555i	− 0.545089i	0.962143	−	0.272545i	$-0.0878652\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	223.023i	0.982481i	0.871024	+	0.491241i	$0.163456\pi$
$227$	223.023i	0.982481i	−0.871024	+	0.491241i	$0.836544\pi$
$228$	0	0
$229$	139.599	0.609605	0.304802	−	0.952416i	$-0.401410\pi$
$229$	139.599	0.609605	0.304802	+	0.952416i	$0.401410\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−103.682	−0.444986	−0.222493	−	0.974934i	$-0.571419\pi$
$233$	−103.682	−0.444986	−0.222493	+	0.974934i	$0.571419\pi$
$234$	0	0
$235$	− 516.061i	− 2.19601i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	110.844i	0.463783i	0.972742	+	0.231892i	$0.0744915\pi$
$239$	110.844i	0.463783i	−0.972742	+	0.231892i	$0.925508\pi$
$240$	0	0
$241$	−80.6426	−0.334616	−0.167308	−	0.985905i	$-0.553508\pi$
$241$	−80.6426	−0.334616	−0.167308	+	0.985905i	$0.553508\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−186.001	−0.759188
$246$	0	0
$247$	341.526i	1.38270i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	471.180i	1.87721i	0.344990	+	0.938606i	$0.387882\pi$
$251$	471.180i	1.87721i	−0.344990	+	0.938606i	$0.612118\pi$
$252$	0	0
$253$	12.1067	0.0478526
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−330.588	−1.28633	−0.643167	−	0.765726i	$-0.722380\pi$
$257$	−330.588	−1.28633	−0.643167	+	0.765726i	$0.722380\pi$
$258$	0	0
$259$	276.988i	1.06945i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 494.228i	− 1.87919i	−0.342283	−	0.939597i	$-0.611200\pi$
$263$	− 494.228i	− 1.87919i	0.342283	−	0.939597i	$-0.388800\pi$
$264$	0	0
$265$	414.014	1.56232
$266$	0	0
$267$	0	0
$268$	0	0
$269$	214.548	0.797576	0.398788	−	0.917043i	$-0.369431\pi$
$269$	214.548	0.797576	0.398788	+	0.917043i	$0.369431\pi$
$270$	0	0
$271$	− 283.519i	− 1.04620i	−0.852272	−	0.523099i	$-0.824776\pi$
$271$	− 283.519i	− 1.04620i	0.852272	−	0.523099i	$-0.175224\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 21.4890i	− 0.0781419i
$276$	0	0
$277$	−107.432	−0.387842	−0.193921	−	0.981017i	$-0.562121\pi$
$277$	−107.432	−0.387842	−0.193921	+	0.981017i	$0.562121\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−27.7120	−0.0986193	−0.0493096	−	0.998784i	$-0.515702\pi$
$281$	−27.7120	−0.0986193	−0.0493096	+	0.998784i	$0.515702\pi$
$282$	0	0
$283$	311.956i	1.10232i	0.834400	+	0.551159i	$0.185814\pi$
$283$	311.956i	1.10232i	−0.834400	+	0.551159i	$0.814186\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 40.2147i	− 0.140121i
$288$	0	0
$289$	−260.630	−0.901834
$290$	0	0
$291$	0	0
$292$	0	0
$293$	536.810	1.83212	0.916058	−	0.401045i	$-0.131353\pi$
$293$	536.810	1.83212	0.916058	+	0.401045i	$0.131353\pi$
$294$	0	0
$295$	719.789i	2.43996i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 141.421i	− 0.472982i
$300$	0	0
$301$	−132.800	−0.441195
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−438.668	−1.43826
$306$	0	0
$307$	− 296.075i	− 0.964415i	−0.876057	−	0.482208i	$-0.839835\pi$
$307$	− 296.075i	− 0.964415i	0.876057	−	0.482208i	$-0.160165\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	274.720i	0.883345i	0.897176	+	0.441673i	$0.145615\pi$
$311$	274.720i	0.883345i	−0.897176	+	0.441673i	$0.854385\pi$
$312$	0	0
$313$	204.294	0.652697	0.326348	−	0.945250i	$-0.394182\pi$
$313$	204.294	0.652697	0.326348	+	0.945250i	$0.394182\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.6889	0.0873466	0.0436733	−	0.999046i	$-0.486094\pi$
$317$	27.6889	0.0873466	0.0436733	+	0.999046i	$0.486094\pi$
$318$	0	0
$319$	54.5847i	0.171112i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	134.171i	0.415390i
$324$	0	0
$325$	−251.018	−0.772364
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−356.821	−1.08456
$330$	0	0
$331$	− 77.6640i	− 0.234634i	−0.993094	−	0.117317i	$-0.962571\pi$
$331$	− 77.6640i	− 0.234634i	0.993094	−	0.117317i	$-0.0374294\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 540.709i	− 1.61406i
$336$	0	0
$337$	314.619	0.933588	0.466794	−	0.884366i	$-0.345409\pi$
$337$	314.619	0.933588	0.466794	+	0.884366i	$0.345409\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−25.9828	−0.0761960
$342$	0	0
$343$	352.099i	1.02653i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	485.244i	1.39840i	0.714928	+	0.699198i	$0.246460\pi$
$347$	485.244i	1.39840i	−0.714928	+	0.699198i	$0.753540\pi$
$348$	0	0
$349$	61.2165	0.175405	0.0877027	−	0.996147i	$-0.472047\pi$
$349$	61.2165	0.175405	0.0877027	+	0.996147i	$0.472047\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	274.644	0.778028	0.389014	−	0.921232i	$-0.372816\pi$
$353$	274.644	0.778028	0.389014	+	0.921232i	$0.372816\pi$
$354$	0	0
$355$	268.662i	0.756793i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	406.229i	1.13156i	0.824557	+	0.565779i	$0.191424\pi$
$359$	406.229i	1.13156i	−0.824557	+	0.565779i	$0.808576\pi$
$360$	0	0
$361$	−273.538	−0.757722
$362$	0	0
$363$	0	0
$364$	0	0
$365$	28.3012	0.0775375
$366$	0	0
$367$	− 678.650i	− 1.84918i	−0.380959	−	0.924592i	$-0.624406\pi$
$367$	− 678.650i	− 1.84918i	0.380959	−	0.924592i	$-0.375594\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 286.263i	− 0.771597i
$372$	0	0
$373$	−17.1447	−0.0459645	−0.0229822	−	0.999736i	$-0.507316\pi$
$373$	−17.1447	−0.0459645	−0.0229822	+	0.999736i	$0.507316\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	637.617	1.69129
$378$	0	0
$379$	− 669.773i	− 1.76721i	−0.468231	−	0.883606i	$-0.655109\pi$
$379$	− 669.773i	− 1.76721i	0.468231	−	0.883606i	$-0.344891\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 282.840i	− 0.738486i	−0.929333	−	0.369243i	$-0.879617\pi$
$383$	− 282.840i	− 0.738486i	0.929333	−	0.369243i	$-0.120383\pi$
$384$	0	0
$385$	−34.9212	−0.0907043
$386$	0	0
$387$	0	0
$388$	0	0
$389$	197.725	0.508290	0.254145	−	0.967166i	$-0.418206\pi$
$389$	197.725	0.508290	0.254145	+	0.967166i	$0.418206\pi$
$390$	0	0
$391$	− 55.5584i	− 0.142093i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	34.3265i	0.0869026i
$396$	0	0
$397$	546.375	1.37626	0.688129	−	0.725588i	$-0.258432\pi$
$397$	546.375	1.37626	0.688129	+	0.725588i	$0.258432\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	586.152	1.46173	0.730863	−	0.682525i	$-0.239118\pi$
$401$	586.152	1.46173	0.730863	+	0.682525i	$0.239118\pi$
$402$	0	0
$403$	303.512i	0.753131i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 70.4857i	− 0.173184i
$408$	0	0
$409$	−614.056	−1.50136	−0.750679	−	0.660667i	$-0.770274\pi$
$409$	−614.056	−1.50136	−0.750679	+	0.660667i	$0.770274\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	497.685	1.20505
$414$	0	0
$415$	− 759.354i	− 1.82977i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 509.128i	− 1.21510i	−0.794281	−	0.607551i	$-0.792152\pi$
$419$	− 509.128i	− 1.21510i	0.794281	−	0.607551i	$-0.207848\pi$
$420$	0	0
$421$	−478.025	−1.13545	−0.567725	−	0.823218i	$-0.692176\pi$
$421$	−478.025	−1.13545	−0.567725	+	0.823218i	$0.692176\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−98.6144	−0.232034
$426$	0	0
$427$	303.309i	0.710325i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 158.810i	− 0.368468i	−0.982882	−	0.184234i	$-0.941020\pi$
$431$	− 158.810i	− 0.368468i	0.982882	−	0.184234i	$-0.0589804\pi$
$432$	0	0
$433$	−367.803	−0.849429	−0.424715	−	0.905327i	$-0.639626\pi$
$433$	−367.803	−0.849429	−0.424715	+	0.905327i	$0.639626\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	262.754	0.601267
$438$	0	0
$439$	145.096i	0.330514i	0.986251	+	0.165257i	$0.0528453\pi$
$439$	145.096i	0.330514i	−0.986251	+	0.165257i	$0.947155\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	771.496i	1.74153i	0.491703	+	0.870763i	$0.336374\pi$
$443$	771.496i	1.74153i	−0.491703	+	0.870763i	$0.663626\pi$
$444$	0	0
$445$	−933.436	−2.09761
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−56.2630	−0.125307	−0.0626536	−	0.998035i	$-0.519956\pi$
$449$	−56.2630	−0.125307	−0.0626536	+	0.998035i	$0.519956\pi$
$450$	0	0
$451$	10.2335i	0.0226908i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	407.923i	0.896533i
$456$	0	0
$457$	−804.889	−1.76125	−0.880623	−	0.473817i	$-0.842876\pi$
$457$	−804.889	−1.76125	−0.880623	+	0.473817i	$0.842876\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−46.1301	−0.100065	−0.0500327	−	0.998748i	$-0.515933\pi$
$461$	−46.1301	−0.100065	−0.0500327	+	0.998748i	$0.515933\pi$
$462$	0	0
$463$	273.418i	0.590536i	0.955414	+	0.295268i	$0.0954090\pi$
$463$	273.418i	0.590536i	−0.955414	+	0.295268i	$0.904591\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 109.131i	− 0.233684i	−0.993150	−	0.116842i	$-0.962723\pi$
$467$	− 109.131i	− 0.233684i	0.993150	−	0.116842i	$-0.0372772\pi$
$468$	0	0
$469$	−373.863	−0.797150
$470$	0	0
$471$	0	0
$472$	0	0
$473$	33.7939	0.0714459
$474$	0	0
$475$	− 466.379i	− 0.981850i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 42.5978i	− 0.0889306i	−0.999011	−	0.0444653i	$-0.985842\pi$
$479$	− 42.5978i	− 0.0889306i	0.999011	−	0.0444653i	$-0.0141584\pi$
$480$	0	0
$481$	−823.361	−1.71177
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−902.202	−1.86021
$486$	0	0
$487$	− 445.984i	− 0.915779i	−0.889009	−	0.457889i	$-0.848606\pi$
$487$	− 445.984i	− 0.915779i	0.889009	−	0.457889i	$-0.151394\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 565.533i	− 1.15180i	−0.817521	−	0.575899i	$-0.804652\pi$
$491$	− 565.533i	− 1.15180i	0.817521	−	0.575899i	$-0.195348\pi$
$492$	0	0
$493$	250.492	0.508098
$494$	0	0
$495$	0	0
$496$	0	0
$497$	185.761	0.373765
$498$	0	0
$499$	923.261i	1.85022i	0.379696	+	0.925111i	$0.376029\pi$
$499$	923.261i	1.85022i	−0.379696	+	0.925111i	$0.623971\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	406.646i	0.808441i	0.914662	+	0.404220i	$0.132457\pi$
$503$	406.646i	0.808441i	−0.914662	+	0.404220i	$0.867543\pi$
$504$	0	0
$505$	298.394	0.590880
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−674.031	−1.32423	−0.662113	−	0.749404i	$-0.730340\pi$
$509$	−674.031	−1.32423	−0.662113	+	0.749404i	$0.730340\pi$
$510$	0	0
$511$	− 19.5683i	− 0.0382942i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 755.988i	− 1.46794i
$516$	0	0
$517$	90.8011	0.175631
$518$	0	0
$519$	0	0
$520$	0	0
$521$	513.618	0.985831	0.492916	−	0.870077i	$-0.335931\pi$
$521$	513.618	0.985831	0.492916	+	0.870077i	$0.335931\pi$
$522$	0	0
$523$	− 381.030i	− 0.728547i	−0.931292	−	0.364274i	$-0.881317\pi$
$523$	− 381.030i	− 0.728547i	0.931292	−	0.364274i	$-0.118683\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	119.237i	0.226256i
$528$	0	0
$529$	420.197	0.794324
$530$	0	0
$531$	0	0
$532$	0	0
$533$	119.540	0.224278
$534$	0	0
$535$	1285.87i	2.40349i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 32.7270i	− 0.0607179i
$540$	0	0
$541$	−626.913	−1.15880	−0.579402	−	0.815042i	$-0.696714\pi$
$541$	−626.913	−1.15880	−0.579402	+	0.815042i	$0.696714\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	41.8980	0.0768770
$546$	0	0
$547$	18.9275i	0.0346023i	0.999850	+	0.0173012i	$0.00550740\pi$
$547$	18.9275i	0.0346023i	−0.999850	+	0.0173012i	$0.994493\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1184.66i	2.15001i
$552$	0	0
$553$	23.7344	0.0429194
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.9003	0.0482949	0.0241475	−	0.999708i	$-0.492313\pi$
$557$	26.9003	0.0482949	0.0241475	+	0.999708i	$0.492313\pi$
$558$	0	0
$559$	− 394.755i	− 0.706180i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	488.414i	0.867520i	0.901029	+	0.433760i	$0.142813\pi$
$563$	488.414i	0.867520i	−0.901029	+	0.433760i	$0.857187\pi$
$564$	0	0
$565$	−321.297	−0.568667
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−266.712	−0.468739	−0.234369	−	0.972148i	$-0.575302\pi$
$569$	−266.712	−0.468739	−0.234369	+	0.972148i	$0.575302\pi$
$570$	0	0
$571$	− 880.359i	− 1.54178i	−0.636966	−	0.770892i	$-0.719811\pi$
$571$	− 880.359i	− 1.54178i	0.636966	−	0.770892i	$-0.280189\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	193.121i	0.335863i
$576$	0	0
$577$	82.9805	0.143814	0.0719069	−	0.997411i	$-0.477092\pi$
$577$	82.9805	0.143814	0.0719069	+	0.997411i	$0.477092\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−525.041	−0.903685
$582$	0	0
$583$	72.8460i	0.124950i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 965.562i	− 1.64491i	−0.568831	−	0.822455i	$-0.692604\pi$
$587$	− 965.562i	− 1.64491i	0.568831	−	0.822455i	$-0.307396\pi$
$588$	0	0
$589$	−563.909	−0.957400
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−289.460	−0.488128	−0.244064	−	0.969759i	$-0.578481\pi$
$593$	−289.460	−0.488128	−0.244064	+	0.969759i	$0.578481\pi$
$594$	0	0
$595$	160.255i	0.269337i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 448.245i	− 0.748322i	−0.927364	−	0.374161i	$-0.877931\pi$
$599$	− 448.245i	− 0.748322i	0.927364	−	0.374161i	$-0.122069\pi$
$600$	0	0
$601$	−229.348	−0.381610	−0.190805	−	0.981628i	$-0.561110\pi$
$601$	−229.348	−0.381610	−0.190805	+	0.981628i	$0.561110\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−789.296	−1.30462
$606$	0	0
$607$	− 85.4214i	− 0.140727i	−0.997521	−	0.0703636i	$-0.977584\pi$
$607$	− 85.4214i	− 0.140727i	0.997521	−	0.0703636i	$-0.0224160\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1060.67i	− 1.73596i
$612$	0	0
$613$	−534.037	−0.871187	−0.435593	−	0.900144i	$-0.643461\pi$
$613$	−534.037	−0.871187	−0.435593	+	0.900144i	$0.643461\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	151.146	0.244970	0.122485	−	0.992470i	$-0.460914\pi$
$617$	151.146	0.244970	0.122485	+	0.992470i	$0.460914\pi$
$618$	0	0
$619$	− 59.2078i	− 0.0956507i	−0.998856	−	0.0478253i	$-0.984771\pi$
$619$	− 59.2078i	− 0.0956507i	0.998856	−	0.0478253i	$-0.0152291\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	645.407i	1.03597i
$624$	0	0
$625$	−745.077	−1.19212
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−323.463	−0.514250
$630$	0	0
$631$	70.0390i	0.110997i	0.998459	+	0.0554984i	$0.0176748\pi$
$631$	70.0390i	0.110997i	−0.998459	+	0.0554984i	$0.982325\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	569.618i	0.897036i
$636$	0	0
$637$	−382.292	−0.600144
$638$	0	0
$639$	0	0
$640$	0	0
$641$	195.338	0.304739	0.152370	−	0.988324i	$-0.451310\pi$
$641$	195.338	0.304739	0.152370	+	0.988324i	$0.451310\pi$
$642$	0	0
$643$	346.902i	0.539505i	0.962930	+	0.269753i	$0.0869419\pi$
$643$	346.902i	0.539505i	−0.962930	+	0.269753i	$0.913058\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 746.221i	− 1.15335i	−0.816972	−	0.576677i	$-0.804349\pi$
$647$	− 746.221i	− 1.15335i	0.816972	−	0.576677i	$-0.195651\pi$
$648$	0	0
$649$	−126.647	−0.195142
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−850.046	−1.30176	−0.650878	−	0.759183i	$-0.725599\pi$
$653$	−850.046	−1.30176	−0.650878	+	0.759183i	$0.725599\pi$
$654$	0	0
$655$	598.271i	0.913391i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	817.574i	1.24063i	0.784354	+	0.620314i	$0.212995\pi$
$659$	817.574i	1.24063i	−0.784354	+	0.620314i	$0.787005\pi$
$660$	0	0
$661$	670.001	1.01362	0.506809	−	0.862058i	$-0.330825\pi$
$661$	670.001	1.01362	0.506809	+	0.862058i	$0.330825\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−757.898	−1.13970
$666$	0	0
$667$	− 490.551i	− 0.735459i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 77.1838i	− 0.115028i
$672$	0	0
$673$	−41.6868	−0.0619418	−0.0309709	−	0.999520i	$-0.509860\pi$
$673$	−41.6868	−0.0619418	−0.0309709	+	0.999520i	$0.509860\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−71.8534	−0.106135	−0.0530675	−	0.998591i	$-0.516900\pi$
$677$	−71.8534	−0.106135	−0.0530675	+	0.998591i	$0.516900\pi$
$678$	0	0
$679$	623.811i	0.918720i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 486.468i	− 0.712252i	−0.934438	−	0.356126i	$-0.884097\pi$
$683$	− 486.468i	− 0.712252i	0.934438	−	0.356126i	$-0.115903\pi$
$684$	0	0
$685$	−978.632	−1.42866
$686$	0	0
$687$	0	0
$688$	0	0
$689$	850.931	1.23502
$690$	0	0
$691$	1153.88i	1.66988i	0.550344	+	0.834938i	$0.314497\pi$
$691$	1153.88i	1.66988i	−0.550344	+	0.834938i	$0.685503\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	669.062i	0.962679i
$696$	0	0
$697$	46.9623	0.0673778
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−278.673	−0.397537	−0.198768	−	0.980047i	$-0.563694\pi$
$701$	−278.673	−0.397537	−0.198768	+	0.980047i	$0.563694\pi$
$702$	0	0
$703$	− 1529.76i	− 2.17605i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 206.319i	− 0.291824i
$708$	0	0
$709$	128.606	0.181390	0.0906950	−	0.995879i	$-0.471091\pi$
$709$	128.606	0.181390	0.0906950	+	0.995879i	$0.471091\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	233.507	0.327499
$714$	0	0
$715$	− 103.805i	− 0.145182i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	99.8703i	0.138902i	0.997585	+	0.0694508i	$0.0221247\pi$
$719$	99.8703i	0.138902i	−0.997585	+	0.0694508i	$0.977875\pi$
$720$	0	0
$721$	−522.714	−0.724984
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−870.712	−1.20098
$726$	0	0
$727$	605.619i	0.833039i	0.909127	+	0.416519i	$0.136750\pi$
$727$	605.619i	0.833039i	−0.909127	+	0.416519i	$0.863250\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 155.082i	− 0.212151i
$732$	0	0
$733$	−11.5244	−0.0157223	−0.00786113	−	0.999969i	$-0.502502\pi$
$733$	−11.5244	−0.0157223	−0.00786113	+	0.999969i	$0.502502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	95.1379	0.129088
$738$	0	0
$739$	241.471i	0.326753i	0.986564	+	0.163377i	$0.0522386\pi$
$739$	241.471i	0.326753i	−0.986564	+	0.163377i	$0.947761\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 660.096i	− 0.888420i	−0.895923	−	0.444210i	$-0.853484\pi$
$743$	− 660.096i	− 0.888420i	0.895923	−	0.444210i	$-0.146516\pi$
$744$	0	0
$745$	−1446.10	−1.94107
$746$	0	0
$747$	0	0
$748$	0	0
$749$	889.088	1.18703
$750$	0	0
$751$	− 624.342i	− 0.831347i	−0.909514	−	0.415674i	$-0.863546\pi$
$751$	− 624.342i	− 0.831347i	0.909514	−	0.415674i	$-0.136454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 172.590i	− 0.228597i
$756$	0	0
$757$	291.603	0.385209	0.192604	−	0.981276i	$-0.438307\pi$
$757$	291.603	0.385209	0.192604	+	0.981276i	$0.438307\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1333.83	1.75274	0.876369	−	0.481640i	$-0.159959\pi$
$761$	1333.83	1.75274	0.876369	+	0.481640i	$0.159959\pi$
$762$	0	0
$763$	− 28.9696i	− 0.0379680i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1479.40i	1.92881i
$768$	0	0
$769$	−75.3889	−0.0980349	−0.0490175	−	0.998798i	$-0.515609\pi$
$769$	−75.3889	−0.0980349	−0.0490175	+	0.998798i	$0.515609\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1331.18	1.72210	0.861049	−	0.508521i	$-0.169808\pi$
$773$	1331.18	1.72210	0.861049	+	0.508521i	$0.169808\pi$
$774$	0	0
$775$	− 414.467i	− 0.534796i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	222.100i	0.285109i
$780$	0	0
$781$	−47.2711	−0.0605264
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1700.03	−2.16564
$786$	0	0
$787$	1099.10i	1.39657i	0.715820	+	0.698285i	$0.246053\pi$
$787$	1099.10i	1.39657i	−0.715820	+	0.698285i	$0.753947\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	222.155i	0.280853i
$792$	0	0
$793$	−901.603	−1.13695
$794$	0	0
$795$	0	0
$796$	0	0
$797$	640.713	0.803906	0.401953	−	0.915660i	$-0.368332\pi$
$797$	640.713	0.803906	0.401953	+	0.915660i	$0.368332\pi$
$798$	0	0
$799$	− 416.692i	− 0.521517i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.97960i	0.00620125i
$804$	0	0
$805$	313.835	0.389858
$806$	0	0
$807$	0	0
$808$	0	0
$809$	733.593	0.906790	0.453395	−	0.891310i	$-0.350213\pi$
$809$	733.593	0.906790	0.453395	+	0.891310i	$0.350213\pi$
$810$	0	0
$811$	− 1149.16i	− 1.41697i	−0.705728	−	0.708483i	$-0.749380\pi$
$811$	− 1149.16i	− 1.41697i	0.705728	−	0.708483i	$-0.250620\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 925.560i	− 1.13566i
$816$	0	0
$817$	733.433	0.897715
$818$	0	0
$819$	0	0
$820$	0	0
$821$	293.112	0.357019	0.178509	−	0.983938i	$-0.442873\pi$
$821$	293.112	0.357019	0.178509	+	0.983938i	$0.442873\pi$
$822$	0	0
$823$	466.668i	0.567033i	0.958967	+	0.283517i	$0.0915011\pi$
$823$	466.668i	0.567033i	−0.958967	+	0.283517i	$0.908499\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 340.554i	− 0.411794i	−0.978574	−	0.205897i	$-0.933989\pi$
$827$	− 340.554i	− 0.411794i	0.978574	−	0.205897i	$-0.0660112\pi$
$828$	0	0
$829$	1368.33	1.65058	0.825289	−	0.564710i	$-0.191012\pi$
$829$	1368.33	1.65058	0.825289	+	0.564710i	$0.191012\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−150.186	−0.180295
$834$	0	0
$835$	− 186.309i	− 0.223125i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 297.555i	− 0.354654i	−0.984152	−	0.177327i	$-0.943255\pi$
$839$	− 297.555i	− 0.354654i	0.984152	−	0.177327i	$-0.0567450\pi$
$840$	0	0
$841$	1370.71	1.62986
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−97.7562	−0.115688
$846$	0	0
$847$	545.744i	0.644325i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	633.453i	0.744363i
$852$	0	0
$853$	−941.528	−1.10378	−0.551892	−	0.833916i	$-0.686094\pi$
$853$	−941.528	−1.10378	−0.551892	+	0.833916i	$0.686094\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−458.626	−0.535153	−0.267576	−	0.963537i	$-0.586223\pi$
$857$	−458.626	−0.535153	−0.267576	+	0.963537i	$0.586223\pi$
$858$	0	0
$859$	40.4325i	0.0470693i	0.999723	+	0.0235346i	$0.00749200\pi$
$859$	40.4325i	0.0470693i	−0.999723	+	0.0235346i	$0.992508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1233.37i	− 1.42917i	−0.699548	−	0.714585i	$-0.746615\pi$
$863$	− 1233.37i	− 1.42917i	0.699548	−	0.714585i	$-0.253385\pi$
$864$	0	0
$865$	841.268	0.972565
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.03976	−0.00695025
$870$	0	0
$871$	− 1111.33i	− 1.27592i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	195.133i	0.223009i
$876$	0	0
$877$	−584.678	−0.666680	−0.333340	−	0.942807i	$-0.608176\pi$
$877$	−584.678	−0.666680	−0.333340	+	0.942807i	$0.608176\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	733.997	0.833140	0.416570	−	0.909104i	$-0.363232\pi$
$881$	733.997	0.833140	0.416570	+	0.909104i	$0.363232\pi$
$882$	0	0
$883$	− 855.769i	− 0.969161i	−0.874747	−	0.484580i	$-0.838972\pi$
$883$	− 855.769i	− 0.969161i	0.874747	−	0.484580i	$-0.161028\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 767.876i	− 0.865700i	−0.901466	−	0.432850i	$-0.857508\pi$
$887$	− 767.876i	− 0.865700i	0.901466	−	0.432850i	$-0.142492\pi$
$888$	0	0
$889$	393.852	0.443028
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1970.67	2.20680
$894$	0	0
$895$	− 1103.30i	− 1.23273i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1052.80i	1.17107i
$900$	0	0
$901$	334.294	0.371026
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1100.58	−1.21611
$906$	0	0
$907$	− 957.563i	− 1.05575i	−0.849323	−	0.527874i	$-0.822989\pi$
$907$	− 957.563i	− 1.05575i	0.849323	−	0.527874i	$-0.177011\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	413.981i	0.454425i	0.973845	+	0.227212i	$0.0729611\pi$
$911$	413.981i	0.454425i	−0.973845	+	0.227212i	$0.927039\pi$
$912$	0	0
$913$	133.608	0.146340
$914$	0	0
$915$	0	0
$916$	0	0
$917$	413.663	0.451105
$918$	0	0
$919$	1703.44i	1.85358i	0.375579	+	0.926790i	$0.377444\pi$
$919$	1703.44i	1.85358i	−0.375579	+	0.926790i	$0.622556\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	552.185i	0.598250i
$924$	0	0
$925$	1124.36	1.21552
$926$	0	0
$927$	0	0
$928$	0	0
$929$	645.209	0.694520	0.347260	−	0.937769i	$-0.387112\pi$
$929$	645.209	0.694520	0.347260	+	0.937769i	$0.387112\pi$
$930$	0	0
$931$	− 710.277i	− 0.762918i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 40.7806i	− 0.0436156i
$936$	0	0
$937$	360.662	0.384911	0.192456	−	0.981306i	$-0.438355\pi$
$937$	360.662	0.384911	0.192456	+	0.981306i	$0.438355\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	111.816	0.118827	0.0594134	−	0.998233i	$-0.481077\pi$
$941$	111.816	0.118827	0.0594134	+	0.998233i	$0.481077\pi$
$942$	0	0
$943$	− 91.9685i	− 0.0975275i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24.7456i	− 0.0261306i	−0.999915	−	0.0130653i	$-0.995841\pi$
$947$	− 24.7456i	− 0.0261306i	0.999915	−	0.0130653i	$-0.00415893\pi$
$948$	0	0
$949$	58.1679	0.0612939
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1569.10	−1.64649	−0.823245	−	0.567686i	$-0.807839\pi$
$953$	−1569.10	−1.64649	−0.823245	+	0.567686i	$0.807839\pi$
$954$	0	0
$955$	1076.25i	1.12696i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	676.657i	0.705586i
$960$	0	0
$961$	459.859	0.478521
$962$	0	0
$963$	0	0
$964$	0	0
$965$	907.757	0.940681
$966$	0	0
$967$	129.015i	0.133418i	0.997772	+	0.0667090i	$0.0212499\pi$
$967$	129.015i	0.133418i	−0.997772	+	0.0667090i	$0.978750\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 595.591i	− 0.613379i	−0.951810	−	0.306689i	$-0.900779\pi$
$971$	− 595.591i	− 0.613379i	0.951810	−	0.306689i	$-0.0992212\pi$
$972$	0	0
$973$	462.610	0.475447
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1232.40	1.26141	0.630706	−	0.776022i	$-0.282765\pi$
$977$	1232.40	1.26141	0.630706	+	0.776022i	$0.282765\pi$
$978$	0	0
$979$	− 164.238i	− 0.167761i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1335.14i	− 1.35823i	−0.734031	−	0.679115i	$-0.762364\pi$
$983$	− 1335.14i	− 1.35823i	0.734031	−	0.679115i	$-0.237636\pi$
$984$	0	0
$985$	877.198	0.890556
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−303.705	−0.307083
$990$	0	0
$991$	617.773i	0.623384i	0.950183	+	0.311692i	$0.100896\pi$
$991$	617.773i	0.623384i	−0.950183	+	0.311692i	$0.899104\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2447.96i	2.46026i
$996$	0	0
$997$	−622.311	−0.624183	−0.312092	−	0.950052i	$-0.601030\pi$
$997$	−622.311	−0.624183	−0.312092	+	0.950052i	$0.601030\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.m.703.2		8
3.2	odd	2		1728.3.g.j.703.8		8
4.3	odd	2	inner	1728.3.g.m.703.1		8
8.3	odd	2		864.3.g.b.703.7	✓	8
8.5	even	2		864.3.g.b.703.8	yes	8
12.11	even	2		1728.3.g.j.703.7		8
24.5	odd	2		864.3.g.d.703.2	yes	8
24.11	even	2		864.3.g.d.703.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.b.703.7	✓	8	8.3	odd	2
864.3.g.b.703.8	yes	8	8.5	even	2
864.3.g.d.703.1	yes	8	24.11	even	2
864.3.g.d.703.2	yes	8	24.5	odd	2
1728.3.g.j.703.7		8	12.11	even	2
1728.3.g.j.703.8		8	3.2	odd	2
1728.3.g.m.703.1		8	4.3	odd	2	inner
1728.3.g.m.703.2		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.59655 q^{5} +4.56106i q^{7} +O(q^{10})\)	\(q-6.59655 q^{5} +4.56106i q^{7} -1.16066i q^{11} -13.5580 q^{13} -5.32636 q^{17} -25.1900i q^{19} +10.4309i q^{23} +18.5144 q^{25} -47.0288 q^{29} -22.3862i q^{31} -30.0872i q^{35} +60.7288 q^{37} -8.81696 q^{41} +29.1160i q^{43} +78.2321i q^{47} +28.1968 q^{49} -62.7623 q^{53} +7.65637i q^{55} -109.116i q^{59} +66.4997 q^{61} +89.4360 q^{65} +81.9685i q^{67} -40.7276i q^{71} -4.29030 q^{73} +5.29386 q^{77} -5.20371i q^{79} +115.114i q^{83} +35.1356 q^{85} +141.504 q^{89} -61.8388i q^{91} +166.167i q^{95} +136.769 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} - 8 q^{13} - 24 q^{17} + 24 q^{25} - 128 q^{29} - 24 q^{37} - 160 q^{41} - 144 q^{49} - 48 q^{53} + 136 q^{61} + 280 q^{65} + 72 q^{73} + 520 q^{77} - 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 - 8 * q^13 - 24 * q^17 + 24 * q^25 - 128 * q^29 - 24 * q^37 - 160 * q^41 - 144 * q^49 - 48 * q^53 + 136 * q^61 + 280 * q^65 + 72 * q^73 + 520 * q^77 - 96 * q^85 + 168 * q^89 + 104 * q^97

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