LMFDB - Embedded newform 900.3.b.b.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(449,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("900.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.b (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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	900.449
Dual form			900.3.b.b.449.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.4868i q^{7} +O(q^{10})$	$q-13.4868i q^{7} +17.6590i q^{11} +7.48683i q^{13} +16.9706 q^{17} +10.9737 q^{19} +21.9017 q^{23} -47.3575i q^{29} -16.9737 q^{31} -5.53950i q^{37} -66.3936i q^{41} -38.9737i q^{43} +32.5642 q^{47} -132.895 q^{49} -11.2392 q^{53} -31.8757i q^{59} +46.9210 q^{61} -76.0000i q^{67} +77.7445i q^{71} -94.9210i q^{73} +238.165 q^{77} +6.92100 q^{79} +62.1509 q^{83} -62.2626i q^{89} +100.974 q^{91} -124.974i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 64 q^{19} + 16 q^{31} - 456 q^{49} - 80 q^{61} - 400 q^{79} + 656 q^{91}+O(q^{100}) $ 8 * q - 64 * q^19 + 16 * q^31 - 456 * q^49 - 80 * q^61 - 400 * q^79 + 656 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 13.4868i	− 1.92669i	−0.268265	−	0.963345i	$-0.586450\pi$
$7$	− 13.4868i	− 1.92669i	0.268265	−	0.963345i	$-0.413550\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	17.6590i	1.60537i	0.596405	+	0.802684i	$0.296595\pi$
$11$	17.6590i	1.60537i	−0.596405	+	0.802684i	$0.703405\pi$
$12$	0	0
$13$	7.48683i	0.575910i	0.957644	+	0.287955i	$0.0929754\pi$
$13$	7.48683i	0.575910i	−0.957644	+	0.287955i	$0.907025\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	16.9706	0.998268	0.499134	−	0.866525i	$-0.333651\pi$
$17$	16.9706	0.998268	0.499134	+	0.866525i	$0.333651\pi$
$18$	0	0
$19$	10.9737	0.577561	0.288781	−	0.957395i	$-0.406750\pi$
$19$	10.9737	0.577561	0.288781	+	0.957395i	$0.406750\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	21.9017	0.952247	0.476124	−	0.879378i	$-0.342041\pi$
$23$	21.9017	0.952247	0.476124	+	0.879378i	$0.342041\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 47.3575i	− 1.63302i	−0.577332	−	0.816509i	$-0.695906\pi$
$29$	− 47.3575i	− 1.63302i	0.577332	−	0.816509i	$-0.304094\pi$
$30$	0	0
$31$	−16.9737	−0.547538	−0.273769	−	0.961796i	$-0.588270\pi$
$31$	−16.9737	−0.547538	−0.273769	+	0.961796i	$0.588270\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.53950i	− 0.149716i	−0.997194	−	0.0748581i	$-0.976150\pi$
$37$	− 5.53950i	− 0.149716i	0.997194	−	0.0748581i	$-0.0238504\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 66.3936i	− 1.61935i	−0.586875	−	0.809677i	$-0.699642\pi$
$41$	− 66.3936i	− 1.61935i	0.586875	−	0.809677i	$-0.300358\pi$
$42$	0	0
$43$	− 38.9737i	− 0.906364i	−0.891418	−	0.453182i	$-0.850289\pi$
$43$	− 38.9737i	− 0.906364i	0.891418	−	0.453182i	$-0.149711\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	32.5642	0.692854	0.346427	−	0.938077i	$-0.387395\pi$
$47$	32.5642	0.692854	0.346427	+	0.938077i	$0.387395\pi$
$48$	0	0
$49$	−132.895	−2.71214
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.2392	−0.212061	−0.106030	−	0.994363i	$-0.533814\pi$
$53$	−11.2392	−0.212061	−0.106030	+	0.994363i	$0.533814\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 31.8757i	− 0.540266i	−0.962823	−	0.270133i	$-0.912932\pi$
$59$	− 31.8757i	− 0.540266i	0.962823	−	0.270133i	$-0.0870676\pi$
$60$	0	0
$61$	46.9210	0.769197	0.384598	−	0.923084i	$-0.374340\pi$
$61$	46.9210	0.769197	0.384598	+	0.923084i	$0.374340\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 76.0000i	− 1.13433i	−0.823605	−	0.567164i	$-0.808040\pi$
$67$	− 76.0000i	− 1.13433i	0.823605	−	0.567164i	$-0.191960\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	77.7445i	1.09499i	0.836808	+	0.547497i	$0.184419\pi$
$71$	77.7445i	1.09499i	−0.836808	+	0.547497i	$0.815581\pi$
$72$	0	0
$73$	− 94.9210i	− 1.30029i	−0.759811	−	0.650144i	$-0.774709\pi$
$73$	− 94.9210i	− 1.30029i	0.759811	−	0.650144i	$-0.225291\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	238.165	3.09305
$78$	0	0
$79$	6.92100	0.0876076	0.0438038	−	0.999040i	$-0.486052\pi$
$79$	6.92100	0.0876076	0.0438038	+	0.999040i	$0.486052\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	62.1509	0.748806	0.374403	−	0.927266i	$-0.377848\pi$
$83$	62.1509	0.748806	0.374403	+	0.927266i	$0.377848\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 62.2626i	− 0.699580i	−0.936828	−	0.349790i	$-0.886253\pi$
$89$	− 62.2626i	− 0.699580i	0.936828	−	0.349790i	$-0.113747\pi$
$90$	0	0
$91$	100.974	1.10960
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 124.974i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$97$	− 124.974i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	112.262i	1.11151i	0.831347	+	0.555754i	$0.187571\pi$
$101$	112.262i	1.11151i	−0.831347	+	0.555754i	$0.812429\pi$
$102$	0	0
$103$	170.302i	1.65342i	0.562627	+	0.826711i	$0.309791\pi$
$103$	170.302i	1.65342i	−0.562627	+	0.826711i	$0.690209\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	93.3381	0.872319	0.436159	−	0.899869i	$-0.356338\pi$
$107$	93.3381	0.872319	0.436159	+	0.899869i	$0.356338\pi$
$108$	0	0
$109$	68.8683	0.631820	0.315910	−	0.948789i	$-0.397690\pi$
$109$	68.8683	0.631820	0.315910	+	0.948789i	$0.397690\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	110.309	0.976183	0.488091	−	0.872793i	$-0.337693\pi$
$113$	110.309	0.976183	0.488091	+	0.872793i	$0.337693\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 228.879i	− 1.92335i
$120$	0	0
$121$	−190.842	−1.57721
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 65.3815i	− 0.514815i	−0.966303	−	0.257407i	$-0.917132\pi$
$127$	− 65.3815i	− 0.514815i	0.966303	−	0.257407i	$-0.0828683\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 47.2458i	− 0.360655i	−0.983607	−	0.180328i	$-0.942284\pi$
$131$	− 47.2458i	− 0.360655i	0.983607	−	0.180328i	$-0.0577158\pi$
$132$	0	0
$133$	− 148.000i	− 1.11278i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−216.039	−1.57693	−0.788465	−	0.615079i	$-0.789124\pi$
$137$	−216.039	−1.57693	−0.788465	+	0.615079i	$0.789124\pi$
$138$	0	0
$139$	−211.842	−1.52404	−0.762022	−	0.647552i	$-0.775793\pi$
$139$	−211.842	−1.52404	−0.762022	+	0.647552i	$0.775793\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−132.210	−0.924548
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	92.5379i	0.621060i	0.950564	+	0.310530i	$0.100506\pi$
$149$	92.5379i	0.621060i	−0.950564	+	0.310530i	$0.899494\pi$
$150$	0	0
$151$	−123.842	−0.820146	−0.410073	−	0.912053i	$-0.634497\pi$
$151$	−123.842	−0.820146	−0.410073	+	0.912053i	$0.634497\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 31.4868i	− 0.200553i	−0.994960	−	0.100277i	$-0.968027\pi$
$157$	− 31.4868i	− 0.200553i	0.994960	−	0.100277i	$-0.0319727\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 295.384i	− 1.83469i
$162$	0	0
$163$	101.132i	0.620440i	0.950665	+	0.310220i	$0.100403\pi$
$163$	101.132i	0.620440i	−0.950665	+	0.310220i	$0.899597\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−147.804	−0.885054	−0.442527	−	0.896755i	$-0.645918\pi$
$167$	−147.804	−0.885054	−0.442527	+	0.896755i	$0.645918\pi$
$168$	0	0
$169$	112.947	0.668327
$170$	0	0
$171$	0	0
$172$	0	0
$173$	332.079	1.91953	0.959767	−	0.280797i	$-0.0905986\pi$
$173$	332.079	1.91953	0.959767	+	0.280797i	$0.0905986\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	45.8688i	0.256250i	0.991758	+	0.128125i	$0.0408959\pi$
$179$	45.8688i	0.256250i	−0.991758	+	0.128125i	$0.959104\pi$
$180$	0	0
$181$	132.868	0.734079	0.367040	−	0.930205i	$-0.380371\pi$
$181$	132.868	0.734079	0.367040	+	0.930205i	$0.380371\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	299.684i	1.60259i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	163.974i	0.858504i	0.903185	+	0.429252i	$0.141223\pi$
$191$	163.974i	0.858504i	−0.903185	+	0.429252i	$0.858777\pi$
$192$	0	0
$193$	− 110.000i	− 0.569948i	−0.958535	−	0.284974i	$-0.908015\pi$
$193$	− 110.000i	− 0.569948i	0.958535	−	0.284974i	$-0.0919850\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	211.108	1.07162	0.535808	−	0.844340i	$-0.320007\pi$
$197$	211.108	1.07162	0.535808	+	0.844340i	$0.320007\pi$
$198$	0	0
$199$	18.1053	0.0909816	0.0454908	−	0.998965i	$-0.485515\pi$
$199$	18.1053	0.0909816	0.0454908	+	0.998965i	$0.485515\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−638.703	−3.14632
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	193.785i	0.927199i
$210$	0	0
$211$	341.579	1.61886	0.809428	−	0.587219i	$-0.199777\pi$
$211$	341.579	1.61886	0.809428	+	0.587219i	$0.199777\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	228.921i	1.05494i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	127.056i	0.574913i
$222$	0	0
$223$	− 59.3288i	− 0.266049i	−0.991113	−	0.133024i	$-0.957531\pi$
$223$	− 59.3288i	− 0.266049i	0.991113	−	0.133024i	$-0.0424688\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	115.593	0.509221	0.254610	−	0.967044i	$-0.418053\pi$
$227$	115.593	0.509221	0.254610	+	0.967044i	$0.418053\pi$
$228$	0	0
$229$	253.895	1.10871	0.554355	−	0.832280i	$-0.312965\pi$
$229$	253.895	1.10871	0.554355	+	0.832280i	$0.312965\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	206.401	0.885840	0.442920	−	0.896561i	$-0.353943\pi$
$233$	206.401	0.885840	0.442920	+	0.896561i	$0.353943\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	346.296i	1.44894i	0.689308	+	0.724469i	$0.257915\pi$
$239$	346.296i	1.44894i	−0.689308	+	0.724469i	$0.742085\pi$
$240$	0	0
$241$	−182.053	−0.755405	−0.377703	−	0.925927i	$-0.623286\pi$
$241$	−182.053	−0.755405	−0.377703	+	0.925927i	$0.623286\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	82.1580i	0.332623i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	140.807i	0.560985i	0.959856	+	0.280493i	$0.0904979\pi$
$251$	140.807i	0.560985i	−0.959856	+	0.280493i	$0.909502\pi$
$252$	0	0
$253$	386.763i	1.52871i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−292.630	−1.13864	−0.569320	−	0.822116i	$-0.692793\pi$
$257$	−292.630	−1.13864	−0.569320	+	0.822116i	$0.692793\pi$
$258$	0	0
$259$	−74.7103	−0.288457
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−270.952	−1.03024	−0.515118	−	0.857119i	$-0.672252\pi$
$263$	−270.952	−1.03024	−0.515118	+	0.857119i	$0.672252\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	275.083i	1.02261i	0.859398	+	0.511307i	$0.170838\pi$
$269$	275.083i	1.02261i	−0.859398	+	0.511307i	$0.829162\pi$
$270$	0	0
$271$	248.158	0.915712	0.457856	−	0.889026i	$-0.348617\pi$
$271$	248.158	0.915712	0.457856	+	0.889026i	$0.348617\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 342.039i	− 1.23480i	−0.786650	−	0.617399i	$-0.788186\pi$
$277$	− 342.039i	− 1.23480i	0.786650	−	0.617399i	$-0.211814\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 352.139i	− 1.25316i	−0.779355	−	0.626582i	$-0.784453\pi$
$281$	− 352.139i	− 1.25316i	0.779355	−	0.626582i	$-0.215547\pi$
$282$	0	0
$283$	− 249.631i	− 0.882089i	−0.897485	−	0.441045i	$-0.854608\pi$
$283$	− 249.631i	− 0.882089i	0.897485	−	0.441045i	$-0.145392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−895.439	−3.12000
$288$	0	0
$289$	−1.00000	−0.00346021
$290$	0	0
$291$	0	0
$292$	0	0
$293$	37.4953	0.127970	0.0639851	−	0.997951i	$-0.479619\pi$
$293$	37.4953	0.127970	0.0639851	+	0.997951i	$0.479619\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	163.974i	0.548409i
$300$	0	0
$301$	−525.631	−1.74628
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	457.842i	1.49134i	0.666314	+	0.745671i	$0.267871\pi$
$307$	457.842i	1.49134i	−0.666314	+	0.745671i	$0.732129\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.48528i	0.0272839i	0.999907	+	0.0136419i	$0.00434250\pi$
$311$	8.48528i	0.0272839i	−0.999907	+	0.0136419i	$0.995658\pi$
$312$	0	0
$313$	1.57866i	0.00504363i	0.999997	+	0.00252181i	$0.000802719\pi$
$313$	1.57866i	0.00504363i	−0.999997	+	0.00252181i	$0.999197\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−529.195	−1.66938	−0.834692	−	0.550717i	$-0.814354\pi$
$317$	−529.195	−1.66938	−0.834692	+	0.550717i	$0.814354\pi$
$318$	0	0
$319$	836.289	2.62160
$320$	0	0
$321$	0	0
$322$	0	0
$323$	186.229	0.576561
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 439.187i	− 1.33492i
$330$	0	0
$331$	−390.763	−1.18055	−0.590276	−	0.807201i	$-0.700981\pi$
$331$	−390.763	−1.18055	−0.590276	+	0.807201i	$0.700981\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	414.710i	1.23059i	0.788295	+	0.615297i	$0.210964\pi$
$337$	414.710i	1.23059i	−0.788295	+	0.615297i	$0.789036\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 299.739i	− 0.878999i
$342$	0	0
$343$	1131.47i	3.29876i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−128.656	−0.370767	−0.185384	−	0.982666i	$-0.559353\pi$
$347$	−128.656	−0.370767	−0.185384	+	0.982666i	$0.559353\pi$
$348$	0	0
$349$	37.0790	0.106244	0.0531218	−	0.998588i	$-0.483083\pi$
$349$	37.0790	0.106244	0.0531218	+	0.998588i	$0.483083\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−190.584	−0.539897	−0.269949	−	0.962875i	$-0.587007\pi$
$353$	−190.584	−0.539897	−0.269949	+	0.962875i	$0.587007\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	254.782i	0.709699i	0.934923	+	0.354849i	$0.115468\pi$
$359$	254.782i	0.709699i	−0.934923	+	0.354849i	$0.884532\pi$
$360$	0	0
$361$	−240.579	−0.666423
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	300.460i	0.818693i	0.912379	+	0.409347i	$0.134243\pi$
$367$	300.460i	0.818693i	−0.912379	+	0.409347i	$0.865757\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	151.582i	0.408576i
$372$	0	0
$373$	− 704.092i	− 1.88765i	−0.330452	−	0.943823i	$-0.607201\pi$
$373$	− 704.092i	− 1.88765i	0.330452	−	0.943823i	$-0.392799\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	354.558	0.940472
$378$	0	0
$379$	−333.789	−0.880711	−0.440355	−	0.897824i	$-0.645148\pi$
$379$	−333.789	−0.880711	−0.440355	+	0.897824i	$0.645148\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	260.996	0.681453	0.340726	−	0.940163i	$-0.389327\pi$
$383$	260.996	0.681453	0.340726	+	0.940163i	$0.389327\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	35.8949i	0.0922747i	0.998935	+	0.0461373i	$0.0146912\pi$
$389$	35.8949i	0.0922747i	−0.998935	+	0.0461373i	$0.985309\pi$
$390$	0	0
$391$	371.684	0.950598
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 90.3552i	− 0.227595i	−0.993504	−	0.113797i	$-0.963699\pi$
$397$	− 90.3552i	− 0.227595i	0.993504	−	0.113797i	$-0.0363015\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	43.6917i	0.108957i	0.998515	+	0.0544784i	$0.0173496\pi$
$401$	43.6917i	0.108957i	−0.998515	+	0.0544784i	$0.982650\pi$
$402$	0	0
$403$	− 127.079i	− 0.315333i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	97.8223	0.240350
$408$	0	0
$409$	260.053	0.635826	0.317913	−	0.948120i	$-0.397018\pi$
$409$	260.053	0.635826	0.317913	+	0.948120i	$0.397018\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−429.902	−1.04092
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	263.285i	0.628366i	0.949362	+	0.314183i	$0.101731\pi$
$419$	263.285i	0.628366i	−0.949362	+	0.314183i	$0.898269\pi$
$420$	0	0
$421$	367.210	0.872233	0.436116	−	0.899890i	$-0.356354\pi$
$421$	367.210	0.872233	0.436116	+	0.899890i	$0.356354\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 632.816i	− 1.48200i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	690.768i	1.60271i	0.598189	+	0.801355i	$0.295887\pi$
$431$	690.768i	1.60271i	−0.598189	+	0.801355i	$0.704113\pi$
$432$	0	0
$433$	117.500i	0.271363i	0.990752	+	0.135682i	$0.0433224\pi$
$433$	117.500i	0.271363i	−0.990752	+	0.135682i	$0.956678\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	240.342	0.549981
$438$	0	0
$439$	−50.0000	−0.113895	−0.0569476	−	0.998377i	$-0.518137\pi$
$439$	−50.0000	−0.113895	−0.0569476	+	0.998377i	$0.518137\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−526.794	−1.18915	−0.594576	−	0.804040i	$-0.702680\pi$
$443$	−526.794	−1.18915	−0.594576	+	0.804040i	$0.702680\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	321.882i	0.716887i	0.933552	+	0.358443i	$0.116692\pi$
$449$	321.882i	0.716887i	−0.933552	+	0.358443i	$0.883308\pi$
$450$	0	0
$451$	1172.45	2.59966
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 433.079i	− 0.947656i	−0.880617	−	0.473828i	$-0.842872\pi$
$457$	− 433.079i	− 0.947656i	0.880617	−	0.473828i	$-0.157128\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 486.768i	− 1.05590i	−0.849277	−	0.527948i	$-0.822961\pi$
$461$	− 486.768i	− 1.05590i	0.849277	−	0.527948i	$-0.177039\pi$
$462$	0	0
$463$	− 423.223i	− 0.914090i	−0.889444	−	0.457045i	$-0.848908\pi$
$463$	− 423.223i	− 0.914090i	0.889444	−	0.457045i	$-0.151092\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	295.831	0.633472	0.316736	−	0.948514i	$-0.397413\pi$
$467$	295.831	0.633472	0.316736	+	0.948514i	$0.397413\pi$
$468$	0	0
$469$	−1025.00	−2.18550
$470$	0	0
$471$	0	0
$472$	0	0
$473$	688.238	1.45505
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	266.728i	0.556843i	0.960459	+	0.278421i	$0.0898112\pi$
$479$	266.728i	0.556843i	−0.960459	+	0.278421i	$0.910189\pi$
$480$	0	0
$481$	41.4733	0.0862231
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	247.750i	0.508727i	0.967109	+	0.254364i	$0.0818660\pi$
$487$	247.750i	0.508727i	−0.967109	+	0.254364i	$0.918134\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	170.394i	0.347035i	0.984831	+	0.173517i	$0.0555133\pi$
$491$	170.394i	0.347035i	−0.984831	+	0.173517i	$0.944487\pi$
$492$	0	0
$493$	− 803.684i	− 1.63019i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1048.53	2.10971
$498$	0	0
$499$	−47.3950	−0.0949800	−0.0474900	−	0.998872i	$-0.515122\pi$
$499$	−47.3950	−0.0949800	−0.0474900	+	0.998872i	$0.515122\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−321.064	−0.638298	−0.319149	−	0.947705i	$-0.603397\pi$
$503$	−321.064	−0.638298	−0.319149	+	0.947705i	$0.603397\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 122.125i	− 0.239931i	−0.992778	−	0.119965i	$-0.961722\pi$
$509$	− 122.125i	− 0.239931i	0.992778	−	0.119965i	$-0.0382783\pi$
$510$	0	0
$511$	−1280.18	−2.50525
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	575.052i	1.11229i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	79.4567i	0.152508i	0.997088	+	0.0762540i	$0.0242960\pi$
$521$	79.4567i	0.152508i	−0.997088	+	0.0762540i	$0.975704\pi$
$522$	0	0
$523$	864.605i	1.65316i	0.562816	+	0.826582i	$0.309718\pi$
$523$	864.605i	1.65316i	−0.562816	+	0.826582i	$0.690282\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−288.053	−0.546589
$528$	0	0
$529$	−49.3160	−0.0932250
$530$	0	0
$531$	0	0
$532$	0	0
$533$	497.077	0.932603
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2346.79i	− 4.35398i
$540$	0	0
$541$	33.6057	0.0621177	0.0310589	−	0.999518i	$-0.490112\pi$
$541$	33.6057	0.0621177	0.0310589	+	0.999518i	$0.490112\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 559.079i	− 1.02208i	−0.859556	−	0.511041i	$-0.829260\pi$
$547$	− 559.079i	− 1.02208i	0.859556	−	0.511041i	$-0.170740\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 519.686i	− 0.943168i
$552$	0	0
$553$	− 93.3423i	− 0.168793i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−310.308	−0.557105	−0.278553	−	0.960421i	$-0.589855\pi$
$557$	−310.308	−0.557105	−0.278553	+	0.960421i	$0.589855\pi$
$558$	0	0
$559$	291.789	0.521984
$560$	0	0
$561$	0	0
$562$	0	0
$563$	566.690	1.00655	0.503277	−	0.864125i	$-0.332128\pi$
$563$	566.690	1.00655	0.503277	+	0.864125i	$0.332128\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	134.052i	0.235593i	0.993038	+	0.117796i	$0.0375830\pi$
$569$	134.052i	0.235593i	−0.993038	+	0.117796i	$0.962417\pi$
$570$	0	0
$571$	−661.920	−1.15923	−0.579615	−	0.814890i	$-0.696797\pi$
$571$	−661.920	−1.15923	−0.579615	+	0.814890i	$0.696797\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	76.7630i	0.133038i	0.997785	+	0.0665191i	$0.0211893\pi$
$577$	76.7630i	0.133038i	−0.997785	+	0.0665191i	$0.978811\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 838.219i	− 1.44272i
$582$	0	0
$583$	− 198.474i	− 0.340436i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	684.554	1.16619	0.583095	−	0.812404i	$-0.301841\pi$
$587$	684.554	1.16619	0.583095	+	0.812404i	$0.301841\pi$
$588$	0	0
$589$	−186.263	−0.316237
$590$	0	0
$591$	0	0
$592$	0	0
$593$	592.146	0.998560	0.499280	−	0.866441i	$-0.333598\pi$
$593$	592.146	0.998560	0.499280	+	0.866441i	$0.333598\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1012.76i	− 1.69075i	−0.534169	−	0.845377i	$-0.679376\pi$
$599$	− 1012.76i	− 1.69075i	0.534169	−	0.845377i	$-0.320624\pi$
$600$	0	0
$601$	323.579	0.538400	0.269200	−	0.963084i	$-0.413241\pi$
$601$	323.579	0.538400	0.269200	+	0.963084i	$0.413241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 799.828i	− 1.31767i	−0.752285	−	0.658837i	$-0.771049\pi$
$607$	− 799.828i	− 1.31767i	0.752285	−	0.658837i	$-0.228951\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	243.802i	0.399022i
$612$	0	0
$613$	680.302i	1.10979i	0.831920	+	0.554896i	$0.187242\pi$
$613$	680.302i	1.10979i	−0.831920	+	0.554896i	$0.812758\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	264.868	0.429283	0.214641	−	0.976693i	$-0.431142\pi$
$617$	264.868	0.429283	0.214641	+	0.976693i	$0.431142\pi$
$618$	0	0
$619$	−535.842	−0.865658	−0.432829	−	0.901476i	$-0.642485\pi$
$619$	−535.842	−0.865658	−0.432829	+	0.901476i	$0.642485\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−839.726	−1.34787
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 94.0085i	− 0.149457i
$630$	0	0
$631$	307.026	0.486571	0.243286	−	0.969955i	$-0.421775\pi$
$631$	307.026	0.486571	0.243286	+	0.969955i	$0.421775\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 994.960i	− 1.56195i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1246.33i	1.94435i	0.234248	+	0.972177i	$0.424737\pi$
$641$	1246.33i	1.94435i	−0.234248	+	0.972177i	$0.575263\pi$
$642$	0	0
$643$	− 155.395i	− 0.241672i	−0.992672	−	0.120836i	$-0.961443\pi$
$643$	− 155.395i	− 0.241672i	0.992672	−	0.120836i	$-0.0385575\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−142.649	−0.220478	−0.110239	−	0.993905i	$-0.535162\pi$
$647$	−142.649	−0.220478	−0.110239	+	0.993905i	$0.535162\pi$
$648$	0	0
$649$	562.894	0.867325
$650$	0	0
$651$	0	0
$652$	0	0
$653$	924.579	1.41589	0.707947	−	0.706266i	$-0.249622\pi$
$653$	924.579	1.41589	0.707947	+	0.706266i	$0.249622\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	352.046i	0.534212i	0.963667	+	0.267106i	$0.0860674\pi$
$659$	352.046i	0.534212i	−0.963667	+	0.267106i	$0.913933\pi$
$660$	0	0
$661$	−642.921	−0.972649	−0.486325	−	0.873778i	$-0.661663\pi$
$661$	−642.921	−0.972649	−0.486325	+	0.873778i	$0.661663\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1037.21i	− 1.55504i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	828.580i	1.23484i
$672$	0	0
$673$	− 982.605i	− 1.46004i	−0.683427	−	0.730019i	$-0.739511\pi$
$673$	− 982.605i	− 1.46004i	0.683427	−	0.730019i	$-0.260489\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	386.769	0.571298	0.285649	−	0.958334i	$-0.407791\pi$
$677$	386.769	0.571298	0.285649	+	0.958334i	$0.407791\pi$
$678$	0	0
$679$	−1685.50	−2.48233
$680$	0	0
$681$	0	0
$682$	0	0
$683$	504.539	0.738710	0.369355	−	0.929288i	$-0.379579\pi$
$683$	504.539	0.738710	0.369355	+	0.929288i	$0.379579\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 84.1462i	− 0.122128i
$690$	0	0
$691$	−271.079	−0.392300	−0.196150	−	0.980574i	$-0.562844\pi$
$691$	−271.079	−0.392300	−0.196150	+	0.980574i	$0.562844\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1126.74i	− 1.61655i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	358.559i	0.511496i	0.966743	+	0.255748i	$0.0823218\pi$
$701$	358.559i	0.511496i	−0.966743	+	0.255748i	$0.917678\pi$
$702$	0	0
$703$	− 60.7886i	− 0.0864703i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1514.06	2.14153
$708$	0	0
$709$	733.579	1.03467	0.517333	−	0.855784i	$-0.326925\pi$
$709$	733.579	1.03467	0.517333	+	0.855784i	$0.326925\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−371.752	−0.521391
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1108.37i	1.54155i	0.637110	+	0.770773i	$0.280130\pi$
$719$	1108.37i	1.54155i	−0.637110	+	0.770773i	$0.719870\pi$
$720$	0	0
$721$	2296.84	3.18563
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1265.04i	1.74008i	0.492981	+	0.870040i	$0.335907\pi$
$727$	1265.04i	1.74008i	−0.492981	+	0.870040i	$0.664093\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 661.405i	− 0.904795i
$732$	0	0
$733$	586.749i	0.800477i	0.916411	+	0.400238i	$0.131073\pi$
$733$	586.749i	0.800477i	−0.916411	+	0.400238i	$0.868927\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1342.09	1.82101
$738$	0	0
$739$	215.973	0.292250	0.146125	−	0.989266i	$-0.453320\pi$
$739$	215.973	0.292250	0.146125	+	0.989266i	$0.453320\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−599.961	−0.807484	−0.403742	−	0.914873i	$-0.632291\pi$
$743$	−599.961	−0.807484	−0.403742	+	0.914873i	$0.632291\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1258.84i	− 1.68069i
$750$	0	0
$751$	813.657	1.08343	0.541716	−	0.840562i	$-0.317775\pi$
$751$	813.657	1.08343	0.541716	+	0.840562i	$0.317775\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1070.43i	1.41405i	0.707190	+	0.707023i	$0.249962\pi$
$757$	1070.43i	1.41405i	−0.707190	+	0.707023i	$0.750038\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	654.892i	0.860567i	0.902694	+	0.430284i	$0.141586\pi$
$761$	654.892i	0.860567i	−0.902694	+	0.430284i	$0.858414\pi$
$762$	0	0
$763$	− 928.816i	− 1.21732i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	238.648	0.311144
$768$	0	0
$769$	−682.631	−0.887686	−0.443843	−	0.896105i	$-0.646385\pi$
$769$	−682.631	−0.887686	−0.443843	+	0.896105i	$0.646385\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−770.690	−0.997012	−0.498506	−	0.866886i	$-0.666118\pi$
$773$	−770.690	−0.997012	−0.498506	+	0.866886i	$0.666118\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 728.581i	− 0.935277i
$780$	0	0
$781$	−1372.89	−1.75787
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 253.869i	− 0.322578i	−0.986907	−	0.161289i	$-0.948435\pi$
$787$	− 253.869i	− 0.322578i	0.986907	−	0.161289i	$-0.0515652\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1487.71i	− 1.88080i
$792$	0	0
$793$	351.290i	0.442988i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1055.86	−1.32479	−0.662396	−	0.749154i	$-0.730460\pi$
$797$	−1055.86	−1.32479	−0.662396	+	0.749154i	$0.730460\pi$
$798$	0	0
$799$	552.632	0.691655
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1676.21	2.08744
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	506.698i	0.626326i	0.949699	+	0.313163i	$0.101389\pi$
$809$	506.698i	0.626326i	−0.949699	+	0.313163i	$0.898611\pi$
$810$	0	0
$811$	537.473	0.662729	0.331365	−	0.943503i	$-0.392491\pi$
$811$	537.473	0.662729	0.331365	+	0.943503i	$0.392491\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 427.684i	− 0.523481i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 257.219i	− 0.313299i	−0.987654	−	0.156650i	$-0.949931\pi$
$821$	− 257.219i	− 0.313299i	0.987654	−	0.156650i	$-0.0500694\pi$
$822$	0	0
$823$	367.013i	0.445945i	0.974825	+	0.222973i	$0.0715760\pi$
$823$	367.013i	0.445945i	−0.974825	+	0.222973i	$0.928424\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−122.254	−0.147829	−0.0739144	−	0.997265i	$-0.523549\pi$
$827$	−122.254	−0.147829	−0.0739144	+	0.997265i	$0.523549\pi$
$828$	0	0
$829$	−335.237	−0.404387	−0.202194	−	0.979346i	$-0.564807\pi$
$829$	−335.237	−0.404387	−0.202194	+	0.979346i	$0.564807\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2255.30	−2.70744
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	774.021i	0.922552i	0.887257	+	0.461276i	$0.152608\pi$
$839$	774.021i	0.922552i	−0.887257	+	0.461276i	$0.847392\pi$
$840$	0	0
$841$	−1401.74	−1.66675
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2573.85i	3.03879i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 121.324i	− 0.142567i
$852$	0	0
$853$	− 764.591i	− 0.896356i	−0.893944	−	0.448178i	$-0.852073\pi$
$853$	− 764.591i	− 0.896356i	0.893944	−	0.448178i	$-0.147927\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1516.15	−1.76913	−0.884567	−	0.466413i	$-0.845546\pi$
$857$	−1516.15	−1.76913	−0.884567	+	0.466413i	$0.845546\pi$
$858$	0	0
$859$	−433.684	−0.504871	−0.252435	−	0.967614i	$-0.581232\pi$
$859$	−433.684	−0.504871	−0.252435	+	0.967614i	$0.581232\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1255.50	1.45481	0.727407	−	0.686206i	$-0.240725\pi$
$863$	1255.50	1.45481	0.727407	+	0.686206i	$0.240725\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	122.218i	0.140642i
$870$	0	0
$871$	568.999	0.653271
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1003.59i	1.14435i	0.820133	+	0.572173i	$0.193900\pi$
$877$	1003.59i	1.14435i	−0.820133	+	0.572173i	$0.806100\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 737.214i	− 0.836792i	−0.908265	−	0.418396i	$-0.862592\pi$
$881$	− 737.214i	− 0.836792i	0.908265	−	0.418396i	$-0.137408\pi$
$882$	0	0
$883$	− 336.394i	− 0.380968i	−0.981690	−	0.190484i	$-0.938994\pi$
$883$	− 336.394i	− 0.380968i	0.981690	−	0.190484i	$-0.0610057\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−793.132	−0.894174	−0.447087	−	0.894491i	$-0.647539\pi$
$887$	−793.132	−0.894174	−0.447087	+	0.894491i	$0.647539\pi$
$888$	0	0
$889$	−881.789	−0.991889
$890$	0	0
$891$	0	0
$892$	0	0
$893$	357.348	0.400166
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	803.831i	0.894139i
$900$	0	0
$901$	−190.736	−0.211694
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1079.66i	− 1.19036i	−0.803592	−	0.595180i	$-0.797081\pi$
$907$	− 1079.66i	− 1.19036i	0.803592	−	0.595180i	$-0.202919\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1473.69i	− 1.61766i	−0.588045	−	0.808828i	$-0.700102\pi$
$911$	− 1473.69i	− 1.61766i	0.588045	−	0.808828i	$-0.299898\pi$
$912$	0	0
$913$	1097.53i	1.20211i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−637.196	−0.694871
$918$	0	0
$919$	1602.92	1.74420	0.872101	−	0.489327i	$-0.162757\pi$
$919$	1602.92	1.74420	0.872101	+	0.489327i	$0.162757\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−582.060	−0.630618
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1033.98i	− 1.11300i	−0.830848	−	0.556499i	$-0.812144\pi$
$929$	− 1033.98i	− 1.11300i	0.830848	−	0.556499i	$-0.187856\pi$
$930$	0	0
$931$	−1458.34	−1.56642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1258.05i	− 1.34264i	−0.741169	−	0.671319i	$-0.765728\pi$
$937$	− 1258.05i	− 1.34264i	0.741169	−	0.671319i	$-0.234272\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	762.875i	0.810707i	0.914160	+	0.405353i	$0.132852\pi$
$941$	762.875i	0.810707i	−0.914160	+	0.405353i	$0.867148\pi$
$942$	0	0
$943$	− 1454.13i	− 1.54203i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1023.55	−1.08084	−0.540419	−	0.841396i	$-0.681734\pi$
$947$	−1023.55	−1.08084	−0.540419	+	0.841396i	$0.681734\pi$
$948$	0	0
$949$	710.658	0.748849
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−63.7513	−0.0668954	−0.0334477	−	0.999440i	$-0.510649\pi$
$953$	−63.7513	−0.0668954	−0.0334477	+	0.999440i	$0.510649\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2913.69i	3.03826i
$960$	0	0
$961$	−672.895	−0.700203
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 505.986i	− 0.523254i	−0.965169	−	0.261627i	$-0.915741\pi$
$967$	− 505.986i	− 0.523254i	0.965169	−	0.261627i	$-0.0842590\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1803.51i	1.85738i	0.370862	+	0.928688i	$0.379062\pi$
$971$	1803.51i	1.85738i	−0.370862	+	0.928688i	$0.620938\pi$
$972$	0	0
$973$	2857.08i	2.93636i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1844.54	−1.88797	−0.943983	−	0.329994i	$-0.892953\pi$
$977$	−1844.54	−1.88797	−0.943983	+	0.329994i	$0.892953\pi$
$978$	0	0
$979$	1099.50	1.12308
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−592.146	−0.602386	−0.301193	−	0.953563i	$-0.597385\pi$
$983$	−592.146	−0.602386	−0.301193	+	0.953563i	$0.597385\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 853.589i	− 0.863083i
$990$	0	0
$991$	961.684	0.970418	0.485209	−	0.874398i	$-0.338744\pi$
$991$	961.684	0.970418	0.485209	+	0.874398i	$0.338744\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	899.670i	0.902378i	0.892429	+	0.451189i	$0.149000\pi$
$997$	899.670i	0.902378i	−0.892429	+	0.451189i	$0.851000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.b.b.449.2		8
3.2	odd	2	inner	900.3.b.b.449.1		8
4.3	odd	2		3600.3.c.k.449.7		8
5.2	odd	4		900.3.g.d.701.4		4
5.3	odd	4		180.3.g.a.161.3	yes	4
5.4	even	2	inner	900.3.b.b.449.8		8
12.11	even	2		3600.3.c.k.449.8		8
15.2	even	4		900.3.g.d.701.3		4
15.8	even	4		180.3.g.a.161.1	✓	4
15.14	odd	2	inner	900.3.b.b.449.7		8
20.3	even	4		720.3.l.c.161.4		4
20.7	even	4		3600.3.l.n.1601.1		4
20.19	odd	2		3600.3.c.k.449.1		8
40.3	even	4		2880.3.l.f.1601.2		4
40.13	odd	4		2880.3.l.b.1601.1		4
45.13	odd	12		1620.3.o.f.1241.4		8
45.23	even	12		1620.3.o.f.1241.2		8
45.38	even	12		1620.3.o.f.701.4		8
45.43	odd	12		1620.3.o.f.701.2		8
60.23	odd	4		720.3.l.c.161.2		4
60.47	odd	4		3600.3.l.n.1601.2		4
60.59	even	2		3600.3.c.k.449.2		8
120.53	even	4		2880.3.l.b.1601.3		4
120.83	odd	4		2880.3.l.f.1601.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.g.a.161.1	✓	4	15.8	even	4
180.3.g.a.161.3	yes	4	5.3	odd	4
720.3.l.c.161.2		4	60.23	odd	4
720.3.l.c.161.4		4	20.3	even	4
900.3.b.b.449.1		8	3.2	odd	2	inner
900.3.b.b.449.2		8	1.1	even	1	trivial
900.3.b.b.449.7		8	15.14	odd	2	inner
900.3.b.b.449.8		8	5.4	even	2	inner
900.3.g.d.701.3		4	15.2	even	4
900.3.g.d.701.4		4	5.2	odd	4
1620.3.o.f.701.2		8	45.43	odd	12
1620.3.o.f.701.4		8	45.38	even	12
1620.3.o.f.1241.2		8	45.23	even	12
1620.3.o.f.1241.4		8	45.13	odd	12
2880.3.l.b.1601.1		4	40.13	odd	4
2880.3.l.b.1601.3		4	120.53	even	4
2880.3.l.f.1601.2		4	40.3	even	4
2880.3.l.f.1601.4		4	120.83	odd	4
3600.3.c.k.449.1		8	20.19	odd	2
3600.3.c.k.449.2		8	60.59	even	2
3600.3.c.k.449.7		8	4.3	odd	2
3600.3.c.k.449.8		8	12.11	even	2
3600.3.l.n.1601.1		4	20.7	even	4
3600.3.l.n.1601.2		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-13.4868i q^{7} +O(q^{10})\)	\(q-13.4868i q^{7} +17.6590i q^{11} +7.48683i q^{13} +16.9706 q^{17} +10.9737 q^{19} +21.9017 q^{23} -47.3575i q^{29} -16.9737 q^{31} -5.53950i q^{37} -66.3936i q^{41} -38.9737i q^{43} +32.5642 q^{47} -132.895 q^{49} -11.2392 q^{53} -31.8757i q^{59} +46.9210 q^{61} -76.0000i q^{67} +77.7445i q^{71} -94.9210i q^{73} +238.165 q^{77} +6.92100 q^{79} +62.1509 q^{83} -62.2626i q^{89} +100.974 q^{91} -124.974i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 64 q^{19} + 16 q^{31} - 456 q^{49} - 80 q^{61} - 400 q^{79} + 656 q^{91}+O(q^{100}) \) 8 * q - 64 * q^19 + 16 * q^31 - 456 * q^49 - 80 * q^61 - 400 * q^79 + 656 * q^91

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