LMFDB - Embedded newform 400.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

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:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+22.0000 q^{3} +218.000 q^{7} +241.000 q^{9} +O(q^{10})$

$q+22.0000 q^{3} +218.000 q^{7} +241.000 q^{9} +480.000 q^{11} +622.000 q^{13} -186.000 q^{17} +1204.00 q^{19} +4796.00 q^{21} -3186.00 q^{23} -44.0000 q^{27} +5526.00 q^{29} -9356.00 q^{31} +10560.0 q^{33} -5618.00 q^{37} +13684.0 q^{39} -14394.0 q^{41} -370.000 q^{43} +16146.0 q^{47} +30717.0 q^{49} -4092.00 q^{51} +4374.00 q^{53} +26488.0 q^{57} +11748.0 q^{59} +13202.0 q^{61} +52538.0 q^{63} -11542.0 q^{67} -70092.0 q^{69} +29532.0 q^{71} -33698.0 q^{73} +104640. q^{77} -31208.0 q^{79} -59531.0 q^{81} -38466.0 q^{83} +121572. q^{87} +119514. q^{89} +135596. q^{91} -205832. q^{93} -94658.0 q^{97} +115680. q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	22.0000	1.41130	0.705650	−	0.708560i	$-0.250655\pi$
$3$	22.0000	1.41130	0.705650	+	0.708560i	$0.250655\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	218.000	1.68156	0.840778	−	0.541380i	$-0.182098\pi$
$7$	218.000	1.68156	0.840778	+	0.541380i	$0.182098\pi$
$8$	0	0
$9$	241.000	0.991770
$10$	0	0
$11$	480.000	1.19608	0.598039	−	0.801467i	$-0.295947\pi$
$11$	480.000	1.19608	0.598039	+	0.801467i	$0.295947\pi$
$12$	0	0
$13$	622.000	1.02078	0.510390	−	0.859943i	$-0.329501\pi$
$13$	622.000	1.02078	0.510390	+	0.859943i	$0.329501\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−186.000	−0.156096	−0.0780478	−	0.996950i	$-0.524869\pi$
$17$	−186.000	−0.156096	−0.0780478	+	0.996950i	$0.524869\pi$
$18$	0	0
$19$	1204.00	0.765143	0.382571	−	0.923926i	$-0.375039\pi$
$19$	1204.00	0.765143	0.382571	+	0.923926i	$0.375039\pi$
$20$	0	0
$21$	4796.00	2.37318
$22$	0	0
$23$	−3186.00	−1.25582	−0.627908	−	0.778287i	$-0.716089\pi$
$23$	−3186.00	−1.25582	−0.627908	+	0.778287i	$0.716089\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−44.0000	−0.0116156
$28$	0	0
$29$	5526.00	1.22016	0.610079	−	0.792341i	$-0.291138\pi$
$29$	5526.00	1.22016	0.610079	+	0.792341i	$0.291138\pi$
$30$	0	0
$31$	−9356.00	−1.74858	−0.874291	−	0.485402i	$-0.838673\pi$
$31$	−9356.00	−1.74858	−0.874291	+	0.485402i	$0.838673\pi$
$32$	0	0
$33$	10560.0	1.68803
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5618.00	−0.674648	−0.337324	−	0.941389i	$-0.609522\pi$
$37$	−5618.00	−0.674648	−0.337324	+	0.941389i	$0.609522\pi$
$38$	0	0
$39$	13684.0	1.44063
$40$	0	0
$41$	−14394.0	−1.33728	−0.668639	−	0.743587i	$-0.733123\pi$
$41$	−14394.0	−1.33728	−0.668639	+	0.743587i	$0.733123\pi$
$42$	0	0
$43$	−370.000	−0.0305162	−0.0152581	−	0.999884i	$-0.504857\pi$
$43$	−370.000	−0.0305162	−0.0152581	+	0.999884i	$0.504857\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16146.0	1.06615	0.533077	−	0.846066i	$-0.321035\pi$
$47$	16146.0	1.06615	0.533077	+	0.846066i	$0.321035\pi$
$48$	0	0
$49$	30717.0	1.82763
$50$	0	0
$51$	−4092.00	−0.220298
$52$	0	0
$53$	4374.00	0.213889	0.106945	−	0.994265i	$-0.465893\pi$
$53$	4374.00	0.213889	0.106945	+	0.994265i	$0.465893\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	26488.0	1.07985
$58$	0	0
$59$	11748.0	0.439374	0.219687	−	0.975570i	$-0.429496\pi$
$59$	11748.0	0.439374	0.219687	+	0.975570i	$0.429496\pi$
$60$	0	0
$61$	13202.0	0.454271	0.227136	−	0.973863i	$-0.427064\pi$
$61$	13202.0	0.454271	0.227136	+	0.973863i	$0.427064\pi$
$62$	0	0
$63$	52538.0	1.66772
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11542.0	−0.314119	−0.157059	−	0.987589i	$-0.550201\pi$
$67$	−11542.0	−0.314119	−0.157059	+	0.987589i	$0.550201\pi$
$68$	0	0
$69$	−70092.0	−1.77233
$70$	0	0
$71$	29532.0	0.695260	0.347630	−	0.937632i	$-0.386987\pi$
$71$	29532.0	0.695260	0.347630	+	0.937632i	$0.386987\pi$
$72$	0	0
$73$	−33698.0	−0.740111	−0.370056	−	0.929010i	$-0.620661\pi$
$73$	−33698.0	−0.740111	−0.370056	+	0.929010i	$0.620661\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	104640.	2.01127
$78$	0	0
$79$	−31208.0	−0.562598	−0.281299	−	0.959620i	$-0.590765\pi$
$79$	−31208.0	−0.562598	−0.281299	+	0.959620i	$0.590765\pi$
$80$	0	0
$81$	−59531.0	−1.00816
$82$	0	0
$83$	−38466.0	−0.612889	−0.306444	−	0.951889i	$-0.599139\pi$
$83$	−38466.0	−0.612889	−0.306444	+	0.951889i	$0.599139\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	121572.	1.72201
$88$	0	0
$89$	119514.	1.59935	0.799675	−	0.600432i	$-0.205005\pi$
$89$	119514.	1.59935	0.799675	+	0.600432i	$0.205005\pi$
$90$	0	0
$91$	135596.	1.71650
$92$	0	0
$93$	−205832.	−2.46777
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−94658.0	−1.02148	−0.510738	−	0.859737i	$-0.670628\pi$
$97$	−94658.0	−1.02148	−0.510738	+	0.859737i	$0.670628\pi$
$98$	0	0
$99$	115680.	1.18623
$100$	0	0
$101$	101046.	0.985634	0.492817	−	0.870133i	$-0.335967\pi$
$101$	101046.	0.985634	0.492817	+	0.870133i	$0.335967\pi$
$102$	0	0
$103$	−143434.	−1.33217	−0.666084	−	0.745877i	$-0.732031\pi$
$103$	−143434.	−1.33217	−0.666084	+	0.745877i	$0.732031\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−57054.0	−0.481755	−0.240878	−	0.970555i	$-0.577435\pi$
$107$	−57054.0	−0.481755	−0.240878	+	0.970555i	$0.577435\pi$
$108$	0	0
$109$	−3118.00	−0.0251368	−0.0125684	−	0.999921i	$-0.504001\pi$
$109$	−3118.00	−0.0251368	−0.0125684	+	0.999921i	$0.504001\pi$
$110$	0	0
$111$	−123596.	−0.952132
$112$	0	0
$113$	54534.0	0.401764	0.200882	−	0.979615i	$-0.435619\pi$
$113$	54534.0	0.401764	0.200882	+	0.979615i	$0.435619\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	149902.	1.01238
$118$	0	0
$119$	−40548.0	−0.262484
$120$	0	0
$121$	69349.0	0.430603
$122$	0	0
$123$	−316668.	−1.88730
$124$	0	0
$125$	0	0
$126$	0	0
$127$	24698.0	0.135879	0.0679395	−	0.997689i	$-0.478358\pi$
$127$	24698.0	0.135879	0.0679395	+	0.997689i	$0.478358\pi$
$128$	0	0
$129$	−8140.00	−0.0430675
$130$	0	0
$131$	−236640.	−1.20479	−0.602393	−	0.798200i	$-0.705786\pi$
$131$	−236640.	−1.20479	−0.602393	+	0.798200i	$0.705786\pi$
$132$	0	0
$133$	262472.	1.28663
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22158.0	0.100862	0.0504312	−	0.998728i	$-0.483940\pi$
$137$	22158.0	0.100862	0.0504312	+	0.998728i	$0.483940\pi$
$138$	0	0
$139$	193204.	0.848163	0.424081	−	0.905624i	$-0.360597\pi$
$139$	193204.	0.848163	0.424081	+	0.905624i	$0.360597\pi$
$140$	0	0
$141$	355212.	1.50467
$142$	0	0
$143$	298560.	1.22093
$144$	0	0
$145$	0	0
$146$	0	0
$147$	675774.	2.57934
$148$	0	0
$149$	448554.	1.65519	0.827597	−	0.561322i	$-0.189707\pi$
$149$	448554.	1.65519	0.827597	+	0.561322i	$0.189707\pi$
$150$	0	0
$151$	140860.	0.502742	0.251371	−	0.967891i	$-0.419119\pi$
$151$	140860.	0.502742	0.251371	+	0.967891i	$0.419119\pi$
$152$	0	0
$153$	−44826.0	−0.154811
$154$	0	0
$155$	0	0
$156$	0	0
$157$	335878.	1.08751	0.543754	−	0.839245i	$-0.317002\pi$
$157$	335878.	1.08751	0.543754	+	0.839245i	$0.317002\pi$
$158$	0	0
$159$	96228.0	0.301862
$160$	0	0
$161$	−694548.	−2.11173
$162$	0	0
$163$	−101650.	−0.299667	−0.149833	−	0.988711i	$-0.547874\pi$
$163$	−101650.	−0.299667	−0.149833	+	0.988711i	$0.547874\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	139242.	0.386348	0.193174	−	0.981164i	$-0.438122\pi$
$167$	139242.	0.386348	0.193174	+	0.981164i	$0.438122\pi$
$168$	0	0
$169$	15591.0	0.0419911
$170$	0	0
$171$	290164.	0.758845
$172$	0	0
$173$	265014.	0.673215	0.336607	−	0.941645i	$-0.390721\pi$
$173$	265014.	0.673215	0.336607	+	0.941645i	$0.390721\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	258456.	0.620088
$178$	0	0
$179$	142812.	0.333144	0.166572	−	0.986029i	$-0.446730\pi$
$179$	142812.	0.333144	0.166572	+	0.986029i	$0.446730\pi$
$180$	0	0
$181$	109670.	0.248824	0.124412	−	0.992231i	$-0.460296\pi$
$181$	109670.	0.248824	0.124412	+	0.992231i	$0.460296\pi$
$182$	0	0
$183$	290444.	0.641113
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−89280.0	−0.186703
$188$	0	0
$189$	−9592.00	−0.0195324
$190$	0	0
$191$	−294948.	−0.585008	−0.292504	−	0.956264i	$-0.594489\pi$
$191$	−294948.	−0.585008	−0.292504	+	0.956264i	$0.594489\pi$
$192$	0	0
$193$	−1.00303e6	−1.93831	−0.969153	−	0.246459i	$-0.920733\pi$
$193$	−1.00303e6	−1.93831	−0.969153	+	0.246459i	$0.920733\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	823998.	1.51273	0.756364	−	0.654151i	$-0.226974\pi$
$197$	823998.	1.51273	0.756364	+	0.654151i	$0.226974\pi$
$198$	0	0
$199$	906712.	1.62307	0.811534	−	0.584305i	$-0.198633\pi$
$199$	906712.	1.62307	0.811534	+	0.584305i	$0.198633\pi$
$200$	0	0
$201$	−253924.	−0.443316
$202$	0	0
$203$	1.20467e6	2.05176
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−767826.	−1.24548
$208$	0	0
$209$	577920.	0.915170
$210$	0	0
$211$	−506384.	−0.783022	−0.391511	−	0.920173i	$-0.628047\pi$
$211$	−506384.	−0.783022	−0.391511	+	0.920173i	$0.628047\pi$
$212$	0	0
$213$	649704.	0.981220
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.03961e6	−2.94034
$218$	0	0
$219$	−741356.	−1.04452
$220$	0	0
$221$	−115692.	−0.159339
$222$	0	0
$223$	−542050.	−0.729923	−0.364962	−	0.931023i	$-0.618918\pi$
$223$	−542050.	−0.729923	−0.364962	+	0.931023i	$0.618918\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.44857e6	−1.86585	−0.932924	−	0.360075i	$-0.882751\pi$
$227$	−1.44857e6	−1.86585	−0.932924	+	0.360075i	$0.882751\pi$
$228$	0	0
$229$	−478786.	−0.603327	−0.301663	−	0.953414i	$-0.597542\pi$
$229$	−478786.	−0.603327	−0.301663	+	0.953414i	$0.597542\pi$
$230$	0	0
$231$	2.30208e6	2.83851
$232$	0	0
$233$	−374106.	−0.451445	−0.225723	−	0.974192i	$-0.572474\pi$
$233$	−374106.	−0.451445	−0.225723	+	0.974192i	$0.572474\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−686576.	−0.793995
$238$	0	0
$239$	−169416.	−0.191849	−0.0959245	−	0.995389i	$-0.530581\pi$
$239$	−169416.	−0.191849	−0.0959245	+	0.995389i	$0.530581\pi$
$240$	0	0
$241$	−353746.	−0.392328	−0.196164	−	0.980571i	$-0.562848\pi$
$241$	−353746.	−0.392328	−0.196164	+	0.980571i	$0.562848\pi$
$242$	0	0
$243$	−1.29899e6	−1.41121
$244$	0	0
$245$	0	0
$246$	0	0
$247$	748888.	0.781042
$248$	0	0
$249$	−846252.	−0.864971
$250$	0	0
$251$	−1.25520e6	−1.25756	−0.628780	−	0.777583i	$-0.716445\pi$
$251$	−1.25520e6	−1.25756	−0.628780	+	0.777583i	$0.716445\pi$
$252$	0	0
$253$	−1.52928e6	−1.50205
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.12877e6	1.06604	0.533021	−	0.846102i	$-0.321057\pi$
$257$	1.12877e6	1.06604	0.533021	+	0.846102i	$0.321057\pi$
$258$	0	0
$259$	−1.22472e6	−1.13446
$260$	0	0
$261$	1.33177e6	1.21012
$262$	0	0
$263$	−263082.	−0.234532	−0.117266	−	0.993101i	$-0.537413\pi$
$263$	−263082.	−0.234532	−0.117266	+	0.993101i	$0.537413\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.62931e6	2.25717
$268$	0	0
$269$	−1.18774e6	−1.00079	−0.500393	−	0.865798i	$-0.666811\pi$
$269$	−1.18774e6	−1.00079	−0.500393	+	0.865798i	$0.666811\pi$
$270$	0	0
$271$	−431300.	−0.356744	−0.178372	−	0.983963i	$-0.557083\pi$
$271$	−431300.	−0.356744	−0.178372	+	0.983963i	$0.557083\pi$
$272$	0	0
$273$	2.98311e6	2.42250
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−743114.	−0.581910	−0.290955	−	0.956737i	$-0.593973\pi$
$277$	−743114.	−0.581910	−0.290955	+	0.956737i	$0.593973\pi$
$278$	0	0
$279$	−2.25480e6	−1.73419
$280$	0	0
$281$	1.92193e6	1.45201	0.726007	−	0.687687i	$-0.241374\pi$
$281$	1.92193e6	1.45201	0.726007	+	0.687687i	$0.241374\pi$
$282$	0	0
$283$	−1.63071e6	−1.21035	−0.605176	−	0.796092i	$-0.706897\pi$
$283$	−1.63071e6	−1.21035	−0.605176	+	0.796092i	$0.706897\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.13789e6	−2.24871
$288$	0	0
$289$	−1.38526e6	−0.975634
$290$	0	0
$291$	−2.08248e6	−1.44161
$292$	0	0
$293$	−71250.0	−0.0484859	−0.0242430	−	0.999706i	$-0.507718\pi$
$293$	−71250.0	−0.0484859	−0.0242430	+	0.999706i	$0.507718\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−21120.0	−0.0138932
$298$	0	0
$299$	−1.98169e6	−1.28191
$300$	0	0
$301$	−80660.0	−0.0513147
$302$	0	0
$303$	2.22301e6	1.39103
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.61762e6	−0.979560	−0.489780	−	0.871846i	$-0.662923\pi$
$307$	−1.61762e6	−0.979560	−0.489780	+	0.871846i	$0.662923\pi$
$308$	0	0
$309$	−3.15555e6	−1.88009
$310$	0	0
$311$	682788.	0.400299	0.200150	−	0.979765i	$-0.435857\pi$
$311$	682788.	0.400299	0.200150	+	0.979765i	$0.435857\pi$
$312$	0	0
$313$	2.70444e6	1.56033	0.780165	−	0.625574i	$-0.215135\pi$
$313$	2.70444e6	1.56033	0.780165	+	0.625574i	$0.215135\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.60347e6	−1.45514	−0.727568	−	0.686035i	$-0.759350\pi$
$317$	−2.60347e6	−1.45514	−0.727568	+	0.686035i	$0.759350\pi$
$318$	0	0
$319$	2.65248e6	1.45940
$320$	0	0
$321$	−1.25519e6	−0.679902
$322$	0	0
$323$	−223944.	−0.119435
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−68596.0	−0.0354756
$328$	0	0
$329$	3.51983e6	1.79280
$330$	0	0
$331$	661432.	0.331830	0.165915	−	0.986140i	$-0.446942\pi$
$331$	661432.	0.331830	0.165915	+	0.986140i	$0.446942\pi$
$332$	0	0
$333$	−1.35394e6	−0.669096
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.71706e6	−0.823588	−0.411794	−	0.911277i	$-0.635098\pi$
$337$	−1.71706e6	−0.823588	−0.411794	+	0.911277i	$0.635098\pi$
$338$	0	0
$339$	1.19975e6	0.567010
$340$	0	0
$341$	−4.49088e6	−2.09144
$342$	0	0
$343$	3.03238e6	1.39171
$344$	0	0
$345$	0	0
$346$	0	0
$347$	131370.	0.0585696	0.0292848	−	0.999571i	$-0.490677\pi$
$347$	131370.	0.0585696	0.0292848	+	0.999571i	$0.490677\pi$
$348$	0	0
$349$	3.50951e6	1.54235	0.771175	−	0.636623i	$-0.219669\pi$
$349$	3.50951e6	1.54235	0.771175	+	0.636623i	$0.219669\pi$
$350$	0	0
$351$	−27368.0	−0.0118570
$352$	0	0
$353$	−2.21992e6	−0.948202	−0.474101	−	0.880470i	$-0.657227\pi$
$353$	−2.21992e6	−0.948202	−0.474101	+	0.880470i	$0.657227\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−892056.	−0.370443
$358$	0	0
$359$	−4.39730e6	−1.80074	−0.900369	−	0.435128i	$-0.856703\pi$
$359$	−4.39730e6	−1.80074	−0.900369	+	0.435128i	$0.856703\pi$
$360$	0	0
$361$	−1.02648e6	−0.414557
$362$	0	0
$363$	1.52568e6	0.607710
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.29824e6	−0.890697	−0.445348	−	0.895357i	$-0.646920\pi$
$367$	−2.29824e6	−0.890697	−0.445348	+	0.895357i	$0.646920\pi$
$368$	0	0
$369$	−3.46895e6	−1.32627
$370$	0	0
$371$	953532.	0.359667
$372$	0	0
$373$	1.73561e6	0.645920	0.322960	−	0.946413i	$-0.395322\pi$
$373$	1.73561e6	0.645920	0.322960	+	0.946413i	$0.395322\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.43717e6	1.24551
$378$	0	0
$379$	5.39115e6	1.92789	0.963947	−	0.266094i	$-0.0857331\pi$
$379$	5.39115e6	1.92789	0.963947	+	0.266094i	$0.0857331\pi$
$380$	0	0
$381$	543356.	0.191766
$382$	0	0
$383$	3.27281e6	1.14005	0.570026	−	0.821627i	$-0.306933\pi$
$383$	3.27281e6	1.14005	0.570026	+	0.821627i	$0.306933\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−89170.0	−0.0302650
$388$	0	0
$389$	603114.	0.202081	0.101040	−	0.994882i	$-0.467783\pi$
$389$	603114.	0.202081	0.101040	+	0.994882i	$0.467783\pi$
$390$	0	0
$391$	592596.	0.196027
$392$	0	0
$393$	−5.20608e6	−1.70032
$394$	0	0
$395$	0	0
$396$	0	0
$397$	749422.	0.238644	0.119322	−	0.992856i	$-0.461928\pi$
$397$	749422.	0.238644	0.119322	+	0.992856i	$0.461928\pi$
$398$	0	0
$399$	5.77438e6	1.81582
$400$	0	0
$401$	5.31357e6	1.65016	0.825079	−	0.565018i	$-0.191131\pi$
$401$	5.31357e6	1.65016	0.825079	+	0.565018i	$0.191131\pi$
$402$	0	0
$403$	−5.81943e6	−1.78492
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.69664e6	−0.806932
$408$	0	0
$409$	999326.	0.295392	0.147696	−	0.989033i	$-0.452814\pi$
$409$	999326.	0.295392	0.147696	+	0.989033i	$0.452814\pi$
$410$	0	0
$411$	487476.	0.142347
$412$	0	0
$413$	2.56106e6	0.738831
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.25049e6	1.19701
$418$	0	0
$419$	−2.03740e6	−0.566944	−0.283472	−	0.958980i	$-0.591486\pi$
$419$	−2.03740e6	−0.566944	−0.283472	+	0.958980i	$0.591486\pi$
$420$	0	0
$421$	−5.11461e6	−1.40640	−0.703198	−	0.710994i	$-0.748245\pi$
$421$	−5.11461e6	−1.40640	−0.703198	+	0.710994i	$0.748245\pi$
$422$	0	0
$423$	3.89119e6	1.05738
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.87804e6	0.763882
$428$	0	0
$429$	6.56832e6	1.72310
$430$	0	0
$431$	3.30404e6	0.856747	0.428374	−	0.903602i	$-0.359087\pi$
$431$	3.30404e6	0.856747	0.428374	+	0.903602i	$0.359087\pi$
$432$	0	0
$433$	2.01638e6	0.516836	0.258418	−	0.966033i	$-0.416799\pi$
$433$	2.01638e6	0.516836	0.258418	+	0.966033i	$0.416799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.83594e6	−0.960879
$438$	0	0
$439$	−6.58321e6	−1.63033	−0.815166	−	0.579227i	$-0.803355\pi$
$439$	−6.58321e6	−1.63033	−0.815166	+	0.579227i	$0.803355\pi$
$440$	0	0
$441$	7.40280e6	1.81259
$442$	0	0
$443$	−4.81783e6	−1.16638	−0.583192	−	0.812334i	$-0.698197\pi$
$443$	−4.81783e6	−1.16638	−0.583192	+	0.812334i	$0.698197\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.86819e6	2.33598
$448$	0	0
$449$	−6.20399e6	−1.45230	−0.726149	−	0.687538i	$-0.758692\pi$
$449$	−6.20399e6	−1.45230	−0.726149	+	0.687538i	$0.758692\pi$
$450$	0	0
$451$	−6.90912e6	−1.59949
$452$	0	0
$453$	3.09892e6	0.709520
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.84383e6	−0.636962	−0.318481	−	0.947929i	$-0.603173\pi$
$457$	−2.84383e6	−0.636962	−0.318481	+	0.947929i	$0.603173\pi$
$458$	0	0
$459$	8184.00	0.00181315
$460$	0	0
$461$	−1.75605e6	−0.384844	−0.192422	−	0.981312i	$-0.561634\pi$
$461$	−1.75605e6	−0.384844	−0.192422	+	0.981312i	$0.561634\pi$
$462$	0	0
$463$	7.66857e6	1.66250	0.831250	−	0.555899i	$-0.187626\pi$
$463$	7.66857e6	1.66250	0.831250	+	0.555899i	$0.187626\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.35903e6	−0.288361	−0.144181	−	0.989551i	$-0.546055\pi$
$467$	−1.35903e6	−0.288361	−0.144181	+	0.989551i	$0.546055\pi$
$468$	0	0
$469$	−2.51616e6	−0.528209
$470$	0	0
$471$	7.38932e6	1.53480
$472$	0	0
$473$	−177600.	−0.0364998
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.05413e6	0.212129
$478$	0	0
$479$	2.02706e6	0.403672	0.201836	−	0.979419i	$-0.435309\pi$
$479$	2.02706e6	0.403672	0.201836	+	0.979419i	$0.435309\pi$
$480$	0	0
$481$	−3.49440e6	−0.688667
$482$	0	0
$483$	−1.52801e7	−2.98028
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.46427e6	0.470833	0.235416	−	0.971895i	$-0.424355\pi$
$487$	2.46427e6	0.470833	0.235416	+	0.971895i	$0.424355\pi$
$488$	0	0
$489$	−2.23630e6	−0.422920
$490$	0	0
$491$	−1.03848e7	−1.94399	−0.971996	−	0.234998i	$-0.924492\pi$
$491$	−1.03848e7	−1.94399	−0.971996	+	0.234998i	$0.924492\pi$
$492$	0	0
$493$	−1.02784e6	−0.190461
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.43798e6	1.16912
$498$	0	0
$499$	−6.49416e6	−1.16754	−0.583769	−	0.811919i	$-0.698423\pi$
$499$	−6.49416e6	−1.16754	−0.583769	+	0.811919i	$0.698423\pi$
$500$	0	0
$501$	3.06332e6	0.545254
$502$	0	0
$503$	−1.03565e7	−1.82513	−0.912565	−	0.408931i	$-0.865902\pi$
$503$	−1.03565e7	−1.82513	−0.912565	+	0.408931i	$0.865902\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	343002.	0.0592621
$508$	0	0
$509$	5.87305e6	1.00478	0.502388	−	0.864643i	$-0.332455\pi$
$509$	5.87305e6	1.00478	0.502388	+	0.864643i	$0.332455\pi$
$510$	0	0
$511$	−7.34616e6	−1.24454
$512$	0	0
$513$	−52976.0	−0.00888763
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.75008e6	1.27520
$518$	0	0
$519$	5.83031e6	0.950108
$520$	0	0
$521$	2.17295e6	0.350717	0.175358	−	0.984505i	$-0.443892\pi$
$521$	2.17295e6	0.350717	0.175358	+	0.984505i	$0.443892\pi$
$522$	0	0
$523$	1.07361e6	0.171629	0.0858145	−	0.996311i	$-0.472651\pi$
$523$	1.07361e6	0.171629	0.0858145	+	0.996311i	$0.472651\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.74022e6	0.272946
$528$	0	0
$529$	3.71425e6	0.577075
$530$	0	0
$531$	2.83127e6	0.435757
$532$	0	0
$533$	−8.95307e6	−1.36507
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.14186e6	0.470166
$538$	0	0
$539$	1.47442e7	2.18599
$540$	0	0
$541$	7.09033e6	1.04153	0.520767	−	0.853699i	$-0.325646\pi$
$541$	7.09033e6	1.04153	0.520767	+	0.853699i	$0.325646\pi$
$542$	0	0
$543$	2.41274e6	0.351165
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.69763e6	0.957091	0.478545	−	0.878063i	$-0.341164\pi$
$547$	6.69763e6	0.957091	0.478545	+	0.878063i	$0.341164\pi$
$548$	0	0
$549$	3.18168e6	0.450532
$550$	0	0
$551$	6.65330e6	0.933595
$552$	0	0
$553$	−6.80334e6	−0.946040
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.19008e7	−1.62532	−0.812662	−	0.582735i	$-0.801982\pi$
$557$	−1.19008e7	−1.62532	−0.812662	+	0.582735i	$0.801982\pi$
$558$	0	0
$559$	−230140.	−0.0311503
$560$	0	0
$561$	−1.96416e6	−0.263493
$562$	0	0
$563$	8.75636e6	1.16427	0.582133	−	0.813093i	$-0.302218\pi$
$563$	8.75636e6	1.16427	0.582133	+	0.813093i	$0.302218\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.29778e7	−1.69528
$568$	0	0
$569$	−1.15677e6	−0.149784	−0.0748922	−	0.997192i	$-0.523861\pi$
$569$	−1.15677e6	−0.149784	−0.0748922	+	0.997192i	$0.523861\pi$
$570$	0	0
$571$	7.07807e6	0.908500	0.454250	−	0.890874i	$-0.349907\pi$
$571$	7.07807e6	0.908500	0.454250	+	0.890874i	$0.349907\pi$
$572$	0	0
$573$	−6.48886e6	−0.825623
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.13404e6	0.391890	0.195945	−	0.980615i	$-0.437223\pi$
$577$	3.13404e6	0.391890	0.195945	+	0.980615i	$0.437223\pi$
$578$	0	0
$579$	−2.20667e7	−2.73553
$580$	0	0
$581$	−8.38559e6	−1.03061
$582$	0	0
$583$	2.09952e6	0.255828
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.13833e7	1.36355	0.681776	−	0.731561i	$-0.261208\pi$
$587$	1.13833e7	1.36355	0.681776	+	0.731561i	$0.261208\pi$
$588$	0	0
$589$	−1.12646e7	−1.33791
$590$	0	0
$591$	1.81280e7	2.13491
$592$	0	0
$593$	1.58655e7	1.85275	0.926376	−	0.376599i	$-0.122906\pi$
$593$	1.58655e7	1.85275	0.926376	+	0.376599i	$0.122906\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.99477e7	2.29064
$598$	0	0
$599$	1.50998e7	1.71951	0.859756	−	0.510705i	$-0.170615\pi$
$599$	1.50998e7	1.71951	0.859756	+	0.510705i	$0.170615\pi$
$600$	0	0
$601$	−8.08705e6	−0.913280	−0.456640	−	0.889652i	$-0.650947\pi$
$601$	−8.08705e6	−0.913280	−0.456640	+	0.889652i	$0.650947\pi$
$602$	0	0
$603$	−2.78162e6	−0.311534
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−710398.	−0.0782582	−0.0391291	−	0.999234i	$-0.512458\pi$
$607$	−710398.	−0.0782582	−0.0391291	+	0.999234i	$0.512458\pi$
$608$	0	0
$609$	2.65027e7	2.89566
$610$	0	0
$611$	1.00428e7	1.08831
$612$	0	0
$613$	−5.96434e6	−0.641078	−0.320539	−	0.947235i	$-0.603864\pi$
$613$	−5.96434e6	−0.641078	−0.320539	+	0.947235i	$0.603864\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.48432e7	−1.56970	−0.784848	−	0.619689i	$-0.787259\pi$
$617$	−1.48432e7	−1.56970	−0.784848	+	0.619689i	$0.787259\pi$
$618$	0	0
$619$	1.82042e7	1.90961	0.954807	−	0.297227i	$-0.0960620\pi$
$619$	1.82042e7	1.90961	0.954807	+	0.297227i	$0.0960620\pi$
$620$	0	0
$621$	140184.	0.0145871
$622$	0	0
$623$	2.60541e7	2.68940
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.27142e7	1.29158
$628$	0	0
$629$	1.04495e6	0.105310
$630$	0	0
$631$	−1.09461e6	−0.109443	−0.0547214	−	0.998502i	$-0.517427\pi$
$631$	−1.09461e6	−0.109443	−0.0547214	+	0.998502i	$0.517427\pi$
$632$	0	0
$633$	−1.11404e7	−1.10508
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.91060e7	1.86561
$638$	0	0
$639$	7.11721e6	0.689537
$640$	0	0
$641$	7.44046e6	0.715245	0.357622	−	0.933866i	$-0.383587\pi$
$641$	7.44046e6	0.715245	0.357622	+	0.933866i	$0.383587\pi$
$642$	0	0
$643$	−1.07915e7	−1.02933	−0.514665	−	0.857391i	$-0.672084\pi$
$643$	−1.07915e7	−1.02933	−0.514665	+	0.857391i	$0.672084\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.62998e6	−0.904409	−0.452204	−	0.891914i	$-0.649362\pi$
$647$	−9.62998e6	−0.904409	−0.452204	+	0.891914i	$0.649362\pi$
$648$	0	0
$649$	5.63904e6	0.525525
$650$	0	0
$651$	−4.48714e7	−4.14970
$652$	0	0
$653$	1.00019e7	0.917905	0.458953	−	0.888461i	$-0.348225\pi$
$653$	1.00019e7	0.917905	0.458953	+	0.888461i	$0.348225\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.12122e6	−0.734020
$658$	0	0
$659$	4.01060e6	0.359746	0.179873	−	0.983690i	$-0.442431\pi$
$659$	4.01060e6	0.359746	0.179873	+	0.983690i	$0.442431\pi$
$660$	0	0
$661$	1.20338e7	1.07127	0.535636	−	0.844449i	$-0.320072\pi$
$661$	1.20338e7	1.07127	0.535636	+	0.844449i	$0.320072\pi$
$662$	0	0
$663$	−2.54522e6	−0.224876
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.76058e7	−1.53229
$668$	0	0
$669$	−1.19251e7	−1.03014
$670$	0	0
$671$	6.33696e6	0.543344
$672$	0	0
$673$	−2.01231e6	−0.171260	−0.0856301	−	0.996327i	$-0.527290\pi$
$673$	−2.01231e6	−0.171260	−0.0856301	+	0.996327i	$0.527290\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.62410e7	−1.36188	−0.680942	−	0.732337i	$-0.738429\pi$
$677$	−1.62410e7	−1.36188	−0.680942	+	0.732337i	$0.738429\pi$
$678$	0	0
$679$	−2.06354e7	−1.71767
$680$	0	0
$681$	−3.18686e7	−2.63327
$682$	0	0
$683$	4.62910e6	0.379704	0.189852	−	0.981813i	$-0.439199\pi$
$683$	4.62910e6	0.379704	0.189852	+	0.981813i	$0.439199\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.05333e7	−0.851476
$688$	0	0
$689$	2.72063e6	0.218334
$690$	0	0
$691$	−1.16794e7	−0.930517	−0.465258	−	0.885175i	$-0.654039\pi$
$691$	−1.16794e7	−0.930517	−0.465258	+	0.885175i	$0.654039\pi$
$692$	0	0
$693$	2.52182e7	1.99472
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.67728e6	0.208743
$698$	0	0
$699$	−8.23033e6	−0.637125
$700$	0	0
$701$	1.99543e7	1.53370	0.766851	−	0.641825i	$-0.221822\pi$
$701$	1.99543e7	1.53370	0.766851	+	0.641825i	$0.221822\pi$
$702$	0	0
$703$	−6.76407e6	−0.516202
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.20280e7	1.65740
$708$	0	0
$709$	−4.88331e6	−0.364837	−0.182419	−	0.983221i	$-0.558393\pi$
$709$	−4.88331e6	−0.364837	−0.182419	+	0.983221i	$0.558393\pi$
$710$	0	0
$711$	−7.52113e6	−0.557968
$712$	0	0
$713$	2.98082e7	2.19590
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.72715e6	−0.270757
$718$	0	0
$719$	1.35778e7	0.979505	0.489753	−	0.871861i	$-0.337087\pi$
$719$	1.35778e7	0.979505	0.489753	+	0.871861i	$0.337087\pi$
$720$	0	0
$721$	−3.12686e7	−2.24012
$722$	0	0
$723$	−7.78241e6	−0.553692
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.42411e6	0.450792	0.225396	−	0.974267i	$-0.427632\pi$
$727$	6.42411e6	0.450792	0.225396	+	0.974267i	$0.427632\pi$
$728$	0	0
$729$	−1.41117e7	−0.983472
$730$	0	0
$731$	68820.0	0.00476345
$732$	0	0
$733$	−9.08556e6	−0.624585	−0.312293	−	0.949986i	$-0.601097\pi$
$733$	−9.08556e6	−0.624585	−0.312293	+	0.949986i	$0.601097\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.54016e6	−0.375711
$738$	0	0
$739$	−2.02457e7	−1.36371	−0.681854	−	0.731488i	$-0.738826\pi$
$739$	−2.02457e7	−1.36371	−0.681854	+	0.731488i	$0.738826\pi$
$740$	0	0
$741$	1.64755e7	1.10229
$742$	0	0
$743$	−5.44831e6	−0.362067	−0.181034	−	0.983477i	$-0.557944\pi$
$743$	−5.44831e6	−0.362067	−0.181034	+	0.983477i	$0.557944\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.27031e6	−0.607845
$748$	0	0
$749$	−1.24378e7	−0.810099
$750$	0	0
$751$	1.14072e6	0.0738041	0.0369021	−	0.999319i	$-0.488251\pi$
$751$	1.14072e6	0.0738041	0.0369021	+	0.999319i	$0.488251\pi$
$752$	0	0
$753$	−2.76144e7	−1.77479
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.90153e7	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$757$	−1.90153e7	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$758$	0	0
$759$	−3.36442e7	−2.11985
$760$	0	0
$761$	2.23551e7	1.39931	0.699656	−	0.714480i	$-0.253337\pi$
$761$	2.23551e7	1.39931	0.699656	+	0.714480i	$0.253337\pi$
$762$	0	0
$763$	−679724.	−0.0422689
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.30726e6	0.448504
$768$	0	0
$769$	−1.00704e7	−0.614088	−0.307044	−	0.951695i	$-0.599340\pi$
$769$	−1.00704e7	−0.614088	−0.307044	+	0.951695i	$0.599340\pi$
$770$	0	0
$771$	2.48330e7	1.50451
$772$	0	0
$773$	4.05963e6	0.244364	0.122182	−	0.992508i	$-0.461011\pi$
$773$	4.05963e6	0.244364	0.122182	+	0.992508i	$0.461011\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.69439e7	−1.60106
$778$	0	0
$779$	−1.73304e7	−1.02321
$780$	0	0
$781$	1.41754e7	0.831585
$782$	0	0
$783$	−243144.	−0.0141729
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.72256e7	−0.991372	−0.495686	−	0.868502i	$-0.665083\pi$
$787$	−1.72256e7	−0.991372	−0.495686	+	0.868502i	$0.665083\pi$
$788$	0	0
$789$	−5.78780e6	−0.330995
$790$	0	0
$791$	1.18884e7	0.675589
$792$	0	0
$793$	8.21164e6	0.463711
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.10793e7	1.17547	0.587733	−	0.809055i	$-0.300020\pi$
$797$	2.10793e7	1.17547	0.587733	+	0.809055i	$0.300020\pi$
$798$	0	0
$799$	−3.00316e6	−0.166422
$800$	0	0
$801$	2.88029e7	1.58619
$802$	0	0
$803$	−1.61750e7	−0.885231
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.61303e7	−1.41241
$808$	0	0
$809$	−1.87877e7	−1.00926	−0.504629	−	0.863336i	$-0.668371\pi$
$809$	−1.87877e7	−1.00926	−0.504629	+	0.863336i	$0.668371\pi$
$810$	0	0
$811$	1.32456e7	0.707164	0.353582	−	0.935404i	$-0.384964\pi$
$811$	1.32456e7	0.707164	0.353582	+	0.935404i	$0.384964\pi$
$812$	0	0
$813$	−9.48860e6	−0.503473
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−445480.	−0.0233493
$818$	0	0
$819$	3.26786e7	1.70237
$820$	0	0
$821$	−7.66925e6	−0.397096	−0.198548	−	0.980091i	$-0.563623\pi$
$821$	−7.66925e6	−0.397096	−0.198548	+	0.980091i	$0.563623\pi$
$822$	0	0
$823$	−8.82786e6	−0.454314	−0.227157	−	0.973858i	$-0.572943\pi$
$823$	−8.82786e6	−0.454314	−0.227157	+	0.973858i	$0.572943\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.06923e7	−1.56051	−0.780254	−	0.625463i	$-0.784910\pi$
$827$	−3.06923e7	−1.56051	−0.780254	+	0.625463i	$0.784910\pi$
$828$	0	0
$829$	3.28414e7	1.65972	0.829860	−	0.557972i	$-0.188420\pi$
$829$	3.28414e7	1.65972	0.829860	+	0.557972i	$0.188420\pi$
$830$	0	0
$831$	−1.63485e7	−0.821250
$832$	0	0
$833$	−5.71336e6	−0.285285
$834$	0	0
$835$	0	0
$836$	0	0
$837$	411664.	0.0203109
$838$	0	0
$839$	8.42117e6	0.413017	0.206508	−	0.978445i	$-0.433790\pi$
$839$	8.42117e6	0.413017	0.206508	+	0.978445i	$0.433790\pi$
$840$	0	0
$841$	1.00255e7	0.488784
$842$	0	0
$843$	4.22824e7	2.04923
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.51181e7	0.724083
$848$	0	0
$849$	−3.58757e7	−1.70817
$850$	0	0
$851$	1.78989e7	0.847234
$852$	0	0
$853$	−2.35126e7	−1.10644	−0.553221	−	0.833035i	$-0.686601\pi$
$853$	−2.35126e7	−1.10644	−0.553221	+	0.833035i	$0.686601\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.13050e7	0.525799	0.262900	−	0.964823i	$-0.415321\pi$
$857$	1.13050e7	0.525799	0.262900	+	0.964823i	$0.415321\pi$
$858$	0	0
$859$	−1.00078e7	−0.462758	−0.231379	−	0.972864i	$-0.574324\pi$
$859$	−1.00078e7	−0.462758	−0.231379	+	0.972864i	$0.574324\pi$
$860$	0	0
$861$	−6.90336e7	−3.17360
$862$	0	0
$863$	2.61429e7	1.19489	0.597443	−	0.801911i	$-0.296183\pi$
$863$	2.61429e7	1.19489	0.597443	+	0.801911i	$0.296183\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.04757e7	−1.37691
$868$	0	0
$869$	−1.49798e7	−0.672911
$870$	0	0
$871$	−7.17912e6	−0.320646
$872$	0	0
$873$	−2.28126e7	−1.01307
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.92041e6	0.0843129	0.0421565	−	0.999111i	$-0.486577\pi$
$877$	1.92041e6	0.0843129	0.0421565	+	0.999111i	$0.486577\pi$
$878$	0	0
$879$	−1.56750e6	−0.0684282
$880$	0	0
$881$	−2.56594e7	−1.11380	−0.556899	−	0.830580i	$-0.688009\pi$
$881$	−2.56594e7	−1.11380	−0.556899	+	0.830580i	$0.688009\pi$
$882$	0	0
$883$	−2.05643e7	−0.887590	−0.443795	−	0.896128i	$-0.646368\pi$
$883$	−2.05643e7	−0.887590	−0.443795	+	0.896128i	$0.646368\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.16868e7	1.35229	0.676143	−	0.736770i	$-0.263650\pi$
$887$	3.16868e7	1.35229	0.676143	+	0.736770i	$0.263650\pi$
$888$	0	0
$889$	5.38416e6	0.228488
$890$	0	0
$891$	−2.85749e7	−1.20584
$892$	0	0
$893$	1.94398e7	0.815761
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.35972e7	−1.80916
$898$	0	0
$899$	−5.17013e7	−2.13355
$900$	0	0
$901$	−813564.	−0.0333872
$902$	0	0
$903$	−1.77452e6	−0.0724205
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.96963e6	0.160225	0.0801127	−	0.996786i	$-0.474472\pi$
$907$	3.96963e6	0.160225	0.0801127	+	0.996786i	$0.474472\pi$
$908$	0	0
$909$	2.43521e7	0.977522
$910$	0	0
$911$	1.37945e7	0.550692	0.275346	−	0.961345i	$-0.411208\pi$
$911$	1.37945e7	0.550692	0.275346	+	0.961345i	$0.411208\pi$
$912$	0	0
$913$	−1.84637e7	−0.733063
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.15875e7	−2.02592
$918$	0	0
$919$	−8.08126e6	−0.315639	−0.157819	−	0.987468i	$-0.550446\pi$
$919$	−8.08126e6	−0.315639	−0.157819	+	0.987468i	$0.550446\pi$
$920$	0	0
$921$	−3.55877e7	−1.38245
$922$	0	0
$923$	1.83689e7	0.709707
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.45676e7	−1.32120
$928$	0	0
$929$	2.99956e7	1.14030	0.570150	−	0.821541i	$-0.306885\pi$
$929$	2.99956e7	1.14030	0.570150	+	0.821541i	$0.306885\pi$
$930$	0	0
$931$	3.69833e7	1.39840
$932$	0	0
$933$	1.50213e7	0.564943
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.07620e7	0.772540	0.386270	−	0.922386i	$-0.373763\pi$
$937$	2.07620e7	0.772540	0.386270	+	0.922386i	$0.373763\pi$
$938$	0	0
$939$	5.94976e7	2.20209
$940$	0	0
$941$	−3.47642e6	−0.127985	−0.0639923	−	0.997950i	$-0.520383\pi$
$941$	−3.47642e6	−0.127985	−0.0639923	+	0.997950i	$0.520383\pi$
$942$	0	0
$943$	4.58593e7	1.67938
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.86700e6	0.0676503	0.0338252	−	0.999428i	$-0.489231\pi$
$947$	1.86700e6	0.0676503	0.0338252	+	0.999428i	$0.489231\pi$
$948$	0	0
$949$	−2.09602e7	−0.755490
$950$	0	0
$951$	−5.72763e7	−2.05364
$952$	0	0
$953$	3.85501e7	1.37497	0.687484	−	0.726199i	$-0.258715\pi$
$953$	3.85501e7	1.37497	0.687484	+	0.726199i	$0.258715\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.83546e7	2.05966
$958$	0	0
$959$	4.83044e6	0.169606
$960$	0	0
$961$	5.89056e7	2.05754
$962$	0	0
$963$	−1.37500e7	−0.477790
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.64875e7	0.567008	0.283504	−	0.958971i	$-0.408503\pi$
$967$	1.64875e7	0.567008	0.283504	+	0.958971i	$0.408503\pi$
$968$	0	0
$969$	−4.92677e6	−0.168559
$970$	0	0
$971$	2.36976e7	0.806597	0.403299	−	0.915068i	$-0.367864\pi$
$971$	2.36976e7	0.806597	0.403299	+	0.915068i	$0.367864\pi$
$972$	0	0
$973$	4.21185e7	1.42623
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.77590e7	1.93590	0.967950	−	0.251143i	$-0.0808064\pi$
$977$	5.77590e7	1.93590	0.967950	+	0.251143i	$0.0808064\pi$
$978$	0	0
$979$	5.73667e7	1.91295
$980$	0	0
$981$	−751438.	−0.0249299
$982$	0	0
$983$	−1.10103e7	−0.363425	−0.181712	−	0.983352i	$-0.558164\pi$
$983$	−1.10103e7	−0.363425	−0.181712	+	0.983352i	$0.558164\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.74362e7	2.53018
$988$	0	0
$989$	1.17882e6	0.0383228
$990$	0	0
$991$	−3.70807e7	−1.19940	−0.599700	−	0.800225i	$-0.704713\pi$
$991$	−3.70807e7	−1.19940	−0.599700	+	0.800225i	$0.704713\pi$
$992$	0	0
$993$	1.45515e7	0.468311
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.52935e6	−0.144311	−0.0721553	−	0.997393i	$-0.522988\pi$
$997$	−4.52935e6	−0.144311	−0.0721553	+	0.997393i	$0.522988\pi$
$998$	0	0
$999$	247192.	0.00783647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.m.1.1		1
4.3	odd	2		100.6.a.a.1.1		1
5.2	odd	4		400.6.c.c.49.1		2
5.3	odd	4		400.6.c.c.49.2		2
5.4	even	2		80.6.a.b.1.1		1
12.11	even	2		900.6.a.b.1.1		1
15.14	odd	2		720.6.a.l.1.1		1
20.3	even	4		100.6.c.a.49.1		2
20.7	even	4		100.6.c.a.49.2		2
20.19	odd	2		20.6.a.a.1.1	✓	1
40.19	odd	2		320.6.a.c.1.1		1
40.29	even	2		320.6.a.n.1.1		1
60.23	odd	4		900.6.d.h.649.2		2
60.47	odd	4		900.6.d.h.649.1		2
60.59	even	2		180.6.a.e.1.1		1
140.139	even	2		980.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.a.a.1.1	✓	1	20.19	odd	2
80.6.a.b.1.1		1	5.4	even	2
100.6.a.a.1.1		1	4.3	odd	2
100.6.c.a.49.1		2	20.3	even	4
100.6.c.a.49.2		2	20.7	even	4
180.6.a.e.1.1		1	60.59	even	2
320.6.a.c.1.1		1	40.19	odd	2
320.6.a.n.1.1		1	40.29	even	2
400.6.a.m.1.1		1	1.1	even	1	trivial
400.6.c.c.49.1		2	5.2	odd	4
400.6.c.c.49.2		2	5.3	odd	4
720.6.a.l.1.1		1	15.14	odd	2
900.6.a.b.1.1		1	12.11	even	2
900.6.d.h.649.1		2	60.47	odd	4
900.6.d.h.649.2		2	60.23	odd	4
980.6.a.b.1.1		1	140.139	even	2

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

Introduction

L-functions

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