LMFDB - Embedded newform 441.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+6.00000 q^{2} +4.00000 q^{4} +78.0000 q^{5} -168.000 q^{8} +O(q^{10})$

$q+6.00000 q^{2} +4.00000 q^{4} +78.0000 q^{5} -168.000 q^{8} +468.000 q^{10} -444.000 q^{11} +442.000 q^{13} -1136.00 q^{16} -126.000 q^{17} -2684.00 q^{19} +312.000 q^{20} -2664.00 q^{22} -4200.00 q^{23} +2959.00 q^{25} +2652.00 q^{26} +5442.00 q^{29} -80.0000 q^{31} -1440.00 q^{32} -756.000 q^{34} -5434.00 q^{37} -16104.0 q^{38} -13104.0 q^{40} +7962.00 q^{41} -11524.0 q^{43} -1776.00 q^{44} -25200.0 q^{46} -13920.0 q^{47} +17754.0 q^{50} +1768.00 q^{52} +9594.00 q^{53} -34632.0 q^{55} +32652.0 q^{58} +27492.0 q^{59} -49478.0 q^{61} -480.000 q^{62} +27712.0 q^{64} +34476.0 q^{65} -59356.0 q^{67} -504.000 q^{68} -32040.0 q^{71} +61846.0 q^{73} -32604.0 q^{74} -10736.0 q^{76} -65776.0 q^{79} -88608.0 q^{80} +47772.0 q^{82} +40188.0 q^{83} -9828.00 q^{85} -69144.0 q^{86} +74592.0 q^{88} -7974.00 q^{89} -16800.0 q^{92} -83520.0 q^{94} -209352. q^{95} +143662. q^{97} +O(q^{100})$

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 $

$2$	6.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	6.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	0.125000
$5$	78.0000	1.39531	0.697653	−	0.716436i	$-0.254228\pi$
$5$	78.0000	1.39531	0.697653	+	0.716436i	$0.254228\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−168.000	−0.928078
$9$	0	0
$10$	468.000	1.47995
$11$	−444.000	−1.10637	−0.553186	−	0.833058i	$-0.686588\pi$
$11$	−444.000	−1.10637	−0.553186	+	0.833058i	$0.686588\pi$
$12$	0	0
$13$	442.000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	442.000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	0	0
$15$	0	0
$16$	−1136.00	−1.10938
$17$	−126.000	−0.105742	−0.0528711	−	0.998601i	$-0.516837\pi$
$17$	−126.000	−0.105742	−0.0528711	+	0.998601i	$0.516837\pi$
$18$	0	0
$19$	−2684.00	−1.70568	−0.852842	−	0.522169i	$-0.825123\pi$
$19$	−2684.00	−1.70568	−0.852842	+	0.522169i	$0.825123\pi$
$20$	312.000	0.174413
$21$	0	0
$22$	−2664.00	−1.17348
$23$	−4200.00	−1.65550	−0.827751	−	0.561096i	$-0.810380\pi$
$23$	−4200.00	−1.65550	−0.827751	+	0.561096i	$0.810380\pi$
$24$	0	0
$25$	2959.00	0.946880
$26$	2652.00	0.769379
$27$	0	0
$28$	0	0
$29$	5442.00	1.20161	0.600805	−	0.799396i	$-0.294847\pi$
$29$	5442.00	1.20161	0.600805	+	0.799396i	$0.294847\pi$
$30$	0	0
$31$	−80.0000	−0.0149515	−0.00747577	−	0.999972i	$-0.502380\pi$
$31$	−80.0000	−0.0149515	−0.00747577	+	0.999972i	$0.502380\pi$
$32$	−1440.00	−0.248592
$33$	0	0
$34$	−756.000	−0.112157
$35$	0	0
$36$	0	0
$37$	−5434.00	−0.652552	−0.326276	−	0.945274i	$-0.605794\pi$
$37$	−5434.00	−0.652552	−0.326276	+	0.945274i	$0.605794\pi$
$38$	−16104.0	−1.80915
$39$	0	0
$40$	−13104.0	−1.29495
$41$	7962.00	0.739712	0.369856	−	0.929089i	$-0.379407\pi$
$41$	7962.00	0.739712	0.369856	+	0.929089i	$0.379407\pi$
$42$	0	0
$43$	−11524.0	−0.950456	−0.475228	−	0.879863i	$-0.657634\pi$
$43$	−11524.0	−0.950456	−0.475228	+	0.879863i	$0.657634\pi$
$44$	−1776.00	−0.138297
$45$	0	0
$46$	−25200.0	−1.75592
$47$	−13920.0	−0.919167	−0.459584	−	0.888134i	$-0.652001\pi$
$47$	−13920.0	−0.919167	−0.459584	+	0.888134i	$0.652001\pi$
$48$	0	0
$49$	0	0
$50$	17754.0	1.00432
$51$	0	0
$52$	1768.00	0.0906721
$53$	9594.00	0.469148	0.234574	−	0.972098i	$-0.424630\pi$
$53$	9594.00	0.469148	0.234574	+	0.972098i	$0.424630\pi$
$54$	0	0
$55$	−34632.0	−1.54373
$56$	0	0
$57$	0	0
$58$	32652.0	1.27450
$59$	27492.0	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$59$	27492.0	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$60$	0	0
$61$	−49478.0	−1.70250	−0.851251	−	0.524759i	$-0.824155\pi$
$61$	−49478.0	−1.70250	−0.851251	+	0.524759i	$0.824155\pi$
$62$	−480.000	−0.0158585
$63$	0	0
$64$	27712.0	0.845703
$65$	34476.0	1.01212
$66$	0	0
$67$	−59356.0	−1.61539	−0.807695	−	0.589600i	$-0.799285\pi$
$67$	−59356.0	−1.61539	−0.807695	+	0.589600i	$0.799285\pi$
$68$	−504.000	−0.0132178
$69$	0	0
$70$	0	0
$71$	−32040.0	−0.754304	−0.377152	−	0.926151i	$-0.623097\pi$
$71$	−32040.0	−0.754304	−0.377152	+	0.926151i	$0.623097\pi$
$72$	0	0
$73$	61846.0	1.35833	0.679164	−	0.733987i	$-0.262343\pi$
$73$	61846.0	1.35833	0.679164	+	0.733987i	$0.262343\pi$
$74$	−32604.0	−0.692136
$75$	0	0
$76$	−10736.0	−0.213210
$77$	0	0
$78$	0	0
$79$	−65776.0	−1.18577	−0.592884	−	0.805288i	$-0.702011\pi$
$79$	−65776.0	−1.18577	−0.592884	+	0.805288i	$0.702011\pi$
$80$	−88608.0	−1.54792
$81$	0	0
$82$	47772.0	0.784583
$83$	40188.0	0.640326	0.320163	−	0.947362i	$-0.396262\pi$
$83$	40188.0	0.640326	0.320163	+	0.947362i	$0.396262\pi$
$84$	0	0
$85$	−9828.00	−0.147543
$86$	−69144.0	−1.00811
$87$	0	0
$88$	74592.0	1.02680
$89$	−7974.00	−0.106709	−0.0533545	−	0.998576i	$-0.516991\pi$
$89$	−7974.00	−0.106709	−0.0533545	+	0.998576i	$0.516991\pi$
$90$	0	0
$91$	0	0
$92$	−16800.0	−0.206938
$93$	0	0
$94$	−83520.0	−0.974924
$95$	−209352.	−2.37995
$96$	0	0
$97$	143662.	1.55029	0.775144	−	0.631784i	$-0.217677\pi$
$97$	143662.	1.55029	0.775144	+	0.631784i	$0.217677\pi$
$98$	0	0
$99$	0	0
$100$	11836.0	0.118360
$101$	−2706.00	−0.0263952	−0.0131976	−	0.999913i	$-0.504201\pi$
$101$	−2706.00	−0.0263952	−0.0131976	+	0.999913i	$0.504201\pi$
$102$	0	0
$103$	−131768.	−1.22382	−0.611909	−	0.790928i	$-0.709598\pi$
$103$	−131768.	−1.22382	−0.611909	+	0.790928i	$0.709598\pi$
$104$	−74256.0	−0.673206
$105$	0	0
$106$	57564.0	0.497607
$107$	128916.	1.08855	0.544274	−	0.838908i	$-0.316805\pi$
$107$	128916.	1.08855	0.544274	+	0.838908i	$0.316805\pi$
$108$	0	0
$109$	−100978.	−0.814068	−0.407034	−	0.913413i	$-0.633437\pi$
$109$	−100978.	−0.814068	−0.407034	+	0.913413i	$0.633437\pi$
$110$	−207792.	−1.63737
$111$	0	0
$112$	0	0
$113$	−220146.	−1.62186	−0.810932	−	0.585140i	$-0.801040\pi$
$113$	−220146.	−1.62186	−0.810932	+	0.585140i	$0.801040\pi$
$114$	0	0
$115$	−327600.	−2.30993
$116$	21768.0	0.150201
$117$	0	0
$118$	164952.	1.09057
$119$	0	0
$120$	0	0
$121$	36085.0	0.224059
$122$	−296868.	−1.80578
$123$	0	0
$124$	−320.000	−0.00186894
$125$	−12948.0	−0.0741187
$126$	0	0
$127$	−74320.0	−0.408880	−0.204440	−	0.978879i	$-0.565537\pi$
$127$	−74320.0	−0.408880	−0.204440	+	0.978879i	$0.565537\pi$
$128$	212352.	1.14560
$129$	0	0
$130$	206856.	1.07352
$131$	−155316.	−0.790748	−0.395374	−	0.918520i	$-0.629385\pi$
$131$	−155316.	−0.790748	−0.395374	+	0.918520i	$0.629385\pi$
$132$	0	0
$133$	0	0
$134$	−356136.	−1.71338
$135$	0	0
$136$	21168.0	0.0981369
$137$	264246.	1.20284	0.601419	−	0.798934i	$-0.294602\pi$
$137$	264246.	1.20284	0.601419	+	0.798934i	$0.294602\pi$
$138$	0	0
$139$	−224612.	−0.986043	−0.493022	−	0.870017i	$-0.664108\pi$
$139$	−224612.	−0.986043	−0.493022	+	0.870017i	$0.664108\pi$
$140$	0	0
$141$	0	0
$142$	−192240.	−0.800061
$143$	−196248.	−0.802537
$144$	0	0
$145$	424476.	1.67661
$146$	371076.	1.44072
$147$	0	0
$148$	−21736.0	−0.0815690
$149$	82074.0	0.302859	0.151429	−	0.988468i	$-0.451612\pi$
$149$	82074.0	0.302859	0.151429	+	0.988468i	$0.451612\pi$
$150$	0	0
$151$	−287032.	−1.02444	−0.512222	−	0.858853i	$-0.671177\pi$
$151$	−287032.	−1.02444	−0.512222	+	0.858853i	$0.671177\pi$
$152$	450912.	1.58301
$153$	0	0
$154$	0	0
$155$	−6240.00	−0.0208620
$156$	0	0
$157$	−129878.	−0.420520	−0.210260	−	0.977646i	$-0.567431\pi$
$157$	−129878.	−0.420520	−0.210260	+	0.977646i	$0.567431\pi$
$158$	−394656.	−1.25770
$159$	0	0
$160$	−112320.	−0.346862
$161$	0	0
$162$	0	0
$163$	555284.	1.63699	0.818495	−	0.574513i	$-0.194809\pi$
$163$	555284.	1.63699	0.818495	+	0.574513i	$0.194809\pi$
$164$	31848.0	0.0924640
$165$	0	0
$166$	241128.	0.679168
$167$	43512.0	0.120731	0.0603654	−	0.998176i	$-0.480773\pi$
$167$	43512.0	0.120731	0.0603654	+	0.998176i	$0.480773\pi$
$168$	0	0
$169$	−175929.	−0.473828
$170$	−58968.0	−0.156493
$171$	0	0
$172$	−46096.0	−0.118807
$173$	−18330.0	−0.0465637	−0.0232818	−	0.999729i	$-0.507412\pi$
$173$	−18330.0	−0.0465637	−0.0232818	+	0.999729i	$0.507412\pi$
$174$	0	0
$175$	0	0
$176$	504384.	1.22738
$177$	0	0
$178$	−47844.0	−0.113182
$179$	153324.	0.357666	0.178833	−	0.983879i	$-0.442768\pi$
$179$	153324.	0.357666	0.178833	+	0.983879i	$0.442768\pi$
$180$	0	0
$181$	382066.	0.866846	0.433423	−	0.901191i	$-0.357306\pi$
$181$	382066.	0.866846	0.433423	+	0.901191i	$0.357306\pi$
$182$	0	0
$183$	0	0
$184$	705600.	1.53643
$185$	−423852.	−0.910510
$186$	0	0
$187$	55944.0	0.116990
$188$	−55680.0	−0.114896
$189$	0	0
$190$	−1.25611e6	−2.52432
$191$	273408.	0.542285	0.271143	−	0.962539i	$-0.412598\pi$
$191$	273408.	0.542285	0.271143	+	0.962539i	$0.412598\pi$
$192$	0	0
$193$	153602.	0.296827	0.148414	−	0.988925i	$-0.452583\pi$
$193$	153602.	0.296827	0.148414	+	0.988925i	$0.452583\pi$
$194$	861972.	1.64433
$195$	0	0
$196$	0	0
$197$	−154422.	−0.283494	−0.141747	−	0.989903i	$-0.545272\pi$
$197$	−154422.	−0.283494	−0.141747	+	0.989903i	$0.545272\pi$
$198$	0	0
$199$	366856.	0.656694	0.328347	−	0.944557i	$-0.393508\pi$
$199$	366856.	0.656694	0.328347	+	0.944557i	$0.393508\pi$
$200$	−497112.	−0.878778
$201$	0	0
$202$	−16236.0	−0.0279963
$203$	0	0
$204$	0	0
$205$	621036.	1.03212
$206$	−790608.	−1.29806
$207$	0	0
$208$	−502112.	−0.804715
$209$	1.19170e6	1.88712
$210$	0	0
$211$	520244.	0.804453	0.402227	−	0.915540i	$-0.368236\pi$
$211$	520244.	0.804453	0.402227	+	0.915540i	$0.368236\pi$
$212$	38376.0	0.0586435
$213$	0	0
$214$	773496.	1.15458
$215$	−898872.	−1.32618
$216$	0	0
$217$	0	0
$218$	−605868.	−0.863449
$219$	0	0
$220$	−138528.	−0.192966
$221$	−55692.0	−0.0767030
$222$	0	0
$223$	−304736.	−0.410357	−0.205178	−	0.978725i	$-0.565777\pi$
$223$	−304736.	−0.410357	−0.205178	+	0.978725i	$0.565777\pi$
$224$	0	0
$225$	0	0
$226$	−1.32088e6	−1.72025
$227$	288588.	0.371718	0.185859	−	0.982576i	$-0.440493\pi$
$227$	288588.	0.371718	0.185859	+	0.982576i	$0.440493\pi$
$228$	0	0
$229$	−772190.	−0.973051	−0.486525	−	0.873666i	$-0.661736\pi$
$229$	−772190.	−0.973051	−0.486525	+	0.873666i	$0.661736\pi$
$230$	−1.96560e6	−2.45005
$231$	0	0
$232$	−914256.	−1.11519
$233$	−252234.	−0.304378	−0.152189	−	0.988351i	$-0.548632\pi$
$233$	−252234.	−0.304378	−0.152189	+	0.988351i	$0.548632\pi$
$234$	0	0
$235$	−1.08576e6	−1.28252
$236$	109968.	0.128525
$237$	0	0
$238$	0	0
$239$	1.45114e6	1.64329	0.821643	−	0.570002i	$-0.193058\pi$
$239$	1.45114e6	1.64329	0.821643	+	0.570002i	$0.193058\pi$
$240$	0	0
$241$	146398.	0.162365	0.0811825	−	0.996699i	$-0.474130\pi$
$241$	146398.	0.162365	0.0811825	+	0.996699i	$0.474130\pi$
$242$	216510.	0.237651
$243$	0	0
$244$	−197912.	−0.212813
$245$	0	0
$246$	0	0
$247$	−1.18633e6	−1.23726
$248$	13440.0	0.0138762
$249$	0	0
$250$	−77688.0	−0.0786147
$251$	607860.	0.609003	0.304501	−	0.952512i	$-0.401510\pi$
$251$	607860.	0.609003	0.304501	+	0.952512i	$0.401510\pi$
$252$	0	0
$253$	1.86480e6	1.83160
$254$	−445920.	−0.433683
$255$	0	0
$256$	387328.	0.369385
$257$	95586.0	0.0902737	0.0451369	−	0.998981i	$-0.485628\pi$
$257$	95586.0	0.0902737	0.0451369	+	0.998981i	$0.485628\pi$
$258$	0	0
$259$	0	0
$260$	137904.	0.126515
$261$	0	0
$262$	−931896.	−0.838715
$263$	2.20034e6	1.96156	0.980779	−	0.195121i	$-0.0625100\pi$
$263$	2.20034e6	1.96156	0.980779	+	0.195121i	$0.0625100\pi$
$264$	0	0
$265$	748332.	0.654605
$266$	0	0
$267$	0	0
$268$	−237424.	−0.201924
$269$	1.77025e6	1.49160	0.745801	−	0.666169i	$-0.232067\pi$
$269$	1.77025e6	1.49160	0.745801	+	0.666169i	$0.232067\pi$
$270$	0	0
$271$	223504.	0.184868	0.0924341	−	0.995719i	$-0.470535\pi$
$271$	223504.	0.184868	0.0924341	+	0.995719i	$0.470535\pi$
$272$	143136.	0.117308
$273$	0	0
$274$	1.58548e6	1.27580
$275$	−1.31380e6	−1.04760
$276$	0	0
$277$	−342778.	−0.268419	−0.134210	−	0.990953i	$-0.542850\pi$
$277$	−342778.	−0.268419	−0.134210	+	0.990953i	$0.542850\pi$
$278$	−1.34767e6	−1.04586
$279$	0	0
$280$	0	0
$281$	−480378.	−0.362925	−0.181463	−	0.983398i	$-0.558083\pi$
$281$	−480378.	−0.362925	−0.181463	+	0.983398i	$0.558083\pi$
$282$	0	0
$283$	29980.0	0.0222518	0.0111259	−	0.999938i	$-0.496458\pi$
$283$	29980.0	0.0222518	0.0111259	+	0.999938i	$0.496458\pi$
$284$	−128160.	−0.0942880
$285$	0	0
$286$	−1.17749e6	−0.851219
$287$	0	0
$288$	0	0
$289$	−1.40398e6	−0.988819
$290$	2.54686e6	1.77832
$291$	0	0
$292$	247384.	0.169791
$293$	−198066.	−0.134785	−0.0673924	−	0.997727i	$-0.521468\pi$
$293$	−198066.	−0.134785	−0.0673924	+	0.997727i	$0.521468\pi$
$294$	0	0
$295$	2.14438e6	1.43465
$296$	912912.	0.605619
$297$	0	0
$298$	492444.	0.321230
$299$	−1.85640e6	−1.20086
$300$	0	0
$301$	0	0
$302$	−1.72219e6	−1.08659
$303$	0	0
$304$	3.04902e6	1.89224
$305$	−3.85928e6	−2.37551
$306$	0	0
$307$	1.04564e6	0.633191	0.316595	−	0.948561i	$-0.397460\pi$
$307$	1.04564e6	0.633191	0.316595	+	0.948561i	$0.397460\pi$
$308$	0	0
$309$	0	0
$310$	−37440.0	−0.0221275
$311$	1.83718e6	1.07708	0.538542	−	0.842598i	$-0.318975\pi$
$311$	1.83718e6	1.07708	0.538542	+	0.842598i	$0.318975\pi$
$312$	0	0
$313$	365494.	0.210872	0.105436	−	0.994426i	$-0.466376\pi$
$313$	365494.	0.210872	0.105436	+	0.994426i	$0.466376\pi$
$314$	−779268.	−0.446029
$315$	0	0
$316$	−263104.	−0.148221
$317$	28338.0	0.0158388	0.00791938	−	0.999969i	$-0.497479\pi$
$317$	28338.0	0.0158388	0.00791938	+	0.999969i	$0.497479\pi$
$318$	0	0
$319$	−2.41625e6	−1.32943
$320$	2.16154e6	1.18001
$321$	0	0
$322$	0	0
$323$	338184.	0.180363
$324$	0	0
$325$	1.30788e6	0.686845
$326$	3.33170e6	1.73629
$327$	0	0
$328$	−1.33762e6	−0.686510
$329$	0	0
$330$	0	0
$331$	1.93392e6	0.970214	0.485107	−	0.874455i	$-0.338781\pi$
$331$	1.93392e6	0.970214	0.485107	+	0.874455i	$0.338781\pi$
$332$	160752.	0.0800408
$333$	0	0
$334$	261072.	0.128054
$335$	−4.62977e6	−2.25397
$336$	0	0
$337$	−1.88817e6	−0.905664	−0.452832	−	0.891596i	$-0.649586\pi$
$337$	−1.88817e6	−0.905664	−0.452832	+	0.891596i	$0.649586\pi$
$338$	−1.05557e6	−0.502570
$339$	0	0
$340$	−39312.0	−0.0184428
$341$	35520.0	0.0165420
$342$	0	0
$343$	0	0
$344$	1.93603e6	0.882097
$345$	0	0
$346$	−109980.	−0.0493882
$347$	−2.91937e6	−1.30156	−0.650782	−	0.759264i	$-0.725559\pi$
$347$	−2.91937e6	−1.30156	−0.650782	+	0.759264i	$0.725559\pi$
$348$	0	0
$349$	780682.	0.343092	0.171546	−	0.985176i	$-0.445124\pi$
$349$	780682.	0.343092	0.171546	+	0.985176i	$0.445124\pi$
$350$	0	0
$351$	0	0
$352$	639360.	0.275036
$353$	1.33437e6	0.569954	0.284977	−	0.958534i	$-0.408014\pi$
$353$	1.33437e6	0.569954	0.284977	+	0.958534i	$0.408014\pi$
$354$	0	0
$355$	−2.49912e6	−1.05249
$356$	−31896.0	−0.0133386
$357$	0	0
$358$	919944.	0.379362
$359$	−1.01743e6	−0.416648	−0.208324	−	0.978060i	$-0.566801\pi$
$359$	−1.01743e6	−0.416648	−0.208324	+	0.978060i	$0.566801\pi$
$360$	0	0
$361$	4.72776e6	1.90936
$362$	2.29240e6	0.919429
$363$	0	0
$364$	0	0
$365$	4.82399e6	1.89528
$366$	0	0
$367$	−837680.	−0.324648	−0.162324	−	0.986737i	$-0.551899\pi$
$367$	−837680.	−0.324648	−0.162324	+	0.986737i	$0.551899\pi$
$368$	4.77120e6	1.83657
$369$	0	0
$370$	−2.54311e6	−0.965742
$371$	0	0
$372$	0	0
$373$	−1.51993e6	−0.565655	−0.282827	−	0.959171i	$-0.591272\pi$
$373$	−1.51993e6	−0.565655	−0.282827	+	0.959171i	$0.591272\pi$
$374$	335664.	0.124087
$375$	0	0
$376$	2.33856e6	0.853059
$377$	2.40536e6	0.871620
$378$	0	0
$379$	2.64465e6	0.945737	0.472869	−	0.881133i	$-0.343219\pi$
$379$	2.64465e6	0.945737	0.472869	+	0.881133i	$0.343219\pi$
$380$	−837408.	−0.297494
$381$	0	0
$382$	1.64045e6	0.575180
$383$	2.01336e6	0.701333	0.350667	−	0.936500i	$-0.385955\pi$
$383$	2.01336e6	0.701333	0.350667	+	0.936500i	$0.385955\pi$
$384$	0	0
$385$	0	0
$386$	921612.	0.314833
$387$	0	0
$388$	574648.	0.193786
$389$	726234.	0.243334	0.121667	−	0.992571i	$-0.461176\pi$
$389$	726234.	0.243334	0.121667	+	0.992571i	$0.461176\pi$
$390$	0	0
$391$	529200.	0.175056
$392$	0	0
$393$	0	0
$394$	−926532.	−0.300691
$395$	−5.13053e6	−1.65451
$396$	0	0
$397$	−4.57578e6	−1.45710	−0.728549	−	0.684993i	$-0.759805\pi$
$397$	−4.57578e6	−1.45710	−0.728549	+	0.684993i	$0.759805\pi$
$398$	2.20114e6	0.696529
$399$	0	0
$400$	−3.36142e6	−1.05045
$401$	33870.0	0.0105185	0.00525926	−	0.999986i	$-0.498326\pi$
$401$	33870.0	0.0105185	0.00525926	+	0.999986i	$0.498326\pi$
$402$	0	0
$403$	−35360.0	−0.0108455
$404$	−10824.0	−0.00329940
$405$	0	0
$406$	0	0
$407$	2.41270e6	0.721966
$408$	0	0
$409$	5.86178e6	1.73269	0.866346	−	0.499444i	$-0.166462\pi$
$409$	5.86178e6	1.73269	0.866346	+	0.499444i	$0.166462\pi$
$410$	3.72622e6	1.09473
$411$	0	0
$412$	−527072.	−0.152977
$413$	0	0
$414$	0	0
$415$	3.13466e6	0.893451
$416$	−636480.	−0.180323
$417$	0	0
$418$	7.15018e6	2.00159
$419$	302748.	0.0842454	0.0421227	−	0.999112i	$-0.486588\pi$
$419$	302748.	0.0842454	0.0421227	+	0.999112i	$0.486588\pi$
$420$	0	0
$421$	−5.36708e6	−1.47582	−0.737909	−	0.674900i	$-0.764187\pi$
$421$	−5.36708e6	−1.47582	−0.737909	+	0.674900i	$0.764187\pi$
$422$	3.12146e6	0.853252
$423$	0	0
$424$	−1.61179e6	−0.435406
$425$	−372834.	−0.100125
$426$	0	0
$427$	0	0
$428$	515664.	0.136068
$429$	0	0
$430$	−5.39323e6	−1.40662
$431$	−1.17706e6	−0.305214	−0.152607	−	0.988287i	$-0.548767\pi$
$431$	−1.17706e6	−0.305214	−0.152607	+	0.988287i	$0.548767\pi$
$432$	0	0
$433$	3.66249e6	0.938766	0.469383	−	0.882995i	$-0.344476\pi$
$433$	3.66249e6	0.938766	0.469383	+	0.882995i	$0.344476\pi$
$434$	0	0
$435$	0	0
$436$	−403912.	−0.101758
$437$	1.12728e7	2.82376
$438$	0	0
$439$	2.53674e6	0.628225	0.314113	−	0.949386i	$-0.398293\pi$
$439$	2.53674e6	0.628225	0.314113	+	0.949386i	$0.398293\pi$
$440$	5.81818e6	1.43270
$441$	0	0
$442$	−334152.	−0.0813558
$443$	−6.01504e6	−1.45623	−0.728113	−	0.685457i	$-0.759603\pi$
$443$	−6.01504e6	−1.45623	−0.728113	+	0.685457i	$0.759603\pi$
$444$	0	0
$445$	−621972.	−0.148892
$446$	−1.82842e6	−0.435249
$447$	0	0
$448$	0	0
$449$	−5.65965e6	−1.32487	−0.662436	−	0.749119i	$-0.730477\pi$
$449$	−5.65965e6	−1.32487	−0.662436	+	0.749119i	$0.730477\pi$
$450$	0	0
$451$	−3.53513e6	−0.818397
$452$	−880584.	−0.202733
$453$	0	0
$454$	1.73153e6	0.394267
$455$	0	0
$456$	0	0
$457$	−6.46159e6	−1.44727	−0.723634	−	0.690184i	$-0.757530\pi$
$457$	−6.46159e6	−1.44727	−0.723634	+	0.690184i	$0.757530\pi$
$458$	−4.63314e6	−1.03208
$459$	0	0
$460$	−1.31040e6	−0.288742
$461$	−3.37353e6	−0.739320	−0.369660	−	0.929167i	$-0.620526\pi$
$461$	−3.37353e6	−0.739320	−0.369660	+	0.929167i	$0.620526\pi$
$462$	0	0
$463$	−4.54974e6	−0.986358	−0.493179	−	0.869928i	$-0.664165\pi$
$463$	−4.54974e6	−0.986358	−0.493179	+	0.869928i	$0.664165\pi$
$464$	−6.18211e6	−1.33304
$465$	0	0
$466$	−1.51340e6	−0.322842
$467$	2.01136e6	0.426773	0.213386	−	0.976968i	$-0.431551\pi$
$467$	2.01136e6	0.426773	0.213386	+	0.976968i	$0.431551\pi$
$468$	0	0
$469$	0	0
$470$	−6.51456e6	−1.36032
$471$	0	0
$472$	−4.61866e6	−0.954247
$473$	5.11666e6	1.05156
$474$	0	0
$475$	−7.94196e6	−1.61508
$476$	0	0
$477$	0	0
$478$	8.70682e6	1.74297
$479$	−7.60402e6	−1.51427	−0.757137	−	0.653257i	$-0.773402\pi$
$479$	−7.60402e6	−1.51427	−0.757137	+	0.653257i	$0.773402\pi$
$480$	0	0
$481$	−2.40183e6	−0.473347
$482$	878388.	0.172214
$483$	0	0
$484$	144340.	0.0280074
$485$	1.12056e7	2.16313
$486$	0	0
$487$	673112.	0.128607	0.0643035	−	0.997930i	$-0.479517\pi$
$487$	673112.	0.128607	0.0643035	+	0.997930i	$0.479517\pi$
$488$	8.31230e6	1.58005
$489$	0	0
$490$	0	0
$491$	2.47170e6	0.462692	0.231346	−	0.972872i	$-0.425687\pi$
$491$	2.47170e6	0.462692	0.231346	+	0.972872i	$0.425687\pi$
$492$	0	0
$493$	−685692.	−0.127061
$494$	−7.11797e6	−1.31232
$495$	0	0
$496$	90880.0	0.0165869
$497$	0	0
$498$	0	0
$499$	6.08152e6	1.09335	0.546677	−	0.837343i	$-0.315892\pi$
$499$	6.08152e6	1.09335	0.546677	+	0.837343i	$0.315892\pi$
$500$	−51792.0	−0.00926483
$501$	0	0
$502$	3.64716e6	0.645945
$503$	−846216.	−0.149129	−0.0745644	−	0.997216i	$-0.523757\pi$
$503$	−846216.	−0.149129	−0.0745644	+	0.997216i	$0.523757\pi$
$504$	0	0
$505$	−211068.	−0.0368293
$506$	1.11888e7	1.94271
$507$	0	0
$508$	−297280.	−0.0511101
$509$	−7.66785e6	−1.31183	−0.655917	−	0.754833i	$-0.727718\pi$
$509$	−7.66785e6	−1.31183	−0.655917	+	0.754833i	$0.727718\pi$
$510$	0	0
$511$	0	0
$512$	−4.47130e6	−0.753804
$513$	0	0
$514$	573516.	0.0957498
$515$	−1.02779e7	−1.70760
$516$	0	0
$517$	6.18048e6	1.01694
$518$	0	0
$519$	0	0
$520$	−5.79197e6	−0.939329
$521$	−9.68938e6	−1.56387	−0.781937	−	0.623357i	$-0.785768\pi$
$521$	−9.68938e6	−1.56387	−0.781937	+	0.623357i	$0.785768\pi$
$522$	0	0
$523$	7.51678e6	1.20165	0.600824	−	0.799381i	$-0.294839\pi$
$523$	7.51678e6	1.20165	0.600824	+	0.799381i	$0.294839\pi$
$524$	−621264.	−0.0988435
$525$	0	0
$526$	1.32021e7	2.08055
$527$	10080.0	0.00158101
$528$	0	0
$529$	1.12037e7	1.74069
$530$	4.48999e6	0.694314
$531$	0	0
$532$	0	0
$533$	3.51920e6	0.536570
$534$	0	0
$535$	1.00554e7	1.51886
$536$	9.97181e6	1.49921
$537$	0	0
$538$	1.06215e7	1.58208
$539$	0	0
$540$	0	0
$541$	7.34325e6	1.07869	0.539343	−	0.842086i	$-0.318673\pi$
$541$	7.34325e6	1.07869	0.539343	+	0.842086i	$0.318673\pi$
$542$	1.34102e6	0.196082
$543$	0	0
$544$	181440.	0.0262867
$545$	−7.87628e6	−1.13587
$546$	0	0
$547$	2.18296e6	0.311945	0.155973	−	0.987761i	$-0.450149\pi$
$547$	2.18296e6	0.311945	0.155973	+	0.987761i	$0.450149\pi$
$548$	1.05698e6	0.150355
$549$	0	0
$550$	−7.88278e6	−1.11115
$551$	−1.46063e7	−2.04957
$552$	0	0
$553$	0	0
$554$	−2.05667e6	−0.284702
$555$	0	0
$556$	−898448.	−0.123255
$557$	−1.25466e7	−1.71351	−0.856755	−	0.515724i	$-0.827523\pi$
$557$	−1.25466e7	−1.71351	−0.856755	+	0.515724i	$0.827523\pi$
$558$	0	0
$559$	−5.09361e6	−0.689439
$560$	0	0
$561$	0	0
$562$	−2.88227e6	−0.384940
$563$	5.15972e6	0.686050	0.343025	−	0.939326i	$-0.388549\pi$
$563$	5.15972e6	0.686050	0.343025	+	0.939326i	$0.388549\pi$
$564$	0	0
$565$	−1.71714e7	−2.26300
$566$	179880.	0.0236016
$567$	0	0
$568$	5.38272e6	0.700053
$569$	−1.17452e7	−1.52083	−0.760414	−	0.649439i	$-0.775004\pi$
$569$	−1.17452e7	−1.52083	−0.760414	+	0.649439i	$0.775004\pi$
$570$	0	0
$571$	−7.54728e6	−0.968725	−0.484362	−	0.874867i	$-0.660948\pi$
$571$	−7.54728e6	−0.968725	−0.484362	+	0.874867i	$0.660948\pi$
$572$	−784992.	−0.100317
$573$	0	0
$574$	0	0
$575$	−1.24278e7	−1.56756
$576$	0	0
$577$	−9.28483e6	−1.16101	−0.580503	−	0.814258i	$-0.697144\pi$
$577$	−9.28483e6	−1.16101	−0.580503	+	0.814258i	$0.697144\pi$
$578$	−8.42389e6	−1.04880
$579$	0	0
$580$	1.69790e6	0.209577
$581$	0	0
$582$	0	0
$583$	−4.25974e6	−0.519053
$584$	−1.03901e7	−1.26063
$585$	0	0
$586$	−1.18840e6	−0.142961
$587$	1.47623e6	0.176831	0.0884155	−	0.996084i	$-0.471820\pi$
$587$	1.47623e6	0.176831	0.0884155	+	0.996084i	$0.471820\pi$
$588$	0	0
$589$	214720.	0.0255026
$590$	1.28663e7	1.52168
$591$	0	0
$592$	6.17302e6	0.723925
$593$	−1.24007e7	−1.44813	−0.724067	−	0.689729i	$-0.757730\pi$
$593$	−1.24007e7	−1.44813	−0.724067	+	0.689729i	$0.757730\pi$
$594$	0	0
$595$	0	0
$596$	328296.	0.0378573
$597$	0	0
$598$	−1.11384e7	−1.27371
$599$	3.69127e6	0.420348	0.210174	−	0.977664i	$-0.432597\pi$
$599$	3.69127e6	0.420348	0.210174	+	0.977664i	$0.432597\pi$
$600$	0	0
$601$	−9.12223e6	−1.03018	−0.515092	−	0.857135i	$-0.672242\pi$
$601$	−9.12223e6	−1.03018	−0.515092	+	0.857135i	$0.672242\pi$
$602$	0	0
$603$	0	0
$604$	−1.14813e6	−0.128055
$605$	2.81463e6	0.312632
$606$	0	0
$607$	5.67914e6	0.625620	0.312810	−	0.949816i	$-0.398730\pi$
$607$	5.67914e6	0.625620	0.312810	+	0.949816i	$0.398730\pi$
$608$	3.86496e6	0.424020
$609$	0	0
$610$	−2.31557e7	−2.51961
$611$	−6.15264e6	−0.666743
$612$	0	0
$613$	−1.40106e7	−1.50593	−0.752966	−	0.658060i	$-0.771377\pi$
$613$	−1.40106e7	−1.50593	−0.752966	+	0.658060i	$0.771377\pi$
$614$	6.27382e6	0.671600
$615$	0	0
$616$	0	0
$617$	253686.	0.0268277	0.0134139	−	0.999910i	$-0.495730\pi$
$617$	253686.	0.0268277	0.0134139	+	0.999910i	$0.495730\pi$
$618$	0	0
$619$	−4.30034e6	−0.451103	−0.225552	−	0.974231i	$-0.572418\pi$
$619$	−4.30034e6	−0.451103	−0.225552	+	0.974231i	$0.572418\pi$
$620$	−24960.0	−0.00260775
$621$	0	0
$622$	1.10231e7	1.14242
$623$	0	0
$624$	0	0
$625$	−1.02568e7	−1.05030
$626$	2.19296e6	0.223664
$627$	0	0
$628$	−519512.	−0.0525650
$629$	684684.	0.0690023
$630$	0	0
$631$	1.04150e7	1.04132	0.520662	−	0.853763i	$-0.325685\pi$
$631$	1.04150e7	1.04132	0.520662	+	0.853763i	$0.325685\pi$
$632$	1.10504e7	1.10048
$633$	0	0
$634$	170028.	0.0167995
$635$	−5.79696e6	−0.570514
$636$	0	0
$637$	0	0
$638$	−1.44975e7	−1.41007
$639$	0	0
$640$	1.65635e7	1.59846
$641$	−4.52714e6	−0.435190	−0.217595	−	0.976039i	$-0.569821\pi$
$641$	−4.52714e6	−0.435190	−0.217595	+	0.976039i	$0.569821\pi$
$642$	0	0
$643$	−1.49687e7	−1.42776	−0.713882	−	0.700266i	$-0.753065\pi$
$643$	−1.49687e7	−1.42776	−0.713882	+	0.700266i	$0.753065\pi$
$644$	0	0
$645$	0	0
$646$	2.02910e6	0.191304
$647$	−1.73020e7	−1.62493	−0.812465	−	0.583010i	$-0.801875\pi$
$647$	−1.73020e7	−1.62493	−0.812465	+	0.583010i	$0.801875\pi$
$648$	0	0
$649$	−1.22064e7	−1.13757
$650$	7.84727e6	0.728509
$651$	0	0
$652$	2.22114e6	0.204624
$653$	−4.07470e6	−0.373949	−0.186975	−	0.982365i	$-0.559868\pi$
$653$	−4.07470e6	−0.373949	−0.186975	+	0.982365i	$0.559868\pi$
$654$	0	0
$655$	−1.21146e7	−1.10334
$656$	−9.04483e6	−0.820618
$657$	0	0
$658$	0	0
$659$	3.79475e6	0.340384	0.170192	−	0.985411i	$-0.445561\pi$
$659$	3.79475e6	0.340384	0.170192	+	0.985411i	$0.445561\pi$
$660$	0	0
$661$	−1.64261e7	−1.46228	−0.731142	−	0.682225i	$-0.761012\pi$
$661$	−1.64261e7	−1.46228	−0.731142	+	0.682225i	$0.761012\pi$
$662$	1.16035e7	1.02907
$663$	0	0
$664$	−6.75158e6	−0.594272
$665$	0	0
$666$	0	0
$667$	−2.28564e7	−1.98927
$668$	174048.	0.0150913
$669$	0	0
$670$	−2.77786e7	−2.39069
$671$	2.19682e7	1.88360
$672$	0	0
$673$	5.50675e6	0.468660	0.234330	−	0.972157i	$-0.424710\pi$
$673$	5.50675e6	0.468660	0.234330	+	0.972157i	$0.424710\pi$
$674$	−1.13290e7	−0.960602
$675$	0	0
$676$	−703716.	−0.0592285
$677$	1.83957e7	1.54257	0.771286	−	0.636488i	$-0.219614\pi$
$677$	1.83957e7	1.54257	0.771286	+	0.636488i	$0.219614\pi$
$678$	0	0
$679$	0	0
$680$	1.65110e6	0.136931
$681$	0	0
$682$	213120.	0.0175454
$683$	−1.75835e6	−0.144229	−0.0721146	−	0.997396i	$-0.522975\pi$
$683$	−1.75835e6	−0.144229	−0.0721146	+	0.997396i	$0.522975\pi$
$684$	0	0
$685$	2.06112e7	1.67833
$686$	0	0
$687$	0	0
$688$	1.30913e7	1.05441
$689$	4.24055e6	0.340309
$690$	0	0
$691$	5.36314e6	0.427291	0.213646	−	0.976911i	$-0.431466\pi$
$691$	5.36314e6	0.427291	0.213646	+	0.976911i	$0.431466\pi$
$692$	−73320.0	−0.00582046
$693$	0	0
$694$	−1.75162e7	−1.38052
$695$	−1.75197e7	−1.37583
$696$	0	0
$697$	−1.00321e6	−0.0782187
$698$	4.68409e6	0.363904
$699$	0	0
$700$	0	0
$701$	2.12606e7	1.63411	0.817054	−	0.576561i	$-0.195606\pi$
$701$	2.12606e7	1.63411	0.817054	+	0.576561i	$0.195606\pi$
$702$	0	0
$703$	1.45849e7	1.11305
$704$	−1.23041e7	−0.935662
$705$	0	0
$706$	8.00622e6	0.604527
$707$	0	0
$708$	0	0
$709$	2.07729e6	0.155196	0.0775980	−	0.996985i	$-0.475275\pi$
$709$	2.07729e6	0.155196	0.0775980	+	0.996985i	$0.475275\pi$
$710$	−1.49947e7	−1.11633
$711$	0	0
$712$	1.33963e6	0.0990343
$713$	336000.	0.0247523
$714$	0	0
$715$	−1.53073e7	−1.11979
$716$	613296.	0.0447082
$717$	0	0
$718$	−6.10459e6	−0.441922
$719$	4.23619e6	0.305600	0.152800	−	0.988257i	$-0.451171\pi$
$719$	4.23619e6	0.305600	0.152800	+	0.988257i	$0.451171\pi$
$720$	0	0
$721$	0	0
$722$	2.83665e7	2.02518
$723$	0	0
$724$	1.52826e6	0.108356
$725$	1.61029e7	1.13778
$726$	0	0
$727$	−2.14524e7	−1.50536	−0.752678	−	0.658389i	$-0.771238\pi$
$727$	−2.14524e7	−1.50536	−0.752678	+	0.658389i	$0.771238\pi$
$728$	0	0
$729$	0	0
$730$	2.89439e7	2.01025
$731$	1.45202e6	0.100503
$732$	0	0
$733$	1.48892e7	1.02355	0.511777	−	0.859118i	$-0.328987\pi$
$733$	1.48892e7	1.02355	0.511777	+	0.859118i	$0.328987\pi$
$734$	−5.02608e6	−0.344341
$735$	0	0
$736$	6.04800e6	0.411545
$737$	2.63541e7	1.78722
$738$	0	0
$739$	6.99324e6	0.471050	0.235525	−	0.971868i	$-0.424319\pi$
$739$	6.99324e6	0.471050	0.235525	+	0.971868i	$0.424319\pi$
$740$	−1.69541e6	−0.113814
$741$	0	0
$742$	0	0
$743$	−1.90428e6	−0.126549	−0.0632745	−	0.997996i	$-0.520154\pi$
$743$	−1.90428e6	−0.126549	−0.0632745	+	0.997996i	$0.520154\pi$
$744$	0	0
$745$	6.40177e6	0.422581
$746$	−9.11958e6	−0.599968
$747$	0	0
$748$	223776.	0.0146238
$749$	0	0
$750$	0	0
$751$	1.95361e7	1.26398	0.631988	−	0.774978i	$-0.282239\pi$
$751$	1.95361e7	1.26398	0.631988	+	0.774978i	$0.282239\pi$
$752$	1.58131e7	1.01970
$753$	0	0
$754$	1.44322e7	0.924493
$755$	−2.23885e7	−1.42941
$756$	0	0
$757$	1.25183e6	0.0793973	0.0396986	−	0.999212i	$-0.487360\pi$
$757$	1.25183e6	0.0793973	0.0396986	+	0.999212i	$0.487360\pi$
$758$	1.58679e7	1.00311
$759$	0	0
$760$	3.51711e7	2.20878
$761$	2.04472e7	1.27989	0.639944	−	0.768422i	$-0.278958\pi$
$761$	2.04472e7	1.27989	0.639944	+	0.768422i	$0.278958\pi$
$762$	0	0
$763$	0	0
$764$	1.09363e6	0.0677857
$765$	0	0
$766$	1.20802e7	0.743876
$767$	1.21515e7	0.745831
$768$	0	0
$769$	−2.21064e6	−0.134804	−0.0674020	−	0.997726i	$-0.521471\pi$
$769$	−2.21064e6	−0.134804	−0.0674020	+	0.997726i	$0.521471\pi$
$770$	0	0
$771$	0	0
$772$	614408.	0.0371034
$773$	1.29151e7	0.777405	0.388703	−	0.921363i	$-0.372923\pi$
$773$	1.29151e7	0.777405	0.388703	+	0.921363i	$0.372923\pi$
$774$	0	0
$775$	−236720.	−0.0141573
$776$	−2.41352e7	−1.43879
$777$	0	0
$778$	4.35740e6	0.258095
$779$	−2.13700e7	−1.26171
$780$	0	0
$781$	1.42258e7	0.834541
$782$	3.17520e6	0.185675
$783$	0	0
$784$	0	0
$785$	−1.01305e7	−0.586754
$786$	0	0
$787$	1.35499e7	0.779830	0.389915	−	0.920851i	$-0.372504\pi$
$787$	1.35499e7	0.779830	0.389915	+	0.920851i	$0.372504\pi$
$788$	−617688.	−0.0354367
$789$	0	0
$790$	−3.07832e7	−1.75487
$791$	0	0
$792$	0	0
$793$	−2.18693e7	−1.23496
$794$	−2.74547e7	−1.54549
$795$	0	0
$796$	1.46742e6	0.0820867
$797$	−2.45956e7	−1.37155	−0.685776	−	0.727813i	$-0.740537\pi$
$797$	−2.45956e7	−1.37155	−0.685776	+	0.727813i	$0.740537\pi$
$798$	0	0
$799$	1.75392e6	0.0971948
$800$	−4.26096e6	−0.235387
$801$	0	0
$802$	203220.	0.0111566
$803$	−2.74596e7	−1.50282
$804$	0	0
$805$	0	0
$806$	−212160.	−0.0115034
$807$	0	0
$808$	454608.	0.0244968
$809$	−1.55237e7	−0.833920	−0.416960	−	0.908925i	$-0.636905\pi$
$809$	−1.55237e7	−0.833920	−0.416960	+	0.908925i	$0.636905\pi$
$810$	0	0
$811$	2.66262e7	1.42153	0.710766	−	0.703429i	$-0.248349\pi$
$811$	2.66262e7	1.42153	0.710766	+	0.703429i	$0.248349\pi$
$812$	0	0
$813$	0	0
$814$	1.44762e7	0.765760
$815$	4.33122e7	2.28410
$816$	0	0
$817$	3.09304e7	1.62118
$818$	3.51707e7	1.83780
$819$	0	0
$820$	2.48414e6	0.129016
$821$	1.23891e7	0.641477	0.320739	−	0.947168i	$-0.396069\pi$
$821$	1.23891e7	0.641477	0.320739	+	0.947168i	$0.396069\pi$
$822$	0	0
$823$	−3.65630e6	−0.188166	−0.0940831	−	0.995564i	$-0.529992\pi$
$823$	−3.65630e6	−0.188166	−0.0940831	+	0.995564i	$0.529992\pi$
$824$	2.21370e7	1.13580
$825$	0	0
$826$	0	0
$827$	−2.80463e7	−1.42597	−0.712987	−	0.701178i	$-0.752658\pi$
$827$	−2.80463e7	−1.42597	−0.712987	+	0.701178i	$0.752658\pi$
$828$	0	0
$829$	−2.11153e7	−1.06712	−0.533558	−	0.845763i	$-0.679145\pi$
$829$	−2.11153e7	−1.06712	−0.533558	+	0.845763i	$0.679145\pi$
$830$	1.88080e7	0.947648
$831$	0	0
$832$	1.22487e7	0.613454
$833$	0	0
$834$	0	0
$835$	3.39394e6	0.168456
$836$	4.76678e6	0.235890
$837$	0	0
$838$	1.81649e6	0.0893557
$839$	1.33947e7	0.656944	0.328472	−	0.944514i	$-0.393466\pi$
$839$	1.33947e7	0.656944	0.328472	+	0.944514i	$0.393466\pi$
$840$	0	0
$841$	9.10422e6	0.443867
$842$	−3.22025e7	−1.56534
$843$	0	0
$844$	2.08098e6	0.100557
$845$	−1.37225e7	−0.661135
$846$	0	0
$847$	0	0
$848$	−1.08988e7	−0.520461
$849$	0	0
$850$	−2.23700e6	−0.106199
$851$	2.28228e7	1.08030
$852$	0	0
$853$	−3.01513e7	−1.41884	−0.709420	−	0.704786i	$-0.751043\pi$
$853$	−3.01513e7	−1.41884	−0.709420	+	0.704786i	$0.751043\pi$
$854$	0	0
$855$	0	0
$856$	−2.16579e7	−1.01026
$857$	2.39894e7	1.11575	0.557875	−	0.829925i	$-0.311617\pi$
$857$	2.39894e7	1.11575	0.557875	+	0.829925i	$0.311617\pi$
$858$	0	0
$859$	8.87576e6	0.410414	0.205207	−	0.978719i	$-0.434213\pi$
$859$	8.87576e6	0.410414	0.205207	+	0.978719i	$0.434213\pi$
$860$	−3.59549e6	−0.165772
$861$	0	0
$862$	−7.06234e6	−0.323728
$863$	8.71286e6	0.398230	0.199115	−	0.979976i	$-0.436193\pi$
$863$	8.71286e6	0.398230	0.199115	+	0.979976i	$0.436193\pi$
$864$	0	0
$865$	−1.42974e6	−0.0649706
$866$	2.19750e7	0.995711
$867$	0	0
$868$	0	0
$869$	2.92045e7	1.31190
$870$	0	0
$871$	−2.62354e7	−1.17177
$872$	1.69643e7	0.755518
$873$	0	0
$874$	6.76368e7	2.99505
$875$	0	0
$876$	0	0
$877$	−2.95788e7	−1.29862	−0.649310	−	0.760524i	$-0.724942\pi$
$877$	−2.95788e7	−1.29862	−0.649310	+	0.760524i	$0.724942\pi$
$878$	1.52205e7	0.666333
$879$	0	0
$880$	3.93420e7	1.71257
$881$	2.45670e7	1.06638	0.533190	−	0.845995i	$-0.320993\pi$
$881$	2.45670e7	1.06638	0.533190	+	0.845995i	$0.320993\pi$
$882$	0	0
$883$	1.45682e7	0.628788	0.314394	−	0.949293i	$-0.398199\pi$
$883$	1.45682e7	0.628788	0.314394	+	0.949293i	$0.398199\pi$
$884$	−222768.	−0.00958787
$885$	0	0
$886$	−3.60902e7	−1.54456
$887$	1.61714e7	0.690141	0.345070	−	0.938577i	$-0.387855\pi$
$887$	1.61714e7	0.690141	0.345070	+	0.938577i	$0.387855\pi$
$888$	0	0
$889$	0	0
$890$	−3.73183e6	−0.157924
$891$	0	0
$892$	−1.21894e6	−0.0512946
$893$	3.73613e7	1.56781
$894$	0	0
$895$	1.19593e7	0.499054
$896$	0	0
$897$	0	0
$898$	−3.39579e7	−1.40524
$899$	−435360.	−0.0179659
$900$	0	0
$901$	−1.20884e6	−0.0496087
$902$	−2.12108e7	−0.868041
$903$	0	0
$904$	3.69845e7	1.50522
$905$	2.98011e7	1.20952
$906$	0	0
$907$	3.14446e7	1.26919	0.634596	−	0.772844i	$-0.281167\pi$
$907$	3.14446e7	1.26919	0.634596	+	0.772844i	$0.281167\pi$
$908$	1.15435e6	0.0464648
$909$	0	0
$910$	0	0
$911$	−1.51427e7	−0.604514	−0.302257	−	0.953227i	$-0.597740\pi$
$911$	−1.51427e7	−0.604514	−0.302257	+	0.953227i	$0.597740\pi$
$912$	0	0
$913$	−1.78435e7	−0.708439
$914$	−3.87695e7	−1.53506
$915$	0	0
$916$	−3.08876e6	−0.121631
$917$	0	0
$918$	0	0
$919$	4.14876e7	1.62043	0.810214	−	0.586134i	$-0.199351\pi$
$919$	4.14876e7	1.62043	0.810214	+	0.586134i	$0.199351\pi$
$920$	5.50368e7	2.14380
$921$	0	0
$922$	−2.02412e7	−0.784167
$923$	−1.41617e7	−0.547155
$924$	0	0
$925$	−1.60792e7	−0.617889
$926$	−2.72985e7	−1.04619
$927$	0	0
$928$	−7.83648e6	−0.298711
$929$	−1.78495e7	−0.678556	−0.339278	−	0.940686i	$-0.610183\pi$
$929$	−1.78495e7	−0.678556	−0.339278	+	0.940686i	$0.610183\pi$
$930$	0	0
$931$	0	0
$932$	−1.00894e6	−0.0380473
$933$	0	0
$934$	1.20681e7	0.452661
$935$	4.36363e6	0.163237
$936$	0	0
$937$	−2.96399e7	−1.10288	−0.551439	−	0.834215i	$-0.685921\pi$
$937$	−2.96399e7	−1.10288	−0.551439	+	0.834215i	$0.685921\pi$
$938$	0	0
$939$	0	0
$940$	−4.34304e6	−0.160315
$941$	−3.22282e7	−1.18648	−0.593242	−	0.805024i	$-0.702152\pi$
$941$	−3.22282e7	−1.18648	−0.593242	+	0.805024i	$0.702152\pi$
$942$	0	0
$943$	−3.34404e7	−1.22459
$944$	−3.12309e7	−1.14066
$945$	0	0
$946$	3.06999e7	1.11535
$947$	−4.84885e7	−1.75697	−0.878484	−	0.477772i	$-0.841444\pi$
$947$	−4.84885e7	−1.75697	−0.878484	+	0.477772i	$0.841444\pi$
$948$	0	0
$949$	2.73359e7	0.985300
$950$	−4.76517e7	−1.71305
$951$	0	0
$952$	0	0
$953$	2.03264e7	0.724983	0.362491	−	0.931987i	$-0.381926\pi$
$953$	2.03264e7	0.724983	0.362491	+	0.931987i	$0.381926\pi$
$954$	0	0
$955$	2.13258e7	0.756654
$956$	5.80454e6	0.205411
$957$	0	0
$958$	−4.56241e7	−1.60613
$959$	0	0
$960$	0	0
$961$	−2.86228e7	−0.999776
$962$	−1.44110e7	−0.502060
$963$	0	0
$964$	585592.	0.0202956
$965$	1.19810e7	0.414165
$966$	0	0
$967$	−3.66292e6	−0.125968	−0.0629841	−	0.998015i	$-0.520062\pi$
$967$	−3.66292e6	−0.125968	−0.0629841	+	0.998015i	$0.520062\pi$
$968$	−6.06228e6	−0.207945
$969$	0	0
$970$	6.72338e7	2.29434
$971$	1.48741e6	0.0506271	0.0253136	−	0.999680i	$-0.491942\pi$
$971$	1.48741e6	0.0506271	0.0253136	+	0.999680i	$0.491942\pi$
$972$	0	0
$973$	0	0
$974$	4.03867e6	0.136408
$975$	0	0
$976$	5.62070e7	1.88871
$977$	−4.07930e7	−1.36725	−0.683627	−	0.729831i	$-0.739599\pi$
$977$	−4.07930e7	−1.36725	−0.683627	+	0.729831i	$0.739599\pi$
$978$	0	0
$979$	3.54046e6	0.118060
$980$	0	0
$981$	0	0
$982$	1.48302e7	0.490759
$983$	−9.26326e6	−0.305759	−0.152880	−	0.988245i	$-0.548855\pi$
$983$	−9.26326e6	−0.305759	−0.152880	+	0.988245i	$0.548855\pi$
$984$	0	0
$985$	−1.20449e7	−0.395561
$986$	−4.11415e6	−0.134768
$987$	0	0
$988$	−4.74531e6	−0.154658
$989$	4.84008e7	1.57348
$990$	0	0
$991$	−5.22051e7	−1.68861	−0.844303	−	0.535866i	$-0.819985\pi$
$991$	−5.22051e7	−1.68861	−0.844303	+	0.535866i	$0.819985\pi$
$992$	115200.	0.00371684
$993$	0	0
$994$	0	0
$995$	2.86148e7	0.916289
$996$	0	0
$997$	1.86609e7	0.594560	0.297280	−	0.954790i	$-0.403921\pi$
$997$	1.86609e7	0.594560	0.297280	+	0.954790i	$0.403921\pi$
$998$	3.64891e7	1.15968
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.j.1.1		1
3.2	odd	2		147.6.a.b.1.1		1
7.6	odd	2		63.6.a.d.1.1		1
21.2	odd	6		147.6.e.i.67.1		2
21.5	even	6		147.6.e.j.67.1		2
21.11	odd	6		147.6.e.i.79.1		2
21.17	even	6		147.6.e.j.79.1		2
21.20	even	2		21.6.a.a.1.1	✓	1
28.27	even	2		1008.6.a.c.1.1		1
84.83	odd	2		336.6.a.r.1.1		1
105.62	odd	4		525.6.d.b.274.1		2
105.83	odd	4		525.6.d.b.274.2		2
105.104	even	2		525.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.a.1.1	✓	1	21.20	even	2
63.6.a.d.1.1		1	7.6	odd	2
147.6.a.b.1.1		1	3.2	odd	2
147.6.e.i.67.1		2	21.2	odd	6
147.6.e.i.79.1		2	21.11	odd	6
147.6.e.j.67.1		2	21.5	even	6
147.6.e.j.79.1		2	21.17	even	6
336.6.a.r.1.1		1	84.83	odd	2
441.6.a.j.1.1		1	1.1	even	1	trivial
525.6.a.d.1.1		1	105.104	even	2
525.6.d.b.274.1		2	105.62	odd	4
525.6.d.b.274.2		2	105.83	odd	4
1008.6.a.c.1.1		1	28.27	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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