LMFDB - Embedded newform 2205.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -7.00000 q^{4} +5.00000 q^{5} +15.0000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -7.00000 q^{4} +5.00000 q^{5} +15.0000 q^{8} -5.00000 q^{10} -10.0000 q^{11} +6.00000 q^{13} +41.0000 q^{16} -84.0000 q^{17} +48.0000 q^{19} -35.0000 q^{20} +10.0000 q^{22} -56.0000 q^{23} +25.0000 q^{25} -6.00000 q^{26} +232.000 q^{29} -6.00000 q^{31} -161.000 q^{32} +84.0000 q^{34} -48.0000 q^{37} -48.0000 q^{38} +75.0000 q^{40} +150.000 q^{41} -426.000 q^{43} +70.0000 q^{44} +56.0000 q^{46} +18.0000 q^{47} -25.0000 q^{50} -42.0000 q^{52} +58.0000 q^{53} -50.0000 q^{55} -232.000 q^{58} -348.000 q^{59} +882.000 q^{61} +6.00000 q^{62} -167.000 q^{64} +30.0000 q^{65} -182.000 q^{67} +588.000 q^{68} +524.000 q^{71} -690.000 q^{73} +48.0000 q^{74} -336.000 q^{76} -1024.00 q^{79} +205.000 q^{80} -150.000 q^{82} -384.000 q^{83} -420.000 q^{85} +426.000 q^{86} -150.000 q^{88} +246.000 q^{89} +392.000 q^{92} -18.0000 q^{94} +240.000 q^{95} +1122.00 q^{97} +O(q^{100})$

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 $

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	15.0000	0.662913
$9$	0	0
$10$	−5.00000	−0.158114
$11$	−10.0000	−0.274101	−0.137051	−	0.990564i	$-0.543762\pi$
$11$	−10.0000	−0.274101	−0.137051	+	0.990564i	$0.543762\pi$
$12$	0	0
$13$	6.00000	0.128008	0.0640039	−	0.997950i	$-0.479613\pi$
$13$	6.00000	0.128008	0.0640039	+	0.997950i	$0.479613\pi$
$14$	0	0
$15$	0	0
$16$	41.0000	0.640625
$17$	−84.0000	−1.19841	−0.599206	−	0.800595i	$-0.704517\pi$
$17$	−84.0000	−1.19841	−0.599206	+	0.800595i	$0.704517\pi$
$18$	0	0
$19$	48.0000	0.579577	0.289788	−	0.957091i	$-0.406415\pi$
$19$	48.0000	0.579577	0.289788	+	0.957091i	$0.406415\pi$
$20$	−35.0000	−0.391312
$21$	0	0
$22$	10.0000	0.0969094
$23$	−56.0000	−0.507687	−0.253844	−	0.967245i	$-0.581695\pi$
$23$	−56.0000	−0.507687	−0.253844	+	0.967245i	$0.581695\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−6.00000	−0.0452576
$27$	0	0
$28$	0	0
$29$	232.000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	232.000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−6.00000	−0.0347623	−0.0173812	−	0.999849i	$-0.505533\pi$
$31$	−6.00000	−0.0347623	−0.0173812	+	0.999849i	$0.505533\pi$
$32$	−161.000	−0.889408
$33$	0	0
$34$	84.0000	0.423702
$35$	0	0
$36$	0	0
$37$	−48.0000	−0.213274	−0.106637	−	0.994298i	$-0.534008\pi$
$37$	−48.0000	−0.213274	−0.106637	+	0.994298i	$0.534008\pi$
$38$	−48.0000	−0.204911
$39$	0	0
$40$	75.0000	0.296464
$41$	150.000	0.571367	0.285684	−	0.958324i	$-0.407779\pi$
$41$	150.000	0.571367	0.285684	+	0.958324i	$0.407779\pi$
$42$	0	0
$43$	−426.000	−1.51080	−0.755400	−	0.655264i	$-0.772557\pi$
$43$	−426.000	−1.51080	−0.755400	+	0.655264i	$0.772557\pi$
$44$	70.0000	0.239839
$45$	0	0
$46$	56.0000	0.179495
$47$	18.0000	0.0558632	0.0279316	−	0.999610i	$-0.491108\pi$
$47$	18.0000	0.0558632	0.0279316	+	0.999610i	$0.491108\pi$
$48$	0	0
$49$	0	0
$50$	−25.0000	−0.0707107
$51$	0	0
$52$	−42.0000	−0.112007
$53$	58.0000	0.150319	0.0751596	−	0.997172i	$-0.476053\pi$
$53$	58.0000	0.150319	0.0751596	+	0.997172i	$0.476053\pi$
$54$	0	0
$55$	−50.0000	−0.122582
$56$	0	0
$57$	0	0
$58$	−232.000	−0.525226
$59$	−348.000	−0.767894	−0.383947	−	0.923355i	$-0.625435\pi$
$59$	−348.000	−0.767894	−0.383947	+	0.923355i	$0.625435\pi$
$60$	0	0
$61$	882.000	1.85129	0.925644	−	0.378396i	$-0.123524\pi$
$61$	882.000	1.85129	0.925644	+	0.378396i	$0.123524\pi$
$62$	6.00000	0.0122903
$63$	0	0
$64$	−167.000	−0.326172
$65$	30.0000	0.0572468
$66$	0	0
$67$	−182.000	−0.331863	−0.165932	−	0.986137i	$-0.553063\pi$
$67$	−182.000	−0.331863	−0.165932	+	0.986137i	$0.553063\pi$
$68$	588.000	1.04861
$69$	0	0
$70$	0	0
$71$	524.000	0.875878	0.437939	−	0.899005i	$-0.355709\pi$
$71$	524.000	0.875878	0.437939	+	0.899005i	$0.355709\pi$
$72$	0	0
$73$	−690.000	−1.10628	−0.553140	−	0.833089i	$-0.686570\pi$
$73$	−690.000	−1.10628	−0.553140	+	0.833089i	$0.686570\pi$
$74$	48.0000	0.0754039
$75$	0	0
$76$	−336.000	−0.507130
$77$	0	0
$78$	0	0
$79$	−1024.00	−1.45834	−0.729171	−	0.684332i	$-0.760094\pi$
$79$	−1024.00	−1.45834	−0.729171	+	0.684332i	$0.760094\pi$
$80$	205.000	0.286496
$81$	0	0
$82$	−150.000	−0.202009
$83$	−384.000	−0.507825	−0.253913	−	0.967227i	$-0.581718\pi$
$83$	−384.000	−0.507825	−0.253913	+	0.967227i	$0.581718\pi$
$84$	0	0
$85$	−420.000	−0.535946
$86$	426.000	0.534148
$87$	0	0
$88$	−150.000	−0.181705
$89$	246.000	0.292988	0.146494	−	0.989212i	$-0.453201\pi$
$89$	246.000	0.292988	0.146494	+	0.989212i	$0.453201\pi$
$90$	0	0
$91$	0	0
$92$	392.000	0.444226
$93$	0	0
$94$	−18.0000	−0.0197506
$95$	240.000	0.259195
$96$	0	0
$97$	1122.00	1.17445	0.587226	−	0.809423i	$-0.300220\pi$
$97$	1122.00	1.17445	0.587226	+	0.809423i	$0.300220\pi$
$98$	0	0
$99$	0	0
$100$	−175.000	−0.175000
$101$	1386.00	1.36547	0.682733	−	0.730668i	$-0.260791\pi$
$101$	1386.00	1.36547	0.682733	+	0.730668i	$0.260791\pi$
$102$	0	0
$103$	−108.000	−0.103316	−0.0516580	−	0.998665i	$-0.516451\pi$
$103$	−108.000	−0.103316	−0.0516580	+	0.998665i	$0.516451\pi$
$104$	90.0000	0.0848579
$105$	0	0
$106$	−58.0000	−0.0531458
$107$	860.000	0.777003	0.388502	−	0.921448i	$-0.372993\pi$
$107$	860.000	0.777003	0.388502	+	0.921448i	$0.372993\pi$
$108$	0	0
$109$	822.000	0.722324	0.361162	−	0.932503i	$-0.382380\pi$
$109$	822.000	0.722324	0.361162	+	0.932503i	$0.382380\pi$
$110$	50.0000	0.0433392
$111$	0	0
$112$	0	0
$113$	1870.00	1.55677	0.778384	−	0.627788i	$-0.216040\pi$
$113$	1870.00	1.55677	0.778384	+	0.627788i	$0.216040\pi$
$114$	0	0
$115$	−280.000	−0.227045
$116$	−1624.00	−1.29987
$117$	0	0
$118$	348.000	0.271491
$119$	0	0
$120$	0	0
$121$	−1231.00	−0.924869
$122$	−882.000	−0.654529
$123$	0	0
$124$	42.0000	0.0304170
$125$	125.000	0.0894427
$126$	0	0
$127$	−1676.00	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$127$	−1676.00	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$128$	1455.00	1.00473
$129$	0	0
$130$	−30.0000	−0.0202398
$131$	732.000	0.488207	0.244104	−	0.969749i	$-0.421506\pi$
$131$	732.000	0.488207	0.244104	+	0.969749i	$0.421506\pi$
$132$	0	0
$133$	0	0
$134$	182.000	0.117331
$135$	0	0
$136$	−1260.00	−0.794442
$137$	−1454.00	−0.906742	−0.453371	−	0.891322i	$-0.649779\pi$
$137$	−1454.00	−0.906742	−0.453371	+	0.891322i	$0.649779\pi$
$138$	0	0
$139$	−972.000	−0.593122	−0.296561	−	0.955014i	$-0.595840\pi$
$139$	−972.000	−0.593122	−0.296561	+	0.955014i	$0.595840\pi$
$140$	0	0
$141$	0	0
$142$	−524.000	−0.309670
$143$	−60.0000	−0.0350871
$144$	0	0
$145$	1160.00	0.664364
$146$	690.000	0.391129
$147$	0	0
$148$	336.000	0.186615
$149$	−280.000	−0.153950	−0.0769748	−	0.997033i	$-0.524526\pi$
$149$	−280.000	−0.153950	−0.0769748	+	0.997033i	$0.524526\pi$
$150$	0	0
$151$	1200.00	0.646719	0.323360	−	0.946276i	$-0.395188\pi$
$151$	1200.00	0.646719	0.323360	+	0.946276i	$0.395188\pi$
$152$	720.000	0.384209
$153$	0	0
$154$	0	0
$155$	−30.0000	−0.0155462
$156$	0	0
$157$	−1134.00	−0.576453	−0.288226	−	0.957562i	$-0.593066\pi$
$157$	−1134.00	−0.576453	−0.288226	+	0.957562i	$0.593066\pi$
$158$	1024.00	0.515602
$159$	0	0
$160$	−805.000	−0.397755
$161$	0	0
$162$	0	0
$163$	2734.00	1.31376	0.656882	−	0.753994i	$-0.271875\pi$
$163$	2734.00	1.31376	0.656882	+	0.753994i	$0.271875\pi$
$164$	−1050.00	−0.499946
$165$	0	0
$166$	384.000	0.179543
$167$	−3174.00	−1.47073	−0.735364	−	0.677673i	$-0.762989\pi$
$167$	−3174.00	−1.47073	−0.735364	+	0.677673i	$0.762989\pi$
$168$	0	0
$169$	−2161.00	−0.983614
$170$	420.000	0.189485
$171$	0	0
$172$	2982.00	1.32195
$173$	1986.00	0.872791	0.436395	−	0.899755i	$-0.356255\pi$
$173$	1986.00	0.872791	0.436395	+	0.899755i	$0.356255\pi$
$174$	0	0
$175$	0	0
$176$	−410.000	−0.175596
$177$	0	0
$178$	−246.000	−0.103587
$179$	−22.0000	−0.00918635	−0.00459318	−	0.999989i	$-0.501462\pi$
$179$	−22.0000	−0.00918635	−0.00459318	+	0.999989i	$0.501462\pi$
$180$	0	0
$181$	−3378.00	−1.38721	−0.693604	−	0.720356i	$-0.743978\pi$
$181$	−3378.00	−1.38721	−0.693604	+	0.720356i	$0.743978\pi$
$182$	0	0
$183$	0	0
$184$	−840.000	−0.336552
$185$	−240.000	−0.0953792
$186$	0	0
$187$	840.000	0.328486
$188$	−126.000	−0.0488803
$189$	0	0
$190$	−240.000	−0.0916391
$191$	−1472.00	−0.557645	−0.278822	−	0.960343i	$-0.589944\pi$
$191$	−1472.00	−0.557645	−0.278822	+	0.960343i	$0.589944\pi$
$192$	0	0
$193$	−1410.00	−0.525876	−0.262938	−	0.964813i	$-0.584691\pi$
$193$	−1410.00	−0.525876	−0.262938	+	0.964813i	$0.584691\pi$
$194$	−1122.00	−0.415231
$195$	0	0
$196$	0	0
$197$	−1882.00	−0.680644	−0.340322	−	0.940309i	$-0.610536\pi$
$197$	−1882.00	−0.680644	−0.340322	+	0.940309i	$0.610536\pi$
$198$	0	0
$199$	894.000	0.318462	0.159231	−	0.987241i	$-0.449099\pi$
$199$	894.000	0.318462	0.159231	+	0.987241i	$0.449099\pi$
$200$	375.000	0.132583
$201$	0	0
$202$	−1386.00	−0.482765
$203$	0	0
$204$	0	0
$205$	750.000	0.255523
$206$	108.000	0.0365278
$207$	0	0
$208$	246.000	0.0820050
$209$	−480.000	−0.158863
$210$	0	0
$211$	2196.00	0.716488	0.358244	−	0.933628i	$-0.383376\pi$
$211$	2196.00	0.716488	0.358244	+	0.933628i	$0.383376\pi$
$212$	−406.000	−0.131529
$213$	0	0
$214$	−860.000	−0.274712
$215$	−2130.00	−0.675650
$216$	0	0
$217$	0	0
$218$	−822.000	−0.255380
$219$	0	0
$220$	350.000	0.107259
$221$	−504.000	−0.153406
$222$	0	0
$223$	−4800.00	−1.44140	−0.720699	−	0.693248i	$-0.756179\pi$
$223$	−4800.00	−1.44140	−0.720699	+	0.693248i	$0.756179\pi$
$224$	0	0
$225$	0	0
$226$	−1870.00	−0.550401
$227$	228.000	0.0666647	0.0333324	−	0.999444i	$-0.489388\pi$
$227$	228.000	0.0666647	0.0333324	+	0.999444i	$0.489388\pi$
$228$	0	0
$229$	4830.00	1.39378	0.696889	−	0.717179i	$-0.254567\pi$
$229$	4830.00	1.39378	0.696889	+	0.717179i	$0.254567\pi$
$230$	280.000	0.0802724
$231$	0	0
$232$	3480.00	0.984798
$233$	2078.00	0.584267	0.292134	−	0.956377i	$-0.405635\pi$
$233$	2078.00	0.584267	0.292134	+	0.956377i	$0.405635\pi$
$234$	0	0
$235$	90.0000	0.0249828
$236$	2436.00	0.671907
$237$	0	0
$238$	0	0
$239$	−472.000	−0.127745	−0.0638727	−	0.997958i	$-0.520345\pi$
$239$	−472.000	−0.127745	−0.0638727	+	0.997958i	$0.520345\pi$
$240$	0	0
$241$	−6636.00	−1.77370	−0.886851	−	0.462055i	$-0.847112\pi$
$241$	−6636.00	−1.77370	−0.886851	+	0.462055i	$0.847112\pi$
$242$	1231.00	0.326990
$243$	0	0
$244$	−6174.00	−1.61988
$245$	0	0
$246$	0	0
$247$	288.000	0.0741903
$248$	−90.0000	−0.0230444
$249$	0	0
$250$	−125.000	−0.0316228
$251$	−4404.00	−1.10748	−0.553741	−	0.832689i	$-0.686800\pi$
$251$	−4404.00	−1.10748	−0.553741	+	0.832689i	$0.686800\pi$
$252$	0	0
$253$	560.000	0.139158
$254$	1676.00	0.414022
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−180.000	−0.0436891	−0.0218445	−	0.999761i	$-0.506954\pi$
$257$	−180.000	−0.0436891	−0.0218445	+	0.999761i	$0.506954\pi$
$258$	0	0
$259$	0	0
$260$	−210.000	−0.0500910
$261$	0	0
$262$	−732.000	−0.172607
$263$	6400.00	1.50054	0.750268	−	0.661134i	$-0.229925\pi$
$263$	6400.00	1.50054	0.750268	+	0.661134i	$0.229925\pi$
$264$	0	0
$265$	290.000	0.0672247
$266$	0	0
$267$	0	0
$268$	1274.00	0.290380
$269$	−7410.00	−1.67954	−0.839769	−	0.542944i	$-0.817310\pi$
$269$	−7410.00	−1.67954	−0.839769	+	0.542944i	$0.817310\pi$
$270$	0	0
$271$	−5370.00	−1.20371	−0.601853	−	0.798607i	$-0.705571\pi$
$271$	−5370.00	−1.20371	−0.601853	+	0.798607i	$0.705571\pi$
$272$	−3444.00	−0.767732
$273$	0	0
$274$	1454.00	0.320582
$275$	−250.000	−0.0548202
$276$	0	0
$277$	−7396.00	−1.60427	−0.802135	−	0.597143i	$-0.796302\pi$
$277$	−7396.00	−1.60427	−0.802135	+	0.597143i	$0.796302\pi$
$278$	972.000	0.209700
$279$	0	0
$280$	0	0
$281$	722.000	0.153277	0.0766386	−	0.997059i	$-0.475581\pi$
$281$	722.000	0.153277	0.0766386	+	0.997059i	$0.475581\pi$
$282$	0	0
$283$	−7140.00	−1.49975	−0.749874	−	0.661580i	$-0.769886\pi$
$283$	−7140.00	−1.49975	−0.749874	+	0.661580i	$0.769886\pi$
$284$	−3668.00	−0.766394
$285$	0	0
$286$	60.0000	0.0124052
$287$	0	0
$288$	0	0
$289$	2143.00	0.436190
$290$	−1160.00	−0.234888
$291$	0	0
$292$	4830.00	0.967994
$293$	−4698.00	−0.936724	−0.468362	−	0.883537i	$-0.655156\pi$
$293$	−4698.00	−0.936724	−0.468362	+	0.883537i	$0.655156\pi$
$294$	0	0
$295$	−1740.00	−0.343413
$296$	−720.000	−0.141382
$297$	0	0
$298$	280.000	0.0544294
$299$	−336.000	−0.0649879
$300$	0	0
$301$	0	0
$302$	−1200.00	−0.228650
$303$	0	0
$304$	1968.00	0.371291
$305$	4410.00	0.827921
$306$	0	0
$307$	−5676.00	−1.05520	−0.527600	−	0.849493i	$-0.676908\pi$
$307$	−5676.00	−1.05520	−0.527600	+	0.849493i	$0.676908\pi$
$308$	0	0
$309$	0	0
$310$	30.0000	0.00549640
$311$	−5748.00	−1.04804	−0.524018	−	0.851707i	$-0.675568\pi$
$311$	−5748.00	−1.04804	−0.524018	+	0.851707i	$0.675568\pi$
$312$	0	0
$313$	−7662.00	−1.38365	−0.691824	−	0.722066i	$-0.743193\pi$
$313$	−7662.00	−1.38365	−0.691824	+	0.722066i	$0.743193\pi$
$314$	1134.00	0.203807
$315$	0	0
$316$	7168.00	1.27605
$317$	−10142.0	−1.79694	−0.898472	−	0.439030i	$-0.855322\pi$
$317$	−10142.0	−1.79694	−0.898472	+	0.439030i	$0.855322\pi$
$318$	0	0
$319$	−2320.00	−0.407195
$320$	−835.000	−0.145868
$321$	0	0
$322$	0	0
$323$	−4032.00	−0.694571
$324$	0	0
$325$	150.000	0.0256015
$326$	−2734.00	−0.464485
$327$	0	0
$328$	2250.00	0.378767
$329$	0	0
$330$	0	0
$331$	6708.00	1.11391	0.556956	−	0.830542i	$-0.311969\pi$
$331$	6708.00	1.11391	0.556956	+	0.830542i	$0.311969\pi$
$332$	2688.00	0.444347
$333$	0	0
$334$	3174.00	0.519981
$335$	−910.000	−0.148414
$336$	0	0
$337$	−306.000	−0.0494626	−0.0247313	−	0.999694i	$-0.507873\pi$
$337$	−306.000	−0.0494626	−0.0247313	+	0.999694i	$0.507873\pi$
$338$	2161.00	0.347760
$339$	0	0
$340$	2940.00	0.468953
$341$	60.0000	0.00952839
$342$	0	0
$343$	0	0
$344$	−6390.00	−1.00153
$345$	0	0
$346$	−1986.00	−0.308578
$347$	3836.00	0.593450	0.296725	−	0.954963i	$-0.404105\pi$
$347$	3836.00	0.593450	0.296725	+	0.954963i	$0.404105\pi$
$348$	0	0
$349$	−10050.0	−1.54144	−0.770722	−	0.637171i	$-0.780104\pi$
$349$	−10050.0	−1.54144	−0.770722	+	0.637171i	$0.780104\pi$
$350$	0	0
$351$	0	0
$352$	1610.00	0.243788
$353$	9672.00	1.45832	0.729162	−	0.684341i	$-0.239910\pi$
$353$	9672.00	1.45832	0.729162	+	0.684341i	$0.239910\pi$
$354$	0	0
$355$	2620.00	0.391705
$356$	−1722.00	−0.256365
$357$	0	0
$358$	22.0000	0.00324787
$359$	−9164.00	−1.34724	−0.673618	−	0.739080i	$-0.735260\pi$
$359$	−9164.00	−1.34724	−0.673618	+	0.739080i	$0.735260\pi$
$360$	0	0
$361$	−4555.00	−0.664091
$362$	3378.00	0.490452
$363$	0	0
$364$	0	0
$365$	−3450.00	−0.494743
$366$	0	0
$367$	−11748.0	−1.67096	−0.835478	−	0.549524i	$-0.814809\pi$
$367$	−11748.0	−1.67096	−0.835478	+	0.549524i	$0.814809\pi$
$368$	−2296.00	−0.325237
$369$	0	0
$370$	240.000	0.0337216
$371$	0	0
$372$	0	0
$373$	6676.00	0.926730	0.463365	−	0.886168i	$-0.346642\pi$
$373$	6676.00	0.926730	0.463365	+	0.886168i	$0.346642\pi$
$374$	−840.000	−0.116137
$375$	0	0
$376$	270.000	0.0370324
$377$	1392.00	0.190164
$378$	0	0
$379$	1248.00	0.169144	0.0845718	−	0.996417i	$-0.473048\pi$
$379$	1248.00	0.169144	0.0845718	+	0.996417i	$0.473048\pi$
$380$	−1680.00	−0.226795
$381$	0	0
$382$	1472.00	0.197157
$383$	11718.0	1.56335	0.781673	−	0.623688i	$-0.214366\pi$
$383$	11718.0	1.56335	0.781673	+	0.623688i	$0.214366\pi$
$384$	0	0
$385$	0	0
$386$	1410.00	0.185925
$387$	0	0
$388$	−7854.00	−1.02765
$389$	2816.00	0.367036	0.183518	−	0.983016i	$-0.441251\pi$
$389$	2816.00	0.367036	0.183518	+	0.983016i	$0.441251\pi$
$390$	0	0
$391$	4704.00	0.608418
$392$	0	0
$393$	0	0
$394$	1882.00	0.240644
$395$	−5120.00	−0.652190
$396$	0	0
$397$	2394.00	0.302648	0.151324	−	0.988484i	$-0.451646\pi$
$397$	2394.00	0.302648	0.151324	+	0.988484i	$0.451646\pi$
$398$	−894.000	−0.112593
$399$	0	0
$400$	1025.00	0.128125
$401$	−14690.0	−1.82939	−0.914693	−	0.404150i	$-0.867567\pi$
$401$	−14690.0	−1.82939	−0.914693	+	0.404150i	$0.867567\pi$
$402$	0	0
$403$	−36.0000	−0.00444985
$404$	−9702.00	−1.19478
$405$	0	0
$406$	0	0
$407$	480.000	0.0584588
$408$	0	0
$409$	−624.000	−0.0754396	−0.0377198	−	0.999288i	$-0.512009\pi$
$409$	−624.000	−0.0754396	−0.0377198	+	0.999288i	$0.512009\pi$
$410$	−750.000	−0.0903411
$411$	0	0
$412$	756.000	0.0904016
$413$	0	0
$414$	0	0
$415$	−1920.00	−0.227106
$416$	−966.000	−0.113851
$417$	0	0
$418$	480.000	0.0561664
$419$	14772.0	1.72234	0.861169	−	0.508319i	$-0.169733\pi$
$419$	14772.0	1.72234	0.861169	+	0.508319i	$0.169733\pi$
$420$	0	0
$421$	862.000	0.0997893	0.0498947	−	0.998754i	$-0.484111\pi$
$421$	862.000	0.0997893	0.0498947	+	0.998754i	$0.484111\pi$
$422$	−2196.00	−0.253317
$423$	0	0
$424$	870.000	0.0996484
$425$	−2100.00	−0.239682
$426$	0	0
$427$	0	0
$428$	−6020.00	−0.679878
$429$	0	0
$430$	2130.00	0.238878
$431$	−140.000	−0.0156463	−0.00782316	−	0.999969i	$-0.502490\pi$
$431$	−140.000	−0.0156463	−0.00782316	+	0.999969i	$0.502490\pi$
$432$	0	0
$433$	−1662.00	−0.184459	−0.0922294	−	0.995738i	$-0.529399\pi$
$433$	−1662.00	−0.184459	−0.0922294	+	0.995738i	$0.529399\pi$
$434$	0	0
$435$	0	0
$436$	−5754.00	−0.632034
$437$	−2688.00	−0.294244
$438$	0	0
$439$	16386.0	1.78146	0.890730	−	0.454532i	$-0.150194\pi$
$439$	16386.0	1.78146	0.890730	+	0.454532i	$0.150194\pi$
$440$	−750.000	−0.0812610
$441$	0	0
$442$	504.000	0.0542372
$443$	9068.00	0.972537	0.486268	−	0.873810i	$-0.338358\pi$
$443$	9068.00	0.972537	0.486268	+	0.873810i	$0.338358\pi$
$444$	0	0
$445$	1230.00	0.131028
$446$	4800.00	0.509611
$447$	0	0
$448$	0	0
$449$	13114.0	1.37837	0.689185	−	0.724586i	$-0.257969\pi$
$449$	13114.0	1.37837	0.689185	+	0.724586i	$0.257969\pi$
$450$	0	0
$451$	−1500.00	−0.156613
$452$	−13090.0	−1.36217
$453$	0	0
$454$	−228.000	−0.0235695
$455$	0	0
$456$	0	0
$457$	−7386.00	−0.756023	−0.378011	−	0.925801i	$-0.623392\pi$
$457$	−7386.00	−0.756023	−0.378011	+	0.925801i	$0.623392\pi$
$458$	−4830.00	−0.492775
$459$	0	0
$460$	1960.00	0.198664
$461$	−7866.00	−0.794699	−0.397349	−	0.917667i	$-0.630070\pi$
$461$	−7866.00	−0.794699	−0.397349	+	0.917667i	$0.630070\pi$
$462$	0	0
$463$	−1168.00	−0.117239	−0.0586194	−	0.998280i	$-0.518670\pi$
$463$	−1168.00	−0.117239	−0.0586194	+	0.998280i	$0.518670\pi$
$464$	9512.00	0.951689
$465$	0	0
$466$	−2078.00	−0.206570
$467$	−10248.0	−1.01546	−0.507731	−	0.861516i	$-0.669516\pi$
$467$	−10248.0	−1.01546	−0.507731	+	0.861516i	$0.669516\pi$
$468$	0	0
$469$	0	0
$470$	−90.0000	−0.00883275
$471$	0	0
$472$	−5220.00	−0.509047
$473$	4260.00	0.414112
$474$	0	0
$475$	1200.00	0.115915
$476$	0	0
$477$	0	0
$478$	472.000	0.0451648
$479$	−17916.0	−1.70898	−0.854492	−	0.519465i	$-0.826131\pi$
$479$	−17916.0	−1.70898	−0.854492	+	0.519465i	$0.826131\pi$
$480$	0	0
$481$	−288.000	−0.0273008
$482$	6636.00	0.627099
$483$	0	0
$484$	8617.00	0.809260
$485$	5610.00	0.525231
$486$	0	0
$487$	−3396.00	−0.315991	−0.157995	−	0.987440i	$-0.550503\pi$
$487$	−3396.00	−0.315991	−0.157995	+	0.987440i	$0.550503\pi$
$488$	13230.0	1.22724
$489$	0	0
$490$	0	0
$491$	8674.00	0.797255	0.398627	−	0.917113i	$-0.369487\pi$
$491$	8674.00	0.797255	0.398627	+	0.917113i	$0.369487\pi$
$492$	0	0
$493$	−19488.0	−1.78032
$494$	−288.000	−0.0262302
$495$	0	0
$496$	−246.000	−0.0222696
$497$	0	0
$498$	0	0
$499$	9792.00	0.878457	0.439229	−	0.898375i	$-0.355252\pi$
$499$	9792.00	0.878457	0.439229	+	0.898375i	$0.355252\pi$
$500$	−875.000	−0.0782624
$501$	0	0
$502$	4404.00	0.391554
$503$	−4878.00	−0.432404	−0.216202	−	0.976349i	$-0.569367\pi$
$503$	−4878.00	−0.432404	−0.216202	+	0.976349i	$0.569367\pi$
$504$	0	0
$505$	6930.00	0.610655
$506$	−560.000	−0.0491997
$507$	0	0
$508$	11732.0	1.02465
$509$	−20886.0	−1.81877	−0.909387	−	0.415952i	$-0.863449\pi$
$509$	−20886.0	−1.81877	−0.909387	+	0.415952i	$0.863449\pi$
$510$	0	0
$511$	0	0
$512$	−11521.0	−0.994455
$513$	0	0
$514$	180.000	0.0154464
$515$	−540.000	−0.0462044
$516$	0	0
$517$	−180.000	−0.0153122
$518$	0	0
$519$	0	0
$520$	450.000	0.0379496
$521$	−1098.00	−0.0923306	−0.0461653	−	0.998934i	$-0.514700\pi$
$521$	−1098.00	−0.0923306	−0.0461653	+	0.998934i	$0.514700\pi$
$522$	0	0
$523$	−16692.0	−1.39558	−0.697792	−	0.716301i	$-0.745834\pi$
$523$	−16692.0	−1.39558	−0.697792	+	0.716301i	$0.745834\pi$
$524$	−5124.00	−0.427181
$525$	0	0
$526$	−6400.00	−0.530520
$527$	504.000	0.0416596
$528$	0	0
$529$	−9031.00	−0.742254
$530$	−290.000	−0.0237675
$531$	0	0
$532$	0	0
$533$	900.000	0.0731395
$534$	0	0
$535$	4300.00	0.347486
$536$	−2730.00	−0.219996
$537$	0	0
$538$	7410.00	0.593806
$539$	0	0
$540$	0	0
$541$	−20950.0	−1.66490	−0.832450	−	0.554100i	$-0.813062\pi$
$541$	−20950.0	−1.66490	−0.832450	+	0.554100i	$0.813062\pi$
$542$	5370.00	0.425574
$543$	0	0
$544$	13524.0	1.06588
$545$	4110.00	0.323033
$546$	0	0
$547$	−20662.0	−1.61507	−0.807535	−	0.589820i	$-0.799199\pi$
$547$	−20662.0	−1.61507	−0.807535	+	0.589820i	$0.799199\pi$
$548$	10178.0	0.793399
$549$	0	0
$550$	250.000	0.0193819
$551$	11136.0	0.860997
$552$	0	0
$553$	0	0
$554$	7396.00	0.567195
$555$	0	0
$556$	6804.00	0.518982
$557$	12746.0	0.969597	0.484798	−	0.874626i	$-0.338893\pi$
$557$	12746.0	0.969597	0.484798	+	0.874626i	$0.338893\pi$
$558$	0	0
$559$	−2556.00	−0.193394
$560$	0	0
$561$	0	0
$562$	−722.000	−0.0541917
$563$	−4632.00	−0.346742	−0.173371	−	0.984857i	$-0.555466\pi$
$563$	−4632.00	−0.346742	−0.173371	+	0.984857i	$0.555466\pi$
$564$	0	0
$565$	9350.00	0.696208
$566$	7140.00	0.530241
$567$	0	0
$568$	7860.00	0.580631
$569$	23510.0	1.73214	0.866072	−	0.499918i	$-0.166637\pi$
$569$	23510.0	1.73214	0.866072	+	0.499918i	$0.166637\pi$
$570$	0	0
$571$	7684.00	0.563162	0.281581	−	0.959537i	$-0.409141\pi$
$571$	7684.00	0.563162	0.281581	+	0.959537i	$0.409141\pi$
$572$	420.000	0.0307012
$573$	0	0
$574$	0	0
$575$	−1400.00	−0.101537
$576$	0	0
$577$	−6342.00	−0.457575	−0.228788	−	0.973476i	$-0.573476\pi$
$577$	−6342.00	−0.457575	−0.228788	+	0.973476i	$0.573476\pi$
$578$	−2143.00	−0.154216
$579$	0	0
$580$	−8120.00	−0.581318
$581$	0	0
$582$	0	0
$583$	−580.000	−0.0412027
$584$	−10350.0	−0.733367
$585$	0	0
$586$	4698.00	0.331182
$587$	−480.000	−0.0337508	−0.0168754	−	0.999858i	$-0.505372\pi$
$587$	−480.000	−0.0337508	−0.0168754	+	0.999858i	$0.505372\pi$
$588$	0	0
$589$	−288.000	−0.0201474
$590$	1740.00	0.121415
$591$	0	0
$592$	−1968.00	−0.136629
$593$	−7668.00	−0.531007	−0.265503	−	0.964110i	$-0.585538\pi$
$593$	−7668.00	−0.531007	−0.265503	+	0.964110i	$0.585538\pi$
$594$	0	0
$595$	0	0
$596$	1960.00	0.134706
$597$	0	0
$598$	336.000	0.0229767
$599$	−14624.0	−0.997530	−0.498765	−	0.866737i	$-0.666213\pi$
$599$	−14624.0	−0.997530	−0.498765	+	0.866737i	$0.666213\pi$
$600$	0	0
$601$	2028.00	0.137644	0.0688218	−	0.997629i	$-0.478076\pi$
$601$	2028.00	0.137644	0.0688218	+	0.997629i	$0.478076\pi$
$602$	0	0
$603$	0	0
$604$	−8400.00	−0.565879
$605$	−6155.00	−0.413614
$606$	0	0
$607$	−26904.0	−1.79901	−0.899505	−	0.436909i	$-0.856073\pi$
$607$	−26904.0	−1.79901	−0.899505	+	0.436909i	$0.856073\pi$
$608$	−7728.00	−0.515480
$609$	0	0
$610$	−4410.00	−0.292714
$611$	108.000	0.00715092
$612$	0	0
$613$	11680.0	0.769577	0.384789	−	0.923005i	$-0.374274\pi$
$613$	11680.0	0.769577	0.384789	+	0.923005i	$0.374274\pi$
$614$	5676.00	0.373070
$615$	0	0
$616$	0	0
$617$	−9142.00	−0.596504	−0.298252	−	0.954487i	$-0.596404\pi$
$617$	−9142.00	−0.596504	−0.298252	+	0.954487i	$0.596404\pi$
$618$	0	0
$619$	11112.0	0.721533	0.360767	−	0.932656i	$-0.382515\pi$
$619$	11112.0	0.721533	0.360767	+	0.932656i	$0.382515\pi$
$620$	210.000	0.0136029
$621$	0	0
$622$	5748.00	0.370537
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	7662.00	0.489193
$627$	0	0
$628$	7938.00	0.504396
$629$	4032.00	0.255590
$630$	0	0
$631$	−12584.0	−0.793916	−0.396958	−	0.917837i	$-0.629934\pi$
$631$	−12584.0	−0.793916	−0.396958	+	0.917837i	$0.629934\pi$
$632$	−15360.0	−0.966753
$633$	0	0
$634$	10142.0	0.635316
$635$	−8380.00	−0.523701
$636$	0	0
$637$	0	0
$638$	2320.00	0.143965
$639$	0	0
$640$	7275.00	0.449328
$641$	13382.0	0.824582	0.412291	−	0.911052i	$-0.364729\pi$
$641$	13382.0	0.824582	0.412291	+	0.911052i	$0.364729\pi$
$642$	0	0
$643$	10692.0	0.655756	0.327878	−	0.944720i	$-0.393666\pi$
$643$	10692.0	0.655756	0.327878	+	0.944720i	$0.393666\pi$
$644$	0	0
$645$	0	0
$646$	4032.00	0.245568
$647$	−13578.0	−0.825048	−0.412524	−	0.910947i	$-0.635353\pi$
$647$	−13578.0	−0.825048	−0.412524	+	0.910947i	$0.635353\pi$
$648$	0	0
$649$	3480.00	0.210481
$650$	−150.000	−0.00905151
$651$	0	0
$652$	−19138.0	−1.14954
$653$	15862.0	0.950579	0.475289	−	0.879829i	$-0.342343\pi$
$653$	15862.0	0.950579	0.475289	+	0.879829i	$0.342343\pi$
$654$	0	0
$655$	3660.00	0.218333
$656$	6150.00	0.366032
$657$	0	0
$658$	0	0
$659$	23386.0	1.38238	0.691191	−	0.722673i	$-0.257087\pi$
$659$	23386.0	1.38238	0.691191	+	0.722673i	$0.257087\pi$
$660$	0	0
$661$	−14142.0	−0.832163	−0.416082	−	0.909327i	$-0.636597\pi$
$661$	−14142.0	−0.832163	−0.416082	+	0.909327i	$0.636597\pi$
$662$	−6708.00	−0.393828
$663$	0	0
$664$	−5760.00	−0.336644
$665$	0	0
$666$	0	0
$667$	−12992.0	−0.754201
$668$	22218.0	1.28689
$669$	0	0
$670$	910.000	0.0524722
$671$	−8820.00	−0.507440
$672$	0	0
$673$	−20790.0	−1.19078	−0.595390	−	0.803436i	$-0.703003\pi$
$673$	−20790.0	−1.19078	−0.595390	+	0.803436i	$0.703003\pi$
$674$	306.000	0.0174877
$675$	0	0
$676$	15127.0	0.860662
$677$	18594.0	1.05558	0.527788	−	0.849376i	$-0.323021\pi$
$677$	18594.0	1.05558	0.527788	+	0.849376i	$0.323021\pi$
$678$	0	0
$679$	0	0
$680$	−6300.00	−0.355285
$681$	0	0
$682$	−60.0000	−0.00336880
$683$	−15248.0	−0.854244	−0.427122	−	0.904194i	$-0.640472\pi$
$683$	−15248.0	−0.854244	−0.427122	+	0.904194i	$0.640472\pi$
$684$	0	0
$685$	−7270.00	−0.405507
$686$	0	0
$687$	0	0
$688$	−17466.0	−0.967856
$689$	348.000	0.0192420
$690$	0	0
$691$	−18036.0	−0.992940	−0.496470	−	0.868054i	$-0.665371\pi$
$691$	−18036.0	−0.992940	−0.496470	+	0.868054i	$0.665371\pi$
$692$	−13902.0	−0.763692
$693$	0	0
$694$	−3836.00	−0.209816
$695$	−4860.00	−0.265252
$696$	0	0
$697$	−12600.0	−0.684733
$698$	10050.0	0.544983
$699$	0	0
$700$	0	0
$701$	26968.0	1.45302	0.726510	−	0.687156i	$-0.241141\pi$
$701$	26968.0	1.45302	0.726510	+	0.687156i	$0.241141\pi$
$702$	0	0
$703$	−2304.00	−0.123609
$704$	1670.00	0.0894041
$705$	0	0
$706$	−9672.00	−0.515596
$707$	0	0
$708$	0	0
$709$	12114.0	0.641680	0.320840	−	0.947133i	$-0.396035\pi$
$709$	12114.0	0.641680	0.320840	+	0.947133i	$0.396035\pi$
$710$	−2620.00	−0.138489
$711$	0	0
$712$	3690.00	0.194226
$713$	336.000	0.0176484
$714$	0	0
$715$	−300.000	−0.0156914
$716$	154.000	0.00803806
$717$	0	0
$718$	9164.00	0.476320
$719$	−37152.0	−1.92703	−0.963516	−	0.267651i	$-0.913752\pi$
$719$	−37152.0	−1.92703	−0.963516	+	0.267651i	$0.913752\pi$
$720$	0	0
$721$	0	0
$722$	4555.00	0.234792
$723$	0	0
$724$	23646.0	1.21381
$725$	5800.00	0.297113
$726$	0	0
$727$	1872.00	0.0955002	0.0477501	−	0.998859i	$-0.484795\pi$
$727$	1872.00	0.0955002	0.0477501	+	0.998859i	$0.484795\pi$
$728$	0	0
$729$	0	0
$730$	3450.00	0.174918
$731$	35784.0	1.81056
$732$	0	0
$733$	−24774.0	−1.24836	−0.624180	−	0.781280i	$-0.714567\pi$
$733$	−24774.0	−1.24836	−0.624180	+	0.781280i	$0.714567\pi$
$734$	11748.0	0.590772
$735$	0	0
$736$	9016.00	0.451541
$737$	1820.00	0.0909641
$738$	0	0
$739$	−19816.0	−0.986392	−0.493196	−	0.869918i	$-0.664171\pi$
$739$	−19816.0	−0.986392	−0.493196	+	0.869918i	$0.664171\pi$
$740$	1680.00	0.0834568
$741$	0	0
$742$	0	0
$743$	19264.0	0.951181	0.475591	−	0.879667i	$-0.342234\pi$
$743$	19264.0	0.951181	0.475591	+	0.879667i	$0.342234\pi$
$744$	0	0
$745$	−1400.00	−0.0688484
$746$	−6676.00	−0.327648
$747$	0	0
$748$	−5880.00	−0.287425
$749$	0	0
$750$	0	0
$751$	7416.00	0.360338	0.180169	−	0.983636i	$-0.442336\pi$
$751$	7416.00	0.360338	0.180169	+	0.983636i	$0.442336\pi$
$752$	738.000	0.0357874
$753$	0	0
$754$	−1392.00	−0.0672330
$755$	6000.00	0.289222
$756$	0	0
$757$	−5388.00	−0.258692	−0.129346	−	0.991599i	$-0.541288\pi$
$757$	−5388.00	−0.258692	−0.129346	+	0.991599i	$0.541288\pi$
$758$	−1248.00	−0.0598013
$759$	0	0
$760$	3600.00	0.171823
$761$	−7926.00	−0.377552	−0.188776	−	0.982020i	$-0.560452\pi$
$761$	−7926.00	−0.377552	−0.188776	+	0.982020i	$0.560452\pi$
$762$	0	0
$763$	0	0
$764$	10304.0	0.487939
$765$	0	0
$766$	−11718.0	−0.552727
$767$	−2088.00	−0.0982964
$768$	0	0
$769$	14880.0	0.697772	0.348886	−	0.937165i	$-0.386560\pi$
$769$	14880.0	0.697772	0.348886	+	0.937165i	$0.386560\pi$
$770$	0	0
$771$	0	0
$772$	9870.00	0.460141
$773$	28902.0	1.34480	0.672401	−	0.740187i	$-0.265263\pi$
$773$	28902.0	1.34480	0.672401	+	0.740187i	$0.265263\pi$
$774$	0	0
$775$	−150.000	−0.00695246
$776$	16830.0	0.778559
$777$	0	0
$778$	−2816.00	−0.129767
$779$	7200.00	0.331151
$780$	0	0
$781$	−5240.00	−0.240079
$782$	−4704.00	−0.215108
$783$	0	0
$784$	0	0
$785$	−5670.00	−0.257797
$786$	0	0
$787$	39084.0	1.77026	0.885130	−	0.465344i	$-0.154070\pi$
$787$	39084.0	1.77026	0.885130	+	0.465344i	$0.154070\pi$
$788$	13174.0	0.595564
$789$	0	0
$790$	5120.00	0.230584
$791$	0	0
$792$	0	0
$793$	5292.00	0.236979
$794$	−2394.00	−0.107002
$795$	0	0
$796$	−6258.00	−0.278654
$797$	17370.0	0.771991	0.385996	−	0.922501i	$-0.373858\pi$
$797$	17370.0	0.771991	0.385996	+	0.922501i	$0.373858\pi$
$798$	0	0
$799$	−1512.00	−0.0669471
$800$	−4025.00	−0.177882
$801$	0	0
$802$	14690.0	0.646785
$803$	6900.00	0.303233
$804$	0	0
$805$	0	0
$806$	36.0000	0.00157326
$807$	0	0
$808$	20790.0	0.905185
$809$	−15622.0	−0.678913	−0.339456	−	0.940622i	$-0.610243\pi$
$809$	−15622.0	−0.678913	−0.339456	+	0.940622i	$0.610243\pi$
$810$	0	0
$811$	23532.0	1.01889	0.509445	−	0.860503i	$-0.329851\pi$
$811$	23532.0	1.01889	0.509445	+	0.860503i	$0.329851\pi$
$812$	0	0
$813$	0	0
$814$	−480.000	−0.0206683
$815$	13670.0	0.587533
$816$	0	0
$817$	−20448.0	−0.875624
$818$	624.000	0.0266719
$819$	0	0
$820$	−5250.00	−0.223583
$821$	19040.0	0.809380	0.404690	−	0.914454i	$-0.367380\pi$
$821$	19040.0	0.809380	0.404690	+	0.914454i	$0.367380\pi$
$822$	0	0
$823$	−17008.0	−0.720366	−0.360183	−	0.932882i	$-0.617286\pi$
$823$	−17008.0	−0.720366	−0.360183	+	0.932882i	$0.617286\pi$
$824$	−1620.00	−0.0684895
$825$	0	0
$826$	0	0
$827$	−37664.0	−1.58368	−0.791841	−	0.610727i	$-0.790877\pi$
$827$	−37664.0	−1.58368	−0.791841	+	0.610727i	$0.790877\pi$
$828$	0	0
$829$	28578.0	1.19729	0.598646	−	0.801014i	$-0.295706\pi$
$829$	28578.0	1.19729	0.598646	+	0.801014i	$0.295706\pi$
$830$	1920.00	0.0802942
$831$	0	0
$832$	−1002.00	−0.0417525
$833$	0	0
$834$	0	0
$835$	−15870.0	−0.657729
$836$	3360.00	0.139005
$837$	0	0
$838$	−14772.0	−0.608938
$839$	−24036.0	−0.989052	−0.494526	−	0.869163i	$-0.664658\pi$
$839$	−24036.0	−0.989052	−0.494526	+	0.869163i	$0.664658\pi$
$840$	0	0
$841$	29435.0	1.20690
$842$	−862.000	−0.0352809
$843$	0	0
$844$	−15372.0	−0.626927
$845$	−10805.0	−0.439886
$846$	0	0
$847$	0	0
$848$	2378.00	0.0962982
$849$	0	0
$850$	2100.00	0.0847405
$851$	2688.00	0.108277
$852$	0	0
$853$	19410.0	0.779116	0.389558	−	0.921002i	$-0.372628\pi$
$853$	19410.0	0.779116	0.389558	+	0.921002i	$0.372628\pi$
$854$	0	0
$855$	0	0
$856$	12900.0	0.515085
$857$	13860.0	0.552449	0.276224	−	0.961093i	$-0.410917\pi$
$857$	13860.0	0.552449	0.276224	+	0.961093i	$0.410917\pi$
$858$	0	0
$859$	−1452.00	−0.0576736	−0.0288368	−	0.999584i	$-0.509180\pi$
$859$	−1452.00	−0.0576736	−0.0288368	+	0.999584i	$0.509180\pi$
$860$	14910.0	0.591194
$861$	0	0
$862$	140.000	0.00553181
$863$	−13544.0	−0.534233	−0.267117	−	0.963664i	$-0.586071\pi$
$863$	−13544.0	−0.534233	−0.267117	+	0.963664i	$0.586071\pi$
$864$	0	0
$865$	9930.00	0.390324
$866$	1662.00	0.0652160
$867$	0	0
$868$	0	0
$869$	10240.0	0.399733
$870$	0	0
$871$	−1092.00	−0.0424811
$872$	12330.0	0.478838
$873$	0	0
$874$	2688.00	0.104031
$875$	0	0
$876$	0	0
$877$	−12076.0	−0.464969	−0.232484	−	0.972600i	$-0.574685\pi$
$877$	−12076.0	−0.464969	−0.232484	+	0.972600i	$0.574685\pi$
$878$	−16386.0	−0.629841
$879$	0	0
$880$	−2050.00	−0.0785290
$881$	−6822.00	−0.260884	−0.130442	−	0.991456i	$-0.541640\pi$
$881$	−6822.00	−0.260884	−0.130442	+	0.991456i	$0.541640\pi$
$882$	0	0
$883$	12810.0	0.488212	0.244106	−	0.969749i	$-0.421506\pi$
$883$	12810.0	0.488212	0.244106	+	0.969749i	$0.421506\pi$
$884$	3528.00	0.134230
$885$	0	0
$886$	−9068.00	−0.343844
$887$	21270.0	0.805160	0.402580	−	0.915385i	$-0.368114\pi$
$887$	21270.0	0.805160	0.402580	+	0.915385i	$0.368114\pi$
$888$	0	0
$889$	0	0
$890$	−1230.00	−0.0463255
$891$	0	0
$892$	33600.0	1.26122
$893$	864.000	0.0323770
$894$	0	0
$895$	−110.000	−0.00410826
$896$	0	0
$897$	0	0
$898$	−13114.0	−0.487327
$899$	−1392.00	−0.0516416
$900$	0	0
$901$	−4872.00	−0.180144
$902$	1500.00	0.0553709
$903$	0	0
$904$	28050.0	1.03200
$905$	−16890.0	−0.620379
$906$	0	0
$907$	7506.00	0.274788	0.137394	−	0.990516i	$-0.456127\pi$
$907$	7506.00	0.274788	0.137394	+	0.990516i	$0.456127\pi$
$908$	−1596.00	−0.0583316
$909$	0	0
$910$	0	0
$911$	−18352.0	−0.667430	−0.333715	−	0.942674i	$-0.608302\pi$
$911$	−18352.0	−0.667430	−0.333715	+	0.942674i	$0.608302\pi$
$912$	0	0
$913$	3840.00	0.139195
$914$	7386.00	0.267294
$915$	0	0
$916$	−33810.0	−1.21956
$917$	0	0
$918$	0	0
$919$	29496.0	1.05874	0.529371	−	0.848391i	$-0.322428\pi$
$919$	29496.0	1.05874	0.529371	+	0.848391i	$0.322428\pi$
$920$	−4200.00	−0.150511
$921$	0	0
$922$	7866.00	0.280968
$923$	3144.00	0.112119
$924$	0	0
$925$	−1200.00	−0.0426549
$926$	1168.00	0.0414502
$927$	0	0
$928$	−37352.0	−1.32127
$929$	5082.00	0.179478	0.0897390	−	0.995965i	$-0.471397\pi$
$929$	5082.00	0.179478	0.0897390	+	0.995965i	$0.471397\pi$
$930$	0	0
$931$	0	0
$932$	−14546.0	−0.511234
$933$	0	0
$934$	10248.0	0.359020
$935$	4200.00	0.146903
$936$	0	0
$937$	3726.00	0.129907	0.0649536	−	0.997888i	$-0.479310\pi$
$937$	3726.00	0.129907	0.0649536	+	0.997888i	$0.479310\pi$
$938$	0	0
$939$	0	0
$940$	−630.000	−0.0218599
$941$	−18054.0	−0.625445	−0.312722	−	0.949845i	$-0.601241\pi$
$941$	−18054.0	−0.625445	−0.312722	+	0.949845i	$0.601241\pi$
$942$	0	0
$943$	−8400.00	−0.290076
$944$	−14268.0	−0.491932
$945$	0	0
$946$	−4260.00	−0.146411
$947$	−36472.0	−1.25151	−0.625755	−	0.780019i	$-0.715209\pi$
$947$	−36472.0	−1.25151	−0.625755	+	0.780019i	$0.715209\pi$
$948$	0	0
$949$	−4140.00	−0.141612
$950$	−1200.00	−0.0409823
$951$	0	0
$952$	0	0
$953$	−5398.00	−0.183482	−0.0917410	−	0.995783i	$-0.529243\pi$
$953$	−5398.00	−0.183482	−0.0917410	+	0.995783i	$0.529243\pi$
$954$	0	0
$955$	−7360.00	−0.249386
$956$	3304.00	0.111777
$957$	0	0
$958$	17916.0	0.604217
$959$	0	0
$960$	0	0
$961$	−29755.0	−0.998792
$962$	288.000	0.00965228
$963$	0	0
$964$	46452.0	1.55199
$965$	−7050.00	−0.235179
$966$	0	0
$967$	−18304.0	−0.608704	−0.304352	−	0.952560i	$-0.598440\pi$
$967$	−18304.0	−0.608704	−0.304352	+	0.952560i	$0.598440\pi$
$968$	−18465.0	−0.613107
$969$	0	0
$970$	−5610.00	−0.185697
$971$	−59940.0	−1.98102	−0.990508	−	0.137457i	$-0.956107\pi$
$971$	−59940.0	−1.98102	−0.990508	+	0.137457i	$0.956107\pi$
$972$	0	0
$973$	0	0
$974$	3396.00	0.111720
$975$	0	0
$976$	36162.0	1.18598
$977$	40958.0	1.34121	0.670605	−	0.741814i	$-0.266035\pi$
$977$	40958.0	1.34121	0.670605	+	0.741814i	$0.266035\pi$
$978$	0	0
$979$	−2460.00	−0.0803084
$980$	0	0
$981$	0	0
$982$	−8674.00	−0.281872
$983$	32634.0	1.05886	0.529432	−	0.848352i	$-0.322405\pi$
$983$	32634.0	1.05886	0.529432	+	0.848352i	$0.322405\pi$
$984$	0	0
$985$	−9410.00	−0.304393
$986$	19488.0	0.629436
$987$	0	0
$988$	−2016.00	−0.0649165
$989$	23856.0	0.767014
$990$	0	0
$991$	48696.0	1.56093	0.780464	−	0.625201i	$-0.214983\pi$
$991$	48696.0	1.56093	0.780464	+	0.625201i	$0.214983\pi$
$992$	966.000	0.0309179
$993$	0	0
$994$	0	0
$995$	4470.00	0.142421
$996$	0	0
$997$	31458.0	0.999283	0.499641	−	0.866232i	$-0.333465\pi$
$997$	31458.0	0.999283	0.499641	+	0.866232i	$0.333465\pi$
$998$	−9792.00	−0.310582
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.m.1.1		1
3.2	odd	2		735.4.a.d.1.1	✓	1
7.6	odd	2		2205.4.a.j.1.1		1
21.20	even	2		735.4.a.f.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.d.1.1	✓	1	3.2	odd	2
735.4.a.f.1.1	yes	1	21.20	even	2
2205.4.a.j.1.1		1	7.6	odd	2
2205.4.a.m.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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