LMFDB - Embedded newform 1860.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1860,4,Mod(1,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1860.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1860.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$109.743552611$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1860.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +5.00000 q^{5} -8.00000 q^{7} +9.00000 q^{9} -20.0000 q^{11} -34.0000 q^{13} -15.0000 q^{15} -6.00000 q^{17} +12.0000 q^{19} +24.0000 q^{21} +120.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +246.000 q^{29} +31.0000 q^{31} +60.0000 q^{33} -40.0000 q^{35} +310.000 q^{37} +102.000 q^{39} -518.000 q^{41} +92.0000 q^{43} +45.0000 q^{45} -88.0000 q^{47} -279.000 q^{49} +18.0000 q^{51} -738.000 q^{53} -100.000 q^{55} -36.0000 q^{57} +268.000 q^{59} +366.000 q^{61} -72.0000 q^{63} -170.000 q^{65} +220.000 q^{67} -360.000 q^{69} -512.000 q^{71} -758.000 q^{73} -75.0000 q^{75} +160.000 q^{77} +160.000 q^{79} +81.0000 q^{81} +1348.00 q^{83} -30.0000 q^{85} -738.000 q^{87} +18.0000 q^{89} +272.000 q^{91} -93.0000 q^{93} +60.0000 q^{95} +1634.00 q^{97} -180.000 q^{99} +O(q^{100})$

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−6.00000	−0.0856008	−0.0428004	−	0.999084i	$-0.513628\pi$
$17$	−6.00000	−0.0856008	−0.0428004	+	0.999084i	$0.513628\pi$
$18$	0	0
$19$	12.0000	0.144894	0.0724471	−	0.997372i	$-0.476919\pi$
$19$	12.0000	0.144894	0.0724471	+	0.997372i	$0.476919\pi$
$20$	0	0
$21$	24.0000	0.249392
$22$	0	0
$23$	120.000	1.08790	0.543951	−	0.839117i	$-0.316928\pi$
$23$	120.000	1.08790	0.543951	+	0.839117i	$0.316928\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	246.000	1.57521	0.787604	−	0.616181i	$-0.211321\pi$
$29$	246.000	1.57521	0.787604	+	0.616181i	$0.211321\pi$
$30$	0	0
$31$	31.0000	0.179605
$31$	31.0000	0.179605
$32$	0	0
$33$	60.0000	0.316505
$34$	0	0
$35$	−40.0000	−0.193178
$36$	0	0
$37$	310.000	1.37740	0.688698	−	0.725048i	$-0.258182\pi$
$37$	310.000	1.37740	0.688698	+	0.725048i	$0.258182\pi$
$38$	0	0
$39$	102.000	0.418797
$40$	0	0
$41$	−518.000	−1.97312	−0.986561	−	0.163393i	$-0.947756\pi$
$41$	−518.000	−1.97312	−0.986561	+	0.163393i	$0.947756\pi$
$42$	0	0
$43$	92.0000	0.326276	0.163138	−	0.986603i	$-0.447838\pi$
$43$	92.0000	0.326276	0.163138	+	0.986603i	$0.447838\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−88.0000	−0.273109	−0.136554	−	0.990633i	$-0.543603\pi$
$47$	−88.0000	−0.273109	−0.136554	+	0.990633i	$0.543603\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	18.0000	0.0494217
$52$	0	0
$53$	−738.000	−1.91268	−0.956341	−	0.292255i	$-0.905595\pi$
$53$	−738.000	−1.91268	−0.956341	+	0.292255i	$0.905595\pi$
$54$	0	0
$55$	−100.000	−0.245164
$56$	0	0
$57$	−36.0000	−0.0836547
$58$	0	0
$59$	268.000	0.591367	0.295683	−	0.955286i	$-0.404453\pi$
$59$	268.000	0.591367	0.295683	+	0.955286i	$0.404453\pi$
$60$	0	0
$61$	366.000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	366.000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−72.0000	−0.143986
$64$	0	0
$65$	−170.000	−0.324399
$66$	0	0
$67$	220.000	0.401153	0.200577	−	0.979678i	$-0.435718\pi$
$67$	220.000	0.401153	0.200577	+	0.979678i	$0.435718\pi$
$68$	0	0
$69$	−360.000	−0.628100
$70$	0	0
$71$	−512.000	−0.855820	−0.427910	−	0.903821i	$-0.640750\pi$
$71$	−512.000	−0.855820	−0.427910	+	0.903821i	$0.640750\pi$
$72$	0	0
$73$	−758.000	−1.21530	−0.607652	−	0.794203i	$-0.707888\pi$
$73$	−758.000	−1.21530	−0.607652	+	0.794203i	$0.707888\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	160.000	0.236801
$78$	0	0
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1348.00	1.78268	0.891339	−	0.453338i	$-0.149767\pi$
$83$	1348.00	1.78268	0.891339	+	0.453338i	$0.149767\pi$
$84$	0	0
$85$	−30.0000	−0.0382818
$86$	0	0
$87$	−738.000	−0.909447
$88$	0	0
$89$	18.0000	0.0214382	0.0107191	−	0.999943i	$-0.496588\pi$
$89$	18.0000	0.0214382	0.0107191	+	0.999943i	$0.496588\pi$
$90$	0	0
$91$	272.000	0.313333
$92$	0	0
$93$	−93.0000	−0.103695
$94$	0	0
$95$	60.0000	0.0647986
$96$	0	0
$97$	1634.00	1.71039	0.855194	−	0.518309i	$-0.173438\pi$
$97$	1634.00	1.71039	0.855194	+	0.518309i	$0.173438\pi$
$98$	0	0
$99$	−180.000	−0.182734
$100$	0	0
$101$	382.000	0.376341	0.188170	−	0.982136i	$-0.439744\pi$
$101$	382.000	0.376341	0.188170	+	0.982136i	$0.439744\pi$
$102$	0	0
$103$	−936.000	−0.895406	−0.447703	−	0.894182i	$-0.647758\pi$
$103$	−936.000	−0.895406	−0.447703	+	0.894182i	$0.647758\pi$
$104$	0	0
$105$	120.000	0.111531
$106$	0	0
$107$	−84.0000	−0.0758933	−0.0379467	−	0.999280i	$-0.512082\pi$
$107$	−84.0000	−0.0758933	−0.0379467	+	0.999280i	$0.512082\pi$
$108$	0	0
$109$	−1226.00	−1.07733	−0.538667	−	0.842518i	$-0.681072\pi$
$109$	−1226.00	−1.07733	−0.538667	+	0.842518i	$0.681072\pi$
$110$	0	0
$111$	−930.000	−0.795240
$112$	0	0
$113$	1090.00	0.907421	0.453711	−	0.891149i	$-0.350100\pi$
$113$	1090.00	0.907421	0.453711	+	0.891149i	$0.350100\pi$
$114$	0	0
$115$	600.000	0.486524
$116$	0	0
$117$	−306.000	−0.241792
$118$	0	0
$119$	48.0000	0.0369761
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	1554.00	1.13918
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−672.000	−0.469530	−0.234765	−	0.972052i	$-0.575432\pi$
$127$	−672.000	−0.469530	−0.234765	+	0.972052i	$0.575432\pi$
$128$	0	0
$129$	−276.000	−0.188376
$130$	0	0
$131$	−940.000	−0.626933	−0.313466	−	0.949599i	$-0.601490\pi$
$131$	−940.000	−0.626933	−0.313466	+	0.949599i	$0.601490\pi$
$132$	0	0
$133$	−96.0000	−0.0625884
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−414.000	−0.258178	−0.129089	−	0.991633i	$-0.541205\pi$
$137$	−414.000	−0.258178	−0.129089	+	0.991633i	$0.541205\pi$
$138$	0	0
$139$	−468.000	−0.285577	−0.142789	−	0.989753i	$-0.545607\pi$
$139$	−468.000	−0.285577	−0.142789	+	0.989753i	$0.545607\pi$
$140$	0	0
$141$	264.000	0.157680
$142$	0	0
$143$	680.000	0.397654
$144$	0	0
$145$	1230.00	0.704455
$146$	0	0
$147$	837.000	0.469623
$148$	0	0
$149$	−1426.00	−0.784043	−0.392022	−	0.919956i	$-0.628224\pi$
$149$	−1426.00	−0.784043	−0.392022	+	0.919956i	$0.628224\pi$
$150$	0	0
$151$	1192.00	0.642408	0.321204	−	0.947010i	$-0.395913\pi$
$151$	1192.00	0.642408	0.321204	+	0.947010i	$0.395913\pi$
$152$	0	0
$153$	−54.0000	−0.0285336
$154$	0	0
$155$	155.000	0.0803219
$156$	0	0
$157$	−1338.00	−0.680153	−0.340077	−	0.940398i	$-0.610453\pi$
$157$	−1338.00	−0.680153	−0.340077	+	0.940398i	$0.610453\pi$
$158$	0	0
$159$	2214.00	1.10429
$160$	0	0
$161$	−960.000	−0.469929
$162$	0	0
$163$	924.000	0.444008	0.222004	−	0.975046i	$-0.428740\pi$
$163$	924.000	0.444008	0.222004	+	0.975046i	$0.428740\pi$
$164$	0	0
$165$	300.000	0.141545
$166$	0	0
$167$	−1064.00	−0.493023	−0.246511	−	0.969140i	$-0.579284\pi$
$167$	−1064.00	−0.493023	−0.246511	+	0.969140i	$0.579284\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	108.000	0.0482980
$172$	0	0
$173$	−2698.00	−1.18569	−0.592847	−	0.805315i	$-0.701996\pi$
$173$	−2698.00	−1.18569	−0.592847	+	0.805315i	$0.701996\pi$
$174$	0	0
$175$	−200.000	−0.0863919
$176$	0	0
$177$	−804.000	−0.341426
$178$	0	0
$179$	−796.000	−0.332379	−0.166189	−	0.986094i	$-0.553146\pi$
$179$	−796.000	−0.332379	−0.166189	+	0.986094i	$0.553146\pi$
$180$	0	0
$181$	−2810.00	−1.15395	−0.576977	−	0.816760i	$-0.695768\pi$
$181$	−2810.00	−1.15395	−0.576977	+	0.816760i	$0.695768\pi$
$182$	0	0
$183$	−1098.00	−0.443533
$184$	0	0
$185$	1550.00	0.615991
$186$	0	0
$187$	120.000	0.0469266
$188$	0	0
$189$	216.000	0.0831306
$190$	0	0
$191$	−3912.00	−1.48200	−0.741001	−	0.671504i	$-0.765649\pi$
$191$	−3912.00	−1.48200	−0.741001	+	0.671504i	$0.765649\pi$
$192$	0	0
$193$	290.000	0.108159	0.0540794	−	0.998537i	$-0.482778\pi$
$193$	290.000	0.108159	0.0540794	+	0.998537i	$0.482778\pi$
$194$	0	0
$195$	510.000	0.187292
$196$	0	0
$197$	−4434.00	−1.60360	−0.801801	−	0.597592i	$-0.796124\pi$
$197$	−4434.00	−1.60360	−0.801801	+	0.597592i	$0.796124\pi$
$198$	0	0
$199$	1064.00	0.379020	0.189510	−	0.981879i	$-0.439310\pi$
$199$	1064.00	0.379020	0.189510	+	0.981879i	$0.439310\pi$
$200$	0	0
$201$	−660.000	−0.231606
$202$	0	0
$203$	−1968.00	−0.680426
$204$	0	0
$205$	−2590.00	−0.882407
$206$	0	0
$207$	1080.00	0.362634
$208$	0	0
$209$	−240.000	−0.0794313
$210$	0	0
$211$	716.000	0.233609	0.116804	−	0.993155i	$-0.462735\pi$
$211$	716.000	0.233609	0.116804	+	0.993155i	$0.462735\pi$
$212$	0	0
$213$	1536.00	0.494108
$214$	0	0
$215$	460.000	0.145915
$216$	0	0
$217$	−248.000	−0.0775822
$218$	0	0
$219$	2274.00	0.701656
$220$	0	0
$221$	204.000	0.0620929
$222$	0	0
$223$	1328.00	0.398787	0.199393	−	0.979920i	$-0.436103\pi$
$223$	1328.00	0.398787	0.199393	+	0.979920i	$0.436103\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	692.000	0.202333	0.101167	−	0.994869i	$-0.467742\pi$
$227$	692.000	0.202333	0.101167	+	0.994869i	$0.467742\pi$
$228$	0	0
$229$	−5594.00	−1.61424	−0.807122	−	0.590385i	$-0.798976\pi$
$229$	−5594.00	−1.61424	−0.807122	+	0.590385i	$0.798976\pi$
$230$	0	0
$231$	−480.000	−0.136717
$232$	0	0
$233$	−2294.00	−0.645000	−0.322500	−	0.946570i	$-0.604523\pi$
$233$	−2294.00	−0.645000	−0.322500	+	0.946570i	$0.604523\pi$
$234$	0	0
$235$	−440.000	−0.122138
$236$	0	0
$237$	−480.000	−0.131558
$238$	0	0
$239$	4256.00	1.15187	0.575937	−	0.817494i	$-0.304637\pi$
$239$	4256.00	1.15187	0.575937	+	0.817494i	$0.304637\pi$
$240$	0	0
$241$	−622.000	−0.166251	−0.0831256	−	0.996539i	$-0.526490\pi$
$241$	−622.000	−0.166251	−0.0831256	+	0.996539i	$0.526490\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−1395.00	−0.363768
$246$	0	0
$247$	−408.000	−0.105103
$248$	0	0
$249$	−4044.00	−1.02923
$250$	0	0
$251$	−1604.00	−0.403361	−0.201680	−	0.979451i	$-0.564640\pi$
$251$	−1604.00	−0.403361	−0.201680	+	0.979451i	$0.564640\pi$
$252$	0	0
$253$	−2400.00	−0.596390
$254$	0	0
$255$	90.0000	0.0221020
$256$	0	0
$257$	1842.00	0.447085	0.223542	−	0.974694i	$-0.428238\pi$
$257$	1842.00	0.447085	0.223542	+	0.974694i	$0.428238\pi$
$258$	0	0
$259$	−2480.00	−0.594980
$260$	0	0
$261$	2214.00	0.525070
$262$	0	0
$263$	−5224.00	−1.22481	−0.612406	−	0.790543i	$-0.709798\pi$
$263$	−5224.00	−1.22481	−0.612406	+	0.790543i	$0.709798\pi$
$264$	0	0
$265$	−3690.00	−0.855377
$266$	0	0
$267$	−54.0000	−0.0123773
$268$	0	0
$269$	2406.00	0.545340	0.272670	−	0.962108i	$-0.412093\pi$
$269$	2406.00	0.545340	0.272670	+	0.962108i	$0.412093\pi$
$270$	0	0
$271$	−912.000	−0.204428	−0.102214	−	0.994762i	$-0.532593\pi$
$271$	−912.000	−0.204428	−0.102214	+	0.994762i	$0.532593\pi$
$272$	0	0
$273$	−816.000	−0.180903
$274$	0	0
$275$	−500.000	−0.109640
$276$	0	0
$277$	982.000	0.213006	0.106503	−	0.994312i	$-0.466035\pi$
$277$	982.000	0.213006	0.106503	+	0.994312i	$0.466035\pi$
$278$	0	0
$279$	279.000	0.0598684
$280$	0	0
$281$	−5478.00	−1.16295	−0.581477	−	0.813563i	$-0.697525\pi$
$281$	−5478.00	−1.16295	−0.581477	+	0.813563i	$0.697525\pi$
$282$	0	0
$283$	4772.00	1.00235	0.501177	−	0.865345i	$-0.332901\pi$
$283$	4772.00	1.00235	0.501177	+	0.865345i	$0.332901\pi$
$284$	0	0
$285$	−180.000	−0.0374115
$286$	0	0
$287$	4144.00	0.852309
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	−4902.00	−0.987493
$292$	0	0
$293$	−418.000	−0.0833441	−0.0416720	−	0.999131i	$-0.513268\pi$
$293$	−418.000	−0.0833441	−0.0416720	+	0.999131i	$0.513268\pi$
$294$	0	0
$295$	1340.00	0.264467
$296$	0	0
$297$	540.000	0.105502
$298$	0	0
$299$	−4080.00	−0.789139
$300$	0	0
$301$	−736.000	−0.140938
$302$	0	0
$303$	−1146.00	−0.217280
$304$	0	0
$305$	1830.00	0.343559
$306$	0	0
$307$	−9412.00	−1.74974	−0.874872	−	0.484355i	$-0.839054\pi$
$307$	−9412.00	−1.74974	−0.874872	+	0.484355i	$0.839054\pi$
$308$	0	0
$309$	2808.00	0.516963
$310$	0	0
$311$	9744.00	1.77663	0.888314	−	0.459236i	$-0.151877\pi$
$311$	9744.00	1.77663	0.888314	+	0.459236i	$0.151877\pi$
$312$	0	0
$313$	−1382.00	−0.249570	−0.124785	−	0.992184i	$-0.539824\pi$
$313$	−1382.00	−0.249570	−0.124785	+	0.992184i	$0.539824\pi$
$314$	0	0
$315$	−360.000	−0.0643927
$316$	0	0
$317$	582.000	0.103118	0.0515590	−	0.998670i	$-0.483581\pi$
$317$	582.000	0.103118	0.0515590	+	0.998670i	$0.483581\pi$
$318$	0	0
$319$	−4920.00	−0.863533
$320$	0	0
$321$	252.000	0.0438170
$322$	0	0
$323$	−72.0000	−0.0124031
$324$	0	0
$325$	−850.000	−0.145075
$326$	0	0
$327$	3678.00	0.622000
$328$	0	0
$329$	704.000	0.117972
$330$	0	0
$331$	−9204.00	−1.52839	−0.764196	−	0.644984i	$-0.776864\pi$
$331$	−9204.00	−1.52839	−0.764196	+	0.644984i	$0.776864\pi$
$332$	0	0
$333$	2790.00	0.459132
$334$	0	0
$335$	1100.00	0.179401
$336$	0	0
$337$	2242.00	0.362402	0.181201	−	0.983446i	$-0.442002\pi$
$337$	2242.00	0.362402	0.181201	+	0.983446i	$0.442002\pi$
$338$	0	0
$339$	−3270.00	−0.523900
$340$	0	0
$341$	−620.000	−0.0984601
$342$	0	0
$343$	4976.00	0.783320
$344$	0	0
$345$	−1800.00	−0.280895
$346$	0	0
$347$	−5412.00	−0.837266	−0.418633	−	0.908155i	$-0.637491\pi$
$347$	−5412.00	−0.837266	−0.418633	+	0.908155i	$0.637491\pi$
$348$	0	0
$349$	−11690.0	−1.79298	−0.896492	−	0.443060i	$-0.853893\pi$
$349$	−11690.0	−1.79298	−0.896492	+	0.443060i	$0.853893\pi$
$350$	0	0
$351$	918.000	0.139599
$352$	0	0
$353$	7770.00	1.17154	0.585772	−	0.810476i	$-0.300791\pi$
$353$	7770.00	1.17154	0.585772	+	0.810476i	$0.300791\pi$
$354$	0	0
$355$	−2560.00	−0.382734
$356$	0	0
$357$	−144.000	−0.0213481
$358$	0	0
$359$	−9936.00	−1.46073	−0.730365	−	0.683057i	$-0.760650\pi$
$359$	−9936.00	−1.46073	−0.730365	+	0.683057i	$0.760650\pi$
$360$	0	0
$361$	−6715.00	−0.979006
$362$	0	0
$363$	2793.00	0.403842
$364$	0	0
$365$	−3790.00	−0.543500
$366$	0	0
$367$	−4480.00	−0.637205	−0.318602	−	0.947888i	$-0.603213\pi$
$367$	−4480.00	−0.637205	−0.318602	+	0.947888i	$0.603213\pi$
$368$	0	0
$369$	−4662.00	−0.657707
$370$	0	0
$371$	5904.00	0.826201
$372$	0	0
$373$	−786.000	−0.109109	−0.0545543	−	0.998511i	$-0.517374\pi$
$373$	−786.000	−0.109109	−0.0545543	+	0.998511i	$0.517374\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−8364.00	−1.14262
$378$	0	0
$379$	−6844.00	−0.927580	−0.463790	−	0.885945i	$-0.653511\pi$
$379$	−6844.00	−0.927580	−0.463790	+	0.885945i	$0.653511\pi$
$380$	0	0
$381$	2016.00	0.271083
$382$	0	0
$383$	−7776.00	−1.03743	−0.518714	−	0.854948i	$-0.673589\pi$
$383$	−7776.00	−1.03743	−0.518714	+	0.854948i	$0.673589\pi$
$384$	0	0
$385$	800.000	0.105901
$386$	0	0
$387$	828.000	0.108759
$388$	0	0
$389$	−1634.00	−0.212975	−0.106487	−	0.994314i	$-0.533960\pi$
$389$	−1634.00	−0.212975	−0.106487	+	0.994314i	$0.533960\pi$
$390$	0	0
$391$	−720.000	−0.0931252
$392$	0	0
$393$	2820.00	0.361960
$394$	0	0
$395$	800.000	0.101905
$396$	0	0
$397$	13542.0	1.71197	0.855987	−	0.516998i	$-0.172950\pi$
$397$	13542.0	1.71197	0.855987	+	0.516998i	$0.172950\pi$
$398$	0	0
$399$	288.000	0.0361354
$400$	0	0
$401$	7626.00	0.949686	0.474843	−	0.880070i	$-0.342505\pi$
$401$	7626.00	0.949686	0.474843	+	0.880070i	$0.342505\pi$
$402$	0	0
$403$	−1054.00	−0.130282
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−6200.00	−0.755092
$408$	0	0
$409$	4362.00	0.527352	0.263676	−	0.964611i	$-0.415065\pi$
$409$	4362.00	0.527352	0.263676	+	0.964611i	$0.415065\pi$
$410$	0	0
$411$	1242.00	0.149059
$412$	0	0
$413$	−2144.00	−0.255446
$414$	0	0
$415$	6740.00	0.797238
$416$	0	0
$417$	1404.00	0.164878
$418$	0	0
$419$	660.000	0.0769525	0.0384763	−	0.999260i	$-0.487750\pi$
$419$	660.000	0.0769525	0.0384763	+	0.999260i	$0.487750\pi$
$420$	0	0
$421$	−5970.00	−0.691116	−0.345558	−	0.938397i	$-0.612310\pi$
$421$	−5970.00	−0.691116	−0.345558	+	0.938397i	$0.612310\pi$
$422$	0	0
$423$	−792.000	−0.0910363
$424$	0	0
$425$	−150.000	−0.0171202
$426$	0	0
$427$	−2928.00	−0.331840
$428$	0	0
$429$	−2040.00	−0.229585
$430$	0	0
$431$	9176.00	1.02550	0.512752	−	0.858537i	$-0.328626\pi$
$431$	9176.00	1.02550	0.512752	+	0.858537i	$0.328626\pi$
$432$	0	0
$433$	−4382.00	−0.486341	−0.243170	−	0.969984i	$-0.578187\pi$
$433$	−4382.00	−0.486341	−0.243170	+	0.969984i	$0.578187\pi$
$434$	0	0
$435$	−3690.00	−0.406717
$436$	0	0
$437$	1440.00	0.157631
$438$	0	0
$439$	13176.0	1.43247	0.716237	−	0.697857i	$-0.245863\pi$
$439$	13176.0	1.43247	0.716237	+	0.697857i	$0.245863\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	10860.0	1.16473	0.582364	−	0.812928i	$-0.302128\pi$
$443$	10860.0	1.16473	0.582364	+	0.812928i	$0.302128\pi$
$444$	0	0
$445$	90.0000	0.00958744
$446$	0	0
$447$	4278.00	0.452668
$448$	0	0
$449$	−14502.0	−1.52426	−0.762129	−	0.647425i	$-0.775846\pi$
$449$	−14502.0	−1.52426	−0.762129	+	0.647425i	$0.775846\pi$
$450$	0	0
$451$	10360.0	1.08167
$452$	0	0
$453$	−3576.00	−0.370894
$454$	0	0
$455$	1360.00	0.140127
$456$	0	0
$457$	−8998.00	−0.921026	−0.460513	−	0.887653i	$-0.652334\pi$
$457$	−8998.00	−0.921026	−0.460513	+	0.887653i	$0.652334\pi$
$458$	0	0
$459$	162.000	0.0164739
$460$	0	0
$461$	−16650.0	−1.68214	−0.841071	−	0.540924i	$-0.818075\pi$
$461$	−16650.0	−1.68214	−0.841071	+	0.540924i	$0.818075\pi$
$462$	0	0
$463$	−15600.0	−1.56586	−0.782930	−	0.622109i	$-0.786276\pi$
$463$	−15600.0	−1.56586	−0.782930	+	0.622109i	$0.786276\pi$
$464$	0	0
$465$	−465.000	−0.0463739
$466$	0	0
$467$	−6076.00	−0.602064	−0.301032	−	0.953614i	$-0.597331\pi$
$467$	−6076.00	−0.602064	−0.301032	+	0.953614i	$0.597331\pi$
$468$	0	0
$469$	−1760.00	−0.173282
$470$	0	0
$471$	4014.00	0.392687
$472$	0	0
$473$	−1840.00	−0.178865
$474$	0	0
$475$	300.000	0.0289788
$476$	0	0
$477$	−6642.00	−0.637560
$478$	0	0
$479$	9800.00	0.934809	0.467404	−	0.884044i	$-0.345189\pi$
$479$	9800.00	0.934809	0.467404	+	0.884044i	$0.345189\pi$
$480$	0	0
$481$	−10540.0	−0.999132
$482$	0	0
$483$	2880.00	0.271314
$484$	0	0
$485$	8170.00	0.764908
$486$	0	0
$487$	−8840.00	−0.822543	−0.411272	−	0.911513i	$-0.634915\pi$
$487$	−8840.00	−0.822543	−0.411272	+	0.911513i	$0.634915\pi$
$488$	0	0
$489$	−2772.00	−0.256348
$490$	0	0
$491$	4716.00	0.433463	0.216731	−	0.976231i	$-0.430460\pi$
$491$	4716.00	0.433463	0.216731	+	0.976231i	$0.430460\pi$
$492$	0	0
$493$	−1476.00	−0.134839
$494$	0	0
$495$	−900.000	−0.0817212
$496$	0	0
$497$	4096.00	0.369679
$498$	0	0
$499$	−4748.00	−0.425951	−0.212976	−	0.977058i	$-0.568315\pi$
$499$	−4748.00	−0.425951	−0.212976	+	0.977058i	$0.568315\pi$
$500$	0	0
$501$	3192.00	0.284647
$502$	0	0
$503$	16160.0	1.43248	0.716241	−	0.697853i	$-0.245861\pi$
$503$	16160.0	1.43248	0.716241	+	0.697853i	$0.245861\pi$
$504$	0	0
$505$	1910.00	0.168305
$506$	0	0
$507$	3123.00	0.273565
$508$	0	0
$509$	12886.0	1.12213	0.561063	−	0.827773i	$-0.310393\pi$
$509$	12886.0	1.12213	0.561063	+	0.827773i	$0.310393\pi$
$510$	0	0
$511$	6064.00	0.524962
$512$	0	0
$513$	−324.000	−0.0278849
$514$	0	0
$515$	−4680.00	−0.400438
$516$	0	0
$517$	1760.00	0.149719
$518$	0	0
$519$	8094.00	0.684561
$520$	0	0
$521$	20058.0	1.68667	0.843337	−	0.537385i	$-0.180588\pi$
$521$	20058.0	1.68667	0.843337	+	0.537385i	$0.180588\pi$
$522$	0	0
$523$	8188.00	0.684582	0.342291	−	0.939594i	$-0.388797\pi$
$523$	8188.00	0.684582	0.342291	+	0.939594i	$0.388797\pi$
$524$	0	0
$525$	600.000	0.0498784
$526$	0	0
$527$	−186.000	−0.0153744
$528$	0	0
$529$	2233.00	0.183529
$530$	0	0
$531$	2412.00	0.197122
$532$	0	0
$533$	17612.0	1.43126
$534$	0	0
$535$	−420.000	−0.0339405
$536$	0	0
$537$	2388.00	0.191899
$538$	0	0
$539$	5580.00	0.445914
$540$	0	0
$541$	−10362.0	−0.823470	−0.411735	−	0.911304i	$-0.635077\pi$
$541$	−10362.0	−0.823470	−0.411735	+	0.911304i	$0.635077\pi$
$542$	0	0
$543$	8430.00	0.666236
$544$	0	0
$545$	−6130.00	−0.481799
$546$	0	0
$547$	−16020.0	−1.25222	−0.626111	−	0.779734i	$-0.715354\pi$
$547$	−16020.0	−1.25222	−0.626111	+	0.779734i	$0.715354\pi$
$548$	0	0
$549$	3294.00	0.256074
$550$	0	0
$551$	2952.00	0.228239
$552$	0	0
$553$	−1280.00	−0.0984288
$554$	0	0
$555$	−4650.00	−0.355642
$556$	0	0
$557$	−10410.0	−0.791896	−0.395948	−	0.918273i	$-0.629584\pi$
$557$	−10410.0	−0.791896	−0.395948	+	0.918273i	$0.629584\pi$
$558$	0	0
$559$	−3128.00	−0.236673
$560$	0	0
$561$	−360.000	−0.0270931
$562$	0	0
$563$	20724.0	1.55135	0.775677	−	0.631130i	$-0.217409\pi$
$563$	20724.0	1.55135	0.775677	+	0.631130i	$0.217409\pi$
$564$	0	0
$565$	5450.00	0.405811
$566$	0	0
$567$	−648.000	−0.0479955
$568$	0	0
$569$	6594.00	0.485826	0.242913	−	0.970048i	$-0.421897\pi$
$569$	6594.00	0.485826	0.242913	+	0.970048i	$0.421897\pi$
$570$	0	0
$571$	−18596.0	−1.36290	−0.681452	−	0.731863i	$-0.738651\pi$
$571$	−18596.0	−1.36290	−0.681452	+	0.731863i	$0.738651\pi$
$572$	0	0
$573$	11736.0	0.855634
$574$	0	0
$575$	3000.00	0.217580
$576$	0	0
$577$	1218.00	0.0878787	0.0439393	−	0.999034i	$-0.486009\pi$
$577$	1218.00	0.0878787	0.0439393	+	0.999034i	$0.486009\pi$
$578$	0	0
$579$	−870.000	−0.0624455
$580$	0	0
$581$	−10784.0	−0.770044
$582$	0	0
$583$	14760.0	1.04854
$584$	0	0
$585$	−1530.00	−0.108133
$586$	0	0
$587$	17868.0	1.25637	0.628187	−	0.778063i	$-0.283797\pi$
$587$	17868.0	1.25637	0.628187	+	0.778063i	$0.283797\pi$
$588$	0	0
$589$	372.000	0.0260238
$590$	0	0
$591$	13302.0	0.925840
$592$	0	0
$593$	21394.0	1.48153	0.740764	−	0.671765i	$-0.234464\pi$
$593$	21394.0	1.48153	0.740764	+	0.671765i	$0.234464\pi$
$594$	0	0
$595$	240.000	0.0165362
$596$	0	0
$597$	−3192.00	−0.218827
$598$	0	0
$599$	−20736.0	−1.41444	−0.707220	−	0.706993i	$-0.750051\pi$
$599$	−20736.0	−1.41444	−0.707220	+	0.706993i	$0.750051\pi$
$600$	0	0
$601$	−14998.0	−1.01794	−0.508969	−	0.860785i	$-0.669973\pi$
$601$	−14998.0	−1.01794	−0.508969	+	0.860785i	$0.669973\pi$
$602$	0	0
$603$	1980.00	0.133718
$604$	0	0
$605$	−4655.00	−0.312814
$606$	0	0
$607$	−9664.00	−0.646210	−0.323105	−	0.946363i	$-0.604727\pi$
$607$	−9664.00	−0.646210	−0.323105	+	0.946363i	$0.604727\pi$
$608$	0	0
$609$	5904.00	0.392844
$610$	0	0
$611$	2992.00	0.198107
$612$	0	0
$613$	15654.0	1.03142	0.515709	−	0.856764i	$-0.327529\pi$
$613$	15654.0	1.03142	0.515709	+	0.856764i	$0.327529\pi$
$614$	0	0
$615$	7770.00	0.509458
$616$	0	0
$617$	14202.0	0.926663	0.463331	−	0.886185i	$-0.346654\pi$
$617$	14202.0	0.926663	0.463331	+	0.886185i	$0.346654\pi$
$618$	0	0
$619$	−8132.00	−0.528033	−0.264017	−	0.964518i	$-0.585047\pi$
$619$	−8132.00	−0.528033	−0.264017	+	0.964518i	$0.585047\pi$
$620$	0	0
$621$	−3240.00	−0.209367
$622$	0	0
$623$	−144.000	−0.00926041
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	720.000	0.0458597
$628$	0	0
$629$	−1860.00	−0.117906
$630$	0	0
$631$	28232.0	1.78114	0.890569	−	0.454848i	$-0.150306\pi$
$631$	28232.0	1.78114	0.890569	+	0.454848i	$0.150306\pi$
$632$	0	0
$633$	−2148.00	−0.134874
$634$	0	0
$635$	−3360.00	−0.209980
$636$	0	0
$637$	9486.00	0.590030
$638$	0	0
$639$	−4608.00	−0.285273
$640$	0	0
$641$	−11446.0	−0.705288	−0.352644	−	0.935757i	$-0.614717\pi$
$641$	−11446.0	−0.705288	−0.352644	+	0.935757i	$0.614717\pi$
$642$	0	0
$643$	8132.00	0.498748	0.249374	−	0.968407i	$-0.419775\pi$
$643$	8132.00	0.498748	0.249374	+	0.968407i	$0.419775\pi$
$644$	0	0
$645$	−1380.00	−0.0842441
$646$	0	0
$647$	4536.00	0.275624	0.137812	−	0.990458i	$-0.455993\pi$
$647$	4536.00	0.275624	0.137812	+	0.990458i	$0.455993\pi$
$648$	0	0
$649$	−5360.00	−0.324189
$650$	0	0
$651$	744.000	0.0447921
$652$	0	0
$653$	6582.00	0.394446	0.197223	−	0.980359i	$-0.436808\pi$
$653$	6582.00	0.394446	0.197223	+	0.980359i	$0.436808\pi$
$654$	0	0
$655$	−4700.00	−0.280373
$656$	0	0
$657$	−6822.00	−0.405101
$658$	0	0
$659$	15460.0	0.913864	0.456932	−	0.889502i	$-0.348948\pi$
$659$	15460.0	0.913864	0.456932	+	0.889502i	$0.348948\pi$
$660$	0	0
$661$	15662.0	0.921605	0.460803	−	0.887503i	$-0.347562\pi$
$661$	15662.0	0.921605	0.460803	+	0.887503i	$0.347562\pi$
$662$	0	0
$663$	−612.000	−0.0358493
$664$	0	0
$665$	−480.000	−0.0279904
$666$	0	0
$667$	29520.0	1.71367
$668$	0	0
$669$	−3984.00	−0.230240
$670$	0	0
$671$	−7320.00	−0.421141
$672$	0	0
$673$	−29326.0	−1.67969	−0.839847	−	0.542823i	$-0.817355\pi$
$673$	−29326.0	−1.67969	−0.839847	+	0.542823i	$0.817355\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	702.000	0.0398524	0.0199262	−	0.999801i	$-0.493657\pi$
$677$	702.000	0.0398524	0.0199262	+	0.999801i	$0.493657\pi$
$678$	0	0
$679$	−13072.0	−0.738818
$680$	0	0
$681$	−2076.00	−0.116817
$682$	0	0
$683$	28076.0	1.57291	0.786456	−	0.617647i	$-0.211914\pi$
$683$	28076.0	1.57291	0.786456	+	0.617647i	$0.211914\pi$
$684$	0	0
$685$	−2070.00	−0.115461
$686$	0	0
$687$	16782.0	0.931984
$688$	0	0
$689$	25092.0	1.38742
$690$	0	0
$691$	−29188.0	−1.60689	−0.803447	−	0.595376i	$-0.797003\pi$
$691$	−29188.0	−1.60689	−0.803447	+	0.595376i	$0.797003\pi$
$692$	0	0
$693$	1440.00	0.0789337
$694$	0	0
$695$	−2340.00	−0.127714
$696$	0	0
$697$	3108.00	0.168901
$698$	0	0
$699$	6882.00	0.372391
$700$	0	0
$701$	8966.00	0.483083	0.241541	−	0.970390i	$-0.422347\pi$
$701$	8966.00	0.483083	0.241541	+	0.970390i	$0.422347\pi$
$702$	0	0
$703$	3720.00	0.199577
$704$	0	0
$705$	1320.00	0.0705164
$706$	0	0
$707$	−3056.00	−0.162564
$708$	0	0
$709$	−31178.0	−1.65150	−0.825751	−	0.564035i	$-0.809248\pi$
$709$	−31178.0	−1.65150	−0.825751	+	0.564035i	$0.809248\pi$
$710$	0	0
$711$	1440.00	0.0759553
$712$	0	0
$713$	3720.00	0.195393
$714$	0	0
$715$	3400.00	0.177836
$716$	0	0
$717$	−12768.0	−0.665034
$718$	0	0
$719$	−37632.0	−1.95193	−0.975964	−	0.217930i	$-0.930070\pi$
$719$	−37632.0	−1.95193	−0.975964	+	0.217930i	$0.930070\pi$
$720$	0	0
$721$	7488.00	0.386779
$722$	0	0
$723$	1866.00	0.0959852
$724$	0	0
$725$	6150.00	0.315042
$726$	0	0
$727$	26424.0	1.34802	0.674011	−	0.738721i	$-0.264570\pi$
$727$	26424.0	1.34802	0.674011	+	0.738721i	$0.264570\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−552.000	−0.0279295
$732$	0	0
$733$	35926.0	1.81031	0.905155	−	0.425082i	$-0.139755\pi$
$733$	35926.0	1.81031	0.905155	+	0.425082i	$0.139755\pi$
$734$	0	0
$735$	4185.00	0.210022
$736$	0	0
$737$	−4400.00	−0.219913
$738$	0	0
$739$	4292.00	0.213645	0.106823	−	0.994278i	$-0.465932\pi$
$739$	4292.00	0.213645	0.106823	+	0.994278i	$0.465932\pi$
$740$	0	0
$741$	1224.00	0.0606812
$742$	0	0
$743$	−20648.0	−1.01952	−0.509759	−	0.860317i	$-0.670265\pi$
$743$	−20648.0	−1.01952	−0.509759	+	0.860317i	$0.670265\pi$
$744$	0	0
$745$	−7130.00	−0.350635
$746$	0	0
$747$	12132.0	0.594226
$748$	0	0
$749$	672.000	0.0327828
$750$	0	0
$751$	−21920.0	−1.06508	−0.532538	−	0.846406i	$-0.678762\pi$
$751$	−21920.0	−1.06508	−0.532538	+	0.846406i	$0.678762\pi$
$752$	0	0
$753$	4812.00	0.232881
$754$	0	0
$755$	5960.00	0.287294
$756$	0	0
$757$	−22842.0	−1.09671	−0.548353	−	0.836247i	$-0.684745\pi$
$757$	−22842.0	−1.09671	−0.548353	+	0.836247i	$0.684745\pi$
$758$	0	0
$759$	7200.00	0.344326
$760$	0	0
$761$	29666.0	1.41313	0.706565	−	0.707648i	$-0.250244\pi$
$761$	29666.0	1.41313	0.706565	+	0.707648i	$0.250244\pi$
$762$	0	0
$763$	9808.00	0.465365
$764$	0	0
$765$	−270.000	−0.0127606
$766$	0	0
$767$	−9112.00	−0.428964
$768$	0	0
$769$	−18718.0	−0.877748	−0.438874	−	0.898549i	$-0.644623\pi$
$769$	−18718.0	−0.877748	−0.438874	+	0.898549i	$0.644623\pi$
$770$	0	0
$771$	−5526.00	−0.258125
$772$	0	0
$773$	−38290.0	−1.78162	−0.890812	−	0.454372i	$-0.849864\pi$
$773$	−38290.0	−1.78162	−0.890812	+	0.454372i	$0.849864\pi$
$774$	0	0
$775$	775.000	0.0359211
$776$	0	0
$777$	7440.00	0.343512
$778$	0	0
$779$	−6216.00	−0.285894
$780$	0	0
$781$	10240.0	0.469163
$782$	0	0
$783$	−6642.00	−0.303149
$784$	0	0
$785$	−6690.00	−0.304174
$786$	0	0
$787$	−30268.0	−1.37095	−0.685475	−	0.728096i	$-0.740405\pi$
$787$	−30268.0	−1.37095	−0.685475	+	0.728096i	$0.740405\pi$
$788$	0	0
$789$	15672.0	0.707146
$790$	0	0
$791$	−8720.00	−0.391969
$792$	0	0
$793$	−12444.0	−0.557250
$794$	0	0
$795$	11070.0	0.493852
$796$	0	0
$797$	−27386.0	−1.21714	−0.608571	−	0.793500i	$-0.708257\pi$
$797$	−27386.0	−1.21714	−0.608571	+	0.793500i	$0.708257\pi$
$798$	0	0
$799$	528.000	0.0233783
$800$	0	0
$801$	162.000	0.00714605
$802$	0	0
$803$	15160.0	0.666233
$804$	0	0
$805$	−4800.00	−0.210159
$806$	0	0
$807$	−7218.00	−0.314852
$808$	0	0
$809$	6498.00	0.282395	0.141197	−	0.989981i	$-0.454905\pi$
$809$	6498.00	0.282395	0.141197	+	0.989981i	$0.454905\pi$
$810$	0	0
$811$	5636.00	0.244028	0.122014	−	0.992528i	$-0.461065\pi$
$811$	5636.00	0.244028	0.122014	+	0.992528i	$0.461065\pi$
$812$	0	0
$813$	2736.00	0.118027
$814$	0	0
$815$	4620.00	0.198566
$816$	0	0
$817$	1104.00	0.0472755
$818$	0	0
$819$	2448.00	0.104444
$820$	0	0
$821$	−31266.0	−1.32910	−0.664550	−	0.747244i	$-0.731377\pi$
$821$	−31266.0	−1.32910	−0.664550	+	0.747244i	$0.731377\pi$
$822$	0	0
$823$	−19560.0	−0.828455	−0.414228	−	0.910173i	$-0.635948\pi$
$823$	−19560.0	−0.828455	−0.414228	+	0.910173i	$0.635948\pi$
$824$	0	0
$825$	1500.00	0.0633010
$826$	0	0
$827$	908.000	0.0381793	0.0190896	−	0.999818i	$-0.493923\pi$
$827$	908.000	0.0381793	0.0190896	+	0.999818i	$0.493923\pi$
$828$	0	0
$829$	190.000	0.00796016	0.00398008	−	0.999992i	$-0.498733\pi$
$829$	190.000	0.00796016	0.00398008	+	0.999992i	$0.498733\pi$
$830$	0	0
$831$	−2946.00	−0.122979
$832$	0	0
$833$	1674.00	0.0696286
$834$	0	0
$835$	−5320.00	−0.220486
$836$	0	0
$837$	−837.000	−0.0345651
$838$	0	0
$839$	32048.0	1.31874	0.659368	−	0.751820i	$-0.270824\pi$
$839$	32048.0	1.31874	0.659368	+	0.751820i	$0.270824\pi$
$840$	0	0
$841$	36127.0	1.48128
$842$	0	0
$843$	16434.0	0.671432
$844$	0	0
$845$	−5205.00	−0.211902
$846$	0	0
$847$	7448.00	0.302144
$848$	0	0
$849$	−14316.0	−0.578709
$850$	0	0
$851$	37200.0	1.49847
$852$	0	0
$853$	−9666.00	−0.387992	−0.193996	−	0.981002i	$-0.562145\pi$
$853$	−9666.00	−0.387992	−0.193996	+	0.981002i	$0.562145\pi$
$854$	0	0
$855$	540.000	0.0215995
$856$	0	0
$857$	−11270.0	−0.449213	−0.224607	−	0.974450i	$-0.572110\pi$
$857$	−11270.0	−0.449213	−0.224607	+	0.974450i	$0.572110\pi$
$858$	0	0
$859$	11116.0	0.441529	0.220764	−	0.975327i	$-0.429145\pi$
$859$	11116.0	0.441529	0.220764	+	0.975327i	$0.429145\pi$
$860$	0	0
$861$	−12432.0	−0.492081
$862$	0	0
$863$	26352.0	1.03944	0.519718	−	0.854338i	$-0.326037\pi$
$863$	26352.0	1.03944	0.519718	+	0.854338i	$0.326037\pi$
$864$	0	0
$865$	−13490.0	−0.530259
$866$	0	0
$867$	14631.0	0.573120
$868$	0	0
$869$	−3200.00	−0.124917
$870$	0	0
$871$	−7480.00	−0.290988
$872$	0	0
$873$	14706.0	0.570129
$874$	0	0
$875$	−1000.00	−0.0386356
$876$	0	0
$877$	16326.0	0.628609	0.314304	−	0.949322i	$-0.398229\pi$
$877$	16326.0	0.628609	0.314304	+	0.949322i	$0.398229\pi$
$878$	0	0
$879$	1254.00	0.0481187
$880$	0	0
$881$	38714.0	1.48049	0.740243	−	0.672340i	$-0.234711\pi$
$881$	38714.0	1.48049	0.740243	+	0.672340i	$0.234711\pi$
$882$	0	0
$883$	36228.0	1.38071	0.690356	−	0.723470i	$-0.257454\pi$
$883$	36228.0	1.38071	0.690356	+	0.723470i	$0.257454\pi$
$884$	0	0
$885$	−4020.00	−0.152690
$886$	0	0
$887$	43600.0	1.65045	0.825223	−	0.564808i	$-0.191050\pi$
$887$	43600.0	1.65045	0.825223	+	0.564808i	$0.191050\pi$
$888$	0	0
$889$	5376.00	0.202818
$890$	0	0
$891$	−1620.00	−0.0609114
$892$	0	0
$893$	−1056.00	−0.0395719
$894$	0	0
$895$	−3980.00	−0.148644
$896$	0	0
$897$	12240.0	0.455609
$898$	0	0
$899$	7626.00	0.282916
$900$	0	0
$901$	4428.00	0.163727
$902$	0	0
$903$	2208.00	0.0813706
$904$	0	0
$905$	−14050.0	−0.516064
$906$	0	0
$907$	19204.0	0.703041	0.351521	−	0.936180i	$-0.385665\pi$
$907$	19204.0	0.703041	0.351521	+	0.936180i	$0.385665\pi$
$908$	0	0
$909$	3438.00	0.125447
$910$	0	0
$911$	7168.00	0.260688	0.130344	−	0.991469i	$-0.458392\pi$
$911$	7168.00	0.260688	0.130344	+	0.991469i	$0.458392\pi$
$912$	0	0
$913$	−26960.0	−0.977268
$914$	0	0
$915$	−5490.00	−0.198354
$916$	0	0
$917$	7520.00	0.270809
$918$	0	0
$919$	−26264.0	−0.942731	−0.471365	−	0.881938i	$-0.656239\pi$
$919$	−26264.0	−0.942731	−0.471365	+	0.881938i	$0.656239\pi$
$920$	0	0
$921$	28236.0	1.01021
$922$	0	0
$923$	17408.0	0.620792
$924$	0	0
$925$	7750.00	0.275479
$926$	0	0
$927$	−8424.00	−0.298469
$928$	0	0
$929$	−51990.0	−1.83610	−0.918050	−	0.396465i	$-0.870237\pi$
$929$	−51990.0	−1.83610	−0.918050	+	0.396465i	$0.870237\pi$
$930$	0	0
$931$	−3348.00	−0.117859
$932$	0	0
$933$	−29232.0	−1.02574
$934$	0	0
$935$	600.000	0.0209862
$936$	0	0
$937$	−1926.00	−0.0671501	−0.0335751	−	0.999436i	$-0.510689\pi$
$937$	−1926.00	−0.0671501	−0.0335751	+	0.999436i	$0.510689\pi$
$938$	0	0
$939$	4146.00	0.144089
$940$	0	0
$941$	34550.0	1.19692	0.598458	−	0.801154i	$-0.295780\pi$
$941$	34550.0	1.19692	0.598458	+	0.801154i	$0.295780\pi$
$942$	0	0
$943$	−62160.0	−2.14656
$944$	0	0
$945$	1080.00	0.0371771
$946$	0	0
$947$	43716.0	1.50008	0.750042	−	0.661390i	$-0.230033\pi$
$947$	43716.0	1.50008	0.750042	+	0.661390i	$0.230033\pi$
$948$	0	0
$949$	25772.0	0.881554
$950$	0	0
$951$	−1746.00	−0.0595352
$952$	0	0
$953$	−25134.0	−0.854323	−0.427162	−	0.904175i	$-0.640486\pi$
$953$	−25134.0	−0.854323	−0.427162	+	0.904175i	$0.640486\pi$
$954$	0	0
$955$	−19560.0	−0.662771
$956$	0	0
$957$	14760.0	0.498561
$958$	0	0
$959$	3312.00	0.111522
$960$	0	0
$961$	961.000	0.0322581
$962$	0	0
$963$	−756.000	−0.0252978
$964$	0	0
$965$	1450.00	0.0483701
$966$	0	0
$967$	−1336.00	−0.0444290	−0.0222145	−	0.999753i	$-0.507072\pi$
$967$	−1336.00	−0.0444290	−0.0222145	+	0.999753i	$0.507072\pi$
$968$	0	0
$969$	216.000	0.00716091
$970$	0	0
$971$	−23508.0	−0.776939	−0.388469	−	0.921462i	$-0.626996\pi$
$971$	−23508.0	−0.776939	−0.388469	+	0.921462i	$0.626996\pi$
$972$	0	0
$973$	3744.00	0.123358
$974$	0	0
$975$	2550.00	0.0837593
$976$	0	0
$977$	−22254.0	−0.728729	−0.364365	−	0.931256i	$-0.618714\pi$
$977$	−22254.0	−0.728729	−0.364365	+	0.931256i	$0.618714\pi$
$978$	0	0
$979$	−360.000	−0.0117525
$980$	0	0
$981$	−11034.0	−0.359112
$982$	0	0
$983$	35816.0	1.16211	0.581054	−	0.813865i	$-0.302640\pi$
$983$	35816.0	1.16211	0.581054	+	0.813865i	$0.302640\pi$
$984$	0	0
$985$	−22170.0	−0.717152
$986$	0	0
$987$	−2112.00	−0.0681111
$988$	0	0
$989$	11040.0	0.354956
$990$	0	0
$991$	51760.0	1.65914	0.829571	−	0.558401i	$-0.188585\pi$
$991$	51760.0	1.65914	0.829571	+	0.558401i	$0.188585\pi$
$992$	0	0
$993$	27612.0	0.882417
$994$	0	0
$995$	5320.00	0.169503
$996$	0	0
$997$	−21666.0	−0.688234	−0.344117	−	0.938927i	$-0.611822\pi$
$997$	−21666.0	−0.688234	−0.344117	+	0.938927i	$0.611822\pi$
$998$	0	0
$999$	−8370.00	−0.265080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1860.4.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.4.a.a.1.1	✓	1	1.1	even	1	trivial

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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1860.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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