LMFDB - Embedded newform 690.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,4,Mod(1,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, 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("690.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	690.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:	$40.7113179040$
Analytic rank:	$0$
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	$\chi$	$=$	690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -5.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -5.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} -46.0000 q^{11} +12.0000 q^{12} -66.0000 q^{13} +10.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} +79.0000 q^{17} -18.0000 q^{18} -30.0000 q^{19} -20.0000 q^{20} -15.0000 q^{21} +92.0000 q^{22} +23.0000 q^{23} -24.0000 q^{24} +25.0000 q^{25} +132.000 q^{26} +27.0000 q^{27} -20.0000 q^{28} +79.0000 q^{29} +30.0000 q^{30} +225.000 q^{31} -32.0000 q^{32} -138.000 q^{33} -158.000 q^{34} +25.0000 q^{35} +36.0000 q^{36} -41.0000 q^{37} +60.0000 q^{38} -198.000 q^{39} +40.0000 q^{40} +237.000 q^{41} +30.0000 q^{42} +104.000 q^{43} -184.000 q^{44} -45.0000 q^{45} -46.0000 q^{46} +276.000 q^{47} +48.0000 q^{48} -318.000 q^{49} -50.0000 q^{50} +237.000 q^{51} -264.000 q^{52} +579.000 q^{53} -54.0000 q^{54} +230.000 q^{55} +40.0000 q^{56} -90.0000 q^{57} -158.000 q^{58} +345.000 q^{59} -60.0000 q^{60} -104.000 q^{61} -450.000 q^{62} -45.0000 q^{63} +64.0000 q^{64} +330.000 q^{65} +276.000 q^{66} -61.0000 q^{67} +316.000 q^{68} +69.0000 q^{69} -50.0000 q^{70} -253.000 q^{71} -72.0000 q^{72} -498.000 q^{73} +82.0000 q^{74} +75.0000 q^{75} -120.000 q^{76} +230.000 q^{77} +396.000 q^{78} +356.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -474.000 q^{82} +283.000 q^{83} -60.0000 q^{84} -395.000 q^{85} -208.000 q^{86} +237.000 q^{87} +368.000 q^{88} +176.000 q^{89} +90.0000 q^{90} +330.000 q^{91} +92.0000 q^{92} +675.000 q^{93} -552.000 q^{94} +150.000 q^{95} -96.0000 q^{96} +930.000 q^{97} +636.000 q^{98} -414.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−6.00000	−0.408248
$7$	−5.00000	−0.269975	−0.134987	−	0.990847i	$-0.543099\pi$
$7$	−5.00000	−0.269975	−0.134987	+	0.990847i	$0.543099\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	−46.0000	−1.26087	−0.630433	−	0.776244i	$-0.717123\pi$
$11$	−46.0000	−1.26087	−0.630433	+	0.776244i	$0.717123\pi$
$12$	12.0000	0.288675
$13$	−66.0000	−1.40809	−0.704043	−	0.710158i	$-0.748624\pi$
$13$	−66.0000	−1.40809	−0.704043	+	0.710158i	$0.748624\pi$
$14$	10.0000	0.190901
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	79.0000	1.12708	0.563539	−	0.826090i	$-0.309440\pi$
$17$	79.0000	1.12708	0.563539	+	0.826090i	$0.309440\pi$
$18$	−18.0000	−0.235702
$19$	−30.0000	−0.362235	−0.181118	−	0.983461i	$-0.557971\pi$
$19$	−30.0000	−0.362235	−0.181118	+	0.983461i	$0.557971\pi$
$20$	−20.0000	−0.223607
$21$	−15.0000	−0.155870
$22$	92.0000	0.891567
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	132.000	0.995667
$27$	27.0000	0.192450
$28$	−20.0000	−0.134987
$29$	79.0000	0.505860	0.252930	−	0.967485i	$-0.418606\pi$
$29$	79.0000	0.505860	0.252930	+	0.967485i	$0.418606\pi$
$30$	30.0000	0.182574
$31$	225.000	1.30359	0.651793	−	0.758397i	$-0.274017\pi$
$31$	225.000	1.30359	0.651793	+	0.758397i	$0.274017\pi$
$32$	−32.0000	−0.176777
$33$	−138.000	−0.727961
$34$	−158.000	−0.796964
$35$	25.0000	0.120736
$36$	36.0000	0.166667
$37$	−41.0000	−0.182172	−0.0910859	−	0.995843i	$-0.529034\pi$
$37$	−41.0000	−0.182172	−0.0910859	+	0.995843i	$0.529034\pi$
$38$	60.0000	0.256139
$39$	−198.000	−0.812958
$40$	40.0000	0.158114
$41$	237.000	0.902761	0.451380	−	0.892332i	$-0.350932\pi$
$41$	237.000	0.902761	0.451380	+	0.892332i	$0.350932\pi$
$42$	30.0000	0.110217
$43$	104.000	0.368834	0.184417	−	0.982848i	$-0.440960\pi$
$43$	104.000	0.368834	0.184417	+	0.982848i	$0.440960\pi$
$44$	−184.000	−0.630433
$45$	−45.0000	−0.149071
$46$	−46.0000	−0.147442
$47$	276.000	0.856569	0.428284	−	0.903644i	$-0.359118\pi$
$47$	276.000	0.856569	0.428284	+	0.903644i	$0.359118\pi$
$48$	48.0000	0.144338
$49$	−318.000	−0.927114
$50$	−50.0000	−0.141421
$51$	237.000	0.650718
$52$	−264.000	−0.704043
$53$	579.000	1.50060	0.750300	−	0.661098i	$-0.229909\pi$
$53$	579.000	1.50060	0.750300	+	0.661098i	$0.229909\pi$
$54$	−54.0000	−0.136083
$55$	230.000	0.563876
$56$	40.0000	0.0954504
$57$	−90.0000	−0.209137
$58$	−158.000	−0.357697
$59$	345.000	0.761274	0.380637	−	0.924725i	$-0.375705\pi$
$59$	345.000	0.761274	0.380637	+	0.924725i	$0.375705\pi$
$60$	−60.0000	−0.129099
$61$	−104.000	−0.218292	−0.109146	−	0.994026i	$-0.534812\pi$
$61$	−104.000	−0.218292	−0.109146	+	0.994026i	$0.534812\pi$
$62$	−450.000	−0.921775
$63$	−45.0000	−0.0899915
$64$	64.0000	0.125000
$65$	330.000	0.629715
$66$	276.000	0.514746
$67$	−61.0000	−0.111229	−0.0556144	−	0.998452i	$-0.517712\pi$
$67$	−61.0000	−0.111229	−0.0556144	+	0.998452i	$0.517712\pi$
$68$	316.000	0.563539
$69$	69.0000	0.120386
$70$	−50.0000	−0.0853735
$71$	−253.000	−0.422895	−0.211448	−	0.977389i	$-0.567818\pi$
$71$	−253.000	−0.422895	−0.211448	+	0.977389i	$0.567818\pi$
$72$	−72.0000	−0.117851
$73$	−498.000	−0.798445	−0.399223	−	0.916854i	$-0.630720\pi$
$73$	−498.000	−0.798445	−0.399223	+	0.916854i	$0.630720\pi$
$74$	82.0000	0.128815
$75$	75.0000	0.115470
$76$	−120.000	−0.181118
$77$	230.000	0.340402
$78$	396.000	0.574848
$79$	356.000	0.507002	0.253501	−	0.967335i	$-0.418418\pi$
$79$	356.000	0.507002	0.253501	+	0.967335i	$0.418418\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−474.000	−0.638348
$83$	283.000	0.374256	0.187128	−	0.982335i	$-0.440082\pi$
$83$	283.000	0.374256	0.187128	+	0.982335i	$0.440082\pi$
$84$	−60.0000	−0.0779350
$85$	−395.000	−0.504044
$86$	−208.000	−0.260805
$87$	237.000	0.292058
$88$	368.000	0.445783
$89$	176.000	0.209618	0.104809	−	0.994492i	$-0.466577\pi$
$89$	176.000	0.209618	0.104809	+	0.994492i	$0.466577\pi$
$90$	90.0000	0.105409
$91$	330.000	0.380147
$92$	92.0000	0.104257
$93$	675.000	0.752626
$94$	−552.000	−0.605686
$95$	150.000	0.161997
$96$	−96.0000	−0.102062
$97$	930.000	0.973476	0.486738	−	0.873548i	$-0.338187\pi$
$97$	930.000	0.973476	0.486738	+	0.873548i	$0.338187\pi$
$98$	636.000	0.655568
$99$	−414.000	−0.420289
$100$	100.000	0.100000
$101$	−1149.00	−1.13198	−0.565989	−	0.824413i	$-0.691506\pi$
$101$	−1149.00	−1.13198	−0.565989	+	0.824413i	$0.691506\pi$
$102$	−474.000	−0.460127
$103$	784.000	0.749998	0.374999	−	0.927025i	$-0.377643\pi$
$103$	784.000	0.749998	0.374999	+	0.927025i	$0.377643\pi$
$104$	528.000	0.497833
$105$	75.0000	0.0697071
$106$	−1158.00	−1.06108
$107$	27.0000	0.0243943	0.0121971	−	0.999926i	$-0.496117\pi$
$107$	27.0000	0.0243943	0.0121971	+	0.999926i	$0.496117\pi$
$108$	108.000	0.0962250
$109$	−538.000	−0.472762	−0.236381	−	0.971660i	$-0.575961\pi$
$109$	−538.000	−0.472762	−0.236381	+	0.971660i	$0.575961\pi$
$110$	−460.000	−0.398721
$111$	−123.000	−0.105177
$112$	−80.0000	−0.0674937
$113$	1347.00	1.12137	0.560686	−	0.828028i	$-0.310537\pi$
$113$	1347.00	1.12137	0.560686	+	0.828028i	$0.310537\pi$
$114$	180.000	0.147882
$115$	−115.000	−0.0932505
$116$	316.000	0.252930
$117$	−594.000	−0.469362
$118$	−690.000	−0.538302
$119$	−395.000	−0.304282
$120$	120.000	0.0912871
$121$	785.000	0.589782
$122$	208.000	0.154356
$123$	711.000	0.521209
$124$	900.000	0.651793
$125$	−125.000	−0.0894427
$126$	90.0000	0.0636336
$127$	−1692.00	−1.18221	−0.591105	−	0.806594i	$-0.701308\pi$
$127$	−1692.00	−1.18221	−0.591105	+	0.806594i	$0.701308\pi$
$128$	−128.000	−0.0883883
$129$	312.000	0.212946
$130$	−660.000	−0.445276
$131$	−1380.00	−0.920391	−0.460195	−	0.887818i	$-0.652221\pi$
$131$	−1380.00	−0.920391	−0.460195	+	0.887818i	$0.652221\pi$
$132$	−552.000	−0.363981
$133$	150.000	0.0977944
$134$	122.000	0.0786507
$135$	−135.000	−0.0860663
$136$	−632.000	−0.398482
$137$	1742.00	1.08634	0.543172	−	0.839622i	$-0.317223\pi$
$137$	1742.00	1.08634	0.543172	+	0.839622i	$0.317223\pi$
$138$	−138.000	−0.0851257
$139$	2411.00	1.47121	0.735606	−	0.677410i	$-0.236898\pi$
$139$	2411.00	1.47121	0.735606	+	0.677410i	$0.236898\pi$
$140$	100.000	0.0603682
$141$	828.000	0.494540
$142$	506.000	0.299032
$143$	3036.00	1.77541
$144$	144.000	0.0833333
$145$	−395.000	−0.226227
$146$	996.000	0.564586
$147$	−954.000	−0.535269
$148$	−164.000	−0.0910859
$149$	3200.00	1.75942	0.879712	−	0.475507i	$-0.157735\pi$
$149$	3200.00	1.75942	0.879712	+	0.475507i	$0.157735\pi$
$150$	−150.000	−0.0816497
$151$	3324.00	1.79141	0.895706	−	0.444646i	$-0.146671\pi$
$151$	3324.00	1.79141	0.895706	+	0.444646i	$0.146671\pi$
$152$	240.000	0.128070
$153$	711.000	0.375692
$154$	−460.000	−0.240700
$155$	−1125.00	−0.582982
$156$	−792.000	−0.406479
$157$	797.000	0.405143	0.202572	−	0.979267i	$-0.435070\pi$
$157$	797.000	0.405143	0.202572	+	0.979267i	$0.435070\pi$
$158$	−712.000	−0.358504
$159$	1737.00	0.866371
$160$	160.000	0.0790569
$161$	−115.000	−0.0562936
$162$	−162.000	−0.0785674
$163$	−790.000	−0.379617	−0.189809	−	0.981821i	$-0.560787\pi$
$163$	−790.000	−0.379617	−0.189809	+	0.981821i	$0.560787\pi$
$164$	948.000	0.451380
$165$	690.000	0.325554
$166$	−566.000	−0.264639
$167$	3036.00	1.40678	0.703391	−	0.710803i	$-0.251668\pi$
$167$	3036.00	1.40678	0.703391	+	0.710803i	$0.251668\pi$
$168$	120.000	0.0551083
$169$	2159.00	0.982704
$170$	790.000	0.356413
$171$	−270.000	−0.120745
$172$	416.000	0.184417
$173$	2148.00	0.943985	0.471993	−	0.881603i	$-0.343535\pi$
$173$	2148.00	0.943985	0.471993	+	0.881603i	$0.343535\pi$
$174$	−474.000	−0.206516
$175$	−125.000	−0.0539949
$176$	−736.000	−0.315216
$177$	1035.00	0.439522
$178$	−352.000	−0.148222
$179$	248.000	0.103555	0.0517776	−	0.998659i	$-0.483511\pi$
$179$	248.000	0.103555	0.0517776	+	0.998659i	$0.483511\pi$
$180$	−180.000	−0.0745356
$181$	−1284.00	−0.527287	−0.263644	−	0.964620i	$-0.584924\pi$
$181$	−1284.00	−0.527287	−0.263644	+	0.964620i	$0.584924\pi$
$182$	−660.000	−0.268805
$183$	−312.000	−0.126031
$184$	−184.000	−0.0737210
$185$	205.000	0.0814697
$186$	−1350.00	−0.532187
$187$	−3634.00	−1.42109
$188$	1104.00	0.428284
$189$	−135.000	−0.0519566
$190$	−300.000	−0.114549
$191$	5076.00	1.92297	0.961483	−	0.274865i	$-0.0886331\pi$
$191$	5076.00	1.92297	0.961483	+	0.274865i	$0.0886331\pi$
$192$	192.000	0.0721688
$193$	−428.000	−0.159628	−0.0798138	−	0.996810i	$-0.525433\pi$
$193$	−428.000	−0.159628	−0.0798138	+	0.996810i	$0.525433\pi$
$194$	−1860.00	−0.688352
$195$	990.000	0.363566
$196$	−1272.00	−0.463557
$197$	2406.00	0.870154	0.435077	−	0.900393i	$-0.356721\pi$
$197$	2406.00	0.870154	0.435077	+	0.900393i	$0.356721\pi$
$198$	828.000	0.297189
$199$	−1766.00	−0.629088	−0.314544	−	0.949243i	$-0.601852\pi$
$199$	−1766.00	−0.629088	−0.314544	+	0.949243i	$0.601852\pi$
$200$	−200.000	−0.0707107
$201$	−183.000	−0.0642180
$202$	2298.00	0.800429
$203$	−395.000	−0.136569
$204$	948.000	0.325359
$205$	−1185.00	−0.403727
$206$	−1568.00	−0.530329
$207$	207.000	0.0695048
$208$	−1056.00	−0.352021
$209$	1380.00	0.456730
$210$	−150.000	−0.0492904
$211$	−3071.00	−1.00197	−0.500987	−	0.865455i	$-0.667029\pi$
$211$	−3071.00	−1.00197	−0.500987	+	0.865455i	$0.667029\pi$
$212$	2316.00	0.750300
$213$	−759.000	−0.244159
$214$	−54.0000	−0.0172494
$215$	−520.000	−0.164947
$216$	−216.000	−0.0680414
$217$	−1125.00	−0.351935
$218$	1076.00	0.334293
$219$	−1494.00	−0.460982
$220$	920.000	0.281938
$221$	−5214.00	−1.58702
$222$	246.000	0.0743713
$223$	1838.00	0.551935	0.275968	−	0.961167i	$-0.411002\pi$
$223$	1838.00	0.551935	0.275968	+	0.961167i	$0.411002\pi$
$224$	160.000	0.0477252
$225$	225.000	0.0666667
$226$	−2694.00	−0.792930
$227$	−5484.00	−1.60346	−0.801731	−	0.597685i	$-0.796087\pi$
$227$	−5484.00	−1.60346	−0.801731	+	0.597685i	$0.796087\pi$
$228$	−360.000	−0.104568
$229$	−3164.00	−0.913026	−0.456513	−	0.889717i	$-0.650902\pi$
$229$	−3164.00	−0.913026	−0.456513	+	0.889717i	$0.650902\pi$
$230$	230.000	0.0659380
$231$	690.000	0.196531
$232$	−632.000	−0.178848
$233$	3212.00	0.903112	0.451556	−	0.892243i	$-0.350869\pi$
$233$	3212.00	0.903112	0.451556	+	0.892243i	$0.350869\pi$
$234$	1188.00	0.331889
$235$	−1380.00	−0.383069
$236$	1380.00	0.380637
$237$	1068.00	0.292718
$238$	790.000	0.215160
$239$	−6523.00	−1.76543	−0.882715	−	0.469909i	$-0.844287\pi$
$239$	−6523.00	−1.76543	−0.882715	+	0.469909i	$0.844287\pi$
$240$	−240.000	−0.0645497
$241$	−3842.00	−1.02691	−0.513454	−	0.858117i	$-0.671634\pi$
$241$	−3842.00	−1.02691	−0.513454	+	0.858117i	$0.671634\pi$
$242$	−1570.00	−0.417039
$243$	243.000	0.0641500
$244$	−416.000	−0.109146
$245$	1590.00	0.414618
$246$	−1422.00	−0.368550
$247$	1980.00	0.510058
$248$	−1800.00	−0.460888
$249$	849.000	0.216077
$250$	250.000	0.0632456
$251$	−2298.00	−0.577882	−0.288941	−	0.957347i	$-0.593303\pi$
$251$	−2298.00	−0.577882	−0.288941	+	0.957347i	$0.593303\pi$
$252$	−180.000	−0.0449958
$253$	−1058.00	−0.262909
$254$	3384.00	0.835949
$255$	−1185.00	−0.291010
$256$	256.000	0.0625000
$257$	2044.00	0.496114	0.248057	−	0.968745i	$-0.420208\pi$
$257$	2044.00	0.496114	0.248057	+	0.968745i	$0.420208\pi$
$258$	−624.000	−0.150576
$259$	205.000	0.0491818
$260$	1320.00	0.314857
$261$	711.000	0.168620
$262$	2760.00	0.650814
$263$	3749.00	0.878986	0.439493	−	0.898246i	$-0.355158\pi$
$263$	3749.00	0.878986	0.439493	+	0.898246i	$0.355158\pi$
$264$	1104.00	0.257373
$265$	−2895.00	−0.671088
$266$	−300.000	−0.0691511
$267$	528.000	0.121023
$268$	−244.000	−0.0556144
$269$	−2787.00	−0.631697	−0.315848	−	0.948810i	$-0.602289\pi$
$269$	−2787.00	−0.631697	−0.315848	+	0.948810i	$0.602289\pi$
$270$	270.000	0.0608581
$271$	287.000	0.0643321	0.0321661	−	0.999483i	$-0.489759\pi$
$271$	287.000	0.0643321	0.0321661	+	0.999483i	$0.489759\pi$
$272$	1264.00	0.281769
$273$	990.000	0.219478
$274$	−3484.00	−0.768161
$275$	−1150.00	−0.252173
$276$	276.000	0.0601929
$277$	−2482.00	−0.538372	−0.269186	−	0.963088i	$-0.586755\pi$
$277$	−2482.00	−0.538372	−0.269186	+	0.963088i	$0.586755\pi$
$278$	−4822.00	−1.04030
$279$	2025.00	0.434529
$280$	−200.000	−0.0426867
$281$	−7730.00	−1.64104	−0.820522	−	0.571616i	$-0.806317\pi$
$281$	−7730.00	−1.64104	−0.820522	+	0.571616i	$0.806317\pi$
$282$	−1656.00	−0.349693
$283$	5669.00	1.19077	0.595384	−	0.803442i	$-0.297000\pi$
$283$	5669.00	1.19077	0.595384	+	0.803442i	$0.297000\pi$
$284$	−1012.00	−0.211448
$285$	450.000	0.0935288
$286$	−6072.00	−1.25540
$287$	−1185.00	−0.243722
$288$	−288.000	−0.0589256
$289$	1328.00	0.270303
$290$	790.000	0.159967
$291$	2790.00	0.562037
$292$	−1992.00	−0.399223
$293$	−1919.00	−0.382625	−0.191313	−	0.981529i	$-0.561274\pi$
$293$	−1919.00	−0.382625	−0.191313	+	0.981529i	$0.561274\pi$
$294$	1908.00	0.378493
$295$	−1725.00	−0.340452
$296$	328.000	0.0644075
$297$	−1242.00	−0.242654
$298$	−6400.00	−1.24410
$299$	−1518.00	−0.293606
$300$	300.000	0.0577350
$301$	−520.000	−0.0995758
$302$	−6648.00	−1.26672
$303$	−3447.00	−0.653548
$304$	−480.000	−0.0905588
$305$	520.000	0.0976233
$306$	−1422.00	−0.265655
$307$	−4946.00	−0.919489	−0.459745	−	0.888051i	$-0.652059\pi$
$307$	−4946.00	−0.919489	−0.459745	+	0.888051i	$0.652059\pi$
$308$	920.000	0.170201
$309$	2352.00	0.433012
$310$	2250.00	0.412230
$311$	368.000	0.0670976	0.0335488	−	0.999437i	$-0.489319\pi$
$311$	368.000	0.0670976	0.0335488	+	0.999437i	$0.489319\pi$
$312$	1584.00	0.287424
$313$	4421.00	0.798370	0.399185	−	0.916870i	$-0.369293\pi$
$313$	4421.00	0.798370	0.399185	+	0.916870i	$0.369293\pi$
$314$	−1594.00	−0.286480
$315$	225.000	0.0402454
$316$	1424.00	0.253501
$317$	508.000	0.0900067	0.0450033	−	0.998987i	$-0.485670\pi$
$317$	508.000	0.0900067	0.0450033	+	0.998987i	$0.485670\pi$
$318$	−3474.00	−0.612617
$319$	−3634.00	−0.637821
$320$	−320.000	−0.0559017
$321$	81.0000	0.0140840
$322$	230.000	0.0398056
$323$	−2370.00	−0.408267
$324$	324.000	0.0555556
$325$	−1650.00	−0.281617
$326$	1580.00	0.268430
$327$	−1614.00	−0.272949
$328$	−1896.00	−0.319174
$329$	−1380.00	−0.231252
$330$	−1380.00	−0.230202
$331$	−7267.00	−1.20674	−0.603369	−	0.797462i	$-0.706175\pi$
$331$	−7267.00	−1.20674	−0.603369	+	0.797462i	$0.706175\pi$
$332$	1132.00	0.187128
$333$	−369.000	−0.0607240
$334$	−6072.00	−0.994746
$335$	305.000	0.0497431
$336$	−240.000	−0.0389675
$337$	−1726.00	−0.278995	−0.139497	−	0.990222i	$-0.544549\pi$
$337$	−1726.00	−0.278995	−0.139497	+	0.990222i	$0.544549\pi$
$338$	−4318.00	−0.694876
$339$	4041.00	0.647425
$340$	−1580.00	−0.252022
$341$	−10350.0	−1.64365
$342$	540.000	0.0853797
$343$	3305.00	0.520272
$344$	−832.000	−0.130402
$345$	−345.000	−0.0538382
$346$	−4296.00	−0.667498
$347$	3704.00	0.573029	0.286515	−	0.958076i	$-0.407503\pi$
$347$	3704.00	0.573029	0.286515	+	0.958076i	$0.407503\pi$
$348$	948.000	0.146029
$349$	4767.00	0.731151	0.365575	−	0.930782i	$-0.380872\pi$
$349$	4767.00	0.731151	0.365575	+	0.930782i	$0.380872\pi$
$350$	250.000	0.0381802
$351$	−1782.00	−0.270986
$352$	1472.00	0.222892
$353$	1060.00	0.159825	0.0799123	−	0.996802i	$-0.474536\pi$
$353$	1060.00	0.159825	0.0799123	+	0.996802i	$0.474536\pi$
$354$	−2070.00	−0.310789
$355$	1265.00	0.189125
$356$	704.000	0.104809
$357$	−1185.00	−0.175677
$358$	−496.000	−0.0732246
$359$	5460.00	0.802696	0.401348	−	0.915926i	$-0.368542\pi$
$359$	5460.00	0.802696	0.401348	+	0.915926i	$0.368542\pi$
$360$	360.000	0.0527046
$361$	−5959.00	−0.868786
$362$	2568.00	0.372848
$363$	2355.00	0.340511
$364$	1320.00	0.190074
$365$	2490.00	0.357075
$366$	624.000	0.0891175
$367$	2857.00	0.406360	0.203180	−	0.979141i	$-0.434872\pi$
$367$	2857.00	0.406360	0.203180	+	0.979141i	$0.434872\pi$
$368$	368.000	0.0521286
$369$	2133.00	0.300920
$370$	−410.000	−0.0576078
$371$	−2895.00	−0.405124
$372$	2700.00	0.376313
$373$	11630.0	1.61442	0.807210	−	0.590265i	$-0.200977\pi$
$373$	11630.0	1.61442	0.807210	+	0.590265i	$0.200977\pi$
$374$	7268.00	1.00486
$375$	−375.000	−0.0516398
$376$	−2208.00	−0.302843
$377$	−5214.00	−0.712294
$378$	270.000	0.0367389
$379$	−8048.00	−1.09076	−0.545380	−	0.838189i	$-0.683615\pi$
$379$	−8048.00	−1.09076	−0.545380	+	0.838189i	$0.683615\pi$
$380$	600.000	0.0809983
$381$	−5076.00	−0.682549
$382$	−10152.0	−1.35974
$383$	−483.000	−0.0644390	−0.0322195	−	0.999481i	$-0.510258\pi$
$383$	−483.000	−0.0644390	−0.0322195	+	0.999481i	$0.510258\pi$
$384$	−384.000	−0.0510310
$385$	−1150.00	−0.152232
$386$	856.000	0.112874
$387$	936.000	0.122945
$388$	3720.00	0.486738
$389$	14594.0	1.90217	0.951086	−	0.308925i	$-0.0999693\pi$
$389$	14594.0	1.90217	0.951086	+	0.308925i	$0.0999693\pi$
$390$	−1980.00	−0.257080
$391$	1817.00	0.235012
$392$	2544.00	0.327784
$393$	−4140.00	−0.531388
$394$	−4812.00	−0.615292
$395$	−1780.00	−0.226738
$396$	−1656.00	−0.210144
$397$	2720.00	0.343861	0.171931	−	0.985109i	$-0.445000\pi$
$397$	2720.00	0.343861	0.171931	+	0.985109i	$0.445000\pi$
$398$	3532.00	0.444832
$399$	450.000	0.0564616
$400$	400.000	0.0500000
$401$	4218.00	0.525279	0.262639	−	0.964894i	$-0.415407\pi$
$401$	4218.00	0.525279	0.262639	+	0.964894i	$0.415407\pi$
$402$	366.000	0.0454090
$403$	−14850.0	−1.83556
$404$	−4596.00	−0.565989
$405$	−405.000	−0.0496904
$406$	790.000	0.0965691
$407$	1886.00	0.229694
$408$	−1896.00	−0.230064
$409$	−8755.00	−1.05845	−0.529226	−	0.848481i	$-0.677518\pi$
$409$	−8755.00	−1.05845	−0.529226	+	0.848481i	$0.677518\pi$
$410$	2370.00	0.285478
$411$	5226.00	0.627201
$412$	3136.00	0.374999
$413$	−1725.00	−0.205525
$414$	−414.000	−0.0491473
$415$	−1415.00	−0.167373
$416$	2112.00	0.248917
$417$	7233.00	0.849404
$418$	−2760.00	−0.322957
$419$	782.000	0.0911771	0.0455885	−	0.998960i	$-0.485484\pi$
$419$	782.000	0.0911771	0.0455885	+	0.998960i	$0.485484\pi$
$420$	300.000	0.0348536
$421$	−15342.0	−1.77606	−0.888032	−	0.459781i	$-0.847928\pi$
$421$	−15342.0	−1.77606	−0.888032	+	0.459781i	$0.847928\pi$
$422$	6142.00	0.708502
$423$	2484.00	0.285523
$424$	−4632.00	−0.530542
$425$	1975.00	0.225415
$426$	1518.00	0.172646
$427$	520.000	0.0589334
$428$	108.000	0.0121971
$429$	9108.00	1.02503
$430$	1040.00	0.116635
$431$	−2208.00	−0.246765	−0.123382	−	0.992359i	$-0.539374\pi$
$431$	−2208.00	−0.246765	−0.123382	+	0.992359i	$0.539374\pi$
$432$	432.000	0.0481125
$433$	2127.00	0.236067	0.118034	−	0.993010i	$-0.462341\pi$
$433$	2127.00	0.236067	0.118034	+	0.993010i	$0.462341\pi$
$434$	2250.00	0.248856
$435$	−1185.00	−0.130612
$436$	−2152.00	−0.236381
$437$	−690.000	−0.0755313
$438$	2988.00	0.325964
$439$	14000.0	1.52206	0.761029	−	0.648718i	$-0.224694\pi$
$439$	14000.0	1.52206	0.761029	+	0.648718i	$0.224694\pi$
$440$	−1840.00	−0.199360
$441$	−2862.00	−0.309038
$442$	10428.0	1.12219
$443$	−5632.00	−0.604028	−0.302014	−	0.953303i	$-0.597659\pi$
$443$	−5632.00	−0.604028	−0.302014	+	0.953303i	$0.597659\pi$
$444$	−492.000	−0.0525885
$445$	−880.000	−0.0937438
$446$	−3676.00	−0.390277
$447$	9600.00	1.01580
$448$	−320.000	−0.0337468
$449$	319.000	0.0335290	0.0167645	−	0.999859i	$-0.494663\pi$
$449$	319.000	0.0335290	0.0167645	+	0.999859i	$0.494663\pi$
$450$	−450.000	−0.0471405
$451$	−10902.0	−1.13826
$452$	5388.00	0.560686
$453$	9972.00	1.03427
$454$	10968.0	1.13382
$455$	−1650.00	−0.170007
$456$	720.000	0.0739410
$457$	3441.00	0.352217	0.176109	−	0.984371i	$-0.443649\pi$
$457$	3441.00	0.352217	0.176109	+	0.984371i	$0.443649\pi$
$458$	6328.00	0.645607
$459$	2133.00	0.216906
$460$	−460.000	−0.0466252
$461$	−16130.0	−1.62961	−0.814804	−	0.579737i	$-0.803155\pi$
$461$	−16130.0	−1.62961	−0.814804	+	0.579737i	$0.803155\pi$
$462$	−1380.00	−0.138968
$463$	−1806.00	−0.181278	−0.0906392	−	0.995884i	$-0.528891\pi$
$463$	−1806.00	−0.181278	−0.0906392	+	0.995884i	$0.528891\pi$
$464$	1264.00	0.126465
$465$	−3375.00	−0.336585
$466$	−6424.00	−0.638597
$467$	14307.0	1.41766	0.708832	−	0.705377i	$-0.249222\pi$
$467$	14307.0	1.41766	0.708832	+	0.705377i	$0.249222\pi$
$468$	−2376.00	−0.234681
$469$	305.000	0.0300290
$470$	2760.00	0.270871
$471$	2391.00	0.233910
$472$	−2760.00	−0.269151
$473$	−4784.00	−0.465050
$474$	−2136.00	−0.206983
$475$	−750.000	−0.0724471
$476$	−1580.00	−0.152141
$477$	5211.00	0.500200
$478$	13046.0	1.24835
$479$	−12416.0	−1.18435	−0.592173	−	0.805811i	$-0.701730\pi$
$479$	−12416.0	−1.18435	−0.592173	+	0.805811i	$0.701730\pi$
$480$	480.000	0.0456435
$481$	2706.00	0.256513
$482$	7684.00	0.726134
$483$	−345.000	−0.0325011
$484$	3140.00	0.294891
$485$	−4650.00	−0.435352
$486$	−486.000	−0.0453609
$487$	4212.00	0.391918	0.195959	−	0.980612i	$-0.437218\pi$
$487$	4212.00	0.391918	0.195959	+	0.980612i	$0.437218\pi$
$488$	832.000	0.0771780
$489$	−2370.00	−0.219172
$490$	−3180.00	−0.293179
$491$	13923.0	1.27971	0.639854	−	0.768497i	$-0.278995\pi$
$491$	13923.0	1.27971	0.639854	+	0.768497i	$0.278995\pi$
$492$	2844.00	0.260605
$493$	6241.00	0.570143
$494$	−3960.00	−0.360666
$495$	2070.00	0.187959
$496$	3600.00	0.325897
$497$	1265.00	0.114171
$498$	−1698.00	−0.152790
$499$	13943.0	1.25085	0.625425	−	0.780284i	$-0.284926\pi$
$499$	13943.0	1.25085	0.625425	+	0.780284i	$0.284926\pi$
$500$	−500.000	−0.0447214
$501$	9108.00	0.812206
$502$	4596.00	0.408625
$503$	7265.00	0.643997	0.321998	−	0.946740i	$-0.395645\pi$
$503$	7265.00	0.643997	0.321998	+	0.946740i	$0.395645\pi$
$504$	360.000	0.0318168
$505$	5745.00	0.506236
$506$	2116.00	0.185904
$507$	6477.00	0.567364
$508$	−6768.00	−0.591105
$509$	−10414.0	−0.906861	−0.453431	−	0.891292i	$-0.649800\pi$
$509$	−10414.0	−0.906861	−0.453431	+	0.891292i	$0.649800\pi$
$510$	2370.00	0.205775
$511$	2490.00	0.215560
$512$	−512.000	−0.0441942
$513$	−810.000	−0.0697122
$514$	−4088.00	−0.350805
$515$	−3920.00	−0.335409
$516$	1248.00	0.106473
$517$	−12696.0	−1.08002
$518$	−410.000	−0.0347768
$519$	6444.00	0.545010
$520$	−2640.00	−0.222638
$521$	−17762.0	−1.49360	−0.746802	−	0.665047i	$-0.768412\pi$
$521$	−17762.0	−1.49360	−0.746802	+	0.665047i	$0.768412\pi$
$522$	−1422.00	−0.119232
$523$	8620.00	0.720700	0.360350	−	0.932817i	$-0.382657\pi$
$523$	8620.00	0.720700	0.360350	+	0.932817i	$0.382657\pi$
$524$	−5520.00	−0.460195
$525$	−375.000	−0.0311740
$526$	−7498.00	−0.621537
$527$	17775.0	1.46924
$528$	−2208.00	−0.181990
$529$	529.000	0.0434783
$530$	5790.00	0.474531
$531$	3105.00	0.253758
$532$	600.000	0.0488972
$533$	−15642.0	−1.27116
$534$	−1056.00	−0.0855760
$535$	−135.000	−0.0109095
$536$	488.000	0.0393254
$537$	744.000	0.0597877
$538$	5574.00	0.446677
$539$	14628.0	1.16897
$540$	−540.000	−0.0430331
$541$	1766.00	0.140344	0.0701722	−	0.997535i	$-0.477645\pi$
$541$	1766.00	0.140344	0.0701722	+	0.997535i	$0.477645\pi$
$542$	−574.000	−0.0454897
$543$	−3852.00	−0.304429
$544$	−2528.00	−0.199241
$545$	2690.00	0.211426
$546$	−1980.00	−0.155194
$547$	2808.00	0.219491	0.109745	−	0.993960i	$-0.464996\pi$
$547$	2808.00	0.219491	0.109745	+	0.993960i	$0.464996\pi$
$548$	6968.00	0.543172
$549$	−936.000	−0.0727641
$550$	2300.00	0.178313
$551$	−2370.00	−0.183240
$552$	−552.000	−0.0425628
$553$	−1780.00	−0.136878
$554$	4964.00	0.380686
$555$	615.000	0.0470366
$556$	9644.00	0.735606
$557$	−9317.00	−0.708750	−0.354375	−	0.935103i	$-0.615306\pi$
$557$	−9317.00	−0.708750	−0.354375	+	0.935103i	$0.615306\pi$
$558$	−4050.00	−0.307258
$559$	−6864.00	−0.519349
$560$	400.000	0.0301841
$561$	−10902.0	−0.820468
$562$	15460.0	1.16039
$563$	19165.0	1.43465	0.717325	−	0.696738i	$-0.245366\pi$
$563$	19165.0	1.43465	0.717325	+	0.696738i	$0.245366\pi$
$564$	3312.00	0.247270
$565$	−6735.00	−0.501493
$566$	−11338.0	−0.841999
$567$	−405.000	−0.0299972
$568$	2024.00	0.149516
$569$	−7422.00	−0.546830	−0.273415	−	0.961896i	$-0.588153\pi$
$569$	−7422.00	−0.546830	−0.273415	+	0.961896i	$0.588153\pi$
$570$	−900.000	−0.0661348
$571$	−1920.00	−0.140717	−0.0703586	−	0.997522i	$-0.522414\pi$
$571$	−1920.00	−0.140717	−0.0703586	+	0.997522i	$0.522414\pi$
$572$	12144.0	0.887703
$573$	15228.0	1.11022
$574$	2370.00	0.172338
$575$	575.000	0.0417029
$576$	576.000	0.0416667
$577$	10400.0	0.750360	0.375180	−	0.926952i	$-0.377581\pi$
$577$	10400.0	0.750360	0.375180	+	0.926952i	$0.377581\pi$
$578$	−2656.00	−0.191133
$579$	−1284.00	−0.0921610
$580$	−1580.00	−0.113114
$581$	−1415.00	−0.101040
$582$	−5580.00	−0.397420
$583$	−26634.0	−1.89205
$584$	3984.00	0.282293
$585$	2970.00	0.209905
$586$	3838.00	0.270557
$587$	−12794.0	−0.899599	−0.449800	−	0.893129i	$-0.648505\pi$
$587$	−12794.0	−0.899599	−0.449800	+	0.893129i	$0.648505\pi$
$588$	−3816.00	−0.267635
$589$	−6750.00	−0.472205
$590$	3450.00	0.240736
$591$	7218.00	0.502384
$592$	−656.000	−0.0455430
$593$	−23196.0	−1.60632	−0.803158	−	0.595766i	$-0.796849\pi$
$593$	−23196.0	−1.60632	−0.803158	+	0.595766i	$0.796849\pi$
$594$	2484.00	0.171582
$595$	1975.00	0.136079
$596$	12800.0	0.879712
$597$	−5298.00	−0.363204
$598$	3036.00	0.207611
$599$	19296.0	1.31622	0.658108	−	0.752924i	$-0.271357\pi$
$599$	19296.0	1.31622	0.658108	+	0.752924i	$0.271357\pi$
$600$	−600.000	−0.0408248
$601$	14171.0	0.961809	0.480904	−	0.876773i	$-0.340308\pi$
$601$	14171.0	0.961809	0.480904	+	0.876773i	$0.340308\pi$
$602$	1040.00	0.0704107
$603$	−549.000	−0.0370763
$604$	13296.0	0.895706
$605$	−3925.00	−0.263759
$606$	6894.00	0.462128
$607$	11464.0	0.766572	0.383286	−	0.923630i	$-0.374792\pi$
$607$	11464.0	0.766572	0.383286	+	0.923630i	$0.374792\pi$
$608$	960.000	0.0640348
$609$	−1185.00	−0.0788483
$610$	−1040.00	−0.0690301
$611$	−18216.0	−1.20612
$612$	2844.00	0.187846
$613$	−4570.00	−0.301110	−0.150555	−	0.988602i	$-0.548106\pi$
$613$	−4570.00	−0.301110	−0.150555	+	0.988602i	$0.548106\pi$
$614$	9892.00	0.650177
$615$	−3555.00	−0.233092
$616$	−1840.00	−0.120350
$617$	8321.00	0.542935	0.271467	−	0.962448i	$-0.412491\pi$
$617$	8321.00	0.542935	0.271467	+	0.962448i	$0.412491\pi$
$618$	−4704.00	−0.306185
$619$	−2112.00	−0.137138	−0.0685690	−	0.997646i	$-0.521843\pi$
$619$	−2112.00	−0.137138	−0.0685690	+	0.997646i	$0.521843\pi$
$620$	−4500.00	−0.291491
$621$	621.000	0.0401286
$622$	−736.000	−0.0474452
$623$	−880.000	−0.0565914
$624$	−3168.00	−0.203240
$625$	625.000	0.0400000
$626$	−8842.00	−0.564533
$627$	4140.00	0.263693
$628$	3188.00	0.202572
$629$	−3239.00	−0.205322
$630$	−450.000	−0.0284578
$631$	25160.0	1.58733	0.793664	−	0.608357i	$-0.208171\pi$
$631$	25160.0	1.58733	0.793664	+	0.608357i	$0.208171\pi$
$632$	−2848.00	−0.179252
$633$	−9213.00	−0.578490
$634$	−1016.00	−0.0636443
$635$	8460.00	0.528701
$636$	6948.00	0.433186
$637$	20988.0	1.30546
$638$	7268.00	0.451008
$639$	−2277.00	−0.140965
$640$	640.000	0.0395285
$641$	−10582.0	−0.652050	−0.326025	−	0.945361i	$-0.605709\pi$
$641$	−10582.0	−0.652050	−0.326025	+	0.945361i	$0.605709\pi$
$642$	−162.000	−0.00995893
$643$	19439.0	1.19222	0.596111	−	0.802902i	$-0.296712\pi$
$643$	19439.0	1.19222	0.596111	+	0.802902i	$0.296712\pi$
$644$	−460.000	−0.0281468
$645$	−1560.00	−0.0952325
$646$	4740.00	0.288689
$647$	−10278.0	−0.624528	−0.312264	−	0.949995i	$-0.601087\pi$
$647$	−10278.0	−0.624528	−0.312264	+	0.949995i	$0.601087\pi$
$648$	−648.000	−0.0392837
$649$	−15870.0	−0.959864
$650$	3300.00	0.199133
$651$	−3375.00	−0.203190
$652$	−3160.00	−0.189809
$653$	10190.0	0.610667	0.305333	−	0.952245i	$-0.401232\pi$
$653$	10190.0	0.610667	0.305333	+	0.952245i	$0.401232\pi$
$654$	3228.00	0.193004
$655$	6900.00	0.411611
$656$	3792.00	0.225690
$657$	−4482.00	−0.266148
$658$	2760.00	0.163520
$659$	7758.00	0.458587	0.229293	−	0.973357i	$-0.426358\pi$
$659$	7758.00	0.458587	0.229293	+	0.973357i	$0.426358\pi$
$660$	2760.00	0.162777
$661$	27930.0	1.64350	0.821748	−	0.569851i	$-0.192999\pi$
$661$	27930.0	1.64350	0.821748	+	0.569851i	$0.192999\pi$
$662$	14534.0	0.853293
$663$	−15642.0	−0.916267
$664$	−2264.00	−0.132320
$665$	−750.000	−0.0437350
$666$	738.000	0.0429383
$667$	1817.00	0.105479
$668$	12144.0	0.703391
$669$	5514.00	0.318660
$670$	−610.000	−0.0351737
$671$	4784.00	0.275237
$672$	480.000	0.0275542
$673$	432.000	0.0247435	0.0123718	−	0.999923i	$-0.496062\pi$
$673$	432.000	0.0247435	0.0123718	+	0.999923i	$0.496062\pi$
$674$	3452.00	0.197279
$675$	675.000	0.0384900
$676$	8636.00	0.491352
$677$	−16643.0	−0.944819	−0.472409	−	0.881379i	$-0.656616\pi$
$677$	−16643.0	−0.944819	−0.472409	+	0.881379i	$0.656616\pi$
$678$	−8082.00	−0.457798
$679$	−4650.00	−0.262814
$680$	3160.00	0.178207
$681$	−16452.0	−0.925759
$682$	20700.0	1.16223
$683$	2292.00	0.128405	0.0642027	−	0.997937i	$-0.479550\pi$
$683$	2292.00	0.128405	0.0642027	+	0.997937i	$0.479550\pi$
$684$	−1080.00	−0.0603726
$685$	−8710.00	−0.485828
$686$	−6610.00	−0.367888
$687$	−9492.00	−0.527136
$688$	1664.00	0.0922084
$689$	−38214.0	−2.11297
$690$	690.000	0.0380693
$691$	23596.0	1.29904	0.649518	−	0.760346i	$-0.274971\pi$
$691$	23596.0	1.29904	0.649518	+	0.760346i	$0.274971\pi$
$692$	8592.00	0.471993
$693$	2070.00	0.113467
$694$	−7408.00	−0.405193
$695$	−12055.0	−0.657946
$696$	−1896.00	−0.103258
$697$	18723.0	1.01748
$698$	−9534.00	−0.517002
$699$	9636.00	0.521412
$700$	−500.000	−0.0269975
$701$	−21732.0	−1.17091	−0.585454	−	0.810706i	$-0.699084\pi$
$701$	−21732.0	−1.17091	−0.585454	+	0.810706i	$0.699084\pi$
$702$	3564.00	0.191616
$703$	1230.00	0.0659891
$704$	−2944.00	−0.157608
$705$	−4140.00	−0.221165
$706$	−2120.00	−0.113013
$707$	5745.00	0.305605
$708$	4140.00	0.219761
$709$	26620.0	1.41006	0.705032	−	0.709176i	$-0.250933\pi$
$709$	26620.0	1.41006	0.705032	+	0.709176i	$0.250933\pi$
$710$	−2530.00	−0.133731
$711$	3204.00	0.169001
$712$	−1408.00	−0.0741110
$713$	5175.00	0.271817
$714$	2370.00	0.124223
$715$	−15180.0	−0.793986
$716$	992.000	0.0517776
$717$	−19569.0	−1.01927
$718$	−10920.0	−0.567592
$719$	16629.0	0.862527	0.431264	−	0.902226i	$-0.358068\pi$
$719$	16629.0	0.862527	0.431264	+	0.902226i	$0.358068\pi$
$720$	−720.000	−0.0372678
$721$	−3920.00	−0.202480
$722$	11918.0	0.614324
$723$	−11526.0	−0.592886
$724$	−5136.00	−0.263644
$725$	1975.00	0.101172
$726$	−4710.00	−0.240778
$727$	−24151.0	−1.23206	−0.616032	−	0.787721i	$-0.711261\pi$
$727$	−24151.0	−1.23206	−0.616032	+	0.787721i	$0.711261\pi$
$728$	−2640.00	−0.134402
$729$	729.000	0.0370370
$730$	−4980.00	−0.252491
$731$	8216.00	0.415704
$732$	−1248.00	−0.0630156
$733$	−16977.0	−0.855470	−0.427735	−	0.903904i	$-0.640688\pi$
$733$	−16977.0	−0.855470	−0.427735	+	0.903904i	$0.640688\pi$
$734$	−5714.00	−0.287340
$735$	4770.00	0.239380
$736$	−736.000	−0.0368605
$737$	2806.00	0.140245
$738$	−4266.00	−0.212783
$739$	−24375.0	−1.21333	−0.606664	−	0.794959i	$-0.707493\pi$
$739$	−24375.0	−1.21333	−0.606664	+	0.794959i	$0.707493\pi$
$740$	820.000	0.0407349
$741$	5940.00	0.294482
$742$	5790.00	0.286466
$743$	13448.0	0.664010	0.332005	−	0.943278i	$-0.392275\pi$
$743$	13448.0	0.664010	0.332005	+	0.943278i	$0.392275\pi$
$744$	−5400.00	−0.266094
$745$	−16000.0	−0.786838
$746$	−23260.0	−1.14157
$747$	2547.00	0.124752
$748$	−14536.0	−0.710547
$749$	−135.000	−0.00658584
$750$	750.000	0.0365148
$751$	4966.00	0.241294	0.120647	−	0.992695i	$-0.461503\pi$
$751$	4966.00	0.241294	0.120647	+	0.992695i	$0.461503\pi$
$752$	4416.00	0.214142
$753$	−6894.00	−0.333641
$754$	10428.0	0.503668
$755$	−16620.0	−0.801144
$756$	−540.000	−0.0259783
$757$	−7295.00	−0.350253	−0.175126	−	0.984546i	$-0.556033\pi$
$757$	−7295.00	−0.350253	−0.175126	+	0.984546i	$0.556033\pi$
$758$	16096.0	0.771284
$759$	−3174.00	−0.151790
$760$	−1200.00	−0.0572744
$761$	−27863.0	−1.32724	−0.663622	−	0.748068i	$-0.730982\pi$
$761$	−27863.0	−1.32724	−0.663622	+	0.748068i	$0.730982\pi$
$762$	10152.0	0.482635
$763$	2690.00	0.127634
$764$	20304.0	0.961483
$765$	−3555.00	−0.168015
$766$	966.000	0.0455653
$767$	−22770.0	−1.07194
$768$	768.000	0.0360844
$769$	−14700.0	−0.689331	−0.344666	−	0.938726i	$-0.612008\pi$
$769$	−14700.0	−0.689331	−0.344666	+	0.938726i	$0.612008\pi$
$770$	2300.00	0.107644
$771$	6132.00	0.286431
$772$	−1712.00	−0.0798138
$773$	1278.00	0.0594650	0.0297325	−	0.999558i	$-0.490534\pi$
$773$	1278.00	0.0594650	0.0297325	+	0.999558i	$0.490534\pi$
$774$	−1872.00	−0.0869349
$775$	5625.00	0.260717
$776$	−7440.00	−0.344176
$777$	615.000	0.0283951
$778$	−29188.0	−1.34504
$779$	−7110.00	−0.327012
$780$	3960.00	0.181783
$781$	11638.0	0.533214
$782$	−3634.00	−0.166178
$783$	2133.00	0.0973527
$784$	−5088.00	−0.231778
$785$	−3985.00	−0.181186
$786$	8280.00	0.375748
$787$	19967.0	0.904380	0.452190	−	0.891922i	$-0.350643\pi$
$787$	19967.0	0.904380	0.452190	+	0.891922i	$0.350643\pi$
$788$	9624.00	0.435077
$789$	11247.0	0.507483
$790$	3560.00	0.160328
$791$	−6735.00	−0.302742
$792$	3312.00	0.148594
$793$	6864.00	0.307374
$794$	−5440.00	−0.243147
$795$	−8685.00	−0.387453
$796$	−7064.00	−0.314544
$797$	−5579.00	−0.247953	−0.123976	−	0.992285i	$-0.539565\pi$
$797$	−5579.00	−0.247953	−0.123976	+	0.992285i	$0.539565\pi$
$798$	−900.000	−0.0399244
$799$	21804.0	0.965419
$800$	−800.000	−0.0353553
$801$	1584.00	0.0698725
$802$	−8436.00	−0.371428
$803$	22908.0	1.00673
$804$	−732.000	−0.0321090
$805$	575.000	0.0251753
$806$	29700.0	1.29794
$807$	−8361.00	−0.364710
$808$	9192.00	0.400215
$809$	41667.0	1.81080	0.905398	−	0.424564i	$-0.139573\pi$
$809$	41667.0	1.81080	0.905398	+	0.424564i	$0.139573\pi$
$810$	810.000	0.0351364
$811$	−9983.00	−0.432245	−0.216122	−	0.976366i	$-0.569341\pi$
$811$	−9983.00	−0.432245	−0.216122	+	0.976366i	$0.569341\pi$
$812$	−1580.00	−0.0682846
$813$	861.000	0.0371422
$814$	−3772.00	−0.162418
$815$	3950.00	0.169770
$816$	3792.00	0.162680
$817$	−3120.00	−0.133605
$818$	17510.0	0.748439
$819$	2970.00	0.126716
$820$	−4740.00	−0.201863
$821$	−41342.0	−1.75742	−0.878712	−	0.477352i	$-0.841597\pi$
$821$	−41342.0	−1.75742	−0.878712	+	0.477352i	$0.841597\pi$
$822$	−10452.0	−0.443498
$823$	10388.0	0.439979	0.219990	−	0.975502i	$-0.429398\pi$
$823$	10388.0	0.439979	0.219990	+	0.975502i	$0.429398\pi$
$824$	−6272.00	−0.265164
$825$	−3450.00	−0.145592
$826$	3450.00	0.145328
$827$	4105.00	0.172606	0.0863028	−	0.996269i	$-0.472495\pi$
$827$	4105.00	0.172606	0.0863028	+	0.996269i	$0.472495\pi$
$828$	828.000	0.0347524
$829$	−5183.00	−0.217145	−0.108572	−	0.994089i	$-0.534628\pi$
$829$	−5183.00	−0.217145	−0.108572	+	0.994089i	$0.534628\pi$
$830$	2830.00	0.118350
$831$	−7446.00	−0.310829
$832$	−4224.00	−0.176011
$833$	−25122.0	−1.04493
$834$	−14466.0	−0.600619
$835$	−15180.0	−0.629132
$836$	5520.00	0.228365
$837$	6075.00	0.250875
$838$	−1564.00	−0.0644719
$839$	33614.0	1.38318	0.691588	−	0.722293i	$-0.256912\pi$
$839$	33614.0	1.38318	0.691588	+	0.722293i	$0.256912\pi$
$840$	−600.000	−0.0246452
$841$	−18148.0	−0.744106
$842$	30684.0	1.25587
$843$	−23190.0	−0.947457
$844$	−12284.0	−0.500987
$845$	−10795.0	−0.439478
$846$	−4968.00	−0.201895
$847$	−3925.00	−0.159226
$848$	9264.00	0.375150
$849$	17007.0	0.687490
$850$	−3950.00	−0.159393
$851$	−943.000	−0.0379855
$852$	−3036.00	−0.122079
$853$	11270.0	0.452377	0.226188	−	0.974084i	$-0.427373\pi$
$853$	11270.0	0.452377	0.226188	+	0.974084i	$0.427373\pi$
$854$	−1040.00	−0.0416722
$855$	1350.00	0.0539989
$856$	−216.000	−0.00862468
$857$	3342.00	0.133210	0.0666048	−	0.997779i	$-0.478783\pi$
$857$	3342.00	0.133210	0.0666048	+	0.997779i	$0.478783\pi$
$858$	−18216.0	−0.724807
$859$	−12843.0	−0.510125	−0.255063	−	0.966925i	$-0.582096\pi$
$859$	−12843.0	−0.510125	−0.255063	+	0.966925i	$0.582096\pi$
$860$	−2080.00	−0.0824737
$861$	−3555.00	−0.140713
$862$	4416.00	0.174489
$863$	−28830.0	−1.13718	−0.568589	−	0.822622i	$-0.692511\pi$
$863$	−28830.0	−1.13718	−0.568589	+	0.822622i	$0.692511\pi$
$864$	−864.000	−0.0340207
$865$	−10740.0	−0.422163
$866$	−4254.00	−0.166925
$867$	3984.00	0.156060
$868$	−4500.00	−0.175968
$869$	−16376.0	−0.639261
$870$	2370.00	0.0923569
$871$	4026.00	0.156620
$872$	4304.00	0.167147
$873$	8370.00	0.324492
$874$	1380.00	0.0534087
$875$	625.000	0.0241473
$876$	−5976.00	−0.230491
$877$	−25202.0	−0.970366	−0.485183	−	0.874413i	$-0.661247\pi$
$877$	−25202.0	−0.970366	−0.485183	+	0.874413i	$0.661247\pi$
$878$	−28000.0	−1.07626
$879$	−5757.00	−0.220909
$880$	3680.00	0.140969
$881$	−7048.00	−0.269527	−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−7048.00	−0.269527	−0.134763	+	0.990878i	$0.543027\pi$
$882$	5724.00	0.218523
$883$	32256.0	1.22933	0.614666	−	0.788787i	$-0.289291\pi$
$883$	32256.0	1.22933	0.614666	+	0.788787i	$0.289291\pi$
$884$	−20856.0	−0.793510
$885$	−5175.00	−0.196560
$886$	11264.0	0.427112
$887$	−37674.0	−1.42612	−0.713060	−	0.701103i	$-0.752691\pi$
$887$	−37674.0	−1.42612	−0.713060	+	0.701103i	$0.752691\pi$
$888$	984.000	0.0371857
$889$	8460.00	0.319167
$890$	1760.00	0.0662869
$891$	−3726.00	−0.140096
$892$	7352.00	0.275968
$893$	−8280.00	−0.310280
$894$	−19200.0	−0.718282
$895$	−1240.00	−0.0463113
$896$	640.000	0.0238626
$897$	−4554.00	−0.169514
$898$	−638.000	−0.0237086
$899$	17775.0	0.659432
$900$	900.000	0.0333333
$901$	45741.0	1.69129
$902$	21804.0	0.804871
$903$	−1560.00	−0.0574901
$904$	−10776.0	−0.396465
$905$	6420.00	0.235810
$906$	−19944.0	−0.731341
$907$	30915.0	1.13177	0.565885	−	0.824484i	$-0.308535\pi$
$907$	30915.0	1.13177	0.565885	+	0.824484i	$0.308535\pi$
$908$	−21936.0	−0.801731
$909$	−10341.0	−0.377326
$910$	3300.00	0.120213
$911$	−1500.00	−0.0545524	−0.0272762	−	0.999628i	$-0.508683\pi$
$911$	−1500.00	−0.0545524	−0.0272762	+	0.999628i	$0.508683\pi$
$912$	−1440.00	−0.0522842
$913$	−13018.0	−0.471887
$914$	−6882.00	−0.249055
$915$	1560.00	0.0563629
$916$	−12656.0	−0.456513
$917$	6900.00	0.248482
$918$	−4266.00	−0.153376
$919$	38104.0	1.36772	0.683860	−	0.729613i	$-0.260300\pi$
$919$	38104.0	1.36772	0.683860	+	0.729613i	$0.260300\pi$
$920$	920.000	0.0329690
$921$	−14838.0	−0.530867
$922$	32260.0	1.15231
$923$	16698.0	0.595473
$924$	2760.00	0.0982655
$925$	−1025.00	−0.0364344
$926$	3612.00	0.128183
$927$	7056.00	0.249999
$928$	−2528.00	−0.0894242
$929$	2775.00	0.0980030	0.0490015	−	0.998799i	$-0.484396\pi$
$929$	2775.00	0.0980030	0.0490015	+	0.998799i	$0.484396\pi$
$930$	6750.00	0.238001
$931$	9540.00	0.335833
$932$	12848.0	0.451556
$933$	1104.00	0.0387388
$934$	−28614.0	−1.00244
$935$	18170.0	0.635532
$936$	4752.00	0.165944
$937$	26754.0	0.932780	0.466390	−	0.884579i	$-0.345554\pi$
$937$	26754.0	0.932780	0.466390	+	0.884579i	$0.345554\pi$
$938$	−610.000	−0.0212337
$939$	13263.0	0.460939
$940$	−5520.00	−0.191535
$941$	51760.0	1.79312	0.896561	−	0.442920i	$-0.146058\pi$
$941$	51760.0	1.79312	0.896561	+	0.442920i	$0.146058\pi$
$942$	−4782.00	−0.165399
$943$	5451.00	0.188239
$944$	5520.00	0.190319
$945$	675.000	0.0232357
$946$	9568.00	0.328840
$947$	3364.00	0.115433	0.0577166	−	0.998333i	$-0.481618\pi$
$947$	3364.00	0.115433	0.0577166	+	0.998333i	$0.481618\pi$
$948$	4272.00	0.146359
$949$	32868.0	1.12428
$950$	1500.00	0.0512278
$951$	1524.00	0.0519654
$952$	3160.00	0.107580
$953$	32122.0	1.09185	0.545925	−	0.837834i	$-0.316178\pi$
$953$	32122.0	1.09185	0.545925	+	0.837834i	$0.316178\pi$
$954$	−10422.0	−0.353695
$955$	−25380.0	−0.859976
$956$	−26092.0	−0.882715
$957$	−10902.0	−0.368246
$958$	24832.0	0.837459
$959$	−8710.00	−0.293285
$960$	−960.000	−0.0322749
$961$	20834.0	0.699339
$962$	−5412.00	−0.181382
$963$	243.000	0.00813143
$964$	−15368.0	−0.513454
$965$	2140.00	0.0713876
$966$	690.000	0.0229818
$967$	43696.0	1.45312	0.726561	−	0.687102i	$-0.241118\pi$
$967$	43696.0	1.45312	0.726561	+	0.687102i	$0.241118\pi$
$968$	−6280.00	−0.208519
$969$	−7110.00	−0.235713
$970$	9300.00	0.307840
$971$	−25916.0	−0.856523	−0.428262	−	0.903655i	$-0.640874\pi$
$971$	−25916.0	−0.856523	−0.428262	+	0.903655i	$0.640874\pi$
$972$	972.000	0.0320750
$973$	−12055.0	−0.397190
$974$	−8424.00	−0.277128
$975$	−4950.00	−0.162592
$976$	−1664.00	−0.0545731
$977$	−56283.0	−1.84304	−0.921521	−	0.388328i	$-0.873053\pi$
$977$	−56283.0	−1.84304	−0.921521	+	0.388328i	$0.873053\pi$
$978$	4740.00	0.154978
$979$	−8096.00	−0.264300
$980$	6360.00	0.207309
$981$	−4842.00	−0.157587
$982$	−27846.0	−0.904890
$983$	−7377.00	−0.239359	−0.119679	−	0.992813i	$-0.538187\pi$
$983$	−7377.00	−0.239359	−0.119679	+	0.992813i	$0.538187\pi$
$984$	−5688.00	−0.184275
$985$	−12030.0	−0.389145
$986$	−12482.0	−0.403152
$987$	−4140.00	−0.133513
$988$	7920.00	0.255029
$989$	2392.00	0.0769072
$990$	−4140.00	−0.132907
$991$	−43019.0	−1.37895	−0.689477	−	0.724308i	$-0.742160\pi$
$991$	−43019.0	−1.37895	−0.689477	+	0.724308i	$0.742160\pi$
$992$	−7200.00	−0.230444
$993$	−21801.0	−0.696711
$994$	−2530.00	−0.0807311
$995$	8830.00	0.281337
$996$	3396.00	0.108039
$997$	−2552.00	−0.0810658	−0.0405329	−	0.999178i	$-0.512906\pi$
$997$	−2552.00	−0.0810658	−0.0405329	+	0.999178i	$0.512906\pi$
$998$	−27886.0	−0.884485
$999$	−1107.00	−0.0350590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.4.a.e.1.1	✓	1
3.2	odd	2		2070.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.4.a.e.1.1	✓	1	1.1	even	1	trivial
2070.4.a.l.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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