LMFDB - Embedded newform 65.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,4,Mod(1,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("65.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 65 = 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	65.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:	$3.83512415037$
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	$\chi$	$=$	65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+5.00000 q^{2} +2.00000 q^{3} +17.0000 q^{4} -5.00000 q^{5} +10.0000 q^{6} -12.0000 q^{7} +45.0000 q^{8} -23.0000 q^{9} +O(q^{10})$

\(q+5.00000 q^{2} +2.00000 q^{3} +17.0000 q^{4} -5.00000 q^{5} +10.0000 q^{6} -12.0000 q^{7} +45.0000 q^{8} -23.0000 q^{9} -25.0000 q^{10} +14.0000 q^{11} +34.0000 q^{12} -13.0000 q^{13} -60.0000 q^{14} -10.0000 q^{15} +89.0000 q^{16} +98.0000 q^{17} -115.000 q^{18} -26.0000 q^{19} -85.0000 q^{20} -24.0000 q^{21} +70.0000 q^{22} -114.000 q^{23} +90.0000 q^{24} +25.0000 q^{25} -65.0000 q^{26} -100.000 q^{27} -204.000 q^{28} +58.0000 q^{29} -50.0000 q^{30} +306.000 q^{31} +85.0000 q^{32} +28.0000 q^{33} +490.000 q^{34} +60.0000 q^{35} -391.000 q^{36} +86.0000 q^{37} -130.000 q^{38} -26.0000 q^{39} -225.000 q^{40} -374.000 q^{41} -120.000 q^{42} -314.000 q^{43} +238.000 q^{44} +115.000 q^{45} -570.000 q^{46} +620.000 q^{47} +178.000 q^{48} -199.000 q^{49} +125.000 q^{50} +196.000 q^{51} -221.000 q^{52} +362.000 q^{53} -500.000 q^{54} -70.0000 q^{55} -540.000 q^{56} -52.0000 q^{57} +290.000 q^{58} +266.000 q^{59} -170.000 q^{60} +634.000 q^{61} +1530.00 q^{62} +276.000 q^{63} -287.000 q^{64} +65.0000 q^{65} +140.000 q^{66} +612.000 q^{67} +1666.00 q^{68} -228.000 q^{69} +300.000 q^{70} -686.000 q^{71} -1035.00 q^{72} +202.000 q^{73} +430.000 q^{74} +50.0000 q^{75} -442.000 q^{76} -168.000 q^{77} -130.000 q^{78} -516.000 q^{79} -445.000 q^{80} +421.000 q^{81} -1870.00 q^{82} +48.0000 q^{83} -408.000 q^{84} -490.000 q^{85} -1570.00 q^{86} +116.000 q^{87} +630.000 q^{88} -1230.00 q^{89} +575.000 q^{90} +156.000 q^{91} -1938.00 q^{92} +612.000 q^{93} +3100.00 q^{94} +130.000 q^{95} +170.000 q^{96} +350.000 q^{97} -995.000 q^{98} -322.000 q^{99} +O(q^{100})\)

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 $

$2$	5.00000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	5.00000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	17.0000	2.12500
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	10.0000	0.680414
$7$	−12.0000	−0.647939	−0.323970	−	0.946068i	$-0.605018\pi$
$7$	−12.0000	−0.647939	−0.323970	+	0.946068i	$0.605018\pi$
$8$	45.0000	1.98874
$9$	−23.0000	−0.851852
$10$	−25.0000	−0.790569
$11$	14.0000	0.383742	0.191871	−	0.981420i	$-0.438545\pi$
$11$	14.0000	0.383742	0.191871	+	0.981420i	$0.438545\pi$
$12$	34.0000	0.817913
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−60.0000	−1.14541
$15$	−10.0000	−0.172133
$16$	89.0000	1.39062
$17$	98.0000	1.39815	0.699073	−	0.715050i	$-0.253596\pi$
$17$	98.0000	1.39815	0.699073	+	0.715050i	$0.253596\pi$
$18$	−115.000	−1.50588
$19$	−26.0000	−0.313937	−0.156969	−	0.987604i	$-0.550172\pi$
$19$	−26.0000	−0.313937	−0.156969	+	0.987604i	$0.550172\pi$
$20$	−85.0000	−0.950329
$21$	−24.0000	−0.249392
$22$	70.0000	0.678366
$23$	−114.000	−1.03351	−0.516753	−	0.856134i	$-0.672859\pi$
$23$	−114.000	−1.03351	−0.516753	+	0.856134i	$0.672859\pi$
$24$	90.0000	0.765466
$25$	25.0000	0.200000
$26$	−65.0000	−0.490290
$27$	−100.000	−0.712778
$28$	−204.000	−1.37687
$29$	58.0000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	58.0000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−50.0000	−0.304290
$31$	306.000	1.77288	0.886439	−	0.462845i	$-0.153171\pi$
$31$	306.000	1.77288	0.886439	+	0.462845i	$0.153171\pi$
$32$	85.0000	0.469563
$33$	28.0000	0.147702
$34$	490.000	2.47160
$35$	60.0000	0.289767
$36$	−391.000	−1.81019
$37$	86.0000	0.382117	0.191058	−	0.981579i	$-0.438808\pi$
$37$	86.0000	0.382117	0.191058	+	0.981579i	$0.438808\pi$
$38$	−130.000	−0.554968
$39$	−26.0000	−0.106752
$40$	−225.000	−0.889391
$41$	−374.000	−1.42461	−0.712305	−	0.701870i	$-0.752349\pi$
$41$	−374.000	−1.42461	−0.712305	+	0.701870i	$0.752349\pi$
$42$	−120.000	−0.440867
$43$	−314.000	−1.11359	−0.556797	−	0.830649i	$-0.687970\pi$
$43$	−314.000	−1.11359	−0.556797	+	0.830649i	$0.687970\pi$
$44$	238.000	0.815451
$45$	115.000	0.380960
$46$	−570.000	−1.82700
$47$	620.000	1.92418	0.962088	−	0.272738i	$-0.0879293\pi$
$47$	620.000	1.92418	0.962088	+	0.272738i	$0.0879293\pi$
$48$	178.000	0.535252
$49$	−199.000	−0.580175
$50$	125.000	0.353553
$51$	196.000	0.538147
$52$	−221.000	−0.589369
$53$	362.000	0.938199	0.469099	−	0.883145i	$-0.344579\pi$
$53$	362.000	0.938199	0.469099	+	0.883145i	$0.344579\pi$
$54$	−500.000	−1.26003
$55$	−70.0000	−0.171615
$56$	−540.000	−1.28858
$57$	−52.0000	−0.120835
$58$	290.000	0.656532
$59$	266.000	0.586953	0.293477	−	0.955966i	$-0.405188\pi$
$59$	266.000	0.586953	0.293477	+	0.955966i	$0.405188\pi$
$60$	−170.000	−0.365782
$61$	634.000	1.33074	0.665372	−	0.746512i	$-0.268273\pi$
$61$	634.000	1.33074	0.665372	+	0.746512i	$0.268273\pi$
$62$	1530.00	3.13404
$63$	276.000	0.551948
$64$	−287.000	−0.560547
$65$	65.0000	0.124035
$66$	140.000	0.261103
$67$	612.000	1.11594	0.557968	−	0.829863i	$-0.311581\pi$
$67$	612.000	1.11594	0.557968	+	0.829863i	$0.311581\pi$
$68$	1666.00	2.97106
$69$	−228.000	−0.397797
$70$	300.000	0.512241
$71$	−686.000	−1.14667	−0.573333	−	0.819323i	$-0.694350\pi$
$71$	−686.000	−1.14667	−0.573333	+	0.819323i	$0.694350\pi$
$72$	−1035.00	−1.69411
$73$	202.000	0.323867	0.161934	−	0.986802i	$-0.448227\pi$
$73$	202.000	0.323867	0.161934	+	0.986802i	$0.448227\pi$
$74$	430.000	0.675493
$75$	50.0000	0.0769800
$76$	−442.000	−0.667117
$77$	−168.000	−0.248641
$78$	−130.000	−0.188713
$79$	−516.000	−0.734868	−0.367434	−	0.930050i	$-0.619764\pi$
$79$	−516.000	−0.734868	−0.367434	+	0.930050i	$0.619764\pi$
$80$	−445.000	−0.621906
$81$	421.000	0.577503
$82$	−1870.00	−2.51838
$83$	48.0000	0.0634781	0.0317391	−	0.999496i	$-0.489895\pi$
$83$	48.0000	0.0634781	0.0317391	+	0.999496i	$0.489895\pi$
$84$	−408.000	−0.529958
$85$	−490.000	−0.625270
$86$	−1570.00	−1.96858
$87$	116.000	0.142948
$88$	630.000	0.763162
$89$	−1230.00	−1.46494	−0.732470	−	0.680799i	$-0.761633\pi$
$89$	−1230.00	−1.46494	−0.732470	+	0.680799i	$0.761633\pi$
$90$	575.000	0.673448
$91$	156.000	0.179706
$92$	−1938.00	−2.19620
$93$	612.000	0.682381
$94$	3100.00	3.40150
$95$	130.000	0.140397
$96$	170.000	0.180735
$97$	350.000	0.366362	0.183181	−	0.983079i	$-0.441361\pi$
$97$	350.000	0.366362	0.183181	+	0.983079i	$0.441361\pi$
$98$	−995.000	−1.02561
$99$	−322.000	−0.326891
$100$	425.000	0.425000
$101$	−1530.00	−1.50733	−0.753667	−	0.657257i	$-0.771717\pi$
$101$	−1530.00	−1.50733	−0.753667	+	0.657257i	$0.771717\pi$
$102$	980.000	0.951318
$103$	−58.0000	−0.0554846	−0.0277423	−	0.999615i	$-0.508832\pi$
$103$	−58.0000	−0.0554846	−0.0277423	+	0.999615i	$0.508832\pi$
$104$	−585.000	−0.551577
$105$	120.000	0.111531
$106$	1810.00	1.65852
$107$	−186.000	−0.168050	−0.0840248	−	0.996464i	$-0.526777\pi$
$107$	−186.000	−0.168050	−0.0840248	+	0.996464i	$0.526777\pi$
$108$	−1700.00	−1.51465
$109$	−1342.00	−1.17927	−0.589634	−	0.807670i	$-0.700728\pi$
$109$	−1342.00	−1.17927	−0.589634	+	0.807670i	$0.700728\pi$
$110$	−350.000	−0.303374
$111$	172.000	0.147077
$112$	−1068.00	−0.901040
$113$	−1966.00	−1.63669	−0.818344	−	0.574729i	$-0.805108\pi$
$113$	−1966.00	−1.63669	−0.818344	+	0.574729i	$0.805108\pi$
$114$	−260.000	−0.213607
$115$	570.000	0.462198
$116$	986.000	0.789205
$117$	299.000	0.236261
$118$	1330.00	1.03760
$119$	−1176.00	−0.905914
$120$	−450.000	−0.342327
$121$	−1135.00	−0.852742
$122$	3170.00	2.35245
$123$	−748.000	−0.548332
$124$	5202.00	3.76737
$125$	−125.000	−0.0894427
$126$	1380.00	0.975716
$127$	1738.00	1.21435	0.607175	−	0.794568i	$-0.292303\pi$
$127$	1738.00	1.21435	0.607175	+	0.794568i	$0.292303\pi$
$128$	−2115.00	−1.46048
$129$	−628.000	−0.428623
$130$	325.000	0.219265
$131$	780.000	0.520221	0.260110	−	0.965579i	$-0.416241\pi$
$131$	780.000	0.520221	0.260110	+	0.965579i	$0.416241\pi$
$132$	476.000	0.313867
$133$	312.000	0.203412
$134$	3060.00	1.97271
$135$	500.000	0.318764
$136$	4410.00	2.78055
$137$	−1074.00	−0.669767	−0.334883	−	0.942260i	$-0.608697\pi$
$137$	−1074.00	−0.669767	−0.334883	+	0.942260i	$0.608697\pi$
$138$	−1140.00	−0.703212
$139$	416.000	0.253846	0.126923	−	0.991913i	$-0.459490\pi$
$139$	416.000	0.253846	0.126923	+	0.991913i	$0.459490\pi$
$140$	1020.00	0.615755
$141$	1240.00	0.740616
$142$	−3430.00	−2.02704
$143$	−182.000	−0.106431
$144$	−2047.00	−1.18461
$145$	−290.000	−0.166091
$146$	1010.00	0.572522
$147$	−398.000	−0.223309
$148$	1462.00	0.811998
$149$	−2054.00	−1.12933	−0.564665	−	0.825320i	$-0.690995\pi$
$149$	−2054.00	−1.12933	−0.564665	+	0.825320i	$0.690995\pi$
$150$	250.000	0.136083
$151$	2302.00	1.24062	0.620312	−	0.784355i	$-0.287006\pi$
$151$	2302.00	1.24062	0.620312	+	0.784355i	$0.287006\pi$
$152$	−1170.00	−0.624339
$153$	−2254.00	−1.19101
$154$	−840.000	−0.439540
$155$	−1530.00	−0.792855
$156$	−442.000	−0.226848
$157$	−2350.00	−1.19459	−0.597294	−	0.802022i	$-0.703758\pi$
$157$	−2350.00	−1.19459	−0.597294	+	0.802022i	$0.703758\pi$
$158$	−2580.00	−1.29907
$159$	724.000	0.361113
$160$	−425.000	−0.209995
$161$	1368.00	0.669649
$162$	2105.00	1.02089
$163$	2924.00	1.40506	0.702532	−	0.711652i	$-0.252053\pi$
$163$	2924.00	1.40506	0.702532	+	0.711652i	$0.252053\pi$
$164$	−6358.00	−3.02730
$165$	−140.000	−0.0660545
$166$	240.000	0.112215
$167$	3036.00	1.40678	0.703391	−	0.710803i	$-0.251668\pi$
$167$	3036.00	1.40678	0.703391	+	0.710803i	$0.251668\pi$
$168$	−1080.00	−0.495975
$169$	169.000	0.0769231
$170$	−2450.00	−1.10533
$171$	598.000	0.267428
$172$	−5338.00	−2.36639
$173$	2226.00	0.978264	0.489132	−	0.872210i	$-0.337314\pi$
$173$	2226.00	0.978264	0.489132	+	0.872210i	$0.337314\pi$
$174$	580.000	0.252699
$175$	−300.000	−0.129588
$176$	1246.00	0.533641
$177$	532.000	0.225918
$178$	−6150.00	−2.58967
$179$	−3244.00	−1.35457	−0.677285	−	0.735721i	$-0.736843\pi$
$179$	−3244.00	−1.35457	−0.677285	+	0.735721i	$0.736843\pi$
$180$	1955.00	0.809539
$181$	−1678.00	−0.689087	−0.344544	−	0.938770i	$-0.611966\pi$
$181$	−1678.00	−0.689087	−0.344544	+	0.938770i	$0.611966\pi$
$182$	780.000	0.317678
$183$	1268.00	0.512204
$184$	−5130.00	−2.05537
$185$	−430.000	−0.170888
$186$	3060.00	1.20629
$187$	1372.00	0.536527
$188$	10540.0	4.08888
$189$	1200.00	0.461837
$190$	650.000	0.248189
$191$	−3968.00	−1.50322	−0.751608	−	0.659610i	$-0.770722\pi$
$191$	−3968.00	−1.50322	−0.751608	+	0.659610i	$0.770722\pi$
$192$	−574.000	−0.215755
$193$	3998.00	1.49110	0.745550	−	0.666450i	$-0.232187\pi$
$193$	3998.00	1.49110	0.745550	+	0.666450i	$0.232187\pi$
$194$	1750.00	0.647643
$195$	130.000	0.0477410
$196$	−3383.00	−1.23287
$197$	4194.00	1.51680	0.758401	−	0.651788i	$-0.225981\pi$
$197$	4194.00	1.51680	0.758401	+	0.651788i	$0.225981\pi$
$198$	−1610.00	−0.577867
$199$	4240.00	1.51038	0.755190	−	0.655506i	$-0.227545\pi$
$199$	4240.00	1.51038	0.755190	+	0.655506i	$0.227545\pi$
$200$	1125.00	0.397748
$201$	1224.00	0.429524
$202$	−7650.00	−2.66461
$203$	−696.000	−0.240639
$204$	3332.00	1.14356
$205$	1870.00	0.637105
$206$	−290.000	−0.0980838
$207$	2622.00	0.880394
$208$	−1157.00	−0.385690
$209$	−364.000	−0.120471
$210$	600.000	0.197162
$211$	596.000	0.194457	0.0972283	−	0.995262i	$-0.469002\pi$
$211$	596.000	0.194457	0.0972283	+	0.995262i	$0.469002\pi$
$212$	6154.00	1.99367
$213$	−1372.00	−0.441352
$214$	−930.000	−0.297072
$215$	1570.00	0.498014
$216$	−4500.00	−1.41753
$217$	−3672.00	−1.14872
$218$	−6710.00	−2.08467
$219$	404.000	0.124657
$220$	−1190.00	−0.364681
$221$	−1274.00	−0.387776
$222$	860.000	0.259997
$223$	4268.00	1.28164	0.640822	−	0.767690i	$-0.278594\pi$
$223$	4268.00	1.28164	0.640822	+	0.767690i	$0.278594\pi$
$224$	−1020.00	−0.304248
$225$	−575.000	−0.170370
$226$	−9830.00	−2.89328
$227$	−5924.00	−1.73211	−0.866057	−	0.499946i	$-0.833353\pi$
$227$	−5924.00	−1.73211	−0.866057	+	0.499946i	$0.833353\pi$
$228$	−884.000	−0.256773
$229$	−750.000	−0.216425	−0.108213	−	0.994128i	$-0.534513\pi$
$229$	−750.000	−0.216425	−0.108213	+	0.994128i	$0.534513\pi$
$230$	2850.00	0.817058
$231$	−336.000	−0.0957021
$232$	2610.00	0.738599
$233$	474.000	0.133274	0.0666369	−	0.997777i	$-0.478773\pi$
$233$	474.000	0.133274	0.0666369	+	0.997777i	$0.478773\pi$
$234$	1495.00	0.417655
$235$	−3100.00	−0.860518
$236$	4522.00	1.24728
$237$	−1032.00	−0.282851
$238$	−5880.00	−1.60144
$239$	1598.00	0.432494	0.216247	−	0.976339i	$-0.430618\pi$
$239$	1598.00	0.432494	0.216247	+	0.976339i	$0.430618\pi$
$240$	−890.000	−0.239372
$241$	2410.00	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	2410.00	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−5675.00	−1.50745
$243$	3542.00	0.935059
$244$	10778.0	2.82783
$245$	995.000	0.259462
$246$	−3740.00	−0.969324
$247$	338.000	0.0870705
$248$	13770.0	3.52579
$249$	96.0000	0.0244327
$250$	−625.000	−0.158114
$251$	−312.000	−0.0784592	−0.0392296	−	0.999230i	$-0.512490\pi$
$251$	−312.000	−0.0784592	−0.0392296	+	0.999230i	$0.512490\pi$
$252$	4692.00	1.17289
$253$	−1596.00	−0.396599
$254$	8690.00	2.14669
$255$	−980.000	−0.240667
$256$	−8279.00	−2.02124
$257$	−5886.00	−1.42863	−0.714316	−	0.699823i	$-0.753262\pi$
$257$	−5886.00	−1.42863	−0.714316	+	0.699823i	$0.753262\pi$
$258$	−3140.00	−0.757705
$259$	−1032.00	−0.247588
$260$	1105.00	0.263574
$261$	−1334.00	−0.316370
$262$	3900.00	0.919629
$263$	1258.00	0.294949	0.147475	−	0.989066i	$-0.452886\pi$
$263$	1258.00	0.294949	0.147475	+	0.989066i	$0.452886\pi$
$264$	1260.00	0.293741
$265$	−1810.00	−0.419575
$266$	1560.00	0.359585
$267$	−2460.00	−0.563856
$268$	10404.0	2.37136
$269$	2414.00	0.547153	0.273577	−	0.961850i	$-0.411793\pi$
$269$	2414.00	0.547153	0.273577	+	0.961850i	$0.411793\pi$
$270$	2500.00	0.563501
$271$	598.000	0.134044	0.0670220	−	0.997751i	$-0.478650\pi$
$271$	598.000	0.134044	0.0670220	+	0.997751i	$0.478650\pi$
$272$	8722.00	1.94430
$273$	312.000	0.0691689
$274$	−5370.00	−1.18399
$275$	350.000	0.0767483
$276$	−3876.00	−0.845318
$277$	−4710.00	−1.02165	−0.510824	−	0.859685i	$-0.670660\pi$
$277$	−4710.00	−1.02165	−0.510824	+	0.859685i	$0.670660\pi$
$278$	2080.00	0.448741
$279$	−7038.00	−1.51023
$280$	2700.00	0.576271
$281$	4266.00	0.905652	0.452826	−	0.891599i	$-0.350416\pi$
$281$	4266.00	0.905652	0.452826	+	0.891599i	$0.350416\pi$
$282$	6200.00	1.30924
$283$	−978.000	−0.205428	−0.102714	−	0.994711i	$-0.532753\pi$
$283$	−978.000	−0.205428	−0.102714	+	0.994711i	$0.532753\pi$
$284$	−11662.0	−2.43666
$285$	260.000	0.0540388
$286$	−910.000	−0.188145
$287$	4488.00	0.923060
$288$	−1955.00	−0.399998
$289$	4691.00	0.954814
$290$	−1450.00	−0.293610
$291$	700.000	0.141013
$292$	3434.00	0.688218
$293$	−882.000	−0.175860	−0.0879300	−	0.996127i	$-0.528025\pi$
$293$	−882.000	−0.175860	−0.0879300	+	0.996127i	$0.528025\pi$
$294$	−1990.00	−0.394759
$295$	−1330.00	−0.262494
$296$	3870.00	0.759930
$297$	−1400.00	−0.273523
$298$	−10270.0	−1.99639
$299$	1482.00	0.286643
$300$	850.000	0.163583
$301$	3768.00	0.721541
$302$	11510.0	2.19313
$303$	−3060.00	−0.580173
$304$	−2314.00	−0.436569
$305$	−3170.00	−0.595127
$306$	−11270.0	−2.10543
$307$	1816.00	0.337605	0.168802	−	0.985650i	$-0.446010\pi$
$307$	1816.00	0.337605	0.168802	+	0.985650i	$0.446010\pi$
$308$	−2856.00	−0.528363
$309$	−116.000	−0.0213560
$310$	−7650.00	−1.40158
$311$	2292.00	0.417902	0.208951	−	0.977926i	$-0.432995\pi$
$311$	2292.00	0.417902	0.208951	+	0.977926i	$0.432995\pi$
$312$	−1170.00	−0.212302
$313$	8474.00	1.53028	0.765142	−	0.643862i	$-0.222669\pi$
$313$	8474.00	1.53028	0.765142	+	0.643862i	$0.222669\pi$
$314$	−11750.0	−2.11175
$315$	−1380.00	−0.246839
$316$	−8772.00	−1.56159
$317$	−4586.00	−0.812541	−0.406270	−	0.913753i	$-0.633171\pi$
$317$	−4586.00	−0.812541	−0.406270	+	0.913753i	$0.633171\pi$
$318$	3620.00	0.638363
$319$	812.000	0.142518
$320$	1435.00	0.250684
$321$	−372.000	−0.0646823
$322$	6840.00	1.18378
$323$	−2548.00	−0.438930
$324$	7157.00	1.22719
$325$	−325.000	−0.0554700
$326$	14620.0	2.48382
$327$	−2684.00	−0.453901
$328$	−16830.0	−2.83317
$329$	−7440.00	−1.24675
$330$	−700.000	−0.116769
$331$	−2046.00	−0.339753	−0.169877	−	0.985465i	$-0.554337\pi$
$331$	−2046.00	−0.339753	−0.169877	+	0.985465i	$0.554337\pi$
$332$	816.000	0.134891
$333$	−1978.00	−0.325507
$334$	15180.0	2.48686
$335$	−3060.00	−0.499062
$336$	−2136.00	−0.346811
$337$	−1286.00	−0.207872	−0.103936	−	0.994584i	$-0.533144\pi$
$337$	−1286.00	−0.207872	−0.103936	+	0.994584i	$0.533144\pi$
$338$	845.000	0.135982
$339$	−3932.00	−0.629961
$340$	−8330.00	−1.32870
$341$	4284.00	0.680327
$342$	2990.00	0.472751
$343$	6504.00	1.02386
$344$	−14130.0	−2.21465
$345$	1140.00	0.177900
$346$	11130.0	1.72934
$347$	−634.000	−0.0980833	−0.0490416	−	0.998797i	$-0.515617\pi$
$347$	−634.000	−0.0980833	−0.0490416	+	0.998797i	$0.515617\pi$
$348$	1972.00	0.303765
$349$	2266.00	0.347554	0.173777	−	0.984785i	$-0.444403\pi$
$349$	2266.00	0.347554	0.173777	+	0.984785i	$0.444403\pi$
$350$	−1500.00	−0.229081
$351$	1300.00	0.197689
$352$	1190.00	0.180191
$353$	210.000	0.0316634	0.0158317	−	0.999875i	$-0.494960\pi$
$353$	210.000	0.0316634	0.0158317	+	0.999875i	$0.494960\pi$
$354$	2660.00	0.399371
$355$	3430.00	0.512804
$356$	−20910.0	−3.11300
$357$	−2352.00	−0.348686
$358$	−16220.0	−2.39456
$359$	162.000	0.0238162	0.0119081	−	0.999929i	$-0.496209\pi$
$359$	162.000	0.0238162	0.0119081	+	0.999929i	$0.496209\pi$
$360$	5175.00	0.757629
$361$	−6183.00	−0.901443
$362$	−8390.00	−1.21815
$363$	−2270.00	−0.328221
$364$	2652.00	0.381875
$365$	−1010.00	−0.144838
$366$	6340.00	0.905457
$367$	678.000	0.0964341	0.0482170	−	0.998837i	$-0.484646\pi$
$367$	678.000	0.0964341	0.0482170	+	0.998837i	$0.484646\pi$
$368$	−10146.0	−1.43722
$369$	8602.00	1.21356
$370$	−2150.00	−0.302090
$371$	−4344.00	−0.607896
$372$	10404.0	1.45006
$373$	−6902.00	−0.958102	−0.479051	−	0.877787i	$-0.659019\pi$
$373$	−6902.00	−0.958102	−0.479051	+	0.877787i	$0.659019\pi$
$374$	6860.00	0.948455
$375$	−250.000	−0.0344265
$376$	27900.0	3.82668
$377$	−754.000	−0.103005
$378$	6000.00	0.816420
$379$	8998.00	1.21952	0.609758	−	0.792588i	$-0.291267\pi$
$379$	8998.00	1.21952	0.609758	+	0.792588i	$0.291267\pi$
$380$	2210.00	0.298344
$381$	3476.00	0.467404
$382$	−19840.0	−2.65734
$383$	132.000	0.0176107	0.00880533	−	0.999961i	$-0.497197\pi$
$383$	132.000	0.0176107	0.00880533	+	0.999961i	$0.497197\pi$
$384$	−4230.00	−0.562139
$385$	840.000	0.111196
$386$	19990.0	2.63592
$387$	7222.00	0.948617
$388$	5950.00	0.778519
$389$	11694.0	1.52419	0.762094	−	0.647466i	$-0.224171\pi$
$389$	11694.0	1.52419	0.762094	+	0.647466i	$0.224171\pi$
$390$	650.000	0.0843949
$391$	−11172.0	−1.44499
$392$	−8955.00	−1.15382
$393$	1560.00	0.200233
$394$	20970.0	2.68135
$395$	2580.00	0.328643
$396$	−5474.00	−0.694644
$397$	13678.0	1.72917	0.864583	−	0.502490i	$-0.167582\pi$
$397$	13678.0	1.72917	0.864583	+	0.502490i	$0.167582\pi$
$398$	21200.0	2.67000
$399$	624.000	0.0782934
$400$	2225.00	0.278125
$401$	8490.00	1.05728	0.528641	−	0.848845i	$-0.322702\pi$
$401$	8490.00	1.05728	0.528641	+	0.848845i	$0.322702\pi$
$402$	6120.00	0.759298
$403$	−3978.00	−0.491708
$404$	−26010.0	−3.20308
$405$	−2105.00	−0.258267
$406$	−3480.00	−0.425393
$407$	1204.00	0.146634
$408$	8820.00	1.07023
$409$	−9982.00	−1.20679	−0.603396	−	0.797442i	$-0.706186\pi$
$409$	−9982.00	−1.20679	−0.603396	+	0.797442i	$0.706186\pi$
$410$	9350.00	1.12625
$411$	−2148.00	−0.257793
$412$	−986.000	−0.117905
$413$	−3192.00	−0.380310
$414$	13110.0	1.55633
$415$	−240.000	−0.0283883
$416$	−1105.00	−0.130233
$417$	832.000	0.0977056
$418$	−1820.00	−0.212964
$419$	−9848.00	−1.14823	−0.574113	−	0.818776i	$-0.694653\pi$
$419$	−9848.00	−1.14823	−0.574113	+	0.818776i	$0.694653\pi$
$420$	2040.00	0.237004
$421$	7666.00	0.887454	0.443727	−	0.896162i	$-0.353656\pi$
$421$	7666.00	0.887454	0.443727	+	0.896162i	$0.353656\pi$
$422$	2980.00	0.343754
$423$	−14260.0	−1.63911
$424$	16290.0	1.86583
$425$	2450.00	0.279629
$426$	−6860.00	−0.780207
$427$	−7608.00	−0.862241
$428$	−3162.00	−0.357105
$429$	−364.000	−0.0409652
$430$	7850.00	0.880374
$431$	−8742.00	−0.977001	−0.488500	−	0.872564i	$-0.662456\pi$
$431$	−8742.00	−0.977001	−0.488500	+	0.872564i	$0.662456\pi$
$432$	−8900.00	−0.991207
$433$	−5654.00	−0.627515	−0.313757	−	0.949503i	$-0.601588\pi$
$433$	−5654.00	−0.627515	−0.313757	+	0.949503i	$0.601588\pi$
$434$	−18360.0	−2.03066
$435$	−580.000	−0.0639284
$436$	−22814.0	−2.50595
$437$	2964.00	0.324456
$438$	2020.00	0.220364
$439$	6976.00	0.758420	0.379210	−	0.925311i	$-0.376196\pi$
$439$	6976.00	0.758420	0.379210	+	0.925311i	$0.376196\pi$
$440$	−3150.00	−0.341296
$441$	4577.00	0.494223
$442$	−6370.00	−0.685498
$443$	66.0000	0.00707845	0.00353923	−	0.999994i	$-0.498873\pi$
$443$	66.0000	0.00707845	0.00353923	+	0.999994i	$0.498873\pi$
$444$	2924.00	0.312538
$445$	6150.00	0.655141
$446$	21340.0	2.26565
$447$	−4108.00	−0.434679
$448$	3444.00	0.363200
$449$	2426.00	0.254989	0.127494	−	0.991839i	$-0.459306\pi$
$449$	2426.00	0.254989	0.127494	+	0.991839i	$0.459306\pi$
$450$	−2875.00	−0.301175
$451$	−5236.00	−0.546682
$452$	−33422.0	−3.47796
$453$	4604.00	0.477516
$454$	−29620.0	−3.06197
$455$	−780.000	−0.0803670
$456$	−2340.00	−0.240308
$457$	−2186.00	−0.223757	−0.111878	−	0.993722i	$-0.535687\pi$
$457$	−2186.00	−0.223757	−0.111878	+	0.993722i	$0.535687\pi$
$458$	−3750.00	−0.382590
$459$	−9800.00	−0.996568
$460$	9690.00	0.982171
$461$	2370.00	0.239440	0.119720	−	0.992808i	$-0.461800\pi$
$461$	2370.00	0.239440	0.119720	+	0.992808i	$0.461800\pi$
$462$	−1680.00	−0.169179
$463$	11832.0	1.18765	0.593823	−	0.804596i	$-0.297618\pi$
$463$	11832.0	1.18765	0.593823	+	0.804596i	$0.297618\pi$
$464$	5162.00	0.516465
$465$	−3060.00	−0.305170
$466$	2370.00	0.235597
$467$	−10070.0	−0.997824	−0.498912	−	0.866653i	$-0.666267\pi$
$467$	−10070.0	−0.997824	−0.498912	+	0.866653i	$0.666267\pi$
$468$	5083.00	0.502055
$469$	−7344.00	−0.723058
$470$	−15500.0	−1.52120
$471$	−4700.00	−0.459797
$472$	11970.0	1.16730
$473$	−4396.00	−0.427333
$474$	−5160.00	−0.500014
$475$	−650.000	−0.0627875
$476$	−19992.0	−1.92507
$477$	−8326.00	−0.799206
$478$	7990.00	0.764548
$479$	15402.0	1.46918	0.734588	−	0.678513i	$-0.237375\pi$
$479$	15402.0	1.46918	0.734588	+	0.678513i	$0.237375\pi$
$480$	−850.000	−0.0808271
$481$	−1118.00	−0.105980
$482$	12050.0	1.13872
$483$	2736.00	0.257748
$484$	−19295.0	−1.81208
$485$	−1750.00	−0.163842
$486$	17710.0	1.65297
$487$	−760.000	−0.0707164	−0.0353582	−	0.999375i	$-0.511257\pi$
$487$	−760.000	−0.0707164	−0.0353582	+	0.999375i	$0.511257\pi$
$488$	28530.0	2.64650
$489$	5848.00	0.540809
$490$	4975.00	0.458669
$491$	−13448.0	−1.23605	−0.618024	−	0.786159i	$-0.712067\pi$
$491$	−13448.0	−1.23605	−0.618024	+	0.786159i	$0.712067\pi$
$492$	−12716.0	−1.16521
$493$	5684.00	0.519259
$494$	1690.00	0.153920
$495$	1610.00	0.146190
$496$	27234.0	2.46541
$497$	8232.00	0.742969
$498$	480.000	0.0431914
$499$	−170.000	−0.0152510	−0.00762550	−	0.999971i	$-0.502427\pi$
$499$	−170.000	−0.0152510	−0.00762550	+	0.999971i	$0.502427\pi$
$500$	−2125.00	−0.190066
$501$	6072.00	0.541471
$502$	−1560.00	−0.138698
$503$	298.000	0.0264158	0.0132079	−	0.999913i	$-0.495796\pi$
$503$	298.000	0.0264158	0.0132079	+	0.999913i	$0.495796\pi$
$504$	12420.0	1.09768
$505$	7650.00	0.674100
$506$	−7980.00	−0.701095
$507$	338.000	0.0296077
$508$	29546.0	2.58050
$509$	−14446.0	−1.25797	−0.628986	−	0.777417i	$-0.716530\pi$
$509$	−14446.0	−1.25797	−0.628986	+	0.777417i	$0.716530\pi$
$510$	−4900.00	−0.425442
$511$	−2424.00	−0.209846
$512$	−24475.0	−2.11260
$513$	2600.00	0.223768
$514$	−29430.0	−2.52549
$515$	290.000	0.0248135
$516$	−10676.0	−0.910823
$517$	8680.00	0.738387
$518$	−5160.00	−0.437678
$519$	4452.00	0.376534
$520$	2925.00	0.246673
$521$	8278.00	0.696096	0.348048	−	0.937477i	$-0.386845\pi$
$521$	8278.00	0.696096	0.348048	+	0.937477i	$0.386845\pi$
$522$	−6670.00	−0.559268
$523$	−12950.0	−1.08272	−0.541361	−	0.840790i	$-0.682091\pi$
$523$	−12950.0	−1.08272	−0.541361	+	0.840790i	$0.682091\pi$
$524$	13260.0	1.10547
$525$	−600.000	−0.0498784
$526$	6290.00	0.521401
$527$	29988.0	2.47874
$528$	2492.00	0.205398
$529$	829.000	0.0681351
$530$	−9050.00	−0.741711
$531$	−6118.00	−0.499997
$532$	5304.00	0.432251
$533$	4862.00	0.395116
$534$	−12300.0	−0.996766
$535$	930.000	0.0751540
$536$	27540.0	2.21930
$537$	−6488.00	−0.521374
$538$	12070.0	0.967239
$539$	−2786.00	−0.222637
$540$	8500.00	0.677374
$541$	−11838.0	−0.940768	−0.470384	−	0.882462i	$-0.655885\pi$
$541$	−11838.0	−0.940768	−0.470384	+	0.882462i	$0.655885\pi$
$542$	2990.00	0.236958
$543$	−3356.00	−0.265230
$544$	8330.00	0.656518
$545$	6710.00	0.527385
$546$	1560.00	0.122274
$547$	11590.0	0.905946	0.452973	−	0.891524i	$-0.350363\pi$
$547$	11590.0	0.905946	0.452973	+	0.891524i	$0.350363\pi$
$548$	−18258.0	−1.42325
$549$	−14582.0	−1.13360
$550$	1750.00	0.135673
$551$	−1508.00	−0.116593
$552$	−10260.0	−0.791113
$553$	6192.00	0.476149
$554$	−23550.0	−1.80604
$555$	−860.000	−0.0657747
$556$	7072.00	0.539424
$557$	−13170.0	−1.00185	−0.500925	−	0.865491i	$-0.667007\pi$
$557$	−13170.0	−1.00185	−0.500925	+	0.865491i	$0.667007\pi$
$558$	−35190.0	−2.66973
$559$	4082.00	0.308855
$560$	5340.00	0.402957
$561$	2744.00	0.206509
$562$	21330.0	1.60098
$563$	−16074.0	−1.20327	−0.601633	−	0.798773i	$-0.705483\pi$
$563$	−16074.0	−1.20327	−0.601633	+	0.798773i	$0.705483\pi$
$564$	21080.0	1.57381
$565$	9830.00	0.731949
$566$	−4890.00	−0.363148
$567$	−5052.00	−0.374187
$568$	−30870.0	−2.28042
$569$	17622.0	1.29834	0.649168	−	0.760645i	$-0.275117\pi$
$569$	17622.0	1.29834	0.649168	+	0.760645i	$0.275117\pi$
$570$	1300.00	0.0955281
$571$	12872.0	0.943391	0.471696	−	0.881761i	$-0.343642\pi$
$571$	12872.0	0.943391	0.471696	+	0.881761i	$0.343642\pi$
$572$	−3094.00	−0.226165
$573$	−7936.00	−0.578588
$574$	22440.0	1.63176
$575$	−2850.00	−0.206701
$576$	6601.00	0.477503
$577$	−1918.00	−0.138384	−0.0691918	−	0.997603i	$-0.522042\pi$
$577$	−1918.00	−0.138384	−0.0691918	+	0.997603i	$0.522042\pi$
$578$	23455.0	1.68789
$579$	7996.00	0.573925
$580$	−4930.00	−0.352943
$581$	−576.000	−0.0411300
$582$	3500.00	0.249278
$583$	5068.00	0.360026
$584$	9090.00	0.644087
$585$	−1495.00	−0.105659
$586$	−4410.00	−0.310880
$587$	20844.0	1.46563	0.732814	−	0.680429i	$-0.238206\pi$
$587$	20844.0	1.46563	0.732814	+	0.680429i	$0.238206\pi$
$588$	−6766.00	−0.474533
$589$	−7956.00	−0.556573
$590$	−6650.00	−0.464027
$591$	8388.00	0.583818
$592$	7654.00	0.531381
$593$	−16754.0	−1.16021	−0.580105	−	0.814542i	$-0.696988\pi$
$593$	−16754.0	−1.16021	−0.580105	+	0.814542i	$0.696988\pi$
$594$	−7000.00	−0.483524
$595$	5880.00	0.405137
$596$	−34918.0	−2.39983
$597$	8480.00	0.581346
$598$	7410.00	0.506718
$599$	17140.0	1.16915	0.584575	−	0.811339i	$-0.301261\pi$
$599$	17140.0	1.16915	0.584575	+	0.811339i	$0.301261\pi$
$600$	2250.00	0.153093
$601$	−12550.0	−0.851789	−0.425894	−	0.904773i	$-0.640041\pi$
$601$	−12550.0	−0.851789	−0.425894	+	0.904773i	$0.640041\pi$
$602$	18840.0	1.27552
$603$	−14076.0	−0.950612
$604$	39134.0	2.63632
$605$	5675.00	0.381358
$606$	−15300.0	−1.02561
$607$	−3434.00	−0.229624	−0.114812	−	0.993387i	$-0.536627\pi$
$607$	−3434.00	−0.229624	−0.114812	+	0.993387i	$0.536627\pi$
$608$	−2210.00	−0.147413
$609$	−1392.00	−0.0926218
$610$	−15850.0	−1.05205
$611$	−8060.00	−0.533671
$612$	−38318.0	−2.53090
$613$	−6110.00	−0.402578	−0.201289	−	0.979532i	$-0.564513\pi$
$613$	−6110.00	−0.402578	−0.201289	+	0.979532i	$0.564513\pi$
$614$	9080.00	0.596806
$615$	3740.00	0.245222
$616$	−7560.00	−0.494482
$617$	9702.00	0.633043	0.316522	−	0.948585i	$-0.397485\pi$
$617$	9702.00	0.633043	0.316522	+	0.948585i	$0.397485\pi$
$618$	−580.000	−0.0377525
$619$	−18574.0	−1.20606	−0.603031	−	0.797718i	$-0.706040\pi$
$619$	−18574.0	−1.20606	−0.603031	+	0.797718i	$0.706040\pi$
$620$	−26010.0	−1.68482
$621$	11400.0	0.736661
$622$	11460.0	0.738753
$623$	14760.0	0.949192
$624$	−2314.00	−0.148452
$625$	625.000	0.0400000
$626$	42370.0	2.70518
$627$	−728.000	−0.0463692
$628$	−39950.0	−2.53850
$629$	8428.00	0.534255
$630$	−6900.00	−0.436353
$631$	−3530.00	−0.222705	−0.111353	−	0.993781i	$-0.535518\pi$
$631$	−3530.00	−0.222705	−0.111353	+	0.993781i	$0.535518\pi$
$632$	−23220.0	−1.46146
$633$	1192.00	0.0748464
$634$	−22930.0	−1.43638
$635$	−8690.00	−0.543074
$636$	12308.0	0.767365
$637$	2587.00	0.160912
$638$	4060.00	0.251939
$639$	15778.0	0.976789
$640$	10575.0	0.653146
$641$	21330.0	1.31433	0.657164	−	0.753748i	$-0.271756\pi$
$641$	21330.0	1.31433	0.657164	+	0.753748i	$0.271756\pi$
$642$	−1860.00	−0.114343
$643$	−18768.0	−1.15107	−0.575535	−	0.817777i	$-0.695206\pi$
$643$	−18768.0	−1.15107	−0.575535	+	0.817777i	$0.695206\pi$
$644$	23256.0	1.42300
$645$	3140.00	0.191686
$646$	−12740.0	−0.775927
$647$	5618.00	0.341370	0.170685	−	0.985326i	$-0.445402\pi$
$647$	5618.00	0.341370	0.170685	+	0.985326i	$0.445402\pi$
$648$	18945.0	1.14850
$649$	3724.00	0.225239
$650$	−1625.00	−0.0980581
$651$	−7344.00	−0.442141
$652$	49708.0	2.98576
$653$	15730.0	0.942668	0.471334	−	0.881955i	$-0.343773\pi$
$653$	15730.0	0.942668	0.471334	+	0.881955i	$0.343773\pi$
$654$	−13420.0	−0.802391
$655$	−3900.00	−0.232650
$656$	−33286.0	−1.98110
$657$	−4646.00	−0.275887
$658$	−37200.0	−2.20396
$659$	−2080.00	−0.122952	−0.0614759	−	0.998109i	$-0.519581\pi$
$659$	−2080.00	−0.122952	−0.0614759	+	0.998109i	$0.519581\pi$
$660$	−2380.00	−0.140366
$661$	25066.0	1.47497	0.737484	−	0.675364i	$-0.236014\pi$
$661$	25066.0	1.47497	0.737484	+	0.675364i	$0.236014\pi$
$662$	−10230.0	−0.600605
$663$	−2548.00	−0.149255
$664$	2160.00	0.126241
$665$	−1560.00	−0.0909687
$666$	−9890.00	−0.575420
$667$	−6612.00	−0.383835
$668$	51612.0	2.98941
$669$	8536.00	0.493305
$670$	−15300.0	−0.882225
$671$	8876.00	0.510662
$672$	−2040.00	−0.117105
$673$	12642.0	0.724091	0.362046	−	0.932160i	$-0.382078\pi$
$673$	12642.0	0.724091	0.362046	+	0.932160i	$0.382078\pi$
$674$	−6430.00	−0.367469
$675$	−2500.00	−0.142556
$676$	2873.00	0.163462
$677$	11490.0	0.652284	0.326142	−	0.945321i	$-0.394251\pi$
$677$	11490.0	0.652284	0.326142	+	0.945321i	$0.394251\pi$
$678$	−19660.0	−1.11363
$679$	−4200.00	−0.237380
$680$	−22050.0	−1.24350
$681$	−11848.0	−0.666691
$682$	21420.0	1.20266
$683$	13716.0	0.768416	0.384208	−	0.923247i	$-0.374475\pi$
$683$	13716.0	0.768416	0.384208	+	0.923247i	$0.374475\pi$
$684$	10166.0	0.568285
$685$	5370.00	0.299529
$686$	32520.0	1.80994
$687$	−1500.00	−0.0833021
$688$	−27946.0	−1.54859
$689$	−4706.00	−0.260209
$690$	5700.00	0.314486
$691$	−15350.0	−0.845067	−0.422534	−	0.906347i	$-0.638859\pi$
$691$	−15350.0	−0.845067	−0.422534	+	0.906347i	$0.638859\pi$
$692$	37842.0	2.07881
$693$	3864.00	0.211806
$694$	−3170.00	−0.173388
$695$	−2080.00	−0.113524
$696$	5220.00	0.284287
$697$	−36652.0	−1.99181
$698$	11330.0	0.614394
$699$	948.000	0.0512971
$700$	−5100.00	−0.275374
$701$	−30098.0	−1.62166	−0.810832	−	0.585280i	$-0.800985\pi$
$701$	−30098.0	−1.62166	−0.810832	+	0.585280i	$0.800985\pi$
$702$	6500.00	0.349468
$703$	−2236.00	−0.119961
$704$	−4018.00	−0.215105
$705$	−6200.00	−0.331213
$706$	1050.00	0.0559735
$707$	18360.0	0.976660
$708$	9044.00	0.480077
$709$	−32270.0	−1.70934	−0.854672	−	0.519168i	$-0.826242\pi$
$709$	−32270.0	−1.70934	−0.854672	+	0.519168i	$0.826242\pi$
$710$	17150.0	0.906518
$711$	11868.0	0.625998
$712$	−55350.0	−2.91338
$713$	−34884.0	−1.83228
$714$	−11760.0	−0.616396
$715$	910.000	0.0475973
$716$	−55148.0	−2.87846
$717$	3196.00	0.166467
$718$	810.000	0.0421016
$719$	−1632.00	−0.0846500	−0.0423250	−	0.999104i	$-0.513476\pi$
$719$	−1632.00	−0.0846500	−0.0423250	+	0.999104i	$0.513476\pi$
$720$	10235.0	0.529772
$721$	696.000	0.0359506
$722$	−30915.0	−1.59354
$723$	4820.00	0.247936
$724$	−28526.0	−1.46431
$725$	1450.00	0.0742781
$726$	−11350.0	−0.580218
$727$	29354.0	1.49750	0.748748	−	0.662855i	$-0.230655\pi$
$727$	29354.0	1.49750	0.748748	+	0.662855i	$0.230655\pi$
$728$	7020.00	0.357388
$729$	−4283.00	−0.217599
$730$	−5050.00	−0.256040
$731$	−30772.0	−1.55697
$732$	21556.0	1.08843
$733$	18650.0	0.939773	0.469886	−	0.882727i	$-0.344295\pi$
$733$	18650.0	0.939773	0.469886	+	0.882727i	$0.344295\pi$
$734$	3390.00	0.170473
$735$	1990.00	0.0998670
$736$	−9690.00	−0.485296
$737$	8568.00	0.428231
$738$	43010.0	2.14528
$739$	−28550.0	−1.42115	−0.710574	−	0.703622i	$-0.751565\pi$
$739$	−28550.0	−1.42115	−0.710574	+	0.703622i	$0.751565\pi$
$740$	−7310.00	−0.363136
$741$	676.000	0.0335135
$742$	−21720.0	−1.07462
$743$	−27556.0	−1.36061	−0.680304	−	0.732930i	$-0.738152\pi$
$743$	−27556.0	−1.36061	−0.680304	+	0.732930i	$0.738152\pi$
$744$	27540.0	1.35708
$745$	10270.0	0.505052
$746$	−34510.0	−1.69370
$747$	−1104.00	−0.0540740
$748$	23324.0	1.14012
$749$	2232.00	0.108886
$750$	−1250.00	−0.0608581
$751$	−23732.0	−1.15312	−0.576560	−	0.817055i	$-0.695605\pi$
$751$	−23732.0	−1.15312	−0.576560	+	0.817055i	$0.695605\pi$
$752$	55180.0	2.67581
$753$	−624.000	−0.0301990
$754$	−3770.00	−0.182089
$755$	−11510.0	−0.554824
$756$	20400.0	0.981403
$757$	−30806.0	−1.47908	−0.739540	−	0.673113i	$-0.764957\pi$
$757$	−30806.0	−1.47908	−0.739540	+	0.673113i	$0.764957\pi$
$758$	44990.0	2.15582
$759$	−3192.00	−0.152651
$760$	5850.00	0.279213
$761$	24642.0	1.17381	0.586907	−	0.809655i	$-0.300346\pi$
$761$	24642.0	1.17381	0.586907	+	0.809655i	$0.300346\pi$
$762$	17380.0	0.826261
$763$	16104.0	0.764094
$764$	−67456.0	−3.19434
$765$	11270.0	0.532638
$766$	660.000	0.0311316
$767$	−3458.00	−0.162792
$768$	−16558.0	−0.777976
$769$	−14566.0	−0.683047	−0.341524	−	0.939873i	$-0.610943\pi$
$769$	−14566.0	−0.683047	−0.341524	+	0.939873i	$0.610943\pi$
$770$	4200.00	0.196568
$771$	−11772.0	−0.549881
$772$	67966.0	3.16859
$773$	−1386.00	−0.0644902	−0.0322451	−	0.999480i	$-0.510266\pi$
$773$	−1386.00	−0.0644902	−0.0322451	+	0.999480i	$0.510266\pi$
$774$	36110.0	1.67693
$775$	7650.00	0.354576
$776$	15750.0	0.728598
$777$	−2064.00	−0.0952968
$778$	58470.0	2.69441
$779$	9724.00	0.447238
$780$	2210.00	0.101450
$781$	−9604.00	−0.440023
$782$	−55860.0	−2.55441
$783$	−5800.00	−0.264719
$784$	−17711.0	−0.806806
$785$	11750.0	0.534236
$786$	7800.00	0.353965
$787$	−26984.0	−1.22221	−0.611103	−	0.791551i	$-0.709274\pi$
$787$	−26984.0	−1.22221	−0.611103	+	0.791551i	$0.709274\pi$
$788$	71298.0	3.22321
$789$	2516.00	0.113526
$790$	12900.0	0.580964
$791$	23592.0	1.06047
$792$	−14490.0	−0.650101
$793$	−8242.00	−0.369082
$794$	68390.0	3.05676
$795$	−3620.00	−0.161495
$796$	72080.0	3.20956
$797$	−29438.0	−1.30834	−0.654170	−	0.756347i	$-0.726982\pi$
$797$	−29438.0	−1.30834	−0.654170	+	0.756347i	$0.726982\pi$
$798$	3120.00	0.138405
$799$	60760.0	2.69028
$800$	2125.00	0.0939126
$801$	28290.0	1.24791
$802$	42450.0	1.86903
$803$	2828.00	0.124281
$804$	20808.0	0.912738
$805$	−6840.00	−0.299476
$806$	−19890.0	−0.869225
$807$	4828.00	0.210599
$808$	−68850.0	−2.99769
$809$	40462.0	1.75843	0.879214	−	0.476427i	$-0.158068\pi$
$809$	40462.0	1.75843	0.879214	+	0.476427i	$0.158068\pi$
$810$	−10525.0	−0.456557
$811$	−1830.00	−0.0792355	−0.0396178	−	0.999215i	$-0.512614\pi$
$811$	−1830.00	−0.0792355	−0.0396178	+	0.999215i	$0.512614\pi$
$812$	−11832.0	−0.511357
$813$	1196.00	0.0515935
$814$	6020.00	0.259215
$815$	−14620.0	−0.628364
$816$	17444.0	0.748360
$817$	8164.00	0.349599
$818$	−49910.0	−2.13333
$819$	−3588.00	−0.153083
$820$	31790.0	1.35385
$821$	−42726.0	−1.81626	−0.908129	−	0.418691i	$-0.862489\pi$
$821$	−42726.0	−1.81626	−0.908129	+	0.418691i	$0.862489\pi$
$822$	−10740.0	−0.455718
$823$	11798.0	0.499699	0.249850	−	0.968285i	$-0.419619\pi$
$823$	11798.0	0.499699	0.249850	+	0.968285i	$0.419619\pi$
$824$	−2610.00	−0.110344
$825$	700.000	0.0295405
$826$	−15960.0	−0.672300
$827$	−22528.0	−0.947249	−0.473625	−	0.880727i	$-0.657055\pi$
$827$	−22528.0	−0.947249	−0.473625	+	0.880727i	$0.657055\pi$
$828$	44574.0	1.87084
$829$	−8934.00	−0.374295	−0.187148	−	0.982332i	$-0.559924\pi$
$829$	−8934.00	−0.374295	−0.187148	+	0.982332i	$0.559924\pi$
$830$	−1200.00	−0.0501839
$831$	−9420.00	−0.393232
$832$	3731.00	0.155468
$833$	−19502.0	−0.811170
$834$	4160.00	0.172721
$835$	−15180.0	−0.629132
$836$	−6188.00	−0.256001
$837$	−30600.0	−1.26367
$838$	−49240.0	−2.02979
$839$	−13454.0	−0.553616	−0.276808	−	0.960925i	$-0.589277\pi$
$839$	−13454.0	−0.553616	−0.276808	+	0.960925i	$0.589277\pi$
$840$	5400.00	0.221807
$841$	−21025.0	−0.862069
$842$	38330.0	1.56881
$843$	8532.00	0.348586
$844$	10132.0	0.413220
$845$	−845.000	−0.0344010
$846$	−71300.0	−2.89757
$847$	13620.0	0.552525
$848$	32218.0	1.30468
$849$	−1956.00	−0.0790692
$850$	12250.0	0.494319
$851$	−9804.00	−0.394920
$852$	−23324.0	−0.937872
$853$	−21282.0	−0.854258	−0.427129	−	0.904191i	$-0.640475\pi$
$853$	−21282.0	−0.854258	−0.427129	+	0.904191i	$0.640475\pi$
$854$	−38040.0	−1.52424
$855$	−2990.00	−0.119597
$856$	−8370.00	−0.334206
$857$	12674.0	0.505176	0.252588	−	0.967574i	$-0.418718\pi$
$857$	12674.0	0.505176	0.252588	+	0.967574i	$0.418718\pi$
$858$	−1820.00	−0.0724170
$859$	19000.0	0.754682	0.377341	−	0.926074i	$-0.376839\pi$
$859$	19000.0	0.754682	0.377341	+	0.926074i	$0.376839\pi$
$860$	26690.0	1.05828
$861$	8976.00	0.355286
$862$	−43710.0	−1.72711
$863$	−6908.00	−0.272481	−0.136240	−	0.990676i	$-0.543502\pi$
$863$	−6908.00	−0.272481	−0.136240	+	0.990676i	$0.543502\pi$
$864$	−8500.00	−0.334694
$865$	−11130.0	−0.437493
$866$	−28270.0	−1.10930
$867$	9382.00	0.367508
$868$	−62424.0	−2.44102
$869$	−7224.00	−0.281999
$870$	−2900.00	−0.113011
$871$	−7956.00	−0.309505
$872$	−60390.0	−2.34526
$873$	−8050.00	−0.312086
$874$	14820.0	0.573563
$875$	1500.00	0.0579534
$876$	6868.00	0.264895
$877$	−13014.0	−0.501085	−0.250543	−	0.968106i	$-0.580609\pi$
$877$	−13014.0	−0.501085	−0.250543	+	0.968106i	$0.580609\pi$
$878$	34880.0	1.34071
$879$	−1764.00	−0.0676886
$880$	−6230.00	−0.238651
$881$	17318.0	0.662268	0.331134	−	0.943584i	$-0.392569\pi$
$881$	17318.0	0.662268	0.331134	+	0.943584i	$0.392569\pi$
$882$	22885.0	0.873671
$883$	42038.0	1.60214	0.801071	−	0.598569i	$-0.204264\pi$
$883$	42038.0	1.60214	0.801071	+	0.598569i	$0.204264\pi$
$884$	−21658.0	−0.824024
$885$	−2660.00	−0.101034
$886$	330.000	0.0125131
$887$	−24882.0	−0.941889	−0.470945	−	0.882163i	$-0.656087\pi$
$887$	−24882.0	−0.941889	−0.470945	+	0.882163i	$0.656087\pi$
$888$	7740.00	0.292497
$889$	−20856.0	−0.786825
$890$	30750.0	1.15814
$891$	5894.00	0.221612
$892$	72556.0	2.72349
$893$	−16120.0	−0.604071
$894$	−20540.0	−0.768412
$895$	16220.0	0.605782
$896$	25380.0	0.946302
$897$	2964.00	0.110329
$898$	12130.0	0.450761
$899$	17748.0	0.658430
$900$	−9775.00	−0.362037
$901$	35476.0	1.31174
$902$	−26180.0	−0.966406
$903$	7536.00	0.277721
$904$	−88470.0	−3.25494
$905$	8390.00	0.308169
$906$	23020.0	0.844137
$907$	24018.0	0.879277	0.439639	−	0.898175i	$-0.355106\pi$
$907$	24018.0	0.879277	0.439639	+	0.898175i	$0.355106\pi$
$908$	−100708.	−3.68074
$909$	35190.0	1.28402
$910$	−3900.00	−0.142070
$911$	9584.00	0.348553	0.174277	−	0.984697i	$-0.444241\pi$
$911$	9584.00	0.348553	0.174277	+	0.984697i	$0.444241\pi$
$912$	−4628.00	−0.168036
$913$	672.000	0.0243592
$914$	−10930.0	−0.395550
$915$	−6340.00	−0.229064
$916$	−12750.0	−0.459904
$917$	−9360.00	−0.337071
$918$	−49000.0	−1.76170
$919$	49252.0	1.76787	0.883936	−	0.467609i	$-0.154884\pi$
$919$	49252.0	1.76787	0.883936	+	0.467609i	$0.154884\pi$
$920$	25650.0	0.919191
$921$	3632.00	0.129944
$922$	11850.0	0.423274
$923$	8918.00	0.318028
$924$	−5712.00	−0.203367
$925$	2150.00	0.0764233
$926$	59160.0	2.09948
$927$	1334.00	0.0472646
$928$	4930.00	0.174391
$929$	−48614.0	−1.71687	−0.858436	−	0.512921i	$-0.828563\pi$
$929$	−48614.0	−1.71687	−0.858436	+	0.512921i	$0.828563\pi$
$930$	−15300.0	−0.539470
$931$	5174.00	0.182139
$932$	8058.00	0.283207
$933$	4584.00	0.160850
$934$	−50350.0	−1.76392
$935$	−6860.00	−0.239942
$936$	13455.0	0.469862
$937$	−37326.0	−1.30137	−0.650687	−	0.759346i	$-0.725519\pi$
$937$	−37326.0	−1.30137	−0.650687	+	0.759346i	$0.725519\pi$
$938$	−36720.0	−1.27820
$939$	16948.0	0.589006
$940$	−52700.0	−1.82860
$941$	10474.0	0.362851	0.181425	−	0.983405i	$-0.441929\pi$
$941$	10474.0	0.362851	0.181425	+	0.983405i	$0.441929\pi$
$942$	−23500.0	−0.812815
$943$	42636.0	1.47234
$944$	23674.0	0.816232
$945$	−6000.00	−0.206540
$946$	−21980.0	−0.755424
$947$	−15096.0	−0.518009	−0.259004	−	0.965876i	$-0.583394\pi$
$947$	−15096.0	−0.518009	−0.259004	+	0.965876i	$0.583394\pi$
$948$	−17544.0	−0.601058
$949$	−2626.00	−0.0898246
$950$	−3250.00	−0.110994
$951$	−9172.00	−0.312747
$952$	−52920.0	−1.80163
$953$	−3262.00	−0.110878	−0.0554389	−	0.998462i	$-0.517656\pi$
$953$	−3262.00	−0.110878	−0.0554389	+	0.998462i	$0.517656\pi$
$954$	−41630.0	−1.41281
$955$	19840.0	0.672259
$956$	27166.0	0.919049
$957$	1624.00	0.0548552
$958$	77010.0	2.59716
$959$	12888.0	0.433968
$960$	2870.00	0.0964884
$961$	63845.0	2.14310
$962$	−5590.00	−0.187348
$963$	4278.00	0.143153
$964$	40970.0	1.36883
$965$	−19990.0	−0.666840
$966$	13680.0	0.455638
$967$	48872.0	1.62525	0.812625	−	0.582786i	$-0.198038\pi$
$967$	48872.0	1.62525	0.812625	+	0.582786i	$0.198038\pi$
$968$	−51075.0	−1.69588
$969$	−5096.00	−0.168944
$970$	−8750.00	−0.289635
$971$	43932.0	1.45195	0.725976	−	0.687720i	$-0.241388\pi$
$971$	43932.0	1.45195	0.725976	+	0.687720i	$0.241388\pi$
$972$	60214.0	1.98700
$973$	−4992.00	−0.164477
$974$	−3800.00	−0.125010
$975$	−650.000	−0.0213504
$976$	56426.0	1.85057
$977$	−3406.00	−0.111533	−0.0557664	−	0.998444i	$-0.517760\pi$
$977$	−3406.00	−0.111533	−0.0557664	+	0.998444i	$0.517760\pi$
$978$	29240.0	0.956025
$979$	−17220.0	−0.562159
$980$	16915.0	0.551357
$981$	30866.0	1.00456
$982$	−67240.0	−2.18505
$983$	35496.0	1.15173	0.575863	−	0.817546i	$-0.304666\pi$
$983$	35496.0	1.15173	0.575863	+	0.817546i	$0.304666\pi$
$984$	−33660.0	−1.09049
$985$	−20970.0	−0.678335
$986$	28420.0	0.917928
$987$	−14880.0	−0.479874
$988$	5746.00	0.185025
$989$	35796.0	1.15091
$990$	8050.00	0.258430
$991$	−20464.0	−0.655964	−0.327982	−	0.944684i	$-0.606369\pi$
$991$	−20464.0	−0.655964	−0.327982	+	0.944684i	$0.606369\pi$
$992$	26010.0	0.832478
$993$	−4092.00	−0.130771
$994$	41160.0	1.31340
$995$	−21200.0	−0.675462
$996$	1632.00	0.0519196
$997$	−30318.0	−0.963070	−0.481535	−	0.876427i	$-0.659921\pi$
$997$	−30318.0	−0.963070	−0.481535	+	0.876427i	$0.659921\pi$
$998$	−850.000	−0.0269602
$999$	−8600.00	−0.272364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.4.a.a.1.1	✓	1
3.2	odd	2		585.4.a.a.1.1		1
4.3	odd	2		1040.4.a.a.1.1		1
5.2	odd	4		325.4.b.a.274.2		2
5.3	odd	4		325.4.b.a.274.1		2
5.4	even	2		325.4.a.a.1.1		1
13.12	even	2		845.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.4.a.a.1.1	✓	1	1.1	even	1	trivial
325.4.a.a.1.1		1	5.4	even	2
325.4.b.a.274.1		2	5.3	odd	4
325.4.b.a.274.2		2	5.2	odd	4
585.4.a.a.1.1		1	3.2	odd	2
845.4.a.a.1.1		1	13.12	even	2
1040.4.a.a.1.1		1	4.3	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:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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