LMFDB - Embedded newform 74.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,4,Mod(1,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("74.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 74 = 2 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	74.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:	$4.36614134042$
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	$\chi$	$=$	74.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -5.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} -10.0000 q^{6} -19.0000 q^{7} +8.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -5.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} -10.0000 q^{6} -19.0000 q^{7} +8.00000 q^{8} -2.00000 q^{9} -28.0000 q^{10} +5.00000 q^{11} -20.0000 q^{12} +6.00000 q^{13} -38.0000 q^{14} +70.0000 q^{15} +16.0000 q^{16} -72.0000 q^{17} -4.00000 q^{18} -44.0000 q^{19} -56.0000 q^{20} +95.0000 q^{21} +10.0000 q^{22} +182.000 q^{23} -40.0000 q^{24} +71.0000 q^{25} +12.0000 q^{26} +145.000 q^{27} -76.0000 q^{28} +10.0000 q^{29} +140.000 q^{30} -244.000 q^{31} +32.0000 q^{32} -25.0000 q^{33} -144.000 q^{34} +266.000 q^{35} -8.00000 q^{36} -37.0000 q^{37} -88.0000 q^{38} -30.0000 q^{39} -112.000 q^{40} -225.000 q^{41} +190.000 q^{42} -2.00000 q^{43} +20.0000 q^{44} +28.0000 q^{45} +364.000 q^{46} +221.000 q^{47} -80.0000 q^{48} +18.0000 q^{49} +142.000 q^{50} +360.000 q^{51} +24.0000 q^{52} -659.000 q^{53} +290.000 q^{54} -70.0000 q^{55} -152.000 q^{56} +220.000 q^{57} +20.0000 q^{58} +156.000 q^{59} +280.000 q^{60} -620.000 q^{61} -488.000 q^{62} +38.0000 q^{63} +64.0000 q^{64} -84.0000 q^{65} -50.0000 q^{66} +416.000 q^{67} -288.000 q^{68} -910.000 q^{69} +532.000 q^{70} -1125.00 q^{71} -16.0000 q^{72} -641.000 q^{73} -74.0000 q^{74} -355.000 q^{75} -176.000 q^{76} -95.0000 q^{77} -60.0000 q^{78} -484.000 q^{79} -224.000 q^{80} -671.000 q^{81} -450.000 q^{82} +1239.00 q^{83} +380.000 q^{84} +1008.00 q^{85} -4.00000 q^{86} -50.0000 q^{87} +40.0000 q^{88} +1304.00 q^{89} +56.0000 q^{90} -114.000 q^{91} +728.000 q^{92} +1220.00 q^{93} +442.000 q^{94} +616.000 q^{95} -160.000 q^{96} -560.000 q^{97} +36.0000 q^{98} -10.0000 q^{99} +O(q^{100})\)

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
$3$	−5.00000	−0.962250	−0.481125	−	0.876652i	$-0.659772\pi$
$3$	−5.00000	−0.962250	−0.481125	+	0.876652i	$0.659772\pi$
$4$	4.00000	0.500000
$5$	−14.0000	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$5$	−14.0000	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$6$	−10.0000	−0.680414
$7$	−19.0000	−1.02590	−0.512952	−	0.858417i	$-0.671448\pi$
$7$	−19.0000	−1.02590	−0.512952	+	0.858417i	$0.671448\pi$
$8$	8.00000	0.353553
$9$	−2.00000	−0.0740741
$10$	−28.0000	−0.885438
$11$	5.00000	0.137051	0.0685253	−	0.997649i	$-0.478171\pi$
$11$	5.00000	0.137051	0.0685253	+	0.997649i	$0.478171\pi$
$12$	−20.0000	−0.481125
$13$	6.00000	0.128008	0.0640039	−	0.997950i	$-0.479613\pi$
$13$	6.00000	0.128008	0.0640039	+	0.997950i	$0.479613\pi$
$14$	−38.0000	−0.725423
$15$	70.0000	1.20493
$16$	16.0000	0.250000
$17$	−72.0000	−1.02721	−0.513605	−	0.858027i	$-0.671690\pi$
$17$	−72.0000	−1.02721	−0.513605	+	0.858027i	$0.671690\pi$
$18$	−4.00000	−0.0523783
$19$	−44.0000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−44.0000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	−56.0000	−0.626099
$21$	95.0000	0.987176
$22$	10.0000	0.0969094
$23$	182.000	1.64998	0.824992	−	0.565145i	$-0.191180\pi$
$23$	182.000	1.64998	0.824992	+	0.565145i	$0.191180\pi$
$24$	−40.0000	−0.340207
$25$	71.0000	0.568000
$26$	12.0000	0.0905151
$27$	145.000	1.03353
$28$	−76.0000	−0.512952
$29$	10.0000	0.0640329	0.0320164	−	0.999487i	$-0.489807\pi$
$29$	10.0000	0.0640329	0.0320164	+	0.999487i	$0.489807\pi$
$30$	140.000	0.852013
$31$	−244.000	−1.41367	−0.706834	−	0.707380i	$-0.749877\pi$
$31$	−244.000	−1.41367	−0.706834	+	0.707380i	$0.749877\pi$
$32$	32.0000	0.176777
$33$	−25.0000	−0.131877
$34$	−144.000	−0.726347
$35$	266.000	1.28463
$36$	−8.00000	−0.0370370
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	−88.0000	−0.375671
$39$	−30.0000	−0.123176
$40$	−112.000	−0.442719
$41$	−225.000	−0.857051	−0.428526	−	0.903530i	$-0.640967\pi$
$41$	−225.000	−0.857051	−0.428526	+	0.903530i	$0.640967\pi$
$42$	190.000	0.698039
$43$	−2.00000	−0.00709296	−0.00354648	−	0.999994i	$-0.501129\pi$
$43$	−2.00000	−0.00709296	−0.00354648	+	0.999994i	$0.501129\pi$
$44$	20.0000	0.0685253
$45$	28.0000	0.0927554
$46$	364.000	1.16671
$47$	221.000	0.685876	0.342938	−	0.939358i	$-0.388578\pi$
$47$	221.000	0.685876	0.342938	+	0.939358i	$0.388578\pi$
$48$	−80.0000	−0.240563
$49$	18.0000	0.0524781
$50$	142.000	0.401637
$51$	360.000	0.988433
$52$	24.0000	0.0640039
$53$	−659.000	−1.70794	−0.853968	−	0.520325i	$-0.825811\pi$
$53$	−659.000	−1.70794	−0.853968	+	0.520325i	$0.825811\pi$
$54$	290.000	0.730815
$55$	−70.0000	−0.171615
$56$	−152.000	−0.362712
$57$	220.000	0.511223
$58$	20.0000	0.0452781
$59$	156.000	0.344228	0.172114	−	0.985077i	$-0.444940\pi$
$59$	156.000	0.344228	0.172114	+	0.985077i	$0.444940\pi$
$60$	280.000	0.602464
$61$	−620.000	−1.30136	−0.650679	−	0.759353i	$-0.725516\pi$
$61$	−620.000	−1.30136	−0.650679	+	0.759353i	$0.725516\pi$
$62$	−488.000	−0.999614
$63$	38.0000	0.0759929
$64$	64.0000	0.125000
$65$	−84.0000	−0.160291
$66$	−50.0000	−0.0932511
$67$	416.000	0.758545	0.379272	−	0.925285i	$-0.376174\pi$
$67$	416.000	0.758545	0.379272	+	0.925285i	$0.376174\pi$
$68$	−288.000	−0.513605
$69$	−910.000	−1.58770
$70$	532.000	0.908374
$71$	−1125.00	−1.88046	−0.940232	−	0.340535i	$-0.889392\pi$
$71$	−1125.00	−1.88046	−0.940232	+	0.340535i	$0.889392\pi$
$72$	−16.0000	−0.0261891
$73$	−641.000	−1.02772	−0.513859	−	0.857875i	$-0.671784\pi$
$73$	−641.000	−1.02772	−0.513859	+	0.857875i	$0.671784\pi$
$74$	−74.0000	−0.116248
$75$	−355.000	−0.546558
$76$	−176.000	−0.265639
$77$	−95.0000	−0.140601
$78$	−60.0000	−0.0870982
$79$	−484.000	−0.689294	−0.344647	−	0.938732i	$-0.612001\pi$
$79$	−484.000	−0.689294	−0.344647	+	0.938732i	$0.612001\pi$
$80$	−224.000	−0.313050
$81$	−671.000	−0.920439
$82$	−450.000	−0.606027
$83$	1239.00	1.63853	0.819265	−	0.573416i	$-0.194382\pi$
$83$	1239.00	1.63853	0.819265	+	0.573416i	$0.194382\pi$
$84$	380.000	0.493588
$85$	1008.00	1.28627
$86$	−4.00000	−0.00501548
$87$	−50.0000	−0.0616157
$88$	40.0000	0.0484547
$89$	1304.00	1.55308	0.776538	−	0.630071i	$-0.216974\pi$
$89$	1304.00	1.55308	0.776538	+	0.630071i	$0.216974\pi$
$90$	56.0000	0.0655880
$91$	−114.000	−0.131324
$92$	728.000	0.824992
$93$	1220.00	1.36030
$94$	442.000	0.484987
$95$	616.000	0.665266
$96$	−160.000	−0.170103
$97$	−560.000	−0.586179	−0.293090	−	0.956085i	$-0.594683\pi$
$97$	−560.000	−0.586179	−0.293090	+	0.956085i	$0.594683\pi$
$98$	36.0000	0.0371076
$99$	−10.0000	−0.0101519
$100$	284.000	0.284000
$101$	735.000	0.724111	0.362056	−	0.932156i	$-0.382075\pi$
$101$	735.000	0.724111	0.362056	+	0.932156i	$0.382075\pi$
$102$	720.000	0.698928
$103$	−650.000	−0.621810	−0.310905	−	0.950441i	$-0.600632\pi$
$103$	−650.000	−0.621810	−0.310905	+	0.950441i	$0.600632\pi$
$104$	48.0000	0.0452576
$105$	−1330.00	−1.23614
$106$	−1318.00	−1.20769
$107$	1956.00	1.76723	0.883615	−	0.468214i	$-0.155102\pi$
$107$	1956.00	1.76723	0.883615	+	0.468214i	$0.155102\pi$
$108$	580.000	0.516764
$109$	1620.00	1.42356	0.711779	−	0.702403i	$-0.247890\pi$
$109$	1620.00	1.42356	0.711779	+	0.702403i	$0.247890\pi$
$110$	−140.000	−0.121350
$111$	185.000	0.158193
$112$	−304.000	−0.256476
$113$	−934.000	−0.777552	−0.388776	−	0.921332i	$-0.627102\pi$
$113$	−934.000	−0.777552	−0.388776	+	0.921332i	$0.627102\pi$
$114$	440.000	0.361489
$115$	−2548.00	−2.06611
$116$	40.0000	0.0320164
$117$	−12.0000	−0.00948205
$118$	312.000	0.243406
$119$	1368.00	1.05382
$120$	560.000	0.426006
$121$	−1306.00	−0.981217
$122$	−1240.00	−0.920199
$123$	1125.00	0.824698
$124$	−976.000	−0.706834
$125$	756.000	0.540950
$126$	76.0000	0.0537351
$127$	1899.00	1.32684	0.663421	−	0.748246i	$-0.269104\pi$
$127$	1899.00	1.32684	0.663421	+	0.748246i	$0.269104\pi$
$128$	128.000	0.0883883
$129$	10.0000	0.00682520
$130$	−168.000	−0.113343
$131$	−236.000	−0.157400	−0.0787001	−	0.996898i	$-0.525077\pi$
$131$	−236.000	−0.157400	−0.0787001	+	0.996898i	$0.525077\pi$
$132$	−100.000	−0.0659385
$133$	836.000	0.545041
$134$	832.000	0.536372
$135$	−2030.00	−1.29418
$136$	−576.000	−0.363173
$137$	1610.00	1.00403	0.502013	−	0.864860i	$-0.332593\pi$
$137$	1610.00	1.00403	0.502013	+	0.864860i	$0.332593\pi$
$138$	−1820.00	−1.12267
$139$	−500.000	−0.305104	−0.152552	−	0.988295i	$-0.548749\pi$
$139$	−500.000	−0.305104	−0.152552	+	0.988295i	$0.548749\pi$
$140$	1064.00	0.642317
$141$	−1105.00	−0.659984
$142$	−2250.00	−1.32969
$143$	30.0000	0.0175435
$144$	−32.0000	−0.0185185
$145$	−140.000	−0.0801818
$146$	−1282.00	−0.726706
$147$	−90.0000	−0.0504971
$148$	−148.000	−0.0821995
$149$	1751.00	0.962735	0.481367	−	0.876519i	$-0.340140\pi$
$149$	1751.00	0.962735	0.481367	+	0.876519i	$0.340140\pi$
$150$	−710.000	−0.386475
$151$	−2704.00	−1.45727	−0.728637	−	0.684900i	$-0.759846\pi$
$151$	−2704.00	−1.45727	−0.728637	+	0.684900i	$0.759846\pi$
$152$	−352.000	−0.187835
$153$	144.000	0.0760896
$154$	−190.000	−0.0994197
$155$	3416.00	1.77019
$156$	−120.000	−0.0615878
$157$	−2661.00	−1.35268	−0.676341	−	0.736589i	$-0.736435\pi$
$157$	−2661.00	−1.35268	−0.676341	+	0.736589i	$0.736435\pi$
$158$	−968.000	−0.487405
$159$	3295.00	1.64346
$160$	−448.000	−0.221359
$161$	−3458.00	−1.69272
$162$	−1342.00	−0.650849
$163$	1854.00	0.890899	0.445449	−	0.895307i	$-0.353044\pi$
$163$	1854.00	0.890899	0.445449	+	0.895307i	$0.353044\pi$
$164$	−900.000	−0.428526
$165$	350.000	0.165136
$166$	2478.00	1.15862
$167$	−1808.00	−0.837768	−0.418884	−	0.908040i	$-0.637579\pi$
$167$	−1808.00	−0.837768	−0.418884	+	0.908040i	$0.637579\pi$
$168$	760.000	0.349019
$169$	−2161.00	−0.983614
$170$	2016.00	0.909530
$171$	88.0000	0.0393540
$172$	−8.00000	−0.00354648
$173$	−3619.00	−1.59045	−0.795224	−	0.606316i	$-0.792647\pi$
$173$	−3619.00	−1.59045	−0.795224	+	0.606316i	$0.792647\pi$
$174$	−100.000	−0.0435689
$175$	−1349.00	−0.582713
$176$	80.0000	0.0342627
$177$	−780.000	−0.331234
$178$	2608.00	1.09819
$179$	34.0000	0.0141971	0.00709855	−	0.999975i	$-0.497740\pi$
$179$	34.0000	0.0141971	0.00709855	+	0.999975i	$0.497740\pi$
$180$	112.000	0.0463777
$181$	737.000	0.302656	0.151328	−	0.988484i	$-0.451645\pi$
$181$	737.000	0.302656	0.151328	+	0.988484i	$0.451645\pi$
$182$	−228.000	−0.0928598
$183$	3100.00	1.25223
$184$	1456.00	0.583357
$185$	518.000	0.205860
$186$	2440.00	0.961879
$187$	−360.000	−0.140780
$188$	884.000	0.342938
$189$	−2755.00	−1.06030
$190$	1232.00	0.470414
$191$	1704.00	0.645535	0.322767	−	0.946478i	$-0.395387\pi$
$191$	1704.00	0.645535	0.322767	+	0.946478i	$0.395387\pi$
$192$	−320.000	−0.120281
$193$	690.000	0.257343	0.128672	−	0.991687i	$-0.458929\pi$
$193$	690.000	0.257343	0.128672	+	0.991687i	$0.458929\pi$
$194$	−1120.00	−0.414491
$195$	420.000	0.154240
$196$	72.0000	0.0262391
$197$	175.000	0.0632905	0.0316453	−	0.999499i	$-0.489925\pi$
$197$	175.000	0.0632905	0.0316453	+	0.999499i	$0.489925\pi$
$198$	−20.0000	−0.00717848
$199$	−3590.00	−1.27884	−0.639418	−	0.768859i	$-0.720825\pi$
$199$	−3590.00	−1.27884	−0.639418	+	0.768859i	$0.720825\pi$
$200$	568.000	0.200818
$201$	−2080.00	−0.729910
$202$	1470.00	0.512024
$203$	−190.000	−0.0656916
$204$	1440.00	0.494217
$205$	3150.00	1.07320
$206$	−1300.00	−0.439686
$207$	−364.000	−0.122221
$208$	96.0000	0.0320019
$209$	−220.000	−0.0728120
$210$	−2660.00	−0.874083
$211$	−1011.00	−0.329858	−0.164929	−	0.986305i	$-0.552740\pi$
$211$	−1011.00	−0.329858	−0.164929	+	0.986305i	$0.552740\pi$
$212$	−2636.00	−0.853968
$213$	5625.00	1.80948
$214$	3912.00	1.24962
$215$	28.0000	0.00888179
$216$	1160.00	0.365407
$217$	4636.00	1.45029
$218$	3240.00	1.00661
$219$	3205.00	0.988922
$220$	−280.000	−0.0858073
$221$	−432.000	−0.131491
$222$	370.000	0.111859
$223$	−3243.00	−0.973845	−0.486922	−	0.873445i	$-0.661881\pi$
$223$	−3243.00	−0.973845	−0.486922	+	0.873445i	$0.661881\pi$
$224$	−608.000	−0.181356
$225$	−142.000	−0.0420741
$226$	−1868.00	−0.549812
$227$	−5480.00	−1.60229	−0.801146	−	0.598469i	$-0.795776\pi$
$227$	−5480.00	−1.60229	−0.801146	+	0.598469i	$0.795776\pi$
$228$	880.000	0.255612
$229$	2203.00	0.635713	0.317857	−	0.948139i	$-0.397037\pi$
$229$	2203.00	0.635713	0.317857	+	0.948139i	$0.397037\pi$
$230$	−5096.00	−1.46096
$231$	475.000	0.135293
$232$	80.0000	0.0226390
$233$	3718.00	1.04538	0.522692	−	0.852522i	$-0.324928\pi$
$233$	3718.00	1.04538	0.522692	+	0.852522i	$0.324928\pi$
$234$	−24.0000	−0.00670483
$235$	−3094.00	−0.858852
$236$	624.000	0.172114
$237$	2420.00	0.663274
$238$	2736.00	0.745162
$239$	−1038.00	−0.280932	−0.140466	−	0.990086i	$-0.544860\pi$
$239$	−1038.00	−0.280932	−0.140466	+	0.990086i	$0.544860\pi$
$240$	1120.00	0.301232
$241$	2050.00	0.547934	0.273967	−	0.961739i	$-0.411664\pi$
$241$	2050.00	0.547934	0.273967	+	0.961739i	$0.411664\pi$
$242$	−2612.00	−0.693825
$243$	−560.000	−0.147835
$244$	−2480.00	−0.650679
$245$	−252.000	−0.0657130
$246$	2250.00	0.583149
$247$	−264.000	−0.0680078
$248$	−1952.00	−0.499807
$249$	−6195.00	−1.57668
$250$	1512.00	0.382509
$251$	702.000	0.176533	0.0882666	−	0.996097i	$-0.471867\pi$
$251$	702.000	0.176533	0.0882666	+	0.996097i	$0.471867\pi$
$252$	152.000	0.0379964
$253$	910.000	0.226131
$254$	3798.00	0.938219
$255$	−5040.00	−1.23771
$256$	256.000	0.0625000
$257$	5340.00	1.29611	0.648055	−	0.761594i	$-0.275583\pi$
$257$	5340.00	1.29611	0.648055	+	0.761594i	$0.275583\pi$
$258$	20.0000	0.00482615
$259$	703.000	0.168658
$260$	−336.000	−0.0801455
$261$	−20.0000	−0.00474318
$262$	−472.000	−0.111299
$263$	−4791.00	−1.12329	−0.561646	−	0.827378i	$-0.689832\pi$
$263$	−4791.00	−1.12329	−0.561646	+	0.827378i	$0.689832\pi$
$264$	−200.000	−0.0466256
$265$	9226.00	2.13867
$266$	1672.00	0.385402
$267$	−6520.00	−1.49445
$268$	1664.00	0.379272
$269$	−6030.00	−1.36675	−0.683375	−	0.730068i	$-0.739488\pi$
$269$	−6030.00	−1.36675	−0.683375	+	0.730068i	$0.739488\pi$
$270$	−4060.00	−0.915125
$271$	−2173.00	−0.487086	−0.243543	−	0.969890i	$-0.578310\pi$
$271$	−2173.00	−0.487086	−0.243543	+	0.969890i	$0.578310\pi$
$272$	−1152.00	−0.256802
$273$	570.000	0.126366
$274$	3220.00	0.709954
$275$	355.000	0.0778447
$276$	−3640.00	−0.793849
$277$	−5996.00	−1.30059	−0.650297	−	0.759680i	$-0.725356\pi$
$277$	−5996.00	−1.30059	−0.650297	+	0.759680i	$0.725356\pi$
$278$	−1000.00	−0.215741
$279$	488.000	0.104716
$280$	2128.00	0.454187
$281$	3636.00	0.771906	0.385953	−	0.922518i	$-0.373873\pi$
$281$	3636.00	0.771906	0.385953	+	0.922518i	$0.373873\pi$
$282$	−2210.00	−0.466679
$283$	−2616.00	−0.549488	−0.274744	−	0.961517i	$-0.588593\pi$
$283$	−2616.00	−0.549488	−0.274744	+	0.961517i	$0.588593\pi$
$284$	−4500.00	−0.940232
$285$	−3080.00	−0.640152
$286$	60.0000	0.0124052
$287$	4275.00	0.879252
$288$	−64.0000	−0.0130946
$289$	271.000	0.0551598
$290$	−280.000	−0.0566971
$291$	2800.00	0.564051
$292$	−2564.00	−0.513859
$293$	8238.00	1.64256	0.821278	−	0.570528i	$-0.193261\pi$
$293$	8238.00	1.64256	0.821278	+	0.570528i	$0.193261\pi$
$294$	−180.000	−0.0357068
$295$	−2184.00	−0.431042
$296$	−296.000	−0.0581238
$297$	725.000	0.141646
$298$	3502.00	0.680756
$299$	1092.00	0.211211
$300$	−1420.00	−0.273279
$301$	38.0000	0.00727669
$302$	−5408.00	−1.03045
$303$	−3675.00	−0.696776
$304$	−704.000	−0.132820
$305$	8680.00	1.62956
$306$	288.000	0.0538035
$307$	8633.00	1.60492	0.802461	−	0.596704i	$-0.203523\pi$
$307$	8633.00	1.60492	0.802461	+	0.596704i	$0.203523\pi$
$308$	−380.000	−0.0703004
$309$	3250.00	0.598337
$310$	6832.00	1.25171
$311$	−1180.00	−0.215150	−0.107575	−	0.994197i	$-0.534309\pi$
$311$	−1180.00	−0.215150	−0.107575	+	0.994197i	$0.534309\pi$
$312$	−240.000	−0.0435491
$313$	238.000	0.0429794	0.0214897	−	0.999769i	$-0.493159\pi$
$313$	238.000	0.0429794	0.0214897	+	0.999769i	$0.493159\pi$
$314$	−5322.00	−0.956490
$315$	−532.000	−0.0951581
$316$	−1936.00	−0.344647
$317$	−7482.00	−1.32565	−0.662825	−	0.748774i	$-0.730643\pi$
$317$	−7482.00	−1.32565	−0.662825	+	0.748774i	$0.730643\pi$
$318$	6590.00	1.16210
$319$	50.0000	0.00877574
$320$	−896.000	−0.156525
$321$	−9780.00	−1.70052
$322$	−6916.00	−1.19694
$323$	3168.00	0.545734
$324$	−2684.00	−0.460219
$325$	426.000	0.0727084
$326$	3708.00	0.629961
$327$	−8100.00	−1.36982
$328$	−1800.00	−0.303013
$329$	−4199.00	−0.703642
$330$	700.000	0.116769
$331$	−2894.00	−0.480570	−0.240285	−	0.970702i	$-0.577241\pi$
$331$	−2894.00	−0.480570	−0.240285	+	0.970702i	$0.577241\pi$
$332$	4956.00	0.819265
$333$	74.0000	0.0121777
$334$	−3616.00	−0.592391
$335$	−5824.00	−0.949848
$336$	1520.00	0.246794
$337$	7631.00	1.23349	0.616746	−	0.787162i	$-0.288450\pi$
$337$	7631.00	1.23349	0.616746	+	0.787162i	$0.288450\pi$
$338$	−4322.00	−0.695520
$339$	4670.00	0.748199
$340$	4032.00	0.643135
$341$	−1220.00	−0.193744
$342$	176.000	0.0278275
$343$	6175.00	0.972066
$344$	−16.0000	−0.00250774
$345$	12740.0	1.98811
$346$	−7238.00	−1.12462
$347$	−9858.00	−1.52509	−0.762543	−	0.646937i	$-0.776050\pi$
$347$	−9858.00	−1.52509	−0.762543	+	0.646937i	$0.776050\pi$
$348$	−200.000	−0.0308078
$349$	−1290.00	−0.197857	−0.0989285	−	0.995095i	$-0.531542\pi$
$349$	−1290.00	−0.197857	−0.0989285	+	0.995095i	$0.531542\pi$
$350$	−2698.00	−0.412040
$351$	870.000	0.132300
$352$	160.000	0.0242274
$353$	−5384.00	−0.811789	−0.405894	−	0.913920i	$-0.633040\pi$
$353$	−5384.00	−0.811789	−0.405894	+	0.913920i	$0.633040\pi$
$354$	−1560.00	−0.234218
$355$	15750.0	2.35471
$356$	5216.00	0.776538
$357$	−6840.00	−1.01404
$358$	68.0000	0.0100389
$359$	−4197.00	−0.617017	−0.308509	−	0.951222i	$-0.599830\pi$
$359$	−4197.00	−0.617017	−0.308509	+	0.951222i	$0.599830\pi$
$360$	224.000	0.0327940
$361$	−4923.00	−0.717743
$362$	1474.00	0.214010
$363$	6530.00	0.944177
$364$	−456.000	−0.0656618
$365$	8974.00	1.28691
$366$	6200.00	0.885462
$367$	10848.0	1.54295	0.771473	−	0.636262i	$-0.219520\pi$
$367$	10848.0	1.54295	0.771473	+	0.636262i	$0.219520\pi$
$368$	2912.00	0.412496
$369$	450.000	0.0634853
$370$	1036.00	0.145565
$371$	12521.0	1.75218
$372$	4880.00	0.680151
$373$	6833.00	0.948524	0.474262	−	0.880384i	$-0.342715\pi$
$373$	6833.00	0.948524	0.474262	+	0.880384i	$0.342715\pi$
$374$	−720.000	−0.0995463
$375$	−3780.00	−0.520529
$376$	1768.00	0.242494
$377$	60.0000	0.00819670
$378$	−5510.00	−0.749746
$379$	12553.0	1.70133	0.850665	−	0.525708i	$-0.176199\pi$
$379$	12553.0	1.70133	0.850665	+	0.525708i	$0.176199\pi$
$380$	2464.00	0.332633
$381$	−9495.00	−1.27675
$382$	3408.00	0.456462
$383$	4476.00	0.597162	0.298581	−	0.954384i	$-0.403487\pi$
$383$	4476.00	0.597162	0.298581	+	0.954384i	$0.403487\pi$
$384$	−640.000	−0.0850517
$385$	1330.00	0.176060
$386$	1380.00	0.181969
$387$	4.00000	0.000525404 0
$388$	−2240.00	−0.293090
$389$	−11244.0	−1.46554	−0.732768	−	0.680479i	$-0.761772\pi$
$389$	−11244.0	−1.46554	−0.732768	+	0.680479i	$0.761772\pi$
$390$	840.000	0.109064
$391$	−13104.0	−1.69488
$392$	144.000	0.0185538
$393$	1180.00	0.151458
$394$	350.000	0.0447532
$395$	6776.00	0.863133
$396$	−40.0000	−0.00507595
$397$	14359.0	1.81526	0.907629	−	0.419773i	$-0.137890\pi$
$397$	14359.0	1.81526	0.907629	+	0.419773i	$0.137890\pi$
$398$	−7180.00	−0.904274
$399$	−4180.00	−0.524466
$400$	1136.00	0.142000
$401$	−8702.00	−1.08368	−0.541842	−	0.840480i	$-0.682273\pi$
$401$	−8702.00	−1.08368	−0.541842	+	0.840480i	$0.682273\pi$
$402$	−4160.00	−0.516124
$403$	−1464.00	−0.180960
$404$	2940.00	0.362056
$405$	9394.00	1.15257
$406$	−380.000	−0.0464509
$407$	−185.000	−0.0225310
$408$	2880.00	0.349464
$409$	3376.00	0.408148	0.204074	−	0.978955i	$-0.434582\pi$
$409$	3376.00	0.408148	0.204074	+	0.978955i	$0.434582\pi$
$410$	6300.00	0.758865
$411$	−8050.00	−0.966125
$412$	−2600.00	−0.310905
$413$	−2964.00	−0.353145
$414$	−728.000	−0.0864233
$415$	−17346.0	−2.05176
$416$	192.000	0.0226288
$417$	2500.00	0.293586
$418$	−440.000	−0.0514859
$419$	−6295.00	−0.733964	−0.366982	−	0.930228i	$-0.619609\pi$
$419$	−6295.00	−0.733964	−0.366982	+	0.930228i	$0.619609\pi$
$420$	−5320.00	−0.618070
$421$	−13884.0	−1.60728	−0.803640	−	0.595116i	$-0.797106\pi$
$421$	−13884.0	−1.60728	−0.803640	+	0.595116i	$0.797106\pi$
$422$	−2022.00	−0.233245
$423$	−442.000	−0.0508056
$424$	−5272.00	−0.603847
$425$	−5112.00	−0.583455
$426$	11250.0	1.27949
$427$	11780.0	1.33507
$428$	7824.00	0.883615
$429$	−150.000	−0.0168813
$430$	56.0000	0.00628037
$431$	−8130.00	−0.908604	−0.454302	−	0.890848i	$-0.650111\pi$
$431$	−8130.00	−0.908604	−0.454302	+	0.890848i	$0.650111\pi$
$432$	2320.00	0.258382
$433$	−5983.00	−0.664029	−0.332015	−	0.943274i	$-0.607728\pi$
$433$	−5983.00	−0.664029	−0.332015	+	0.943274i	$0.607728\pi$
$434$	9272.00	1.02551
$435$	700.000	0.0771550
$436$	6480.00	0.711779
$437$	−8008.00	−0.876601
$438$	6410.00	0.699273
$439$	−7664.00	−0.833218	−0.416609	−	0.909086i	$-0.636782\pi$
$439$	−7664.00	−0.833218	−0.416609	+	0.909086i	$0.636782\pi$
$440$	−560.000	−0.0606749
$441$	−36.0000	−0.00388727
$442$	−864.000	−0.0929780
$443$	−2577.00	−0.276381	−0.138191	−	0.990406i	$-0.544129\pi$
$443$	−2577.00	−0.276381	−0.138191	+	0.990406i	$0.544129\pi$
$444$	740.000	0.0790965
$445$	−18256.0	−1.94476
$446$	−6486.00	−0.688612
$447$	−8755.00	−0.926392
$448$	−1216.00	−0.128238
$449$	12080.0	1.26969	0.634845	−	0.772640i	$-0.281064\pi$
$449$	12080.0	1.26969	0.634845	+	0.772640i	$0.281064\pi$
$450$	−284.000	−0.0297509
$451$	−1125.00	−0.117459
$452$	−3736.00	−0.388776
$453$	13520.0	1.40226
$454$	−10960.0	−1.13299
$455$	1596.00	0.164443
$456$	1760.00	0.180745
$457$	−10490.0	−1.07375	−0.536873	−	0.843663i	$-0.680394\pi$
$457$	−10490.0	−1.07375	−0.536873	+	0.843663i	$0.680394\pi$
$458$	4406.00	0.449517
$459$	−10440.0	−1.06165
$460$	−10192.0	−1.03305
$461$	−14810.0	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$461$	−14810.0	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$462$	950.000	0.0956667
$463$	15578.0	1.56365	0.781826	−	0.623496i	$-0.214288\pi$
$463$	15578.0	1.56365	0.781826	+	0.623496i	$0.214288\pi$
$464$	160.000	0.0160082
$465$	−17080.0	−1.70337
$466$	7436.00	0.739198
$467$	−2062.00	−0.204321	−0.102161	−	0.994768i	$-0.532576\pi$
$467$	−2062.00	−0.204321	−0.102161	+	0.994768i	$0.532576\pi$
$468$	−48.0000	−0.00474103
$469$	−7904.00	−0.778194
$470$	−6188.00	−0.607300
$471$	13305.0	1.30162
$472$	1248.00	0.121703
$473$	−10.0000	−0.000972094 0
$474$	4840.00	0.469005
$475$	−3124.00	−0.301766
$476$	5472.00	0.526909
$477$	1318.00	0.126514
$478$	−2076.00	−0.198649
$479$	3438.00	0.327946	0.163973	−	0.986465i	$-0.447569\pi$
$479$	3438.00	0.327946	0.163973	+	0.986465i	$0.447569\pi$
$480$	2240.00	0.213003
$481$	−222.000	−0.0210443
$482$	4100.00	0.387448
$483$	17290.0	1.62882
$484$	−5224.00	−0.490609
$485$	7840.00	0.734012
$486$	−1120.00	−0.104535
$487$	−19592.0	−1.82299	−0.911497	−	0.411306i	$-0.865073\pi$
$487$	−19592.0	−1.82299	−0.911497	+	0.411306i	$0.865073\pi$
$488$	−4960.00	−0.460100
$489$	−9270.00	−0.857268
$490$	−504.000	−0.0464661
$491$	4444.00	0.408462	0.204231	−	0.978923i	$-0.434531\pi$
$491$	4444.00	0.408462	0.204231	+	0.978923i	$0.434531\pi$
$492$	4500.00	0.412349
$493$	−720.000	−0.0657752
$494$	−528.000	−0.0480888
$495$	140.000	0.0127122
$496$	−3904.00	−0.353417
$497$	21375.0	1.92917
$498$	−12390.0	−1.11488
$499$	−1140.00	−0.102271	−0.0511357	−	0.998692i	$-0.516284\pi$
$499$	−1140.00	−0.102271	−0.0511357	+	0.998692i	$0.516284\pi$
$500$	3024.00	0.270475
$501$	9040.00	0.806143
$502$	1404.00	0.124828
$503$	−3180.00	−0.281887	−0.140944	−	0.990018i	$-0.545014\pi$
$503$	−3180.00	−0.281887	−0.140944	+	0.990018i	$0.545014\pi$
$504$	304.000	0.0268675
$505$	−10290.0	−0.906731
$506$	1820.00	0.159899
$507$	10805.0	0.946483
$508$	7596.00	0.663421
$509$	15213.0	1.32476	0.662382	−	0.749167i	$-0.269546\pi$
$509$	15213.0	1.32476	0.662382	+	0.749167i	$0.269546\pi$
$510$	−10080.0	−0.875196
$511$	12179.0	1.05434
$512$	512.000	0.0441942
$513$	−6380.00	−0.549091
$514$	10680.0	0.916488
$515$	9100.00	0.778629
$516$	40.0000	0.00341260
$517$	1105.00	0.0939997
$518$	1406.00	0.119259
$519$	18095.0	1.53041
$520$	−672.000	−0.0566714
$521$	16271.0	1.36823	0.684113	−	0.729376i	$-0.260190\pi$
$521$	16271.0	1.36823	0.684113	+	0.729376i	$0.260190\pi$
$522$	−40.0000	−0.00335393
$523$	16322.0	1.36465	0.682324	−	0.731050i	$-0.260969\pi$
$523$	16322.0	1.36465	0.682324	+	0.731050i	$0.260969\pi$
$524$	−944.000	−0.0787001
$525$	6745.00	0.560716
$526$	−9582.00	−0.794287
$527$	17568.0	1.45213
$528$	−400.000	−0.0329693
$529$	20957.0	1.72245
$530$	18452.0	1.51227
$531$	−312.000	−0.0254984
$532$	3344.00	0.272520
$533$	−1350.00	−0.109709
$534$	−13040.0	−1.05673
$535$	−27384.0	−2.21292
$536$	3328.00	0.268186
$537$	−170.000	−0.0136612
$538$	−12060.0	−0.966438
$539$	90.0000	0.00719216
$540$	−8120.00	−0.647091
$541$	−11876.0	−0.943788	−0.471894	−	0.881655i	$-0.656429\pi$
$541$	−11876.0	−0.943788	−0.471894	+	0.881655i	$0.656429\pi$
$542$	−4346.00	−0.344422
$543$	−3685.00	−0.291231
$544$	−2304.00	−0.181587
$545$	−22680.0	−1.78258
$546$	1140.00	0.0893544
$547$	3824.00	0.298908	0.149454	−	0.988769i	$-0.452248\pi$
$547$	3824.00	0.298908	0.149454	+	0.988769i	$0.452248\pi$
$548$	6440.00	0.502013
$549$	1240.00	0.0963969
$550$	710.000	0.0550445
$551$	−440.000	−0.0340193
$552$	−7280.00	−0.561336
$553$	9196.00	0.707150
$554$	−11992.0	−0.919659
$555$	−2590.00	−0.198089
$556$	−2000.00	−0.152552
$557$	−16410.0	−1.24832	−0.624160	−	0.781297i	$-0.714559\pi$
$557$	−16410.0	−1.24832	−0.624160	+	0.781297i	$0.714559\pi$
$558$	976.000	0.0740455
$559$	−12.0000	−0.000907953 0
$560$	4256.00	0.321159
$561$	1800.00	0.135465
$562$	7272.00	0.545820
$563$	12798.0	0.958031	0.479015	−	0.877806i	$-0.340994\pi$
$563$	12798.0	0.958031	0.479015	+	0.877806i	$0.340994\pi$
$564$	−4420.00	−0.329992
$565$	13076.0	0.973649
$566$	−5232.00	−0.388547
$567$	12749.0	0.944282
$568$	−9000.00	−0.664844
$569$	−22716.0	−1.67365	−0.836823	−	0.547474i	$-0.815590\pi$
$569$	−22716.0	−1.67365	−0.836823	+	0.547474i	$0.815590\pi$
$570$	−6160.00	−0.452656
$571$	−3151.00	−0.230937	−0.115469	−	0.993311i	$-0.536837\pi$
$571$	−3151.00	−0.230937	−0.115469	+	0.993311i	$0.536837\pi$
$572$	120.000	0.00877177
$573$	−8520.00	−0.621166
$574$	8550.00	0.621725
$575$	12922.0	0.937191
$576$	−128.000	−0.00925926
$577$	−3556.00	−0.256565	−0.128283	−	0.991738i	$-0.540946\pi$
$577$	−3556.00	−0.256565	−0.128283	+	0.991738i	$0.540946\pi$
$578$	542.000	0.0390039
$579$	−3450.00	−0.247629
$580$	−560.000	−0.0400909
$581$	−23541.0	−1.68097
$582$	5600.00	0.398844
$583$	−3295.00	−0.234074
$584$	−5128.00	−0.363353
$585$	168.000	0.0118734
$586$	16476.0	1.16146
$587$	13564.0	0.953741	0.476871	−	0.878973i	$-0.341771\pi$
$587$	13564.0	0.953741	0.476871	+	0.878973i	$0.341771\pi$
$588$	−360.000	−0.0252486
$589$	10736.0	0.751051
$590$	−4368.00	−0.304793
$591$	−875.000	−0.0609013
$592$	−592.000	−0.0410997
$593$	427.000	0.0295696	0.0147848	−	0.999891i	$-0.495294\pi$
$593$	427.000	0.0295696	0.0147848	+	0.999891i	$0.495294\pi$
$594$	1450.00	0.100159
$595$	−19152.0	−1.31959
$596$	7004.00	0.481367
$597$	17950.0	1.23056
$598$	2184.00	0.149348
$599$	18603.0	1.26894	0.634472	−	0.772945i	$-0.281217\pi$
$599$	18603.0	1.26894	0.634472	+	0.772945i	$0.281217\pi$
$600$	−2840.00	−0.193238
$601$	−20406.0	−1.38499	−0.692494	−	0.721423i	$-0.743488\pi$
$601$	−20406.0	−1.38499	−0.692494	+	0.721423i	$0.743488\pi$
$602$	76.0000	0.00514540
$603$	−832.000	−0.0561885
$604$	−10816.0	−0.728637
$605$	18284.0	1.22868
$606$	−7350.00	−0.492695
$607$	1028.00	0.0687401	0.0343700	−	0.999409i	$-0.489058\pi$
$607$	1028.00	0.0687401	0.0343700	+	0.999409i	$0.489058\pi$
$608$	−1408.00	−0.0939177
$609$	950.000	0.0632117
$610$	17360.0	1.15227
$611$	1326.00	0.0877974
$612$	576.000	0.0380448
$613$	21875.0	1.44131	0.720655	−	0.693294i	$-0.243841\pi$
$613$	21875.0	1.44131	0.720655	+	0.693294i	$0.243841\pi$
$614$	17266.0	1.13485
$615$	−15750.0	−1.03269
$616$	−760.000	−0.0497099
$617$	−8431.00	−0.550112	−0.275056	−	0.961428i	$-0.588696\pi$
$617$	−8431.00	−0.550112	−0.275056	+	0.961428i	$0.588696\pi$
$618$	6500.00	0.423088
$619$	1217.00	0.0790232	0.0395116	−	0.999219i	$-0.487420\pi$
$619$	1217.00	0.0790232	0.0395116	+	0.999219i	$0.487420\pi$
$620$	13664.0	0.885096
$621$	26390.0	1.70530
$622$	−2360.00	−0.152134
$623$	−24776.0	−1.59331
$624$	−480.000	−0.0307939
$625$	−19459.0	−1.24538
$626$	476.000	0.0303910
$627$	1100.00	0.0700634
$628$	−10644.0	−0.676341
$629$	2664.00	0.168872
$630$	−1064.00	−0.0672869
$631$	−7408.00	−0.467366	−0.233683	−	0.972313i	$-0.575078\pi$
$631$	−7408.00	−0.467366	−0.233683	+	0.972313i	$0.575078\pi$
$632$	−3872.00	−0.243702
$633$	5055.00	0.317406
$634$	−14964.0	−0.937376
$635$	−26586.0	−1.66147
$636$	13180.0	0.821731
$637$	108.000	0.00671761
$638$	100.000	0.00620539
$639$	2250.00	0.139294
$640$	−1792.00	−0.110680
$641$	14559.0	0.897108	0.448554	−	0.893756i	$-0.351939\pi$
$641$	14559.0	0.897108	0.448554	+	0.893756i	$0.351939\pi$
$642$	−19560.0	−1.20245
$643$	−12122.0	−0.743460	−0.371730	−	0.928341i	$-0.621235\pi$
$643$	−12122.0	−0.743460	−0.371730	+	0.928341i	$0.621235\pi$
$644$	−13832.0	−0.846362
$645$	−140.000	−0.00854650
$646$	6336.00	0.385893
$647$	22648.0	1.37617	0.688087	−	0.725628i	$-0.258451\pi$
$647$	22648.0	1.37617	0.688087	+	0.725628i	$0.258451\pi$
$648$	−5368.00	−0.325424
$649$	780.000	0.0471767
$650$	852.000	0.0514126
$651$	−23180.0	−1.39554
$652$	7416.00	0.445449
$653$	−18368.0	−1.10076	−0.550379	−	0.834915i	$-0.685517\pi$
$653$	−18368.0	−1.10076	−0.550379	+	0.834915i	$0.685517\pi$
$654$	−16200.0	−0.968609
$655$	3304.00	0.197096
$656$	−3600.00	−0.214263
$657$	1282.00	0.0761272
$658$	−8398.00	−0.497550
$659$	1327.00	0.0784409	0.0392205	−	0.999231i	$-0.487513\pi$
$659$	1327.00	0.0784409	0.0392205	+	0.999231i	$0.487513\pi$
$660$	1400.00	0.0825681
$661$	−13768.0	−0.810156	−0.405078	−	0.914282i	$-0.632756\pi$
$661$	−13768.0	−0.810156	−0.405078	+	0.914282i	$0.632756\pi$
$662$	−5788.00	−0.339814
$663$	2160.00	0.126527
$664$	9912.00	0.579308
$665$	−11704.0	−0.682499
$666$	148.000	0.00861094
$667$	1820.00	0.105653
$668$	−7232.00	−0.418884
$669$	16215.0	0.937082
$670$	−11648.0	−0.671644
$671$	−3100.00	−0.178352
$672$	3040.00	0.174510
$673$	−12773.0	−0.731594	−0.365797	−	0.930695i	$-0.619204\pi$
$673$	−12773.0	−0.731594	−0.365797	+	0.930695i	$0.619204\pi$
$674$	15262.0	0.872211
$675$	10295.0	0.587044
$676$	−8644.00	−0.491807
$677$	−22575.0	−1.28158	−0.640789	−	0.767717i	$-0.721392\pi$
$677$	−22575.0	−1.28158	−0.640789	+	0.767717i	$0.721392\pi$
$678$	9340.00	0.529057
$679$	10640.0	0.601363
$680$	8064.00	0.454765
$681$	27400.0	1.54181
$682$	−2440.00	−0.136998
$683$	−11770.0	−0.659395	−0.329697	−	0.944087i	$-0.606947\pi$
$683$	−11770.0	−0.659395	−0.329697	+	0.944087i	$0.606947\pi$
$684$	352.000	0.0196770
$685$	−22540.0	−1.25724
$686$	12350.0	0.687355
$687$	−11015.0	−0.611715
$688$	−32.0000	−0.00177324
$689$	−3954.00	−0.218629
$690$	25480.0	1.40581
$691$	21404.0	1.17836	0.589180	−	0.808002i	$-0.299451\pi$
$691$	21404.0	1.17836	0.589180	+	0.808002i	$0.299451\pi$
$692$	−14476.0	−0.795224
$693$	190.000	0.0104149
$694$	−19716.0	−1.07840
$695$	7000.00	0.382051
$696$	−400.000	−0.0217844
$697$	16200.0	0.880371
$698$	−2580.00	−0.139906
$699$	−18590.0	−1.00592
$700$	−5396.00	−0.291357
$701$	416.000	0.0224138	0.0112069	−	0.999937i	$-0.496433\pi$
$701$	416.000	0.0224138	0.0112069	+	0.999937i	$0.496433\pi$
$702$	1740.00	0.0935500
$703$	1628.00	0.0873417
$704$	320.000	0.0171313
$705$	15470.0	0.826431
$706$	−10768.0	−0.574021
$707$	−13965.0	−0.742868
$708$	−3120.00	−0.165617
$709$	14960.0	0.792432	0.396216	−	0.918157i	$-0.370323\pi$
$709$	14960.0	0.792432	0.396216	+	0.918157i	$0.370323\pi$
$710$	31500.0	1.66503
$711$	968.000	0.0510588
$712$	10432.0	0.549095
$713$	−44408.0	−2.33253
$714$	−13680.0	−0.717032
$715$	−420.000	−0.0219680
$716$	136.000	0.00709855
$717$	5190.00	0.270327
$718$	−8394.00	−0.436297
$719$	22501.0	1.16710	0.583551	−	0.812077i	$-0.301663\pi$
$719$	22501.0	1.16710	0.583551	+	0.812077i	$0.301663\pi$
$720$	448.000	0.0231889
$721$	12350.0	0.637917
$722$	−9846.00	−0.507521
$723$	−10250.0	−0.527250
$724$	2948.00	0.151328
$725$	710.000	0.0363707
$726$	13060.0	0.667634
$727$	4616.00	0.235486	0.117743	−	0.993044i	$-0.462434\pi$
$727$	4616.00	0.235486	0.117743	+	0.993044i	$0.462434\pi$
$728$	−912.000	−0.0464299
$729$	20917.0	1.06269
$730$	17948.0	0.909980
$731$	144.000	0.00728595
$732$	12400.0	0.626116
$733$	3251.00	0.163818	0.0819089	−	0.996640i	$-0.473898\pi$
$733$	3251.00	0.163818	0.0819089	+	0.996640i	$0.473898\pi$
$734$	21696.0	1.09103
$735$	1260.00	0.0632324
$736$	5824.00	0.291679
$737$	2080.00	0.103959
$738$	900.000	0.0448909
$739$	−7319.00	−0.364322	−0.182161	−	0.983269i	$-0.558309\pi$
$739$	−7319.00	−0.364322	−0.182161	+	0.983269i	$0.558309\pi$
$740$	2072.00	0.102930
$741$	1320.00	0.0654405
$742$	25042.0	1.23898
$743$	−19177.0	−0.946885	−0.473443	−	0.880825i	$-0.656989\pi$
$743$	−19177.0	−0.946885	−0.473443	+	0.880825i	$0.656989\pi$
$744$	9760.00	0.480939
$745$	−24514.0	−1.20553
$746$	13666.0	0.670708
$747$	−2478.00	−0.121373
$748$	−1440.00	−0.0703899
$749$	−37164.0	−1.81301
$750$	−7560.00	−0.368070
$751$	21203.0	1.03024	0.515119	−	0.857119i	$-0.327748\pi$
$751$	21203.0	1.03024	0.515119	+	0.857119i	$0.327748\pi$
$752$	3536.00	0.171469
$753$	−3510.00	−0.169869
$754$	120.000	0.00579594
$755$	37856.0	1.82480
$756$	−11020.0	−0.530150
$757$	7938.00	0.381125	0.190562	−	0.981675i	$-0.438969\pi$
$757$	7938.00	0.381125	0.190562	+	0.981675i	$0.438969\pi$
$758$	25106.0	1.20302
$759$	−4550.00	−0.217595
$760$	4928.00	0.235207
$761$	−17811.0	−0.848421	−0.424210	−	0.905564i	$-0.639448\pi$
$761$	−17811.0	−0.848421	−0.424210	+	0.905564i	$0.639448\pi$
$762$	−18990.0	−0.902802
$763$	−30780.0	−1.46043
$764$	6816.00	0.322767
$765$	−2016.00	−0.0952793
$766$	8952.00	0.422257
$767$	936.000	0.0440639
$768$	−1280.00	−0.0601407
$769$	40790.0	1.91278	0.956388	−	0.292099i	$-0.0943536\pi$
$769$	40790.0	1.91278	0.956388	+	0.292099i	$0.0943536\pi$
$770$	2660.00	0.124493
$771$	−26700.0	−1.24718
$772$	2760.00	0.128672
$773$	3123.00	0.145312	0.0726562	−	0.997357i	$-0.476852\pi$
$773$	3123.00	0.145312	0.0726562	+	0.997357i	$0.476852\pi$
$774$	8.00000	0.000371517 0
$775$	−17324.0	−0.802963
$776$	−4480.00	−0.207246
$777$	−3515.00	−0.162291
$778$	−22488.0	−1.03629
$779$	9900.00	0.455333
$780$	1680.00	0.0771201
$781$	−5625.00	−0.257719
$782$	−26208.0	−1.19846
$783$	1450.00	0.0661798
$784$	288.000	0.0131195
$785$	37254.0	1.69382
$786$	2360.00	0.107097
$787$	4685.00	0.212201	0.106101	−	0.994355i	$-0.466163\pi$
$787$	4685.00	0.212201	0.106101	+	0.994355i	$0.466163\pi$
$788$	700.000	0.0316453
$789$	23955.0	1.08089
$790$	13552.0	0.610327
$791$	17746.0	0.797693
$792$	−80.0000	−0.00358924
$793$	−3720.00	−0.166584
$794$	28718.0	1.28358
$795$	−46130.0	−2.05794
$796$	−14360.0	−0.639418
$797$	−36848.0	−1.63767	−0.818835	−	0.574029i	$-0.805380\pi$
$797$	−36848.0	−1.63767	−0.818835	+	0.574029i	$0.805380\pi$
$798$	−8360.00	−0.370853
$799$	−15912.0	−0.704538
$800$	2272.00	0.100409
$801$	−2608.00	−0.115043
$802$	−17404.0	−0.766280
$803$	−3205.00	−0.140849
$804$	−8320.00	−0.364955
$805$	48412.0	2.11963
$806$	−2928.00	−0.127958
$807$	30150.0	1.31516
$808$	5880.00	0.256012
$809$	12946.0	0.562617	0.281308	−	0.959617i	$-0.409232\pi$
$809$	12946.0	0.562617	0.281308	+	0.959617i	$0.409232\pi$
$810$	18788.0	0.814991
$811$	31489.0	1.36341	0.681707	−	0.731626i	$-0.261238\pi$
$811$	31489.0	1.36341	0.681707	+	0.731626i	$0.261238\pi$
$812$	−760.000	−0.0328458
$813$	10865.0	0.468699
$814$	−370.000	−0.0159318
$815$	−25956.0	−1.11558
$816$	5760.00	0.247108
$817$	88.0000	0.00376834
$818$	6752.00	0.288604
$819$	228.000	0.00972767
$820$	12600.0	0.536599
$821$	−11835.0	−0.503099	−0.251550	−	0.967844i	$-0.580940\pi$
$821$	−11835.0	−0.503099	−0.251550	+	0.967844i	$0.580940\pi$
$822$	−16100.0	−0.683153
$823$	−19728.0	−0.835571	−0.417785	−	0.908546i	$-0.637194\pi$
$823$	−19728.0	−0.835571	−0.417785	+	0.908546i	$0.637194\pi$
$824$	−5200.00	−0.219843
$825$	−1775.00	−0.0749061
$826$	−5928.00	−0.249711
$827$	22642.0	0.952043	0.476021	−	0.879434i	$-0.342078\pi$
$827$	22642.0	0.952043	0.476021	+	0.879434i	$0.342078\pi$
$828$	−1456.00	−0.0611105
$829$	24280.0	1.01722	0.508612	−	0.860996i	$-0.330159\pi$
$829$	24280.0	1.01722	0.508612	+	0.860996i	$0.330159\pi$
$830$	−34692.0	−1.45082
$831$	29980.0	1.25150
$832$	384.000	0.0160010
$833$	−1296.00	−0.0539060
$834$	5000.00	0.207597
$835$	25312.0	1.04905
$836$	−880.000	−0.0364060
$837$	−35380.0	−1.46107
$838$	−12590.0	−0.518991
$839$	16508.0	0.679284	0.339642	−	0.940555i	$-0.389694\pi$
$839$	16508.0	0.679284	0.339642	+	0.940555i	$0.389694\pi$
$840$	−10640.0	−0.437042
$841$	−24289.0	−0.995900
$842$	−27768.0	−1.13652
$843$	−18180.0	−0.742767
$844$	−4044.00	−0.164929
$845$	30254.0	1.23168
$846$	−884.000	−0.0359250
$847$	24814.0	1.00663
$848$	−10544.0	−0.426984
$849$	13080.0	0.528745
$850$	−10224.0	−0.412565
$851$	−6734.00	−0.271256
$852$	22500.0	0.904739
$853$	−19990.0	−0.802397	−0.401198	−	0.915991i	$-0.631406\pi$
$853$	−19990.0	−0.802397	−0.401198	+	0.915991i	$0.631406\pi$
$854$	23560.0	0.944036
$855$	−1232.00	−0.0492790
$856$	15648.0	0.624810
$857$	−3744.00	−0.149233	−0.0746165	−	0.997212i	$-0.523773\pi$
$857$	−3744.00	−0.149233	−0.0746165	+	0.997212i	$0.523773\pi$
$858$	−300.000	−0.0119369
$859$	−3140.00	−0.124721	−0.0623605	−	0.998054i	$-0.519863\pi$
$859$	−3140.00	−0.124721	−0.0623605	+	0.998054i	$0.519863\pi$
$860$	112.000	0.00444089
$861$	−21375.0	−0.846060
$862$	−16260.0	−0.642480
$863$	−13168.0	−0.519402	−0.259701	−	0.965689i	$-0.583624\pi$
$863$	−13168.0	−0.519402	−0.259701	+	0.965689i	$0.583624\pi$
$864$	4640.00	0.182704
$865$	50666.0	1.99156
$866$	−11966.0	−0.469540
$867$	−1355.00	−0.0530775
$868$	18544.0	0.725143
$869$	−2420.00	−0.0944682
$870$	1400.00	0.0545568
$871$	2496.00	0.0970996
$872$	12960.0	0.503304
$873$	1120.00	0.0434207
$874$	−16016.0	−0.619850
$875$	−14364.0	−0.554962
$876$	12820.0	0.494461
$877$	35002.0	1.34770	0.673850	−	0.738868i	$-0.264639\pi$
$877$	35002.0	1.34770	0.673850	+	0.738868i	$0.264639\pi$
$878$	−15328.0	−0.589174
$879$	−41190.0	−1.58055
$880$	−1120.00	−0.0429036
$881$	−17142.0	−0.655538	−0.327769	−	0.944758i	$-0.606297\pi$
$881$	−17142.0	−0.655538	−0.327769	+	0.944758i	$0.606297\pi$
$882$	−72.0000	−0.00274871
$883$	−22340.0	−0.851417	−0.425708	−	0.904860i	$-0.639975\pi$
$883$	−22340.0	−0.851417	−0.425708	+	0.904860i	$0.639975\pi$
$884$	−1728.00	−0.0657454
$885$	10920.0	0.414770
$886$	−5154.00	−0.195431
$887$	−22605.0	−0.855695	−0.427848	−	0.903851i	$-0.640728\pi$
$887$	−22605.0	−0.855695	−0.427848	+	0.903851i	$0.640728\pi$
$888$	1480.00	0.0559297
$889$	−36081.0	−1.36121
$890$	−36512.0	−1.37515
$891$	−3355.00	−0.126147
$892$	−12972.0	−0.486922
$893$	−9724.00	−0.364391
$894$	−17510.0	−0.655058
$895$	−476.000	−0.0177776
$896$	−2432.00	−0.0906779
$897$	−5460.00	−0.203238
$898$	24160.0	0.897806
$899$	−2440.00	−0.0905212
$900$	−568.000	−0.0210370
$901$	47448.0	1.75441
$902$	−2250.00	−0.0830563
$903$	−190.000	−0.00700200
$904$	−7472.00	−0.274906
$905$	−10318.0	−0.378986
$906$	27040.0	0.991549
$907$	−13880.0	−0.508134	−0.254067	−	0.967187i	$-0.581768\pi$
$907$	−13880.0	−0.508134	−0.254067	+	0.967187i	$0.581768\pi$
$908$	−21920.0	−0.801146
$909$	−1470.00	−0.0536379
$910$	3192.00	0.116279
$911$	52582.0	1.91232	0.956158	−	0.292852i	$-0.0946044\pi$
$911$	52582.0	1.91232	0.956158	+	0.292852i	$0.0946044\pi$
$912$	3520.00	0.127806
$913$	6195.00	0.224561
$914$	−20980.0	−0.759252
$915$	−43400.0	−1.56804
$916$	8812.00	0.317857
$917$	4484.00	0.161477
$918$	−20880.0	−0.750700
$919$	−19298.0	−0.692690	−0.346345	−	0.938107i	$-0.612577\pi$
$919$	−19298.0	−0.692690	−0.346345	+	0.938107i	$0.612577\pi$
$920$	−20384.0	−0.730479
$921$	−43165.0	−1.54434
$922$	−29620.0	−1.05801
$923$	−6750.00	−0.240714
$924$	1900.00	0.0676465
$925$	−2627.00	−0.0933786
$926$	31156.0	1.10567
$927$	1300.00	0.0460600
$928$	320.000	0.0113195
$929$	8658.00	0.305769	0.152885	−	0.988244i	$-0.451144\pi$
$929$	8658.00	0.305769	0.152885	+	0.988244i	$0.451144\pi$
$930$	−34160.0	−1.20446
$931$	−792.000	−0.0278805
$932$	14872.0	0.522692
$933$	5900.00	0.207028
$934$	−4124.00	−0.144477
$935$	5040.00	0.176284
$936$	−96.0000	−0.00335241
$937$	−22427.0	−0.781919	−0.390960	−	0.920408i	$-0.627857\pi$
$937$	−22427.0	−0.781919	−0.390960	+	0.920408i	$0.627857\pi$
$938$	−15808.0	−0.550266
$939$	−1190.00	−0.0413570
$940$	−12376.0	−0.429426
$941$	48158.0	1.66834	0.834169	−	0.551509i	$-0.185948\pi$
$941$	48158.0	1.66834	0.834169	+	0.551509i	$0.185948\pi$
$942$	26610.0	0.920383
$943$	−40950.0	−1.41412
$944$	2496.00	0.0860571
$945$	38570.0	1.32771
$946$	−20.0000	−0.000687374 0
$947$	13696.0	0.469969	0.234984	−	0.971999i	$-0.424496\pi$
$947$	13696.0	0.469969	0.234984	+	0.971999i	$0.424496\pi$
$948$	9680.00	0.331637
$949$	−3846.00	−0.131556
$950$	−6248.00	−0.213381
$951$	37410.0	1.27561
$952$	10944.0	0.372581
$953$	33293.0	1.13165	0.565827	−	0.824524i	$-0.308557\pi$
$953$	33293.0	1.13165	0.565827	+	0.824524i	$0.308557\pi$
$954$	2636.00	0.0894588
$955$	−23856.0	−0.808337
$956$	−4152.00	−0.140466
$957$	−250.000	−0.00844446
$958$	6876.00	0.231893
$959$	−30590.0	−1.03003
$960$	4480.00	0.150616
$961$	29745.0	0.998456
$962$	−444.000	−0.0148806
$963$	−3912.00	−0.130906
$964$	8200.00	0.273967
$965$	−9660.00	−0.322245
$966$	34580.0	1.15175
$967$	1194.00	0.0397068	0.0198534	−	0.999803i	$-0.493680\pi$
$967$	1194.00	0.0397068	0.0198534	+	0.999803i	$0.493680\pi$
$968$	−10448.0	−0.346913
$969$	−15840.0	−0.525133
$970$	15680.0	0.519025
$971$	30912.0	1.02164	0.510820	−	0.859687i	$-0.329342\pi$
$971$	30912.0	1.02164	0.510820	+	0.859687i	$0.329342\pi$
$972$	−2240.00	−0.0739177
$973$	9500.00	0.313007
$974$	−39184.0	−1.28905
$975$	−2130.00	−0.0699637
$976$	−9920.00	−0.325340
$977$	−31404.0	−1.02836	−0.514178	−	0.857684i	$-0.671903\pi$
$977$	−31404.0	−1.02836	−0.514178	+	0.857684i	$0.671903\pi$
$978$	−18540.0	−0.606180
$979$	6520.00	0.212850
$980$	−1008.00	−0.0328565
$981$	−3240.00	−0.105449
$982$	8888.00	0.288826
$983$	40411.0	1.31120	0.655601	−	0.755108i	$-0.272416\pi$
$983$	40411.0	1.31120	0.655601	+	0.755108i	$0.272416\pi$
$984$	9000.00	0.291575
$985$	−2450.00	−0.0792523
$986$	−1440.00	−0.0465101
$987$	20995.0	0.677080
$988$	−1056.00	−0.0340039
$989$	−364.000	−0.0117033
$990$	280.000	0.00898887
$991$	−11002.0	−0.352664	−0.176332	−	0.984331i	$-0.556423\pi$
$991$	−11002.0	−0.352664	−0.176332	+	0.984331i	$0.556423\pi$
$992$	−7808.00	−0.249903
$993$	14470.0	0.462429
$994$	42750.0	1.36413
$995$	50260.0	1.60136
$996$	−24780.0	−0.788338
$997$	6618.00	0.210225	0.105112	−	0.994460i	$-0.466480\pi$
$997$	6618.00	0.210225	0.105112	+	0.994460i	$0.466480\pi$
$998$	−2280.00	−0.0723168
$999$	−5365.00	−0.169911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.a.b.1.1	✓	1
3.2	odd	2		666.4.a.c.1.1		1
4.3	odd	2		592.4.a.a.1.1		1
5.4	even	2		1850.4.a.d.1.1		1
8.3	odd	2		2368.4.a.b.1.1		1
8.5	even	2		2368.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.a.b.1.1	✓	1	1.1	even	1	trivial
592.4.a.a.1.1		1	4.3	odd	2
666.4.a.c.1.1		1	3.2	odd	2
1850.4.a.d.1.1		1	5.4	even	2
2368.4.a.b.1.1		1	8.3	odd	2
2368.4.a.d.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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