LMFDB - Embedded newform 605.8.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [605,8,Mod(1,605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("605.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	605.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,14,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$188.992940418$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	605.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+14.0000 q^{2} -48.0000 q^{3} +68.0000 q^{4} +125.000 q^{5} -672.000 q^{6} +1644.00 q^{7} -840.000 q^{8} +117.000 q^{9} +1750.00 q^{10} -3264.00 q^{12} -3862.00 q^{13} +23016.0 q^{14} -6000.00 q^{15} -20464.0 q^{16} +12254.0 q^{17} +1638.00 q^{18} +25940.0 q^{19} +8500.00 q^{20} -78912.0 q^{21} +12972.0 q^{23} +40320.0 q^{24} +15625.0 q^{25} -54068.0 q^{26} +99360.0 q^{27} +111792. q^{28} +81610.0 q^{29} -84000.0 q^{30} -156888. q^{31} -178976. q^{32} +171556. q^{34} +205500. q^{35} +7956.00 q^{36} +110126. q^{37} +363160. q^{38} +185376. q^{39} -105000. q^{40} -467882. q^{41} -1.10477e6 q^{42} +499208. q^{43} +14625.0 q^{45} +181608. q^{46} -396884. q^{47} +982272. q^{48} +1.87919e6 q^{49} +218750. q^{50} -588192. q^{51} -262616. q^{52} -1.28050e6 q^{53} +1.39104e6 q^{54} -1.38096e6 q^{56} -1.24512e6 q^{57} +1.14254e6 q^{58} -1.33742e6 q^{59} -408000. q^{60} +923978. q^{61} -2.19643e6 q^{62} +192348. q^{63} +113728. q^{64} -482750. q^{65} -797304. q^{67} +833272. q^{68} -622656. q^{69} +2.87700e6 q^{70} +5.10339e6 q^{71} -98280.0 q^{72} +4.26748e6 q^{73} +1.54176e6 q^{74} -750000. q^{75} +1.76392e6 q^{76} +2.59526e6 q^{78} +960.000 q^{79} -2.55800e6 q^{80} -5.02516e6 q^{81} -6.55035e6 q^{82} -6.14083e6 q^{83} -5.36602e6 q^{84} +1.53175e6 q^{85} +6.98891e6 q^{86} -3.91728e6 q^{87} +2.01057e6 q^{89} +204750. q^{90} -6.34913e6 q^{91} +882096. q^{92} +7.53062e6 q^{93} -5.55638e6 q^{94} +3.24250e6 q^{95} +8.59085e6 q^{96} -4.88193e6 q^{97} +2.63087e7 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	14.0000	1.23744	0.618718	−	0.785613i	$-0.287652\pi$
$2$	14.0000	1.23744	0.618718	+	0.785613i	$0.287652\pi$
$3$	−48.0000	−1.02640	−0.513200	−	0.858269i	$-0.671540\pi$
$3$	−48.0000	−1.02640	−0.513200	+	0.858269i	$0.671540\pi$
$4$	68.0000	0.531250
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	−672.000	−1.27011
$7$	1644.00	1.81158	0.905792	−	0.423722i	$-0.139277\pi$
$7$	1644.00	1.81158	0.905792	+	0.423722i	$0.139277\pi$
$8$	−840.000	−0.580049
$9$	117.000	0.0534979
$10$	1750.00	0.553399
$11$	0	0
$11$	0	0
$12$	−3264.00	−0.545275
$13$	−3862.00	−0.487540	−0.243770	−	0.969833i	$-0.578384\pi$
$13$	−3862.00	−0.487540	−0.243770	+	0.969833i	$0.578384\pi$
$14$	23016.0	2.24172
$15$	−6000.00	−0.459020
$16$	−20464.0	−1.24902
$17$	12254.0	0.604932	0.302466	−	0.953160i	$-0.402190\pi$
$17$	12254.0	0.604932	0.302466	+	0.953160i	$0.402190\pi$
$18$	1638.00	0.0662003
$19$	25940.0	0.867626	0.433813	−	0.901003i	$-0.357168\pi$
$19$	25940.0	0.867626	0.433813	+	0.901003i	$0.357168\pi$
$20$	8500.00	0.237582
$21$	−78912.0	−1.85941
$22$	0	0
$23$	12972.0	0.222310	0.111155	−	0.993803i	$-0.464545\pi$
$23$	12972.0	0.222310	0.111155	+	0.993803i	$0.464545\pi$
$24$	40320.0	0.595362
$25$	15625.0	0.200000
$26$	−54068.0	−0.603300
$27$	99360.0	0.971490
$28$	111792.	0.962404
$29$	81610.0	0.621370	0.310685	−	0.950513i	$-0.399442\pi$
$29$	81610.0	0.621370	0.310685	+	0.950513i	$0.399442\pi$
$30$	−84000.0	−0.568009
$31$	−156888.	−0.945853	−0.472927	−	0.881102i	$-0.656802\pi$
$31$	−156888.	−0.945853	−0.472927	+	0.881102i	$0.656802\pi$
$32$	−178976.	−0.965539
$33$	0	0
$34$	171556.	0.748565
$35$	205500.	0.810165
$36$	7956.00	0.0284208
$37$	110126.	0.357424	0.178712	−	0.983901i	$-0.442807\pi$
$37$	110126.	0.357424	0.178712	+	0.983901i	$0.442807\pi$
$38$	363160.	1.07363
$39$	185376.	0.500412
$40$	−105000.	−0.259406
$41$	−467882.	−1.06021	−0.530106	−	0.847931i	$-0.677848\pi$
$41$	−467882.	−1.06021	−0.530106	+	0.847931i	$0.677848\pi$
$42$	−1.10477e6	−2.30090
$43$	499208.	0.957507	0.478753	−	0.877949i	$-0.341089\pi$
$43$	499208.	0.957507	0.478753	+	0.877949i	$0.341089\pi$
$44$	0	0
$45$	14625.0	0.0239250
$46$	181608.	0.275095
$47$	−396884.	−0.557598	−0.278799	−	0.960349i	$-0.589936\pi$
$47$	−396884.	−0.557598	−0.278799	+	0.960349i	$0.589936\pi$
$48$	982272.	1.28200
$49$	1.87919e6	2.28184
$50$	218750.	0.247487
$51$	−588192.	−0.620903
$52$	−262616.	−0.259006
$53$	−1.28050e6	−1.18144	−0.590722	−	0.806875i	$-0.701157\pi$
$53$	−1.28050e6	−1.18144	−0.590722	+	0.806875i	$0.701157\pi$
$54$	1.39104e6	1.20216
$55$	0	0
$56$	−1.38096e6	−1.05081
$57$	−1.24512e6	−0.890531
$58$	1.14254e6	0.768906
$59$	−1.33742e6	−0.847785	−0.423893	−	0.905712i	$-0.639337\pi$
$59$	−1.33742e6	−0.847785	−0.423893	+	0.905712i	$0.639337\pi$
$60$	−408000.	−0.243855
$61$	923978.	0.521203	0.260602	−	0.965446i	$-0.416079\pi$
$61$	923978.	0.521203	0.260602	+	0.965446i	$0.416079\pi$
$62$	−2.19643e6	−1.17043
$63$	192348.	0.0969161
$64$	113728.	0.0542297
$65$	−482750.	−0.218035
$66$	0	0
$67$	−797304.	−0.323864	−0.161932	−	0.986802i	$-0.551772\pi$
$67$	−797304.	−0.323864	−0.161932	+	0.986802i	$0.551772\pi$
$68$	833272.	0.321370
$69$	−622656.	−0.228179
$70$	2.87700e6	1.00253
$71$	5.10339e6	1.69221	0.846106	−	0.533015i	$-0.178941\pi$
$71$	5.10339e6	1.69221	0.846106	+	0.533015i	$0.178941\pi$
$72$	−98280.0	−0.0310314
$73$	4.26748e6	1.28393	0.641965	−	0.766734i	$-0.278119\pi$
$73$	4.26748e6	1.28393	0.641965	+	0.766734i	$0.278119\pi$
$74$	1.54176e6	0.442290
$75$	−750000.	−0.205280
$76$	1.76392e6	0.460926
$77$	0	0
$78$	2.59526e6	0.619228
$79$	960.000	0.000219067 0	0.000109533	−	1.00000i	$-0.499965\pi$
$79$	960.000	0.000219067 0	0.000109533		1.00000i	$0.499965\pi$
$80$	−2.55800e6	−0.558580
$81$	−5.02516e6	−1.05064
$82$	−6.55035e6	−1.31195
$83$	−6.14083e6	−1.17884	−0.589419	−	0.807828i	$-0.700643\pi$
$83$	−6.14083e6	−1.17884	−0.589419	+	0.807828i	$0.700643\pi$
$84$	−5.36602e6	−0.987812
$85$	1.53175e6	0.270534
$86$	6.98891e6	1.18485
$87$	−3.91728e6	−0.637775
$88$	0	0
$89$	2.01057e6	0.302311	0.151156	−	0.988510i	$-0.451701\pi$
$89$	2.01057e6	0.302311	0.151156	+	0.988510i	$0.451701\pi$
$90$	204750.	0.0296057
$91$	−6.34913e6	−0.883221
$92$	882096.	0.118102
$93$	7.53062e6	0.970824
$94$	−5.55638e6	−0.689992
$95$	3.24250e6	0.388014
$96$	8.59085e6	0.991030
$97$	−4.88193e6	−0.543114	−0.271557	−	0.962422i	$-0.587539\pi$
$97$	−4.88193e6	−0.543114	−0.271557	+	0.962422i	$0.587539\pi$
$98$	2.63087e7	2.82363
$99$	0	0
$100$	1.06250e6	0.106250
$101$	−9.72670e6	−0.939379	−0.469689	−	0.882832i	$-0.655634\pi$
$101$	−9.72670e6	−0.939379	−0.469689	+	0.882832i	$0.655634\pi$
$102$	−8.23469e6	−0.768328
$103$	1.63151e7	1.47115	0.735577	−	0.677441i	$-0.236911\pi$
$103$	1.63151e7	1.47115	0.735577	+	0.677441i	$0.236911\pi$
$104$	3.24408e6	0.282797
$105$	−9.86400e6	−0.831554
$106$	−1.79270e7	−1.46196
$107$	4.08974e6	0.322740	0.161370	−	0.986894i	$-0.448409\pi$
$107$	4.08974e6	0.322740	0.161370	+	0.986894i	$0.448409\pi$
$108$	6.75648e6	0.516104
$109$	2.68318e7	1.98453	0.992263	−	0.124158i	$-0.0396228\pi$
$109$	2.68318e7	1.98453	0.992263	+	0.124158i	$0.0396228\pi$
$110$	0	0
$111$	−5.28605e6	−0.366860
$112$	−3.36428e7	−2.26271
$113$	−1.74810e7	−1.13971	−0.569853	−	0.821747i	$-0.693000\pi$
$113$	−1.74810e7	−1.13971	−0.569853	+	0.821747i	$0.693000\pi$
$114$	−1.74317e7	−1.10198
$115$	1.62150e6	0.0994202
$116$	5.54948e6	0.330103
$117$	−451854.	−0.0260824
$118$	−1.87239e7	−1.04908
$119$	2.01456e7	1.09589
$120$	5.04000e6	0.266254
$121$	0	0
$122$	1.29357e7	0.644956
$123$	2.24583e7	1.08820
$124$	−1.06684e7	−0.502485
$125$	1.95312e6	0.0894427
$126$	2.69287e6	0.119928
$127$	1.25018e7	0.541575	0.270787	−	0.962639i	$-0.412716\pi$
$127$	1.25018e7	0.541575	0.270787	+	0.962639i	$0.412716\pi$
$128$	2.45011e7	1.03264
$129$	−2.39620e7	−0.982786
$130$	−6.75850e6	−0.269804
$131$	7.75619e6	0.301439	0.150719	−	0.988577i	$-0.451841\pi$
$131$	7.75619e6	0.301439	0.150719	+	0.988577i	$0.451841\pi$
$132$	0	0
$133$	4.26454e7	1.57178
$134$	−1.11623e7	−0.400761
$135$	1.24200e7	0.434464
$136$	−1.02934e7	−0.350890
$137$	3.61720e7	1.20185	0.600926	−	0.799305i	$-0.294799\pi$
$137$	3.61720e7	1.20185	0.600926	+	0.799305i	$0.294799\pi$
$138$	−8.71718e6	−0.282358
$139$	−1.09092e7	−0.344542	−0.172271	−	0.985050i	$-0.555110\pi$
$139$	−1.09092e7	−0.344542	−0.172271	+	0.985050i	$0.555110\pi$
$140$	1.39740e7	0.430400
$141$	1.90504e7	0.572319
$142$	7.14475e7	2.09401
$143$	0	0
$144$	−2.39429e6	−0.0668202
$145$	1.02012e7	0.277885
$146$	5.97447e7	1.58878
$147$	−9.02013e7	−2.34208
$148$	7.48857e6	0.189882
$149$	3.64580e7	0.902904	0.451452	−	0.892295i	$-0.350906\pi$
$149$	3.64580e7	0.902904	0.451452	+	0.892295i	$0.350906\pi$
$150$	−1.05000e7	−0.254021
$151$	−7.18955e6	−0.169935	−0.0849674	−	0.996384i	$-0.527079\pi$
$151$	−7.18955e6	−0.169935	−0.0849674	+	0.996384i	$0.527079\pi$
$152$	−2.17896e7	−0.503265
$153$	1.43372e6	0.0323626
$154$	0	0
$155$	−1.96110e7	−0.422998
$156$	1.26056e7	0.265844
$157$	8.79932e7	1.81468	0.907341	−	0.420396i	$-0.138109\pi$
$157$	8.79932e7	1.81468	0.907341	+	0.420396i	$0.138109\pi$
$158$	13440.0	0.000271081 0
$159$	6.14639e7	1.21264
$160$	−2.23720e7	−0.431802
$161$	2.13260e7	0.402734
$162$	−7.03522e7	−1.30010
$163$	−5.48875e7	−0.992697	−0.496349	−	0.868123i	$-0.665326\pi$
$163$	−5.48875e7	−0.992697	−0.496349	+	0.868123i	$0.665326\pi$
$164$	−3.18160e7	−0.563238
$165$	0	0
$166$	−8.59716e7	−1.45874
$167$	−8.61460e6	−0.143129	−0.0715644	−	0.997436i	$-0.522799\pi$
$167$	−8.61460e6	−0.143129	−0.0715644	+	0.997436i	$0.522799\pi$
$168$	6.62861e7	1.07855
$169$	−4.78335e7	−0.762304
$170$	2.14445e7	0.334769
$171$	3.03498e6	0.0464162
$172$	3.39461e7	0.508676
$173$	5.12524e7	0.752580	0.376290	−	0.926502i	$-0.377200\pi$
$173$	5.12524e7	0.752580	0.376290	+	0.926502i	$0.377200\pi$
$174$	−5.48419e7	−0.789206
$175$	2.56875e7	0.362317
$176$	0	0
$177$	6.41962e7	0.870167
$178$	2.81480e7	0.374091
$179$	−5.01627e7	−0.653725	−0.326862	−	0.945072i	$-0.605991\pi$
$179$	−5.01627e7	−0.653725	−0.326862	+	0.945072i	$0.605991\pi$
$180$	994500.	0.0127102
$181$	6.90817e7	0.865940	0.432970	−	0.901408i	$-0.357466\pi$
$181$	6.90817e7	0.865940	0.432970	+	0.901408i	$0.357466\pi$
$182$	−8.88878e7	−1.09293
$183$	−4.43509e7	−0.534963
$184$	−1.08965e7	−0.128951
$185$	1.37658e7	0.159845
$186$	1.05429e8	1.20133
$187$	0	0
$188$	−2.69881e7	−0.296224
$189$	1.63348e8	1.75994
$190$	4.53950e7	0.480143
$191$	1.54745e8	1.60695	0.803473	−	0.595342i	$-0.202983\pi$
$191$	1.54745e8	1.60695	0.803473	+	0.595342i	$0.202983\pi$
$192$	−5.45894e6	−0.0556614
$193$	1.59406e7	0.159607	0.0798037	−	0.996811i	$-0.474571\pi$
$193$	1.59406e7	0.159607	0.0798037	+	0.996811i	$0.474571\pi$
$194$	−6.83471e7	−0.672069
$195$	2.31720e7	0.223791
$196$	1.27785e8	1.21223
$197$	1.68188e8	1.56734	0.783670	−	0.621177i	$-0.213345\pi$
$197$	1.68188e8	1.56734	0.783670	+	0.621177i	$0.213345\pi$
$198$	0	0
$199$	1.77773e8	1.59911	0.799556	−	0.600591i	$-0.205068\pi$
$199$	1.77773e8	1.59911	0.799556	+	0.600591i	$0.205068\pi$
$200$	−1.31250e7	−0.116010
$201$	3.82706e7	0.332414
$202$	−1.36174e8	−1.16242
$203$	1.34167e8	1.12566
$204$	−3.99971e7	−0.329855
$205$	−5.84852e7	−0.474141
$206$	2.28411e8	1.82046
$207$	1.51772e6	0.0118931
$208$	7.90320e7	0.608949
$209$	0	0
$210$	−1.38096e8	−1.02900
$211$	1.61996e8	1.18718	0.593590	−	0.804767i	$-0.297710\pi$
$211$	1.61996e8	1.18718	0.593590	+	0.804767i	$0.297710\pi$
$212$	−8.70739e7	−0.627642
$213$	−2.44963e8	−1.73689
$214$	5.72564e7	0.399370
$215$	6.24010e7	0.428210
$216$	−8.34624e7	−0.563511
$217$	−2.57924e8	−1.71349
$218$	3.75645e8	2.45572
$219$	−2.04839e8	−1.31783
$220$	0	0
$221$	−4.73249e7	−0.294929
$222$	−7.40047e7	−0.453966
$223$	1.75932e7	0.106237	0.0531187	−	0.998588i	$-0.483084\pi$
$223$	1.75932e7	0.106237	0.0531187	+	0.998588i	$0.483084\pi$
$224$	−2.94237e8	−1.74916
$225$	1.82812e6	0.0106996
$226$	−2.44735e8	−1.41031
$227$	2.03036e8	1.15208	0.576039	−	0.817422i	$-0.304597\pi$
$227$	2.03036e8	1.15208	0.576039	+	0.817422i	$0.304597\pi$
$228$	−8.46682e7	−0.473095
$229$	1.59559e8	0.878005	0.439003	−	0.898486i	$-0.355332\pi$
$229$	1.59559e8	0.878005	0.439003	+	0.898486i	$0.355332\pi$
$230$	2.27010e7	0.123026
$231$	0	0
$232$	−6.85524e7	−0.360425
$233$	1.94985e7	0.100985	0.0504924	−	0.998724i	$-0.483921\pi$
$233$	1.94985e7	0.100985	0.0504924	+	0.998724i	$0.483921\pi$
$234$	−6.32596e6	−0.0322753
$235$	−4.96105e7	−0.249365
$236$	−9.09446e7	−0.450386
$237$	−46080.0	−0.000224850 0
$238$	2.82038e8	1.35609
$239$	1.60220e8	0.759146	0.379573	−	0.925162i	$-0.376071\pi$
$239$	1.60220e8	0.759146	0.379573	+	0.925162i	$0.376071\pi$
$240$	1.22784e8	0.573327
$241$	3.78779e8	1.74311	0.871557	−	0.490294i	$-0.163110\pi$
$241$	3.78779e8	1.74311	0.871557	+	0.490294i	$0.163110\pi$
$242$	0	0
$243$	2.39073e7	0.106883
$244$	6.28305e7	0.276889
$245$	2.34899e8	1.02047
$246$	3.14417e8	1.34658
$247$	−1.00180e8	−0.423002
$248$	1.31786e8	0.548641
$249$	2.94760e8	1.20996
$250$	2.73438e7	0.110680
$251$	1.61304e8	0.643855	0.321927	−	0.946764i	$-0.395669\pi$
$251$	1.61304e8	0.643855	0.321927	+	0.946764i	$0.395669\pi$
$252$	1.30797e7	0.0514867
$253$	0	0
$254$	1.75025e8	0.670164
$255$	−7.35240e7	−0.277676
$256$	3.28458e8	1.22360
$257$	2.27387e8	0.835603	0.417801	−	0.908538i	$-0.362801\pi$
$257$	2.27387e8	0.835603	0.417801	+	0.908538i	$0.362801\pi$
$258$	−3.35468e8	−1.21614
$259$	1.81047e8	0.647504
$260$	−3.28270e7	−0.115831
$261$	9.54837e6	0.0332420
$262$	1.08587e8	0.373011
$263$	−4.57728e8	−1.55154	−0.775768	−	0.631018i	$-0.782637\pi$
$263$	−4.57728e8	−1.55154	−0.775768	+	0.631018i	$0.782637\pi$
$264$	0	0
$265$	−1.60062e8	−0.528358
$266$	5.97035e8	1.94498
$267$	−9.65074e7	−0.310292
$268$	−5.42167e7	−0.172053
$269$	4.67286e8	1.46369	0.731847	−	0.681469i	$-0.238659\pi$
$269$	4.67286e8	1.46369	0.731847	+	0.681469i	$0.238659\pi$
$270$	1.73880e8	0.537621
$271$	4.45932e7	0.136106	0.0680528	−	0.997682i	$-0.478321\pi$
$271$	4.45932e7	0.136106	0.0680528	+	0.997682i	$0.478321\pi$
$272$	−2.50766e8	−0.755574
$273$	3.04758e8	0.906538
$274$	5.06408e8	1.48722
$275$	0	0
$276$	−4.23406e7	−0.121220
$277$	−3.16657e8	−0.895179	−0.447590	−	0.894239i	$-0.647717\pi$
$277$	−3.16657e8	−0.895179	−0.447590	+	0.894239i	$0.647717\pi$
$278$	−1.52729e8	−0.426349
$279$	−1.83559e7	−0.0506012
$280$	−1.72620e8	−0.469935
$281$	2.25818e8	0.607136	0.303568	−	0.952810i	$-0.401822\pi$
$281$	2.25818e8	0.607136	0.303568	+	0.952810i	$0.401822\pi$
$282$	2.66706e8	0.708208
$283$	−2.08210e7	−0.0546072	−0.0273036	−	0.999627i	$-0.508692\pi$
$283$	−2.08210e7	−0.0546072	−0.0273036	+	0.999627i	$0.508692\pi$
$284$	3.47031e8	0.898987
$285$	−1.55640e8	−0.398258
$286$	0	0
$287$	−7.69198e8	−1.92066
$288$	−2.09402e7	−0.0516544
$289$	−2.60178e8	−0.634057
$290$	1.42818e8	0.343865
$291$	2.34333e8	0.557452
$292$	2.90189e8	0.682088
$293$	1.78825e8	0.415329	0.207665	−	0.978200i	$-0.433414\pi$
$293$	1.78825e8	0.415329	0.207665	+	0.978200i	$0.433414\pi$
$294$	−1.26282e9	−2.89818
$295$	−1.67178e8	−0.379141
$296$	−9.25058e7	−0.207323
$297$	0	0
$298$	5.10413e8	1.11729
$299$	−5.00979e7	−0.108385
$300$	−5.10000e7	−0.109055
$301$	8.20698e8	1.73461
$302$	−1.00654e8	−0.210284
$303$	4.66882e8	0.964179
$304$	−5.30836e8	−1.08368
$305$	1.15497e8	0.233089
$306$	2.00721e7	0.0400467
$307$	8.55159e7	0.168680	0.0843398	−	0.996437i	$-0.473122\pi$
$307$	8.55159e7	0.168680	0.0843398	+	0.996437i	$0.473122\pi$
$308$	0	0
$309$	−7.83122e8	−1.50999
$310$	−2.74554e8	−0.523434
$311$	−4.84706e8	−0.913728	−0.456864	−	0.889537i	$-0.651027\pi$
$311$	−4.84706e8	−0.913728	−0.456864	+	0.889537i	$0.651027\pi$
$312$	−1.55716e8	−0.290263
$313$	5.70821e8	1.05219	0.526096	−	0.850425i	$-0.323655\pi$
$313$	5.70821e8	1.05219	0.526096	+	0.850425i	$0.323655\pi$
$314$	1.23190e9	2.24555
$315$	2.40435e7	0.0433422
$316$	65280.0	0.000116379 0
$317$	−5.50191e8	−0.970076	−0.485038	−	0.874493i	$-0.661194\pi$
$317$	−5.50191e8	−0.970076	−0.485038	+	0.874493i	$0.661194\pi$
$318$	8.60495e8	1.50056
$319$	0	0
$320$	1.42160e7	0.0242523
$321$	−1.96308e8	−0.331261
$322$	2.98564e8	0.498358
$323$	3.17869e8	0.524855
$324$	−3.41711e8	−0.558150
$325$	−6.03438e7	−0.0975081
$326$	−7.68425e8	−1.22840
$327$	−1.28792e9	−2.03692
$328$	3.93021e8	0.614975
$329$	−6.52477e8	−1.01014
$330$	0	0
$331$	−9.39839e8	−1.42448	−0.712238	−	0.701938i	$-0.752319\pi$
$331$	−9.39839e8	−1.42448	−0.712238	+	0.701938i	$0.752319\pi$
$332$	−4.17577e8	−0.626257
$333$	1.28847e7	0.0191215
$334$	−1.20604e8	−0.177113
$335$	−9.96630e7	−0.144836
$336$	1.61486e9	2.32245
$337$	−5.33632e8	−0.759516	−0.379758	−	0.925086i	$-0.623993\pi$
$337$	−5.33632e8	−0.759516	−0.379758	+	0.925086i	$0.623993\pi$
$338$	−6.69669e8	−0.943304
$339$	8.39090e8	1.16979
$340$	1.04159e8	0.143721
$341$	0	0
$342$	4.24897e7	0.0574371
$343$	1.73549e9	2.32216
$344$	−4.19335e8	−0.555401
$345$	−7.78320e7	−0.102045
$346$	7.17533e8	0.931270
$347$	−1.07934e9	−1.38677	−0.693385	−	0.720567i	$-0.743882\pi$
$347$	−1.07934e9	−1.38677	−0.693385	+	0.720567i	$0.743882\pi$
$348$	−2.66375e8	−0.338818
$349$	4.27217e8	0.537972	0.268986	−	0.963144i	$-0.413311\pi$
$349$	4.27217e8	0.537972	0.268986	+	0.963144i	$0.413311\pi$
$350$	3.59625e8	0.448344
$351$	−3.83728e8	−0.473641
$352$	0	0
$353$	−1.48966e9	−1.80250	−0.901250	−	0.433299i	$-0.857350\pi$
$353$	−1.48966e9	−1.80250	−0.901250	+	0.433299i	$0.857350\pi$
$354$	8.98746e8	1.07678
$355$	6.37924e8	0.756780
$356$	1.36719e8	0.160603
$357$	−9.66988e8	−1.12482
$358$	−7.02277e8	−0.808943
$359$	−8.41275e8	−0.959638	−0.479819	−	0.877367i	$-0.659298\pi$
$359$	−8.41275e8	−0.959638	−0.479819	+	0.877367i	$0.659298\pi$
$360$	−1.22850e7	−0.0138777
$361$	−2.20988e8	−0.247226
$362$	9.67143e8	1.07155
$363$	0	0
$364$	−4.31741e8	−0.469211
$365$	5.33435e8	0.574191
$366$	−6.20913e8	−0.661983
$367$	−7.50462e8	−0.792496	−0.396248	−	0.918143i	$-0.629688\pi$
$367$	−7.50462e8	−0.792496	−0.396248	+	0.918143i	$0.629688\pi$
$368$	−2.65459e8	−0.277671
$369$	−5.47422e7	−0.0567192
$370$	1.92720e8	0.197798
$371$	−2.10514e9	−2.14029
$372$	5.12082e8	0.515750
$373$	−1.71074e8	−0.170688	−0.0853439	−	0.996352i	$-0.527199\pi$
$373$	−1.71074e8	−0.170688	−0.0853439	+	0.996352i	$0.527199\pi$
$374$	0	0
$375$	−9.37500e7	−0.0918040
$376$	3.33383e8	0.323434
$377$	−3.15178e8	−0.302943
$378$	2.28687e9	2.17781
$379$	4.66239e7	0.0439918	0.0219959	−	0.999758i	$-0.492998\pi$
$379$	4.66239e7	0.0439918	0.0219959	+	0.999758i	$0.492998\pi$
$380$	2.20490e8	0.206132
$381$	−6.00085e8	−0.555872
$382$	2.16644e9	1.98849
$383$	−4.42266e8	−0.402242	−0.201121	−	0.979566i	$-0.564458\pi$
$383$	−4.42266e8	−0.402242	−0.201121	+	0.979566i	$0.564458\pi$
$384$	−1.17605e9	−1.05991
$385$	0	0
$386$	2.23168e8	0.197504
$387$	5.84073e7	0.0512247
$388$	−3.31972e8	−0.288529
$389$	−4.64033e8	−0.399691	−0.199846	−	0.979827i	$-0.564044\pi$
$389$	−4.64033e8	−0.399691	−0.199846	+	0.979827i	$0.564044\pi$
$390$	3.24408e8	0.276927
$391$	1.58959e8	0.134483
$392$	−1.57852e9	−1.32358
$393$	−3.72297e8	−0.309397
$394$	2.35463e9	1.93948
$395$	120000.	9.79696e−5 0
$396$	0	0
$397$	−3.17792e8	−0.254904	−0.127452	−	0.991845i	$-0.540680\pi$
$397$	−3.17792e8	−0.254904	−0.127452	+	0.991845i	$0.540680\pi$
$398$	2.48882e9	1.97880
$399$	−2.04698e9	−1.61327
$400$	−3.19750e8	−0.249805
$401$	−1.19563e9	−0.925958	−0.462979	−	0.886369i	$-0.653219\pi$
$401$	−1.19563e9	−0.925958	−0.462979	+	0.886369i	$0.653219\pi$
$402$	5.35788e8	0.411341
$403$	6.05901e8	0.461142
$404$	−6.61416e8	−0.499045
$405$	−6.28145e8	−0.469859
$406$	1.87834e9	1.39294
$407$	0	0
$408$	4.94081e8	0.360154
$409$	−2.21305e9	−1.59941	−0.799704	−	0.600395i	$-0.795010\pi$
$409$	−2.21305e9	−1.59941	−0.799704	+	0.600395i	$0.795010\pi$
$410$	−8.18794e8	−0.586720
$411$	−1.73626e9	−1.23358
$412$	1.10942e9	0.781551
$413$	−2.19872e9	−1.53583
$414$	2.12481e7	0.0147170
$415$	−7.67604e8	−0.527192
$416$	6.91205e8	0.470739
$417$	5.23643e8	0.353638
$418$	0	0
$419$	−8.02299e8	−0.532828	−0.266414	−	0.963859i	$-0.585839\pi$
$419$	−8.02299e8	−0.532828	−0.266414	+	0.963859i	$0.585839\pi$
$420$	−6.70752e8	−0.441763
$421$	3.44713e7	0.0225149	0.0112575	−	0.999937i	$-0.496417\pi$
$421$	3.44713e7	0.0225149	0.0112575	+	0.999937i	$0.496417\pi$
$422$	2.26795e9	1.46906
$423$	−4.64354e7	−0.0298303
$424$	1.07562e9	0.685295
$425$	1.91469e8	0.120986
$426$	−3.42948e9	−2.14929
$427$	1.51902e9	0.944204
$428$	2.78103e8	0.171456
$429$	0	0
$430$	8.73614e8	0.529883
$431$	−1.72692e9	−1.03897	−0.519485	−	0.854480i	$-0.673876\pi$
$431$	−1.72692e9	−1.03897	−0.519485	+	0.854480i	$0.673876\pi$
$432$	−2.03330e9	−1.21341
$433$	4.88308e8	0.289059	0.144529	−	0.989501i	$-0.453833\pi$
$433$	4.88308e8	0.289059	0.144529	+	0.989501i	$0.453833\pi$
$434$	−3.61093e9	−2.12034
$435$	−4.89660e8	−0.285221
$436$	1.82456e9	1.05428
$437$	3.36494e8	0.192882
$438$	−2.86775e9	−1.63073
$439$	2.88640e9	1.62828	0.814142	−	0.580665i	$-0.197207\pi$
$439$	2.88640e9	1.62828	0.814142	+	0.580665i	$0.197207\pi$
$440$	0	0
$441$	2.19866e8	0.122074
$442$	−6.62549e8	−0.364956
$443$	9.26583e8	0.506374	0.253187	−	0.967417i	$-0.418521\pi$
$443$	9.26583e8	0.506374	0.253187	+	0.967417i	$0.418521\pi$
$444$	−3.59451e8	−0.194895
$445$	2.51321e8	0.135198
$446$	2.46304e8	0.131462
$447$	−1.74999e9	−0.926741
$448$	1.86969e8	0.0982418
$449$	1.35535e9	0.706627	0.353313	−	0.935505i	$-0.385055\pi$
$449$	1.35535e9	0.706627	0.353313	+	0.935505i	$0.385055\pi$
$450$	2.55938e7	0.0132401
$451$	0	0
$452$	−1.18871e9	−0.605469
$453$	3.45098e8	0.174421
$454$	2.84250e9	1.42563
$455$	−7.93641e8	−0.394988
$456$	1.04590e9	0.516551
$457$	−4.63429e7	−0.0227131	−0.0113566	−	0.999936i	$-0.503615\pi$
$457$	−4.63429e7	−0.0227131	−0.0113566	+	0.999936i	$0.503615\pi$
$458$	2.23383e9	1.08648
$459$	1.21756e9	0.587686
$460$	1.10262e8	0.0528170
$461$	1.52117e8	0.0723144	0.0361572	−	0.999346i	$-0.488488\pi$
$461$	1.52117e8	0.0723144	0.0361572	+	0.999346i	$0.488488\pi$
$462$	0	0
$463$	1.63450e9	0.765337	0.382668	−	0.923886i	$-0.375005\pi$
$463$	1.63450e9	0.765337	0.382668	+	0.923886i	$0.375005\pi$
$464$	−1.67007e9	−0.776106
$465$	9.41328e8	0.434166
$466$	2.72979e8	0.124962
$467$	−1.11380e9	−0.506057	−0.253029	−	0.967459i	$-0.581427\pi$
$467$	−1.11380e9	−0.506057	−0.253029	+	0.967459i	$0.581427\pi$
$468$	−3.07261e7	−0.0138563
$469$	−1.31077e9	−0.586706
$470$	−6.94547e8	−0.308574
$471$	−4.22367e9	−1.86259
$472$	1.12343e9	0.491756
$473$	0	0
$474$	−645120.	−0.000278238 0
$475$	4.05312e8	0.173525
$476$	1.36990e9	0.582189
$477$	−1.49818e8	−0.0632049
$478$	2.24309e9	0.939395
$479$	−1.27745e9	−0.531091	−0.265546	−	0.964098i	$-0.585552\pi$
$479$	−1.27745e9	−0.531091	−0.265546	+	0.964098i	$0.585552\pi$
$480$	1.07386e9	0.443202
$481$	−4.25307e8	−0.174259
$482$	5.30290e9	2.15699
$483$	−1.02365e9	−0.413366
$484$	0	0
$485$	−6.10242e8	−0.242888
$486$	3.34702e8	0.132261
$487$	9.79673e8	0.384352	0.192176	−	0.981360i	$-0.438445\pi$
$487$	9.79673e8	0.384352	0.192176	+	0.981360i	$0.438445\pi$
$488$	−7.76142e8	−0.302323
$489$	2.63460e9	1.01890
$490$	3.28859e9	1.26277
$491$	4.92125e9	1.87625	0.938124	−	0.346298i	$-0.112562\pi$
$491$	4.92125e9	1.87625	0.938124	+	0.346298i	$0.112562\pi$
$492$	1.52717e9	0.578108
$493$	1.00005e9	0.375887
$494$	−1.40252e9	−0.523439
$495$	0	0
$496$	3.21056e9	1.18139
$497$	8.38998e9	3.06559
$498$	4.12664e9	1.49725
$499$	−3.65786e9	−1.31788	−0.658940	−	0.752196i	$-0.728995\pi$
$499$	−3.65786e9	−1.31788	−0.658940	+	0.752196i	$0.728995\pi$
$500$	1.32812e8	0.0475164
$501$	4.13501e8	0.146908
$502$	2.25826e9	0.796729
$503$	3.88358e9	1.36064	0.680322	−	0.732914i	$-0.261840\pi$
$503$	3.88358e9	1.36064	0.680322	+	0.732914i	$0.261840\pi$
$504$	−1.61572e8	−0.0562160
$505$	−1.21584e9	−0.420103
$506$	0	0
$507$	2.29601e9	0.782430
$508$	8.50120e8	0.287711
$509$	3.90072e9	1.31109	0.655545	−	0.755156i	$-0.272439\pi$
$509$	3.90072e9	1.31109	0.655545	+	0.755156i	$0.272439\pi$
$510$	−1.02934e9	−0.343607
$511$	7.01573e9	2.32595
$512$	1.46228e9	0.481487
$513$	2.57740e9	0.842890
$514$	3.18342e9	1.03401
$515$	2.03938e9	0.657920
$516$	−1.62941e9	−0.522105
$517$	0	0
$518$	2.53466e9	0.801245
$519$	−2.46011e9	−0.772448
$520$	4.05510e8	0.126471
$521$	2.88399e9	0.893431	0.446716	−	0.894676i	$-0.352594\pi$
$521$	2.88399e9	0.893431	0.446716	+	0.894676i	$0.352594\pi$
$522$	1.33677e8	0.0411349
$523$	−8.77188e8	−0.268125	−0.134062	−	0.990973i	$-0.542802\pi$
$523$	−8.77188e8	−0.268125	−0.134062	+	0.990973i	$0.542802\pi$
$524$	5.27421e8	0.160139
$525$	−1.23300e9	−0.371882
$526$	−6.40819e9	−1.91993
$527$	−1.92251e9	−0.572177
$528$	0	0
$529$	−3.23655e9	−0.950578
$530$	−2.24087e9	−0.653810
$531$	−1.56478e8	−0.0453548
$532$	2.89988e9	0.835007
$533$	1.80696e9	0.516896
$534$	−1.35110e9	−0.383967
$535$	5.11218e8	0.144334
$536$	6.69735e8	0.187857
$537$	2.40781e9	0.670983
$538$	6.54201e9	1.81123
$539$	0	0
$540$	8.44560e8	0.230809
$541$	6.53485e8	0.177437	0.0887187	−	0.996057i	$-0.471723\pi$
$541$	6.53485e8	0.177437	0.0887187	+	0.996057i	$0.471723\pi$
$542$	6.24305e8	0.168422
$543$	−3.31592e9	−0.888801
$544$	−2.19317e9	−0.584086
$545$	3.35397e9	0.887507
$546$	4.26661e9	1.12178
$547$	4.59299e9	1.19988	0.599942	−	0.800043i	$-0.295190\pi$
$547$	4.59299e9	1.19988	0.599942	+	0.800043i	$0.295190\pi$
$548$	2.45970e9	0.638484
$549$	1.08105e8	0.0278833
$550$	0	0
$551$	2.11696e9	0.539117
$552$	5.23031e8	0.132355
$553$	1.57824e6	0.000396858 0
$554$	−4.43320e9	−1.10773
$555$	−6.60756e8	−0.164065
$556$	−7.41827e8	−0.183038
$557$	−6.83164e9	−1.67507	−0.837533	−	0.546387i	$-0.816003\pi$
$557$	−6.83164e9	−1.67507	−0.837533	+	0.546387i	$0.816003\pi$
$558$	−2.56983e8	−0.0626158
$559$	−1.92794e9	−0.466823
$560$	−4.20535e9	−1.01192
$561$	0	0
$562$	3.16145e9	0.751292
$563$	−3.42509e9	−0.808897	−0.404449	−	0.914561i	$-0.632537\pi$
$563$	−3.42509e9	−0.808897	−0.404449	+	0.914561i	$0.632537\pi$
$564$	1.29543e9	0.304044
$565$	−2.18513e9	−0.509692
$566$	−2.91494e8	−0.0675729
$567$	−8.26136e9	−1.90332
$568$	−4.28685e9	−0.981565
$569$	7.50930e9	1.70886	0.854430	−	0.519566i	$-0.173906\pi$
$569$	7.50930e9	1.70886	0.854430	+	0.519566i	$0.173906\pi$
$570$	−2.17896e9	−0.492819
$571$	1.35841e8	0.0305355	0.0152677	−	0.999883i	$-0.495140\pi$
$571$	1.35841e8	0.0305355	0.0152677	+	0.999883i	$0.495140\pi$
$572$	0	0
$573$	−7.42778e9	−1.64937
$574$	−1.07688e10	−2.37670
$575$	2.02688e8	0.0444621
$576$	1.33062e7	0.00290118
$577$	1.63775e9	0.354922	0.177461	−	0.984128i	$-0.443212\pi$
$577$	1.63775e9	0.354922	0.177461	+	0.984128i	$0.443212\pi$
$578$	−3.64249e9	−0.784606
$579$	−7.65147e8	−0.163821
$580$	6.93685e8	0.147627
$581$	−1.00955e10	−2.13556
$582$	3.28066e9	0.689812
$583$	0	0
$584$	−3.58468e9	−0.744742
$585$	−5.64818e7	−0.0116644
$586$	2.50356e9	0.513944
$587$	−5.97205e9	−1.21868	−0.609341	−	0.792909i	$-0.708566\pi$
$587$	−5.97205e9	−1.21868	−0.609341	+	0.792909i	$0.708566\pi$
$588$	−6.13369e9	−1.24423
$589$	−4.06967e9	−0.820647
$590$	−2.34048e9	−0.469163
$591$	−8.07303e9	−1.60872
$592$	−2.25362e9	−0.446431
$593$	−8.31347e9	−1.63716	−0.818579	−	0.574394i	$-0.805238\pi$
$593$	−8.31347e9	−1.63716	−0.818579	+	0.574394i	$0.805238\pi$
$594$	0	0
$595$	2.51820e9	0.490095
$596$	2.47915e9	0.479668
$597$	−8.53308e9	−1.64133
$598$	−7.01370e8	−0.134120
$599$	9.78368e9	1.85998	0.929990	−	0.367585i	$-0.119815\pi$
$599$	9.78368e9	1.85998	0.929990	+	0.367585i	$0.119815\pi$
$600$	6.30000e8	0.119072
$601$	−5.40159e9	−1.01499	−0.507494	−	0.861655i	$-0.669428\pi$
$601$	−5.40159e9	−1.01499	−0.507494	+	0.861655i	$0.669428\pi$
$602$	1.14898e10	2.14646
$603$	−9.32846e7	−0.0173260
$604$	−4.88890e8	−0.0902779
$605$	0	0
$606$	6.53634e9	1.19311
$607$	2.84439e9	0.516214	0.258107	−	0.966116i	$-0.416901\pi$
$607$	2.84439e9	0.516214	0.258107	+	0.966116i	$0.416901\pi$
$608$	−4.64264e9	−0.837726
$609$	−6.44001e9	−1.15538
$610$	1.61696e9	0.288433
$611$	1.53277e9	0.271851
$612$	9.74928e7	0.0171926
$613$	7.02106e9	1.23109	0.615547	−	0.788101i	$-0.288935\pi$
$613$	7.02106e9	1.23109	0.615547	+	0.788101i	$0.288935\pi$
$614$	1.19722e9	0.208730
$615$	2.80729e9	0.486659
$616$	0	0
$617$	3.35166e9	0.574462	0.287231	−	0.957861i	$-0.407265\pi$
$617$	3.35166e9	0.574462	0.287231	+	0.957861i	$0.407265\pi$
$618$	−1.09637e10	−1.86852
$619$	−3.92362e9	−0.664921	−0.332461	−	0.943117i	$-0.607879\pi$
$619$	−3.92362e9	−0.664921	−0.332461	+	0.943117i	$0.607879\pi$
$620$	−1.33355e9	−0.224718
$621$	1.28890e9	0.215972
$622$	−6.78588e9	−1.13068
$623$	3.30538e9	0.547662
$624$	−3.79353e9	−0.625026
$625$	2.44141e8	0.0400000
$626$	7.99150e9	1.30202
$627$	0	0
$628$	5.98354e9	0.964049
$629$	1.34948e9	0.216217
$630$	3.36609e8	0.0536332
$631$	−6.81545e8	−0.107992	−0.0539960	−	0.998541i	$-0.517196\pi$
$631$	−6.81545e8	−0.107992	−0.0539960	+	0.998541i	$0.517196\pi$
$632$	−806400.	−0.000127069 0
$633$	−7.77583e9	−1.21852
$634$	−7.70267e9	−1.20041
$635$	1.56272e9	0.242200
$636$	4.17955e9	0.644213
$637$	−7.25744e9	−1.11249
$638$	0	0
$639$	5.97097e8	0.0905298
$640$	3.06264e9	0.461813
$641$	9.65199e9	1.44748	0.723742	−	0.690071i	$-0.242421\pi$
$641$	9.65199e9	1.44748	0.723742	+	0.690071i	$0.242421\pi$
$642$	−2.74831e9	−0.409914
$643$	−5.07826e9	−0.753315	−0.376657	−	0.926353i	$-0.622927\pi$
$643$	−5.07826e9	−0.753315	−0.376657	+	0.926353i	$0.622927\pi$
$644$	1.45017e9	0.213952
$645$	−2.99525e9	−0.439515
$646$	4.45016e9	0.649474
$647$	−2.08330e9	−0.302404	−0.151202	−	0.988503i	$-0.548314\pi$
$647$	−2.08330e9	−0.302404	−0.151202	+	0.988503i	$0.548314\pi$
$648$	4.22113e9	0.609420
$649$	0	0
$650$	−8.44812e8	−0.120660
$651$	1.23803e10	1.75873
$652$	−3.73235e9	−0.527370
$653$	6.84881e9	0.962540	0.481270	−	0.876572i	$-0.340176\pi$
$653$	6.84881e9	0.962540	0.481270	+	0.876572i	$0.340176\pi$
$654$	−1.80309e10	−2.52056
$655$	9.69524e8	0.134807
$656$	9.57474e9	1.32423
$657$	4.99295e8	0.0686876
$658$	−9.13468e9	−1.24998
$659$	−6.99913e9	−0.952676	−0.476338	−	0.879262i	$-0.658036\pi$
$659$	−6.99913e9	−0.952676	−0.476338	+	0.879262i	$0.658036\pi$
$660$	0	0
$661$	−1.99594e9	−0.268809	−0.134404	−	0.990927i	$-0.542912\pi$
$661$	−1.99594e9	−0.268809	−0.134404	+	0.990927i	$0.542912\pi$
$662$	−1.31577e10	−1.76270
$663$	2.27160e9	0.302715
$664$	5.15830e9	0.683783
$665$	5.33067e9	0.702920
$666$	1.80386e8	0.0236616
$667$	1.05864e9	0.138137
$668$	−5.85793e8	−0.0760372
$669$	−8.44472e8	−0.109042
$670$	−1.39528e9	−0.179226
$671$	0	0
$672$	1.41234e10	1.79533
$673$	1.17939e10	1.49144	0.745718	−	0.666261i	$-0.232106\pi$
$673$	1.17939e10	1.49144	0.745718	+	0.666261i	$0.232106\pi$
$674$	−7.47085e9	−0.939853
$675$	1.55250e9	0.194298
$676$	−3.25268e9	−0.404974
$677$	−7.80222e9	−0.966402	−0.483201	−	0.875509i	$-0.660526\pi$
$677$	−7.80222e9	−0.966402	−0.483201	+	0.875509i	$0.660526\pi$
$678$	1.17473e10	1.44755
$679$	−8.02590e9	−0.983897
$680$	−1.28667e9	−0.156923
$681$	−9.74572e9	−1.18249
$682$	0	0
$683$	4.51153e9	0.541815	0.270908	−	0.962605i	$-0.412676\pi$
$683$	4.51153e9	0.541815	0.270908	+	0.962605i	$0.412676\pi$
$684$	2.06379e8	0.0246586
$685$	4.52150e9	0.537484
$686$	2.42968e10	2.87353
$687$	−7.65883e9	−0.901185
$688$	−1.02158e10	−1.19595
$689$	4.94528e9	0.576002
$690$	−1.08965e9	−0.126274
$691$	−1.06331e10	−1.22599	−0.612997	−	0.790086i	$-0.710036\pi$
$691$	−1.06331e10	−1.22599	−0.612997	+	0.790086i	$0.710036\pi$
$692$	3.48516e9	0.399808
$693$	0	0
$694$	−1.51107e10	−1.71604
$695$	−1.36365e9	−0.154084
$696$	3.29052e9	0.369940
$697$	−5.73343e9	−0.641356
$698$	5.98104e9	0.665707
$699$	−9.35929e8	−0.103651
$700$	1.74675e9	0.192481
$701$	4.38514e9	0.480807	0.240403	−	0.970673i	$-0.422720\pi$
$701$	4.38514e9	0.480807	0.240403	+	0.970673i	$0.422720\pi$
$702$	−5.37220e9	−0.586100
$703$	2.85667e9	0.310110
$704$	0	0
$705$	2.38130e9	0.255949
$706$	−2.08552e10	−2.23048
$707$	−1.59907e10	−1.70176
$708$	4.36534e9	0.462276
$709$	−5.98805e9	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$709$	−5.98805e9	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$710$	8.93094e9	0.936468
$711$	112320.	1.17196e−5 0
$712$	−1.68888e9	−0.175355
$713$	−2.03515e9	−0.210273
$714$	−1.35378e10	−1.39189
$715$	0	0
$716$	−3.41106e9	−0.347291
$717$	−7.69058e9	−0.779188
$718$	−1.17779e10	−1.18749
$719$	−1.17768e10	−1.18161	−0.590807	−	0.806813i	$-0.701191\pi$
$719$	−1.17768e10	−1.18161	−0.590807	+	0.806813i	$0.701191\pi$
$720$	−2.99286e8	−0.0298829
$721$	2.68219e10	2.66512
$722$	−3.09383e9	−0.305926
$723$	−1.81814e10	−1.78913
$724$	4.69755e9	0.460031
$725$	1.27516e9	0.124274
$726$	0	0
$727$	8.41051e9	0.811805	0.405902	−	0.913916i	$-0.366957\pi$
$727$	8.41051e9	0.811805	0.405902	+	0.913916i	$0.366957\pi$
$728$	5.33327e9	0.512311
$729$	9.84247e9	0.940931
$730$	7.46809e9	0.710525
$731$	6.11729e9	0.579227
$732$	−3.01586e9	−0.284199
$733$	−1.44084e10	−1.35130	−0.675650	−	0.737223i	$-0.736137\pi$
$733$	−1.44084e10	−1.35130	−0.675650	+	0.737223i	$0.736137\pi$
$734$	−1.05065e10	−0.980664
$735$	−1.12752e10	−1.04741
$736$	−2.32168e9	−0.214649
$737$	0	0
$738$	−7.66391e8	−0.0701864
$739$	−8.21708e9	−0.748966	−0.374483	−	0.927234i	$-0.622180\pi$
$739$	−8.21708e9	−0.748966	−0.374483	+	0.927234i	$0.622180\pi$
$740$	9.36071e8	0.0849176
$741$	4.80865e9	0.434170
$742$	−2.94719e10	−2.64847
$743$	1.72531e10	1.54314	0.771570	−	0.636144i	$-0.219472\pi$
$743$	1.72531e10	1.54314	0.771570	+	0.636144i	$0.219472\pi$
$744$	−6.32572e9	−0.563125
$745$	4.55726e9	0.403791
$746$	−2.39503e9	−0.211215
$747$	−7.18477e8	−0.0630654
$748$	0	0
$749$	6.72354e9	0.584671
$750$	−1.31250e9	−0.113602
$751$	1.58498e10	1.36548	0.682739	−	0.730662i	$-0.260789\pi$
$751$	1.58498e10	1.36548	0.682739	+	0.730662i	$0.260789\pi$
$752$	8.12183e9	0.696453
$753$	−7.74260e9	−0.660853
$754$	−4.41249e9	−0.374873
$755$	−8.98694e8	−0.0759972
$756$	1.11077e10	0.934966
$757$	7.13856e9	0.598102	0.299051	−	0.954237i	$-0.403330\pi$
$757$	7.13856e9	0.598102	0.299051	+	0.954237i	$0.403330\pi$
$758$	6.52735e8	0.0544371
$759$	0	0
$760$	−2.72370e9	−0.225067
$761$	2.59993e9	0.213853	0.106926	−	0.994267i	$-0.465899\pi$
$761$	2.59993e9	0.213853	0.106926	+	0.994267i	$0.465899\pi$
$762$	−8.40119e9	−0.687857
$763$	4.41114e10	3.59514
$764$	1.05227e10	0.853690
$765$	1.79215e8	0.0144730
$766$	−6.19172e9	−0.497749
$767$	5.16512e9	0.413329
$768$	−1.57660e10	−1.25591
$769$	4.96477e9	0.393692	0.196846	−	0.980434i	$-0.436930\pi$
$769$	4.96477e9	0.393692	0.196846	+	0.980434i	$0.436930\pi$
$770$	0	0
$771$	−1.09146e10	−0.857663
$772$	1.08396e9	0.0847914
$773$	−1.49681e10	−1.16557	−0.582786	−	0.812626i	$-0.698037\pi$
$773$	−1.49681e10	−1.16557	−0.582786	+	0.812626i	$0.698037\pi$
$774$	8.17703e8	0.0633873
$775$	−2.45138e9	−0.189171
$776$	4.10082e9	0.315032
$777$	−8.69026e9	−0.664598
$778$	−6.49646e9	−0.494593
$779$	−1.21369e10	−0.919867
$780$	1.57570e9	0.118889
$781$	0	0
$782$	2.22542e9	0.166414
$783$	8.10877e9	0.603655
$784$	−3.84558e10	−2.85007
$785$	1.09992e10	0.811550
$786$	−5.21216e9	−0.382859
$787$	9.79990e9	0.716655	0.358328	−	0.933596i	$-0.383347\pi$
$787$	9.79990e9	0.716655	0.358328	+	0.933596i	$0.383347\pi$
$788$	1.14368e10	0.832650
$789$	2.19709e10	1.59250
$790$	1.68000e6	0.000121231 0
$791$	−2.87388e10	−2.06467
$792$	0	0
$793$	−3.56840e9	−0.254108
$794$	−4.44909e9	−0.315428
$795$	7.68299e9	0.542307
$796$	1.20885e10	0.849529
$797$	3.93169e9	0.275090	0.137545	−	0.990495i	$-0.456079\pi$
$797$	3.93169e9	0.275090	0.137545	+	0.990495i	$0.456079\pi$
$798$	−2.86577e10	−1.99632
$799$	−4.86342e9	−0.337309
$800$	−2.79650e9	−0.193108
$801$	2.35237e8	0.0161730
$802$	−1.67388e10	−1.14581
$803$	0	0
$804$	2.60240e9	0.176595
$805$	2.66575e9	0.180108
$806$	8.48262e9	0.570634
$807$	−2.24297e10	−1.50234
$808$	8.17043e9	0.544885
$809$	−1.26324e10	−0.838816	−0.419408	−	0.907798i	$-0.637762\pi$
$809$	−1.26324e10	−0.838816	−0.419408	+	0.907798i	$0.637762\pi$
$810$	−8.79403e9	−0.581420
$811$	−1.16653e10	−0.767934	−0.383967	−	0.923347i	$-0.625442\pi$
$811$	−1.16653e10	−0.767934	−0.383967	+	0.923347i	$0.625442\pi$
$812$	9.12335e9	0.598009
$813$	−2.14047e9	−0.139699
$814$	0	0
$815$	−6.86094e9	−0.443948
$816$	1.20368e10	0.775522
$817$	1.29495e10	0.830758
$818$	−3.09827e10	−1.97917
$819$	−7.42848e8	−0.0472505
$820$	−3.97700e9	−0.251888
$821$	8.17500e9	0.515569	0.257784	−	0.966202i	$-0.417008\pi$
$821$	8.17500e9	0.515569	0.257784	+	0.966202i	$0.417008\pi$
$822$	−2.43076e10	−1.52648
$823$	−1.75211e10	−1.09563	−0.547813	−	0.836601i	$-0.684540\pi$
$823$	−1.75211e10	−1.09563	−0.547813	+	0.836601i	$0.684540\pi$
$824$	−1.37046e10	−0.853341
$825$	0	0
$826$	−3.07821e10	−1.90050
$827$	−1.22225e10	−0.751437	−0.375718	−	0.926734i	$-0.622604\pi$
$827$	−1.22225e10	−0.751437	−0.375718	+	0.926734i	$0.622604\pi$
$828$	1.03205e8	0.00631823
$829$	−1.06634e10	−0.650063	−0.325032	−	0.945703i	$-0.605375\pi$
$829$	−1.06634e10	−0.650063	−0.325032	+	0.945703i	$0.605375\pi$
$830$	−1.07465e10	−0.652367
$831$	1.51995e10	0.918812
$832$	−4.39218e8	−0.0264392
$833$	2.30276e10	1.38036
$834$	7.33100e9	0.437605
$835$	−1.07682e9	−0.0640092
$836$	0	0
$837$	−1.55884e10	−0.918887
$838$	−1.12322e10	−0.659341
$839$	−2.31400e9	−0.135268	−0.0676342	−	0.997710i	$-0.521545\pi$
$839$	−2.31400e9	−0.135268	−0.0676342	+	0.997710i	$0.521545\pi$
$840$	8.28576e9	0.482342
$841$	−1.05897e10	−0.613899
$842$	4.82599e8	0.0278608
$843$	−1.08392e10	−0.623164
$844$	1.10158e10	0.630690
$845$	−5.97918e9	−0.340913
$846$	−6.50096e8	−0.0369132
$847$	0	0
$848$	2.62041e10	1.47565
$849$	9.99410e8	0.0560488
$850$	2.68056e9	0.149713
$851$	1.42855e9	0.0794590
$852$	−1.66575e10	−0.922721
$853$	4.22377e9	0.233012	0.116506	−	0.993190i	$-0.462831\pi$
$853$	4.22377e9	0.233012	0.116506	+	0.993190i	$0.462831\pi$
$854$	2.12663e10	1.16839
$855$	3.79372e8	0.0207579
$856$	−3.43538e9	−0.187205
$857$	3.52104e9	0.191090	0.0955450	−	0.995425i	$-0.469541\pi$
$857$	3.52104e9	0.191090	0.0955450	+	0.995425i	$0.469541\pi$
$858$	0	0
$859$	2.44930e10	1.31846	0.659229	−	0.751943i	$-0.270883\pi$
$859$	2.44930e10	1.31846	0.659229	+	0.751943i	$0.270883\pi$
$860$	4.24327e9	0.227487
$861$	3.69215e10	1.97137
$862$	−2.41769e10	−1.28566
$863$	−5.40573e9	−0.286297	−0.143148	−	0.989701i	$-0.545723\pi$
$863$	−5.40573e9	−0.286297	−0.143148	+	0.989701i	$0.545723\pi$
$864$	−1.77831e10	−0.938012
$865$	6.40655e9	0.336564
$866$	6.83631e9	0.357692
$867$	1.24886e10	0.650797
$868$	−1.75388e10	−0.910293
$869$	0	0
$870$	−6.85524e9	−0.352944
$871$	3.07919e9	0.157897
$872$	−2.25387e10	−1.15112
$873$	−5.71186e8	−0.0290555
$874$	4.71091e9	0.238679
$875$	3.21094e9	0.162033
$876$	−1.39290e10	−0.700095
$877$	2.89155e10	1.44755	0.723773	−	0.690039i	$-0.242406\pi$
$877$	2.89155e10	1.44755	0.723773	+	0.690039i	$0.242406\pi$
$878$	4.04096e10	2.01490
$879$	−8.58362e9	−0.426294
$880$	0	0
$881$	7.80643e8	0.0384624	0.0192312	−	0.999815i	$-0.493878\pi$
$881$	7.80643e8	0.0384624	0.0192312	+	0.999815i	$0.493878\pi$
$882$	3.07812e9	0.151059
$883$	−1.36907e10	−0.669211	−0.334605	−	0.942358i	$-0.608603\pi$
$883$	−1.36907e10	−0.669211	−0.334605	+	0.942358i	$0.608603\pi$
$884$	−3.21810e9	−0.156681
$885$	8.02452e9	0.389151
$886$	1.29722e10	0.626606
$887$	−3.58403e9	−0.172441	−0.0862203	−	0.996276i	$-0.527479\pi$
$887$	−3.58403e9	−0.172441	−0.0862203	+	0.996276i	$0.527479\pi$
$888$	4.44028e9	0.212797
$889$	2.05529e10	0.981108
$890$	3.51850e9	0.167299
$891$	0	0
$892$	1.19634e9	0.0564386
$893$	−1.02952e10	−0.483786
$894$	−2.44998e10	−1.14678
$895$	−6.27033e9	−0.292355
$896$	4.02798e10	1.87072
$897$	2.40470e9	0.111247
$898$	1.89749e10	0.874406
$899$	−1.28036e10	−0.587725
$900$	1.24312e8	0.00568416
$901$	−1.56912e10	−0.714694
$902$	0	0
$903$	−3.93935e10	−1.78040
$904$	1.46841e10	0.661085
$905$	8.63521e9	0.387260
$906$	4.83138e9	0.215835
$907$	−3.01108e10	−1.33998	−0.669988	−	0.742372i	$-0.733701\pi$
$907$	−3.01108e10	−1.33998	−0.669988	+	0.742372i	$0.733701\pi$
$908$	1.38064e10	0.612042
$909$	−1.13802e9	−0.0502548
$910$	−1.11110e10	−0.488773
$911$	−2.48800e10	−1.09027	−0.545137	−	0.838347i	$-0.683523\pi$
$911$	−2.48800e10	−1.09027	−0.545137	+	0.838347i	$0.683523\pi$
$912$	2.54801e10	1.11229
$913$	0	0
$914$	−6.48801e8	−0.0281060
$915$	−5.54387e9	−0.239243
$916$	1.08500e10	0.466440
$917$	1.27512e10	0.546082
$918$	1.70458e10	0.727224
$919$	−1.08420e10	−0.460794	−0.230397	−	0.973097i	$-0.574003\pi$
$919$	−1.08420e10	−0.460794	−0.230397	+	0.973097i	$0.574003\pi$
$920$	−1.36206e9	−0.0576685
$921$	−4.10477e9	−0.173133
$922$	2.12964e9	0.0894846
$923$	−1.97093e10	−0.825021
$924$	0	0
$925$	1.72072e9	0.0714848
$926$	2.28831e10	0.947056
$927$	1.90886e9	0.0787037
$928$	−1.46062e10	−0.599957
$929$	1.07045e10	0.438038	0.219019	−	0.975721i	$-0.429714\pi$
$929$	1.07045e10	0.438038	0.219019	+	0.975721i	$0.429714\pi$
$930$	1.31786e10	0.537253
$931$	4.87463e10	1.97978
$932$	1.32590e9	0.0536482
$933$	2.32659e10	0.937851
$934$	−1.55933e10	−0.626214
$935$	0	0
$936$	3.79557e8	0.0151291
$937$	−3.42787e10	−1.36124	−0.680621	−	0.732635i	$-0.738290\pi$
$937$	−3.42787e10	−1.36124	−0.680621	+	0.732635i	$0.738290\pi$
$938$	−1.83507e10	−0.726012
$939$	−2.73994e10	−1.07997
$940$	−3.37351e9	−0.132475
$941$	3.73695e9	0.146202	0.0731010	−	0.997325i	$-0.476710\pi$
$941$	3.73695e9	0.146202	0.0731010	+	0.997325i	$0.476710\pi$
$942$	−5.91314e10	−2.30484
$943$	−6.06937e9	−0.235696
$944$	2.73690e10	1.05890
$945$	2.04185e10	0.787068
$946$	0	0
$947$	3.32150e10	1.27089	0.635447	−	0.772145i	$-0.280816\pi$
$947$	3.32150e10	1.27089	0.635447	+	0.772145i	$0.280816\pi$
$948$	−3.13344e6	−0.000119452 0
$949$	−1.64810e10	−0.625968
$950$	5.67438e9	0.214726
$951$	2.64091e10	0.995686
$952$	−1.69223e10	−0.635667
$953$	−4.69895e10	−1.75864	−0.879318	−	0.476235i	$-0.842001\pi$
$953$	−4.69895e10	−1.75864	−0.879318	+	0.476235i	$0.842001\pi$
$954$	−2.09746e9	−0.0782120
$955$	1.93432e10	0.718648
$956$	1.08950e10	0.403296
$957$	0	0
$958$	−1.78843e10	−0.657192
$959$	5.94668e10	2.17726
$960$	−6.82368e8	−0.0248925
$961$	−2.89877e9	−0.105361
$962$	−5.95429e9	−0.215634
$963$	4.78500e8	0.0172659
$964$	2.57570e10	0.926030
$965$	1.99257e9	0.0713786
$966$	−1.43311e10	−0.511515
$967$	1.42294e10	0.506050	0.253025	−	0.967460i	$-0.418575\pi$
$967$	1.42294e10	0.506050	0.253025	+	0.967460i	$0.418575\pi$
$968$	0	0
$969$	−1.52577e10	−0.538711
$970$	−8.54338e9	−0.300558
$971$	5.45474e8	0.0191208	0.00956041	−	0.999954i	$-0.496957\pi$
$971$	5.45474e8	0.0191208	0.00956041	+	0.999954i	$0.496957\pi$
$972$	1.62570e9	0.0567816
$973$	−1.79348e10	−0.624167
$974$	1.37154e10	0.475612
$975$	2.89650e9	0.100082
$976$	−1.89083e10	−0.650995
$977$	−1.97916e10	−0.678968	−0.339484	−	0.940612i	$-0.610252\pi$
$977$	−1.97916e10	−0.678968	−0.339484	+	0.940612i	$0.610252\pi$
$978$	3.68844e10	1.26083
$979$	0	0
$980$	1.59731e10	0.542125
$981$	3.13932e9	0.106168
$982$	6.88975e10	2.32174
$983$	−4.71503e10	−1.58324	−0.791620	−	0.611013i	$-0.790762\pi$
$983$	−4.71503e10	−1.58324	−0.791620	+	0.611013i	$0.790762\pi$
$984$	−1.88650e10	−0.631210
$985$	2.10235e10	0.700936
$986$	1.40007e10	0.465136
$987$	3.13189e10	1.03680
$988$	−6.81226e9	−0.224720
$989$	6.47573e9	0.212864
$990$	0	0
$991$	3.87968e10	1.26631	0.633153	−	0.774027i	$-0.281761\pi$
$991$	3.87968e10	1.26631	0.633153	+	0.774027i	$0.281761\pi$
$992$	2.80792e10	0.913258
$993$	4.51123e10	1.46208
$994$	1.17460e11	3.79347
$995$	2.22216e10	0.715145
$996$	2.00437e10	0.642791
$997$	5.66394e10	1.81003	0.905015	−	0.425380i	$-0.139860\pi$
$997$	5.66394e10	1.81003	0.905015	+	0.425380i	$0.139860\pi$
$998$	−5.12101e10	−1.63079
$999$	1.09421e10	0.347234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.8.a.c.1.1		1
11.10	odd	2		5.8.a.a.1.1	✓	1
33.32	even	2		45.8.a.f.1.1		1
44.43	even	2		80.8.a.d.1.1		1
55.32	even	4		25.8.b.a.24.1		2
55.43	even	4		25.8.b.a.24.2		2
55.54	odd	2		25.8.a.a.1.1		1
77.76	even	2		245.8.a.a.1.1		1
88.21	odd	2		320.8.a.h.1.1		1
88.43	even	2		320.8.a.a.1.1		1
165.32	odd	4		225.8.b.b.199.2		2
165.98	odd	4		225.8.b.b.199.1		2
165.164	even	2		225.8.a.b.1.1		1
220.43	odd	4		400.8.c.e.49.2		2
220.87	odd	4		400.8.c.e.49.1		2
220.219	even	2		400.8.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.a.1.1	✓	1	11.10	odd	2
25.8.a.a.1.1		1	55.54	odd	2
25.8.b.a.24.1		2	55.32	even	4
25.8.b.a.24.2		2	55.43	even	4
45.8.a.f.1.1		1	33.32	even	2
80.8.a.d.1.1		1	44.43	even	2
225.8.a.b.1.1		1	165.164	even	2
225.8.b.b.199.1		2	165.98	odd	4
225.8.b.b.199.2		2	165.32	odd	4
245.8.a.a.1.1		1	77.76	even	2
320.8.a.a.1.1		1	88.43	even	2
320.8.a.h.1.1		1	88.21	odd	2
400.8.a.e.1.1		1	220.219	even	2
400.8.c.e.49.1		2	220.87	odd	4
400.8.c.e.49.2		2	220.43	odd	4
605.8.a.c.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more