LMFDB - Embedded newform 605.8.a.a.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,-6,-70] 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:	yes
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)

$f(q)$

$=$

\(q-6.00000 q^{2} -70.0000 q^{3} -92.0000 q^{4} -125.000 q^{5} +420.000 q^{6} -624.000 q^{7} +1320.00 q^{8} +2713.00 q^{9} +750.000 q^{10} +6440.00 q^{12} -6822.00 q^{13} +3744.00 q^{14} +8750.00 q^{15} +3856.00 q^{16} -36246.0 q^{17} -16278.0 q^{18} +34080.0 q^{19} +11500.0 q^{20} +43680.0 q^{21} +16290.0 q^{23} -92400.0 q^{24} +15625.0 q^{25} +40932.0 q^{26} -36820.0 q^{27} +57408.0 q^{28} +158760. q^{29} -52500.0 q^{30} +194620. q^{31} -192096. q^{32} +217476. q^{34} +78000.0 q^{35} -249596. q^{36} +371590. q^{37} -204480. q^{38} +477540. q^{39} -165000. q^{40} +562980. q^{41} -262080. q^{42} -234852. q^{43} -339125. q^{45} -97740.0 q^{46} -832530. q^{47} -269920. q^{48} -434167. q^{49} -93750.0 q^{50} +2.53722e6 q^{51} +627624. q^{52} -227010. q^{53} +220920. q^{54} -823680. q^{56} -2.38560e6 q^{57} -952560. q^{58} -462624. q^{59} -805000. q^{60} -1.08276e6 q^{61} -1.16772e6 q^{62} -1.69291e6 q^{63} +659008. q^{64} +852750. q^{65} -2.58791e6 q^{67} +3.33463e6 q^{68} -1.14030e6 q^{69} -468000. q^{70} -3.09799e6 q^{71} +3.58116e6 q^{72} -2.72242e6 q^{73} -2.22954e6 q^{74} -1.09375e6 q^{75} -3.13536e6 q^{76} -2.86524e6 q^{78} -211620. q^{79} -482000. q^{80} -3.35593e6 q^{81} -3.37788e6 q^{82} -5.21663e6 q^{83} -4.01856e6 q^{84} +4.53075e6 q^{85} +1.40911e6 q^{86} -1.11132e7 q^{87} -9.07713e6 q^{89} +2.03475e6 q^{90} +4.25693e6 q^{91} -1.49868e6 q^{92} -1.36234e7 q^{93} +4.99518e6 q^{94} -4.26000e6 q^{95} +1.34467e7 q^{96} +1.47348e7 q^{97} +2.60500e6 q^{98} +O(q^{100})\)

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 $

$2$	−6.00000	−0.530330	−0.265165	−	0.964203i	$-0.585426\pi$
$2$	−6.00000	−0.530330	−0.265165	+	0.964203i	$0.585426\pi$
$3$	−70.0000	−1.49683	−0.748417	−	0.663228i	$-0.769186\pi$
$3$	−70.0000	−1.49683	−0.748417	+	0.663228i	$0.769186\pi$
$4$	−92.0000	−0.718750
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	420.000	0.793816
$7$	−624.000	−0.687609	−0.343804	−	0.939041i	$-0.611716\pi$
$7$	−624.000	−0.687609	−0.343804	+	0.939041i	$0.611716\pi$
$8$	1320.00	0.911505
$9$	2713.00	1.24051
$10$	750.000	0.237171
$11$	0	0
$11$	0	0
$12$	6440.00	1.07585
$13$	−6822.00	−0.861212	−0.430606	−	0.902540i	$-0.641700\pi$
$13$	−6822.00	−0.861212	−0.430606	+	0.902540i	$0.641700\pi$
$14$	3744.00	0.364660
$15$	8750.00	0.669405
$16$	3856.00	0.235352
$17$	−36246.0	−1.78932	−0.894662	−	0.446744i	$-0.852583\pi$
$17$	−36246.0	−1.78932	−0.894662	+	0.446744i	$0.852583\pi$
$18$	−16278.0	−0.657881
$19$	34080.0	1.13989	0.569944	−	0.821684i	$-0.306965\pi$
$19$	34080.0	1.13989	0.569944	+	0.821684i	$0.306965\pi$
$20$	11500.0	0.321435
$21$	43680.0	1.02924
$22$	0	0
$23$	16290.0	0.279173	0.139587	−	0.990210i	$-0.455423\pi$
$23$	16290.0	0.279173	0.139587	+	0.990210i	$0.455423\pi$
$24$	−92400.0	−1.36437
$25$	15625.0	0.200000
$26$	40932.0	0.456727
$27$	−36820.0	−0.360007
$28$	57408.0	0.494219
$29$	158760.	1.20878	0.604391	−	0.796688i	$-0.293416\pi$
$29$	158760.	1.20878	0.604391	+	0.796688i	$0.293416\pi$
$30$	−52500.0	−0.355005
$31$	194620.	1.17333	0.586667	−	0.809828i	$-0.300440\pi$
$31$	194620.	1.17333	0.586667	+	0.809828i	$0.300440\pi$
$32$	−192096.	−1.03632
$33$	0	0
$34$	217476.	0.948932
$35$	78000.0	0.307508
$36$	−249596.	−0.891618
$37$	371590.	1.20603	0.603015	−	0.797730i	$-0.293966\pi$
$37$	371590.	1.20603	0.603015	+	0.797730i	$0.293966\pi$
$38$	−204480.	−0.604517
$39$	477540.	1.28909
$40$	−165000.	−0.407637
$41$	562980.	1.27570	0.637851	−	0.770160i	$-0.279823\pi$
$41$	562980.	1.27570	0.637851	+	0.770160i	$0.279823\pi$
$42$	−262080.	−0.545835
$43$	−234852.	−0.450458	−0.225229	−	0.974306i	$-0.572313\pi$
$43$	−234852.	−0.450458	−0.225229	+	0.974306i	$0.572313\pi$
$44$	0	0
$45$	−339125.	−0.554774
$46$	−97740.0	−0.148054
$47$	−832530.	−1.16965	−0.584827	−	0.811158i	$-0.698838\pi$
$47$	−832530.	−1.16965	−0.584827	+	0.811158i	$0.698838\pi$
$48$	−269920.	−0.352282
$49$	−434167.	−0.527194
$50$	−93750.0	−0.106066
$51$	2.53722e6	2.67832
$52$	627624.	0.618996
$53$	−227010.	−0.209450	−0.104725	−	0.994501i	$-0.533396\pi$
$53$	−227010.	−0.209450	−0.104725	+	0.994501i	$0.533396\pi$
$54$	220920.	0.190922
$55$	0	0
$56$	−823680.	−0.626759
$57$	−2.38560e6	−1.70622
$58$	−952560.	−0.641054
$59$	−462624.	−0.293255	−0.146628	−	0.989192i	$-0.546842\pi$
$59$	−462624.	−0.293255	−0.146628	+	0.989192i	$0.546842\pi$
$60$	−805000.	−0.481135
$61$	−1.08276e6	−0.610770	−0.305385	−	0.952229i	$-0.598785\pi$
$61$	−1.08276e6	−0.610770	−0.305385	+	0.952229i	$0.598785\pi$
$62$	−1.16772e6	−0.622254
$63$	−1.69291e6	−0.852987
$64$	659008.	0.314240
$65$	852750.	0.385146
$66$	0	0
$67$	−2.58791e6	−1.05120	−0.525602	−	0.850730i	$-0.676160\pi$
$67$	−2.58791e6	−1.05120	−0.525602	+	0.850730i	$0.676160\pi$
$68$	3.33463e6	1.28608
$69$	−1.14030e6	−0.417876
$70$	−468000.	−0.163081
$71$	−3.09799e6	−1.02725	−0.513625	−	0.858015i	$-0.671698\pi$
$71$	−3.09799e6	−1.02725	−0.513625	+	0.858015i	$0.671698\pi$
$72$	3.58116e6	1.13073
$73$	−2.72242e6	−0.819078	−0.409539	−	0.912293i	$-0.634310\pi$
$73$	−2.72242e6	−0.819078	−0.409539	+	0.912293i	$0.634310\pi$
$74$	−2.22954e6	−0.639594
$75$	−1.09375e6	−0.299367
$76$	−3.13536e6	−0.819294
$77$	0	0
$78$	−2.86524e6	−0.683644
$79$	−211620.	−0.0482905	−0.0241453	−	0.999708i	$-0.507686\pi$
$79$	−211620.	−0.0482905	−0.0241453	+	0.999708i	$0.507686\pi$
$80$	−482000.	−0.105252
$81$	−3.35593e6	−0.701642
$82$	−3.37788e6	−0.676544
$83$	−5.21663e6	−1.00142	−0.500710	−	0.865615i	$-0.666928\pi$
$83$	−5.21663e6	−1.00142	−0.500710	+	0.865615i	$0.666928\pi$
$84$	−4.01856e6	−0.739764
$85$	4.53075e6	0.800210
$86$	1.40911e6	0.238892
$87$	−1.11132e7	−1.80935
$88$	0	0
$89$	−9.07713e6	−1.36485	−0.682423	−	0.730958i	$-0.739074\pi$
$89$	−9.07713e6	−1.36485	−0.682423	+	0.730958i	$0.739074\pi$
$90$	2.03475e6	0.294213
$91$	4.25693e6	0.592177
$92$	−1.49868e6	−0.200656
$93$	−1.36234e7	−1.75629
$94$	4.99518e6	0.620303
$95$	−4.26000e6	−0.509773
$96$	1.34467e7	1.55120
$97$	1.47348e7	1.63924	0.819621	−	0.572907i	$-0.194184\pi$
$97$	1.47348e7	1.63924	0.819621	+	0.572907i	$0.194184\pi$
$98$	2.60500e6	0.279587
$99$	0	0
$100$	−1.43750e6	−0.143750
$101$	8.33946e6	0.805403	0.402701	−	0.915331i	$-0.368071\pi$
$101$	8.33946e6	0.805403	0.402701	+	0.915331i	$0.368071\pi$
$102$	−1.52233e7	−1.42039
$103$	1.28659e7	1.16014	0.580070	−	0.814567i	$-0.303025\pi$
$103$	1.28659e7	1.16014	0.580070	+	0.814567i	$0.303025\pi$
$104$	−9.00504e6	−0.784999
$105$	−5.46000e6	−0.460288
$106$	1.36206e6	0.111077
$107$	−1.85560e7	−1.46434	−0.732170	−	0.681121i	$-0.761493\pi$
$107$	−1.85560e7	−1.46434	−0.732170	+	0.681121i	$0.761493\pi$
$108$	3.38744e6	0.258755
$109$	1.92621e7	1.42466	0.712329	−	0.701845i	$-0.247640\pi$
$109$	1.92621e7	1.42466	0.712329	+	0.701845i	$0.247640\pi$
$110$	0	0
$111$	−2.60113e7	−1.80523
$112$	−2.40614e6	−0.161830
$113$	−1.89964e7	−1.23851	−0.619253	−	0.785192i	$-0.712564\pi$
$113$	−1.89964e7	−1.23851	−0.619253	+	0.785192i	$0.712564\pi$
$114$	1.43136e7	0.904861
$115$	−2.03625e6	−0.124850
$116$	−1.46059e7	−0.868812
$117$	−1.85081e7	−1.06834
$118$	2.77574e6	0.155522
$119$	2.26175e7	1.23035
$120$	1.15500e7	0.610165
$121$	0	0
$122$	6.49656e6	0.323910
$123$	−3.94086e7	−1.90952
$124$	−1.79050e7	−0.843334
$125$	−1.95312e6	−0.0894427
$126$	1.01575e7	0.452365
$127$	5.89018e6	0.255162	0.127581	−	0.991828i	$-0.459279\pi$
$127$	5.89018e6	0.255162	0.127581	+	0.991828i	$0.459279\pi$
$128$	2.06342e7	0.869668
$129$	1.64396e7	0.674261
$130$	−5.11650e6	−0.204254
$131$	−2.07798e6	−0.0807592	−0.0403796	−	0.999184i	$-0.512857\pi$
$131$	−2.07798e6	−0.0807592	−0.0403796	+	0.999184i	$0.512857\pi$
$132$	0	0
$133$	−2.12659e7	−0.783797
$134$	1.55275e7	0.557486
$135$	4.60250e6	0.161000
$136$	−4.78447e7	−1.63098
$137$	2.29433e7	0.762314	0.381157	−	0.924510i	$-0.375526\pi$
$137$	2.29433e7	0.762314	0.381157	+	0.924510i	$0.375526\pi$
$138$	6.84180e6	0.221612
$139$	−4.45852e7	−1.40812	−0.704059	−	0.710141i	$-0.748631\pi$
$139$	−4.45852e7	−1.40812	−0.704059	+	0.710141i	$0.748631\pi$
$140$	−7.17600e6	−0.221021
$141$	5.82771e7	1.75078
$142$	1.85880e7	0.544781
$143$	0	0
$144$	1.04613e7	0.291956
$145$	−1.98450e7	−0.540584
$146$	1.63345e7	0.434382
$147$	3.03917e7	0.789122
$148$	−3.41863e7	−0.866834
$149$	−5.13947e7	−1.27282	−0.636410	−	0.771351i	$-0.719581\pi$
$149$	−5.13947e7	−1.27282	−0.636410	+	0.771351i	$0.719581\pi$
$150$	6.56250e6	0.158763
$151$	2.93506e7	0.693741	0.346870	−	0.937913i	$-0.387244\pi$
$151$	2.93506e7	0.693741	0.346870	+	0.937913i	$0.387244\pi$
$152$	4.49856e7	1.03901
$153$	−9.83354e7	−2.21968
$154$	0	0
$155$	−2.43275e7	−0.524731
$156$	−4.39337e7	−0.926534
$157$	3.52934e7	0.727856	0.363928	−	0.931427i	$-0.381435\pi$
$157$	3.52934e7	0.727856	0.363928	+	0.931427i	$0.381435\pi$
$158$	1.26972e6	0.0256099
$159$	1.58907e7	0.313511
$160$	2.40120e7	0.463456
$161$	−1.01650e7	−0.191962
$162$	2.01356e7	0.372102
$163$	−6.16529e7	−1.11506	−0.557529	−	0.830158i	$-0.688250\pi$
$163$	−6.16529e7	−1.11506	−0.557529	+	0.830158i	$0.688250\pi$
$164$	−5.17942e7	−0.916911
$165$	0	0
$166$	3.12998e7	0.531084
$167$	−9.89798e6	−0.164452	−0.0822260	−	0.996614i	$-0.526203\pi$
$167$	−9.89798e6	−0.164452	−0.0822260	+	0.996614i	$0.526203\pi$
$168$	5.76576e7	0.938154
$169$	−1.62088e7	−0.258314
$170$	−2.71845e7	−0.424375
$171$	9.24590e7	1.41404
$172$	2.16064e7	0.323767
$173$	8.63034e7	1.26726	0.633631	−	0.773636i	$-0.281564\pi$
$173$	8.63034e7	1.26726	0.633631	+	0.773636i	$0.281564\pi$
$174$	6.66792e7	0.959551
$175$	−9.75000e6	−0.137522
$176$	0	0
$177$	3.23837e7	0.438955
$178$	5.44628e7	0.723819
$179$	1.03058e8	1.34306	0.671530	−	0.740977i	$-0.265637\pi$
$179$	1.03058e8	1.34306	0.671530	+	0.740977i	$0.265637\pi$
$180$	3.11995e7	0.398744
$181$	−2.95251e7	−0.370098	−0.185049	−	0.982729i	$-0.559244\pi$
$181$	−2.95251e7	−0.370098	−0.185049	+	0.982729i	$0.559244\pi$
$182$	−2.55416e7	−0.314049
$183$	7.57932e7	0.914221
$184$	2.15028e7	0.254468
$185$	−4.64488e7	−0.539353
$186$	8.17404e7	0.931411
$187$	0	0
$188$	7.65928e7	0.840689
$189$	2.29757e7	0.247544
$190$	2.55600e7	0.270348
$191$	−7.62419e7	−0.791730	−0.395865	−	0.918309i	$-0.629555\pi$
$191$	−7.62419e7	−0.791730	−0.395865	+	0.918309i	$0.629555\pi$
$192$	−4.61306e7	−0.470364
$193$	3.51875e7	0.352321	0.176160	−	0.984361i	$-0.443632\pi$
$193$	3.51875e7	0.352321	0.176160	+	0.984361i	$0.443632\pi$
$194$	−8.84087e7	−0.869339
$195$	−5.96925e7	−0.576499
$196$	3.99434e7	0.378921
$197$	−1.93388e8	−1.80217	−0.901087	−	0.433638i	$-0.857230\pi$
$197$	−1.93388e8	−1.80217	−0.901087	+	0.433638i	$0.857230\pi$
$198$	0	0
$199$	5.29594e7	0.476384	0.238192	−	0.971218i	$-0.423445\pi$
$199$	5.29594e7	0.476384	0.238192	+	0.971218i	$0.423445\pi$
$200$	2.06250e7	0.182301
$201$	1.81154e8	1.57348
$202$	−5.00368e7	−0.427129
$203$	−9.90662e7	−0.831169
$204$	−2.33424e8	−1.92504
$205$	−7.03725e7	−0.570512
$206$	−7.71955e7	−0.615257
$207$	4.41948e7	0.346318
$208$	−2.63056e7	−0.202688
$209$	0	0
$210$	3.27600e7	0.244105
$211$	−2.14234e8	−1.57000	−0.784999	−	0.619497i	$-0.787337\pi$
$211$	−2.14234e8	−1.57000	−0.784999	+	0.619497i	$0.787337\pi$
$212$	2.08849e7	0.150542
$213$	2.16859e8	1.53762
$214$	1.11336e8	0.776584
$215$	2.93565e7	0.201451
$216$	−4.86024e7	−0.328148
$217$	−1.21443e8	−0.806795
$218$	−1.15573e8	−0.755539
$219$	1.90570e8	1.22602
$220$	0	0
$221$	2.47270e8	1.54099
$222$	1.56068e8	0.957366
$223$	1.01628e8	0.613685	0.306843	−	0.951760i	$-0.400728\pi$
$223$	1.01628e8	0.613685	0.306843	+	0.951760i	$0.400728\pi$
$224$	1.19868e8	0.712582
$225$	4.23906e7	0.248102
$226$	1.13979e8	0.656817
$227$	−9.08156e7	−0.515311	−0.257656	−	0.966237i	$-0.582950\pi$
$227$	−9.08156e7	−0.515311	−0.257656	+	0.966237i	$0.582950\pi$
$228$	2.19475e8	1.22635
$229$	−1.60654e8	−0.884030	−0.442015	−	0.897008i	$-0.645736\pi$
$229$	−1.60654e8	−0.884030	−0.442015	+	0.897008i	$0.645736\pi$
$230$	1.22175e7	0.0662117
$231$	0	0
$232$	2.09563e8	1.10181
$233$	3.36627e8	1.74342	0.871712	−	0.490019i	$-0.163010\pi$
$233$	3.36627e8	1.74342	0.871712	+	0.490019i	$0.163010\pi$
$234$	1.11049e8	0.566575
$235$	1.04066e8	0.523085
$236$	4.25614e7	0.210777
$237$	1.48134e7	0.0722829
$238$	−1.35705e8	−0.652494
$239$	−3.33763e7	−0.158141	−0.0790706	−	0.996869i	$-0.525195\pi$
$239$	−3.33763e7	−0.158141	−0.0790706	+	0.996869i	$0.525195\pi$
$240$	3.37400e7	0.157545
$241$	−1.55254e8	−0.714469	−0.357235	−	0.934015i	$-0.616280\pi$
$241$	−1.55254e8	−0.714469	−0.357235	+	0.934015i	$0.616280\pi$
$242$	0	0
$243$	3.15441e8	1.41025
$244$	9.96139e7	0.438991
$245$	5.42709e7	0.235768
$246$	2.36452e8	1.01267
$247$	−2.32494e8	−0.981685
$248$	2.56898e8	1.06950
$249$	3.65164e8	1.49896
$250$	1.17187e7	0.0474342
$251$	1.49603e8	0.597149	0.298574	−	0.954386i	$-0.403489\pi$
$251$	1.49603e8	0.597149	0.298574	+	0.954386i	$0.403489\pi$
$252$	1.55748e8	0.613084
$253$	0	0
$254$	−3.53411e7	−0.135320
$255$	−3.17152e8	−1.19778
$256$	−2.08158e8	−0.775451
$257$	−1.03773e8	−0.381346	−0.190673	−	0.981654i	$-0.561067\pi$
$257$	−1.03773e8	−0.381346	−0.190673	+	0.981654i	$0.561067\pi$
$258$	−9.86378e7	−0.357581
$259$	−2.31872e8	−0.829277
$260$	−7.84530e7	−0.276823
$261$	4.30716e8	1.49951
$262$	1.24679e7	0.0428290
$263$	3.33664e8	1.13100	0.565502	−	0.824747i	$-0.308682\pi$
$263$	3.33664e8	1.13100	0.565502	+	0.824747i	$0.308682\pi$
$264$	0	0
$265$	2.83762e7	0.0936687
$266$	1.27596e8	0.415671
$267$	6.35399e8	2.04295
$268$	2.38088e8	0.755553
$269$	−3.54676e8	−1.11096	−0.555480	−	0.831530i	$-0.687465\pi$
$269$	−3.54676e8	−1.11096	−0.555480	+	0.831530i	$0.687465\pi$
$270$	−2.76150e7	−0.0853831
$271$	1.95202e8	0.595789	0.297894	−	0.954599i	$-0.403716\pi$
$271$	1.95202e8	0.595789	0.297894	+	0.954599i	$0.403716\pi$
$272$	−1.39765e8	−0.421120
$273$	−2.97985e8	−0.886390
$274$	−1.37660e8	−0.404278
$275$	0	0
$276$	1.04908e8	0.300348
$277$	−6.42046e8	−1.81504	−0.907521	−	0.420006i	$-0.862028\pi$
$277$	−6.42046e8	−1.81504	−0.907521	+	0.420006i	$0.862028\pi$
$278$	2.67511e8	0.746767
$279$	5.28004e8	1.45553
$280$	1.02960e8	0.280295
$281$	−6.61226e7	−0.177778	−0.0888889	−	0.996042i	$-0.528332\pi$
$281$	−6.61226e7	−0.177778	−0.0888889	+	0.996042i	$0.528332\pi$
$282$	−3.49663e8	−0.928490
$283$	−5.25066e8	−1.37709	−0.688544	−	0.725195i	$-0.741750\pi$
$283$	−5.25066e8	−1.37709	−0.688544	+	0.725195i	$0.741750\pi$
$284$	2.85015e8	0.738336
$285$	2.98200e8	0.763046
$286$	0	0
$287$	−3.51300e8	−0.877184
$288$	−5.21156e8	−1.28557
$289$	9.03434e8	2.20168
$290$	1.19070e8	0.286688
$291$	−1.03144e9	−2.45367
$292$	2.50463e8	0.588713
$293$	−5.21337e7	−0.121083	−0.0605413	−	0.998166i	$-0.519283\pi$
$293$	−5.21337e7	−0.121083	−0.0605413	+	0.998166i	$0.519283\pi$
$294$	−1.82350e8	−0.418495
$295$	5.78280e7	0.131148
$296$	4.90499e8	1.09930
$297$	0	0
$298$	3.08368e8	0.675014
$299$	−1.11130e8	−0.240427
$300$	1.00625e8	0.215170
$301$	1.46548e8	0.309739
$302$	−1.76103e8	−0.367911
$303$	−5.83762e8	−1.20555
$304$	1.31412e8	0.268274
$305$	1.35345e8	0.273145
$306$	5.90012e8	1.17716
$307$	−8.27710e8	−1.63265	−0.816327	−	0.577591i	$-0.803993\pi$
$307$	−8.27710e8	−1.63265	−0.816327	+	0.577591i	$0.803993\pi$
$308$	0	0
$309$	−9.00614e8	−1.73654
$310$	1.45965e8	0.278281
$311$	−2.67487e8	−0.504244	−0.252122	−	0.967695i	$-0.581128\pi$
$311$	−2.67487e8	−0.504244	−0.252122	+	0.967695i	$0.581128\pi$
$312$	6.30353e8	1.17501
$313$	−8.15540e8	−1.50328	−0.751640	−	0.659574i	$-0.770737\pi$
$313$	−8.15540e8	−1.50328	−0.751640	+	0.659574i	$0.770737\pi$
$314$	−2.11761e8	−0.386004
$315$	2.11614e8	0.381467
$316$	1.94690e7	0.0347088
$317$	7.72975e7	0.136288	0.0681441	−	0.997675i	$-0.478292\pi$
$317$	7.72975e7	0.136288	0.0681441	+	0.997675i	$0.478292\pi$
$318$	−9.53442e7	−0.166264
$319$	0	0
$320$	−8.23760e7	−0.140532
$321$	1.29892e9	2.19188
$322$	6.09898e7	0.101803
$323$	−1.23526e9	−2.03963
$324$	3.08746e8	0.504305
$325$	−1.06594e8	−0.172242
$326$	3.69918e8	0.591349
$327$	−1.34835e9	−2.13248
$328$	7.43134e8	1.16281
$329$	5.19499e8	0.804264
$330$	0	0
$331$	−7.38027e8	−1.11860	−0.559299	−	0.828966i	$-0.688930\pi$
$331$	−7.38027e8	−1.11860	−0.559299	+	0.828966i	$0.688930\pi$
$332$	4.79930e8	0.719771
$333$	1.00812e9	1.49609
$334$	5.93879e7	0.0872138
$335$	3.23489e8	0.470113
$336$	1.68430e8	0.242232
$337$	5.51843e8	0.785436	0.392718	−	0.919659i	$-0.371535\pi$
$337$	5.51843e8	0.785436	0.392718	+	0.919659i	$0.371535\pi$
$338$	9.72530e7	0.136992
$339$	1.32975e9	1.85384
$340$	−4.16829e8	−0.575151
$341$	0	0
$342$	−5.54754e8	−0.749910
$343$	7.84811e8	1.05011
$344$	−3.10005e8	−0.410595
$345$	1.42538e8	0.186880
$346$	−5.17820e8	−0.672067
$347$	1.26438e9	1.62452	0.812260	−	0.583295i	$-0.198237\pi$
$347$	1.26438e9	1.62452	0.812260	+	0.583295i	$0.198237\pi$
$348$	1.02241e9	1.30047
$349$	9.91561e8	1.24862	0.624311	−	0.781176i	$-0.285380\pi$
$349$	9.91561e8	1.24862	0.624311	+	0.781176i	$0.285380\pi$
$350$	5.85000e7	0.0729319
$351$	2.51186e8	0.310042
$352$	0	0
$353$	1.22157e9	1.47810	0.739052	−	0.673648i	$-0.235274\pi$
$353$	1.22157e9	1.47810	0.739052	+	0.673648i	$0.235274\pi$
$354$	−1.94302e8	−0.232791
$355$	3.87249e8	0.459400
$356$	8.35096e8	0.980983
$357$	−1.58323e9	−1.84164
$358$	−6.18347e8	−0.712265
$359$	1.27682e8	0.145646	0.0728230	−	0.997345i	$-0.476799\pi$
$359$	1.27682e8	0.145646	0.0728230	+	0.997345i	$0.476799\pi$
$360$	−4.47645e8	−0.505679
$361$	2.67575e8	0.299343
$362$	1.77151e8	0.196274
$363$	0	0
$364$	−3.91637e8	−0.425627
$365$	3.40303e8	0.366303
$366$	−4.54759e8	−0.484839
$367$	−9.76752e8	−1.03146	−0.515731	−	0.856751i	$-0.672480\pi$
$367$	−9.76752e8	−1.03146	−0.515731	+	0.856751i	$0.672480\pi$
$368$	6.28142e7	0.0657038
$369$	1.52736e9	1.58252
$370$	2.78692e8	0.286035
$371$	1.41654e8	0.144019
$372$	1.25335e9	1.26233
$373$	−7.12879e8	−0.711271	−0.355635	−	0.934625i	$-0.615735\pi$
$373$	−7.12879e8	−0.711271	−0.355635	+	0.934625i	$0.615735\pi$
$374$	0	0
$375$	1.36719e8	0.133881
$376$	−1.09894e9	−1.06615
$377$	−1.08306e9	−1.04102
$378$	−1.37854e8	−0.131280
$379$	9.14903e8	0.863252	0.431626	−	0.902053i	$-0.357940\pi$
$379$	9.14903e8	0.863252	0.431626	+	0.902053i	$0.357940\pi$
$380$	3.91920e8	0.366399
$381$	−4.12312e8	−0.381935
$382$	4.57452e8	0.419878
$383$	1.35613e9	1.23340	0.616701	−	0.787198i	$-0.288469\pi$
$383$	1.35613e9	1.23340	0.616701	+	0.787198i	$0.288469\pi$
$384$	−1.44440e9	−1.30175
$385$	0	0
$386$	−2.11125e8	−0.186846
$387$	−6.37153e8	−0.558799
$388$	−1.35560e9	−1.17820
$389$	1.89773e9	1.63460	0.817300	−	0.576212i	$-0.195470\pi$
$389$	1.89773e9	1.63460	0.817300	+	0.576212i	$0.195470\pi$
$390$	3.58155e8	0.305735
$391$	−5.90447e8	−0.499531
$392$	−5.73100e8	−0.480540
$393$	1.45459e8	0.120883
$394$	1.16033e9	0.955747
$395$	2.64525e7	0.0215962
$396$	0	0
$397$	−5.33541e8	−0.427958	−0.213979	−	0.976838i	$-0.568642\pi$
$397$	−5.33541e8	−0.427958	−0.213979	+	0.976838i	$0.568642\pi$
$398$	−3.17756e8	−0.252641
$399$	1.48861e9	1.17321
$400$	6.02500e7	0.0470703
$401$	−2.18925e9	−1.69547	−0.847734	−	0.530421i	$-0.822034\pi$
$401$	−2.18925e9	−1.69547	−0.847734	+	0.530421i	$0.822034\pi$
$402$	−1.08692e9	−0.834463
$403$	−1.32770e9	−1.01049
$404$	−7.67230e8	−0.578883
$405$	4.19491e8	0.313784
$406$	5.94397e8	0.440794
$407$	0	0
$408$	3.34913e9	2.44130
$409$	1.68692e9	1.21916	0.609582	−	0.792723i	$-0.291337\pi$
$409$	1.68692e9	1.21916	0.609582	+	0.792723i	$0.291337\pi$
$410$	4.22235e8	0.302559
$411$	−1.60603e9	−1.14106
$412$	−1.18366e9	−0.833850
$413$	2.88677e8	0.201645
$414$	−2.65169e8	−0.183663
$415$	6.52078e8	0.447849
$416$	1.31048e9	0.892490
$417$	3.12097e9	2.10772
$418$	0	0
$419$	−2.66666e9	−1.77100	−0.885500	−	0.464638i	$-0.846184\pi$
$419$	−2.66666e9	−1.77100	−0.885500	+	0.464638i	$0.846184\pi$
$420$	5.02320e8	0.330832
$421$	1.30931e9	0.855175	0.427588	−	0.903974i	$-0.359364\pi$
$421$	1.30931e9	0.855175	0.427588	+	0.903974i	$0.359364\pi$
$422$	1.28540e9	0.832617
$423$	−2.25865e9	−1.45097
$424$	−2.99653e8	−0.190914
$425$	−5.66344e8	−0.357865
$426$	−1.30116e9	−0.815447
$427$	6.75642e8	0.419971
$428$	1.70716e9	1.05249
$429$	0	0
$430$	−1.76139e8	−0.106836
$431$	4.65959e8	0.280335	0.140168	−	0.990128i	$-0.455236\pi$
$431$	4.65959e8	0.280335	0.140168	+	0.990128i	$0.455236\pi$
$432$	−1.41978e8	−0.0847281
$433$	−1.95790e9	−1.15900	−0.579498	−	0.814974i	$-0.696751\pi$
$433$	−1.95790e9	−1.15900	−0.579498	+	0.814974i	$0.696751\pi$
$434$	7.28657e8	0.427867
$435$	1.38915e9	0.809164
$436$	−1.77211e9	−1.02397
$437$	5.55163e8	0.318226
$438$	−1.14342e9	−0.650198
$439$	−7.52365e7	−0.0424427	−0.0212213	−	0.999775i	$-0.506755\pi$
$439$	−7.52365e7	−0.0424427	−0.0212213	+	0.999775i	$0.506755\pi$
$440$	0	0
$441$	−1.17790e9	−0.653991
$442$	−1.48362e9	−0.817232
$443$	1.55425e9	0.849390	0.424695	−	0.905337i	$-0.360381\pi$
$443$	1.55425e9	0.849390	0.424695	+	0.905337i	$0.360381\pi$
$444$	2.39304e9	1.29751
$445$	1.13464e9	0.610378
$446$	−6.09767e8	−0.325456
$447$	3.59763e9	1.90520
$448$	−4.11221e8	−0.216074
$449$	6.00322e8	0.312984	0.156492	−	0.987679i	$-0.449981\pi$
$449$	6.00322e8	0.312984	0.156492	+	0.987679i	$0.449981\pi$
$450$	−2.54344e8	−0.131576
$451$	0	0
$452$	1.74767e9	0.890176
$453$	−2.05454e9	−1.03841
$454$	5.44893e8	0.273285
$455$	−5.32116e8	−0.264830
$456$	−3.14899e9	−1.55523
$457$	−3.63508e9	−1.78159	−0.890794	−	0.454408i	$-0.849851\pi$
$457$	−3.63508e9	−1.78159	−0.890794	+	0.454408i	$0.849851\pi$
$458$	9.63923e8	0.468827
$459$	1.33458e9	0.644168
$460$	1.87335e8	0.0897360
$461$	−1.64873e9	−0.783786	−0.391893	−	0.920011i	$-0.628180\pi$
$461$	−1.64873e9	−0.783786	−0.391893	+	0.920011i	$0.628180\pi$
$462$	0	0
$463$	3.65413e9	1.71100	0.855502	−	0.517800i	$-0.173249\pi$
$463$	3.65413e9	1.71100	0.855502	+	0.517800i	$0.173249\pi$
$464$	6.12179e8	0.284489
$465$	1.70292e9	0.785435
$466$	−2.01976e9	−0.924590
$467$	2.68522e9	1.22003	0.610015	−	0.792390i	$-0.291163\pi$
$467$	2.68522e9	1.22003	0.610015	+	0.792390i	$0.291163\pi$
$468$	1.70274e9	0.767872
$469$	1.61486e9	0.722818
$470$	−6.24398e8	−0.277408
$471$	−2.47054e9	−1.08948
$472$	−6.10664e8	−0.267304
$473$	0	0
$474$	−8.88804e7	−0.0383338
$475$	5.32500e8	0.227977
$476$	−2.08081e9	−0.884317
$477$	−6.15878e8	−0.259825
$478$	2.00258e8	0.0838670
$479$	2.00478e9	0.833473	0.416736	−	0.909027i	$-0.363174\pi$
$479$	2.00478e9	0.833473	0.416736	+	0.909027i	$0.363174\pi$
$480$	−1.68084e9	−0.693717
$481$	−2.53499e9	−1.03865
$482$	9.31525e8	0.378905
$483$	7.11547e8	0.287335
$484$	0	0
$485$	−1.84185e9	−0.733091
$486$	−1.89264e9	−0.747897
$487$	2.16023e9	0.847516	0.423758	−	0.905776i	$-0.360711\pi$
$487$	2.16023e9	0.847516	0.423758	+	0.905776i	$0.360711\pi$
$488$	−1.42924e9	−0.556720
$489$	4.31571e9	1.66906
$490$	−3.25625e8	−0.125035
$491$	−2.65170e9	−1.01097	−0.505486	−	0.862835i	$-0.668687\pi$
$491$	−2.65170e9	−1.01097	−0.505486	+	0.862835i	$0.668687\pi$
$492$	3.62559e9	1.37246
$493$	−5.75441e9	−2.16290
$494$	1.39496e9	0.520617
$495$	0	0
$496$	7.50455e8	0.276146
$497$	1.93315e9	0.706346
$498$	−2.19098e9	−0.794944
$499$	2.31290e9	0.833306	0.416653	−	0.909066i	$-0.363203\pi$
$499$	2.31290e9	0.833306	0.416653	+	0.909066i	$0.363203\pi$
$500$	1.79688e8	0.0642870
$501$	6.92859e8	0.246157
$502$	−8.97619e8	−0.316686
$503$	8.00265e8	0.280379	0.140190	−	0.990125i	$-0.455229\pi$
$503$	8.00265e8	0.280379	0.140190	+	0.990125i	$0.455229\pi$
$504$	−2.23464e9	−0.777502
$505$	−1.04243e9	−0.360187
$506$	0	0
$507$	1.13462e9	0.386653
$508$	−5.41896e8	−0.183397
$509$	4.23695e9	1.42410	0.712050	−	0.702128i	$-0.247767\pi$
$509$	4.23695e9	1.42410	0.712050	+	0.702128i	$0.247767\pi$
$510$	1.90292e9	0.635219
$511$	1.69879e9	0.563205
$512$	−1.39223e9	−0.458423
$513$	−1.25483e9	−0.410367
$514$	6.22639e8	0.202239
$515$	−1.60824e9	−0.518830
$516$	−1.51245e9	−0.484625
$517$	0	0
$518$	1.39123e9	0.439790
$519$	−6.04124e9	−1.89688
$520$	1.12563e9	0.351062
$521$	−1.37846e9	−0.427033	−0.213516	−	0.976940i	$-0.568492\pi$
$521$	−1.37846e9	−0.427033	−0.213516	+	0.976940i	$0.568492\pi$
$522$	−2.58430e9	−0.795235
$523$	−3.06487e9	−0.936820	−0.468410	−	0.883511i	$-0.655173\pi$
$523$	−3.06487e9	−0.936820	−0.468410	+	0.883511i	$0.655173\pi$
$524$	1.91174e8	0.0580457
$525$	6.82500e8	0.205847
$526$	−2.00198e9	−0.599806
$527$	−7.05420e9	−2.09947
$528$	0	0
$529$	−3.13946e9	−0.922062
$530$	−1.70258e8	−0.0496753
$531$	−1.25510e9	−0.363787
$532$	1.95646e9	0.563354
$533$	−3.84065e9	−1.09865
$534$	−3.81239e9	−1.08344
$535$	2.31951e9	0.654873
$536$	−3.41604e9	−0.958178
$537$	−7.21405e9	−2.01034
$538$	2.12805e9	0.589175
$539$	0	0
$540$	−4.23430e8	−0.115719
$541$	6.54325e9	1.77665	0.888327	−	0.459211i	$-0.151868\pi$
$541$	6.54325e9	1.77665	0.888327	+	0.459211i	$0.151868\pi$
$542$	−1.17121e9	−0.315965
$543$	2.06676e9	0.553975
$544$	6.96271e9	1.85431
$545$	−2.40776e9	−0.637127
$546$	1.78791e9	0.470080
$547$	4.84121e9	1.26473	0.632366	−	0.774670i	$-0.282084\pi$
$547$	4.84121e9	1.26473	0.632366	+	0.774670i	$0.282084\pi$
$548$	−2.11078e9	−0.547913
$549$	−2.93753e9	−0.757668
$550$	0	0
$551$	5.41054e9	1.37788
$552$	−1.50520e9	−0.380896
$553$	1.32051e8	0.0332050
$554$	3.85228e9	0.962572
$555$	3.25141e9	0.807322
$556$	4.10184e9	1.01208
$557$	−3.28124e9	−0.804536	−0.402268	−	0.915522i	$-0.631778\pi$
$557$	−3.28124e9	−0.804536	−0.402268	+	0.915522i	$0.631778\pi$
$558$	−3.16802e9	−0.771914
$559$	1.60216e9	0.387940
$560$	3.00768e8	0.0723725
$561$	0	0
$562$	3.96735e8	0.0942809
$563$	−6.42362e8	−0.151705	−0.0758526	−	0.997119i	$-0.524168\pi$
$563$	−6.42362e8	−0.151705	−0.0758526	+	0.997119i	$0.524168\pi$
$564$	−5.36149e9	−1.25837
$565$	2.37456e9	0.553876
$566$	3.15040e9	0.730311
$567$	2.09410e9	0.482455
$568$	−4.08935e9	−0.936343
$569$	−5.64269e9	−1.28408	−0.642042	−	0.766670i	$-0.721912\pi$
$569$	−5.64269e9	−1.28408	−0.642042	+	0.766670i	$0.721912\pi$
$570$	−1.78920e9	−0.404666
$571$	−9.20027e8	−0.206811	−0.103406	−	0.994639i	$-0.532974\pi$
$571$	−9.20027e8	−0.206811	−0.103406	+	0.994639i	$0.532974\pi$
$572$	0	0
$573$	5.33694e9	1.18509
$574$	2.10780e9	0.465197
$575$	2.54531e8	0.0558346
$576$	1.78789e9	0.389818
$577$	1.60588e9	0.348015	0.174008	−	0.984744i	$-0.444328\pi$
$577$	1.60588e9	0.348015	0.174008	+	0.984744i	$0.444328\pi$
$578$	−5.42060e9	−1.16762
$579$	−2.46313e9	−0.527366
$580$	1.82574e9	0.388545
$581$	3.25518e9	0.688586
$582$	6.18861e9	1.30126
$583$	0	0
$584$	−3.59360e9	−0.746594
$585$	2.31351e9	0.477778
$586$	3.12802e8	0.0642138
$587$	−6.51895e9	−1.33028	−0.665141	−	0.746717i	$-0.731629\pi$
$587$	−6.51895e9	−1.33028	−0.665141	+	0.746717i	$0.731629\pi$
$588$	−2.79604e9	−0.567181
$589$	6.63265e9	1.33747
$590$	−3.46968e8	−0.0695516
$591$	1.35371e10	2.69756
$592$	1.43285e9	0.283841
$593$	−9.40228e9	−1.85158	−0.925788	−	0.378042i	$-0.876597\pi$
$593$	−9.40228e9	−1.85158	−0.925788	+	0.378042i	$0.876597\pi$
$594$	0	0
$595$	−2.82719e9	−0.550231
$596$	4.72832e9	0.914839
$597$	−3.70716e9	−0.713067
$598$	6.66782e8	0.127506
$599$	8.71970e9	1.65771	0.828854	−	0.559466i	$-0.188994\pi$
$599$	8.71970e9	1.65771	0.828854	+	0.559466i	$0.188994\pi$
$600$	−1.44375e9	−0.272874
$601$	−6.30737e9	−1.18519	−0.592594	−	0.805501i	$-0.701896\pi$
$601$	−6.30737e9	−1.18519	−0.592594	+	0.805501i	$0.701896\pi$
$602$	−8.79286e8	−0.164264
$603$	−7.02100e9	−1.30403
$604$	−2.70025e9	−0.498626
$605$	0	0
$606$	3.50257e9	0.639342
$607$	−6.39261e9	−1.16016	−0.580080	−	0.814560i	$-0.696979\pi$
$607$	−6.39261e9	−1.16016	−0.580080	+	0.814560i	$0.696979\pi$
$608$	−6.54663e9	−1.18129
$609$	6.93464e9	1.24412
$610$	−8.12070e8	−0.144857
$611$	5.67952e9	1.00732
$612$	9.04686e9	1.59539
$613$	4.49574e8	0.0788296	0.0394148	−	0.999223i	$-0.487451\pi$
$613$	4.49574e8	0.0788296	0.0394148	+	0.999223i	$0.487451\pi$
$614$	4.96626e9	0.865845
$615$	4.92608e9	0.853961
$616$	0	0
$617$	−2.90693e9	−0.498237	−0.249119	−	0.968473i	$-0.580141\pi$
$617$	−2.90693e9	−0.498237	−0.249119	+	0.968473i	$0.580141\pi$
$618$	5.40368e9	0.920937
$619$	7.09482e9	1.20233	0.601166	−	0.799125i	$-0.294703\pi$
$619$	7.09482e9	1.20233	0.601166	+	0.799125i	$0.294703\pi$
$620$	2.23813e9	0.377150
$621$	−5.99798e8	−0.100504
$622$	1.60492e9	0.267416
$623$	5.66413e9	0.938480
$624$	1.84139e9	0.303390
$625$	2.44141e8	0.0400000
$626$	4.89324e9	0.797235
$627$	0	0
$628$	−3.24700e9	−0.523146
$629$	−1.34687e10	−2.15798
$630$	−1.26968e9	−0.202304
$631$	−1.25767e10	−1.99279	−0.996397	−	0.0848059i	$-0.972973\pi$
$631$	−1.25767e10	−1.99279	−0.996397	+	0.0848059i	$0.972973\pi$
$632$	−2.79338e8	−0.0440170
$633$	1.49964e10	2.35003
$634$	−4.63785e8	−0.0722777
$635$	−7.36272e8	−0.114112
$636$	−1.46194e9	−0.225336
$637$	2.96189e9	0.454026
$638$	0	0
$639$	−8.40485e9	−1.27432
$640$	−2.57928e9	−0.388927
$641$	−7.07538e9	−1.06108	−0.530538	−	0.847661i	$-0.678010\pi$
$641$	−7.07538e9	−1.06108	−0.530538	+	0.847661i	$0.678010\pi$
$642$	−7.79354e9	−1.16242
$643$	−2.32114e9	−0.344320	−0.172160	−	0.985069i	$-0.555075\pi$
$643$	−2.32114e9	−0.344320	−0.172160	+	0.985069i	$0.555075\pi$
$644$	9.35176e8	0.137973
$645$	−2.05496e9	−0.301539
$646$	7.41158e9	1.08168
$647$	−2.62294e9	−0.380736	−0.190368	−	0.981713i	$-0.560968\pi$
$647$	−2.62294e9	−0.380736	−0.190368	+	0.981713i	$0.560968\pi$
$648$	−4.42983e9	−0.639550
$649$	0	0
$650$	6.39562e8	0.0913453
$651$	8.50100e9	1.20764
$652$	5.67207e9	0.801448
$653$	−9.32088e9	−1.30997	−0.654984	−	0.755642i	$-0.727325\pi$
$653$	−9.32088e9	−1.30997	−0.654984	+	0.755642i	$0.727325\pi$
$654$	8.09008e9	1.13092
$655$	2.59747e8	0.0361166
$656$	2.17085e9	0.300239
$657$	−7.38593e9	−1.01608
$658$	−3.11699e9	−0.426526
$659$	3.94097e9	0.536420	0.268210	−	0.963361i	$-0.413568\pi$
$659$	3.94097e9	0.536420	0.268210	+	0.963361i	$0.413568\pi$
$660$	0	0
$661$	6.40869e9	0.863105	0.431553	−	0.902088i	$-0.357966\pi$
$661$	6.40869e9	0.863105	0.431553	+	0.902088i	$0.357966\pi$
$662$	4.42816e9	0.593226
$663$	−1.73089e10	−2.30660
$664$	−6.88595e9	−0.912800
$665$	2.65824e9	0.350525
$666$	−6.04874e9	−0.793424
$667$	2.58620e9	0.337460
$668$	9.10615e8	0.118200
$669$	−7.11395e9	−0.918585
$670$	−1.94093e9	−0.249315
$671$	0	0
$672$	−8.39075e9	−1.06662
$673$	−5.60126e9	−0.708326	−0.354163	−	0.935184i	$-0.615234\pi$
$673$	−5.60126e9	−0.708326	−0.354163	+	0.935184i	$0.615234\pi$
$674$	−3.31106e9	−0.416541
$675$	−5.75312e8	−0.0720013
$676$	1.49121e9	0.185663
$677$	−7.64936e9	−0.947468	−0.473734	−	0.880668i	$-0.657094\pi$
$677$	−7.64936e9	−0.947468	−0.473734	+	0.880668i	$0.657094\pi$
$678$	−7.97851e9	−0.983146
$679$	−9.19451e9	−1.12716
$680$	5.98059e9	0.729395
$681$	6.35709e9	0.771336
$682$	0	0
$683$	8.67638e8	0.104200	0.0520998	−	0.998642i	$-0.483409\pi$
$683$	8.67638e8	0.104200	0.0520998	+	0.998642i	$0.483409\pi$
$684$	−8.50623e9	−1.01634
$685$	−2.86791e9	−0.340917
$686$	−4.70887e9	−0.556906
$687$	1.12458e10	1.32325
$688$	−9.05589e8	−0.106016
$689$	1.54866e9	0.180380
$690$	−8.55225e8	−0.0991080
$691$	4.46630e9	0.514962	0.257481	−	0.966283i	$-0.417108\pi$
$691$	4.46630e9	0.514962	0.257481	+	0.966283i	$0.417108\pi$
$692$	−7.93991e9	−0.910844
$693$	0	0
$694$	−7.58629e9	−0.861532
$695$	5.57315e9	0.629730
$696$	−1.46694e10	−1.64923
$697$	−2.04058e10	−2.28264
$698$	−5.94937e9	−0.662181
$699$	−2.35639e10	−2.60962
$700$	8.97000e8	0.0988438
$701$	−6.65205e9	−0.729361	−0.364680	−	0.931133i	$-0.618822\pi$
$701$	−6.65205e9	−0.729361	−0.364680	+	0.931133i	$0.618822\pi$
$702$	−1.50712e9	−0.164425
$703$	1.26638e10	1.37474
$704$	0	0
$705$	−7.28464e9	−0.782972
$706$	−7.32939e9	−0.783883
$707$	−5.20382e9	−0.553802
$708$	−2.97930e9	−0.315499
$709$	−1.98561e9	−0.209234	−0.104617	−	0.994513i	$-0.533362\pi$
$709$	−1.98561e9	−0.209234	−0.104617	+	0.994513i	$0.533362\pi$
$710$	−2.32349e9	−0.243634
$711$	−5.74125e8	−0.0599050
$712$	−1.19818e10	−1.24406
$713$	3.17036e9	0.327563
$714$	9.49935e9	0.976675
$715$	0	0
$716$	−9.48132e9	−0.965325
$717$	2.33634e9	0.236711
$718$	−7.66091e8	−0.0772405
$719$	3.05968e9	0.306991	0.153495	−	0.988149i	$-0.450947\pi$
$719$	3.05968e9	0.306991	0.153495	+	0.988149i	$0.450947\pi$
$720$	−1.30767e9	−0.130567
$721$	−8.02833e9	−0.797722
$722$	−1.60545e9	−0.158751
$723$	1.08678e10	1.06944
$724$	2.71631e9	0.266008
$725$	2.48062e9	0.241756
$726$	0	0
$727$	−4.91772e9	−0.474672	−0.237336	−	0.971428i	$-0.576274\pi$
$727$	−4.91772e9	−0.474672	−0.237336	+	0.971428i	$0.576274\pi$
$728$	5.61914e9	0.539772
$729$	−1.47414e10	−1.40927
$730$	−2.04182e9	−0.194261
$731$	8.51245e9	0.806016
$732$	−6.97297e9	−0.657097
$733$	1.05544e10	0.989852	0.494926	−	0.868935i	$-0.335195\pi$
$733$	1.05544e10	0.989852	0.494926	+	0.868935i	$0.335195\pi$
$734$	5.86051e9	0.547015
$735$	−3.79896e9	−0.352906
$736$	−3.12924e9	−0.289312
$737$	0	0
$738$	−9.16419e9	−0.839260
$739$	1.97875e10	1.80358	0.901789	−	0.432176i	$-0.142254\pi$
$739$	1.97875e10	1.80358	0.901789	+	0.432176i	$0.142254\pi$
$740$	4.27328e9	0.387660
$741$	1.62746e10	1.46942
$742$	−8.49925e8	−0.0763778
$743$	1.23627e10	1.10574	0.552868	−	0.833269i	$-0.313534\pi$
$743$	1.23627e10	1.10574	0.552868	+	0.833269i	$0.313534\pi$
$744$	−1.79829e10	−1.60086
$745$	6.42434e9	0.569222
$746$	4.27727e9	0.377208
$747$	−1.41527e10	−1.24227
$748$	0	0
$749$	1.15790e10	1.00689
$750$	−8.20312e8	−0.0710011
$751$	−4.30863e9	−0.371193	−0.185596	−	0.982626i	$-0.559422\pi$
$751$	−4.30863e9	−0.371193	−0.185596	+	0.982626i	$0.559422\pi$
$752$	−3.21024e9	−0.275280
$753$	−1.04722e10	−0.893833
$754$	6.49836e9	0.552083
$755$	−3.66882e9	−0.310250
$756$	−2.11376e9	−0.177922
$757$	−5.39714e9	−0.452197	−0.226099	−	0.974104i	$-0.572597\pi$
$757$	−5.39714e9	−0.452197	−0.226099	+	0.974104i	$0.572597\pi$
$758$	−5.48942e9	−0.457809
$759$	0	0
$760$	−5.62320e9	−0.464661
$761$	−1.12446e10	−0.924903	−0.462451	−	0.886645i	$-0.653030\pi$
$761$	−1.12446e10	−0.924903	−0.462451	+	0.886645i	$0.653030\pi$
$762$	2.47387e9	0.202551
$763$	−1.20196e10	−0.979608
$764$	7.01426e9	0.569056
$765$	1.22919e10	0.992670
$766$	−8.13676e9	−0.654110
$767$	3.15602e9	0.252555
$768$	1.45711e10	1.16072
$769$	−1.82198e10	−1.44478	−0.722391	−	0.691485i	$-0.756957\pi$
$769$	−1.82198e10	−1.44478	−0.722391	+	0.691485i	$0.756957\pi$
$770$	0	0
$771$	7.26412e9	0.570812
$772$	−3.23725e9	−0.253230
$773$	−1.96076e10	−1.52685	−0.763425	−	0.645896i	$-0.776484\pi$
$773$	−1.96076e10	−1.52685	−0.763425	+	0.645896i	$0.776484\pi$
$774$	3.82292e9	0.296348
$775$	3.04094e9	0.234667
$776$	1.94499e10	1.49418
$777$	1.62311e10	1.24129
$778$	−1.13864e10	−0.866878
$779$	1.91864e10	1.45416
$780$	5.49171e9	0.414359
$781$	0	0
$782$	3.54268e9	0.264916
$783$	−5.84554e9	−0.435170
$784$	−1.67415e9	−0.124076
$785$	−4.41168e9	−0.325507
$786$	−8.72752e8	−0.0641079
$787$	−5.46749e9	−0.399831	−0.199916	−	0.979813i	$-0.564067\pi$
$787$	−5.46749e9	−0.399831	−0.199916	+	0.979813i	$0.564067\pi$
$788$	1.77917e10	1.29531
$789$	−2.33565e10	−1.69293
$790$	−1.58715e8	−0.0114531
$791$	1.18538e10	0.851607
$792$	0	0
$793$	7.38659e9	0.526002
$794$	3.20125e9	0.226959
$795$	−1.98634e9	−0.140206
$796$	−4.87226e9	−0.342401
$797$	−6.29413e9	−0.440384	−0.220192	−	0.975457i	$-0.570668\pi$
$797$	−6.29413e9	−0.440384	−0.220192	+	0.975457i	$0.570668\pi$
$798$	−8.93169e9	−0.622190
$799$	3.01759e10	2.09289
$800$	−3.00150e9	−0.207264
$801$	−2.46263e10	−1.69311
$802$	1.31355e10	0.899158
$803$	0	0
$804$	−1.66661e10	−1.13094
$805$	1.27062e9	0.0858480
$806$	7.96619e9	0.535893
$807$	2.48273e10	1.66292
$808$	1.10081e10	0.734129
$809$	−1.72859e10	−1.14782	−0.573909	−	0.818919i	$-0.694574\pi$
$809$	−1.72859e10	−1.14782	−0.573909	+	0.818919i	$0.694574\pi$
$810$	−2.51695e9	−0.166409
$811$	−2.36949e10	−1.55985	−0.779923	−	0.625876i	$-0.784742\pi$
$811$	−2.36949e10	−1.55985	−0.779923	+	0.625876i	$0.784742\pi$
$812$	9.11409e9	0.597403
$813$	−1.36642e10	−0.891797
$814$	0	0
$815$	7.70662e9	0.498669
$816$	9.78352e9	0.630347
$817$	−8.00376e9	−0.513472
$818$	−1.01215e10	−0.646559
$819$	1.15490e10	0.734603
$820$	6.47427e9	0.410055
$821$	7.83177e9	0.493923	0.246961	−	0.969025i	$-0.420568\pi$
$821$	7.83177e9	0.493923	0.246961	+	0.969025i	$0.420568\pi$
$822$	9.63619e9	0.605137
$823$	1.70184e10	1.06419	0.532094	−	0.846685i	$-0.321405\pi$
$823$	1.70184e10	1.06419	0.532094	+	0.846685i	$0.321405\pi$
$824$	1.69830e10	1.05747
$825$	0	0
$826$	−1.73206e9	−0.106938
$827$	−2.06817e10	−1.27150	−0.635752	−	0.771893i	$-0.719310\pi$
$827$	−2.06817e10	−1.27150	−0.635752	+	0.771893i	$0.719310\pi$
$828$	−4.06592e9	−0.248916
$829$	2.02037e10	1.23166	0.615829	−	0.787880i	$-0.288821\pi$
$829$	2.02037e10	1.23166	0.615829	+	0.787880i	$0.288821\pi$
$830$	−3.91247e9	−0.237508
$831$	4.49432e10	2.71682
$832$	−4.49575e9	−0.270627
$833$	1.57368e10	0.943321
$834$	−1.87258e10	−1.11779
$835$	1.23725e9	0.0735452
$836$	0	0
$837$	−7.16591e9	−0.422408
$838$	1.60000e10	0.939215
$839$	−8.30945e9	−0.485742	−0.242871	−	0.970059i	$-0.578089\pi$
$839$	−8.30945e9	−0.485742	−0.242871	+	0.970059i	$0.578089\pi$
$840$	−7.20720e9	−0.419555
$841$	7.95486e9	0.461155
$842$	−7.85586e9	−0.453525
$843$	4.62858e9	0.266104
$844$	1.97095e10	1.12844
$845$	2.02610e9	0.115522
$846$	1.35519e10	0.769493
$847$	0	0
$848$	−8.75351e8	−0.0492943
$849$	3.67546e10	2.06127
$850$	3.39806e9	0.189786
$851$	6.05320e9	0.336691
$852$	−1.99511e10	−1.10517
$853$	−2.49408e10	−1.37590	−0.687952	−	0.725756i	$-0.741490\pi$
$853$	−2.49408e10	−1.37590	−0.687952	+	0.725756i	$0.741490\pi$
$854$	−4.05385e9	−0.222723
$855$	−1.15574e10	−0.632380
$856$	−2.44940e10	−1.33475
$857$	−1.90455e10	−1.03362	−0.516808	−	0.856101i	$-0.672880\pi$
$857$	−1.90455e10	−1.03362	−0.516808	+	0.856101i	$0.672880\pi$
$858$	0	0
$859$	2.41484e10	1.29991	0.649953	−	0.759975i	$-0.274789\pi$
$859$	2.41484e10	1.29991	0.649953	+	0.759975i	$0.274789\pi$
$860$	−2.70080e9	−0.144793
$861$	2.45910e10	1.31300
$862$	−2.79576e9	−0.148670
$863$	3.30113e10	1.74833	0.874167	−	0.485626i	$-0.161408\pi$
$863$	3.30113e10	1.74833	0.874167	+	0.485626i	$0.161408\pi$
$864$	7.07297e9	0.373082
$865$	−1.07879e10	−0.566737
$866$	1.17474e10	0.614651
$867$	−6.32404e10	−3.29555
$868$	1.11727e10	0.579884
$869$	0	0
$870$	−8.33490e9	−0.429124
$871$	1.76547e10	0.905310
$872$	2.54260e10	1.29858
$873$	3.99755e10	2.03350
$874$	−3.33098e9	−0.168765
$875$	1.21875e9	0.0615016
$876$	−1.75324e10	−0.881205
$877$	−7.62731e8	−0.0381832	−0.0190916	−	0.999818i	$-0.506077\pi$
$877$	−7.62731e8	−0.0381832	−0.0190916	+	0.999818i	$0.506077\pi$
$878$	4.51419e8	0.0225086
$879$	3.64936e9	0.181241
$880$	0	0
$881$	6.41792e9	0.316212	0.158106	−	0.987422i	$-0.449461\pi$
$881$	6.41792e9	0.316212	0.158106	+	0.987422i	$0.449461\pi$
$882$	7.06737e9	0.346831
$883$	1.80563e10	0.882605	0.441303	−	0.897358i	$-0.354517\pi$
$883$	1.80563e10	0.882605	0.441303	+	0.897358i	$0.354517\pi$
$884$	−2.27489e10	−1.10758
$885$	−4.04796e9	−0.196307
$886$	−9.32548e9	−0.450457
$887$	−1.50984e10	−0.726438	−0.363219	−	0.931704i	$-0.618322\pi$
$887$	−1.50984e10	−0.726438	−0.363219	+	0.931704i	$0.618322\pi$
$888$	−3.43349e10	−1.64547
$889$	−3.67547e9	−0.175451
$890$	−6.80785e9	−0.323702
$891$	0	0
$892$	−9.34976e9	−0.441086
$893$	−2.83726e10	−1.33327
$894$	−2.15858e10	−1.01038
$895$	−1.28822e10	−0.600635
$896$	−1.28758e10	−0.597992
$897$	7.77913e9	0.359880
$898$	−3.60193e9	−0.165985
$899$	3.08979e10	1.41831
$900$	−3.89994e9	−0.178324
$901$	8.22820e9	0.374773
$902$	0	0
$903$	−1.02583e10	−0.463628
$904$	−2.50753e10	−1.12890
$905$	3.69064e9	0.165513
$906$	1.23272e10	0.550702
$907$	2.30530e10	1.02589	0.512946	−	0.858421i	$-0.328554\pi$
$907$	2.30530e10	1.02589	0.512946	+	0.858421i	$0.328554\pi$
$908$	8.35503e9	0.370380
$909$	2.26250e10	0.999112
$910$	3.19270e9	0.140447
$911$	−2.94065e10	−1.28863	−0.644315	−	0.764760i	$-0.722858\pi$
$911$	−2.94065e10	−1.28863	−0.644315	+	0.764760i	$0.722858\pi$
$912$	−9.19887e9	−0.401562
$913$	0	0
$914$	2.18105e10	0.944829
$915$	−9.47415e9	−0.408852
$916$	1.47801e10	0.635396
$917$	1.29666e9	0.0555307
$918$	−8.00747e9	−0.341622
$919$	4.12092e10	1.75142	0.875708	−	0.482841i	$-0.160395\pi$
$919$	4.12092e10	1.75142	0.875708	+	0.482841i	$0.160395\pi$
$920$	−2.68785e9	−0.113801
$921$	5.79397e10	2.44381
$922$	9.89241e9	0.415665
$923$	2.11345e10	0.884680
$924$	0	0
$925$	5.80609e9	0.241206
$926$	−2.19248e10	−0.907397
$927$	3.49052e10	1.43917
$928$	−3.04972e10	−1.25268
$929$	−1.78647e10	−0.731038	−0.365519	−	0.930804i	$-0.619108\pi$
$929$	−1.78647e10	−0.731038	−0.365519	+	0.930804i	$0.619108\pi$
$930$	−1.02176e10	−0.416540
$931$	−1.47964e10	−0.600942
$932$	−3.09697e10	−1.25309
$933$	1.87241e10	0.754769
$934$	−1.61113e10	−0.647019
$935$	0	0
$936$	−2.44307e10	−0.973800
$937$	1.16972e10	0.464507	0.232253	−	0.972655i	$-0.425390\pi$
$937$	1.16972e10	0.464507	0.232253	+	0.972655i	$0.425390\pi$
$938$	−9.68914e9	−0.383332
$939$	5.70878e10	2.25016
$940$	−9.57410e9	−0.375967
$941$	−2.88582e9	−0.112903	−0.0564514	−	0.998405i	$-0.517979\pi$
$941$	−2.88582e9	−0.112903	−0.0564514	+	0.998405i	$0.517979\pi$
$942$	1.48232e10	0.577784
$943$	9.17094e9	0.356142
$944$	−1.78388e9	−0.0690181
$945$	−2.87196e9	−0.110705
$946$	0	0
$947$	2.38952e10	0.914295	0.457148	−	0.889391i	$-0.348871\pi$
$947$	2.38952e10	0.914295	0.457148	+	0.889391i	$0.348871\pi$
$948$	−1.36283e9	−0.0519533
$949$	1.85724e10	0.705400
$950$	−3.19500e9	−0.120903
$951$	−5.41082e9	−0.204001
$952$	2.98551e10	1.12147
$953$	−2.13647e10	−0.799599	−0.399800	−	0.916603i	$-0.630920\pi$
$953$	−2.13647e10	−0.799599	−0.399800	+	0.916603i	$0.630920\pi$
$954$	3.69527e9	0.137793
$955$	9.53024e9	0.354072
$956$	3.07062e9	0.113664
$957$	0	0
$958$	−1.20287e10	−0.442016
$959$	−1.43166e10	−0.524174
$960$	5.76632e9	0.210353
$961$	1.03643e10	0.376712
$962$	1.52099e10	0.550826
$963$	−5.03425e10	−1.81653
$964$	1.42834e10	0.513525
$965$	−4.39844e9	−0.157563
$966$	−4.26928e9	−0.152382
$967$	1.08933e10	0.387407	0.193704	−	0.981060i	$-0.437950\pi$
$967$	1.08933e10	0.387407	0.193704	+	0.981060i	$0.437950\pi$
$968$	0	0
$969$	8.64685e10	3.05298
$970$	1.10511e10	0.388780
$971$	2.81392e10	0.986381	0.493190	−	0.869921i	$-0.335831\pi$
$971$	2.81392e10	0.986381	0.493190	+	0.869921i	$0.335831\pi$
$972$	−2.90205e10	−1.01362
$973$	2.78212e10	0.968235
$974$	−1.29614e10	−0.449463
$975$	7.46156e9	0.257818
$976$	−4.17512e9	−0.143746
$977$	2.39654e9	0.0822156	0.0411078	−	0.999155i	$-0.486911\pi$
$977$	2.39654e9	0.0822156	0.0411078	+	0.999155i	$0.486911\pi$
$978$	−2.58942e10	−0.885151
$979$	0	0
$980$	−4.99292e9	−0.169459
$981$	5.22581e10	1.76731
$982$	1.59102e10	0.536149
$983$	5.23911e10	1.75922	0.879611	−	0.475694i	$-0.157803\pi$
$983$	5.23911e10	1.75922	0.879611	+	0.475694i	$0.157803\pi$
$984$	−5.20194e10	−1.74053
$985$	2.41734e10	0.805957
$986$	3.45265e10	1.14705
$987$	−3.63649e10	−1.20385
$988$	2.13894e10	0.705586
$989$	−3.82574e9	−0.125756
$990$	0	0
$991$	1.48174e10	0.483631	0.241816	−	0.970322i	$-0.422257\pi$
$991$	1.48174e10	0.483631	0.241816	+	0.970322i	$0.422257\pi$
$992$	−3.73857e10	−1.21595
$993$	5.16619e10	1.67436
$994$	−1.15989e10	−0.374597
$995$	−6.61992e9	−0.213045
$996$	−3.35951e10	−1.07738
$997$	2.74765e10	0.878069	0.439034	−	0.898470i	$-0.355321\pi$
$997$	2.74765e10	0.878069	0.439034	+	0.898470i	$0.355321\pi$
$998$	−1.38774e10	−0.441927
$999$	−1.36819e10	−0.434179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.8.a.a.1.1	✓	1
11.10	odd	2		605.8.a.b.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.8.a.a.1.1	✓	1	1.1	even	1	trivial
605.8.a.b.1.1	yes	1	11.10	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 $\F_{q}$
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