LMFDB - Embedded newform 2057.4.a.t.1.40

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2057,4,Mod(1,2057)] 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("2057.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2057, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2057 = 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2057.a (trivial)

Newform invariants

comment:select newform

sage:traces = [44,1,-28,139,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$121.366928882$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 187)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.40
Character	$\chi$	$=$	2057.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.73079 q^{2} -1.21482 q^{3} +14.3803 q^{4} -15.1061 q^{5} -5.74704 q^{6} +9.37827 q^{7} +30.1840 q^{8} -25.5242 q^{9} -71.4635 q^{10} -17.4695 q^{12} +74.3698 q^{13} +44.3666 q^{14} +18.3511 q^{15} +27.7515 q^{16} +17.0000 q^{17} -120.750 q^{18} -3.05838 q^{19} -217.230 q^{20} -11.3929 q^{21} -16.3508 q^{23} -36.6681 q^{24} +103.193 q^{25} +351.828 q^{26} +63.8073 q^{27} +134.863 q^{28} +148.700 q^{29} +86.8151 q^{30} -15.4541 q^{31} -110.186 q^{32} +80.4234 q^{34} -141.669 q^{35} -367.047 q^{36} -19.7730 q^{37} -14.4685 q^{38} -90.3458 q^{39} -455.961 q^{40} -255.122 q^{41} -53.8973 q^{42} -277.874 q^{43} +385.570 q^{45} -77.3520 q^{46} -149.172 q^{47} -33.7130 q^{48} -255.048 q^{49} +488.183 q^{50} -20.6519 q^{51} +1069.46 q^{52} +232.610 q^{53} +301.859 q^{54} +283.074 q^{56} +3.71537 q^{57} +703.470 q^{58} -341.102 q^{59} +263.895 q^{60} -703.020 q^{61} -73.1100 q^{62} -239.373 q^{63} -743.278 q^{64} -1123.43 q^{65} -115.186 q^{67} +244.466 q^{68} +19.8632 q^{69} -670.204 q^{70} -646.835 q^{71} -770.424 q^{72} -276.780 q^{73} -93.5418 q^{74} -125.360 q^{75} -43.9805 q^{76} -427.407 q^{78} -727.486 q^{79} -419.215 q^{80} +611.640 q^{81} -1206.93 q^{82} -1305.51 q^{83} -163.834 q^{84} -256.803 q^{85} -1314.56 q^{86} -180.644 q^{87} +1062.02 q^{89} +1824.05 q^{90} +697.460 q^{91} -235.130 q^{92} +18.7739 q^{93} -705.702 q^{94} +46.2000 q^{95} +133.856 q^{96} -1475.26 q^{97} -1206.58 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} - 24 q^{6} - 2 q^{7} - 63 q^{8} + 356 q^{9} - 65 q^{10} - 337 q^{12} + 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} + 748 q^{17} - 55 q^{18} - 36 q^{19}+ \cdots + 2302 q^{98}+O(q^{100}) $ 44 * q + q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 - 24 * q^6 - 2 * q^7 - 63 * q^8 + 356 * q^9 - 65 * q^10 - 337 * q^12 + 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 + 748 * q^17 - 55 * q^18 - 36 * q^19 - 209 * q^20 - 240 * q^21 - 656 * q^23 - 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 + 63 * q^28 - 698 * q^29 + 304 * q^30 - 1418 * q^31 - 83 * q^32 + 17 * q^34 + 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 + 114 * q^39 - 281 * q^40 + 54 * q^41 - 1943 * q^42 + 382 * q^43 - 1226 * q^45 + 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 + 1307 * q^50 - 476 * q^51 + 3112 * q^52 - 2492 * q^53 + 2400 * q^54 + 27 * q^56 + 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 - 964 * q^61 - 3256 * q^62 + 2322 * q^63 - 3653 * q^64 - 152 * q^65 - 3442 * q^67 + 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 - 1281 * q^72 - 2240 * q^73 + 3494 * q^74 - 1892 * q^75 + 1261 * q^76 + 1120 * q^78 - 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 - 1656 * q^83 - 5401 * q^84 - 408 * q^85 - 1373 * q^86 + 1968 * q^87 - 2992 * q^89 + 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 - 694 * q^94 - 8280 * q^95 + 2203 * q^96 - 4746 * q^97 + 2302 * q^98

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 $

$2$	4.73079	1.67259	0.836293	−	0.548283i	$-0.184718\pi$
$2$	4.73079	1.67259	0.836293	+	0.548283i	$0.184718\pi$
$3$	−1.21482	−0.233792	−0.116896	−	0.993144i	$-0.537294\pi$
$3$	−1.21482	−0.233792	−0.116896	+	0.993144i	$0.537294\pi$
$4$	14.3803	1.79754
$5$	−15.1061	−1.35113	−0.675563	−	0.737302i	$-0.736099\pi$
$5$	−15.1061	−1.35113	−0.675563	+	0.737302i	$0.736099\pi$
$6$	−5.74704	−0.391037
$7$	9.37827	0.506379	0.253189	−	0.967417i	$-0.418520\pi$
$7$	9.37827	0.506379	0.253189	+	0.967417i	$0.418520\pi$
$8$	30.1840	1.33396
$9$	−25.5242	−0.945341
$10$	−71.4635	−2.25987
$11$	0	0
$11$	0	0
$12$	−17.4695	−0.420251
$13$	74.3698	1.58665	0.793326	−	0.608797i	$-0.208347\pi$
$13$	74.3698	1.58665	0.793326	+	0.608797i	$0.208347\pi$
$14$	44.3666	0.846962
$15$	18.3511	0.315882
$16$	27.7515	0.433617
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−120.750	−1.58116
$19$	−3.05838	−0.0369284	−0.0184642	−	0.999830i	$-0.505878\pi$
$19$	−3.05838	−0.0369284	−0.0184642	+	0.999830i	$0.505878\pi$
$20$	−217.230	−2.42871
$21$	−11.3929	−0.118387
$22$	0	0
$23$	−16.3508	−0.148234	−0.0741168	−	0.997250i	$-0.523614\pi$
$23$	−16.3508	−0.148234	−0.0741168	+	0.997250i	$0.523614\pi$
$24$	−36.6681	−0.311868
$25$	103.193	0.825542
$26$	351.828	2.65381
$27$	63.8073	0.454805
$28$	134.863	0.910238
$29$	148.700	0.952172	0.476086	−	0.879399i	$-0.342055\pi$
$29$	148.700	0.952172	0.476086	+	0.879399i	$0.342055\pi$
$30$	86.8151	0.528340
$31$	−15.4541	−0.0895367	−0.0447683	−	0.998997i	$-0.514255\pi$
$31$	−15.4541	−0.0895367	−0.0447683	+	0.998997i	$0.514255\pi$
$32$	−110.186	−0.608697
$33$	0	0
$34$	80.4234	0.405662
$35$	−141.669	−0.684182
$36$	−367.047	−1.69929
$37$	−19.7730	−0.0878556	−0.0439278	−	0.999035i	$-0.513987\pi$
$37$	−19.7730	−0.0878556	−0.0439278	+	0.999035i	$0.513987\pi$
$38$	−14.4685	−0.0617660
$39$	−90.3458	−0.370946
$40$	−455.961	−1.80235
$41$	−255.122	−0.971791	−0.485896	−	0.874017i	$-0.661506\pi$
$41$	−255.122	−0.971791	−0.485896	+	0.874017i	$0.661506\pi$
$42$	−53.8973	−0.198013
$43$	−277.874	−0.985475	−0.492738	−	0.870178i	$-0.664004\pi$
$43$	−277.874	−0.985475	−0.492738	+	0.870178i	$0.664004\pi$
$44$	0	0
$45$	385.570	1.27728
$46$	−77.3520	−0.247933
$47$	−149.172	−0.462958	−0.231479	−	0.972840i	$-0.574356\pi$
$47$	−149.172	−0.462958	−0.231479	+	0.972840i	$0.574356\pi$
$48$	−33.7130	−0.101376
$49$	−255.048	−0.743580
$50$	488.183	1.38079
$51$	−20.6519	−0.0567028
$52$	1069.46	2.85208
$53$	232.610	0.602856	0.301428	−	0.953489i	$-0.402537\pi$
$53$	232.610	0.602856	0.301428	+	0.953489i	$0.402537\pi$
$54$	301.859	0.760700
$55$	0	0
$56$	283.074	0.675488
$57$	3.71537	0.00863356
$58$	703.470	1.59259
$59$	−341.102	−0.752673	−0.376337	−	0.926483i	$-0.622816\pi$
$59$	−341.102	−0.752673	−0.376337	+	0.926483i	$0.622816\pi$
$60$	263.895	0.567812
$61$	−703.020	−1.47561	−0.737807	−	0.675012i	$-0.764138\pi$
$61$	−703.020	−1.47561	−0.737807	+	0.675012i	$0.764138\pi$
$62$	−73.1100	−0.149758
$63$	−239.373	−0.478701
$64$	−743.278	−1.45171
$65$	−1123.43	−2.14377
$66$	0	0
$67$	−115.186	−0.210034	−0.105017	−	0.994470i	$-0.533490\pi$
$67$	−115.186	−0.210034	−0.105017	+	0.994470i	$0.533490\pi$
$68$	244.466	0.435968
$69$	19.8632	0.0346558
$70$	−670.204	−1.14435
$71$	−646.835	−1.08120	−0.540600	−	0.841280i	$-0.681803\pi$
$71$	−646.835	−1.08120	−0.540600	+	0.841280i	$0.681803\pi$
$72$	−770.424	−1.26105
$73$	−276.780	−0.443763	−0.221882	−	0.975074i	$-0.571220\pi$
$73$	−276.780	−0.443763	−0.221882	+	0.975074i	$0.571220\pi$
$74$	−93.5418	−0.146946
$75$	−125.360	−0.193005
$76$	−43.9805	−0.0663804
$77$	0	0
$78$	−427.407	−0.620439
$79$	−727.486	−1.03606	−0.518029	−	0.855363i	$-0.673334\pi$
$79$	−727.486	−1.03606	−0.518029	+	0.855363i	$0.673334\pi$
$80$	−419.215	−0.585871
$81$	611.640	0.839012
$82$	−1206.93	−1.62540
$83$	−1305.51	−1.72648	−0.863242	−	0.504790i	$-0.831570\pi$
$83$	−1305.51	−1.72648	−0.863242	+	0.504790i	$0.831570\pi$
$84$	−163.834	−0.212806
$85$	−256.803	−0.327696
$86$	−1314.56	−1.64829
$87$	−180.644	−0.222610
$88$	0	0
$89$	1062.02	1.26488	0.632438	−	0.774611i	$-0.282054\pi$
$89$	1062.02	1.26488	0.632438	+	0.774611i	$0.282054\pi$
$90$	1824.05	2.13635
$91$	697.460	0.803447
$92$	−235.130	−0.266456
$93$	18.7739	0.0209329
$94$	−705.702	−0.774337
$95$	46.2000	0.0498950
$96$	133.856	0.142308
$97$	−1475.26	−1.54423	−0.772113	−	0.635486i	$-0.780800\pi$
$97$	−1475.26	−1.54423	−0.772113	+	0.635486i	$0.780800\pi$
$98$	−1206.58	−1.24370
$99$	0	0
$100$	1483.95	1.48395
$101$	1411.38	1.39047	0.695234	−	0.718784i	$-0.255301\pi$
$101$	1411.38	1.39047	0.695234	+	0.718784i	$0.255301\pi$
$102$	−97.6997	−0.0948403
$103$	−97.4268	−0.0932014	−0.0466007	−	0.998914i	$-0.514839\pi$
$103$	−97.4268	−0.0932014	−0.0466007	+	0.998914i	$0.514839\pi$
$104$	2244.78	2.11653
$105$	172.102	0.159956
$106$	1100.43	1.00833
$107$	−762.495	−0.688909	−0.344454	−	0.938803i	$-0.611936\pi$
$107$	−762.495	−0.688909	−0.344454	+	0.938803i	$0.611936\pi$
$108$	917.571	0.817531
$109$	1551.22	1.36312	0.681558	−	0.731764i	$-0.261303\pi$
$109$	1551.22	1.36312	0.681558	+	0.731764i	$0.261303\pi$
$110$	0	0
$111$	24.0206	0.0205399
$112$	260.261	0.219574
$113$	−754.694	−0.628280	−0.314140	−	0.949377i	$-0.601716\pi$
$113$	−754.694	−0.628280	−0.314140	+	0.949377i	$0.601716\pi$
$114$	17.5766	0.0144404
$115$	246.996	0.200282
$116$	2138.36	1.71157
$117$	−1898.23	−1.49993
$118$	−1613.68	−1.25891
$119$	159.431	0.122815
$120$	553.910	0.421374
$121$	0	0
$122$	−3325.84	−2.46809
$123$	309.927	0.227197
$124$	−222.235	−0.160946
$125$	329.421	0.235714
$126$	−1132.42	−0.800668
$127$	2245.95	1.56926	0.784629	−	0.619966i	$-0.212854\pi$
$127$	2245.95	1.56926	0.784629	+	0.619966i	$0.212854\pi$
$128$	−2634.80	−1.81942
$129$	337.567	0.230396
$130$	−5314.73	−3.58564
$131$	−181.227	−0.120869	−0.0604346	−	0.998172i	$-0.519249\pi$
$131$	−181.227	−0.120869	−0.0604346	+	0.998172i	$0.519249\pi$
$132$	0	0
$133$	−28.6823	−0.0186998
$134$	−544.922	−0.351299
$135$	−963.877	−0.614499
$136$	513.128	0.323532
$137$	2369.81	1.47786	0.738928	−	0.673784i	$-0.235332\pi$
$137$	2369.81	1.47786	0.738928	+	0.673784i	$0.235332\pi$
$138$	93.9685	0.0579647
$139$	−2476.11	−1.51094	−0.755470	−	0.655183i	$-0.772591\pi$
$139$	−2476.11	−1.51094	−0.755470	+	0.655183i	$0.772591\pi$
$140$	−2037.24	−1.22985
$141$	181.217	0.108236
$142$	−3060.04	−1.80840
$143$	0	0
$144$	−708.334	−0.409916
$145$	−2246.28	−1.28650
$146$	−1309.39	−0.742232
$147$	309.837	0.173843
$148$	−284.342	−0.157924
$149$	−2603.50	−1.43145	−0.715727	−	0.698380i	$-0.753905\pi$
$149$	−2603.50	−1.43145	−0.715727	+	0.698380i	$0.753905\pi$
$150$	−593.053	−0.322817
$151$	−3000.60	−1.61712	−0.808560	−	0.588414i	$-0.799753\pi$
$151$	−3000.60	−1.61712	−0.808560	+	0.588414i	$0.799753\pi$
$152$	−92.3142	−0.0492610
$153$	−433.912	−0.229279
$154$	0	0
$155$	233.450	0.120975
$156$	−1299.20	−0.666792
$157$	−2381.27	−1.21048	−0.605242	−	0.796042i	$-0.706924\pi$
$157$	−2381.27	−1.21048	−0.605242	+	0.796042i	$0.706924\pi$
$158$	−3441.58	−1.73290
$159$	−282.578	−0.140943
$160$	1664.48	0.822427
$161$	−153.342	−0.0750623
$162$	2893.54	1.40332
$163$	−107.379	−0.0515985	−0.0257993	−	0.999667i	$-0.508213\pi$
$163$	−107.379	−0.0515985	−0.0257993	+	0.999667i	$0.508213\pi$
$164$	−3668.75	−1.74684
$165$	0	0
$166$	−6176.08	−2.88769
$167$	−1677.71	−0.777394	−0.388697	−	0.921366i	$-0.627075\pi$
$167$	−1677.71	−0.777394	−0.388697	+	0.921366i	$0.627075\pi$
$168$	−343.883	−0.157924
$169$	3333.87	1.51747
$170$	−1214.88	−0.548100
$171$	78.0627	0.0349100
$172$	−3995.93	−1.77143
$173$	750.859	0.329981	0.164991	−	0.986295i	$-0.447241\pi$
$173$	750.859	0.329981	0.164991	+	0.986295i	$0.447241\pi$
$174$	−854.588	−0.372334
$175$	967.770	0.418037
$176$	0	0
$177$	414.377	0.175969
$178$	5024.19	2.11561
$179$	−158.994	−0.0663896	−0.0331948	−	0.999449i	$-0.510568\pi$
$179$	−158.994	−0.0663896	−0.0331948	+	0.999449i	$0.510568\pi$
$180$	5544.63	2.29596
$181$	−1761.26	−0.723277	−0.361638	−	0.932318i	$-0.617783\pi$
$181$	−1761.26	−0.723277	−0.361638	+	0.932318i	$0.617783\pi$
$182$	3299.54	1.34383
$183$	854.040	0.344986
$184$	−493.532	−0.197737
$185$	298.692	0.118704
$186$	88.8153	0.0350121
$187$	0	0
$188$	−2145.15	−0.832186
$189$	598.402	0.230304
$190$	218.562	0.0834536
$191$	3696.38	1.40032	0.700158	−	0.713988i	$-0.253113\pi$
$191$	3696.38	1.40032	0.700158	+	0.713988i	$0.253113\pi$
$192$	902.947	0.339399
$193$	3726.91	1.38999	0.694997	−	0.719012i	$-0.255406\pi$
$193$	3726.91	1.38999	0.694997	+	0.719012i	$0.255406\pi$
$194$	−6979.13	−2.58285
$195$	1364.77	0.501195
$196$	−3667.68	−1.33662
$197$	3744.19	1.35412	0.677062	−	0.735926i	$-0.263253\pi$
$197$	3744.19	1.35412	0.677062	+	0.735926i	$0.263253\pi$
$198$	0	0
$199$	2175.62	0.775004	0.387502	−	0.921869i	$-0.373338\pi$
$199$	2175.62	0.775004	0.387502	+	0.921869i	$0.373338\pi$
$200$	3114.77	1.10124
$201$	139.930	0.0491041
$202$	6676.92	2.32568
$203$	1394.55	0.482160
$204$	−296.981	−0.101926
$205$	3853.89	1.31301
$206$	−460.905	−0.155887
$207$	417.341	0.140131
$208$	2063.87	0.687999
$209$	0	0
$210$	814.175	0.267540
$211$	−4637.23	−1.51299	−0.756493	−	0.654002i	$-0.773089\pi$
$211$	−4637.23	−1.51299	−0.756493	+	0.654002i	$0.773089\pi$
$212$	3345.01	1.08366
$213$	785.786	0.252775
$214$	−3607.20	−1.15226
$215$	4197.58	1.33150
$216$	1925.96	0.606690
$217$	−144.933	−0.0453395
$218$	7338.47	2.27993
$219$	336.238	0.103748
$220$	0	0
$221$	1264.29	0.384820
$222$	113.636	0.0343548
$223$	2094.00	0.628809	0.314404	−	0.949289i	$-0.398195\pi$
$223$	2094.00	0.628809	0.314404	+	0.949289i	$0.398195\pi$
$224$	−1033.35	−0.308231
$225$	−2633.92	−0.780419
$226$	−3570.29	−1.05085
$227$	−4297.27	−1.25648	−0.628238	−	0.778021i	$-0.716223\pi$
$227$	−4297.27	−1.25648	−0.628238	+	0.778021i	$0.716223\pi$
$228$	53.4283	0.0155192
$229$	−2283.29	−0.658881	−0.329441	−	0.944176i	$-0.606860\pi$
$229$	−2283.29	−0.658881	−0.329441	+	0.944176i	$0.606860\pi$
$230$	1168.48	0.334989
$231$	0	0
$232$	4488.38	1.27016
$233$	−4491.52	−1.26287	−0.631436	−	0.775428i	$-0.717534\pi$
$233$	−4491.52	−1.26287	−0.631436	+	0.775428i	$0.717534\pi$
$234$	−8980.13	−2.50876
$235$	2253.40	0.625514
$236$	−4905.17	−1.35296
$237$	883.763	0.242222
$238$	754.232	0.205418
$239$	−5282.16	−1.42960	−0.714801	−	0.699328i	$-0.753483\pi$
$239$	−5282.16	−1.42960	−0.714801	+	0.699328i	$0.753483\pi$
$240$	509.270	0.136972
$241$	−371.992	−0.0994278	−0.0497139	−	0.998763i	$-0.515831\pi$
$241$	−371.992	−0.0994278	−0.0497139	+	0.998763i	$0.515831\pi$
$242$	0	0
$243$	−2465.83	−0.650959
$244$	−10109.7	−2.65248
$245$	3852.77	1.00467
$246$	1466.20	0.380006
$247$	−227.451	−0.0585926
$248$	−466.467	−0.119438
$249$	1585.96	0.403638
$250$	1558.42	0.394252
$251$	5802.13	1.45907	0.729536	−	0.683942i	$-0.239736\pi$
$251$	5802.13	1.45907	0.729536	+	0.683942i	$0.239736\pi$
$252$	−3442.26	−0.860485
$253$	0	0
$254$	10625.1	2.62472
$255$	311.969	0.0766127
$256$	−6518.46	−1.59142
$257$	2506.95	0.608479	0.304240	−	0.952596i	$-0.401598\pi$
$257$	2506.95	0.608479	0.304240	+	0.952596i	$0.401598\pi$
$258$	1596.96	0.385357
$259$	−185.436	−0.0444882
$260$	−16155.4	−3.85351
$261$	−3795.46	−0.900127
$262$	−857.345	−0.202164
$263$	4285.24	1.00471	0.502356	−	0.864661i	$-0.332467\pi$
$263$	4285.24	1.00471	0.502356	+	0.864661i	$0.332467\pi$
$264$	0	0
$265$	−3513.81	−0.814535
$266$	−135.690	−0.0312770
$267$	−1290.16	−0.295717
$268$	−1656.42	−0.377544
$269$	1905.08	0.431801	0.215901	−	0.976415i	$-0.430731\pi$
$269$	1905.08	0.431801	0.215901	+	0.976415i	$0.430731\pi$
$270$	−4559.90	−1.02780
$271$	3449.04	0.773116	0.386558	−	0.922265i	$-0.373664\pi$
$271$	3449.04	0.773116	0.386558	+	0.922265i	$0.373664\pi$
$272$	471.775	0.105167
$273$	−847.287	−0.187839
$274$	11211.1	2.47184
$275$	0	0
$276$	285.640	0.0622952
$277$	1524.50	0.330679	0.165339	−	0.986237i	$-0.447128\pi$
$277$	1524.50	0.330679	0.165339	+	0.986237i	$0.447128\pi$
$278$	−11713.9	−2.52718
$279$	394.454	0.0846427
$280$	−4276.13	−0.912670
$281$	−1521.46	−0.323000	−0.161500	−	0.986873i	$-0.551633\pi$
$281$	−1521.46	−0.323000	−0.161500	+	0.986873i	$0.551633\pi$
$282$	857.299	0.181033
$283$	−5816.59	−1.22177	−0.610884	−	0.791720i	$-0.709186\pi$
$283$	−5816.59	−1.22177	−0.610884	+	0.791720i	$0.709186\pi$
$284$	−9301.70	−1.94350
$285$	−56.1246	−0.0116650
$286$	0	0
$287$	−2392.61	−0.492095
$288$	2812.41	0.575427
$289$	289.000	0.0588235
$290$	−10626.7	−2.15179
$291$	1792.17	0.361027
$292$	−3980.20	−0.797683
$293$	3998.19	0.797191	0.398595	−	0.917127i	$-0.369498\pi$
$293$	3998.19	0.797191	0.398595	+	0.917127i	$0.369498\pi$
$294$	1465.77	0.290767
$295$	5152.71	1.01696
$296$	−596.828	−0.117196
$297$	0	0
$298$	−12316.6	−2.39423
$299$	−1216.00	−0.235195
$300$	−1802.73	−0.346935
$301$	−2605.98	−0.499024
$302$	−14195.2	−2.70477
$303$	−1714.57	−0.325080
$304$	−84.8745	−0.0160128
$305$	10619.8	1.99374
$306$	−2052.74	−0.383489
$307$	8679.98	1.61366	0.806829	−	0.590785i	$-0.201182\pi$
$307$	8679.98	1.61366	0.806829	+	0.590785i	$0.201182\pi$
$308$	0	0
$309$	118.356	0.0217897
$310$	1104.40	0.202342
$311$	7271.08	1.32574	0.662870	−	0.748734i	$-0.269338\pi$
$311$	7271.08	1.32574	0.662870	+	0.748734i	$0.269338\pi$
$312$	−2727.00	−0.494827
$313$	−107.036	−0.0193291	−0.00966454	−	0.999953i	$-0.503076\pi$
$313$	−107.036	−0.0193291	−0.00966454	+	0.999953i	$0.503076\pi$
$314$	−11265.3	−2.02464
$315$	3615.98	0.646785
$316$	−10461.5	−1.86236
$317$	−7857.41	−1.39216	−0.696082	−	0.717962i	$-0.745075\pi$
$317$	−7857.41	−1.39216	−0.696082	+	0.717962i	$0.745075\pi$
$318$	−1336.82	−0.235739
$319$	0	0
$320$	11228.0	1.96145
$321$	926.293	0.161061
$322$	−725.428	−0.125548
$323$	−51.9924	−0.00895646
$324$	8795.59	1.50816
$325$	7674.43	1.30985
$326$	−507.986	−0.0863030
$327$	−1884.44	−0.318685
$328$	−7700.62	−1.29633
$329$	−1398.98	−0.234432
$330$	0	0
$331$	−4599.42	−0.763767	−0.381883	−	0.924210i	$-0.624724\pi$
$331$	−4599.42	−0.763767	−0.381883	+	0.924210i	$0.624724\pi$
$332$	−18773.7	−3.10343
$333$	504.690	0.0830536
$334$	−7936.87	−1.30026
$335$	1740.01	0.283782
$336$	−316.169	−0.0513347
$337$	10001.6	1.61668	0.808342	−	0.588713i	$-0.200365\pi$
$337$	10001.6	1.61668	0.808342	+	0.588713i	$0.200365\pi$
$338$	15771.8	2.53809
$339$	916.815	0.146887
$340$	−3692.91	−0.589048
$341$	0	0
$342$	369.298	0.0583899
$343$	−5608.66	−0.882912
$344$	−8387.36	−1.31458
$345$	−300.054	−0.0468243
$346$	3552.16	0.551922
$347$	−9009.87	−1.39388	−0.696939	−	0.717131i	$-0.745455\pi$
$347$	−9009.87	−1.39388	−0.696939	+	0.717131i	$0.745455\pi$
$348$	−2597.72	−0.400151
$349$	3822.52	0.586289	0.293145	−	0.956068i	$-0.405298\pi$
$349$	3822.52	0.586289	0.293145	+	0.956068i	$0.405298\pi$
$350$	4578.31	0.699203
$351$	4745.34	0.721617
$352$	0	0
$353$	−6424.02	−0.968601	−0.484301	−	0.874902i	$-0.660926\pi$
$353$	−6424.02	−0.968601	−0.484301	+	0.874902i	$0.660926\pi$
$354$	1960.33	0.294323
$355$	9771.12	1.46084
$356$	15272.2	2.27367
$357$	−193.679	−0.0287131
$358$	−752.165	−0.111042
$359$	−3170.70	−0.466137	−0.233068	−	0.972460i	$-0.574877\pi$
$359$	−3170.70	−0.466137	−0.233068	+	0.972460i	$0.574877\pi$
$360$	11638.1	1.70383
$361$	−6849.65	−0.998636
$362$	−8332.13	−1.20974
$363$	0	0
$364$	10029.7	1.44423
$365$	4181.06	0.599580
$366$	4040.28	0.577019
$367$	−5181.48	−0.736978	−0.368489	−	0.929632i	$-0.620125\pi$
$367$	−5181.48	−0.736978	−0.368489	+	0.929632i	$0.620125\pi$
$368$	−453.758	−0.0642765
$369$	6511.80	0.918674
$370$	1413.05	0.198543
$371$	2181.48	0.305274
$372$	269.975	0.0376278
$373$	552.488	0.0766936	0.0383468	−	0.999264i	$-0.487791\pi$
$373$	552.488	0.0766936	0.0383468	+	0.999264i	$0.487791\pi$
$374$	0	0
$375$	−400.186	−0.0551080
$376$	−4502.62	−0.617566
$377$	11058.8	1.51077
$378$	2830.91	0.385202
$379$	−5889.14	−0.798165	−0.399083	−	0.916915i	$-0.630671\pi$
$379$	−5889.14	−0.798165	−0.399083	+	0.916915i	$0.630671\pi$
$380$	664.372	0.0896883
$381$	−2728.42	−0.366879
$382$	17486.8	2.34215
$383$	6268.16	0.836261	0.418130	−	0.908387i	$-0.362686\pi$
$383$	6268.16	0.836261	0.418130	+	0.908387i	$0.362686\pi$
$384$	3200.80	0.425365
$385$	0	0
$386$	17631.2	2.32488
$387$	7092.52	0.931610
$388$	−21214.7	−2.77581
$389$	9789.98	1.27602	0.638010	−	0.770028i	$-0.279758\pi$
$389$	9789.98	1.27602	0.638010	+	0.770028i	$0.279758\pi$
$390$	6456.43	0.838292
$391$	−277.963	−0.0359519
$392$	−7698.38	−0.991905
$393$	220.157	0.0282582
$394$	17712.9	2.26489
$395$	10989.4	1.39984
$396$	0	0
$397$	−12072.6	−1.52621	−0.763103	−	0.646276i	$-0.776325\pi$
$397$	−12072.6	−1.52621	−0.763103	+	0.646276i	$0.776325\pi$
$398$	10292.4	1.29626
$399$	34.8438	0.00437185
$400$	2863.75	0.357969
$401$	−7906.87	−0.984664	−0.492332	−	0.870408i	$-0.663855\pi$
$401$	−7906.87	−0.984664	−0.492332	+	0.870408i	$0.663855\pi$
$402$	661.981	0.0821308
$403$	−1149.32	−0.142064
$404$	20296.1	2.49943
$405$	−9239.46	−1.13361
$406$	6597.33	0.806453
$407$	0	0
$408$	−623.357	−0.0756392
$409$	6786.60	0.820479	0.410240	−	0.911978i	$-0.365445\pi$
$409$	6786.60	0.820479	0.410240	+	0.911978i	$0.365445\pi$
$410$	18231.9	2.19613
$411$	−2878.88	−0.345511
$412$	−1401.03	−0.167533
$413$	−3198.95	−0.381138
$414$	1974.35	0.234382
$415$	19721.1	2.33270
$416$	−8194.51	−0.965791
$417$	3008.02	0.353245
$418$	0	0
$419$	−8453.14	−0.985592	−0.492796	−	0.870145i	$-0.664025\pi$
$419$	−8453.14	−0.985592	−0.492796	+	0.870145i	$0.664025\pi$
$420$	2474.88	0.287528
$421$	−5918.54	−0.685159	−0.342580	−	0.939489i	$-0.611301\pi$
$421$	−5918.54	−0.685159	−0.342580	+	0.939489i	$0.611301\pi$
$422$	−21937.7	−2.53060
$423$	3807.51	0.437653
$424$	7021.10	0.804185
$425$	1754.28	0.200223
$426$	3717.39	0.422789
$427$	−6593.11	−0.747220
$428$	−10964.9	−1.23834
$429$	0	0
$430$	19857.9	2.22705
$431$	9869.66	1.10303	0.551514	−	0.834166i	$-0.314050\pi$
$431$	9869.66	1.10303	0.551514	+	0.834166i	$0.314050\pi$
$432$	1770.75	0.197211
$433$	4697.07	0.521309	0.260654	−	0.965432i	$-0.416062\pi$
$433$	4697.07	0.521309	0.260654	+	0.965432i	$0.416062\pi$
$434$	−685.645	−0.0758342
$435$	2728.82	0.300774
$436$	22307.0	2.45026
$437$	50.0068	0.00547403
$438$	1590.67	0.173528
$439$	11226.2	1.22049	0.610246	−	0.792212i	$-0.291071\pi$
$439$	11226.2	1.22049	0.610246	+	0.792212i	$0.291071\pi$
$440$	0	0
$441$	6509.90	0.702937
$442$	5981.07	0.643644
$443$	−18172.0	−1.94893	−0.974467	−	0.224529i	$-0.927916\pi$
$443$	−18172.0	−1.94893	−0.974467	+	0.224529i	$0.927916\pi$
$444$	345.424	0.0369214
$445$	−16042.9	−1.70901
$446$	9906.25	1.05174
$447$	3162.77	0.334662
$448$	−6970.66	−0.735118
$449$	1078.61	0.113369	0.0566847	−	0.998392i	$-0.481947\pi$
$449$	1078.61	0.113369	0.0566847	+	0.998392i	$0.481947\pi$
$450$	−12460.5	−1.30532
$451$	0	0
$452$	−10852.7	−1.12936
$453$	3645.18	0.378069
$454$	−20329.5	−2.10156
$455$	−10535.9	−1.08556
$456$	112.145	0.0115168
$457$	−7130.96	−0.729918	−0.364959	−	0.931024i	$-0.618917\pi$
$457$	−7130.96	−0.729918	−0.364959	+	0.931024i	$0.618917\pi$
$458$	−10801.7	−1.10204
$459$	1084.72	0.110306
$460$	3551.88	0.360016
$461$	4180.76	0.422380	0.211190	−	0.977445i	$-0.432266\pi$
$461$	4180.76	0.422380	0.211190	+	0.977445i	$0.432266\pi$
$462$	0	0
$463$	14236.4	1.42899	0.714495	−	0.699641i	$-0.246657\pi$
$463$	14236.4	1.42899	0.714495	+	0.699641i	$0.246657\pi$
$464$	4126.66	0.412877
$465$	−283.599	−0.0282830
$466$	−21248.4	−2.11226
$467$	1646.29	0.163129	0.0815643	−	0.996668i	$-0.474008\pi$
$467$	1646.29	0.163129	0.0815643	+	0.996668i	$0.474008\pi$
$468$	−27297.2	−2.69619
$469$	−1080.25	−0.106357
$470$	10660.4	1.04623
$471$	2892.81	0.283001
$472$	−10295.8	−1.00403
$473$	0	0
$474$	4180.89	0.405137
$475$	−315.603	−0.0304860
$476$	2292.67	0.220765
$477$	−5937.18	−0.569905
$478$	−24988.8	−2.39113
$479$	8856.11	0.844773	0.422386	−	0.906416i	$-0.361193\pi$
$479$	8856.11	0.844773	0.422386	+	0.906416i	$0.361193\pi$
$480$	−2022.03	−0.192277
$481$	−1470.51	−0.139396
$482$	−1759.81	−0.166302
$483$	186.282	0.0175489
$484$	0	0
$485$	22285.3	2.08644
$486$	−11665.3	−1.08878
$487$	4234.58	0.394018	0.197009	−	0.980402i	$-0.436877\pi$
$487$	4234.58	0.394018	0.197009	+	0.980402i	$0.436877\pi$
$488$	−21220.0	−1.96841
$489$	130.446	0.0120633
$490$	18226.6	1.68040
$491$	−14812.4	−1.36145	−0.680726	−	0.732538i	$-0.738336\pi$
$491$	−14812.4	−1.36145	−0.680726	+	0.732538i	$0.738336\pi$
$492$	4456.86	0.408396
$493$	2527.91	0.230936
$494$	−1076.02	−0.0980011
$495$	0	0
$496$	−428.874	−0.0388246
$497$	−6066.19	−0.547497
$498$	7502.82	0.675119
$499$	−9301.35	−0.834440	−0.417220	−	0.908806i	$-0.636996\pi$
$499$	−9301.35	−0.834440	−0.417220	+	0.908806i	$0.636996\pi$
$500$	4737.18	0.423706
$501$	2038.11	0.181748
$502$	27448.6	2.44042
$503$	16109.2	1.42798	0.713991	−	0.700155i	$-0.246886\pi$
$503$	16109.2	1.42798	0.713991	+	0.700155i	$0.246886\pi$
$504$	−7225.24	−0.638567
$505$	−21320.3	−1.87870
$506$	0	0
$507$	−4050.05	−0.354771
$508$	32297.5	2.82081
$509$	12174.1	1.06013	0.530067	−	0.847956i	$-0.322167\pi$
$509$	12174.1	1.06013	0.530067	+	0.847956i	$0.322167\pi$
$510$	1475.86	0.128141
$511$	−2595.72	−0.224712
$512$	−9759.03	−0.842368
$513$	−195.147	−0.0167952
$514$	11859.8	1.01773
$515$	1471.73	0.125927
$516$	4854.32	0.414146
$517$	0	0
$518$	−877.260	−0.0744104
$519$	−912.157	−0.0771469
$520$	−33909.8	−2.85970
$521$	−17043.3	−1.43317	−0.716585	−	0.697500i	$-0.754296\pi$
$521$	−17043.3	−1.43317	−0.716585	+	0.697500i	$0.754296\pi$
$522$	−17955.5	−1.50554
$523$	21496.6	1.79728	0.898642	−	0.438684i	$-0.144555\pi$
$523$	21496.6	1.79728	0.898642	+	0.438684i	$0.144555\pi$
$524$	−2606.10	−0.217267
$525$	−1175.66	−0.0977337
$526$	20272.5	1.68047
$527$	−262.720	−0.0217158
$528$	0	0
$529$	−11899.7	−0.978027
$530$	−16623.1	−1.36238
$531$	8706.37	0.711533
$532$	−412.461	−0.0336136
$533$	−18973.4	−1.54189
$534$	−6103.47	−0.494613
$535$	11518.3	0.930803
$536$	−3476.79	−0.280176
$537$	193.148	0.0155214
$538$	9012.51	0.722224
$539$	0	0
$540$	−13860.9	−1.10459
$541$	−18718.8	−1.48759	−0.743793	−	0.668410i	$-0.766975\pi$
$541$	−18718.8	−1.48759	−0.743793	+	0.668410i	$0.766975\pi$
$542$	16316.7	1.29310
$543$	2139.60	0.169096
$544$	−1873.16	−0.147631
$545$	−23432.8	−1.84174
$546$	−4008.33	−0.314177
$547$	−15414.5	−1.20490	−0.602448	−	0.798158i	$-0.705808\pi$
$547$	−15414.5	−1.20490	−0.602448	+	0.798158i	$0.705808\pi$
$548$	34078.6	2.65651
$549$	17944.0	1.39496
$550$	0	0
$551$	−454.782	−0.0351622
$552$	599.551	0.0462293
$553$	−6822.56	−0.524638
$554$	7212.06	0.553089
$555$	−362.856	−0.0277520
$556$	−35607.3	−2.71598
$557$	7138.14	0.543003	0.271502	−	0.962438i	$-0.412480\pi$
$557$	7138.14	0.543003	0.271502	+	0.962438i	$0.412480\pi$
$558$	1866.08	0.141572
$559$	−20665.5	−1.56361
$560$	−3931.51	−0.296673
$561$	0	0
$562$	−7197.72	−0.540245
$563$	6563.19	0.491306	0.245653	−	0.969358i	$-0.420998\pi$
$563$	6563.19	0.491306	0.245653	+	0.969358i	$0.420998\pi$
$564$	2605.96	0.194558
$565$	11400.4	0.848885
$566$	−27517.0	−2.04351
$567$	5736.12	0.424858
$568$	−19524.1	−1.44227
$569$	−17000.2	−1.25252	−0.626262	−	0.779613i	$-0.715416\pi$
$569$	−17000.2	−1.25252	−0.626262	+	0.779613i	$0.715416\pi$
$570$	−265.513	−0.0195108
$571$	5317.38	0.389712	0.194856	−	0.980832i	$-0.437576\pi$
$571$	5317.38	0.389712	0.194856	+	0.980832i	$0.437576\pi$
$572$	0	0
$573$	−4490.42	−0.327382
$574$	−11318.9	−0.823070
$575$	−1687.28	−0.122373
$576$	18971.6	1.37237
$577$	−20855.3	−1.50471	−0.752353	−	0.658760i	$-0.771081\pi$
$577$	−20855.3	−1.50471	−0.752353	+	0.658760i	$0.771081\pi$
$578$	1367.20	0.0983874
$579$	−4527.52	−0.324969
$580$	−32302.2	−2.31255
$581$	−12243.4	−0.874256
$582$	8478.37	0.603849
$583$	0	0
$584$	−8354.35	−0.591961
$585$	28674.8	2.02659
$586$	18914.6	1.33337
$587$	12109.2	0.851448	0.425724	−	0.904853i	$-0.360019\pi$
$587$	12109.2	0.851448	0.425724	+	0.904853i	$0.360019\pi$
$588$	4455.56	0.312490
$589$	47.2645	0.00330645
$590$	24376.4	1.70095
$591$	−4548.50	−0.316583
$592$	−548.729	−0.0380957
$593$	18341.2	1.27012	0.635060	−	0.772463i	$-0.280975\pi$
$593$	18341.2	1.27012	0.635060	+	0.772463i	$0.280975\pi$
$594$	0	0
$595$	−2408.37	−0.165938
$596$	−37439.1	−2.57310
$597$	−2642.99	−0.181190
$598$	−5752.66	−0.393384
$599$	11973.5	0.816732	0.408366	−	0.912818i	$-0.366099\pi$
$599$	11973.5	0.816732	0.408366	+	0.912818i	$0.366099\pi$
$600$	−3783.88	−0.257461
$601$	4316.20	0.292948	0.146474	−	0.989215i	$-0.453208\pi$
$601$	4316.20	0.292948	0.146474	+	0.989215i	$0.453208\pi$
$602$	−12328.3	−0.834660
$603$	2940.04	0.198553
$604$	−43149.6	−2.90684
$605$	0	0
$606$	−8111.24	−0.543724
$607$	−23784.3	−1.59041	−0.795203	−	0.606344i	$-0.792636\pi$
$607$	−23784.3	−1.59041	−0.795203	+	0.606344i	$0.792636\pi$
$608$	336.990	0.0224782
$609$	−1694.13	−0.112725
$610$	50240.2	3.33470
$611$	−11093.9	−0.734553
$612$	−6239.80	−0.412139
$613$	20172.2	1.32912	0.664559	−	0.747236i	$-0.268620\pi$
$613$	20172.2	1.32912	0.664559	+	0.747236i	$0.268620\pi$
$614$	41063.2	2.69898
$615$	−4681.78	−0.306971
$616$	0	0
$617$	27515.2	1.79533	0.897667	−	0.440674i	$-0.145261\pi$
$617$	27515.2	1.79533	0.897667	+	0.440674i	$0.145261\pi$
$618$	559.916	0.0364452
$619$	16484.3	1.07037	0.535187	−	0.844734i	$-0.320241\pi$
$619$	16484.3	1.07037	0.535187	+	0.844734i	$0.320241\pi$
$620$	3357.09	0.217458
$621$	−1043.30	−0.0674173
$622$	34397.9	2.21741
$623$	9959.91	0.640506
$624$	−2507.23	−0.160848
$625$	−17875.3	−1.14402
$626$	−506.362	−0.0323296
$627$	0	0
$628$	−34243.4	−2.17590
$629$	−336.141	−0.0213081
$630$	17106.4	1.08180
$631$	−2385.42	−0.150494	−0.0752472	−	0.997165i	$-0.523975\pi$
$631$	−2385.42	−0.150494	−0.0752472	+	0.997165i	$0.523975\pi$
$632$	−21958.5	−1.38206
$633$	5633.39	0.353723
$634$	−37171.7	−2.32851
$635$	−33927.4	−2.12026
$636$	−4063.57	−0.253351
$637$	−18967.9	−1.17980
$638$	0	0
$639$	16509.9	1.02210
$640$	39801.5	2.45827
$641$	−3037.05	−0.187139	−0.0935697	−	0.995613i	$-0.529828\pi$
$641$	−3037.05	−0.187139	−0.0935697	+	0.995613i	$0.529828\pi$
$642$	4382.09	0.269389
$643$	25746.1	1.57904	0.789522	−	0.613722i	$-0.210328\pi$
$643$	25746.1	1.57904	0.789522	+	0.613722i	$0.210328\pi$
$644$	−2205.11	−0.134928
$645$	−5099.30	−0.311294
$646$	−245.965	−0.0149804
$647$	9022.58	0.548244	0.274122	−	0.961695i	$-0.411613\pi$
$647$	9022.58	0.548244	0.274122	+	0.961695i	$0.411613\pi$
$648$	18461.7	1.11921
$649$	0	0
$650$	36306.1	2.19083
$651$	176.067	0.0106000
$652$	−1544.14	−0.0927505
$653$	4560.66	0.273312	0.136656	−	0.990619i	$-0.456365\pi$
$653$	4560.66	0.273312	0.136656	+	0.990619i	$0.456365\pi$
$654$	−8914.90	−0.533028
$655$	2737.62	0.163310
$656$	−7080.02	−0.421385
$657$	7064.60	0.419508
$658$	−6618.27	−0.392108
$659$	6296.60	0.372201	0.186101	−	0.982531i	$-0.440415\pi$
$659$	6296.60	0.372201	0.186101	+	0.982531i	$0.440415\pi$
$660$	0	0
$661$	17023.8	1.00174	0.500870	−	0.865522i	$-0.333013\pi$
$661$	17023.8	1.00174	0.500870	+	0.865522i	$0.333013\pi$
$662$	−21758.9	−1.27747
$663$	−1535.88	−0.0899677
$664$	−39405.5	−2.30306
$665$	433.276	0.0252658
$666$	2387.58	0.138914
$667$	−2431.37	−0.141144
$668$	−24126.0	−1.39740
$669$	−2543.82	−0.147010
$670$	8231.62	0.474650
$671$	0	0
$672$	1255.34	0.0720620
$673$	−18760.0	−1.07451	−0.537256	−	0.843419i	$-0.680539\pi$
$673$	−18760.0	−1.07451	−0.537256	+	0.843419i	$0.680539\pi$
$674$	47315.5	2.70404
$675$	6584.46	0.375461
$676$	47942.2	2.72771
$677$	17467.6	0.991632	0.495816	−	0.868428i	$-0.334869\pi$
$677$	17467.6	0.991632	0.495816	+	0.868428i	$0.334869\pi$
$678$	4337.26	0.245680
$679$	−13835.4	−0.781963
$680$	−7751.34	−0.437133
$681$	5220.40	0.293754
$682$	0	0
$683$	−13898.3	−0.778630	−0.389315	−	0.921105i	$-0.627288\pi$
$683$	−13898.3	−0.778630	−0.389315	+	0.921105i	$0.627288\pi$
$684$	1122.57	0.0627522
$685$	−35798.4	−1.99677
$686$	−26533.4	−1.47675
$687$	2773.78	0.154041
$688$	−7711.42	−0.427318
$689$	17299.1	0.956524
$690$	−1419.49	−0.0783177
$691$	21052.6	1.15901	0.579507	−	0.814967i	$-0.303245\pi$
$691$	21052.6	1.15901	0.579507	+	0.814967i	$0.303245\pi$
$692$	10797.6	0.593156
$693$	0	0
$694$	−42623.8	−2.33138
$695$	37404.2	2.04147
$696$	−5452.56	−0.296952
$697$	−4337.08	−0.235694
$698$	18083.5	0.980619
$699$	5456.37	0.295249
$700$	13916.9	0.751440
$701$	−1890.63	−0.101866	−0.0509331	−	0.998702i	$-0.516220\pi$
$701$	−1890.63	−0.101866	−0.0509331	+	0.998702i	$0.516220\pi$
$702$	22449.2	1.20697
$703$	60.4733	0.00324437
$704$	0	0
$705$	−2737.48	−0.146240
$706$	−30390.7	−1.62007
$707$	13236.3	0.704104
$708$	5958.88	0.316311
$709$	−11087.8	−0.587323	−0.293662	−	0.955909i	$-0.594874\pi$
$709$	−11087.8	−0.587323	−0.293662	+	0.955909i	$0.594874\pi$
$710$	46225.1	2.44337
$711$	18568.5	0.979428
$712$	32056.0	1.68729
$713$	252.686	0.0132723
$714$	−916.254	−0.0480251
$715$	0	0
$716$	−2286.38	−0.119338
$717$	6416.86	0.334229
$718$	−14999.9	−0.779654
$719$	17864.1	0.926591	0.463296	−	0.886204i	$-0.346667\pi$
$719$	17864.1	0.926591	0.463296	+	0.886204i	$0.346667\pi$
$720$	10700.1	0.553848
$721$	−913.694	−0.0471952
$722$	−32404.2	−1.67030
$723$	451.902	0.0232454
$724$	−25327.5	−1.30012
$725$	15344.8	0.786058
$726$	0	0
$727$	−31914.5	−1.62812	−0.814060	−	0.580781i	$-0.802747\pi$
$727$	−31914.5	−1.62812	−0.814060	+	0.580781i	$0.802747\pi$
$728$	21052.2	1.07177
$729$	−13518.7	−0.686823
$730$	19779.7	1.00285
$731$	−4723.86	−0.239013
$732$	12281.4	0.620127
$733$	−8147.34	−0.410544	−0.205272	−	0.978705i	$-0.565808\pi$
$733$	−8147.34	−0.410544	−0.205272	+	0.978705i	$0.565808\pi$
$734$	−24512.5	−1.23266
$735$	−4680.41	−0.234884
$736$	1801.63	0.0902293
$737$	0	0
$738$	30805.9	1.53656
$739$	−36042.0	−1.79408	−0.897041	−	0.441948i	$-0.854288\pi$
$739$	−36042.0	−1.79408	−0.897041	+	0.441948i	$0.854288\pi$
$740$	4295.29	0.213376
$741$	276.312	0.0136985
$742$	10320.1	0.510597
$743$	−23296.4	−1.15028	−0.575142	−	0.818053i	$-0.695053\pi$
$743$	−23296.4	−1.15028	−0.575142	+	0.818053i	$0.695053\pi$
$744$	566.672	0.0279237
$745$	39328.5	1.93408
$746$	2613.70	0.128277
$747$	33322.1	1.63212
$748$	0	0
$749$	−7150.89	−0.348849
$750$	−1893.19	−0.0921729
$751$	−21029.5	−1.02181	−0.510903	−	0.859638i	$-0.670689\pi$
$751$	−21029.5	−1.02181	−0.510903	+	0.859638i	$0.670689\pi$
$752$	−4139.75	−0.200746
$753$	−7048.53	−0.341119
$754$	52317.0	2.52689
$755$	45327.2	2.18493
$756$	8605.23	0.413980
$757$	13708.1	0.658165	0.329082	−	0.944301i	$-0.393261\pi$
$757$	13708.1	0.658165	0.329082	+	0.944301i	$0.393261\pi$
$758$	−27860.3	−1.33500
$759$	0	0
$760$	1394.50	0.0665578
$761$	1992.12	0.0948941	0.0474471	−	0.998874i	$-0.484891\pi$
$761$	1992.12	0.0948941	0.0474471	+	0.998874i	$0.484891\pi$
$762$	−12907.6	−0.613637
$763$	14547.7	0.690253
$764$	53155.2	2.51713
$765$	6554.69	0.309785
$766$	29653.3	1.39872
$767$	−25367.7	−1.19423
$768$	7918.74	0.372061
$769$	−29040.4	−1.36180	−0.680899	−	0.732377i	$-0.738411\pi$
$769$	−29040.4	−1.36180	−0.680899	+	0.732377i	$0.738411\pi$
$770$	0	0
$771$	−3045.48	−0.142257
$772$	53594.2	2.49857
$773$	33225.6	1.54598	0.772991	−	0.634418i	$-0.218760\pi$
$773$	33225.6	1.54598	0.772991	+	0.634418i	$0.218760\pi$
$774$	33553.2	1.55820
$775$	−1594.75	−0.0739163
$776$	−44529.3	−2.05993
$777$	225.271	0.0104010
$778$	46314.3	2.13425
$779$	780.261	0.0358867
$780$	19625.8	0.900920
$781$	0	0
$782$	−1314.98	−0.0601326
$783$	9488.18	0.433052
$784$	−7077.96	−0.322429
$785$	35971.6	1.63552
$786$	1041.52	0.0472643
$787$	7750.27	0.351038	0.175519	−	0.984476i	$-0.443840\pi$
$787$	7750.27	0.351038	0.175519	+	0.984476i	$0.443840\pi$
$788$	53842.7	2.43409
$789$	−5205.78	−0.234893
$790$	51988.7	2.34136
$791$	−7077.72	−0.318148
$792$	0	0
$793$	−52283.5	−2.34129
$794$	−57112.7	−2.55271
$795$	4268.64	0.190432
$796$	31286.2	1.39310
$797$	−2604.81	−0.115768	−0.0578841	−	0.998323i	$-0.518435\pi$
$797$	−2604.81	−0.115768	−0.0578841	+	0.998323i	$0.518435\pi$
$798$	164.838	0.00731230
$799$	−2535.93	−0.112284
$800$	−11370.4	−0.502505
$801$	−27107.2	−1.19574
$802$	−37405.7	−1.64693
$803$	0	0
$804$	2012.25	0.0882667
$805$	2316.39	0.101419
$806$	−5437.18	−0.237613
$807$	−2314.32	−0.100952
$808$	42601.0	1.85483
$809$	4489.12	0.195091	0.0975457	−	0.995231i	$-0.468901\pi$
$809$	4489.12	0.195091	0.0975457	+	0.995231i	$0.468901\pi$
$810$	−43709.9	−1.89606
$811$	−24225.5	−1.04892	−0.524460	−	0.851435i	$-0.675733\pi$
$811$	−24225.5	−1.04892	−0.524460	+	0.851435i	$0.675733\pi$
$812$	20054.1	0.866703
$813$	−4189.96	−0.180748
$814$	0	0
$815$	1622.07	0.0697161
$816$	−573.120	−0.0245873
$817$	849.845	0.0363920
$818$	32106.0	1.37232
$819$	−17802.1	−0.759532
$820$	55420.3	2.36020
$821$	16743.1	0.711739	0.355870	−	0.934536i	$-0.384185\pi$
$821$	16743.1	0.711739	0.355870	+	0.934536i	$0.384185\pi$
$822$	−13619.4	−0.577896
$823$	−21611.2	−0.915331	−0.457665	−	0.889124i	$-0.651314\pi$
$823$	−21611.2	−0.915331	−0.457665	+	0.889124i	$0.651314\pi$
$824$	−2940.73	−0.124327
$825$	0	0
$826$	−15133.5	−0.637486
$827$	−5126.94	−0.215576	−0.107788	−	0.994174i	$-0.534377\pi$
$827$	−5126.94	−0.215576	−0.107788	+	0.994174i	$0.534377\pi$
$828$	6001.50	0.251892
$829$	−16626.3	−0.696570	−0.348285	−	0.937389i	$-0.613236\pi$
$829$	−16626.3	−0.696570	−0.348285	+	0.937389i	$0.613236\pi$
$830$	93296.3	3.90164
$831$	−1851.98	−0.0773100
$832$	−55277.5	−2.30337
$833$	−4335.82	−0.180345
$834$	14230.3	0.590833
$835$	25343.5	1.05036
$836$	0	0
$837$	−986.084	−0.0407217
$838$	−39990.0	−1.64849
$839$	3374.89	0.138873	0.0694363	−	0.997586i	$-0.477880\pi$
$839$	3374.89	0.138873	0.0694363	+	0.997586i	$0.477880\pi$
$840$	5194.72	0.213375
$841$	−2277.17	−0.0933688
$842$	−27999.4	−1.14599
$843$	1848.30	0.0755147
$844$	−66684.9	−2.71966
$845$	−50361.7	−2.05029
$846$	18012.5	0.732012
$847$	0	0
$848$	6455.26	0.261409
$849$	7066.10	0.285639
$850$	8299.11	0.334891
$851$	323.303	0.0130231
$852$	11299.9	0.454375
$853$	8375.87	0.336207	0.168103	−	0.985769i	$-0.446236\pi$
$853$	8375.87	0.336207	0.168103	+	0.985769i	$0.446236\pi$
$854$	−31190.6	−1.24979
$855$	−1179.22	−0.0471678
$856$	−23015.2	−0.918975
$857$	44120.2	1.75860	0.879298	−	0.476273i	$-0.158013\pi$
$857$	44120.2	1.75860	0.879298	+	0.476273i	$0.158013\pi$
$858$	0	0
$859$	16898.3	0.671203	0.335601	−	0.942004i	$-0.391060\pi$
$859$	16898.3	0.671203	0.335601	+	0.942004i	$0.391060\pi$
$860$	60362.7	2.39343
$861$	2906.58	0.115048
$862$	46691.2	1.84491
$863$	4152.99	0.163812	0.0819058	−	0.996640i	$-0.473899\pi$
$863$	4152.99	0.163812	0.0819058	+	0.996640i	$0.473899\pi$
$864$	−7030.67	−0.276838
$865$	−11342.5	−0.445847
$866$	22220.8	0.871933
$867$	−351.082	−0.0137525
$868$	−2084.18	−0.0814996
$869$	0	0
$870$	12909.4	0.503070
$871$	−8566.39	−0.333250
$872$	46821.9	1.81834
$873$	37654.8	1.45982
$874$	236.572	0.00915578
$875$	3089.40	0.119361
$876$	4835.21	0.186492
$877$	25779.1	0.992585	0.496293	−	0.868155i	$-0.334694\pi$
$877$	25779.1	0.992585	0.496293	+	0.868155i	$0.334694\pi$
$878$	53108.6	2.04138
$879$	−4857.08	−0.186377
$880$	0	0
$881$	41808.9	1.59884	0.799420	−	0.600773i	$-0.205140\pi$
$881$	41808.9	1.59884	0.799420	+	0.600773i	$0.205140\pi$
$882$	30797.0	1.17572
$883$	−7314.84	−0.278781	−0.139391	−	0.990237i	$-0.544514\pi$
$883$	−7314.84	−0.278781	−0.139391	+	0.990237i	$0.544514\pi$
$884$	18180.9	0.691730
$885$	−6259.60	−0.237756
$886$	−85967.9	−3.25976
$887$	7807.55	0.295549	0.147775	−	0.989021i	$-0.452789\pi$
$887$	7807.55	0.295549	0.147775	+	0.989021i	$0.452789\pi$
$888$	725.037	0.0273994
$889$	21063.1	0.794639
$890$	−75895.7	−2.85846
$891$	0	0
$892$	30112.4	1.13031
$893$	456.225	0.0170963
$894$	14962.4	0.559751
$895$	2401.77	0.0897008
$896$	−24709.9	−0.921316
$897$	1477.22	0.0549867
$898$	5102.68	0.189620
$899$	−2298.03	−0.0852543
$900$	−37876.6	−1.40284
$901$	3954.36	0.146214
$902$	0	0
$903$	3165.79	0.116668
$904$	−22779.7	−0.838099
$905$	26605.6	0.977239
$906$	17244.6	0.632353
$907$	−15301.5	−0.560175	−0.280087	−	0.959975i	$-0.590363\pi$
$907$	−15301.5	−0.560175	−0.280087	+	0.959975i	$0.590363\pi$
$908$	−61796.2	−2.25857
$909$	−36024.3	−1.31447
$910$	−49843.0	−1.81569
$911$	−19515.5	−0.709744	−0.354872	−	0.934915i	$-0.615475\pi$
$911$	−19515.5	−0.709744	−0.354872	+	0.934915i	$0.615475\pi$
$912$	103.107	0.00374366
$913$	0	0
$914$	−33735.1	−1.22085
$915$	−12901.2	−0.466120
$916$	−32834.4	−1.18437
$917$	−1699.59	−0.0612056
$918$	5131.60	0.184497
$919$	7054.60	0.253221	0.126610	−	0.991953i	$-0.459590\pi$
$919$	7054.60	0.253221	0.126610	+	0.991953i	$0.459590\pi$
$920$	7455.32	0.267168
$921$	−10544.6	−0.377260
$922$	19778.3	0.706467
$923$	−48105.0	−1.71549
$924$	0	0
$925$	−2040.43	−0.0725286
$926$	67349.4	2.39011
$927$	2486.74	0.0881071
$928$	−16384.7	−0.579584
$929$	−12597.3	−0.444891	−0.222445	−	0.974945i	$-0.571404\pi$
$929$	−12597.3	−0.444891	−0.222445	+	0.974945i	$0.571404\pi$
$930$	−1341.65	−0.0473058
$931$	780.034	0.0274593
$932$	−64589.6	−2.27007
$933$	−8833.04	−0.309947
$934$	7788.23	0.272847
$935$	0	0
$936$	−57296.3	−2.00084
$937$	−18890.2	−0.658608	−0.329304	−	0.944224i	$-0.606814\pi$
$937$	−18890.2	−0.658608	−0.329304	+	0.944224i	$0.606814\pi$
$938$	−5110.43	−0.177891
$939$	130.029	0.00451898
$940$	32404.7	1.12439
$941$	−31591.7	−1.09443	−0.547216	−	0.836991i	$-0.684312\pi$
$941$	−31591.7	−1.09443	−0.547216	+	0.836991i	$0.684312\pi$
$942$	13685.2	0.473343
$943$	4171.45	0.144052
$944$	−9466.09	−0.326372
$945$	−9039.50	−0.311169
$946$	0	0
$947$	48580.1	1.66699	0.833495	−	0.552527i	$-0.186336\pi$
$947$	48580.1	1.66699	0.833495	+	0.552527i	$0.186336\pi$
$948$	12708.8	0.435404
$949$	−20584.1	−0.704098
$950$	−1493.05	−0.0509904
$951$	9545.32	0.325476
$952$	4812.26	0.163830
$953$	−30330.4	−1.03095	−0.515477	−	0.856904i	$-0.672385\pi$
$953$	−30330.4	−1.03095	−0.515477	+	0.856904i	$0.672385\pi$
$954$	−28087.5	−0.953215
$955$	−55837.7	−1.89200
$956$	−75959.3	−2.56977
$957$	0	0
$958$	41896.4	1.41295
$959$	22224.7	0.748355
$960$	−13640.0	−0.458571
$961$	−29552.2	−0.991983
$962$	−6956.69	−0.233152
$963$	19462.1	0.651254
$964$	−5349.37	−0.178726
$965$	−56298.9	−1.87806
$966$	881.262	0.0293521
$967$	3638.39	0.120996	0.0604978	−	0.998168i	$-0.480731\pi$
$967$	3638.39	0.120996	0.0604978	+	0.998168i	$0.480731\pi$
$968$	0	0
$969$	63.1613	0.00209395
$970$	105427.	3.48976
$971$	45444.2	1.50193	0.750964	−	0.660343i	$-0.229589\pi$
$971$	45444.2	1.50193	0.750964	+	0.660343i	$0.229589\pi$
$972$	−35459.5	−1.17013
$973$	−23221.6	−0.765108
$974$	20032.9	0.659030
$975$	−9323.04	−0.306232
$976$	−19509.8	−0.639851
$977$	−57757.6	−1.89133	−0.945666	−	0.325140i	$-0.894588\pi$
$977$	−57757.6	−1.89133	−0.945666	+	0.325140i	$0.894588\pi$
$978$	617.111	0.0201769
$979$	0	0
$980$	55404.1	1.80594
$981$	−39593.6	−1.28861
$982$	−70074.2	−2.27715
$983$	20371.0	0.660971	0.330486	−	0.943811i	$-0.392787\pi$
$983$	20371.0	0.660971	0.330486	+	0.943811i	$0.392787\pi$
$984$	9354.85	0.303071
$985$	−56559.9	−1.82959
$986$	11959.0	0.386260
$987$	1699.50	0.0548083
$988$	−3270.82	−0.105323
$989$	4543.46	0.146080
$990$	0	0
$991$	−26839.8	−0.860336	−0.430168	−	0.902749i	$-0.641546\pi$
$991$	−26839.8	−0.860336	−0.430168	+	0.902749i	$0.641546\pi$
$992$	1702.82	0.0545007
$993$	5587.45	0.178562
$994$	−28697.8	−0.915735
$995$	−32865.1	−1.04713
$996$	22806.6	0.725556
$997$	7516.18	0.238756	0.119378	−	0.992849i	$-0.461910\pi$
$997$	7516.18	0.238756	0.119378	+	0.992849i	$0.461910\pi$
$998$	−44002.7	−1.39567
$999$	−1261.66	−0.0399572

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.t.1.40		44
11.3	even	5		187.4.g.a.86.20	✓	88
11.4	even	5		187.4.g.a.137.20	yes	88
11.10	odd	2		2057.4.a.s.1.5		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.a.86.20	✓	88	11.3	even	5
187.4.g.a.137.20	yes	88	11.4	even	5
2057.4.a.s.1.5		44	11.10	odd	2
2057.4.a.t.1.40		44	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:	\( N \)	\(=\)	\( 2057 = 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2057.a (trivial)

\(f(q)\)	\(=\)	\(q+4.73079 q^{2} -1.21482 q^{3} +14.3803 q^{4} -15.1061 q^{5} -5.74704 q^{6} +9.37827 q^{7} +30.1840 q^{8} -25.5242 q^{9} -71.4635 q^{10} -17.4695 q^{12} +74.3698 q^{13} +44.3666 q^{14} +18.3511 q^{15} +27.7515 q^{16} +17.0000 q^{17} -120.750 q^{18} -3.05838 q^{19} -217.230 q^{20} -11.3929 q^{21} -16.3508 q^{23} -36.6681 q^{24} +103.193 q^{25} +351.828 q^{26} +63.8073 q^{27} +134.863 q^{28} +148.700 q^{29} +86.8151 q^{30} -15.4541 q^{31} -110.186 q^{32} +80.4234 q^{34} -141.669 q^{35} -367.047 q^{36} -19.7730 q^{37} -14.4685 q^{38} -90.3458 q^{39} -455.961 q^{40} -255.122 q^{41} -53.8973 q^{42} -277.874 q^{43} +385.570 q^{45} -77.3520 q^{46} -149.172 q^{47} -33.7130 q^{48} -255.048 q^{49} +488.183 q^{50} -20.6519 q^{51} +1069.46 q^{52} +232.610 q^{53} +301.859 q^{54} +283.074 q^{56} +3.71537 q^{57} +703.470 q^{58} -341.102 q^{59} +263.895 q^{60} -703.020 q^{61} -73.1100 q^{62} -239.373 q^{63} -743.278 q^{64} -1123.43 q^{65} -115.186 q^{67} +244.466 q^{68} +19.8632 q^{69} -670.204 q^{70} -646.835 q^{71} -770.424 q^{72} -276.780 q^{73} -93.5418 q^{74} -125.360 q^{75} -43.9805 q^{76} -427.407 q^{78} -727.486 q^{79} -419.215 q^{80} +611.640 q^{81} -1206.93 q^{82} -1305.51 q^{83} -163.834 q^{84} -256.803 q^{85} -1314.56 q^{86} -180.644 q^{87} +1062.02 q^{89} +1824.05 q^{90} +697.460 q^{91} -235.130 q^{92} +18.7739 q^{93} -705.702 q^{94} +46.2000 q^{95} +133.856 q^{96} -1475.26 q^{97} -1206.58 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} - 24 q^{6} - 2 q^{7} - 63 q^{8} + 356 q^{9} - 65 q^{10} - 337 q^{12} + 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} + 748 q^{17} - 55 q^{18} - 36 q^{19}+ \cdots + 2302 q^{98}+O(q^{100}) \) 44 * q + q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 - 24 * q^6 - 2 * q^7 - 63 * q^8 + 356 * q^9 - 65 * q^10 - 337 * q^12 + 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 + 748 * q^17 - 55 * q^18 - 36 * q^19 - 209 * q^20 - 240 * q^21 - 656 * q^23 - 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 + 63 * q^28 - 698 * q^29 + 304 * q^30 - 1418 * q^31 - 83 * q^32 + 17 * q^34 + 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 + 114 * q^39 - 281 * q^40 + 54 * q^41 - 1943 * q^42 + 382 * q^43 - 1226 * q^45 + 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 + 1307 * q^50 - 476 * q^51 + 3112 * q^52 - 2492 * q^53 + 2400 * q^54 + 27 * q^56 + 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 - 964 * q^61 - 3256 * q^62 + 2322 * q^63 - 3653 * q^64 - 152 * q^65 - 3442 * q^67 + 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 - 1281 * q^72 - 2240 * q^73 + 3494 * q^74 - 1892 * q^75 + 1261 * q^76 + 1120 * q^78 - 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 - 1656 * q^83 - 5401 * q^84 - 408 * q^85 - 1373 * q^86 + 1968 * q^87 - 2992 * q^89 + 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 - 694 * q^94 - 8280 * q^95 + 2203 * q^96 - 4746 * q^97 + 2302 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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