LMFDB - Embedded newform 2057.4.a.s.1.5

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			1.5
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	$-0.815282\pi$
$2$	−4.73079	−1.67259	−0.836293	+	0.548283i	$0.815282\pi$
$3$	−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	$-0.581480\pi$
$7$	−9.37827	−0.506379	−0.253189	+	0.967417i	$0.581480\pi$
$8$	−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	$-0.791653\pi$
$13$	−74.3698	−1.58665	−0.793326	+	0.608797i	$0.791653\pi$
$14$	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	$-0.494122\pi$
$19$	3.05838	0.0369284	0.0184642	+	0.999830i	$0.494122\pi$
$20$	−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	$-0.657945\pi$
$29$	−148.700	−0.952172	−0.476086	+	0.879399i	$0.657945\pi$
$30$	−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	$-0.338494\pi$
$41$	255.122	0.971791	0.485896	+	0.874017i	$0.338494\pi$
$42$	−53.8973	−0.198013
$43$	277.874	0.985475	0.492738	−	0.870178i	$-0.335996\pi$
$43$	277.874	0.985475	0.492738	+	0.870178i	$0.335996\pi$
$44$	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	$-0.235862\pi$
$61$	703.020	1.47561	0.737807	+	0.675012i	$0.235862\pi$
$62$	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	$-0.428780\pi$
$73$	276.780	0.443763	0.221882	+	0.975074i	$0.428780\pi$
$74$	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	$-0.326666\pi$
$79$	727.486	1.03606	0.518029	+	0.855363i	$0.326666\pi$
$80$	−419.215	−0.585871
$81$	611.640	0.839012
$82$	−1206.93	−1.62540
$83$	1305.51	1.72648	0.863242	−	0.504790i	$-0.168430\pi$
$83$	1305.51	1.72648	0.863242	+	0.504790i	$0.168430\pi$
$84$	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	$-0.744699\pi$
$101$	−1411.38	−1.39047	−0.695234	+	0.718784i	$0.744699\pi$
$102$	−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	$-0.388064\pi$
$107$	762.495	0.688909	0.344454	+	0.938803i	$0.388064\pi$
$108$	917.571	0.817531
$109$	−1551.22	−1.36312	−0.681558	−	0.731764i	$-0.738697\pi$
$109$	−1551.22	−1.36312	−0.681558	+	0.731764i	$0.738697\pi$
$110$	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	$-0.787146\pi$
$127$	−2245.95	−1.56926	−0.784629	+	0.619966i	$0.787146\pi$
$128$	2634.80	1.81942
$129$	−337.567	−0.230396
$130$	−5314.73	−3.58564
$131$	181.227	0.120869	0.0604346	−	0.998172i	$-0.480751\pi$
$131$	181.227	0.120869	0.0604346	+	0.998172i	$0.480751\pi$
$132$	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	$-0.227409\pi$
$139$	2476.11	1.51094	0.755470	+	0.655183i	$0.227409\pi$
$140$	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	$-0.246095\pi$
$149$	2603.50	1.43145	0.715727	+	0.698380i	$0.246095\pi$
$150$	593.053	0.322817
$151$	3000.60	1.61712	0.808560	−	0.588414i	$-0.200247\pi$
$151$	3000.60	1.61712	0.808560	+	0.588414i	$0.200247\pi$
$152$	−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	$-0.372925\pi$
$167$	1677.71	0.777394	0.388697	+	0.921366i	$0.372925\pi$
$168$	−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	$-0.552759\pi$
$173$	−750.859	−0.329981	−0.164991	+	0.986295i	$0.552759\pi$
$174$	−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	$-0.744594\pi$
$193$	−3726.91	−1.38999	−0.694997	+	0.719012i	$0.744594\pi$
$194$	6979.13	2.58285
$195$	−1364.77	−0.501195
$196$	−3667.68	−1.33662
$197$	−3744.19	−1.35412	−0.677062	−	0.735926i	$-0.736747\pi$
$197$	−3744.19	−1.35412	−0.677062	+	0.735926i	$0.736747\pi$
$198$	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	$-0.226911\pi$
$211$	4637.23	1.51299	0.756493	+	0.654002i	$0.226911\pi$
$212$	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	$-0.283777\pi$
$227$	4297.27	1.25648	0.628238	+	0.778021i	$0.283777\pi$
$228$	−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	$-0.282466\pi$
$233$	4491.52	1.26287	0.631436	+	0.775428i	$0.282466\pi$
$234$	−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	$-0.246517\pi$
$239$	5282.16	1.42960	0.714801	+	0.699328i	$0.246517\pi$
$240$	509.270	0.136972
$241$	371.992	0.0994278	0.0497139	−	0.998763i	$-0.484169\pi$
$241$	371.992	0.0994278	0.0497139	+	0.998763i	$0.484169\pi$
$242$	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	$-0.667533\pi$
$263$	−4285.24	−1.00471	−0.502356	+	0.864661i	$0.667533\pi$
$264$	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	$-0.626336\pi$
$271$	−3449.04	−0.773116	−0.386558	+	0.922265i	$0.626336\pi$
$272$	−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	$-0.552872\pi$
$277$	−1524.50	−0.330679	−0.165339	+	0.986237i	$0.552872\pi$
$278$	−11713.9	−2.52718
$279$	394.454	0.0846427
$280$	−4276.13	−0.912670
$281$	1521.46	0.323000	0.161500	−	0.986873i	$-0.448367\pi$
$281$	1521.46	0.323000	0.161500	+	0.986873i	$0.448367\pi$
$282$	−857.299	−0.181033
$283$	5816.59	1.22177	0.610884	−	0.791720i	$-0.290814\pi$
$283$	5816.59	1.22177	0.610884	+	0.791720i	$0.290814\pi$
$284$	−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	$-0.630502\pi$
$293$	−3998.19	−0.797191	−0.398595	+	0.917127i	$0.630502\pi$
$294$	−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	$-0.798818\pi$
$307$	−8679.98	−1.61366	−0.806829	+	0.590785i	$0.798818\pi$
$308$	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	$-0.799635\pi$
$337$	−10001.6	−1.61668	−0.808342	+	0.588713i	$0.799635\pi$
$338$	−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	$-0.254545\pi$
$347$	9009.87	1.39388	0.696939	+	0.717131i	$0.254545\pi$
$348$	2597.72	0.400151
$349$	−3822.52	−0.586289	−0.293145	−	0.956068i	$-0.594702\pi$
$349$	−3822.52	−0.586289	−0.293145	+	0.956068i	$0.594702\pi$
$350$	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	$-0.425123\pi$
$359$	3170.70	0.466137	0.233068	+	0.972460i	$0.425123\pi$
$360$	−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	$-0.512209\pi$
$373$	−552.488	−0.0766936	−0.0383468	+	0.999264i	$0.512209\pi$
$374$	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	$-0.634555\pi$
$409$	−6786.60	−0.820479	−0.410240	+	0.911978i	$0.634555\pi$
$410$	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	$-0.685950\pi$
$431$	−9869.66	−1.10303	−0.551514	+	0.834166i	$0.685950\pi$
$432$	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	$-0.708929\pi$
$439$	−11226.2	−1.22049	−0.610246	+	0.792212i	$0.708929\pi$
$440$	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	$-0.381083\pi$
$457$	7130.96	0.729918	0.364959	+	0.931024i	$0.381083\pi$
$458$	10801.7	1.10204
$459$	−1084.72	−0.110306
$460$	3551.88	0.360016
$461$	−4180.76	−0.422380	−0.211190	−	0.977445i	$-0.567734\pi$
$461$	−4180.76	−0.422380	−0.211190	+	0.977445i	$0.567734\pi$
$462$	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	$-0.638807\pi$
$479$	−8856.11	−0.844773	−0.422386	+	0.906416i	$0.638807\pi$
$480$	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	$-0.261664\pi$
$491$	14812.4	1.36145	0.680726	+	0.732538i	$0.261664\pi$
$492$	−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	$-0.753114\pi$
$503$	−16109.2	−1.42798	−0.713991	+	0.700155i	$0.753114\pi$
$504$	−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	$-0.855445\pi$
$523$	−21496.6	−1.79728	−0.898642	+	0.438684i	$0.855445\pi$
$524$	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	$-0.233025\pi$
$541$	18718.8	1.48759	0.743793	+	0.668410i	$0.233025\pi$
$542$	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	$-0.294192\pi$
$547$	15414.5	1.20490	0.602448	+	0.798158i	$0.294192\pi$
$548$	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	$-0.587520\pi$
$557$	−7138.14	−0.543003	−0.271502	+	0.962438i	$0.587520\pi$
$558$	−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	$-0.579002\pi$
$563$	−6563.19	−0.491306	−0.245653	+	0.969358i	$0.579002\pi$
$564$	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	$-0.284584\pi$
$569$	17000.2	1.25252	0.626262	+	0.779613i	$0.284584\pi$
$570$	−265.513	−0.0195108
$571$	−5317.38	−0.389712	−0.194856	−	0.980832i	$-0.562424\pi$
$571$	−5317.38	−0.389712	−0.194856	+	0.980832i	$0.562424\pi$
$572$	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	$-0.719025\pi$
$593$	−18341.2	−1.27012	−0.635060	+	0.772463i	$0.719025\pi$
$594$	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	$-0.546792\pi$
$601$	−4316.20	−0.292948	−0.146474	+	0.989215i	$0.546792\pi$
$602$	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	$-0.207364\pi$
$607$	23784.3	1.59041	0.795203	+	0.606344i	$0.207364\pi$
$608$	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	$-0.731380\pi$
$613$	−20172.2	−1.32912	−0.664559	+	0.747236i	$0.731380\pi$
$614$	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	$-0.559585\pi$
$659$	−6296.60	−0.372201	−0.186101	+	0.982531i	$0.559585\pi$
$660$	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	$-0.319461\pi$
$673$	18760.0	1.07451	0.537256	+	0.843419i	$0.319461\pi$
$674$	47315.5	2.70404
$675$	6584.46	0.375461
$676$	47942.2	2.72771
$677$	−17467.6	−0.991632	−0.495816	−	0.868428i	$-0.665131\pi$
$677$	−17467.6	−0.991632	−0.495816	+	0.868428i	$0.665131\pi$
$678$	−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	$-0.483780\pi$
$701$	1890.63	0.101866	0.0509331	+	0.998702i	$0.483780\pi$
$702$	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	$-0.434192\pi$
$733$	8147.34	0.410544	0.205272	+	0.978705i	$0.434192\pi$
$734$	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	$-0.145712\pi$
$739$	36042.0	1.79408	0.897041	+	0.441948i	$0.145712\pi$
$740$	4295.29	0.213376
$741$	276.312	0.0136985
$742$	10320.1	0.510597
$743$	23296.4	1.15028	0.575142	−	0.818053i	$-0.304947\pi$
$743$	23296.4	1.15028	0.575142	+	0.818053i	$0.304947\pi$
$744$	−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	$-0.515109\pi$
$761$	−1992.12	−0.0948941	−0.0474471	+	0.998874i	$0.515109\pi$
$762$	−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	$-0.261589\pi$
$769$	29040.4	1.36180	0.680899	+	0.732377i	$0.261589\pi$
$770$	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	$-0.556160\pi$
$787$	−7750.27	−0.351038	−0.175519	+	0.984476i	$0.556160\pi$
$788$	−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	$-0.531099\pi$
$809$	−4489.12	−0.195091	−0.0975457	+	0.995231i	$0.531099\pi$
$810$	43709.9	1.89606
$811$	24225.5	1.04892	0.524460	−	0.851435i	$-0.324267\pi$
$811$	24225.5	1.04892	0.524460	+	0.851435i	$0.324267\pi$
$812$	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	$-0.615815\pi$
$821$	−16743.1	−0.711739	−0.355870	+	0.934536i	$0.615815\pi$
$822$	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	$-0.465623\pi$
$827$	5126.94	0.215576	0.107788	+	0.994174i	$0.465623\pi$
$828$	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	$-0.553764\pi$
$853$	−8375.87	−0.336207	−0.168103	+	0.985769i	$0.553764\pi$
$854$	31190.6	1.24979
$855$	1179.22	0.0471678
$856$	−23015.2	−0.918975
$857$	−44120.2	−1.75860	−0.879298	−	0.476273i	$-0.841987\pi$
$857$	−44120.2	−1.75860	−0.879298	+	0.476273i	$0.841987\pi$
$858$	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	$-0.665306\pi$
$877$	−25779.1	−0.992585	−0.496293	+	0.868155i	$0.665306\pi$
$878$	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	$-0.547211\pi$
$887$	−7807.55	−0.295549	−0.147775	+	0.989021i	$0.547211\pi$
$888$	−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	$-0.540410\pi$
$919$	−7054.60	−0.253221	−0.126610	+	0.991953i	$0.540410\pi$
$920$	−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	$-0.393186\pi$
$937$	18890.2	0.658608	0.329304	+	0.944224i	$0.393186\pi$
$938$	−5110.43	−0.177891
$939$	130.029	0.00451898
$940$	32404.7	1.12439
$941$	31591.7	1.09443	0.547216	−	0.836991i	$-0.315688\pi$
$941$	31591.7	1.09443	0.547216	+	0.836991i	$0.315688\pi$
$942$	−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	$-0.327615\pi$
$953$	30330.4	1.03095	0.515477	+	0.856904i	$0.327615\pi$
$954$	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	$-0.519269\pi$
$967$	−3638.39	−0.120996	−0.0604978	+	0.998168i	$0.519269\pi$
$968$	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	$-0.538090\pi$
$997$	−7516.18	−0.238756	−0.119378	+	0.992849i	$0.538090\pi$
$998$	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.s.1.5		44
11.7	odd	10		187.4.g.a.137.20	yes	88
11.8	odd	10		187.4.g.a.86.20	✓	88
11.10	odd	2		2057.4.a.t.1.40		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.a.86.20	✓	88	11.8	odd	10
187.4.g.a.137.20	yes	88	11.7	odd	10
2057.4.a.s.1.5		44	1.1	even	1	trivial
2057.4.a.t.1.40		44	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:	\( N \)	\(=\)	\( 2057 = 11^{2} \cdot 17 \)
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

Related objects

Downloads

Learn more