LMFDB - Embedded newform 2057.4.a.u.1.7

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 = [52,-1,20,235,40] 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:	$0$
Dimension:	$52$
Twist minimal:	no (minimal twist has level 187)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	\(q-4.59808 q^{2} -2.80460 q^{3} +13.1424 q^{4} +1.26573 q^{5} +12.8958 q^{6} -24.2666 q^{7} -23.6449 q^{8} -19.1342 q^{9} -5.81993 q^{10} -36.8591 q^{12} +8.37678 q^{13} +111.580 q^{14} -3.54987 q^{15} +3.58258 q^{16} +17.0000 q^{17} +87.9806 q^{18} -80.3701 q^{19} +16.6347 q^{20} +68.0583 q^{21} +34.5851 q^{23} +66.3147 q^{24} -123.398 q^{25} -38.5171 q^{26} +129.388 q^{27} -318.921 q^{28} -131.037 q^{29} +16.3226 q^{30} +313.539 q^{31} +172.687 q^{32} -78.1674 q^{34} -30.7150 q^{35} -251.468 q^{36} -118.221 q^{37} +369.548 q^{38} -23.4935 q^{39} -29.9281 q^{40} -326.298 q^{41} -312.938 q^{42} -482.045 q^{43} -24.2187 q^{45} -159.025 q^{46} -575.911 q^{47} -10.0477 q^{48} +245.870 q^{49} +567.394 q^{50} -47.6783 q^{51} +110.091 q^{52} +387.949 q^{53} -594.937 q^{54} +573.783 q^{56} +225.406 q^{57} +602.517 q^{58} +481.128 q^{59} -46.6537 q^{60} -489.261 q^{61} -1441.68 q^{62} +464.323 q^{63} -822.688 q^{64} +10.6027 q^{65} -547.688 q^{67} +223.420 q^{68} -96.9975 q^{69} +141.230 q^{70} -227.013 q^{71} +452.427 q^{72} -794.129 q^{73} +543.589 q^{74} +346.082 q^{75} -1056.25 q^{76} +108.025 q^{78} -277.963 q^{79} +4.53459 q^{80} +153.741 q^{81} +1500.35 q^{82} +30.8937 q^{83} +894.446 q^{84} +21.5174 q^{85} +2216.48 q^{86} +367.506 q^{87} -195.375 q^{89} +111.360 q^{90} -203.276 q^{91} +454.530 q^{92} -879.352 q^{93} +2648.08 q^{94} -101.727 q^{95} -484.317 q^{96} +1634.86 q^{97} -1130.53 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q - q^{2} + 20 q^{3} + 235 q^{4} + 40 q^{5} + 24 q^{6} + 42 q^{7} - 45 q^{8} + 572 q^{9} - 33 q^{10} + 233 q^{12} - 12 q^{13} + 73 q^{14} + 400 q^{15} + 1223 q^{16} + 884 q^{17} - 201 q^{18} + 44 q^{19}+ \cdots + 4610 q^{98}+O(q^{100}) $ 52 * q - q^2 + 20 * q^3 + 235 * q^4 + 40 * q^5 + 24 * q^6 + 42 * q^7 - 45 * q^8 + 572 * q^9 - 33 * q^10 + 233 * q^12 - 12 * q^13 + 73 * q^14 + 400 * q^15 + 1223 * q^16 + 884 * q^17 - 201 * q^18 + 44 * q^19 + 655 * q^20 - 260 * q^21 + 572 * q^23 + 104 * q^24 + 1858 * q^25 + 465 * q^26 + 1070 * q^27 + 577 * q^28 - 322 * q^29 + 320 * q^30 + 1110 * q^31 - 481 * q^32 - 17 * q^34 - 102 * q^35 + 2507 * q^36 + 1678 * q^37 - 360 * q^38 + 1282 * q^39 - 1791 * q^40 + 826 * q^41 + 2133 * q^42 - 270 * q^43 + 710 * q^45 - 2158 * q^46 + 2464 * q^47 + 2201 * q^48 + 3224 * q^49 - 2379 * q^50 + 340 * q^51 + 3664 * q^52 + 992 * q^53 - 1202 * q^54 + 1731 * q^56 - 1016 * q^57 + 1358 * q^58 + 1442 * q^59 + 1444 * q^60 + 140 * q^61 - 464 * q^62 + 766 * q^63 + 8427 * q^64 + 1268 * q^65 + 5766 * q^67 + 3995 * q^68 + 2460 * q^69 + 2422 * q^70 + 2704 * q^71 - 5455 * q^72 - 4 * q^73 + 4008 * q^74 + 5204 * q^75 + 1935 * q^76 + 4092 * q^78 + 2180 * q^79 + 5040 * q^80 + 7192 * q^81 + 3197 * q^82 - 4200 * q^83 - 7951 * q^84 + 680 * q^85 + 3091 * q^86 + 752 * q^87 - 240 * q^89 + 4495 * q^90 + 5494 * q^91 + 6902 * q^92 + 6266 * q^93 - 5990 * q^94 - 3168 * q^95 + 9467 * q^96 + 5322 * q^97 + 4610 * 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 $

$2$	−4.59808	−1.62567	−0.812834	−	0.582496i	$-0.802076\pi$
$2$	−4.59808	−1.62567	−0.812834	+	0.582496i	$0.802076\pi$
$3$	−2.80460	−0.539746	−0.269873	−	0.962896i	$-0.586982\pi$
$3$	−2.80460	−0.539746	−0.269873	+	0.962896i	$0.586982\pi$
$4$	13.1424	1.64279
$5$	1.26573	0.113210	0.0566052	−	0.998397i	$-0.481972\pi$
$5$	1.26573	0.113210	0.0566052	+	0.998397i	$0.481972\pi$
$6$	12.8958	0.877448
$7$	−24.2666	−1.31028	−0.655138	−	0.755510i	$-0.727389\pi$
$7$	−24.2666	−1.31028	−0.655138	+	0.755510i	$0.727389\pi$
$8$	−23.6449	−1.04497
$9$	−19.1342	−0.708674
$10$	−5.81993	−0.184042
$11$	0	0
$11$	0	0
$12$	−36.8591	−0.886692
$13$	8.37678	0.178715	0.0893577	−	0.996000i	$-0.471519\pi$
$13$	8.37678	0.178715	0.0893577	+	0.996000i	$0.471519\pi$
$14$	111.580	2.13007
$15$	−3.54987	−0.0611049
$16$	3.58258	0.0559779
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	87.9806	1.15207
$19$	−80.3701	−0.970429	−0.485215	−	0.874395i	$-0.661259\pi$
$19$	−80.3701	−0.970429	−0.485215	+	0.874395i	$0.661259\pi$
$20$	16.6347	0.185981
$21$	68.0583	0.707216
$22$	0	0
$23$	34.5851	0.313543	0.156772	−	0.987635i	$-0.449891\pi$
$23$	34.5851	0.313543	0.156772	+	0.987635i	$0.449891\pi$
$24$	66.3147	0.564018
$25$	−123.398	−0.987183
$26$	−38.5171	−0.290532
$27$	129.388	0.922250
$28$	−318.921	−2.15251
$29$	−131.037	−0.839066	−0.419533	−	0.907740i	$-0.637806\pi$
$29$	−131.037	−0.839066	−0.419533	+	0.907740i	$0.637806\pi$
$30$	16.3226	0.0993362
$31$	313.539	1.81656	0.908278	−	0.418368i	$-0.137398\pi$
$31$	313.539	1.81656	0.908278	+	0.418368i	$0.137398\pi$
$32$	172.687	0.953968
$33$	0	0
$34$	−78.1674	−0.394282
$35$	−30.7150	−0.148337
$36$	−251.468	−1.16421
$37$	−118.221	−0.525280	−0.262640	−	0.964894i	$-0.584593\pi$
$37$	−118.221	−0.525280	−0.262640	+	0.964894i	$0.584593\pi$
$38$	369.548	1.57760
$39$	−23.4935	−0.0964609
$40$	−29.9281	−0.118301
$41$	−326.298	−1.24291	−0.621454	−	0.783451i	$-0.713458\pi$
$41$	−326.298	−1.24291	−0.621454	+	0.783451i	$0.713458\pi$
$42$	−312.938	−1.14970
$43$	−482.045	−1.70956	−0.854780	−	0.518990i	$-0.826308\pi$
$43$	−482.045	−1.70956	−0.854780	+	0.518990i	$0.826308\pi$
$44$	0	0
$45$	−24.2187	−0.0802293
$46$	−159.025	−0.509717
$47$	−575.911	−1.78735	−0.893673	−	0.448720i	$-0.851880\pi$
$47$	−575.911	−1.78735	−0.893673	+	0.448720i	$0.851880\pi$
$48$	−10.0477	−0.0302138
$49$	245.870	0.716821
$50$	567.394	1.60483
$51$	−47.6783	−0.130908
$52$	110.091	0.293593
$53$	387.949	1.00545	0.502726	−	0.864446i	$-0.332331\pi$
$53$	387.949	1.00545	0.502726	+	0.864446i	$0.332331\pi$
$54$	−594.937	−1.49927
$55$	0	0
$56$	573.783	1.36920
$57$	225.406	0.523786
$58$	602.517	1.36404
$59$	481.128	1.06165	0.530827	−	0.847480i	$-0.321881\pi$
$59$	481.128	1.06165	0.530827	+	0.847480i	$0.321881\pi$
$60$	−46.6537	−0.100383
$61$	−489.261	−1.02694	−0.513471	−	0.858107i	$-0.671641\pi$
$61$	−489.261	−1.02694	−0.513471	+	0.858107i	$0.671641\pi$
$62$	−1441.68	−2.95311
$63$	464.323	0.928558
$64$	−822.688	−1.60681
$65$	10.6027	0.0202324
$66$	0	0
$67$	−547.688	−0.998667	−0.499334	−	0.866410i	$-0.666422\pi$
$67$	−547.688	−0.998667	−0.499334	+	0.866410i	$0.666422\pi$
$68$	223.420	0.398436
$69$	−96.9975	−0.169234
$70$	141.230	0.241146
$71$	−227.013	−0.379457	−0.189728	−	0.981837i	$-0.560761\pi$
$71$	−227.013	−0.379457	−0.189728	+	0.981837i	$0.560761\pi$
$72$	452.427	0.740542
$73$	−794.129	−1.27323	−0.636615	−	0.771182i	$-0.719666\pi$
$73$	−794.129	−1.27323	−0.636615	+	0.771182i	$0.719666\pi$
$74$	543.589	0.853931
$75$	346.082	0.532828
$76$	−1056.25	−1.59422
$77$	0	0
$78$	108.025	0.156813
$79$	−277.963	−0.395864	−0.197932	−	0.980216i	$-0.563423\pi$
$79$	−277.963	−0.395864	−0.197932	+	0.980216i	$0.563423\pi$
$80$	4.53459	0.00633728
$81$	153.741	0.210893
$82$	1500.35	2.02056
$83$	30.8937	0.0408558	0.0204279	−	0.999791i	$-0.493497\pi$
$83$	30.8937	0.0408558	0.0204279	+	0.999791i	$0.493497\pi$
$84$	894.446	1.16181
$85$	21.5174	0.0274575
$86$	2216.48	2.77918
$87$	367.506	0.452882
$88$	0	0
$89$	−195.375	−0.232694	−0.116347	−	0.993209i	$-0.537118\pi$
$89$	−195.375	−0.232694	−0.116347	+	0.993209i	$0.537118\pi$
$90$	111.360	0.130426
$91$	−203.276	−0.234166
$92$	454.530	0.515087
$93$	−879.352	−0.980479
$94$	2648.08	2.90563
$95$	−101.727	−0.109863
$96$	−484.317	−0.514900
$97$	1634.86	1.71129	0.855644	−	0.517565i	$-0.173162\pi$
$97$	1634.86	1.71129	0.855644	+	0.517565i	$0.173162\pi$
$98$	−1130.53	−1.16531
$99$	0	0
$100$	−1621.74	−1.62174
$101$	895.835	0.882563	0.441282	−	0.897369i	$-0.354524\pi$
$101$	895.835	0.882563	0.441282	+	0.897369i	$0.354524\pi$
$102$	219.228	0.212812
$103$	−711.884	−0.681010	−0.340505	−	0.940243i	$-0.610598\pi$
$103$	−711.884	−0.681010	−0.340505	+	0.940243i	$0.610598\pi$
$104$	−198.068	−0.186752
$105$	86.1434	0.0800642
$106$	−1783.82	−1.63453
$107$	197.173	0.178144	0.0890720	−	0.996025i	$-0.471610\pi$
$107$	197.173	0.178144	0.0890720	+	0.996025i	$0.471610\pi$
$108$	1700.46	1.51507
$109$	−145.629	−0.127970	−0.0639852	−	0.997951i	$-0.520381\pi$
$109$	−145.629	−0.127970	−0.0639852	+	0.997951i	$0.520381\pi$
$110$	0	0
$111$	331.562	0.283518
$112$	−86.9373	−0.0733464
$113$	−2093.40	−1.74275	−0.871376	−	0.490617i	$-0.836772\pi$
$113$	−2093.40	−1.74275	−0.871376	+	0.490617i	$0.836772\pi$
$114$	−1036.44	−0.851501
$115$	43.7754	0.0354963
$116$	−1722.13	−1.37841
$117$	−160.283	−0.126651
$118$	−2212.27	−1.72590
$119$	−412.533	−0.317788
$120$	83.9365	0.0638527
$121$	0	0
$122$	2249.66	1.66947
$123$	915.137	0.670855
$124$	4120.64	2.98423
$125$	−314.405	−0.224970
$126$	−2134.99	−1.50953
$127$	−1560.20	−1.09012	−0.545060	−	0.838397i	$-0.683493\pi$
$127$	−1560.20	−1.09012	−0.545060	+	0.838397i	$0.683493\pi$
$128$	2401.29	1.65817
$129$	1351.94	0.922729
$130$	−48.7523	−0.0328912
$131$	−1233.27	−0.822528	−0.411264	−	0.911516i	$-0.634913\pi$
$131$	−1233.27	−0.822528	−0.411264	+	0.911516i	$0.634913\pi$
$132$	0	0
$133$	1950.31	1.27153
$134$	2518.31	1.62350
$135$	163.770	0.104408
$136$	−401.964	−0.253442
$137$	1987.87	1.23967	0.619836	−	0.784732i	$-0.287199\pi$
$137$	1987.87	1.23967	0.619836	+	0.784732i	$0.287199\pi$
$138$	446.002	0.275118
$139$	−623.314	−0.380351	−0.190175	−	0.981750i	$-0.560906\pi$
$139$	−623.314	−0.380351	−0.190175	+	0.981750i	$0.560906\pi$
$140$	−403.668	−0.243687
$141$	1615.20	0.964713
$142$	1043.82	0.616871
$143$	0	0
$144$	−68.5499	−0.0396701
$145$	−165.857	−0.0949909
$146$	3651.47	2.06985
$147$	−689.567	−0.386901
$148$	−1553.70	−0.862927
$149$	−1898.79	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$149$	−1898.79	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$150$	−1591.31	−0.866202
$151$	−1963.03	−1.05794	−0.528969	−	0.848641i	$-0.677421\pi$
$151$	−1963.03	−1.05794	−0.528969	+	0.848641i	$0.677421\pi$
$152$	1900.35	1.01407
$153$	−325.281	−0.171879
$154$	0	0
$155$	396.855	0.205653
$156$	−308.760	−0.158465
$157$	1546.62	0.786204	0.393102	−	0.919495i	$-0.371402\pi$
$157$	1546.62	0.786204	0.393102	+	0.919495i	$0.371402\pi$
$158$	1278.10	0.643543
$159$	−1088.04	−0.542689
$160$	218.575	0.107999
$161$	−839.264	−0.410828
$162$	−706.914	−0.342842
$163$	−2832.34	−1.36102	−0.680508	−	0.732740i	$-0.738241\pi$
$163$	−2832.34	−1.36102	−0.680508	+	0.732740i	$0.738241\pi$
$164$	−4288.33	−2.04184
$165$	0	0
$166$	−142.052	−0.0664179
$167$	−1227.70	−0.568874	−0.284437	−	0.958695i	$-0.591807\pi$
$167$	−1227.70	−0.568874	−0.284437	+	0.958695i	$0.591807\pi$
$168$	−1609.23	−0.739019
$169$	−2126.83	−0.968061
$170$	−98.9388	−0.0446368
$171$	1537.82	0.687718
$172$	−6335.20	−2.80846
$173$	−1445.39	−0.635208	−0.317604	−	0.948223i	$-0.602878\pi$
$173$	−1445.39	−0.635208	−0.317604	+	0.948223i	$0.602878\pi$
$174$	−1689.82	−0.736236
$175$	2994.45	1.29348
$176$	0	0
$177$	−1349.37	−0.573024
$178$	898.352	0.378283
$179$	−4488.13	−1.87407	−0.937034	−	0.349237i	$-0.886441\pi$
$179$	−4488.13	−1.87407	−0.937034	+	0.349237i	$0.886441\pi$
$180$	−318.291	−0.131800
$181$	−2527.90	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$181$	−2527.90	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$182$	934.681	0.380677
$183$	1372.18	0.554288
$184$	−817.763	−0.327643
$185$	−149.636	−0.0594672
$186$	4043.33	1.59393
$187$	0	0
$188$	−7568.82	−2.93624
$189$	−3139.81	−1.20840
$190$	467.748	0.178600
$191$	−3435.79	−1.30160	−0.650799	−	0.759250i	$-0.725566\pi$
$191$	−3435.79	−1.30160	−0.650799	+	0.759250i	$0.725566\pi$
$192$	2307.31	0.867270
$193$	−621.559	−0.231818	−0.115909	−	0.993260i	$-0.536978\pi$
$193$	−621.559	−0.231818	−0.115909	+	0.993260i	$0.536978\pi$
$194$	−7517.22	−2.78198
$195$	−29.7365	−0.0109204
$196$	3231.31	1.17759
$197$	−3911.12	−1.41450	−0.707248	−	0.706965i	$-0.750064\pi$
$197$	−3911.12	−1.41450	−0.707248	+	0.706965i	$0.750064\pi$
$198$	0	0
$199$	828.102	0.294988	0.147494	−	0.989063i	$-0.452879\pi$
$199$	828.102	0.294988	0.147494	+	0.989063i	$0.452879\pi$
$200$	2917.74	1.03158
$201$	1536.05	0.539027
$202$	−4119.12	−1.43475
$203$	3179.82	1.09941
$204$	−626.604	−0.215054
$205$	−413.006	−0.140710
$206$	3273.30	1.10710
$207$	−661.758	−0.222200
$208$	30.0105	0.0100041
$209$	0	0
$210$	−396.095	−0.130158
$211$	−4967.39	−1.62071	−0.810353	−	0.585942i	$-0.800725\pi$
$211$	−4967.39	−1.62071	−0.810353	+	0.585942i	$0.800725\pi$
$212$	5098.57	1.65175
$213$	636.680	0.204810
$214$	−906.616	−0.289603
$215$	−610.138	−0.193540
$216$	−3059.38	−0.963723
$217$	−7608.53	−2.38019
$218$	669.616	0.208037
$219$	2227.22	0.687221
$220$	0	0
$221$	142.405	0.0433448
$222$	−1524.55	−0.460906
$223$	3972.86	1.19302	0.596508	−	0.802607i	$-0.296554\pi$
$223$	3972.86	1.19302	0.596508	+	0.802607i	$0.296554\pi$
$224$	−4190.52	−1.24996
$225$	2361.12	0.699591
$226$	9625.64	2.83313
$227$	798.165	0.233375	0.116687	−	0.993169i	$-0.462772\pi$
$227$	798.165	0.233375	0.116687	+	0.993169i	$0.462772\pi$
$228$	2962.37	0.860472
$229$	−1080.36	−0.311757	−0.155878	−	0.987776i	$-0.549821\pi$
$229$	−1080.36	−0.311757	−0.155878	+	0.987776i	$0.549821\pi$
$230$	−201.283	−0.0577052
$231$	0	0
$232$	3098.36	0.876798
$233$	−919.521	−0.258540	−0.129270	−	0.991609i	$-0.541263\pi$
$233$	−919.521	−0.258540	−0.129270	+	0.991609i	$0.541263\pi$
$234$	736.994	0.205892
$235$	−728.948	−0.202346
$236$	6323.16	1.74408
$237$	779.575	0.213666
$238$	1896.86	0.516618
$239$	319.095	0.0863621	0.0431810	−	0.999067i	$-0.486251\pi$
$239$	319.095	0.0863621	0.0431810	+	0.999067i	$0.486251\pi$
$240$	−12.7177	−0.00342052
$241$	357.428	0.0955352	0.0477676	−	0.998858i	$-0.484789\pi$
$241$	357.428	0.0955352	0.0477676	+	0.998858i	$0.484789\pi$
$242$	0	0
$243$	−3924.66	−1.03608
$244$	−6430.04	−1.68705
$245$	311.205	0.0811516
$246$	−4207.88	−1.09059
$247$	−673.242	−0.173431
$248$	−7413.61	−1.89824
$249$	−86.6447	−0.0220517
$250$	1445.66	0.365726
$251$	2797.57	0.703510	0.351755	−	0.936092i	$-0.385585\pi$
$251$	2797.57	0.703510	0.351755	+	0.936092i	$0.385585\pi$
$252$	6102.29	1.52543
$253$	0	0
$254$	7173.92	1.77217
$255$	−60.3478	−0.0148201
$256$	−4459.83	−1.08883
$257$	2971.84	0.721316	0.360658	−	0.932698i	$-0.382552\pi$
$257$	2971.84	0.721316	0.360658	+	0.932698i	$0.382552\pi$
$258$	−6216.35	−1.50005
$259$	2868.82	0.688262
$260$	139.345	0.0332377
$261$	2507.28	0.594624
$262$	5670.67	1.33716
$263$	3076.65	0.721348	0.360674	−	0.932692i	$-0.382547\pi$
$263$	3076.65	0.721348	0.360674	+	0.932692i	$0.382547\pi$
$264$	0	0
$265$	491.039	0.113828
$266$	−8967.69	−2.06708
$267$	547.951	0.125596
$268$	−7197.90	−1.64060
$269$	5486.97	1.24367	0.621834	−	0.783149i	$-0.286388\pi$
$269$	5486.97	1.24367	0.621834	+	0.783149i	$0.286388\pi$
$270$	−753.030	−0.169733
$271$	4423.26	0.991491	0.495746	−	0.868468i	$-0.334895\pi$
$271$	4423.26	0.991491	0.495746	+	0.868468i	$0.334895\pi$
$272$	60.9039	0.0135766
$273$	570.109	0.126390
$274$	−9140.38	−2.01529
$275$	0	0
$276$	−1274.77	−0.278016
$277$	1125.05	0.244036	0.122018	−	0.992528i	$-0.461063\pi$
$277$	1125.05	0.244036	0.122018	+	0.992528i	$0.461063\pi$
$278$	2866.05	0.618324
$279$	−5999.31	−1.28735
$280$	726.255	0.155007
$281$	3802.52	0.807258	0.403629	−	0.914923i	$-0.367749\pi$
$281$	3802.52	0.807258	0.403629	+	0.914923i	$0.367749\pi$
$282$	−7426.83	−1.56830
$283$	5454.36	1.14568	0.572841	−	0.819667i	$-0.305841\pi$
$283$	5454.36	1.14568	0.572841	+	0.819667i	$0.305841\pi$
$284$	−2983.48	−0.623369
$285$	285.303	0.0592980
$286$	0	0
$287$	7918.16	1.62855
$288$	−3304.22	−0.676052
$289$	289.000	0.0588235
$290$	762.625	0.154424
$291$	−4585.13	−0.923661
$292$	−10436.7	−2.09165
$293$	−9409.06	−1.87605	−0.938026	−	0.346564i	$-0.887348\pi$
$293$	−9409.06	−1.87605	−0.938026	+	0.346564i	$0.887348\pi$
$294$	3170.68	0.628973
$295$	608.979	0.120190
$296$	2795.32	0.548902
$297$	0	0
$298$	8730.78	1.69718
$299$	289.712	0.0560350
$300$	4548.33	0.875327
$301$	11697.6	2.23999
$302$	9026.15	1.71986
$303$	−2512.46	−0.476360
$304$	−287.933	−0.0543226
$305$	−619.273	−0.116260
$306$	1495.67	0.279418
$307$	8256.58	1.53494	0.767472	−	0.641082i	$-0.221514\pi$
$307$	8256.58	1.53494	0.767472	+	0.641082i	$0.221514\pi$
$308$	0	0
$309$	1996.55	0.367572
$310$	−1824.77	−0.334323
$311$	−9790.65	−1.78513	−0.892567	−	0.450914i	$-0.851098\pi$
$311$	−9790.65	−1.78513	−0.892567	+	0.450914i	$0.851098\pi$
$312$	555.503	0.100799
$313$	3000.48	0.541844	0.270922	−	0.962601i	$-0.412671\pi$
$313$	3000.48	0.541844	0.270922	+	0.962601i	$0.412671\pi$
$314$	−7111.50	−1.27811
$315$	587.707	0.105122
$316$	−3653.08	−0.650323
$317$	1881.65	0.333387	0.166694	−	0.986009i	$-0.446691\pi$
$317$	1881.65	0.333387	0.166694	+	0.986009i	$0.446691\pi$
$318$	5002.91	0.882231
$319$	0	0
$320$	−1041.30	−0.181908
$321$	−552.991	−0.0961525
$322$	3859.00	0.667869
$323$	−1366.29	−0.235364
$324$	2020.52	0.346454
$325$	−1033.68	−0.176425
$326$	13023.3	2.21256
$327$	408.433	0.0690715
$328$	7715.31	1.29880
$329$	13975.4	2.34191
$330$	0	0
$331$	1730.96	0.287439	0.143720	−	0.989618i	$-0.454094\pi$
$331$	1730.96	0.287439	0.143720	+	0.989618i	$0.454094\pi$
$332$	406.016	0.0671176
$333$	2262.06	0.372253
$334$	5645.04	0.924800
$335$	−693.225	−0.113059
$336$	243.825	0.0395884
$337$	−2631.37	−0.425341	−0.212671	−	0.977124i	$-0.568216\pi$
$337$	−2631.37	−0.425341	−0.212671	+	0.977124i	$0.568216\pi$
$338$	9779.34	1.57374
$339$	5871.17	0.940643
$340$	282.789	0.0451071
$341$	0	0
$342$	−7071.01	−1.11800
$343$	2357.03	0.371042
$344$	11397.9	1.78644
$345$	−122.773	−0.0191590
$346$	6646.02	1.03264
$347$	−301.909	−0.0467069	−0.0233535	−	0.999727i	$-0.507434\pi$
$347$	−301.909	−0.0467069	−0.0233535	+	0.999727i	$0.507434\pi$
$348$	4829.89	0.743992
$349$	2778.26	0.426122	0.213061	−	0.977039i	$-0.431657\pi$
$349$	2778.26	0.426122	0.213061	+	0.977039i	$0.431657\pi$
$350$	−13768.7	−2.10277
$351$	1083.86	0.164820
$352$	0	0
$353$	−6498.55	−0.979839	−0.489919	−	0.871768i	$-0.662974\pi$
$353$	−6498.55	−0.979839	−0.489919	+	0.871768i	$0.662974\pi$
$354$	6204.53	0.931546
$355$	−287.337	−0.0429585
$356$	−2567.69	−0.382268
$357$	1156.99	0.171525
$358$	20636.8	3.04661
$359$	−101.524	−0.0149254	−0.00746270	−	0.999972i	$-0.502375\pi$
$359$	−101.524	−0.0149254	−0.00746270	+	0.999972i	$0.502375\pi$
$360$	572.651	0.0838371
$361$	−399.651	−0.0582666
$362$	11623.5	1.68762
$363$	0	0
$364$	−2671.53	−0.384687
$365$	−1005.15	−0.144143
$366$	−6309.41	−0.901087
$367$	5884.56	0.836979	0.418490	−	0.908222i	$-0.362560\pi$
$367$	5884.56	0.836979	0.418490	+	0.908222i	$0.362560\pi$
$368$	123.904	0.0175515
$369$	6243.46	0.880817
$370$	688.037	0.0966738
$371$	−9414.22	−1.31742
$372$	−11556.7	−1.61072
$373$	2403.84	0.333689	0.166845	−	0.985983i	$-0.446642\pi$
$373$	2403.84	0.333689	0.166845	+	0.985983i	$0.446642\pi$
$374$	0	0
$375$	881.781	0.121427
$376$	13617.4	1.86772
$377$	−1097.67	−0.149954
$378$	14437.1	1.96446
$379$	2211.94	0.299788	0.149894	−	0.988702i	$-0.452107\pi$
$379$	2211.94	0.299788	0.149894	+	0.988702i	$0.452107\pi$
$380$	−1336.93	−0.180482
$381$	4375.74	0.588388
$382$	15798.1	2.11597
$383$	−4141.62	−0.552550	−0.276275	−	0.961079i	$-0.589100\pi$
$383$	−4141.62	−0.552550	−0.276275	+	0.961079i	$0.589100\pi$
$384$	−6734.67	−0.894993
$385$	0	0
$386$	2857.98	0.376858
$387$	9223.54	1.21152
$388$	21485.9	2.81129
$389$	3884.11	0.506252	0.253126	−	0.967433i	$-0.418541\pi$
$389$	3884.11	0.506252	0.253126	+	0.967433i	$0.418541\pi$
$390$	136.731	0.0177529
$391$	587.947	0.0760454
$392$	−5813.58	−0.749056
$393$	3458.83	0.443956
$394$	17983.7	2.29950
$395$	−351.826	−0.0448159
$396$	0	0
$397$	−12783.9	−1.61614	−0.808070	−	0.589087i	$-0.799488\pi$
$397$	−12783.9	−1.61614	−0.808070	+	0.589087i	$0.799488\pi$
$398$	−3807.68	−0.479552
$399$	−5469.85	−0.686303
$400$	−442.083	−0.0552604
$401$	−9299.04	−1.15803	−0.579017	−	0.815315i	$-0.696564\pi$
$401$	−9299.04	−1.15803	−0.579017	+	0.815315i	$0.696564\pi$
$402$	−7062.87	−0.876278
$403$	2626.44	0.324646
$404$	11773.4	1.44987
$405$	194.595	0.0238753
$406$	−14621.1	−1.78727
$407$	0	0
$408$	1127.35	0.136794
$409$	−1300.05	−0.157172	−0.0785858	−	0.996907i	$-0.525040\pi$
$409$	−1300.05	−0.157172	−0.0785858	+	0.996907i	$0.525040\pi$
$410$	1899.03	0.228748
$411$	−5575.18	−0.669108
$412$	−9355.83	−1.11876
$413$	−11675.4	−1.39106
$414$	3042.82	0.361223
$415$	39.1032	0.00462530
$416$	1446.56	0.170489
$417$	1748.15	0.205293
$418$	0	0
$419$	−7209.60	−0.840601	−0.420301	−	0.907385i	$-0.638075\pi$
$419$	−7209.60	−0.840601	−0.420301	+	0.907385i	$0.638075\pi$
$420$	1132.13	0.131529
$421$	1214.75	0.140625	0.0703124	−	0.997525i	$-0.477600\pi$
$421$	1214.75	0.140625	0.0703124	+	0.997525i	$0.477600\pi$
$422$	22840.4	2.63473
$423$	11019.6	1.26665
$424$	−9173.04	−1.05067
$425$	−2097.76	−0.239427
$426$	−2927.51	−0.332953
$427$	11872.7	1.34558
$428$	2591.31	0.292654
$429$	0	0
$430$	2805.47	0.314632
$431$	−921.527	−0.102989	−0.0514947	−	0.998673i	$-0.516399\pi$
$431$	−921.527	−0.102989	−0.0514947	+	0.998673i	$0.516399\pi$
$432$	463.544	0.0516256
$433$	−1266.30	−0.140541	−0.0702706	−	0.997528i	$-0.522386\pi$
$433$	−1266.30	−0.140541	−0.0702706	+	0.997528i	$0.522386\pi$
$434$	34984.6	3.86939
$435$	465.163	0.0512710
$436$	−1913.91	−0.210229
$437$	−2779.61	−0.304271
$438$	−10240.9	−1.11719
$439$	−2452.90	−0.266675	−0.133338	−	0.991071i	$-0.542569\pi$
$439$	−2452.90	−0.266675	−0.133338	+	0.991071i	$0.542569\pi$
$440$	0	0
$441$	−4704.52	−0.507993
$442$	−654.791	−0.0704643
$443$	1653.48	0.177335	0.0886674	−	0.996061i	$-0.471739\pi$
$443$	1653.48	0.177335	0.0886674	+	0.996061i	$0.471739\pi$
$444$	4357.51	0.465762
$445$	−247.293	−0.0263434
$446$	−18267.5	−1.93945
$447$	5325.35	0.563491
$448$	19963.9	2.10537
$449$	16.2585	0.00170887	0.000854437	−	1.00000i	$-0.499728\pi$
$449$	16.2585	0.00170887	0.000854437		1.00000i	$0.499728\pi$
$450$	−10856.6	−1.13730
$451$	0	0
$452$	−27512.2	−2.86298
$453$	5505.51	0.571018
$454$	−3670.03	−0.379390
$455$	−257.293	−0.0265101
$456$	−5329.72	−0.547340
$457$	17074.0	1.74768	0.873838	−	0.486217i	$-0.161624\pi$
$457$	17074.0	1.74768	0.873838	+	0.486217i	$0.161624\pi$
$458$	4967.59	0.506813
$459$	2199.60	0.223679
$460$	575.312	0.0583132
$461$	−7894.51	−0.797579	−0.398790	−	0.917042i	$-0.630570\pi$
$461$	−7894.51	−0.797579	−0.398790	+	0.917042i	$0.630570\pi$
$462$	0	0
$463$	16165.7	1.62264	0.811322	−	0.584600i	$-0.198749\pi$
$463$	16165.7	1.62264	0.811322	+	0.584600i	$0.198749\pi$
$464$	−469.450	−0.0469691
$465$	−1113.02	−0.111000
$466$	4228.03	0.420300
$467$	14249.5	1.41196	0.705981	−	0.708230i	$-0.250506\pi$
$467$	14249.5	1.41196	0.705981	+	0.708230i	$0.250506\pi$
$468$	−2106.49	−0.208061
$469$	13290.5	1.30853
$470$	3351.76	0.328947
$471$	−4337.67	−0.424350
$472$	−11376.3	−1.10940
$473$	0	0
$474$	−3584.55	−0.347350
$475$	9917.50	0.957992
$476$	−5421.65	−0.522061
$477$	−7423.10	−0.712537
$478$	−1467.22	−0.140396
$479$	469.881	0.0448214	0.0224107	−	0.999749i	$-0.492866\pi$
$479$	469.881	0.0448214	0.0224107	+	0.999749i	$0.492866\pi$
$480$	−613.015	−0.0582921
$481$	−990.309	−0.0938757
$482$	−1643.48	−0.155308
$483$	2353.80	0.221743
$484$	0	0
$485$	2069.29	0.193735
$486$	18045.9	1.68432
$487$	6295.49	0.585782	0.292891	−	0.956146i	$-0.405383\pi$
$487$	6295.49	0.585782	0.292891	+	0.956146i	$0.405383\pi$
$488$	11568.6	1.07312
$489$	7943.58	0.734603
$490$	−1430.94	−0.131926
$491$	7874.64	0.723783	0.361891	−	0.932220i	$-0.382131\pi$
$491$	7874.64	0.723783	0.361891	+	0.932220i	$0.382131\pi$
$492$	12027.1	1.10208
$493$	−2227.62	−0.203503
$494$	3095.62	0.281941
$495$	0	0
$496$	1123.28	0.101687
$497$	5508.83	0.497193
$498$	398.399	0.0358488
$499$	2861.27	0.256689	0.128345	−	0.991730i	$-0.459034\pi$
$499$	2861.27	0.256689	0.128345	+	0.991730i	$0.459034\pi$
$500$	−4132.02	−0.369579
$501$	3443.20	0.307047
$502$	−12863.4	−1.14367
$503$	18084.3	1.60306	0.801528	−	0.597957i	$-0.204021\pi$
$503$	18084.3	1.60306	0.801528	+	0.597957i	$0.204021\pi$
$504$	−10978.9	−0.970315
$505$	1133.89	0.0999153
$506$	0	0
$507$	5964.91	0.522507
$508$	−20504.7	−1.79084
$509$	−12454.4	−1.08454	−0.542271	−	0.840204i	$-0.682435\pi$
$509$	−12454.4	−1.08454	−0.542271	+	0.840204i	$0.682435\pi$
$510$	277.484	0.0240926
$511$	19270.8	1.66828
$512$	1296.34	0.111896
$513$	−10398.9	−0.894979
$514$	−13664.8	−1.17262
$515$	−901.053	−0.0770974
$516$	17767.7	1.51585
$517$	0	0
$518$	−13191.1	−1.11888
$519$	4053.74	0.342851
$520$	−250.701	−0.0211423
$521$	−14191.1	−1.19333	−0.596664	−	0.802491i	$-0.703507\pi$
$521$	−14191.1	−1.19333	−0.596664	+	0.802491i	$0.703507\pi$
$522$	−11528.7	−0.966661
$523$	7489.34	0.626168	0.313084	−	0.949725i	$-0.398638\pi$
$523$	7489.34	0.626168	0.313084	+	0.949725i	$0.398638\pi$
$524$	−16208.0	−1.35124
$525$	−8398.25	−0.698152
$526$	−14146.7	−1.17267
$527$	5330.16	0.440579
$528$	0	0
$529$	−10970.9	−0.901691
$530$	−2257.84	−0.185046
$531$	−9206.01	−0.752367
$532$	25631.7	2.08886
$533$	−2733.33	−0.222127
$534$	−2519.52	−0.204177
$535$	249.567	0.0201677
$536$	12950.0	1.04358
$537$	12587.4	1.01152
$538$	−25229.6	−2.02179
$539$	0	0
$540$	2152.33	0.171521
$541$	−12985.7	−1.03198	−0.515988	−	0.856596i	$-0.672575\pi$
$541$	−12985.7	−1.03198	−0.515988	+	0.856596i	$0.672575\pi$
$542$	−20338.5	−1.61183
$543$	7089.76	0.560314
$544$	2935.67	0.231371
$545$	−184.328	−0.0144876
$546$	−2621.41	−0.205469
$547$	20019.1	1.56481	0.782407	−	0.622767i	$-0.213992\pi$
$547$	20019.1	1.56481	0.782407	+	0.622767i	$0.213992\pi$
$548$	26125.3	2.03652
$549$	9361.62	0.727767
$550$	0	0
$551$	10531.4	0.814254
$552$	2293.50	0.176844
$553$	6745.22	0.518691
$554$	−5173.09	−0.396721
$555$	419.668	0.0320972
$556$	−8191.81	−0.624838
$557$	9043.18	0.687921	0.343960	−	0.938984i	$-0.388231\pi$
$557$	9043.18	0.687921	0.343960	+	0.938984i	$0.388231\pi$
$558$	27585.3	2.09280
$559$	−4037.98	−0.305525
$560$	−110.039	−0.00830358
$561$	0	0
$562$	−17484.3	−1.31233
$563$	−7596.86	−0.568685	−0.284343	−	0.958723i	$-0.591775\pi$
$563$	−7596.86	−0.568685	−0.284343	+	0.958723i	$0.591775\pi$
$564$	21227.5	1.58482
$565$	−2649.69	−0.197298
$566$	−25079.6	−1.86250
$567$	−3730.78	−0.276328
$568$	5367.70	0.396521
$569$	7260.64	0.534942	0.267471	−	0.963566i	$-0.413812\pi$
$569$	7260.64	0.534942	0.267471	+	0.963566i	$0.413812\pi$
$570$	−1311.85	−0.0963987
$571$	14813.9	1.08571	0.542856	−	0.839826i	$-0.317343\pi$
$571$	14813.9	1.08571	0.542856	+	0.839826i	$0.317343\pi$
$572$	0	0
$573$	9636.03	0.702532
$574$	−36408.4	−2.64748
$575$	−4267.73	−0.309525
$576$	15741.5	1.13871
$577$	23782.6	1.71591	0.857956	−	0.513722i	$-0.171734\pi$
$577$	23782.6	1.71591	0.857956	+	0.513722i	$0.171734\pi$
$578$	−1328.85	−0.0956275
$579$	1743.23	0.125123
$580$	−2179.75	−0.156051
$581$	−749.687	−0.0535323
$582$	21082.8	1.50156
$583$	0	0
$584$	18777.1	1.33049
$585$	−202.875	−0.0143382
$586$	43263.6	3.04984
$587$	1428.67	0.100456	0.0502280	−	0.998738i	$-0.484005\pi$
$587$	1428.67	0.100456	0.0502280	+	0.998738i	$0.484005\pi$
$588$	−9062.53	−0.635599
$589$	−25199.1	−1.76284
$590$	−2800.13	−0.195389
$591$	10969.1	0.763469
$592$	−423.536	−0.0294041
$593$	7166.64	0.496288	0.248144	−	0.968723i	$-0.420179\pi$
$593$	7166.64	0.496288	0.248144	+	0.968723i	$0.420179\pi$
$594$	0	0
$595$	−522.155	−0.0359769
$596$	−24954.5	−1.71506
$597$	−2322.50	−0.159219
$598$	−1332.12	−0.0910942
$599$	−21168.7	−1.44396	−0.721979	−	0.691915i	$-0.756767\pi$
$599$	−21168.7	−1.44396	−0.721979	+	0.691915i	$0.756767\pi$
$600$	−8183.10	−0.556789
$601$	23433.7	1.59048	0.795242	−	0.606292i	$-0.207344\pi$
$601$	23433.7	1.59048	0.795242	+	0.606292i	$0.207344\pi$
$602$	−53786.5	−3.64149
$603$	10479.6	0.707730
$604$	−25798.8	−1.73798
$605$	0	0
$606$	11552.5	0.774403
$607$	23906.4	1.59857	0.799286	−	0.600951i	$-0.205211\pi$
$607$	23906.4	1.59857	0.799286	+	0.600951i	$0.205211\pi$
$608$	−13878.8	−0.925758
$609$	−8918.13	−0.593401
$610$	2847.47	0.189001
$611$	−4824.28	−0.319426
$612$	−4274.96	−0.282361
$613$	−17833.2	−1.17500	−0.587499	−	0.809225i	$-0.699888\pi$
$613$	−17833.2	−1.17500	−0.587499	+	0.809225i	$0.699888\pi$
$614$	−37964.4	−2.49531
$615$	1158.32	0.0759477
$616$	0	0
$617$	−17112.6	−1.11658	−0.558289	−	0.829646i	$-0.688542\pi$
$617$	−17112.6	−1.11658	−0.558289	+	0.829646i	$0.688542\pi$
$618$	−9180.31	−0.597550
$619$	11497.8	0.746584	0.373292	−	0.927714i	$-0.378229\pi$
$619$	11497.8	0.746584	0.373292	+	0.927714i	$0.378229\pi$
$620$	5215.61	0.337845
$621$	4474.90	0.289165
$622$	45018.2	2.90204
$623$	4741.11	0.304893
$624$	−84.1676	−0.00539968
$625$	15026.8	0.961714
$626$	−13796.4	−0.880858
$627$	0	0
$628$	20326.3	1.29157
$629$	−2009.75	−0.127399
$630$	−2702.33	−0.170894
$631$	−8291.05	−0.523077	−0.261538	−	0.965193i	$-0.584230\pi$
$631$	−8291.05	−0.523077	−0.261538	+	0.965193i	$0.584230\pi$
$632$	6572.42	0.413666
$633$	13931.5	0.874770
$634$	−8651.96	−0.541977
$635$	−1974.79	−0.123413
$636$	−14299.5	−0.891525
$637$	2059.60	0.128107
$638$	0	0
$639$	4343.70	0.268911
$640$	3039.39	0.187722
$641$	−25537.9	−1.57361	−0.786806	−	0.617201i	$-0.788267\pi$
$641$	−25537.9	−1.57361	−0.786806	+	0.617201i	$0.788267\pi$
$642$	2542.70	0.156312
$643$	13151.4	0.806595	0.403298	−	0.915069i	$-0.367864\pi$
$643$	13151.4	0.806595	0.403298	+	0.915069i	$0.367864\pi$
$644$	−11029.9	−0.674905
$645$	1711.20	0.104462
$646$	6282.32	0.382623
$647$	−12210.5	−0.741952	−0.370976	−	0.928642i	$-0.620977\pi$
$647$	−12210.5	−0.741952	−0.370976	+	0.928642i	$0.620977\pi$
$648$	−3635.20	−0.220377
$649$	0	0
$650$	4752.93	0.286808
$651$	21338.9	1.28470
$652$	−37223.6	−2.23587
$653$	−15932.3	−0.954792	−0.477396	−	0.878688i	$-0.658419\pi$
$653$	−15932.3	−0.954792	−0.477396	+	0.878688i	$0.658419\pi$
$654$	−1878.01	−0.112287
$655$	−1560.99	−0.0931187
$656$	−1168.99	−0.0695754
$657$	15195.0	0.902305
$658$	−64260.1	−3.80717
$659$	−3405.11	−0.201281	−0.100641	−	0.994923i	$-0.532089\pi$
$659$	−3405.11	−0.201281	−0.100641	+	0.994923i	$0.532089\pi$
$660$	0	0
$661$	−19433.1	−1.14351	−0.571754	−	0.820425i	$-0.693737\pi$
$661$	−19433.1	−1.14351	−0.571754	+	0.820425i	$0.693737\pi$
$662$	−7959.11	−0.467281
$663$	−399.390	−0.0233952
$664$	−730.481	−0.0426930
$665$	2468.57	0.143950
$666$	−10401.1	−0.605159
$667$	−4531.92	−0.263083
$668$	−16134.8	−0.934543
$669$	−11142.3	−0.643926
$670$	3187.50	0.183797
$671$	0	0
$672$	11752.8	0.674661
$673$	−28929.2	−1.65696	−0.828482	−	0.560015i	$-0.810795\pi$
$673$	−28929.2	−1.65696	−0.828482	+	0.560015i	$0.810795\pi$
$674$	12099.3	0.691463
$675$	−15966.2	−0.910430
$676$	−27951.5	−1.59032
$677$	19371.9	1.09974	0.549868	−	0.835252i	$-0.314678\pi$
$677$	19371.9	1.09974	0.549868	+	0.835252i	$0.314678\pi$
$678$	−26996.1	−1.52917
$679$	−39672.5	−2.24226
$680$	−508.778	−0.0286923
$681$	−2238.53	−0.125963
$682$	0	0
$683$	26909.6	1.50757	0.753783	−	0.657124i	$-0.228227\pi$
$683$	26909.6	1.50757	0.753783	+	0.657124i	$0.228227\pi$
$684$	20210.5	1.12978
$685$	2516.10	0.140344
$686$	−10837.8	−0.603191
$687$	3029.99	0.168270
$688$	−1726.96	−0.0956976
$689$	3249.76	0.179690
$690$	564.519	0.0311462
$691$	11678.2	0.642922	0.321461	−	0.946923i	$-0.395826\pi$
$691$	11678.2	0.642922	0.321461	+	0.946923i	$0.395826\pi$
$692$	−18995.8	−1.04352
$693$	0	0
$694$	1388.20	0.0759299
$695$	−788.947	−0.0430597
$696$	−8689.66	−0.473248
$697$	−5547.07	−0.301450
$698$	−12774.6	−0.692733
$699$	2578.89	0.139546
$700$	39354.1	2.12492
$701$	14386.8	0.775155	0.387577	−	0.921837i	$-0.373312\pi$
$701$	14386.8	0.775155	0.387577	+	0.921837i	$0.373312\pi$
$702$	−4983.66	−0.267943
$703$	9501.41	0.509747
$704$	0	0
$705$	2044.41	0.109215
$706$	29880.9	1.59289
$707$	−21738.9	−1.15640
$708$	−17734.0	−0.941360
$709$	−20825.4	−1.10312	−0.551561	−	0.834134i	$-0.685968\pi$
$709$	−20825.4	−1.10312	−0.551561	+	0.834134i	$0.685968\pi$
$710$	1321.20	0.0698361
$711$	5318.60	0.280539
$712$	4619.64	0.243158
$713$	10843.8	0.569568
$714$	−5319.94	−0.278843
$715$	0	0
$716$	−58984.5	−3.07871
$717$	−894.935	−0.0466136
$718$	466.814	0.0242637
$719$	8891.30	0.461182	0.230591	−	0.973051i	$-0.425934\pi$
$719$	8891.30	0.461182	0.230591	+	0.973051i	$0.425934\pi$
$720$	−86.7657	−0.00449106
$721$	17275.0	0.892310
$722$	1837.63	0.0947222
$723$	−1002.44	−0.0515647
$724$	−33222.5	−1.70540
$725$	16169.7	0.828312
$726$	0	0
$727$	13923.8	0.710322	0.355161	−	0.934805i	$-0.384426\pi$
$727$	13923.8	0.710322	0.355161	+	0.934805i	$0.384426\pi$
$728$	4806.46	0.244697
$729$	6856.11	0.348326
$730$	4621.78	0.234328
$731$	−8194.76	−0.414629
$732$	18033.7	0.910581
$733$	11035.8	0.556095	0.278047	−	0.960567i	$-0.410313\pi$
$733$	11035.8	0.556095	0.278047	+	0.960567i	$0.410313\pi$
$734$	−27057.7	−1.36065
$735$	−872.806	−0.0438013
$736$	5972.38	0.299110
$737$	0	0
$738$	−28707.9	−1.43192
$739$	668.067	0.0332547	0.0166274	−	0.999862i	$-0.494707\pi$
$739$	668.067	0.0332547	0.0166274	+	0.999862i	$0.494707\pi$
$740$	−1966.56	−0.0976923
$741$	1888.18	0.0936085
$742$	43287.4	2.14168
$743$	32628.3	1.61106	0.805530	−	0.592555i	$-0.201881\pi$
$743$	32628.3	1.61106	0.805530	+	0.592555i	$0.201881\pi$
$744$	20792.2	1.02457
$745$	−2403.35	−0.118191
$746$	−11053.1	−0.542468
$747$	−591.127	−0.0289534
$748$	0	0
$749$	−4784.72	−0.233418
$750$	−4054.50	−0.197399
$751$	−8925.21	−0.433669	−0.216835	−	0.976208i	$-0.569573\pi$
$751$	−8925.21	−0.433669	−0.216835	+	0.976208i	$0.569573\pi$
$752$	−2063.25	−0.100052
$753$	−7846.07	−0.379717
$754$	5047.15	0.243775
$755$	−2484.66	−0.119770
$756$	−41264.5	−1.98515
$757$	18711.4	0.898386	0.449193	−	0.893435i	$-0.351712\pi$
$757$	18711.4	0.898386	0.449193	+	0.893435i	$0.351712\pi$
$758$	−10170.7	−0.487355
$759$	0	0
$760$	2405.33	0.114803
$761$	−13360.9	−0.636441	−0.318221	−	0.948017i	$-0.603085\pi$
$761$	−13360.9	−0.636441	−0.318221	+	0.948017i	$0.603085\pi$
$762$	−20120.0	−0.956523
$763$	3533.94	0.167676
$764$	−45154.4	−2.13826
$765$	−411.719	−0.0194585
$766$	19043.5	0.898263
$767$	4030.31	0.189734
$768$	12508.1	0.587690
$769$	24912.0	1.16821	0.584103	−	0.811680i	$-0.301446\pi$
$769$	24912.0	1.16821	0.584103	+	0.811680i	$0.301446\pi$
$770$	0	0
$771$	−8334.83	−0.389328
$772$	−8168.75	−0.380829
$773$	−4859.00	−0.226088	−0.113044	−	0.993590i	$-0.536060\pi$
$773$	−4859.00	−0.226088	−0.113044	+	0.993590i	$0.536060\pi$
$774$	−42410.6	−1.96953
$775$	−38690.0	−1.79327
$776$	−38656.2	−1.78824
$777$	−8045.90	−0.371487
$778$	−17859.4	−0.822997
$779$	26224.6	1.20615
$780$	−390.807	−0.0179399
$781$	0	0
$782$	−2703.43	−0.123624
$783$	−16954.6	−0.773828
$784$	880.849	0.0401261
$785$	1957.61	0.0890064
$786$	−15904.0	−0.721725
$787$	15402.0	0.697614	0.348807	−	0.937195i	$-0.386587\pi$
$787$	15402.0	0.697614	0.348807	+	0.937195i	$0.386587\pi$
$788$	−51401.3	−2.32373
$789$	−8628.79	−0.389345
$790$	1617.72	0.0728558
$791$	50799.9	2.28348
$792$	0	0
$793$	−4098.43	−0.183530
$794$	58781.6	2.62731
$795$	−1377.17	−0.0614380
$796$	10883.2	0.484604
$797$	16061.6	0.713842	0.356921	−	0.934135i	$-0.383826\pi$
$797$	16061.6	0.713842	0.356921	+	0.934135i	$0.383826\pi$
$798$	25150.8	1.11570
$799$	−9790.48	−0.433495
$800$	−21309.2	−0.941741
$801$	3738.35	0.164904
$802$	42757.7	1.88258
$803$	0	0
$804$	20187.3	0.885510
$805$	−1062.28	−0.0465100
$806$	−12076.6	−0.527767
$807$	−15388.8	−0.671265
$808$	−21182.0	−0.922251
$809$	−16822.2	−0.731072	−0.365536	−	0.930797i	$-0.619114\pi$
$809$	−16822.2	−0.731072	−0.365536	+	0.930797i	$0.619114\pi$
$810$	−894.763	−0.0388133
$811$	6976.69	0.302077	0.151039	−	0.988528i	$-0.451738\pi$
$811$	6976.69	0.302077	0.151039	+	0.988528i	$0.451738\pi$
$812$	41790.3	1.80610
$813$	−12405.5	−0.535153
$814$	0	0
$815$	−3584.97	−0.154081
$816$	−170.811	−0.00732793
$817$	38742.0	1.65901
$818$	5977.73	0.255509
$819$	3889.53	0.165948
$820$	−5427.87	−0.231158
$821$	25465.2	1.08251	0.541256	−	0.840858i	$-0.317949\pi$
$821$	25465.2	1.08251	0.541256	+	0.840858i	$0.317949\pi$
$822$	25635.1	1.08775
$823$	−29665.3	−1.25646	−0.628231	−	0.778027i	$-0.716221\pi$
$823$	−29665.3	−1.25646	−0.628231	+	0.778027i	$0.716221\pi$
$824$	16832.5	0.711634
$825$	0	0
$826$	53684.3	2.26140
$827$	−39608.8	−1.66546	−0.832728	−	0.553682i	$-0.813222\pi$
$827$	−39608.8	−1.66546	−0.832728	+	0.553682i	$0.813222\pi$
$828$	−8697.06	−0.365029
$829$	−25188.5	−1.05529	−0.527643	−	0.849466i	$-0.676924\pi$
$829$	−25188.5	−1.05529	−0.527643	+	0.849466i	$0.676924\pi$
$830$	−179.799	−0.00751919
$831$	−3155.33	−0.131717
$832$	−6891.47	−0.287162
$833$	4179.78	0.173855
$834$	−8038.13	−0.333738
$835$	−1553.93	−0.0644024
$836$	0	0
$837$	40568.2	1.67532
$838$	33150.3	1.36654
$839$	29492.4	1.21358	0.606789	−	0.794863i	$-0.292457\pi$
$839$	29492.4	1.21358	0.606789	+	0.794863i	$0.292457\pi$
$840$	−2036.86	−0.0836646
$841$	−7218.38	−0.295969
$842$	−5585.50	−0.228609
$843$	−10664.6	−0.435714
$844$	−65283.1	−2.66249
$845$	−2691.99	−0.109595
$846$	−50669.0	−2.05914
$847$	0	0
$848$	1389.86	0.0562830
$849$	−15297.3	−0.618377
$850$	9645.69	0.389229
$851$	−4088.68	−0.164698
$852$	8367.47	0.336461
$853$	−10281.9	−0.412714	−0.206357	−	0.978477i	$-0.566161\pi$
$853$	−10281.9	−0.412714	−0.206357	+	0.978477i	$0.566161\pi$
$854$	−54591.7	−2.18746
$855$	1946.46	0.0778568
$856$	−4662.14	−0.186155
$857$	−5568.43	−0.221953	−0.110977	−	0.993823i	$-0.535398\pi$
$857$	−5568.43	−0.221953	−0.110977	+	0.993823i	$0.535398\pi$
$858$	0	0
$859$	44572.2	1.77041	0.885205	−	0.465201i	$-0.154018\pi$
$859$	44572.2	1.77041	0.885205	+	0.465201i	$0.154018\pi$
$860$	−8018.65	−0.317946
$861$	−22207.3	−0.879005
$862$	4237.26	0.167426
$863$	−37406.1	−1.47545	−0.737727	−	0.675099i	$-0.764101\pi$
$863$	−37406.1	−1.47545	−0.737727	+	0.675099i	$0.764101\pi$
$864$	22343.6	0.879797
$865$	−1829.47	−0.0719121
$866$	5822.53	0.228473
$867$	−810.530	−0.0317498
$868$	−99994.0	−3.91016
$869$	0	0
$870$	−2138.86	−0.0833496
$871$	−4587.86	−0.178477
$872$	3443.40	0.133725
$873$	−31281.7	−1.21275
$874$	12780.9	0.494644
$875$	7629.55	0.294772
$876$	29270.9	1.12896
$877$	−23478.3	−0.903996	−0.451998	−	0.892019i	$-0.649289\pi$
$877$	−23478.3	−0.903996	−0.451998	+	0.892019i	$0.649289\pi$
$878$	11278.6	0.433525
$879$	26388.7	1.01259
$880$	0	0
$881$	−27217.9	−1.04086	−0.520428	−	0.853905i	$-0.674228\pi$
$881$	−27217.9	−1.04086	−0.520428	+	0.853905i	$0.674228\pi$
$882$	21631.8	0.825827
$883$	47283.9	1.80207	0.901036	−	0.433744i	$-0.142808\pi$
$883$	47283.9	1.80207	0.901036	+	0.433744i	$0.142808\pi$
$884$	1871.54	0.0712066
$885$	−1707.94	−0.0648722
$886$	−7602.85	−0.288287
$887$	46665.8	1.76650	0.883249	−	0.468904i	$-0.155351\pi$
$887$	46665.8	1.76650	0.883249	+	0.468904i	$0.155351\pi$
$888$	−7839.77	−0.296267
$889$	37860.8	1.42836
$890$	1137.07	0.0428255
$891$	0	0
$892$	52212.8	1.95988
$893$	46286.0	1.73449
$894$	−24486.4	−0.916048
$895$	−5680.76	−0.212164
$896$	−58271.3	−2.17266
$897$	−812.526	−0.0302447
$898$	−74.7577	−0.00277806
$899$	−41085.1	−1.52421
$900$	31030.7	1.14928
$901$	6595.14	0.243858
$902$	0	0
$903$	−32807.1	−1.20903
$904$	49498.4	1.82112
$905$	−3199.64	−0.117524
$906$	−25314.8	−0.928286
$907$	−14387.2	−0.526703	−0.263351	−	0.964700i	$-0.584828\pi$
$907$	−14387.2	−0.526703	−0.263351	+	0.964700i	$0.584828\pi$
$908$	10489.8	0.383386
$909$	−17141.1	−0.625450
$910$	1183.05	0.0430965
$911$	41117.5	1.49537	0.747686	−	0.664053i	$-0.231165\pi$
$911$	41117.5	1.49537	0.747686	+	0.664053i	$0.231165\pi$
$912$	807.536	0.0293204
$913$	0	0
$914$	−78507.6	−2.84114
$915$	1736.81	0.0627511
$916$	−14198.5	−0.512152
$917$	29927.3	1.07774
$918$	−10113.9	−0.363627
$919$	−34403.6	−1.23490	−0.617448	−	0.786612i	$-0.711833\pi$
$919$	−34403.6	−1.23490	−0.617448	+	0.786612i	$0.711833\pi$
$920$	−1035.07	−0.0370926
$921$	−23156.4	−0.828480
$922$	36299.6	1.29660
$923$	−1901.63	−0.0678148
$924$	0	0
$925$	14588.2	0.518548
$926$	−74331.2	−2.63788
$927$	13621.3	0.482614
$928$	−22628.3	−0.800441
$929$	2144.59	0.0757390	0.0378695	−	0.999283i	$-0.487943\pi$
$929$	2144.59	0.0757390	0.0378695	+	0.999283i	$0.487943\pi$
$930$	5117.77	0.180450
$931$	−19760.6	−0.695624
$932$	−12084.7	−0.424728
$933$	27458.9	0.963520
$934$	−65520.2	−2.29538
$935$	0	0
$936$	3789.88	0.132346
$937$	−5196.06	−0.181161	−0.0905804	−	0.995889i	$-0.528872\pi$
$937$	−5196.06	−0.181161	−0.0905804	+	0.995889i	$0.528872\pi$
$938$	−61111.0	−2.12723
$939$	−8415.15	−0.292458
$940$	−9580.09	−0.332413
$941$	−28748.0	−0.995916	−0.497958	−	0.867201i	$-0.665917\pi$
$941$	−28748.0	−0.995916	−0.497958	+	0.867201i	$0.665917\pi$
$942$	19944.9	0.689853
$943$	−11285.1	−0.389705
$944$	1723.68	0.0594291
$945$	−3974.16	−0.136804
$946$	0	0
$947$	41745.1	1.43245	0.716226	−	0.697868i	$-0.245868\pi$
$947$	41745.1	1.43245	0.716226	+	0.697868i	$0.245868\pi$
$948$	10245.5	0.351009
$949$	−6652.24	−0.227546
$950$	−45601.5	−1.55738
$951$	−5277.27	−0.179944
$952$	9754.32	0.332079
$953$	−26758.5	−0.909540	−0.454770	−	0.890609i	$-0.650278\pi$
$953$	−26758.5	−0.909540	−0.454770	+	0.890609i	$0.650278\pi$
$954$	34132.0	1.15835
$955$	−4348.79	−0.147354
$956$	4193.66	0.141875
$957$	0	0
$958$	−2160.55	−0.0728646
$959$	−48238.9	−1.62431
$960$	2920.44	0.0981840
$961$	68515.5	2.29987
$962$	4553.52	0.152611
$963$	−3772.74	−0.126246
$964$	4697.45	0.156945
$965$	−786.726	−0.0262442
$966$	−10823.0	−0.360480
$967$	−40523.8	−1.34763	−0.673814	−	0.738901i	$-0.735345\pi$
$967$	−40523.8	−1.34763	−0.673814	+	0.738901i	$0.735345\pi$
$968$	0	0
$969$	3831.90	0.127037
$970$	−9514.77	−0.314949
$971$	−49992.1	−1.65224	−0.826118	−	0.563497i	$-0.809456\pi$
$971$	−49992.1	−1.65224	−0.826118	+	0.563497i	$0.809456\pi$
$972$	−51579.3	−1.70206
$973$	15125.7	0.498364
$974$	−28947.2	−0.952287
$975$	2899.05	0.0952246
$976$	−1752.82	−0.0574860
$977$	32141.2	1.05249	0.526247	−	0.850331i	$-0.323599\pi$
$977$	32141.2	1.05249	0.526247	+	0.850331i	$0.323599\pi$
$978$	−36525.2	−1.19422
$979$	0	0
$980$	4089.96	0.133315
$981$	2786.50	0.0906893
$982$	−36208.2	−1.17663
$983$	14468.9	0.469468	0.234734	−	0.972060i	$-0.424578\pi$
$983$	14468.9	0.469468	0.234734	+	0.972060i	$0.424578\pi$
$984$	−21638.4	−0.701023
$985$	−4950.42	−0.160136
$986$	10242.8	0.330829
$987$	−39195.5	−1.26404
$988$	−8847.99	−0.284911
$989$	−16671.6	−0.536021
$990$	0	0
$991$	538.242	0.0172531	0.00862655	−	0.999963i	$-0.497254\pi$
$991$	538.242	0.0172531	0.00862655	+	0.999963i	$0.497254\pi$
$992$	54143.9	1.73293
$993$	−4854.67	−0.155144
$994$	−25330.1	−0.808270
$995$	1048.15	0.0333957
$996$	−1138.71	−0.0362265
$997$	58718.6	1.86523	0.932615	−	0.360872i	$-0.117521\pi$
$997$	58718.6	1.86523	0.932615	+	0.360872i	$0.117521\pi$
$998$	−13156.3	−0.417291
$999$	−15296.4	−0.484440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.u.1.7		52
11.7	odd	10		187.4.g.b.137.23	yes	104
11.8	odd	10		187.4.g.b.86.23	✓	104
11.10	odd	2		2057.4.a.v.1.46		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.b.86.23	✓	104	11.8	odd	10
187.4.g.b.137.23	yes	104	11.7	odd	10
2057.4.a.u.1.7		52	1.1	even	1	trivial
2057.4.a.v.1.46		52	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)

\(f(q)\)	\(=\)	\(q-4.59808 q^{2} -2.80460 q^{3} +13.1424 q^{4} +1.26573 q^{5} +12.8958 q^{6} -24.2666 q^{7} -23.6449 q^{8} -19.1342 q^{9} -5.81993 q^{10} -36.8591 q^{12} +8.37678 q^{13} +111.580 q^{14} -3.54987 q^{15} +3.58258 q^{16} +17.0000 q^{17} +87.9806 q^{18} -80.3701 q^{19} +16.6347 q^{20} +68.0583 q^{21} +34.5851 q^{23} +66.3147 q^{24} -123.398 q^{25} -38.5171 q^{26} +129.388 q^{27} -318.921 q^{28} -131.037 q^{29} +16.3226 q^{30} +313.539 q^{31} +172.687 q^{32} -78.1674 q^{34} -30.7150 q^{35} -251.468 q^{36} -118.221 q^{37} +369.548 q^{38} -23.4935 q^{39} -29.9281 q^{40} -326.298 q^{41} -312.938 q^{42} -482.045 q^{43} -24.2187 q^{45} -159.025 q^{46} -575.911 q^{47} -10.0477 q^{48} +245.870 q^{49} +567.394 q^{50} -47.6783 q^{51} +110.091 q^{52} +387.949 q^{53} -594.937 q^{54} +573.783 q^{56} +225.406 q^{57} +602.517 q^{58} +481.128 q^{59} -46.6537 q^{60} -489.261 q^{61} -1441.68 q^{62} +464.323 q^{63} -822.688 q^{64} +10.6027 q^{65} -547.688 q^{67} +223.420 q^{68} -96.9975 q^{69} +141.230 q^{70} -227.013 q^{71} +452.427 q^{72} -794.129 q^{73} +543.589 q^{74} +346.082 q^{75} -1056.25 q^{76} +108.025 q^{78} -277.963 q^{79} +4.53459 q^{80} +153.741 q^{81} +1500.35 q^{82} +30.8937 q^{83} +894.446 q^{84} +21.5174 q^{85} +2216.48 q^{86} +367.506 q^{87} -195.375 q^{89} +111.360 q^{90} -203.276 q^{91} +454.530 q^{92} -879.352 q^{93} +2648.08 q^{94} -101.727 q^{95} -484.317 q^{96} +1634.86 q^{97} -1130.53 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - q^{2} + 20 q^{3} + 235 q^{4} + 40 q^{5} + 24 q^{6} + 42 q^{7} - 45 q^{8} + 572 q^{9} - 33 q^{10} + 233 q^{12} - 12 q^{13} + 73 q^{14} + 400 q^{15} + 1223 q^{16} + 884 q^{17} - 201 q^{18} + 44 q^{19}+ \cdots + 4610 q^{98}+O(q^{100}) \) 52 * q - q^2 + 20 * q^3 + 235 * q^4 + 40 * q^5 + 24 * q^6 + 42 * q^7 - 45 * q^8 + 572 * q^9 - 33 * q^10 + 233 * q^12 - 12 * q^13 + 73 * q^14 + 400 * q^15 + 1223 * q^16 + 884 * q^17 - 201 * q^18 + 44 * q^19 + 655 * q^20 - 260 * q^21 + 572 * q^23 + 104 * q^24 + 1858 * q^25 + 465 * q^26 + 1070 * q^27 + 577 * q^28 - 322 * q^29 + 320 * q^30 + 1110 * q^31 - 481 * q^32 - 17 * q^34 - 102 * q^35 + 2507 * q^36 + 1678 * q^37 - 360 * q^38 + 1282 * q^39 - 1791 * q^40 + 826 * q^41 + 2133 * q^42 - 270 * q^43 + 710 * q^45 - 2158 * q^46 + 2464 * q^47 + 2201 * q^48 + 3224 * q^49 - 2379 * q^50 + 340 * q^51 + 3664 * q^52 + 992 * q^53 - 1202 * q^54 + 1731 * q^56 - 1016 * q^57 + 1358 * q^58 + 1442 * q^59 + 1444 * q^60 + 140 * q^61 - 464 * q^62 + 766 * q^63 + 8427 * q^64 + 1268 * q^65 + 5766 * q^67 + 3995 * q^68 + 2460 * q^69 + 2422 * q^70 + 2704 * q^71 - 5455 * q^72 - 4 * q^73 + 4008 * q^74 + 5204 * q^75 + 1935 * q^76 + 4092 * q^78 + 2180 * q^79 + 5040 * q^80 + 7192 * q^81 + 3197 * q^82 - 4200 * q^83 - 7951 * q^84 + 680 * q^85 + 3091 * q^86 + 752 * q^87 - 240 * q^89 + 4495 * q^90 + 5494 * q^91 + 6902 * q^92 + 6266 * q^93 - 5990 * q^94 - 3168 * q^95 + 9467 * q^96 + 5322 * q^97 + 4610 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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