LMFDB - Embedded newform 1859.4.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,4,Mod(1,1859)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1859.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1859 = 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1859.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$109.684550701$
Analytic rank:	$1$
Dimension:	$39$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	1859.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.86773 q^{2} -7.77085 q^{3} +15.6948 q^{4} -6.79003 q^{5} +37.8264 q^{6} +12.1965 q^{7} -37.4563 q^{8} +33.3861 q^{9} +33.0520 q^{10} -11.0000 q^{11} -121.962 q^{12} -59.3695 q^{14} +52.7643 q^{15} +56.7686 q^{16} +31.0773 q^{17} -162.514 q^{18} +110.731 q^{19} -106.568 q^{20} -94.7775 q^{21} +53.5450 q^{22} +84.4283 q^{23} +291.067 q^{24} -78.8955 q^{25} -49.6251 q^{27} +191.422 q^{28} +21.9448 q^{29} -256.842 q^{30} -83.3657 q^{31} +23.3160 q^{32} +85.4793 q^{33} -151.276 q^{34} -82.8149 q^{35} +523.988 q^{36} -303.283 q^{37} -539.010 q^{38} +254.329 q^{40} +242.328 q^{41} +461.351 q^{42} -325.927 q^{43} -172.643 q^{44} -226.692 q^{45} -410.975 q^{46} +474.115 q^{47} -441.140 q^{48} -194.244 q^{49} +384.042 q^{50} -241.497 q^{51} -267.590 q^{53} +241.562 q^{54} +74.6903 q^{55} -456.837 q^{56} -860.475 q^{57} -106.821 q^{58} -564.798 q^{59} +828.125 q^{60} +60.8398 q^{61} +405.802 q^{62} +407.195 q^{63} -567.644 q^{64} -416.090 q^{66} -851.817 q^{67} +487.752 q^{68} -656.080 q^{69} +403.121 q^{70} -743.888 q^{71} -1250.52 q^{72} +1049.77 q^{73} +1476.30 q^{74} +613.085 q^{75} +1737.90 q^{76} -134.162 q^{77} -739.430 q^{79} -385.460 q^{80} -515.795 q^{81} -1179.59 q^{82} +161.060 q^{83} -1487.51 q^{84} -211.016 q^{85} +1586.53 q^{86} -170.529 q^{87} +412.019 q^{88} +922.681 q^{89} +1103.48 q^{90} +1325.09 q^{92} +647.822 q^{93} -2307.87 q^{94} -751.868 q^{95} -181.185 q^{96} +1639.37 q^{97} +945.529 q^{98} -367.247 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q - 23 q^{3} + 114 q^{4} + 23 q^{5} + 77 q^{6} - 4 q^{7} - 21 q^{8} + 260 q^{9} - 158 q^{10} - 429 q^{11} - 351 q^{12} - 176 q^{14} + 30 q^{15} + 230 q^{16} - 244 q^{17} + 21 q^{18} - 70 q^{19} + 366 q^{20}+ \cdots - 2860 q^{99}+O(q^{100}) $ 39 * q - 23 * q^3 + 114 * q^4 + 23 * q^5 + 77 * q^6 - 4 * q^7 - 21 * q^8 + 260 * q^9 - 158 * q^10 - 429 * q^11 - 351 * q^12 - 176 * q^14 + 30 * q^15 + 230 * q^16 - 244 * q^17 + 21 * q^18 - 70 * q^19 + 366 * q^20 - 142 * q^21 - 47 * q^23 + 846 * q^24 + 322 * q^25 - 416 * q^27 + 1131 * q^28 - 838 * q^29 - 293 * q^30 + 507 * q^31 - 1433 * q^32 + 253 * q^33 + 166 * q^34 - 498 * q^35 + 815 * q^36 + 89 * q^37 + 81 * q^38 - 2917 * q^40 + 618 * q^41 - 318 * q^42 - 1064 * q^43 - 1254 * q^44 + 238 * q^45 - 1331 * q^46 + 1499 * q^47 - 1460 * q^48 - 413 * q^49 - 2459 * q^50 - 2350 * q^51 - 2745 * q^53 - 845 * q^54 - 253 * q^55 - 2904 * q^56 + 1450 * q^57 - 2509 * q^58 + 2285 * q^59 - 3566 * q^60 - 6218 * q^61 - 911 * q^62 - 1930 * q^63 + 67 * q^64 - 847 * q^66 + 546 * q^67 - 170 * q^68 - 5254 * q^69 - 2195 * q^70 - 263 * q^71 - 2393 * q^72 - 1148 * q^73 + 775 * q^74 - 5385 * q^75 - 7247 * q^76 + 44 * q^77 - 3666 * q^79 + 5594 * q^80 - 1901 * q^81 - 4414 * q^82 + 2722 * q^83 - 9971 * q^84 + 1858 * q^85 + 2478 * q^86 - 2284 * q^87 + 231 * q^88 + 13 * q^89 - 6771 * q^90 - 2232 * q^92 - 1082 * q^93 - 7330 * q^94 - 2352 * q^95 + 5770 * q^96 - 1197 * q^97 + 6813 * q^98 - 2860 * q^99

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 $

$2$	−4.86773	−1.72100	−0.860501	−	0.509448i	$-0.829850\pi$
$2$	−4.86773	−1.72100	−0.860501	+	0.509448i	$0.829850\pi$
$3$	−7.77085	−1.49550	−0.747750	−	0.663980i	$-0.768866\pi$
$3$	−7.77085	−1.49550	−0.747750	+	0.663980i	$0.768866\pi$
$4$	15.6948	1.96185
$5$	−6.79003	−0.607319	−0.303659	−	0.952781i	$-0.598208\pi$
$5$	−6.79003	−0.607319	−0.303659	+	0.952781i	$0.598208\pi$
$6$	37.8264	2.57376
$7$	12.1965	0.658552	0.329276	−	0.944234i	$-0.393195\pi$
$7$	12.1965	0.658552	0.329276	+	0.944234i	$0.393195\pi$
$8$	−37.4563	−1.65535
$9$	33.3861	1.23652
$10$	33.0520	1.04520
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−121.962	−2.93395
$13$	0	0
$13$	0	0
$14$	−59.3695	−1.13337
$15$	52.7643	0.908245
$16$	56.7686	0.887009
$17$	31.0773	0.443373	0.221687	−	0.975118i	$-0.428844\pi$
$17$	31.0773	0.443373	0.221687	+	0.975118i	$0.428844\pi$
$18$	−162.514	−2.12806
$19$	110.731	1.33702	0.668512	−	0.743701i	$-0.266931\pi$
$19$	110.731	1.33702	0.668512	+	0.743701i	$0.266931\pi$
$20$	−106.568	−1.19147
$21$	−94.7775	−0.984864
$22$	53.5450	0.518902
$23$	84.4283	0.765414	0.382707	−	0.923870i	$-0.374992\pi$
$23$	84.4283	0.765414	0.382707	+	0.923870i	$0.374992\pi$
$24$	291.067	2.47557
$25$	−78.8955	−0.631164
$26$	0	0
$27$	−49.6251	−0.353717
$28$	191.422	1.29198
$29$	21.9448	0.140519	0.0702593	−	0.997529i	$-0.477617\pi$
$29$	21.9448	0.140519	0.0702593	+	0.997529i	$0.477617\pi$
$30$	−256.842	−1.56309
$31$	−83.3657	−0.482997	−0.241499	−	0.970401i	$-0.577639\pi$
$31$	−83.3657	−0.482997	−0.241499	+	0.970401i	$0.577639\pi$
$32$	23.3160	0.128804
$33$	85.4793	0.450910
$34$	−151.276	−0.763047
$35$	−82.8149	−0.399951
$36$	523.988	2.42587
$37$	−303.283	−1.34755	−0.673775	−	0.738937i	$-0.735328\pi$
$37$	−303.283	−1.34755	−0.673775	+	0.738937i	$0.735328\pi$
$38$	−539.010	−2.30102
$39$	0	0
$40$	254.329	1.00532
$41$	242.328	0.923056	0.461528	−	0.887126i	$-0.347301\pi$
$41$	242.328	0.923056	0.461528	+	0.887126i	$0.347301\pi$
$42$	461.351	1.69495
$43$	−325.927	−1.15589	−0.577947	−	0.816074i	$-0.696146\pi$
$43$	−325.927	−1.15589	−0.577947	+	0.816074i	$0.696146\pi$
$44$	−172.643	−0.591520
$45$	−226.692	−0.750962
$46$	−410.975	−1.31728
$47$	474.115	1.47142	0.735711	−	0.677296i	$-0.236848\pi$
$47$	474.115	1.47142	0.735711	+	0.677296i	$0.236848\pi$
$48$	−441.140	−1.32652
$49$	−194.244	−0.566310
$50$	384.042	1.08623
$51$	−241.497	−0.663065
$52$	0	0
$53$	−267.590	−0.693515	−0.346757	−	0.937955i	$-0.612717\pi$
$53$	−267.590	−0.693515	−0.346757	+	0.937955i	$0.612717\pi$
$54$	241.562	0.608748
$55$	74.6903	0.183114
$56$	−456.837	−1.09013
$57$	−860.475	−1.99952
$58$	−106.821	−0.241833
$59$	−564.798	−1.24628	−0.623139	−	0.782111i	$-0.714143\pi$
$59$	−564.798	−1.24628	−0.623139	+	0.782111i	$0.714143\pi$
$60$	828.125	1.78184
$61$	60.8398	0.127701	0.0638503	−	0.997959i	$-0.479662\pi$
$61$	60.8398	0.127701	0.0638503	+	0.997959i	$0.479662\pi$
$62$	405.802	0.831240
$63$	407.195	0.814313
$64$	−567.644	−1.10868
$65$	0	0
$66$	−416.090	−0.776018
$67$	−851.817	−1.55322	−0.776612	−	0.629980i	$-0.783063\pi$
$67$	−851.817	−1.55322	−0.776612	+	0.629980i	$0.783063\pi$
$68$	487.752	0.869832
$69$	−656.080	−1.14468
$70$	403.121	0.688316
$71$	−743.888	−1.24343	−0.621713	−	0.783245i	$-0.713563\pi$
$71$	−743.888	−1.24343	−0.621713	+	0.783245i	$0.713563\pi$
$72$	−1250.52	−2.04687
$73$	1049.77	1.68311	0.841553	−	0.540175i	$-0.181642\pi$
$73$	1049.77	1.68311	0.841553	+	0.540175i	$0.181642\pi$
$74$	1476.30	2.31914
$75$	613.085	0.943906
$76$	1737.90	2.62304
$77$	−134.162	−0.198561
$78$	0	0
$79$	−739.430	−1.05307	−0.526534	−	0.850154i	$-0.676509\pi$
$79$	−739.430	−1.05307	−0.526534	+	0.850154i	$0.676509\pi$
$80$	−385.460	−0.538697
$81$	−515.795	−0.707537
$82$	−1179.59	−1.58858
$83$	161.060	0.212996	0.106498	−	0.994313i	$-0.466036\pi$
$83$	161.060	0.212996	0.106498	+	0.994313i	$0.466036\pi$
$84$	−1487.51	−1.93216
$85$	−211.016	−0.269269
$86$	1586.53	1.98930
$87$	−170.529	−0.210146
$88$	412.019	0.499106
$89$	922.681	1.09892	0.549460	−	0.835520i	$-0.314833\pi$
$89$	922.681	1.09892	0.549460	+	0.835520i	$0.314833\pi$
$90$	1103.48	1.29241
$91$	0	0
$92$	1325.09	1.50163
$93$	647.822	0.722323
$94$	−2307.87	−2.53232
$95$	−751.868	−0.812000
$96$	−181.185	−0.192626
$97$	1639.37	1.71601	0.858005	−	0.513642i	$-0.171704\pi$
$97$	1639.37	1.71601	0.858005	+	0.513642i	$0.171704\pi$
$98$	945.529	0.974621
$99$	−367.247	−0.372825
$100$	−1238.25	−1.23825
$101$	−285.763	−0.281529	−0.140765	−	0.990043i	$-0.544956\pi$
$101$	−285.763	−0.281529	−0.140765	+	0.990043i	$0.544956\pi$
$102$	1175.54	1.14114
$103$	−533.704	−0.510558	−0.255279	−	0.966867i	$-0.582167\pi$
$103$	−533.704	−0.510558	−0.255279	+	0.966867i	$0.582167\pi$
$104$	0	0
$105$	643.542	0.598126
$106$	1302.56	1.19354
$107$	1512.59	1.36661	0.683306	−	0.730132i	$-0.260541\pi$
$107$	1512.59	1.36661	0.683306	+	0.730132i	$0.260541\pi$
$108$	−778.856	−0.693940
$109$	−150.230	−0.132013	−0.0660064	−	0.997819i	$-0.521026\pi$
$109$	−150.230	−0.132013	−0.0660064	+	0.997819i	$0.521026\pi$
$110$	−363.573	−0.315139
$111$	2356.76	2.01526
$112$	692.381	0.584141
$113$	−273.090	−0.227347	−0.113673	−	0.993518i	$-0.536262\pi$
$113$	−273.090	−0.227347	−0.113673	+	0.993518i	$0.536262\pi$
$114$	4188.56	3.44118
$115$	−573.271	−0.464850
$116$	344.419	0.275677
$117$	0	0
$118$	2749.28	2.14485
$119$	379.035	0.291984
$120$	−1976.35	−1.50346
$121$	121.000	0.0909091
$122$	−296.152	−0.219773
$123$	−1883.09	−1.38043
$124$	−1308.41	−0.947569
$125$	1384.46	0.990636
$126$	−1982.11	−1.40143
$127$	−1639.81	−1.14574	−0.572872	−	0.819645i	$-0.694171\pi$
$127$	−1639.81	−1.14574	−0.572872	+	0.819645i	$0.694171\pi$
$128$	2576.61	1.77924
$129$	2532.73	1.72864
$130$	0	0
$131$	1478.05	0.985783	0.492891	−	0.870091i	$-0.335940\pi$
$131$	1478.05	0.985783	0.492891	+	0.870091i	$0.335940\pi$
$132$	1341.58	0.884619
$133$	1350.54	0.880500
$134$	4146.41	2.67310
$135$	336.956	0.214819
$136$	−1164.04	−0.733937
$137$	2959.08	1.84534	0.922670	−	0.385591i	$-0.126002\pi$
$137$	2959.08	1.84534	0.922670	+	0.385591i	$0.126002\pi$
$138$	3193.62	1.96999
$139$	665.710	0.406222	0.203111	−	0.979156i	$-0.434895\pi$
$139$	665.710	0.406222	0.203111	+	0.979156i	$0.434895\pi$
$140$	−1299.76	−0.784644
$141$	−3684.28	−2.20051
$142$	3621.05	2.13994
$143$	0	0
$144$	1895.28	1.09680
$145$	−149.006	−0.0853396
$146$	−5110.02	−2.89663
$147$	1509.44	0.846916
$148$	−4759.96	−2.64369
$149$	2466.98	1.35639	0.678197	−	0.734880i	$-0.262762\pi$
$149$	2466.98	1.35639	0.678197	+	0.734880i	$0.262762\pi$
$150$	−2984.33	−1.62446
$151$	455.869	0.245683	0.122841	−	0.992426i	$-0.460799\pi$
$151$	455.869	0.245683	0.122841	+	0.992426i	$0.460799\pi$
$152$	−4147.58	−2.21324
$153$	1037.55	0.548240
$154$	653.065	0.341724
$155$	566.056	0.293333
$156$	0	0
$157$	−2164.39	−1.10024	−0.550119	−	0.835086i	$-0.685418\pi$
$157$	−2164.39	−1.10024	−0.550119	+	0.835086i	$0.685418\pi$
$158$	3599.35	1.81233
$159$	2079.40	1.03715
$160$	−158.316	−0.0782250
$161$	1029.73	0.504065
$162$	2510.75	1.21767
$163$	−883.355	−0.424477	−0.212238	−	0.977218i	$-0.568075\pi$
$163$	−883.355	−0.424477	−0.212238	+	0.977218i	$0.568075\pi$
$164$	3803.29	1.81090
$165$	−580.407	−0.273846
$166$	−783.999	−0.366567
$167$	−2156.89	−0.999430	−0.499715	−	0.866190i	$-0.666562\pi$
$167$	−2156.89	−0.999430	−0.499715	+	0.866190i	$0.666562\pi$
$168$	3550.01	1.63029
$169$	0	0
$170$	1027.17	0.463413
$171$	3696.88	1.65326
$172$	−5115.37	−2.26769
$173$	634.892	0.279017	0.139508	−	0.990221i	$-0.455448\pi$
$173$	634.892	0.279017	0.139508	+	0.990221i	$0.455448\pi$
$174$	830.091	0.361661
$175$	−962.252	−0.415654
$176$	−624.454	−0.267443
$177$	4388.96	1.86381
$178$	−4491.36	−1.89125
$179$	−917.912	−0.383285	−0.191642	−	0.981465i	$-0.561381\pi$
$179$	−917.912	−0.383285	−0.191642	+	0.981465i	$0.561381\pi$
$180$	−3557.89	−1.47328
$181$	−3805.60	−1.56281	−0.781404	−	0.624026i	$-0.785496\pi$
$181$	−3805.60	−1.56281	−0.781404	+	0.624026i	$0.785496\pi$
$182$	0	0
$183$	−472.776	−0.190976
$184$	−3162.37	−1.26703
$185$	2059.30	0.818392
$186$	−3153.42	−1.24312
$187$	−341.850	−0.133682
$188$	7441.15	2.88671
$189$	−605.255	−0.232941
$190$	3659.89	1.39745
$191$	−4327.49	−1.63940	−0.819702	−	0.572790i	$-0.805861\pi$
$191$	−4327.49	−1.63940	−0.819702	+	0.572790i	$0.805861\pi$
$192$	4411.08	1.65803
$193$	826.347	0.308196	0.154098	−	0.988056i	$-0.450753\pi$
$193$	826.347	0.308196	0.154098	+	0.988056i	$0.450753\pi$
$194$	−7980.02	−2.95326
$195$	0	0
$196$	−3048.63	−1.11102
$197$	2643.51	0.956052	0.478026	−	0.878346i	$-0.341352\pi$
$197$	2643.51	0.956052	0.478026	+	0.878346i	$0.341352\pi$
$198$	1787.66	0.641633
$199$	2737.38	0.975116	0.487558	−	0.873091i	$-0.337888\pi$
$199$	2737.38	0.975116	0.487558	+	0.873091i	$0.337888\pi$
$200$	2955.13	1.04480
$201$	6619.34	2.32285
$202$	1391.02	0.484513
$203$	267.650	0.0925388
$204$	−3790.24	−1.30083
$205$	−1645.41	−0.560589
$206$	2597.93	0.878671
$207$	2818.73	0.946451
$208$	0	0
$209$	−1218.04	−0.403128
$210$	−3132.59	−1.02938
$211$	5313.85	1.73375	0.866874	−	0.498528i	$-0.166126\pi$
$211$	5313.85	1.73375	0.866874	+	0.498528i	$0.166126\pi$
$212$	−4199.77	−1.36057
$213$	5780.64	1.85954
$214$	−7362.88	−2.35194
$215$	2213.06	0.701996
$216$	1858.77	0.585525
$217$	−1016.77	−0.318079
$218$	731.278	0.227194
$219$	−8157.63	−2.51708
$220$	1172.25	0.359241
$221$	0	0
$222$	−11472.1	−3.46827
$223$	−479.582	−0.144014	−0.0720071	−	0.997404i	$-0.522940\pi$
$223$	−479.582	−0.144014	−0.0720071	+	0.997404i	$0.522940\pi$
$224$	284.375	0.0848240
$225$	−2634.01	−0.780447
$226$	1329.33	0.391264
$227$	3870.68	1.13175	0.565873	−	0.824493i	$-0.308539\pi$
$227$	3870.68	1.13175	0.565873	+	0.824493i	$0.308539\pi$
$228$	−13505.0	−3.92276
$229$	−943.994	−0.272406	−0.136203	−	0.990681i	$-0.543490\pi$
$229$	−943.994	−0.272406	−0.136203	+	0.990681i	$0.543490\pi$
$230$	2790.53	0.800009
$231$	1042.55	0.296948
$232$	−821.969	−0.232607
$233$	−2082.41	−0.585506	−0.292753	−	0.956188i	$-0.594571\pi$
$233$	−2082.41	−0.585506	−0.292753	+	0.956188i	$0.594571\pi$
$234$	0	0
$235$	−3219.26	−0.893622
$236$	−8864.40	−2.44501
$237$	5746.00	1.57486
$238$	−1845.04	−0.502506
$239$	6593.08	1.78440	0.892199	−	0.451643i	$-0.149162\pi$
$239$	6593.08	1.78440	0.892199	+	0.451643i	$0.149162\pi$
$240$	2995.35	0.805622
$241$	788.446	0.210740	0.105370	−	0.994433i	$-0.466397\pi$
$241$	788.446	0.210740	0.105370	+	0.994433i	$0.466397\pi$
$242$	−588.996	−0.156455
$243$	5348.04	1.41184
$244$	954.868	0.250529
$245$	1318.92	0.343931
$246$	9166.40	2.37572
$247$	0	0
$248$	3122.57	0.799529
$249$	−1251.58	−0.318536
$250$	−6739.16	−1.70489
$251$	−84.4314	−0.0212321	−0.0106161	−	0.999944i	$-0.503379\pi$
$251$	−84.4314	−0.0212321	−0.0106161	+	0.999944i	$0.503379\pi$
$252$	6390.84	1.59756
$253$	−928.712	−0.230781
$254$	7982.14	1.97183
$255$	1639.77	0.402692
$256$	−8001.10	−1.95339
$257$	6409.97	1.55581	0.777904	−	0.628383i	$-0.216283\pi$
$257$	6409.97	1.55581	0.777904	+	0.628383i	$0.216283\pi$
$258$	−12328.7	−2.97499
$259$	−3699.00	−0.887431
$260$	0	0
$261$	732.649	0.173754
$262$	−7194.73	−1.69653
$263$	3669.52	0.860351	0.430176	−	0.902745i	$-0.358452\pi$
$263$	3669.52	0.860351	0.430176	+	0.902745i	$0.358452\pi$
$264$	−3201.74	−0.746414
$265$	1816.94	0.421185
$266$	−6574.05	−1.51534
$267$	−7170.01	−1.64344
$268$	−13369.1	−3.04719
$269$	−8172.06	−1.85226	−0.926132	−	0.377199i	$-0.876887\pi$
$269$	−8172.06	−1.85226	−0.926132	+	0.377199i	$0.876887\pi$
$270$	−1640.21	−0.369704
$271$	1854.49	0.415692	0.207846	−	0.978162i	$-0.433355\pi$
$271$	1854.49	0.415692	0.207846	+	0.978162i	$0.433355\pi$
$272$	1764.21	0.393276
$273$	0	0
$274$	−14404.0	−3.17584
$275$	867.850	0.190303
$276$	−10297.0	−2.24569
$277$	3214.95	0.697355	0.348678	−	0.937243i	$-0.386631\pi$
$277$	3214.95	0.697355	0.348678	+	0.937243i	$0.386631\pi$
$278$	−3240.50	−0.699109
$279$	−2783.25	−0.597236
$280$	3101.94	0.662058
$281$	−8553.21	−1.81581	−0.907904	−	0.419178i	$-0.862318\pi$
$281$	−8553.21	−1.81581	−0.907904	+	0.419178i	$0.862318\pi$
$282$	17934.1	3.78709
$283$	−813.492	−0.170873	−0.0854366	−	0.996344i	$-0.527229\pi$
$283$	−813.492	−0.170873	−0.0854366	+	0.996344i	$0.527229\pi$
$284$	−11675.2	−2.43942
$285$	5842.65	1.21435
$286$	0	0
$287$	2955.57	0.607880
$288$	778.429	0.159269
$289$	−3947.20	−0.803420
$290$	725.319	0.146870
$291$	−12739.3	−2.56629
$292$	16476.0	3.30200
$293$	−3994.34	−0.796422	−0.398211	−	0.917294i	$-0.630369\pi$
$293$	−3994.34	−0.796422	−0.398211	+	0.917294i	$0.630369\pi$
$294$	−7347.56	−1.45755
$295$	3835.00	0.756888
$296$	11359.8	2.23066
$297$	545.876	0.106650
$298$	−12008.6	−2.33436
$299$	0	0
$300$	9622.25	1.85180
$301$	−3975.19	−0.761216
$302$	−2219.05	−0.422821
$303$	2220.62	0.421027
$304$	6286.05	1.18595
$305$	−413.104	−0.0775549
$306$	−5050.50	−0.943523
$307$	8207.46	1.52581	0.762906	−	0.646510i	$-0.223772\pi$
$307$	8207.46	1.52581	0.762906	+	0.646510i	$0.223772\pi$
$308$	−2105.65	−0.389547
$309$	4147.33	0.763539
$310$	−2755.41	−0.504828
$311$	−3572.46	−0.651369	−0.325684	−	0.945479i	$-0.605595\pi$
$311$	−3572.46	−0.651369	−0.325684	+	0.945479i	$0.605595\pi$
$312$	0	0
$313$	3214.48	0.580490	0.290245	−	0.956952i	$-0.406263\pi$
$313$	3214.48	0.580490	0.290245	+	0.956952i	$0.406263\pi$
$314$	10535.7	1.89351
$315$	−2764.86	−0.494547
$316$	−11605.2	−2.06596
$317$	−9457.67	−1.67570	−0.837848	−	0.545903i	$-0.816187\pi$
$317$	−9457.67	−1.67570	−0.837848	+	0.545903i	$0.816187\pi$
$318$	−10122.0	−1.78494
$319$	−241.392	−0.0423680
$320$	3854.32	0.673323
$321$	−11754.1	−2.04377
$322$	−5012.47	−0.867497
$323$	3441.22	0.592801
$324$	−8095.30	−1.38808
$325$	0	0
$326$	4299.93	0.730526
$327$	1167.41	0.197425
$328$	−9076.70	−1.52798
$329$	5782.57	0.969007
$330$	2825.27	0.471290
$331$	−1982.64	−0.329231	−0.164616	−	0.986358i	$-0.552638\pi$
$331$	−1982.64	−0.329231	−0.164616	+	0.986358i	$0.552638\pi$
$332$	2527.81	0.417867
$333$	−10125.4	−1.66627
$334$	10499.1	1.72002
$335$	5783.86	0.943302
$336$	−5380.38	−0.873583
$337$	4257.10	0.688128	0.344064	−	0.938946i	$-0.388196\pi$
$337$	4257.10	0.688128	0.344064	+	0.938946i	$0.388196\pi$
$338$	0	0
$339$	2122.14	0.339997
$340$	−3311.85	−0.528266
$341$	917.023	0.145629
$342$	−17995.4	−2.84526
$343$	−6552.52	−1.03150
$344$	12208.0	1.91341
$345$	4454.80	0.695184
$346$	−3090.48	−0.480189
$347$	−2527.84	−0.391070	−0.195535	−	0.980697i	$-0.562644\pi$
$347$	−2527.84	−0.391070	−0.195535	+	0.980697i	$0.562644\pi$
$348$	−2676.43	−0.412274
$349$	−4817.10	−0.738835	−0.369418	−	0.929264i	$-0.620443\pi$
$349$	−4817.10	−0.738835	−0.369418	+	0.929264i	$0.620443\pi$
$350$	4683.99	0.715342
$351$	0	0
$352$	−256.476	−0.0388358
$353$	−3702.54	−0.558262	−0.279131	−	0.960253i	$-0.590046\pi$
$353$	−3702.54	−0.558262	−0.279131	+	0.960253i	$0.590046\pi$
$354$	−21364.3	−3.20762
$355$	5051.02	0.755156
$356$	14481.3	2.15592
$357$	−2945.43	−0.436662
$358$	4468.15	0.659634
$359$	−4293.48	−0.631201	−0.315600	−	0.948892i	$-0.602206\pi$
$359$	−4293.48	−0.631201	−0.315600	+	0.948892i	$0.602206\pi$
$360$	8491.05	1.24310
$361$	5402.39	0.787635
$362$	18524.6	2.68960
$363$	−940.272	−0.135955
$364$	0	0
$365$	−7127.99	−1.02218
$366$	2301.35	0.328671
$367$	9320.93	1.32575	0.662873	−	0.748732i	$-0.269337\pi$
$367$	9320.93	1.32575	0.662873	+	0.748732i	$0.269337\pi$
$368$	4792.88	0.678929
$369$	8090.38	1.14138
$370$	−10024.1	−1.40846
$371$	−3263.67	−0.456715
$372$	10167.4	1.41709
$373$	6547.67	0.908915	0.454458	−	0.890768i	$-0.349833\pi$
$373$	6547.67	0.908915	0.454458	+	0.890768i	$0.349833\pi$
$374$	1664.03	0.230067
$375$	−10758.4	−1.48150
$376$	−17758.6	−2.43572
$377$	0	0
$378$	2946.22	0.400892
$379$	−7238.33	−0.981024	−0.490512	−	0.871434i	$-0.663190\pi$
$379$	−7238.33	−0.981024	−0.490512	+	0.871434i	$0.663190\pi$
$380$	−11800.4	−1.59302
$381$	12742.7	1.71346
$382$	21065.1	2.82142
$383$	755.065	0.100736	0.0503682	−	0.998731i	$-0.483961\pi$
$383$	755.065	0.100736	0.0503682	+	0.998731i	$0.483961\pi$
$384$	−20022.5	−2.66085
$385$	910.964	0.120590
$386$	−4022.43	−0.530405
$387$	−10881.4	−1.42929
$388$	25729.6	3.36655
$389$	−4097.38	−0.534050	−0.267025	−	0.963690i	$-0.586041\pi$
$389$	−4097.38	−0.534050	−0.267025	+	0.963690i	$0.586041\pi$
$390$	0	0
$391$	2623.80	0.339364
$392$	7275.66	0.937440
$393$	−11485.7	−1.47424
$394$	−12867.9	−1.64537
$395$	5020.75	0.639548
$396$	−5763.87	−0.731427
$397$	10460.5	1.32241	0.661203	−	0.750207i	$-0.270046\pi$
$397$	10460.5	1.32241	0.661203	+	0.750207i	$0.270046\pi$
$398$	−13324.9	−1.67818
$399$	−10494.8	−1.31679
$400$	−4478.78	−0.559848
$401$	−6046.58	−0.752997	−0.376499	−	0.926417i	$-0.622872\pi$
$401$	−6046.58	−0.752997	−0.376499	+	0.926417i	$0.622872\pi$
$402$	−32221.2	−3.99762
$403$	0	0
$404$	−4485.00	−0.552319
$405$	3502.26	0.429701
$406$	−1302.85	−0.159259
$407$	3336.11	0.406302
$408$	9045.56	1.09760
$409$	−6524.91	−0.788841	−0.394421	−	0.918930i	$-0.629055\pi$
$409$	−6524.91	−0.788841	−0.394421	+	0.918930i	$0.629055\pi$
$410$	8009.44	0.964775
$411$	−22994.6	−2.75971
$412$	−8376.39	−1.00164
$413$	−6888.59	−0.820739
$414$	−13720.8	−1.62884
$415$	−1093.61	−0.129357
$416$	0	0
$417$	−5173.13	−0.607505
$418$	5929.10	0.693785
$419$	−13509.6	−1.57514	−0.787572	−	0.616223i	$-0.788662\pi$
$419$	−13509.6	−1.57514	−0.787572	+	0.616223i	$0.788662\pi$
$420$	10100.3	1.17344
$421$	−8784.45	−1.01693	−0.508465	−	0.861082i	$-0.669787\pi$
$421$	−8784.45	−1.01693	−0.508465	+	0.861082i	$0.669787\pi$
$422$	−25866.4	−2.98378
$423$	15828.8	1.81944
$424$	10022.9	1.14801
$425$	−2451.86	−0.279841
$426$	−28138.6	−3.20028
$427$	742.035	0.0840974
$428$	23739.8	2.68109
$429$	0	0
$430$	−10772.6	−1.20814
$431$	−7124.99	−0.796285	−0.398142	−	0.917324i	$-0.630345\pi$
$431$	−7124.99	−0.796285	−0.398142	+	0.917324i	$0.630345\pi$
$432$	−2817.15	−0.313750
$433$	−6400.92	−0.710412	−0.355206	−	0.934788i	$-0.615589\pi$
$433$	−6400.92	−0.710412	−0.355206	+	0.934788i	$0.615589\pi$
$434$	4949.38	0.547415
$435$	1157.90	0.127625
$436$	−2357.83	−0.258990
$437$	9348.85	1.02338
$438$	39709.1	4.33191
$439$	7615.19	0.827912	0.413956	−	0.910297i	$-0.364147\pi$
$439$	7615.19	0.827912	0.413956	+	0.910297i	$0.364147\pi$
$440$	−2797.62	−0.303117
$441$	−6485.05	−0.700254
$442$	0	0
$443$	−403.957	−0.0433241	−0.0216620	−	0.999765i	$-0.506896\pi$
$443$	−403.957	−0.0433241	−0.0216620	+	0.999765i	$0.506896\pi$
$444$	36988.9	3.95364
$445$	−6265.03	−0.667395
$446$	2334.47	0.247849
$447$	−19170.5	−2.02849
$448$	−6923.30	−0.730123
$449$	12544.9	1.31855	0.659275	−	0.751902i	$-0.270863\pi$
$449$	12544.9	1.31855	0.659275	+	0.751902i	$0.270863\pi$
$450$	12821.6	1.34315
$451$	−2665.61	−0.278312
$452$	−4286.10	−0.446020
$453$	−3542.49	−0.367419
$454$	−18841.4	−1.94774
$455$	0	0
$456$	32230.2	3.30990
$457$	−15475.2	−1.58403	−0.792013	−	0.610504i	$-0.790967\pi$
$457$	−15475.2	−1.58403	−0.792013	+	0.610504i	$0.790967\pi$
$458$	4595.11	0.468811
$459$	−1542.21	−0.156829
$460$	−8997.38	−0.911967
$461$	−15071.8	−1.52269	−0.761347	−	0.648344i	$-0.775462\pi$
$461$	−15071.8	−1.52269	−0.761347	+	0.648344i	$0.775462\pi$
$462$	−5074.87	−0.511048
$463$	2467.06	0.247633	0.123817	−	0.992305i	$-0.460487\pi$
$463$	2467.06	0.247633	0.123817	+	0.992305i	$0.460487\pi$
$464$	1245.77	0.124641
$465$	−4398.73	−0.438680
$466$	10136.6	1.00766
$467$	−10906.3	−1.08070	−0.540348	−	0.841442i	$-0.681707\pi$
$467$	−10906.3	−1.08070	−0.540348	+	0.841442i	$0.681707\pi$
$468$	0	0
$469$	−10389.2	−1.02288
$470$	15670.5	1.53793
$471$	16819.2	1.64541
$472$	21155.2	2.06303
$473$	3585.20	0.348515
$474$	−27970.0	−2.71035
$475$	−8736.19	−0.843882
$476$	5948.89	0.572829
$477$	−8933.77	−0.857545
$478$	−32093.3	−3.07095
$479$	9696.41	0.924927	0.462464	−	0.886638i	$-0.346966\pi$
$479$	9696.41	0.924927	0.462464	+	0.886638i	$0.346966\pi$
$480$	1230.25	0.116985
$481$	0	0
$482$	−3837.94	−0.362684
$483$	−8001.91	−0.753829
$484$	1899.07	0.178350
$485$	−11131.4	−1.04216
$486$	−26032.8	−2.42978
$487$	−8851.15	−0.823581	−0.411790	−	0.911279i	$-0.635096\pi$
$487$	−8851.15	−0.823581	−0.411790	+	0.911279i	$0.635096\pi$
$488$	−2278.83	−0.211389
$489$	6864.42	0.634805
$490$	−6420.17	−0.591905
$491$	−6581.76	−0.604950	−0.302475	−	0.953157i	$-0.597813\pi$
$491$	−6581.76	−0.604950	−0.302475	+	0.953157i	$0.597813\pi$
$492$	−29554.8	−2.70820
$493$	681.983	0.0623022
$494$	0	0
$495$	2493.62	0.226424
$496$	−4732.55	−0.428423
$497$	−9072.87	−0.818861
$498$	6092.33	0.548201
$499$	9264.63	0.831146	0.415573	−	0.909560i	$-0.363581\pi$
$499$	9264.63	0.831146	0.415573	+	0.909560i	$0.363581\pi$
$500$	21728.8	1.94348
$501$	16760.8	1.49465
$502$	410.990	0.0365406
$503$	−1265.32	−0.112163	−0.0560814	−	0.998426i	$-0.517861\pi$
$503$	−1265.32	−0.112163	−0.0560814	+	0.998426i	$0.517861\pi$
$504$	−15252.0	−1.34797
$505$	1940.34	0.170978
$506$	4520.72	0.397175
$507$	0	0
$508$	−25736.5	−2.24778
$509$	12585.4	1.09595	0.547976	−	0.836494i	$-0.315399\pi$
$509$	12585.4	1.09595	0.547976	+	0.836494i	$0.315399\pi$
$510$	−7981.96	−0.693034
$511$	12803.6	1.10841
$512$	18334.3	1.58256
$513$	−5495.04	−0.472928
$514$	−31202.0	−2.67755
$515$	3623.87	0.310071
$516$	39750.7	3.39133
$517$	−5215.27	−0.443650
$518$	18005.7	1.52727
$519$	−4933.65	−0.417270
$520$	0	0
$521$	6595.77	0.554637	0.277318	−	0.960778i	$-0.410554\pi$
$521$	6595.77	0.554637	0.277318	+	0.960778i	$0.410554\pi$
$522$	−3566.34	−0.299031
$523$	18286.4	1.52889	0.764445	−	0.644688i	$-0.223013\pi$
$523$	18286.4	1.52889	0.764445	+	0.644688i	$0.223013\pi$
$524$	23197.7	1.93396
$525$	7477.52	0.621611
$526$	−17862.2	−1.48067
$527$	−2590.78	−0.214148
$528$	4852.54	0.399961
$529$	−5038.85	−0.414141
$530$	−8844.39	−0.724860
$531$	−18856.4	−1.54105
$532$	21196.4	1.72741
$533$	0	0
$534$	34901.7	2.82836
$535$	−10270.5	−0.829970
$536$	31905.9	2.57113
$537$	7132.95	0.573202
$538$	39779.4	3.18775
$539$	2136.69	0.170749
$540$	5288.46	0.421443
$541$	−6494.33	−0.516106	−0.258053	−	0.966131i	$-0.583081\pi$
$541$	−6494.33	−0.516106	−0.258053	+	0.966131i	$0.583081\pi$
$542$	−9027.18	−0.715407
$543$	29572.8	2.33718
$544$	724.597	0.0571082
$545$	1020.06	0.0801739
$546$	0	0
$547$	−7095.02	−0.554591	−0.277295	−	0.960785i	$-0.589438\pi$
$547$	−7095.02	−0.554591	−0.277295	+	0.960785i	$0.589438\pi$
$548$	46442.3	3.62028
$549$	2031.20	0.157904
$550$	−4224.46	−0.327512
$551$	2429.97	0.187877
$552$	24574.3	1.89484
$553$	−9018.49	−0.693500
$554$	−15649.5	−1.20015
$555$	−16002.5	−1.22391
$556$	10448.2	0.796947
$557$	−13753.5	−1.04624	−0.523120	−	0.852259i	$-0.675232\pi$
$557$	−13753.5	−1.04624	−0.523120	+	0.852259i	$0.675232\pi$
$558$	13548.1	1.02785
$559$	0	0
$560$	−4701.28	−0.354760
$561$	2656.46	0.199922
$562$	41634.7	3.12501
$563$	13735.7	1.02822	0.514112	−	0.857723i	$-0.328122\pi$
$563$	13735.7	1.02822	0.514112	+	0.857723i	$0.328122\pi$
$564$	−57824.0	−4.31708
$565$	1854.29	0.138072
$566$	3959.86	0.294073
$567$	−6290.91	−0.465950
$568$	27863.3	2.05830
$569$	21279.1	1.56778	0.783890	−	0.620899i	$-0.213232\pi$
$569$	21279.1	1.56778	0.783890	+	0.620899i	$0.213232\pi$
$570$	−28440.5	−2.08989
$571$	9365.84	0.686424	0.343212	−	0.939258i	$-0.388485\pi$
$571$	9365.84	0.686424	0.343212	+	0.939258i	$0.388485\pi$
$572$	0	0
$573$	33628.3	2.45173
$574$	−14386.9	−1.04616
$575$	−6661.02	−0.483102
$576$	−18951.4	−1.37091
$577$	22068.2	1.59222	0.796110	−	0.605153i	$-0.206888\pi$
$577$	22068.2	1.59222	0.796110	+	0.605153i	$0.206888\pi$
$578$	19213.9	1.38269
$579$	−6421.41	−0.460907
$580$	−2338.61	−0.167424
$581$	1964.38	0.140269
$582$	62011.5	4.41660
$583$	2943.49	0.209103
$584$	−39320.6	−2.78613
$585$	0	0
$586$	19443.4	1.37064
$587$	11150.9	0.784065	0.392033	−	0.919951i	$-0.371772\pi$
$587$	11150.9	0.784065	0.392033	+	0.919951i	$0.371772\pi$
$588$	23690.4	1.66152
$589$	−9231.18	−0.645780
$590$	−18667.7	−1.30261
$591$	−20542.3	−1.42978
$592$	−17216.9	−1.19529
$593$	−16730.2	−1.15856	−0.579280	−	0.815129i	$-0.696666\pi$
$593$	−16730.2	−1.15856	−0.579280	+	0.815129i	$0.696666\pi$
$594$	−2657.18	−0.183544
$595$	−2573.66	−0.177327
$596$	38718.7	2.66104
$597$	−21271.8	−1.45829
$598$	0	0
$599$	21934.5	1.49619	0.748097	−	0.663590i	$-0.230968\pi$
$599$	21934.5	1.49619	0.748097	+	0.663590i	$0.230968\pi$
$600$	−22963.9	−1.56249
$601$	4434.24	0.300959	0.150479	−	0.988613i	$-0.451918\pi$
$601$	4434.24	0.300959	0.150479	+	0.988613i	$0.451918\pi$
$602$	19350.1	1.31005
$603$	−28438.8	−1.92059
$604$	7154.78	0.481993
$605$	−821.594	−0.0552108
$606$	−10809.4	−0.724589
$607$	22581.3	1.50996	0.754982	−	0.655745i	$-0.227646\pi$
$607$	22581.3	1.50996	0.754982	+	0.655745i	$0.227646\pi$
$608$	2581.81	0.172214
$609$	−2079.87	−0.138392
$610$	2010.88	0.133472
$611$	0	0
$612$	16284.1	1.07557
$613$	−24639.4	−1.62345	−0.811727	−	0.584037i	$-0.801472\pi$
$613$	−24639.4	−1.62345	−0.811727	+	0.584037i	$0.801472\pi$
$614$	−39951.7	−2.62593
$615$	12786.3	0.838361
$616$	5025.21	0.328687
$617$	9756.74	0.636615	0.318308	−	0.947988i	$-0.396886\pi$
$617$	9756.74	0.636615	0.318308	+	0.947988i	$0.396886\pi$
$618$	−20188.1	−1.31405
$619$	18965.9	1.23151	0.615755	−	0.787937i	$-0.288851\pi$
$619$	18965.9	1.23151	0.615755	+	0.787937i	$0.288851\pi$
$620$	8884.14	0.575477
$621$	−4189.76	−0.270740
$622$	17389.8	1.12101
$623$	11253.5	0.723696
$624$	0	0
$625$	461.433	0.0295317
$626$	−15647.2	−0.999025
$627$	9465.22	0.602878
$628$	−33969.8	−2.15850
$629$	−9425.19	−0.597468
$630$	13458.6	0.851118
$631$	−15836.3	−0.999102	−0.499551	−	0.866285i	$-0.666502\pi$
$631$	−15836.3	−0.999102	−0.499551	+	0.866285i	$0.666502\pi$
$632$	27696.3	1.74320
$633$	−41293.1	−2.59282
$634$	46037.4	2.88388
$635$	11134.3	0.695832
$636$	32635.8	2.03474
$637$	0	0
$638$	1175.03	0.0729154
$639$	−24835.5	−1.53752
$640$	−17495.3	−1.08057
$641$	3727.95	0.229711	0.114856	−	0.993382i	$-0.463359\pi$
$641$	3727.95	0.229711	0.114856	+	0.993382i	$0.463359\pi$
$642$	57215.8	3.51733
$643$	16971.4	1.04088	0.520440	−	0.853898i	$-0.325768\pi$
$643$	16971.4	1.04088	0.520440	+	0.853898i	$0.325768\pi$
$644$	16161.5	0.988900
$645$	−17197.3	−1.04984
$646$	−16750.9	−1.02021
$647$	2460.07	0.149483	0.0747414	−	0.997203i	$-0.476187\pi$
$647$	2460.07	0.149483	0.0747414	+	0.997203i	$0.476187\pi$
$648$	19319.7	1.17122
$649$	6212.78	0.375767
$650$	0	0
$651$	7901.19	0.475687
$652$	−13864.1	−0.832760
$653$	−7568.89	−0.453589	−0.226794	−	0.973943i	$-0.572825\pi$
$653$	−7568.89	−0.453589	−0.226794	+	0.973943i	$0.572825\pi$
$654$	−5682.65	−0.339769
$655$	−10036.0	−0.598684
$656$	13756.6	0.818758
$657$	35047.8	2.08119
$658$	−28148.0	−1.66766
$659$	1352.39	0.0799415	0.0399708	−	0.999201i	$-0.487274\pi$
$659$	1352.39	0.0799415	0.0399708	+	0.999201i	$0.487274\pi$
$660$	−9109.38	−0.537246
$661$	−3444.05	−0.202660	−0.101330	−	0.994853i	$-0.532310\pi$
$661$	−3444.05	−0.202660	−0.101330	+	0.994853i	$0.532310\pi$
$662$	9650.93	0.566608
$663$	0	0
$664$	−6032.72	−0.352583
$665$	−9170.19	−0.534744
$666$	49287.8	2.86766
$667$	1852.76	0.107555
$668$	−33851.9	−1.96073
$669$	3726.76	0.215373
$670$	−28154.3	−1.62343
$671$	−669.237	−0.0385032
$672$	−2209.83	−0.126854
$673$	−21330.5	−1.22174	−0.610869	−	0.791732i	$-0.709180\pi$
$673$	−21330.5	−1.22174	−0.610869	+	0.791732i	$0.709180\pi$
$674$	−20722.4	−1.18427
$675$	3915.20	0.223253
$676$	0	0
$677$	−17214.8	−0.977278	−0.488639	−	0.872486i	$-0.662506\pi$
$677$	−17214.8	−0.977278	−0.488639	+	0.872486i	$0.662506\pi$
$678$	−10330.0	−0.585136
$679$	19994.7	1.13008
$680$	7903.86	0.445734
$681$	−30078.5	−1.69253
$682$	−4463.82	−0.250628
$683$	26914.9	1.50786	0.753930	−	0.656955i	$-0.228156\pi$
$683$	26914.9	1.50786	0.753930	+	0.656955i	$0.228156\pi$
$684$	58021.8	3.24345
$685$	−20092.3	−1.12071
$686$	31895.9	1.77521
$687$	7335.64	0.407383
$688$	−18502.4	−1.02529
$689$	0	0
$690$	−21684.8	−1.19641
$691$	−29556.4	−1.62717	−0.813587	−	0.581443i	$-0.802488\pi$
$691$	−29556.4	−1.62717	−0.813587	+	0.581443i	$0.802488\pi$
$692$	9964.51	0.547390
$693$	−4479.14	−0.245525
$694$	12304.8	0.673033
$695$	−4520.19	−0.246706
$696$	6387.39	0.347864
$697$	7530.89	0.409258
$698$	23448.4	1.27154
$699$	16182.1	0.875625
$700$	−15102.4	−0.815451
$701$	17864.1	0.962507	0.481253	−	0.876582i	$-0.340182\pi$
$701$	17864.1	0.962507	0.481253	+	0.876582i	$0.340182\pi$
$702$	0	0
$703$	−33582.8	−1.80171
$704$	6244.09	0.334280
$705$	25016.4	1.33641
$706$	18023.0	0.960770
$707$	−3485.32	−0.185402
$708$	68883.9	3.65652
$709$	−30283.0	−1.60409	−0.802046	−	0.597263i	$-0.796255\pi$
$709$	−30283.0	−1.60409	−0.802046	+	0.597263i	$0.796255\pi$
$710$	−24587.0	−1.29963
$711$	−24686.7	−1.30214
$712$	−34560.2	−1.81910
$713$	−7038.43	−0.369693
$714$	14337.5	0.751497
$715$	0	0
$716$	−14406.5	−0.751948
$717$	−51233.8	−2.66857
$718$	20899.5	1.08630
$719$	−6678.03	−0.346382	−0.173191	−	0.984888i	$-0.555408\pi$
$719$	−6678.03	−0.346382	−0.173191	+	0.984888i	$0.555408\pi$
$720$	−12869.0	−0.666110
$721$	−6509.35	−0.336229
$722$	−26297.4	−1.35552
$723$	−6126.89	−0.315161
$724$	−59728.2	−3.06600
$725$	−1731.34	−0.0886903
$726$	4576.99	0.233978
$727$	15670.1	0.799409	0.399705	−	0.916644i	$-0.369113\pi$
$727$	15670.1	0.799409	0.399705	+	0.916644i	$0.369113\pi$
$728$	0	0
$729$	−27632.3	−1.40387
$730$	34697.2	1.75918
$731$	−10128.9	−0.512492
$732$	−7420.14	−0.374667
$733$	9590.61	0.483270	0.241635	−	0.970367i	$-0.422316\pi$
$733$	9590.61	0.483270	0.241635	+	0.970367i	$0.422316\pi$
$734$	−45371.8	−2.28161
$735$	−10249.2	−0.514348
$736$	1968.53	0.0985883
$737$	9369.98	0.468314
$738$	−39381.8	−1.96431
$739$	12190.9	0.606834	0.303417	−	0.952858i	$-0.401872\pi$
$739$	12190.9	0.606834	0.303417	+	0.952858i	$0.401872\pi$
$740$	32320.3	1.60556
$741$	0	0
$742$	15886.7	0.786008
$743$	−5903.11	−0.291473	−0.145736	−	0.989323i	$-0.546555\pi$
$743$	−5903.11	−0.291473	−0.145736	+	0.989323i	$0.546555\pi$
$744$	−24265.0	−1.19570
$745$	−16750.9	−0.823764
$746$	−31872.3	−1.56425
$747$	5377.17	0.263374
$748$	−5365.27	−0.262264
$749$	18448.4	0.899985
$750$	52369.0	2.54966
$751$	−11178.9	−0.543173	−0.271587	−	0.962414i	$-0.587548\pi$
$751$	−11178.9	−0.543173	−0.271587	+	0.962414i	$0.587548\pi$
$752$	26914.8	1.30516
$753$	656.104	0.0317527
$754$	0	0
$755$	−3095.37	−0.149208
$756$	−9499.36	−0.456995
$757$	−21033.9	−1.00990	−0.504948	−	0.863150i	$-0.668488\pi$
$757$	−21033.9	−1.00990	−0.504948	+	0.863150i	$0.668488\pi$
$758$	35234.3	1.68835
$759$	7216.88	0.345133
$760$	28162.2	1.34414
$761$	882.086	0.0420179	0.0210089	−	0.999779i	$-0.493312\pi$
$761$	882.086	0.0420179	0.0210089	+	0.999779i	$0.493312\pi$
$762$	−62028.0	−2.94887
$763$	−1832.28	−0.0869373
$764$	−67919.1	−3.21627
$765$	−7044.98	−0.332957
$766$	−3675.45	−0.173367
$767$	0	0
$768$	62175.4	2.92130
$769$	26048.5	1.22150	0.610751	−	0.791823i	$-0.290868\pi$
$769$	26048.5	1.22150	0.610751	+	0.791823i	$0.290868\pi$
$770$	−4434.33	−0.207535
$771$	−49810.9	−2.32671
$772$	12969.4	0.604634
$773$	−4753.72	−0.221189	−0.110595	−	0.993866i	$-0.535276\pi$
$773$	−4753.72	−0.221189	−0.110595	+	0.993866i	$0.535276\pi$
$774$	52967.9	2.45981
$775$	6577.18	0.304851
$776$	−61404.7	−2.84059
$777$	28744.4	1.32715
$778$	19944.9	0.919101
$779$	26833.3	1.23415
$780$	0	0
$781$	8182.77	0.374907
$782$	−12772.0	−0.584047
$783$	−1089.01	−0.0497038
$784$	−11027.0	−0.502322
$785$	14696.3	0.668196
$786$	55909.2	2.53717
$787$	34950.3	1.58303	0.791514	−	0.611151i	$-0.209293\pi$
$787$	34950.3	1.58303	0.791514	+	0.611151i	$0.209293\pi$
$788$	41489.4	1.87563
$789$	−28515.3	−1.28666
$790$	−24439.7	−1.10066
$791$	−3330.76	−0.149720
$792$	13755.7	0.617155
$793$	0	0
$794$	−50918.7	−2.27586
$795$	−14119.2	−0.629882
$796$	42962.7	1.91303
$797$	2517.67	0.111895	0.0559476	−	0.998434i	$-0.482182\pi$
$797$	2517.67	0.111895	0.0559476	+	0.998434i	$0.482182\pi$
$798$	51086.0	2.26620
$799$	14734.2	0.652389
$800$	−1839.53	−0.0812963
$801$	30804.7	1.35884
$802$	29433.1	1.29591
$803$	−11547.5	−0.507475
$804$	103889.	4.55708
$805$	−6991.93	−0.306128
$806$	0	0
$807$	63503.8	2.77006
$808$	10703.6	0.466029
$809$	−1840.70	−0.0799946	−0.0399973	−	0.999200i	$-0.512735\pi$
$809$	−1840.70	−0.0799946	−0.0399973	+	0.999200i	$0.512735\pi$
$810$	−17048.1	−0.739516
$811$	−36003.3	−1.55888	−0.779438	−	0.626480i	$-0.784495\pi$
$811$	−36003.3	−1.55888	−0.779438	+	0.626480i	$0.784495\pi$
$812$	4200.72	0.181547
$813$	−14411.0	−0.621667
$814$	−16239.3	−0.699246
$815$	5998.01	0.257793
$816$	−13709.4	−0.588144
$817$	−36090.3	−1.54546
$818$	31761.5	1.35760
$819$	0	0
$820$	−25824.5	−1.09979
$821$	39115.2	1.66276	0.831382	−	0.555701i	$-0.187550\pi$
$821$	39115.2	1.66276	0.831382	+	0.555701i	$0.187550\pi$
$822$	111931.	4.74946
$823$	13571.6	0.574820	0.287410	−	0.957808i	$-0.407206\pi$
$823$	13571.6	0.574820	0.287410	+	0.957808i	$0.407206\pi$
$824$	19990.6	0.845151
$825$	−6743.93	−0.284598
$826$	33531.8	1.41249
$827$	32945.2	1.38527	0.692634	−	0.721289i	$-0.256450\pi$
$827$	32945.2	1.38527	0.692634	+	0.721289i	$0.256450\pi$
$828$	44239.4	1.85680
$829$	−11996.8	−0.502613	−0.251306	−	0.967908i	$-0.580860\pi$
$829$	−11996.8	−0.502613	−0.251306	+	0.967908i	$0.580860\pi$
$830$	5323.38	0.222623
$831$	−24982.9	−1.04290
$832$	0	0
$833$	−6036.58	−0.251087
$834$	25181.4	1.04552
$835$	14645.3	0.606973
$836$	−19116.9	−0.790877
$837$	4137.03	0.170844
$838$	65760.9	2.71083
$839$	21000.6	0.864148	0.432074	−	0.901838i	$-0.357782\pi$
$839$	21000.6	0.864148	0.432074	+	0.901838i	$0.357782\pi$
$840$	−24104.7	−0.990108
$841$	−23907.4	−0.980255
$842$	42760.3	1.75014
$843$	66465.7	2.71554
$844$	83399.9	3.40135
$845$	0	0
$846$	−77050.5	−3.13127
$847$	1475.78	0.0598683
$848$	−15190.7	−0.615154
$849$	6321.52	0.255541
$850$	11935.0	0.481607
$851$	−25605.6	−1.03143
$852$	90726.1	3.64815
$853$	−40511.4	−1.62612	−0.813061	−	0.582178i	$-0.802201\pi$
$853$	−40511.4	−1.62612	−0.813061	+	0.582178i	$0.802201\pi$
$854$	−3612.03	−0.144732
$855$	−25101.9	−1.00406
$856$	−56656.0	−2.26222
$857$	−19156.1	−0.763546	−0.381773	−	0.924256i	$-0.624686\pi$
$857$	−19156.1	−0.763546	−0.381773	+	0.924256i	$0.624686\pi$
$858$	0	0
$859$	−38727.4	−1.53825	−0.769127	−	0.639096i	$-0.779309\pi$
$859$	−38727.4	−1.53825	−0.769127	+	0.639096i	$0.779309\pi$
$860$	34733.5	1.37721
$861$	−22967.2	−0.909084
$862$	34682.5	1.37041
$863$	−36280.3	−1.43105	−0.715524	−	0.698588i	$-0.753812\pi$
$863$	−36280.3	−1.43105	−0.715524	+	0.698588i	$0.753812\pi$
$864$	−1157.06	−0.0455601
$865$	−4310.94	−0.169452
$866$	31158.0	1.22262
$867$	30673.1	1.20152
$868$	−15958.1	−0.624023
$869$	8133.73	0.317512
$870$	−5636.34	−0.219644
$871$	0	0
$872$	5627.04	0.218527
$873$	54732.1	2.12188
$874$	−45507.7	−1.76124
$875$	16885.6	0.652385
$876$	−128032.	−4.93814
$877$	−51747.7	−1.99247	−0.996234	−	0.0867073i	$-0.972365\pi$
$877$	−51747.7	−1.99247	−0.996234	+	0.0867073i	$0.972365\pi$
$878$	−37068.7	−1.42484
$879$	31039.4	1.19105
$880$	4240.06	0.162423
$881$	−24397.8	−0.933009	−0.466505	−	0.884519i	$-0.654487\pi$
$881$	−24397.8	−0.933009	−0.466505	+	0.884519i	$0.654487\pi$
$882$	31567.5	1.20514
$883$	−28537.3	−1.08761	−0.543804	−	0.839212i	$-0.683016\pi$
$883$	−28537.3	−1.08761	−0.543804	+	0.839212i	$0.683016\pi$
$884$	0	0
$885$	−29801.2	−1.13193
$886$	1966.35	0.0745609
$887$	−29577.4	−1.11963	−0.559814	−	0.828618i	$-0.689128\pi$
$887$	−29577.4	−1.11963	−0.559814	+	0.828618i	$0.689128\pi$
$888$	−88275.5	−3.33596
$889$	−20000.0	−0.754531
$890$	30496.5	1.14859
$891$	5673.74	0.213331
$892$	−7526.94	−0.282534
$893$	52499.3	1.96733
$894$	93316.9	3.49103
$895$	6232.65	0.232776
$896$	31425.8	1.17172
$897$	0	0
$898$	−61065.0	−2.26923
$899$	−1829.44	−0.0678701
$900$	−41340.3	−1.53112
$901$	−8315.96	−0.307486
$902$	12975.5	0.478975
$903$	30890.6	1.13840
$904$	10228.9	0.376338
$905$	25840.2	0.949122
$906$	17243.9	0.632329
$907$	3029.84	0.110920	0.0554599	−	0.998461i	$-0.482338\pi$
$907$	3029.84	0.110920	0.0554599	+	0.998461i	$0.482338\pi$
$908$	60749.6	2.22032
$909$	−9540.50	−0.348117
$910$	0	0
$911$	−27762.9	−1.00969	−0.504844	−	0.863211i	$-0.668450\pi$
$911$	−27762.9	−1.00969	−0.504844	+	0.863211i	$0.668450\pi$
$912$	−48847.9	−1.77359
$913$	−1771.66	−0.0642207
$914$	75329.2	2.72611
$915$	3210.17	0.115983
$916$	−14815.8	−0.534419
$917$	18027.1	0.649189
$918$	7507.07	0.269902
$919$	−4346.97	−0.156032	−0.0780159	−	0.996952i	$-0.524858\pi$
$919$	−4346.97	−0.156032	−0.0780159	+	0.996952i	$0.524858\pi$
$920$	21472.6	0.769490
$921$	−63778.9	−2.28185
$922$	73365.3	2.62056
$923$	0	0
$924$	16362.7	0.582567
$925$	23927.6	0.850525
$926$	−12009.0	−0.426178
$927$	−17818.3	−0.631315
$928$	511.664	0.0180993
$929$	−2228.75	−0.0787115	−0.0393558	−	0.999225i	$-0.512531\pi$
$929$	−2228.75	−0.0787115	−0.0393558	+	0.999225i	$0.512531\pi$
$930$	21411.8	0.754970
$931$	−21508.9	−0.757170
$932$	−32683.0	−1.14868
$933$	27761.1	0.974122
$934$	53089.1	1.85988
$935$	2321.17	0.0811876
$936$	0	0
$937$	−3718.86	−0.129658	−0.0648291	−	0.997896i	$-0.520650\pi$
$937$	−3718.86	−0.129658	−0.0648291	+	0.997896i	$0.520650\pi$
$938$	50571.9	1.76038
$939$	−24979.3	−0.868123
$940$	−50525.6	−1.75315
$941$	−38302.8	−1.32693	−0.663463	−	0.748209i	$-0.730914\pi$
$941$	−38302.8	−1.32693	−0.663463	+	0.748209i	$0.730914\pi$
$942$	−81871.3	−2.83175
$943$	20459.4	0.706520
$944$	−32062.8	−1.10546
$945$	4109.70	0.141469
$946$	−17451.8	−0.599796
$947$	−12761.7	−0.437907	−0.218954	−	0.975735i	$-0.570264\pi$
$947$	−12761.7	−0.437907	−0.218954	+	0.975735i	$0.570264\pi$
$948$	90182.3	3.08965
$949$	0	0
$950$	42525.4	1.45232
$951$	73494.1	2.50600
$952$	−14197.2	−0.483336
$953$	28280.0	0.961258	0.480629	−	0.876924i	$-0.340408\pi$
$953$	28280.0	0.961258	0.480629	+	0.876924i	$0.340408\pi$
$954$	43487.2	1.47584
$955$	29383.8	0.995641
$956$	103477.	3.50072
$957$	1875.82	0.0633613
$958$	−47199.5	−1.59180
$959$	36090.6	1.21525
$960$	−29951.4	−1.00695
$961$	−22841.2	−0.766713
$962$	0	0
$963$	50499.4	1.68985
$964$	12374.5	0.413440
$965$	−5610.92	−0.187173
$966$	38951.1	1.29734
$967$	−43554.8	−1.44842	−0.724212	−	0.689577i	$-0.757796\pi$
$967$	−43554.8	−1.44842	−0.724212	+	0.689577i	$0.757796\pi$
$968$	−4532.21	−0.150486
$969$	−26741.2	−0.886534
$970$	54184.6	1.79357
$971$	38867.6	1.28457	0.642286	−	0.766465i	$-0.277986\pi$
$971$	38867.6	1.28457	0.642286	+	0.766465i	$0.277986\pi$
$972$	83936.4	2.76982
$973$	8119.37	0.267518
$974$	43085.0	1.41739
$975$	0	0
$976$	3453.79	0.113272
$977$	−25776.3	−0.844071	−0.422036	−	0.906579i	$-0.638684\pi$
$977$	−25776.3	−0.844071	−0.422036	+	0.906579i	$0.638684\pi$
$978$	−33414.1	−1.09250
$979$	−10149.5	−0.331337
$980$	20700.3	0.674741
$981$	−5015.58	−0.163237
$982$	32038.2	1.04112
$983$	10135.6	0.328867	0.164433	−	0.986388i	$-0.447420\pi$
$983$	10135.6	0.328867	0.164433	+	0.986388i	$0.447420\pi$
$984$	70533.7	2.28509
$985$	−17949.5	−0.580629
$986$	−3319.71	−0.107222
$987$	−44935.5	−1.44915
$988$	0	0
$989$	−27517.5	−0.884738
$990$	−12138.3	−0.389676
$991$	34291.6	1.09920	0.549601	−	0.835427i	$-0.314780\pi$
$991$	34291.6	1.09920	0.549601	+	0.835427i	$0.314780\pi$
$992$	−1943.75	−0.0622119
$993$	15406.8	0.492365
$994$	44164.3	1.40926
$995$	−18586.9	−0.592206
$996$	−19643.2	−0.624920
$997$	−11783.3	−0.374305	−0.187153	−	0.982331i	$-0.559926\pi$
$997$	−11783.3	−0.374305	−0.187153	+	0.982331i	$0.559926\pi$
$998$	−45097.7	−1.43040
$999$	15050.4	0.476651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1859.4.a.o.1.4	yes	39
13.12	even	2		1859.4.a.n.1.36	✓	39

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1859.4.a.n.1.36	✓	39	13.12	even	2
1859.4.a.o.1.4	yes	39	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1859.a (trivial)

\(f(q)\)	\(=\)	\(q-4.86773 q^{2} -7.77085 q^{3} +15.6948 q^{4} -6.79003 q^{5} +37.8264 q^{6} +12.1965 q^{7} -37.4563 q^{8} +33.3861 q^{9} +33.0520 q^{10} -11.0000 q^{11} -121.962 q^{12} -59.3695 q^{14} +52.7643 q^{15} +56.7686 q^{16} +31.0773 q^{17} -162.514 q^{18} +110.731 q^{19} -106.568 q^{20} -94.7775 q^{21} +53.5450 q^{22} +84.4283 q^{23} +291.067 q^{24} -78.8955 q^{25} -49.6251 q^{27} +191.422 q^{28} +21.9448 q^{29} -256.842 q^{30} -83.3657 q^{31} +23.3160 q^{32} +85.4793 q^{33} -151.276 q^{34} -82.8149 q^{35} +523.988 q^{36} -303.283 q^{37} -539.010 q^{38} +254.329 q^{40} +242.328 q^{41} +461.351 q^{42} -325.927 q^{43} -172.643 q^{44} -226.692 q^{45} -410.975 q^{46} +474.115 q^{47} -441.140 q^{48} -194.244 q^{49} +384.042 q^{50} -241.497 q^{51} -267.590 q^{53} +241.562 q^{54} +74.6903 q^{55} -456.837 q^{56} -860.475 q^{57} -106.821 q^{58} -564.798 q^{59} +828.125 q^{60} +60.8398 q^{61} +405.802 q^{62} +407.195 q^{63} -567.644 q^{64} -416.090 q^{66} -851.817 q^{67} +487.752 q^{68} -656.080 q^{69} +403.121 q^{70} -743.888 q^{71} -1250.52 q^{72} +1049.77 q^{73} +1476.30 q^{74} +613.085 q^{75} +1737.90 q^{76} -134.162 q^{77} -739.430 q^{79} -385.460 q^{80} -515.795 q^{81} -1179.59 q^{82} +161.060 q^{83} -1487.51 q^{84} -211.016 q^{85} +1586.53 q^{86} -170.529 q^{87} +412.019 q^{88} +922.681 q^{89} +1103.48 q^{90} +1325.09 q^{92} +647.822 q^{93} -2307.87 q^{94} -751.868 q^{95} -181.185 q^{96} +1639.37 q^{97} +945.529 q^{98} -367.247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q - 23 q^{3} + 114 q^{4} + 23 q^{5} + 77 q^{6} - 4 q^{7} - 21 q^{8} + 260 q^{9} - 158 q^{10} - 429 q^{11} - 351 q^{12} - 176 q^{14} + 30 q^{15} + 230 q^{16} - 244 q^{17} + 21 q^{18} - 70 q^{19} + 366 q^{20}+ \cdots - 2860 q^{99}+O(q^{100}) \) 39 * q - 23 * q^3 + 114 * q^4 + 23 * q^5 + 77 * q^6 - 4 * q^7 - 21 * q^8 + 260 * q^9 - 158 * q^10 - 429 * q^11 - 351 * q^12 - 176 * q^14 + 30 * q^15 + 230 * q^16 - 244 * q^17 + 21 * q^18 - 70 * q^19 + 366 * q^20 - 142 * q^21 - 47 * q^23 + 846 * q^24 + 322 * q^25 - 416 * q^27 + 1131 * q^28 - 838 * q^29 - 293 * q^30 + 507 * q^31 - 1433 * q^32 + 253 * q^33 + 166 * q^34 - 498 * q^35 + 815 * q^36 + 89 * q^37 + 81 * q^38 - 2917 * q^40 + 618 * q^41 - 318 * q^42 - 1064 * q^43 - 1254 * q^44 + 238 * q^45 - 1331 * q^46 + 1499 * q^47 - 1460 * q^48 - 413 * q^49 - 2459 * q^50 - 2350 * q^51 - 2745 * q^53 - 845 * q^54 - 253 * q^55 - 2904 * q^56 + 1450 * q^57 - 2509 * q^58 + 2285 * q^59 - 3566 * q^60 - 6218 * q^61 - 911 * q^62 - 1930 * q^63 + 67 * q^64 - 847 * q^66 + 546 * q^67 - 170 * q^68 - 5254 * q^69 - 2195 * q^70 - 263 * q^71 - 2393 * q^72 - 1148 * q^73 + 775 * q^74 - 5385 * q^75 - 7247 * q^76 + 44 * q^77 - 3666 * q^79 + 5594 * q^80 - 1901 * q^81 - 4414 * q^82 + 2722 * q^83 - 9971 * q^84 + 1858 * q^85 + 2478 * q^86 - 2284 * q^87 + 231 * q^88 + 13 * q^89 - 6771 * q^90 - 2232 * q^92 - 1082 * q^93 - 7330 * q^94 - 2352 * q^95 + 5770 * q^96 - 1197 * q^97 + 6813 * q^98 - 2860 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more