LMFDB - Embedded newform 2057.4.a.s.1.11

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)

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")

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);

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	\(q-3.20225 q^{2} -7.51392 q^{3} +2.25442 q^{4} -15.1991 q^{5} +24.0615 q^{6} +8.04966 q^{7} +18.3988 q^{8} +29.4590 q^{9} +48.6713 q^{10} -16.9395 q^{12} +64.4012 q^{13} -25.7770 q^{14} +114.205 q^{15} -76.9529 q^{16} -17.0000 q^{17} -94.3353 q^{18} -147.839 q^{19} -34.2651 q^{20} -60.4845 q^{21} +134.378 q^{23} -138.247 q^{24} +106.012 q^{25} -206.229 q^{26} -18.4771 q^{27} +18.1473 q^{28} -149.069 q^{29} -365.713 q^{30} -30.5687 q^{31} +99.2322 q^{32} +54.4383 q^{34} -122.348 q^{35} +66.4130 q^{36} -169.244 q^{37} +473.418 q^{38} -483.906 q^{39} -279.645 q^{40} -68.6591 q^{41} +193.687 q^{42} +46.4505 q^{43} -447.751 q^{45} -430.312 q^{46} -58.3594 q^{47} +578.218 q^{48} -278.203 q^{49} -339.478 q^{50} +127.737 q^{51} +145.187 q^{52} -422.473 q^{53} +59.1683 q^{54} +148.104 q^{56} +1110.85 q^{57} +477.357 q^{58} +674.907 q^{59} +257.465 q^{60} -393.087 q^{61} +97.8887 q^{62} +237.135 q^{63} +297.857 q^{64} -978.840 q^{65} +852.901 q^{67} -38.3251 q^{68} -1009.70 q^{69} +391.788 q^{70} -682.814 q^{71} +542.011 q^{72} +1024.32 q^{73} +541.960 q^{74} -796.569 q^{75} -333.291 q^{76} +1549.59 q^{78} -830.974 q^{79} +1169.61 q^{80} -656.559 q^{81} +219.864 q^{82} -138.563 q^{83} -136.357 q^{84} +258.385 q^{85} -148.746 q^{86} +1120.09 q^{87} +277.221 q^{89} +1433.81 q^{90} +518.408 q^{91} +302.944 q^{92} +229.691 q^{93} +186.882 q^{94} +2247.02 q^{95} -745.623 q^{96} -402.754 q^{97} +890.876 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) $ 44 * q - q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 + 24 * q^6 + 2 * q^7 + 63 * q^8 + 356 * q^9 + 65 * q^10 - 337 * q^12 - 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 - 748 * q^17 + 55 * q^18 + 36 * q^19 - 209 * q^20 + 240 * q^21 - 656 * q^23 + 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 - 63 * q^28 + 698 * q^29 - 304 * q^30 - 1418 * q^31 + 83 * q^32 + 17 * q^34 - 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 - 114 * q^39 + 281 * q^40 - 54 * q^41 - 1943 * q^42 - 382 * q^43 - 1226 * q^45 - 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 - 1307 * q^50 + 476 * q^51 - 3112 * q^52 - 2492 * q^53 - 2400 * q^54 + 27 * q^56 - 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 + 964 * q^61 + 3256 * q^62 - 2322 * q^63 - 3653 * q^64 + 152 * q^65 - 3442 * q^67 - 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 + 1281 * q^72 + 2240 * q^73 - 3494 * q^74 - 1892 * q^75 - 1261 * q^76 + 1120 * q^78 + 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 + 1656 * q^83 + 5401 * q^84 + 408 * q^85 - 1373 * q^86 - 1968 * q^87 - 2992 * q^89 - 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 + 694 * q^94 + 8280 * q^95 - 2203 * q^96 - 4746 * q^97 - 2302 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.20225	−1.13217	−0.566083	−	0.824348i	$-0.691542\pi$
$2$	−3.20225	−1.13217	−0.566083	+	0.824348i	$0.691542\pi$
$3$	−7.51392	−1.44606	−0.723028	−	0.690819i	$-0.757250\pi$
$3$	−7.51392	−1.44606	−0.723028	+	0.690819i	$0.757250\pi$
$4$	2.25442	0.281802
$5$	−15.1991	−1.35945	−0.679724	−	0.733468i	$-0.737900\pi$
$5$	−15.1991	−1.35945	−0.679724	+	0.733468i	$0.737900\pi$
$6$	24.0615	1.63718
$7$	8.04966	0.434641	0.217320	−	0.976100i	$-0.430268\pi$
$7$	8.04966	0.434641	0.217320	+	0.976100i	$0.430268\pi$
$8$	18.3988	0.813120
$9$	29.4590	1.09108
$10$	48.6713	1.53912
$11$	0	0
$11$	0	0
$12$	−16.9395	−0.407501
$13$	64.4012	1.37398	0.686988	−	0.726669i	$-0.258933\pi$
$13$	64.4012	1.37398	0.686988	+	0.726669i	$0.258933\pi$
$14$	−25.7770	−0.492086
$15$	114.205	1.96584
$16$	−76.9529	−1.20239
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−94.3353	−1.23528
$19$	−147.839	−1.78509	−0.892543	−	0.450961i	$-0.851081\pi$
$19$	−147.839	−1.78509	−0.892543	+	0.450961i	$0.851081\pi$
$20$	−34.2651	−0.383095
$21$	−60.4845	−0.628515
$22$	0	0
$23$	134.378	1.21825	0.609124	−	0.793075i	$-0.291521\pi$
$23$	134.378	1.21825	0.609124	+	0.793075i	$0.291521\pi$
$24$	−138.247	−1.17582
$25$	106.012	0.848099
$26$	−206.229	−1.55557
$27$	−18.4771	−0.131701
$28$	18.1473	0.122483
$29$	−149.069	−0.954532	−0.477266	−	0.878759i	$-0.658372\pi$
$29$	−149.069	−0.954532	−0.477266	+	0.878759i	$0.658372\pi$
$30$	−365.713	−2.22566
$31$	−30.5687	−0.177107	−0.0885533	−	0.996071i	$-0.528224\pi$
$31$	−30.5687	−0.177107	−0.0885533	+	0.996071i	$0.528224\pi$
$32$	99.2322	0.548186
$33$	0	0
$34$	54.4383	0.274591
$35$	−122.348	−0.590872
$36$	66.4130	0.307467
$37$	−169.244	−0.751985	−0.375993	−	0.926623i	$-0.622698\pi$
$37$	−169.244	−0.751985	−0.375993	+	0.926623i	$0.622698\pi$
$38$	473.418	2.02102
$39$	−483.906	−1.98684
$40$	−279.645	−1.10539
$41$	−68.6591	−0.261530	−0.130765	−	0.991413i	$-0.541743\pi$
$41$	−68.6591	−0.261530	−0.130765	+	0.991413i	$0.541743\pi$
$42$	193.687	0.711584
$43$	46.4505	0.164736	0.0823678	−	0.996602i	$-0.473752\pi$
$43$	46.4505	0.164736	0.0823678	+	0.996602i	$0.473752\pi$
$44$	0	0
$45$	−447.751	−1.48326
$46$	−430.312	−1.37926
$47$	−58.3594	−0.181119	−0.0905596	−	0.995891i	$-0.528866\pi$
$47$	−58.3594	−0.181119	−0.0905596	+	0.995891i	$0.528866\pi$
$48$	578.218	1.73872
$49$	−278.203	−0.811087
$50$	−339.478	−0.960190
$51$	127.737	0.350720
$52$	145.187	0.387189
$53$	−422.473	−1.09493	−0.547463	−	0.836830i	$-0.684406\pi$
$53$	−422.473	−1.09493	−0.547463	+	0.836830i	$0.684406\pi$
$54$	59.1683	0.149107
$55$	0	0
$56$	148.104	0.353415
$57$	1110.85	2.58133
$58$	477.357	1.08069
$59$	674.907	1.48924	0.744622	−	0.667486i	$-0.232630\pi$
$59$	674.907	1.48924	0.744622	+	0.667486i	$0.232630\pi$
$60$	257.465	0.553977
$61$	−393.087	−0.825077	−0.412538	−	0.910940i	$-0.635358\pi$
$61$	−393.087	−0.825077	−0.412538	+	0.910940i	$0.635358\pi$
$62$	97.8887	0.200514
$63$	237.135	0.474226
$64$	297.857	0.581752
$65$	−978.840	−1.86785
$66$	0	0
$67$	852.901	1.55520	0.777600	−	0.628759i	$-0.216437\pi$
$67$	852.901	1.55520	0.777600	+	0.628759i	$0.216437\pi$
$68$	−38.3251	−0.0683470
$69$	−1009.70	−1.76165
$70$	391.788	0.668966
$71$	−682.814	−1.14134	−0.570670	−	0.821179i	$-0.693316\pi$
$71$	−682.814	−1.14134	−0.570670	+	0.821179i	$0.693316\pi$
$72$	542.011	0.887176
$73$	1024.32	1.64230	0.821151	−	0.570711i	$-0.193333\pi$
$73$	1024.32	1.64230	0.821151	+	0.570711i	$0.193333\pi$
$74$	541.960	0.851373
$75$	−796.569	−1.22640
$76$	−333.291	−0.503041
$77$	0	0
$78$	1549.59	2.24944
$79$	−830.974	−1.18344	−0.591721	−	0.806143i	$-0.701551\pi$
$79$	−830.974	−1.18344	−0.591721	+	0.806143i	$0.701551\pi$
$80$	1169.61	1.63459
$81$	−656.559	−0.900629
$82$	219.864	0.296096
$83$	−138.563	−0.183244	−0.0916221	−	0.995794i	$-0.529205\pi$
$83$	−138.563	−0.183244	−0.0916221	+	0.995794i	$0.529205\pi$
$84$	−136.357	−0.177117
$85$	258.385	0.329715
$86$	−148.746	−0.186508
$87$	1120.09	1.38031
$88$	0	0
$89$	277.221	0.330172	0.165086	−	0.986279i	$-0.447210\pi$
$89$	277.221	0.330172	0.165086	+	0.986279i	$0.447210\pi$
$90$	1433.81	1.67930
$91$	518.408	0.597186
$92$	302.944	0.343305
$93$	229.691	0.256106
$94$	186.882	0.205057
$95$	2247.02	2.42673
$96$	−745.623	−0.792707
$97$	−402.754	−0.421582	−0.210791	−	0.977531i	$-0.567604\pi$
$97$	−402.754	−0.421582	−0.210791	+	0.977531i	$0.567604\pi$
$98$	890.876	0.918286
$99$	0	0
$100$	238.996	0.238996
$101$	172.035	0.169486	0.0847429	−	0.996403i	$-0.472993\pi$
$101$	172.035	0.169486	0.0847429	+	0.996403i	$0.472993\pi$
$102$	−409.045	−0.397074
$103$	1431.96	1.36985	0.684927	−	0.728612i	$-0.259834\pi$
$103$	1431.96	1.36985	0.684927	+	0.728612i	$0.259834\pi$
$104$	1184.91	1.11721
$105$	919.310	0.854433
$106$	1352.86	1.23964
$107$	2004.52	1.81107	0.905534	−	0.424273i	$-0.139470\pi$
$107$	2004.52	1.81107	0.905534	+	0.424273i	$0.139470\pi$
$108$	−41.6550	−0.0371135
$109$	1389.48	1.22099	0.610495	−	0.792020i	$-0.290971\pi$
$109$	1389.48	1.22099	0.610495	+	0.792020i	$0.290971\pi$
$110$	0	0
$111$	1271.68	1.08741
$112$	−619.445	−0.522608
$113$	1171.49	0.975264	0.487632	−	0.873049i	$-0.337861\pi$
$113$	1171.49	0.975264	0.487632	+	0.873049i	$0.337861\pi$
$114$	−3557.23	−2.92250
$115$	−2042.42	−1.65615
$116$	−336.064	−0.268989
$117$	1897.20	1.49911
$118$	−2161.22	−1.68607
$119$	−136.844	−0.105416
$120$	2101.23	1.59846
$121$	0	0
$122$	1258.76	0.934124
$123$	515.899	0.378187
$124$	−68.9146	−0.0499090
$125$	288.594	0.206501
$126$	−759.367	−0.536903
$127$	−2072.96	−1.44839	−0.724196	−	0.689595i	$-0.757789\pi$
$127$	−2072.96	−1.44839	−0.724196	+	0.689595i	$0.757789\pi$
$128$	−1747.67	−1.20683
$129$	−349.025	−0.238217
$130$	3134.49	2.11472
$131$	575.088	0.383555	0.191777	−	0.981438i	$-0.438575\pi$
$131$	575.088	0.383555	0.191777	+	0.981438i	$0.438575\pi$
$132$	0	0
$133$	−1190.06	−0.775872
$134$	−2731.20	−1.76075
$135$	280.835	0.179040
$136$	−312.780	−0.197211
$137$	−1329.55	−0.829134	−0.414567	−	0.910019i	$-0.636067\pi$
$137$	−1329.55	−0.829134	−0.414567	+	0.910019i	$0.636067\pi$
$138$	3233.33	1.99449
$139$	−501.046	−0.305742	−0.152871	−	0.988246i	$-0.548852\pi$
$139$	−501.046	−0.305742	−0.152871	+	0.988246i	$0.548852\pi$
$140$	−275.822	−0.166509
$141$	438.508	0.261908
$142$	2186.54	1.29219
$143$	0	0
$144$	−2266.96	−1.31190
$145$	2265.71	1.29764
$146$	−3280.14	−1.85936
$147$	2090.40	1.17288
$148$	−381.545	−0.211911
$149$	−866.548	−0.476446	−0.238223	−	0.971211i	$-0.576565\pi$
$149$	−866.548	−0.476446	−0.238223	+	0.971211i	$0.576565\pi$
$150$	2550.81	1.38849
$151$	−2707.54	−1.45918	−0.729591	−	0.683884i	$-0.760290\pi$
$151$	−2707.54	−1.45918	−0.729591	+	0.683884i	$0.760290\pi$
$152$	−2720.07	−1.45149
$153$	−500.804	−0.264625
$154$	0	0
$155$	464.617	0.240767
$156$	−1090.93	−0.559897
$157$	437.490	0.222392	0.111196	−	0.993799i	$-0.464532\pi$
$157$	437.490	0.222392	0.111196	+	0.993799i	$0.464532\pi$
$158$	2660.99	1.33985
$159$	3174.43	1.58332
$160$	−1508.24	−0.745230
$161$	1081.70	0.529501
$162$	2102.47	1.01966
$163$	1095.90	0.526612	0.263306	−	0.964712i	$-0.415187\pi$
$163$	1095.90	0.526612	0.263306	+	0.964712i	$0.415187\pi$
$164$	−154.786	−0.0736998
$165$	0	0
$166$	443.714	0.207463
$167$	−802.601	−0.371899	−0.185950	−	0.982559i	$-0.559536\pi$
$167$	−802.601	−0.371899	−0.185950	+	0.982559i	$0.559536\pi$
$168$	−1112.84	−0.511058
$169$	1950.52	0.887808
$170$	−827.412	−0.373292
$171$	−4355.20	−1.94766
$172$	104.719	0.0464228
$173$	27.3369	0.0120138	0.00600689	−	0.999982i	$-0.498088\pi$
$173$	27.3369	0.0120138	0.00600689	+	0.999982i	$0.498088\pi$
$174$	−3586.82	−1.56274
$175$	853.364	0.368619
$176$	0	0
$177$	−5071.20	−2.15353
$178$	−887.730	−0.373810
$179$	2128.07	0.888600	0.444300	−	0.895878i	$-0.353452\pi$
$179$	2128.07	0.888600	0.444300	+	0.895878i	$0.353452\pi$
$180$	−1009.42	−0.417986
$181$	2233.48	0.917202	0.458601	−	0.888642i	$-0.348351\pi$
$181$	2233.48	0.917202	0.458601	+	0.888642i	$0.348351\pi$
$182$	−1660.07	−0.676114
$183$	2953.63	1.19311
$184$	2472.39	0.990582
$185$	2572.35	1.02229
$186$	−735.528	−0.289955
$187$	0	0
$188$	−131.566	−0.0510397
$189$	−148.734	−0.0572425
$190$	−7195.53	−2.74747
$191$	−2175.84	−0.824286	−0.412143	−	0.911119i	$-0.635220\pi$
$191$	−2175.84	−0.824286	−0.412143	+	0.911119i	$0.635220\pi$
$192$	−2238.07	−0.841245
$193$	3118.62	1.16313	0.581563	−	0.813502i	$-0.302441\pi$
$193$	3118.62	1.16313	0.581563	+	0.813502i	$0.302441\pi$
$194$	1289.72	0.477301
$195$	7354.93	2.70101
$196$	−627.185	−0.228566
$197$	4103.79	1.48418	0.742089	−	0.670302i	$-0.233835\pi$
$197$	4103.79	1.48418	0.742089	+	0.670302i	$0.233835\pi$
$198$	0	0
$199$	−5469.06	−1.94820	−0.974099	−	0.226121i	$-0.927395\pi$
$199$	−5469.06	−1.94820	−0.974099	+	0.226121i	$0.927395\pi$
$200$	1950.50	0.689606
$201$	−6408.63	−2.24891
$202$	−550.898	−0.191886
$203$	−1199.96	−0.414879
$204$	287.972	0.0988336
$205$	1043.56	0.355537
$206$	−4585.49	−1.55090
$207$	3958.64	1.32920
$208$	−4955.86	−1.65205
$209$	0	0
$210$	−2943.86	−0.967361
$211$	1296.43	0.422987	0.211493	−	0.977379i	$-0.432167\pi$
$211$	1296.43	0.422987	0.211493	+	0.977379i	$0.432167\pi$
$212$	−952.430	−0.308553
$213$	5130.61	1.65044
$214$	−6418.98	−2.05043
$215$	−706.005	−0.223949
$216$	−339.956	−0.107088
$217$	−246.068	−0.0769778
$218$	−4449.46	−1.38236
$219$	−7696.69	−2.37486
$220$	0	0
$221$	−1094.82	−0.333238
$222$	−4072.25	−1.23113
$223$	2869.87	0.861796	0.430898	−	0.902401i	$-0.358197\pi$
$223$	2869.87	0.861796	0.430898	+	0.902401i	$0.358197\pi$
$224$	798.786	0.238264
$225$	3123.02	0.925340
$226$	−3751.42	−1.10416
$227$	5269.34	1.54070	0.770349	−	0.637622i	$-0.220082\pi$
$227$	5269.34	1.54070	0.770349	+	0.637622i	$0.220082\pi$
$228$	2504.32	0.727425
$229$	−1994.80	−0.575632	−0.287816	−	0.957686i	$-0.592929\pi$
$229$	−1994.80	−0.575632	−0.287816	+	0.957686i	$0.592929\pi$
$230$	6540.35	1.87503
$231$	0	0
$232$	−2742.69	−0.776149
$233$	4054.91	1.14011	0.570055	−	0.821606i	$-0.306922\pi$
$233$	4054.91	1.14011	0.570055	+	0.821606i	$0.306922\pi$
$234$	−6075.31	−1.69724
$235$	887.011	0.246222
$236$	1521.52	0.419672
$237$	6243.87	1.71132
$238$	438.210	0.119348
$239$	−355.429	−0.0961958	−0.0480979	−	0.998843i	$-0.515316\pi$
$239$	−355.429	−0.0961958	−0.0480979	+	0.998843i	$0.515316\pi$
$240$	−8788.40	−2.36370
$241$	−6077.20	−1.62434	−0.812171	−	0.583419i	$-0.801715\pi$
$241$	−6077.20	−1.62434	−0.812171	+	0.583419i	$0.801715\pi$
$242$	0	0
$243$	5432.21	1.43406
$244$	−886.182	−0.232508
$245$	4228.43	1.10263
$246$	−1652.04	−0.428171
$247$	−9521.03	−2.45267
$248$	−562.428	−0.144009
$249$	1041.15	0.264981
$250$	−924.152	−0.233794
$251$	−917.973	−0.230844	−0.115422	−	0.993317i	$-0.536822\pi$
$251$	−917.973	−0.230844	−0.115422	+	0.993317i	$0.536822\pi$
$252$	534.602	0.133638
$253$	0	0
$254$	6638.15	1.63982
$255$	−1941.48	−0.476786
$256$	3213.63	0.784577
$257$	326.089	0.0791473	0.0395736	−	0.999217i	$-0.487400\pi$
$257$	326.089	0.0791473	0.0395736	+	0.999217i	$0.487400\pi$
$258$	1117.67	0.269701
$259$	−1362.35	−0.326844
$260$	−2206.71	−0.526363
$261$	−4391.43	−1.04147
$262$	−1841.58	−0.434248
$263$	3792.65	0.889221	0.444610	−	0.895724i	$-0.353342\pi$
$263$	3792.65	0.889221	0.444610	+	0.895724i	$0.353342\pi$
$264$	0	0
$265$	6421.20	1.48850
$266$	3810.86	0.878416
$267$	−2083.01	−0.477447
$268$	1922.79	0.438259
$269$	2995.42	0.678937	0.339469	−	0.940617i	$-0.389753\pi$
$269$	2995.42	0.678937	0.339469	+	0.940617i	$0.389753\pi$
$270$	−899.304	−0.202703
$271$	−1454.70	−0.326077	−0.163038	−	0.986620i	$-0.552129\pi$
$271$	−1454.70	−0.326077	−0.163038	+	0.986620i	$0.552129\pi$
$272$	1308.20	0.291622
$273$	−3895.28	−0.863564
$274$	4257.56	0.938718
$275$	0	0
$276$	−2276.29	−0.496438
$277$	5196.24	1.12712	0.563560	−	0.826075i	$-0.309431\pi$
$277$	5196.24	1.12712	0.563560	+	0.826075i	$0.309431\pi$
$278$	1604.48	0.346151
$279$	−900.525	−0.193237
$280$	−2251.05	−0.480450
$281$	4076.28	0.865376	0.432688	−	0.901544i	$-0.357565\pi$
$281$	4076.28	0.865376	0.432688	+	0.901544i	$0.357565\pi$
$282$	−1404.21	−0.296524
$283$	−669.972	−0.140727	−0.0703634	−	0.997521i	$-0.522416\pi$
$283$	−669.972	−0.140727	−0.0703634	+	0.997521i	$0.522416\pi$
$284$	−1539.35	−0.321632
$285$	−16884.0	−3.50919
$286$	0	0
$287$	−552.682	−0.113672
$288$	2923.29	0.598112
$289$	289.000	0.0588235
$290$	−7255.39	−1.46914
$291$	3026.26	0.609630
$292$	2309.25	0.462804
$293$	−891.425	−0.177739	−0.0888696	−	0.996043i	$-0.528325\pi$
$293$	−891.425	−0.177739	−0.0888696	+	0.996043i	$0.528325\pi$
$294$	−6693.97	−1.32789
$295$	−10258.0	−2.02455
$296$	−3113.88	−0.611454
$297$	0	0
$298$	2774.91	0.539416
$299$	8654.09	1.67384
$300$	−1795.80	−0.345602
$301$	373.910	0.0716008
$302$	8670.23	1.65204
$303$	−1292.65	−0.245086
$304$	11376.7	2.14637
$305$	5974.57	1.12165
$306$	1603.70	0.299599
$307$	−9723.45	−1.80764	−0.903822	−	0.427909i	$-0.859250\pi$
$307$	−9723.45	−1.80764	−0.903822	+	0.427909i	$0.859250\pi$
$308$	0	0
$309$	−10759.6	−1.98088
$310$	−1487.82	−0.272589
$311$	1755.25	0.320036	0.160018	−	0.987114i	$-0.448845\pi$
$311$	1755.25	0.320036	0.160018	+	0.987114i	$0.448845\pi$
$312$	−8903.29	−1.61554
$313$	−10965.5	−1.98022	−0.990109	−	0.140300i	$-0.955193\pi$
$313$	−10965.5	−1.98022	−0.990109	+	0.140300i	$0.955193\pi$
$314$	−1400.95	−0.251784
$315$	−3604.24	−0.644686
$316$	−1873.36	−0.333496
$317$	−1546.79	−0.274057	−0.137029	−	0.990567i	$-0.543755\pi$
$317$	−1546.79	−0.274057	−0.137029	+	0.990567i	$0.543755\pi$
$318$	−10165.3	−1.79259
$319$	0	0
$320$	−4527.15	−0.790861
$321$	−15061.8	−2.61891
$322$	−3463.86	−0.599483
$323$	2513.27	0.432947
$324$	−1480.16	−0.253799
$325$	6827.33	1.16527
$326$	−3509.36	−0.596213
$327$	−10440.4	−1.76562
$328$	−1263.25	−0.212656
$329$	−469.774	−0.0787218
$330$	0	0
$331$	3002.55	0.498595	0.249298	−	0.968427i	$-0.419800\pi$
$331$	3002.55	0.498595	0.249298	+	0.968427i	$0.419800\pi$
$332$	−312.379	−0.0516386
$333$	−4985.75	−0.820473
$334$	2570.13	0.421052
$335$	−12963.3	−2.11421
$336$	4654.46	0.755720
$337$	−9086.09	−1.46870	−0.734348	−	0.678773i	$-0.762512\pi$
$337$	−9086.09	−1.46870	−0.734348	+	0.678773i	$0.762512\pi$
$338$	−6246.04	−1.00515
$339$	−8802.51	−1.41029
$340$	582.506	0.0929143
$341$	0	0
$342$	13946.5	2.20508
$343$	−5000.47	−0.787173
$344$	854.633	0.133950
$345$	15346.6	2.39488
$346$	−87.5395	−0.0136016
$347$	11357.3	1.75704	0.878519	−	0.477708i	$-0.158532\pi$
$347$	11357.3	1.75704	0.878519	+	0.477708i	$0.158532\pi$
$348$	2525.16	0.388973
$349$	5164.14	0.792064	0.396032	−	0.918237i	$-0.370387\pi$
$349$	5164.14	0.792064	0.396032	+	0.918237i	$0.370387\pi$
$350$	−2732.69	−0.417338
$351$	−1189.95	−0.180953
$352$	0	0
$353$	9220.27	1.39021	0.695107	−	0.718906i	$-0.255357\pi$
$353$	9220.27	1.39021	0.695107	+	0.718906i	$0.255357\pi$
$354$	16239.3	2.43815
$355$	10378.2	1.55159
$356$	624.971	0.0930432
$357$	1028.24	0.152437
$358$	−6814.62	−1.00604
$359$	−6305.61	−0.927012	−0.463506	−	0.886094i	$-0.653409\pi$
$359$	−6305.61	−0.927012	−0.463506	+	0.886094i	$0.653409\pi$
$360$	−8238.08	−1.20607
$361$	14997.4	2.18653
$362$	−7152.18	−1.03843
$363$	0	0
$364$	1168.71	0.168288
$365$	−15568.8	−2.23262
$366$	−9458.26	−1.35080
$367$	537.232	0.0764122	0.0382061	−	0.999270i	$-0.487836\pi$
$367$	537.232	0.0764122	0.0382061	+	0.999270i	$0.487836\pi$
$368$	−10340.8	−1.46481
$369$	−2022.63	−0.285350
$370$	−8237.30	−1.15740
$371$	−3400.76	−0.475900
$372$	517.819	0.0721712
$373$	−11240.0	−1.56029	−0.780144	−	0.625601i	$-0.784854\pi$
$373$	−11240.0	−1.56029	−0.780144	+	0.625601i	$0.784854\pi$
$374$	0	0
$375$	−2168.48	−0.298612
$376$	−1073.74	−0.147272
$377$	−9600.23	−1.31150
$378$	476.284	0.0648080
$379$	14042.0	1.90314	0.951572	−	0.307426i	$-0.0994676\pi$
$379$	14042.0	1.90314	0.951572	+	0.307426i	$0.0994676\pi$
$380$	5065.72	0.683858
$381$	15576.1	2.09445
$382$	6967.60	0.933229
$383$	−507.916	−0.0677631	−0.0338816	−	0.999426i	$-0.510787\pi$
$383$	−507.916	−0.0677631	−0.0338816	+	0.999426i	$0.510787\pi$
$384$	13131.9	1.74514
$385$	0	0
$386$	−9986.61	−1.31685
$387$	1368.39	0.179739
$388$	−907.974	−0.118803
$389$	6993.32	0.911505	0.455752	−	0.890107i	$-0.349370\pi$
$389$	6993.32	0.911505	0.455752	+	0.890107i	$0.349370\pi$
$390$	−23552.3	−3.05800
$391$	−2284.42	−0.295469
$392$	−5118.60	−0.659511
$393$	−4321.17	−0.554641
$394$	−13141.4	−1.68034
$395$	12630.0	1.60883
$396$	0	0
$397$	−7432.72	−0.939641	−0.469821	−	0.882762i	$-0.655681\pi$
$397$	−7432.72	−0.939641	−0.469821	+	0.882762i	$0.655681\pi$
$398$	17513.3	2.20569
$399$	8941.99	1.12195
$400$	−8157.97	−1.01975
$401$	−7302.47	−0.909396	−0.454698	−	0.890646i	$-0.650253\pi$
$401$	−7302.47	−0.909396	−0.454698	+	0.890646i	$0.650253\pi$
$402$	20522.1	2.54614
$403$	−1968.66	−0.243340
$404$	387.837	0.0477615
$405$	9979.10	1.22436
$406$	3842.56	0.469712
$407$	0	0
$408$	2350.20	0.285177
$409$	10931.5	1.32159	0.660793	−	0.750568i	$-0.270220\pi$
$409$	10931.5	1.32159	0.660793	+	0.750568i	$0.270220\pi$
$410$	−3341.73	−0.402527
$411$	9990.16	1.19897
$412$	3228.23	0.386028
$413$	5432.77	0.647286
$414$	−12676.6	−1.50488
$415$	2106.03	0.249111
$416$	6390.68	0.753194
$417$	3764.82	0.442120
$418$	0	0
$419$	12419.2	1.44801	0.724007	−	0.689793i	$-0.242298\pi$
$419$	12419.2	1.44801	0.724007	+	0.689793i	$0.242298\pi$
$420$	2072.51	0.240781
$421$	−11806.9	−1.36682	−0.683410	−	0.730035i	$-0.739504\pi$
$421$	−11806.9	−1.36682	−0.683410	+	0.730035i	$0.739504\pi$
$422$	−4151.51	−0.478892
$423$	−1719.21	−0.197615
$424$	−7773.00	−0.890307
$425$	−1802.21	−0.205694
$426$	−16429.5	−1.86857
$427$	−3164.22	−0.358612
$428$	4519.03	0.510363
$429$	0	0
$430$	2260.80	0.253548
$431$	−3024.31	−0.337995	−0.168998	−	0.985616i	$-0.554053\pi$
$431$	−3024.31	−0.337995	−0.168998	+	0.985616i	$0.554053\pi$
$432$	1421.87	0.158355
$433$	8328.14	0.924307	0.462154	−	0.886800i	$-0.347077\pi$
$433$	8328.14	0.924307	0.462154	+	0.886800i	$0.347077\pi$
$434$	787.971	0.0871517
$435$	−17024.4	−1.87645
$436$	3132.46	0.344077
$437$	−19866.3	−2.17468
$438$	24646.7	2.68874
$439$	−2482.62	−0.269907	−0.134953	−	0.990852i	$-0.543088\pi$
$439$	−2482.62	−0.269907	−0.134953	+	0.990852i	$0.543088\pi$
$440$	0	0
$441$	−8195.59	−0.884958
$442$	3505.89	0.377281
$443$	3962.85	0.425013	0.212506	−	0.977160i	$-0.431837\pi$
$443$	3962.85	0.425013	0.212506	+	0.977160i	$0.431837\pi$
$444$	2866.90	0.306435
$445$	−4213.50	−0.448852
$446$	−9190.03	−0.975696
$447$	6511.18	0.688967
$448$	2397.65	0.252853
$449$	−1760.15	−0.185003	−0.0925016	−	0.995713i	$-0.529486\pi$
$449$	−1760.15	−0.185003	−0.0925016	+	0.995713i	$0.529486\pi$
$450$	−10000.7	−1.04764
$451$	0	0
$452$	2641.03	0.274831
$453$	20344.3	2.11006
$454$	−16873.8	−1.74433
$455$	−7879.33	−0.811843
$456$	20438.4	2.09893
$457$	15574.7	1.59421	0.797106	−	0.603839i	$-0.206363\pi$
$457$	15574.7	1.59421	0.797106	+	0.603839i	$0.206363\pi$
$458$	6387.84	0.651712
$459$	314.110	0.0319421
$460$	−4604.47	−0.466705
$461$	−11769.0	−1.18901	−0.594507	−	0.804091i	$-0.702653\pi$
$461$	−11769.0	−1.18901	−0.594507	+	0.804091i	$0.702653\pi$
$462$	0	0
$463$	2075.70	0.208350	0.104175	−	0.994559i	$-0.466780\pi$
$463$	2075.70	0.208350	0.104175	+	0.994559i	$0.466780\pi$
$464$	11471.3	1.14772
$465$	−3491.10	−0.348163
$466$	−12984.8	−1.29080
$467$	12345.4	1.22329	0.611644	−	0.791133i	$-0.290509\pi$
$467$	12345.4	1.22329	0.611644	+	0.791133i	$0.290509\pi$
$468$	4277.07	0.422453
$469$	6865.56	0.675954
$470$	−2840.43	−0.278765
$471$	−3287.27	−0.321591
$472$	12417.5	1.21093
$473$	0	0
$474$	−19994.5	−1.93750
$475$	−15672.8	−1.51393
$476$	−308.504	−0.0297064
$477$	−12445.6	−1.19465
$478$	1138.17	0.108910
$479$	−1774.24	−0.169242	−0.0846210	−	0.996413i	$-0.526968\pi$
$479$	−1774.24	−0.169242	−0.0846210	+	0.996413i	$0.526968\pi$
$480$	11332.8	1.07764
$481$	−10899.5	−1.03321
$482$	19460.7	1.83903
$483$	−8127.78	−0.765687
$484$	0	0
$485$	6121.49	0.573118
$486$	−17395.3	−1.62360
$487$	−5260.84	−0.489510	−0.244755	−	0.969585i	$-0.578707\pi$
$487$	−5260.84	−0.489510	−0.244755	+	0.969585i	$0.578707\pi$
$488$	−7232.34	−0.670886
$489$	−8234.53	−0.761510
$490$	−13540.5	−1.24836
$491$	12836.1	1.17980	0.589901	−	0.807475i	$-0.299167\pi$
$491$	12836.1	1.17980	0.589901	+	0.807475i	$0.299167\pi$
$492$	1163.05	0.106574
$493$	2534.17	0.231508
$494$	30488.7	2.77683
$495$	0	0
$496$	2352.35	0.212951
$497$	−5496.43	−0.496073
$498$	−3334.03	−0.300003
$499$	−11178.9	−1.00288	−0.501440	−	0.865192i	$-0.667196\pi$
$499$	−11178.9	−1.00288	−0.501440	+	0.865192i	$0.667196\pi$
$500$	650.612	0.0581925
$501$	6030.68	0.537787
$502$	2939.58	0.261354
$503$	−3189.45	−0.282724	−0.141362	−	0.989958i	$-0.545148\pi$
$503$	−3189.45	−0.282724	−0.141362	+	0.989958i	$0.545148\pi$
$504$	4363.01	0.385603
$505$	−2614.77	−0.230407
$506$	0	0
$507$	−14656.0	−1.28382
$508$	−4673.32	−0.408160
$509$	−2937.27	−0.255780	−0.127890	−	0.991788i	$-0.540820\pi$
$509$	−2937.27	−0.255780	−0.127890	+	0.991788i	$0.540820\pi$
$510$	6217.11	0.539801
$511$	8245.46	0.713811
$512$	3690.52	0.318554
$513$	2731.64	0.235097
$514$	−1044.22	−0.0896079
$515$	−21764.4	−1.86225
$516$	−786.848	−0.0671299
$517$	0	0
$518$	4362.60	0.370042
$519$	−205.407	−0.0173726
$520$	−18009.5	−1.51878
$521$	−1090.04	−0.0916615	−0.0458307	−	0.998949i	$-0.514593\pi$
$521$	−1090.04	−0.0916615	−0.0458307	+	0.998949i	$0.514593\pi$
$522$	14062.5	1.17911
$523$	4135.78	0.345784	0.172892	−	0.984941i	$-0.444689\pi$
$523$	4135.78	0.345784	0.172892	+	0.984941i	$0.444689\pi$
$524$	1296.49	0.108086
$525$	−6412.11	−0.533043
$526$	−12145.0	−1.00675
$527$	519.668	0.0429547
$528$	0	0
$529$	5890.40	0.484129
$530$	−20562.3	−1.68523
$531$	19882.1	1.62488
$532$	−2682.88	−0.218642
$533$	−4421.73	−0.359336
$534$	6670.34	0.540550
$535$	−30466.9	−2.46205
$536$	15692.4	1.26456
$537$	−15990.2	−1.28497
$538$	−9592.10	−0.768670
$539$	0	0
$540$	633.119	0.0504539
$541$	4737.32	0.376476	0.188238	−	0.982123i	$-0.439722\pi$
$541$	4737.32	0.376476	0.188238	+	0.982123i	$0.439722\pi$
$542$	4658.32	0.369173
$543$	−16782.2	−1.32632
$544$	−1686.95	−0.132955
$545$	−21118.8	−1.65987
$546$	12473.7	0.977698
$547$	−14315.5	−1.11899	−0.559494	−	0.828834i	$-0.689004\pi$
$547$	−14315.5	−1.11899	−0.559494	+	0.828834i	$0.689004\pi$
$548$	−2997.37	−0.233652
$549$	−11580.0	−0.900221
$550$	0	0
$551$	22038.3	1.70392
$552$	−18577.4	−1.43244
$553$	−6689.06	−0.514372
$554$	−16639.7	−1.27609
$555$	−19328.4	−1.47828
$556$	−1129.57	−0.0861588
$557$	−3777.06	−0.287324	−0.143662	−	0.989627i	$-0.545888\pi$
$557$	−3777.06	−0.287324	−0.143662	+	0.989627i	$0.545888\pi$
$558$	2883.71	0.218776
$559$	2991.47	0.226343
$560$	9415.00	0.710458
$561$	0	0
$562$	−13053.3	−0.979750
$563$	−9768.35	−0.731238	−0.365619	−	0.930765i	$-0.619143\pi$
$563$	−9768.35	−0.731238	−0.365619	+	0.930765i	$0.619143\pi$
$564$	988.581	0.0738063
$565$	−17805.6	−1.32582
$566$	2145.42	0.159326
$567$	−5285.08	−0.391450
$568$	−12563.0	−0.928047
$569$	−43.8685	−0.00323210	−0.00161605	−	0.999999i	$-0.500514\pi$
$569$	−43.8685	−0.00323210	−0.00161605	+	0.999999i	$0.500514\pi$
$570$	54066.7	3.97299
$571$	21264.9	1.55851	0.779255	−	0.626707i	$-0.215598\pi$
$571$	21264.9	1.55851	0.779255	+	0.626707i	$0.215598\pi$
$572$	0	0
$573$	16349.1	1.19196
$574$	1769.83	0.128695
$575$	14245.7	1.03320
$576$	8774.58	0.634735
$577$	−22057.3	−1.59143	−0.795717	−	0.605669i	$-0.792906\pi$
$577$	−22057.3	−1.59143	−0.795717	+	0.605669i	$0.792906\pi$
$578$	−925.451	−0.0665981
$579$	−23433.1	−1.68194
$580$	5107.86	0.365677
$581$	−1115.39	−0.0796454
$582$	−9690.84	−0.690203
$583$	0	0
$584$	18846.3	1.33539
$585$	−28835.7	−2.03796
$586$	2854.57	0.201231
$587$	−20263.7	−1.42483	−0.712413	−	0.701761i	$-0.752397\pi$
$587$	−20263.7	−1.42483	−0.712413	+	0.701761i	$0.752397\pi$
$588$	4712.62	0.330519
$589$	4519.26	0.316151
$590$	32848.6	2.29213
$591$	−30835.6	−2.14620
$592$	13023.8	0.904180
$593$	431.868	0.0299067	0.0149534	−	0.999888i	$-0.495240\pi$
$593$	431.868	0.0299067	0.0149534	+	0.999888i	$0.495240\pi$
$594$	0	0
$595$	2079.91	0.143307
$596$	−1953.56	−0.134263
$597$	41094.1	2.81720
$598$	−27712.6	−1.89507
$599$	−20230.0	−1.37993	−0.689964	−	0.723844i	$-0.742374\pi$
$599$	−20230.0	−1.37993	−0.689964	+	0.723844i	$0.742374\pi$
$600$	−14655.9	−0.997209
$601$	−84.8689	−0.00576019	−0.00288010	−	0.999996i	$-0.500917\pi$
$601$	−84.8689	−0.00576019	−0.00288010	+	0.999996i	$0.500917\pi$
$602$	−1197.36	−0.0810641
$603$	25125.6	1.69684
$604$	−6103.92	−0.411201
$605$	0	0
$606$	4139.40	0.277478
$607$	16945.2	1.13309	0.566545	−	0.824031i	$-0.308280\pi$
$607$	16945.2	1.13309	0.566545	+	0.824031i	$0.308280\pi$
$608$	−14670.4	−0.978559
$609$	9016.37	0.599938
$610$	−19132.1	−1.26989
$611$	−3758.42	−0.248853
$612$	−1129.02	−0.0745718
$613$	16231.7	1.06948	0.534740	−	0.845017i	$-0.320410\pi$
$613$	16231.7	1.06948	0.534740	+	0.845017i	$0.320410\pi$
$614$	31136.9	2.04655
$615$	−7841.20	−0.514126
$616$	0	0
$617$	25976.6	1.69494	0.847469	−	0.530845i	$-0.178125\pi$
$617$	25976.6	1.69494	0.847469	+	0.530845i	$0.178125\pi$
$618$	34455.0	2.24269
$619$	−605.381	−0.0393091	−0.0196545	−	0.999807i	$-0.506257\pi$
$619$	−605.381	−0.0393091	−0.0196545	+	0.999807i	$0.506257\pi$
$620$	1047.44	0.0678487
$621$	−2482.91	−0.160444
$622$	−5620.76	−0.362334
$623$	2231.53	0.143506
$624$	37238.0	2.38896
$625$	−17637.9	−1.12883
$626$	35114.4	2.24194
$627$	0	0
$628$	986.284	0.0626704
$629$	2877.14	0.182383
$630$	11541.7	0.729892
$631$	−4607.64	−0.290693	−0.145346	−	0.989381i	$-0.546430\pi$
$631$	−4607.64	−0.290693	−0.145346	+	0.989381i	$0.546430\pi$
$632$	−15288.9	−0.962280
$633$	−9741.31	−0.611662
$634$	4953.20	0.310279
$635$	31507.1	1.96901
$636$	7156.48	0.446184
$637$	−17916.6	−1.11441
$638$	0	0
$639$	−20115.1	−1.24529
$640$	26563.0	1.64062
$641$	11220.7	0.691407	0.345704	−	0.938344i	$-0.387640\pi$
$641$	11220.7	0.691407	0.345704	+	0.938344i	$0.387640\pi$
$642$	48231.7	2.96504
$643$	6792.82	0.416614	0.208307	−	0.978064i	$-0.433205\pi$
$643$	6792.82	0.416614	0.208307	+	0.978064i	$0.433205\pi$
$644$	2438.59	0.149214
$645$	5304.87	0.323843
$646$	−8048.11	−0.490168
$647$	−1187.77	−0.0721729	−0.0360865	−	0.999349i	$-0.511489\pi$
$647$	−1187.77	−0.0721729	−0.0360865	+	0.999349i	$0.511489\pi$
$648$	−12079.9	−0.732320
$649$	0	0
$650$	−21862.8	−1.31928
$651$	1848.94	0.111314
$652$	2470.62	0.148400
$653$	−11524.5	−0.690639	−0.345320	−	0.938485i	$-0.612229\pi$
$653$	−11524.5	−0.690639	−0.345320	+	0.938485i	$0.612229\pi$
$654$	33432.9	1.99898
$655$	−8740.81	−0.521423
$656$	5283.52	0.314461
$657$	30175.6	1.79188
$658$	1504.33	0.0891262
$659$	−2262.41	−0.133735	−0.0668673	−	0.997762i	$-0.521300\pi$
$659$	−2262.41	−0.133735	−0.0668673	+	0.997762i	$0.521300\pi$
$660$	0	0
$661$	7569.36	0.445407	0.222704	−	0.974886i	$-0.428512\pi$
$661$	7569.36	0.445407	0.222704	+	0.974886i	$0.428512\pi$
$662$	−9614.92	−0.564493
$663$	8226.40	0.481881
$664$	−2549.39	−0.149000
$665$	18087.8	1.05476
$666$	15965.6	0.928913
$667$	−20031.6	−1.16286
$668$	−1809.40	−0.104802
$669$	−21563.9	−1.24620
$670$	41511.8	2.39364
$671$	0	0
$672$	−6002.02	−0.344543
$673$	12090.3	0.692490	0.346245	−	0.938144i	$-0.387457\pi$
$673$	12090.3	0.692490	0.346245	+	0.938144i	$0.387457\pi$
$674$	29095.9	1.66281
$675$	−1958.80	−0.111695
$676$	4397.27	0.250186
$677$	−18348.0	−1.04161	−0.520806	−	0.853675i	$-0.674368\pi$
$677$	−18348.0	−1.04161	−0.520806	+	0.853675i	$0.674368\pi$
$678$	28187.9	1.59668
$679$	−3242.03	−0.183237
$680$	4753.97	0.268098
$681$	−39593.4	−2.22794
$682$	0	0
$683$	20163.7	1.12964	0.564819	−	0.825215i	$-0.308946\pi$
$683$	20163.7	1.12964	0.564819	+	0.825215i	$0.308946\pi$
$684$	−9818.44	−0.548856
$685$	20208.0	1.12716
$686$	16012.8	0.891211
$687$	14988.7	0.832396
$688$	−3574.50	−0.198076
$689$	−27207.8	−1.50440
$690$	−49143.7	−2.71140
$691$	9204.28	0.506725	0.253363	−	0.967371i	$-0.418463\pi$
$691$	9204.28	0.506725	0.253363	+	0.967371i	$0.418463\pi$
$692$	61.6287	0.00338551
$693$	0	0
$694$	−36368.9	−1.98926
$695$	7615.45	0.415641
$696$	20608.4	1.12235
$697$	1167.20	0.0634304
$698$	−16536.9	−0.896748
$699$	−30468.3	−1.64866
$700$	1923.84	0.103877
$701$	−5532.69	−0.298098	−0.149049	−	0.988830i	$-0.547621\pi$
$701$	−5532.69	−0.298098	−0.149049	+	0.988830i	$0.547621\pi$
$702$	3810.51	0.204869
$703$	25020.8	1.34236
$704$	0	0
$705$	−6664.93	−0.356051
$706$	−29525.6	−1.57395
$707$	1384.82	0.0736655
$708$	−11432.6	−0.606869
$709$	−36674.6	−1.94265	−0.971327	−	0.237746i	$-0.923591\pi$
$709$	−36674.6	−1.94265	−0.971327	+	0.237746i	$0.923591\pi$
$710$	−33233.5	−1.75666
$711$	−24479.7	−1.29122
$712$	5100.53	0.268470
$713$	−4107.76	−0.215760
$714$	−3292.67	−0.172584
$715$	0	0
$716$	4797.56	0.250409
$717$	2670.67	0.139104
$718$	20192.2	1.04953
$719$	35385.4	1.83540	0.917699	−	0.397275i	$-0.130044\pi$
$719$	35385.4	1.83540	0.917699	+	0.397275i	$0.130044\pi$
$720$	34455.7	1.78346
$721$	11526.8	0.595394
$722$	−48025.6	−2.47552
$723$	45663.6	2.34889
$724$	5035.20	0.258469
$725$	−15803.2	−0.809538
$726$	0	0
$727$	−29981.9	−1.52953	−0.764763	−	0.644311i	$-0.777144\pi$
$727$	−29981.9	−1.52953	−0.764763	+	0.644311i	$0.777144\pi$
$728$	9538.09	0.485584
$729$	−23090.2	−1.17310
$730$	49855.2	2.52770
$731$	−789.658	−0.0399542
$732$	6658.71	0.336220
$733$	−8092.56	−0.407784	−0.203892	−	0.978993i	$-0.565359\pi$
$733$	−8092.56	−0.407784	−0.203892	+	0.978993i	$0.565359\pi$
$734$	−1720.35	−0.0865113
$735$	−31772.1	−1.59447
$736$	13334.6	0.667826
$737$	0	0
$738$	6476.97	0.323063
$739$	−18934.7	−0.942520	−0.471260	−	0.881994i	$-0.656201\pi$
$739$	−18934.7	−0.942520	−0.471260	+	0.881994i	$0.656201\pi$
$740$	5799.14	0.288082
$741$	71540.3	3.54669
$742$	10890.1	0.538798
$743$	−5254.38	−0.259441	−0.129720	−	0.991551i	$-0.541408\pi$
$743$	−5254.38	−0.259441	−0.129720	+	0.991551i	$0.541408\pi$
$744$	4226.04	0.208245
$745$	13170.7	0.647703
$746$	35993.4	1.76651
$747$	−4081.94	−0.199933
$748$	0	0
$749$	16135.7	0.787165
$750$	6944.00	0.338079
$751$	−23450.8	−1.13946	−0.569728	−	0.821834i	$-0.692951\pi$
$751$	−23450.8	−1.13946	−0.569728	+	0.821834i	$0.692951\pi$
$752$	4490.93	0.217776
$753$	6897.58	0.333814
$754$	30742.3	1.48484
$755$	41152.2	1.98368
$756$	−335.309	−0.0161310
$757$	−3004.72	−0.144265	−0.0721323	−	0.997395i	$-0.522980\pi$
$757$	−3004.72	−0.144265	−0.0721323	+	0.997395i	$0.522980\pi$
$758$	−44966.2	−2.15468
$759$	0	0
$760$	41342.5	1.97323
$761$	−3501.96	−0.166815	−0.0834074	−	0.996516i	$-0.526580\pi$
$761$	−3501.96	−0.166815	−0.0834074	+	0.996516i	$0.526580\pi$
$762$	−49878.5	−2.37127
$763$	11184.8	0.530692
$764$	−4905.26	−0.232285
$765$	7611.76	0.359744
$766$	1626.47	0.0767192
$767$	43464.8	2.04618
$768$	−24146.9	−1.13454
$769$	12838.6	0.602042	0.301021	−	0.953617i	$-0.402673\pi$
$769$	12838.6	0.602042	0.301021	+	0.953617i	$0.402673\pi$
$770$	0	0
$771$	−2450.21	−0.114451
$772$	7030.67	0.327771
$773$	18940.3	0.881289	0.440645	−	0.897682i	$-0.354750\pi$
$773$	18940.3	0.881289	0.440645	+	0.897682i	$0.354750\pi$
$774$	−4381.92	−0.203494
$775$	−3240.66	−0.150204
$776$	−7410.18	−0.342797
$777$	10236.6	0.472634
$778$	−22394.4	−1.03198
$779$	10150.5	0.466855
$780$	16581.1	0.761151
$781$	0	0
$782$	7315.30	0.334520
$783$	2754.36	0.125712
$784$	21408.5	0.975243
$785$	−6649.45	−0.302330
$786$	13837.5	0.627947
$787$	23712.2	1.07401	0.537006	−	0.843579i	$-0.319555\pi$
$787$	23712.2	1.07401	0.537006	+	0.843579i	$0.319555\pi$
$788$	9251.65	0.418244
$789$	−28497.7	−1.28586
$790$	−40444.6	−1.82146
$791$	9430.12	0.423890
$792$	0	0
$793$	−25315.3	−1.13363
$794$	23801.4	1.06383
$795$	−48248.4	−2.15245
$796$	−12329.5	−0.549006
$797$	6432.08	0.285867	0.142933	−	0.989732i	$-0.454347\pi$
$797$	6432.08	0.285867	0.142933	+	0.989732i	$0.454347\pi$
$798$	−28634.5	−1.27024
$799$	992.111	0.0439278
$800$	10519.8	0.464916
$801$	8166.66	0.360243
$802$	23384.3	1.02959
$803$	0	0
$804$	−14447.7	−0.633746
$805$	−16440.8	−0.719829
$806$	6304.15	0.275502
$807$	−22507.4	−0.981781
$808$	3165.23	0.137812
$809$	23868.4	1.03729	0.518645	−	0.854990i	$-0.326437\pi$
$809$	23868.4	1.03729	0.518645	+	0.854990i	$0.326437\pi$
$810$	−31955.6	−1.38618
$811$	−7398.37	−0.320335	−0.160168	−	0.987090i	$-0.551203\pi$
$811$	−7398.37	−0.320335	−0.160168	+	0.987090i	$0.551203\pi$
$812$	−2705.20	−0.116914
$813$	10930.5	0.471525
$814$	0	0
$815$	−16656.7	−0.715902
$816$	−9829.71	−0.421702
$817$	−6867.20	−0.294067
$818$	−35005.5	−1.49626
$819$	15271.8	0.651575
$820$	2352.61	0.100191
$821$	11135.5	0.473362	0.236681	−	0.971587i	$-0.423940\pi$
$821$	11135.5	0.473362	0.236681	+	0.971587i	$0.423940\pi$
$822$	−31991.0	−1.35744
$823$	−30744.2	−1.30216	−0.651079	−	0.759010i	$-0.725683\pi$
$823$	−30744.2	−1.30216	−0.651079	+	0.759010i	$0.725683\pi$
$824$	26346.3	1.11386
$825$	0	0
$826$	−17397.1	−0.732836
$827$	−36921.4	−1.55246	−0.776230	−	0.630450i	$-0.782870\pi$
$827$	−36921.4	−1.55246	−0.776230	+	0.630450i	$0.782870\pi$
$828$	8924.43	0.374572
$829$	34719.1	1.45458	0.727288	−	0.686333i	$-0.240781\pi$
$829$	34719.1	1.45458	0.727288	+	0.686333i	$0.240781\pi$
$830$	−6744.05	−0.282035
$831$	−39044.2	−1.62988
$832$	19182.3	0.799313
$833$	4729.45	0.196718
$834$	−12055.9	−0.500554
$835$	12198.8	0.505578
$836$	0	0
$837$	564.821	0.0233250
$838$	−39769.4	−1.63939
$839$	27476.2	1.13061	0.565306	−	0.824881i	$-0.308758\pi$
$839$	27476.2	1.13061	0.565306	+	0.824881i	$0.308758\pi$
$840$	16914.2	0.694757
$841$	−2167.41	−0.0888685
$842$	37808.5	1.54747
$843$	−30628.9	−1.25138
$844$	2922.70	0.119199
$845$	−29646.1	−1.20693
$846$	5505.35	0.223733
$847$	0	0
$848$	32510.5	1.31653
$849$	5034.12	0.203499
$850$	5771.13	0.232880
$851$	−22742.6	−0.916105
$852$	11566.5	0.465098
$853$	30769.3	1.23508	0.617539	−	0.786540i	$-0.288130\pi$
$853$	30769.3	1.23508	0.617539	+	0.786540i	$0.288130\pi$
$854$	10132.6	0.406009
$855$	66195.1	2.64775
$856$	36880.8	1.47262
$857$	29390.4	1.17148	0.585739	−	0.810500i	$-0.300804\pi$
$857$	29390.4	1.17148	0.585739	+	0.810500i	$0.300804\pi$
$858$	0	0
$859$	−17187.3	−0.682680	−0.341340	−	0.939940i	$-0.610881\pi$
$859$	−17187.3	−0.682680	−0.341340	+	0.939940i	$0.610881\pi$
$860$	−1591.63	−0.0631094
$861$	4152.81	0.164376
$862$	9684.61	0.382667
$863$	−29747.6	−1.17337	−0.586685	−	0.809815i	$-0.699567\pi$
$863$	−29747.6	−1.17337	−0.586685	+	0.809815i	$0.699567\pi$
$864$	−1833.52	−0.0721964
$865$	−415.495	−0.0163321
$866$	−26668.8	−1.04647
$867$	−2171.52	−0.0850621
$868$	−554.739	−0.0216925
$869$	0	0
$870$	54516.4	2.12446
$871$	54927.8	2.13681
$872$	25564.7	0.992811
$873$	−11864.7	−0.459978
$874$	63616.9	2.46210
$875$	2323.09	0.0897539
$876$	−17351.5	−0.669240
$877$	−13793.9	−0.531113	−0.265557	−	0.964095i	$-0.585556\pi$
$877$	−13793.9	−0.531113	−0.265557	+	0.964095i	$0.585556\pi$
$878$	7949.98	0.305580
$879$	6698.10	0.257021
$880$	0	0
$881$	−203.857	−0.00779584	−0.00389792	−	0.999992i	$-0.501241\pi$
$881$	−203.857	−0.00779584	−0.00389792	+	0.999992i	$0.501241\pi$
$882$	26244.4	1.00192
$883$	−37137.3	−1.41537	−0.707684	−	0.706529i	$-0.750260\pi$
$883$	−37137.3	−1.41537	−0.707684	+	0.706529i	$0.750260\pi$
$884$	−2468.18	−0.0939071
$885$	77077.6	2.92761
$886$	−12690.0	−0.481186
$887$	−2336.32	−0.0884396	−0.0442198	−	0.999022i	$-0.514080\pi$
$887$	−2336.32	−0.0884396	−0.0442198	+	0.999022i	$0.514080\pi$
$888$	23397.4	0.884197
$889$	−16686.6	−0.629530
$890$	13492.7	0.508175
$891$	0	0
$892$	6469.87	0.242856
$893$	8627.82	0.323313
$894$	−20850.4	−0.780025
$895$	−32344.7	−1.20801
$896$	−14068.2	−0.524536
$897$	−65026.2	−2.42047
$898$	5636.43	0.209455
$899$	4556.85	0.169054
$900$	7040.60	0.260763
$901$	7182.04	0.265559
$902$	0	0
$903$	−2809.53	−0.103539
$904$	21554.1	0.793006
$905$	−33946.9	−1.24689
$906$	−65147.4	−2.38894
$907$	37160.4	1.36041	0.680204	−	0.733023i	$-0.261891\pi$
$907$	37160.4	1.36041	0.680204	+	0.733023i	$0.261891\pi$
$908$	11879.3	0.434172
$909$	5067.97	0.184922
$910$	25231.6	0.919142
$911$	−13365.0	−0.486061	−0.243030	−	0.970019i	$-0.578141\pi$
$911$	−13365.0	−0.486061	−0.243030	+	0.970019i	$0.578141\pi$
$912$	−85483.4	−3.10377
$913$	0	0
$914$	−49874.2	−1.80491
$915$	−44892.5	−1.62197
$916$	−4497.10	−0.162214
$917$	4629.26	0.166709
$918$	−1005.86	−0.0361638
$919$	28237.4	1.01357	0.506783	−	0.862074i	$-0.330835\pi$
$919$	28237.4	1.01357	0.506783	+	0.862074i	$0.330835\pi$
$920$	−37578.1	−1.34664
$921$	73061.3	2.61395
$922$	37687.2	1.34616
$923$	−43974.1	−1.56817
$924$	0	0
$925$	−17941.9	−0.637758
$926$	−6646.92	−0.235887
$927$	42184.1	1.49461
$928$	−14792.5	−0.523261
$929$	−1805.88	−0.0637773	−0.0318887	−	0.999491i	$-0.510152\pi$
$929$	−1805.88	−0.0637773	−0.0318887	+	0.999491i	$0.510152\pi$
$930$	11179.4	0.394178
$931$	41129.3	1.44786
$932$	9141.45	0.321285
$933$	−13188.8	−0.462790
$934$	−39533.0	−1.38497
$935$	0	0
$936$	34906.2	1.21896
$937$	17968.8	0.626485	0.313243	−	0.949673i	$-0.398585\pi$
$937$	17968.8	0.626485	0.313243	+	0.949673i	$0.398585\pi$
$938$	−21985.3	−0.765292
$939$	82394.1	2.86350
$940$	1999.69	0.0693859
$941$	35610.4	1.23365	0.616826	−	0.787100i	$-0.288418\pi$
$941$	35610.4	1.23365	0.616826	+	0.787100i	$0.288418\pi$
$942$	10526.7	0.364094
$943$	−9226.26	−0.318609
$944$	−51936.1	−1.79065
$945$	2260.63	0.0778181
$946$	0	0
$947$	−34102.4	−1.17020	−0.585099	−	0.810962i	$-0.698944\pi$
$947$	−34102.4	−1.17020	−0.585099	+	0.810962i	$0.698944\pi$
$948$	14076.3	0.482254
$949$	65967.7	2.25648
$950$	50188.2	1.71402
$951$	11622.4	0.396302
$952$	−2517.77	−0.0857158
$953$	−24539.1	−0.834101	−0.417050	−	0.908883i	$-0.636936\pi$
$953$	−24539.1	−0.834101	−0.417050	+	0.908883i	$0.636936\pi$
$954$	39854.1	1.35254
$955$	33070.9	1.12057
$956$	−801.285	−0.0271082
$957$	0	0
$958$	5681.55	0.191610
$959$	−10702.4	−0.360376
$960$	34016.7	1.14363
$961$	−28856.6	−0.968633
$962$	34902.9	1.16977
$963$	59051.3	1.97601
$964$	−13700.5	−0.457743
$965$	−47400.2	−1.58121
$966$	26027.2	0.866886
$967$	10527.3	0.350090	0.175045	−	0.984560i	$-0.443993\pi$
$967$	10527.3	0.350090	0.175045	+	0.984560i	$0.443993\pi$
$968$	0	0
$969$	−18884.5	−0.626065
$970$	−19602.5	−0.648866
$971$	−28288.6	−0.934937	−0.467469	−	0.884010i	$-0.654834\pi$
$971$	−28288.6	−0.934937	−0.467469	+	0.884010i	$0.654834\pi$
$972$	12246.5	0.404121
$973$	−4033.25	−0.132888
$974$	16846.5	0.554207
$975$	−51300.0	−1.68504
$976$	30249.2	0.992063
$977$	−9510.74	−0.311439	−0.155719	−	0.987801i	$-0.549770\pi$
$977$	−9510.74	−0.311439	−0.155719	+	0.987801i	$0.549770\pi$
$978$	26369.1	0.862157
$979$	0	0
$980$	9532.65	0.310724
$981$	40932.7	1.33219
$982$	−41104.3	−1.33573
$983$	−48169.6	−1.56294	−0.781470	−	0.623943i	$-0.785530\pi$
$983$	−48169.6	−1.56294	−0.781470	+	0.623943i	$0.785530\pi$
$984$	9491.93	0.307512
$985$	−62373.9	−2.01766
$986$	−8115.06	−0.262106
$987$	3529.84	0.113836
$988$	−21464.4	−0.691166
$989$	6241.91	0.200689
$990$	0	0
$991$	−45023.8	−1.44322	−0.721609	−	0.692301i	$-0.756597\pi$
$991$	−45023.8	−1.44322	−0.721609	+	0.692301i	$0.756597\pi$
$992$	−3033.40	−0.0970873
$993$	−22560.9	−0.720996
$994$	17600.9	0.561638
$995$	83124.8	2.64847
$996$	2347.19	0.0746723
$997$	41968.5	1.33315	0.666577	−	0.745436i	$-0.267759\pi$
$997$	41968.5	1.33315	0.666577	+	0.745436i	$0.267759\pi$
$998$	35797.7	1.13543
$999$	3127.13	0.0990369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.s.1.11		44
11.7	odd	10		187.4.g.a.137.17	yes	88
11.8	odd	10		187.4.g.a.86.17	✓	88
11.10	odd	2		2057.4.a.t.1.34		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.a.86.17	✓	88	11.8	odd	10
187.4.g.a.137.17	yes	88	11.7	odd	10
2057.4.a.s.1.11		44	1.1	even	1	trivial
2057.4.a.t.1.34		44	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.20225 q^{2} -7.51392 q^{3} +2.25442 q^{4} -15.1991 q^{5} +24.0615 q^{6} +8.04966 q^{7} +18.3988 q^{8} +29.4590 q^{9} +48.6713 q^{10} -16.9395 q^{12} +64.4012 q^{13} -25.7770 q^{14} +114.205 q^{15} -76.9529 q^{16} -17.0000 q^{17} -94.3353 q^{18} -147.839 q^{19} -34.2651 q^{20} -60.4845 q^{21} +134.378 q^{23} -138.247 q^{24} +106.012 q^{25} -206.229 q^{26} -18.4771 q^{27} +18.1473 q^{28} -149.069 q^{29} -365.713 q^{30} -30.5687 q^{31} +99.2322 q^{32} +54.4383 q^{34} -122.348 q^{35} +66.4130 q^{36} -169.244 q^{37} +473.418 q^{38} -483.906 q^{39} -279.645 q^{40} -68.6591 q^{41} +193.687 q^{42} +46.4505 q^{43} -447.751 q^{45} -430.312 q^{46} -58.3594 q^{47} +578.218 q^{48} -278.203 q^{49} -339.478 q^{50} +127.737 q^{51} +145.187 q^{52} -422.473 q^{53} +59.1683 q^{54} +148.104 q^{56} +1110.85 q^{57} +477.357 q^{58} +674.907 q^{59} +257.465 q^{60} -393.087 q^{61} +97.8887 q^{62} +237.135 q^{63} +297.857 q^{64} -978.840 q^{65} +852.901 q^{67} -38.3251 q^{68} -1009.70 q^{69} +391.788 q^{70} -682.814 q^{71} +542.011 q^{72} +1024.32 q^{73} +541.960 q^{74} -796.569 q^{75} -333.291 q^{76} +1549.59 q^{78} -830.974 q^{79} +1169.61 q^{80} -656.559 q^{81} +219.864 q^{82} -138.563 q^{83} -136.357 q^{84} +258.385 q^{85} -148.746 q^{86} +1120.09 q^{87} +277.221 q^{89} +1433.81 q^{90} +518.408 q^{91} +302.944 q^{92} +229.691 q^{93} +186.882 q^{94} +2247.02 q^{95} -745.623 q^{96} -402.754 q^{97} +890.876 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) \) 44 * q - q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 + 24 * q^6 + 2 * q^7 + 63 * q^8 + 356 * q^9 + 65 * q^10 - 337 * q^12 - 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 - 748 * q^17 + 55 * q^18 + 36 * q^19 - 209 * q^20 + 240 * q^21 - 656 * q^23 + 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 - 63 * q^28 + 698 * q^29 - 304 * q^30 - 1418 * q^31 + 83 * q^32 + 17 * q^34 - 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 - 114 * q^39 + 281 * q^40 - 54 * q^41 - 1943 * q^42 - 382 * q^43 - 1226 * q^45 - 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 - 1307 * q^50 + 476 * q^51 - 3112 * q^52 - 2492 * q^53 - 2400 * q^54 + 27 * q^56 - 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 + 964 * q^61 + 3256 * q^62 - 2322 * q^63 - 3653 * q^64 + 152 * q^65 - 3442 * q^67 - 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 + 1281 * q^72 + 2240 * q^73 - 3494 * q^74 - 1892 * q^75 - 1261 * q^76 + 1120 * q^78 + 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 + 1656 * q^83 + 5401 * q^84 + 408 * q^85 - 1373 * q^86 - 1968 * q^87 - 2992 * q^89 - 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 + 694 * q^94 + 8280 * q^95 - 2203 * q^96 - 4746 * q^97 - 2302 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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