LMFDB - Embedded newform 1899.4.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1899,4,Mod(1,1899)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1899, 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("1899.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1899 = 3^{2} \cdot 211 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1899.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:	$112.044627101$
Analytic rank:	$1$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 633)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	1899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.470819 q^{2} -7.77833 q^{4} +2.59445 q^{5} -14.7664 q^{7} +7.42874 q^{8} +O(q^{10})$	\(q-0.470819 q^{2} -7.77833 q^{4} +2.59445 q^{5} -14.7664 q^{7} +7.42874 q^{8} -1.22152 q^{10} +23.6635 q^{11} -45.0179 q^{13} +6.95230 q^{14} +58.7290 q^{16} +19.9654 q^{17} -112.439 q^{19} -20.1805 q^{20} -11.1412 q^{22} +171.334 q^{23} -118.269 q^{25} +21.1953 q^{26} +114.858 q^{28} +116.976 q^{29} +100.551 q^{31} -87.0807 q^{32} -9.40009 q^{34} -38.3107 q^{35} +284.675 q^{37} +52.9383 q^{38} +19.2735 q^{40} -0.372163 q^{41} +9.96988 q^{43} -184.062 q^{44} -80.6673 q^{46} -419.682 q^{47} -124.954 q^{49} +55.6832 q^{50} +350.164 q^{52} +512.098 q^{53} +61.3938 q^{55} -109.696 q^{56} -55.0748 q^{58} -378.356 q^{59} +297.222 q^{61} -47.3411 q^{62} -428.833 q^{64} -116.797 q^{65} +519.734 q^{67} -155.297 q^{68} +18.0374 q^{70} -33.2035 q^{71} +20.0491 q^{73} -134.031 q^{74} +874.585 q^{76} -349.424 q^{77} +544.451 q^{79} +152.370 q^{80} +0.175222 q^{82} +763.226 q^{83} +51.7992 q^{85} -4.69401 q^{86} +175.790 q^{88} -695.667 q^{89} +664.752 q^{91} -1332.69 q^{92} +197.594 q^{94} -291.717 q^{95} -1621.19 q^{97} +58.8306 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 6 q^{2} + 108 q^{4} - 25 q^{5} + 50 q^{7} - 57 q^{8}+O(q^{10}) $ 26 * q - 6 * q^2 + 108 * q^4 - 25 * q^5 + 50 * q^7 - 57 * q^8	$ 26 q - 6 q^{2} + 108 q^{4} - 25 q^{5} + 50 q^{7} - 57 q^{8} - 32 q^{10} - 73 q^{11} - 61 q^{13} - 243 q^{14} + 440 q^{16} - 96 q^{17} + 71 q^{19} - 277 q^{20} + 206 q^{22} - 678 q^{23} + 597 q^{25} - 335 q^{26} + 354 q^{28} - 224 q^{29} - 188 q^{31} - 502 q^{32} - 261 q^{34} - 624 q^{35} + 223 q^{37} - 737 q^{38} - 562 q^{40} - 106 q^{41} + 607 q^{43} - 767 q^{44} - 529 q^{46} - 1657 q^{47} + 848 q^{49} - 906 q^{50} + 156 q^{52} - 1902 q^{53} + 1215 q^{55} - 3540 q^{56} + 2871 q^{58} - 2790 q^{59} - 1952 q^{61} - 3134 q^{62} + 1659 q^{64} - 3653 q^{65} + 2932 q^{67} - 7363 q^{68} + 4509 q^{70} - 4157 q^{71} + 94 q^{73} - 6271 q^{74} + 2757 q^{76} - 3564 q^{77} + 2365 q^{79} - 6249 q^{80} + 2481 q^{82} - 4852 q^{83} + 640 q^{85} - 5710 q^{86} + 5291 q^{88} - 2504 q^{89} - 812 q^{91} - 9719 q^{92} - 252 q^{94} - 4999 q^{95} - 302 q^{97} - 14279 q^{98}+O(q^{100}) $ 26 * q - 6 * q^2 + 108 * q^4 - 25 * q^5 + 50 * q^7 - 57 * q^8 - 32 * q^10 - 73 * q^11 - 61 * q^13 - 243 * q^14 + 440 * q^16 - 96 * q^17 + 71 * q^19 - 277 * q^20 + 206 * q^22 - 678 * q^23 + 597 * q^25 - 335 * q^26 + 354 * q^28 - 224 * q^29 - 188 * q^31 - 502 * q^32 - 261 * q^34 - 624 * q^35 + 223 * q^37 - 737 * q^38 - 562 * q^40 - 106 * q^41 + 607 * q^43 - 767 * q^44 - 529 * q^46 - 1657 * q^47 + 848 * q^49 - 906 * q^50 + 156 * q^52 - 1902 * q^53 + 1215 * q^55 - 3540 * q^56 + 2871 * q^58 - 2790 * q^59 - 1952 * q^61 - 3134 * q^62 + 1659 * q^64 - 3653 * q^65 + 2932 * q^67 - 7363 * q^68 + 4509 * q^70 - 4157 * q^71 + 94 * q^73 - 6271 * q^74 + 2757 * q^76 - 3564 * q^77 + 2365 * q^79 - 6249 * q^80 + 2481 * q^82 - 4852 * q^83 + 640 * q^85 - 5710 * q^86 + 5291 * q^88 - 2504 * q^89 - 812 * q^91 - 9719 * q^92 - 252 * q^94 - 4999 * q^95 - 302 * q^97 - 14279 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−0.470819	−0.166460	−0.0832299	−	0.996530i	$-0.526524\pi$
$2$	−0.470819	−0.166460	−0.0832299	+	0.996530i	$0.526524\pi$
$3$	0	0
$3$	0	0
$4$	−7.77833	−0.972291
$5$	2.59445	0.232055	0.116027	−	0.993246i	$-0.462984\pi$
$5$	2.59445	0.232055	0.116027	+	0.993246i	$0.462984\pi$
$6$	0	0
$7$	−14.7664	−0.797310	−0.398655	−	0.917101i	$-0.630523\pi$
$7$	−14.7664	−0.797310	−0.398655	+	0.917101i	$0.630523\pi$
$8$	7.42874	0.328307
$9$	0	0
$10$	−1.22152	−0.0386278
$11$	23.6635	0.648619	0.324310	−	0.945951i	$-0.394868\pi$
$11$	23.6635	0.648619	0.324310	+	0.945951i	$0.394868\pi$
$12$	0	0
$13$	−45.0179	−0.960440	−0.480220	−	0.877148i	$-0.659443\pi$
$13$	−45.0179	−0.960440	−0.480220	+	0.877148i	$0.659443\pi$
$14$	6.95230	0.132720
$15$	0	0
$16$	58.7290	0.917641
$17$	19.9654	0.284842	0.142421	−	0.989806i	$-0.454511\pi$
$17$	19.9654	0.284842	0.142421	+	0.989806i	$0.454511\pi$
$18$	0	0
$19$	−112.439	−1.35764	−0.678821	−	0.734304i	$-0.737509\pi$
$19$	−112.439	−1.35764	−0.678821	+	0.734304i	$0.737509\pi$
$20$	−20.1805	−0.225625
$21$	0	0
$22$	−11.1412	−0.107969
$23$	171.334	1.55329	0.776643	−	0.629941i	$-0.216921\pi$
$23$	171.334	1.55329	0.776643	+	0.629941i	$0.216921\pi$
$24$	0	0
$25$	−118.269	−0.946151
$26$	21.1953	0.159875
$27$	0	0
$28$	114.858	0.775218
$29$	116.976	0.749034	0.374517	−	0.927220i	$-0.377809\pi$
$29$	116.976	0.749034	0.374517	+	0.927220i	$0.377809\pi$
$30$	0	0
$31$	100.551	0.582562	0.291281	−	0.956638i	$-0.405919\pi$
$31$	100.551	0.582562	0.291281	+	0.956638i	$0.405919\pi$
$32$	−87.0807	−0.481057
$33$	0	0
$34$	−9.40009	−0.0474148
$35$	−38.3107	−0.185020
$36$	0	0
$37$	284.675	1.26487	0.632437	−	0.774612i	$-0.282055\pi$
$37$	284.675	1.26487	0.632437	+	0.774612i	$0.282055\pi$
$38$	52.9383	0.225993
$39$	0	0
$40$	19.2735	0.0761852
$41$	−0.372163	−0.00141761	−0.000708806	−	1.00000i	$-0.500226\pi$
$41$	−0.372163	−0.00141761	−0.000708806		1.00000i	$0.500226\pi$
$42$	0	0
$43$	9.96988	0.0353580	0.0176790	−	0.999844i	$-0.494372\pi$
$43$	9.96988	0.0353580	0.0176790	+	0.999844i	$0.494372\pi$
$44$	−184.062	−0.630647
$45$	0	0
$46$	−80.6673	−0.258560
$47$	−419.682	−1.30249	−0.651244	−	0.758868i	$-0.725753\pi$
$47$	−419.682	−1.30249	−0.651244	+	0.758868i	$0.725753\pi$
$48$	0	0
$49$	−124.954	−0.364296
$50$	55.6832	0.157496
$51$	0	0
$52$	350.164	0.933827
$53$	512.098	1.32721	0.663604	−	0.748084i	$-0.269026\pi$
$53$	512.098	1.32721	0.663604	+	0.748084i	$0.269026\pi$
$54$	0	0
$55$	61.3938	0.150515
$56$	−109.696	−0.261763
$57$	0	0
$58$	−55.0748	−0.124684
$59$	−378.356	−0.834878	−0.417439	−	0.908705i	$-0.637072\pi$
$59$	−378.356	−0.834878	−0.417439	+	0.908705i	$0.637072\pi$
$60$	0	0
$61$	297.222	0.623858	0.311929	−	0.950105i	$-0.399025\pi$
$61$	297.222	0.623858	0.311929	+	0.950105i	$0.399025\pi$
$62$	−47.3411	−0.0969730
$63$	0	0
$64$	−428.833	−0.837565
$65$	−116.797	−0.222875
$66$	0	0
$67$	519.734	0.947696	0.473848	−	0.880607i	$-0.342865\pi$
$67$	519.734	0.947696	0.473848	+	0.880607i	$0.342865\pi$
$68$	−155.297	−0.276950
$69$	0	0
$70$	18.0374	0.0307983
$71$	−33.2035	−0.0555004	−0.0277502	−	0.999615i	$-0.508834\pi$
$71$	−33.2035	−0.0555004	−0.0277502	+	0.999615i	$0.508834\pi$
$72$	0	0
$73$	20.0491	0.0321448	0.0160724	−	0.999871i	$-0.494884\pi$
$73$	20.0491	0.0321448	0.0160724	+	0.999871i	$0.494884\pi$
$74$	−134.031	−0.210551
$75$	0	0
$76$	874.585	1.32002
$77$	−349.424	−0.517151
$78$	0	0
$79$	544.451	0.775387	0.387693	−	0.921788i	$-0.373272\pi$
$79$	544.451	0.775387	0.387693	+	0.921788i	$0.373272\pi$
$80$	152.370	0.212943
$81$	0	0
$82$	0.175222	0.000235975 0
$83$	763.226	1.00934	0.504668	−	0.863313i	$-0.331615\pi$
$83$	763.226	1.00934	0.504668	+	0.863313i	$0.331615\pi$
$84$	0	0
$85$	51.7992	0.0660990
$86$	−4.69401	−0.00588568
$87$	0	0
$88$	175.790	0.212946
$89$	−695.667	−0.828546	−0.414273	−	0.910153i	$-0.635964\pi$
$89$	−695.667	−0.828546	−0.414273	+	0.910153i	$0.635964\pi$
$90$	0	0
$91$	664.752	0.765768
$92$	−1332.69	−1.51025
$93$	0	0
$94$	197.594	0.216812
$95$	−291.717	−0.315047
$96$	0	0
$97$	−1621.19	−1.69698	−0.848490	−	0.529212i	$-0.822488\pi$
$97$	−1621.19	−1.69698	−0.848490	+	0.529212i	$0.822488\pi$
$98$	58.8306	0.0606407
$99$	0	0
$100$	919.934	0.919934
$101$	1874.72	1.84694	0.923472	−	0.383667i	$-0.125339\pi$
$101$	1874.72	1.84694	0.923472	+	0.383667i	$0.125339\pi$
$102$	0	0
$103$	1276.35	1.22099	0.610496	−	0.792019i	$-0.290970\pi$
$103$	1276.35	1.22099	0.610496	+	0.792019i	$0.290970\pi$
$104$	−334.426	−0.315319
$105$	0	0
$106$	−241.106	−0.220927
$107$	−181.493	−0.163977	−0.0819887	−	0.996633i	$-0.526127\pi$
$107$	−181.493	−0.163977	−0.0819887	+	0.996633i	$0.526127\pi$
$108$	0	0
$109$	707.703	0.621887	0.310944	−	0.950428i	$-0.399355\pi$
$109$	707.703	0.621887	0.310944	+	0.950428i	$0.399355\pi$
$110$	−28.9054	−0.0250547
$111$	0	0
$112$	−867.216	−0.731645
$113$	−1201.96	−1.00063	−0.500316	−	0.865843i	$-0.666783\pi$
$113$	−1201.96	−1.00063	−0.500316	+	0.865843i	$0.666783\pi$
$114$	0	0
$115$	444.517	0.360447
$116$	−909.882	−0.728279
$117$	0	0
$118$	178.137	0.138974
$119$	−294.817	−0.227108
$120$	0	0
$121$	−771.039	−0.579293
$122$	−139.938	−0.103847
$123$	0	0
$124$	−782.115	−0.566419
$125$	−631.149	−0.451613
$126$	0	0
$127$	−1228.77	−0.858549	−0.429274	−	0.903174i	$-0.641231\pi$
$127$	−1228.77	−0.858549	−0.429274	+	0.903174i	$0.641231\pi$
$128$	898.548	0.620478
$129$	0	0
$130$	54.9901	0.0370996
$131$	−1535.27	−1.02395	−0.511974	−	0.859001i	$-0.671085\pi$
$131$	−1535.27	−1.02395	−0.511974	+	0.859001i	$0.671085\pi$
$132$	0	0
$133$	1660.31	1.08246
$134$	−244.701	−0.157753
$135$	0	0
$136$	148.318	0.0935158
$137$	−1883.37	−1.17450	−0.587252	−	0.809404i	$-0.699790\pi$
$137$	−1883.37	−1.17450	−0.587252	+	0.809404i	$0.699790\pi$
$138$	0	0
$139$	1669.88	1.01897	0.509487	−	0.860479i	$-0.329835\pi$
$139$	1669.88	1.01897	0.509487	+	0.860479i	$0.329835\pi$
$140$	297.993	0.179893
$141$	0	0
$142$	15.6328	0.00923857
$143$	−1065.28	−0.622960
$144$	0	0
$145$	303.490	0.173817
$146$	−9.43950	−0.00535081
$147$	0	0
$148$	−2214.30	−1.22983
$149$	3424.29	1.88274	0.941371	−	0.337374i	$-0.109539\pi$
$149$	3424.29	1.88274	0.941371	+	0.337374i	$0.109539\pi$
$150$	0	0
$151$	−2901.46	−1.56369	−0.781846	−	0.623472i	$-0.785722\pi$
$151$	−2901.46	−1.56369	−0.781846	+	0.623472i	$0.785722\pi$
$152$	−835.278	−0.445723
$153$	0	0
$154$	164.516	0.0860848
$155$	260.873	0.135186
$156$	0	0
$157$	−979.274	−0.497800	−0.248900	−	0.968529i	$-0.580069\pi$
$157$	−979.274	−0.497800	−0.248900	+	0.968529i	$0.580069\pi$
$158$	−256.338	−0.129071
$159$	0	0
$160$	−225.927	−0.111632
$161$	−2529.98	−1.23845
$162$	0	0
$163$	102.525	0.0492661	0.0246330	−	0.999697i	$-0.492158\pi$
$163$	102.525	0.0492661	0.0246330	+	0.999697i	$0.492158\pi$
$164$	2.89481	0.00137833
$165$	0	0
$166$	−359.341	−0.168014
$167$	−2028.09	−0.939751	−0.469876	−	0.882733i	$-0.655701\pi$
$167$	−2028.09	−0.939751	−0.469876	+	0.882733i	$0.655701\pi$
$168$	0	0
$169$	−170.389	−0.0775555
$170$	−24.3881	−0.0110028
$171$	0	0
$172$	−77.5490	−0.0343782
$173$	−1218.29	−0.535406	−0.267703	−	0.963502i	$-0.586265\pi$
$173$	−1218.29	−0.535406	−0.267703	+	0.963502i	$0.586265\pi$
$174$	0	0
$175$	1746.40	0.754376
$176$	1389.73	0.595200
$177$	0	0
$178$	327.534	0.137920
$179$	2804.88	1.17121	0.585605	−	0.810597i	$-0.300857\pi$
$179$	2804.88	1.17121	0.585605	+	0.810597i	$0.300857\pi$
$180$	0	0
$181$	−1396.76	−0.573595	−0.286797	−	0.957991i	$-0.592591\pi$
$181$	−1396.76	−0.573595	−0.286797	+	0.957991i	$0.592591\pi$
$182$	−312.978	−0.127470
$183$	0	0
$184$	1272.79	0.509955
$185$	738.576	0.293520
$186$	0	0
$187$	472.451	0.184754
$188$	3264.43	1.26640
$189$	0	0
$190$	137.346	0.0524427
$191$	1318.73	0.499583	0.249791	−	0.968300i	$-0.419638\pi$
$191$	1318.73	0.499583	0.249791	+	0.968300i	$0.419638\pi$
$192$	0	0
$193$	−2098.62	−0.782703	−0.391351	−	0.920241i	$-0.627992\pi$
$193$	−2098.62	−0.782703	−0.391351	+	0.920241i	$0.627992\pi$
$194$	763.288	0.282479
$195$	0	0
$196$	971.931	0.354202
$197$	−3039.16	−1.09914	−0.549572	−	0.835447i	$-0.685209\pi$
$197$	−3039.16	−1.09914	−0.549572	+	0.835447i	$0.685209\pi$
$198$	0	0
$199$	−5091.27	−1.81362	−0.906810	−	0.421539i	$-0.861490\pi$
$199$	−5091.27	−1.81362	−0.906810	+	0.421539i	$0.861490\pi$
$200$	−878.588	−0.310628
$201$	0	0
$202$	−882.653	−0.307442
$203$	−1727.32	−0.597213
$204$	0	0
$205$	−0.965559	−0.000328964 0
$206$	−600.929	−0.203246
$207$	0	0
$208$	−2643.86	−0.881339
$209$	−2660.69	−0.880593
$210$	0	0
$211$	−211.000	−0.0688428
$211$	−211.000	−0.0688428
$212$	−3983.27	−1.29043
$213$	0	0
$214$	85.4503	0.0272956
$215$	25.8664	0.00820498
$216$	0	0
$217$	−1484.77	−0.464482
$218$	−333.200	−0.103519
$219$	0	0
$220$	−477.541	−0.146345
$221$	−898.800	−0.273574
$222$	0	0
$223$	739.350	0.222020	0.111010	−	0.993819i	$-0.464591\pi$
$223$	739.350	0.222020	0.111010	+	0.993819i	$0.464591\pi$
$224$	1285.87	0.383552
$225$	0	0
$226$	565.908	0.166565
$227$	−1454.66	−0.425327	−0.212664	−	0.977125i	$-0.568214\pi$
$227$	−1454.66	−0.425327	−0.212664	+	0.977125i	$0.568214\pi$
$228$	0	0
$229$	275.559	0.0795173	0.0397586	−	0.999209i	$-0.487341\pi$
$229$	275.559	0.0795173	0.0397586	+	0.999209i	$0.487341\pi$
$230$	−209.287	−0.0600000
$231$	0	0
$232$	868.988	0.245913
$233$	−865.146	−0.243252	−0.121626	−	0.992576i	$-0.538811\pi$
$233$	−865.146	−0.243252	−0.121626	+	0.992576i	$0.538811\pi$
$234$	0	0
$235$	−1088.85	−0.302249
$236$	2942.98	0.811744
$237$	0	0
$238$	138.805	0.0378043
$239$	−2856.24	−0.773032	−0.386516	−	0.922283i	$-0.626322\pi$
$239$	−2856.24	−0.773032	−0.386516	+	0.922283i	$0.626322\pi$
$240$	0	0
$241$	990.785	0.264822	0.132411	−	0.991195i	$-0.457728\pi$
$241$	990.785	0.264822	0.132411	+	0.991195i	$0.457728\pi$
$242$	363.020	0.0964290
$243$	0	0
$244$	−2311.89	−0.606572
$245$	−324.186	−0.0845367
$246$	0	0
$247$	5061.75	1.30393
$248$	746.964	0.191259
$249$	0	0
$250$	297.157	0.0751755
$251$	−2082.80	−0.523766	−0.261883	−	0.965100i	$-0.584344\pi$
$251$	−2082.80	−0.523766	−0.261883	+	0.965100i	$0.584344\pi$
$252$	0	0
$253$	4054.36	1.00749
$254$	578.528	0.142914
$255$	0	0
$256$	3007.61	0.734280
$257$	687.284	0.166816	0.0834078	−	0.996515i	$-0.473420\pi$
$257$	687.284	0.166816	0.0834078	+	0.996515i	$0.473420\pi$
$258$	0	0
$259$	−4203.63	−1.00850
$260$	908.483	0.216699
$261$	0	0
$262$	722.834	0.170446
$263$	−6232.14	−1.46118	−0.730589	−	0.682817i	$-0.760755\pi$
$263$	−6232.14	−1.46118	−0.730589	+	0.682817i	$0.760755\pi$
$264$	0	0
$265$	1328.61	0.307985
$266$	−781.707	−0.180186
$267$	0	0
$268$	−4042.66	−0.921436
$269$	1831.24	0.415067	0.207533	−	0.978228i	$-0.433456\pi$
$269$	1831.24	0.415067	0.207533	+	0.978228i	$0.433456\pi$
$270$	0	0
$271$	−2272.49	−0.509388	−0.254694	−	0.967022i	$-0.581975\pi$
$271$	−2272.49	−0.509388	−0.254694	+	0.967022i	$0.581975\pi$
$272$	1172.55	0.261383
$273$	0	0
$274$	886.725	0.195507
$275$	−2798.65	−0.613691
$276$	0	0
$277$	138.717	0.0300891	0.0150445	−	0.999887i	$-0.495211\pi$
$277$	138.717	0.0300891	0.0150445	+	0.999887i	$0.495211\pi$
$278$	−786.211	−0.169618
$279$	0	0
$280$	−284.600	−0.0607433
$281$	−5624.51	−1.19406	−0.597029	−	0.802219i	$-0.703652\pi$
$281$	−5624.51	−1.19406	−0.597029	+	0.802219i	$0.703652\pi$
$282$	0	0
$283$	−3808.08	−0.799883	−0.399941	−	0.916541i	$-0.630969\pi$
$283$	−3808.08	−0.799883	−0.399941	+	0.916541i	$0.630969\pi$
$284$	258.267	0.0539625
$285$	0	0
$286$	501.554	0.103698
$287$	5.49551	0.00113028
$288$	0	0
$289$	−4514.38	−0.918865
$290$	−142.889	−0.0289335
$291$	0	0
$292$	−155.948	−0.0312541
$293$	6200.38	1.23628	0.618140	−	0.786068i	$-0.287887\pi$
$293$	6200.38	1.23628	0.618140	+	0.786068i	$0.287887\pi$
$294$	0	0
$295$	−981.627	−0.193737
$296$	2114.78	0.415267
$297$	0	0
$298$	−1612.22	−0.313401
$299$	−7713.09	−1.49184
$300$	0	0
$301$	−147.219	−0.0281913
$302$	1366.06	0.260292
$303$	0	0
$304$	−6603.41	−1.24583
$305$	771.127	0.144769
$306$	0	0
$307$	−9350.14	−1.73824	−0.869121	−	0.494599i	$-0.835315\pi$
$307$	−9350.14	−1.73824	−0.869121	+	0.494599i	$0.835315\pi$
$308$	2717.94	0.502821
$309$	0	0
$310$	−122.824	−0.0225031
$311$	−1734.20	−0.316198	−0.158099	−	0.987423i	$-0.550537\pi$
$311$	−1734.20	−0.316198	−0.158099	+	0.987423i	$0.550537\pi$
$312$	0	0
$313$	−7860.42	−1.41948	−0.709740	−	0.704464i	$-0.751187\pi$
$313$	−7860.42	−1.41948	−0.709740	+	0.704464i	$0.751187\pi$
$314$	461.061	0.0828636
$315$	0	0
$316$	−4234.92	−0.753902
$317$	9073.13	1.60756	0.803782	−	0.594924i	$-0.202818\pi$
$317$	9073.13	1.60756	0.803782	+	0.594924i	$0.202818\pi$
$318$	0	0
$319$	2768.07	0.485838
$320$	−1112.59	−0.194361
$321$	0	0
$322$	1191.16	0.206152
$323$	−2244.88	−0.386714
$324$	0	0
$325$	5324.21	0.908721
$326$	−48.2707	−0.00820082
$327$	0	0
$328$	−2.76470	−0.000465412 0
$329$	6197.19	1.03849
$330$	0	0
$331$	2232.73	0.370762	0.185381	−	0.982667i	$-0.440648\pi$
$331$	2232.73	0.370762	0.185381	+	0.982667i	$0.440648\pi$
$332$	−5936.62	−0.981369
$333$	0	0
$334$	954.865	0.156431
$335$	1348.42	0.219917
$336$	0	0
$337$	3378.18	0.546057	0.273029	−	0.962006i	$-0.411975\pi$
$337$	3378.18	0.546057	0.273029	+	0.962006i	$0.411975\pi$
$338$	80.2226	0.0129099
$339$	0	0
$340$	−402.912	−0.0642675
$341$	2379.38	0.377861
$342$	0	0
$343$	6909.99	1.08777
$344$	74.0637	0.0116083
$345$	0	0
$346$	573.596	0.0891235
$347$	−2401.38	−0.371507	−0.185754	−	0.982596i	$-0.559473\pi$
$347$	−2401.38	−0.371507	−0.185754	+	0.982596i	$0.559473\pi$
$348$	0	0
$349$	−2696.01	−0.413507	−0.206754	−	0.978393i	$-0.566290\pi$
$349$	−2696.01	−0.413507	−0.206754	+	0.978393i	$0.566290\pi$
$350$	−822.240	−0.125573
$351$	0	0
$352$	−2060.63	−0.312023
$353$	−4715.70	−0.711024	−0.355512	−	0.934672i	$-0.615693\pi$
$353$	−4715.70	−0.711024	−0.355512	+	0.934672i	$0.615693\pi$
$354$	0	0
$355$	−86.1447	−0.0128791
$356$	5411.13	0.805588
$357$	0	0
$358$	−1320.59	−0.194959
$359$	2554.44	0.375538	0.187769	−	0.982213i	$-0.439874\pi$
$359$	2554.44	0.375538	0.187769	+	0.982213i	$0.439874\pi$
$360$	0	0
$361$	5783.45	0.843192
$362$	657.623	0.0954804
$363$	0	0
$364$	−5170.66	−0.744550
$365$	52.0164	0.00745935
$366$	0	0
$367$	−13223.1	−1.88077	−0.940385	−	0.340112i	$-0.889535\pi$
$367$	−13223.1	−1.88077	−0.940385	+	0.340112i	$0.889535\pi$
$368$	10062.3	1.42536
$369$	0	0
$370$	−347.736	−0.0488593
$371$	−7561.84	−1.05820
$372$	0	0
$373$	−3765.65	−0.522730	−0.261365	−	0.965240i	$-0.584173\pi$
$373$	−3765.65	−0.522730	−0.261365	+	0.965240i	$0.584173\pi$
$374$	−222.439	−0.0307541
$375$	0	0
$376$	−3117.71	−0.427616
$377$	−5266.04	−0.719402
$378$	0	0
$379$	7243.36	0.981705	0.490853	−	0.871243i	$-0.336685\pi$
$379$	7243.36	0.981705	0.490853	+	0.871243i	$0.336685\pi$
$380$	2269.07	0.306318
$381$	0	0
$382$	−620.886	−0.0831604
$383$	2603.55	0.347350	0.173675	−	0.984803i	$-0.444436\pi$
$383$	2603.55	0.347350	0.173675	+	0.984803i	$0.444436\pi$
$384$	0	0
$385$	−906.564	−0.120007
$386$	988.068	0.130288
$387$	0	0
$388$	12610.2	1.64996
$389$	−920.666	−0.119999	−0.0599995	−	0.998198i	$-0.519110\pi$
$389$	−920.666	−0.119999	−0.0599995	+	0.998198i	$0.519110\pi$
$390$	0	0
$391$	3420.75	0.442442
$392$	−928.248	−0.119601
$393$	0	0
$394$	1430.90	0.182963
$395$	1412.55	0.179932
$396$	0	0
$397$	9979.81	1.26164	0.630822	−	0.775928i	$-0.282718\pi$
$397$	9979.81	1.26164	0.630822	+	0.775928i	$0.282718\pi$
$398$	2397.07	0.301895
$399$	0	0
$400$	−6945.81	−0.868227
$401$	−4979.05	−0.620054	−0.310027	−	0.950728i	$-0.600338\pi$
$401$	−4979.05	−0.620054	−0.310027	+	0.950728i	$0.600338\pi$
$402$	0	0
$403$	−4526.57	−0.559515
$404$	−14582.2	−1.79577
$405$	0	0
$406$	813.256	0.0994118
$407$	6736.41	0.820422
$408$	0	0
$409$	−3849.94	−0.465445	−0.232723	−	0.972543i	$-0.574763\pi$
$409$	−3849.94	−0.465445	−0.232723	+	0.972543i	$0.574763\pi$
$410$	0.454604	5.47592e−5 0
$411$	0	0
$412$	−9927.84	−1.18716
$413$	5586.96	0.665657
$414$	0	0
$415$	1980.15	0.234221
$416$	3920.19	0.462027
$417$	0	0
$418$	1252.70	0.146583
$419$	−1490.12	−0.173740	−0.0868701	−	0.996220i	$-0.527687\pi$
$419$	−1490.12	−0.173740	−0.0868701	+	0.996220i	$0.527687\pi$
$420$	0	0
$421$	−12512.8	−1.44854	−0.724271	−	0.689515i	$-0.757824\pi$
$421$	−12512.8	−1.44854	−0.724271	+	0.689515i	$0.757824\pi$
$422$	99.3429	0.0114596
$423$	0	0
$424$	3804.24	0.435732
$425$	−2361.28	−0.269504
$426$	0	0
$427$	−4388.89	−0.497409
$428$	1411.71	0.159434
$429$	0	0
$430$	−12.1784	−0.00136580
$431$	−9637.17	−1.07704	−0.538522	−	0.842611i	$-0.681017\pi$
$431$	−9637.17	−1.07704	−0.538522	+	0.842611i	$0.681017\pi$
$432$	0	0
$433$	3496.92	0.388109	0.194055	−	0.980991i	$-0.437836\pi$
$433$	3496.92	0.388109	0.194055	+	0.980991i	$0.437836\pi$
$434$	699.058	0.0773176
$435$	0	0
$436$	−5504.75	−0.604655
$437$	−19264.5	−2.10881
$438$	0	0
$439$	−1579.18	−0.171686	−0.0858429	−	0.996309i	$-0.527358\pi$
$439$	−1579.18	−0.171686	−0.0858429	+	0.996309i	$0.527358\pi$
$440$	456.078	0.0494152
$441$	0	0
$442$	423.172	0.0455390
$443$	−2910.11	−0.312107	−0.156054	−	0.987749i	$-0.549877\pi$
$443$	−2910.11	−0.312107	−0.156054	+	0.987749i	$0.549877\pi$
$444$	0	0
$445$	−1804.88	−0.192268
$446$	−348.100	−0.0369574
$447$	0	0
$448$	6332.32	0.667799
$449$	−9077.19	−0.954074	−0.477037	−	0.878883i	$-0.658289\pi$
$449$	−9077.19	−0.954074	−0.477037	+	0.878883i	$0.658289\pi$
$450$	0	0
$451$	−8.80667	−0.000919490 0
$452$	9349.28	0.972905
$453$	0	0
$454$	684.883	0.0707999
$455$	1724.67	0.177700
$456$	0	0
$457$	−5089.43	−0.520949	−0.260474	−	0.965481i	$-0.583879\pi$
$457$	−5089.43	−0.520949	−0.260474	+	0.965481i	$0.583879\pi$
$458$	−129.739	−0.0132364
$459$	0	0
$460$	−3457.60	−0.350460
$461$	−8581.26	−0.866962	−0.433481	−	0.901163i	$-0.642715\pi$
$461$	−8581.26	−0.866962	−0.433481	+	0.901163i	$0.642715\pi$
$462$	0	0
$463$	14153.6	1.42067	0.710337	−	0.703861i	$-0.248542\pi$
$463$	14153.6	1.42067	0.710337	+	0.703861i	$0.248542\pi$
$464$	6869.92	0.687345
$465$	0	0
$466$	407.327	0.0404916
$467$	−18600.0	−1.84305	−0.921525	−	0.388320i	$-0.873056\pi$
$467$	−18600.0	−1.84305	−0.921525	+	0.388320i	$0.873056\pi$
$468$	0	0
$469$	−7674.60	−0.755608
$470$	512.649	0.0503122
$471$	0	0
$472$	−2810.71	−0.274096
$473$	235.922	0.0229339
$474$	0	0
$475$	13298.0	1.28453
$476$	2293.18	0.220815
$477$	0	0
$478$	1344.77	0.128679
$479$	−10332.1	−0.985568	−0.492784	−	0.870152i	$-0.664021\pi$
$479$	−10332.1	−0.985568	−0.492784	+	0.870152i	$0.664021\pi$
$480$	0	0
$481$	−12815.5	−1.21484
$482$	−466.481	−0.0440822
$483$	0	0
$484$	5997.40	0.563242
$485$	−4206.10	−0.393792
$486$	0	0
$487$	20561.6	1.91322	0.956608	−	0.291378i	$-0.0941138\pi$
$487$	20561.6	1.91322	0.956608	+	0.291378i	$0.0941138\pi$
$488$	2207.98	0.204817
$489$	0	0
$490$	152.633	0.0140720
$491$	11772.6	1.08205	0.541027	−	0.841005i	$-0.318036\pi$
$491$	11772.6	1.08205	0.541027	+	0.841005i	$0.318036\pi$
$492$	0	0
$493$	2335.48	0.213357
$494$	−2383.17	−0.217052
$495$	0	0
$496$	5905.24	0.534583
$497$	490.295	0.0442510
$498$	0	0
$499$	11455.2	1.02767	0.513834	−	0.857889i	$-0.328225\pi$
$499$	11455.2	1.02767	0.513834	+	0.857889i	$0.328225\pi$
$500$	4909.29	0.439100
$501$	0	0
$502$	980.624	0.0871860
$503$	19472.8	1.72614	0.863072	−	0.505081i	$-0.168538\pi$
$503$	19472.8	1.72614	0.863072	+	0.505081i	$0.168538\pi$
$504$	0	0
$505$	4863.86	0.428592
$506$	−1908.87	−0.167707
$507$	0	0
$508$	9557.78	0.834759
$509$	−13060.8	−1.13735	−0.568675	−	0.822562i	$-0.692544\pi$
$509$	−13060.8	−1.13735	−0.568675	+	0.822562i	$0.692544\pi$
$510$	0	0
$511$	−296.053	−0.0256294
$512$	−8604.43	−0.742706
$513$	0	0
$514$	−323.587	−0.0277681
$515$	3311.42	0.283337
$516$	0	0
$517$	−9931.15	−0.844819
$518$	1979.15	0.167874
$519$	0	0
$520$	−867.653	−0.0731713
$521$	12383.1	1.04129	0.520646	−	0.853773i	$-0.325691\pi$
$521$	12383.1	1.04129	0.520646	+	0.853773i	$0.325691\pi$
$522$	0	0
$523$	3200.43	0.267581	0.133790	−	0.991010i	$-0.457285\pi$
$523$	3200.43	0.267581	0.133790	+	0.991010i	$0.457285\pi$
$524$	11941.8	0.995575
$525$	0	0
$526$	2934.21	0.243227
$527$	2007.53	0.165938
$528$	0	0
$529$	17188.3	1.41270
$530$	−625.537	−0.0512671
$531$	0	0
$532$	−12914.5	−1.05247
$533$	16.7540	0.00136153
$534$	0	0
$535$	−470.874	−0.0380517
$536$	3860.97	0.311135
$537$	0	0
$538$	−862.185	−0.0690919
$539$	−2956.84	−0.236290
$540$	0	0
$541$	−9151.70	−0.727287	−0.363644	−	0.931538i	$-0.618467\pi$
$541$	−9151.70	−0.727287	−0.363644	+	0.931538i	$0.618467\pi$
$542$	1069.93	0.0847925
$543$	0	0
$544$	−1738.60	−0.137026
$545$	1836.10	0.144312
$546$	0	0
$547$	−20476.8	−1.60059	−0.800295	−	0.599606i	$-0.795324\pi$
$547$	−20476.8	−1.60059	−0.800295	+	0.599606i	$0.795324\pi$
$548$	14649.5	1.14196
$549$	0	0
$550$	1317.66	0.102155
$551$	−13152.7	−1.01692
$552$	0	0
$553$	−8039.58	−0.618224
$554$	−65.3104	−0.00500862
$555$	0	0
$556$	−12988.9	−0.990739
$557$	17361.9	1.32073	0.660364	−	0.750945i	$-0.270402\pi$
$557$	17361.9	1.32073	0.660364	+	0.750945i	$0.270402\pi$
$558$	0	0
$559$	−448.823	−0.0339592
$560$	−2249.95	−0.169782
$561$	0	0
$562$	2648.13	0.198763
$563$	−4423.76	−0.331153	−0.165576	−	0.986197i	$-0.552948\pi$
$563$	−4423.76	−0.331153	−0.165576	+	0.986197i	$0.552948\pi$
$564$	0	0
$565$	−3118.44	−0.232201
$566$	1792.92	0.133148
$567$	0	0
$568$	−246.660	−0.0182212
$569$	−3862.53	−0.284579	−0.142290	−	0.989825i	$-0.545446\pi$
$569$	−3862.53	−0.284579	−0.142290	+	0.989825i	$0.545446\pi$
$570$	0	0
$571$	9226.09	0.676182	0.338091	−	0.941113i	$-0.390219\pi$
$571$	9226.09	0.676182	0.338091	+	0.941113i	$0.390219\pi$
$572$	8286.10	0.605698
$573$	0	0
$574$	−2.58739	−0.000188146 0
$575$	−20263.5	−1.46964
$576$	0	0
$577$	20440.3	1.47477	0.737384	−	0.675474i	$-0.236061\pi$
$577$	20440.3	1.47477	0.737384	+	0.675474i	$0.236061\pi$
$578$	2125.46	0.152954
$579$	0	0
$580$	−2360.64	−0.169001
$581$	−11270.1	−0.804754
$582$	0	0
$583$	12118.0	0.860853
$584$	148.939	0.0105534
$585$	0	0
$586$	−2919.26	−0.205791
$587$	10807.8	0.759944	0.379972	−	0.924998i	$-0.375934\pi$
$587$	10807.8	0.759944	0.379972	+	0.924998i	$0.375934\pi$
$588$	0	0
$589$	−11305.8	−0.790910
$590$	462.169	0.0322495
$591$	0	0
$592$	16718.7	1.16070
$593$	−5848.37	−0.404998	−0.202499	−	0.979282i	$-0.564906\pi$
$593$	−5848.37	−0.404998	−0.202499	+	0.979282i	$0.564906\pi$
$594$	0	0
$595$	−764.888	−0.0527014
$596$	−26635.2	−1.83057
$597$	0	0
$598$	3631.47	0.248331
$599$	−12144.3	−0.828387	−0.414194	−	0.910189i	$-0.635936\pi$
$599$	−12144.3	−0.828387	−0.414194	+	0.910189i	$0.635936\pi$
$600$	0	0
$601$	−10250.3	−0.695702	−0.347851	−	0.937550i	$-0.613089\pi$
$601$	−10250.3	−0.695702	−0.347851	+	0.937550i	$0.613089\pi$
$602$	69.3136	0.00469271
$603$	0	0
$604$	22568.5	1.52036
$605$	−2000.42	−0.134428
$606$	0	0
$607$	19692.7	1.31681	0.658403	−	0.752666i	$-0.271232\pi$
$607$	19692.7	1.31681	0.658403	+	0.752666i	$0.271232\pi$
$608$	9791.24	0.653104
$609$	0	0
$610$	−363.062	−0.0240983
$611$	18893.2	1.25096
$612$	0	0
$613$	19346.8	1.27473	0.637366	−	0.770561i	$-0.280024\pi$
$613$	19346.8	1.27473	0.637366	+	0.770561i	$0.280024\pi$
$614$	4402.22	0.289347
$615$	0	0
$616$	−2595.78	−0.169784
$617$	−6923.88	−0.451775	−0.225887	−	0.974153i	$-0.572528\pi$
$617$	−6923.88	−0.451775	−0.225887	+	0.974153i	$0.572528\pi$
$618$	0	0
$619$	9587.94	0.622571	0.311286	−	0.950316i	$-0.399240\pi$
$619$	9587.94	0.622571	0.311286	+	0.950316i	$0.399240\pi$
$620$	−2029.16	−0.131440
$621$	0	0
$622$	816.497	0.0526343
$623$	10272.5	0.660608
$624$	0	0
$625$	13146.1	0.841352
$626$	3700.84	0.236286
$627$	0	0
$628$	7617.11	0.484006
$629$	5683.66	0.360290
$630$	0	0
$631$	10817.6	0.682478	0.341239	−	0.939977i	$-0.389153\pi$
$631$	10817.6	0.682478	0.341239	+	0.939977i	$0.389153\pi$
$632$	4044.59	0.254565
$633$	0	0
$634$	−4271.80	−0.267595
$635$	−3187.98	−0.199230
$636$	0	0
$637$	5625.15	0.349885
$638$	−1303.26	−0.0808724
$639$	0	0
$640$	2331.24	0.143985
$641$	1736.80	0.107019	0.0535097	−	0.998567i	$-0.482959\pi$
$641$	1736.80	0.107019	0.0535097	+	0.998567i	$0.482959\pi$
$642$	0	0
$643$	−30998.3	−1.90117	−0.950585	−	0.310464i	$-0.899516\pi$
$643$	−30998.3	−1.90117	−0.950585	+	0.310464i	$0.899516\pi$
$644$	19679.0	1.20413
$645$	0	0
$646$	1056.93	0.0643723
$647$	−25584.0	−1.55458	−0.777289	−	0.629144i	$-0.783405\pi$
$647$	−25584.0	−1.55458	−0.777289	+	0.629144i	$0.783405\pi$
$648$	0	0
$649$	−8953.23	−0.541518
$650$	−2506.74	−0.151265
$651$	0	0
$652$	−797.472	−0.0479010
$653$	−24899.4	−1.49217	−0.746085	−	0.665850i	$-0.768069\pi$
$653$	−24899.4	−1.49217	−0.746085	+	0.665850i	$0.768069\pi$
$654$	0	0
$655$	−3983.18	−0.237612
$656$	−21.8568	−0.00130086
$657$	0	0
$658$	−2917.76	−0.172866
$659$	23403.0	1.38339	0.691693	−	0.722191i	$-0.256865\pi$
$659$	23403.0	1.38339	0.691693	+	0.722191i	$0.256865\pi$
$660$	0	0
$661$	26295.7	1.54733	0.773665	−	0.633595i	$-0.218421\pi$
$661$	26295.7	1.54733	0.773665	+	0.633595i	$0.218421\pi$
$662$	−1051.21	−0.0617169
$663$	0	0
$664$	5669.81	0.331372
$665$	4307.60	0.251190
$666$	0	0
$667$	20042.0	1.16346
$668$	15775.2	0.913712
$669$	0	0
$670$	−634.864	−0.0366074
$671$	7033.30	0.404646
$672$	0	0
$673$	−29118.8	−1.66783	−0.833913	−	0.551895i	$-0.813905\pi$
$673$	−29118.8	−1.66783	−0.833913	+	0.551895i	$0.813905\pi$
$674$	−1590.51	−0.0908965
$675$	0	0
$676$	1325.34	0.0754065
$677$	−30595.8	−1.73691	−0.868457	−	0.495765i	$-0.834888\pi$
$677$	−30595.8	−1.73691	−0.868457	+	0.495765i	$0.834888\pi$
$678$	0	0
$679$	23939.1	1.35302
$680$	384.803	0.0217008
$681$	0	0
$682$	−1120.26	−0.0628986
$683$	−4235.87	−0.237308	−0.118654	−	0.992936i	$-0.537858\pi$
$683$	−4235.87	−0.237308	−0.118654	+	0.992936i	$0.537858\pi$
$684$	0	0
$685$	−4886.30	−0.272549
$686$	−3253.35	−0.181069
$687$	0	0
$688$	585.522	0.0324459
$689$	−23053.6	−1.27470
$690$	0	0
$691$	−29467.7	−1.62229	−0.811147	−	0.584843i	$-0.801156\pi$
$691$	−29467.7	−1.62229	−0.811147	+	0.584843i	$0.801156\pi$
$692$	9476.29	0.520570
$693$	0	0
$694$	1130.62	0.0618410
$695$	4332.42	0.236458
$696$	0	0
$697$	−7.43038	−0.000403796 0
$698$	1269.33	0.0688323
$699$	0	0
$700$	−13584.1	−0.733473
$701$	33093.4	1.78305	0.891526	−	0.452969i	$-0.149635\pi$
$701$	33093.4	1.78305	0.891526	+	0.452969i	$0.149635\pi$
$702$	0	0
$703$	−32008.5	−1.71725
$704$	−10147.7	−0.543260
$705$	0	0
$706$	2220.24	0.118357
$707$	−27682.8	−1.47259
$708$	0	0
$709$	9474.99	0.501891	0.250946	−	0.968001i	$-0.419258\pi$
$709$	9474.99	0.501891	0.250946	+	0.968001i	$0.419258\pi$
$710$	40.5586	0.00214386
$711$	0	0
$712$	−5167.93	−0.272018
$713$	17227.7	0.904885
$714$	0	0
$715$	−2763.82	−0.144561
$716$	−21817.3	−1.13876
$717$	0	0
$718$	−1202.68	−0.0625119
$719$	20633.4	1.07023	0.535116	−	0.844778i	$-0.320268\pi$
$719$	20633.4	1.07023	0.535116	+	0.844778i	$0.320268\pi$
$720$	0	0
$721$	−18847.0	−0.973509
$722$	−2722.96	−0.140358
$723$	0	0
$724$	10864.5	0.557701
$725$	−13834.7	−0.708699
$726$	0	0
$727$	15242.7	0.777609	0.388804	−	0.921320i	$-0.372888\pi$
$727$	15242.7	0.777609	0.388804	+	0.921320i	$0.372888\pi$
$728$	4938.27	0.251407
$729$	0	0
$730$	−24.4903	−0.00124168
$731$	199.053	0.0100714
$732$	0	0
$733$	−410.243	−0.0206721	−0.0103361	−	0.999947i	$-0.503290\pi$
$733$	−410.243	−0.0206721	−0.0103361	+	0.999947i	$0.503290\pi$
$734$	6225.71	0.313072
$735$	0	0
$736$	−14919.9	−0.747220
$737$	12298.7	0.614694
$738$	0	0
$739$	5903.42	0.293858	0.146929	−	0.989147i	$-0.453061\pi$
$739$	5903.42	0.293858	0.146929	+	0.989147i	$0.453061\pi$
$740$	−5744.89	−0.285387
$741$	0	0
$742$	3560.26	0.176147
$743$	−24034.2	−1.18671	−0.593357	−	0.804939i	$-0.702198\pi$
$743$	−24034.2	−1.18671	−0.593357	+	0.804939i	$0.702198\pi$
$744$	0	0
$745$	8884.14	0.436899
$746$	1772.94	0.0870134
$747$	0	0
$748$	−3674.88	−0.179635
$749$	2680.00	0.130741
$750$	0	0
$751$	−28102.1	−1.36546	−0.682729	−	0.730671i	$-0.739207\pi$
$751$	−28102.1	−1.36546	−0.682729	+	0.730671i	$0.739207\pi$
$752$	−24647.5	−1.19522
$753$	0	0
$754$	2479.35	0.119751
$755$	−7527.69	−0.362862
$756$	0	0
$757$	−19021.3	−0.913265	−0.456633	−	0.889655i	$-0.650945\pi$
$757$	−19021.3	−0.913265	−0.456633	+	0.889655i	$0.650945\pi$
$758$	−3410.31	−0.163414
$759$	0	0
$760$	−2167.09	−0.103432
$761$	−9650.71	−0.459708	−0.229854	−	0.973225i	$-0.573825\pi$
$761$	−9650.71	−0.459708	−0.229854	+	0.973225i	$0.573825\pi$
$762$	0	0
$763$	−10450.2	−0.495837
$764$	−10257.6	−0.485740
$765$	0	0
$766$	−1225.80	−0.0578198
$767$	17032.8	0.801850
$768$	0	0
$769$	4539.80	0.212886	0.106443	−	0.994319i	$-0.466054\pi$
$769$	4539.80	0.212886	0.106443	+	0.994319i	$0.466054\pi$
$770$	426.828	0.0199764
$771$	0	0
$772$	16323.7	0.761015
$773$	9944.46	0.462713	0.231357	−	0.972869i	$-0.425684\pi$
$773$	9944.46	0.462713	0.231357	+	0.972869i	$0.425684\pi$
$774$	0	0
$775$	−11892.0	−0.551191
$776$	−12043.4	−0.557130
$777$	0	0
$778$	433.467	0.0199750
$779$	41.8455	0.00192461
$780$	0	0
$781$	−785.710	−0.0359986
$782$	−1610.55	−0.0736487
$783$	0	0
$784$	−7338.41	−0.334293
$785$	−2540.68	−0.115517
$786$	0	0
$787$	−5185.30	−0.234861	−0.117431	−	0.993081i	$-0.537466\pi$
$787$	−5185.30	−0.234861	−0.117431	+	0.993081i	$0.537466\pi$
$788$	23639.6	1.06869
$789$	0	0
$790$	−665.057	−0.0299515
$791$	17748.7	0.797813
$792$	0	0
$793$	−13380.3	−0.599178
$794$	−4698.69	−0.210013
$795$	0	0
$796$	39601.6	1.76337
$797$	4013.08	0.178357	0.0891786	−	0.996016i	$-0.471576\pi$
$797$	4013.08	0.178357	0.0891786	+	0.996016i	$0.471576\pi$
$798$	0	0
$799$	−8379.12	−0.371004
$800$	10298.9	0.455153
$801$	0	0
$802$	2344.23	0.103214
$803$	474.431	0.0208497
$804$	0	0
$805$	−6563.92	−0.287388
$806$	2131.20	0.0931368
$807$	0	0
$808$	13926.8	0.606364
$809$	−9454.31	−0.410872	−0.205436	−	0.978671i	$-0.565861\pi$
$809$	−9454.31	−0.410872	−0.205436	+	0.978671i	$0.565861\pi$
$810$	0	0
$811$	−5626.83	−0.243631	−0.121815	−	0.992553i	$-0.538872\pi$
$811$	−5626.83	−0.243631	−0.121815	+	0.992553i	$0.538872\pi$
$812$	13435.7	0.580665
$813$	0	0
$814$	−3171.63	−0.136567
$815$	265.996	0.0114324
$816$	0	0
$817$	−1121.00	−0.0480035
$818$	1812.62	0.0774779
$819$	0	0
$820$	7.51043	0.000319848 0
$821$	33248.6	1.41338	0.706689	−	0.707524i	$-0.250188\pi$
$821$	33248.6	1.41338	0.706689	+	0.707524i	$0.250188\pi$
$822$	0	0
$823$	44732.9	1.89464	0.947321	−	0.320286i	$-0.103779\pi$
$823$	44732.9	1.89464	0.947321	+	0.320286i	$0.103779\pi$
$824$	9481.65	0.400860
$825$	0	0
$826$	−2630.45	−0.110805
$827$	−2003.19	−0.0842296	−0.0421148	−	0.999113i	$-0.513410\pi$
$827$	−2003.19	−0.0842296	−0.0421148	+	0.999113i	$0.513410\pi$
$828$	0	0
$829$	−19946.2	−0.835656	−0.417828	−	0.908526i	$-0.637208\pi$
$829$	−19946.2	−0.835656	−0.417828	+	0.908526i	$0.637208\pi$
$830$	−932.294	−0.0389884
$831$	0	0
$832$	19305.2	0.804430
$833$	−2494.75	−0.103767
$834$	0	0
$835$	−5261.79	−0.218074
$836$	20695.7	0.856192
$837$	0	0
$838$	701.577	0.0289207
$839$	−2564.19	−0.105514	−0.0527568	−	0.998607i	$-0.516801\pi$
$839$	−2564.19	−0.105514	−0.0527568	+	0.998607i	$0.516801\pi$
$840$	0	0
$841$	−10705.5	−0.438948
$842$	5891.26	0.241124
$843$	0	0
$844$	1641.23	0.0669353
$845$	−442.067	−0.0179971
$846$	0	0
$847$	11385.5	0.461876
$848$	30075.0	1.21790
$849$	0	0
$850$	1111.74	0.0448615
$851$	48774.5	1.96471
$852$	0	0
$853$	42484.4	1.70532	0.852660	−	0.522467i	$-0.174988\pi$
$853$	42484.4	1.70532	0.852660	+	0.522467i	$0.174988\pi$
$854$	2066.38	0.0827985
$855$	0	0
$856$	−1348.26	−0.0538349
$857$	−12275.9	−0.489308	−0.244654	−	0.969610i	$-0.578674\pi$
$857$	−12275.9	−0.489308	−0.244654	+	0.969610i	$0.578674\pi$
$858$	0	0
$859$	20538.7	0.815798	0.407899	−	0.913027i	$-0.366262\pi$
$859$	20538.7	0.815798	0.407899	+	0.913027i	$0.366262\pi$
$860$	−201.197	−0.00797763
$861$	0	0
$862$	4537.36	0.179284
$863$	25803.6	1.01780	0.508901	−	0.860825i	$-0.330052\pi$
$863$	25803.6	1.01780	0.508901	+	0.860825i	$0.330052\pi$
$864$	0	0
$865$	−3160.80	−0.124243
$866$	−1646.42	−0.0646045
$867$	0	0
$868$	11549.0	0.451612
$869$	12883.6	0.502931
$870$	0	0
$871$	−23397.3	−0.910205
$872$	5257.34	0.204170
$873$	0	0
$874$	9070.12	0.351031
$875$	9319.79	0.360076
$876$	0	0
$877$	−29087.4	−1.11997	−0.559983	−	0.828504i	$-0.689192\pi$
$877$	−29087.4	−1.11997	−0.559983	+	0.828504i	$0.689192\pi$
$878$	743.507	0.0285788
$879$	0	0
$880$	3605.60	0.138119
$881$	−4391.17	−0.167926	−0.0839628	−	0.996469i	$-0.526758\pi$
$881$	−4391.17	−0.167926	−0.0839628	+	0.996469i	$0.526758\pi$
$882$	0	0
$883$	−24568.3	−0.936340	−0.468170	−	0.883638i	$-0.655087\pi$
$883$	−24568.3	−0.936340	−0.468170	+	0.883638i	$0.655087\pi$
$884$	6991.16	0.265993
$885$	0	0
$886$	1370.14	0.0519533
$887$	33359.5	1.26280	0.631399	−	0.775458i	$-0.282481\pi$
$887$	33359.5	1.26280	0.631399	+	0.775458i	$0.282481\pi$
$888$	0	0
$889$	18144.5	0.684530
$890$	849.770	0.0320049
$891$	0	0
$892$	−5750.90	−0.215868
$893$	47188.5	1.76831
$894$	0	0
$895$	7277.12	0.271785
$896$	−13268.3	−0.494714
$897$	0	0
$898$	4273.72	0.158815
$899$	11762.0	0.436359
$900$	0	0
$901$	10224.2	0.378045
$902$	4.14635	0.000153058 0
$903$	0	0
$904$	−8929.08	−0.328514
$905$	−3623.83	−0.133105
$906$	0	0
$907$	−21486.7	−0.786608	−0.393304	−	0.919408i	$-0.628668\pi$
$907$	−21486.7	−0.786608	−0.393304	+	0.919408i	$0.628668\pi$
$908$	11314.8	0.413542
$909$	0	0
$910$	−812.006	−0.0295799
$911$	−15681.8	−0.570321	−0.285161	−	0.958480i	$-0.592047\pi$
$911$	−15681.8	−0.570321	−0.285161	+	0.958480i	$0.592047\pi$
$912$	0	0
$913$	18060.6	0.654675
$914$	2396.20	0.0867170
$915$	0	0
$916$	−2143.39	−0.0773139
$917$	22670.4	0.816403
$918$	0	0
$919$	17508.6	0.628462	0.314231	−	0.949346i	$-0.398253\pi$
$919$	17508.6	0.628462	0.314231	+	0.949346i	$0.398253\pi$
$920$	3302.20	0.118337
$921$	0	0
$922$	4040.22	0.144314
$923$	1494.75	0.0533047
$924$	0	0
$925$	−33668.2	−1.19676
$926$	−6663.78	−0.236485
$927$	0	0
$928$	−10186.4	−0.360328
$929$	28902.1	1.02072	0.510359	−	0.859961i	$-0.329512\pi$
$929$	28902.1	1.02072	0.510359	+	0.859961i	$0.329512\pi$
$930$	0	0
$931$	14049.6	0.494584
$932$	6729.39	0.236511
$933$	0	0
$934$	8757.22	0.306794
$935$	1225.75	0.0428731
$936$	0	0
$937$	−28984.6	−1.01055	−0.505275	−	0.862959i	$-0.668609\pi$
$937$	−28984.6	−1.01055	−0.505275	+	0.862959i	$0.668609\pi$
$938$	3613.35	0.125778
$939$	0	0
$940$	8469.40	0.293874
$941$	49928.0	1.72965	0.864827	−	0.502070i	$-0.167428\pi$
$941$	49928.0	1.72965	0.864827	+	0.502070i	$0.167428\pi$
$942$	0	0
$943$	−63.7641	−0.00220196
$944$	−22220.5	−0.766118
$945$	0	0
$946$	−111.077	−0.00381756
$947$	19792.9	0.679179	0.339590	−	0.940574i	$-0.389712\pi$
$947$	19792.9	0.679179	0.339590	+	0.940574i	$0.389712\pi$
$948$	0	0
$949$	−902.568	−0.0308731
$950$	−6260.95	−0.213823
$951$	0	0
$952$	−2190.12	−0.0745611
$953$	54051.0	1.83723	0.918617	−	0.395148i	$-0.129307\pi$
$953$	54051.0	1.83723	0.918617	+	0.395148i	$0.129307\pi$
$954$	0	0
$955$	3421.39	0.115931
$956$	22216.8	0.751612
$957$	0	0
$958$	4864.57	0.164057
$959$	27810.5	0.936443
$960$	0	0
$961$	−19680.6	−0.660622
$962$	6033.78	0.202221
$963$	0	0
$964$	−7706.65	−0.257484
$965$	−5444.75	−0.181630
$966$	0	0
$967$	39369.4	1.30924	0.654619	−	0.755959i	$-0.272829\pi$
$967$	39369.4	1.30924	0.654619	+	0.755959i	$0.272829\pi$
$968$	−5727.85	−0.190186
$969$	0	0
$970$	1980.31	0.0655505
$971$	−5477.93	−0.181045	−0.0905227	−	0.995894i	$-0.528854\pi$
$971$	−5477.93	−0.181045	−0.0905227	+	0.995894i	$0.528854\pi$
$972$	0	0
$973$	−24658.1	−0.812438
$974$	−9680.81	−0.318473
$975$	0	0
$976$	17455.5	0.572478
$977$	11180.6	0.366120	0.183060	−	0.983102i	$-0.441400\pi$
$977$	11180.6	0.366120	0.183060	+	0.983102i	$0.441400\pi$
$978$	0	0
$979$	−16461.9	−0.537411
$980$	2521.63	0.0821943
$981$	0	0
$982$	−5542.75	−0.180119
$983$	665.057	0.0215788	0.0107894	−	0.999942i	$-0.496566\pi$
$983$	665.057	0.0215788	0.0107894	+	0.999942i	$0.496566\pi$
$984$	0	0
$985$	−7884.95	−0.255061
$986$	−1099.59	−0.0355153
$987$	0	0
$988$	−39372.0	−1.26780
$989$	1708.18	0.0549210
$990$	0	0
$991$	37191.7	1.19216	0.596081	−	0.802924i	$-0.296724\pi$
$991$	37191.7	1.19216	0.596081	+	0.802924i	$0.296724\pi$
$992$	−8756.01	−0.280246
$993$	0	0
$994$	−230.840	−0.00736601
$995$	−13209.1	−0.420859
$996$	0	0
$997$	23302.5	0.740220	0.370110	−	0.928988i	$-0.379320\pi$
$997$	23302.5	0.740220	0.370110	+	0.928988i	$0.379320\pi$
$998$	−5393.34	−0.171065
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1899.4.a.d.1.13		26
3.2	odd	2		633.4.a.d.1.14	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
633.4.a.d.1.14	✓	26	3.2	odd	2
1899.4.a.d.1.13		26	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:	\( N \)	\(=\)	\( 1899 = 3^{2} \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1899.a (trivial)

\(f(q)\)	\(=\)	\(q-0.470819 q^{2} -7.77833 q^{4} +2.59445 q^{5} -14.7664 q^{7} +7.42874 q^{8} +O(q^{10})\)	\(q-0.470819 q^{2} -7.77833 q^{4} +2.59445 q^{5} -14.7664 q^{7} +7.42874 q^{8} -1.22152 q^{10} +23.6635 q^{11} -45.0179 q^{13} +6.95230 q^{14} +58.7290 q^{16} +19.9654 q^{17} -112.439 q^{19} -20.1805 q^{20} -11.1412 q^{22} +171.334 q^{23} -118.269 q^{25} +21.1953 q^{26} +114.858 q^{28} +116.976 q^{29} +100.551 q^{31} -87.0807 q^{32} -9.40009 q^{34} -38.3107 q^{35} +284.675 q^{37} +52.9383 q^{38} +19.2735 q^{40} -0.372163 q^{41} +9.96988 q^{43} -184.062 q^{44} -80.6673 q^{46} -419.682 q^{47} -124.954 q^{49} +55.6832 q^{50} +350.164 q^{52} +512.098 q^{53} +61.3938 q^{55} -109.696 q^{56} -55.0748 q^{58} -378.356 q^{59} +297.222 q^{61} -47.3411 q^{62} -428.833 q^{64} -116.797 q^{65} +519.734 q^{67} -155.297 q^{68} +18.0374 q^{70} -33.2035 q^{71} +20.0491 q^{73} -134.031 q^{74} +874.585 q^{76} -349.424 q^{77} +544.451 q^{79} +152.370 q^{80} +0.175222 q^{82} +763.226 q^{83} +51.7992 q^{85} -4.69401 q^{86} +175.790 q^{88} -695.667 q^{89} +664.752 q^{91} -1332.69 q^{92} +197.594 q^{94} -291.717 q^{95} -1621.19 q^{97} +58.8306 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 6 q^{2} + 108 q^{4} - 25 q^{5} + 50 q^{7} - 57 q^{8}+O(q^{10}) \) 26 * q - 6 * q^2 + 108 * q^4 - 25 * q^5 + 50 * q^7 - 57 * q^8	\( 26 q - 6 q^{2} + 108 q^{4} - 25 q^{5} + 50 q^{7} - 57 q^{8} - 32 q^{10} - 73 q^{11} - 61 q^{13} - 243 q^{14} + 440 q^{16} - 96 q^{17} + 71 q^{19} - 277 q^{20} + 206 q^{22} - 678 q^{23} + 597 q^{25} - 335 q^{26} + 354 q^{28} - 224 q^{29} - 188 q^{31} - 502 q^{32} - 261 q^{34} - 624 q^{35} + 223 q^{37} - 737 q^{38} - 562 q^{40} - 106 q^{41} + 607 q^{43} - 767 q^{44} - 529 q^{46} - 1657 q^{47} + 848 q^{49} - 906 q^{50} + 156 q^{52} - 1902 q^{53} + 1215 q^{55} - 3540 q^{56} + 2871 q^{58} - 2790 q^{59} - 1952 q^{61} - 3134 q^{62} + 1659 q^{64} - 3653 q^{65} + 2932 q^{67} - 7363 q^{68} + 4509 q^{70} - 4157 q^{71} + 94 q^{73} - 6271 q^{74} + 2757 q^{76} - 3564 q^{77} + 2365 q^{79} - 6249 q^{80} + 2481 q^{82} - 4852 q^{83} + 640 q^{85} - 5710 q^{86} + 5291 q^{88} - 2504 q^{89} - 812 q^{91} - 9719 q^{92} - 252 q^{94} - 4999 q^{95} - 302 q^{97} - 14279 q^{98}+O(q^{100}) \) 26 * q - 6 * q^2 + 108 * q^4 - 25 * q^5 + 50 * q^7 - 57 * q^8 - 32 * q^10 - 73 * q^11 - 61 * q^13 - 243 * q^14 + 440 * q^16 - 96 * q^17 + 71 * q^19 - 277 * q^20 + 206 * q^22 - 678 * q^23 + 597 * q^25 - 335 * q^26 + 354 * q^28 - 224 * q^29 - 188 * q^31 - 502 * q^32 - 261 * q^34 - 624 * q^35 + 223 * q^37 - 737 * q^38 - 562 * q^40 - 106 * q^41 + 607 * q^43 - 767 * q^44 - 529 * q^46 - 1657 * q^47 + 848 * q^49 - 906 * q^50 + 156 * q^52 - 1902 * q^53 + 1215 * q^55 - 3540 * q^56 + 2871 * q^58 - 2790 * q^59 - 1952 * q^61 - 3134 * q^62 + 1659 * q^64 - 3653 * q^65 + 2932 * q^67 - 7363 * q^68 + 4509 * q^70 - 4157 * q^71 + 94 * q^73 - 6271 * q^74 + 2757 * q^76 - 3564 * q^77 + 2365 * q^79 - 6249 * q^80 + 2481 * q^82 - 4852 * q^83 + 640 * q^85 - 5710 * q^86 + 5291 * q^88 - 2504 * q^89 - 812 * q^91 - 9719 * q^92 - 252 * q^94 - 4999 * q^95 - 302 * q^97 - 14279 * q^98

Introduction

L-functions

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