LMFDB - Embedded newform 538.4.a.c.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [538,4,Mod(1,538)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("538.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(538, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	538.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.7430275831$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 299 x^{16} + 2191 x^{15} + 37840 x^{14} - 247598 x^{13} - 2647415 x^{12} + \cdots - 88752454191 $ x^18 - 8x^17 - 299x^16 + 2191x^15 + 37840x^14 - 247598x^13 - 2647415x^12 + 14946954x^11 + 112143744x^10 - 520080033x^9 - 2953528791x^8 + 10430700798x^7 + 47410674521x^6 - 111852486489x^5 - 425662048215x^4 + 497952330084x^3 + 1645078311180x^2 - 37397915784*x - 88752454191
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.230427$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -1.23043 q^{3} +4.00000 q^{4} -2.13428 q^{5} +2.46085 q^{6} +8.15129 q^{7} -8.00000 q^{8} -25.4860 q^{9} +4.26856 q^{10} -16.5349 q^{11} -4.92171 q^{12} +59.2303 q^{13} -16.3026 q^{14} +2.62608 q^{15} +16.0000 q^{16} +95.2369 q^{17} +50.9721 q^{18} -37.4417 q^{19} -8.53713 q^{20} -10.0296 q^{21} +33.0698 q^{22} -118.852 q^{23} +9.84342 q^{24} -120.445 q^{25} -118.461 q^{26} +64.5802 q^{27} +32.6052 q^{28} -165.730 q^{29} -5.25216 q^{30} +139.847 q^{31} -32.0000 q^{32} +20.3450 q^{33} -190.474 q^{34} -17.3971 q^{35} -101.944 q^{36} +122.795 q^{37} +74.8834 q^{38} -72.8786 q^{39} +17.0743 q^{40} +263.594 q^{41} +20.0591 q^{42} +341.053 q^{43} -66.1397 q^{44} +54.3944 q^{45} +237.703 q^{46} -264.528 q^{47} -19.6868 q^{48} -276.557 q^{49} +240.890 q^{50} -117.182 q^{51} +236.921 q^{52} +369.221 q^{53} -129.160 q^{54} +35.2902 q^{55} -65.2103 q^{56} +46.0693 q^{57} +331.461 q^{58} -664.882 q^{59} +10.5043 q^{60} -113.669 q^{61} -279.694 q^{62} -207.744 q^{63} +64.0000 q^{64} -126.414 q^{65} -40.6900 q^{66} -314.601 q^{67} +380.948 q^{68} +146.238 q^{69} +34.7943 q^{70} -995.728 q^{71} +203.888 q^{72} -1102.49 q^{73} -245.591 q^{74} +148.199 q^{75} -149.767 q^{76} -134.781 q^{77} +145.757 q^{78} -235.836 q^{79} -34.1485 q^{80} +608.662 q^{81} -527.188 q^{82} +2.23481 q^{83} -40.1183 q^{84} -203.262 q^{85} -682.106 q^{86} +203.919 q^{87} +132.279 q^{88} -1256.72 q^{89} -108.789 q^{90} +482.804 q^{91} -475.407 q^{92} -172.071 q^{93} +529.056 q^{94} +79.9111 q^{95} +39.3737 q^{96} +208.162 q^{97} +553.113 q^{98} +421.410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 36 q^{2} - 10 q^{3} + 72 q^{4} - 26 q^{5} + 20 q^{6} - 9 q^{7} - 144 q^{8} + 178 q^{9} + 52 q^{10} - 130 q^{11} - 40 q^{12} - 27 q^{13} + 18 q^{14} - 143 q^{15} + 288 q^{16} - 110 q^{17} - 356 q^{18}+ \cdots - 9455 q^{99}+O(q^{100}) $ 18 * q - 36 * q^2 - 10 * q^3 + 72 * q^4 - 26 * q^5 + 20 * q^6 - 9 * q^7 - 144 * q^8 + 178 * q^9 + 52 * q^10 - 130 * q^11 - 40 * q^12 - 27 * q^13 + 18 * q^14 - 143 * q^15 + 288 * q^16 - 110 * q^17 - 356 * q^18 - 47 * q^19 - 104 * q^20 + 69 * q^21 + 260 * q^22 - 329 * q^23 + 80 * q^24 + 442 * q^25 + 54 * q^26 - 325 * q^27 - 36 * q^28 - 3 * q^29 + 286 * q^30 + 97 * q^31 - 576 * q^32 - 129 * q^33 + 220 * q^34 - 834 * q^35 + 712 * q^36 - 104 * q^37 + 94 * q^38 - 242 * q^39 + 208 * q^40 + 196 * q^41 - 138 * q^42 - 547 * q^43 - 520 * q^44 - 481 * q^45 + 658 * q^46 - 1074 * q^47 - 160 * q^48 + 1315 * q^49 - 884 * q^50 - 1475 * q^51 - 108 * q^52 - 1681 * q^53 + 650 * q^54 + 163 * q^55 + 72 * q^56 - 1178 * q^57 + 6 * q^58 - 924 * q^59 - 572 * q^60 + 640 * q^61 - 194 * q^62 - 445 * q^63 + 1152 * q^64 - 2052 * q^65 + 258 * q^66 - 1685 * q^67 - 440 * q^68 - 1787 * q^69 + 1668 * q^70 - 3635 * q^71 - 1424 * q^72 - 1136 * q^73 + 208 * q^74 - 3332 * q^75 - 188 * q^76 - 5512 * q^77 + 484 * q^78 - 406 * q^79 - 416 * q^80 - 3194 * q^81 - 392 * q^82 - 6854 * q^83 + 276 * q^84 - 1227 * q^85 + 1094 * q^86 - 8831 * q^87 + 1040 * q^88 - 4690 * q^89 + 962 * q^90 - 5743 * q^91 - 1316 * q^92 - 4097 * q^93 + 2148 * q^94 - 3332 * q^95 + 320 * q^96 - 4132 * q^97 - 2630 * q^98 - 9455 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−1.23043	−0.236796	−0.118398	−	0.992966i	$-0.537776\pi$
$3$	−1.23043	−0.236796	−0.118398	+	0.992966i	$0.537776\pi$
$4$	4.00000	0.500000
$5$	−2.13428	−0.190896	−0.0954480	−	0.995434i	$-0.530428\pi$
$5$	−2.13428	−0.190896	−0.0954480	+	0.995434i	$0.530428\pi$
$6$	2.46085	0.167440
$7$	8.15129	0.440128	0.220064	−	0.975485i	$-0.429373\pi$
$7$	8.15129	0.440128	0.220064	+	0.975485i	$0.429373\pi$
$8$	−8.00000	−0.353553
$9$	−25.4860	−0.943928
$10$	4.26856	0.134984
$11$	−16.5349	−0.453224	−0.226612	−	0.973985i	$-0.572765\pi$
$11$	−16.5349	−0.453224	−0.226612	+	0.973985i	$0.572765\pi$
$12$	−4.92171	−0.118398
$13$	59.2303	1.26366	0.631829	−	0.775108i	$-0.282305\pi$
$13$	59.2303	1.26366	0.631829	+	0.775108i	$0.282305\pi$
$14$	−16.3026	−0.311218
$15$	2.62608	0.0452034
$16$	16.0000	0.250000
$17$	95.2369	1.35873	0.679363	−	0.733802i	$-0.262256\pi$
$17$	95.2369	1.35873	0.679363	+	0.733802i	$0.262256\pi$
$18$	50.9721	0.667458
$19$	−37.4417	−0.452090	−0.226045	−	0.974117i	$-0.572580\pi$
$19$	−37.4417	−0.452090	−0.226045	+	0.974117i	$0.572580\pi$
$20$	−8.53713	−0.0954480
$21$	−10.0296	−0.104220
$22$	33.0698	0.320478
$23$	−118.852	−1.07749	−0.538745	−	0.842469i	$-0.681101\pi$
$23$	−118.852	−1.07749	−0.538745	+	0.842469i	$0.681101\pi$
$24$	9.84342	0.0837199
$25$	−120.445	−0.963559
$26$	−118.461	−0.893541
$27$	64.5802	0.460314
$28$	32.6052	0.220064
$29$	−165.730	−1.06122	−0.530610	−	0.847616i	$-0.678037\pi$
$29$	−165.730	−1.06122	−0.530610	+	0.847616i	$0.678037\pi$
$30$	−5.25216	−0.0319636
$31$	139.847	0.810234	0.405117	−	0.914265i	$-0.367231\pi$
$31$	139.847	0.810234	0.405117	+	0.914265i	$0.367231\pi$
$32$	−32.0000	−0.176777
$33$	20.3450	0.107322
$34$	−190.474	−0.960765
$35$	−17.3971	−0.0840187
$36$	−101.944	−0.471964
$37$	122.795	0.545607	0.272803	−	0.962070i	$-0.412049\pi$
$37$	122.795	0.545607	0.272803	+	0.962070i	$0.412049\pi$
$38$	74.8834	0.319676
$39$	−72.8786	−0.299229
$40$	17.0743	0.0674919
$41$	263.594	1.00406	0.502030	−	0.864850i	$-0.332587\pi$
$41$	263.594	1.00406	0.502030	+	0.864850i	$0.332587\pi$
$42$	20.0591	0.0736950
$43$	341.053	1.20954	0.604769	−	0.796401i	$-0.293265\pi$
$43$	341.053	1.20954	0.604769	+	0.796401i	$0.293265\pi$
$44$	−66.1397	−0.226612
$45$	54.3944	0.180192
$46$	237.703	0.761901
$47$	−264.528	−0.820965	−0.410483	−	0.911868i	$-0.634640\pi$
$47$	−264.528	−0.820965	−0.410483	+	0.911868i	$0.634640\pi$
$48$	−19.6868	−0.0591989
$49$	−276.557	−0.806287
$50$	240.890	0.681339
$51$	−117.182	−0.321741
$52$	236.921	0.631829
$53$	369.221	0.956912	0.478456	−	0.878112i	$-0.341197\pi$
$53$	369.221	0.956912	0.478456	+	0.878112i	$0.341197\pi$
$54$	−129.160	−0.325491
$55$	35.2902	0.0865186
$56$	−65.2103	−0.155609
$57$	46.0693	0.107053
$58$	331.461	0.750396
$59$	−664.882	−1.46712	−0.733561	−	0.679623i	$-0.762143\pi$
$59$	−664.882	−1.46712	−0.733561	+	0.679623i	$0.762143\pi$
$60$	10.5043	0.0226017
$61$	−113.669	−0.238586	−0.119293	−	0.992859i	$-0.538063\pi$
$61$	−113.669	−0.238586	−0.119293	+	0.992859i	$0.538063\pi$
$62$	−279.694	−0.572922
$63$	−207.744	−0.415449
$64$	64.0000	0.125000
$65$	−126.414	−0.241227
$66$	−40.6900	−0.0758878
$67$	−314.601	−0.573651	−0.286825	−	0.957983i	$-0.592600\pi$
$67$	−314.601	−0.573651	−0.286825	+	0.957983i	$0.592600\pi$
$68$	380.948	0.679363
$69$	146.238	0.255145
$70$	34.7943	0.0594102
$71$	−995.728	−1.66438	−0.832191	−	0.554489i	$-0.812914\pi$
$71$	−995.728	−1.66438	−0.832191	+	0.554489i	$0.812914\pi$
$72$	203.888	0.333729
$73$	−1102.49	−1.76762	−0.883810	−	0.467847i	$-0.845030\pi$
$73$	−1102.49	−1.76762	−0.883810	+	0.467847i	$0.845030\pi$
$74$	−245.591	−0.385802
$75$	148.199	0.228167
$76$	−149.767	−0.226045
$77$	−134.781	−0.199477
$78$	145.757	0.211587
$79$	−235.836	−0.335868	−0.167934	−	0.985798i	$-0.553710\pi$
$79$	−235.836	−0.335868	−0.167934	+	0.985798i	$0.553710\pi$
$80$	−34.1485	−0.0477240
$81$	608.662	0.834927
$82$	−527.188	−0.709978
$83$	2.23481	0.00295546	0.00147773	−	0.999999i	$-0.499530\pi$
$83$	2.23481	0.00295546	0.00147773	+	0.999999i	$0.499530\pi$
$84$	−40.1183	−0.0521102
$85$	−203.262	−0.259375
$86$	−682.106	−0.855272
$87$	203.919	0.251292
$88$	132.279	0.160239
$89$	−1256.72	−1.49677	−0.748384	−	0.663266i	$-0.769170\pi$
$89$	−1256.72	−1.49677	−0.748384	+	0.663266i	$0.769170\pi$
$90$	−108.789	−0.127415
$91$	482.804	0.556171
$92$	−475.407	−0.538745
$93$	−172.071	−0.191860
$94$	529.056	0.580510
$95$	79.9111	0.0863022
$96$	39.3737	0.0418600
$97$	208.162	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$97$	208.162	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$98$	553.113	0.570131
$99$	421.410	0.427811
$100$	−481.779	−0.481779
$101$	−935.092	−0.921239	−0.460619	−	0.887598i	$-0.652373\pi$
$101$	−935.092	−0.921239	−0.460619	+	0.887598i	$0.652373\pi$
$102$	234.364	0.227505
$103$	739.654	0.707575	0.353788	−	0.935326i	$-0.384894\pi$
$103$	739.654	0.707575	0.353788	+	0.935326i	$0.384894\pi$
$104$	−473.843	−0.446770
$105$	21.4059	0.0198953
$106$	−738.441	−0.676639
$107$	−1281.40	−1.15773	−0.578867	−	0.815422i	$-0.696505\pi$
$107$	−1281.40	−1.15773	−0.578867	+	0.815422i	$0.696505\pi$
$108$	258.321	0.230157
$109$	−1507.80	−1.32496	−0.662482	−	0.749078i	$-0.730497\pi$
$109$	−1507.80	−1.32496	−0.662482	+	0.749078i	$0.730497\pi$
$110$	−70.5803	−0.0611779
$111$	−151.091	−0.129197
$112$	130.421	0.110032
$113$	−425.931	−0.354586	−0.177293	−	0.984158i	$-0.556734\pi$
$113$	−425.931	−0.354586	−0.177293	+	0.984158i	$0.556734\pi$
$114$	−92.1386	−0.0756979
$115$	253.663	0.205689
$116$	−662.922	−0.530610
$117$	−1509.55	−1.19280
$118$	1329.76	1.03741
$119$	776.304	0.598014
$120$	−21.0086	−0.0159818
$121$	−1057.60	−0.794588
$122$	227.337	0.168706
$123$	−324.333	−0.237757
$124$	559.388	0.405117
$125$	523.848	0.374835
$126$	415.488	0.293767
$127$	1079.91	0.754541	0.377270	−	0.926103i	$-0.376863\pi$
$127$	1079.91	0.754541	0.377270	+	0.926103i	$0.376863\pi$
$128$	−128.000	−0.0883883
$129$	−419.641	−0.286413
$130$	252.828	0.170573
$131$	−2272.32	−1.51553	−0.757763	−	0.652530i	$-0.773708\pi$
$131$	−2272.32	−1.51553	−0.757763	+	0.652530i	$0.773708\pi$
$132$	81.3800	0.0536608
$133$	−305.198	−0.198978
$134$	629.202	0.405632
$135$	−137.832	−0.0878721
$136$	−761.895	−0.480382
$137$	1819.26	1.13452	0.567262	−	0.823538i	$-0.308003\pi$
$137$	1819.26	1.13452	0.567262	+	0.823538i	$0.308003\pi$
$138$	−292.477	−0.180415
$139$	25.0859	0.0153076	0.00765381	−	0.999971i	$-0.497564\pi$
$139$	25.0859	0.0153076	0.00765381	+	0.999971i	$0.497564\pi$
$140$	−69.5886	−0.0420093
$141$	325.482	0.194401
$142$	1991.46	1.17690
$143$	−979.369	−0.572720
$144$	−407.777	−0.235982
$145$	353.715	0.202583
$146$	2204.97	1.24990
$147$	340.283	0.190925
$148$	491.182	0.272803
$149$	−1957.09	−1.07605	−0.538023	−	0.842930i	$-0.680828\pi$
$149$	−1957.09	−1.07605	−0.538023	+	0.842930i	$0.680828\pi$
$150$	−296.397	−0.161338
$151$	643.252	0.346669	0.173335	−	0.984863i	$-0.444546\pi$
$151$	643.252	0.346669	0.173335	+	0.984863i	$0.444546\pi$
$152$	299.534	0.159838
$153$	−2427.21	−1.28254
$154$	269.562	0.141051
$155$	−298.473	−0.154670
$156$	−291.514	−0.149614
$157$	−2697.68	−1.37132	−0.685662	−	0.727920i	$-0.740487\pi$
$157$	−2697.68	−1.37132	−0.685662	+	0.727920i	$0.740487\pi$
$158$	471.671	0.237495
$159$	−454.299	−0.226593
$160$	68.2970	0.0337460
$161$	−968.794	−0.474234
$162$	−1217.32	−0.590383
$163$	650.005	0.312346	0.156173	−	0.987730i	$-0.450084\pi$
$163$	650.005	0.312346	0.156173	+	0.987730i	$0.450084\pi$
$164$	1054.38	0.502030
$165$	−43.4220	−0.0204872
$166$	−4.46963	−0.00208982
$167$	−2899.78	−1.34366	−0.671830	−	0.740705i	$-0.734492\pi$
$167$	−2899.78	−1.34366	−0.671830	+	0.740705i	$0.734492\pi$
$168$	80.2365	0.0368475
$169$	1311.23	0.596829
$170$	406.525	0.183406
$171$	954.241	0.426741
$172$	1364.21	0.604769
$173$	−383.602	−0.168582	−0.0842911	−	0.996441i	$-0.526863\pi$
$173$	−383.602	−0.168582	−0.0842911	+	0.996441i	$0.526863\pi$
$174$	−407.838	−0.177691
$175$	−981.781	−0.424089
$176$	−264.559	−0.113306
$177$	818.089	0.347409
$178$	2513.44	1.05837
$179$	3302.17	1.37886	0.689430	−	0.724353i	$-0.257861\pi$
$179$	3302.17	1.37886	0.689430	+	0.724353i	$0.257861\pi$
$180$	217.578	0.0900960
$181$	3337.10	1.37041	0.685206	−	0.728349i	$-0.259712\pi$
$181$	3337.10	1.37041	0.685206	+	0.728349i	$0.259712\pi$
$182$	−965.607	−0.393272
$183$	139.861	0.0564962
$184$	950.813	0.380950
$185$	−262.080	−0.104154
$186$	344.143	0.135666
$187$	−1574.73	−0.615808
$188$	−1058.11	−0.410483
$189$	526.412	0.202597
$190$	−159.822	−0.0610249
$191$	−1331.17	−0.504293	−0.252147	−	0.967689i	$-0.581137\pi$
$191$	−1331.17	−0.504293	−0.252147	+	0.967689i	$0.581137\pi$
$192$	−78.7473	−0.0295995
$193$	3529.85	1.31650	0.658249	−	0.752800i	$-0.271297\pi$
$193$	3529.85	1.31650	0.658249	+	0.752800i	$0.271297\pi$
$194$	−416.325	−0.154074
$195$	155.543	0.0571215
$196$	−1106.23	−0.403144
$197$	3747.40	1.35529	0.677643	−	0.735391i	$-0.263002\pi$
$197$	3747.40	1.35529	0.677643	+	0.735391i	$0.263002\pi$
$198$	−842.819	−0.302508
$199$	−4548.90	−1.62042	−0.810209	−	0.586142i	$-0.800646\pi$
$199$	−4548.90	−1.62042	−0.810209	+	0.586142i	$0.800646\pi$
$200$	963.559	0.340669
$201$	387.093	0.135838
$202$	1870.18	0.651414
$203$	−1350.92	−0.467073
$204$	−468.728	−0.160870
$205$	−562.584	−0.191671
$206$	−1479.31	−0.500331
$207$	3029.06	1.01707
$208$	947.686	0.315914
$209$	619.095	0.204898
$210$	−42.8118	−0.0140681
$211$	−156.427	−0.0510373	−0.0255186	−	0.999674i	$-0.508124\pi$
$211$	−156.427	−0.0510373	−0.0255186	+	0.999674i	$0.508124\pi$
$212$	1476.88	0.478456
$213$	1225.17	0.394119
$214$	2562.80	0.818641
$215$	−727.904	−0.230896
$216$	−516.642	−0.162746
$217$	1139.93	0.356607
$218$	3015.60	0.936891
$219$	1356.53	0.418565
$220$	141.161	0.0432593
$221$	5640.92	1.71696
$222$	302.182	0.0913564
$223$	2522.13	0.757372	0.378686	−	0.925525i	$-0.376376\pi$
$223$	2522.13	0.757372	0.378686	+	0.925525i	$0.376376\pi$
$224$	−260.841	−0.0778044
$225$	3069.66	0.909530
$226$	851.862	0.250730
$227$	−847.591	−0.247826	−0.123913	−	0.992293i	$-0.539544\pi$
$227$	−847.591	−0.247826	−0.123913	+	0.992293i	$0.539544\pi$
$228$	184.277	0.0535265
$229$	−1771.22	−0.511115	−0.255557	−	0.966794i	$-0.582259\pi$
$229$	−1771.22	−0.511115	−0.255557	+	0.966794i	$0.582259\pi$
$230$	−507.326	−0.145444
$231$	165.838	0.0472352
$232$	1325.84	0.375198
$233$	2763.35	0.776965	0.388482	−	0.921456i	$-0.372999\pi$
$233$	2763.35	0.776965	0.388482	+	0.921456i	$0.372999\pi$
$234$	3019.10	0.843438
$235$	564.577	0.156719
$236$	−2659.53	−0.733561
$237$	290.178	0.0795321
$238$	−1552.61	−0.422860
$239$	661.640	0.179071	0.0895354	−	0.995984i	$-0.471462\pi$
$239$	661.640	0.179071	0.0895354	+	0.995984i	$0.471462\pi$
$240$	42.0172	0.0113008
$241$	1545.53	0.413096	0.206548	−	0.978436i	$-0.433777\pi$
$241$	1545.53	0.413096	0.206548	+	0.978436i	$0.433777\pi$
$242$	2115.19	0.561859
$243$	−2492.58	−0.658021
$244$	−454.674	−0.119293
$245$	590.249	0.153917
$246$	648.667	0.168120
$247$	−2217.68	−0.571287
$248$	−1118.78	−0.286461
$249$	−2.74978	−0.000699839 0
$250$	−1047.70	−0.265049
$251$	4065.59	1.02238	0.511191	−	0.859467i	$-0.329204\pi$
$251$	4065.59	1.02238	0.511191	+	0.859467i	$0.329204\pi$
$252$	−830.976	−0.207725
$253$	1965.20	0.488345
$254$	−2159.82	−0.533541
$255$	250.100	0.0614190
$256$	256.000	0.0625000
$257$	5024.75	1.21959	0.609796	−	0.792558i	$-0.291251\pi$
$257$	5024.75	1.21959	0.609796	+	0.792558i	$0.291251\pi$
$258$	839.282	0.202525
$259$	1000.94	0.240137
$260$	−505.657	−0.120614
$261$	4223.81	1.00171
$262$	4544.65	1.07164
$263$	−8270.68	−1.93913	−0.969567	−	0.244826i	$-0.921269\pi$
$263$	−8270.68	−1.93913	−0.969567	+	0.244826i	$0.921269\pi$
$264$	−162.760	−0.0379439
$265$	−788.021	−0.182671
$266$	610.396	0.140698
$267$	1546.31	0.354428
$268$	−1258.40	−0.286825
$269$	269.000	0.0609711
$269$	269.000	0.0609711
$270$	275.665	0.0621349
$271$	−7038.15	−1.57763	−0.788813	−	0.614633i	$-0.789304\pi$
$271$	−7038.15	−1.57763	−0.788813	+	0.614633i	$0.789304\pi$
$272$	1523.79	0.339682
$273$	−594.055	−0.131699
$274$	−3638.52	−0.802229
$275$	1991.55	0.436708
$276$	584.953	0.127573
$277$	−1318.88	−0.286078	−0.143039	−	0.989717i	$-0.545687\pi$
$277$	−1318.88	−0.286078	−0.143039	+	0.989717i	$0.545687\pi$
$278$	−50.1718	−0.0108241
$279$	−3564.15	−0.764802
$280$	139.177	0.0297051
$281$	4708.52	0.999597	0.499798	−	0.866142i	$-0.333407\pi$
$281$	4708.52	0.999597	0.499798	+	0.866142i	$0.333407\pi$
$282$	−650.965	−0.137462
$283$	2780.25	0.583989	0.291995	−	0.956420i	$-0.405681\pi$
$283$	2780.25	0.583989	0.291995	+	0.956420i	$0.405681\pi$
$284$	−3982.91	−0.832191
$285$	−98.3248	−0.0204360
$286$	1958.74	0.404974
$287$	2148.63	0.441915
$288$	815.554	0.166864
$289$	4157.07	0.846137
$290$	−707.431	−0.143247
$291$	−256.129	−0.0515963
$292$	−4409.94	−0.883810
$293$	−8323.53	−1.65961	−0.829805	−	0.558053i	$-0.811548\pi$
$293$	−8323.53	−1.65961	−0.829805	+	0.558053i	$0.811548\pi$
$294$	−680.565	−0.135005
$295$	1419.05	0.280068
$296$	−982.364	−0.192901
$297$	−1067.83	−0.208625
$298$	3914.17	0.760879
$299$	−7039.62	−1.36158
$300$	592.794	0.114083
$301$	2780.02	0.532352
$302$	−1286.50	−0.245132
$303$	1150.56	0.218145
$304$	−599.067	−0.113023
$305$	242.601	0.0455452
$306$	4854.43	0.906892
$307$	−2964.43	−0.551105	−0.275552	−	0.961286i	$-0.588861\pi$
$307$	−2964.43	−0.551105	−0.275552	+	0.961286i	$0.588861\pi$
$308$	−539.123	−0.0997383
$309$	−910.090	−0.167551
$310$	596.946	0.109368
$311$	2667.86	0.486432	0.243216	−	0.969972i	$-0.421798\pi$
$311$	2667.86	0.486432	0.243216	+	0.969972i	$0.421798\pi$
$312$	583.029	0.105793
$313$	4965.94	0.896778	0.448389	−	0.893839i	$-0.351998\pi$
$313$	4965.94	0.896778	0.448389	+	0.893839i	$0.351998\pi$
$314$	5395.35	0.969673
$315$	443.384	0.0793076
$316$	−943.342	−0.167934
$317$	−8792.08	−1.55777	−0.778884	−	0.627168i	$-0.784214\pi$
$317$	−8792.08	−1.55777	−0.778884	+	0.627168i	$0.784214\pi$
$318$	908.598	0.160225
$319$	2740.34	0.480970
$320$	−136.594	−0.0238620
$321$	1576.67	0.274146
$322$	1937.59	0.335334
$323$	−3565.83	−0.614267
$324$	2434.65	0.417464
$325$	−7133.99	−1.21761
$326$	−1300.01	−0.220862
$327$	1855.24	0.313746
$328$	−2108.75	−0.354989
$329$	−2156.24	−0.361330
$330$	86.8439	0.0144867
$331$	11409.5	1.89463	0.947314	−	0.320305i	$-0.103786\pi$
$331$	11409.5	1.89463	0.947314	+	0.320305i	$0.103786\pi$
$332$	8.93926	0.00147773
$333$	−3129.57	−0.515014
$334$	5799.55	0.950112
$335$	671.447	0.109508
$336$	−160.473	−0.0260551
$337$	2233.26	0.360989	0.180495	−	0.983576i	$-0.442230\pi$
$337$	2233.26	0.360989	0.180495	+	0.983576i	$0.442230\pi$
$338$	−2622.47	−0.422022
$339$	524.077	0.0839645
$340$	−813.050	−0.129688
$341$	−2312.36	−0.367218
$342$	−1908.48	−0.301751
$343$	−5050.18	−0.794998
$344$	−2728.43	−0.427636
$345$	−312.114	−0.0487062
$346$	767.204	0.119206
$347$	−10816.1	−1.67330	−0.836651	−	0.547736i	$-0.815490\pi$
$347$	−10816.1	−1.67330	−0.836651	+	0.547736i	$0.815490\pi$
$348$	815.677	0.125646
$349$	10411.5	1.59689	0.798445	−	0.602068i	$-0.205656\pi$
$349$	10411.5	1.59689	0.798445	+	0.602068i	$0.205656\pi$
$350$	1963.56	0.299876
$351$	3825.11	0.581679
$352$	529.117	0.0801195
$353$	7610.41	1.14748	0.573742	−	0.819036i	$-0.305491\pi$
$353$	7610.41	1.14748	0.573742	+	0.819036i	$0.305491\pi$
$354$	−1636.18	−0.245655
$355$	2125.16	0.317724
$356$	−5026.89	−0.748384
$357$	−955.185	−0.141607
$358$	−6604.34	−0.975001
$359$	2551.23	0.375067	0.187533	−	0.982258i	$-0.439951\pi$
$359$	2551.23	0.375067	0.187533	+	0.982258i	$0.439951\pi$
$360$	−435.155	−0.0637075
$361$	−5457.12	−0.795614
$362$	−6674.20	−0.969028
$363$	1301.30	0.188155
$364$	1931.21	0.278086
$365$	2353.02	0.337431
$366$	−279.722	−0.0399489
$367$	−4050.30	−0.576087	−0.288044	−	0.957617i	$-0.593005\pi$
$367$	−4050.30	−0.576087	−0.288044	+	0.957617i	$0.593005\pi$
$368$	−1901.63	−0.269373
$369$	−6717.97	−0.947761
$370$	524.160	0.0736481
$371$	3009.62	0.421164
$372$	−688.286	−0.0959300
$373$	5732.09	0.795701	0.397851	−	0.917450i	$-0.369756\pi$
$373$	5732.09	0.795701	0.397851	+	0.917450i	$0.369756\pi$
$374$	3149.47	0.435442
$375$	−644.557	−0.0887594
$376$	2116.22	0.290255
$377$	−9816.27	−1.34102
$378$	−1052.82	−0.143258
$379$	195.715	0.0265257	0.0132628	−	0.999912i	$-0.495778\pi$
$379$	195.715	0.0265257	0.0132628	+	0.999912i	$0.495778\pi$
$380$	319.645	0.0431511
$381$	−1328.75	−0.178672
$382$	2662.34	0.356589
$383$	−2145.44	−0.286233	−0.143116	−	0.989706i	$-0.545712\pi$
$383$	−2145.44	−0.286233	−0.143116	+	0.989706i	$0.545712\pi$
$384$	157.495	0.0209300
$385$	287.660	0.0380793
$386$	−7059.70	−0.930904
$387$	−8692.10	−1.14172
$388$	832.650	0.108947
$389$	7821.65	1.01947	0.509734	−	0.860332i	$-0.329744\pi$
$389$	7821.65	1.01947	0.509734	+	0.860332i	$0.329744\pi$
$390$	−311.087	−0.0403910
$391$	−11319.1	−1.46401
$392$	2212.45	0.285066
$393$	2795.93	0.358870
$394$	−7494.80	−0.958331
$395$	503.339	0.0641158
$396$	1685.64	0.213905
$397$	6997.18	0.884580	0.442290	−	0.896872i	$-0.354166\pi$
$397$	6997.18	0.884580	0.442290	+	0.896872i	$0.354166\pi$
$398$	9097.80	1.14581
$399$	375.524	0.0471171
$400$	−1927.12	−0.240890
$401$	−7213.18	−0.898277	−0.449139	−	0.893462i	$-0.648269\pi$
$401$	−7213.18	−0.898277	−0.449139	+	0.893462i	$0.648269\pi$
$402$	−774.187	−0.0960520
$403$	8283.18	1.02386
$404$	−3740.37	−0.460619
$405$	−1299.06	−0.159384
$406$	2701.83	0.330270
$407$	−2030.41	−0.247282
$408$	937.457	0.113753
$409$	11412.1	1.37968	0.689842	−	0.723960i	$-0.257680\pi$
$409$	11412.1	1.37968	0.689842	+	0.723960i	$0.257680\pi$
$410$	1125.17	0.135532
$411$	−2238.46	−0.268650
$412$	2958.62	0.353788
$413$	−5419.64	−0.645722
$414$	−6058.12	−0.719179
$415$	−4.76972	−0.000564184 0
$416$	−1895.37	−0.223385
$417$	−30.8664	−0.00362478
$418$	−1238.19	−0.144885
$419$	3380.29	0.394124	0.197062	−	0.980391i	$-0.436860\pi$
$419$	3380.29	0.394124	0.197062	+	0.980391i	$0.436860\pi$
$420$	85.6237	0.00994763
$421$	12950.8	1.49925	0.749623	−	0.661865i	$-0.230235\pi$
$421$	12950.8	1.49925	0.749623	+	0.661865i	$0.230235\pi$
$422$	312.854	0.0360888
$423$	6741.77	0.774932
$424$	−2953.76	−0.338320
$425$	−11470.8	−1.30921
$426$	−2450.34	−0.278684
$427$	−926.545	−0.105009
$428$	−5125.59	−0.578867
$429$	1205.04	0.135618
$430$	1455.81	0.163268
$431$	834.923	0.0933105	0.0466553	−	0.998911i	$-0.485144\pi$
$431$	834.923	0.0933105	0.0466553	+	0.998911i	$0.485144\pi$
$432$	1033.28	0.115078
$433$	−10245.5	−1.13711	−0.568553	−	0.822647i	$-0.692497\pi$
$433$	−10245.5	−1.13711	−0.568553	+	0.822647i	$0.692497\pi$
$434$	−2279.87	−0.252159
$435$	−435.221	−0.0479707
$436$	−6031.20	−0.662482
$437$	4450.01	0.487123
$438$	−2713.06	−0.295970
$439$	12080.6	1.31338	0.656690	−	0.754161i	$-0.271956\pi$
$439$	12080.6	1.31338	0.656690	+	0.754161i	$0.271956\pi$
$440$	−282.321	−0.0305890
$441$	7048.33	0.761077
$442$	−11281.8	−1.21408
$443$	−8754.41	−0.938905	−0.469452	−	0.882958i	$-0.655549\pi$
$443$	−8754.41	−0.938905	−0.469452	+	0.882958i	$0.655549\pi$
$444$	−604.364	−0.0645987
$445$	2682.20	0.285727
$446$	−5044.25	−0.535543
$447$	2408.05	0.254803
$448$	521.682	0.0550160
$449$	13118.5	1.37884	0.689419	−	0.724362i	$-0.257866\pi$
$449$	13118.5	1.37884	0.689419	+	0.724362i	$0.257866\pi$
$450$	−6139.33	−0.643135
$451$	−4358.51	−0.455065
$452$	−1703.72	−0.177293
$453$	−791.474	−0.0820898
$454$	1695.18	0.175240
$455$	−1030.44	−0.106171
$456$	−368.554	−0.0378490
$457$	17317.2	1.77257	0.886285	−	0.463140i	$-0.153277\pi$
$457$	17317.2	1.77257	0.886285	+	0.463140i	$0.153277\pi$
$458$	3542.43	0.361413
$459$	6150.42	0.625441
$460$	1014.65	0.102844
$461$	−12118.4	−1.22431	−0.612157	−	0.790736i	$-0.709698\pi$
$461$	−12118.4	−1.22431	−0.612157	+	0.790736i	$0.709698\pi$
$462$	−331.676	−0.0334004
$463$	−10836.1	−1.08768	−0.543841	−	0.839188i	$-0.683031\pi$
$463$	−10836.1	−1.08768	−0.543841	+	0.839188i	$0.683031\pi$
$464$	−2651.69	−0.265305
$465$	367.249	0.0366253
$466$	−5526.69	−0.549397
$467$	−17119.4	−1.69634	−0.848172	−	0.529721i	$-0.822297\pi$
$467$	−17119.4	−1.69634	−0.848172	+	0.529721i	$0.822297\pi$
$468$	−6038.19	−0.596401
$469$	−2564.40	−0.252480
$470$	−1129.15	−0.110817
$471$	3319.29	0.324724
$472$	5319.06	0.518706
$473$	−5639.29	−0.548192
$474$	−580.357	−0.0562377
$475$	4509.66	0.435616
$476$	3105.21	0.299007
$477$	−9409.97	−0.903256
$478$	−1323.28	−0.126622
$479$	4707.84	0.449075	0.224537	−	0.974465i	$-0.427913\pi$
$479$	4707.84	0.449075	0.224537	+	0.974465i	$0.427913\pi$
$480$	−84.0345	−0.00799090
$481$	7273.22	0.689460
$482$	−3091.05	−0.292103
$483$	1192.03	0.112297
$484$	−4230.39	−0.397294
$485$	−444.277	−0.0415950
$486$	4985.16	0.465291
$487$	−15440.3	−1.43669	−0.718344	−	0.695688i	$-0.755100\pi$
$487$	−15440.3	−1.43669	−0.718344	+	0.695688i	$0.755100\pi$
$488$	909.349	0.0843530
$489$	−799.784	−0.0739621
$490$	−1180.50	−0.108836
$491$	−10836.9	−0.996056	−0.498028	−	0.867161i	$-0.665942\pi$
$491$	−10836.9	−0.996056	−0.498028	+	0.867161i	$0.665942\pi$
$492$	−1297.33	−0.118879
$493$	−15783.7	−1.44191
$494$	4435.37	0.403961
$495$	−899.407	−0.0816674
$496$	2237.55	0.202559
$497$	−8116.46	−0.732542
$498$	5.49955	0.000494861 0
$499$	−7847.68	−0.704029	−0.352014	−	0.935995i	$-0.614503\pi$
$499$	−7847.68	−0.704029	−0.352014	+	0.935995i	$0.614503\pi$
$500$	2095.39	0.187418
$501$	3567.96	0.318173
$502$	−8131.19	−0.722933
$503$	−15376.6	−1.36304	−0.681521	−	0.731798i	$-0.738681\pi$
$503$	−15376.6	−1.36304	−0.681521	+	0.731798i	$0.738681\pi$
$504$	1661.95	0.146883
$505$	1995.75	0.175861
$506$	−3930.40	−0.345312
$507$	−1613.38	−0.141327
$508$	4319.65	0.377270
$509$	−6273.95	−0.546342	−0.273171	−	0.961966i	$-0.588072\pi$
$509$	−6273.95	−0.546342	−0.273171	+	0.961966i	$0.588072\pi$
$510$	−500.199	−0.0434298
$511$	−8986.68	−0.777979
$512$	−512.000	−0.0441942
$513$	−2417.99	−0.208103
$514$	−10049.5	−0.862382
$515$	−1578.63	−0.135073
$516$	−1678.56	−0.143207
$517$	4373.95	0.372081
$518$	−2001.88	−0.169802
$519$	471.994	0.0399196
$520$	1011.31	0.0852866
$521$	3752.22	0.315524	0.157762	−	0.987477i	$-0.449572\pi$
$521$	3752.22	0.315524	0.157762	+	0.987477i	$0.449572\pi$
$522$	−8447.63	−0.708319
$523$	−10006.4	−0.836615	−0.418308	−	0.908305i	$-0.637377\pi$
$523$	−10006.4	−0.836615	−0.418308	+	0.908305i	$0.637377\pi$
$524$	−9089.30	−0.757763
$525$	1208.01	0.100423
$526$	16541.4	1.37117
$527$	13318.6	1.10089
$528$	325.520	0.0268304
$529$	1958.72	0.160986
$530$	1576.04	0.129168
$531$	16945.2	1.38486
$532$	−1220.79	−0.0994888
$533$	15612.8	1.26879
$534$	−3092.61	−0.250619
$535$	2734.87	0.221007
$536$	2516.81	0.202816
$537$	−4063.08	−0.326508
$538$	−538.000	−0.0431131
$539$	4572.84	0.365429
$540$	−551.330	−0.0439360
$541$	−14744.0	−1.17170	−0.585852	−	0.810418i	$-0.699240\pi$
$541$	−14744.0	−1.17170	−0.585852	+	0.810418i	$0.699240\pi$
$542$	14076.3	1.11555
$543$	−4106.06	−0.324508
$544$	−3047.58	−0.240191
$545$	3218.07	0.252930
$546$	1188.11	0.0931252
$547$	16000.4	1.25069	0.625347	−	0.780347i	$-0.284958\pi$
$547$	16000.4	1.25069	0.625347	+	0.780347i	$0.284958\pi$
$548$	7277.03	0.567262
$549$	2896.96	0.225208
$550$	−3983.09	−0.308799
$551$	6205.23	0.479767
$552$	−1169.91	−0.0902075
$553$	−1922.36	−0.147825
$554$	2637.76	0.202288
$555$	322.471	0.0246633
$556$	100.344	0.00765381
$557$	9128.35	0.694400	0.347200	−	0.937791i	$-0.387133\pi$
$557$	9128.35	0.694400	0.347200	+	0.937791i	$0.387133\pi$
$558$	7128.29	0.540797
$559$	20200.7	1.52844
$560$	−278.354	−0.0210047
$561$	1937.60	0.145821
$562$	−9417.04	−0.706822
$563$	1825.29	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$563$	1825.29	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$564$	1301.93	0.0972005
$565$	909.057	0.0676890
$566$	−5560.51	−0.412943
$567$	4961.38	0.367475
$568$	7965.82	0.588448
$569$	−7517.79	−0.553888	−0.276944	−	0.960886i	$-0.589322\pi$
$569$	−7517.79	−0.553888	−0.276944	+	0.960886i	$0.589322\pi$
$570$	196.650	0.0144504
$571$	12799.0	0.938044	0.469022	−	0.883186i	$-0.344606\pi$
$571$	12799.0	0.938044	0.469022	+	0.883186i	$0.344606\pi$
$572$	−3917.48	−0.286360
$573$	1637.91	0.119415
$574$	−4297.26	−0.312481
$575$	14315.1	1.03823
$576$	−1631.11	−0.117991
$577$	−15147.2	−1.09287	−0.546435	−	0.837501i	$-0.684016\pi$
$577$	−15147.2	−1.09287	−0.546435	+	0.837501i	$0.684016\pi$
$578$	−8314.15	−0.598309
$579$	−4343.22	−0.311741
$580$	1414.86	0.101291
$581$	18.2166	0.00130078
$582$	512.257	0.0364841
$583$	−6105.03	−0.433696
$584$	8819.89	0.624948
$585$	3221.80	0.227701
$586$	16647.1	1.17352
$587$	−12803.4	−0.900258	−0.450129	−	0.892964i	$-0.648622\pi$
$587$	−12803.4	−0.900258	−0.450129	+	0.892964i	$0.648622\pi$
$588$	1361.13	0.0954627
$589$	−5236.11	−0.366299
$590$	−2838.09	−0.198038
$591$	−4610.90	−0.320926
$592$	1964.73	0.136402
$593$	23807.7	1.64868	0.824338	−	0.566098i	$-0.191548\pi$
$593$	23807.7	1.64868	0.824338	+	0.566098i	$0.191548\pi$
$594$	2135.66	0.147520
$595$	−1656.85	−0.114158
$596$	−7828.34	−0.538023
$597$	5597.09	0.383708
$598$	14079.2	0.962781
$599$	7322.84	0.499505	0.249752	−	0.968310i	$-0.419651\pi$
$599$	7322.84	0.499505	0.249752	+	0.968310i	$0.419651\pi$
$600$	−1185.59	−0.0806691
$601$	−3911.17	−0.265457	−0.132729	−	0.991152i	$-0.542374\pi$
$601$	−3911.17	−0.265457	−0.132729	+	0.991152i	$0.542374\pi$
$602$	−5560.05	−0.376430
$603$	8017.93	0.541485
$604$	2573.01	0.173335
$605$	2257.21	0.151684
$606$	−2301.12	−0.154252
$607$	−21163.3	−1.41514	−0.707572	−	0.706642i	$-0.750209\pi$
$607$	−21163.3	−1.41514	−0.707572	+	0.706642i	$0.750209\pi$
$608$	1198.13	0.0799190
$609$	1662.20	0.110601
$610$	−485.201	−0.0322053
$611$	−15668.1	−1.03742
$612$	−9708.85	−0.641270
$613$	3201.80	0.210962	0.105481	−	0.994421i	$-0.466362\pi$
$613$	3201.80	0.210962	0.105481	+	0.994421i	$0.466362\pi$
$614$	5928.87	0.389690
$615$	692.219	0.0453869
$616$	1078.25	0.0705257
$617$	−1824.73	−0.119061	−0.0595306	−	0.998226i	$-0.518960\pi$
$617$	−1824.73	−0.119061	−0.0595306	+	0.998226i	$0.518960\pi$
$618$	1820.18	0.118476
$619$	−15545.2	−1.00939	−0.504696	−	0.863297i	$-0.668395\pi$
$619$	−15545.2	−1.00939	−0.504696	+	0.863297i	$0.668395\pi$
$620$	−1193.89	−0.0773352
$621$	−7675.47	−0.495984
$622$	−5335.72	−0.343959
$623$	−10243.9	−0.658770
$624$	−1166.06	−0.0748072
$625$	13937.6	0.892004
$626$	−9931.88	−0.634118
$627$	−761.752	−0.0485190
$628$	−10790.7	−0.685662
$629$	11694.7	0.741330
$630$	−886.769	−0.0560789
$631$	18701.6	1.17987	0.589934	−	0.807451i	$-0.299154\pi$
$631$	18701.6	1.17987	0.589934	+	0.807451i	$0.299154\pi$
$632$	1886.68	0.118747
$633$	192.472	0.0120854
$634$	17584.2	1.10151
$635$	−2304.84	−0.144039
$636$	−1817.20	−0.113296
$637$	−16380.5	−1.01887
$638$	−5480.68	−0.340097
$639$	25377.2	1.57106
$640$	273.188	0.0168730
$641$	−2403.44	−0.148097	−0.0740484	−	0.997255i	$-0.523592\pi$
$641$	−2403.44	−0.148097	−0.0740484	+	0.997255i	$0.523592\pi$
$642$	−3153.34	−0.193851
$643$	−9633.29	−0.590824	−0.295412	−	0.955370i	$-0.595457\pi$
$643$	−9633.29	−0.590824	−0.295412	+	0.955370i	$0.595457\pi$
$644$	−3875.18	−0.237117
$645$	895.632	0.0546752
$646$	7131.67	0.434352
$647$	−8284.08	−0.503370	−0.251685	−	0.967809i	$-0.580985\pi$
$647$	−8284.08	−0.503370	−0.251685	+	0.967809i	$0.580985\pi$
$648$	−4869.30	−0.295191
$649$	10993.8	0.664935
$650$	14268.0	0.860979
$651$	−1402.60	−0.0844430
$652$	2600.02	0.156173
$653$	2412.64	0.144585	0.0722924	−	0.997383i	$-0.476969\pi$
$653$	2412.64	0.144585	0.0722924	+	0.997383i	$0.476969\pi$
$654$	−3710.48	−0.221852
$655$	4849.78	0.289308
$656$	4217.51	0.251015
$657$	28098.0	1.66850
$658$	4312.49	0.255499
$659$	−26634.0	−1.57437	−0.787187	−	0.616715i	$-0.788463\pi$
$659$	−26634.0	−1.57437	−0.787187	+	0.616715i	$0.788463\pi$
$660$	−173.688	−0.0102436
$661$	25991.1	1.52940	0.764702	−	0.644385i	$-0.222886\pi$
$661$	25991.1	1.52940	0.764702	+	0.644385i	$0.222886\pi$
$662$	−22819.0	−1.33970
$663$	−6940.74	−0.406570
$664$	−17.8785	−0.00104491
$665$	651.379	0.0379840
$666$	6259.14	0.364170
$667$	19697.3	1.14345
$668$	−11599.1	−0.671830
$669$	−3103.29	−0.179343
$670$	−1342.89	−0.0774336
$671$	1879.50	0.108133
$672$	320.946	0.0184238
$673$	−12627.3	−0.723251	−0.361626	−	0.932323i	$-0.617778\pi$
$673$	−12627.3	−0.723251	−0.361626	+	0.932323i	$0.617778\pi$
$674$	−4466.52	−0.255258
$675$	−7778.36	−0.443539
$676$	5244.94	0.298415
$677$	27501.3	1.56124	0.780622	−	0.625003i	$-0.214902\pi$
$677$	27501.3	1.56124	0.780622	+	0.625003i	$0.214902\pi$
$678$	−1048.15	−0.0593718
$679$	1696.79	0.0959012
$680$	1626.10	0.0917030
$681$	1042.90	0.0586842
$682$	4624.72	0.259662
$683$	−28416.2	−1.59197	−0.795986	−	0.605315i	$-0.793047\pi$
$683$	−28416.2	−1.59197	−0.795986	+	0.605315i	$0.793047\pi$
$684$	3816.96	0.213370
$685$	−3882.81	−0.216576
$686$	10100.4	0.562148
$687$	2179.35	0.121030
$688$	5456.85	0.302384
$689$	21869.1	1.20921
$690$	624.227	0.0344405
$691$	20800.5	1.14513	0.572567	−	0.819858i	$-0.305948\pi$
$691$	20800.5	1.14513	0.572567	+	0.819858i	$0.305948\pi$
$692$	−1534.41	−0.0842911
$693$	3435.03	0.188292
$694$	21632.1	1.18320
$695$	−53.5404	−0.00292216
$696$	−1631.35	−0.0888453
$697$	25103.9	1.36424
$698$	−20823.0	−1.12917
$699$	−3400.10	−0.183982
$700$	−3927.12	−0.212045
$701$	−11944.7	−0.643573	−0.321786	−	0.946812i	$-0.604283\pi$
$701$	−11944.7	−0.643573	−0.321786	+	0.946812i	$0.604283\pi$
$702$	−7650.22	−0.411309
$703$	−4597.67	−0.246664
$704$	−1058.23	−0.0566530
$705$	−694.671	−0.0371104
$706$	−15220.8	−0.811393
$707$	−7622.20	−0.405463
$708$	3272.35	0.173704
$709$	−2907.44	−0.154007	−0.0770037	−	0.997031i	$-0.524535\pi$
$709$	−2907.44	−0.154007	−0.0770037	+	0.997031i	$0.524535\pi$
$710$	−4250.33	−0.224665
$711$	6010.52	0.317035
$712$	10053.8	0.529187
$713$	−16621.0	−0.873020
$714$	1910.37	0.100131
$715$	2090.25	0.109330
$716$	13208.7	0.689430
$717$	−814.100	−0.0424032
$718$	−5102.47	−0.265212
$719$	5091.46	0.264088	0.132044	−	0.991244i	$-0.457846\pi$
$719$	5091.46	0.264088	0.132044	+	0.991244i	$0.457846\pi$
$720$	870.310	0.0450480
$721$	6029.13	0.311424
$722$	10914.2	0.562584
$723$	−1901.66	−0.0978194
$724$	13348.4	0.685206
$725$	19961.4	1.02255
$726$	−2602.59	−0.133046
$727$	4145.87	0.211502	0.105751	−	0.994393i	$-0.466275\pi$
$727$	4145.87	0.211502	0.105751	+	0.994393i	$0.466275\pi$
$728$	−3862.43	−0.196636
$729$	−13366.9	−0.679111
$730$	−4706.03	−0.238600
$731$	32480.9	1.64343
$732$	559.443	0.0282481
$733$	−7814.79	−0.393787	−0.196893	−	0.980425i	$-0.563085\pi$
$733$	−7814.79	−0.393787	−0.196893	+	0.980425i	$0.563085\pi$
$734$	8100.61	0.407355
$735$	−726.259	−0.0364469
$736$	3803.25	0.190475
$737$	5201.90	0.259992
$738$	13435.9	0.670168
$739$	−32097.9	−1.59775	−0.798877	−	0.601494i	$-0.794572\pi$
$739$	−32097.9	−1.59775	−0.798877	+	0.601494i	$0.794572\pi$
$740$	−1048.32	−0.0520771
$741$	2728.70	0.135278
$742$	−6019.25	−0.297808
$743$	18737.8	0.925199	0.462599	−	0.886567i	$-0.346917\pi$
$743$	18737.8	0.925199	0.462599	+	0.886567i	$0.346917\pi$
$744$	1376.57	0.0678328
$745$	4176.97	0.205413
$746$	−11464.2	−0.562646
$747$	−56.9566	−0.00278974
$748$	−6298.94	−0.307904
$749$	−10445.0	−0.509551
$750$	1289.11	0.0627624
$751$	26251.7	1.27555	0.637776	−	0.770222i	$-0.279855\pi$
$751$	26251.7	1.27555	0.637776	+	0.770222i	$0.279855\pi$
$752$	−4232.45	−0.205241
$753$	−5002.42	−0.242096
$754$	19632.5	0.948243
$755$	−1372.88	−0.0661778
$756$	2105.65	0.101299
$757$	33356.3	1.60152	0.800762	−	0.598982i	$-0.204428\pi$
$757$	33356.3	1.60152	0.800762	+	0.598982i	$0.204428\pi$
$758$	−391.431	−0.0187565
$759$	−2418.04	−0.115638
$760$	−639.289	−0.0305124
$761$	−23455.3	−1.11729	−0.558643	−	0.829408i	$-0.688678\pi$
$761$	−23455.3	−1.11729	−0.558643	+	0.829408i	$0.688678\pi$
$762$	2657.50	0.126340
$763$	−12290.5	−0.583154
$764$	−5324.68	−0.252147
$765$	5180.36	0.244832
$766$	4290.89	0.202397
$767$	−39381.2	−1.85394
$768$	−314.989	−0.0147997
$769$	22391.0	1.04999	0.524993	−	0.851107i	$-0.324068\pi$
$769$	22391.0	1.04999	0.524993	+	0.851107i	$0.324068\pi$
$770$	−575.321	−0.0269261
$771$	−6182.59	−0.288794
$772$	14119.4	0.658249
$773$	35148.6	1.63546	0.817728	−	0.575604i	$-0.195233\pi$
$773$	35148.6	1.63546	0.817728	+	0.575604i	$0.195233\pi$
$774$	17384.2	0.807315
$775$	−16843.8	−0.780708
$776$	−1665.30	−0.0770371
$777$	−1231.59	−0.0568634
$778$	−15643.3	−0.720873
$779$	−9869.41	−0.453926
$780$	622.174	0.0285608
$781$	16464.3	0.754338
$782$	22638.1	1.03521
$783$	−10702.9	−0.488494
$784$	−4424.90	−0.201572
$785$	5757.60	0.261780
$786$	−5591.86	−0.253759
$787$	−25387.0	−1.14987	−0.574936	−	0.818198i	$-0.694973\pi$
$787$	−25387.0	−1.14987	−0.574936	+	0.818198i	$0.694973\pi$
$788$	14989.6	0.677643
$789$	10176.5	0.459179
$790$	−1006.68	−0.0453367
$791$	−3471.89	−0.156063
$792$	−3371.28	−0.151254
$793$	−6732.63	−0.301491
$794$	−13994.4	−0.625493
$795$	969.602	0.0432556
$796$	−18195.6	−0.810209
$797$	−5392.73	−0.239674	−0.119837	−	0.992794i	$-0.538237\pi$
$797$	−5392.73	−0.239674	−0.119837	+	0.992794i	$0.538237\pi$
$798$	−751.048	−0.0333168
$799$	−25192.8	−1.11547
$800$	3854.23	0.170335
$801$	32028.9	1.41284
$802$	14426.4	0.635178
$803$	18229.5	0.801128
$804$	1548.37	0.0679190
$805$	2067.68	0.0905293
$806$	−16566.4	−0.723977
$807$	−330.985	−0.0144377
$808$	7480.73	0.325707
$809$	−44240.2	−1.92262	−0.961312	−	0.275461i	$-0.911170\pi$
$809$	−44240.2	−1.92262	−0.961312	+	0.275461i	$0.911170\pi$
$810$	2598.11	0.112702
$811$	7346.89	0.318106	0.159053	−	0.987270i	$-0.449156\pi$
$811$	7346.89	0.318106	0.159053	+	0.987270i	$0.449156\pi$
$812$	−5403.67	−0.233536
$813$	8659.92	0.373575
$814$	4060.83	0.174855
$815$	−1387.29	−0.0596255
$816$	−1874.91	−0.0804352
$817$	−12769.6	−0.546820
$818$	−22824.1	−0.975584
$819$	−12304.8	−0.524985
$820$	−2250.34	−0.0958356
$821$	−16510.9	−0.701870	−0.350935	−	0.936400i	$-0.614136\pi$
$821$	−16510.9	−0.701870	−0.350935	+	0.936400i	$0.614136\pi$
$822$	4476.93	0.189964
$823$	6466.03	0.273866	0.136933	−	0.990580i	$-0.456276\pi$
$823$	6466.03	0.273866	0.136933	+	0.990580i	$0.456276\pi$
$824$	−5917.23	−0.250166
$825$	−2450.45	−0.103411
$826$	10839.3	0.456595
$827$	−8438.91	−0.354836	−0.177418	−	0.984136i	$-0.556774\pi$
$827$	−8438.91	−0.354836	−0.177418	+	0.984136i	$0.556774\pi$
$828$	12116.2	0.508537
$829$	−35505.6	−1.48753	−0.743764	−	0.668442i	$-0.766962\pi$
$829$	−35505.6	−1.48753	−0.743764	+	0.668442i	$0.766962\pi$
$830$	9.53945	0.000398939 0
$831$	1622.78	0.0677421
$832$	3790.74	0.157957
$833$	−26338.4	−1.09552
$834$	61.7328	0.00256311
$835$	6188.94	0.256499
$836$	2476.38	0.102449
$837$	9031.35	0.372962
$838$	−6760.58	−0.278688
$839$	33317.6	1.37098	0.685489	−	0.728083i	$-0.259589\pi$
$839$	33317.6	1.37098	0.685489	+	0.728083i	$0.259589\pi$
$840$	−171.247	−0.00703404
$841$	3077.58	0.126187
$842$	−25901.6	−1.06013
$843$	−5793.49	−0.236700
$844$	−625.707	−0.0255186
$845$	−2798.54	−0.113932
$846$	−13483.5	−0.547959
$847$	−8620.77	−0.349721
$848$	5907.53	0.239228
$849$	−3420.90	−0.138286
$850$	22941.6	0.925753
$851$	−14594.4	−0.587886
$852$	4900.68	0.197059
$853$	−4221.70	−0.169459	−0.0847293	−	0.996404i	$-0.527003\pi$
$853$	−4221.70	−0.169459	−0.0847293	+	0.996404i	$0.527003\pi$
$854$	1853.09	0.0742523
$855$	−2036.62	−0.0814630
$856$	10251.2	0.409321
$857$	−1594.32	−0.0635483	−0.0317742	−	0.999495i	$-0.510116\pi$
$857$	−1594.32	−0.0635483	−0.0317742	+	0.999495i	$0.510116\pi$
$858$	−2410.08	−0.0958962
$859$	−19987.9	−0.793923	−0.396961	−	0.917835i	$-0.629935\pi$
$859$	−19987.9	−0.793923	−0.396961	+	0.917835i	$0.629935\pi$
$860$	−2911.61	−0.115448
$861$	−2643.73	−0.104644
$862$	−1669.85	−0.0659805
$863$	37458.6	1.47753	0.738763	−	0.673966i	$-0.235410\pi$
$863$	37458.6	1.47753	0.738763	+	0.673966i	$0.235410\pi$
$864$	−2066.57	−0.0813728
$865$	818.715	0.0321817
$866$	20491.0	0.804056
$867$	−5114.97	−0.200362
$868$	4559.73	0.178303
$869$	3899.52	0.152223
$870$	870.442	0.0339204
$871$	−18633.9	−0.724898
$872$	12062.4	0.468446
$873$	−5305.24	−0.205676
$874$	−8900.02	−0.344448
$875$	4270.04	0.164976
$876$	5426.11	0.209282
$877$	13588.7	0.523215	0.261607	−	0.965174i	$-0.415747\pi$
$877$	13588.7	0.523215	0.261607	+	0.965174i	$0.415747\pi$
$878$	−24161.1	−0.928699
$879$	10241.5	0.392989
$880$	564.643	0.0216297
$881$	−7580.82	−0.289903	−0.144951	−	0.989439i	$-0.546303\pi$
$881$	−7580.82	−0.289903	−0.144951	+	0.989439i	$0.546303\pi$
$882$	−14096.7	−0.538163
$883$	4915.42	0.187335	0.0936677	−	0.995604i	$-0.470141\pi$
$883$	4915.42	0.187335	0.0936677	+	0.995604i	$0.470141\pi$
$884$	22563.7	0.858482
$885$	−1746.03	−0.0663189
$886$	17508.8	0.663906
$887$	44126.4	1.67037	0.835186	−	0.549968i	$-0.185360\pi$
$887$	44126.4	1.67037	0.835186	+	0.549968i	$0.185360\pi$
$888$	1208.73	0.0456782
$889$	8802.67	0.332095
$890$	−5364.40	−0.202039
$891$	−10064.2	−0.378409
$892$	10088.5	0.378686
$893$	9904.37	0.371150
$894$	−4816.10	−0.180173
$895$	−7047.76	−0.263219
$896$	−1043.36	−0.0389022
$897$	8661.74	0.322416
$898$	−26236.9	−0.974986
$899$	−23176.9	−0.859836
$900$	12278.7	0.454765
$901$	35163.4	1.30018
$902$	8717.02	0.321779
$903$	−3420.62	−0.126059
$904$	3407.45	0.125365
$905$	−7122.31	−0.261606
$906$	1582.95	0.0580463
$907$	37834.0	1.38507	0.692535	−	0.721384i	$-0.256494\pi$
$907$	37834.0	1.38507	0.692535	+	0.721384i	$0.256494\pi$
$908$	−3390.36	−0.123913
$909$	23831.8	0.869583
$910$	2060.88	0.0750741
$911$	33705.5	1.22581	0.612905	−	0.790156i	$-0.290001\pi$
$911$	33705.5	1.22581	0.612905	+	0.790156i	$0.290001\pi$
$912$	737.108	0.0267633
$913$	−36.9525	−0.00133948
$914$	−34634.4	−1.25340
$915$	−298.502	−0.0107849
$916$	−7084.86	−0.255557
$917$	−18522.4	−0.667026
$918$	−12300.8	−0.442253
$919$	−5349.66	−0.192023	−0.0960114	−	0.995380i	$-0.530609\pi$
$919$	−5349.66	−0.192023	−0.0960114	+	0.995380i	$0.530609\pi$
$920$	−2029.30	−0.0727219
$921$	3647.52	0.130499
$922$	24236.7	0.865721
$923$	−58977.3	−2.10321
$924$	663.352	0.0236176
$925$	−14790.1	−0.525724
$926$	21672.2	0.769108
$927$	−18850.9	−0.667900
$928$	5303.37	0.187599
$929$	−18521.5	−0.654112	−0.327056	−	0.945005i	$-0.606057\pi$
$929$	−18521.5	−0.654112	−0.327056	+	0.945005i	$0.606057\pi$
$930$	−734.498	−0.0258980
$931$	10354.7	0.364515
$932$	11053.4	0.388482
$933$	−3282.61	−0.115185
$934$	34238.8	1.19950
$935$	3360.93	0.117555
$936$	12076.4	0.421719
$937$	25150.1	0.876859	0.438429	−	0.898766i	$-0.355535\pi$
$937$	25150.1	0.876859	0.438429	+	0.898766i	$0.355535\pi$
$938$	5128.80	0.178530
$939$	−6110.23	−0.212353
$940$	2258.31	0.0783595
$941$	−17136.6	−0.593664	−0.296832	−	0.954930i	$-0.595930\pi$
$941$	−17136.6	−0.593664	−0.296832	+	0.954930i	$0.595930\pi$
$942$	−6638.59	−0.229614
$943$	−31328.6	−1.08187
$944$	−10638.1	−0.366781
$945$	−1123.51	−0.0386750
$946$	11278.6	0.387630
$947$	−36524.5	−1.25331	−0.626656	−	0.779296i	$-0.715577\pi$
$947$	−36524.5	−1.25331	−0.626656	+	0.779296i	$0.715577\pi$
$948$	1160.71	0.0397661
$949$	−65300.6	−2.23366
$950$	−9019.32	−0.308027
$951$	10818.0	0.368873
$952$	−6210.43	−0.211430
$953$	22894.0	0.778185	0.389092	−	0.921199i	$-0.372789\pi$
$953$	22894.0	0.778185	0.389092	+	0.921199i	$0.372789\pi$
$954$	18819.9	0.638698
$955$	2841.09	0.0962675
$956$	2646.56	0.0895354
$957$	−3371.79	−0.113892
$958$	−9415.68	−0.317544
$959$	14829.3	0.499336
$960$	168.069	0.00565042
$961$	−10233.8	−0.343521
$962$	−14546.4	−0.487522
$963$	32657.8	1.09282
$964$	6182.11	0.206548
$965$	−7533.69	−0.251314
$966$	−2384.06	−0.0794057
$967$	−3358.80	−0.111698	−0.0558489	−	0.998439i	$-0.517787\pi$
$967$	−3358.80	−0.111698	−0.0558489	+	0.998439i	$0.517787\pi$
$968$	8460.77	0.280929
$969$	4387.50	0.145456
$970$	888.554	0.0294121
$971$	−4021.97	−0.132926	−0.0664629	−	0.997789i	$-0.521171\pi$
$971$	−4021.97	−0.132926	−0.0664629	+	0.997789i	$0.521171\pi$
$972$	−9970.32	−0.329011
$973$	204.483	0.00673732
$974$	30880.6	1.01589
$975$	8777.85	0.288324
$976$	−1818.70	−0.0596466
$977$	34172.6	1.11901	0.559507	−	0.828825i	$-0.310990\pi$
$977$	34172.6	1.11901	0.559507	+	0.828825i	$0.310990\pi$
$978$	1599.57	0.0522991
$979$	20779.8	0.678371
$980$	2361.00	0.0769585
$981$	38427.9	1.25067
$982$	21673.8	0.704318
$983$	24587.3	0.797774	0.398887	−	0.917000i	$-0.369396\pi$
$983$	24587.3	0.797774	0.398887	+	0.917000i	$0.369396\pi$
$984$	2594.67	0.0840599
$985$	−7998.01	−0.258718
$986$	31567.3	1.01958
$987$	2653.10	0.0855614
$988$	−8870.74	−0.285644
$989$	−40534.7	−1.30327
$990$	1798.81	0.0577475
$991$	1181.56	0.0378742	0.0189371	−	0.999821i	$-0.493972\pi$
$991$	1181.56	0.0378742	0.0189371	+	0.999821i	$0.493972\pi$
$992$	−4475.10	−0.143231
$993$	−14038.5	−0.448640
$994$	16232.9	0.517985
$995$	9708.63	0.309331
$996$	−10.9991	−0.000349920 0
$997$	53886.2	1.71173	0.855864	−	0.517200i	$-0.173026\pi$
$997$	53886.2	1.71173	0.855864	+	0.517200i	$0.173026\pi$
$998$	15695.4	0.497824
$999$	7930.16	0.251150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.4.a.c.1.9	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.4.a.c.1.9	✓	18	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -1.23043 q^{3} +4.00000 q^{4} -2.13428 q^{5} +2.46085 q^{6} +8.15129 q^{7} -8.00000 q^{8} -25.4860 q^{9} +4.26856 q^{10} -16.5349 q^{11} -4.92171 q^{12} +59.2303 q^{13} -16.3026 q^{14} +2.62608 q^{15} +16.0000 q^{16} +95.2369 q^{17} +50.9721 q^{18} -37.4417 q^{19} -8.53713 q^{20} -10.0296 q^{21} +33.0698 q^{22} -118.852 q^{23} +9.84342 q^{24} -120.445 q^{25} -118.461 q^{26} +64.5802 q^{27} +32.6052 q^{28} -165.730 q^{29} -5.25216 q^{30} +139.847 q^{31} -32.0000 q^{32} +20.3450 q^{33} -190.474 q^{34} -17.3971 q^{35} -101.944 q^{36} +122.795 q^{37} +74.8834 q^{38} -72.8786 q^{39} +17.0743 q^{40} +263.594 q^{41} +20.0591 q^{42} +341.053 q^{43} -66.1397 q^{44} +54.3944 q^{45} +237.703 q^{46} -264.528 q^{47} -19.6868 q^{48} -276.557 q^{49} +240.890 q^{50} -117.182 q^{51} +236.921 q^{52} +369.221 q^{53} -129.160 q^{54} +35.2902 q^{55} -65.2103 q^{56} +46.0693 q^{57} +331.461 q^{58} -664.882 q^{59} +10.5043 q^{60} -113.669 q^{61} -279.694 q^{62} -207.744 q^{63} +64.0000 q^{64} -126.414 q^{65} -40.6900 q^{66} -314.601 q^{67} +380.948 q^{68} +146.238 q^{69} +34.7943 q^{70} -995.728 q^{71} +203.888 q^{72} -1102.49 q^{73} -245.591 q^{74} +148.199 q^{75} -149.767 q^{76} -134.781 q^{77} +145.757 q^{78} -235.836 q^{79} -34.1485 q^{80} +608.662 q^{81} -527.188 q^{82} +2.23481 q^{83} -40.1183 q^{84} -203.262 q^{85} -682.106 q^{86} +203.919 q^{87} +132.279 q^{88} -1256.72 q^{89} -108.789 q^{90} +482.804 q^{91} -475.407 q^{92} -172.071 q^{93} +529.056 q^{94} +79.9111 q^{95} +39.3737 q^{96} +208.162 q^{97} +553.113 q^{98} +421.410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 36 q^{2} - 10 q^{3} + 72 q^{4} - 26 q^{5} + 20 q^{6} - 9 q^{7} - 144 q^{8} + 178 q^{9} + 52 q^{10} - 130 q^{11} - 40 q^{12} - 27 q^{13} + 18 q^{14} - 143 q^{15} + 288 q^{16} - 110 q^{17} - 356 q^{18}+ \cdots - 9455 q^{99}+O(q^{100}) \) 18 * q - 36 * q^2 - 10 * q^3 + 72 * q^4 - 26 * q^5 + 20 * q^6 - 9 * q^7 - 144 * q^8 + 178 * q^9 + 52 * q^10 - 130 * q^11 - 40 * q^12 - 27 * q^13 + 18 * q^14 - 143 * q^15 + 288 * q^16 - 110 * q^17 - 356 * q^18 - 47 * q^19 - 104 * q^20 + 69 * q^21 + 260 * q^22 - 329 * q^23 + 80 * q^24 + 442 * q^25 + 54 * q^26 - 325 * q^27 - 36 * q^28 - 3 * q^29 + 286 * q^30 + 97 * q^31 - 576 * q^32 - 129 * q^33 + 220 * q^34 - 834 * q^35 + 712 * q^36 - 104 * q^37 + 94 * q^38 - 242 * q^39 + 208 * q^40 + 196 * q^41 - 138 * q^42 - 547 * q^43 - 520 * q^44 - 481 * q^45 + 658 * q^46 - 1074 * q^47 - 160 * q^48 + 1315 * q^49 - 884 * q^50 - 1475 * q^51 - 108 * q^52 - 1681 * q^53 + 650 * q^54 + 163 * q^55 + 72 * q^56 - 1178 * q^57 + 6 * q^58 - 924 * q^59 - 572 * q^60 + 640 * q^61 - 194 * q^62 - 445 * q^63 + 1152 * q^64 - 2052 * q^65 + 258 * q^66 - 1685 * q^67 - 440 * q^68 - 1787 * q^69 + 1668 * q^70 - 3635 * q^71 - 1424 * q^72 - 1136 * q^73 + 208 * q^74 - 3332 * q^75 - 188 * q^76 - 5512 * q^77 + 484 * q^78 - 406 * q^79 - 416 * q^80 - 3194 * q^81 - 392 * q^82 - 6854 * q^83 + 276 * q^84 - 1227 * q^85 + 1094 * q^86 - 8831 * q^87 + 1040 * q^88 - 4690 * q^89 + 962 * q^90 - 5743 * q^91 - 1316 * q^92 - 4097 * q^93 + 2148 * q^94 - 3332 * q^95 + 320 * q^96 - 4132 * q^97 - 2630 * q^98 - 9455 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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