LMFDB - Embedded newform 2013.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.14443 q^{2} +3.00000 q^{3} +18.4652 q^{4} +9.21599 q^{5} -15.4333 q^{6} +10.6441 q^{7} -53.8373 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.14443 q^{2} +3.00000 q^{3} +18.4652 q^{4} +9.21599 q^{5} -15.4333 q^{6} +10.6441 q^{7} -53.8373 q^{8} +9.00000 q^{9} -47.4110 q^{10} -11.0000 q^{11} +55.3955 q^{12} +19.6835 q^{13} -54.7581 q^{14} +27.6480 q^{15} +129.241 q^{16} +4.05826 q^{17} -46.2999 q^{18} +35.5678 q^{19} +170.175 q^{20} +31.9324 q^{21} +56.5887 q^{22} -169.095 q^{23} -161.512 q^{24} -40.0656 q^{25} -101.260 q^{26} +27.0000 q^{27} +196.546 q^{28} -182.950 q^{29} -142.233 q^{30} -123.911 q^{31} -234.173 q^{32} -33.0000 q^{33} -20.8775 q^{34} +98.0963 q^{35} +166.186 q^{36} +204.851 q^{37} -182.976 q^{38} +59.0504 q^{39} -496.164 q^{40} +40.2809 q^{41} -164.274 q^{42} +131.215 q^{43} -203.117 q^{44} +82.9439 q^{45} +869.899 q^{46} +569.474 q^{47} +387.723 q^{48} -229.702 q^{49} +206.115 q^{50} +12.1748 q^{51} +363.459 q^{52} -687.505 q^{53} -138.900 q^{54} -101.376 q^{55} -573.052 q^{56} +106.703 q^{57} +941.172 q^{58} +11.8598 q^{59} +510.524 q^{60} -61.0000 q^{61} +637.450 q^{62} +95.7973 q^{63} +170.758 q^{64} +181.403 q^{65} +169.766 q^{66} -163.233 q^{67} +74.9365 q^{68} -507.286 q^{69} -504.650 q^{70} -982.004 q^{71} -484.536 q^{72} -26.2907 q^{73} -1053.84 q^{74} -120.197 q^{75} +656.764 q^{76} -117.086 q^{77} -303.781 q^{78} -819.312 q^{79} +1191.08 q^{80} +81.0000 q^{81} -207.222 q^{82} -298.545 q^{83} +589.638 q^{84} +37.4009 q^{85} -675.025 q^{86} -548.849 q^{87} +592.211 q^{88} -74.7850 q^{89} -426.699 q^{90} +209.514 q^{91} -3122.37 q^{92} -371.732 q^{93} -2929.62 q^{94} +327.792 q^{95} -702.519 q^{96} -456.935 q^{97} +1181.69 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9}+O(q^{10}) $ 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9} - 41 q^{10} - 407 q^{11} + 390 q^{12} + 51 q^{13} - 228 q^{14} - 105 q^{15} + 462 q^{16} - 190 q^{17} - 72 q^{18} - 51 q^{19} - 720 q^{20} - 105 q^{21} + 88 q^{22} - 583 q^{23} - 351 q^{24} + 598 q^{25} - 1019 q^{26} + 999 q^{27} - 498 q^{28} - 566 q^{29} - 123 q^{30} - 696 q^{31} - 859 q^{32} - 1221 q^{33} - 348 q^{34} - 1102 q^{35} + 1170 q^{36} - 1022 q^{37} - 455 q^{38} + 153 q^{39} - 503 q^{40} - 790 q^{41} - 684 q^{42} - 87 q^{43} - 1430 q^{44} - 315 q^{45} - 303 q^{46} - 1603 q^{47} + 1386 q^{48} + 110 q^{49} - 1926 q^{50} - 570 q^{51} + 736 q^{52} - 2619 q^{53} - 216 q^{54} + 385 q^{55} - 4937 q^{56} - 153 q^{57} - 1099 q^{58} - 2471 q^{59} - 2160 q^{60} - 2257 q^{61} - 2909 q^{62} - 315 q^{63} - 265 q^{64} - 1970 q^{65} + 264 q^{66} - 3033 q^{67} - 1956 q^{68} - 1749 q^{69} + 2410 q^{70} - 3891 q^{71} - 1053 q^{72} + 391 q^{73} - 532 q^{74} + 1794 q^{75} + 1554 q^{76} + 385 q^{77} - 3057 q^{78} + 67 q^{79} - 5111 q^{80} + 2997 q^{81} - 4818 q^{82} - 5315 q^{83} - 1494 q^{84} - 2747 q^{85} - 5195 q^{86} - 1698 q^{87} + 1287 q^{88} - 8945 q^{89} - 369 q^{90} - 4432 q^{91} - 4701 q^{92} - 2088 q^{93} - 372 q^{94} - 3388 q^{95} - 2577 q^{96} - 3784 q^{97} - 4502 q^{98} - 3663 q^{99}+O(q^{100}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9 - 41 * q^10 - 407 * q^11 + 390 * q^12 + 51 * q^13 - 228 * q^14 - 105 * q^15 + 462 * q^16 - 190 * q^17 - 72 * q^18 - 51 * q^19 - 720 * q^20 - 105 * q^21 + 88 * q^22 - 583 * q^23 - 351 * q^24 + 598 * q^25 - 1019 * q^26 + 999 * q^27 - 498 * q^28 - 566 * q^29 - 123 * q^30 - 696 * q^31 - 859 * q^32 - 1221 * q^33 - 348 * q^34 - 1102 * q^35 + 1170 * q^36 - 1022 * q^37 - 455 * q^38 + 153 * q^39 - 503 * q^40 - 790 * q^41 - 684 * q^42 - 87 * q^43 - 1430 * q^44 - 315 * q^45 - 303 * q^46 - 1603 * q^47 + 1386 * q^48 + 110 * q^49 - 1926 * q^50 - 570 * q^51 + 736 * q^52 - 2619 * q^53 - 216 * q^54 + 385 * q^55 - 4937 * q^56 - 153 * q^57 - 1099 * q^58 - 2471 * q^59 - 2160 * q^60 - 2257 * q^61 - 2909 * q^62 - 315 * q^63 - 265 * q^64 - 1970 * q^65 + 264 * q^66 - 3033 * q^67 - 1956 * q^68 - 1749 * q^69 + 2410 * q^70 - 3891 * q^71 - 1053 * q^72 + 391 * q^73 - 532 * q^74 + 1794 * q^75 + 1554 * q^76 + 385 * q^77 - 3057 * q^78 + 67 * q^79 - 5111 * q^80 + 2997 * q^81 - 4818 * q^82 - 5315 * q^83 - 1494 * q^84 - 2747 * q^85 - 5195 * q^86 - 1698 * q^87 + 1287 * q^88 - 8945 * q^89 - 369 * q^90 - 4432 * q^91 - 4701 * q^92 - 2088 * q^93 - 372 * q^94 - 3388 * q^95 - 2577 * q^96 - 3784 * q^97 - 4502 * q^98 - 3663 * q^99

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 $

$2$	−5.14443	−1.81883	−0.909415	−	0.415889i	$-0.863471\pi$
$2$	−5.14443	−1.81883	−0.909415	+	0.415889i	$0.863471\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	18.4652	2.30815
$5$	9.21599	0.824303	0.412151	−	0.911115i	$-0.364777\pi$
$5$	9.21599	0.824303	0.412151	+	0.911115i	$0.364777\pi$
$6$	−15.4333	−1.05010
$7$	10.6441	0.574730	0.287365	−	0.957821i	$-0.407221\pi$
$7$	10.6441	0.574730	0.287365	+	0.957821i	$0.407221\pi$
$8$	−53.8373	−2.37930
$9$	9.00000	0.333333
$10$	−47.4110	−1.49927
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	55.3955	1.33261
$13$	19.6835	0.419939	0.209970	−	0.977708i	$-0.432663\pi$
$13$	19.6835	0.419939	0.209970	+	0.977708i	$0.432663\pi$
$14$	−54.7581	−1.04534
$15$	27.6480	0.475912
$16$	129.241	2.01939
$17$	4.05826	0.0578984	0.0289492	−	0.999581i	$-0.490784\pi$
$17$	4.05826	0.0578984	0.0289492	+	0.999581i	$0.490784\pi$
$18$	−46.2999	−0.606277
$19$	35.5678	0.429463	0.214732	−	0.976673i	$-0.431112\pi$
$19$	35.5678	0.429463	0.214732	+	0.976673i	$0.431112\pi$
$20$	170.175	1.90261
$21$	31.9324	0.331820
$22$	56.5887	0.548398
$23$	−169.095	−1.53299	−0.766495	−	0.642250i	$-0.778001\pi$
$23$	−169.095	−1.53299	−0.766495	+	0.642250i	$0.778001\pi$
$24$	−161.512	−1.37369
$25$	−40.0656	−0.320525
$26$	−101.260	−0.763799
$27$	27.0000	0.192450
$28$	196.546	1.32656
$29$	−182.950	−1.17148	−0.585740	−	0.810499i	$-0.699196\pi$
$29$	−182.950	−1.17148	−0.585740	+	0.810499i	$0.699196\pi$
$30$	−142.233	−0.865603
$31$	−123.911	−0.717904	−0.358952	−	0.933356i	$-0.616866\pi$
$31$	−123.911	−0.717904	−0.358952	+	0.933356i	$0.616866\pi$
$32$	−234.173	−1.29364
$33$	−33.0000	−0.174078
$34$	−20.8775	−0.105307
$35$	98.0963	0.473751
$36$	166.186	0.769382
$37$	204.851	0.910198	0.455099	−	0.890441i	$-0.349604\pi$
$37$	204.851	0.910198	0.455099	+	0.890441i	$0.349604\pi$
$38$	−182.976	−0.781121
$39$	59.0504	0.242452
$40$	−496.164	−1.96126
$41$	40.2809	0.153434	0.0767172	−	0.997053i	$-0.475556\pi$
$41$	40.2809	0.153434	0.0767172	+	0.997053i	$0.475556\pi$
$42$	−164.274	−0.603525
$43$	131.215	0.465350	0.232675	−	0.972555i	$-0.425252\pi$
$43$	131.215	0.465350	0.232675	+	0.972555i	$0.425252\pi$
$44$	−203.117	−0.695932
$45$	82.9439	0.274768
$46$	869.899	2.78825
$47$	569.474	1.76737	0.883684	−	0.468085i	$-0.155056\pi$
$47$	569.474	1.76737	0.883684	+	0.468085i	$0.155056\pi$
$48$	387.723	1.16590
$49$	−229.702	−0.669686
$50$	206.115	0.582980
$51$	12.1748	0.0334277
$52$	363.459	0.969282
$53$	−687.505	−1.78181	−0.890906	−	0.454188i	$-0.849930\pi$
$53$	−687.505	−1.78181	−0.890906	+	0.454188i	$0.849930\pi$
$54$	−138.900	−0.350034
$55$	−101.376	−0.248537
$56$	−573.052	−1.36745
$57$	106.703	0.247951
$58$	941.172	2.13072
$59$	11.8598	0.0261697	0.0130849	−	0.999914i	$-0.495835\pi$
$59$	11.8598	0.0261697	0.0130849	+	0.999914i	$0.495835\pi$
$60$	510.524	1.09847
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	637.450	1.30575
$63$	95.7973	0.191577
$64$	170.758	0.333512
$65$	181.403	0.346157
$66$	169.766	0.316618
$67$	−163.233	−0.297644	−0.148822	−	0.988864i	$-0.547548\pi$
$67$	−163.233	−0.297644	−0.148822	+	0.988864i	$0.547548\pi$
$68$	74.9365	0.133638
$69$	−507.286	−0.885073
$70$	−504.650	−0.861674
$71$	−982.004	−1.64144	−0.820721	−	0.571329i	$-0.806428\pi$
$71$	−982.004	−1.64144	−0.820721	+	0.571329i	$0.806428\pi$
$72$	−484.536	−0.793099
$73$	−26.2907	−0.0421519	−0.0210760	−	0.999778i	$-0.506709\pi$
$73$	−26.2907	−0.0421519	−0.0210760	+	0.999778i	$0.506709\pi$
$74$	−1053.84	−1.65550
$75$	−120.197	−0.185055
$76$	656.764	0.991264
$77$	−117.086	−0.173288
$78$	−303.781	−0.440979
$79$	−819.312	−1.16683	−0.583417	−	0.812173i	$-0.698285\pi$
$79$	−819.312	−1.16683	−0.583417	+	0.812173i	$0.698285\pi$
$80$	1191.08	1.66459
$81$	81.0000	0.111111
$82$	−207.222	−0.279071
$83$	−298.545	−0.394814	−0.197407	−	0.980322i	$-0.563252\pi$
$83$	−298.545	−0.394814	−0.197407	+	0.980322i	$0.563252\pi$
$84$	589.638	0.765890
$85$	37.4009	0.0477259
$86$	−675.025	−0.846393
$87$	−548.849	−0.676354
$88$	592.211	0.717385
$89$	−74.7850	−0.0890696	−0.0445348	−	0.999008i	$-0.514181\pi$
$89$	−74.7850	−0.0890696	−0.0445348	+	0.999008i	$0.514181\pi$
$90$	−426.699	−0.499756
$91$	209.514	0.241352
$92$	−3122.37	−3.53837
$93$	−371.732	−0.414482
$94$	−2929.62	−3.21454
$95$	327.792	0.354008
$96$	−702.519	−0.746881
$97$	−456.935	−0.478296	−0.239148	−	0.970983i	$-0.576868\pi$
$97$	−456.935	−0.478296	−0.239148	+	0.970983i	$0.576868\pi$
$98$	1181.69	1.21805
$99$	−99.0000	−0.100504
$100$	−739.818	−0.739818
$101$	−1720.84	−1.69534	−0.847672	−	0.530520i	$-0.821997\pi$
$101$	−1720.84	−1.69534	−0.847672	+	0.530520i	$0.821997\pi$
$102$	−62.6324	−0.0607993
$103$	−462.475	−0.442417	−0.221209	−	0.975226i	$-0.571000\pi$
$103$	−462.475	−0.442417	−0.221209	+	0.975226i	$0.571000\pi$
$104$	−1059.71	−0.999160
$105$	294.289	0.273521
$106$	3536.82	3.24081
$107$	206.816	0.186857	0.0934284	−	0.995626i	$-0.470217\pi$
$107$	206.816	0.186857	0.0934284	+	0.995626i	$0.470217\pi$
$108$	498.559	0.444203
$109$	835.515	0.734200	0.367100	−	0.930181i	$-0.380351\pi$
$109$	835.515	0.734200	0.367100	+	0.930181i	$0.380351\pi$
$110$	521.521	0.452046
$111$	614.553	0.525503
$112$	1375.66	1.16060
$113$	−1319.64	−1.09860	−0.549298	−	0.835626i	$-0.685105\pi$
$113$	−1319.64	−1.09860	−0.549298	+	0.835626i	$0.685105\pi$
$114$	−548.927	−0.450980
$115$	−1558.38	−1.26365
$116$	−3378.20	−2.70395
$117$	177.151	0.139980
$118$	−61.0119	−0.0475983
$119$	43.1967	0.0332760
$120$	−1488.49	−1.13233
$121$	121.000	0.0909091
$122$	313.810	0.232877
$123$	120.843	0.0885854
$124$	−2288.03	−1.65703
$125$	−1521.24	−1.08851
$126$	−492.823	−0.348445
$127$	567.833	0.396748	0.198374	−	0.980126i	$-0.436434\pi$
$127$	567.833	0.396748	0.198374	+	0.980126i	$0.436434\pi$
$128$	994.929	0.687033
$129$	393.644	0.268670
$130$	−933.213	−0.629602
$131$	1110.67	0.740763	0.370382	−	0.928880i	$-0.379227\pi$
$131$	1110.67	0.740763	0.370382	+	0.928880i	$0.379227\pi$
$132$	−609.350	−0.401797
$133$	378.588	0.246825
$134$	839.743	0.541364
$135$	248.832	0.158637
$136$	−218.486	−0.137758
$137$	−1677.12	−1.04588	−0.522940	−	0.852369i	$-0.675165\pi$
$137$	−1677.12	−1.04588	−0.522940	+	0.852369i	$0.675165\pi$
$138$	2609.70	1.60980
$139$	−2188.81	−1.33563	−0.667815	−	0.744327i	$-0.732770\pi$
$139$	−2188.81	−1.33563	−0.667815	+	0.744327i	$0.732770\pi$
$140$	1811.36	1.09349
$141$	1708.42	1.02039
$142$	5051.85	2.98551
$143$	−216.518	−0.126617
$144$	1163.17	0.673130
$145$	−1686.06	−0.965654
$146$	135.250	0.0766672
$147$	−689.107	−0.386643
$148$	3782.61	2.10087
$149$	1174.59	0.645814	0.322907	−	0.946431i	$-0.395340\pi$
$149$	1174.59	0.645814	0.322907	+	0.946431i	$0.395340\pi$
$150$	618.344	0.336584
$151$	2711.59	1.46136	0.730682	−	0.682718i	$-0.239202\pi$
$151$	2711.59	1.46136	0.730682	+	0.682718i	$0.239202\pi$
$152$	−1914.87	−1.02182
$153$	36.5244	0.0192995
$154$	602.339	0.315181
$155$	−1141.96	−0.591770
$156$	1090.38	0.559615
$157$	200.752	0.102050	0.0510248	−	0.998697i	$-0.483751\pi$
$157$	200.752	0.102050	0.0510248	+	0.998697i	$0.483751\pi$
$158$	4214.90	2.12227
$159$	−2062.51	−1.02873
$160$	−2158.14	−1.06635
$161$	−1799.87	−0.881055
$162$	−416.699	−0.202092
$163$	−681.671	−0.327562	−0.163781	−	0.986497i	$-0.552369\pi$
$163$	−681.671	−0.327562	−0.163781	+	0.986497i	$0.552369\pi$
$164$	743.793	0.354149
$165$	−304.128	−0.143493
$166$	1535.84	0.718099
$167$	232.611	0.107784	0.0538922	−	0.998547i	$-0.482837\pi$
$167$	232.611	0.107784	0.0538922	+	0.998547i	$0.482837\pi$
$168$	−1719.16	−0.789499
$169$	−1809.56	−0.823651
$170$	−192.406	−0.0868053
$171$	320.110	0.143154
$172$	2422.90	1.07410
$173$	−3859.15	−1.69599	−0.847994	−	0.530005i	$-0.822190\pi$
$173$	−3859.15	−1.69599	−0.847994	+	0.530005i	$0.822190\pi$
$174$	2823.52	1.23017
$175$	−426.464	−0.184215
$176$	−1421.65	−0.608869
$177$	35.5794	0.0151091
$178$	384.726	0.162003
$179$	−136.948	−0.0571842	−0.0285921	−	0.999591i	$-0.509102\pi$
$179$	−136.948	−0.0571842	−0.0285921	+	0.999591i	$0.509102\pi$
$180$	1531.57	0.634204
$181$	2477.40	1.01737	0.508684	−	0.860953i	$-0.330132\pi$
$181$	2477.40	1.01737	0.508684	+	0.860953i	$0.330132\pi$
$182$	−1077.83	−0.438978
$183$	−183.000	−0.0739221
$184$	9103.63	3.64744
$185$	1887.91	0.750279
$186$	1912.35	0.753873
$187$	−44.6409	−0.0174570
$188$	10515.4	4.07934
$189$	287.392	0.110607
$190$	−1686.30	−0.643880
$191$	1246.92	0.472375	0.236188	−	0.971707i	$-0.424102\pi$
$191$	1246.92	0.472375	0.236188	+	0.971707i	$0.424102\pi$
$192$	512.275	0.192553
$193$	5132.59	1.91426	0.957129	−	0.289662i	$-0.0935429\pi$
$193$	5132.59	1.91426	0.957129	+	0.289662i	$0.0935429\pi$
$194$	2350.67	0.869940
$195$	544.208	0.199854
$196$	−4241.49	−1.54573
$197$	−1363.13	−0.492990	−0.246495	−	0.969144i	$-0.579279\pi$
$197$	−1363.13	−0.492990	−0.246495	+	0.969144i	$0.579279\pi$
$198$	509.299	0.182799
$199$	−799.116	−0.284663	−0.142331	−	0.989819i	$-0.545460\pi$
$199$	−799.116	−0.284663	−0.142331	+	0.989819i	$0.545460\pi$
$200$	2157.02	0.762623
$201$	−489.700	−0.171845
$202$	8852.73	3.08354
$203$	−1947.34	−0.673284
$204$	224.810	0.0771560
$205$	371.228	0.126477
$206$	2379.17	0.804683
$207$	−1521.86	−0.510997
$208$	2543.91	0.848022
$209$	−391.245	−0.129488
$210$	−1513.95	−0.497488
$211$	1133.59	0.369856	0.184928	−	0.982752i	$-0.440795\pi$
$211$	1133.59	0.369856	0.184928	+	0.982752i	$0.440795\pi$
$212$	−12694.9	−4.11268
$213$	−2946.01	−0.947688
$214$	−1063.95	−0.339861
$215$	1209.27	0.383590
$216$	−1453.61	−0.457896
$217$	−1318.92	−0.412601
$218$	−4298.25	−1.33539
$219$	−78.8720	−0.0243364
$220$	−1871.92	−0.573659
$221$	79.8807	0.0243138
$222$	−3161.53	−0.955801
$223$	−3573.80	−1.07318	−0.536590	−	0.843843i	$-0.680288\pi$
$223$	−3573.80	−1.07318	−0.536590	+	0.843843i	$0.680288\pi$
$224$	−2492.57	−0.743491
$225$	−360.590	−0.106842
$226$	6788.80	1.99816
$227$	−2309.47	−0.675263	−0.337632	−	0.941278i	$-0.609626\pi$
$227$	−2309.47	−0.675263	−0.337632	+	0.941278i	$0.609626\pi$
$228$	1970.29	0.572306
$229$	5221.17	1.50666	0.753329	−	0.657644i	$-0.228447\pi$
$229$	5221.17	1.50666	0.753329	+	0.657644i	$0.228447\pi$
$230$	8016.97	2.29836
$231$	−351.257	−0.100048
$232$	9849.52	2.78730
$233$	−215.283	−0.0605307	−0.0302653	−	0.999542i	$-0.509635\pi$
$233$	−215.283	−0.0605307	−0.0302653	+	0.999542i	$0.509635\pi$
$234$	−911.342	−0.254600
$235$	5248.26	1.45685
$236$	218.993	0.0604035
$237$	−2457.94	−0.673672
$238$	−222.223	−0.0605233
$239$	795.262	0.215235	0.107618	−	0.994192i	$-0.465678\pi$
$239$	795.262	0.215235	0.107618	+	0.994192i	$0.465678\pi$
$240$	3573.25	0.961052
$241$	2633.25	0.703827	0.351913	−	0.936033i	$-0.385531\pi$
$241$	2633.25	0.703827	0.351913	+	0.936033i	$0.385531\pi$
$242$	−622.476	−0.165348
$243$	243.000	0.0641500
$244$	−1126.38	−0.295528
$245$	−2116.93	−0.552024
$246$	−621.666	−0.161122
$247$	700.097	0.180349
$248$	6671.02	1.70811
$249$	−895.634	−0.227946
$250$	7825.93	1.97982
$251$	−2088.70	−0.525250	−0.262625	−	0.964898i	$-0.584588\pi$
$251$	−2088.70	−0.525250	−0.262625	+	0.964898i	$0.584588\pi$
$252$	1768.91	0.442187
$253$	1860.05	0.462214
$254$	−2921.18	−0.721618
$255$	112.203	0.0275545
$256$	−6484.41	−1.58311
$257$	−360.200	−0.0874266	−0.0437133	−	0.999044i	$-0.513919\pi$
$257$	−360.200	−0.0874266	−0.0437133	+	0.999044i	$0.513919\pi$
$258$	−2025.07	−0.488665
$259$	2180.46	0.523118
$260$	3349.63	0.798982
$261$	−1646.55	−0.390493
$262$	−5713.78	−1.34732
$263$	−1178.27	−0.276255	−0.138127	−	0.990414i	$-0.544108\pi$
$263$	−1178.27	−0.276255	−0.138127	+	0.990414i	$0.544108\pi$
$264$	1776.63	0.414182
$265$	−6336.03	−1.46875
$266$	−1947.62	−0.448934
$267$	−224.355	−0.0514244
$268$	−3014.13	−0.687006
$269$	1760.50	0.399032	0.199516	−	0.979895i	$-0.436063\pi$
$269$	1760.50	0.399032	0.199516	+	0.979895i	$0.436063\pi$
$270$	−1280.10	−0.288534
$271$	6672.16	1.49559	0.747795	−	0.663930i	$-0.231113\pi$
$271$	6672.16	1.49559	0.747795	+	0.663930i	$0.231113\pi$
$272$	524.494	0.116920
$273$	628.541	0.139344
$274$	8627.80	1.90228
$275$	440.721	0.0966418
$276$	−9367.11	−2.04288
$277$	3267.64	0.708784	0.354392	−	0.935097i	$-0.384688\pi$
$277$	3267.64	0.708784	0.354392	+	0.935097i	$0.384688\pi$
$278$	11260.2	2.42929
$279$	−1115.20	−0.239301
$280$	−5281.24	−1.12719
$281$	8491.97	1.80281	0.901403	−	0.432982i	$-0.142538\pi$
$281$	8491.97	1.80281	0.901403	+	0.432982i	$0.142538\pi$
$282$	−8788.85	−1.85592
$283$	−3824.97	−0.803430	−0.401715	−	0.915765i	$-0.631586\pi$
$283$	−3824.97	−0.803430	−0.401715	+	0.915765i	$0.631586\pi$
$284$	−18132.9	−3.78869
$285$	983.376	0.204387
$286$	1113.86	0.230294
$287$	428.755	0.0881834
$288$	−2107.56	−0.431212
$289$	−4896.53	−0.996648
$290$	8673.83	1.75636
$291$	−1370.80	−0.276144
$292$	−485.461	−0.0972927
$293$	4459.87	0.889243	0.444621	−	0.895719i	$-0.353338\pi$
$293$	4459.87	0.889243	0.444621	+	0.895719i	$0.353338\pi$
$294$	3545.06	0.703239
$295$	109.300	0.0215718
$296$	−11028.6	−2.16563
$297$	−297.000	−0.0580259
$298$	−6042.61	−1.17463
$299$	−3328.38	−0.643763
$300$	−2219.45	−0.427134
$301$	1396.67	0.267451
$302$	−13949.6	−2.65797
$303$	−5162.51	−0.978808
$304$	4596.81	0.867254
$305$	−562.175	−0.105541
$306$	−187.897	−0.0351025
$307$	1209.81	0.224911	0.112455	−	0.993657i	$-0.464128\pi$
$307$	1209.81	0.224911	0.112455	+	0.993657i	$0.464128\pi$
$308$	−2162.00	−0.399973
$309$	−1387.42	−0.255430
$310$	5874.73	1.07633
$311$	−35.9336	−0.00655179	−0.00327590	−	0.999995i	$-0.501043\pi$
$311$	−35.9336	−0.00655179	−0.00327590	+	0.999995i	$0.501043\pi$
$312$	−3179.12	−0.576865
$313$	7827.82	1.41359	0.706796	−	0.707417i	$-0.250140\pi$
$313$	7827.82	1.41359	0.706796	+	0.707417i	$0.250140\pi$
$314$	−1032.76	−0.185611
$315$	882.867	0.157917
$316$	−15128.7	−2.69322
$317$	−656.367	−0.116294	−0.0581471	−	0.998308i	$-0.518519\pi$
$317$	−656.367	−0.116294	−0.0581471	+	0.998308i	$0.518519\pi$
$318$	10610.5	1.87109
$319$	2012.45	0.353214
$320$	1573.71	0.274915
$321$	620.449	0.107882
$322$	9259.33	1.60249
$323$	144.343	0.0248653
$324$	1495.68	0.256461
$325$	−788.630	−0.134601
$326$	3506.81	0.595779
$327$	2506.54	0.423891
$328$	−2168.61	−0.365066
$329$	6061.56	1.01576
$330$	1564.56	0.260989
$331$	9906.79	1.64510	0.822548	−	0.568696i	$-0.192552\pi$
$331$	9906.79	1.64510	0.822548	+	0.568696i	$0.192552\pi$
$332$	−5512.68	−0.911287
$333$	1843.66	0.303399
$334$	−1196.65	−0.196042
$335$	−1504.36	−0.245349
$336$	4126.98	0.670075
$337$	−487.200	−0.0787521	−0.0393761	−	0.999224i	$-0.512537\pi$
$337$	−487.200	−0.0787521	−0.0393761	+	0.999224i	$0.512537\pi$
$338$	9309.16	1.49808
$339$	−3958.92	−0.634275
$340$	690.614	0.110158
$341$	1363.02	0.216456
$342$	−1646.78	−0.260374
$343$	−6095.92	−0.959618
$344$	−7064.25	−1.10721
$345$	−4675.14	−0.729568
$346$	19853.1	3.08472
$347$	−6974.74	−1.07903	−0.539515	−	0.841976i	$-0.681392\pi$
$347$	−6974.74	−1.07903	−0.539515	+	0.841976i	$0.681392\pi$
$348$	−10134.6	−1.56112
$349$	−3565.34	−0.546843	−0.273422	−	0.961894i	$-0.588155\pi$
$349$	−3565.34	−0.546843	−0.273422	+	0.961894i	$0.588155\pi$
$350$	2193.91	0.335056
$351$	531.454	0.0808174
$352$	2575.90	0.390046
$353$	−5367.16	−0.809249	−0.404624	−	0.914483i	$-0.632598\pi$
$353$	−5367.16	−0.809249	−0.404624	+	0.914483i	$0.632598\pi$
$354$	−183.036	−0.0274809
$355$	−9050.14	−1.35305
$356$	−1380.92	−0.205586
$357$	129.590	0.0192119
$358$	704.519	0.104008
$359$	−2423.95	−0.356355	−0.178177	−	0.983998i	$-0.557020\pi$
$359$	−2423.95	−0.356355	−0.178177	+	0.983998i	$0.557020\pi$
$360$	−4465.48	−0.653754
$361$	−5593.94	−0.815561
$362$	−12744.8	−1.85042
$363$	363.000	0.0524864
$364$	3868.71	0.557075
$365$	−242.294	−0.0347459
$366$	941.431	0.134452
$367$	−10508.2	−1.49461	−0.747307	−	0.664479i	$-0.768654\pi$
$367$	−10508.2	−1.49461	−0.747307	+	0.664479i	$0.768654\pi$
$368$	−21854.0	−3.09571
$369$	362.528	0.0511448
$370$	−9712.20	−1.36463
$371$	−7317.90	−1.02406
$372$	−6864.09	−0.956685
$373$	9969.56	1.38393	0.691963	−	0.721933i	$-0.256746\pi$
$373$	9969.56	1.38393	0.691963	+	0.721933i	$0.256746\pi$
$374$	229.652	0.0317514
$375$	−4563.73	−0.628453
$376$	−30658.9	−4.20509
$377$	−3601.09	−0.491951
$378$	−1478.47	−0.201175
$379$	−11541.2	−1.56420	−0.782102	−	0.623150i	$-0.785853\pi$
$379$	−11541.2	−1.56420	−0.782102	+	0.623150i	$0.785853\pi$
$380$	6052.73	0.817102
$381$	1703.50	0.229063
$382$	−6414.67	−0.859171
$383$	−96.9548	−0.0129351	−0.00646757	−	0.999979i	$-0.502059\pi$
$383$	−96.9548	−0.0129351	−0.00646757	+	0.999979i	$0.502059\pi$
$384$	2984.79	0.396658
$385$	−1079.06	−0.142841
$386$	−26404.2	−3.48171
$387$	1180.93	0.155117
$388$	−8437.38	−1.10398
$389$	−11262.7	−1.46797	−0.733984	−	0.679166i	$-0.762341\pi$
$389$	−11262.7	−1.46797	−0.733984	+	0.679166i	$0.762341\pi$
$390$	−2799.64	−0.363501
$391$	−686.233	−0.0887578
$392$	12366.6	1.59338
$393$	3332.02	0.427680
$394$	7012.54	0.896666
$395$	−7550.77	−0.961824
$396$	−1828.05	−0.231977
$397$	8245.31	1.04237	0.521184	−	0.853444i	$-0.325491\pi$
$397$	8245.31	1.04237	0.521184	+	0.853444i	$0.325491\pi$
$398$	4111.00	0.517753
$399$	1135.76	0.142505
$400$	−5178.12	−0.647265
$401$	4716.01	0.587297	0.293649	−	0.955913i	$-0.405130\pi$
$401$	4716.01	0.587297	0.293649	+	0.955913i	$0.405130\pi$
$402$	2519.23	0.312557
$403$	−2438.99	−0.301476
$404$	−31775.6	−3.91310
$405$	746.495	0.0915892
$406$	10018.0	1.22459
$407$	−2253.36	−0.274435
$408$	−655.458	−0.0795343
$409$	−15734.6	−1.90227	−0.951133	−	0.308782i	$-0.900079\pi$
$409$	−15734.6	−1.90227	−0.951133	+	0.308782i	$0.900079\pi$
$410$	−1909.76	−0.230039
$411$	−5031.35	−0.603840
$412$	−8539.68	−1.02116
$413$	126.237	0.0150405
$414$	7829.09	0.929417
$415$	−2751.38	−0.325446
$416$	−4609.34	−0.543248
$417$	−6566.44	−0.771127
$418$	2012.73	0.235517
$419$	−5341.33	−0.622770	−0.311385	−	0.950284i	$-0.600793\pi$
$419$	−5341.33	−0.622770	−0.311385	+	0.950284i	$0.600793\pi$
$420$	5434.09	0.631325
$421$	−2600.12	−0.301003	−0.150501	−	0.988610i	$-0.548089\pi$
$421$	−2600.12	−0.301003	−0.150501	+	0.988610i	$0.548089\pi$
$422$	−5831.67	−0.672705
$423$	5125.26	0.589122
$424$	37013.4	4.23946
$425$	−162.597	−0.0185579
$426$	15155.6	1.72368
$427$	−649.293	−0.0735866
$428$	3818.90	0.431293
$429$	−649.555	−0.0731021
$430$	−6221.02	−0.697684
$431$	−7002.76	−0.782624	−0.391312	−	0.920258i	$-0.627979\pi$
$431$	−7002.76	−0.782624	−0.391312	+	0.920258i	$0.627979\pi$
$432$	3489.51	0.388632
$433$	−4993.52	−0.554211	−0.277105	−	0.960840i	$-0.589375\pi$
$433$	−4993.52	−0.554211	−0.277105	+	0.960840i	$0.589375\pi$
$434$	6785.11	0.750451
$435$	−5058.19	−0.557521
$436$	15427.9	1.69464
$437$	−6014.34	−0.658363
$438$	405.751	0.0442638
$439$	5146.96	0.559569	0.279785	−	0.960063i	$-0.409737\pi$
$439$	5146.96	0.559569	0.279785	+	0.960063i	$0.409737\pi$
$440$	5457.80	0.591342
$441$	−2067.32	−0.223229
$442$	−410.941	−0.0442228
$443$	9941.72	1.06624	0.533121	−	0.846039i	$-0.321019\pi$
$443$	9941.72	1.06624	0.533121	+	0.846039i	$0.321019\pi$
$444$	11347.8	1.21294
$445$	−689.218	−0.0734204
$446$	18385.2	1.95193
$447$	3523.78	0.372861
$448$	1817.58	0.191679
$449$	−11823.9	−1.24278	−0.621388	−	0.783503i	$-0.713431\pi$
$449$	−11823.9	−1.24278	−0.621388	+	0.783503i	$0.713431\pi$
$450$	1855.03	0.194327
$451$	−443.090	−0.0462622
$452$	−24367.4	−2.53572
$453$	8134.77	0.843719
$454$	11880.9	1.22819
$455$	1930.88	0.198947
$456$	−5744.62	−0.589948
$457$	−3678.36	−0.376512	−0.188256	−	0.982120i	$-0.560284\pi$
$457$	−3678.36	−0.376512	−0.188256	+	0.982120i	$0.560284\pi$
$458$	−26860.0	−2.74036
$459$	109.573	0.0111426
$460$	−28775.7	−2.91669
$461$	9740.91	0.984120	0.492060	−	0.870561i	$-0.336244\pi$
$461$	9740.91	0.984120	0.492060	+	0.870561i	$0.336244\pi$
$462$	1807.02	0.181970
$463$	−780.346	−0.0783278	−0.0391639	−	0.999233i	$-0.512469\pi$
$463$	−780.346	−0.0783278	−0.0391639	+	0.999233i	$0.512469\pi$
$464$	−23644.6	−2.36568
$465$	−3425.88	−0.341659
$466$	1107.51	0.110095
$467$	10717.7	1.06201	0.531004	−	0.847369i	$-0.321815\pi$
$467$	10717.7	1.06201	0.531004	+	0.847369i	$0.321815\pi$
$468$	3271.13	0.323094
$469$	−1737.48	−0.171065
$470$	−26999.3	−2.64976
$471$	602.257	0.0589184
$472$	−638.499	−0.0622655
$473$	−1443.36	−0.140308
$474$	12644.7	1.22529
$475$	−1425.04	−0.137654
$476$	797.635	0.0768058
$477$	−6187.54	−0.593937
$478$	−4091.17	−0.391477
$479$	14657.3	1.39814	0.699072	−	0.715052i	$-0.253597\pi$
$479$	14657.3	1.39814	0.699072	+	0.715052i	$0.253597\pi$
$480$	−6474.41	−0.615656
$481$	4032.18	0.382228
$482$	−13546.6	−1.28014
$483$	−5399.62	−0.508678
$484$	2234.29	0.209831
$485$	−4211.11	−0.394261
$486$	−1250.10	−0.116678
$487$	5843.22	0.543700	0.271850	−	0.962340i	$-0.412365\pi$
$487$	5843.22	0.543700	0.271850	+	0.962340i	$0.412365\pi$
$488$	3284.08	0.304638
$489$	−2045.01	−0.189118
$490$	10890.4	1.00404
$491$	−8294.18	−0.762344	−0.381172	−	0.924504i	$-0.624480\pi$
$491$	−8294.18	−0.762344	−0.381172	+	0.924504i	$0.624480\pi$
$492$	2231.38	0.204468
$493$	−742.458	−0.0678269
$494$	−3601.60	−0.328024
$495$	−912.383	−0.0828456
$496$	−16014.3	−1.44973
$497$	−10452.6	−0.943386
$498$	4607.53	0.414595
$499$	−12383.6	−1.11095	−0.555476	−	0.831532i	$-0.687464\pi$
$499$	−12383.6	−1.11095	−0.555476	+	0.831532i	$0.687464\pi$
$500$	−28090.0	−2.51245
$501$	697.834	0.0622294
$502$	10745.2	0.955340
$503$	−6397.88	−0.567132	−0.283566	−	0.958953i	$-0.591517\pi$
$503$	−6397.88	−0.567132	−0.283566	+	0.958953i	$0.591517\pi$
$504$	−5157.47	−0.455817
$505$	−15859.2	−1.39748
$506$	−9568.88	−0.840689
$507$	−5428.68	−0.475535
$508$	10485.1	0.915753
$509$	−146.925	−0.0127943	−0.00639717	−	0.999980i	$-0.502036\pi$
$509$	−146.925	−0.0127943	−0.00639717	+	0.999980i	$0.502036\pi$
$510$	−577.219	−0.0501170
$511$	−279.842	−0.0242260
$512$	25399.2	2.19237
$513$	960.329	0.0826502
$514$	1853.02	0.159014
$515$	−4262.16	−0.364686
$516$	7268.70	0.620130
$517$	−6264.21	−0.532881
$518$	−11217.3	−0.951463
$519$	−11577.5	−0.979180
$520$	−9766.23	−0.823611
$521$	−7308.97	−0.614610	−0.307305	−	0.951611i	$-0.599427\pi$
$521$	−7308.97	−0.614610	−0.307305	+	0.951611i	$0.599427\pi$
$522$	8470.55	0.710241
$523$	−15241.0	−1.27427	−0.637136	−	0.770752i	$-0.719881\pi$
$523$	−15241.0	−1.27427	−0.637136	+	0.770752i	$0.719881\pi$
$524$	20508.8	1.70979
$525$	−1279.39	−0.106357
$526$	6061.51	0.502461
$527$	−502.862	−0.0415655
$528$	−4264.95	−0.351531
$529$	16426.2	1.35006
$530$	32595.3	2.67141
$531$	106.738	0.00872324
$532$	6990.70	0.569709
$533$	792.867	0.0644332
$534$	1154.18	0.0935322
$535$	1906.02	0.154027
$536$	8788.05	0.708183
$537$	−410.844	−0.0330153
$538$	−9056.77	−0.725772
$539$	2526.72	0.201918
$540$	4594.72	0.366158
$541$	14229.6	1.13082	0.565412	−	0.824808i	$-0.308717\pi$
$541$	14229.6	1.13082	0.565412	+	0.824808i	$0.308717\pi$
$542$	−34324.5	−2.72022
$543$	7432.20	0.587378
$544$	−950.336	−0.0748995
$545$	7700.10	0.605203
$546$	−3233.49	−0.253444
$547$	−9307.19	−0.727508	−0.363754	−	0.931495i	$-0.618505\pi$
$547$	−9307.19	−0.727508	−0.363754	+	0.931495i	$0.618505\pi$
$548$	−30968.2	−2.41405
$549$	−549.000	−0.0426790
$550$	−2267.26	−0.175775
$551$	−6507.11	−0.503108
$552$	27310.9	2.10585
$553$	−8720.88	−0.670614
$554$	−16810.1	−1.28916
$555$	5663.72	0.433174
$556$	−40416.8	−3.08283
$557$	2252.82	0.171373	0.0856867	−	0.996322i	$-0.472692\pi$
$557$	2252.82	0.171373	0.0856867	+	0.996322i	$0.472692\pi$
$558$	5737.05	0.435248
$559$	2582.76	0.195419
$560$	12678.1	0.956690
$561$	−133.923	−0.0100788
$562$	−43686.3	−3.27900
$563$	11987.6	0.897364	0.448682	−	0.893692i	$-0.351894\pi$
$563$	11987.6	0.897364	0.448682	+	0.893692i	$0.351894\pi$
$564$	31546.3	2.35521
$565$	−12161.8	−0.905577
$566$	19677.3	1.46130
$567$	862.176	0.0638589
$568$	52868.5	3.90548
$569$	20926.6	1.54181	0.770904	−	0.636951i	$-0.219805\pi$
$569$	20926.6	1.54181	0.770904	+	0.636951i	$0.219805\pi$
$570$	−5058.91	−0.371745
$571$	17662.8	1.29451	0.647256	−	0.762273i	$-0.275916\pi$
$571$	17662.8	1.29451	0.647256	+	0.762273i	$0.275916\pi$
$572$	−3998.04	−0.292249
$573$	3740.75	0.272726
$574$	−2205.70	−0.160391
$575$	6774.90	0.491361
$576$	1536.82	0.111171
$577$	617.022	0.0445181	0.0222591	−	0.999752i	$-0.492914\pi$
$577$	617.022	0.0445181	0.0222591	+	0.999752i	$0.492914\pi$
$578$	25189.9	1.81273
$579$	15397.8	1.10520
$580$	−31133.4	−2.22887
$581$	−3177.75	−0.226911
$582$	7052.01	0.502260
$583$	7562.55	0.537237
$584$	1415.42	0.100292
$585$	1632.62	0.115386
$586$	−22943.5	−1.61738
$587$	23213.4	1.63223	0.816116	−	0.577888i	$-0.196123\pi$
$587$	23213.4	1.63223	0.816116	+	0.577888i	$0.196123\pi$
$588$	−12724.5	−0.892429
$589$	−4407.22	−0.308313
$590$	−562.285	−0.0392354
$591$	−4089.40	−0.284628
$592$	26475.2	1.83805
$593$	−27725.0	−1.91995	−0.959976	−	0.280083i	$-0.909638\pi$
$593$	−27725.0	−1.91995	−0.959976	+	0.280083i	$0.909638\pi$
$594$	1527.90	0.105539
$595$	398.101	0.0274295
$596$	21689.0	1.49063
$597$	−2397.35	−0.164350
$598$	17122.6	1.17090
$599$	−20460.6	−1.39566	−0.697828	−	0.716265i	$-0.745850\pi$
$599$	−20460.6	−1.39566	−0.697828	+	0.716265i	$0.745850\pi$
$600$	6471.07	0.440301
$601$	−10860.3	−0.737104	−0.368552	−	0.929607i	$-0.620146\pi$
$601$	−10860.3	−0.737104	−0.368552	+	0.929607i	$0.620146\pi$
$602$	−7185.06	−0.486447
$603$	−1469.10	−0.0992146
$604$	50069.9	3.37304
$605$	1115.13	0.0749366
$606$	26558.2	1.78029
$607$	−15612.4	−1.04396	−0.521982	−	0.852957i	$-0.674807\pi$
$607$	−15612.4	−1.04396	−0.521982	+	0.852957i	$0.674807\pi$
$608$	−8329.01	−0.555569
$609$	−5842.03	−0.388721
$610$	2892.07	0.191962
$611$	11209.2	0.742187
$612$	674.429	0.0445460
$613$	−17488.4	−1.15229	−0.576143	−	0.817349i	$-0.695443\pi$
$613$	−17488.4	−1.15229	−0.576143	+	0.817349i	$0.695443\pi$
$614$	−6223.80	−0.409075
$615$	1113.68	0.0730212
$616$	6303.57	0.412302
$617$	5641.83	0.368123	0.184061	−	0.982915i	$-0.441075\pi$
$617$	5641.83	0.368123	0.184061	+	0.982915i	$0.441075\pi$
$618$	7137.51	0.464584
$619$	−14863.7	−0.965140	−0.482570	−	0.875858i	$-0.660297\pi$
$619$	−14863.7	−0.965140	−0.482570	+	0.875858i	$0.660297\pi$
$620$	−21086.5	−1.36589
$621$	−4565.57	−0.295024
$622$	184.858	0.0119166
$623$	−796.023	−0.0511910
$624$	7631.74	0.489606
$625$	−9011.55	−0.576739
$626$	−40269.7	−2.57109
$627$	−1173.74	−0.0747600
$628$	3706.93	0.235545
$629$	831.340	0.0526990
$630$	−4541.85	−0.287225
$631$	5113.30	0.322595	0.161297	−	0.986906i	$-0.448432\pi$
$631$	5113.30	0.322595	0.161297	+	0.986906i	$0.448432\pi$
$632$	44109.6	2.77624
$633$	3400.77	0.213536
$634$	3376.64	0.211519
$635$	5233.14	0.327041
$636$	−38084.7	−2.37446
$637$	−4521.34	−0.281227
$638$	−10352.9	−0.642437
$639$	−8838.04	−0.547148
$640$	9169.26	0.566323
$641$	26127.7	1.60996	0.804978	−	0.593304i	$-0.202177\pi$
$641$	26127.7	1.60996	0.804978	+	0.593304i	$0.202177\pi$
$642$	−3191.86	−0.196219
$643$	16021.8	0.982642	0.491321	−	0.870979i	$-0.336514\pi$
$643$	16021.8	0.982642	0.491321	+	0.870979i	$0.336514\pi$
$644$	−33235.0	−2.03360
$645$	3627.82	0.221466
$646$	−742.564	−0.0452257
$647$	−5768.64	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$647$	−5768.64	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$648$	−4360.82	−0.264366
$649$	−130.458	−0.00789047
$650$	4057.05	0.244816
$651$	−3956.77	−0.238215
$652$	−12587.2	−0.756060
$653$	−20113.4	−1.20536	−0.602679	−	0.797984i	$-0.705900\pi$
$653$	−20113.4	−1.20536	−0.602679	+	0.797984i	$0.705900\pi$
$654$	−12894.7	−0.770985
$655$	10236.0	0.610613
$656$	5205.94	0.309844
$657$	−236.616	−0.0140506
$658$	−31183.3	−1.84749
$659$	24872.2	1.47023	0.735116	−	0.677941i	$-0.237128\pi$
$659$	24872.2	1.47023	0.735116	+	0.677941i	$0.237128\pi$
$660$	−5615.77	−0.331202
$661$	11201.8	0.659152	0.329576	−	0.944129i	$-0.393094\pi$
$661$	11201.8	0.659152	0.329576	+	0.944129i	$0.393094\pi$
$662$	−50964.8	−2.99215
$663$	239.642	0.0140376
$664$	16072.8	0.939379
$665$	3489.06	0.203459
$666$	−9484.58	−0.551832
$667$	30935.9	1.79587
$668$	4295.21	0.248782
$669$	−10721.4	−0.619601
$670$	7739.06	0.446248
$671$	671.000	0.0386046
$672$	−7477.71	−0.429255
$673$	3954.56	0.226504	0.113252	−	0.993566i	$-0.463873\pi$
$673$	3954.56	0.226504	0.113252	+	0.993566i	$0.463873\pi$
$674$	2506.37	0.143237
$675$	−1081.77	−0.0616850
$676$	−33413.8	−1.90111
$677$	4240.88	0.240754	0.120377	−	0.992728i	$-0.461590\pi$
$677$	4240.88	0.240754	0.120377	+	0.992728i	$0.461590\pi$
$678$	20366.4	1.15364
$679$	−4863.68	−0.274891
$680$	−2013.56	−0.113554
$681$	−6928.40	−0.389863
$682$	−7011.95	−0.393697
$683$	−25069.6	−1.40448	−0.702240	−	0.711940i	$-0.747817\pi$
$683$	−25069.6	−1.40448	−0.702240	+	0.711940i	$0.747817\pi$
$684$	5910.88	0.330421
$685$	−15456.3	−0.862123
$686$	31360.1	1.74538
$687$	15663.5	0.869870
$688$	16958.3	0.939724
$689$	−13532.5	−0.748253
$690$	24050.9	1.32696
$691$	−20009.7	−1.10160	−0.550800	−	0.834638i	$-0.685677\pi$
$691$	−20009.7	−1.10160	−0.550800	+	0.834638i	$0.685677\pi$
$692$	−71259.9	−3.91459
$693$	−1053.77	−0.0577625
$694$	35881.0	1.96257
$695$	−20172.1	−1.10096
$696$	29548.6	1.60925
$697$	163.470	0.00888362
$698$	18341.6	0.994616
$699$	−645.849	−0.0349474
$700$	−7874.72	−0.425195
$701$	−13868.6	−0.747232	−0.373616	−	0.927583i	$-0.621882\pi$
$701$	−13868.6	−0.747232	−0.373616	+	0.927583i	$0.621882\pi$
$702$	−2734.03	−0.146993
$703$	7286.09	0.390896
$704$	−1878.34	−0.100558
$705$	15744.8	0.841111
$706$	27611.0	1.47189
$707$	−18316.8	−0.974365
$708$	656.979	0.0348740
$709$	12276.6	0.650290	0.325145	−	0.945664i	$-0.394587\pi$
$709$	12276.6	0.650290	0.325145	+	0.945664i	$0.394587\pi$
$710$	46557.8	2.46096
$711$	−7373.81	−0.388945
$712$	4026.23	0.211923
$713$	20952.7	1.10054
$714$	−666.668	−0.0349432
$715$	−1995.43	−0.104370
$716$	−2528.77	−0.131989
$717$	2385.79	0.124266
$718$	12469.9	0.648149
$719$	1565.70	0.0812108	0.0406054	−	0.999175i	$-0.487071\pi$
$719$	1565.70	0.0812108	0.0406054	+	0.999175i	$0.487071\pi$
$720$	10719.8	0.554863
$721$	−4922.65	−0.254271
$722$	28777.6	1.48337
$723$	7899.74	0.406355
$724$	45745.6	2.34823
$725$	7329.99	0.375488
$726$	−1867.43	−0.0954639
$727$	15109.9	0.770832	0.385416	−	0.922743i	$-0.374058\pi$
$727$	15109.9	0.770832	0.385416	+	0.922743i	$0.374058\pi$
$728$	−11279.7	−0.574247
$729$	729.000	0.0370370
$730$	1246.47	0.0631970
$731$	532.504	0.0269430
$732$	−3379.13	−0.170623
$733$	15275.7	0.769744	0.384872	−	0.922970i	$-0.374246\pi$
$733$	15275.7	0.769744	0.384872	+	0.922970i	$0.374246\pi$
$734$	54058.7	2.71845
$735$	−6350.80	−0.318711
$736$	39597.5	1.98313
$737$	1795.57	0.0897430
$738$	−1865.00	−0.0930238
$739$	490.027	0.0243923	0.0121962	−	0.999926i	$-0.496118\pi$
$739$	490.027	0.0243923	0.0121962	+	0.999926i	$0.496118\pi$
$740$	34860.5	1.73175
$741$	2100.29	0.104124
$742$	37646.4	1.86259
$743$	−4322.86	−0.213446	−0.106723	−	0.994289i	$-0.534036\pi$
$743$	−4322.86	−0.213446	−0.106723	+	0.994289i	$0.534036\pi$
$744$	20013.1	0.986175
$745$	10825.0	0.532347
$746$	−51287.7	−2.51713
$747$	−2686.90	−0.131605
$748$	−824.302	−0.0402934
$749$	2201.38	0.107392
$750$	23477.8	1.14305
$751$	−38263.1	−1.85917	−0.929587	−	0.368603i	$-0.879836\pi$
$751$	−38263.1	−1.85917	−0.929587	+	0.368603i	$0.879836\pi$
$752$	73599.4	3.56901
$753$	−6266.10	−0.303253
$754$	18525.5	0.894775
$755$	24990.0	1.20461
$756$	5306.74	0.255297
$757$	−18656.7	−0.895756	−0.447878	−	0.894095i	$-0.647820\pi$
$757$	−18656.7	−0.895756	−0.447878	+	0.894095i	$0.647820\pi$
$758$	59373.1	2.84502
$759$	5580.14	0.266859
$760$	−17647.4	−0.842289
$761$	−679.523	−0.0323689	−0.0161844	−	0.999869i	$-0.505152\pi$
$761$	−679.523	−0.0323689	−0.0161844	+	0.999869i	$0.505152\pi$
$762$	−8763.54	−0.416627
$763$	8893.34	0.421967
$764$	23024.5	1.09031
$765$	336.608	0.0159086
$766$	498.777	0.0235268
$767$	233.442	0.0109897
$768$	−19453.2	−0.914008
$769$	−40596.6	−1.90371	−0.951854	−	0.306552i	$-0.900825\pi$
$769$	−40596.6	−1.90371	−0.951854	+	0.306552i	$0.900825\pi$
$770$	5551.15	0.259804
$771$	−1080.60	−0.0504758
$772$	94774.1	4.41839
$773$	18078.1	0.841171	0.420585	−	0.907253i	$-0.361825\pi$
$773$	18078.1	0.841171	0.420585	+	0.907253i	$0.361825\pi$
$774$	−6075.22	−0.282131
$775$	4964.55	0.230106
$776$	24600.2	1.13801
$777$	6541.39	0.302022
$778$	57940.0	2.66999
$779$	1432.70	0.0658945
$780$	10048.9	0.461292
$781$	10802.0	0.494914
$782$	3530.28	0.161435
$783$	−4939.64	−0.225451
$784$	−29686.9	−1.35236
$785$	1850.13	0.0841198
$786$	−17141.3	−0.777877
$787$	−30009.7	−1.35925	−0.679626	−	0.733559i	$-0.737858\pi$
$787$	−30009.7	−1.35925	−0.679626	+	0.733559i	$0.737858\pi$
$788$	−25170.5	−1.13789
$789$	−3534.80	−0.159496
$790$	38844.4	1.74940
$791$	−14046.5	−0.631396
$792$	5329.89	0.239128
$793$	−1200.69	−0.0537677
$794$	−42417.4	−1.89589
$795$	−19008.1	−0.847985
$796$	−14755.8	−0.657043
$797$	−19066.3	−0.847382	−0.423691	−	0.905807i	$-0.639266\pi$
$797$	−19066.3	−0.847382	−0.423691	+	0.905807i	$0.639266\pi$
$798$	−5842.86	−0.259192
$799$	2311.07	0.102328
$800$	9382.27	0.414642
$801$	−673.065	−0.0296899
$802$	−24261.2	−1.06819
$803$	289.197	0.0127093
$804$	−9042.40	−0.396643
$805$	−16587.6	−0.726257
$806$	12547.2	0.548334
$807$	5281.50	0.230381
$808$	92645.3	4.03373
$809$	−16772.8	−0.728926	−0.364463	−	0.931218i	$-0.618748\pi$
$809$	−16772.8	−0.728926	−0.364463	+	0.931218i	$0.618748\pi$
$810$	−3840.29	−0.166585
$811$	31810.3	1.37732	0.688662	−	0.725083i	$-0.258198\pi$
$811$	31810.3	1.37732	0.688662	+	0.725083i	$0.258198\pi$
$812$	−35958.0	−1.55404
$813$	20016.5	0.863479
$814$	11592.3	0.499151
$815$	−6282.27	−0.270010
$816$	1573.48	0.0675036
$817$	4667.01	0.199851
$818$	80945.6	3.45990
$819$	1885.62	0.0804506
$820$	6854.79	0.291926
$821$	−20977.6	−0.891746	−0.445873	−	0.895096i	$-0.647107\pi$
$821$	−20977.6	−0.891746	−0.445873	+	0.895096i	$0.647107\pi$
$822$	25883.4	1.09828
$823$	−20075.4	−0.850285	−0.425143	−	0.905126i	$-0.639776\pi$
$823$	−20075.4	−0.850285	−0.425143	+	0.905126i	$0.639776\pi$
$824$	24898.4	1.05264
$825$	1322.16	0.0557962
$826$	−649.419	−0.0273562
$827$	−21038.2	−0.884607	−0.442304	−	0.896865i	$-0.645839\pi$
$827$	−21038.2	−0.884607	−0.442304	+	0.896865i	$0.645839\pi$
$828$	−28101.3	−1.17946
$829$	42251.1	1.77014	0.885068	−	0.465462i	$-0.154112\pi$
$829$	42251.1	1.77014	0.885068	+	0.465462i	$0.154112\pi$
$830$	14154.3	0.591931
$831$	9802.91	0.409217
$832$	3361.12	0.140055
$833$	−932.192	−0.0387738
$834$	33780.6	1.40255
$835$	2143.74	0.0888470
$836$	−7224.41	−0.298877
$837$	−3345.59	−0.138161
$838$	27478.1	1.13271
$839$	13681.3	0.562970	0.281485	−	0.959566i	$-0.409173\pi$
$839$	13681.3	0.562970	0.281485	+	0.959566i	$0.409173\pi$
$840$	−15843.7	−0.650786
$841$	9081.61	0.372365
$842$	13376.1	0.547473
$843$	25475.9	1.04085
$844$	20931.9	0.853681
$845$	−16676.9	−0.678938
$846$	−26366.6	−1.07151
$847$	1287.94	0.0522482
$848$	−88853.8	−3.59818
$849$	−11474.9	−0.463861
$850$	836.467	0.0337536
$851$	−34639.3	−1.39532
$852$	−54398.6	−2.18740
$853$	36361.4	1.45954	0.729772	−	0.683691i	$-0.239626\pi$
$853$	36361.4	1.45954	0.729772	+	0.683691i	$0.239626\pi$
$854$	3340.24	0.133842
$855$	2950.13	0.118003
$856$	−11134.4	−0.444588
$857$	−20297.6	−0.809045	−0.404522	−	0.914528i	$-0.632562\pi$
$857$	−20297.6	−0.809045	−0.404522	+	0.914528i	$0.632562\pi$
$858$	3341.59	0.132960
$859$	25290.0	1.00452	0.502260	−	0.864717i	$-0.332502\pi$
$859$	25290.0	1.00452	0.502260	+	0.864717i	$0.332502\pi$
$860$	22329.4	0.885381
$861$	1286.27	0.0509127
$862$	36025.2	1.42346
$863$	−16420.5	−0.647695	−0.323848	−	0.946109i	$-0.604977\pi$
$863$	−16420.5	−0.647695	−0.323848	+	0.946109i	$0.604977\pi$
$864$	−6322.67	−0.248960
$865$	−35565.9	−1.39801
$866$	25688.8	1.00802
$867$	−14689.6	−0.575415
$868$	−24354.1	−0.952342
$869$	9012.44	0.351814
$870$	26021.5	1.01404
$871$	−3213.00	−0.124992
$872$	−44981.9	−1.74688
$873$	−4112.41	−0.159432
$874$	30940.3	1.19745
$875$	−16192.3	−0.625600
$876$	−1456.38	−0.0561720
$877$	−33488.7	−1.28943	−0.644716	−	0.764422i	$-0.723024\pi$
$877$	−33488.7	−1.28943	−0.644716	+	0.764422i	$0.723024\pi$
$878$	−26478.2	−1.01776
$879$	13379.6	0.513405
$880$	−13101.9	−0.501893
$881$	13778.3	0.526904	0.263452	−	0.964673i	$-0.415139\pi$
$881$	13778.3	0.526904	0.263452	+	0.964673i	$0.415139\pi$
$882$	10635.2	0.406015
$883$	−46334.0	−1.76587	−0.882936	−	0.469493i	$-0.844437\pi$
$883$	−46334.0	−1.76587	−0.882936	+	0.469493i	$0.844437\pi$
$884$	1475.01	0.0561199
$885$	327.899	0.0124545
$886$	−51144.5	−1.93931
$887$	−16143.8	−0.611111	−0.305556	−	0.952174i	$-0.598842\pi$
$887$	−16143.8	−0.611111	−0.305556	+	0.952174i	$0.598842\pi$
$888$	−33085.9	−1.25033
$889$	6044.10	0.228023
$890$	3545.63	0.133539
$891$	−891.000	−0.0335013
$892$	−65990.8	−2.47706
$893$	20254.9	0.759019
$894$	−18127.8	−0.678171
$895$	−1262.11	−0.0471371
$896$	10590.2	0.394858
$897$	−9985.14	−0.371677
$898$	60827.5	2.26040
$899$	22669.4	0.841010
$900$	−6658.36	−0.246606
$901$	−2790.07	−0.103164
$902$	2279.44	0.0841432
$903$	4190.00	0.154413
$904$	71046.0	2.61389
$905$	22831.7	0.838620
$906$	−41848.7	−1.53458
$907$	46508.0	1.70262	0.851308	−	0.524667i	$-0.175810\pi$
$907$	46508.0	1.70262	0.851308	+	0.524667i	$0.175810\pi$
$908$	−42644.7	−1.55861
$909$	−15487.5	−0.565115
$910$	−9933.26	−0.361851
$911$	5240.13	0.190575	0.0952873	−	0.995450i	$-0.469623\pi$
$911$	5240.13	0.190575	0.0952873	+	0.995450i	$0.469623\pi$
$912$	13790.4	0.500710
$913$	3283.99	0.119041
$914$	18923.0	0.684813
$915$	−1686.53	−0.0609342
$916$	96409.8	3.47759
$917$	11822.2	0.425739
$918$	−563.691	−0.0202664
$919$	−43107.1	−1.54731	−0.773653	−	0.633610i	$-0.781572\pi$
$919$	−43107.1	−1.54731	−0.773653	+	0.633610i	$0.781572\pi$
$920$	83899.0	3.00659
$921$	3629.44	0.129852
$922$	−50111.4	−1.78995
$923$	−19329.3	−0.689307
$924$	−6486.01	−0.230924
$925$	−8207.48	−0.291741
$926$	4014.44	0.142465
$927$	−4162.27	−0.147472
$928$	42841.9	1.51547
$929$	49974.6	1.76492	0.882461	−	0.470385i	$-0.155885\pi$
$929$	49974.6	1.76492	0.882461	+	0.470385i	$0.155885\pi$
$930$	17624.2	0.621419
$931$	−8169.99	−0.287605
$932$	−3975.23	−0.139714
$933$	−107.801	−0.00378268
$934$	−55136.7	−1.93161
$935$	−411.410	−0.0143899
$936$	−9537.35	−0.333053
$937$	28929.2	1.00862	0.504310	−	0.863523i	$-0.331747\pi$
$937$	28929.2	1.00862	0.504310	+	0.863523i	$0.331747\pi$
$938$	8938.35	0.311138
$939$	23483.5	0.816138
$940$	96910.0	3.36261
$941$	14004.2	0.485147	0.242574	−	0.970133i	$-0.422008\pi$
$941$	14004.2	0.485147	0.242574	+	0.970133i	$0.422008\pi$
$942$	−3098.27	−0.107163
$943$	−6811.30	−0.235214
$944$	1532.77	0.0528469
$945$	2648.60	0.0911735
$946$	7425.27	0.255197
$947$	9858.99	0.338304	0.169152	−	0.985590i	$-0.445897\pi$
$947$	9858.99	0.338304	0.169152	+	0.985590i	$0.445897\pi$
$948$	−45386.2	−1.55493
$949$	−517.491	−0.0177012
$950$	7331.03	0.250369
$951$	−1969.10	−0.0671425
$952$	−2325.60	−0.0791733
$953$	46549.0	1.58223	0.791117	−	0.611664i	$-0.209500\pi$
$953$	46549.0	1.58223	0.791117	+	0.611664i	$0.209500\pi$
$954$	31831.4	1.08027
$955$	11491.6	0.389380
$956$	14684.7	0.496795
$957$	6037.34	0.203928
$958$	−75403.6	−2.54299
$959$	−17851.5	−0.601099
$960$	4721.12	0.158722
$961$	−14437.1	−0.484614
$962$	−20743.3	−0.695208
$963$	1861.35	0.0622856
$964$	48623.3	1.62454
$965$	47301.9	1.57793
$966$	27778.0	0.925199
$967$	−55564.7	−1.84782	−0.923908	−	0.382614i	$-0.875024\pi$
$967$	−55564.7	−1.84782	−0.923908	+	0.382614i	$0.875024\pi$
$968$	−6514.32	−0.216300
$969$	433.030	0.0143560
$970$	21663.7	0.717094
$971$	5981.43	0.197686	0.0988431	−	0.995103i	$-0.468486\pi$
$971$	5981.43	0.197686	0.0988431	+	0.995103i	$0.468486\pi$
$972$	4487.04	0.148068
$973$	−23298.0	−0.767627
$974$	−30060.1	−0.988898
$975$	−2365.89	−0.0777119
$976$	−7883.70	−0.258557
$977$	24604.4	0.805697	0.402848	−	0.915267i	$-0.368020\pi$
$977$	24604.4	0.805697	0.402848	+	0.915267i	$0.368020\pi$
$978$	10520.4	0.343973
$979$	822.635	0.0268555
$980$	−39089.5	−1.27415
$981$	7519.63	0.244733
$982$	42668.8	1.38658
$983$	−0.0564529	−1.83171e−6 0	−9.15853e−7	−	1.00000i	$-0.500000\pi$
$983$	−0.0564529	−1.83171e−6 0	−9.15853e−7		1.00000i	$0.500000\pi$
$984$	−6505.84	−0.210771
$985$	−12562.6	−0.406373
$986$	3819.52	0.123366
$987$	18184.7	0.586449
$988$	12927.4	0.416271
$989$	−22187.8	−0.713377
$990$	4693.69	0.150682
$991$	−46877.5	−1.50264	−0.751318	−	0.659941i	$-0.770581\pi$
$991$	−46877.5	−1.50264	−0.751318	+	0.659941i	$0.770581\pi$
$992$	29016.5	0.928706
$993$	29720.4	0.949796
$994$	53772.6	1.71586
$995$	−7364.65	−0.234648
$996$	−16538.0	−0.526132
$997$	24439.0	0.776321	0.388161	−	0.921592i	$-0.373111\pi$
$997$	24439.0	0.776321	0.388161	+	0.921592i	$0.373111\pi$
$998$	63706.5	2.02063
$999$	5530.98	0.175168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.c.1.3	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.c.1.3	✓	37	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-5.14443 q^{2} +3.00000 q^{3} +18.4652 q^{4} +9.21599 q^{5} -15.4333 q^{6} +10.6441 q^{7} -53.8373 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.14443 q^{2} +3.00000 q^{3} +18.4652 q^{4} +9.21599 q^{5} -15.4333 q^{6} +10.6441 q^{7} -53.8373 q^{8} +9.00000 q^{9} -47.4110 q^{10} -11.0000 q^{11} +55.3955 q^{12} +19.6835 q^{13} -54.7581 q^{14} +27.6480 q^{15} +129.241 q^{16} +4.05826 q^{17} -46.2999 q^{18} +35.5678 q^{19} +170.175 q^{20} +31.9324 q^{21} +56.5887 q^{22} -169.095 q^{23} -161.512 q^{24} -40.0656 q^{25} -101.260 q^{26} +27.0000 q^{27} +196.546 q^{28} -182.950 q^{29} -142.233 q^{30} -123.911 q^{31} -234.173 q^{32} -33.0000 q^{33} -20.8775 q^{34} +98.0963 q^{35} +166.186 q^{36} +204.851 q^{37} -182.976 q^{38} +59.0504 q^{39} -496.164 q^{40} +40.2809 q^{41} -164.274 q^{42} +131.215 q^{43} -203.117 q^{44} +82.9439 q^{45} +869.899 q^{46} +569.474 q^{47} +387.723 q^{48} -229.702 q^{49} +206.115 q^{50} +12.1748 q^{51} +363.459 q^{52} -687.505 q^{53} -138.900 q^{54} -101.376 q^{55} -573.052 q^{56} +106.703 q^{57} +941.172 q^{58} +11.8598 q^{59} +510.524 q^{60} -61.0000 q^{61} +637.450 q^{62} +95.7973 q^{63} +170.758 q^{64} +181.403 q^{65} +169.766 q^{66} -163.233 q^{67} +74.9365 q^{68} -507.286 q^{69} -504.650 q^{70} -982.004 q^{71} -484.536 q^{72} -26.2907 q^{73} -1053.84 q^{74} -120.197 q^{75} +656.764 q^{76} -117.086 q^{77} -303.781 q^{78} -819.312 q^{79} +1191.08 q^{80} +81.0000 q^{81} -207.222 q^{82} -298.545 q^{83} +589.638 q^{84} +37.4009 q^{85} -675.025 q^{86} -548.849 q^{87} +592.211 q^{88} -74.7850 q^{89} -426.699 q^{90} +209.514 q^{91} -3122.37 q^{92} -371.732 q^{93} -2929.62 q^{94} +327.792 q^{95} -702.519 q^{96} -456.935 q^{97} +1181.69 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9}+O(q^{10}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9} - 41 q^{10} - 407 q^{11} + 390 q^{12} + 51 q^{13} - 228 q^{14} - 105 q^{15} + 462 q^{16} - 190 q^{17} - 72 q^{18} - 51 q^{19} - 720 q^{20} - 105 q^{21} + 88 q^{22} - 583 q^{23} - 351 q^{24} + 598 q^{25} - 1019 q^{26} + 999 q^{27} - 498 q^{28} - 566 q^{29} - 123 q^{30} - 696 q^{31} - 859 q^{32} - 1221 q^{33} - 348 q^{34} - 1102 q^{35} + 1170 q^{36} - 1022 q^{37} - 455 q^{38} + 153 q^{39} - 503 q^{40} - 790 q^{41} - 684 q^{42} - 87 q^{43} - 1430 q^{44} - 315 q^{45} - 303 q^{46} - 1603 q^{47} + 1386 q^{48} + 110 q^{49} - 1926 q^{50} - 570 q^{51} + 736 q^{52} - 2619 q^{53} - 216 q^{54} + 385 q^{55} - 4937 q^{56} - 153 q^{57} - 1099 q^{58} - 2471 q^{59} - 2160 q^{60} - 2257 q^{61} - 2909 q^{62} - 315 q^{63} - 265 q^{64} - 1970 q^{65} + 264 q^{66} - 3033 q^{67} - 1956 q^{68} - 1749 q^{69} + 2410 q^{70} - 3891 q^{71} - 1053 q^{72} + 391 q^{73} - 532 q^{74} + 1794 q^{75} + 1554 q^{76} + 385 q^{77} - 3057 q^{78} + 67 q^{79} - 5111 q^{80} + 2997 q^{81} - 4818 q^{82} - 5315 q^{83} - 1494 q^{84} - 2747 q^{85} - 5195 q^{86} - 1698 q^{87} + 1287 q^{88} - 8945 q^{89} - 369 q^{90} - 4432 q^{91} - 4701 q^{92} - 2088 q^{93} - 372 q^{94} - 3388 q^{95} - 2577 q^{96} - 3784 q^{97} - 4502 q^{98} - 3663 q^{99}+O(q^{100}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9 - 41 * q^10 - 407 * q^11 + 390 * q^12 + 51 * q^13 - 228 * q^14 - 105 * q^15 + 462 * q^16 - 190 * q^17 - 72 * q^18 - 51 * q^19 - 720 * q^20 - 105 * q^21 + 88 * q^22 - 583 * q^23 - 351 * q^24 + 598 * q^25 - 1019 * q^26 + 999 * q^27 - 498 * q^28 - 566 * q^29 - 123 * q^30 - 696 * q^31 - 859 * q^32 - 1221 * q^33 - 348 * q^34 - 1102 * q^35 + 1170 * q^36 - 1022 * q^37 - 455 * q^38 + 153 * q^39 - 503 * q^40 - 790 * q^41 - 684 * q^42 - 87 * q^43 - 1430 * q^44 - 315 * q^45 - 303 * q^46 - 1603 * q^47 + 1386 * q^48 + 110 * q^49 - 1926 * q^50 - 570 * q^51 + 736 * q^52 - 2619 * q^53 - 216 * q^54 + 385 * q^55 - 4937 * q^56 - 153 * q^57 - 1099 * q^58 - 2471 * q^59 - 2160 * q^60 - 2257 * q^61 - 2909 * q^62 - 315 * q^63 - 265 * q^64 - 1970 * q^65 + 264 * q^66 - 3033 * q^67 - 1956 * q^68 - 1749 * q^69 + 2410 * q^70 - 3891 * q^71 - 1053 * q^72 + 391 * q^73 - 532 * q^74 + 1794 * q^75 + 1554 * q^76 + 385 * q^77 - 3057 * q^78 + 67 * q^79 - 5111 * q^80 + 2997 * q^81 - 4818 * q^82 - 5315 * q^83 - 1494 * q^84 - 2747 * q^85 - 5195 * q^86 - 1698 * q^87 + 1287 * q^88 - 8945 * q^89 - 369 * q^90 - 4432 * q^91 - 4701 * q^92 - 2088 * q^93 - 372 * q^94 - 3388 * q^95 - 2577 * q^96 - 3784 * q^97 - 4502 * q^98 - 3663 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more