LMFDB - Embedded newform 1045.4.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58489 q^{2} -6.85881 q^{3} -5.48811 q^{4} -5.00000 q^{5} +10.8705 q^{6} +9.66724 q^{7} +21.3772 q^{8} +20.0432 q^{9} +O(q^{10})$	\(q-1.58489 q^{2} -6.85881 q^{3} -5.48811 q^{4} -5.00000 q^{5} +10.8705 q^{6} +9.66724 q^{7} +21.3772 q^{8} +20.0432 q^{9} +7.92447 q^{10} -11.0000 q^{11} +37.6419 q^{12} +78.5908 q^{13} -15.3216 q^{14} +34.2940 q^{15} +10.0243 q^{16} +81.7208 q^{17} -31.7664 q^{18} -19.0000 q^{19} +27.4406 q^{20} -66.3058 q^{21} +17.4338 q^{22} +71.6196 q^{23} -146.622 q^{24} +25.0000 q^{25} -124.558 q^{26} +47.7152 q^{27} -53.0549 q^{28} -81.6983 q^{29} -54.3524 q^{30} -96.0788 q^{31} -186.905 q^{32} +75.4469 q^{33} -129.519 q^{34} -48.3362 q^{35} -109.999 q^{36} +198.018 q^{37} +30.1130 q^{38} -539.039 q^{39} -106.886 q^{40} -55.4800 q^{41} +105.088 q^{42} +11.3487 q^{43} +60.3692 q^{44} -100.216 q^{45} -113.510 q^{46} +130.722 q^{47} -68.7544 q^{48} -249.544 q^{49} -39.6223 q^{50} -560.507 q^{51} -431.315 q^{52} -407.239 q^{53} -75.6235 q^{54} +55.0000 q^{55} +206.659 q^{56} +130.317 q^{57} +129.483 q^{58} +886.975 q^{59} -188.209 q^{60} +242.630 q^{61} +152.275 q^{62} +193.763 q^{63} +216.031 q^{64} -392.954 q^{65} -119.575 q^{66} -852.725 q^{67} -448.493 q^{68} -491.225 q^{69} +76.6078 q^{70} +541.369 q^{71} +428.468 q^{72} -193.188 q^{73} -313.837 q^{74} -171.470 q^{75} +104.274 q^{76} -106.340 q^{77} +854.320 q^{78} +564.204 q^{79} -50.1213 q^{80} -868.436 q^{81} +87.9299 q^{82} -400.040 q^{83} +363.893 q^{84} -408.604 q^{85} -17.9864 q^{86} +560.353 q^{87} -235.149 q^{88} -835.514 q^{89} +158.832 q^{90} +759.757 q^{91} -393.057 q^{92} +658.986 q^{93} -207.180 q^{94} +95.0000 q^{95} +1281.95 q^{96} -1625.97 q^{97} +395.501 q^{98} -220.475 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * 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 $

$2$	−1.58489	−0.560345	−0.280172	−	0.959950i	$-0.590392\pi$
$2$	−1.58489	−0.560345	−0.280172	+	0.959950i	$0.590392\pi$
$3$	−6.85881	−1.31998	−0.659989	−	0.751275i	$-0.729439\pi$
$3$	−6.85881	−1.31998	−0.659989	+	0.751275i	$0.729439\pi$
$4$	−5.48811	−0.686014
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	10.8705	0.739642
$7$	9.66724	0.521982	0.260991	−	0.965341i	$-0.415951\pi$
$7$	9.66724	0.521982	0.260991	+	0.965341i	$0.415951\pi$
$8$	21.3772	0.944749
$9$	20.0432	0.742342
$10$	7.92447	0.250594
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	37.6419	0.905523
$13$	78.5908	1.67671	0.838353	−	0.545128i	$-0.183519\pi$
$13$	78.5908	1.67671	0.838353	+	0.545128i	$0.183519\pi$
$14$	−15.3216	−0.292490
$15$	34.2940	0.590312
$16$	10.0243	0.156629
$17$	81.7208	1.16590	0.582948	−	0.812510i	$-0.301899\pi$
$17$	81.7208	1.16590	0.582948	+	0.812510i	$0.301899\pi$
$18$	−31.7664	−0.415967
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	27.4406	0.306795
$21$	−66.3058	−0.689005
$22$	17.4338	0.168950
$23$	71.6196	0.649292	0.324646	−	0.945836i	$-0.394755\pi$
$23$	71.6196	0.649292	0.324646	+	0.945836i	$0.394755\pi$
$24$	−146.622	−1.24705
$25$	25.0000	0.200000
$26$	−124.558	−0.939533
$27$	47.7152	0.340103
$28$	−53.0549	−0.358087
$29$	−81.6983	−0.523138	−0.261569	−	0.965185i	$-0.584240\pi$
$29$	−81.6983	−0.523138	−0.261569	+	0.965185i	$0.584240\pi$
$30$	−54.3524	−0.330778
$31$	−96.0788	−0.556654	−0.278327	−	0.960486i	$-0.589780\pi$
$31$	−96.0788	−0.556654	−0.278327	+	0.960486i	$0.589780\pi$
$32$	−186.905	−1.03252
$33$	75.4469	0.397988
$34$	−129.519	−0.653303
$35$	−48.3362	−0.233438
$36$	−109.999	−0.509257
$37$	198.018	0.879836	0.439918	−	0.898038i	$-0.355008\pi$
$37$	198.018	0.879836	0.439918	+	0.898038i	$0.355008\pi$
$38$	30.1130	0.128552
$39$	−539.039	−2.21321
$40$	−106.886	−0.422505
$41$	−55.4800	−0.211330	−0.105665	−	0.994402i	$-0.533697\pi$
$41$	−55.4800	−0.211330	−0.105665	+	0.994402i	$0.533697\pi$
$42$	105.088	0.386080
$43$	11.3487	0.0402478	0.0201239	−	0.999797i	$-0.493594\pi$
$43$	11.3487	0.0402478	0.0201239	+	0.999797i	$0.493594\pi$
$44$	60.3692	0.206841
$45$	−100.216	−0.331985
$46$	−113.510	−0.363827
$47$	130.722	0.405696	0.202848	−	0.979210i	$-0.434980\pi$
$47$	130.722	0.405696	0.202848	+	0.979210i	$0.434980\pi$
$48$	−68.7544	−0.206747
$49$	−249.544	−0.727535
$50$	−39.6223	−0.112069
$51$	−560.507	−1.53896
$52$	−431.315	−1.15024
$53$	−407.239	−1.05545	−0.527723	−	0.849417i	$-0.676954\pi$
$53$	−407.239	−1.05545	−0.527723	+	0.849417i	$0.676954\pi$
$54$	−75.6235	−0.190575
$55$	55.0000	0.134840
$56$	206.659	0.493142
$57$	130.317	0.302824
$58$	129.483	0.293137
$59$	886.975	1.95719	0.978596	−	0.205789i	$-0.0659761\pi$
$59$	886.975	1.95719	0.978596	+	0.205789i	$0.0659761\pi$
$60$	−188.209	−0.404962
$61$	242.630	0.509271	0.254636	−	0.967037i	$-0.418044\pi$
$61$	242.630	0.509271	0.254636	+	0.967037i	$0.418044\pi$
$62$	152.275	0.311918
$63$	193.763	0.387489
$64$	216.031	0.421935
$65$	−392.954	−0.749845
$66$	−119.575	−0.223011
$67$	−852.725	−1.55488	−0.777440	−	0.628958i	$-0.783482\pi$
$67$	−852.725	−1.55488	−0.777440	+	0.628958i	$0.783482\pi$
$68$	−448.493	−0.799820
$69$	−491.225	−0.857052
$70$	76.6078	0.130805
$71$	541.369	0.904910	0.452455	−	0.891787i	$-0.350548\pi$
$71$	541.369	0.904910	0.452455	+	0.891787i	$0.350548\pi$
$72$	428.468	0.701326
$73$	−193.188	−0.309739	−0.154870	−	0.987935i	$-0.549496\pi$
$73$	−193.188	−0.309739	−0.154870	+	0.987935i	$0.549496\pi$
$74$	−313.837	−0.493011
$75$	−171.470	−0.263996
$76$	104.274	0.157382
$77$	−106.340	−0.157384
$78$	854.320	1.24016
$79$	564.204	0.803517	0.401759	−	0.915746i	$-0.368399\pi$
$79$	564.204	0.803517	0.401759	+	0.915746i	$0.368399\pi$
$80$	−50.1213	−0.0700466
$81$	−868.436	−1.19127
$82$	87.9299	0.118417
$83$	−400.040	−0.529038	−0.264519	−	0.964381i	$-0.585213\pi$
$83$	−400.040	−0.529038	−0.264519	+	0.964381i	$0.585213\pi$
$84$	363.893	0.472667
$85$	−408.604	−0.521404
$86$	−17.9864	−0.0225526
$87$	560.353	0.690530
$88$	−235.149	−0.284852
$89$	−835.514	−0.995105	−0.497553	−	0.867434i	$-0.665768\pi$
$89$	−835.514	−0.995105	−0.497553	+	0.867434i	$0.665768\pi$
$90$	158.832	0.186026
$91$	759.757	0.875210
$92$	−393.057	−0.445424
$93$	658.986	0.734770
$94$	−207.180	−0.227329
$95$	95.0000	0.102598
$96$	1281.95	1.36290
$97$	−1625.97	−1.70198	−0.850992	−	0.525179i	$-0.823999\pi$
$97$	−1625.97	−1.70198	−0.850992	+	0.525179i	$0.823999\pi$
$98$	395.501	0.407670
$99$	−220.475	−0.223824
$100$	−137.203	−0.137203
$101$	646.654	0.637074	0.318537	−	0.947910i	$-0.396809\pi$
$101$	646.654	0.637074	0.318537	+	0.947910i	$0.396809\pi$
$102$	888.345	0.862345
$103$	766.605	0.733358	0.366679	−	0.930348i	$-0.380495\pi$
$103$	766.605	0.733358	0.366679	+	0.930348i	$0.380495\pi$
$104$	1680.05	1.58407
$105$	331.529	0.308132
$106$	645.431	0.591413
$107$	1509.07	1.36343	0.681715	−	0.731618i	$-0.261235\pi$
$107$	1509.07	1.36343	0.681715	+	0.731618i	$0.261235\pi$
$108$	−261.866	−0.233316
$109$	283.046	0.248724	0.124362	−	0.992237i	$-0.460312\pi$
$109$	283.046	0.248724	0.124362	+	0.992237i	$0.460312\pi$
$110$	−87.1692	−0.0755569
$111$	−1358.17	−1.16136
$112$	96.9070	0.0817576
$113$	−890.144	−0.741041	−0.370521	−	0.928824i	$-0.620821\pi$
$113$	−890.144	−0.741041	−0.370521	+	0.928824i	$0.620821\pi$
$114$	−206.539	−0.169686
$115$	−358.098	−0.290372
$116$	448.369	0.358880
$117$	1575.21	1.24469
$118$	−1405.76	−1.09670
$119$	790.015	0.608576
$120$	733.111	0.557697
$121$	121.000	0.0909091
$122$	−384.542	−0.285368
$123$	380.526	0.278951
$124$	527.291	0.381872
$125$	−125.000	−0.0894427
$126$	−307.093	−0.217127
$127$	1951.07	1.36322	0.681611	−	0.731715i	$-0.261280\pi$
$127$	1951.07	1.36322	0.681611	+	0.731715i	$0.261280\pi$
$128$	1152.86	0.796086
$129$	−77.8382	−0.0531262
$130$	622.790	0.420172
$131$	1136.20	0.757790	0.378895	−	0.925440i	$-0.376304\pi$
$131$	1136.20	0.757790	0.378895	+	0.925440i	$0.376304\pi$
$132$	−414.061	−0.273026
$133$	−183.678	−0.119751
$134$	1351.48	0.871268
$135$	−238.576	−0.152099
$136$	1746.96	1.10148
$137$	1157.64	0.721924	0.360962	−	0.932581i	$-0.382448\pi$
$137$	1157.64	0.721924	0.360962	+	0.932581i	$0.382448\pi$
$138$	778.540	0.480244
$139$	−2977.28	−1.81676	−0.908378	−	0.418149i	$-0.862679\pi$
$139$	−2977.28	−1.81676	−0.908378	+	0.418149i	$0.862679\pi$
$140$	265.275	0.160141
$141$	−896.594	−0.535509
$142$	−858.012	−0.507062
$143$	−864.499	−0.505546
$144$	200.918	0.116272
$145$	408.491	0.233954
$146$	306.183	0.173561
$147$	1711.58	0.960330
$148$	−1086.74	−0.603580
$149$	551.030	0.302967	0.151484	−	0.988460i	$-0.451595\pi$
$149$	551.030	0.302967	0.151484	+	0.988460i	$0.451595\pi$
$150$	271.762	0.147928
$151$	3193.85	1.72127	0.860636	−	0.509221i	$-0.170066\pi$
$151$	3193.85	1.72127	0.860636	+	0.509221i	$0.170066\pi$
$152$	−406.167	−0.216740
$153$	1637.95	0.865492
$154$	168.537	0.0881890
$155$	480.394	0.248943
$156$	2958.31	1.51830
$157$	3083.16	1.56728	0.783640	−	0.621215i	$-0.213361\pi$
$157$	3083.16	1.56728	0.783640	+	0.621215i	$0.213361\pi$
$158$	−894.203	−0.450247
$159$	2793.18	1.39317
$160$	934.526	0.461755
$161$	692.364	0.338919
$162$	1376.38	0.667522
$163$	972.849	0.467481	0.233741	−	0.972299i	$-0.424903\pi$
$163$	972.849	0.467481	0.233741	+	0.972299i	$0.424903\pi$
$164$	304.480	0.144975
$165$	−377.234	−0.177986
$166$	634.022	0.296443
$167$	−1532.64	−0.710174	−0.355087	−	0.934833i	$-0.615549\pi$
$167$	−1532.64	−0.710174	−0.355087	+	0.934833i	$0.615549\pi$
$168$	−1417.43	−0.650937
$169$	3979.51	1.81134
$170$	647.594	0.292166
$171$	−380.821	−0.170305
$172$	−62.2827	−0.0276105
$173$	−754.487	−0.331576	−0.165788	−	0.986161i	$-0.553017\pi$
$173$	−754.487	−0.331576	−0.165788	+	0.986161i	$0.553017\pi$
$174$	−888.100	−0.386935
$175$	241.681	0.104396
$176$	−110.267	−0.0472254
$177$	−6083.59	−2.58345
$178$	1324.20	0.557602
$179$	−3608.69	−1.50685	−0.753425	−	0.657534i	$-0.771600\pi$
$179$	−3608.69	−1.50685	−0.753425	+	0.657534i	$0.771600\pi$
$180$	549.997	0.227747
$181$	−2645.89	−1.08656	−0.543281	−	0.839551i	$-0.682818\pi$
$181$	−2645.89	−1.08656	−0.543281	+	0.839551i	$0.682818\pi$
$182$	−1204.13	−0.490419
$183$	−1664.15	−0.672227
$184$	1531.03	0.613418
$185$	−990.089	−0.393475
$186$	−1044.42	−0.411725
$187$	−898.929	−0.351531
$188$	−717.414	−0.278313
$189$	461.274	0.177528
$190$	−150.565	−0.0574901
$191$	4986.08	1.88890	0.944451	−	0.328653i	$-0.106595\pi$
$191$	4986.08	1.88890	0.944451	+	0.328653i	$0.106595\pi$
$192$	−1481.71	−0.556945
$193$	−491.839	−0.183437	−0.0917186	−	0.995785i	$-0.529236\pi$
$193$	−491.839	−0.183437	−0.0917186	+	0.995785i	$0.529236\pi$
$194$	2576.99	0.953698
$195$	2695.20	0.989779
$196$	1369.53	0.499099
$197$	1250.70	0.452330	0.226165	−	0.974089i	$-0.427381\pi$
$197$	1250.70	0.452330	0.226165	+	0.974089i	$0.427381\pi$
$198$	349.430	0.125419
$199$	−3315.14	−1.18092	−0.590462	−	0.807065i	$-0.701055\pi$
$199$	−3315.14	−1.18092	−0.590462	+	0.807065i	$0.701055\pi$
$200$	534.431	0.188950
$201$	5848.67	2.05241
$202$	−1024.88	−0.356981
$203$	−789.797	−0.273068
$204$	3076.13	1.05575
$205$	277.400	0.0945095
$206$	−1214.99	−0.410933
$207$	1435.49	0.481997
$208$	787.815	0.262621
$209$	209.000	0.0691714
$210$	−525.438	−0.172660
$211$	−2206.45	−0.719897	−0.359948	−	0.932972i	$-0.617206\pi$
$211$	−2206.45	−0.719897	−0.359948	+	0.932972i	$0.617206\pi$
$212$	2234.98	0.724051
$213$	−3713.14	−1.19446
$214$	−2391.71	−0.763990
$215$	−56.7433	−0.0179993
$216$	1020.02	0.321312
$217$	−928.817	−0.290563
$218$	−448.598	−0.139371
$219$	1325.04	0.408849
$220$	−301.846	−0.0925021
$221$	6422.51	1.95486
$222$	2152.55	0.650764
$223$	−5141.07	−1.54382	−0.771910	−	0.635732i	$-0.780698\pi$
$223$	−5141.07	−1.54382	−0.771910	+	0.635732i	$0.780698\pi$
$224$	−1806.86	−0.538954
$225$	501.081	0.148468
$226$	1410.78	0.415239
$227$	2310.03	0.675428	0.337714	−	0.941249i	$-0.390346\pi$
$227$	2310.03	0.675428	0.337714	+	0.941249i	$0.390346\pi$
$228$	−715.196	−0.207741
$229$	−3907.18	−1.12748	−0.563742	−	0.825951i	$-0.690639\pi$
$229$	−3907.18	−1.12748	−0.563742	+	0.825951i	$0.690639\pi$
$230$	567.548	0.162709
$231$	729.363	0.207743
$232$	−1746.48	−0.494234
$233$	1464.28	0.411709	0.205854	−	0.978583i	$-0.434003\pi$
$233$	1464.28	0.411709	0.205854	+	0.978583i	$0.434003\pi$
$234$	−2496.55	−0.697454
$235$	−653.608	−0.181433
$236$	−4867.82	−1.34266
$237$	−3869.76	−1.06063
$238$	−1252.09	−0.341012
$239$	−3519.09	−0.952431	−0.476216	−	0.879328i	$-0.657992\pi$
$239$	−3519.09	−0.952431	−0.476216	+	0.879328i	$0.657992\pi$
$240$	343.772	0.0924600
$241$	1456.14	0.389204	0.194602	−	0.980882i	$-0.437659\pi$
$241$	1456.14	0.389204	0.194602	+	0.980882i	$0.437659\pi$
$242$	−191.772	−0.0509404
$243$	4668.13	1.23235
$244$	−1331.58	−0.349367
$245$	1247.72	0.325363
$246$	−603.094	−0.156308
$247$	−1493.23	−0.384663
$248$	−2053.90	−0.525898
$249$	2743.80	0.698318
$250$	198.112	0.0501187
$251$	1551.31	0.390112	0.195056	−	0.980792i	$-0.437511\pi$
$251$	1551.31	0.390112	0.195056	+	0.980792i	$0.437511\pi$
$252$	−1063.39	−0.265823
$253$	−787.816	−0.195769
$254$	−3092.23	−0.763874
$255$	2802.54	0.688242
$256$	−3555.40	−0.868018
$257$	−1638.98	−0.397808	−0.198904	−	0.980019i	$-0.563738\pi$
$257$	−1638.98	−0.397808	−0.198904	+	0.980019i	$0.563738\pi$
$258$	123.365	0.0297690
$259$	1914.29	0.459259
$260$	2156.58	0.514404
$261$	−1637.50	−0.388347
$262$	−1800.76	−0.424623
$263$	−1743.96	−0.408886	−0.204443	−	0.978879i	$-0.565538\pi$
$263$	−1743.96	−0.408886	−0.204443	+	0.978879i	$0.565538\pi$
$264$	1612.84	0.375999
$265$	2036.20	0.472010
$266$	291.110	0.0671018
$267$	5730.63	1.31352
$268$	4679.85	1.06667
$269$	−1810.69	−0.410407	−0.205204	−	0.978719i	$-0.565786\pi$
$269$	−1810.69	−0.410407	−0.205204	+	0.978719i	$0.565786\pi$
$270$	378.118	0.0852278
$271$	4991.50	1.11886	0.559432	−	0.828877i	$-0.311019\pi$
$271$	4991.50	1.11886	0.559432	+	0.828877i	$0.311019\pi$
$272$	819.191	0.182613
$273$	−5211.02	−1.15526
$274$	−1834.73	−0.404526
$275$	−275.000	−0.0603023
$276$	2695.90	0.587949
$277$	4129.25	0.895677	0.447838	−	0.894115i	$-0.352194\pi$
$277$	4129.25	0.895677	0.447838	+	0.894115i	$0.352194\pi$
$278$	4718.67	1.01801
$279$	−1925.73	−0.413227
$280$	−1033.29	−0.220540
$281$	3352.98	0.711822	0.355911	−	0.934520i	$-0.384171\pi$
$281$	3352.98	0.711822	0.355911	+	0.934520i	$0.384171\pi$
$282$	1421.01	0.300070
$283$	1505.65	0.316261	0.158130	−	0.987418i	$-0.449453\pi$
$283$	1505.65	0.316261	0.158130	+	0.987418i	$0.449453\pi$
$284$	−2971.09	−0.620781
$285$	−651.587	−0.135427
$286$	1370.14	0.283280
$287$	−536.339	−0.110310
$288$	−3746.18	−0.766479
$289$	1765.30	0.359311
$290$	−647.416	−0.131095
$291$	11152.2	2.24658
$292$	1060.24	0.212485
$293$	−1788.96	−0.356697	−0.178349	−	0.983967i	$-0.557076\pi$
$293$	−1788.96	−0.356697	−0.178349	+	0.983967i	$0.557076\pi$
$294$	−2712.67	−0.538116
$295$	−4434.88	−0.875283
$296$	4233.07	0.831224
$297$	−524.867	−0.102545
$298$	−873.323	−0.169766
$299$	5628.64	1.08867
$300$	941.047	0.181105
$301$	109.710	0.0210086
$302$	−5061.92	−0.964505
$303$	−4435.27	−0.840923
$304$	−190.461	−0.0359332
$305$	−1213.15	−0.227753
$306$	−2595.98	−0.484974
$307$	6501.91	1.20874	0.604370	−	0.796704i	$-0.293425\pi$
$307$	6501.91	1.20874	0.604370	+	0.796704i	$0.293425\pi$
$308$	583.604	0.107967
$309$	−5258.00	−0.968016
$310$	−761.373	−0.139494
$311$	8020.80	1.46244	0.731219	−	0.682143i	$-0.238952\pi$
$311$	8020.80	1.46244	0.731219	+	0.682143i	$0.238952\pi$
$312$	−11523.2	−2.09093
$313$	−5891.41	−1.06390	−0.531952	−	0.846774i	$-0.678541\pi$
$313$	−5891.41	−1.06390	−0.531952	+	0.846774i	$0.678541\pi$
$314$	−4886.48	−0.878217
$315$	−968.814	−0.173290
$316$	−3096.41	−0.551224
$317$	947.466	0.167871	0.0839354	−	0.996471i	$-0.473251\pi$
$317$	947.466	0.167871	0.0839354	+	0.996471i	$0.473251\pi$
$318$	−4426.89	−0.780653
$319$	898.681	0.157732
$320$	−1080.15	−0.188695
$321$	−10350.4	−1.79970
$322$	−1097.32	−0.189911
$323$	−1552.70	−0.267475
$324$	4766.07	0.817228
$325$	1964.77	0.335341
$326$	−1541.86	−0.261951
$327$	−1941.36	−0.328310
$328$	−1186.01	−0.199653
$329$	1263.72	0.211766
$330$	597.876	0.0997334
$331$	−8979.87	−1.49117	−0.745586	−	0.666409i	$-0.767830\pi$
$331$	−8979.87	−1.49117	−0.745586	+	0.666409i	$0.767830\pi$
$332$	2195.47	0.362927
$333$	3968.91	0.653139
$334$	2429.07	0.397942
$335$	4263.62	0.695363
$336$	−664.666	−0.107918
$337$	8773.91	1.41824	0.709118	−	0.705090i	$-0.249093\pi$
$337$	8773.91	1.41824	0.709118	+	0.705090i	$0.249093\pi$
$338$	−6307.11	−1.01497
$339$	6105.32	0.978158
$340$	2242.47	0.357690
$341$	1056.87	0.167837
$342$	603.561	0.0954294
$343$	−5728.27	−0.901742
$344$	242.603	0.0380240
$345$	2456.13	0.383285
$346$	1195.78	0.185797
$347$	3458.56	0.535058	0.267529	−	0.963550i	$-0.413793\pi$
$347$	3458.56	0.535058	0.267529	+	0.963550i	$0.413793\pi$
$348$	−3075.28	−0.473713
$349$	10969.7	1.68250	0.841250	−	0.540647i	$-0.181820\pi$
$349$	10969.7	1.68250	0.841250	+	0.540647i	$0.181820\pi$
$350$	−383.039	−0.0584980
$351$	3749.97	0.570253
$352$	2055.96	0.311315
$353$	−10610.7	−1.59985	−0.799927	−	0.600097i	$-0.795129\pi$
$353$	−10610.7	−1.59985	−0.799927	+	0.600097i	$0.795129\pi$
$354$	9641.85	1.44762
$355$	−2706.84	−0.404688
$356$	4585.40	0.682656
$357$	−5418.56	−0.803307
$358$	5719.39	0.844355
$359$	−7581.95	−1.11465	−0.557326	−	0.830294i	$-0.688173\pi$
$359$	−7581.95	−1.11465	−0.557326	+	0.830294i	$0.688173\pi$
$360$	−2142.34	−0.313643
$361$	361.000	0.0526316
$362$	4193.46	0.608849
$363$	−829.916	−0.119998
$364$	−4169.63	−0.600406
$365$	965.940	0.138520
$366$	2637.50	0.376679
$367$	−10414.8	−1.48133	−0.740665	−	0.671875i	$-0.765489\pi$
$367$	−10414.8	−1.48133	−0.740665	+	0.671875i	$0.765489\pi$
$368$	717.934	0.101698
$369$	−1112.00	−0.156879
$370$	1569.19	0.220481
$371$	−3936.88	−0.550924
$372$	−3616.59	−0.504063
$373$	7639.28	1.06045	0.530224	−	0.847858i	$-0.322108\pi$
$373$	7639.28	1.06045	0.530224	+	0.847858i	$0.322108\pi$
$374$	1424.71	0.196978
$375$	857.351	0.118062
$376$	2794.46	0.383280
$377$	−6420.73	−0.877148
$378$	−731.071	−0.0994768
$379$	6624.23	0.897794	0.448897	−	0.893583i	$-0.351817\pi$
$379$	6624.23	0.897794	0.448897	+	0.893583i	$0.351817\pi$
$380$	−521.371	−0.0703835
$381$	−13382.0	−1.79942
$382$	−7902.41	−1.05844
$383$	9261.09	1.23556	0.617780	−	0.786351i	$-0.288032\pi$
$383$	9261.09	1.23556	0.617780	+	0.786351i	$0.288032\pi$
$384$	−7907.21	−1.05082
$385$	531.698	0.0703841
$386$	779.513	0.102788
$387$	227.464	0.0298776
$388$	8923.52	1.16758
$389$	−12712.6	−1.65695	−0.828476	−	0.560024i	$-0.810792\pi$
$389$	−12712.6	−1.65695	−0.828476	+	0.560024i	$0.810792\pi$
$390$	−4271.60	−0.554617
$391$	5852.82	0.757007
$392$	−5334.57	−0.687337
$393$	−7792.99	−1.00027
$394$	−1982.23	−0.253461
$395$	−2821.02	−0.359344
$396$	1209.99	0.153547
$397$	−9196.62	−1.16263	−0.581316	−	0.813678i	$-0.697462\pi$
$397$	−9196.62	−1.16263	−0.581316	+	0.813678i	$0.697462\pi$
$398$	5254.14	0.661725
$399$	1259.81	0.158069
$400$	250.606	0.0313258
$401$	−10113.0	−1.25940	−0.629699	−	0.776839i	$-0.716822\pi$
$401$	−10113.0	−1.25940	−0.629699	+	0.776839i	$0.716822\pi$
$402$	−9269.53	−1.15005
$403$	−7550.91	−0.933344
$404$	−3548.91	−0.437041
$405$	4342.18	0.532752
$406$	1251.74	0.153012
$407$	−2178.20	−0.265280
$408$	−11982.1	−1.45393
$409$	6324.91	0.764661	0.382331	−	0.924026i	$-0.375122\pi$
$409$	6324.91	0.764661	0.382331	+	0.924026i	$0.375122\pi$
$410$	−439.649	−0.0529579
$411$	−7940.00	−0.952923
$412$	−4207.22	−0.503094
$413$	8574.61	1.02162
$414$	−2275.10	−0.270084
$415$	2000.20	0.236593
$416$	−14689.0	−1.73122
$417$	20420.6	2.39808
$418$	−331.243	−0.0387598
$419$	10100.2	1.17763	0.588814	−	0.808268i	$-0.299595\pi$
$419$	10100.2	1.17763	0.588814	+	0.808268i	$0.299595\pi$
$420$	−1819.47	−0.211383
$421$	9775.93	1.13171	0.565855	−	0.824505i	$-0.308546\pi$
$421$	9775.93	1.13171	0.565855	+	0.824505i	$0.308546\pi$
$422$	3496.99	0.403390
$423$	2620.08	0.301165
$424$	−8705.65	−0.997131
$425$	2043.02	0.233179
$426$	5884.94	0.669310
$427$	2345.56	0.265831
$428$	−8281.92	−0.935332
$429$	5929.43	0.667309
$430$	89.9321	0.0100858
$431$	1837.65	0.205375	0.102687	−	0.994714i	$-0.467256\pi$
$431$	1837.65	0.205375	0.102687	+	0.994714i	$0.467256\pi$
$432$	478.309	0.0532701
$433$	−4785.69	−0.531145	−0.265572	−	0.964091i	$-0.585561\pi$
$433$	−4785.69	−0.531145	−0.265572	+	0.964091i	$0.585561\pi$
$434$	1472.08	0.162816
$435$	−2801.76	−0.308814
$436$	−1553.39	−0.170628
$437$	−1360.77	−0.148958
$438$	−2100.05	−0.229096
$439$	1759.61	0.191302	0.0956509	−	0.995415i	$-0.469507\pi$
$439$	1759.61	0.191302	0.0956509	+	0.995415i	$0.469507\pi$
$440$	1175.75	0.127390
$441$	−5001.67	−0.540079
$442$	−10179.0	−1.09540
$443$	4239.36	0.454668	0.227334	−	0.973817i	$-0.426999\pi$
$443$	4239.36	0.454668	0.227334	+	0.973817i	$0.426999\pi$
$444$	7453.76	0.796712
$445$	4177.57	0.445025
$446$	8148.05	0.865071
$447$	−3779.41	−0.399910
$448$	2088.42	0.220243
$449$	−6353.94	−0.667842	−0.333921	−	0.942601i	$-0.608372\pi$
$449$	−6353.94	−0.667842	−0.333921	+	0.942601i	$0.608372\pi$
$450$	−794.160	−0.0831934
$451$	610.280	0.0637183
$452$	4885.21	0.508365
$453$	−21906.0	−2.27204
$454$	−3661.15	−0.378472
$455$	−3798.78	−0.391406
$456$	2785.82	0.286092
$457$	−1627.02	−0.166540	−0.0832701	−	0.996527i	$-0.526536\pi$
$457$	−1627.02	−0.166540	−0.0832701	+	0.996527i	$0.526536\pi$
$458$	6192.47	0.631780
$459$	3899.33	0.396525
$460$	1965.28	0.199200
$461$	−593.190	−0.0599297	−0.0299649	−	0.999551i	$-0.509540\pi$
$461$	−593.190	−0.0599297	−0.0299649	+	0.999551i	$0.509540\pi$
$462$	−1155.96	−0.116408
$463$	553.090	0.0555168	0.0277584	−	0.999615i	$-0.491163\pi$
$463$	553.090	0.0555168	0.0277584	+	0.999615i	$0.491163\pi$
$464$	−818.965	−0.0819385
$465$	−3294.93	−0.328599
$466$	−2320.73	−0.230699
$467$	19779.4	1.95991	0.979957	−	0.199210i	$-0.0638376\pi$
$467$	19779.4	1.95991	0.979957	+	0.199210i	$0.0638376\pi$
$468$	−8644.94	−0.853873
$469$	−8243.50	−0.811619
$470$	1035.90	0.101665
$471$	−21146.8	−2.06877
$472$	18961.1	1.84906
$473$	−124.835	−0.0121352
$474$	6133.17	0.594316
$475$	−475.000	−0.0458831
$476$	−4335.69	−0.417492
$477$	−8162.39	−0.783501
$478$	5577.39	0.533690
$479$	6114.89	0.583291	0.291646	−	0.956526i	$-0.405797\pi$
$479$	6114.89	0.583291	0.291646	+	0.956526i	$0.405797\pi$
$480$	−6409.73	−0.609506
$481$	15562.4	1.47523
$482$	−2307.82	−0.218088
$483$	−4748.79	−0.447366
$484$	−664.061	−0.0623649
$485$	8129.86	0.761150
$486$	−7398.48	−0.690539
$487$	2460.21	0.228917	0.114459	−	0.993428i	$-0.463487\pi$
$487$	2460.21	0.228917	0.114459	+	0.993428i	$0.463487\pi$
$488$	5186.75	0.481134
$489$	−6672.58	−0.617065
$490$	−1977.51	−0.182316
$491$	−704.521	−0.0647547	−0.0323774	−	0.999476i	$-0.510308\pi$
$491$	−704.521	−0.0647547	−0.0323774	+	0.999476i	$0.510308\pi$
$492$	−2088.37	−0.191364
$493$	−6676.45	−0.609924
$494$	2366.60	0.215544
$495$	1102.38	0.100097
$496$	−963.119	−0.0871881
$497$	5233.54	0.472347
$498$	−4348.63	−0.391299
$499$	4657.32	0.417817	0.208908	−	0.977935i	$-0.433009\pi$
$499$	4657.32	0.417817	0.208908	+	0.977935i	$0.433009\pi$
$500$	686.014	0.0613590
$501$	10512.1	0.937413
$502$	−2458.67	−0.218597
$503$	6439.02	0.570779	0.285389	−	0.958412i	$-0.407877\pi$
$503$	6439.02	0.570779	0.285389	+	0.958412i	$0.407877\pi$
$504$	4142.11	0.366080
$505$	−3233.27	−0.284908
$506$	1248.60	0.109698
$507$	−27294.7	−2.39093
$508$	−10707.7	−0.935189
$509$	20961.6	1.82536	0.912679	−	0.408677i	$-0.134010\pi$
$509$	20961.6	1.82536	0.912679	+	0.408677i	$0.134010\pi$
$510$	−4441.72	−0.385653
$511$	−1867.60	−0.161678
$512$	−3587.91	−0.309697
$513$	−906.589	−0.0780251
$514$	2597.61	0.222910
$515$	−3833.03	−0.327968
$516$	427.185	0.0364453
$517$	−1437.94	−0.122322
$518$	−3033.94	−0.257343
$519$	5174.88	0.437672
$520$	−8400.27	−0.708416
$521$	1324.26	0.111357	0.0556785	−	0.998449i	$-0.482268\pi$
$521$	1324.26	0.111357	0.0556785	+	0.998449i	$0.482268\pi$
$522$	2595.26	0.217608
$523$	−9359.11	−0.782496	−0.391248	−	0.920285i	$-0.627957\pi$
$523$	−9359.11	−0.782496	−0.391248	+	0.920285i	$0.627957\pi$
$524$	−6235.60	−0.519854
$525$	−1657.64	−0.137801
$526$	2763.98	0.229117
$527$	−7851.64	−0.649000
$528$	756.299	0.0623365
$529$	−7037.63	−0.578419
$530$	−3227.16	−0.264488
$531$	17777.8	1.45291
$532$	1008.04	0.0821508
$533$	−4360.22	−0.354338
$534$	−9082.44	−0.736022
$535$	−7545.33	−0.609744
$536$	−18228.9	−1.46897
$537$	24751.3	1.98901
$538$	2869.75	0.229970
$539$	2744.99	0.219360
$540$	1309.33	0.104342
$541$	2837.60	0.225505	0.112752	−	0.993623i	$-0.464033\pi$
$541$	2837.60	0.225505	0.112752	+	0.993623i	$0.464033\pi$
$542$	−7910.99	−0.626949
$543$	18147.7	1.43424
$544$	−15274.0	−1.20380
$545$	−1415.23	−0.111233
$546$	8258.92	0.647343
$547$	4044.79	0.316166	0.158083	−	0.987426i	$-0.449469\pi$
$547$	4044.79	0.316166	0.158083	+	0.987426i	$0.449469\pi$
$548$	−6353.24	−0.495250
$549$	4863.08	0.378053
$550$	435.846	0.0337901
$551$	1552.27	0.120016
$552$	−10501.0	−0.809698
$553$	5454.30	0.419422
$554$	−6544.42	−0.501888
$555$	6790.83	0.519378
$556$	16339.6	1.24632
$557$	8804.00	0.669726	0.334863	−	0.942267i	$-0.391310\pi$
$557$	8804.00	0.669726	0.334863	+	0.942267i	$0.391310\pi$
$558$	3052.08	0.231550
$559$	891.900	0.0674836
$560$	−484.535	−0.0365631
$561$	6165.58	0.464013
$562$	−5314.12	−0.398866
$563$	14608.5	1.09356	0.546781	−	0.837276i	$-0.315853\pi$
$563$	14608.5	1.09356	0.546781	+	0.837276i	$0.315853\pi$
$564$	4920.61	0.367367
$565$	4450.72	0.331404
$566$	−2386.30	−0.177215
$567$	−8395.39	−0.621822
$568$	11573.0	0.854913
$569$	25805.2	1.90125	0.950624	−	0.310346i	$-0.100445\pi$
$569$	25805.2	1.90125	0.950624	+	0.310346i	$0.100445\pi$
$570$	1032.70	0.0758857
$571$	−24976.1	−1.83051	−0.915253	−	0.402880i	$-0.868009\pi$
$571$	−24976.1	−1.83051	−0.915253	+	0.402880i	$0.868009\pi$
$572$	4744.47	0.346811
$573$	−34198.6	−2.49331
$574$	850.040	0.0618118
$575$	1790.49	0.129858
$576$	4329.95	0.313220
$577$	−6448.37	−0.465250	−0.232625	−	0.972567i	$-0.574731\pi$
$577$	−6448.37	−0.465250	−0.232625	+	0.972567i	$0.574731\pi$
$578$	−2797.81	−0.201338
$579$	3373.43	0.242133
$580$	−2241.85	−0.160496
$581$	−3867.29	−0.276148
$582$	−17675.1	−1.25886
$583$	4479.63	0.318229
$584$	−4129.82	−0.292626
$585$	−7876.07	−0.556641
$586$	2835.32	0.199873
$587$	−15619.8	−1.09830	−0.549148	−	0.835725i	$-0.685048\pi$
$587$	−15619.8	−1.09830	−0.549148	+	0.835725i	$0.685048\pi$
$588$	−9393.32	−0.658800
$589$	1825.50	0.127705
$590$	7028.81	0.490460
$591$	−8578.34	−0.597065
$592$	1984.98	0.137808
$593$	5603.58	0.388046	0.194023	−	0.980997i	$-0.437846\pi$
$593$	5603.58	0.388046	0.194023	+	0.980997i	$0.437846\pi$
$594$	831.859	0.0574606
$595$	−3950.08	−0.272164
$596$	−3024.11	−0.207840
$597$	22737.9	1.55879
$598$	−8920.80	−0.610031
$599$	7082.45	0.483107	0.241554	−	0.970387i	$-0.422343\pi$
$599$	7082.45	0.483107	0.241554	+	0.970387i	$0.422343\pi$
$600$	−3665.56	−0.249410
$601$	−14241.7	−0.966606	−0.483303	−	0.875453i	$-0.660563\pi$
$601$	−14241.7	−0.966606	−0.483303	+	0.875453i	$0.660563\pi$
$602$	−173.879	−0.0117721
$603$	−17091.4	−1.15425
$604$	−17528.2	−1.18082
$605$	−605.000	−0.0406558
$606$	7029.44	0.471207
$607$	18995.9	1.27021	0.635107	−	0.772424i	$-0.280956\pi$
$607$	18995.9	1.27021	0.635107	+	0.772424i	$0.280956\pi$
$608$	3551.20	0.236875
$609$	5417.07	0.360444
$610$	1922.71	0.127620
$611$	10273.5	0.680232
$612$	−8989.25	−0.593740
$613$	7710.29	0.508019	0.254009	−	0.967202i	$-0.418251\pi$
$613$	7710.29	0.508019	0.254009	+	0.967202i	$0.418251\pi$
$614$	−10304.8	−0.677311
$615$	−1902.63	−0.124750
$616$	−2273.25	−0.148688
$617$	−14981.7	−0.977537	−0.488769	−	0.872413i	$-0.662554\pi$
$617$	−14981.7	−0.977537	−0.488769	+	0.872413i	$0.662554\pi$
$618$	8333.37	0.542423
$619$	21539.9	1.39865	0.699323	−	0.714806i	$-0.253485\pi$
$619$	21539.9	1.39865	0.699323	+	0.714806i	$0.253485\pi$
$620$	−2636.46	−0.170778
$621$	3417.34	0.220827
$622$	−12712.1	−0.819469
$623$	−8077.12	−0.519427
$624$	−5403.47	−0.346654
$625$	625.000	0.0400000
$626$	9337.25	0.596153
$627$	−1433.49	−0.0913048
$628$	−16920.7	−1.07518
$629$	16182.2	1.02580
$630$	1535.47	0.0971023
$631$	843.249	0.0532000	0.0266000	−	0.999646i	$-0.491532\pi$
$631$	843.249	0.0532000	0.0266000	+	0.999646i	$0.491532\pi$
$632$	12061.1	0.759122
$633$	15133.6	0.950248
$634$	−1501.63	−0.0940654
$635$	−9755.33	−0.609651
$636$	−15329.3	−0.955731
$637$	−19611.9	−1.21986
$638$	−1424.31	−0.0883842
$639$	10850.8	0.671752
$640$	−5764.28	−0.356020
$641$	17125.5	1.05525	0.527625	−	0.849477i	$-0.323083\pi$
$641$	17125.5	1.05525	0.527625	+	0.849477i	$0.323083\pi$
$642$	16404.3	1.00845
$643$	−23094.1	−1.41640	−0.708199	−	0.706013i	$-0.750492\pi$
$643$	−23094.1	−1.41640	−0.708199	+	0.706013i	$0.750492\pi$
$644$	−3799.77	−0.232503
$645$	389.191	0.0237587
$646$	2460.86	0.149878
$647$	4333.82	0.263339	0.131669	−	0.991294i	$-0.457966\pi$
$647$	4333.82	0.263339	0.131669	+	0.991294i	$0.457966\pi$
$648$	−18564.8	−1.12545
$649$	−9756.73	−0.590116
$650$	−3113.95	−0.187907
$651$	6370.58	0.383537
$652$	−5339.10	−0.320699
$653$	−21218.9	−1.27161	−0.635803	−	0.771851i	$-0.719331\pi$
$653$	−21218.9	−1.27161	−0.635803	+	0.771851i	$0.719331\pi$
$654$	3076.85	0.183967
$655$	−5681.01	−0.338894
$656$	−556.146	−0.0331004
$657$	−3872.11	−0.229932
$658$	−2002.86	−0.118662
$659$	31938.7	1.88794	0.943972	−	0.330026i	$-0.107057\pi$
$659$	31938.7	1.88794	0.943972	+	0.330026i	$0.107057\pi$
$660$	2070.30	0.122101
$661$	19903.4	1.17118	0.585591	−	0.810607i	$-0.300862\pi$
$661$	19903.4	1.17118	0.585591	+	0.810607i	$0.300862\pi$
$662$	14232.1	0.835571
$663$	−44050.7	−2.58037
$664$	−8551.75	−0.499808
$665$	918.388	0.0535542
$666$	−6290.31	−0.365983
$667$	−5851.20	−0.339669
$668$	8411.28	0.487189
$669$	35261.6	2.03781
$670$	−6757.39	−0.389643
$671$	−2668.93	−0.153551
$672$	12392.9	0.711408
$673$	1652.46	0.0946472	0.0473236	−	0.998880i	$-0.484931\pi$
$673$	1652.46	0.0946472	0.0473236	+	0.998880i	$0.484931\pi$
$674$	−13905.7	−0.794701
$675$	1192.88	0.0680207
$676$	−21840.0	−1.24260
$677$	22526.3	1.27881	0.639405	−	0.768870i	$-0.279181\pi$
$677$	22526.3	1.27881	0.639405	+	0.768870i	$0.279181\pi$
$678$	−9676.29	−0.548106
$679$	−15718.7	−0.888405
$680$	−8734.82	−0.492596
$681$	−15844.0	−0.891549
$682$	−1675.02	−0.0940468
$683$	21256.8	1.19088	0.595439	−	0.803400i	$-0.296978\pi$
$683$	21256.8	1.19088	0.595439	+	0.803400i	$0.296978\pi$
$684$	2089.99	0.116831
$685$	−5788.18	−0.322854
$686$	9078.70	0.505286
$687$	26798.6	1.48826
$688$	113.762	0.00630397
$689$	−32005.3	−1.76967
$690$	−3892.70	−0.214772
$691$	−3211.72	−0.176816	−0.0884078	−	0.996084i	$-0.528178\pi$
$691$	−3211.72	−0.176816	−0.0884078	+	0.996084i	$0.528178\pi$
$692$	4140.71	0.227465
$693$	−2131.39	−0.116832
$694$	−5481.45	−0.299817
$695$	14886.4	0.812478
$696$	11978.8	0.652377
$697$	−4533.87	−0.246388
$698$	−17385.7	−0.942780
$699$	−10043.2	−0.543447
$700$	−1326.37	−0.0716174
$701$	33771.6	1.81959	0.909797	−	0.415053i	$-0.136237\pi$
$701$	33771.6	1.81959	0.909797	+	0.415053i	$0.136237\pi$
$702$	−5943.31	−0.319538
$703$	−3762.34	−0.201848
$704$	−2376.34	−0.127218
$705$	4482.97	0.239487
$706$	16816.8	0.896470
$707$	6251.36	0.332541
$708$	33387.4	1.77228
$709$	6021.06	0.318936	0.159468	−	0.987203i	$-0.449022\pi$
$709$	6021.06	0.318936	0.159468	+	0.987203i	$0.449022\pi$
$710$	4290.06	0.226765
$711$	11308.5	0.596484
$712$	−17861.0	−0.940124
$713$	−6881.13	−0.361431
$714$	8587.85	0.450129
$715$	4322.49	0.226087
$716$	19804.9	1.03372
$717$	24136.8	1.25719
$718$	12016.6	0.624589
$719$	26100.7	1.35381	0.676907	−	0.736069i	$-0.263320\pi$
$719$	26100.7	1.35381	0.676907	+	0.736069i	$0.263320\pi$
$720$	−1004.59	−0.0519985
$721$	7410.96	0.382800
$722$	−572.147	−0.0294918
$723$	−9987.36	−0.513740
$724$	14521.0	0.745396
$725$	−2042.46	−0.104628
$726$	1315.33	0.0672402
$727$	−17260.9	−0.880566	−0.440283	−	0.897859i	$-0.645122\pi$
$727$	−17260.9	−0.880566	−0.440283	+	0.897859i	$0.645122\pi$
$728$	16241.5	0.826854
$729$	−8569.99	−0.435401
$730$	−1530.91	−0.0776187
$731$	927.422	0.0469247
$732$	9133.04	0.461157
$733$	18569.2	0.935703	0.467852	−	0.883807i	$-0.345028\pi$
$733$	18569.2	0.935703	0.467852	+	0.883807i	$0.345028\pi$
$734$	16506.3	0.830055
$735$	−8557.88	−0.429472
$736$	−13386.1	−0.670404
$737$	9379.97	0.468814
$738$	1762.40	0.0879062
$739$	−14952.1	−0.744278	−0.372139	−	0.928177i	$-0.621376\pi$
$739$	−14952.1	−0.744278	−0.372139	+	0.928177i	$0.621376\pi$
$740$	5433.72	0.269929
$741$	10241.7	0.507746
$742$	6239.54	0.308707
$743$	−15714.1	−0.775899	−0.387949	−	0.921681i	$-0.626816\pi$
$743$	−15714.1	−0.775899	−0.387949	+	0.921681i	$0.626816\pi$
$744$	14087.3	0.694173
$745$	−2755.15	−0.135491
$746$	−12107.4	−0.594216
$747$	−8018.10	−0.392727
$748$	4933.42	0.241155
$749$	14588.5	0.711686
$750$	−1358.81	−0.0661556
$751$	−1291.84	−0.0627696	−0.0313848	−	0.999507i	$-0.509992\pi$
$751$	−1291.84	−0.0627696	−0.0313848	+	0.999507i	$0.509992\pi$
$752$	1310.39	0.0635437
$753$	−10640.2	−0.514939
$754$	10176.2	0.491505
$755$	−15969.3	−0.769776
$756$	−2531.53	−0.121787
$757$	−18367.6	−0.881879	−0.440939	−	0.897537i	$-0.645355\pi$
$757$	−18367.6	−0.881879	−0.440939	+	0.897537i	$0.645355\pi$
$758$	−10498.7	−0.503074
$759$	5403.48	0.258411
$760$	2030.84	0.0969292
$761$	334.026	0.0159112	0.00795560	−	0.999968i	$-0.497468\pi$
$761$	334.026	0.0159112	0.00795560	+	0.999968i	$0.497468\pi$
$762$	21209.0	1.00830
$763$	2736.28	0.129829
$764$	−27364.2	−1.29581
$765$	−8189.75	−0.387060
$766$	−14677.8	−0.692339
$767$	69708.1	3.28164
$768$	24385.8	1.14576
$769$	−23364.5	−1.09564	−0.547819	−	0.836597i	$-0.684542\pi$
$769$	−23364.5	−1.09564	−0.547819	+	0.836597i	$0.684542\pi$
$770$	−842.686	−0.0394393
$771$	11241.4	0.525098
$772$	2699.27	0.125840
$773$	12485.1	0.580930	0.290465	−	0.956886i	$-0.406190\pi$
$773$	12485.1	0.580930	0.290465	+	0.956886i	$0.406190\pi$
$774$	−360.506	−0.0167417
$775$	−2401.97	−0.111331
$776$	−34758.8	−1.60795
$777$	−13129.7	−0.606211
$778$	20148.1	0.928464
$779$	1054.12	0.0484824
$780$	−14791.5	−0.679002
$781$	−5955.05	−0.272841
$782$	−9276.09	−0.424185
$783$	−3898.25	−0.177921
$784$	−2501.50	−0.113953
$785$	−15415.8	−0.700909
$786$	12351.1	0.560493
$787$	33165.2	1.50217	0.751086	−	0.660204i	$-0.229530\pi$
$787$	33165.2	1.50217	0.751086	+	0.660204i	$0.229530\pi$
$788$	−6864.00	−0.310305
$789$	11961.5	0.539720
$790$	4471.02	0.201356
$791$	−8605.24	−0.386810
$792$	−4713.15	−0.211458
$793$	19068.5	0.853898
$794$	14575.7	0.651475
$795$	−13965.9	−0.623042
$796$	18193.9	0.810131
$797$	12840.0	0.570658	0.285329	−	0.958430i	$-0.407897\pi$
$797$	12840.0	0.570658	0.285329	+	0.958430i	$0.407897\pi$
$798$	−1996.66	−0.0885729
$799$	10682.7	0.472999
$800$	−4672.63	−0.206503
$801$	−16746.4	−0.738708
$802$	16028.0	0.705697
$803$	2125.07	0.0933898
$804$	−32098.2	−1.40798
$805$	−3461.82	−0.151569
$806$	11967.4	0.522994
$807$	12419.2	0.541729
$808$	13823.7	0.601875
$809$	−22493.4	−0.977535	−0.488768	−	0.872414i	$-0.662553\pi$
$809$	−22493.4	−0.977535	−0.488768	+	0.872414i	$0.662553\pi$
$810$	−6881.90	−0.298525
$811$	13651.3	0.591076	0.295538	−	0.955331i	$-0.404501\pi$
$811$	13651.3	0.591076	0.295538	+	0.955331i	$0.404501\pi$
$812$	4334.50	0.187329
$813$	−34235.7	−1.47687
$814$	3452.21	0.148648
$815$	−4864.25	−0.209064
$816$	−5618.67	−0.241045
$817$	−215.624	−0.00923347
$818$	−10024.3	−0.428474
$819$	15228.0	0.649705
$820$	−1522.40	−0.0648348
$821$	37501.2	1.59415	0.797077	−	0.603877i	$-0.206378\pi$
$821$	37501.2	1.59415	0.797077	+	0.603877i	$0.206378\pi$
$822$	12584.1	0.533965
$823$	−34172.6	−1.44737	−0.723683	−	0.690133i	$-0.757552\pi$
$823$	−34172.6	−1.44737	−0.723683	+	0.690133i	$0.757552\pi$
$824$	16387.9	0.692839
$825$	1886.17	0.0795977
$826$	−13589.8	−0.572459
$827$	5046.07	0.212176	0.106088	−	0.994357i	$-0.466168\pi$
$827$	5046.07	0.212176	0.106088	+	0.994357i	$0.466168\pi$
$828$	−7878.12	−0.330656
$829$	−18888.3	−0.791337	−0.395669	−	0.918393i	$-0.629487\pi$
$829$	−18888.3	−0.791337	−0.395669	+	0.918393i	$0.629487\pi$
$830$	−3170.11	−0.132574
$831$	−28321.7	−1.18227
$832$	16978.0	0.707461
$833$	−20393.0	−0.848229
$834$	−32364.4	−1.34375
$835$	7663.18	0.317599
$836$	−1147.02	−0.0474526
$837$	−4584.42	−0.189320
$838$	−16007.7	−0.659878
$839$	47607.8	1.95900	0.979502	−	0.201433i	$-0.0645597\pi$
$839$	47607.8	1.95900	0.979502	+	0.201433i	$0.0645597\pi$
$840$	7087.17	0.291108
$841$	−17714.4	−0.726327
$842$	−15493.8	−0.634147
$843$	−22997.5	−0.939590
$844$	12109.2	0.493859
$845$	−19897.6	−0.810056
$846$	−4152.55	−0.168756
$847$	1169.74	0.0474529
$848$	−4082.27	−0.165313
$849$	−10327.0	−0.417457
$850$	−3237.97	−0.130661
$851$	14182.0	0.571271
$852$	20378.1	0.819417
$853$	−5687.95	−0.228314	−0.114157	−	0.993463i	$-0.536417\pi$
$853$	−5687.95	−0.228314	−0.114157	+	0.993463i	$0.536417\pi$
$854$	−3717.47	−0.148957
$855$	1904.11	0.0761626
$856$	32259.7	1.28810
$857$	29038.0	1.15743	0.578716	−	0.815529i	$-0.303554\pi$
$857$	29038.0	1.15743	0.578716	+	0.815529i	$0.303554\pi$
$858$	−9397.52	−0.373923
$859$	−32626.7	−1.29594	−0.647968	−	0.761667i	$-0.724381\pi$
$859$	−32626.7	−1.29594	−0.647968	+	0.761667i	$0.724381\pi$
$860$	311.413	0.0123478
$861$	3678.64	0.145607
$862$	−2912.48	−0.115081
$863$	23958.6	0.945031	0.472515	−	0.881322i	$-0.343346\pi$
$863$	23958.6	0.945031	0.472515	+	0.881322i	$0.343346\pi$
$864$	−8918.22	−0.351162
$865$	3772.43	0.148285
$866$	7584.81	0.297624
$867$	−12107.8	−0.474283
$868$	5097.45	0.199330
$869$	−6206.24	−0.242270
$870$	4440.50	0.173043
$871$	−67016.3	−2.60707
$872$	6050.74	0.234982
$873$	−32589.7	−1.26345
$874$	2156.68	0.0834678
$875$	−1208.41	−0.0466875
$876$	−7271.96	−0.280476
$877$	45381.5	1.74735	0.873675	−	0.486510i	$-0.161730\pi$
$877$	45381.5	1.74735	0.873675	+	0.486510i	$0.161730\pi$
$878$	−2788.79	−0.107195
$879$	12270.2	0.470833
$880$	551.334	0.0211199
$881$	−6515.27	−0.249154	−0.124577	−	0.992210i	$-0.539757\pi$
$881$	−6515.27	−0.249154	−0.124577	+	0.992210i	$0.539757\pi$
$882$	7927.12	0.302630
$883$	27168.7	1.03545	0.517724	−	0.855548i	$-0.326779\pi$
$883$	27168.7	1.03545	0.517724	+	0.855548i	$0.326779\pi$
$884$	−35247.4	−1.34106
$885$	30418.0	1.15535
$886$	−6718.93	−0.254771
$887$	34754.6	1.31561	0.657804	−	0.753189i	$-0.271485\pi$
$887$	34754.6	1.31561	0.657804	+	0.753189i	$0.271485\pi$
$888$	−29033.8	−1.09720
$889$	18861.4	0.711577
$890$	−6621.01	−0.249367
$891$	9552.80	0.359182
$892$	28214.8	1.05908
$893$	−2483.71	−0.0930730
$894$	5989.96	0.224087
$895$	18043.5	0.673884
$896$	11144.9	0.415543
$897$	−38605.8	−1.43702
$898$	10070.3	0.374221
$899$	7849.47	0.291206
$900$	−2749.99	−0.101851
$901$	−33279.9	−1.23054
$902$	−967.229	−0.0357042
$903$	−752.481	−0.0277309
$904$	−19028.8	−0.700098
$905$	13229.5	0.485925
$906$	34718.7	1.27313
$907$	25932.5	0.949366	0.474683	−	0.880157i	$-0.342563\pi$
$907$	25932.5	0.949366	0.474683	+	0.880157i	$0.342563\pi$
$908$	−12677.7	−0.463353
$909$	12961.0	0.472926
$910$	6020.67	0.219322
$911$	−10568.9	−0.384371	−0.192185	−	0.981359i	$-0.561558\pi$
$911$	−10568.9	−0.384371	−0.192185	+	0.981359i	$0.561558\pi$
$912$	1306.33	0.0474310
$913$	4400.44	0.159511
$914$	2578.66	0.0933199
$915$	8320.75	0.300629
$916$	21443.1	0.773470
$917$	10983.9	0.395553
$918$	−6180.02	−0.222191
$919$	2368.21	0.0850053	0.0425027	−	0.999096i	$-0.486467\pi$
$919$	2368.21	0.0850053	0.0425027	+	0.999096i	$0.486467\pi$
$920$	−7655.14	−0.274329
$921$	−44595.3	−1.59551
$922$	940.143	0.0335813
$923$	42546.6	1.51727
$924$	−4002.83	−0.142514
$925$	4950.44	0.175967
$926$	−876.589	−0.0311085
$927$	15365.2	0.544402
$928$	15269.8	0.540147
$929$	6316.67	0.223082	0.111541	−	0.993760i	$-0.464421\pi$
$929$	6316.67	0.223082	0.111541	+	0.993760i	$0.464421\pi$
$930$	5222.11	0.184129
$931$	4741.34	0.166908
$932$	−8036.13	−0.282438
$933$	−55013.1	−1.93039
$934$	−31348.2	−1.09823
$935$	4494.65	0.157209
$936$	33673.7	1.17592
$937$	−40477.1	−1.41124	−0.705618	−	0.708592i	$-0.749331\pi$
$937$	−40477.1	−1.41124	−0.705618	+	0.708592i	$0.749331\pi$
$938$	13065.1	0.454786
$939$	40408.0	1.40433
$940$	3587.07	0.124465
$941$	22314.3	0.773036	0.386518	−	0.922282i	$-0.373678\pi$
$941$	22314.3	0.773036	0.386518	+	0.922282i	$0.373678\pi$
$942$	33515.4	1.15923
$943$	−3973.46	−0.137215
$944$	8891.27	0.306553
$945$	−2306.37	−0.0793929
$946$	197.851	0.00679987
$947$	33814.8	1.16033	0.580165	−	0.814499i	$-0.302988\pi$
$947$	33814.8	1.16033	0.580165	+	0.814499i	$0.302988\pi$
$948$	21237.7	0.727604
$949$	−15182.8	−0.519341
$950$	752.825	0.0257104
$951$	−6498.49	−0.221586
$952$	16888.3	0.574952
$953$	−29536.4	−1.00396	−0.501981	−	0.864878i	$-0.667395\pi$
$953$	−29536.4	−1.00396	−0.501981	+	0.864878i	$0.667395\pi$
$954$	12936.5	0.439031
$955$	−24930.4	−0.844742
$956$	19313.2	0.653381
$957$	−6163.88	−0.208203
$958$	−9691.46	−0.326844
$959$	11191.2	0.376831
$960$	7408.57	0.249073
$961$	−20559.9	−0.690137
$962$	−24664.7	−0.826635
$963$	30246.6	1.01213
$964$	−7991.44	−0.266999
$965$	2459.20	0.0820356
$966$	7526.33	0.250679
$967$	48841.7	1.62424	0.812122	−	0.583488i	$-0.198313\pi$
$967$	48841.7	1.62424	0.812122	+	0.583488i	$0.198313\pi$
$968$	2586.64	0.0858863
$969$	10649.6	0.353061
$970$	−12885.0	−0.426507
$971$	49032.8	1.62053	0.810267	−	0.586062i	$-0.199322\pi$
$971$	49032.8	1.62053	0.810267	+	0.586062i	$0.199322\pi$
$972$	−25619.2	−0.845407
$973$	−28782.0	−0.948315
$974$	−3899.17	−0.128272
$975$	−13476.0	−0.442643
$976$	2432.18	0.0797667
$977$	−55063.8	−1.80312	−0.901560	−	0.432655i	$-0.857577\pi$
$977$	−55063.8	−1.80312	−0.901560	+	0.432655i	$0.857577\pi$
$978$	10575.3	0.345769
$979$	9190.66	0.300035
$980$	−6847.64	−0.223204
$981$	5673.16	0.184638
$982$	1116.59	0.0362850
$983$	29674.1	0.962826	0.481413	−	0.876494i	$-0.340124\pi$
$983$	29674.1	0.962826	0.481413	+	0.876494i	$0.340124\pi$
$984$	8134.60	0.263538
$985$	−6253.52	−0.202288
$986$	10581.5	0.341767
$987$	−8667.59	−0.279526
$988$	8194.99	0.263884
$989$	812.786	0.0261326
$990$	−1747.15	−0.0560890
$991$	43854.2	1.40573	0.702863	−	0.711326i	$-0.251905\pi$
$991$	43854.2	1.40573	0.702863	+	0.711326i	$0.251905\pi$
$992$	17957.6	0.574753
$993$	61591.2	1.96831
$994$	−8294.61	−0.264677
$995$	16575.7	0.528126
$996$	−15058.3	−0.479056
$997$	−25298.5	−0.803624	−0.401812	−	0.915722i	$-0.631619\pi$
$997$	−25298.5	−0.803624	−0.401812	+	0.915722i	$0.631619\pi$
$998$	−7381.36	−0.234121
$999$	9448.46	0.299235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.8	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.8	✓	23	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.58489 q^{2} -6.85881 q^{3} -5.48811 q^{4} -5.00000 q^{5} +10.8705 q^{6} +9.66724 q^{7} +21.3772 q^{8} +20.0432 q^{9} +O(q^{10})\)	\(q-1.58489 q^{2} -6.85881 q^{3} -5.48811 q^{4} -5.00000 q^{5} +10.8705 q^{6} +9.66724 q^{7} +21.3772 q^{8} +20.0432 q^{9} +7.92447 q^{10} -11.0000 q^{11} +37.6419 q^{12} +78.5908 q^{13} -15.3216 q^{14} +34.2940 q^{15} +10.0243 q^{16} +81.7208 q^{17} -31.7664 q^{18} -19.0000 q^{19} +27.4406 q^{20} -66.3058 q^{21} +17.4338 q^{22} +71.6196 q^{23} -146.622 q^{24} +25.0000 q^{25} -124.558 q^{26} +47.7152 q^{27} -53.0549 q^{28} -81.6983 q^{29} -54.3524 q^{30} -96.0788 q^{31} -186.905 q^{32} +75.4469 q^{33} -129.519 q^{34} -48.3362 q^{35} -109.999 q^{36} +198.018 q^{37} +30.1130 q^{38} -539.039 q^{39} -106.886 q^{40} -55.4800 q^{41} +105.088 q^{42} +11.3487 q^{43} +60.3692 q^{44} -100.216 q^{45} -113.510 q^{46} +130.722 q^{47} -68.7544 q^{48} -249.544 q^{49} -39.6223 q^{50} -560.507 q^{51} -431.315 q^{52} -407.239 q^{53} -75.6235 q^{54} +55.0000 q^{55} +206.659 q^{56} +130.317 q^{57} +129.483 q^{58} +886.975 q^{59} -188.209 q^{60} +242.630 q^{61} +152.275 q^{62} +193.763 q^{63} +216.031 q^{64} -392.954 q^{65} -119.575 q^{66} -852.725 q^{67} -448.493 q^{68} -491.225 q^{69} +76.6078 q^{70} +541.369 q^{71} +428.468 q^{72} -193.188 q^{73} -313.837 q^{74} -171.470 q^{75} +104.274 q^{76} -106.340 q^{77} +854.320 q^{78} +564.204 q^{79} -50.1213 q^{80} -868.436 q^{81} +87.9299 q^{82} -400.040 q^{83} +363.893 q^{84} -408.604 q^{85} -17.9864 q^{86} +560.353 q^{87} -235.149 q^{88} -835.514 q^{89} +158.832 q^{90} +759.757 q^{91} -393.057 q^{92} +658.986 q^{93} -207.180 q^{94} +95.0000 q^{95} +1281.95 q^{96} -1625.97 q^{97} +395.501 q^{98} -220.475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * q^99

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