LMFDB - Embedded newform 1045.4.a.g.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.48215 q^{2} +3.78622 q^{3} -1.83894 q^{4} -5.00000 q^{5} -9.39797 q^{6} +1.80512 q^{7} +24.4217 q^{8} -12.6645 q^{9} +O(q^{10})$	\(q-2.48215 q^{2} +3.78622 q^{3} -1.83894 q^{4} -5.00000 q^{5} -9.39797 q^{6} +1.80512 q^{7} +24.4217 q^{8} -12.6645 q^{9} +12.4107 q^{10} -11.0000 q^{11} -6.96264 q^{12} -79.4575 q^{13} -4.48057 q^{14} -18.9311 q^{15} -45.9068 q^{16} -138.818 q^{17} +31.4352 q^{18} -19.0000 q^{19} +9.19470 q^{20} +6.83458 q^{21} +27.3036 q^{22} +24.6171 q^{23} +92.4661 q^{24} +25.0000 q^{25} +197.225 q^{26} -150.179 q^{27} -3.31951 q^{28} +198.290 q^{29} +46.9898 q^{30} +189.328 q^{31} -81.4263 q^{32} -41.6485 q^{33} +344.566 q^{34} -9.02560 q^{35} +23.2893 q^{36} +336.290 q^{37} +47.1608 q^{38} -300.844 q^{39} -122.109 q^{40} -241.773 q^{41} -16.9645 q^{42} -254.240 q^{43} +20.2283 q^{44} +63.3225 q^{45} -61.1034 q^{46} +515.912 q^{47} -173.813 q^{48} -339.742 q^{49} -62.0537 q^{50} -525.595 q^{51} +146.118 q^{52} -399.017 q^{53} +372.766 q^{54} +55.0000 q^{55} +44.0841 q^{56} -71.9383 q^{57} -492.185 q^{58} +224.755 q^{59} +34.8132 q^{60} -700.355 q^{61} -469.939 q^{62} -22.8609 q^{63} +569.366 q^{64} +397.288 q^{65} +103.378 q^{66} -395.829 q^{67} +255.278 q^{68} +93.2060 q^{69} +22.4029 q^{70} -716.837 q^{71} -309.289 q^{72} +1025.92 q^{73} -834.722 q^{74} +94.6556 q^{75} +34.9399 q^{76} -19.8563 q^{77} +746.739 q^{78} +551.919 q^{79} +229.534 q^{80} -226.668 q^{81} +600.116 q^{82} +335.410 q^{83} -12.5684 q^{84} +694.089 q^{85} +631.062 q^{86} +750.770 q^{87} -268.639 q^{88} -423.417 q^{89} -157.176 q^{90} -143.430 q^{91} -45.2695 q^{92} +716.837 q^{93} -1280.57 q^{94} +95.0000 q^{95} -308.298 q^{96} -619.565 q^{97} +843.289 q^{98} +139.310 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.48215	−0.877572	−0.438786	−	0.898592i	$-0.644591\pi$
$2$	−2.48215	−0.877572	−0.438786	+	0.898592i	$0.644591\pi$
$3$	3.78622	0.728659	0.364330	−	0.931270i	$-0.381298\pi$
$3$	3.78622	0.728659	0.364330	+	0.931270i	$0.381298\pi$
$4$	−1.83894	−0.229868
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−9.39797	−0.639451
$7$	1.80512	0.0974673	0.0487336	−	0.998812i	$-0.484481\pi$
$7$	1.80512	0.0974673	0.0487336	+	0.998812i	$0.484481\pi$
$8$	24.4217	1.07930
$9$	−12.6645	−0.469056
$10$	12.4107	0.392462
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−6.96264	−0.167495
$13$	−79.4575	−1.69520	−0.847598	−	0.530639i	$-0.821952\pi$
$13$	−79.4575	−1.69520	−0.847598	+	0.530639i	$0.821952\pi$
$14$	−4.48057	−0.0855345
$15$	−18.9311	−0.325866
$16$	−45.9068	−0.717293
$17$	−138.818	−1.98049	−0.990243	−	0.139353i	$-0.955498\pi$
$17$	−138.818	−1.98049	−0.990243	+	0.139353i	$0.955498\pi$
$18$	31.4352	0.411630
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	9.19470	0.102800
$21$	6.83458	0.0710204
$22$	27.3036	0.264598
$23$	24.6171	0.223175	0.111588	−	0.993755i	$-0.464406\pi$
$23$	24.6171	0.223175	0.111588	+	0.993755i	$0.464406\pi$
$24$	92.4661	0.786440
$25$	25.0000	0.200000
$26$	197.225	1.48766
$27$	−150.179	−1.07044
$28$	−3.31951	−0.0224046
$29$	198.290	1.26971	0.634854	−	0.772632i	$-0.281060\pi$
$29$	198.290	1.26971	0.634854	+	0.772632i	$0.281060\pi$
$30$	46.9898	0.285971
$31$	189.328	1.09691	0.548455	−	0.836180i	$-0.315216\pi$
$31$	189.328	1.09691	0.548455	+	0.836180i	$0.315216\pi$
$32$	−81.4263	−0.449821
$33$	−41.6485	−0.219699
$34$	344.566	1.73802
$35$	−9.02560	−0.0435887
$36$	23.2893	0.107821
$37$	336.290	1.49421	0.747105	−	0.664706i	$-0.231443\pi$
$37$	336.290	1.49421	0.747105	+	0.664706i	$0.231443\pi$
$38$	47.1608	0.201329
$39$	−300.844	−1.23522
$40$	−122.109	−0.482676
$41$	−241.773	−0.920942	−0.460471	−	0.887675i	$-0.652319\pi$
$41$	−241.773	−0.920942	−0.460471	+	0.887675i	$0.652319\pi$
$42$	−16.9645	−0.0623255
$43$	−254.240	−0.901658	−0.450829	−	0.892610i	$-0.648872\pi$
$43$	−254.240	−0.901658	−0.450829	+	0.892610i	$0.648872\pi$
$44$	20.2283	0.0693077
$45$	63.3225	0.209768
$46$	−61.1034	−0.195852
$47$	515.912	1.60114	0.800569	−	0.599241i	$-0.204531\pi$
$47$	515.912	1.60114	0.800569	+	0.599241i	$0.204531\pi$
$48$	−173.813	−0.522662
$49$	−339.742	−0.990500
$50$	−62.0537	−0.175514
$51$	−525.595	−1.44310
$52$	146.118	0.389671
$53$	−399.017	−1.03414	−0.517068	−	0.855945i	$-0.672976\pi$
$53$	−399.017	−1.03414	−0.517068	+	0.855945i	$0.672976\pi$
$54$	372.766	0.939389
$55$	55.0000	0.134840
$56$	44.0841	0.105196
$57$	−71.9383	−0.167166
$58$	−492.185	−1.11426
$59$	224.755	0.495942	0.247971	−	0.968767i	$-0.420236\pi$
$59$	224.755	0.495942	0.247971	+	0.968767i	$0.420236\pi$
$60$	34.8132	0.0749061
$61$	−700.355	−1.47002	−0.735010	−	0.678056i	$-0.762823\pi$
$61$	−700.355	−1.47002	−0.735010	+	0.678056i	$0.762823\pi$
$62$	−469.939	−0.962618
$63$	−22.8609	−0.0457176
$64$	569.366	1.11204
$65$	397.288	0.758115
$66$	103.378	0.192802
$67$	−395.829	−0.721764	−0.360882	−	0.932611i	$-0.617524\pi$
$67$	−395.829	−0.721764	−0.360882	+	0.932611i	$0.617524\pi$
$68$	255.278	0.455249
$69$	93.2060	0.162619
$70$	22.4029	0.0382522
$71$	−716.837	−1.19821	−0.599105	−	0.800670i	$-0.704477\pi$
$71$	−716.837	−1.19821	−0.599105	+	0.800670i	$0.704477\pi$
$72$	−309.289	−0.506251
$73$	1025.92	1.64486	0.822428	−	0.568869i	$-0.192619\pi$
$73$	1025.92	1.64486	0.822428	+	0.568869i	$0.192619\pi$
$74$	−834.722	−1.31128
$75$	94.6556	0.145732
$76$	34.9399	0.0527352
$77$	−19.8563	−0.0293875
$78$	746.739	1.08399
$79$	551.919	0.786021	0.393011	−	0.919534i	$-0.371434\pi$
$79$	551.919	0.786021	0.393011	+	0.919534i	$0.371434\pi$
$80$	229.534	0.320783
$81$	−226.668	−0.310931
$82$	600.116	0.808192
$83$	335.410	0.443566	0.221783	−	0.975096i	$-0.428812\pi$
$83$	335.410	0.443566	0.221783	+	0.975096i	$0.428812\pi$
$84$	−12.5684	−0.0163253
$85$	694.089	0.885700
$86$	631.062	0.791270
$87$	750.770	0.925184
$88$	−268.639	−0.325420
$89$	−423.417	−0.504294	−0.252147	−	0.967689i	$-0.581137\pi$
$89$	−423.417	−0.504294	−0.252147	+	0.967689i	$0.581137\pi$
$90$	−157.176	−0.184087
$91$	−143.430	−0.165226
$92$	−45.2695	−0.0513008
$93$	716.837	0.799274
$94$	−1280.57	−1.40511
$95$	95.0000	0.102598
$96$	−308.298	−0.327766
$97$	−619.565	−0.648529	−0.324264	−	0.945966i	$-0.605117\pi$
$97$	−619.565	−0.648529	−0.324264	+	0.945966i	$0.605117\pi$
$98$	843.289	0.869235
$99$	139.310	0.141426
$100$	−45.9735	−0.0459735
$101$	771.841	0.760406	0.380203	−	0.924903i	$-0.375854\pi$
$101$	771.841	0.760406	0.380203	+	0.924903i	$0.375854\pi$
$102$	1304.61	1.26642
$103$	91.8480	0.0878646	0.0439323	−	0.999035i	$-0.486011\pi$
$103$	91.8480	0.0878646	0.0439323	+	0.999035i	$0.486011\pi$
$104$	−1940.49	−1.82962
$105$	−34.1729	−0.0317613
$106$	990.419	0.907528
$107$	602.508	0.544361	0.272181	−	0.962246i	$-0.412255\pi$
$107$	602.508	0.544361	0.272181	+	0.962246i	$0.412255\pi$
$108$	276.170	0.246060
$109$	1013.56	0.890658	0.445329	−	0.895367i	$-0.353087\pi$
$109$	1013.56	0.890658	0.445329	+	0.895367i	$0.353087\pi$
$110$	−136.518	−0.118332
$111$	1273.27	1.08877
$112$	−82.8672	−0.0699126
$113$	1602.25	1.33387	0.666935	−	0.745116i	$-0.267606\pi$
$113$	1602.25	1.33387	0.666935	+	0.745116i	$0.267606\pi$
$114$	178.561	0.146700
$115$	−123.086	−0.0998070
$116$	−364.643	−0.291865
$117$	1006.29	0.795142
$118$	−557.875	−0.435225
$119$	−250.583	−0.193033
$120$	−462.330	−0.351707
$121$	121.000	0.0909091
$122$	1738.38	1.29005
$123$	−915.407	−0.671052
$124$	−348.162	−0.252144
$125$	−125.000	−0.0894427
$126$	56.7443	0.0401205
$127$	1836.08	1.28288	0.641439	−	0.767174i	$-0.278338\pi$
$127$	1836.08	1.28288	0.641439	+	0.767174i	$0.278338\pi$
$128$	−761.841	−0.526077
$129$	−962.611	−0.657001
$130$	−986.127	−0.665300
$131$	−1711.79	−1.14168	−0.570839	−	0.821062i	$-0.693382\pi$
$131$	−1711.79	−1.14168	−0.570839	+	0.821062i	$0.693382\pi$
$132$	76.5891	0.0505017
$133$	−34.2973	−0.0223605
$134$	982.506	0.633400
$135$	750.894	0.478716
$136$	−3390.17	−2.13753
$137$	−1859.83	−1.15983	−0.579913	−	0.814678i	$-0.696913\pi$
$137$	−1859.83	−1.15983	−0.579913	+	0.814678i	$0.696913\pi$
$138$	−231.351	−0.142710
$139$	2071.57	1.26409	0.632045	−	0.774932i	$-0.282216\pi$
$139$	2071.57	1.26409	0.632045	+	0.774932i	$0.282216\pi$
$140$	16.5975	0.0100196
$141$	1953.36	1.16668
$142$	1779.30	1.05152
$143$	874.033	0.511121
$144$	581.387	0.336451
$145$	−991.450	−0.567830
$146$	−2546.48	−1.44348
$147$	−1286.34	−0.721737
$148$	−618.418	−0.343470
$149$	2649.29	1.45663	0.728316	−	0.685241i	$-0.240303\pi$
$149$	2649.29	1.45663	0.728316	+	0.685241i	$0.240303\pi$
$150$	−234.949	−0.127890
$151$	−817.152	−0.440390	−0.220195	−	0.975456i	$-0.570669\pi$
$151$	−817.152	−0.440390	−0.220195	+	0.975456i	$0.570669\pi$
$152$	−464.012	−0.247608
$153$	1758.06	0.928958
$154$	49.2863	0.0257896
$155$	−946.638	−0.490553
$156$	553.234	0.283937
$157$	314.407	0.159824	0.0799121	−	0.996802i	$-0.474536\pi$
$157$	314.407	0.159824	0.0799121	+	0.996802i	$0.474536\pi$
$158$	−1369.94	−0.689790
$159$	−1510.77	−0.753532
$160$	407.131	0.201166
$161$	44.4369	0.0217523
$162$	562.625	0.272864
$163$	−1677.11	−0.805900	−0.402950	−	0.915222i	$-0.632015\pi$
$163$	−1677.11	−0.805900	−0.402950	+	0.915222i	$0.632015\pi$
$164$	444.606	0.211695
$165$	208.242	0.0982524
$166$	−832.537	−0.389261
$167$	−3304.18	−1.53105	−0.765525	−	0.643407i	$-0.777520\pi$
$167$	−3304.18	−1.53105	−0.765525	+	0.643407i	$0.777520\pi$
$168$	166.912	0.0766521
$169$	4116.50	1.87369
$170$	−1722.83	−0.777265
$171$	240.626	0.107609
$172$	467.533	0.207262
$173$	−1496.40	−0.657627	−0.328814	−	0.944395i	$-0.606649\pi$
$173$	−1496.40	−0.657627	−0.328814	+	0.944395i	$0.606649\pi$
$174$	−1863.52	−0.811915
$175$	45.1280	0.0194935
$176$	504.974	0.216272
$177$	850.972	0.361373
$178$	1050.98	0.442554
$179$	4321.82	1.80463	0.902313	−	0.431081i	$-0.141868\pi$
$179$	4321.82	1.80463	0.902313	+	0.431081i	$0.141868\pi$
$180$	−116.446	−0.0482189
$181$	4114.09	1.68949	0.844745	−	0.535169i	$-0.179752\pi$
$181$	4114.09	1.68949	0.844745	+	0.535169i	$0.179752\pi$
$182$	356.015	0.144998
$183$	−2651.70	−1.07114
$184$	601.193	0.240872
$185$	−1681.45	−0.668231
$186$	−1779.29	−0.701420
$187$	1527.00	0.597139
$188$	−948.731	−0.368050
$189$	−271.090	−0.104333
$190$	−235.804	−0.0900370
$191$	1447.73	0.548450	0.274225	−	0.961666i	$-0.411579\pi$
$191$	1447.73	0.548450	0.274225	+	0.961666i	$0.411579\pi$
$192$	2155.75	0.810301
$193$	−1572.83	−0.586605	−0.293302	−	0.956020i	$-0.594754\pi$
$193$	−1572.83	−0.586605	−0.293302	+	0.956020i	$0.594754\pi$
$194$	1537.85	0.569131
$195$	1504.22	0.552407
$196$	624.765	0.227684
$197$	4358.88	1.57643	0.788217	−	0.615397i	$-0.211004\pi$
$197$	4358.88	1.57643	0.788217	+	0.615397i	$0.211004\pi$
$198$	−345.787	−0.124111
$199$	1806.77	0.643612	0.321806	−	0.946806i	$-0.395710\pi$
$199$	1806.77	0.643612	0.321806	+	0.946806i	$0.395710\pi$
$200$	610.543	0.215859
$201$	−1498.70	−0.525920
$202$	−1915.82	−0.667311
$203$	357.937	0.123755
$204$	966.538	0.331722
$205$	1208.87	0.411858
$206$	−227.980	−0.0771075
$207$	−311.764	−0.104682
$208$	3647.64	1.21595
$209$	209.000	0.0691714
$210$	84.8223	0.0278728
$211$	−5596.74	−1.82604	−0.913022	−	0.407911i	$-0.866257\pi$
$211$	−5596.74	−1.82604	−0.913022	+	0.407911i	$0.866257\pi$
$212$	733.768	0.237714
$213$	−2714.11	−0.873087
$214$	−1495.51	−0.477716
$215$	1271.20	0.403234
$216$	−3667.62	−1.15532
$217$	341.759	0.106913
$218$	−2515.81	−0.781616
$219$	3884.35	1.19854
$220$	−101.142	−0.0309953
$221$	11030.1	3.35731
$222$	−3160.44	−0.955474
$223$	2006.76	0.602613	0.301307	−	0.953527i	$-0.402577\pi$
$223$	2006.76	0.602613	0.301307	+	0.953527i	$0.402577\pi$
$224$	−146.984	−0.0438428
$225$	−316.613	−0.0938112
$226$	−3977.03	−1.17057
$227$	−2688.14	−0.785982	−0.392991	−	0.919542i	$-0.628560\pi$
$227$	−2688.14	−0.785982	−0.392991	+	0.919542i	$0.628560\pi$
$228$	132.290	0.0384260
$229$	−3563.89	−1.02842	−0.514211	−	0.857664i	$-0.671915\pi$
$229$	−3563.89	−1.02842	−0.514211	+	0.857664i	$0.671915\pi$
$230$	305.517	0.0875878
$231$	−75.1804	−0.0214135
$232$	4842.58	1.37039
$233$	1517.89	0.426781	0.213391	−	0.976967i	$-0.431549\pi$
$233$	1517.89	0.426781	0.213391	+	0.976967i	$0.431549\pi$
$234$	−2497.76	−0.697794
$235$	−2579.56	−0.716051
$236$	−413.311	−0.114001
$237$	2089.69	0.572742
$238$	621.983	0.169400
$239$	−1728.11	−0.467708	−0.233854	−	0.972272i	$-0.575134\pi$
$239$	−1728.11	−0.467708	−0.233854	+	0.972272i	$0.575134\pi$
$240$	869.067	0.233742
$241$	−2334.39	−0.623948	−0.311974	−	0.950091i	$-0.600990\pi$
$241$	−2334.39	−0.623948	−0.311974	+	0.950091i	$0.600990\pi$
$242$	−300.340	−0.0797793
$243$	3196.61	0.843878
$244$	1287.91	0.337910
$245$	1698.71	0.442965
$246$	2272.18	0.588897
$247$	1509.69	0.388905
$248$	4623.70	1.18389
$249$	1269.94	0.323209
$250$	310.269	0.0784924
$251$	6528.70	1.64178	0.820892	−	0.571083i	$-0.193477\pi$
$251$	6528.70	1.64178	0.820892	+	0.571083i	$0.193477\pi$
$252$	42.0399	0.0105090
$253$	−270.789	−0.0672899
$254$	−4557.41	−1.12582
$255$	2627.98	0.645373
$256$	−2663.93	−0.650373
$257$	7700.35	1.86901	0.934503	−	0.355955i	$-0.115845\pi$
$257$	7700.35	1.86901	0.934503	+	0.355955i	$0.115845\pi$
$258$	2389.34	0.576566
$259$	607.044	0.145637
$260$	−730.588	−0.174266
$261$	−2511.24	−0.595564
$262$	4248.92	1.00190
$263$	−2024.61	−0.474687	−0.237344	−	0.971426i	$-0.576277\pi$
$263$	−2024.61	−0.474687	−0.237344	+	0.971426i	$0.576277\pi$
$264$	−1017.13	−0.237121
$265$	1995.08	0.462479
$266$	85.1309	0.0196230
$267$	−1603.15	−0.367458
$268$	727.906	0.165910
$269$	5525.21	1.25233	0.626167	−	0.779689i	$-0.284623\pi$
$269$	5525.21	1.25233	0.626167	+	0.779689i	$0.284623\pi$
$270$	−1863.83	−0.420107
$271$	2588.06	0.580122	0.290061	−	0.957008i	$-0.406324\pi$
$271$	2588.06	0.580122	0.290061	+	0.957008i	$0.406324\pi$
$272$	6372.68	1.42059
$273$	−543.059	−0.120394
$274$	4616.38	1.01783
$275$	−275.000	−0.0603023
$276$	−171.400	−0.0373808
$277$	4468.51	0.969266	0.484633	−	0.874718i	$-0.338953\pi$
$277$	4468.51	0.969266	0.484633	+	0.874718i	$0.338953\pi$
$278$	−5141.95	−1.10933
$279$	−2397.74	−0.514512
$280$	−220.420	−0.0470451
$281$	−2314.19	−0.491292	−0.245646	−	0.969360i	$-0.579000\pi$
$281$	−2314.19	−0.491292	−0.245646	+	0.969360i	$0.579000\pi$
$282$	−4848.52	−1.02385
$283$	−7786.11	−1.63546	−0.817731	−	0.575600i	$-0.804768\pi$
$283$	−7786.11	−1.63546	−0.817731	+	0.575600i	$0.804768\pi$
$284$	1318.22	0.275430
$285$	359.691	0.0747588
$286$	−2169.48	−0.448545
$287$	−436.429	−0.0897617
$288$	1031.22	0.210991
$289$	14357.4	2.92232
$290$	2460.92	0.498312
$291$	−2345.81	−0.472557
$292$	−1886.60	−0.378099
$293$	2205.20	0.439690	0.219845	−	0.975535i	$-0.429445\pi$
$293$	2205.20	0.439690	0.219845	+	0.975535i	$0.429445\pi$
$294$	3192.88	0.633376
$295$	−1123.77	−0.221792
$296$	8212.78	1.61270
$297$	1651.97	0.322750
$298$	−6575.93	−1.27830
$299$	−1956.02	−0.378326
$300$	−174.066	−0.0334990
$301$	−458.934	−0.0878821
$302$	2028.29	0.386474
$303$	2922.36	0.554077
$304$	872.229	0.164558
$305$	3501.78	0.657413
$306$	−4363.76	−0.815228
$307$	−7270.06	−1.35154	−0.675772	−	0.737111i	$-0.736190\pi$
$307$	−7270.06	−1.35154	−0.675772	+	0.737111i	$0.736190\pi$
$308$	36.5146	0.00675523
$309$	347.757	0.0640234
$310$	2349.70	0.430496
$311$	2792.13	0.509091	0.254546	−	0.967061i	$-0.418074\pi$
$311$	2792.13	0.509091	0.254546	+	0.967061i	$0.418074\pi$
$312$	−7347.12	−1.33317
$313$	95.3199	0.0172134	0.00860671	−	0.999963i	$-0.497260\pi$
$313$	95.3199	0.0172134	0.00860671	+	0.999963i	$0.497260\pi$
$314$	−780.405	−0.140257
$315$	114.305	0.0204455
$316$	−1014.95	−0.180681
$317$	215.430	0.0381697	0.0190848	−	0.999818i	$-0.493925\pi$
$317$	215.430	0.0381697	0.0190848	+	0.999818i	$0.493925\pi$
$318$	3749.95	0.661278
$319$	−2181.19	−0.382831
$320$	−2846.83	−0.497321
$321$	2281.23	0.396654
$322$	−110.299	−0.0190892
$323$	2637.54	0.454355
$324$	416.830	0.0714729
$325$	−1986.44	−0.339039
$326$	4162.85	0.707236
$327$	3837.58	0.648986
$328$	−5904.51	−0.993970
$329$	931.282	0.156059
$330$	−516.888	−0.0862235
$331$	−6854.33	−1.13821	−0.569106	−	0.822264i	$-0.692711\pi$
$331$	−6854.33	−1.13821	−0.569106	+	0.822264i	$0.692711\pi$
$332$	−616.799	−0.101962
$333$	−4258.95	−0.700868
$334$	8201.47	1.34361
$335$	1979.14	0.322783
$336$	−313.754	−0.0509425
$337$	−3950.91	−0.638635	−0.319317	−	0.947648i	$-0.603454\pi$
$337$	−3950.91	−0.638635	−0.319317	+	0.947648i	$0.603454\pi$
$338$	−10217.8	−1.64430
$339$	6066.49	0.971937
$340$	−1276.39	−0.203594
$341$	−2082.60	−0.330731
$342$	−597.269	−0.0944345
$343$	−1232.43	−0.194009
$344$	−6208.98	−0.973157
$345$	−466.030	−0.0727253
$346$	3714.30	0.577115
$347$	−6670.13	−1.03191	−0.515953	−	0.856617i	$-0.672562\pi$
$347$	−6670.13	−1.03191	−0.515953	+	0.856617i	$0.672562\pi$
$348$	−1380.62	−0.212670
$349$	6380.01	0.978551	0.489275	−	0.872129i	$-0.337261\pi$
$349$	6380.01	0.978551	0.489275	+	0.872129i	$0.337261\pi$
$350$	−112.014	−0.0171069
$351$	11932.8	1.81461
$352$	895.689	0.135626
$353$	−3557.78	−0.536434	−0.268217	−	0.963358i	$-0.586434\pi$
$353$	−3557.78	−0.536434	−0.268217	+	0.963358i	$0.586434\pi$
$354$	−2112.24	−0.317131
$355$	3584.19	0.535856
$356$	778.640	0.115921
$357$	−948.762	−0.140655
$358$	−10727.4	−1.58369
$359$	779.055	0.114532	0.0572660	−	0.998359i	$-0.481762\pi$
$359$	779.055	0.114532	0.0572660	+	0.998359i	$0.481762\pi$
$360$	1546.44	0.226402
$361$	361.000	0.0526316
$362$	−10211.8	−1.48265
$363$	458.133	0.0662417
$364$	263.760	0.0379801
$365$	−5129.58	−0.735602
$366$	6581.91	0.940006
$367$	850.537	0.120975	0.0604873	−	0.998169i	$-0.480735\pi$
$367$	850.537	0.120975	0.0604873	+	0.998169i	$0.480735\pi$
$368$	−1130.09	−0.160082
$369$	3061.94	0.431973
$370$	4173.61	0.586421
$371$	−720.273	−0.100794
$372$	−1318.22	−0.183727
$373$	−4857.78	−0.674333	−0.337167	−	0.941445i	$-0.609469\pi$
$373$	−4857.78	−0.674333	−0.337167	+	0.941445i	$0.609469\pi$
$374$	−3790.23	−0.524032
$375$	−473.278	−0.0651733
$376$	12599.4	1.72810
$377$	−15755.6	−2.15240
$378$	672.887	0.0915597
$379$	−2698.42	−0.365722	−0.182861	−	0.983139i	$-0.558536\pi$
$379$	−2698.42	−0.365722	−0.182861	+	0.983139i	$0.558536\pi$
$380$	−174.699	−0.0235839
$381$	6951.80	0.934780
$382$	−3593.48	−0.481304
$383$	13743.7	1.83360	0.916801	−	0.399345i	$-0.130762\pi$
$383$	13743.7	1.83360	0.916801	+	0.399345i	$0.130762\pi$
$384$	−2884.50	−0.383331
$385$	99.2815	0.0131425
$386$	3903.99	0.514788
$387$	3219.83	0.422928
$388$	1139.34	0.149076
$389$	1337.00	0.174263	0.0871317	−	0.996197i	$-0.472230\pi$
$389$	1337.00	0.174263	0.0871317	+	0.996197i	$0.472230\pi$
$390$	−3733.70	−0.484777
$391$	−3417.30	−0.441995
$392$	−8297.07	−1.06904
$393$	−6481.22	−0.831894
$394$	−10819.4	−1.38343
$395$	−2759.59	−0.351519
$396$	−256.182	−0.0325092
$397$	10717.4	1.35489	0.677446	−	0.735573i	$-0.263087\pi$
$397$	10717.4	1.35489	0.677446	+	0.735573i	$0.263087\pi$
$398$	−4484.68	−0.564816
$399$	−129.857	−0.0162932
$400$	−1147.67	−0.143459
$401$	−15687.9	−1.95366	−0.976831	−	0.214014i	$-0.931346\pi$
$401$	−15687.9	−1.95366	−0.976831	+	0.214014i	$0.931346\pi$
$402$	3719.99	0.461533
$403$	−15043.5	−1.85948
$404$	−1419.37	−0.174793
$405$	1133.34	0.139052
$406$	−888.452	−0.108604
$407$	−3699.19	−0.450521
$408$	−12835.9	−1.55753
$409$	−1036.86	−0.125353	−0.0626763	−	0.998034i	$-0.519964\pi$
$409$	−1036.86	−0.125353	−0.0626763	+	0.998034i	$0.519964\pi$
$410$	−3000.58	−0.361435
$411$	−7041.74	−0.845118
$412$	−168.903	−0.0201972
$413$	405.709	0.0483381
$414$	773.845	0.0918657
$415$	−1677.05	−0.198369
$416$	6469.93	0.762534
$417$	7843.43	0.921090
$418$	−518.769	−0.0607029
$419$	8665.65	1.01037	0.505184	−	0.863011i	$-0.331424\pi$
$419$	8665.65	1.01037	0.505184	+	0.863011i	$0.331424\pi$
$420$	62.8420	0.00730089
$421$	−15813.2	−1.83061	−0.915305	−	0.402761i	$-0.868051\pi$
$421$	−15813.2	−1.83061	−0.915305	+	0.402761i	$0.868051\pi$
$422$	13891.9	1.60248
$423$	−6533.77	−0.751023
$424$	−9744.67	−1.11614
$425$	−3470.44	−0.396097
$426$	6736.81	0.766196
$427$	−1264.22	−0.143279
$428$	−1107.98	−0.125131
$429$	3309.28	0.372433
$430$	−3155.31	−0.353867
$431$	3629.03	0.405578	0.202789	−	0.979222i	$-0.434999\pi$
$431$	3629.03	0.405578	0.202789	+	0.979222i	$0.434999\pi$
$432$	6894.22	0.767820
$433$	−1343.72	−0.149134	−0.0745670	−	0.997216i	$-0.523757\pi$
$433$	−1343.72	−0.149134	−0.0745670	+	0.997216i	$0.523757\pi$
$434$	−848.296	−0.0938238
$435$	−3753.85	−0.413755
$436$	−1863.88	−0.204733
$437$	−467.726	−0.0511999
$438$	−9641.54	−1.05180
$439$	−2823.27	−0.306941	−0.153471	−	0.988153i	$-0.549045\pi$
$439$	−2823.27	−0.306941	−0.153471	+	0.988153i	$0.549045\pi$
$440$	1343.19	0.145532
$441$	4302.66	0.464600
$442$	−27378.4	−2.94628
$443$	−9202.63	−0.986976	−0.493488	−	0.869753i	$-0.664278\pi$
$443$	−9202.63	−0.986976	−0.493488	+	0.869753i	$0.664278\pi$
$444$	−2341.47	−0.250273
$445$	2117.09	0.225527
$446$	−4981.08	−0.528836
$447$	10030.8	1.06139
$448$	1027.77	0.108388
$449$	−14182.5	−1.49067	−0.745336	−	0.666689i	$-0.767711\pi$
$449$	−14182.5	−1.49067	−0.745336	+	0.666689i	$0.767711\pi$
$450$	785.880	0.0823261
$451$	2659.50	0.277674
$452$	−2946.45	−0.306614
$453$	−3093.92	−0.320894
$454$	6672.36	0.689756
$455$	717.151	0.0738914
$456$	−1756.86	−0.180422
$457$	12093.7	1.23790	0.618951	−	0.785430i	$-0.287558\pi$
$457$	12093.7	1.23790	0.618951	+	0.785430i	$0.287558\pi$
$458$	8846.11	0.902515
$459$	20847.5	2.11999
$460$	226.347	0.0229424
$461$	11783.4	1.19047	0.595236	−	0.803551i	$-0.297059\pi$
$461$	11783.4	1.19047	0.595236	+	0.803551i	$0.297059\pi$
$462$	186.609	0.0187919
$463$	3422.35	0.343521	0.171761	−	0.985139i	$-0.445054\pi$
$463$	3422.35	0.343521	0.171761	+	0.985139i	$0.445054\pi$
$464$	−9102.85	−0.910753
$465$	−3584.18	−0.357446
$466$	−3767.62	−0.374531
$467$	−2559.93	−0.253660	−0.126830	−	0.991924i	$-0.540480\pi$
$467$	−2559.93	−0.253660	−0.126830	+	0.991924i	$0.540480\pi$
$468$	−1850.51	−0.182777
$469$	−714.518	−0.0703484
$470$	6402.85	0.628386
$471$	1190.42	0.116457
$472$	5488.90	0.535269
$473$	2796.64	0.271860
$474$	−5186.91	−0.502622
$475$	−475.000	−0.0458831
$476$	460.807	0.0443719
$477$	5053.35	0.485067
$478$	4289.43	0.410448
$479$	7776.49	0.741789	0.370894	−	0.928675i	$-0.379051\pi$
$479$	7776.49	0.741789	0.370894	+	0.928675i	$0.379051\pi$
$480$	1541.49	0.146581
$481$	−26720.8	−2.53298
$482$	5794.31	0.547559
$483$	168.248	0.0158500
$484$	−222.512	−0.0208971
$485$	3097.83	0.290031
$486$	−7934.45	−0.740564
$487$	−3688.73	−0.343229	−0.171614	−	0.985164i	$-0.554898\pi$
$487$	−3688.73	−0.343229	−0.171614	+	0.985164i	$0.554898\pi$
$488$	−17103.9	−1.58659
$489$	−6349.93	−0.587227
$490$	−4216.44	−0.388734
$491$	13237.3	1.21668	0.608342	−	0.793675i	$-0.291835\pi$
$491$	13237.3	1.21668	0.608342	+	0.793675i	$0.291835\pi$
$492$	1683.38	0.154253
$493$	−27526.2	−2.51464
$494$	−3747.28	−0.341292
$495$	−696.548	−0.0632475
$496$	−8691.42	−0.786807
$497$	−1293.98	−0.116786
$498$	−3152.17	−0.283639
$499$	−7873.11	−0.706310	−0.353155	−	0.935565i	$-0.614891\pi$
$499$	−7873.11	−0.706310	−0.353155	+	0.935565i	$0.614891\pi$
$500$	229.868	0.0205600
$501$	−12510.4	−1.11561
$502$	−16205.2	−1.44078
$503$	6998.47	0.620371	0.310185	−	0.950676i	$-0.399609\pi$
$503$	6998.47	0.620371	0.310185	+	0.950676i	$0.399609\pi$
$504$	−558.303	−0.0493429
$505$	−3859.20	−0.340064
$506$	672.137	0.0590517
$507$	15586.0	1.36528
$508$	−3376.44	−0.294892
$509$	1998.60	0.174040	0.0870199	−	0.996207i	$-0.472266\pi$
$509$	1998.60	0.174040	0.0870199	+	0.996207i	$0.472266\pi$
$510$	−6523.03	−0.566362
$511$	1851.90	0.160320
$512$	12707.0	1.09683
$513$	2853.40	0.245576
$514$	−19113.4	−1.64019
$515$	−459.240	−0.0392943
$516$	1770.18	0.151023
$517$	−5675.03	−0.482761
$518$	−1506.77	−0.127807
$519$	−5665.72	−0.479186
$520$	9702.44	0.818231
$521$	5485.47	0.461273	0.230636	−	0.973040i	$-0.425919\pi$
$521$	5485.47	0.461273	0.230636	+	0.973040i	$0.425919\pi$
$522$	6233.28	0.522650
$523$	−16871.1	−1.41056	−0.705280	−	0.708929i	$-0.749179\pi$
$523$	−16871.1	−1.41056	−0.705280	+	0.708929i	$0.749179\pi$
$524$	3147.88	0.262435
$525$	170.865	0.0142041
$526$	5025.38	0.416572
$527$	−26282.0	−2.17242
$528$	1911.95	0.157589
$529$	−11561.0	−0.950193
$530$	−4952.09	−0.405859
$531$	−2846.41	−0.232625
$532$	63.0706	0.00513996
$533$	19210.7	1.56118
$534$	3979.26	0.322471
$535$	−3012.54	−0.243446
$536$	−9666.82	−0.778998
$537$	16363.4	1.31496
$538$	−13714.4	−1.09901
$539$	3737.16	0.298647
$540$	−1380.85	−0.110041
$541$	−3121.25	−0.248046	−0.124023	−	0.992279i	$-0.539580\pi$
$541$	−3121.25	−0.248046	−0.124023	+	0.992279i	$0.539580\pi$
$542$	−6423.94	−0.509099
$543$	15576.9	1.23106
$544$	11303.4	0.890864
$545$	−5067.81	−0.398314
$546$	1347.95	0.105654
$547$	−12448.7	−0.973071	−0.486536	−	0.873661i	$-0.661740\pi$
$547$	−12448.7	−0.973071	−0.486536	+	0.873661i	$0.661740\pi$
$548$	3420.12	0.266606
$549$	8869.65	0.689522
$550$	682.591	0.0529196
$551$	−3767.51	−0.291291
$552$	2276.25	0.175514
$553$	996.279	0.0766114
$554$	−11091.5	−0.850600
$555$	−6366.35	−0.486913
$556$	−3809.50	−0.290573
$557$	−2256.75	−0.171673	−0.0858363	−	0.996309i	$-0.527356\pi$
$557$	−2256.75	−0.171673	−0.0858363	+	0.996309i	$0.527356\pi$
$558$	5951.55	0.451522
$559$	20201.3	1.52849
$560$	414.336	0.0312659
$561$	5781.55	0.435111
$562$	5744.16	0.431144
$563$	1147.04	0.0858646	0.0429323	−	0.999078i	$-0.486330\pi$
$563$	1147.04	0.0858646	0.0429323	+	0.999078i	$0.486330\pi$
$564$	−3592.11	−0.268183
$565$	−8011.27	−0.596525
$566$	19326.3	1.43524
$567$	−409.164	−0.0303056
$568$	−17506.4	−1.29322
$569$	5999.75	0.442043	0.221022	−	0.975269i	$-0.429061\pi$
$569$	5999.75	0.442043	0.221022	+	0.975269i	$0.429061\pi$
$570$	−892.807	−0.0656063
$571$	9072.40	0.664918	0.332459	−	0.943118i	$-0.392122\pi$
$571$	9072.40	0.664918	0.332459	+	0.943118i	$0.392122\pi$
$572$	−1607.29	−0.117490
$573$	5481.42	0.399633
$574$	1083.28	0.0787723
$575$	615.429	0.0446350
$576$	−7210.74	−0.521610
$577$	22518.3	1.62469	0.812346	−	0.583176i	$-0.198190\pi$
$577$	22518.3	1.62469	0.812346	+	0.583176i	$0.198190\pi$
$578$	−35637.1	−2.56455
$579$	−5955.08	−0.427435
$580$	1823.22	0.130526
$581$	605.455	0.0432332
$582$	5822.65	0.414702
$583$	4389.18	0.311803
$584$	25054.6	1.77529
$585$	−5031.45	−0.355598
$586$	−5473.63	−0.385859
$587$	24029.6	1.68962	0.844810	−	0.535066i	$-0.179713\pi$
$587$	24029.6	1.68962	0.844810	+	0.535066i	$0.179713\pi$
$588$	2365.50	0.165904
$589$	−3597.22	−0.251649
$590$	2789.37	0.194639
$591$	16503.7	1.14868
$592$	−15438.0	−1.07179
$593$	4440.24	0.307485	0.153743	−	0.988111i	$-0.450867\pi$
$593$	4440.24	0.307485	0.153743	+	0.988111i	$0.450867\pi$
$594$	−4100.42	−0.283236
$595$	1252.91	0.0863268
$596$	−4871.89	−0.334833
$597$	6840.85	0.468974
$598$	4855.12	0.332008
$599$	4213.25	0.287394	0.143697	−	0.989622i	$-0.454101\pi$
$599$	4213.25	0.287394	0.143697	+	0.989622i	$0.454101\pi$
$600$	2311.65	0.157288
$601$	−5531.34	−0.375421	−0.187710	−	0.982224i	$-0.560107\pi$
$601$	−5531.34	−0.375421	−0.187710	+	0.982224i	$0.560107\pi$
$602$	1139.14	0.0771229
$603$	5012.98	0.338548
$604$	1502.69	0.101231
$605$	−605.000	−0.0406558
$606$	−7253.73	−0.486242
$607$	−14100.0	−0.942838	−0.471419	−	0.881909i	$-0.656258\pi$
$607$	−14100.0	−0.942838	−0.471419	+	0.881909i	$0.656258\pi$
$608$	1547.10	0.103196
$609$	1355.23	0.0901751
$610$	−8691.92	−0.576927
$611$	−40993.1	−2.71424
$612$	−3232.97	−0.213537
$613$	−10834.8	−0.713889	−0.356944	−	0.934126i	$-0.616181\pi$
$613$	−10834.8	−0.713889	−0.356944	+	0.934126i	$0.616181\pi$
$614$	18045.4	1.18608
$615$	4577.03	0.300104
$616$	−484.925	−0.0317178
$617$	−19155.9	−1.24990	−0.624949	−	0.780666i	$-0.714880\pi$
$617$	−19155.9	−1.24990	−0.624949	+	0.780666i	$0.714880\pi$
$618$	−863.185	−0.0561851
$619$	15290.0	0.992824	0.496412	−	0.868087i	$-0.334650\pi$
$619$	15290.0	0.992824	0.496412	+	0.868087i	$0.334650\pi$
$620$	1740.81	0.112762
$621$	−3696.97	−0.238896
$622$	−6930.49	−0.446764
$623$	−764.319	−0.0491521
$624$	13810.8	0.886015
$625$	625.000	0.0400000
$626$	−236.598	−0.0151060
$627$	791.321	0.0504024
$628$	−578.176	−0.0367384
$629$	−46683.1	−2.95926
$630$	−283.721	−0.0179424
$631$	−24818.4	−1.56578	−0.782888	−	0.622163i	$-0.786254\pi$
$631$	−24818.4	−1.56578	−0.782888	+	0.622163i	$0.786254\pi$
$632$	13478.8	0.848351
$633$	−21190.5	−1.33056
$634$	−534.730	−0.0334966
$635$	−9180.38	−0.573720
$636$	2778.21	0.173213
$637$	26995.0	1.67909
$638$	5414.03	0.335962
$639$	9078.39	0.562027
$640$	3809.21	0.235269
$641$	24445.6	1.50631	0.753153	−	0.657845i	$-0.228532\pi$
$641$	24445.6	1.50631	0.753153	+	0.657845i	$0.228532\pi$
$642$	−5662.35	−0.348092
$643$	−20539.5	−1.25972	−0.629859	−	0.776710i	$-0.716887\pi$
$643$	−20539.5	−1.25972	−0.629859	+	0.776710i	$0.716887\pi$
$644$	−81.7168	−0.00500014
$645$	4813.06	0.293820
$646$	−6546.76	−0.398729
$647$	12539.2	0.761927	0.380963	−	0.924590i	$-0.375592\pi$
$647$	12539.2	0.761927	0.380963	+	0.924590i	$0.375592\pi$
$648$	−5535.63	−0.335587
$649$	−2472.30	−0.149532
$650$	4930.63	0.297531
$651$	1293.98	0.0779031
$652$	3084.12	0.185250
$653$	1056.47	0.0633123	0.0316562	−	0.999499i	$-0.489922\pi$
$653$	1056.47	0.0633123	0.0316562	+	0.999499i	$0.489922\pi$
$654$	−9525.43	−0.569532
$655$	8558.95	0.510574
$656$	11099.0	0.660585
$657$	−12992.7	−0.771529
$658$	−2311.58	−0.136953
$659$	4869.61	0.287850	0.143925	−	0.989589i	$-0.454028\pi$
$659$	4869.61	0.287850	0.143925	+	0.989589i	$0.454028\pi$
$660$	−382.945	−0.0225850
$661$	20331.4	1.19637	0.598184	−	0.801359i	$-0.295889\pi$
$661$	20331.4	1.19637	0.598184	+	0.801359i	$0.295889\pi$
$662$	17013.5	0.998863
$663$	41762.5	2.44634
$664$	8191.28	0.478740
$665$	171.486	0.00999993
$666$	10571.3	0.615062
$667$	4881.33	0.283367
$668$	6076.19	0.351939
$669$	7598.05	0.439100
$670$	−4912.53	−0.283265
$671$	7703.91	0.443228
$672$	−556.515	−0.0319465
$673$	619.560	0.0354863	0.0177432	−	0.999843i	$-0.494352\pi$
$673$	619.560	0.0354863	0.0177432	+	0.999843i	$0.494352\pi$
$674$	9806.75	0.560448
$675$	−3754.47	−0.214088
$676$	−7569.99	−0.430701
$677$	4867.38	0.276320	0.138160	−	0.990410i	$-0.455881\pi$
$677$	4867.38	0.276320	0.138160	+	0.990410i	$0.455881\pi$
$678$	−15057.9	−0.852944
$679$	−1118.39	−0.0632103
$680$	16950.8	0.955934
$681$	−10177.9	−0.572713
$682$	5169.33	0.290240
$683$	−22236.9	−1.24578	−0.622891	−	0.782308i	$-0.714042\pi$
$683$	−22236.9	−1.24578	−0.622891	+	0.782308i	$0.714042\pi$
$684$	−442.496	−0.0247358
$685$	9299.16	0.518690
$686$	3059.07	0.170256
$687$	−13493.7	−0.749369
$688$	11671.4	0.646753
$689$	31704.9	1.75306
$690$	1156.76	0.0638217
$691$	13200.5	0.726732	0.363366	−	0.931647i	$-0.381627\pi$
$691$	13200.5	0.726732	0.363366	+	0.931647i	$0.381627\pi$
$692$	2751.80	0.151167
$693$	251.470	0.0137844
$694$	16556.2	0.905571
$695$	−10357.9	−0.565318
$696$	18335.1	0.998548
$697$	33562.4	1.82391
$698$	−15836.1	−0.858749
$699$	5747.05	0.310978
$700$	−82.9877	−0.00448091
$701$	7754.53	0.417810	0.208905	−	0.977936i	$-0.433010\pi$
$701$	7754.53	0.417810	0.208905	+	0.977936i	$0.433010\pi$
$702$	−29619.0	−1.59245
$703$	−6389.51	−0.342795
$704$	−6263.03	−0.335294
$705$	−9766.79	−0.521757
$706$	8830.93	0.470760
$707$	1393.26	0.0741147
$708$	−1564.89	−0.0830679
$709$	4981.10	0.263849	0.131925	−	0.991260i	$-0.457884\pi$
$709$	4981.10	0.263849	0.131925	+	0.991260i	$0.457884\pi$
$710$	−8896.48	−0.470252
$711$	−6989.78	−0.368688
$712$	−10340.6	−0.544283
$713$	4660.70	0.244803
$714$	2354.97	0.123435
$715$	−4370.16	−0.228580
$716$	−7947.58	−0.414825
$717$	−6543.02	−0.340800
$718$	−1933.73	−0.100510
$719$	−32583.8	−1.69008	−0.845042	−	0.534700i	$-0.820425\pi$
$719$	−32583.8	−1.69008	−0.845042	+	0.534700i	$0.820425\pi$
$720$	−2906.93	−0.150465
$721$	165.797	0.00856392
$722$	−896.055	−0.0461880
$723$	−8838.53	−0.454645
$724$	−7565.56	−0.388359
$725$	4957.25	0.253941
$726$	−1137.15	−0.0581319
$727$	−31854.2	−1.62505	−0.812523	−	0.582929i	$-0.801907\pi$
$727$	−31854.2	−1.62505	−0.812523	+	0.582929i	$0.801907\pi$
$728$	−3502.81	−0.178328
$729$	18223.1	0.925830
$730$	12732.4	0.645544
$731$	35293.1	1.78572
$732$	4876.32	0.246221
$733$	−28799.2	−1.45119	−0.725596	−	0.688121i	$-0.758436\pi$
$733$	−28799.2	−1.45119	−0.725596	+	0.688121i	$0.758436\pi$
$734$	−2111.16	−0.106164
$735$	6431.69	0.322771
$736$	−2004.48	−0.100389
$737$	4354.12	0.217620
$738$	−7600.18	−0.379087
$739$	−797.424	−0.0396938	−0.0198469	−	0.999803i	$-0.506318\pi$
$739$	−797.424	−0.0396938	−0.0198469	+	0.999803i	$0.506318\pi$
$740$	3092.09	0.153605
$741$	5716.03	0.283379
$742$	1787.82	0.0884543
$743$	−5624.92	−0.277737	−0.138868	−	0.990311i	$-0.544346\pi$
$743$	−5624.92	−0.277737	−0.138868	+	0.990311i	$0.544346\pi$
$744$	17506.4	0.862654
$745$	−13246.4	−0.651426
$746$	12057.7	0.591776
$747$	−4247.80	−0.208057
$748$	−2808.05	−0.137263
$749$	1087.60	0.0530574
$750$	1174.75	0.0571942
$751$	4090.10	0.198735	0.0993674	−	0.995051i	$-0.468318\pi$
$751$	4090.10	0.198735	0.0993674	+	0.995051i	$0.468318\pi$
$752$	−23683.8	−1.14849
$753$	24719.1	1.19630
$754$	39107.8	1.88889
$755$	4085.76	0.196948
$756$	498.519	0.0239828
$757$	−6291.45	−0.302070	−0.151035	−	0.988528i	$-0.548261\pi$
$757$	−6291.45	−0.302070	−0.151035	+	0.988528i	$0.548261\pi$
$758$	6697.88	0.320947
$759$	−1025.27	−0.0490314
$760$	2320.06	0.110734
$761$	25476.5	1.21356	0.606781	−	0.794869i	$-0.292460\pi$
$761$	25476.5	1.21356	0.606781	+	0.794869i	$0.292460\pi$
$762$	−17255.4	−0.820337
$763$	1829.60	0.0868100
$764$	−2662.29	−0.126071
$765$	−8790.29	−0.415443
$766$	−34113.9	−1.60912
$767$	−17858.5	−0.840720
$768$	−10086.2	−0.473900
$769$	24999.7	1.17232	0.586159	−	0.810196i	$-0.300639\pi$
$769$	24999.7	1.17232	0.586159	+	0.810196i	$0.300639\pi$
$770$	−246.432	−0.0115335
$771$	29155.2	1.36187
$772$	2892.34	0.134841
$773$	21406.4	0.996034	0.498017	−	0.867167i	$-0.334062\pi$
$773$	21406.4	0.996034	0.498017	+	0.867167i	$0.334062\pi$
$774$	−7992.09	−0.371150
$775$	4733.19	0.219382
$776$	−15130.8	−0.699955
$777$	2298.40	0.106119
$778$	−3318.63	−0.152929
$779$	4593.69	0.211278
$780$	−2766.17	−0.126981
$781$	7885.21	0.361274
$782$	8482.24	0.387883
$783$	−29778.9	−1.35915
$784$	15596.4	0.710479
$785$	−1572.03	−0.0714756
$786$	16087.4	0.730047
$787$	13901.8	0.629662	0.314831	−	0.949148i	$-0.398052\pi$
$787$	13901.8	0.629662	0.314831	+	0.949148i	$0.398052\pi$
$788$	−8015.73	−0.362371
$789$	−7665.63	−0.345885
$790$	6849.72	0.308484
$791$	2892.26	0.130009
$792$	3402.18	0.152640
$793$	55648.5	2.49197
$794$	−26602.2	−1.18901
$795$	7553.83	0.336990
$796$	−3322.55	−0.147946
$797$	−13091.1	−0.581822	−0.290911	−	0.956750i	$-0.593958\pi$
$797$	−13091.1	−0.581822	−0.290911	+	0.956750i	$0.593958\pi$
$798$	322.325	0.0142985
$799$	−71617.7	−3.17103
$800$	−2035.66	−0.0899642
$801$	5362.37	0.236542
$802$	38939.8	1.71448
$803$	−11285.1	−0.495943
$804$	2756.01	0.120892
$805$	−222.184	−0.00972792
$806$	37340.2	1.63183
$807$	20919.7	0.912524
$808$	18849.7	0.820704
$809$	−23316.9	−1.01332	−0.506661	−	0.862145i	$-0.669120\pi$
$809$	−23316.9	−1.01332	−0.506661	+	0.862145i	$0.669120\pi$
$810$	−2813.12	−0.122029
$811$	34945.0	1.51305	0.756525	−	0.653965i	$-0.226896\pi$
$811$	34945.0	1.51305	0.756525	+	0.653965i	$0.226896\pi$
$812$	−658.225	−0.0284472
$813$	9798.96	0.422712
$814$	9181.94	0.395365
$815$	8385.57	0.360410
$816$	24128.4	1.03513
$817$	4830.57	0.206855
$818$	2573.63	0.110006
$819$	1816.47	0.0775003
$820$	−2223.03	−0.0946727
$821$	32407.5	1.37762	0.688812	−	0.724940i	$-0.258132\pi$
$821$	32407.5	1.37762	0.688812	+	0.724940i	$0.258132\pi$
$822$	17478.6	0.741652
$823$	−13527.8	−0.572965	−0.286483	−	0.958085i	$-0.592486\pi$
$823$	−13527.8	−0.572965	−0.286483	+	0.958085i	$0.592486\pi$
$824$	2243.09	0.0948320
$825$	−1041.21	−0.0439398
$826$	−1007.03	−0.0424202
$827$	11237.4	0.472508	0.236254	−	0.971691i	$-0.424080\pi$
$827$	11237.4	0.472508	0.236254	+	0.971691i	$0.424080\pi$
$828$	573.316	0.0240629
$829$	9025.92	0.378146	0.189073	−	0.981963i	$-0.439452\pi$
$829$	9025.92	0.378146	0.189073	+	0.981963i	$0.439452\pi$
$830$	4162.68	0.174083
$831$	16918.8	0.706264
$832$	−45240.4	−1.88513
$833$	47162.2	1.96167
$834$	−19468.6	−0.808323
$835$	16520.9	0.684706
$836$	−384.339	−0.0159003
$837$	−28433.0	−1.17418
$838$	−21509.4	−0.886671
$839$	36393.4	1.49754	0.748772	−	0.662828i	$-0.230644\pi$
$839$	36393.4	1.49754	0.748772	+	0.662828i	$0.230644\pi$
$840$	−834.561	−0.0342799
$841$	14929.9	0.612157
$842$	39250.6	1.60649
$843$	−8762.04	−0.357984
$844$	10292.1	0.419748
$845$	−20582.5	−0.837940
$846$	16217.8	0.659077
$847$	218.419	0.00886066
$848$	18317.6	0.741778
$849$	−29479.9	−1.19169
$850$	8614.16	0.347604
$851$	8278.51	0.333471
$852$	4991.08	0.200694
$853$	14994.5	0.601877	0.300938	−	0.953644i	$-0.402700\pi$
$853$	14994.5	0.601877	0.300938	+	0.953644i	$0.402700\pi$
$854$	3137.99	0.125738
$855$	−1203.13	−0.0481241
$856$	14714.3	0.587528
$857$	37444.2	1.49250	0.746248	−	0.665668i	$-0.231853\pi$
$857$	37444.2	1.49250	0.746248	+	0.665668i	$0.231853\pi$
$858$	−8214.13	−0.326837
$859$	−25902.5	−1.02885	−0.514426	−	0.857535i	$-0.671995\pi$
$859$	−25902.5	−1.02885	−0.514426	+	0.857535i	$0.671995\pi$
$860$	−2337.67	−0.0926904
$861$	−1652.42	−0.0654056
$862$	−9007.78	−0.355924
$863$	13752.7	0.542465	0.271233	−	0.962514i	$-0.412569\pi$
$863$	13752.7	0.542465	0.271233	+	0.962514i	$0.412569\pi$
$864$	12228.5	0.481507
$865$	7482.02	0.294100
$866$	3335.31	0.130876
$867$	54360.2	2.12938
$868$	−628.474	−0.0245758
$869$	−6071.10	−0.236994
$870$	9317.61	0.363100
$871$	31451.6	1.22353
$872$	24752.9	0.961285
$873$	7846.49	0.304196
$874$	1160.96	0.0449316
$875$	−225.640	−0.00871774
$876$	−7143.09	−0.275505
$877$	−40483.9	−1.55877	−0.779386	−	0.626544i	$-0.784469\pi$
$877$	−40483.9	−1.55877	−0.779386	+	0.626544i	$0.784469\pi$
$878$	7007.76	0.269363
$879$	8349.38	0.320384
$880$	−2524.87	−0.0967198
$881$	−48629.3	−1.85966	−0.929831	−	0.367987i	$-0.880047\pi$
$881$	−48629.3	−1.85966	−0.929831	+	0.367987i	$0.880047\pi$
$882$	−10679.8	−0.407720
$883$	−2651.18	−0.101041	−0.0505205	−	0.998723i	$-0.516088\pi$
$883$	−2651.18	−0.101041	−0.0505205	+	0.998723i	$0.516088\pi$
$884$	−20283.7	−0.771737
$885$	−4254.86	−0.161611
$886$	22842.3	0.866142
$887$	−49726.7	−1.88237	−0.941184	−	0.337895i	$-0.890285\pi$
$887$	−49726.7	−1.88237	−0.941184	+	0.337895i	$0.890285\pi$
$888$	31095.4	1.17511
$889$	3314.34	0.125039
$890$	−5254.92	−0.197916
$891$	2493.35	0.0937491
$892$	−3690.32	−0.138521
$893$	−9802.32	−0.367326
$894$	−24897.9	−0.931445
$895$	−21609.1	−0.807054
$896$	−1375.21	−0.0512753
$897$	−7405.92	−0.275671
$898$	35203.0	1.30817
$899$	37541.7	1.39276
$900$	582.232	0.0215642
$901$	55390.6	2.04809
$902$	−6601.28	−0.243679
$903$	−1737.63	−0.0640361
$904$	39129.8	1.43964
$905$	−20570.4	−0.755563
$906$	7679.57	0.281608
$907$	46870.7	1.71589	0.857947	−	0.513739i	$-0.171740\pi$
$907$	46870.7	1.71589	0.857947	+	0.513739i	$0.171740\pi$
$908$	4943.33	0.180672
$909$	−9774.98	−0.356673
$910$	−1780.08	−0.0648450
$911$	30107.7	1.09497	0.547483	−	0.836817i	$-0.315586\pi$
$911$	30107.7	1.09497	0.547483	+	0.836817i	$0.315586\pi$
$912$	3302.45	0.119907
$913$	−3689.51	−0.133740
$914$	−30018.4	−1.08635
$915$	13258.5	0.479030
$916$	6553.79	0.236401
$917$	−3089.99	−0.111276
$918$	−51746.5	−1.86045
$919$	−26043.9	−0.934829	−0.467414	−	0.884038i	$-0.654814\pi$
$919$	−26043.9	−0.934829	−0.467414	+	0.884038i	$0.654814\pi$
$920$	−3005.96	−0.107721
$921$	−27526.1	−0.984815
$922$	−29248.1	−1.04472
$923$	56958.1	2.03120
$924$	138.252	0.00492226
$925$	8407.25	0.298842
$926$	−8494.79	−0.301465
$927$	−1163.21	−0.0412134
$928$	−16146.0	−0.571141
$929$	−1495.32	−0.0528093	−0.0264046	−	0.999651i	$-0.508406\pi$
$929$	−1495.32	−0.0528093	−0.0264046	+	0.999651i	$0.508406\pi$
$930$	8896.47	0.313685
$931$	6455.09	0.227236
$932$	−2791.30	−0.0981031
$933$	10571.6	0.370954
$934$	6354.12	0.222605
$935$	−7634.98	−0.267049
$936$	24575.3	0.858194
$937$	−29981.3	−1.04530	−0.522649	−	0.852548i	$-0.675056\pi$
$937$	−29981.3	−1.04530	−0.522649	+	0.852548i	$0.675056\pi$
$938$	1773.54	0.0617357
$939$	360.902	0.0125427
$940$	4743.66	0.164597
$941$	−45172.8	−1.56492	−0.782461	−	0.622699i	$-0.786036\pi$
$941$	−45172.8	−1.56492	−0.782461	+	0.622699i	$0.786036\pi$
$942$	−2954.79	−0.102200
$943$	−5951.76	−0.205531
$944$	−10317.8	−0.355736
$945$	1355.45	0.0466591
$946$	−6941.69	−0.238577
$947$	−33322.7	−1.14345	−0.571723	−	0.820447i	$-0.693725\pi$
$947$	−33322.7	−1.14345	−0.571723	+	0.820447i	$0.693725\pi$
$948$	−3842.81	−0.131655
$949$	−81516.8	−2.78835
$950$	1179.02	0.0402658
$951$	815.668	0.0278127
$952$	−6119.66	−0.208339
$953$	−34960.4	−1.18833	−0.594165	−	0.804343i	$-0.702517\pi$
$953$	−34960.4	−1.18833	−0.594165	+	0.804343i	$0.702517\pi$
$954$	−12543.2	−0.425681
$955$	−7238.64	−0.245274
$956$	3177.90	0.107511
$957$	−8258.47	−0.278953
$958$	−19302.4	−0.650973
$959$	−3357.22	−0.113045
$960$	−10778.7	−0.362377
$961$	6053.93	0.203213
$962$	66324.9	2.22287
$963$	−7630.47	−0.255336
$964$	4292.81	0.143425
$965$	7864.14	0.262338
$966$	−417.616	−0.0139095
$967$	−55164.0	−1.83449	−0.917247	−	0.398320i	$-0.869593\pi$
$967$	−55164.0	−1.83449	−0.917247	+	0.398320i	$0.869593\pi$
$968$	2955.03	0.0981179
$969$	9986.31	0.331070
$970$	−7689.26	−0.254523
$971$	36079.2	1.19242	0.596208	−	0.802830i	$-0.296673\pi$
$971$	36079.2	1.19242	0.596208	+	0.802830i	$0.296673\pi$
$972$	−5878.37	−0.193980
$973$	3739.43	0.123207
$974$	9155.97	0.301208
$975$	−7521.10	−0.247044
$976$	32151.0	1.05444
$977$	23125.6	0.757271	0.378635	−	0.925546i	$-0.376393\pi$
$977$	23125.6	0.757271	0.378635	+	0.925546i	$0.376393\pi$
$978$	15761.5	0.515334
$979$	4657.59	0.152050
$980$	−3123.82	−0.101823
$981$	−12836.3	−0.417768
$982$	−32857.0	−1.06773
$983$	−35545.2	−1.15332	−0.576661	−	0.816983i	$-0.695645\pi$
$983$	−35545.2	−1.15332	−0.576661	+	0.816983i	$0.695645\pi$
$984$	−22355.8	−0.724265
$985$	−21794.4	−0.705003
$986$	68324.0	2.20677
$987$	3526.04	0.113713
$988$	−2776.24	−0.0893966
$989$	−6258.67	−0.201228
$990$	1728.94	0.0555042
$991$	−11935.3	−0.382581	−0.191290	−	0.981533i	$-0.561267\pi$
$991$	−11935.3	−0.382581	−0.191290	+	0.981533i	$0.561267\pi$
$992$	−15416.2	−0.493413
$993$	−25952.0	−0.829369
$994$	3211.84	0.102488
$995$	−9033.87	−0.287832
$996$	−2335.34	−0.0742952
$997$	−43334.5	−1.37655	−0.688273	−	0.725452i	$-0.741631\pi$
$997$	−43334.5	−1.37655	−0.688273	+	0.725452i	$0.741631\pi$
$998$	19542.2	0.619838
$999$	−50503.6	−1.59946

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.7	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-2.48215 q^{2} +3.78622 q^{3} -1.83894 q^{4} -5.00000 q^{5} -9.39797 q^{6} +1.80512 q^{7} +24.4217 q^{8} -12.6645 q^{9} +O(q^{10})\)	\(q-2.48215 q^{2} +3.78622 q^{3} -1.83894 q^{4} -5.00000 q^{5} -9.39797 q^{6} +1.80512 q^{7} +24.4217 q^{8} -12.6645 q^{9} +12.4107 q^{10} -11.0000 q^{11} -6.96264 q^{12} -79.4575 q^{13} -4.48057 q^{14} -18.9311 q^{15} -45.9068 q^{16} -138.818 q^{17} +31.4352 q^{18} -19.0000 q^{19} +9.19470 q^{20} +6.83458 q^{21} +27.3036 q^{22} +24.6171 q^{23} +92.4661 q^{24} +25.0000 q^{25} +197.225 q^{26} -150.179 q^{27} -3.31951 q^{28} +198.290 q^{29} +46.9898 q^{30} +189.328 q^{31} -81.4263 q^{32} -41.6485 q^{33} +344.566 q^{34} -9.02560 q^{35} +23.2893 q^{36} +336.290 q^{37} +47.1608 q^{38} -300.844 q^{39} -122.109 q^{40} -241.773 q^{41} -16.9645 q^{42} -254.240 q^{43} +20.2283 q^{44} +63.3225 q^{45} -61.1034 q^{46} +515.912 q^{47} -173.813 q^{48} -339.742 q^{49} -62.0537 q^{50} -525.595 q^{51} +146.118 q^{52} -399.017 q^{53} +372.766 q^{54} +55.0000 q^{55} +44.0841 q^{56} -71.9383 q^{57} -492.185 q^{58} +224.755 q^{59} +34.8132 q^{60} -700.355 q^{61} -469.939 q^{62} -22.8609 q^{63} +569.366 q^{64} +397.288 q^{65} +103.378 q^{66} -395.829 q^{67} +255.278 q^{68} +93.2060 q^{69} +22.4029 q^{70} -716.837 q^{71} -309.289 q^{72} +1025.92 q^{73} -834.722 q^{74} +94.6556 q^{75} +34.9399 q^{76} -19.8563 q^{77} +746.739 q^{78} +551.919 q^{79} +229.534 q^{80} -226.668 q^{81} +600.116 q^{82} +335.410 q^{83} -12.5684 q^{84} +694.089 q^{85} +631.062 q^{86} +750.770 q^{87} -268.639 q^{88} -423.417 q^{89} -157.176 q^{90} -143.430 q^{91} -45.2695 q^{92} +716.837 q^{93} -1280.57 q^{94} +95.0000 q^{95} -308.298 q^{96} -619.565 q^{97} +843.289 q^{98} +139.310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 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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