LMFDB - Embedded newform 1045.4.a.g.1.20

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			1.20
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)

$f(q)$	$=$	$q+4.39002 q^{2} -6.92776 q^{3} +11.2723 q^{4} -5.00000 q^{5} -30.4130 q^{6} -34.5356 q^{7} +14.3653 q^{8} +20.9939 q^{9} +O(q^{10})$	\(q+4.39002 q^{2} -6.92776 q^{3} +11.2723 q^{4} -5.00000 q^{5} -30.4130 q^{6} -34.5356 q^{7} +14.3653 q^{8} +20.9939 q^{9} -21.9501 q^{10} -11.0000 q^{11} -78.0916 q^{12} -62.8275 q^{13} -151.612 q^{14} +34.6388 q^{15} -27.1142 q^{16} -4.27264 q^{17} +92.1636 q^{18} -19.0000 q^{19} -56.3613 q^{20} +239.255 q^{21} -48.2902 q^{22} +25.5501 q^{23} -99.5194 q^{24} +25.0000 q^{25} -275.814 q^{26} +41.6089 q^{27} -389.295 q^{28} +240.038 q^{29} +152.065 q^{30} +312.033 q^{31} -233.954 q^{32} +76.2054 q^{33} -18.7570 q^{34} +172.678 q^{35} +236.649 q^{36} -73.5102 q^{37} -83.4104 q^{38} +435.254 q^{39} -71.8265 q^{40} -387.467 q^{41} +1050.33 q^{42} -524.238 q^{43} -123.995 q^{44} -104.969 q^{45} +112.165 q^{46} +262.427 q^{47} +187.840 q^{48} +849.710 q^{49} +109.750 q^{50} +29.5998 q^{51} -708.209 q^{52} -93.1791 q^{53} +182.664 q^{54} +55.0000 q^{55} -496.115 q^{56} +131.627 q^{57} +1053.77 q^{58} +187.284 q^{59} +390.458 q^{60} -119.249 q^{61} +1369.83 q^{62} -725.037 q^{63} -810.150 q^{64} +314.138 q^{65} +334.543 q^{66} -129.635 q^{67} -48.1623 q^{68} -177.005 q^{69} +758.060 q^{70} -878.625 q^{71} +301.584 q^{72} -798.326 q^{73} -322.711 q^{74} -173.194 q^{75} -214.173 q^{76} +379.892 q^{77} +1910.77 q^{78} +353.389 q^{79} +135.571 q^{80} -855.092 q^{81} -1700.99 q^{82} +1276.18 q^{83} +2696.94 q^{84} +21.3632 q^{85} -2301.41 q^{86} -1662.92 q^{87} -158.018 q^{88} -648.944 q^{89} -460.818 q^{90} +2169.79 q^{91} +288.007 q^{92} -2161.69 q^{93} +1152.06 q^{94} +95.0000 q^{95} +1620.78 q^{96} +1034.99 q^{97} +3730.24 q^{98} -230.933 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.39002	1.55211	0.776053	−	0.630668i	$-0.217219\pi$
$2$	4.39002	1.55211	0.776053	+	0.630668i	$0.217219\pi$
$3$	−6.92776	−1.33325	−0.666624	−	0.745394i	$-0.732261\pi$
$3$	−6.92776	−1.33325	−0.666624	+	0.745394i	$0.732261\pi$
$4$	11.2723	1.40903
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−30.4130	−2.06934
$7$	−34.5356	−1.86475	−0.932374	−	0.361494i	$-0.882267\pi$
$7$	−34.5356	−1.86475	−0.932374	+	0.361494i	$0.882267\pi$
$8$	14.3653	0.634863
$9$	20.9939	0.777552
$10$	−21.9501	−0.694123
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−78.0916	−1.87859
$13$	−62.8275	−1.34040	−0.670201	−	0.742180i	$-0.733792\pi$
$13$	−62.8275	−1.34040	−0.670201	+	0.742180i	$0.733792\pi$
$14$	−151.612	−2.89429
$15$	34.6388	0.596247
$16$	−27.1142	−0.423659
$17$	−4.27264	−0.0609569	−0.0304785	−	0.999535i	$-0.509703\pi$
$17$	−4.27264	−0.0609569	−0.0304785	+	0.999535i	$0.509703\pi$
$18$	92.1636	1.20684
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−56.3613	−0.630139
$21$	239.255	2.48617
$22$	−48.2902	−0.467978
$23$	25.5501	0.231633	0.115816	−	0.993271i	$-0.463052\pi$
$23$	25.5501	0.231633	0.115816	+	0.993271i	$0.463052\pi$
$24$	−99.5194	−0.846430
$25$	25.0000	0.200000
$26$	−275.814	−2.08045
$27$	41.6089	0.296579
$28$	−389.295	−2.62749
$29$	240.038	1.53703	0.768515	−	0.639832i	$-0.220996\pi$
$29$	240.038	1.53703	0.768515	+	0.639832i	$0.220996\pi$
$30$	152.065	0.925438
$31$	312.033	1.80783	0.903917	−	0.427708i	$-0.140679\pi$
$31$	312.033	1.80783	0.903917	+	0.427708i	$0.140679\pi$
$32$	−233.954	−1.29243
$33$	76.2054	0.401990
$34$	−18.7570	−0.0946116
$35$	172.678	0.833941
$36$	236.649	1.09560
$37$	−73.5102	−0.326622	−0.163311	−	0.986575i	$-0.552217\pi$
$37$	−73.5102	−0.326622	−0.163311	+	0.986575i	$0.552217\pi$
$38$	−83.4104	−0.356078
$39$	435.254	1.78709
$40$	−71.8265	−0.283919
$41$	−387.467	−1.47591	−0.737954	−	0.674851i	$-0.764208\pi$
$41$	−387.467	−1.47591	−0.737954	+	0.674851i	$0.764208\pi$
$42$	1050.33	3.85880
$43$	−524.238	−1.85920	−0.929599	−	0.368572i	$-0.879847\pi$
$43$	−524.238	−1.85920	−0.929599	+	0.368572i	$0.879847\pi$
$44$	−123.995	−0.424839
$45$	−104.969	−0.347732
$46$	112.165	0.359519
$47$	262.427	0.814446	0.407223	−	0.913329i	$-0.366497\pi$
$47$	262.427	0.814446	0.407223	+	0.913329i	$0.366497\pi$
$48$	187.840	0.564843
$49$	849.710	2.47729
$50$	109.750	0.310421
$51$	29.5998	0.0812707
$52$	−708.209	−1.88867
$53$	−93.1791	−0.241493	−0.120747	−	0.992683i	$-0.538529\pi$
$53$	−93.1791	−0.241493	−0.120747	+	0.992683i	$0.538529\pi$
$54$	182.664	0.460322
$55$	55.0000	0.134840
$56$	−496.115	−1.18386
$57$	131.627	0.305868
$58$	1053.77	2.38563
$59$	187.284	0.413258	0.206629	−	0.978419i	$-0.433751\pi$
$59$	187.284	0.413258	0.206629	+	0.978419i	$0.433751\pi$
$60$	390.458	0.840132
$61$	−119.249	−0.250299	−0.125150	−	0.992138i	$-0.539941\pi$
$61$	−119.249	−0.250299	−0.125150	+	0.992138i	$0.539941\pi$
$62$	1369.83	2.80595
$63$	−725.037	−1.44994
$64$	−810.150	−1.58232
$65$	314.138	0.599446
$66$	334.543	0.623930
$67$	−129.635	−0.236380	−0.118190	−	0.992991i	$-0.537709\pi$
$67$	−129.635	−0.236380	−0.118190	+	0.992991i	$0.537709\pi$
$68$	−48.1623	−0.0858903
$69$	−177.005	−0.308824
$70$	758.060	1.29436
$71$	−878.625	−1.46864	−0.734321	−	0.678802i	$-0.762499\pi$
$71$	−878.625	−1.46864	−0.734321	+	0.678802i	$0.762499\pi$
$72$	301.584	0.493639
$73$	−798.326	−1.27996	−0.639980	−	0.768392i	$-0.721057\pi$
$73$	−798.326	−1.27996	−0.639980	+	0.768392i	$0.721057\pi$
$74$	−322.711	−0.506951
$75$	−173.194	−0.266650
$76$	−214.173	−0.323254
$77$	379.892	0.562243
$78$	1910.77	2.77375
$79$	353.389	0.503283	0.251642	−	0.967820i	$-0.419030\pi$
$79$	353.389	0.503283	0.251642	+	0.967820i	$0.419030\pi$
$80$	135.571	0.189466
$81$	−855.092	−1.17297
$82$	−1700.99	−2.29077
$83$	1276.18	1.68770	0.843848	−	0.536582i	$-0.180285\pi$
$83$	1276.18	1.68770	0.843848	+	0.536582i	$0.180285\pi$
$84$	2696.94	3.50310
$85$	21.3632	0.0272608
$86$	−2301.41	−2.88567
$87$	−1662.92	−2.04924
$88$	−158.018	−0.191418
$89$	−648.944	−0.772899	−0.386449	−	0.922311i	$-0.626299\pi$
$89$	−648.944	−0.772899	−0.386449	+	0.922311i	$0.626299\pi$
$90$	−460.818	−0.539716
$91$	2169.79	2.49951
$92$	288.007	0.326378
$93$	−2161.69	−2.41029
$94$	1152.06	1.26411
$95$	95.0000	0.102598
$96$	1620.78	1.72313
$97$	1034.99	1.08337	0.541686	−	0.840581i	$-0.317786\pi$
$97$	1034.99	1.08337	0.541686	+	0.840581i	$0.317786\pi$
$98$	3730.24	3.84501
$99$	−230.933	−0.234441
$100$	281.807	0.281807
$101$	1926.54	1.89800	0.949002	−	0.315271i	$-0.102095\pi$
$101$	1926.54	1.89800	0.949002	+	0.315271i	$0.102095\pi$
$102$	129.944	0.126141
$103$	1434.88	1.37265	0.686327	−	0.727294i	$-0.259222\pi$
$103$	1434.88	1.37265	0.686327	+	0.727294i	$0.259222\pi$
$104$	−902.537	−0.850971
$105$	−1196.27	−1.11185
$106$	−409.058	−0.374823
$107$	−673.428	−0.608437	−0.304218	−	0.952602i	$-0.598395\pi$
$107$	−673.428	−0.608437	−0.304218	+	0.952602i	$0.598395\pi$
$108$	469.026	0.417890
$109$	1453.86	1.27757	0.638783	−	0.769387i	$-0.279438\pi$
$109$	1453.86	1.27757	0.638783	+	0.769387i	$0.279438\pi$
$110$	241.451	0.209286
$111$	509.261	0.435468
$112$	936.405	0.790017
$113$	−351.170	−0.292348	−0.146174	−	0.989259i	$-0.546696\pi$
$113$	−351.170	−0.292348	−0.146174	+	0.989259i	$0.546696\pi$
$114$	577.847	0.474740
$115$	−127.750	−0.103589
$116$	2705.77	2.16573
$117$	−1318.99	−1.04223
$118$	822.178	0.641421
$119$	147.558	0.113669
$120$	497.597	0.378535
$121$	121.000	0.0909091
$122$	−523.505	−0.388491
$123$	2684.28	1.96775
$124$	3517.32	2.54730
$125$	−125.000	−0.0894427
$126$	−3182.93	−2.25046
$127$	−2163.82	−1.51187	−0.755937	−	0.654644i	$-0.772818\pi$
$127$	−2163.82	−1.51187	−0.755937	+	0.654644i	$0.772818\pi$
$128$	−1684.94	−1.16351
$129$	3631.80	2.47877
$130$	1379.07	0.930404
$131$	1294.21	0.863172	0.431586	−	0.902072i	$-0.357954\pi$
$131$	1294.21	0.863172	0.431586	+	0.902072i	$0.357954\pi$
$132$	859.007	0.566417
$133$	656.177	0.427803
$134$	−569.101	−0.366887
$135$	−208.044	−0.132634
$136$	−61.3778	−0.0386993
$137$	139.608	0.0870619	0.0435310	−	0.999052i	$-0.486139\pi$
$137$	139.608	0.0870619	0.0435310	+	0.999052i	$0.486139\pi$
$138$	−777.054	−0.479328
$139$	−830.033	−0.506493	−0.253246	−	0.967402i	$-0.581498\pi$
$139$	−830.033	−0.506493	−0.253246	+	0.967402i	$0.581498\pi$
$140$	1946.47	1.17505
$141$	−1818.03	−1.08586
$142$	−3857.18	−2.27949
$143$	691.103	0.404146
$144$	−569.232	−0.329417
$145$	−1200.19	−0.687381
$146$	−3504.67	−1.98663
$147$	−5886.59	−3.30284
$148$	−828.626	−0.460221
$149$	−1766.33	−0.971163	−0.485582	−	0.874191i	$-0.661392\pi$
$149$	−1766.33	−0.971163	−0.485582	+	0.874191i	$0.661392\pi$
$150$	−760.325	−0.413869
$151$	−1025.36	−0.552598	−0.276299	−	0.961072i	$-0.589108\pi$
$151$	−1025.36	−0.552598	−0.276299	+	0.961072i	$0.589108\pi$
$152$	−272.941	−0.145647
$153$	−89.6994	−0.0473971
$154$	1667.73	0.872661
$155$	−1560.17	−0.808488
$156$	4906.30	2.51807
$157$	−1248.37	−0.634589	−0.317295	−	0.948327i	$-0.602774\pi$
$157$	−1248.37	−0.634589	−0.317295	+	0.948327i	$0.602774\pi$
$158$	1551.38	0.781149
$159$	645.522	0.321970
$160$	1169.77	0.577991
$161$	−882.387	−0.431937
$162$	−3753.87	−1.82057
$163$	2296.71	1.10363	0.551816	−	0.833966i	$-0.313935\pi$
$163$	2296.71	1.10363	0.551816	+	0.833966i	$0.313935\pi$
$164$	−4367.63	−2.07960
$165$	−381.027	−0.179775
$166$	5602.45	2.61948
$167$	759.115	0.351749	0.175875	−	0.984413i	$-0.443725\pi$
$167$	759.115	0.351749	0.175875	+	0.984413i	$0.443725\pi$
$168$	3436.97	1.57838
$169$	1750.30	0.796677
$170$	93.7849	0.0423116
$171$	−398.884	−0.178383
$172$	−5909.35	−2.61967
$173$	−3052.41	−1.34145	−0.670724	−	0.741707i	$-0.734017\pi$
$173$	−3052.41	−1.34145	−0.670724	+	0.741707i	$0.734017\pi$
$174$	−7300.27	−3.18064
$175$	−863.391	−0.372950
$176$	298.256	0.127738
$177$	−1297.46	−0.550976
$178$	−2848.88	−1.19962
$179$	−346.245	−0.144579	−0.0722893	−	0.997384i	$-0.523031\pi$
$179$	−346.245	−0.144579	−0.0722893	+	0.997384i	$0.523031\pi$
$180$	−1183.24	−0.489965
$181$	−3136.03	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$181$	−3136.03	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$182$	9525.41	3.87951
$183$	826.128	0.333711
$184$	367.034	0.147055
$185$	367.551	0.146070
$186$	−9489.87	−3.74103
$187$	46.9990	0.0183792
$188$	2958.15	1.14758
$189$	−1436.99	−0.553045
$190$	417.052	0.159243
$191$	3290.01	1.24637	0.623185	−	0.782074i	$-0.285838\pi$
$191$	3290.01	1.24637	0.623185	+	0.782074i	$0.285838\pi$
$192$	5612.52	2.10963
$193$	1960.54	0.731207	0.365603	−	0.930771i	$-0.380863\pi$
$193$	1960.54	0.731207	0.365603	+	0.930771i	$0.380863\pi$
$194$	4543.62	1.68151
$195$	−2176.27	−0.799210
$196$	9578.15	3.49058
$197$	−3335.93	−1.20647	−0.603236	−	0.797563i	$-0.706122\pi$
$197$	−3335.93	−1.20647	−0.603236	+	0.797563i	$0.706122\pi$
$198$	−1013.80	−0.363877
$199$	2836.27	1.01034	0.505170	−	0.863020i	$-0.331430\pi$
$199$	2836.27	1.01034	0.505170	+	0.863020i	$0.331430\pi$
$200$	359.133	0.126973
$201$	898.082	0.315153
$202$	8457.57	2.94590
$203$	−8289.85	−2.86617
$204$	333.657	0.114513
$205$	1937.34	0.660046
$206$	6299.17	2.13050
$207$	536.395	0.180106
$208$	1703.52	0.567873
$209$	209.000	0.0691714
$210$	−5251.66	−1.72571
$211$	2596.96	0.847307	0.423654	−	0.905824i	$-0.360747\pi$
$211$	2596.96	0.847307	0.423654	+	0.905824i	$0.360747\pi$
$212$	−1050.34	−0.340272
$213$	6086.90	1.95806
$214$	−2956.36	−0.944358
$215$	2621.19	0.831459
$216$	597.724	0.188287
$217$	−10776.3	−3.37116
$218$	6382.48	1.98292
$219$	5530.62	1.70650
$220$	619.975	0.189994
$221$	268.439	0.0817068
$222$	2235.67	0.675892
$223$	3057.31	0.918085	0.459042	−	0.888414i	$-0.348193\pi$
$223$	3057.31	0.918085	0.459042	+	0.888414i	$0.348193\pi$
$224$	8079.75	2.41005
$225$	524.847	0.155510
$226$	−1541.64	−0.453754
$227$	−3280.94	−0.959311	−0.479656	−	0.877457i	$-0.659238\pi$
$227$	−3280.94	−0.959311	−0.479656	+	0.877457i	$0.659238\pi$
$228$	1483.74	0.430978
$229$	3867.90	1.11615	0.558074	−	0.829791i	$-0.311540\pi$
$229$	3867.90	1.11615	0.558074	+	0.829791i	$0.311540\pi$
$230$	−560.826	−0.160782
$231$	−2631.80	−0.749609
$232$	3448.21	0.975803
$233$	−2534.48	−0.712615	−0.356307	−	0.934369i	$-0.615964\pi$
$233$	−2534.48	−0.712615	−0.356307	+	0.934369i	$0.615964\pi$
$234$	−5790.41	−1.61765
$235$	−1312.14	−0.364231
$236$	2111.11	0.582295
$237$	−2448.20	−0.671002
$238$	647.784	0.176427
$239$	−5991.70	−1.62164	−0.810818	−	0.585298i	$-0.800977\pi$
$239$	−5991.70	−1.62164	−0.810818	+	0.585298i	$0.800977\pi$
$240$	−939.202	−0.252605
$241$	−3556.46	−0.950589	−0.475295	−	0.879827i	$-0.657659\pi$
$241$	−3556.46	−0.950589	−0.475295	+	0.879827i	$0.657659\pi$
$242$	531.192	0.141101
$243$	4800.43	1.26728
$244$	−1344.21	−0.352680
$245$	−4248.55	−1.10788
$246$	11784.0	3.05416
$247$	1193.72	0.307509
$248$	4482.45	1.14773
$249$	−8841.06	−2.25012
$250$	−548.752	−0.138825
$251$	−2467.88	−0.620602	−0.310301	−	0.950638i	$-0.600430\pi$
$251$	−2467.88	−0.620602	−0.310301	+	0.950638i	$0.600430\pi$
$252$	−8172.81	−2.04301
$253$	−281.051	−0.0698399
$254$	−9499.21	−2.34659
$255$	−147.999	−0.0363454
$256$	−915.718	−0.223564
$257$	1141.06	0.276954	0.138477	−	0.990366i	$-0.455779\pi$
$257$	1141.06	0.276954	0.138477	+	0.990366i	$0.455779\pi$
$258$	15943.7	3.84732
$259$	2538.72	0.609067
$260$	3541.04	0.844639
$261$	5039.32	1.19512
$262$	5681.60	1.33973
$263$	1492.95	0.350034	0.175017	−	0.984565i	$-0.444002\pi$
$263$	1492.95	0.350034	0.175017	+	0.984565i	$0.444002\pi$
$264$	1094.71	0.255208
$265$	465.895	0.107999
$266$	2880.63	0.663995
$267$	4495.73	1.03047
$268$	−1461.28	−0.333067
$269$	2576.58	0.584003	0.292002	−	0.956418i	$-0.405679\pi$
$269$	2576.58	0.584003	0.292002	+	0.956418i	$0.405679\pi$
$270$	−913.319	−0.205862
$271$	6056.95	1.35769	0.678844	−	0.734283i	$-0.262481\pi$
$271$	6056.95	1.35769	0.678844	+	0.734283i	$0.262481\pi$
$272$	115.849	0.0258249
$273$	−15031.8	−3.33247
$274$	612.880	0.135129
$275$	−275.000	−0.0603023
$276$	−1995.24	−0.435143
$277$	8912.71	1.93326	0.966629	−	0.256179i	$-0.0824636\pi$
$277$	8912.71	1.93326	0.966629	+	0.256179i	$0.0824636\pi$
$278$	−3643.86	−0.786130
$279$	6550.80	1.40568
$280$	2480.57	0.529438
$281$	−1675.79	−0.355761	−0.177881	−	0.984052i	$-0.556924\pi$
$281$	−1675.79	−0.355761	−0.177881	+	0.984052i	$0.556924\pi$
$282$	−7981.21	−1.68537
$283$	2200.68	0.462251	0.231126	−	0.972924i	$-0.425759\pi$
$283$	2200.68	0.462251	0.231126	+	0.972924i	$0.425759\pi$
$284$	−9904.09	−2.06937
$285$	−658.137	−0.136788
$286$	3033.95	0.627278
$287$	13381.4	2.75220
$288$	−4911.61	−1.00493
$289$	−4894.74	−0.996284
$290$	−5268.85	−1.06689
$291$	−7170.15	−1.44440
$292$	−8998.95	−1.80351
$293$	3199.39	0.637919	0.318959	−	0.947768i	$-0.396667\pi$
$293$	3199.39	0.637919	0.318959	+	0.947768i	$0.396667\pi$
$294$	−25842.2	−5.12636
$295$	−936.418	−0.184815
$296$	−1056.00	−0.207360
$297$	−457.698	−0.0894219
$298$	−7754.22	−1.50735
$299$	−1605.25	−0.310481
$300$	−1952.29	−0.375718
$301$	18104.9	3.46694
$302$	−4501.33	−0.857691
$303$	−13346.6	−2.53051
$304$	515.169	0.0971940
$305$	596.244	0.111937
$306$	−393.782	−0.0735654
$307$	5051.56	0.939112	0.469556	−	0.882903i	$-0.344414\pi$
$307$	5051.56	0.939112	0.469556	+	0.882903i	$0.344414\pi$
$308$	4282.24	0.792219
$309$	−9940.53	−1.83009
$310$	−6849.16	−1.25486
$311$	5737.04	1.04604	0.523019	−	0.852321i	$-0.324806\pi$
$311$	5737.04	1.04604	0.523019	+	0.852321i	$0.324806\pi$
$312$	6252.56	1.13456
$313$	−4004.20	−0.723101	−0.361550	−	0.932353i	$-0.617752\pi$
$313$	−4004.20	−0.723101	−0.361550	+	0.932353i	$0.617752\pi$
$314$	−5480.35	−0.984950
$315$	3625.19	0.648432
$316$	3983.49	0.709143
$317$	−7318.62	−1.29670	−0.648351	−	0.761342i	$-0.724541\pi$
$317$	−7318.62	−1.29670	−0.648351	+	0.761342i	$0.724541\pi$
$318$	2833.86	0.499732
$319$	−2640.41	−0.463432
$320$	4050.75	0.707637
$321$	4665.35	0.811197
$322$	−3873.70	−0.670412
$323$	81.1802	0.0139845
$324$	−9638.82	−1.65275
$325$	−1570.69	−0.268080
$326$	10082.6	1.71296
$327$	−10072.0	−1.70331
$328$	−5566.08	−0.936999
$329$	−9063.10	−1.51874
$330$	−1672.72	−0.279030
$331$	1729.54	0.287202	0.143601	−	0.989636i	$-0.454132\pi$
$331$	1729.54	0.287202	0.143601	+	0.989636i	$0.454132\pi$
$332$	14385.4	2.37802
$333$	−1543.27	−0.253965
$334$	3332.53	0.545952
$335$	648.176	0.105712
$336$	−6487.19	−1.05329
$337$	5360.97	0.866560	0.433280	−	0.901259i	$-0.357356\pi$
$337$	5360.97	0.866560	0.433280	+	0.901259i	$0.357356\pi$
$338$	7683.85	1.23653
$339$	2432.82	0.389772
$340$	240.812	0.0384113
$341$	−3432.37	−0.545082
$342$	−1751.11	−0.276869
$343$	−17499.5	−2.75477
$344$	−7530.84	−1.18034
$345$	885.024	0.138110
$346$	−13400.1	−2.08207
$347$	−7833.01	−1.21181	−0.605905	−	0.795537i	$-0.707189\pi$
$347$	−7833.01	−1.21181	−0.605905	+	0.795537i	$0.707189\pi$
$348$	−18744.9	−2.88745
$349$	−3736.01	−0.573021	−0.286510	−	0.958077i	$-0.592495\pi$
$349$	−3736.01	−0.573021	−0.286510	+	0.958077i	$0.592495\pi$
$350$	−3790.30	−0.578858
$351$	−2614.18	−0.397535
$352$	2573.50	0.389681
$353$	6090.09	0.918251	0.459125	−	0.888371i	$-0.348163\pi$
$353$	6090.09	0.918251	0.459125	+	0.888371i	$0.348163\pi$
$354$	−5695.86	−0.855173
$355$	4393.12	0.656797
$356$	−7315.07	−1.08904
$357$	−1022.25	−0.151549
$358$	−1520.02	−0.224401
$359$	−1415.00	−0.208024	−0.104012	−	0.994576i	$-0.533168\pi$
$359$	−1415.00	−0.208024	−0.104012	+	0.994576i	$0.533168\pi$
$360$	−1507.92	−0.220762
$361$	361.000	0.0526316
$362$	−13767.2	−1.99887
$363$	−838.259	−0.121204
$364$	24458.4	3.52190
$365$	3991.63	0.572415
$366$	3626.72	0.517955
$367$	−6910.49	−0.982901	−0.491451	−	0.870905i	$-0.663533\pi$
$367$	−6910.49	−0.982901	−0.491451	+	0.870905i	$0.663533\pi$
$368$	−692.768	−0.0981333
$369$	−8134.45	−1.14759
$370$	1613.56	0.226716
$371$	3218.00	0.450324
$372$	−24367.2	−3.39618
$373$	3039.70	0.421956	0.210978	−	0.977491i	$-0.432335\pi$
$373$	3039.70	0.421956	0.210978	+	0.977491i	$0.432335\pi$
$374$	206.327	0.0285265
$375$	865.970	0.119249
$376$	3769.85	0.517062
$377$	−15081.0	−2.06024
$378$	−6308.41	−0.858385
$379$	2543.33	0.344701	0.172351	−	0.985036i	$-0.444864\pi$
$379$	2543.33	0.344701	0.172351	+	0.985036i	$0.444864\pi$
$380$	1070.87	0.144564
$381$	14990.4	2.01570
$382$	14443.2	1.93450
$383$	1274.29	0.170008	0.0850042	−	0.996381i	$-0.472910\pi$
$383$	1274.29	0.170008	0.0850042	+	0.996381i	$0.472910\pi$
$384$	11672.9	1.55125
$385$	−1899.46	−0.251443
$386$	8606.82	1.13491
$387$	−11005.8	−1.44562
$388$	11666.7	1.52651
$389$	7522.10	0.980426	0.490213	−	0.871603i	$-0.336919\pi$
$389$	7522.10	0.980426	0.490213	+	0.871603i	$0.336919\pi$
$390$	−9553.87	−1.24046
$391$	−109.166	−0.0141196
$392$	12206.3	1.57274
$393$	−8965.97	−1.15082
$394$	−14644.8	−1.87257
$395$	−1766.95	−0.225075
$396$	−2603.14	−0.330335
$397$	−7815.87	−0.988079	−0.494040	−	0.869439i	$-0.664480\pi$
$397$	−7815.87	−0.988079	−0.494040	+	0.869439i	$0.664480\pi$
$398$	12451.3	1.56816
$399$	−4545.84	−0.570367
$400$	−677.854	−0.0847318
$401$	7710.02	0.960150	0.480075	−	0.877227i	$-0.340609\pi$
$401$	7710.02	0.960150	0.480075	+	0.877227i	$0.340609\pi$
$402$	3942.60	0.489152
$403$	−19604.3	−2.42322
$404$	21716.5	2.67435
$405$	4275.46	0.524566
$406$	−36392.6	−4.44861
$407$	808.612	0.0984801
$408$	425.211	0.0515957
$409$	1944.27	0.235056	0.117528	−	0.993070i	$-0.462503\pi$
$409$	1944.27	0.235056	0.117528	+	0.993070i	$0.462503\pi$
$410$	8504.94	1.02446
$411$	−967.168	−0.116075
$412$	16174.4	1.93411
$413$	−6467.96	−0.770623
$414$	2354.78	0.279544
$415$	−6380.89	−0.754760
$416$	14698.8	1.73237
$417$	5750.27	0.675281
$418$	917.514	0.107361
$419$	14814.7	1.72731	0.863655	−	0.504083i	$-0.168169\pi$
$419$	14814.7	1.72731	0.863655	+	0.504083i	$0.168169\pi$
$420$	−13484.7	−1.56663
$421$	11615.7	1.34469	0.672346	−	0.740237i	$-0.265287\pi$
$421$	11615.7	1.34469	0.672346	+	0.740237i	$0.265287\pi$
$422$	11400.7	1.31511
$423$	5509.37	0.633274
$424$	−1338.55	−0.153315
$425$	−106.816	−0.0121914
$426$	26721.6	3.03912
$427$	4118.34	0.466745
$428$	−7591.06	−0.857308
$429$	−4787.80	−0.538828
$430$	11507.1	1.29051
$431$	−2955.69	−0.330326	−0.165163	−	0.986266i	$-0.552815\pi$
$431$	−2955.69	−0.330326	−0.165163	+	0.986266i	$0.552815\pi$
$432$	−1128.19	−0.125648
$433$	−11523.8	−1.27898	−0.639489	−	0.768801i	$-0.720854\pi$
$433$	−11523.8	−1.27898	−0.639489	+	0.768801i	$0.720854\pi$
$434$	−47308.0	−5.23239
$435$	8314.62	0.916449
$436$	16388.3	1.80013
$437$	−485.451	−0.0531402
$438$	24279.5	2.64868
$439$	−11145.9	−1.21176	−0.605882	−	0.795555i	$-0.707180\pi$
$439$	−11145.9	−1.21176	−0.605882	+	0.795555i	$0.707180\pi$
$440$	790.092	0.0856049
$441$	17838.7	1.92622
$442$	1178.45	0.126818
$443$	16278.1	1.74582	0.872909	−	0.487883i	$-0.162231\pi$
$443$	16278.1	1.74582	0.872909	+	0.487883i	$0.162231\pi$
$444$	5740.53	0.613589
$445$	3244.72	0.345651
$446$	13421.7	1.42497
$447$	12236.7	1.29480
$448$	27979.0	2.95064
$449$	−13998.8	−1.47137	−0.735683	−	0.677326i	$-0.763139\pi$
$449$	−13998.8	−1.47137	−0.735683	+	0.677326i	$0.763139\pi$
$450$	2304.09	0.241369
$451$	4262.14	0.445003
$452$	−3958.48	−0.411927
$453$	7103.43	0.736751
$454$	−14403.4	−1.48895
$455$	−10848.9	−1.11782
$456$	1890.87	0.194184
$457$	−10168.0	−1.04079	−0.520395	−	0.853926i	$-0.674215\pi$
$457$	−10168.0	−1.04079	−0.520395	+	0.853926i	$0.674215\pi$
$458$	16980.2	1.73238
$459$	−177.780	−0.0180785
$460$	−1440.03	−0.145961
$461$	−14843.4	−1.49963	−0.749814	−	0.661649i	$-0.769857\pi$
$461$	−14843.4	−1.49963	−0.749814	+	0.661649i	$0.769857\pi$
$462$	−11553.7	−1.16347
$463$	11737.8	1.17819	0.589097	−	0.808062i	$-0.299484\pi$
$463$	11737.8	1.17819	0.589097	+	0.808062i	$0.299484\pi$
$464$	−6508.42	−0.651176
$465$	10808.5	1.07792
$466$	−11126.4	−1.10605
$467$	3826.69	0.379182	0.189591	−	0.981863i	$-0.439284\pi$
$467$	3826.69	0.379182	0.189591	+	0.981863i	$0.439284\pi$
$468$	−14868.1	−1.46854
$469$	4477.04	0.440790
$470$	−5760.31	−0.565326
$471$	8648.39	0.846065
$472$	2690.39	0.262362
$473$	5766.62	0.560569
$474$	−10747.6	−1.04147
$475$	−475.000	−0.0458831
$476$	1663.32	0.160164
$477$	−1956.19	−0.187773
$478$	−26303.7	−2.51695
$479$	5260.57	0.501799	0.250900	−	0.968013i	$-0.419274\pi$
$479$	5260.57	0.501799	0.250900	+	0.968013i	$0.419274\pi$
$480$	−8103.89	−0.770605
$481$	4618.46	0.437804
$482$	−15612.9	−1.47542
$483$	6112.97	0.575879
$484$	1363.94	0.128094
$485$	−5174.94	−0.484499
$486$	21074.0	1.96695
$487$	19168.4	1.78358	0.891790	−	0.452449i	$-0.149449\pi$
$487$	19168.4	1.78358	0.891790	+	0.452449i	$0.149449\pi$
$488$	−1713.05	−0.158906
$489$	−15911.1	−1.47142
$490$	−18651.2	−1.71954
$491$	7603.76	0.698885	0.349443	−	0.936958i	$-0.386371\pi$
$491$	7603.76	0.698885	0.349443	+	0.936958i	$0.386371\pi$
$492$	30257.9	2.77263
$493$	−1025.59	−0.0936926
$494$	5240.47	0.477287
$495$	1154.66	0.104845
$496$	−8460.52	−0.765905
$497$	30343.9	2.73865
$498$	−38812.4	−3.49242
$499$	4565.63	0.409591	0.204795	−	0.978805i	$-0.434347\pi$
$499$	4565.63	0.409591	0.204795	+	0.978805i	$0.434347\pi$
$500$	−1409.03	−0.126028
$501$	−5258.97	−0.468969
$502$	−10834.0	−0.963240
$503$	−4813.51	−0.426687	−0.213344	−	0.976977i	$-0.568435\pi$
$503$	−4813.51	−0.426687	−0.213344	+	0.976977i	$0.568435\pi$
$504$	−10415.4	−0.920512
$505$	−9632.72	−0.848813
$506$	−1233.82	−0.108399
$507$	−12125.7	−1.06217
$508$	−24391.2	−2.13028
$509$	10888.2	0.948153	0.474076	−	0.880484i	$-0.342782\pi$
$509$	10888.2	0.948153	0.474076	+	0.880484i	$0.342782\pi$
$510$	−649.719	−0.0564119
$511$	27570.7	2.38680
$512$	9459.50	0.816513
$513$	−790.569	−0.0680399
$514$	5009.26	0.429861
$515$	−7174.42	−0.613869
$516$	40938.6	3.49267
$517$	−2886.70	−0.245565
$518$	11145.0	0.945337
$519$	21146.4	1.78848
$520$	4512.68	0.380566
$521$	−9537.74	−0.802027	−0.401014	−	0.916072i	$-0.631342\pi$
$521$	−9537.74	−0.802027	−0.401014	+	0.916072i	$0.631342\pi$
$522$	22122.7	1.85495
$523$	10962.5	0.916550	0.458275	−	0.888810i	$-0.348467\pi$
$523$	10962.5	0.916550	0.458275	+	0.888810i	$0.348467\pi$
$524$	14588.7	1.21624
$525$	5981.37	0.497235
$526$	6554.07	0.543291
$527$	−1333.21	−0.110200
$528$	−2066.25	−0.170306
$529$	−11514.2	−0.946346
$530$	2045.29	0.167626
$531$	3931.81	0.321330
$532$	7396.60	0.602788
$533$	24343.6	1.97831
$534$	19736.4	1.59939
$535$	3367.14	0.272101
$536$	−1862.25	−0.150069
$537$	2398.70	0.192759
$538$	11311.2	0.906435
$539$	−9346.81	−0.746930
$540$	−2345.13	−0.186886
$541$	134.964	0.0107256	0.00536282	−	0.999986i	$-0.498293\pi$
$541$	134.964	0.0107256	0.00536282	+	0.999986i	$0.498293\pi$
$542$	26590.1	2.10728
$543$	21725.7	1.71701
$544$	999.602	0.0787823
$545$	−7269.31	−0.571345
$546$	−65989.8	−5.17235
$547$	−16373.2	−1.27983	−0.639914	−	0.768446i	$-0.721030\pi$
$547$	−16373.2	−1.27983	−0.639914	+	0.768446i	$0.721030\pi$
$548$	1573.69	0.122673
$549$	−2503.50	−0.194621
$550$	−1207.26	−0.0935955
$551$	−4560.71	−0.352619
$552$	−2542.73	−0.196061
$553$	−12204.5	−0.938497
$554$	39127.0	3.00062
$555$	−2546.31	−0.194747
$556$	−9356.35	−0.713665
$557$	259.529	0.0197426	0.00987129	−	0.999951i	$-0.496858\pi$
$557$	259.529	0.0197426	0.00987129	+	0.999951i	$0.496858\pi$
$558$	28758.1	2.18177
$559$	32936.6	2.49207
$560$	−4682.02	−0.353306
$561$	−325.598	−0.0245040
$562$	−7356.73	−0.552179
$563$	14510.2	1.08620	0.543102	−	0.839667i	$-0.317250\pi$
$563$	14510.2	1.08620	0.543102	+	0.839667i	$0.317250\pi$
$564$	−20493.4	−1.53001
$565$	1755.85	0.130742
$566$	9661.05	0.717463
$567$	29531.1	2.18729
$568$	−12621.7	−0.932386
$569$	−2549.92	−0.187870	−0.0939350	−	0.995578i	$-0.529945\pi$
$569$	−2549.92	−0.187870	−0.0939350	+	0.995578i	$0.529945\pi$
$570$	−2889.24	−0.212310
$571$	−4437.59	−0.325232	−0.162616	−	0.986689i	$-0.551993\pi$
$571$	−4437.59	−0.325232	−0.162616	+	0.986689i	$0.551993\pi$
$572$	7790.29	0.569456
$573$	−22792.4	−1.66172
$574$	58744.7	4.27170
$575$	638.751	0.0463266
$576$	−17008.2	−1.23034
$577$	19032.7	1.37321	0.686603	−	0.727032i	$-0.259101\pi$
$577$	19032.7	1.37321	0.686603	+	0.727032i	$0.259101\pi$
$578$	−21488.0	−1.54634
$579$	−13582.2	−0.974880
$580$	−13528.8	−0.968542
$581$	−44073.6	−3.14713
$582$	−31477.1	−2.24187
$583$	1024.97	0.0728129
$584$	−11468.2	−0.812599
$585$	6594.97	0.466100
$586$	14045.4	0.990118
$587$	−7598.31	−0.534269	−0.267134	−	0.963659i	$-0.586077\pi$
$587$	−7598.31	−0.534269	−0.267134	+	0.963659i	$0.586077\pi$
$588$	−66355.2	−4.65381
$589$	−5928.63	−0.414746
$590$	−4110.89	−0.286852
$591$	23110.5	1.60853
$592$	1993.17	0.138376
$593$	−380.688	−0.0263625	−0.0131813	−	0.999913i	$-0.504196\pi$
$593$	−380.688	−0.0263625	−0.0131813	+	0.999913i	$0.504196\pi$
$594$	−2009.30	−0.138792
$595$	−737.792	−0.0508345
$596$	−19910.5	−1.36840
$597$	−19649.0	−1.34703
$598$	−7047.06	−0.481899
$599$	−979.061	−0.0667836	−0.0333918	−	0.999442i	$-0.510631\pi$
$599$	−979.061	−0.0667836	−0.0333918	+	0.999442i	$0.510631\pi$
$600$	−2487.99	−0.169286
$601$	1420.25	0.0963946	0.0481973	−	0.998838i	$-0.484652\pi$
$601$	1420.25	0.0963946	0.0481973	+	0.998838i	$0.484652\pi$
$602$	79480.8	5.38106
$603$	−2721.55	−0.183798
$604$	−11558.1	−0.778629
$605$	−605.000	−0.0406558
$606$	−58592.0	−3.92762
$607$	−14208.8	−0.950110	−0.475055	−	0.879956i	$-0.657572\pi$
$607$	−14208.8	−0.950110	−0.475055	+	0.879956i	$0.657572\pi$
$608$	4445.13	0.296503
$609$	57430.1	3.82132
$610$	2617.52	0.173738
$611$	−16487.7	−1.09169
$612$	−1011.11	−0.0667841
$613$	27444.7	1.80829	0.904145	−	0.427226i	$-0.140509\pi$
$613$	27444.7	1.80829	0.904145	+	0.427226i	$0.140509\pi$
$614$	22176.4	1.45760
$615$	−13421.4	−0.880005
$616$	5457.26	0.356947
$617$	11037.7	0.720193	0.360097	−	0.932915i	$-0.382744\pi$
$617$	11037.7	0.720193	0.360097	+	0.932915i	$0.382744\pi$
$618$	−43639.1	−2.84049
$619$	−892.920	−0.0579798	−0.0289899	−	0.999580i	$-0.509229\pi$
$619$	−892.920	−0.0579798	−0.0289899	+	0.999580i	$0.509229\pi$
$620$	−17586.6	−1.13919
$621$	1063.11	0.0686974
$622$	25185.7	1.62356
$623$	22411.7	1.44126
$624$	−11801.6	−0.757116
$625$	625.000	0.0400000
$626$	−17578.5	−1.12233
$627$	−1447.90	−0.0922227
$628$	−14071.9	−0.894157
$629$	314.083	0.0199098
$630$	15914.6	1.00644
$631$	25301.6	1.59626	0.798130	−	0.602486i	$-0.205823\pi$
$631$	25301.6	1.59626	0.798130	+	0.602486i	$0.205823\pi$
$632$	5076.54	0.319516
$633$	−17991.1	−1.12967
$634$	−32128.9	−2.01262
$635$	10819.1	0.676131
$636$	7276.50	0.453667
$637$	−53385.2	−3.32056
$638$	−11591.5	−0.719295
$639$	−18445.8	−1.14195
$640$	8424.70	0.520337
$641$	−26736.7	−1.64748	−0.823741	−	0.566966i	$-0.808117\pi$
$641$	−26736.7	−1.64748	−0.823741	+	0.566966i	$0.808117\pi$
$642$	20481.0	1.25906
$643$	−19440.8	−1.19233	−0.596165	−	0.802862i	$-0.703310\pi$
$643$	−19440.8	−1.19233	−0.596165	+	0.802862i	$0.703310\pi$
$644$	−9946.50	−0.608613
$645$	−18159.0	−1.10854
$646$	356.382	0.0217054
$647$	19374.4	1.17726	0.588629	−	0.808403i	$-0.299668\pi$
$647$	19374.4	1.17726	0.588629	+	0.808403i	$0.299668\pi$
$648$	−12283.6	−0.744672
$649$	−2060.12	−0.124602
$650$	−6895.35	−0.416089
$651$	74655.4	4.49459
$652$	25889.1	1.55506
$653$	−15446.4	−0.925672	−0.462836	−	0.886444i	$-0.653168\pi$
$653$	−15446.4	−0.925672	−0.462836	+	0.886444i	$0.653168\pi$
$654$	−44216.3	−2.64372
$655$	−6471.04	−0.386022
$656$	10505.8	0.625281
$657$	−16760.0	−0.995235
$658$	−39787.2	−2.35724
$659$	15215.0	0.899380	0.449690	−	0.893185i	$-0.351534\pi$
$659$	15215.0	0.899380	0.449690	+	0.893185i	$0.351534\pi$
$660$	−4295.04	−0.253309
$661$	−17728.4	−1.04320	−0.521600	−	0.853190i	$-0.674665\pi$
$661$	−17728.4	−1.04320	−0.521600	+	0.853190i	$0.674665\pi$
$662$	7592.70	0.445768
$663$	−1859.68	−0.108935
$664$	18332.7	1.07146
$665$	−3280.88	−0.191319
$666$	−6774.96	−0.394181
$667$	6132.97	0.356027
$668$	8556.95	0.495626
$669$	−21180.4	−1.22404
$670$	2845.51	0.164077
$671$	1311.74	0.0754681
$672$	−55974.6	−3.21320
$673$	21045.4	1.20541	0.602705	−	0.797964i	$-0.294089\pi$
$673$	21045.4	1.20541	0.602705	+	0.797964i	$0.294089\pi$
$674$	23534.8	1.34499
$675$	1040.22	0.0593158
$676$	19729.8	1.12254
$677$	26691.2	1.51525	0.757627	−	0.652688i	$-0.226359\pi$
$677$	26691.2	1.51525	0.757627	+	0.652688i	$0.226359\pi$
$678$	10680.1	0.604967
$679$	−35744.0	−2.02022
$680$	306.889	0.0173068
$681$	22729.6	1.27900
$682$	−15068.2	−0.846026
$683$	19865.4	1.11293	0.556464	−	0.830872i	$-0.312158\pi$
$683$	19865.4	1.11293	0.556464	+	0.830872i	$0.312158\pi$
$684$	−4496.33	−0.251347
$685$	−698.038	−0.0389353
$686$	−76823.3	−4.27570
$687$	−26795.9	−1.48810
$688$	14214.3	0.787666
$689$	5854.21	0.323698
$690$	3885.27	0.214362
$691$	35302.7	1.94353	0.971765	−	0.235951i	$-0.0758205\pi$
$691$	35302.7	1.94353	0.971765	+	0.235951i	$0.0758205\pi$
$692$	−34407.6	−1.89015
$693$	7975.41	0.437173
$694$	−34387.1	−1.88086
$695$	4150.16	0.226510
$696$	−23888.4	−1.30099
$697$	1655.51	0.0899668
$698$	−16401.2	−0.889389
$699$	17558.3	0.950092
$700$	−9732.37	−0.525499
$701$	−29822.8	−1.60684	−0.803418	−	0.595415i	$-0.796988\pi$
$701$	−29822.8	−1.60684	−0.803418	+	0.595415i	$0.796988\pi$
$702$	−11476.3	−0.617016
$703$	1396.69	0.0749321
$704$	8911.65	0.477089
$705$	9090.17	0.485611
$706$	26735.6	1.42522
$707$	−66534.4	−3.53930
$708$	−14625.3	−0.776344
$709$	−32540.7	−1.72369	−0.861843	−	0.507175i	$-0.830690\pi$
$709$	−32540.7	−1.72369	−0.861843	+	0.507175i	$0.830690\pi$
$710$	19285.9	1.01942
$711$	7419.01	0.391329
$712$	−9322.28	−0.490684
$713$	7972.47	0.418754
$714$	−4487.69	−0.235221
$715$	−3455.51	−0.180740
$716$	−3902.97	−0.203716
$717$	41509.1	2.16204
$718$	−6211.86	−0.322875
$719$	−2548.11	−0.132167	−0.0660837	−	0.997814i	$-0.521050\pi$
$719$	−2548.11	−0.132167	−0.0660837	+	0.997814i	$0.521050\pi$
$720$	2846.16	0.147320
$721$	−49554.6	−2.55965
$722$	1584.80	0.0816898
$723$	24638.3	1.26737
$724$	−35350.2	−1.81461
$725$	6000.94	0.307406
$726$	−3679.97	−0.188122
$727$	18722.0	0.955104	0.477552	−	0.878603i	$-0.341524\pi$
$727$	18722.0	0.955104	0.477552	+	0.878603i	$0.341524\pi$
$728$	31169.7	1.58685
$729$	−10168.8	−0.516627
$730$	17523.3	0.888449
$731$	2239.88	0.113331
$732$	9312.33	0.470210
$733$	25163.0	1.26796	0.633980	−	0.773349i	$-0.281420\pi$
$733$	25163.0	1.26796	0.633980	+	0.773349i	$0.281420\pi$
$734$	−30337.2	−1.52557
$735$	29432.9	1.47707
$736$	−5977.54	−0.299368
$737$	1425.99	0.0712713
$738$	−35710.4	−1.78119
$739$	13988.4	0.696308	0.348154	−	0.937437i	$-0.386809\pi$
$739$	13988.4	0.696308	0.348154	+	0.937437i	$0.386809\pi$
$740$	4143.13	0.205817
$741$	−8269.83	−0.409986
$742$	14127.1	0.698950
$743$	26339.8	1.30056	0.650279	−	0.759695i	$-0.274652\pi$
$743$	26339.8	1.30056	0.650279	+	0.759695i	$0.274652\pi$
$744$	−31053.4	−1.53020
$745$	8831.65	0.434317
$746$	13344.3	0.654921
$747$	26791.9	1.31227
$748$	529.786	0.0258969
$749$	23257.3	1.13458
$750$	3801.63	0.185088
$751$	−14293.5	−0.694509	−0.347254	−	0.937771i	$-0.612886\pi$
$751$	−14293.5	−0.694509	−0.347254	+	0.937771i	$0.612886\pi$
$752$	−7115.50	−0.345047
$753$	17096.9	0.827417
$754$	−66205.7	−3.19771
$755$	5126.78	0.247129
$756$	−16198.1	−0.779259
$757$	2859.22	0.137279	0.0686393	−	0.997642i	$-0.478134\pi$
$757$	2859.22	0.137279	0.0686393	+	0.997642i	$0.478134\pi$
$758$	11165.2	0.535013
$759$	1947.05	0.0931140
$760$	1364.70	0.0651355
$761$	1435.10	0.0683605	0.0341802	−	0.999416i	$-0.489118\pi$
$761$	1435.10	0.0683605	0.0341802	+	0.999416i	$0.489118\pi$
$762$	65808.3	3.12859
$763$	−50210.0	−2.38234
$764$	37085.8	1.75618
$765$	448.497	0.0211966
$766$	5594.16	0.263871
$767$	−11766.6	−0.553932
$768$	6343.87	0.298066
$769$	40823.9	1.91436	0.957182	−	0.289485i	$-0.0934842\pi$
$769$	40823.9	1.91436	0.957182	+	0.289485i	$0.0934842\pi$
$770$	−8338.66	−0.390266
$771$	−7904.96	−0.369248
$772$	22099.7	1.03029
$773$	28782.4	1.33924	0.669619	−	0.742705i	$-0.266457\pi$
$773$	28782.4	1.33924	0.669619	+	0.742705i	$0.266457\pi$
$774$	−48315.6	−2.24376
$775$	7800.83	0.361567
$776$	14867.9	0.687793
$777$	−17587.7	−0.812038
$778$	33022.2	1.52173
$779$	7361.88	0.338596
$780$	−24531.5	−1.12611
$781$	9664.87	0.442812
$782$	−479.242	−0.0219151
$783$	9987.70	0.455851
$784$	−23039.2	−1.04952
$785$	6241.83	0.283797
$786$	−39360.8	−1.78620
$787$	−7405.35	−0.335416	−0.167708	−	0.985837i	$-0.553637\pi$
$787$	−7405.35	−0.335416	−0.167708	+	0.985837i	$0.553637\pi$
$788$	−37603.5	−1.69996
$789$	−10342.8	−0.466683
$790$	−7756.92	−0.349340
$791$	12127.9	0.545155
$792$	−3317.42	−0.148838
$793$	7492.11	0.335502
$794$	−34311.8	−1.53360
$795$	−3227.61	−0.143989
$796$	31971.2	1.42360
$797$	−16434.6	−0.730418	−0.365209	−	0.930926i	$-0.619002\pi$
$797$	−16434.6	−0.730418	−0.365209	+	0.930926i	$0.619002\pi$
$798$	−19956.3	−0.885271
$799$	−1121.26	−0.0496461
$800$	−5848.85	−0.258485
$801$	−13623.9	−0.600969
$802$	33847.1	1.49025
$803$	8781.59	0.385922
$804$	10123.4	0.444062
$805$	4411.94	0.193168
$806$	−86063.2	−3.76110
$807$	−17849.9	−0.778621
$808$	27675.4	1.20497
$809$	−9523.44	−0.413877	−0.206938	−	0.978354i	$-0.566350\pi$
$809$	−9523.44	−0.413877	−0.206938	+	0.978354i	$0.566350\pi$
$810$	18769.3	0.814182
$811$	25451.5	1.10200	0.551001	−	0.834504i	$-0.314246\pi$
$811$	25451.5	1.10200	0.551001	+	0.834504i	$0.314246\pi$
$812$	−93445.4	−4.03853
$813$	−41961.1	−1.81014
$814$	3549.82	0.152852
$815$	−11483.5	−0.493560
$816$	−802.575	−0.0344311
$817$	9960.52	0.426529
$818$	8535.39	0.364832
$819$	45552.3	1.94350
$820$	21838.2	0.930027
$821$	−153.828	−0.00653916	−0.00326958	−	0.999995i	$-0.501041\pi$
$821$	−153.828	−0.00653916	−0.00326958	+	0.999995i	$0.501041\pi$
$822$	−4245.89	−0.180161
$823$	22586.1	0.956624	0.478312	−	0.878190i	$-0.341249\pi$
$823$	22586.1	0.956624	0.478312	+	0.878190i	$0.341249\pi$
$824$	20612.5	0.871446
$825$	1905.13	0.0803979
$826$	−28394.5	−1.19609
$827$	−25573.5	−1.07530	−0.537652	−	0.843167i	$-0.680688\pi$
$827$	−25573.5	−1.07530	−0.537652	+	0.843167i	$0.680688\pi$
$828$	6046.39	0.253776
$829$	30483.1	1.27711	0.638553	−	0.769578i	$-0.279533\pi$
$829$	30483.1	1.27711	0.638553	+	0.769578i	$0.279533\pi$
$830$	−28012.2	−1.17147
$831$	−61745.1	−2.57751
$832$	50899.7	2.12095
$833$	−3630.50	−0.151008
$834$	25243.8	1.04811
$835$	−3795.58	−0.157307
$836$	2355.90	0.0974649
$837$	12983.4	0.536166
$838$	65036.6	2.68097
$839$	−6751.28	−0.277807	−0.138904	−	0.990306i	$-0.544358\pi$
$839$	−6751.28	−0.277807	−0.138904	+	0.990306i	$0.544358\pi$
$840$	−17184.8	−0.705872
$841$	33229.1	1.36246
$842$	50993.2	2.08710
$843$	11609.4	0.474318
$844$	29273.6	1.19388
$845$	−8751.50	−0.356285
$846$	24186.3	0.982908
$847$	−4178.81	−0.169523
$848$	2526.47	0.102311
$849$	−15245.8	−0.616296
$850$	−468.924	−0.0189223
$851$	−1878.19	−0.0756563
$852$	68613.2	2.75898
$853$	3768.01	0.151248	0.0756238	−	0.997136i	$-0.475905\pi$
$853$	3768.01	0.151248	0.0756238	+	0.997136i	$0.475905\pi$
$854$	18079.6	0.724438
$855$	1994.42	0.0797751
$856$	−9674.00	−0.386274
$857$	−17135.2	−0.682996	−0.341498	−	0.939882i	$-0.610934\pi$
$857$	−17135.2	−0.682996	−0.341498	+	0.939882i	$0.610934\pi$
$858$	−21018.5	−0.836317
$859$	−28946.2	−1.14975	−0.574873	−	0.818243i	$-0.694948\pi$
$859$	−28946.2	−1.14975	−0.574873	+	0.818243i	$0.694948\pi$
$860$	29546.7	1.17155
$861$	−92703.3	−3.66936
$862$	−12975.5	−0.512701
$863$	−21089.9	−0.831875	−0.415937	−	0.909393i	$-0.636546\pi$
$863$	−21089.9	−0.831875	−0.415937	+	0.909393i	$0.636546\pi$
$864$	−9734.57	−0.383306
$865$	15262.1	0.599914
$866$	−50589.6	−1.98511
$867$	33909.6	1.32829
$868$	−121473.	−4.75007
$869$	−3887.28	−0.151746
$870$	36501.3	1.42243
$871$	8144.66	0.316844
$872$	20885.2	0.811079
$873$	21728.4	0.842378
$874$	−2131.14	−0.0824792
$875$	4316.95	0.166788
$876$	62342.6	2.40452
$877$	−18225.6	−0.701750	−0.350875	−	0.936422i	$-0.614116\pi$
$877$	−18225.6	−0.701750	−0.350875	+	0.936422i	$0.614116\pi$
$878$	−48930.7	−1.88079
$879$	−22164.6	−0.850504
$880$	−1491.28	−0.0571261
$881$	15006.4	0.573870	0.286935	−	0.957950i	$-0.407364\pi$
$881$	15006.4	0.573870	0.286935	+	0.957950i	$0.407364\pi$
$882$	78312.3	2.98970
$883$	11704.8	0.446089	0.223045	−	0.974808i	$-0.428400\pi$
$883$	11704.8	0.446089	0.223045	+	0.974808i	$0.428400\pi$
$884$	3025.92	0.115128
$885$	6487.28	0.246404
$886$	71461.3	2.70969
$887$	−29331.4	−1.11032	−0.555159	−	0.831744i	$-0.687343\pi$
$887$	−29331.4	−1.11032	−0.555159	+	0.831744i	$0.687343\pi$
$888$	7315.69	0.276462
$889$	74728.9	2.81926
$890$	14244.4	0.536487
$891$	9406.01	0.353662
$892$	34462.9	1.29361
$893$	−4986.12	−0.186847
$894$	53719.4	2.00967
$895$	1731.23	0.0646576
$896$	58190.4	2.16965
$897$	11120.8	0.413948
$898$	−61454.9	−2.28372
$899$	74899.7	2.77869
$900$	5916.22	0.219119
$901$	398.121	0.0147207
$902$	18710.9	0.690692
$903$	−125426.	−4.62229
$904$	−5044.66	−0.185601
$905$	15680.1	0.575940
$906$	31184.2	1.14352
$907$	−43513.4	−1.59299	−0.796493	−	0.604647i	$-0.793314\pi$
$907$	−43513.4	−1.59299	−0.796493	+	0.604647i	$0.793314\pi$
$908$	−36983.6	−1.35170
$909$	40445.7	1.47580
$910$	−47627.1	−1.73497
$911$	−15674.2	−0.570042	−0.285021	−	0.958521i	$-0.592000\pi$
$911$	−15674.2	−0.570042	−0.285021	+	0.958521i	$0.592000\pi$
$912$	−3568.97	−0.129584
$913$	−14038.0	−0.508859
$914$	−44637.9	−1.61542
$915$	−4130.64	−0.149240
$916$	43600.0	1.57269
$917$	−44696.3	−1.60960
$918$	−780.457	−0.0280598
$919$	−38178.6	−1.37040	−0.685200	−	0.728355i	$-0.740285\pi$
$919$	−38178.6	−1.37040	−0.685200	+	0.728355i	$0.740285\pi$
$920$	−1835.17	−0.0657650
$921$	−34996.0	−1.25207
$922$	−65163.0	−2.32758
$923$	55201.8	1.96857
$924$	−29666.4	−1.05622
$925$	−1837.75	−0.0653243
$926$	51529.3	1.82868
$927$	30123.8	1.06731
$928$	−56157.8	−1.98650
$929$	52850.7	1.86650	0.933249	−	0.359231i	$-0.116961\pi$
$929$	52850.7	1.86650	0.933249	+	0.359231i	$0.116961\pi$
$930$	47449.4	1.67304
$931$	−16144.5	−0.568329
$932$	−28569.3	−1.00410
$933$	−39744.8	−1.39463
$934$	16799.2	0.588531
$935$	−234.995	−0.00821943
$936$	−18947.8	−0.661674
$937$	−20057.6	−0.699309	−0.349655	−	0.936879i	$-0.613701\pi$
$937$	−20057.6	−0.699309	−0.349655	+	0.936879i	$0.613701\pi$
$938$	19654.3	0.684152
$939$	27740.1	0.964073
$940$	−14790.8	−0.513214
$941$	1691.92	0.0586132	0.0293066	−	0.999570i	$-0.490670\pi$
$941$	1691.92	0.0586132	0.0293066	+	0.999570i	$0.490670\pi$
$942$	37966.6	1.31318
$943$	−9899.81	−0.341869
$944$	−5078.04	−0.175081
$945$	7184.94	0.247329
$946$	25315.6	0.870063
$947$	−1848.14	−0.0634175	−0.0317088	−	0.999497i	$-0.510095\pi$
$947$	−1848.14	−0.0634175	−0.0317088	+	0.999497i	$0.510095\pi$
$948$	−27596.7	−0.945463
$949$	50156.9	1.71566
$950$	−2085.26	−0.0712155
$951$	50701.7	1.72883
$952$	2119.72	0.0721644
$953$	19079.9	0.648539	0.324270	−	0.945965i	$-0.394881\pi$
$953$	19079.9	0.648539	0.324270	+	0.945965i	$0.394881\pi$
$954$	−8587.72	−0.291444
$955$	−16450.0	−0.557394
$956$	−67540.1	−2.28494
$957$	18292.2	0.617870
$958$	23094.0	0.778845
$959$	−4821.44	−0.162349
$960$	−28062.6	−0.943455
$961$	67573.8	2.26826
$962$	20275.1	0.679519
$963$	−14137.9	−0.473091
$964$	−40089.4	−1.33941
$965$	−9802.71	−0.327006
$966$	26836.0	0.893826
$967$	2731.48	0.0908362	0.0454181	−	0.998968i	$-0.485538\pi$
$967$	2731.48	0.0908362	0.0454181	+	0.998968i	$0.485538\pi$
$968$	1738.20	0.0577148
$969$	−562.397	−0.0186448
$970$	−22718.1	−0.751994
$971$	50099.2	1.65578	0.827889	−	0.560892i	$-0.189542\pi$
$971$	50099.2	1.65578	0.827889	+	0.560892i	$0.189542\pi$
$972$	54111.7	1.78563
$973$	28665.7	0.944481
$974$	84149.7	2.76831
$975$	10881.4	0.357418
$976$	3233.33	0.106042
$977$	−52382.7	−1.71532	−0.857662	−	0.514214i	$-0.828084\pi$
$977$	−52382.7	−1.71532	−0.857662	+	0.514214i	$0.828084\pi$
$978$	−69849.8	−2.28380
$979$	7138.39	0.233038
$980$	−47890.8	−1.56103
$981$	30522.2	0.993374
$982$	33380.6	1.08474
$983$	−27842.5	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$983$	−27842.5	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$984$	38560.5	1.24925
$985$	16679.6	0.539551
$986$	−4502.38	−0.145421
$987$	62787.0	2.02485
$988$	13456.0	0.433291
$989$	−13394.3	−0.430651
$990$	5069.00	0.162731
$991$	49902.5	1.59960	0.799801	−	0.600265i	$-0.204938\pi$
$991$	49902.5	1.59960	0.799801	+	0.600265i	$0.204938\pi$
$992$	−73001.5	−2.33649
$993$	−11981.8	−0.382912
$994$	133210.	4.25067
$995$	−14181.3	−0.451838
$996$	−99658.8	−3.17049
$997$	7314.77	0.232358	0.116179	−	0.993228i	$-0.462935\pi$
$997$	7314.77	0.232358	0.116179	+	0.993228i	$0.462935\pi$
$998$	20043.2	0.635728
$999$	−3058.68	−0.0968691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.20	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.20	✓	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)

\(f(q)\)	\(=\)	\(q+4.39002 q^{2} -6.92776 q^{3} +11.2723 q^{4} -5.00000 q^{5} -30.4130 q^{6} -34.5356 q^{7} +14.3653 q^{8} +20.9939 q^{9} +O(q^{10})\)	\(q+4.39002 q^{2} -6.92776 q^{3} +11.2723 q^{4} -5.00000 q^{5} -30.4130 q^{6} -34.5356 q^{7} +14.3653 q^{8} +20.9939 q^{9} -21.9501 q^{10} -11.0000 q^{11} -78.0916 q^{12} -62.8275 q^{13} -151.612 q^{14} +34.6388 q^{15} -27.1142 q^{16} -4.27264 q^{17} +92.1636 q^{18} -19.0000 q^{19} -56.3613 q^{20} +239.255 q^{21} -48.2902 q^{22} +25.5501 q^{23} -99.5194 q^{24} +25.0000 q^{25} -275.814 q^{26} +41.6089 q^{27} -389.295 q^{28} +240.038 q^{29} +152.065 q^{30} +312.033 q^{31} -233.954 q^{32} +76.2054 q^{33} -18.7570 q^{34} +172.678 q^{35} +236.649 q^{36} -73.5102 q^{37} -83.4104 q^{38} +435.254 q^{39} -71.8265 q^{40} -387.467 q^{41} +1050.33 q^{42} -524.238 q^{43} -123.995 q^{44} -104.969 q^{45} +112.165 q^{46} +262.427 q^{47} +187.840 q^{48} +849.710 q^{49} +109.750 q^{50} +29.5998 q^{51} -708.209 q^{52} -93.1791 q^{53} +182.664 q^{54} +55.0000 q^{55} -496.115 q^{56} +131.627 q^{57} +1053.77 q^{58} +187.284 q^{59} +390.458 q^{60} -119.249 q^{61} +1369.83 q^{62} -725.037 q^{63} -810.150 q^{64} +314.138 q^{65} +334.543 q^{66} -129.635 q^{67} -48.1623 q^{68} -177.005 q^{69} +758.060 q^{70} -878.625 q^{71} +301.584 q^{72} -798.326 q^{73} -322.711 q^{74} -173.194 q^{75} -214.173 q^{76} +379.892 q^{77} +1910.77 q^{78} +353.389 q^{79} +135.571 q^{80} -855.092 q^{81} -1700.99 q^{82} +1276.18 q^{83} +2696.94 q^{84} +21.3632 q^{85} -2301.41 q^{86} -1662.92 q^{87} -158.018 q^{88} -648.944 q^{89} -460.818 q^{90} +2169.79 q^{91} +288.007 q^{92} -2161.69 q^{93} +1152.06 q^{94} +95.0000 q^{95} +1620.78 q^{96} +1034.99 q^{97} +3730.24 q^{98} -230.933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 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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