LMFDB - Embedded newform 1849.4.a.j.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.07922 q^{2} -5.77516 q^{3} +1.48163 q^{4} -13.4757 q^{5} +17.7830 q^{6} +34.2458 q^{7} +20.0715 q^{8} +6.35250 q^{9} +O(q^{10})$	\(q-3.07922 q^{2} -5.77516 q^{3} +1.48163 q^{4} -13.4757 q^{5} +17.7830 q^{6} +34.2458 q^{7} +20.0715 q^{8} +6.35250 q^{9} +41.4947 q^{10} +24.2940 q^{11} -8.55663 q^{12} +11.8136 q^{13} -105.451 q^{14} +77.8243 q^{15} -73.6578 q^{16} +110.941 q^{17} -19.5608 q^{18} +45.6852 q^{19} -19.9659 q^{20} -197.775 q^{21} -74.8068 q^{22} -214.470 q^{23} -115.916 q^{24} +56.5941 q^{25} -36.3769 q^{26} +119.243 q^{27} +50.7395 q^{28} -57.4470 q^{29} -239.638 q^{30} -181.469 q^{31} +66.2366 q^{32} -140.302 q^{33} -341.613 q^{34} -461.486 q^{35} +9.41203 q^{36} +156.106 q^{37} -140.675 q^{38} -68.2257 q^{39} -270.478 q^{40} +0.775433 q^{41} +608.995 q^{42} +35.9947 q^{44} -85.6043 q^{45} +660.401 q^{46} -146.038 q^{47} +425.386 q^{48} +829.777 q^{49} -174.266 q^{50} -640.704 q^{51} +17.5034 q^{52} -282.368 q^{53} -367.175 q^{54} -327.379 q^{55} +687.367 q^{56} -263.840 q^{57} +176.892 q^{58} -698.452 q^{59} +115.306 q^{60} +98.7249 q^{61} +558.783 q^{62} +217.547 q^{63} +385.305 q^{64} -159.197 q^{65} +432.022 q^{66} -517.724 q^{67} +164.373 q^{68} +1238.60 q^{69} +1421.02 q^{70} -29.8847 q^{71} +127.505 q^{72} +75.4773 q^{73} -480.686 q^{74} -326.840 q^{75} +67.6884 q^{76} +831.970 q^{77} +210.082 q^{78} +882.904 q^{79} +992.589 q^{80} -860.163 q^{81} -2.38773 q^{82} +716.095 q^{83} -293.029 q^{84} -1495.01 q^{85} +331.766 q^{87} +487.619 q^{88} -142.143 q^{89} +263.595 q^{90} +404.568 q^{91} -317.764 q^{92} +1048.01 q^{93} +449.685 q^{94} -615.640 q^{95} -382.527 q^{96} -352.135 q^{97} -2555.07 q^{98} +154.328 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * 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 $

$2$	−3.07922	−1.08867	−0.544335	−	0.838868i	$-0.683218\pi$
$2$	−3.07922	−1.08867	−0.544335	+	0.838868i	$0.683218\pi$
$3$	−5.77516	−1.11143	−0.555715	−	0.831373i	$-0.687556\pi$
$3$	−5.77516	−1.11143	−0.555715	+	0.831373i	$0.687556\pi$
$4$	1.48163	0.185203
$5$	−13.4757	−1.20530	−0.602651	−	0.798005i	$-0.705889\pi$
$5$	−13.4757	−1.20530	−0.602651	+	0.798005i	$0.705889\pi$
$6$	17.7830	1.20998
$7$	34.2458	1.84910	0.924551	−	0.381059i	$-0.124441\pi$
$7$	34.2458	1.84910	0.924551	+	0.381059i	$0.124441\pi$
$8$	20.0715	0.887045
$9$	6.35250	0.235278
$10$	41.4947	1.31218
$11$	24.2940	0.665903	0.332951	−	0.942944i	$-0.391955\pi$
$11$	24.2940	0.665903	0.332951	+	0.942944i	$0.391955\pi$
$12$	−8.55663	−0.205840
$13$	11.8136	0.252040	0.126020	−	0.992028i	$-0.459780\pi$
$13$	11.8136	0.252040	0.126020	+	0.992028i	$0.459780\pi$
$14$	−105.451	−2.01306
$15$	77.8243	1.33961
$16$	−73.6578	−1.15090
$17$	110.941	1.58278	0.791389	−	0.611313i	$-0.209359\pi$
$17$	110.941	1.58278	0.791389	+	0.611313i	$0.209359\pi$
$18$	−19.5608	−0.256140
$19$	45.6852	0.551627	0.275813	−	0.961211i	$-0.411053\pi$
$19$	45.6852	0.551627	0.275813	+	0.961211i	$0.411053\pi$
$20$	−19.9659	−0.223226
$21$	−197.775	−2.05515
$22$	−74.8068	−0.724949
$23$	−214.470	−1.94435	−0.972176	−	0.234254i	$-0.924735\pi$
$23$	−214.470	−1.94435	−0.972176	+	0.234254i	$0.924735\pi$
$24$	−115.916	−0.985889
$25$	56.5941	0.452753
$26$	−36.3769	−0.274388
$27$	119.243	0.849936
$28$	50.7395	0.342459
$29$	−57.4470	−0.367850	−0.183925	−	0.982940i	$-0.558880\pi$
$29$	−57.4470	−0.367850	−0.183925	+	0.982940i	$0.558880\pi$
$30$	−239.638	−1.45839
$31$	−181.469	−1.05138	−0.525689	−	0.850677i	$-0.676193\pi$
$31$	−181.469	−1.05138	−0.525689	+	0.850677i	$0.676193\pi$
$32$	66.2366	0.365909
$33$	−140.302	−0.740105
$34$	−341.613	−1.72312
$35$	−461.486	−2.22873
$36$	9.41203	0.0435742
$37$	156.106	0.693613	0.346806	−	0.937937i	$-0.387266\pi$
$37$	156.106	0.693613	0.346806	+	0.937937i	$0.387266\pi$
$38$	−140.675	−0.600540
$39$	−68.2257	−0.280124
$40$	−270.478	−1.06916
$41$	0.775433	0.00295371	0.00147686	−	0.999999i	$-0.499530\pi$
$41$	0.775433	0.00295371	0.00147686	+	0.999999i	$0.499530\pi$
$42$	608.995	2.23738
$43$	0	0
$43$	0	0
$44$	35.9947	0.123327
$45$	−85.6043	−0.283581
$46$	660.401	2.11676
$47$	−146.038	−0.453232	−0.226616	−	0.973984i	$-0.572766\pi$
$47$	−146.038	−0.453232	−0.226616	+	0.973984i	$0.572766\pi$
$48$	425.386	1.27915
$49$	829.777	2.41918
$50$	−174.266	−0.492898
$51$	−640.704	−1.75915
$52$	17.5034	0.0466785
$53$	−282.368	−0.731815	−0.365908	−	0.930651i	$-0.619241\pi$
$53$	−282.368	−0.731815	−0.365908	+	0.930651i	$0.619241\pi$
$54$	−367.175	−0.925300
$55$	−327.379	−0.802614
$56$	687.367	1.64024
$57$	−263.840	−0.613095
$58$	176.892	0.400467
$59$	−698.452	−1.54120	−0.770599	−	0.637320i	$-0.780043\pi$
$59$	−698.452	−1.54120	−0.770599	+	0.637320i	$0.780043\pi$
$60$	115.306	0.248100
$61$	98.7249	0.207220	0.103610	−	0.994618i	$-0.466961\pi$
$61$	98.7249	0.207220	0.103610	+	0.994618i	$0.466961\pi$
$62$	558.783	1.14461
$63$	217.547	0.435053
$64$	385.305	0.752549
$65$	−159.197	−0.303784
$66$	432.022	0.805730
$67$	−517.724	−0.944031	−0.472015	−	0.881590i	$-0.656473\pi$
$67$	−517.724	−0.944031	−0.472015	+	0.881590i	$0.656473\pi$
$68$	164.373	0.293135
$69$	1238.60	2.16101
$70$	1421.02	2.42635
$71$	−29.8847	−0.0499529	−0.0249765	−	0.999688i	$-0.507951\pi$
$71$	−29.8847	−0.0499529	−0.0249765	+	0.999688i	$0.507951\pi$
$72$	127.505	0.208702
$73$	75.4773	0.121013	0.0605065	−	0.998168i	$-0.480728\pi$
$73$	75.4773	0.121013	0.0605065	+	0.998168i	$0.480728\pi$
$74$	−480.686	−0.755116
$75$	−326.840	−0.503203
$76$	67.6884	0.102163
$77$	831.970	1.23132
$78$	210.082	0.304963
$79$	882.904	1.25740	0.628699	−	0.777649i	$-0.283588\pi$
$79$	882.904	1.25740	0.628699	+	0.777649i	$0.283588\pi$
$80$	992.589	1.38719
$81$	−860.163	−1.17992
$82$	−2.38773	−0.00321562
$83$	716.095	0.947008	0.473504	−	0.880792i	$-0.342989\pi$
$83$	716.095	0.947008	0.473504	+	0.880792i	$0.342989\pi$
$84$	−293.029	−0.380620
$85$	−1495.01	−1.90772
$86$	0	0
$87$	331.766	0.408839
$88$	487.619	0.590686
$89$	−142.143	−0.169293	−0.0846466	−	0.996411i	$-0.526976\pi$
$89$	−142.143	−0.169293	−0.0846466	+	0.996411i	$0.526976\pi$
$90$	263.595	0.308726
$91$	404.568	0.466047
$92$	−317.764	−0.360100
$93$	1048.01	1.16853
$94$	449.685	0.493420
$95$	−615.640	−0.664877
$96$	−382.527	−0.406682
$97$	−352.135	−0.368597	−0.184299	−	0.982870i	$-0.559001\pi$
$97$	−352.135	−0.368597	−0.184299	+	0.982870i	$0.559001\pi$
$98$	−2555.07	−2.63369
$99$	154.328	0.156672
$100$	83.8512	0.0838512
$101$	1101.97	1.08564	0.542820	−	0.839849i	$-0.317357\pi$
$101$	1101.97	1.08564	0.542820	+	0.839849i	$0.317357\pi$
$102$	1972.87	1.91513
$103$	−909.666	−0.870214	−0.435107	−	0.900379i	$-0.643290\pi$
$103$	−909.666	−0.870214	−0.435107	+	0.900379i	$0.643290\pi$
$104$	237.118	0.223570
$105$	2665.16	2.47707
$106$	869.474	0.796706
$107$	−1868.64	−1.68830	−0.844150	−	0.536107i	$-0.819894\pi$
$107$	−1868.64	−1.68830	−0.844150	+	0.536107i	$0.819894\pi$
$108$	176.673	0.157411
$109$	1049.08	0.921869	0.460935	−	0.887434i	$-0.347514\pi$
$109$	1049.08	0.921869	0.460935	+	0.887434i	$0.347514\pi$
$110$	1008.07	0.873782
$111$	−901.538	−0.770903
$112$	−2522.47	−2.12814
$113$	739.032	0.615242	0.307621	−	0.951509i	$-0.400467\pi$
$113$	739.032	0.615242	0.307621	+	0.951509i	$0.400467\pi$
$114$	812.421	0.667458
$115$	2890.13	2.34353
$116$	−85.1149	−0.0681269
$117$	75.0462	0.0592993
$118$	2150.69	1.67786
$119$	3799.28	2.92672
$120$	1562.05	1.18829
$121$	−740.799	−0.556573
$122$	−303.996	−0.225594
$123$	−4.47825	−0.00328285
$124$	−268.869	−0.194719
$125$	921.817	0.659598
$126$	−669.876	−0.473629
$127$	1516.44	1.05955	0.529773	−	0.848140i	$-0.322277\pi$
$127$	1516.44	1.05955	0.529773	+	0.848140i	$0.322277\pi$
$128$	−1716.33	−1.18519
$129$	0	0
$130$	490.203	0.330720
$131$	1383.75	0.922893	0.461446	−	0.887168i	$-0.347331\pi$
$131$	1383.75	0.922893	0.461446	+	0.887168i	$0.347331\pi$
$132$	−207.875	−0.137070
$133$	1564.53	1.02001
$134$	1594.19	1.02774
$135$	−1606.88	−1.02443
$136$	2226.76	1.40399
$137$	−1180.51	−0.736190	−0.368095	−	0.929788i	$-0.619990\pi$
$137$	−1180.51	−0.736190	−0.368095	+	0.929788i	$0.619990\pi$
$138$	−3813.92	−2.35263
$139$	−2967.86	−1.81101	−0.905505	−	0.424336i	$-0.860507\pi$
$139$	−2967.86	−1.81101	−0.905505	+	0.424336i	$0.860507\pi$
$140$	−683.749	−0.412767
$141$	843.396	0.503736
$142$	92.0216	0.0543823
$143$	287.001	0.167834
$144$	−467.911	−0.270782
$145$	774.138	0.443370
$146$	−232.412	−0.131743
$147$	−4792.10	−2.68875
$148$	231.291	0.128459
$149$	568.440	0.312540	0.156270	−	0.987714i	$-0.450053\pi$
$149$	568.440	0.312540	0.156270	+	0.987714i	$0.450053\pi$
$150$	1006.41	0.547822
$151$	1740.05	0.937772	0.468886	−	0.883259i	$-0.344656\pi$
$151$	1740.05	0.937772	0.468886	+	0.883259i	$0.344656\pi$
$152$	916.973	0.489318
$153$	704.755	0.372393
$154$	−2561.82	−1.34050
$155$	2445.42	1.26723
$156$	−101.085	−0.0518799
$157$	−1742.13	−0.885587	−0.442793	−	0.896624i	$-0.646012\pi$
$157$	−1742.13	−0.885587	−0.442793	+	0.896624i	$0.646012\pi$
$158$	−2718.66	−1.36889
$159$	1630.72	0.813362
$160$	−892.583	−0.441030
$161$	−7344.70	−3.59530
$162$	2648.64	1.28455
$163$	−1870.73	−0.898939	−0.449469	−	0.893296i	$-0.648387\pi$
$163$	−1870.73	−0.898939	−0.449469	+	0.893296i	$0.648387\pi$
$164$	1.14890	0.000547037 0
$165$	1890.67	0.892050
$166$	−2205.02	−1.03098
$167$	295.044	0.136714	0.0683568	−	0.997661i	$-0.478224\pi$
$167$	295.044	0.136714	0.0683568	+	0.997661i	$0.478224\pi$
$168$	−3969.65	−1.82301
$169$	−2057.44	−0.936476
$170$	4603.47	2.07688
$171$	290.216	0.129786
$172$	0	0
$173$	−1615.88	−0.710132	−0.355066	−	0.934841i	$-0.615542\pi$
$173$	−1615.88	−0.710132	−0.355066	+	0.934841i	$0.615542\pi$
$174$	−1021.58	−0.445091
$175$	1938.11	0.837186
$176$	−1789.45	−0.766390
$177$	4033.67	1.71294
$178$	437.689	0.184305
$179$	2117.02	0.883987	0.441993	−	0.897018i	$-0.354271\pi$
$179$	2117.02	0.883987	0.441993	+	0.897018i	$0.354271\pi$
$180$	−126.834	−0.0525201
$181$	297.604	0.122214	0.0611070	−	0.998131i	$-0.480537\pi$
$181$	297.604	0.122214	0.0611070	+	0.998131i	$0.480537\pi$
$182$	−1245.76	−0.507371
$183$	−570.152	−0.230311
$184$	−4304.74	−1.72473
$185$	−2103.64	−0.836013
$186$	−3227.06	−1.27215
$187$	2695.21	1.05398
$188$	−216.374	−0.0839400
$189$	4083.56	1.57162
$190$	1895.69	0.723832
$191$	−4542.23	−1.72075	−0.860377	−	0.509659i	$-0.829772\pi$
$191$	−4542.23	−1.72075	−0.860377	+	0.509659i	$0.829772\pi$
$192$	−2225.20	−0.836406
$193$	3262.95	1.21696	0.608478	−	0.793571i	$-0.291781\pi$
$193$	3262.95	1.21696	0.608478	+	0.793571i	$0.291781\pi$
$194$	1084.30	0.401281
$195$	919.388	0.337635
$196$	1229.42	0.448039
$197$	−868.209	−0.313997	−0.156998	−	0.987599i	$-0.550182\pi$
$197$	−868.209	−0.313997	−0.156998	+	0.987599i	$0.550182\pi$
$198$	−475.211	−0.170564
$199$	4215.53	1.50166	0.750832	−	0.660493i	$-0.229653\pi$
$199$	4215.53	1.50166	0.750832	+	0.660493i	$0.229653\pi$
$200$	1135.93	0.401612
$201$	2989.94	1.04922
$202$	−3393.20	−1.18190
$203$	−1967.32	−0.680191
$204$	−949.283	−0.325800
$205$	−10.4495	−0.00356012
$206$	2801.07	0.947377
$207$	−1362.42	−0.457463
$208$	−870.167	−0.290073
$209$	1109.88	0.367330
$210$	−8206.62	−2.69672
$211$	−2905.47	−0.947964	−0.473982	−	0.880534i	$-0.657184\pi$
$211$	−2905.47	−0.947964	−0.473982	+	0.880534i	$0.657184\pi$
$212$	−418.363	−0.135535
$213$	172.589	0.0555192
$214$	5753.96	1.83800
$215$	0	0
$216$	2393.38	0.753931
$217$	−6214.55	−1.94411
$218$	−3230.36	−1.00361
$219$	−435.894	−0.134498
$220$	−485.053	−0.148647
$221$	1310.62	0.398922
$222$	2776.04	0.839259
$223$	2146.80	0.644667	0.322333	−	0.946626i	$-0.395533\pi$
$223$	2146.80	0.644667	0.322333	+	0.946626i	$0.395533\pi$
$224$	2268.33	0.676602
$225$	359.514	0.106523
$226$	−2275.65	−0.669795
$227$	873.162	0.255303	0.127652	−	0.991819i	$-0.459256\pi$
$227$	873.162	0.255303	0.127652	+	0.991819i	$0.459256\pi$
$228$	−390.911	−0.113547
$229$	3489.37	1.00692	0.503459	−	0.864019i	$-0.332061\pi$
$229$	3489.37	1.00692	0.503459	+	0.864019i	$0.332061\pi$
$230$	−8899.36	−2.55133
$231$	−4804.76	−1.36853
$232$	−1153.05	−0.326299
$233$	−5132.13	−1.44299	−0.721496	−	0.692418i	$-0.756545\pi$
$233$	−5132.13	−1.44299	−0.721496	+	0.692418i	$0.756545\pi$
$234$	−231.084	−0.0645574
$235$	1967.97	0.546281
$236$	−1034.84	−0.285435
$237$	−5098.91	−1.39751
$238$	−11698.8	−3.18623
$239$	3497.81	0.946672	0.473336	−	0.880882i	$-0.343050\pi$
$239$	3497.81	0.946672	0.473336	+	0.880882i	$0.343050\pi$
$240$	−5732.36	−1.54176
$241$	−5168.03	−1.38134	−0.690669	−	0.723171i	$-0.742684\pi$
$241$	−5168.03	−1.38134	−0.690669	+	0.723171i	$0.742684\pi$
$242$	2281.09	0.605925
$243$	1748.03	0.461466
$244$	146.273	0.0383778
$245$	−11181.8	−2.91584
$246$	13.7895	0.00357394
$247$	539.709	0.139032
$248$	−3642.36	−0.932621
$249$	−4135.57	−1.05253
$250$	−2838.48	−0.718085
$251$	2060.88	0.518253	0.259126	−	0.965843i	$-0.416565\pi$
$251$	2060.88	0.518253	0.259126	+	0.965843i	$0.416565\pi$
$252$	322.323	0.0805731
$253$	−5210.34	−1.29475
$254$	−4669.46	−1.15350
$255$	8633.92	2.12030
$256$	2202.54	0.537728
$257$	−2254.70	−0.547254	−0.273627	−	0.961836i	$-0.588223\pi$
$257$	−2254.70	−0.547254	−0.273627	+	0.961836i	$0.588223\pi$
$258$	0	0
$259$	5345.98	1.28256
$260$	−235.870	−0.0562617
$261$	−364.932	−0.0865469
$262$	−4260.88	−1.00473
$263$	−6481.38	−1.51962	−0.759808	−	0.650147i	$-0.774707\pi$
$263$	−6481.38	−1.51962	−0.759808	+	0.650147i	$0.774707\pi$
$264$	−2816.08	−0.656506
$265$	3805.10	0.882059
$266$	−4817.53	−1.11046
$267$	820.897	0.188158
$268$	−767.073	−0.174837
$269$	2971.06	0.673415	0.336708	−	0.941609i	$-0.390687\pi$
$269$	2971.06	0.673415	0.336708	+	0.941609i	$0.390687\pi$
$270$	4947.93	1.11527
$271$	−5738.03	−1.28620	−0.643100	−	0.765782i	$-0.722352\pi$
$271$	−5738.03	−1.28620	−0.643100	+	0.765782i	$0.722352\pi$
$272$	−8171.69	−1.82162
$273$	−2336.45	−0.517979
$274$	3635.06	0.801468
$275$	1374.90	0.301489
$276$	1835.14	0.400226
$277$	1591.55	0.345224	0.172612	−	0.984990i	$-0.444779\pi$
$277$	1591.55	0.345224	0.172612	+	0.984990i	$0.444779\pi$
$278$	9138.70	1.97159
$279$	−1152.78	−0.247366
$280$	−9262.74	−1.97698
$281$	−1279.65	−0.271665	−0.135832	−	0.990732i	$-0.543371\pi$
$281$	−1279.65	−0.271665	−0.135832	+	0.990732i	$0.543371\pi$
$282$	−2597.01	−0.548402
$283$	8416.02	1.76778	0.883888	−	0.467699i	$-0.154917\pi$
$283$	8416.02	1.76778	0.883888	+	0.467699i	$0.154917\pi$
$284$	−44.2779	−0.00925144
$285$	3555.42	0.738964
$286$	−883.741	−0.182716
$287$	26.5553	0.00546171
$288$	420.768	0.0860902
$289$	7394.97	1.50518
$290$	−2383.74	−0.482684
$291$	2033.64	0.409670
$292$	111.829	0.0224120
$293$	−3718.98	−0.741519	−0.370760	−	0.928729i	$-0.620903\pi$
$293$	−3718.98	−0.741519	−0.370760	+	0.928729i	$0.620903\pi$
$294$	14756.0	2.92716
$295$	9412.12	1.85761
$296$	3133.29	0.615266
$297$	2896.89	0.565974
$298$	−1750.35	−0.340253
$299$	−2533.67	−0.490053
$300$	−484.254	−0.0931948
$301$	0	0
$302$	−5358.02	−1.02092
$303$	−6364.03	−1.20661
$304$	−3365.07	−0.634869
$305$	−1330.39	−0.249763
$306$	−2170.10	−0.405413
$307$	6406.06	1.19092	0.595461	−	0.803384i	$-0.296969\pi$
$307$	6406.06	1.19092	0.595461	+	0.803384i	$0.296969\pi$
$308$	1232.67	0.228045
$309$	5253.47	0.967183
$310$	−7529.98	−1.37959
$311$	−3626.35	−0.661194	−0.330597	−	0.943772i	$-0.607250\pi$
$311$	−3626.35	−0.661194	−0.330597	+	0.943772i	$0.607250\pi$
$312$	−1369.39	−0.248483
$313$	6160.29	1.11246	0.556230	−	0.831028i	$-0.312247\pi$
$313$	6160.29	1.11246	0.556230	+	0.831028i	$0.312247\pi$
$314$	5364.41	0.964112
$315$	−2931.59	−0.524370
$316$	1308.13	0.232874
$317$	−1508.62	−0.267294	−0.133647	−	0.991029i	$-0.542669\pi$
$317$	−1508.62	−0.267294	−0.133647	+	0.991029i	$0.542669\pi$
$318$	−5021.36	−0.885483
$319$	−1395.62	−0.244952
$320$	−5192.25	−0.907049
$321$	10791.7	1.87643
$322$	22616.0	3.91410
$323$	5068.38	0.873102
$324$	−1274.44	−0.218525
$325$	668.582	0.114112
$326$	5760.40	0.978648
$327$	−6058.61	−1.02459
$328$	15.5641	0.00262008
$329$	−5001.21	−0.838072
$330$	−5821.79	−0.971148
$331$	2373.17	0.394082	0.197041	−	0.980395i	$-0.436867\pi$
$331$	2373.17	0.394082	0.197041	+	0.980395i	$0.436867\pi$
$332$	1060.98	0.175389
$333$	991.664	0.163192
$334$	−908.506	−0.148836
$335$	6976.69	1.13784
$336$	14567.7	2.36528
$337$	−6631.98	−1.07201	−0.536005	−	0.844215i	$-0.680067\pi$
$337$	−6631.98	−1.07201	−0.536005	+	0.844215i	$0.680067\pi$
$338$	6335.31	1.01951
$339$	−4268.03	−0.683798
$340$	−2215.04	−0.353317
$341$	−4408.61	−0.700116
$342$	−893.639	−0.141294
$343$	16670.1	2.62420
$344$	0	0
$345$	−16691.0	−2.60467
$346$	4975.64	0.773099
$347$	−4392.54	−0.679551	−0.339775	−	0.940507i	$-0.610351\pi$
$347$	−4392.54	−0.679551	−0.339775	+	0.940507i	$0.610351\pi$
$348$	491.553	0.0757183
$349$	1187.57	0.182147	0.0910736	−	0.995844i	$-0.470970\pi$
$349$	1187.57	0.182147	0.0910736	+	0.995844i	$0.470970\pi$
$350$	−5967.88	−0.911419
$351$	1408.69	0.214217
$352$	1609.15	0.243660
$353$	−6116.11	−0.922175	−0.461088	−	0.887355i	$-0.652541\pi$
$353$	−6116.11	−0.922175	−0.461088	+	0.887355i	$0.652541\pi$
$354$	−12420.6	−1.86482
$355$	402.716	0.0602084
$356$	−210.602	−0.0313536
$357$	−21941.4	−3.25284
$358$	−6518.79	−0.962370
$359$	−867.123	−0.127479	−0.0637396	−	0.997967i	$-0.520303\pi$
$359$	−867.123	−0.127479	−0.0637396	+	0.997967i	$0.520303\pi$
$360$	−1718.21	−0.251549
$361$	−4771.86	−0.695708
$362$	−916.390	−0.133051
$363$	4278.24	0.618593
$364$	599.418	0.0863133
$365$	−1017.11	−0.145857
$366$	1755.63	0.250733
$367$	7941.70	1.12957	0.564787	−	0.825237i	$-0.308959\pi$
$367$	7941.70	1.12957	0.564787	+	0.825237i	$0.308959\pi$
$368$	15797.4	2.23776
$369$	4.92594	0.000694943 0
$370$	6477.57	0.910143
$371$	−9669.93	−1.35320
$372$	1552.76	0.216416
$373$	200.374	0.0278149	0.0139075	−	0.999903i	$-0.495573\pi$
$373$	200.374	0.0278149	0.0139075	+	0.999903i	$0.495573\pi$
$374$	−8299.17	−1.14743
$375$	−5323.64	−0.733098
$376$	−2931.22	−0.402037
$377$	−678.658	−0.0927127
$378$	−12574.2	−1.71097
$379$	9566.94	1.29662	0.648312	−	0.761375i	$-0.275475\pi$
$379$	9566.94	1.29662	0.648312	+	0.761375i	$0.275475\pi$
$380$	−912.147	−0.123137
$381$	−8757.69	−1.17761
$382$	13986.5	1.87333
$383$	5373.45	0.716894	0.358447	−	0.933550i	$-0.383306\pi$
$383$	5373.45	0.716894	0.358447	+	0.933550i	$0.383306\pi$
$384$	9912.10	1.31725
$385$	−11211.4	−1.48411
$386$	−10047.4	−1.32486
$387$	0	0
$388$	−521.733	−0.0682654
$389$	7774.81	1.01336	0.506682	−	0.862133i	$-0.330872\pi$
$389$	7774.81	1.01336	0.506682	+	0.862133i	$0.330872\pi$
$390$	−2831.00	−0.367573
$391$	−23793.6	−3.07747
$392$	16654.9	2.14592
$393$	−7991.39	−1.02573
$394$	2673.41	0.341839
$395$	−11897.7	−1.51554
$396$	228.656	0.0290162
$397$	974.462	0.123191	0.0615955	−	0.998101i	$-0.480381\pi$
$397$	974.462	0.123191	0.0615955	+	0.998101i	$0.480381\pi$
$398$	−12980.6	−1.63482
$399$	−9035.41	−1.13367
$400$	−4168.59	−0.521074
$401$	3270.33	0.407263	0.203631	−	0.979048i	$-0.434726\pi$
$401$	3270.33	0.407263	0.203631	+	0.979048i	$0.434726\pi$
$402$	−9206.70	−1.14226
$403$	−2143.81	−0.264989
$404$	1632.70	0.201064
$405$	11591.3	1.42216
$406$	6057.82	0.740504
$407$	3792.45	0.461879
$408$	−12859.9	−1.56044
$409$	−5108.88	−0.617647	−0.308824	−	0.951119i	$-0.599935\pi$
$409$	−5108.88	−0.617647	−0.308824	+	0.951119i	$0.599935\pi$
$410$	32.1763	0.00387579
$411$	6817.65	0.818224
$412$	−1347.78	−0.161166
$413$	−23919.1	−2.84983
$414$	4195.20	0.498026
$415$	−9649.87	−1.14143
$416$	782.495	0.0922235
$417$	17139.9	2.01281
$418$	−3417.57	−0.399901
$419$	4610.75	0.537590	0.268795	−	0.963197i	$-0.413375\pi$
$419$	4610.75	0.537590	0.268795	+	0.963197i	$0.413375\pi$
$420$	3948.76	0.458762
$421$	8130.00	0.941168	0.470584	−	0.882355i	$-0.344043\pi$
$421$	8130.00	0.941168	0.470584	+	0.882355i	$0.344043\pi$
$422$	8946.58	1.03202
$423$	−927.710	−0.106635
$424$	−5667.56	−0.649153
$425$	6278.62	0.716607
$426$	−531.440	−0.0604421
$427$	3380.92	0.383171
$428$	−2768.62	−0.312678
$429$	−1657.48	−0.186536
$430$	0	0
$431$	−10180.0	−1.13771	−0.568853	−	0.822439i	$-0.692613\pi$
$431$	−10180.0	−1.13771	−0.568853	+	0.822439i	$0.692613\pi$
$432$	−8783.15	−0.978193
$433$	3809.91	0.422846	0.211423	−	0.977395i	$-0.432190\pi$
$433$	3809.91	0.422846	0.211423	+	0.977395i	$0.432190\pi$
$434$	19136.0	2.11649
$435$	−4470.77	−0.492775
$436$	1554.35	0.170733
$437$	−9798.11	−1.07256
$438$	1342.21	0.146423
$439$	−13805.3	−1.50090	−0.750448	−	0.660930i	$-0.770162\pi$
$439$	−13805.3	−1.50090	−0.750448	+	0.660930i	$0.770162\pi$
$440$	−6571.00	−0.711955
$441$	5271.16	0.569179
$442$	−4035.69	−0.434295
$443$	−3291.23	−0.352982	−0.176491	−	0.984302i	$-0.556475\pi$
$443$	−3291.23	−0.352982	−0.176491	+	0.984302i	$0.556475\pi$
$444$	−1335.74	−0.142774
$445$	1915.47	0.204049
$446$	−6610.49	−0.701830
$447$	−3282.83	−0.347366
$448$	13195.1	1.39154
$449$	−17505.7	−1.83997	−0.919986	−	0.391952i	$-0.871800\pi$
$449$	−17505.7	−1.83997	−0.919986	+	0.391952i	$0.871800\pi$
$450$	−1107.02	−0.115968
$451$	18.8384	0.00196689
$452$	1094.97	0.113945
$453$	−10049.1	−1.04227
$454$	−2688.66	−0.277941
$455$	−5451.83	−0.561727
$456$	−5295.67	−0.543843
$457$	−6024.93	−0.616706	−0.308353	−	0.951272i	$-0.599778\pi$
$457$	−6024.93	−0.616706	−0.308353	+	0.951272i	$0.599778\pi$
$458$	−10744.6	−1.09620
$459$	13228.9	1.34526
$460$	4282.09	0.434029
$461$	9644.81	0.974411	0.487205	−	0.873287i	$-0.338016\pi$
$461$	9644.81	0.974411	0.487205	+	0.873287i	$0.338016\pi$
$462$	14794.9	1.48988
$463$	8578.11	0.861034	0.430517	−	0.902583i	$-0.358331\pi$
$463$	8578.11	0.861034	0.430517	+	0.902583i	$0.358331\pi$
$464$	4231.42	0.423359
$465$	−14122.7	−1.40844
$466$	15803.0	1.57094
$467$	−3847.04	−0.381199	−0.190599	−	0.981668i	$-0.561043\pi$
$467$	−3847.04	−0.381199	−0.190599	+	0.981668i	$0.561043\pi$
$468$	111.190	0.0109824
$469$	−17729.9	−1.74561
$470$	−6059.82	−0.594720
$471$	10061.1	0.984268
$472$	−14019.0	−1.36711
$473$	0	0
$474$	15700.7	1.52143
$475$	2585.51	0.249750
$476$	5629.10	0.542037
$477$	−1793.74	−0.172180
$478$	−10770.5	−1.03061
$479$	−3588.42	−0.342295	−0.171147	−	0.985245i	$-0.554747\pi$
$479$	−3588.42	−0.342295	−0.171147	+	0.985245i	$0.554747\pi$
$480$	5154.81	0.490175
$481$	1844.18	0.174818
$482$	15913.5	1.50382
$483$	42416.9	3.99593
$484$	−1097.59	−0.103079
$485$	4745.27	0.444271
$486$	−5382.58	−0.502384
$487$	−14142.4	−1.31592	−0.657962	−	0.753051i	$-0.728581\pi$
$487$	−14142.4	−1.31592	−0.657962	+	0.753051i	$0.728581\pi$
$488$	1981.56	0.183814
$489$	10803.8	0.999108
$490$	34431.3	3.17439
$491$	−7516.33	−0.690850	−0.345425	−	0.938446i	$-0.612265\pi$
$491$	−7516.33	−0.690850	−0.345425	+	0.938446i	$0.612265\pi$
$492$	−6.63509	−0.000607993 0
$493$	−6373.24	−0.582224
$494$	−1661.88	−0.151360
$495$	−2079.68	−0.188837
$496$	13366.6	1.21004
$497$	−1023.43	−0.0923680
$498$	12734.3	1.14586
$499$	−7452.63	−0.668588	−0.334294	−	0.942469i	$-0.608498\pi$
$499$	−7452.63	−0.668588	−0.334294	+	0.942469i	$0.608498\pi$
$500$	1365.79	0.122160
$501$	−1703.93	−0.151948
$502$	−6345.90	−0.564206
$503$	−13585.0	−1.20422	−0.602112	−	0.798412i	$-0.705674\pi$
$503$	−13585.0	−1.20422	−0.602112	+	0.798412i	$0.705674\pi$
$504$	4366.50	0.385911
$505$	−14849.7	−1.30852
$506$	16043.8	1.40955
$507$	11882.0	1.04083
$508$	2246.80	0.196231
$509$	−3781.95	−0.329336	−0.164668	−	0.986349i	$-0.552655\pi$
$509$	−3781.95	−0.329336	−0.164668	+	0.986349i	$0.552655\pi$
$510$	−26585.8	−2.30831
$511$	2584.78	0.223765
$512$	6948.56	0.599778
$513$	5447.63	0.468847
$514$	6942.72	0.595779
$515$	12258.4	1.04887
$516$	0	0
$517$	−3547.87	−0.301808
$518$	−16461.5	−1.39629
$519$	9331.94	0.789262
$520$	−3195.33	−0.269470
$521$	−5698.91	−0.479220	−0.239610	−	0.970869i	$-0.577020\pi$
$521$	−5698.91	−0.479220	−0.239610	+	0.970869i	$0.577020\pi$
$522$	1123.71	0.0942211
$523$	−19344.4	−1.61734	−0.808671	−	0.588262i	$-0.799813\pi$
$523$	−19344.4	−1.61734	−0.808671	+	0.588262i	$0.799813\pi$
$524$	2050.20	0.170923
$525$	−11192.9	−0.930474
$526$	19957.6	1.65436
$527$	−20132.4	−1.66410
$528$	10334.3	0.851789
$529$	33830.4	2.78050
$530$	−11716.8	−0.960271
$531$	−4436.92	−0.362610
$532$	2318.04	0.188910
$533$	9.16068	0.000744452 0
$534$	−2527.73	−0.204842
$535$	25181.2	2.03491
$536$	−10391.5	−0.837398
$537$	−12226.1	−0.982490
$538$	−9148.56	−0.733127
$539$	20158.7	1.61094
$540$	−2380.79	−0.189727
$541$	4375.41	0.347714	0.173857	−	0.984771i	$-0.444377\pi$
$541$	4375.41	0.347714	0.173857	+	0.984771i	$0.444377\pi$
$542$	17668.7	1.40025
$543$	−1718.71	−0.135832
$544$	7348.37	0.579152
$545$	−14137.1	−1.11113
$546$	7194.44	0.563908
$547$	−22211.1	−1.73616	−0.868080	−	0.496424i	$-0.834646\pi$
$547$	−22211.1	−1.73616	−0.868080	+	0.496424i	$0.834646\pi$
$548$	−1749.08	−0.136345
$549$	627.150	0.0487543
$550$	−4233.62	−0.328222
$551$	−2624.48	−0.202916
$552$	24860.6	1.91691
$553$	30235.8	2.32506
$554$	−4900.74	−0.375835
$555$	12148.8	0.929170
$556$	−4397.25	−0.335405
$557$	2299.41	0.174918	0.0874590	−	0.996168i	$-0.472125\pi$
$557$	2299.41	0.174918	0.0874590	+	0.996168i	$0.472125\pi$
$558$	3549.67	0.269300
$559$	0	0
$560$	33992.0	2.56505
$561$	−15565.3	−1.17142
$562$	3940.34	0.295753
$563$	10335.3	0.773677	0.386839	−	0.922147i	$-0.373567\pi$
$563$	10335.3	0.773677	0.386839	+	0.922147i	$0.373567\pi$
$564$	1249.60	0.0932935
$565$	−9958.96	−0.741552
$566$	−25914.8	−1.92453
$567$	−29457.0	−2.18180
$568$	−599.831	−0.0443105
$569$	3082.61	0.227117	0.113559	−	0.993531i	$-0.463775\pi$
$569$	3082.61	0.227117	0.113559	+	0.993531i	$0.463775\pi$
$570$	−10947.9	−0.804489
$571$	3788.95	0.277693	0.138846	−	0.990314i	$-0.455661\pi$
$571$	3788.95	0.277693	0.138846	+	0.990314i	$0.455661\pi$
$572$	425.228	0.0310834
$573$	26232.1	1.91250
$574$	−81.7699	−0.00594601
$575$	−12137.7	−0.880310
$576$	2447.65	0.177058
$577$	11757.4	0.848296	0.424148	−	0.905593i	$-0.360574\pi$
$577$	11757.4	0.848296	0.424148	+	0.905593i	$0.360574\pi$
$578$	−22770.8	−1.63865
$579$	−18844.1	−1.35256
$580$	1146.98	0.0821135
$581$	24523.3	1.75111
$582$	−6262.03	−0.445996
$583$	−6859.86	−0.487318
$584$	1514.95	0.107344
$585$	−1011.30	−0.0714736
$586$	11451.6	0.807270
$587$	−9782.37	−0.687839	−0.343919	−	0.938999i	$-0.611755\pi$
$587$	−9782.37	−0.687839	−0.343919	+	0.938999i	$0.611755\pi$
$588$	−7100.09	−0.497964
$589$	−8290.44	−0.579969
$590$	−28982.0	−2.02232
$591$	5014.05	0.348985
$592$	−11498.4	−0.798281
$593$	−19483.1	−1.34920	−0.674600	−	0.738183i	$-0.735684\pi$
$593$	−19483.1	−1.34920	−0.674600	+	0.738183i	$0.735684\pi$
$594$	−8920.17	−0.616160
$595$	−51197.9	−3.52758
$596$	842.215	0.0578833
$597$	−24345.4	−1.66900
$598$	7801.74	0.533507
$599$	−4655.42	−0.317555	−0.158778	−	0.987314i	$-0.550755\pi$
$599$	−4655.42	−0.317555	−0.158778	+	0.987314i	$0.550755\pi$
$600$	−6560.18	−0.446364
$601$	13673.1	0.928014	0.464007	−	0.885832i	$-0.346411\pi$
$601$	13673.1	0.928014	0.464007	+	0.885832i	$0.346411\pi$
$602$	0	0
$603$	−3288.84	−0.222110
$604$	2578.11	0.173678
$605$	9982.78	0.670839
$606$	19596.3	1.31360
$607$	7757.17	0.518705	0.259352	−	0.965783i	$-0.416491\pi$
$607$	7757.17	0.518705	0.259352	+	0.965783i	$0.416491\pi$
$608$	3026.03	0.201845
$609$	11361.6	0.755986
$610$	4096.56	0.271909
$611$	−1725.25	−0.114232
$612$	1044.18	0.0689683
$613$	14088.9	0.928297	0.464148	−	0.885757i	$-0.346360\pi$
$613$	14088.9	0.928297	0.464148	+	0.885757i	$0.346360\pi$
$614$	−19725.7	−1.29652
$615$	60.3475	0.00395682
$616$	16698.9	1.09224
$617$	12906.3	0.842122	0.421061	−	0.907032i	$-0.361658\pi$
$617$	12906.3	0.842122	0.421061	+	0.907032i	$0.361658\pi$
$618$	−16176.6	−1.05294
$619$	−23723.2	−1.54041	−0.770206	−	0.637795i	$-0.779847\pi$
$619$	−23723.2	−1.54041	−0.770206	+	0.637795i	$0.779847\pi$
$620$	3623.19	0.234695
$621$	−25574.0	−1.65257
$622$	11166.3	0.719823
$623$	−4867.80	−0.313040
$624$	5025.35	0.322396
$625$	−19496.4	−1.24777
$626$	−18968.9	−1.21110
$627$	−6409.73	−0.408262
$628$	−2581.18	−0.164013
$629$	17318.6	1.09783
$630$	9027.03	0.570866
$631$	12214.1	0.770578	0.385289	−	0.922796i	$-0.374102\pi$
$631$	12214.1	0.770578	0.385289	+	0.922796i	$0.374102\pi$
$632$	17721.2	1.11537
$633$	16779.5	1.05360
$634$	4645.37	0.290995
$635$	−20435.1	−1.27707
$636$	2416.12	0.150637
$637$	9802.69	0.609728
$638$	4297.43	0.266672
$639$	−189.842	−0.0117528
$640$	23128.8	1.42851
$641$	−16542.4	−1.01932	−0.509662	−	0.860375i	$-0.670229\pi$
$641$	−16542.4	−1.01932	−0.509662	+	0.860375i	$0.670229\pi$
$642$	−33230.0	−2.04281
$643$	6165.28	0.378126	0.189063	−	0.981965i	$-0.439455\pi$
$643$	6165.28	0.378126	0.189063	+	0.981965i	$0.439455\pi$
$644$	−10882.1	−0.665861
$645$	0	0
$646$	−15606.7	−0.950520
$647$	−1033.87	−0.0628217	−0.0314108	−	0.999507i	$-0.510000\pi$
$647$	−1033.87	−0.0628217	−0.0314108	+	0.999507i	$0.510000\pi$
$648$	−17264.8	−1.04664
$649$	−16968.2	−1.02629
$650$	−2058.71	−0.124230
$651$	35890.0	2.16074
$652$	−2771.72	−0.166486
$653$	10363.9	0.621085	0.310543	−	0.950559i	$-0.399489\pi$
$653$	10363.9	0.621085	0.310543	+	0.950559i	$0.399489\pi$
$654$	18655.8	1.11544
$655$	−18647.0	−1.11236
$656$	−57.1166	−0.00339944
$657$	479.470	0.0284717
$658$	15399.8	0.912384
$659$	27813.2	1.64408	0.822039	−	0.569431i	$-0.192837\pi$
$659$	27813.2	1.64408	0.822039	+	0.569431i	$0.192837\pi$
$660$	2801.26	0.165210
$661$	−12139.7	−0.714344	−0.357172	−	0.934039i	$-0.616259\pi$
$661$	−12139.7	−0.714344	−0.357172	+	0.934039i	$0.616259\pi$
$662$	−7307.52	−0.429026
$663$	−7569.05	−0.443375
$664$	14373.1	0.840039
$665$	−21083.1	−1.22942
$666$	−3053.56	−0.177662
$667$	12320.7	0.715229
$668$	437.144	0.0253198
$669$	−12398.1	−0.716502
$670$	−21482.8	−1.23873
$671$	2398.43	0.137988
$672$	−13100.0	−0.751996
$673$	14488.3	0.829842	0.414921	−	0.909857i	$-0.363809\pi$
$673$	14488.3	0.829842	0.414921	+	0.909857i	$0.363809\pi$
$674$	20421.4	1.16706
$675$	6748.43	0.384811
$676$	−3048.35	−0.173438
$677$	6394.53	0.363016	0.181508	−	0.983389i	$-0.441902\pi$
$677$	6394.53	0.363016	0.181508	+	0.983389i	$0.441902\pi$
$678$	13142.2	0.744431
$679$	−12059.2	−0.681574
$680$	−30007.1	−1.69224
$681$	−5042.65	−0.283752
$682$	13575.1	0.762196
$683$	24918.6	1.39602	0.698011	−	0.716087i	$-0.254069\pi$
$683$	24918.6	1.39602	0.698011	+	0.716087i	$0.254069\pi$
$684$	429.991	0.0240367
$685$	15908.2	0.887331
$686$	−51331.0	−2.85689
$687$	−20151.7	−1.11912
$688$	0	0
$689$	−3335.79	−0.184446
$690$	51395.2	2.83563
$691$	26390.3	1.45287	0.726434	−	0.687236i	$-0.241176\pi$
$691$	26390.3	1.45287	0.726434	+	0.687236i	$0.241176\pi$
$692$	−2394.12	−0.131519
$693$	5285.09	0.289703
$694$	13525.6	0.739807
$695$	39993.9	2.18281
$696$	6659.05	0.362659
$697$	86.0275	0.00467507
$698$	−3656.81	−0.198298
$699$	29638.9	1.60379
$700$	2871.55	0.155049
$701$	−16215.7	−0.873693	−0.436846	−	0.899536i	$-0.643905\pi$
$701$	−16215.7	−0.873693	−0.436846	+	0.899536i	$0.643905\pi$
$702$	−4337.67	−0.233212
$703$	7131.74	0.382615
$704$	9360.62	0.501125
$705$	−11365.3	−0.607154
$706$	18832.9	1.00394
$707$	37737.7	2.00746
$708$	5976.39	0.317241
$709$	−4007.85	−0.212296	−0.106148	−	0.994350i	$-0.533852\pi$
$709$	−4007.85	−0.212296	−0.106148	+	0.994350i	$0.533852\pi$
$710$	−1240.05	−0.0655471
$711$	5608.65	0.295838
$712$	−2853.02	−0.150171
$713$	38919.6	2.04425
$714$	67562.6	3.54127
$715$	−3867.54	−0.202290
$716$	3136.63	0.163717
$717$	−20200.4	−1.05216
$718$	2670.07	0.138783
$719$	−27845.1	−1.44429	−0.722147	−	0.691740i	$-0.756844\pi$
$719$	−27845.1	−1.44429	−0.722147	+	0.691740i	$0.756844\pi$
$720$	6305.43	0.326374
$721$	−31152.3	−1.60911
$722$	14693.6	0.757397
$723$	29846.2	1.53526
$724$	440.938	0.0226344
$725$	−3251.16	−0.166545
$726$	−13173.6	−0.673443
$727$	17757.5	0.905899	0.452949	−	0.891536i	$-0.350372\pi$
$727$	17757.5	0.905899	0.452949	+	0.891536i	$0.350372\pi$
$728$	8120.30	0.413405
$729$	13129.2	0.667035
$730$	3131.90	0.158790
$731$	0	0
$732$	−844.752	−0.0426543
$733$	37573.6	1.89333	0.946667	−	0.322212i	$-0.104427\pi$
$733$	37573.6	1.89333	0.946667	+	0.322212i	$0.104427\pi$
$734$	−24454.3	−1.22973
$735$	64576.8	3.24075
$736$	−14205.7	−0.711455
$737$	−12577.6	−0.628633
$738$	−15.1681	−0.000756564 0
$739$	5146.38	0.256174	0.128087	−	0.991763i	$-0.459116\pi$
$739$	5146.38	0.256174	0.128087	+	0.991763i	$0.459116\pi$
$740$	−3116.80	−0.154832
$741$	−3116.91	−0.154524
$742$	29775.9	1.47319
$743$	−14021.8	−0.692342	−0.346171	−	0.938171i	$-0.612518\pi$
$743$	−14021.8	−0.692342	−0.346171	+	0.938171i	$0.612518\pi$
$744$	21035.2	1.03654
$745$	−7660.12	−0.376705
$746$	−616.996	−0.0302813
$747$	4549.00	0.222810
$748$	3993.29	0.195200
$749$	−63993.1	−3.12184
$750$	16392.7	0.798102
$751$	7633.53	0.370907	0.185454	−	0.982653i	$-0.440625\pi$
$751$	7633.53	0.370907	0.185454	+	0.982653i	$0.440625\pi$
$752$	10756.9	0.521626
$753$	−11901.9	−0.576002
$754$	2089.74	0.100934
$755$	−23448.4	−1.13030
$756$	6050.31	0.291068
$757$	9584.30	0.460168	0.230084	−	0.973171i	$-0.426100\pi$
$757$	9584.30	0.460168	0.230084	+	0.973171i	$0.426100\pi$
$758$	−29458.7	−1.41160
$759$	30090.6	1.43902
$760$	−12356.8	−0.589776
$761$	12340.8	0.587849	0.293925	−	0.955829i	$-0.405039\pi$
$761$	12340.8	0.587849	0.293925	+	0.955829i	$0.405039\pi$
$762$	26966.9	1.28203
$763$	35926.7	1.70463
$764$	−6729.87	−0.318689
$765$	−9497.05	−0.448845
$766$	−16546.1	−0.780461
$767$	−8251.26	−0.388443
$768$	−12720.0	−0.597648
$769$	−6270.07	−0.294024	−0.147012	−	0.989135i	$-0.546966\pi$
$769$	−6270.07	−0.294024	−0.147012	+	0.989135i	$0.546966\pi$
$770$	34522.3	1.61571
$771$	13021.2	0.608234
$772$	4834.47	0.225384
$773$	−29116.0	−1.35476	−0.677379	−	0.735634i	$-0.736884\pi$
$773$	−29116.0	−1.35476	−0.677379	+	0.735634i	$0.736884\pi$
$774$	0	0
$775$	−10270.1	−0.476015
$776$	−7067.90	−0.326962
$777$	−30873.9	−1.42548
$778$	−23940.4	−1.10322
$779$	35.4258	0.00162935
$780$	1362.19	0.0625310
$781$	−726.020	−0.0332638
$782$	73265.7	3.35036
$783$	−6850.13	−0.312649
$784$	−61119.6	−2.78424
$785$	23476.4	1.06740
$786$	24607.3	1.11668
$787$	28536.0	1.29250	0.646250	−	0.763126i	$-0.276336\pi$
$787$	28536.0	1.29250	0.646250	+	0.763126i	$0.276336\pi$
$788$	−1286.36	−0.0581531
$789$	37431.0	1.68895
$790$	36635.8	1.64993
$791$	25308.8	1.13764
$792$	3097.60	0.138975
$793$	1166.30	0.0522277
$794$	−3000.59	−0.134114
$795$	−21975.1	−0.980347
$796$	6245.84	0.278113
$797$	−16273.3	−0.723248	−0.361624	−	0.932324i	$-0.617778\pi$
$797$	−16273.3	−0.723248	−0.361624	+	0.932324i	$0.617778\pi$
$798$	27822.0	1.23420
$799$	−16201.7	−0.717365
$800$	3748.60	0.165666
$801$	−902.962	−0.0398310
$802$	−10070.1	−0.443375
$803$	1833.65	0.0805829
$804$	4429.97	0.194320
$805$	98974.9	4.33342
$806$	6601.26	0.288486
$807$	−17158.4	−0.748454
$808$	22118.1	0.963012
$809$	−22873.9	−0.994073	−0.497037	−	0.867730i	$-0.665579\pi$
$809$	−22873.9	−0.994073	−0.497037	+	0.867730i	$0.665579\pi$
$810$	−35692.2	−1.54827
$811$	1107.97	0.0479728	0.0239864	−	0.999712i	$-0.492364\pi$
$811$	1107.97	0.0479728	0.0239864	+	0.999712i	$0.492364\pi$
$812$	−2914.83	−0.125974
$813$	33138.0	1.42952
$814$	−11677.8	−0.502834
$815$	25209.4	1.08349
$816$	47192.8	2.02461
$817$	0	0
$818$	15731.4	0.672414
$819$	2570.02	0.109651
$820$	−15.4822	−0.000659345 0
$821$	−24233.1	−1.03014	−0.515068	−	0.857149i	$-0.672233\pi$
$821$	−24233.1	−1.03014	−0.515068	+	0.857149i	$0.672233\pi$
$822$	−20993.1	−0.890776
$823$	−15663.8	−0.663435	−0.331717	−	0.943379i	$-0.607628\pi$
$823$	−15663.8	−0.663435	−0.331717	+	0.943379i	$0.607628\pi$
$824$	−18258.4	−0.771920
$825$	−7940.27	−0.335084
$826$	73652.2	3.10253
$827$	−13112.5	−0.551349	−0.275674	−	0.961251i	$-0.588901\pi$
$827$	−13112.5	−0.551349	−0.275674	+	0.961251i	$0.588901\pi$
$828$	−2018.60	−0.0847236
$829$	−23726.4	−0.994030	−0.497015	−	0.867742i	$-0.665571\pi$
$829$	−23726.4	−0.994030	−0.497015	+	0.867742i	$0.665571\pi$
$830$	29714.1	1.24264
$831$	−9191.47	−0.383692
$832$	4551.86	0.189672
$833$	92056.6	3.82902
$834$	−52777.5	−2.19129
$835$	−3975.92	−0.164781
$836$	1644.42	0.0680306
$837$	−21638.8	−0.893604
$838$	−14197.5	−0.585258
$839$	−10199.7	−0.419704	−0.209852	−	0.977733i	$-0.567298\pi$
$839$	−10199.7	−0.419704	−0.209852	+	0.977733i	$0.567298\pi$
$840$	53493.8	2.19728
$841$	−21088.8	−0.864687
$842$	−25034.1	−1.02462
$843$	7390.21	0.301936
$844$	−4304.81	−0.175566
$845$	27725.4	1.12874
$846$	2856.63	0.116091
$847$	−25369.3	−1.02916
$848$	20798.6	0.842249
$849$	−48603.9	−1.96476
$850$	−19333.3	−0.780148
$851$	−33480.1	−1.34863
$852$	255.712	0.0102823
$853$	23124.5	0.928214	0.464107	−	0.885779i	$-0.346375\pi$
$853$	23124.5	0.928214	0.464107	+	0.885779i	$0.346375\pi$
$854$	−10410.6	−0.417147
$855$	−3910.85	−0.156431
$856$	−37506.5	−1.49760
$857$	−2642.41	−0.105324	−0.0526621	−	0.998612i	$-0.516771\pi$
$857$	−2642.41	−0.105324	−0.0526621	+	0.998612i	$0.516771\pi$
$858$	5103.75	0.203076
$859$	−20573.1	−0.817166	−0.408583	−	0.912721i	$-0.633977\pi$
$859$	−20573.1	−0.817166	−0.408583	+	0.912721i	$0.633977\pi$
$860$	0	0
$861$	−153.361	−0.00607032
$862$	31346.4	1.23859
$863$	20572.4	0.811465	0.405732	−	0.913992i	$-0.367017\pi$
$863$	20572.4	0.811465	0.405732	+	0.913992i	$0.367017\pi$
$864$	7898.22	0.310999
$865$	21775.0	0.855923
$866$	−11731.6	−0.460340
$867$	−42707.1	−1.67291
$868$	−9207.63	−0.360055
$869$	21449.3	0.837305
$870$	13766.5	0.536469
$871$	−6116.20	−0.237933
$872$	21056.7	0.817740
$873$	−2236.94	−0.0867228
$874$	30170.6	1.16766
$875$	31568.4	1.21966
$876$	−645.831	−0.0249094
$877$	−50009.6	−1.92555	−0.962773	−	0.270313i	$-0.912873\pi$
$877$	−50009.6	−1.92555	−0.962773	+	0.270313i	$0.912873\pi$
$878$	42509.8	1.63398
$879$	21477.7	0.824147
$880$	24114.0	0.923731
$881$	32839.4	1.25583	0.627915	−	0.778282i	$-0.283909\pi$
$881$	32839.4	1.25583	0.627915	+	0.778282i	$0.283909\pi$
$882$	−16231.1	−0.619648
$883$	5095.20	0.194187	0.0970935	−	0.995275i	$-0.469045\pi$
$883$	5095.20	0.194187	0.0970935	+	0.995275i	$0.469045\pi$
$884$	1941.85	0.0738817
$885$	−54356.5	−2.06460
$886$	10134.4	0.384281
$887$	−41889.4	−1.58569	−0.792845	−	0.609423i	$-0.791401\pi$
$887$	−41889.4	−1.58569	−0.792845	+	0.609423i	$0.791401\pi$
$888$	−18095.3	−0.683826
$889$	51931.8	1.95921
$890$	−5898.16	−0.222143
$891$	−20896.9	−0.785714
$892$	3180.76	0.119394
$893$	−6671.80	−0.250015
$894$	10108.6	0.378167
$895$	−28528.3	−1.06547
$896$	−58777.3	−2.19153
$897$	14632.4	0.544660
$898$	53904.1	2.00312
$899$	10424.8	0.386749
$900$	532.665	0.0197283
$901$	−31326.3	−1.15830
$902$	−58.0077	−0.00214129
$903$	0	0
$904$	14833.5	0.545747
$905$	−4010.42	−0.147305
$906$	30943.4	1.13469
$907$	−7205.23	−0.263777	−0.131889	−	0.991265i	$-0.542104\pi$
$907$	−7205.23	−0.263777	−0.131889	+	0.991265i	$0.542104\pi$
$908$	1293.70	0.0472829
$909$	7000.24	0.255427
$910$	16787.4	0.611536
$911$	−11977.2	−0.435591	−0.217796	−	0.975994i	$-0.569887\pi$
$911$	−11977.2	−0.435591	−0.217796	+	0.975994i	$0.569887\pi$
$912$	19433.8	0.705613
$913$	17396.9	0.630615
$914$	18552.1	0.671389
$915$	7683.19	0.277594
$916$	5169.94	0.186484
$917$	47387.7	1.70652
$918$	−40734.9	−1.46454
$919$	−54968.3	−1.97305	−0.986527	−	0.163597i	$-0.947690\pi$
$919$	−54968.3	−1.97305	−0.986527	+	0.163597i	$0.947690\pi$
$920$	58009.3	2.07882
$921$	−36996.0	−1.32363
$922$	−29698.5	−1.06081
$923$	−353.047	−0.0125901
$924$	−7118.86	−0.253456
$925$	8834.68	0.314035
$926$	−26413.9	−0.937382
$927$	−5778.66	−0.204742
$928$	−3805.09	−0.134599
$929$	49645.6	1.75330	0.876652	−	0.481125i	$-0.159772\pi$
$929$	49645.6	1.75330	0.876652	+	0.481125i	$0.159772\pi$
$930$	43486.9	1.53332
$931$	37908.6	1.33448
$932$	−7603.90	−0.267247
$933$	20942.8	0.734872
$934$	11845.9	0.415000
$935$	−36319.8	−1.27036
$936$	1506.29	0.0526012
$937$	4460.11	0.155502	0.0777510	−	0.996973i	$-0.475226\pi$
$937$	4460.11	0.155502	0.0777510	+	0.996973i	$0.475226\pi$
$938$	54594.3	1.90039
$939$	−35576.7	−1.23642
$940$	2915.79	0.101173
$941$	−41928.2	−1.45252	−0.726259	−	0.687421i	$-0.758743\pi$
$941$	−41928.2	−1.45252	−0.726259	+	0.687421i	$0.758743\pi$
$942$	−30980.3	−1.07154
$943$	−166.307	−0.00574305
$944$	51446.4	1.77377
$945$	−55028.8	−1.89427
$946$	0	0
$947$	37529.0	1.28778	0.643890	−	0.765118i	$-0.277320\pi$
$947$	37529.0	1.28778	0.643890	+	0.765118i	$0.277320\pi$
$948$	−7554.68	−0.258823
$949$	891.661	0.0305001
$950$	−7961.37	−0.271896
$951$	8712.50	0.297079
$952$	76257.3	2.59613
$953$	−6351.12	−0.215879	−0.107940	−	0.994157i	$-0.534425\pi$
$953$	−6351.12	−0.215879	−0.107940	+	0.994157i	$0.534425\pi$
$954$	5523.34	0.187447
$955$	61209.6	2.07403
$956$	5182.44	0.175327
$957$	8059.94	0.272247
$958$	11049.6	0.372646
$959$	−40427.6	−1.36129
$960$	29986.1	1.00812
$961$	3139.90	0.105398
$962$	−5678.65	−0.190319
$963$	−11870.5	−0.397220
$964$	−7657.09	−0.255828
$965$	−43970.5	−1.46680
$966$	−130611.	−4.35025
$967$	−15622.6	−0.519535	−0.259768	−	0.965671i	$-0.583646\pi$
$967$	−15622.6	−0.519535	−0.259768	+	0.965671i	$0.583646\pi$
$968$	−14869.0	−0.493706
$969$	−29270.7	−0.970392
$970$	−14611.7	−0.483665
$971$	5920.86	0.195684	0.0978422	−	0.995202i	$-0.468806\pi$
$971$	5920.86	0.195684	0.0978422	+	0.995202i	$0.468806\pi$
$972$	2589.93	0.0854650
$973$	−101637.	−3.34874
$974$	43547.8	1.43261
$975$	−3861.17	−0.126827
$976$	−7271.86	−0.238490
$977$	−36549.5	−1.19685	−0.598424	−	0.801179i	$-0.704206\pi$
$977$	−36549.5	−1.19685	−0.598424	+	0.801179i	$0.704206\pi$
$978$	−33267.3	−1.08770
$979$	−3453.22	−0.112733
$980$	−16567.3	−0.540022
$981$	6664.29	0.216896
$982$	23144.5	0.752108
$983$	−52748.6	−1.71152	−0.855758	−	0.517377i	$-0.826909\pi$
$983$	−52748.6	−1.71152	−0.855758	+	0.517377i	$0.826909\pi$
$984$	−89.8854	−0.00291203
$985$	11699.7	0.378461
$986$	19624.7	0.633850
$987$	28882.8	0.931459
$988$	799.646	0.0257491
$989$	0	0
$990$	6403.79	0.205582
$991$	−56108.6	−1.79854	−0.899268	−	0.437397i	$-0.855900\pi$
$991$	−56108.6	−1.79854	−0.899268	+	0.437397i	$0.855900\pi$
$992$	−12019.9	−0.384709
$993$	−13705.4	−0.437995
$994$	3151.36	0.100558
$995$	−56807.2	−1.80996
$996$	−6127.36	−0.194933
$997$	−23518.2	−0.747071	−0.373536	−	0.927616i	$-0.621855\pi$
$997$	−23518.2	−0.747071	−0.373536	+	0.927616i	$0.621855\pi$
$998$	22948.3	0.727872
$999$	18614.5	0.589526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.12	yes	50
43.42	odd	2		1849.4.a.i.1.39	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.39	✓	50	43.42	odd	2
1849.4.a.j.1.12	yes	50	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-3.07922 q^{2} -5.77516 q^{3} +1.48163 q^{4} -13.4757 q^{5} +17.7830 q^{6} +34.2458 q^{7} +20.0715 q^{8} +6.35250 q^{9} +O(q^{10})\)	\(q-3.07922 q^{2} -5.77516 q^{3} +1.48163 q^{4} -13.4757 q^{5} +17.7830 q^{6} +34.2458 q^{7} +20.0715 q^{8} +6.35250 q^{9} +41.4947 q^{10} +24.2940 q^{11} -8.55663 q^{12} +11.8136 q^{13} -105.451 q^{14} +77.8243 q^{15} -73.6578 q^{16} +110.941 q^{17} -19.5608 q^{18} +45.6852 q^{19} -19.9659 q^{20} -197.775 q^{21} -74.8068 q^{22} -214.470 q^{23} -115.916 q^{24} +56.5941 q^{25} -36.3769 q^{26} +119.243 q^{27} +50.7395 q^{28} -57.4470 q^{29} -239.638 q^{30} -181.469 q^{31} +66.2366 q^{32} -140.302 q^{33} -341.613 q^{34} -461.486 q^{35} +9.41203 q^{36} +156.106 q^{37} -140.675 q^{38} -68.2257 q^{39} -270.478 q^{40} +0.775433 q^{41} +608.995 q^{42} +35.9947 q^{44} -85.6043 q^{45} +660.401 q^{46} -146.038 q^{47} +425.386 q^{48} +829.777 q^{49} -174.266 q^{50} -640.704 q^{51} +17.5034 q^{52} -282.368 q^{53} -367.175 q^{54} -327.379 q^{55} +687.367 q^{56} -263.840 q^{57} +176.892 q^{58} -698.452 q^{59} +115.306 q^{60} +98.7249 q^{61} +558.783 q^{62} +217.547 q^{63} +385.305 q^{64} -159.197 q^{65} +432.022 q^{66} -517.724 q^{67} +164.373 q^{68} +1238.60 q^{69} +1421.02 q^{70} -29.8847 q^{71} +127.505 q^{72} +75.4773 q^{73} -480.686 q^{74} -326.840 q^{75} +67.6884 q^{76} +831.970 q^{77} +210.082 q^{78} +882.904 q^{79} +992.589 q^{80} -860.163 q^{81} -2.38773 q^{82} +716.095 q^{83} -293.029 q^{84} -1495.01 q^{85} +331.766 q^{87} +487.619 q^{88} -142.143 q^{89} +263.595 q^{90} +404.568 q^{91} -317.764 q^{92} +1048.01 q^{93} +449.685 q^{94} -615.640 q^{95} -382.527 q^{96} -352.135 q^{97} -2555.07 q^{98} +154.328 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * q^99

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