LMFDB - Embedded newform 1849.4.a.j.1.25

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			1.25
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)

$f(q)$	$=$	$q+0.213850 q^{2} -7.23549 q^{3} -7.95427 q^{4} -3.30603 q^{5} -1.54731 q^{6} -30.5623 q^{7} -3.41181 q^{8} +25.3524 q^{9} +O(q^{10})$	\(q+0.213850 q^{2} -7.23549 q^{3} -7.95427 q^{4} -3.30603 q^{5} -1.54731 q^{6} -30.5623 q^{7} -3.41181 q^{8} +25.3524 q^{9} -0.706992 q^{10} -66.6639 q^{11} +57.5531 q^{12} +32.7495 q^{13} -6.53574 q^{14} +23.9207 q^{15} +62.9045 q^{16} -58.7055 q^{17} +5.42159 q^{18} -53.7524 q^{19} +26.2970 q^{20} +221.134 q^{21} -14.2560 q^{22} -169.539 q^{23} +24.6862 q^{24} -114.070 q^{25} +7.00348 q^{26} +11.9214 q^{27} +243.101 q^{28} +67.9267 q^{29} +5.11544 q^{30} -2.72881 q^{31} +40.7466 q^{32} +482.346 q^{33} -12.5541 q^{34} +101.040 q^{35} -201.660 q^{36} +335.946 q^{37} -11.4949 q^{38} -236.959 q^{39} +11.2795 q^{40} +113.357 q^{41} +47.2893 q^{42} +530.262 q^{44} -83.8156 q^{45} -36.2558 q^{46} -24.6905 q^{47} -455.145 q^{48} +591.057 q^{49} -24.3939 q^{50} +424.763 q^{51} -260.499 q^{52} -597.346 q^{53} +2.54938 q^{54} +220.393 q^{55} +104.273 q^{56} +388.925 q^{57} +14.5261 q^{58} +206.000 q^{59} -190.272 q^{60} +392.622 q^{61} -0.583554 q^{62} -774.828 q^{63} -494.523 q^{64} -108.271 q^{65} +103.150 q^{66} +607.078 q^{67} +466.959 q^{68} +1226.70 q^{69} +21.6073 q^{70} -98.1848 q^{71} -86.4976 q^{72} +874.055 q^{73} +71.8418 q^{74} +825.354 q^{75} +427.561 q^{76} +2037.40 q^{77} -50.6736 q^{78} +788.085 q^{79} -207.964 q^{80} -770.771 q^{81} +24.2414 q^{82} -495.384 q^{83} -1758.96 q^{84} +194.082 q^{85} -491.483 q^{87} +227.445 q^{88} -528.746 q^{89} -17.9239 q^{90} -1000.90 q^{91} +1348.56 q^{92} +19.7443 q^{93} -5.28004 q^{94} +177.707 q^{95} -294.822 q^{96} +2.16686 q^{97} +126.397 q^{98} -1690.09 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.213850	0.0756072	0.0378036	−	0.999285i	$-0.487964\pi$
$2$	0.213850	0.0756072	0.0378036	+	0.999285i	$0.487964\pi$
$3$	−7.23549	−1.39247	−0.696236	−	0.717813i	$-0.745143\pi$
$3$	−7.23549	−1.39247	−0.696236	+	0.717813i	$0.745143\pi$
$4$	−7.95427	−0.994284
$5$	−3.30603	−0.295700	−0.147850	−	0.989010i	$-0.547235\pi$
$5$	−3.30603	−0.295700	−0.147850	+	0.989010i	$0.547235\pi$
$6$	−1.54731	−0.105281
$7$	−30.5623	−1.65021	−0.825106	−	0.564979i	$-0.808884\pi$
$7$	−30.5623	−1.65021	−0.825106	+	0.564979i	$0.808884\pi$
$8$	−3.41181	−0.150782
$9$	25.3524	0.938977
$10$	−0.706992	−0.0223571
$11$	−66.6639	−1.82727	−0.913633	−	0.406541i	$-0.866735\pi$
$11$	−66.6639	−1.82727	−0.913633	+	0.406541i	$0.866735\pi$
$12$	57.5531	1.38451
$13$	32.7495	0.698699	0.349350	−	0.936992i	$-0.386403\pi$
$13$	32.7495	0.698699	0.349350	+	0.936992i	$0.386403\pi$
$14$	−6.53574	−0.124768
$15$	23.9207	0.411754
$16$	62.9045	0.982883
$17$	−58.7055	−0.837540	−0.418770	−	0.908092i	$-0.637539\pi$
$17$	−58.7055	−0.837540	−0.418770	+	0.908092i	$0.637539\pi$
$18$	5.42159	0.0709934
$19$	−53.7524	−0.649034	−0.324517	−	0.945880i	$-0.605202\pi$
$19$	−53.7524	−0.649034	−0.324517	+	0.945880i	$0.605202\pi$
$20$	26.2970	0.294010
$21$	221.134	2.29787
$22$	−14.2560	−0.138154
$23$	−169.539	−1.53701	−0.768505	−	0.639843i	$-0.778999\pi$
$23$	−169.539	−1.53701	−0.768505	+	0.639843i	$0.778999\pi$
$24$	24.6862	0.209960
$25$	−114.070	−0.912562
$26$	7.00348	0.0528267
$27$	11.9214	0.0849731
$28$	243.101	1.64078
$29$	67.9267	0.434954	0.217477	−	0.976065i	$-0.430217\pi$
$29$	67.9267	0.434954	0.217477	+	0.976065i	$0.430217\pi$
$30$	5.11544	0.0311316
$31$	−2.72881	−0.0158099	−0.00790497	−	0.999969i	$-0.502516\pi$
$31$	−2.72881	−0.0158099	−0.00790497	+	0.999969i	$0.502516\pi$
$32$	40.7466	0.225095
$33$	482.346	2.54442
$34$	−12.5541	−0.0633241
$35$	101.040	0.487967
$36$	−201.660	−0.933609
$37$	335.946	1.49268	0.746339	−	0.665566i	$-0.231810\pi$
$37$	335.946	1.49268	0.746339	+	0.665566i	$0.231810\pi$
$38$	−11.4949	−0.0490717
$39$	−236.959	−0.972919
$40$	11.2795	0.0445863
$41$	113.357	0.431791	0.215896	−	0.976416i	$-0.430733\pi$
$41$	113.357	0.431791	0.215896	+	0.976416i	$0.430733\pi$
$42$	47.2893	0.173736
$43$	0	0
$43$	0	0
$44$	530.262	1.81682
$45$	−83.8156	−0.277655
$46$	−36.2558	−0.116209
$47$	−24.6905	−0.0766271	−0.0383135	−	0.999266i	$-0.512199\pi$
$47$	−24.6905	−0.0766271	−0.0383135	+	0.999266i	$0.512199\pi$
$48$	−455.145	−1.36864
$49$	591.057	1.72320
$50$	−24.3939	−0.0689963
$51$	424.763	1.16625
$52$	−260.499	−0.694705
$53$	−597.346	−1.54815	−0.774073	−	0.633096i	$-0.781784\pi$
$53$	−597.346	−1.54815	−0.774073	+	0.633096i	$0.781784\pi$
$54$	2.54938	0.00642458
$55$	220.393	0.540322
$56$	104.273	0.248823
$57$	388.925	0.903761
$58$	14.5261	0.0328857
$59$	206.000	0.454557	0.227278	−	0.973830i	$-0.427017\pi$
$59$	206.000	0.454557	0.227278	+	0.973830i	$0.427017\pi$
$60$	−190.272	−0.409400
$61$	392.622	0.824100	0.412050	−	0.911161i	$-0.364813\pi$
$61$	392.622	0.824100	0.412050	+	0.911161i	$0.364813\pi$
$62$	−0.583554	−0.00119535
$63$	−774.828	−1.54951
$64$	−494.523	−0.965864
$65$	−108.271	−0.206605
$66$	103.150	0.192376
$67$	607.078	1.10696	0.553481	−	0.832862i	$-0.313299\pi$
$67$	607.078	1.10696	0.553481	+	0.832862i	$0.313299\pi$
$68$	466.959	0.832752
$69$	1226.70	2.14024
$70$	21.6073	0.0368939
$71$	−98.1848	−0.164118	−0.0820591	−	0.996627i	$-0.526150\pi$
$71$	−98.1848	−0.164118	−0.0820591	+	0.996627i	$0.526150\pi$
$72$	−86.4976	−0.141581
$73$	874.055	1.40138	0.700688	−	0.713468i	$-0.252877\pi$
$73$	874.055	1.40138	0.700688	+	0.713468i	$0.252877\pi$
$74$	71.8418	0.112857
$75$	825.354	1.27072
$76$	427.561	0.645324
$77$	2037.40	3.01537
$78$	−50.6736	−0.0735597
$79$	788.085	1.12236	0.561180	−	0.827694i	$-0.310348\pi$
$79$	788.085	1.12236	0.561180	+	0.827694i	$0.310348\pi$
$80$	−207.964	−0.290639
$81$	−770.771	−1.05730
$82$	24.2414	0.0326466
$83$	−495.384	−0.655126	−0.327563	−	0.944829i	$-0.606227\pi$
$83$	−495.384	−0.655126	−0.327563	+	0.944829i	$0.606227\pi$
$84$	−1758.96	−2.28474
$85$	194.082	0.247660
$86$	0	0
$87$	−491.483	−0.605661
$88$	227.445	0.275519
$89$	−528.746	−0.629741	−0.314870	−	0.949135i	$-0.601961\pi$
$89$	−528.746	−0.629741	−0.314870	+	0.949135i	$0.601961\pi$
$90$	−17.9239	−0.0209928
$91$	−1000.90	−1.15300
$92$	1348.56	1.52822
$93$	19.7443	0.0220149
$94$	−5.28004	−0.00579356
$95$	177.707	0.191919
$96$	−294.822	−0.313439
$97$	2.16686	0.00226815	0.00113408	−	0.999999i	$-0.499639\pi$
$97$	2.16686	0.00226815	0.00113408	+	0.999999i	$0.499639\pi$
$98$	126.397	0.130286
$99$	−1690.09	−1.71576
$100$	907.345	0.907345
$101$	−748.226	−0.737141	−0.368570	−	0.929600i	$-0.620153\pi$
$101$	−748.226	−0.737141	−0.368570	+	0.929600i	$0.620153\pi$
$102$	90.8355	0.0881770
$103$	1963.62	1.87846	0.939231	−	0.343285i	$-0.111540\pi$
$103$	1963.62	1.87846	0.939231	+	0.343285i	$0.111540\pi$
$104$	−111.735	−0.105351
$105$	−731.073	−0.679481
$106$	−127.742	−0.117051
$107$	−1394.58	−1.26000	−0.629998	−	0.776597i	$-0.716944\pi$
$107$	−1394.58	−1.26000	−0.629998	+	0.776597i	$0.716944\pi$
$108$	−94.8260	−0.0844873
$109$	−518.030	−0.455214	−0.227607	−	0.973753i	$-0.573090\pi$
$109$	−518.030	−0.455214	−0.227607	+	0.973753i	$0.573090\pi$
$110$	47.1308	0.0408523
$111$	−2430.73	−2.07851
$112$	−1922.51	−1.62197
$113$	383.171	0.318988	0.159494	−	0.987199i	$-0.449014\pi$
$113$	383.171	0.318988	0.159494	+	0.987199i	$0.449014\pi$
$114$	83.1714	0.0683309
$115$	560.499	0.454494
$116$	−540.307	−0.432468
$117$	830.279	0.656062
$118$	44.0529	0.0343678
$119$	1794.18	1.38212
$120$	−81.6131	−0.0620852
$121$	3113.07	2.33890
$122$	83.9620	0.0623079
$123$	−820.197	−0.601257
$124$	21.7057	0.0157196
$125$	790.372	0.565544
$126$	−165.697	−0.117154
$127$	60.9317	0.0425733	0.0212867	−	0.999773i	$-0.493224\pi$
$127$	60.9317	0.0425733	0.0212867	+	0.999773i	$0.493224\pi$
$128$	−431.726	−0.298122
$129$	0	0
$130$	−23.1537	−0.0156209
$131$	−1069.06	−0.713008	−0.356504	−	0.934294i	$-0.616031\pi$
$131$	−1069.06	−0.713008	−0.356504	+	0.934294i	$0.616031\pi$
$132$	−3836.71	−2.52987
$133$	1642.80	1.07104
$134$	129.823	0.0836943
$135$	−39.4124	−0.0251265
$136$	200.292	0.126286
$137$	−1551.04	−0.967260	−0.483630	−	0.875272i	$-0.660682\pi$
$137$	−1551.04	−0.967260	−0.483630	+	0.875272i	$0.660682\pi$
$138$	262.328	0.161818
$139$	−818.479	−0.499442	−0.249721	−	0.968318i	$-0.580339\pi$
$139$	−818.479	−0.499442	−0.249721	+	0.968318i	$0.580339\pi$
$140$	−803.698	−0.485178
$141$	178.648	0.106701
$142$	−20.9968	−0.0124085
$143$	−2183.21	−1.27671
$144$	1594.78	0.922905
$145$	−224.567	−0.128616
$146$	186.916	0.105954
$147$	−4276.59	−2.39950
$148$	−2672.20	−1.48415
$149$	593.512	0.326325	0.163163	−	0.986599i	$-0.447831\pi$
$149$	593.512	0.326325	0.163163	+	0.986599i	$0.447831\pi$
$150$	176.502	0.0960753
$151$	−1565.64	−0.843772	−0.421886	−	0.906649i	$-0.638632\pi$
$151$	−1565.64	−0.843772	−0.421886	+	0.906649i	$0.638632\pi$
$152$	183.393	0.0978628
$153$	−1488.32	−0.786431
$154$	435.698	0.227984
$155$	9.02150	0.00467500
$156$	1884.84	0.967357
$157$	2808.47	1.42765	0.713823	−	0.700326i	$-0.246962\pi$
$157$	2808.47	1.42765	0.713823	+	0.700326i	$0.246962\pi$
$158$	168.532	0.0848586
$159$	4322.09	2.15575
$160$	−134.709	−0.0665607
$161$	5181.50	2.53639
$162$	−164.829	−0.0799395
$163$	−1654.13	−0.794856	−0.397428	−	0.917633i	$-0.630097\pi$
$163$	−1654.13	−0.794856	−0.397428	+	0.917633i	$0.630097\pi$
$164$	−901.675	−0.429323
$165$	−1594.65	−0.752383
$166$	−105.938	−0.0495323
$167$	−820.480	−0.380183	−0.190092	−	0.981766i	$-0.560879\pi$
$167$	−820.480	−0.380183	−0.190092	+	0.981766i	$0.560879\pi$
$168$	−754.467	−0.346478
$169$	−1124.47	−0.511819
$170$	41.5043	0.0187249
$171$	−1362.75	−0.609428
$172$	0	0
$173$	−3090.35	−1.35812	−0.679060	−	0.734083i	$-0.737612\pi$
$173$	−3090.35	−1.35812	−0.679060	+	0.734083i	$0.737612\pi$
$174$	−105.103	−0.0457924
$175$	3486.25	1.50592
$176$	−4193.46	−1.79599
$177$	−1490.51	−0.632958
$178$	−113.072	−0.0476130
$179$	−443.168	−0.185050	−0.0925249	−	0.995710i	$-0.529494\pi$
$179$	−443.168	−0.185050	−0.0925249	+	0.995710i	$0.529494\pi$
$180$	666.692	0.276068
$181$	2436.76	1.00068	0.500339	−	0.865830i	$-0.333209\pi$
$181$	2436.76	1.00068	0.500339	+	0.865830i	$0.333209\pi$
$182$	−214.043	−0.0871753
$183$	−2840.81	−1.14754
$184$	578.434	0.231754
$185$	−1110.64	−0.441385
$186$	4.22230	0.00166448
$187$	3913.54	1.53041
$188$	196.395	0.0761891
$189$	−364.346	−0.140224
$190$	38.0025	0.0145105
$191$	3887.50	1.47272	0.736360	−	0.676590i	$-0.236543\pi$
$191$	3887.50	1.47272	0.736360	+	0.676590i	$0.236543\pi$
$192$	3578.12	1.34494
$193$	−984.557	−0.367202	−0.183601	−	0.983001i	$-0.558775\pi$
$193$	−984.557	−0.367202	−0.183601	+	0.983001i	$0.558775\pi$
$194$	0.463381	0.000171489 0
$195$	783.393	0.287692
$196$	−4701.42	−1.71335
$197$	4103.62	1.48412	0.742058	−	0.670336i	$-0.233850\pi$
$197$	4103.62	1.48412	0.742058	+	0.670336i	$0.233850\pi$
$198$	−361.425	−0.129724
$199$	−1387.37	−0.494212	−0.247106	−	0.968988i	$-0.579480\pi$
$199$	−1387.37	−0.494212	−0.247106	+	0.968988i	$0.579480\pi$
$200$	389.186	0.137598
$201$	−4392.51	−1.54141
$202$	−160.008	−0.0557332
$203$	−2076.00	−0.717766
$204$	−3378.68	−1.15958
$205$	−374.762	−0.127681
$206$	419.920	0.142025
$207$	−4298.21	−1.44322
$208$	2060.10	0.686740
$209$	3583.34	1.18596
$210$	−156.340	−0.0513736
$211$	−1794.07	−0.585349	−0.292675	−	0.956212i	$-0.594545\pi$
$211$	−1794.07	−0.585349	−0.292675	+	0.956212i	$0.594545\pi$
$212$	4751.45	1.53930
$213$	710.416	0.228530
$214$	−298.231	−0.0952648
$215$	0	0
$216$	−40.6736	−0.0128124
$217$	83.3987	0.0260897
$218$	−110.780	−0.0344174
$219$	−6324.22	−1.95138
$220$	−1753.06	−0.537233
$221$	−1922.58	−0.585189
$222$	−519.811	−0.157151
$223$	3494.65	1.04941	0.524706	−	0.851284i	$-0.324175\pi$
$223$	3494.65	1.04941	0.524706	+	0.851284i	$0.324175\pi$
$224$	−1245.31	−0.371455
$225$	−2891.95	−0.856874
$226$	81.9409	0.0241178
$227$	−5846.94	−1.70958	−0.854790	−	0.518974i	$-0.826314\pi$
$227$	−5846.94	−1.70958	−0.854790	+	0.518974i	$0.826314\pi$
$228$	−3093.61	−0.898595
$229$	243.067	0.0701410	0.0350705	−	0.999385i	$-0.488834\pi$
$229$	243.067	0.0701410	0.0350705	+	0.999385i	$0.488834\pi$
$230$	119.862	0.0343630
$231$	−14741.6	−4.19882
$232$	−231.753	−0.0655834
$233$	−459.249	−0.129126	−0.0645630	−	0.997914i	$-0.520565\pi$
$233$	−459.249	−0.129126	−0.0645630	+	0.997914i	$0.520565\pi$
$234$	177.555	0.0496031
$235$	81.6273	0.0226586
$236$	−1638.58	−0.451958
$237$	−5702.18	−1.56285
$238$	383.684	0.104498
$239$	−5216.79	−1.41191	−0.705954	−	0.708258i	$-0.749481\pi$
$239$	−5216.79	−1.41191	−0.705954	+	0.708258i	$0.749481\pi$
$240$	1504.72	0.404706
$241$	4016.94	1.07367	0.536833	−	0.843688i	$-0.319620\pi$
$241$	4016.94	1.07367	0.536833	+	0.843688i	$0.319620\pi$
$242$	665.730	0.176838
$243$	5255.03	1.38729
$244$	−3123.02	−0.819389
$245$	−1954.05	−0.509549
$246$	−175.399	−0.0454594
$247$	−1760.37	−0.453479
$248$	9.31018	0.00238386
$249$	3584.35	0.912245
$250$	169.021	0.0427592
$251$	−6018.47	−1.51348	−0.756739	−	0.653718i	$-0.773208\pi$
$251$	−6018.47	−1.51348	−0.756739	+	0.653718i	$0.773208\pi$
$252$	6163.19	1.54065
$253$	11302.1	2.80853
$254$	13.0302	0.00321885
$255$	−1404.28	−0.344860
$256$	3863.86	0.943324
$257$	7266.87	1.76379	0.881897	−	0.471442i	$-0.156266\pi$
$257$	7266.87	1.76379	0.881897	+	0.471442i	$0.156266\pi$
$258$	0	0
$259$	−10267.3	−2.46324
$260$	861.215	0.205424
$261$	1722.10	0.408412
$262$	−228.618	−0.0539086
$263$	2259.66	0.529796	0.264898	−	0.964276i	$-0.414662\pi$
$263$	2259.66	0.529796	0.264898	+	0.964276i	$0.414662\pi$
$264$	−1645.68	−0.383653
$265$	1974.84	0.457787
$266$	351.312	0.0809786
$267$	3825.74	0.876896
$268$	−4828.86	−1.10063
$269$	5104.54	1.15699	0.578493	−	0.815687i	$-0.303641\pi$
$269$	5104.54	1.15699	0.578493	+	0.815687i	$0.303641\pi$
$270$	−8.42833	−0.00189975
$271$	−2818.35	−0.631744	−0.315872	−	0.948802i	$-0.602297\pi$
$271$	−2818.35	−0.631744	−0.315872	+	0.948802i	$0.602297\pi$
$272$	−3692.84	−0.823204
$273$	7242.03	1.60552
$274$	−331.690	−0.0731319
$275$	7604.36	1.66749
$276$	−9757.47	−2.12801
$277$	−3983.20	−0.863998	−0.431999	−	0.901874i	$-0.642192\pi$
$277$	−3983.20	−0.863998	−0.431999	+	0.901874i	$0.642192\pi$
$278$	−175.031	−0.0377614
$279$	−69.1817	−0.0148452
$280$	−344.729	−0.0735768
$281$	−3607.30	−0.765813	−0.382906	−	0.923787i	$-0.625077\pi$
$281$	−3607.30	−0.765813	−0.382906	+	0.923787i	$0.625077\pi$
$282$	38.2037	0.00806737
$283$	8143.20	1.71047	0.855235	−	0.518240i	$-0.173413\pi$
$283$	8143.20	1.71047	0.855235	+	0.518240i	$0.173413\pi$
$284$	780.988	0.163180
$285$	−1285.80	−0.267242
$286$	−466.879	−0.0965284
$287$	−3464.47	−0.712547
$288$	1033.02	0.211359
$289$	−1466.66	−0.298527
$290$	−48.0236	−0.00972429
$291$	−15.6783	−0.00315834
$292$	−6952.47	−1.39336
$293$	−5015.12	−0.999954	−0.499977	−	0.866039i	$-0.666658\pi$
$293$	−5015.12	−0.999954	−0.499977	+	0.866039i	$0.666658\pi$
$294$	−914.546	−0.181420
$295$	−681.040	−0.134412
$296$	−1146.18	−0.225069
$297$	−794.726	−0.155268
$298$	126.922	0.0246725
$299$	−5552.31	−1.07391
$300$	−6565.09	−1.26345
$301$	0	0
$302$	−334.810	−0.0637953
$303$	5413.78	1.02645
$304$	−3381.27	−0.637924
$305$	−1298.02	−0.243686
$306$	−318.277	−0.0594598
$307$	−4183.41	−0.777720	−0.388860	−	0.921297i	$-0.627131\pi$
$307$	−4183.41	−0.777720	−0.388860	+	0.921297i	$0.627131\pi$
$308$	−16206.1	−2.99814
$309$	−14207.8	−2.61571
$310$	1.92924	0.000353464 0
$311$	10090.2	1.83975	0.919875	−	0.392211i	$-0.128290\pi$
$311$	10090.2	1.83975	0.919875	+	0.392211i	$0.128290\pi$
$312$	808.460	0.146699
$313$	7623.64	1.37672	0.688360	−	0.725369i	$-0.258331\pi$
$313$	7623.64	1.37672	0.688360	+	0.725369i	$0.258331\pi$
$314$	600.591	0.107940
$315$	2561.60	0.458190
$316$	−6268.64	−1.11594
$317$	−2555.28	−0.452740	−0.226370	−	0.974041i	$-0.572686\pi$
$317$	−2555.28	−0.452740	−0.226370	+	0.974041i	$0.572686\pi$
$318$	924.277	0.162990
$319$	−4528.26	−0.794777
$320$	1634.90	0.285606
$321$	10090.5	1.75451
$322$	1108.06	0.191770
$323$	3155.56	0.543592
$324$	6130.92	1.05126
$325$	−3735.75	−0.637606
$326$	−353.735	−0.0600969
$327$	3748.20	0.633872
$328$	−386.754	−0.0651065
$329$	754.598	0.126451
$330$	−341.015	−0.0568856
$331$	4133.00	0.686315	0.343157	−	0.939278i	$-0.388504\pi$
$331$	4133.00	0.686315	0.343157	+	0.939278i	$0.388504\pi$
$332$	3940.42	0.651381
$333$	8517.02	1.40159
$334$	−175.459	−0.0287446
$335$	−2007.02	−0.327328
$336$	13910.3	2.25854
$337$	−1330.25	−0.215024	−0.107512	−	0.994204i	$-0.534288\pi$
$337$	−1330.25	−0.215024	−0.107512	+	0.994204i	$0.534288\pi$
$338$	−240.467	−0.0386972
$339$	−2772.43	−0.444182
$340$	−1543.78	−0.246245
$341$	181.913	0.0288890
$342$	−291.424	−0.0460771
$343$	−7581.19	−1.19343
$344$	0	0
$345$	−4055.49	−0.632870
$346$	−660.869	−0.102684
$347$	−8891.64	−1.37559	−0.687793	−	0.725907i	$-0.741420\pi$
$347$	−8891.64	−1.37559	−0.687793	+	0.725907i	$0.741420\pi$
$348$	3909.39	0.602199
$349$	−6098.16	−0.935320	−0.467660	−	0.883908i	$-0.654903\pi$
$349$	−6098.16	−0.935320	−0.467660	+	0.883908i	$0.654903\pi$
$350$	745.533	0.113858
$351$	390.420	0.0593706
$352$	−2716.33	−0.411309
$353$	7243.90	1.09222	0.546110	−	0.837713i	$-0.316108\pi$
$353$	7243.90	1.09222	0.546110	+	0.837713i	$0.316108\pi$
$354$	−318.745	−0.0478562
$355$	324.601	0.0485297
$356$	4205.79	0.626141
$357$	−12981.8	−1.92456
$358$	−94.7712	−0.0139911
$359$	−496.292	−0.0729618	−0.0364809	−	0.999334i	$-0.511615\pi$
$359$	−496.292	−0.0729618	−0.0364809	+	0.999334i	$0.511615\pi$
$360$	285.963	0.0418655
$361$	−3969.68	−0.578755
$362$	521.099	0.0756585
$363$	−22524.6	−3.25685
$364$	7961.45	1.14641
$365$	−2889.65	−0.414387
$366$	−607.507	−0.0867620
$367$	−1824.67	−0.259529	−0.129765	−	0.991545i	$-0.541422\pi$
$367$	−1824.67	−0.259529	−0.129765	+	0.991545i	$0.541422\pi$
$368$	−10664.7	−1.51070
$369$	2873.88	0.405442
$370$	−237.511	−0.0333719
$371$	18256.3	2.55477
$372$	−157.051	−0.0218890
$373$	−9254.89	−1.28472	−0.642359	−	0.766404i	$-0.722044\pi$
$373$	−9254.89	−1.28472	−0.642359	+	0.766404i	$0.722044\pi$
$374$	836.908	0.115710
$375$	−5718.73	−0.787504
$376$	84.2392	0.0115540
$377$	2224.57	0.303902
$378$	−77.9152	−0.0106019
$379$	−1365.96	−0.185131	−0.0925653	−	0.995707i	$-0.529507\pi$
$379$	−1365.96	−0.185131	−0.0925653	+	0.995707i	$0.529507\pi$
$380$	−1413.53	−0.190822
$381$	−440.871	−0.0592822
$382$	831.339	0.111348
$383$	5408.14	0.721522	0.360761	−	0.932658i	$-0.382517\pi$
$383$	5408.14	0.721522	0.360761	+	0.932658i	$0.382517\pi$
$384$	3123.75	0.415126
$385$	−6735.71	−0.891646
$386$	−210.547	−0.0277631
$387$	0	0
$388$	−17.2358	−0.00225519
$389$	−4751.47	−0.619304	−0.309652	−	0.950850i	$-0.600212\pi$
$389$	−4751.47	−0.619304	−0.309652	+	0.950850i	$0.600212\pi$
$390$	167.528	0.0217516
$391$	9952.85	1.28731
$392$	−2016.57	−0.259828
$393$	7735.17	0.992843
$394$	877.558	0.112210
$395$	−2605.43	−0.331882
$396$	13443.4	1.70595
$397$	1510.65	0.190975	0.0954876	−	0.995431i	$-0.469559\pi$
$397$	1510.65	0.190975	0.0954876	+	0.995431i	$0.469559\pi$
$398$	−296.689	−0.0373660
$399$	−11886.5	−1.49140
$400$	−7175.53	−0.896942
$401$	−4681.87	−0.583046	−0.291523	−	0.956564i	$-0.594162\pi$
$401$	−4681.87	−0.583046	−0.291523	+	0.956564i	$0.594162\pi$
$402$	−939.337	−0.116542
$403$	−89.3672	−0.0110464
$404$	5951.59	0.732927
$405$	2548.19	0.312643
$406$	−443.951	−0.0542683
$407$	−22395.4	−2.72752
$408$	−1449.21	−0.175850
$409$	−6753.27	−0.816449	−0.408224	−	0.912882i	$-0.633852\pi$
$409$	−6753.27	−0.816449	−0.408224	+	0.912882i	$0.633852\pi$
$410$	−80.1428	−0.00965358
$411$	11222.6	1.34688
$412$	−15619.2	−1.86772
$413$	−6295.83	−0.750115
$414$	−919.169	−0.109118
$415$	1637.75	0.193721
$416$	1334.43	0.157274
$417$	5922.10	0.695459
$418$	766.296	0.0896669
$419$	10883.5	1.26896	0.634481	−	0.772939i	$-0.281214\pi$
$419$	10883.5	1.26896	0.634481	+	0.772939i	$0.281214\pi$
$420$	5815.15	0.675596
$421$	12444.8	1.44067	0.720334	−	0.693628i	$-0.243989\pi$
$421$	12444.8	1.44067	0.720334	+	0.693628i	$0.243989\pi$
$422$	−383.661	−0.0442566
$423$	−625.962	−0.0719511
$424$	2038.03	0.233433
$425$	6696.55	0.764307
$426$	151.922	0.0172785
$427$	−11999.4	−1.35994
$428$	11092.9	1.25279
$429$	15796.6	1.77778
$430$	0	0
$431$	7870.01	0.879547	0.439774	−	0.898109i	$-0.355059\pi$
$431$	7870.01	0.879547	0.439774	+	0.898109i	$0.355059\pi$
$432$	749.910	0.0835186
$433$	2427.39	0.269407	0.134703	−	0.990886i	$-0.456992\pi$
$433$	2427.39	0.269407	0.134703	+	0.990886i	$0.456992\pi$
$434$	17.8348	0.00197257
$435$	1624.86	0.179094
$436$	4120.55	0.452611
$437$	9113.10	0.997572
$438$	−1352.43	−0.147538
$439$	−9708.09	−1.05545	−0.527724	−	0.849416i	$-0.676955\pi$
$439$	−9708.09	−1.05545	−0.527724	+	0.849416i	$0.676955\pi$
$440$	−751.938	−0.0814710
$441$	14984.7	1.61804
$442$	−411.143	−0.0442445
$443$	−1853.67	−0.198805	−0.0994023	−	0.995047i	$-0.531693\pi$
$443$	−1853.67	−0.198805	−0.0994023	+	0.995047i	$0.531693\pi$
$444$	19334.7	2.06663
$445$	1748.05	0.186214
$446$	747.329	0.0793431
$447$	−4294.36	−0.454398
$448$	15113.8	1.59388
$449$	12703.8	1.33525	0.667627	−	0.744496i	$-0.267310\pi$
$449$	12703.8	1.33525	0.667627	+	0.744496i	$0.267310\pi$
$450$	−618.442	−0.0647859
$451$	−7556.84	−0.788997
$452$	−3047.84	−0.317165
$453$	11328.1	1.17493
$454$	−1250.36	−0.129257
$455$	3309.01	0.340942
$456$	−1326.94	−0.136271
$457$	3614.91	0.370018	0.185009	−	0.982737i	$-0.440769\pi$
$457$	3614.91	0.370018	0.185009	+	0.982737i	$0.440769\pi$
$458$	51.9797	0.00530317
$459$	−699.852	−0.0711683
$460$	−4458.36	−0.451896
$461$	−1454.92	−0.146990	−0.0734948	−	0.997296i	$-0.523415\pi$
$461$	−1454.92	−0.146990	−0.0734948	+	0.997296i	$0.523415\pi$
$462$	−3152.49	−0.317461
$463$	1007.60	0.101139	0.0505694	−	0.998721i	$-0.483896\pi$
$463$	1007.60	0.101139	0.0505694	+	0.998721i	$0.483896\pi$
$464$	4272.90	0.427509
$465$	−65.2750	−0.00650980
$466$	−98.2101	−0.00976286
$467$	12311.1	1.21989	0.609946	−	0.792443i	$-0.291191\pi$
$467$	12311.1	1.21989	0.609946	+	0.792443i	$0.291191\pi$
$468$	−6604.26	−0.652312
$469$	−18553.7	−1.82672
$470$	17.4560	0.00171316
$471$	−20320.7	−1.98796
$472$	−702.832	−0.0685391
$473$	0	0
$474$	−1219.41	−0.118163
$475$	6131.54	0.592283
$476$	−14271.4	−1.37422
$477$	−15144.1	−1.45367
$478$	−1115.61	−0.106750
$479$	15343.4	1.46359	0.731795	−	0.681524i	$-0.238683\pi$
$479$	15343.4	1.46359	0.731795	+	0.681524i	$0.238683\pi$
$480$	974.689	0.0926838
$481$	11002.1	1.04293
$482$	859.020	0.0811770
$483$	−37490.7	−3.53185
$484$	−24762.2	−2.32553
$485$	−7.16368	−0.000670693 0
$486$	1123.79	0.104889
$487$	1291.92	0.120211	0.0601053	−	0.998192i	$-0.480856\pi$
$487$	1291.92	0.120211	0.0601053	+	0.998192i	$0.480856\pi$
$488$	−1339.55	−0.124260
$489$	11968.5	1.10681
$490$	−417.872	−0.0385256
$491$	−8952.69	−0.822870	−0.411435	−	0.911439i	$-0.634972\pi$
$491$	−8952.69	−0.822870	−0.411435	+	0.911439i	$0.634972\pi$
$492$	6524.06	0.597820
$493$	−3987.67	−0.364291
$494$	−376.454	−0.0342863
$495$	5587.47	0.507350
$496$	−171.654	−0.0155393
$497$	3000.76	0.270830
$498$	766.512	0.0689723
$499$	18751.4	1.68222	0.841110	−	0.540864i	$-0.181903\pi$
$499$	18751.4	1.68222	0.841110	+	0.540864i	$0.181903\pi$
$500$	−6286.83	−0.562311
$501$	5936.57	0.529394
$502$	−1287.05	−0.114430
$503$	1320.15	0.117023	0.0585116	−	0.998287i	$-0.481365\pi$
$503$	1320.15	0.117023	0.0585116	+	0.998287i	$0.481365\pi$
$504$	2643.57	0.233639
$505$	2473.65	0.217972
$506$	2416.95	0.212345
$507$	8136.07	0.712694
$508$	−484.667	−0.0423300
$509$	15917.4	1.38610	0.693049	−	0.720890i	$-0.256267\pi$
$509$	15917.4	1.38610	0.693049	+	0.720890i	$0.256267\pi$
$510$	−300.304	−0.0260739
$511$	−26713.2	−2.31257
$512$	4280.09	0.369444
$513$	−640.803	−0.0551504
$514$	1554.02	0.133356
$515$	−6491.79	−0.555461
$516$	0	0
$517$	1645.96	0.140018
$518$	−2195.65	−0.186238
$519$	22360.2	1.89114
$520$	369.400	0.0311524
$521$	−12988.8	−1.09223	−0.546115	−	0.837710i	$-0.683894\pi$
$521$	−12988.8	−1.09223	−0.546115	+	0.837710i	$0.683894\pi$
$522$	368.271	0.0308789
$523$	−6216.75	−0.519770	−0.259885	−	0.965640i	$-0.583685\pi$
$523$	−6216.75	−0.519770	−0.259885	+	0.965640i	$0.583685\pi$
$524$	8503.58	0.708932
$525$	−25224.8	−2.09695
$526$	483.227	0.0400564
$527$	160.196	0.0132415
$528$	30341.8	2.50086
$529$	16576.3	1.36240
$530$	422.319	0.0346120
$531$	5222.58	0.426818
$532$	−13067.3	−1.06492
$533$	3712.40	0.301692
$534$	818.132	0.0662997
$535$	4610.53	0.372580
$536$	−2071.24	−0.166910
$537$	3206.54	0.257677
$538$	1091.60	0.0874765
$539$	−39402.1	−3.14874
$540$	313.497	0.0249829
$541$	5366.36	0.426465	0.213233	−	0.977001i	$-0.431601\pi$
$541$	5366.36	0.426465	0.213233	+	0.977001i	$0.431601\pi$
$542$	−602.703	−0.0477644
$543$	−17631.1	−1.39341
$544$	−2392.05	−0.188526
$545$	1712.62	0.134607
$546$	1548.70	0.121389
$547$	8933.86	0.698326	0.349163	−	0.937062i	$-0.386466\pi$
$547$	8933.86	0.698326	0.349163	+	0.937062i	$0.386466\pi$
$548$	12337.4	0.961731
$549$	9953.90	0.773810
$550$	1626.19	0.126074
$551$	−3651.22	−0.282300
$552$	−4185.26	−0.322711
$553$	−24085.7	−1.85213
$554$	−851.806	−0.0653245
$555$	8036.06	0.614616
$556$	6510.40	0.496587
$557$	11275.9	0.857762	0.428881	−	0.903361i	$-0.358908\pi$
$557$	11275.9	0.857762	0.428881	+	0.903361i	$0.358908\pi$
$558$	−14.7945	−0.00112240
$559$	0	0
$560$	6355.87	0.479615
$561$	−28316.4	−2.13105
$562$	−771.419	−0.0579010
$563$	−1065.62	−0.0797704	−0.0398852	−	0.999204i	$-0.512699\pi$
$563$	−1065.62	−0.0797704	−0.0398852	+	0.999204i	$0.512699\pi$
$564$	−1421.01	−0.106091
$565$	−1266.77	−0.0943248
$566$	1741.42	0.129324
$567$	23556.6	1.74477
$568$	334.988	0.0247461
$569$	14804.4	1.09074	0.545370	−	0.838196i	$-0.316389\pi$
$569$	14804.4	1.09074	0.545370	+	0.838196i	$0.316389\pi$
$570$	−274.967	−0.0202054
$571$	−16641.0	−1.21962	−0.609812	−	0.792546i	$-0.708755\pi$
$571$	−16641.0	−1.21962	−0.609812	+	0.792546i	$0.708755\pi$
$572$	17365.9	1.26941
$573$	−28128.0	−2.05072
$574$	−740.875	−0.0538737
$575$	19339.3	1.40262
$576$	−12537.3	−0.906924
$577$	−12019.2	−0.867183	−0.433591	−	0.901110i	$-0.642754\pi$
$577$	−12019.2	−0.867183	−0.433591	+	0.901110i	$0.642754\pi$
$578$	−313.645	−0.0225708
$579$	7123.76	0.511318
$580$	1786.27	0.127881
$581$	15140.1	1.08110
$582$	−3.35279	−0.000238793 0
$583$	39821.4	2.82887
$584$	−2982.11	−0.211303
$585$	−2744.92	−0.193998
$586$	−1072.48	−0.0756038
$587$	14904.6	1.04801	0.524004	−	0.851716i	$-0.324438\pi$
$587$	14904.6	1.04801	0.524004	+	0.851716i	$0.324438\pi$
$588$	34017.1	2.38579
$589$	146.680	0.0102612
$590$	−145.640	−0.0101626
$591$	−29691.7	−2.06659
$592$	21132.5	1.46713
$593$	11518.4	0.797649	0.398824	−	0.917027i	$-0.369418\pi$
$593$	11518.4	0.797649	0.398824	+	0.917027i	$0.369418\pi$
$594$	−169.952	−0.0117394
$595$	−5931.60	−0.408692
$596$	−4720.96	−0.324460
$597$	10038.3	0.688177
$598$	−1187.36	−0.0811952
$599$	4433.16	0.302394	0.151197	−	0.988504i	$-0.451687\pi$
$599$	4433.16	0.302394	0.151197	+	0.988504i	$0.451687\pi$
$600$	−2815.95	−0.191601
$601$	25109.7	1.70424	0.852119	−	0.523347i	$-0.175317\pi$
$601$	25109.7	1.70424	0.852119	+	0.523347i	$0.175317\pi$
$602$	0	0
$603$	15390.9	1.03941
$604$	12453.5	0.838949
$605$	−10291.9	−0.691612
$606$	1157.73	0.0776069
$607$	2512.35	0.167995	0.0839976	−	0.996466i	$-0.473231\pi$
$607$	2512.35	0.167995	0.0839976	+	0.996466i	$0.473231\pi$
$608$	−2190.23	−0.146094
$609$	15020.9	0.999469
$610$	−277.581	−0.0184244
$611$	−808.601	−0.0535393
$612$	11838.5	0.781935
$613$	−19653.8	−1.29496	−0.647479	−	0.762083i	$-0.724177\pi$
$613$	−19653.8	−1.29496	−0.647479	+	0.762083i	$0.724177\pi$
$614$	−894.621	−0.0588013
$615$	2711.59	0.177792
$616$	−6951.24	−0.454665
$617$	−2093.92	−0.136626	−0.0683129	−	0.997664i	$-0.521762\pi$
$617$	−2093.92	−0.136626	−0.0683129	+	0.997664i	$0.521762\pi$
$618$	−3038.33	−0.197766
$619$	13916.5	0.903635	0.451817	−	0.892110i	$-0.350776\pi$
$619$	13916.5	0.903635	0.451817	+	0.892110i	$0.350776\pi$
$620$	−71.7595	−0.00464827
$621$	−2021.14	−0.130605
$622$	2157.78	0.139098
$623$	16159.7	1.03921
$624$	−14905.8	−0.956266
$625$	11645.8	0.745330
$626$	1630.31	0.104090
$627$	−25927.3	−1.65141
$628$	−22339.3	−1.41949
$629$	−19721.9	−1.25018
$630$	547.797	0.0346425
$631$	8609.66	0.543178	0.271589	−	0.962413i	$-0.412451\pi$
$631$	8609.66	0.543178	0.271589	+	0.962413i	$0.412451\pi$
$632$	−2688.80	−0.169232
$633$	12981.0	0.815082
$634$	−546.444	−0.0342304
$635$	−201.442	−0.0125889
$636$	−34379.1	−2.14343
$637$	19356.8	1.20400
$638$	−968.366	−0.0600909
$639$	−2489.22	−0.154103
$640$	1427.30	0.0881546
$641$	13316.9	0.820570	0.410285	−	0.911957i	$-0.365429\pi$
$641$	13316.9	0.820570	0.410285	+	0.911957i	$0.365429\pi$
$642$	2157.85	0.132653
$643$	17434.2	1.06927	0.534633	−	0.845084i	$-0.320450\pi$
$643$	17434.2	1.06927	0.534633	+	0.845084i	$0.320450\pi$
$644$	−41215.0	−2.52189
$645$	0	0
$646$	674.815	0.0410995
$647$	12415.4	0.754402	0.377201	−	0.926131i	$-0.376887\pi$
$647$	12415.4	0.754402	0.377201	+	0.926131i	$0.376887\pi$
$648$	2629.73	0.159422
$649$	−13732.7	−0.830596
$650$	−798.888	−0.0482076
$651$	−603.431	−0.0363292
$652$	13157.4	0.790312
$653$	13796.2	0.826781	0.413390	−	0.910554i	$-0.364345\pi$
$653$	13796.2	0.826781	0.413390	+	0.910554i	$0.364345\pi$
$654$	801.552	0.0479253
$655$	3534.33	0.210836
$656$	7130.69	0.424401
$657$	22159.4	1.31586
$658$	161.370	0.00956060
$659$	20453.4	1.20903	0.604516	−	0.796593i	$-0.293366\pi$
$659$	20453.4	1.20903	0.604516	+	0.796593i	$0.293366\pi$
$660$	12684.3	0.748082
$661$	6563.71	0.386231	0.193116	−	0.981176i	$-0.438141\pi$
$661$	6563.71	0.386231	0.193116	+	0.981176i	$0.438141\pi$
$662$	883.840	0.0518904
$663$	13910.8	0.814858
$664$	1690.16	0.0987814
$665$	−5431.13	−0.316707
$666$	1821.36	0.105970
$667$	−11516.2	−0.668529
$668$	6526.31	0.378010
$669$	−25285.5	−1.46128
$670$	−429.200	−0.0247484
$671$	−26173.7	−1.50585
$672$	9010.45	0.517240
$673$	−11849.9	−0.678720	−0.339360	−	0.940657i	$-0.610210\pi$
$673$	−11849.9	−0.678720	−0.339360	+	0.940657i	$0.610210\pi$
$674$	−284.473	−0.0162574
$675$	−1359.88	−0.0775432
$676$	8944.31	0.508894
$677$	−4684.29	−0.265926	−0.132963	−	0.991121i	$-0.542449\pi$
$677$	−4684.29	−0.265926	−0.132963	+	0.991121i	$0.542449\pi$
$678$	−592.883	−0.0335834
$679$	−66.2242	−0.00374293
$680$	−662.171	−0.0373428
$681$	42305.5	2.38054
$682$	38.9020	0.00218421
$683$	5928.82	0.332152	0.166076	−	0.986113i	$-0.446890\pi$
$683$	5928.82	0.332152	0.166076	+	0.986113i	$0.446890\pi$
$684$	10839.7	0.605944
$685$	5127.79	0.286019
$686$	−1621.23	−0.0902318
$687$	−1758.71	−0.0976694
$688$	0	0
$689$	−19562.8	−1.08169
$690$	−867.264	−0.0478495
$691$	−19182.3	−1.05605	−0.528025	−	0.849229i	$-0.677067\pi$
$691$	−19182.3	−1.05605	−0.528025	+	0.849229i	$0.677067\pi$
$692$	24581.4	1.35036
$693$	51653.0	2.83137
$694$	−1901.47	−0.104004
$695$	2705.91	0.147685
$696$	1676.85	0.0913230
$697$	−6654.70	−0.361643
$698$	−1304.09	−0.0707170
$699$	3322.89	0.179804
$700$	−27730.6	−1.49731
$701$	14967.8	0.806457	0.403229	−	0.915099i	$-0.367888\pi$
$701$	14967.8	0.806457	0.403229	+	0.915099i	$0.367888\pi$
$702$	83.4912	0.00448885
$703$	−18057.9	−0.968799
$704$	32966.8	1.76489
$705$	−590.614	−0.0315515
$706$	1549.10	0.0825798
$707$	22867.5	1.21644
$708$	11855.9	0.629339
$709$	−3735.54	−0.197872	−0.0989360	−	0.995094i	$-0.531544\pi$
$709$	−3735.54	−0.197872	−0.0989360	+	0.995094i	$0.531544\pi$
$710$	69.4159	0.00366920
$711$	19979.8	1.05387
$712$	1803.98	0.0949538
$713$	462.638	0.0243000
$714$	−2776.14	−0.145511
$715$	7217.76	0.377523
$716$	3525.07	0.183992
$717$	37746.0	1.96604
$718$	−106.132	−0.00551644
$719$	−21766.2	−1.12899	−0.564493	−	0.825438i	$-0.690928\pi$
$719$	−21766.2	−1.12899	−0.564493	+	0.825438i	$0.690928\pi$
$720$	−5272.38	−0.272903
$721$	−60012.9	−3.09986
$722$	−848.915	−0.0437581
$723$	−29064.5	−1.49505
$724$	−19382.6	−0.994957
$725$	−7748.41	−0.396922
$726$	−4816.88	−0.246241
$727$	−13634.0	−0.695537	−0.347769	−	0.937580i	$-0.613061\pi$
$727$	−13634.0	−0.695537	−0.347769	+	0.937580i	$0.613061\pi$
$728$	3414.89	0.173852
$729$	−17211.9	−0.874457
$730$	−617.950	−0.0313306
$731$	0	0
$732$	22596.6	1.14098
$733$	6863.64	0.345859	0.172929	−	0.984934i	$-0.444677\pi$
$733$	6863.64	0.345859	0.172929	+	0.984934i	$0.444677\pi$
$734$	−390.206	−0.0196223
$735$	14138.5	0.709533
$736$	−6908.12	−0.345974
$737$	−40470.2	−2.02271
$738$	614.578	0.0306544
$739$	−22700.3	−1.12996	−0.564982	−	0.825103i	$-0.691117\pi$
$739$	−22700.3	−1.12996	−0.564982	+	0.825103i	$0.691117\pi$
$740$	8834.37	0.438862
$741$	12737.1	0.631457
$742$	3904.10	0.193159
$743$	−33576.5	−1.65788	−0.828939	−	0.559339i	$-0.811055\pi$
$743$	−33576.5	−1.65788	−0.828939	+	0.559339i	$0.811055\pi$
$744$	−67.3637	−0.00331945
$745$	−1962.17	−0.0964943
$746$	−1979.15	−0.0971340
$747$	−12559.2	−0.615148
$748$	−31129.3	−1.52166
$749$	42621.7	2.07926
$750$	−1222.95	−0.0595410
$751$	−24851.7	−1.20753	−0.603763	−	0.797164i	$-0.706333\pi$
$751$	−24851.7	−1.20753	−0.603763	+	0.797164i	$0.706333\pi$
$752$	−1553.14	−0.0753155
$753$	43546.6	2.10747
$754$	475.723	0.0229772
$755$	5176.03	0.249503
$756$	2898.10	0.139422
$757$	7571.57	0.363532	0.181766	−	0.983342i	$-0.441819\pi$
$757$	7571.57	0.363532	0.181766	+	0.983342i	$0.441819\pi$
$758$	−292.109	−0.0139972
$759$	−81776.3	−3.91079
$760$	−606.302	−0.0289380
$761$	−9150.62	−0.435887	−0.217943	−	0.975961i	$-0.569935\pi$
$761$	−9150.62	−0.435887	−0.217943	+	0.975961i	$0.569935\pi$
$762$	−94.2801	−0.00448216
$763$	15832.2	0.751198
$764$	−30922.2	−1.46430
$765$	4920.44	0.232547
$766$	1156.53	0.0545523
$767$	6746.39	0.317599
$768$	−27956.9	−1.31355
$769$	−14125.8	−0.662404	−0.331202	−	0.943560i	$-0.607454\pi$
$769$	−14125.8	−0.662404	−0.331202	+	0.943560i	$0.607454\pi$
$770$	−1440.43	−0.0674149
$771$	−52579.4	−2.45603
$772$	7831.43	0.365103
$773$	12974.0	0.603678	0.301839	−	0.953359i	$-0.402399\pi$
$773$	12974.0	0.603678	0.301839	+	0.953359i	$0.402399\pi$
$774$	0	0
$775$	311.275	0.0144275
$776$	−7.39291	−0.000341997 0
$777$	74288.9	3.42998
$778$	−1016.10	−0.0468238
$779$	−6093.23	−0.280247
$780$	−6231.32	−0.286047
$781$	6545.38	0.299888
$782$	2128.41	0.0973298
$783$	809.781	0.0369594
$784$	37180.1	1.69370
$785$	−9284.88	−0.422155
$786$	1654.16	0.0750662
$787$	14982.2	0.678600	0.339300	−	0.940678i	$-0.389810\pi$
$787$	14982.2	0.678600	0.339300	+	0.940678i	$0.389810\pi$
$788$	−32641.3	−1.47563
$789$	−16349.7	−0.737726
$790$	−557.170	−0.0250927
$791$	−11710.6	−0.526398
$792$	5766.26	0.258706
$793$	12858.2	0.575798
$794$	323.051	0.0144391
$795$	−14288.9	−0.637455
$796$	11035.5	0.491387
$797$	6159.90	0.273770	0.136885	−	0.990587i	$-0.456291\pi$
$797$	6159.90	0.273770	0.136885	+	0.990587i	$0.456291\pi$
$798$	−2541.91	−0.112760
$799$	1449.47	0.0641782
$800$	−4647.97	−0.205413
$801$	−13405.0	−0.591312
$802$	−1001.22	−0.0440825
$803$	−58267.9	−2.56068
$804$	34939.2	1.53260
$805$	−17130.2	−0.750011
$806$	−19.1111	−0.000835187 0
$807$	−36933.9	−1.61107
$808$	2552.81	0.111148
$809$	1822.38	0.0791982	0.0395991	−	0.999216i	$-0.487392\pi$
$809$	1822.38	0.0791982	0.0395991	+	0.999216i	$0.487392\pi$
$810$	544.929	0.0236381
$811$	11549.1	0.500053	0.250027	−	0.968239i	$-0.419561\pi$
$811$	11549.1	0.500053	0.250027	+	0.968239i	$0.419561\pi$
$812$	16513.0	0.713663
$813$	20392.2	0.879685
$814$	−4789.25	−0.206220
$815$	5468.60	0.235039
$816$	26719.5	1.14629
$817$	0	0
$818$	−1444.18	−0.0617294
$819$	−25375.3	−1.08264
$820$	2980.96	0.126951
$821$	18127.6	0.770593	0.385296	−	0.922793i	$-0.374099\pi$
$821$	18127.6	0.770593	0.385296	+	0.922793i	$0.374099\pi$
$822$	2399.94	0.101834
$823$	−5421.11	−0.229609	−0.114804	−	0.993388i	$-0.536624\pi$
$823$	−5421.11	−0.229609	−0.114804	+	0.993388i	$0.536624\pi$
$824$	−6699.52	−0.283239
$825$	−55021.3	−2.32194
$826$	−1346.36	−0.0567141
$827$	−19267.7	−0.810160	−0.405080	−	0.914281i	$-0.632756\pi$
$827$	−19267.7	−0.810160	−0.405080	+	0.914281i	$0.632756\pi$
$828$	34189.1	1.43497
$829$	−4914.30	−0.205887	−0.102944	−	0.994687i	$-0.532826\pi$
$829$	−4914.30	−0.205887	−0.102944	+	0.994687i	$0.532826\pi$
$830$	350.233	0.0146467
$831$	28820.4	1.20309
$832$	−16195.4	−0.674849
$833$	−34698.3	−1.44325
$834$	1266.44	0.0525817
$835$	2712.53	0.112420
$836$	−28502.9	−1.17918
$837$	−32.5312	−0.00134342
$838$	2327.44	0.0959427
$839$	32743.9	1.34737	0.673687	−	0.739017i	$-0.264710\pi$
$839$	32743.9	1.34737	0.673687	+	0.739017i	$0.264710\pi$
$840$	2494.29	0.102454
$841$	−19775.0	−0.810815
$842$	2661.31	0.108925
$843$	26100.6	1.06637
$844$	14270.5	0.582003
$845$	3717.52	0.151345
$846$	−133.862	−0.00544002
$847$	−95142.8	−3.85968
$848$	−37575.8	−1.52165
$849$	−58920.1	−2.38178
$850$	1432.05	0.0577871
$851$	−56955.7	−2.29426
$852$	−5650.84	−0.227224
$853$	−12007.0	−0.481959	−0.240979	−	0.970530i	$-0.577469\pi$
$853$	−12007.0	−0.481959	−0.240979	+	0.970530i	$0.577469\pi$
$854$	−2566.08	−0.102821
$855$	4505.29	0.180208
$856$	4758.06	0.189985
$857$	38581.8	1.53784	0.768921	−	0.639344i	$-0.220794\pi$
$857$	38581.8	1.53784	0.768921	+	0.639344i	$0.220794\pi$
$858$	3378.10	0.134413
$859$	−6919.06	−0.274826	−0.137413	−	0.990514i	$-0.543879\pi$
$859$	−6919.06	−0.274826	−0.137413	+	0.990514i	$0.543879\pi$
$860$	0	0
$861$	25067.1	0.992201
$862$	1683.00	0.0665002
$863$	−12317.1	−0.485841	−0.242920	−	0.970046i	$-0.578105\pi$
$863$	−12317.1	−0.485841	−0.242920	+	0.970046i	$0.578105\pi$
$864$	485.756	0.0191270
$865$	10216.8	0.401596
$866$	519.097	0.0203691
$867$	10612.0	0.415690
$868$	−663.376	−0.0259406
$869$	−52536.8	−2.05085
$870$	347.475	0.0135408
$871$	19881.5	0.773433
$872$	1767.42	0.0686381
$873$	54.9349	0.00212974
$874$	1948.83	0.0754236
$875$	−24155.6	−0.933267
$876$	50304.5	1.94022
$877$	18339.0	0.706115	0.353058	−	0.935602i	$-0.385142\pi$
$877$	18339.0	0.706115	0.353058	+	0.935602i	$0.385142\pi$
$878$	−2076.07	−0.0797996
$879$	36286.9	1.39241
$880$	13863.7	0.531074
$881$	−22751.1	−0.870040	−0.435020	−	0.900421i	$-0.643259\pi$
$881$	−22751.1	−0.870040	−0.435020	+	0.900421i	$0.643259\pi$
$882$	3204.47	0.122336
$883$	22792.2	0.868649	0.434325	−	0.900756i	$-0.356987\pi$
$883$	22792.2	0.868649	0.434325	+	0.900756i	$0.356987\pi$
$884$	15292.7	0.581843
$885$	4927.66	0.187165
$886$	−396.406	−0.0150311
$887$	−15601.2	−0.590572	−0.295286	−	0.955409i	$-0.595415\pi$
$887$	−15601.2	−0.590572	−0.295286	+	0.955409i	$0.595415\pi$
$888$	8293.20	0.313403
$889$	−1862.22	−0.0702550
$890$	373.819	0.0140792
$891$	51382.6	1.93197
$892$	−27797.4	−1.04341
$893$	1327.17	0.0497336
$894$	−918.346	−0.0343558
$895$	1465.12	0.0547192
$896$	13194.6	0.491964
$897$	40173.7	1.49539
$898$	2716.70	0.100955
$899$	−185.359	−0.00687660
$900$	23003.3	0.851976
$901$	35067.5	1.29663
$902$	−1616.03	−0.0596539
$903$	0	0
$904$	−1307.31	−0.0480978
$905$	−8055.98	−0.295900
$906$	2422.52	0.0888331
$907$	8836.95	0.323513	0.161756	−	0.986831i	$-0.448284\pi$
$907$	8836.95	0.323513	0.161756	+	0.986831i	$0.448284\pi$
$908$	46508.1	1.69981
$909$	−18969.3	−0.692158
$910$	707.630	0.0257777
$911$	−18084.8	−0.657711	−0.328856	−	0.944380i	$-0.606663\pi$
$911$	−18084.8	−0.657711	−0.328856	+	0.944380i	$0.606663\pi$
$912$	24465.1	0.888292
$913$	33024.2	1.19709
$914$	773.047	0.0279761
$915$	9391.80	0.339326
$916$	−1933.42	−0.0697401
$917$	32672.9	1.17661
$918$	−149.663	−0.00538084
$919$	12681.9	0.455210	0.227605	−	0.973754i	$-0.426911\pi$
$919$	12681.9	0.455210	0.227605	+	0.973754i	$0.426911\pi$
$920$	−1912.32	−0.0685296
$921$	30269.1	1.08295
$922$	−311.133	−0.0111135
$923$	−3215.51	−0.114669
$924$	117259.	4.17482
$925$	−38321.4	−1.36216
$926$	215.476	0.00764683
$927$	49782.5	1.76383
$928$	2767.78	0.0979062
$929$	47824.0	1.68897	0.844486	−	0.535577i	$-0.179906\pi$
$929$	47824.0	1.68897	0.844486	+	0.535577i	$0.179906\pi$
$930$	−13.9590	−0.000492188 0
$931$	−31770.7	−1.11841
$932$	3652.99	0.128388
$933$	−73007.5	−2.56180
$934$	2632.72	0.0922327
$935$	−12938.3	−0.452541
$936$	−2832.76	−0.0989226
$937$	−21397.1	−0.746011	−0.373006	−	0.927829i	$-0.621673\pi$
$937$	−21397.1	−0.746011	−0.373006	+	0.927829i	$0.621673\pi$
$938$	−3967.71	−0.138113
$939$	−55160.8	−1.91704
$940$	−649.285	−0.0225291
$941$	40280.0	1.39542	0.697710	−	0.716381i	$-0.254203\pi$
$941$	40280.0	1.39542	0.697710	+	0.716381i	$0.254203\pi$
$942$	−4345.57	−0.150304
$943$	−19218.4	−0.663668
$944$	12958.3	0.446776
$945$	1204.54	0.0414641
$946$	0	0
$947$	9.77531	0.000335433 0	0.000167716	−	1.00000i	$-0.499947\pi$
$947$	9.77531	0.000335433 0	0.000167716		1.00000i	$0.499947\pi$
$948$	45356.7	1.55392
$949$	28624.9	0.979140
$950$	1311.23	0.0447809
$951$	18488.7	0.630427
$952$	−6121.40	−0.208399
$953$	−42128.3	−1.43197	−0.715985	−	0.698115i	$-0.754022\pi$
$953$	−42128.3	−1.43197	−0.715985	+	0.698115i	$0.754022\pi$
$954$	−3238.57	−0.109908
$955$	−12852.2	−0.435483
$956$	41495.7	1.40384
$957$	32764.2	1.10670
$958$	3281.19	0.110658
$959$	47403.5	1.59618
$960$	−11829.3	−0.397698
$961$	−29783.6	−0.999750
$962$	2352.79	0.0788533
$963$	−35356.0	−1.18311
$964$	−31951.8	−1.06753
$965$	3254.97	0.108582
$966$	−8017.37	−0.267034
$967$	33902.0	1.12742	0.563710	−	0.825973i	$-0.309373\pi$
$967$	33902.0	1.12742	0.563710	+	0.825973i	$0.309373\pi$
$968$	−10621.2	−0.352665
$969$	−22832.0	−0.756936
$970$	−1.53195	−5.07092e−5 0
$971$	8561.31	0.282951	0.141476	−	0.989942i	$-0.454815\pi$
$971$	8561.31	0.282951	0.141476	+	0.989942i	$0.454815\pi$
$972$	−41799.9	−1.37936
$973$	25014.6	0.824185
$974$	276.277	0.00908878
$975$	27030.0	0.887848
$976$	24697.7	0.809994
$977$	−42596.6	−1.39487	−0.697434	−	0.716649i	$-0.745675\pi$
$977$	−42596.6	−1.39487	−0.697434	+	0.716649i	$0.745675\pi$
$978$	2559.45	0.0836832
$979$	35248.3	1.15070
$980$	15543.0	0.506636
$981$	−13133.3	−0.427435
$982$	−1914.53	−0.0622150
$983$	−20090.7	−0.651875	−0.325938	−	0.945391i	$-0.605680\pi$
$983$	−20090.7	−0.651875	−0.325938	+	0.945391i	$0.605680\pi$
$984$	2798.36	0.0906589
$985$	−13566.7	−0.438853
$986$	−852.762	−0.0275431
$987$	−5459.89	−0.176079
$988$	14002.4	0.450887
$989$	0	0
$990$	1194.88	0.0383593
$991$	14315.0	0.458861	0.229430	−	0.973325i	$-0.426314\pi$
$991$	14315.0	0.458861	0.229430	+	0.973325i	$0.426314\pi$
$992$	−111.190	−0.00355874
$993$	−29904.3	−0.955674
$994$	641.711	0.0204767
$995$	4586.69	0.146139
$996$	−28510.9	−0.907030
$997$	49560.0	1.57430	0.787151	−	0.616760i	$-0.211555\pi$
$997$	49560.0	1.57430	0.787151	+	0.616760i	$0.211555\pi$
$998$	4009.98	0.127188
$999$	4004.94	0.126837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.25	yes	50
43.42	odd	2		1849.4.a.i.1.26	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.26	✓	50	43.42	odd	2
1849.4.a.j.1.25	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)

\(f(q)\)	\(=\)	\(q+0.213850 q^{2} -7.23549 q^{3} -7.95427 q^{4} -3.30603 q^{5} -1.54731 q^{6} -30.5623 q^{7} -3.41181 q^{8} +25.3524 q^{9} +O(q^{10})\)	\(q+0.213850 q^{2} -7.23549 q^{3} -7.95427 q^{4} -3.30603 q^{5} -1.54731 q^{6} -30.5623 q^{7} -3.41181 q^{8} +25.3524 q^{9} -0.706992 q^{10} -66.6639 q^{11} +57.5531 q^{12} +32.7495 q^{13} -6.53574 q^{14} +23.9207 q^{15} +62.9045 q^{16} -58.7055 q^{17} +5.42159 q^{18} -53.7524 q^{19} +26.2970 q^{20} +221.134 q^{21} -14.2560 q^{22} -169.539 q^{23} +24.6862 q^{24} -114.070 q^{25} +7.00348 q^{26} +11.9214 q^{27} +243.101 q^{28} +67.9267 q^{29} +5.11544 q^{30} -2.72881 q^{31} +40.7466 q^{32} +482.346 q^{33} -12.5541 q^{34} +101.040 q^{35} -201.660 q^{36} +335.946 q^{37} -11.4949 q^{38} -236.959 q^{39} +11.2795 q^{40} +113.357 q^{41} +47.2893 q^{42} +530.262 q^{44} -83.8156 q^{45} -36.2558 q^{46} -24.6905 q^{47} -455.145 q^{48} +591.057 q^{49} -24.3939 q^{50} +424.763 q^{51} -260.499 q^{52} -597.346 q^{53} +2.54938 q^{54} +220.393 q^{55} +104.273 q^{56} +388.925 q^{57} +14.5261 q^{58} +206.000 q^{59} -190.272 q^{60} +392.622 q^{61} -0.583554 q^{62} -774.828 q^{63} -494.523 q^{64} -108.271 q^{65} +103.150 q^{66} +607.078 q^{67} +466.959 q^{68} +1226.70 q^{69} +21.6073 q^{70} -98.1848 q^{71} -86.4976 q^{72} +874.055 q^{73} +71.8418 q^{74} +825.354 q^{75} +427.561 q^{76} +2037.40 q^{77} -50.6736 q^{78} +788.085 q^{79} -207.964 q^{80} -770.771 q^{81} +24.2414 q^{82} -495.384 q^{83} -1758.96 q^{84} +194.082 q^{85} -491.483 q^{87} +227.445 q^{88} -528.746 q^{89} -17.9239 q^{90} -1000.90 q^{91} +1348.56 q^{92} +19.7443 q^{93} -5.28004 q^{94} +177.707 q^{95} -294.822 q^{96} +2.16686 q^{97} +126.397 q^{98} -1690.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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