LMFDB - Embedded newform 1849.4.a.l.1.18

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:	$0$
Dimension:	$60$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q-2.21102 q^{2} +3.63995 q^{3} -3.11137 q^{4} +17.7911 q^{5} -8.04801 q^{6} +23.5136 q^{7} +24.5675 q^{8} -13.7508 q^{9} +O(q^{10})$	\(q-2.21102 q^{2} +3.63995 q^{3} -3.11137 q^{4} +17.7911 q^{5} -8.04801 q^{6} +23.5136 q^{7} +24.5675 q^{8} -13.7508 q^{9} -39.3366 q^{10} +35.9150 q^{11} -11.3252 q^{12} -18.5179 q^{13} -51.9892 q^{14} +64.7587 q^{15} -29.4284 q^{16} +110.163 q^{17} +30.4033 q^{18} -120.442 q^{19} -55.3547 q^{20} +85.5884 q^{21} -79.4090 q^{22} -113.773 q^{23} +89.4244 q^{24} +191.524 q^{25} +40.9435 q^{26} -148.331 q^{27} -73.1596 q^{28} +194.960 q^{29} -143.183 q^{30} +61.9051 q^{31} -131.473 q^{32} +130.729 q^{33} -243.572 q^{34} +418.334 q^{35} +42.7838 q^{36} +193.936 q^{37} +266.300 q^{38} -67.4041 q^{39} +437.083 q^{40} +28.5481 q^{41} -189.238 q^{42} -111.745 q^{44} -244.642 q^{45} +251.554 q^{46} +401.882 q^{47} -107.118 q^{48} +209.891 q^{49} -423.464 q^{50} +400.986 q^{51} +57.6160 q^{52} -496.002 q^{53} +327.963 q^{54} +638.968 q^{55} +577.672 q^{56} -438.401 q^{57} -431.062 q^{58} +509.007 q^{59} -201.488 q^{60} +245.610 q^{61} -136.874 q^{62} -323.331 q^{63} +526.118 q^{64} -329.454 q^{65} -289.044 q^{66} +418.358 q^{67} -342.757 q^{68} -414.126 q^{69} -924.946 q^{70} -112.882 q^{71} -337.823 q^{72} +117.169 q^{73} -428.798 q^{74} +697.136 q^{75} +374.739 q^{76} +844.493 q^{77} +149.032 q^{78} +252.731 q^{79} -523.564 q^{80} -168.645 q^{81} -63.1204 q^{82} +250.132 q^{83} -266.297 q^{84} +1959.92 q^{85} +709.646 q^{87} +882.343 q^{88} +1252.96 q^{89} +540.909 q^{90} -435.422 q^{91} +353.989 q^{92} +225.331 q^{93} -888.571 q^{94} -2142.79 q^{95} -478.555 q^{96} -125.052 q^{97} -464.074 q^{98} -493.860 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9}+O(q^{10}) $ 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} + 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} + 625 q^{18} + 610 q^{19} + 345 q^{20} + 611 q^{21} + 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} + 1071 q^{26} + 1609 q^{27} + 46 q^{28} + 773 q^{29} + 375 q^{30} - 97 q^{31} + 1967 q^{32} + 500 q^{33} + 217 q^{34} + 247 q^{35} + 175 q^{36} + 228 q^{37} + 1253 q^{38} + 1493 q^{39} + 2220 q^{40} - 951 q^{41} + 2643 q^{42} - 1378 q^{44} + 1086 q^{45} - 565 q^{46} - 2 q^{47} + 2303 q^{48} + 1264 q^{49} + 3273 q^{50} + 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} + 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} + 2999 q^{61} + 5569 q^{62} + 2377 q^{63} + 2082 q^{64} + 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} + 1817 q^{69} + 2738 q^{70} + 8003 q^{71} + 1412 q^{72} - 1011 q^{73} - 1413 q^{74} + 7457 q^{75} + 5516 q^{76} + 4052 q^{77} + 1091 q^{78} - 4422 q^{79} + 1610 q^{80} + 2108 q^{81} + 4676 q^{82} - 297 q^{83} - 54 q^{84} + 4333 q^{85} + 1377 q^{87} + 3652 q^{88} + 2480 q^{89} - 1414 q^{90} + 4551 q^{91} - 3286 q^{92} + 4 q^{93} + 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} + 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 + 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 + 625 * q^18 + 610 * q^19 + 345 * q^20 + 611 * q^21 + 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 + 1071 * q^26 + 1609 * q^27 + 46 * q^28 + 773 * q^29 + 375 * q^30 - 97 * q^31 + 1967 * q^32 + 500 * q^33 + 217 * q^34 + 247 * q^35 + 175 * q^36 + 228 * q^37 + 1253 * q^38 + 1493 * q^39 + 2220 * q^40 - 951 * q^41 + 2643 * q^42 - 1378 * q^44 + 1086 * q^45 - 565 * q^46 - 2 * q^47 + 2303 * q^48 + 1264 * q^49 + 3273 * q^50 + 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 + 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 + 2999 * q^61 + 5569 * q^62 + 2377 * q^63 + 2082 * q^64 + 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 + 1817 * q^69 + 2738 * q^70 + 8003 * q^71 + 1412 * q^72 - 1011 * q^73 - 1413 * q^74 + 7457 * q^75 + 5516 * q^76 + 4052 * q^77 + 1091 * q^78 - 4422 * q^79 + 1610 * q^80 + 2108 * q^81 + 4676 * q^82 - 297 * q^83 - 54 * q^84 + 4333 * q^85 + 1377 * q^87 + 3652 * q^88 + 2480 * q^89 - 1414 * q^90 + 4551 * q^91 - 3286 * q^92 + 4 * q^93 + 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 + 753 * q^98 + 5072 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.21102	−0.781715	−0.390858	−	0.920451i	$-0.627822\pi$
$2$	−2.21102	−0.781715	−0.390858	+	0.920451i	$0.627822\pi$
$3$	3.63995	0.700508	0.350254	−	0.936655i	$-0.386095\pi$
$3$	3.63995	0.700508	0.350254	+	0.936655i	$0.386095\pi$
$4$	−3.11137	−0.388921
$5$	17.7911	1.59129	0.795643	−	0.605766i	$-0.207133\pi$
$5$	17.7911	1.59129	0.795643	+	0.605766i	$0.207133\pi$
$6$	−8.04801	−0.547598
$7$	23.5136	1.26962	0.634808	−	0.772670i	$-0.281079\pi$
$7$	23.5136	1.26962	0.634808	+	0.772670i	$0.281079\pi$
$8$	24.5675	1.08574
$9$	−13.7508	−0.509288
$10$	−39.3366	−1.24393
$11$	35.9150	0.984435	0.492218	−	0.870472i	$-0.336186\pi$
$11$	35.9150	0.984435	0.492218	+	0.870472i	$0.336186\pi$
$12$	−11.3252	−0.272442
$13$	−18.5179	−0.395072	−0.197536	−	0.980296i	$-0.563294\pi$
$13$	−18.5179	−0.395072	−0.197536	+	0.980296i	$0.563294\pi$
$14$	−51.9892	−0.992479
$15$	64.7587	1.11471
$16$	−29.4284	−0.459819
$17$	110.163	1.57167	0.785834	−	0.618438i	$-0.212234\pi$
$17$	110.163	1.57167	0.785834	+	0.618438i	$0.212234\pi$
$18$	30.4033	0.398119
$19$	−120.442	−1.45427	−0.727137	−	0.686492i	$-0.759150\pi$
$19$	−120.442	−1.45427	−0.727137	+	0.686492i	$0.759150\pi$
$20$	−55.3547	−0.618885
$21$	85.5884	0.889377
$22$	−79.4090	−0.769548
$23$	−113.773	−1.03144	−0.515722	−	0.856756i	$-0.672476\pi$
$23$	−113.773	−1.03144	−0.515722	+	0.856756i	$0.672476\pi$
$24$	89.4244	0.760570
$25$	191.524	1.53219
$26$	40.9435	0.308834
$27$	−148.331	−1.05727
$28$	−73.1596	−0.493781
$29$	194.960	1.24839	0.624194	−	0.781269i	$-0.285428\pi$
$29$	194.960	1.24839	0.624194	+	0.781269i	$0.285428\pi$
$30$	−143.183	−0.871385
$31$	61.9051	0.358661	0.179330	−	0.983789i	$-0.442607\pi$
$31$	61.9051	0.358661	0.179330	+	0.983789i	$0.442607\pi$
$32$	−131.473	−0.726293
$33$	130.729	0.689605
$34$	−243.572	−1.22860
$35$	418.334	2.02032
$36$	42.7838	0.198073
$37$	193.936	0.861700	0.430850	−	0.902423i	$-0.358214\pi$
$37$	193.936	0.861700	0.430850	+	0.902423i	$0.358214\pi$
$38$	266.300	1.13683
$39$	−67.4041	−0.276751
$40$	437.083	1.72772
$41$	28.5481	0.108743	0.0543714	−	0.998521i	$-0.482684\pi$
$41$	28.5481	0.108743	0.0543714	+	0.998521i	$0.482684\pi$
$42$	−189.238	−0.695239
$43$	0	0
$43$	0	0
$44$	−111.745	−0.382868
$45$	−244.642	−0.810423
$46$	251.554	0.806296
$47$	401.882	1.24724	0.623622	−	0.781726i	$-0.285660\pi$
$47$	401.882	1.24724	0.623622	+	0.781726i	$0.285660\pi$
$48$	−107.118	−0.322107
$49$	209.891	0.611927
$50$	−423.464	−1.19774
$51$	400.986	1.10097
$52$	57.6160	0.153652
$53$	−496.002	−1.28549	−0.642747	−	0.766079i	$-0.722205\pi$
$53$	−496.002	−1.28549	−0.642747	+	0.766079i	$0.722205\pi$
$54$	327.963	0.826483
$55$	638.968	1.56652
$56$	577.672	1.37848
$57$	−438.401	−1.01873
$58$	−431.062	−0.975884
$59$	509.007	1.12317	0.561585	−	0.827419i	$-0.310192\pi$
$59$	509.007	1.12317	0.561585	+	0.827419i	$0.310192\pi$
$60$	−201.488	−0.433534
$61$	245.610	0.515526	0.257763	−	0.966208i	$-0.417015\pi$
$61$	245.610	0.515526	0.257763	+	0.966208i	$0.417015\pi$
$62$	−136.874	−0.280371
$63$	−323.331	−0.646601
$64$	526.118	1.02757
$65$	−329.454	−0.628672
$66$	−289.044	−0.539074
$67$	418.358	0.762844	0.381422	−	0.924401i	$-0.375434\pi$
$67$	418.358	0.762844	0.381422	+	0.924401i	$0.375434\pi$
$68$	−342.757	−0.611255
$69$	−414.126	−0.722535
$70$	−924.946	−1.57932
$71$	−112.882	−0.188685	−0.0943427	−	0.995540i	$-0.530075\pi$
$71$	−112.882	−0.188685	−0.0943427	+	0.995540i	$0.530075\pi$
$72$	−337.823	−0.552955
$73$	117.169	0.187858	0.0939289	−	0.995579i	$-0.470057\pi$
$73$	117.169	0.187858	0.0939289	+	0.995579i	$0.470057\pi$
$74$	−428.798	−0.673604
$75$	697.136	1.07331
$76$	374.739	0.565598
$77$	844.493	1.24986
$78$	149.032	0.216340
$79$	252.731	0.359929	0.179965	−	0.983673i	$-0.442402\pi$
$79$	252.731	0.359929	0.179965	+	0.983673i	$0.442402\pi$
$80$	−523.564	−0.731703
$81$	−168.645	−0.231337
$82$	−63.1204	−0.0850059
$83$	250.132	0.330789	0.165395	−	0.986227i	$-0.447110\pi$
$83$	250.132	0.330789	0.165395	+	0.986227i	$0.447110\pi$
$84$	−266.297	−0.345898
$85$	1959.92	2.50097
$86$	0	0
$87$	709.646	0.874506
$88$	882.343	1.06884
$89$	1252.96	1.49228	0.746140	−	0.665789i	$-0.231905\pi$
$89$	1252.96	1.49228	0.746140	+	0.665789i	$0.231905\pi$
$90$	540.909	0.633520
$91$	−435.422	−0.501590
$92$	353.989	0.401151
$93$	225.331	0.251245
$94$	−888.571	−0.974990
$95$	−2142.79	−2.31417
$96$	−478.555	−0.508774
$97$	−125.052	−0.130898	−0.0654491	−	0.997856i	$-0.520848\pi$
$97$	−125.052	−0.130898	−0.0654491	+	0.997856i	$0.520848\pi$
$98$	−464.074	−0.478353
$99$	−493.860	−0.501361
$100$	−595.901	−0.595901
$101$	−527.050	−0.519242	−0.259621	−	0.965711i	$-0.583598\pi$
$101$	−527.050	−0.519242	−0.259621	+	0.965711i	$0.583598\pi$
$102$	−886.590	−0.860642
$103$	80.1552	0.0766789	0.0383394	−	0.999265i	$-0.487793\pi$
$103$	80.1552	0.0766789	0.0383394	+	0.999265i	$0.487793\pi$
$104$	−454.938	−0.428946
$105$	1522.71	1.41525
$106$	1096.67	1.00489
$107$	−680.056	−0.614425	−0.307213	−	0.951641i	$-0.599396\pi$
$107$	−680.056	−0.614425	−0.307213	+	0.951641i	$0.599396\pi$
$108$	461.512	0.411194
$109$	−84.5094	−0.0742618	−0.0371309	−	0.999310i	$-0.511822\pi$
$109$	−84.5094	−0.0742618	−0.0371309	+	0.999310i	$0.511822\pi$
$110$	−1412.77	−1.22457
$111$	705.917	0.603628
$112$	−691.969	−0.583794
$113$	−281.012	−0.233941	−0.116971	−	0.993135i	$-0.537318\pi$
$113$	−281.012	−0.233941	−0.116971	+	0.993135i	$0.537318\pi$
$114$	969.316	0.796358
$115$	−2024.14	−1.64132
$116$	−606.594	−0.485525
$117$	254.635	0.201206
$118$	−1125.43	−0.877999
$119$	2590.32	1.99542
$120$	1590.96	1.21028
$121$	−41.1115	−0.0308877
$122$	−543.049	−0.402995
$123$	103.913	0.0761752
$124$	−192.610	−0.139491
$125$	1183.53	0.846867
$126$	714.893	0.505458
$127$	1235.02	0.862913	0.431457	−	0.902134i	$-0.358000\pi$
$127$	1235.02	0.862913	0.431457	+	0.902134i	$0.358000\pi$
$128$	−111.474	−0.0769765
$129$	0	0
$130$	728.430	0.491443
$131$	−1986.18	−1.32468	−0.662339	−	0.749204i	$-0.730436\pi$
$131$	−1986.18	−1.32468	−0.662339	+	0.749204i	$0.730436\pi$
$132$	−406.746	−0.268202
$133$	−2832.02	−1.84637
$134$	−925.000	−0.596327
$135$	−2638.97	−1.68242
$136$	2706.42	1.70642
$137$	−2981.61	−1.85939	−0.929694	−	0.368332i	$-0.879929\pi$
$137$	−2981.61	−1.85939	−0.929694	+	0.368332i	$0.879929\pi$
$138$	915.643	0.564817
$139$	2571.95	1.56942	0.784712	−	0.619861i	$-0.212811\pi$
$139$	2571.95	1.56942	0.784712	+	0.619861i	$0.212811\pi$
$140$	−1301.59	−0.785747
$141$	1462.83	0.873705
$142$	249.585	0.147498
$143$	−665.070	−0.388923
$144$	404.664	0.234180
$145$	3468.56	1.98654
$146$	−259.064	−0.146851
$147$	763.992	0.428660
$148$	−603.407	−0.335134
$149$	970.595	0.533653	0.266826	−	0.963745i	$-0.414025\pi$
$149$	970.595	0.533653	0.266826	+	0.963745i	$0.414025\pi$
$150$	−1541.39	−0.839024
$151$	2041.77	1.10038	0.550190	−	0.835040i	$-0.314555\pi$
$151$	2041.77	1.10038	0.550190	+	0.835040i	$0.314555\pi$
$152$	−2958.95	−1.57897
$153$	−1514.82	−0.800432
$154$	−1867.19	−0.977031
$155$	1101.36	0.570732
$156$	209.719	0.107634
$157$	−3711.58	−1.88673	−0.943365	−	0.331756i	$-0.892359\pi$
$157$	−3711.58	−1.88673	−0.943365	+	0.331756i	$0.892359\pi$
$158$	−558.793	−0.281362
$159$	−1805.42	−0.900498
$160$	−2339.05	−1.15574
$161$	−2675.21	−1.30954
$162$	372.877	0.180839
$163$	1430.93	0.687603	0.343802	−	0.939042i	$-0.388285\pi$
$163$	1430.93	0.687603	0.343802	+	0.939042i	$0.388285\pi$
$164$	−88.8236	−0.0422924
$165$	2325.81	1.09736
$166$	−553.047	−0.258583
$167$	−976.053	−0.452271	−0.226136	−	0.974096i	$-0.572609\pi$
$167$	−976.053	−0.452271	−0.226136	+	0.974096i	$0.572609\pi$
$168$	2102.69	0.965633
$169$	−1854.09	−0.843918
$170$	−4333.42	−1.95505
$171$	1656.17	0.740645
$172$	0	0
$173$	−2291.59	−1.00709	−0.503545	−	0.863969i	$-0.667971\pi$
$173$	−2291.59	−1.00709	−0.503545	+	0.863969i	$0.667971\pi$
$174$	−1569.04	−0.683614
$175$	4503.42	1.94529
$176$	−1056.92	−0.452662
$177$	1852.76	0.786790
$178$	−2770.31	−1.16654
$179$	−542.480	−0.226519	−0.113259	−	0.993565i	$-0.536129\pi$
$179$	−542.480	−0.226519	−0.113259	+	0.993565i	$0.536129\pi$
$180$	761.171	0.315191
$181$	3353.59	1.37718	0.688591	−	0.725150i	$-0.258229\pi$
$181$	3353.59	1.37718	0.688591	+	0.725150i	$0.258229\pi$
$182$	962.730	0.392100
$183$	894.007	0.361130
$184$	−2795.11	−1.11988
$185$	3450.34	1.37121
$186$	−498.213	−0.196402
$187$	3956.49	1.54720
$188$	−1250.40	−0.485080
$189$	−3487.79	−1.34233
$190$	4737.77	1.80902
$191$	2178.01	0.825108	0.412554	−	0.910933i	$-0.364637\pi$
$191$	2178.01	0.825108	0.412554	+	0.910933i	$0.364637\pi$
$192$	1915.04	0.719824
$193$	263.127	0.0981364	0.0490682	−	0.998795i	$-0.484375\pi$
$193$	263.127	0.0981364	0.0490682	+	0.998795i	$0.484375\pi$
$194$	276.493	0.102325
$195$	−1199.19	−0.440390
$196$	−653.049	−0.237992
$197$	4892.82	1.76954	0.884768	−	0.466031i	$-0.154317\pi$
$197$	4892.82	1.76954	0.884768	+	0.466031i	$0.154317\pi$
$198$	1091.94	0.391922
$199$	−1865.82	−0.664644	−0.332322	−	0.943166i	$-0.607832\pi$
$199$	−1865.82	−0.664644	−0.332322	+	0.943166i	$0.607832\pi$
$200$	4705.26	1.66356
$201$	1522.80	0.534378
$202$	1165.32	0.405900
$203$	4584.23	1.58497
$204$	−1247.62	−0.428189
$205$	507.902	0.173041
$206$	−177.225	−0.0599410
$207$	1564.46	0.525303
$208$	544.952	0.181662
$209$	−4325.67	−1.43164
$210$	−3366.76	−1.10632
$211$	779.136	0.254208	0.127104	−	0.991889i	$-0.459432\pi$
$211$	779.136	0.254208	0.127104	+	0.991889i	$0.459432\pi$
$212$	1543.25	0.499956
$213$	−410.885	−0.132176
$214$	1503.62	0.480306
$215$	0	0
$216$	−3644.12	−1.14792
$217$	1455.61	0.455362
$218$	186.852	0.0580516
$219$	426.490	0.131596
$220$	−1988.07	−0.609252
$221$	−2039.98	−0.620922
$222$	−1560.80	−0.471865
$223$	−1588.21	−0.476927	−0.238463	−	0.971151i	$-0.576644\pi$
$223$	−1588.21	−0.476927	−0.238463	+	0.971151i	$0.576644\pi$
$224$	−3091.41	−0.922115
$225$	−2633.60	−0.780327
$226$	621.324	0.182876
$227$	−2820.58	−0.824707	−0.412354	−	0.911024i	$-0.635293\pi$
$227$	−2820.58	−0.824707	−0.412354	+	0.911024i	$0.635293\pi$
$228$	1364.03	0.396206
$229$	−4656.13	−1.34361	−0.671804	−	0.740729i	$-0.734480\pi$
$229$	−4656.13	−1.34361	−0.671804	+	0.740729i	$0.734480\pi$
$230$	4475.43	1.28305
$231$	3073.91	0.875534
$232$	4789.69	1.35543
$233$	−3606.72	−1.01409	−0.507047	−	0.861918i	$-0.669263\pi$
$233$	−3606.72	−1.01409	−0.507047	+	0.861918i	$0.669263\pi$
$234$	−563.005	−0.157285
$235$	7149.93	1.98472
$236$	−1583.71	−0.436825
$237$	919.926	0.252133
$238$	−5727.27	−1.55985
$239$	3960.21	1.07182	0.535910	−	0.844275i	$-0.319969\pi$
$239$	3960.21	1.07182	0.535910	+	0.844275i	$0.319969\pi$
$240$	−1905.75	−0.512564
$241$	1397.44	0.373514	0.186757	−	0.982406i	$-0.440202\pi$
$241$	1397.44	0.373514	0.186757	+	0.982406i	$0.440202\pi$
$242$	90.8986	0.0241454
$243$	3391.07	0.895215
$244$	−764.183	−0.200499
$245$	3734.20	0.973751
$246$	−229.755	−0.0595473
$247$	2230.32	0.574543
$248$	1520.86	0.389413
$249$	910.465	0.231720
$250$	−2616.82	−0.662009
$251$	−410.525	−0.103235	−0.0516177	−	0.998667i	$-0.516438\pi$
$251$	−410.525	−0.103235	−0.0516177	+	0.998667i	$0.516438\pi$
$252$	1006.00	0.251477
$253$	−4086.14	−1.01539
$254$	−2730.65	−0.674553
$255$	7133.99	1.75195
$256$	−3962.47	−0.967400
$257$	6317.48	1.53336	0.766680	−	0.642029i	$-0.221907\pi$
$257$	6317.48	1.53336	0.766680	+	0.642029i	$0.221907\pi$
$258$	0	0
$259$	4560.14	1.09403
$260$	1025.05	0.244504
$261$	−2680.86	−0.635790
$262$	4391.48	1.03552
$263$	−4447.88	−1.04284	−0.521422	−	0.853299i	$-0.674598\pi$
$263$	−4447.88	−1.04284	−0.521422	+	0.853299i	$0.674598\pi$
$264$	3211.68	0.748732
$265$	−8824.43	−2.04559
$266$	6261.67	1.44334
$267$	4560.69	1.04535
$268$	−1301.67	−0.296686
$269$	4750.56	1.07675	0.538377	−	0.842704i	$-0.319038\pi$
$269$	4750.56	1.07675	0.538377	+	0.842704i	$0.319038\pi$
$270$	5834.82	1.31517
$271$	−2331.03	−0.522509	−0.261255	−	0.965270i	$-0.584136\pi$
$271$	−2331.03	−0.522509	−0.261255	+	0.965270i	$0.584136\pi$
$272$	−3241.91	−0.722683
$273$	−1584.91	−0.351368
$274$	6592.41	1.45351
$275$	6878.58	1.50834
$276$	1288.50	0.281009
$277$	6817.93	1.47888	0.739439	−	0.673223i	$-0.235091\pi$
$277$	6817.93	1.47888	0.739439	+	0.673223i	$0.235091\pi$
$278$	−5686.64	−1.22684
$279$	−851.244	−0.182662
$280$	10277.4	2.19355
$281$	−4283.01	−0.909264	−0.454632	−	0.890679i	$-0.650229\pi$
$281$	−4283.01	−0.909264	−0.454632	+	0.890679i	$0.650229\pi$
$282$	−3234.35	−0.682988
$283$	2617.96	0.549900	0.274950	−	0.961459i	$-0.411339\pi$
$283$	2617.96	0.549900	0.274950	+	0.961459i	$0.411339\pi$
$284$	351.218	0.0733837
$285$	−7799.65	−1.62109
$286$	1470.49	0.304027
$287$	671.269	0.138062
$288$	1807.86	0.369893
$289$	7222.80	1.47014
$290$	−7669.08	−1.55291
$291$	−455.183	−0.0916952
$292$	−364.557	−0.0730619
$293$	−4126.02	−0.822677	−0.411339	−	0.911483i	$-0.634939\pi$
$293$	−4126.02	−0.822677	−0.411339	+	0.911483i	$0.634939\pi$
$294$	−1689.21	−0.335090
$295$	9055.80	1.78728
$296$	4764.53	0.935583
$297$	−5327.30	−1.04081
$298$	−2146.01	−0.417165
$299$	2106.83	0.407495
$300$	−2169.05	−0.417434
$301$	0	0
$302$	−4514.41	−0.860183
$303$	−1918.44	−0.363733
$304$	3544.41	0.668703
$305$	4369.67	0.820350
$306$	3349.31	0.625710
$307$	−5500.13	−1.02251	−0.511253	−	0.859431i	$-0.670818\pi$
$307$	−5500.13	−1.02251	−0.511253	+	0.859431i	$0.670818\pi$
$308$	−2627.53	−0.486095
$309$	291.761	0.0537142
$310$	−2435.14	−0.446150
$311$	5901.38	1.07600	0.538001	−	0.842944i	$-0.319180\pi$
$311$	5901.38	1.07600	0.538001	+	0.842944i	$0.319180\pi$
$312$	−1655.95	−0.300480
$313$	5572.77	1.00636	0.503182	−	0.864181i	$-0.332163\pi$
$313$	5572.77	1.00636	0.503182	+	0.864181i	$0.332163\pi$
$314$	8206.40	1.47489
$315$	−5752.42	−1.02893
$316$	−786.338	−0.139984
$317$	−5228.23	−0.926330	−0.463165	−	0.886272i	$-0.653286\pi$
$317$	−5228.23	−0.926330	−0.463165	+	0.886272i	$0.653286\pi$
$318$	3991.83	0.703933
$319$	7002.01	1.22896
$320$	9360.22	1.63516
$321$	−2475.37	−0.430410
$322$	5914.95	1.02369
$323$	−13268.2	−2.28564
$324$	524.716	0.0899718
$325$	−3546.61	−0.605325
$326$	−3163.83	−0.537510
$327$	−307.610	−0.0520210
$328$	701.355	0.118067
$329$	9449.70	1.58352
$330$	−5142.42	−0.857821
$331$	2835.39	0.470837	0.235418	−	0.971894i	$-0.424354\pi$
$331$	2835.39	0.470837	0.235418	+	0.971894i	$0.424354\pi$
$332$	−778.252	−0.128651
$333$	−2666.78	−0.438854
$334$	2158.08	0.353547
$335$	7443.05	1.21390
$336$	−2518.73	−0.408952
$337$	−6319.36	−1.02148	−0.510738	−	0.859737i	$-0.670628\pi$
$337$	−6319.36	−1.02148	−0.510738	+	0.859737i	$0.670628\pi$
$338$	4099.44	0.659704
$339$	−1022.87	−0.163878
$340$	−6098.02	−0.972681
$341$	2223.32	0.353078
$342$	−3661.83	−0.578974
$343$	−3129.88	−0.492704
$344$	0	0
$345$	−7367.77	−1.14976
$346$	5066.77	0.787258
$347$	6108.05	0.944950	0.472475	−	0.881344i	$-0.343361\pi$
$347$	6108.05	0.944950	0.472475	+	0.881344i	$0.343361\pi$
$348$	−2207.97	−0.340114
$349$	−7402.33	−1.13535	−0.567676	−	0.823252i	$-0.692157\pi$
$349$	−7402.33	−1.13535	−0.567676	+	0.823252i	$0.692157\pi$
$350$	−9957.17	−1.52067
$351$	2746.77	0.417697
$352$	−4721.86	−0.714989
$353$	7681.82	1.15825	0.579124	−	0.815239i	$-0.303395\pi$
$353$	7681.82	1.15825	0.579124	+	0.815239i	$0.303395\pi$
$354$	−4096.49	−0.615045
$355$	−2008.30	−0.300252
$356$	−3898.41	−0.580380
$357$	9428.64	1.39781
$358$	1199.44	0.177073
$359$	−10010.2	−1.47164	−0.735820	−	0.677177i	$-0.763203\pi$
$359$	−10010.2	−1.47164	−0.735820	+	0.677177i	$0.763203\pi$
$360$	−6010.24	−0.879910
$361$	7647.20	1.11491
$362$	−7414.86	−1.07656
$363$	−149.644	−0.0216371
$364$	1354.76	0.195079
$365$	2084.57	0.298935
$366$	−1976.67	−0.282301
$367$	1546.41	0.219951	0.109976	−	0.993934i	$-0.464923\pi$
$367$	1546.41	0.219951	0.109976	+	0.993934i	$0.464923\pi$
$368$	3348.15	0.474278
$369$	−392.558	−0.0553815
$370$	−7628.79	−1.07190
$371$	−11662.8	−1.63208
$372$	−701.089	−0.0977145
$373$	11437.4	1.58768	0.793842	−	0.608124i	$-0.208078\pi$
$373$	11437.4	1.58768	0.793842	+	0.608124i	$0.208078\pi$
$374$	−8747.90	−1.20947
$375$	4307.99	0.593237
$376$	9873.24	1.35418
$377$	−3610.25	−0.493203
$378$	7711.60	1.04932
$379$	431.574	0.0584920	0.0292460	−	0.999572i	$-0.490689\pi$
$379$	431.574	0.0584920	0.0292460	+	0.999572i	$0.490689\pi$
$380$	6667.02	0.900029
$381$	4495.39	0.604478
$382$	−4815.64	−0.644999
$383$	3155.02	0.420924	0.210462	−	0.977602i	$-0.432503\pi$
$383$	3155.02	0.420924	0.210462	+	0.977602i	$0.432503\pi$
$384$	−405.759	−0.0539226
$385$	15024.5	1.98888
$386$	−581.781	−0.0767147
$387$	0	0
$388$	389.084	0.0509091
$389$	599.586	0.0781497	0.0390748	−	0.999236i	$-0.487559\pi$
$389$	599.586	0.0781497	0.0390748	+	0.999236i	$0.487559\pi$
$390$	2651.45	0.344259
$391$	−12533.5	−1.62109
$392$	5156.50	0.664394
$393$	−7229.57	−0.927948
$394$	−10818.1	−1.38327
$395$	4496.36	0.572750
$396$	1536.58	0.194990
$397$	12632.5	1.59699	0.798497	−	0.601999i	$-0.205629\pi$
$397$	12632.5	1.59699	0.798497	+	0.601999i	$0.205629\pi$
$398$	4125.37	0.519563
$399$	−10308.4	−1.29340
$400$	−5636.24	−0.704530
$401$	−9936.30	−1.23739	−0.618697	−	0.785630i	$-0.712339\pi$
$401$	−9936.30	−1.23739	−0.618697	+	0.785630i	$0.712339\pi$
$402$	−3366.95	−0.417732
$403$	−1146.35	−0.141697
$404$	1639.85	0.201944
$405$	−3000.37	−0.368123
$406$	−10135.8	−1.23900
$407$	6965.22	0.848288
$408$	9851.23	1.19536
$409$	13310.6	1.60921	0.804603	−	0.593813i	$-0.202378\pi$
$409$	13310.6	1.60921	0.804603	+	0.593813i	$0.202378\pi$
$410$	−1122.98	−0.135269
$411$	−10852.9	−1.30252
$412$	−249.392	−0.0298220
$413$	11968.6	1.42600
$414$	−3459.07	−0.410637
$415$	4450.12	0.526380
$416$	2434.60	0.286938
$417$	9361.76	1.09939
$418$	9564.15	1.11913
$419$	−4229.58	−0.493147	−0.246574	−	0.969124i	$-0.579305\pi$
$419$	−4229.58	−0.493147	−0.246574	+	0.969124i	$0.579305\pi$
$420$	−4737.72	−0.550422
$421$	6537.84	0.756853	0.378426	−	0.925631i	$-0.376465\pi$
$421$	6537.84	0.756853	0.378426	+	0.925631i	$0.376465\pi$
$422$	−1722.69	−0.198719
$423$	−5526.19	−0.635207
$424$	−12185.5	−1.39571
$425$	21098.8	2.40809
$426$	908.478	0.103324
$427$	5775.18	0.654521
$428$	2115.91	0.238963
$429$	−2420.82	−0.272443
$430$	0	0
$431$	1630.38	0.182210	0.0911052	−	0.995841i	$-0.470960\pi$
$431$	1630.38	0.182210	0.0911052	+	0.995841i	$0.470960\pi$
$432$	4365.14	0.486152
$433$	−10072.0	−1.11785	−0.558925	−	0.829219i	$-0.688786\pi$
$433$	−10072.0	−1.11785	−0.558925	+	0.829219i	$0.688786\pi$
$434$	−3218.40	−0.355963
$435$	12625.4	1.39159
$436$	262.940	0.0288820
$437$	13703.0	1.50000
$438$	−942.979	−0.102871
$439$	−9889.34	−1.07515	−0.537577	−	0.843215i	$-0.680660\pi$
$439$	−9889.34	−1.07515	−0.537577	+	0.843215i	$0.680660\pi$
$440$	15697.9	1.70083
$441$	−2886.17	−0.311647
$442$	4510.44	0.485384
$443$	1747.99	0.187471	0.0937356	−	0.995597i	$-0.470119\pi$
$443$	1747.99	0.187471	0.0937356	+	0.995597i	$0.470119\pi$
$444$	−2196.37	−0.234764
$445$	22291.5	2.37464
$446$	3511.58	0.372821
$447$	3532.92	0.373828
$448$	12370.9	1.30462
$449$	881.367	0.0926376	0.0463188	−	0.998927i	$-0.485251\pi$
$449$	881.367	0.0926376	0.0463188	+	0.998927i	$0.485251\pi$
$450$	5822.96	0.609993
$451$	1025.30	0.107050
$452$	874.332	0.0909848
$453$	7431.95	0.770824
$454$	6236.37	0.644686
$455$	−7746.65	−0.798173
$456$	−10770.4	−1.10608
$457$	14950.3	1.53030	0.765151	−	0.643851i	$-0.222664\pi$
$457$	14950.3	1.53030	0.765151	+	0.643851i	$0.222664\pi$
$458$	10294.8	1.05032
$459$	−16340.5	−1.66168
$460$	6297.85	0.638346
$461$	−4395.29	−0.444054	−0.222027	−	0.975040i	$-0.571267\pi$
$461$	−4395.29	−0.444054	−0.222027	+	0.975040i	$0.571267\pi$
$462$	−6796.49	−0.684418
$463$	−7855.66	−0.788518	−0.394259	−	0.918999i	$-0.628999\pi$
$463$	−7855.66	−0.788518	−0.394259	+	0.918999i	$0.628999\pi$
$464$	−5737.38	−0.574032
$465$	4008.90	0.399802
$466$	7974.54	0.792733
$467$	5626.20	0.557494	0.278747	−	0.960365i	$-0.410081\pi$
$467$	5626.20	0.557494	0.278747	+	0.960365i	$0.410081\pi$
$468$	−792.265	−0.0782531
$469$	9837.12	0.968520
$470$	−15808.7	−1.55149
$471$	−13510.0	−1.32167
$472$	12505.0	1.21947
$473$	0	0
$474$	−2033.98	−0.197096
$475$	−23067.4	−2.22823
$476$	−8059.45	−0.776060
$477$	6820.42	0.654687
$478$	−8756.13	−0.837858
$479$	15831.1	1.51010	0.755052	−	0.655665i	$-0.227612\pi$
$479$	15831.1	1.51010	0.755052	+	0.655665i	$0.227612\pi$
$480$	−8514.03	−0.809605
$481$	−3591.29	−0.340434
$482$	−3089.77	−0.291981
$483$	−9737.61	−0.917343
$484$	127.913	0.0120129
$485$	−2224.82	−0.208296
$486$	−7497.74	−0.699804
$487$	20451.8	1.90300	0.951498	−	0.307656i	$-0.0995447\pi$
$487$	20451.8	1.90300	0.951498	+	0.307656i	$0.0995447\pi$
$488$	6034.02	0.559728
$489$	5208.52	0.481672
$490$	−8256.40	−0.761196
$491$	10455.2	0.960968	0.480484	−	0.877003i	$-0.340461\pi$
$491$	10455.2	0.960968	0.480484	+	0.877003i	$0.340461\pi$
$492$	−323.313	−0.0296262
$493$	21477.3	1.96205
$494$	−4931.30	−0.449129
$495$	−8786.32	−0.797809
$496$	−1821.77	−0.164919
$497$	−2654.27	−0.239558
$498$	−2013.06	−0.181139
$499$	−630.152	−0.0565320	−0.0282660	−	0.999600i	$-0.508999\pi$
$499$	−630.152	−0.0565320	−0.0282660	+	0.999600i	$0.508999\pi$
$500$	−3682.41	−0.329364
$501$	−3552.78	−0.316820
$502$	907.680	0.0807007
$503$	−7721.37	−0.684451	−0.342226	−	0.939618i	$-0.611181\pi$
$503$	−7721.37	−0.684451	−0.342226	+	0.939618i	$0.611181\pi$
$504$	−7943.44	−0.702041
$505$	−9376.81	−0.826263
$506$	9034.57	0.793746
$507$	−6748.78	−0.591172
$508$	−3842.59	−0.335605
$509$	−8570.98	−0.746369	−0.373185	−	0.927757i	$-0.621734\pi$
$509$	−8570.98	−0.746369	−0.373185	+	0.927757i	$0.621734\pi$
$510$	−15773.4	−1.36953
$511$	2755.07	0.238507
$512$	9652.91	0.833208
$513$	17865.2	1.53756
$514$	−13968.1	−1.19865
$515$	1426.05	0.122018
$516$	0	0
$517$	14433.6	1.22783
$518$	−10082.6	−0.855219
$519$	−8341.28	−0.705475
$520$	−8093.86	−0.682575
$521$	4891.04	0.411286	0.205643	−	0.978627i	$-0.434071\pi$
$521$	4891.04	0.411286	0.205643	+	0.978627i	$0.434071\pi$
$522$	5927.45	0.497006
$523$	−9198.83	−0.769095	−0.384547	−	0.923105i	$-0.625642\pi$
$523$	−9198.83	−0.769095	−0.384547	+	0.923105i	$0.625642\pi$
$524$	6179.73	0.515196
$525$	16392.2	1.36269
$526$	9834.37	0.815207
$527$	6819.63	0.563696
$528$	−3847.14	−0.317093
$529$	777.205	0.0638781
$530$	19511.0	1.59907
$531$	−6999.24	−0.572018
$532$	8811.47	0.718093
$533$	−528.649	−0.0429612
$534$	−10083.8	−0.817170
$535$	−12099.0	−0.977726
$536$	10278.0	0.828251
$537$	−1974.60	−0.158678
$538$	−10503.6	−0.841714
$539$	7538.24	0.602403
$540$	8210.81	0.654328
$541$	13591.8	1.08015	0.540073	−	0.841618i	$-0.318397\pi$
$541$	13591.8	1.08015	0.540073	+	0.841618i	$0.318397\pi$
$542$	5153.97	0.408454
$543$	12206.9	0.964727
$544$	−14483.4	−1.14149
$545$	−1503.52	−0.118172
$546$	3504.29	0.274670
$547$	4650.60	0.363519	0.181760	−	0.983343i	$-0.441821\pi$
$547$	4650.60	0.363519	0.181760	+	0.983343i	$0.441821\pi$
$548$	9276.90	0.723156
$549$	−3377.33	−0.262552
$550$	−15208.7	−1.17909
$551$	−23481.4	−1.81550
$552$	−10174.1	−0.784486
$553$	5942.61	0.456972
$554$	−15074.6	−1.15606
$555$	12559.1	0.960545
$556$	−8002.29	−0.610382
$557$	−15039.1	−1.14404	−0.572018	−	0.820241i	$-0.693839\pi$
$557$	−15039.1	−1.14404	−0.572018	+	0.820241i	$0.693839\pi$
$558$	1882.12	0.142790
$559$	0	0
$560$	−12310.9	−0.928983
$561$	14401.4	1.08383
$562$	9469.85	0.710785
$563$	−6522.84	−0.488286	−0.244143	−	0.969739i	$-0.578507\pi$
$563$	−6522.84	−0.488286	−0.244143	+	0.969739i	$0.578507\pi$
$564$	−4551.40	−0.339802
$565$	−4999.52	−0.372268
$566$	−5788.38	−0.429865
$567$	−3965.45	−0.293709
$568$	−2773.24	−0.204863
$569$	−16872.5	−1.24311	−0.621557	−	0.783369i	$-0.713499\pi$
$569$	−16872.5	−1.24311	−0.621557	+	0.783369i	$0.713499\pi$
$570$	17245.2	1.26723
$571$	−7242.05	−0.530771	−0.265385	−	0.964142i	$-0.585499\pi$
$571$	−7242.05	−0.530771	−0.265385	+	0.964142i	$0.585499\pi$
$572$	2069.28	0.151260
$573$	7927.86	0.577995
$574$	−1484.19	−0.107925
$575$	−21790.2	−1.58037
$576$	−7234.53	−0.523331
$577$	6375.53	0.459994	0.229997	−	0.973191i	$-0.426128\pi$
$577$	6375.53	0.459994	0.229997	+	0.973191i	$0.426128\pi$
$578$	−15969.8	−1.14923
$579$	957.770	0.0687454
$580$	−10792.0	−0.772608
$581$	5881.50	0.419976
$582$	1006.42	0.0716796
$583$	−17813.9	−1.26548
$584$	2878.56	0.203965
$585$	4530.25	0.320175
$586$	9122.72	0.643099
$587$	−16732.5	−1.17653	−0.588265	−	0.808668i	$-0.700189\pi$
$587$	−16732.5	−1.17653	−0.588265	+	0.808668i	$0.700189\pi$
$588$	−2377.06	−0.166715
$589$	−7455.96	−0.521592
$590$	−20022.6	−1.39715
$591$	17809.6	1.23957
$592$	−5707.23	−0.396226
$593$	15900.4	1.10110	0.550550	−	0.834802i	$-0.314418\pi$
$593$	15900.4	1.10110	0.550550	+	0.834802i	$0.314418\pi$
$594$	11778.8	0.813619
$595$	46084.7	3.17528
$596$	−3019.88	−0.207549
$597$	−6791.47	−0.465589
$598$	−4658.25	−0.318545
$599$	18912.3	1.29004	0.645020	−	0.764166i	$-0.276849\pi$
$599$	18912.3	1.29004	0.645020	+	0.764166i	$0.276849\pi$
$600$	17126.9	1.16534
$601$	−11023.2	−0.748164	−0.374082	−	0.927396i	$-0.622042\pi$
$601$	−11023.2	−0.748164	−0.374082	+	0.927396i	$0.622042\pi$
$602$	0	0
$603$	−5752.75	−0.388508
$604$	−6352.72	−0.427961
$605$	−731.420	−0.0491511
$606$	4241.71	0.284336
$607$	1802.86	0.120553	0.0602767	−	0.998182i	$-0.480802\pi$
$607$	1802.86	0.120553	0.0602767	+	0.998182i	$0.480802\pi$
$608$	15834.8	1.05623
$609$	16686.3	1.11029
$610$	−9661.45	−0.641280
$611$	−7442.00	−0.492751
$612$	4713.17	0.311305
$613$	−11618.0	−0.765492	−0.382746	−	0.923854i	$-0.625022\pi$
$613$	−11618.0	−0.765492	−0.382746	+	0.923854i	$0.625022\pi$
$614$	12160.9	0.799308
$615$	1848.74	0.121217
$616$	20747.1	1.35702
$617$	10241.1	0.668221	0.334111	−	0.942534i	$-0.391564\pi$
$617$	10241.1	0.668221	0.334111	+	0.942534i	$0.391564\pi$
$618$	−645.090	−0.0419892
$619$	−20093.5	−1.30473	−0.652364	−	0.757906i	$-0.726223\pi$
$619$	−20093.5	−1.30473	−0.652364	+	0.757906i	$0.726223\pi$
$620$	−3426.74	−0.221970
$621$	16876.0	1.09051
$622$	−13048.1	−0.841127
$623$	29461.5	1.89462
$624$	1983.59	0.127255
$625$	−2884.12	−0.184583
$626$	−12321.5	−0.786690
$627$	−15745.2	−1.00287
$628$	11548.1	0.733790
$629$	21364.5	1.35431
$630$	12718.7	0.804328
$631$	13927.9	0.878705	0.439353	−	0.898315i	$-0.355208\pi$
$631$	13927.9	0.878705	0.439353	+	0.898315i	$0.355208\pi$
$632$	6208.96	0.390790
$633$	2836.02	0.178075
$634$	11559.7	0.724126
$635$	21972.3	1.37314
$636$	5617.33	0.350223
$637$	−3886.74	−0.241755
$638$	−15481.6	−0.960694
$639$	1552.22	0.0960953
$640$	−1983.24	−0.122492
$641$	−13156.6	−0.810693	−0.405346	−	0.914163i	$-0.632849\pi$
$641$	−13156.6	−0.810693	−0.405346	+	0.914163i	$0.632849\pi$
$642$	5473.10	0.336458
$643$	18419.6	1.12970	0.564850	−	0.825194i	$-0.308934\pi$
$643$	18419.6	1.12970	0.564850	+	0.825194i	$0.308934\pi$
$644$	8323.56	0.509308
$645$	0	0
$646$	29336.2	1.78672
$647$	−15425.8	−0.937330	−0.468665	−	0.883376i	$-0.655265\pi$
$647$	−15425.8	−0.937330	−0.468665	+	0.883376i	$0.655265\pi$
$648$	−4143.18	−0.251172
$649$	18281.0	1.10569
$650$	7841.65	0.473192
$651$	5298.36	0.318985
$652$	−4452.16	−0.267424
$653$	−874.442	−0.0524036	−0.0262018	−	0.999657i	$-0.508341\pi$
$653$	−874.442	−0.0524036	−0.0262018	+	0.999657i	$0.508341\pi$
$654$	680.133	0.0406656
$655$	−35336.3	−2.10794
$656$	−840.124	−0.0500020
$657$	−1611.17	−0.0956738
$658$	−20893.5	−1.23786
$659$	−33139.9	−1.95895	−0.979474	−	0.201569i	$-0.935396\pi$
$659$	−33139.9	−1.95895	−0.979474	+	0.201569i	$0.935396\pi$
$660$	−7236.46	−0.426786
$661$	−20496.3	−1.20607	−0.603037	−	0.797713i	$-0.706043\pi$
$661$	−20496.3	−1.20607	−0.603037	+	0.797713i	$0.706043\pi$
$662$	−6269.11	−0.368060
$663$	−7425.41	−0.434961
$664$	6145.11	0.359151
$665$	−50384.8	−2.93811
$666$	5896.31	0.343059
$667$	−22181.2	−1.28764
$668$	3036.86	0.175898
$669$	−5781.02	−0.334091
$670$	−16456.8	−0.948926
$671$	8821.08	0.507502
$672$	−11252.6	−0.645949
$673$	−27630.6	−1.58259	−0.791294	−	0.611436i	$-0.790592\pi$
$673$	−27630.6	−1.58259	−0.791294	+	0.611436i	$0.790592\pi$
$674$	13972.3	0.798503
$675$	−28408.9	−1.61994
$676$	5768.76	0.328218
$677$	−9559.39	−0.542684	−0.271342	−	0.962483i	$-0.587467\pi$
$677$	−9559.39	−0.542684	−0.271342	+	0.962483i	$0.587467\pi$
$678$	2261.59	0.128106
$679$	−2940.43	−0.166191
$680$	48150.2	2.71541
$681$	−10266.8	−0.577714
$682$	−4915.82	−0.276007
$683$	16330.7	0.914899	0.457449	−	0.889236i	$-0.348763\pi$
$683$	16330.7	0.914899	0.457449	+	0.889236i	$0.348763\pi$
$684$	−5152.95	−0.288053
$685$	−53046.2	−2.95882
$686$	6920.23	0.385154
$687$	−16948.1	−0.941208
$688$	0	0
$689$	9184.90	0.507862
$690$	16290.3	0.898785
$691$	−410.309	−0.0225888	−0.0112944	−	0.999936i	$-0.503595\pi$
$691$	−410.309	−0.0225888	−0.0112944	+	0.999936i	$0.503595\pi$
$692$	7129.99	0.391679
$693$	−11612.4	−0.636537
$694$	−13505.1	−0.738682
$695$	45757.9	2.49740
$696$	17434.2	0.949487
$697$	3144.93	0.170908
$698$	16366.7	0.887522
$699$	−13128.3	−0.710381
$700$	−14011.8	−0.756567
$701$	−19208.4	−1.03494	−0.517468	−	0.855703i	$-0.673125\pi$
$701$	−19208.4	−1.03494	−0.517468	+	0.855703i	$0.673125\pi$
$702$	−6073.17	−0.326520
$703$	−23358.0	−1.25315
$704$	18895.5	1.01158
$705$	26025.4	1.39031
$706$	−16984.7	−0.905421
$707$	−12392.9	−0.659239
$708$	−5764.61	−0.305999
$709$	−34220.8	−1.81268	−0.906338	−	0.422553i	$-0.861134\pi$
$709$	−34220.8	−1.81268	−0.906338	+	0.422553i	$0.861134\pi$
$710$	4440.40	0.234712
$711$	−3475.24	−0.183308
$712$	30782.0	1.62023
$713$	−7043.11	−0.369939
$714$	−20846.9	−1.09269
$715$	−11832.3	−0.618887
$716$	1687.86	0.0880980
$717$	14415.0	0.750819
$718$	22132.8	1.15040
$719$	−2084.14	−0.108102	−0.0540510	−	0.998538i	$-0.517213\pi$
$719$	−2084.14	−0.108102	−0.0540510	+	0.998538i	$0.517213\pi$
$720$	7199.42	0.372648
$721$	1884.74	0.0973528
$722$	−16908.1	−0.871546
$723$	5086.60	0.261649
$724$	−10434.2	−0.535616
$725$	37339.6	1.91277
$726$	330.866	0.0169140
$727$	−2674.32	−0.136430	−0.0682152	−	0.997671i	$-0.521730\pi$
$727$	−2674.32	−0.136430	−0.0682152	+	0.997671i	$0.521730\pi$
$728$	−10697.2	−0.544597
$729$	16896.7	0.858442
$730$	−4609.04	−0.233682
$731$	0	0
$732$	−2781.59	−0.140451
$733$	35700.6	1.79895	0.899475	−	0.436972i	$-0.143949\pi$
$733$	35700.6	1.79895	0.899475	+	0.436972i	$0.143949\pi$
$734$	−3419.15	−0.171939
$735$	13592.3	0.682120
$736$	14958.0	0.749132
$737$	15025.3	0.750970
$738$	867.956	0.0432925
$739$	−36565.4	−1.82014	−0.910069	−	0.414457i	$-0.863972\pi$
$739$	−36565.4	−1.82014	−0.910069	+	0.414457i	$0.863972\pi$
$740$	−10735.3	−0.533293
$741$	8118.26	0.402472
$742$	25786.8	1.27582
$743$	14478.5	0.714892	0.357446	−	0.933934i	$-0.383648\pi$
$743$	14478.5	0.714892	0.357446	+	0.933934i	$0.383648\pi$
$744$	5535.83	0.272787
$745$	17268.0	0.849194
$746$	−25288.4	−1.24112
$747$	−3439.51	−0.168467
$748$	−12310.1	−0.601741
$749$	−15990.6	−0.780085
$750$	−9525.08	−0.463742
$751$	−22881.9	−1.11181	−0.555907	−	0.831245i	$-0.687629\pi$
$751$	−22881.9	−1.11181	−0.555907	+	0.831245i	$0.687629\pi$
$752$	−11826.7	−0.573507
$753$	−1494.29	−0.0723172
$754$	7982.36	0.385544
$755$	36325.4	1.75102
$756$	10851.8	0.522059
$757$	−2346.61	−0.112667	−0.0563336	−	0.998412i	$-0.517941\pi$
$757$	−2346.61	−0.112667	−0.0563336	+	0.998412i	$0.517941\pi$
$758$	−954.221	−0.0457241
$759$	−14873.3	−0.711289
$760$	−52643.1	−2.51259
$761$	2589.59	0.123354	0.0616772	−	0.998096i	$-0.480355\pi$
$761$	2589.59	0.123354	0.0616772	+	0.998096i	$0.480355\pi$
$762$	−9939.43	−0.472530
$763$	−1987.12	−0.0942840
$764$	−6776.61	−0.320902
$765$	−26950.4	−1.27372
$766$	−6975.82	−0.329043
$767$	−9425.72	−0.443733
$768$	−14423.2	−0.677671
$769$	−4652.86	−0.218188	−0.109094	−	0.994031i	$-0.534795\pi$
$769$	−4652.86	−0.218188	−0.109094	+	0.994031i	$0.534795\pi$
$770$	−33219.5	−1.55474
$771$	22995.3	1.07413
$772$	−818.687	−0.0381673
$773$	−32809.2	−1.52661	−0.763303	−	0.646041i	$-0.776424\pi$
$773$	−32809.2	−1.52661	−0.763303	+	0.646041i	$0.776424\pi$
$774$	0	0
$775$	11856.3	0.549537
$776$	−3072.22	−0.142122
$777$	16598.7	0.766376
$778$	−1325.70	−0.0610908
$779$	−3438.38	−0.158142
$780$	3731.13	0.171277
$781$	−4054.17	−0.185748
$782$	27711.8	1.26723
$783$	−28918.6	−1.31988
$784$	−6176.76	−0.281376
$785$	−66033.2	−3.00233
$786$	15984.8	0.725391
$787$	8812.27	0.399140	0.199570	−	0.979884i	$-0.436045\pi$
$787$	8812.27	0.399140	0.199570	+	0.979884i	$0.436045\pi$
$788$	−15223.4	−0.688210
$789$	−16190.0	−0.730520
$790$	−9941.56	−0.447728
$791$	−6607.61	−0.297016
$792$	−12132.9	−0.544349
$793$	−4548.17	−0.203670
$794$	−27930.7	−1.24839
$795$	−32120.5	−1.43295
$796$	5805.25	0.258494
$797$	−2591.35	−0.115170	−0.0575849	−	0.998341i	$-0.518340\pi$
$797$	−2591.35	−0.115170	−0.0575849	+	0.998341i	$0.518340\pi$
$798$	22792.1	1.01107
$799$	44272.3	1.96025
$800$	−25180.2	−1.11282
$801$	−17229.1	−0.760001
$802$	21969.4	0.967290
$803$	4208.13	0.184934
$804$	−4738.00	−0.207831
$805$	−47594.9	−2.08385
$806$	2534.61	0.110767
$807$	17291.8	0.754274
$808$	−12948.3	−0.563763
$809$	26425.8	1.14843	0.574217	−	0.818703i	$-0.305307\pi$
$809$	26425.8	1.14843	0.574217	+	0.818703i	$0.305307\pi$
$810$	6633.90	0.287767
$811$	1270.91	0.0550282	0.0275141	−	0.999621i	$-0.491241\pi$
$811$	1270.91	0.0550282	0.0275141	+	0.999621i	$0.491241\pi$
$812$	−14263.2	−0.616430
$813$	−8484.83	−0.366022
$814$	−15400.3	−0.663120
$815$	25457.9	1.09417
$816$	−11800.4	−0.506245
$817$	0	0
$818$	−29430.0	−1.25794
$819$	5987.40	0.255454
$820$	−1580.27	−0.0672993
$821$	−6574.06	−0.279460	−0.139730	−	0.990190i	$-0.544623\pi$
$821$	−6574.06	−0.279460	−0.139730	+	0.990190i	$0.544623\pi$
$822$	23996.0	1.01820
$823$	1387.09	0.0587496	0.0293748	−	0.999568i	$-0.490648\pi$
$823$	1387.09	0.0587496	0.0293748	+	0.999568i	$0.490648\pi$
$824$	1969.21	0.0832534
$825$	25037.7	1.05661
$826$	−26462.9	−1.11472
$827$	−42201.7	−1.77448	−0.887241	−	0.461307i	$-0.847381\pi$
$827$	−42201.7	−1.77448	−0.887241	+	0.461307i	$0.847381\pi$
$828$	−4867.62	−0.204301
$829$	31216.5	1.30783	0.653916	−	0.756567i	$-0.273125\pi$
$829$	31216.5	1.30783	0.653916	+	0.756567i	$0.273125\pi$
$830$	−9839.32	−0.411479
$831$	24816.9	1.03597
$832$	−9742.58	−0.405965
$833$	23122.1	0.961746
$834$	−20699.1	−0.859413
$835$	−17365.1	−0.719693
$836$	13458.7	0.556795
$837$	−9182.43	−0.379201
$838$	9351.71	0.385501
$839$	−8957.21	−0.368578	−0.184289	−	0.982872i	$-0.558998\pi$
$839$	−8957.21	−0.368578	−0.184289	+	0.982872i	$0.558998\pi$
$840$	37409.3	1.53660
$841$	13620.6	0.558472
$842$	−14455.3	−0.591643
$843$	−15589.9	−0.636947
$844$	−2424.18	−0.0988670
$845$	−32986.3	−1.34292
$846$	12218.5	0.496551
$847$	−966.681	−0.0392155
$848$	14596.6	0.591094
$849$	9529.24	0.385209
$850$	−46649.9	−1.88244
$851$	−22064.6	−0.888796
$852$	1278.42	0.0514059
$853$	6229.24	0.250041	0.125021	−	0.992154i	$-0.460100\pi$
$853$	6229.24	0.250041	0.125021	+	0.992154i	$0.460100\pi$
$854$	−12769.1	−0.511649
$855$	29465.1	1.17858
$856$	−16707.3	−0.667107
$857$	1759.08	0.0701154	0.0350577	−	0.999385i	$-0.488838\pi$
$857$	1759.08	0.0701154	0.0350577	+	0.999385i	$0.488838\pi$
$858$	5352.49	0.212973
$859$	34126.5	1.35551	0.677754	−	0.735289i	$-0.262953\pi$
$859$	34126.5	1.35551	0.677754	+	0.735289i	$0.262953\pi$
$860$	0	0
$861$	2443.38	0.0967134
$862$	−3604.81	−0.142437
$863$	−5588.78	−0.220445	−0.110223	−	0.993907i	$-0.535156\pi$
$863$	−5588.78	−0.220445	−0.110223	+	0.993907i	$0.535156\pi$
$864$	19501.5	0.767887
$865$	−40770.0	−1.60257
$866$	22269.4	0.873840
$867$	26290.6	1.02984
$868$	−4528.96	−0.177100
$869$	9076.82	0.354327
$870$	−27915.0	−1.08783
$871$	−7747.10	−0.301378
$872$	−2076.19	−0.0806291
$873$	1719.57	0.0666649
$874$	−30297.6	−1.17258
$875$	27829.1	1.07520
$876$	−1326.97	−0.0511804
$877$	−15627.1	−0.601700	−0.300850	−	0.953672i	$-0.597270\pi$
$877$	−15627.1	−0.601700	−0.300850	+	0.953672i	$0.597270\pi$
$878$	21865.6	0.840464
$879$	−15018.5	−0.576292
$880$	−18803.8	−0.720314
$881$	6494.75	0.248370	0.124185	−	0.992259i	$-0.460368\pi$
$881$	6494.75	0.248370	0.124185	+	0.992259i	$0.460368\pi$
$882$	6381.39	0.243620
$883$	20625.9	0.786091	0.393045	−	0.919519i	$-0.371422\pi$
$883$	20625.9	0.786091	0.393045	+	0.919519i	$0.371422\pi$
$884$	6347.12	0.241490
$885$	32962.6	1.25201
$886$	−3864.86	−0.146549
$887$	27953.0	1.05814	0.529070	−	0.848578i	$-0.322541\pi$
$887$	27953.0	1.05814	0.529070	+	0.848578i	$0.322541\pi$
$888$	17342.6	0.655384
$889$	29039.7	1.09557
$890$	−49287.0	−1.85630
$891$	−6056.87	−0.227736
$892$	4941.52	0.185487
$893$	−48403.3	−1.81384
$894$	−7811.36	−0.292227
$895$	−9651.33	−0.360456
$896$	−2621.16	−0.0977307
$897$	7668.74	0.285453
$898$	−1948.72	−0.0724162
$899$	12069.1	0.447748
$900$	8194.11	0.303486
$901$	−54640.9	−2.02037
$902$	−2266.97	−0.0836828
$903$	0	0
$904$	−6903.76	−0.254000
$905$	59664.0	2.19149
$906$	−16432.2	−0.602565
$907$	−50444.5	−1.84673	−0.923365	−	0.383924i	$-0.874572\pi$
$907$	−50444.5	−1.84673	−0.923365	+	0.383924i	$0.874572\pi$
$908$	8775.87	0.320746
$909$	7247.36	0.264444
$910$	17128.0	0.623944
$911$	17096.5	0.621770	0.310885	−	0.950448i	$-0.399375\pi$
$911$	17096.5	0.621770	0.310885	+	0.950448i	$0.399375\pi$
$912$	12901.5	0.468432
$913$	8983.48	0.325640
$914$	−33055.6	−1.19626
$915$	15905.4	0.574662
$916$	14487.0	0.522557
$917$	−46702.2	−1.68183
$918$	36129.2	1.29896
$919$	3815.06	0.136939	0.0684697	−	0.997653i	$-0.478188\pi$
$919$	3815.06	0.136939	0.0684697	+	0.997653i	$0.478188\pi$
$920$	−49728.1	−1.78205
$921$	−20020.2	−0.716273
$922$	9718.09	0.347124
$923$	2090.34	0.0745443
$924$	−9564.07	−0.340514
$925$	37143.4	1.32029
$926$	17369.1	0.616396
$927$	−1102.20	−0.0390517
$928$	−25632.1	−0.906696
$929$	4321.76	0.152629	0.0763144	−	0.997084i	$-0.475685\pi$
$929$	4321.76	0.152629	0.0763144	+	0.997084i	$0.475685\pi$
$930$	−8863.77	−0.312532
$931$	−25279.6	−0.889910
$932$	11221.8	0.394403
$933$	21480.7	0.753748
$934$	−12439.7	−0.435801
$935$	70390.4	2.46205
$936$	6255.76	0.218457
$937$	−21679.8	−0.755869	−0.377935	−	0.925832i	$-0.623366\pi$
$937$	−21679.8	−0.755869	−0.377935	+	0.925832i	$0.623366\pi$
$938$	−21750.1	−0.757107
$939$	20284.6	0.704966
$940$	−22246.1	−0.771901
$941$	−9544.57	−0.330652	−0.165326	−	0.986239i	$-0.552868\pi$
$941$	−9544.57	−0.330652	−0.165326	+	0.986239i	$0.552868\pi$
$942$	29870.9	1.03317
$943$	−3247.99	−0.112162
$944$	−14979.3	−0.516455
$945$	−62051.7	−2.13602
$946$	0	0
$947$	23883.0	0.819529	0.409764	−	0.912191i	$-0.365611\pi$
$947$	23883.0	0.819529	0.409764	+	0.912191i	$0.365611\pi$
$948$	−2862.23	−0.0980600
$949$	−2169.72	−0.0742173
$950$	51002.7	1.74184
$951$	−19030.5	−0.648902
$952$	63637.8	2.16651
$953$	−14159.4	−0.481288	−0.240644	−	0.970613i	$-0.577359\pi$
$953$	−14159.4	−0.481288	−0.240644	+	0.970613i	$0.577359\pi$
$954$	−15080.1	−0.511779
$955$	38749.3	1.31298
$956$	−12321.7	−0.416854
$957$	25486.9	0.860894
$958$	−35002.9	−1.18047
$959$	−70108.5	−2.36071
$960$	34070.7	1.14544
$961$	−25958.8	−0.871362
$962$	7940.42	0.266122
$963$	9351.31	0.312920
$964$	−4347.94	−0.145267
$965$	4681.33	0.156163
$966$	21530.1	0.717101
$967$	899.639	0.0299177	0.0149589	−	0.999888i	$-0.495238\pi$
$967$	899.639	0.0299177	0.0149589	+	0.999888i	$0.495238\pi$
$968$	−1010.01	−0.0335360
$969$	−48295.4	−1.60111
$970$	4919.13	0.162828
$971$	−2024.50	−0.0669097	−0.0334549	−	0.999440i	$-0.510651\pi$
$971$	−2024.50	−0.0669097	−0.0334549	+	0.999440i	$0.510651\pi$
$972$	−10550.9	−0.348168
$973$	60475.9	1.99257
$974$	−45219.4	−1.48760
$975$	−12909.5	−0.424035
$976$	−7227.91	−0.237049
$977$	58871.4	1.92780	0.963901	−	0.266262i	$-0.0857887\pi$
$977$	58871.4	1.92780	0.963901	+	0.266262i	$0.0857887\pi$
$978$	−11516.2	−0.376530
$979$	44999.9	1.46905
$980$	−11618.5	−0.378713
$981$	1162.07	0.0378207
$982$	−23116.7	−0.751204
$983$	−30098.3	−0.976588	−0.488294	−	0.872679i	$-0.662381\pi$
$983$	−30098.3	−0.976588	−0.488294	+	0.872679i	$0.662381\pi$
$984$	2552.89	0.0827066
$985$	87048.6	2.81584
$986$	−47486.9	−1.53377
$987$	34396.4	1.10927
$988$	−6939.36	−0.223452
$989$	0	0
$990$	19426.8	0.623660
$991$	23059.0	0.739144	0.369572	−	0.929202i	$-0.379504\pi$
$991$	23059.0	0.739144	0.369572	+	0.929202i	$0.379504\pi$
$992$	−8138.86	−0.260493
$993$	10320.7	0.329825
$994$	5868.66	0.187266
$995$	−33195.0	−1.05764
$996$	−2832.79	−0.0901210
$997$	−33136.5	−1.05260	−0.526301	−	0.850299i	$-0.676421\pi$
$997$	−33136.5	−1.05260	−0.526301	+	0.850299i	$0.676421\pi$
$998$	1393.28	0.0441919
$999$	−28766.7	−0.911049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.l.1.18		60
43.9	even	21		43.4.g.a.38.8	yes	120
43.24	even	21		43.4.g.a.17.8	✓	120
43.42	odd	2		1849.4.a.k.1.43		60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.17.8	✓	120	43.24	even	21
43.4.g.a.38.8	yes	120	43.9	even	21
1849.4.a.k.1.43		60	43.42	odd	2
1849.4.a.l.1.18		60	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)

\(f(q)\)	\(=\)	\(q-2.21102 q^{2} +3.63995 q^{3} -3.11137 q^{4} +17.7911 q^{5} -8.04801 q^{6} +23.5136 q^{7} +24.5675 q^{8} -13.7508 q^{9} +O(q^{10})\)	\(q-2.21102 q^{2} +3.63995 q^{3} -3.11137 q^{4} +17.7911 q^{5} -8.04801 q^{6} +23.5136 q^{7} +24.5675 q^{8} -13.7508 q^{9} -39.3366 q^{10} +35.9150 q^{11} -11.3252 q^{12} -18.5179 q^{13} -51.9892 q^{14} +64.7587 q^{15} -29.4284 q^{16} +110.163 q^{17} +30.4033 q^{18} -120.442 q^{19} -55.3547 q^{20} +85.5884 q^{21} -79.4090 q^{22} -113.773 q^{23} +89.4244 q^{24} +191.524 q^{25} +40.9435 q^{26} -148.331 q^{27} -73.1596 q^{28} +194.960 q^{29} -143.183 q^{30} +61.9051 q^{31} -131.473 q^{32} +130.729 q^{33} -243.572 q^{34} +418.334 q^{35} +42.7838 q^{36} +193.936 q^{37} +266.300 q^{38} -67.4041 q^{39} +437.083 q^{40} +28.5481 q^{41} -189.238 q^{42} -111.745 q^{44} -244.642 q^{45} +251.554 q^{46} +401.882 q^{47} -107.118 q^{48} +209.891 q^{49} -423.464 q^{50} +400.986 q^{51} +57.6160 q^{52} -496.002 q^{53} +327.963 q^{54} +638.968 q^{55} +577.672 q^{56} -438.401 q^{57} -431.062 q^{58} +509.007 q^{59} -201.488 q^{60} +245.610 q^{61} -136.874 q^{62} -323.331 q^{63} +526.118 q^{64} -329.454 q^{65} -289.044 q^{66} +418.358 q^{67} -342.757 q^{68} -414.126 q^{69} -924.946 q^{70} -112.882 q^{71} -337.823 q^{72} +117.169 q^{73} -428.798 q^{74} +697.136 q^{75} +374.739 q^{76} +844.493 q^{77} +149.032 q^{78} +252.731 q^{79} -523.564 q^{80} -168.645 q^{81} -63.1204 q^{82} +250.132 q^{83} -266.297 q^{84} +1959.92 q^{85} +709.646 q^{87} +882.343 q^{88} +1252.96 q^{89} +540.909 q^{90} -435.422 q^{91} +353.989 q^{92} +225.331 q^{93} -888.571 q^{94} -2142.79 q^{95} -478.555 q^{96} -125.052 q^{97} -464.074 q^{98} -493.860 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9}+O(q^{10}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} + 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} + 625 q^{18} + 610 q^{19} + 345 q^{20} + 611 q^{21} + 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} + 1071 q^{26} + 1609 q^{27} + 46 q^{28} + 773 q^{29} + 375 q^{30} - 97 q^{31} + 1967 q^{32} + 500 q^{33} + 217 q^{34} + 247 q^{35} + 175 q^{36} + 228 q^{37} + 1253 q^{38} + 1493 q^{39} + 2220 q^{40} - 951 q^{41} + 2643 q^{42} - 1378 q^{44} + 1086 q^{45} - 565 q^{46} - 2 q^{47} + 2303 q^{48} + 1264 q^{49} + 3273 q^{50} + 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} + 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} + 2999 q^{61} + 5569 q^{62} + 2377 q^{63} + 2082 q^{64} + 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} + 1817 q^{69} + 2738 q^{70} + 8003 q^{71} + 1412 q^{72} - 1011 q^{73} - 1413 q^{74} + 7457 q^{75} + 5516 q^{76} + 4052 q^{77} + 1091 q^{78} - 4422 q^{79} + 1610 q^{80} + 2108 q^{81} + 4676 q^{82} - 297 q^{83} - 54 q^{84} + 4333 q^{85} + 1377 q^{87} + 3652 q^{88} + 2480 q^{89} - 1414 q^{90} + 4551 q^{91} - 3286 q^{92} + 4 q^{93} + 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} + 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 + 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 + 625 * q^18 + 610 * q^19 + 345 * q^20 + 611 * q^21 + 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 + 1071 * q^26 + 1609 * q^27 + 46 * q^28 + 773 * q^29 + 375 * q^30 - 97 * q^31 + 1967 * q^32 + 500 * q^33 + 217 * q^34 + 247 * q^35 + 175 * q^36 + 228 * q^37 + 1253 * q^38 + 1493 * q^39 + 2220 * q^40 - 951 * q^41 + 2643 * q^42 - 1378 * q^44 + 1086 * q^45 - 565 * q^46 - 2 * q^47 + 2303 * q^48 + 1264 * q^49 + 3273 * q^50 + 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 + 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 + 2999 * q^61 + 5569 * q^62 + 2377 * q^63 + 2082 * q^64 + 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 + 1817 * q^69 + 2738 * q^70 + 8003 * q^71 + 1412 * q^72 - 1011 * q^73 - 1413 * q^74 + 7457 * q^75 + 5516 * q^76 + 4052 * q^77 + 1091 * q^78 - 4422 * q^79 + 1610 * q^80 + 2108 * q^81 + 4676 * q^82 - 297 * q^83 - 54 * q^84 + 4333 * q^85 + 1377 * q^87 + 3652 * q^88 + 2480 * q^89 - 1414 * q^90 + 4551 * q^91 - 3286 * q^92 + 4 * q^93 + 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 + 753 * q^98 + 5072 * q^99

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