LMFDB - Embedded newform 1045.4.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.6569959560$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.10270 q^{2} +7.55803 q^{3} +1.62677 q^{4} +5.00000 q^{5} -23.4503 q^{6} +17.4690 q^{7} +19.7742 q^{8} +30.1238 q^{9} +O(q^{10})$	\(q-3.10270 q^{2} +7.55803 q^{3} +1.62677 q^{4} +5.00000 q^{5} -23.4503 q^{6} +17.4690 q^{7} +19.7742 q^{8} +30.1238 q^{9} -15.5135 q^{10} +11.0000 q^{11} +12.2952 q^{12} +0.157040 q^{13} -54.2011 q^{14} +37.7901 q^{15} -74.3678 q^{16} +21.4377 q^{17} -93.4652 q^{18} -19.0000 q^{19} +8.13384 q^{20} +132.031 q^{21} -34.1297 q^{22} +69.9536 q^{23} +149.454 q^{24} +25.0000 q^{25} -0.487250 q^{26} +23.6096 q^{27} +28.4180 q^{28} +106.845 q^{29} -117.252 q^{30} +182.544 q^{31} +72.5472 q^{32} +83.1383 q^{33} -66.5150 q^{34} +87.3449 q^{35} +49.0044 q^{36} +373.311 q^{37} +58.9514 q^{38} +1.18692 q^{39} +98.8712 q^{40} -268.068 q^{41} -409.653 q^{42} -36.4071 q^{43} +17.8945 q^{44} +150.619 q^{45} -217.045 q^{46} +181.342 q^{47} -562.074 q^{48} -37.8346 q^{49} -77.5676 q^{50} +162.027 q^{51} +0.255469 q^{52} -362.533 q^{53} -73.2537 q^{54} +55.0000 q^{55} +345.436 q^{56} -143.603 q^{57} -331.509 q^{58} -278.927 q^{59} +61.4758 q^{60} +679.259 q^{61} -566.381 q^{62} +526.232 q^{63} +369.850 q^{64} +0.785202 q^{65} -257.954 q^{66} -589.528 q^{67} +34.8743 q^{68} +528.711 q^{69} -271.005 q^{70} +732.175 q^{71} +595.675 q^{72} -166.925 q^{73} -1158.27 q^{74} +188.951 q^{75} -30.9086 q^{76} +192.159 q^{77} -3.68265 q^{78} +779.826 q^{79} -371.839 q^{80} -634.900 q^{81} +831.737 q^{82} -565.667 q^{83} +214.784 q^{84} +107.189 q^{85} +112.960 q^{86} +807.538 q^{87} +217.517 q^{88} +246.185 q^{89} -467.326 q^{90} +2.74334 q^{91} +113.798 q^{92} +1379.68 q^{93} -562.652 q^{94} -95.0000 q^{95} +548.314 q^{96} -17.9354 q^{97} +117.390 q^{98} +331.362 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) $ 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9} + 60 q^{10} + 242 q^{11} + 164 q^{12} + 207 q^{13} + 129 q^{14} + 105 q^{15} + 604 q^{16} + 209 q^{17} + 869 q^{18} - 418 q^{19} + 480 q^{20} + 45 q^{21} + 132 q^{22} + 191 q^{23} + 458 q^{24} + 550 q^{25} + 363 q^{26} + 45 q^{27} + 577 q^{28} + 389 q^{29} + 135 q^{30} + 198 q^{31} + 1149 q^{32} + 231 q^{33} + 467 q^{34} + 465 q^{35} + 1315 q^{36} + 312 q^{37} - 228 q^{38} + 137 q^{39} + 570 q^{40} + 632 q^{41} - 1794 q^{42} + 1584 q^{43} + 1056 q^{44} + 1045 q^{45} + 681 q^{46} - 54 q^{47} + 2491 q^{48} + 1063 q^{49} + 300 q^{50} + 37 q^{51} + 1246 q^{52} + 343 q^{53} + 1078 q^{54} + 1210 q^{55} - 87 q^{56} - 399 q^{57} - 1424 q^{58} + 2787 q^{59} + 820 q^{60} + 2070 q^{61} - 446 q^{62} + 1696 q^{63} + 1758 q^{64} + 1035 q^{65} + 297 q^{66} + 2423 q^{67} - 524 q^{68} - 997 q^{69} + 645 q^{70} + 2538 q^{71} + 6010 q^{72} + 1397 q^{73} - 1977 q^{74} + 525 q^{75} - 1824 q^{76} + 1023 q^{77} + 202 q^{78} + 878 q^{79} + 3020 q^{80} + 2030 q^{81} - 190 q^{82} + 4932 q^{83} - 4580 q^{84} + 1045 q^{85} - 3394 q^{86} + 6009 q^{87} + 1254 q^{88} + 1812 q^{89} + 4345 q^{90} + 4349 q^{91} - 788 q^{92} - 4848 q^{93} - 2152 q^{94} - 2090 q^{95} + 4032 q^{96} + 988 q^{97} + 1366 q^{98} + 2299 q^{99}+O(q^{100}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9 + 60 * q^10 + 242 * q^11 + 164 * q^12 + 207 * q^13 + 129 * q^14 + 105 * q^15 + 604 * q^16 + 209 * q^17 + 869 * q^18 - 418 * q^19 + 480 * q^20 + 45 * q^21 + 132 * q^22 + 191 * q^23 + 458 * q^24 + 550 * q^25 + 363 * q^26 + 45 * q^27 + 577 * q^28 + 389 * q^29 + 135 * q^30 + 198 * q^31 + 1149 * q^32 + 231 * q^33 + 467 * q^34 + 465 * q^35 + 1315 * q^36 + 312 * q^37 - 228 * q^38 + 137 * q^39 + 570 * q^40 + 632 * q^41 - 1794 * q^42 + 1584 * q^43 + 1056 * q^44 + 1045 * q^45 + 681 * q^46 - 54 * q^47 + 2491 * q^48 + 1063 * q^49 + 300 * q^50 + 37 * q^51 + 1246 * q^52 + 343 * q^53 + 1078 * q^54 + 1210 * q^55 - 87 * q^56 - 399 * q^57 - 1424 * q^58 + 2787 * q^59 + 820 * q^60 + 2070 * q^61 - 446 * q^62 + 1696 * q^63 + 1758 * q^64 + 1035 * q^65 + 297 * q^66 + 2423 * q^67 - 524 * q^68 - 997 * q^69 + 645 * q^70 + 2538 * q^71 + 6010 * q^72 + 1397 * q^73 - 1977 * q^74 + 525 * q^75 - 1824 * q^76 + 1023 * q^77 + 202 * q^78 + 878 * q^79 + 3020 * q^80 + 2030 * q^81 - 190 * q^82 + 4932 * q^83 - 4580 * q^84 + 1045 * q^85 - 3394 * q^86 + 6009 * q^87 + 1254 * q^88 + 1812 * q^89 + 4345 * q^90 + 4349 * q^91 - 788 * q^92 - 4848 * q^93 - 2152 * q^94 - 2090 * q^95 + 4032 * q^96 + 988 * q^97 + 1366 * q^98 + 2299 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.10270	−1.09697	−0.548486	−	0.836160i	$-0.684795\pi$
$2$	−3.10270	−1.09697	−0.548486	+	0.836160i	$0.684795\pi$
$3$	7.55803	1.45454	0.727272	−	0.686350i	$-0.240788\pi$
$3$	7.55803	1.45454	0.727272	+	0.686350i	$0.240788\pi$
$4$	1.62677	0.203346
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−23.4503	−1.59559
$7$	17.4690	0.943236	0.471618	−	0.881803i	$-0.343670\pi$
$7$	17.4690	0.943236	0.471618	+	0.881803i	$0.343670\pi$
$8$	19.7742	0.873906
$9$	30.1238	1.11570
$10$	−15.5135	−0.490580
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	12.2952	0.295776
$13$	0.157040	0.00335040	0.00167520	−	0.999999i	$-0.499467\pi$
$13$	0.157040	0.00335040	0.00167520	+	0.999999i	$0.499467\pi$
$14$	−54.2011	−1.03470
$15$	37.7901	0.650491
$16$	−74.3678	−1.16200
$17$	21.4377	0.305848	0.152924	−	0.988238i	$-0.451131\pi$
$17$	21.4377	0.305848	0.152924	+	0.988238i	$0.451131\pi$
$18$	−93.4652	−1.22389
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	8.13384	0.0909392
$21$	132.031	1.37198
$22$	−34.1297	−0.330749
$23$	69.9536	0.634188	0.317094	−	0.948394i	$-0.397293\pi$
$23$	69.9536	0.634188	0.317094	+	0.948394i	$0.397293\pi$
$24$	149.454	1.27113
$25$	25.0000	0.200000
$26$	−0.487250	−0.00367529
$27$	23.6096	0.168284
$28$	28.4180	0.191803
$29$	106.845	0.684160	0.342080	−	0.939671i	$-0.388869\pi$
$29$	106.845	0.684160	0.342080	+	0.939671i	$0.388869\pi$
$30$	−117.252	−0.713570
$31$	182.544	1.05761	0.528805	−	0.848743i	$-0.322640\pi$
$31$	182.544	1.05761	0.528805	+	0.848743i	$0.322640\pi$
$32$	72.5472	0.400770
$33$	83.1383	0.438561
$34$	−66.5150	−0.335507
$35$	87.3449	0.421828
$36$	49.0044	0.226872
$37$	373.311	1.65870	0.829352	−	0.558727i	$-0.188710\pi$
$37$	373.311	1.65870	0.829352	+	0.558727i	$0.188710\pi$
$38$	58.9514	0.251662
$39$	1.18692	0.00487330
$40$	98.8712	0.390823
$41$	−268.068	−1.02110	−0.510552	−	0.859847i	$-0.670559\pi$
$41$	−268.068	−1.02110	−0.510552	+	0.859847i	$0.670559\pi$
$42$	−409.653	−1.50502
$43$	−36.4071	−0.129117	−0.0645585	−	0.997914i	$-0.520564\pi$
$43$	−36.4071	−0.129117	−0.0645585	+	0.997914i	$0.520564\pi$
$44$	17.8945	0.0613112
$45$	150.619	0.498954
$46$	−217.045	−0.695687
$47$	181.342	0.562798	0.281399	−	0.959591i	$-0.409202\pi$
$47$	181.342	0.562798	0.281399	+	0.959591i	$0.409202\pi$
$48$	−562.074	−1.69017
$49$	−37.8346	−0.110305
$50$	−77.5676	−0.219394
$51$	162.027	0.444869
$52$	0.255469	0.000681291 0
$53$	−362.533	−0.939580	−0.469790	−	0.882778i	$-0.655670\pi$
$53$	−362.533	−0.939580	−0.469790	+	0.882778i	$0.655670\pi$
$54$	−73.2537	−0.184603
$55$	55.0000	0.134840
$56$	345.436	0.824300
$57$	−143.603	−0.333695
$58$	−331.509	−0.750503
$59$	−278.927	−0.615478	−0.307739	−	0.951471i	$-0.599572\pi$
$59$	−278.927	−0.615478	−0.307739	+	0.951471i	$0.599572\pi$
$60$	61.4758	0.132275
$61$	679.259	1.42574	0.712870	−	0.701296i	$-0.247395\pi$
$61$	679.259	1.42574	0.712870	+	0.701296i	$0.247395\pi$
$62$	−566.381	−1.16017
$63$	526.232	1.05236
$64$	369.850	0.722363
$65$	0.785202	0.00149834
$66$	−257.954	−0.481089
$67$	−589.528	−1.07496	−0.537480	−	0.843277i	$-0.680623\pi$
$67$	−589.528	−1.07496	−0.537480	+	0.843277i	$0.680623\pi$
$68$	34.8743	0.0621930
$69$	528.711	0.922454
$70$	−271.005	−0.462733
$71$	732.175	1.22385	0.611924	−	0.790916i	$-0.290396\pi$
$71$	732.175	1.22385	0.611924	+	0.790916i	$0.290396\pi$
$72$	595.675	0.975014
$73$	−166.925	−0.267631	−0.133816	−	0.991006i	$-0.542723\pi$
$73$	−166.925	−0.267631	−0.133816	+	0.991006i	$0.542723\pi$
$74$	−1158.27	−1.81955
$75$	188.951	0.290909
$76$	−30.9086	−0.0466508
$77$	192.159	0.284396
$78$	−3.68265	−0.00534587
$79$	779.826	1.11060	0.555299	−	0.831651i	$-0.312604\pi$
$79$	779.826	1.11060	0.555299	+	0.831651i	$0.312604\pi$
$80$	−371.839	−0.519661
$81$	−634.900	−0.870919
$82$	831.737	1.12012
$83$	−565.667	−0.748073	−0.374036	−	0.927414i	$-0.622026\pi$
$83$	−565.667	−0.748073	−0.374036	+	0.927414i	$0.622026\pi$
$84$	214.784	0.278986
$85$	107.189	0.136779
$86$	112.960	0.141638
$87$	807.538	0.995140
$88$	217.517	0.263493
$89$	246.185	0.293208	0.146604	−	0.989195i	$-0.453166\pi$
$89$	246.185	0.293208	0.146604	+	0.989195i	$0.453166\pi$
$90$	−467.326	−0.547339
$91$	2.74334	0.00316022
$92$	113.798	0.128960
$93$	1379.68	1.53834
$94$	−562.652	−0.617373
$95$	−95.0000	−0.102598
$96$	548.314	0.582938
$97$	−17.9354	−0.0187739	−0.00938694	−	0.999956i	$-0.502988\pi$
$97$	−17.9354	−0.0187739	−0.00938694	+	0.999956i	$0.502988\pi$
$98$	117.390	0.121001
$99$	331.362	0.336395
$100$	40.6692	0.0406692
$101$	736.788	0.725872	0.362936	−	0.931814i	$-0.381774\pi$
$101$	736.788	0.725872	0.362936	+	0.931814i	$0.381774\pi$
$102$	−502.722	−0.488009
$103$	−1361.25	−1.30221	−0.651106	−	0.758986i	$-0.725695\pi$
$103$	−1361.25	−1.30221	−0.651106	+	0.758986i	$0.725695\pi$
$104$	3.10536	0.00292794
$105$	660.155	0.613567
$106$	1124.83	1.03069
$107$	1374.33	1.24170	0.620848	−	0.783931i	$-0.286789\pi$
$107$	1374.33	1.24170	0.620848	+	0.783931i	$0.286789\pi$
$108$	38.4074	0.0342200
$109$	−68.5301	−0.0602201	−0.0301100	−	0.999547i	$-0.509586\pi$
$109$	−68.5301	−0.0602201	−0.0301100	+	0.999547i	$0.509586\pi$
$110$	−170.649	−0.147916
$111$	2821.50	2.41266
$112$	−1299.13	−1.09604
$113$	−2050.52	−1.70705	−0.853527	−	0.521048i	$-0.825541\pi$
$113$	−2050.52	−1.70705	−0.853527	+	0.521048i	$0.825541\pi$
$114$	445.556	0.366054
$115$	349.768	0.283618
$116$	173.812	0.139121
$117$	4.73065	0.00373803
$118$	865.428	0.675162
$119$	374.496	0.288487
$120$	747.271	0.568469
$121$	121.000	0.0909091
$122$	−2107.54	−1.56400
$123$	−2026.07	−1.48524
$124$	296.958	0.215061
$125$	125.000	0.0894427
$126$	−1632.74	−1.15441
$127$	788.531	0.550951	0.275476	−	0.961308i	$-0.411165\pi$
$127$	788.531	0.550951	0.275476	+	0.961308i	$0.411165\pi$
$128$	−1727.91	−1.19318
$129$	−275.166	−0.187806
$130$	−2.43625	−0.00164364
$131$	669.101	0.446257	0.223128	−	0.974789i	$-0.428373\pi$
$131$	669.101	0.446257	0.223128	+	0.974789i	$0.428373\pi$
$132$	135.247	0.0891797
$133$	−331.911	−0.216393
$134$	1829.13	1.17920
$135$	118.048	0.0752590
$136$	423.915	0.267283
$137$	−126.774	−0.0790589	−0.0395294	−	0.999218i	$-0.512586\pi$
$137$	−126.774	−0.0790589	−0.0395294	+	0.999218i	$0.512586\pi$
$138$	−1640.43	−1.01191
$139$	−2757.15	−1.68243	−0.841217	−	0.540697i	$-0.818161\pi$
$139$	−2757.15	−1.68243	−0.841217	+	0.540697i	$0.818161\pi$
$140$	142.090	0.0857771
$141$	1370.59	0.818614
$142$	−2271.72	−1.34253
$143$	1.72745	0.00101018
$144$	−2240.24	−1.29643
$145$	534.225	0.305965
$146$	517.919	0.293584
$147$	−285.955	−0.160443
$148$	607.292	0.337291
$149$	94.5027	0.0519595	0.0259797	−	0.999662i	$-0.491729\pi$
$149$	94.5027	0.0519595	0.0259797	+	0.999662i	$0.491729\pi$
$150$	−586.258	−0.319118
$151$	849.796	0.457983	0.228991	−	0.973428i	$-0.426457\pi$
$151$	849.796	0.457983	0.228991	+	0.973428i	$0.426457\pi$
$152$	−375.711	−0.200488
$153$	645.786	0.341233
$154$	−596.212	−0.311975
$155$	912.722	0.472978
$156$	1.93084	0.000990967 0
$157$	−464.968	−0.236360	−0.118180	−	0.992992i	$-0.537706\pi$
$157$	−464.968	−0.236360	−0.118180	+	0.992992i	$0.537706\pi$
$158$	−2419.57	−1.21829
$159$	−2740.03	−1.36666
$160$	362.736	0.179230
$161$	1222.02	0.598190
$162$	1969.91	0.955373
$163$	723.011	0.347427	0.173713	−	0.984796i	$-0.444423\pi$
$163$	723.011	0.347427	0.173713	+	0.984796i	$0.444423\pi$
$164$	−436.085	−0.207637
$165$	415.692	0.196131
$166$	1755.10	0.820614
$167$	1737.00	0.804870	0.402435	−	0.915448i	$-0.368164\pi$
$167$	1737.00	0.804870	0.402435	+	0.915448i	$0.368164\pi$
$168$	2610.81	1.19898
$169$	−2196.98	−0.999989
$170$	−332.575	−0.150043
$171$	−572.352	−0.255958
$172$	−59.2259	−0.0262554
$173$	824.111	0.362174	0.181087	−	0.983467i	$-0.442039\pi$
$173$	824.111	0.362174	0.181087	+	0.983467i	$0.442039\pi$
$174$	−2505.55	−1.09164
$175$	436.725	0.188647
$176$	−818.046	−0.350355
$177$	−2108.14	−0.895239
$178$	−763.839	−0.321641
$179$	−441.400	−0.184312	−0.0921559	−	0.995745i	$-0.529376\pi$
$179$	−441.400	−0.184312	−0.0921559	+	0.995745i	$0.529376\pi$
$180$	245.022	0.101460
$181$	3176.82	1.30459	0.652297	−	0.757964i	$-0.273806\pi$
$181$	3176.82	1.30459	0.652297	+	0.757964i	$0.273806\pi$
$182$	−8.51176	−0.00346667
$183$	5133.86	2.07380
$184$	1383.28	0.554221
$185$	1866.56	0.741795
$186$	−4280.72	−1.68752
$187$	235.815	0.0922167
$188$	295.002	0.114443
$189$	412.436	0.158732
$190$	294.757	0.112547
$191$	2936.57	1.11247	0.556237	−	0.831024i	$-0.312245\pi$
$191$	2936.57	1.11247	0.556237	+	0.831024i	$0.312245\pi$
$192$	2795.33	1.05071
$193$	3787.56	1.41261	0.706307	−	0.707906i	$-0.250360\pi$
$193$	3787.56	1.41261	0.706307	+	0.707906i	$0.250360\pi$
$194$	55.6483	0.0205944
$195$	5.93458	0.00217941
$196$	−61.5482	−0.0224301
$197$	−2631.52	−0.951715	−0.475858	−	0.879522i	$-0.657862\pi$
$197$	−2631.52	−0.951715	−0.475858	+	0.879522i	$0.657862\pi$
$198$	−1028.12	−0.369016
$199$	−1027.14	−0.365889	−0.182945	−	0.983123i	$-0.558563\pi$
$199$	−1027.14	−0.365889	−0.182945	+	0.983123i	$0.558563\pi$
$200$	494.356	0.174781
$201$	−4455.67	−1.56357
$202$	−2286.03	−0.796261
$203$	1866.47	0.645324
$204$	263.581	0.0904624
$205$	−1340.34	−0.456651
$206$	4223.55	1.42849
$207$	2107.27	0.707561
$208$	−11.6788	−0.00389315
$209$	−209.000	−0.0691714
$210$	−2048.27	−0.673066
$211$	−1573.89	−0.513513	−0.256756	−	0.966476i	$-0.582654\pi$
$211$	−1573.89	−0.513513	−0.256756	+	0.966476i	$0.582654\pi$
$212$	−589.757	−0.191060
$213$	5533.80	1.78014
$214$	−4264.13	−1.36210
$215$	−182.036	−0.0577429
$216$	466.863	0.147065
$217$	3188.86	0.997577
$218$	212.628	0.0660597
$219$	−1261.62	−0.389281
$220$	89.4723	0.0274192
$221$	3.36659	0.00102471
$222$	−8754.27	−2.64661
$223$	−4296.09	−1.29008	−0.645039	−	0.764150i	$-0.723159\pi$
$223$	−4296.09	−1.29008	−0.645039	+	0.764150i	$0.723159\pi$
$224$	1267.33	0.378021
$225$	753.095	0.223139
$226$	6362.17	1.87259
$227$	4687.94	1.37070	0.685352	−	0.728212i	$-0.259649\pi$
$227$	4687.94	1.37070	0.685352	+	0.728212i	$0.259649\pi$
$228$	−233.608	−0.0678556
$229$	3232.21	0.932709	0.466355	−	0.884598i	$-0.345567\pi$
$229$	3232.21	0.932709	0.466355	+	0.884598i	$0.345567\pi$
$230$	−1085.23	−0.311120
$231$	1452.34	0.413667
$232$	2112.78	0.597892
$233$	2324.13	0.653472	0.326736	−	0.945116i	$-0.394051\pi$
$233$	2324.13	0.653472	0.326736	+	0.945116i	$0.394051\pi$
$234$	−14.6778	−0.00410051
$235$	906.712	0.251691
$236$	−453.750	−0.125155
$237$	5893.94	1.61541
$238$	−1161.95	−0.316462
$239$	6830.22	1.84858	0.924289	−	0.381692i	$-0.124659\pi$
$239$	6830.22	1.84858	0.924289	+	0.381692i	$0.124659\pi$
$240$	−2810.37	−0.755869
$241$	1588.25	0.424514	0.212257	−	0.977214i	$-0.431919\pi$
$241$	1588.25	0.424514	0.212257	+	0.977214i	$0.431919\pi$
$242$	−375.427	−0.0997247
$243$	−5436.05	−1.43507
$244$	1105.00	0.289919
$245$	−189.173	−0.0493299
$246$	6286.29	1.62926
$247$	−2.98377	−0.000768634 0
$248$	3609.68	0.924253
$249$	−4275.33	−1.08810
$250$	−387.838	−0.0981161
$251$	−2653.34	−0.667240	−0.333620	−	0.942708i	$-0.608270\pi$
$251$	−2653.34	−0.667240	−0.333620	+	0.942708i	$0.608270\pi$
$252$	856.058	0.213994
$253$	769.490	0.191215
$254$	−2446.58	−0.604378
$255$	810.135	0.198952
$256$	2402.40	0.586523
$257$	5964.11	1.44759	0.723796	−	0.690014i	$-0.242396\pi$
$257$	5964.11	1.44759	0.723796	+	0.690014i	$0.242396\pi$
$258$	853.758	0.206018
$259$	6521.37	1.56455
$260$	1.27734	0.000304683 0
$261$	3218.58	0.763314
$262$	−2076.02	−0.489531
$263$	6305.99	1.47850	0.739248	−	0.673434i	$-0.235181\pi$
$263$	6305.99	1.47850	0.739248	+	0.673434i	$0.235181\pi$
$264$	1644.00	0.383262
$265$	−1812.66	−0.420193
$266$	1029.82	0.237377
$267$	1860.67	0.426484
$268$	−959.025	−0.218589
$269$	548.524	0.124327	0.0621637	−	0.998066i	$-0.480200\pi$
$269$	548.524	0.124327	0.0621637	+	0.998066i	$0.480200\pi$
$270$	−366.268	−0.0825570
$271$	−3334.86	−0.747522	−0.373761	−	0.927525i	$-0.621932\pi$
$271$	−3334.86	−0.747522	−0.373761	+	0.927525i	$0.621932\pi$
$272$	−1594.28	−0.355394
$273$	20.7342	0.00459667
$274$	393.343	0.0867253
$275$	275.000	0.0603023
$276$	860.091	0.187578
$277$	4151.35	0.900472	0.450236	−	0.892910i	$-0.351340\pi$
$277$	4151.35	0.900472	0.450236	+	0.892910i	$0.351340\pi$
$278$	8554.62	1.84558
$279$	5498.93	1.17997
$280$	1727.18	0.368638
$281$	3668.75	0.778859	0.389430	−	0.921056i	$-0.372672\pi$
$281$	3668.75	0.778859	0.389430	+	0.921056i	$0.372672\pi$
$282$	−4252.54	−0.897996
$283$	6270.41	1.31709	0.658546	−	0.752540i	$-0.271172\pi$
$283$	6270.41	1.31709	0.658546	+	0.752540i	$0.271172\pi$
$284$	1191.08	0.248865
$285$	−718.013	−0.149233
$286$	−5.35975	−0.00110814
$287$	−4682.88	−0.963142
$288$	2185.40	0.447138
$289$	−4453.42	−0.906457
$290$	−1657.54	−0.335635
$291$	−135.556	−0.0273074
$292$	−271.548	−0.0544218
$293$	707.517	0.141070	0.0705352	−	0.997509i	$-0.477529\pi$
$293$	707.517	0.141070	0.0705352	+	0.997509i	$0.477529\pi$
$294$	887.234	0.176002
$295$	−1394.63	−0.275250
$296$	7381.95	1.44955
$297$	259.706	0.0507396
$298$	−293.214	−0.0569980
$299$	10.9855	0.00212478
$300$	307.379	0.0591551
$301$	−635.995	−0.121788
$302$	−2636.66	−0.502394
$303$	5568.66	1.05581
$304$	1412.99	0.266580
$305$	3396.29	0.637610
$306$	−2003.68	−0.374323
$307$	−6746.49	−1.25421	−0.627105	−	0.778935i	$-0.715760\pi$
$307$	−6746.49	−1.25421	−0.627105	+	0.778935i	$0.715760\pi$
$308$	312.598	0.0578309
$309$	−10288.4	−1.89412
$310$	−2831.91	−0.518843
$311$	−4197.24	−0.765286	−0.382643	−	0.923896i	$-0.624986\pi$
$311$	−4197.24	−0.765286	−0.382643	+	0.923896i	$0.624986\pi$
$312$	23.4704	0.00425881
$313$	4268.73	0.770873	0.385436	−	0.922734i	$-0.374051\pi$
$313$	4268.73	0.770873	0.385436	+	0.922734i	$0.374051\pi$
$314$	1442.66	0.259280
$315$	2631.16	0.470632
$316$	1268.60	0.225836
$317$	−6793.43	−1.20365	−0.601825	−	0.798628i	$-0.705559\pi$
$317$	−6793.43	−1.20365	−0.601825	+	0.798628i	$0.705559\pi$
$318$	8501.51	1.49919
$319$	1175.30	0.206282
$320$	1849.25	0.323050
$321$	10387.2	1.80610
$322$	−3791.56	−0.656197
$323$	−407.317	−0.0701664
$324$	−1032.84	−0.177098
$325$	3.92601	0.000670080 0
$326$	−2243.29	−0.381117
$327$	−517.952	−0.0875927
$328$	−5300.85	−0.892349
$329$	3167.87	0.530851
$330$	−1289.77	−0.215150
$331$	3057.10	0.507654	0.253827	−	0.967250i	$-0.418311\pi$
$331$	3057.10	0.507654	0.253827	+	0.967250i	$0.418311\pi$
$332$	−920.210	−0.152118
$333$	11245.6	1.85061
$334$	−5389.41	−0.882920
$335$	−2947.64	−0.480736
$336$	−9818.86	−1.59423
$337$	−7882.07	−1.27408	−0.637038	−	0.770833i	$-0.719841\pi$
$337$	−7882.07	−1.27408	−0.637038	+	0.770833i	$0.719841\pi$
$338$	6816.56	1.09696
$339$	−15497.9	−2.48298
$340$	174.371	0.0278136
$341$	2007.99	0.318882
$342$	1775.84	0.280779
$343$	−6652.79	−1.04728
$344$	−719.923	−0.112836
$345$	2643.56	0.412534
$346$	−2556.97	−0.397294
$347$	−1021.04	−0.157961	−0.0789805	−	0.996876i	$-0.525166\pi$
$347$	−1021.04	−0.157961	−0.0789805	+	0.996876i	$0.525166\pi$
$348$	1313.68	0.202358
$349$	6184.05	0.948494	0.474247	−	0.880392i	$-0.342720\pi$
$349$	6184.05	0.948494	0.474247	+	0.880392i	$0.342720\pi$
$350$	−1355.03	−0.206941
$351$	3.70767	0.000563820 0
$352$	798.019	0.120837
$353$	−5363.87	−0.808754	−0.404377	−	0.914592i	$-0.632512\pi$
$353$	−5363.87	−0.808754	−0.404377	+	0.914592i	$0.632512\pi$
$354$	6540.93	0.982052
$355$	3660.88	0.547322
$356$	400.486	0.0596228
$357$	2830.45	0.419617
$358$	1369.53	0.202185
$359$	−4189.03	−0.615846	−0.307923	−	0.951411i	$-0.599634\pi$
$359$	−4189.03	−0.615846	−0.307923	+	0.951411i	$0.599634\pi$
$360$	2978.38	0.436039
$361$	361.000	0.0526316
$362$	−9856.74	−1.43110
$363$	914.521	0.132231
$364$	4.46278	0.000642618 0
$365$	−834.625	−0.119688
$366$	−15928.8	−2.27490
$367$	−846.808	−0.120444	−0.0602221	−	0.998185i	$-0.519181\pi$
$367$	−846.808	−0.120444	−0.0602221	+	0.998185i	$0.519181\pi$
$368$	−5202.29	−0.736925
$369$	−8075.23	−1.13924
$370$	−5791.37	−0.813728
$371$	−6333.08	−0.886246
$372$	2244.41	0.312816
$373$	5896.12	0.818470	0.409235	−	0.912429i	$-0.365796\pi$
$373$	5896.12	0.818470	0.409235	+	0.912429i	$0.365796\pi$
$374$	−731.665	−0.101159
$375$	944.753	0.130098
$376$	3585.91	0.491833
$377$	16.7790	0.00229221
$378$	−1279.67	−0.174124
$379$	−6686.67	−0.906257	−0.453128	−	0.891445i	$-0.649692\pi$
$379$	−6686.67	−0.906257	−0.453128	+	0.891445i	$0.649692\pi$
$380$	−154.543	−0.0208629
$381$	5959.74	0.801383
$382$	−9111.30	−1.22035
$383$	−3638.31	−0.485402	−0.242701	−	0.970101i	$-0.578033\pi$
$383$	−3638.31	−0.485402	−0.242701	+	0.970101i	$0.578033\pi$
$384$	−13059.6	−1.73553
$385$	960.794	0.127186
$386$	−11751.7	−1.54960
$387$	−1096.72	−0.144055
$388$	−29.1768	−0.00381760
$389$	−280.555	−0.0365673	−0.0182836	−	0.999833i	$-0.505820\pi$
$389$	−280.555	−0.0365673	−0.0182836	+	0.999833i	$0.505820\pi$
$390$	−18.4132	−0.00239075
$391$	1499.65	0.193965
$392$	−748.151	−0.0963963
$393$	5057.08	0.649100
$394$	8164.82	1.04400
$395$	3899.13	0.496675
$396$	539.049	0.0684046
$397$	−13919.8	−1.75973	−0.879865	−	0.475223i	$-0.842367\pi$
$397$	−13919.8	−1.75973	−0.879865	+	0.475223i	$0.842367\pi$
$398$	3186.91	0.401370
$399$	−2508.59	−0.314753
$400$	−1859.19	−0.232399
$401$	−8674.33	−1.08024	−0.540119	−	0.841589i	$-0.681621\pi$
$401$	−8674.33	−1.08024	−0.540119	+	0.841589i	$0.681621\pi$
$402$	13824.6	1.71520
$403$	28.6669	0.00354342
$404$	1198.58	0.147603
$405$	−3174.50	−0.389487
$406$	−5791.12	−0.707902
$407$	4106.43	0.500118
$408$	3203.96	0.388774
$409$	−2346.79	−0.283720	−0.141860	−	0.989887i	$-0.545308\pi$
$409$	−2346.79	−0.283720	−0.141860	+	0.989887i	$0.545308\pi$
$410$	4158.68	0.500934
$411$	−958.164	−0.114995
$412$	−2214.44	−0.264800
$413$	−4872.57	−0.580541
$414$	−6538.23	−0.776174
$415$	−2828.34	−0.334548
$416$	11.3928	0.00134274
$417$	−20838.6	−2.44717
$418$	648.465	0.0758791
$419$	−8106.68	−0.945196	−0.472598	−	0.881278i	$-0.656684\pi$
$419$	−8106.68	−0.945196	−0.472598	+	0.881278i	$0.656684\pi$
$420$	1073.92	0.124767
$421$	−10028.9	−1.16100	−0.580499	−	0.814261i	$-0.697143\pi$
$421$	−10028.9	−1.16100	−0.580499	+	0.814261i	$0.697143\pi$
$422$	4883.32	0.563309
$423$	5462.72	0.627911
$424$	−7168.82	−0.821105
$425$	535.944	0.0611696
$426$	−17169.7	−1.95276
$427$	11866.0	1.34481
$428$	2235.72	0.252494
$429$	13.0561	0.00146936
$430$	564.802	0.0633423
$431$	8970.71	1.00256	0.501281	−	0.865285i	$-0.332862\pi$
$431$	8970.71	1.00256	0.501281	+	0.865285i	$0.332862\pi$
$432$	−1755.80	−0.195546
$433$	9077.94	1.00752	0.503762	−	0.863843i	$-0.331949\pi$
$433$	9077.94	1.00752	0.503762	+	0.863843i	$0.331949\pi$
$434$	−9894.10	−1.09431
$435$	4037.69	0.445040
$436$	−111.483	−0.0122455
$437$	−1329.12	−0.145493
$438$	3914.44	0.427030
$439$	−15674.6	−1.70412	−0.852058	−	0.523447i	$-0.824646\pi$
$439$	−15674.6	−1.70412	−0.852058	+	0.523447i	$0.824646\pi$
$440$	1087.58	0.117838
$441$	−1139.72	−0.123067
$442$	−10.4455	−0.00112408
$443$	1631.30	0.174956	0.0874781	−	0.996166i	$-0.472119\pi$
$443$	1631.30	0.174956	0.0874781	+	0.996166i	$0.472119\pi$
$444$	4589.93	0.490604
$445$	1230.92	0.131127
$446$	13329.5	1.41518
$447$	714.254	0.0755773
$448$	6460.90	0.681359
$449$	7724.09	0.811854	0.405927	−	0.913906i	$-0.366949\pi$
$449$	7724.09	0.811854	0.405927	+	0.913906i	$0.366949\pi$
$450$	−2336.63	−0.244777
$451$	−2948.75	−0.307874
$452$	−3335.73	−0.347123
$453$	6422.78	0.666156
$454$	−14545.3	−1.50362
$455$	13.7167	0.00141329
$456$	−2839.63	−0.291618
$457$	−1013.20	−0.103710	−0.0518549	−	0.998655i	$-0.516513\pi$
$457$	−1013.20	−0.103710	−0.0518549	+	0.998655i	$0.516513\pi$
$458$	−10028.6	−1.02316
$459$	506.137	0.0514694
$460$	568.992	0.0576726
$461$	−1331.41	−0.134511	−0.0672557	−	0.997736i	$-0.521424\pi$
$461$	−1331.41	−0.134511	−0.0672557	+	0.997736i	$0.521424\pi$
$462$	−4506.19	−0.453781
$463$	9839.31	0.987628	0.493814	−	0.869568i	$-0.335602\pi$
$463$	9839.31	0.987628	0.493814	+	0.869568i	$0.335602\pi$
$464$	−7945.83	−0.794991
$465$	6898.38	0.687967
$466$	−7211.09	−0.716840
$467$	13924.4	1.37975	0.689876	−	0.723928i	$-0.257665\pi$
$467$	13924.4	1.37975	0.689876	+	0.723928i	$0.257665\pi$
$468$	7.69568	0.000760113 0
$469$	−10298.4	−1.01394
$470$	−2813.26	−0.276098
$471$	−3514.24	−0.343795
$472$	−5515.57	−0.537870
$473$	−400.478	−0.0389302
$474$	−18287.2	−1.77206
$475$	−475.000	−0.0458831
$476$	609.218	0.0586627
$477$	−10920.9	−1.04829
$478$	−21192.2	−2.02784
$479$	−14605.2	−1.39317	−0.696585	−	0.717474i	$-0.745298\pi$
$479$	−14605.2	−1.39317	−0.696585	+	0.717474i	$0.745298\pi$
$480$	2741.57	0.260698
$481$	58.6250	0.00555732
$482$	−4927.86	−0.465680
$483$	9236.05	0.870093
$484$	196.839	0.0184860
$485$	−89.6771	−0.00839593
$486$	16866.5	1.57423
$487$	−15814.2	−1.47148	−0.735739	−	0.677265i	$-0.763165\pi$
$487$	−15814.2	−1.47148	−0.735739	+	0.677265i	$0.763165\pi$
$488$	13431.8	1.24596
$489$	5464.54	0.505347
$490$	586.948	0.0541135
$491$	5733.98	0.527028	0.263514	−	0.964656i	$-0.415118\pi$
$491$	5733.98	0.527028	0.263514	+	0.964656i	$0.415118\pi$
$492$	−3295.95	−0.302018
$493$	2290.52	0.209249
$494$	9.25775	0.000843170 0
$495$	1656.81	0.150440
$496$	−13575.4	−1.22894
$497$	12790.4	1.15438
$498$	13265.1	1.19362
$499$	−14977.0	−1.34361	−0.671806	−	0.740727i	$-0.734481\pi$
$499$	−14977.0	−1.34361	−0.671806	+	0.740727i	$0.734481\pi$
$500$	203.346	0.0181878
$501$	13128.3	1.17072
$502$	8232.53	0.731943
$503$	−2931.13	−0.259826	−0.129913	−	0.991525i	$-0.541470\pi$
$503$	−2931.13	−0.259826	−0.129913	+	0.991525i	$0.541470\pi$
$504$	10405.8	0.919668
$505$	3683.94	0.324620
$506$	−2387.50	−0.209757
$507$	−16604.8	−1.45453
$508$	1282.76	0.112034
$509$	−3174.19	−0.276411	−0.138206	−	0.990404i	$-0.544133\pi$
$509$	−3174.19	−0.276411	−0.138206	+	0.990404i	$0.544133\pi$
$510$	−2513.61	−0.218244
$511$	−2916.01	−0.252440
$512$	6369.36	0.549782
$513$	−448.583	−0.0386071
$514$	−18504.9	−1.58797
$515$	−6806.25	−0.582367
$516$	−447.631	−0.0381897
$517$	1994.77	0.169690
$518$	−20233.9	−1.71627
$519$	6228.66	0.526797
$520$	15.5268	0.00130941
$521$	7564.75	0.636118	0.318059	−	0.948071i	$-0.396969\pi$
$521$	7564.75	0.636118	0.318059	+	0.948071i	$0.396969\pi$
$522$	−9986.29	−0.837333
$523$	7407.66	0.619339	0.309670	−	0.950844i	$-0.399782\pi$
$523$	7407.66	0.619339	0.309670	+	0.950844i	$0.399782\pi$
$524$	1088.47	0.0907446
$525$	3300.78	0.274396
$526$	−19565.6	−1.62187
$527$	3913.34	0.323468
$528$	−6182.81	−0.509607
$529$	−7273.49	−0.597805
$530$	5624.16	0.460940
$531$	−8402.33	−0.686686
$532$	−539.942	−0.0440027
$533$	−42.0976	−0.00342111
$534$	−5773.11	−0.467841
$535$	6871.64	0.555303
$536$	−11657.5	−0.939414
$537$	−3336.12	−0.268089
$538$	−1701.91	−0.136384
$539$	−416.181	−0.0332582
$540$	192.037	0.0153036
$541$	8935.67	0.710119	0.355059	−	0.934844i	$-0.384461\pi$
$541$	8935.67	0.710119	0.355059	+	0.934844i	$0.384461\pi$
$542$	10347.1	0.820010
$543$	24010.5	1.89759
$544$	1555.25	0.122575
$545$	−342.650	−0.0269312
$546$	−64.3321	−0.00504242
$547$	4727.32	0.369516	0.184758	−	0.982784i	$-0.440850\pi$
$547$	4727.32	0.369516	0.184758	+	0.982784i	$0.440850\pi$
$548$	−206.233	−0.0160763
$549$	20461.8	1.59069
$550$	−853.243	−0.0661499
$551$	−2030.06	−0.156957
$552$	10454.9	0.806139
$553$	13622.8	1.04756
$554$	−12880.4	−0.987792
$555$	14107.5	1.07897
$556$	−4485.25	−0.342117
$557$	4050.29	0.308108	0.154054	−	0.988062i	$-0.450767\pi$
$557$	4050.29	0.308108	0.154054	+	0.988062i	$0.450767\pi$
$558$	−17061.5	−1.29440
$559$	−5.71739	−0.000432594 0
$560$	−6495.65	−0.490163
$561$	1782.30	0.134133
$562$	−11383.1	−0.854386
$563$	−1482.77	−0.110997	−0.0554984	−	0.998459i	$-0.517675\pi$
$563$	−1482.77	−0.110997	−0.0554984	+	0.998459i	$0.517675\pi$
$564$	2229.63	0.166462
$565$	−10252.6	−0.763418
$566$	−19455.2	−1.44481
$567$	−11091.1	−0.821482
$568$	14478.2	1.06953
$569$	−14970.3	−1.10296	−0.551481	−	0.834187i	$-0.685937\pi$
$569$	−14970.3	−1.10296	−0.551481	+	0.834187i	$0.685937\pi$
$570$	2227.78	0.163704
$571$	−15665.7	−1.14814	−0.574071	−	0.818806i	$-0.694636\pi$
$571$	−15665.7	−1.14814	−0.574071	+	0.818806i	$0.694636\pi$
$572$	2.81015	0.000205417 0
$573$	22194.7	1.61814
$574$	14529.6	1.05654
$575$	1748.84	0.126838
$576$	11141.3	0.805937
$577$	8499.99	0.613274	0.306637	−	0.951826i	$-0.400796\pi$
$577$	8499.99	0.613274	0.306637	+	0.951826i	$0.400796\pi$
$578$	13817.7	0.994357
$579$	28626.5	2.05471
$580$	869.061	0.0622169
$581$	−9881.63	−0.705610
$582$	420.591	0.0299555
$583$	−3987.86	−0.283294
$584$	−3300.82	−0.233885
$585$	23.6533	0.00167170
$586$	−2195.22	−0.154750
$587$	12866.6	0.904702	0.452351	−	0.891840i	$-0.350585\pi$
$587$	12866.6	0.904702	0.452351	+	0.891840i	$0.350585\pi$
$588$	−465.183	−0.0326255
$589$	−3468.34	−0.242633
$590$	4327.14	0.301941
$591$	−19889.1	−1.38431
$592$	−27762.3	−1.92741
$593$	−23658.9	−1.63837	−0.819186	−	0.573527i	$-0.805575\pi$
$593$	−23658.9	−1.63837	−0.819186	+	0.573527i	$0.805575\pi$
$594$	−805.791	−0.0556599
$595$	1872.48	0.129015
$596$	153.734	0.0105658
$597$	−7763.14	−0.532201
$598$	−34.0849	−0.00233083
$599$	6041.51	0.412102	0.206051	−	0.978541i	$-0.433939\pi$
$599$	6041.51	0.412102	0.206051	+	0.978541i	$0.433939\pi$
$600$	3736.36	0.254227
$601$	−9451.98	−0.641521	−0.320761	−	0.947160i	$-0.603939\pi$
$601$	−9451.98	−0.641521	−0.320761	+	0.947160i	$0.603939\pi$
$602$	1973.30	0.133598
$603$	−17758.8	−1.19933
$604$	1382.42	0.0931290
$605$	605.000	0.0406558
$606$	−17277.9	−1.15820
$607$	−24447.4	−1.63475	−0.817373	−	0.576109i	$-0.804570\pi$
$607$	−24447.4	−1.63475	−0.817373	+	0.576109i	$0.804570\pi$
$608$	−1378.40	−0.0919430
$609$	14106.9	0.938652
$610$	−10537.7	−0.699440
$611$	28.4781	0.00188560
$612$	1050.54	0.0693885
$613$	317.077	0.0208917	0.0104458	−	0.999945i	$-0.496675\pi$
$613$	317.077	0.0208917	0.0104458	+	0.999945i	$0.496675\pi$
$614$	20932.3	1.37583
$615$	−10130.3	−0.664219
$616$	3799.80	0.248536
$617$	14423.9	0.941144	0.470572	−	0.882361i	$-0.344048\pi$
$617$	14423.9	0.941144	0.470572	+	0.882361i	$0.344048\pi$
$618$	31921.7	2.07780
$619$	−16366.3	−1.06271	−0.531354	−	0.847150i	$-0.678317\pi$
$619$	−16366.3	−1.06271	−0.531354	+	0.847150i	$0.678317\pi$
$620$	1484.79	0.0961782
$621$	1651.58	0.106724
$622$	13022.8	0.839496
$623$	4300.60	0.276565
$624$	−88.2683	−0.00566276
$625$	625.000	0.0400000
$626$	−13244.6	−0.845625
$627$	−1579.63	−0.100613
$628$	−756.395	−0.0480628
$629$	8002.96	0.507311
$630$	−8163.71	−0.516270
$631$	−21501.6	−1.35652	−0.678260	−	0.734822i	$-0.737266\pi$
$631$	−21501.6	−1.35652	−0.678260	+	0.734822i	$0.737266\pi$
$632$	15420.5	0.970559
$633$	−11895.5	−0.746926
$634$	21078.0	1.32037
$635$	3942.66	0.246393
$636$	−4457.40	−0.277905
$637$	−5.94157	−0.000369566 0
$638$	−3646.59	−0.226285
$639$	22055.9	1.36544
$640$	−8639.56	−0.533607
$641$	−12086.5	−0.744754	−0.372377	−	0.928082i	$-0.621457\pi$
$641$	−12086.5	−0.744754	−0.372377	+	0.928082i	$0.621457\pi$
$642$	−32228.4	−1.98124
$643$	−23259.7	−1.42655	−0.713277	−	0.700882i	$-0.752790\pi$
$643$	−23259.7	−1.42655	−0.713277	+	0.700882i	$0.752790\pi$
$644$	1987.94	0.121640
$645$	−1375.83	−0.0839895
$646$	1263.78	0.0769705
$647$	26757.8	1.62590	0.812949	−	0.582334i	$-0.197861\pi$
$647$	26757.8	1.62590	0.812949	+	0.582334i	$0.197861\pi$
$648$	−12554.7	−0.761102
$649$	−3068.20	−0.185574
$650$	−12.1813	−0.000735058 0
$651$	24101.5	1.45102
$652$	1176.17	0.0706479
$653$	−18867.6	−1.13070	−0.565350	−	0.824851i	$-0.691259\pi$
$653$	−18867.6	−1.13070	−0.565350	+	0.824851i	$0.691259\pi$
$654$	1607.05	0.0960867
$655$	3345.50	0.199572
$656$	19935.7	1.18652
$657$	−5028.41	−0.298595
$658$	−9828.95	−0.582329
$659$	−19872.1	−1.17467	−0.587335	−	0.809344i	$-0.699823\pi$
$659$	−19872.1	−1.17467	−0.587335	+	0.809344i	$0.699823\pi$
$660$	676.234	0.0398824
$661$	−23264.0	−1.36893	−0.684467	−	0.729044i	$-0.739965\pi$
$661$	−23264.0	−1.36893	−0.684467	+	0.729044i	$0.739965\pi$
$662$	−9485.29	−0.556882
$663$	25.4448	0.00149049
$664$	−11185.6	−0.653746
$665$	−1659.55	−0.0967740
$666$	−34891.6	−2.03006
$667$	7474.20	0.433886
$668$	2825.70	0.163667
$669$	−32470.0	−1.87647
$670$	9145.65	0.527354
$671$	7471.84	0.429877
$672$	9578.48	0.549848
$673$	−3405.97	−0.195083	−0.0975413	−	0.995231i	$-0.531098\pi$
$673$	−3405.97	−0.195083	−0.0975413	+	0.995231i	$0.531098\pi$
$674$	24455.7	1.39762
$675$	590.241	0.0336569
$676$	−3573.97	−0.203344
$677$	10741.2	0.609775	0.304888	−	0.952388i	$-0.401381\pi$
$677$	10741.2	0.609775	0.304888	+	0.952388i	$0.401381\pi$
$678$	48085.5	2.72376
$679$	−313.314	−0.0177082
$680$	2119.58	0.119532
$681$	35431.6	1.99375
$682$	−6230.19	−0.349804
$683$	−9418.28	−0.527644	−0.263822	−	0.964571i	$-0.584983\pi$
$683$	−9418.28	−0.527644	−0.263822	+	0.964571i	$0.584983\pi$
$684$	−931.084	−0.0520481
$685$	−633.872	−0.0353562
$686$	20641.6	1.14884
$687$	24429.1	1.35667
$688$	2707.52	0.150034
$689$	−56.9324	−0.00314797
$690$	−8202.17	−0.452538
$691$	−4713.39	−0.259487	−0.129744	−	0.991548i	$-0.541415\pi$
$691$	−4713.39	−0.259487	−0.129744	+	0.991548i	$0.541415\pi$
$692$	1340.64	0.0736466
$693$	5788.55	0.317300
$694$	3167.99	0.173279
$695$	−13785.8	−0.752408
$696$	15968.5	0.869659
$697$	−5746.78	−0.312303
$698$	−19187.3	−1.04047
$699$	17565.9	0.950503
$700$	710.450	0.0383607
$701$	19989.8	1.07704	0.538520	−	0.842613i	$-0.318984\pi$
$701$	19989.8	1.07704	0.538520	+	0.842613i	$0.318984\pi$
$702$	−11.5038	−0.000618494 0
$703$	−7092.92	−0.380533
$704$	4068.35	0.217801
$705$	6852.95	0.366095
$706$	16642.5	0.887180
$707$	12870.9	0.684669
$708$	−3429.45	−0.182043
$709$	12678.0	0.671556	0.335778	−	0.941941i	$-0.391001\pi$
$709$	12678.0	0.671556	0.335778	+	0.941941i	$0.391001\pi$
$710$	−11358.6	−0.600396
$711$	23491.3	1.23909
$712$	4868.12	0.256237
$713$	12769.6	0.670725
$714$	−8782.04	−0.460308
$715$	8.63723	0.000451768 0
$716$	−718.056	−0.0374791
$717$	51623.0	2.68884
$718$	12997.3	0.675565
$719$	10960.1	0.568490	0.284245	−	0.958752i	$-0.408257\pi$
$719$	10960.1	0.568490	0.284245	+	0.958752i	$0.408257\pi$
$720$	−11201.2	−0.579783
$721$	−23779.6	−1.22829
$722$	−1120.08	−0.0577353
$723$	12004.0	0.617475
$724$	5167.96	0.265284
$725$	2671.13	0.136832
$726$	−2837.49	−0.145054
$727$	−24833.5	−1.26688	−0.633442	−	0.773790i	$-0.718358\pi$
$727$	−24833.5	−1.26688	−0.633442	+	0.773790i	$0.718358\pi$
$728$	54.2474	0.00276174
$729$	−23943.5	−1.21646
$730$	2589.59	0.131295
$731$	−780.486	−0.0394902
$732$	8351.60	0.421699
$733$	−18171.9	−0.915684	−0.457842	−	0.889034i	$-0.651377\pi$
$733$	−18171.9	−0.915684	−0.457842	+	0.889034i	$0.651377\pi$
$734$	2627.39	0.132124
$735$	−1429.78	−0.0717525
$736$	5074.94	0.254164
$737$	−6484.80	−0.324112
$738$	25055.1	1.24971
$739$	−30121.2	−1.49936	−0.749678	−	0.661802i	$-0.769792\pi$
$739$	−30121.2	−1.49936	−0.749678	+	0.661802i	$0.769792\pi$
$740$	3036.46	0.150841
$741$	−22.5514	−0.00111801
$742$	19649.7	0.972187
$743$	−8266.17	−0.408151	−0.204076	−	0.978955i	$-0.565419\pi$
$743$	−8266.17	−0.408151	−0.204076	+	0.978955i	$0.565419\pi$
$744$	27282.0	1.34437
$745$	472.513	0.0232370
$746$	−18293.9	−0.897838
$747$	−17040.0	−0.834622
$748$	383.617	0.0187519
$749$	24008.1	1.17121
$750$	−2931.29	−0.142714
$751$	9565.32	0.464772	0.232386	−	0.972624i	$-0.425347\pi$
$751$	9565.32	0.464772	0.232386	+	0.972624i	$0.425347\pi$
$752$	−13486.0	−0.653969
$753$	−20054.0	−0.970530
$754$	−52.0603	−0.00251449
$755$	4248.98	0.204816
$756$	670.939	0.0322775
$757$	−13231.7	−0.635291	−0.317645	−	0.948210i	$-0.602892\pi$
$757$	−13231.7	−0.635291	−0.317645	+	0.948210i	$0.602892\pi$
$758$	20746.8	0.994138
$759$	5815.82	0.278130
$760$	−1878.55	−0.0896609
$761$	−8007.63	−0.381441	−0.190720	−	0.981644i	$-0.561082\pi$
$761$	−8007.63	−0.381441	−0.190720	+	0.981644i	$0.561082\pi$
$762$	−18491.3	−0.879094
$763$	−1197.15	−0.0568018
$764$	4777.12	0.226217
$765$	3228.93	0.152604
$766$	11288.6	0.532472
$767$	−43.8028	−0.00206210
$768$	18157.4	0.853123
$769$	14561.6	0.682840	0.341420	−	0.939911i	$-0.389092\pi$
$769$	14561.6	0.682840	0.341420	+	0.939911i	$0.389092\pi$
$770$	−2981.06	−0.139519
$771$	45076.9	2.10559
$772$	6161.48	0.287250
$773$	6478.66	0.301450	0.150725	−	0.988576i	$-0.451839\pi$
$773$	6478.66	0.301450	0.150725	+	0.988576i	$0.451839\pi$
$774$	3402.80	0.158025
$775$	4563.61	0.211522
$776$	−354.659	−0.0164066
$777$	49288.7	2.27570
$778$	870.478	0.0401133
$779$	5093.30	0.234257
$780$	9.65419	0.000443174 0
$781$	8053.93	0.369004
$782$	−4652.96	−0.212774
$783$	2522.57	0.115133
$784$	2813.68	0.128174
$785$	−2324.84	−0.105703
$786$	−15690.6	−0.712044
$787$	20900.7	0.946668	0.473334	−	0.880883i	$-0.343050\pi$
$787$	20900.7	0.946668	0.473334	+	0.880883i	$0.343050\pi$
$788$	−4280.87	−0.193528
$789$	47660.9	2.15054
$790$	−12097.8	−0.544838
$791$	−35820.6	−1.61016
$792$	6552.43	0.293978
$793$	106.671	0.00477680
$794$	43188.9	1.93037
$795$	−13700.2	−0.611189
$796$	−1670.92	−0.0744021
$797$	27522.5	1.22321	0.611604	−	0.791164i	$-0.290524\pi$
$797$	27522.5	1.22321	0.611604	+	0.791164i	$0.290524\pi$
$798$	7783.41	0.345275
$799$	3887.57	0.172131
$800$	1813.68	0.0801541
$801$	7416.02	0.327131
$802$	26913.9	1.18499
$803$	−1836.17	−0.0806939
$804$	−7248.34	−0.317947
$805$	6110.09	0.267519
$806$	−88.9448	−0.00388703
$807$	4145.76	0.180840
$808$	14569.4	0.634345
$809$	18976.4	0.824689	0.412344	−	0.911028i	$-0.364710\pi$
$809$	18976.4	0.824689	0.412344	+	0.911028i	$0.364710\pi$
$810$	9849.53	0.427256
$811$	6995.18	0.302878	0.151439	−	0.988467i	$-0.451609\pi$
$811$	6995.18	0.302878	0.151439	+	0.988467i	$0.451609\pi$
$812$	3036.32	0.131224
$813$	−25205.0	−1.08730
$814$	−12741.0	−0.548615
$815$	3615.05	0.155374
$816$	−12049.6	−0.516936
$817$	691.735	0.0296215
$818$	7281.40	0.311233
$819$	82.6397	0.00352584
$820$	−2180.43	−0.0928583
$821$	5419.59	0.230384	0.115192	−	0.993343i	$-0.463252\pi$
$821$	5419.59	0.230384	0.115192	+	0.993343i	$0.463252\pi$
$822$	2972.90	0.126146
$823$	−14701.9	−0.622692	−0.311346	−	0.950297i	$-0.600780\pi$
$823$	−14701.9	−0.622692	−0.311346	+	0.950297i	$0.600780\pi$
$824$	−26917.7	−1.13801
$825$	2078.46	0.0877122
$826$	15118.1	0.636837
$827$	11556.8	0.485935	0.242968	−	0.970034i	$-0.421879\pi$
$827$	11556.8	0.485935	0.242968	+	0.970034i	$0.421879\pi$
$828$	3428.04	0.143880
$829$	40058.0	1.67825	0.839126	−	0.543937i	$-0.183067\pi$
$829$	40058.0	1.67825	0.839126	+	0.543937i	$0.183067\pi$
$830$	8775.49	0.366990
$831$	31376.0	1.30978
$832$	58.0814	0.00242020
$833$	−811.089	−0.0337366
$834$	64656.1	2.68448
$835$	8685.02	0.359949
$836$	−339.995	−0.0140657
$837$	4309.81	0.177979
$838$	25152.6	1.03685
$839$	−13147.9	−0.541020	−0.270510	−	0.962717i	$-0.587192\pi$
$839$	−13147.9	−0.541020	−0.270510	+	0.962717i	$0.587192\pi$
$840$	13054.1	0.536200
$841$	−12973.1	−0.531926
$842$	31116.8	1.27358
$843$	27728.5	1.13288
$844$	−2560.36	−0.104421
$845$	−10984.9	−0.447209
$846$	−16949.2	−0.688801
$847$	2113.75	0.0857488
$848$	26960.8	1.09179
$849$	47391.9	1.91577
$850$	−1662.87	−0.0671013
$851$	26114.5	1.05193
$852$	9002.22	0.361985
$853$	34045.3	1.36658	0.683289	−	0.730148i	$-0.260549\pi$
$853$	34045.3	1.36658	0.683289	+	0.730148i	$0.260549\pi$
$854$	−36816.5	−1.47522
$855$	−2861.76	−0.114468
$856$	27176.3	1.08513
$857$	−43287.8	−1.72542	−0.862709	−	0.505701i	$-0.831234\pi$
$857$	−43287.8	−1.72542	−0.862709	+	0.505701i	$0.831234\pi$
$858$	−40.5091	−0.00161184
$859$	904.768	0.0359375	0.0179687	−	0.999839i	$-0.494280\pi$
$859$	904.768	0.0359375	0.0179687	+	0.999839i	$0.494280\pi$
$860$	−296.130	−0.0117418
$861$	−35393.4	−1.40093
$862$	−27833.5	−1.09978
$863$	−37306.0	−1.47151	−0.735754	−	0.677249i	$-0.763172\pi$
$863$	−37306.0	−1.47151	−0.735754	+	0.677249i	$0.763172\pi$
$864$	1712.81	0.0674434
$865$	4120.56	0.161969
$866$	−28166.1	−1.10522
$867$	−33659.1	−1.31848
$868$	5187.55	0.202853
$869$	8578.08	0.334858
$870$	−12527.8	−0.488196
$871$	−92.5797	−0.00360154
$872$	−1355.13	−0.0526267
$873$	−540.283	−0.0209459
$874$	4123.86	0.159601
$875$	2183.62	0.0843656
$876$	−2052.37	−0.0791589
$877$	−43092.2	−1.65920	−0.829600	−	0.558358i	$-0.811432\pi$
$877$	−43092.2	−1.65920	−0.829600	+	0.558358i	$0.811432\pi$
$878$	48633.6	1.86937
$879$	5347.44	0.205193
$880$	−4090.23	−0.156684
$881$	37400.4	1.43025	0.715125	−	0.698996i	$-0.246370\pi$
$881$	37400.4	1.43025	0.715125	+	0.698996i	$0.246370\pi$
$882$	3536.22	0.135001
$883$	−32680.9	−1.24553	−0.622763	−	0.782410i	$-0.713990\pi$
$883$	−32680.9	−1.24553	−0.622763	+	0.782410i	$0.713990\pi$
$884$	5.47667	0.000208371 0
$885$	−10540.7	−0.400363
$886$	−5061.45	−0.191922
$887$	1349.06	0.0510675	0.0255338	−	0.999674i	$-0.491871\pi$
$887$	1349.06	0.0510675	0.0255338	+	0.999674i	$0.491871\pi$
$888$	55793.0	2.10844
$889$	13774.8	0.519677
$890$	−3819.19	−0.143842
$891$	−6983.90	−0.262592
$892$	−6988.74	−0.262332
$893$	−3445.50	−0.129115
$894$	−2216.12	−0.0829061
$895$	−2207.00	−0.0824267
$896$	−30184.9	−1.12545
$897$	83.0291	0.00309059
$898$	−23965.6	−0.890580
$899$	19504.0	0.723575
$900$	1225.11	0.0453745
$901$	−7771.89	−0.287369
$902$	9149.10	0.337729
$903$	−4806.87	−0.177146
$904$	−40547.6	−1.49181
$905$	15884.1	0.583432
$906$	−19928.0	−0.730754
$907$	−26515.0	−0.970689	−0.485344	−	0.874323i	$-0.661306\pi$
$907$	−26515.0	−0.970689	−0.485344	+	0.874323i	$0.661306\pi$
$908$	7626.20	0.278727
$909$	22194.8	0.809853
$910$	−42.5588	−0.00155034
$911$	−29893.7	−1.08718	−0.543592	−	0.839350i	$-0.682936\pi$
$911$	−29893.7	−1.08718	−0.543592	+	0.839350i	$0.682936\pi$
$912$	10679.4	0.387753
$913$	−6222.34	−0.225552
$914$	3143.65	0.113767
$915$	25669.3	0.927432
$916$	5258.06	0.189663
$917$	11688.5	0.420926
$918$	−1570.39	−0.0564605
$919$	−34663.8	−1.24424	−0.622118	−	0.782923i	$-0.713728\pi$
$919$	−34663.8	−1.24424	−0.622118	+	0.782923i	$0.713728\pi$
$920$	6916.40	0.247855
$921$	−50990.1	−1.82430
$922$	4130.96	0.147555
$923$	114.981	0.00410038
$924$	2362.62	0.0841176
$925$	9332.79	0.331741
$926$	−30528.5	−1.08340
$927$	−41006.0	−1.45287
$928$	7751.31	0.274191
$929$	−21937.8	−0.774766	−0.387383	−	0.921919i	$-0.626621\pi$
$929$	−21937.8	−0.774766	−0.387383	+	0.921919i	$0.626621\pi$
$930$	−21403.6	−0.754680
$931$	718.858	0.0253057
$932$	3780.83	0.132881
$933$	−31722.9	−1.11314
$934$	−43203.2	−1.51355
$935$	1179.08	0.0412405
$936$	93.5451	0.00326669
$937$	−28103.1	−0.979818	−0.489909	−	0.871774i	$-0.662970\pi$
$937$	−28103.1	−0.979818	−0.489909	+	0.871774i	$0.662970\pi$
$938$	31953.0	1.11226
$939$	32263.2	1.12127
$940$	1475.01	0.0511804
$941$	12584.3	0.435959	0.217980	−	0.975953i	$-0.430053\pi$
$941$	12584.3	0.435959	0.217980	+	0.975953i	$0.430053\pi$
$942$	10903.6	0.377133
$943$	−18752.4	−0.647572
$944$	20743.2	0.715183
$945$	2062.18	0.0709871
$946$	1242.56	0.0427054
$947$	46408.4	1.59247	0.796236	−	0.604986i	$-0.206821\pi$
$947$	46408.4	1.59247	0.796236	+	0.604986i	$0.206821\pi$
$948$	9588.09	0.328488
$949$	−26.2140	−0.000896672 0
$950$	1473.78	0.0503325
$951$	−51344.9	−1.75076
$952$	7405.37	0.252111
$953$	−7264.82	−0.246937	−0.123468	−	0.992349i	$-0.539402\pi$
$953$	−7264.82	−0.246937	−0.123468	+	0.992349i	$0.539402\pi$
$954$	33884.2	1.14994
$955$	14682.8	0.497514
$956$	11111.2	0.375901
$957$	8882.92	0.300046
$958$	45315.6	1.52827
$959$	−2214.62	−0.0745712
$960$	13976.7	0.469891
$961$	3531.45	0.118541
$962$	−181.896	−0.00609622
$963$	41400.0	1.38535
$964$	2583.71	0.0863234
$965$	18937.8	0.631740
$966$	−28656.7	−0.954467
$967$	36043.7	1.19864	0.599321	−	0.800509i	$-0.295438\pi$
$967$	36043.7	1.19864	0.599321	+	0.800509i	$0.295438\pi$
$968$	2392.68	0.0794460
$969$	−3078.51	−0.102060
$970$	278.241	0.00921010
$971$	−30190.8	−0.997807	−0.498903	−	0.866658i	$-0.666264\pi$
$971$	−30190.8	−0.997807	−0.498903	+	0.866658i	$0.666264\pi$
$972$	−8843.20	−0.291817
$973$	−48164.6	−1.58693
$974$	49066.8	1.61417
$975$	29.6729	0.000974660 0
$976$	−50514.9	−1.65671
$977$	−2002.03	−0.0655583	−0.0327792	−	0.999463i	$-0.510436\pi$
$977$	−2002.03	−0.0655583	−0.0327792	+	0.999463i	$0.510436\pi$
$978$	−16954.8	−0.554352
$979$	2708.03	0.0884056
$980$	−307.741	−0.0100310
$981$	−2064.38	−0.0671873
$982$	−17790.8	−0.578135
$983$	27924.5	0.906055	0.453028	−	0.891496i	$-0.350344\pi$
$983$	27924.5	0.906055	0.453028	+	0.891496i	$0.350344\pi$
$984$	−40064.0	−1.29796
$985$	−13157.6	−0.425620
$986$	−7106.80	−0.229540
$987$	23942.8	0.772146
$988$	−4.85390	−0.000156299 0
$989$	−2546.81	−0.0818845
$990$	−5140.58	−0.165029
$991$	−34611.7	−1.10946	−0.554731	−	0.832030i	$-0.687179\pi$
$991$	−34611.7	−1.10946	−0.554731	+	0.832030i	$0.687179\pi$
$992$	13243.1	0.423859
$993$	23105.7	0.738405
$994$	−39684.7	−1.26632
$995$	−5135.69	−0.163631
$996$	−6954.97	−0.221262
$997$	25905.1	0.822892	0.411446	−	0.911434i	$-0.365024\pi$
$997$	25905.1	0.822892	0.411446	+	0.911434i	$0.365024\pi$
$998$	46469.1	1.47390
$999$	8813.75	0.279134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.e.1.5	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.e.1.5	✓	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-3.10270 q^{2} +7.55803 q^{3} +1.62677 q^{4} +5.00000 q^{5} -23.4503 q^{6} +17.4690 q^{7} +19.7742 q^{8} +30.1238 q^{9} +O(q^{10})\)	\(q-3.10270 q^{2} +7.55803 q^{3} +1.62677 q^{4} +5.00000 q^{5} -23.4503 q^{6} +17.4690 q^{7} +19.7742 q^{8} +30.1238 q^{9} -15.5135 q^{10} +11.0000 q^{11} +12.2952 q^{12} +0.157040 q^{13} -54.2011 q^{14} +37.7901 q^{15} -74.3678 q^{16} +21.4377 q^{17} -93.4652 q^{18} -19.0000 q^{19} +8.13384 q^{20} +132.031 q^{21} -34.1297 q^{22} +69.9536 q^{23} +149.454 q^{24} +25.0000 q^{25} -0.487250 q^{26} +23.6096 q^{27} +28.4180 q^{28} +106.845 q^{29} -117.252 q^{30} +182.544 q^{31} +72.5472 q^{32} +83.1383 q^{33} -66.5150 q^{34} +87.3449 q^{35} +49.0044 q^{36} +373.311 q^{37} +58.9514 q^{38} +1.18692 q^{39} +98.8712 q^{40} -268.068 q^{41} -409.653 q^{42} -36.4071 q^{43} +17.8945 q^{44} +150.619 q^{45} -217.045 q^{46} +181.342 q^{47} -562.074 q^{48} -37.8346 q^{49} -77.5676 q^{50} +162.027 q^{51} +0.255469 q^{52} -362.533 q^{53} -73.2537 q^{54} +55.0000 q^{55} +345.436 q^{56} -143.603 q^{57} -331.509 q^{58} -278.927 q^{59} +61.4758 q^{60} +679.259 q^{61} -566.381 q^{62} +526.232 q^{63} +369.850 q^{64} +0.785202 q^{65} -257.954 q^{66} -589.528 q^{67} +34.8743 q^{68} +528.711 q^{69} -271.005 q^{70} +732.175 q^{71} +595.675 q^{72} -166.925 q^{73} -1158.27 q^{74} +188.951 q^{75} -30.9086 q^{76} +192.159 q^{77} -3.68265 q^{78} +779.826 q^{79} -371.839 q^{80} -634.900 q^{81} +831.737 q^{82} -565.667 q^{83} +214.784 q^{84} +107.189 q^{85} +112.960 q^{86} +807.538 q^{87} +217.517 q^{88} +246.185 q^{89} -467.326 q^{90} +2.74334 q^{91} +113.798 q^{92} +1379.68 q^{93} -562.652 q^{94} -95.0000 q^{95} +548.314 q^{96} -17.9354 q^{97} +117.390 q^{98} +331.362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9} + 60 q^{10} + 242 q^{11} + 164 q^{12} + 207 q^{13} + 129 q^{14} + 105 q^{15} + 604 q^{16} + 209 q^{17} + 869 q^{18} - 418 q^{19} + 480 q^{20} + 45 q^{21} + 132 q^{22} + 191 q^{23} + 458 q^{24} + 550 q^{25} + 363 q^{26} + 45 q^{27} + 577 q^{28} + 389 q^{29} + 135 q^{30} + 198 q^{31} + 1149 q^{32} + 231 q^{33} + 467 q^{34} + 465 q^{35} + 1315 q^{36} + 312 q^{37} - 228 q^{38} + 137 q^{39} + 570 q^{40} + 632 q^{41} - 1794 q^{42} + 1584 q^{43} + 1056 q^{44} + 1045 q^{45} + 681 q^{46} - 54 q^{47} + 2491 q^{48} + 1063 q^{49} + 300 q^{50} + 37 q^{51} + 1246 q^{52} + 343 q^{53} + 1078 q^{54} + 1210 q^{55} - 87 q^{56} - 399 q^{57} - 1424 q^{58} + 2787 q^{59} + 820 q^{60} + 2070 q^{61} - 446 q^{62} + 1696 q^{63} + 1758 q^{64} + 1035 q^{65} + 297 q^{66} + 2423 q^{67} - 524 q^{68} - 997 q^{69} + 645 q^{70} + 2538 q^{71} + 6010 q^{72} + 1397 q^{73} - 1977 q^{74} + 525 q^{75} - 1824 q^{76} + 1023 q^{77} + 202 q^{78} + 878 q^{79} + 3020 q^{80} + 2030 q^{81} - 190 q^{82} + 4932 q^{83} - 4580 q^{84} + 1045 q^{85} - 3394 q^{86} + 6009 q^{87} + 1254 q^{88} + 1812 q^{89} + 4345 q^{90} + 4349 q^{91} - 788 q^{92} - 4848 q^{93} - 2152 q^{94} - 2090 q^{95} + 4032 q^{96} + 988 q^{97} + 1366 q^{98} + 2299 q^{99}+O(q^{100}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9 + 60 * q^10 + 242 * q^11 + 164 * q^12 + 207 * q^13 + 129 * q^14 + 105 * q^15 + 604 * q^16 + 209 * q^17 + 869 * q^18 - 418 * q^19 + 480 * q^20 + 45 * q^21 + 132 * q^22 + 191 * q^23 + 458 * q^24 + 550 * q^25 + 363 * q^26 + 45 * q^27 + 577 * q^28 + 389 * q^29 + 135 * q^30 + 198 * q^31 + 1149 * q^32 + 231 * q^33 + 467 * q^34 + 465 * q^35 + 1315 * q^36 + 312 * q^37 - 228 * q^38 + 137 * q^39 + 570 * q^40 + 632 * q^41 - 1794 * q^42 + 1584 * q^43 + 1056 * q^44 + 1045 * q^45 + 681 * q^46 - 54 * q^47 + 2491 * q^48 + 1063 * q^49 + 300 * q^50 + 37 * q^51 + 1246 * q^52 + 343 * q^53 + 1078 * q^54 + 1210 * q^55 - 87 * q^56 - 399 * q^57 - 1424 * q^58 + 2787 * q^59 + 820 * q^60 + 2070 * q^61 - 446 * q^62 + 1696 * q^63 + 1758 * q^64 + 1035 * q^65 + 297 * q^66 + 2423 * q^67 - 524 * q^68 - 997 * q^69 + 645 * q^70 + 2538 * q^71 + 6010 * q^72 + 1397 * q^73 - 1977 * q^74 + 525 * q^75 - 1824 * q^76 + 1023 * q^77 + 202 * q^78 + 878 * q^79 + 3020 * q^80 + 2030 * q^81 - 190 * q^82 + 4932 * q^83 - 4580 * q^84 + 1045 * q^85 - 3394 * q^86 + 6009 * q^87 + 1254 * q^88 + 1812 * q^89 + 4345 * q^90 + 4349 * q^91 - 788 * q^92 - 4848 * q^93 - 2152 * q^94 - 2090 * q^95 + 4032 * q^96 + 988 * q^97 + 1366 * q^98 + 2299 * q^99

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