LMFDB - Embedded newform 1045.4.a.e.1.16

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			1.16
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)

$f(q)$	$=$	$q+2.94017 q^{2} -3.08997 q^{3} +0.644599 q^{4} +5.00000 q^{5} -9.08503 q^{6} +21.8209 q^{7} -21.6261 q^{8} -17.4521 q^{9} +O(q^{10})$	\(q+2.94017 q^{2} -3.08997 q^{3} +0.644599 q^{4} +5.00000 q^{5} -9.08503 q^{6} +21.8209 q^{7} -21.6261 q^{8} -17.4521 q^{9} +14.7008 q^{10} +11.0000 q^{11} -1.99179 q^{12} +82.8764 q^{13} +64.1571 q^{14} -15.4498 q^{15} -68.7413 q^{16} +12.6851 q^{17} -51.3121 q^{18} -19.0000 q^{19} +3.22300 q^{20} -67.4258 q^{21} +32.3419 q^{22} -184.087 q^{23} +66.8240 q^{24} +25.0000 q^{25} +243.671 q^{26} +137.356 q^{27} +14.0657 q^{28} +107.759 q^{29} -45.4252 q^{30} -12.9216 q^{31} -29.1020 q^{32} -33.9896 q^{33} +37.2964 q^{34} +109.104 q^{35} -11.2496 q^{36} -57.5023 q^{37} -55.8632 q^{38} -256.085 q^{39} -108.131 q^{40} +146.230 q^{41} -198.243 q^{42} +304.603 q^{43} +7.09059 q^{44} -87.2605 q^{45} -541.246 q^{46} +480.149 q^{47} +212.408 q^{48} +133.151 q^{49} +73.5042 q^{50} -39.1966 q^{51} +53.4220 q^{52} -581.027 q^{53} +403.849 q^{54} +55.0000 q^{55} -471.901 q^{56} +58.7094 q^{57} +316.829 q^{58} -420.422 q^{59} -9.95895 q^{60} +357.265 q^{61} -37.9917 q^{62} -380.820 q^{63} +464.365 q^{64} +414.382 q^{65} -99.9353 q^{66} +584.906 q^{67} +8.17681 q^{68} +568.822 q^{69} +320.785 q^{70} +1175.38 q^{71} +377.421 q^{72} +333.142 q^{73} -169.067 q^{74} -77.2492 q^{75} -12.2474 q^{76} +240.030 q^{77} -752.934 q^{78} +786.816 q^{79} -343.706 q^{80} +46.7824 q^{81} +429.943 q^{82} +87.8607 q^{83} -43.4626 q^{84} +63.4255 q^{85} +895.585 q^{86} -332.971 q^{87} -237.887 q^{88} +1457.01 q^{89} -256.561 q^{90} +1808.43 q^{91} -118.662 q^{92} +39.9274 q^{93} +1411.72 q^{94} -95.0000 q^{95} +89.9243 q^{96} -1386.43 q^{97} +391.486 q^{98} -191.973 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.94017	1.03951	0.519754	−	0.854316i	$-0.326024\pi$
$2$	2.94017	1.03951	0.519754	+	0.854316i	$0.326024\pi$
$3$	−3.08997	−0.594665	−0.297332	−	0.954774i	$-0.596097\pi$
$3$	−3.08997	−0.594665	−0.297332	+	0.954774i	$0.596097\pi$
$4$	0.644599	0.0805749
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−9.08503	−0.618158
$7$	21.8209	1.17822	0.589108	−	0.808054i	$-0.299479\pi$
$7$	21.8209	1.17822	0.589108	+	0.808054i	$0.299479\pi$
$8$	−21.6261	−0.955749
$9$	−17.4521	−0.646374
$10$	14.7008	0.464882
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−1.99179	−0.0479150
$13$	82.8764	1.76814	0.884068	−	0.467359i	$-0.154794\pi$
$13$	82.8764	1.76814	0.884068	+	0.467359i	$0.154794\pi$
$14$	64.1571	1.22476
$15$	−15.4498	−0.265942
$16$	−68.7413	−1.07408
$17$	12.6851	0.180976	0.0904879	−	0.995898i	$-0.471157\pi$
$17$	12.6851	0.180976	0.0904879	+	0.995898i	$0.471157\pi$
$18$	−51.3121	−0.671910
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	3.22300	0.0360342
$21$	−67.4258	−0.700644
$22$	32.3419	0.313423
$23$	−184.087	−1.66890	−0.834451	−	0.551083i	$-0.814215\pi$
$23$	−184.087	−1.66890	−0.834451	+	0.551083i	$0.814215\pi$
$24$	66.8240	0.568350
$25$	25.0000	0.200000
$26$	243.671	1.83799
$27$	137.356	0.979040
$28$	14.0657	0.0949347
$29$	107.759	0.690010	0.345005	−	0.938601i	$-0.387877\pi$
$29$	107.759	0.690010	0.345005	+	0.938601i	$0.387877\pi$
$30$	−45.4252	−0.276449
$31$	−12.9216	−0.0748642	−0.0374321	−	0.999299i	$-0.511918\pi$
$31$	−12.9216	−0.0748642	−0.0374321	+	0.999299i	$0.511918\pi$
$32$	−29.1020	−0.160768
$33$	−33.9896	−0.179298
$34$	37.2964	0.188126
$35$	109.104	0.526915
$36$	−11.2496	−0.0520815
$37$	−57.5023	−0.255495	−0.127748	−	0.991807i	$-0.540775\pi$
$37$	−57.5023	−0.255495	−0.127748	+	0.991807i	$0.540775\pi$
$38$	−55.8632	−0.238479
$39$	−256.085	−1.05145
$40$	−108.131	−0.427424
$41$	146.230	0.557009	0.278504	−	0.960435i	$-0.410161\pi$
$41$	146.230	0.557009	0.278504	+	0.960435i	$0.410161\pi$
$42$	−198.243	−0.728324
$43$	304.603	1.08027	0.540134	−	0.841579i	$-0.318373\pi$
$43$	304.603	1.08027	0.540134	+	0.841579i	$0.318373\pi$
$44$	7.09059	0.0242942
$45$	−87.2605	−0.289067
$46$	−541.246	−1.73483
$47$	480.149	1.49015	0.745074	−	0.666981i	$-0.232414\pi$
$47$	480.149	1.49015	0.745074	+	0.666981i	$0.232414\pi$
$48$	212.408	0.638719
$49$	133.151	0.388195
$50$	73.5042	0.207901
$51$	−39.1966	−0.107620
$52$	53.4220	0.142467
$53$	−581.027	−1.50585	−0.752927	−	0.658105i	$-0.771358\pi$
$53$	−581.027	−1.50585	−0.752927	+	0.658105i	$0.771358\pi$
$54$	403.849	1.01772
$55$	55.0000	0.134840
$56$	−471.901	−1.12608
$57$	58.7094	0.136425
$58$	316.829	0.717271
$59$	−420.422	−0.927699	−0.463849	−	0.885914i	$-0.653532\pi$
$59$	−420.422	−0.927699	−0.463849	+	0.885914i	$0.653532\pi$
$60$	−9.95895	−0.0214283
$61$	357.265	0.749886	0.374943	−	0.927048i	$-0.377662\pi$
$61$	357.265	0.749886	0.374943	+	0.927048i	$0.377662\pi$
$62$	−37.9917	−0.0778218
$63$	−380.820	−0.761569
$64$	464.365	0.906964
$65$	414.382	0.790734
$66$	−99.9353	−0.186382
$67$	584.906	1.06653	0.533266	−	0.845948i	$-0.320964\pi$
$67$	584.906	1.06653	0.533266	+	0.845948i	$0.320964\pi$
$68$	8.17681	0.0145821
$69$	568.822	0.992437
$70$	320.785	0.547731
$71$	1175.38	1.96467	0.982334	−	0.187138i	$-0.0599212\pi$
$71$	1175.38	1.96467	0.982334	+	0.187138i	$0.0599212\pi$
$72$	377.421	0.617771
$73$	333.142	0.534127	0.267064	−	0.963679i	$-0.413947\pi$
$73$	333.142	0.534127	0.267064	+	0.963679i	$0.413947\pi$
$74$	−169.067	−0.265589
$75$	−77.2492	−0.118933
$76$	−12.2474	−0.0184851
$77$	240.030	0.355246
$78$	−752.934	−1.09299
$79$	786.816	1.12055	0.560277	−	0.828305i	$-0.310695\pi$
$79$	786.816	1.12055	0.560277	+	0.828305i	$0.310695\pi$
$80$	−343.706	−0.480344
$81$	46.7824	0.0641734
$82$	429.943	0.579015
$83$	87.8607	0.116192	0.0580962	−	0.998311i	$-0.481497\pi$
$83$	87.8607	0.116192	0.0580962	+	0.998311i	$0.481497\pi$
$84$	−43.4626	−0.0564543
$85$	63.4255	0.0809349
$86$	895.585	1.12295
$87$	−332.971	−0.410325
$88$	−237.887	−0.288169
$89$	1457.01	1.73531	0.867655	−	0.497166i	$-0.165626\pi$
$89$	1457.01	1.73531	0.867655	+	0.497166i	$0.165626\pi$
$90$	−256.561	−0.300487
$91$	1808.43	2.08325
$92$	−118.662	−0.134472
$93$	39.9274	0.0445191
$94$	1411.72	1.54902
$95$	−95.0000	−0.102598
$96$	89.9243	0.0956027
$97$	−1386.43	−1.45124	−0.725622	−	0.688093i	$-0.758448\pi$
$97$	−1386.43	−1.45124	−0.725622	+	0.688093i	$0.758448\pi$
$98$	391.486	0.403531
$99$	−191.973	−0.194889
$100$	16.1150	0.0161150
$101$	−240.220	−0.236661	−0.118331	−	0.992974i	$-0.537754\pi$
$101$	−240.220	−0.236661	−0.118331	+	0.992974i	$0.537754\pi$
$102$	−115.245	−0.111872
$103$	−1423.35	−1.36162	−0.680811	−	0.732460i	$-0.738372\pi$
$103$	−1423.35	−1.36162	−0.680811	+	0.732460i	$0.738372\pi$
$104$	−1792.29	−1.68989
$105$	−337.129	−0.313337
$106$	−1708.32	−1.56534
$107$	1606.85	1.45177	0.725886	−	0.687815i	$-0.241430\pi$
$107$	1606.85	1.45177	0.725886	+	0.687815i	$0.241430\pi$
$108$	88.5393	0.0788861
$109$	−75.4645	−0.0663136	−0.0331568	−	0.999450i	$-0.510556\pi$
$109$	−75.4645	−0.0663136	−0.0331568	+	0.999450i	$0.510556\pi$
$110$	161.709	0.140167
$111$	177.680	0.151934
$112$	−1500.00	−1.26550
$113$	1725.80	1.43672	0.718362	−	0.695670i	$-0.244892\pi$
$113$	1725.80	1.43672	0.718362	+	0.695670i	$0.244892\pi$
$114$	172.616	0.141815
$115$	−920.434	−0.746355
$116$	69.4612	0.0555975
$117$	−1446.37	−1.14288
$118$	−1236.11	−0.964349
$119$	276.800	0.213229
$120$	334.120	0.254174
$121$	121.000	0.0909091
$122$	1050.42	0.779512
$123$	−451.848	−0.331233
$124$	−8.32926	−0.00603217
$125$	125.000	0.0894427
$126$	−1119.68	−0.791656
$127$	−1067.22	−0.745674	−0.372837	−	0.927897i	$-0.621615\pi$
$127$	−1067.22	−0.745674	−0.372837	+	0.927897i	$0.621615\pi$
$128$	1598.13	1.10356
$129$	−941.214	−0.642397
$130$	1218.35	0.821974
$131$	1183.85	0.789571	0.394785	−	0.918773i	$-0.370819\pi$
$131$	1183.85	0.789571	0.394785	+	0.918773i	$0.370819\pi$
$132$	−21.9097	−0.0144469
$133$	−414.597	−0.270301
$134$	1719.72	1.10867
$135$	686.778	0.437840
$136$	−274.330	−0.172968
$137$	1744.35	1.08781	0.543906	−	0.839146i	$-0.316945\pi$
$137$	1744.35	1.08781	0.543906	+	0.839146i	$0.316945\pi$
$138$	1672.43	1.03164
$139$	2178.10	1.32909	0.664546	−	0.747248i	$-0.268625\pi$
$139$	2178.10	1.32909	0.664546	+	0.747248i	$0.268625\pi$
$140$	70.3286	0.0424561
$141$	−1483.65	−0.886139
$142$	3455.80	2.04229
$143$	911.640	0.533113
$144$	1199.68	0.694259
$145$	538.794	0.308582
$146$	979.493	0.555229
$147$	−411.432	−0.230846
$148$	−37.0659	−0.0205865
$149$	887.005	0.487693	0.243846	−	0.969814i	$-0.421591\pi$
$149$	887.005	0.487693	0.243846	+	0.969814i	$0.421591\pi$
$150$	−227.126	−0.123632
$151$	−2532.97	−1.36510	−0.682551	−	0.730838i	$-0.739130\pi$
$151$	−2532.97	−1.36510	−0.682551	+	0.730838i	$0.739130\pi$
$152$	410.896	0.219264
$153$	−221.382	−0.116978
$154$	705.728	0.369280
$155$	−64.6080	−0.0334803
$156$	−165.072	−0.0847203
$157$	−146.243	−0.0743403	−0.0371701	−	0.999309i	$-0.511834\pi$
$157$	−146.243	−0.0743403	−0.0371701	+	0.999309i	$0.511834\pi$
$158$	2313.37	1.16482
$159$	1795.36	0.895477
$160$	−145.510	−0.0718974
$161$	−4016.93	−1.96633
$162$	137.548	0.0667087
$163$	397.980	0.191241	0.0956204	−	0.995418i	$-0.469517\pi$
$163$	397.980	0.191241	0.0956204	+	0.995418i	$0.469517\pi$
$164$	94.2600	0.0448809
$165$	−169.948	−0.0801846
$166$	258.325	0.120783
$167$	−695.452	−0.322249	−0.161125	−	0.986934i	$-0.551512\pi$
$167$	−695.452	−0.322249	−0.161125	+	0.986934i	$0.551512\pi$
$168$	1458.16	0.669640
$169$	4671.49	2.12630
$170$	186.482	0.0841324
$171$	331.590	0.148288
$172$	196.347	0.0870425
$173$	−2828.81	−1.24318	−0.621590	−	0.783343i	$-0.713513\pi$
$173$	−2828.81	−1.24318	−0.621590	+	0.783343i	$0.713513\pi$
$174$	−978.992	−0.426535
$175$	545.522	0.235643
$176$	−756.154	−0.323848
$177$	1299.09	0.551670
$178$	4283.85	1.80387
$179$	−145.927	−0.0609337	−0.0304668	−	0.999536i	$-0.509699\pi$
$179$	−145.927	−0.0609337	−0.0304668	+	0.999536i	$0.509699\pi$
$180$	−56.2480	−0.0232916
$181$	−1727.72	−0.709507	−0.354753	−	0.934960i	$-0.615435\pi$
$181$	−1727.72	−0.709507	−0.354753	+	0.934960i	$0.615435\pi$
$182$	5317.11	2.16555
$183$	−1103.94	−0.445931
$184$	3981.08	1.59505
$185$	−287.512	−0.114261
$186$	117.393	0.0462779
$187$	139.536	0.0545663
$188$	309.504	0.120069
$189$	2997.22	1.15352
$190$	−279.316	−0.106651
$191$	4736.98	1.79453	0.897267	−	0.441487i	$-0.145549\pi$
$191$	4736.98	1.79453	0.897267	+	0.441487i	$0.145549\pi$
$192$	−1434.87	−0.539339
$193$	−2955.35	−1.10223	−0.551117	−	0.834428i	$-0.685798\pi$
$193$	−2955.35	−1.10223	−0.551117	+	0.834428i	$0.685798\pi$
$194$	−4076.34	−1.50858
$195$	−1280.43	−0.470222
$196$	85.8289	0.0312787
$197$	1011.20	0.365712	0.182856	−	0.983140i	$-0.441466\pi$
$197$	1011.20	0.365712	0.182856	+	0.983140i	$0.441466\pi$
$198$	−564.433	−0.202589
$199$	−1817.59	−0.647466	−0.323733	−	0.946148i	$-0.604938\pi$
$199$	−1817.59	−0.647466	−0.323733	+	0.946148i	$0.604938\pi$
$200$	−540.653	−0.191150
$201$	−1807.34	−0.634229
$202$	−706.287	−0.246011
$203$	2351.39	0.812982
$204$	−25.2661	−0.00867146
$205$	731.152	0.249102
$206$	−4184.89	−1.41541
$207$	3212.70	1.07873
$208$	−5697.03	−1.89912
$209$	−209.000	−0.0691714
$210$	−991.217	−0.325716
$211$	−5692.91	−1.85742	−0.928710	−	0.370806i	$-0.879081\pi$
$211$	−5692.91	−1.85742	−0.928710	+	0.370806i	$0.879081\pi$
$212$	−374.530	−0.121334
$213$	−3631.87	−1.16832
$214$	4724.40	1.50913
$215$	1523.02	0.483111
$216$	−2970.47	−0.935717
$217$	−281.961	−0.0882062
$218$	−221.878	−0.0689335
$219$	−1029.40	−0.317627
$220$	35.4529	0.0108647
$221$	1051.30	0.319990
$222$	522.410	0.157936
$223$	−2615.17	−0.785313	−0.392657	−	0.919685i	$-0.628444\pi$
$223$	−2615.17	−0.785313	−0.392657	+	0.919685i	$0.628444\pi$
$224$	−635.032	−0.189419
$225$	−436.302	−0.129275
$226$	5074.15	1.49348
$227$	5467.18	1.59854	0.799271	−	0.600970i	$-0.205219\pi$
$227$	5467.18	1.59854	0.799271	+	0.600970i	$0.205219\pi$
$228$	37.8440	0.0109925
$229$	4785.51	1.38094	0.690470	−	0.723361i	$-0.257404\pi$
$229$	4785.51	1.38094	0.690470	+	0.723361i	$0.257404\pi$
$230$	−2706.23	−0.775842
$231$	−741.684	−0.211252
$232$	−2330.41	−0.659477
$233$	4229.41	1.18918	0.594588	−	0.804030i	$-0.297315\pi$
$233$	4229.41	1.18918	0.594588	+	0.804030i	$0.297315\pi$
$234$	−4252.56	−1.18803
$235$	2400.75	0.666415
$236$	−271.003	−0.0747492
$237$	−2431.24	−0.666354
$238$	813.840	0.221653
$239$	−2837.19	−0.767876	−0.383938	−	0.923359i	$-0.625432\pi$
$239$	−2837.19	−0.767876	−0.383938	+	0.923359i	$0.625432\pi$
$240$	1062.04	0.285644
$241$	−477.897	−0.127735	−0.0638674	−	0.997958i	$-0.520343\pi$
$241$	−477.897	−0.127735	−0.0638674	+	0.997958i	$0.520343\pi$
$242$	355.761	0.0945006
$243$	−3853.16	−1.01720
$244$	230.293	0.0604220
$245$	665.754	0.173606
$246$	−1328.51	−0.344320
$247$	−1574.65	−0.405638
$248$	279.444	0.0715513
$249$	−271.487	−0.0690954
$250$	367.521	0.0929763
$251$	−1770.84	−0.445317	−0.222658	−	0.974897i	$-0.571473\pi$
$251$	−1770.84	−0.445317	−0.222658	+	0.974897i	$0.571473\pi$
$252$	−245.476	−0.0613633
$253$	−2024.95	−0.503193
$254$	−3137.81	−0.775134
$255$	−195.983	−0.0481291
$256$	983.849	0.240197
$257$	−320.501	−0.0777912	−0.0388956	−	0.999243i	$-0.512384\pi$
$257$	−320.501	−0.0777912	−0.0388956	+	0.999243i	$0.512384\pi$
$258$	−2767.33	−0.667777
$259$	−1254.75	−0.301029
$260$	267.110	0.0637133
$261$	−1880.62	−0.446005
$262$	3480.73	0.820764
$263$	1521.20	0.356659	0.178330	−	0.983971i	$-0.442931\pi$
$263$	1521.20	0.356659	0.178330	+	0.983971i	$0.442931\pi$
$264$	735.064	0.171364
$265$	−2905.14	−0.673438
$266$	−1218.98	−0.280980
$267$	−4502.11	−1.03193
$268$	377.030	0.0859357
$269$	2302.47	0.521874	0.260937	−	0.965356i	$-0.415968\pi$
$269$	2302.47	0.521874	0.260937	+	0.965356i	$0.415968\pi$
$270$	2019.24	0.455138
$271$	25.9115	0.00580815	0.00290408	−	0.999996i	$-0.499076\pi$
$271$	25.9115	0.00580815	0.00290408	+	0.999996i	$0.499076\pi$
$272$	−871.990	−0.194383
$273$	−5588.01	−1.23883
$274$	5128.70	1.13079
$275$	275.000	0.0603023
$276$	366.662	0.0799655
$277$	−8735.66	−1.89486	−0.947428	−	0.319969i	$-0.896327\pi$
$277$	−8735.66	−1.89486	−0.947428	+	0.319969i	$0.896327\pi$
$278$	6403.97	1.38160
$279$	225.509	0.0483903
$280$	−2359.51	−0.503598
$281$	4364.25	0.926510	0.463255	−	0.886225i	$-0.346681\pi$
$281$	4364.25	0.926510	0.463255	+	0.886225i	$0.346681\pi$
$282$	−4362.17	−0.921147
$283$	−783.795	−0.164635	−0.0823176	−	0.996606i	$-0.526232\pi$
$283$	−783.795	−0.164635	−0.0823176	+	0.996606i	$0.526232\pi$
$284$	757.646	0.158303
$285$	293.547	0.0610113
$286$	2680.38	0.554175
$287$	3190.88	0.656277
$288$	507.891	0.103916
$289$	−4752.09	−0.967248
$290$	1584.15	0.320773
$291$	4284.03	0.863004
$292$	214.743	0.0430372
$293$	−6732.87	−1.34245	−0.671226	−	0.741253i	$-0.734232\pi$
$293$	−6732.87	−1.34245	−0.671226	+	0.741253i	$0.734232\pi$
$294$	−1209.68	−0.239966
$295$	−2102.11	−0.414879
$296$	1243.55	0.244189
$297$	1510.91	0.295192
$298$	2607.94	0.506960
$299$	−15256.4	−2.95084
$300$	−49.7948	−0.00958301
$301$	6646.71	1.27279
$302$	−7447.37	−1.41903
$303$	742.272	0.140734
$304$	1306.08	0.246411
$305$	1786.32	0.335359
$306$	−650.900	−0.121600
$307$	7821.78	1.45411	0.727056	−	0.686578i	$-0.240888\pi$
$307$	7821.78	1.45411	0.727056	+	0.686578i	$0.240888\pi$
$308$	154.723	0.0286239
$309$	4398.11	0.809708
$310$	−189.959	−0.0348030
$311$	−10799.4	−1.96906	−0.984530	−	0.175217i	$-0.943937\pi$
$311$	−10799.4	−1.96906	−0.984530	+	0.175217i	$0.943937\pi$
$312$	5538.13	1.00492
$313$	−642.518	−0.116030	−0.0580148	−	0.998316i	$-0.518477\pi$
$313$	−642.518	−0.116030	−0.0580148	+	0.998316i	$0.518477\pi$
$314$	−429.978	−0.0772772
$315$	−1904.10	−0.340584
$316$	507.181	0.0902885
$317$	2431.05	0.430730	0.215365	−	0.976534i	$-0.430906\pi$
$317$	2431.05	0.430730	0.215365	+	0.976534i	$0.430906\pi$
$318$	5278.65	0.930855
$319$	1185.35	0.208046
$320$	2321.83	0.405606
$321$	−4965.10	−0.863318
$322$	−11810.5	−2.04401
$323$	−241.017	−0.0415187
$324$	30.1559	0.00517077
$325$	2071.91	0.353627
$326$	1170.13	0.198796
$327$	233.183	0.0394344
$328$	−3162.40	−0.532361
$329$	10477.3	1.75572
$330$	−499.677	−0.0833524
$331$	3153.61	0.523679	0.261840	−	0.965111i	$-0.415671\pi$
$331$	3153.61	0.523679	0.261840	+	0.965111i	$0.415671\pi$
$332$	56.6349	0.00936218
$333$	1003.54	0.165145
$334$	−2044.75	−0.334981
$335$	2924.53	0.476968
$336$	4634.94	0.752549
$337$	1193.36	0.192898	0.0964488	−	0.995338i	$-0.469252\pi$
$337$	1193.36	0.192898	0.0964488	+	0.995338i	$0.469252\pi$
$338$	13735.0	2.21031
$339$	−5332.67	−0.854369
$340$	40.8840	0.00652132
$341$	−142.138	−0.0225724
$342$	974.931	0.154147
$343$	−4579.09	−0.720839
$344$	−6587.39	−1.03247
$345$	2844.11	0.443831
$346$	−8317.17	−1.29229
$347$	−1258.16	−0.194644	−0.0973221	−	0.995253i	$-0.531028\pi$
$347$	−1258.16	−0.194644	−0.0973221	+	0.995253i	$0.531028\pi$
$348$	−214.633	−0.0330619
$349$	−6922.63	−1.06178	−0.530888	−	0.847442i	$-0.678142\pi$
$349$	−6922.63	−1.06178	−0.530888	+	0.847442i	$0.678142\pi$
$350$	1603.93	0.244953
$351$	11383.5	1.73108
$352$	−320.122	−0.0484732
$353$	2587.38	0.390120	0.195060	−	0.980791i	$-0.437510\pi$
$353$	2587.38	0.390120	0.195060	+	0.980791i	$0.437510\pi$
$354$	3819.54	0.573464
$355$	5876.88	0.878626
$356$	939.186	0.139822
$357$	−855.304	−0.126800
$358$	−429.051	−0.0633410
$359$	−1631.26	−0.239817	−0.119909	−	0.992785i	$-0.538260\pi$
$359$	−1631.26	−0.239817	−0.119909	+	0.992785i	$0.538260\pi$
$360$	1887.11	0.276276
$361$	361.000	0.0526316
$362$	−5079.80	−0.737537
$363$	−373.886	−0.0540604
$364$	1165.72	0.167857
$365$	1665.71	0.238869
$366$	−3245.76	−0.463548
$367$	−2523.46	−0.358920	−0.179460	−	0.983765i	$-0.557435\pi$
$367$	−2523.46	−0.358920	−0.179460	+	0.983765i	$0.557435\pi$
$368$	12654.4	1.79254
$369$	−2552.03	−0.360036
$370$	−845.333	−0.118775
$371$	−12678.5	−1.77422
$372$	25.7371	0.00358712
$373$	−875.284	−0.121503	−0.0607513	−	0.998153i	$-0.519350\pi$
$373$	−875.284	−0.121503	−0.0607513	+	0.998153i	$0.519350\pi$
$374$	410.260	0.0567220
$375$	−386.246	−0.0531884
$376$	−10383.8	−1.42421
$377$	8930.65	1.22003
$378$	8812.33	1.19909
$379$	−5941.38	−0.805246	−0.402623	−	0.915366i	$-0.631901\pi$
$379$	−5941.38	−0.805246	−0.402623	+	0.915366i	$0.631901\pi$
$380$	−61.2369	−0.00826681
$381$	3297.68	0.443426
$382$	13927.5	1.86543
$383$	−14285.2	−1.90585	−0.952925	−	0.303207i	$-0.901943\pi$
$383$	−14285.2	−1.90585	−0.952925	+	0.303207i	$0.901943\pi$
$384$	−4938.17	−0.656250
$385$	1200.15	0.158871
$386$	−8689.24	−1.14578
$387$	−5315.96	−0.698257
$388$	−893.692	−0.116934
$389$	−13490.6	−1.75835	−0.879176	−	0.476497i	$-0.841906\pi$
$389$	−13490.6	−1.75835	−0.879176	+	0.476497i	$0.841906\pi$
$390$	−3764.67	−0.488799
$391$	−2335.16	−0.302031
$392$	−2879.54	−0.371017
$393$	−3658.07	−0.469530
$394$	2973.11	0.380160
$395$	3934.08	0.501127
$396$	−123.746	−0.0157032
$397$	−7587.31	−0.959184	−0.479592	−	0.877492i	$-0.659215\pi$
$397$	−7587.31	−0.959184	−0.479592	+	0.877492i	$0.659215\pi$
$398$	−5344.03	−0.673046
$399$	1281.09	0.160739
$400$	−1718.53	−0.214817
$401$	12212.1	1.52081	0.760403	−	0.649452i	$-0.225002\pi$
$401$	12212.1	1.52081	0.760403	+	0.649452i	$0.225002\pi$
$402$	−5313.89	−0.659285
$403$	−1070.90	−0.132370
$404$	−154.846	−0.0190689
$405$	233.912	0.0286992
$406$	6913.49	0.845100
$407$	−632.525	−0.0770347
$408$	847.670	0.102858
$409$	1413.49	0.170887	0.0854434	−	0.996343i	$-0.472769\pi$
$409$	1413.49	0.170887	0.0854434	+	0.996343i	$0.472769\pi$
$410$	2149.71	0.258943
$411$	−5390.00	−0.646884
$412$	−917.491	−0.109712
$413$	−9173.97	−1.09303
$414$	9445.88	1.12135
$415$	439.303	0.0519628
$416$	−2411.87	−0.284259
$417$	−6730.25	−0.790364
$418$	−614.496	−0.0719042
$419$	8680.09	1.01205	0.506027	−	0.862518i	$-0.331114\pi$
$419$	8680.09	1.01205	0.506027	+	0.862518i	$0.331114\pi$
$420$	−217.313	−0.0252471
$421$	−9564.46	−1.10723	−0.553614	−	0.832773i	$-0.686752\pi$
$421$	−9564.46	−1.10723	−0.553614	+	0.832773i	$0.686752\pi$
$422$	−16738.1	−1.93080
$423$	−8379.62	−0.963194
$424$	12565.4	1.43922
$425$	317.128	0.0361952
$426$	−10678.3	−1.21447
$427$	7795.83	0.883529
$428$	1035.77	0.116976
$429$	−2816.94	−0.317023
$430$	4477.92	0.502197
$431$	−4544.44	−0.507884	−0.253942	−	0.967219i	$-0.581727\pi$
$431$	−4544.44	−0.507884	−0.253942	+	0.967219i	$0.581727\pi$
$432$	−9442.00	−1.05157
$433$	11866.8	1.31704	0.658522	−	0.752562i	$-0.271182\pi$
$433$	11866.8	1.31704	0.658522	+	0.752562i	$0.271182\pi$
$434$	−829.013	−0.0916910
$435$	−1664.86	−0.183503
$436$	−48.6443	−0.00534321
$437$	3497.65	0.382872
$438$	−3026.60	−0.330175
$439$	−1986.84	−0.216006	−0.108003	−	0.994151i	$-0.534446\pi$
$439$	−1986.84	−0.216006	−0.108003	+	0.994151i	$0.534446\pi$
$440$	−1189.44	−0.128873
$441$	−2323.76	−0.250919
$442$	3090.99	0.332632
$443$	1530.89	0.164187	0.0820936	−	0.996625i	$-0.473839\pi$
$443$	1530.89	0.164187	0.0820936	+	0.996625i	$0.473839\pi$
$444$	114.533	0.0122421
$445$	7285.04	0.776054
$446$	−7689.05	−0.816339
$447$	−2740.82	−0.290014
$448$	10132.9	1.06860
$449$	6675.68	0.701658	0.350829	−	0.936439i	$-0.385900\pi$
$449$	6675.68	0.701658	0.350829	+	0.936439i	$0.385900\pi$
$450$	−1282.80	−0.134382
$451$	1608.54	0.167945
$452$	1112.45	0.115764
$453$	7826.81	0.811778
$454$	16074.4	1.66170
$455$	9042.17	0.931656
$456$	−1269.66	−0.130388
$457$	−12178.0	−1.24652	−0.623262	−	0.782013i	$-0.714193\pi$
$457$	−12178.0	−1.24652	−0.623262	+	0.782013i	$0.714193\pi$
$458$	14070.2	1.43550
$459$	1742.37	0.177183
$460$	−593.311	−0.0601375
$461$	−12157.0	−1.22822	−0.614109	−	0.789221i	$-0.710485\pi$
$461$	−12157.0	−1.22822	−0.614109	+	0.789221i	$0.710485\pi$
$462$	−2180.68	−0.219598
$463$	11546.9	1.15903	0.579515	−	0.814962i	$-0.303242\pi$
$463$	11546.9	1.15903	0.579515	+	0.814962i	$0.303242\pi$
$464$	−7407.48	−0.741128
$465$	199.637	0.0199095
$466$	12435.2	1.23616
$467$	−922.458	−0.0914053	−0.0457027	−	0.998955i	$-0.514553\pi$
$467$	−922.458	−0.0914053	−0.0457027	+	0.998955i	$0.514553\pi$
$468$	−932.326	−0.0920872
$469$	12763.2	1.25661
$470$	7058.60	0.692743
$471$	451.885	0.0442075
$472$	9092.09	0.886647
$473$	3350.63	0.325713
$474$	−7148.25	−0.692679
$475$	−475.000	−0.0458831
$476$	178.425	0.0171809
$477$	10140.1	0.973344
$478$	−8341.82	−0.798213
$479$	6385.63	0.609116	0.304558	−	0.952494i	$-0.401491\pi$
$479$	6385.63	0.609116	0.304558	+	0.952494i	$0.401491\pi$
$480$	449.622	0.0427548
$481$	−4765.58	−0.451750
$482$	−1405.10	−0.132781
$483$	12412.2	1.16931
$484$	77.9965	0.00732499
$485$	−6932.16	−0.649016
$486$	−11328.9	−1.05739
$487$	−2794.43	−0.260016	−0.130008	−	0.991513i	$-0.541500\pi$
$487$	−2794.43	−0.260016	−0.130008	+	0.991513i	$0.541500\pi$
$488$	−7726.25	−0.716703
$489$	−1229.75	−0.113724
$490$	1957.43	0.180465
$491$	−2556.60	−0.234985	−0.117493	−	0.993074i	$-0.537486\pi$
$491$	−2556.60	−0.234985	−0.117493	+	0.993074i	$0.537486\pi$
$492$	−291.260	−0.0266891
$493$	1366.93	0.124875
$494$	−4629.74	−0.421664
$495$	−959.865	−0.0871571
$496$	888.248	0.0804103
$497$	25647.7	2.31480
$498$	−798.217	−0.0718252
$499$	−2490.89	−0.223462	−0.111731	−	0.993738i	$-0.535639\pi$
$499$	−2490.89	−0.223462	−0.111731	+	0.993738i	$0.535639\pi$
$500$	80.5749	0.00720684
$501$	2148.92	0.191630
$502$	−5206.58	−0.462910
$503$	15499.1	1.37390	0.686949	−	0.726706i	$-0.258950\pi$
$503$	15499.1	1.37390	0.686949	+	0.726706i	$0.258950\pi$
$504$	8235.67	0.727868
$505$	−1201.10	−0.105838
$506$	−5953.71	−0.523072
$507$	−14434.8	−1.26444
$508$	−687.930	−0.0600826
$509$	5686.01	0.495144	0.247572	−	0.968870i	$-0.420367\pi$
$509$	5686.01	0.495144	0.247572	+	0.968870i	$0.420367\pi$
$510$	−576.223	−0.0500305
$511$	7269.45	0.629318
$512$	−9892.35	−0.853876
$513$	−2609.76	−0.224607
$514$	−942.329	−0.0808645
$515$	−7116.76	−0.608935
$516$	−606.706	−0.0517611
$517$	5281.64	0.449297
$518$	−3689.18	−0.312921
$519$	8740.92	0.739275
$520$	−8961.47	−0.755743
$521$	−20203.0	−1.69887	−0.849434	−	0.527694i	$-0.823057\pi$
$521$	−20203.0	−1.69887	−0.849434	+	0.527694i	$0.823057\pi$
$522$	−5529.33	−0.463625
$523$	18328.0	1.53236	0.766181	−	0.642624i	$-0.222154\pi$
$523$	18328.0	1.53236	0.766181	+	0.642624i	$0.222154\pi$
$524$	763.111	0.0636196
$525$	−1685.65	−0.140129
$526$	4472.60	0.370750
$527$	−163.912	−0.0135486
$528$	2336.49	0.192581
$529$	21720.9	1.78523
$530$	−8541.59	−0.700044
$531$	7337.24	0.599640
$532$	−267.249	−0.0217795
$533$	12119.1	0.984867
$534$	−13237.0	−1.07270
$535$	8034.23	0.649252
$536$	−12649.3	−1.01934
$537$	450.911	0.0362351
$538$	6769.66	0.542492
$539$	1464.66	0.117045
$540$	442.696	0.0352789
$541$	−19588.6	−1.55671	−0.778355	−	0.627824i	$-0.783946\pi$
$541$	−19588.6	−1.55671	−0.778355	+	0.627824i	$0.783946\pi$
$542$	76.1841	0.00603761
$543$	5338.61	0.421919
$544$	−369.162	−0.0290950
$545$	−377.322	−0.0296564
$546$	−16429.7	−1.28778
$547$	−14959.4	−1.16932	−0.584658	−	0.811280i	$-0.698771\pi$
$547$	−14959.4	−1.16932	−0.584658	+	0.811280i	$0.698771\pi$
$548$	1124.41	0.0876504
$549$	−6235.02	−0.484707
$550$	808.547	0.0626846
$551$	−2047.42	−0.158299
$552$	−12301.4	−0.948520
$553$	17169.0	1.32026
$554$	−25684.3	−1.96972
$555$	888.401	0.0679469
$556$	1404.00	0.107091
$557$	−10061.8	−0.765405	−0.382702	−	0.923872i	$-0.625007\pi$
$557$	−10061.8	−0.765405	−0.382702	+	0.923872i	$0.625007\pi$
$558$	663.035	0.0503020
$559$	25244.4	1.91006
$560$	−7499.98	−0.565950
$561$	−431.162	−0.0324486
$562$	12831.6	0.963114
$563$	−9704.63	−0.726468	−0.363234	−	0.931698i	$-0.618327\pi$
$563$	−9704.63	−0.726468	−0.363234	+	0.931698i	$0.618327\pi$
$564$	−956.357	−0.0714005
$565$	8629.01	0.642522
$566$	−2304.49	−0.171139
$567$	1020.83	0.0756102
$568$	−25418.8	−1.87773
$569$	14562.6	1.07293	0.536465	−	0.843923i	$-0.319759\pi$
$569$	14562.6	1.07293	0.536465	+	0.843923i	$0.319759\pi$
$570$	863.078	0.0634217
$571$	−1144.47	−0.0838781	−0.0419391	−	0.999120i	$-0.513354\pi$
$571$	−1144.47	−0.0838781	−0.0419391	+	0.999120i	$0.513354\pi$
$572$	587.642	0.0429555
$573$	−14637.1	−1.06715
$574$	9381.72	0.682205
$575$	−4602.17	−0.333780
$576$	−8104.15	−0.586238
$577$	561.315	0.0404989	0.0202494	−	0.999795i	$-0.493554\pi$
$577$	561.315	0.0404989	0.0202494	+	0.999795i	$0.493554\pi$
$578$	−13971.9	−1.00546
$579$	9131.95	0.655459
$580$	347.306	0.0248640
$581$	1917.20	0.136900
$582$	12595.8	0.897099
$583$	−6391.30	−0.454032
$584$	−7204.57	−0.510492
$585$	−7231.83	−0.511110
$586$	−19795.8	−1.39549
$587$	−8964.02	−0.630297	−0.315149	−	0.949042i	$-0.602054\pi$
$587$	−8964.02	−0.630297	−0.315149	+	0.949042i	$0.602054\pi$
$588$	−265.208	−0.0186004
$589$	245.511	0.0171750
$590$	−6180.55	−0.431270
$591$	−3124.59	−0.217476
$592$	3952.78	0.274423
$593$	8309.40	0.575424	0.287712	−	0.957717i	$-0.407105\pi$
$593$	8309.40	0.575424	0.287712	+	0.957717i	$0.407105\pi$
$594$	4442.34	0.306854
$595$	1384.00	0.0953588
$596$	571.762	0.0392958
$597$	5616.30	0.385025
$598$	−44856.5	−3.06742
$599$	26926.0	1.83667	0.918334	−	0.395806i	$-0.129535\pi$
$599$	26926.0	1.83667	0.918334	+	0.395806i	$0.129535\pi$
$600$	1670.60	0.113670
$601$	18019.1	1.22299	0.611493	−	0.791250i	$-0.290569\pi$
$601$	18019.1	1.22299	0.611493	+	0.791250i	$0.290569\pi$
$602$	19542.5	1.32307
$603$	−10207.8	−0.689379
$604$	−1632.75	−0.109993
$605$	605.000	0.0406558
$606$	2182.40	0.146294
$607$	21839.8	1.46038	0.730191	−	0.683243i	$-0.239431\pi$
$607$	21839.8	1.46038	0.730191	+	0.683243i	$0.239431\pi$
$608$	552.939	0.0368826
$609$	−7265.72	−0.483451
$610$	5252.10	0.348608
$611$	39793.0	2.63479
$612$	−142.702	−0.00942550
$613$	−12203.0	−0.804035	−0.402017	−	0.915632i	$-0.631691\pi$
$613$	−12203.0	−0.804035	−0.402017	+	0.915632i	$0.631691\pi$
$614$	22997.3	1.51156
$615$	−2259.24	−0.148132
$616$	−5190.91	−0.339526
$617$	12510.8	0.816314	0.408157	−	0.912912i	$-0.366172\pi$
$617$	12510.8	0.816314	0.408157	+	0.912912i	$0.366172\pi$
$618$	12931.2	0.841697
$619$	15556.5	1.01013	0.505064	−	0.863082i	$-0.331469\pi$
$619$	15556.5	1.01013	0.505064	+	0.863082i	$0.331469\pi$
$620$	−41.6463	−0.00269767
$621$	−25285.3	−1.63392
$622$	−31752.1	−2.04685
$623$	31793.2	2.04457
$624$	17603.6	1.12934
$625$	625.000	0.0400000
$626$	−1889.11	−0.120614
$627$	645.803	0.0411338
$628$	−94.2678	−0.00598996
$629$	−729.423	−0.0462385
$630$	−5598.38	−0.354039
$631$	−1603.68	−0.101175	−0.0505875	−	0.998720i	$-0.516109\pi$
$631$	−1603.68	−0.101175	−0.0505875	+	0.998720i	$0.516109\pi$
$632$	−17015.8	−1.07097
$633$	17590.9	1.10454
$634$	7147.70	0.447747
$635$	−5336.11	−0.333476
$636$	1157.28	0.0721530
$637$	11035.1	0.686381
$638$	3485.12	0.216265
$639$	−20512.8	−1.26991
$640$	7990.65	0.493528
$641$	−12261.0	−0.755510	−0.377755	−	0.925905i	$-0.623304\pi$
$641$	−12261.0	−0.755510	−0.377755	+	0.925905i	$0.623304\pi$
$642$	−14598.2	−0.897425
$643$	−28745.1	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$643$	−28745.1	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$644$	−2589.31	−0.158437
$645$	−4706.07	−0.287289
$646$	−708.631	−0.0431590
$647$	15707.2	0.954427	0.477213	−	0.878787i	$-0.341647\pi$
$647$	15707.2	0.954427	0.477213	+	0.878787i	$0.341647\pi$
$648$	−1011.72	−0.0613337
$649$	−4624.64	−0.279712
$650$	6091.76	0.367598
$651$	871.250	0.0524531
$652$	256.538	0.0154092
$653$	−11883.6	−0.712162	−0.356081	−	0.934455i	$-0.615887\pi$
$653$	−11883.6	−0.712162	−0.356081	+	0.934455i	$0.615887\pi$
$654$	685.597	0.0409923
$655$	5919.27	0.353107
$656$	−10052.1	−0.598274
$657$	−5814.02	−0.345246
$658$	30805.0	1.82508
$659$	−17499.5	−1.03442	−0.517210	−	0.855859i	$-0.673029\pi$
$659$	−17499.5	−1.03442	−0.517210	+	0.855859i	$0.673029\pi$
$660$	−109.548	−0.00646086
$661$	16939.9	0.996802	0.498401	−	0.866947i	$-0.333921\pi$
$661$	16939.9	0.996802	0.498401	+	0.866947i	$0.333921\pi$
$662$	9272.14	0.544368
$663$	−3248.47	−0.190287
$664$	−1900.09	−0.111051
$665$	−2072.98	−0.120882
$666$	2950.57	0.171670
$667$	−19837.0	−1.15156
$668$	−448.287	−0.0259652
$669$	8080.80	0.466998
$670$	8598.62	0.495811
$671$	3929.91	0.226099
$672$	1962.23	0.112641
$673$	−6887.42	−0.394488	−0.197244	−	0.980354i	$-0.563199\pi$
$673$	−6887.42	−0.394488	−0.197244	+	0.980354i	$0.563199\pi$
$674$	3508.68	0.200518
$675$	3433.89	0.195808
$676$	3011.24	0.171327
$677$	−4913.89	−0.278961	−0.139480	−	0.990225i	$-0.544543\pi$
$677$	−4913.89	−0.278961	−0.139480	+	0.990225i	$0.544543\pi$
$678$	−15679.0	−0.888122
$679$	−30253.1	−1.70988
$680$	−1371.65	−0.0773534
$681$	−16893.4	−0.950597
$682$	−417.909	−0.0234642
$683$	21558.3	1.20777	0.603885	−	0.797071i	$-0.293619\pi$
$683$	21558.3	1.20777	0.603885	+	0.797071i	$0.293619\pi$
$684$	213.743	0.0119483
$685$	8721.77	0.486484
$686$	−13463.3	−0.749317
$687$	−14787.1	−0.821196
$688$	−20938.8	−1.16030
$689$	−48153.4	−2.66255
$690$	8362.17	0.461366
$691$	23332.4	1.28452	0.642262	−	0.766485i	$-0.277996\pi$
$691$	23332.4	1.28452	0.642262	+	0.766485i	$0.277996\pi$
$692$	−1823.45	−0.100169
$693$	−4189.02	−0.229622
$694$	−3699.20	−0.202334
$695$	10890.5	0.594388
$696$	7200.88	0.392167
$697$	1854.95	0.100805
$698$	−20353.7	−1.10372
$699$	−13068.8	−0.707161
$700$	351.643	0.0189869
$701$	−31262.1	−1.68439	−0.842193	−	0.539176i	$-0.818736\pi$
$701$	−31262.1	−1.68439	−0.842193	+	0.539176i	$0.818736\pi$
$702$	33469.5	1.79947
$703$	1092.54	0.0586146
$704$	5108.02	0.273460
$705$	−7418.23	−0.396293
$706$	7607.33	0.405532
$707$	−5241.81	−0.278838
$708$	837.391	0.0444507
$709$	35610.9	1.88631	0.943157	−	0.332346i	$-0.107840\pi$
$709$	35610.9	1.88631	0.943157	+	0.332346i	$0.107840\pi$
$710$	17279.0	0.913338
$711$	−13731.6	−0.724297
$712$	−31509.5	−1.65852
$713$	2378.70	0.124941
$714$	−2514.74	−0.131809
$715$	4558.20	0.238415
$716$	−94.0646	−0.00490972
$717$	8766.82	0.456629
$718$	−4796.17	−0.249292
$719$	−35627.7	−1.84797	−0.923983	−	0.382433i	$-0.875086\pi$
$719$	−35627.7	−1.84797	−0.923983	+	0.382433i	$0.875086\pi$
$720$	5998.40	0.310482
$721$	−31058.8	−1.60428
$722$	1061.40	0.0547109
$723$	1476.69	0.0759593
$724$	−1113.69	−0.0571684
$725$	2693.97	0.138002
$726$	−1099.29	−0.0561962
$727$	−5429.12	−0.276967	−0.138484	−	0.990365i	$-0.544223\pi$
$727$	−5429.12	−0.276967	−0.138484	+	0.990365i	$0.544223\pi$
$728$	−39109.4	−1.99106
$729$	10643.0	0.540721
$730$	4897.47	0.248306
$731$	3863.92	0.195503
$732$	−711.597	−0.0359308
$733$	18129.2	0.913528	0.456764	−	0.889588i	$-0.349008\pi$
$733$	18129.2	0.913528	0.456764	+	0.889588i	$0.349008\pi$
$734$	−7419.41	−0.373100
$735$	−2057.16	−0.103237
$736$	5357.30	0.268305
$737$	6433.97	0.321571
$738$	−7503.40	−0.374260
$739$	−18529.5	−0.922351	−0.461175	−	0.887309i	$-0.652572\pi$
$739$	−18529.5	−0.922351	−0.461175	+	0.887309i	$0.652572\pi$
$740$	−185.330	−0.00920656
$741$	4865.62	0.241219
$742$	−37277.0	−1.84432
$743$	37434.9	1.84839	0.924195	−	0.381920i	$-0.124737\pi$
$743$	37434.9	1.84839	0.924195	+	0.381920i	$0.124737\pi$
$744$	−863.474	−0.0425491
$745$	4435.02	0.218103
$746$	−2573.48	−0.126303
$747$	−1533.35	−0.0751037
$748$	89.9449	0.00439667
$749$	35062.8	1.71050
$750$	−1135.63	−0.0552897
$751$	39632.9	1.92573	0.962866	−	0.269979i	$-0.0870166\pi$
$751$	39632.9	1.92573	0.962866	+	0.269979i	$0.0870166\pi$
$752$	−33006.1	−1.60054
$753$	5471.84	0.264814
$754$	26257.6	1.26823
$755$	−12664.9	−0.610492
$756$	1932.00	0.0929449
$757$	16486.3	0.791552	0.395776	−	0.918347i	$-0.370476\pi$
$757$	16486.3	0.791552	0.395776	+	0.918347i	$0.370476\pi$
$758$	−17468.7	−0.837059
$759$	6257.04	0.299231
$760$	2054.48	0.0980578
$761$	−1768.18	−0.0842265	−0.0421132	−	0.999113i	$-0.513409\pi$
$761$	−1768.18	−0.0842265	−0.0421132	+	0.999113i	$0.513409\pi$
$762$	9695.75	0.460945
$763$	−1646.70	−0.0781318
$764$	3053.46	0.144594
$765$	−1106.91	−0.0523142
$766$	−42001.0	−1.98114
$767$	−34843.0	−1.64030
$768$	−3040.06	−0.142837
$769$	−32291.3	−1.51424	−0.757122	−	0.653274i	$-0.773395\pi$
$769$	−32291.3	−1.51424	−0.757122	+	0.653274i	$0.773395\pi$
$770$	3528.64	0.165147
$771$	990.339	0.0462597
$772$	−1905.02	−0.0888123
$773$	3981.66	0.185266	0.0926328	−	0.995700i	$-0.470472\pi$
$773$	3981.66	0.185266	0.0926328	+	0.995700i	$0.470472\pi$
$774$	−15629.8	−0.725844
$775$	−323.040	−0.0149728
$776$	29983.1	1.38703
$777$	3877.14	0.179011
$778$	−39664.6	−1.82782
$779$	−2778.38	−0.127787
$780$	−825.362	−0.0378881
$781$	12929.1	0.592369
$782$	−6865.77	−0.313963
$783$	14801.3	0.675548
$784$	−9152.96	−0.416953
$785$	−731.213	−0.0332460
$786$	−10755.3	−0.488079
$787$	−24108.5	−1.09196	−0.545982	−	0.837797i	$-0.683843\pi$
$787$	−24108.5	−1.09196	−0.545982	+	0.837797i	$0.683843\pi$
$788$	651.821	0.0294672
$789$	−4700.47	−0.212093
$790$	11566.9	0.520925
$791$	37658.5	1.69277
$792$	4151.63	0.186265
$793$	29608.8	1.32590
$794$	−22308.0	−0.997078
$795$	8976.78	0.400470
$796$	−1171.62	−0.0521695
$797$	−14658.6	−0.651488	−0.325744	−	0.945458i	$-0.605615\pi$
$797$	−14658.6	−0.651488	−0.325744	+	0.945458i	$0.605615\pi$
$798$	3766.62	0.167089
$799$	6090.75	0.269681
$800$	−727.551	−0.0321535
$801$	−25427.9	−1.12166
$802$	35905.6	1.58089
$803$	3664.56	0.161045
$804$	−1165.01	−0.0511029
$805$	−20084.7	−0.879368
$806$	−3148.62	−0.137600
$807$	−7114.56	−0.310340
$808$	5195.03	0.226189
$809$	−24551.9	−1.06700	−0.533498	−	0.845801i	$-0.679123\pi$
$809$	−24551.9	−1.06700	−0.533498	+	0.845801i	$0.679123\pi$
$810$	687.741	0.0298331
$811$	16129.3	0.698368	0.349184	−	0.937054i	$-0.386459\pi$
$811$	16129.3	0.698368	0.349184	+	0.937054i	$0.386459\pi$
$812$	1515.70	0.0655059
$813$	−80.0656	−0.00345390
$814$	−1859.73	−0.0800781
$815$	1989.90	0.0855254
$816$	2694.42	0.115593
$817$	−5787.46	−0.247831
$818$	4155.91	0.177638
$819$	−31561.0	−1.34656
$820$	471.300	0.0200714
$821$	−80.7939	−0.00343450	−0.00171725	−	0.999999i	$-0.500547\pi$
$821$	−80.7939	−0.00343450	−0.00171725	+	0.999999i	$0.500547\pi$
$822$	−15847.5	−0.672440
$823$	42488.6	1.79959	0.899794	−	0.436315i	$-0.143717\pi$
$823$	42488.6	1.79959	0.899794	+	0.436315i	$0.143717\pi$
$824$	30781.6	1.30137
$825$	−849.741	−0.0358596
$826$	−26973.0	−1.13621
$827$	−18322.4	−0.770414	−0.385207	−	0.922830i	$-0.625870\pi$
$827$	−18322.4	−0.770414	−0.385207	+	0.922830i	$0.625870\pi$
$828$	2070.90	0.0869189
$829$	−22311.0	−0.934733	−0.467367	−	0.884064i	$-0.654797\pi$
$829$	−22311.0	−0.934733	−0.467367	+	0.884064i	$0.654797\pi$
$830$	1291.63	0.0540157
$831$	26992.9	1.12680
$832$	38484.9	1.60363
$833$	1689.03	0.0702539
$834$	−19788.1	−0.821588
$835$	−3477.26	−0.144114
$836$	−134.721	−0.00557348
$837$	−1774.85	−0.0732950
$838$	25520.9	1.05204
$839$	27721.8	1.14072	0.570360	−	0.821395i	$-0.306804\pi$
$839$	27721.8	1.14072	0.570360	+	0.821395i	$0.306804\pi$
$840$	7290.80	0.299472
$841$	−12777.0	−0.523886
$842$	−28121.1	−1.15097
$843$	−13485.4	−0.550963
$844$	−3669.64	−0.149661
$845$	23357.5	0.950912
$846$	−24637.5	−1.00125
$847$	2640.33	0.107111
$848$	39940.6	1.61741
$849$	2421.90	0.0979027
$850$	932.409	0.0376251
$851$	10585.4	0.426396
$852$	−2341.10	−0.0941371
$853$	−1036.14	−0.0415905	−0.0207952	−	0.999784i	$-0.506620\pi$
$853$	−1036.14	−0.0415905	−0.0207952	+	0.999784i	$0.506620\pi$
$854$	22921.1	0.918434
$855$	1657.95	0.0663166
$856$	−34749.9	−1.38753
$857$	15796.8	0.629646	0.314823	−	0.949150i	$-0.398055\pi$
$857$	15796.8	0.629646	0.314823	+	0.949150i	$0.398055\pi$
$858$	−8282.28	−0.329548
$859$	−4654.86	−0.184891	−0.0924457	−	0.995718i	$-0.529468\pi$
$859$	−4654.86	−0.184891	−0.0924457	+	0.995718i	$0.529468\pi$
$860$	981.734	0.0389266
$861$	−9859.71	−0.390265
$862$	−13361.4	−0.527949
$863$	20998.8	0.828283	0.414142	−	0.910212i	$-0.364082\pi$
$863$	20998.8	0.828283	0.414142	+	0.910212i	$0.364082\pi$
$864$	−3997.33	−0.157398
$865$	−14144.0	−0.555967
$866$	34890.3	1.36908
$867$	14683.8	0.575188
$868$	−181.752	−0.00710721
$869$	8654.98	0.337860
$870$	−4894.96	−0.190752
$871$	48474.9	1.88577
$872$	1632.00	0.0633792
$873$	24196.1	0.938047
$874$	10283.7	0.397998
$875$	2727.61	0.105383
$876$	−663.549	−0.0255927
$877$	19013.9	0.732101	0.366050	−	0.930595i	$-0.380710\pi$
$877$	19013.9	0.732101	0.366050	+	0.930595i	$0.380710\pi$
$878$	−5841.65	−0.224540
$879$	20804.4	0.798309
$880$	−3780.77	−0.144829
$881$	−28553.4	−1.09193	−0.545964	−	0.837809i	$-0.683836\pi$
$881$	−28553.4	−1.09193	−0.545964	+	0.837809i	$0.683836\pi$
$882$	−6832.25	−0.260832
$883$	9568.17	0.364660	0.182330	−	0.983237i	$-0.441636\pi$
$883$	9568.17	0.364660	0.182330	+	0.983237i	$0.441636\pi$
$884$	677.664	0.0257832
$885$	6495.44	0.246714
$886$	4501.08	0.170674
$887$	41810.3	1.58270	0.791348	−	0.611366i	$-0.209380\pi$
$887$	41810.3	1.58270	0.791348	+	0.611366i	$0.209380\pi$
$888$	−3842.54	−0.145211
$889$	−23287.7	−0.878566
$890$	21419.3	0.806714
$891$	514.607	0.0193490
$892$	−1685.74	−0.0632765
$893$	−9122.84	−0.341864
$894$	−8058.46	−0.301471
$895$	−729.637	−0.0272504
$896$	34872.6	1.30024
$897$	47141.9	1.75476
$898$	19627.6	0.729379
$899$	−1392.42	−0.0516571
$900$	−281.240	−0.0104163
$901$	−7370.39	−0.272523
$902$	4729.37	0.174580
$903$	−20538.1	−0.756883
$904$	−37322.4	−1.37315
$905$	−8638.62	−0.317301
$906$	23012.1	0.843849
$907$	46854.1	1.71529	0.857643	−	0.514246i	$-0.171928\pi$
$907$	46854.1	1.71529	0.857643	+	0.514246i	$0.171928\pi$
$908$	3524.14	0.128802
$909$	4192.34	0.152972
$910$	26585.5	0.968463
$911$	21698.7	0.789143	0.394571	−	0.918865i	$-0.370893\pi$
$911$	21698.7	0.789143	0.394571	+	0.918865i	$0.370893\pi$
$912$	−4035.76	−0.146532
$913$	966.467	0.0350333
$914$	−35805.3	−1.29577
$915$	−5519.68	−0.199426
$916$	3084.73	0.111269
$917$	25832.7	0.930285
$918$	5122.86	0.184183
$919$	−39565.3	−1.42017	−0.710086	−	0.704114i	$-0.751344\pi$
$919$	−39565.3	−1.42017	−0.710086	+	0.704114i	$0.751344\pi$
$920$	19905.4	0.713328
$921$	−24169.0	−0.864709
$922$	−35743.7	−1.27674
$923$	97410.8	3.47380
$924$	−478.089	−0.0170216
$925$	−1437.56	−0.0510990
$926$	33949.9	1.20482
$927$	24840.5	0.880117
$928$	−3136.00	−0.110931
$929$	−27038.3	−0.954894	−0.477447	−	0.878661i	$-0.658438\pi$
$929$	−27038.3	−0.954894	−0.477447	+	0.878661i	$0.658438\pi$
$930$	586.966	0.0206961
$931$	−2529.87	−0.0890580
$932$	2726.28	0.0958177
$933$	33369.8	1.17093
$934$	−2712.18	−0.0950165
$935$	697.681	0.0244028
$936$	31279.3	1.09230
$937$	−38358.7	−1.33738	−0.668689	−	0.743542i	$-0.733144\pi$
$937$	−38358.7	−1.33738	−0.668689	+	0.743542i	$0.733144\pi$
$938$	37525.9	1.30625
$939$	1985.36	0.0689987
$940$	1547.52	0.0536963
$941$	28383.3	0.983284	0.491642	−	0.870797i	$-0.336397\pi$
$941$	28383.3	0.983284	0.491642	+	0.870797i	$0.336397\pi$
$942$	1328.62	0.0459540
$943$	−26919.1	−0.929593
$944$	28900.3	0.996425
$945$	14986.1	0.515871
$946$	9851.43	0.338581
$947$	6104.68	0.209478	0.104739	−	0.994500i	$-0.466599\pi$
$947$	6104.68	0.209478	0.104739	+	0.994500i	$0.466599\pi$
$948$	−1567.17	−0.0536914
$949$	27609.6	0.944410
$950$	−1396.58	−0.0476959
$951$	−7511.87	−0.256140
$952$	−5986.12	−0.203793
$953$	6516.61	0.221505	0.110752	−	0.993848i	$-0.464674\pi$
$953$	6516.61	0.221505	0.110752	+	0.993848i	$0.464674\pi$
$954$	29813.7	1.01180
$955$	23684.9	0.802540
$956$	−1828.85	−0.0618715
$957$	−3662.68	−0.123718
$958$	18774.8	0.633181
$959$	38063.4	1.28168
$960$	−7174.37	−0.241200
$961$	−29624.0	−0.994395
$962$	−14011.6	−0.469597
$963$	−28042.8	−0.938388
$964$	−308.052	−0.0102922
$965$	−14776.8	−0.492934
$966$	36494.0	1.21550
$967$	8518.96	0.283300	0.141650	−	0.989917i	$-0.454759\pi$
$967$	8518.96	0.283300	0.141650	+	0.989917i	$0.454759\pi$
$968$	−2616.76	−0.0868863
$969$	744.735	0.0246897
$970$	−20381.7	−0.674657
$971$	13177.0	0.435501	0.217750	−	0.976005i	$-0.430128\pi$
$971$	13177.0	0.435501	0.217750	+	0.976005i	$0.430128\pi$
$972$	−2483.74	−0.0819609
$973$	47528.0	1.56596
$974$	−8216.09	−0.270288
$975$	−6402.13	−0.210290
$976$	−24558.8	−0.805440
$977$	−10075.1	−0.329919	−0.164960	−	0.986300i	$-0.552749\pi$
$977$	−10075.1	−0.329919	−0.164960	+	0.986300i	$0.552749\pi$
$978$	−3615.66	−0.118217
$979$	16027.1	0.523216
$980$	429.144	0.0139883
$981$	1317.01	0.0428634
$982$	−7516.84	−0.244269
$983$	−57333.9	−1.86029	−0.930146	−	0.367191i	$-0.880320\pi$
$983$	−57333.9	−1.86029	−0.930146	+	0.367191i	$0.880320\pi$
$984$	9771.71	0.316576
$985$	5056.02	0.163551
$986$	4019.01	0.129809
$987$	−32374.5	−1.04406
$988$	−1015.02	−0.0326842
$989$	−56073.4	−1.80286
$990$	−2822.17	−0.0906004
$991$	−18725.5	−0.600238	−0.300119	−	0.953902i	$-0.597026\pi$
$991$	−18725.5	−0.600238	−0.300119	+	0.953902i	$0.597026\pi$
$992$	376.045	0.0120357
$993$	−9744.54	−0.311414
$994$	75408.7	2.40625
$995$	−9087.97	−0.289556
$996$	−175.000	−0.00556736
$997$	−10627.7	−0.337597	−0.168798	−	0.985651i	$-0.553989\pi$
$997$	−10627.7	−0.337597	−0.168798	+	0.985651i	$0.553989\pi$
$998$	−7323.63	−0.232290
$999$	−7898.26	−0.250140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.e.1.16	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.e.1.16	✓	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)

\(f(q)\)	\(=\)	\(q+2.94017 q^{2} -3.08997 q^{3} +0.644599 q^{4} +5.00000 q^{5} -9.08503 q^{6} +21.8209 q^{7} -21.6261 q^{8} -17.4521 q^{9} +O(q^{10})\)	\(q+2.94017 q^{2} -3.08997 q^{3} +0.644599 q^{4} +5.00000 q^{5} -9.08503 q^{6} +21.8209 q^{7} -21.6261 q^{8} -17.4521 q^{9} +14.7008 q^{10} +11.0000 q^{11} -1.99179 q^{12} +82.8764 q^{13} +64.1571 q^{14} -15.4498 q^{15} -68.7413 q^{16} +12.6851 q^{17} -51.3121 q^{18} -19.0000 q^{19} +3.22300 q^{20} -67.4258 q^{21} +32.3419 q^{22} -184.087 q^{23} +66.8240 q^{24} +25.0000 q^{25} +243.671 q^{26} +137.356 q^{27} +14.0657 q^{28} +107.759 q^{29} -45.4252 q^{30} -12.9216 q^{31} -29.1020 q^{32} -33.9896 q^{33} +37.2964 q^{34} +109.104 q^{35} -11.2496 q^{36} -57.5023 q^{37} -55.8632 q^{38} -256.085 q^{39} -108.131 q^{40} +146.230 q^{41} -198.243 q^{42} +304.603 q^{43} +7.09059 q^{44} -87.2605 q^{45} -541.246 q^{46} +480.149 q^{47} +212.408 q^{48} +133.151 q^{49} +73.5042 q^{50} -39.1966 q^{51} +53.4220 q^{52} -581.027 q^{53} +403.849 q^{54} +55.0000 q^{55} -471.901 q^{56} +58.7094 q^{57} +316.829 q^{58} -420.422 q^{59} -9.95895 q^{60} +357.265 q^{61} -37.9917 q^{62} -380.820 q^{63} +464.365 q^{64} +414.382 q^{65} -99.9353 q^{66} +584.906 q^{67} +8.17681 q^{68} +568.822 q^{69} +320.785 q^{70} +1175.38 q^{71} +377.421 q^{72} +333.142 q^{73} -169.067 q^{74} -77.2492 q^{75} -12.2474 q^{76} +240.030 q^{77} -752.934 q^{78} +786.816 q^{79} -343.706 q^{80} +46.7824 q^{81} +429.943 q^{82} +87.8607 q^{83} -43.4626 q^{84} +63.4255 q^{85} +895.585 q^{86} -332.971 q^{87} -237.887 q^{88} +1457.01 q^{89} -256.561 q^{90} +1808.43 q^{91} -118.662 q^{92} +39.9274 q^{93} +1411.72 q^{94} -95.0000 q^{95} +89.9243 q^{96} -1386.43 q^{97} +391.486 q^{98} -191.973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) \) 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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