LMFDB - Embedded newform 1045.4.a.f.1.9

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:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-2.04967 q^{2} -4.19821 q^{3} -3.79884 q^{4} -5.00000 q^{5} +8.60496 q^{6} -17.0006 q^{7} +24.1838 q^{8} -9.37502 q^{9} +O(q^{10})$	\(q-2.04967 q^{2} -4.19821 q^{3} -3.79884 q^{4} -5.00000 q^{5} +8.60496 q^{6} -17.0006 q^{7} +24.1838 q^{8} -9.37502 q^{9} +10.2484 q^{10} +11.0000 q^{11} +15.9483 q^{12} -56.8351 q^{13} +34.8456 q^{14} +20.9911 q^{15} -19.1780 q^{16} -31.2236 q^{17} +19.2157 q^{18} -19.0000 q^{19} +18.9942 q^{20} +71.3720 q^{21} -22.5464 q^{22} +121.111 q^{23} -101.529 q^{24} +25.0000 q^{25} +116.493 q^{26} +152.710 q^{27} +64.5825 q^{28} -28.6751 q^{29} -43.0248 q^{30} +69.2583 q^{31} -154.161 q^{32} -46.1803 q^{33} +63.9982 q^{34} +85.0029 q^{35} +35.6142 q^{36} -274.720 q^{37} +38.9438 q^{38} +238.606 q^{39} -120.919 q^{40} -209.355 q^{41} -146.289 q^{42} +475.485 q^{43} -41.7873 q^{44} +46.8751 q^{45} -248.239 q^{46} +578.585 q^{47} +80.5135 q^{48} -53.9802 q^{49} -51.2418 q^{50} +131.083 q^{51} +215.908 q^{52} +602.622 q^{53} -313.006 q^{54} -55.0000 q^{55} -411.138 q^{56} +79.7660 q^{57} +58.7745 q^{58} +640.061 q^{59} -79.7417 q^{60} +4.96918 q^{61} -141.957 q^{62} +159.381 q^{63} +469.405 q^{64} +284.176 q^{65} +94.6545 q^{66} +263.636 q^{67} +118.614 q^{68} -508.451 q^{69} -174.228 q^{70} +648.282 q^{71} -226.723 q^{72} -718.618 q^{73} +563.085 q^{74} -104.955 q^{75} +72.1780 q^{76} -187.006 q^{77} -489.064 q^{78} -1022.23 q^{79} +95.8902 q^{80} -387.983 q^{81} +429.110 q^{82} -79.8926 q^{83} -271.131 q^{84} +156.118 q^{85} -974.589 q^{86} +120.384 q^{87} +266.021 q^{88} +897.727 q^{89} -96.0786 q^{90} +966.230 q^{91} -460.083 q^{92} -290.761 q^{93} -1185.91 q^{94} +95.0000 q^{95} +647.202 q^{96} -1090.53 q^{97} +110.642 q^{98} -103.125 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * 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 $

$2$	−2.04967	−0.724669	−0.362334	−	0.932048i	$-0.618020\pi$
$2$	−2.04967	−0.724669	−0.362334	+	0.932048i	$0.618020\pi$
$3$	−4.19821	−0.807946	−0.403973	−	0.914771i	$-0.632371\pi$
$3$	−4.19821	−0.807946	−0.403973	+	0.914771i	$0.632371\pi$
$4$	−3.79884	−0.474855
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	8.60496	0.585493
$7$	−17.0006	−0.917945	−0.458973	−	0.888450i	$-0.651782\pi$
$7$	−17.0006	−0.917945	−0.458973	+	0.888450i	$0.651782\pi$
$8$	24.1838	1.06878
$9$	−9.37502	−0.347223
$10$	10.2484	0.324082
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	15.9483	0.383658
$13$	−56.8351	−1.21256	−0.606278	−	0.795253i	$-0.707338\pi$
$13$	−56.8351	−1.21256	−0.606278	+	0.795253i	$0.707338\pi$
$14$	34.8456	0.665206
$15$	20.9911	0.361324
$16$	−19.1780	−0.299657
$17$	−31.2236	−0.445461	−0.222731	−	0.974880i	$-0.571497\pi$
$17$	−31.2236	−0.445461	−0.222731	+	0.974880i	$0.571497\pi$
$18$	19.2157	0.251622
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	18.9942	0.212362
$21$	71.3720	0.741650
$22$	−22.5464	−0.218496
$23$	121.111	1.09798	0.548989	−	0.835830i	$-0.315013\pi$
$23$	121.111	1.09798	0.548989	+	0.835830i	$0.315013\pi$
$24$	−101.529	−0.863518
$25$	25.0000	0.200000
$26$	116.493	0.878701
$27$	152.710	1.08848
$28$	64.5825	0.435891
$29$	−28.6751	−0.183615	−0.0918074	−	0.995777i	$-0.529264\pi$
$29$	−28.6751	−0.183615	−0.0918074	+	0.995777i	$0.529264\pi$
$30$	−43.0248	−0.261840
$31$	69.2583	0.401263	0.200632	−	0.979667i	$-0.435701\pi$
$31$	69.2583	0.401263	0.200632	+	0.979667i	$0.435701\pi$
$32$	−154.161	−0.851629
$33$	−46.1803	−0.243605
$34$	63.9982	0.322812
$35$	85.0029	0.410518
$36$	35.6142	0.164881
$37$	−274.720	−1.22064	−0.610319	−	0.792156i	$-0.708959\pi$
$37$	−274.720	−1.22064	−0.610319	+	0.792156i	$0.708959\pi$
$38$	38.9438	0.166250
$39$	238.606	0.979680
$40$	−120.919	−0.477974
$41$	−209.355	−0.797458	−0.398729	−	0.917069i	$-0.630549\pi$
$41$	−209.355	−0.797458	−0.398729	+	0.917069i	$0.630549\pi$
$42$	−146.289	−0.537451
$43$	475.485	1.68630	0.843149	−	0.537679i	$-0.180699\pi$
$43$	475.485	1.68630	0.843149	+	0.537679i	$0.180699\pi$
$44$	−41.7873	−0.143174
$45$	46.8751	0.155283
$46$	−248.239	−0.795670
$47$	578.585	1.79564	0.897822	−	0.440358i	$-0.145149\pi$
$47$	578.585	1.79564	0.897822	+	0.440358i	$0.145149\pi$
$48$	80.5135	0.242107
$49$	−53.9802	−0.157377
$50$	−51.2418	−0.144934
$51$	131.083	0.359909
$52$	215.908	0.575789
$53$	602.622	1.56182	0.780910	−	0.624644i	$-0.214756\pi$
$53$	602.622	1.56182	0.780910	+	0.624644i	$0.214756\pi$
$54$	−313.006	−0.788790
$55$	−55.0000	−0.134840
$56$	−411.138	−0.981083
$57$	79.7660	0.185356
$58$	58.7745	0.133060
$59$	640.061	1.41235	0.706177	−	0.708035i	$-0.250418\pi$
$59$	640.061	1.41235	0.706177	+	0.708035i	$0.250418\pi$
$60$	−79.7417	−0.171577
$61$	4.96918	0.0104301	0.00521507	−	0.999986i	$-0.498340\pi$
$61$	4.96918	0.0104301	0.00521507	+	0.999986i	$0.498340\pi$
$62$	−141.957	−0.290783
$63$	159.381	0.318732
$64$	469.405	0.916806
$65$	284.176	0.542271
$66$	94.6545	0.176533
$67$	263.636	0.480720	0.240360	−	0.970684i	$-0.422734\pi$
$67$	263.636	0.480720	0.240360	+	0.970684i	$0.422734\pi$
$68$	118.614	0.211530
$69$	−508.451	−0.887106
$70$	−174.228	−0.297489
$71$	648.282	1.08362	0.541809	−	0.840501i	$-0.317740\pi$
$71$	648.282	1.08362	0.541809	+	0.840501i	$0.317740\pi$
$72$	−226.723	−0.371106
$73$	−718.618	−1.15216	−0.576082	−	0.817392i	$-0.695419\pi$
$73$	−718.618	−1.15216	−0.576082	+	0.817392i	$0.695419\pi$
$74$	563.085	0.884558
$75$	−104.955	−0.161589
$76$	72.1780	0.108939
$77$	−187.006	−0.276771
$78$	−489.064	−0.709943
$79$	−1022.23	−1.45582	−0.727912	−	0.685671i	$-0.759509\pi$
$79$	−1022.23	−1.45582	−0.727912	+	0.685671i	$0.759509\pi$
$80$	95.8902	0.134011
$81$	−387.983	−0.532213
$82$	429.110	0.577893
$83$	−79.8926	−0.105655	−0.0528274	−	0.998604i	$-0.516823\pi$
$83$	−79.8926	−0.105655	−0.0528274	+	0.998604i	$0.516823\pi$
$84$	−271.131	−0.352177
$85$	156.118	0.199216
$86$	−974.589	−1.22201
$87$	120.384	0.148351
$88$	266.021	0.322250
$89$	897.727	1.06920	0.534600	−	0.845105i	$-0.320462\pi$
$89$	897.727	1.06920	0.534600	+	0.845105i	$0.320462\pi$
$90$	−96.0786	−0.112529
$91$	966.230	1.11306
$92$	−460.083	−0.521380
$93$	−290.761	−0.324199
$94$	−1185.91	−1.30125
$95$	95.0000	0.102598
$96$	647.202	0.688071
$97$	−1090.53	−1.14151	−0.570756	−	0.821120i	$-0.693350\pi$
$97$	−1090.53	−1.14151	−0.570756	+	0.821120i	$0.693350\pi$
$98$	110.642	0.114046
$99$	−103.125	−0.104692
$100$	−94.9711	−0.0949711
$101$	−582.230	−0.573605	−0.286802	−	0.957990i	$-0.592592\pi$
$101$	−582.230	−0.573605	−0.286802	+	0.957990i	$0.592592\pi$
$102$	−268.678	−0.260814
$103$	−151.761	−0.145179	−0.0725896	−	0.997362i	$-0.523126\pi$
$103$	−151.761	−0.145179	−0.0725896	+	0.997362i	$0.523126\pi$
$104$	−1374.49	−1.29596
$105$	−356.860	−0.331676
$106$	−1235.18	−1.13180
$107$	−54.9329	−0.0496314	−0.0248157	−	0.999692i	$-0.507900\pi$
$107$	−54.9329	−0.0496314	−0.0248157	+	0.999692i	$0.507900\pi$
$108$	−580.121	−0.516872
$109$	621.124	0.545806	0.272903	−	0.962042i	$-0.412016\pi$
$109$	621.124	0.545806	0.272903	+	0.962042i	$0.412016\pi$
$110$	112.732	0.0977143
$111$	1153.33	0.986210
$112$	326.038	0.275069
$113$	1596.23	1.32886	0.664429	−	0.747351i	$-0.268675\pi$
$113$	1596.23	1.32886	0.664429	+	0.747351i	$0.268675\pi$
$114$	−163.494	−0.134321
$115$	−605.557	−0.491030
$116$	108.932	0.0871905
$117$	532.831	0.421027
$118$	−1311.92	−1.02349
$119$	530.820	0.408909
$120$	507.643	0.386177
$121$	121.000	0.0909091
$122$	−10.1852	−0.00755840
$123$	878.917	0.644303
$124$	−263.101	−0.190542
$125$	−125.000	−0.0894427
$126$	−326.679	−0.230975
$127$	−324.516	−0.226741	−0.113371	−	0.993553i	$-0.536165\pi$
$127$	−324.516	−0.226741	−0.113371	+	0.993553i	$0.536165\pi$
$128$	271.165	0.187249
$129$	−1996.19	−1.36244
$130$	−582.467	−0.392967
$131$	733.042	0.488902	0.244451	−	0.969662i	$-0.421392\pi$
$131$	733.042	0.488902	0.244451	+	0.969662i	$0.421392\pi$
$132$	175.432	0.115677
$133$	323.011	0.210591
$134$	−540.367	−0.348363
$135$	−763.550	−0.486785
$136$	−755.104	−0.476101
$137$	−1641.26	−1.02352	−0.511760	−	0.859129i	$-0.671006\pi$
$137$	−1641.26	−1.02352	−0.511760	+	0.859129i	$0.671006\pi$
$138$	1042.16	0.642858
$139$	−2541.80	−1.55103	−0.775514	−	0.631331i	$-0.782509\pi$
$139$	−2541.80	−1.55103	−0.775514	+	0.631331i	$0.782509\pi$
$140$	−322.913	−0.194936
$141$	−2429.02	−1.45078
$142$	−1328.77	−0.785264
$143$	−625.186	−0.365599
$144$	179.795	0.104048
$145$	143.375	0.0821151
$146$	1472.93	0.834937
$147$	226.620	0.127152
$148$	1043.62	0.579627
$149$	1972.67	1.08461	0.542307	−	0.840180i	$-0.317551\pi$
$149$	1972.67	1.08461	0.542307	+	0.840180i	$0.317551\pi$
$150$	215.124	0.117099
$151$	−3008.95	−1.62162	−0.810812	−	0.585307i	$-0.800974\pi$
$151$	−3008.95	−1.62162	−0.810812	+	0.585307i	$0.800974\pi$
$152$	−459.491	−0.245195
$153$	292.722	0.154674
$154$	383.302	0.200567
$155$	−346.291	−0.179450
$156$	−906.426	−0.465206
$157$	−1273.05	−0.647138	−0.323569	−	0.946205i	$-0.604883\pi$
$157$	−1273.05	−0.647138	−0.323569	+	0.946205i	$0.604883\pi$
$158$	2095.24	1.05499
$159$	−2529.93	−1.26187
$160$	770.807	0.380860
$161$	−2058.96	−1.00788
$162$	795.239	0.385678
$163$	716.484	0.344291	0.172145	−	0.985072i	$-0.444930\pi$
$163$	716.484	0.344291	0.172145	+	0.985072i	$0.444930\pi$
$164$	795.308	0.378677
$165$	230.902	0.108943
$166$	163.754	0.0765648
$167$	1372.19	0.635830	0.317915	−	0.948119i	$-0.397017\pi$
$167$	1372.19	0.635830	0.317915	+	0.948119i	$0.397017\pi$
$168$	1726.04	0.792662
$169$	1033.23	0.470291
$170$	−319.991	−0.144366
$171$	178.125	0.0796584
$172$	−1806.29	−0.800748
$173$	402.052	0.176690	0.0883452	−	0.996090i	$-0.471842\pi$
$173$	402.052	0.176690	0.0883452	+	0.996090i	$0.471842\pi$
$174$	−246.748	−0.107505
$175$	−425.015	−0.183589
$176$	−210.958	−0.0903500
$177$	−2687.11	−1.14111
$178$	−1840.05	−0.774816
$179$	−4228.98	−1.76586	−0.882929	−	0.469507i	$-0.844432\pi$
$179$	−4228.98	−1.76586	−0.882929	+	0.469507i	$0.844432\pi$
$180$	−178.071	−0.0737369
$181$	−641.505	−0.263440	−0.131720	−	0.991287i	$-0.542050\pi$
$181$	−641.505	−0.263440	−0.131720	+	0.991287i	$0.542050\pi$
$182$	−1980.45	−0.806599
$183$	−20.8617	−0.00842699
$184$	2928.93	1.17350
$185$	1373.60	0.545886
$186$	595.965	0.234937
$187$	−343.460	−0.134312
$188$	−2197.95	−0.852672
$189$	−2596.16	−0.999168
$190$	−194.719	−0.0743494
$191$	−1083.14	−0.410332	−0.205166	−	0.978727i	$-0.565773\pi$
$191$	−1083.14	−0.410332	−0.205166	+	0.978727i	$0.565773\pi$
$192$	−1970.66	−0.740730
$193$	1864.42	0.695358	0.347679	−	0.937614i	$-0.386970\pi$
$193$	1864.42	0.695358	0.347679	+	0.937614i	$0.386970\pi$
$194$	2235.23	0.827218
$195$	−1193.03	−0.438126
$196$	205.062	0.0747312
$197$	265.335	0.0959612	0.0479806	−	0.998848i	$-0.484721\pi$
$197$	265.335	0.0959612	0.0479806	+	0.998848i	$0.484721\pi$
$198$	211.373	0.0758668
$199$	40.3173	0.0143619	0.00718094	−	0.999974i	$-0.497714\pi$
$199$	40.3173	0.0143619	0.00718094	+	0.999974i	$0.497714\pi$
$200$	604.594	0.213756
$201$	−1106.80	−0.388396
$202$	1193.38	0.415673
$203$	487.493	0.168548
$204$	−497.965	−0.170905
$205$	1046.78	0.356634
$206$	311.060	0.105207
$207$	−1135.42	−0.381243
$208$	1089.99	0.363351
$209$	−209.000	−0.0691714
$210$	731.446	0.240355
$211$	−187.649	−0.0612240	−0.0306120	−	0.999531i	$-0.509746\pi$
$211$	−187.649	−0.0612240	−0.0306120	+	0.999531i	$0.509746\pi$
$212$	−2289.27	−0.741639
$213$	−2721.62	−0.875505
$214$	112.594	0.0359663
$215$	−2377.43	−0.754136
$216$	3693.10	1.16335
$217$	−1177.43	−0.368337
$218$	−1273.10	−0.395528
$219$	3016.91	0.930886
$220$	208.936	0.0640295
$221$	1774.60	0.540146
$222$	−2363.95	−0.714675
$223$	3423.39	1.02801	0.514007	−	0.857786i	$-0.328161\pi$
$223$	3423.39	1.02801	0.514007	+	0.857786i	$0.328161\pi$
$224$	2620.83	0.781749
$225$	−234.376	−0.0694446
$226$	−3271.76	−0.962982
$227$	−2588.67	−0.756898	−0.378449	−	0.925622i	$-0.623542\pi$
$227$	−2588.67	−0.756898	−0.378449	+	0.925622i	$0.623542\pi$
$228$	−303.019	−0.0880171
$229$	−4031.20	−1.16327	−0.581636	−	0.813449i	$-0.697587\pi$
$229$	−4031.20	−1.16327	−0.581636	+	0.813449i	$0.697587\pi$
$230$	1241.19	0.355834
$231$	785.092	0.223616
$232$	−693.472	−0.196244
$233$	−1507.48	−0.423856	−0.211928	−	0.977285i	$-0.567974\pi$
$233$	−1507.48	−0.423856	−0.211928	+	0.977285i	$0.567974\pi$
$234$	−1092.13	−0.305105
$235$	−2892.92	−0.803037
$236$	−2431.49	−0.670664
$237$	4291.54	1.17623
$238$	−1088.01	−0.296323
$239$	−672.818	−0.182096	−0.0910481	−	0.995846i	$-0.529022\pi$
$239$	−672.818	−0.182096	−0.0910481	+	0.995846i	$0.529022\pi$
$240$	−402.567	−0.108273
$241$	4637.19	1.23945	0.619725	−	0.784819i	$-0.287244\pi$
$241$	4637.19	1.23945	0.619725	+	0.784819i	$0.287244\pi$
$242$	−248.010	−0.0658790
$243$	−2494.34	−0.658484
$244$	−18.8771	−0.00495281
$245$	269.901	0.0703810
$246$	−1801.49	−0.466906
$247$	1079.87	0.278179
$248$	1674.93	0.428863
$249$	335.406	0.0853634
$250$	256.209	0.0648163
$251$	−3768.12	−0.947576	−0.473788	−	0.880639i	$-0.657114\pi$
$251$	−3768.12	−0.947576	−0.473788	+	0.880639i	$0.657114\pi$
$252$	−605.463	−0.151352
$253$	1332.23	0.331053
$254$	665.151	0.164312
$255$	−655.417	−0.160956
$256$	−4311.04	−1.05250
$257$	5466.48	1.32681	0.663404	−	0.748262i	$-0.269111\pi$
$257$	5466.48	1.32681	0.663404	+	0.748262i	$0.269111\pi$
$258$	4091.53	0.987316
$259$	4670.39	1.12048
$260$	−1079.54	−0.257500
$261$	268.830	0.0637553
$262$	−1502.50	−0.354292
$263$	−1755.62	−0.411621	−0.205811	−	0.978592i	$-0.565983\pi$
$263$	−1755.62	−0.411621	−0.205811	+	0.978592i	$0.565983\pi$
$264$	−1116.81	−0.260360
$265$	−3013.11	−0.698467
$266$	−662.067	−0.152609
$267$	−3768.85	−0.863857
$268$	−1001.51	−0.228273
$269$	−4289.91	−0.972343	−0.486171	−	0.873864i	$-0.661607\pi$
$269$	−4289.91	−0.972343	−0.486171	+	0.873864i	$0.661607\pi$
$270$	1565.03	0.352758
$271$	−6355.23	−1.42455	−0.712274	−	0.701901i	$-0.752335\pi$
$271$	−6355.23	−1.42455	−0.712274	+	0.701901i	$0.752335\pi$
$272$	598.808	0.133485
$273$	−4056.44	−0.899292
$274$	3364.04	0.741712
$275$	275.000	0.0603023
$276$	1931.53	0.421247
$277$	5483.56	1.18944	0.594721	−	0.803932i	$-0.297263\pi$
$277$	5483.56	1.18944	0.594721	+	0.803932i	$0.297263\pi$
$278$	5209.86	1.12398
$279$	−649.298	−0.139328
$280$	2055.69	0.438754
$281$	856.017	0.181728	0.0908642	−	0.995863i	$-0.471037\pi$
$281$	856.017	0.181728	0.0908642	+	0.995863i	$0.471037\pi$
$282$	4978.70	1.05134
$283$	7215.32	1.51557	0.757785	−	0.652504i	$-0.226282\pi$
$283$	7215.32	1.51557	0.757785	+	0.652504i	$0.226282\pi$
$284$	−2462.72	−0.514562
$285$	−398.830	−0.0828935
$286$	1281.43	0.264938
$287$	3559.16	0.732023
$288$	1445.27	0.295705
$289$	−3938.09	−0.801564
$290$	−293.873	−0.0595062
$291$	4578.28	0.922281
$292$	2729.92	0.547111
$293$	−5446.10	−1.08588	−0.542942	−	0.839770i	$-0.682690\pi$
$293$	−5446.10	−1.08588	−0.542942	+	0.839770i	$0.682690\pi$
$294$	−464.497	−0.0921430
$295$	−3200.31	−0.631624
$296$	−6643.75	−1.30460
$297$	1679.81	0.328190
$298$	−4043.33	−0.785986
$299$	−6883.38	−1.33136
$300$	398.709	0.0767315
$301$	−8083.53	−1.54793
$302$	6167.37	1.17514
$303$	2444.32	0.463442
$304$	364.383	0.0687460
$305$	−24.8459	−0.00466450
$306$	−599.984	−0.112088
$307$	5456.30	1.01436	0.507178	−	0.861841i	$-0.330689\pi$
$307$	5456.30	1.01436	0.507178	+	0.861841i	$0.330689\pi$
$308$	710.408	0.131426
$309$	637.125	0.117297
$310$	709.784	0.130042
$311$	1461.20	0.266422	0.133211	−	0.991088i	$-0.457471\pi$
$311$	1461.20	0.266422	0.133211	+	0.991088i	$0.457471\pi$
$312$	5770.39	1.04706
$313$	7762.95	1.40188	0.700939	−	0.713221i	$-0.252764\pi$
$313$	7762.95	1.40188	0.700939	+	0.713221i	$0.252764\pi$
$314$	2609.34	0.468960
$315$	−796.904	−0.142541
$316$	3883.30	0.691306
$317$	1695.01	0.300320	0.150160	−	0.988662i	$-0.452021\pi$
$317$	1695.01	0.300320	0.150160	+	0.988662i	$0.452021\pi$
$318$	5185.53	0.914435
$319$	−315.426	−0.0553620
$320$	−2347.02	−0.410008
$321$	230.620	0.0400995
$322$	4220.20	0.730381
$323$	593.249	0.102196
$324$	1473.89	0.252724
$325$	−1420.88	−0.242511
$326$	−1468.56	−0.249497
$327$	−2607.61	−0.440982
$328$	−5063.00	−0.852309
$329$	−9836.28	−1.64830
$330$	−473.273	−0.0789479
$331$	−4615.42	−0.766424	−0.383212	−	0.923661i	$-0.625182\pi$
$331$	−4615.42	−0.766424	−0.383212	+	0.923661i	$0.625182\pi$
$332$	303.500	0.0501708
$333$	2575.50	0.423834
$334$	−2812.55	−0.460766
$335$	−1318.18	−0.214985
$336$	−1368.78	−0.222241
$337$	−1661.07	−0.268500	−0.134250	−	0.990948i	$-0.542862\pi$
$337$	−1661.07	−0.268500	−0.134250	+	0.990948i	$0.542862\pi$
$338$	−2117.78	−0.340805
$339$	−6701.32	−1.07365
$340$	−593.068	−0.0945989
$341$	761.841	0.120985
$342$	−365.099	−0.0577260
$343$	6748.89	1.06241
$344$	11499.0	1.80228
$345$	2542.26	0.396726
$346$	−824.075	−0.128042
$347$	5392.99	0.834326	0.417163	−	0.908832i	$-0.363024\pi$
$347$	5392.99	0.834326	0.417163	+	0.908832i	$0.363024\pi$
$348$	−457.320	−0.0704452
$349$	1661.98	0.254911	0.127455	−	0.991844i	$-0.459319\pi$
$349$	1661.98	0.254911	0.127455	+	0.991844i	$0.459319\pi$
$350$	871.141	0.133041
$351$	−8679.29	−1.31985
$352$	−1695.78	−0.256776
$353$	−6166.35	−0.929749	−0.464875	−	0.885377i	$-0.653901\pi$
$353$	−6166.35	−0.929749	−0.464875	+	0.885377i	$0.653901\pi$
$354$	5507.70	0.826924
$355$	−3241.41	−0.484609
$356$	−3410.32	−0.507716
$357$	−2228.49	−0.330376
$358$	8668.02	1.27966
$359$	9170.44	1.34818	0.674091	−	0.738648i	$-0.264536\pi$
$359$	9170.44	1.34818	0.674091	+	0.738648i	$0.264536\pi$
$360$	1133.62	0.165963
$361$	361.000	0.0526316
$362$	1314.88	0.190907
$363$	−507.984	−0.0734496
$364$	−3670.56	−0.528542
$365$	3593.09	0.515263
$366$	42.7596	0.00610678
$367$	9728.89	1.38377	0.691886	−	0.722007i	$-0.256780\pi$
$367$	9728.89	1.38377	0.691886	+	0.722007i	$0.256780\pi$
$368$	−2322.68	−0.329016
$369$	1962.71	0.276896
$370$	−2815.43	−0.395586
$371$	−10244.9	−1.43366
$372$	1104.56	0.153948
$373$	−2161.57	−0.300058	−0.150029	−	0.988682i	$-0.547937\pi$
$373$	−2161.57	−0.300058	−0.150029	+	0.988682i	$0.547937\pi$
$374$	703.980	0.0973314
$375$	524.776	0.0722649
$376$	13992.4	1.91915
$377$	1629.75	0.222643
$378$	5321.28	0.724066
$379$	5197.39	0.704412	0.352206	−	0.935923i	$-0.385432\pi$
$379$	5197.39	0.704412	0.352206	+	0.935923i	$0.385432\pi$
$380$	−360.890	−0.0487191
$381$	1362.39	0.183195
$382$	2220.09	0.297355
$383$	544.117	0.0725929	0.0362964	−	0.999341i	$-0.488444\pi$
$383$	544.117	0.0725929	0.0362964	+	0.999341i	$0.488444\pi$
$384$	−1138.41	−0.151287
$385$	935.032	0.123776
$386$	−3821.46	−0.503904
$387$	−4457.69	−0.585522
$388$	4142.76	0.542053
$389$	10416.1	1.35762	0.678811	−	0.734313i	$-0.262496\pi$
$389$	10416.1	1.35762	0.678811	+	0.734313i	$0.262496\pi$
$390$	2445.32	0.317496
$391$	−3781.54	−0.489106
$392$	−1305.44	−0.168201
$393$	−3077.46	−0.395006
$394$	−543.850	−0.0695401
$395$	5111.16	0.651064
$396$	391.757	0.0497134
$397$	−10811.7	−1.36680	−0.683402	−	0.730042i	$-0.739500\pi$
$397$	−10811.7	−1.36680	−0.683402	+	0.730042i	$0.739500\pi$
$398$	−82.6372	−0.0104076
$399$	−1356.07	−0.170146
$400$	−479.451	−0.0599314
$401$	−10826.4	−1.34824	−0.674118	−	0.738624i	$-0.735476\pi$
$401$	−10826.4	−1.34824	−0.674118	+	0.738624i	$0.735476\pi$
$402$	2268.58	0.281458
$403$	−3936.30	−0.486554
$404$	2211.80	0.272379
$405$	1939.92	0.238013
$406$	−999.202	−0.122142
$407$	−3021.92	−0.368036
$408$	3170.09	0.384664
$409$	6787.02	0.820529	0.410264	−	0.911967i	$-0.365436\pi$
$409$	6787.02	0.820529	0.410264	+	0.911967i	$0.365436\pi$
$410$	−2145.55	−0.258442
$411$	6890.35	0.826949
$412$	576.516	0.0689391
$413$	−10881.4	−1.29646
$414$	2327.24	0.276275
$415$	399.463	0.0472503
$416$	8761.78	1.03265
$417$	10671.0	1.25315
$418$	428.382	0.0501264
$419$	3990.45	0.465266	0.232633	−	0.972565i	$-0.425266\pi$
$419$	3990.45	0.465266	0.232633	+	0.972565i	$0.425266\pi$
$420$	1355.66	0.157498
$421$	702.945	0.0813763	0.0406881	−	0.999172i	$-0.487045\pi$
$421$	702.945	0.0813763	0.0406881	+	0.999172i	$0.487045\pi$
$422$	384.618	0.0443671
$423$	−5424.25	−0.623489
$424$	14573.7	1.66924
$425$	−780.590	−0.0890922
$426$	5578.44	0.634451
$427$	−84.4790	−0.00957430
$428$	208.681	0.0235678
$429$	2624.66	0.295385
$430$	4872.95	0.546498
$431$	−14329.4	−1.60145	−0.800723	−	0.599035i	$-0.795551\pi$
$431$	−14329.4	−1.60145	−0.800723	+	0.599035i	$0.795551\pi$
$432$	−2928.68	−0.326172
$433$	7971.93	0.884773	0.442386	−	0.896825i	$-0.354132\pi$
$433$	7971.93	0.884773	0.442386	+	0.896825i	$0.354132\pi$
$434$	2413.35	0.266923
$435$	−601.921	−0.0663446
$436$	−2359.55	−0.259179
$437$	−2301.12	−0.251893
$438$	−6183.68	−0.674584
$439$	−12004.3	−1.30509	−0.652543	−	0.757752i	$-0.726298\pi$
$439$	−12004.3	−1.30509	−0.652543	+	0.757752i	$0.726298\pi$
$440$	−1330.11	−0.144114
$441$	506.066	0.0546448
$442$	−3637.34	−0.391427
$443$	17369.4	1.86285	0.931426	−	0.363930i	$-0.118565\pi$
$443$	17369.4	1.86285	0.931426	+	0.363930i	$0.118565\pi$
$444$	−4381.32	−0.468307
$445$	−4488.63	−0.478161
$446$	−7016.82	−0.744969
$447$	−8281.70	−0.876310
$448$	−7980.15	−0.841578
$449$	6786.82	0.713341	0.356670	−	0.934230i	$-0.383912\pi$
$449$	6786.82	0.713341	0.356670	+	0.934230i	$0.383912\pi$
$450$	480.393	0.0503243
$451$	−2302.91	−0.240443
$452$	−6063.84	−0.631016
$453$	12632.2	1.31018
$454$	5305.92	0.548500
$455$	−4831.15	−0.497775
$456$	1929.04	0.198105
$457$	−9868.67	−1.01015	−0.505073	−	0.863076i	$-0.668535\pi$
$457$	−9868.67	−1.01015	−0.505073	+	0.863076i	$0.668535\pi$
$458$	8262.64	0.842986
$459$	−4768.16	−0.484877
$460$	2300.42	0.233168
$461$	−924.718	−0.0934239	−0.0467119	−	0.998908i	$-0.514874\pi$
$461$	−924.718	−0.0934239	−0.0467119	+	0.998908i	$0.514874\pi$
$462$	−1609.18	−0.162047
$463$	17260.6	1.73255	0.866273	−	0.499571i	$-0.166509\pi$
$463$	17260.6	1.73255	0.866273	+	0.499571i	$0.166509\pi$
$464$	549.932	0.0550215
$465$	1453.80	0.144986
$466$	3089.85	0.307155
$467$	−2001.21	−0.198298	−0.0991488	−	0.995073i	$-0.531612\pi$
$467$	−2001.21	−0.198298	−0.0991488	+	0.995073i	$0.531612\pi$
$468$	−2024.14	−0.199927
$469$	−4481.96	−0.441275
$470$	5929.55	0.581935
$471$	5344.54	0.522852
$472$	15479.1	1.50950
$473$	5230.34	0.508438
$474$	−8796.26	−0.852375
$475$	−475.000	−0.0458831
$476$	−2016.50	−0.194173
$477$	−5649.59	−0.542300
$478$	1379.06	0.131959
$479$	−6121.65	−0.583936	−0.291968	−	0.956428i	$-0.594310\pi$
$479$	−6121.65	−0.583936	−0.291968	+	0.956428i	$0.594310\pi$
$480$	−3236.01	−0.307715
$481$	15613.7	1.48009
$482$	−9504.71	−0.898190
$483$	8643.97	0.814315
$484$	−459.660	−0.0431687
$485$	5452.66	0.510500
$486$	5112.57	0.477183
$487$	6957.84	0.647413	0.323706	−	0.946158i	$-0.395071\pi$
$487$	6957.84	0.647413	0.323706	+	0.946158i	$0.395071\pi$
$488$	120.174	0.0111475
$489$	−3007.95	−0.278168
$490$	−553.209	−0.0510029
$491$	10135.1	0.931552	0.465776	−	0.884903i	$-0.345775\pi$
$491$	10135.1	0.931552	0.465776	+	0.884903i	$0.345775\pi$
$492$	−3338.87	−0.305951
$493$	895.340	0.0817933
$494$	−2213.37	−0.201588
$495$	515.626	0.0468196
$496$	−1328.24	−0.120241
$497$	−11021.2	−0.994702
$498$	−687.473	−0.0618602
$499$	−13117.3	−1.17677	−0.588387	−	0.808579i	$-0.700237\pi$
$499$	−13117.3	−1.17677	−0.588387	+	0.808579i	$0.700237\pi$
$500$	474.855	0.0424724
$501$	−5760.76	−0.513716
$502$	7723.41	0.686679
$503$	20539.0	1.82065	0.910327	−	0.413889i	$-0.135830\pi$
$503$	20539.0	1.82065	0.910327	+	0.413889i	$0.135830\pi$
$504$	3854.43	0.340655
$505$	2911.15	0.256524
$506$	−2730.63	−0.239903
$507$	−4337.72	−0.379970
$508$	1232.79	0.107669
$509$	359.444	0.0313008	0.0156504	−	0.999878i	$-0.495018\pi$
$509$	359.444	0.0313008	0.0156504	+	0.999878i	$0.495018\pi$
$510$	1343.39	0.116640
$511$	12216.9	1.05762
$512$	6666.89	0.575464
$513$	−2901.49	−0.249715
$514$	−11204.5	−0.961495
$515$	758.805	0.0649261
$516$	7583.21	0.646961
$517$	6364.43	0.541407
$518$	−9572.77	−0.811976
$519$	−1687.90	−0.142756
$520$	6872.43	0.579570
$521$	3231.24	0.271714	0.135857	−	0.990728i	$-0.456621\pi$
$521$	3231.24	0.271714	0.135857	+	0.990728i	$0.456621\pi$
$522$	−551.013	−0.0462015
$523$	−11830.8	−0.989148	−0.494574	−	0.869136i	$-0.664676\pi$
$523$	−11830.8	−0.989148	−0.494574	+	0.869136i	$0.664676\pi$
$524$	−2784.71	−0.232158
$525$	1784.30	0.148330
$526$	3598.45	0.298289
$527$	−2162.49	−0.178747
$528$	885.648	0.0729979
$529$	2500.97	0.205554
$530$	6175.88	0.506157
$531$	−6000.59	−0.490402
$532$	−1227.07	−0.100000
$533$	11898.7	0.966963
$534$	7724.90	0.626010
$535$	274.664	0.0221959
$536$	6375.71	0.513785
$537$	17754.1	1.42672
$538$	8792.90	0.704626
$539$	−593.782	−0.0474509
$540$	2900.61	0.231152
$541$	15766.4	1.25296	0.626480	−	0.779437i	$-0.284495\pi$
$541$	15766.4	1.25296	0.626480	+	0.779437i	$0.284495\pi$
$542$	13026.1	1.03233
$543$	2693.18	0.212846
$544$	4813.48	0.379368
$545$	−3105.62	−0.244092
$546$	8314.37	0.651689
$547$	4544.17	0.355200	0.177600	−	0.984103i	$-0.443167\pi$
$547$	4544.17	0.355200	0.177600	+	0.984103i	$0.443167\pi$
$548$	6234.88	0.486024
$549$	−46.5862	−0.00362159
$550$	−563.660	−0.0436992
$551$	544.827	0.0421241
$552$	−12296.3	−0.948123
$553$	17378.5	1.33637
$554$	−11239.5	−0.861951
$555$	−5766.65	−0.441046
$556$	9655.90	0.736514
$557$	−4499.19	−0.342257	−0.171128	−	0.985249i	$-0.554741\pi$
$557$	−4499.19	−0.342257	−0.171128	+	0.985249i	$0.554741\pi$
$558$	1330.85	0.100966
$559$	−27024.3	−2.04473
$560$	−1630.19	−0.123014
$561$	1441.92	0.108517
$562$	−1754.55	−0.131693
$563$	−16692.4	−1.24956	−0.624780	−	0.780801i	$-0.714811\pi$
$563$	−16692.4	−1.24956	−0.624780	+	0.780801i	$0.714811\pi$
$564$	9227.47	0.688913
$565$	−7981.17	−0.594284
$566$	−14789.0	−1.09829
$567$	6595.94	0.488542
$568$	15677.9	1.15815
$569$	−8069.67	−0.594549	−0.297274	−	0.954792i	$-0.596078\pi$
$569$	−8069.67	−0.594549	−0.297274	+	0.954792i	$0.596078\pi$
$570$	817.471	0.0600703
$571$	5482.59	0.401820	0.200910	−	0.979610i	$-0.435610\pi$
$571$	5482.59	0.401820	0.200910	+	0.979610i	$0.435610\pi$
$572$	2374.98	0.173607
$573$	4547.26	0.331526
$574$	−7295.11	−0.530474
$575$	3027.79	0.219595
$576$	−4400.68	−0.318336
$577$	−16541.2	−1.19345	−0.596724	−	0.802447i	$-0.703531\pi$
$577$	−16541.2	−1.19345	−0.596724	+	0.802447i	$0.703531\pi$
$578$	8071.79	0.580869
$579$	−7827.25	−0.561812
$580$	−544.661	−0.0389928
$581$	1358.22	0.0969854
$582$	−9383.98	−0.668348
$583$	6628.84	0.470906
$584$	−17378.9	−1.23141
$585$	−2664.15	−0.188289
$586$	11162.7	0.786907
$587$	−17054.7	−1.19919	−0.599593	−	0.800305i	$-0.704671\pi$
$587$	−17054.7	−1.19919	−0.599593	+	0.800305i	$0.704671\pi$
$588$	−860.895	−0.0603788
$589$	−1315.91	−0.0920561
$590$	6559.58	0.457718
$591$	−1113.93	−0.0775315
$592$	5268.58	0.365773
$593$	−4904.87	−0.339661	−0.169831	−	0.985473i	$-0.554322\pi$
$593$	−4904.87	−0.339661	−0.169831	+	0.985473i	$0.554322\pi$
$594$	−3443.06	−0.237829
$595$	−2654.10	−0.182870
$596$	−7493.87	−0.515035
$597$	−169.260	−0.0116036
$598$	14108.7	0.964794
$599$	8010.14	0.546386	0.273193	−	0.961959i	$-0.411920\pi$
$599$	8010.14	0.546386	0.273193	+	0.961959i	$0.411920\pi$
$600$	−2538.21	−0.172704
$601$	12369.7	0.839550	0.419775	−	0.907628i	$-0.362109\pi$
$601$	12369.7	0.839550	0.419775	+	0.907628i	$0.362109\pi$
$602$	16568.6	1.12174
$603$	−2471.59	−0.166917
$604$	11430.5	0.770037
$605$	−605.000	−0.0406558
$606$	−5010.06	−0.335842
$607$	−15121.2	−1.01112	−0.505560	−	0.862791i	$-0.668714\pi$
$607$	−15121.2	−1.01112	−0.505560	+	0.862791i	$0.668714\pi$
$608$	2929.07	0.195377
$609$	−2046.60	−0.136178
$610$	50.9260	0.00338022
$611$	−32883.9	−2.17732
$612$	−1112.01	−0.0734480
$613$	13120.6	0.864498	0.432249	−	0.901754i	$-0.357720\pi$
$613$	13120.6	0.864498	0.432249	+	0.901754i	$0.357720\pi$
$614$	−11183.6	−0.735072
$615$	−4394.59	−0.288141
$616$	−4522.52	−0.295808
$617$	−4148.62	−0.270693	−0.135346	−	0.990798i	$-0.543215\pi$
$617$	−4148.62	−0.270693	−0.135346	+	0.990798i	$0.543215\pi$
$618$	−1305.90	−0.0850014
$619$	−4772.73	−0.309906	−0.154953	−	0.987922i	$-0.549523\pi$
$619$	−4772.73	−0.309906	−0.154953	+	0.987922i	$0.549523\pi$
$620$	1315.51	0.0852129
$621$	18494.9	1.19513
$622$	−2994.99	−0.193068
$623$	−15261.9	−0.981468
$624$	−4575.99	−0.293568
$625$	625.000	0.0400000
$626$	−15911.5	−1.01590
$627$	877.426	0.0558868
$628$	4836.13	0.307297
$629$	8577.74	0.543747
$630$	1633.39	0.103295
$631$	−5268.71	−0.332399	−0.166200	−	0.986092i	$-0.553150\pi$
$631$	−5268.71	−0.332399	−0.166200	+	0.986092i	$0.553150\pi$
$632$	−24721.4	−1.55596
$633$	787.789	0.0494657
$634$	−3474.22	−0.217632
$635$	1622.58	0.101402
$636$	9610.82	0.599204
$637$	3067.97	0.190828
$638$	646.520	0.0401191
$639$	−6077.66	−0.376257
$640$	−1355.83	−0.0837403
$641$	17731.8	1.09261	0.546307	−	0.837585i	$-0.316033\pi$
$641$	17731.8	1.09261	0.546307	+	0.837585i	$0.316033\pi$
$642$	−472.695	−0.0290589
$643$	24063.8	1.47587	0.737935	−	0.674872i	$-0.235801\pi$
$643$	24063.8	1.47587	0.737935	+	0.674872i	$0.235801\pi$
$644$	7821.68	0.478599
$645$	9980.94	0.609301
$646$	−1215.97	−0.0740581
$647$	−19468.2	−1.18296	−0.591478	−	0.806321i	$-0.701455\pi$
$647$	−19468.2	−1.18296	−0.591478	+	0.806321i	$0.701455\pi$
$648$	−9382.90	−0.568819
$649$	7040.68	0.425841
$650$	2912.33	0.175740
$651$	4943.10	0.297597
$652$	−2721.81	−0.163488
$653$	22300.1	1.33640	0.668201	−	0.743981i	$-0.267065\pi$
$653$	22300.1	1.33640	0.668201	+	0.743981i	$0.267065\pi$
$654$	5344.74	0.319566
$655$	−3665.21	−0.218644
$656$	4015.02	0.238964
$657$	6737.07	0.400058
$658$	20161.2	1.19447
$659$	5359.59	0.316813	0.158407	−	0.987374i	$-0.449364\pi$
$659$	5359.59	0.316813	0.158407	+	0.987374i	$0.449364\pi$
$660$	−877.159	−0.0517324
$661$	−11907.3	−0.700669	−0.350334	−	0.936625i	$-0.613932\pi$
$661$	−11907.3	−0.700669	−0.350334	+	0.936625i	$0.613932\pi$
$662$	9460.09	0.555403
$663$	−7450.13	−0.436409
$664$	−1932.10	−0.112922
$665$	−1615.06	−0.0941792
$666$	−5278.94	−0.307139
$667$	−3472.88	−0.201605
$668$	−5212.75	−0.301927
$669$	−14372.1	−0.830579
$670$	2701.84	0.155793
$671$	54.6610	0.00314481
$672$	−11002.8	−0.631611
$673$	544.623	0.0311942	0.0155971	−	0.999878i	$-0.495035\pi$
$673$	544.623	0.0311942	0.0155971	+	0.999878i	$0.495035\pi$
$674$	3404.65	0.194573
$675$	3817.75	0.217697
$676$	−3925.08	−0.223320
$677$	−22803.1	−1.29453	−0.647263	−	0.762267i	$-0.724086\pi$
$677$	−22803.1	−1.29453	−0.647263	+	0.762267i	$0.724086\pi$
$678$	13735.5	0.778037
$679$	18539.7	1.04785
$680$	3775.52	0.212919
$681$	10867.8	0.611533
$682$	−1561.52	−0.0876743
$683$	−11474.4	−0.642837	−0.321418	−	0.946937i	$-0.604160\pi$
$683$	−11474.4	−0.642837	−0.321418	+	0.946937i	$0.604160\pi$
$684$	−676.671	−0.0378262
$685$	8206.29	0.457732
$686$	−13833.0	−0.769894
$687$	16923.8	0.939860
$688$	−9118.88	−0.505311
$689$	−34250.1	−1.89379
$690$	−5210.79	−0.287495
$691$	−23366.8	−1.28642	−0.643210	−	0.765690i	$-0.722398\pi$
$691$	−23366.8	−1.28642	−0.643210	+	0.765690i	$0.722398\pi$
$692$	−1527.33	−0.0839024
$693$	1753.19	0.0961012
$694$	−11053.9	−0.604610
$695$	12709.0	0.693640
$696$	2911.34	0.158555
$697$	6536.83	0.355237
$698$	−3406.52	−0.184726
$699$	6328.73	0.342453
$700$	1614.56	0.0871782
$701$	−15914.7	−0.857475	−0.428738	−	0.903429i	$-0.641042\pi$
$701$	−15914.7	−0.857475	−0.428738	+	0.903429i	$0.641042\pi$
$702$	17789.7	0.956452
$703$	5219.67	0.280034
$704$	5163.45	0.276427
$705$	12145.1	0.648810
$706$	12639.0	0.673760
$707$	9898.25	0.526538
$708$	10207.9	0.541860
$709$	−22098.7	−1.17057	−0.585285	−	0.810827i	$-0.699017\pi$
$709$	−22098.7	−1.17057	−0.585285	+	0.810827i	$0.699017\pi$
$710$	6643.83	0.351181
$711$	9583.45	0.505496
$712$	21710.4	1.14274
$713$	8387.97	0.440578
$714$	4567.68	0.239413
$715$	3125.93	0.163501
$716$	16065.2	0.838527
$717$	2824.63	0.147124
$718$	−18796.4	−0.976985
$719$	−10624.5	−0.551079	−0.275540	−	0.961290i	$-0.588857\pi$
$719$	−10624.5	−0.551079	−0.275540	+	0.961290i	$0.588857\pi$
$720$	−898.973	−0.0465316
$721$	2580.03	0.133267
$722$	−739.932	−0.0381405
$723$	−19467.9	−1.00141
$724$	2436.98	0.125096
$725$	−716.877	−0.0367230
$726$	1041.20	0.0532267
$727$	−22439.0	−1.14473	−0.572364	−	0.820000i	$-0.693974\pi$
$727$	−22439.0	−1.14473	−0.572364	+	0.820000i	$0.693974\pi$
$728$	23367.1	1.18962
$729$	20947.3	1.06423
$730$	−7364.66	−0.373395
$731$	−14846.4	−0.751180
$732$	79.2502	0.00400160
$733$	−17254.7	−0.869461	−0.434731	−	0.900561i	$-0.643156\pi$
$733$	−17254.7	−0.869461	−0.434731	+	0.900561i	$0.643156\pi$
$734$	−19941.0	−1.00278
$735$	−1133.10	−0.0568641
$736$	−18670.7	−0.935070
$737$	2899.99	0.144943
$738$	−4022.91	−0.200658
$739$	−20622.8	−1.02655	−0.513276	−	0.858223i	$-0.671568\pi$
$739$	−20622.8	−1.02655	−0.513276	+	0.858223i	$0.671568\pi$
$740$	−5218.08	−0.259217
$741$	−4533.51	−0.224754
$742$	20998.7	1.03893
$743$	−23446.0	−1.15767	−0.578836	−	0.815444i	$-0.696493\pi$
$743$	−23446.0	−1.15767	−0.578836	+	0.815444i	$0.696493\pi$
$744$	−7031.69	−0.346498
$745$	−9863.36	−0.485055
$746$	4430.50	0.217443
$747$	748.995	0.0366858
$748$	1304.75	0.0637786
$749$	933.891	0.0455589
$750$	−1075.62	−0.0523681
$751$	17883.2	0.868930	0.434465	−	0.900689i	$-0.356937\pi$
$751$	17883.2	0.868930	0.434465	+	0.900689i	$0.356937\pi$
$752$	−11096.1	−0.538077
$753$	15819.4	0.765590
$754$	−3340.46	−0.161343
$755$	15044.8	0.725212
$756$	9862.40	0.474460
$757$	−12064.7	−0.579257	−0.289629	−	0.957139i	$-0.593532\pi$
$757$	−12064.7	−0.579257	−0.289629	+	0.957139i	$0.593532\pi$
$758$	−10652.9	−0.510465
$759$	−5592.96	−0.267473
$760$	2297.46	0.109655
$761$	5548.07	0.264280	0.132140	−	0.991231i	$-0.457815\pi$
$761$	5548.07	0.264280	0.132140	+	0.991231i	$0.457815\pi$
$762$	−2792.45	−0.132755
$763$	−10559.5	−0.501020
$764$	4114.69	0.194848
$765$	−1463.61	−0.0691725
$766$	−1115.26	−0.0526058
$767$	−36378.0	−1.71256
$768$	18098.6	0.850363
$769$	2663.36	0.124894	0.0624469	−	0.998048i	$-0.480110\pi$
$769$	2663.36	0.124894	0.0624469	+	0.998048i	$0.480110\pi$
$770$	−1916.51	−0.0896964
$771$	−22949.4	−1.07199
$772$	−7082.66	−0.330195
$773$	−7863.00	−0.365864	−0.182932	−	0.983126i	$-0.558559\pi$
$773$	−7863.00	−0.365864	−0.182932	+	0.983126i	$0.558559\pi$
$774$	9136.80	0.424309
$775$	1731.46	0.0802526
$776$	−26373.2	−1.22003
$777$	−19607.3	−0.905287
$778$	−21349.5	−0.983826
$779$	3977.75	0.182949
$780$	4532.13	0.208047
$781$	7131.10	0.326723
$782$	7750.91	0.354440
$783$	−4378.97	−0.199862
$784$	1035.23	0.0471590
$785$	6365.26	0.289409
$786$	6307.79	0.286249
$787$	−9533.54	−0.431809	−0.215905	−	0.976414i	$-0.569270\pi$
$787$	−9533.54	−0.431809	−0.215905	+	0.976414i	$0.569270\pi$
$788$	−1007.97	−0.0455677
$789$	7370.48	0.332568
$790$	−10476.2	−0.471806
$791$	−27136.9	−1.21982
$792$	−2493.96	−0.111893
$793$	−282.424	−0.0126471
$794$	22160.3	0.990480
$795$	12649.7	0.564324
$796$	−153.159	−0.00681982
$797$	1879.07	0.0835133	0.0417566	−	0.999128i	$-0.486705\pi$
$797$	1879.07	0.0835133	0.0417566	+	0.999128i	$0.486705\pi$
$798$	2779.50	0.123300
$799$	−18065.5	−0.799890
$800$	−3854.04	−0.170326
$801$	−8416.21	−0.371251
$802$	22190.5	0.977024
$803$	−7904.80	−0.347390
$804$	4204.56	0.184432
$805$	10294.8	0.450739
$806$	8068.13	0.352590
$807$	18009.9	0.785600
$808$	−14080.5	−0.613058
$809$	−27290.0	−1.18599	−0.592994	−	0.805207i	$-0.702054\pi$
$809$	−27290.0	−1.18599	−0.592994	+	0.805207i	$0.702054\pi$
$810$	−3976.19	−0.172480
$811$	−22669.9	−0.981565	−0.490782	−	0.871282i	$-0.663289\pi$
$811$	−22669.9	−0.981565	−0.490782	+	0.871282i	$0.663289\pi$
$812$	−1851.91	−0.0800361
$813$	26680.6	1.15096
$814$	6193.94	0.266704
$815$	−3582.42	−0.153972
$816$	−2513.92	−0.107849
$817$	−9034.22	−0.386863
$818$	−13911.2	−0.594612
$819$	−9058.43	−0.386480
$820$	−3976.54	−0.169350
$821$	20472.9	0.870292	0.435146	−	0.900360i	$-0.356697\pi$
$821$	20472.9	0.870292	0.435146	+	0.900360i	$0.356697\pi$
$822$	−14123.0	−0.599264
$823$	44823.8	1.89849	0.949247	−	0.314531i	$-0.101847\pi$
$823$	44823.8	1.89849	0.949247	+	0.314531i	$0.101847\pi$
$824$	−3670.15	−0.155165
$825$	−1154.51	−0.0487210
$826$	22303.3	0.939507
$827$	9937.58	0.417852	0.208926	−	0.977931i	$-0.433003\pi$
$827$	9937.58	0.417852	0.208926	+	0.977931i	$0.433003\pi$
$828$	4313.29	0.181035
$829$	−16721.5	−0.700556	−0.350278	−	0.936646i	$-0.613913\pi$
$829$	−16721.5	−0.700556	−0.350278	+	0.936646i	$0.613913\pi$
$830$	−818.768	−0.0342408
$831$	−23021.2	−0.961005
$832$	−26678.7	−1.11168
$833$	1685.46	0.0701052
$834$	−21872.1	−0.908116
$835$	−6860.97	−0.284352
$836$	793.958	0.0328464
$837$	10576.4	0.436768
$838$	−8179.12	−0.337164
$839$	15121.8	0.622244	0.311122	−	0.950370i	$-0.399295\pi$
$839$	15121.8	0.622244	0.311122	+	0.950370i	$0.399295\pi$
$840$	−8630.22	−0.354489
$841$	−23566.7	−0.966286
$842$	−1440.81	−0.0589708
$843$	−3593.74	−0.146827
$844$	712.848	0.0290726
$845$	−5166.15	−0.210321
$846$	11117.9	0.451823
$847$	−2057.07	−0.0834496
$848$	−11557.1	−0.468010
$849$	−30291.4	−1.22450
$850$	1599.95	0.0645623
$851$	−33271.7	−1.34023
$852$	10339.0	0.415738
$853$	5126.83	0.205790	0.102895	−	0.994692i	$-0.467189\pi$
$853$	5126.83	0.205790	0.102895	+	0.994692i	$0.467189\pi$
$854$	173.154	0.00693819
$855$	−890.627	−0.0356243
$856$	−1328.48	−0.0530452
$857$	−34104.6	−1.35938	−0.679692	−	0.733498i	$-0.737886\pi$
$857$	−34104.6	−1.35938	−0.679692	+	0.733498i	$0.737886\pi$
$858$	−5379.70	−0.214056
$859$	23863.7	0.947869	0.473934	−	0.880560i	$-0.342833\pi$
$859$	23863.7	0.947869	0.473934	+	0.880560i	$0.342833\pi$
$860$	9031.47	0.358105
$861$	−14942.1	−0.591435
$862$	29370.6	1.16052
$863$	25304.3	0.998107	0.499054	−	0.866571i	$-0.333681\pi$
$863$	25304.3	0.998107	0.499054	+	0.866571i	$0.333681\pi$
$864$	−23542.0	−0.926985
$865$	−2010.26	−0.0790184
$866$	−16339.8	−0.641167
$867$	16532.9	0.647621
$868$	4472.88	0.174907
$869$	−11244.5	−0.438947
$870$	1233.74	0.0480778
$871$	−14983.8	−0.582900
$872$	15021.1	0.583347
$873$	10223.8	0.396360
$874$	4716.54	0.182539
$875$	2125.07	0.0821035
$876$	−11460.8	−0.442036
$877$	−9545.44	−0.367533	−0.183767	−	0.982970i	$-0.558829\pi$
$877$	−9545.44	−0.367533	−0.183767	+	0.982970i	$0.558829\pi$
$878$	24604.8	0.945754
$879$	22863.9	0.877336
$880$	1054.79	0.0404057
$881$	−16148.7	−0.617550	−0.308775	−	0.951135i	$-0.599919\pi$
$881$	−16148.7	−0.617550	−0.308775	+	0.951135i	$0.599919\pi$
$882$	−1037.27	−0.0395994
$883$	42251.3	1.61027	0.805136	−	0.593091i	$-0.202092\pi$
$883$	42251.3	1.61027	0.805136	+	0.593091i	$0.202092\pi$
$884$	−6741.42	−0.256491
$885$	13435.6	0.510318
$886$	−35601.5	−1.34995
$887$	−44491.6	−1.68419	−0.842097	−	0.539325i	$-0.818679\pi$
$887$	−44491.6	−1.68419	−0.842097	+	0.539325i	$0.818679\pi$
$888$	27891.9	1.05404
$889$	5516.96	0.208136
$890$	9200.23	0.346508
$891$	−4267.82	−0.160468
$892$	−13004.9	−0.488158
$893$	−10993.1	−0.411949
$894$	16974.8	0.635035
$895$	21144.9	0.789716
$896$	−4609.97	−0.171884
$897$	28897.9	1.07567
$898$	−13910.8	−0.516936
$899$	−1985.99	−0.0736779
$900$	890.356	0.0329762
$901$	−18816.0	−0.695730
$902$	4720.21	0.174241
$903$	33936.4	1.25064
$904$	38602.9	1.42026
$905$	3207.53	0.117814
$906$	−25891.9	−0.949450
$907$	−15222.8	−0.557293	−0.278646	−	0.960394i	$-0.589886\pi$
$907$	−15222.8	−0.557293	−0.278646	+	0.960394i	$0.589886\pi$
$908$	9833.94	0.359417
$909$	5458.42	0.199169
$910$	9902.27	0.360722
$911$	8470.71	0.308065	0.154032	−	0.988066i	$-0.450774\pi$
$911$	8470.71	0.308065	0.154032	+	0.988066i	$0.450774\pi$
$912$	−1529.76	−0.0555431
$913$	−878.819	−0.0318561
$914$	20227.5	0.732022
$915$	104.308	0.00376867
$916$	15313.9	0.552386
$917$	−12462.1	−0.448785
$918$	9773.16	0.351375
$919$	−11535.5	−0.414058	−0.207029	−	0.978335i	$-0.566380\pi$
$919$	−11535.5	−0.414058	−0.207029	+	0.978335i	$0.566380\pi$
$920$	−14644.6	−0.524804
$921$	−22906.7	−0.819545
$922$	1895.37	0.0677013
$923$	−36845.2	−1.31395
$924$	−2982.44	−0.106185
$925$	−6867.99	−0.244128
$926$	−35378.6	−1.25552
$927$	1422.76	0.0504096
$928$	4420.59	0.156372
$929$	5253.96	0.185551	0.0927755	−	0.995687i	$-0.470426\pi$
$929$	5253.96	0.185551	0.0927755	+	0.995687i	$0.470426\pi$
$930$	−2979.82	−0.105067
$931$	1025.62	0.0361047
$932$	5726.69	0.201271
$933$	−6134.44	−0.215255
$934$	4101.83	0.143700
$935$	1717.30	0.0600660
$936$	12885.8	0.449986
$937$	−20648.9	−0.719926	−0.359963	−	0.932967i	$-0.617211\pi$
$937$	−20648.9	−0.719926	−0.359963	+	0.932967i	$0.617211\pi$
$938$	9186.56	0.319778
$939$	−32590.5	−1.13264
$940$	10989.8	0.381326
$941$	47106.9	1.63192	0.815962	−	0.578105i	$-0.196208\pi$
$941$	47106.9	1.63192	0.815962	+	0.578105i	$0.196208\pi$
$942$	−10954.6	−0.378895
$943$	−25355.3	−0.875591
$944$	−12275.1	−0.423222
$945$	12980.8	0.446842
$946$	−10720.5	−0.368449
$947$	−21422.3	−0.735091	−0.367546	−	0.930005i	$-0.619802\pi$
$947$	−21422.3	−0.735091	−0.367546	+	0.930005i	$0.619802\pi$
$948$	−16302.9	−0.558538
$949$	40842.8	1.39706
$950$	973.594	0.0332501
$951$	−7116.02	−0.242642
$952$	12837.2	0.437034
$953$	−49983.9	−1.69899	−0.849495	−	0.527596i	$-0.823094\pi$
$953$	−49983.9	−1.69899	−0.849495	+	0.527596i	$0.823094\pi$
$954$	11579.8	0.392988
$955$	5415.71	0.183506
$956$	2555.93	0.0864694
$957$	1324.23	0.0447295
$958$	12547.4	0.423160
$959$	27902.3	0.939535
$960$	9853.30	0.331264
$961$	−24994.3	−0.838988
$962$	−32003.0	−1.07258
$963$	514.997	0.0172332
$964$	−17615.9	−0.588559
$965$	−9322.12	−0.310974
$966$	−17717.3	−0.590108
$967$	55645.2	1.85049	0.925247	−	0.379365i	$-0.123857\pi$
$967$	55645.2	1.85049	0.925247	+	0.379365i	$0.123857\pi$
$968$	2926.24	0.0971619
$969$	−2490.58	−0.0825687
$970$	−11176.2	−0.369943
$971$	40916.6	1.35229	0.676146	−	0.736768i	$-0.263649\pi$
$971$	40916.6	1.35229	0.676146	+	0.736768i	$0.263649\pi$
$972$	9475.59	0.312685
$973$	43212.1	1.42376
$974$	−14261.3	−0.469160
$975$	5965.14	0.195936
$976$	−95.2992	−0.00312546
$977$	−31483.2	−1.03095	−0.515474	−	0.856905i	$-0.672384\pi$
$977$	−31483.2	−1.03095	−0.515474	+	0.856905i	$0.672384\pi$
$978$	6165.32	0.201580
$979$	9875.00	0.322376
$980$	−1025.31	−0.0334208
$981$	−5823.05	−0.189516
$982$	−20773.7	−0.675067
$983$	−57346.7	−1.86071	−0.930354	−	0.366664i	$-0.880500\pi$
$983$	−57346.7	−1.86071	−0.930354	+	0.366664i	$0.880500\pi$
$984$	21255.5	0.688619
$985$	−1326.68	−0.0429152
$986$	−1835.15	−0.0592730
$987$	41294.8	1.33174
$988$	−4102.25	−0.132095
$989$	57586.7	1.85152
$990$	−1056.87	−0.0339287
$991$	8067.20	0.258590	0.129295	−	0.991606i	$-0.458729\pi$
$991$	8067.20	0.258590	0.129295	+	0.991606i	$0.458729\pi$
$992$	−10677.0	−0.341727
$993$	19376.5	0.619229
$994$	22589.8	0.720830
$995$	−201.586	−0.00642283
$996$	−1274.16	−0.0405353
$997$	−19099.4	−0.606705	−0.303352	−	0.952878i	$-0.598106\pi$
$997$	−19099.4	−0.606705	−0.303352	+	0.952878i	$0.598106\pi$
$998$	26886.1	0.852772
$999$	−41952.4	−1.32864

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.f.1.9	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.f.1.9	✓	23	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)

\(f(q)\)	\(=\)	\(q-2.04967 q^{2} -4.19821 q^{3} -3.79884 q^{4} -5.00000 q^{5} +8.60496 q^{6} -17.0006 q^{7} +24.1838 q^{8} -9.37502 q^{9} +O(q^{10})\)	\(q-2.04967 q^{2} -4.19821 q^{3} -3.79884 q^{4} -5.00000 q^{5} +8.60496 q^{6} -17.0006 q^{7} +24.1838 q^{8} -9.37502 q^{9} +10.2484 q^{10} +11.0000 q^{11} +15.9483 q^{12} -56.8351 q^{13} +34.8456 q^{14} +20.9911 q^{15} -19.1780 q^{16} -31.2236 q^{17} +19.2157 q^{18} -19.0000 q^{19} +18.9942 q^{20} +71.3720 q^{21} -22.5464 q^{22} +121.111 q^{23} -101.529 q^{24} +25.0000 q^{25} +116.493 q^{26} +152.710 q^{27} +64.5825 q^{28} -28.6751 q^{29} -43.0248 q^{30} +69.2583 q^{31} -154.161 q^{32} -46.1803 q^{33} +63.9982 q^{34} +85.0029 q^{35} +35.6142 q^{36} -274.720 q^{37} +38.9438 q^{38} +238.606 q^{39} -120.919 q^{40} -209.355 q^{41} -146.289 q^{42} +475.485 q^{43} -41.7873 q^{44} +46.8751 q^{45} -248.239 q^{46} +578.585 q^{47} +80.5135 q^{48} -53.9802 q^{49} -51.2418 q^{50} +131.083 q^{51} +215.908 q^{52} +602.622 q^{53} -313.006 q^{54} -55.0000 q^{55} -411.138 q^{56} +79.7660 q^{57} +58.7745 q^{58} +640.061 q^{59} -79.7417 q^{60} +4.96918 q^{61} -141.957 q^{62} +159.381 q^{63} +469.405 q^{64} +284.176 q^{65} +94.6545 q^{66} +263.636 q^{67} +118.614 q^{68} -508.451 q^{69} -174.228 q^{70} +648.282 q^{71} -226.723 q^{72} -718.618 q^{73} +563.085 q^{74} -104.955 q^{75} +72.1780 q^{76} -187.006 q^{77} -489.064 q^{78} -1022.23 q^{79} +95.8902 q^{80} -387.983 q^{81} +429.110 q^{82} -79.8926 q^{83} -271.131 q^{84} +156.118 q^{85} -974.589 q^{86} +120.384 q^{87} +266.021 q^{88} +897.727 q^{89} -96.0786 q^{90} +966.230 q^{91} -460.083 q^{92} -290.761 q^{93} -1185.91 q^{94} +95.0000 q^{95} +647.202 q^{96} -1090.53 q^{97} +110.642 q^{98} -103.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * q^99

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