LMFDB - Embedded newform 1045.4.a.f.1.12

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			1.12
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)

$f(q)$	$=$	$q-0.277460 q^{2} -1.24426 q^{3} -7.92302 q^{4} -5.00000 q^{5} +0.345233 q^{6} -34.3401 q^{7} +4.41800 q^{8} -25.4518 q^{9} +O(q^{10})$	\(q-0.277460 q^{2} -1.24426 q^{3} -7.92302 q^{4} -5.00000 q^{5} +0.345233 q^{6} -34.3401 q^{7} +4.41800 q^{8} -25.4518 q^{9} +1.38730 q^{10} +11.0000 q^{11} +9.85832 q^{12} +62.9677 q^{13} +9.52799 q^{14} +6.22132 q^{15} +62.1583 q^{16} +72.5591 q^{17} +7.06186 q^{18} -19.0000 q^{19} +39.6151 q^{20} +42.7281 q^{21} -3.05206 q^{22} +42.2005 q^{23} -5.49716 q^{24} +25.0000 q^{25} -17.4710 q^{26} +65.2639 q^{27} +272.077 q^{28} +165.592 q^{29} -1.72617 q^{30} -28.0562 q^{31} -52.5904 q^{32} -13.6869 q^{33} -20.1322 q^{34} +171.700 q^{35} +201.655 q^{36} -129.169 q^{37} +5.27174 q^{38} -78.3484 q^{39} -22.0900 q^{40} +280.172 q^{41} -11.8553 q^{42} -79.1892 q^{43} -87.1532 q^{44} +127.259 q^{45} -11.7090 q^{46} -441.283 q^{47} -77.3413 q^{48} +836.240 q^{49} -6.93650 q^{50} -90.2826 q^{51} -498.894 q^{52} -173.112 q^{53} -18.1081 q^{54} -55.0000 q^{55} -151.714 q^{56} +23.6410 q^{57} -45.9452 q^{58} -351.076 q^{59} -49.2916 q^{60} -331.437 q^{61} +7.78447 q^{62} +874.017 q^{63} -482.675 q^{64} -314.839 q^{65} +3.79757 q^{66} -705.428 q^{67} -574.887 q^{68} -52.5086 q^{69} -47.6400 q^{70} -24.9913 q^{71} -112.446 q^{72} -209.747 q^{73} +35.8392 q^{74} -31.1066 q^{75} +150.537 q^{76} -377.741 q^{77} +21.7386 q^{78} -768.197 q^{79} -310.792 q^{80} +605.993 q^{81} -77.7366 q^{82} +906.319 q^{83} -338.535 q^{84} -362.795 q^{85} +21.9718 q^{86} -206.040 q^{87} +48.5980 q^{88} -338.224 q^{89} -35.3093 q^{90} -2162.32 q^{91} -334.356 q^{92} +34.9093 q^{93} +122.438 q^{94} +95.0000 q^{95} +65.4364 q^{96} +824.720 q^{97} -232.023 q^{98} -279.970 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.277460	−0.0980969	−0.0490485	−	0.998796i	$-0.515619\pi$
$2$	−0.277460	−0.0980969	−0.0490485	+	0.998796i	$0.515619\pi$
$3$	−1.24426	−0.239459	−0.119729	−	0.992807i	$-0.538203\pi$
$3$	−1.24426	−0.239459	−0.119729	+	0.992807i	$0.538203\pi$
$4$	−7.92302	−0.990377
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0.345233	0.0234902
$7$	−34.3401	−1.85419	−0.927095	−	0.374827i	$-0.877702\pi$
$7$	−34.3401	−1.85419	−0.927095	+	0.374827i	$0.877702\pi$
$8$	4.41800	0.195250
$9$	−25.4518	−0.942660
$10$	1.38730	0.0438703
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	9.85832	0.237154
$13$	62.9677	1.34339	0.671696	−	0.740827i	$-0.265566\pi$
$13$	62.9677	1.34339	0.671696	+	0.740827i	$0.265566\pi$
$14$	9.52799	0.181890
$15$	6.22132	0.107089
$16$	62.1583	0.971224
$17$	72.5591	1.03519	0.517593	−	0.855627i	$-0.326828\pi$
$17$	72.5591	1.03519	0.517593	+	0.855627i	$0.326828\pi$
$18$	7.06186	0.0924720
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	39.6151	0.442910
$21$	42.7281	0.444002
$22$	−3.05206	−0.0295773
$23$	42.2005	0.382584	0.191292	−	0.981533i	$-0.438732\pi$
$23$	42.2005	0.382584	0.191292	+	0.981533i	$0.438732\pi$
$24$	−5.49716	−0.0467543
$25$	25.0000	0.200000
$26$	−17.4710	−0.131783
$27$	65.2639	0.465187
$28$	272.077	1.83635
$29$	165.592	1.06033	0.530167	−	0.847893i	$-0.322129\pi$
$29$	165.592	1.06033	0.530167	+	0.847893i	$0.322129\pi$
$30$	−1.72617	−0.0105051
$31$	−28.0562	−0.162550	−0.0812749	−	0.996692i	$-0.525899\pi$
$31$	−28.0562	−0.162550	−0.0812749	+	0.996692i	$0.525899\pi$
$32$	−52.5904	−0.290524
$33$	−13.6869	−0.0721995
$34$	−20.1322	−0.101549
$35$	171.700	0.829219
$36$	201.655	0.933588
$37$	−129.169	−0.573926	−0.286963	−	0.957942i	$-0.592646\pi$
$37$	−129.169	−0.573926	−0.286963	+	0.957942i	$0.592646\pi$
$38$	5.27174	0.0225050
$39$	−78.3484	−0.321687
$40$	−22.0900	−0.0873184
$41$	280.172	1.06721	0.533604	−	0.845734i	$-0.320837\pi$
$41$	280.172	1.06721	0.533604	+	0.845734i	$0.320837\pi$
$42$	−11.8553	−0.0435552
$43$	−79.1892	−0.280843	−0.140421	−	0.990092i	$-0.544846\pi$
$43$	−79.1892	−0.280843	−0.140421	+	0.990092i	$0.544846\pi$
$44$	−87.1532	−0.298610
$45$	127.259	0.421570
$46$	−11.7090	−0.0375303
$47$	−441.283	−1.36953	−0.684763	−	0.728766i	$-0.740094\pi$
$47$	−441.283	−1.36953	−0.684763	+	0.728766i	$0.740094\pi$
$48$	−77.3413	−0.232568
$49$	836.240	2.43802
$50$	−6.93650	−0.0196194
$51$	−90.2826	−0.247884
$52$	−498.894	−1.33046
$53$	−173.112	−0.448657	−0.224329	−	0.974514i	$-0.572019\pi$
$53$	−173.112	−0.448657	−0.224329	+	0.974514i	$0.572019\pi$
$54$	−18.1081	−0.0456334
$55$	−55.0000	−0.134840
$56$	−151.714	−0.362030
$57$	23.6410	0.0549356
$58$	−45.9452	−0.104015
$59$	−351.076	−0.774681	−0.387340	−	0.921937i	$-0.626606\pi$
$59$	−351.076	−0.774681	−0.387340	+	0.921937i	$0.626606\pi$
$60$	−49.2916	−0.106059
$61$	−331.437	−0.695675	−0.347837	−	0.937555i	$-0.613084\pi$
$61$	−331.437	−0.695675	−0.347837	+	0.937555i	$0.613084\pi$
$62$	7.78447	0.0159456
$63$	874.017	1.74787
$64$	−482.675	−0.942724
$65$	−314.839	−0.600783
$66$	3.79757	0.00708255
$67$	−705.428	−1.28630	−0.643148	−	0.765742i	$-0.722372\pi$
$67$	−705.428	−1.28630	−0.643148	+	0.765742i	$0.722372\pi$
$68$	−574.887	−1.02522
$69$	−52.5086	−0.0916129
$70$	−47.6400	−0.0813438
$71$	−24.9913	−0.0417736	−0.0208868	−	0.999782i	$-0.506649\pi$
$71$	−24.9913	−0.0417736	−0.0208868	+	0.999782i	$0.506649\pi$
$72$	−112.446	−0.184054
$73$	−209.747	−0.336288	−0.168144	−	0.985762i	$-0.553777\pi$
$73$	−209.747	−0.336288	−0.168144	+	0.985762i	$0.553777\pi$
$74$	35.8392	0.0563004
$75$	−31.1066	−0.0478917
$76$	150.537	0.227208
$77$	−377.741	−0.559059
$78$	21.7386	0.0315565
$79$	−768.197	−1.09404	−0.547019	−	0.837120i	$-0.684237\pi$
$79$	−768.197	−1.09404	−0.547019	+	0.837120i	$0.684237\pi$
$80$	−310.792	−0.434344
$81$	605.993	0.831267
$82$	−77.7366	−0.104690
$83$	906.319	1.19857	0.599286	−	0.800535i	$-0.295451\pi$
$83$	906.319	1.19857	0.599286	+	0.800535i	$0.295451\pi$
$84$	−338.535	−0.439729
$85$	−362.795	−0.462949
$86$	21.9718	0.0275498
$87$	−206.040	−0.253906
$88$	48.5980	0.0588700
$89$	−338.224	−0.402827	−0.201414	−	0.979506i	$-0.564554\pi$
$89$	−338.224	−0.402827	−0.201414	+	0.979506i	$0.564554\pi$
$90$	−35.3093	−0.0413547
$91$	−2162.32	−2.49090
$92$	−334.356	−0.378902
$93$	34.9093	0.0389239
$94$	122.438	0.134346
$95$	95.0000	0.102598
$96$	65.4364	0.0695685
$97$	824.720	0.863274	0.431637	−	0.902047i	$-0.357936\pi$
$97$	824.720	0.863274	0.431637	+	0.902047i	$0.357936\pi$
$98$	−232.023	−0.239162
$99$	−279.970	−0.284223
$100$	−198.075	−0.198075
$101$	1732.30	1.70664	0.853318	−	0.521391i	$-0.174587\pi$
$101$	1732.30	1.70664	0.853318	+	0.521391i	$0.174587\pi$
$102$	25.0498	0.0243167
$103$	1153.41	1.10339	0.551693	−	0.834047i	$-0.313982\pi$
$103$	1153.41	1.10339	0.551693	+	0.834047i	$0.313982\pi$
$104$	278.191	0.262297
$105$	−213.640	−0.198564
$106$	48.0318	0.0440119
$107$	1845.35	1.66726	0.833629	−	0.552325i	$-0.186259\pi$
$107$	1845.35	1.66726	0.833629	+	0.552325i	$0.186259\pi$
$108$	−517.087	−0.460710
$109$	347.839	0.305660	0.152830	−	0.988252i	$-0.451161\pi$
$109$	347.839	0.305660	0.152830	+	0.988252i	$0.451161\pi$
$110$	15.2603	0.0132274
$111$	160.720	0.137431
$112$	−2134.52	−1.80083
$113$	1810.49	1.50723	0.753614	−	0.657317i	$-0.228309\pi$
$113$	1810.49	1.50723	0.753614	+	0.657317i	$0.228309\pi$
$114$	−6.55943	−0.00538901
$115$	−211.003	−0.171097
$116$	−1311.99	−1.05013
$117$	−1602.64	−1.26636
$118$	97.4095	0.0759938
$119$	−2491.68	−1.91943
$120$	27.4858	0.0209091
$121$	121.000	0.0909091
$122$	91.9605	0.0682435
$123$	−348.608	−0.255552
$124$	222.290	0.160986
$125$	−125.000	−0.0894427
$126$	−242.505	−0.171461
$127$	2029.33	1.41790	0.708952	−	0.705257i	$-0.249168\pi$
$127$	2029.33	1.41790	0.708952	+	0.705257i	$0.249168\pi$
$128$	554.646	0.383002
$129$	98.5323	0.0672503
$130$	87.3551	0.0589350
$131$	−699.549	−0.466564	−0.233282	−	0.972409i	$-0.574947\pi$
$131$	−699.549	−0.466564	−0.233282	+	0.972409i	$0.574947\pi$
$132$	108.442	0.0715047
$133$	652.461	0.425380
$134$	195.728	0.126182
$135$	−326.319	−0.208038
$136$	320.566	0.202120
$137$	−2422.75	−1.51088	−0.755438	−	0.655221i	$-0.772576\pi$
$137$	−2422.75	−1.51088	−0.755438	+	0.655221i	$0.772576\pi$
$138$	14.5690	0.00898695
$139$	−742.864	−0.453302	−0.226651	−	0.973976i	$-0.572778\pi$
$139$	−742.864	−0.453302	−0.226651	+	0.973976i	$0.572778\pi$
$140$	−1360.38	−0.821239
$141$	549.072	0.327945
$142$	6.93409	0.00409786
$143$	692.645	0.405048
$144$	−1582.04	−0.915533
$145$	−827.960	−0.474196
$146$	58.1964	0.0329888
$147$	−1040.50	−0.583804
$148$	1023.41	0.568403
$149$	1457.83	0.801544	0.400772	−	0.916178i	$-0.368742\pi$
$149$	1457.83	0.801544	0.400772	+	0.916178i	$0.368742\pi$
$150$	8.63083	0.00469803
$151$	−489.690	−0.263910	−0.131955	−	0.991256i	$-0.542125\pi$
$151$	−489.690	−0.263910	−0.131955	+	0.991256i	$0.542125\pi$
$152$	−83.9420	−0.0447934
$153$	−1846.76	−0.975828
$154$	104.808	0.0548420
$155$	140.281	0.0726945
$156$	620.756	0.318591
$157$	−3505.78	−1.78211	−0.891056	−	0.453894i	$-0.850034\pi$
$157$	−3505.78	−1.78211	−0.891056	+	0.453894i	$0.850034\pi$
$158$	213.144	0.107322
$159$	215.398	0.107435
$160$	262.952	0.129926
$161$	−1449.17	−0.709382
$162$	−168.139	−0.0815447
$163$	−3371.49	−1.62009	−0.810046	−	0.586366i	$-0.800558\pi$
$163$	−3371.49	−1.62009	−0.810046	+	0.586366i	$0.800558\pi$
$164$	−2219.81	−1.05694
$165$	68.4345	0.0322886
$166$	−251.467	−0.117576
$167$	−832.509	−0.385757	−0.192879	−	0.981223i	$-0.561782\pi$
$167$	−832.509	−0.385757	−0.192879	+	0.981223i	$0.561782\pi$
$168$	188.773	0.0866913
$169$	1767.93	0.804703
$170$	100.661	0.0454139
$171$	483.584	0.216261
$172$	627.418	0.278140
$173$	1449.84	0.637164	0.318582	−	0.947895i	$-0.396793\pi$
$173$	1449.84	0.637164	0.318582	+	0.947895i	$0.396793\pi$
$174$	57.1679	0.0249074
$175$	−858.502	−0.370838
$176$	683.741	0.292835
$177$	436.831	0.185504
$178$	93.8436	0.0395161
$179$	−1294.76	−0.540642	−0.270321	−	0.962770i	$-0.587130\pi$
$179$	−1294.76	−0.540642	−0.270321	+	0.962770i	$0.587130\pi$
$180$	−1008.28	−0.417513
$181$	4032.60	1.65603	0.828013	−	0.560709i	$-0.189471\pi$
$181$	4032.60	1.65603	0.828013	+	0.560709i	$0.189471\pi$
$182$	599.956	0.244350
$183$	412.395	0.166585
$184$	186.442	0.0746994
$185$	645.845	0.256667
$186$	−9.68594	−0.00381832
$187$	798.150	0.312120
$188$	3496.29	1.35635
$189$	−2241.17	−0.862544
$190$	−26.3587	−0.0100645
$191$	554.961	0.210239	0.105119	−	0.994460i	$-0.466478\pi$
$191$	554.961	0.210239	0.105119	+	0.994460i	$0.466478\pi$
$192$	600.575	0.225743
$193$	−4660.56	−1.73821	−0.869105	−	0.494627i	$-0.835305\pi$
$193$	−4660.56	−1.73821	−0.869105	+	0.494627i	$0.835305\pi$
$194$	−228.827	−0.0846846
$195$	391.742	0.143863
$196$	−6625.54	−2.41456
$197$	−1042.36	−0.376979	−0.188490	−	0.982075i	$-0.560359\pi$
$197$	−1042.36	−0.376979	−0.188490	+	0.982075i	$0.560359\pi$
$198$	77.6804	0.0278814
$199$	2624.09	0.934758	0.467379	−	0.884057i	$-0.345198\pi$
$199$	2624.09	0.934758	0.467379	+	0.884057i	$0.345198\pi$
$200$	110.450	0.0390500
$201$	877.739	0.308015
$202$	−480.644	−0.167416
$203$	−5686.44	−1.96606
$204$	715.311	0.245499
$205$	−1400.86	−0.477270
$206$	−320.025	−0.108239
$207$	−1074.08	−0.360646
$208$	3913.97	1.30473
$209$	−209.000	−0.0691714
$210$	59.2767	0.0194785
$211$	−6046.56	−1.97281	−0.986404	−	0.164336i	$-0.947452\pi$
$211$	−6046.56	−1.97281	−0.986404	+	0.164336i	$0.947452\pi$
$212$	1371.57	0.444340
$213$	31.0958	0.0100030
$214$	−512.010	−0.163553
$215$	395.946	0.125597
$216$	288.336	0.0908276
$217$	963.452	0.301398
$218$	−96.5115	−0.0299843
$219$	260.981	0.0805271
$220$	435.766	0.133542
$221$	4568.88	1.39066
$222$	−44.5935	−0.0134816
$223$	−1259.73	−0.378286	−0.189143	−	0.981950i	$-0.560571\pi$
$223$	−1259.73	−0.378286	−0.189143	+	0.981950i	$0.560571\pi$
$224$	1805.96	0.538686
$225$	−636.295	−0.188532
$226$	−502.339	−0.147854
$227$	3660.93	1.07041	0.535207	−	0.844721i	$-0.320233\pi$
$227$	3660.93	1.07041	0.535207	+	0.844721i	$0.320233\pi$
$228$	−187.308	−0.0544069
$229$	−4877.41	−1.40746	−0.703730	−	0.710468i	$-0.748483\pi$
$229$	−4877.41	−1.40746	−0.703730	+	0.710468i	$0.748483\pi$
$230$	58.5448	0.0167840
$231$	470.009	0.133872
$232$	731.586	0.207030
$233$	2561.10	0.720101	0.360051	−	0.932933i	$-0.382759\pi$
$233$	2561.10	0.720101	0.360051	+	0.932933i	$0.382759\pi$
$234$	444.669	0.124226
$235$	2206.41	0.612470
$236$	2781.58	0.767226
$237$	955.840	0.261977
$238$	691.343	0.188290
$239$	923.981	0.250073	0.125036	−	0.992152i	$-0.460095\pi$
$239$	923.981	0.250073	0.125036	+	0.992152i	$0.460095\pi$
$240$	386.707	0.104008
$241$	−7185.66	−1.92062	−0.960309	−	0.278938i	$-0.910018\pi$
$241$	−7185.66	−1.92062	−0.960309	+	0.278938i	$0.910018\pi$
$242$	−33.5727	−0.00891790
$243$	−2516.14	−0.664241
$244$	2625.98	0.688980
$245$	−4181.20	−1.09031
$246$	96.7248	0.0250689
$247$	−1196.39	−0.308195
$248$	−123.952	−0.0317378
$249$	−1127.70	−0.287008
$250$	34.6825	0.00877406
$251$	−6666.29	−1.67638	−0.838192	−	0.545375i	$-0.816387\pi$
$251$	−6666.29	−1.67638	−0.838192	+	0.545375i	$0.816387\pi$
$252$	−6924.85	−1.73105
$253$	464.206	0.115353
$254$	−563.057	−0.139092
$255$	451.413	0.110857
$256$	3707.51	0.905153
$257$	813.163	0.197369	0.0986843	−	0.995119i	$-0.468537\pi$
$257$	813.163	0.197369	0.0986843	+	0.995119i	$0.468537\pi$
$258$	−27.3388	−0.00659704
$259$	4435.67	1.06417
$260$	2494.47	0.595002
$261$	−4214.62	−0.999534
$262$	194.097	0.0457685
$263$	6845.70	1.60503	0.802517	−	0.596630i	$-0.203494\pi$
$263$	6845.70	1.60503	0.802517	+	0.596630i	$0.203494\pi$
$264$	−60.4687	−0.0140969
$265$	865.562	0.200646
$266$	−181.032	−0.0417285
$267$	420.839	0.0964605
$268$	5589.12	1.27392
$269$	8515.92	1.93020	0.965102	−	0.261875i	$-0.0843408\pi$
$269$	8515.92	1.93020	0.965102	+	0.261875i	$0.0843408\pi$
$270$	90.5406	0.0204079
$271$	−6437.71	−1.44304	−0.721519	−	0.692395i	$-0.756556\pi$
$271$	−6437.71	−1.44304	−0.721519	+	0.692395i	$0.756556\pi$
$272$	4510.15	1.00540
$273$	2690.49	0.596468
$274$	672.217	0.148212
$275$	275.000	0.0603023
$276$	416.026	0.0907313
$277$	−1474.50	−0.319834	−0.159917	−	0.987130i	$-0.551123\pi$
$277$	−1474.50	−0.319834	−0.159917	+	0.987130i	$0.551123\pi$
$278$	206.115	0.0444675
$279$	714.081	0.153229
$280$	758.572	0.161905
$281$	−725.346	−0.153988	−0.0769938	−	0.997032i	$-0.524532\pi$
$281$	−725.346	−0.153988	−0.0769938	+	0.997032i	$0.524532\pi$
$282$	−152.345	−0.0321704
$283$	−2807.58	−0.589729	−0.294865	−	0.955539i	$-0.595275\pi$
$283$	−2807.58	−0.589729	−0.294865	+	0.955539i	$0.595275\pi$
$284$	198.007	0.0413716
$285$	−118.205	−0.0245679
$286$	−192.181	−0.0397340
$287$	−9621.13	−1.97881
$288$	1338.52	0.273865
$289$	351.823	0.0716106
$290$	229.726	0.0465171
$291$	−1026.17	−0.206718
$292$	1661.83	0.333052
$293$	6486.79	1.29339	0.646694	−	0.762750i	$-0.276151\pi$
$293$	6486.79	1.29339	0.646694	+	0.762750i	$0.276151\pi$
$294$	288.698	0.0572694
$295$	1755.38	0.346448
$296$	−570.669	−0.112059
$297$	717.903	0.140259
$298$	−404.489	−0.0786290
$299$	2657.27	0.513960
$300$	246.458	0.0474309
$301$	2719.36	0.520736
$302$	135.869	0.0258887
$303$	−2155.44	−0.408669
$304$	−1181.01	−0.222814
$305$	1657.18	0.311115
$306$	512.402	0.0957257
$307$	10224.8	1.90085	0.950423	−	0.310961i	$-0.100651\pi$
$307$	10224.8	1.90085	0.950423	+	0.310961i	$0.100651\pi$
$308$	2992.85	0.553679
$309$	−1435.15	−0.264215
$310$	−38.9224	−0.00713110
$311$	−8790.10	−1.60270	−0.801351	−	0.598194i	$-0.795885\pi$
$311$	−8790.10	−1.60270	−0.801351	+	0.598194i	$0.795885\pi$
$312$	−346.143	−0.0628093
$313$	−9249.00	−1.67024	−0.835119	−	0.550070i	$-0.814601\pi$
$313$	−9249.00	−1.67024	−0.835119	+	0.550070i	$0.814601\pi$
$314$	972.713	0.174820
$315$	−4370.08	−0.781671
$316$	6086.44	1.08351
$317$	−5544.99	−0.982454	−0.491227	−	0.871032i	$-0.663451\pi$
$317$	−5544.99	−0.982454	−0.491227	+	0.871032i	$0.663451\pi$
$318$	−59.7642	−0.0105390
$319$	1821.51	0.319703
$320$	2413.37	0.421599
$321$	−2296.10	−0.399239
$322$	402.087	0.0695882
$323$	−1378.62	−0.237488
$324$	−4801.30	−0.823267
$325$	1574.19	0.268678
$326$	935.452	0.158926
$327$	−432.804	−0.0731930
$328$	1237.80	0.208372
$329$	15153.7	2.53936
$330$	−18.9878	−0.00316741
$331$	−5652.79	−0.938688	−0.469344	−	0.883015i	$-0.655510\pi$
$331$	−5652.79	−0.938688	−0.469344	+	0.883015i	$0.655510\pi$
$332$	−7180.78	−1.18704
$333$	3287.58	0.541017
$334$	230.988	0.0378416
$335$	3527.14	0.575249
$336$	2655.91	0.431225
$337$	−2778.10	−0.449059	−0.224530	−	0.974467i	$-0.572085\pi$
$337$	−2778.10	−0.449059	−0.224530	+	0.974467i	$0.572085\pi$
$338$	−490.530	−0.0789389
$339$	−2252.73	−0.360919
$340$	2874.43	0.458494
$341$	−308.618	−0.0490106
$342$	−134.175	−0.0212145
$343$	−16937.9	−2.66636
$344$	−349.858	−0.0548345
$345$	262.543	0.0409706
$346$	−402.273	−0.0625038
$347$	−6417.60	−0.992838	−0.496419	−	0.868083i	$-0.665352\pi$
$347$	−6417.60	−0.992838	−0.496419	+	0.868083i	$0.665352\pi$
$348$	1632.46	0.251463
$349$	−5542.91	−0.850158	−0.425079	−	0.905156i	$-0.639754\pi$
$349$	−5542.91	−0.850158	−0.425079	+	0.905156i	$0.639754\pi$
$350$	238.200	0.0363781
$351$	4109.52	0.624928
$352$	−578.495	−0.0875963
$353$	−3902.39	−0.588395	−0.294197	−	0.955745i	$-0.595052\pi$
$353$	−3902.39	−0.588395	−0.294197	+	0.955745i	$0.595052\pi$
$354$	−121.203	−0.0181974
$355$	124.957	0.0186817
$356$	2679.75	0.398951
$357$	3100.31	0.459624
$358$	359.244	0.0530353
$359$	1731.42	0.254542	0.127271	−	0.991868i	$-0.459378\pi$
$359$	1731.42	0.254542	0.127271	+	0.991868i	$0.459378\pi$
$360$	562.230	0.0823115
$361$	361.000	0.0526316
$362$	−1118.88	−0.162451
$363$	−150.556	−0.0217690
$364$	17132.1	2.46693
$365$	1048.74	0.150393
$366$	−114.423	−0.0163415
$367$	−2591.71	−0.368628	−0.184314	−	0.982867i	$-0.559006\pi$
$367$	−2591.71	−0.368628	−0.184314	+	0.982867i	$0.559006\pi$
$368$	2623.11	0.371574
$369$	−7130.89	−1.00601
$370$	−179.196	−0.0251783
$371$	5944.69	0.831895
$372$	−276.587	−0.0385494
$373$	8529.34	1.18400	0.592001	−	0.805937i	$-0.298338\pi$
$373$	8529.34	1.18400	0.592001	+	0.805937i	$0.298338\pi$
$374$	−221.455	−0.0306181
$375$	155.533	0.0214178
$376$	−1949.59	−0.267400
$377$	10427.0	1.42444
$378$	621.834	0.0846129
$379$	−930.701	−0.126140	−0.0630698	−	0.998009i	$-0.520089\pi$
$379$	−930.701	−0.126140	−0.0630698	+	0.998009i	$0.520089\pi$
$380$	−752.687	−0.101611
$381$	−2525.02	−0.339529
$382$	−153.980	−0.0206238
$383$	−3181.52	−0.424459	−0.212230	−	0.977220i	$-0.568072\pi$
$383$	−3181.52	−0.424459	−0.212230	+	0.977220i	$0.568072\pi$
$384$	−690.126	−0.0917132
$385$	1888.70	0.250019
$386$	1293.12	0.170513
$387$	2015.51	0.264739
$388$	−6534.27	−0.854967
$389$	−11957.9	−1.55859	−0.779294	−	0.626658i	$-0.784422\pi$
$389$	−11957.9	−1.55859	−0.779294	+	0.626658i	$0.784422\pi$
$390$	−108.693	−0.0141125
$391$	3062.03	0.396045
$392$	3694.51	0.476023
$393$	870.424	0.111723
$394$	289.212	0.0369805
$395$	3840.99	0.489268
$396$	2218.21	0.281487
$397$	11378.1	1.43842	0.719209	−	0.694794i	$-0.244505\pi$
$397$	11378.1	1.43842	0.719209	+	0.694794i	$0.244505\pi$
$398$	−728.080	−0.0916969
$399$	−811.834	−0.101861
$400$	1553.96	0.194245
$401$	7229.02	0.900249	0.450125	−	0.892966i	$-0.351380\pi$
$401$	7229.02	0.900249	0.450125	+	0.892966i	$0.351380\pi$
$402$	−243.537	−0.0302153
$403$	−1766.63	−0.218368
$404$	−13725.0	−1.69021
$405$	−3029.97	−0.371754
$406$	1577.76	0.192864
$407$	−1420.86	−0.173045
$408$	−398.869	−0.0483994
$409$	7599.79	0.918790	0.459395	−	0.888232i	$-0.348066\pi$
$409$	7599.79	0.918790	0.459395	+	0.888232i	$0.348066\pi$
$410$	388.683	0.0468187
$411$	3014.55	0.361792
$412$	−9138.48	−1.09277
$413$	12056.0	1.43640
$414$	298.014	0.0353783
$415$	−4531.60	−0.536018
$416$	−3311.50	−0.390288
$417$	924.319	0.108547
$418$	57.9891	0.00678551
$419$	−5246.50	−0.611714	−0.305857	−	0.952077i	$-0.598943\pi$
$419$	−5246.50	−0.611714	−0.305857	+	0.952077i	$0.598943\pi$
$420$	1692.68	0.196653
$421$	−2532.96	−0.293228	−0.146614	−	0.989194i	$-0.546838\pi$
$421$	−2532.96	−0.293228	−0.146614	+	0.989194i	$0.546838\pi$
$422$	1677.68	0.193526
$423$	11231.4	1.29100
$424$	−764.811	−0.0876002
$425$	1813.98	0.207037
$426$	−8.62784	−0.000981268 0
$427$	11381.6	1.28991
$428$	−14620.7	−1.65121
$429$	−861.833	−0.0969922
$430$	−109.859	−0.0123207
$431$	−3320.46	−0.371093	−0.185546	−	0.982636i	$-0.559405\pi$
$431$	−3320.46	−0.371093	−0.185546	+	0.982636i	$0.559405\pi$
$432$	4056.69	0.451800
$433$	−7763.43	−0.861632	−0.430816	−	0.902440i	$-0.641774\pi$
$433$	−7763.43	−0.861632	−0.430816	+	0.902440i	$0.641774\pi$
$434$	−267.319	−0.0295662
$435$	1030.20	0.113550
$436$	−2755.94	−0.302719
$437$	−801.810	−0.0877707
$438$	−72.4117	−0.00789946
$439$	4048.50	0.440146	0.220073	−	0.975483i	$-0.429370\pi$
$439$	4048.50	0.440146	0.220073	+	0.975483i	$0.429370\pi$
$440$	−242.990	−0.0263275
$441$	−21283.8	−2.29822
$442$	−1267.68	−0.136420
$443$	14323.9	1.53623	0.768114	−	0.640313i	$-0.221195\pi$
$443$	14323.9	1.53623	0.768114	+	0.640313i	$0.221195\pi$
$444$	−1273.39	−0.136109
$445$	1691.12	0.180150
$446$	349.525	0.0371087
$447$	−1813.92	−0.191937
$448$	16575.1	1.74799
$449$	−17054.0	−1.79249	−0.896247	−	0.443555i	$-0.853717\pi$
$449$	−17054.0	−1.79249	−0.896247	+	0.443555i	$0.853717\pi$
$450$	176.546	0.0184944
$451$	3081.89	0.321776
$452$	−14344.6	−1.49272
$453$	609.303	0.0631955
$454$	−1015.76	−0.105004
$455$	10811.6	1.11397
$456$	104.446	0.0107262
$457$	−8314.84	−0.851098	−0.425549	−	0.904935i	$-0.639919\pi$
$457$	−8314.84	−0.851098	−0.425549	+	0.904935i	$0.639919\pi$
$458$	1353.29	0.138067
$459$	4735.49	0.481555
$460$	1671.78	0.169450
$461$	−7588.78	−0.766691	−0.383346	−	0.923605i	$-0.625228\pi$
$461$	−7588.78	−0.766691	−0.383346	+	0.923605i	$0.625228\pi$
$462$	−130.409	−0.0131324
$463$	−6672.78	−0.669785	−0.334893	−	0.942256i	$-0.608700\pi$
$463$	−6672.78	−0.669785	−0.334893	+	0.942256i	$0.608700\pi$
$464$	10292.9	1.02982
$465$	−174.547	−0.0174073
$466$	−710.604	−0.0706397
$467$	−1748.30	−0.173237	−0.0866185	−	0.996242i	$-0.527606\pi$
$467$	−1748.30	−0.173237	−0.0866185	+	0.996242i	$0.527606\pi$
$468$	12697.8	1.25418
$469$	24224.5	2.38503
$470$	−612.191	−0.0600815
$471$	4362.11	0.426742
$472$	−1551.05	−0.151256
$473$	−871.082	−0.0846773
$474$	−265.207	−0.0256991
$475$	−475.000	−0.0458831
$476$	19741.7	1.90096
$477$	4406.03	0.422931
$478$	−256.368	−0.0245313
$479$	−7220.27	−0.688731	−0.344366	−	0.938836i	$-0.611906\pi$
$479$	−7220.27	−0.688731	−0.344366	+	0.938836i	$0.611906\pi$
$480$	−327.182	−0.0311120
$481$	−8133.48	−0.771007
$482$	1993.73	0.188407
$483$	1803.15	0.169868
$484$	−958.685	−0.0900343
$485$	−4123.60	−0.386068
$486$	698.128	0.0651600
$487$	13939.7	1.29706	0.648531	−	0.761188i	$-0.275384\pi$
$487$	13939.7	1.29706	0.648531	+	0.761188i	$0.275384\pi$
$488$	−1464.29	−0.135830
$489$	4195.02	0.387945
$490$	1160.12	0.106957
$491$	−4766.47	−0.438101	−0.219051	−	0.975713i	$-0.570296\pi$
$491$	−4766.47	−0.438101	−0.219051	+	0.975713i	$0.570296\pi$
$492$	2762.03	0.253093
$493$	12015.2	1.09764
$494$	331.949	0.0302330
$495$	1399.85	0.127108
$496$	−1743.93	−0.157872
$497$	858.204	0.0774561
$498$	312.892	0.0281546
$499$	1887.01	0.169287	0.0846434	−	0.996411i	$-0.473025\pi$
$499$	1887.01	0.169287	0.0846434	+	0.996411i	$0.473025\pi$
$500$	990.377	0.0885820
$501$	1035.86	0.0923729
$502$	1849.63	0.164448
$503$	4691.91	0.415908	0.207954	−	0.978139i	$-0.433320\pi$
$503$	4691.91	0.415908	0.207954	+	0.978139i	$0.433320\pi$
$504$	3861.41	0.341271
$505$	−8661.49	−0.763231
$506$	−128.799	−0.0113158
$507$	−2199.77	−0.192693
$508$	−16078.4	−1.40426
$509$	13382.2	1.16534	0.582668	−	0.812710i	$-0.302009\pi$
$509$	13382.2	1.16534	0.582668	+	0.812710i	$0.302009\pi$
$510$	−125.249	−0.0108748
$511$	7202.73	0.623542
$512$	−5465.86	−0.471795
$513$	−1240.01	−0.106721
$514$	−225.620	−0.0193613
$515$	−5767.05	−0.493449
$516$	−780.673	−0.0666031
$517$	−4854.11	−0.412927
$518$	−1230.72	−0.104392
$519$	−1803.98	−0.152574
$520$	−1390.96	−0.117303
$521$	8523.06	0.716703	0.358351	−	0.933587i	$-0.383339\pi$
$521$	8523.06	0.716703	0.358351	+	0.933587i	$0.383339\pi$
$522$	1169.39	0.0980512
$523$	18360.6	1.53509	0.767546	−	0.640994i	$-0.221478\pi$
$523$	18360.6	1.53509	0.767546	+	0.640994i	$0.221478\pi$
$524$	5542.54	0.462074
$525$	1068.20	0.0888003
$526$	−1899.41	−0.157449
$527$	−2035.73	−0.168269
$528$	−850.755	−0.0701219
$529$	−10386.1	−0.853630
$530$	−240.159	−0.0196827
$531$	8935.51	0.730260
$532$	−5169.46	−0.421287
$533$	17641.8	1.43368
$534$	−116.766	−0.00946248
$535$	−9226.74	−0.745620
$536$	−3116.58	−0.251149
$537$	1611.02	0.129461
$538$	−2362.83	−0.189347
$539$	9198.64	0.735090
$540$	2585.43	0.206036
$541$	−14937.6	−1.18709	−0.593545	−	0.804801i	$-0.702272\pi$
$541$	−14937.6	−1.18709	−0.593545	+	0.804801i	$0.702272\pi$
$542$	1786.21	0.141558
$543$	−5017.61	−0.396550
$544$	−3815.92	−0.300746
$545$	−1739.20	−0.136695
$546$	−746.503	−0.0585117
$547$	14173.9	1.10792	0.553960	−	0.832543i	$-0.313116\pi$
$547$	14173.9	1.10792	0.553960	+	0.832543i	$0.313116\pi$
$548$	19195.5	1.49634
$549$	8435.67	0.655784
$550$	−76.3015	−0.00591547
$551$	−3146.25	−0.243257
$552$	−231.983	−0.0178874
$553$	26379.9	2.02855
$554$	409.114	0.0313747
$555$	−803.601	−0.0614612
$556$	5885.73	0.448940
$557$	−6370.34	−0.484596	−0.242298	−	0.970202i	$-0.577901\pi$
$557$	−6370.34	−0.484596	−0.242298	+	0.970202i	$0.577901\pi$
$558$	−198.129	−0.0150313
$559$	−4986.36	−0.377282
$560$	10672.6	0.805357
$561$	−993.109	−0.0747399
$562$	201.254	0.0151057
$563$	23845.8	1.78504	0.892522	−	0.451004i	$-0.148934\pi$
$563$	23845.8	1.78504	0.892522	+	0.451004i	$0.148934\pi$
$564$	−4350.31	−0.324789
$565$	−9052.46	−0.674053
$566$	778.991	0.0578506
$567$	−20809.9	−1.54133
$568$	−110.412	−0.00815629
$569$	−13257.5	−0.976775	−0.488387	−	0.872627i	$-0.662415\pi$
$569$	−13257.5	−0.976775	−0.488387	+	0.872627i	$0.662415\pi$
$570$	32.7972	0.00241004
$571$	5465.92	0.400599	0.200299	−	0.979735i	$-0.435809\pi$
$571$	5465.92	0.400599	0.200299	+	0.979735i	$0.435809\pi$
$572$	−5487.84	−0.401150
$573$	−690.518	−0.0503435
$574$	2669.48	0.194115
$575$	1055.01	0.0765167
$576$	12284.9	0.888668
$577$	12296.8	0.887214	0.443607	−	0.896221i	$-0.353699\pi$
$577$	12296.8	0.887214	0.443607	+	0.896221i	$0.353699\pi$
$578$	−97.6167	−0.00702478
$579$	5798.97	0.416230
$580$	6559.94	0.469632
$581$	−31123.1	−2.22238
$582$	284.721	0.0202784
$583$	−1904.24	−0.135275
$584$	−926.662	−0.0656602
$585$	8013.21	0.566334
$586$	−1799.83	−0.126877
$587$	1133.94	0.0797321	0.0398661	−	0.999205i	$-0.487307\pi$
$587$	1133.94	0.0797321	0.0398661	+	0.999205i	$0.487307\pi$
$588$	8243.92	0.578186
$589$	533.068	0.0372915
$590$	−487.047	−0.0339855
$591$	1296.97	0.0902709
$592$	−8028.93	−0.557410
$593$	−15963.5	−1.10547	−0.552733	−	0.833359i	$-0.686415\pi$
$593$	−15963.5	−1.10547	−0.552733	+	0.833359i	$0.686415\pi$
$594$	−199.189	−0.0137590
$595$	12458.4	0.858396
$596$	−11550.4	−0.793830
$597$	−3265.06	−0.223836
$598$	−737.287	−0.0504179
$599$	−23006.7	−1.56933	−0.784665	−	0.619920i	$-0.787165\pi$
$599$	−23006.7	−1.56933	−0.784665	+	0.619920i	$0.787165\pi$
$600$	−137.429	−0.00935085
$601$	−1231.29	−0.0835694	−0.0417847	−	0.999127i	$-0.513304\pi$
$601$	−1231.29	−0.0835694	−0.0417847	+	0.999127i	$0.513304\pi$
$602$	−754.515	−0.0510826
$603$	17954.4	1.21254
$604$	3879.82	0.261370
$605$	−605.000	−0.0406558
$606$	598.047	0.0400891
$607$	15332.5	1.02525	0.512626	−	0.858612i	$-0.328673\pi$
$607$	15332.5	1.02525	0.512626	+	0.858612i	$0.328673\pi$
$608$	999.218	0.0666508
$609$	7075.43	0.470790
$610$	−459.802	−0.0305194
$611$	−27786.6	−1.83981
$612$	14631.9	0.966438
$613$	−16754.9	−1.10396	−0.551978	−	0.833859i	$-0.686127\pi$
$613$	−16754.9	−1.10396	−0.551978	+	0.833859i	$0.686127\pi$
$614$	−2836.97	−0.186467
$615$	1743.04	0.114286
$616$	−1668.86	−0.109156
$617$	8735.57	0.569985	0.284992	−	0.958530i	$-0.408009\pi$
$617$	8735.57	0.569985	0.284992	+	0.958530i	$0.408009\pi$
$618$	398.195	0.0259187
$619$	−29289.3	−1.90184	−0.950919	−	0.309439i	$-0.899859\pi$
$619$	−29289.3	−1.90184	−0.950919	+	0.309439i	$0.899859\pi$
$620$	−1111.45	−0.0719949
$621$	2754.17	0.177973
$622$	2438.90	0.157220
$623$	11614.6	0.746918
$624$	−4870.01	−0.312430
$625$	625.000	0.0400000
$626$	2566.23	0.163845
$627$	260.051	0.0165637
$628$	27776.3	1.76496
$629$	−9372.39	−0.594120
$630$	1212.52	0.0766795
$631$	−9684.94	−0.611016	−0.305508	−	0.952189i	$-0.598826\pi$
$631$	−9684.94	−0.611016	−0.305508	+	0.952189i	$0.598826\pi$
$632$	−3393.90	−0.213611
$633$	7523.52	0.472406
$634$	1538.51	0.0963757
$635$	−10146.6	−0.634106
$636$	−1706.60	−0.106401
$637$	52656.1	3.27521
$638$	−505.397	−0.0313618
$639$	636.074	0.0393783
$640$	−2773.23	−0.171284
$641$	−565.332	−0.0348350	−0.0174175	−	0.999848i	$-0.505544\pi$
$641$	−565.332	−0.0348350	−0.0174175	+	0.999848i	$0.505544\pi$
$642$	637.076	0.0391641
$643$	9509.95	0.583259	0.291630	−	0.956531i	$-0.405803\pi$
$643$	9509.95	0.583259	0.291630	+	0.956531i	$0.405803\pi$
$644$	11481.8	0.702556
$645$	−492.661	−0.0300752
$646$	382.513	0.0232968
$647$	8970.39	0.545073	0.272537	−	0.962145i	$-0.412137\pi$
$647$	8970.39	0.545073	0.272537	+	0.962145i	$0.412137\pi$
$648$	2677.28	0.162305
$649$	−3861.83	−0.233575
$650$	−436.776	−0.0263565
$651$	−1198.79	−0.0721724
$652$	26712.3	1.60450
$653$	−12080.8	−0.723979	−0.361989	−	0.932182i	$-0.617902\pi$
$653$	−12080.8	−0.723979	−0.361989	+	0.932182i	$0.617902\pi$
$654$	120.086	0.00718001
$655$	3497.75	0.208654
$656$	17415.0	1.03650
$657$	5338.44	0.317005
$658$	−4204.54	−0.249103
$659$	−7321.03	−0.432757	−0.216379	−	0.976310i	$-0.569425\pi$
$659$	−7321.03	−0.432757	−0.216379	+	0.976310i	$0.569425\pi$
$660$	−542.208	−0.0319779
$661$	15868.7	0.933770	0.466885	−	0.884318i	$-0.345376\pi$
$661$	15868.7	0.933770	0.466885	+	0.884318i	$0.345376\pi$
$662$	1568.42	0.0920824
$663$	−5684.89	−0.333006
$664$	4004.12	0.234021
$665$	−3262.31	−0.190236
$666$	−912.173	−0.0530721
$667$	6988.07	0.405666
$668$	6595.98	0.382045
$669$	1567.44	0.0905839
$670$	−978.641	−0.0564301
$671$	−3645.81	−0.209754
$672$	−2247.09	−0.128993
$673$	10824.7	0.620001	0.310001	−	0.950736i	$-0.399671\pi$
$673$	10824.7	0.620001	0.310001	+	0.950736i	$0.399671\pi$
$674$	770.813	0.0440513
$675$	1631.60	0.0930373
$676$	−14007.4	−0.796959
$677$	−20965.9	−1.19023	−0.595114	−	0.803642i	$-0.702893\pi$
$677$	−20965.9	−1.19023	−0.595114	+	0.803642i	$0.702893\pi$
$678$	625.042	0.0354050
$679$	−28320.9	−1.60067
$680$	−1602.83	−0.0903908
$681$	−4555.16	−0.256320
$682$	85.6292	0.00480779
$683$	−33814.2	−1.89439	−0.947193	−	0.320663i	$-0.896094\pi$
$683$	−33814.2	−1.89439	−0.947193	+	0.320663i	$0.896094\pi$
$684$	−3831.45	−0.214180
$685$	12113.8	0.675684
$686$	4699.59	0.261561
$687$	6068.78	0.337028
$688$	−4922.27	−0.272761
$689$	−10900.5	−0.602723
$690$	−72.8452	−0.00401909
$691$	4841.57	0.266544	0.133272	−	0.991079i	$-0.457452\pi$
$691$	4841.57	0.266544	0.133272	+	0.991079i	$0.457452\pi$
$692$	−11487.1	−0.631033
$693$	9614.18	0.527002
$694$	1780.63	0.0973943
$695$	3714.32	0.202723
$696$	−910.285	−0.0495751
$697$	20329.0	1.10476
$698$	1537.94	0.0833979
$699$	−3186.69	−0.172434
$700$	6801.92	0.367269
$701$	17244.2	0.929106	0.464553	−	0.885545i	$-0.346215\pi$
$701$	17244.2	0.929106	0.464553	+	0.885545i	$0.346215\pi$
$702$	−1140.23	−0.0613035
$703$	2454.21	0.131668
$704$	−5309.42	−0.284242
$705$	−2745.36	−0.146661
$706$	1082.76	0.0577197
$707$	−59487.3	−3.16443
$708$	−3461.02	−0.183719
$709$	176.327	0.00934007	0.00467004	−	0.999989i	$-0.498513\pi$
$709$	176.327	0.00934007	0.00467004	+	0.999989i	$0.498513\pi$
$710$	−34.6705	−0.00183262
$711$	19552.0	1.03130
$712$	−1494.27	−0.0786520
$713$	−1183.99	−0.0621889
$714$	−860.213	−0.0450877
$715$	−3463.22	−0.181143
$716$	10258.4	0.535439
$717$	−1149.68	−0.0598820
$718$	−480.398	−0.0249698
$719$	−21675.1	−1.12426	−0.562131	−	0.827048i	$-0.690018\pi$
$719$	−21675.1	−1.12426	−0.562131	+	0.827048i	$0.690018\pi$
$720$	7910.21	0.409439
$721$	−39608.1	−2.04589
$722$	−100.163	−0.00516300
$723$	8940.85	0.459909
$724$	−31950.3	−1.64009
$725$	4139.80	0.212067
$726$	41.7732	0.00213547
$727$	−30383.5	−1.55002	−0.775008	−	0.631951i	$-0.782254\pi$
$727$	−30383.5	−1.55002	−0.775008	+	0.631951i	$0.782254\pi$
$728$	−9553.11	−0.486349
$729$	−13231.1	−0.672208
$730$	−290.982	−0.0147531
$731$	−5745.90	−0.290725
$732$	−3267.41	−0.164982
$733$	29096.6	1.46618	0.733089	−	0.680132i	$-0.238078\pi$
$733$	29096.6	1.46618	0.733089	+	0.680132i	$0.238078\pi$
$734$	719.097	0.0361612
$735$	5202.52	0.261085
$736$	−2219.35	−0.111150
$737$	−7759.71	−0.387833
$738$	1978.54	0.0986869
$739$	25449.5	1.26681	0.633406	−	0.773819i	$-0.281656\pi$
$739$	25449.5	1.26681	0.633406	+	0.773819i	$0.281656\pi$
$740$	−5117.04	−0.254197
$741$	1488.62	0.0738000
$742$	−1649.41	−0.0816064
$743$	−20570.2	−1.01567	−0.507837	−	0.861453i	$-0.669555\pi$
$743$	−20570.2	−1.01567	−0.507837	+	0.861453i	$0.669555\pi$
$744$	154.229	0.00759990
$745$	−7289.15	−0.358461
$746$	−2366.55	−0.116147
$747$	−23067.5	−1.12985
$748$	−6323.76	−0.309117
$749$	−63369.4	−3.09141
$750$	−43.1542	−0.00210102
$751$	−15996.3	−0.777246	−0.388623	−	0.921397i	$-0.627049\pi$
$751$	−15996.3	−0.777246	−0.388623	+	0.921397i	$0.627049\pi$
$752$	−27429.4	−1.33012
$753$	8294.62	0.401425
$754$	−2893.06	−0.139734
$755$	2448.45	0.118024
$756$	17756.8	0.854244
$757$	−18471.0	−0.886841	−0.443421	−	0.896314i	$-0.646235\pi$
$757$	−18471.0	−0.886841	−0.443421	+	0.896314i	$0.646235\pi$
$758$	258.232	0.0123739
$759$	−577.595	−0.0276223
$760$	419.710	0.0200322
$761$	12087.9	0.575803	0.287902	−	0.957660i	$-0.407042\pi$
$761$	12087.9	0.575803	0.287902	+	0.957660i	$0.407042\pi$
$762$	700.592	0.0333068
$763$	−11944.8	−0.566752
$764$	−4396.97	−0.208216
$765$	9233.80	0.436404
$766$	882.744	0.0416381
$767$	−22106.4	−1.04070
$768$	−4613.11	−0.216747
$769$	14862.0	0.696929	0.348464	−	0.937322i	$-0.386703\pi$
$769$	14862.0	0.696929	0.348464	+	0.937322i	$0.386703\pi$
$770$	−524.040	−0.0245261
$771$	−1011.79	−0.0472616
$772$	36925.7	1.72148
$773$	−23779.9	−1.10647	−0.553237	−	0.833024i	$-0.686608\pi$
$773$	−23779.9	−1.10647	−0.553237	+	0.833024i	$0.686608\pi$
$774$	−559.223	−0.0259701
$775$	−701.405	−0.0325100
$776$	3643.61	0.168554
$777$	−5519.15	−0.254824
$778$	3317.85	0.152893
$779$	−5323.27	−0.244834
$780$	−3103.78	−0.142478
$781$	−274.905	−0.0125952
$782$	−849.592	−0.0388508
$783$	10807.2	0.493253
$784$	51979.3	2.36786
$785$	17528.9	0.796984
$786$	−241.508	−0.0109597
$787$	−28593.0	−1.29508	−0.647542	−	0.762030i	$-0.724203\pi$
$787$	−28593.0	−1.29508	−0.647542	+	0.762030i	$0.724203\pi$
$788$	8258.61	0.373351
$789$	−8517.85	−0.384339
$790$	−1065.72	−0.0479957
$791$	−62172.4	−2.79469
$792$	−1236.91	−0.0554944
$793$	−20869.8	−0.934564
$794$	−3156.97	−0.141104
$795$	−1076.99	−0.0480463
$796$	−20790.7	−0.925763
$797$	7864.59	0.349533	0.174767	−	0.984610i	$-0.444083\pi$
$797$	7864.59	0.349533	0.174767	+	0.984610i	$0.444083\pi$
$798$	225.251	0.00999225
$799$	−32019.1	−1.41771
$800$	−1314.76	−0.0581048
$801$	8608.41	0.379729
$802$	−2005.76	−0.0883117
$803$	−2307.22	−0.101395
$804$	−6954.34	−0.305051
$805$	7245.85	0.317245
$806$	490.171	0.0214212
$807$	−10596.0	−0.462204
$808$	7653.30	0.333220
$809$	−19204.0	−0.834584	−0.417292	−	0.908773i	$-0.637021\pi$
$809$	−19204.0	−0.834584	−0.417292	+	0.908773i	$0.637021\pi$
$810$	840.695	0.0364679
$811$	−1801.23	−0.0779897	−0.0389949	−	0.999239i	$-0.512416\pi$
$811$	−1801.23	−0.0779897	−0.0389949	+	0.999239i	$0.512416\pi$
$812$	45053.8	1.94714
$813$	8010.21	0.345548
$814$	394.232	0.0169752
$815$	16857.4	0.724527
$816$	−5611.82	−0.240751
$817$	1504.60	0.0644298
$818$	−2108.64	−0.0901305
$819$	55034.8	2.34807
$820$	11099.0	0.472677
$821$	−9833.01	−0.417996	−0.208998	−	0.977916i	$-0.567020\pi$
$821$	−9833.01	−0.417996	−0.208998	+	0.977916i	$0.567020\pi$
$822$	−836.416	−0.0354907
$823$	25636.2	1.08581	0.542904	−	0.839795i	$-0.317325\pi$
$823$	25636.2	1.08581	0.542904	+	0.839795i	$0.317325\pi$
$824$	5095.76	0.215436
$825$	−342.172	−0.0144399
$826$	−3345.05	−0.140907
$827$	−13535.4	−0.569133	−0.284566	−	0.958656i	$-0.591850\pi$
$827$	−13535.4	−0.569133	−0.284566	+	0.958656i	$0.591850\pi$
$828$	8509.95	0.357176
$829$	−6585.03	−0.275884	−0.137942	−	0.990440i	$-0.544049\pi$
$829$	−6585.03	−0.275884	−0.137942	+	0.990440i	$0.544049\pi$
$830$	1257.34	0.0525817
$831$	1834.66	0.0765870
$832$	−30392.9	−1.26645
$833$	60676.8	2.52380
$834$	−256.462	−0.0106481
$835$	4162.54	0.172516
$836$	1655.91	0.0685058
$837$	−1831.06	−0.0756160
$838$	1455.69	0.0600073
$839$	16874.4	0.694363	0.347181	−	0.937798i	$-0.387139\pi$
$839$	16874.4	0.694363	0.347181	+	0.937798i	$0.387139\pi$
$840$	−943.864	−0.0387695
$841$	3031.73	0.124307
$842$	702.796	0.0287648
$843$	902.521	0.0368736
$844$	47907.0	1.95382
$845$	−8839.66	−0.359874
$846$	−3116.28	−0.126643
$847$	−4155.15	−0.168563
$848$	−10760.4	−0.435746
$849$	3493.37	0.141216
$850$	−503.306	−0.0203097
$851$	−5451.00	−0.219575
$852$	−246.372	−0.00990678
$853$	4115.81	0.165208	0.0826040	−	0.996582i	$-0.473676\pi$
$853$	4115.81	0.165208	0.0826040	+	0.996582i	$0.473676\pi$
$854$	−3157.93	−0.126536
$855$	−2417.92	−0.0967148
$856$	8152.75	0.325532
$857$	26946.1	1.07405	0.537024	−	0.843567i	$-0.319548\pi$
$857$	26946.1	1.07405	0.537024	+	0.843567i	$0.319548\pi$
$858$	239.124	0.00951464
$859$	5347.52	0.212404	0.106202	−	0.994345i	$-0.466131\pi$
$859$	5347.52	0.212404	0.106202	+	0.994345i	$0.466131\pi$
$860$	−3137.09	−0.124388
$861$	11971.2	0.473842
$862$	921.295	0.0364030
$863$	−18653.0	−0.735752	−0.367876	−	0.929875i	$-0.619915\pi$
$863$	−18653.0	−0.735752	−0.367876	+	0.929875i	$0.619915\pi$
$864$	−3432.26	−0.135148
$865$	−7249.21	−0.284948
$866$	2154.04	0.0845234
$867$	−437.760	−0.0171478
$868$	−7633.44	−0.298498
$869$	−8450.17	−0.329865
$870$	−285.840	−0.0111389
$871$	−44419.2	−1.72800
$872$	1536.75	0.0596801
$873$	−20990.6	−0.813774
$874$	222.470	0.00861003
$875$	4292.51	0.165844
$876$	−2067.75	−0.0797522
$877$	27877.9	1.07340	0.536698	−	0.843774i	$-0.319671\pi$
$877$	27877.9	1.07340	0.536698	+	0.843774i	$0.319671\pi$
$878$	−1123.30	−0.0431770
$879$	−8071.28	−0.309713
$880$	−3418.71	−0.130960
$881$	20637.8	0.789224	0.394612	−	0.918848i	$-0.370879\pi$
$881$	20637.8	0.789224	0.394612	+	0.918848i	$0.370879\pi$
$882$	5905.41	0.225448
$883$	47240.9	1.80044	0.900218	−	0.435440i	$-0.143407\pi$
$883$	47240.9	1.80044	0.900218	+	0.435440i	$0.143407\pi$
$884$	−36199.3	−1.37728
$885$	−2184.15	−0.0829599
$886$	−3974.31	−0.150699
$887$	−25681.2	−0.972143	−0.486071	−	0.873919i	$-0.661571\pi$
$887$	−25681.2	−0.972143	−0.486071	+	0.873919i	$0.661571\pi$
$888$	710.062	0.0268335
$889$	−69687.3	−2.62906
$890$	−469.218	−0.0176722
$891$	6665.93	0.250636
$892$	9980.87	0.374646
$893$	8384.37	0.314191
$894$	503.291	0.0188284
$895$	6473.80	0.241782
$896$	−19046.6	−0.710159
$897$	−3306.35	−0.123072
$898$	4731.81	0.175838
$899$	−4645.88	−0.172357
$900$	5041.38	0.186718
$901$	−12560.9	−0.464444
$902$	−855.103	−0.0315652
$903$	−3383.60	−0.124695
$904$	7998.75	0.294286
$905$	−20163.0	−0.740597
$906$	−169.057	−0.00619928
$907$	11789.7	0.431612	0.215806	−	0.976436i	$-0.430762\pi$
$907$	11789.7	0.431612	0.215806	+	0.976436i	$0.430762\pi$
$908$	−29005.6	−1.06011
$909$	−44090.1	−1.60878
$910$	−2999.78	−0.109277
$911$	11401.2	0.414640	0.207320	−	0.978273i	$-0.433526\pi$
$911$	11401.2	0.414640	0.207320	+	0.978273i	$0.433526\pi$
$912$	1469.49	0.0533547
$913$	9969.51	0.361383
$914$	2307.03	0.0834901
$915$	−2061.97	−0.0744992
$916$	38643.8	1.39392
$917$	24022.6	0.865098
$918$	−1313.91	−0.0472390
$919$	−28747.5	−1.03187	−0.515937	−	0.856626i	$-0.672556\pi$
$919$	−28747.5	−1.03187	−0.515937	+	0.856626i	$0.672556\pi$
$920$	−932.210	−0.0334066
$921$	−12722.3	−0.455174
$922$	2105.58	0.0752101
$923$	−1573.65	−0.0561183
$924$	−3723.89	−0.132583
$925$	−3229.22	−0.114785
$926$	1851.43	0.0657039
$927$	−29356.3	−1.04012
$928$	−8708.56	−0.308052
$929$	22140.3	0.781915	0.390958	−	0.920409i	$-0.372144\pi$
$929$	22140.3	0.781915	0.390958	+	0.920409i	$0.372144\pi$
$930$	48.4297	0.00170760
$931$	−15888.6	−0.559320
$932$	−20291.7	−0.713172
$933$	10937.2	0.383781
$934$	485.083	0.0169940
$935$	−3990.75	−0.139584
$936$	−7080.47	−0.247257
$937$	12102.2	0.421944	0.210972	−	0.977492i	$-0.432337\pi$
$937$	12102.2	0.421944	0.210972	+	0.977492i	$0.432337\pi$
$938$	−6721.32	−0.233965
$939$	11508.2	0.399953
$940$	−17481.4	−0.606576
$941$	19192.2	0.664877	0.332438	−	0.943125i	$-0.392129\pi$
$941$	19192.2	0.664877	0.332438	+	0.943125i	$0.392129\pi$
$942$	−1210.31	−0.0418621
$943$	11823.4	0.408297
$944$	−21822.3	−0.752388
$945$	11205.8	0.385741
$946$	241.690	0.00830658
$947$	45996.3	1.57833	0.789165	−	0.614181i	$-0.210513\pi$
$947$	45996.3	1.57833	0.789165	+	0.614181i	$0.210513\pi$
$948$	−7573.14	−0.259456
$949$	−13207.3	−0.451767
$950$	131.794	0.00450100
$951$	6899.44	0.235257
$952$	−11008.3	−0.374769
$953$	110.181	0.00374512	0.00187256	−	0.999998i	$-0.499404\pi$
$953$	110.181	0.00374512	0.00187256	+	0.999998i	$0.499404\pi$
$954$	−1222.50	−0.0414882
$955$	−2774.81	−0.0940216
$956$	−7320.71	−0.247666
$957$	−2266.44	−0.0765555
$958$	2003.33	0.0675624
$959$	83197.5	2.80145
$960$	−3002.87	−0.100956
$961$	−29003.8	−0.973578
$962$	2256.71	0.0756335
$963$	−46967.5	−1.57166
$964$	56932.1	1.90214
$965$	23302.8	0.777351
$966$	−500.302	−0.0166635
$967$	−29817.6	−0.991592	−0.495796	−	0.868439i	$-0.665124\pi$
$967$	−29817.6	−0.991592	−0.495796	+	0.868439i	$0.665124\pi$
$968$	534.578	0.0177500
$969$	1715.37	0.0568686
$970$	1144.13	0.0378721
$971$	45095.7	1.49041	0.745205	−	0.666835i	$-0.232351\pi$
$971$	45095.7	1.49041	0.745205	+	0.666835i	$0.232351\pi$
$972$	19935.4	0.657849
$973$	25510.0	0.840507
$974$	−3867.72	−0.127238
$975$	−1958.71	−0.0643374
$976$	−20601.6	−0.675656
$977$	17531.5	0.574085	0.287043	−	0.957918i	$-0.407328\pi$
$977$	17531.5	0.574085	0.287043	+	0.957918i	$0.407328\pi$
$978$	−1163.95	−0.0380562
$979$	−3720.46	−0.121457
$980$	33127.7	1.07982
$981$	−8853.14	−0.288134
$982$	1322.50	0.0429764
$983$	35759.4	1.16027	0.580137	−	0.814519i	$-0.302999\pi$
$983$	35759.4	1.16027	0.580137	+	0.814519i	$0.302999\pi$
$984$	−1540.15	−0.0498966
$985$	5211.79	0.168590
$986$	−3333.74	−0.107675
$987$	−18855.2	−0.608071
$988$	9478.99	0.305230
$989$	−3341.83	−0.107446
$990$	−388.402	−0.0124689
$991$	−23513.7	−0.753719	−0.376860	−	0.926270i	$-0.622996\pi$
$991$	−23513.7	−0.753719	−0.376860	+	0.926270i	$0.622996\pi$
$992$	1475.49	0.0472246
$993$	7033.56	0.224777
$994$	−238.117	−0.00759821
$995$	−13120.5	−0.418037
$996$	8934.79	0.284247
$997$	−46340.7	−1.47204	−0.736020	−	0.676959i	$-0.763297\pi$
$997$	−46340.7	−1.47204	−0.736020	+	0.676959i	$0.763297\pi$
$998$	−523.569	−0.0166065
$999$	−8430.07	−0.266983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.f.1.12	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.f.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-0.277460 q^{2} -1.24426 q^{3} -7.92302 q^{4} -5.00000 q^{5} +0.345233 q^{6} -34.3401 q^{7} +4.41800 q^{8} -25.4518 q^{9} +O(q^{10})\)	\(q-0.277460 q^{2} -1.24426 q^{3} -7.92302 q^{4} -5.00000 q^{5} +0.345233 q^{6} -34.3401 q^{7} +4.41800 q^{8} -25.4518 q^{9} +1.38730 q^{10} +11.0000 q^{11} +9.85832 q^{12} +62.9677 q^{13} +9.52799 q^{14} +6.22132 q^{15} +62.1583 q^{16} +72.5591 q^{17} +7.06186 q^{18} -19.0000 q^{19} +39.6151 q^{20} +42.7281 q^{21} -3.05206 q^{22} +42.2005 q^{23} -5.49716 q^{24} +25.0000 q^{25} -17.4710 q^{26} +65.2639 q^{27} +272.077 q^{28} +165.592 q^{29} -1.72617 q^{30} -28.0562 q^{31} -52.5904 q^{32} -13.6869 q^{33} -20.1322 q^{34} +171.700 q^{35} +201.655 q^{36} -129.169 q^{37} +5.27174 q^{38} -78.3484 q^{39} -22.0900 q^{40} +280.172 q^{41} -11.8553 q^{42} -79.1892 q^{43} -87.1532 q^{44} +127.259 q^{45} -11.7090 q^{46} -441.283 q^{47} -77.3413 q^{48} +836.240 q^{49} -6.93650 q^{50} -90.2826 q^{51} -498.894 q^{52} -173.112 q^{53} -18.1081 q^{54} -55.0000 q^{55} -151.714 q^{56} +23.6410 q^{57} -45.9452 q^{58} -351.076 q^{59} -49.2916 q^{60} -331.437 q^{61} +7.78447 q^{62} +874.017 q^{63} -482.675 q^{64} -314.839 q^{65} +3.79757 q^{66} -705.428 q^{67} -574.887 q^{68} -52.5086 q^{69} -47.6400 q^{70} -24.9913 q^{71} -112.446 q^{72} -209.747 q^{73} +35.8392 q^{74} -31.1066 q^{75} +150.537 q^{76} -377.741 q^{77} +21.7386 q^{78} -768.197 q^{79} -310.792 q^{80} +605.993 q^{81} -77.7366 q^{82} +906.319 q^{83} -338.535 q^{84} -362.795 q^{85} +21.9718 q^{86} -206.040 q^{87} +48.5980 q^{88} -338.224 q^{89} -35.3093 q^{90} -2162.32 q^{91} -334.356 q^{92} +34.9093 q^{93} +122.438 q^{94} +95.0000 q^{95} +65.4364 q^{96} +824.720 q^{97} -232.023 q^{98} -279.970 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) 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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