LMFDB - Embedded newform 1045.4.a.e.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+2.70597 q^{2} +6.22305 q^{3} -0.677713 q^{4} +5.00000 q^{5} +16.8394 q^{6} -21.7619 q^{7} -23.4817 q^{8} +11.7264 q^{9} +O(q^{10})$	\(q+2.70597 q^{2} +6.22305 q^{3} -0.677713 q^{4} +5.00000 q^{5} +16.8394 q^{6} -21.7619 q^{7} -23.4817 q^{8} +11.7264 q^{9} +13.5299 q^{10} +11.0000 q^{11} -4.21744 q^{12} +67.9069 q^{13} -58.8871 q^{14} +31.1153 q^{15} -58.1190 q^{16} +62.9522 q^{17} +31.7313 q^{18} -19.0000 q^{19} -3.38856 q^{20} -135.425 q^{21} +29.7657 q^{22} +183.264 q^{23} -146.128 q^{24} +25.0000 q^{25} +183.754 q^{26} -95.0484 q^{27} +14.7483 q^{28} +136.111 q^{29} +84.1971 q^{30} +283.272 q^{31} +30.5848 q^{32} +68.4536 q^{33} +170.347 q^{34} -108.809 q^{35} -7.94714 q^{36} -30.2449 q^{37} -51.4135 q^{38} +422.588 q^{39} -117.408 q^{40} +64.7299 q^{41} -366.457 q^{42} -0.464383 q^{43} -7.45484 q^{44} +58.6320 q^{45} +495.907 q^{46} -4.97760 q^{47} -361.678 q^{48} +130.580 q^{49} +67.6493 q^{50} +391.755 q^{51} -46.0214 q^{52} -89.4905 q^{53} -257.198 q^{54} +55.0000 q^{55} +511.005 q^{56} -118.238 q^{57} +368.314 q^{58} -152.566 q^{59} -21.0872 q^{60} +224.225 q^{61} +766.525 q^{62} -255.189 q^{63} +547.714 q^{64} +339.534 q^{65} +185.234 q^{66} +2.06225 q^{67} -42.6635 q^{68} +1140.46 q^{69} -294.435 q^{70} +487.714 q^{71} -275.355 q^{72} +1029.94 q^{73} -81.8417 q^{74} +155.576 q^{75} +12.8765 q^{76} -239.381 q^{77} +1143.51 q^{78} -375.283 q^{79} -290.595 q^{80} -908.104 q^{81} +175.157 q^{82} +392.068 q^{83} +91.7795 q^{84} +314.761 q^{85} -1.25661 q^{86} +847.029 q^{87} -258.298 q^{88} +460.573 q^{89} +158.657 q^{90} -1477.78 q^{91} -124.200 q^{92} +1762.81 q^{93} -13.4692 q^{94} -95.0000 q^{95} +190.331 q^{96} -244.011 q^{97} +353.345 q^{98} +128.990 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) $ 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9} + 60 q^{10} + 242 q^{11} + 164 q^{12} + 207 q^{13} + 129 q^{14} + 105 q^{15} + 604 q^{16} + 209 q^{17} + 869 q^{18} - 418 q^{19} + 480 q^{20} + 45 q^{21} + 132 q^{22} + 191 q^{23} + 458 q^{24} + 550 q^{25} + 363 q^{26} + 45 q^{27} + 577 q^{28} + 389 q^{29} + 135 q^{30} + 198 q^{31} + 1149 q^{32} + 231 q^{33} + 467 q^{34} + 465 q^{35} + 1315 q^{36} + 312 q^{37} - 228 q^{38} + 137 q^{39} + 570 q^{40} + 632 q^{41} - 1794 q^{42} + 1584 q^{43} + 1056 q^{44} + 1045 q^{45} + 681 q^{46} - 54 q^{47} + 2491 q^{48} + 1063 q^{49} + 300 q^{50} + 37 q^{51} + 1246 q^{52} + 343 q^{53} + 1078 q^{54} + 1210 q^{55} - 87 q^{56} - 399 q^{57} - 1424 q^{58} + 2787 q^{59} + 820 q^{60} + 2070 q^{61} - 446 q^{62} + 1696 q^{63} + 1758 q^{64} + 1035 q^{65} + 297 q^{66} + 2423 q^{67} - 524 q^{68} - 997 q^{69} + 645 q^{70} + 2538 q^{71} + 6010 q^{72} + 1397 q^{73} - 1977 q^{74} + 525 q^{75} - 1824 q^{76} + 1023 q^{77} + 202 q^{78} + 878 q^{79} + 3020 q^{80} + 2030 q^{81} - 190 q^{82} + 4932 q^{83} - 4580 q^{84} + 1045 q^{85} - 3394 q^{86} + 6009 q^{87} + 1254 q^{88} + 1812 q^{89} + 4345 q^{90} + 4349 q^{91} - 788 q^{92} - 4848 q^{93} - 2152 q^{94} - 2090 q^{95} + 4032 q^{96} + 988 q^{97} + 1366 q^{98} + 2299 q^{99}+O(q^{100}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9 + 60 * q^10 + 242 * q^11 + 164 * q^12 + 207 * q^13 + 129 * q^14 + 105 * q^15 + 604 * q^16 + 209 * q^17 + 869 * q^18 - 418 * q^19 + 480 * q^20 + 45 * q^21 + 132 * q^22 + 191 * q^23 + 458 * q^24 + 550 * q^25 + 363 * q^26 + 45 * q^27 + 577 * q^28 + 389 * q^29 + 135 * q^30 + 198 * q^31 + 1149 * q^32 + 231 * q^33 + 467 * q^34 + 465 * q^35 + 1315 * q^36 + 312 * q^37 - 228 * q^38 + 137 * q^39 + 570 * q^40 + 632 * q^41 - 1794 * q^42 + 1584 * q^43 + 1056 * q^44 + 1045 * q^45 + 681 * q^46 - 54 * q^47 + 2491 * q^48 + 1063 * q^49 + 300 * q^50 + 37 * q^51 + 1246 * q^52 + 343 * q^53 + 1078 * q^54 + 1210 * q^55 - 87 * q^56 - 399 * q^57 - 1424 * q^58 + 2787 * q^59 + 820 * q^60 + 2070 * q^61 - 446 * q^62 + 1696 * q^63 + 1758 * q^64 + 1035 * q^65 + 297 * q^66 + 2423 * q^67 - 524 * q^68 - 997 * q^69 + 645 * q^70 + 2538 * q^71 + 6010 * q^72 + 1397 * q^73 - 1977 * q^74 + 525 * q^75 - 1824 * q^76 + 1023 * q^77 + 202 * q^78 + 878 * q^79 + 3020 * q^80 + 2030 * q^81 - 190 * q^82 + 4932 * q^83 - 4580 * q^84 + 1045 * q^85 - 3394 * q^86 + 6009 * q^87 + 1254 * q^88 + 1812 * q^89 + 4345 * q^90 + 4349 * q^91 - 788 * q^92 - 4848 * q^93 - 2152 * q^94 - 2090 * q^95 + 4032 * q^96 + 988 * q^97 + 1366 * q^98 + 2299 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.70597	0.956706	0.478353	−	0.878168i	$-0.341234\pi$
$2$	2.70597	0.956706	0.478353	+	0.878168i	$0.341234\pi$
$3$	6.22305	1.19763	0.598814	−	0.800888i	$-0.295639\pi$
$3$	6.22305	1.19763	0.598814	+	0.800888i	$0.295639\pi$
$4$	−0.677713	−0.0847141
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	16.8394	1.14578
$7$	−21.7619	−1.17503	−0.587516	−	0.809213i	$-0.699894\pi$
$7$	−21.7619	−1.17503	−0.587516	+	0.809213i	$0.699894\pi$
$8$	−23.4817	−1.03775
$9$	11.7264	0.434311
$10$	13.5299	0.427852
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−4.21744	−0.101456
$13$	67.9069	1.44877	0.724384	−	0.689397i	$-0.242124\pi$
$13$	67.9069	1.44877	0.724384	+	0.689397i	$0.242124\pi$
$14$	−58.8871	−1.12416
$15$	31.1153	0.535595
$16$	−58.1190	−0.908109
$17$	62.9522	0.898127	0.449063	−	0.893500i	$-0.351758\pi$
$17$	62.9522	0.898127	0.449063	+	0.893500i	$0.351758\pi$
$18$	31.7313	0.415508
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−3.38856	−0.0378853
$21$	−135.425	−1.40725
$22$	29.7657	0.288458
$23$	183.264	1.66144	0.830720	−	0.556690i	$-0.187929\pi$
$23$	183.264	1.66144	0.830720	+	0.556690i	$0.187929\pi$
$24$	−146.128	−1.24284
$25$	25.0000	0.200000
$26$	183.754	1.38604
$27$	−95.0484	−0.677484
$28$	14.7483	0.0995417
$29$	136.111	0.871561	0.435780	−	0.900053i	$-0.356472\pi$
$29$	136.111	0.871561	0.435780	+	0.900053i	$0.356472\pi$
$30$	84.1971	0.512407
$31$	283.272	1.64120	0.820598	−	0.571506i	$-0.193640\pi$
$31$	283.272	1.64120	0.820598	+	0.571506i	$0.193640\pi$
$32$	30.5848	0.168959
$33$	68.4536	0.361098
$34$	170.347	0.859243
$35$	−108.809	−0.525490
$36$	−7.94714	−0.0367923
$37$	−30.2449	−0.134384	−0.0671922	−	0.997740i	$-0.521404\pi$
$37$	−30.2449	−0.134384	−0.0671922	+	0.997740i	$0.521404\pi$
$38$	−51.4135	−0.219483
$39$	422.588	1.73508
$40$	−117.408	−0.464097
$41$	64.7299	0.246564	0.123282	−	0.992372i	$-0.460658\pi$
$41$	64.7299	0.246564	0.123282	+	0.992372i	$0.460658\pi$
$42$	−366.457	−1.34632
$43$	−0.464383	−0.00164692	−0.000823462	−	1.00000i	$-0.500262\pi$
$43$	−0.464383	−0.00164692	−0.000823462		1.00000i	$0.500262\pi$
$44$	−7.45484	−0.0255423
$45$	58.6320	0.194230
$46$	495.907	1.58951
$47$	−4.97760	−0.0154480	−0.00772401	−	0.999970i	$-0.502459\pi$
$47$	−4.97760	−0.0154480	−0.00772401	+	0.999970i	$0.502459\pi$
$48$	−361.678	−1.08758
$49$	130.580	0.380698
$50$	67.6493	0.191341
$51$	391.755	1.07562
$52$	−46.0214	−0.122731
$53$	−89.4905	−0.231933	−0.115967	−	0.993253i	$-0.536997\pi$
$53$	−89.4905	−0.231933	−0.115967	+	0.993253i	$0.536997\pi$
$54$	−257.198	−0.648153
$55$	55.0000	0.134840
$56$	511.005	1.21939
$57$	−118.238	−0.274755
$58$	368.314	0.833827
$59$	−152.566	−0.336651	−0.168325	−	0.985731i	$-0.553836\pi$
$59$	−152.566	−0.336651	−0.168325	+	0.985731i	$0.553836\pi$
$60$	−21.0872	−0.0453725
$61$	224.225	0.470640	0.235320	−	0.971918i	$-0.424386\pi$
$61$	224.225	0.470640	0.235320	+	0.971918i	$0.424386\pi$
$62$	766.525	1.57014
$63$	−255.189	−0.510329
$64$	547.714	1.06975
$65$	339.534	0.647908
$66$	185.234	0.345465
$67$	2.06225	0.00376036	0.00188018	−	0.999998i	$-0.499402\pi$
$67$	2.06225	0.00376036	0.00188018	+	0.999998i	$0.499402\pi$
$68$	−42.6635	−0.0760840
$69$	1140.46	1.98979
$70$	−294.435	−0.502739
$71$	487.714	0.815226	0.407613	−	0.913155i	$-0.366361\pi$
$71$	487.714	0.815226	0.407613	+	0.913155i	$0.366361\pi$
$72$	−275.355	−0.450708
$73$	1029.94	1.65130	0.825651	−	0.564182i	$-0.190808\pi$
$73$	1029.94	1.65130	0.825651	+	0.564182i	$0.190808\pi$
$74$	−81.8417	−0.128566
$75$	155.576	0.239525
$76$	12.8765	0.0194347
$77$	−239.381	−0.354285
$78$	1143.51	1.65996
$79$	−375.283	−0.534463	−0.267232	−	0.963632i	$-0.586109\pi$
$79$	−375.283	−0.534463	−0.267232	+	0.963632i	$0.586109\pi$
$80$	−290.595	−0.406119
$81$	−908.104	−1.24569
$82$	175.157	0.235889
$83$	392.068	0.518494	0.259247	−	0.965811i	$-0.416526\pi$
$83$	392.068	0.518494	0.259247	+	0.965811i	$0.416526\pi$
$84$	91.7795	0.119214
$85$	314.761	0.401655
$86$	−1.25661	−0.00157562
$87$	847.029	1.04381
$88$	−258.298	−0.312894
$89$	460.573	0.548546	0.274273	−	0.961652i	$-0.411563\pi$
$89$	460.573	0.548546	0.274273	+	0.961652i	$0.411563\pi$
$90$	158.657	0.185821
$91$	−1477.78	−1.70235
$92$	−124.200	−0.140747
$93$	1762.81	1.96554
$94$	−13.4692	−0.0147792
$95$	−95.0000	−0.102598
$96$	190.331	0.202350
$97$	−244.011	−0.255418	−0.127709	−	0.991812i	$-0.540762\pi$
$97$	−244.011	−0.255418	−0.127709	+	0.991812i	$0.540762\pi$
$98$	353.345	0.364216
$99$	128.990	0.130950
$100$	−16.9428	−0.0169428
$101$	−1197.91	−1.18017	−0.590083	−	0.807343i	$-0.700905\pi$
$101$	−1197.91	−1.18017	−0.590083	+	0.807343i	$0.700905\pi$
$102$	1060.08	1.02905
$103$	259.111	0.247873	0.123937	−	0.992290i	$-0.460448\pi$
$103$	259.111	0.247873	0.123937	+	0.992290i	$0.460448\pi$
$104$	−1594.57	−1.50346
$105$	−677.127	−0.629341
$106$	−242.159	−0.221892
$107$	−309.073	−0.279245	−0.139623	−	0.990205i	$-0.544589\pi$
$107$	−309.073	−0.279245	−0.139623	+	0.990205i	$0.544589\pi$
$108$	64.4155	0.0573925
$109$	70.1973	0.0616851	0.0308426	−	0.999524i	$-0.490181\pi$
$109$	70.1973	0.0616851	0.0308426	+	0.999524i	$0.490181\pi$
$110$	148.828	0.129002
$111$	−188.215	−0.160942
$112$	1264.78	1.06706
$113$	−1152.61	−0.959545	−0.479772	−	0.877393i	$-0.659281\pi$
$113$	−1152.61	−0.959545	−0.479772	+	0.877393i	$0.659281\pi$
$114$	−319.949	−0.262859
$115$	916.319	0.743019
$116$	−92.2445	−0.0738335
$117$	796.304	0.629216
$118$	−412.839	−0.322076
$119$	−1369.96	−1.05533
$120$	−730.638	−0.555815
$121$	121.000	0.0909091
$122$	606.746	0.450264
$123$	402.818	0.295291
$124$	−191.977	−0.139032
$125$	125.000	0.0894427
$126$	−690.534	−0.488235
$127$	−653.725	−0.456762	−0.228381	−	0.973572i	$-0.573343\pi$
$127$	−653.725	−0.456762	−0.228381	+	0.973572i	$0.573343\pi$
$128$	1237.42	0.854480
$129$	−2.88988	−0.00197240
$130$	918.770	0.619858
$131$	1745.17	1.16394	0.581971	−	0.813209i	$-0.302282\pi$
$131$	1745.17	1.16394	0.581971	+	0.813209i	$0.302282\pi$
$132$	−46.3919	−0.0305901
$133$	413.476	0.269571
$134$	5.58039	0.00359756
$135$	−475.242	−0.302980
$136$	−1478.22	−0.932033
$137$	−1601.93	−0.998995	−0.499498	−	0.866315i	$-0.666482\pi$
$137$	−1601.93	−0.998995	−0.499498	+	0.866315i	$0.666482\pi$
$138$	3086.05	1.90364
$139$	−2148.42	−1.31098	−0.655491	−	0.755203i	$-0.727538\pi$
$139$	−2148.42	−1.31098	−0.655491	+	0.755203i	$0.727538\pi$
$140$	73.7415	0.0445164
$141$	−30.9759	−0.0185010
$142$	1319.74	0.779931
$143$	746.975	0.436820
$144$	−681.527	−0.394402
$145$	680.557	0.389774
$146$	2786.98	1.57981
$147$	812.603	0.455935
$148$	20.4973	0.0113843
$149$	−1701.64	−0.935598	−0.467799	−	0.883835i	$-0.654953\pi$
$149$	−1701.64	−0.935598	−0.467799	+	0.883835i	$0.654953\pi$
$150$	420.985	0.229155
$151$	−1034.41	−0.557476	−0.278738	−	0.960367i	$-0.589916\pi$
$151$	−1034.41	−0.557476	−0.278738	+	0.960367i	$0.589916\pi$
$152$	446.151	0.238077
$153$	738.203	0.390067
$154$	−647.758	−0.338947
$155$	1416.36	0.733965
$156$	−286.393	−0.146986
$157$	−831.222	−0.422540	−0.211270	−	0.977428i	$-0.567760\pi$
$157$	−831.222	−0.422540	−0.211270	+	0.977428i	$0.567760\pi$
$158$	−1015.50	−0.511324
$159$	−556.904	−0.277770
$160$	152.924	0.0755606
$161$	−3988.16	−1.95224
$162$	−2457.31	−1.19175
$163$	−3449.20	−1.65744	−0.828718	−	0.559666i	$-0.810929\pi$
$163$	−3449.20	−1.65744	−0.828718	+	0.559666i	$0.810929\pi$
$164$	−43.8683	−0.0208874
$165$	342.268	0.161488
$166$	1060.92	0.496046
$167$	1994.41	0.924143	0.462071	−	0.886843i	$-0.347106\pi$
$167$	1994.41	0.924143	0.462071	+	0.886843i	$0.347106\pi$
$168$	3180.01	1.46038
$169$	2414.34	1.09893
$170$	851.735	0.384265
$171$	−222.802	−0.0996379
$172$	0.314718	0.000139518 0
$173$	1595.69	0.701260	0.350630	−	0.936514i	$-0.385968\pi$
$173$	1595.69	0.701260	0.350630	+	0.936514i	$0.385968\pi$
$174$	2292.04	0.998614
$175$	−544.047	−0.235006
$176$	−639.309	−0.273805
$177$	−949.426	−0.403182
$178$	1246.30	0.524797
$179$	4507.10	1.88199	0.940996	−	0.338417i	$-0.109891\pi$
$179$	4507.10	1.88199	0.940996	+	0.338417i	$0.109891\pi$
$180$	−39.7357	−0.0164540
$181$	−4035.42	−1.65719	−0.828593	−	0.559851i	$-0.810858\pi$
$181$	−4035.42	−1.65719	−0.828593	+	0.559851i	$0.810858\pi$
$182$	−3998.83	−1.62864
$183$	1395.36	0.563651
$184$	−4303.33	−1.72416
$185$	−151.224	−0.0600985
$186$	4770.13	1.88045
$187$	692.475	0.270795
$188$	3.37338	0.00130867
$189$	2068.43	0.796065
$190$	−257.067	−0.0981559
$191$	619.053	0.234519	0.117259	−	0.993101i	$-0.462589\pi$
$191$	619.053	0.234519	0.117259	+	0.993101i	$0.462589\pi$
$192$	3408.45	1.28117
$193$	271.998	0.101445	0.0507225	−	0.998713i	$-0.483848\pi$
$193$	271.998	0.101445	0.0507225	+	0.998713i	$0.483848\pi$
$194$	−660.288	−0.244360
$195$	2112.94	0.775953
$196$	−88.4954	−0.0322505
$197$	−175.726	−0.0635531	−0.0317766	−	0.999495i	$-0.510116\pi$
$197$	−175.726	−0.0635531	−0.0317766	+	0.999495i	$0.510116\pi$
$198$	349.045	0.125280
$199$	4347.60	1.54871	0.774354	−	0.632753i	$-0.218075\pi$
$199$	4347.60	1.54871	0.774354	+	0.632753i	$0.218075\pi$
$200$	−587.041	−0.207550
$201$	12.8335	0.00450351
$202$	−3241.52	−1.12907
$203$	−2962.04	−1.02411
$204$	−265.498	−0.0911203
$205$	323.649	0.110267
$206$	701.147	0.237142
$207$	2149.03	0.721582
$208$	−3946.68	−1.31564
$209$	−209.000	−0.0691714
$210$	−1832.29	−0.602094
$211$	1980.10	0.646045	0.323023	−	0.946391i	$-0.395301\pi$
$211$	1980.10	0.646045	0.323023	+	0.946391i	$0.395301\pi$
$212$	60.6488	0.0196480
$213$	3035.07	0.976337
$214$	−836.344	−0.267156
$215$	−2.32191	−0.000736526 0
$216$	2231.89	0.703061
$217$	−6164.52	−1.92846
$218$	189.952	0.0590145
$219$	6409.35	1.97764
$220$	−37.2742	−0.0114228
$221$	4274.89	1.30118
$222$	−509.306	−0.153975
$223$	4439.00	1.33299	0.666496	−	0.745509i	$-0.267793\pi$
$223$	4439.00	1.33299	0.666496	+	0.745509i	$0.267793\pi$
$224$	−665.583	−0.198532
$225$	293.160	0.0868623
$226$	−3118.93	−0.918002
$227$	−3377.94	−0.987672	−0.493836	−	0.869555i	$-0.664406\pi$
$227$	−3377.94	−0.987672	−0.493836	+	0.869555i	$0.664406\pi$
$228$	80.1314	0.0232756
$229$	2212.61	0.638485	0.319243	−	0.947673i	$-0.396572\pi$
$229$	2212.61	0.638485	0.319243	+	0.947673i	$0.396572\pi$
$230$	2479.53	0.710850
$231$	−1489.68	−0.424302
$232$	−3196.12	−0.904464
$233$	−2235.91	−0.628666	−0.314333	−	0.949313i	$-0.601781\pi$
$233$	−2235.91	−0.628666	−0.314333	+	0.949313i	$0.601781\pi$
$234$	2154.78	0.601975
$235$	−24.8880	−0.00690857
$236$	103.396	0.0285191
$237$	−2335.40	−0.640088
$238$	−3707.07	−1.00964
$239$	−2355.16	−0.637416	−0.318708	−	0.947853i	$-0.603249\pi$
$239$	−2355.16	−0.637416	−0.318708	+	0.947853i	$0.603249\pi$
$240$	−1808.39	−0.486379
$241$	3578.16	0.956388	0.478194	−	0.878254i	$-0.341291\pi$
$241$	3578.16	0.956388	0.478194	+	0.878254i	$0.341291\pi$
$242$	327.423	0.0869733
$243$	−3084.88	−0.814382
$244$	−151.960	−0.0398698
$245$	652.898	0.170253
$246$	1090.01	0.282507
$247$	−1290.23	−0.332370
$248$	−6651.69	−1.70316
$249$	2439.86	0.620963
$250$	338.247	0.0855704
$251$	4870.10	1.22469	0.612346	−	0.790590i	$-0.290226\pi$
$251$	4870.10	1.22469	0.612346	+	0.790590i	$0.290226\pi$
$252$	172.945	0.0432321
$253$	2015.90	0.500943
$254$	−1768.96	−0.436987
$255$	1958.78	0.481033
$256$	−1033.29	−0.252267
$257$	−7577.01	−1.83907	−0.919535	−	0.393008i	$-0.871434\pi$
$257$	−7577.01	−1.83907	−0.919535	+	0.393008i	$0.871434\pi$
$258$	−7.81993	−0.00188701
$259$	658.185	0.157906
$260$	−230.107	−0.0548870
$261$	1596.10	0.378529
$262$	4722.39	1.11355
$263$	671.217	0.157373	0.0786863	−	0.996899i	$-0.474927\pi$
$263$	671.217	0.157373	0.0786863	+	0.996899i	$0.474927\pi$
$264$	−1607.40	−0.374731
$265$	−447.452	−0.103724
$266$	1118.85	0.257900
$267$	2866.17	0.656954
$268$	−1.39761	−0.000318555 0
$269$	−6670.97	−1.51203	−0.756016	−	0.654553i	$-0.772857\pi$
$269$	−6670.97	−1.51203	−0.756016	+	0.654553i	$0.772857\pi$
$270$	−1285.99	−0.289863
$271$	2626.22	0.588678	0.294339	−	0.955701i	$-0.404901\pi$
$271$	2626.22	0.588678	0.294339	+	0.955701i	$0.404901\pi$
$272$	−3658.72	−0.815598
$273$	−9196.31	−2.03878
$274$	−4334.79	−0.955745
$275$	275.000	0.0603023
$276$	−772.904	−0.168563
$277$	8176.21	1.77351	0.886753	−	0.462245i	$-0.152956\pi$
$277$	8176.21	1.77351	0.886753	+	0.462245i	$0.152956\pi$
$278$	−5813.56	−1.25422
$279$	3321.76	0.712790
$280$	2555.02	0.545328
$281$	−5795.44	−1.23034	−0.615172	−	0.788393i	$-0.710913\pi$
$281$	−5795.44	−1.23034	−0.615172	+	0.788393i	$0.710913\pi$
$282$	−83.8198	−0.0177000
$283$	−2926.17	−0.614640	−0.307320	−	0.951606i	$-0.599432\pi$
$283$	−2926.17	−0.614640	−0.307320	+	0.951606i	$0.599432\pi$
$284$	−330.530	−0.0690611
$285$	−591.190	−0.122874
$286$	2021.30	0.417908
$287$	−1408.64	−0.289720
$288$	358.650	0.0733807
$289$	−950.017	−0.193368
$290$	1841.57	0.372899
$291$	−1518.50	−0.305896
$292$	−698.001	−0.139889
$293$	4885.06	0.974021	0.487011	−	0.873396i	$-0.338087\pi$
$293$	4885.06	0.974021	0.487011	+	0.873396i	$0.338087\pi$
$294$	2198.88	0.436195
$295$	−762.829	−0.150555
$296$	710.199	0.139458
$297$	−1045.53	−0.204269
$298$	−4604.60	−0.895092
$299$	12444.9	2.40704
$300$	−105.436	−0.0202912
$301$	10.1058	0.00193519
$302$	−2799.08	−0.533341
$303$	−7454.67	−1.41340
$304$	1104.26	0.208335
$305$	1121.12	0.210476
$306$	1997.56	0.373179
$307$	−1691.93	−0.314539	−0.157270	−	0.987556i	$-0.550269\pi$
$307$	−1691.93	−0.314539	−0.157270	+	0.987556i	$0.550269\pi$
$308$	162.231	0.0300130
$309$	1612.46	0.296860
$310$	3832.63	0.702189
$311$	−916.766	−0.167154	−0.0835772	−	0.996501i	$-0.526635\pi$
$311$	−916.766	−0.167154	−0.0835772	+	0.996501i	$0.526635\pi$
$312$	−9923.07	−1.80059
$313$	−3317.96	−0.599177	−0.299588	−	0.954069i	$-0.596849\pi$
$313$	−3317.96	−0.599177	−0.299588	+	0.954069i	$0.596849\pi$
$314$	−2249.26	−0.404246
$315$	−1275.94	−0.228226
$316$	254.334	0.0452766
$317$	2757.47	0.488565	0.244283	−	0.969704i	$-0.421448\pi$
$317$	2757.47	0.488565	0.244283	+	0.969704i	$0.421448\pi$
$318$	−1506.97	−0.265744
$319$	1497.23	0.262785
$320$	2738.57	0.478408
$321$	−1923.38	−0.334432
$322$	−10791.9	−1.86772
$323$	−1196.09	−0.206044
$324$	615.434	0.105527
$325$	1697.67	0.289753
$326$	−9333.44	−1.58568
$327$	436.841	0.0738758
$328$	−1519.96	−0.255872
$329$	108.322	0.0181519
$330$	926.168	0.154497
$331$	7317.38	1.21510	0.607552	−	0.794280i	$-0.292152\pi$
$331$	7317.38	1.21510	0.607552	+	0.794280i	$0.292152\pi$
$332$	−265.709	−0.0439238
$333$	−354.663	−0.0583647
$334$	5396.81	0.884133
$335$	10.3113	0.00168168
$336$	7870.79	1.27794
$337$	10507.6	1.69847	0.849233	−	0.528018i	$-0.177065\pi$
$337$	10507.6	1.69847	0.849233	+	0.528018i	$0.177065\pi$
$338$	6533.14	1.05135
$339$	−7172.76	−1.14918
$340$	−213.318	−0.0340258
$341$	3115.99	0.494839
$342$	−602.895	−0.0953241
$343$	4622.67	0.727699
$344$	10.9045	0.00170910
$345$	5702.30	0.889860
$346$	4317.89	0.670899
$347$	7038.36	1.08887	0.544436	−	0.838802i	$-0.316744\pi$
$347$	7038.36	1.08887	0.544436	+	0.838802i	$0.316744\pi$
$348$	−574.042	−0.0884250
$349$	3377.25	0.517994	0.258997	−	0.965878i	$-0.416608\pi$
$349$	3377.25	0.517994	0.258997	+	0.965878i	$0.416608\pi$
$350$	−1472.18	−0.224832
$351$	−6454.44	−0.981517
$352$	336.433	0.0509430
$353$	−9080.02	−1.36907	−0.684534	−	0.728981i	$-0.739994\pi$
$353$	−9080.02	−1.36907	−0.684534	+	0.728981i	$0.739994\pi$
$354$	−2569.12	−0.385727
$355$	2438.57	0.364580
$356$	−312.136	−0.0464696
$357$	−8525.33	−1.26389
$358$	12196.1	1.80051
$359$	−985.855	−0.144934	−0.0724672	−	0.997371i	$-0.523087\pi$
$359$	−985.855	−0.144934	−0.0724672	+	0.997371i	$0.523087\pi$
$360$	−1376.78	−0.201563
$361$	361.000	0.0526316
$362$	−10919.7	−1.58544
$363$	752.990	0.108875
$364$	1001.51	0.144213
$365$	5149.68	0.738484
$366$	3775.81	0.539248
$367$	−7760.48	−1.10380	−0.551899	−	0.833911i	$-0.686097\pi$
$367$	−7760.48	−1.10380	−0.551899	+	0.833911i	$0.686097\pi$
$368$	−10651.1	−1.50877
$369$	759.049	0.107085
$370$	−409.209	−0.0574966
$371$	1947.48	0.272529
$372$	−1194.68	−0.166509
$373$	7880.25	1.09390	0.546949	−	0.837166i	$-0.315789\pi$
$373$	7880.25	1.09390	0.546949	+	0.837166i	$0.315789\pi$
$374$	1873.82	0.259072
$375$	777.882	0.107119
$376$	116.882	0.0160312
$377$	9242.90	1.26269
$378$	5597.12	0.761600
$379$	10978.7	1.48796	0.743981	−	0.668201i	$-0.232935\pi$
$379$	10978.7	1.48796	0.743981	+	0.668201i	$0.232935\pi$
$380$	64.3827	0.00869148
$381$	−4068.17	−0.547031
$382$	1675.14	0.224365
$383$	−14643.5	−1.95365	−0.976827	−	0.214028i	$-0.931342\pi$
$383$	−14643.5	−1.95365	−0.976827	+	0.214028i	$0.931342\pi$
$384$	7700.53	1.02335
$385$	−1196.90	−0.158441
$386$	736.020	0.0970530
$387$	−5.44554	−0.000715277 0
$388$	165.370	0.0216375
$389$	3938.22	0.513305	0.256652	−	0.966504i	$-0.417380\pi$
$389$	3938.22	0.513305	0.256652	+	0.966504i	$0.417380\pi$
$390$	5717.56	0.742359
$391$	11536.9	1.49218
$392$	−3066.22	−0.395070
$393$	10860.3	1.39397
$394$	−475.510	−0.0608017
$395$	−1876.41	−0.239019
$396$	−87.4185	−0.0110933
$397$	−4838.90	−0.611731	−0.305866	−	0.952075i	$-0.598946\pi$
$397$	−4838.90	−0.611731	−0.305866	+	0.952075i	$0.598946\pi$
$398$	11764.5	1.48166
$399$	2573.08	0.322845
$400$	−1452.98	−0.181622
$401$	−12300.0	−1.53175	−0.765874	−	0.642990i	$-0.777694\pi$
$401$	−12300.0	−1.53175	−0.765874	+	0.642990i	$0.777694\pi$
$402$	34.7271	0.00430853
$403$	19236.1	2.37771
$404$	811.840	0.0999766
$405$	−4540.52	−0.557087
$406$	−8015.20	−0.979773
$407$	−332.693	−0.0405184
$408$	−9199.06	−1.11623
$409$	3105.84	0.375487	0.187743	−	0.982218i	$-0.439883\pi$
$409$	3105.84	0.375487	0.187743	+	0.982218i	$0.439883\pi$
$410$	875.786	0.105493
$411$	−9968.92	−1.19642
$412$	−175.603	−0.0209984
$413$	3320.12	0.395575
$414$	5815.20	0.690342
$415$	1960.34	0.231878
$416$	2076.92	0.244782
$417$	−13369.7	−1.57007
$418$	−565.548	−0.0661767
$419$	−9392.27	−1.09509	−0.547545	−	0.836776i	$-0.684437\pi$
$419$	−9392.27	−1.09509	−0.547545	+	0.836776i	$0.684437\pi$
$420$	458.898	0.0533141
$421$	−13567.6	−1.57065	−0.785325	−	0.619083i	$-0.787504\pi$
$421$	−13567.6	−1.57065	−0.785325	+	0.619083i	$0.787504\pi$
$422$	5358.09	0.618075
$423$	−58.3693	−0.00670925
$424$	2101.38	0.240689
$425$	1573.81	0.179625
$426$	8212.82	0.934067
$427$	−4879.55	−0.553016
$428$	209.463	0.0236560
$429$	4648.47	0.523147
$430$	−6.28303	−0.000704639 0
$431$	−3448.54	−0.385407	−0.192704	−	0.981257i	$-0.561726\pi$
$431$	−3448.54	−0.385407	−0.192704	+	0.981257i	$0.561726\pi$
$432$	5524.12	0.615230
$433$	949.374	0.105367	0.0526836	−	0.998611i	$-0.483223\pi$
$433$	949.374	0.105367	0.0526836	+	0.998611i	$0.483223\pi$
$434$	−16681.0	−1.84497
$435$	4235.15	0.466804
$436$	−47.5736	−0.00522560
$437$	−3482.01	−0.381161
$438$	17343.5	1.89202
$439$	16677.1	1.81310	0.906552	−	0.422094i	$-0.138705\pi$
$439$	16677.1	1.81310	0.906552	+	0.422094i	$0.138705\pi$
$440$	−1291.49	−0.139930
$441$	1531.23	0.165342
$442$	11567.7	1.24484
$443$	2070.03	0.222010	0.111005	−	0.993820i	$-0.464593\pi$
$443$	2070.03	0.222010	0.111005	+	0.993820i	$0.464593\pi$
$444$	127.556	0.0136341
$445$	2302.86	0.245317
$446$	12011.8	1.27528
$447$	−10589.4	−1.12050
$448$	−11919.3	−1.25699
$449$	−13330.9	−1.40117	−0.700583	−	0.713571i	$-0.747076\pi$
$449$	−13330.9	−1.40117	−0.700583	+	0.713571i	$0.747076\pi$
$450$	793.283	0.0831016
$451$	712.029	0.0743417
$452$	781.140	0.0812870
$453$	−6437.18	−0.667649
$454$	−9140.61	−0.944912
$455$	−7388.91	−0.761313
$456$	2776.42	0.285127
$457$	−15493.1	−1.58586	−0.792928	−	0.609316i	$-0.791444\pi$
$457$	−15493.1	−1.58586	−0.792928	+	0.609316i	$0.791444\pi$
$458$	5987.25	0.610842
$459$	−5983.51	−0.608467
$460$	−621.001	−0.0629442
$461$	−687.024	−0.0694098	−0.0347049	−	0.999398i	$-0.511049\pi$
$461$	−687.024	−0.0694098	−0.0347049	+	0.999398i	$0.511049\pi$
$462$	−4031.03	−0.405932
$463$	−9857.16	−0.989419	−0.494710	−	0.869058i	$-0.664726\pi$
$463$	−9857.16	−0.989419	−0.494710	+	0.869058i	$0.664726\pi$
$464$	−7910.66	−0.791473
$465$	8814.07	0.879017
$466$	−6050.30	−0.601448
$467$	−2526.27	−0.250325	−0.125163	−	0.992136i	$-0.539945\pi$
$467$	−2526.27	−0.250325	−0.125163	+	0.992136i	$0.539945\pi$
$468$	−539.665	−0.0533035
$469$	−44.8785	−0.00441854
$470$	−67.3462	−0.00660946
$471$	−5172.74	−0.506045
$472$	3582.50	0.349360
$473$	−5.10821	−0.000496566 0
$474$	−6319.54	−0.612376
$475$	−475.000	−0.0458831
$476$	928.439	0.0894011
$477$	−1049.40	−0.100731
$478$	−6372.99	−0.609819
$479$	1113.19	0.106186	0.0530929	−	0.998590i	$-0.483092\pi$
$479$	1113.19	0.106186	0.0530929	+	0.998590i	$0.483092\pi$
$480$	951.654	0.0904935
$481$	−2053.83	−0.194692
$482$	9682.41	0.914982
$483$	−24818.6	−2.33806
$484$	−82.0033	−0.00770128
$485$	−1220.06	−0.114227
$486$	−8347.59	−0.779124
$487$	12488.7	1.16204	0.581022	−	0.813888i	$-0.302653\pi$
$487$	12488.7	1.16204	0.581022	+	0.813888i	$0.302653\pi$
$488$	−5265.16	−0.488407
$489$	−21464.5	−1.98499
$490$	1766.72	0.162882
$491$	−685.256	−0.0629841	−0.0314920	−	0.999504i	$-0.510026\pi$
$491$	−685.256	−0.0629841	−0.0314920	+	0.999504i	$0.510026\pi$
$492$	−272.995	−0.0250153
$493$	8568.52	0.782772
$494$	−3491.33	−0.317980
$495$	644.952	0.0585625
$496$	−16463.5	−1.49039
$497$	−10613.6	−0.957916
$498$	6602.19	0.594079
$499$	15992.6	1.43472	0.717361	−	0.696702i	$-0.245350\pi$
$499$	15992.6	1.43472	0.717361	+	0.696702i	$0.245350\pi$
$500$	−84.7141	−0.00757706
$501$	12411.3	1.10678
$502$	13178.3	1.17167
$503$	1020.46	0.0904575	0.0452288	−	0.998977i	$-0.485598\pi$
$503$	1020.46	0.0904575	0.0452288	+	0.998977i	$0.485598\pi$
$504$	5992.25	0.529595
$505$	−5989.56	−0.527786
$506$	5454.97	0.479255
$507$	15024.6	1.31610
$508$	443.038	0.0386942
$509$	−4353.05	−0.379068	−0.189534	−	0.981874i	$-0.560698\pi$
$509$	−4353.05	−0.379068	−0.189534	+	0.981874i	$0.560698\pi$
$510$	5300.39	0.460207
$511$	−22413.4	−1.94033
$512$	−12695.4	−1.09583
$513$	1805.92	0.155426
$514$	−20503.2	−1.75945
$515$	1295.55	0.110852
$516$	1.95851	0.000167090 0
$517$	−54.7536	−0.00465775
$518$	1781.03	0.151069
$519$	9930.06	0.839848
$520$	−7972.83	−0.672368
$521$	−746.579	−0.0627797	−0.0313898	−	0.999507i	$-0.509993\pi$
$521$	−746.579	−0.0627797	−0.0313898	+	0.999507i	$0.509993\pi$
$522$	4319.00	0.362141
$523$	−18639.8	−1.55843	−0.779217	−	0.626754i	$-0.784383\pi$
$523$	−18639.8	−1.55843	−0.779217	+	0.626754i	$0.784383\pi$
$524$	−1182.73	−0.0986023
$525$	−3385.63	−0.281450
$526$	1816.29	0.150559
$527$	17832.6	1.47400
$528$	−3978.45	−0.327917
$529$	21418.6	1.76038
$530$	−1210.79	−0.0992330
$531$	−1789.05	−0.146211
$532$	−280.218	−0.0228364
$533$	4395.60	0.357213
$534$	7755.77	0.628511
$535$	−1545.37	−0.124882
$536$	−48.4251	−0.00390232
$537$	28047.9	2.25393
$538$	−18051.5	−1.44657
$539$	1436.37	0.114785
$540$	322.078	0.0256667
$541$	−16583.7	−1.31791	−0.658955	−	0.752182i	$-0.729001\pi$
$541$	−16583.7	−1.31791	−0.658955	+	0.752182i	$0.729001\pi$
$542$	7106.49	0.563191
$543$	−25112.7	−1.98469
$544$	1925.38	0.151746
$545$	350.986	0.0275864
$546$	−24885.0	−1.95051
$547$	−12820.3	−1.00211	−0.501056	−	0.865415i	$-0.667055\pi$
$547$	−12820.3	−1.00211	−0.501056	+	0.865415i	$0.667055\pi$
$548$	1085.65	0.0846290
$549$	2629.35	0.204404
$550$	744.142	0.0576915
$551$	−2586.12	−0.199950
$552$	−26779.9	−2.06491
$553$	8166.86	0.628011
$554$	22124.6	1.69672
$555$	−941.077	−0.0719756
$556$	1456.01	0.111059
$557$	−1994.26	−0.151705	−0.0758524	−	0.997119i	$-0.524168\pi$
$557$	−1994.26	−0.151705	−0.0758524	+	0.997119i	$0.524168\pi$
$558$	8988.59	0.681931
$559$	−31.5348	−0.00238601
$560$	6323.89	0.477202
$561$	4309.31	0.324312
$562$	−15682.3	−1.17708
$563$	2591.77	0.194015	0.0970073	−	0.995284i	$-0.469073\pi$
$563$	2591.77	0.194015	0.0970073	+	0.995284i	$0.469073\pi$
$564$	20.9927	0.00156729
$565$	−5763.06	−0.429121
$566$	−7918.15	−0.588029
$567$	19762.1	1.46372
$568$	−11452.3	−0.846003
$569$	−5513.35	−0.406207	−0.203103	−	0.979157i	$-0.565103\pi$
$569$	−5513.35	−0.406207	−0.203103	+	0.979157i	$0.565103\pi$
$570$	−1599.74	−0.117554
$571$	−11509.6	−0.843543	−0.421771	−	0.906702i	$-0.638592\pi$
$571$	−11509.6	−0.843543	−0.421771	+	0.906702i	$0.638592\pi$
$572$	−506.235	−0.0370048
$573$	3852.40	0.280866
$574$	−3811.75	−0.277177
$575$	4581.59	0.332288
$576$	6422.71	0.464606
$577$	−20295.5	−1.46432	−0.732160	−	0.681133i	$-0.761488\pi$
$577$	−20295.5	−1.46432	−0.732160	+	0.681133i	$0.761488\pi$
$578$	−2570.72	−0.184996
$579$	1692.66	0.121493
$580$	−461.222	−0.0330193
$581$	−8532.13	−0.609247
$582$	−4109.01	−0.292653
$583$	−984.395	−0.0699305
$584$	−24184.6	−1.71364
$585$	3981.52	0.281394
$586$	13218.8	0.931852
$587$	−23530.5	−1.65453	−0.827265	−	0.561812i	$-0.810105\pi$
$587$	−23530.5	−1.65453	−0.827265	+	0.561812i	$0.810105\pi$
$588$	−550.712	−0.0386241
$589$	−5382.16	−0.376516
$590$	−2064.20	−0.144037
$591$	−1093.55	−0.0761130
$592$	1757.80	0.122036
$593$	−2626.68	−0.181897	−0.0909485	−	0.995856i	$-0.528990\pi$
$593$	−2626.68	−0.181897	−0.0909485	+	0.995856i	$0.528990\pi$
$594$	−2829.18	−0.195425
$595$	−6849.80	−0.471957
$596$	1153.23	0.0792584
$597$	27055.3	1.85478
$598$	33675.5	2.30283
$599$	5722.93	0.390371	0.195186	−	0.980766i	$-0.437469\pi$
$599$	5722.93	0.390371	0.195186	+	0.980766i	$0.437469\pi$
$600$	−3653.19	−0.248568
$601$	−12519.6	−0.849726	−0.424863	−	0.905258i	$-0.639678\pi$
$601$	−12519.6	−0.849726	−0.424863	+	0.905258i	$0.639678\pi$
$602$	27.3461	0.00185140
$603$	24.1828	0.00163317
$604$	701.032	0.0472261
$605$	605.000	0.0406558
$606$	−20172.1	−1.35221
$607$	5468.44	0.365663	0.182831	−	0.983144i	$-0.441474\pi$
$607$	5468.44	0.365663	0.182831	+	0.983144i	$0.441474\pi$
$608$	−581.111	−0.0387618
$609$	−18432.9	−1.22650
$610$	3033.73	0.201364
$611$	−338.013	−0.0223806
$612$	−500.290	−0.0330442
$613$	−14620.1	−0.963293	−0.481646	−	0.876366i	$-0.659961\pi$
$613$	−14620.1	−0.963293	−0.481646	+	0.876366i	$0.659961\pi$
$614$	−4578.32	−0.300922
$615$	2014.09	0.132058
$616$	5621.05	0.367660
$617$	5802.55	0.378609	0.189305	−	0.981918i	$-0.439377\pi$
$617$	5802.55	0.378609	0.189305	+	0.981918i	$0.439377\pi$
$618$	4363.28	0.284008
$619$	5476.12	0.355579	0.177790	−	0.984068i	$-0.443105\pi$
$619$	5476.12	0.355579	0.177790	+	0.984068i	$0.443105\pi$
$620$	−959.884	−0.0621772
$621$	−17418.9	−1.12560
$622$	−2480.74	−0.159918
$623$	−10022.9	−0.644559
$624$	−24560.4	−1.57565
$625$	625.000	0.0400000
$626$	−8978.31	−0.573236
$627$	−1300.62	−0.0828416
$628$	563.330	0.0357951
$629$	−1903.98	−0.120694
$630$	−3452.67	−0.218345
$631$	−12886.6	−0.813004	−0.406502	−	0.913650i	$-0.633252\pi$
$631$	−12886.6	−0.813004	−0.406502	+	0.913650i	$0.633252\pi$
$632$	8812.26	0.554641
$633$	12322.3	0.773721
$634$	7461.65	0.467413
$635$	−3268.63	−0.204270
$636$	377.421	0.0235310
$637$	8867.24	0.551543
$638$	4051.45	0.251408
$639$	5719.14	0.354062
$640$	6187.10	0.382135
$641$	2340.30	0.144206	0.0721032	−	0.997397i	$-0.477029\pi$
$641$	2340.30	0.144206	0.0721032	+	0.997397i	$0.477029\pi$
$642$	−5204.61	−0.319953
$643$	5938.74	0.364232	0.182116	−	0.983277i	$-0.441705\pi$
$643$	5938.74	0.364232	0.182116	+	0.983277i	$0.441705\pi$
$644$	2702.83	0.165383
$645$	−14.4494	−0.000882084 0
$646$	−3236.59	−0.197124
$647$	−9741.76	−0.591945	−0.295972	−	0.955196i	$-0.595644\pi$
$647$	−9741.76	−0.591945	−0.295972	+	0.955196i	$0.595644\pi$
$648$	21323.8	1.29271
$649$	−1678.22	−0.101504
$650$	4593.85	0.277209
$651$	−38362.2	−2.30957
$652$	2337.57	0.140408
$653$	14532.6	0.870914	0.435457	−	0.900210i	$-0.356587\pi$
$653$	14532.6	0.870914	0.435457	+	0.900210i	$0.356587\pi$
$654$	1182.08	0.0706774
$655$	8725.86	0.520531
$656$	−3762.04	−0.223907
$657$	12077.5	0.717179
$658$	293.116	0.0173660
$659$	2485.05	0.146895	0.0734476	−	0.997299i	$-0.476600\pi$
$659$	2485.05	0.146895	0.0734476	+	0.997299i	$0.476600\pi$
$660$	−231.959	−0.0136803
$661$	2645.19	0.155652	0.0778261	−	0.996967i	$-0.475202\pi$
$661$	2645.19	0.155652	0.0778261	+	0.996967i	$0.475202\pi$
$662$	19800.6	1.16250
$663$	26602.9	1.55833
$664$	−9206.39	−0.538068
$665$	2067.38	0.120556
$666$	−959.710	−0.0558378
$667$	24944.3	1.44805
$668$	−1351.64	−0.0782879
$669$	27624.1	1.59643
$670$	27.9020	0.00160888
$671$	2466.47	0.141903
$672$	−4141.96	−0.237767
$673$	24280.7	1.39072	0.695359	−	0.718663i	$-0.255245\pi$
$673$	24280.7	1.39072	0.695359	+	0.718663i	$0.255245\pi$
$674$	28433.2	1.62493
$675$	−2376.21	−0.135497
$676$	−1636.23	−0.0930946
$677$	33279.9	1.88929	0.944644	−	0.328096i	$-0.106407\pi$
$677$	33279.9	1.88929	0.944644	+	0.328096i	$0.106407\pi$
$678$	−19409.3	−1.09942
$679$	5310.14	0.300125
$680$	−7391.11	−0.416818
$681$	−21021.1	−1.18286
$682$	8431.78	0.473416
$683$	32023.8	1.79408	0.897039	−	0.441951i	$-0.145713\pi$
$683$	32023.8	1.79408	0.897039	+	0.441951i	$0.145713\pi$
$684$	150.996	0.00844073
$685$	−8009.67	−0.446764
$686$	12508.8	0.696194
$687$	13769.2	0.764667
$688$	26.9895	0.00149559
$689$	−6077.02	−0.336017
$690$	15430.3	0.851334
$691$	−17202.5	−0.947054	−0.473527	−	0.880779i	$-0.657019\pi$
$691$	−17202.5	−0.947054	−0.473527	+	0.880779i	$0.657019\pi$
$692$	−1081.42	−0.0594066
$693$	−2807.08	−0.153870
$694$	19045.6	1.04173
$695$	−10742.1	−0.586289
$696$	−19889.6	−1.08321
$697$	4074.89	0.221445
$698$	9138.73	0.495568
$699$	−13914.2	−0.752907
$700$	368.708	0.0199083
$701$	35264.9	1.90005	0.950025	−	0.312174i	$-0.101057\pi$
$701$	35264.9	1.90005	0.950025	+	0.312174i	$0.101057\pi$
$702$	−17465.5	−0.939023
$703$	574.652	0.0308299
$704$	6024.85	0.322543
$705$	−154.879	−0.00827389
$706$	−24570.3	−1.30979
$707$	26068.8	1.38673
$708$	643.438	0.0341552
$709$	32228.9	1.70717	0.853583	−	0.520957i	$-0.174425\pi$
$709$	32228.9	1.70717	0.853583	+	0.520957i	$0.174425\pi$
$710$	6598.71	0.348796
$711$	−4400.72	−0.232124
$712$	−10815.0	−0.569255
$713$	51913.4	2.72675
$714$	−23069.3	−1.20917
$715$	3734.88	0.195352
$716$	−3054.52	−0.159431
$717$	−14656.3	−0.763386
$718$	−2667.70	−0.138660
$719$	−23010.7	−1.19354	−0.596770	−	0.802412i	$-0.703549\pi$
$719$	−23010.7	−1.19354	−0.596770	+	0.802412i	$0.703549\pi$
$720$	−3407.64	−0.176382
$721$	−5638.74	−0.291259
$722$	976.856	0.0503529
$723$	22267.1	1.14540
$724$	2734.86	0.140387
$725$	3402.79	0.174312
$726$	2037.57	0.104162
$727$	−15844.8	−0.808325	−0.404162	−	0.914687i	$-0.632437\pi$
$727$	−15844.8	−0.808325	−0.404162	+	0.914687i	$0.632437\pi$
$728$	34700.7	1.76661
$729$	5321.47	0.270358
$730$	13934.9	0.706512
$731$	−29.2339	−0.00147915
$732$	−945.655	−0.0477492
$733$	−10833.4	−0.545896	−0.272948	−	0.962029i	$-0.587999\pi$
$733$	−10833.4	−0.545896	−0.272948	+	0.962029i	$0.587999\pi$
$734$	−20999.6	−1.05601
$735$	4063.02	0.203900
$736$	5605.08	0.280715
$737$	22.6848	0.00113379
$738$	2053.97	0.102449
$739$	34209.6	1.70287	0.851435	−	0.524460i	$-0.175733\pi$
$739$	34209.6	1.70287	0.851435	+	0.524460i	$0.175733\pi$
$740$	102.487	0.00509119
$741$	−8029.17	−0.398055
$742$	5269.83	0.260730
$743$	−32130.4	−1.58647	−0.793237	−	0.608913i	$-0.791606\pi$
$743$	−32130.4	−1.58647	−0.793237	+	0.608913i	$0.791606\pi$
$744$	−41393.8	−2.03975
$745$	−8508.22	−0.418412
$746$	21323.7	1.04654
$747$	4597.54	0.225188
$748$	−469.299	−0.0229402
$749$	6726.02	0.328122
$750$	2104.93	0.102481
$751$	−24940.0	−1.21181	−0.605907	−	0.795536i	$-0.707190\pi$
$751$	−24940.0	−1.21181	−0.605907	+	0.795536i	$0.707190\pi$
$752$	289.293	0.0140285
$753$	30306.9	1.46673
$754$	25011.0	1.20802
$755$	−5172.04	−0.249311
$756$	−1401.80	−0.0674379
$757$	−24710.7	−1.18643	−0.593214	−	0.805045i	$-0.702141\pi$
$757$	−24710.7	−1.18643	−0.593214	+	0.805045i	$0.702141\pi$
$758$	29708.0	1.42354
$759$	12545.1	0.599943
$760$	2230.76	0.106471
$761$	13699.2	0.652556	0.326278	−	0.945274i	$-0.394205\pi$
$761$	13699.2	0.652556	0.326278	+	0.945274i	$0.394205\pi$
$762$	−11008.4	−0.523347
$763$	−1527.62	−0.0724819
$764$	−419.540	−0.0198670
$765$	3691.02	0.174443
$766$	−39625.0	−1.86907
$767$	−10360.3	−0.487728
$768$	−6430.19	−0.302122
$769$	9226.48	0.432660	0.216330	−	0.976320i	$-0.430591\pi$
$769$	9226.48	0.432660	0.216330	+	0.976320i	$0.430591\pi$
$770$	−3238.79	−0.151582
$771$	−47152.2	−2.20252
$772$	−184.337	−0.00859382
$773$	−21385.4	−0.995058	−0.497529	−	0.867447i	$-0.665759\pi$
$773$	−21385.4	−0.995058	−0.497529	+	0.867447i	$0.665759\pi$
$774$	−14.7355	−0.000684310 0
$775$	7081.79	0.328239
$776$	5729.79	0.265061
$777$	4095.92	0.189112
$778$	10656.7	0.491082
$779$	−1229.87	−0.0565656
$780$	−1431.97	−0.0657342
$781$	5364.86	0.245800
$782$	31218.4	1.42758
$783$	−12937.2	−0.590469
$784$	−7589.15	−0.345716
$785$	−4156.11	−0.188965
$786$	29387.7	1.33362
$787$	15966.0	0.723159	0.361579	−	0.932341i	$-0.382238\pi$
$787$	15966.0	0.723159	0.361579	+	0.932341i	$0.382238\pi$
$788$	119.092	0.00538385
$789$	4177.02	0.188474
$790$	−5077.52	−0.228671
$791$	25083.0	1.12750
$792$	−3028.91	−0.135893
$793$	15226.4	0.681847
$794$	−13093.9	−0.585247
$795$	−2784.52	−0.124222
$796$	−2946.42	−0.131197
$797$	−28417.6	−1.26299	−0.631496	−	0.775379i	$-0.717559\pi$
$797$	−28417.6	−1.26299	−0.631496	+	0.775379i	$0.717559\pi$
$798$	6962.69	0.308868
$799$	−313.351	−0.0138743
$800$	764.620	0.0337917
$801$	5400.86	0.238240
$802$	−33283.4	−1.46543
$803$	11329.3	0.497886
$804$	−8.69743	−0.000381511 0
$805$	−19940.8	−0.873070
$806$	52052.3	2.27477
$807$	−41513.8	−1.81085
$808$	28128.9	1.22472
$809$	18091.2	0.786220	0.393110	−	0.919491i	$-0.371399\pi$
$809$	18091.2	0.786220	0.393110	+	0.919491i	$0.371399\pi$
$810$	−12286.5	−0.532969
$811$	−11284.5	−0.488599	−0.244300	−	0.969700i	$-0.578558\pi$
$811$	−11284.5	−0.488599	−0.244300	+	0.969700i	$0.578558\pi$
$812$	2007.41	0.0867567
$813$	16343.1	0.705017
$814$	−900.259	−0.0387642
$815$	−17246.0	−0.741228
$816$	−22768.4	−0.976782
$817$	8.82327	0.000377830 0
$818$	8404.33	0.359230
$819$	−17329.1	−0.739349
$820$	−219.341	−0.00934114
$821$	−9274.62	−0.394259	−0.197129	−	0.980377i	$-0.563162\pi$
$821$	−9274.62	−0.394259	−0.197129	+	0.980377i	$0.563162\pi$
$822$	−26975.6	−1.14463
$823$	37624.5	1.59357	0.796785	−	0.604263i	$-0.206532\pi$
$823$	37624.5	1.59357	0.796785	+	0.604263i	$0.206532\pi$
$824$	−6084.35	−0.257231
$825$	1711.34	0.0722196
$826$	8984.16	0.378449
$827$	−44586.7	−1.87477	−0.937383	−	0.348301i	$-0.886759\pi$
$827$	−44586.7	−1.87477	−0.937383	+	0.348301i	$0.886759\pi$
$828$	−1456.42	−0.0611282
$829$	−9154.99	−0.383554	−0.191777	−	0.981439i	$-0.561425\pi$
$829$	−9154.99	−0.383554	−0.191777	+	0.981439i	$0.561425\pi$
$830$	5304.62	0.221839
$831$	50881.0	2.12400
$832$	37193.5	1.54982
$833$	8220.27	0.341915
$834$	−36178.1	−1.50209
$835$	9972.04	0.413289
$836$	141.642	0.00585980
$837$	−26924.5	−1.11188
$838$	−25415.2	−1.04768
$839$	−14922.8	−0.614056	−0.307028	−	0.951701i	$-0.599334\pi$
$839$	−14922.8	−0.614056	−0.307028	+	0.951701i	$0.599334\pi$
$840$	15900.1	0.653100
$841$	−5862.67	−0.240382
$842$	−36713.5	−1.50265
$843$	−36065.3	−1.47349
$844$	−1341.94	−0.0547291
$845$	12071.7	0.491455
$846$	−157.946	−0.00641878
$847$	−2633.19	−0.106821
$848$	5201.10	0.210621
$849$	−18209.7	−0.736109
$850$	4258.68	0.171849
$851$	−5542.78	−0.223272
$852$	−2056.91	−0.0827095
$853$	39681.9	1.59283	0.796413	−	0.604753i	$-0.206728\pi$
$853$	39681.9	1.59283	0.796413	+	0.604753i	$0.206728\pi$
$854$	−13203.9	−0.529074
$855$	−1114.01	−0.0445594
$856$	7257.55	0.289787
$857$	−8889.53	−0.354330	−0.177165	−	0.984181i	$-0.556693\pi$
$857$	−8889.53	−0.354330	−0.177165	+	0.984181i	$0.556693\pi$
$858$	12578.6	0.500498
$859$	13044.1	0.518114	0.259057	−	0.965862i	$-0.416588\pi$
$859$	13044.1	0.518114	0.259057	+	0.965862i	$0.416588\pi$
$860$	1.57359	6.23942e−5 0
$861$	−8766.07	−0.346977
$862$	−9331.67	−0.368721
$863$	−17105.6	−0.674719	−0.337359	−	0.941376i	$-0.609534\pi$
$863$	−17105.6	−0.674719	−0.337359	+	0.941376i	$0.609534\pi$
$864$	−2907.04	−0.114467
$865$	7978.44	0.313613
$866$	2568.98	0.100805
$867$	−5912.01	−0.231583
$868$	4177.78	0.163368
$869$	−4128.11	−0.161147
$870$	11460.2	0.446594
$871$	140.041	0.00544788
$872$	−1648.35	−0.0640139
$873$	−2861.37	−0.110931
$874$	−9422.23	−0.364658
$875$	−2720.24	−0.105098
$876$	−4343.70	−0.167534
$877$	−30836.3	−1.18731	−0.593654	−	0.804721i	$-0.702315\pi$
$877$	−30836.3	−1.18731	−0.593654	+	0.804721i	$0.702315\pi$
$878$	45127.7	1.73461
$879$	30400.0	1.16651
$880$	−3196.55	−0.122449
$881$	−18795.1	−0.718756	−0.359378	−	0.933192i	$-0.617011\pi$
$881$	−18795.1	−0.718756	−0.359378	+	0.933192i	$0.617011\pi$
$882$	4143.46	0.158183
$883$	−7217.94	−0.275088	−0.137544	−	0.990496i	$-0.543921\pi$
$883$	−7217.94	−0.275088	−0.137544	+	0.990496i	$0.543921\pi$
$884$	−2897.15	−0.110228
$885$	−4747.13	−0.180308
$886$	5601.45	0.212398
$887$	−8876.99	−0.336032	−0.168016	−	0.985784i	$-0.553736\pi$
$887$	−8876.99	−0.336032	−0.168016	+	0.985784i	$0.553736\pi$
$888$	4419.61	0.167018
$889$	14226.3	0.536709
$890$	6231.48	0.234696
$891$	−9989.15	−0.375588
$892$	−3008.36	−0.112923
$893$	94.5743	0.00354402
$894$	−28654.7	−1.07199
$895$	22535.5	0.841653
$896$	−26928.6	−1.00404
$897$	77445.1	2.88274
$898$	−36073.0	−1.34050
$899$	38556.5	1.43040
$900$	−198.678	−0.00735846
$901$	−5633.62	−0.208305
$902$	1926.73	0.0711232
$903$	62.8892	0.00231763
$904$	27065.2	0.995770
$905$	−20177.1	−0.741116
$906$	−17418.8	−0.638744
$907$	22542.9	0.825276	0.412638	−	0.910895i	$-0.364607\pi$
$907$	22542.9	0.825276	0.412638	+	0.910895i	$0.364607\pi$
$908$	2289.27	0.0836698
$909$	−14047.2	−0.512559
$910$	−19994.2	−0.728352
$911$	50310.4	1.82970	0.914851	−	0.403792i	$-0.132308\pi$
$911$	50310.4	1.82970	0.914851	+	0.403792i	$0.132308\pi$
$912$	6871.88	0.249507
$913$	4312.74	0.156332
$914$	−41923.9	−1.51720
$915$	6976.81	0.252072
$916$	−1499.51	−0.0540887
$917$	−37978.2	−1.36767
$918$	−16191.2	−0.582124
$919$	−32573.4	−1.16920	−0.584602	−	0.811320i	$-0.698749\pi$
$919$	−32573.4	−1.16920	−0.584602	+	0.811320i	$0.698749\pi$
$920$	−21516.7	−0.771069
$921$	−10529.0	−0.376701
$922$	−1859.07	−0.0664047
$923$	33119.2	1.18107
$924$	1009.57	0.0359443
$925$	−756.121	−0.0268769
$926$	−26673.2	−0.946583
$927$	3038.44	0.107654
$928$	4162.94	0.147258
$929$	38493.0	1.35943	0.679716	−	0.733475i	$-0.262103\pi$
$929$	38493.0	1.35943	0.679716	+	0.733475i	$0.262103\pi$
$930$	23850.6	0.840961
$931$	−2481.01	−0.0873382
$932$	1515.30	0.0532569
$933$	−5705.08	−0.200189
$934$	−6836.02	−0.239488
$935$	3462.37	0.121103
$936$	−18698.5	−0.652970
$937$	35840.2	1.24957	0.624786	−	0.780796i	$-0.285186\pi$
$937$	35840.2	1.24957	0.624786	+	0.780796i	$0.285186\pi$
$938$	−121.440	−0.00422724
$939$	−20647.9	−0.717590
$940$	16.8669	0.000585253 0
$941$	19442.4	0.673544	0.336772	−	0.941586i	$-0.390665\pi$
$941$	19442.4	0.673544	0.336772	+	0.941586i	$0.390665\pi$
$942$	−13997.3	−0.484136
$943$	11862.6	0.409651
$944$	8866.98	0.305716
$945$	10342.2	0.356011
$946$	−13.8227	−0.000475068 0
$947$	10298.9	0.353399	0.176700	−	0.984265i	$-0.443458\pi$
$947$	10298.9	0.353399	0.176700	+	0.984265i	$0.443458\pi$
$948$	1582.73	0.0542245
$949$	69939.8	2.39235
$950$	−1285.34	−0.0438967
$951$	17159.9	0.585119
$952$	32168.9	1.09517
$953$	−9191.67	−0.312432	−0.156216	−	0.987723i	$-0.549930\pi$
$953$	−9191.67	−0.312432	−0.156216	+	0.987723i	$0.549930\pi$
$954$	−2839.65	−0.0963701
$955$	3095.26	0.104880
$956$	1596.12	0.0539981
$957$	9317.32	0.314719
$958$	3012.27	0.101589
$959$	34861.1	1.17385
$960$	17042.3	0.572955
$961$	50451.8	1.69353
$962$	−5557.62	−0.186263
$963$	−3624.32	−0.121279
$964$	−2424.97	−0.0810196
$965$	1359.99	0.0453676
$966$	−67158.3	−2.23684
$967$	25503.5	0.848127	0.424063	−	0.905633i	$-0.360603\pi$
$967$	25503.5	0.848127	0.424063	+	0.905633i	$0.360603\pi$
$968$	−2841.28	−0.0943411
$969$	−7443.35	−0.246764
$970$	−3301.44	−0.109281
$971$	12757.9	0.421649	0.210824	−	0.977524i	$-0.432385\pi$
$971$	12757.9	0.421649	0.210824	+	0.977524i	$0.432385\pi$
$972$	2090.66	0.0689897
$973$	46753.6	1.54044
$974$	33794.0	1.11173
$975$	10564.7	0.347017
$976$	−13031.7	−0.427392
$977$	13781.3	0.451281	0.225641	−	0.974211i	$-0.427552\pi$
$977$	13781.3	0.451281	0.225641	+	0.974211i	$0.427552\pi$
$978$	−58082.5	−1.89905
$979$	5066.30	0.165393
$980$	−442.477	−0.0144229
$981$	823.162	0.0267905
$982$	−1854.28	−0.0602572
$983$	−10537.7	−0.341913	−0.170957	−	0.985279i	$-0.554686\pi$
$983$	−10537.7	−0.341913	−0.170957	+	0.985279i	$0.554686\pi$
$984$	−9458.82	−0.306439
$985$	−878.631	−0.0284218
$986$	23186.2	0.748883
$987$	674.093	0.0217392
$988$	874.406	0.0281564
$989$	−85.1045	−0.00273626
$990$	1745.22	0.0560271
$991$	−41865.3	−1.34197	−0.670987	−	0.741469i	$-0.734129\pi$
$991$	−41865.3	−1.34197	−0.670987	+	0.741469i	$0.734129\pi$
$992$	8663.81	0.277294
$993$	45536.4	1.45524
$994$	−28720.1	−0.916444
$995$	21738.0	0.692603
$996$	−1653.52	−0.0526043
$997$	827.325	0.0262805	0.0131402	−	0.999914i	$-0.495817\pi$
$997$	827.325	0.0262805	0.0131402	+	0.999914i	$0.495817\pi$
$998$	43275.5	1.37261
$999$	2874.72	0.0910433

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.e.1.15	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.e.1.15	✓	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.70597 q^{2} +6.22305 q^{3} -0.677713 q^{4} +5.00000 q^{5} +16.8394 q^{6} -21.7619 q^{7} -23.4817 q^{8} +11.7264 q^{9} +O(q^{10})\)	\(q+2.70597 q^{2} +6.22305 q^{3} -0.677713 q^{4} +5.00000 q^{5} +16.8394 q^{6} -21.7619 q^{7} -23.4817 q^{8} +11.7264 q^{9} +13.5299 q^{10} +11.0000 q^{11} -4.21744 q^{12} +67.9069 q^{13} -58.8871 q^{14} +31.1153 q^{15} -58.1190 q^{16} +62.9522 q^{17} +31.7313 q^{18} -19.0000 q^{19} -3.38856 q^{20} -135.425 q^{21} +29.7657 q^{22} +183.264 q^{23} -146.128 q^{24} +25.0000 q^{25} +183.754 q^{26} -95.0484 q^{27} +14.7483 q^{28} +136.111 q^{29} +84.1971 q^{30} +283.272 q^{31} +30.5848 q^{32} +68.4536 q^{33} +170.347 q^{34} -108.809 q^{35} -7.94714 q^{36} -30.2449 q^{37} -51.4135 q^{38} +422.588 q^{39} -117.408 q^{40} +64.7299 q^{41} -366.457 q^{42} -0.464383 q^{43} -7.45484 q^{44} +58.6320 q^{45} +495.907 q^{46} -4.97760 q^{47} -361.678 q^{48} +130.580 q^{49} +67.6493 q^{50} +391.755 q^{51} -46.0214 q^{52} -89.4905 q^{53} -257.198 q^{54} +55.0000 q^{55} +511.005 q^{56} -118.238 q^{57} +368.314 q^{58} -152.566 q^{59} -21.0872 q^{60} +224.225 q^{61} +766.525 q^{62} -255.189 q^{63} +547.714 q^{64} +339.534 q^{65} +185.234 q^{66} +2.06225 q^{67} -42.6635 q^{68} +1140.46 q^{69} -294.435 q^{70} +487.714 q^{71} -275.355 q^{72} +1029.94 q^{73} -81.8417 q^{74} +155.576 q^{75} +12.8765 q^{76} -239.381 q^{77} +1143.51 q^{78} -375.283 q^{79} -290.595 q^{80} -908.104 q^{81} +175.157 q^{82} +392.068 q^{83} +91.7795 q^{84} +314.761 q^{85} -1.25661 q^{86} +847.029 q^{87} -258.298 q^{88} +460.573 q^{89} +158.657 q^{90} -1477.78 q^{91} -124.200 q^{92} +1762.81 q^{93} -13.4692 q^{94} -95.0000 q^{95} +190.331 q^{96} -244.011 q^{97} +353.345 q^{98} +128.990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9} + 60 q^{10} + 242 q^{11} + 164 q^{12} + 207 q^{13} + 129 q^{14} + 105 q^{15} + 604 q^{16} + 209 q^{17} + 869 q^{18} - 418 q^{19} + 480 q^{20} + 45 q^{21} + 132 q^{22} + 191 q^{23} + 458 q^{24} + 550 q^{25} + 363 q^{26} + 45 q^{27} + 577 q^{28} + 389 q^{29} + 135 q^{30} + 198 q^{31} + 1149 q^{32} + 231 q^{33} + 467 q^{34} + 465 q^{35} + 1315 q^{36} + 312 q^{37} - 228 q^{38} + 137 q^{39} + 570 q^{40} + 632 q^{41} - 1794 q^{42} + 1584 q^{43} + 1056 q^{44} + 1045 q^{45} + 681 q^{46} - 54 q^{47} + 2491 q^{48} + 1063 q^{49} + 300 q^{50} + 37 q^{51} + 1246 q^{52} + 343 q^{53} + 1078 q^{54} + 1210 q^{55} - 87 q^{56} - 399 q^{57} - 1424 q^{58} + 2787 q^{59} + 820 q^{60} + 2070 q^{61} - 446 q^{62} + 1696 q^{63} + 1758 q^{64} + 1035 q^{65} + 297 q^{66} + 2423 q^{67} - 524 q^{68} - 997 q^{69} + 645 q^{70} + 2538 q^{71} + 6010 q^{72} + 1397 q^{73} - 1977 q^{74} + 525 q^{75} - 1824 q^{76} + 1023 q^{77} + 202 q^{78} + 878 q^{79} + 3020 q^{80} + 2030 q^{81} - 190 q^{82} + 4932 q^{83} - 4580 q^{84} + 1045 q^{85} - 3394 q^{86} + 6009 q^{87} + 1254 q^{88} + 1812 q^{89} + 4345 q^{90} + 4349 q^{91} - 788 q^{92} - 4848 q^{93} - 2152 q^{94} - 2090 q^{95} + 4032 q^{96} + 988 q^{97} + 1366 q^{98} + 2299 q^{99}+O(q^{100}) \) 22 * q + 12 * q^2 + 21 * q^3 + 96 * q^4 + 110 * q^5 + 27 * q^6 + 93 * q^7 + 114 * q^8 + 209 * q^9 + 60 * q^10 + 242 * q^11 + 164 * q^12 + 207 * q^13 + 129 * q^14 + 105 * q^15 + 604 * q^16 + 209 * q^17 + 869 * q^18 - 418 * q^19 + 480 * q^20 + 45 * q^21 + 132 * q^22 + 191 * q^23 + 458 * q^24 + 550 * q^25 + 363 * q^26 + 45 * q^27 + 577 * q^28 + 389 * q^29 + 135 * q^30 + 198 * q^31 + 1149 * q^32 + 231 * q^33 + 467 * q^34 + 465 * q^35 + 1315 * q^36 + 312 * q^37 - 228 * q^38 + 137 * q^39 + 570 * q^40 + 632 * q^41 - 1794 * q^42 + 1584 * q^43 + 1056 * q^44 + 1045 * q^45 + 681 * q^46 - 54 * q^47 + 2491 * q^48 + 1063 * q^49 + 300 * q^50 + 37 * q^51 + 1246 * q^52 + 343 * q^53 + 1078 * q^54 + 1210 * q^55 - 87 * q^56 - 399 * q^57 - 1424 * q^58 + 2787 * q^59 + 820 * q^60 + 2070 * q^61 - 446 * q^62 + 1696 * q^63 + 1758 * q^64 + 1035 * q^65 + 297 * q^66 + 2423 * q^67 - 524 * q^68 - 997 * q^69 + 645 * q^70 + 2538 * q^71 + 6010 * q^72 + 1397 * q^73 - 1977 * q^74 + 525 * q^75 - 1824 * q^76 + 1023 * q^77 + 202 * q^78 + 878 * q^79 + 3020 * q^80 + 2030 * q^81 - 190 * q^82 + 4932 * q^83 - 4580 * q^84 + 1045 * q^85 - 3394 * q^86 + 6009 * q^87 + 1254 * q^88 + 1812 * q^89 + 4345 * q^90 + 4349 * q^91 - 788 * q^92 - 4848 * q^93 - 2152 * q^94 - 2090 * q^95 + 4032 * q^96 + 988 * q^97 + 1366 * q^98 + 2299 * q^99

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