LMFDB - Embedded newform 1045.4.a.e.1.1

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			1.1
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)

$f(q)$	$=$	$q-5.28344 q^{2} +5.20066 q^{3} +19.9148 q^{4} +5.00000 q^{5} -27.4774 q^{6} +6.49388 q^{7} -62.9511 q^{8} +0.0468967 q^{9} +O(q^{10})$	\(q-5.28344 q^{2} +5.20066 q^{3} +19.9148 q^{4} +5.00000 q^{5} -27.4774 q^{6} +6.49388 q^{7} -62.9511 q^{8} +0.0468967 q^{9} -26.4172 q^{10} +11.0000 q^{11} +103.570 q^{12} +33.3334 q^{13} -34.3100 q^{14} +26.0033 q^{15} +173.280 q^{16} +0.464360 q^{17} -0.247776 q^{18} -19.0000 q^{19} +99.5739 q^{20} +33.7725 q^{21} -58.1179 q^{22} -53.3527 q^{23} -327.388 q^{24} +25.0000 q^{25} -176.115 q^{26} -140.174 q^{27} +129.324 q^{28} +284.737 q^{29} -137.387 q^{30} -327.625 q^{31} -411.909 q^{32} +57.2073 q^{33} -2.45342 q^{34} +32.4694 q^{35} +0.933938 q^{36} -50.8384 q^{37} +100.385 q^{38} +173.356 q^{39} -314.756 q^{40} +290.836 q^{41} -178.435 q^{42} +250.903 q^{43} +219.063 q^{44} +0.234484 q^{45} +281.886 q^{46} +365.127 q^{47} +901.173 q^{48} -300.830 q^{49} -132.086 q^{50} +2.41498 q^{51} +663.827 q^{52} +175.020 q^{53} +740.602 q^{54} +55.0000 q^{55} -408.797 q^{56} -98.8126 q^{57} -1504.39 q^{58} +863.269 q^{59} +517.850 q^{60} +450.612 q^{61} +1730.99 q^{62} +0.304542 q^{63} +790.053 q^{64} +166.667 q^{65} -302.252 q^{66} +760.807 q^{67} +9.24763 q^{68} -277.469 q^{69} -171.550 q^{70} -932.711 q^{71} -2.95220 q^{72} -344.721 q^{73} +268.602 q^{74} +130.017 q^{75} -378.381 q^{76} +71.4327 q^{77} -915.915 q^{78} +505.893 q^{79} +866.402 q^{80} -730.264 q^{81} -1536.61 q^{82} +898.002 q^{83} +672.572 q^{84} +2.32180 q^{85} -1325.63 q^{86} +1480.82 q^{87} -692.462 q^{88} +1023.21 q^{89} -1.23888 q^{90} +216.463 q^{91} -1062.51 q^{92} -1703.87 q^{93} -1929.13 q^{94} -95.0000 q^{95} -2142.20 q^{96} -650.503 q^{97} +1589.42 q^{98} +0.515864 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.28344	−1.86798	−0.933990	−	0.357299i	$-0.883698\pi$
$2$	−5.28344	−1.86798	−0.933990	+	0.357299i	$0.883698\pi$
$3$	5.20066	1.00087	0.500434	−	0.865775i	$-0.333174\pi$
$3$	5.20066	1.00087	0.500434	+	0.865775i	$0.333174\pi$
$4$	19.9148	2.48935
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−27.4774	−1.86960
$7$	6.49388	0.350637	0.175318	−	0.984512i	$-0.443905\pi$
$7$	6.49388	0.350637	0.175318	+	0.984512i	$0.443905\pi$
$8$	−62.9511	−2.78207
$9$	0.0468967	0.00173692
$10$	−26.4172	−0.835386
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	103.570	2.49151
$13$	33.3334	0.711155	0.355577	−	0.934647i	$-0.384284\pi$
$13$	33.3334	0.711155	0.355577	+	0.934647i	$0.384284\pi$
$14$	−34.3100	−0.654982
$15$	26.0033	0.447602
$16$	173.280	2.70751
$17$	0.464360	0.00662493	0.00331247	−	0.999995i	$-0.498946\pi$
$17$	0.464360	0.00662493	0.00331247	+	0.999995i	$0.498946\pi$
$18$	−0.247776	−0.00324452
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	99.5739	1.11327
$21$	33.7725	0.350941
$22$	−58.1179	−0.563217
$23$	−53.3527	−0.483687	−0.241844	−	0.970315i	$-0.577752\pi$
$23$	−53.3527	−0.483687	−0.241844	+	0.970315i	$0.577752\pi$
$24$	−327.388	−2.78449
$25$	25.0000	0.200000
$26$	−176.115	−1.32842
$27$	−140.174	−0.999130
$28$	129.324	0.872856
$29$	284.737	1.82325	0.911627	−	0.411019i	$-0.134827\pi$
$29$	284.737	1.82325	0.911627	+	0.411019i	$0.134827\pi$
$30$	−137.387	−0.836111
$31$	−327.625	−1.89817	−0.949084	−	0.315023i	$-0.897988\pi$
$31$	−327.625	−1.89817	−0.949084	+	0.315023i	$0.897988\pi$
$32$	−411.909	−2.27550
$33$	57.2073	0.301773
$34$	−2.45342	−0.0123752
$35$	32.4694	0.156809
$36$	0.933938	0.00432379
$37$	−50.8384	−0.225886	−0.112943	−	0.993601i	$-0.536028\pi$
$37$	−50.8384	−0.225886	−0.112943	+	0.993601i	$0.536028\pi$
$38$	100.385	0.428544
$39$	173.356	0.711772
$40$	−314.756	−1.24418
$41$	290.836	1.10783	0.553914	−	0.832574i	$-0.313134\pi$
$41$	290.836	1.10783	0.553914	+	0.832574i	$0.313134\pi$
$42$	−178.435	−0.655550
$43$	250.903	0.889822	0.444911	−	0.895575i	$-0.353235\pi$
$43$	250.903	0.889822	0.444911	+	0.895575i	$0.353235\pi$
$44$	219.063	0.750567
$45$	0.234484	0.000776773 0
$46$	281.886	0.903518
$47$	365.127	1.13317	0.566587	−	0.824002i	$-0.308263\pi$
$47$	365.127	1.13317	0.566587	+	0.824002i	$0.308263\pi$
$48$	901.173	2.70986
$49$	−300.830	−0.877054
$50$	−132.086	−0.373596
$51$	2.41498	0.00663068
$52$	663.827	1.77031
$53$	175.020	0.453600	0.226800	−	0.973941i	$-0.427174\pi$
$53$	175.020	0.453600	0.226800	+	0.973941i	$0.427174\pi$
$54$	740.602	1.86635
$55$	55.0000	0.134840
$56$	−408.797	−0.975496
$57$	−98.8126	−0.229615
$58$	−1504.39	−3.40580
$59$	863.269	1.90488	0.952441	−	0.304723i	$-0.0985638\pi$
$59$	863.269	1.90488	0.952441	+	0.304723i	$0.0985638\pi$
$60$	517.850	1.11424
$61$	450.612	0.945819	0.472909	−	0.881111i	$-0.343204\pi$
$61$	450.612	0.945819	0.472909	+	0.881111i	$0.343204\pi$
$62$	1730.99	3.54574
$63$	0.304542	0.000609026 0
$64$	790.053	1.54307
$65$	166.667	0.318038
$66$	−302.252	−0.563706
$67$	760.807	1.38727	0.693637	−	0.720324i	$-0.256007\pi$
$67$	760.807	1.38727	0.693637	+	0.720324i	$0.256007\pi$
$68$	9.24763	0.0164918
$69$	−277.469	−0.484107
$70$	−171.550	−0.292917
$71$	−932.711	−1.55905	−0.779524	−	0.626372i	$-0.784539\pi$
$71$	−932.711	−1.55905	−0.779524	+	0.626372i	$0.784539\pi$
$72$	−2.95220	−0.00483223
$73$	−344.721	−0.552692	−0.276346	−	0.961058i	$-0.589124\pi$
$73$	−344.721	−0.552692	−0.276346	+	0.961058i	$0.589124\pi$
$74$	268.602	0.421950
$75$	130.017	0.200174
$76$	−378.381	−0.571096
$77$	71.4327	0.105721
$78$	−915.915	−1.32958
$79$	505.893	0.720473	0.360237	−	0.932861i	$-0.382696\pi$
$79$	505.893	0.720473	0.360237	+	0.932861i	$0.382696\pi$
$80$	866.402	1.21083
$81$	−730.264	−1.00173
$82$	−1536.61	−2.06940
$83$	898.002	1.18757	0.593786	−	0.804623i	$-0.297632\pi$
$83$	898.002	1.18757	0.593786	+	0.804623i	$0.297632\pi$
$84$	672.572	0.873614
$85$	2.32180	0.00296276
$86$	−1325.63	−1.66217
$87$	1480.82	1.82484
$88$	−692.462	−0.838826
$89$	1023.21	1.21866	0.609328	−	0.792918i	$-0.291439\pi$
$89$	1023.21	1.21866	0.609328	+	0.792918i	$0.291439\pi$
$90$	−1.23888	−0.00145100
$91$	216.463	0.249357
$92$	−1062.51	−1.20407
$93$	−1703.87	−1.89982
$94$	−1929.13	−2.11675
$95$	−95.0000	−0.102598
$96$	−2142.20	−2.27747
$97$	−650.503	−0.680914	−0.340457	−	0.940260i	$-0.610582\pi$
$97$	−650.503	−0.680914	−0.340457	+	0.940260i	$0.610582\pi$
$98$	1589.42	1.63832
$99$	0.515864	0.000523700 0
$100$	497.870	0.497870
$101$	−544.501	−0.536434	−0.268217	−	0.963358i	$-0.586434\pi$
$101$	−544.501	−0.536434	−0.268217	+	0.963358i	$0.586434\pi$
$102$	−12.7594	−0.0123860
$103$	954.500	0.913104	0.456552	−	0.889697i	$-0.349084\pi$
$103$	954.500	0.913104	0.456552	+	0.889697i	$0.349084\pi$
$104$	−2098.37	−1.97848
$105$	168.862	0.156946
$106$	−924.706	−0.847315
$107$	1082.11	0.977681	0.488841	−	0.872373i	$-0.337420\pi$
$107$	1082.11	0.977681	0.488841	+	0.872373i	$0.337420\pi$
$108$	−2791.54	−2.48718
$109$	−1739.61	−1.52866	−0.764331	−	0.644824i	$-0.776931\pi$
$109$	−1739.61	−1.52866	−0.764331	+	0.644824i	$0.776931\pi$
$110$	−290.589	−0.251878
$111$	−264.393	−0.226082
$112$	1125.26	0.949351
$113$	−1725.59	−1.43654	−0.718272	−	0.695762i	$-0.755067\pi$
$113$	−1725.59	−1.43654	−0.718272	+	0.695762i	$0.755067\pi$
$114$	522.071	0.428916
$115$	−266.763	−0.216311
$116$	5670.48	4.53871
$117$	1.56323	0.00123522
$118$	−4561.03	−3.55828
$119$	3.01550	0.00232294
$120$	−1636.94	−1.24526
$121$	121.000	0.0909091
$122$	−2380.78	−1.76677
$123$	1512.54	1.10879
$124$	−6524.59	−4.72520
$125$	125.000	0.0894427
$126$	−1.60903	−0.00113765
$127$	2023.79	1.41404	0.707018	−	0.707195i	$-0.250040\pi$
$127$	2023.79	1.41404	0.707018	+	0.707195i	$0.250040\pi$
$128$	−878.932	−0.606933
$129$	1304.86	0.890594
$130$	−880.575	−0.594089
$131$	−128.165	−0.0854798	−0.0427399	−	0.999086i	$-0.513609\pi$
$131$	−128.165	−0.0854798	−0.0427399	+	0.999086i	$0.513609\pi$
$132$	1139.27	0.751218
$133$	−123.384	−0.0804415
$134$	−4019.68	−2.59140
$135$	−700.870	−0.446824
$136$	−29.2320	−0.0184310
$137$	2973.15	1.85411	0.927057	−	0.374920i	$-0.122330\pi$
$137$	2973.15	1.85411	0.927057	+	0.374920i	$0.122330\pi$
$138$	1465.99	0.904302
$139$	−3091.64	−1.88655	−0.943273	−	0.332019i	$-0.892270\pi$
$139$	−3091.64	−1.88655	−0.943273	+	0.332019i	$0.892270\pi$
$140$	646.621	0.390353
$141$	1898.90	1.13416
$142$	4927.93	2.91227
$143$	366.667	0.214421
$144$	8.12629	0.00470271
$145$	1423.69	0.815384
$146$	1821.31	1.03242
$147$	−1564.51	−0.877815
$148$	−1012.44	−0.562309
$149$	−366.339	−0.201421	−0.100710	−	0.994916i	$-0.532112\pi$
$149$	−366.339	−0.201421	−0.100710	+	0.994916i	$0.532112\pi$
$150$	−686.935	−0.373920
$151$	339.782	0.183120	0.0915598	−	0.995800i	$-0.470815\pi$
$151$	339.782	0.183120	0.0915598	+	0.995800i	$0.470815\pi$
$152$	1196.07	0.638251
$153$	0.0217770	1.15070e−5 0
$154$	−377.411	−0.197484
$155$	−1638.13	−0.848887
$156$	3452.34	1.77185
$157$	−2459.22	−1.25011	−0.625055	−	0.780581i	$-0.714923\pi$
$157$	−2459.22	−1.25011	−0.625055	+	0.780581i	$0.714923\pi$
$158$	−2672.86	−1.34583
$159$	910.218	0.453993
$160$	−2059.54	−1.01763
$161$	−346.466	−0.169598
$162$	3858.31	1.87122
$163$	1878.26	0.902555	0.451277	−	0.892384i	$-0.350968\pi$
$163$	1878.26	0.902555	0.451277	+	0.892384i	$0.350968\pi$
$164$	5791.93	2.75777
$165$	286.036	0.134957
$166$	−4744.55	−2.21836
$167$	−1175.18	−0.544541	−0.272271	−	0.962221i	$-0.587775\pi$
$167$	−1175.18	−0.544541	−0.272271	+	0.962221i	$0.587775\pi$
$168$	−2126.02	−0.976343
$169$	−1085.89	−0.494259
$170$	−12.2671	−0.00553438
$171$	−0.891038	−0.000398476 0
$172$	4996.68	2.21508
$173$	−264.428	−0.116208	−0.0581042	−	0.998311i	$-0.518506\pi$
$173$	−264.428	−0.116208	−0.0581042	+	0.998311i	$0.518506\pi$
$174$	−7823.84	−3.40876
$175$	162.347	0.0701273
$176$	1906.08	0.816344
$177$	4489.57	1.90654
$178$	−5406.09	−2.27643
$179$	−1953.60	−0.815748	−0.407874	−	0.913038i	$-0.633730\pi$
$179$	−1953.60	−0.815748	−0.407874	+	0.913038i	$0.633730\pi$
$180$	4.66969	0.00193366
$181$	1702.87	0.699302	0.349651	−	0.936880i	$-0.386300\pi$
$181$	1702.87	0.699302	0.349651	+	0.936880i	$0.386300\pi$
$182$	−1143.67	−0.465793
$183$	2343.48	0.946640
$184$	3358.61	1.34565
$185$	−254.192	−0.101019
$186$	9002.29	3.54882
$187$	5.10796	0.00199749
$188$	7271.42	2.82087
$189$	−910.273	−0.350331
$190$	501.927	0.191651
$191$	1431.28	0.542219	0.271109	−	0.962549i	$-0.412610\pi$
$191$	1431.28	0.542219	0.271109	+	0.962549i	$0.412610\pi$
$192$	4108.80	1.54441
$193$	3197.86	1.19268	0.596339	−	0.802733i	$-0.296621\pi$
$193$	3197.86	1.19268	0.596339	+	0.802733i	$0.296621\pi$
$194$	3436.90	1.27193
$195$	866.778	0.318314
$196$	−5990.96	−2.18329
$197$	5179.30	1.87315	0.936573	−	0.350472i	$-0.113979\pi$
$197$	5179.30	1.87315	0.936573	+	0.350472i	$0.113979\pi$
$198$	−2.72554	−0.000978261 0
$199$	2153.76	0.767217	0.383608	−	0.923496i	$-0.374681\pi$
$199$	2153.76	0.767217	0.383608	+	0.923496i	$0.374681\pi$
$200$	−1573.78	−0.556415
$201$	3956.70	1.38848
$202$	2876.84	1.00205
$203$	1849.05	0.639299
$204$	48.0938	0.0165061
$205$	1454.18	0.495435
$206$	−5043.05	−1.70566
$207$	−2.50207	−0.000840124 0
$208$	5776.02	1.92546
$209$	−209.000	−0.0691714
$210$	−892.175	−0.293171
$211$	3599.27	1.17433	0.587165	−	0.809467i	$-0.300244\pi$
$211$	3599.27	1.17433	0.587165	+	0.809467i	$0.300244\pi$
$212$	3485.48	1.12917
$213$	−4850.72	−1.56040
$214$	−5717.29	−1.82629
$215$	1254.51	0.397940
$216$	8824.11	2.77965
$217$	−2127.56	−0.665567
$218$	9191.12	2.85551
$219$	−1792.78	−0.553172
$220$	1095.31	0.335664
$221$	15.4787	0.00471135
$222$	1396.91	0.422317
$223$	1144.63	0.343724	0.171862	−	0.985121i	$-0.445022\pi$
$223$	1144.63	0.343724	0.171862	+	0.985121i	$0.445022\pi$
$224$	−2674.88	−0.797872
$225$	1.17242	0.000347383 0
$226$	9117.04	2.68344
$227$	−1416.21	−0.414086	−0.207043	−	0.978332i	$-0.566384\pi$
$227$	−1416.21	−0.414086	−0.207043	+	0.978332i	$0.566384\pi$
$228$	−1967.83	−0.571591
$229$	−272.042	−0.0785023	−0.0392511	−	0.999229i	$-0.512497\pi$
$229$	−272.042	−0.0785023	−0.0392511	+	0.999229i	$0.512497\pi$
$230$	1409.43	0.404065
$231$	371.497	0.105813
$232$	−17924.5	−5.07242
$233$	−927.221	−0.260705	−0.130353	−	0.991468i	$-0.541611\pi$
$233$	−927.221	−0.260705	−0.130353	+	0.991468i	$0.541611\pi$
$234$	−8.25922	−0.00230736
$235$	1825.63	0.506771
$236$	17191.8	4.74192
$237$	2630.98	0.721099
$238$	−15.9322	−0.00433921
$239$	−4816.60	−1.30360	−0.651800	−	0.758391i	$-0.725986\pi$
$239$	−4816.60	−1.30360	−0.651800	+	0.758391i	$0.725986\pi$
$240$	4505.87	1.21188
$241$	2242.89	0.599491	0.299746	−	0.954019i	$-0.403098\pi$
$241$	2242.89	0.599491	0.299746	+	0.954019i	$0.403098\pi$
$242$	−639.297	−0.169816
$243$	−13.1588	−0.00347383
$244$	8973.84	2.35447
$245$	−1504.15	−0.392230
$246$	−7991.42	−2.07120
$247$	−633.334	−0.163150
$248$	20624.4	5.28084
$249$	4670.21	1.18860
$250$	−660.431	−0.167077
$251$	−7231.07	−1.81841	−0.909205	−	0.416349i	$-0.863310\pi$
$251$	−7231.07	−1.81841	−0.909205	+	0.416349i	$0.863310\pi$
$252$	6.06488	0.00151608
$253$	−586.879	−0.145837
$254$	−10692.6	−2.64139
$255$	12.0749	0.00296533
$256$	−1676.63	−0.409335
$257$	1870.73	0.454059	0.227030	−	0.973888i	$-0.427099\pi$
$257$	1870.73	0.454059	0.227030	+	0.973888i	$0.427099\pi$
$258$	−6894.17	−1.66361
$259$	−330.138	−0.0792038
$260$	3319.13	0.791707
$261$	13.3532	0.00316684
$262$	677.154	0.159674
$263$	5315.62	1.24629	0.623146	−	0.782105i	$-0.285854\pi$
$263$	5315.62	1.24629	0.623146	+	0.782105i	$0.285854\pi$
$264$	−3601.26	−0.839555
$265$	875.098	0.202856
$266$	651.891	0.150263
$267$	5321.39	1.21971
$268$	15151.3	3.45341
$269$	5517.39	1.25056	0.625281	−	0.780399i	$-0.284984\pi$
$269$	5517.39	1.25056	0.625281	+	0.780399i	$0.284984\pi$
$270$	3703.01	0.834659
$271$	1744.05	0.390934	0.195467	−	0.980710i	$-0.437378\pi$
$271$	1744.05	0.390934	0.195467	+	0.980710i	$0.437378\pi$
$272$	80.4645	0.0179371
$273$	1125.75	0.249573
$274$	−15708.5	−3.46345
$275$	275.000	0.0603023
$276$	−5525.74	−1.20511
$277$	1599.94	0.347044	0.173522	−	0.984830i	$-0.444485\pi$
$277$	1599.94	0.347044	0.173522	+	0.984830i	$0.444485\pi$
$278$	16334.5	3.52403
$279$	−15.3646	−0.00329696
$280$	−2043.98	−0.436255
$281$	−1948.37	−0.413630	−0.206815	−	0.978380i	$-0.566310\pi$
$281$	−1948.37	−0.413630	−0.206815	+	0.978380i	$0.566310\pi$
$282$	−10032.7	−2.11858
$283$	−6326.13	−1.32880	−0.664398	−	0.747379i	$-0.731312\pi$
$283$	−6326.13	−1.32880	−0.664398	+	0.747379i	$0.731312\pi$
$284$	−18574.7	−3.88101
$285$	−494.063	−0.102687
$286$	−1937.26	−0.400534
$287$	1888.65	0.388445
$288$	−19.3172	−0.00395234
$289$	−4912.78	−0.999956
$290$	−7521.96	−1.52312
$291$	−3383.05	−0.681505
$292$	−6865.04	−1.37584
$293$	−1714.48	−0.341847	−0.170924	−	0.985284i	$-0.554675\pi$
$293$	−1714.48	−0.341847	−0.170924	+	0.985284i	$0.554675\pi$
$294$	8266.02	1.63974
$295$	4316.34	0.851889
$296$	3200.33	0.628431
$297$	−1541.91	−0.301249
$298$	1935.53	0.376249
$299$	−1778.42	−0.343976
$300$	2589.25	0.498302
$301$	1629.33	0.312004
$302$	−1795.22	−0.342064
$303$	−2831.77	−0.536900
$304$	−3292.33	−0.621145
$305$	2253.06	0.422983
$306$	−0.115057	−2.14948e−5 0
$307$	387.965	0.0721249	0.0360624	−	0.999350i	$-0.488518\pi$
$307$	387.965	0.0721249	0.0360624	+	0.999350i	$0.488518\pi$
$308$	1422.57	0.263176
$309$	4964.03	0.913896
$310$	8654.95	1.58570
$311$	5528.60	1.00803	0.504016	−	0.863694i	$-0.331855\pi$
$311$	5528.60	1.00803	0.504016	+	0.863694i	$0.331855\pi$
$312$	−10912.9	−1.98020
$313$	6235.48	1.12604	0.563020	−	0.826443i	$-0.309639\pi$
$313$	6235.48	1.12604	0.563020	+	0.826443i	$0.309639\pi$
$314$	12993.2	2.33518
$315$	1.52271	0.000272365 0
$316$	10074.7	1.79351
$317$	10255.0	1.81696	0.908479	−	0.417931i	$-0.137245\pi$
$317$	10255.0	1.81696	0.908479	+	0.417931i	$0.137245\pi$
$318$	−4809.08	−0.848050
$319$	3132.11	0.549732
$320$	3950.27	0.690083
$321$	5627.71	0.978530
$322$	1830.53	0.316806
$323$	−8.82284	−0.00151986
$324$	−14543.1	−2.49366
$325$	833.334	0.142231
$326$	−9923.67	−1.68595
$327$	−9047.12	−1.52999
$328$	−18308.4	−3.08206
$329$	2371.09	0.397332
$330$	−1511.26	−0.252097
$331$	−3073.25	−0.510335	−0.255168	−	0.966897i	$-0.582131\pi$
$331$	−3073.25	−0.510335	−0.255168	+	0.966897i	$0.582131\pi$
$332$	17883.5	2.95628
$333$	−2.38415	−0.000392345 0
$334$	6209.01	1.01719
$335$	3804.04	0.620408
$336$	5852.11	0.950175
$337$	7429.70	1.20095	0.600477	−	0.799642i	$-0.294977\pi$
$337$	7429.70	1.20095	0.600477	+	0.799642i	$0.294977\pi$
$338$	5737.22	0.923266
$339$	−8974.19	−1.43779
$340$	46.2382	0.00737534
$341$	−3603.88	−0.572319
$342$	4.70775	0.000744345 0
$343$	−4180.95	−0.658164
$344$	−15794.6	−2.47555
$345$	−1387.35	−0.216499
$346$	1397.09	0.217075
$347$	136.369	0.0210970	0.0105485	−	0.999944i	$-0.496642\pi$
$347$	136.369	0.0210970	0.0105485	+	0.999944i	$0.496642\pi$
$348$	29490.2	4.54265
$349$	2394.96	0.367334	0.183667	−	0.982989i	$-0.441203\pi$
$349$	2394.96	0.367334	0.183667	+	0.982989i	$0.441203\pi$
$350$	−857.751	−0.130996
$351$	−4672.47	−0.710536
$352$	−4531.00	−0.686088
$353$	−3022.58	−0.455738	−0.227869	−	0.973692i	$-0.573176\pi$
$353$	−3022.58	−0.455738	−0.227869	+	0.973692i	$0.573176\pi$
$354$	−23720.4	−3.56137
$355$	−4663.55	−0.697228
$356$	20377.1	3.03366
$357$	15.6826	0.00232496
$358$	10321.7	1.52380
$359$	−4226.69	−0.621382	−0.310691	−	0.950511i	$-0.600560\pi$
$359$	−4226.69	−0.621382	−0.310691	+	0.950511i	$0.600560\pi$
$360$	−14.7610	−0.00216104
$361$	361.000	0.0526316
$362$	−8997.04	−1.30628
$363$	629.280	0.0909880
$364$	4310.81	0.620736
$365$	−1723.60	−0.247171
$366$	−12381.7	−1.76830
$367$	−10326.9	−1.46882	−0.734411	−	0.678705i	$-0.762542\pi$
$367$	−10326.9	−1.46882	−0.734411	+	0.678705i	$0.762542\pi$
$368$	−9244.98	−1.30959
$369$	13.6392	0.00192420
$370$	1343.01	0.188702
$371$	1136.56	0.159049
$372$	−33932.2	−4.72930
$373$	−13146.8	−1.82498	−0.912488	−	0.409102i	$-0.865842\pi$
$373$	−13146.8	−1.82498	−0.912488	+	0.409102i	$0.865842\pi$
$374$	−26.9876	−0.00373128
$375$	650.083	0.0895204
$376$	−22985.1	−3.15257
$377$	9491.24	1.29662
$378$	4809.38	0.654412
$379$	12474.7	1.69072	0.845362	−	0.534194i	$-0.179385\pi$
$379$	12474.7	1.69072	0.845362	+	0.534194i	$0.179385\pi$
$380$	−1891.90	−0.255402
$381$	10525.1	1.41526
$382$	−7562.09	−1.01285
$383$	9232.46	1.23174	0.615870	−	0.787848i	$-0.288805\pi$
$383$	9232.46	1.23174	0.615870	+	0.787848i	$0.288805\pi$
$384$	−4571.03	−0.607459
$385$	357.163	0.0472798
$386$	−16895.7	−2.22790
$387$	11.7665	0.00154555
$388$	−12954.6	−1.69503
$389$	2020.39	0.263336	0.131668	−	0.991294i	$-0.457967\pi$
$389$	2020.39	0.263336	0.131668	+	0.991294i	$0.457967\pi$
$390$	−4579.57	−0.594604
$391$	−24.7749	−0.00320439
$392$	18937.6	2.44003
$393$	−666.544	−0.0855540
$394$	−27364.5	−3.49900
$395$	2529.46	0.322205
$396$	10.2733	0.00130367
$397$	9899.42	1.25148	0.625740	−	0.780032i	$-0.284797\pi$
$397$	9899.42	1.25148	0.625740	+	0.780032i	$0.284797\pi$
$398$	−11379.3	−1.43315
$399$	−641.677	−0.0805114
$400$	4332.01	0.541501
$401$	7919.42	0.986227	0.493113	−	0.869965i	$-0.335859\pi$
$401$	7919.42	0.986227	0.493113	+	0.869965i	$0.335859\pi$
$402$	−20905.0	−2.59365
$403$	−10920.8	−1.34989
$404$	−10843.6	−1.33537
$405$	−3651.32	−0.447989
$406$	−9769.34	−1.19420
$407$	−559.222	−0.0681072
$408$	−152.026	−0.0184470
$409$	−10207.4	−1.23405	−0.617023	−	0.786945i	$-0.711661\pi$
$409$	−10207.4	−1.23405	−0.617023	+	0.786945i	$0.711661\pi$
$410$	−7683.07	−0.925463
$411$	15462.4	1.85572
$412$	19008.7	2.27303
$413$	5605.96	0.667921
$414$	13.2195	0.00156933
$415$	4490.01	0.531099
$416$	−13730.3	−1.61823
$417$	−16078.6	−1.88818
$418$	1104.24	0.129211
$419$	3977.40	0.463744	0.231872	−	0.972746i	$-0.425515\pi$
$419$	3977.40	0.463744	0.231872	+	0.972746i	$0.425515\pi$
$420$	3362.86	0.390692
$421$	−1117.55	−0.129373	−0.0646863	−	0.997906i	$-0.520605\pi$
$421$	−1117.55	−0.129373	−0.0646863	+	0.997906i	$0.520605\pi$
$422$	−19016.5	−2.19363
$423$	17.1233	0.00196823
$424$	−11017.7	−1.26195
$425$	11.6090	0.00132499
$426$	25628.5	2.91480
$427$	2926.22	0.331639
$428$	21550.1	2.43379
$429$	1906.91	0.214607
$430$	−6628.16	−0.743345
$431$	10560.3	1.18021	0.590105	−	0.807327i	$-0.299086\pi$
$431$	10560.3	1.18021	0.590105	+	0.807327i	$0.299086\pi$
$432$	−24289.4	−2.70515
$433$	−684.524	−0.0759726	−0.0379863	−	0.999278i	$-0.512094\pi$
$433$	−684.524	−0.0759726	−0.0379863	+	0.999278i	$0.512094\pi$
$434$	11240.8	1.24327
$435$	7404.11	0.816091
$436$	−34643.9	−3.80537
$437$	1013.70	0.110965
$438$	9472.03	1.03331
$439$	7771.74	0.844931	0.422466	−	0.906379i	$-0.361165\pi$
$439$	7771.74	0.844931	0.422466	+	0.906379i	$0.361165\pi$
$440$	−3462.31	−0.375135
$441$	−14.1079	−0.00152337
$442$	−81.7808	−0.00880071
$443$	−4286.71	−0.459746	−0.229873	−	0.973221i	$-0.573831\pi$
$443$	−4286.71	−0.459746	−0.229873	+	0.973221i	$0.573831\pi$
$444$	−5265.34	−0.562797
$445$	5116.07	0.545000
$446$	−6047.61	−0.642069
$447$	−1905.21	−0.201595
$448$	5130.51	0.541057
$449$	−16227.7	−1.70564	−0.852820	−	0.522204i	$-0.825110\pi$
$449$	−16227.7	−1.70564	−0.852820	+	0.522204i	$0.825110\pi$
$450$	−6.19441	−0.000648905 0
$451$	3199.19	0.334022
$452$	−34364.7	−3.57606
$453$	1767.09	0.183279
$454$	7482.49	0.773504
$455$	1082.31	0.111516
$456$	6220.36	0.638805
$457$	−16706.0	−1.71001	−0.855005	−	0.518620i	$-0.826446\pi$
$457$	−16706.0	−1.71001	−0.855005	+	0.518620i	$0.826446\pi$
$458$	1437.32	0.146641
$459$	−65.0912	−0.00661917
$460$	−5312.54	−0.538475
$461$	−1567.31	−0.158345	−0.0791726	−	0.996861i	$-0.525228\pi$
$461$	−1567.31	−0.158345	−0.0791726	+	0.996861i	$0.525228\pi$
$462$	−1962.79	−0.197656
$463$	−4615.55	−0.463289	−0.231644	−	0.972801i	$-0.574411\pi$
$463$	−4615.55	−0.463289	−0.231644	+	0.972801i	$0.574411\pi$
$464$	49339.4	4.93647
$465$	−8519.34	−0.849624
$466$	4898.92	0.486992
$467$	3285.20	0.325526	0.162763	−	0.986665i	$-0.447959\pi$
$467$	3285.20	0.325526	0.162763	+	0.986665i	$0.447959\pi$
$468$	31.1313	0.00307488
$469$	4940.59	0.486429
$470$	−9645.63	−0.946638
$471$	−12789.6	−1.25119
$472$	−54343.7	−5.29952
$473$	2759.93	0.268291
$474$	−13900.6	−1.34700
$475$	−475.000	−0.0458831
$476$	60.0530	0.00578262
$477$	8.20784	0.000787865 0
$478$	25448.3	2.43510
$479$	5636.68	0.537675	0.268838	−	0.963185i	$-0.413360\pi$
$479$	5636.68	0.537675	0.268838	+	0.963185i	$0.413360\pi$
$480$	−10711.0	−1.01852
$481$	−1694.61	−0.160640
$482$	−11850.2	−1.11984
$483$	−1801.85	−0.169746
$484$	2409.69	0.226304
$485$	−3252.52	−0.304514
$486$	69.5240	0.00648904
$487$	13466.3	1.25301	0.626507	−	0.779416i	$-0.284484\pi$
$487$	13466.3	1.25301	0.626507	+	0.779416i	$0.284484\pi$
$488$	−28366.5	−2.63134
$489$	9768.18	0.903338
$490$	7947.08	0.732679
$491$	2140.38	0.196729	0.0983645	−	0.995150i	$-0.468639\pi$
$491$	2140.38	0.196729	0.0983645	+	0.995150i	$0.468639\pi$
$492$	30121.9	2.76016
$493$	132.221	0.0120789
$494$	3346.18	0.304761
$495$	2.57932	0.000234206 0
$496$	−56771.0	−5.13930
$497$	−6056.91	−0.546659
$498$	−24674.8	−2.22029
$499$	−3914.47	−0.351174	−0.175587	−	0.984464i	$-0.556182\pi$
$499$	−3914.47	−0.351174	−0.175587	+	0.984464i	$0.556182\pi$
$500$	2489.35	0.222654
$501$	−6111.73	−0.545014
$502$	38204.9	3.39675
$503$	13595.5	1.20516	0.602579	−	0.798059i	$-0.294140\pi$
$503$	13595.5	1.20516	0.602579	+	0.798059i	$0.294140\pi$
$504$	−19.1712	−0.00169436
$505$	−2722.51	−0.239901
$506$	3100.75	0.272421
$507$	−5647.33	−0.494688
$508$	40303.4	3.52003
$509$	−16044.2	−1.39714	−0.698572	−	0.715540i	$-0.746181\pi$
$509$	−16044.2	−1.39714	−0.698572	+	0.715540i	$0.746181\pi$
$510$	−63.7971	−0.00553918
$511$	−2238.57	−0.193794
$512$	15889.9	1.37156
$513$	2663.31	0.229216
$514$	−9883.92	−0.848173
$515$	4772.50	0.408352
$516$	25986.0	2.21700
$517$	4016.39	0.341665
$518$	1744.27	0.147951
$519$	−1375.20	−0.116309
$520$	−10491.9	−0.884805
$521$	12706.5	1.06849	0.534245	−	0.845329i	$-0.320596\pi$
$521$	12706.5	1.06849	0.534245	+	0.845329i	$0.320596\pi$
$522$	−70.5511	−0.00591559
$523$	−8337.38	−0.697071	−0.348535	−	0.937296i	$-0.613321\pi$
$523$	−8337.38	−0.697071	−0.348535	+	0.937296i	$0.613321\pi$
$524$	−2552.38	−0.212789
$525$	844.312	0.0701882
$526$	−28084.8	−2.32805
$527$	−152.136	−0.0125752
$528$	9912.90	0.817053
$529$	−9320.49	−0.766047
$530$	−4623.53	−0.378931
$531$	40.4845	0.00330862
$532$	−2457.16	−0.200247
$533$	9694.53	0.787837
$534$	−28115.3	−2.27840
$535$	5410.57	0.437232
$536$	−47893.7	−3.85950
$537$	−10160.0	−0.816456
$538$	−29150.8	−2.33603
$539$	−3309.12	−0.264442
$540$	−13957.7	−1.11230
$541$	−18308.9	−1.45501	−0.727504	−	0.686103i	$-0.759320\pi$
$541$	−18308.9	−1.45501	−0.727504	+	0.686103i	$0.759320\pi$
$542$	−9214.57	−0.730258
$543$	8856.07	0.699909
$544$	−191.274	−0.0150750
$545$	−8698.04	−0.683639
$546$	−5947.84	−0.466198
$547$	20343.8	1.59019	0.795097	−	0.606482i	$-0.207420\pi$
$547$	20343.8	1.59019	0.795097	+	0.606482i	$0.207420\pi$
$548$	59209.7	4.61554
$549$	21.1322	0.00164281
$550$	−1452.95	−0.112643
$551$	−5410.00	−0.418283
$552$	17467.0	1.34682
$553$	3285.21	0.252624
$554$	−8453.21	−0.648272
$555$	−1321.97	−0.101107
$556$	−61569.4	−4.69627
$557$	22588.2	1.71830	0.859150	−	0.511724i	$-0.170993\pi$
$557$	22588.2	1.71830	0.859150	+	0.511724i	$0.170993\pi$
$558$	81.1778	0.00615865
$559$	8363.44	0.632801
$560$	5626.31	0.424563
$561$	26.5648	0.00199923
$562$	10294.1	0.772653
$563$	10084.8	0.754928	0.377464	−	0.926024i	$-0.376796\pi$
$563$	10084.8	0.754928	0.377464	+	0.926024i	$0.376796\pi$
$564$	37816.2	2.82332
$565$	−8627.93	−0.642442
$566$	33423.7	2.48216
$567$	−4742.25	−0.351244
$568$	58715.2	4.33739
$569$	18825.3	1.38699	0.693497	−	0.720460i	$-0.256069\pi$
$569$	18825.3	1.38699	0.693497	+	0.720460i	$0.256069\pi$
$570$	2610.35	0.191817
$571$	−3567.32	−0.261449	−0.130725	−	0.991419i	$-0.541730\pi$
$571$	−3567.32	−0.261449	−0.130725	+	0.991419i	$0.541730\pi$
$572$	7302.09	0.533769
$573$	7443.60	0.542689
$574$	−9978.59	−0.725607
$575$	−1333.82	−0.0967374
$576$	37.0509	0.00268019
$577$	−7747.17	−0.558958	−0.279479	−	0.960152i	$-0.590162\pi$
$577$	−7747.17	−0.558958	−0.279479	+	0.960152i	$0.590162\pi$
$578$	25956.4	1.86790
$579$	16631.0	1.19371
$580$	28352.4	2.02977
$581$	5831.52	0.416406
$582$	17874.2	1.27304
$583$	1925.21	0.136765
$584$	21700.6	1.53763
$585$	7.81613	0.000552405 0
$586$	9058.38	0.638564
$587$	19825.1	1.39399	0.696994	−	0.717077i	$-0.254520\pi$
$587$	19825.1	1.39399	0.696994	+	0.717077i	$0.254520\pi$
$588$	−31156.9	−2.18519
$589$	6224.88	0.435470
$590$	−22805.2	−1.59131
$591$	26935.8	1.87477
$592$	−8809.30	−0.611588
$593$	14278.6	0.988787	0.494394	−	0.869238i	$-0.335390\pi$
$593$	14278.6	0.988787	0.494394	+	0.869238i	$0.335390\pi$
$594$	8146.62	0.562727
$595$	15.0775	0.00103885
$596$	−7295.56	−0.501406
$597$	11201.0	0.767883
$598$	9396.20	0.642541
$599$	−24380.0	−1.66301	−0.831504	−	0.555519i	$-0.812519\pi$
$599$	−24380.0	−1.66301	−0.831504	+	0.555519i	$0.812519\pi$
$600$	−8184.69	−0.556898
$601$	22741.5	1.54350	0.771751	−	0.635925i	$-0.219381\pi$
$601$	22741.5	1.54350	0.771751	+	0.635925i	$0.219381\pi$
$602$	−8608.49	−0.582817
$603$	35.6794	0.00240958
$604$	6766.68	0.455848
$605$	605.000	0.0406558
$606$	14961.5	1.00292
$607$	13930.6	0.931508	0.465754	−	0.884914i	$-0.345783\pi$
$607$	13930.6	0.931508	0.465754	+	0.884914i	$0.345783\pi$
$608$	7826.26	0.522034
$609$	9616.28	0.639854
$610$	−11903.9	−0.790124
$611$	12170.9	0.805862
$612$	0.433684	2.86448e−5 0
$613$	−4657.47	−0.306873	−0.153437	−	0.988158i	$-0.549034\pi$
$613$	−4657.47	−0.306873	−0.153437	+	0.988158i	$0.549034\pi$
$614$	−2049.79	−0.134728
$615$	7562.69	0.495865
$616$	−4496.77	−0.294123
$617$	−9765.79	−0.637206	−0.318603	−	0.947888i	$-0.603214\pi$
$617$	−9765.79	−0.637206	−0.318603	+	0.947888i	$0.603214\pi$
$618$	−26227.2	−1.70714
$619$	−16345.6	−1.06137	−0.530684	−	0.847570i	$-0.678065\pi$
$619$	−16345.6	−1.06137	−0.530684	+	0.847570i	$0.678065\pi$
$620$	−32622.9	−2.11317
$621$	7478.66	0.483266
$622$	−29210.0	−1.88298
$623$	6644.62	0.427305
$624$	30039.1	1.92713
$625$	625.000	0.0400000
$626$	−32944.8	−2.10342
$627$	−1086.94	−0.0692315
$628$	−48974.8	−3.11196
$629$	−23.6073	−0.00149648
$630$	−8.04515	−0.000508772 0
$631$	5660.43	0.357113	0.178556	−	0.983930i	$-0.442857\pi$
$631$	5660.43	0.357113	0.178556	+	0.983930i	$0.442857\pi$
$632$	−31846.5	−2.00441
$633$	18718.6	1.17535
$634$	−54181.5	−3.39404
$635$	10119.0	0.632376
$636$	18126.8	1.13015
$637$	−10027.7	−0.623721
$638$	−16548.3	−1.02689
$639$	−43.7411	−0.00270794
$640$	−4394.66	−0.271429
$641$	−18589.9	−1.14549	−0.572743	−	0.819735i	$-0.694121\pi$
$641$	−18589.9	−1.14549	−0.572743	+	0.819735i	$0.694121\pi$
$642$	−29733.7	−1.82787
$643$	−13477.9	−0.826622	−0.413311	−	0.910590i	$-0.635628\pi$
$643$	−13477.9	−0.826622	−0.413311	+	0.910590i	$0.635628\pi$
$644$	−6899.79	−0.422189
$645$	6524.31	0.398286
$646$	46.6150	0.00283907
$647$	−4021.66	−0.244370	−0.122185	−	0.992507i	$-0.538990\pi$
$647$	−4021.66	−0.244370	−0.122185	+	0.992507i	$0.538990\pi$
$648$	45970.9	2.78690
$649$	9495.96	0.574344
$650$	−4402.87	−0.265684
$651$	−11064.7	−0.666145
$652$	37405.1	2.24677
$653$	−6799.73	−0.407494	−0.203747	−	0.979024i	$-0.565312\pi$
$653$	−6799.73	−0.407494	−0.203747	+	0.979024i	$0.565312\pi$
$654$	47799.9	2.85799
$655$	−640.826	−0.0382277
$656$	50396.1	2.99945
$657$	−16.1663	−0.000959979 0
$658$	−12527.5	−0.742209
$659$	−17542.9	−1.03699	−0.518494	−	0.855082i	$-0.673507\pi$
$659$	−17542.9	−1.03699	−0.518494	+	0.855082i	$0.673507\pi$
$660$	5696.36	0.335955
$661$	6797.60	0.399994	0.199997	−	0.979797i	$-0.435907\pi$
$661$	6797.60	0.399994	0.199997	+	0.979797i	$0.435907\pi$
$662$	16237.3	0.953296
$663$	80.4994	0.00471544
$664$	−56530.2	−3.30391
$665$	−616.918	−0.0359745
$666$	12.5965	0.000732892 0
$667$	−15191.5	−0.881884
$668$	−23403.5	−1.35555
$669$	5952.86	0.344022
$670$	−20098.4	−1.15891
$671$	4956.73	0.285175
$672$	−13911.2	−0.798564
$673$	−9648.14	−0.552613	−0.276307	−	0.961070i	$-0.589111\pi$
$673$	−9648.14	−0.552613	−0.276307	+	0.961070i	$0.589111\pi$
$674$	−39254.4	−2.24336
$675$	−3504.35	−0.199826
$676$	−21625.2	−1.23038
$677$	−9499.05	−0.539259	−0.269629	−	0.962964i	$-0.586901\pi$
$677$	−9499.05	−0.539259	−0.269629	+	0.962964i	$0.586901\pi$
$678$	47414.7	2.68577
$679$	−4224.29	−0.238753
$680$	−146.160	−0.00824261
$681$	−7365.26	−0.414445
$682$	19040.9	1.06908
$683$	−6217.99	−0.348353	−0.174176	−	0.984714i	$-0.555726\pi$
$683$	−6217.99	−0.348353	−0.174176	+	0.984714i	$0.555726\pi$
$684$	−17.7448	−0.000991945 0
$685$	14865.8	0.829185
$686$	22089.8	1.22944
$687$	−1414.80	−0.0785704
$688$	43476.6	2.40920
$689$	5833.99	0.322579
$690$	7329.97	0.404416
$691$	−14819.4	−0.815858	−0.407929	−	0.913014i	$-0.633749\pi$
$691$	−14819.4	−0.815858	−0.407929	+	0.913014i	$0.633749\pi$
$692$	−5266.02	−0.289283
$693$	3.34996	0.000183628 0
$694$	−720.496	−0.0394087
$695$	−15458.2	−0.843689
$696$	−93219.4	−5.07683
$697$	135.053	0.00733928
$698$	−12653.7	−0.686172
$699$	−4822.17	−0.260931
$700$	3233.11	0.174571
$701$	−10145.9	−0.546656	−0.273328	−	0.961921i	$-0.588124\pi$
$701$	−10145.9	−0.546656	−0.273328	+	0.961921i	$0.588124\pi$
$702$	24686.7	1.32727
$703$	965.929	0.0518218
$704$	8690.58	0.465254
$705$	9494.51	0.507211
$706$	15969.6	0.851310
$707$	−3535.92	−0.188093
$708$	89408.9	4.74603
$709$	−14314.8	−0.758257	−0.379129	−	0.925344i	$-0.623776\pi$
$709$	−14314.8	−0.758257	−0.379129	+	0.925344i	$0.623776\pi$
$710$	24639.6	1.30241
$711$	23.7247	0.00125140
$712$	−64412.4	−3.39039
$713$	17479.7	0.918119
$714$	−82.8581	−0.00434298
$715$	1833.33	0.0958921
$716$	−38905.5	−2.03068
$717$	−25049.5	−1.30473
$718$	22331.5	1.16073
$719$	−25846.7	−1.34064	−0.670318	−	0.742074i	$-0.733842\pi$
$719$	−25846.7	−1.34064	−0.670318	+	0.742074i	$0.733842\pi$
$720$	40.6314	0.00210312
$721$	6198.41	0.320167
$722$	−1907.32	−0.0983147
$723$	11664.5	0.600012
$724$	33912.4	1.74081
$725$	7118.43	0.364651
$726$	−3324.77	−0.169964
$727$	17646.0	0.900211	0.450105	−	0.892975i	$-0.351386\pi$
$727$	17646.0	0.900211	0.450105	+	0.892975i	$0.351386\pi$
$728$	−13626.6	−0.693729
$729$	19648.7	0.998257
$730$	9106.56	0.461711
$731$	116.509	0.00589501
$732$	46669.9	2.35652
$733$	−15316.0	−0.771771	−0.385885	−	0.922547i	$-0.626104\pi$
$733$	−15316.0	−0.771771	−0.385885	+	0.922547i	$0.626104\pi$
$734$	54561.4	2.74373
$735$	−7822.57	−0.392571
$736$	21976.4	1.10063
$737$	8368.88	0.418279
$738$	−72.0622	−0.00359437
$739$	27023.9	1.34518	0.672590	−	0.740015i	$-0.265182\pi$
$739$	27023.9	1.34518	0.672590	+	0.740015i	$0.265182\pi$
$740$	−5062.18	−0.251472
$741$	−3293.76	−0.163292
$742$	−6004.93	−0.297100
$743$	−11964.3	−0.590751	−0.295376	−	0.955381i	$-0.595445\pi$
$743$	−11964.3	−0.590751	−0.295376	+	0.955381i	$0.595445\pi$
$744$	107260.	5.28543
$745$	−1831.69	−0.0900780
$746$	69460.5	3.40902
$747$	42.1134	0.00206271
$748$	101.724	0.00497245
$749$	7027.12	0.342811
$750$	−3434.68	−0.167222
$751$	−25446.6	−1.23643	−0.618216	−	0.786008i	$-0.712144\pi$
$751$	−25446.6	−1.23643	−0.618216	+	0.786008i	$0.712144\pi$
$752$	63269.3	3.06808
$753$	−37606.3	−1.81999
$754$	−50146.5	−2.42205
$755$	1698.91	0.0818936
$756$	−18127.9	−0.872097
$757$	−13722.5	−0.658856	−0.329428	−	0.944181i	$-0.606856\pi$
$757$	−13722.5	−0.658856	−0.329428	+	0.944181i	$0.606856\pi$
$758$	−65909.6	−3.15824
$759$	−3052.16	−0.145964
$760$	5980.36	0.285435
$761$	3443.27	0.164019	0.0820094	−	0.996632i	$-0.473866\pi$
$761$	3443.27	0.164019	0.0820094	+	0.996632i	$0.473866\pi$
$762$	−55608.6	−2.64368
$763$	−11296.8	−0.536005
$764$	28503.6	1.34977
$765$	0.108885	5.14607e−6 0
$766$	−48779.2	−2.30087
$767$	28775.7	1.35467
$768$	−8719.61	−0.409690
$769$	33476.3	1.56981	0.784906	−	0.619614i	$-0.212711\pi$
$769$	33476.3	1.56981	0.784906	+	0.619614i	$0.212711\pi$
$770$	−1887.05	−0.0883177
$771$	9729.06	0.454453
$772$	63684.6	2.96899
$773$	−20941.3	−0.974392	−0.487196	−	0.873293i	$-0.661980\pi$
$773$	−20941.3	−0.974392	−0.487196	+	0.873293i	$0.661980\pi$
$774$	−62.1678	−0.00288705
$775$	−8190.63	−0.379634
$776$	40949.9	1.89435
$777$	−1716.94	−0.0792726
$778$	−10674.6	−0.491907
$779$	−5525.88	−0.254153
$780$	17261.7	0.792395
$781$	−10259.8	−0.470071
$782$	130.897	0.00598574
$783$	−39912.7	−1.82167
$784$	−52127.9	−2.37463
$785$	−12296.1	−0.559066
$786$	3521.65	0.159813
$787$	7396.17	0.335000	0.167500	−	0.985872i	$-0.446431\pi$
$787$	7396.17	0.335000	0.167500	+	0.985872i	$0.446431\pi$
$788$	103145.	4.66291
$789$	27644.7	1.24737
$790$	−13364.3	−0.601873
$791$	−11205.8	−0.503705
$792$	−32.4742	−0.00145697
$793$	15020.4	0.672624
$794$	−52303.0	−2.33774
$795$	4551.09	0.203032
$796$	42891.7	1.90987
$797$	13465.0	0.598436	0.299218	−	0.954185i	$-0.403274\pi$
$797$	13465.0	0.598436	0.299218	+	0.954185i	$0.403274\pi$
$798$	3390.27	0.150394
$799$	169.550	0.00750721
$800$	−10297.7	−0.455099
$801$	47.9854	0.00211670
$802$	−41841.8	−1.84225
$803$	−3791.93	−0.166643
$804$	78796.9	3.45641
$805$	−1732.33	−0.0758467
$806$	57699.7	2.52157
$807$	28694.1	1.25165
$808$	34276.9	1.49240
$809$	−37001.8	−1.60805	−0.804027	−	0.594593i	$-0.797313\pi$
$809$	−37001.8	−1.60805	−0.804027	+	0.594593i	$0.797313\pi$
$810$	19291.5	0.836834
$811$	−233.315	−0.0101021	−0.00505105	−	0.999987i	$-0.501608\pi$
$811$	−233.315	−0.0101021	−0.00505105	+	0.999987i	$0.501608\pi$
$812$	36823.4	1.59144
$813$	9070.19	0.391274
$814$	2954.62	0.127223
$815$	9391.29	0.403635
$816$	418.469	0.0179526
$817$	−4767.16	−0.204139
$818$	53930.4	2.30517
$819$	10.1514	0.000433112 0
$820$	28959.7	1.23331
$821$	−3678.44	−0.156368	−0.0781842	−	0.996939i	$-0.524912\pi$
$821$	−3678.44	−0.156368	−0.0781842	+	0.996939i	$0.524912\pi$
$822$	−81694.6	−3.46645
$823$	−18693.6	−0.791757	−0.395879	−	0.918303i	$-0.629560\pi$
$823$	−18693.6	−0.791757	−0.395879	+	0.918303i	$0.629560\pi$
$824$	−60086.8	−2.54032
$825$	1430.18	0.0603546
$826$	−29618.8	−1.24766
$827$	−21055.1	−0.885318	−0.442659	−	0.896690i	$-0.645965\pi$
$827$	−21055.1	−0.885318	−0.442659	+	0.896690i	$0.645965\pi$
$828$	−49.8281	−0.00209136
$829$	−35047.0	−1.46831	−0.734156	−	0.678980i	$-0.762422\pi$
$829$	−35047.0	−1.46831	−0.734156	+	0.678980i	$0.762422\pi$
$830$	−23722.7	−0.992082
$831$	8320.77	0.347346
$832$	26335.1	1.09736
$833$	−139.693	−0.00581042
$834$	84950.4	3.52709
$835$	−5875.91	−0.243526
$836$	−4162.19	−0.172192
$837$	45924.5	1.89652
$838$	−21014.4	−0.866265
$839$	48090.9	1.97888	0.989441	−	0.144933i	$-0.0462966\pi$
$839$	48090.9	1.97888	0.989441	+	0.144933i	$0.0462966\pi$
$840$	−10630.1	−0.436634
$841$	56686.2	2.32425
$842$	5904.50	0.241666
$843$	−10132.8	−0.413989
$844$	71678.6	2.92332
$845$	−5429.44	−0.221039
$846$	−90.4698	−0.00367661
$847$	785.759	0.0318760
$848$	30327.5	1.22812
$849$	−32900.1	−1.32995
$850$	−61.3355	−0.00247505
$851$	2712.36	0.109258
$852$	−96601.0	−3.88438
$853$	20537.5	0.824372	0.412186	−	0.911100i	$-0.364765\pi$
$853$	20537.5	0.824372	0.412186	+	0.911100i	$0.364765\pi$
$854$	−15460.5	−0.619494
$855$	−4.45519	−0.000178204 0
$856$	−68120.3	−2.71998
$857$	−25977.2	−1.03543	−0.517716	−	0.855553i	$-0.673218\pi$
$857$	−25977.2	−1.03543	−0.517716	+	0.855553i	$0.673218\pi$
$858$	−10075.1	−0.400882
$859$	−9562.08	−0.379807	−0.189903	−	0.981803i	$-0.560817\pi$
$859$	−9562.08	−0.379807	−0.189903	+	0.981803i	$0.560817\pi$
$860$	24983.4	0.990612
$861$	9822.24	0.388782
$862$	−55794.6	−2.20461
$863$	−29562.4	−1.16607	−0.583033	−	0.812448i	$-0.698134\pi$
$863$	−29562.4	−1.16607	−0.583033	+	0.812448i	$0.698134\pi$
$864$	57738.9	2.27351
$865$	−1322.14	−0.0519700
$866$	3616.65	0.141915
$867$	−25549.7	−1.00082
$868$	−42369.9	−1.65683
$869$	5564.82	0.217231
$870$	−39119.2	−1.52444
$871$	25360.3	0.986567
$872$	109510.	4.25285
$873$	−30.5065	−0.00118269
$874$	−5355.83	−0.207281
$875$	811.735	0.0313619
$876$	−35702.8	−1.37704
$877$	−12298.0	−0.473516	−0.236758	−	0.971569i	$-0.576085\pi$
$877$	−12298.0	−0.473516	−0.236758	+	0.971569i	$0.576085\pi$
$878$	−41061.5	−1.57831
$879$	−8916.45	−0.342144
$880$	9530.42	0.365080
$881$	−18020.2	−0.689120	−0.344560	−	0.938764i	$-0.611972\pi$
$881$	−18020.2	−0.689120	−0.344560	+	0.938764i	$0.611972\pi$
$882$	74.5384	0.00284562
$883$	−15568.5	−0.593343	−0.296672	−	0.954980i	$-0.595877\pi$
$883$	−15568.5	−0.593343	−0.296672	+	0.954980i	$0.595877\pi$
$884$	308.255	0.0117282
$885$	22447.9	0.852629
$886$	22648.6	0.858796
$887$	−28124.1	−1.06462	−0.532308	−	0.846551i	$-0.678675\pi$
$887$	−28124.1	−1.06462	−0.532308	+	0.846551i	$0.678675\pi$
$888$	16643.9	0.628976
$889$	13142.3	0.495813
$890$	−27030.5	−1.01805
$891$	−8032.90	−0.302034
$892$	22795.2	0.855648
$893$	−6937.41	−0.259968
$894$	10066.0	0.376576
$895$	−9768.00	−0.364813
$896$	−5707.68	−0.212813
$897$	−9248.98	−0.344275
$898$	85738.2	3.18610
$899$	−93287.0	−3.46084
$900$	23.3485	0.000864758 0
$901$	81.2721	0.00300507
$902$	−16902.8	−0.623947
$903$	8473.61	0.312275
$904$	108628.	3.99657
$905$	8514.37	0.312737
$906$	−9336.33	−0.342361
$907$	−5100.64	−0.186730	−0.0933649	−	0.995632i	$-0.529762\pi$
$907$	−5100.64	−0.186730	−0.0933649	+	0.995632i	$0.529762\pi$
$908$	−28203.6	−1.03080
$909$	−25.5353	−0.000931742 0
$910$	−5718.35	−0.208309
$911$	8934.44	0.324930	0.162465	−	0.986714i	$-0.448056\pi$
$911$	8934.44	0.324930	0.162465	+	0.986714i	$0.448056\pi$
$912$	−17122.3	−0.621684
$913$	9878.02	0.358067
$914$	88265.3	3.19426
$915$	11717.4	0.423350
$916$	−5417.65	−0.195420
$917$	−832.290	−0.0299723
$918$	343.906	0.0123645
$919$	−30353.2	−1.08951	−0.544755	−	0.838595i	$-0.683377\pi$
$919$	−30353.2	−1.08951	−0.544755	+	0.838595i	$0.683377\pi$
$920$	16793.1	0.601794
$921$	2017.68	0.0721875
$922$	8280.82	0.295785
$923$	−31090.4	−1.10872
$924$	7398.29	0.263405
$925$	−1270.96	−0.0451772
$926$	24386.0	0.865414
$927$	44.7629	0.00158598
$928$	−117286.	−4.14880
$929$	−29544.1	−1.04339	−0.521696	−	0.853132i	$-0.674700\pi$
$929$	−29544.1	−1.04339	−0.521696	+	0.853132i	$0.674700\pi$
$930$	45011.5	1.58708
$931$	5715.76	0.201210
$932$	−18465.4	−0.648986
$933$	28752.4	1.00891
$934$	−17357.2	−0.608077
$935$	25.5398	0.000893306 0
$936$	−98.4068	−0.00343646
$937$	11183.0	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$937$	11183.0	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$938$	−26103.3	−0.908640
$939$	32428.7	1.12702
$940$	36357.1	1.26153
$941$	25213.0	0.873453	0.436727	−	0.899594i	$-0.356138\pi$
$941$	25213.0	0.873453	0.436727	+	0.899594i	$0.356138\pi$
$942$	67573.0	2.33721
$943$	−15516.9	−0.535842
$944$	149588.	5.15748
$945$	−4551.37	−0.156673
$946$	−14582.0	−0.501163
$947$	−38067.6	−1.30626	−0.653132	−	0.757244i	$-0.726545\pi$
$947$	−38067.6	−1.30626	−0.653132	+	0.757244i	$0.726545\pi$
$948$	52395.4	1.79507
$949$	−11490.7	−0.393049
$950$	2509.64	0.0857088
$951$	53332.6	1.81853
$952$	−189.829	−0.00646260
$953$	−20104.7	−0.683375	−0.341687	−	0.939814i	$-0.610998\pi$
$953$	−20104.7	−0.683375	−0.341687	+	0.939814i	$0.610998\pi$
$954$	−43.3657	−0.00147171
$955$	7156.40	0.242488
$956$	−95921.7	−3.24511
$957$	16289.0	0.550209
$958$	−29781.1	−1.00437
$959$	19307.3	0.650120
$960$	20544.0	0.690682
$961$	77547.3	2.60304
$962$	8953.40	0.300072
$963$	50.7476	0.00169815
$964$	44666.7	1.49234
$965$	15989.3	0.533382
$966$	9519.99	0.317081
$967$	−33790.7	−1.12372	−0.561858	−	0.827233i	$-0.689913\pi$
$967$	−33790.7	−1.12372	−0.561858	+	0.827233i	$0.689913\pi$
$968$	−7617.08	−0.252916
$969$	−45.8846	−0.00152118
$970$	17184.5	0.568826
$971$	−54354.0	−1.79640	−0.898200	−	0.439588i	$-0.855125\pi$
$971$	−54354.0	−1.79640	−0.898200	+	0.439588i	$0.855125\pi$
$972$	−262.056	−0.00864757
$973$	−20076.8	−0.661492
$974$	−71148.6	−2.34060
$975$	4333.89	0.142354
$976$	78082.2	2.56081
$977$	−41296.5	−1.35230	−0.676148	−	0.736766i	$-0.736352\pi$
$977$	−41296.5	−1.35230	−0.676148	+	0.736766i	$0.736352\pi$
$978$	−51609.7	−1.68742
$979$	11255.3	0.367439
$980$	−29954.8	−0.976398
$981$	−81.5820	−0.00265516
$982$	−11308.6	−0.367486
$983$	1830.83	0.0594043	0.0297022	−	0.999559i	$-0.490544\pi$
$983$	1830.83	0.0594043	0.0297022	+	0.999559i	$0.490544\pi$
$984$	−95216.0	−3.08473
$985$	25896.5	0.837697
$986$	−698.580	−0.0225632
$987$	12331.2	0.397677
$988$	−12612.7	−0.406137
$989$	−13386.3	−0.430395
$990$	−13.6277	−0.000437492 0
$991$	41921.7	1.34378	0.671890	−	0.740651i	$-0.265483\pi$
$991$	41921.7	1.34378	0.671890	+	0.740651i	$0.265483\pi$
$992$	134952.	4.31927
$993$	−15982.9	−0.510778
$994$	32001.4	1.02115
$995$	10768.8	0.343110
$996$	93006.2	2.95885
$997$	−23495.3	−0.746344	−0.373172	−	0.927762i	$-0.621730\pi$
$997$	−23495.3	−0.746344	−0.373172	+	0.927762i	$0.621730\pi$
$998$	20681.9	0.655985
$999$	7126.22	0.225689

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.e.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-5.28344 q^{2} +5.20066 q^{3} +19.9148 q^{4} +5.00000 q^{5} -27.4774 q^{6} +6.49388 q^{7} -62.9511 q^{8} +0.0468967 q^{9} +O(q^{10})\)	\(q-5.28344 q^{2} +5.20066 q^{3} +19.9148 q^{4} +5.00000 q^{5} -27.4774 q^{6} +6.49388 q^{7} -62.9511 q^{8} +0.0468967 q^{9} -26.4172 q^{10} +11.0000 q^{11} +103.570 q^{12} +33.3334 q^{13} -34.3100 q^{14} +26.0033 q^{15} +173.280 q^{16} +0.464360 q^{17} -0.247776 q^{18} -19.0000 q^{19} +99.5739 q^{20} +33.7725 q^{21} -58.1179 q^{22} -53.3527 q^{23} -327.388 q^{24} +25.0000 q^{25} -176.115 q^{26} -140.174 q^{27} +129.324 q^{28} +284.737 q^{29} -137.387 q^{30} -327.625 q^{31} -411.909 q^{32} +57.2073 q^{33} -2.45342 q^{34} +32.4694 q^{35} +0.933938 q^{36} -50.8384 q^{37} +100.385 q^{38} +173.356 q^{39} -314.756 q^{40} +290.836 q^{41} -178.435 q^{42} +250.903 q^{43} +219.063 q^{44} +0.234484 q^{45} +281.886 q^{46} +365.127 q^{47} +901.173 q^{48} -300.830 q^{49} -132.086 q^{50} +2.41498 q^{51} +663.827 q^{52} +175.020 q^{53} +740.602 q^{54} +55.0000 q^{55} -408.797 q^{56} -98.8126 q^{57} -1504.39 q^{58} +863.269 q^{59} +517.850 q^{60} +450.612 q^{61} +1730.99 q^{62} +0.304542 q^{63} +790.053 q^{64} +166.667 q^{65} -302.252 q^{66} +760.807 q^{67} +9.24763 q^{68} -277.469 q^{69} -171.550 q^{70} -932.711 q^{71} -2.95220 q^{72} -344.721 q^{73} +268.602 q^{74} +130.017 q^{75} -378.381 q^{76} +71.4327 q^{77} -915.915 q^{78} +505.893 q^{79} +866.402 q^{80} -730.264 q^{81} -1536.61 q^{82} +898.002 q^{83} +672.572 q^{84} +2.32180 q^{85} -1325.63 q^{86} +1480.82 q^{87} -692.462 q^{88} +1023.21 q^{89} -1.23888 q^{90} +216.463 q^{91} -1062.51 q^{92} -1703.87 q^{93} -1929.13 q^{94} -95.0000 q^{95} -2142.20 q^{96} -650.503 q^{97} +1589.42 q^{98} +0.515864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) \) 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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