LMFDB - Embedded newform 1045.4.a.e.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.620206 q^{2} +0.467922 q^{3} -7.61534 q^{4} +5.00000 q^{5} +0.290208 q^{6} -15.6454 q^{7} -9.68474 q^{8} -26.7810 q^{9} +O(q^{10})$	\(q+0.620206 q^{2} +0.467922 q^{3} -7.61534 q^{4} +5.00000 q^{5} +0.290208 q^{6} -15.6454 q^{7} -9.68474 q^{8} -26.7810 q^{9} +3.10103 q^{10} +11.0000 q^{11} -3.56339 q^{12} +20.4353 q^{13} -9.70338 q^{14} +2.33961 q^{15} +54.9162 q^{16} -120.529 q^{17} -16.6098 q^{18} -19.0000 q^{19} -38.0767 q^{20} -7.32083 q^{21} +6.82227 q^{22} -40.8460 q^{23} -4.53170 q^{24} +25.0000 q^{25} +12.6741 q^{26} -25.1654 q^{27} +119.145 q^{28} +104.374 q^{29} +1.45104 q^{30} -204.672 q^{31} +111.537 q^{32} +5.14714 q^{33} -74.7527 q^{34} -78.2270 q^{35} +203.947 q^{36} +108.882 q^{37} -11.7839 q^{38} +9.56213 q^{39} -48.4237 q^{40} -232.959 q^{41} -4.54043 q^{42} +143.702 q^{43} -83.7688 q^{44} -133.905 q^{45} -25.3330 q^{46} +18.2054 q^{47} +25.6965 q^{48} -98.2216 q^{49} +15.5052 q^{50} -56.3981 q^{51} -155.622 q^{52} +579.388 q^{53} -15.6077 q^{54} +55.0000 q^{55} +151.522 q^{56} -8.89052 q^{57} +64.7331 q^{58} +446.949 q^{59} -17.8169 q^{60} -83.3093 q^{61} -126.939 q^{62} +419.000 q^{63} -370.154 q^{64} +102.176 q^{65} +3.19229 q^{66} -715.708 q^{67} +917.867 q^{68} -19.1128 q^{69} -48.5169 q^{70} +791.794 q^{71} +259.367 q^{72} +498.506 q^{73} +67.5294 q^{74} +11.6981 q^{75} +144.692 q^{76} -172.099 q^{77} +5.93049 q^{78} +572.357 q^{79} +274.581 q^{80} +711.313 q^{81} -144.483 q^{82} +865.848 q^{83} +55.7506 q^{84} -602.643 q^{85} +89.1251 q^{86} +48.8387 q^{87} -106.532 q^{88} -240.529 q^{89} -83.0489 q^{90} -319.718 q^{91} +311.056 q^{92} -95.7706 q^{93} +11.2911 q^{94} -95.0000 q^{95} +52.1908 q^{96} +700.110 q^{97} -60.9177 q^{98} -294.592 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.620206	0.219276	0.109638	−	0.993972i	$-0.465031\pi$
$2$	0.620206	0.219276	0.109638	+	0.993972i	$0.465031\pi$
$3$	0.467922	0.0900517	0.0450258	−	0.998986i	$-0.485663\pi$
$3$	0.467922	0.0900517	0.0450258	+	0.998986i	$0.485663\pi$
$4$	−7.61534	−0.951918
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0.290208	0.0197462
$7$	−15.6454	−0.844772	−0.422386	−	0.906416i	$-0.638807\pi$
$7$	−15.6454	−0.844772	−0.422386	+	0.906416i	$0.638807\pi$
$8$	−9.68474	−0.428009
$9$	−26.7810	−0.991891
$10$	3.10103	0.0980633
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−3.56339	−0.0857218
$13$	20.4353	0.435979	0.217990	−	0.975951i	$-0.430050\pi$
$13$	20.4353	0.435979	0.217990	+	0.975951i	$0.430050\pi$
$14$	−9.70338	−0.185238
$15$	2.33961	0.0402723
$16$	54.9162	0.858066
$17$	−120.529	−1.71956	−0.859779	−	0.510666i	$-0.829399\pi$
$17$	−120.529	−1.71956	−0.859779	+	0.510666i	$0.829399\pi$
$18$	−16.6098	−0.217498
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−38.0767	−0.425711
$21$	−7.32083	−0.0760731
$22$	6.82227	0.0661142
$23$	−40.8460	−0.370304	−0.185152	−	0.982710i	$-0.559278\pi$
$23$	−40.8460	−0.370304	−0.185152	+	0.982710i	$0.559278\pi$
$24$	−4.53170	−0.0385429
$25$	25.0000	0.200000
$26$	12.6741	0.0955998
$27$	−25.1654	−0.179373
$28$	119.145	0.804154
$29$	104.374	0.668334	0.334167	−	0.942514i	$-0.391545\pi$
$29$	104.374	0.668334	0.334167	+	0.942514i	$0.391545\pi$
$30$	1.45104	0.00883076
$31$	−204.672	−1.18581	−0.592906	−	0.805272i	$-0.702019\pi$
$31$	−204.672	−1.18581	−0.592906	+	0.805272i	$0.702019\pi$
$32$	111.537	0.616162
$33$	5.14714	0.0271516
$34$	−74.7527	−0.377058
$35$	−78.2270	−0.377794
$36$	203.947	0.944199
$37$	108.882	0.483787	0.241893	−	0.970303i	$-0.422232\pi$
$37$	108.882	0.483787	0.241893	+	0.970303i	$0.422232\pi$
$38$	−11.7839	−0.0503054
$39$	9.56213	0.0392607
$40$	−48.4237	−0.191411
$41$	−232.959	−0.887369	−0.443685	−	0.896183i	$-0.646329\pi$
$41$	−232.959	−0.887369	−0.443685	+	0.896183i	$0.646329\pi$
$42$	−4.54043	−0.0166810
$43$	143.702	0.509637	0.254819	−	0.966989i	$-0.417984\pi$
$43$	143.702	0.509637	0.254819	+	0.966989i	$0.417984\pi$
$44$	−83.7688	−0.287014
$45$	−133.905	−0.443587
$46$	−25.3330	−0.0811987
$47$	18.2054	0.0565006	0.0282503	−	0.999601i	$-0.491006\pi$
$47$	18.2054	0.0565006	0.0282503	+	0.999601i	$0.491006\pi$
$48$	25.6965	0.0772703
$49$	−98.2216	−0.286360
$50$	15.5052	0.0438552
$51$	−56.3981	−0.154849
$52$	−155.622	−0.415016
$53$	579.388	1.50160	0.750802	−	0.660527i	$-0.229667\pi$
$53$	579.388	1.50160	0.750802	+	0.660527i	$0.229667\pi$
$54$	−15.6077	−0.0393322
$55$	55.0000	0.134840
$56$	151.522	0.361570
$57$	−8.89052	−0.0206593
$58$	64.7331	0.146550
$59$	446.949	0.986234	0.493117	−	0.869963i	$-0.335857\pi$
$59$	446.949	0.986234	0.493117	+	0.869963i	$0.335857\pi$
$60$	−17.8169	−0.0383360
$61$	−83.3093	−0.174863	−0.0874317	−	0.996171i	$-0.527866\pi$
$61$	−83.3093	−0.174863	−0.0874317	+	0.996171i	$0.527866\pi$
$62$	−126.939	−0.260020
$63$	419.000	0.837921
$64$	−370.154	−0.722956
$65$	102.176	0.194976
$66$	3.19229	0.00595370
$67$	−715.708	−1.30504	−0.652520	−	0.757772i	$-0.726288\pi$
$67$	−715.708	−1.30504	−0.652520	+	0.757772i	$0.726288\pi$
$68$	917.867	1.63688
$69$	−19.1128	−0.0333465
$70$	−48.5169	−0.0828411
$71$	791.794	1.32350	0.661751	−	0.749724i	$-0.269814\pi$
$71$	791.794	1.32350	0.661751	+	0.749724i	$0.269814\pi$
$72$	259.367	0.424538
$73$	498.506	0.799257	0.399628	−	0.916677i	$-0.369139\pi$
$73$	498.506	0.799257	0.399628	+	0.916677i	$0.369139\pi$
$74$	67.5294	0.106083
$75$	11.6981	0.0180103
$76$	144.692	0.218385
$77$	−172.099	−0.254708
$78$	5.93049	0.00860892
$79$	572.357	0.815129	0.407565	−	0.913176i	$-0.366378\pi$
$79$	572.357	0.815129	0.407565	+	0.913176i	$0.366378\pi$
$80$	274.581	0.383739
$81$	711.313	0.975738
$82$	−144.483	−0.194579
$83$	865.848	1.14505	0.572525	−	0.819887i	$-0.305964\pi$
$83$	865.848	1.14505	0.572525	+	0.819887i	$0.305964\pi$
$84$	55.7506	0.0724154
$85$	−602.643	−0.769010
$86$	89.1251	0.111751
$87$	48.8387	0.0601846
$88$	−106.532	−0.129050
$89$	−240.529	−0.286472	−0.143236	−	0.989689i	$-0.545751\pi$
$89$	−240.529	−0.286472	−0.143236	+	0.989689i	$0.545751\pi$
$90$	−83.0489	−0.0972680
$91$	−319.718	−0.368303
$92$	311.056	0.352499
$93$	−95.7706	−0.106784
$94$	11.2911	0.0123892
$95$	−95.0000	−0.102598
$96$	52.1908	0.0554865
$97$	700.110	0.732840	0.366420	−	0.930450i	$-0.380583\pi$
$97$	700.110	0.732840	0.366420	+	0.930450i	$0.380583\pi$
$98$	−60.9177	−0.0627920
$99$	−294.592	−0.299066
$100$	−190.384	−0.190384
$101$	−953.912	−0.939780	−0.469890	−	0.882725i	$-0.655706\pi$
$101$	−953.912	−0.939780	−0.469890	+	0.882725i	$0.655706\pi$
$102$	−34.9784	−0.0339547
$103$	−100.754	−0.0963840	−0.0481920	−	0.998838i	$-0.515346\pi$
$103$	−100.754	−0.0963840	−0.0481920	+	0.998838i	$0.515346\pi$
$104$	−197.910	−0.186603
$105$	−36.6041	−0.0340209
$106$	359.340	0.329266
$107$	149.828	0.135368	0.0676842	−	0.997707i	$-0.478439\pi$
$107$	149.828	0.135368	0.0676842	+	0.997707i	$0.478439\pi$
$108$	191.643	0.170748
$109$	2031.43	1.78510	0.892549	−	0.450950i	$-0.148915\pi$
$109$	2031.43	1.78510	0.892549	+	0.450950i	$0.148915\pi$
$110$	34.1114	0.0295672
$111$	50.9484	0.0435658
$112$	−859.186	−0.724870
$113$	−1047.97	−0.872428	−0.436214	−	0.899843i	$-0.643681\pi$
$113$	−1047.97	−0.872428	−0.436214	+	0.899843i	$0.643681\pi$
$114$	−5.51396	−0.00453008
$115$	−204.230	−0.165605
$116$	−794.840	−0.636199
$117$	−547.278	−0.432444
$118$	277.201	0.216258
$119$	1885.72	1.45263
$120$	−22.6585	−0.0172369
$121$	121.000	0.0909091
$122$	−51.6690	−0.0383434
$123$	−109.007	−0.0799091
$124$	1558.65	1.12880
$125$	125.000	0.0894427
$126$	259.867	0.183736
$127$	−704.211	−0.492036	−0.246018	−	0.969265i	$-0.579122\pi$
$127$	−704.211	−0.492036	−0.246018	+	0.969265i	$0.579122\pi$
$128$	−1121.87	−0.774689
$129$	67.2415	0.0458937
$130$	63.3705	0.0427535
$131$	633.636	0.422603	0.211302	−	0.977421i	$-0.432230\pi$
$131$	633.636	0.422603	0.211302	+	0.977421i	$0.432230\pi$
$132$	−39.1973	−0.0258461
$133$	297.263	0.193804
$134$	−443.887	−0.286164
$135$	−125.827	−0.0802181
$136$	1167.29	0.735987
$137$	−1749.15	−1.09081	−0.545403	−	0.838174i	$-0.683623\pi$
$137$	−1749.15	−1.09081	−0.545403	+	0.838174i	$0.683623\pi$
$138$	−11.8539	−0.00731208
$139$	947.190	0.577983	0.288992	−	0.957332i	$-0.406680\pi$
$139$	947.190	0.577983	0.288992	+	0.957332i	$0.406680\pi$
$140$	595.725	0.359628
$141$	8.51871	0.00508798
$142$	491.076	0.290212
$143$	224.788	0.131453
$144$	−1470.71	−0.851108
$145$	521.868	0.298888
$146$	309.177	0.175258
$147$	−45.9601	−0.0257872
$148$	−829.175	−0.460525
$149$	1577.32	0.867241	0.433620	−	0.901096i	$-0.357236\pi$
$149$	1577.32	0.867241	0.433620	+	0.901096i	$0.357236\pi$
$150$	7.25521	0.00394924
$151$	−1326.04	−0.714648	−0.357324	−	0.933981i	$-0.616311\pi$
$151$	−1326.04	−0.714648	−0.357324	+	0.933981i	$0.616311\pi$
$152$	184.010	0.0981920
$153$	3227.88	1.70561
$154$	−106.737	−0.0558514
$155$	−1023.36	−0.530311
$156$	−72.8189	−0.0373729
$157$	3411.05	1.73396	0.866978	−	0.498346i	$-0.166059\pi$
$157$	3411.05	1.73396	0.866978	+	0.498346i	$0.166059\pi$
$158$	354.980	0.178738
$159$	271.109	0.135222
$160$	557.686	0.275556
$161$	639.052	0.312822
$162$	441.161	0.213956
$163$	2262.21	1.08706	0.543528	−	0.839391i	$-0.317088\pi$
$163$	2262.21	1.08706	0.543528	+	0.839391i	$0.317088\pi$
$164$	1774.07	0.844703
$165$	25.7357	0.0121426
$166$	537.004	0.251082
$167$	2986.34	1.38377	0.691886	−	0.722007i	$-0.256780\pi$
$167$	2986.34	1.38377	0.691886	+	0.722007i	$0.256780\pi$
$168$	70.9003	0.0325600
$169$	−1779.40	−0.809922
$170$	−373.763	−0.168626
$171$	508.840	0.227555
$172$	−1094.34	−0.485133
$173$	594.135	0.261106	0.130553	−	0.991441i	$-0.458325\pi$
$173$	594.135	0.261106	0.130553	+	0.991441i	$0.458325\pi$
$174$	30.2901	0.0131970
$175$	−391.135	−0.168954
$176$	604.078	0.258717
$177$	209.137	0.0888120
$178$	−149.178	−0.0628165
$179$	−403.919	−0.168661	−0.0843305	−	0.996438i	$-0.526875\pi$
$179$	−403.919	−0.168661	−0.0843305	+	0.996438i	$0.526875\pi$
$180$	1019.73	0.422258
$181$	3694.87	1.51733	0.758667	−	0.651479i	$-0.225851\pi$
$181$	3694.87	1.51733	0.758667	+	0.651479i	$0.225851\pi$
$182$	−198.291	−0.0807600
$183$	−38.9823	−0.0157467
$184$	395.583	0.158493
$185$	544.410	0.216356
$186$	−59.3975	−0.0234153
$187$	−1325.82	−0.518466
$188$	−138.640	−0.0537840
$189$	393.722	0.151529
$190$	−58.9196	−0.0224973
$191$	−1476.91	−0.559505	−0.279753	−	0.960072i	$-0.590252\pi$
$191$	−1476.91	−0.559505	−0.279753	+	0.960072i	$0.590252\pi$
$192$	−173.203	−0.0651034
$193$	−2338.34	−0.872112	−0.436056	−	0.899920i	$-0.643625\pi$
$193$	−2338.34	−0.872112	−0.436056	+	0.899920i	$0.643625\pi$
$194$	434.213	0.160694
$195$	47.8106	0.0175579
$196$	747.991	0.272592
$197$	−780.284	−0.282197	−0.141099	−	0.989996i	$-0.545064\pi$
$197$	−780.284	−0.282197	−0.141099	+	0.989996i	$0.545064\pi$
$198$	−182.708	−0.0655781
$199$	3381.39	1.20453	0.602263	−	0.798298i	$-0.294266\pi$
$199$	3381.39	1.20453	0.602263	+	0.798298i	$0.294266\pi$
$200$	−242.118	−0.0856018
$201$	−334.896	−0.117521
$202$	−591.622	−0.206071
$203$	−1632.97	−0.564590
$204$	429.491	0.147404
$205$	−1164.80	−0.396844
$206$	−62.4880	−0.0211347
$207$	1093.90	0.367301
$208$	1122.23	0.374099
$209$	−209.000	−0.0691714
$210$	−22.7021	−0.00745998
$211$	1280.94	0.417930	0.208965	−	0.977923i	$-0.432990\pi$
$211$	1280.94	0.417930	0.208965	+	0.977923i	$0.432990\pi$
$212$	−4412.24	−1.42940
$213$	370.498	0.119184
$214$	92.9242	0.0296830
$215$	718.511	0.227917
$216$	243.720	0.0767733
$217$	3202.17	1.00174
$218$	1259.91	0.391429
$219$	233.262	0.0719744
$220$	−418.844	−0.128357
$221$	−2463.04	−0.749692
$222$	31.5985	0.00955294
$223$	−3751.78	−1.12663	−0.563314	−	0.826243i	$-0.690474\pi$
$223$	−3751.78	−1.12663	−0.563314	+	0.826243i	$0.690474\pi$
$224$	−1745.05	−0.520517
$225$	−669.526	−0.198378
$226$	−649.955	−0.191303
$227$	1369.73	0.400494	0.200247	−	0.979745i	$-0.435826\pi$
$227$	1369.73	0.400494	0.200247	+	0.979745i	$0.435826\pi$
$228$	67.7044	0.0196659
$229$	1711.05	0.493753	0.246876	−	0.969047i	$-0.420596\pi$
$229$	1711.05	0.493753	0.246876	+	0.969047i	$0.420596\pi$
$230$	−126.665	−0.0363132
$231$	−80.5291	−0.0229369
$232$	−1010.83	−0.286053
$233$	6729.54	1.89213	0.946066	−	0.323974i	$-0.105019\pi$
$233$	6729.54	1.89213	0.946066	+	0.323974i	$0.105019\pi$
$234$	−339.426	−0.0948246
$235$	91.0270	0.0252679
$236$	−3403.67	−0.938814
$237$	267.819	0.0734038
$238$	1169.53	0.318528
$239$	−3092.28	−0.836915	−0.418458	−	0.908236i	$-0.637429\pi$
$239$	−3092.28	−0.836915	−0.418458	+	0.908236i	$0.637429\pi$
$240$	128.483	0.0345563
$241$	552.186	0.147591	0.0737955	−	0.997273i	$-0.476489\pi$
$241$	552.186	0.147591	0.0737955	+	0.997273i	$0.476489\pi$
$242$	75.0450	0.0199342
$243$	1012.30	0.267240
$244$	634.429	0.166456
$245$	−491.108	−0.128064
$246$	−67.6068	−0.0175222
$247$	−388.270	−0.100020
$248$	1982.19	0.507538
$249$	405.150	0.103114
$250$	77.5258	0.0196127
$251$	−4617.15	−1.16108	−0.580542	−	0.814230i	$-0.697159\pi$
$251$	−4617.15	−1.16108	−0.580542	+	0.814230i	$0.697159\pi$
$252$	−3190.83	−0.797632
$253$	−449.306	−0.111651
$254$	−436.756	−0.107892
$255$	−281.990	−0.0692506
$256$	2265.44	0.553085
$257$	−2673.29	−0.648852	−0.324426	−	0.945911i	$-0.605171\pi$
$257$	−2673.29	−0.648852	−0.324426	+	0.945911i	$0.605171\pi$
$258$	41.7036	0.0100634
$259$	−1703.50	−0.408689
$260$	−778.109	−0.185601
$261$	−2795.23	−0.662914
$262$	392.985	0.0926668
$263$	−567.189	−0.132982	−0.0664912	−	0.997787i	$-0.521180\pi$
$263$	−567.189	−0.132982	−0.0664912	+	0.997787i	$0.521180\pi$
$264$	−49.8487	−0.0116211
$265$	2896.94	0.671538
$266$	184.364	0.0424966
$267$	−112.549	−0.0257973
$268$	5450.36	1.24229
$269$	−7127.17	−1.61543	−0.807716	−	0.589571i	$-0.799297\pi$
$269$	−7127.17	−1.61543	−0.807716	+	0.589571i	$0.799297\pi$
$270$	−78.0386	−0.0175899
$271$	2026.68	0.454288	0.227144	−	0.973861i	$-0.427061\pi$
$271$	2026.68	0.454288	0.227144	+	0.973861i	$0.427061\pi$
$272$	−6618.98	−1.47549
$273$	−149.603	−0.0331663
$274$	−1084.84	−0.239187
$275$	275.000	0.0603023
$276$	145.550	0.0317431
$277$	4459.77	0.967371	0.483686	−	0.875242i	$-0.339298\pi$
$277$	4459.77	0.967371	0.483686	+	0.875242i	$0.339298\pi$
$278$	587.454	0.126738
$279$	5481.33	1.17620
$280$	757.608	0.161699
$281$	−5260.97	−1.11688	−0.558439	−	0.829545i	$-0.688600\pi$
$281$	−5260.97	−1.11688	−0.558439	+	0.829545i	$0.688600\pi$
$282$	5.28336	0.00111567
$283$	−5645.74	−1.18588	−0.592941	−	0.805246i	$-0.702033\pi$
$283$	−5645.74	−1.18588	−0.592941	+	0.805246i	$0.702033\pi$
$284$	−6029.78	−1.25987
$285$	−44.4526	−0.00923911
$286$	139.415	0.0288244
$287$	3644.74	0.749625
$288$	−2987.09	−0.611166
$289$	9614.16	1.95688
$290$	323.666	0.0655390
$291$	327.597	0.0659934
$292$	−3796.30	−0.760827
$293$	−1450.04	−0.289121	−0.144560	−	0.989496i	$-0.546177\pi$
$293$	−1450.04	−0.289121	−0.144560	+	0.989496i	$0.546177\pi$
$294$	−28.5047	−0.00565452
$295$	2234.74	0.441057
$296$	−1054.49	−0.207065
$297$	−276.819	−0.0540830
$298$	978.262	0.190165
$299$	−834.700	−0.161445
$300$	−89.0847	−0.0171444
$301$	−2248.28	−0.430527
$302$	−822.420	−0.156705
$303$	−446.357	−0.0846287
$304$	−1043.41	−0.196854
$305$	−416.547	−0.0782013
$306$	2001.95	0.374000
$307$	−8303.31	−1.54363	−0.771816	−	0.635846i	$-0.780651\pi$
$307$	−8303.31	−1.54363	−0.771816	+	0.635846i	$0.780651\pi$
$308$	1310.60	0.242461
$309$	−47.1449	−0.00867954
$310$	−634.695	−0.116285
$311$	4875.33	0.888922	0.444461	−	0.895798i	$-0.353395\pi$
$311$	4875.33	0.888922	0.444461	+	0.895798i	$0.353395\pi$
$312$	−92.6067	−0.0168039
$313$	3031.37	0.547423	0.273711	−	0.961812i	$-0.411749\pi$
$313$	3031.37	0.547423	0.273711	+	0.961812i	$0.411749\pi$
$314$	2115.55	0.380215
$315$	2095.00	0.374730
$316$	−4358.70	−0.775936
$317$	5615.48	0.994942	0.497471	−	0.867481i	$-0.334262\pi$
$317$	5615.48	0.994942	0.497471	+	0.867481i	$0.334262\pi$
$318$	168.143	0.0296510
$319$	1148.11	0.201510
$320$	−1850.77	−0.323316
$321$	70.1078	0.0121901
$322$	396.344	0.0685944
$323$	2290.04	0.394494
$324$	−5416.89	−0.928822
$325$	510.882	0.0871958
$326$	1403.04	0.238365
$327$	950.551	0.160751
$328$	2256.15	0.379802
$329$	−284.831	−0.0477301
$330$	15.9615	0.00266257
$331$	390.623	0.0648658	0.0324329	−	0.999474i	$-0.489674\pi$
$331$	390.623	0.0648658	0.0324329	+	0.999474i	$0.489674\pi$
$332$	−6593.73	−1.08999
$333$	−2915.98	−0.479863
$334$	1852.15	0.303428
$335$	−3578.54	−0.583631
$336$	−402.032	−0.0652758
$337$	111.421	0.0180103	0.00900515	−	0.999959i	$-0.497134\pi$
$337$	111.421	0.0180103	0.00900515	+	0.999959i	$0.497134\pi$
$338$	−1103.59	−0.177597
$339$	−490.367	−0.0785636
$340$	4589.34	0.732034
$341$	−2251.39	−0.357536
$342$	315.586	0.0498974
$343$	6903.09	1.08668
$344$	−1391.72	−0.218129
$345$	−95.5638	−0.0149130
$346$	368.487	0.0572542
$347$	−7131.73	−1.10332	−0.551659	−	0.834070i	$-0.686005\pi$
$347$	−7131.73	−1.10332	−0.551659	+	0.834070i	$0.686005\pi$
$348$	−371.924	−0.0572908
$349$	9055.35	1.38889	0.694444	−	0.719547i	$-0.255650\pi$
$349$	9055.35	1.38889	0.694444	+	0.719547i	$0.255650\pi$
$350$	−242.584	−0.0370477
$351$	−514.261	−0.0782029
$352$	1226.91	0.185780
$353$	−3832.29	−0.577825	−0.288913	−	0.957355i	$-0.593294\pi$
$353$	−3832.29	−0.577825	−0.288913	+	0.957355i	$0.593294\pi$
$354$	129.708	0.0194744
$355$	3958.97	0.591888
$356$	1831.71	0.272698
$357$	882.370	0.130812
$358$	−250.513	−0.0369833
$359$	883.473	0.129883	0.0649414	−	0.997889i	$-0.479314\pi$
$359$	883.473	0.129883	0.0649414	+	0.997889i	$0.479314\pi$
$360$	1296.84	0.189859
$361$	361.000	0.0526316
$362$	2291.58	0.332715
$363$	56.6186	0.00818652
$364$	2434.76	0.350594
$365$	2492.53	0.357439
$366$	−24.1771	−0.00345288
$367$	6096.33	0.867101	0.433550	−	0.901129i	$-0.357261\pi$
$367$	6096.33	0.867101	0.433550	+	0.901129i	$0.357261\pi$
$368$	−2243.11	−0.317745
$369$	6238.90	0.880173
$370$	337.647	0.0474417
$371$	−9064.75	−1.26851
$372$	729.326	0.101650
$373$	−9652.16	−1.33987	−0.669933	−	0.742422i	$-0.733677\pi$
$373$	−9652.16	−1.33987	−0.669933	+	0.742422i	$0.733677\pi$
$374$	−822.279	−0.113687
$375$	58.4903	0.00805447
$376$	−176.314	−0.0241828
$377$	2132.90	0.291380
$378$	244.189	0.0332268
$379$	−9120.77	−1.23615	−0.618077	−	0.786117i	$-0.712088\pi$
$379$	−9120.77	−1.23615	−0.618077	+	0.786117i	$0.712088\pi$
$380$	723.458	0.0976647
$381$	−329.516	−0.0443087
$382$	−915.990	−0.122686
$383$	12661.6	1.68923	0.844615	−	0.535374i	$-0.179829\pi$
$383$	12661.6	1.68923	0.844615	+	0.535374i	$0.179829\pi$
$384$	−524.948	−0.0697621
$385$	−860.497	−0.113909
$386$	−1450.26	−0.191233
$387$	−3848.50	−0.505504
$388$	−5331.58	−0.697603
$389$	−8462.30	−1.10297	−0.551485	−	0.834185i	$-0.685939\pi$
$389$	−8462.30	−1.10297	−0.551485	+	0.834185i	$0.685939\pi$
$390$	29.6525	0.00385003
$391$	4923.12	0.636759
$392$	951.250	0.122565
$393$	296.492	0.0380561
$394$	−483.937	−0.0618792
$395$	2861.79	0.364537
$396$	2243.42	0.284687
$397$	−8686.54	−1.09815	−0.549074	−	0.835773i	$-0.685020\pi$
$397$	−8686.54	−1.09815	−0.549074	+	0.835773i	$0.685020\pi$
$398$	2097.16	0.264124
$399$	139.096	0.0174524
$400$	1372.91	0.171613
$401$	2151.67	0.267953	0.133976	−	0.990985i	$-0.457225\pi$
$401$	2151.67	0.267953	0.133976	+	0.990985i	$0.457225\pi$
$402$	−207.704	−0.0257695
$403$	−4182.53	−0.516989
$404$	7264.37	0.894593
$405$	3556.56	0.436363
$406$	−1012.78	−0.123801
$407$	1197.70	0.145867
$408$	546.200	0.0662768
$409$	−5660.61	−0.684350	−0.342175	−	0.939636i	$-0.611164\pi$
$409$	−5660.61	−0.684350	−0.342175	+	0.939636i	$0.611164\pi$
$410$	−722.415	−0.0870183
$411$	−818.468	−0.0982288
$412$	767.273	0.0917496
$413$	−6992.69	−0.833143
$414$	678.443	0.0805403
$415$	4329.24	0.512082
$416$	2279.30	0.268634
$417$	443.211	0.0520483
$418$	−129.623	−0.0151676
$419$	13507.0	1.57484	0.787422	−	0.616414i	$-0.211415\pi$
$419$	13507.0	1.57484	0.787422	+	0.616414i	$0.211415\pi$
$420$	278.753	0.0323851
$421$	−9456.11	−1.09469	−0.547343	−	0.836908i	$-0.684361\pi$
$421$	−9456.11	−1.09469	−0.547343	+	0.836908i	$0.684361\pi$
$422$	794.445	0.0916422
$423$	−487.560	−0.0560425
$424$	−5611.22	−0.642700
$425$	−3013.22	−0.343912
$426$	229.785	0.0261341
$427$	1303.41	0.147720
$428$	−1140.99	−0.128860
$429$	105.183	0.0118375
$430$	445.625	0.0499767
$431$	15176.4	1.69610	0.848051	−	0.529915i	$-0.177776\pi$
$431$	15176.4	1.69610	0.848051	+	0.529915i	$0.177776\pi$
$432$	−1381.99	−0.153914
$433$	1167.64	0.129592	0.0647958	−	0.997899i	$-0.479360\pi$
$433$	1167.64	0.129592	0.0647958	+	0.997899i	$0.479360\pi$
$434$	1986.01	0.219658
$435$	244.194	0.0269154
$436$	−15470.0	−1.69927
$437$	776.074	0.0849535
$438$	144.671	0.0157823
$439$	−2116.52	−0.230105	−0.115053	−	0.993359i	$-0.536704\pi$
$439$	−2116.52	−0.230105	−0.115053	+	0.993359i	$0.536704\pi$
$440$	−532.661	−0.0577127
$441$	2630.48	0.284038
$442$	−1527.59	−0.164389
$443$	3972.64	0.426063	0.213032	−	0.977045i	$-0.431666\pi$
$443$	3972.64	0.426063	0.213032	+	0.977045i	$0.431666\pi$
$444$	−387.989	−0.0414711
$445$	−1202.65	−0.128114
$446$	−2326.88	−0.247043
$447$	738.062	0.0780965
$448$	5791.20	0.610733
$449$	−5798.49	−0.609460	−0.304730	−	0.952439i	$-0.598566\pi$
$449$	−5798.49	−0.609460	−0.304730	+	0.952439i	$0.598566\pi$
$450$	−415.244	−0.0434996
$451$	−2562.55	−0.267552
$452$	7980.62	0.830480
$453$	−620.485	−0.0643552
$454$	849.514	0.0878187
$455$	−1598.59	−0.164710
$456$	86.1024	0.00884235
$457$	−46.2277	−0.00473182	−0.00236591	−	0.999997i	$-0.500753\pi$
$457$	−46.2277	−0.00473182	−0.00236591	+	0.999997i	$0.500753\pi$
$458$	1061.20	0.108268
$459$	3033.15	0.308443
$460$	1555.28	0.157642
$461$	−4489.07	−0.453529	−0.226765	−	0.973950i	$-0.572815\pi$
$461$	−4489.07	−0.453529	−0.226765	+	0.973950i	$0.572815\pi$
$462$	−49.9447	−0.00502952
$463$	5583.41	0.560438	0.280219	−	0.959936i	$-0.409593\pi$
$463$	5583.41	0.560438	0.280219	+	0.959936i	$0.409593\pi$
$464$	5731.80	0.573474
$465$	−478.853	−0.0477554
$466$	4173.70	0.414899
$467$	−8272.61	−0.819723	−0.409862	−	0.912148i	$-0.634423\pi$
$467$	−8272.61	−0.819723	−0.409862	+	0.912148i	$0.634423\pi$
$468$	4167.71	0.411651
$469$	11197.5	1.10246
$470$	56.4555	0.00554064
$471$	1596.10	0.156146
$472$	−4328.58	−0.422117
$473$	1580.73	0.153661
$474$	166.103	0.0160957
$475$	−475.000	−0.0458831
$476$	−14360.4	−1.38279
$477$	−15516.6	−1.48943
$478$	−1917.85	−0.183516
$479$	7289.16	0.695303	0.347652	−	0.937624i	$-0.386979\pi$
$479$	7289.16	0.695303	0.347652	+	0.937624i	$0.386979\pi$
$480$	260.954	0.0248143
$481$	2225.04	0.210921
$482$	342.469	0.0323632
$483$	299.027	0.0281702
$484$	−921.457	−0.0865380
$485$	3500.55	0.327736
$486$	627.837	0.0585993
$487$	−8424.92	−0.783921	−0.391960	−	0.919982i	$-0.628203\pi$
$487$	−8424.92	−0.783921	−0.391960	+	0.919982i	$0.628203\pi$
$488$	806.829	0.0748431
$489$	1058.54	0.0978912
$490$	−304.588	−0.0280814
$491$	−14563.6	−1.33859	−0.669295	−	0.742997i	$-0.733404\pi$
$491$	−14563.6	−1.33859	−0.669295	+	0.742997i	$0.733404\pi$
$492$	830.125	0.0760669
$493$	−12580.0	−1.14924
$494$	−240.808	−0.0219321
$495$	−1472.96	−0.133747
$496$	−11239.8	−1.01750
$497$	−12387.9	−1.11806
$498$	251.276	0.0226104
$499$	−16042.9	−1.43924	−0.719618	−	0.694370i	$-0.755683\pi$
$499$	−16042.9	−1.43924	−0.719618	+	0.694370i	$0.755683\pi$
$500$	−951.918	−0.0851421
$501$	1397.37	0.124611
$502$	−2863.59	−0.254598
$503$	−16118.4	−1.42879	−0.714396	−	0.699742i	$-0.753298\pi$
$503$	−16118.4	−1.42879	−0.714396	+	0.699742i	$0.753298\pi$
$504$	−4057.91	−0.358638
$505$	−4769.56	−0.420282
$506$	−278.663	−0.0244823
$507$	−832.620	−0.0729349
$508$	5362.81	0.468378
$509$	854.944	0.0744494	0.0372247	−	0.999307i	$-0.488148\pi$
$509$	854.944	0.0744494	0.0372247	+	0.999307i	$0.488148\pi$
$510$	−174.892	−0.0151850
$511$	−7799.33	−0.675190
$512$	10380.0	0.895968
$513$	478.142	0.0411510
$514$	−1657.99	−0.142278
$515$	−503.768	−0.0431042
$516$	−512.067	−0.0436870
$517$	200.259	0.0170356
$518$	−1056.52	−0.0896158
$519$	278.009	0.0235130
$520$	−989.552	−0.0834514
$521$	−7583.38	−0.637685	−0.318843	−	0.947808i	$-0.603294\pi$
$521$	−7583.38	−0.637685	−0.318843	+	0.947808i	$0.603294\pi$
$522$	−1733.62	−0.145361
$523$	1531.17	0.128018	0.0640090	−	0.997949i	$-0.479611\pi$
$523$	1531.17	0.128018	0.0640090	+	0.997949i	$0.479611\pi$
$524$	−4825.35	−0.402284
$525$	−183.021	−0.0152146
$526$	−351.774	−0.0291598
$527$	24668.8	2.03907
$528$	282.662	0.0232979
$529$	−10498.6	−0.862875
$530$	1796.70	0.147252
$531$	−11969.8	−0.978236
$532$	−2263.76	−0.184485
$533$	−4760.59	−0.386874
$534$	−69.8036	−0.00565674
$535$	749.140	0.0605386
$536$	6931.44	0.558569
$537$	−189.003	−0.0151882
$538$	−4420.32	−0.354226
$539$	−1080.44	−0.0863409
$540$	958.214	0.0763610
$541$	−12332.8	−0.980087	−0.490044	−	0.871698i	$-0.663019\pi$
$541$	−12332.8	−0.980087	−0.490044	+	0.871698i	$0.663019\pi$
$542$	1256.96	0.0996145
$543$	1728.91	0.136638
$544$	−13443.4	−1.05953
$545$	10157.2	0.798320
$546$	−92.7849	−0.00727258
$547$	17511.5	1.36881	0.684405	−	0.729102i	$-0.260062\pi$
$547$	17511.5	1.36881	0.684405	+	0.729102i	$0.260062\pi$
$548$	13320.4	1.03836
$549$	2231.11	0.173445
$550$	170.557	0.0132228
$551$	−1983.10	−0.153326
$552$	185.102	0.0142726
$553$	−8954.75	−0.688598
$554$	2765.98	0.212121
$555$	254.742	0.0194832
$556$	−7213.18	−0.550192
$557$	−11816.1	−0.898858	−0.449429	−	0.893316i	$-0.648373\pi$
$557$	−11816.1	−0.898858	−0.449429	+	0.893316i	$0.648373\pi$
$558$	3399.56	0.257912
$559$	2936.60	0.222191
$560$	−4295.93	−0.324172
$561$	−620.379	−0.0466888
$562$	−3262.88	−0.244905
$563$	13697.8	1.02539	0.512695	−	0.858571i	$-0.328647\pi$
$563$	13697.8	1.02539	0.512695	+	0.858571i	$0.328647\pi$
$564$	−64.8729	−0.00484334
$565$	−5239.83	−0.390162
$566$	−3501.53	−0.260036
$567$	−11128.8	−0.824276
$568$	−7668.32	−0.566471
$569$	−6206.10	−0.457247	−0.228623	−	0.973515i	$-0.573422\pi$
$569$	−6206.10	−0.457247	−0.228623	+	0.973515i	$0.573422\pi$
$570$	−27.5698	−0.00202592
$571$	−1657.10	−0.121449	−0.0607245	−	0.998155i	$-0.519341\pi$
$571$	−1657.10	−0.121449	−0.0607245	+	0.998155i	$0.519341\pi$
$572$	−1711.84	−0.125132
$573$	−691.080	−0.0503844
$574$	2260.49	0.164375
$575$	−1021.15	−0.0740607
$576$	9913.10	0.717094
$577$	−1778.71	−0.128334	−0.0641670	−	0.997939i	$-0.520439\pi$
$577$	−1778.71	−0.128334	−0.0641670	+	0.997939i	$0.520439\pi$
$578$	5962.76	0.429097
$579$	−1094.16	−0.0785351
$580$	−3974.20	−0.284517
$581$	−13546.5	−0.967306
$582$	203.178	0.0144708
$583$	6373.27	0.452751
$584$	−4827.90	−0.342089
$585$	−2736.39	−0.193395
$586$	−899.325	−0.0633972
$587$	−6545.19	−0.460219	−0.230110	−	0.973165i	$-0.573909\pi$
$587$	−6545.19	−0.460219	−0.230110	+	0.973165i	$0.573909\pi$
$588$	350.002	0.0245473
$589$	3888.77	0.272044
$590$	1386.00	0.0967133
$591$	−365.112	−0.0254124
$592$	5979.39	0.415121
$593$	20783.5	1.43925	0.719625	−	0.694363i	$-0.244314\pi$
$593$	20783.5	1.43925	0.719625	+	0.694363i	$0.244314\pi$
$594$	−171.685	−0.0118591
$595$	9428.59	0.649638
$596$	−12011.8	−0.825542
$597$	1582.23	0.108470
$598$	−517.686	−0.0354009
$599$	−18284.9	−1.24724	−0.623622	−	0.781726i	$-0.714340\pi$
$599$	−18284.9	−1.24724	−0.623622	+	0.781726i	$0.714340\pi$
$600$	−113.293	−0.00770859
$601$	−20973.1	−1.42348	−0.711738	−	0.702445i	$-0.752092\pi$
$601$	−20973.1	−1.42348	−0.711738	+	0.702445i	$0.752092\pi$
$602$	−1394.40	−0.0944043
$603$	19167.4	1.29446
$604$	10098.3	0.680286
$605$	605.000	0.0406558
$606$	−276.833	−0.0185571
$607$	20837.5	1.39336	0.696680	−	0.717382i	$-0.254660\pi$
$607$	20837.5	1.39336	0.696680	+	0.717382i	$0.254660\pi$
$608$	−2119.21	−0.141357
$609$	−764.101	−0.0508422
$610$	−258.345	−0.0171477
$611$	372.032	0.0246331
$612$	−24581.4	−1.62360
$613$	24765.6	1.63176	0.815882	−	0.578219i	$-0.196252\pi$
$613$	24765.6	1.63176	0.815882	+	0.578219i	$0.196252\pi$
$614$	−5149.77	−0.338482
$615$	−545.034	−0.0357364
$616$	1666.74	0.109017
$617$	20462.9	1.33518	0.667591	−	0.744528i	$-0.267326\pi$
$617$	20462.9	1.33518	0.667591	+	0.744528i	$0.267326\pi$
$618$	−29.2395	−0.00190322
$619$	5995.46	0.389302	0.194651	−	0.980873i	$-0.437643\pi$
$619$	5995.46	0.389302	0.194651	+	0.980873i	$0.437643\pi$
$620$	7793.24	0.504813
$621$	1027.90	0.0664225
$622$	3023.71	0.194919
$623$	3763.17	0.242004
$624$	525.116	0.0336882
$625$	625.000	0.0400000
$626$	1880.08	0.120037
$627$	−97.7958	−0.00622901
$628$	−25976.3	−1.65058
$629$	−13123.4	−0.831900
$630$	1299.33	0.0821693
$631$	21980.5	1.38673	0.693367	−	0.720584i	$-0.256126\pi$
$631$	21980.5	1.38673	0.693367	+	0.720584i	$0.256126\pi$
$632$	−5543.13	−0.348883
$633$	599.379	0.0376353
$634$	3482.76	0.218167
$635$	−3521.06	−0.220045
$636$	−2064.58	−0.128720
$637$	−2007.19	−0.124847
$638$	712.065	0.0441864
$639$	−21205.1	−1.31277
$640$	−5609.35	−0.346452
$641$	−610.205	−0.0376001	−0.0188000	−	0.999823i	$-0.505985\pi$
$641$	−610.205	−0.0376001	−0.0188000	+	0.999823i	$0.505985\pi$
$642$	43.4813	0.00267301
$643$	29134.2	1.78684	0.893421	−	0.449220i	$-0.148298\pi$
$643$	29134.2	1.78684	0.893421	+	0.449220i	$0.148298\pi$
$644$	−4866.60	−0.297781
$645$	336.207	0.0205243
$646$	1420.30	0.0865031
$647$	8455.29	0.513774	0.256887	−	0.966441i	$-0.417303\pi$
$647$	8455.29	0.513774	0.256887	+	0.966441i	$0.417303\pi$
$648$	−6888.88	−0.417625
$649$	4916.44	0.297361
$650$	316.852	0.0191200
$651$	1498.37	0.0902085
$652$	−17227.5	−1.03479
$653$	−11470.0	−0.687375	−0.343688	−	0.939084i	$-0.611676\pi$
$653$	−11470.0	−0.687375	−0.343688	+	0.939084i	$0.611676\pi$
$654$	589.538	0.0352489
$655$	3168.18	0.188994
$656$	−12793.2	−0.761421
$657$	−13350.5	−0.792775
$658$	−176.654	−0.0104661
$659$	1140.79	0.0674336	0.0337168	−	0.999431i	$-0.489266\pi$
$659$	1140.79	0.0674336	0.0337168	+	0.999431i	$0.489266\pi$
$660$	−195.986	−0.0115587
$661$	−26659.5	−1.56874	−0.784369	−	0.620295i	$-0.787013\pi$
$661$	−26659.5	−1.56874	−0.784369	+	0.620295i	$0.787013\pi$
$662$	242.267	0.0142235
$663$	−1152.51	−0.0675110
$664$	−8385.51	−0.490092
$665$	1486.31	0.0866718
$666$	−1808.51	−0.105223
$667$	−4263.24	−0.247486
$668$	−22742.0	−1.31724
$669$	−1755.54	−0.101455
$670$	−2219.43	−0.127976
$671$	−916.402	−0.0527233
$672$	−816.545	−0.0468734
$673$	−270.534	−0.0154953	−0.00774763	−	0.999970i	$-0.502466\pi$
$673$	−270.534	−0.0154953	−0.00774763	+	0.999970i	$0.502466\pi$
$674$	69.1038	0.00394923
$675$	−629.134	−0.0358746
$676$	13550.7	0.770980
$677$	7961.22	0.451956	0.225978	−	0.974132i	$-0.427442\pi$
$677$	7961.22	0.451956	0.225978	+	0.974132i	$0.427442\pi$
$678$	−304.128	−0.0172271
$679$	−10953.5	−0.619082
$680$	5836.44	0.329143
$681$	640.926	0.0360651
$682$	−1396.33	−0.0783991
$683$	22371.9	1.25335	0.626674	−	0.779281i	$-0.284416\pi$
$683$	22371.9	1.25335	0.626674	+	0.779281i	$0.284416\pi$
$684$	−3874.99	−0.216614
$685$	−8745.77	−0.487823
$686$	4281.34	0.238283
$687$	800.638	0.0444633
$688$	7891.59	0.437302
$689$	11840.0	0.654668
$690$	−59.2693	−0.00327006
$691$	25634.9	1.41128	0.705641	−	0.708570i	$-0.250659\pi$
$691$	25634.9	1.41128	0.705641	+	0.708570i	$0.250659\pi$
$692$	−4524.54	−0.248551
$693$	4609.00	0.252643
$694$	−4423.15	−0.241931
$695$	4735.95	0.258482
$696$	−472.990	−0.0257595
$697$	28078.3	1.52588
$698$	5616.19	0.304550
$699$	3148.90	0.170390
$700$	2978.63	0.160831
$701$	31256.2	1.68407	0.842034	−	0.539425i	$-0.181358\pi$
$701$	31256.2	1.68407	0.842034	+	0.539425i	$0.181358\pi$
$702$	−318.948	−0.0171480
$703$	−2068.76	−0.110988
$704$	−4071.69	−0.217979
$705$	42.5935	0.00227541
$706$	−2376.81	−0.126703
$707$	14924.3	0.793900
$708$	−1592.65	−0.0845418
$709$	−1533.32	−0.0812201	−0.0406100	−	0.999175i	$-0.512930\pi$
$709$	−1533.32	−0.0812201	−0.0406100	+	0.999175i	$0.512930\pi$
$710$	2455.38	0.129787
$711$	−15328.3	−0.808519
$712$	2329.46	0.122613
$713$	8360.04	0.439111
$714$	547.251	0.0286840
$715$	1123.94	0.0587874
$716$	3075.98	0.160551
$717$	−1446.95	−0.0753656
$718$	547.936	0.0284802
$719$	4073.93	0.211310	0.105655	−	0.994403i	$-0.466306\pi$
$719$	4073.93	0.211310	0.105655	+	0.994403i	$0.466306\pi$
$720$	−7353.57	−0.380627
$721$	1576.33	0.0814225
$722$	223.895	0.0115408
$723$	258.380	0.0132908
$724$	−28137.7	−1.44438
$725$	2609.34	0.133667
$726$	35.1152	0.00179511
$727$	11295.4	0.576236	0.288118	−	0.957595i	$-0.406970\pi$
$727$	11295.4	0.576236	0.288118	+	0.957595i	$0.406970\pi$
$728$	3096.39	0.157637
$729$	−18731.8	−0.951672
$730$	1545.88	0.0783777
$731$	−17320.2	−0.876351
$732$	296.863	0.0149896
$733$	38966.7	1.96353	0.981766	−	0.190095i	$-0.0608796\pi$
$733$	38966.7	1.96353	0.981766	+	0.190095i	$0.0608796\pi$
$734$	3780.98	0.190134
$735$	−229.800	−0.0115324
$736$	−4555.85	−0.228167
$737$	−7872.79	−0.393484
$738$	3869.40	0.193001
$739$	−23392.5	−1.16442	−0.582210	−	0.813038i	$-0.697812\pi$
$739$	−23392.5	−1.16442	−0.582210	+	0.813038i	$0.697812\pi$
$740$	−4145.87	−0.205953
$741$	−181.680	−0.00900701
$742$	−5622.02	−0.278155
$743$	33004.6	1.62964	0.814819	−	0.579716i	$-0.196837\pi$
$743$	33004.6	1.62964	0.814819	+	0.579716i	$0.196837\pi$
$744$	927.513	0.0457047
$745$	7886.58	0.387842
$746$	−5986.33	−0.293801
$747$	−23188.3	−1.13576
$748$	10096.5	0.493538
$749$	−2344.12	−0.114355
$750$	36.2761	0.00176615
$751$	19597.5	0.952226	0.476113	−	0.879384i	$-0.342045\pi$
$751$	19597.5	0.952226	0.476113	+	0.879384i	$0.342045\pi$
$752$	999.771	0.0484813
$753$	−2160.47	−0.104558
$754$	1322.84	0.0638926
$755$	−6630.21	−0.319600
$756$	−2998.33	−0.144244
$757$	30230.3	1.45144	0.725718	−	0.687992i	$-0.241508\pi$
$757$	30230.3	1.45144	0.725718	+	0.687992i	$0.241508\pi$
$758$	−5656.76	−0.271059
$759$	−210.240	−0.0100543
$760$	920.050	0.0439128
$761$	−12883.0	−0.613678	−0.306839	−	0.951761i	$-0.599271\pi$
$761$	−12883.0	−0.613678	−0.306839	+	0.951761i	$0.599271\pi$
$762$	−204.368	−0.00971584
$763$	−31782.5	−1.50800
$764$	11247.2	0.532603
$765$	16139.4	0.762774
$766$	7852.78	0.370408
$767$	9133.53	0.429977
$768$	1060.05	0.0498063
$769$	6359.79	0.298231	0.149116	−	0.988820i	$-0.452357\pi$
$769$	6359.79	0.298231	0.149116	+	0.988820i	$0.452357\pi$
$770$	−533.686	−0.0249775
$771$	−1250.89	−0.0584302
$772$	17807.3	0.830179
$773$	22365.3	1.04065	0.520325	−	0.853968i	$-0.325811\pi$
$773$	22365.3	1.04065	0.520325	+	0.853968i	$0.325811\pi$
$774$	−2386.86	−0.110845
$775$	−5116.80	−0.237162
$776$	−6780.39	−0.313662
$777$	−797.107	−0.0368032
$778$	−5248.37	−0.241855
$779$	4426.23	0.203576
$780$	−364.094	−0.0167137
$781$	8709.73	0.399051
$782$	3053.35	0.139626
$783$	−2626.60	−0.119881
$784$	−5393.96	−0.245716
$785$	17055.2	0.775449
$786$	183.886	0.00834480
$787$	−4234.70	−0.191805	−0.0959027	−	0.995391i	$-0.530574\pi$
$787$	−4234.70	−0.191805	−0.0959027	+	0.995391i	$0.530574\pi$
$788$	5942.13	0.268629
$789$	−265.400	−0.0119753
$790$	1774.90	0.0799342
$791$	16395.8	0.737002
$792$	2853.04	0.128003
$793$	−1702.45	−0.0762368
$794$	−5387.45	−0.240798
$795$	1355.54	0.0604731
$796$	−25750.5	−1.14661
$797$	3683.35	0.163703	0.0818513	−	0.996645i	$-0.473917\pi$
$797$	3683.35	0.163703	0.0818513	+	0.996645i	$0.473917\pi$
$798$	86.2681	0.00382689
$799$	−2194.27	−0.0971562
$800$	2788.43	0.123232
$801$	6441.62	0.284149
$802$	1334.48	0.0587557
$803$	5483.57	0.240985
$804$	2550.35	0.111870
$805$	3195.26	0.139898
$806$	−2594.03	−0.113363
$807$	−3334.96	−0.145472
$808$	9238.38	0.402234
$809$	33536.8	1.45747	0.728734	−	0.684796i	$-0.240109\pi$
$809$	33536.8	1.45747	0.728734	+	0.684796i	$0.240109\pi$
$810$	2205.80	0.0956840
$811$	−28113.8	−1.21727	−0.608637	−	0.793449i	$-0.708284\pi$
$811$	−28113.8	−1.21727	−0.608637	+	0.793449i	$0.708284\pi$
$812$	12435.6	0.537443
$813$	948.329	0.0409094
$814$	742.823	0.0319852
$815$	11311.1	0.486146
$816$	−3097.17	−0.132871
$817$	−2730.34	−0.116919
$818$	−3510.75	−0.150062
$819$	8562.39	0.365316
$820$	8870.33	0.377763
$821$	26294.1	1.11775	0.558873	−	0.829254i	$-0.311234\pi$
$821$	26294.1	1.11775	0.558873	+	0.829254i	$0.311234\pi$
$822$	−507.619	−0.0215392
$823$	9721.20	0.411737	0.205869	−	0.978580i	$-0.433998\pi$
$823$	9721.20	0.411737	0.205869	+	0.978580i	$0.433998\pi$
$824$	975.772	0.0412532
$825$	128.679	0.00543032
$826$	−4336.91	−0.182688
$827$	19765.3	0.831083	0.415541	−	0.909574i	$-0.363592\pi$
$827$	19765.3	0.831083	0.415541	+	0.909574i	$0.363592\pi$
$828$	−8330.42	−0.349640
$829$	5916.79	0.247887	0.123944	−	0.992289i	$-0.460446\pi$
$829$	5916.79	0.247887	0.123944	+	0.992289i	$0.460446\pi$
$830$	2685.02	0.112287
$831$	2086.83	0.0871134
$832$	−7564.19	−0.315194
$833$	11838.5	0.492413
$834$	274.883	0.0114130
$835$	14931.7	0.618841
$836$	1591.61	0.0658455
$837$	5150.64	0.212703
$838$	8377.12	0.345326
$839$	3110.65	0.128000	0.0639998	−	0.997950i	$-0.479614\pi$
$839$	3110.65	0.128000	0.0639998	+	0.997950i	$0.479614\pi$
$840$	354.502	0.0145613
$841$	−13495.2	−0.553330
$842$	−5864.74	−0.240038
$843$	−2461.72	−0.100577
$844$	−9754.77	−0.397835
$845$	−8897.00	−0.362208
$846$	−302.388	−0.0122888
$847$	−1893.09	−0.0767975
$848$	31817.8	1.28848
$849$	−2641.77	−0.106791
$850$	−1868.82	−0.0754116
$851$	−4447.40	−0.179148
$852$	−2821.47	−0.113453
$853$	13594.4	0.545677	0.272838	−	0.962060i	$-0.412038\pi$
$853$	13594.4	0.545677	0.272838	+	0.962060i	$0.412038\pi$
$854$	808.382	0.0323914
$855$	2544.20	0.101766
$856$	−1451.04	−0.0579389
$857$	−9133.21	−0.364043	−0.182021	−	0.983295i	$-0.558264\pi$
$857$	−9133.21	−0.364043	−0.182021	+	0.983295i	$0.558264\pi$
$858$	65.2354	0.00259569
$859$	31964.2	1.26962	0.634810	−	0.772668i	$-0.281078\pi$
$859$	31964.2	1.26962	0.634810	+	0.772668i	$0.281078\pi$
$860$	−5471.71	−0.216958
$861$	1705.46	0.0675050
$862$	9412.48	0.371915
$863$	25707.8	1.01403	0.507013	−	0.861938i	$-0.330749\pi$
$863$	25707.8	1.01403	0.507013	+	0.861938i	$0.330749\pi$
$864$	−2806.87	−0.110523
$865$	2970.68	0.116770
$866$	724.178	0.0284163
$867$	4498.68	0.176221
$868$	−24385.7	−0.953575
$869$	6295.93	0.245771
$870$	151.450	0.00590190
$871$	−14625.7	−0.568970
$872$	−19673.9	−0.764038
$873$	−18749.7	−0.726897
$874$	481.326	0.0186283
$875$	−1955.67	−0.0755587
$876$	−1776.37	−0.0685137
$877$	−11070.7	−0.426262	−0.213131	−	0.977024i	$-0.568366\pi$
$877$	−11070.7	−0.426262	−0.213131	+	0.977024i	$0.568366\pi$
$878$	−1312.68	−0.0504565
$879$	−678.507	−0.0260358
$880$	3020.39	0.115702
$881$	33497.0	1.28098	0.640489	−	0.767967i	$-0.278732\pi$
$881$	33497.0	1.28098	0.640489	+	0.767967i	$0.278732\pi$
$882$	1631.44	0.0622828
$883$	8500.55	0.323971	0.161985	−	0.986793i	$-0.448210\pi$
$883$	8500.55	0.323971	0.161985	+	0.986793i	$0.448210\pi$
$884$	18756.9	0.713645
$885$	1045.69	0.0397179
$886$	2463.86	0.0934255
$887$	−579.848	−0.0219497	−0.0109748	−	0.999940i	$-0.503493\pi$
$887$	−579.848	−0.0219497	−0.0109748	+	0.999940i	$0.503493\pi$
$888$	−493.421	−0.0186466
$889$	11017.7	0.415659
$890$	−745.889	−0.0280924
$891$	7824.44	0.294196
$892$	28571.1	1.07246
$893$	−345.902	−0.0129621
$894$	457.751	0.0171247
$895$	−2019.59	−0.0754275
$896$	17552.1	0.654436
$897$	−390.575	−0.0145384
$898$	−3596.26	−0.133640
$899$	−21362.3	−0.792518
$900$	5098.67	0.188840
$901$	−69832.9	−2.58210
$902$	−1589.31	−0.0586677
$903$	−1052.02	−0.0387697
$904$	10149.3	0.373407
$905$	18474.3	0.678572
$906$	−384.829	−0.0141116
$907$	−27666.2	−1.01283	−0.506417	−	0.862289i	$-0.669030\pi$
$907$	−27666.2	−1.01283	−0.506417	+	0.862289i	$0.669030\pi$
$908$	−10430.9	−0.381237
$909$	25546.8	0.932159
$910$	−991.456	−0.0361170
$911$	−41070.8	−1.49367	−0.746836	−	0.665008i	$-0.768428\pi$
$911$	−41070.8	−1.49367	−0.746836	+	0.665008i	$0.768428\pi$
$912$	−488.234	−0.0177270
$913$	9524.33	0.345246
$914$	−28.6707	−0.00103758
$915$	−194.911	−0.00704216
$916$	−13030.2	−0.470012
$917$	−9913.48	−0.357003
$918$	1881.18	0.0676341
$919$	11381.4	0.408527	0.204263	−	0.978916i	$-0.434520\pi$
$919$	11381.4	0.408527	0.204263	+	0.978916i	$0.434520\pi$
$920$	1977.91	0.0708803
$921$	−3885.30	−0.139007
$922$	−2784.15	−0.0994481
$923$	16180.5	0.577019
$924$	613.257	0.0218341
$925$	2722.05	0.0967573
$926$	3462.86	0.122891
$927$	2698.29	0.0956024
$928$	11641.5	0.411802
$929$	−34608.1	−1.22223	−0.611116	−	0.791541i	$-0.709279\pi$
$929$	−34608.1	−1.22223	−0.611116	+	0.791541i	$0.709279\pi$
$930$	−296.988	−0.0104716
$931$	1866.21	0.0656956
$932$	−51247.7	−1.80115
$933$	2281.28	0.0800489
$934$	−5130.73	−0.179746
$935$	−6629.08	−0.231865
$936$	5300.25	0.185090
$937$	2128.09	0.0741961	0.0370980	−	0.999312i	$-0.488189\pi$
$937$	2128.09	0.0741961	0.0370980	+	0.999312i	$0.488189\pi$
$938$	6944.78	0.241743
$939$	1418.45	0.0492963
$940$	−693.202	−0.0240529
$941$	46759.0	1.61987	0.809937	−	0.586517i	$-0.199501\pi$
$941$	46759.0	1.61987	0.809937	+	0.586517i	$0.199501\pi$
$942$	989.914	0.0342390
$943$	9515.46	0.328596
$944$	24544.7	0.846254
$945$	1968.61	0.0677660
$946$	980.376	0.0336943
$947$	−15027.4	−0.515654	−0.257827	−	0.966191i	$-0.583006\pi$
$947$	−15027.4	−0.515654	−0.257827	+	0.966191i	$0.583006\pi$
$948$	−2039.53	−0.0698744
$949$	10187.1	0.348459
$950$	−294.598	−0.0100611
$951$	2627.61	0.0895962
$952$	−18262.7	−0.621741
$953$	−15085.2	−0.512757	−0.256379	−	0.966576i	$-0.582529\pi$
$953$	−15085.2	−0.512757	−0.256379	+	0.966576i	$0.582529\pi$
$954$	−9623.51	−0.326596
$955$	−7384.56	−0.250218
$956$	23548.8	0.796675
$957$	537.226	0.0181463
$958$	4520.78	0.152463
$959$	27366.2	0.921481
$960$	−866.015	−0.0291151
$961$	12099.6	0.406151
$962$	1379.98	0.0462499
$963$	−4012.55	−0.134271
$964$	−4205.09	−0.140495
$965$	−11691.7	−0.390020
$966$	185.458	0.00617704
$967$	−1741.65	−0.0579190	−0.0289595	−	0.999581i	$-0.509219\pi$
$967$	−1741.65	−0.0579190	−0.0289595	+	0.999581i	$0.509219\pi$
$968$	−1171.85	−0.0389099
$969$	1071.56	0.0355248
$970$	2171.07	0.0718646
$971$	76.7491	0.00253656	0.00126828	−	0.999999i	$-0.499596\pi$
$971$	76.7491	0.00253656	0.00126828	+	0.999999i	$0.499596\pi$
$972$	−7709.04	−0.254391
$973$	−14819.2	−0.488264
$974$	−5225.19	−0.171895
$975$	239.053	0.00785213
$976$	−4575.03	−0.150044
$977$	32118.1	1.05174	0.525869	−	0.850565i	$-0.323740\pi$
$977$	32118.1	1.05174	0.525869	+	0.850565i	$0.323740\pi$
$978$	656.513	0.0214652
$979$	−2645.82	−0.0863747
$980$	3739.96	0.121907
$981$	−54403.8	−1.77062
$982$	−9032.46	−0.293521
$983$	30608.2	0.993133	0.496566	−	0.867999i	$-0.334594\pi$
$983$	30608.2	0.993133	0.496566	+	0.867999i	$0.334594\pi$
$984$	1055.70	0.0342018
$985$	−3901.42	−0.126203
$986$	−7802.20	−0.252001
$987$	−133.279	−0.00429818
$988$	2956.81	0.0952113
$989$	−5869.67	−0.188720
$990$	−913.538	−0.0293274
$991$	−11030.2	−0.353568	−0.176784	−	0.984250i	$-0.556569\pi$
$991$	−11030.2	−0.353568	−0.176784	+	0.984250i	$0.556569\pi$
$992$	−22828.6	−0.730653
$993$	182.781	0.00584128
$994$	−7683.07	−0.245163
$995$	16907.0	0.538680
$996$	−3085.35	−0.0981558
$997$	−5229.71	−0.166125	−0.0830625	−	0.996544i	$-0.526470\pi$
$997$	−5229.71	−0.166125	−0.0830625	+	0.996544i	$0.526470\pi$
$998$	−9949.91	−0.315590
$999$	−2740.06	−0.0867783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.e.1.12	✓	22

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

\(f(q)\)	\(=\)	\(q+0.620206 q^{2} +0.467922 q^{3} -7.61534 q^{4} +5.00000 q^{5} +0.290208 q^{6} -15.6454 q^{7} -9.68474 q^{8} -26.7810 q^{9} +O(q^{10})\)	\(q+0.620206 q^{2} +0.467922 q^{3} -7.61534 q^{4} +5.00000 q^{5} +0.290208 q^{6} -15.6454 q^{7} -9.68474 q^{8} -26.7810 q^{9} +3.10103 q^{10} +11.0000 q^{11} -3.56339 q^{12} +20.4353 q^{13} -9.70338 q^{14} +2.33961 q^{15} +54.9162 q^{16} -120.529 q^{17} -16.6098 q^{18} -19.0000 q^{19} -38.0767 q^{20} -7.32083 q^{21} +6.82227 q^{22} -40.8460 q^{23} -4.53170 q^{24} +25.0000 q^{25} +12.6741 q^{26} -25.1654 q^{27} +119.145 q^{28} +104.374 q^{29} +1.45104 q^{30} -204.672 q^{31} +111.537 q^{32} +5.14714 q^{33} -74.7527 q^{34} -78.2270 q^{35} +203.947 q^{36} +108.882 q^{37} -11.7839 q^{38} +9.56213 q^{39} -48.4237 q^{40} -232.959 q^{41} -4.54043 q^{42} +143.702 q^{43} -83.7688 q^{44} -133.905 q^{45} -25.3330 q^{46} +18.2054 q^{47} +25.6965 q^{48} -98.2216 q^{49} +15.5052 q^{50} -56.3981 q^{51} -155.622 q^{52} +579.388 q^{53} -15.6077 q^{54} +55.0000 q^{55} +151.522 q^{56} -8.89052 q^{57} +64.7331 q^{58} +446.949 q^{59} -17.8169 q^{60} -83.3093 q^{61} -126.939 q^{62} +419.000 q^{63} -370.154 q^{64} +102.176 q^{65} +3.19229 q^{66} -715.708 q^{67} +917.867 q^{68} -19.1128 q^{69} -48.5169 q^{70} +791.794 q^{71} +259.367 q^{72} +498.506 q^{73} +67.5294 q^{74} +11.6981 q^{75} +144.692 q^{76} -172.099 q^{77} +5.93049 q^{78} +572.357 q^{79} +274.581 q^{80} +711.313 q^{81} -144.483 q^{82} +865.848 q^{83} +55.7506 q^{84} -602.643 q^{85} +89.1251 q^{86} +48.8387 q^{87} -106.532 q^{88} -240.529 q^{89} -83.0489 q^{90} -319.718 q^{91} +311.056 q^{92} -95.7706 q^{93} +11.2911 q^{94} -95.0000 q^{95} +52.1908 q^{96} +700.110 q^{97} -60.9177 q^{98} -294.592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 12 q^{2} + 21 q^{3} + 96 q^{4} + 110 q^{5} + 27 q^{6} + 93 q^{7} + 114 q^{8} + 209 q^{9}+O(q^{10}) \) 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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