LMFDB - Embedded newform 1849.4.a.i.1.47

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.47
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.75171 q^{2} -7.02039 q^{3} +14.5788 q^{4} -4.22415 q^{5} -33.3589 q^{6} -11.1787 q^{7} +31.2605 q^{8} +22.2859 q^{9} +O(q^{10})$	\(q+4.75171 q^{2} -7.02039 q^{3} +14.5788 q^{4} -4.22415 q^{5} -33.3589 q^{6} -11.1787 q^{7} +31.2605 q^{8} +22.2859 q^{9} -20.0719 q^{10} -43.0911 q^{11} -102.349 q^{12} +62.9713 q^{13} -53.1182 q^{14} +29.6552 q^{15} +31.9107 q^{16} +136.464 q^{17} +105.896 q^{18} +108.392 q^{19} -61.5829 q^{20} +78.4792 q^{21} -204.757 q^{22} -142.408 q^{23} -219.461 q^{24} -107.157 q^{25} +299.222 q^{26} +33.0947 q^{27} -162.973 q^{28} +149.025 q^{29} +140.913 q^{30} -29.2265 q^{31} -98.4535 q^{32} +302.516 q^{33} +648.439 q^{34} +47.2206 q^{35} +324.902 q^{36} +69.6966 q^{37} +515.047 q^{38} -442.084 q^{39} -132.049 q^{40} +128.292 q^{41} +372.911 q^{42} -628.216 q^{44} -94.1390 q^{45} -676.684 q^{46} -95.1118 q^{47} -224.026 q^{48} -218.036 q^{49} -509.177 q^{50} -958.032 q^{51} +918.046 q^{52} -416.499 q^{53} +157.256 q^{54} +182.023 q^{55} -349.453 q^{56} -760.953 q^{57} +708.123 q^{58} -644.210 q^{59} +432.336 q^{60} -228.604 q^{61} -138.876 q^{62} -249.129 q^{63} -723.109 q^{64} -266.000 q^{65} +1437.47 q^{66} -877.104 q^{67} +1989.48 q^{68} +999.763 q^{69} +224.379 q^{70} -258.534 q^{71} +696.669 q^{72} -396.446 q^{73} +331.179 q^{74} +752.281 q^{75} +1580.22 q^{76} +481.704 q^{77} -2100.65 q^{78} +916.890 q^{79} -134.796 q^{80} -834.058 q^{81} +609.605 q^{82} -66.9221 q^{83} +1144.13 q^{84} -576.445 q^{85} -1046.21 q^{87} -1347.05 q^{88} +365.655 q^{89} -447.322 q^{90} -703.941 q^{91} -2076.14 q^{92} +205.182 q^{93} -451.944 q^{94} -457.863 q^{95} +691.182 q^{96} +1014.49 q^{97} -1036.04 q^{98} -960.325 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * 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 $

$2$	4.75171	1.67998	0.839992	−	0.542598i	$-0.182559\pi$
$2$	4.75171	1.67998	0.839992	+	0.542598i	$0.182559\pi$
$3$	−7.02039	−1.35108	−0.675538	−	0.737325i	$-0.736088\pi$
$3$	−7.02039	−1.35108	−0.675538	+	0.737325i	$0.736088\pi$
$4$	14.5788	1.82235
$5$	−4.22415	−0.377819	−0.188910	−	0.981994i	$-0.560495\pi$
$5$	−4.22415	−0.377819	−0.188910	+	0.981994i	$0.560495\pi$
$6$	−33.3589	−2.26979
$7$	−11.1787	−0.603595	−0.301798	−	0.953372i	$-0.597587\pi$
$7$	−11.1787	−0.603595	−0.301798	+	0.953372i	$0.597587\pi$
$8$	31.2605	1.38153
$9$	22.2859	0.825405
$10$	−20.0719	−0.634730
$11$	−43.0911	−1.18113	−0.590566	−	0.806989i	$-0.701095\pi$
$11$	−43.0911	−1.18113	−0.590566	+	0.806989i	$0.701095\pi$
$12$	−102.349	−2.46213
$13$	62.9713	1.34347	0.671735	−	0.740792i	$-0.265549\pi$
$13$	62.9713	1.34347	0.671735	+	0.740792i	$0.265549\pi$
$14$	−53.1182	−1.01403
$15$	29.6552	0.510462
$16$	31.9107	0.498605
$17$	136.464	1.94691	0.973454	−	0.228884i	$-0.0735077\pi$
$17$	136.464	1.94691	0.973454	+	0.228884i	$0.0735077\pi$
$18$	105.896	1.38667
$19$	108.392	1.30878	0.654389	−	0.756158i	$-0.272926\pi$
$19$	108.392	1.30878	0.654389	+	0.756158i	$0.272926\pi$
$20$	−61.5829	−0.688518
$21$	78.4792	0.815503
$22$	−204.757	−1.98428
$23$	−142.408	−1.29105	−0.645526	−	0.763738i	$-0.723362\pi$
$23$	−142.408	−1.29105	−0.645526	+	0.763738i	$0.723362\pi$
$24$	−219.461	−1.86655
$25$	−107.157	−0.857253
$26$	299.222	2.25701
$27$	33.0947	0.235892
$28$	−162.973	−1.09996
$29$	149.025	0.954249	0.477124	−	0.878836i	$-0.341679\pi$
$29$	149.025	0.954249	0.477124	+	0.878836i	$0.341679\pi$
$30$	140.913	0.857568
$31$	−29.2265	−0.169330	−0.0846652	−	0.996409i	$-0.526982\pi$
$31$	−29.2265	−0.169330	−0.0846652	+	0.996409i	$0.526982\pi$
$32$	−98.4535	−0.543884
$33$	302.516	1.59580
$34$	648.439	3.27077
$35$	47.2206	0.228050
$36$	324.902	1.50417
$37$	69.6966	0.309677	0.154839	−	0.987940i	$-0.450514\pi$
$37$	69.6966	0.309677	0.154839	+	0.987940i	$0.450514\pi$
$38$	515.047	2.19873
$39$	−442.084	−1.81513
$40$	−132.049	−0.521969
$41$	128.292	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$41$	128.292	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$42$	372.911	1.37003
$43$	0	0
$43$	0	0
$44$	−628.216	−2.15243
$45$	−94.1390	−0.311854
$46$	−676.684	−2.16895
$47$	−95.1118	−0.295180	−0.147590	−	0.989049i	$-0.547152\pi$
$47$	−95.1118	−0.295180	−0.147590	+	0.989049i	$0.547152\pi$
$48$	−224.026	−0.673653
$49$	−218.036	−0.635673
$50$	−509.177	−1.44017
$51$	−958.032	−2.63042
$52$	918.046	2.44827
$53$	−416.499	−1.07944	−0.539722	−	0.841843i	$-0.681471\pi$
$53$	−416.499	−1.07944	−0.539722	+	0.841843i	$0.681471\pi$
$54$	157.256	0.396294
$55$	182.023	0.446254
$56$	−349.453	−0.833887
$57$	−760.953	−1.76826
$58$	708.123	1.60312
$59$	−644.210	−1.42151	−0.710754	−	0.703440i	$-0.751646\pi$
$59$	−644.210	−1.42151	−0.710754	+	0.703440i	$0.751646\pi$
$60$	432.336	0.930240
$61$	−228.604	−0.479832	−0.239916	−	0.970794i	$-0.577120\pi$
$61$	−228.604	−0.479832	−0.239916	+	0.970794i	$0.577120\pi$
$62$	−138.876	−0.284472
$63$	−249.129	−0.498210
$64$	−723.109	−1.41232
$65$	−266.000	−0.507589
$66$	1437.47	2.68092
$67$	−877.104	−1.59933	−0.799667	−	0.600444i	$-0.794990\pi$
$67$	−877.104	−1.59933	−0.799667	+	0.600444i	$0.794990\pi$
$68$	1989.48	3.54794
$69$	999.763	1.74431
$70$	224.379	0.383120
$71$	−258.534	−0.432145	−0.216073	−	0.976377i	$-0.569325\pi$
$71$	−258.534	−0.432145	−0.216073	+	0.976377i	$0.569325\pi$
$72$	696.669	1.14032
$73$	−396.446	−0.635624	−0.317812	−	0.948154i	$-0.602948\pi$
$73$	−396.446	−0.635624	−0.317812	+	0.948154i	$0.602948\pi$
$74$	331.179	0.520253
$75$	752.281	1.15821
$76$	1580.22	2.38505
$77$	481.704	0.712926
$78$	−2100.65	−3.04939
$79$	916.890	1.30580	0.652900	−	0.757444i	$-0.273552\pi$
$79$	916.890	1.30580	0.652900	+	0.757444i	$0.273552\pi$
$80$	−134.796	−0.188382
$81$	−834.058	−1.14411
$82$	609.605	0.820971
$83$	−66.9221	−0.0885019	−0.0442510	−	0.999020i	$-0.514090\pi$
$83$	−66.9221	−0.0885019	−0.0442510	+	0.999020i	$0.514090\pi$
$84$	1144.13	1.48613
$85$	−576.445	−0.735579
$86$	0	0
$87$	−1046.21	−1.28926
$88$	−1347.05	−1.63177
$89$	365.655	0.435498	0.217749	−	0.976005i	$-0.430129\pi$
$89$	365.655	0.435498	0.217749	+	0.976005i	$0.430129\pi$
$90$	−447.322	−0.523909
$91$	−703.941	−0.810912
$92$	−2076.14	−2.35275
$93$	205.182	0.228778
$94$	−451.944	−0.495899
$95$	−457.863	−0.494481
$96$	691.182	0.734828
$97$	1014.49	1.06192	0.530958	−	0.847398i	$-0.321832\pi$
$97$	1014.49	1.06192	0.530958	+	0.847398i	$0.321832\pi$
$98$	−1036.04	−1.06792
$99$	−960.325	−0.974912
$100$	−1562.21	−1.56221
$101$	−1840.07	−1.81281	−0.906406	−	0.422408i	$-0.861185\pi$
$101$	−1840.07	−1.81281	−0.906406	+	0.422408i	$0.861185\pi$
$102$	−4552.30	−4.41906
$103$	−1091.88	−1.04452	−0.522261	−	0.852786i	$-0.674911\pi$
$103$	−1091.88	−1.04452	−0.522261	+	0.852786i	$0.674911\pi$
$104$	1968.52	1.85605
$105$	−331.508	−0.308113
$106$	−1979.09	−1.81345
$107$	−110.329	−0.0996812	−0.0498406	−	0.998757i	$-0.515871\pi$
$107$	−110.329	−0.0996812	−0.0498406	+	0.998757i	$0.515871\pi$
$108$	482.480	0.429876
$109$	−743.706	−0.653524	−0.326762	−	0.945107i	$-0.605958\pi$
$109$	−743.706	−0.653524	−0.326762	+	0.945107i	$0.605958\pi$
$110$	864.921	0.749700
$111$	−489.298	−0.418397
$112$	−356.722	−0.300956
$113$	1133.22	0.943402	0.471701	−	0.881759i	$-0.343640\pi$
$113$	1133.22	0.943402	0.471701	+	0.881759i	$0.343640\pi$
$114$	−3615.83	−2.97064
$115$	601.554	0.487784
$116$	2172.60	1.73897
$117$	1403.37	1.10891
$118$	−3061.10	−2.38811
$119$	−1525.50	−1.17514
$120$	927.036	0.705220
$121$	525.842	0.395073
$122$	−1086.26	−0.806110
$123$	−900.657	−0.660240
$124$	−426.087	−0.308579
$125$	980.663	0.701706
$126$	−1183.79	−0.836986
$127$	220.566	0.154111	0.0770555	−	0.997027i	$-0.475448\pi$
$127$	220.566	0.154111	0.0770555	+	0.997027i	$0.475448\pi$
$128$	−2648.38	−1.82879
$129$	0	0
$130$	−1263.96	−0.852741
$131$	1305.39	0.870628	0.435314	−	0.900279i	$-0.356637\pi$
$131$	1305.39	0.870628	0.435314	+	0.900279i	$0.356637\pi$
$132$	4410.32	2.90810
$133$	−1211.68	−0.789972
$134$	−4167.75	−2.68686
$135$	−139.797	−0.0891243
$136$	4265.94	2.68972
$137$	2023.98	1.26219	0.631096	−	0.775705i	$-0.282605\pi$
$137$	2023.98	1.26219	0.631096	+	0.775705i	$0.282605\pi$
$138$	4750.59	2.93041
$139$	324.650	0.198104	0.0990521	−	0.995082i	$-0.468419\pi$
$139$	324.650	0.198104	0.0990521	+	0.995082i	$0.468419\pi$
$140$	688.420	0.415586
$141$	667.722	0.398811
$142$	−1228.48	−0.725997
$143$	−2713.50	−1.58682
$144$	711.160	0.411551
$145$	−629.502	−0.360533
$146$	−1883.80	−1.06784
$147$	1530.70	0.858842
$148$	1016.09	0.564340
$149$	−1524.69	−0.838308	−0.419154	−	0.907915i	$-0.637673\pi$
$149$	−1524.69	−0.838308	−0.419154	+	0.907915i	$0.637673\pi$
$150$	3574.63	1.94578
$151$	−3148.37	−1.69676	−0.848380	−	0.529388i	$-0.822421\pi$
$151$	−3148.37	−1.69676	−0.848380	+	0.529388i	$0.822421\pi$
$152$	3388.38	1.80812
$153$	3041.23	1.60699
$154$	2288.92	1.19770
$155$	123.457	0.0639762
$156$	−6445.04	−3.30780
$157$	−1808.52	−0.919335	−0.459667	−	0.888091i	$-0.652031\pi$
$157$	−1808.52	−0.919335	−0.459667	+	0.888091i	$0.652031\pi$
$158$	4356.80	2.19372
$159$	2923.99	1.45841
$160$	415.882	0.205490
$161$	1591.95	0.779273
$162$	−3963.20	−1.92209
$163$	−668.967	−0.321457	−0.160729	−	0.986999i	$-0.551384\pi$
$163$	−668.967	−0.321457	−0.160729	+	0.986999i	$0.551384\pi$
$164$	1870.34	0.890541
$165$	−1277.87	−0.602923
$166$	−317.995	−0.148682
$167$	−2632.15	−1.21965	−0.609825	−	0.792536i	$-0.708760\pi$
$167$	−2632.15	−1.21965	−0.609825	+	0.792536i	$0.708760\pi$
$168$	2453.30	1.12664
$169$	1768.39	0.804911
$170$	−2739.10	−1.23576
$171$	2415.61	1.08027
$172$	0	0
$173$	2225.67	0.978119	0.489060	−	0.872250i	$-0.337340\pi$
$173$	2225.67	0.978119	0.489060	+	0.872250i	$0.337340\pi$
$174$	−4971.30	−2.16594
$175$	1197.88	0.517434
$176$	−1375.07	−0.588918
$177$	4522.61	1.92057
$178$	1737.49	0.731630
$179$	−1594.43	−0.665774	−0.332887	−	0.942967i	$-0.608023\pi$
$179$	−1594.43	−0.665774	−0.332887	+	0.942967i	$0.608023\pi$
$180$	−1372.43	−0.568306
$181$	3965.16	1.62833	0.814166	−	0.580632i	$-0.197194\pi$
$181$	3965.16	1.62833	0.814166	+	0.580632i	$0.197194\pi$
$182$	−3344.92	−1.36232
$183$	1604.89	0.648289
$184$	−4451.76	−1.78363
$185$	−294.409	−0.117002
$186$	974.965	0.384344
$187$	−5880.39	−2.29955
$188$	−1386.61	−0.537922
$189$	−369.957	−0.142383
$190$	−2175.63	−0.830721
$191$	2496.73	0.945848	0.472924	−	0.881103i	$-0.343199\pi$
$191$	2496.73	0.945848	0.472924	+	0.881103i	$0.343199\pi$
$192$	5076.51	1.90815
$193$	−2258.43	−0.842307	−0.421154	−	0.906989i	$-0.638375\pi$
$193$	−2258.43	−0.842307	−0.421154	+	0.906989i	$0.638375\pi$
$194$	4820.56	1.78400
$195$	1867.43	0.685790
$196$	−3178.70	−1.15842
$197$	−1867.33	−0.675338	−0.337669	−	0.941265i	$-0.609638\pi$
$197$	−1867.33	−0.675338	−0.337669	+	0.941265i	$0.609638\pi$
$198$	−4563.19	−1.63784
$199$	−3008.91	−1.07184	−0.535920	−	0.844269i	$-0.680035\pi$
$199$	−3008.91	−1.07184	−0.535920	+	0.844269i	$0.680035\pi$
$200$	−3349.77	−1.18432
$201$	6157.62	2.16082
$202$	−8743.50	−3.04550
$203$	−1665.91	−0.575980
$204$	−13966.9	−4.79354
$205$	−541.922	−0.184632
$206$	−5188.28	−1.75478
$207$	−3173.70	−1.06564
$208$	2009.46	0.669861
$209$	−4670.72	−1.54584
$210$	−1575.23	−0.517624
$211$	−253.754	−0.0827923	−0.0413961	−	0.999143i	$-0.513181\pi$
$211$	−253.754	−0.0827923	−0.0413961	+	0.999143i	$0.513181\pi$
$212$	−6072.05	−1.96712
$213$	1815.01	0.583861
$214$	−524.251	−0.167463
$215$	0	0
$216$	1034.56	0.325892
$217$	326.716	0.102207
$218$	−3533.88	−1.09791
$219$	2783.21	0.858776
$220$	2653.68	0.813231
$221$	8593.33	2.61561
$222$	−2325.00	−0.702901
$223$	−2211.05	−0.663960	−0.331980	−	0.943286i	$-0.607717\pi$
$223$	−2211.05	−0.663960	−0.331980	+	0.943286i	$0.607717\pi$
$224$	1100.59	0.328286
$225$	−2388.08	−0.707580
$226$	5384.74	1.58490
$227$	−4508.16	−1.31814	−0.659068	−	0.752084i	$-0.729049\pi$
$227$	−4508.16	−1.31814	−0.659068	+	0.752084i	$0.729049\pi$
$228$	−11093.8	−3.22238
$229$	4592.16	1.32515	0.662573	−	0.748997i	$-0.269464\pi$
$229$	4592.16	1.32515	0.662573	+	0.748997i	$0.269464\pi$
$230$	2858.41	0.819470
$231$	−3381.75	−0.963217
$232$	4658.59	1.31833
$233$	−5822.36	−1.63706	−0.818531	−	0.574463i	$-0.805211\pi$
$233$	−5822.36	−1.63706	−0.818531	+	0.574463i	$0.805211\pi$
$234$	6668.44	1.86295
$235$	401.766	0.111525
$236$	−9391.80	−2.59048
$237$	−6436.93	−1.76423
$238$	−7248.73	−1.97422
$239$	−2551.34	−0.690512	−0.345256	−	0.938509i	$-0.612208\pi$
$239$	−2551.34	−0.690512	−0.345256	+	0.938509i	$0.612208\pi$
$240$	946.318	0.254519
$241$	2606.50	0.696679	0.348339	−	0.937369i	$-0.386746\pi$
$241$	2606.50	0.696679	0.348339	+	0.937369i	$0.386746\pi$
$242$	2498.65	0.663716
$243$	4961.86	1.30989
$244$	−3332.77	−0.874421
$245$	921.015	0.240169
$246$	−4279.67	−1.10919
$247$	6825.57	1.75830
$248$	−913.636	−0.233935
$249$	469.820	0.119573
$250$	4659.83	1.17885
$251$	−936.255	−0.235442	−0.117721	−	0.993047i	$-0.537559\pi$
$251$	−936.255	−0.235442	−0.117721	+	0.993047i	$0.537559\pi$
$252$	−3631.99	−0.907913
$253$	6136.53	1.52490
$254$	1048.07	0.258904
$255$	4046.87	0.993822
$256$	−6799.46	−1.66003
$257$	−6041.18	−1.46630	−0.733149	−	0.680068i	$-0.761950\pi$
$257$	−6041.18	−1.46630	−0.733149	+	0.680068i	$0.761950\pi$
$258$	0	0
$259$	−779.121	−0.186920
$260$	−3877.96	−0.925003
$261$	3321.16	0.787641
$262$	6202.83	1.46264
$263$	5628.61	1.31968	0.659839	−	0.751407i	$-0.270625\pi$
$263$	5628.61	1.31968	0.659839	+	0.751407i	$0.270625\pi$
$264$	9456.82	2.20465
$265$	1759.35	0.407835
$266$	−5757.57	−1.32714
$267$	−2567.04	−0.588391
$268$	−12787.1	−2.91454
$269$	−4773.16	−1.08188	−0.540938	−	0.841062i	$-0.681931\pi$
$269$	−4773.16	−1.08188	−0.540938	+	0.841062i	$0.681931\pi$
$270$	−664.274	−0.149727
$271$	2753.55	0.617219	0.308609	−	0.951189i	$-0.400136\pi$
$271$	2753.55	0.617219	0.308609	+	0.951189i	$0.400136\pi$
$272$	4354.67	0.970738
$273$	4941.94	1.09560
$274$	9617.38	2.12046
$275$	4617.49	1.01253
$276$	14575.3	3.17874
$277$	1587.11	0.344260	0.172130	−	0.985074i	$-0.444935\pi$
$277$	1587.11	0.344260	0.172130	+	0.985074i	$0.444935\pi$
$278$	1542.65	0.332812
$279$	−651.340	−0.139766
$280$	1476.14	0.315058
$281$	3096.65	0.657405	0.328702	−	0.944434i	$-0.393389\pi$
$281$	3096.65	0.657405	0.328702	+	0.944434i	$0.393389\pi$
$282$	3172.82	0.669996
$283$	2960.70	0.621891	0.310946	−	0.950428i	$-0.399354\pi$
$283$	2960.70	0.621891	0.310946	+	0.950428i	$0.399354\pi$
$284$	−3769.11	−0.787519
$285$	3214.38	0.668081
$286$	−12893.8	−2.66583
$287$	−1434.14	−0.294964
$288$	−2194.13	−0.448924
$289$	13709.5	2.79045
$290$	−2991.22	−0.605690
$291$	−7122.12	−1.43473
$292$	−5779.71	−1.15833
$293$	4920.02	0.980992	0.490496	−	0.871443i	$-0.336816\pi$
$293$	4920.02	0.980992	0.490496	+	0.871443i	$0.336816\pi$
$294$	7273.43	1.44284
$295$	2721.24	0.537073
$296$	2178.75	0.427829
$297$	−1426.08	−0.278619
$298$	−7244.91	−1.40834
$299$	−8967.65	−1.73449
$300$	10967.4	2.11067
$301$	0	0
$302$	−14960.1	−2.85053
$303$	12918.0	2.44925
$304$	3458.86	0.652563
$305$	965.657	0.181290
$306$	14451.1	2.69971
$307$	−9438.56	−1.75468	−0.877340	−	0.479869i	$-0.840684\pi$
$307$	−9438.56	−1.75468	−0.877340	+	0.479869i	$0.840684\pi$
$308$	7022.66	1.29920
$309$	7665.40	1.41123
$310$	586.633	0.107479
$311$	−3537.86	−0.645059	−0.322530	−	0.946559i	$-0.604533\pi$
$311$	−3537.86	−0.645059	−0.322530	+	0.946559i	$0.604533\pi$
$312$	−13819.8	−2.50766
$313$	1481.13	0.267471	0.133736	−	0.991017i	$-0.457303\pi$
$313$	1481.13	0.267471	0.133736	+	0.991017i	$0.457303\pi$
$314$	−8593.57	−1.54447
$315$	1052.36	0.188233
$316$	13367.1	2.37962
$317$	−732.044	−0.129702	−0.0648512	−	0.997895i	$-0.520657\pi$
$317$	−732.044	−0.129702	−0.0648512	+	0.997895i	$0.520657\pi$
$318$	13894.0	2.45011
$319$	−6421.64	−1.12709
$320$	3054.52	0.533602
$321$	774.552	0.134677
$322$	7564.48	1.30917
$323$	14791.6	2.54807
$324$	−12159.5	−2.08497
$325$	−6747.79	−1.15169
$326$	−3178.74	−0.540043
$327$	5221.11	0.882961
$328$	4010.46	0.675124
$329$	1063.23	0.178170
$330$	−6072.09	−1.01290
$331$	−417.381	−0.0693092	−0.0346546	−	0.999399i	$-0.511033\pi$
$331$	−417.381	−0.0693092	−0.0346546	+	0.999399i	$0.511033\pi$
$332$	−975.643	−0.161281
$333$	1553.25	0.255609
$334$	−12507.2	−2.04899
$335$	3705.02	0.604259
$336$	2504.33	0.406614
$337$	−2917.52	−0.471594	−0.235797	−	0.971802i	$-0.575770\pi$
$337$	−2917.52	−0.471594	−0.235797	+	0.971802i	$0.575770\pi$
$338$	8402.88	1.35224
$339$	−7955.65	−1.27461
$340$	−8403.86	−1.34048
$341$	1259.40	0.200001
$342$	11478.3	1.81484
$343$	6271.67	0.987284
$344$	0	0
$345$	−4223.14	−0.659033
$346$	10575.8	1.64323
$347$	−2114.38	−0.327106	−0.163553	−	0.986535i	$-0.552295\pi$
$347$	−2114.38	−0.327106	−0.163553	+	0.986535i	$0.552295\pi$
$348$	−15252.5	−2.34948
$349$	−2209.08	−0.338823	−0.169412	−	0.985545i	$-0.554187\pi$
$349$	−2209.08	−0.338823	−0.169412	+	0.985545i	$0.554187\pi$
$350$	5691.96	0.869281
$351$	2084.02	0.316913
$352$	4242.47	0.642398
$353$	−9680.14	−1.45955	−0.729776	−	0.683686i	$-0.760376\pi$
$353$	−9680.14	−1.45955	−0.729776	+	0.683686i	$0.760376\pi$
$354$	21490.1	3.22652
$355$	1092.08	0.163273
$356$	5330.80	0.793629
$357$	10709.6	1.58771
$358$	−7576.30	−1.11849
$359$	−6934.75	−1.01951	−0.509753	−	0.860321i	$-0.670263\pi$
$359$	−6934.75	−1.01951	−0.509753	+	0.860321i	$0.670263\pi$
$360$	−2942.83	−0.430836
$361$	4889.77	0.712899
$362$	18841.3	2.73557
$363$	−3691.62	−0.533773
$364$	−10262.6	−1.47776
$365$	1674.65	0.240151
$366$	7625.98	1.08912
$367$	−2151.01	−0.305945	−0.152972	−	0.988230i	$-0.548885\pi$
$367$	−2151.01	−0.305945	−0.152972	+	0.988230i	$0.548885\pi$
$368$	−4544.35	−0.643725
$369$	2859.10	0.403357
$370$	−1398.95	−0.196562
$371$	4655.94	0.651548
$372$	2991.30	0.416913
$373$	−2182.22	−0.302925	−0.151463	−	0.988463i	$-0.548398\pi$
$373$	−2182.22	−0.302925	−0.151463	+	0.988463i	$0.548398\pi$
$374$	−27941.9	−3.86322
$375$	−6884.64	−0.948057
$376$	−2973.24	−0.407801
$377$	9384.29	1.28200
$378$	−1757.93	−0.239201
$379$	−1718.17	−0.232866	−0.116433	−	0.993199i	$-0.537146\pi$
$379$	−1718.17	−0.232866	−0.116433	+	0.993199i	$0.537146\pi$
$380$	−6675.08	−0.901117
$381$	−1548.46	−0.208216
$382$	11863.7	1.58901
$383$	435.777	0.0581388	0.0290694	−	0.999577i	$-0.490746\pi$
$383$	435.777	0.0581388	0.0290694	+	0.999577i	$0.490746\pi$
$384$	18592.6	2.47084
$385$	−2034.79	−0.269357
$386$	−10731.4	−1.41506
$387$	0	0
$388$	14790.0	1.93518
$389$	−2870.76	−0.374173	−0.187087	−	0.982343i	$-0.559904\pi$
$389$	−2870.76	−0.374173	−0.187087	+	0.982343i	$0.559904\pi$
$390$	8873.47	1.15212
$391$	−19433.6	−2.51356
$392$	−6815.91	−0.878202
$393$	−9164.34	−1.17628
$394$	−8873.01	−1.13456
$395$	−3873.08	−0.493356
$396$	−14000.4	−1.77663
$397$	−9081.90	−1.14813	−0.574065	−	0.818810i	$-0.694634\pi$
$397$	−9081.90	−1.14813	−0.574065	+	0.818810i	$0.694634\pi$
$398$	−14297.5	−1.80067
$399$	8506.50	1.06731
$400$	−3419.44	−0.427430
$401$	4802.23	0.598035	0.299017	−	0.954248i	$-0.403341\pi$
$401$	4802.23	0.598035	0.299017	+	0.954248i	$0.403341\pi$
$402$	29259.2	3.63014
$403$	−1840.43	−0.227490
$404$	−26826.0	−3.30357
$405$	3523.18	0.432267
$406$	−7915.93	−0.967638
$407$	−3003.30	−0.365770
$408$	−29948.6	−3.63401
$409$	3318.18	0.401158	0.200579	−	0.979678i	$-0.435718\pi$
$409$	3318.18	0.401158	0.200579	+	0.979678i	$0.435718\pi$
$410$	−2575.06	−0.310178
$411$	−14209.1	−1.70532
$412$	−15918.2	−1.90348
$413$	7201.46	0.858016
$414$	−15080.5	−1.79026
$415$	282.689	0.0334377
$416$	−6199.75	−0.730691
$417$	−2279.17	−0.267654
$418$	−22193.9	−2.59699
$419$	9133.20	1.06488	0.532442	−	0.846467i	$-0.321275\pi$
$419$	9133.20	1.06488	0.532442	+	0.846467i	$0.321275\pi$
$420$	−4832.98	−0.561488
$421$	−12874.9	−1.49046	−0.745232	−	0.666806i	$-0.767661\pi$
$421$	−12874.9	−1.49046	−0.745232	+	0.666806i	$0.767661\pi$
$422$	−1205.77	−0.139090
$423$	−2119.65	−0.243643
$424$	−13020.0	−1.49129
$425$	−14623.0	−1.66899
$426$	8624.40	0.980877
$427$	2555.51	0.289624
$428$	−1608.46	−0.181654
$429$	19049.9	2.14391
$430$	0	0
$431$	5433.33	0.607226	0.303613	−	0.952795i	$-0.401807\pi$
$431$	5433.33	0.607226	0.303613	+	0.952795i	$0.401807\pi$
$432$	1056.07	0.117617
$433$	8137.92	0.903195	0.451598	−	0.892222i	$-0.350854\pi$
$433$	8137.92	0.903195	0.451598	+	0.892222i	$0.350854\pi$
$434$	1552.46	0.171706
$435$	4419.35	0.487108
$436$	−10842.3	−1.19095
$437$	−15435.9	−1.68970
$438$	13225.0	1.44273
$439$	2828.77	0.307540	0.153770	−	0.988107i	$-0.450859\pi$
$439$	2828.77	0.307540	0.153770	+	0.988107i	$0.450859\pi$
$440$	5690.13	0.616515
$441$	−4859.13	−0.524687
$442$	40833.1	4.39419
$443$	−1724.64	−0.184966	−0.0924830	−	0.995714i	$-0.529480\pi$
$443$	−1724.64	−0.184966	−0.0924830	+	0.995714i	$0.529480\pi$
$444$	−7133.37	−0.762466
$445$	−1544.58	−0.164540
$446$	−10506.3	−1.11544
$447$	10704.0	1.13262
$448$	8083.44	0.852471
$449$	−18393.7	−1.93331	−0.966653	−	0.256090i	$-0.917566\pi$
$449$	−18393.7	−1.93331	−0.966653	+	0.256090i	$0.917566\pi$
$450$	−11347.5	−1.18872
$451$	−5528.22	−0.577193
$452$	16521.0	1.71921
$453$	22102.8	2.29245
$454$	−21421.5	−2.21445
$455$	2973.55	0.306378
$456$	−23787.8	−2.44290
$457$	−2439.34	−0.249689	−0.124844	−	0.992176i	$-0.539843\pi$
$457$	−2439.34	−0.249689	−0.124844	+	0.992176i	$0.539843\pi$
$458$	21820.6	2.22622
$459$	4516.24	0.459259
$460$	8769.92	0.888913
$461$	11301.0	1.14173	0.570866	−	0.821043i	$-0.306608\pi$
$461$	11301.0	1.14173	0.570866	+	0.821043i	$0.306608\pi$
$462$	−16069.1	−1.61819
$463$	10842.6	1.08833	0.544165	−	0.838979i	$-0.316847\pi$
$463$	10842.6	1.08833	0.544165	+	0.838979i	$0.316847\pi$
$464$	4755.49	0.475793
$465$	−866.718	−0.0864367
$466$	−27666.2	−2.75024
$467$	6330.48	0.627280	0.313640	−	0.949542i	$-0.398452\pi$
$467$	6330.48	0.627280	0.313640	+	0.949542i	$0.398452\pi$
$468$	20459.5	2.02081
$469$	9804.92	0.965350
$470$	1909.08	0.187360
$471$	12696.5	1.24209
$472$	−20138.3	−1.96386
$473$	0	0
$474$	−30586.4	−2.96389
$475$	−11614.9	−1.12195
$476$	−22239.9	−2.14152
$477$	−9282.07	−0.890979
$478$	−12123.2	−1.16005
$479$	−8678.52	−0.827832	−0.413916	−	0.910315i	$-0.635839\pi$
$479$	−8678.52	−0.827832	−0.413916	+	0.910315i	$0.635839\pi$
$480$	−2919.65	−0.277632
$481$	4388.89	0.416042
$482$	12385.4	1.17041
$483$	−11176.1	−1.05286
$484$	7666.14	0.719960
$485$	−4285.35	−0.401212
$486$	23577.3	2.20059
$487$	11021.8	1.02556	0.512778	−	0.858521i	$-0.328616\pi$
$487$	11021.8	1.02556	0.512778	+	0.858521i	$0.328616\pi$
$488$	−7146.28	−0.662903
$489$	4696.41	0.434313
$490$	4376.40	0.403481
$491$	5786.44	0.531850	0.265925	−	0.963994i	$-0.414323\pi$
$491$	5786.44	0.531850	0.265925	+	0.963994i	$0.414323\pi$
$492$	−13130.5	−1.20319
$493$	20336.5	1.85783
$494$	32433.2	2.95392
$495$	4056.55	0.368340
$496$	−932.640	−0.0844289
$497$	2890.08	0.260841
$498$	2232.45	0.200880
$499$	−5756.39	−0.516415	−0.258208	−	0.966089i	$-0.583132\pi$
$499$	−5756.39	−0.516415	−0.258208	+	0.966089i	$0.583132\pi$
$500$	14296.9	1.27875
$501$	18478.7	1.64784
$502$	−4448.82	−0.395539
$503$	−3112.75	−0.275926	−0.137963	−	0.990437i	$-0.544055\pi$
$503$	−3112.75	−0.275926	−0.137963	+	0.990437i	$0.544055\pi$
$504$	−7787.89	−0.688294
$505$	7772.73	0.684915
$506$	29159.0	2.56181
$507$	−12414.8	−1.08750
$508$	3215.59	0.280844
$509$	−17859.5	−1.55522	−0.777611	−	0.628746i	$-0.783569\pi$
$509$	−17859.5	−1.55522	−0.777611	+	0.628746i	$0.783569\pi$
$510$	19229.6	1.66961
$511$	4431.77	0.383660
$512$	−11122.1	−0.960022
$513$	3587.19	0.308730
$514$	−28706.0	−2.46336
$515$	4612.24	0.394640
$516$	0	0
$517$	4098.47	0.348647
$518$	−3702.16	−0.314022
$519$	−15625.1	−1.32151
$520$	−8315.30	−0.701250
$521$	13837.7	1.16361	0.581803	−	0.813329i	$-0.302347\pi$
$521$	13837.7	1.16361	0.581803	+	0.813329i	$0.302347\pi$
$522$	15781.2	1.32323
$523$	21318.7	1.78241	0.891204	−	0.453602i	$-0.149861\pi$
$523$	21318.7	1.78241	0.891204	+	0.453602i	$0.149861\pi$
$524$	19031.0	1.58659
$525$	−8409.56	−0.699092
$526$	26745.6	2.21704
$527$	−3988.37	−0.329670
$528$	9653.52	0.795673
$529$	8113.15	0.666816
$530$	8359.95	0.685156
$531$	−14356.8	−1.17332
$532$	−17664.9	−1.43960
$533$	8078.69	0.656524
$534$	−12197.8	−0.988488
$535$	466.045	0.0376615
$536$	−27418.7	−2.20953
$537$	11193.6	0.899511
$538$	−22680.7	−1.81754
$539$	9395.40	0.750813
$540$	−2038.07	−0.162416
$541$	4358.16	0.346343	0.173172	−	0.984892i	$-0.444598\pi$
$541$	4358.16	0.346343	0.173172	+	0.984892i	$0.444598\pi$
$542$	13084.1	1.03692
$543$	−27837.0	−2.20000
$544$	−13435.4	−1.05889
$545$	3141.52	0.246914
$546$	23482.7	1.84060
$547$	−17185.3	−1.34331	−0.671657	−	0.740862i	$-0.734417\pi$
$547$	−17185.3	−1.34331	−0.671657	+	0.740862i	$0.734417\pi$
$548$	29507.2	2.30015
$549$	−5094.65	−0.396055
$550$	21941.0	1.70103
$551$	16153.1	1.24890
$552$	31253.1	2.40982
$553$	−10249.7	−0.788175
$554$	7541.47	0.578351
$555$	2066.87	0.158079
$556$	4733.01	0.361015
$557$	−2037.15	−0.154967	−0.0774836	−	0.996994i	$-0.524689\pi$
$557$	−2037.15	−0.154967	−0.0774836	+	0.996994i	$0.524689\pi$
$558$	−3094.98	−0.234805
$559$	0	0
$560$	1506.84	0.113707
$561$	41282.7	3.10687
$562$	14714.4	1.10443
$563$	−4997.46	−0.374099	−0.187049	−	0.982351i	$-0.559892\pi$
$563$	−4997.46	−0.374099	−0.187049	+	0.982351i	$0.559892\pi$
$564$	9734.58	0.726773
$565$	−4786.89	−0.356435
$566$	14068.4	1.04477
$567$	9323.71	0.690581
$568$	−8081.90	−0.597023
$569$	−14634.1	−1.07819	−0.539096	−	0.842244i	$-0.681234\pi$
$569$	−14634.1	−1.07819	−0.539096	+	0.842244i	$0.681234\pi$
$570$	15273.8	1.12237
$571$	−24429.4	−1.79044	−0.895219	−	0.445626i	$-0.852981\pi$
$571$	−24429.4	−1.79044	−0.895219	+	0.445626i	$0.852981\pi$
$572$	−39559.6	−2.89173
$573$	−17528.0	−1.27791
$574$	−6814.62	−0.495534
$575$	15260.0	1.10676
$576$	−16115.1	−1.16574
$577$	7423.87	0.535632	0.267816	−	0.963470i	$-0.413698\pi$
$577$	7423.87	0.535632	0.267816	+	0.963470i	$0.413698\pi$
$578$	65143.5	4.68791
$579$	15855.1	1.13802
$580$	−9177.38	−0.657017
$581$	748.105	0.0534193
$582$	−33842.3	−2.41032
$583$	17947.4	1.27497
$584$	−12393.1	−0.878135
$585$	−5928.06	−0.418966
$586$	23378.5	1.64805
$587$	3113.36	0.218913	0.109457	−	0.993992i	$-0.465089\pi$
$587$	3113.36	0.218913	0.109457	+	0.993992i	$0.465089\pi$
$588$	22315.7	1.56511
$589$	−3167.92	−0.221616
$590$	12930.5	0.902275
$591$	13109.4	0.912433
$592$	2224.07	0.154407
$593$	950.921	0.0658510	0.0329255	−	0.999458i	$-0.489518\pi$
$593$	950.921	0.0658510	0.0329255	+	0.999458i	$0.489518\pi$
$594$	−6776.35	−0.468076
$595$	6443.93	0.443992
$596$	−22228.2	−1.52769
$597$	21123.8	1.44814
$598$	−42611.7	−2.91392
$599$	−20463.9	−1.39588	−0.697941	−	0.716155i	$-0.745900\pi$
$599$	−20463.9	−1.39588	−0.697941	+	0.716155i	$0.745900\pi$
$600$	23516.7	1.60011
$601$	11078.2	0.751899	0.375949	−	0.926640i	$-0.377317\pi$
$601$	11078.2	0.751899	0.375949	+	0.926640i	$0.377317\pi$
$602$	0	0
$603$	−19547.1	−1.32010
$604$	−45899.4	−3.09209
$605$	−2221.23	−0.149266
$606$	61382.8	4.11470
$607$	−14486.3	−0.968670	−0.484335	−	0.874883i	$-0.660938\pi$
$607$	−14486.3	−0.968670	−0.484335	+	0.874883i	$0.660938\pi$
$608$	−10671.5	−0.711823
$609$	11695.3	0.778192
$610$	4588.52	0.304564
$611$	−5989.32	−0.396566
$612$	44337.5	2.92849
$613$	12062.1	0.794753	0.397376	−	0.917656i	$-0.369921\pi$
$613$	12062.1	0.794753	0.397376	+	0.917656i	$0.369921\pi$
$614$	−44849.3	−2.94784
$615$	3804.51	0.249451
$616$	15058.3	0.984930
$617$	10781.2	0.703458	0.351729	−	0.936102i	$-0.385594\pi$
$617$	10781.2	0.703458	0.351729	+	0.936102i	$0.385594\pi$
$618$	36423.8	2.37084
$619$	24086.1	1.56398	0.781990	−	0.623291i	$-0.214205\pi$
$619$	24086.1	1.56398	0.781990	+	0.623291i	$0.214205\pi$
$620$	1799.86	0.116587
$621$	−4712.96	−0.304548
$622$	−16810.9	−1.08369
$623$	−4087.56	−0.262865
$624$	−14107.2	−0.905032
$625$	9252.11	0.592135
$626$	7037.91	0.449347
$627$	32790.3	2.08855
$628$	−26366.0	−1.67535
$629$	9511.10	0.602913
$630$	5000.49	0.316229
$631$	−213.204	−0.0134509	−0.00672545	−	0.999977i	$-0.502141\pi$
$631$	−213.204	−0.0134509	−0.00672545	+	0.999977i	$0.502141\pi$
$632$	28662.5	1.80401
$633$	1781.46	0.111859
$634$	−3478.46	−0.217898
$635$	−931.705	−0.0582261
$636$	42628.2	2.65773
$637$	−13730.0	−0.854007
$638$	−30513.8	−1.89350
$639$	−5761.66	−0.356695
$640$	11187.1	0.690953
$641$	19848.9	1.22306	0.611532	−	0.791219i	$-0.290553\pi$
$641$	19848.9	1.22306	0.611532	+	0.791219i	$0.290553\pi$
$642$	3680.45	0.226255
$643$	7733.15	0.474286	0.237143	−	0.971475i	$-0.423789\pi$
$643$	7733.15	0.474286	0.237143	+	0.971475i	$0.423789\pi$
$644$	23208.7	1.42011
$645$	0	0
$646$	70285.4	4.28072
$647$	8440.19	0.512856	0.256428	−	0.966563i	$-0.417454\pi$
$647$	8440.19	0.512856	0.256428	+	0.966563i	$0.417454\pi$
$648$	−26073.1	−1.58063
$649$	27759.7	1.67899
$650$	−32063.6	−1.93483
$651$	−2293.67	−0.138089
$652$	−9752.72	−0.585807
$653$	−11887.4	−0.712390	−0.356195	−	0.934412i	$-0.615926\pi$
$653$	−11887.4	−0.712390	−0.356195	+	0.934412i	$0.615926\pi$
$654$	24809.2	1.48336
$655$	−5514.15	−0.328940
$656$	4093.88	0.243657
$657$	−8835.18	−0.524647
$658$	5052.17	0.299322
$659$	16141.8	0.954164	0.477082	−	0.878859i	$-0.341694\pi$
$659$	16141.8	0.954164	0.477082	+	0.878859i	$0.341694\pi$
$660$	−18629.8	−1.09874
$661$	−12054.7	−0.709338	−0.354669	−	0.934992i	$-0.615407\pi$
$661$	−12054.7	−0.709338	−0.354669	+	0.934992i	$0.615407\pi$
$662$	−1983.28	−0.116438
$663$	−60328.6	−3.53389
$664$	−2092.02	−0.122268
$665$	5118.33	0.298467
$666$	7380.62	0.429419
$667$	−21222.4	−1.23198
$668$	−38373.5	−2.22263
$669$	15522.5	0.897060
$670$	17605.2	1.01515
$671$	9850.80	0.566745
$672$	−7726.55	−0.443539
$673$	15294.5	0.876017	0.438008	−	0.898971i	$-0.355684\pi$
$673$	15294.5	0.876017	0.438008	+	0.898971i	$0.355684\pi$
$674$	−13863.2	−0.792271
$675$	−3546.31	−0.202219
$676$	25781.0	1.46683
$677$	−10935.3	−0.620797	−0.310398	−	0.950607i	$-0.600462\pi$
$677$	−10935.3	−0.620797	−0.310398	+	0.950607i	$0.600462\pi$
$678$	−37803.0	−2.14132
$679$	−11340.7	−0.640967
$680$	−18020.0	−1.01623
$681$	31649.0	1.78090
$682$	5984.32	0.335999
$683$	−27098.4	−1.51814	−0.759070	−	0.651009i	$-0.774346\pi$
$683$	−27098.4	−1.51814	−0.759070	+	0.651009i	$0.774346\pi$
$684$	35216.7	1.96863
$685$	−8549.59	−0.476880
$686$	29801.2	1.65862
$687$	−32238.8	−1.79037
$688$	0	0
$689$	−26227.5	−1.45020
$690$	−20067.2	−1.10717
$691$	15040.8	0.828046	0.414023	−	0.910266i	$-0.364123\pi$
$691$	15040.8	0.828046	0.414023	+	0.910266i	$0.364123\pi$
$692$	32447.6	1.78247
$693$	10735.2	0.588452
$694$	−10046.9	−0.549532
$695$	−1371.37	−0.0748476
$696$	−32705.1	−1.78116
$697$	17507.2	0.951410
$698$	−10496.9	−0.569218
$699$	40875.2	2.21179
$700$	17463.6	0.942945
$701$	−7504.62	−0.404344	−0.202172	−	0.979350i	$-0.564800\pi$
$701$	−7504.62	−0.404344	−0.202172	+	0.979350i	$0.564800\pi$
$702$	9902.65	0.532409
$703$	7554.54	0.405299
$704$	31159.5	1.66814
$705$	−2820.56	−0.150678
$706$	−45997.3	−2.45203
$707$	20569.7	1.09421
$708$	65934.2	3.49994
$709$	28975.7	1.53484	0.767421	−	0.641143i	$-0.221539\pi$
$709$	28975.7	1.53484	0.767421	+	0.641143i	$0.221539\pi$
$710$	5189.27	0.274296
$711$	20433.7	1.07781
$712$	11430.6	0.601655
$713$	4162.10	0.218614
$714$	50888.9	2.66733
$715$	11462.2	0.599529
$716$	−23244.9	−1.21327
$717$	17911.4	0.932934
$718$	−32952.0	−1.71275
$719$	13465.7	0.698448	0.349224	−	0.937039i	$-0.386445\pi$
$719$	13465.7	0.698448	0.349224	+	0.937039i	$0.386445\pi$
$720$	−3004.04	−0.155492
$721$	12205.8	0.630468
$722$	23234.8	1.19766
$723$	−18298.7	−0.941265
$724$	57807.3	2.96739
$725$	−15969.0	−0.818032
$726$	−17541.5	−0.896731
$727$	7095.20	0.361962	0.180981	−	0.983487i	$-0.442073\pi$
$727$	7095.20	0.361962	0.180981	+	0.983487i	$0.442073\pi$
$728$	−22005.5	−1.12030
$729$	−12314.6	−0.625648
$730$	7957.45	0.403450
$731$	0	0
$732$	23397.3	1.18141
$733$	−11709.6	−0.590048	−0.295024	−	0.955490i	$-0.595328\pi$
$733$	−11709.6	−0.590048	−0.295024	+	0.955490i	$0.595328\pi$
$734$	−10221.0	−0.513982
$735$	−6465.89	−0.324487
$736$	14020.6	0.702182
$737$	37795.4	1.88902
$738$	13585.6	0.677633
$739$	33371.8	1.66117	0.830584	−	0.556894i	$-0.188007\pi$
$739$	33371.8	1.66117	0.830584	+	0.556894i	$0.188007\pi$
$740$	−4292.12	−0.213218
$741$	−47918.2	−2.37560
$742$	22123.7	1.09459
$743$	−11255.4	−0.555745	−0.277873	−	0.960618i	$-0.589629\pi$
$743$	−11255.4	−0.555745	−0.277873	+	0.960618i	$0.589629\pi$
$744$	6414.09	0.316064
$745$	6440.53	0.316729
$746$	−10369.3	−0.508909
$747$	−1491.42	−0.0730499
$748$	−85729.0	−4.19059
$749$	1233.34	0.0601671
$750$	−32713.9	−1.59272
$751$	28744.1	1.39665	0.698327	−	0.715779i	$-0.253928\pi$
$751$	28744.1	1.39665	0.698327	+	0.715779i	$0.253928\pi$
$752$	−3035.09	−0.147178
$753$	6572.88	0.318100
$754$	44591.5	2.15375
$755$	13299.2	0.641068
$756$	−5393.52	−0.259471
$757$	963.643	0.0462671	0.0231336	−	0.999732i	$-0.492636\pi$
$757$	963.643	0.0462671	0.0231336	+	0.999732i	$0.492636\pi$
$758$	−8164.24	−0.391212
$759$	−43080.9	−2.06026
$760$	−14313.0	−0.683142
$761$	18973.6	0.903799	0.451899	−	0.892069i	$-0.350747\pi$
$761$	18973.6	0.903799	0.451899	+	0.892069i	$0.350747\pi$
$762$	−7357.85	−0.349799
$763$	8313.70	0.394464
$764$	36399.3	1.72366
$765$	−12846.6	−0.607150
$766$	2070.69	0.0976723
$767$	−40566.8	−1.90975
$768$	47734.9	2.24282
$769$	−24386.0	−1.14354	−0.571770	−	0.820414i	$-0.693743\pi$
$769$	−24386.0	−1.14354	−0.571770	+	0.820414i	$0.693743\pi$
$770$	−9668.74	−0.452516
$771$	42411.5	1.98108
$772$	−32925.2	−1.53498
$773$	−6779.29	−0.315439	−0.157719	−	0.987484i	$-0.550414\pi$
$773$	−6779.29	−0.315439	−0.157719	+	0.987484i	$0.550414\pi$
$774$	0	0
$775$	3131.82	0.145159
$776$	31713.5	1.46707
$777$	5469.74	0.252543
$778$	−13641.0	−0.628605
$779$	13905.7	0.639570
$780$	27224.8	1.24975
$781$	11140.5	0.510421
$782$	−92343.1	−4.22274
$783$	4931.92	0.225099
$784$	−6957.68	−0.316950
$785$	7639.45	0.347342
$786$	−43546.3	−1.97614
$787$	24019.6	1.08794	0.543968	−	0.839106i	$-0.316921\pi$
$787$	24019.6	1.08794	0.543968	+	0.839106i	$0.316921\pi$
$788$	−27223.4	−1.23070
$789$	−39515.1	−1.78298
$790$	−18403.8	−0.828831
$791$	−12668.0	−0.569433
$792$	−30020.2	−1.34687
$793$	−14395.5	−0.644640
$794$	−43154.6	−1.92884
$795$	−12351.4	−0.551016
$796$	−43866.3	−1.95327
$797$	13544.4	0.601966	0.300983	−	0.953629i	$-0.402685\pi$
$797$	13544.4	0.601966	0.300983	+	0.953629i	$0.402685\pi$
$798$	40420.4	1.79307
$799$	−12979.4	−0.574689
$800$	10549.9	0.466246
$801$	8148.96	0.359462
$802$	22818.8	1.00469
$803$	17083.3	0.750756
$804$	89770.6	3.93777
$805$	−6724.62	−0.294424
$806$	−8745.22	−0.382180
$807$	33509.5	1.46170
$808$	−57521.6	−2.50446
$809$	−26389.6	−1.14686	−0.573430	−	0.819255i	$-0.694388\pi$
$809$	−26389.6	−1.14686	−0.573430	+	0.819255i	$0.694388\pi$
$810$	16741.1	0.726202
$811$	−21060.8	−0.911892	−0.455946	−	0.890007i	$-0.650699\pi$
$811$	−21060.8	−0.911892	−0.455946	+	0.890007i	$0.650699\pi$
$812$	−24286.9	−1.04964
$813$	−19331.0	−0.833909
$814$	−14270.8	−0.614487
$815$	2825.81	0.121453
$816$	−30571.5	−1.31154
$817$	0	0
$818$	15767.0	0.673939
$819$	−15688.0	−0.669331
$820$	−7900.57	−0.336463
$821$	612.058	0.0260182	0.0130091	−	0.999915i	$-0.495859\pi$
$821$	612.058	0.0260182	0.0130091	+	0.999915i	$0.495859\pi$
$822$	−67517.8	−2.86491
$823$	−21153.4	−0.895944	−0.447972	−	0.894048i	$-0.647854\pi$
$823$	−21153.4	−0.895944	−0.447972	+	0.894048i	$0.647854\pi$
$824$	−34132.6	−1.44304
$825$	−32416.6	−1.36800
$826$	34219.3	1.44145
$827$	22165.3	0.931999	0.465999	−	0.884785i	$-0.345695\pi$
$827$	22165.3	0.931999	0.465999	+	0.884785i	$0.345695\pi$
$828$	−46268.7	−1.94197
$829$	42815.7	1.79379	0.896895	−	0.442243i	$-0.145817\pi$
$829$	42815.7	1.79379	0.896895	+	0.442243i	$0.145817\pi$
$830$	1343.26	0.0561748
$831$	−11142.1	−0.465121
$832$	−45535.1	−1.89741
$833$	−29754.1	−1.23760
$834$	−10830.0	−0.449654
$835$	11118.6	0.460807
$836$	−68093.4	−2.81706
$837$	−967.242	−0.0399436
$838$	43398.4	1.78899
$839$	−28252.1	−1.16254	−0.581270	−	0.813711i	$-0.697444\pi$
$839$	−28252.1	−1.16254	−0.581270	+	0.813711i	$0.697444\pi$
$840$	−10363.1	−0.425668
$841$	−2180.61	−0.0894097
$842$	−61177.9	−2.50396
$843$	−21739.7	−0.888203
$844$	−3699.43	−0.150876
$845$	−7469.94	−0.304111
$846$	−10072.0	−0.409317
$847$	−5878.25	−0.238464
$848$	−13290.8	−0.538217
$849$	−20785.3	−0.840222
$850$	−69484.5	−2.80388
$851$	−9925.39	−0.399810
$852$	26460.6	1.06400
$853$	17130.3	0.687609	0.343804	−	0.939041i	$-0.388284\pi$
$853$	17130.3	0.687609	0.343804	+	0.939041i	$0.388284\pi$
$854$	12143.0	0.486564
$855$	−10203.9	−0.408147
$856$	−3448.94	−0.137713
$857$	−36446.6	−1.45273	−0.726366	−	0.687309i	$-0.758792\pi$
$857$	−36446.6	−1.45273	−0.726366	+	0.687309i	$0.758792\pi$
$858$	90519.5	3.60173
$859$	−34690.3	−1.37790	−0.688951	−	0.724808i	$-0.741929\pi$
$859$	−34690.3	−1.37790	−0.688951	+	0.724808i	$0.741929\pi$
$860$	0	0
$861$	10068.2	0.398518
$862$	25817.6	1.02013
$863$	−32454.0	−1.28012	−0.640062	−	0.768323i	$-0.721091\pi$
$863$	−32454.0	−1.28012	−0.640062	+	0.768323i	$0.721091\pi$
$864$	−3258.28	−0.128298
$865$	−9401.56	−0.369552
$866$	38669.1	1.51735
$867$	−96245.9	−3.77011
$868$	4763.12	0.186257
$869$	−39509.8	−1.54232
$870$	20999.5	0.818333
$871$	−55232.4	−2.14866
$872$	−23248.6	−0.902865
$873$	22608.8	0.876510
$874$	−73347.0	−2.83867
$875$	−10962.6	−0.423546
$876$	40575.8	1.56499
$877$	26856.7	1.03408	0.517038	−	0.855962i	$-0.327034\pi$
$877$	26856.7	1.03408	0.517038	+	0.855962i	$0.327034\pi$
$878$	13441.5	0.516662
$879$	−34540.5	−1.32539
$880$	5808.49	0.222505
$881$	−17343.7	−0.663250	−0.331625	−	0.943411i	$-0.607597\pi$
$881$	−17343.7	−0.663250	−0.331625	+	0.943411i	$0.607597\pi$
$882$	−23089.2	−0.881466
$883$	8366.36	0.318857	0.159428	−	0.987210i	$-0.449035\pi$
$883$	8366.36	0.318857	0.159428	+	0.987210i	$0.449035\pi$
$884$	125280.	4.76656
$885$	−19104.2	−0.725626
$886$	−8194.98	−0.310740
$887$	−13819.8	−0.523137	−0.261568	−	0.965185i	$-0.584240\pi$
$887$	−13819.8	−0.523137	−0.261568	+	0.965185i	$0.584240\pi$
$888$	−15295.7	−0.578029
$889$	−2465.66	−0.0930207
$890$	−7339.40	−0.276424
$891$	35940.4	1.35135
$892$	−32234.5	−1.20997
$893$	−10309.3	−0.386326
$894$	50862.2	1.90278
$895$	6735.12	0.251542
$896$	29605.5	1.10385
$897$	62956.4	2.34343
$898$	−87401.8	−3.24792
$899$	−4355.48	−0.161583
$900$	−34815.4	−1.28946
$901$	−56837.2	−2.10158
$902$	−26268.5	−0.969675
$903$	0	0
$904$	35425.1	1.30334
$905$	−16749.4	−0.615215
$906$	105026.	3.85128
$907$	−38464.8	−1.40816	−0.704082	−	0.710119i	$-0.748641\pi$
$907$	−38464.8	−1.40816	−0.704082	+	0.710119i	$0.748641\pi$
$908$	−65723.4	−2.40210
$909$	−41007.7	−1.49630
$910$	14129.4	0.514711
$911$	31120.3	1.13179	0.565896	−	0.824476i	$-0.308530\pi$
$911$	31120.3	1.13179	0.565896	+	0.824476i	$0.308530\pi$
$912$	−24282.6	−0.881662
$913$	2883.75	0.104532
$914$	−11591.1	−0.419473
$915$	−6779.29	−0.244936
$916$	66948.1	2.41488
$917$	−14592.6	−0.525507
$918$	21459.9	0.771548
$919$	−30589.6	−1.09799	−0.548997	−	0.835824i	$-0.684990\pi$
$919$	−30589.6	−1.09799	−0.548997	+	0.835824i	$0.684990\pi$
$920$	18804.9	0.673890
$921$	66262.4	2.37071
$922$	53699.0	1.91809
$923$	−16280.2	−0.580574
$924$	−49301.9	−1.75532
$925$	−7468.46	−0.265472
$926$	51520.7	1.82838
$927$	−24333.5	−0.862153
$928$	−14672.0	−0.519000
$929$	33705.0	1.19034	0.595170	−	0.803600i	$-0.297085\pi$
$929$	33705.0	1.19034	0.595170	+	0.803600i	$0.297085\pi$
$930$	−4118.39	−0.145212
$931$	−23633.3	−0.831954
$932$	−84882.9	−2.98330
$933$	24837.1	0.871524
$934$	30080.6	1.05382
$935$	24839.6	0.868816
$936$	43870.2	1.53199
$937$	−9240.51	−0.322171	−0.161086	−	0.986940i	$-0.551500\pi$
$937$	−9240.51	−0.322171	−0.161086	+	0.986940i	$0.551500\pi$
$938$	46590.2	1.62177
$939$	−10398.1	−0.361374
$940$	5857.26	0.203237
$941$	2959.70	0.102533	0.0512665	−	0.998685i	$-0.483674\pi$
$941$	2959.70	0.102533	0.0512665	+	0.998685i	$0.483674\pi$
$942$	60330.2	2.08669
$943$	−18269.8	−0.630908
$944$	−20557.2	−0.708771
$945$	1562.75	0.0537950
$946$	0	0
$947$	−8960.28	−0.307466	−0.153733	−	0.988112i	$-0.549130\pi$
$947$	−8960.28	−0.307466	−0.153733	+	0.988112i	$0.549130\pi$
$948$	−93842.6	−3.21505
$949$	−24964.8	−0.853942
$950$	−55190.6	−1.88486
$951$	5139.23	0.175238
$952$	−47687.8	−1.62350
$953$	−29389.4	−0.998969	−0.499484	−	0.866323i	$-0.666477\pi$
$953$	−29389.4	−0.998969	−0.499484	+	0.866323i	$0.666477\pi$
$954$	−44105.7	−1.49683
$955$	−10546.5	−0.357359
$956$	−37195.4	−1.25835
$957$	45082.4	1.52279
$958$	−41237.9	−1.39075
$959$	−22625.6	−0.761854
$960$	−21443.9	−0.720936
$961$	−28936.8	−0.971327
$962$	20854.8	0.698944
$963$	−2458.78	−0.0822774
$964$	37999.6	1.26959
$965$	9539.94	0.318240
$966$	−53105.6	−1.76878
$967$	26171.5	0.870341	0.435170	−	0.900348i	$-0.356688\pi$
$967$	26171.5	0.870341	0.435170	+	0.900348i	$0.356688\pi$
$968$	16438.1	0.545806
$969$	−103843.	−3.44263
$970$	−20362.8	−0.674030
$971$	27259.4	0.900922	0.450461	−	0.892796i	$-0.351260\pi$
$971$	27259.4	0.900922	0.450461	+	0.892796i	$0.351260\pi$
$972$	72337.8	2.38708
$973$	−3629.18	−0.119575
$974$	52372.5	1.72292
$975$	47372.2	1.55602
$976$	−7294.92	−0.239247
$977$	19534.3	0.639669	0.319834	−	0.947473i	$-0.396373\pi$
$977$	19534.3	0.639669	0.319834	+	0.947473i	$0.396373\pi$
$978$	22316.0	0.729639
$979$	−15756.5	−0.514381
$980$	13427.3	0.437672
$981$	−16574.2	−0.539422
$982$	27495.5	0.893500
$983$	48920.6	1.58731	0.793655	−	0.608368i	$-0.208176\pi$
$983$	48920.6	1.58731	0.793655	+	0.608368i	$0.208176\pi$
$984$	−28155.0	−0.912143
$985$	7887.87	0.255156
$986$	96633.4	3.12113
$987$	−7464.29	−0.240720
$988$	99508.6	3.20424
$989$	0	0
$990$	19275.6	0.618806
$991$	47979.4	1.53796	0.768979	−	0.639274i	$-0.220765\pi$
$991$	47979.4	1.53796	0.768979	+	0.639274i	$0.220765\pi$
$992$	2877.45	0.0920960
$993$	2930.18	0.0936420
$994$	13732.8	0.438209
$995$	12710.1	0.404962
$996$	6849.40	0.217903
$997$	−19997.6	−0.635235	−0.317617	−	0.948219i	$-0.602883\pi$
$997$	−19997.6	−0.635235	−0.317617	+	0.948219i	$0.602883\pi$
$998$	−27352.7	−0.867570
$999$	2306.59	0.0730502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.47	✓	50
43.42	odd	2		1849.4.a.j.1.4	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.47	✓	50	1.1	even	1	trivial
1849.4.a.j.1.4	yes	50	43.42	odd	2

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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+4.75171 q^{2} -7.02039 q^{3} +14.5788 q^{4} -4.22415 q^{5} -33.3589 q^{6} -11.1787 q^{7} +31.2605 q^{8} +22.2859 q^{9} +O(q^{10})\)	\(q+4.75171 q^{2} -7.02039 q^{3} +14.5788 q^{4} -4.22415 q^{5} -33.3589 q^{6} -11.1787 q^{7} +31.2605 q^{8} +22.2859 q^{9} -20.0719 q^{10} -43.0911 q^{11} -102.349 q^{12} +62.9713 q^{13} -53.1182 q^{14} +29.6552 q^{15} +31.9107 q^{16} +136.464 q^{17} +105.896 q^{18} +108.392 q^{19} -61.5829 q^{20} +78.4792 q^{21} -204.757 q^{22} -142.408 q^{23} -219.461 q^{24} -107.157 q^{25} +299.222 q^{26} +33.0947 q^{27} -162.973 q^{28} +149.025 q^{29} +140.913 q^{30} -29.2265 q^{31} -98.4535 q^{32} +302.516 q^{33} +648.439 q^{34} +47.2206 q^{35} +324.902 q^{36} +69.6966 q^{37} +515.047 q^{38} -442.084 q^{39} -132.049 q^{40} +128.292 q^{41} +372.911 q^{42} -628.216 q^{44} -94.1390 q^{45} -676.684 q^{46} -95.1118 q^{47} -224.026 q^{48} -218.036 q^{49} -509.177 q^{50} -958.032 q^{51} +918.046 q^{52} -416.499 q^{53} +157.256 q^{54} +182.023 q^{55} -349.453 q^{56} -760.953 q^{57} +708.123 q^{58} -644.210 q^{59} +432.336 q^{60} -228.604 q^{61} -138.876 q^{62} -249.129 q^{63} -723.109 q^{64} -266.000 q^{65} +1437.47 q^{66} -877.104 q^{67} +1989.48 q^{68} +999.763 q^{69} +224.379 q^{70} -258.534 q^{71} +696.669 q^{72} -396.446 q^{73} +331.179 q^{74} +752.281 q^{75} +1580.22 q^{76} +481.704 q^{77} -2100.65 q^{78} +916.890 q^{79} -134.796 q^{80} -834.058 q^{81} +609.605 q^{82} -66.9221 q^{83} +1144.13 q^{84} -576.445 q^{85} -1046.21 q^{87} -1347.05 q^{88} +365.655 q^{89} -447.322 q^{90} -703.941 q^{91} -2076.14 q^{92} +205.182 q^{93} -451.944 q^{94} -457.863 q^{95} +691.182 q^{96} +1014.49 q^{97} -1036.04 q^{98} -960.325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * q^99

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