LMFDB - Embedded newform 1849.4.a.i.1.43

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			1.43
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)

$f(q)$	$=$	$q+4.15048 q^{2} -4.65912 q^{3} +9.22650 q^{4} -19.9023 q^{5} -19.3376 q^{6} +27.8552 q^{7} +5.09055 q^{8} -5.29261 q^{9} +O(q^{10})$	\(q+4.15048 q^{2} -4.65912 q^{3} +9.22650 q^{4} -19.9023 q^{5} -19.3376 q^{6} +27.8552 q^{7} +5.09055 q^{8} -5.29261 q^{9} -82.6043 q^{10} -0.733230 q^{11} -42.9873 q^{12} +67.6888 q^{13} +115.613 q^{14} +92.7274 q^{15} -52.6837 q^{16} -45.0640 q^{17} -21.9669 q^{18} +96.4948 q^{19} -183.629 q^{20} -129.781 q^{21} -3.04326 q^{22} -76.2235 q^{23} -23.7175 q^{24} +271.103 q^{25} +280.941 q^{26} +150.455 q^{27} +257.006 q^{28} -182.776 q^{29} +384.863 q^{30} -37.9792 q^{31} -259.387 q^{32} +3.41621 q^{33} -187.037 q^{34} -554.384 q^{35} -48.8323 q^{36} +95.8006 q^{37} +400.500 q^{38} -315.370 q^{39} -101.314 q^{40} +33.5766 q^{41} -538.653 q^{42} -6.76515 q^{44} +105.335 q^{45} -316.364 q^{46} +533.353 q^{47} +245.460 q^{48} +432.913 q^{49} +1125.21 q^{50} +209.958 q^{51} +624.530 q^{52} -0.134055 q^{53} +624.461 q^{54} +14.5930 q^{55} +141.798 q^{56} -449.581 q^{57} -758.607 q^{58} -110.795 q^{59} +855.549 q^{60} +533.253 q^{61} -157.632 q^{62} -147.427 q^{63} -655.112 q^{64} -1347.16 q^{65} +14.1789 q^{66} -411.431 q^{67} -415.783 q^{68} +355.134 q^{69} -2300.96 q^{70} -216.076 q^{71} -26.9423 q^{72} -136.316 q^{73} +397.619 q^{74} -1263.10 q^{75} +890.309 q^{76} -20.4243 q^{77} -1308.94 q^{78} +81.5918 q^{79} +1048.53 q^{80} -558.088 q^{81} +139.359 q^{82} -926.163 q^{83} -1197.42 q^{84} +896.879 q^{85} +851.573 q^{87} -3.73254 q^{88} -1125.82 q^{89} +437.193 q^{90} +1885.49 q^{91} -703.276 q^{92} +176.950 q^{93} +2213.67 q^{94} -1920.47 q^{95} +1208.52 q^{96} -1512.02 q^{97} +1796.80 q^{98} +3.88071 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.15048	1.46742	0.733708	−	0.679465i	$-0.237788\pi$
$2$	4.15048	1.46742	0.733708	+	0.679465i	$0.237788\pi$
$3$	−4.65912	−0.896648	−0.448324	−	0.893871i	$-0.647979\pi$
$3$	−4.65912	−0.896648	−0.448324	+	0.893871i	$0.647979\pi$
$4$	9.22650	1.15331
$5$	−19.9023	−1.78012	−0.890060	−	0.455844i	$-0.849338\pi$
$5$	−19.9023	−1.78012	−0.890060	+	0.455844i	$0.849338\pi$
$6$	−19.3376	−1.31576
$7$	27.8552	1.50404	0.752020	−	0.659140i	$-0.229079\pi$
$7$	27.8552	1.50404	0.752020	+	0.659140i	$0.229079\pi$
$8$	5.09055	0.224973
$9$	−5.29261	−0.196023
$10$	−82.6043	−2.61218
$11$	−0.733230	−0.0200979	−0.0100490	−	0.999950i	$-0.503199\pi$
$11$	−0.733230	−0.0200979	−0.0100490	+	0.999950i	$0.503199\pi$
$12$	−42.9873	−1.03411
$13$	67.6888	1.44411	0.722057	−	0.691834i	$-0.243197\pi$
$13$	67.6888	1.44411	0.722057	+	0.691834i	$0.243197\pi$
$14$	115.613	2.20705
$15$	92.7274	1.59614
$16$	−52.6837	−0.823183
$17$	−45.0640	−0.642919	−0.321460	−	0.946923i	$-0.604173\pi$
$17$	−45.0640	−0.642919	−0.321460	+	0.946923i	$0.604173\pi$
$18$	−21.9669	−0.287647
$19$	96.4948	1.16513	0.582564	−	0.812785i	$-0.302050\pi$
$19$	96.4948	1.16513	0.582564	+	0.812785i	$0.302050\pi$
$20$	−183.629	−2.05303
$21$	−129.781	−1.34859
$22$	−3.04326	−0.0294920
$23$	−76.2235	−0.691031	−0.345515	−	0.938413i	$-0.612296\pi$
$23$	−76.2235	−0.691031	−0.345515	+	0.938413i	$0.612296\pi$
$24$	−23.7175	−0.201721
$25$	271.103	2.16883
$26$	280.941	2.11912
$27$	150.455	1.07241
$28$	257.006	1.73463
$29$	−182.776	−1.17036	−0.585182	−	0.810902i	$-0.698977\pi$
$29$	−182.776	−1.17036	−0.585182	+	0.810902i	$0.698977\pi$
$30$	384.863	2.34220
$31$	−37.9792	−0.220041	−0.110020	−	0.993929i	$-0.535092\pi$
$31$	−37.9792	−0.220041	−0.110020	+	0.993929i	$0.535092\pi$
$32$	−259.387	−1.43293
$33$	3.41621	0.0180208
$34$	−187.037	−0.943430
$35$	−554.384	−2.67737
$36$	−48.8323	−0.226075
$37$	95.8006	0.425663	0.212831	−	0.977089i	$-0.431731\pi$
$37$	95.8006	0.425663	0.212831	+	0.977089i	$0.431731\pi$
$38$	400.500	1.70973
$39$	−315.370	−1.29486
$40$	−101.314	−0.400478
$41$	33.5766	0.127897	0.0639485	−	0.997953i	$-0.479631\pi$
$41$	33.5766	0.127897	0.0639485	+	0.997953i	$0.479631\pi$
$42$	−538.653	−1.97895
$43$	0	0
$43$	0	0
$44$	−6.76515	−0.0231792
$45$	105.335	0.348944
$46$	−316.364	−1.01403
$47$	533.353	1.65527	0.827634	−	0.561268i	$-0.189686\pi$
$47$	533.353	1.65527	0.827634	+	0.561268i	$0.189686\pi$
$48$	245.460	0.738106
$49$	432.913	1.26214
$50$	1125.21	3.18257
$51$	209.958	0.576472
$52$	624.530	1.66551
$53$	−0.134055	−0.000347430 0	−0.000173715	−	1.00000i	$-0.500055\pi$
$53$	−0.134055	−0.000347430 0	−0.000173715		1.00000i	$0.500055\pi$
$54$	624.461	1.57367
$55$	14.5930	0.0357767
$56$	141.798	0.338368
$57$	−449.581	−1.04471
$58$	−758.607	−1.71741
$59$	−110.795	−0.244480	−0.122240	−	0.992501i	$-0.539008\pi$
$59$	−110.795	−0.244480	−0.122240	+	0.992501i	$0.539008\pi$
$60$	855.549	1.84085
$61$	533.253	1.11928	0.559640	−	0.828736i	$-0.310939\pi$
$61$	533.253	1.11928	0.559640	+	0.828736i	$0.310939\pi$
$62$	−157.632	−0.322891
$63$	−147.427	−0.294826
$64$	−655.112	−1.27952
$65$	−1347.16	−2.57070
$66$	14.1789	0.0264440
$67$	−411.431	−0.750214	−0.375107	−	0.926982i	$-0.622394\pi$
$67$	−411.431	−0.750214	−0.375107	+	0.926982i	$0.622394\pi$
$68$	−415.783	−0.741486
$69$	355.134	0.619611
$70$	−2300.96	−3.92882
$71$	−216.076	−0.361176	−0.180588	−	0.983559i	$-0.557800\pi$
$71$	−216.076	−0.361176	−0.180588	+	0.983559i	$0.557800\pi$
$72$	−26.9423	−0.0440997
$73$	−136.316	−0.218557	−0.109278	−	0.994011i	$-0.534854\pi$
$73$	−136.316	−0.218557	−0.109278	+	0.994011i	$0.534854\pi$
$74$	397.619	0.624625
$75$	−1263.10	−1.94467
$76$	890.309	1.34376
$77$	−20.4243	−0.0302281
$78$	−1308.94	−1.90010
$79$	81.5918	0.116200	0.0581000	−	0.998311i	$-0.481496\pi$
$79$	81.5918	0.116200	0.0581000	+	0.998311i	$0.481496\pi$
$80$	1048.53	1.46536
$81$	−558.088	−0.765552
$82$	139.359	0.187678
$83$	−926.163	−1.22481	−0.612407	−	0.790542i	$-0.709799\pi$
$83$	−926.163	−1.22481	−0.612407	+	0.790542i	$0.709799\pi$
$84$	−1197.42	−1.55535
$85$	896.879	1.14447
$86$	0	0
$87$	851.573	1.04941
$88$	−3.73254	−0.00452148
$89$	−1125.82	−1.34086	−0.670429	−	0.741974i	$-0.733890\pi$
$89$	−1125.82	−1.34086	−0.670429	+	0.741974i	$0.733890\pi$
$90$	437.193	0.512046
$91$	1885.49	2.17201
$92$	−703.276	−0.796974
$93$	176.950	0.197299
$94$	2213.67	2.42897
$95$	−1920.47	−2.07407
$96$	1208.52	1.28483
$97$	−1512.02	−1.58270	−0.791352	−	0.611361i	$-0.790622\pi$
$97$	−1512.02	−1.58270	−0.791352	+	0.611361i	$0.790622\pi$
$98$	1796.80	1.85208
$99$	3.88071	0.00393965
$100$	2501.33	2.50133
$101$	−1324.79	−1.30517	−0.652583	−	0.757717i	$-0.726315\pi$
$101$	−1324.79	−1.30517	−0.652583	+	0.757717i	$0.726315\pi$
$102$	871.429	0.845925
$103$	−1115.34	−1.06696	−0.533482	−	0.845811i	$-0.679117\pi$
$103$	−1115.34	−1.06696	−0.533482	+	0.845811i	$0.679117\pi$
$104$	344.573	0.324886
$105$	2582.94	2.40066
$106$	−0.556391	−0.000509825 0
$107$	−647.960	−0.585427	−0.292713	−	0.956200i	$-0.594558\pi$
$107$	−647.960	−0.585427	−0.292713	+	0.956200i	$0.594558\pi$
$108$	1388.17	1.23682
$109$	−2058.34	−1.80875	−0.904374	−	0.426740i	$-0.859662\pi$
$109$	−2058.34	−1.80875	−0.904374	+	0.426740i	$0.859662\pi$
$110$	60.5680	0.0524994
$111$	−446.346	−0.381670
$112$	−1467.52	−1.23810
$113$	911.539	0.758853	0.379426	−	0.925222i	$-0.376121\pi$
$113$	911.539	0.758853	0.379426	+	0.925222i	$0.376121\pi$
$114$	−1865.98	−1.53302
$115$	1517.03	1.23012
$116$	−1686.38	−1.34980
$117$	−358.250	−0.283079
$118$	−459.854	−0.358754
$119$	−1255.27	−0.966976
$120$	472.033	0.359088
$121$	−1330.46	−0.999596
$122$	2213.26	1.64245
$123$	−156.437	−0.114679
$124$	−350.415	−0.253776
$125$	−2907.79	−2.08065
$126$	−611.893	−0.432633
$127$	−375.272	−0.262205	−0.131102	−	0.991369i	$-0.541852\pi$
$127$	−375.272	−0.262205	−0.131102	+	0.991369i	$0.541852\pi$
$128$	−643.932	−0.444657
$129$	0	0
$130$	−5591.38	−3.77228
$131$	−2701.28	−1.80161	−0.900807	−	0.434219i	$-0.857024\pi$
$131$	−2701.28	−1.80161	−0.900807	+	0.434219i	$0.857024\pi$
$132$	31.5196	0.0207836
$133$	2687.88	1.75240
$134$	−1707.64	−1.10088
$135$	−2994.41	−1.90902
$136$	−229.400	−0.144639
$137$	−2022.29	−1.26114	−0.630568	−	0.776134i	$-0.717178\pi$
$137$	−2022.29	−1.26114	−0.630568	+	0.776134i	$0.717178\pi$
$138$	1473.98	0.909228
$139$	−564.430	−0.344419	−0.172210	−	0.985060i	$-0.555091\pi$
$139$	−564.430	−0.344419	−0.172210	+	0.985060i	$0.555091\pi$
$140$	−5115.02	−3.08785
$141$	−2484.96	−1.48419
$142$	−896.820	−0.529996
$143$	−49.6314	−0.0290237
$144$	278.835	0.161363
$145$	3637.66	2.08339
$146$	−565.779	−0.320714
$147$	−2017.00	−1.13169
$148$	883.904	0.490922
$149$	−754.273	−0.414714	−0.207357	−	0.978265i	$-0.566486\pi$
$149$	−754.273	−0.414714	−0.207357	+	0.978265i	$0.566486\pi$
$150$	−5242.48	−2.85364
$151$	2555.27	1.37712	0.688560	−	0.725179i	$-0.258243\pi$
$151$	2555.27	1.37712	0.688560	+	0.725179i	$0.258243\pi$
$152$	491.211	0.262122
$153$	238.506	0.126027
$154$	−84.7707	−0.0443572
$155$	755.875	0.391699
$156$	−2909.76	−1.49338
$157$	−1811.53	−0.920864	−0.460432	−	0.887695i	$-0.652305\pi$
$157$	−1811.53	−0.920864	−0.460432	+	0.887695i	$0.652305\pi$
$158$	338.645	0.170514
$159$	0.624576	0.000311523 0
$160$	5162.41	2.55078
$161$	−2123.22	−1.03934
$162$	−2316.33	−1.12338
$163$	2865.26	1.37684	0.688419	−	0.725313i	$-0.258305\pi$
$163$	2865.26	1.37684	0.688419	+	0.725313i	$0.258305\pi$
$164$	309.794	0.147505
$165$	−67.9905	−0.0320791
$166$	−3844.02	−1.79731
$167$	14.5741	0.00675318	0.00337659	−	0.999994i	$-0.498925\pi$
$167$	14.5741	0.00675318	0.00337659	+	0.999994i	$0.498925\pi$
$168$	−660.655	−0.303397
$169$	2384.77	1.08547
$170$	3722.48	1.67942
$171$	−510.710	−0.228392
$172$	0	0
$173$	1036.88	0.455678	0.227839	−	0.973699i	$-0.426834\pi$
$173$	1036.88	0.455678	0.227839	+	0.973699i	$0.426834\pi$
$174$	3534.44	1.53991
$175$	7551.64	3.26200
$176$	38.6293	0.0165443
$177$	516.209	0.219213
$178$	−4672.68	−1.96760
$179$	1266.66	0.528909	0.264455	−	0.964398i	$-0.414808\pi$
$179$	1266.66	0.528909	0.264455	+	0.964398i	$0.414808\pi$
$180$	971.877	0.402441
$181$	−60.5440	−0.0248630	−0.0124315	−	0.999923i	$-0.503957\pi$
$181$	−60.5440	−0.0248630	−0.0124315	+	0.999923i	$0.503957\pi$
$182$	7825.67	3.18724
$183$	−2484.49	−1.00360
$184$	−388.019	−0.155463
$185$	−1906.66	−0.757731
$186$	734.426	0.289520
$187$	33.0423	0.0129213
$188$	4920.98	1.90904
$189$	4190.96	1.61295
$190$	−7970.89	−3.04352
$191$	2237.34	0.847581	0.423791	−	0.905760i	$-0.360699\pi$
$191$	2237.34	0.847581	0.423791	+	0.905760i	$0.360699\pi$
$192$	3052.24	1.14728
$193$	−1150.33	−0.429029	−0.214515	−	0.976721i	$-0.568817\pi$
$193$	−1150.33	−0.429029	−0.214515	+	0.976721i	$0.568817\pi$
$194$	−6275.60	−2.32249
$195$	6276.60	2.30501
$196$	3994.27	1.45564
$197$	3565.84	1.28962	0.644811	−	0.764342i	$-0.276936\pi$
$197$	3565.84	1.28962	0.644811	+	0.764342i	$0.276936\pi$
$198$	16.1068	0.00578111
$199$	−2313.80	−0.824226	−0.412113	−	0.911133i	$-0.635209\pi$
$199$	−2313.80	−0.824226	−0.412113	+	0.911133i	$0.635209\pi$
$200$	1380.06	0.487926
$201$	1916.91	0.672678
$202$	−5498.52	−1.91522
$203$	−5091.26	−1.76028
$204$	1937.18	0.664852
$205$	−668.252	−0.227672
$206$	−4629.18	−1.56568
$207$	403.422	0.135458
$208$	−3566.10	−1.18877
$209$	−70.7529	−0.0234167
$210$	10720.5	3.52277
$211$	1691.58	0.551912	0.275956	−	0.961170i	$-0.411006\pi$
$211$	1691.58	0.551912	0.275956	+	0.961170i	$0.411006\pi$
$212$	−1.23685	−0.000400695 0
$213$	1006.72	0.323848
$214$	−2689.34	−0.859065
$215$	0	0
$216$	765.899	0.241263
$217$	−1057.92	−0.330950
$218$	−8543.12	−2.65419
$219$	635.115	0.195968
$220$	134.642	0.0412617
$221$	−3050.33	−0.928448
$222$	−1852.55	−0.560068
$223$	−406.066	−0.121938	−0.0609691	−	0.998140i	$-0.519419\pi$
$223$	−406.066	−0.121938	−0.0609691	+	0.998140i	$0.519419\pi$
$224$	−7225.29	−2.15518
$225$	−1434.84	−0.425139
$226$	3783.32	1.11355
$227$	4463.18	1.30499	0.652493	−	0.757795i	$-0.273723\pi$
$227$	4463.18	1.30499	0.652493	+	0.757795i	$0.273723\pi$
$228$	−4148.06	−1.20488
$229$	−5509.43	−1.58984	−0.794920	−	0.606714i	$-0.792487\pi$
$229$	−5509.43	−1.58984	−0.794920	+	0.606714i	$0.792487\pi$
$230$	6296.39	1.80509
$231$	95.1592	0.0271040
$232$	−930.428	−0.263300
$233$	4222.96	1.18736	0.593680	−	0.804701i	$-0.297674\pi$
$233$	4222.96	1.18736	0.593680	+	0.804701i	$0.297674\pi$
$234$	−1486.91	−0.415395
$235$	−10615.0	−2.94657
$236$	−1022.25	−0.281962
$237$	−380.146	−0.104190
$238$	−5209.96	−1.41896
$239$	−876.821	−0.237309	−0.118654	−	0.992936i	$-0.537858\pi$
$239$	−876.821	−0.237309	−0.118654	+	0.992936i	$0.537858\pi$
$240$	−4885.22	−1.31392
$241$	3169.86	0.847257	0.423628	−	0.905836i	$-0.360756\pi$
$241$	3169.86	0.847257	0.423628	+	0.905836i	$0.360756\pi$
$242$	−5522.06	−1.46682
$243$	−1462.09	−0.385980
$244$	4920.06	1.29088
$245$	−8615.99	−2.24676
$246$	−649.290	−0.168281
$247$	6531.61	1.68258
$248$	−193.335	−0.0495031
$249$	4315.10	1.09823
$250$	−12068.7	−3.05318
$251$	3990.59	1.00352	0.501761	−	0.865006i	$-0.332686\pi$
$251$	3990.59	1.00352	0.501761	+	0.865006i	$0.332686\pi$
$252$	−1360.23	−0.340027
$253$	55.8894	0.0138883
$254$	−1557.56	−0.384763
$255$	−4178.67	−1.02619
$256$	2568.27	0.627018
$257$	−4425.70	−1.07419	−0.537096	−	0.843521i	$-0.680479\pi$
$257$	−4425.70	−1.07419	−0.537096	+	0.843521i	$0.680479\pi$
$258$	0	0
$259$	2668.55	0.640214
$260$	−12429.6	−2.96481
$261$	967.361	0.229418
$262$	−11211.6	−2.64372
$263$	−72.8494	−0.0170802	−0.00854009	−	0.999964i	$-0.502718\pi$
$263$	−72.8494	−0.0170802	−0.00854009	+	0.999964i	$0.502718\pi$
$264$	17.3904	0.00405418
$265$	2.66800	0.000618467 0
$266$	11156.0	2.57150
$267$	5245.32	1.20228
$268$	−3796.07	−0.865231
$269$	3884.39	0.880430	0.440215	−	0.897892i	$-0.354902\pi$
$269$	3884.39	0.880430	0.440215	+	0.897892i	$0.354902\pi$
$270$	−12428.2	−2.80133
$271$	6716.63	1.50556	0.752779	−	0.658273i	$-0.228713\pi$
$271$	6716.63	1.50556	0.752779	+	0.658273i	$0.228713\pi$
$272$	2374.14	0.529240
$273$	−8784.70	−1.94752
$274$	−8393.46	−1.85061
$275$	−198.781	−0.0435889
$276$	3276.65	0.714605
$277$	−2043.39	−0.443233	−0.221616	−	0.975134i	$-0.571133\pi$
$277$	−2043.39	−0.443233	−0.221616	+	0.975134i	$0.571133\pi$
$278$	−2342.66	−0.505407
$279$	201.009	0.0431330
$280$	−2822.12	−0.602335
$281$	−2091.71	−0.444060	−0.222030	−	0.975040i	$-0.571268\pi$
$281$	−2091.71	−0.444060	−0.222030	+	0.975040i	$0.571268\pi$
$282$	−10313.8	−2.17793
$283$	−4108.63	−0.863013	−0.431507	−	0.902110i	$-0.642018\pi$
$283$	−4108.63	−0.863013	−0.431507	+	0.902110i	$0.642018\pi$
$284$	−1993.63	−0.416549
$285$	8947.71	1.85971
$286$	−205.994	−0.0425899
$287$	935.283	0.192362
$288$	1372.84	0.280886
$289$	−2882.24	−0.586655
$290$	15098.1	3.05720
$291$	7044.67	1.41913
$292$	−1257.72	−0.252064
$293$	−771.427	−0.153813	−0.0769066	−	0.997038i	$-0.524504\pi$
$293$	−771.427	−0.153813	−0.0769066	+	0.997038i	$0.524504\pi$
$294$	−8371.50	−1.66067
$295$	2205.09	0.435204
$296$	487.678	0.0957624
$297$	−110.318	−0.0215532
$298$	−3130.60	−0.608559
$299$	−5159.48	−0.997927
$300$	−11654.0	−2.24281
$301$	0	0
$302$	10605.6	2.02081
$303$	6172.36	1.17027
$304$	−5083.71	−0.959114
$305$	−10613.0	−1.99245
$306$	989.916	0.184934
$307$	−6628.00	−1.23218	−0.616092	−	0.787675i	$-0.711285\pi$
$307$	−6628.00	−1.23218	−0.616092	+	0.787675i	$0.711285\pi$
$308$	−188.445	−0.0348624
$309$	5196.49	0.956692
$310$	3137.24	0.574785
$311$	4804.84	0.876069	0.438035	−	0.898958i	$-0.355675\pi$
$311$	4804.84	0.876069	0.438035	+	0.898958i	$0.355675\pi$
$312$	−1605.41	−0.291308
$313$	−969.070	−0.175000	−0.0875001	−	0.996165i	$-0.527888\pi$
$313$	−969.070	−0.175000	−0.0875001	+	0.996165i	$0.527888\pi$
$314$	−7518.71	−1.35129
$315$	2934.14	0.524826
$316$	752.806	0.134015
$317$	−3279.49	−0.581055	−0.290528	−	0.956867i	$-0.593831\pi$
$317$	−3279.49	−0.581055	−0.290528	+	0.956867i	$0.593831\pi$
$318$	2.59229	0.000457133 0
$319$	134.017	0.0235219
$320$	13038.3	2.27769
$321$	3018.92	0.524921
$322$	−8812.40	−1.52514
$323$	−4348.44	−0.749083
$324$	−5149.19	−0.882921
$325$	18350.6	3.13203
$326$	11892.2	2.02040
$327$	9590.07	1.62181
$328$	170.923	0.0287733
$329$	14856.7	2.48959
$330$	−282.193	−0.0470734
$331$	6633.82	1.10159	0.550797	−	0.834639i	$-0.314324\pi$
$331$	6633.82	1.10159	0.550797	+	0.834639i	$0.314324\pi$
$332$	−8545.24	−1.41259
$333$	−507.036	−0.0834396
$334$	60.4897	0.00990973
$335$	8188.45	1.33547
$336$	6837.34	1.11014
$337$	−4061.37	−0.656489	−0.328244	−	0.944593i	$-0.606457\pi$
$337$	−4061.37	−0.656489	−0.328244	+	0.944593i	$0.606457\pi$
$338$	9897.93	1.59283
$339$	−4246.97	−0.680424
$340$	8275.05	1.31993
$341$	27.8475	0.00442236
$342$	−2119.69	−0.335146
$343$	2504.56	0.394267
$344$	0	0
$345$	−7068.01	−1.10298
$346$	4303.54	0.668669
$347$	7916.21	1.22468	0.612341	−	0.790594i	$-0.290228\pi$
$347$	7916.21	1.22468	0.612341	+	0.790594i	$0.290228\pi$
$348$	7857.04	1.21029
$349$	−12863.0	−1.97290	−0.986448	−	0.164077i	$-0.947535\pi$
$349$	−12863.0	−1.97290	−0.986448	+	0.164077i	$0.947535\pi$
$350$	31342.9	4.78672
$351$	10184.1	1.54868
$352$	190.191	0.0287988
$353$	−7949.47	−1.19861	−0.599303	−	0.800522i	$-0.704555\pi$
$353$	−7949.47	−1.19861	−0.599303	+	0.800522i	$0.704555\pi$
$354$	2142.52	0.321676
$355$	4300.42	0.642937
$356$	−10387.3	−1.54643
$357$	5848.44	0.867037
$358$	5257.26	0.776130
$359$	−3114.11	−0.457817	−0.228909	−	0.973448i	$-0.573516\pi$
$359$	−3114.11	−0.457817	−0.228909	+	0.973448i	$0.573516\pi$
$360$	536.215	0.0785028
$361$	2452.25	0.357523
$362$	−251.287	−0.0364843
$363$	6198.78	0.896286
$364$	17396.4	2.50500
$365$	2713.02	0.389057
$366$	−10311.8	−1.47270
$367$	−4292.76	−0.610573	−0.305287	−	0.952260i	$-0.598752\pi$
$367$	−4292.76	−0.610573	−0.305287	+	0.952260i	$0.598752\pi$
$368$	4015.74	0.568845
$369$	−177.708	−0.0250707
$370$	−7913.54	−1.11191
$371$	−3.73412	−0.000522549 0
$372$	1632.62	0.227547
$373$	−2868.12	−0.398138	−0.199069	−	0.979985i	$-0.563792\pi$
$373$	−2868.12	−0.398138	−0.199069	+	0.979985i	$0.563792\pi$
$374$	137.141	0.0189610
$375$	13547.8	1.86561
$376$	2715.06	0.372390
$377$	−12371.9	−1.69014
$378$	17394.5	2.36687
$379$	5154.61	0.698613	0.349307	−	0.937009i	$-0.386417\pi$
$379$	5154.61	0.698613	0.349307	+	0.937009i	$0.386417\pi$
$380$	−17719.2	−2.39205
$381$	1748.44	0.235105
$382$	9286.03	1.24376
$383$	4471.13	0.596512	0.298256	−	0.954486i	$-0.403595\pi$
$383$	4471.13	0.596512	0.298256	+	0.954486i	$0.403595\pi$
$384$	3000.16	0.398701
$385$	406.491	0.0538096
$386$	−4774.43	−0.629565
$387$	0	0
$388$	−13950.6	−1.82535
$389$	−8980.63	−1.17053	−0.585265	−	0.810842i	$-0.699010\pi$
$389$	−8980.63	−1.17053	−0.585265	+	0.810842i	$0.699010\pi$
$390$	26050.9	3.38241
$391$	3434.94	0.444277
$392$	2203.77	0.283946
$393$	12585.6	1.61541
$394$	14800.0	1.89241
$395$	−1623.87	−0.206850
$396$	35.8053	0.00454365
$397$	6091.78	0.770120	0.385060	−	0.922892i	$-0.374181\pi$
$397$	6091.78	0.770120	0.385060	+	0.922892i	$0.374181\pi$
$398$	−9603.39	−1.20948
$399$	−12523.2	−1.57129
$400$	−14282.7	−1.78534
$401$	−6404.00	−0.797507	−0.398753	−	0.917058i	$-0.630557\pi$
$401$	−6404.00	−0.797507	−0.398753	+	0.917058i	$0.630557\pi$
$402$	7956.09	0.987099
$403$	−2570.76	−0.317764
$404$	−12223.2	−1.50526
$405$	11107.3	1.36277
$406$	−21131.2	−2.58306
$407$	−70.2439	−0.00855494
$408$	1068.80	0.129690
$409$	12843.8	1.55277	0.776386	−	0.630258i	$-0.217051\pi$
$409$	12843.8	1.55277	0.776386	+	0.630258i	$0.217051\pi$
$410$	−2773.57	−0.334090
$411$	9422.07	1.13079
$412$	−10290.6	−1.23054
$413$	−3086.23	−0.367708
$414$	1674.39	0.198773
$415$	18432.8	2.18032
$416$	−17557.6	−2.06931
$417$	2629.75	0.308823
$418$	−293.659	−0.0343620
$419$	−8367.04	−0.975552	−0.487776	−	0.872969i	$-0.662192\pi$
$419$	−8367.04	−0.975552	−0.487776	+	0.872969i	$0.662192\pi$
$420$	23831.5	2.76871
$421$	−2176.45	−0.251956	−0.125978	−	0.992033i	$-0.540207\pi$
$421$	−2176.45	−0.251956	−0.125978	+	0.992033i	$0.540207\pi$
$422$	7020.88	0.809884
$423$	−2822.83	−0.324470
$424$	−0.682411	−7.81623e−5 0
$425$	−12217.0	−1.39438
$426$	4178.39	0.475220
$427$	14853.9	1.68344
$428$	−5978.40	−0.675179
$429$	231.239	0.0260240
$430$	0	0
$431$	2491.13	0.278408	0.139204	−	0.990264i	$-0.455546\pi$
$431$	2491.13	0.278408	0.139204	+	0.990264i	$0.455546\pi$
$432$	−7926.54	−0.882791
$433$	−15117.3	−1.67781	−0.838906	−	0.544277i	$-0.816804\pi$
$433$	−15117.3	−1.67781	−0.838906	+	0.544277i	$0.816804\pi$
$434$	−4390.87	−0.485642
$435$	−16948.3	−1.86807
$436$	−18991.3	−2.08605
$437$	−7355.18	−0.805139
$438$	2636.03	0.287567
$439$	2947.73	0.320473	0.160237	−	0.987079i	$-0.448774\pi$
$439$	2947.73	0.320473	0.160237	+	0.987079i	$0.448774\pi$
$440$	74.2864	0.00804878
$441$	−2291.24	−0.247408
$442$	−12660.3	−1.36242
$443$	−6749.23	−0.723850	−0.361925	−	0.932207i	$-0.617880\pi$
$443$	−6749.23	−0.723850	−0.361925	+	0.932207i	$0.617880\pi$
$444$	−4118.21	−0.440184
$445$	22406.4	2.38689
$446$	−1685.37	−0.178934
$447$	3514.25	0.371853
$448$	−18248.3	−1.92444
$449$	−5120.14	−0.538162	−0.269081	−	0.963118i	$-0.586720\pi$
$449$	−5120.14	−0.538162	−0.269081	+	0.963118i	$0.586720\pi$
$450$	−5955.29	−0.623856
$451$	−24.6194	−0.00257047
$452$	8410.31	0.875194
$453$	−11905.3	−1.23479
$454$	18524.4	1.91496
$455$	−37525.6	−3.86643
$456$	−2288.61	−0.235031
$457$	11669.3	1.19446	0.597230	−	0.802070i	$-0.296268\pi$
$457$	11669.3	1.19446	0.597230	+	0.802070i	$0.296268\pi$
$458$	−22866.8	−2.33296
$459$	−6780.11	−0.689474
$460$	13996.8	1.41871
$461$	−12772.3	−1.29038	−0.645188	−	0.764024i	$-0.723221\pi$
$461$	−12772.3	−1.29038	−0.645188	+	0.764024i	$0.723221\pi$
$462$	394.957	0.0397728
$463$	6707.16	0.673235	0.336618	−	0.941641i	$-0.390717\pi$
$463$	6707.16	0.673235	0.336618	+	0.941641i	$0.390717\pi$
$464$	9629.30	0.963425
$465$	−3521.71	−0.351216
$466$	17527.3	1.74235
$467$	13188.3	1.30681	0.653407	−	0.757007i	$-0.273339\pi$
$467$	13188.3	1.30681	0.653407	+	0.757007i	$0.273339\pi$
$468$	−3305.40	−0.326479
$469$	−11460.5	−1.12835
$470$	−44057.3	−4.32385
$471$	8440.12	0.825691
$472$	−564.009	−0.0550013
$473$	0	0
$474$	−1577.79	−0.152891
$475$	26160.0	2.52696
$476$	−11581.7	−1.11523
$477$	0.709499	6.81042e−5 0
$478$	−3639.23	−0.348231
$479$	2983.64	0.284606	0.142303	−	0.989823i	$-0.454549\pi$
$479$	2983.64	0.284606	0.142303	+	0.989823i	$0.454549\pi$
$480$	−24052.3	−2.28715
$481$	6484.62	0.614706
$482$	13156.5	1.24328
$483$	9892.35	0.931920
$484$	−12275.5	−1.15285
$485$	30092.7	2.81740
$486$	−6068.38	−0.566394
$487$	903.861	0.0841024	0.0420512	−	0.999115i	$-0.486611\pi$
$487$	903.861	0.0841024	0.0420512	+	0.999115i	$0.486611\pi$
$488$	2714.55	0.251807
$489$	−13349.6	−1.23454
$490$	−35760.5	−3.29693
$491$	14745.6	1.35531	0.677656	−	0.735379i	$-0.262996\pi$
$491$	14745.6	1.35531	0.677656	+	0.735379i	$0.262996\pi$
$492$	−1443.37	−0.132260
$493$	8236.60	0.752450
$494$	27109.3	2.46904
$495$	−77.2351	−0.00701305
$496$	2000.89	0.181134
$497$	−6018.85	−0.543224
$498$	17909.8	1.61156
$499$	846.074	0.0759028	0.0379514	−	0.999280i	$-0.487917\pi$
$499$	846.074	0.0759028	0.0379514	+	0.999280i	$0.487917\pi$
$500$	−26828.8	−2.39964
$501$	−67.9026	−0.00605522
$502$	16562.9	1.47258
$503$	239.890	0.0212647	0.0106324	−	0.999943i	$-0.496616\pi$
$503$	239.890	0.0212647	0.0106324	+	0.999943i	$0.496616\pi$
$504$	−750.484	−0.0663278
$505$	26366.5	2.32335
$506$	231.968	0.0203799
$507$	−11110.9	−0.973280
$508$	−3462.44	−0.302404
$509$	5624.56	0.489792	0.244896	−	0.969549i	$-0.421246\pi$
$509$	5624.56	0.489792	0.244896	+	0.969549i	$0.421246\pi$
$510$	−17343.5	−1.50585
$511$	−3797.13	−0.328718
$512$	15811.0	1.36475
$513$	14518.1	1.24950
$514$	−18368.8	−1.57629
$515$	22197.8	1.89932
$516$	0	0
$517$	−391.071	−0.0332675
$518$	11075.8	0.939461
$519$	−4830.93	−0.408582
$520$	−6857.80	−0.578336
$521$	−12416.2	−1.04408	−0.522038	−	0.852922i	$-0.674828\pi$
$521$	−12416.2	−1.04408	−0.522038	+	0.852922i	$0.674828\pi$
$522$	4015.01	0.336652
$523$	−8470.79	−0.708225	−0.354112	−	0.935203i	$-0.615217\pi$
$523$	−8470.79	−0.708225	−0.354112	+	0.935203i	$0.615217\pi$
$524$	−24923.3	−2.07782
$525$	−35184.0	−2.92487
$526$	−302.360	−0.0250637
$527$	1711.49	0.141468
$528$	−179.979	−0.0148344
$529$	−6356.97	−0.522477
$530$	11.0735	0.000907549 0
$531$	586.397	0.0479237
$532$	24799.8	2.02106
$533$	2272.76	0.184698
$534$	21770.6	1.76424
$535$	12895.9	1.04213
$536$	−2094.41	−0.168778
$537$	−5901.53	−0.474245
$538$	16122.1	1.29196
$539$	−317.425	−0.0253664
$540$	−27627.9	−2.20170
$541$	1109.59	0.0881792	0.0440896	−	0.999028i	$-0.485961\pi$
$541$	1109.59	0.0881792	0.0440896	+	0.999028i	$0.485961\pi$
$542$	27877.3	2.20928
$543$	282.082	0.0222933
$544$	11689.0	0.921255
$545$	40965.9	3.21979
$546$	−36460.7	−2.85783
$547$	−13123.6	−1.02582	−0.512912	−	0.858441i	$-0.671433\pi$
$547$	−13123.6	−1.02582	−0.512912	+	0.858441i	$0.671433\pi$
$548$	−18658.6	−1.45448
$549$	−2822.30	−0.219404
$550$	−825.037	−0.0639631
$551$	−17636.9	−1.36362
$552$	1807.83	0.139395
$553$	2272.76	0.174769
$554$	−8481.06	−0.650407
$555$	8883.34	0.679417
$556$	−5207.71	−0.397223
$557$	16074.4	1.22279	0.611395	−	0.791325i	$-0.290608\pi$
$557$	16074.4	1.22279	0.611395	+	0.791325i	$0.290608\pi$
$558$	834.285	0.0632941
$559$	0	0
$560$	29207.0	2.20397
$561$	−153.948	−0.0115859
$562$	−8681.60	−0.651621
$563$	−13329.8	−0.997839	−0.498919	−	0.866648i	$-0.666270\pi$
$563$	−13329.8	−0.997839	−0.498919	+	0.866648i	$0.666270\pi$
$564$	−22927.4	−1.71174
$565$	−18141.8	−1.35085
$566$	−17052.8	−1.26640
$567$	−15545.7	−1.15142
$568$	−1099.95	−0.0812548
$569$	−17313.5	−1.27561	−0.637803	−	0.770199i	$-0.720157\pi$
$569$	−17313.5	−1.27561	−0.637803	+	0.770199i	$0.720157\pi$
$570$	37137.3	2.72897
$571$	8514.71	0.624045	0.312022	−	0.950075i	$-0.398994\pi$
$571$	8514.71	0.624045	0.312022	+	0.950075i	$0.398994\pi$
$572$	−457.924	−0.0334734
$573$	−10424.0	−0.759982
$574$	3881.87	0.282276
$575$	−20664.4	−1.49872
$576$	3467.26	0.250814
$577$	−23224.4	−1.67564	−0.837820	−	0.545947i	$-0.816170\pi$
$577$	−23224.4	−1.67564	−0.837820	+	0.545947i	$0.816170\pi$
$578$	−11962.7	−0.860868
$579$	5359.53	0.384688
$580$	33562.9	2.40280
$581$	−25798.5	−1.84217
$582$	29238.8	2.08245
$583$	0.0982928	6.98263e−6 0
$584$	−693.926	−0.0491693
$585$	7130.02	0.503915
$586$	−3201.79	−0.225708
$587$	−15872.8	−1.11608	−0.558042	−	0.829813i	$-0.688447\pi$
$587$	−15872.8	−1.11608	−0.558042	+	0.829813i	$0.688447\pi$
$588$	−18609.8	−1.30520
$589$	−3664.79	−0.256376
$590$	9152.18	0.638626
$591$	−16613.7	−1.15634
$592$	−5047.13	−0.350399
$593$	−21818.4	−1.51092	−0.755461	−	0.655194i	$-0.772587\pi$
$593$	−21818.4	−1.51092	−0.755461	+	0.655194i	$0.772587\pi$
$594$	−457.874	−0.0316276
$595$	24982.8	1.72133
$596$	−6959.30	−0.478295
$597$	10780.3	0.739041
$598$	−21414.3	−1.46437
$599$	−10835.2	−0.739087	−0.369544	−	0.929213i	$-0.620486\pi$
$599$	−10835.2	−0.739087	−0.369544	+	0.929213i	$0.620486\pi$
$600$	−6429.88	−0.437498
$601$	7125.29	0.483605	0.241803	−	0.970325i	$-0.422261\pi$
$601$	7125.29	0.483605	0.241803	+	0.970325i	$0.422261\pi$
$602$	0	0
$603$	2177.55	0.147059
$604$	23576.2	1.58825
$605$	26479.3	1.77940
$606$	25618.3	1.71728
$607$	10703.4	0.715710	0.357855	−	0.933777i	$-0.383508\pi$
$607$	10703.4	0.715710	0.357855	+	0.933777i	$0.383508\pi$
$608$	−25029.5	−1.66954
$609$	23720.8	1.57835
$610$	−44049.0	−2.92376
$611$	36102.0	2.39040
$612$	2200.58	0.145348
$613$	−3774.44	−0.248692	−0.124346	−	0.992239i	$-0.539683\pi$
$613$	−3774.44	−0.248692	−0.124346	+	0.992239i	$0.539683\pi$
$614$	−27509.4	−1.80813
$615$	3113.47	0.204142
$616$	−103.971	−0.00680049
$617$	−19188.0	−1.25199	−0.625997	−	0.779826i	$-0.715308\pi$
$617$	−19188.0	−1.25199	−0.625997	+	0.779826i	$0.715308\pi$
$618$	21567.9	1.40387
$619$	−19479.3	−1.26485	−0.632424	−	0.774622i	$-0.717940\pi$
$619$	−19479.3	−1.26485	−0.632424	+	0.774622i	$0.717940\pi$
$620$	6974.07	0.451751
$621$	−11468.2	−0.741069
$622$	19942.4	1.28556
$623$	−31359.9	−2.01671
$624$	16614.9	1.06591
$625$	23984.0	1.53498
$626$	−4022.11	−0.256798
$627$	329.646	0.0209965
$628$	−16714.0	−1.06204
$629$	−4317.16	−0.273667
$630$	12178.1	0.770138
$631$	−7063.66	−0.445642	−0.222821	−	0.974859i	$-0.571527\pi$
$631$	−7063.66	−0.445642	−0.222821	+	0.974859i	$0.571527\pi$
$632$	415.347	0.0261418
$633$	−7881.28	−0.494870
$634$	−13611.5	−0.852650
$635$	7468.79	0.466756
$636$	5.76265	0.000359283 0
$637$	29303.4	1.82267
$638$	556.234	0.0345165
$639$	1143.61	0.0707988
$640$	12815.8	0.791543
$641$	8274.88	0.509888	0.254944	−	0.966956i	$-0.417943\pi$
$641$	8274.88	0.509888	0.254944	+	0.966956i	$0.417943\pi$
$642$	12530.0	0.770278
$643$	18880.9	1.15799	0.578996	−	0.815331i	$-0.303445\pi$
$643$	18880.9	1.15799	0.578996	+	0.815331i	$0.303445\pi$
$644$	−19589.9	−1.19868
$645$	0	0
$646$	−18048.1	−1.09922
$647$	−13808.8	−0.839075	−0.419537	−	0.907738i	$-0.637808\pi$
$647$	−13808.8	−0.839075	−0.419537	+	0.907738i	$0.637808\pi$
$648$	−2840.97	−0.172228
$649$	81.2386	0.00491355
$650$	76164.0	4.59599
$651$	4928.97	0.296746
$652$	26436.3	1.58792
$653$	30854.7	1.84906	0.924530	−	0.381109i	$-0.124458\pi$
$653$	30854.7	1.84906	0.924530	+	0.381109i	$0.124458\pi$
$654$	39803.4	2.37987
$655$	53761.7	3.20709
$656$	−1768.94	−0.105283
$657$	721.471	0.0428421
$658$	61662.4	3.65327
$659$	−11484.2	−0.678848	−0.339424	−	0.940633i	$-0.610232\pi$
$659$	−11484.2	−0.678848	−0.339424	+	0.940633i	$0.610232\pi$
$660$	−627.314	−0.0369972
$661$	6233.04	0.366773	0.183387	−	0.983041i	$-0.441294\pi$
$661$	6233.04	0.366773	0.183387	+	0.983041i	$0.441294\pi$
$662$	27533.5	1.61650
$663$	14211.8	0.832491
$664$	−4714.68	−0.275550
$665$	−53495.2	−3.11948
$666$	−2104.44	−0.122441
$667$	13931.8	0.808758
$668$	134.468	0.00778852
$669$	1891.91	0.109336
$670$	33986.0	1.95969
$671$	−390.997	−0.0224952
$672$	33663.5	1.93244
$673$	−27119.3	−1.55330	−0.776651	−	0.629931i	$-0.783083\pi$
$673$	−27119.3	−1.55330	−0.776651	+	0.629931i	$0.783083\pi$
$674$	−16856.6	−0.963343
$675$	40788.9	2.32587
$676$	22003.0	1.25188
$677$	22288.2	1.26530	0.632649	−	0.774439i	$-0.281968\pi$
$677$	22288.2	1.26530	0.632649	+	0.774439i	$0.281968\pi$
$678$	−17627.0	−0.998465
$679$	−42117.6	−2.38045
$680$	4565.60	0.257475
$681$	−20794.5	−1.17011
$682$	115.580	0.00648945
$683$	2868.96	0.160729	0.0803644	−	0.996766i	$-0.474392\pi$
$683$	2868.96	0.160729	0.0803644	+	0.996766i	$0.474392\pi$
$684$	−4712.06	−0.263407
$685$	40248.2	2.24497
$686$	10395.1	0.578554
$687$	25669.1	1.42553
$688$	0	0
$689$	−9.07398	−0.000501729 0
$690$	−29335.6	−1.61853
$691$	32355.3	1.78126	0.890631	−	0.454726i	$-0.150263\pi$
$691$	32355.3	1.78126	0.890631	+	0.454726i	$0.150263\pi$
$692$	9566.74	0.525539
$693$	108.098	0.00592540
$694$	32856.1	1.79712
$695$	11233.5	0.613108
$696$	4334.97	0.236087
$697$	−1513.09	−0.0822274
$698$	−53387.6	−2.89506
$699$	−19675.2	−1.06464
$700$	69675.2	3.76211
$701$	−5159.16	−0.277973	−0.138986	−	0.990294i	$-0.544384\pi$
$701$	−5159.16	−0.277973	−0.138986	+	0.990294i	$0.544384\pi$
$702$	42269.0	2.27256
$703$	9244.26	0.495952
$704$	480.348	0.0257156
$705$	49456.5	2.64204
$706$	−32994.1	−1.75885
$707$	−36902.4	−1.96302
$708$	4762.80	0.252821
$709$	20468.6	1.08422	0.542112	−	0.840306i	$-0.317625\pi$
$709$	20468.6	1.08422	0.542112	+	0.840306i	$0.317625\pi$
$710$	17848.8	0.943457
$711$	−431.834	−0.0227778
$712$	−5731.03	−0.301656
$713$	2894.91	0.152055
$714$	24273.8	1.27230
$715$	987.782	0.0516657
$716$	11686.9	0.609997
$717$	4085.21	0.212782
$718$	−12925.0	−0.671809
$719$	−16111.8	−0.835702	−0.417851	−	0.908515i	$-0.637217\pi$
$719$	−16111.8	−0.835702	−0.417851	+	0.908515i	$0.637217\pi$
$720$	−5549.46	−0.287245
$721$	−31068.0	−1.60476
$722$	10178.0	0.524635
$723$	−14768.8	−0.759691
$724$	−558.609	−0.0286748
$725$	−49551.1	−2.53832
$726$	25727.9	1.31522
$727$	−9681.53	−0.493904	−0.246952	−	0.969028i	$-0.579429\pi$
$727$	−9681.53	−0.493904	−0.246952	+	0.969028i	$0.579429\pi$
$728$	9598.15	0.488642
$729$	21880.4	1.11164
$730$	11260.3	0.570909
$731$	0	0
$732$	−22923.1	−1.15746
$733$	−18580.8	−0.936288	−0.468144	−	0.883652i	$-0.655077\pi$
$733$	−18580.8	−0.936288	−0.468144	+	0.883652i	$0.655077\pi$
$734$	−17817.0	−0.895966
$735$	40142.9	2.01455
$736$	19771.4	0.990196
$737$	301.674	0.0150778
$738$	−737.573	−0.0367892
$739$	27864.0	1.38700	0.693502	−	0.720455i	$-0.256067\pi$
$739$	27864.0	1.38700	0.693502	+	0.720455i	$0.256067\pi$
$740$	−17591.8	−0.873900
$741$	−30431.6	−1.50868
$742$	−15.4984	−0.000766798 0
$743$	−9787.89	−0.483288	−0.241644	−	0.970365i	$-0.577687\pi$
$743$	−9787.89	−0.483288	−0.241644	+	0.970365i	$0.577687\pi$
$744$	900.770	0.0443869
$745$	15011.8	0.738241
$746$	−11904.1	−0.584235
$747$	4901.82	0.240091
$748$	304.864	0.0149023
$749$	−18049.1	−0.880505
$750$	56229.7	2.73763
$751$	−19515.4	−0.948239	−0.474119	−	0.880461i	$-0.657234\pi$
$751$	−19515.4	−0.948239	−0.474119	+	0.880461i	$0.657234\pi$
$752$	−28099.1	−1.36259
$753$	−18592.6	−0.899805
$754$	−51349.2	−2.48014
$755$	−50855.9	−2.45144
$756$	38667.9	1.86023
$757$	−23470.3	−1.12687	−0.563436	−	0.826160i	$-0.690521\pi$
$757$	−23470.3	−1.12687	−0.563436	+	0.826160i	$0.690521\pi$
$758$	21394.1	1.02516
$759$	−260.395	−0.0124529
$760$	−9776.26	−0.466608
$761$	24891.6	1.18570	0.592851	−	0.805312i	$-0.298002\pi$
$761$	24891.6	1.18570	0.592851	+	0.805312i	$0.298002\pi$
$762$	7256.85	0.344997
$763$	−57335.6	−2.72043
$764$	20642.8	0.977526
$765$	−4746.83	−0.224343
$766$	18557.4	0.875332
$767$	−7499.60	−0.353057
$768$	−11965.9	−0.562215
$769$	−32481.7	−1.52317	−0.761586	−	0.648064i	$-0.775579\pi$
$769$	−32481.7	−1.52317	−0.761586	+	0.648064i	$0.775579\pi$
$770$	1687.13	0.0789612
$771$	20619.8	0.963172
$772$	−10613.5	−0.494805
$773$	−1476.22	−0.0686879	−0.0343440	−	0.999410i	$-0.510934\pi$
$773$	−1476.22	−0.0686879	−0.0343440	+	0.999410i	$0.510934\pi$
$774$	0	0
$775$	−10296.3	−0.477230
$776$	−7697.00	−0.356065
$777$	−12433.1	−0.574047
$778$	−37274.0	−1.71766
$779$	3239.96	0.149016
$780$	57911.0	2.65839
$781$	158.434	0.00725890
$782$	14256.6	0.651939
$783$	−27499.5	−1.25511
$784$	−22807.5	−1.03897
$785$	36053.6	1.63925
$786$	52236.1	2.37049
$787$	7837.58	0.354993	0.177496	−	0.984121i	$-0.443200\pi$
$787$	7837.58	0.354993	0.177496	+	0.984121i	$0.443200\pi$
$788$	32900.2	1.48734
$789$	339.414	0.0153149
$790$	−6739.83	−0.303535
$791$	25391.1	1.14135
$792$	19.7549	0.000886313 0
$793$	36095.2	1.61637
$794$	25283.8	1.13009
$795$	−12.4305	−0.000554547 0
$796$	−21348.3	−0.950590
$797$	10321.0	0.458706	0.229353	−	0.973343i	$-0.426339\pi$
$797$	10321.0	0.458706	0.229353	+	0.973343i	$0.426339\pi$
$798$	−51977.2	−2.30573
$799$	−24035.0	−1.06420
$800$	−70320.7	−3.10777
$801$	5958.52	0.262839
$802$	−26579.7	−1.17028
$803$	99.9514	0.00439254
$804$	17686.3	0.775807
$805$	42257.1	1.85015
$806$	−10669.9	−0.466292
$807$	−18097.8	−0.789435
$808$	−6743.92	−0.293626
$809$	30215.4	1.31312	0.656561	−	0.754273i	$-0.272010\pi$
$809$	30215.4	1.31312	0.656561	+	0.754273i	$0.272010\pi$
$810$	46100.4	1.99976
$811$	43448.1	1.88122	0.940611	−	0.339487i	$-0.110253\pi$
$811$	43448.1	1.88122	0.940611	+	0.339487i	$0.110253\pi$
$812$	−46974.5	−2.03015
$813$	−31293.6	−1.34996
$814$	−291.546	−0.0125537
$815$	−57025.4	−2.45094
$816$	−11061.4	−0.474542
$817$	0	0
$818$	53307.8	2.27856
$819$	−9979.15	−0.425763
$820$	−6165.63	−0.262577
$821$	28087.5	1.19398	0.596991	−	0.802248i	$-0.296363\pi$
$821$	28087.5	1.19398	0.596991	+	0.802248i	$0.296363\pi$
$822$	39106.1	1.65935
$823$	−44076.2	−1.86683	−0.933413	−	0.358803i	$-0.883185\pi$
$823$	−44076.2	−1.86683	−0.933413	+	0.358803i	$0.883185\pi$
$824$	−5677.67	−0.240038
$825$	926.144	0.0390839
$826$	−12809.3	−0.539581
$827$	6095.24	0.256291	0.128145	−	0.991755i	$-0.459098\pi$
$827$	6095.24	0.256291	0.128145	+	0.991755i	$0.459098\pi$
$828$	3722.17	0.156225
$829$	−8358.62	−0.350189	−0.175095	−	0.984552i	$-0.556023\pi$
$829$	−8358.62	−0.350189	−0.175095	+	0.984552i	$0.556023\pi$
$830$	76505.0	3.19943
$831$	9520.40	0.397424
$832$	−44343.7	−1.84777
$833$	−19508.8	−0.811453
$834$	10914.7	0.453172
$835$	−290.059	−0.0120215
$836$	−652.802	−0.0270067
$837$	−5714.16	−0.235974
$838$	−34727.2	−1.43154
$839$	−12358.3	−0.508530	−0.254265	−	0.967135i	$-0.581834\pi$
$839$	−12358.3	−0.508530	−0.254265	+	0.967135i	$0.581834\pi$
$840$	13148.6	0.540082
$841$	9017.93	0.369754
$842$	−9033.31	−0.369725
$843$	9745.52	0.398165
$844$	15607.4	0.636526
$845$	−47462.4	−1.93226
$846$	−11716.1	−0.476133
$847$	−37060.3	−1.50343
$848$	7.06249	0.000285999 0
$849$	19142.6	0.773819
$850$	−50706.4	−2.04614
$851$	−7302.26	−0.294146
$852$	9288.54	0.373498
$853$	−17084.9	−0.685786	−0.342893	−	0.939374i	$-0.611407\pi$
$853$	−17084.9	−0.685786	−0.342893	+	0.939374i	$0.611407\pi$
$854$	61650.8	2.47031
$855$	10164.3	0.406564
$856$	−3298.47	−0.131705
$857$	−16684.6	−0.665037	−0.332519	−	0.943097i	$-0.607898\pi$
$857$	−16684.6	−0.665037	−0.332519	+	0.943097i	$0.607898\pi$
$858$	959.752	0.0381881
$859$	48813.4	1.93887	0.969436	−	0.245343i	$-0.0789005\pi$
$859$	48813.4	1.93887	0.969436	+	0.245343i	$0.0789005\pi$
$860$	0	0
$861$	−4357.59	−0.172481
$862$	10339.4	0.408540
$863$	−38260.4	−1.50915	−0.754577	−	0.656212i	$-0.772158\pi$
$863$	−38260.4	−1.50915	−0.754577	+	0.656212i	$0.772158\pi$
$864$	−39026.1	−1.53669
$865$	−20636.3	−0.811161
$866$	−62744.2	−2.46205
$867$	13428.7	0.526023
$868$	−9760.88	−0.381689
$869$	−59.8256	−0.00233538
$870$	−70343.6	−2.74123
$871$	−27849.3	−1.08339
$872$	−10478.1	−0.406919
$873$	8002.53	0.310246
$874$	−30527.5	−1.18147
$875$	−80997.3	−3.12938
$876$	5859.88	0.226013
$877$	29665.4	1.14222	0.571112	−	0.820872i	$-0.306512\pi$
$877$	29665.4	1.14222	0.571112	+	0.820872i	$0.306512\pi$
$878$	12234.5	0.470268
$879$	3594.17	0.137916
$880$	−768.814	−0.0294508
$881$	3377.77	0.129171	0.0645857	−	0.997912i	$-0.479427\pi$
$881$	3377.77	0.129171	0.0645857	+	0.997912i	$0.479427\pi$
$882$	−9509.77	−0.363050
$883$	32125.9	1.22437	0.612186	−	0.790713i	$-0.290290\pi$
$883$	32125.9	1.22437	0.612186	+	0.790713i	$0.290290\pi$
$884$	−28143.8	−1.07079
$885$	−10273.8	−0.390225
$886$	−28012.6	−1.06219
$887$	−25106.0	−0.950368	−0.475184	−	0.879886i	$-0.657618\pi$
$887$	−25106.0	−0.950368	−0.475184	+	0.879886i	$0.657618\pi$
$888$	−2272.15	−0.0858652
$889$	−10453.3	−0.394366
$890$	92997.3	3.50256
$891$	409.207	0.0153860
$892$	−3746.57	−0.140633
$893$	51465.8	1.92860
$894$	14585.8	0.545663
$895$	−25209.5	−0.941522
$896$	−17936.9	−0.668782
$897$	24038.6	0.894789
$898$	−21251.1	−0.789707
$899$	6941.67	0.257528
$900$	−13238.6	−0.490318
$901$	6.04103	0.000223370 0
$902$	−102.182	−0.00377194
$903$	0	0
$904$	4640.23	0.170721
$905$	1204.97	0.0442591
$906$	−49412.8	−1.81195
$907$	17495.5	0.640496	0.320248	−	0.947334i	$-0.396234\pi$
$907$	17495.5	0.640496	0.320248	+	0.947334i	$0.396234\pi$
$908$	41179.5	1.50506
$909$	7011.61	0.255842
$910$	−155749.	−5.67366
$911$	−44170.3	−1.60639	−0.803197	−	0.595713i	$-0.796870\pi$
$911$	−44170.3	−1.60639	−0.803197	+	0.595713i	$0.796870\pi$
$912$	23685.6	0.859987
$913$	679.091	0.0246162
$914$	48433.3	1.75277
$915$	49447.2	1.78653
$916$	−50832.7	−1.83358
$917$	−75244.6	−2.70970
$918$	−28140.7	−1.01174
$919$	2965.31	0.106438	0.0532191	−	0.998583i	$-0.483052\pi$
$919$	2965.31	0.106438	0.0532191	+	0.998583i	$0.483052\pi$
$920$	7722.50	0.276743
$921$	30880.7	1.10483
$922$	−53011.0	−1.89352
$923$	−14625.9	−0.521580
$924$	877.986	0.0312593
$925$	25971.8	0.923188
$926$	27837.9	0.987917
$927$	5903.05	0.209149
$928$	47409.7	1.67705
$929$	−34759.3	−1.22757	−0.613786	−	0.789472i	$-0.710354\pi$
$929$	−34759.3	−1.22757	−0.613786	+	0.789472i	$0.710354\pi$
$930$	−14616.8	−0.515380
$931$	41773.9	1.47055
$932$	38963.1	1.36940
$933$	−22386.3	−0.785526
$934$	54737.8	1.91764
$935$	−657.619	−0.0230015
$936$	−1823.69	−0.0636850
$937$	27998.7	0.976178	0.488089	−	0.872794i	$-0.337694\pi$
$937$	27998.7	0.976178	0.488089	+	0.872794i	$0.337694\pi$
$938$	−47566.6	−1.65576
$939$	4515.01	0.156914
$940$	−97939.1	−3.39832
$941$	−23821.3	−0.825242	−0.412621	−	0.910903i	$-0.635387\pi$
$941$	−23821.3	−0.825242	−0.412621	+	0.910903i	$0.635387\pi$
$942$	35030.6	1.21163
$943$	−2559.32	−0.0883808
$944$	5837.12	0.201252
$945$	−83409.9	−2.87124
$946$	0	0
$947$	−15304.0	−0.525145	−0.262573	−	0.964912i	$-0.584571\pi$
$947$	−15304.0	−0.525145	−0.262573	+	0.964912i	$0.584571\pi$
$948$	−3507.41	−0.120164
$949$	−9227.09	−0.315621
$950$	108577.	3.70810
$951$	15279.5	0.521002
$952$	−6390.00	−0.217543
$953$	528.850	0.0179760	0.00898800	−	0.999960i	$-0.497139\pi$
$953$	528.850	0.0179760	0.00898800	+	0.999960i	$0.497139\pi$
$954$	2.94476	9.99373e−5 0
$955$	−44528.2	−1.50880
$956$	−8089.98	−0.273691
$957$	−624.399	−0.0210909
$958$	12383.6	0.417635
$959$	−56331.2	−1.89680
$960$	−60746.8	−2.04229
$961$	−28348.6	−0.951582
$962$	26914.3	0.902029
$963$	3429.40	0.114757
$964$	29246.7	0.977151
$965$	22894.3	0.763724
$966$	41058.0	1.36752
$967$	22602.1	0.751639	0.375819	−	0.926693i	$-0.377361\pi$
$967$	22602.1	0.751639	0.375819	+	0.926693i	$0.377361\pi$
$968$	−6772.78	−0.224882
$969$	20259.9	0.671663
$970$	124899.	4.13430
$971$	23322.2	0.770797	0.385398	−	0.922750i	$-0.374064\pi$
$971$	23322.2	0.770797	0.385398	+	0.922750i	$0.374064\pi$
$972$	−13490.0	−0.445156
$973$	−15722.3	−0.518021
$974$	3751.46	0.123413
$975$	−85497.8	−2.80833
$976$	−28093.8	−0.921373
$977$	27151.1	0.889090	0.444545	−	0.895757i	$-0.353365\pi$
$977$	27151.1	0.889090	0.444545	+	0.895757i	$0.353365\pi$
$978$	−55407.2	−1.81158
$979$	825.483	0.0269485
$980$	−79495.4	−2.59121
$981$	10894.0	0.354556
$982$	61201.2	1.98881
$983$	−16794.4	−0.544922	−0.272461	−	0.962167i	$-0.587837\pi$
$983$	−16794.4	−0.544922	−0.272461	+	0.962167i	$0.587837\pi$
$984$	−796.351	−0.0257995
$985$	−70968.6	−2.29568
$986$	34185.9	1.10416
$987$	−69219.0	−2.23229
$988$	60263.9	1.94054
$989$	0	0
$990$	−320.563	−0.0102911
$991$	6415.98	0.205661	0.102831	−	0.994699i	$-0.467210\pi$
$991$	6415.98	0.205661	0.102831	+	0.994699i	$0.467210\pi$
$992$	9851.32	0.315302
$993$	−30907.7	−0.987742
$994$	−24981.1	−0.797136
$995$	46050.1	1.46722
$996$	39813.3	1.26660
$997$	−38417.5	−1.22036	−0.610178	−	0.792264i	$-0.708902\pi$
$997$	−38417.5	−1.22036	−0.610178	+	0.792264i	$0.708902\pi$
$998$	3511.62	0.111381
$999$	14413.7	0.456486

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.43	✓	50
43.42	odd	2		1849.4.a.j.1.8	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.43	✓	50	1.1	even	1	trivial
1849.4.a.j.1.8	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)

\(f(q)\)	\(=\)	\(q+4.15048 q^{2} -4.65912 q^{3} +9.22650 q^{4} -19.9023 q^{5} -19.3376 q^{6} +27.8552 q^{7} +5.09055 q^{8} -5.29261 q^{9} +O(q^{10})\)	\(q+4.15048 q^{2} -4.65912 q^{3} +9.22650 q^{4} -19.9023 q^{5} -19.3376 q^{6} +27.8552 q^{7} +5.09055 q^{8} -5.29261 q^{9} -82.6043 q^{10} -0.733230 q^{11} -42.9873 q^{12} +67.6888 q^{13} +115.613 q^{14} +92.7274 q^{15} -52.6837 q^{16} -45.0640 q^{17} -21.9669 q^{18} +96.4948 q^{19} -183.629 q^{20} -129.781 q^{21} -3.04326 q^{22} -76.2235 q^{23} -23.7175 q^{24} +271.103 q^{25} +280.941 q^{26} +150.455 q^{27} +257.006 q^{28} -182.776 q^{29} +384.863 q^{30} -37.9792 q^{31} -259.387 q^{32} +3.41621 q^{33} -187.037 q^{34} -554.384 q^{35} -48.8323 q^{36} +95.8006 q^{37} +400.500 q^{38} -315.370 q^{39} -101.314 q^{40} +33.5766 q^{41} -538.653 q^{42} -6.76515 q^{44} +105.335 q^{45} -316.364 q^{46} +533.353 q^{47} +245.460 q^{48} +432.913 q^{49} +1125.21 q^{50} +209.958 q^{51} +624.530 q^{52} -0.134055 q^{53} +624.461 q^{54} +14.5930 q^{55} +141.798 q^{56} -449.581 q^{57} -758.607 q^{58} -110.795 q^{59} +855.549 q^{60} +533.253 q^{61} -157.632 q^{62} -147.427 q^{63} -655.112 q^{64} -1347.16 q^{65} +14.1789 q^{66} -411.431 q^{67} -415.783 q^{68} +355.134 q^{69} -2300.96 q^{70} -216.076 q^{71} -26.9423 q^{72} -136.316 q^{73} +397.619 q^{74} -1263.10 q^{75} +890.309 q^{76} -20.4243 q^{77} -1308.94 q^{78} +81.5918 q^{79} +1048.53 q^{80} -558.088 q^{81} +139.359 q^{82} -926.163 q^{83} -1197.42 q^{84} +896.879 q^{85} +851.573 q^{87} -3.73254 q^{88} -1125.82 q^{89} +437.193 q^{90} +1885.49 q^{91} -703.276 q^{92} +176.950 q^{93} +2213.67 q^{94} -1920.47 q^{95} +1208.52 q^{96} -1512.02 q^{97} +1796.80 q^{98} +3.88071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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