LMFDB - Embedded newform 1849.4.a.i.1.14

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			1.14
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)

$f(q)$	$=$	$q-3.22072 q^{2} -8.71909 q^{3} +2.37305 q^{4} +13.5152 q^{5} +28.0818 q^{6} -24.0687 q^{7} +18.1228 q^{8} +49.0226 q^{9} +O(q^{10})$	\(q-3.22072 q^{2} -8.71909 q^{3} +2.37305 q^{4} +13.5152 q^{5} +28.0818 q^{6} -24.0687 q^{7} +18.1228 q^{8} +49.0226 q^{9} -43.5288 q^{10} -71.6649 q^{11} -20.6909 q^{12} +26.7691 q^{13} +77.5186 q^{14} -117.841 q^{15} -77.3530 q^{16} +35.6223 q^{17} -157.888 q^{18} +88.4072 q^{19} +32.0724 q^{20} +209.857 q^{21} +230.813 q^{22} -64.6600 q^{23} -158.015 q^{24} +57.6615 q^{25} -86.2157 q^{26} -192.017 q^{27} -57.1164 q^{28} +106.675 q^{29} +379.532 q^{30} -7.27119 q^{31} +104.150 q^{32} +624.853 q^{33} -114.730 q^{34} -325.294 q^{35} +116.333 q^{36} -195.473 q^{37} -284.735 q^{38} -233.402 q^{39} +244.934 q^{40} -404.629 q^{41} -675.892 q^{42} -170.065 q^{44} +662.552 q^{45} +208.252 q^{46} +31.1933 q^{47} +674.449 q^{48} +236.303 q^{49} -185.712 q^{50} -310.594 q^{51} +63.5244 q^{52} +34.4808 q^{53} +618.434 q^{54} -968.567 q^{55} -436.193 q^{56} -770.831 q^{57} -343.572 q^{58} -486.120 q^{59} -279.642 q^{60} +491.320 q^{61} +23.4185 q^{62} -1179.91 q^{63} +283.386 q^{64} +361.790 q^{65} -2012.48 q^{66} +102.537 q^{67} +84.5337 q^{68} +563.777 q^{69} +1047.68 q^{70} -119.002 q^{71} +888.429 q^{72} +123.974 q^{73} +629.565 q^{74} -502.756 q^{75} +209.795 q^{76} +1724.88 q^{77} +751.723 q^{78} +34.0155 q^{79} -1045.44 q^{80} +350.606 q^{81} +1303.20 q^{82} -681.077 q^{83} +498.003 q^{84} +481.444 q^{85} -930.114 q^{87} -1298.77 q^{88} +1563.77 q^{89} -2133.90 q^{90} -644.297 q^{91} -153.442 q^{92} +63.3982 q^{93} -100.465 q^{94} +1194.84 q^{95} -908.094 q^{96} +93.1966 q^{97} -761.066 q^{98} -3513.20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.22072	−1.13870	−0.569349	−	0.822096i	$-0.692805\pi$
$2$	−3.22072	−1.13870	−0.569349	+	0.822096i	$0.692805\pi$
$3$	−8.71909	−1.67799	−0.838995	−	0.544139i	$-0.816857\pi$
$3$	−8.71909	−1.67799	−0.838995	+	0.544139i	$0.816857\pi$
$4$	2.37305	0.296632
$5$	13.5152	1.20884	0.604420	−	0.796666i	$-0.293405\pi$
$5$	13.5152	1.20884	0.604420	+	0.796666i	$0.293405\pi$
$6$	28.0818	1.91072
$7$	−24.0687	−1.29959	−0.649794	−	0.760110i	$-0.725145\pi$
$7$	−24.0687	−1.29959	−0.649794	+	0.760110i	$0.725145\pi$
$8$	18.1228	0.800924
$9$	49.0226	1.81565
$10$	−43.5288	−1.37650
$11$	−71.6649	−1.96434	−0.982171	−	0.187987i	$-0.939804\pi$
$11$	−71.6649	−1.96434	−0.982171	+	0.187987i	$0.939804\pi$
$12$	−20.6909	−0.497745
$13$	26.7691	0.571108	0.285554	−	0.958363i	$-0.407822\pi$
$13$	26.7691	0.571108	0.285554	+	0.958363i	$0.407822\pi$
$14$	77.5186	1.47984
$15$	−117.841	−2.02842
$16$	−77.3530	−1.20864
$17$	35.6223	0.508217	0.254108	−	0.967176i	$-0.418218\pi$
$17$	35.6223	0.508217	0.254108	+	0.967176i	$0.418218\pi$
$18$	−157.888	−2.06748
$19$	88.4072	1.06747	0.533737	−	0.845650i	$-0.320787\pi$
$19$	88.4072	1.06747	0.533737	+	0.845650i	$0.320787\pi$
$20$	32.0724	0.358580
$21$	209.857	2.18070
$22$	230.813	2.23679
$23$	−64.6600	−0.586197	−0.293099	−	0.956082i	$-0.594686\pi$
$23$	−64.6600	−0.586197	−0.293099	+	0.956082i	$0.594686\pi$
$24$	−158.015	−1.34394
$25$	57.6615	0.461292
$26$	−86.2157	−0.650319
$27$	−192.017	−1.36866
$28$	−57.1164	−0.385499
$29$	106.675	0.683074	0.341537	−	0.939868i	$-0.389053\pi$
$29$	106.675	0.683074	0.341537	+	0.939868i	$0.389053\pi$
$30$	379.532	2.30976
$31$	−7.27119	−0.0421272	−0.0210636	−	0.999778i	$-0.506705\pi$
$31$	−7.27119	−0.0421272	−0.0210636	+	0.999778i	$0.506705\pi$
$32$	104.150	0.575353
$33$	624.853	3.29615
$34$	−114.730	−0.578705
$35$	−325.294	−1.57099
$36$	116.333	0.538580
$37$	−195.473	−0.868529	−0.434265	−	0.900785i	$-0.642992\pi$
$37$	−195.473	−0.868529	−0.434265	+	0.900785i	$0.642992\pi$
$38$	−284.735	−1.21553
$39$	−233.402	−0.958314
$40$	244.934	0.968188
$41$	−404.629	−1.54128	−0.770639	−	0.637272i	$-0.780063\pi$
$41$	−404.629	−1.54128	−0.770639	+	0.637272i	$0.780063\pi$
$42$	−675.892	−2.48315
$43$	0	0
$43$	0	0
$44$	−170.065	−0.582687
$45$	662.552	2.19483
$46$	208.252	0.667502
$47$	31.1933	0.0968087	0.0484043	−	0.998828i	$-0.484586\pi$
$47$	31.1933	0.0968087	0.0484043	+	0.998828i	$0.484586\pi$
$48$	674.449	2.02809
$49$	236.303	0.688930
$50$	−185.712	−0.525272
$51$	−310.594	−0.852783
$52$	63.5244	0.169409
$53$	34.4808	0.0893641	0.0446820	−	0.999001i	$-0.485773\pi$
$53$	34.4808	0.0893641	0.0446820	+	0.999001i	$0.485773\pi$
$54$	618.434	1.55849
$55$	−968.567	−2.37457
$56$	−436.193	−1.04087
$57$	−770.831	−1.79121
$58$	−343.572	−0.777814
$59$	−486.120	−1.07267	−0.536335	−	0.844005i	$-0.680191\pi$
$59$	−486.120	−1.07267	−0.536335	+	0.844005i	$0.680191\pi$
$60$	−279.642	−0.601694
$61$	491.320	1.03126	0.515632	−	0.856810i	$-0.327557\pi$
$61$	491.320	1.03126	0.515632	+	0.856810i	$0.327557\pi$
$62$	23.4185	0.0479701
$63$	−1179.91	−2.35960
$64$	283.386	0.553488
$65$	361.790	0.690378
$66$	−2012.48	−3.75332
$67$	102.537	0.186969	0.0934843	−	0.995621i	$-0.470200\pi$
$67$	102.537	0.186969	0.0934843	+	0.995621i	$0.470200\pi$
$68$	84.5337	0.150753
$69$	563.777	0.983634
$70$	1047.68	1.78889
$71$	−119.002	−0.198915	−0.0994573	−	0.995042i	$-0.531711\pi$
$71$	−119.002	−0.198915	−0.0994573	+	0.995042i	$0.531711\pi$
$72$	888.429	1.45420
$73$	123.974	0.198768	0.0993840	−	0.995049i	$-0.468313\pi$
$73$	123.974	0.198768	0.0993840	+	0.995049i	$0.468313\pi$
$74$	629.565	0.988992
$75$	−502.756	−0.774044
$76$	209.795	0.316647
$77$	1724.88	2.55284
$78$	751.723	1.09123
$79$	34.0155	0.0484436	0.0242218	−	0.999707i	$-0.492289\pi$
$79$	34.0155	0.0484436	0.0242218	+	0.999707i	$0.492289\pi$
$80$	−1045.44	−1.46105
$81$	350.606	0.480941
$82$	1303.20	1.75505
$83$	−681.077	−0.900698	−0.450349	−	0.892853i	$-0.648700\pi$
$83$	−681.077	−0.900698	−0.450349	+	0.892853i	$0.648700\pi$
$84$	498.003	0.646864
$85$	481.444	0.614352
$86$	0	0
$87$	−930.114	−1.14619
$88$	−1298.77	−1.57329
$89$	1563.77	1.86247	0.931234	−	0.364422i	$-0.118733\pi$
$89$	1563.77	1.86247	0.931234	+	0.364422i	$0.118733\pi$
$90$	−2133.90	−2.49925
$91$	−644.297	−0.742205
$92$	−153.442	−0.173885
$93$	63.3982	0.0706891
$94$	−100.465	−0.110236
$95$	1194.84	1.29040
$96$	−908.094	−0.965437
$97$	93.1966	0.0975535	0.0487767	−	0.998810i	$-0.484468\pi$
$97$	93.1966	0.0975535	0.0487767	+	0.998810i	$0.484468\pi$
$98$	−761.066	−0.784483
$99$	−3513.20	−3.56656
$100$	136.834	0.136834
$101$	−358.186	−0.352880	−0.176440	−	0.984311i	$-0.556458\pi$
$101$	−358.186	−0.352880	−0.176440	+	0.984311i	$0.556458\pi$
$102$	1000.34	0.971062
$103$	−280.395	−0.268234	−0.134117	−	0.990965i	$-0.542820\pi$
$103$	−280.395	−0.268234	−0.134117	+	0.990965i	$0.542820\pi$
$104$	485.131	0.457414
$105$	2836.27	2.63611
$106$	−111.053	−0.101759
$107$	2033.32	1.83709	0.918543	−	0.395322i	$-0.129367\pi$
$107$	2033.32	1.83709	0.918543	+	0.395322i	$0.129367\pi$
$108$	−455.667	−0.405987
$109$	1575.03	1.38404	0.692022	−	0.721876i	$-0.256720\pi$
$109$	1575.03	1.38404	0.692022	+	0.721876i	$0.256720\pi$
$110$	3119.49	2.70392
$111$	1704.35	1.45738
$112$	1861.79	1.57074
$113$	939.934	0.782492	0.391246	−	0.920286i	$-0.372044\pi$
$113$	939.934	0.782492	0.391246	+	0.920286i	$0.372044\pi$
$114$	2482.63	2.03965
$115$	−873.895	−0.708618
$116$	253.147	0.202621
$117$	1312.29	1.03693
$118$	1565.66	1.22145
$119$	−857.384	−0.660473
$120$	−2135.61	−1.62461
$121$	3804.85	2.85864
$122$	−1582.41	−1.17430
$123$	3528.00	2.58625
$124$	−17.2549	−0.0124963
$125$	−910.095	−0.651211
$126$	3800.17	2.68687
$127$	622.014	0.434605	0.217302	−	0.976104i	$-0.430274\pi$
$127$	622.014	0.434605	0.217302	+	0.976104i	$0.430274\pi$
$128$	−1745.91	−1.20561
$129$	0	0
$130$	−1165.23	−0.786131
$131$	2763.95	1.84342	0.921708	−	0.387884i	$-0.126794\pi$
$131$	2763.95	1.84342	0.921708	+	0.387884i	$0.126794\pi$
$132$	1482.81	0.977743
$133$	−2127.85	−1.38728
$134$	−330.243	−0.212901
$135$	−2595.16	−1.65449
$136$	645.577	0.407043
$137$	−685.313	−0.427374	−0.213687	−	0.976902i	$-0.568547\pi$
$137$	−685.313	−0.427374	−0.213687	+	0.976902i	$0.568547\pi$
$138$	−1815.77	−1.12006
$139$	−57.4962	−0.0350846	−0.0175423	−	0.999846i	$-0.505584\pi$
$139$	−57.4962	−0.0350846	−0.0175423	+	0.999846i	$0.505584\pi$
$140$	−771.941	−0.466007
$141$	−271.977	−0.162444
$142$	383.272	0.226503
$143$	−1918.40	−1.12185
$144$	−3792.05	−2.19447
$145$	1441.74	0.825726
$146$	−399.286	−0.226337
$147$	−2060.35	−1.15602
$148$	−463.868	−0.257633
$149$	−648.505	−0.356561	−0.178281	−	0.983980i	$-0.557053\pi$
$149$	−648.505	−0.356561	−0.178281	+	0.983980i	$0.557053\pi$
$150$	1619.24	0.881402
$151$	2521.32	1.35882	0.679410	−	0.733759i	$-0.262236\pi$
$151$	2521.32	1.35882	0.679410	+	0.733759i	$0.262236\pi$
$152$	1602.19	0.854965
$153$	1746.30	0.922745
$154$	−5555.36	−2.90691
$155$	−98.2718	−0.0509250
$156$	−553.876	−0.284266
$157$	−892.524	−0.453702	−0.226851	−	0.973930i	$-0.572843\pi$
$157$	−892.524	−0.453702	−0.226851	+	0.973930i	$0.572843\pi$
$158$	−109.554	−0.0551626
$159$	−300.641	−0.149952
$160$	1407.61	0.695509
$161$	1556.28	0.761815
$162$	−1129.21	−0.547647
$163$	−3013.59	−1.44811	−0.724057	−	0.689740i	$-0.757725\pi$
$163$	−3013.59	−1.44811	−0.724057	+	0.689740i	$0.757725\pi$
$164$	−960.206	−0.457192
$165$	8445.03	3.98451
$166$	2193.56	1.02562
$167$	3783.09	1.75296	0.876480	−	0.481439i	$-0.159886\pi$
$167$	3783.09	1.75296	0.876480	+	0.481439i	$0.159886\pi$
$168$	3803.21	1.74657
$169$	−1480.42	−0.673836
$170$	−1550.60	−0.699561
$171$	4333.95	1.93816
$172$	0	0
$173$	1859.79	0.817326	0.408663	−	0.912685i	$-0.365995\pi$
$173$	1859.79	0.817326	0.408663	+	0.912685i	$0.365995\pi$
$174$	2995.64	1.30517
$175$	−1387.84	−0.599490
$176$	5543.50	2.37419
$177$	4238.53	1.79993
$178$	−5036.48	−2.12079
$179$	−949.068	−0.396294	−0.198147	−	0.980172i	$-0.563492\pi$
$179$	−949.068	−0.396294	−0.198147	+	0.980172i	$0.563492\pi$
$180$	1572.27	0.651057
$181$	−3778.60	−1.55172	−0.775859	−	0.630906i	$-0.782683\pi$
$181$	−3778.60	−1.55172	−0.775859	+	0.630906i	$0.782683\pi$
$182$	2075.10	0.845147
$183$	−4283.87	−1.73045
$184$	−1171.82	−0.469499
$185$	−2641.86	−1.04991
$186$	−204.188	−0.0804934
$187$	−2552.87	−0.998312
$188$	74.0233	0.0287165
$189$	4621.61	1.77869
$190$	−3848.26	−1.46938
$191$	2236.60	0.847300	0.423650	−	0.905826i	$-0.360749\pi$
$191$	2236.60	0.847300	0.423650	+	0.905826i	$0.360749\pi$
$192$	−2470.87	−0.928748
$193$	−1426.80	−0.532142	−0.266071	−	0.963953i	$-0.585726\pi$
$193$	−1426.80	−0.532142	−0.266071	+	0.963953i	$0.585726\pi$
$194$	−300.161	−0.111084
$195$	−3154.48	−1.15845
$196$	560.760	0.204358
$197$	3733.48	1.35025	0.675125	−	0.737703i	$-0.264090\pi$
$197$	3733.48	1.35025	0.675125	+	0.737703i	$0.264090\pi$
$198$	11315.0	4.06124
$199$	2509.13	0.893808	0.446904	−	0.894582i	$-0.352527\pi$
$199$	2509.13	0.893808	0.446904	+	0.894582i	$0.352527\pi$
$200$	1044.99	0.369460
$201$	−894.030	−0.313731
$202$	1153.62	0.401823
$203$	−2567.54	−0.887715
$204$	−737.057	−0.252962
$205$	−5468.65	−1.86316
$206$	903.074	0.305438
$207$	−3169.80	−1.06433
$208$	−2070.67	−0.690265
$209$	−6335.69	−2.09689
$210$	−9134.84	−3.00173
$211$	−1516.64	−0.494833	−0.247416	−	0.968909i	$-0.579582\pi$
$211$	−1516.64	−0.494833	−0.247416	+	0.968909i	$0.579582\pi$
$212$	81.8247	0.0265082
$213$	1037.59	0.333777
$214$	−6548.75	−2.09188
$215$	0	0
$216$	−3479.90	−1.09619
$217$	175.008	0.0547480
$218$	−5072.74	−1.57601
$219$	−1080.94	−0.333531
$220$	−2298.46	−0.704374
$221$	953.576	0.290247
$222$	−5489.23	−1.65952
$223$	2914.69	0.875256	0.437628	−	0.899156i	$-0.355819\pi$
$223$	2914.69	0.875256	0.437628	+	0.899156i	$0.355819\pi$
$224$	−2506.76	−0.747722
$225$	2826.72	0.837546
$226$	−3027.27	−0.891021
$227$	3001.30	0.877549	0.438774	−	0.898597i	$-0.355413\pi$
$227$	3001.30	0.877549	0.438774	+	0.898597i	$0.355413\pi$
$228$	−1829.22	−0.531330
$229$	−576.669	−0.166408	−0.0832038	−	0.996533i	$-0.526515\pi$
$229$	−576.669	−0.166408	−0.0832038	+	0.996533i	$0.526515\pi$
$230$	2814.57	0.806902
$231$	−15039.4	−4.28364
$232$	1933.26	0.547090
$233$	5444.61	1.53085	0.765425	−	0.643525i	$-0.222529\pi$
$233$	5444.61	1.53085	0.765425	+	0.643525i	$0.222529\pi$
$234$	−4226.52	−1.18075
$235$	421.584	0.117026
$236$	−1153.59	−0.318188
$237$	−296.584	−0.0812879
$238$	2761.39	0.752078
$239$	−1814.68	−0.491138	−0.245569	−	0.969379i	$-0.578975\pi$
$239$	−1814.68	−0.491138	−0.245569	+	0.969379i	$0.578975\pi$
$240$	9115.33	2.45163
$241$	−3802.66	−1.01639	−0.508196	−	0.861241i	$-0.669688\pi$
$241$	−3802.66	−1.01639	−0.508196	+	0.861241i	$0.669688\pi$
$242$	−12254.4	−3.25513
$243$	2127.50	0.561642
$244$	1165.93	0.305906
$245$	3193.69	0.832805
$246$	−11362.7	−2.94496
$247$	2366.58	0.609643
$248$	−131.774	−0.0337407
$249$	5938.38	1.51136
$250$	2931.16	0.741532
$251$	589.071	0.148135	0.0740673	−	0.997253i	$-0.476402\pi$
$251$	589.071	0.148135	0.0740673	+	0.997253i	$0.476402\pi$
$252$	−2799.99	−0.699933
$253$	4633.85	1.15149
$254$	−2003.33	−0.494883
$255$	−4197.76	−1.03088
$256$	3356.00	0.819335
$257$	−1674.00	−0.406308	−0.203154	−	0.979147i	$-0.565119\pi$
$257$	−1674.00	−0.406308	−0.203154	+	0.979147i	$0.565119\pi$
$258$	0	0
$259$	4704.79	1.12873
$260$	858.548	0.204788
$261$	5229.51	1.24022
$262$	−8901.92	−2.09909
$263$	−5843.39	−1.37003	−0.685017	−	0.728527i	$-0.740205\pi$
$263$	−5843.39	−1.37003	−0.685017	+	0.728527i	$0.740205\pi$
$264$	11324.1	2.63996
$265$	466.016	0.108027
$266$	6853.21	1.57969
$267$	−13634.7	−3.12520
$268$	243.326	0.0554608
$269$	−1405.83	−0.318643	−0.159322	−	0.987227i	$-0.550931\pi$
$269$	−1405.83	−0.318643	−0.159322	+	0.987227i	$0.550931\pi$
$270$	8358.28	1.88396
$271$	3263.18	0.731455	0.365727	−	0.930722i	$-0.380820\pi$
$271$	3263.18	0.731455	0.365727	+	0.930722i	$0.380820\pi$
$272$	−2755.50	−0.614252
$273$	5617.69	1.24541
$274$	2207.20	0.486650
$275$	−4132.31	−0.906136
$276$	1337.87	0.291777
$277$	−706.424	−0.153231	−0.0766153	−	0.997061i	$-0.524411\pi$
$277$	−706.424	−0.153231	−0.0766153	+	0.997061i	$0.524411\pi$
$278$	185.179	0.0399507
$279$	−356.453	−0.0764884
$280$	−5895.25	−1.25825
$281$	−7794.16	−1.65466	−0.827332	−	0.561714i	$-0.810142\pi$
$281$	−7794.16	−1.65466	−0.827332	+	0.561714i	$0.810142\pi$
$282$	875.963	0.184975
$283$	−4923.94	−1.03427	−0.517134	−	0.855905i	$-0.673001\pi$
$283$	−4923.94	−1.03427	−0.517134	+	0.855905i	$0.673001\pi$
$284$	−282.398	−0.0590044
$285$	−10418.0	−2.16529
$286$	6178.64	1.27745
$287$	9738.89	2.00303
$288$	5105.71	1.04464
$289$	−3644.05	−0.741716
$290$	−4643.46	−0.940252
$291$	−812.590	−0.163694
$292$	294.197	0.0589609
$293$	5279.68	1.05270	0.526352	−	0.850267i	$-0.323560\pi$
$293$	5279.68	1.05270	0.526352	+	0.850267i	$0.323560\pi$
$294$	6635.81	1.31635
$295$	−6570.03	−1.29668
$296$	−3542.53	−0.695626
$297$	13760.9	2.68851
$298$	2088.66	0.406015
$299$	−1730.89	−0.334782
$300$	−1193.07	−0.229606
$301$	0	0
$302$	−8120.46	−1.54729
$303$	3123.06	0.592129
$304$	−6838.57	−1.29019
$305$	6640.31	1.24663
$306$	−5624.35	−1.05073
$307$	5804.15	1.07902	0.539512	−	0.841978i	$-0.318609\pi$
$307$	5804.15	1.07902	0.539512	+	0.841978i	$0.318609\pi$
$308$	4093.24	0.757253
$309$	2444.79	0.450095
$310$	316.506	0.0579882
$311$	−8304.16	−1.51410	−0.757051	−	0.653356i	$-0.773361\pi$
$311$	−8304.16	−1.51410	−0.757051	+	0.653356i	$0.773361\pi$
$312$	−4229.91	−0.767536
$313$	2.78390	0.000502733 0	0.000251367	−	1.00000i	$-0.499920\pi$
$313$	2.78390	0.000502733 0	0.000251367		1.00000i	$0.499920\pi$
$314$	2874.57	0.516629
$315$	−15946.8	−2.85238
$316$	80.7206	0.0143699
$317$	3284.80	0.581997	0.290998	−	0.956724i	$-0.406013\pi$
$317$	3284.80	0.581997	0.290998	+	0.956724i	$0.406013\pi$
$318$	968.281	0.170750
$319$	−7644.89	−1.34179
$320$	3830.03	0.669078
$321$	−17728.7	−3.08261
$322$	−5012.36	−0.867477
$323$	3149.27	0.542508
$324$	832.007	0.142662
$325$	1543.54	0.263448
$326$	9705.94	1.64896
$327$	−13732.9	−2.32241
$328$	−7333.02	−1.23445
$329$	−750.782	−0.125811
$330$	−27199.1	−4.53716
$331$	−7993.89	−1.32744	−0.663722	−	0.747979i	$-0.731024\pi$
$331$	−7993.89	−1.32744	−0.663722	+	0.747979i	$0.731024\pi$
$332$	−1616.23	−0.267176
$333$	−9582.60	−1.57695
$334$	−12184.3	−1.99609
$335$	1385.81	0.226015
$336$	−16233.1	−2.63568
$337$	7345.18	1.18729	0.593646	−	0.804726i	$-0.297688\pi$
$337$	7345.18	1.18729	0.593646	+	0.804726i	$0.297688\pi$
$338$	4768.01	0.767295
$339$	−8195.37	−1.31301
$340$	1142.49	0.182236
$341$	521.089	0.0827523
$342$	−13958.5	−2.20698
$343$	2568.06	0.404263
$344$	0	0
$345$	7619.57	1.18906
$346$	−5989.88	−0.930687
$347$	−8573.21	−1.32632	−0.663162	−	0.748476i	$-0.730786\pi$
$347$	−8573.21	−1.32632	−0.663162	+	0.748476i	$0.730786\pi$
$348$	−2207.21	−0.339997
$349$	−4612.19	−0.707407	−0.353704	−	0.935358i	$-0.615078\pi$
$349$	−4612.19	−0.707407	−0.353704	+	0.935358i	$0.615078\pi$
$350$	4469.84	0.682638
$351$	−5140.12	−0.781651
$352$	−7463.90	−1.13019
$353$	7509.11	1.13221	0.566104	−	0.824334i	$-0.308450\pi$
$353$	7509.11	1.13221	0.566104	+	0.824334i	$0.308450\pi$
$354$	−13651.1	−2.04957
$355$	−1608.34	−0.240456
$356$	3710.92	0.552467
$357$	7475.61	1.10827
$358$	3056.69	0.451259
$359$	−10602.9	−1.55877	−0.779384	−	0.626547i	$-0.784468\pi$
$359$	−10602.9	−1.55877	−0.779384	+	0.626547i	$0.784468\pi$
$360$	12007.3	1.75789
$361$	956.838	0.139501
$362$	12169.8	1.76694
$363$	−33174.9	−4.79678
$364$	−1528.95	−0.220162
$365$	1675.54	0.240278
$366$	13797.2	1.97046
$367$	10146.4	1.44315	0.721576	−	0.692335i	$-0.243418\pi$
$367$	10146.4	1.44315	0.721576	+	0.692335i	$0.243418\pi$
$368$	5001.65	0.708503
$369$	−19836.0	−2.79842
$370$	8508.71	1.19553
$371$	−829.908	−0.116137
$372$	150.447	0.0209686
$373$	5897.78	0.818702	0.409351	−	0.912377i	$-0.365755\pi$
$373$	5897.78	0.818702	0.409351	+	0.912377i	$0.365755\pi$
$374$	8222.09	1.13678
$375$	7935.21	1.09273
$376$	565.311	0.0775363
$377$	2855.60	0.390109
$378$	−14884.9	−2.02539
$379$	−1623.92	−0.220093	−0.110047	−	0.993926i	$-0.535100\pi$
$379$	−1623.92	−0.220093	−0.110047	+	0.993926i	$0.535100\pi$
$380$	2835.43	0.382775
$381$	−5423.40	−0.729263
$382$	−7203.45	−0.964819
$383$	−8375.64	−1.11743	−0.558715	−	0.829360i	$-0.688705\pi$
$383$	−8375.64	−1.11743	−0.558715	+	0.829360i	$0.688705\pi$
$384$	15222.7	2.02300
$385$	23312.2	3.08597
$386$	4595.33	0.605949
$387$	0	0
$388$	221.161	0.0289375
$389$	−13745.0	−1.79151	−0.895757	−	0.444544i	$-0.853366\pi$
$389$	−13745.0	−1.79151	−0.895757	+	0.444544i	$0.853366\pi$
$390$	10159.7	1.31912
$391$	−2303.34	−0.297915
$392$	4282.48	0.551780
$393$	−24099.2	−3.09323
$394$	−12024.5	−1.53753
$395$	459.727	0.0585605
$396$	−8337.01	−1.05796
$397$	6849.80	0.865949	0.432975	−	0.901406i	$-0.357464\pi$
$397$	6849.80	0.865949	0.432975	+	0.901406i	$0.357464\pi$
$398$	−8081.22	−1.01778
$399$	18552.9	2.32784
$400$	−4460.29	−0.557537
$401$	1819.07	0.226533	0.113267	−	0.993565i	$-0.463869\pi$
$401$	1819.07	0.226533	0.113267	+	0.993565i	$0.463869\pi$
$402$	2879.42	0.357245
$403$	−194.643	−0.0240592
$404$	−849.995	−0.104675
$405$	4738.52	0.581381
$406$	8269.34	1.01084
$407$	14008.6	1.70609
$408$	−5628.85	−0.683014
$409$	6571.50	0.794474	0.397237	−	0.917716i	$-0.369969\pi$
$409$	6571.50	0.794474	0.397237	+	0.917716i	$0.369969\pi$
$410$	17613.0	2.12157
$411$	5975.31	0.717130
$412$	−665.392	−0.0795668
$413$	11700.3	1.39403
$414$	10209.1	1.21195
$415$	−9204.92	−1.08880
$416$	2788.00	0.328589
$417$	501.314	0.0588716
$418$	20405.5	2.38772
$419$	12326.7	1.43723	0.718613	−	0.695410i	$-0.244777\pi$
$419$	12326.7	1.43723	0.718613	+	0.695410i	$0.244777\pi$
$420$	6730.63	0.781955
$421$	−5434.97	−0.629179	−0.314590	−	0.949228i	$-0.601867\pi$
$421$	−5434.97	−0.629179	−0.314590	+	0.949228i	$0.601867\pi$
$422$	4884.67	0.563465
$423$	1529.18	0.175771
$424$	624.889	0.0715738
$425$	2054.04	0.234436
$426$	−3341.79	−0.380071
$427$	−11825.4	−1.34022
$428$	4825.17	0.544938
$429$	16726.7	1.88246
$430$	0	0
$431$	−5331.48	−0.595843	−0.297922	−	0.954590i	$-0.596293\pi$
$431$	−5331.48	−0.595843	−0.297922	+	0.954590i	$0.596293\pi$
$432$	14853.1	1.65422
$433$	3987.83	0.442593	0.221297	−	0.975207i	$-0.428971\pi$
$433$	3987.83	0.442593	0.221297	+	0.975207i	$0.428971\pi$
$434$	−563.652	−0.0623414
$435$	−12570.7	−1.38556
$436$	3737.64	0.410551
$437$	−5716.41	−0.625751
$438$	3481.41	0.379791
$439$	1810.69	0.196856	0.0984278	−	0.995144i	$-0.468619\pi$
$439$	1810.69	0.196856	0.0984278	+	0.995144i	$0.468619\pi$
$440$	−17553.2	−1.90185
$441$	11584.2	1.25086
$442$	−3071.21	−0.330503
$443$	−10313.9	−1.10616	−0.553078	−	0.833129i	$-0.686547\pi$
$443$	−10313.9	−1.10616	−0.553078	+	0.833129i	$0.686547\pi$
$444$	4044.51	0.432306
$445$	21134.8	2.25142
$446$	−9387.41	−0.996651
$447$	5654.38	0.598307
$448$	−6820.74	−0.719307
$449$	−2382.45	−0.250411	−0.125206	−	0.992131i	$-0.539959\pi$
$449$	−2382.45	−0.250411	−0.125206	+	0.992131i	$0.539959\pi$
$450$	−9104.08	−0.953712
$451$	28997.7	3.02760
$452$	2230.51	0.232112
$453$	−21983.6	−2.28009
$454$	−9666.37	−0.999262
$455$	−8707.82	−0.897207
$456$	−13969.6	−1.43462
$457$	−968.005	−0.0990840	−0.0495420	−	0.998772i	$-0.515776\pi$
$457$	−968.005	−0.0990840	−0.0495420	+	0.998772i	$0.515776\pi$
$458$	1857.29	0.189488
$459$	−6840.10	−0.695574
$460$	−2073.80	−0.210199
$461$	5755.46	0.581472	0.290736	−	0.956803i	$-0.406100\pi$
$461$	5755.46	0.581472	0.290736	+	0.956803i	$0.406100\pi$
$462$	48437.7	4.87777
$463$	−14706.8	−1.47621	−0.738104	−	0.674687i	$-0.764279\pi$
$463$	−14706.8	−1.47621	−0.738104	+	0.674687i	$0.764279\pi$
$464$	−8251.67	−0.825591
$465$	856.841	0.0854517
$466$	−17535.6	−1.74317
$467$	4670.26	0.462770	0.231385	−	0.972862i	$-0.425674\pi$
$467$	4670.26	0.462770	0.231385	+	0.972862i	$0.425674\pi$
$468$	3114.13	0.307587
$469$	−2467.93	−0.242982
$470$	−1357.81	−0.133257
$471$	7782.00	0.761307
$472$	−8809.88	−0.859126
$473$	0	0
$474$	955.216	0.0925623
$475$	5097.70	0.492417
$476$	−2034.62	−0.195917
$477$	1690.34	0.162254
$478$	5844.59	0.559258
$479$	1645.32	0.156944	0.0784722	−	0.996916i	$-0.474996\pi$
$479$	1645.32	0.156944	0.0784722	+	0.996916i	$0.474996\pi$
$480$	−12273.1	−1.16706
$481$	−5232.63	−0.496024
$482$	12247.3	1.15736
$483$	−13569.4	−1.27832
$484$	9029.13	0.847964
$485$	1259.57	0.117926
$486$	−6852.08	−0.639541
$487$	15977.1	1.48664	0.743318	−	0.668938i	$-0.233251\pi$
$487$	15977.1	1.48664	0.743318	+	0.668938i	$0.233251\pi$
$488$	8904.11	0.825964
$489$	26275.8	2.42992
$490$	−10286.0	−0.948313
$491$	−6890.84	−0.633359	−0.316679	−	0.948533i	$-0.602568\pi$
$491$	−6890.84	−0.633359	−0.316679	+	0.948533i	$0.602568\pi$
$492$	8372.13	0.767164
$493$	3800.03	0.347150
$494$	−7622.09	−0.694199
$495$	−47481.7	−4.31140
$496$	562.448	0.0509167
$497$	2864.22	0.258507
$498$	−19125.9	−1.72098
$499$	5724.01	0.513511	0.256756	−	0.966476i	$-0.417346\pi$
$499$	5724.01	0.513511	0.256756	+	0.966476i	$0.417346\pi$
$500$	−2159.71	−0.193170
$501$	−32985.1	−2.94145
$502$	−1897.23	−0.168681
$503$	19916.3	1.76546	0.882728	−	0.469885i	$-0.155705\pi$
$503$	19916.3	1.76546	0.882728	+	0.469885i	$0.155705\pi$
$504$	−21383.3	−1.88986
$505$	−4840.97	−0.426575
$506$	−14924.3	−1.31120
$507$	12907.9	1.13069
$508$	1476.07	0.128918
$509$	−12749.9	−1.11028	−0.555138	−	0.831758i	$-0.687335\pi$
$509$	−12749.9	−1.11028	−0.555138	+	0.831758i	$0.687335\pi$
$510$	13519.8	1.17386
$511$	−2983.89	−0.258317
$512$	3158.53	0.272634
$513$	−16975.7	−1.46101
$514$	5391.49	0.462662
$515$	−3789.60	−0.324252
$516$	0	0
$517$	−2235.46	−0.190165
$518$	−15152.8	−1.28528
$519$	−16215.7	−1.37147
$520$	6556.66	0.552940
$521$	−3723.80	−0.313134	−0.156567	−	0.987667i	$-0.550043\pi$
$521$	−3723.80	−0.313134	−0.156567	+	0.987667i	$0.550043\pi$
$522$	−16842.8	−1.41224
$523$	−13767.0	−1.15103	−0.575517	−	0.817790i	$-0.695199\pi$
$523$	−13767.0	−1.15103	−0.575517	+	0.817790i	$0.695199\pi$
$524$	6559.01	0.546816
$525$	12100.7	1.00594
$526$	18819.9	1.56005
$527$	−259.017	−0.0214098
$528$	−48334.3	−3.98386
$529$	−7986.08	−0.656373
$530$	−1500.91	−0.123010
$531$	−23830.9	−1.94759
$532$	−5049.50	−0.411510
$533$	−10831.5	−0.880236
$534$	43913.5	3.55866
$535$	27480.7	2.22074
$536$	1858.26	0.149748
$537$	8275.02	0.664978
$538$	4527.79	0.362838
$539$	−16934.6	−1.35329
$540$	−6158.45	−0.490773
$541$	−14720.0	−1.16980	−0.584899	−	0.811106i	$-0.698866\pi$
$541$	−14720.0	−1.16980	−0.584899	+	0.811106i	$0.698866\pi$
$542$	−10509.8	−0.832906
$543$	32946.0	2.60377
$544$	3710.07	0.292404
$545$	21286.9	1.67309
$546$	−18093.0	−1.41815
$547$	−19914.4	−1.55663	−0.778315	−	0.627874i	$-0.783925\pi$
$547$	−19914.4	−1.55663	−0.778315	+	0.627874i	$0.783925\pi$
$548$	−1626.29	−0.126773
$549$	24085.8	1.87242
$550$	13309.0	1.03181
$551$	9430.88	0.729164
$552$	10217.2	0.787816
$553$	−818.709	−0.0629567
$554$	2275.20	0.174483
$555$	23034.7	1.76174
$556$	−136.441	−0.0104072
$557$	2514.93	0.191313	0.0956563	−	0.995414i	$-0.469505\pi$
$557$	2514.93	0.191313	0.0956563	+	0.995414i	$0.469505\pi$
$558$	1148.03	0.0870971
$559$	0	0
$560$	25162.5	1.89877
$561$	22258.7	1.67516
$562$	25102.8	1.88416
$563$	−10579.1	−0.791932	−0.395966	−	0.918265i	$-0.629590\pi$
$563$	−10579.1	−0.791932	−0.395966	+	0.918265i	$0.629590\pi$
$564$	−645.417	−0.0481861
$565$	12703.4	0.945907
$566$	15858.6	1.17772
$567$	−8438.64	−0.625026
$568$	−2156.65	−0.159315
$569$	−8023.71	−0.591162	−0.295581	−	0.955318i	$-0.595513\pi$
$569$	−8023.71	−0.591162	−0.295581	+	0.955318i	$0.595513\pi$
$570$	33553.4	2.46561
$571$	14485.2	1.06162	0.530812	−	0.847490i	$-0.321887\pi$
$571$	14485.2	1.06162	0.530812	+	0.847490i	$0.321887\pi$
$572$	−4552.47	−0.332777
$573$	−19501.1	−1.42176
$574$	−31366.3	−2.28084
$575$	−3728.39	−0.270408
$576$	13892.3	1.00494
$577$	10236.0	0.738529	0.369264	−	0.929324i	$-0.379610\pi$
$577$	10236.0	0.738529	0.369264	+	0.929324i	$0.379610\pi$
$578$	11736.5	0.844590
$579$	12440.4	0.892930
$580$	3421.34	0.244937
$581$	16392.7	1.17054
$582$	2617.13	0.186398
$583$	−2471.06	−0.175542
$584$	2246.76	0.159198
$585$	17735.9	1.25349
$586$	−17004.4	−1.19871
$587$	−16260.3	−1.14333	−0.571665	−	0.820487i	$-0.693702\pi$
$587$	−16260.3	−1.14333	−0.571665	+	0.820487i	$0.693702\pi$
$588$	−4889.32	−0.342912
$589$	−642.825	−0.0449697
$590$	21160.2	1.47653
$591$	−32552.5	−2.26571
$592$	15120.4	1.04974
$593$	7324.06	0.507189	0.253594	−	0.967311i	$-0.418387\pi$
$593$	7324.06	0.507189	0.253594	+	0.967311i	$0.418387\pi$
$594$	−44320.0	−3.06140
$595$	−11587.7	−0.798405
$596$	−1538.94	−0.105767
$597$	−21877.4	−1.49980
$598$	5574.71	0.381215
$599$	−12995.7	−0.886462	−0.443231	−	0.896407i	$-0.646168\pi$
$599$	−12995.7	−0.886462	−0.443231	+	0.896407i	$0.646168\pi$
$600$	−9111.37	−0.619950
$601$	−18095.7	−1.22819	−0.614093	−	0.789233i	$-0.710478\pi$
$601$	−18095.7	−1.22819	−0.614093	+	0.789233i	$0.710478\pi$
$602$	0	0
$603$	5026.63	0.339470
$604$	5983.22	0.403069
$605$	51423.5	3.45564
$606$	−10058.5	−0.674255
$607$	12834.4	0.858207	0.429103	−	0.903255i	$-0.358830\pi$
$607$	12834.4	0.858207	0.429103	+	0.903255i	$0.358830\pi$
$608$	9207.62	0.614175
$609$	22386.6	1.48958
$610$	−21386.6	−1.41954
$611$	835.015	0.0552882
$612$	4144.06	0.273715
$613$	−11358.5	−0.748391	−0.374196	−	0.927350i	$-0.622081\pi$
$613$	−11358.5	−0.748391	−0.374196	+	0.927350i	$0.622081\pi$
$614$	−18693.6	−1.22868
$615$	47681.7	3.12636
$616$	31259.7	2.04463
$617$	9745.84	0.635904	0.317952	−	0.948107i	$-0.397005\pi$
$617$	9745.84	0.635904	0.317952	+	0.948107i	$0.397005\pi$
$618$	−7873.99	−0.512521
$619$	−19796.2	−1.28543	−0.642713	−	0.766107i	$-0.722191\pi$
$619$	−19796.2	−1.28543	−0.642713	+	0.766107i	$0.722191\pi$
$620$	−233.204	−0.0151060
$621$	12415.8	0.802303
$622$	26745.4	1.72410
$623$	−37638.0	−2.42044
$624$	18054.4	1.15826
$625$	−19507.8	−1.24850
$626$	−8.96618	−0.000572461 0
$627$	55241.5	3.51855
$628$	−2118.01	−0.134582
$629$	−6963.21	−0.441401
$630$	51360.1	3.24800
$631$	15835.2	0.999031	0.499516	−	0.866305i	$-0.333511\pi$
$631$	15835.2	0.999031	0.499516	+	0.866305i	$0.333511\pi$
$632$	616.457	0.0387996
$633$	13223.7	0.830325
$634$	−10579.4	−0.662718
$635$	8406.66	0.525367
$636$	−713.437	−0.0444806
$637$	6325.61	0.393453
$638$	24622.1	1.52789
$639$	−5833.79	−0.361160
$640$	−23596.4	−1.45739
$641$	−7595.24	−0.468009	−0.234005	−	0.972236i	$-0.575183\pi$
$641$	−7595.24	−0.468009	−0.234005	+	0.972236i	$0.575183\pi$
$642$	57099.2	3.51016
$643$	1315.59	0.0806873	0.0403437	−	0.999186i	$-0.487155\pi$
$643$	1315.59	0.0806873	0.0403437	+	0.999186i	$0.487155\pi$
$644$	3693.14	0.225979
$645$	0	0
$646$	−10142.9	−0.617753
$647$	−15048.3	−0.914386	−0.457193	−	0.889367i	$-0.651145\pi$
$647$	−15048.3	−0.914386	−0.457193	+	0.889367i	$0.651145\pi$
$648$	6353.98	0.385197
$649$	34837.8	2.10709
$650$	−4971.33	−0.299987
$651$	−1525.91	−0.0918667
$652$	−7151.41	−0.429557
$653$	−15506.6	−0.929282	−0.464641	−	0.885499i	$-0.653817\pi$
$653$	−15506.6	−0.929282	−0.464641	+	0.885499i	$0.653817\pi$
$654$	44229.7	2.64453
$655$	37355.4	2.22839
$656$	31299.3	1.86285
$657$	6077.53	0.360894
$658$	2418.06	0.143261
$659$	−1105.73	−0.0653611	−0.0326806	−	0.999466i	$-0.510404\pi$
$659$	−1105.73	−0.0653611	−0.0326806	+	0.999466i	$0.510404\pi$
$660$	20040.5	1.18193
$661$	25902.8	1.52421	0.762104	−	0.647455i	$-0.224167\pi$
$661$	25902.8	1.52421	0.762104	+	0.647455i	$0.224167\pi$
$662$	25746.1	1.51156
$663$	−8314.32	−0.487031
$664$	−12343.0	−0.721390
$665$	−28758.4	−1.67699
$666$	30862.9	1.79567
$667$	−6897.64	−0.400416
$668$	8977.47	0.519983
$669$	−25413.5	−1.46867
$670$	−4463.32	−0.257363
$671$	−35210.4	−2.02576
$672$	21856.7	1.25467
$673$	27050.7	1.54937	0.774685	−	0.632347i	$-0.217908\pi$
$673$	27050.7	1.54937	0.774685	+	0.632347i	$0.217908\pi$
$674$	−23656.8	−1.35197
$675$	−11072.0	−0.631351
$676$	−3513.11	−0.199881
$677$	22070.9	1.25296	0.626479	−	0.779438i	$-0.284495\pi$
$677$	22070.9	1.25296	0.626479	+	0.779438i	$0.284495\pi$
$678$	26395.0	1.49513
$679$	−2243.12	−0.126779
$680$	8725.13	0.492049
$681$	−26168.7	−1.47252
$682$	−1678.28	−0.0942298
$683$	−6638.47	−0.371909	−0.185955	−	0.982558i	$-0.559538\pi$
$683$	−6638.47	−0.371909	−0.185955	+	0.982558i	$0.559538\pi$
$684$	10284.7	0.574920
$685$	−9262.17	−0.516627
$686$	−8271.01	−0.460333
$687$	5028.03	0.279230
$688$	0	0
$689$	923.018	0.0510365
$690$	−24540.5	−1.35397
$691$	−29895.5	−1.64584	−0.822922	−	0.568154i	$-0.807658\pi$
$691$	−29895.5	−1.64584	−0.822922	+	0.568154i	$0.807658\pi$
$692$	4413.39	0.242445
$693$	84558.2	4.63507
$694$	27611.9	1.51028
$695$	−777.074	−0.0424116
$696$	−16856.3	−0.918012
$697$	−14413.8	−0.783303
$698$	14854.6	0.805523
$699$	−47472.0	−2.56875
$700$	−3293.42	−0.177828
$701$	−28435.1	−1.53207	−0.766034	−	0.642800i	$-0.777773\pi$
$701$	−28435.1	−1.53207	−0.766034	+	0.642800i	$0.777773\pi$
$702$	16554.9	0.890064
$703$	−17281.2	−0.927133
$704$	−20308.8	−1.08724
$705$	−3675.83	−0.196369
$706$	−24184.7	−1.28924
$707$	8621.08	0.458598
$708$	10058.3	0.533916
$709$	2985.59	0.158147	0.0790735	−	0.996869i	$-0.474804\pi$
$709$	2985.59	0.158147	0.0790735	+	0.996869i	$0.474804\pi$
$710$	5180.01	0.273806
$711$	1667.53	0.0879567
$712$	28340.0	1.49169
$713$	470.155	0.0246949
$714$	−24076.9	−1.26198
$715$	−25927.6	−1.35614
$716$	−2252.19	−0.117554
$717$	15822.4	0.824125
$718$	34148.9	1.77496
$719$	13.7437	0.000712871 0	0.000356436	−	1.00000i	$-0.499887\pi$
$719$	13.7437	0.000712871 0	0.000356436		1.00000i	$0.499887\pi$
$720$	−51250.4	−2.65276
$721$	6748.74	0.348594
$722$	−3081.71	−0.158850
$723$	33155.7	1.70550
$724$	−8966.82	−0.460289
$725$	6151.07	0.315097
$726$	106847.	5.46208
$727$	−25528.1	−1.30232	−0.651158	−	0.758942i	$-0.725717\pi$
$727$	−25528.1	−1.30232	−0.651158	+	0.758942i	$0.725717\pi$
$728$	−11676.5	−0.594450
$729$	−28016.2	−1.42337
$730$	−5396.44	−0.273604
$731$	0	0
$732$	−10165.9	−0.513307
$733$	20204.9	1.01812	0.509062	−	0.860730i	$-0.329992\pi$
$733$	20204.9	1.01812	0.509062	+	0.860730i	$0.329992\pi$
$734$	−32678.7	−1.64331
$735$	−27846.1	−1.39744
$736$	−6734.34	−0.337271
$737$	−7348.30	−0.367270
$738$	63886.1	3.18656
$739$	20377.5	1.01434	0.507172	−	0.861845i	$-0.330691\pi$
$739$	20377.5	1.01434	0.507172	+	0.861845i	$0.330691\pi$
$740$	−6269.29	−0.311437
$741$	−20634.4	−1.02298
$742$	2672.90	0.132244
$743$	−31394.6	−1.55014	−0.775071	−	0.631874i	$-0.782286\pi$
$743$	−31394.6	−1.55014	−0.775071	+	0.631874i	$0.782286\pi$
$744$	1148.95	0.0566165
$745$	−8764.70	−0.431025
$746$	−18995.1	−0.932253
$747$	−33388.2	−1.63535
$748$	−6058.10	−0.296131
$749$	−48939.3	−2.38745
$750$	−25557.1	−1.24428
$751$	−765.609	−0.0372004	−0.0186002	−	0.999827i	$-0.505921\pi$
$751$	−765.609	−0.0372004	−0.0186002	+	0.999827i	$0.505921\pi$
$752$	−2412.90	−0.117007
$753$	−5136.16	−0.248569
$754$	−9197.10	−0.444216
$755$	34076.2	1.64260
$756$	10967.3	0.527616
$757$	−20525.6	−0.985491	−0.492745	−	0.870174i	$-0.664007\pi$
$757$	−20525.6	−0.985491	−0.492745	+	0.870174i	$0.664007\pi$
$758$	5230.20	0.250619
$759$	−40403.0	−1.93219
$760$	21654.0	1.03352
$761$	−16665.0	−0.793829	−0.396915	−	0.917856i	$-0.629919\pi$
$761$	−16665.0	−0.793829	−0.396915	+	0.917856i	$0.629919\pi$
$762$	17467.3	0.830410
$763$	−37909.0	−1.79869
$764$	5307.56	0.251336
$765$	23601.6	1.11545
$766$	26975.6	1.27241
$767$	−13013.0	−0.612610
$768$	−29261.3	−1.37484
$769$	16234.8	0.761303	0.380652	−	0.924719i	$-0.375700\pi$
$769$	16234.8	0.761303	0.380652	+	0.924719i	$0.375700\pi$
$770$	−75082.0	−3.51399
$771$	14595.8	0.681781
$772$	−3385.88	−0.157850
$773$	−10674.0	−0.496658	−0.248329	−	0.968676i	$-0.579881\pi$
$773$	−10674.0	−0.496658	−0.248329	+	0.968676i	$0.579881\pi$
$774$	0	0
$775$	−419.268	−0.0194330
$776$	1688.99	0.0781329
$777$	−41021.5	−1.89400
$778$	44268.8	2.03999
$779$	−35772.1	−1.64527
$780$	−7485.76	−0.343632
$781$	8528.26	0.390736
$782$	7418.42	0.339235
$783$	−20483.5	−0.934894
$784$	−18278.8	−0.832669
$785$	−12062.7	−0.548452
$786$	77616.7	3.52226
$787$	−5413.70	−0.245206	−0.122603	−	0.992456i	$-0.539124\pi$
$787$	−5413.70	−0.245206	−0.122603	+	0.992456i	$0.539124\pi$
$788$	8859.74	0.400527
$789$	50949.1	2.29890
$790$	−1480.65	−0.0666827
$791$	−22623.0	−1.01692
$792$	−63669.1	−2.85655
$793$	13152.2	0.588963
$794$	−22061.3	−0.986054
$795$	−4063.23	−0.181268
$796$	5954.31	0.265132
$797$	25337.3	1.12609	0.563044	−	0.826427i	$-0.309630\pi$
$797$	25337.3	1.12609	0.563044	+	0.826427i	$0.309630\pi$
$798$	−59753.8	−2.65070
$799$	1111.18	0.0491998
$800$	6005.45	0.265406
$801$	76660.2	3.38159
$802$	−5858.70	−0.257953
$803$	−8884.58	−0.390448
$804$	−2121.58	−0.0930627
$805$	21033.5	0.920912
$806$	626.891	0.0273961
$807$	12257.6	0.534680
$808$	−6491.34	−0.282630
$809$	−12894.1	−0.560360	−0.280180	−	0.959948i	$-0.590394\pi$
$809$	−12894.1	−0.560360	−0.280180	+	0.959948i	$0.590394\pi$
$810$	−15261.5	−0.662017
$811$	8554.95	0.370413	0.185207	−	0.982700i	$-0.440705\pi$
$811$	8554.95	0.370413	0.185207	+	0.982700i	$0.440705\pi$
$812$	−6092.92	−0.263324
$813$	−28452.0	−1.22737
$814$	−45117.7	−1.94272
$815$	−40729.4	−1.75054
$816$	24025.4	1.03071
$817$	0	0
$818$	−21165.0	−0.904665
$819$	−31585.1	−1.34759
$820$	−12977.4	−0.552672
$821$	−30480.2	−1.29569	−0.647847	−	0.761770i	$-0.724331\pi$
$821$	−30480.2	−1.29569	−0.647847	+	0.761770i	$0.724331\pi$
$822$	−19244.8	−0.816594
$823$	−13207.1	−0.559380	−0.279690	−	0.960090i	$-0.590232\pi$
$823$	−13207.1	−0.559380	−0.279690	+	0.960090i	$0.590232\pi$
$824$	−5081.55	−0.214835
$825$	36030.0	1.52049
$826$	−37683.4	−1.58738
$827$	33164.4	1.39449	0.697243	−	0.716835i	$-0.254410\pi$
$827$	33164.4	1.39449	0.697243	+	0.716835i	$0.254410\pi$
$828$	−7522.11	−0.315714
$829$	−39344.0	−1.64834	−0.824169	−	0.566343i	$-0.808358\pi$
$829$	−39344.0	−1.64834	−0.824169	+	0.566343i	$0.808358\pi$
$830$	29646.5	1.23981
$831$	6159.38	0.257120
$832$	7585.98	0.316101
$833$	8417.66	0.350126
$834$	−1614.59	−0.0670370
$835$	51129.3	2.11905
$836$	−15034.9	−0.622003
$837$	1396.19	0.0576577
$838$	−39700.8	−1.63657
$839$	−11863.9	−0.488186	−0.244093	−	0.969752i	$-0.578490\pi$
$839$	−11863.9	−0.488186	−0.244093	+	0.969752i	$0.578490\pi$
$840$	51401.3	2.11132
$841$	−13009.3	−0.533410
$842$	17504.5	0.716445
$843$	67958.0	2.77651
$844$	−3599.07	−0.146783
$845$	−20008.2	−0.814559
$846$	−4925.05	−0.200150
$847$	−91577.9	−3.71506
$848$	−2667.19	−0.108009
$849$	42932.3	1.73549
$850$	−6615.49	−0.266952
$851$	12639.3	0.509130
$852$	2462.26	0.0990088
$853$	−15340.4	−0.615760	−0.307880	−	0.951425i	$-0.599620\pi$
$853$	−15340.4	−0.615760	−0.307880	+	0.951425i	$0.599620\pi$
$854$	38086.5	1.52610
$855$	58574.4	2.34293
$856$	36849.4	1.47136
$857$	2347.71	0.0935778	0.0467889	−	0.998905i	$-0.485101\pi$
$857$	2347.71	0.0935778	0.0467889	+	0.998905i	$0.485101\pi$
$858$	−53872.1	−2.14355
$859$	38761.4	1.53961	0.769803	−	0.638281i	$-0.220354\pi$
$859$	38761.4	1.53961	0.769803	+	0.638281i	$0.220354\pi$
$860$	0	0
$861$	−84914.3	−3.36106
$862$	17171.2	0.678485
$863$	−33659.3	−1.32767	−0.663834	−	0.747880i	$-0.731072\pi$
$863$	−33659.3	−1.32767	−0.663834	+	0.747880i	$0.731072\pi$
$864$	−19998.6	−0.787461
$865$	25135.5	0.988016
$866$	−12843.7	−0.503980
$867$	31772.8	1.24459
$868$	415.304	0.0162400
$869$	−2437.72	−0.0951598
$870$	40486.7	1.57773
$871$	2744.82	0.106779
$872$	28544.1	1.10851
$873$	4568.74	0.177123
$874$	18411.0	0.712541
$875$	21904.8	0.846306
$876$	−2565.13	−0.0989358
$877$	26492.9	1.02007	0.510034	−	0.860154i	$-0.329633\pi$
$877$	26492.9	1.02007	0.510034	+	0.860154i	$0.329633\pi$
$878$	−5831.74	−0.224159
$879$	−46034.0	−1.76643
$880$	74921.6	2.87001
$881$	−34532.1	−1.32056	−0.660281	−	0.751018i	$-0.729563\pi$
$881$	−34532.1	−1.32056	−0.660281	+	0.751018i	$0.729563\pi$
$882$	−37309.5	−1.42435
$883$	8248.77	0.314375	0.157187	−	0.987569i	$-0.449757\pi$
$883$	8248.77	0.314375	0.157187	+	0.987569i	$0.449757\pi$
$884$	2262.89	0.0860963
$885$	57284.7	2.17582
$886$	33218.2	1.25958
$887$	33247.6	1.25856	0.629282	−	0.777177i	$-0.283349\pi$
$887$	33247.6	1.25856	0.629282	+	0.777177i	$0.283349\pi$
$888$	30887.6	1.16725
$889$	−14971.1	−0.564807
$890$	−68069.2	−2.56369
$891$	−25126.1	−0.944734
$892$	6916.72	0.259629
$893$	2757.71	0.103341
$894$	−18211.2	−0.681290
$895$	−12826.9	−0.479056
$896$	42021.8	1.56680
$897$	15091.8	0.561761
$898$	7673.20	0.285142
$899$	−775.657	−0.0287760
$900$	6707.96	0.248443
$901$	1228.29	0.0454163
$902$	−93393.4	−3.44752
$903$	0	0
$904$	17034.3	0.626716
$905$	−51068.7	−1.87578
$906$	70803.1	2.59633
$907$	11992.7	0.439042	0.219521	−	0.975608i	$-0.429551\pi$
$907$	11992.7	0.439042	0.219521	+	0.975608i	$0.429551\pi$
$908$	7122.26	0.260309
$909$	−17559.2	−0.640707
$910$	28045.5	1.02165
$911$	33624.5	1.22287	0.611433	−	0.791296i	$-0.290593\pi$
$911$	33624.5	1.22287	0.611433	+	0.791296i	$0.290593\pi$
$912$	59626.1	2.16493
$913$	48809.3	1.76928
$914$	3117.68	0.112827
$915$	−57897.5	−2.09184
$916$	−1368.47	−0.0493618
$917$	−66524.7	−2.39568
$918$	22030.1	0.792049
$919$	−21470.7	−0.770678	−0.385339	−	0.922775i	$-0.625915\pi$
$919$	−21470.7	−0.770678	−0.385339	+	0.922775i	$0.625915\pi$
$920$	−15837.5	−0.567549
$921$	−50606.9	−1.81059
$922$	−18536.7	−0.662120
$923$	−3185.57	−0.113602
$924$	−35689.3	−1.27066
$925$	−11271.3	−0.400646
$926$	47366.6	1.68095
$927$	−13745.7	−0.487020
$928$	11110.3	0.393009
$929$	7941.69	0.280472	0.140236	−	0.990118i	$-0.455214\pi$
$929$	7941.69	0.280472	0.140236	+	0.990118i	$0.455214\pi$
$930$	−2759.65	−0.0973036
$931$	20890.9	0.735415
$932$	12920.3	0.454099
$933$	72404.8	2.54065
$934$	−15041.6	−0.526955
$935$	−34502.6	−1.20680
$936$	23782.4	0.830504
$937$	−43111.2	−1.50307	−0.751537	−	0.659691i	$-0.770687\pi$
$937$	−43111.2	−1.50307	−0.751537	+	0.659691i	$0.770687\pi$
$938$	7948.53	0.276683
$939$	−24.2731	−0.000843581 0
$940$	1000.44	0.0347137
$941$	−16162.2	−0.559906	−0.279953	−	0.960014i	$-0.590319\pi$
$941$	−16162.2	−0.559906	−0.279953	+	0.960014i	$0.590319\pi$
$942$	−25063.7	−0.866898
$943$	26163.3	0.903493
$944$	37602.9	1.29647
$945$	62462.1	2.15015
$946$	0	0
$947$	7629.78	0.261811	0.130905	−	0.991395i	$-0.458212\pi$
$947$	7629.78	0.261811	0.130905	+	0.991395i	$0.458212\pi$
$948$	−703.811	−0.0241126
$949$	3318.67	0.113518
$950$	−16418.3	−0.560715
$951$	−28640.5	−0.976585
$952$	−15538.2	−0.528988
$953$	−12245.8	−0.416243	−0.208122	−	0.978103i	$-0.566735\pi$
$953$	−12245.8	−0.416243	−0.208122	+	0.978103i	$0.566735\pi$
$954$	−5444.11	−0.184758
$955$	30228.1	1.02425
$956$	−4306.34	−0.145687
$957$	66656.5	2.25151
$958$	−5299.10	−0.178712
$959$	16494.6	0.555410
$960$	−33394.4	−1.12271
$961$	−29738.1	−0.998225
$962$	16852.9	0.564821
$963$	99678.5	3.33551
$964$	−9023.91	−0.301494
$965$	−19283.6	−0.643274
$966$	43703.2	1.45562
$967$	−9078.20	−0.301898	−0.150949	−	0.988542i	$-0.548233\pi$
$967$	−9078.20	−0.301898	−0.150949	+	0.988542i	$0.548233\pi$
$968$	68954.7	2.28955
$969$	−27458.8	−0.910324
$970$	−4056.74	−0.134283
$971$	5977.14	0.197544	0.0987722	−	0.995110i	$-0.468508\pi$
$971$	5977.14	0.197544	0.0987722	+	0.995110i	$0.468508\pi$
$972$	5048.67	0.166601
$973$	1383.86	0.0455955
$974$	−51457.8	−1.69283
$975$	−13458.3	−0.442063
$976$	−38005.1	−1.24643
$977$	−45996.0	−1.50619	−0.753093	−	0.657914i	$-0.771439\pi$
$977$	−45996.0	−1.50619	−0.753093	+	0.657914i	$0.771439\pi$
$978$	−84627.0	−2.76695
$979$	−112068.	−3.65853
$980$	7578.80	0.247037
$981$	77212.2	2.51294
$982$	22193.5	0.721204
$983$	−8525.77	−0.276633	−0.138316	−	0.990388i	$-0.544169\pi$
$983$	−8525.77	−0.276633	−0.138316	+	0.990388i	$0.544169\pi$
$984$	63937.3	2.07139
$985$	50458.8	1.63224
$986$	−12238.8	−0.395298
$987$	6546.14	0.211110
$988$	5616.02	0.180839
$989$	0	0
$990$	152925.	4.90938
$991$	−12323.8	−0.395033	−0.197516	−	0.980300i	$-0.563288\pi$
$991$	−12323.8	−0.395033	−0.197516	+	0.980300i	$0.563288\pi$
$992$	−757.294	−0.0242380
$993$	69699.5	2.22744
$994$	−9224.87	−0.294361
$995$	33911.5	1.08047
$996$	14092.1	0.448318
$997$	3674.25	0.116715	0.0583574	−	0.998296i	$-0.481414\pi$
$997$	3674.25	0.116715	0.0583574	+	0.998296i	$0.481414\pi$
$998$	−18435.5	−0.584734
$999$	37534.2	1.18872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.14	✓	50
43.42	odd	2		1849.4.a.j.1.37	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.14	✓	50	1.1	even	1	trivial
1849.4.a.j.1.37	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)

\(f(q)\)	\(=\)	\(q-3.22072 q^{2} -8.71909 q^{3} +2.37305 q^{4} +13.5152 q^{5} +28.0818 q^{6} -24.0687 q^{7} +18.1228 q^{8} +49.0226 q^{9} +O(q^{10})\)	\(q-3.22072 q^{2} -8.71909 q^{3} +2.37305 q^{4} +13.5152 q^{5} +28.0818 q^{6} -24.0687 q^{7} +18.1228 q^{8} +49.0226 q^{9} -43.5288 q^{10} -71.6649 q^{11} -20.6909 q^{12} +26.7691 q^{13} +77.5186 q^{14} -117.841 q^{15} -77.3530 q^{16} +35.6223 q^{17} -157.888 q^{18} +88.4072 q^{19} +32.0724 q^{20} +209.857 q^{21} +230.813 q^{22} -64.6600 q^{23} -158.015 q^{24} +57.6615 q^{25} -86.2157 q^{26} -192.017 q^{27} -57.1164 q^{28} +106.675 q^{29} +379.532 q^{30} -7.27119 q^{31} +104.150 q^{32} +624.853 q^{33} -114.730 q^{34} -325.294 q^{35} +116.333 q^{36} -195.473 q^{37} -284.735 q^{38} -233.402 q^{39} +244.934 q^{40} -404.629 q^{41} -675.892 q^{42} -170.065 q^{44} +662.552 q^{45} +208.252 q^{46} +31.1933 q^{47} +674.449 q^{48} +236.303 q^{49} -185.712 q^{50} -310.594 q^{51} +63.5244 q^{52} +34.4808 q^{53} +618.434 q^{54} -968.567 q^{55} -436.193 q^{56} -770.831 q^{57} -343.572 q^{58} -486.120 q^{59} -279.642 q^{60} +491.320 q^{61} +23.4185 q^{62} -1179.91 q^{63} +283.386 q^{64} +361.790 q^{65} -2012.48 q^{66} +102.537 q^{67} +84.5337 q^{68} +563.777 q^{69} +1047.68 q^{70} -119.002 q^{71} +888.429 q^{72} +123.974 q^{73} +629.565 q^{74} -502.756 q^{75} +209.795 q^{76} +1724.88 q^{77} +751.723 q^{78} +34.0155 q^{79} -1045.44 q^{80} +350.606 q^{81} +1303.20 q^{82} -681.077 q^{83} +498.003 q^{84} +481.444 q^{85} -930.114 q^{87} -1298.77 q^{88} +1563.77 q^{89} -2133.90 q^{90} -644.297 q^{91} -153.442 q^{92} +63.3982 q^{93} -100.465 q^{94} +1194.84 q^{95} -908.094 q^{96} +93.1966 q^{97} -761.066 q^{98} -3513.20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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