LMFDB - Embedded newform 2013.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.39202 q^{2} +3.00000 q^{3} +21.0739 q^{4} +2.00966 q^{5} -16.1761 q^{6} -20.9762 q^{7} -70.4947 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.39202 q^{2} +3.00000 q^{3} +21.0739 q^{4} +2.00966 q^{5} -16.1761 q^{6} -20.9762 q^{7} -70.4947 q^{8} +9.00000 q^{9} -10.8361 q^{10} -11.0000 q^{11} +63.2217 q^{12} +88.7001 q^{13} +113.104 q^{14} +6.02897 q^{15} +211.518 q^{16} -53.2357 q^{17} -48.5282 q^{18} +109.368 q^{19} +42.3513 q^{20} -62.9285 q^{21} +59.3122 q^{22} -173.049 q^{23} -211.484 q^{24} -120.961 q^{25} -478.273 q^{26} +27.0000 q^{27} -442.049 q^{28} +119.806 q^{29} -32.5084 q^{30} +169.566 q^{31} -576.551 q^{32} -33.0000 q^{33} +287.048 q^{34} -42.1549 q^{35} +189.665 q^{36} -411.570 q^{37} -589.713 q^{38} +266.100 q^{39} -141.670 q^{40} +344.956 q^{41} +339.312 q^{42} -234.092 q^{43} -231.813 q^{44} +18.0869 q^{45} +933.085 q^{46} -401.338 q^{47} +634.554 q^{48} +96.9994 q^{49} +652.226 q^{50} -159.707 q^{51} +1869.26 q^{52} +553.224 q^{53} -145.585 q^{54} -22.1062 q^{55} +1478.71 q^{56} +328.103 q^{57} -645.997 q^{58} -559.645 q^{59} +127.054 q^{60} -61.0000 q^{61} -914.306 q^{62} -188.785 q^{63} +1416.63 q^{64} +178.257 q^{65} +177.937 q^{66} +826.774 q^{67} -1121.88 q^{68} -519.147 q^{69} +227.300 q^{70} +928.741 q^{71} -634.453 q^{72} +883.581 q^{73} +2219.19 q^{74} -362.884 q^{75} +2304.80 q^{76} +230.738 q^{77} -1434.82 q^{78} -374.811 q^{79} +425.079 q^{80} +81.0000 q^{81} -1860.01 q^{82} -1184.79 q^{83} -1326.15 q^{84} -106.986 q^{85} +1262.23 q^{86} +359.418 q^{87} +775.442 q^{88} -1432.56 q^{89} -97.5251 q^{90} -1860.59 q^{91} -3646.82 q^{92} +508.699 q^{93} +2164.02 q^{94} +219.792 q^{95} -1729.65 q^{96} +1.75309 q^{97} -523.023 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9}+O(q^{10}) $ 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9} - 41 q^{10} - 407 q^{11} + 390 q^{12} + 51 q^{13} - 228 q^{14} - 105 q^{15} + 462 q^{16} - 190 q^{17} - 72 q^{18} - 51 q^{19} - 720 q^{20} - 105 q^{21} + 88 q^{22} - 583 q^{23} - 351 q^{24} + 598 q^{25} - 1019 q^{26} + 999 q^{27} - 498 q^{28} - 566 q^{29} - 123 q^{30} - 696 q^{31} - 859 q^{32} - 1221 q^{33} - 348 q^{34} - 1102 q^{35} + 1170 q^{36} - 1022 q^{37} - 455 q^{38} + 153 q^{39} - 503 q^{40} - 790 q^{41} - 684 q^{42} - 87 q^{43} - 1430 q^{44} - 315 q^{45} - 303 q^{46} - 1603 q^{47} + 1386 q^{48} + 110 q^{49} - 1926 q^{50} - 570 q^{51} + 736 q^{52} - 2619 q^{53} - 216 q^{54} + 385 q^{55} - 4937 q^{56} - 153 q^{57} - 1099 q^{58} - 2471 q^{59} - 2160 q^{60} - 2257 q^{61} - 2909 q^{62} - 315 q^{63} - 265 q^{64} - 1970 q^{65} + 264 q^{66} - 3033 q^{67} - 1956 q^{68} - 1749 q^{69} + 2410 q^{70} - 3891 q^{71} - 1053 q^{72} + 391 q^{73} - 532 q^{74} + 1794 q^{75} + 1554 q^{76} + 385 q^{77} - 3057 q^{78} + 67 q^{79} - 5111 q^{80} + 2997 q^{81} - 4818 q^{82} - 5315 q^{83} - 1494 q^{84} - 2747 q^{85} - 5195 q^{86} - 1698 q^{87} + 1287 q^{88} - 8945 q^{89} - 369 q^{90} - 4432 q^{91} - 4701 q^{92} - 2088 q^{93} - 372 q^{94} - 3388 q^{95} - 2577 q^{96} - 3784 q^{97} - 4502 q^{98} - 3663 q^{99}+O(q^{100}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9 - 41 * q^10 - 407 * q^11 + 390 * q^12 + 51 * q^13 - 228 * q^14 - 105 * q^15 + 462 * q^16 - 190 * q^17 - 72 * q^18 - 51 * q^19 - 720 * q^20 - 105 * q^21 + 88 * q^22 - 583 * q^23 - 351 * q^24 + 598 * q^25 - 1019 * q^26 + 999 * q^27 - 498 * q^28 - 566 * q^29 - 123 * q^30 - 696 * q^31 - 859 * q^32 - 1221 * q^33 - 348 * q^34 - 1102 * q^35 + 1170 * q^36 - 1022 * q^37 - 455 * q^38 + 153 * q^39 - 503 * q^40 - 790 * q^41 - 684 * q^42 - 87 * q^43 - 1430 * q^44 - 315 * q^45 - 303 * q^46 - 1603 * q^47 + 1386 * q^48 + 110 * q^49 - 1926 * q^50 - 570 * q^51 + 736 * q^52 - 2619 * q^53 - 216 * q^54 + 385 * q^55 - 4937 * q^56 - 153 * q^57 - 1099 * q^58 - 2471 * q^59 - 2160 * q^60 - 2257 * q^61 - 2909 * q^62 - 315 * q^63 - 265 * q^64 - 1970 * q^65 + 264 * q^66 - 3033 * q^67 - 1956 * q^68 - 1749 * q^69 + 2410 * q^70 - 3891 * q^71 - 1053 * q^72 + 391 * q^73 - 532 * q^74 + 1794 * q^75 + 1554 * q^76 + 385 * q^77 - 3057 * q^78 + 67 * q^79 - 5111 * q^80 + 2997 * q^81 - 4818 * q^82 - 5315 * q^83 - 1494 * q^84 - 2747 * q^85 - 5195 * q^86 - 1698 * q^87 + 1287 * q^88 - 8945 * q^89 - 369 * q^90 - 4432 * q^91 - 4701 * q^92 - 2088 * q^93 - 372 * q^94 - 3388 * q^95 - 2577 * q^96 - 3784 * q^97 - 4502 * q^98 - 3663 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−5.39202	−1.90637	−0.953184	−	0.302392i	$-0.902215\pi$
$2$	−5.39202	−1.90637	−0.953184	+	0.302392i	$0.902215\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	21.0739	2.63424
$5$	2.00966	0.179749	0.0898746	−	0.995953i	$-0.471353\pi$
$5$	2.00966	0.179749	0.0898746	+	0.995953i	$0.471353\pi$
$6$	−16.1761	−1.10064
$7$	−20.9762	−1.13261	−0.566303	−	0.824197i	$-0.691627\pi$
$7$	−20.9762	−1.13261	−0.566303	+	0.824197i	$0.691627\pi$
$8$	−70.4947	−3.11546
$9$	9.00000	0.333333
$10$	−10.8361	−0.342668
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	63.2217	1.52088
$13$	88.7001	1.89238	0.946191	−	0.323607i	$-0.104896\pi$
$13$	88.7001	1.89238	0.946191	+	0.323607i	$0.104896\pi$
$14$	113.104	2.15916
$15$	6.02897	0.103778
$16$	211.518	3.30497
$17$	−53.2357	−0.759504	−0.379752	−	0.925088i	$-0.623991\pi$
$17$	−53.2357	−0.759504	−0.379752	+	0.925088i	$0.623991\pi$
$18$	−48.5282	−0.635456
$19$	109.368	1.32056	0.660281	−	0.751019i	$-0.270437\pi$
$19$	109.368	1.32056	0.660281	+	0.751019i	$0.270437\pi$
$20$	42.3513	0.473502
$21$	−62.9285	−0.653911
$22$	59.3122	0.574791
$23$	−173.049	−1.56884	−0.784418	−	0.620232i	$-0.787038\pi$
$23$	−173.049	−1.56884	−0.784418	+	0.620232i	$0.787038\pi$
$24$	−211.484	−1.79871
$25$	−120.961	−0.967690
$26$	−478.273	−3.60758
$27$	27.0000	0.192450
$28$	−442.049	−2.98355
$29$	119.806	0.767153	0.383576	−	0.923509i	$-0.374692\pi$
$29$	119.806	0.767153	0.383576	+	0.923509i	$0.374692\pi$
$30$	−32.5084	−0.197840
$31$	169.566	0.982421	0.491210	−	0.871041i	$-0.336555\pi$
$31$	169.566	0.982421	0.491210	+	0.871041i	$0.336555\pi$
$32$	−576.551	−3.18503
$33$	−33.0000	−0.174078
$34$	287.048	1.44789
$35$	−42.1549	−0.203585
$36$	189.665	0.878079
$37$	−411.570	−1.82869	−0.914346	−	0.404933i	$-0.867295\pi$
$37$	−411.570	−1.82869	−0.914346	+	0.404933i	$0.867295\pi$
$38$	−589.713	−2.51748
$39$	266.100	1.09257
$40$	−141.670	−0.560001
$41$	344.956	1.31398	0.656989	−	0.753900i	$-0.271830\pi$
$41$	344.956	1.31398	0.656989	+	0.753900i	$0.271830\pi$
$42$	339.312	1.24659
$43$	−234.092	−0.830203	−0.415102	−	0.909775i	$-0.636254\pi$
$43$	−234.092	−0.830203	−0.415102	+	0.909775i	$0.636254\pi$
$44$	−231.813	−0.794252
$45$	18.0869	0.0599164
$46$	933.085	2.99078
$47$	−401.338	−1.24556	−0.622778	−	0.782398i	$-0.713996\pi$
$47$	−401.338	−1.24556	−0.622778	+	0.782398i	$0.713996\pi$
$48$	634.554	1.90812
$49$	96.9994	0.282797
$50$	652.226	1.84477
$51$	−159.707	−0.438500
$52$	1869.26	4.98498
$53$	553.224	1.43380	0.716898	−	0.697178i	$-0.245561\pi$
$53$	553.224	1.43380	0.716898	+	0.697178i	$0.245561\pi$
$54$	−145.585	−0.366881
$55$	−22.1062	−0.0541964
$56$	1478.71	3.52859
$57$	328.103	0.762427
$58$	−645.997	−1.46248
$59$	−559.645	−1.23491	−0.617454	−	0.786607i	$-0.711836\pi$
$59$	−559.645	−1.23491	−0.617454	+	0.786607i	$0.711836\pi$
$60$	127.054	0.273377
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−914.306	−1.87285
$63$	−188.785	−0.377535
$64$	1416.63	2.76686
$65$	178.257	0.340154
$66$	177.937	0.331856
$67$	826.774	1.50756	0.753780	−	0.657127i	$-0.228229\pi$
$67$	826.774	1.50756	0.753780	+	0.657127i	$0.228229\pi$
$68$	−1121.88	−2.00071
$69$	−519.147	−0.905768
$70$	227.300	0.388108
$71$	928.741	1.55241	0.776207	−	0.630479i	$-0.217141\pi$
$71$	928.741	1.55241	0.776207	+	0.630479i	$0.217141\pi$
$72$	−634.453	−1.03849
$73$	883.581	1.41665	0.708324	−	0.705887i	$-0.249451\pi$
$73$	883.581	1.41665	0.708324	+	0.705887i	$0.249451\pi$
$74$	2219.19	3.48616
$75$	−362.884	−0.558696
$76$	2304.80	3.47867
$77$	230.738	0.341494
$78$	−1434.82	−2.08284
$79$	−374.811	−0.533792	−0.266896	−	0.963725i	$-0.585998\pi$
$79$	−374.811	−0.533792	−0.266896	+	0.963725i	$0.585998\pi$
$80$	425.079	0.594066
$81$	81.0000	0.111111
$82$	−1860.01	−2.50493
$83$	−1184.79	−1.56684	−0.783422	−	0.621490i	$-0.786528\pi$
$83$	−1184.79	−1.56684	−0.783422	+	0.621490i	$0.786528\pi$
$84$	−1326.15	−1.72256
$85$	−106.986	−0.136520
$86$	1262.23	1.58267
$87$	359.418	0.442916
$88$	775.442	0.939345
$89$	−1432.56	−1.70619	−0.853094	−	0.521758i	$-0.825277\pi$
$89$	−1432.56	−1.70619	−0.853094	+	0.521758i	$0.825277\pi$
$90$	−97.5251	−0.114223
$91$	−1860.59	−2.14332
$92$	−3646.82	−4.13269
$93$	508.699	0.567201
$94$	2164.02	2.37449
$95$	219.792	0.237370
$96$	−1729.65	−1.83888
$97$	1.75309	0.00183504	0.000917522	−	1.00000i	$-0.499708\pi$
$97$	1.75309	0.00183504	0.000917522		1.00000i	$0.499708\pi$
$98$	−523.023	−0.539115
$99$	−99.0000	−0.100504
$100$	−2549.13	−2.54913
$101$	426.918	0.420593	0.210297	−	0.977638i	$-0.432557\pi$
$101$	426.918	0.420593	0.210297	+	0.977638i	$0.432557\pi$
$102$	861.145	0.835941
$103$	232.925	0.222823	0.111412	−	0.993774i	$-0.464463\pi$
$103$	232.925	0.222823	0.111412	+	0.993774i	$0.464463\pi$
$104$	−6252.89	−5.89564
$105$	−126.465	−0.117540
$106$	−2983.00	−2.73334
$107$	−1609.91	−1.45454	−0.727269	−	0.686352i	$-0.759211\pi$
$107$	−1609.91	−1.45454	−0.727269	+	0.686352i	$0.759211\pi$
$108$	568.995	0.506959
$109$	−492.625	−0.432889	−0.216444	−	0.976295i	$-0.569446\pi$
$109$	−492.625	−0.432889	−0.216444	+	0.976295i	$0.569446\pi$
$110$	119.197	0.103318
$111$	−1234.71	−1.05580
$112$	−4436.83	−3.74323
$113$	920.997	0.766726	0.383363	−	0.923598i	$-0.374766\pi$
$113$	920.997	0.766726	0.383363	+	0.923598i	$0.374766\pi$
$114$	−1769.14	−1.45347
$115$	−347.770	−0.281997
$116$	2524.78	2.02086
$117$	798.301	0.630794
$118$	3017.62	2.35419
$119$	1116.68	0.860219
$120$	−425.011	−0.323317
$121$	121.000	0.0909091
$122$	328.913	0.244085
$123$	1034.87	0.758626
$124$	3573.43	2.58793
$125$	−494.298	−0.353691
$126$	1017.94	0.719721
$127$	−1349.52	−0.942914	−0.471457	−	0.881889i	$-0.656272\pi$
$127$	−1349.52	−0.942914	−0.471457	+	0.881889i	$0.656272\pi$
$128$	−3026.11	−2.08963
$129$	−702.277	−0.479318
$130$	−961.165	−0.648459
$131$	599.648	0.399935	0.199967	−	0.979803i	$-0.435916\pi$
$131$	599.648	0.399935	0.199967	+	0.979803i	$0.435916\pi$
$132$	−695.439	−0.458562
$133$	−2294.12	−1.49568
$134$	−4457.98	−2.87396
$135$	54.2608	0.0345928
$136$	3752.84	2.36620
$137$	−933.851	−0.582367	−0.291183	−	0.956667i	$-0.594049\pi$
$137$	−933.851	−0.582367	−0.291183	+	0.956667i	$0.594049\pi$
$138$	2799.25	1.72673
$139$	3208.06	1.95758	0.978791	−	0.204863i	$-0.0656748\pi$
$139$	3208.06	1.95758	0.978791	+	0.204863i	$0.0656748\pi$
$140$	−888.368	−0.536292
$141$	−1204.01	−0.719122
$142$	−5007.79	−2.95947
$143$	−975.701	−0.570575
$144$	1903.66	1.10166
$145$	240.769	0.137895
$146$	−4764.29	−2.70065
$147$	290.998	0.163273
$148$	−8673.38	−4.81721
$149$	120.299	0.0661428	0.0330714	−	0.999453i	$-0.489471\pi$
$149$	120.299	0.0661428	0.0330714	+	0.999453i	$0.489471\pi$
$150$	1956.68	1.06508
$151$	942.372	0.507875	0.253938	−	0.967221i	$-0.418274\pi$
$151$	942.372	0.507875	0.253938	+	0.967221i	$0.418274\pi$
$152$	−7709.85	−4.11415
$153$	−479.122	−0.253168
$154$	−1244.14	−0.651012
$155$	340.771	0.176589
$156$	5607.77	2.87808
$157$	−1099.65	−0.558991	−0.279496	−	0.960147i	$-0.590167\pi$
$157$	−1099.65	−0.558991	−0.279496	+	0.960147i	$0.590167\pi$
$158$	2020.99	1.01760
$159$	1659.67	0.827802
$160$	−1158.67	−0.572506
$161$	3629.91	1.77687
$162$	−436.754	−0.211819
$163$	−1138.41	−0.547040	−0.273520	−	0.961866i	$-0.588188\pi$
$163$	−1138.41	−0.547040	−0.273520	+	0.961866i	$0.588188\pi$
$164$	7269.57	3.46133
$165$	−66.3187	−0.0312903
$166$	6388.44	2.98698
$167$	2612.12	1.21037	0.605185	−	0.796085i	$-0.293099\pi$
$167$	2612.12	1.21037	0.605185	+	0.796085i	$0.293099\pi$
$168$	4436.13	2.03723
$169$	5670.70	2.58111
$170$	576.869	0.260258
$171$	984.310	0.440187
$172$	−4933.24	−2.18695
$173$	−461.443	−0.202791	−0.101396	−	0.994846i	$-0.532331\pi$
$173$	−461.443	−0.202791	−0.101396	+	0.994846i	$0.532331\pi$
$174$	−1937.99	−0.844360
$175$	2537.30	1.09601
$176$	−2326.70	−0.996485
$177$	−1678.93	−0.712974
$178$	7724.38	3.25262
$179$	−1009.88	−0.421686	−0.210843	−	0.977520i	$-0.567621\pi$
$179$	−1009.88	−0.421686	−0.210843	+	0.977520i	$0.567621\pi$
$180$	381.162	0.157834
$181$	−882.948	−0.362591	−0.181296	−	0.983429i	$-0.558029\pi$
$181$	−882.948	−0.362591	−0.181296	+	0.983429i	$0.558029\pi$
$182$	10032.3	4.08596
$183$	−183.000	−0.0739221
$184$	12199.1	4.88764
$185$	−827.114	−0.328706
$186$	−2742.92	−1.08129
$187$	585.593	0.228999
$188$	−8457.75	−3.28109
$189$	−566.356	−0.217970
$190$	−1185.12	−0.452515
$191$	−2449.45	−0.927937	−0.463968	−	0.885852i	$-0.653575\pi$
$191$	−2449.45	−0.927937	−0.463968	+	0.885852i	$0.653575\pi$
$192$	4249.90	1.59745
$193$	−711.746	−0.265454	−0.132727	−	0.991153i	$-0.542373\pi$
$193$	−711.746	−0.265454	−0.132727	+	0.991153i	$0.542373\pi$
$194$	−9.45270	−0.00349827
$195$	534.771	0.196388
$196$	2044.16	0.744955
$197$	−30.2626	−0.0109448	−0.00547238	−	0.999985i	$-0.501742\pi$
$197$	−30.2626	−0.0109448	−0.00547238	+	0.999985i	$0.501742\pi$
$198$	533.810	0.191597
$199$	1664.33	0.592871	0.296435	−	0.955053i	$-0.404202\pi$
$199$	1664.33	0.592871	0.296435	+	0.955053i	$0.404202\pi$
$200$	8527.13	3.01480
$201$	2480.32	0.870390
$202$	−2301.95	−0.801805
$203$	−2513.07	−0.868882
$204$	−3365.65	−1.15511
$205$	693.244	0.236187
$206$	−1255.94	−0.424783
$207$	−1557.44	−0.522946
$208$	18761.7	6.25426
$209$	−1203.05	−0.398164
$210$	681.901	0.224074
$211$	1864.04	0.608179	0.304089	−	0.952644i	$-0.401648\pi$
$211$	1864.04	0.608179	0.304089	+	0.952644i	$0.401648\pi$
$212$	11658.6	3.77696
$213$	2786.22	0.896286
$214$	8680.65	2.77288
$215$	−470.446	−0.149228
$216$	−1903.36	−0.599570
$217$	−3556.85	−1.11270
$218$	2656.24	0.825245
$219$	2650.74	0.817903
$220$	−465.865	−0.142766
$221$	−4722.01	−1.43727
$222$	6657.58	2.01274
$223$	−3474.19	−1.04327	−0.521634	−	0.853169i	$-0.674677\pi$
$223$	−3474.19	−1.04327	−0.521634	+	0.853169i	$0.674677\pi$
$224$	12093.8	3.60738
$225$	−1088.65	−0.322563
$226$	−4966.03	−1.46166
$227$	−2411.28	−0.705031	−0.352516	−	0.935806i	$-0.614674\pi$
$227$	−2411.28	−0.705031	−0.352516	+	0.935806i	$0.614674\pi$
$228$	6914.41	2.00841
$229$	−3636.23	−1.04930	−0.524648	−	0.851319i	$-0.675803\pi$
$229$	−3636.23	−1.04930	−0.524648	+	0.851319i	$0.675803\pi$
$230$	1875.18	0.537590
$231$	692.213	0.197161
$232$	−8445.70	−2.39003
$233$	2057.66	0.578548	0.289274	−	0.957246i	$-0.406586\pi$
$233$	2057.66	0.578548	0.289274	+	0.957246i	$0.406586\pi$
$234$	−4304.45	−1.20253
$235$	−806.552	−0.223888
$236$	−11793.9	−3.25304
$237$	−1124.43	−0.308185
$238$	−6021.17	−1.63989
$239$	−7058.78	−1.91044	−0.955219	−	0.295900i	$-0.904381\pi$
$239$	−7058.78	−1.91044	−0.955219	+	0.295900i	$0.904381\pi$
$240$	1275.24	0.342984
$241$	1638.91	0.438056	0.219028	−	0.975719i	$-0.429711\pi$
$241$	1638.91	0.438056	0.219028	+	0.975719i	$0.429711\pi$
$242$	−652.435	−0.173306
$243$	243.000	0.0641500
$244$	−1285.51	−0.337279
$245$	194.936	0.0508326
$246$	−5580.04	−1.44622
$247$	9700.93	2.49901
$248$	−11953.5	−3.06069
$249$	−3554.38	−0.904618
$250$	2665.27	0.674265
$251$	−3544.78	−0.891413	−0.445707	−	0.895179i	$-0.647048\pi$
$251$	−3544.78	−0.891413	−0.445707	+	0.895179i	$0.647048\pi$
$252$	−3978.45	−0.994518
$253$	1903.54	0.473022
$254$	7276.61	1.79754
$255$	−320.957	−0.0788200
$256$	4983.78	1.21674
$257$	−4292.15	−1.04178	−0.520889	−	0.853624i	$-0.674399\pi$
$257$	−4292.15	−1.04178	−0.520889	+	0.853624i	$0.674399\pi$
$258$	3786.69	0.913757
$259$	8633.15	2.07119
$260$	3756.57	0.896047
$261$	1078.25	0.255718
$262$	−3233.31	−0.762423
$263$	4268.66	1.00082	0.500412	−	0.865787i	$-0.333182\pi$
$263$	4268.66	1.00082	0.500412	+	0.865787i	$0.333182\pi$
$264$	2326.33	0.542331
$265$	1111.79	0.257724
$266$	12369.9	2.85131
$267$	−4297.67	−0.985068
$268$	17423.3	3.97127
$269$	−4807.45	−1.08965	−0.544824	−	0.838551i	$-0.683403\pi$
$269$	−4807.45	−1.08965	−0.544824	+	0.838551i	$0.683403\pi$
$270$	−292.575	−0.0659465
$271$	−1675.23	−0.375509	−0.187754	−	0.982216i	$-0.560121\pi$
$271$	−1675.23	−0.375509	−0.187754	+	0.982216i	$0.560121\pi$
$272$	−11260.3	−2.51014
$273$	−5581.76	−1.23745
$274$	5035.34	1.11020
$275$	1330.57	0.291770
$276$	−10940.5	−2.38601
$277$	−1250.55	−0.271257	−0.135628	−	0.990760i	$-0.543305\pi$
$277$	−1250.55	−0.271257	−0.135628	+	0.990760i	$0.543305\pi$
$278$	−17297.9	−3.73187
$279$	1526.10	0.327474
$280$	2971.70	0.634261
$281$	355.753	0.0755247	0.0377623	−	0.999287i	$-0.487977\pi$
$281$	355.753	0.0755247	0.0377623	+	0.999287i	$0.487977\pi$
$282$	6492.07	1.37091
$283$	−3441.23	−0.722825	−0.361413	−	0.932406i	$-0.617705\pi$
$283$	−3441.23	−0.722825	−0.361413	+	0.932406i	$0.617705\pi$
$284$	19572.2	4.08942
$285$	659.375	0.137046
$286$	5261.00	1.08773
$287$	−7235.86	−1.48822
$288$	−5188.96	−1.06168
$289$	−2078.96	−0.423154
$290$	−1298.23	−0.262879
$291$	5.25927	0.00105946
$292$	18620.5	3.73179
$293$	3595.59	0.716916	0.358458	−	0.933546i	$-0.383303\pi$
$293$	3595.59	0.716916	0.358458	+	0.933546i	$0.383303\pi$
$294$	−1569.07	−0.311258
$295$	−1124.69	−0.221974
$296$	29013.5	5.69721
$297$	−297.000	−0.0580259
$298$	−648.655	−0.126092
$299$	−15349.5	−2.96884
$300$	−7647.38	−1.47174
$301$	4910.36	0.940294
$302$	−5081.29	−0.968197
$303$	1280.75	0.242830
$304$	23133.2	4.36442
$305$	−122.589	−0.0230145
$306$	2583.43	0.482631
$307$	404.668	0.0752300	0.0376150	−	0.999292i	$-0.488024\pi$
$307$	404.668	0.0752300	0.0376150	+	0.999292i	$0.488024\pi$
$308$	4862.54	0.899575
$309$	698.776	0.128647
$310$	−1837.44	−0.336644
$311$	3008.06	0.548461	0.274231	−	0.961664i	$-0.411577\pi$
$311$	3008.06	0.548461	0.274231	+	0.961664i	$0.411577\pi$
$312$	−18758.7	−3.40385
$313$	6377.80	1.15174	0.575869	−	0.817542i	$-0.304664\pi$
$313$	6377.80	1.15174	0.575869	+	0.817542i	$0.304664\pi$
$314$	5929.34	1.06564
$315$	−379.394	−0.0678617
$316$	−7898.73	−1.40613
$317$	−1323.63	−0.234519	−0.117260	−	0.993101i	$-0.537411\pi$
$317$	−1323.63	−0.234519	−0.117260	+	0.993101i	$0.537411\pi$
$318$	−8948.99	−1.57810
$319$	−1317.87	−0.231305
$320$	2846.95	0.497342
$321$	−4829.72	−0.839778
$322$	−19572.5	−3.38738
$323$	−5822.27	−1.00297
$324$	1706.99	0.292693
$325$	−10729.3	−1.83124
$326$	6138.36	1.04286
$327$	−1477.87	−0.249928
$328$	−24317.6	−4.09364
$329$	8418.53	1.41072
$330$	357.592	0.0596509
$331$	1127.88	0.187292	0.0936460	−	0.995606i	$-0.470148\pi$
$331$	1127.88	0.187292	0.0936460	+	0.995606i	$0.470148\pi$
$332$	−24968.2	−4.12744
$333$	−3704.13	−0.609564
$334$	−14084.6	−2.30741
$335$	1661.53	0.270983
$336$	−13310.5	−2.16115
$337$	7799.68	1.26076	0.630379	−	0.776287i	$-0.282899\pi$
$337$	7799.68	1.26076	0.630379	+	0.776287i	$0.282899\pi$
$338$	−30576.6	−4.92055
$339$	2762.99	0.442670
$340$	−2254.60	−0.359627
$341$	−1865.23	−0.296211
$342$	−5307.42	−0.839159
$343$	5160.15	0.812309
$344$	16502.3	2.58646
$345$	−1043.31	−0.162811
$346$	2488.11	0.386595
$347$	5961.94	0.922345	0.461172	−	0.887311i	$-0.347429\pi$
$347$	5961.94	0.922345	0.461172	+	0.887311i	$0.347429\pi$
$348$	7574.34	1.16675
$349$	10558.1	1.61938	0.809689	−	0.586860i	$-0.199636\pi$
$349$	10558.1	1.61938	0.809689	+	0.586860i	$0.199636\pi$
$350$	−13681.2	−2.08940
$351$	2394.90	0.364189
$352$	6342.07	0.960322
$353$	−4614.32	−0.695738	−0.347869	−	0.937543i	$-0.613095\pi$
$353$	−4614.32	−0.695738	−0.347869	+	0.937543i	$0.613095\pi$
$354$	9052.85	1.35919
$355$	1866.45	0.279045
$356$	−30189.5	−4.49450
$357$	3350.04	0.496647
$358$	5445.28	0.803888
$359$	−12848.3	−1.88888	−0.944440	−	0.328685i	$-0.893394\pi$
$359$	−12848.3	−1.88888	−0.944440	+	0.328685i	$0.893394\pi$
$360$	−1275.03	−0.186667
$361$	5102.30	0.743885
$362$	4760.88	0.691232
$363$	363.000	0.0524864
$364$	−39209.8	−5.64603
$365$	1775.70	0.254642
$366$	986.740	0.140923
$367$	−8093.57	−1.15117	−0.575587	−	0.817740i	$-0.695226\pi$
$367$	−8093.57	−1.15117	−0.575587	+	0.817740i	$0.695226\pi$
$368$	−36603.0	−5.18495
$369$	3104.61	0.437993
$370$	4459.82	0.626635
$371$	−11604.5	−1.62393
$372$	10720.3	1.49414
$373$	−9115.40	−1.26536	−0.632678	−	0.774415i	$-0.718044\pi$
$373$	−9115.40	−1.26536	−0.632678	+	0.774415i	$0.718044\pi$
$374$	−3157.53	−0.436556
$375$	−1482.89	−0.204204
$376$	28292.2	3.88048
$377$	10626.8	1.45175
$378$	3053.81	0.415531
$379$	−8380.14	−1.13578	−0.567888	−	0.823106i	$-0.692239\pi$
$379$	−8380.14	−1.13578	−0.567888	+	0.823106i	$0.692239\pi$
$380$	4631.87	0.625289
$381$	−4048.55	−0.544392
$382$	13207.5	1.76899
$383$	−7341.88	−0.979511	−0.489755	−	0.871860i	$-0.662914\pi$
$383$	−7341.88	−0.979511	−0.489755	+	0.871860i	$0.662914\pi$
$384$	−9078.33	−1.20645
$385$	463.704	0.0613832
$386$	3837.75	0.506052
$387$	−2106.83	−0.276734
$388$	36.9444	0.00483394
$389$	7081.77	0.923034	0.461517	−	0.887131i	$-0.347305\pi$
$389$	7081.77	0.923034	0.461517	+	0.887131i	$0.347305\pi$
$390$	−2883.49	−0.374388
$391$	9212.40	1.19154
$392$	−6837.95	−0.881042
$393$	1798.94	0.230902
$394$	163.176	0.0208648
$395$	−753.242	−0.0959487
$396$	−2086.32	−0.264751
$397$	−227.166	−0.0287183	−0.0143591	−	0.999897i	$-0.504571\pi$
$397$	−227.166	−0.0287183	−0.0143591	+	0.999897i	$0.504571\pi$
$398$	−8974.11	−1.13023
$399$	−6882.35	−0.863530
$400$	−25585.5	−3.19818
$401$	−5019.55	−0.625099	−0.312549	−	0.949902i	$-0.601183\pi$
$401$	−5019.55	−0.625099	−0.312549	+	0.949902i	$0.601183\pi$
$402$	−13373.9	−1.65928
$403$	15040.6	1.85912
$404$	8996.82	1.10794
$405$	162.782	0.0199721
$406$	13550.5	1.65641
$407$	4527.27	0.551372
$408$	11258.5	1.36613
$409$	−8664.97	−1.04757	−0.523784	−	0.851851i	$-0.675480\pi$
$409$	−8664.97	−1.04757	−0.523784	+	0.851851i	$0.675480\pi$
$410$	−3737.99	−0.450259
$411$	−2801.55	−0.336230
$412$	4908.64	0.586970
$413$	11739.2	1.39866
$414$	8397.76	0.996926
$415$	−2381.03	−0.281639
$416$	−51140.2	−6.02729
$417$	9624.17	1.13021
$418$	6486.85	0.759048
$419$	8737.78	1.01878	0.509390	−	0.860536i	$-0.329871\pi$
$419$	8737.78	1.01878	0.509390	+	0.860536i	$0.329871\pi$
$420$	−2665.10	−0.309628
$421$	−15078.9	−1.74560	−0.872802	−	0.488075i	$-0.837699\pi$
$421$	−15078.9	−1.74560	−0.872802	+	0.488075i	$0.837699\pi$
$422$	−10050.9	−1.15941
$423$	−3612.04	−0.415185
$424$	−38999.4	−4.46693
$425$	6439.46	0.734964
$426$	−15023.4	−1.70865
$427$	1279.55	0.145015
$428$	−33927.0	−3.83160
$429$	−2927.10	−0.329422
$430$	2536.65	0.284484
$431$	−6845.60	−0.765061	−0.382530	−	0.923943i	$-0.624947\pi$
$431$	−6845.60	−0.765061	−0.382530	+	0.923943i	$0.624947\pi$
$432$	5710.98	0.636041
$433$	8920.64	0.990067	0.495033	−	0.868874i	$-0.335156\pi$
$433$	8920.64	0.990067	0.495033	+	0.868874i	$0.335156\pi$
$434$	19178.6	2.12121
$435$	722.308	0.0796138
$436$	−10381.5	−1.14033
$437$	−18926.0	−2.07175
$438$	−14292.9	−1.55922
$439$	−2418.09	−0.262891	−0.131445	−	0.991323i	$-0.541962\pi$
$439$	−2418.09	−0.262891	−0.131445	+	0.991323i	$0.541962\pi$
$440$	1558.37	0.168847
$441$	872.995	0.0942657
$442$	25461.2	2.73997
$443$	1874.71	0.201061	0.100531	−	0.994934i	$-0.467946\pi$
$443$	1874.71	0.201061	0.100531	+	0.994934i	$0.467946\pi$
$444$	−26020.1	−2.78122
$445$	−2878.95	−0.306686
$446$	18732.9	1.98885
$447$	360.897	0.0381876
$448$	−29715.5	−3.13377
$449$	15828.9	1.66372	0.831860	−	0.554985i	$-0.187276\pi$
$449$	15828.9	1.66372	0.831860	+	0.554985i	$0.187276\pi$
$450$	5870.03	0.614924
$451$	−3794.52	−0.396179
$452$	19409.0	2.01974
$453$	2827.12	0.293222
$454$	13001.7	1.34405
$455$	−3739.14	−0.385261
$456$	−23129.5	−2.37531
$457$	−15463.7	−1.58285	−0.791424	−	0.611267i	$-0.790660\pi$
$457$	−15463.7	−1.58285	−0.791424	+	0.611267i	$0.790660\pi$
$458$	19606.6	2.00034
$459$	−1437.36	−0.146167
$460$	−7328.86	−0.742848
$461$	6999.78	0.707185	0.353593	−	0.935400i	$-0.384960\pi$
$461$	6999.78	0.707185	0.353593	+	0.935400i	$0.384960\pi$
$462$	−3732.43	−0.375862
$463$	−7154.34	−0.718121	−0.359061	−	0.933314i	$-0.616903\pi$
$463$	−7154.34	−0.718121	−0.359061	+	0.933314i	$0.616903\pi$
$464$	25341.1	2.53542
$465$	1022.31	0.101954
$466$	−11094.9	−1.10292
$467$	−6151.29	−0.609524	−0.304762	−	0.952429i	$-0.598577\pi$
$467$	−6151.29	−0.609524	−0.304762	+	0.952429i	$0.598577\pi$
$468$	16823.3	1.66166
$469$	−17342.5	−1.70747
$470$	4348.95	0.426813
$471$	−3298.95	−0.322734
$472$	39452.0	3.84730
$473$	2575.02	0.250316
$474$	6062.97	0.587514
$475$	−13229.3	−1.27790
$476$	23532.8	2.26602
$477$	4979.02	0.477932
$478$	38061.1	3.64200
$479$	−8450.93	−0.806123	−0.403061	−	0.915173i	$-0.632054\pi$
$479$	−8450.93	−0.806123	−0.403061	+	0.915173i	$0.632054\pi$
$480$	−3476.01	−0.330537
$481$	−36506.3	−3.46059
$482$	−8837.04	−0.835096
$483$	10889.7	1.02588
$484$	2549.94	0.239476
$485$	3.52311	0.000329848 0
$486$	−1310.26	−0.122294
$487$	2111.59	0.196479	0.0982396	−	0.995163i	$-0.468679\pi$
$487$	2111.59	0.196479	0.0982396	+	0.995163i	$0.468679\pi$
$488$	4300.18	0.398893
$489$	−3415.24	−0.315834
$490$	−1051.10	−0.0969056
$491$	−15128.2	−1.39048	−0.695239	−	0.718779i	$-0.744701\pi$
$491$	−15128.2	−1.39048	−0.695239	+	0.718779i	$0.744701\pi$
$492$	21808.7	1.99840
$493$	−6377.97	−0.582655
$494$	−52307.6	−4.76403
$495$	−198.956	−0.0180655
$496$	35866.4	3.24687
$497$	−19481.4	−1.75827
$498$	19165.3	1.72453
$499$	−12627.3	−1.13282	−0.566409	−	0.824124i	$-0.691668\pi$
$499$	−12627.3	−1.13282	−0.566409	+	0.824124i	$0.691668\pi$
$500$	−10416.8	−0.931706
$501$	7836.36	0.698808
$502$	19113.6	1.69936
$503$	−7845.00	−0.695410	−0.347705	−	0.937604i	$-0.613039\pi$
$503$	−7845.00	−0.695410	−0.347705	+	0.937604i	$0.613039\pi$
$504$	13308.4	1.17620
$505$	857.959	0.0756013
$506$	−10263.9	−0.901754
$507$	17012.1	1.49021
$508$	−28439.5	−2.48386
$509$	10666.3	0.928831	0.464415	−	0.885617i	$-0.346264\pi$
$509$	10666.3	0.928831	0.464415	+	0.885617i	$0.346264\pi$
$510$	1730.61	0.150260
$511$	−18534.1	−1.60451
$512$	−2663.76	−0.229927
$513$	2952.93	0.254142
$514$	23143.4	1.98601
$515$	468.100	0.0400524
$516$	−14799.7	−1.26264
$517$	4414.72	0.375549
$518$	−46550.1	−3.94845
$519$	−1384.33	−0.117082
$520$	−12566.2	−1.05974
$521$	−9302.42	−0.782239	−0.391119	−	0.920340i	$-0.627912\pi$
$521$	−9302.42	−0.782239	−0.391119	+	0.920340i	$0.627912\pi$
$522$	−5813.97	−0.487492
$523$	−3281.14	−0.274329	−0.137165	−	0.990548i	$-0.543799\pi$
$523$	−3281.14	−0.274329	−0.137165	+	0.990548i	$0.543799\pi$
$524$	12636.9	1.05352
$525$	7611.91	0.632783
$526$	−23016.7	−1.90794
$527$	−9027.00	−0.746152
$528$	−6980.09	−0.575321
$529$	17779.0	1.46125
$530$	−5994.80	−0.491316
$531$	−5036.80	−0.411636
$532$	−48346.0	−3.93997
$533$	30597.7	2.48655
$534$	23173.1	1.87790
$535$	−3235.36	−0.261452
$536$	−58283.2	−4.69674
$537$	−3029.63	−0.243460
$538$	25921.9	2.07727
$539$	−1066.99	−0.0852665
$540$	1143.49	0.0911255
$541$	−20352.6	−1.61742	−0.808711	−	0.588206i	$-0.799835\pi$
$541$	−20352.6	−1.61742	−0.808711	+	0.588206i	$0.799835\pi$
$542$	9032.87	0.715858
$543$	−2648.84	−0.209342
$544$	30693.1	2.41904
$545$	−990.007	−0.0778114
$546$	30097.0	2.35903
$547$	8509.84	0.665182	0.332591	−	0.943071i	$-0.392077\pi$
$547$	8509.84	0.665182	0.332591	+	0.943071i	$0.392077\pi$
$548$	−19679.9	−1.53409
$549$	−549.000	−0.0426790
$550$	−7174.48	−0.556220
$551$	13102.9	1.01307
$552$	36597.2	2.82188
$553$	7862.10	0.604576
$554$	6742.97	0.517115
$555$	−2481.34	−0.189779
$556$	67606.3	5.15673
$557$	7405.91	0.563372	0.281686	−	0.959507i	$-0.409106\pi$
$557$	7405.91	0.563372	0.281686	+	0.959507i	$0.409106\pi$
$558$	−8228.76	−0.624285
$559$	−20764.0	−1.57106
$560$	−8916.52	−0.672842
$561$	1756.78	0.132213
$562$	−1918.23	−0.143978
$563$	−8964.55	−0.671067	−0.335533	−	0.942028i	$-0.608917\pi$
$563$	−8964.55	−0.671067	−0.335533	+	0.942028i	$0.608917\pi$
$564$	−25373.3	−1.89434
$565$	1850.89	0.137819
$566$	18555.2	1.37797
$567$	−1699.07	−0.125845
$568$	−65471.4	−4.83647
$569$	−18895.1	−1.39213	−0.696065	−	0.717978i	$-0.745068\pi$
$569$	−18895.1	−1.39213	−0.696065	+	0.717978i	$0.745068\pi$
$570$	−3555.37	−0.261259
$571$	5508.47	0.403717	0.201858	−	0.979415i	$-0.435302\pi$
$571$	5508.47	0.403717	0.201858	+	0.979415i	$0.435302\pi$
$572$	−20561.8	−1.50303
$573$	−7348.35	−0.535745
$574$	39015.9	2.83710
$575$	20932.2	1.51815
$576$	12749.7	0.922288
$577$	−5928.87	−0.427768	−0.213884	−	0.976859i	$-0.568611\pi$
$577$	−5928.87	−0.427768	−0.213884	+	0.976859i	$0.568611\pi$
$578$	11209.8	0.806687
$579$	−2135.24	−0.153260
$580$	5073.95	0.363249
$581$	24852.4	1.77462
$582$	−28.3581	−0.00201973
$583$	−6085.46	−0.432306
$584$	−62287.8	−4.41351
$585$	1604.31	0.113385
$586$	−19387.5	−1.36671
$587$	−18476.8	−1.29918	−0.649592	−	0.760283i	$-0.725060\pi$
$587$	−18476.8	−1.29918	−0.649592	+	0.760283i	$0.725060\pi$
$588$	6132.47	0.430100
$589$	18545.1	1.29735
$590$	6064.38	0.423163
$591$	−90.7877	−0.00631896
$592$	−87054.4	−6.04377
$593$	24632.6	1.70580	0.852902	−	0.522071i	$-0.174840\pi$
$593$	24632.6	1.70580	0.852902	+	0.522071i	$0.174840\pi$
$594$	1601.43	0.110619
$595$	2244.15	0.154624
$596$	2535.17	0.174236
$597$	4992.99	0.342294
$598$	82764.7	5.65970
$599$	4593.53	0.313333	0.156667	−	0.987652i	$-0.449925\pi$
$599$	4593.53	0.313333	0.156667	+	0.987652i	$0.449925\pi$
$600$	25581.4	1.74059
$601$	−9600.70	−0.651615	−0.325808	−	0.945436i	$-0.605636\pi$
$601$	−9600.70	−0.651615	−0.325808	+	0.945436i	$0.605636\pi$
$602$	−26476.8	−1.79255
$603$	7440.96	0.502520
$604$	19859.4	1.33786
$605$	243.169	0.0163408
$606$	−6905.85	−0.462922
$607$	−10313.2	−0.689624	−0.344812	−	0.938672i	$-0.612057\pi$
$607$	−10313.2	−0.689624	−0.344812	+	0.938672i	$0.612057\pi$
$608$	−63056.1	−4.20603
$609$	−7539.22	−0.501649
$610$	661.003	0.0438742
$611$	−35598.7	−2.35707
$612$	−10097.0	−0.666904
$613$	−9020.81	−0.594367	−0.297184	−	0.954820i	$-0.596047\pi$
$613$	−9020.81	−0.594367	−0.297184	+	0.954820i	$0.596047\pi$
$614$	−2181.98	−0.143416
$615$	2079.73	0.136362
$616$	−16265.8	−1.06391
$617$	−13529.2	−0.882764	−0.441382	−	0.897319i	$-0.645512\pi$
$617$	−13529.2	−0.882764	−0.441382	+	0.897319i	$0.645512\pi$
$618$	−3767.82	−0.245249
$619$	−19997.0	−1.29846	−0.649231	−	0.760591i	$-0.724909\pi$
$619$	−19997.0	−1.29846	−0.649231	+	0.760591i	$0.724909\pi$
$620$	7181.37	0.465178
$621$	−4672.33	−0.301923
$622$	−16219.5	−1.04557
$623$	30049.5	1.93244
$624$	56285.0	3.61090
$625$	14126.8	0.904115
$626$	−34389.2	−2.19564
$627$	−3609.14	−0.229880
$628$	−23173.9	−1.47252
$629$	21910.2	1.38890
$630$	2045.70	0.129369
$631$	−22948.4	−1.44780	−0.723899	−	0.689906i	$-0.757652\pi$
$631$	−22948.4	−1.44780	−0.723899	+	0.689906i	$0.757652\pi$
$632$	26422.2	1.66301
$633$	5592.12	0.351132
$634$	7137.05	0.447080
$635$	−2712.06	−0.169488
$636$	34975.8	2.18063
$637$	8603.86	0.535160
$638$	7105.97	0.440953
$639$	8358.67	0.517471
$640$	−6081.45	−0.375610
$641$	−21232.0	−1.30829	−0.654145	−	0.756369i	$-0.726972\pi$
$641$	−21232.0	−1.30829	−0.654145	+	0.756369i	$0.726972\pi$
$642$	26042.0	1.60093
$643$	−13355.5	−0.819114	−0.409557	−	0.912285i	$-0.634317\pi$
$643$	−13355.5	−0.819114	−0.409557	+	0.912285i	$0.634317\pi$
$644$	76496.3	4.68071
$645$	−1411.34	−0.0861571
$646$	31393.8	1.91203
$647$	−21869.7	−1.32888	−0.664440	−	0.747341i	$-0.731330\pi$
$647$	−21869.7	−1.32888	−0.664440	+	0.747341i	$0.731330\pi$
$648$	−5710.07	−0.346162
$649$	6156.09	0.372339
$650$	57852.5	3.49102
$651$	−10670.6	−0.642415
$652$	−23990.8	−1.44103
$653$	22137.6	1.32666	0.663331	−	0.748326i	$-0.269142\pi$
$653$	22137.6	1.32666	0.663331	+	0.748326i	$0.269142\pi$
$654$	7968.73	0.476455
$655$	1205.09	0.0718880
$656$	72964.4	4.34266
$657$	7952.23	0.472216
$658$	−45392.9	−2.68936
$659$	−388.026	−0.0229368	−0.0114684	−	0.999934i	$-0.503651\pi$
$659$	−388.026	−0.0229368	−0.0114684	+	0.999934i	$0.503651\pi$
$660$	−1397.59	−0.0824262
$661$	−4478.36	−0.263522	−0.131761	−	0.991282i	$-0.542063\pi$
$661$	−4478.36	−0.263522	−0.131761	+	0.991282i	$0.542063\pi$
$662$	−6081.53	−0.357047
$663$	−14166.0	−0.829809
$664$	83521.7	4.88143
$665$	−4610.39	−0.268847
$666$	19972.7	1.16205
$667$	−20732.3	−1.20354
$668$	55047.5	3.18840
$669$	−10422.6	−0.602331
$670$	−8959.02	−0.516593
$671$	671.000	0.0386046
$672$	36281.5	2.08272
$673$	8251.95	0.472644	0.236322	−	0.971675i	$-0.424058\pi$
$673$	8251.95	0.472644	0.236322	+	0.971675i	$0.424058\pi$
$674$	−42056.0	−2.40347
$675$	−3265.95	−0.186232
$676$	119504.	6.79926
$677$	3595.77	0.204131	0.102066	−	0.994778i	$-0.467455\pi$
$677$	3595.77	0.204131	0.102066	+	0.994778i	$0.467455\pi$
$678$	−14898.1	−0.843891
$679$	−36.7731	−0.00207838
$680$	7541.92	0.425323
$681$	−7233.83	−0.407050
$682$	10057.4	0.564687
$683$	−15929.9	−0.892449	−0.446224	−	0.894921i	$-0.647232\pi$
$683$	−15929.9	−0.892449	−0.446224	+	0.894921i	$0.647232\pi$
$684$	20743.2	1.15956
$685$	−1876.72	−0.104680
$686$	−27823.6	−1.54856
$687$	−10908.7	−0.605811
$688$	−49514.7	−2.74380
$689$	49071.0	2.71329
$690$	5625.54	0.310378
$691$	21013.9	1.15688	0.578442	−	0.815724i	$-0.303661\pi$
$691$	21013.9	1.15688	0.578442	+	0.815724i	$0.303661\pi$
$692$	−9724.41	−0.534200
$693$	2076.64	0.113831
$694$	−32146.9	−1.75833
$695$	6447.10	0.351874
$696$	−25337.1	−1.37989
$697$	−18364.0	−0.997972
$698$	−56929.5	−3.08713
$699$	6172.97	0.334025
$700$	53470.9	2.88716
$701$	1972.79	0.106293	0.0531465	−	0.998587i	$-0.483075\pi$
$701$	1972.79	0.106293	0.0531465	+	0.998587i	$0.483075\pi$
$702$	−12913.4	−0.694279
$703$	−45012.4	−2.41490
$704$	−15583.0	−0.834241
$705$	−2419.66	−0.129262
$706$	24880.5	1.32633
$707$	−8955.10	−0.476367
$708$	−35381.7	−1.87814
$709$	12361.7	0.654802	0.327401	−	0.944886i	$-0.393827\pi$
$709$	12361.7	0.654802	0.327401	+	0.944886i	$0.393827\pi$
$710$	−10064.0	−0.531963
$711$	−3373.30	−0.177931
$712$	100988.	5.31555
$713$	−29343.3	−1.54126
$714$	−18063.5	−0.946793
$715$	−1960.83	−0.102560
$716$	−21282.0	−1.11082
$717$	−21176.3	−1.10299
$718$	69278.3	3.60090
$719$	29008.2	1.50462	0.752311	−	0.658809i	$-0.228939\pi$
$719$	29008.2	1.50462	0.752311	+	0.658809i	$0.228939\pi$
$720$	3825.71	0.198022
$721$	−4885.88	−0.252371
$722$	−27511.7	−1.41812
$723$	4916.73	0.252912
$724$	−18607.2	−0.955151
$725$	−14491.9	−0.742366
$726$	−1957.30	−0.100058
$727$	27071.1	1.38103	0.690517	−	0.723316i	$-0.257383\pi$
$727$	27071.1	1.38103	0.690517	+	0.723316i	$0.257383\pi$
$728$	131162.	6.67743
$729$	729.000	0.0370370
$730$	−9574.59	−0.485440
$731$	12462.1	0.630543
$732$	−3856.52	−0.194728
$733$	5923.93	0.298506	0.149253	−	0.988799i	$-0.452313\pi$
$733$	5923.93	0.298506	0.149253	+	0.988799i	$0.452313\pi$
$734$	43640.7	2.19456
$735$	584.807	0.0293482
$736$	99771.7	4.99679
$737$	−9094.51	−0.454546
$738$	−16740.1	−0.834975
$739$	−1518.05	−0.0755648	−0.0377824	−	0.999286i	$-0.512029\pi$
$739$	−1518.05	−0.0755648	−0.0377824	+	0.999286i	$0.512029\pi$
$740$	−17430.5	−0.865890
$741$	29102.8	1.44280
$742$	62571.8	3.09580
$743$	36636.3	1.80896	0.904479	−	0.426518i	$-0.140260\pi$
$743$	36636.3	1.80896	0.904479	+	0.426518i	$0.140260\pi$
$744$	−35860.6	−1.76709
$745$	241.760	0.0118891
$746$	49150.4	2.41223
$747$	−10663.1	−0.522281
$748$	12340.7	0.603238
$749$	33769.7	1.64742
$750$	7995.80	0.389287
$751$	−5736.32	−0.278723	−0.139362	−	0.990242i	$-0.544505\pi$
$751$	−5736.32	−0.278723	−0.139362	+	0.990242i	$0.544505\pi$
$752$	−84890.2	−4.11652
$753$	−10634.4	−0.514658
$754$	−57300.0	−2.76756
$755$	1893.85	0.0912902
$756$	−11935.3	−0.574185
$757$	9071.93	0.435568	0.217784	−	0.975997i	$-0.430117\pi$
$757$	9071.93	0.435568	0.217784	+	0.975997i	$0.430117\pi$
$758$	45185.9	2.16521
$759$	5710.62	0.273099
$760$	−15494.2	−0.739516
$761$	−34947.0	−1.66469	−0.832343	−	0.554261i	$-0.813001\pi$
$761$	−34947.0	−1.66469	−0.832343	+	0.554261i	$0.813001\pi$
$762$	21829.8	1.03781
$763$	10333.4	0.490293
$764$	−51619.4	−2.44441
$765$	−962.871	−0.0455067
$766$	39587.6	1.86731
$767$	−49640.5	−2.33692
$768$	14951.3	0.702487
$769$	−17461.9	−0.818845	−0.409423	−	0.912345i	$-0.634270\pi$
$769$	−17461.9	−0.818845	−0.409423	+	0.912345i	$0.634270\pi$
$770$	−2500.30	−0.117019
$771$	−12876.4	−0.601471
$772$	−14999.3	−0.699268
$773$	−5265.68	−0.245011	−0.122505	−	0.992468i	$-0.539093\pi$
$773$	−5265.68	−0.245011	−0.122505	+	0.992468i	$0.539093\pi$
$774$	11360.1	0.527558
$775$	−20511.0	−0.950679
$776$	−123.584	−0.00571700
$777$	25899.5	1.19580
$778$	−38185.1	−1.75964
$779$	37727.1	1.73519
$780$	11269.7	0.517333
$781$	−10216.2	−0.468070
$782$	−49673.5	−2.27151
$783$	3234.76	0.147639
$784$	20517.1	0.934635
$785$	−2209.92	−0.100478
$786$	−9699.94	−0.440185
$787$	−467.831	−0.0211898	−0.0105949	−	0.999944i	$-0.503373\pi$
$787$	−467.831	−0.0211898	−0.0105949	+	0.999944i	$0.503373\pi$
$788$	−637.750	−0.0288311
$789$	12806.0	0.577826
$790$	4061.50	0.182913
$791$	−19319.0	−0.868399
$792$	6978.98	0.313115
$793$	−5410.71	−0.242295
$794$	1224.89	0.0547475
$795$	3335.37	0.148797
$796$	35073.9	1.56176
$797$	18844.2	0.837509	0.418755	−	0.908099i	$-0.362467\pi$
$797$	18844.2	0.837509	0.418755	+	0.908099i	$0.362467\pi$
$798$	37109.8	1.64620
$799$	21365.5	0.946005
$800$	69740.4	3.08212
$801$	−12893.0	−0.568729
$802$	27065.5	1.19167
$803$	−9719.39	−0.427136
$804$	52270.0	2.29281
$805$	7294.87	0.319392
$806$	−81099.0	−3.54416
$807$	−14422.3	−0.629108
$808$	−30095.5	−1.31034
$809$	9756.63	0.424011	0.212006	−	0.977268i	$-0.432001\pi$
$809$	9756.63	0.424011	0.212006	+	0.977268i	$0.432001\pi$
$810$	−877.726	−0.0380742
$811$	−44361.8	−1.92078	−0.960391	−	0.278656i	$-0.910111\pi$
$811$	−44361.8	−1.92078	−0.960391	+	0.278656i	$0.910111\pi$
$812$	−52960.2	−2.28884
$813$	−5025.69	−0.216800
$814$	−24411.1	−1.05112
$815$	−2287.82	−0.0983300
$816$	−33780.9	−1.44923
$817$	−25602.2	−1.09634
$818$	46721.7	1.99705
$819$	−16745.3	−0.714442
$820$	14609.4	0.622172
$821$	−14739.8	−0.626583	−0.313291	−	0.949657i	$-0.601432\pi$
$821$	−14739.8	−0.626583	−0.313291	+	0.949657i	$0.601432\pi$
$822$	15106.0	0.640977
$823$	16385.4	0.693995	0.346997	−	0.937866i	$-0.387201\pi$
$823$	16385.4	0.693995	0.346997	+	0.937866i	$0.387201\pi$
$824$	−16420.0	−0.694197
$825$	3991.72	0.168453
$826$	−63298.0	−2.66637
$827$	−16870.6	−0.709367	−0.354684	−	0.934986i	$-0.615411\pi$
$827$	−16870.6	−0.709367	−0.354684	+	0.934986i	$0.615411\pi$
$828$	−32821.4	−1.37756
$829$	−23186.2	−0.971400	−0.485700	−	0.874126i	$-0.661435\pi$
$829$	−23186.2	−0.971400	−0.485700	+	0.874126i	$0.661435\pi$
$830$	12838.6	0.536908
$831$	−3751.64	−0.156610
$832$	125656.	5.23596
$833$	−5163.83	−0.214785
$834$	−51893.7	−2.15460
$835$	5249.47	0.217563
$836$	−25352.8	−1.04886
$837$	4578.30	0.189067
$838$	−47114.3	−1.94217
$839$	−26015.0	−1.07049	−0.535243	−	0.844698i	$-0.679780\pi$
$839$	−26015.0	−1.07049	−0.535243	+	0.844698i	$0.679780\pi$
$840$	8915.10	0.366191
$841$	−10035.5	−0.411477
$842$	81305.6	3.32776
$843$	1067.26	0.0436042
$844$	39282.6	1.60209
$845$	11396.2	0.463953
$846$	19476.2	0.791496
$847$	−2538.12	−0.102964
$848$	117017.	4.73865
$849$	−10323.7	−0.417323
$850$	−34721.7	−1.40111
$851$	71221.8	2.86892
$852$	58716.6	2.36103
$853$	−5439.61	−0.218346	−0.109173	−	0.994023i	$-0.534820\pi$
$853$	−5439.61	−0.218346	−0.109173	+	0.994023i	$0.534820\pi$
$854$	−6899.34	−0.276453
$855$	1978.13	0.0791234
$856$	113490.	4.53155
$857$	−41519.8	−1.65495	−0.827474	−	0.561504i	$-0.810223\pi$
$857$	−41519.8	−1.65495	−0.827474	+	0.561504i	$0.810223\pi$
$858$	15783.0	0.627999
$859$	−46921.2	−1.86371	−0.931857	−	0.362825i	$-0.881812\pi$
$859$	−46921.2	−1.86371	−0.931857	+	0.362825i	$0.881812\pi$
$860$	−9914.12	−0.393103
$861$	−21707.6	−0.859224
$862$	36911.6	1.45849
$863$	44995.9	1.77483	0.887416	−	0.460970i	$-0.152498\pi$
$863$	44995.9	1.77483	0.887416	+	0.460970i	$0.152498\pi$
$864$	−15566.9	−0.612959
$865$	−927.343	−0.0364516
$866$	−48100.3	−1.88743
$867$	−6236.87	−0.244308
$868$	−74956.8	−2.93110
$869$	4122.92	0.160944
$870$	−3894.70	−0.151773
$871$	73334.9	2.85288
$872$	34727.4	1.34865
$873$	15.7778	0.000611682 0
$874$	102049.	3.94951
$875$	10368.5	0.400593
$876$	55861.5	2.15455
$877$	−49948.4	−1.92319	−0.961595	−	0.274471i	$-0.911497\pi$
$877$	−49948.4	−1.92319	−0.961595	+	0.274471i	$0.911497\pi$
$878$	13038.4	0.501166
$879$	10786.8	0.413912
$880$	−4675.87	−0.179118
$881$	23517.3	0.899339	0.449669	−	0.893195i	$-0.351542\pi$
$881$	23517.3	0.899339	0.449669	+	0.893195i	$0.351542\pi$
$882$	−4707.21	−0.179705
$883$	33238.6	1.26678	0.633391	−	0.773832i	$-0.281662\pi$
$883$	33238.6	1.26678	0.633391	+	0.773832i	$0.281662\pi$
$884$	−99511.2	−3.78611
$885$	−3374.08	−0.128157
$886$	−10108.5	−0.383297
$887$	15713.4	0.594820	0.297410	−	0.954750i	$-0.403877\pi$
$887$	15713.4	0.594820	0.297410	+	0.954750i	$0.403877\pi$
$888$	87040.5	3.28929
$889$	28307.6	1.06795
$890$	15523.4	0.584656
$891$	−891.000	−0.0335013
$892$	−73214.7	−2.74822
$893$	−43893.4	−1.64483
$894$	−1945.96	−0.0727995
$895$	−2029.51	−0.0757977
$896$	63476.2	2.36673
$897$	−46048.4	−1.71406
$898$	−85349.6	−3.17166
$899$	20315.1	0.753667
$900$	−22942.1	−0.849708
$901$	−29451.3	−1.08897
$902$	20460.1	0.755264
$903$	14731.1	0.542879
$904$	−64925.4	−2.38870
$905$	−1774.42	−0.0651755
$906$	−15243.9	−0.558989
$907$	6654.09	0.243600	0.121800	−	0.992555i	$-0.461133\pi$
$907$	6654.09	0.243600	0.121800	+	0.992555i	$0.461133\pi$
$908$	−50815.0	−1.85722
$909$	3842.26	0.140198
$910$	20161.5	0.734449
$911$	−43953.1	−1.59850	−0.799249	−	0.601000i	$-0.794769\pi$
$911$	−43953.1	−1.59850	−0.799249	+	0.601000i	$0.794769\pi$
$912$	69399.7	2.51980
$913$	13032.7	0.472421
$914$	83380.7	3.01749
$915$	−367.767	−0.0132874
$916$	−76629.5	−2.76409
$917$	−12578.3	−0.452969
$918$	7750.30	0.278647
$919$	−40401.8	−1.45020	−0.725100	−	0.688644i	$-0.758206\pi$
$919$	−40401.8	−1.45020	−0.725100	+	0.688644i	$0.758206\pi$
$920$	24515.9	0.878550
$921$	1214.00	0.0434341
$922$	−37743.0	−1.34815
$923$	82379.4	2.93776
$924$	14587.6	0.519370
$925$	49784.0	1.76961
$926$	38576.3	1.36900
$927$	2096.33	0.0742745
$928$	−69074.4	−2.44340
$929$	18115.4	0.639770	0.319885	−	0.947456i	$-0.396356\pi$
$929$	18115.4	0.639770	0.319885	+	0.947456i	$0.396356\pi$
$930$	−5512.33	−0.194362
$931$	10608.6	0.373451
$932$	43362.9	1.52403
$933$	9024.18	0.316654
$934$	33167.9	1.16198
$935$	1176.84	0.0411624
$936$	−56276.0	−1.96521
$937$	−13058.4	−0.455282	−0.227641	−	0.973745i	$-0.573101\pi$
$937$	−13058.4	−0.455282	−0.227641	+	0.973745i	$0.573101\pi$
$938$	93511.4	3.25507
$939$	19133.4	0.664957
$940$	−16997.2	−0.589774
$941$	−15633.7	−0.541599	−0.270799	−	0.962636i	$-0.587288\pi$
$941$	−15633.7	−0.541599	−0.270799	+	0.962636i	$0.587288\pi$
$942$	17788.0	0.615249
$943$	−59694.4	−2.06142
$944$	−118375.	−4.08133
$945$	−1138.18	−0.0391800
$946$	−13884.5	−0.477194
$947$	−19084.4	−0.654867	−0.327434	−	0.944874i	$-0.606184\pi$
$947$	−19084.4	−0.654867	−0.327434	+	0.944874i	$0.606184\pi$
$948$	−23696.2	−0.811832
$949$	78373.7	2.68084
$950$	71332.5	2.43614
$951$	−3970.90	−0.135400
$952$	−78720.2	−2.67997
$953$	9115.33	0.309837	0.154918	−	0.987927i	$-0.450489\pi$
$953$	9115.33	0.309837	0.154918	+	0.987927i	$0.450489\pi$
$954$	−26847.0	−0.911114
$955$	−4922.56	−0.166796
$956$	−148756.	−5.03255
$957$	−3953.60	−0.133544
$958$	45567.6	1.53677
$959$	19588.6	0.659592
$960$	8540.85	0.287140
$961$	−1038.20	−0.0348496
$962$	196843.	6.59715
$963$	−14489.2	−0.484846
$964$	34538.2	1.15394
$965$	−1430.37	−0.0477151
$966$	−58717.6	−1.95570
$967$	−48201.3	−1.60295	−0.801473	−	0.598030i	$-0.795950\pi$
$967$	−48201.3	−1.60295	−0.801473	+	0.598030i	$0.795950\pi$
$968$	−8529.86	−0.283223
$969$	−17466.8	−0.579066
$970$	−18.9967	−0.000628812 0
$971$	51553.8	1.70385	0.851926	−	0.523661i	$-0.175434\pi$
$971$	51553.8	1.70385	0.851926	+	0.523661i	$0.175434\pi$
$972$	5120.96	0.168986
$973$	−67292.7	−2.21717
$974$	−11385.7	−0.374561
$975$	−32187.8	−1.05727
$976$	−12902.6	−0.423158
$977$	−39878.2	−1.30585	−0.652926	−	0.757422i	$-0.726459\pi$
$977$	−39878.2	−1.30585	−0.652926	+	0.757422i	$0.726459\pi$
$978$	18415.1	0.602095
$979$	15758.1	0.514435
$980$	4108.05	0.133905
$981$	−4433.62	−0.144296
$982$	81571.4	2.65076
$983$	−52395.9	−1.70007	−0.850036	−	0.526725i	$-0.823420\pi$
$983$	−52395.9	−1.70007	−0.850036	+	0.526725i	$0.823420\pi$
$984$	−72952.8	−2.36347
$985$	−60.8174	−0.00196731
$986$	34390.1	1.11076
$987$	25255.6	0.814482
$988$	204436.	6.58298
$989$	40509.5	1.30245
$990$	1072.78	0.0344394
$991$	−60333.8	−1.93397	−0.966986	−	0.254830i	$-0.917980\pi$
$991$	−60333.8	−1.93397	−0.966986	+	0.254830i	$0.917980\pi$
$992$	−97763.8	−3.12904
$993$	3383.63	0.108133
$994$	105044.	3.35191
$995$	3344.74	0.106568
$996$	−74904.7	−2.38298
$997$	32960.8	1.04702	0.523509	−	0.852020i	$-0.324622\pi$
$997$	32960.8	1.04702	0.523509	+	0.852020i	$0.324622\pi$
$998$	68086.7	2.15957
$999$	−11112.4	−0.351932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.c.1.2	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.c.1.2	✓	37	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-5.39202 q^{2} +3.00000 q^{3} +21.0739 q^{4} +2.00966 q^{5} -16.1761 q^{6} -20.9762 q^{7} -70.4947 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.39202 q^{2} +3.00000 q^{3} +21.0739 q^{4} +2.00966 q^{5} -16.1761 q^{6} -20.9762 q^{7} -70.4947 q^{8} +9.00000 q^{9} -10.8361 q^{10} -11.0000 q^{11} +63.2217 q^{12} +88.7001 q^{13} +113.104 q^{14} +6.02897 q^{15} +211.518 q^{16} -53.2357 q^{17} -48.5282 q^{18} +109.368 q^{19} +42.3513 q^{20} -62.9285 q^{21} +59.3122 q^{22} -173.049 q^{23} -211.484 q^{24} -120.961 q^{25} -478.273 q^{26} +27.0000 q^{27} -442.049 q^{28} +119.806 q^{29} -32.5084 q^{30} +169.566 q^{31} -576.551 q^{32} -33.0000 q^{33} +287.048 q^{34} -42.1549 q^{35} +189.665 q^{36} -411.570 q^{37} -589.713 q^{38} +266.100 q^{39} -141.670 q^{40} +344.956 q^{41} +339.312 q^{42} -234.092 q^{43} -231.813 q^{44} +18.0869 q^{45} +933.085 q^{46} -401.338 q^{47} +634.554 q^{48} +96.9994 q^{49} +652.226 q^{50} -159.707 q^{51} +1869.26 q^{52} +553.224 q^{53} -145.585 q^{54} -22.1062 q^{55} +1478.71 q^{56} +328.103 q^{57} -645.997 q^{58} -559.645 q^{59} +127.054 q^{60} -61.0000 q^{61} -914.306 q^{62} -188.785 q^{63} +1416.63 q^{64} +178.257 q^{65} +177.937 q^{66} +826.774 q^{67} -1121.88 q^{68} -519.147 q^{69} +227.300 q^{70} +928.741 q^{71} -634.453 q^{72} +883.581 q^{73} +2219.19 q^{74} -362.884 q^{75} +2304.80 q^{76} +230.738 q^{77} -1434.82 q^{78} -374.811 q^{79} +425.079 q^{80} +81.0000 q^{81} -1860.01 q^{82} -1184.79 q^{83} -1326.15 q^{84} -106.986 q^{85} +1262.23 q^{86} +359.418 q^{87} +775.442 q^{88} -1432.56 q^{89} -97.5251 q^{90} -1860.59 q^{91} -3646.82 q^{92} +508.699 q^{93} +2164.02 q^{94} +219.792 q^{95} -1729.65 q^{96} +1.75309 q^{97} -523.023 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9}+O(q^{10}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9	\( 37 q - 8 q^{2} + 111 q^{3} + 130 q^{4} - 35 q^{5} - 24 q^{6} - 35 q^{7} - 117 q^{8} + 333 q^{9} - 41 q^{10} - 407 q^{11} + 390 q^{12} + 51 q^{13} - 228 q^{14} - 105 q^{15} + 462 q^{16} - 190 q^{17} - 72 q^{18} - 51 q^{19} - 720 q^{20} - 105 q^{21} + 88 q^{22} - 583 q^{23} - 351 q^{24} + 598 q^{25} - 1019 q^{26} + 999 q^{27} - 498 q^{28} - 566 q^{29} - 123 q^{30} - 696 q^{31} - 859 q^{32} - 1221 q^{33} - 348 q^{34} - 1102 q^{35} + 1170 q^{36} - 1022 q^{37} - 455 q^{38} + 153 q^{39} - 503 q^{40} - 790 q^{41} - 684 q^{42} - 87 q^{43} - 1430 q^{44} - 315 q^{45} - 303 q^{46} - 1603 q^{47} + 1386 q^{48} + 110 q^{49} - 1926 q^{50} - 570 q^{51} + 736 q^{52} - 2619 q^{53} - 216 q^{54} + 385 q^{55} - 4937 q^{56} - 153 q^{57} - 1099 q^{58} - 2471 q^{59} - 2160 q^{60} - 2257 q^{61} - 2909 q^{62} - 315 q^{63} - 265 q^{64} - 1970 q^{65} + 264 q^{66} - 3033 q^{67} - 1956 q^{68} - 1749 q^{69} + 2410 q^{70} - 3891 q^{71} - 1053 q^{72} + 391 q^{73} - 532 q^{74} + 1794 q^{75} + 1554 q^{76} + 385 q^{77} - 3057 q^{78} + 67 q^{79} - 5111 q^{80} + 2997 q^{81} - 4818 q^{82} - 5315 q^{83} - 1494 q^{84} - 2747 q^{85} - 5195 q^{86} - 1698 q^{87} + 1287 q^{88} - 8945 q^{89} - 369 q^{90} - 4432 q^{91} - 4701 q^{92} - 2088 q^{93} - 372 q^{94} - 3388 q^{95} - 2577 q^{96} - 3784 q^{97} - 4502 q^{98} - 3663 q^{99}+O(q^{100}) \) 37 * q - 8 * q^2 + 111 * q^3 + 130 * q^4 - 35 * q^5 - 24 * q^6 - 35 * q^7 - 117 * q^8 + 333 * q^9 - 41 * q^10 - 407 * q^11 + 390 * q^12 + 51 * q^13 - 228 * q^14 - 105 * q^15 + 462 * q^16 - 190 * q^17 - 72 * q^18 - 51 * q^19 - 720 * q^20 - 105 * q^21 + 88 * q^22 - 583 * q^23 - 351 * q^24 + 598 * q^25 - 1019 * q^26 + 999 * q^27 - 498 * q^28 - 566 * q^29 - 123 * q^30 - 696 * q^31 - 859 * q^32 - 1221 * q^33 - 348 * q^34 - 1102 * q^35 + 1170 * q^36 - 1022 * q^37 - 455 * q^38 + 153 * q^39 - 503 * q^40 - 790 * q^41 - 684 * q^42 - 87 * q^43 - 1430 * q^44 - 315 * q^45 - 303 * q^46 - 1603 * q^47 + 1386 * q^48 + 110 * q^49 - 1926 * q^50 - 570 * q^51 + 736 * q^52 - 2619 * q^53 - 216 * q^54 + 385 * q^55 - 4937 * q^56 - 153 * q^57 - 1099 * q^58 - 2471 * q^59 - 2160 * q^60 - 2257 * q^61 - 2909 * q^62 - 315 * q^63 - 265 * q^64 - 1970 * q^65 + 264 * q^66 - 3033 * q^67 - 1956 * q^68 - 1749 * q^69 + 2410 * q^70 - 3891 * q^71 - 1053 * q^72 + 391 * q^73 - 532 * q^74 + 1794 * q^75 + 1554 * q^76 + 385 * q^77 - 3057 * q^78 + 67 * q^79 - 5111 * q^80 + 2997 * q^81 - 4818 * q^82 - 5315 * q^83 - 1494 * q^84 - 2747 * q^85 - 5195 * q^86 - 1698 * q^87 + 1287 * q^88 - 8945 * q^89 - 369 * q^90 - 4432 * q^91 - 4701 * q^92 - 2088 * q^93 - 372 * q^94 - 3388 * q^95 - 2577 * q^96 - 3784 * q^97 - 4502 * q^98 - 3663 * q^99

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