LMFDB - Embedded newform 538.6.a.b.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,6,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("538.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	538.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:	$86.2864950594$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} -8.18103 q^{3} +16.0000 q^{4} -4.49984 q^{5} +32.7241 q^{6} +129.673 q^{7} -64.0000 q^{8} -176.071 q^{9} +17.9994 q^{10} -320.988 q^{11} -130.896 q^{12} -526.959 q^{13} -518.693 q^{14} +36.8133 q^{15} +256.000 q^{16} -111.314 q^{17} +704.283 q^{18} -559.106 q^{19} -71.9975 q^{20} -1060.86 q^{21} +1283.95 q^{22} +1967.84 q^{23} +523.586 q^{24} -3104.75 q^{25} +2107.83 q^{26} +3428.43 q^{27} +2074.77 q^{28} -558.996 q^{29} -147.253 q^{30} -1366.88 q^{31} -1024.00 q^{32} +2626.01 q^{33} +445.256 q^{34} -583.509 q^{35} -2817.13 q^{36} +8325.86 q^{37} +2236.42 q^{38} +4311.06 q^{39} +287.990 q^{40} +16577.7 q^{41} +4243.44 q^{42} -8136.63 q^{43} -5135.80 q^{44} +792.291 q^{45} -7871.37 q^{46} -24739.9 q^{47} -2094.34 q^{48} +8.16140 q^{49} +12419.0 q^{50} +910.662 q^{51} -8431.34 q^{52} -33249.8 q^{53} -13713.7 q^{54} +1444.39 q^{55} -8299.09 q^{56} +4574.06 q^{57} +2235.99 q^{58} +10297.0 q^{59} +589.013 q^{60} +905.539 q^{61} +5467.51 q^{62} -22831.7 q^{63} +4096.00 q^{64} +2371.23 q^{65} -10504.0 q^{66} -4482.56 q^{67} -1781.02 q^{68} -16099.0 q^{69} +2334.04 q^{70} -66494.9 q^{71} +11268.5 q^{72} +77206.4 q^{73} -33303.4 q^{74} +25400.1 q^{75} -8945.69 q^{76} -41623.5 q^{77} -17244.3 q^{78} -48991.9 q^{79} -1151.96 q^{80} +14737.1 q^{81} -66310.9 q^{82} +18670.4 q^{83} -16973.8 q^{84} +500.895 q^{85} +32546.5 q^{86} +4573.17 q^{87} +20543.2 q^{88} +133418. q^{89} -3169.16 q^{90} -68332.4 q^{91} +31485.5 q^{92} +11182.5 q^{93} +98959.6 q^{94} +2515.89 q^{95} +8377.37 q^{96} -78905.2 q^{97} -32.6456 q^{98} +56516.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9} - 556 q^{10} + 1241 q^{11} + 528 q^{12} - 202 q^{13} + 100 q^{14} + 1786 q^{15} + 6912 q^{16} + 1550 q^{17}+ \cdots + 335378 q^{99}+O(q^{100}) $ 27 * q - 108 * q^2 + 33 * q^3 + 432 * q^4 + 139 * q^5 - 132 * q^6 - 25 * q^7 - 1728 * q^8 + 2346 * q^9 - 556 * q^10 + 1241 * q^11 + 528 * q^12 - 202 * q^13 + 100 * q^14 + 1786 * q^15 + 6912 * q^16 + 1550 * q^17 - 9384 * q^18 - 66 * q^19 + 2224 * q^20 - 1217 * q^21 - 4964 * q^22 + 10568 * q^23 - 2112 * q^24 + 15880 * q^25 + 808 * q^26 + 8256 * q^27 - 400 * q^28 - 2122 * q^29 - 7144 * q^30 - 1621 * q^31 - 27648 * q^32 + 18369 * q^33 - 6200 * q^34 + 23123 * q^35 + 37536 * q^36 + 8542 * q^37 + 264 * q^38 + 589 * q^39 - 8896 * q^40 + 10859 * q^41 + 4868 * q^42 + 23759 * q^43 + 19856 * q^44 + 28006 * q^45 - 42272 * q^46 + 38457 * q^47 + 8448 * q^48 + 57376 * q^49 - 63520 * q^50 + 58237 * q^51 - 3232 * q^52 + 73586 * q^53 - 33024 * q^54 + 25923 * q^55 + 1600 * q^56 + 52534 * q^57 + 8488 * q^58 + 68927 * q^59 + 28576 * q^60 - 76016 * q^61 + 6484 * q^62 - 20738 * q^63 + 110592 * q^64 + 74867 * q^65 - 73476 * q^66 + 94787 * q^67 + 24800 * q^68 - 90601 * q^69 - 92492 * q^70 + 15523 * q^71 - 150144 * q^72 - 96279 * q^73 - 34168 * q^74 - 123786 * q^75 - 1056 * q^76 - 199866 * q^77 - 2356 * q^78 - 246217 * q^79 + 35584 * q^80 - 54473 * q^81 - 43436 * q^82 - 56658 * q^83 - 19472 * q^84 - 120541 * q^85 - 95036 * q^86 - 352954 * q^87 - 79424 * q^88 - 83070 * q^89 - 112024 * q^90 - 132348 * q^91 + 169088 * q^92 + 40727 * q^93 - 153828 * q^94 - 72090 * q^95 - 33792 * q^96 - 36273 * q^97 - 229504 * q^98 + 335378 * q^99

Coefficient data

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

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

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

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−8.18103	−0.524813	−0.262407	−	0.964957i	$-0.584516\pi$
$3$	−8.18103	−0.524813	−0.262407	+	0.964957i	$0.584516\pi$
$4$	16.0000	0.500000
$5$	−4.49984	−0.0804956	−0.0402478	−	0.999190i	$-0.512815\pi$
$5$	−4.49984	−0.0804956	−0.0402478	+	0.999190i	$0.512815\pi$
$6$	32.7241	0.371099
$7$	129.673	1.00024	0.500121	−	0.865955i	$-0.333289\pi$
$7$	129.673	1.00024	0.500121	+	0.865955i	$0.333289\pi$
$8$	−64.0000	−0.353553
$9$	−176.071	−0.724571
$10$	17.9994	0.0569190
$11$	−320.988	−0.799846	−0.399923	−	0.916549i	$-0.630963\pi$
$11$	−320.988	−0.799846	−0.399923	+	0.916549i	$0.630963\pi$
$12$	−130.896	−0.262407
$13$	−526.959	−0.864805	−0.432402	−	0.901681i	$-0.642334\pi$
$13$	−526.959	−0.864805	−0.432402	+	0.901681i	$0.642334\pi$
$14$	−518.693	−0.707278
$15$	36.8133	0.0422452
$16$	256.000	0.250000
$17$	−111.314	−0.0934173	−0.0467086	−	0.998909i	$-0.514873\pi$
$17$	−111.314	−0.0934173	−0.0467086	+	0.998909i	$0.514873\pi$
$18$	704.283	0.512349
$19$	−559.106	−0.355312	−0.177656	−	0.984093i	$-0.556851\pi$
$19$	−559.106	−0.355312	−0.177656	+	0.984093i	$0.556851\pi$
$20$	−71.9975	−0.0402478
$21$	−1060.86	−0.524941
$22$	1283.95	0.565577
$23$	1967.84	0.775659	0.387829	−	0.921731i	$-0.373225\pi$
$23$	1967.84	0.775659	0.387829	+	0.921731i	$0.373225\pi$
$24$	523.586	0.185550
$25$	−3104.75	−0.993520
$26$	2107.83	0.611509
$27$	3428.43	0.905078
$28$	2074.77	0.500121
$29$	−558.996	−0.123428	−0.0617140	−	0.998094i	$-0.519657\pi$
$29$	−558.996	−0.123428	−0.0617140	+	0.998094i	$0.519657\pi$
$30$	−147.253	−0.0298718
$31$	−1366.88	−0.255461	−0.127731	−	0.991809i	$-0.540769\pi$
$31$	−1366.88	−0.255461	−0.127731	+	0.991809i	$0.540769\pi$
$32$	−1024.00	−0.176777
$33$	2626.01	0.419770
$34$	445.256	0.0660560
$35$	−583.509	−0.0805152
$36$	−2817.13	−0.362286
$37$	8325.86	0.999827	0.499913	−	0.866075i	$-0.333365\pi$
$37$	8325.86	0.999827	0.499913	+	0.866075i	$0.333365\pi$
$38$	2236.42	0.251244
$39$	4311.06	0.453861
$40$	287.990	0.0284595
$41$	16577.7	1.54016	0.770079	−	0.637948i	$-0.220217\pi$
$41$	16577.7	1.54016	0.770079	+	0.637948i	$0.220217\pi$
$42$	4243.44	0.371189
$43$	−8136.63	−0.671079	−0.335539	−	0.942026i	$-0.608919\pi$
$43$	−8136.63	−0.671079	−0.335539	+	0.942026i	$0.608919\pi$
$44$	−5135.80	−0.399923
$45$	792.291	0.0583248
$46$	−7871.37	−0.548474
$47$	−24739.9	−1.63363	−0.816814	−	0.576901i	$-0.804262\pi$
$47$	−24739.9	−1.63363	−0.816814	+	0.576901i	$0.804262\pi$
$48$	−2094.34	−0.131203
$49$	8.16140	0.000485596 0
$50$	12419.0	0.702525
$51$	910.662	0.0490266
$52$	−8431.34	−0.432402
$53$	−33249.8	−1.62592	−0.812961	−	0.582318i	$-0.802146\pi$
$53$	−33249.8	−1.62592	−0.812961	+	0.582318i	$0.802146\pi$
$54$	−13713.7	−0.639987
$55$	1444.39	0.0643841
$56$	−8299.09	−0.353639
$57$	4574.06	0.186472
$58$	2235.99	0.0872768
$59$	10297.0	0.385106	0.192553	−	0.981287i	$-0.438323\pi$
$59$	10297.0	0.385106	0.192553	+	0.981287i	$0.438323\pi$
$60$	589.013	0.0211226
$61$	905.539	0.0311589	0.0155795	−	0.999879i	$-0.495041\pi$
$61$	905.539	0.0311589	0.0155795	+	0.999879i	$0.495041\pi$
$62$	5467.51	0.180638
$63$	−22831.7	−0.724747
$64$	4096.00	0.125000
$65$	2371.23	0.0696130
$66$	−10504.0	−0.296822
$67$	−4482.56	−0.121994	−0.0609971	−	0.998138i	$-0.519428\pi$
$67$	−4482.56	−0.121994	−0.0609971	+	0.998138i	$0.519428\pi$
$68$	−1781.02	−0.0467086
$69$	−16099.0	−0.407076
$70$	2334.04	0.0569328
$71$	−66494.9	−1.56546	−0.782731	−	0.622360i	$-0.786174\pi$
$71$	−66494.9	−1.56546	−0.782731	+	0.622360i	$0.786174\pi$
$72$	11268.5	0.256175
$73$	77206.4	1.69569	0.847844	−	0.530246i	$-0.177900\pi$
$73$	77206.4	1.69569	0.847844	+	0.530246i	$0.177900\pi$
$74$	−33303.4	−0.706984
$75$	25400.1	0.521413
$76$	−8945.69	−0.177656
$77$	−41623.5	−0.800040
$78$	−17244.3	−0.320928
$79$	−48991.9	−0.883194	−0.441597	−	0.897213i	$-0.645588\pi$
$79$	−48991.9	−0.883194	−0.441597	+	0.897213i	$0.645588\pi$
$80$	−1151.96	−0.0201239
$81$	14737.1	0.249574
$82$	−66310.9	−1.08906
$83$	18670.4	0.297481	0.148740	−	0.988876i	$-0.452478\pi$
$83$	18670.4	0.297481	0.148740	+	0.988876i	$0.452478\pi$
$84$	−16973.8	−0.262470
$85$	500.895	0.00751968
$86$	32546.5	0.474524
$87$	4573.17	0.0647767
$88$	20543.2	0.282788
$89$	133418.	1.78541	0.892706	−	0.450639i	$-0.148804\pi$
$89$	133418.	1.78541	0.892706	+	0.450639i	$0.148804\pi$
$90$	−3169.16	−0.0412419
$91$	−68332.4	−0.865015
$92$	31485.5	0.387829
$93$	11182.5	0.134069
$94$	98959.6	1.15515
$95$	2515.89	0.0286011
$96$	8377.37	0.0927748
$97$	−78905.2	−0.851484	−0.425742	−	0.904845i	$-0.639987\pi$
$97$	−78905.2	−0.851484	−0.425742	+	0.904845i	$0.639987\pi$
$98$	−32.6456	−0.000343368 0
$99$	56516.5	0.579545
$100$	−49676.0	−0.496760
$101$	178715.	1.74324	0.871619	−	0.490184i	$-0.163071\pi$
$101$	178715.	1.74324	0.871619	+	0.490184i	$0.163071\pi$
$102$	−3642.65	−0.0346671
$103$	−104450.	−0.970098	−0.485049	−	0.874487i	$-0.661198\pi$
$103$	−104450.	−0.970098	−0.485049	+	0.874487i	$0.661198\pi$
$104$	33725.3	0.305755
$105$	4773.71	0.0422554
$106$	132999.	1.14970
$107$	147033.	1.24152	0.620761	−	0.784000i	$-0.286824\pi$
$107$	147033.	1.24152	0.620761	+	0.784000i	$0.286824\pi$
$108$	54854.9	0.452539
$109$	−133770.	−1.07843	−0.539215	−	0.842168i	$-0.681279\pi$
$109$	−133770.	−1.07843	−0.539215	+	0.842168i	$0.681279\pi$
$110$	−5777.57	−0.0455264
$111$	−68114.1	−0.524722
$112$	33196.4	0.250061
$113$	−84037.9	−0.619126	−0.309563	−	0.950879i	$-0.600183\pi$
$113$	−84037.9	−0.619126	−0.309563	+	0.950879i	$0.600183\pi$
$114$	−18296.2	−0.131856
$115$	−8854.98	−0.0624371
$116$	−8943.94	−0.0617140
$117$	92782.0	0.626612
$118$	−41188.0	−0.272311
$119$	−14434.4	−0.0934399
$120$	−2356.05	−0.0149359
$121$	−58018.0	−0.360246
$122$	−3622.16	−0.0220327
$123$	−135623.	−0.808296
$124$	−21870.0	−0.127731
$125$	28032.9	0.160470
$126$	91326.7	0.512473
$127$	230568.	1.26850	0.634249	−	0.773129i	$-0.281309\pi$
$127$	230568.	1.26850	0.634249	+	0.773129i	$0.281309\pi$
$128$	−16384.0	−0.0883883
$129$	66566.0	0.352191
$130$	−9484.92	−0.0492238
$131$	175408.	0.893039	0.446520	−	0.894774i	$-0.352663\pi$
$131$	175408.	0.893039	0.446520	+	0.894774i	$0.352663\pi$
$132$	42016.1	0.209885
$133$	−72501.1	−0.355398
$134$	17930.3	0.0862630
$135$	−15427.4	−0.0728548
$136$	7124.09	0.0330280
$137$	−106195.	−0.483395	−0.241697	−	0.970352i	$-0.577704\pi$
$137$	−106195.	−0.483395	−0.241697	+	0.970352i	$0.577704\pi$
$138$	64395.9	0.287846
$139$	246545.	1.08233	0.541165	−	0.840916i	$-0.317983\pi$
$139$	246545.	1.08233	0.541165	+	0.840916i	$0.317983\pi$
$140$	−9336.15	−0.0402576
$141$	202398.	0.857350
$142$	265980.	1.10695
$143$	169147.	0.691711
$144$	−45074.1	−0.181143
$145$	2515.40	0.00993542
$146$	−308825.	−1.19903
$147$	−66.7687	−0.000254847 0
$148$	133214.	0.499913
$149$	90173.2	0.332745	0.166373	−	0.986063i	$-0.446795\pi$
$149$	90173.2	0.332745	0.166373	+	0.986063i	$0.446795\pi$
$150$	−101600.	−0.368694
$151$	5899.00	0.0210541	0.0105270	−	0.999945i	$-0.496649\pi$
$151$	5899.00	0.0210541	0.0105270	+	0.999945i	$0.496649\pi$
$152$	35782.8	0.125622
$153$	19599.1	0.0676874
$154$	166494.	0.565714
$155$	6150.73	0.0205635
$156$	68977.0	0.226930
$157$	−306248.	−0.991570	−0.495785	−	0.868445i	$-0.665120\pi$
$157$	−306248.	−0.991570	−0.495785	+	0.868445i	$0.665120\pi$
$158$	195968.	0.624513
$159$	272018.	0.853306
$160$	4607.84	0.0142297
$161$	255177.	0.775847
$162$	−58948.4	−0.176476
$163$	−273207.	−0.805422	−0.402711	−	0.915327i	$-0.631932\pi$
$163$	−273207.	−0.805422	−0.402711	+	0.915327i	$0.631932\pi$
$164$	265244.	0.770079
$165$	−11816.6	−0.0337896
$166$	−74681.7	−0.210351
$167$	579511.	1.60794	0.803971	−	0.594668i	$-0.202716\pi$
$167$	579511.	1.60794	0.803971	+	0.594668i	$0.202716\pi$
$168$	67895.1	0.185595
$169$	−93607.7	−0.252113
$170$	−2003.58	−0.00531722
$171$	98442.2	0.257449
$172$	−130186.	−0.335539
$173$	327012.	0.830708	0.415354	−	0.909660i	$-0.363658\pi$
$173$	327012.	0.830708	0.415354	+	0.909660i	$0.363658\pi$
$174$	−18292.7	−0.0458040
$175$	−402603.	−0.993762
$176$	−82172.8	−0.199962
$177$	−84240.0	−0.202109
$178$	−533671.	−1.26248
$179$	252475.	0.588960	0.294480	−	0.955658i	$-0.404854\pi$
$179$	252475.	0.588960	0.294480	+	0.955658i	$0.404854\pi$
$180$	12676.6	0.0291624
$181$	561278.	1.27345	0.636724	−	0.771092i	$-0.280289\pi$
$181$	561278.	1.27345	0.636724	+	0.771092i	$0.280289\pi$
$182$	273330.	0.611658
$183$	−7408.24	−0.0163526
$184$	−125942.	−0.274237
$185$	−37465.1	−0.0804817
$186$	−44729.8	−0.0948014
$187$	35730.4	0.0747194
$188$	−395838.	−0.816814
$189$	444576.	0.905297
$190$	−10063.5	−0.0202240
$191$	44227.2	0.0877215	0.0438608	−	0.999038i	$-0.486034\pi$
$191$	44227.2	0.0877215	0.0438608	+	0.999038i	$0.486034\pi$
$192$	−33509.5	−0.0656017
$193$	832064.	1.60792	0.803958	−	0.594686i	$-0.202724\pi$
$193$	832064.	1.60792	0.803958	+	0.594686i	$0.202724\pi$
$194$	315621.	0.602090
$195$	−19399.1	−0.0365338
$196$	130.582	0.000242798 0
$197$	923407.	1.69523	0.847613	−	0.530614i	$-0.178039\pi$
$197$	923407.	1.69523	0.847613	+	0.530614i	$0.178039\pi$
$198$	−226066.	−0.409800
$199$	589613.	1.05544	0.527721	−	0.849418i	$-0.323047\pi$
$199$	589613.	1.05544	0.527721	+	0.849418i	$0.323047\pi$
$200$	198704.	0.351263
$201$	36672.0	0.0640242
$202$	−714858.	−1.23266
$203$	−72486.9	−0.123458
$204$	14570.6	0.0245133
$205$	−74597.2	−0.123976
$206$	417800.	0.685963
$207$	−346479.	−0.562020
$208$	−134901.	−0.216201
$209$	179466.	0.284195
$210$	−19094.8	−0.0298791
$211$	367136.	0.567702	0.283851	−	0.958868i	$-0.408388\pi$
$211$	367136.	0.567702	0.283851	+	0.958868i	$0.408388\pi$
$212$	−531997.	−0.812961
$213$	543997.	0.821575
$214$	−588131.	−0.877889
$215$	36613.6	0.0540189
$216$	−219420.	−0.319993
$217$	−177247.	−0.255523
$218$	535080.	0.762566
$219$	−631627.	−0.889920
$220$	23110.3	0.0321921
$221$	58657.8	0.0807877
$222$	272456.	0.371035
$223$	402082.	0.541443	0.270722	−	0.962658i	$-0.412738\pi$
$223$	402082.	0.541443	0.270722	+	0.962658i	$0.412738\pi$
$224$	−132785.	−0.176820
$225$	546656.	0.719876
$226$	336152.	0.437788
$227$	−964086.	−1.24180	−0.620899	−	0.783890i	$-0.713232\pi$
$227$	−964086.	−1.24180	−0.620899	+	0.783890i	$0.713232\pi$
$228$	73185.0	0.0932362
$229$	1.31061e6	1.65152	0.825760	−	0.564021i	$-0.190746\pi$
$229$	1.31061e6	1.65152	0.825760	+	0.564021i	$0.190746\pi$
$230$	35419.9	0.0441497
$231$	340523.	0.419872
$232$	35775.8	0.0436384
$233$	881268.	1.06345	0.531726	−	0.846916i	$-0.321544\pi$
$233$	881268.	1.06345	0.531726	+	0.846916i	$0.321544\pi$
$234$	−371128.	−0.443082
$235$	111326.	0.131500
$236$	164752.	0.192553
$237$	400804.	0.463512
$238$	57737.8	0.0660720
$239$	1.53737e6	1.74094	0.870469	−	0.492224i	$-0.163816\pi$
$239$	1.53737e6	1.74094	0.870469	+	0.492224i	$0.163816\pi$
$240$	9424.21	0.0105613
$241$	−303134.	−0.336195	−0.168098	−	0.985770i	$-0.553762\pi$
$241$	−303134.	−0.336195	−0.168098	+	0.985770i	$0.553762\pi$
$242$	232072.	0.254732
$243$	−953673.	−1.03606
$244$	14488.6	0.0155795
$245$	−36.7250	−3.90883e−5 0
$246$	542492.	0.571551
$247$	294626.	0.307276
$248$	87480.1	0.0903192
$249$	−152743.	−0.156122
$250$	−112132.	−0.113469
$251$	1.12436e6	1.12648	0.563238	−	0.826295i	$-0.309555\pi$
$251$	1.12436e6	1.12648	0.563238	+	0.826295i	$0.309555\pi$
$252$	−365307.	−0.362373
$253$	−631653.	−0.620408
$254$	−922273.	−0.896964
$255$	−4097.84	−0.00394643
$256$	65536.0	0.0625000
$257$	−213101.	−0.201258	−0.100629	−	0.994924i	$-0.532085\pi$
$257$	−213101.	−0.201258	−0.100629	+	0.994924i	$0.532085\pi$
$258$	−266264.	−0.249037
$259$	1.07964e6	1.00007
$260$	37939.7	0.0348065
$261$	98422.9	0.0894324
$262$	−701631.	−0.631474
$263$	−1.99320e6	−1.77690	−0.888448	−	0.458978i	$-0.848216\pi$
$263$	−1.99320e6	−1.77690	−0.888448	+	0.458978i	$0.848216\pi$
$264$	−168065.	−0.148411
$265$	149619.	0.130880
$266$	290004.	0.251305
$267$	−1.09149e6	−0.937008
$268$	−71721.0	−0.0609971
$269$	72361.0	0.0609711
$269$	72361.0	0.0609711
$270$	61709.6	0.0515161
$271$	−1.48527e6	−1.22852	−0.614262	−	0.789102i	$-0.710546\pi$
$271$	−1.48527e6	−1.22852	−0.614262	+	0.789102i	$0.710546\pi$
$272$	−28496.4	−0.0233543
$273$	559030.	0.453971
$274$	424779.	0.341812
$275$	996587.	0.794664
$276$	−257584.	−0.203538
$277$	−1.17388e6	−0.919227	−0.459613	−	0.888119i	$-0.652012\pi$
$277$	−1.17388e6	−0.919227	−0.459613	+	0.888119i	$0.652012\pi$
$278$	−986182.	−0.765323
$279$	240667.	0.185100
$280$	37344.6	0.0284664
$281$	−528685.	−0.399421	−0.199710	−	0.979855i	$-0.564000\pi$
$281$	−528685.	−0.399421	−0.199710	+	0.979855i	$0.564000\pi$
$282$	−809591.	−0.606238
$283$	851227.	0.631799	0.315900	−	0.948793i	$-0.397694\pi$
$283$	851227.	0.631799	0.315900	+	0.948793i	$0.397694\pi$
$284$	−1.06392e6	−0.782731
$285$	−20582.5	−0.0150102
$286$	−676588.	−0.489113
$287$	2.14969e6	1.54053
$288$	180296.	0.128087
$289$	−1.40747e6	−0.991273
$290$	−10061.6	−0.00702540
$291$	645526.	0.446870
$292$	1.23530e6	0.847844
$293$	−1.06424e6	−0.724220	−0.362110	−	0.932135i	$-0.617944\pi$
$293$	−1.06424e6	−0.724220	−0.362110	+	0.932135i	$0.617944\pi$
$294$	267.075	0.000180204 0
$295$	−46334.8	−0.0309993
$296$	−532855.	−0.353492
$297$	−1.10048e6	−0.723923
$298$	−360693.	−0.235286
$299$	−1.03697e6	−0.670793
$300$	406401.	0.260706
$301$	−1.05510e6	−0.671242
$302$	−23596.0	−0.0148875
$303$	−1.46207e6	−0.914874
$304$	−143131.	−0.0888280
$305$	−4074.78	−0.00250816
$306$	−78396.5	−0.0478622
$307$	326834.	0.197916	0.0989581	−	0.995092i	$-0.468449\pi$
$307$	326834.	0.197916	0.0989581	+	0.995092i	$0.468449\pi$
$308$	−665976.	−0.400020
$309$	854509.	0.509120
$310$	−24602.9	−0.0145406
$311$	−120503.	−0.0706473	−0.0353237	−	0.999376i	$-0.511246\pi$
$311$	−120503.	−0.0706473	−0.0353237	+	0.999376i	$0.511246\pi$
$312$	−275908.	−0.160464
$313$	−1.42853e6	−0.824190	−0.412095	−	0.911141i	$-0.635203\pi$
$313$	−1.42853e6	−0.824190	−0.412095	+	0.911141i	$0.635203\pi$
$314$	1.22499e6	0.701146
$315$	102739.	0.0583389
$316$	−783870.	−0.441597
$317$	−144666.	−0.0808570	−0.0404285	−	0.999182i	$-0.512872\pi$
$317$	−144666.	−0.0808570	−0.0404285	+	0.999182i	$0.512872\pi$
$318$	−1.08807e6	−0.603378
$319$	179431.	0.0987235
$320$	−18431.4	−0.0100620
$321$	−1.20288e6	−0.651567
$322$	−1.02071e6	−0.548607
$323$	62236.2	0.0331923
$324$	235794.	0.124787
$325$	1.63608e6	0.859201
$326$	1.09283e6	0.569519
$327$	1.09438e6	0.565975
$328$	−1.06097e6	−0.544528
$329$	−3.20810e6	−1.63402
$330$	47266.5	0.0238929
$331$	2.03741e6	1.02214	0.511068	−	0.859540i	$-0.329250\pi$
$331$	2.03741e6	1.02214	0.511068	+	0.859540i	$0.329250\pi$
$332$	298727.	0.148740
$333$	−1.46594e6	−0.724446
$334$	−2.31804e6	−1.13699
$335$	20170.8	0.00982000
$336$	−271580.	−0.131235
$337$	2.62479e6	1.25898	0.629492	−	0.777007i	$-0.283263\pi$
$337$	2.62479e6	1.25898	0.629492	+	0.777007i	$0.283263\pi$
$338$	374431.	0.178271
$339$	687517.	0.324926
$340$	8014.32	0.00375984
$341$	438750.	0.204330
$342$	−393769.	−0.182044
$343$	−2.17836e6	−0.999757
$344$	520744.	0.237262
$345$	72442.8	0.0327678
$346$	−1.30805e6	−0.587399
$347$	4.35017e6	1.93947	0.969733	−	0.244169i	$-0.0785151\pi$
$347$	4.35017e6	1.93947	0.969733	+	0.244169i	$0.0785151\pi$
$348$	73170.7	0.0323884
$349$	−12156.8	−0.00534265	−0.00267133	−	0.999996i	$-0.500850\pi$
$349$	−12156.8	−0.00534265	−0.00267133	+	0.999996i	$0.500850\pi$
$350$	1.61041e6	0.702696
$351$	−1.80664e6	−0.782716
$352$	328691.	0.141394
$353$	2.72752e6	1.16502	0.582508	−	0.812825i	$-0.302072\pi$
$353$	2.72752e6	1.16502	0.582508	+	0.812825i	$0.302072\pi$
$354$	336960.	0.142912
$355$	299217.	0.126013
$356$	2.13468e6	0.892706
$357$	118089.	0.0490385
$358$	−1.00990e6	−0.416458
$359$	854521.	0.349935	0.174967	−	0.984574i	$-0.444018\pi$
$359$	854521.	0.349935	0.174967	+	0.984574i	$0.444018\pi$
$360$	−50706.6	−0.0206209
$361$	−2.16350e6	−0.873753
$362$	−2.24511e6	−0.900464
$363$	474647.	0.189062
$364$	−1.09332e6	−0.432507
$365$	−347416.	−0.136495
$366$	29633.0	0.0115631
$367$	4.48843e6	1.73952	0.869760	−	0.493475i	$-0.164274\pi$
$367$	4.48843e6	1.73952	0.869760	+	0.493475i	$0.164274\pi$
$368$	503768.	0.193915
$369$	−2.91885e6	−1.11595
$370$	149860.	0.0569091
$371$	−4.31162e6	−1.62632
$372$	178919.	0.0670347
$373$	3.36839e6	1.25358	0.626788	−	0.779190i	$-0.284369\pi$
$373$	3.36839e6	1.25358	0.626788	+	0.779190i	$0.284369\pi$
$374$	−142922.	−0.0528346
$375$	−229338.	−0.0842166
$376$	1.58335e6	0.577575
$377$	294568.	0.106741
$378$	−1.77830e6	−0.640142
$379$	−2.60226e6	−0.930578	−0.465289	−	0.885159i	$-0.654050\pi$
$379$	−2.60226e6	−0.930578	−0.465289	+	0.885159i	$0.654050\pi$
$380$	40254.2	0.0143005
$381$	−1.88628e6	−0.665725
$382$	−176909.	−0.0620285
$383$	−1.12236e6	−0.390964	−0.195482	−	0.980707i	$-0.562627\pi$
$383$	−1.12236e6	−0.390964	−0.195482	+	0.980707i	$0.562627\pi$
$384$	134038.	0.0463874
$385$	187299.	0.0643997
$386$	−3.32825e6	−1.13697
$387$	1.43262e6	0.486244
$388$	−1.26248e6	−0.425742
$389$	−1.53614e6	−0.514704	−0.257352	−	0.966318i	$-0.582850\pi$
$389$	−1.53614e6	−0.514704	−0.257352	+	0.966318i	$0.582850\pi$
$390$	77596.4	0.0258333
$391$	−219048.	−0.0724599
$392$	−522.330	−0.000171684 0
$393$	−1.43502e6	−0.468679
$394$	−3.69363e6	−1.19871
$395$	220456.	0.0710933
$396$	904264.	0.289773
$397$	3.78536e6	1.20540	0.602700	−	0.797968i	$-0.294091\pi$
$397$	3.78536e6	1.20540	0.602700	+	0.797968i	$0.294091\pi$
$398$	−2.35845e6	−0.746310
$399$	593133.	0.186518
$400$	−794816.	−0.248380
$401$	1.97513e6	0.613387	0.306693	−	0.951808i	$-0.400777\pi$
$401$	1.97513e6	0.613387	0.306693	+	0.951808i	$0.400777\pi$
$402$	−146688.	−0.0452720
$403$	720287.	0.220924
$404$	2.85943e6	0.871619
$405$	−66314.6	−0.0200896
$406$	289948.	0.0872980
$407$	−2.67250e6	−0.799708
$408$	−58282.4	−0.0173335
$409$	−2.50677e6	−0.740979	−0.370489	−	0.928837i	$-0.620810\pi$
$409$	−2.50677e6	−0.740979	−0.370489	+	0.928837i	$0.620810\pi$
$410$	298389.	0.0876643
$411$	868783.	0.253692
$412$	−1.67120e6	−0.485049
$413$	1.33524e6	0.385200
$414$	1.38592e6	0.397408
$415$	−84014.0	−0.0239459
$416$	539606.	0.152877
$417$	−2.01700e6	−0.568022
$418$	−717864.	−0.200956
$419$	−242450.	−0.0674663	−0.0337332	−	0.999431i	$-0.510740\pi$
$419$	−242450.	−0.0674663	−0.0337332	+	0.999431i	$0.510740\pi$
$420$	76379.3	0.0211277
$421$	−2.55168e6	−0.701651	−0.350826	−	0.936441i	$-0.614099\pi$
$421$	−2.55168e6	−0.701651	−0.350826	+	0.936441i	$0.614099\pi$
$422$	−1.46854e6	−0.401426
$423$	4.35597e6	1.18368
$424$	2.12799e6	0.574850
$425$	345602.	0.0928120
$426$	−2.17599e6	−0.580941
$427$	117424.	0.0311665
$428$	2.35252e6	0.620761
$429$	−1.38380e6	−0.363019
$430$	−146454.	−0.0381971
$431$	5.86782e6	1.52154	0.760771	−	0.649020i	$-0.224821\pi$
$431$	5.86782e6	1.52154	0.760771	+	0.649020i	$0.224821\pi$
$432$	877678.	0.226269
$433$	1.35364e6	0.346963	0.173482	−	0.984837i	$-0.444498\pi$
$433$	1.35364e6	0.346963	0.173482	+	0.984837i	$0.444498\pi$
$434$	708990.	0.180682
$435$	−20578.5	−0.00521424
$436$	−2.14032e6	−0.539215
$437$	−1.10023e6	−0.275601
$438$	2.52651e6	0.629268
$439$	−3.15308e6	−0.780861	−0.390431	−	0.920632i	$-0.627674\pi$
$439$	−3.15308e6	−0.780861	−0.390431	+	0.920632i	$0.627674\pi$
$440$	−92441.2	−0.0227632
$441$	−1436.98	−0.000351848 0
$442$	−234631.	−0.0571255
$443$	1.51996e6	0.367978	0.183989	−	0.982928i	$-0.441099\pi$
$443$	1.51996e6	0.367978	0.183989	+	0.982928i	$0.441099\pi$
$444$	−1.08983e6	−0.262361
$445$	−600359.	−0.143718
$446$	−1.60833e6	−0.382858
$447$	−737709.	−0.174629
$448$	531142.	0.125030
$449$	−3.18720e6	−0.746093	−0.373047	−	0.927813i	$-0.621687\pi$
$449$	−3.18720e6	−0.746093	−0.373047	+	0.927813i	$0.621687\pi$
$450$	−2.18662e6	−0.509029
$451$	−5.32125e6	−1.23189
$452$	−1.34461e6	−0.309563
$453$	−48259.9	−0.0110495
$454$	3.85634e6	0.878084
$455$	307485.	0.0696299
$456$	−292740.	−0.0659280
$457$	−5.00524e6	−1.12107	−0.560537	−	0.828129i	$-0.689405\pi$
$457$	−5.00524e6	−1.12107	−0.560537	+	0.828129i	$0.689405\pi$
$458$	−5.24243e6	−1.16780
$459$	−381632.	−0.0845499
$460$	−141680.	−0.0312186
$461$	−6.49152e6	−1.42264	−0.711319	−	0.702869i	$-0.751902\pi$
$461$	−6.49152e6	−1.42264	−0.711319	+	0.702869i	$0.751902\pi$
$462$	−1.36209e6	−0.296894
$463$	−7.69883e6	−1.66906	−0.834531	−	0.550961i	$-0.814261\pi$
$463$	−7.69883e6	−1.66906	−0.834531	+	0.550961i	$0.814261\pi$
$464$	−143103.	−0.0308570
$465$	−50319.3	−0.0107920
$466$	−3.52507e6	−0.751974
$467$	−99627.5	−0.0211391	−0.0105696	−	0.999944i	$-0.503364\pi$
$467$	−99627.5	−0.0211391	−0.0105696	+	0.999944i	$0.503364\pi$
$468$	1.48451e6	0.313306
$469$	−581269.	−0.122024
$470$	−445302.	−0.0929845
$471$	2.50542e6	0.520389
$472$	−659008.	−0.136156
$473$	2.61176e6	0.536760
$474$	−1.60322e6	−0.327753
$475$	1.73588e6	0.353010
$476$	−230951.	−0.0467200
$477$	5.85432e6	1.17810
$478$	−6.14947e6	−1.23103
$479$	8.29383e6	1.65164	0.825822	−	0.563931i	$-0.190712\pi$
$479$	8.29383e6	1.65164	0.825822	+	0.563931i	$0.190712\pi$
$480$	−37696.9	−0.00746796
$481$	−4.38738e6	−0.864655
$482$	1.21253e6	0.237726
$483$	−2.08761e6	−0.407175
$484$	−928288.	−0.180123
$485$	355061.	0.0685407
$486$	3.81469e6	0.732603
$487$	−295103.	−0.0563835	−0.0281917	−	0.999603i	$-0.508975\pi$
$487$	−295103.	−0.0563835	−0.0281917	+	0.999603i	$0.508975\pi$
$488$	−57954.5	−0.0110163
$489$	2.23512e6	0.422696
$490$	146.900	2.76396e−5 0
$491$	1.05479e7	1.97452	0.987260	−	0.159117i	$-0.0508648\pi$
$491$	1.05479e7	1.97452	0.987260	+	0.159117i	$0.0508648\pi$
$492$	−2.16997e6	−0.404148
$493$	62224.1	0.0115303
$494$	−1.17850e6	−0.217277
$495$	−254315.	−0.0466509
$496$	−349920.	−0.0638653
$497$	−8.62261e6	−1.56584
$498$	610973.	0.110395
$499$	−5.86674e6	−1.05474	−0.527370	−	0.849636i	$-0.676822\pi$
$499$	−5.86674e6	−1.05474	−0.527370	+	0.849636i	$0.676822\pi$
$500$	448526.	0.0802348
$501$	−4.74100e6	−0.843870
$502$	−4.49745e6	−0.796539
$503$	−1.06665e7	−1.87976	−0.939882	−	0.341498i	$-0.889066\pi$
$503$	−1.06665e7	−1.87976	−0.939882	+	0.341498i	$0.889066\pi$
$504$	1.46123e6	0.256237
$505$	−804187.	−0.140323
$506$	2.52661e6	0.438694
$507$	765808.	0.132312
$508$	3.68909e6	0.634249
$509$	−2.03629e6	−0.348374	−0.174187	−	0.984713i	$-0.555730\pi$
$509$	−2.03629e6	−0.348374	−0.174187	+	0.984713i	$0.555730\pi$
$510$	16391.3	0.00279055
$511$	1.00116e7	1.69610
$512$	−262144.	−0.0441942
$513$	−1.91685e6	−0.321585
$514$	852404.	0.142311
$515$	470009.	0.0780886
$516$	1.06506e6	0.176096
$517$	7.94120e6	1.30665
$518$	−4.31857e6	−0.707156
$519$	−2.67529e6	−0.435966
$520$	−151759.	−0.0246119
$521$	−3.22728e6	−0.520886	−0.260443	−	0.965489i	$-0.583869\pi$
$521$	−3.22728e6	−0.520886	−0.260443	+	0.965489i	$0.583869\pi$
$522$	−393692.	−0.0632383
$523$	8.12883e6	1.29949	0.649746	−	0.760151i	$-0.274875\pi$
$523$	8.12883e6	1.29949	0.649746	+	0.760151i	$0.274875\pi$
$524$	2.80652e6	0.446520
$525$	3.29371e6	0.521539
$526$	7.97281e6	1.25645
$527$	152152.	0.0238645
$528$	672258.	0.104942
$529$	−2.56394e6	−0.398354
$530$	−598476.	−0.0925459
$531$	−1.81300e6	−0.279037
$532$	−1.16002e6	−0.177699
$533$	−8.73578e6	−1.33194
$534$	4.36598e6	0.662565
$535$	−661624.	−0.0999371
$536$	286884.	0.0431315
$537$	−2.06550e6	−0.309094
$538$	−289444.	−0.0431131
$539$	−2619.71	−0.000388402 0
$540$	−246838.	−0.0364274
$541$	1.27666e7	1.87535	0.937676	−	0.347510i	$-0.112973\pi$
$541$	1.27666e7	1.87535	0.937676	+	0.347510i	$0.112973\pi$
$542$	5.94110e6	0.868697
$543$	−4.59183e6	−0.668323
$544$	113985.	0.0165140
$545$	601943.	0.0868089
$546$	−2.23612e6	−0.321006
$547$	6.38665e6	0.912651	0.456325	−	0.889813i	$-0.349165\pi$
$547$	6.38665e6	0.912651	0.456325	+	0.889813i	$0.349165\pi$
$548$	−1.69912e6	−0.241697
$549$	−159439.	−0.0225769
$550$	−3.98635e6	−0.561912
$551$	312538.	0.0438555
$552$	1.03033e6	0.143923
$553$	−6.35294e6	−0.883409
$554$	4.69550e6	0.649992
$555$	306503.	0.0422379
$556$	3.94473e6	0.541165
$557$	1.37092e7	1.87230	0.936148	−	0.351606i	$-0.114364\pi$
$557$	1.37092e7	1.87230	0.936148	+	0.351606i	$0.114364\pi$
$558$	−962668.	−0.130885
$559$	4.28767e6	0.580352
$560$	−149378.	−0.0201288
$561$	−292311.	−0.0392138
$562$	2.11474e6	0.282433
$563$	−3.63178e6	−0.482890	−0.241445	−	0.970414i	$-0.577621\pi$
$563$	−3.63178e6	−0.482890	−0.241445	+	0.970414i	$0.577621\pi$
$564$	3.23836e6	0.428675
$565$	378157.	0.0498369
$566$	−3.40491e6	−0.446750
$567$	1.91101e6	0.249635
$568$	4.25567e6	0.553474
$569$	1.48701e7	1.92545	0.962726	−	0.270479i	$-0.0871821\pi$
$569$	1.48701e7	1.92545	0.962726	+	0.270479i	$0.0871821\pi$
$570$	82330.2	0.0106138
$571$	−8.82127e6	−1.13225	−0.566123	−	0.824321i	$-0.691557\pi$
$571$	−8.82127e6	−1.13225	−0.566123	+	0.824321i	$0.691557\pi$
$572$	2.70635e6	0.345855
$573$	−361824.	−0.0460374
$574$	−8.59876e6	−1.08932
$575$	−6.10966e6	−0.770633
$576$	−721186.	−0.0905714
$577$	−4.07125e6	−0.509082	−0.254541	−	0.967062i	$-0.581924\pi$
$577$	−4.07125e6	−0.509082	−0.254541	+	0.967062i	$0.581924\pi$
$578$	5.62986e6	0.700936
$579$	−6.80714e6	−0.843855
$580$	40246.3	0.00496771
$581$	2.42106e6	0.297553
$582$	−2.58210e6	−0.315985
$583$	1.06728e7	1.30049
$584$	−4.94121e6	−0.599516
$585$	−417504.	−0.0504396
$586$	4.25696e6	0.512101
$587$	9.65348e6	1.15635	0.578174	−	0.815913i	$-0.303765\pi$
$587$	9.65348e6	1.15635	0.578174	+	0.815913i	$0.303765\pi$
$588$	−1068.30	−0.000127423 0
$589$	764229.	0.0907685
$590$	185339.	0.0219199
$591$	−7.55442e6	−0.889677
$592$	2.13142e6	0.249957
$593$	−1.58606e7	−1.85218	−0.926090	−	0.377302i	$-0.876852\pi$
$593$	−1.58606e7	−1.85218	−0.926090	+	0.377302i	$0.876852\pi$
$594$	4.40193e6	0.511891
$595$	64952.7	0.00752150
$596$	1.44277e6	0.166373
$597$	−4.82364e6	−0.553910
$598$	4.14788e6	0.474322
$599$	−9.80937e6	−1.11705	−0.558527	−	0.829486i	$-0.688633\pi$
$599$	−9.80937e6	−1.11705	−0.558527	+	0.829486i	$0.688633\pi$
$600$	−1.62560e6	−0.184347
$601$	−5.41631e6	−0.611670	−0.305835	−	0.952084i	$-0.598936\pi$
$601$	−5.41631e6	−0.611670	−0.305835	+	0.952084i	$0.598936\pi$
$602$	4.22042e6	0.474640
$603$	789248.	0.0883935
$604$	94384.1	0.0105270
$605$	261072.	0.0289982
$606$	5.84828e6	0.646914
$607$	2.21422e6	0.243921	0.121960	−	0.992535i	$-0.461082\pi$
$607$	2.21422e6	0.243921	0.121960	+	0.992535i	$0.461082\pi$
$608$	572524.	0.0628109
$609$	593017.	0.0647924
$610$	16299.1	0.00177354
$611$	1.30369e7	1.41277
$612$	313586.	0.0338437
$613$	−7.86427e6	−0.845293	−0.422646	−	0.906295i	$-0.638899\pi$
$613$	−7.86427e6	−0.845293	−0.422646	+	0.906295i	$0.638899\pi$
$614$	−1.30734e6	−0.139948
$615$	610282.	0.0650643
$616$	2.66390e6	0.282857
$617$	−4.00230e6	−0.423250	−0.211625	−	0.977351i	$-0.567876\pi$
$617$	−4.00230e6	−0.423250	−0.211625	+	0.977351i	$0.567876\pi$
$618$	−3.41804e6	−0.360002
$619$	1.13493e7	1.19053	0.595267	−	0.803528i	$-0.297046\pi$
$619$	1.13493e7	1.19053	0.595267	+	0.803528i	$0.297046\pi$
$620$	98411.7	0.0102818
$621$	6.74661e6	0.702031
$622$	482011.	0.0499552
$623$	1.73007e7	1.78585
$624$	1.10363e6	0.113465
$625$	9.57620e6	0.980603
$626$	5.71410e6	0.582790
$627$	−1.46822e6	−0.149149
$628$	−4.89996e6	−0.495785
$629$	−926784.	−0.0934011
$630$	−410956.	−0.0412519
$631$	7.52545e6	0.752418	0.376209	−	0.926535i	$-0.377227\pi$
$631$	7.52545e6	0.752418	0.376209	+	0.926535i	$0.377227\pi$
$632$	3.13548e6	0.312256
$633$	−3.00355e6	−0.297938
$634$	578663.	0.0571745
$635$	−1.03752e6	−0.102109
$636$	4.35229e6	0.426653
$637$	−4300.72	−0.000419945 0
$638$	−717724.	−0.0698080
$639$	1.17078e7	1.13429
$640$	73725.4	0.00711487
$641$	8.19920e6	0.788181	0.394091	−	0.919072i	$-0.371060\pi$
$641$	8.19920e6	0.788181	0.394091	+	0.919072i	$0.371060\pi$
$642$	4.81152e6	0.460728
$643$	1.15319e7	1.09995	0.549976	−	0.835180i	$-0.314637\pi$
$643$	1.15319e7	1.09995	0.549976	+	0.835180i	$0.314637\pi$
$644$	4.08282e6	0.387923
$645$	−299537.	−0.0283498
$646$	−248945.	−0.0234705
$647$	8.38640e6	0.787617	0.393808	−	0.919193i	$-0.371157\pi$
$647$	8.38640e6	0.787617	0.393808	+	0.919193i	$0.371157\pi$
$648$	−943175.	−0.0882378
$649$	−3.30521e6	−0.308026
$650$	−6.54430e6	−0.607547
$651$	1.45007e6	0.134102
$652$	−4.37132e6	−0.402711
$653$	−3.71980e6	−0.341379	−0.170690	−	0.985325i	$-0.554600\pi$
$653$	−3.71980e6	−0.341379	−0.170690	+	0.985325i	$0.554600\pi$
$654$	−4.37750e6	−0.400205
$655$	−789307.	−0.0718857
$656$	4.24390e6	0.385040
$657$	−1.35938e7	−1.22865
$658$	1.28324e7	1.15543
$659$	1.48109e6	0.132852	0.0664259	−	0.997791i	$-0.478840\pi$
$659$	1.48109e6	0.132852	0.0664259	+	0.997791i	$0.478840\pi$
$660$	−189066.	−0.0168948
$661$	−1.58550e7	−1.41144	−0.705719	−	0.708492i	$-0.749376\pi$
$661$	−1.58550e7	−1.41144	−0.705719	+	0.708492i	$0.749376\pi$
$662$	−8.14965e6	−0.722760
$663$	−479881.	−0.0423984
$664$	−1.19491e6	−0.105175
$665$	326243.	0.0286080
$666$	5.86376e6	0.512260
$667$	−1.10002e6	−0.0957381
$668$	9.27218e6	0.803971
$669$	−3.28945e6	−0.284157
$670$	−80683.3	−0.00694379
$671$	−290667.	−0.0249224
$672$	1.08632e6	0.0927973
$673$	6.42447e6	0.546764	0.273382	−	0.961906i	$-0.411858\pi$
$673$	6.42447e6	0.546764	0.273382	+	0.961906i	$0.411858\pi$
$674$	−1.04992e7	−0.890237
$675$	−1.06444e7	−0.899213
$676$	−1.49772e6	−0.126056
$677$	1.96627e7	1.64881	0.824407	−	0.565998i	$-0.191509\pi$
$677$	1.96627e7	1.64881	0.824407	+	0.565998i	$0.191509\pi$
$678$	−2.75007e6	−0.229757
$679$	−1.02319e7	−0.851691
$680$	−32057.3	−0.00265861
$681$	7.88722e6	0.651712
$682$	−1.75500e6	−0.144483
$683$	9.72816e6	0.797956	0.398978	−	0.916960i	$-0.369365\pi$
$683$	9.72816e6	0.797956	0.398978	+	0.916960i	$0.369365\pi$
$684$	1.57507e6	0.128724
$685$	477860.	0.0389111
$686$	8.71344e6	0.706935
$687$	−1.07221e7	−0.866740
$688$	−2.08298e6	−0.167770
$689$	1.75213e7	1.40611
$690$	−289771.	−0.0231704
$691$	−1.34584e7	−1.07226	−0.536128	−	0.844136i	$-0.680114\pi$
$691$	−1.34584e7	−1.07226	−0.536128	+	0.844136i	$0.680114\pi$
$692$	5.23219e6	0.415354
$693$	7.32868e6	0.579686
$694$	−1.74007e7	−1.37141
$695$	−1.10942e6	−0.0871229
$696$	−292683.	−0.0229020
$697$	−1.84533e6	−0.143877
$698$	48627.3	0.00377783
$699$	−7.20968e6	−0.558114
$700$	−6.44165e6	−0.496881
$701$	−1.31637e7	−1.01177	−0.505885	−	0.862601i	$-0.668834\pi$
$701$	−1.31637e7	−1.01177	−0.505885	+	0.862601i	$0.668834\pi$
$702$	7.22656e6	0.553463
$703$	−4.65504e6	−0.355251
$704$	−1.31477e6	−0.0999808
$705$	−910758.	−0.0690129
$706$	−1.09101e7	−0.823790
$707$	2.31745e7	1.74366
$708$	−1.34784e6	−0.101054
$709$	−8.66178e6	−0.647130	−0.323565	−	0.946206i	$-0.604881\pi$
$709$	−8.66178e6	−0.647130	−0.323565	+	0.946206i	$0.604881\pi$
$710$	−1.19687e6	−0.0891045
$711$	8.62604e6	0.639937
$712$	−8.53874e6	−0.631239
$713$	−2.68980e6	−0.198151
$714$	−472354.	−0.0346755
$715$	−761135.	−0.0556797
$716$	4.03960e6	0.294480
$717$	−1.25773e7	−0.913667
$718$	−3.41809e6	−0.247441
$719$	1.63398e7	1.17876	0.589379	−	0.807857i	$-0.299372\pi$
$719$	1.63398e7	1.17876	0.589379	+	0.807857i	$0.299372\pi$
$720$	202826.	0.0145812
$721$	−1.35444e7	−0.970334
$722$	8.65400e6	0.617837
$723$	2.47995e6	0.176440
$724$	8.98044e6	0.636724
$725$	1.73554e6	0.122628
$726$	−1.89859e6	−0.133687
$727$	1.83055e7	1.28453	0.642266	−	0.766482i	$-0.277994\pi$
$727$	1.83055e7	1.28453	0.642266	+	0.766482i	$0.277994\pi$
$728$	4.37328e6	0.305829
$729$	4.22091e6	0.294163
$730$	1.38967e6	0.0965169
$731$	905720.	0.0626903
$732$	−118532.	−0.00817631
$733$	5.38425e6	0.370139	0.185070	−	0.982725i	$-0.440749\pi$
$733$	5.38425e6	0.370139	0.185070	+	0.982725i	$0.440749\pi$
$734$	−1.79537e7	−1.23003
$735$	300.449	2.05141e−5 0
$736$	−2.01507e6	−0.137118
$737$	1.43885e6	0.0975766
$738$	1.16754e7	0.789099
$739$	−2.43585e7	−1.64074	−0.820370	−	0.571832i	$-0.806233\pi$
$739$	−2.43585e7	−1.64074	−0.820370	+	0.571832i	$0.806233\pi$
$740$	−599441.	−0.0402408
$741$	−2.41034e6	−0.161262
$742$	1.72465e7	1.14998
$743$	1.19506e7	0.794179	0.397090	−	0.917780i	$-0.370020\pi$
$743$	1.19506e7	0.794179	0.397090	+	0.917780i	$0.370020\pi$
$744$	−715677.	−0.0474007
$745$	−405765.	−0.0267845
$746$	−1.34736e7	−0.886412
$747$	−3.28732e6	−0.215546
$748$	571686.	0.0373597
$749$	1.90662e7	1.24182
$750$	917352.	0.0595501
$751$	−4.25982e6	−0.275608	−0.137804	−	0.990460i	$-0.544004\pi$
$751$	−4.25982e6	−0.275608	−0.137804	+	0.990460i	$0.544004\pi$
$752$	−6.33341e6	−0.408407
$753$	−9.19844e6	−0.591190
$754$	−1.17827e6	−0.0754774
$755$	−26544.6	−0.00169476
$756$	7.11321e6	0.452649
$757$	2.82685e7	1.79293	0.896465	−	0.443114i	$-0.146126\pi$
$757$	2.82685e7	1.79293	0.896465	+	0.443114i	$0.146126\pi$
$758$	1.04090e7	0.658018
$759$	5.16757e6	0.325598
$760$	−161017.	−0.0101120
$761$	2.14111e7	1.34023	0.670113	−	0.742260i	$-0.266246\pi$
$761$	2.14111e7	1.34023	0.670113	+	0.742260i	$0.266246\pi$
$762$	7.54514e6	0.470739
$763$	−1.73464e7	−1.07869
$764$	707636.	0.0438608
$765$	−88192.9	−0.00544854
$766$	4.48945e6	0.276453
$767$	−5.42609e6	−0.333042
$768$	−536152.	−0.0328008
$769$	−1.66000e7	−1.01226	−0.506130	−	0.862457i	$-0.668924\pi$
$769$	−1.66000e7	−1.01226	−0.506130	+	0.862457i	$0.668924\pi$
$770$	−749197.	−0.0455375
$771$	1.74338e6	0.105623
$772$	1.33130e7	0.803958
$773$	−6.99308e6	−0.420939	−0.210470	−	0.977600i	$-0.567499\pi$
$773$	−6.99308e6	−0.420939	−0.210470	+	0.977600i	$0.567499\pi$
$774$	−5.73049e6	−0.343827
$775$	4.24381e6	0.253806
$776$	5.04994e6	0.301045
$777$	−8.83258e6	−0.524850
$778$	6.14457e6	0.363951
$779$	−9.26871e6	−0.547237
$780$	−310386.	−0.0182669
$781$	2.13440e7	1.25213
$782$	876193.	0.0512369
$783$	−1.91648e6	−0.111712
$784$	2089.32	0.000121399 0
$785$	1.37807e6	0.0798171
$786$	5.74006e6	0.331406
$787$	577701.	0.0332481	0.0166240	−	0.999862i	$-0.494708\pi$
$787$	577701.	0.0332481	0.0166240	+	0.999862i	$0.494708\pi$
$788$	1.47745e7	0.847613
$789$	1.63064e7	0.932538
$790$	−881823.	−0.0502705
$791$	−1.08975e7	−0.619276
$792$	−3.61706e6	−0.204900
$793$	−477182.	−0.0269464
$794$	−1.51415e7	−0.852347
$795$	−1.22404e6	−0.0686874
$796$	9.43381e6	0.527721
$797$	−2.18951e7	−1.22096	−0.610480	−	0.792031i	$-0.709024\pi$
$797$	−2.18951e7	−1.22096	−0.610480	+	0.792031i	$0.709024\pi$
$798$	−2.37253e6	−0.131888
$799$	2.75389e6	0.152609
$800$	3.17927e6	0.175631
$801$	−2.34910e7	−1.29366
$802$	−7.90052e6	−0.433730
$803$	−2.47823e7	−1.35629
$804$	586752.	0.0320121
$805$	−1.14825e6	−0.0624523
$806$	−2.88115e6	−0.156217
$807$	−591987.	−0.0319984
$808$	−1.14377e7	−0.616328
$809$	3.20759e7	1.72309	0.861543	−	0.507685i	$-0.169499\pi$
$809$	3.20759e7	1.72309	0.861543	+	0.507685i	$0.169499\pi$
$810$	265259.	0.0142055
$811$	9.64590e6	0.514980	0.257490	−	0.966281i	$-0.417104\pi$
$811$	9.64590e6	0.514980	0.257490	+	0.966281i	$0.417104\pi$
$812$	−1.15979e6	−0.0617290
$813$	1.21511e7	0.644745
$814$	1.06900e7	0.565479
$815$	1.22939e6	0.0648329
$816$	233130.	0.0122567
$817$	4.54924e6	0.238442
$818$	1.00271e7	0.523951
$819$	1.20313e7	0.626765
$820$	−1.19355e6	−0.0619880
$821$	−2.78407e7	−1.44152	−0.720762	−	0.693183i	$-0.756208\pi$
$821$	−2.78407e7	−1.44152	−0.720762	+	0.693183i	$0.756208\pi$
$822$	−3.47513e6	−0.179387
$823$	−2.02943e7	−1.04442	−0.522210	−	0.852817i	$-0.674892\pi$
$823$	−2.02943e7	−1.04442	−0.522210	+	0.852817i	$0.674892\pi$
$824$	6.68480e6	0.342981
$825$	−8.15310e6	−0.417050
$826$	−5.34098e6	−0.272377
$827$	−2.50241e7	−1.27231	−0.636157	−	0.771560i	$-0.719477\pi$
$827$	−2.50241e7	−1.27231	−0.636157	+	0.771560i	$0.719477\pi$
$828$	−5.54367e6	−0.281010
$829$	5.70901e6	0.288519	0.144260	−	0.989540i	$-0.453920\pi$
$829$	5.70901e6	0.288519	0.144260	+	0.989540i	$0.453920\pi$
$830$	336056.	0.0169323
$831$	9.60351e6	0.482422
$832$	−2.15842e6	−0.108101
$833$	−908.478	−4.53630e−5 0
$834$	8.06798e6	0.401652
$835$	−2.60771e6	−0.129432
$836$	2.87146e6	0.142097
$837$	−4.68624e6	−0.231212
$838$	969800.	0.0477059
$839$	−4.39868e6	−0.215734	−0.107867	−	0.994165i	$-0.534402\pi$
$839$	−4.39868e6	−0.215734	−0.107867	+	0.994165i	$0.534402\pi$
$840$	−305517.	−0.0149395
$841$	−2.01987e7	−0.984766
$842$	1.02067e7	0.496143
$843$	4.32518e6	0.209621
$844$	5.87418e6	0.283851
$845$	421220.	0.0202940
$846$	−1.74239e7	−0.836988
$847$	−7.52338e6	−0.360334
$848$	−8.51196e6	−0.406481
$849$	−6.96391e6	−0.331577
$850$	−1.38241e6	−0.0656280
$851$	1.63840e7	0.775524
$852$	8.70395e6	0.410788
$853$	6.09800e6	0.286956	0.143478	−	0.989654i	$-0.454171\pi$
$853$	6.09800e6	0.286956	0.143478	+	0.989654i	$0.454171\pi$
$854$	−469697.	−0.0220380
$855$	−442974.	−0.0207235
$856$	−9.41010e6	−0.438944
$857$	−2.39347e7	−1.11321	−0.556604	−	0.830778i	$-0.687896\pi$
$857$	−2.39347e7	−1.11321	−0.556604	+	0.830778i	$0.687896\pi$
$858$	5.53519e6	0.256693
$859$	−3.26808e7	−1.51116	−0.755579	−	0.655058i	$-0.772644\pi$
$859$	−3.26808e7	−1.51116	−0.755579	+	0.655058i	$0.772644\pi$
$860$	585817.	0.0270095
$861$	−1.75867e7	−0.808492
$862$	−2.34713e7	−1.07589
$863$	−2.12310e6	−0.0970382	−0.0485191	−	0.998822i	$-0.515450\pi$
$863$	−2.12310e6	−0.0970382	−0.0485191	+	0.998822i	$0.515450\pi$
$864$	−3.51071e6	−0.159997
$865$	−1.47150e6	−0.0668683
$866$	−5.41456e6	−0.245340
$867$	1.15145e7	0.520233
$868$	−2.83596e6	−0.127762
$869$	1.57258e7	0.706420
$870$	82314.1	0.00368702
$871$	2.36212e6	0.105501
$872$	8.56128e6	0.381283
$873$	1.38929e7	0.616961
$874$	4.40093e6	0.194879
$875$	3.63512e6	0.160509
$876$	−1.01060e7	−0.444960
$877$	−2.14470e7	−0.941601	−0.470801	−	0.882240i	$-0.656035\pi$
$877$	−2.14470e7	−0.941601	−0.470801	+	0.882240i	$0.656035\pi$
$878$	1.26123e7	0.552152
$879$	8.70658e6	0.380080
$880$	369765.	0.0160960
$881$	5.27396e6	0.228927	0.114463	−	0.993427i	$-0.463485\pi$
$881$	5.27396e6	0.228927	0.114463	+	0.993427i	$0.463485\pi$
$882$	5747.94	0.000248794 0
$883$	−4.42714e7	−1.91083	−0.955414	−	0.295268i	$-0.904591\pi$
$883$	−4.42714e7	−1.91083	−0.955414	+	0.295268i	$0.904591\pi$
$884$	938525.	0.0403938
$885$	379067.	0.0162689
$886$	−6.07983e6	−0.260200
$887$	−2.07642e7	−0.886146	−0.443073	−	0.896485i	$-0.646112\pi$
$887$	−2.07642e7	−0.886146	−0.443073	+	0.896485i	$0.646112\pi$
$888$	4.35930e6	0.185517
$889$	2.98985e7	1.26881
$890$	2.40143e6	0.101624
$891$	−4.73043e6	−0.199621
$892$	6.43332e6	0.270722
$893$	1.38322e7	0.580448
$894$	2.95084e6	0.123481
$895$	−1.13610e6	−0.0474087
$896$	−2.12457e6	−0.0884098
$897$	8.48349e6	0.352041
$898$	1.27488e7	0.527568
$899$	764079.	0.0315311
$900$	8.74650e6	0.359938
$901$	3.70117e6	0.151889
$902$	2.12850e7	0.871078
$903$	8.63184e6	0.352277
$904$	5.37843e6	0.218894
$905$	−2.52566e6	−0.102507
$906$	193040.	0.00781315
$907$	−3.50219e7	−1.41358	−0.706792	−	0.707422i	$-0.749858\pi$
$907$	−3.50219e7	−1.41358	−0.706792	+	0.707422i	$0.749858\pi$
$908$	−1.54254e7	−0.620899
$909$	−3.14664e7	−1.26310
$910$	−1.22994e6	−0.0492358
$911$	−3.27024e7	−1.30552	−0.652761	−	0.757564i	$-0.726389\pi$
$911$	−3.27024e7	−1.30552	−0.652761	+	0.757564i	$0.726389\pi$
$912$	1.17096e6	0.0466181
$913$	−5.99297e6	−0.237939
$914$	2.00209e7	0.792719
$915$	33335.9	0.00131631
$916$	2.09697e7	0.825760
$917$	2.27457e7	0.893256
$918$	1.52653e6	0.0597858
$919$	−5.55289e6	−0.216885	−0.108443	−	0.994103i	$-0.534586\pi$
$919$	−5.55289e6	−0.216885	−0.108443	+	0.994103i	$0.534586\pi$
$920$	566719.	0.0220749
$921$	−2.67384e6	−0.103869
$922$	2.59661e7	1.00596
$923$	3.50401e7	1.35382
$924$	5.44837e6	0.209936
$925$	−2.58497e7	−0.993348
$926$	3.07953e7	1.18021
$927$	1.83906e7	0.702905
$928$	572412.	0.0218192
$929$	−1.98243e7	−0.753631	−0.376815	−	0.926288i	$-0.622981\pi$
$929$	−1.98243e7	−0.753631	−0.376815	+	0.926288i	$0.622981\pi$
$930$	201277.	0.00763110
$931$	−4563.09	−0.000172538 0
$932$	1.41003e7	0.531726
$933$	985836.	0.0370766
$934$	398510.	0.0149476
$935$	−160781.	−0.00601459
$936$	−5.93805e6	−0.221541
$937$	2.56390e7	0.954008	0.477004	−	0.878901i	$-0.341723\pi$
$937$	2.56390e7	0.954008	0.477004	+	0.878901i	$0.341723\pi$
$938$	2.32507e6	0.0862839
$939$	1.16868e7	0.432546
$940$	1.78121e6	0.0657499
$941$	−2.00098e7	−0.736663	−0.368331	−	0.929695i	$-0.620071\pi$
$941$	−2.00098e7	−0.736663	−0.368331	+	0.929695i	$0.620071\pi$
$942$	−1.00217e7	−0.367971
$943$	3.26224e7	1.19464
$944$	2.63603e6	0.0962765
$945$	−2.00052e6	−0.0728725
$946$	−1.04470e7	−0.379547
$947$	1.73797e7	0.629750	0.314875	−	0.949133i	$-0.398037\pi$
$947$	1.73797e7	0.629750	0.314875	+	0.949133i	$0.398037\pi$
$948$	6.41286e6	0.231756
$949$	−4.06845e7	−1.46644
$950$	−6.94354e6	−0.249616
$951$	1.18351e6	0.0424348
$952$	923804.	0.0330360
$953$	−4.04381e7	−1.44231	−0.721154	−	0.692775i	$-0.756388\pi$
$953$	−4.04381e7	−1.44231	−0.721154	+	0.692775i	$0.756388\pi$
$954$	−2.34173e7	−0.833040
$955$	−199015.	−0.00706120
$956$	2.45979e7	0.870469
$957$	−1.46793e6	−0.0518114
$958$	−3.31753e7	−1.16789
$959$	−1.37706e7	−0.483512
$960$	150787.	0.00528065
$961$	−2.67608e7	−0.934740
$962$	1.75495e7	0.611403
$963$	−2.58882e7	−0.899571
$964$	−4.85014e6	−0.168098
$965$	−3.74415e6	−0.129430
$966$	8.35043e6	0.287916
$967$	1.26654e7	0.435566	0.217783	−	0.975997i	$-0.430118\pi$
$967$	1.26654e7	0.435566	0.217783	+	0.975997i	$0.430118\pi$
$968$	3.71315e6	0.127366
$969$	−509156.	−0.0174197
$970$	−1.42024e6	−0.0484656
$971$	5.26371e7	1.79161	0.895806	−	0.444446i	$-0.146599\pi$
$971$	5.26371e7	1.79161	0.895806	+	0.444446i	$0.146599\pi$
$972$	−1.52588e7	−0.518029
$973$	3.19704e7	1.08259
$974$	1.18041e6	0.0398691
$975$	−1.33848e7	−0.450920
$976$	231818.	0.00778974
$977$	−2.32884e7	−0.780553	−0.390277	−	0.920698i	$-0.627621\pi$
$977$	−2.32884e7	−0.780553	−0.390277	+	0.920698i	$0.627621\pi$
$978$	−8.94047e6	−0.298891
$979$	−4.28254e7	−1.42806
$980$	−587.600	−1.95442e−5 0
$981$	2.35530e7	0.781400
$982$	−4.21915e7	−1.39620
$983$	4.53666e7	1.49745	0.748725	−	0.662881i	$-0.230666\pi$
$983$	4.53666e7	1.49745	0.748725	+	0.662881i	$0.230666\pi$
$984$	8.67987e6	0.285776
$985$	−4.15519e6	−0.136458
$986$	−248896.	−0.00815316
$987$	2.62456e7	0.857558
$988$	4.71401e6	0.153638
$989$	−1.60116e7	−0.520528
$990$	1.01726e6	0.0329871
$991$	−3.18589e6	−0.103050	−0.0515249	−	0.998672i	$-0.516408\pi$
$991$	−3.18589e6	−0.103050	−0.0515249	+	0.998672i	$0.516408\pi$
$992$	1.39968e6	0.0451596
$993$	−1.66681e7	−0.536431
$994$	3.44905e7	1.10722
$995$	−2.65316e6	−0.0849584
$996$	−2.44389e6	−0.0780610
$997$	2.31939e7	0.738985	0.369492	−	0.929234i	$-0.379532\pi$
$997$	2.31939e7	0.738985	0.369492	+	0.929234i	$0.379532\pi$
$998$	2.34670e7	0.745814
$999$	2.85446e7	0.904921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.b.1.10	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.b.1.10	✓	27	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:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -8.18103 q^{3} +16.0000 q^{4} -4.49984 q^{5} +32.7241 q^{6} +129.673 q^{7} -64.0000 q^{8} -176.071 q^{9} +17.9994 q^{10} -320.988 q^{11} -130.896 q^{12} -526.959 q^{13} -518.693 q^{14} +36.8133 q^{15} +256.000 q^{16} -111.314 q^{17} +704.283 q^{18} -559.106 q^{19} -71.9975 q^{20} -1060.86 q^{21} +1283.95 q^{22} +1967.84 q^{23} +523.586 q^{24} -3104.75 q^{25} +2107.83 q^{26} +3428.43 q^{27} +2074.77 q^{28} -558.996 q^{29} -147.253 q^{30} -1366.88 q^{31} -1024.00 q^{32} +2626.01 q^{33} +445.256 q^{34} -583.509 q^{35} -2817.13 q^{36} +8325.86 q^{37} +2236.42 q^{38} +4311.06 q^{39} +287.990 q^{40} +16577.7 q^{41} +4243.44 q^{42} -8136.63 q^{43} -5135.80 q^{44} +792.291 q^{45} -7871.37 q^{46} -24739.9 q^{47} -2094.34 q^{48} +8.16140 q^{49} +12419.0 q^{50} +910.662 q^{51} -8431.34 q^{52} -33249.8 q^{53} -13713.7 q^{54} +1444.39 q^{55} -8299.09 q^{56} +4574.06 q^{57} +2235.99 q^{58} +10297.0 q^{59} +589.013 q^{60} +905.539 q^{61} +5467.51 q^{62} -22831.7 q^{63} +4096.00 q^{64} +2371.23 q^{65} -10504.0 q^{66} -4482.56 q^{67} -1781.02 q^{68} -16099.0 q^{69} +2334.04 q^{70} -66494.9 q^{71} +11268.5 q^{72} +77206.4 q^{73} -33303.4 q^{74} +25400.1 q^{75} -8945.69 q^{76} -41623.5 q^{77} -17244.3 q^{78} -48991.9 q^{79} -1151.96 q^{80} +14737.1 q^{81} -66310.9 q^{82} +18670.4 q^{83} -16973.8 q^{84} +500.895 q^{85} +32546.5 q^{86} +4573.17 q^{87} +20543.2 q^{88} +133418. q^{89} -3169.16 q^{90} -68332.4 q^{91} +31485.5 q^{92} +11182.5 q^{93} +98959.6 q^{94} +2515.89 q^{95} +8377.37 q^{96} -78905.2 q^{97} -32.6456 q^{98} +56516.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 108 q^{2} + 33 q^{3} + 432 q^{4} + 139 q^{5} - 132 q^{6} - 25 q^{7} - 1728 q^{8} + 2346 q^{9} - 556 q^{10} + 1241 q^{11} + 528 q^{12} - 202 q^{13} + 100 q^{14} + 1786 q^{15} + 6912 q^{16} + 1550 q^{17}+ \cdots + 335378 q^{99}+O(q^{100}) \) 27 * q - 108 * q^2 + 33 * q^3 + 432 * q^4 + 139 * q^5 - 132 * q^6 - 25 * q^7 - 1728 * q^8 + 2346 * q^9 - 556 * q^10 + 1241 * q^11 + 528 * q^12 - 202 * q^13 + 100 * q^14 + 1786 * q^15 + 6912 * q^16 + 1550 * q^17 - 9384 * q^18 - 66 * q^19 + 2224 * q^20 - 1217 * q^21 - 4964 * q^22 + 10568 * q^23 - 2112 * q^24 + 15880 * q^25 + 808 * q^26 + 8256 * q^27 - 400 * q^28 - 2122 * q^29 - 7144 * q^30 - 1621 * q^31 - 27648 * q^32 + 18369 * q^33 - 6200 * q^34 + 23123 * q^35 + 37536 * q^36 + 8542 * q^37 + 264 * q^38 + 589 * q^39 - 8896 * q^40 + 10859 * q^41 + 4868 * q^42 + 23759 * q^43 + 19856 * q^44 + 28006 * q^45 - 42272 * q^46 + 38457 * q^47 + 8448 * q^48 + 57376 * q^49 - 63520 * q^50 + 58237 * q^51 - 3232 * q^52 + 73586 * q^53 - 33024 * q^54 + 25923 * q^55 + 1600 * q^56 + 52534 * q^57 + 8488 * q^58 + 68927 * q^59 + 28576 * q^60 - 76016 * q^61 + 6484 * q^62 - 20738 * q^63 + 110592 * q^64 + 74867 * q^65 - 73476 * q^66 + 94787 * q^67 + 24800 * q^68 - 90601 * q^69 - 92492 * q^70 + 15523 * q^71 - 150144 * q^72 - 96279 * q^73 - 34168 * q^74 - 123786 * q^75 - 1056 * q^76 - 199866 * q^77 - 2356 * q^78 - 246217 * q^79 + 35584 * q^80 - 54473 * q^81 - 43436 * q^82 - 56658 * q^83 - 19472 * q^84 - 120541 * q^85 - 95036 * q^86 - 352954 * q^87 - 79424 * q^88 - 83070 * q^89 - 112024 * q^90 - 132348 * q^91 + 169088 * q^92 + 40727 * q^93 - 153828 * q^94 - 72090 * q^95 - 33792 * q^96 - 36273 * q^97 - 229504 * q^98 + 335378 * q^99

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