LMFDB - Embedded newform 538.6.a.c.1.13

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:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	\(q-4.00000 q^{2} -7.00462 q^{3} +16.0000 q^{4} +30.1411 q^{5} +28.0185 q^{6} -141.413 q^{7} -64.0000 q^{8} -193.935 q^{9} -120.565 q^{10} +146.526 q^{11} -112.074 q^{12} -116.391 q^{13} +565.653 q^{14} -211.127 q^{15} +256.000 q^{16} +782.764 q^{17} +775.741 q^{18} +1775.40 q^{19} +482.258 q^{20} +990.547 q^{21} -586.103 q^{22} -132.024 q^{23} +448.296 q^{24} -2216.51 q^{25} +465.565 q^{26} +3060.57 q^{27} -2262.61 q^{28} -254.793 q^{29} +844.510 q^{30} +7579.28 q^{31} -1024.00 q^{32} -1026.36 q^{33} -3131.05 q^{34} -4262.36 q^{35} -3102.96 q^{36} -10076.2 q^{37} -7101.60 q^{38} +815.277 q^{39} -1929.03 q^{40} +19015.2 q^{41} -3962.19 q^{42} -10210.4 q^{43} +2344.41 q^{44} -5845.43 q^{45} +528.095 q^{46} -13642.4 q^{47} -1793.18 q^{48} +3190.71 q^{49} +8866.05 q^{50} -5482.97 q^{51} -1862.26 q^{52} -3407.93 q^{53} -12242.3 q^{54} +4416.45 q^{55} +9050.45 q^{56} -12436.0 q^{57} +1019.17 q^{58} +22157.1 q^{59} -3378.04 q^{60} +20595.6 q^{61} -30317.1 q^{62} +27425.0 q^{63} +4096.00 q^{64} -3508.17 q^{65} +4105.43 q^{66} +26682.3 q^{67} +12524.2 q^{68} +924.777 q^{69} +17049.4 q^{70} +45901.5 q^{71} +12411.9 q^{72} -49954.2 q^{73} +40305.0 q^{74} +15525.8 q^{75} +28406.4 q^{76} -20720.7 q^{77} -3261.11 q^{78} -35619.0 q^{79} +7716.13 q^{80} +25688.1 q^{81} -76061.0 q^{82} +7121.80 q^{83} +15848.7 q^{84} +23593.4 q^{85} +40841.4 q^{86} +1784.73 q^{87} -9377.65 q^{88} -76444.8 q^{89} +23381.7 q^{90} +16459.3 q^{91} -2112.38 q^{92} -53090.0 q^{93} +54569.6 q^{94} +53512.6 q^{95} +7172.74 q^{96} +91550.6 q^{97} -12762.8 q^{98} -28416.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 120 q^{2} - 30 q^{3} + 480 q^{4} - 136 q^{5} + 120 q^{6} - 123 q^{7} - 1920 q^{8} + 2670 q^{9} + 544 q^{10} - 1058 q^{11} - 480 q^{12} - 371 q^{13} + 492 q^{14} - 1364 q^{15} + 7680 q^{16} - 1918 q^{17}+ \cdots - 78063 q^{99}+O(q^{100}) $ 30 * q - 120 * q^2 - 30 * q^3 + 480 * q^4 - 136 * q^5 + 120 * q^6 - 123 * q^7 - 1920 * q^8 + 2670 * q^9 + 544 * q^10 - 1058 * q^11 - 480 * q^12 - 371 * q^13 + 492 * q^14 - 1364 * q^15 + 7680 * q^16 - 1918 * q^17 - 10680 * q^18 - 3315 * q^19 - 2176 * q^20 - 1217 * q^21 + 4232 * q^22 - 9534 * q^23 + 1920 * q^24 + 22130 * q^25 + 1484 * q^26 - 12156 * q^27 - 1968 * q^28 - 8009 * q^29 + 5456 * q^30 - 3543 * q^31 - 30720 * q^32 - 3411 * q^33 + 7672 * q^34 - 11177 * q^35 + 42720 * q^36 - 6517 * q^37 + 13260 * q^38 - 2453 * q^39 + 8704 * q^40 - 5951 * q^41 + 4868 * q^42 - 18768 * q^43 - 16928 * q^44 - 22619 * q^45 + 38136 * q^46 - 41067 * q^47 - 7680 * q^48 + 107797 * q^49 - 88520 * q^50 - 4187 * q^51 - 5936 * q^52 - 64055 * q^53 + 48624 * q^54 + 31973 * q^55 + 7872 * q^56 - 25442 * q^57 + 32036 * q^58 - 66832 * q^59 - 21824 * q^60 + 46777 * q^61 + 14172 * q^62 - 115994 * q^63 + 122880 * q^64 - 68783 * q^65 + 13644 * q^66 - 107218 * q^67 - 30688 * q^68 + 228589 * q^69 + 44708 * q^70 + 4733 * q^71 - 170880 * q^72 + 150385 * q^73 + 26068 * q^74 + 133325 * q^75 - 53040 * q^76 + 292020 * q^77 + 9812 * q^78 + 31827 * q^79 - 34816 * q^80 + 425214 * q^81 + 23804 * q^82 + 107301 * q^83 - 19472 * q^84 + 338625 * q^85 + 75072 * q^86 + 288484 * q^87 + 67712 * q^88 - 99690 * q^89 + 90476 * q^90 - 109140 * q^91 - 152544 * q^92 - 114545 * q^93 + 164268 * q^94 - 155668 * q^95 + 30720 * q^96 - 79959 * q^97 - 431188 * q^98 - 78063 * 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
$3$	−7.00462	−0.449347	−0.224673	−	0.974434i	$-0.572132\pi$
$3$	−7.00462	−0.449347	−0.224673	+	0.974434i	$0.572132\pi$
$4$	16.0000	0.500000
$5$	30.1411	0.539181	0.269591	−	0.962975i	$-0.413112\pi$
$5$	30.1411	0.539181	0.269591	+	0.962975i	$0.413112\pi$
$6$	28.0185	0.317736
$7$	−141.413	−1.09080	−0.545400	−	0.838176i	$-0.683622\pi$
$7$	−141.413	−1.09080	−0.545400	+	0.838176i	$0.683622\pi$
$8$	−64.0000	−0.353553
$9$	−193.935	−0.798087
$10$	−120.565	−0.381259
$11$	146.526	0.365117	0.182559	−	0.983195i	$-0.441562\pi$
$11$	146.526	0.365117	0.182559	+	0.983195i	$0.441562\pi$
$12$	−112.074	−0.224673
$13$	−116.391	−0.191013	−0.0955063	−	0.995429i	$-0.530447\pi$
$13$	−116.391	−0.191013	−0.0955063	+	0.995429i	$0.530447\pi$
$14$	565.653	0.771312
$15$	−211.127	−0.242279
$16$	256.000	0.250000
$17$	782.764	0.656914	0.328457	−	0.944519i	$-0.393471\pi$
$17$	782.764	0.656914	0.328457	+	0.944519i	$0.393471\pi$
$18$	775.741	0.564333
$19$	1775.40	1.12827	0.564134	−	0.825683i	$-0.309210\pi$
$19$	1775.40	1.12827	0.564134	+	0.825683i	$0.309210\pi$
$20$	482.258	0.269591
$21$	990.547	0.490147
$22$	−586.103	−0.258177
$23$	−132.024	−0.0520394	−0.0260197	−	0.999661i	$-0.508283\pi$
$23$	−132.024	−0.0520394	−0.0260197	+	0.999661i	$0.508283\pi$
$24$	448.296	0.158868
$25$	−2216.51	−0.709284
$26$	465.565	0.135066
$27$	3060.57	0.807965
$28$	−2262.61	−0.545400
$29$	−254.793	−0.0562590	−0.0281295	−	0.999604i	$-0.508955\pi$
$29$	−254.793	−0.0562590	−0.0281295	+	0.999604i	$0.508955\pi$
$30$	844.510	0.171317
$31$	7579.28	1.41652	0.708261	−	0.705950i	$-0.249480\pi$
$31$	7579.28	1.41652	0.708261	+	0.705950i	$0.249480\pi$
$32$	−1024.00	−0.176777
$33$	−1026.36	−0.164064
$34$	−3131.05	−0.464508
$35$	−4262.36	−0.588139
$36$	−3102.96	−0.399044
$37$	−10076.2	−1.21002	−0.605012	−	0.796216i	$-0.706832\pi$
$37$	−10076.2	−1.21002	−0.605012	+	0.796216i	$0.706832\pi$
$38$	−7101.60	−0.797806
$39$	815.277	0.0858309
$40$	−1929.03	−0.190629
$41$	19015.2	1.76662	0.883308	−	0.468793i	$-0.155311\pi$
$41$	19015.2	1.76662	0.883308	+	0.468793i	$0.155311\pi$
$42$	−3962.19	−0.346587
$43$	−10210.4	−0.842111	−0.421056	−	0.907035i	$-0.638340\pi$
$43$	−10210.4	−0.842111	−0.421056	+	0.907035i	$0.638340\pi$
$44$	2344.41	0.182559
$45$	−5845.43	−0.430314
$46$	528.095	0.0367974
$47$	−13642.4	−0.900836	−0.450418	−	0.892818i	$-0.648725\pi$
$47$	−13642.4	−0.900836	−0.450418	+	0.892818i	$0.648725\pi$
$48$	−1793.18	−0.112337
$49$	3190.71	0.189844
$50$	8866.05	0.501539
$51$	−5482.97	−0.295182
$52$	−1862.26	−0.0955063
$53$	−3407.93	−0.166649	−0.0833243	−	0.996522i	$-0.526554\pi$
$53$	−3407.93	−0.166649	−0.0833243	+	0.996522i	$0.526554\pi$
$54$	−12242.3	−0.571317
$55$	4416.45	0.196864
$56$	9050.45	0.385656
$57$	−12436.0	−0.506984
$58$	1019.17	0.0397811
$59$	22157.1	0.828672	0.414336	−	0.910124i	$-0.364014\pi$
$59$	22157.1	0.828672	0.414336	+	0.910124i	$0.364014\pi$
$60$	−3378.04	−0.121140
$61$	20595.6	0.708680	0.354340	−	0.935117i	$-0.384706\pi$
$61$	20595.6	0.708680	0.354340	+	0.935117i	$0.384706\pi$
$62$	−30317.1	−1.00163
$63$	27425.0	0.870554
$64$	4096.00	0.125000
$65$	−3508.17	−0.102990
$66$	4105.43	0.116011
$67$	26682.3	0.726166	0.363083	−	0.931757i	$-0.381724\pi$
$67$	26682.3	0.726166	0.363083	+	0.931757i	$0.381724\pi$
$68$	12524.2	0.328457
$69$	924.777	0.0233838
$70$	17049.4	0.415877
$71$	45901.5	1.08064	0.540321	−	0.841459i	$-0.318303\pi$
$71$	45901.5	1.08064	0.540321	+	0.841459i	$0.318303\pi$
$72$	12411.9	0.282167
$73$	−49954.2	−1.09715	−0.548574	−	0.836102i	$-0.684829\pi$
$73$	−49954.2	−1.09715	−0.548574	+	0.836102i	$0.684829\pi$
$74$	40305.0	0.855616
$75$	15525.8	0.318714
$76$	28406.4	0.564134
$77$	−20720.7	−0.398270
$78$	−3261.11	−0.0606916
$79$	−35619.0	−0.642117	−0.321058	−	0.947059i	$-0.604039\pi$
$79$	−35619.0	−0.642117	−0.321058	+	0.947059i	$0.604039\pi$
$80$	7716.13	0.134795
$81$	25688.1	0.435031
$82$	−76061.0	−1.24919
$83$	7121.80	0.113473	0.0567367	−	0.998389i	$-0.481930\pi$
$83$	7121.80	0.113473	0.0567367	+	0.998389i	$0.481930\pi$
$84$	15848.7	0.245074
$85$	23593.4	0.354195
$86$	40841.4	0.595463
$87$	1784.73	0.0252798
$88$	−9377.65	−0.129088
$89$	−76444.8	−1.02299	−0.511497	−	0.859285i	$-0.670909\pi$
$89$	−76444.8	−1.02299	−0.511497	+	0.859285i	$0.670909\pi$
$90$	23381.7	0.304278
$91$	16459.3	0.208356
$92$	−2112.38	−0.0260197
$93$	−53090.0	−0.636510
$94$	54569.6	0.636987
$95$	53512.6	0.608341
$96$	7172.74	0.0794341
$97$	91550.6	0.987943	0.493972	−	0.869478i	$-0.335545\pi$
$97$	91550.6	0.987943	0.493972	+	0.869478i	$0.335545\pi$
$98$	−12762.8	−0.134240
$99$	−28416.5	−0.291395
$100$	−35464.2	−0.354642
$101$	124214.	1.21162	0.605809	−	0.795610i	$-0.292850\pi$
$101$	124214.	1.21162	0.605809	+	0.795610i	$0.292850\pi$
$102$	21931.9	0.208725
$103$	−103888.	−0.964880	−0.482440	−	0.875929i	$-0.660249\pi$
$103$	−103888.	−0.964880	−0.482440	+	0.875929i	$0.660249\pi$
$104$	7449.04	0.0675331
$105$	29856.2	0.264278
$106$	13631.7	0.117838
$107$	−107447.	−0.907269	−0.453635	−	0.891188i	$-0.649873\pi$
$107$	−107447.	−0.907269	−0.453635	+	0.891188i	$0.649873\pi$
$108$	48969.1	0.403982
$109$	28106.3	0.226588	0.113294	−	0.993561i	$-0.463860\pi$
$109$	28106.3	0.226588	0.113294	+	0.993561i	$0.463860\pi$
$110$	−17665.8	−0.139204
$111$	70580.3	0.543721
$112$	−36201.8	−0.272700
$113$	173201.	1.27601	0.638004	−	0.770033i	$-0.279760\pi$
$113$	173201.	1.27601	0.638004	+	0.770033i	$0.279760\pi$
$114$	49744.0	0.358492
$115$	−3979.35	−0.0280587
$116$	−4076.68	−0.0281295
$117$	22572.4	0.152445
$118$	−88628.3	−0.585959
$119$	−110693.	−0.716561
$120$	13512.2	0.0856587
$121$	−139581.	−0.866689
$122$	−82382.4	−0.501112
$123$	−133195.	−0.793823
$124$	121268.	0.708261
$125$	−160999.	−0.921614
$126$	−109700.	−0.615574
$127$	−182020.	−1.00141	−0.500703	−	0.865619i	$-0.666925\pi$
$127$	−182020.	−1.00141	−0.500703	+	0.865619i	$0.666925\pi$
$128$	−16384.0	−0.0883883
$129$	71519.7	0.378400
$130$	14032.7	0.0728252
$131$	−376457.	−1.91663	−0.958313	−	0.285720i	$-0.907767\pi$
$131$	−376457.	−1.91663	−0.958313	+	0.285720i	$0.907767\pi$
$132$	−16421.7	−0.0820321
$133$	−251065.	−1.23071
$134$	−106729.	−0.513477
$135$	92249.0	0.435639
$136$	−50096.9	−0.232254
$137$	−109528.	−0.498568	−0.249284	−	0.968430i	$-0.580195\pi$
$137$	−109528.	−0.498568	−0.249284	+	0.968430i	$0.580195\pi$
$138$	−3699.11	−0.0165348
$139$	−204804.	−0.899087	−0.449543	−	0.893259i	$-0.648413\pi$
$139$	−204804.	−0.899087	−0.449543	+	0.893259i	$0.648413\pi$
$140$	−68197.7	−0.294069
$141$	95559.8	0.404788
$142$	−183606.	−0.764129
$143$	−17054.3	−0.0697420
$144$	−49647.4	−0.199522
$145$	−7679.74	−0.0303338
$146$	199817.	0.775800
$147$	−22349.7	−0.0853059
$148$	−161220.	−0.605012
$149$	262832.	0.969867	0.484933	−	0.874551i	$-0.338844\pi$
$149$	262832.	0.969867	0.484933	+	0.874551i	$0.338844\pi$
$150$	−62103.3	−0.225365
$151$	−368740.	−1.31607	−0.658034	−	0.752989i	$-0.728612\pi$
$151$	−368740.	−1.31607	−0.658034	+	0.752989i	$0.728612\pi$
$152$	−113626.	−0.398903
$153$	−151805.	−0.524275
$154$	82882.7	0.281619
$155$	228448.	0.763762
$156$	13044.4	0.0429155
$157$	−313590.	−1.01534	−0.507672	−	0.861550i	$-0.669494\pi$
$157$	−313590.	−1.01534	−0.507672	+	0.861550i	$0.669494\pi$
$158$	142476.	0.454045
$159$	23871.3	0.0748830
$160$	−30864.5	−0.0953147
$161$	18669.9	0.0567646
$162$	−102753.	−0.307613
$163$	−16471.0	−0.0485569	−0.0242785	−	0.999705i	$-0.507729\pi$
$163$	−16471.0	−0.0485569	−0.0242785	+	0.999705i	$0.507729\pi$
$164$	304244.	0.883308
$165$	−30935.6	−0.0884604
$166$	−28487.2	−0.0802379
$167$	194218.	0.538888	0.269444	−	0.963016i	$-0.413160\pi$
$167$	194218.	0.538888	0.269444	+	0.963016i	$0.413160\pi$
$168$	−63395.0	−0.173293
$169$	−357746.	−0.963514
$170$	−94373.6	−0.250454
$171$	−344313.	−0.900456
$172$	−163366.	−0.421056
$173$	−321551.	−0.816836	−0.408418	−	0.912795i	$-0.633919\pi$
$173$	−321551.	−0.816836	−0.408418	+	0.912795i	$0.633919\pi$
$174$	−7138.91	−0.0178755
$175$	313444.	0.773687
$176$	37510.6	0.0912793
$177$	−155202.	−0.372361
$178$	305779.	0.723365
$179$	−589979.	−1.37627	−0.688136	−	0.725582i	$-0.741571\pi$
$179$	−589979.	−1.37627	−0.688136	+	0.725582i	$0.741571\pi$
$180$	−93526.9	−0.215157
$181$	−108550.	−0.246282	−0.123141	−	0.992389i	$-0.539297\pi$
$181$	−108550.	−0.246282	−0.123141	+	0.992389i	$0.539297\pi$
$182$	−65837.1	−0.147330
$183$	−144264.	−0.318443
$184$	8449.52	0.0183987
$185$	−303709.	−0.652422
$186$	212360.	0.450081
$187$	114695.	0.239850
$188$	−218278.	−0.450418
$189$	−432805.	−0.881328
$190$	−214050.	−0.430162
$191$	491532.	0.974918	0.487459	−	0.873146i	$-0.337924\pi$
$191$	491532.	0.974918	0.487459	+	0.873146i	$0.337924\pi$
$192$	−28690.9	−0.0561684
$193$	600081.	1.15962	0.579811	−	0.814751i	$-0.303126\pi$
$193$	600081.	1.15962	0.579811	+	0.814751i	$0.303126\pi$
$194$	−366202.	−0.698581
$195$	24573.4	0.0462784
$196$	51051.4	0.0949221
$197$	71656.5	0.131550	0.0657749	−	0.997834i	$-0.479048\pi$
$197$	71656.5	0.131550	0.0657749	+	0.997834i	$0.479048\pi$
$198$	113666.	0.206048
$199$	−504730.	−0.903496	−0.451748	−	0.892146i	$-0.649199\pi$
$199$	−504730.	−0.903496	−0.451748	+	0.892146i	$0.649199\pi$
$200$	141857.	0.250770
$201$	−186899.	−0.326300
$202$	−496854.	−0.856743
$203$	36031.1	0.0613673
$204$	−87727.4	−0.147591
$205$	573141.	0.952526
$206$	415553.	0.682273
$207$	25604.1	0.0415320
$208$	−29796.2	−0.0477531
$209$	260142.	0.411950
$210$	−119425.	−0.186873
$211$	238538.	0.368852	0.184426	−	0.982846i	$-0.440957\pi$
$211$	238538.	0.368852	0.184426	+	0.982846i	$0.440957\pi$
$212$	−54526.9	−0.0833243
$213$	−321523.	−0.485583
$214$	429789.	0.641536
$215$	−307752.	−0.454051
$216$	−195876.	−0.285659
$217$	−1.07181e6	−1.54514
$218$	−112425.	−0.160222
$219$	349910.	0.493000
$220$	70663.3	0.0984322
$221$	−91106.8	−0.125479
$222$	−282321.	−0.384469
$223$	103013.	0.138717	0.0693584	−	0.997592i	$-0.477905\pi$
$223$	103013.	0.138717	0.0693584	+	0.997592i	$0.477905\pi$
$224$	144807.	0.192828
$225$	429860.	0.566070
$226$	−692802.	−0.902273
$227$	−416223.	−0.536119	−0.268059	−	0.963402i	$-0.586382\pi$
$227$	−416223.	−0.536119	−0.268059	+	0.963402i	$0.586382\pi$
$228$	−198976.	−0.253492
$229$	732743.	0.923343	0.461672	−	0.887051i	$-0.347250\pi$
$229$	732743.	0.923343	0.461672	+	0.887051i	$0.347250\pi$
$230$	15917.4	0.0198405
$231$	145141.	0.178961
$232$	16306.7	0.0198906
$233$	406445.	0.490470	0.245235	−	0.969464i	$-0.421135\pi$
$233$	406445.	0.490470	0.245235	+	0.969464i	$0.421135\pi$
$234$	−90289.5	−0.107795
$235$	−411197.	−0.485714
$236$	354513.	0.414336
$237$	249498.	0.288533
$238$	442773.	0.506685
$239$	−1.49130e6	−1.68877	−0.844383	−	0.535741i	$-0.820032\pi$
$239$	−1.49130e6	−1.68877	−0.844383	+	0.535741i	$0.820032\pi$
$240$	−54048.6	−0.0605698
$241$	−1.23609e6	−1.37091	−0.685455	−	0.728115i	$-0.740397\pi$
$241$	−1.23609e6	−1.37091	−0.685455	+	0.728115i	$0.740397\pi$
$242$	558325.	0.612842
$243$	−923654.	−1.00344
$244$	329530.	0.354340
$245$	96171.7	0.102360
$246$	532779.	0.561318
$247$	−206641.	−0.215513
$248$	−485074.	−0.500816
$249$	−49885.5	−0.0509890
$250$	643997.	0.651679
$251$	−1.38010e6	−1.38269	−0.691347	−	0.722522i	$-0.742983\pi$
$251$	−1.38010e6	−1.38269	−0.691347	+	0.722522i	$0.742983\pi$
$252$	438800.	0.435277
$253$	−19344.9	−0.0190005
$254$	728081.	0.708101
$255$	−165263.	−0.159157
$256$	65536.0	0.0625000
$257$	9837.81	0.00929106	0.00464553	−	0.999989i	$-0.498521\pi$
$257$	9837.81	0.00929106	0.00464553	+	0.999989i	$0.498521\pi$
$258$	−286079.	−0.267569
$259$	1.42491e6	1.31989
$260$	−56130.6	−0.0514952
$261$	49413.3	0.0448996
$262$	1.50583e6	1.35526
$263$	174206.	0.155301	0.0776506	−	0.996981i	$-0.475258\pi$
$263$	174206.	0.155301	0.0776506	+	0.996981i	$0.475258\pi$
$264$	65686.9	0.0580055
$265$	−102719.	−0.0898537
$266$	1.00426e6	0.870246
$267$	535467.	0.459679
$268$	426916.	0.363083
$269$	−72361.0	−0.0609711
$269$	−72361.0	−0.0609711
$270$	−368996.	−0.308044
$271$	−1.75856e6	−1.45457	−0.727283	−	0.686338i	$-0.759217\pi$
$271$	−1.75856e6	−1.45457	−0.727283	+	0.686338i	$0.759217\pi$
$272$	200387.	0.164228
$273$	−115291.	−0.0936243
$274$	438113.	0.352541
$275$	−324776.	−0.258972
$276$	14796.4	0.0116919
$277$	1.92015e6	1.50361	0.751807	−	0.659383i	$-0.229182\pi$
$277$	1.92015e6	1.50361	0.751807	+	0.659383i	$0.229182\pi$
$278$	819216.	0.635750
$279$	−1.46989e6	−1.13051
$280$	272791.	0.207938
$281$	283251.	0.213996	0.106998	−	0.994259i	$-0.465876\pi$
$281$	283251.	0.213996	0.106998	+	0.994259i	$0.465876\pi$
$282$	−382239.	−0.286228
$283$	−1.67620e6	−1.24411	−0.622056	−	0.782973i	$-0.713702\pi$
$283$	−1.67620e6	−1.24411	−0.622056	+	0.782973i	$0.713702\pi$
$284$	734425.	0.540321
$285$	−374836.	−0.273356
$286$	68217.3	0.0493150
$287$	−2.68901e6	−1.92702
$288$	198590.	0.141083
$289$	−807138.	−0.568464
$290$	30719.0	0.0214492
$291$	−641278.	−0.443929
$292$	−799267.	−0.548574
$293$	−1.08332e6	−0.737203	−0.368602	−	0.929587i	$-0.620163\pi$
$293$	−1.08332e6	−0.737203	−0.368602	+	0.929587i	$0.620163\pi$
$294$	89398.9	0.0603204
$295$	667840.	0.446804
$296$	644879.	0.427808
$297$	448452.	0.295002
$298$	−1.05133e6	−0.685799
$299$	15366.4	0.00994019
$300$	248413.	0.159357
$301$	1.44388e6	0.918575
$302$	1.47496e6	0.930600
$303$	−870070.	−0.544437
$304$	454502.	0.282067
$305$	620775.	0.382107
$306$	607222.	0.370718
$307$	1.09219e6	0.661383	0.330691	−	0.943739i	$-0.392718\pi$
$307$	1.09219e6	0.661383	0.330691	+	0.943739i	$0.392718\pi$
$308$	−331531.	−0.199135
$309$	727698.	0.433566
$310$	−913792.	−0.540061
$311$	−1.33500e6	−0.782672	−0.391336	−	0.920248i	$-0.627987\pi$
$311$	−1.33500e6	−0.782672	−0.391336	+	0.920248i	$0.627987\pi$
$312$	−52177.7	−0.0303458
$313$	−1.91972e6	−1.10759	−0.553794	−	0.832654i	$-0.686820\pi$
$313$	−1.91972e6	−1.10759	−0.553794	+	0.832654i	$0.686820\pi$
$314$	1.25436e6	0.717957
$315$	826621.	0.469386
$316$	−569904.	−0.321058
$317$	2.25945e6	1.26286	0.631429	−	0.775434i	$-0.282469\pi$
$317$	2.25945e6	1.26286	0.631429	+	0.775434i	$0.282469\pi$
$318$	−95485.2	−0.0529503
$319$	−37333.7	−0.0205411
$320$	123458.	0.0673976
$321$	752628.	0.407679
$322$	−74679.7	−0.0401386
$323$	1.38972e6	0.741175
$324$	411010.	0.217515
$325$	257983.	0.135482
$326$	65884.0	0.0343349
$327$	−196874.	−0.101817
$328$	−1.21698e6	−0.624593
$329$	1.92921e6	0.982632
$330$	123742.	0.0625509
$331$	1.34558e6	0.675054	0.337527	−	0.941316i	$-0.390410\pi$
$331$	1.34558e6	0.675054	0.337527	+	0.941316i	$0.390410\pi$
$332$	113949.	0.0567367
$333$	1.95414e6	0.965705
$334$	−776872.	−0.381051
$335$	804234.	0.391535
$336$	253580.	0.122537
$337$	−1.87630e6	−0.899969	−0.449985	−	0.893036i	$-0.648571\pi$
$337$	−1.87630e6	−0.899969	−0.449985	+	0.893036i	$0.648571\pi$
$338$	1.43098e6	0.681307
$339$	−1.21320e6	−0.573370
$340$	377494.	0.177098
$341$	1.11056e6	0.517197
$342$	1.37725e6	0.636719
$343$	1.92552e6	0.883718
$344$	653462.	0.297731
$345$	27873.8	0.0126081
$346$	1.28620e6	0.577590
$347$	−1.35820e6	−0.605534	−0.302767	−	0.953065i	$-0.597910\pi$
$347$	−1.35820e6	−0.605534	−0.302767	+	0.953065i	$0.597910\pi$
$348$	28555.6	0.0126399
$349$	59218.0	0.0260250	0.0130125	−	0.999915i	$-0.495858\pi$
$349$	59218.0	0.0260250	0.0130125	+	0.999915i	$0.495858\pi$
$350$	−1.25378e6	−0.547079
$351$	−356223.	−0.154331
$352$	−150042.	−0.0645442
$353$	−2.57774e6	−1.10104	−0.550519	−	0.834823i	$-0.685570\pi$
$353$	−2.57774e6	−1.10104	−0.550519	+	0.834823i	$0.685570\pi$
$354$	620808.	0.263299
$355$	1.38353e6	0.582661
$356$	−1.22312e6	−0.511497
$357$	775364.	0.321985
$358$	2.35992e6	0.973171
$359$	−4.01621e6	−1.64468	−0.822338	−	0.568999i	$-0.807331\pi$
$359$	−4.01621e6	−1.64468	−0.822338	+	0.568999i	$0.807331\pi$
$360$	374107.	0.152139
$361$	675946.	0.272988
$362$	434200.	0.174148
$363$	977714.	0.389444
$364$	263348.	0.104178
$365$	−1.50568e6	−0.591561
$366$	577058.	0.225173
$367$	2.27437e6	0.881448	0.440724	−	0.897643i	$-0.354722\pi$
$367$	2.27437e6	0.881448	0.440724	+	0.897643i	$0.354722\pi$
$368$	−33798.1	−0.0130099
$369$	−3.68773e6	−1.40991
$370$	1.21484e6	0.461332
$371$	481927.	0.181780
$372$	−849440.	−0.318255
$373$	−6160.18	−0.00229256	−0.00114628	−	0.999999i	$-0.500365\pi$
$373$	−6160.18	−0.00229256	−0.00114628	+	0.999999i	$0.500365\pi$
$374$	−458780.	−0.169600
$375$	1.12774e6	0.414124
$376$	873113.	0.318494
$377$	29655.6	0.0107462
$378$	1.73122e6	0.623193
$379$	4.74069e6	1.69529	0.847644	−	0.530565i	$-0.178020\pi$
$379$	4.74069e6	1.69529	0.847644	+	0.530565i	$0.178020\pi$
$380$	856201.	0.304170
$381$	1.27498e6	0.449979
$382$	−1.96613e6	−0.689371
$383$	−5.36562e6	−1.86906	−0.934530	−	0.355885i	$-0.884180\pi$
$383$	−5.36562e6	−1.86906	−0.934530	+	0.355885i	$0.884180\pi$
$384$	114764.	0.0397170
$385$	−624545.	−0.214740
$386$	−2.40032e6	−0.819977
$387$	1.98015e6	0.672078
$388$	1.46481e6	0.493972
$389$	2.40076e6	0.804404	0.402202	−	0.915551i	$-0.368245\pi$
$389$	2.40076e6	0.804404	0.402202	+	0.915551i	$0.368245\pi$
$390$	−98293.5	−0.0327238
$391$	−103343.	−0.0341854
$392$	−204206.	−0.0671201
$393$	2.63694e6	0.861230
$394$	−286626.	−0.0930197
$395$	−1.07360e6	−0.346217
$396$	−454664.	−0.145698
$397$	2.43127e6	0.774206	0.387103	−	0.922036i	$-0.373476\pi$
$397$	2.43127e6	0.774206	0.387103	+	0.922036i	$0.373476\pi$
$398$	2.01892e6	0.638868
$399$	1.75862e6	0.553018
$400$	−567427.	−0.177321
$401$	5.72777e6	1.77879	0.889395	−	0.457140i	$-0.151126\pi$
$401$	5.72777e6	1.77879	0.889395	+	0.457140i	$0.151126\pi$
$402$	747597.	0.230729
$403$	−882161.	−0.270574
$404$	1.98742e6	0.605809
$405$	774270.	0.234560
$406$	−144124.	−0.0433932
$407$	−1.47643e6	−0.441801
$408$	350910.	0.104363
$409$	−285844.	−0.0844931	−0.0422465	−	0.999107i	$-0.513451\pi$
$409$	−285844.	−0.0844931	−0.0422465	+	0.999107i	$0.513451\pi$
$410$	−2.29256e6	−0.673538
$411$	767204.	0.224030
$412$	−1.66221e6	−0.482440
$413$	−3.13330e6	−0.903915
$414$	−102416.	−0.0293676
$415$	214659.	0.0611828
$416$	119185.	0.0337666
$417$	1.43458e6	0.404002
$418$	−1.04057e6	−0.291293
$419$	−2.79748e6	−0.778451	−0.389226	−	0.921142i	$-0.627257\pi$
$419$	−2.79748e6	−0.778451	−0.389226	+	0.921142i	$0.627257\pi$
$420$	477699.	0.132139
$421$	4.02156e6	1.10583	0.552916	−	0.833237i	$-0.313515\pi$
$421$	4.02156e6	1.10583	0.552916	+	0.833237i	$0.313515\pi$
$422$	−954153.	−0.260818
$423$	2.64574e6	0.718946
$424$	218108.	0.0589192
$425$	−1.73500e6	−0.465938
$426$	1.28609e6	0.343359
$427$	−2.91249e6	−0.773028
$428$	−1.71916e6	−0.453635
$429$	119459.	0.0313383
$430$	1.23101e6	0.321062
$431$	4.12311e6	1.06913	0.534567	−	0.845126i	$-0.320475\pi$
$431$	4.12311e6	1.06913	0.534567	+	0.845126i	$0.320475\pi$
$432$	783505.	0.201991
$433$	−715562.	−0.183412	−0.0917060	−	0.995786i	$-0.529232\pi$
$433$	−715562.	−0.183412	−0.0917060	+	0.995786i	$0.529232\pi$
$434$	4.28724e6	1.09258
$435$	53793.7	0.0136304
$436$	449701.	0.113294
$437$	−234395.	−0.0587144
$438$	−1.39964e6	−0.348603
$439$	6.01411e6	1.48939	0.744697	−	0.667402i	$-0.232594\pi$
$439$	6.01411e6	1.48939	0.744697	+	0.667402i	$0.232594\pi$
$440$	−282653.	−0.0696020
$441$	−618791.	−0.151512
$442$	364427.	0.0887269
$443$	−4.56997e6	−1.10638	−0.553190	−	0.833055i	$-0.686590\pi$
$443$	−4.56997e6	−1.10638	−0.553190	+	0.833055i	$0.686590\pi$
$444$	1.12928e6	0.271860
$445$	−2.30413e6	−0.551579
$446$	−412051.	−0.0980875
$447$	−1.84104e6	−0.435807
$448$	−579229.	−0.136350
$449$	−70469.6	−0.0164963	−0.00824814	−	0.999966i	$-0.502625\pi$
$449$	−70469.6	−0.0164963	−0.00824814	+	0.999966i	$0.502625\pi$
$450$	−1.71944e6	−0.400272
$451$	2.78622e6	0.645022
$452$	2.77121e6	0.638004
$453$	2.58289e6	0.591371
$454$	1.66489e6	0.379093
$455$	496101.	0.112342
$456$	795905.	0.179246
$457$	−5.00593e6	−1.12123	−0.560614	−	0.828077i	$-0.689435\pi$
$457$	−5.00593e6	−1.12123	−0.560614	+	0.828077i	$0.689435\pi$
$458$	−2.93097e6	−0.652902
$459$	2.39570e6	0.530763
$460$	−63669.6	−0.0140293
$461$	6.35210e6	1.39208	0.696042	−	0.718001i	$-0.254943\pi$
$461$	6.35210e6	1.39208	0.696042	+	0.718001i	$0.254943\pi$
$462$	−580563.	−0.126545
$463$	6.69522e6	1.45148	0.725742	−	0.687967i	$-0.241496\pi$
$463$	6.69522e6	1.45148	0.725742	+	0.687967i	$0.241496\pi$
$464$	−65226.9	−0.0140647
$465$	−1.60019e6	−0.343194
$466$	−1.62578e6	−0.346814
$467$	1.02905e6	0.218346	0.109173	−	0.994023i	$-0.465180\pi$
$467$	1.02905e6	0.218346	0.109173	+	0.994023i	$0.465180\pi$
$468$	361158.	0.0762224
$469$	−3.77323e6	−0.792101
$470$	1.64479e6	0.343452
$471$	2.19658e6	0.456242
$472$	−1.41805e6	−0.292980
$473$	−1.49608e6	−0.307469
$474$	−997991.	−0.204024
$475$	−3.93519e6	−0.800262
$476$	−1.77109e6	−0.358281
$477$	660918.	0.133000
$478$	5.96519e6	1.19414
$479$	−6.18377e6	−1.23144	−0.615722	−	0.787963i	$-0.711136\pi$
$479$	−6.18377e6	−1.23144	−0.615722	+	0.787963i	$0.711136\pi$
$480$	216194.	0.0428293
$481$	1.17279e6	0.231130
$482$	4.94438e6	0.969380
$483$	−130776.	−0.0255070
$484$	−2.23330e6	−0.433345
$485$	2.75944e6	0.532680
$486$	3.69461e6	0.709543
$487$	−7.35384e6	−1.40505	−0.702525	−	0.711659i	$-0.747944\pi$
$487$	−7.35384e6	−1.40505	−0.702525	+	0.711659i	$0.747944\pi$
$488$	−1.31812e6	−0.250556
$489$	115373.	0.0218189
$490$	−384687.	−0.0723797
$491$	−482664.	−0.0903528	−0.0451764	−	0.998979i	$-0.514385\pi$
$491$	−482664.	−0.0903528	−0.0451764	+	0.998979i	$0.514385\pi$
$492$	−2.13111e6	−0.396912
$493$	−199442.	−0.0369573
$494$	826564.	0.152391
$495$	−856506.	−0.157115
$496$	1.94029e6	0.354131
$497$	−6.49109e6	−1.17876
$498$	199542.	0.0360546
$499$	2.21552e6	0.398312	0.199156	−	0.979968i	$-0.436180\pi$
$499$	2.21552e6	0.398312	0.199156	+	0.979968i	$0.436180\pi$
$500$	−2.57599e6	−0.460807
$501$	−1.36042e6	−0.242148
$502$	5.52040e6	0.977713
$503$	−5.33618e6	−0.940395	−0.470197	−	0.882561i	$-0.655817\pi$
$503$	−5.33618e6	−0.940395	−0.470197	+	0.882561i	$0.655817\pi$
$504$	−1.75520e6	−0.307787
$505$	3.74394e6	0.653282
$506$	77379.6	0.0134354
$507$	2.50588e6	0.432952
$508$	−2.91232e6	−0.500703
$509$	1.11435e6	0.190646	0.0953230	−	0.995446i	$-0.469612\pi$
$509$	1.11435e6	0.190646	0.0953230	+	0.995446i	$0.469612\pi$
$510$	661051.	0.112541
$511$	7.06419e6	1.19677
$512$	−262144.	−0.0441942
$513$	5.43373e6	0.911601
$514$	−39351.2	−0.00656977
$515$	−3.13131e6	−0.520245
$516$	1.14431e6	0.189200
$517$	−1.99896e6	−0.328911
$518$	−5.69966e6	−0.933306
$519$	2.25235e6	0.367043
$520$	224523.	0.0364126
$521$	3.67291e6	0.592811	0.296405	−	0.955062i	$-0.404212\pi$
$521$	3.67291e6	0.592811	0.296405	+	0.955062i	$0.404212\pi$
$522$	−197653.	−0.0317488
$523$	3.82421e6	0.611347	0.305674	−	0.952136i	$-0.401118\pi$
$523$	3.82421e6	0.611347	0.305674	+	0.952136i	$0.401118\pi$
$524$	−6.02332e6	−0.958313
$525$	−2.19556e6	−0.347654
$526$	−696826.	−0.109815
$527$	5.93278e6	0.930533
$528$	−262748.	−0.0410161
$529$	−6.41891e6	−0.997292
$530$	410876.	0.0635362
$531$	−4.29704e6	−0.661352
$532$	−4.01704e6	−0.615357
$533$	−2.21321e6	−0.337446
$534$	−2.14187e6	−0.325042
$535$	−3.23858e6	−0.489182
$536$	−1.70767e6	−0.256738
$537$	4.13258e6	0.618423
$538$	289444.	0.0431131
$539$	467521.	0.0693154
$540$	1.47598e6	0.217820
$541$	−210553.	−0.0309292	−0.0154646	−	0.999880i	$-0.504923\pi$
$541$	−210553.	−0.0309292	−0.0154646	+	0.999880i	$0.504923\pi$
$542$	7.03423e6	1.02853
$543$	760351.	0.110666
$544$	−801550.	−0.116127
$545$	847156.	0.122172
$546$	461164.	0.0662024
$547$	−2.11250e6	−0.301875	−0.150938	−	0.988543i	$-0.548229\pi$
$547$	−2.11250e6	−0.301875	−0.150938	+	0.988543i	$0.548229\pi$
$548$	−1.75245e6	−0.249284
$549$	−3.99421e6	−0.565588
$550$	1.29910e6	0.183121
$551$	−452359.	−0.0634752
$552$	−59185.7	−0.00826741
$553$	5.03700e6	0.700421
$554$	−7.68061e6	−1.06322
$555$	2.12737e6	0.293164
$556$	−3.27687e6	−0.449543
$557$	−393675.	−0.0537651	−0.0268825	−	0.999639i	$-0.508558\pi$
$557$	−393675.	−0.0537651	−0.0268825	+	0.999639i	$0.508558\pi$
$558$	5.87955e6	0.799390
$559$	1.18840e6	0.160854
$560$	−1.09116e6	−0.147035
$561$	−803396.	−0.107776
$562$	−1.13301e6	−0.151318
$563$	4.75844e6	0.632694	0.316347	−	0.948644i	$-0.397544\pi$
$563$	4.75844e6	0.632694	0.316347	+	0.948644i	$0.397544\pi$
$564$	1.52896e6	0.202394
$565$	5.22046e6	0.687999
$566$	6.70480e6	0.879720
$567$	−3.63264e6	−0.474532
$568$	−2.93770e6	−0.382064
$569$	−1.36827e6	−0.177171	−0.0885854	−	0.996069i	$-0.528235\pi$
$569$	−1.36827e6	−0.177171	−0.0885854	+	0.996069i	$0.528235\pi$
$570$	1.49934e6	0.193292
$571$	−6.56779e6	−0.843003	−0.421501	−	0.906828i	$-0.638497\pi$
$571$	−6.56779e6	−0.843003	−0.421501	+	0.906828i	$0.638497\pi$
$572$	−272869.	−0.0348710
$573$	−3.44300e6	−0.438076
$574$	1.07560e7	1.36261
$575$	292632.	0.0369107
$576$	−794359.	−0.0997609
$577$	−1.13941e7	−1.42475	−0.712375	−	0.701799i	$-0.752381\pi$
$577$	−1.13941e7	−1.42475	−0.712375	+	0.701799i	$0.752381\pi$
$578$	3.22855e6	0.401965
$579$	−4.20334e6	−0.521073
$580$	−122876.	−0.0151669
$581$	−1.00712e6	−0.123777
$582$	2.56511e6	0.313905
$583$	−499350.	−0.0608462
$584$	3.19707e6	0.387900
$585$	680357.	0.0821953
$586$	4.33328e6	0.521281
$587$	−5.49443e6	−0.658154	−0.329077	−	0.944303i	$-0.606738\pi$
$587$	−5.49443e6	−0.658154	−0.329077	+	0.944303i	$0.606738\pi$
$588$	−357596.	−0.0426529
$589$	1.34562e7	1.59822
$590$	−2.67136e6	−0.315938
$591$	−501927.	−0.0591115
$592$	−2.57952e6	−0.302506
$593$	−9.03344e6	−1.05491	−0.527456	−	0.849582i	$-0.676854\pi$
$593$	−9.03344e6	−1.05491	−0.527456	+	0.849582i	$0.676854\pi$
$594$	−1.79381e6	−0.208598
$595$	−3.33642e6	−0.386356
$596$	4.20531e6	0.484933
$597$	3.53544e6	0.405983
$598$	−61465.7	−0.00702878
$599$	1.25886e6	0.143354	0.0716770	−	0.997428i	$-0.477165\pi$
$599$	1.25886e6	0.143354	0.0716770	+	0.997428i	$0.477165\pi$
$600$	−993653.	−0.112683
$601$	−2.03254e6	−0.229537	−0.114768	−	0.993392i	$-0.536613\pi$
$601$	−2.03254e6	−0.229537	−0.114768	+	0.993392i	$0.536613\pi$
$602$	−5.77552e6	−0.649531
$603$	−5.17463e6	−0.579544
$604$	−5.89984e6	−0.658034
$605$	−4.20714e6	−0.467303
$606$	3.48028e6	0.384975
$607$	7.39381e6	0.814510	0.407255	−	0.913314i	$-0.366486\pi$
$607$	7.39381e6	0.814510	0.407255	+	0.913314i	$0.366486\pi$
$608$	−1.81801e6	−0.199451
$609$	−252384.	−0.0275752
$610$	−2.48310e6	−0.270190
$611$	1.58785e6	0.172071
$612$	−2.42889e6	−0.262137
$613$	−6.20300e6	−0.666731	−0.333365	−	0.942798i	$-0.608184\pi$
$613$	−6.20300e6	−0.666731	−0.333365	+	0.942798i	$0.608184\pi$
$614$	−4.36877e6	−0.467668
$615$	−4.01464e6	−0.428015
$616$	1.32612e6	0.140810
$617$	3.24055e6	0.342693	0.171347	−	0.985211i	$-0.445188\pi$
$617$	3.24055e6	0.342693	0.171347	+	0.985211i	$0.445188\pi$
$618$	−2.91079e6	−0.306577
$619$	1.31608e7	1.38056	0.690279	−	0.723543i	$-0.257488\pi$
$619$	1.31608e7	1.38056	0.690279	+	0.723543i	$0.257488\pi$
$620$	3.65517e6	0.381881
$621$	−404068.	−0.0420461
$622$	5.33999e6	0.553433
$623$	1.08103e7	1.11588
$624$	208711.	0.0214577
$625$	2.07390e6	0.212367
$626$	7.67890e6	0.783183
$627$	−1.82220e6	−0.185108
$628$	−5.01744e6	−0.507672
$629$	−7.88731e6	−0.794882
$630$	−3.30649e6	−0.331906
$631$	9.19773e6	0.919618	0.459809	−	0.888018i	$-0.347918\pi$
$631$	9.19773e6	0.919618	0.459809	+	0.888018i	$0.347918\pi$
$632$	2.27962e6	0.227023
$633$	−1.67087e6	−0.165742
$634$	−9.03780e6	−0.892976
$635$	−5.48630e6	−0.539940
$636$	381941.	0.0374415
$637$	−371371.	−0.0362626
$638$	149335.	0.0145248
$639$	−8.90193e6	−0.862446
$640$	−493832.	−0.0476573
$641$	−1.86531e7	−1.79310	−0.896551	−	0.442941i	$-0.853935\pi$
$641$	−1.86531e7	−1.79310	−0.896551	+	0.442941i	$0.853935\pi$
$642$	−3.01051e6	−0.288272
$643$	2.83581e6	0.270489	0.135244	−	0.990812i	$-0.456818\pi$
$643$	2.83581e6	0.270489	0.135244	+	0.990812i	$0.456818\pi$
$644$	298719.	0.0283823
$645$	2.15568e6	0.204026
$646$	−5.55887e6	−0.524090
$647$	−6.43163e6	−0.604033	−0.302016	−	0.953303i	$-0.597660\pi$
$647$	−6.43163e6	−0.604033	−0.302016	+	0.953303i	$0.597660\pi$
$648$	−1.64404e6	−0.153807
$649$	3.24658e6	0.302562
$650$	−1.03193e6	−0.0958003
$651$	7.50763e6	0.694305
$652$	−263536.	−0.0242785
$653$	−1.58350e7	−1.45323	−0.726616	−	0.687044i	$-0.758908\pi$
$653$	−1.58350e7	−1.45323	−0.726616	+	0.687044i	$0.758908\pi$
$654$	787496.	0.0719953
$655$	−1.13469e7	−1.03341
$656$	4.86790e6	0.441654
$657$	9.68788e6	0.875619
$658$	−7.71686e6	−0.694826
$659$	−8.51499e6	−0.763784	−0.381892	−	0.924207i	$-0.624727\pi$
$659$	−8.51499e6	−0.763784	−0.381892	+	0.924207i	$0.624727\pi$
$660$	−494970.	−0.0442302
$661$	−1.00810e6	−0.0897427	−0.0448713	−	0.998993i	$-0.514288\pi$
$661$	−1.00810e6	−0.0897427	−0.0448713	+	0.998993i	$0.514288\pi$
$662$	−5.38231e6	−0.477335
$663$	638169.	0.0563835
$664$	−455795.	−0.0401189
$665$	−7.56739e6	−0.663578
$666$	−7.81655e6	−0.682857
$667$	33638.7	0.00292769
$668$	3.10749e6	0.269444
$669$	−721566.	−0.0623319
$670$	−3.21694e6	−0.276857
$671$	3.01779e6	0.258751
$672$	−1.01432e6	−0.0866467
$673$	9.47555e6	0.806430	0.403215	−	0.915105i	$-0.367893\pi$
$673$	9.47555e6	0.806430	0.403215	+	0.915105i	$0.367893\pi$
$674$	7.50520e6	0.636375
$675$	−6.78378e6	−0.573076
$676$	−5.72394e6	−0.481757
$677$	−1.37544e7	−1.15337	−0.576686	−	0.816966i	$-0.695654\pi$
$677$	−1.37544e7	−1.15337	−0.576686	+	0.816966i	$0.695654\pi$
$678$	4.85282e6	0.405434
$679$	−1.29465e7	−1.07765
$680$	−1.50998e6	−0.125227
$681$	2.91548e6	0.240903
$682$	−4.44224e6	−0.365713
$683$	−2.18232e6	−0.179005	−0.0895026	−	0.995987i	$-0.528528\pi$
$683$	−2.18232e6	−0.179005	−0.0895026	+	0.995987i	$0.528528\pi$
$684$	−5.50900e6	−0.450228
$685$	−3.30131e6	−0.268819
$686$	−7.70210e6	−0.624883
$687$	−5.13259e6	−0.414901
$688$	−2.61385e6	−0.210528
$689$	396654.	0.0318320
$690$	−111495.	−0.00891526
$691$	8.97674e6	0.715193	0.357597	−	0.933876i	$-0.383596\pi$
$691$	8.97674e6	0.715193	0.357597	+	0.933876i	$0.383596\pi$
$692$	−5.14482e6	−0.408418
$693$	4.01847e6	0.317854
$694$	5.43278e6	0.428177
$695$	−6.17303e6	−0.484771
$696$	−114223.	−0.00893776
$697$	1.48844e7	1.16051
$698$	−236872.	−0.0184024
$699$	−2.84699e6	−0.220391
$700$	5.01511e6	0.386843
$701$	−8.46411e6	−0.650558	−0.325279	−	0.945618i	$-0.605458\pi$
$701$	−8.46411e6	−0.650558	−0.325279	+	0.945618i	$0.605458\pi$
$702$	1.42489e6	0.109129
$703$	−1.78894e7	−1.36523
$704$	600170.	0.0456397
$705$	2.88028e6	0.218254
$706$	1.03110e7	0.778552
$707$	−1.75655e7	−1.32163
$708$	−2.48323e6	−0.186180
$709$	2.51142e6	0.187631	0.0938153	−	0.995590i	$-0.470094\pi$
$709$	2.51142e6	0.187631	0.0938153	+	0.995590i	$0.470094\pi$
$710$	−5.53410e6	−0.412004
$711$	6.90778e6	0.512465
$712$	4.89246e6	0.361683
$713$	−1.00064e6	−0.0737151
$714$	−3.10146e6	−0.227677
$715$	−514037.	−0.0376036
$716$	−9.43967e6	−0.688136
$717$	1.04460e7	0.758841
$718$	1.60649e7	1.16296
$719$	−1.94814e7	−1.40539	−0.702697	−	0.711489i	$-0.748021\pi$
$719$	−1.94814e7	−1.40539	−0.702697	+	0.711489i	$0.748021\pi$
$720$	−1.49643e6	−0.107578
$721$	1.46912e7	1.05249
$722$	−2.70379e6	−0.193032
$723$	8.65838e6	0.616015
$724$	−1.73680e6	−0.123141
$725$	564751.	0.0399036
$726$	−3.91086e6	−0.275379
$727$	7.75969e6	0.544513	0.272257	−	0.962225i	$-0.412230\pi$
$727$	7.75969e6	0.544513	0.272257	+	0.962225i	$0.412230\pi$
$728$	−1.05339e6	−0.0736651
$729$	227629.	0.0158639
$730$	6.02271e6	0.418297
$731$	−7.99229e6	−0.553195
$732$	−2.30823e6	−0.159222
$733$	−9.39411e6	−0.645796	−0.322898	−	0.946434i	$-0.604657\pi$
$733$	−9.39411e6	−0.645796	−0.322898	+	0.946434i	$0.604657\pi$
$734$	−9.09749e6	−0.623278
$735$	−673647.	−0.0459953
$736$	135192.	0.00919936
$737$	3.90964e6	0.265136
$738$	1.47509e7	0.996960
$739$	1.83497e7	1.23600	0.618000	−	0.786178i	$-0.287943\pi$
$739$	1.83497e7	1.23600	0.618000	+	0.786178i	$0.287943\pi$
$740$	−4.85935e6	−0.326211
$741$	1.44744e6	0.0968403
$742$	−1.92771e6	−0.128538
$743$	1.77016e7	1.17636	0.588181	−	0.808729i	$-0.299844\pi$
$743$	1.77016e7	1.17636	0.588181	+	0.808729i	$0.299844\pi$
$744$	3.39776e6	0.225040
$745$	7.92205e6	0.522934
$746$	24640.7	0.00162109
$747$	−1.38117e6	−0.0905618
$748$	1.83512e6	0.119925
$749$	1.51945e7	0.989649
$750$	−4.51096e6	−0.292830
$751$	−4.72045e6	−0.305410	−0.152705	−	0.988272i	$-0.548798\pi$
$751$	−4.72045e6	−0.305410	−0.152705	+	0.988272i	$0.548798\pi$
$752$	−3.49245e6	−0.225209
$753$	9.66708e6	0.621310
$754$	−118623.	−0.00759869
$755$	−1.11142e7	−0.709599
$756$	−6.92488e6	−0.440664
$757$	1.13581e7	0.720388	0.360194	−	0.932877i	$-0.382711\pi$
$757$	1.13581e7	0.720388	0.360194	+	0.932877i	$0.382711\pi$
$758$	−1.89628e7	−1.19875
$759$	135504.	0.00853781
$760$	−3.42481e6	−0.215081
$761$	−202337.	−0.0126652	−0.00633261	−	0.999980i	$-0.502016\pi$
$761$	−202337.	−0.0126652	−0.00633261	+	0.999980i	$0.502016\pi$
$762$	−5.09993e6	−0.318183
$763$	−3.97460e6	−0.247162
$764$	7.86451e6	0.487459
$765$	−4.57559e6	−0.282679
$766$	2.14625e7	1.32162
$767$	−2.57889e6	−0.158287
$768$	−459055.	−0.0280842
$769$	1.62109e7	0.988533	0.494267	−	0.869310i	$-0.335437\pi$
$769$	1.62109e7	0.988533	0.494267	+	0.869310i	$0.335437\pi$
$770$	2.49818e6	0.151844
$771$	−68910.1	−0.00417491
$772$	9.60129e6	0.579811
$773$	9.18039e6	0.552602	0.276301	−	0.961071i	$-0.410891\pi$
$773$	9.18039e6	0.552602	0.276301	+	0.961071i	$0.410891\pi$
$774$	−7.92059e6	−0.475231
$775$	−1.67996e7	−1.00472
$776$	−5.85924e6	−0.349291
$777$	−9.98099e6	−0.593090
$778$	−9.60303e6	−0.568799
$779$	3.37597e7	1.99322
$780$	393174.	0.0231392
$781$	6.72576e6	0.394561
$782$	413374.	0.0241727
$783$	−779810.	−0.0454553
$784$	816822.	0.0474610
$785$	−9.45197e6	−0.547455
$786$	−1.05478e7	−0.608982
$787$	3.46106e7	1.99192	0.995960	−	0.0897972i	$-0.0286219\pi$
$787$	3.46106e7	1.99192	0.995960	+	0.0897972i	$0.0286219\pi$
$788$	1.14650e6	0.0657749
$789$	−1.22025e6	−0.0697841
$790$	4.29439e6	0.244813
$791$	−2.44929e7	−1.39187
$792$	1.81866e6	0.103024
$793$	−2.39715e6	−0.135367
$794$	−9.72507e6	−0.547446
$795$	719508.	0.0403755
$796$	−8.07568e6	−0.451748
$797$	1.67718e7	0.935264	0.467632	−	0.883923i	$-0.345107\pi$
$797$	1.67718e7	0.935264	0.467632	+	0.883923i	$0.345107\pi$
$798$	−7.03447e6	−0.391043
$799$	−1.06788e7	−0.591772
$800$	2.26971e6	0.125385
$801$	1.48253e7	0.816438
$802$	−2.29111e7	−1.25779
$803$	−7.31958e6	−0.400587
$804$	−2.99039e6	−0.163150
$805$	562733.	0.0306064
$806$	3.52865e6	0.191324
$807$	506862.	0.0273972
$808$	−7.94967e6	−0.428372
$809$	355816.	0.0191141	0.00955706	−	0.999954i	$-0.496958\pi$
$809$	355816.	0.0191141	0.00955706	+	0.999954i	$0.496958\pi$
$810$	−3.09708e6	−0.165859
$811$	2.29373e7	1.22459	0.612295	−	0.790629i	$-0.290246\pi$
$811$	2.29373e7	1.22459	0.612295	+	0.790629i	$0.290246\pi$
$812$	576497.	0.0306836
$813$	1.23180e7	0.653604
$814$	5.90571e6	0.312400
$815$	−496455.	−0.0261810
$816$	−1.40364e6	−0.0737955
$817$	−1.81275e7	−0.950127
$818$	1.14338e6	0.0597456
$819$	−3.19203e6	−0.166287
$820$	9.17026e6	0.476263
$821$	2.23270e7	1.15604	0.578018	−	0.816024i	$-0.303826\pi$
$821$	2.23270e7	1.15604	0.578018	+	0.816024i	$0.303826\pi$
$822$	−3.06882e6	−0.158413
$823$	−1.82423e7	−0.938815	−0.469408	−	0.882982i	$-0.655533\pi$
$823$	−1.82423e7	−0.938815	−0.469408	+	0.882982i	$0.655533\pi$
$824$	6.64885e6	0.341137
$825$	2.27493e6	0.116368
$826$	1.25332e7	0.639164
$827$	2.12085e7	1.07832	0.539158	−	0.842205i	$-0.318743\pi$
$827$	2.12085e7	1.07832	0.539158	+	0.842205i	$0.318743\pi$
$828$	409665.	0.0207660
$829$	3.84252e7	1.94191	0.970957	−	0.239254i	$-0.0769029\pi$
$829$	3.84252e7	1.94191	0.970957	+	0.239254i	$0.0769029\pi$
$830$	−858637.	−0.0432627
$831$	−1.34500e7	−0.675644
$832$	−476739.	−0.0238766
$833$	2.49757e6	0.124711
$834$	−5.73830e6	−0.285672
$835$	5.85396e6	0.290558
$836$	4.16227e6	0.205975
$837$	2.31969e7	1.14450
$838$	1.11899e7	0.550448
$839$	−4.33208e6	−0.212467	−0.106234	−	0.994341i	$-0.533879\pi$
$839$	−4.33208e6	−0.212467	−0.106234	+	0.994341i	$0.533879\pi$
$840$	−1.91080e6	−0.0934365
$841$	−2.04462e7	−0.996835
$842$	−1.60862e7	−0.781941
$843$	−1.98407e6	−0.0961586
$844$	3.81661e6	0.184426
$845$	−1.07829e7	−0.519509
$846$	−1.05830e7	−0.508372
$847$	1.97386e7	0.945385
$848$	−872431.	−0.0416621
$849$	1.17411e7	0.559038
$850$	6.94002e6	0.329468
$851$	1.33030e6	0.0629690
$852$	−5.14437e6	−0.242791
$853$	−2.60078e7	−1.22386	−0.611930	−	0.790912i	$-0.709607\pi$
$853$	−2.60078e7	−1.22386	−0.611930	+	0.790912i	$0.709607\pi$
$854$	1.16500e7	0.546613
$855$	−1.03780e7	−0.485509
$856$	6.87663e6	0.320768
$857$	−1.06641e6	−0.0495988	−0.0247994	−	0.999692i	$-0.507895\pi$
$857$	−1.06641e6	−0.0495988	−0.0247994	+	0.999692i	$0.507895\pi$
$858$	−477836.	−0.0221596
$859$	4.24985e6	0.196513	0.0982563	−	0.995161i	$-0.468674\pi$
$859$	4.24985e6	0.196513	0.0982563	+	0.995161i	$0.468674\pi$
$860$	−4.92403e6	−0.227025
$861$	1.88355e7	0.865903
$862$	−1.64924e7	−0.755992
$863$	2.64597e7	1.20936	0.604682	−	0.796467i	$-0.293300\pi$
$863$	2.64597e7	1.20936	0.604682	+	0.796467i	$0.293300\pi$
$864$	−3.13402e6	−0.142829
$865$	−9.69192e6	−0.440423
$866$	2.86225e6	0.129692
$867$	5.65370e6	0.255438
$868$	−1.71490e7	−0.772571
$869$	−5.21910e6	−0.234448
$870$	−215175.	−0.00963814
$871$	−3.10558e6	−0.138707
$872$	−1.79880e6	−0.0801111
$873$	−1.77549e7	−0.788465
$874$	937580.	0.0415174
$875$	2.27674e7	1.00530
$876$	5.59857e6	0.246500
$877$	−1.93576e7	−0.849872	−0.424936	−	0.905224i	$-0.639703\pi$
$877$	−1.93576e7	−0.849872	−0.424936	+	0.905224i	$0.639703\pi$
$878$	−2.40564e7	−1.05316
$879$	7.58824e6	0.331260
$880$	1.13061e6	0.0492161
$881$	2.05942e7	0.893936	0.446968	−	0.894550i	$-0.352504\pi$
$881$	2.05942e7	0.893936	0.446968	+	0.894550i	$0.352504\pi$
$882$	2.47517e6	0.107135
$883$	−3.12635e7	−1.34939	−0.674693	−	0.738098i	$-0.735724\pi$
$883$	−3.12635e7	−1.34939	−0.674693	+	0.738098i	$0.735724\pi$
$884$	−1.45771e6	−0.0627394
$885$	−4.67797e6	−0.200770
$886$	1.82799e7	0.782329
$887$	−2.00863e7	−0.857217	−0.428608	−	0.903490i	$-0.640996\pi$
$887$	−2.00863e7	−0.857217	−0.428608	+	0.903490i	$0.640996\pi$
$888$	−4.51714e6	−0.192234
$889$	2.57401e7	1.09233
$890$	9.21653e6	0.390025
$891$	3.76397e6	0.158837
$892$	1.64820e6	0.0693584
$893$	−2.42207e7	−1.01638
$894$	7.36415e6	0.308162
$895$	−1.77827e7	−0.742060
$896$	2.31691e6	0.0964140
$897$	−107636.	−0.00446659
$898$	281878.	0.0116646
$899$	−1.93114e6	−0.0796921
$900$	6.87775e6	0.283035
$901$	−2.66761e6	−0.109474
$902$	−1.11449e7	−0.456099
$903$	−1.01138e7	−0.412759
$904$	−1.10848e7	−0.451137
$905$	−3.27182e6	−0.132791
$906$	−1.03315e7	−0.418162
$907$	−3.69898e7	−1.49301	−0.746507	−	0.665377i	$-0.768271\pi$
$907$	−3.69898e7	−1.49301	−0.746507	+	0.665377i	$0.768271\pi$
$908$	−6.65956e6	−0.268059
$909$	−2.40894e7	−0.966977
$910$	−1.98440e6	−0.0794377
$911$	−3.45672e7	−1.37997	−0.689983	−	0.723826i	$-0.742382\pi$
$911$	−3.45672e7	−1.37997	−0.689983	+	0.723826i	$0.742382\pi$
$912$	−3.18362e6	−0.126746
$913$	1.04353e6	0.0414311
$914$	2.00237e7	0.792828
$915$	−4.34830e6	−0.171698
$916$	1.17239e7	0.461672
$917$	5.32361e7	2.09066
$918$	−9.58280e6	−0.375306
$919$	−4.89308e7	−1.91115	−0.955573	−	0.294756i	$-0.904762\pi$
$919$	−4.89308e7	−1.91115	−0.955573	+	0.294756i	$0.904762\pi$
$920$	254678.	0.00992025
$921$	−7.65039e6	−0.297190
$922$	−2.54084e7	−0.984352
$923$	−5.34254e6	−0.206416
$924$	2.32225e6	0.0894806
$925$	2.23341e7	0.858251
$926$	−2.67809e7	−1.02635
$927$	2.01476e7	0.770059
$928$	260908.	0.00994528
$929$	4.11751e7	1.56529	0.782645	−	0.622468i	$-0.213870\pi$
$929$	4.11751e7	1.56529	0.782645	+	0.622468i	$0.213870\pi$
$930$	6.40077e6	0.242675
$931$	5.66479e6	0.214195
$932$	6.50312e6	0.245235
$933$	9.35116e6	0.351691
$934$	−4.11620e6	−0.154394
$935$	3.45704e6	0.129323
$936$	−1.44463e6	−0.0538974
$937$	4.52601e7	1.68410	0.842048	−	0.539403i	$-0.181350\pi$
$937$	4.52601e7	1.68410	0.842048	+	0.539403i	$0.181350\pi$
$938$	1.50929e7	0.560100
$939$	1.34469e7	0.497691
$940$	−6.57916e6	−0.242857
$941$	2.77391e7	1.02122	0.510609	−	0.859813i	$-0.329420\pi$
$941$	2.77391e7	1.02122	0.510609	+	0.859813i	$0.329420\pi$
$942$	−8.78633e6	−0.322612
$943$	−2.51046e6	−0.0919337
$944$	5.67221e6	0.207168
$945$	−1.30452e7	−0.475195
$946$	5.98432e6	0.217414
$947$	−3.74649e7	−1.35753	−0.678765	−	0.734356i	$-0.737484\pi$
$947$	−3.74649e7	−1.35753	−0.678765	+	0.734356i	$0.737484\pi$
$948$	3.99196e6	0.144267
$949$	5.81423e6	0.209569
$950$	1.57408e7	0.565871
$951$	−1.58266e7	−0.567461
$952$	7.08436e6	0.253343
$953$	−2.66749e7	−0.951418	−0.475709	−	0.879603i	$-0.657808\pi$
$953$	−2.66749e7	−0.951418	−0.475709	+	0.879603i	$0.657808\pi$
$954$	−2.64367e6	−0.0940453
$955$	1.48153e7	0.525658
$956$	−2.38607e7	−0.844383
$957$	261508.	0.00923009
$958$	2.47351e7	0.870763
$959$	1.54887e7	0.543838
$960$	−864778.	−0.0302849
$961$	2.88163e7	1.00654
$962$	−4.69114e6	−0.163434
$963$	2.08378e7	0.724080
$964$	−1.97775e7	−0.685455
$965$	1.80871e7	0.625246
$966$	523103.	0.0180362
$967$	−5.25408e6	−0.180688	−0.0903442	−	0.995911i	$-0.528797\pi$
$967$	−5.25408e6	−0.180688	−0.0903442	+	0.995911i	$0.528797\pi$
$968$	8.93320e6	0.306421
$969$	−9.73446e6	−0.333045
$970$	−1.10378e7	−0.376662
$971$	2.46081e7	0.837588	0.418794	−	0.908081i	$-0.362453\pi$
$971$	2.46081e7	0.837588	0.418794	+	0.908081i	$0.362453\pi$
$972$	−1.47785e7	−0.501722
$973$	2.89620e7	0.980724
$974$	2.94154e7	0.993521
$975$	−1.80707e6	−0.0608785
$976$	5.27248e6	0.177170
$977$	−2.19749e7	−0.736529	−0.368264	−	0.929721i	$-0.620048\pi$
$977$	−2.19749e7	−0.736529	−0.368264	+	0.929721i	$0.620048\pi$
$978$	−461493.	−0.0154283
$979$	−1.12011e7	−0.373512
$980$	1.53875e6	0.0511802
$981$	−5.45080e6	−0.180837
$982$	1.93066e6	0.0638891
$983$	5.51108e7	1.81908	0.909542	−	0.415613i	$-0.136433\pi$
$983$	5.51108e7	1.81908	0.909542	+	0.415613i	$0.136433\pi$
$984$	8.52446e6	0.280659
$985$	2.15981e6	0.0709292
$986$	797770.	0.0261328
$987$	−1.35134e7	−0.441543
$988$	−3.30626e6	−0.107757
$989$	1.34801e6	0.0438230
$990$	3.42602e6	0.111097
$991$	−1.52842e7	−0.494377	−0.247188	−	0.968967i	$-0.579507\pi$
$991$	−1.52842e7	−0.494377	−0.247188	+	0.968967i	$0.579507\pi$
$992$	−7.76118e6	−0.250408
$993$	−9.42526e6	−0.303334
$994$	2.59644e7	0.833511
$995$	−1.52131e7	−0.487148
$996$	−798168.	−0.0254945
$997$	2.50421e7	0.797873	0.398936	−	0.916979i	$-0.369379\pi$
$997$	2.50421e7	0.797873	0.398936	+	0.916979i	$0.369379\pi$
$998$	−8.86207e6	−0.281649
$999$	−3.08390e7	−0.977657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.c.1.13	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.c.1.13	✓	30	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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -7.00462 q^{3} +16.0000 q^{4} +30.1411 q^{5} +28.0185 q^{6} -141.413 q^{7} -64.0000 q^{8} -193.935 q^{9} -120.565 q^{10} +146.526 q^{11} -112.074 q^{12} -116.391 q^{13} +565.653 q^{14} -211.127 q^{15} +256.000 q^{16} +782.764 q^{17} +775.741 q^{18} +1775.40 q^{19} +482.258 q^{20} +990.547 q^{21} -586.103 q^{22} -132.024 q^{23} +448.296 q^{24} -2216.51 q^{25} +465.565 q^{26} +3060.57 q^{27} -2262.61 q^{28} -254.793 q^{29} +844.510 q^{30} +7579.28 q^{31} -1024.00 q^{32} -1026.36 q^{33} -3131.05 q^{34} -4262.36 q^{35} -3102.96 q^{36} -10076.2 q^{37} -7101.60 q^{38} +815.277 q^{39} -1929.03 q^{40} +19015.2 q^{41} -3962.19 q^{42} -10210.4 q^{43} +2344.41 q^{44} -5845.43 q^{45} +528.095 q^{46} -13642.4 q^{47} -1793.18 q^{48} +3190.71 q^{49} +8866.05 q^{50} -5482.97 q^{51} -1862.26 q^{52} -3407.93 q^{53} -12242.3 q^{54} +4416.45 q^{55} +9050.45 q^{56} -12436.0 q^{57} +1019.17 q^{58} +22157.1 q^{59} -3378.04 q^{60} +20595.6 q^{61} -30317.1 q^{62} +27425.0 q^{63} +4096.00 q^{64} -3508.17 q^{65} +4105.43 q^{66} +26682.3 q^{67} +12524.2 q^{68} +924.777 q^{69} +17049.4 q^{70} +45901.5 q^{71} +12411.9 q^{72} -49954.2 q^{73} +40305.0 q^{74} +15525.8 q^{75} +28406.4 q^{76} -20720.7 q^{77} -3261.11 q^{78} -35619.0 q^{79} +7716.13 q^{80} +25688.1 q^{81} -76061.0 q^{82} +7121.80 q^{83} +15848.7 q^{84} +23593.4 q^{85} +40841.4 q^{86} +1784.73 q^{87} -9377.65 q^{88} -76444.8 q^{89} +23381.7 q^{90} +16459.3 q^{91} -2112.38 q^{92} -53090.0 q^{93} +54569.6 q^{94} +53512.6 q^{95} +7172.74 q^{96} +91550.6 q^{97} -12762.8 q^{98} -28416.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 120 q^{2} - 30 q^{3} + 480 q^{4} - 136 q^{5} + 120 q^{6} - 123 q^{7} - 1920 q^{8} + 2670 q^{9} + 544 q^{10} - 1058 q^{11} - 480 q^{12} - 371 q^{13} + 492 q^{14} - 1364 q^{15} + 7680 q^{16} - 1918 q^{17}+ \cdots - 78063 q^{99}+O(q^{100}) \) 30 * q - 120 * q^2 - 30 * q^3 + 480 * q^4 - 136 * q^5 + 120 * q^6 - 123 * q^7 - 1920 * q^8 + 2670 * q^9 + 544 * q^10 - 1058 * q^11 - 480 * q^12 - 371 * q^13 + 492 * q^14 - 1364 * q^15 + 7680 * q^16 - 1918 * q^17 - 10680 * q^18 - 3315 * q^19 - 2176 * q^20 - 1217 * q^21 + 4232 * q^22 - 9534 * q^23 + 1920 * q^24 + 22130 * q^25 + 1484 * q^26 - 12156 * q^27 - 1968 * q^28 - 8009 * q^29 + 5456 * q^30 - 3543 * q^31 - 30720 * q^32 - 3411 * q^33 + 7672 * q^34 - 11177 * q^35 + 42720 * q^36 - 6517 * q^37 + 13260 * q^38 - 2453 * q^39 + 8704 * q^40 - 5951 * q^41 + 4868 * q^42 - 18768 * q^43 - 16928 * q^44 - 22619 * q^45 + 38136 * q^46 - 41067 * q^47 - 7680 * q^48 + 107797 * q^49 - 88520 * q^50 - 4187 * q^51 - 5936 * q^52 - 64055 * q^53 + 48624 * q^54 + 31973 * q^55 + 7872 * q^56 - 25442 * q^57 + 32036 * q^58 - 66832 * q^59 - 21824 * q^60 + 46777 * q^61 + 14172 * q^62 - 115994 * q^63 + 122880 * q^64 - 68783 * q^65 + 13644 * q^66 - 107218 * q^67 - 30688 * q^68 + 228589 * q^69 + 44708 * q^70 + 4733 * q^71 - 170880 * q^72 + 150385 * q^73 + 26068 * q^74 + 133325 * q^75 - 53040 * q^76 + 292020 * q^77 + 9812 * q^78 + 31827 * q^79 - 34816 * q^80 + 425214 * q^81 + 23804 * q^82 + 107301 * q^83 - 19472 * q^84 + 338625 * q^85 + 75072 * q^86 + 288484 * q^87 + 67712 * q^88 - 99690 * q^89 + 90476 * q^90 - 109140 * q^91 - 152544 * q^92 - 114545 * q^93 + 164268 * q^94 - 155668 * q^95 + 30720 * q^96 - 79959 * q^97 - 431188 * q^98 - 78063 * q^99

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