LMFDB - Embedded newform 531.8.a.g.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,8,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.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:	$165.876448532$
Analytic rank:	$1$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.306188 q^{2} -127.906 q^{4} +108.774 q^{5} +1364.63 q^{7} +78.3553 q^{8} +O(q^{10})$	\(q-0.306188 q^{2} -127.906 q^{4} +108.774 q^{5} +1364.63 q^{7} +78.3553 q^{8} -33.3052 q^{10} +1380.42 q^{11} +3791.36 q^{13} -417.832 q^{14} +16348.0 q^{16} -34931.9 q^{17} +11151.7 q^{19} -13912.9 q^{20} -422.666 q^{22} -31517.5 q^{23} -66293.2 q^{25} -1160.87 q^{26} -174544. q^{28} -106066. q^{29} +21299.3 q^{31} -15035.0 q^{32} +10695.7 q^{34} +148436. q^{35} +228684. q^{37} -3414.50 q^{38} +8523.02 q^{40} +569395. q^{41} -968608. q^{43} -176564. q^{44} +9650.26 q^{46} -678969. q^{47} +1.03866e6 q^{49} +20298.2 q^{50} -484939. q^{52} -1.04628e6 q^{53} +150153. q^{55} +106926. q^{56} +32476.1 q^{58} +205379. q^{59} +2.29622e6 q^{61} -6521.57 q^{62} -2.08794e6 q^{64} +412402. q^{65} +1.89799e6 q^{67} +4.46800e6 q^{68} -45449.2 q^{70} +3.15394e6 q^{71} -3.58952e6 q^{73} -70020.3 q^{74} -1.42637e6 q^{76} +1.88375e6 q^{77} -1.93077e6 q^{79} +1.77824e6 q^{80} -174342. q^{82} +7.35502e6 q^{83} -3.79968e6 q^{85} +296576. q^{86} +108163. q^{88} -8.45678e6 q^{89} +5.17379e6 q^{91} +4.03128e6 q^{92} +207892. q^{94} +1.21301e6 q^{95} -739735. q^{97} -318026. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 24 q^{2} + 1776 q^{4} - 1000 q^{5} + 154 q^{7} - 4608 q^{8}+O(q^{10}) $ 33 * q - 24 * q^2 + 1776 * q^4 - 1000 * q^5 + 154 * q^7 - 4608 * q^8	\( 33 q - 24 q^{2} + 1776 q^{4} - 1000 q^{5} + 154 q^{7} - 4608 q^{8} + 5042 q^{10} - 13310 q^{11} - 14172 q^{13} - 16464 q^{14} + 95772 q^{16} - 39304 q^{17} - 56302 q^{19} + 17936 q^{20} + 152764 q^{22} - 17988 q^{23} + 468523 q^{25} + 27624 q^{26} - 59896 q^{28} - 474564 q^{29} + 188186 q^{31} - 251068 q^{32} + 169888 q^{34} - 514500 q^{35} + 1148200 q^{37} - 446594 q^{38} + 501214 q^{40} - 1246152 q^{41} + 62268 q^{43} - 2555520 q^{44} - 1289942 q^{46} - 1485654 q^{47} + 3829555 q^{49} - 6430160 q^{50} - 2624804 q^{52} - 4086740 q^{53} - 1119118 q^{55} - 8448352 q^{56} + 2966706 q^{58} + 6777507 q^{59} + 2436146 q^{61} - 9005952 q^{62} + 11117562 q^{64} - 16730354 q^{65} - 2652248 q^{67} - 15929124 q^{68} + 3359254 q^{70} - 7356324 q^{71} + 1900454 q^{73} - 25386964 q^{74} - 16047360 q^{76} - 20774826 q^{77} - 5912712 q^{79} - 13568404 q^{80} + 1579434 q^{82} + 1052766 q^{83} + 18372730 q^{85} - 43499960 q^{86} + 18209214 q^{88} - 174788 q^{89} - 18891512 q^{91} - 46033270 q^{92} + 365448 q^{94} - 31505580 q^{95} + 14418540 q^{97} + 15964056 q^{98}+O(q^{100}) \) 33 * q - 24 * q^2 + 1776 * q^4 - 1000 * q^5 + 154 * q^7 - 4608 * q^8 + 5042 * q^10 - 13310 * q^11 - 14172 * q^13 - 16464 * q^14 + 95772 * q^16 - 39304 * q^17 - 56302 * q^19 + 17936 * q^20 + 152764 * q^22 - 17988 * q^23 + 468523 * q^25 + 27624 * q^26 - 59896 * q^28 - 474564 * q^29 + 188186 * q^31 - 251068 * q^32 + 169888 * q^34 - 514500 * q^35 + 1148200 * q^37 - 446594 * q^38 + 501214 * q^40 - 1246152 * q^41 + 62268 * q^43 - 2555520 * q^44 - 1289942 * q^46 - 1485654 * q^47 + 3829555 * q^49 - 6430160 * q^50 - 2624804 * q^52 - 4086740 * q^53 - 1119118 * q^55 - 8448352 * q^56 + 2966706 * q^58 + 6777507 * q^59 + 2436146 * q^61 - 9005952 * q^62 + 11117562 * q^64 - 16730354 * q^65 - 2652248 * q^67 - 15929124 * q^68 + 3359254 * q^70 - 7356324 * q^71 + 1900454 * q^73 - 25386964 * q^74 - 16047360 * q^76 - 20774826 * q^77 - 5912712 * q^79 - 13568404 * q^80 + 1579434 * q^82 + 1052766 * q^83 + 18372730 * q^85 - 43499960 * q^86 + 18209214 * q^88 - 174788 * q^89 - 18891512 * q^91 - 46033270 * q^92 + 365448 * q^94 - 31505580 * q^95 + 14418540 * q^97 + 15964056 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.306188	−0.0270634	−0.0135317	−	0.999908i	$-0.504307\pi$
$2$	−0.306188	−0.0270634	−0.0135317	+	0.999908i	$0.504307\pi$
$3$	0	0
$3$	0	0
$4$	−127.906	−0.999268
$5$	108.774	0.389162	0.194581	−	0.980886i	$-0.437665\pi$
$5$	108.774	0.389162	0.194581	+	0.980886i	$0.437665\pi$
$6$	0	0
$7$	1364.63	1.50373	0.751866	−	0.659315i	$-0.229154\pi$
$7$	1364.63	1.50373	0.751866	+	0.659315i	$0.229154\pi$
$8$	78.3553	0.0541070
$9$	0	0
$10$	−33.3052	−0.0105320
$11$	1380.42	0.312706	0.156353	−	0.987701i	$-0.450026\pi$
$11$	1380.42	0.312706	0.156353	+	0.987701i	$0.450026\pi$
$12$	0	0
$13$	3791.36	0.478623	0.239311	−	0.970943i	$-0.423078\pi$
$13$	3791.36	0.478623	0.239311	+	0.970943i	$0.423078\pi$
$14$	−417.832	−0.0406961
$15$	0	0
$16$	16348.0	0.997803
$17$	−34931.9	−1.72445	−0.862225	−	0.506526i	$-0.830929\pi$
$17$	−34931.9	−1.72445	−0.862225	+	0.506526i	$0.830929\pi$
$18$	0	0
$19$	11151.7	0.372994	0.186497	−	0.982456i	$-0.440287\pi$
$19$	11151.7	0.372994	0.186497	+	0.982456i	$0.440287\pi$
$20$	−13912.9	−0.388877
$21$	0	0
$22$	−422.666	−0.00846288
$23$	−31517.5	−0.540137	−0.270069	−	0.962841i	$-0.587046\pi$
$23$	−31517.5	−0.540137	−0.270069	+	0.962841i	$0.587046\pi$
$24$	0	0
$25$	−66293.2	−0.848553
$26$	−1160.87	−0.0129532
$27$	0	0
$28$	−174544.	−1.50263
$29$	−106066.	−0.807577	−0.403788	−	0.914852i	$-0.632307\pi$
$29$	−106066.	−0.807577	−0.403788	+	0.914852i	$0.632307\pi$
$30$	0	0
$31$	21299.3	0.128410	0.0642050	−	0.997937i	$-0.479549\pi$
$31$	21299.3	0.128410	0.0642050	+	0.997937i	$0.479549\pi$
$32$	−15035.0	−0.0811110
$33$	0	0
$34$	10695.7	0.0466695
$35$	148436.	0.585195
$36$	0	0
$37$	228684.	0.742216	0.371108	−	0.928590i	$-0.378978\pi$
$37$	228684.	0.742216	0.371108	+	0.928590i	$0.378978\pi$
$38$	−3414.50	−0.0100945
$39$	0	0
$40$	8523.02	0.0210564
$41$	569395.	1.29024	0.645119	−	0.764082i	$-0.276808\pi$
$41$	569395.	1.29024	0.645119	+	0.764082i	$0.276808\pi$
$42$	0	0
$43$	−968608.	−1.85784	−0.928921	−	0.370279i	$-0.879262\pi$
$43$	−968608.	−1.85784	−0.928921	+	0.370279i	$0.879262\pi$
$44$	−176564.	−0.312477
$45$	0	0
$46$	9650.26	0.0146180
$47$	−678969.	−0.953910	−0.476955	−	0.878928i	$-0.658260\pi$
$47$	−678969.	−0.953910	−0.476955	+	0.878928i	$0.658260\pi$
$48$	0	0
$49$	1.03866e6	1.26121
$50$	20298.2	0.0229647
$51$	0	0
$52$	−484939.	−0.478272
$53$	−1.04628e6	−0.965345	−0.482672	−	0.875801i	$-0.660334\pi$
$53$	−1.04628e6	−0.965345	−0.482672	+	0.875801i	$0.660334\pi$
$54$	0	0
$55$	150153.	0.121693
$56$	106926.	0.0813625
$57$	0	0
$58$	32476.1	0.0218558
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	0	0
$61$	2.29622e6	1.29527	0.647633	−	0.761952i	$-0.275759\pi$
$61$	2.29622e6	1.29527	0.647633	+	0.761952i	$0.275759\pi$
$62$	−6521.57	−0.00347521
$63$	0	0
$64$	−2.08794e6	−0.995608
$65$	412402.	0.186262
$66$	0	0
$67$	1.89799e6	0.770961	0.385481	−	0.922716i	$-0.374036\pi$
$67$	1.89799e6	0.770961	0.385481	+	0.922716i	$0.374036\pi$
$68$	4.46800e6	1.72319
$69$	0	0
$70$	−45449.2	−0.0158374
$71$	3.15394e6	1.04580	0.522901	−	0.852393i	$-0.324850\pi$
$71$	3.15394e6	1.04580	0.522901	+	0.852393i	$0.324850\pi$
$72$	0	0
$73$	−3.58952e6	−1.07996	−0.539978	−	0.841679i	$-0.681568\pi$
$73$	−3.58952e6	−1.07996	−0.539978	+	0.841679i	$0.681568\pi$
$74$	−70020.3	−0.0200869
$75$	0	0
$76$	−1.42637e6	−0.372721
$77$	1.88375e6	0.470226
$78$	0	0
$79$	−1.93077e6	−0.440591	−0.220295	−	0.975433i	$-0.570702\pi$
$79$	−1.93077e6	−0.440591	−0.220295	+	0.975433i	$0.570702\pi$
$80$	1.77824e6	0.388307
$81$	0	0
$82$	−174342.	−0.0349183
$83$	7.35502e6	1.41192	0.705961	−	0.708251i	$-0.250516\pi$
$83$	7.35502e6	1.41192	0.705961	+	0.708251i	$0.250516\pi$
$84$	0	0
$85$	−3.79968e6	−0.671090
$86$	296576.	0.0502795
$87$	0	0
$88$	108163.	0.0169196
$89$	−8.45678e6	−1.27157	−0.635785	−	0.771866i	$-0.719323\pi$
$89$	−8.45678e6	−1.27157	−0.635785	+	0.771866i	$0.719323\pi$
$90$	0	0
$91$	5.17379e6	0.719721
$92$	4.03128e6	0.539742
$93$	0	0
$94$	207892.	0.0258161
$95$	1.21301e6	0.145155
$96$	0	0
$97$	−739735.	−0.0822953	−0.0411477	−	0.999153i	$-0.513101\pi$
$97$	−739735.	−0.0822953	−0.0411477	+	0.999153i	$0.513101\pi$
$98$	−318026.	−0.0341327
$99$	0	0
$100$	8.47932e6	0.847932
$101$	1.86284e6	0.179908	0.0899538	−	0.995946i	$-0.471328\pi$
$101$	1.86284e6	0.179908	0.0899538	+	0.995946i	$0.471328\pi$
$102$	0	0
$103$	−5.01489e6	−0.452201	−0.226101	−	0.974104i	$-0.572598\pi$
$103$	−5.01489e6	−0.452201	−0.226101	+	0.974104i	$0.572598\pi$
$104$	297073.	0.0258969
$105$	0	0
$106$	320358.	0.0261255
$107$	2.20579e7	1.74069	0.870345	−	0.492442i	$-0.163896\pi$
$107$	2.20579e7	1.74069	0.870345	+	0.492442i	$0.163896\pi$
$108$	0	0
$109$	6.02192e6	0.445392	0.222696	−	0.974888i	$-0.428514\pi$
$109$	6.02192e6	0.445392	0.222696	+	0.974888i	$0.428514\pi$
$110$	−45975.1	−0.00329343
$111$	0	0
$112$	2.23089e7	1.50043
$113$	−2.00653e7	−1.30819	−0.654096	−	0.756412i	$-0.726951\pi$
$113$	−2.00653e7	−1.30819	−0.654096	+	0.756412i	$0.726951\pi$
$114$	0	0
$115$	−3.42828e6	−0.210201
$116$	1.35665e7	0.806985
$117$	0	0
$118$	−62884.5	−0.00352336
$119$	−4.76689e7	−2.59311
$120$	0	0
$121$	−1.75816e7	−0.902215
$122$	−703074.	−0.0350543
$123$	0	0
$124$	−2.72431e6	−0.128316
$125$	−1.57089e7	−0.719386
$126$	0	0
$127$	7.58366e6	0.328523	0.164262	−	0.986417i	$-0.447476\pi$
$127$	7.58366e6	0.328523	0.164262	+	0.986417i	$0.447476\pi$
$128$	2.56379e6	0.108056
$129$	0	0
$130$	−126272.	−0.00504088
$131$	−3.80169e7	−1.47750	−0.738749	−	0.673980i	$-0.764583\pi$
$131$	−3.80169e7	−1.47750	−0.738749	+	0.673980i	$0.764583\pi$
$132$	0	0
$133$	1.52179e7	0.560884
$134$	−581142.	−0.0208648
$135$	0	0
$136$	−2.73710e6	−0.0933048
$137$	−5.80500e7	−1.92877	−0.964385	−	0.264502i	$-0.914792\pi$
$137$	−5.80500e7	−1.92877	−0.964385	+	0.264502i	$0.914792\pi$
$138$	0	0
$139$	1.53939e7	0.486178	0.243089	−	0.970004i	$-0.421839\pi$
$139$	1.53939e7	0.486178	0.243089	+	0.970004i	$0.421839\pi$
$140$	−1.89859e7	−0.584767
$141$	0	0
$142$	−965698.	−0.0283030
$143$	5.23366e6	0.149668
$144$	0	0
$145$	−1.15372e7	−0.314278
$146$	1.09907e6	0.0292273
$147$	0	0
$148$	−2.92501e7	−0.741672
$149$	−1.41821e7	−0.351226	−0.175613	−	0.984459i	$-0.556191\pi$
$149$	−1.41821e7	−0.351226	−0.175613	+	0.984459i	$0.556191\pi$
$150$	0	0
$151$	−4.80991e7	−1.13689	−0.568444	−	0.822722i	$-0.692454\pi$
$151$	−4.80991e7	−1.13689	−0.568444	+	0.822722i	$0.692454\pi$
$152$	873792.	0.0201816
$153$	0	0
$154$	−576782.	−0.0127259
$155$	2.31681e6	0.0499722
$156$	0	0
$157$	4.29244e7	0.885229	0.442614	−	0.896712i	$-0.354051\pi$
$157$	4.29244e7	0.885229	0.442614	+	0.896712i	$0.354051\pi$
$158$	591177.	0.0119239
$159$	0	0
$160$	−1.63542e6	−0.0315653
$161$	−4.30096e7	−0.812222
$162$	0	0
$163$	1.74523e7	0.315643	0.157822	−	0.987468i	$-0.449553\pi$
$163$	1.74523e7	0.315643	0.157822	+	0.987468i	$0.449553\pi$
$164$	−7.28292e7	−1.28929
$165$	0	0
$166$	−2.25202e6	−0.0382114
$167$	−8.03770e7	−1.33544	−0.667720	−	0.744413i	$-0.732729\pi$
$167$	−8.03770e7	−1.33544	−0.667720	+	0.744413i	$0.732729\pi$
$168$	0	0
$169$	−4.83741e7	−0.770920
$170$	1.16341e6	0.0181620
$171$	0	0
$172$	1.23891e8	1.85648
$173$	9.87388e7	1.44986	0.724931	−	0.688822i	$-0.241872\pi$
$173$	9.87388e7	1.44986	0.724931	+	0.688822i	$0.241872\pi$
$174$	0	0
$175$	−9.04655e7	−1.27600
$176$	2.25671e7	0.312019
$177$	0	0
$178$	2.58936e6	0.0344130
$179$	−1.31592e8	−1.71492	−0.857460	−	0.514550i	$-0.827959\pi$
$179$	−1.31592e8	−1.71492	−0.857460	+	0.514550i	$0.827959\pi$
$180$	0	0
$181$	6.14014e7	0.769668	0.384834	−	0.922986i	$-0.374259\pi$
$181$	6.14014e7	0.769668	0.384834	+	0.922986i	$0.374259\pi$
$182$	−1.58415e6	−0.0194781
$183$	0	0
$184$	−2.46956e6	−0.0292252
$185$	2.48749e7	0.288842
$186$	0	0
$187$	−4.82205e7	−0.539245
$188$	8.68444e7	0.953212
$189$	0	0
$190$	−371409.	−0.00392839
$191$	2.72360e7	0.282831	0.141415	−	0.989950i	$-0.454835\pi$
$191$	2.72360e7	0.282831	0.141415	+	0.989950i	$0.454835\pi$
$192$	0	0
$193$	8.33561e7	0.834616	0.417308	−	0.908765i	$-0.362974\pi$
$193$	8.33561e7	0.834616	0.417308	+	0.908765i	$0.362974\pi$
$194$	226498.	0.00222719
$195$	0	0
$196$	−1.32851e8	−1.26029
$197$	1.66599e7	0.155253	0.0776267	−	0.996982i	$-0.475266\pi$
$197$	1.66599e7	0.155253	0.0776267	+	0.996982i	$0.475266\pi$
$198$	0	0
$199$	−9.50050e7	−0.854596	−0.427298	−	0.904111i	$-0.640534\pi$
$199$	−9.50050e7	−0.854596	−0.427298	+	0.904111i	$0.640534\pi$
$200$	−5.19443e6	−0.0459127
$201$	0	0
$202$	−570377.	−0.00486892
$203$	−1.44741e8	−1.21438
$204$	0	0
$205$	6.19354e7	0.502111
$206$	1.53550e6	0.0122381
$207$	0	0
$208$	6.19812e7	0.477572
$209$	1.53939e7	0.116637
$210$	0	0
$211$	7.24057e7	0.530620	0.265310	−	0.964163i	$-0.414526\pi$
$211$	7.24057e7	0.530620	0.265310	+	0.964163i	$0.414526\pi$
$212$	1.33826e8	0.964638
$213$	0	0
$214$	−6.75386e6	−0.0471090
$215$	−1.05359e8	−0.723001
$216$	0	0
$217$	2.90655e7	0.193094
$218$	−1.84384e6	−0.0120538
$219$	0	0
$220$	−1.92056e7	−0.121604
$221$	−1.32439e8	−0.825361
$222$	0	0
$223$	−1.58178e8	−0.955168	−0.477584	−	0.878586i	$-0.658487\pi$
$223$	−1.58178e8	−0.955168	−0.477584	+	0.878586i	$0.658487\pi$
$224$	−2.05172e7	−0.121969
$225$	0	0
$226$	6.14375e6	0.0354041
$227$	−1.06050e8	−0.601756	−0.300878	−	0.953663i	$-0.597280\pi$
$227$	−1.06050e8	−0.601756	−0.300878	+	0.953663i	$0.597280\pi$
$228$	0	0
$229$	−1.31366e8	−0.722866	−0.361433	−	0.932398i	$-0.617712\pi$
$229$	−1.31366e8	−0.722866	−0.361433	+	0.932398i	$0.617712\pi$
$230$	1.04970e6	0.00568875
$231$	0	0
$232$	−8.31085e6	−0.0436956
$233$	3.11522e8	1.61340	0.806701	−	0.590960i	$-0.201251\pi$
$233$	3.11522e8	1.61340	0.806701	+	0.590960i	$0.201251\pi$
$234$	0	0
$235$	−7.38542e7	−0.371225
$236$	−2.62693e7	−0.130094
$237$	0	0
$238$	1.45956e7	0.0701784
$239$	−2.16486e8	−1.02574	−0.512870	−	0.858466i	$-0.671418\pi$
$239$	−2.16486e8	−1.02574	−0.512870	+	0.858466i	$0.671418\pi$
$240$	0	0
$241$	3.42282e8	1.57516	0.787581	−	0.616212i	$-0.211333\pi$
$241$	3.42282e8	1.57516	0.787581	+	0.616212i	$0.211333\pi$
$242$	5.38327e6	0.0244170
$243$	0	0
$244$	−2.93701e8	−1.29432
$245$	1.12980e8	0.490816
$246$	0	0
$247$	4.22800e7	0.178524
$248$	1.66891e6	0.00694788
$249$	0	0
$250$	4.80988e6	0.0194690
$251$	6.01836e7	0.240226	0.120113	−	0.992760i	$-0.461674\pi$
$251$	6.01836e7	0.240226	0.120113	+	0.992760i	$0.461674\pi$
$252$	0	0
$253$	−4.35072e7	−0.168904
$254$	−2.32202e6	−0.00889096
$255$	0	0
$256$	2.66472e8	0.992684
$257$	−4.62939e8	−1.70121	−0.850605	−	0.525806i	$-0.823764\pi$
$257$	−4.62939e8	−1.70121	−0.850605	+	0.525806i	$0.823764\pi$
$258$	0	0
$259$	3.12069e8	1.11609
$260$	−5.27487e7	−0.186125
$261$	0	0
$262$	1.16403e7	0.0399862
$263$	−4.95242e8	−1.67870	−0.839349	−	0.543593i	$-0.817064\pi$
$263$	−4.95242e8	−1.67870	−0.839349	+	0.543593i	$0.817064\pi$
$264$	0	0
$265$	−1.13808e8	−0.375675
$266$	−4.65952e6	−0.0151794
$267$	0	0
$268$	−2.42765e8	−0.770397
$269$	−3.83775e8	−1.20211	−0.601055	−	0.799208i	$-0.705253\pi$
$269$	−3.83775e8	−1.20211	−0.601055	+	0.799208i	$0.705253\pi$
$270$	0	0
$271$	−1.34781e8	−0.411374	−0.205687	−	0.978618i	$-0.565943\pi$
$271$	−1.34781e8	−0.411374	−0.205687	+	0.978618i	$0.565943\pi$
$272$	−5.71066e8	−1.72066
$273$	0	0
$274$	1.77742e7	0.0521991
$275$	−9.15122e7	−0.265347
$276$	0	0
$277$	2.92128e8	0.825837	0.412919	−	0.910768i	$-0.364509\pi$
$277$	2.92128e8	0.825837	0.412919	+	0.910768i	$0.364509\pi$
$278$	−4.71341e6	−0.0131576
$279$	0	0
$280$	1.16307e7	0.0316632
$281$	−1.94745e8	−0.523594	−0.261797	−	0.965123i	$-0.584315\pi$
$281$	−1.94745e8	−0.523594	−0.261797	+	0.965123i	$0.584315\pi$
$282$	0	0
$283$	1.70866e7	0.0448128	0.0224064	−	0.999749i	$-0.492867\pi$
$283$	1.70866e7	0.0448128	0.0224064	+	0.999749i	$0.492867\pi$
$284$	−4.03409e8	−1.04504
$285$	0	0
$286$	−1.60248e6	−0.00405053
$287$	7.77011e8	1.94017
$288$	0	0
$289$	8.09896e8	1.97372
$290$	3.53256e6	0.00850543
$291$	0	0
$292$	4.59122e8	1.07917
$293$	−4.26160e8	−0.989774	−0.494887	−	0.868957i	$-0.664791\pi$
$293$	−4.26160e8	−0.989774	−0.494887	+	0.868957i	$0.664791\pi$
$294$	0	0
$295$	2.23399e7	0.0506645
$296$	1.79186e7	0.0401591
$297$	0	0
$298$	4.34237e6	0.00950539
$299$	−1.19494e8	−0.258522
$300$	0	0
$301$	−1.32179e9	−2.79370
$302$	1.47273e7	0.0307680
$303$	0	0
$304$	1.82308e8	0.372175
$305$	2.49769e8	0.504068
$306$	0	0
$307$	−7.48385e8	−1.47619	−0.738093	−	0.674699i	$-0.764273\pi$
$307$	−7.48385e8	−1.47619	−0.738093	+	0.674699i	$0.764273\pi$
$308$	−2.40944e8	−0.469881
$309$	0	0
$310$	−709377.	−0.00135242
$311$	−4.87291e8	−0.918602	−0.459301	−	0.888281i	$-0.651900\pi$
$311$	−4.87291e8	−0.918602	−0.459301	+	0.888281i	$0.651900\pi$
$312$	0	0
$313$	−3.91139e8	−0.720984	−0.360492	−	0.932762i	$-0.617391\pi$
$313$	−3.91139e8	−0.720984	−0.360492	+	0.932762i	$0.617391\pi$
$314$	−1.31429e7	−0.0239573
$315$	0	0
$316$	2.46957e8	0.440268
$317$	2.79172e8	0.492227	0.246113	−	0.969241i	$-0.420846\pi$
$317$	2.79172e8	0.492227	0.246113	+	0.969241i	$0.420846\pi$
$318$	0	0
$319$	−1.46415e8	−0.252534
$320$	−2.27114e8	−0.387453
$321$	0	0
$322$	1.31690e7	0.0219815
$323$	−3.89548e8	−0.643210
$324$	0	0
$325$	−2.51342e8	−0.406137
$326$	−5.34368e6	−0.00854238
$327$	0	0
$328$	4.46151e7	0.0698110
$329$	−9.26539e8	−1.43443
$330$	0	0
$331$	−5.78922e8	−0.877449	−0.438725	−	0.898622i	$-0.644570\pi$
$331$	−5.78922e8	−0.877449	−0.438725	+	0.898622i	$0.644570\pi$
$332$	−9.40753e8	−1.41089
$333$	0	0
$334$	2.46104e7	0.0361415
$335$	2.06452e8	0.300029
$336$	0	0
$337$	7.64970e8	1.08878	0.544389	−	0.838833i	$-0.316761\pi$
$337$	7.64970e8	1.08878	0.544389	+	0.838833i	$0.316761\pi$
$338$	1.48115e7	0.0208637
$339$	0	0
$340$	4.86002e8	0.670598
$341$	2.94019e7	0.0401545
$342$	0	0
$343$	2.93558e8	0.392794
$344$	−7.58956e7	−0.100522
$345$	0	0
$346$	−3.02326e7	−0.0392382
$347$	4.28957e8	0.551139	0.275569	−	0.961281i	$-0.411134\pi$
$347$	4.28957e8	0.551139	0.275569	+	0.961281i	$0.411134\pi$
$348$	0	0
$349$	1.29124e9	1.62599	0.812995	−	0.582271i	$-0.197836\pi$
$349$	1.29124e9	1.62599	0.812995	+	0.582271i	$0.197836\pi$
$350$	2.76994e7	0.0345328
$351$	0	0
$352$	−2.07546e7	−0.0253639
$353$	−7.25527e8	−0.877894	−0.438947	−	0.898513i	$-0.644648\pi$
$353$	−7.25527e8	−0.877894	−0.438947	+	0.898513i	$0.644648\pi$
$354$	0	0
$355$	3.43067e8	0.406986
$356$	1.08168e9	1.27064
$357$	0	0
$358$	4.02918e7	0.0464116
$359$	7.97374e8	0.909560	0.454780	−	0.890604i	$-0.349718\pi$
$359$	7.97374e8	0.909560	0.454780	+	0.890604i	$0.349718\pi$
$360$	0	0
$361$	−7.69512e8	−0.860875
$362$	−1.88004e7	−0.0208298
$363$	0	0
$364$	−6.61761e8	−0.719194
$365$	−3.90446e8	−0.420278
$366$	0	0
$367$	−1.65925e8	−0.175218	−0.0876092	−	0.996155i	$-0.527923\pi$
$367$	−1.65925e8	−0.175218	−0.0876092	+	0.996155i	$0.527923\pi$
$368$	−5.15248e8	−0.538951
$369$	0	0
$370$	−7.61638e6	−0.00781705
$371$	−1.42778e9	−1.45162
$372$	0	0
$373$	5.83358e8	0.582041	0.291021	−	0.956717i	$-0.406005\pi$
$373$	5.83358e8	0.582041	0.291021	+	0.956717i	$0.406005\pi$
$374$	1.47645e7	0.0145938
$375$	0	0
$376$	−5.32008e7	−0.0516132
$377$	−4.02135e8	−0.386525
$378$	0	0
$379$	−8.69851e8	−0.820744	−0.410372	−	0.911918i	$-0.634601\pi$
$379$	−8.69851e8	−0.820744	−0.410372	+	0.911918i	$0.634601\pi$
$380$	−1.55152e8	−0.145049
$381$	0	0
$382$	−8.33933e6	−0.00765437
$383$	−4.96463e8	−0.451535	−0.225768	−	0.974181i	$-0.572489\pi$
$383$	−4.96463e8	−0.451535	−0.225768	+	0.974181i	$0.572489\pi$
$384$	0	0
$385$	2.04903e8	0.182994
$386$	−2.55226e7	−0.0225876
$387$	0	0
$388$	9.46168e7	0.0822351
$389$	1.15071e9	0.991160	0.495580	−	0.868562i	$-0.334955\pi$
$389$	1.15071e9	0.991160	0.495580	+	0.868562i	$0.334955\pi$
$390$	0	0
$391$	1.10096e9	0.931439
$392$	8.13847e7	0.0682404
$393$	0	0
$394$	−5.10106e6	−0.00420169
$395$	−2.10017e8	−0.171461
$396$	0	0
$397$	−2.11573e9	−1.69705	−0.848523	−	0.529159i	$-0.822507\pi$
$397$	−2.11573e9	−1.69705	−0.848523	+	0.529159i	$0.822507\pi$
$398$	2.90894e7	0.0231283
$399$	0	0
$400$	−1.08376e9	−0.846689
$401$	−1.68557e9	−1.30539	−0.652697	−	0.757619i	$-0.726363\pi$
$401$	−1.68557e9	−1.30539	−0.652697	+	0.757619i	$0.726363\pi$
$402$	0	0
$403$	8.07532e7	0.0614600
$404$	−2.38268e8	−0.179776
$405$	0	0
$406$	4.43178e7	0.0328653
$407$	3.15679e8	0.232095
$408$	0	0
$409$	−3.25943e8	−0.235565	−0.117782	−	0.993039i	$-0.537579\pi$
$409$	−3.25943e8	−0.235565	−0.117782	+	0.993039i	$0.537579\pi$
$410$	−1.89638e7	−0.0135889
$411$	0	0
$412$	6.41436e8	0.451870
$413$	2.80266e8	0.195769
$414$	0	0
$415$	8.00035e8	0.549466
$416$	−5.70033e7	−0.0388216
$417$	0	0
$418$	−4.71343e6	−0.00315661
$419$	−1.62609e9	−1.07993	−0.539964	−	0.841688i	$-0.681562\pi$
$419$	−1.62609e9	−1.07993	−0.539964	+	0.841688i	$0.681562\pi$
$420$	0	0
$421$	−2.63161e9	−1.71884	−0.859418	−	0.511273i	$-0.829174\pi$
$421$	−2.63161e9	−1.71884	−0.859418	+	0.511273i	$0.829174\pi$
$422$	−2.21697e7	−0.0143604
$423$	0	0
$424$	−8.19816e7	−0.0522319
$425$	2.31574e9	1.46329
$426$	0	0
$427$	3.13348e9	1.94773
$428$	−2.82135e9	−1.73942
$429$	0	0
$430$	3.22597e7	0.0195669
$431$	4.09233e8	0.246207	0.123103	−	0.992394i	$-0.460715\pi$
$431$	4.09233e8	0.246207	0.123103	+	0.992394i	$0.460715\pi$
$432$	0	0
$433$	1.94621e9	1.15208	0.576038	−	0.817423i	$-0.304598\pi$
$433$	1.94621e9	1.15208	0.576038	+	0.817423i	$0.304598\pi$
$434$	−8.89951e6	−0.00522579
$435$	0	0
$436$	−7.70241e8	−0.445066
$437$	−3.51472e8	−0.201468
$438$	0	0
$439$	−3.31628e9	−1.87079	−0.935396	−	0.353602i	$-0.884957\pi$
$439$	−3.31628e9	−1.87079	−0.935396	+	0.353602i	$0.884957\pi$
$440$	1.17653e7	0.00658445
$441$	0	0
$442$	4.05513e7	0.0223371
$443$	1.95922e9	1.07070	0.535352	−	0.844629i	$-0.320179\pi$
$443$	1.95922e9	1.07070	0.535352	+	0.844629i	$0.320179\pi$
$444$	0	0
$445$	−9.19878e8	−0.494846
$446$	4.84322e7	0.0258501
$447$	0	0
$448$	−2.84926e9	−1.49713
$449$	1.35503e9	0.706458	0.353229	−	0.935537i	$-0.385084\pi$
$449$	1.35503e9	0.706458	0.353229	+	0.935537i	$0.385084\pi$
$450$	0	0
$451$	7.86002e8	0.403465
$452$	2.56648e9	1.30723
$453$	0	0
$454$	3.24712e7	0.0162856
$455$	5.62774e8	0.280088
$456$	0	0
$457$	3.60604e9	1.76736	0.883679	−	0.468094i	$-0.155059\pi$
$457$	3.60604e9	1.76736	0.883679	+	0.468094i	$0.155059\pi$
$458$	4.02225e7	0.0195632
$459$	0	0
$460$	4.38499e8	0.210047
$461$	−1.13023e8	−0.0537298	−0.0268649	−	0.999639i	$-0.508552\pi$
$461$	−1.13023e8	−0.0537298	−0.0268649	+	0.999639i	$0.508552\pi$
$462$	0	0
$463$	9.93503e8	0.465196	0.232598	−	0.972573i	$-0.425277\pi$
$463$	9.93503e8	0.465196	0.232598	+	0.972573i	$0.425277\pi$
$464$	−1.73397e9	−0.805803
$465$	0	0
$466$	−9.53841e7	−0.0436642
$467$	−9.29833e7	−0.0422470	−0.0211235	−	0.999777i	$-0.506724\pi$
$467$	−9.29833e7	−0.0422470	−0.0211235	+	0.999777i	$0.506724\pi$
$468$	0	0
$469$	2.59005e9	1.15932
$470$	2.26132e7	0.0100466
$471$	0	0
$472$	1.60925e7	0.00704413
$473$	−1.33708e9	−0.580957
$474$	0	0
$475$	−7.39280e8	−0.316505
$476$	6.09715e9	2.59121
$477$	0	0
$478$	6.62853e7	0.0277600
$479$	−2.20001e8	−0.0914639	−0.0457319	−	0.998954i	$-0.514562\pi$
$479$	−2.20001e8	−0.0914639	−0.0457319	+	0.998954i	$0.514562\pi$
$480$	0	0
$481$	8.67025e8	0.355241
$482$	−1.04803e8	−0.0426292
$483$	0	0
$484$	2.24880e9	0.901554
$485$	−8.04640e7	−0.0320262
$486$	0	0
$487$	−1.55645e9	−0.610639	−0.305319	−	0.952250i	$-0.598763\pi$
$487$	−1.55645e9	−0.610639	−0.305319	+	0.952250i	$0.598763\pi$
$488$	1.79921e8	0.0700830
$489$	0	0
$490$	−3.45929e7	−0.0132831
$491$	−5.08698e9	−1.93943	−0.969716	−	0.244236i	$-0.921463\pi$
$491$	−5.08698e9	−1.93943	−0.969716	+	0.244236i	$0.921463\pi$
$492$	0	0
$493$	3.70509e9	1.39263
$494$	−1.29456e7	−0.00483146
$495$	0	0
$496$	3.48201e8	0.128128
$497$	4.30396e9	1.57261
$498$	0	0
$499$	5.45378e8	0.196492	0.0982462	−	0.995162i	$-0.468677\pi$
$499$	5.45378e8	0.196492	0.0982462	+	0.995162i	$0.468677\pi$
$500$	2.00927e9	0.718859
$501$	0	0
$502$	−1.84275e7	−0.00650134
$503$	3.73639e9	1.30907	0.654537	−	0.756030i	$-0.272863\pi$
$503$	3.73639e9	1.30907	0.654537	+	0.756030i	$0.272863\pi$
$504$	0	0
$505$	2.02628e8	0.0700132
$506$	1.33214e7	0.00457112
$507$	0	0
$508$	−9.69998e8	−0.328283
$509$	3.31062e9	1.11275	0.556374	−	0.830932i	$-0.312192\pi$
$509$	3.31062e9	1.11275	0.556374	+	0.830932i	$0.312192\pi$
$510$	0	0
$511$	−4.89835e9	−1.62397
$512$	−4.09755e8	−0.134921
$513$	0	0
$514$	1.41746e8	0.0460405
$515$	−5.45490e8	−0.175979
$516$	0	0
$517$	−9.37260e8	−0.298293
$518$	−9.55515e7	−0.0302053
$519$	0	0
$520$	3.23139e7	0.0100781
$521$	−7.72545e8	−0.239327	−0.119664	−	0.992815i	$-0.538182\pi$
$521$	−7.72545e8	−0.239327	−0.119664	+	0.992815i	$0.538182\pi$
$522$	0	0
$523$	3.87820e9	1.18542	0.592712	−	0.805414i	$-0.298057\pi$
$523$	3.87820e9	1.18542	0.592712	+	0.805414i	$0.298057\pi$
$524$	4.86260e9	1.47642
$525$	0	0
$526$	1.51637e8	0.0454313
$527$	−7.44023e8	−0.221436
$528$	0	0
$529$	−2.41147e9	−0.708252
$530$	3.48466e7	0.0101671
$531$	0	0
$532$	−1.94646e9	−0.560473
$533$	2.15878e9	0.617538
$534$	0	0
$535$	2.39933e9	0.677410
$536$	1.48718e8	0.0417144
$537$	0	0
$538$	1.17507e8	0.0325332
$539$	1.43379e9	0.394388
$540$	0	0
$541$	−3.37493e9	−0.916376	−0.458188	−	0.888855i	$-0.651501\pi$
$541$	−3.37493e9	−0.916376	−0.458188	+	0.888855i	$0.651501\pi$
$542$	4.12683e7	0.0111332
$543$	0	0
$544$	5.25202e8	0.139872
$545$	6.55028e8	0.173329
$546$	0	0
$547$	2.44940e9	0.639887	0.319943	−	0.947437i	$-0.396336\pi$
$547$	2.44940e9	0.639887	0.319943	+	0.947437i	$0.396336\pi$
$548$	7.42496e9	1.92736
$549$	0	0
$550$	2.80199e7	0.00718121
$551$	−1.18281e9	−0.301221
$552$	0	0
$553$	−2.63478e9	−0.662531
$554$	−8.94460e7	−0.0223500
$555$	0	0
$556$	−1.96897e9	−0.485822
$557$	3.56801e9	0.874848	0.437424	−	0.899255i	$-0.355891\pi$
$557$	3.56801e9	0.874848	0.437424	+	0.899255i	$0.355891\pi$
$558$	0	0
$559$	−3.67234e9	−0.889205
$560$	2.42663e9	0.583910
$561$	0	0
$562$	5.96286e7	0.0141702
$563$	−2.93642e9	−0.693489	−0.346744	−	0.937960i	$-0.612713\pi$
$563$	−2.93642e9	−0.693489	−0.346744	+	0.937960i	$0.612713\pi$
$564$	0	0
$565$	−2.18258e9	−0.509098
$566$	−5.23169e6	−0.00121279
$567$	0	0
$568$	2.47128e8	0.0565852
$569$	9.92744e8	0.225915	0.112957	−	0.993600i	$-0.463968\pi$
$569$	9.92744e8	0.225915	0.112957	+	0.993600i	$0.463968\pi$
$570$	0	0
$571$	2.90579e9	0.653188	0.326594	−	0.945165i	$-0.394099\pi$
$571$	2.90579e9	0.653188	0.326594	+	0.945165i	$0.394099\pi$
$572$	−6.69418e8	−0.149558
$573$	0	0
$574$	−2.37911e8	−0.0525077
$575$	2.08940e9	0.458335
$576$	0	0
$577$	4.52464e9	0.980547	0.490273	−	0.871569i	$-0.336897\pi$
$577$	4.52464e9	0.980547	0.490273	+	0.871569i	$0.336897\pi$
$578$	−2.47980e8	−0.0534157
$579$	0	0
$580$	1.47569e9	0.314048
$581$	1.00369e10	2.12315
$582$	0	0
$583$	−1.44430e9	−0.301869
$584$	−2.81258e8	−0.0584332
$585$	0	0
$586$	1.30485e8	0.0267867
$587$	−4.81657e9	−0.982890	−0.491445	−	0.870909i	$-0.663531\pi$
$587$	−4.81657e9	−0.982890	−0.491445	+	0.870909i	$0.663531\pi$
$588$	0	0
$589$	2.37522e8	0.0478962
$590$	−6.84020e6	−0.00137116
$591$	0	0
$592$	3.73853e9	0.740585
$593$	6.21778e7	0.0122446	0.00612229	−	0.999981i	$-0.498051\pi$
$593$	6.21778e7	0.0122446	0.00612229	+	0.999981i	$0.498051\pi$
$594$	0	0
$595$	−5.18514e9	−1.00914
$596$	1.81397e9	0.350969
$597$	0	0
$598$	3.65876e7	0.00699649
$599$	1.74655e9	0.332037	0.166019	−	0.986123i	$-0.446909\pi$
$599$	1.74655e9	0.332037	0.166019	+	0.986123i	$0.446909\pi$
$600$	0	0
$601$	−9.84017e9	−1.84902	−0.924511	−	0.381156i	$-0.875526\pi$
$601$	−9.84017e9	−1.84902	−0.924511	+	0.381156i	$0.875526\pi$
$602$	4.04715e8	0.0756070
$603$	0	0
$604$	6.15217e9	1.13605
$605$	−1.91242e9	−0.351108
$606$	0	0
$607$	5.49249e9	0.996802	0.498401	−	0.866947i	$-0.333921\pi$
$607$	5.49249e9	0.996802	0.498401	+	0.866947i	$0.333921\pi$
$608$	−1.67666e8	−0.0302539
$609$	0	0
$610$	−7.64762e7	−0.0136418
$611$	−2.57422e9	−0.456563
$612$	0	0
$613$	−8.81582e9	−1.54579	−0.772896	−	0.634533i	$-0.781193\pi$
$613$	−8.81582e9	−1.54579	−0.772896	+	0.634533i	$0.781193\pi$
$614$	2.29146e8	0.0399506
$615$	0	0
$616$	1.47602e8	0.0254425
$617$	−2.75940e9	−0.472952	−0.236476	−	0.971637i	$-0.575992\pi$
$617$	−2.75940e9	−0.472952	−0.236476	+	0.971637i	$0.575992\pi$
$618$	0	0
$619$	4.79478e9	0.812552	0.406276	−	0.913750i	$-0.366827\pi$
$619$	4.79478e9	0.812552	0.406276	+	0.913750i	$0.366827\pi$
$620$	−2.96334e8	−0.0499356
$621$	0	0
$622$	1.49202e8	0.0248605
$623$	−1.15404e10	−1.91210
$624$	0	0
$625$	3.47043e9	0.568596
$626$	1.19762e8	0.0195123
$627$	0	0
$628$	−5.49030e9	−0.884581
$629$	−7.98836e9	−1.27991
$630$	0	0
$631$	2.33005e9	0.369201	0.184600	−	0.982814i	$-0.440901\pi$
$631$	2.33005e9	0.369201	0.184600	+	0.982814i	$0.440901\pi$
$632$	−1.51286e8	−0.0238390
$633$	0	0
$634$	−8.54791e7	−0.0133213
$635$	8.24905e8	0.127849
$636$	0	0
$637$	3.93795e9	0.603645
$638$	4.48306e7	0.00683443
$639$	0	0
$640$	2.78873e8	0.0420511
$641$	9.93268e9	1.48958	0.744789	−	0.667299i	$-0.232550\pi$
$641$	9.93268e9	1.48958	0.744789	+	0.667299i	$0.232550\pi$
$642$	0	0
$643$	1.45897e9	0.216425	0.108213	−	0.994128i	$-0.465487\pi$
$643$	1.45897e9	0.216425	0.108213	+	0.994128i	$0.465487\pi$
$644$	5.50120e9	0.811627
$645$	0	0
$646$	1.19275e8	0.0174074
$647$	−8.32239e9	−1.20804	−0.604022	−	0.796967i	$-0.706436\pi$
$647$	−8.32239e9	−1.20804	−0.604022	+	0.796967i	$0.706436\pi$
$648$	0	0
$649$	2.83509e8	0.0407108
$650$	7.69577e7	0.0109915
$651$	0	0
$652$	−2.23226e9	−0.315412
$653$	−1.06438e10	−1.49589	−0.747946	−	0.663760i	$-0.768960\pi$
$653$	−1.06438e10	−1.49589	−0.747946	+	0.663760i	$0.768960\pi$
$654$	0	0
$655$	−4.13525e9	−0.574986
$656$	9.30847e9	1.28740
$657$	0	0
$658$	2.83695e8	0.0388205
$659$	−5.23489e9	−0.712539	−0.356269	−	0.934383i	$-0.615951\pi$
$659$	−5.23489e9	−0.712539	−0.356269	+	0.934383i	$0.615951\pi$
$660$	0	0
$661$	5.75185e9	0.774645	0.387322	−	0.921944i	$-0.373400\pi$
$661$	5.75185e9	0.774645	0.387322	+	0.921944i	$0.373400\pi$
$662$	1.77259e8	0.0237468
$663$	0	0
$664$	5.76305e8	0.0763948
$665$	1.65531e9	0.218274
$666$	0	0
$667$	3.34294e9	0.436202
$668$	1.02807e10	1.33446
$669$	0	0
$670$	−6.32131e7	−0.00811980
$671$	3.16974e9	0.405037
$672$	0	0
$673$	−3.16609e9	−0.400379	−0.200189	−	0.979757i	$-0.564156\pi$
$673$	−3.16609e9	−0.400379	−0.200189	+	0.979757i	$0.564156\pi$
$674$	−2.34224e8	−0.0294661
$675$	0	0
$676$	6.18735e9	0.770355
$677$	8.47144e9	1.04929	0.524646	−	0.851320i	$-0.324198\pi$
$677$	8.47144e9	1.04929	0.524646	+	0.851320i	$0.324198\pi$
$678$	0	0
$679$	−1.00946e9	−0.123750
$680$	−2.97725e8	−0.0363106
$681$	0	0
$682$	−9.00248e6	−0.00108672
$683$	−3.14022e9	−0.377127	−0.188564	−	0.982061i	$-0.560383\pi$
$683$	−3.14022e9	−0.377127	−0.188564	+	0.982061i	$0.560383\pi$
$684$	0	0
$685$	−6.31433e9	−0.750603
$686$	−8.98839e7	−0.0106303
$687$	0	0
$688$	−1.58348e10	−1.85376
$689$	−3.96683e9	−0.462036
$690$	0	0
$691$	9.26785e9	1.06858	0.534289	−	0.845302i	$-0.320580\pi$
$691$	9.26785e9	1.06858	0.534289	+	0.845302i	$0.320580\pi$
$692$	−1.26293e10	−1.44880
$693$	0	0
$694$	−1.31341e8	−0.0149157
$695$	1.67445e9	0.189202
$696$	0	0
$697$	−1.98900e10	−2.22495
$698$	−3.95362e8	−0.0440048
$699$	0	0
$700$	1.15711e10	1.27506
$701$	3.59286e9	0.393938	0.196969	−	0.980410i	$-0.436890\pi$
$701$	3.59286e9	0.393938	0.196969	+	0.980410i	$0.436890\pi$
$702$	0	0
$703$	2.55021e9	0.276842
$704$	−2.88223e9	−0.311332
$705$	0	0
$706$	2.22147e8	0.0237588
$707$	2.54208e9	0.270533
$708$	0	0
$709$	1.73640e9	0.182973	0.0914866	−	0.995806i	$-0.470838\pi$
$709$	1.73640e9	0.182973	0.0914866	+	0.995806i	$0.470838\pi$
$710$	−1.05043e8	−0.0110144
$711$	0	0
$712$	−6.62634e8	−0.0688008
$713$	−6.71299e8	−0.0693590
$714$	0	0
$715$	5.69286e8	0.0582451
$716$	1.68314e10	1.71366
$717$	0	0
$718$	−2.44146e8	−0.0246158
$719$	−1.67233e9	−0.167792	−0.0838959	−	0.996475i	$-0.526736\pi$
$719$	−1.67233e9	−0.167792	−0.0838959	+	0.996475i	$0.526736\pi$
$720$	0	0
$721$	−6.84346e9	−0.679990
$722$	2.35615e8	0.0232982
$723$	0	0
$724$	−7.85363e9	−0.769104
$725$	7.03147e9	0.685272
$726$	0	0
$727$	1.81130e9	0.174832	0.0874158	−	0.996172i	$-0.472139\pi$
$727$	1.81130e9	0.174832	0.0874158	+	0.996172i	$0.472139\pi$
$728$	4.05394e8	0.0389420
$729$	0	0
$730$	1.19550e8	0.0113741
$731$	3.38353e10	3.20375
$732$	0	0
$733$	1.92456e10	1.80496	0.902482	−	0.430727i	$-0.141743\pi$
$733$	1.92456e10	1.80496	0.902482	+	0.430727i	$0.141743\pi$
$734$	5.08041e7	0.00474201
$735$	0	0
$736$	4.73866e8	0.0438110
$737$	2.62002e9	0.241084
$738$	0	0
$739$	−9.89116e9	−0.901553	−0.450777	−	0.892637i	$-0.648853\pi$
$739$	−9.89116e9	−0.901553	−0.450777	+	0.892637i	$0.648853\pi$
$740$	−3.18165e9	−0.288630
$741$	0	0
$742$	4.37169e8	0.0392858
$743$	2.88542e9	0.258077	0.129038	−	0.991640i	$-0.458811\pi$
$743$	2.88542e9	0.258077	0.129038	+	0.991640i	$0.458811\pi$
$744$	0	0
$745$	−1.54264e9	−0.136684
$746$	−1.78617e8	−0.0157520
$747$	0	0
$748$	6.16770e9	0.538850
$749$	3.01008e10	2.61753
$750$	0	0
$751$	1.64428e10	1.41656	0.708282	−	0.705930i	$-0.249471\pi$
$751$	1.64428e10	1.41656	0.708282	+	0.705930i	$0.249471\pi$
$752$	−1.10998e10	−0.951815
$753$	0	0
$754$	1.23129e8	0.0104607
$755$	−5.23193e9	−0.442433
$756$	0	0
$757$	1.55974e9	0.130682	0.0653410	−	0.997863i	$-0.479186\pi$
$757$	1.55974e9	0.130682	0.0653410	+	0.997863i	$0.479186\pi$
$758$	2.66337e8	0.0222121
$759$	0	0
$760$	9.50459e7	0.00785390
$761$	1.24541e10	1.02439	0.512194	−	0.858870i	$-0.328833\pi$
$761$	1.24541e10	1.02439	0.512194	+	0.858870i	$0.328833\pi$
$762$	0	0
$763$	8.21767e9	0.669751
$764$	−3.48366e9	−0.282624
$765$	0	0
$766$	1.52011e8	0.0122201
$767$	7.78666e8	0.0623114
$768$	0	0
$769$	−2.13765e10	−1.69509	−0.847547	−	0.530720i	$-0.821921\pi$
$769$	−2.13765e10	−1.69509	−0.847547	+	0.530720i	$0.821921\pi$
$770$	−6.27389e7	−0.00495244
$771$	0	0
$772$	−1.06618e10	−0.834005
$773$	8.61165e9	0.670592	0.335296	−	0.942113i	$-0.391164\pi$
$773$	8.61165e9	0.670592	0.335296	+	0.942113i	$0.391164\pi$
$774$	0	0
$775$	−1.41200e9	−0.108963
$776$	−5.79622e7	−0.00445275
$777$	0	0
$778$	−3.52334e8	−0.0268242
$779$	6.34970e9	0.481252
$780$	0	0
$781$	4.35376e9	0.327028
$782$	−3.37101e8	−0.0252079
$783$	0	0
$784$	1.69801e10	1.25844
$785$	4.66906e9	0.344497
$786$	0	0
$787$	1.17520e10	0.859407	0.429703	−	0.902970i	$-0.358618\pi$
$787$	1.17520e10	0.859407	0.429703	+	0.902970i	$0.358618\pi$
$788$	−2.13091e9	−0.155140
$789$	0	0
$790$	6.43047e7	0.00464032
$791$	−2.73817e10	−1.96717
$792$	0	0
$793$	8.70580e9	0.619944
$794$	6.47810e8	0.0459278
$795$	0	0
$796$	1.21517e10	0.853970
$797$	1.41996e10	0.993506	0.496753	−	0.867892i	$-0.334525\pi$
$797$	1.41996e10	0.993506	0.496753	+	0.867892i	$0.334525\pi$
$798$	0	0
$799$	2.37176e10	1.64497
$800$	9.96721e8	0.0688270
$801$	0	0
$802$	5.16101e8	0.0353284
$803$	−4.95503e9	−0.337709
$804$	0	0
$805$	−4.67832e9	−0.316086
$806$	−2.47256e7	−0.00166332
$807$	0	0
$808$	1.45963e8	0.00973427
$809$	−1.55261e10	−1.03096	−0.515482	−	0.856900i	$-0.672387\pi$
$809$	−1.55261e10	−1.03096	−0.515482	+	0.856900i	$0.672387\pi$
$810$	0	0
$811$	−1.25535e10	−0.826404	−0.413202	−	0.910639i	$-0.635590\pi$
$811$	−1.25535e10	−0.826404	−0.413202	+	0.910639i	$0.635590\pi$
$812$	1.85132e10	1.21349
$813$	0	0
$814$	−9.66571e7	−0.00628128
$815$	1.89836e9	0.122836
$816$	0	0
$817$	−1.08016e10	−0.692964
$818$	9.97998e7	0.00637519
$819$	0	0
$820$	−7.92192e9	−0.501744
$821$	−3.03373e9	−0.191327	−0.0956634	−	0.995414i	$-0.530497\pi$
$821$	−3.03373e9	−0.191327	−0.0956634	+	0.995414i	$0.530497\pi$
$822$	0	0
$823$	−2.40968e10	−1.50681	−0.753407	−	0.657554i	$-0.771591\pi$
$823$	−2.40968e10	−1.50681	−0.753407	+	0.657554i	$0.771591\pi$
$824$	−3.92944e8	−0.0244672
$825$	0	0
$826$	−8.58139e7	−0.00529819
$827$	−7.44605e9	−0.457780	−0.228890	−	0.973452i	$-0.573510\pi$
$827$	−7.44605e9	−0.457780	−0.228890	+	0.973452i	$0.573510\pi$
$828$	0	0
$829$	−9.49662e9	−0.578933	−0.289466	−	0.957188i	$-0.593478\pi$
$829$	−9.49662e9	−0.578933	−0.289466	+	0.957188i	$0.593478\pi$
$830$	−2.44961e8	−0.0148704
$831$	0	0
$832$	−7.91614e9	−0.476521
$833$	−3.62824e10	−2.17490
$834$	0	0
$835$	−8.74293e9	−0.519702
$836$	−1.96898e9	−0.116552
$837$	0	0
$838$	4.97888e8	0.0292265
$839$	−2.40441e10	−1.40554	−0.702768	−	0.711419i	$-0.748053\pi$
$839$	−2.40441e10	−1.40554	−0.702768	+	0.711419i	$0.748053\pi$
$840$	0	0
$841$	−5.99985e9	−0.347820
$842$	8.05767e8	0.0465176
$843$	0	0
$844$	−9.26114e9	−0.530232
$845$	−5.26184e9	−0.300013
$846$	0	0
$847$	−2.39923e10	−1.35669
$848$	−1.71046e10	−0.963224
$849$	0	0
$850$	−7.09052e8	−0.0396015
$851$	−7.20755e9	−0.400898
$852$	0	0
$853$	2.58216e10	1.42450	0.712248	−	0.701928i	$-0.247677\pi$
$853$	2.58216e10	1.42450	0.712248	+	0.701928i	$0.247677\pi$
$854$	−9.59433e8	−0.0527123
$855$	0	0
$856$	1.72836e9	0.0941835
$857$	−1.37614e10	−0.746842	−0.373421	−	0.927662i	$-0.621815\pi$
$857$	−1.37614e10	−0.746842	−0.373421	+	0.927662i	$0.621815\pi$
$858$	0	0
$859$	1.66562e10	0.896605	0.448302	−	0.893882i	$-0.352029\pi$
$859$	1.66562e10	0.896605	0.448302	+	0.893882i	$0.352029\pi$
$860$	1.34761e10	0.722471
$861$	0	0
$862$	−1.25302e8	−0.00666320
$863$	−1.23322e10	−0.653135	−0.326568	−	0.945174i	$-0.605892\pi$
$863$	−1.23322e10	−0.653135	−0.326568	+	0.945174i	$0.605892\pi$
$864$	0	0
$865$	1.07402e10	0.564231
$866$	−5.95904e8	−0.0311791
$867$	0	0
$868$	−3.71766e9	−0.192953
$869$	−2.66526e9	−0.137775
$870$	0	0
$871$	7.19598e9	0.369000
$872$	4.71850e8	0.0240988
$873$	0	0
$874$	1.07616e8	0.00545241
$875$	−2.14368e10	−1.08176
$876$	0	0
$877$	6.87040e9	0.343941	0.171970	−	0.985102i	$-0.444987\pi$
$877$	6.87040e9	0.343941	0.171970	+	0.985102i	$0.444987\pi$
$878$	1.01540e9	0.0506300
$879$	0	0
$880$	2.45471e9	0.121426
$881$	−7.79647e9	−0.384134	−0.192067	−	0.981382i	$-0.561519\pi$
$881$	−7.79647e9	−0.384134	−0.192067	+	0.981382i	$0.561519\pi$
$882$	0	0
$883$	−4.08759e10	−1.99804	−0.999022	−	0.0442243i	$-0.985918\pi$
$883$	−4.08759e10	−1.99804	−0.999022	+	0.0442243i	$0.985918\pi$
$884$	1.69398e10	0.824756
$885$	0	0
$886$	−5.99888e8	−0.0289769
$887$	−2.13201e10	−1.02579	−0.512894	−	0.858452i	$-0.671427\pi$
$887$	−2.13201e10	−1.02579	−0.512894	+	0.858452i	$0.671427\pi$
$888$	0	0
$889$	1.03489e10	0.494011
$890$	2.81655e8	0.0133922
$891$	0	0
$892$	2.02320e10	0.954469
$893$	−7.57163e9	−0.355803
$894$	0	0
$895$	−1.43138e10	−0.667381
$896$	3.49861e9	0.162487
$897$	0	0
$898$	−4.14893e8	−0.0191192
$899$	−2.25913e9	−0.103701
$900$	0	0
$901$	3.65485e10	1.66469
$902$	−2.40664e8	−0.0109191
$903$	0	0
$904$	−1.57222e9	−0.0707823
$905$	6.67888e9	0.299525
$906$	0	0
$907$	−3.96885e10	−1.76620	−0.883099	−	0.469186i	$-0.844547\pi$
$907$	−3.96885e10	−1.76620	−0.883099	+	0.469186i	$0.844547\pi$
$908$	1.35645e10	0.601315
$909$	0	0
$910$	−1.72314e8	−0.00758013
$911$	2.93991e10	1.28831	0.644154	−	0.764896i	$-0.277210\pi$
$911$	2.93991e10	1.28831	0.644154	+	0.764896i	$0.277210\pi$
$912$	0	0
$913$	1.01530e10	0.441516
$914$	−1.10413e9	−0.0478307
$915$	0	0
$916$	1.68025e10	0.722337
$917$	−5.18789e10	−2.22176
$918$	0	0
$919$	2.16809e10	0.921452	0.460726	−	0.887542i	$-0.347589\pi$
$919$	2.16809e10	0.921452	0.460726	+	0.887542i	$0.347589\pi$
$920$	−2.68624e8	−0.0113733
$921$	0	0
$922$	3.46064e7	0.00145411
$923$	1.19577e10	0.500545
$924$	0	0
$925$	−1.51602e10	−0.629809
$926$	−3.04198e8	−0.0125898
$927$	0	0
$928$	1.59471e9	0.0655033
$929$	−2.90038e10	−1.18686	−0.593431	−	0.804885i	$-0.702227\pi$
$929$	−2.90038e10	−1.18686	−0.593431	+	0.804885i	$0.702227\pi$
$930$	0	0
$931$	1.15828e10	0.470425
$932$	−3.98456e10	−1.61222
$933$	0	0
$934$	2.84703e7	0.00114335
$935$	−5.24514e9	−0.209853
$936$	0	0
$937$	5.69933e9	0.226326	0.113163	−	0.993576i	$-0.463902\pi$
$937$	5.69933e9	0.226326	0.113163	+	0.993576i	$0.463902\pi$
$938$	−7.93041e8	−0.0313752
$939$	0	0
$940$	9.44641e9	0.370953
$941$	−3.84796e10	−1.50545	−0.752725	−	0.658335i	$-0.771261\pi$
$941$	−3.84796e10	−1.50545	−0.752725	+	0.658335i	$0.771261\pi$
$942$	0	0
$943$	−1.79459e10	−0.696906
$944$	3.35754e9	0.129903
$945$	0	0
$946$	4.09398e8	0.0157227
$947$	−4.81213e10	−1.84125	−0.920625	−	0.390448i	$-0.872320\pi$
$947$	−4.81213e10	−1.84125	−0.920625	+	0.390448i	$0.872320\pi$
$948$	0	0
$949$	−1.36092e10	−0.516892
$950$	2.26358e8	0.00856572
$951$	0	0
$952$	−3.73511e9	−0.140305
$953$	−7.33140e9	−0.274386	−0.137193	−	0.990544i	$-0.543808\pi$
$953$	−7.33140e9	−0.274386	−0.137193	+	0.990544i	$0.543808\pi$
$954$	0	0
$955$	2.96257e9	0.110067
$956$	2.76899e10	1.02499
$957$	0	0
$958$	6.73615e7	0.00247533
$959$	−7.92166e10	−2.90036
$960$	0	0
$961$	−2.70590e10	−0.983511
$962$	−2.65472e8	−0.00961405
$963$	0	0
$964$	−4.37801e10	−1.57401
$965$	9.06697e9	0.324801
$966$	0	0
$967$	5.30351e9	0.188612	0.0943062	−	0.995543i	$-0.469937\pi$
$967$	5.30351e9	0.188612	0.0943062	+	0.995543i	$0.469937\pi$
$968$	−1.37761e9	−0.0488162
$969$	0	0
$970$	2.46371e7	0.000866738 0
$971$	−5.10894e9	−0.179087	−0.0895433	−	0.995983i	$-0.528541\pi$
$971$	−5.10894e9	−0.179087	−0.0895433	+	0.995983i	$0.528541\pi$
$972$	0	0
$973$	2.10069e10	0.731082
$974$	4.76566e8	0.0165260
$975$	0	0
$976$	3.75386e10	1.29242
$977$	−5.84301e9	−0.200450	−0.100225	−	0.994965i	$-0.531956\pi$
$977$	−5.84301e9	−0.200450	−0.100225	+	0.994965i	$0.531956\pi$
$978$	0	0
$979$	−1.16739e10	−0.397627
$980$	−1.44508e10	−0.490456
$981$	0	0
$982$	1.55757e9	0.0524876
$983$	−3.64242e10	−1.22307	−0.611537	−	0.791216i	$-0.709448\pi$
$983$	−3.64242e10	−1.22307	−0.611537	+	0.791216i	$0.709448\pi$
$984$	0	0
$985$	1.81217e9	0.0604187
$986$	−1.13445e9	−0.0376892
$987$	0	0
$988$	−5.40788e9	−0.178393
$989$	3.05281e10	1.00349
$990$	0	0
$991$	2.36200e10	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$991$	2.36200e10	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$992$	−3.20235e8	−0.0104155
$993$	0	0
$994$	−1.31782e9	−0.0425601
$995$	−1.03341e10	−0.332576
$996$	0	0
$997$	2.98020e10	0.952384	0.476192	−	0.879341i	$-0.342017\pi$
$997$	2.98020e10	0.952384	0.476192	+	0.879341i	$0.342017\pi$
$998$	−1.66988e8	−0.00531775
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.g.1.16	✓	33
3.2	odd	2		531.8.a.h.1.18	yes	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.8.a.g.1.16	✓	33	1.1	even	1	trivial
531.8.a.h.1.18	yes	33	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-0.306188 q^{2} -127.906 q^{4} +108.774 q^{5} +1364.63 q^{7} +78.3553 q^{8} +O(q^{10})\)	\(q-0.306188 q^{2} -127.906 q^{4} +108.774 q^{5} +1364.63 q^{7} +78.3553 q^{8} -33.3052 q^{10} +1380.42 q^{11} +3791.36 q^{13} -417.832 q^{14} +16348.0 q^{16} -34931.9 q^{17} +11151.7 q^{19} -13912.9 q^{20} -422.666 q^{22} -31517.5 q^{23} -66293.2 q^{25} -1160.87 q^{26} -174544. q^{28} -106066. q^{29} +21299.3 q^{31} -15035.0 q^{32} +10695.7 q^{34} +148436. q^{35} +228684. q^{37} -3414.50 q^{38} +8523.02 q^{40} +569395. q^{41} -968608. q^{43} -176564. q^{44} +9650.26 q^{46} -678969. q^{47} +1.03866e6 q^{49} +20298.2 q^{50} -484939. q^{52} -1.04628e6 q^{53} +150153. q^{55} +106926. q^{56} +32476.1 q^{58} +205379. q^{59} +2.29622e6 q^{61} -6521.57 q^{62} -2.08794e6 q^{64} +412402. q^{65} +1.89799e6 q^{67} +4.46800e6 q^{68} -45449.2 q^{70} +3.15394e6 q^{71} -3.58952e6 q^{73} -70020.3 q^{74} -1.42637e6 q^{76} +1.88375e6 q^{77} -1.93077e6 q^{79} +1.77824e6 q^{80} -174342. q^{82} +7.35502e6 q^{83} -3.79968e6 q^{85} +296576. q^{86} +108163. q^{88} -8.45678e6 q^{89} +5.17379e6 q^{91} +4.03128e6 q^{92} +207892. q^{94} +1.21301e6 q^{95} -739735. q^{97} -318026. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 24 q^{2} + 1776 q^{4} - 1000 q^{5} + 154 q^{7} - 4608 q^{8}+O(q^{10}) \) 33 * q - 24 * q^2 + 1776 * q^4 - 1000 * q^5 + 154 * q^7 - 4608 * q^8	\( 33 q - 24 q^{2} + 1776 q^{4} - 1000 q^{5} + 154 q^{7} - 4608 q^{8} + 5042 q^{10} - 13310 q^{11} - 14172 q^{13} - 16464 q^{14} + 95772 q^{16} - 39304 q^{17} - 56302 q^{19} + 17936 q^{20} + 152764 q^{22} - 17988 q^{23} + 468523 q^{25} + 27624 q^{26} - 59896 q^{28} - 474564 q^{29} + 188186 q^{31} - 251068 q^{32} + 169888 q^{34} - 514500 q^{35} + 1148200 q^{37} - 446594 q^{38} + 501214 q^{40} - 1246152 q^{41} + 62268 q^{43} - 2555520 q^{44} - 1289942 q^{46} - 1485654 q^{47} + 3829555 q^{49} - 6430160 q^{50} - 2624804 q^{52} - 4086740 q^{53} - 1119118 q^{55} - 8448352 q^{56} + 2966706 q^{58} + 6777507 q^{59} + 2436146 q^{61} - 9005952 q^{62} + 11117562 q^{64} - 16730354 q^{65} - 2652248 q^{67} - 15929124 q^{68} + 3359254 q^{70} - 7356324 q^{71} + 1900454 q^{73} - 25386964 q^{74} - 16047360 q^{76} - 20774826 q^{77} - 5912712 q^{79} - 13568404 q^{80} + 1579434 q^{82} + 1052766 q^{83} + 18372730 q^{85} - 43499960 q^{86} + 18209214 q^{88} - 174788 q^{89} - 18891512 q^{91} - 46033270 q^{92} + 365448 q^{94} - 31505580 q^{95} + 14418540 q^{97} + 15964056 q^{98}+O(q^{100}) \) 33 * q - 24 * q^2 + 1776 * q^4 - 1000 * q^5 + 154 * q^7 - 4608 * q^8 + 5042 * q^10 - 13310 * q^11 - 14172 * q^13 - 16464 * q^14 + 95772 * q^16 - 39304 * q^17 - 56302 * q^19 + 17936 * q^20 + 152764 * q^22 - 17988 * q^23 + 468523 * q^25 + 27624 * q^26 - 59896 * q^28 - 474564 * q^29 + 188186 * q^31 - 251068 * q^32 + 169888 * q^34 - 514500 * q^35 + 1148200 * q^37 - 446594 * q^38 + 501214 * q^40 - 1246152 * q^41 + 62268 * q^43 - 2555520 * q^44 - 1289942 * q^46 - 1485654 * q^47 + 3829555 * q^49 - 6430160 * q^50 - 2624804 * q^52 - 4086740 * q^53 - 1119118 * q^55 - 8448352 * q^56 + 2966706 * q^58 + 6777507 * q^59 + 2436146 * q^61 - 9005952 * q^62 + 11117562 * q^64 - 16730354 * q^65 - 2652248 * q^67 - 15929124 * q^68 + 3359254 * q^70 - 7356324 * q^71 + 1900454 * q^73 - 25386964 * q^74 - 16047360 * q^76 - 20774826 * q^77 - 5912712 * q^79 - 13568404 * q^80 + 1579434 * q^82 + 1052766 * q^83 + 18372730 * q^85 - 43499960 * q^86 + 18209214 * q^88 - 174788 * q^89 - 18891512 * q^91 - 46033270 * q^92 + 365448 * q^94 - 31505580 * q^95 + 14418540 * q^97 + 15964056 * q^98

Introduction

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