LMFDB - Embedded newform 531.8.a.e.1.1

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$20.8794$ of defining polynomial
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)

$f(q)$	$=$	$q-21.8794 q^{2} +350.709 q^{4} -336.305 q^{5} -653.674 q^{7} -4872.75 q^{8} +O(q^{10})$	\(q-21.8794 q^{2} +350.709 q^{4} -336.305 q^{5} -653.674 q^{7} -4872.75 q^{8} +7358.16 q^{10} -2824.81 q^{11} -5709.07 q^{13} +14302.0 q^{14} +61722.2 q^{16} -27447.6 q^{17} -52668.8 q^{19} -117945. q^{20} +61805.2 q^{22} +112683. q^{23} +34976.1 q^{25} +124911. q^{26} -229250. q^{28} -129138. q^{29} +264379. q^{31} -726735. q^{32} +600538. q^{34} +219834. q^{35} -162719. q^{37} +1.15236e6 q^{38} +1.63873e6 q^{40} +570791. q^{41} +468651. q^{43} -990686. q^{44} -2.46544e6 q^{46} +451069. q^{47} -396253. q^{49} -765257. q^{50} -2.00222e6 q^{52} +1.09397e6 q^{53} +949997. q^{55} +3.18519e6 q^{56} +2.82547e6 q^{58} -205379. q^{59} -1.17187e6 q^{61} -5.78445e6 q^{62} +8.00009e6 q^{64} +1.91999e6 q^{65} -1.24840e6 q^{67} -9.62614e6 q^{68} -4.80984e6 q^{70} -2.90922e6 q^{71} +1.95003e6 q^{73} +3.56019e6 q^{74} -1.84714e7 q^{76} +1.84650e6 q^{77} +529861. q^{79} -2.07575e7 q^{80} -1.24886e7 q^{82} -7.53928e6 q^{83} +9.23078e6 q^{85} -1.02538e7 q^{86} +1.37646e7 q^{88} -6.07893e6 q^{89} +3.73187e6 q^{91} +3.95190e7 q^{92} -9.86913e6 q^{94} +1.77128e7 q^{95} -2.74881e6 q^{97} +8.66979e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) $ 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * 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 $

$2$	−21.8794	−1.93389	−0.966943	−	0.254992i	$-0.917927\pi$
$2$	−21.8794	−1.93389	−0.966943	+	0.254992i	$0.917927\pi$
$3$	0	0
$3$	0	0
$4$	350.709	2.73992
$5$	−336.305	−1.20320	−0.601601	−	0.798797i	$-0.705470\pi$
$5$	−336.305	−1.20320	−0.601601	+	0.798797i	$0.705470\pi$
$6$	0	0
$7$	−653.674	−0.720308	−0.360154	−	0.932893i	$-0.617276\pi$
$7$	−653.674	−0.720308	−0.360154	+	0.932893i	$0.617276\pi$
$8$	−4872.75	−3.36480
$9$	0	0
$10$	7358.16	2.32685
$11$	−2824.81	−0.639903	−0.319952	−	0.947434i	$-0.603667\pi$
$11$	−2824.81	−0.639903	−0.319952	+	0.947434i	$0.603667\pi$
$12$	0	0
$13$	−5709.07	−0.720715	−0.360358	−	0.932814i	$-0.617345\pi$
$13$	−5709.07	−0.720715	−0.360358	+	0.932814i	$0.617345\pi$
$14$	14302.0	1.39299
$15$	0	0
$16$	61722.2	3.76723
$17$	−27447.6	−1.35498	−0.677491	−	0.735531i	$-0.736933\pi$
$17$	−27447.6	−1.35498	−0.677491	+	0.735531i	$0.736933\pi$
$18$	0	0
$19$	−52668.8	−1.76163	−0.880817	−	0.473458i	$-0.843006\pi$
$19$	−52668.8	−1.76163	−0.880817	+	0.473458i	$0.843006\pi$
$20$	−117945.	−3.29667
$21$	0	0
$22$	61805.2	1.23750
$23$	112683.	1.93113	0.965565	−	0.260161i	$-0.0837757\pi$
$23$	112683.	1.93113	0.965565	+	0.260161i	$0.0837757\pi$
$24$	0	0
$25$	34976.1	0.447694
$26$	124911.	1.39378
$27$	0	0
$28$	−229250.	−1.97358
$29$	−129138.	−0.983246	−0.491623	−	0.870808i	$-0.663596\pi$
$29$	−129138.	−0.983246	−0.491623	+	0.870808i	$0.663596\pi$
$30$	0	0
$31$	264379.	1.59390	0.796949	−	0.604047i	$-0.206446\pi$
$31$	264379.	1.59390	0.796949	+	0.604047i	$0.206446\pi$
$32$	−726735.	−3.92059
$33$	0	0
$34$	600538.	2.62038
$35$	219834.	0.866676
$36$	0	0
$37$	−162719.	−0.528118	−0.264059	−	0.964506i	$-0.585061\pi$
$37$	−162719.	−0.528118	−0.264059	+	0.964506i	$0.585061\pi$
$38$	1.15236e6	3.40680
$39$	0	0
$40$	1.63873e6	4.04853
$41$	570791.	1.29340	0.646701	−	0.762743i	$-0.276148\pi$
$41$	570791.	1.29340	0.646701	+	0.762743i	$0.276148\pi$
$42$	0	0
$43$	468651.	0.898898	0.449449	−	0.893306i	$-0.351620\pi$
$43$	468651.	0.898898	0.449449	+	0.893306i	$0.351620\pi$
$44$	−990686.	−1.75328
$45$	0	0
$46$	−2.46544e6	−3.73459
$47$	451069.	0.633724	0.316862	−	0.948472i	$-0.397371\pi$
$47$	451069.	0.633724	0.316862	+	0.948472i	$0.397371\pi$
$48$	0	0
$49$	−396253.	−0.481156
$50$	−765257.	−0.865789
$51$	0	0
$52$	−2.00222e6	−1.97470
$53$	1.09397e6	1.00935	0.504674	−	0.863310i	$-0.331613\pi$
$53$	1.09397e6	1.00935	0.504674	+	0.863310i	$0.331613\pi$
$54$	0	0
$55$	949997.	0.769933
$56$	3.18519e6	2.42369
$57$	0	0
$58$	2.82547e6	1.90149
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	−1.17187e6	−0.661038	−0.330519	−	0.943799i	$-0.607224\pi$
$61$	−1.17187e6	−0.661038	−0.330519	+	0.943799i	$0.607224\pi$
$62$	−5.78445e6	−3.08242
$63$	0	0
$64$	8.00009e6	3.81474
$65$	1.91999e6	0.867166
$66$	0	0
$67$	−1.24840e6	−0.507096	−0.253548	−	0.967323i	$-0.581598\pi$
$67$	−1.24840e6	−0.507096	−0.253548	+	0.967323i	$0.581598\pi$
$68$	−9.62614e6	−3.71254
$69$	0	0
$70$	−4.80984e6	−1.67605
$71$	−2.90922e6	−0.964655	−0.482327	−	0.875991i	$-0.660208\pi$
$71$	−2.90922e6	−0.964655	−0.482327	+	0.875991i	$0.660208\pi$
$72$	0	0
$73$	1.95003e6	0.586694	0.293347	−	0.956006i	$-0.405231\pi$
$73$	1.95003e6	0.586694	0.293347	+	0.956006i	$0.405231\pi$
$74$	3.56019e6	1.02132
$75$	0	0
$76$	−1.84714e7	−4.82673
$77$	1.84650e6	0.460928
$78$	0	0
$79$	529861.	0.120911	0.0604557	−	0.998171i	$-0.480745\pi$
$79$	529861.	0.120911	0.0604557	+	0.998171i	$0.480745\pi$
$80$	−2.07575e7	−4.53273
$81$	0	0
$82$	−1.24886e7	−2.50129
$83$	−7.53928e6	−1.44729	−0.723646	−	0.690171i	$-0.757535\pi$
$83$	−7.53928e6	−1.44729	−0.723646	+	0.690171i	$0.757535\pi$
$84$	0	0
$85$	9.23078e6	1.63032
$86$	−1.02538e7	−1.73837
$87$	0	0
$88$	1.37646e7	2.15315
$89$	−6.07893e6	−0.914034	−0.457017	−	0.889458i	$-0.651082\pi$
$89$	−6.07893e6	−0.914034	−0.457017	+	0.889458i	$0.651082\pi$
$90$	0	0
$91$	3.73187e6	0.519137
$92$	3.95190e7	5.29114
$93$	0	0
$94$	−9.86913e6	−1.22555
$95$	1.77128e7	2.11960
$96$	0	0
$97$	−2.74881e6	−0.305804	−0.152902	−	0.988241i	$-0.548862\pi$
$97$	−2.74881e6	−0.305804	−0.152902	+	0.988241i	$0.548862\pi$
$98$	8.66979e6	0.930502
$99$	0	0
$100$	1.22664e7	1.22664
$101$	1.90082e7	1.83576	0.917880	−	0.396858i	$-0.129900\pi$
$101$	1.90082e7	1.83576	0.917880	+	0.396858i	$0.129900\pi$
$102$	0	0
$103$	1.79174e7	1.61564	0.807818	−	0.589432i	$-0.200648\pi$
$103$	1.79174e7	1.61564	0.807818	+	0.589432i	$0.200648\pi$
$104$	2.78189e7	2.42506
$105$	0	0
$106$	−2.39355e7	−1.95196
$107$	−85176.8	−0.00672168	−0.00336084	−	0.999994i	$-0.501070\pi$
$107$	−85176.8	−0.00672168	−0.00336084	+	0.999994i	$0.501070\pi$
$108$	0	0
$109$	1.58698e7	1.17376	0.586881	−	0.809673i	$-0.300356\pi$
$109$	1.58698e7	1.17376	0.586881	+	0.809673i	$0.300356\pi$
$110$	−2.07854e7	−1.48896
$111$	0	0
$112$	−4.03462e7	−2.71356
$113$	2.00702e7	1.30851	0.654257	−	0.756273i	$-0.272982\pi$
$113$	2.00702e7	1.30851	0.654257	+	0.756273i	$0.272982\pi$
$114$	0	0
$115$	−3.78959e7	−2.32354
$116$	−4.52900e7	−2.69401
$117$	0	0
$118$	4.49357e6	0.251771
$119$	1.79418e7	0.976005
$120$	0	0
$121$	−1.15076e7	−0.590524
$122$	2.56399e7	1.27837
$123$	0	0
$124$	9.27200e7	4.36715
$125$	1.45112e7	0.664536
$126$	0	0
$127$	1.19818e7	0.519048	0.259524	−	0.965737i	$-0.416434\pi$
$127$	1.19818e7	0.519048	0.259524	+	0.965737i	$0.416434\pi$
$128$	−8.20154e7	−3.45669
$129$	0	0
$130$	−4.20083e7	−1.67700
$131$	−8.95447e6	−0.348009	−0.174004	−	0.984745i	$-0.555671\pi$
$131$	−8.95447e6	−0.348009	−0.174004	+	0.984745i	$0.555671\pi$
$132$	0	0
$133$	3.44282e7	1.26892
$134$	2.73142e7	0.980666
$135$	0	0
$136$	1.33745e8	4.55924
$137$	1.70175e7	0.565424	0.282712	−	0.959205i	$-0.408766\pi$
$137$	1.70175e7	0.565424	0.282712	+	0.959205i	$0.408766\pi$
$138$	0	0
$139$	−1.49569e7	−0.472379	−0.236189	−	0.971707i	$-0.575899\pi$
$139$	−1.49569e7	−0.472379	−0.236189	+	0.971707i	$0.575899\pi$
$140$	7.70978e7	2.37462
$141$	0	0
$142$	6.36520e7	1.86553
$143$	1.61270e7	0.461188
$144$	0	0
$145$	4.34299e7	1.18304
$146$	−4.26655e7	−1.13460
$147$	0	0
$148$	−5.70669e7	−1.44700
$149$	5.19387e7	1.28629	0.643145	−	0.765744i	$-0.277629\pi$
$149$	5.19387e7	1.28629	0.643145	+	0.765744i	$0.277629\pi$
$150$	0	0
$151$	−4.79328e6	−0.113296	−0.0566478	−	0.998394i	$-0.518041\pi$
$151$	−4.79328e6	−0.113296	−0.0566478	+	0.998394i	$0.518041\pi$
$152$	2.56642e8	5.92754
$153$	0	0
$154$	−4.04004e7	−0.891381
$155$	−8.89119e7	−1.91778
$156$	0	0
$157$	5.35304e7	1.10396	0.551978	−	0.833859i	$-0.313873\pi$
$157$	5.35304e7	1.10396	0.551978	+	0.833859i	$0.313873\pi$
$158$	−1.15931e7	−0.233829
$159$	0	0
$160$	2.44404e8	4.71725
$161$	−7.36581e7	−1.39101
$162$	0	0
$163$	−8.41640e7	−1.52219	−0.761096	−	0.648639i	$-0.775339\pi$
$163$	−8.41640e7	−1.52219	−0.761096	+	0.648639i	$0.775339\pi$
$164$	2.00182e8	3.54382
$165$	0	0
$166$	1.64955e8	2.79890
$167$	−6.28714e6	−0.104459	−0.0522295	−	0.998635i	$-0.516633\pi$
$167$	−6.28714e6	−0.104459	−0.0522295	+	0.998635i	$0.516633\pi$
$168$	0	0
$169$	−3.01550e7	−0.480569
$170$	−2.01964e8	−3.15285
$171$	0	0
$172$	1.64360e8	2.46290
$173$	4.64860e7	0.682591	0.341296	−	0.939956i	$-0.389134\pi$
$173$	4.64860e7	0.682591	0.341296	+	0.939956i	$0.389134\pi$
$174$	0	0
$175$	−2.28630e7	−0.322477
$176$	−1.74353e8	−2.41066
$177$	0	0
$178$	1.33004e8	1.76764
$179$	−1.12901e8	−1.47134	−0.735671	−	0.677339i	$-0.763133\pi$
$179$	−1.12901e8	−1.47134	−0.735671	+	0.677339i	$0.763133\pi$
$180$	0	0
$181$	−1.04425e8	−1.30897	−0.654487	−	0.756073i	$-0.727115\pi$
$181$	−1.04425e8	−1.30897	−0.654487	+	0.756073i	$0.727115\pi$
$182$	−8.16513e7	−1.00395
$183$	0	0
$184$	−5.49077e8	−6.49787
$185$	5.47231e7	0.635433
$186$	0	0
$187$	7.75343e7	0.867058
$188$	1.58194e8	1.73635
$189$	0	0
$190$	−3.87545e8	−4.09906
$191$	1.38829e8	1.44167	0.720833	−	0.693108i	$-0.243759\pi$
$191$	1.38829e8	1.44167	0.720833	+	0.693108i	$0.243759\pi$
$192$	0	0
$193$	3.06880e7	0.307268	0.153634	−	0.988128i	$-0.450902\pi$
$193$	3.06880e7	0.307268	0.153634	+	0.988128i	$0.450902\pi$
$194$	6.01424e7	0.591391
$195$	0	0
$196$	−1.38970e8	−1.31833
$197$	2.78823e7	0.259834	0.129917	−	0.991525i	$-0.458529\pi$
$197$	2.78823e7	0.259834	0.129917	+	0.991525i	$0.458529\pi$
$198$	0	0
$199$	1.31062e8	1.17894	0.589469	−	0.807791i	$-0.299337\pi$
$199$	1.31062e8	1.17894	0.589469	+	0.807791i	$0.299337\pi$
$200$	−1.70430e8	−1.50640
$201$	0	0
$202$	−4.15888e8	−3.55015
$203$	8.44144e7	0.708240
$204$	0	0
$205$	−1.91960e8	−1.55622
$206$	−3.92021e8	−3.12446
$207$	0	0
$208$	−3.52377e8	−2.71510
$209$	1.48779e8	1.12727
$210$	0	0
$211$	7.00592e7	0.513424	0.256712	−	0.966488i	$-0.417361\pi$
$211$	7.00592e7	0.513424	0.256712	+	0.966488i	$0.417361\pi$
$212$	3.83666e8	2.76553
$213$	0	0
$214$	1.86362e6	0.0129990
$215$	−1.57610e8	−1.08155
$216$	0	0
$217$	−1.72817e8	−1.14810
$218$	−3.47223e8	−2.26992
$219$	0	0
$220$	3.33173e8	2.10955
$221$	1.56700e8	0.976557
$222$	0	0
$223$	1.42923e8	0.863049	0.431525	−	0.902101i	$-0.357976\pi$
$223$	1.42923e8	0.863049	0.431525	+	0.902101i	$0.357976\pi$
$224$	4.75048e8	2.82403
$225$	0	0
$226$	−4.39126e8	−2.53052
$227$	1.16361e8	0.660262	0.330131	−	0.943935i	$-0.392907\pi$
$227$	1.16361e8	0.660262	0.330131	+	0.943935i	$0.392907\pi$
$228$	0	0
$229$	2.48550e8	1.36770	0.683849	−	0.729624i	$-0.260305\pi$
$229$	2.48550e8	1.36770	0.683849	+	0.729624i	$0.260305\pi$
$230$	8.29141e8	4.49346
$231$	0	0
$232$	6.29259e8	3.30842
$233$	1.28332e8	0.664645	0.332323	−	0.943166i	$-0.392168\pi$
$233$	1.28332e8	0.664645	0.332323	+	0.943166i	$0.392168\pi$
$234$	0	0
$235$	−1.51697e8	−0.762498
$236$	−7.20283e7	−0.356707
$237$	0	0
$238$	−3.92556e8	−1.88748
$239$	−1.52392e8	−0.722056	−0.361028	−	0.932555i	$-0.617574\pi$
$239$	−1.52392e8	−0.722056	−0.361028	+	0.932555i	$0.617574\pi$
$240$	0	0
$241$	−2.46545e8	−1.13458	−0.567292	−	0.823517i	$-0.692009\pi$
$241$	−2.46545e8	−1.13458	−0.567292	+	0.823517i	$0.692009\pi$
$242$	2.51780e8	1.14201
$243$	0	0
$244$	−4.10987e8	−1.81119
$245$	1.33262e8	0.578928
$246$	0	0
$247$	3.00690e8	1.26964
$248$	−1.28825e9	−5.36315
$249$	0	0
$250$	−3.17497e8	−1.28514
$251$	1.53947e7	0.0614486	0.0307243	−	0.999528i	$-0.490219\pi$
$251$	1.53947e7	0.0614486	0.0307243	+	0.999528i	$0.490219\pi$
$252$	0	0
$253$	−3.18308e8	−1.23574
$254$	−2.62154e8	−1.00378
$255$	0	0
$256$	7.70438e8	2.87010
$257$	−1.45269e7	−0.0533836	−0.0266918	−	0.999644i	$-0.508497\pi$
$257$	−1.45269e7	−0.0533836	−0.0266918	+	0.999644i	$0.508497\pi$
$258$	0	0
$259$	1.06365e8	0.380408
$260$	6.73358e8	2.37596
$261$	0	0
$262$	1.95919e8	0.673010
$263$	−4.08752e8	−1.38553	−0.692763	−	0.721166i	$-0.743607\pi$
$263$	−4.08752e8	−1.38553	−0.692763	+	0.721166i	$0.743607\pi$
$264$	0	0
$265$	−3.67908e8	−1.21445
$266$	−7.53270e8	−2.45394
$267$	0	0
$268$	−4.37824e8	−1.38940
$269$	−2.03154e6	−0.00636345	−0.00318173	−	0.999995i	$-0.501013\pi$
$269$	−2.03154e6	−0.00636345	−0.00318173	+	0.999995i	$0.501013\pi$
$270$	0	0
$271$	1.02015e8	0.311366	0.155683	−	0.987807i	$-0.450242\pi$
$271$	1.02015e8	0.311366	0.155683	+	0.987807i	$0.450242\pi$
$272$	−1.69413e9	−5.10452
$273$	0	0
$274$	−3.72333e8	−1.09347
$275$	−9.88007e7	−0.286481
$276$	0	0
$277$	−1.80373e8	−0.509907	−0.254954	−	0.966953i	$-0.582060\pi$
$277$	−1.80373e8	−0.509907	−0.254954	+	0.966953i	$0.582060\pi$
$278$	3.27249e8	0.913526
$279$	0	0
$280$	−1.07120e9	−2.91619
$281$	−3.39131e8	−0.911791	−0.455896	−	0.890033i	$-0.650681\pi$
$281$	−3.39131e8	−0.911791	−0.455896	+	0.890033i	$0.650681\pi$
$282$	0	0
$283$	−1.22871e8	−0.322252	−0.161126	−	0.986934i	$-0.551513\pi$
$283$	−1.22871e8	−0.322252	−0.161126	+	0.986934i	$0.551513\pi$
$284$	−1.02029e9	−2.64307
$285$	0	0
$286$	−3.52850e8	−0.891886
$287$	−3.73112e8	−0.931649
$288$	0	0
$289$	3.43034e8	0.835977
$290$	−9.50220e8	−2.28787
$291$	0	0
$292$	6.83894e8	1.60749
$293$	2.73920e8	0.636190	0.318095	−	0.948059i	$-0.396957\pi$
$293$	2.73920e8	0.636190	0.318095	+	0.948059i	$0.396957\pi$
$294$	0	0
$295$	6.90700e7	0.156643
$296$	7.92888e8	1.77701
$297$	0	0
$298$	−1.13639e9	−2.48754
$299$	−6.43316e8	−1.39180
$300$	0	0
$301$	−3.06345e8	−0.647483
$302$	1.04874e8	0.219101
$303$	0	0
$304$	−3.25083e9	−6.63647
$305$	3.94107e8	0.795362
$306$	0	0
$307$	−2.51384e8	−0.495853	−0.247927	−	0.968779i	$-0.579749\pi$
$307$	−2.51384e8	−0.495853	−0.247927	+	0.968779i	$0.579749\pi$
$308$	6.47586e8	1.26290
$309$	0	0
$310$	1.94534e9	3.70877
$311$	−3.46715e8	−0.653599	−0.326799	−	0.945094i	$-0.605970\pi$
$311$	−3.46715e8	−0.653599	−0.326799	+	0.945094i	$0.605970\pi$
$312$	0	0
$313$	8.59091e8	1.58356	0.791779	−	0.610807i	$-0.209155\pi$
$313$	8.59091e8	1.58356	0.791779	+	0.610807i	$0.209155\pi$
$314$	−1.17122e9	−2.13493
$315$	0	0
$316$	1.85827e8	0.331287
$317$	−6.36316e8	−1.12193	−0.560964	−	0.827840i	$-0.689569\pi$
$317$	−6.36316e8	−1.12193	−0.560964	+	0.827840i	$0.689569\pi$
$318$	0	0
$319$	3.64791e8	0.629182
$320$	−2.69047e9	−4.58990
$321$	0	0
$322$	1.61160e9	2.69005
$323$	1.44563e9	2.38698
$324$	0	0
$325$	−1.99681e8	−0.322660
$326$	1.84146e9	2.94375
$327$	0	0
$328$	−2.78132e9	−4.35204
$329$	−2.94852e8	−0.456477
$330$	0	0
$331$	4.08883e8	0.619728	0.309864	−	0.950781i	$-0.399716\pi$
$331$	4.08883e8	0.619728	0.309864	+	0.950781i	$0.399716\pi$
$332$	−2.64409e9	−3.96546
$333$	0	0
$334$	1.37559e8	0.202012
$335$	4.19842e8	0.610139
$336$	0	0
$337$	6.00455e8	0.854626	0.427313	−	0.904104i	$-0.359460\pi$
$337$	6.00455e8	0.854626	0.427313	+	0.904104i	$0.359460\pi$
$338$	6.59774e8	0.929366
$339$	0	0
$340$	3.23732e9	4.46693
$341$	−7.46818e8	−1.01994
$342$	0	0
$343$	7.97349e8	1.06689
$344$	−2.28362e9	−3.02461
$345$	0	0
$346$	−1.01709e9	−1.32005
$347$	−1.81629e8	−0.233363	−0.116681	−	0.993169i	$-0.537226\pi$
$347$	−1.81629e8	−0.233363	−0.116681	+	0.993169i	$0.537226\pi$
$348$	0	0
$349$	−7.06742e8	−0.889963	−0.444981	−	0.895540i	$-0.646790\pi$
$349$	−7.06742e8	−0.889963	−0.444981	+	0.895540i	$0.646790\pi$
$350$	5.00229e8	0.623635
$351$	0	0
$352$	2.05288e9	2.50880
$353$	−2.31567e8	−0.280199	−0.140099	−	0.990137i	$-0.544742\pi$
$353$	−2.31567e8	−0.280199	−0.140099	+	0.990137i	$0.544742\pi$
$354$	0	0
$355$	9.78384e8	1.16067
$356$	−2.13194e9	−2.50438
$357$	0	0
$358$	2.47022e9	2.84541
$359$	−1.13982e9	−1.30019	−0.650093	−	0.759854i	$-0.725270\pi$
$359$	−1.13982e9	−1.30019	−0.650093	+	0.759854i	$0.725270\pi$
$360$	0	0
$361$	1.88013e9	2.10335
$362$	2.28477e9	2.53141
$363$	0	0
$364$	1.30880e9	1.42239
$365$	−6.55805e8	−0.705911
$366$	0	0
$367$	6.44750e8	0.680864	0.340432	−	0.940269i	$-0.389427\pi$
$367$	6.44750e8	0.680864	0.340432	+	0.940269i	$0.389427\pi$
$368$	6.95506e9	7.27500
$369$	0	0
$370$	−1.19731e9	−1.22885
$371$	−7.15101e8	−0.727041
$372$	0	0
$373$	1.69263e8	0.168881	0.0844404	−	0.996429i	$-0.473090\pi$
$373$	1.69263e8	0.168881	0.0844404	+	0.996429i	$0.473090\pi$
$374$	−1.69640e9	−1.67679
$375$	0	0
$376$	−2.19795e9	−2.13236
$377$	7.37260e8	0.708640
$378$	0	0
$379$	−1.01373e9	−0.956503	−0.478252	−	0.878223i	$-0.658729\pi$
$379$	−1.01373e9	−0.956503	−0.478252	+	0.878223i	$0.658729\pi$
$380$	6.21203e9	5.80753
$381$	0	0
$382$	−3.03751e9	−2.78802
$383$	−5.54096e8	−0.503953	−0.251976	−	0.967733i	$-0.581080\pi$
$383$	−5.54096e8	−0.503953	−0.251976	+	0.967733i	$0.581080\pi$
$384$	0	0
$385$	−6.20988e8	−0.554589
$386$	−6.71435e8	−0.594221
$387$	0	0
$388$	−9.64033e8	−0.837879
$389$	−1.17131e9	−1.00890	−0.504450	−	0.863441i	$-0.668305\pi$
$389$	−1.17131e9	−1.00890	−0.504450	+	0.863441i	$0.668305\pi$
$390$	0	0
$391$	−3.09289e9	−2.61665
$392$	1.93084e9	1.61900
$393$	0	0
$394$	−6.10048e8	−0.502490
$395$	−1.78195e8	−0.145481
$396$	0	0
$397$	1.86752e9	1.49796	0.748978	−	0.662594i	$-0.230545\pi$
$397$	1.86752e9	1.49796	0.748978	+	0.662594i	$0.230545\pi$
$398$	−2.86756e9	−2.27993
$399$	0	0
$400$	2.15880e9	1.68656
$401$	−2.23006e9	−1.72708	−0.863539	−	0.504282i	$-0.831757\pi$
$401$	−2.23006e9	−1.72708	−0.863539	+	0.504282i	$0.831757\pi$
$402$	0	0
$403$	−1.50936e9	−1.14875
$404$	6.66635e9	5.02983
$405$	0	0
$406$	−1.84694e9	−1.36965
$407$	4.59649e8	0.337945
$408$	0	0
$409$	−1.03440e9	−0.747575	−0.373788	−	0.927514i	$-0.621941\pi$
$409$	−1.03440e9	−0.747575	−0.373788	+	0.927514i	$0.621941\pi$
$410$	4.19997e9	3.00956
$411$	0	0
$412$	6.28378e9	4.42671
$413$	1.34251e8	0.0937761
$414$	0	0
$415$	2.53550e9	1.74138
$416$	4.14898e9	2.82563
$417$	0	0
$418$	−3.25520e9	−2.18002
$419$	1.07822e9	0.716075	0.358038	−	0.933707i	$-0.383446\pi$
$419$	1.07822e9	0.716075	0.358038	+	0.933707i	$0.383446\pi$
$420$	0	0
$421$	−1.88974e9	−1.23428	−0.617140	−	0.786853i	$-0.711709\pi$
$421$	−1.88974e9	−1.23428	−0.617140	+	0.786853i	$0.711709\pi$
$422$	−1.53285e9	−0.992904
$423$	0	0
$424$	−5.33065e9	−3.39625
$425$	−9.60010e8	−0.606617
$426$	0	0
$427$	7.66024e8	0.476151
$428$	−2.98723e7	−0.0184168
$429$	0	0
$430$	3.44841e9	2.09160
$431$	2.24349e9	1.34975	0.674876	−	0.737931i	$-0.264197\pi$
$431$	2.24349e9	1.34975	0.674876	+	0.737931i	$0.264197\pi$
$432$	0	0
$433$	1.57772e9	0.933948	0.466974	−	0.884271i	$-0.345344\pi$
$433$	1.57772e9	0.933948	0.466974	+	0.884271i	$0.345344\pi$
$434$	3.78115e9	2.22029
$435$	0	0
$436$	5.56570e9	3.21601
$437$	−5.93488e9	−3.40194
$438$	0	0
$439$	2.36005e9	1.33136	0.665681	−	0.746236i	$-0.268141\pi$
$439$	2.36005e9	1.33136	0.665681	+	0.746236i	$0.268141\pi$
$440$	−4.62910e9	−2.59067
$441$	0	0
$442$	−3.42852e9	−1.88855
$443$	4.93538e8	0.269717	0.134858	−	0.990865i	$-0.456942\pi$
$443$	4.93538e8	0.269717	0.134858	+	0.990865i	$0.456942\pi$
$444$	0	0
$445$	2.04438e9	1.09977
$446$	−3.12708e9	−1.66904
$447$	0	0
$448$	−5.22945e9	−2.74779
$449$	2.07688e9	1.08280	0.541401	−	0.840764i	$-0.317894\pi$
$449$	2.07688e9	1.08280	0.541401	+	0.840764i	$0.317894\pi$
$450$	0	0
$451$	−1.61238e9	−0.827653
$452$	7.03882e9	3.58522
$453$	0	0
$454$	−2.54591e9	−1.27687
$455$	−1.25505e9	−0.624627
$456$	0	0
$457$	−1.48289e9	−0.726777	−0.363389	−	0.931638i	$-0.618380\pi$
$457$	−1.48289e9	−0.726777	−0.363389	+	0.931638i	$0.618380\pi$
$458$	−5.43814e9	−2.64497
$459$	0	0
$460$	−1.32905e10	−6.36630
$461$	−2.57673e9	−1.22494	−0.612471	−	0.790493i	$-0.709825\pi$
$461$	−2.57673e9	−1.22494	−0.612471	+	0.790493i	$0.709825\pi$
$462$	0	0
$463$	−4.54013e8	−0.212586	−0.106293	−	0.994335i	$-0.533898\pi$
$463$	−4.54013e8	−0.212586	−0.106293	+	0.994335i	$0.533898\pi$
$464$	−7.97070e9	−3.70411
$465$	0	0
$466$	−2.80784e9	−1.28535
$467$	−3.63008e9	−1.64933	−0.824663	−	0.565624i	$-0.808635\pi$
$467$	−3.63008e9	−1.64933	−0.824663	+	0.565624i	$0.808635\pi$
$468$	0	0
$469$	8.16044e8	0.365265
$470$	3.31904e9	1.47458
$471$	0	0
$472$	1.00076e9	0.438060
$473$	−1.32385e9	−0.575208
$474$	0	0
$475$	−1.84215e9	−0.788672
$476$	6.29236e9	2.67417
$477$	0	0
$478$	3.33426e9	1.39637
$479$	9.05516e7	0.0376463	0.0188231	−	0.999823i	$-0.494008\pi$
$479$	9.05516e7	0.0376463	0.0188231	+	0.999823i	$0.494008\pi$
$480$	0	0
$481$	9.28973e8	0.380623
$482$	5.39427e9	2.19416
$483$	0	0
$484$	−4.03584e9	−1.61799
$485$	9.24439e8	0.367944
$486$	0	0
$487$	2.49820e8	0.0980112	0.0490056	−	0.998799i	$-0.484395\pi$
$487$	2.49820e8	0.0980112	0.0490056	+	0.998799i	$0.484395\pi$
$488$	5.71025e9	2.22426
$489$	0	0
$490$	−2.91569e9	−1.11958
$491$	−4.25867e8	−0.162364	−0.0811818	−	0.996699i	$-0.525869\pi$
$491$	−4.25867e8	−0.162364	−0.0811818	+	0.996699i	$0.525869\pi$
$492$	0	0
$493$	3.54454e9	1.33228
$494$	−6.57892e9	−2.45533
$495$	0	0
$496$	1.63180e10	6.00457
$497$	1.90168e9	0.694849
$498$	0	0
$499$	−4.17328e9	−1.50358	−0.751789	−	0.659404i	$-0.770809\pi$
$499$	−4.17328e9	−1.50358	−0.751789	+	0.659404i	$0.770809\pi$
$500$	5.08921e9	1.82077
$501$	0	0
$502$	−3.36826e8	−0.118835
$503$	−4.99769e9	−1.75098	−0.875491	−	0.483234i	$-0.839462\pi$
$503$	−4.99769e9	−1.75098	−0.875491	+	0.483234i	$0.839462\pi$
$504$	0	0
$505$	−6.39255e9	−2.20879
$506$	6.96440e9	2.38977
$507$	0	0
$508$	4.20212e9	1.42215
$509$	3.79733e9	1.27634	0.638170	−	0.769895i	$-0.279692\pi$
$509$	3.79733e9	1.27634	0.638170	+	0.769895i	$0.279692\pi$
$510$	0	0
$511$	−1.27468e9	−0.422600
$512$	−6.35877e9	−2.09377
$513$	0	0
$514$	3.17841e8	0.103238
$515$	−6.02570e9	−1.94394
$516$	0	0
$517$	−1.27418e9	−0.405522
$518$	−2.32721e9	−0.735666
$519$	0	0
$520$	−9.35563e9	−2.91784
$521$	−4.83903e9	−1.49909	−0.749543	−	0.661956i	$-0.769727\pi$
$521$	−4.83903e9	−1.49909	−0.749543	+	0.661956i	$0.769727\pi$
$522$	0	0
$523$	−1.38035e9	−0.421922	−0.210961	−	0.977494i	$-0.567659\pi$
$523$	−1.38035e9	−0.421922	−0.210961	+	0.977494i	$0.567659\pi$
$524$	−3.14042e9	−0.953515
$525$	0	0
$526$	8.94325e9	2.67945
$527$	−7.25657e9	−2.15970
$528$	0	0
$529$	9.29267e9	2.72926
$530$	8.04962e9	2.34860
$531$	0	0
$532$	1.20743e10	3.47673
$533$	−3.25869e9	−0.932175
$534$	0	0
$535$	2.86454e7	0.00808754
$536$	6.08312e9	1.70628
$537$	0	0
$538$	4.44490e7	0.0123062
$539$	1.11934e9	0.307894
$540$	0	0
$541$	3.09741e9	0.841023	0.420512	−	0.907287i	$-0.361851\pi$
$541$	3.09741e9	0.841023	0.420512	+	0.907287i	$0.361851\pi$
$542$	−2.23203e9	−0.602146
$543$	0	0
$544$	1.99471e10	5.31232
$545$	−5.33711e9	−1.41227
$546$	0	0
$547$	4.36175e9	1.13948	0.569738	−	0.821826i	$-0.307045\pi$
$547$	4.36175e9	1.13948	0.569738	+	0.821826i	$0.307045\pi$
$548$	5.96820e9	1.54921
$549$	0	0
$550$	2.16170e9	0.554021
$551$	6.80155e9	1.73212
$552$	0	0
$553$	−3.46357e8	−0.0870935
$554$	3.94645e9	0.986103
$555$	0	0
$556$	−5.24553e9	−1.29428
$557$	−7.54573e9	−1.85016	−0.925078	−	0.379778i	$-0.876000\pi$
$557$	−7.54573e9	−1.85016	−0.925078	+	0.379778i	$0.876000\pi$
$558$	0	0
$559$	−2.67556e9	−0.647849
$560$	1.35686e10	3.26496
$561$	0	0
$562$	7.41999e9	1.76330
$563$	−5.42932e9	−1.28223	−0.641115	−	0.767445i	$-0.721528\pi$
$563$	−5.42932e9	−1.28223	−0.641115	+	0.767445i	$0.721528\pi$
$564$	0	0
$565$	−6.74973e9	−1.57441
$566$	2.68834e9	0.623199
$567$	0	0
$568$	1.41759e10	3.24587
$569$	7.43477e9	1.69190	0.845950	−	0.533262i	$-0.179034\pi$
$569$	7.43477e9	1.69190	0.845950	+	0.533262i	$0.179034\pi$
$570$	0	0
$571$	−6.57051e9	−1.47697	−0.738487	−	0.674268i	$-0.764459\pi$
$571$	−6.57051e9	−1.47697	−0.738487	+	0.674268i	$0.764459\pi$
$572$	5.65590e9	1.26362
$573$	0	0
$574$	8.16347e9	1.80170
$575$	3.94122e9	0.864555
$576$	0	0
$577$	3.17169e9	0.687347	0.343673	−	0.939089i	$-0.388329\pi$
$577$	3.17169e9	0.687347	0.343673	+	0.939089i	$0.388329\pi$
$578$	−7.50538e9	−1.61668
$579$	0	0
$580$	1.52313e10	3.24144
$581$	4.92823e9	1.04250
$582$	0	0
$583$	−3.09026e9	−0.645885
$584$	−9.50201e9	−1.97411
$585$	0	0
$586$	−5.99321e9	−1.23032
$587$	4.96828e9	1.01385	0.506923	−	0.861991i	$-0.330783\pi$
$587$	4.96828e9	1.01385	0.506923	+	0.861991i	$0.330783\pi$
$588$	0	0
$589$	−1.39245e10	−2.80786
$590$	−1.51121e9	−0.302931
$591$	0	0
$592$	−1.00434e10	−1.98954
$593$	1.11991e9	0.220542	0.110271	−	0.993902i	$-0.464828\pi$
$593$	1.11991e9	0.220542	0.110271	+	0.993902i	$0.464828\pi$
$594$	0	0
$595$	−6.03392e9	−1.17433
$596$	1.82154e10	3.52433
$597$	0	0
$598$	1.40754e10	2.69157
$599$	−1.35680e9	−0.257942	−0.128971	−	0.991648i	$-0.541167\pi$
$599$	−1.35680e9	−0.257942	−0.128971	+	0.991648i	$0.541167\pi$
$600$	0	0
$601$	−8.43823e9	−1.58559	−0.792795	−	0.609489i	$-0.791375\pi$
$601$	−8.43823e9	−1.58559	−0.792795	+	0.609489i	$0.791375\pi$
$602$	6.70266e9	1.25216
$603$	0	0
$604$	−1.68105e9	−0.310420
$605$	3.87008e9	0.710519
$606$	0	0
$607$	2.84233e9	0.515839	0.257920	−	0.966166i	$-0.416963\pi$
$607$	2.84233e9	0.515839	0.257920	+	0.966166i	$0.416963\pi$
$608$	3.82762e10	6.90663
$609$	0	0
$610$	−8.62284e9	−1.53814
$611$	−2.57519e9	−0.456735
$612$	0	0
$613$	−5.43477e8	−0.0952949	−0.0476475	−	0.998864i	$-0.515172\pi$
$613$	−5.43477e8	−0.0952949	−0.0476475	+	0.998864i	$0.515172\pi$
$614$	5.50014e9	0.958924
$615$	0	0
$616$	−8.99755e9	−1.55093
$617$	−5.89919e8	−0.101110	−0.0505550	−	0.998721i	$-0.516099\pi$
$617$	−5.89919e8	−0.101110	−0.0505550	+	0.998721i	$0.516099\pi$
$618$	0	0
$619$	−8.29034e9	−1.40493	−0.702466	−	0.711717i	$-0.747918\pi$
$619$	−8.29034e9	−1.40493	−0.702466	+	0.711717i	$0.747918\pi$
$620$	−3.11822e10	−5.25456
$621$	0	0
$622$	7.58592e9	1.26399
$623$	3.97364e9	0.658386
$624$	0	0
$625$	−7.61270e9	−1.24726
$626$	−1.87964e10	−3.06242
$627$	0	0
$628$	1.87736e10	3.02475
$629$	4.46624e9	0.715591
$630$	0	0
$631$	3.36985e9	0.533959	0.266980	−	0.963702i	$-0.413974\pi$
$631$	3.36985e9	0.533959	0.266980	+	0.963702i	$0.413974\pi$
$632$	−2.58188e9	−0.406843
$633$	0	0
$634$	1.39222e10	2.16968
$635$	−4.02953e9	−0.624520
$636$	0	0
$637$	2.26224e9	0.346777
$638$	−7.98141e9	−1.21677
$639$	0	0
$640$	2.75822e10	4.15909
$641$	6.77698e9	1.01633	0.508163	−	0.861261i	$-0.330325\pi$
$641$	6.77698e9	1.01633	0.508163	+	0.861261i	$0.330325\pi$
$642$	0	0
$643$	5.27093e9	0.781895	0.390948	−	0.920413i	$-0.372147\pi$
$643$	5.27093e9	0.781895	0.390948	+	0.920413i	$0.372147\pi$
$644$	−2.58326e10	−3.81125
$645$	0	0
$646$	−3.16296e10	−4.61615
$647$	1.36234e10	1.97752	0.988762	−	0.149498i	$-0.0477656\pi$
$647$	1.36234e10	1.97752	0.988762	+	0.149498i	$0.0477656\pi$
$648$	0	0
$649$	5.80156e8	0.0833083
$650$	4.36890e9	0.623987
$651$	0	0
$652$	−2.95171e10	−4.17068
$653$	−7.55574e9	−1.06189	−0.530947	−	0.847405i	$-0.678164\pi$
$653$	−7.55574e9	−1.06189	−0.530947	+	0.847405i	$0.678164\pi$
$654$	0	0
$655$	3.01143e9	0.418725
$656$	3.52305e10	4.87254
$657$	0	0
$658$	6.45120e9	0.882774
$659$	−6.65668e9	−0.906063	−0.453032	−	0.891494i	$-0.649658\pi$
$659$	−6.65668e9	−0.906063	−0.453032	+	0.891494i	$0.649658\pi$
$660$	0	0
$661$	−9.57345e9	−1.28933	−0.644664	−	0.764466i	$-0.723003\pi$
$661$	−9.57345e9	−1.28933	−0.644664	+	0.764466i	$0.723003\pi$
$662$	−8.94613e9	−1.19848
$663$	0	0
$664$	3.67370e10	4.86985
$665$	−1.15784e10	−1.52676
$666$	0	0
$667$	−1.45517e10	−1.89878
$668$	−2.20496e9	−0.286209
$669$	0	0
$670$	−9.18589e9	−1.17994
$671$	3.31032e9	0.423001
$672$	0	0
$673$	8.00895e9	1.01280	0.506399	−	0.862299i	$-0.330976\pi$
$673$	8.00895e9	1.01280	0.506399	+	0.862299i	$0.330976\pi$
$674$	−1.31376e10	−1.65275
$675$	0	0
$676$	−1.05756e10	−1.31672
$677$	3.66833e8	0.0454368	0.0227184	−	0.999742i	$-0.492768\pi$
$677$	3.66833e8	0.0454368	0.0227184	+	0.999742i	$0.492768\pi$
$678$	0	0
$679$	1.79683e9	0.220273
$680$	−4.49793e10	−5.48569
$681$	0	0
$682$	1.63400e10	1.97245
$683$	9.58167e9	1.15072	0.575359	−	0.817901i	$-0.304863\pi$
$683$	9.58167e9	1.15072	0.575359	+	0.817901i	$0.304863\pi$
$684$	0	0
$685$	−5.72307e9	−0.680319
$686$	−1.74455e10	−2.06324
$687$	0	0
$688$	2.89262e10	3.38635
$689$	−6.24557e9	−0.727452
$690$	0	0
$691$	1.25570e10	1.44781	0.723907	−	0.689898i	$-0.242345\pi$
$691$	1.25570e10	1.44781	0.723907	+	0.689898i	$0.242345\pi$
$692$	1.63031e10	1.87024
$693$	0	0
$694$	3.97393e9	0.451297
$695$	5.03008e9	0.568367
$696$	0	0
$697$	−1.56669e10	−1.75254
$698$	1.54631e10	1.72109
$699$	0	0
$700$	−8.01825e9	−0.883561
$701$	2.90472e9	0.318487	0.159243	−	0.987239i	$-0.449095\pi$
$701$	2.90472e9	0.318487	0.159243	+	0.987239i	$0.449095\pi$
$702$	0	0
$703$	8.57019e9	0.930351
$704$	−2.25987e10	−2.44107
$705$	0	0
$706$	5.06656e9	0.541872
$707$	−1.24252e10	−1.32231
$708$	0	0
$709$	4.07802e9	0.429721	0.214861	−	0.976645i	$-0.431070\pi$
$709$	4.07802e9	0.429721	0.214861	+	0.976645i	$0.431070\pi$
$710$	−2.14065e10	−2.24461
$711$	0	0
$712$	2.96211e10	3.07554
$713$	2.97910e10	3.07802
$714$	0	0
$715$	−5.42360e9	−0.554902
$716$	−3.95956e10	−4.03136
$717$	0	0
$718$	2.49386e10	2.51441
$719$	6.67220e8	0.0669450	0.0334725	−	0.999440i	$-0.489343\pi$
$719$	6.67220e8	0.0669450	0.0334725	+	0.999440i	$0.489343\pi$
$720$	0	0
$721$	−1.17121e10	−1.16376
$722$	−4.11361e10	−4.06764
$723$	0	0
$724$	−3.66229e10	−3.58648
$725$	−4.51675e9	−0.440193
$726$	0	0
$727$	−1.54917e10	−1.49530	−0.747651	−	0.664092i	$-0.768819\pi$
$727$	−1.54917e10	−1.49530	−0.747651	+	0.664092i	$0.768819\pi$
$728$	−1.81845e10	−1.74679
$729$	0	0
$730$	1.43486e10	1.36515
$731$	−1.28634e10	−1.21799
$732$	0	0
$733$	1.32907e10	1.24647	0.623237	−	0.782033i	$-0.285817\pi$
$733$	1.32907e10	1.24647	0.623237	+	0.782033i	$0.285817\pi$
$734$	−1.41068e10	−1.31671
$735$	0	0
$736$	−8.18908e10	−7.57116
$737$	3.52648e9	0.324493
$738$	0	0
$739$	1.38709e10	1.26430	0.632148	−	0.774847i	$-0.282173\pi$
$739$	1.38709e10	1.26430	0.632148	+	0.774847i	$0.282173\pi$
$740$	1.91919e10	1.74103
$741$	0	0
$742$	1.56460e10	1.40601
$743$	9.01349e9	0.806181	0.403090	−	0.915160i	$-0.367936\pi$
$743$	9.01349e9	0.806181	0.403090	+	0.915160i	$0.367936\pi$
$744$	0	0
$745$	−1.74672e10	−1.54767
$746$	−3.70337e9	−0.326596
$747$	0	0
$748$	2.71920e10	2.37567
$749$	5.56779e7	0.00484168
$750$	0	0
$751$	−6.25646e9	−0.539000	−0.269500	−	0.963000i	$-0.586858\pi$
$751$	−6.25646e9	−0.539000	−0.269500	+	0.963000i	$0.586858\pi$
$752$	2.78410e10	2.38738
$753$	0	0
$754$	−1.61308e10	−1.37043
$755$	1.61200e9	0.136317
$756$	0	0
$757$	−2.50426e9	−0.209819	−0.104909	−	0.994482i	$-0.533455\pi$
$757$	−2.50426e9	−0.209819	−0.104909	+	0.994482i	$0.533455\pi$
$758$	2.21799e10	1.84977
$759$	0	0
$760$	−8.63099e10	−7.13203
$761$	−6.57794e9	−0.541057	−0.270529	−	0.962712i	$-0.587198\pi$
$761$	−6.57794e9	−0.541057	−0.270529	+	0.962712i	$0.587198\pi$
$762$	0	0
$763$	−1.03737e10	−0.845470
$764$	4.86888e10	3.95005
$765$	0	0
$766$	1.21233e10	0.974587
$767$	1.17252e9	0.0938292
$768$	0	0
$769$	5.82071e9	0.461566	0.230783	−	0.973005i	$-0.425871\pi$
$769$	5.82071e9	0.461566	0.230783	+	0.973005i	$0.425871\pi$
$770$	1.35869e10	1.07251
$771$	0	0
$772$	1.07626e10	0.841889
$773$	−5.98583e9	−0.466118	−0.233059	−	0.972463i	$-0.574874\pi$
$773$	−5.98583e9	−0.466118	−0.233059	+	0.972463i	$0.574874\pi$
$774$	0	0
$775$	9.24693e9	0.713578
$776$	1.33943e10	1.02897
$777$	0	0
$778$	2.56276e10	1.95110
$779$	−3.00629e10	−2.27850
$780$	0	0
$781$	8.21798e9	0.617286
$782$	6.76706e10	5.06030
$783$	0	0
$784$	−2.44576e10	−1.81262
$785$	−1.80026e10	−1.32828
$786$	0	0
$787$	−4.85827e9	−0.355279	−0.177640	−	0.984096i	$-0.556846\pi$
$787$	−4.85827e9	−0.355279	−0.177640	+	0.984096i	$0.556846\pi$
$788$	9.77857e9	0.711924
$789$	0	0
$790$	3.89880e9	0.281343
$791$	−1.31194e10	−0.942533
$792$	0	0
$793$	6.69032e9	0.476421
$794$	−4.08603e10	−2.89688
$795$	0	0
$796$	4.59647e10	3.23019
$797$	7.28406e9	0.509647	0.254823	−	0.966988i	$-0.417983\pi$
$797$	7.28406e9	0.509647	0.254823	+	0.966988i	$0.417983\pi$
$798$	0	0
$799$	−1.23808e10	−0.858685
$800$	−2.54183e10	−1.75522
$801$	0	0
$802$	4.87925e10	3.33997
$803$	−5.50846e9	−0.375427
$804$	0	0
$805$	2.47716e10	1.67366
$806$	3.30239e10	2.22155
$807$	0	0
$808$	−9.26222e10	−6.17697
$809$	−3.79381e9	−0.251916	−0.125958	−	0.992036i	$-0.540201\pi$
$809$	−3.79381e9	−0.251916	−0.125958	+	0.992036i	$0.540201\pi$
$810$	0	0
$811$	−4.79601e9	−0.315724	−0.157862	−	0.987461i	$-0.550460\pi$
$811$	−4.79601e9	−0.315724	−0.157862	+	0.987461i	$0.550460\pi$
$812$	2.96049e10	1.94052
$813$	0	0
$814$	−1.00569e10	−0.653547
$815$	2.83048e10	1.83150
$816$	0	0
$817$	−2.46833e10	−1.58353
$818$	2.26320e10	1.44573
$819$	0	0
$820$	−6.73222e10	−4.26392
$821$	−1.01365e10	−0.639272	−0.319636	−	0.947540i	$-0.603561\pi$
$821$	−1.01365e10	−0.639272	−0.319636	+	0.947540i	$0.603561\pi$
$822$	0	0
$823$	−9.27062e9	−0.579708	−0.289854	−	0.957071i	$-0.593607\pi$
$823$	−9.27062e9	−0.579708	−0.289854	+	0.957071i	$0.593607\pi$
$824$	−8.73068e10	−5.43629
$825$	0	0
$826$	−2.93733e9	−0.181352
$827$	1.15153e10	0.707953	0.353977	−	0.935254i	$-0.384829\pi$
$827$	1.15153e10	0.707953	0.353977	+	0.935254i	$0.384829\pi$
$828$	0	0
$829$	−7.01951e9	−0.427923	−0.213962	−	0.976842i	$-0.568637\pi$
$829$	−7.01951e9	−0.427923	−0.213962	+	0.976842i	$0.568637\pi$
$830$	−5.54752e10	−3.36764
$831$	0	0
$832$	−4.56731e10	−2.74934
$833$	1.08762e10	0.651958
$834$	0	0
$835$	2.11440e9	0.125685
$836$	5.21782e10	3.08864
$837$	0	0
$838$	−2.35908e10	−1.38481
$839$	−2.55234e10	−1.49201	−0.746004	−	0.665942i	$-0.768030\pi$
$839$	−2.55234e10	−1.49201	−0.746004	+	0.665942i	$0.768030\pi$
$840$	0	0
$841$	−5.73184e8	−0.0332283
$842$	4.13463e10	2.38696
$843$	0	0
$844$	2.45704e10	1.40674
$845$	1.01413e10	0.578222
$846$	0	0
$847$	7.52225e9	0.425359
$848$	6.75224e10	3.80244
$849$	0	0
$850$	2.10045e10	1.17313
$851$	−1.83357e10	−1.01987
$852$	0	0
$853$	1.30151e10	0.718004	0.359002	−	0.933337i	$-0.383117\pi$
$853$	1.30151e10	0.718004	0.359002	+	0.933337i	$0.383117\pi$
$854$	−1.67602e10	−0.920822
$855$	0	0
$856$	4.15045e8	0.0226171
$857$	7.46263e9	0.405004	0.202502	−	0.979282i	$-0.435093\pi$
$857$	7.46263e9	0.405004	0.202502	+	0.979282i	$0.435093\pi$
$858$	0	0
$859$	2.06753e9	0.111295	0.0556476	−	0.998450i	$-0.482278\pi$
$859$	2.06753e9	0.111295	0.0556476	+	0.998450i	$0.482278\pi$
$860$	−5.52752e10	−2.96337
$861$	0	0
$862$	−4.90863e10	−2.61027
$863$	2.78352e10	1.47420	0.737099	−	0.675785i	$-0.236195\pi$
$863$	2.78352e10	1.47420	0.737099	+	0.675785i	$0.236195\pi$
$864$	0	0
$865$	−1.56335e10	−0.821295
$866$	−3.45196e10	−1.80615
$867$	0	0
$868$	−6.06087e10	−3.14569
$869$	−1.49676e9	−0.0773716
$870$	0	0
$871$	7.12718e9	0.365472
$872$	−7.73298e10	−3.94947
$873$	0	0
$874$	1.29852e11	6.57897
$875$	−9.48560e9	−0.478670
$876$	0	0
$877$	−1.33724e10	−0.669436	−0.334718	−	0.942318i	$-0.608641\pi$
$877$	−1.33724e10	−0.669436	−0.334718	+	0.942318i	$0.608641\pi$
$878$	−5.16366e10	−2.57470
$879$	0	0
$880$	5.86359e10	2.90051
$881$	3.66543e10	1.80596	0.902982	−	0.429678i	$-0.141373\pi$
$881$	3.66543e10	1.80596	0.902982	+	0.429678i	$0.141373\pi$
$882$	0	0
$883$	3.23252e10	1.58008	0.790040	−	0.613055i	$-0.210060\pi$
$883$	3.23252e10	1.58008	0.790040	+	0.613055i	$0.210060\pi$
$884$	5.49563e10	2.67568
$885$	0	0
$886$	−1.07983e10	−0.521602
$887$	−2.43296e10	−1.17058	−0.585292	−	0.810823i	$-0.699020\pi$
$887$	−2.43296e10	−1.17058	−0.585292	+	0.810823i	$0.699020\pi$
$888$	0	0
$889$	−7.83217e9	−0.373875
$890$	−4.47298e10	−2.12682
$891$	0	0
$892$	5.01245e10	2.36468
$893$	−2.37572e10	−1.11639
$894$	0	0
$895$	3.79693e10	1.77032
$896$	5.36113e10	2.48988
$897$	0	0
$898$	−4.54410e10	−2.09402
$899$	−3.41414e10	−1.56719
$900$	0	0
$901$	−3.00269e10	−1.36765
$902$	3.52778e10	1.60059
$903$	0	0
$904$	−9.77973e10	−4.40289
$905$	3.51188e10	1.57496
$906$	0	0
$907$	−1.00143e10	−0.445653	−0.222827	−	0.974858i	$-0.571528\pi$
$907$	−1.00143e10	−0.445653	−0.222827	+	0.974858i	$0.571528\pi$
$908$	4.08088e10	1.80906
$909$	0	0
$910$	2.74597e10	1.20796
$911$	1.43309e10	0.627998	0.313999	−	0.949423i	$-0.398331\pi$
$911$	1.43309e10	0.627998	0.313999	+	0.949423i	$0.398331\pi$
$912$	0	0
$913$	2.12970e10	0.926127
$914$	3.24447e10	1.40550
$915$	0	0
$916$	8.71689e10	3.74738
$917$	5.85331e9	0.250674
$918$	0	0
$919$	5.16709e9	0.219605	0.109802	−	0.993953i	$-0.464978\pi$
$919$	5.16709e9	0.219605	0.109802	+	0.993953i	$0.464978\pi$
$920$	1.84657e11	7.81825
$921$	0	0
$922$	5.63774e10	2.36890
$923$	1.66089e10	0.695242
$924$	0	0
$925$	−5.69126e9	−0.236435
$926$	9.93354e9	0.411117
$927$	0	0
$928$	9.38492e10	3.85490
$929$	3.79974e10	1.55489	0.777444	−	0.628953i	$-0.216516\pi$
$929$	3.79974e10	1.55489	0.777444	+	0.628953i	$0.216516\pi$
$930$	0	0
$931$	2.08701e10	0.847621
$932$	4.50073e10	1.82107
$933$	0	0
$934$	7.94240e10	3.18961
$935$	−2.60752e10	−1.04325
$936$	0	0
$937$	−3.56133e10	−1.41424	−0.707121	−	0.707093i	$-0.750006\pi$
$937$	−3.56133e10	−1.41424	−0.707121	+	0.707093i	$0.750006\pi$
$938$	−1.78546e10	−0.706382
$939$	0	0
$940$	−5.32015e10	−2.08918
$941$	−1.98729e10	−0.777495	−0.388748	−	0.921344i	$-0.627092\pi$
$941$	−1.98729e10	−0.777495	−0.388748	+	0.921344i	$0.627092\pi$
$942$	0	0
$943$	6.43186e10	2.49773
$944$	−1.26764e10	−0.490451
$945$	0	0
$946$	2.89651e10	1.11239
$947$	−1.94063e10	−0.742537	−0.371269	−	0.928525i	$-0.621077\pi$
$947$	−1.94063e10	−0.742537	−0.371269	+	0.928525i	$0.621077\pi$
$948$	0	0
$949$	−1.11329e10	−0.422839
$950$	4.03051e10	1.52520
$951$	0	0
$952$	−8.74260e10	−3.28406
$953$	2.57497e10	0.963711	0.481855	−	0.876251i	$-0.339963\pi$
$953$	2.57497e10	0.963711	0.481855	+	0.876251i	$0.339963\pi$
$954$	0	0
$955$	−4.66891e10	−1.73462
$956$	−5.34454e10	−1.97837
$957$	0	0
$958$	−1.98122e9	−0.0728036
$959$	−1.11239e10	−0.407279
$960$	0	0
$961$	4.23834e10	1.54051
$962$	−2.03254e10	−0.736082
$963$	0	0
$964$	−8.64657e10	−3.10867
$965$	−1.03205e10	−0.369705
$966$	0	0
$967$	−4.00078e10	−1.42283	−0.711413	−	0.702774i	$-0.751945\pi$
$967$	−4.00078e10	−1.42283	−0.711413	+	0.702774i	$0.751945\pi$
$968$	5.60739e10	1.98699
$969$	0	0
$970$	−2.02262e10	−0.711562
$971$	−1.34901e10	−0.472878	−0.236439	−	0.971646i	$-0.575980\pi$
$971$	−1.34901e10	−0.472878	−0.236439	+	0.971646i	$0.575980\pi$
$972$	0	0
$973$	9.77695e9	0.340258
$974$	−5.46592e9	−0.189542
$975$	0	0
$976$	−7.23307e10	−2.49028
$977$	−4.28309e9	−0.146935	−0.0734676	−	0.997298i	$-0.523407\pi$
$977$	−4.28309e9	−0.146935	−0.0734676	+	0.997298i	$0.523407\pi$
$978$	0	0
$979$	1.71718e10	0.584893
$980$	4.67362e10	1.58621
$981$	0	0
$982$	9.31773e9	0.313993
$983$	1.23005e10	0.413034	0.206517	−	0.978443i	$-0.433787\pi$
$983$	1.23005e10	0.413034	0.206517	+	0.978443i	$0.433787\pi$
$984$	0	0
$985$	−9.37695e9	−0.312633
$986$	−7.75525e10	−2.57648
$987$	0	0
$988$	1.05455e11	3.47870
$989$	5.28091e10	1.73589
$990$	0	0
$991$	−2.97106e10	−0.969737	−0.484868	−	0.874587i	$-0.661133\pi$
$991$	−2.97106e10	−0.969737	−0.484868	+	0.874587i	$0.661133\pi$
$992$	−1.92133e11	−6.24901
$993$	0	0
$994$	−4.16077e10	−1.34376
$995$	−4.40768e10	−1.41850
$996$	0	0
$997$	1.27080e10	0.406112	0.203056	−	0.979167i	$-0.434913\pi$
$997$	1.27080e10	0.406112	0.203056	+	0.979167i	$0.434913\pi$
$998$	9.13090e10	2.90775
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.1		18
3.2	odd	2		177.8.a.d.1.18	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.18	✓	18	3.2	odd	2
531.8.a.e.1.1		18	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-21.8794 q^{2} +350.709 q^{4} -336.305 q^{5} -653.674 q^{7} -4872.75 q^{8} +O(q^{10})\)	\(q-21.8794 q^{2} +350.709 q^{4} -336.305 q^{5} -653.674 q^{7} -4872.75 q^{8} +7358.16 q^{10} -2824.81 q^{11} -5709.07 q^{13} +14302.0 q^{14} +61722.2 q^{16} -27447.6 q^{17} -52668.8 q^{19} -117945. q^{20} +61805.2 q^{22} +112683. q^{23} +34976.1 q^{25} +124911. q^{26} -229250. q^{28} -129138. q^{29} +264379. q^{31} -726735. q^{32} +600538. q^{34} +219834. q^{35} -162719. q^{37} +1.15236e6 q^{38} +1.63873e6 q^{40} +570791. q^{41} +468651. q^{43} -990686. q^{44} -2.46544e6 q^{46} +451069. q^{47} -396253. q^{49} -765257. q^{50} -2.00222e6 q^{52} +1.09397e6 q^{53} +949997. q^{55} +3.18519e6 q^{56} +2.82547e6 q^{58} -205379. q^{59} -1.17187e6 q^{61} -5.78445e6 q^{62} +8.00009e6 q^{64} +1.91999e6 q^{65} -1.24840e6 q^{67} -9.62614e6 q^{68} -4.80984e6 q^{70} -2.90922e6 q^{71} +1.95003e6 q^{73} +3.56019e6 q^{74} -1.84714e7 q^{76} +1.84650e6 q^{77} +529861. q^{79} -2.07575e7 q^{80} -1.24886e7 q^{82} -7.53928e6 q^{83} +9.23078e6 q^{85} -1.02538e7 q^{86} +1.37646e7 q^{88} -6.07893e6 q^{89} +3.73187e6 q^{91} +3.95190e7 q^{92} -9.86913e6 q^{94} +1.77128e7 q^{95} -2.74881e6 q^{97} +8.66979e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * q^98

Introduction

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