LMFDB - Embedded newform 531.8.a.e.1.9

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			1.9
Root			$2.93322$ 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)

$f(q)$	$=$	$q-3.93322 q^{2} -112.530 q^{4} +305.648 q^{5} -1406.85 q^{7} +946.057 q^{8} +O(q^{10})$	\(q-3.93322 q^{2} -112.530 q^{4} +305.648 q^{5} -1406.85 q^{7} +946.057 q^{8} -1202.18 q^{10} -322.020 q^{11} -6387.28 q^{13} +5533.44 q^{14} +10682.8 q^{16} +14848.7 q^{17} -12707.1 q^{19} -34394.5 q^{20} +1266.58 q^{22} +17301.0 q^{23} +15295.9 q^{25} +25122.6 q^{26} +158312. q^{28} +68692.0 q^{29} +153138. q^{31} -163113. q^{32} -58403.3 q^{34} -430000. q^{35} +198033. q^{37} +49979.8 q^{38} +289161. q^{40} -419043. q^{41} +104716. q^{43} +36236.8 q^{44} -68048.6 q^{46} +738804. q^{47} +1.15568e6 q^{49} -60162.0 q^{50} +718760. q^{52} -248408. q^{53} -98424.9 q^{55} -1.33096e6 q^{56} -270181. q^{58} -205379. q^{59} +915348. q^{61} -602325. q^{62} -725835. q^{64} -1.95226e6 q^{65} -828193. q^{67} -1.67092e6 q^{68} +1.69129e6 q^{70} +302314. q^{71} -3.36995e6 q^{73} -778909. q^{74} +1.42993e6 q^{76} +453033. q^{77} -608944. q^{79} +3.26517e6 q^{80} +1.64819e6 q^{82} +3.11831e6 q^{83} +4.53849e6 q^{85} -411872. q^{86} -304649. q^{88} +8.70611e6 q^{89} +8.98593e6 q^{91} -1.94688e6 q^{92} -2.90588e6 q^{94} -3.88390e6 q^{95} -1.80756e6 q^{97} -4.54552e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.93322	−0.347651	−0.173825	−	0.984776i	$-0.555613\pi$
$2$	−3.93322	−0.347651	−0.173825	+	0.984776i	$0.555613\pi$
$3$	0	0
$3$	0	0
$4$	−112.530	−0.879139
$5$	305.648	1.09352	0.546760	−	0.837289i	$-0.315861\pi$
$5$	305.648	1.09352	0.546760	+	0.837289i	$0.315861\pi$
$6$	0	0
$7$	−1406.85	−1.55026	−0.775128	−	0.631804i	$-0.782315\pi$
$7$	−1406.85	−1.55026	−0.775128	+	0.631804i	$0.782315\pi$
$8$	946.057	0.653284
$9$	0	0
$10$	−1202.18	−0.380163
$11$	−322.020	−0.0729472	−0.0364736	−	0.999335i	$-0.511612\pi$
$11$	−322.020	−0.0729472	−0.0364736	+	0.999335i	$0.511612\pi$
$12$	0	0
$13$	−6387.28	−0.806333	−0.403167	−	0.915127i	$-0.632090\pi$
$13$	−6387.28	−0.806333	−0.403167	+	0.915127i	$0.632090\pi$
$14$	5533.44	0.538948
$15$	0	0
$16$	10682.8	0.652024
$17$	14848.7	0.733023	0.366512	−	0.930413i	$-0.380552\pi$
$17$	14848.7	0.733023	0.366512	+	0.930413i	$0.380552\pi$
$18$	0	0
$19$	−12707.1	−0.425019	−0.212509	−	0.977159i	$-0.568164\pi$
$19$	−12707.1	−0.425019	−0.212509	+	0.977159i	$0.568164\pi$
$20$	−34394.5	−0.961356
$21$	0	0
$22$	1266.58	0.0253601
$23$	17301.0	0.296499	0.148250	−	0.988950i	$-0.452636\pi$
$23$	17301.0	0.296499	0.148250	+	0.988950i	$0.452636\pi$
$24$	0	0
$25$	15295.9	0.195787
$26$	25122.6	0.280322
$27$	0	0
$28$	158312.	1.36289
$29$	68692.0	0.523013	0.261507	−	0.965202i	$-0.415781\pi$
$29$	68692.0	0.523013	0.261507	+	0.965202i	$0.415781\pi$
$30$	0	0
$31$	153138.	0.923244	0.461622	−	0.887077i	$-0.347268\pi$
$31$	153138.	0.923244	0.461622	+	0.887077i	$0.347268\pi$
$32$	−163113.	−0.879961
$33$	0	0
$34$	−58403.3	−0.254836
$35$	−430000.	−1.69524
$36$	0	0
$37$	198033.	0.642736	0.321368	−	0.946954i	$-0.395857\pi$
$37$	198033.	0.642736	0.321368	+	0.946954i	$0.395857\pi$
$38$	49979.8	0.147758
$39$	0	0
$40$	289161.	0.714380
$41$	−419043.	−0.949544	−0.474772	−	0.880109i	$-0.657469\pi$
$41$	−419043.	−0.949544	−0.474772	+	0.880109i	$0.657469\pi$
$42$	0	0
$43$	104716.	0.200851	0.100426	−	0.994945i	$-0.467980\pi$
$43$	104716.	0.200851	0.100426	+	0.994945i	$0.467980\pi$
$44$	36236.8	0.0641307
$45$	0	0
$46$	−68048.6	−0.103078
$47$	738804.	1.03797	0.518987	−	0.854782i	$-0.326309\pi$
$47$	738804.	1.03797	0.518987	+	0.854782i	$0.326309\pi$
$48$	0	0
$49$	1.15568e6	1.40330
$50$	−60162.0	−0.0680656
$51$	0	0
$52$	718760.	0.708879
$53$	−248408.	−0.229192	−0.114596	−	0.993412i	$-0.536557\pi$
$53$	−248408.	−0.229192	−0.114596	+	0.993412i	$0.536557\pi$
$54$	0	0
$55$	−98424.9	−0.0797692
$56$	−1.33096e6	−1.01276
$57$	0	0
$58$	−270181.	−0.181826
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	915348.	0.516335	0.258168	−	0.966100i	$-0.416881\pi$
$61$	915348.	0.516335	0.258168	+	0.966100i	$0.416881\pi$
$62$	−602325.	−0.320966
$63$	0	0
$64$	−725835.	−0.346105
$65$	−1.95226e6	−0.881742
$66$	0	0
$67$	−828193.	−0.336411	−0.168205	−	0.985752i	$-0.553797\pi$
$67$	−828193.	−0.336411	−0.168205	+	0.985752i	$0.553797\pi$
$68$	−1.67092e6	−0.644429
$69$	0	0
$70$	1.69129e6	0.589351
$71$	302314.	0.100243	0.0501215	−	0.998743i	$-0.484039\pi$
$71$	302314.	0.100243	0.0501215	+	0.998743i	$0.484039\pi$
$72$	0	0
$73$	−3.36995e6	−1.01390	−0.506948	−	0.861977i	$-0.669226\pi$
$73$	−3.36995e6	−1.01390	−0.506948	+	0.861977i	$0.669226\pi$
$74$	−778909.	−0.223448
$75$	0	0
$76$	1.42993e6	0.373651
$77$	453033.	0.113087
$78$	0	0
$79$	−608944.	−0.138958	−0.0694789	−	0.997583i	$-0.522134\pi$
$79$	−608944.	−0.138958	−0.0694789	+	0.997583i	$0.522134\pi$
$80$	3.26517e6	0.713002
$81$	0	0
$82$	1.64819e6	0.330110
$83$	3.11831e6	0.598614	0.299307	−	0.954157i	$-0.403245\pi$
$83$	3.11831e6	0.598614	0.299307	+	0.954157i	$0.403245\pi$
$84$	0	0
$85$	4.53849e6	0.801576
$86$	−411872.	−0.0698260
$87$	0	0
$88$	−304649.	−0.0476552
$89$	8.70611e6	1.30906	0.654529	−	0.756037i	$-0.272867\pi$
$89$	8.70611e6	1.30906	0.654529	+	0.756037i	$0.272867\pi$
$90$	0	0
$91$	8.98593e6	1.25002
$92$	−1.94688e6	−0.260664
$93$	0	0
$94$	−2.90588e6	−0.360852
$95$	−3.88390e6	−0.464767
$96$	0	0
$97$	−1.80756e6	−0.201091	−0.100545	−	0.994932i	$-0.532059\pi$
$97$	−1.80756e6	−0.201091	−0.100545	+	0.994932i	$0.532059\pi$
$98$	−4.54552e6	−0.487857
$99$	0	0
$100$	−1.72124e6	−0.172124
$101$	−8.27630e6	−0.799303	−0.399651	−	0.916667i	$-0.630869\pi$
$101$	−8.27630e6	−0.799303	−0.399651	+	0.916667i	$0.630869\pi$
$102$	0	0
$103$	1.84311e7	1.66197	0.830983	−	0.556298i	$-0.187779\pi$
$103$	1.84311e7	1.66197	0.830983	+	0.556298i	$0.187779\pi$
$104$	−6.04273e6	−0.526765
$105$	0	0
$106$	977042.	0.0796788
$107$	−6.61305e6	−0.521866	−0.260933	−	0.965357i	$-0.584030\pi$
$107$	−6.61305e6	−0.521866	−0.260933	+	0.965357i	$0.584030\pi$
$108$	0	0
$109$	1.30251e7	0.963359	0.481679	−	0.876347i	$-0.340027\pi$
$109$	1.30251e7	0.963359	0.481679	+	0.876347i	$0.340027\pi$
$110$	387127.	0.0277318
$111$	0	0
$112$	−1.50290e7	−1.01080
$113$	7.28274e6	0.474810	0.237405	−	0.971411i	$-0.423703\pi$
$113$	7.28274e6	0.474810	0.237405	+	0.971411i	$0.423703\pi$
$114$	0	0
$115$	5.28802e6	0.324228
$116$	−7.72989e6	−0.459802
$117$	0	0
$118$	807801.	0.0452603
$119$	−2.08899e7	−1.13637
$120$	0	0
$121$	−1.93835e7	−0.994679
$122$	−3.60026e6	−0.179504
$123$	0	0
$124$	−1.72326e7	−0.811660
$125$	−1.92036e7	−0.879423
$126$	0	0
$127$	3.11648e7	1.35006	0.675028	−	0.737792i	$-0.264131\pi$
$127$	3.11648e7	1.35006	0.675028	+	0.737792i	$0.264131\pi$
$128$	2.37333e7	1.00028
$129$	0	0
$130$	7.67868e6	0.306538
$131$	2.99857e7	1.16537	0.582687	−	0.812697i	$-0.302001\pi$
$131$	2.99857e7	1.16537	0.582687	+	0.812697i	$0.302001\pi$
$132$	0	0
$133$	1.78769e7	0.658889
$134$	3.25747e6	0.116953
$135$	0	0
$136$	1.40477e7	0.478873
$137$	−1.65933e7	−0.551329	−0.275664	−	0.961254i	$-0.588898\pi$
$137$	−1.65933e7	−0.551329	−0.275664	+	0.961254i	$0.588898\pi$
$138$	0	0
$139$	−5.13930e7	−1.62313	−0.811563	−	0.584264i	$-0.801383\pi$
$139$	−5.13930e7	−1.62313	−0.811563	+	0.584264i	$0.801383\pi$
$140$	4.83878e7	1.49035
$141$	0	0
$142$	−1.18907e6	−0.0348496
$143$	2.05683e6	0.0588197
$144$	0	0
$145$	2.09956e7	0.571926
$146$	1.32547e7	0.352481
$147$	0	0
$148$	−2.22847e7	−0.565054
$149$	2.55525e7	0.632821	0.316410	−	0.948622i	$-0.397522\pi$
$149$	2.55525e7	0.632821	0.316410	+	0.948622i	$0.397522\pi$
$150$	0	0
$151$	−2.50959e6	−0.0593176	−0.0296588	−	0.999560i	$-0.509442\pi$
$151$	−2.50959e6	−0.0593176	−0.0296588	+	0.999560i	$0.509442\pi$
$152$	−1.20216e7	−0.277658
$153$	0	0
$154$	−1.78188e6	−0.0393147
$155$	4.68063e7	1.00959
$156$	0	0
$157$	−1.51685e7	−0.312819	−0.156409	−	0.987692i	$-0.549992\pi$
$157$	−1.51685e7	−0.312819	−0.156409	+	0.987692i	$0.549992\pi$
$158$	2.39511e6	0.0483088
$159$	0	0
$160$	−4.98552e7	−0.962255
$161$	−2.43398e7	−0.459650
$162$	0	0
$163$	1.10949e7	0.200664	0.100332	−	0.994954i	$-0.468010\pi$
$163$	1.10949e7	0.200664	0.100332	+	0.994954i	$0.468010\pi$
$164$	4.71548e7	0.834781
$165$	0	0
$166$	−1.22650e7	−0.208108
$167$	−8.97494e7	−1.49116	−0.745579	−	0.666417i	$-0.767827\pi$
$167$	−8.97494e7	−1.49116	−0.745579	+	0.666417i	$0.767827\pi$
$168$	0	0
$169$	−2.19511e7	−0.349827
$170$	−1.78509e7	−0.278669
$171$	0	0
$172$	−1.17837e7	−0.176576
$173$	−9.67840e7	−1.42116	−0.710578	−	0.703618i	$-0.751567\pi$
$173$	−9.67840e7	−1.42116	−0.710578	+	0.703618i	$0.751567\pi$
$174$	0	0
$175$	−2.15189e7	−0.303520
$176$	−3.44006e6	−0.0475633
$177$	0	0
$178$	−3.42430e7	−0.455095
$179$	3.95244e7	0.515086	0.257543	−	0.966267i	$-0.417087\pi$
$179$	3.95244e7	0.515086	0.257543	+	0.966267i	$0.417087\pi$
$180$	0	0
$181$	−7.62592e6	−0.0955910	−0.0477955	−	0.998857i	$-0.515220\pi$
$181$	−7.62592e6	−0.0955910	−0.0477955	+	0.998857i	$0.515220\pi$
$182$	−3.53436e7	−0.434572
$183$	0	0
$184$	1.63677e7	0.193698
$185$	6.05286e7	0.702845
$186$	0	0
$187$	−4.78159e6	−0.0534720
$188$	−8.31374e7	−0.912523
$189$	0	0
$190$	1.52762e7	0.161577
$191$	7.01758e7	0.728736	0.364368	−	0.931255i	$-0.381285\pi$
$191$	7.01758e7	0.728736	0.364368	+	0.931255i	$0.381285\pi$
$192$	0	0
$193$	−1.47888e8	−1.48076	−0.740378	−	0.672191i	$-0.765353\pi$
$193$	−1.47888e8	−1.48076	−0.740378	+	0.672191i	$0.765353\pi$
$194$	7.10954e6	0.0699094
$195$	0	0
$196$	−1.30048e8	−1.23369
$197$	2.42261e7	0.225762	0.112881	−	0.993608i	$-0.463992\pi$
$197$	2.42261e7	0.225762	0.112881	+	0.993608i	$0.463992\pi$
$198$	0	0
$199$	−1.19894e8	−1.07847	−0.539237	−	0.842154i	$-0.681287\pi$
$199$	−1.19894e8	−1.07847	−0.539237	+	0.842154i	$0.681287\pi$
$200$	1.44708e7	0.127905
$201$	0	0
$202$	3.25525e7	0.277878
$203$	−9.66391e7	−0.810805
$204$	0	0
$205$	−1.28080e8	−1.03835
$206$	−7.24937e7	−0.577783
$207$	0	0
$208$	−6.82338e7	−0.525749
$209$	4.09194e6	0.0310039
$210$	0	0
$211$	−1.87566e8	−1.37456	−0.687281	−	0.726391i	$-0.741196\pi$
$211$	−1.87566e8	−1.37456	−0.687281	+	0.726391i	$0.741196\pi$
$212$	2.79533e7	0.201492
$213$	0	0
$214$	2.60106e7	0.181427
$215$	3.20063e7	0.219635
$216$	0	0
$217$	−2.15441e8	−1.43127
$218$	−5.12306e7	−0.334912
$219$	0	0
$220$	1.10757e7	0.0701282
$221$	−9.48430e7	−0.591061
$222$	0	0
$223$	5.60154e7	0.338252	0.169126	−	0.985594i	$-0.445905\pi$
$223$	5.60154e7	0.338252	0.169126	+	0.985594i	$0.445905\pi$
$224$	2.29475e8	1.36417
$225$	0	0
$226$	−2.86446e7	−0.165068
$227$	−1.96404e8	−1.11445	−0.557224	−	0.830362i	$-0.688134\pi$
$227$	−1.96404e8	−1.11445	−0.557224	+	0.830362i	$0.688134\pi$
$228$	0	0
$229$	4.38337e7	0.241203	0.120602	−	0.992701i	$-0.461518\pi$
$229$	4.38337e7	0.241203	0.120602	+	0.992701i	$0.461518\pi$
$230$	−2.07989e7	−0.112718
$231$	0	0
$232$	6.49865e7	0.341676
$233$	−1.57505e7	−0.0815732	−0.0407866	−	0.999168i	$-0.512986\pi$
$233$	−1.57505e7	−0.0815732	−0.0407866	+	0.999168i	$0.512986\pi$
$234$	0	0
$235$	2.25814e8	1.13505
$236$	2.31113e7	0.114454
$237$	0	0
$238$	8.21645e7	0.395062
$239$	1.02176e8	0.484126	0.242063	−	0.970261i	$-0.422176\pi$
$239$	1.02176e8	0.484126	0.242063	+	0.970261i	$0.422176\pi$
$240$	0	0
$241$	1.57502e7	0.0724813	0.0362407	−	0.999343i	$-0.488462\pi$
$241$	1.57502e7	0.0724813	0.0362407	+	0.999343i	$0.488462\pi$
$242$	7.62395e7	0.345801
$243$	0	0
$244$	−1.03004e8	−0.453930
$245$	3.53230e8	1.53453
$246$	0	0
$247$	8.11638e7	0.342707
$248$	1.44877e8	0.603141
$249$	0	0
$250$	7.55320e7	0.305732
$251$	−1.52785e8	−0.609848	−0.304924	−	0.952377i	$-0.598631\pi$
$251$	−1.52785e8	−0.609848	−0.304924	+	0.952377i	$0.598631\pi$
$252$	0	0
$253$	−5.57127e6	−0.0216288
$254$	−1.22578e8	−0.469348
$255$	0	0
$256$	−441486.	−0.00164466
$257$	−2.55028e8	−0.937176	−0.468588	−	0.883417i	$-0.655237\pi$
$257$	−2.55028e8	−0.937176	−0.468588	+	0.883417i	$0.655237\pi$
$258$	0	0
$259$	−2.78603e8	−0.996405
$260$	2.19688e8	0.775174
$261$	0	0
$262$	−1.17941e8	−0.405143
$263$	−3.21627e8	−1.09020	−0.545101	−	0.838370i	$-0.683509\pi$
$263$	−3.21627e8	−1.09020	−0.545101	+	0.838370i	$0.683509\pi$
$264$	0	0
$265$	−7.59254e7	−0.250626
$266$	−7.03139e7	−0.229063
$267$	0	0
$268$	9.31964e7	0.295752
$269$	−2.32732e8	−0.728992	−0.364496	−	0.931205i	$-0.618759\pi$
$269$	−2.32732e8	−0.728992	−0.364496	+	0.931205i	$0.618759\pi$
$270$	0	0
$271$	1.17634e8	0.359036	0.179518	−	0.983755i	$-0.442546\pi$
$271$	1.17634e8	0.359036	0.179518	+	0.983755i	$0.442546\pi$
$272$	1.58625e8	0.477949
$273$	0	0
$274$	6.52651e7	0.191670
$275$	−4.92558e6	−0.0142821
$276$	0	0
$277$	2.49236e8	0.704583	0.352291	−	0.935890i	$-0.385403\pi$
$277$	2.49236e8	0.704583	0.352291	+	0.935890i	$0.385403\pi$
$278$	2.02140e8	0.564281
$279$	0	0
$280$	−4.06805e8	−1.10747
$281$	−4.06099e8	−1.09184	−0.545921	−	0.837837i	$-0.683820\pi$
$281$	−4.06099e8	−1.09184	−0.545921	+	0.837837i	$0.683820\pi$
$282$	0	0
$283$	−7.10346e8	−1.86302	−0.931510	−	0.363717i	$-0.881508\pi$
$283$	−7.10346e8	−1.86302	−0.931510	+	0.363717i	$0.881508\pi$
$284$	−3.40194e7	−0.0881276
$285$	0	0
$286$	−8.08998e6	−0.0204487
$287$	5.89529e8	1.47204
$288$	0	0
$289$	−1.89854e8	−0.462677
$290$	−8.25802e7	−0.198831
$291$	0	0
$292$	3.79219e8	0.891355
$293$	−9.92751e7	−0.230571	−0.115285	−	0.993332i	$-0.536778\pi$
$293$	−9.92751e7	−0.230571	−0.115285	+	0.993332i	$0.536778\pi$
$294$	0	0
$295$	−6.27737e7	−0.142364
$296$	1.87351e8	0.419889
$297$	0	0
$298$	−1.00503e8	−0.220001
$299$	−1.10506e8	−0.239077
$300$	0	0
$301$	−1.47320e8	−0.311371
$302$	9.87078e6	0.0206218
$303$	0	0
$304$	−1.35747e8	−0.277123
$305$	2.79775e8	0.564623
$306$	0	0
$307$	9.45615e8	1.86522	0.932610	−	0.360885i	$-0.117525\pi$
$307$	9.45615e8	1.86522	0.932610	+	0.360885i	$0.117525\pi$
$308$	−5.09797e7	−0.0994191
$309$	0	0
$310$	−1.84099e8	−0.350983
$311$	−3.51898e8	−0.663370	−0.331685	−	0.943390i	$-0.607617\pi$
$311$	−3.51898e8	−0.663370	−0.331685	+	0.943390i	$0.607617\pi$
$312$	0	0
$313$	4.30491e8	0.793521	0.396761	−	0.917922i	$-0.370134\pi$
$313$	4.30491e8	0.793521	0.396761	+	0.917922i	$0.370134\pi$
$314$	5.96609e7	0.108752
$315$	0	0
$316$	6.85244e7	0.122163
$317$	−9.07736e8	−1.60049	−0.800243	−	0.599676i	$-0.795296\pi$
$317$	−9.07736e8	−1.60049	−0.800243	+	0.599676i	$0.795296\pi$
$318$	0	0
$319$	−2.21202e7	−0.0381524
$320$	−2.21850e8	−0.378473
$321$	0	0
$322$	9.57340e7	0.159798
$323$	−1.88684e8	−0.311549
$324$	0	0
$325$	−9.76991e7	−0.157870
$326$	−4.36389e7	−0.0697609
$327$	0	0
$328$	−3.96438e8	−0.620322
$329$	−1.03938e9	−1.60913
$330$	0	0
$331$	−6.30152e8	−0.955096	−0.477548	−	0.878606i	$-0.658474\pi$
$331$	−6.30152e8	−0.955096	−0.477548	+	0.878606i	$0.658474\pi$
$332$	−3.50903e8	−0.526264
$333$	0	0
$334$	3.53004e8	0.518402
$335$	−2.53136e8	−0.367872
$336$	0	0
$337$	5.99949e8	0.853905	0.426952	−	0.904274i	$-0.359587\pi$
$337$	5.99949e8	0.853905	0.426952	+	0.904274i	$0.359587\pi$
$338$	8.63386e7	0.121618
$339$	0	0
$340$	−5.10715e8	−0.704697
$341$	−4.93134e7	−0.0673480
$342$	0	0
$343$	−4.67259e8	−0.625213
$344$	9.90674e7	0.131213
$345$	0	0
$346$	3.80673e8	0.494066
$347$	−9.73412e8	−1.25067	−0.625336	−	0.780355i	$-0.715038\pi$
$347$	−9.73412e8	−1.25067	−0.625336	+	0.780355i	$0.715038\pi$
$348$	0	0
$349$	4.28531e7	0.0539627	0.0269813	−	0.999636i	$-0.491411\pi$
$349$	4.28531e7	0.0539627	0.0269813	+	0.999636i	$0.491411\pi$
$350$	8.46388e7	0.105519
$351$	0	0
$352$	5.25256e7	0.0641907
$353$	−1.33381e9	−1.61393	−0.806963	−	0.590603i	$-0.798890\pi$
$353$	−1.33381e9	−1.61393	−0.806963	+	0.590603i	$0.798890\pi$
$354$	0	0
$355$	9.24018e7	0.109618
$356$	−9.79697e8	−1.15084
$357$	0	0
$358$	−1.55458e8	−0.179070
$359$	−5.52883e8	−0.630671	−0.315336	−	0.948980i	$-0.602117\pi$
$359$	−5.52883e8	−0.630671	−0.315336	+	0.948980i	$0.602117\pi$
$360$	0	0
$361$	−7.32402e8	−0.819359
$362$	2.99944e7	0.0332323
$363$	0	0
$364$	−1.01118e9	−1.09894
$365$	−1.03002e9	−1.10872
$366$	0	0
$367$	−8.45526e8	−0.892885	−0.446443	−	0.894812i	$-0.647309\pi$
$367$	−8.45526e8	−0.892885	−0.446443	+	0.894812i	$0.647309\pi$
$368$	1.84822e8	0.193325
$369$	0	0
$370$	−2.38072e8	−0.244345
$371$	3.49471e8	0.355306
$372$	0	0
$373$	−1.80814e9	−1.80406	−0.902031	−	0.431671i	$-0.857924\pi$
$373$	−1.80814e9	−1.80406	−0.902031	+	0.431671i	$0.857924\pi$
$374$	1.88070e7	0.0185896
$375$	0	0
$376$	6.98950e8	0.678092
$377$	−4.38755e8	−0.421723
$378$	0	0
$379$	−1.25199e9	−1.18131	−0.590653	−	0.806925i	$-0.701130\pi$
$379$	−1.25199e9	−1.18131	−0.590653	+	0.806925i	$0.701130\pi$
$380$	4.37054e8	0.408595
$381$	0	0
$382$	−2.76017e8	−0.253346
$383$	−2.12795e6	−0.00193537	−0.000967687	−	1.00000i	$-0.500308\pi$
$383$	−2.12795e6	−0.00193537	−0.000967687		1.00000i	$0.500308\pi$
$384$	0	0
$385$	1.38469e8	0.123663
$386$	5.81677e8	0.514786
$387$	0	0
$388$	2.03405e8	0.176787
$389$	−2.97352e8	−0.256122	−0.128061	−	0.991766i	$-0.540875\pi$
$389$	−2.97352e8	−0.256122	−0.128061	+	0.991766i	$0.540875\pi$
$390$	0	0
$391$	2.56898e8	0.217341
$392$	1.09333e9	0.916751
$393$	0	0
$394$	−9.52865e7	−0.0784865
$395$	−1.86123e8	−0.151953
$396$	0	0
$397$	7.60962e8	0.610374	0.305187	−	0.952292i	$-0.401281\pi$
$397$	7.60962e8	0.610374	0.305187	+	0.952292i	$0.401281\pi$
$398$	4.71568e8	0.374933
$399$	0	0
$400$	1.63402e8	0.127658
$401$	−4.16579e8	−0.322620	−0.161310	−	0.986904i	$-0.551572\pi$
$401$	−4.16579e8	−0.322620	−0.161310	+	0.986904i	$0.551572\pi$
$402$	0	0
$403$	−9.78134e8	−0.744442
$404$	9.31330e8	0.702698
$405$	0	0
$406$	3.80103e8	0.281877
$407$	−6.37707e7	−0.0468858
$408$	0	0
$409$	−2.40984e9	−1.74163	−0.870815	−	0.491611i	$-0.836408\pi$
$409$	−2.40984e9	−1.74163	−0.870815	+	0.491611i	$0.836408\pi$
$410$	5.03766e8	0.360982
$411$	0	0
$412$	−2.07405e9	−1.46110
$413$	2.88937e8	0.201826
$414$	0	0
$415$	9.53107e8	0.654596
$416$	1.04185e9	0.709542
$417$	0	0
$418$	−1.60945e7	−0.0107785
$419$	2.79997e8	0.185953	0.0929767	−	0.995668i	$-0.470362\pi$
$419$	2.79997e8	0.185953	0.0929767	+	0.995668i	$0.470362\pi$
$420$	0	0
$421$	1.28732e9	0.840810	0.420405	−	0.907337i	$-0.361888\pi$
$421$	1.28732e9	0.840810	0.420405	+	0.907337i	$0.361888\pi$
$422$	7.37736e8	0.477868
$423$	0	0
$424$	−2.35008e8	−0.149727
$425$	2.27124e8	0.143517
$426$	0	0
$427$	−1.28775e9	−0.800452
$428$	7.44165e8	0.458792
$429$	0	0
$430$	−1.25888e8	−0.0763562
$431$	1.52813e9	0.919367	0.459684	−	0.888083i	$-0.347963\pi$
$431$	1.52813e9	0.919367	0.459684	+	0.888083i	$0.347963\pi$
$432$	0	0
$433$	3.05014e9	1.80556	0.902780	−	0.430103i	$-0.141523\pi$
$433$	3.05014e9	1.80556	0.902780	+	0.430103i	$0.141523\pi$
$434$	8.47378e8	0.497580
$435$	0	0
$436$	−1.46571e9	−0.846926
$437$	−2.19845e8	−0.126018
$438$	0	0
$439$	1.16263e9	0.655864	0.327932	−	0.944701i	$-0.393648\pi$
$439$	1.16263e9	0.655864	0.327932	+	0.944701i	$0.393648\pi$
$440$	−9.31155e7	−0.0521120
$441$	0	0
$442$	3.73038e8	0.205483
$443$	1.39964e9	0.764900	0.382450	−	0.923976i	$-0.375080\pi$
$443$	1.39964e9	0.764900	0.382450	+	0.923976i	$0.375080\pi$
$444$	0	0
$445$	2.66101e9	1.43148
$446$	−2.20321e8	−0.117594
$447$	0	0
$448$	1.02114e9	0.536552
$449$	9.56624e8	0.498746	0.249373	−	0.968408i	$-0.419776\pi$
$449$	9.56624e8	0.498746	0.249373	+	0.968408i	$0.419776\pi$
$450$	0	0
$451$	1.34940e8	0.0692665
$452$	−8.19525e8	−0.417424
$453$	0	0
$454$	7.72500e8	0.387439
$455$	2.74653e9	1.36693
$456$	0	0
$457$	2.44936e9	1.20045	0.600227	−	0.799830i	$-0.295077\pi$
$457$	2.44936e9	1.20045	0.600227	+	0.799830i	$0.295077\pi$
$458$	−1.72407e8	−0.0838546
$459$	0	0
$460$	−5.95060e8	−0.285041
$461$	−3.14456e9	−1.49488	−0.747441	−	0.664329i	$-0.768718\pi$
$461$	−3.14456e9	−1.49488	−0.747441	+	0.664329i	$0.768718\pi$
$462$	0	0
$463$	3.16804e9	1.48340	0.741699	−	0.670733i	$-0.234020\pi$
$463$	3.16804e9	1.48340	0.741699	+	0.670733i	$0.234020\pi$
$464$	7.33820e8	0.341017
$465$	0	0
$466$	6.19500e7	0.0283590
$467$	1.93980e9	0.881350	0.440675	−	0.897667i	$-0.354739\pi$
$467$	1.93980e9	0.881350	0.440675	+	0.897667i	$0.354739\pi$
$468$	0	0
$469$	1.16514e9	0.521523
$470$	−8.88176e8	−0.394600
$471$	0	0
$472$	−1.94300e8	−0.0850504
$473$	−3.37207e7	−0.0146515
$474$	0	0
$475$	−1.94366e8	−0.0832133
$476$	2.35073e9	0.999031
$477$	0	0
$478$	−4.01882e8	−0.168307
$479$	1.57859e9	0.656287	0.328143	−	0.944628i	$-0.393577\pi$
$479$	1.57859e9	0.656287	0.328143	+	0.944628i	$0.393577\pi$
$480$	0	0
$481$	−1.26490e9	−0.518259
$482$	−6.19490e7	−0.0251982
$483$	0	0
$484$	2.18122e9	0.874461
$485$	−5.52478e8	−0.219897
$486$	0	0
$487$	1.69122e9	0.663512	0.331756	−	0.943365i	$-0.392359\pi$
$487$	1.69122e9	0.663512	0.331756	+	0.943365i	$0.392359\pi$
$488$	8.65971e8	0.337314
$489$	0	0
$490$	−1.38933e9	−0.533482
$491$	−3.03523e9	−1.15720	−0.578598	−	0.815613i	$-0.696400\pi$
$491$	−3.03523e9	−1.15720	−0.578598	+	0.815613i	$0.696400\pi$
$492$	0	0
$493$	1.01999e9	0.383381
$494$	−3.19235e8	−0.119142
$495$	0	0
$496$	1.63593e9	0.601977
$497$	−4.25310e8	−0.155403
$498$	0	0
$499$	−2.19815e9	−0.791966	−0.395983	−	0.918258i	$-0.629596\pi$
$499$	−2.19815e9	−0.791966	−0.395983	+	0.918258i	$0.629596\pi$
$500$	2.16098e9	0.773135
$501$	0	0
$502$	6.00936e8	0.212014
$503$	4.49246e9	1.57397	0.786984	−	0.616973i	$-0.211641\pi$
$503$	4.49246e9	1.57397	0.786984	+	0.616973i	$0.211641\pi$
$504$	0	0
$505$	−2.52964e9	−0.874054
$506$	2.19130e7	0.00751926
$507$	0	0
$508$	−3.50697e9	−1.18689
$509$	4.25095e9	1.42881	0.714403	−	0.699734i	$-0.246698\pi$
$509$	4.25095e9	1.42881	0.714403	+	0.699734i	$0.246698\pi$
$510$	0	0
$511$	4.74100e9	1.57180
$512$	−3.03613e9	−0.999713
$513$	0	0
$514$	1.00308e9	0.325810
$515$	5.63345e9	1.81739
$516$	0	0
$517$	−2.37910e8	−0.0757173
$518$	1.09581e9	0.346401
$519$	0	0
$520$	−1.84695e9	−0.576028
$521$	−6.47770e8	−0.200673	−0.100336	−	0.994954i	$-0.531992\pi$
$521$	−6.47770e8	−0.200673	−0.100336	+	0.994954i	$0.531992\pi$
$522$	0	0
$523$	2.31099e9	0.706386	0.353193	−	0.935551i	$-0.385096\pi$
$523$	2.31099e9	0.706386	0.353193	+	0.935551i	$0.385096\pi$
$524$	−3.37429e9	−1.02453
$525$	0	0
$526$	1.26503e9	0.379010
$527$	2.27390e9	0.676759
$528$	0	0
$529$	−3.10550e9	−0.912088
$530$	2.98631e8	0.0871304
$531$	0	0
$532$	−2.01169e9	−0.579255
$533$	2.67655e9	0.765648
$534$	0	0
$535$	−2.02127e9	−0.570671
$536$	−7.83518e8	−0.219772
$537$	0	0
$538$	9.15386e8	0.253435
$539$	−3.72151e8	−0.102367
$540$	0	0
$541$	2.47612e8	0.0672329	0.0336165	−	0.999435i	$-0.489298\pi$
$541$	2.47612e8	0.0672329	0.0336165	+	0.999435i	$0.489298\pi$
$542$	−4.62678e8	−0.124819
$543$	0	0
$544$	−2.42202e9	−0.645032
$545$	3.98110e9	1.05345
$546$	0	0
$547$	−2.13776e9	−0.558474	−0.279237	−	0.960222i	$-0.590082\pi$
$547$	−2.13776e9	−0.558474	−0.279237	+	0.960222i	$0.590082\pi$
$548$	1.86724e9	0.484694
$549$	0	0
$550$	1.93734e7	0.00496519
$551$	−8.72874e8	−0.222291
$552$	0	0
$553$	8.56691e8	0.215420
$554$	−9.80301e8	−0.244949
$555$	0	0
$556$	5.78325e9	1.42695
$557$	−6.39469e9	−1.56793	−0.783964	−	0.620806i	$-0.786806\pi$
$557$	−6.39469e9	−1.56793	−0.783964	+	0.620806i	$0.786806\pi$
$558$	0	0
$559$	−6.68852e8	−0.161953
$560$	−4.59359e9	−1.10534
$561$	0	0
$562$	1.59728e9	0.379580
$563$	−7.78393e9	−1.83831	−0.919156	−	0.393893i	$-0.871128\pi$
$563$	−7.78393e9	−1.83831	−0.919156	+	0.393893i	$0.871128\pi$
$564$	0	0
$565$	2.22596e9	0.519215
$566$	2.79395e9	0.647680
$567$	0	0
$568$	2.86006e8	0.0654872
$569$	−1.45339e9	−0.330742	−0.165371	−	0.986231i	$-0.552882\pi$
$569$	−1.45339e9	−0.330742	−0.165371	+	0.986231i	$0.552882\pi$
$570$	0	0
$571$	6.80957e9	1.53071	0.765356	−	0.643608i	$-0.222563\pi$
$571$	6.80957e9	1.53071	0.765356	+	0.643608i	$0.222563\pi$
$572$	−2.31455e8	−0.0517107
$573$	0	0
$574$	−2.31875e9	−0.511755
$575$	2.64634e8	0.0580507
$576$	0	0
$577$	−2.47806e9	−0.537027	−0.268513	−	0.963276i	$-0.586532\pi$
$577$	−2.47806e9	−0.537027	−0.268513	+	0.963276i	$0.586532\pi$
$578$	7.46738e8	0.160850
$579$	0	0
$580$	−2.36263e9	−0.502802
$581$	−4.38699e9	−0.928005
$582$	0	0
$583$	7.99922e7	0.0167189
$584$	−3.18816e9	−0.662362
$585$	0	0
$586$	3.90471e8	0.0801580
$587$	1.40629e9	0.286974	0.143487	−	0.989652i	$-0.454168\pi$
$587$	1.40629e9	0.286974	0.143487	+	0.989652i	$0.454168\pi$
$588$	0	0
$589$	−1.94593e9	−0.392396
$590$	2.46903e8	0.0494930
$591$	0	0
$592$	2.11554e9	0.419079
$593$	−1.46308e8	−0.0288123	−0.0144062	−	0.999896i	$-0.504586\pi$
$593$	−1.46308e8	−0.0288123	−0.0144062	+	0.999896i	$0.504586\pi$
$594$	0	0
$595$	−6.38496e9	−1.24265
$596$	−2.87541e9	−0.556337
$597$	0	0
$598$	4.34646e8	0.0831154
$599$	5.20381e9	0.989299	0.494649	−	0.869093i	$-0.335297\pi$
$599$	5.20381e9	0.989299	0.494649	+	0.869093i	$0.335297\pi$
$600$	0	0
$601$	3.88435e9	0.729891	0.364945	−	0.931029i	$-0.381088\pi$
$601$	3.88435e9	0.729891	0.364945	+	0.931029i	$0.381088\pi$
$602$	5.79440e8	0.108248
$603$	0	0
$604$	2.82404e8	0.0521484
$605$	−5.92453e9	−1.08770
$606$	0	0
$607$	9.88692e9	1.79432	0.897161	−	0.441703i	$-0.145626\pi$
$607$	9.88692e9	1.79432	0.897161	+	0.441703i	$0.145626\pi$
$608$	2.07269e9	0.374000
$609$	0	0
$610$	−1.10041e9	−0.196292
$611$	−4.71895e9	−0.836953
$612$	0	0
$613$	3.19394e9	0.560035	0.280018	−	0.959995i	$-0.409660\pi$
$613$	3.19394e9	0.560035	0.280018	+	0.959995i	$0.409660\pi$
$614$	−3.71931e9	−0.648445
$615$	0	0
$616$	4.28595e8	0.0738779
$617$	1.12180e10	1.92272	0.961360	−	0.275295i	$-0.0887754\pi$
$617$	1.12180e10	1.92272	0.961360	+	0.275295i	$0.0887754\pi$
$618$	0	0
$619$	7.74489e9	1.31250	0.656248	−	0.754545i	$-0.272142\pi$
$619$	7.74489e9	1.31250	0.656248	+	0.754545i	$0.272142\pi$
$620$	−5.26710e9	−0.887566
$621$	0	0
$622$	1.38409e9	0.230621
$623$	−1.22482e10	−2.02938
$624$	0	0
$625$	−7.06454e9	−1.15745
$626$	−1.69321e9	−0.275868
$627$	0	0
$628$	1.70690e9	0.275011
$629$	2.94054e9	0.471140
$630$	0	0
$631$	6.77645e9	1.07374	0.536871	−	0.843665i	$-0.319606\pi$
$631$	6.77645e9	1.07374	0.536871	+	0.843665i	$0.319606\pi$
$632$	−5.76096e8	−0.0907789
$633$	0	0
$634$	3.57032e9	0.556410
$635$	9.52548e9	1.47631
$636$	0	0
$637$	−7.38162e9	−1.13152
$638$	8.70036e7	0.0132637
$639$	0	0
$640$	7.25405e9	1.09383
$641$	−8.18828e7	−0.0122797	−0.00613987	−	0.999981i	$-0.501954\pi$
$641$	−8.18828e7	−0.0122797	−0.00613987	+	0.999981i	$0.501954\pi$
$642$	0	0
$643$	2.87560e9	0.426570	0.213285	−	0.976990i	$-0.431584\pi$
$643$	2.87560e9	0.426570	0.213285	+	0.976990i	$0.431584\pi$
$644$	2.73896e9	0.404096
$645$	0	0
$646$	7.42136e8	0.108310
$647$	4.48392e9	0.650869	0.325434	−	0.945565i	$-0.394490\pi$
$647$	4.48392e9	0.650869	0.325434	+	0.945565i	$0.394490\pi$
$648$	0	0
$649$	6.61362e7	0.00949691
$650$	3.84272e8	0.0548835
$651$	0	0
$652$	−1.24851e9	−0.176411
$653$	5.52310e9	0.776224	0.388112	−	0.921612i	$-0.373127\pi$
$653$	5.52310e9	0.776224	0.388112	+	0.921612i	$0.373127\pi$
$654$	0	0
$655$	9.16509e9	1.27436
$656$	−4.47654e9	−0.619125
$657$	0	0
$658$	4.08812e9	0.559414
$659$	−5.93057e9	−0.807230	−0.403615	−	0.914929i	$-0.632247\pi$
$659$	−5.93057e9	−0.807230	−0.403615	+	0.914929i	$0.632247\pi$
$660$	0	0
$661$	−1.03877e10	−1.39898	−0.699492	−	0.714641i	$-0.746590\pi$
$661$	−1.03877e10	−1.39898	−0.699492	+	0.714641i	$0.746590\pi$
$662$	2.47852e9	0.332040
$663$	0	0
$664$	2.95010e9	0.391065
$665$	5.46405e9	0.720508
$666$	0	0
$667$	1.18844e9	0.155073
$668$	1.00995e10	1.31094
$669$	0	0
$670$	9.95639e8	0.127891
$671$	−2.94760e8	−0.0376652
$672$	0	0
$673$	−1.00404e9	−0.126969	−0.0634843	−	0.997983i	$-0.520221\pi$
$673$	−1.00404e9	−0.126969	−0.0634843	+	0.997983i	$0.520221\pi$
$674$	−2.35973e9	−0.296861
$675$	0	0
$676$	2.47015e9	0.307546
$677$	−4.85982e9	−0.601950	−0.300975	−	0.953632i	$-0.597312\pi$
$677$	−4.85982e9	−0.601950	−0.300975	+	0.953632i	$0.597312\pi$
$678$	0	0
$679$	2.54296e9	0.311742
$680$	4.29366e9	0.523657
$681$	0	0
$682$	1.93961e8	0.0234136
$683$	−1.18996e10	−1.42909	−0.714543	−	0.699592i	$-0.753365\pi$
$683$	−1.18996e10	−1.42909	−0.714543	+	0.699592i	$0.753365\pi$
$684$	0	0
$685$	−5.07171e9	−0.602889
$686$	1.83783e9	0.217356
$687$	0	0
$688$	1.11866e9	0.130960
$689$	1.58665e9	0.184805
$690$	0	0
$691$	3.67203e9	0.423383	0.211691	−	0.977337i	$-0.432103\pi$
$691$	3.67203e9	0.423383	0.211691	+	0.977337i	$0.432103\pi$
$692$	1.08911e10	1.24939
$693$	0	0
$694$	3.82864e9	0.434797
$695$	−1.57082e10	−1.77492
$696$	0	0
$697$	−6.22225e9	−0.696038
$698$	−1.68551e8	−0.0187602
$699$	0	0
$700$	2.42152e9	0.266837
$701$	−1.31260e10	−1.43919	−0.719595	−	0.694394i	$-0.755672\pi$
$701$	−1.31260e10	−1.43919	−0.719595	+	0.694394i	$0.755672\pi$
$702$	0	0
$703$	−2.51643e9	−0.273175
$704$	2.33733e8	0.0252474
$705$	0	0
$706$	5.24618e9	0.561082
$707$	1.16435e10	1.23912
$708$	0	0
$709$	1.38490e10	1.45934	0.729668	−	0.683801i	$-0.239675\pi$
$709$	1.38490e10	1.45934	0.729668	+	0.683801i	$0.239675\pi$
$710$	−3.63437e8	−0.0381087
$711$	0	0
$712$	8.23647e9	0.855187
$713$	2.64944e9	0.273741
$714$	0	0
$715$	6.28668e8	0.0643206
$716$	−4.44767e9	−0.452832
$717$	0	0
$718$	2.17461e9	0.219253
$719$	−5.91612e9	−0.593589	−0.296795	−	0.954941i	$-0.595918\pi$
$719$	−5.91612e9	−0.593589	−0.296795	+	0.954941i	$0.595918\pi$
$720$	0	0
$721$	−2.59298e10	−2.57647
$722$	2.88070e9	0.284851
$723$	0	0
$724$	8.58143e8	0.0840378
$725$	1.05070e9	0.102399
$726$	0	0
$727$	−5.62356e8	−0.0542801	−0.0271401	−	0.999632i	$-0.508640\pi$
$727$	−5.62356e8	−0.0542801	−0.0271401	+	0.999632i	$0.508640\pi$
$728$	8.50120e9	0.816621
$729$	0	0
$730$	4.05129e9	0.385446
$731$	1.55490e9	0.147229
$732$	0	0
$733$	−4.63518e9	−0.434713	−0.217356	−	0.976092i	$-0.569743\pi$
$733$	−4.63518e9	−0.434713	−0.217356	+	0.976092i	$0.569743\pi$
$734$	3.32564e9	0.310412
$735$	0	0
$736$	−2.82201e9	−0.260908
$737$	2.66695e8	0.0245402
$738$	0	0
$739$	8.68070e9	0.791224	0.395612	−	0.918418i	$-0.370533\pi$
$739$	8.68070e9	0.791224	0.395612	+	0.918418i	$0.370533\pi$
$740$	−6.81127e9	−0.617898
$741$	0	0
$742$	−1.37455e9	−0.123523
$743$	7.23578e9	0.647179	0.323589	−	0.946198i	$-0.395110\pi$
$743$	7.23578e9	0.647179	0.323589	+	0.946198i	$0.395110\pi$
$744$	0	0
$745$	7.81006e9	0.692002
$746$	7.11182e9	0.627184
$747$	0	0
$748$	5.38071e8	0.0470093
$749$	9.30355e9	0.809026
$750$	0	0
$751$	−1.53619e10	−1.32344	−0.661722	−	0.749750i	$-0.730174\pi$
$751$	−1.53619e10	−1.32344	−0.661722	+	0.749750i	$0.730174\pi$
$752$	7.89246e9	0.676784
$753$	0	0
$754$	1.72572e9	0.146612
$755$	−7.67053e8	−0.0648651
$756$	0	0
$757$	−1.36460e10	−1.14332	−0.571661	−	0.820490i	$-0.693701\pi$
$757$	−1.36460e10	−1.14332	−0.571661	+	0.820490i	$0.693701\pi$
$758$	4.92434e9	0.410682
$759$	0	0
$760$	−3.67439e9	−0.303625
$761$	−1.05663e10	−0.869112	−0.434556	−	0.900645i	$-0.643095\pi$
$761$	−1.05663e10	−0.869112	−0.434556	+	0.900645i	$0.643095\pi$
$762$	0	0
$763$	−1.83243e10	−1.49345
$764$	−7.89686e9	−0.640660
$765$	0	0
$766$	8.36968e6	0.000672834 0
$767$	1.31181e9	0.104976
$768$	0	0
$769$	−1.72180e10	−1.36534	−0.682668	−	0.730729i	$-0.739180\pi$
$769$	−1.72180e10	−1.36534	−0.682668	+	0.730729i	$0.739180\pi$
$770$	−5.44628e8	−0.0429915
$771$	0	0
$772$	1.66418e10	1.30179
$773$	−1.20767e10	−0.940419	−0.470209	−	0.882555i	$-0.655822\pi$
$773$	−1.20767e10	−0.940419	−0.470209	+	0.882555i	$0.655822\pi$
$774$	0	0
$775$	2.34238e9	0.180759
$776$	−1.71006e9	−0.131369
$777$	0	0
$778$	1.16955e9	0.0890410
$779$	5.32481e9	0.403574
$780$	0	0
$781$	−9.73513e7	−0.00731245
$782$	−1.01043e9	−0.0755587
$783$	0	0
$784$	1.23458e10	0.914983
$785$	−4.63621e9	−0.342074
$786$	0	0
$787$	5.72468e9	0.418639	0.209319	−	0.977847i	$-0.432875\pi$
$787$	5.72468e9	0.418639	0.209319	+	0.977847i	$0.432875\pi$
$788$	−2.72616e9	−0.198476
$789$	0	0
$790$	7.32062e8	0.0528266
$791$	−1.02457e10	−0.736078
$792$	0	0
$793$	−5.84659e9	−0.416338
$794$	−2.99303e9	−0.212197
$795$	0	0
$796$	1.34916e10	0.948129
$797$	1.13444e10	0.793740	0.396870	−	0.917875i	$-0.370096\pi$
$797$	1.13444e10	0.793740	0.396870	+	0.917875i	$0.370096\pi$
$798$	0	0
$799$	1.09703e10	0.760859
$800$	−2.49495e9	−0.172285
$801$	0	0
$802$	1.63850e9	0.112159
$803$	1.08519e9	0.0739608
$804$	0	0
$805$	−7.43943e9	−0.502637
$806$	3.84722e9	0.258806
$807$	0	0
$808$	−7.82984e9	−0.522172
$809$	−2.36003e10	−1.56711	−0.783553	−	0.621325i	$-0.786595\pi$
$809$	−2.36003e10	−1.56711	−0.783553	+	0.621325i	$0.786595\pi$
$810$	0	0
$811$	−5.41251e9	−0.356308	−0.178154	−	0.984003i	$-0.557013\pi$
$811$	−5.41251e9	−0.356308	−0.178154	+	0.984003i	$0.557013\pi$
$812$	1.08748e10	0.712810
$813$	0	0
$814$	2.50824e8	0.0162999
$815$	3.39115e9	0.219430
$816$	0	0
$817$	−1.33064e9	−0.0853655
$818$	9.47841e9	0.605479
$819$	0	0
$820$	1.44128e10	0.912850
$821$	−1.82304e9	−0.114973	−0.0574863	−	0.998346i	$-0.518309\pi$
$821$	−1.82304e9	−0.114973	−0.0574863	+	0.998346i	$0.518309\pi$
$822$	0	0
$823$	−1.51277e10	−0.945960	−0.472980	−	0.881073i	$-0.656822\pi$
$823$	−1.51277e10	−0.945960	−0.472980	+	0.881073i	$0.656822\pi$
$824$	1.74369e10	1.08574
$825$	0	0
$826$	−1.13645e9	−0.0701651
$827$	2.65641e10	1.63315	0.816573	−	0.577242i	$-0.195871\pi$
$827$	2.65641e10	1.63315	0.816573	+	0.577242i	$0.195871\pi$
$828$	0	0
$829$	−2.30465e10	−1.40496	−0.702479	−	0.711704i	$-0.747924\pi$
$829$	−2.30465e10	−1.40496	−0.702479	+	0.711704i	$0.747924\pi$
$830$	−3.74878e9	−0.227571
$831$	0	0
$832$	4.63611e9	0.279076
$833$	1.71603e10	1.02865
$834$	0	0
$835$	−2.74317e10	−1.63061
$836$	−4.60465e8	−0.0272568
$837$	0	0
$838$	−1.10129e9	−0.0646469
$839$	2.55035e10	1.49084	0.745422	−	0.666592i	$-0.232248\pi$
$839$	2.55035e10	1.49084	0.745422	+	0.666592i	$0.232248\pi$
$840$	0	0
$841$	−1.25313e10	−0.726457
$842$	−5.06330e9	−0.292308
$843$	0	0
$844$	2.11067e10	1.20843
$845$	−6.70932e9	−0.382543
$846$	0	0
$847$	2.72696e10	1.54201
$848$	−2.65368e9	−0.149439
$849$	0	0
$850$	−8.93329e8	−0.0498937
$851$	3.42617e9	0.190571
$852$	0	0
$853$	−2.38277e10	−1.31450	−0.657250	−	0.753673i	$-0.728280\pi$
$853$	−2.38277e10	−1.31450	−0.657250	+	0.753673i	$0.728280\pi$
$854$	5.06502e9	0.278278
$855$	0	0
$856$	−6.25632e9	−0.340927
$857$	−3.88776e9	−0.210992	−0.105496	−	0.994420i	$-0.533643\pi$
$857$	−3.88776e9	−0.210992	−0.105496	+	0.994420i	$0.533643\pi$
$858$	0	0
$859$	1.22415e10	0.658961	0.329480	−	0.944162i	$-0.393126\pi$
$859$	1.22415e10	0.658961	0.329480	+	0.944162i	$0.393126\pi$
$860$	−3.60166e9	−0.193089
$861$	0	0
$862$	−6.01046e9	−0.319619
$863$	−1.45480e10	−0.770489	−0.385244	−	0.922815i	$-0.625883\pi$
$863$	−1.45480e10	−0.770489	−0.385244	+	0.922815i	$0.625883\pi$
$864$	0	0
$865$	−2.95818e10	−1.55406
$866$	−1.19969e10	−0.627704
$867$	0	0
$868$	2.42436e10	1.25828
$869$	1.96092e8	0.0101366
$870$	0	0
$871$	5.28991e9	0.271259
$872$	1.23225e10	0.629347
$873$	0	0
$874$	8.64699e8	0.0438102
$875$	2.70165e10	1.36333
$876$	0	0
$877$	−1.79027e10	−0.896231	−0.448115	−	0.893976i	$-0.647905\pi$
$877$	−1.79027e10	−0.896231	−0.448115	+	0.893976i	$0.647905\pi$
$878$	−4.57286e9	−0.228012
$879$	0	0
$880$	−1.05145e9	−0.0520115
$881$	8.23369e9	0.405675	0.202838	−	0.979212i	$-0.434984\pi$
$881$	8.23369e9	0.405675	0.202838	+	0.979212i	$0.434984\pi$
$882$	0	0
$883$	−1.02587e10	−0.501452	−0.250726	−	0.968058i	$-0.580669\pi$
$883$	−1.02587e10	−0.501452	−0.250726	+	0.968058i	$0.580669\pi$
$884$	1.06727e10	0.519625
$885$	0	0
$886$	−5.50511e9	−0.265918
$887$	−8.18874e9	−0.393989	−0.196995	−	0.980405i	$-0.563118\pi$
$887$	−8.18874e9	−0.393989	−0.196995	+	0.980405i	$0.563118\pi$
$888$	0	0
$889$	−4.38442e10	−2.09293
$890$	−1.04663e10	−0.497656
$891$	0	0
$892$	−6.30341e9	−0.297371
$893$	−9.38804e9	−0.441159
$894$	0	0
$895$	1.20806e10	0.563257
$896$	−3.33891e10	−1.55070
$897$	0	0
$898$	−3.76261e9	−0.173389
$899$	1.05193e10	0.482869
$900$	0	0
$901$	−3.68854e9	−0.168003
$902$	−5.30749e8	−0.0240806
$903$	0	0
$904$	6.88988e9	0.310186
$905$	−2.33085e9	−0.104531
$906$	0	0
$907$	−1.90356e10	−0.847110	−0.423555	−	0.905870i	$-0.639218\pi$
$907$	−1.90356e10	−0.847110	−0.423555	+	0.905870i	$0.639218\pi$
$908$	2.21013e10	0.979755
$909$	0	0
$910$	−1.08027e10	−0.475213
$911$	−1.11471e10	−0.488483	−0.244241	−	0.969714i	$-0.578539\pi$
$911$	−1.11471e10	−0.488483	−0.244241	+	0.969714i	$0.578539\pi$
$912$	0	0
$913$	−1.00416e9	−0.0436672
$914$	−9.63386e9	−0.417339
$915$	0	0
$916$	−4.93259e9	−0.212051
$917$	−4.21853e10	−1.80663
$918$	0	0
$919$	9.09748e9	0.386649	0.193324	−	0.981135i	$-0.438073\pi$
$919$	9.09748e9	0.386649	0.193324	+	0.981135i	$0.438073\pi$
$920$	5.00276e9	0.211813
$921$	0	0
$922$	1.23682e10	0.519697
$923$	−1.93097e9	−0.0808293
$924$	0	0
$925$	3.02909e9	0.125839
$926$	−1.24606e10	−0.515705
$927$	0	0
$928$	−1.12045e10	−0.460231
$929$	2.50717e10	1.02596	0.512978	−	0.858402i	$-0.328542\pi$
$929$	2.50717e10	1.02596	0.512978	+	0.858402i	$0.328542\pi$
$930$	0	0
$931$	−1.46853e10	−0.596428
$932$	1.77240e9	0.0717142
$933$	0	0
$934$	−7.62967e9	−0.306402
$935$	−1.46148e9	−0.0584727
$936$	0	0
$937$	4.32100e9	0.171591	0.0857957	−	0.996313i	$-0.472657\pi$
$937$	4.32100e9	0.171591	0.0857957	+	0.996313i	$0.472657\pi$
$938$	−4.58276e9	−0.181308
$939$	0	0
$940$	−2.54108e10	−0.997863
$941$	−1.72767e10	−0.675922	−0.337961	−	0.941160i	$-0.609737\pi$
$941$	−1.72767e10	−0.675922	−0.337961	+	0.941160i	$0.609737\pi$
$942$	0	0
$943$	−7.24986e9	−0.281539
$944$	−2.19402e9	−0.0848863
$945$	0	0
$946$	1.32631e8	0.00509361
$947$	2.35290e10	0.900284	0.450142	−	0.892957i	$-0.351373\pi$
$947$	2.35290e10	0.900284	0.450142	+	0.892957i	$0.351373\pi$
$948$	0	0
$949$	2.15248e10	0.817537
$950$	7.64484e8	0.0289292
$951$	0	0
$952$	−1.97630e10	−0.742376
$953$	−9.37890e9	−0.351016	−0.175508	−	0.984478i	$-0.556157\pi$
$953$	−9.37890e9	−0.351016	−0.175508	+	0.984478i	$0.556157\pi$
$954$	0	0
$955$	2.14491e10	0.796888
$956$	−1.14979e10	−0.425614
$957$	0	0
$958$	−6.20892e9	−0.228159
$959$	2.33442e10	0.854701
$960$	0	0
$961$	−4.06144e9	−0.147621
$962$	4.97511e9	0.180173
$963$	0	0
$964$	−1.77237e9	−0.0637212
$965$	−4.52018e10	−1.61924
$966$	0	0
$967$	4.13184e10	1.46944	0.734718	−	0.678373i	$-0.237315\pi$
$967$	4.13184e10	1.46944	0.734718	+	0.678373i	$0.237315\pi$
$968$	−1.83379e10	−0.649808
$969$	0	0
$970$	2.17302e9	0.0764473
$971$	−6.75904e9	−0.236929	−0.118464	−	0.992958i	$-0.537797\pi$
$971$	−6.75904e9	−0.236929	−0.118464	+	0.992958i	$0.537797\pi$
$972$	0	0
$973$	7.23021e10	2.51626
$974$	−6.65194e9	−0.230671
$975$	0	0
$976$	9.77845e9	0.336663
$977$	−2.83700e10	−0.973258	−0.486629	−	0.873609i	$-0.661774\pi$
$977$	−2.83700e10	−0.973258	−0.486629	+	0.873609i	$0.661774\pi$
$978$	0	0
$979$	−2.80354e9	−0.0954922
$980$	−3.97489e10	−1.34907
$981$	0	0
$982$	1.19382e10	0.402300
$983$	−5.77572e10	−1.93940	−0.969702	−	0.244289i	$-0.921445\pi$
$983$	−5.77572e10	−1.93940	−0.969702	+	0.244289i	$0.921445\pi$
$984$	0	0
$985$	7.40466e9	0.246876
$986$	−4.01184e9	−0.133283
$987$	0	0
$988$	−9.13334e9	−0.301287
$989$	1.81169e9	0.0595522
$990$	0	0
$991$	−2.44660e10	−0.798554	−0.399277	−	0.916830i	$-0.630739\pi$
$991$	−2.44660e10	−0.798554	−0.399277	+	0.916830i	$0.630739\pi$
$992$	−2.49787e10	−0.812418
$993$	0	0
$994$	1.67284e9	0.0540258
$995$	−3.66453e10	−1.17933
$996$	0	0
$997$	−3.98637e10	−1.27393	−0.636964	−	0.770894i	$-0.719810\pi$
$997$	−3.98637e10	−1.27393	−0.636964	+	0.770894i	$0.719810\pi$
$998$	8.64583e9	0.275327
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.9		18
3.2	odd	2		177.8.a.d.1.10	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.10	✓	18	3.2	odd	2
531.8.a.e.1.9		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)

\(f(q)\)	\(=\)	\(q-3.93322 q^{2} -112.530 q^{4} +305.648 q^{5} -1406.85 q^{7} +946.057 q^{8} +O(q^{10})\)	\(q-3.93322 q^{2} -112.530 q^{4} +305.648 q^{5} -1406.85 q^{7} +946.057 q^{8} -1202.18 q^{10} -322.020 q^{11} -6387.28 q^{13} +5533.44 q^{14} +10682.8 q^{16} +14848.7 q^{17} -12707.1 q^{19} -34394.5 q^{20} +1266.58 q^{22} +17301.0 q^{23} +15295.9 q^{25} +25122.6 q^{26} +158312. q^{28} +68692.0 q^{29} +153138. q^{31} -163113. q^{32} -58403.3 q^{34} -430000. q^{35} +198033. q^{37} +49979.8 q^{38} +289161. q^{40} -419043. q^{41} +104716. q^{43} +36236.8 q^{44} -68048.6 q^{46} +738804. q^{47} +1.15568e6 q^{49} -60162.0 q^{50} +718760. q^{52} -248408. q^{53} -98424.9 q^{55} -1.33096e6 q^{56} -270181. q^{58} -205379. q^{59} +915348. q^{61} -602325. q^{62} -725835. q^{64} -1.95226e6 q^{65} -828193. q^{67} -1.67092e6 q^{68} +1.69129e6 q^{70} +302314. q^{71} -3.36995e6 q^{73} -778909. q^{74} +1.42993e6 q^{76} +453033. q^{77} -608944. q^{79} +3.26517e6 q^{80} +1.64819e6 q^{82} +3.11831e6 q^{83} +4.53849e6 q^{85} -411872. q^{86} -304649. q^{88} +8.70611e6 q^{89} +8.98593e6 q^{91} -1.94688e6 q^{92} -2.90588e6 q^{94} -3.88390e6 q^{95} -1.80756e6 q^{97} -4.54552e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 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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