LMFDB - Embedded newform 531.8.a.e.1.3

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			1.3
Root			$17.9457$ 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)

$f(q)$	$=$	$q-18.9457 q^{2} +230.941 q^{4} -260.561 q^{5} +254.948 q^{7} -1950.29 q^{8} +O(q^{10})$	\(q-18.9457 q^{2} +230.941 q^{4} -260.561 q^{5} +254.948 q^{7} -1950.29 q^{8} +4936.51 q^{10} +3600.84 q^{11} +10076.4 q^{13} -4830.18 q^{14} +7389.22 q^{16} -4423.02 q^{17} +16970.9 q^{19} -60174.1 q^{20} -68220.5 q^{22} -63177.4 q^{23} -10233.2 q^{25} -190906. q^{26} +58877.9 q^{28} +27436.7 q^{29} -22223.7 q^{31} +109643. q^{32} +83797.4 q^{34} -66429.5 q^{35} +111541. q^{37} -321525. q^{38} +508168. q^{40} -87545.8 q^{41} -454839. q^{43} +831580. q^{44} +1.19694e6 q^{46} +747408. q^{47} -758544. q^{49} +193875. q^{50} +2.32706e6 q^{52} -1.10581e6 q^{53} -938236. q^{55} -497223. q^{56} -519807. q^{58} -205379. q^{59} -227215. q^{61} +421044. q^{62} -3.02308e6 q^{64} -2.62552e6 q^{65} +1.66699e6 q^{67} -1.02146e6 q^{68} +1.25856e6 q^{70} -1.63889e6 q^{71} -379956. q^{73} -2.11322e6 q^{74} +3.91926e6 q^{76} +918027. q^{77} -1.69232e6 q^{79} -1.92534e6 q^{80} +1.65862e6 q^{82} -3.64673e6 q^{83} +1.15247e6 q^{85} +8.61726e6 q^{86} -7.02267e6 q^{88} +7.90873e6 q^{89} +2.56897e6 q^{91} -1.45902e7 q^{92} -1.41602e7 q^{94} -4.42194e6 q^{95} -905108. q^{97} +1.43712e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−18.9457	−1.67458	−0.837291	−	0.546758i	$-0.815862\pi$
$2$	−18.9457	−1.67458	−0.837291	+	0.546758i	$0.815862\pi$
$3$	0	0
$3$	0	0
$4$	230.941	1.80422
$5$	−260.561	−0.932210	−0.466105	−	0.884729i	$-0.654343\pi$
$5$	−260.561	−0.932210	−0.466105	+	0.884729i	$0.654343\pi$
$6$	0	0
$7$	254.948	0.280937	0.140468	−	0.990085i	$-0.455139\pi$
$7$	254.948	0.280937	0.140468	+	0.990085i	$0.455139\pi$
$8$	−1950.29	−1.34674
$9$	0	0
$10$	4936.51	1.56106
$11$	3600.84	0.815697	0.407849	−	0.913050i	$-0.366279\pi$
$11$	3600.84	0.815697	0.407849	+	0.913050i	$0.366279\pi$
$12$	0	0
$13$	10076.4	1.27205	0.636027	−	0.771667i	$-0.280577\pi$
$13$	10076.4	1.27205	0.636027	+	0.771667i	$0.280577\pi$
$14$	−4830.18	−0.470452
$15$	0	0
$16$	7389.22	0.451002
$17$	−4423.02	−0.218347	−0.109174	−	0.994023i	$-0.534820\pi$
$17$	−4423.02	−0.218347	−0.109174	+	0.994023i	$0.534820\pi$
$18$	0	0
$19$	16970.9	0.567631	0.283815	−	0.958879i	$-0.408400\pi$
$19$	16970.9	0.567631	0.283815	+	0.958879i	$0.408400\pi$
$20$	−60174.1	−1.68192
$21$	0	0
$22$	−68220.5	−1.36595
$23$	−63177.4	−1.08272	−0.541358	−	0.840792i	$-0.682090\pi$
$23$	−63177.4	−1.08272	−0.541358	+	0.840792i	$0.682090\pi$
$24$	0	0
$25$	−10233.2	−0.130984
$26$	−190906.	−2.13016
$27$	0	0
$28$	58877.9	0.506873
$29$	27436.7	0.208900	0.104450	−	0.994530i	$-0.466692\pi$
$29$	27436.7	0.208900	0.104450	+	0.994530i	$0.466692\pi$
$30$	0	0
$31$	−22223.7	−0.133983	−0.0669916	−	0.997754i	$-0.521340\pi$
$31$	−22223.7	−0.133983	−0.0669916	+	0.997754i	$0.521340\pi$
$32$	109643.	0.591500
$33$	0	0
$34$	83797.4	0.365641
$35$	−66429.5	−0.261892
$36$	0	0
$37$	111541.	0.362016	0.181008	−	0.983482i	$-0.442064\pi$
$37$	111541.	0.362016	0.181008	+	0.983482i	$0.442064\pi$
$38$	−321525.	−0.950544
$39$	0	0
$40$	508168.	1.25544
$41$	−87545.8	−0.198377	−0.0991886	−	0.995069i	$-0.531625\pi$
$41$	−87545.8	−0.198377	−0.0991886	+	0.995069i	$0.531625\pi$
$42$	0	0
$43$	−454839.	−0.872405	−0.436202	−	0.899849i	$-0.643677\pi$
$43$	−454839.	−0.872405	−0.436202	+	0.899849i	$0.643677\pi$
$44$	831580.	1.47170
$45$	0	0
$46$	1.19694e6	1.81310
$47$	747408.	1.05006	0.525031	−	0.851083i	$-0.324054\pi$
$47$	747408.	1.05006	0.525031	+	0.851083i	$0.324054\pi$
$48$	0	0
$49$	−758544.	−0.921074
$50$	193875.	0.219344
$51$	0	0
$52$	2.32706e6	2.29507
$53$	−1.10581e6	−1.02027	−0.510136	−	0.860094i	$-0.670405\pi$
$53$	−1.10581e6	−1.02027	−0.510136	+	0.860094i	$0.670405\pi$
$54$	0	0
$55$	−938236.	−0.760401
$56$	−497223.	−0.378349
$57$	0	0
$58$	−519807.	−0.349820
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	−227215.	−0.128169	−0.0640844	−	0.997944i	$-0.520413\pi$
$61$	−227215.	−0.128169	−0.0640844	+	0.997944i	$0.520413\pi$
$62$	421044.	0.224366
$63$	0	0
$64$	−3.02308e6	−1.44152
$65$	−2.62552e6	−1.18582
$66$	0	0
$67$	1.66699e6	0.677128	0.338564	−	0.940943i	$-0.390059\pi$
$67$	1.66699e6	0.677128	0.338564	+	0.940943i	$0.390059\pi$
$68$	−1.02146e6	−0.393948
$69$	0	0
$70$	1.25856e6	0.438560
$71$	−1.63889e6	−0.543432	−0.271716	−	0.962377i	$-0.587591\pi$
$71$	−1.63889e6	−0.543432	−0.271716	+	0.962377i	$0.587591\pi$
$72$	0	0
$73$	−379956.	−0.114315	−0.0571575	−	0.998365i	$-0.518204\pi$
$73$	−379956.	−0.114315	−0.0571575	+	0.998365i	$0.518204\pi$
$74$	−2.11322e6	−0.606225
$75$	0	0
$76$	3.91926e6	1.02413
$77$	918027.	0.229159
$78$	0	0
$79$	−1.69232e6	−0.386179	−0.193090	−	0.981181i	$-0.561851\pi$
$79$	−1.69232e6	−0.386179	−0.193090	+	0.981181i	$0.561851\pi$
$80$	−1.92534e6	−0.420429
$81$	0	0
$82$	1.65862e6	0.332199
$83$	−3.64673e6	−0.700052	−0.350026	−	0.936740i	$-0.613827\pi$
$83$	−3.64673e6	−0.700052	−0.350026	+	0.936740i	$0.613827\pi$
$84$	0	0
$85$	1.15247e6	0.203546
$86$	8.61726e6	1.46091
$87$	0	0
$88$	−7.02267e6	−1.09853
$89$	7.90873e6	1.18916	0.594582	−	0.804035i	$-0.297318\pi$
$89$	7.90873e6	1.18916	0.594582	+	0.804035i	$0.297318\pi$
$90$	0	0
$91$	2.56897e6	0.357367
$92$	−1.45902e7	−1.95346
$93$	0	0
$94$	−1.41602e7	−1.75842
$95$	−4.42194e6	−0.529151
$96$	0	0
$97$	−905108.	−0.100693	−0.0503465	−	0.998732i	$-0.516033\pi$
$97$	−905108.	−0.100693	−0.0503465	+	0.998732i	$0.516033\pi$
$98$	1.43712e7	1.54241
$99$	0	0
$100$	−2.36325e6	−0.236325
$101$	5.51628e6	0.532748	0.266374	−	0.963870i	$-0.414174\pi$
$101$	5.51628e6	0.532748	0.266374	+	0.963870i	$0.414174\pi$
$102$	0	0
$103$	−8.52758e6	−0.768945	−0.384473	−	0.923136i	$-0.625617\pi$
$103$	−8.52758e6	−0.768945	−0.384473	+	0.923136i	$0.625617\pi$
$104$	−1.96520e7	−1.71313
$105$	0	0
$106$	2.09504e7	1.70853
$107$	−6.00654e6	−0.474003	−0.237001	−	0.971509i	$-0.576165\pi$
$107$	−6.00654e6	−0.474003	−0.237001	+	0.971509i	$0.576165\pi$
$108$	0	0
$109$	−4.33355e6	−0.320517	−0.160258	−	0.987075i	$-0.551233\pi$
$109$	−4.33355e6	−0.320517	−0.160258	+	0.987075i	$0.551233\pi$
$110$	1.77756e7	1.27335
$111$	0	0
$112$	1.88387e6	0.126703
$113$	2.30765e7	1.50451	0.752255	−	0.658872i	$-0.228966\pi$
$113$	2.30765e7	1.50451	0.752255	+	0.658872i	$0.228966\pi$
$114$	0	0
$115$	1.64615e7	1.00932
$116$	6.33624e6	0.376902
$117$	0	0
$118$	3.89106e6	0.218012
$119$	−1.12764e6	−0.0613419
$120$	0	0
$121$	−6.52115e6	−0.334638
$122$	4.30475e6	0.214629
$123$	0	0
$124$	−5.13236e6	−0.241736
$125$	2.30227e7	1.05432
$126$	0	0
$127$	8.51811e6	0.369003	0.184502	−	0.982832i	$-0.440933\pi$
$127$	8.51811e6	0.369003	0.184502	+	0.982832i	$0.440933\pi$
$128$	4.32402e7	1.82244
$129$	0	0
$130$	4.97425e7	1.98576
$131$	2.97734e7	1.15712	0.578560	−	0.815640i	$-0.303615\pi$
$131$	2.97734e7	1.15712	0.578560	+	0.815640i	$0.303615\pi$
$132$	0	0
$133$	4.32669e6	0.159469
$134$	−3.15823e7	−1.13391
$135$	0	0
$136$	8.62617e6	0.294057
$137$	−1.89565e7	−0.629849	−0.314924	−	0.949117i	$-0.601979\pi$
$137$	−1.89565e7	−0.629849	−0.314924	+	0.949117i	$0.601979\pi$
$138$	0	0
$139$	2.62226e7	0.828177	0.414088	−	0.910237i	$-0.364100\pi$
$139$	2.62226e7	0.828177	0.414088	+	0.910237i	$0.364100\pi$
$140$	−1.53413e7	−0.472512
$141$	0	0
$142$	3.10500e7	0.910022
$143$	3.62836e7	1.03761
$144$	0	0
$145$	−7.14891e6	−0.194739
$146$	7.19854e6	0.191430
$147$	0	0
$148$	2.57593e7	0.653158
$149$	2.30589e7	0.571066	0.285533	−	0.958369i	$-0.407829\pi$
$149$	2.30589e7	0.571066	0.285533	+	0.958369i	$0.407829\pi$
$150$	0	0
$151$	4.51544e7	1.06728	0.533642	−	0.845710i	$-0.320823\pi$
$151$	4.51544e7	1.06728	0.533642	+	0.845710i	$0.320823\pi$
$152$	−3.30981e7	−0.764451
$153$	0	0
$154$	−1.73927e7	−0.383746
$155$	5.79062e6	0.124900
$156$	0	0
$157$	−3.09695e7	−0.638684	−0.319342	−	0.947640i	$-0.603462\pi$
$157$	−3.09695e7	−0.638684	−0.319342	+	0.947640i	$0.603462\pi$
$158$	3.20623e7	0.646689
$159$	0	0
$160$	−2.85686e7	−0.551402
$161$	−1.61070e7	−0.304175
$162$	0	0
$163$	8.52913e7	1.54258	0.771290	−	0.636483i	$-0.219612\pi$
$163$	8.52913e7	1.54258	0.771290	+	0.636483i	$0.219612\pi$
$164$	−2.02179e7	−0.357917
$165$	0	0
$166$	6.90900e7	1.17229
$167$	4.10631e7	0.682250	0.341125	−	0.940018i	$-0.389192\pi$
$167$	4.10631e7	0.682250	0.341125	+	0.940018i	$0.389192\pi$
$168$	0	0
$169$	3.87862e7	0.618122
$170$	−2.18343e7	−0.340854
$171$	0	0
$172$	−1.05041e8	−1.57401
$173$	−3.80863e7	−0.559252	−0.279626	−	0.960109i	$-0.590210\pi$
$173$	−3.80863e7	−0.559252	−0.279626	+	0.960109i	$0.590210\pi$
$174$	0	0
$175$	−2.60893e6	−0.0367984
$176$	2.66074e7	0.367881
$177$	0	0
$178$	−1.49837e8	−1.99135
$179$	−2.59663e7	−0.338395	−0.169198	−	0.985582i	$-0.554118\pi$
$179$	−2.59663e7	−0.338395	−0.169198	+	0.985582i	$0.554118\pi$
$180$	0	0
$181$	−4.28031e6	−0.0536537	−0.0268269	−	0.999640i	$-0.508540\pi$
$181$	−4.28031e6	−0.0536537	−0.0268269	+	0.999640i	$0.508540\pi$
$182$	−4.86711e7	−0.598440
$183$	0	0
$184$	1.23214e8	1.45814
$185$	−2.90631e7	−0.337475
$186$	0	0
$187$	−1.59266e7	−0.178105
$188$	1.72607e8	1.89455
$189$	0	0
$190$	8.37768e7	0.886107
$191$	−6.15146e6	−0.0638795	−0.0319398	−	0.999490i	$-0.510168\pi$
$191$	−6.15146e6	−0.0638795	−0.0319398	+	0.999490i	$0.510168\pi$
$192$	0	0
$193$	−1.33350e8	−1.33519	−0.667594	−	0.744525i	$-0.732676\pi$
$193$	−1.33350e8	−1.33519	−0.667594	+	0.744525i	$0.732676\pi$
$194$	1.71479e7	0.168619
$195$	0	0
$196$	−1.75179e8	−1.66183
$197$	1.28756e8	1.19987	0.599936	−	0.800048i	$-0.295193\pi$
$197$	1.28756e8	1.19987	0.599936	+	0.800048i	$0.295193\pi$
$198$	0	0
$199$	−1.66821e8	−1.50060	−0.750301	−	0.661096i	$-0.770092\pi$
$199$	−1.66821e8	−1.50060	−0.750301	+	0.661096i	$0.770092\pi$
$200$	1.99576e7	0.176402
$201$	0	0
$202$	−1.04510e8	−0.892130
$203$	6.99493e6	0.0586877
$204$	0	0
$205$	2.28110e7	0.184929
$206$	1.61561e8	1.28766
$207$	0	0
$208$	7.44570e7	0.573699
$209$	6.11093e7	0.463015
$210$	0	0
$211$	2.09952e7	0.153862	0.0769310	−	0.997036i	$-0.475488\pi$
$211$	2.09952e7	0.153862	0.0769310	+	0.997036i	$0.475488\pi$
$212$	−2.55377e8	−1.84080
$213$	0	0
$214$	1.13798e8	0.793757
$215$	1.18513e8	0.813265
$216$	0	0
$217$	−5.66589e6	−0.0376408
$218$	8.21022e7	0.536732
$219$	0	0
$220$	−2.16677e8	−1.37193
$221$	−4.45684e7	−0.277750
$222$	0	0
$223$	−1.74632e8	−1.05453	−0.527263	−	0.849702i	$-0.676782\pi$
$223$	−1.74632e8	−1.05453	−0.527263	+	0.849702i	$0.676782\pi$
$224$	2.79532e7	0.166174
$225$	0	0
$226$	−4.37201e8	−2.51942
$227$	3.02436e8	1.71610	0.858051	−	0.513565i	$-0.171676\pi$
$227$	3.02436e8	1.71610	0.858051	+	0.513565i	$0.171676\pi$
$228$	0	0
$229$	−1.49255e8	−0.821307	−0.410653	−	0.911792i	$-0.634699\pi$
$229$	−1.49255e8	−0.821307	−0.410653	+	0.911792i	$0.634699\pi$
$230$	−3.11876e8	−1.69019
$231$	0	0
$232$	−5.35094e7	−0.281334
$233$	2.80934e8	1.45499	0.727493	−	0.686115i	$-0.240686\pi$
$233$	2.80934e8	1.45499	0.727493	+	0.686115i	$0.240686\pi$
$234$	0	0
$235$	−1.94745e8	−0.978879
$236$	−4.74304e7	−0.234890
$237$	0	0
$238$	2.13640e7	0.102722
$239$	−9.57324e6	−0.0453593	−0.0226796	−	0.999743i	$-0.507220\pi$
$239$	−9.57324e6	−0.0453593	−0.0226796	+	0.999743i	$0.507220\pi$
$240$	0	0
$241$	7.63167e7	0.351204	0.175602	−	0.984461i	$-0.443813\pi$
$241$	7.63167e7	0.351204	0.175602	+	0.984461i	$0.443813\pi$
$242$	1.23548e8	0.560379
$243$	0	0
$244$	−5.24732e7	−0.231245
$245$	1.97647e8	0.858635
$246$	0	0
$247$	1.71006e8	0.722057
$248$	4.33426e7	0.180441
$249$	0	0
$250$	−4.36181e8	−1.76554
$251$	−3.07061e8	−1.22565	−0.612825	−	0.790218i	$-0.709967\pi$
$251$	−3.07061e8	−1.22565	−0.612825	+	0.790218i	$0.709967\pi$
$252$	0	0
$253$	−2.27492e8	−0.883168
$254$	−1.61382e8	−0.617926
$255$	0	0
$256$	−4.32263e8	−1.61031
$257$	−3.52469e8	−1.29525	−0.647626	−	0.761958i	$-0.724238\pi$
$257$	−3.52469e8	−1.29525	−0.647626	+	0.761958i	$0.724238\pi$
$258$	0	0
$259$	2.84371e7	0.101704
$260$	−6.06341e8	−2.13949
$261$	0	0
$262$	−5.64078e8	−1.93769
$263$	−3.39887e8	−1.15210	−0.576048	−	0.817416i	$-0.695406\pi$
$263$	−3.39887e8	−1.15210	−0.576048	+	0.817416i	$0.695406\pi$
$264$	0	0
$265$	2.88131e8	0.951107
$266$	−8.19723e7	−0.267043
$267$	0	0
$268$	3.84975e8	1.22169
$269$	−4.31047e8	−1.35018	−0.675090	−	0.737735i	$-0.735895\pi$
$269$	−4.31047e8	−1.35018	−0.675090	+	0.737735i	$0.735895\pi$
$270$	0	0
$271$	−1.49089e8	−0.455044	−0.227522	−	0.973773i	$-0.573062\pi$
$271$	−1.49089e8	−0.455044	−0.227522	+	0.973773i	$0.573062\pi$
$272$	−3.26827e7	−0.0984752
$273$	0	0
$274$	3.59145e8	1.05473
$275$	−3.68479e7	−0.106844
$276$	0	0
$277$	3.21914e8	0.910041	0.455020	−	0.890481i	$-0.349632\pi$
$277$	3.21914e8	0.910041	0.455020	+	0.890481i	$0.349632\pi$
$278$	−4.96805e8	−1.38685
$279$	0	0
$280$	1.29557e8	0.352701
$281$	−6.09518e8	−1.63876	−0.819378	−	0.573253i	$-0.805681\pi$
$281$	−6.09518e8	−1.63876	−0.819378	+	0.573253i	$0.805681\pi$
$282$	0	0
$283$	−4.38418e8	−1.14984	−0.574918	−	0.818211i	$-0.694966\pi$
$283$	−4.38418e8	−1.14984	−0.574918	+	0.818211i	$0.694966\pi$
$284$	−3.78486e8	−0.980474
$285$	0	0
$286$	−6.87420e8	−1.73756
$287$	−2.23196e7	−0.0557315
$288$	0	0
$289$	−3.90776e8	−0.952324
$290$	1.35441e8	0.326106
$291$	0	0
$292$	−8.77473e7	−0.206250
$293$	4.48586e7	0.104186	0.0520930	−	0.998642i	$-0.483411\pi$
$293$	4.48586e7	0.104186	0.0520930	+	0.998642i	$0.483411\pi$
$294$	0	0
$295$	5.35137e7	0.121363
$296$	−2.17537e8	−0.487541
$297$	0	0
$298$	−4.36868e8	−0.956297
$299$	−6.36604e8	−1.37727
$300$	0	0
$301$	−1.15960e8	−0.245091
$302$	−8.55482e8	−1.78726
$303$	0	0
$304$	1.25401e8	0.256003
$305$	5.92032e7	0.119480
$306$	0	0
$307$	7.78794e7	0.153617	0.0768083	−	0.997046i	$-0.475527\pi$
$307$	7.78794e7	0.153617	0.0768083	+	0.997046i	$0.475527\pi$
$308$	2.12010e8	0.413455
$309$	0	0
$310$	−1.09708e8	−0.209156
$311$	−4.98982e8	−0.940640	−0.470320	−	0.882496i	$-0.655861\pi$
$311$	−4.98982e8	−0.940640	−0.470320	+	0.882496i	$0.655861\pi$
$312$	0	0
$313$	3.57809e8	0.659547	0.329774	−	0.944060i	$-0.393028\pi$
$313$	3.57809e8	0.659547	0.329774	+	0.944060i	$0.393028\pi$
$314$	5.86741e8	1.06953
$315$	0	0
$316$	−3.90827e8	−0.696754
$317$	−1.30653e8	−0.230363	−0.115181	−	0.993344i	$-0.536745\pi$
$317$	−1.30653e8	−0.230363	−0.115181	+	0.993344i	$0.536745\pi$
$318$	0	0
$319$	9.87949e7	0.170399
$320$	7.87696e8	1.34380
$321$	0	0
$322$	3.05158e8	0.509366
$323$	−7.50625e7	−0.123941
$324$	0	0
$325$	−1.03114e8	−0.166619
$326$	−1.61591e9	−2.58318
$327$	0	0
$328$	1.70739e8	0.267162
$329$	1.90550e8	0.295001
$330$	0	0
$331$	9.92186e8	1.50382	0.751909	−	0.659267i	$-0.229133\pi$
$331$	9.92186e8	1.50382	0.751909	+	0.659267i	$0.229133\pi$
$332$	−8.42179e8	−1.26305
$333$	0	0
$334$	−7.77970e8	−1.14248
$335$	−4.34351e8	−0.631225
$336$	0	0
$337$	−2.68966e7	−0.0382818	−0.0191409	−	0.999817i	$-0.506093\pi$
$337$	−2.68966e7	−0.0382818	−0.0191409	+	0.999817i	$0.506093\pi$
$338$	−7.34833e8	−1.03510
$339$	0	0
$340$	2.66151e8	0.367242
$341$	−8.00239e7	−0.109290
$342$	0	0
$343$	−4.03350e8	−0.539701
$344$	8.87067e8	1.17490
$345$	0	0
$346$	7.21573e8	0.936513
$347$	−1.17341e9	−1.50764	−0.753821	−	0.657080i	$-0.771791\pi$
$347$	−1.17341e9	−1.50764	−0.753821	+	0.657080i	$0.771791\pi$
$348$	0	0
$349$	1.13126e9	1.42454	0.712269	−	0.701907i	$-0.247668\pi$
$349$	1.13126e9	1.42454	0.712269	+	0.701907i	$0.247668\pi$
$350$	4.94280e7	0.0616219
$351$	0	0
$352$	3.94805e8	0.482485
$353$	−1.22377e9	−1.48078	−0.740389	−	0.672179i	$-0.765359\pi$
$353$	−1.22377e9	−1.48078	−0.740389	+	0.672179i	$0.765359\pi$
$354$	0	0
$355$	4.27030e8	0.506593
$356$	1.82645e9	2.14552
$357$	0	0
$358$	4.91950e8	0.566670
$359$	6.37571e7	0.0727274	0.0363637	−	0.999339i	$-0.488423\pi$
$359$	6.37571e7	0.0727274	0.0363637	+	0.999339i	$0.488423\pi$
$360$	0	0
$361$	−6.05862e8	−0.677795
$362$	8.10936e7	0.0898476
$363$	0	0
$364$	5.93280e8	0.644770
$365$	9.90015e7	0.106566
$366$	0	0
$367$	1.18246e9	1.24870	0.624348	−	0.781147i	$-0.285365\pi$
$367$	1.18246e9	1.24870	0.624348	+	0.781147i	$0.285365\pi$
$368$	−4.66832e8	−0.488307
$369$	0	0
$370$	5.50622e8	0.565129
$371$	−2.81925e8	−0.286632
$372$	0	0
$373$	5.09652e8	0.508502	0.254251	−	0.967138i	$-0.418171\pi$
$373$	5.09652e8	0.508502	0.254251	+	0.967138i	$0.418171\pi$
$374$	3.01741e8	0.298252
$375$	0	0
$376$	−1.45766e9	−1.41416
$377$	2.76464e8	0.265732
$378$	0	0
$379$	5.82373e8	0.549496	0.274748	−	0.961516i	$-0.411406\pi$
$379$	5.82373e8	0.549496	0.274748	+	0.961516i	$0.411406\pi$
$380$	−1.02121e9	−0.954708
$381$	0	0
$382$	1.16544e8	0.106971
$383$	2.95800e8	0.269031	0.134515	−	0.990912i	$-0.457052\pi$
$383$	2.95800e8	0.269031	0.134515	+	0.990912i	$0.457052\pi$
$384$	0	0
$385$	−2.39202e8	−0.213625
$386$	2.52641e9	2.23588
$387$	0	0
$388$	−2.09026e8	−0.181673
$389$	1.67193e9	1.44011	0.720055	−	0.693917i	$-0.244116\pi$
$389$	1.67193e9	1.44011	0.720055	+	0.693917i	$0.244116\pi$
$390$	0	0
$391$	2.79435e8	0.236408
$392$	1.47938e9	1.24045
$393$	0	0
$394$	−2.43937e9	−2.00928
$395$	4.40953e8	0.360000
$396$	0	0
$397$	3.26565e7	0.0261940	0.0130970	−	0.999914i	$-0.495831\pi$
$397$	3.26565e7	0.0261940	0.0130970	+	0.999914i	$0.495831\pi$
$398$	3.16055e9	2.51288
$399$	0	0
$400$	−7.56150e7	−0.0590742
$401$	−2.13725e7	−0.0165520	−0.00827600	−	0.999966i	$-0.502634\pi$
$401$	−2.13725e7	−0.0165520	−0.00827600	+	0.999966i	$0.502634\pi$
$402$	0	0
$403$	−2.23936e8	−0.170434
$404$	1.27393e9	0.961197
$405$	0	0
$406$	−1.32524e8	−0.0982773
$407$	4.01640e8	0.295295
$408$	0	0
$409$	7.70641e8	0.556955	0.278478	−	0.960443i	$-0.410170\pi$
$409$	7.70641e8	0.556955	0.278478	+	0.960443i	$0.410170\pi$
$410$	−4.32171e8	−0.309679
$411$	0	0
$412$	−1.96937e9	−1.38735
$413$	−5.23610e7	−0.0365749
$414$	0	0
$415$	9.50195e8	0.652596
$416$	1.10481e9	0.752420
$417$	0	0
$418$	−1.15776e9	−0.775356
$419$	−1.32286e9	−0.878547	−0.439274	−	0.898353i	$-0.644764\pi$
$419$	−1.32286e9	−0.878547	−0.439274	+	0.898353i	$0.644764\pi$
$420$	0	0
$421$	−5.11479e8	−0.334073	−0.167036	−	0.985951i	$-0.553420\pi$
$421$	−5.11479e8	−0.334073	−0.167036	+	0.985951i	$0.553420\pi$
$422$	−3.97769e8	−0.257655
$423$	0	0
$424$	2.15665e9	1.37404
$425$	4.52615e7	0.0286001
$426$	0	0
$427$	−5.79280e7	−0.0360073
$428$	−1.38715e9	−0.855208
$429$	0	0
$430$	−2.24532e9	−1.36188
$431$	1.38021e8	0.0830374	0.0415187	−	0.999138i	$-0.486780\pi$
$431$	1.38021e8	0.0830374	0.0415187	+	0.999138i	$0.486780\pi$
$432$	0	0
$433$	−1.42354e9	−0.842680	−0.421340	−	0.906903i	$-0.638440\pi$
$433$	−1.42354e9	−0.842680	−0.421340	+	0.906903i	$0.638440\pi$
$434$	1.07344e8	0.0630326
$435$	0	0
$436$	−1.00079e9	−0.578284
$437$	−1.07217e9	−0.614583
$438$	0	0
$439$	−6.95019e8	−0.392077	−0.196038	−	0.980596i	$-0.562808\pi$
$439$	−6.95019e8	−0.392077	−0.196038	+	0.980596i	$0.562808\pi$
$440$	1.82983e9	1.02406
$441$	0	0
$442$	8.44380e8	0.465115
$443$	−2.52605e9	−1.38047	−0.690237	−	0.723583i	$-0.742494\pi$
$443$	−2.52605e9	−1.38047	−0.690237	+	0.723583i	$0.742494\pi$
$444$	0	0
$445$	−2.06070e9	−1.10855
$446$	3.30853e9	1.76589
$447$	0	0
$448$	−7.70729e8	−0.404976
$449$	−1.92004e9	−1.00103	−0.500517	−	0.865727i	$-0.666857\pi$
$449$	−1.92004e9	−1.00103	−0.500517	+	0.865727i	$0.666857\pi$
$450$	0	0
$451$	−3.15238e8	−0.161816
$452$	5.32930e9	2.71447
$453$	0	0
$454$	−5.72987e9	−2.87375
$455$	−6.69373e8	−0.333141
$456$	0	0
$457$	2.28317e9	1.11900	0.559502	−	0.828829i	$-0.310992\pi$
$457$	2.28317e9	1.11900	0.559502	+	0.828829i	$0.310992\pi$
$458$	2.82775e9	1.37535
$459$	0	0
$460$	3.80164e9	1.82104
$461$	−8.39344e8	−0.399013	−0.199506	−	0.979897i	$-0.563934\pi$
$461$	−8.39344e8	−0.399013	−0.199506	+	0.979897i	$0.563934\pi$
$462$	0	0
$463$	−1.31128e9	−0.613990	−0.306995	−	0.951711i	$-0.599323\pi$
$463$	−1.31128e9	−0.613990	−0.306995	+	0.951711i	$0.599323\pi$
$464$	2.02735e8	0.0942143
$465$	0	0
$466$	−5.32250e9	−2.43649
$467$	−2.02299e9	−0.919145	−0.459572	−	0.888140i	$-0.651997\pi$
$467$	−2.02299e9	−0.919145	−0.459572	+	0.888140i	$0.651997\pi$
$468$	0	0
$469$	4.24996e8	0.190230
$470$	3.68959e9	1.63921
$471$	0	0
$472$	4.00548e8	0.175331
$473$	−1.63780e9	−0.711618
$474$	0	0
$475$	−1.73665e8	−0.0743508
$476$	−2.60419e8	−0.110675
$477$	0	0
$478$	1.81372e8	0.0759578
$479$	−1.74780e9	−0.726635	−0.363318	−	0.931665i	$-0.618356\pi$
$479$	−1.74780e9	−0.726635	−0.363318	+	0.931665i	$0.618356\pi$
$480$	0	0
$481$	1.12393e9	0.460504
$482$	−1.44588e9	−0.588120
$483$	0	0
$484$	−1.50600e9	−0.603762
$485$	2.35836e8	0.0938671
$486$	0	0
$487$	−2.38890e8	−0.0937231	−0.0468616	−	0.998901i	$-0.514922\pi$
$487$	−2.38890e8	−0.0937231	−0.0468616	+	0.998901i	$0.514922\pi$
$488$	4.43134e8	0.172610
$489$	0	0
$490$	−3.74456e9	−1.43785
$491$	−2.15282e9	−0.820772	−0.410386	−	0.911912i	$-0.634606\pi$
$491$	−2.15282e9	−0.820772	−0.410386	+	0.911912i	$0.634606\pi$
$492$	0	0
$493$	−1.21353e8	−0.0456128
$494$	−3.23983e9	−1.20914
$495$	0	0
$496$	−1.64216e8	−0.0604267
$497$	−4.17832e8	−0.152670
$498$	0	0
$499$	9.90364e8	0.356815	0.178407	−	0.983957i	$-0.442906\pi$
$499$	9.90364e8	0.356815	0.178407	+	0.983957i	$0.442906\pi$
$500$	5.31687e9	1.90222
$501$	0	0
$502$	5.81750e9	2.05245
$503$	−1.94133e9	−0.680161	−0.340080	−	0.940396i	$-0.610454\pi$
$503$	−1.94133e9	−0.680161	−0.340080	+	0.940396i	$0.610454\pi$
$504$	0	0
$505$	−1.43733e9	−0.496633
$506$	4.30999e9	1.47894
$507$	0	0
$508$	1.96718e9	0.665764
$509$	−1.30601e9	−0.438970	−0.219485	−	0.975616i	$-0.570438\pi$
$509$	−1.30601e9	−0.438970	−0.219485	+	0.975616i	$0.570438\pi$
$510$	0	0
$511$	−9.68690e7	−0.0321153
$512$	2.65480e9	0.874150
$513$	0	0
$514$	6.67778e9	2.16901
$515$	2.22195e9	0.716818
$516$	0	0
$517$	2.69129e9	0.856533
$518$	−5.38762e8	−0.170311
$519$	0	0
$520$	5.12053e9	1.59699
$521$	−4.39027e9	−1.36006	−0.680031	−	0.733183i	$-0.738034\pi$
$521$	−4.39027e9	−1.36006	−0.680031	+	0.733183i	$0.738034\pi$
$522$	0	0
$523$	3.06969e9	0.938293	0.469146	−	0.883120i	$-0.344562\pi$
$523$	3.06969e9	0.938293	0.469146	+	0.883120i	$0.344562\pi$
$524$	6.87589e9	2.08771
$525$	0	0
$526$	6.43940e9	1.92928
$527$	9.82959e7	0.0292549
$528$	0	0
$529$	5.86560e8	0.172273
$530$	−5.45885e9	−1.59271
$531$	0	0
$532$	9.99209e8	0.287717
$533$	−8.82150e8	−0.252346
$534$	0	0
$535$	1.56507e9	0.441870
$536$	−3.25111e9	−0.911915
$537$	0	0
$538$	8.16650e9	2.26099
$539$	−2.73139e9	−0.751318
$540$	0	0
$541$	−7.85502e7	−0.0213283	−0.0106642	−	0.999943i	$-0.503395\pi$
$541$	−7.85502e7	−0.0213283	−0.0106642	+	0.999943i	$0.503395\pi$
$542$	2.82460e9	0.762009
$543$	0	0
$544$	−4.84952e8	−0.129153
$545$	1.12915e9	0.298789
$546$	0	0
$547$	−6.56046e9	−1.71387	−0.856937	−	0.515422i	$-0.827635\pi$
$547$	−6.56046e9	−1.71387	−0.856937	+	0.515422i	$0.827635\pi$
$548$	−4.37783e9	−1.13639
$549$	0	0
$550$	6.98111e8	0.178918
$551$	4.65623e8	0.118578
$552$	0	0
$553$	−4.31455e8	−0.108492
$554$	−6.09890e9	−1.52394
$555$	0	0
$556$	6.05586e9	1.49422
$557$	5.79150e9	1.42003	0.710015	−	0.704186i	$-0.248688\pi$
$557$	5.79150e9	1.42003	0.710015	+	0.704186i	$0.248688\pi$
$558$	0	0
$559$	−4.58316e9	−1.10975
$560$	−4.90862e8	−0.118114
$561$	0	0
$562$	1.15478e10	2.74423
$563$	−1.04570e9	−0.246961	−0.123480	−	0.992347i	$-0.539406\pi$
$563$	−1.04570e9	−0.246961	−0.123480	+	0.992347i	$0.539406\pi$
$564$	0	0
$565$	−6.01282e9	−1.40252
$566$	8.30614e9	1.92549
$567$	0	0
$568$	3.19631e9	0.731862
$569$	−7.96871e8	−0.181341	−0.0906704	−	0.995881i	$-0.528901\pi$
$569$	−7.96871e8	−0.181341	−0.0906704	+	0.995881i	$0.528901\pi$
$570$	0	0
$571$	1.01872e9	0.228996	0.114498	−	0.993423i	$-0.463474\pi$
$571$	1.01872e9	0.228996	0.114498	+	0.993423i	$0.463474\pi$
$572$	8.37937e9	1.87208
$573$	0	0
$574$	4.22862e8	0.0933269
$575$	6.46504e8	0.141819
$576$	0	0
$577$	−1.71883e9	−0.372492	−0.186246	−	0.982503i	$-0.559632\pi$
$577$	−1.71883e9	−0.372492	−0.186246	+	0.982503i	$0.559632\pi$
$578$	7.40353e9	1.59475
$579$	0	0
$580$	−1.65098e9	−0.351352
$581$	−9.29728e8	−0.196671
$582$	0	0
$583$	−3.98185e9	−0.832233
$584$	7.41023e8	0.153953
$585$	0	0
$586$	−8.49879e8	−0.174468
$587$	4.84773e9	0.989249	0.494624	−	0.869107i	$-0.335306\pi$
$587$	4.84773e9	0.989249	0.494624	+	0.869107i	$0.335306\pi$
$588$	0	0
$589$	−3.77155e8	−0.0760530
$590$	−1.01386e9	−0.203233
$591$	0	0
$592$	8.24199e8	0.163270
$593$	6.50386e9	1.28080	0.640398	−	0.768043i	$-0.278770\pi$
$593$	6.50386e9	1.28080	0.640398	+	0.768043i	$0.278770\pi$
$594$	0	0
$595$	2.93819e8	0.0571835
$596$	5.32524e9	1.03033
$597$	0	0
$598$	1.20609e10	2.30636
$599$	−3.67574e9	−0.698797	−0.349398	−	0.936974i	$-0.613614\pi$
$599$	−3.67574e9	−0.698797	−0.349398	+	0.936974i	$0.613614\pi$
$600$	0	0
$601$	2.39919e9	0.450821	0.225410	−	0.974264i	$-0.427628\pi$
$601$	2.39919e9	0.450821	0.225410	+	0.974264i	$0.427628\pi$
$602$	2.19695e9	0.410425
$603$	0	0
$604$	1.04280e10	1.92562
$605$	1.69916e9	0.311953
$606$	0	0
$607$	−6.08714e9	−1.10472	−0.552361	−	0.833605i	$-0.686273\pi$
$607$	−6.08714e9	−1.10472	−0.552361	+	0.833605i	$0.686273\pi$
$608$	1.86073e9	0.335754
$609$	0	0
$610$	−1.12165e9	−0.200079
$611$	7.53121e9	1.33574
$612$	0	0
$613$	9.32738e9	1.63549	0.817745	−	0.575581i	$-0.195224\pi$
$613$	9.32738e9	1.63549	0.817745	+	0.575581i	$0.195224\pi$
$614$	−1.47548e9	−0.257244
$615$	0	0
$616$	−1.79042e9	−0.308618
$617$	−6.37210e9	−1.09216	−0.546078	−	0.837734i	$-0.683880\pi$
$617$	−6.37210e9	−1.09216	−0.546078	+	0.837734i	$0.683880\pi$
$618$	0	0
$619$	−4.05521e9	−0.687220	−0.343610	−	0.939112i	$-0.611650\pi$
$619$	−4.05521e9	−0.687220	−0.343610	+	0.939112i	$0.611650\pi$
$620$	1.33729e9	0.225348
$621$	0	0
$622$	9.45358e9	1.57518
$623$	2.01632e9	0.334080
$624$	0	0
$625$	−5.19933e9	−0.851859
$626$	−6.77895e9	−1.10447
$627$	0	0
$628$	−7.15213e9	−1.15233
$629$	−4.93347e8	−0.0790452
$630$	0	0
$631$	−8.06865e9	−1.27849	−0.639246	−	0.769002i	$-0.720754\pi$
$631$	−8.06865e9	−1.27849	−0.639246	+	0.769002i	$0.720754\pi$
$632$	3.30052e9	0.520083
$633$	0	0
$634$	2.47532e9	0.385762
$635$	−2.21948e9	−0.343988
$636$	0	0
$637$	−7.64343e9	−1.17166
$638$	−1.87174e9	−0.285347
$639$	0	0
$640$	−1.12667e10	−1.69890
$641$	3.59478e9	0.539100	0.269550	−	0.962986i	$-0.413125\pi$
$641$	3.59478e9	0.539100	0.269550	+	0.962986i	$0.413125\pi$
$642$	0	0
$643$	−7.51354e9	−1.11457	−0.557283	−	0.830322i	$-0.688156\pi$
$643$	−7.51354e9	−1.11457	−0.557283	+	0.830322i	$0.688156\pi$
$644$	−3.71976e9	−0.548800
$645$	0	0
$646$	1.42211e9	0.207549
$647$	−1.51619e9	−0.220084	−0.110042	−	0.993927i	$-0.535099\pi$
$647$	−1.51619e9	−0.220084	−0.110042	+	0.993927i	$0.535099\pi$
$648$	0	0
$649$	−7.39536e8	−0.106195
$650$	1.95357e9	0.279018
$651$	0	0
$652$	1.96972e10	2.78316
$653$	−1.30271e10	−1.83084	−0.915421	−	0.402497i	$-0.868142\pi$
$653$	−1.30271e10	−1.83084	−0.915421	+	0.402497i	$0.868142\pi$
$654$	0	0
$655$	−7.75777e9	−1.07868
$656$	−6.46895e8	−0.0894685
$657$	0	0
$658$	−3.61011e9	−0.494004
$659$	−8.76958e8	−0.119366	−0.0596829	−	0.998217i	$-0.519009\pi$
$659$	−8.76958e8	−0.119366	−0.0596829	+	0.998217i	$0.519009\pi$
$660$	0	0
$661$	−9.86047e9	−1.32798	−0.663991	−	0.747740i	$-0.731139\pi$
$661$	−9.86047e9	−1.32798	−0.663991	+	0.747740i	$0.731139\pi$
$662$	−1.87977e10	−2.51827
$663$	0	0
$664$	7.11218e9	0.942788
$665$	−1.12736e9	−0.148658
$666$	0	0
$667$	−1.73338e9	−0.226179
$668$	9.48314e9	1.23093
$669$	0	0
$670$	8.22910e9	1.05704
$671$	−8.18163e8	−0.104547
$672$	0	0
$673$	5.88788e9	0.744571	0.372286	−	0.928118i	$-0.378574\pi$
$673$	5.88788e9	0.744571	0.372286	+	0.928118i	$0.378574\pi$
$674$	5.09576e8	0.0641061
$675$	0	0
$676$	8.95732e9	1.11523
$677$	−2.79927e9	−0.346724	−0.173362	−	0.984858i	$-0.555463\pi$
$677$	−2.79927e9	−0.346724	−0.173362	+	0.984858i	$0.555463\pi$
$678$	0	0
$679$	−2.30756e8	−0.0282884
$680$	−2.24764e9	−0.274123
$681$	0	0
$682$	1.51611e9	0.183015
$683$	−2.69530e9	−0.323694	−0.161847	−	0.986816i	$-0.551745\pi$
$683$	−2.69530e9	−0.323694	−0.161847	+	0.986816i	$0.551745\pi$
$684$	0	0
$685$	4.93932e9	0.587152
$686$	7.64177e9	0.903773
$687$	0	0
$688$	−3.36090e9	−0.393456
$689$	−1.11427e10	−1.29784
$690$	0	0
$691$	−6.92452e9	−0.798393	−0.399196	−	0.916865i	$-0.630711\pi$
$691$	−6.92452e9	−0.798393	−0.399196	+	0.916865i	$0.630711\pi$
$692$	−8.79568e9	−1.00902
$693$	0	0
$694$	2.22312e10	2.52467
$695$	−6.83256e9	−0.772035
$696$	0	0
$697$	3.87217e8	0.0433151
$698$	−2.14326e10	−2.38551
$699$	0	0
$700$	−6.02507e8	−0.0663925
$701$	1.39420e10	1.52867	0.764333	−	0.644822i	$-0.223068\pi$
$701$	1.39420e10	1.52867	0.764333	+	0.644822i	$0.223068\pi$
$702$	0	0
$703$	1.89294e9	0.205491
$704$	−1.08856e10	−1.17584
$705$	0	0
$706$	2.31853e10	2.47968
$707$	1.40637e9	0.149669
$708$	0	0
$709$	1.10083e10	1.16000	0.579998	−	0.814618i	$-0.303053\pi$
$709$	1.10083e10	1.16000	0.579998	+	0.814618i	$0.303053\pi$
$710$	−8.09040e9	−0.848332
$711$	0	0
$712$	−1.54243e10	−1.60149
$713$	1.40404e9	0.145066
$714$	0	0
$715$	−9.45408e9	−0.967271
$716$	−5.99667e9	−0.610541
$717$	0	0
$718$	−1.20793e9	−0.121788
$719$	−5.92255e9	−0.594234	−0.297117	−	0.954841i	$-0.596025\pi$
$719$	−5.92255e9	−0.594234	−0.297117	+	0.954841i	$0.596025\pi$
$720$	0	0
$721$	−2.17409e9	−0.216025
$722$	1.14785e10	1.13502
$723$	0	0
$724$	−9.88498e8	−0.0968034
$725$	−2.80764e8	−0.0273626
$726$	0	0
$727$	2.95171e9	0.284907	0.142453	−	0.989802i	$-0.454501\pi$
$727$	2.95171e9	0.284907	0.142453	+	0.989802i	$0.454501\pi$
$728$	−5.01024e9	−0.481281
$729$	0	0
$730$	−1.87566e9	−0.178453
$731$	2.01176e9	0.190487
$732$	0	0
$733$	1.40788e9	0.132038	0.0660192	−	0.997818i	$-0.478970\pi$
$733$	1.40788e9	0.132038	0.0660192	+	0.997818i	$0.478970\pi$
$734$	−2.24026e10	−2.09104
$735$	0	0
$736$	−6.92694e9	−0.640427
$737$	6.00255e9	0.552331
$738$	0	0
$739$	1.57441e10	1.43504	0.717518	−	0.696540i	$-0.245278\pi$
$739$	1.57441e10	1.43504	0.717518	+	0.696540i	$0.245278\pi$
$740$	−6.71186e9	−0.608880
$741$	0	0
$742$	5.34127e9	0.479989
$743$	1.31118e10	1.17274	0.586369	−	0.810044i	$-0.300557\pi$
$743$	1.31118e10	1.17274	0.586369	+	0.810044i	$0.300557\pi$
$744$	0	0
$745$	−6.00824e9	−0.532354
$746$	−9.65572e9	−0.851528
$747$	0	0
$748$	−3.67810e9	−0.321342
$749$	−1.53136e9	−0.133165
$750$	0	0
$751$	−1.05574e10	−0.909532	−0.454766	−	0.890611i	$-0.650277\pi$
$751$	−1.05574e10	−0.909532	−0.454766	+	0.890611i	$0.650277\pi$
$752$	5.52276e9	0.473580
$753$	0	0
$754$	−5.23781e9	−0.444990
$755$	−1.17654e10	−0.994934
$756$	0	0
$757$	7.96333e8	0.0667205	0.0333602	−	0.999443i	$-0.489379\pi$
$757$	7.96333e8	0.0667205	0.0333602	+	0.999443i	$0.489379\pi$
$758$	−1.10335e10	−0.920175
$759$	0	0
$760$	8.62405e9	0.712629
$761$	−2.66159e9	−0.218925	−0.109462	−	0.993991i	$-0.534913\pi$
$761$	−2.66159e9	−0.218925	−0.109462	+	0.993991i	$0.534913\pi$
$762$	0	0
$763$	−1.10483e9	−0.0900450
$764$	−1.42062e9	−0.115253
$765$	0	0
$766$	−5.60414e9	−0.450514
$767$	−2.06949e9	−0.165607
$768$	0	0
$769$	3.74953e9	0.297327	0.148664	−	0.988888i	$-0.452503\pi$
$769$	3.74953e9	0.297327	0.148664	+	0.988888i	$0.452503\pi$
$770$	4.53185e9	0.357732
$771$	0	0
$772$	−3.07960e10	−2.40898
$773$	2.17332e10	1.69237	0.846187	−	0.532886i	$-0.178893\pi$
$773$	2.17332e10	1.69237	0.846187	+	0.532886i	$0.178893\pi$
$774$	0	0
$775$	2.27419e8	0.0175497
$776$	1.76522e9	0.135607
$777$	0	0
$778$	−3.16760e10	−2.41158
$779$	−1.48573e9	−0.112605
$780$	0	0
$781$	−5.90137e9	−0.443276
$782$	−5.29411e9	−0.395885
$783$	0	0
$784$	−5.60505e9	−0.415406
$785$	8.06944e9	0.595388
$786$	0	0
$787$	−5.76067e9	−0.421271	−0.210636	−	0.977565i	$-0.567553\pi$
$787$	−5.76067e9	−0.421271	−0.210636	+	0.977565i	$0.567553\pi$
$788$	2.97350e10	2.16484
$789$	0	0
$790$	−8.35418e9	−0.602850
$791$	5.88331e9	0.422672
$792$	0	0
$793$	−2.28952e9	−0.163038
$794$	−6.18701e8	−0.0438641
$795$	0	0
$796$	−3.85258e10	−2.70742
$797$	1.81058e9	0.126682	0.0633409	−	0.997992i	$-0.479824\pi$
$797$	1.81058e9	0.126682	0.0633409	+	0.997992i	$0.479824\pi$
$798$	0	0
$799$	−3.30580e9	−0.229278
$800$	−1.12199e9	−0.0774773
$801$	0	0
$802$	4.04918e8	0.0277177
$803$	−1.36816e9	−0.0932464
$804$	0	0
$805$	4.19684e9	0.283555
$806$	4.24263e9	0.285405
$807$	0	0
$808$	−1.07583e10	−0.717473
$809$	1.38975e10	0.922823	0.461411	−	0.887186i	$-0.347343\pi$
$809$	1.38975e10	0.922823	0.461411	+	0.887186i	$0.347343\pi$
$810$	0	0
$811$	2.46372e10	1.62188	0.810940	−	0.585129i	$-0.198956\pi$
$811$	2.46372e10	1.62188	0.810940	+	0.585129i	$0.198956\pi$
$812$	1.61541e9	0.105886
$813$	0	0
$814$	−7.60936e9	−0.494496
$815$	−2.22235e10	−1.43801
$816$	0	0
$817$	−7.71900e9	−0.495204
$818$	−1.46004e10	−0.932667
$819$	0	0
$820$	5.26798e9	0.333654
$821$	6.82682e8	0.0430544	0.0215272	−	0.999768i	$-0.493147\pi$
$821$	6.82682e8	0.0430544	0.0215272	+	0.999768i	$0.493147\pi$
$822$	0	0
$823$	−2.59117e10	−1.62030	−0.810151	−	0.586222i	$-0.800615\pi$
$823$	−2.59117e10	−1.62030	−0.810151	+	0.586222i	$0.800615\pi$
$824$	1.66312e10	1.03557
$825$	0	0
$826$	9.92018e8	0.0612476
$827$	1.70204e10	1.04641	0.523203	−	0.852208i	$-0.324737\pi$
$827$	1.70204e10	1.04641	0.523203	+	0.852208i	$0.324737\pi$
$828$	0	0
$829$	−1.88349e10	−1.14821	−0.574107	−	0.818780i	$-0.694651\pi$
$829$	−1.88349e10	−1.14821	−0.574107	+	0.818780i	$0.694651\pi$
$830$	−1.80021e10	−1.09283
$831$	0	0
$832$	−3.04619e10	−1.83369
$833$	3.35506e9	0.201114
$834$	0	0
$835$	−1.06994e10	−0.636001
$836$	1.41126e10	0.835383
$837$	0	0
$838$	2.50626e10	1.47120
$839$	−8.65181e9	−0.505755	−0.252877	−	0.967498i	$-0.581377\pi$
$839$	−8.65181e9	−0.505755	−0.252877	+	0.967498i	$0.581377\pi$
$840$	0	0
$841$	−1.64971e10	−0.956361
$842$	9.69035e9	0.559432
$843$	0	0
$844$	4.84865e9	0.277602
$845$	−1.01062e10	−0.576219
$846$	0	0
$847$	−1.66256e9	−0.0940122
$848$	−8.17109e9	−0.460145
$849$	0	0
$850$	−8.57512e8	−0.0478932
$851$	−7.04685e9	−0.391960
$852$	0	0
$853$	1.84081e10	1.01552	0.507758	−	0.861500i	$-0.330474\pi$
$853$	1.84081e10	1.01552	0.507758	+	0.861500i	$0.330474\pi$
$854$	1.09749e9	0.0602973
$855$	0	0
$856$	1.17145e10	0.638359
$857$	−2.23324e10	−1.21200	−0.605999	−	0.795466i	$-0.707226\pi$
$857$	−2.23324e10	−1.21200	−0.605999	+	0.795466i	$0.707226\pi$
$858$	0	0
$859$	1.81305e10	0.975966	0.487983	−	0.872853i	$-0.337733\pi$
$859$	1.81305e10	0.975966	0.487983	+	0.872853i	$0.337733\pi$
$860$	2.73695e10	1.46731
$861$	0	0
$862$	−2.61490e9	−0.139053
$863$	9.25181e8	0.0489992	0.0244996	−	0.999700i	$-0.492201\pi$
$863$	9.25181e8	0.0489992	0.0244996	+	0.999700i	$0.492201\pi$
$864$	0	0
$865$	9.92379e9	0.521340
$866$	2.69700e10	1.41114
$867$	0	0
$868$	−1.30849e9	−0.0679125
$869$	−6.09378e9	−0.315005
$870$	0	0
$871$	1.67973e10	0.861343
$872$	8.45167e9	0.431653
$873$	0	0
$874$	2.03131e10	1.02917
$875$	5.86959e9	0.296196
$876$	0	0
$877$	−9.58624e9	−0.479899	−0.239949	−	0.970785i	$-0.577131\pi$
$877$	−9.58624e9	−0.479899	−0.239949	+	0.970785i	$0.577131\pi$
$878$	1.31676e10	0.656565
$879$	0	0
$880$	−6.93283e9	−0.342942
$881$	−1.89685e9	−0.0934583	−0.0467291	−	0.998908i	$-0.514880\pi$
$881$	−1.89685e9	−0.0934583	−0.0467291	+	0.998908i	$0.514880\pi$
$882$	0	0
$883$	3.26755e10	1.59720	0.798600	−	0.601862i	$-0.205574\pi$
$883$	3.26755e10	1.59720	0.798600	+	0.601862i	$0.205574\pi$
$884$	−1.02927e10	−0.501123
$885$	0	0
$886$	4.78578e10	2.31172
$887$	−2.36089e10	−1.13591	−0.567955	−	0.823060i	$-0.692265\pi$
$887$	−2.36089e10	−1.13591	−0.567955	+	0.823060i	$0.692265\pi$
$888$	0	0
$889$	2.17168e9	0.103667
$890$	3.90415e10	1.85636
$891$	0	0
$892$	−4.03297e10	−1.90260
$893$	1.26841e10	0.596048
$894$	0	0
$895$	6.76579e9	0.315455
$896$	1.10240e10	0.511991
$897$	0	0
$898$	3.63766e10	1.67631
$899$	−6.09744e8	−0.0279891
$900$	0	0
$901$	4.89103e9	0.222774
$902$	5.97241e9	0.270974
$903$	0	0
$904$	−4.50058e10	−2.02618
$905$	1.11528e9	0.0500166
$906$	0	0
$907$	−1.15848e10	−0.515540	−0.257770	−	0.966206i	$-0.582988\pi$
$907$	−1.15848e10	−0.515540	−0.257770	+	0.966206i	$0.582988\pi$
$908$	6.98448e10	3.09623
$909$	0	0
$910$	1.26818e10	0.557872
$911$	−4.24123e10	−1.85856	−0.929281	−	0.369373i	$-0.879573\pi$
$911$	−4.24123e10	−1.85856	−0.929281	+	0.369373i	$0.879573\pi$
$912$	0	0
$913$	−1.31313e10	−0.571031
$914$	−4.32564e10	−1.87386
$915$	0	0
$916$	−3.44691e10	−1.48182
$917$	7.59067e9	0.325078
$918$	0	0
$919$	−1.45194e10	−0.617085	−0.308543	−	0.951211i	$-0.599841\pi$
$919$	−1.45194e10	−0.617085	−0.308543	+	0.951211i	$0.599841\pi$
$920$	−3.21048e10	−1.35929
$921$	0	0
$922$	1.59020e10	0.668180
$923$	−1.65142e10	−0.691275
$924$	0	0
$925$	−1.14141e9	−0.0474184
$926$	2.48431e10	1.02818
$927$	0	0
$928$	3.00823e9	0.123564
$929$	−1.76962e9	−0.0724145	−0.0362072	−	0.999344i	$-0.511528\pi$
$929$	−1.76962e9	−0.0724145	−0.0362072	+	0.999344i	$0.511528\pi$
$930$	0	0
$931$	−1.28731e10	−0.522830
$932$	6.48791e10	2.62512
$933$	0	0
$934$	3.83270e10	1.53918
$935$	4.14984e9	0.166032
$936$	0	0
$937$	2.29343e10	0.910747	0.455373	−	0.890301i	$-0.349506\pi$
$937$	2.29343e10	0.910747	0.455373	+	0.890301i	$0.349506\pi$
$938$	−8.05185e9	−0.318556
$939$	0	0
$940$	−4.49746e10	−1.76612
$941$	1.81317e10	0.709375	0.354687	−	0.934985i	$-0.384587\pi$
$941$	1.81317e10	0.709375	0.354687	+	0.934985i	$0.384587\pi$
$942$	0	0
$943$	5.53091e9	0.214786
$944$	−1.51759e9	−0.0587155
$945$	0	0
$946$	3.10293e10	1.19166
$947$	4.46859e10	1.70980	0.854901	−	0.518791i	$-0.173618\pi$
$947$	4.46859e10	1.70980	0.854901	+	0.518791i	$0.173618\pi$
$948$	0	0
$949$	−3.82860e9	−0.145415
$950$	3.29022e9	0.124507
$951$	0	0
$952$	2.19923e9	0.0826116
$953$	4.37413e10	1.63707	0.818534	−	0.574458i	$-0.194787\pi$
$953$	4.37413e10	1.63707	0.818534	+	0.574458i	$0.194787\pi$
$954$	0	0
$955$	1.60283e9	0.0595491
$956$	−2.21085e9	−0.0818383
$957$	0	0
$958$	3.31133e10	1.21681
$959$	−4.83293e9	−0.176948
$960$	0	0
$961$	−2.70187e10	−0.982049
$962$	−2.12938e10	−0.771151
$963$	0	0
$964$	1.76246e10	0.633652
$965$	3.47458e10	1.24468
$966$	0	0
$967$	−1.87281e10	−0.666042	−0.333021	−	0.942919i	$-0.608068\pi$
$967$	−1.87281e10	−0.666042	−0.333021	+	0.942919i	$0.608068\pi$
$968$	1.27181e10	0.450671
$969$	0	0
$970$	−4.46808e9	−0.157188
$971$	−4.41287e10	−1.54687	−0.773434	−	0.633877i	$-0.781463\pi$
$971$	−4.41287e10	−1.54687	−0.773434	+	0.633877i	$0.781463\pi$
$972$	0	0
$973$	6.68539e9	0.232666
$974$	4.52595e9	0.156947
$975$	0	0
$976$	−1.67894e9	−0.0578044
$977$	−2.33709e10	−0.801762	−0.400881	−	0.916130i	$-0.631296\pi$
$977$	−2.33709e10	−0.801762	−0.400881	+	0.916130i	$0.631296\pi$
$978$	0	0
$979$	2.84780e10	0.969997
$980$	4.56447e10	1.54917
$981$	0	0
$982$	4.07868e10	1.37445
$983$	−1.07329e10	−0.360396	−0.180198	−	0.983630i	$-0.557674\pi$
$983$	−1.07329e10	−0.360396	−0.180198	+	0.983630i	$0.557674\pi$
$984$	0	0
$985$	−3.35487e10	−1.11853
$986$	2.29912e9	0.0763823
$987$	0	0
$988$	3.94922e10	1.30275
$989$	2.87355e10	0.944566
$990$	0	0
$991$	2.64397e10	0.862975	0.431488	−	0.902119i	$-0.357989\pi$
$991$	2.64397e10	0.862975	0.431488	+	0.902119i	$0.357989\pi$
$992$	−2.43667e9	−0.0792511
$993$	0	0
$994$	7.91614e9	0.255659
$995$	4.34671e10	1.39888
$996$	0	0
$997$	−3.85050e10	−1.23051	−0.615253	−	0.788330i	$-0.710946\pi$
$997$	−3.85050e10	−1.23051	−0.615253	+	0.788330i	$0.710946\pi$
$998$	−1.87632e10	−0.597516
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.3		18
3.2	odd	2		177.8.a.d.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.16	✓	18	3.2	odd	2
531.8.a.e.1.3		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)

\(f(q)\)	\(=\)	\(q-18.9457 q^{2} +230.941 q^{4} -260.561 q^{5} +254.948 q^{7} -1950.29 q^{8} +O(q^{10})\)	\(q-18.9457 q^{2} +230.941 q^{4} -260.561 q^{5} +254.948 q^{7} -1950.29 q^{8} +4936.51 q^{10} +3600.84 q^{11} +10076.4 q^{13} -4830.18 q^{14} +7389.22 q^{16} -4423.02 q^{17} +16970.9 q^{19} -60174.1 q^{20} -68220.5 q^{22} -63177.4 q^{23} -10233.2 q^{25} -190906. q^{26} +58877.9 q^{28} +27436.7 q^{29} -22223.7 q^{31} +109643. q^{32} +83797.4 q^{34} -66429.5 q^{35} +111541. q^{37} -321525. q^{38} +508168. q^{40} -87545.8 q^{41} -454839. q^{43} +831580. q^{44} +1.19694e6 q^{46} +747408. q^{47} -758544. q^{49} +193875. q^{50} +2.32706e6 q^{52} -1.10581e6 q^{53} -938236. q^{55} -497223. q^{56} -519807. q^{58} -205379. q^{59} -227215. q^{61} +421044. q^{62} -3.02308e6 q^{64} -2.62552e6 q^{65} +1.66699e6 q^{67} -1.02146e6 q^{68} +1.25856e6 q^{70} -1.63889e6 q^{71} -379956. q^{73} -2.11322e6 q^{74} +3.91926e6 q^{76} +918027. q^{77} -1.69232e6 q^{79} -1.92534e6 q^{80} +1.65862e6 q^{82} -3.64673e6 q^{83} +1.15247e6 q^{85} +8.61726e6 q^{86} -7.02267e6 q^{88} +7.90873e6 q^{89} +2.56897e6 q^{91} -1.45902e7 q^{92} -1.41602e7 q^{94} -4.42194e6 q^{95} -905108. q^{97} +1.43712e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 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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