LMFDB - Embedded newform 531.8.a.e.1.7

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			1.7
Root			$8.74396$ 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)

$f(q)$	$=$	$q-9.74396 q^{2} -33.0553 q^{4} -321.882 q^{5} -939.921 q^{7} +1569.32 q^{8} +O(q^{10})$	\(q-9.74396 q^{2} -33.0553 q^{4} -321.882 q^{5} -939.921 q^{7} +1569.32 q^{8} +3136.41 q^{10} -4998.14 q^{11} -1233.64 q^{13} +9158.55 q^{14} -11060.3 q^{16} -9783.69 q^{17} +19322.1 q^{19} +10639.9 q^{20} +48701.6 q^{22} -79805.1 q^{23} +25483.2 q^{25} +12020.5 q^{26} +31069.4 q^{28} -144604. q^{29} +148320. q^{31} -93101.7 q^{32} +95331.8 q^{34} +302544. q^{35} +203959. q^{37} -188274. q^{38} -505135. q^{40} +393071. q^{41} +24364.6 q^{43} +165215. q^{44} +777617. q^{46} +18634.3 q^{47} +59908.4 q^{49} -248307. q^{50} +40778.3 q^{52} +228261. q^{53} +1.60881e6 q^{55} -1.47503e6 q^{56} +1.40901e6 q^{58} -205379. q^{59} +2.54932e6 q^{61} -1.44523e6 q^{62} +2.32289e6 q^{64} +397086. q^{65} +2.19172e6 q^{67} +323403. q^{68} -2.94797e6 q^{70} +3.19514e6 q^{71} +2.62084e6 q^{73} -1.98737e6 q^{74} -638699. q^{76} +4.69785e6 q^{77} -2.93367e6 q^{79} +3.56010e6 q^{80} -3.83007e6 q^{82} +5.63931e6 q^{83} +3.14920e6 q^{85} -237407. q^{86} -7.84365e6 q^{88} -8.11136e6 q^{89} +1.15952e6 q^{91} +2.63798e6 q^{92} -181571. q^{94} -6.21944e6 q^{95} +6.86976e6 q^{97} -583745. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.74396	−0.861252	−0.430626	−	0.902530i	$-0.641707\pi$
$2$	−9.74396	−0.861252	−0.430626	+	0.902530i	$0.641707\pi$
$3$	0	0
$3$	0	0
$4$	−33.0553	−0.258245
$5$	−321.882	−1.15160	−0.575800	−	0.817590i	$-0.695309\pi$
$5$	−321.882	−1.15160	−0.575800	+	0.817590i	$0.695309\pi$
$6$	0	0
$7$	−939.921	−1.03573	−0.517867	−	0.855461i	$-0.673274\pi$
$7$	−939.921	−1.03573	−0.517867	+	0.855461i	$0.673274\pi$
$8$	1569.32	1.08367
$9$	0	0
$10$	3136.41	0.991819
$11$	−4998.14	−1.13223	−0.566114	−	0.824327i	$-0.691554\pi$
$11$	−4998.14	−1.13223	−0.566114	+	0.824327i	$0.691554\pi$
$12$	0	0
$13$	−1233.64	−0.155735	−0.0778674	−	0.996964i	$-0.524811\pi$
$13$	−1233.64	−0.155735	−0.0778674	+	0.996964i	$0.524811\pi$
$14$	9158.55	0.892028
$15$	0	0
$16$	−11060.3	−0.675065
$17$	−9783.69	−0.482983	−0.241491	−	0.970403i	$-0.577637\pi$
$17$	−9783.69	−0.482983	−0.241491	+	0.970403i	$0.577637\pi$
$18$	0	0
$19$	19322.1	0.646275	0.323137	−	0.946352i	$-0.395262\pi$
$19$	19322.1	0.646275	0.323137	+	0.946352i	$0.395262\pi$
$20$	10639.9	0.297395
$21$	0	0
$22$	48701.6	0.975133
$23$	−79805.1	−1.36768	−0.683838	−	0.729634i	$-0.739690\pi$
$23$	−79805.1	−1.36768	−0.683838	+	0.729634i	$0.739690\pi$
$24$	0	0
$25$	25483.2	0.326184
$26$	12020.5	0.134127
$27$	0	0
$28$	31069.4	0.267473
$29$	−144604.	−1.10100	−0.550498	−	0.834836i	$-0.685562\pi$
$29$	−144604.	−1.10100	−0.550498	+	0.834836i	$0.685562\pi$
$30$	0	0
$31$	148320.	0.894200	0.447100	−	0.894484i	$-0.352457\pi$
$31$	148320.	0.894200	0.447100	+	0.894484i	$0.352457\pi$
$32$	−93101.7	−0.502265
$33$	0	0
$34$	95331.8	0.415970
$35$	302544.	1.19275
$36$	0	0
$37$	203959.	0.661969	0.330985	−	0.943636i	$-0.392619\pi$
$37$	203959.	0.661969	0.330985	+	0.943636i	$0.392619\pi$
$38$	−188274.	−0.556605
$39$	0	0
$40$	−505135.	−1.24795
$41$	393071.	0.890693	0.445346	−	0.895358i	$-0.353081\pi$
$41$	393071.	0.890693	0.445346	+	0.895358i	$0.353081\pi$
$42$	0	0
$43$	24364.6	0.0467325	0.0233663	−	0.999727i	$-0.492562\pi$
$43$	24364.6	0.0467325	0.0233663	+	0.999727i	$0.492562\pi$
$44$	165215.	0.292392
$45$	0	0
$46$	777617.	1.17791
$47$	18634.3	0.0261800	0.0130900	−	0.999914i	$-0.495833\pi$
$47$	18634.3	0.0261800	0.0130900	+	0.999914i	$0.495833\pi$
$48$	0	0
$49$	59908.4	0.0727448
$50$	−248307.	−0.280927
$51$	0	0
$52$	40778.3	0.0402177
$53$	228261.	0.210603	0.105302	−	0.994440i	$-0.466419\pi$
$53$	228261.	0.210603	0.105302	+	0.994440i	$0.466419\pi$
$54$	0	0
$55$	1.60881e6	1.30387
$56$	−1.47503e6	−1.12239
$57$	0	0
$58$	1.40901e6	0.948235
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	2.54932e6	1.43804	0.719019	−	0.694991i	$-0.244592\pi$
$61$	2.54932e6	1.43804	0.719019	+	0.694991i	$0.244592\pi$
$62$	−1.44523e6	−0.770132
$63$	0	0
$64$	2.32289e6	1.10764
$65$	397086.	0.179344
$66$	0	0
$67$	2.19172e6	0.890271	0.445136	−	0.895463i	$-0.353155\pi$
$67$	2.19172e6	0.890271	0.445136	+	0.895463i	$0.353155\pi$
$68$	323403.	0.124728
$69$	0	0
$70$	−2.94797e6	−1.02726
$71$	3.19514e6	1.05946	0.529732	−	0.848165i	$-0.322293\pi$
$71$	3.19514e6	1.05946	0.529732	+	0.848165i	$0.322293\pi$
$72$	0	0
$73$	2.62084e6	0.788516	0.394258	−	0.919000i	$-0.371002\pi$
$73$	2.62084e6	0.788516	0.394258	+	0.919000i	$0.371002\pi$
$74$	−1.98737e6	−0.570122
$75$	0	0
$76$	−638699.	−0.166897
$77$	4.69785e6	1.17269
$78$	0	0
$79$	−2.93367e6	−0.669448	−0.334724	−	0.942316i	$-0.608643\pi$
$79$	−2.93367e6	−0.669448	−0.334724	+	0.942316i	$0.608643\pi$
$80$	3.56010e6	0.777405
$81$	0	0
$82$	−3.83007e6	−0.767111
$83$	5.63931e6	1.08256	0.541280	−	0.840842i	$-0.317940\pi$
$83$	5.63931e6	1.08256	0.541280	+	0.840842i	$0.317940\pi$
$84$	0	0
$85$	3.14920e6	0.556203
$86$	−237407.	−0.0402485
$87$	0	0
$88$	−7.84365e6	−1.22696
$89$	−8.11136e6	−1.21963	−0.609816	−	0.792543i	$-0.708757\pi$
$89$	−8.11136e6	−1.21963	−0.609816	+	0.792543i	$0.708757\pi$
$90$	0	0
$91$	1.15952e6	0.161300
$92$	2.63798e6	0.353195
$93$	0	0
$94$	−181571.	−0.0225476
$95$	−6.21944e6	−0.744250
$96$	0	0
$97$	6.86976e6	0.764259	0.382130	−	0.924109i	$-0.375191\pi$
$97$	6.86976e6	0.764259	0.382130	+	0.924109i	$0.375191\pi$
$98$	−583745.	−0.0626516
$99$	0	0
$100$	−842354.	−0.0842354
$101$	−2.38664e6	−0.230495	−0.115248	−	0.993337i	$-0.536766\pi$
$101$	−2.38664e6	−0.230495	−0.115248	+	0.993337i	$0.536766\pi$
$102$	0	0
$103$	−1.07956e7	−0.973456	−0.486728	−	0.873553i	$-0.661810\pi$
$103$	−1.07956e7	−0.973456	−0.486728	+	0.873553i	$0.661810\pi$
$104$	−1.93597e6	−0.168765
$105$	0	0
$106$	−2.22416e6	−0.181383
$107$	7.34597e6	0.579703	0.289852	−	0.957072i	$-0.406394\pi$
$107$	7.34597e6	0.579703	0.289852	+	0.957072i	$0.406394\pi$
$108$	0	0
$109$	9.08395e6	0.671865	0.335933	−	0.941886i	$-0.390949\pi$
$109$	9.08395e6	0.671865	0.335933	+	0.941886i	$0.390949\pi$
$110$	−1.56762e7	−1.12296
$111$	0	0
$112$	1.03958e7	0.699188
$113$	−2.23282e7	−1.45573	−0.727863	−	0.685722i	$-0.759487\pi$
$113$	−2.23282e7	−1.45573	−0.727863	+	0.685722i	$0.759487\pi$
$114$	0	0
$115$	2.56878e7	1.57502
$116$	4.77992e6	0.284326
$117$	0	0
$118$	2.00120e6	0.112125
$119$	9.19589e6	0.500241
$120$	0	0
$121$	5.49419e6	0.281939
$122$	−2.48405e7	−1.23851
$123$	0	0
$124$	−4.90277e6	−0.230922
$125$	1.69445e7	0.775967
$126$	0	0
$127$	−6.47924e6	−0.280680	−0.140340	−	0.990103i	$-0.544820\pi$
$127$	−6.47924e6	−0.280680	−0.140340	+	0.990103i	$0.544820\pi$
$128$	−1.07171e7	−0.451694
$129$	0	0
$130$	−3.86919e6	−0.154461
$131$	−2.30891e7	−0.897339	−0.448670	−	0.893698i	$-0.648102\pi$
$131$	−2.30891e7	−0.897339	−0.448670	+	0.893698i	$0.648102\pi$
$132$	0	0
$133$	−1.81613e7	−0.669368
$134$	−2.13560e7	−0.766748
$135$	0	0
$136$	−1.53537e7	−0.523392
$137$	4.31217e7	1.43276	0.716381	−	0.697709i	$-0.245797\pi$
$137$	4.31217e7	1.43276	0.716381	+	0.697709i	$0.245797\pi$
$138$	0	0
$139$	1.97420e7	0.623504	0.311752	−	0.950163i	$-0.399084\pi$
$139$	1.97420e7	0.623504	0.311752	+	0.950163i	$0.399084\pi$
$140$	−1.00007e7	−0.308022
$141$	0	0
$142$	−3.11333e7	−0.912465
$143$	6.16589e6	0.176327
$144$	0	0
$145$	4.65453e7	1.26791
$146$	−2.55374e7	−0.679111
$147$	0	0
$148$	−6.74194e6	−0.170950
$149$	5.76420e7	1.42754	0.713768	−	0.700382i	$-0.246987\pi$
$149$	5.76420e7	1.42754	0.713768	+	0.700382i	$0.246987\pi$
$150$	0	0
$151$	3.07490e7	0.726794	0.363397	−	0.931634i	$-0.381617\pi$
$151$	3.07490e7	0.726794	0.363397	+	0.931634i	$0.381617\pi$
$152$	3.03225e7	0.700346
$153$	0	0
$154$	−4.57757e7	−1.00998
$155$	−4.77417e7	−1.02976
$156$	0	0
$157$	5.96166e6	0.122947	0.0614736	−	0.998109i	$-0.480420\pi$
$157$	5.96166e6	0.122947	0.0614736	+	0.998109i	$0.480420\pi$
$158$	2.85856e7	0.576564
$159$	0	0
$160$	2.99678e7	0.578409
$161$	7.50105e7	1.41655
$162$	0	0
$163$	2.08332e7	0.376790	0.188395	−	0.982093i	$-0.439671\pi$
$163$	2.08332e7	0.376790	0.188395	+	0.982093i	$0.439671\pi$
$164$	−1.29931e7	−0.230017
$165$	0	0
$166$	−5.49491e7	−0.932358
$167$	4.41080e7	0.732840	0.366420	−	0.930449i	$-0.380583\pi$
$167$	4.41080e7	0.732840	0.366420	+	0.930449i	$0.380583\pi$
$168$	0	0
$169$	−6.12267e7	−0.975747
$170$	−3.06856e7	−0.479031
$171$	0	0
$172$	−805379.	−0.0120684
$173$	6.63421e7	0.974154	0.487077	−	0.873359i	$-0.338063\pi$
$173$	6.63421e7	0.974154	0.487077	+	0.873359i	$0.338063\pi$
$174$	0	0
$175$	−2.39522e7	−0.337840
$176$	5.52807e7	0.764327
$177$	0	0
$178$	7.90367e7	1.05041
$179$	1.18508e8	1.54441	0.772206	−	0.635372i	$-0.219153\pi$
$179$	1.18508e8	1.54441	0.772206	+	0.635372i	$0.219153\pi$
$180$	0	0
$181$	−5.40992e7	−0.678134	−0.339067	−	0.940762i	$-0.610111\pi$
$181$	−5.40992e7	−0.678134	−0.339067	+	0.940762i	$0.610111\pi$
$182$	−1.12983e7	−0.138920
$183$	0	0
$184$	−1.25239e8	−1.48210
$185$	−6.56509e7	−0.762324
$186$	0	0
$187$	4.89002e7	0.546846
$188$	−615961.	−0.00676084
$189$	0	0
$190$	6.06020e7	0.640987
$191$	−8.73557e7	−0.907140	−0.453570	−	0.891221i	$-0.649850\pi$
$191$	−8.73557e7	−0.907140	−0.453570	+	0.891221i	$0.649850\pi$
$192$	0	0
$193$	−206564.	−0.00206826	−0.00103413	−	0.999999i	$-0.500329\pi$
$193$	−206564.	−0.00206826	−0.00103413	+	0.999999i	$0.500329\pi$
$194$	−6.69387e7	−0.658220
$195$	0	0
$196$	−1.98029e6	−0.0187859
$197$	1.06894e8	0.996142	0.498071	−	0.867136i	$-0.334042\pi$
$197$	1.06894e8	0.996142	0.498071	+	0.867136i	$0.334042\pi$
$198$	0	0
$199$	−7.19278e7	−0.647010	−0.323505	−	0.946226i	$-0.604861\pi$
$199$	−7.19278e7	−0.647010	−0.323505	+	0.946226i	$0.604861\pi$
$200$	3.99911e7	0.353475
$201$	0	0
$202$	2.32553e7	0.198514
$203$	1.35916e8	1.14034
$204$	0	0
$205$	−1.26523e8	−1.02572
$206$	1.05192e8	0.838391
$207$	0	0
$208$	1.36444e7	0.105131
$209$	−9.65745e7	−0.731730
$210$	0	0
$211$	−1.64503e8	−1.20555	−0.602776	−	0.797910i	$-0.705939\pi$
$211$	−1.64503e8	−1.20555	−0.602776	+	0.797910i	$0.705939\pi$
$212$	−7.54523e6	−0.0543872
$213$	0	0
$214$	−7.15788e7	−0.499271
$215$	−7.84252e6	−0.0538172
$216$	0	0
$217$	−1.39409e8	−0.926153
$218$	−8.85137e7	−0.578645
$219$	0	0
$220$	−5.31798e7	−0.336718
$221$	1.20695e7	0.0752172
$222$	0	0
$223$	−6.38980e7	−0.385852	−0.192926	−	0.981213i	$-0.561798\pi$
$223$	−6.38980e7	−0.385852	−0.192926	+	0.981213i	$0.561798\pi$
$224$	8.75082e7	0.520213
$225$	0	0
$226$	2.17565e8	1.25375
$227$	−2.31309e8	−1.31251	−0.656256	−	0.754539i	$-0.727861\pi$
$227$	−2.31309e8	−1.31251	−0.656256	+	0.754539i	$0.727861\pi$
$228$	0	0
$229$	−2.79302e8	−1.53692	−0.768458	−	0.639900i	$-0.778976\pi$
$229$	−2.79302e8	−1.53692	−0.768458	+	0.639900i	$0.778976\pi$
$230$	−2.50301e8	−1.35649
$231$	0	0
$232$	−2.26929e8	−1.19311
$233$	−2.23510e8	−1.15758	−0.578790	−	0.815477i	$-0.696475\pi$
$233$	−2.23510e8	−1.15758	−0.578790	+	0.815477i	$0.696475\pi$
$234$	0	0
$235$	−5.99803e6	−0.0301489
$236$	6.78887e6	0.0336206
$237$	0	0
$238$	−8.96044e7	−0.430834
$239$	−7.42988e7	−0.352038	−0.176019	−	0.984387i	$-0.556322\pi$
$239$	−7.42988e7	−0.352038	−0.176019	+	0.984387i	$0.556322\pi$
$240$	0	0
$241$	−1.59415e8	−0.733615	−0.366808	−	0.930297i	$-0.619549\pi$
$241$	−1.59415e8	−0.733615	−0.366808	+	0.930297i	$0.619549\pi$
$242$	−5.35351e7	−0.242820
$243$	0	0
$244$	−8.42686e7	−0.371365
$245$	−1.92835e7	−0.0837729
$246$	0	0
$247$	−2.38365e7	−0.100647
$248$	2.32761e8	0.969014
$249$	0	0
$250$	−1.65106e8	−0.668303
$251$	1.11214e8	0.443918	0.221959	−	0.975056i	$-0.428755\pi$
$251$	1.11214e8	0.443918	0.221959	+	0.975056i	$0.428755\pi$
$252$	0	0
$253$	3.98877e8	1.54852
$254$	6.31334e7	0.241736
$255$	0	0
$256$	−1.92903e8	−0.718619
$257$	1.27820e8	0.469712	0.234856	−	0.972030i	$-0.424538\pi$
$257$	1.27820e8	0.469712	0.234856	+	0.972030i	$0.424538\pi$
$258$	0	0
$259$	−1.91706e8	−0.685624
$260$	−1.31258e7	−0.0463147
$261$	0	0
$262$	2.24979e8	0.772836
$263$	−3.37653e8	−1.14453	−0.572263	−	0.820070i	$-0.693934\pi$
$263$	−3.37653e8	−1.14453	−0.572263	+	0.820070i	$0.693934\pi$
$264$	0	0
$265$	−7.34730e7	−0.242531
$266$	1.76963e8	0.576495
$267$	0	0
$268$	−7.24479e7	−0.229908
$269$	3.70415e8	1.16026	0.580130	−	0.814524i	$-0.303002\pi$
$269$	3.70415e8	1.16026	0.580130	+	0.814524i	$0.303002\pi$
$270$	0	0
$271$	2.34791e8	0.716619	0.358309	−	0.933603i	$-0.383353\pi$
$271$	2.34791e8	0.716619	0.358309	+	0.933603i	$0.383353\pi$
$272$	1.08210e8	0.326045
$273$	0	0
$274$	−4.20176e8	−1.23397
$275$	−1.27368e8	−0.369315
$276$	0	0
$277$	−1.55266e8	−0.438932	−0.219466	−	0.975620i	$-0.570432\pi$
$277$	−1.55266e8	−0.438932	−0.219466	+	0.975620i	$0.570432\pi$
$278$	−1.92365e8	−0.536994
$279$	0	0
$280$	4.74787e8	1.29254
$281$	1.15239e8	0.309834	0.154917	−	0.987927i	$-0.450489\pi$
$281$	1.15239e8	0.309834	0.154917	+	0.987927i	$0.450489\pi$
$282$	0	0
$283$	4.45082e8	1.16731	0.583657	−	0.812000i	$-0.301621\pi$
$283$	4.45082e8	1.16731	0.583657	+	0.812000i	$0.301621\pi$
$284$	−1.05616e8	−0.273601
$285$	0	0
$286$	−6.00801e7	−0.151862
$287$	−3.69456e8	−0.922521
$288$	0	0
$289$	−3.14618e8	−0.766728
$290$	−4.53535e8	−1.09199
$291$	0	0
$292$	−8.66327e7	−0.203630
$293$	−7.32530e8	−1.70133	−0.850666	−	0.525707i	$-0.823801\pi$
$293$	−7.32530e8	−1.70133	−0.850666	+	0.525707i	$0.823801\pi$
$294$	0	0
$295$	6.61078e7	0.149926
$296$	3.20077e8	0.717353
$297$	0	0
$298$	−5.61661e8	−1.22947
$299$	9.84505e7	0.212995
$300$	0	0
$301$	−2.29008e7	−0.0484024
$302$	−2.99617e8	−0.625953
$303$	0	0
$304$	−2.13708e8	−0.436277
$305$	−8.20581e8	−1.65604
$306$	0	0
$307$	−2.59871e7	−0.0512594	−0.0256297	−	0.999672i	$-0.508159\pi$
$307$	−2.59871e7	−0.0512594	−0.0256297	+	0.999672i	$0.508159\pi$
$308$	−1.55289e8	−0.302840
$309$	0	0
$310$	4.65193e8	0.886884
$311$	−7.71976e8	−1.45527	−0.727633	−	0.685966i	$-0.759380\pi$
$311$	−7.71976e8	−1.45527	−0.727633	+	0.685966i	$0.759380\pi$
$312$	0	0
$313$	−5.17834e8	−0.954521	−0.477261	−	0.878762i	$-0.658370\pi$
$313$	−5.17834e8	−0.954521	−0.477261	+	0.878762i	$0.658370\pi$
$314$	−5.80901e7	−0.105888
$315$	0	0
$316$	9.69735e7	0.172881
$317$	−5.63749e8	−0.993982	−0.496991	−	0.867756i	$-0.665562\pi$
$317$	−5.63749e8	−0.993982	−0.496991	+	0.867756i	$0.665562\pi$
$318$	0	0
$319$	7.22748e8	1.24658
$320$	−7.47698e8	−1.27556
$321$	0	0
$322$	−7.30899e8	−1.22000
$323$	−1.89042e8	−0.312139
$324$	0	0
$325$	−3.14370e7	−0.0507983
$326$	−2.02998e8	−0.324511
$327$	0	0
$328$	6.16853e8	0.965213
$329$	−1.75147e7	−0.0271155
$330$	0	0
$331$	−4.79482e8	−0.726733	−0.363366	−	0.931646i	$-0.618373\pi$
$331$	−4.79482e8	−0.726733	−0.363366	+	0.931646i	$0.618373\pi$
$332$	−1.86409e8	−0.279566
$333$	0	0
$334$	−4.29786e8	−0.631160
$335$	−7.05474e8	−1.02524
$336$	0	0
$337$	3.39194e8	0.482773	0.241387	−	0.970429i	$-0.422398\pi$
$337$	3.39194e8	0.482773	0.241387	+	0.970429i	$0.422398\pi$
$338$	5.96590e8	0.840364
$339$	0	0
$340$	−1.04098e8	−0.143636
$341$	−7.41325e8	−1.01244
$342$	0	0
$343$	7.17756e8	0.960390
$344$	3.82357e7	0.0506424
$345$	0	0
$346$	−6.46434e8	−0.838992
$347$	−9.78350e8	−1.25702	−0.628508	−	0.777803i	$-0.716334\pi$
$347$	−9.78350e8	−1.25702	−0.628508	+	0.777803i	$0.716334\pi$
$348$	0	0
$349$	−2.36450e8	−0.297749	−0.148875	−	0.988856i	$-0.547565\pi$
$349$	−2.36450e8	−0.297749	−0.148875	+	0.988856i	$0.547565\pi$
$350$	2.33389e8	0.290966
$351$	0	0
$352$	4.65335e8	0.568678
$353$	1.61081e8	0.194910	0.0974548	−	0.995240i	$-0.468930\pi$
$353$	1.61081e8	0.194910	0.0974548	+	0.995240i	$0.468930\pi$
$354$	0	0
$355$	−1.02846e9	−1.22008
$356$	2.68124e8	0.314963
$357$	0	0
$358$	−1.15474e9	−1.33013
$359$	1.10346e9	1.25871	0.629355	−	0.777118i	$-0.283319\pi$
$359$	1.10346e9	1.25871	0.629355	+	0.777118i	$0.283319\pi$
$360$	0	0
$361$	−5.20528e8	−0.582329
$362$	5.27140e8	0.584044
$363$	0	0
$364$	−3.83284e7	−0.0416548
$365$	−8.43602e8	−0.908056
$366$	0	0
$367$	−1.44038e8	−0.152105	−0.0760527	−	0.997104i	$-0.524232\pi$
$367$	−1.44038e8	−0.152105	−0.0760527	+	0.997104i	$0.524232\pi$
$368$	8.82665e8	0.923270
$369$	0	0
$370$	6.39700e8	0.656553
$371$	−2.14547e8	−0.218129
$372$	0	0
$373$	1.06553e9	1.06312	0.531561	−	0.847020i	$-0.321606\pi$
$373$	1.06553e9	1.06312	0.531561	+	0.847020i	$0.321606\pi$
$374$	−4.76481e8	−0.470972
$375$	0	0
$376$	2.92430e7	0.0283704
$377$	1.78388e8	0.171463
$378$	0	0
$379$	1.49139e9	1.40719	0.703597	−	0.710600i	$-0.251576\pi$
$379$	1.49139e9	1.40719	0.703597	+	0.710600i	$0.251576\pi$
$380$	2.05586e8	0.192199
$381$	0	0
$382$	8.51190e8	0.781276
$383$	1.49307e9	1.35795	0.678976	−	0.734160i	$-0.262424\pi$
$383$	1.49307e9	1.35795	0.678976	+	0.734160i	$0.262424\pi$
$384$	0	0
$385$	−1.51216e9	−1.35047
$386$	2.01275e6	0.00178129
$387$	0	0
$388$	−2.27082e8	−0.197366
$389$	−7.33205e8	−0.631541	−0.315771	−	0.948836i	$-0.602263\pi$
$389$	−7.33205e8	−0.631541	−0.315771	+	0.948836i	$0.602263\pi$
$390$	0	0
$391$	7.80788e8	0.660563
$392$	9.40153e7	0.0788310
$393$	0	0
$394$	−1.04157e9	−0.857930
$395$	9.44297e8	0.770937
$396$	0	0
$397$	−2.09263e9	−1.67851	−0.839257	−	0.543735i	$-0.817010\pi$
$397$	−2.09263e9	−1.67851	−0.839257	+	0.543735i	$0.817010\pi$
$398$	7.00861e8	0.557239
$399$	0	0
$400$	−2.81851e8	−0.220196
$401$	1.30799e9	1.01298	0.506489	−	0.862247i	$-0.330943\pi$
$401$	1.30799e9	1.01298	0.506489	+	0.862247i	$0.330943\pi$
$402$	0	0
$403$	−1.82973e8	−0.139258
$404$	7.88910e7	0.0595241
$405$	0	0
$406$	−1.32436e9	−0.982120
$407$	−1.01942e9	−0.749499
$408$	0	0
$409$	2.74486e9	1.98376	0.991878	−	0.127193i	$-0.0405967\pi$
$409$	2.74486e9	1.98376	0.991878	+	0.127193i	$0.0405967\pi$
$410$	1.23283e9	0.883406
$411$	0	0
$412$	3.56852e8	0.251390
$413$	1.93040e8	0.134841
$414$	0	0
$415$	−1.81519e9	−1.24668
$416$	1.14854e8	0.0782201
$417$	0	0
$418$	9.41018e8	0.630204
$419$	6.51049e7	0.0432379	0.0216189	−	0.999766i	$-0.493118\pi$
$419$	6.51049e7	0.0432379	0.0216189	+	0.999766i	$0.493118\pi$
$420$	0	0
$421$	1.48843e9	0.972170	0.486085	−	0.873912i	$-0.338425\pi$
$421$	1.48843e9	0.972170	0.486085	+	0.873912i	$0.338425\pi$
$422$	1.60291e9	1.03828
$423$	0	0
$424$	3.58213e8	0.228224
$425$	−2.49319e8	−0.157541
$426$	0	0
$427$	−2.39616e9	−1.48942
$428$	−2.42823e8	−0.149705
$429$	0	0
$430$	7.64172e7	0.0463502
$431$	−1.85011e9	−1.11308	−0.556540	−	0.830821i	$-0.687871\pi$
$431$	−1.85011e9	−1.11308	−0.556540	+	0.830821i	$0.687871\pi$
$432$	0	0
$433$	6.17068e8	0.365280	0.182640	−	0.983180i	$-0.441536\pi$
$433$	6.17068e8	0.365280	0.182640	+	0.983180i	$0.441536\pi$
$434$	1.35840e9	0.797651
$435$	0	0
$436$	−3.00273e8	−0.173506
$437$	−1.54200e9	−0.883894
$438$	0	0
$439$	−7.44414e8	−0.419941	−0.209971	−	0.977708i	$-0.567337\pi$
$439$	−7.44414e8	−0.419941	−0.209971	+	0.977708i	$0.567337\pi$
$440$	2.52473e9	1.41296
$441$	0	0
$442$	−1.17605e8	−0.0647810
$443$	1.54073e9	0.842004	0.421002	−	0.907060i	$-0.361679\pi$
$443$	1.54073e9	0.842004	0.421002	+	0.907060i	$0.361679\pi$
$444$	0	0
$445$	2.61090e9	1.40453
$446$	6.22619e8	0.332316
$447$	0	0
$448$	−2.18334e9	−1.14722
$449$	3.21711e8	0.167727	0.0838636	−	0.996477i	$-0.473274\pi$
$449$	3.21711e8	0.167727	0.0838636	+	0.996477i	$0.473274\pi$
$450$	0	0
$451$	−1.96462e9	−1.00847
$452$	7.38067e8	0.375934
$453$	0	0
$454$	2.25387e9	1.13040
$455$	−3.73229e8	−0.185753
$456$	0	0
$457$	−1.64590e9	−0.806673	−0.403336	−	0.915052i	$-0.632150\pi$
$457$	−1.64590e9	−0.806673	−0.403336	+	0.915052i	$0.632150\pi$
$458$	2.72151e9	1.32367
$459$	0	0
$460$	−8.49119e8	−0.406740
$461$	−1.86228e9	−0.885301	−0.442650	−	0.896694i	$-0.645962\pi$
$461$	−1.86228e9	−0.885301	−0.442650	+	0.896694i	$0.645962\pi$
$462$	0	0
$463$	1.30026e9	0.608833	0.304417	−	0.952539i	$-0.401538\pi$
$463$	1.30026e9	0.608833	0.304417	+	0.952539i	$0.401538\pi$
$464$	1.59935e9	0.743244
$465$	0	0
$466$	2.17787e9	0.996969
$467$	−2.19765e9	−0.998501	−0.499251	−	0.866458i	$-0.666391\pi$
$467$	−2.19765e9	−0.998501	−0.499251	+	0.866458i	$0.666391\pi$
$468$	0	0
$469$	−2.06004e9	−0.922084
$470$	5.84446e7	0.0259658
$471$	0	0
$472$	−3.22305e8	−0.141081
$473$	−1.21777e8	−0.0529118
$474$	0	0
$475$	4.92389e8	0.210805
$476$	−3.03973e8	−0.129185
$477$	0	0
$478$	7.23964e8	0.303193
$479$	4.13730e9	1.72006	0.860029	−	0.510245i	$-0.170445\pi$
$479$	4.13730e9	1.72006	0.860029	+	0.510245i	$0.170445\pi$
$480$	0	0
$481$	−2.51612e8	−0.103092
$482$	1.55333e9	0.631828
$483$	0	0
$484$	−1.81612e8	−0.0728092
$485$	−2.21125e9	−0.880121
$486$	0	0
$487$	5.06252e9	1.98617	0.993083	−	0.117413i	$-0.0374601\pi$
$487$	5.06252e9	1.98617	0.993083	+	0.117413i	$0.0374601\pi$
$488$	4.00069e9	1.55835
$489$	0	0
$490$	1.87897e8	0.0721496
$491$	−5.29752e8	−0.201970	−0.100985	−	0.994888i	$-0.532199\pi$
$491$	−5.29752e8	−0.201970	−0.100985	+	0.994888i	$0.532199\pi$
$492$	0	0
$493$	1.41476e9	0.531762
$494$	2.32262e8	0.0866828
$495$	0	0
$496$	−1.64046e9	−0.603643
$497$	−3.00318e9	−1.09732
$498$	0	0
$499$	1.14827e9	0.413705	0.206852	−	0.978372i	$-0.433678\pi$
$499$	1.14827e9	0.413705	0.206852	+	0.978372i	$0.433678\pi$
$500$	−5.60105e8	−0.200389
$501$	0	0
$502$	−1.08367e9	−0.382326
$503$	−2.33999e9	−0.819835	−0.409918	−	0.912123i	$-0.634443\pi$
$503$	−2.33999e9	−0.819835	−0.409918	+	0.912123i	$0.634443\pi$
$504$	0	0
$505$	7.68216e8	0.265438
$506$	−3.88664e9	−1.33367
$507$	0	0
$508$	2.14173e8	0.0724840
$509$	−3.44383e9	−1.15752	−0.578762	−	0.815497i	$-0.696464\pi$
$509$	−3.44383e9	−1.15752	−0.578762	+	0.815497i	$0.696464\pi$
$510$	0	0
$511$	−2.46338e9	−0.816693
$512$	3.25143e9	1.07061
$513$	0	0
$514$	−1.24547e9	−0.404540
$515$	3.47491e9	1.12103
$516$	0	0
$517$	−9.31365e7	−0.0296417
$518$	1.86797e9	0.590495
$519$	0	0
$520$	6.23153e8	0.194349
$521$	5.23947e9	1.62314	0.811568	−	0.584258i	$-0.198614\pi$
$521$	5.23947e9	1.62314	0.811568	+	0.584258i	$0.198614\pi$
$522$	0	0
$523$	1.85224e9	0.566164	0.283082	−	0.959096i	$-0.408643\pi$
$523$	1.85224e9	0.566164	0.283082	+	0.959096i	$0.408643\pi$
$524$	7.63216e8	0.231733
$525$	0	0
$526$	3.29008e9	0.985725
$527$	−1.45112e9	−0.431883
$528$	0	0
$529$	2.96402e9	0.870536
$530$	7.15918e8	0.208880
$531$	0	0
$532$	6.00326e8	0.172861
$533$	−4.84908e8	−0.138712
$534$	0	0
$535$	−2.36454e9	−0.667587
$536$	3.43949e9	0.964757
$537$	0	0
$538$	−3.60931e9	−0.999277
$539$	−2.99431e8	−0.0823636
$540$	0	0
$541$	7.04253e8	0.191222	0.0956111	−	0.995419i	$-0.469519\pi$
$541$	7.04253e8	0.191222	0.0956111	+	0.995419i	$0.469519\pi$
$542$	−2.28779e9	−0.617190
$543$	0	0
$544$	9.10878e8	0.242585
$545$	−2.92396e9	−0.773721
$546$	0	0
$547$	3.55125e9	0.927738	0.463869	−	0.885904i	$-0.346461\pi$
$547$	3.55125e9	0.927738	0.463869	+	0.885904i	$0.346461\pi$
$548$	−1.42540e9	−0.370003
$549$	0	0
$550$	1.24107e9	0.318073
$551$	−2.79405e9	−0.711546
$552$	0	0
$553$	2.75742e9	0.693370
$554$	1.51291e9	0.378032
$555$	0	0
$556$	−6.52578e8	−0.161017
$557$	1.10300e9	0.270447	0.135223	−	0.990815i	$-0.456825\pi$
$557$	1.10300e9	0.270447	0.135223	+	0.990815i	$0.456825\pi$
$558$	0	0
$559$	−3.00570e7	−0.00727788
$560$	−3.34622e9	−0.805185
$561$	0	0
$562$	−1.12289e9	−0.266845
$563$	−5.07113e8	−0.119764	−0.0598818	−	0.998205i	$-0.519072\pi$
$563$	−5.07113e8	−0.119764	−0.0598818	+	0.998205i	$0.519072\pi$
$564$	0	0
$565$	7.18706e9	1.67642
$566$	−4.33686e9	−1.00535
$567$	0	0
$568$	5.01419e9	1.14810
$569$	6.60666e9	1.50345	0.751726	−	0.659476i	$-0.229222\pi$
$569$	6.60666e9	1.50345	0.751726	+	0.659476i	$0.229222\pi$
$570$	0	0
$571$	3.31524e9	0.745227	0.372613	−	0.927987i	$-0.378462\pi$
$571$	3.31524e9	0.745227	0.372613	+	0.927987i	$0.378462\pi$
$572$	−2.03815e8	−0.0455356
$573$	0	0
$574$	3.59996e9	0.794523
$575$	−2.03369e9	−0.446114
$576$	0	0
$577$	−6.56723e9	−1.42320	−0.711601	−	0.702584i	$-0.752030\pi$
$577$	−6.56723e9	−1.42320	−0.711601	+	0.702584i	$0.752030\pi$
$578$	3.06562e9	0.660346
$579$	0	0
$580$	−1.53857e9	−0.327431
$581$	−5.30050e9	−1.12125
$582$	0	0
$583$	−1.14088e9	−0.238451
$584$	4.11293e9	0.854488
$585$	0	0
$586$	7.13774e9	1.46528
$587$	2.12865e9	0.434380	0.217190	−	0.976129i	$-0.430311\pi$
$587$	2.12865e9	0.434380	0.217190	+	0.976129i	$0.430311\pi$
$588$	0	0
$589$	2.86586e9	0.577899
$590$	−6.44152e8	−0.129124
$591$	0	0
$592$	−2.25585e9	−0.446872
$593$	−6.79461e7	−0.0133805	−0.00669026	−	0.999978i	$-0.502130\pi$
$593$	−6.79461e7	−0.0133805	−0.00669026	+	0.999978i	$0.502130\pi$
$594$	0	0
$595$	−2.95999e9	−0.576078
$596$	−1.90537e9	−0.368653
$597$	0	0
$598$	−9.59297e8	−0.183442
$599$	3.92845e9	0.746840	0.373420	−	0.927662i	$-0.378185\pi$
$599$	3.92845e9	0.746840	0.373420	+	0.927662i	$0.378185\pi$
$600$	0	0
$601$	−5.25955e9	−0.988298	−0.494149	−	0.869377i	$-0.664520\pi$
$601$	−5.25955e9	−0.988298	−0.494149	+	0.869377i	$0.664520\pi$
$602$	2.23144e8	0.0416867
$603$	0	0
$604$	−1.01642e9	−0.187691
$605$	−1.76848e9	−0.324681
$606$	0	0
$607$	−7.11758e9	−1.29173	−0.645865	−	0.763452i	$-0.723503\pi$
$607$	−7.11758e9	−1.29173	−0.645865	+	0.763452i	$0.723503\pi$
$608$	−1.79892e9	−0.324601
$609$	0	0
$610$	7.99571e9	1.42627
$611$	−2.29879e7	−0.00407714
$612$	0	0
$613$	−8.31463e9	−1.45791	−0.728956	−	0.684561i	$-0.759994\pi$
$613$	−8.31463e9	−1.45791	−0.728956	+	0.684561i	$0.759994\pi$
$614$	2.53217e8	0.0441473
$615$	0	0
$616$	7.37241e9	1.27080
$617$	8.10336e9	1.38889	0.694444	−	0.719547i	$-0.255651\pi$
$617$	8.10336e9	1.38889	0.694444	+	0.719547i	$0.255651\pi$
$618$	0	0
$619$	6.25611e8	0.106020	0.0530099	−	0.998594i	$-0.483119\pi$
$619$	6.25611e8	0.106020	0.0530099	+	0.998594i	$0.483119\pi$
$620$	1.57812e9	0.265930
$621$	0	0
$622$	7.52210e9	1.25335
$623$	7.62404e9	1.26321
$624$	0	0
$625$	−7.44500e9	−1.21979
$626$	5.04575e9	0.822083
$627$	0	0
$628$	−1.97065e8	−0.0317504
$629$	−1.99548e9	−0.319719
$630$	0	0
$631$	−5.93874e9	−0.941004	−0.470502	−	0.882399i	$-0.655927\pi$
$631$	−5.93874e9	−0.941004	−0.470502	+	0.882399i	$0.655927\pi$
$632$	−4.60386e9	−0.725458
$633$	0	0
$634$	5.49315e9	0.856070
$635$	2.08555e9	0.323231
$636$	0	0
$637$	−7.39053e7	−0.0113289
$638$	−7.04242e9	−1.07362
$639$	0	0
$640$	3.44966e9	0.520171
$641$	−7.93154e9	−1.18947	−0.594736	−	0.803921i	$-0.702743\pi$
$641$	−7.93154e9	−1.18947	−0.594736	+	0.803921i	$0.702743\pi$
$642$	0	0
$643$	1.20200e9	0.178306	0.0891530	−	0.996018i	$-0.471584\pi$
$643$	1.20200e9	0.178306	0.0891530	+	0.996018i	$0.471584\pi$
$644$	−2.47949e9	−0.365816
$645$	0	0
$646$	1.84201e9	0.268831
$647$	−1.87158e9	−0.271671	−0.135835	−	0.990731i	$-0.543372\pi$
$647$	−1.87158e9	−0.271671	−0.135835	+	0.990731i	$0.543372\pi$
$648$	0	0
$649$	1.02651e9	0.147403
$650$	3.06321e8	0.0437501
$651$	0	0
$652$	−6.88648e8	−0.0973040
$653$	9.58524e9	1.34712	0.673561	−	0.739132i	$-0.264764\pi$
$653$	9.58524e9	1.34712	0.673561	+	0.739132i	$0.264764\pi$
$654$	0	0
$655$	7.43196e9	1.03338
$656$	−4.34747e9	−0.601276
$657$	0	0
$658$	1.70663e8	0.0233533
$659$	1.22998e10	1.67417	0.837087	−	0.547070i	$-0.184257\pi$
$659$	1.22998e10	1.67417	0.837087	+	0.547070i	$0.184257\pi$
$660$	0	0
$661$	1.28794e9	0.173456	0.0867280	−	0.996232i	$-0.472359\pi$
$661$	1.28794e9	0.173456	0.0867280	+	0.996232i	$0.472359\pi$
$662$	4.67206e9	0.625900
$663$	0	0
$664$	8.84985e9	1.17313
$665$	5.84579e9	0.770845
$666$	0	0
$667$	1.15401e10	1.50581
$668$	−1.45800e9	−0.189252
$669$	0	0
$670$	6.87411e9	0.882988
$671$	−1.27419e10	−1.62818
$672$	0	0
$673$	9.85741e8	0.124655	0.0623275	−	0.998056i	$-0.480148\pi$
$673$	9.85741e8	0.124655	0.0623275	+	0.998056i	$0.480148\pi$
$674$	−3.30509e9	−0.415789
$675$	0	0
$676$	2.02387e9	0.251981
$677$	1.00272e10	1.24200	0.620999	−	0.783812i	$-0.286727\pi$
$677$	1.00272e10	1.24200	0.620999	+	0.783812i	$0.286727\pi$
$678$	0	0
$679$	−6.45703e9	−0.791569
$680$	4.94208e9	0.602738
$681$	0	0
$682$	7.22344e9	0.871964
$683$	2.51763e8	0.0302356	0.0151178	−	0.999886i	$-0.495188\pi$
$683$	2.51763e8	0.0302356	0.0151178	+	0.999886i	$0.495188\pi$
$684$	0	0
$685$	−1.38801e10	−1.64997
$686$	−6.99378e9	−0.827138
$687$	0	0
$688$	−2.69479e8	−0.0315475
$689$	−2.81591e8	−0.0327983
$690$	0	0
$691$	1.20798e10	1.39279	0.696396	−	0.717657i	$-0.254786\pi$
$691$	1.20798e10	1.39279	0.696396	+	0.717657i	$0.254786\pi$
$692$	−2.19296e9	−0.251570
$693$	0	0
$694$	9.53300e9	1.08261
$695$	−6.35460e9	−0.718028
$696$	0	0
$697$	−3.84569e9	−0.430189
$698$	2.30396e9	0.256437
$699$	0	0
$700$	7.91746e8	0.0872455
$701$	−1.78453e10	−1.95664	−0.978321	−	0.207096i	$-0.933599\pi$
$701$	−1.78453e10	−1.95664	−0.978321	+	0.207096i	$0.933599\pi$
$702$	0	0
$703$	3.94093e9	0.427814
$704$	−1.16101e10	−1.25410
$705$	0	0
$706$	−1.56957e9	−0.167866
$707$	2.24325e9	0.238732
$708$	0	0
$709$	−6.60811e9	−0.696331	−0.348165	−	0.937433i	$-0.613195\pi$
$709$	−6.60811e9	−0.696331	−0.348165	+	0.937433i	$0.613195\pi$
$710$	1.00213e10	1.05080
$711$	0	0
$712$	−1.27293e10	−1.32167
$713$	−1.18367e10	−1.22298
$714$	0	0
$715$	−1.98469e9	−0.203059
$716$	−3.91733e9	−0.398836
$717$	0	0
$718$	−1.07521e10	−1.08407
$719$	−1.04223e10	−1.04572	−0.522859	−	0.852419i	$-0.675134\pi$
$719$	−1.04223e10	−1.04572	−0.522859	+	0.852419i	$0.675134\pi$
$720$	0	0
$721$	1.01470e10	1.00824
$722$	5.07200e9	0.501532
$723$	0	0
$724$	1.78826e9	0.175124
$725$	−3.68495e9	−0.359128
$726$	0	0
$727$	−1.07207e10	−1.03479	−0.517394	−	0.855747i	$-0.673098\pi$
$727$	−1.07207e10	−1.03479	−0.517394	+	0.855747i	$0.673098\pi$
$728$	1.81966e9	0.174795
$729$	0	0
$730$	8.22002e9	0.782065
$731$	−2.38375e8	−0.0225710
$732$	0	0
$733$	2.61994e9	0.245712	0.122856	−	0.992424i	$-0.460795\pi$
$733$	2.61994e9	0.245712	0.122856	+	0.992424i	$0.460795\pi$
$734$	1.40350e9	0.131001
$735$	0	0
$736$	7.42999e9	0.686935
$737$	−1.09545e10	−1.00799
$738$	0	0
$739$	1.57747e10	1.43783	0.718913	−	0.695100i	$-0.244640\pi$
$739$	1.57747e10	1.43783	0.718913	+	0.695100i	$0.244640\pi$
$740$	2.17011e9	0.196866
$741$	0	0
$742$	2.09054e9	0.187864
$743$	942965.	8.43402e−5 0	4.21701e−5	−	1.00000i	$-0.499987\pi$
$743$	942965.	8.43402e−5 0	4.21701e−5		1.00000i	$0.499987\pi$
$744$	0	0
$745$	−1.85539e10	−1.64395
$746$	−1.03824e10	−0.915617
$747$	0	0
$748$	−1.61641e9	−0.141220
$749$	−6.90463e9	−0.600418
$750$	0	0
$751$	−1.30768e10	−1.12658	−0.563288	−	0.826261i	$-0.690464\pi$
$751$	−1.30768e10	−1.12658	−0.563288	+	0.826261i	$0.690464\pi$
$752$	−2.06100e8	−0.0176732
$753$	0	0
$754$	−1.73821e9	−0.147673
$755$	−9.89755e9	−0.836976
$756$	0	0
$757$	2.15132e10	1.80247	0.901237	−	0.433326i	$-0.142660\pi$
$757$	2.15132e10	1.80247	0.901237	+	0.433326i	$0.142660\pi$
$758$	−1.45320e10	−1.21195
$759$	0	0
$760$	−9.76027e9	−0.806519
$761$	1.50838e10	1.24069	0.620347	−	0.784328i	$-0.286992\pi$
$761$	1.50838e10	1.24069	0.620347	+	0.784328i	$0.286992\pi$
$762$	0	0
$763$	−8.53820e9	−0.695874
$764$	2.88757e9	0.234264
$765$	0	0
$766$	−1.45484e10	−1.16954
$767$	2.53363e8	0.0202749
$768$	0	0
$769$	−2.21876e9	−0.175941	−0.0879707	−	0.996123i	$-0.528038\pi$
$769$	−2.21876e9	−0.175941	−0.0879707	+	0.996123i	$0.528038\pi$
$770$	1.47344e10	1.16309
$771$	0	0
$772$	6.82804e6	0.000534116 0
$773$	8.81943e9	0.686771	0.343386	−	0.939194i	$-0.388426\pi$
$773$	8.81943e9	0.686771	0.343386	+	0.939194i	$0.388426\pi$
$774$	0	0
$775$	3.77967e9	0.291674
$776$	1.07808e10	0.828202
$777$	0	0
$778$	7.14431e9	0.543916
$779$	7.59497e9	0.575632
$780$	0	0
$781$	−1.59697e10	−1.19955
$782$	−7.60796e9	−0.568912
$783$	0	0
$784$	−6.62603e8	−0.0491074
$785$	−1.91895e9	−0.141586
$786$	0	0
$787$	−1.26134e10	−0.922406	−0.461203	−	0.887295i	$-0.652582\pi$
$787$	−1.26134e10	−0.922406	−0.461203	+	0.887295i	$0.652582\pi$
$788$	−3.53341e9	−0.257248
$789$	0	0
$790$	−9.20119e9	−0.663971
$791$	2.09868e10	1.50775
$792$	0	0
$793$	−3.14494e9	−0.223952
$794$	2.03905e10	1.44562
$795$	0	0
$796$	2.37760e9	0.167087
$797$	−2.61316e9	−0.182836	−0.0914179	−	0.995813i	$-0.529140\pi$
$797$	−2.61316e9	−0.182836	−0.0914179	+	0.995813i	$0.529140\pi$
$798$	0	0
$799$	−1.82312e8	−0.0126445
$800$	−2.37253e9	−0.163831
$801$	0	0
$802$	−1.27450e10	−0.872429
$803$	−1.30993e10	−0.892779
$804$	0	0
$805$	−2.41445e10	−1.63130
$806$	1.78289e9	0.119936
$807$	0	0
$808$	−3.74539e9	−0.249780
$809$	2.83771e10	1.88429	0.942147	−	0.335199i	$-0.108804\pi$
$809$	2.83771e10	1.88429	0.942147	+	0.335199i	$0.108804\pi$
$810$	0	0
$811$	−2.32326e10	−1.52941	−0.764705	−	0.644380i	$-0.777115\pi$
$811$	−2.32326e10	−1.52941	−0.764705	+	0.644380i	$0.777115\pi$
$812$	−4.49274e9	−0.294487
$813$	0	0
$814$	9.93315e9	0.645508
$815$	−6.70584e9	−0.433912
$816$	0	0
$817$	4.70775e8	0.0302020
$818$	−2.67458e10	−1.70851
$819$	0	0
$820$	4.18225e9	0.264887
$821$	−1.23630e10	−0.779691	−0.389845	−	0.920880i	$-0.627472\pi$
$821$	−1.23630e10	−0.779691	−0.389845	+	0.920880i	$0.627472\pi$
$822$	0	0
$823$	1.47567e10	0.922763	0.461382	−	0.887202i	$-0.347354\pi$
$823$	1.47567e10	0.922763	0.461382	+	0.887202i	$0.347354\pi$
$824$	−1.69417e10	−1.05490
$825$	0	0
$826$	−1.88097e9	−0.116132
$827$	−7.08701e9	−0.435706	−0.217853	−	0.975982i	$-0.569905\pi$
$827$	−7.08701e9	−0.435706	−0.217853	+	0.975982i	$0.569905\pi$
$828$	0	0
$829$	−5.40146e7	−0.00329284	−0.00164642	−	0.999999i	$-0.500524\pi$
$829$	−5.40146e7	−0.00329284	−0.00164642	+	0.999999i	$0.500524\pi$
$830$	1.76872e10	1.07370
$831$	0	0
$832$	−2.86561e9	−0.172498
$833$	−5.86126e8	−0.0351345
$834$	0	0
$835$	−1.41976e10	−0.843940
$836$	3.19230e9	0.188965
$837$	0	0
$838$	−6.34379e8	−0.0372387
$839$	4.08313e9	0.238686	0.119343	−	0.992853i	$-0.461921\pi$
$839$	4.08313e9	0.238686	0.119343	+	0.992853i	$0.461921\pi$
$840$	0	0
$841$	3.66030e9	0.212193
$842$	−1.45032e10	−0.837283
$843$	0	0
$844$	5.43771e9	0.311328
$845$	1.97078e10	1.12367
$846$	0	0
$847$	−5.16410e9	−0.292013
$848$	−2.52462e9	−0.142171
$849$	0	0
$850$	2.42936e9	0.135683
$851$	−1.62770e10	−0.905359
$852$	0	0
$853$	4.28167e8	0.0236207	0.0118103	−	0.999930i	$-0.496241\pi$
$853$	4.28167e8	0.0236207	0.0118103	+	0.999930i	$0.496241\pi$
$854$	2.33481e10	1.28277
$855$	0	0
$856$	1.15281e10	0.628205
$857$	7.39852e9	0.401524	0.200762	−	0.979640i	$-0.435658\pi$
$857$	7.39852e9	0.401524	0.200762	+	0.979640i	$0.435658\pi$
$858$	0	0
$859$	−2.72977e9	−0.146943	−0.0734717	−	0.997297i	$-0.523408\pi$
$859$	−2.72977e9	−0.146943	−0.0734717	+	0.997297i	$0.523408\pi$
$860$	2.59237e8	0.0138980
$861$	0	0
$862$	1.80274e10	0.958642
$863$	−1.38149e10	−0.731663	−0.365832	−	0.930681i	$-0.619215\pi$
$863$	−1.38149e10	−0.731663	−0.365832	+	0.930681i	$0.619215\pi$
$864$	0	0
$865$	−2.13543e10	−1.12184
$866$	−6.01268e9	−0.314598
$867$	0	0
$868$	4.60822e9	0.239174
$869$	1.46629e10	0.757968
$870$	0	0
$871$	−2.70378e9	−0.138646
$872$	1.42556e10	0.728078
$873$	0	0
$874$	1.50252e10	0.761255
$875$	−1.59265e10	−0.803695
$876$	0	0
$877$	2.96899e10	1.48631	0.743157	−	0.669117i	$-0.233327\pi$
$877$	2.96899e10	1.48631	0.743157	+	0.669117i	$0.233327\pi$
$878$	7.25354e9	0.361675
$879$	0	0
$880$	−1.77939e10	−0.880200
$881$	3.95312e10	1.94771	0.973855	−	0.227169i	$-0.0729469\pi$
$881$	3.95312e10	1.94771	0.973855	+	0.227169i	$0.0729469\pi$
$882$	0	0
$883$	−6.41631e9	−0.313634	−0.156817	−	0.987628i	$-0.550123\pi$
$883$	−6.41631e9	−0.313634	−0.156817	+	0.987628i	$0.550123\pi$
$884$	−3.98962e8	−0.0194244
$885$	0	0
$886$	−1.50128e10	−0.725177
$887$	3.07401e10	1.47901	0.739507	−	0.673149i	$-0.235059\pi$
$887$	3.07401e10	1.47901	0.739507	+	0.673149i	$0.235059\pi$
$888$	0	0
$889$	6.08997e9	0.290710
$890$	−2.54405e10	−1.20965
$891$	0	0
$892$	2.11217e9	0.0996441
$893$	3.60053e8	0.0169195
$894$	0	0
$895$	−3.81457e10	−1.77855
$896$	1.00733e10	0.467835
$897$	0	0
$898$	−3.13474e9	−0.144455
$899$	−2.14476e10	−0.984511
$900$	0	0
$901$	−2.23323e9	−0.101718
$902$	1.91432e10	0.868544
$903$	0	0
$904$	−3.50401e10	−1.57752
$905$	1.74136e10	0.780940
$906$	0	0
$907$	6.93998e9	0.308839	0.154420	−	0.988005i	$-0.450649\pi$
$907$	6.93998e9	0.308839	0.154420	+	0.988005i	$0.450649\pi$
$908$	7.64601e9	0.338949
$909$	0	0
$910$	3.63673e9	0.159980
$911$	−1.43108e10	−0.627118	−0.313559	−	0.949569i	$-0.601521\pi$
$911$	−1.43108e10	−0.627118	−0.313559	+	0.949569i	$0.601521\pi$
$912$	0	0
$913$	−2.81860e10	−1.22570
$914$	1.60376e10	0.694749
$915$	0	0
$916$	9.23243e9	0.396901
$917$	2.17019e10	0.929405
$918$	0	0
$919$	4.66570e10	1.98295	0.991477	−	0.130280i	$-0.0415876\pi$
$919$	4.66570e10	1.98295	0.991477	+	0.130280i	$0.0415876\pi$
$920$	4.03123e10	1.70679
$921$	0	0
$922$	1.81459e10	0.762467
$923$	−3.94165e9	−0.164995
$924$	0	0
$925$	5.19753e9	0.215924
$926$	−1.26697e10	−0.524359
$927$	0	0
$928$	1.34628e10	0.552992
$929$	2.83056e10	1.15829	0.579146	−	0.815224i	$-0.303386\pi$
$929$	2.83056e10	1.15829	0.579146	+	0.815224i	$0.303386\pi$
$930$	0	0
$931$	1.15756e9	0.0470131
$932$	7.38819e9	0.298939
$933$	0	0
$934$	2.14138e10	0.859962
$935$	−1.57401e10	−0.629748
$936$	0	0
$937$	7.53166e9	0.299090	0.149545	−	0.988755i	$-0.452219\pi$
$937$	7.53166e9	0.299090	0.149545	+	0.988755i	$0.452219\pi$
$938$	2.00729e10	0.794147
$939$	0	0
$940$	1.98267e8	0.00778579
$941$	2.80301e10	1.09663	0.548316	−	0.836271i	$-0.315269\pi$
$941$	2.80301e10	1.09663	0.548316	+	0.836271i	$0.315269\pi$
$942$	0	0
$943$	−3.13691e10	−1.21818
$944$	2.27155e9	0.0878860
$945$	0	0
$946$	1.18659e9	0.0455704
$947$	−5.03691e10	−1.92725	−0.963627	−	0.267250i	$-0.913885\pi$
$947$	−5.03691e10	−1.92725	−0.963627	+	0.267250i	$0.913885\pi$
$948$	0	0
$949$	−3.23317e9	−0.122799
$950$	−4.79781e9	−0.181556
$951$	0	0
$952$	1.44313e10	0.542095
$953$	1.09520e10	0.409890	0.204945	−	0.978773i	$-0.434298\pi$
$953$	1.09520e10	0.409890	0.204945	+	0.978773i	$0.434298\pi$
$954$	0	0
$955$	2.81182e10	1.04466
$956$	2.45597e9	0.0909118
$957$	0	0
$958$	−4.03137e10	−1.48140
$959$	−4.05310e10	−1.48396
$960$	0	0
$961$	−5.51371e9	−0.200407
$962$	2.45170e9	0.0887879
$963$	0	0
$964$	5.26950e9	0.189452
$965$	6.64893e7	0.00238181
$966$	0	0
$967$	−1.15786e10	−0.411778	−0.205889	−	0.978575i	$-0.566009\pi$
$967$	−1.15786e10	−0.411778	−0.205889	+	0.978575i	$0.566009\pi$
$968$	8.62211e9	0.305527
$969$	0	0
$970$	2.15464e10	0.758007
$971$	2.02758e10	0.710739	0.355370	−	0.934726i	$-0.384355\pi$
$971$	2.02758e10	0.710739	0.355370	+	0.934726i	$0.384355\pi$
$972$	0	0
$973$	−1.85559e10	−0.645784
$974$	−4.93290e10	−1.71059
$975$	0	0
$976$	−2.81962e10	−0.970769
$977$	1.76622e10	0.605917	0.302958	−	0.953004i	$-0.402026\pi$
$977$	1.76622e10	0.605917	0.302958	+	0.953004i	$0.402026\pi$
$978$	0	0
$979$	4.05417e10	1.38090
$980$	6.37421e8	0.0216339
$981$	0	0
$982$	5.16188e9	0.173947
$983$	−5.25893e10	−1.76588	−0.882938	−	0.469489i	$-0.844438\pi$
$983$	−5.25893e10	−1.76588	−0.882938	+	0.469489i	$0.844438\pi$
$984$	0	0
$985$	−3.44073e10	−1.14716
$986$	−1.37853e10	−0.457981
$987$	0	0
$988$	7.87923e8	0.0259917
$989$	−1.94442e9	−0.0639149
$990$	0	0
$991$	−2.06461e10	−0.673876	−0.336938	−	0.941527i	$-0.609391\pi$
$991$	−2.06461e10	−0.673876	−0.336938	+	0.941527i	$0.609391\pi$
$992$	−1.38089e10	−0.449125
$993$	0	0
$994$	2.92629e10	0.945071
$995$	2.31523e10	0.745097
$996$	0	0
$997$	−2.76691e9	−0.0884224	−0.0442112	−	0.999022i	$-0.514077\pi$
$997$	−2.76691e9	−0.0884224	−0.0442112	+	0.999022i	$0.514077\pi$
$998$	−1.11887e10	−0.356304
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.7		18
3.2	odd	2		177.8.a.d.1.12	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.12	✓	18	3.2	odd	2
531.8.a.e.1.7		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)

\(f(q)\)	\(=\)	\(q-9.74396 q^{2} -33.0553 q^{4} -321.882 q^{5} -939.921 q^{7} +1569.32 q^{8} +O(q^{10})\)	\(q-9.74396 q^{2} -33.0553 q^{4} -321.882 q^{5} -939.921 q^{7} +1569.32 q^{8} +3136.41 q^{10} -4998.14 q^{11} -1233.64 q^{13} +9158.55 q^{14} -11060.3 q^{16} -9783.69 q^{17} +19322.1 q^{19} +10639.9 q^{20} +48701.6 q^{22} -79805.1 q^{23} +25483.2 q^{25} +12020.5 q^{26} +31069.4 q^{28} -144604. q^{29} +148320. q^{31} -93101.7 q^{32} +95331.8 q^{34} +302544. q^{35} +203959. q^{37} -188274. q^{38} -505135. q^{40} +393071. q^{41} +24364.6 q^{43} +165215. q^{44} +777617. q^{46} +18634.3 q^{47} +59908.4 q^{49} -248307. q^{50} +40778.3 q^{52} +228261. q^{53} +1.60881e6 q^{55} -1.47503e6 q^{56} +1.40901e6 q^{58} -205379. q^{59} +2.54932e6 q^{61} -1.44523e6 q^{62} +2.32289e6 q^{64} +397086. q^{65} +2.19172e6 q^{67} +323403. q^{68} -2.94797e6 q^{70} +3.19514e6 q^{71} +2.62084e6 q^{73} -1.98737e6 q^{74} -638699. q^{76} +4.69785e6 q^{77} -2.93367e6 q^{79} +3.56010e6 q^{80} -3.83007e6 q^{82} +5.63931e6 q^{83} +3.14920e6 q^{85} -237407. q^{86} -7.84365e6 q^{88} -8.11136e6 q^{89} +1.15952e6 q^{91} +2.63798e6 q^{92} -181571. q^{94} -6.21944e6 q^{95} +6.86976e6 q^{97} -583745. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 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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