LMFDB - Embedded newform 531.8.a.d.1.11

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} + \cdots - 58\!\cdots\!76 $ x^17 - 2x^16 - 1639x^15 + 1625x^14 + 1070274x^13 - 274939x^12 - 357079564x^11 - 89298188x^10 + 64650816672x^9 + 33122051904x^8 - 6210397064704x^7 - 2735256748800x^6 + 288860762071040x^5 - 34502173230080x^4 - 5633463408885760x^3 + 4719471961341952x^2 + 37636623107620864x - 58321181718347776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{10}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-4.85375$ 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)

$f(q)$	$=$	$q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +O(q^{10})$	\(q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +1307.20 q^{10} -972.348 q^{11} -1278.77 q^{13} -5478.12 q^{14} +552.586 q^{16} -36467.5 q^{17} +1900.72 q^{19} -15453.9 q^{20} -6664.23 q^{22} -80181.7 q^{23} -41748.1 q^{25} -8764.35 q^{26} +64763.2 q^{28} +201337. q^{29} +106900. q^{31} +187162. q^{32} -249939. q^{34} -152446. q^{35} -246438. q^{37} +13027.1 q^{38} -273238. q^{40} +348348. q^{41} +303791. q^{43} +78785.6 q^{44} -549545. q^{46} -1.12223e6 q^{47} -184682. q^{49} -286131. q^{50} +103614. q^{52} -534787. q^{53} -185453. q^{55} +1.14507e6 q^{56} +1.37991e6 q^{58} +205379. q^{59} -2.14348e6 q^{61} +732666. q^{62} +1.21203e6 q^{64} -243896. q^{65} -158432. q^{67} +2.95482e6 q^{68} -1.04483e6 q^{70} +3.94945e6 q^{71} +3.25842e6 q^{73} -1.68903e6 q^{74} -154008. q^{76} +777186. q^{77} -5.28930e6 q^{79} +105393. q^{80} +2.38749e6 q^{82} +3.63337e6 q^{83} -6.95534e6 q^{85} +2.08211e6 q^{86} +1.39300e6 q^{88} +1.34439e6 q^{89} +1.02210e6 q^{91} +6.49682e6 q^{92} -7.69146e6 q^{94} +362519. q^{95} +2.79048e6 q^{97} -1.26576e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 32 q^{2} + 1166 q^{4} + 1072 q^{5} - 2407 q^{7} + 6645 q^{8}+O(q^{10}) $ 17 * q + 32 * q^2 + 1166 * q^4 + 1072 * q^5 - 2407 * q^7 + 6645 * q^8	\( 17 q + 32 q^{2} + 1166 q^{4} + 1072 q^{5} - 2407 q^{7} + 6645 q^{8} - 6391 q^{10} + 8888 q^{11} - 12702 q^{13} + 17555 q^{14} + 139226 q^{16} + 36167 q^{17} - 71037 q^{19} + 274883 q^{20} - 325182 q^{22} + 269995 q^{23} + 97329 q^{25} + 336906 q^{26} - 901362 q^{28} + 543825 q^{29} - 633109 q^{31} + 837062 q^{32} - 529288 q^{34} + 287621 q^{35} - 867607 q^{37} + 1727169 q^{38} - 815662 q^{40} + 1428939 q^{41} - 477060 q^{43} + 1667926 q^{44} + 5305549 q^{46} + 1217849 q^{47} + 4350738 q^{49} - 4561369 q^{50} + 4175994 q^{52} + 3487068 q^{53} - 960484 q^{55} + 5363196 q^{56} - 3082906 q^{58} + 3491443 q^{59} + 998917 q^{61} + 5742614 q^{62} + 17531621 q^{64} + 6075816 q^{65} - 356026 q^{67} + 16149231 q^{68} - 548798 q^{70} + 12879428 q^{71} - 6176157 q^{73} + 5971906 q^{74} - 17624580 q^{76} - 239687 q^{77} - 18886490 q^{79} + 70463349 q^{80} - 19351611 q^{82} + 22824893 q^{83} - 7973079 q^{85} + 27502196 q^{86} - 62527651 q^{88} + 30609647 q^{89} - 36301521 q^{91} + 41388548 q^{92} + 1010176 q^{94} + 29303629 q^{95} - 26249806 q^{97} + 93110852 q^{98}+O(q^{100}) \) 17 * q + 32 * q^2 + 1166 * q^4 + 1072 * q^5 - 2407 * q^7 + 6645 * q^8 - 6391 * q^10 + 8888 * q^11 - 12702 * q^13 + 17555 * q^14 + 139226 * q^16 + 36167 * q^17 - 71037 * q^19 + 274883 * q^20 - 325182 * q^22 + 269995 * q^23 + 97329 * q^25 + 336906 * q^26 - 901362 * q^28 + 543825 * q^29 - 633109 * q^31 + 837062 * q^32 - 529288 * q^34 + 287621 * q^35 - 867607 * q^37 + 1727169 * q^38 - 815662 * q^40 + 1428939 * q^41 - 477060 * q^43 + 1667926 * q^44 + 5305549 * q^46 + 1217849 * q^47 + 4350738 * q^49 - 4561369 * q^50 + 4175994 * q^52 + 3487068 * q^53 - 960484 * q^55 + 5363196 * q^56 - 3082906 * q^58 + 3491443 * q^59 + 998917 * q^61 + 5742614 * q^62 + 17531621 * q^64 + 6075816 * q^65 - 356026 * q^67 + 16149231 * q^68 - 548798 * q^70 + 12879428 * q^71 - 6176157 * q^73 + 5971906 * q^74 - 17624580 * q^76 - 239687 * q^77 - 18886490 * q^79 + 70463349 * q^80 - 19351611 * q^82 + 22824893 * q^83 - 7973079 * q^85 + 27502196 * q^86 - 62527651 * q^88 + 30609647 * q^89 - 36301521 * q^91 + 41388548 * q^92 + 1010176 * q^94 + 29303629 * q^95 - 26249806 * q^97 + 93110852 * 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 $

$2$	6.85375	0.605791	0.302896	−	0.953024i	$-0.402047\pi$
$2$	6.85375	0.605791	0.302896	+	0.953024i	$0.402047\pi$
$3$	0	0
$3$	0	0
$4$	−81.0262	−0.633017
$5$	190.727	0.682367	0.341183	−	0.939997i	$-0.389172\pi$
$5$	190.727	0.682367	0.341183	+	0.939997i	$0.389172\pi$
$6$	0	0
$7$	−799.288	−0.880765	−0.440383	−	0.897810i	$-0.645157\pi$
$7$	−799.288	−0.880765	−0.440383	+	0.897810i	$0.645157\pi$
$8$	−1432.61	−0.989267
$9$	0	0
$10$	1307.20	0.413372
$11$	−972.348	−0.220266	−0.110133	−	0.993917i	$-0.535128\pi$
$11$	−972.348	−0.220266	−0.110133	+	0.993917i	$0.535128\pi$
$12$	0	0
$13$	−1278.77	−0.161432	−0.0807161	−	0.996737i	$-0.525721\pi$
$13$	−1278.77	−0.161432	−0.0807161	+	0.996737i	$0.525721\pi$
$14$	−5478.12	−0.533560
$15$	0	0
$16$	552.586	0.0337272
$17$	−36467.5	−1.80026	−0.900128	−	0.435625i	$-0.856527\pi$
$17$	−36467.5	−1.80026	−0.900128	+	0.435625i	$0.856527\pi$
$18$	0	0
$19$	1900.72	0.0635742	0.0317871	−	0.999495i	$-0.489880\pi$
$19$	1900.72	0.0635742	0.0317871	+	0.999495i	$0.489880\pi$
$20$	−15453.9	−0.431950
$21$	0	0
$22$	−6664.23	−0.133435
$23$	−80181.7	−1.37413	−0.687065	−	0.726596i	$-0.741101\pi$
$23$	−80181.7	−1.37413	−0.687065	+	0.726596i	$0.741101\pi$
$24$	0	0
$25$	−41748.1	−0.534376
$26$	−8764.35	−0.0977942
$27$	0	0
$28$	64763.2	0.557539
$29$	201337.	1.53296	0.766479	−	0.642270i	$-0.222007\pi$
$29$	201337.	1.53296	0.766479	+	0.642270i	$0.222007\pi$
$30$	0	0
$31$	106900.	0.644484	0.322242	−	0.946657i	$-0.395564\pi$
$31$	106900.	0.644484	0.322242	+	0.946657i	$0.395564\pi$
$32$	187162.	1.00970
$33$	0	0
$34$	−249939.	−1.09058
$35$	−152446.	−0.601005
$36$	0	0
$37$	−246438.	−0.799838	−0.399919	−	0.916550i	$-0.630962\pi$
$37$	−246438.	−0.799838	−0.399919	+	0.916550i	$0.630962\pi$
$38$	13027.1	0.0385127
$39$	0	0
$40$	−273238.	−0.675043
$41$	348348.	0.789350	0.394675	−	0.918821i	$-0.370857\pi$
$41$	348348.	0.789350	0.394675	+	0.918821i	$0.370857\pi$
$42$	0	0
$43$	303791.	0.582687	0.291343	−	0.956619i	$-0.405898\pi$
$43$	303791.	0.582687	0.291343	+	0.956619i	$0.405898\pi$
$44$	78785.6	0.139432
$45$	0	0
$46$	−549545.	−0.832436
$47$	−1.12223e6	−1.57666	−0.788330	−	0.615252i	$-0.789054\pi$
$47$	−1.12223e6	−1.57666	−0.788330	+	0.615252i	$0.789054\pi$
$48$	0	0
$49$	−184682.	−0.224253
$50$	−286131.	−0.323720
$51$	0	0
$52$	103614.	0.102189
$53$	−534787.	−0.493418	−0.246709	−	0.969090i	$-0.579349\pi$
$53$	−534787.	−0.493418	−0.246709	+	0.969090i	$0.579349\pi$
$54$	0	0
$55$	−185453.	−0.150302
$56$	1.14507e6	0.871312
$57$	0	0
$58$	1.37991e6	0.928652
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	0	0
$61$	−2.14348e6	−1.20911	−0.604555	−	0.796564i	$-0.706649\pi$
$61$	−2.14348e6	−1.20911	−0.604555	+	0.796564i	$0.706649\pi$
$62$	732666.	0.390423
$63$	0	0
$64$	1.21203e6	0.577940
$65$	−243896.	−0.110156
$66$	0	0
$67$	−158432.	−0.0643548	−0.0321774	−	0.999482i	$-0.510244\pi$
$67$	−158432.	−0.0643548	−0.0321774	+	0.999482i	$0.510244\pi$
$68$	2.95482e6	1.13959
$69$	0	0
$70$	−1.04483e6	−0.364084
$71$	3.94945e6	1.30958	0.654790	−	0.755811i	$-0.272757\pi$
$71$	3.94945e6	1.30958	0.654790	+	0.755811i	$0.272757\pi$
$72$	0	0
$73$	3.25842e6	0.980342	0.490171	−	0.871626i	$-0.336934\pi$
$73$	3.25842e6	0.980342	0.490171	+	0.871626i	$0.336934\pi$
$74$	−1.68903e6	−0.484535
$75$	0	0
$76$	−154008.	−0.0402435
$77$	777186.	0.194003
$78$	0	0
$79$	−5.28930e6	−1.20699	−0.603494	−	0.797367i	$-0.706225\pi$
$79$	−5.28930e6	−1.20699	−0.603494	+	0.797367i	$0.706225\pi$
$80$	105393.	0.0230143
$81$	0	0
$82$	2.38749e6	0.478181
$83$	3.63337e6	0.697487	0.348744	−	0.937218i	$-0.386608\pi$
$83$	3.63337e6	0.697487	0.348744	+	0.937218i	$0.386608\pi$
$84$	0	0
$85$	−6.95534e6	−1.22844
$86$	2.08211e6	0.352987
$87$	0	0
$88$	1.39300e6	0.217902
$89$	1.34439e6	0.202143	0.101072	−	0.994879i	$-0.467773\pi$
$89$	1.34439e6	0.202143	0.101072	+	0.994879i	$0.467773\pi$
$90$	0	0
$91$	1.02210e6	0.142184
$92$	6.49682e6	0.869848
$93$	0	0
$94$	−7.69146e6	−0.955128
$95$	362519.	0.0433809
$96$	0	0
$97$	2.79048e6	0.310441	0.155220	−	0.987880i	$-0.450391\pi$
$97$	2.79048e6	0.310441	0.155220	+	0.987880i	$0.450391\pi$
$98$	−1.26576e6	−0.135850
$99$	0	0
$100$	3.38269e6	0.338269
$101$	1.43415e7	1.38506	0.692532	−	0.721387i	$-0.256495\pi$
$101$	1.43415e7	1.38506	0.692532	+	0.721387i	$0.256495\pi$
$102$	0	0
$103$	1.62771e6	0.146773	0.0733867	−	0.997304i	$-0.476619\pi$
$103$	1.62771e6	0.146773	0.0733867	+	0.997304i	$0.476619\pi$
$104$	1.83198e6	0.159700
$105$	0	0
$106$	−3.66529e6	−0.298908
$107$	7.86386e6	0.620572	0.310286	−	0.950643i	$-0.399575\pi$
$107$	7.86386e6	0.620572	0.310286	+	0.950643i	$0.399575\pi$
$108$	0	0
$109$	2.63958e6	0.195228	0.0976141	−	0.995224i	$-0.468879\pi$
$109$	2.63958e6	0.195228	0.0976141	+	0.995224i	$0.468879\pi$
$110$	−1.27105e6	−0.0910518
$111$	0	0
$112$	−441675.	−0.0297057
$113$	2.91448e7	1.90015	0.950074	−	0.312026i	$-0.101008\pi$
$113$	2.91448e7	1.90015	0.950074	+	0.312026i	$0.101008\pi$
$114$	0	0
$115$	−1.52928e7	−0.937661
$116$	−1.63135e7	−0.970388
$117$	0	0
$118$	1.40762e6	0.0788673
$119$	2.91480e7	1.58560
$120$	0	0
$121$	−1.85417e7	−0.951483
$122$	−1.46909e7	−0.732468
$123$	0	0
$124$	−8.66170e6	−0.407969
$125$	−2.28631e7	−1.04701
$126$	0	0
$127$	−3.71615e7	−1.60983	−0.804915	−	0.593390i	$-0.797789\pi$
$127$	−3.71615e7	−1.60983	−0.804915	+	0.593390i	$0.797789\pi$
$128$	−1.56498e7	−0.659588
$129$	0	0
$130$	−1.67160e6	−0.0667315
$131$	−1.50104e7	−0.583369	−0.291684	−	0.956515i	$-0.594216\pi$
$131$	−1.50104e7	−0.583369	−0.291684	+	0.956515i	$0.594216\pi$
$132$	0	0
$133$	−1.51922e6	−0.0559939
$134$	−1.08585e6	−0.0389856
$135$	0	0
$136$	5.22437e7	1.78094
$137$	−1.75306e7	−0.582470	−0.291235	−	0.956652i	$-0.594066\pi$
$137$	−1.75306e7	−0.582470	−0.291235	+	0.956652i	$0.594066\pi$
$138$	0	0
$139$	−1.16871e7	−0.369110	−0.184555	−	0.982822i	$-0.559084\pi$
$139$	−1.16871e7	−0.369110	−0.184555	+	0.982822i	$0.559084\pi$
$140$	1.23521e7	0.380446
$141$	0	0
$142$	2.70685e7	0.793332
$143$	1.24341e6	0.0355580
$144$	0	0
$145$	3.84004e7	1.04604
$146$	2.23324e7	0.593883
$147$	0	0
$148$	1.99679e7	0.506311
$149$	6.76391e7	1.67512	0.837559	−	0.546346i	$-0.183982\pi$
$149$	6.76391e7	1.67512	0.837559	+	0.546346i	$0.183982\pi$
$150$	0	0
$151$	4.31830e7	1.02069	0.510345	−	0.859970i	$-0.329518\pi$
$151$	4.31830e7	1.02069	0.510345	+	0.859970i	$0.329518\pi$
$152$	−2.72300e6	−0.0628919
$153$	0	0
$154$	5.32664e6	0.117525
$155$	2.03888e7	0.439775
$156$	0	0
$157$	4.62920e7	0.954679	0.477340	−	0.878719i	$-0.341601\pi$
$157$	4.62920e7	0.954679	0.477340	+	0.878719i	$0.341601\pi$
$158$	−3.62515e7	−0.731183
$159$	0	0
$160$	3.56968e7	0.688985
$161$	6.40883e7	1.21029
$162$	0	0
$163$	7.22501e7	1.30672	0.653359	−	0.757048i	$-0.273359\pi$
$163$	7.22501e7	1.30672	0.653359	+	0.757048i	$0.273359\pi$
$164$	−2.82253e7	−0.499672
$165$	0	0
$166$	2.49022e7	0.422532
$167$	−3.24667e7	−0.539425	−0.269713	−	0.962941i	$-0.586929\pi$
$167$	−3.24667e7	−0.539425	−0.269713	+	0.962941i	$0.586929\pi$
$168$	0	0
$169$	−6.11133e7	−0.973940
$170$	−4.76702e7	−0.744175
$171$	0	0
$172$	−2.46150e7	−0.368851
$173$	−1.31381e7	−0.192918	−0.0964589	−	0.995337i	$-0.530752\pi$
$173$	−1.31381e7	−0.192918	−0.0964589	+	0.995337i	$0.530752\pi$
$174$	0	0
$175$	3.33687e7	0.470659
$176$	−537306.	−0.00742895
$177$	0	0
$178$	9.21409e6	0.122457
$179$	1.99708e7	0.260261	0.130131	−	0.991497i	$-0.458460\pi$
$179$	1.99708e7	0.260261	0.130131	+	0.991497i	$0.458460\pi$
$180$	0	0
$181$	9.38499e7	1.17641	0.588205	−	0.808712i	$-0.299835\pi$
$181$	9.38499e7	1.17641	0.588205	+	0.808712i	$0.299835\pi$
$182$	7.00524e6	0.0861337
$183$	0	0
$184$	1.14869e8	1.35938
$185$	−4.70025e7	−0.545783
$186$	0	0
$187$	3.54591e7	0.396535
$188$	9.09298e7	0.998053
$189$	0	0
$190$	2.48462e6	0.0262798
$191$	7.50591e7	0.779447	0.389724	−	0.920932i	$-0.372571\pi$
$191$	7.50591e7	0.779447	0.389724	+	0.920932i	$0.372571\pi$
$192$	0	0
$193$	−7.13691e7	−0.714594	−0.357297	−	0.933991i	$-0.616302\pi$
$193$	−7.13691e7	−0.714594	−0.357297	+	0.933991i	$0.616302\pi$
$194$	1.91253e7	0.188062
$195$	0	0
$196$	1.49641e7	0.141956
$197$	1.40656e8	1.31077	0.655384	−	0.755296i	$-0.272507\pi$
$197$	1.40656e8	1.31077	0.655384	+	0.755296i	$0.272507\pi$
$198$	0	0
$199$	2.79140e7	0.251094	0.125547	−	0.992088i	$-0.459931\pi$
$199$	2.79140e7	0.251094	0.125547	+	0.992088i	$0.459931\pi$
$200$	5.98088e7	0.528640
$201$	0	0
$202$	9.82931e7	0.839060
$203$	−1.60926e8	−1.35018
$204$	0	0
$205$	6.64394e7	0.538626
$206$	1.11559e7	0.0889140
$207$	0	0
$208$	−706629.	−0.00544465
$209$	−1.84816e6	−0.0140032
$210$	0	0
$211$	1.66531e8	1.22041	0.610207	−	0.792242i	$-0.291086\pi$
$211$	1.66531e8	1.22041	0.610207	+	0.792242i	$0.291086\pi$
$212$	4.33317e7	0.312342
$213$	0	0
$214$	5.38969e7	0.375937
$215$	5.79412e7	0.397606
$216$	0	0
$217$	−8.54439e7	−0.567639
$218$	1.80910e7	0.118268
$219$	0	0
$220$	1.50266e7	0.0951438
$221$	4.66334e7	0.290619
$222$	0	0
$223$	8.94297e6	0.0540026	0.0270013	−	0.999635i	$-0.491404\pi$
$223$	8.94297e6	0.0540026	0.0270013	+	0.999635i	$0.491404\pi$
$224$	−1.49596e8	−0.889308
$225$	0	0
$226$	1.99751e8	1.15109
$227$	6.88172e7	0.390487	0.195243	−	0.980755i	$-0.437450\pi$
$227$	6.88172e7	0.390487	0.195243	+	0.980755i	$0.437450\pi$
$228$	0	0
$229$	1.97507e8	1.08682	0.543411	−	0.839467i	$-0.317132\pi$
$229$	1.97507e8	1.08682	0.543411	+	0.839467i	$0.317132\pi$
$230$	−1.04813e8	−0.568027
$231$	0	0
$232$	−2.88438e8	−1.51651
$233$	−2.46365e8	−1.27595	−0.637973	−	0.770058i	$-0.720227\pi$
$233$	−2.46365e8	−1.27595	−0.637973	+	0.770058i	$0.720227\pi$
$234$	0	0
$235$	−2.14039e8	−1.07586
$236$	−1.66411e7	−0.0824118
$237$	0	0
$238$	1.99773e8	0.960545
$239$	3.78259e8	1.79224	0.896120	−	0.443812i	$-0.146374\pi$
$239$	3.78259e8	1.79224	0.896120	+	0.443812i	$0.146374\pi$
$240$	0	0
$241$	−2.71983e8	−1.25165	−0.625824	−	0.779964i	$-0.715237\pi$
$241$	−2.71983e8	−1.25165	−0.625824	+	0.779964i	$0.715237\pi$
$242$	−1.27080e8	−0.576400
$243$	0	0
$244$	1.73678e8	0.765387
$245$	−3.52239e7	−0.153023
$246$	0	0
$247$	−2.43058e6	−0.0102629
$248$	−1.53146e8	−0.637567
$249$	0	0
$250$	−1.56698e8	−0.634268
$251$	−1.67816e8	−0.669847	−0.334923	−	0.942245i	$-0.608710\pi$
$251$	−1.67816e8	−0.669847	−0.334923	+	0.942245i	$0.608710\pi$
$252$	0	0
$253$	7.79645e7	0.302674
$254$	−2.54695e8	−0.975221
$255$	0	0
$256$	−2.62399e8	−0.977513
$257$	3.77370e8	1.38676	0.693381	−	0.720571i	$-0.256120\pi$
$257$	3.77370e8	1.38676	0.693381	+	0.720571i	$0.256120\pi$
$258$	0	0
$259$	1.96975e8	0.704470
$260$	1.97619e7	0.0697305
$261$	0	0
$262$	−1.02878e8	−0.353400
$263$	2.37395e8	0.804686	0.402343	−	0.915489i	$-0.368196\pi$
$263$	2.37395e8	0.804686	0.402343	+	0.915489i	$0.368196\pi$
$264$	0	0
$265$	−1.01998e8	−0.336692
$266$	−1.04124e7	−0.0339206
$267$	0	0
$268$	1.28371e7	0.0407377
$269$	−2.56851e8	−0.804542	−0.402271	−	0.915521i	$-0.631779\pi$
$269$	−2.56851e8	−0.804542	−0.402271	+	0.915521i	$0.631779\pi$
$270$	0	0
$271$	−2.99436e8	−0.913927	−0.456964	−	0.889485i	$-0.651063\pi$
$271$	−2.99436e8	−0.913927	−0.456964	+	0.889485i	$0.651063\pi$
$272$	−2.01514e7	−0.0607176
$273$	0	0
$274$	−1.20150e8	−0.352855
$275$	4.05937e7	0.117705
$276$	0	0
$277$	−1.07342e8	−0.303453	−0.151727	−	0.988422i	$-0.548483\pi$
$277$	−1.07342e8	−0.303453	−0.151727	+	0.988422i	$0.548483\pi$
$278$	−8.01006e7	−0.223604
$279$	0	0
$280$	2.18396e8	0.594555
$281$	−1.26999e8	−0.341451	−0.170726	−	0.985319i	$-0.554611\pi$
$281$	−1.26999e8	−0.341451	−0.170726	+	0.985319i	$0.554611\pi$
$282$	0	0
$283$	6.79420e8	1.78191	0.890955	−	0.454091i	$-0.150036\pi$
$283$	6.79420e8	1.78191	0.890955	+	0.454091i	$0.150036\pi$
$284$	−3.20009e8	−0.828986
$285$	0	0
$286$	8.52200e6	0.0215407
$287$	−2.78430e8	−0.695232
$288$	0	0
$289$	9.19537e8	2.24092
$290$	2.63187e8	0.633682
$291$	0	0
$292$	−2.64018e8	−0.620573
$293$	−3.54246e8	−0.822752	−0.411376	−	0.911466i	$-0.634952\pi$
$293$	−3.54246e8	−0.822752	−0.411376	+	0.911466i	$0.634952\pi$
$294$	0	0
$295$	3.91714e7	0.0888366
$296$	3.53051e8	0.791254
$297$	0	0
$298$	4.63581e8	1.01477
$299$	1.02534e8	0.221829
$300$	0	0
$301$	−2.42816e8	−0.513210
$302$	2.95966e8	0.618325
$303$	0	0
$304$	1.05031e6	0.00214418
$305$	−4.08821e8	−0.825056
$306$	0	0
$307$	−4.47178e8	−0.882056	−0.441028	−	0.897493i	$-0.645386\pi$
$307$	−4.47178e8	−0.882056	−0.441028	+	0.897493i	$0.645386\pi$
$308$	−6.29724e7	−0.122807
$309$	0	0
$310$	1.39739e8	0.266412
$311$	1.03177e9	1.94501	0.972506	−	0.232879i	$-0.0748145\pi$
$311$	1.03177e9	1.94501	0.972506	+	0.232879i	$0.0748145\pi$
$312$	0	0
$313$	−1.47667e8	−0.272194	−0.136097	−	0.990695i	$-0.543456\pi$
$313$	−1.47667e8	−0.272194	−0.136097	+	0.990695i	$0.543456\pi$
$314$	3.17274e8	0.578336
$315$	0	0
$316$	4.28571e8	0.764044
$317$	8.08129e8	1.42486	0.712431	−	0.701742i	$-0.247594\pi$
$317$	8.08129e8	1.42486	0.712431	+	0.701742i	$0.247594\pi$
$318$	0	0
$319$	−1.95769e8	−0.337658
$320$	2.31167e8	0.394367
$321$	0	0
$322$	4.39245e8	0.733181
$323$	−6.93145e7	−0.114450
$324$	0	0
$325$	5.33861e7	0.0862654
$326$	4.95184e8	0.791599
$327$	0	0
$328$	−4.99047e8	−0.780878
$329$	8.96983e8	1.38867
$330$	0	0
$331$	−1.71934e8	−0.260594	−0.130297	−	0.991475i	$-0.541593\pi$
$331$	−1.71934e8	−0.260594	−0.130297	+	0.991475i	$0.541593\pi$
$332$	−2.94398e8	−0.441521
$333$	0	0
$334$	−2.22519e8	−0.326779
$335$	−3.02173e7	−0.0439136
$336$	0	0
$337$	2.64986e8	0.377154	0.188577	−	0.982058i	$-0.439613\pi$
$337$	2.64986e8	0.377154	0.188577	+	0.982058i	$0.439613\pi$
$338$	−4.18855e8	−0.590004
$339$	0	0
$340$	5.63565e8	0.777620
$341$	−1.03944e8	−0.141958
$342$	0	0
$343$	8.05862e8	1.07828
$344$	−4.35215e8	−0.576433
$345$	0	0
$346$	−9.00455e7	−0.116868
$347$	−5.14421e8	−0.660945	−0.330473	−	0.943816i	$-0.607208\pi$
$347$	−5.14421e8	−0.660945	−0.330473	+	0.943816i	$0.607208\pi$
$348$	0	0
$349$	−6.09218e7	−0.0767157	−0.0383578	−	0.999264i	$-0.512213\pi$
$349$	−6.09218e7	−0.0767157	−0.0383578	+	0.999264i	$0.512213\pi$
$350$	2.28701e8	0.285121
$351$	0	0
$352$	−1.81986e8	−0.222402
$353$	1.90611e8	0.230641	0.115320	−	0.993328i	$-0.463211\pi$
$353$	1.90611e8	0.230641	0.115320	+	0.993328i	$0.463211\pi$
$354$	0	0
$355$	7.53268e8	0.893614
$356$	−1.08931e8	−0.127960
$357$	0	0
$358$	1.36875e8	0.157664
$359$	1.39955e9	1.59646	0.798229	−	0.602354i	$-0.205770\pi$
$359$	1.39955e9	1.59646	0.798229	+	0.602354i	$0.205770\pi$
$360$	0	0
$361$	−8.90259e8	−0.995958
$362$	6.43224e8	0.712659
$363$	0	0
$364$	−8.28171e7	−0.0900047
$365$	6.21471e8	0.668953
$366$	0	0
$367$	−1.50266e9	−1.58683	−0.793413	−	0.608684i	$-0.791698\pi$
$367$	−1.50266e9	−1.58683	−0.793413	+	0.608684i	$0.791698\pi$
$368$	−4.43073e7	−0.0463455
$369$	0	0
$370$	−3.22143e8	−0.330631
$371$	4.27449e8	0.434585
$372$	0	0
$373$	8.89561e7	0.0887554	0.0443777	−	0.999015i	$-0.485869\pi$
$373$	8.89561e7	0.0887554	0.0443777	+	0.999015i	$0.485869\pi$
$374$	2.43027e8	0.240218
$375$	0	0
$376$	1.60772e9	1.55974
$377$	−2.57463e8	−0.247469
$378$	0	0
$379$	−7.27075e8	−0.686028	−0.343014	−	0.939330i	$-0.611448\pi$
$379$	−7.27075e8	−0.686028	−0.343014	+	0.939330i	$0.611448\pi$
$380$	−2.93736e7	−0.0274608
$381$	0	0
$382$	5.14436e8	0.472182
$383$	−2.06745e9	−1.88036	−0.940178	−	0.340683i	$-0.889342\pi$
$383$	−2.06745e9	−1.88036	−0.940178	+	0.340683i	$0.889342\pi$
$384$	0	0
$385$	1.48231e8	0.132381
$386$	−4.89145e8	−0.432895
$387$	0	0
$388$	−2.26102e8	−0.196514
$389$	8.24531e7	0.0710205	0.0355102	−	0.999369i	$-0.488694\pi$
$389$	8.24531e7	0.0710205	0.0355102	+	0.999369i	$0.488694\pi$
$390$	0	0
$391$	2.92402e9	2.47379
$392$	2.64578e8	0.221846
$393$	0	0
$394$	9.64019e8	0.794051
$395$	−1.00881e9	−0.823609
$396$	0	0
$397$	2.60792e8	0.209183	0.104592	−	0.994515i	$-0.466646\pi$
$397$	2.60792e8	0.209183	0.104592	+	0.994515i	$0.466646\pi$
$398$	1.91316e8	0.152111
$399$	0	0
$400$	−2.30694e7	−0.0180230
$401$	5.57165e8	0.431498	0.215749	−	0.976449i	$-0.430781\pi$
$401$	5.57165e8	0.431498	0.215749	+	0.976449i	$0.430781\pi$
$402$	0	0
$403$	−1.36700e8	−0.104040
$404$	−1.16204e9	−0.876769
$405$	0	0
$406$	−1.10295e9	−0.817925
$407$	2.39624e8	0.176177
$408$	0	0
$409$	−1.59001e9	−1.14913	−0.574564	−	0.818460i	$-0.694828\pi$
$409$	−1.59001e9	−1.14913	−0.574564	+	0.818460i	$0.694828\pi$
$410$	4.55359e8	0.326295
$411$	0	0
$412$	−1.31887e8	−0.0929100
$413$	−1.64157e8	−0.114666
$414$	0	0
$415$	6.92983e8	0.475942
$416$	−2.39336e8	−0.162998
$417$	0	0
$418$	−1.26668e7	−0.00848303
$419$	−5.88021e8	−0.390520	−0.195260	−	0.980751i	$-0.562555\pi$
$419$	−5.88021e8	−0.390520	−0.195260	+	0.980751i	$0.562555\pi$
$420$	0	0
$421$	−7.50924e8	−0.490466	−0.245233	−	0.969464i	$-0.578864\pi$
$421$	−7.50924e8	−0.490466	−0.245233	+	0.969464i	$0.578864\pi$
$422$	1.14136e9	0.739316
$423$	0	0
$424$	7.66142e8	0.488123
$425$	1.52245e9	0.962013
$426$	0	0
$427$	1.71326e9	1.06494
$428$	−6.37178e8	−0.392833
$429$	0	0
$430$	3.97114e8	0.240866
$431$	2.36435e9	1.42247	0.711233	−	0.702957i	$-0.248137\pi$
$431$	2.36435e9	1.42247	0.711233	+	0.702957i	$0.248137\pi$
$432$	0	0
$433$	6.97398e7	0.0412832	0.0206416	−	0.999787i	$-0.493429\pi$
$433$	6.97398e7	0.0412832	0.0206416	+	0.999787i	$0.493429\pi$
$434$	−5.85611e8	−0.343871
$435$	0	0
$436$	−2.13875e8	−0.123583
$437$	−1.52403e8	−0.0873592
$438$	0	0
$439$	−2.54774e9	−1.43724	−0.718620	−	0.695403i	$-0.755226\pi$
$439$	−2.54774e9	−1.43724	−0.718620	+	0.695403i	$0.755226\pi$
$440$	2.65683e8	0.148689
$441$	0	0
$442$	3.19614e8	0.176055
$443$	−1.15431e9	−0.630825	−0.315412	−	0.948955i	$-0.602143\pi$
$443$	−1.15431e9	−0.630825	−0.315412	+	0.948955i	$0.602143\pi$
$444$	0	0
$445$	2.56411e8	0.137936
$446$	6.12929e7	0.0327143
$447$	0	0
$448$	−9.68759e8	−0.509029
$449$	−5.55994e8	−0.289873	−0.144937	−	0.989441i	$-0.546298\pi$
$449$	−5.55994e8	−0.289873	−0.144937	+	0.989441i	$0.546298\pi$
$450$	0	0
$451$	−3.38715e8	−0.173867
$452$	−2.36150e9	−1.20283
$453$	0	0
$454$	4.71655e8	0.236554
$455$	1.94943e8	0.0970215
$456$	0	0
$457$	3.93559e8	0.192887	0.0964435	−	0.995338i	$-0.469253\pi$
$457$	3.93559e8	0.192887	0.0964435	+	0.995338i	$0.469253\pi$
$458$	1.35366e9	0.658388
$459$	0	0
$460$	1.23912e9	0.593555
$461$	2.99382e9	1.42322	0.711611	−	0.702574i	$-0.247966\pi$
$461$	2.99382e9	1.42322	0.711611	+	0.702574i	$0.247966\pi$
$462$	0	0
$463$	−1.18369e9	−0.554248	−0.277124	−	0.960834i	$-0.589381\pi$
$463$	−1.18369e9	−0.554248	−0.277124	+	0.960834i	$0.589381\pi$
$464$	1.11256e8	0.0517023
$465$	0	0
$466$	−1.68852e9	−0.772957
$467$	−3.15650e9	−1.43416	−0.717079	−	0.696992i	$-0.754521\pi$
$467$	−3.15650e9	−1.43416	−0.717079	+	0.696992i	$0.754521\pi$
$468$	0	0
$469$	1.26633e8	0.0566815
$470$	−1.46697e9	−0.651747
$471$	0	0
$472$	−2.94228e8	−0.128792
$473$	−2.95390e8	−0.128346
$474$	0	0
$475$	−7.93515e7	−0.0339725
$476$	−2.36175e9	−1.00371
$477$	0	0
$478$	2.59249e9	1.08572
$479$	−2.27050e9	−0.943945	−0.471972	−	0.881613i	$-0.656458\pi$
$479$	−2.27050e9	−0.943945	−0.471972	+	0.881613i	$0.656458\pi$
$480$	0	0
$481$	3.15137e8	0.129120
$482$	−1.86410e9	−0.758238
$483$	0	0
$484$	1.50236e9	0.602305
$485$	5.32222e8	0.211834
$486$	0	0
$487$	2.04995e8	0.0804252	0.0402126	−	0.999191i	$-0.487196\pi$
$487$	2.04995e8	0.0804252	0.0402126	+	0.999191i	$0.487196\pi$
$488$	3.07078e9	1.19613
$489$	0	0
$490$	−2.41416e8	−0.0926999
$491$	−1.08333e9	−0.413023	−0.206512	−	0.978444i	$-0.566211\pi$
$491$	−1.08333e9	−0.413023	−0.206512	+	0.978444i	$0.566211\pi$
$492$	0	0
$493$	−7.34224e9	−2.75972
$494$	−1.66586e7	−0.00621718
$495$	0	0
$496$	5.90715e7	0.0217366
$497$	−3.15675e9	−1.15343
$498$	0	0
$499$	1.23774e9	0.445942	0.222971	−	0.974825i	$-0.428425\pi$
$499$	1.23774e9	0.445942	0.222971	+	0.974825i	$0.428425\pi$
$500$	1.85251e9	0.662773
$501$	0	0
$502$	−1.15017e9	−0.405787
$503$	−3.42069e9	−1.19847	−0.599234	−	0.800574i	$-0.704528\pi$
$503$	−3.42069e9	−1.19847	−0.599234	+	0.800574i	$0.704528\pi$
$504$	0	0
$505$	2.73532e9	0.945122
$506$	5.34349e8	0.183357
$507$	0	0
$508$	3.01105e9	1.01905
$509$	3.62806e9	1.21944	0.609722	−	0.792616i	$-0.291281\pi$
$509$	3.62806e9	1.21944	0.609722	+	0.792616i	$0.291281\pi$
$510$	0	0
$511$	−2.60442e9	−0.863451
$512$	2.04753e8	0.0674195
$513$	0	0
$514$	2.58640e9	0.840088
$515$	3.10449e8	0.100153
$516$	0	0
$517$	1.09120e9	0.347285
$518$	1.35002e9	0.426762
$519$	0	0
$520$	3.49408e8	0.108974
$521$	1.36162e9	0.421817	0.210909	−	0.977506i	$-0.432358\pi$
$521$	1.36162e9	0.421817	0.210909	+	0.977506i	$0.432358\pi$
$522$	0	0
$523$	1.07415e9	0.328329	0.164165	−	0.986433i	$-0.447507\pi$
$523$	1.07415e9	0.328329	0.164165	+	0.986433i	$0.447507\pi$
$524$	1.21624e9	0.369282
$525$	0	0
$526$	1.62705e9	0.487472
$527$	−3.89838e9	−1.16024
$528$	0	0
$529$	3.02428e9	0.888234
$530$	−6.99072e8	−0.203965
$531$	0	0
$532$	1.23097e8	0.0354451
$533$	−4.45456e8	−0.127426
$534$	0	0
$535$	1.49985e9	0.423458
$536$	2.26972e8	0.0636642
$537$	0	0
$538$	−1.76039e9	−0.487385
$539$	1.79575e8	0.0493953
$540$	0	0
$541$	−6.13348e9	−1.66539	−0.832696	−	0.553730i	$-0.813204\pi$
$541$	−6.13348e9	−1.66539	−0.832696	+	0.553730i	$0.813204\pi$
$542$	−2.05226e9	−0.553649
$543$	0	0
$544$	−6.82531e9	−1.81772
$545$	5.03440e8	0.133217
$546$	0	0
$547$	−6.39180e9	−1.66981	−0.834906	−	0.550392i	$-0.814478\pi$
$547$	−6.39180e9	−1.66981	−0.834906	+	0.550392i	$0.814478\pi$
$548$	1.42043e9	0.368713
$549$	0	0
$550$	2.78219e8	0.0713045
$551$	3.82685e8	0.0974565
$552$	0	0
$553$	4.22767e9	1.06307
$554$	−7.35697e8	−0.183829
$555$	0	0
$556$	9.46963e8	0.233653
$557$	−5.20568e9	−1.27639	−0.638197	−	0.769873i	$-0.720319\pi$
$557$	−5.20568e9	−1.27639	−0.638197	+	0.769873i	$0.720319\pi$
$558$	0	0
$559$	−3.88478e8	−0.0940643
$560$	−8.42396e7	−0.0202702
$561$	0	0
$562$	−8.70420e8	−0.206848
$563$	−3.32429e9	−0.785089	−0.392545	−	0.919733i	$-0.628405\pi$
$563$	−3.32429e9	−0.785089	−0.392545	+	0.919733i	$0.628405\pi$
$564$	0	0
$565$	5.55872e9	1.29660
$566$	4.65657e9	1.07947
$567$	0	0
$568$	−5.65803e9	−1.29553
$569$	−1.61858e9	−0.368333	−0.184166	−	0.982895i	$-0.558959\pi$
$569$	−1.61858e9	−0.368333	−0.184166	+	0.982895i	$0.558959\pi$
$570$	0	0
$571$	−6.26853e8	−0.140909	−0.0704546	−	0.997515i	$-0.522445\pi$
$571$	−6.26853e8	−0.140909	−0.0704546	+	0.997515i	$0.522445\pi$
$572$	−1.00749e8	−0.0225088
$573$	0	0
$574$	−1.90829e9	−0.421165
$575$	3.34743e9	0.734302
$576$	0	0
$577$	2.33845e9	0.506772	0.253386	−	0.967365i	$-0.418456\pi$
$577$	2.33845e9	0.506772	0.253386	+	0.967365i	$0.418456\pi$
$578$	6.30228e9	1.35753
$579$	0	0
$580$	−3.11144e9	−0.662161
$581$	−2.90411e9	−0.614323
$582$	0	0
$583$	5.19999e8	0.108683
$584$	−4.66806e9	−0.969820
$585$	0	0
$586$	−2.42791e9	−0.498416
$587$	5.68734e9	1.16058	0.580291	−	0.814409i	$-0.302939\pi$
$587$	5.68734e9	1.16058	0.580291	+	0.814409i	$0.302939\pi$
$588$	0	0
$589$	2.03187e8	0.0409725
$590$	2.68471e8	0.0538164
$591$	0	0
$592$	−1.36178e8	−0.0269763
$593$	6.92748e9	1.36422	0.682109	−	0.731250i	$-0.261063\pi$
$593$	6.92748e9	1.36422	0.682109	+	0.731250i	$0.261063\pi$
$594$	0	0
$595$	5.55932e9	1.08196
$596$	−5.48053e9	−1.06038
$597$	0	0
$598$	7.02741e8	0.134382
$599$	−1.04577e7	−0.00198812	−0.000994062	−	1.00000i	$-0.500316\pi$
$599$	−1.04577e7	−0.00198812	−0.000994062		1.00000i	$0.500316\pi$
$600$	0	0
$601$	5.50968e9	1.03530	0.517649	−	0.855593i	$-0.326807\pi$
$601$	5.50968e9	1.03530	0.517649	+	0.855593i	$0.326807\pi$
$602$	−1.66420e9	−0.310898
$603$	0	0
$604$	−3.49896e9	−0.646114
$605$	−3.53641e9	−0.649260
$606$	0	0
$607$	−1.08515e10	−1.96939	−0.984693	−	0.174298i	$-0.944234\pi$
$607$	−1.08515e10	−1.96939	−0.984693	+	0.174298i	$0.944234\pi$
$608$	3.55742e8	0.0641908
$609$	0	0
$610$	−2.80195e9	−0.499812
$611$	1.43507e9	0.254524
$612$	0	0
$613$	−5.00263e9	−0.877177	−0.438588	−	0.898688i	$-0.644521\pi$
$613$	−5.00263e9	−0.877177	−0.438588	+	0.898688i	$0.644521\pi$
$614$	−3.06485e9	−0.534342
$615$	0	0
$616$	−1.11341e9	−0.191920
$617$	−1.47593e9	−0.252970	−0.126485	−	0.991969i	$-0.540370\pi$
$617$	−1.47593e9	−0.252970	−0.126485	+	0.991969i	$0.540370\pi$
$618$	0	0
$619$	−9.36559e9	−1.58715	−0.793575	−	0.608472i	$-0.791783\pi$
$619$	−9.36559e9	−1.58715	−0.793575	+	0.608472i	$0.791783\pi$
$620$	−1.65202e9	−0.278385
$621$	0	0
$622$	7.07150e9	1.17827
$623$	−1.07455e9	−0.178041
$624$	0	0
$625$	−1.09904e9	−0.180067
$626$	−1.01207e9	−0.164893
$627$	0	0
$628$	−3.75087e9	−0.604328
$629$	8.98698e9	1.43991
$630$	0	0
$631$	−1.19966e10	−1.90088	−0.950438	−	0.310914i	$-0.899365\pi$
$631$	−1.19966e10	−1.90088	−0.950438	+	0.310914i	$0.899365\pi$
$632$	7.57751e9	1.19403
$633$	0	0
$634$	5.53871e9	0.863170
$635$	−7.08771e9	−1.09849
$636$	0	0
$637$	2.36165e8	0.0362016
$638$	−1.34175e9	−0.204551
$639$	0	0
$640$	−2.98484e9	−0.450081
$641$	1.25626e10	1.88398	0.941989	−	0.335644i	$-0.108954\pi$
$641$	1.25626e10	1.88398	0.941989	+	0.335644i	$0.108954\pi$
$642$	0	0
$643$	−4.11211e9	−0.609995	−0.304997	−	0.952353i	$-0.598656\pi$
$643$	−4.11211e9	−0.609995	−0.304997	+	0.952353i	$0.598656\pi$
$644$	−5.19283e9	−0.766131
$645$	0	0
$646$	−4.75064e8	−0.0693327
$647$	8.36038e9	1.21356	0.606780	−	0.794870i	$-0.292461\pi$
$647$	8.36038e9	1.21356	0.606780	+	0.794870i	$0.292461\pi$
$648$	0	0
$649$	−1.99700e8	−0.0286762
$650$	3.65895e8	0.0522588
$651$	0	0
$652$	−5.85415e9	−0.827175
$653$	−5.92578e8	−0.0832817	−0.0416409	−	0.999133i	$-0.513259\pi$
$653$	−5.92578e8	−0.0832817	−0.0416409	+	0.999133i	$0.513259\pi$
$654$	0	0
$655$	−2.86290e9	−0.398072
$656$	1.92492e8	0.0266225
$657$	0	0
$658$	6.14769e9	0.841243
$659$	−3.61916e9	−0.492617	−0.246308	−	0.969192i	$-0.579218\pi$
$659$	−3.61916e9	−0.492617	−0.246308	+	0.969192i	$0.579218\pi$
$660$	0	0
$661$	1.49686e9	0.201593	0.100797	−	0.994907i	$-0.467861\pi$
$661$	1.49686e9	0.201593	0.100797	+	0.994907i	$0.467861\pi$
$662$	−1.17839e9	−0.157866
$663$	0	0
$664$	−5.20521e9	−0.690002
$665$	−2.89757e8	−0.0382084
$666$	0	0
$667$	−1.61435e10	−2.10648
$668$	2.63066e9	0.341465
$669$	0	0
$670$	−2.07102e8	−0.0266025
$671$	2.08421e9	0.266326
$672$	0	0
$673$	1.55009e10	1.96022	0.980109	−	0.198461i	$-0.0635943\pi$
$673$	1.55009e10	1.96022	0.980109	+	0.198461i	$0.0635943\pi$
$674$	1.81615e9	0.228476
$675$	0	0
$676$	4.95177e9	0.616520
$677$	7.97429e9	0.987715	0.493857	−	0.869543i	$-0.335586\pi$
$677$	7.97429e9	0.987715	0.493857	+	0.869543i	$0.335586\pi$
$678$	0	0
$679$	−2.23040e9	−0.273425
$680$	9.96431e9	1.21525
$681$	0	0
$682$	−7.12407e8	−0.0859969
$683$	−7.37364e9	−0.885542	−0.442771	−	0.896635i	$-0.646005\pi$
$683$	−7.37364e9	−0.885542	−0.442771	+	0.896635i	$0.646005\pi$
$684$	0	0
$685$	−3.34356e9	−0.397458
$686$	5.52317e9	0.653212
$687$	0	0
$688$	1.67871e8	0.0196524
$689$	6.83868e8	0.0796535
$690$	0	0
$691$	5.70889e9	0.658232	0.329116	−	0.944290i	$-0.393249\pi$
$691$	5.70889e9	0.658232	0.329116	+	0.944290i	$0.393249\pi$
$692$	1.06453e9	0.122120
$693$	0	0
$694$	−3.52571e9	−0.400395
$695$	−2.22905e9	−0.251868
$696$	0	0
$697$	−1.27034e10	−1.42103
$698$	−4.17543e8	−0.0464737
$699$	0	0
$700$	−2.70374e9	−0.297935
$701$	−1.23928e10	−1.35880	−0.679402	−	0.733767i	$-0.737761\pi$
$701$	−1.23928e10	−1.35880	−0.679402	+	0.733767i	$0.737761\pi$
$702$	0	0
$703$	−4.68410e8	−0.0508491
$704$	−1.17851e9	−0.127300
$705$	0	0
$706$	1.30640e9	0.139720
$707$	−1.14630e10	−1.21992
$708$	0	0
$709$	1.39979e10	1.47504	0.737518	−	0.675328i	$-0.235998\pi$
$709$	1.39979e10	1.47504	0.737518	+	0.675328i	$0.235998\pi$
$710$	5.16271e9	0.541344
$711$	0	0
$712$	−1.92599e9	−0.199974
$713$	−8.57143e9	−0.885605
$714$	0	0
$715$	2.37152e8	0.0242636
$716$	−1.61815e9	−0.164750
$717$	0	0
$718$	9.59215e9	0.967120
$719$	−7.66012e9	−0.768572	−0.384286	−	0.923214i	$-0.625552\pi$
$719$	−7.66012e9	−0.768572	−0.384286	+	0.923214i	$0.625552\pi$
$720$	0	0
$721$	−1.30101e9	−0.129273
$722$	−6.10161e9	−0.603343
$723$	0	0
$724$	−7.60430e9	−0.744688
$725$	−8.40543e9	−0.819175
$726$	0	0
$727$	1.02618e10	0.990493	0.495247	−	0.868752i	$-0.335078\pi$
$727$	1.02618e10	0.990493	0.495247	+	0.868752i	$0.335078\pi$
$728$	−1.46428e9	−0.140658
$729$	0	0
$730$	4.25940e9	0.405246
$731$	−1.10785e10	−1.04899
$732$	0	0
$733$	−1.67714e10	−1.57292	−0.786459	−	0.617643i	$-0.788088\pi$
$733$	−1.67714e10	−1.57292	−0.786459	+	0.617643i	$0.788088\pi$
$734$	−1.02988e10	−0.961285
$735$	0	0
$736$	−1.50069e10	−1.38746
$737$	1.54051e8	0.0141752
$738$	0	0
$739$	5.49527e9	0.500880	0.250440	−	0.968132i	$-0.419425\pi$
$739$	5.49527e9	0.500880	0.250440	+	0.968132i	$0.419425\pi$
$740$	3.80843e9	0.345490
$741$	0	0
$742$	2.92962e9	0.263268
$743$	1.77986e9	0.159193	0.0795966	−	0.996827i	$-0.474637\pi$
$743$	1.77986e9	0.159193	0.0795966	+	0.996827i	$0.474637\pi$
$744$	0	0
$745$	1.29006e10	1.14305
$746$	6.09683e8	0.0537673
$747$	0	0
$748$	−2.87311e9	−0.251013
$749$	−6.28548e9	−0.546578
$750$	0	0
$751$	−7.12062e9	−0.613448	−0.306724	−	0.951798i	$-0.599233\pi$
$751$	−7.12062e9	−0.613448	−0.306724	+	0.951798i	$0.599233\pi$
$752$	−6.20127e8	−0.0531763
$753$	0	0
$754$	−1.76459e9	−0.149914
$755$	8.23619e9	0.696485
$756$	0	0
$757$	6.43143e9	0.538855	0.269428	−	0.963021i	$-0.413166\pi$
$757$	6.43143e9	0.538855	0.269428	+	0.963021i	$0.413166\pi$
$758$	−4.98319e9	−0.415590
$759$	0	0
$760$	−5.19350e8	−0.0429153
$761$	−1.46912e10	−1.20840	−0.604202	−	0.796831i	$-0.706508\pi$
$761$	−1.46912e10	−1.20840	−0.604202	+	0.796831i	$0.706508\pi$
$762$	0	0
$763$	−2.10979e9	−0.171950
$764$	−6.08175e9	−0.493403
$765$	0	0
$766$	−1.41698e10	−1.13910
$767$	−2.62632e8	−0.0210167
$768$	0	0
$769$	1.24049e10	0.983677	0.491838	−	0.870687i	$-0.336325\pi$
$769$	1.24049e10	0.983677	0.491838	+	0.870687i	$0.336325\pi$
$770$	1.01594e9	0.0801952
$771$	0	0
$772$	5.78276e9	0.452350
$773$	2.65186e9	0.206501	0.103250	−	0.994655i	$-0.467076\pi$
$773$	2.65186e9	0.206501	0.103250	+	0.994655i	$0.467076\pi$
$774$	0	0
$775$	−4.46287e9	−0.344396
$776$	−3.99768e9	−0.307109
$777$	0	0
$778$	5.65113e8	0.0430236
$779$	6.62112e8	0.0501823
$780$	0	0
$781$	−3.84024e9	−0.288456
$782$	2.00405e10	1.49860
$783$	0	0
$784$	−1.02053e8	−0.00756342
$785$	8.82916e9	0.651441
$786$	0	0
$787$	−4.05957e9	−0.296871	−0.148436	−	0.988922i	$-0.547424\pi$
$787$	−4.05957e9	−0.296871	−0.148436	+	0.988922i	$0.547424\pi$
$788$	−1.13968e10	−0.829738
$789$	0	0
$790$	−6.91415e9	−0.498935
$791$	−2.32951e10	−1.67358
$792$	0	0
$793$	2.74102e9	0.195189
$794$	1.78740e9	0.126721
$795$	0	0
$796$	−2.26177e9	−0.158947
$797$	−2.36833e10	−1.65706	−0.828530	−	0.559944i	$-0.810823\pi$
$797$	−2.36833e10	−1.65706	−0.828530	+	0.559944i	$0.810823\pi$
$798$	0	0
$799$	4.09248e10	2.83839
$800$	−7.81364e9	−0.539558
$801$	0	0
$802$	3.81867e9	0.261398
$803$	−3.16832e9	−0.215936
$804$	0	0
$805$	1.22234e10	0.825859
$806$	−9.36910e8	−0.0630268
$807$	0	0
$808$	−2.05458e10	−1.37020
$809$	9.66394e9	0.641704	0.320852	−	0.947129i	$-0.396031\pi$
$809$	9.66394e9	0.641704	0.320852	+	0.947129i	$0.396031\pi$
$810$	0	0
$811$	1.27359e10	0.838413	0.419207	−	0.907891i	$-0.362308\pi$
$811$	1.27359e10	0.838413	0.419207	+	0.907891i	$0.362308\pi$
$812$	1.30392e10	0.854684
$813$	0	0
$814$	1.64232e9	0.106727
$815$	1.37801e10	0.891661
$816$	0	0
$817$	5.77422e8	0.0370438
$818$	−1.08975e10	−0.696132
$819$	0	0
$820$	−5.38333e9	−0.340959
$821$	−1.18429e10	−0.746893	−0.373446	−	0.927652i	$-0.621824\pi$
$821$	−1.18429e10	−0.746893	−0.373446	+	0.927652i	$0.621824\pi$
$822$	0	0
$823$	1.12076e10	0.700829	0.350415	−	0.936595i	$-0.386041\pi$
$823$	1.12076e10	0.700829	0.350415	+	0.936595i	$0.386041\pi$
$824$	−2.33188e9	−0.145198
$825$	0	0
$826$	−1.12509e9	−0.0694636
$827$	1.69838e10	1.04416	0.522078	−	0.852898i	$-0.325157\pi$
$827$	1.69838e10	1.04416	0.522078	+	0.852898i	$0.325157\pi$
$828$	0	0
$829$	−4.59293e9	−0.279994	−0.139997	−	0.990152i	$-0.544709\pi$
$829$	−4.59293e9	−0.279994	−0.139997	+	0.990152i	$0.544709\pi$
$830$	4.74953e9	0.288322
$831$	0	0
$832$	−1.54990e9	−0.0932980
$833$	6.73488e9	0.403713
$834$	0	0
$835$	−6.19230e9	−0.368086
$836$	1.49749e8	0.00886428
$837$	0	0
$838$	−4.03015e9	−0.236574
$839$	2.30945e10	1.35002	0.675012	−	0.737806i	$-0.264138\pi$
$839$	2.30945e10	1.35002	0.675012	+	0.737806i	$0.264138\pi$
$840$	0	0
$841$	2.32866e10	1.34996
$842$	−5.14664e9	−0.297120
$843$	0	0
$844$	−1.34934e10	−0.772542
$845$	−1.16560e10	−0.664584
$846$	0	0
$847$	1.48202e10	0.838033
$848$	−2.95516e8	−0.0166416
$849$	0	0
$850$	1.04345e10	0.582779
$851$	1.97598e10	1.09908
$852$	0	0
$853$	2.22496e10	1.22744	0.613721	−	0.789523i	$-0.289672\pi$
$853$	2.22496e10	1.22744	0.613721	+	0.789523i	$0.289672\pi$
$854$	1.17422e10	0.645132
$855$	0	0
$856$	−1.12659e10	−0.613912
$857$	−1.07105e10	−0.581270	−0.290635	−	0.956834i	$-0.593867\pi$
$857$	−1.07105e10	−0.581270	−0.290635	+	0.956834i	$0.593867\pi$
$858$	0	0
$859$	3.19391e10	1.71928	0.859639	−	0.510902i	$-0.170688\pi$
$859$	3.19391e10	1.71928	0.859639	+	0.510902i	$0.170688\pi$
$860$	−4.69475e9	−0.251691
$861$	0	0
$862$	1.62047e10	0.861717
$863$	−2.90663e10	−1.53940	−0.769700	−	0.638406i	$-0.779594\pi$
$863$	−2.90663e10	−1.53940	−0.769700	+	0.638406i	$0.779594\pi$
$864$	0	0
$865$	−2.50580e9	−0.131641
$866$	4.77979e8	0.0250090
$867$	0	0
$868$	6.92319e9	0.359325
$869$	5.14304e9	0.265858
$870$	0	0
$871$	2.02598e8	0.0103889
$872$	−3.78150e9	−0.193133
$873$	0	0
$874$	−1.04453e9	−0.0529215
$875$	1.82742e10	0.922167
$876$	0	0
$877$	−1.87443e9	−0.0938364	−0.0469182	−	0.998899i	$-0.514940\pi$
$877$	−1.87443e9	−0.0938364	−0.0469182	+	0.998899i	$0.514940\pi$
$878$	−1.74616e10	−0.870668
$879$	0	0
$880$	−1.02479e8	−0.00506927
$881$	3.84330e10	1.89360	0.946802	−	0.321818i	$-0.104294\pi$
$881$	3.84330e10	1.89360	0.946802	+	0.321818i	$0.104294\pi$
$882$	0	0
$883$	1.17045e10	0.572125	0.286063	−	0.958211i	$-0.407653\pi$
$883$	1.17045e10	0.572125	0.286063	+	0.958211i	$0.407653\pi$
$884$	−3.77853e9	−0.183967
$885$	0	0
$886$	−7.91134e9	−0.382148
$887$	−3.09354e9	−0.148841	−0.0744205	−	0.997227i	$-0.523711\pi$
$887$	−3.09354e9	−0.148841	−0.0744205	+	0.997227i	$0.523711\pi$
$888$	0	0
$889$	2.97027e10	1.41788
$890$	1.75738e9	0.0835604
$891$	0	0
$892$	−7.24615e8	−0.0341846
$893$	−2.13304e9	−0.100235
$894$	0	0
$895$	3.80897e9	0.177594
$896$	1.25087e10	0.580942
$897$	0	0
$898$	−3.81064e9	−0.175603
$899$	2.15229e10	0.987967
$900$	0	0
$901$	1.95023e10	0.888279
$902$	−2.32147e9	−0.105327
$903$	0	0
$904$	−4.17533e10	−1.87975
$905$	1.78997e10	0.802744
$906$	0	0
$907$	−5.95597e9	−0.265050	−0.132525	−	0.991180i	$-0.542308\pi$
$907$	−5.95597e9	−0.265050	−0.132525	+	0.991180i	$0.542308\pi$
$908$	−5.57599e9	−0.247185
$909$	0	0
$910$	1.33609e9	0.0587748
$911$	4.08258e10	1.78904	0.894521	−	0.447026i	$-0.147517\pi$
$911$	4.08258e10	1.78904	0.894521	+	0.447026i	$0.147517\pi$
$912$	0	0
$913$	−3.53290e9	−0.153633
$914$	2.69735e9	0.116849
$915$	0	0
$916$	−1.60032e10	−0.687977
$917$	1.19976e10	0.513811
$918$	0	0
$919$	−1.12519e10	−0.478214	−0.239107	−	0.970993i	$-0.576855\pi$
$919$	−1.12519e10	−0.478214	−0.239107	+	0.970993i	$0.576855\pi$
$920$	2.19087e10	0.927598
$921$	0	0
$922$	2.05189e10	0.862176
$923$	−5.05043e9	−0.211408
$924$	0	0
$925$	1.02883e10	0.427414
$926$	−8.11271e9	−0.335759
$927$	0	0
$928$	3.76825e10	1.54783
$929$	−1.44429e10	−0.591018	−0.295509	−	0.955340i	$-0.595489\pi$
$929$	−1.44429e10	−0.591018	−0.295509	+	0.955340i	$0.595489\pi$
$930$	0	0
$931$	−3.51029e8	−0.0142567
$932$	1.99620e10	0.807696
$933$	0	0
$934$	−2.16339e10	−0.868801
$935$	6.76301e9	0.270582
$936$	0	0
$937$	2.43718e10	0.967829	0.483915	−	0.875115i	$-0.339215\pi$
$937$	2.43718e10	0.967829	0.483915	+	0.875115i	$0.339215\pi$
$938$	8.67909e8	0.0343372
$939$	0	0
$940$	1.73428e10	0.681038
$941$	1.04613e10	0.409280	0.204640	−	0.978837i	$-0.434398\pi$
$941$	1.04613e10	0.409280	0.204640	+	0.978837i	$0.434398\pi$
$942$	0	0
$943$	−2.79311e10	−1.08467
$944$	1.13490e8	0.00439090
$945$	0	0
$946$	−2.02453e9	−0.0777509
$947$	6.80097e9	0.260223	0.130112	−	0.991499i	$-0.458466\pi$
$947$	6.80097e9	0.260223	0.130112	+	0.991499i	$0.458466\pi$
$948$	0	0
$949$	−4.16677e9	−0.158259
$950$	−5.43855e8	−0.0205802
$951$	0	0
$952$	−4.17578e10	−1.56859
$953$	9.17306e8	0.0343312	0.0171656	−	0.999853i	$-0.494536\pi$
$953$	9.17306e8	0.0343312	0.0171656	+	0.999853i	$0.494536\pi$
$954$	0	0
$955$	1.43158e10	0.531869
$956$	−3.06488e10	−1.13452
$957$	0	0
$958$	−1.55614e10	−0.571833
$959$	1.40120e10	0.513019
$960$	0	0
$961$	−1.60850e10	−0.584640
$962$	2.15987e9	0.0782195
$963$	0	0
$964$	2.20377e10	0.792314
$965$	−1.36120e10	−0.487615
$966$	0	0
$967$	−3.69970e10	−1.31575	−0.657875	−	0.753127i	$-0.728544\pi$
$967$	−3.69970e10	−1.31575	−0.657875	+	0.753127i	$0.728544\pi$
$968$	2.65631e10	0.941271
$969$	0	0
$970$	3.64771e9	0.128327
$971$	1.31210e10	0.459937	0.229968	−	0.973198i	$-0.426138\pi$
$971$	1.31210e10	0.459937	0.229968	+	0.973198i	$0.426138\pi$
$972$	0	0
$973$	9.34138e9	0.325099
$974$	1.40498e9	0.0487209
$975$	0	0
$976$	−1.18446e9	−0.0407798
$977$	3.90962e10	1.34123	0.670615	−	0.741806i	$-0.266030\pi$
$977$	3.90962e10	1.34123	0.670615	+	0.741806i	$0.266030\pi$
$978$	0	0
$979$	−1.30721e9	−0.0445253
$980$	2.85406e9	0.0968660
$981$	0	0
$982$	−7.42485e9	−0.250206
$983$	3.75059e10	1.25940	0.629699	−	0.776840i	$-0.283178\pi$
$983$	3.75059e10	1.25940	0.629699	+	0.776840i	$0.283178\pi$
$984$	0	0
$985$	2.68269e10	0.894424
$986$	−5.03219e10	−1.67181
$987$	0	0
$988$	1.96941e8	0.00649660
$989$	−2.43585e10	−0.800688
$990$	0	0
$991$	5.39247e10	1.76007	0.880036	−	0.474907i	$-0.157518\pi$
$991$	5.39247e10	1.76007	0.880036	+	0.474907i	$0.157518\pi$
$992$	2.00076e10	0.650735
$993$	0	0
$994$	−2.16355e10	−0.698739
$995$	5.32397e9	0.171338
$996$	0	0
$997$	2.01581e10	0.644192	0.322096	−	0.946707i	$-0.395613\pi$
$997$	2.01581e10	0.644192	0.322096	+	0.946707i	$0.395613\pi$
$998$	8.48317e9	0.270148
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.d.1.11		17
3.2	odd	2		177.8.a.b.1.7	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.b.1.7	✓	17	3.2	odd	2
531.8.a.d.1.11		17	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)

\(f(q)\)	\(=\)	\(q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +O(q^{10})\)	\(q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +1307.20 q^{10} -972.348 q^{11} -1278.77 q^{13} -5478.12 q^{14} +552.586 q^{16} -36467.5 q^{17} +1900.72 q^{19} -15453.9 q^{20} -6664.23 q^{22} -80181.7 q^{23} -41748.1 q^{25} -8764.35 q^{26} +64763.2 q^{28} +201337. q^{29} +106900. q^{31} +187162. q^{32} -249939. q^{34} -152446. q^{35} -246438. q^{37} +13027.1 q^{38} -273238. q^{40} +348348. q^{41} +303791. q^{43} +78785.6 q^{44} -549545. q^{46} -1.12223e6 q^{47} -184682. q^{49} -286131. q^{50} +103614. q^{52} -534787. q^{53} -185453. q^{55} +1.14507e6 q^{56} +1.37991e6 q^{58} +205379. q^{59} -2.14348e6 q^{61} +732666. q^{62} +1.21203e6 q^{64} -243896. q^{65} -158432. q^{67} +2.95482e6 q^{68} -1.04483e6 q^{70} +3.94945e6 q^{71} +3.25842e6 q^{73} -1.68903e6 q^{74} -154008. q^{76} +777186. q^{77} -5.28930e6 q^{79} +105393. q^{80} +2.38749e6 q^{82} +3.63337e6 q^{83} -6.95534e6 q^{85} +2.08211e6 q^{86} +1.39300e6 q^{88} +1.34439e6 q^{89} +1.02210e6 q^{91} +6.49682e6 q^{92} -7.69146e6 q^{94} +362519. q^{95} +2.79048e6 q^{97} -1.26576e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 32 q^{2} + 1166 q^{4} + 1072 q^{5} - 2407 q^{7} + 6645 q^{8}+O(q^{10}) \) 17 * q + 32 * q^2 + 1166 * q^4 + 1072 * q^5 - 2407 * q^7 + 6645 * q^8	\( 17 q + 32 q^{2} + 1166 q^{4} + 1072 q^{5} - 2407 q^{7} + 6645 q^{8} - 6391 q^{10} + 8888 q^{11} - 12702 q^{13} + 17555 q^{14} + 139226 q^{16} + 36167 q^{17} - 71037 q^{19} + 274883 q^{20} - 325182 q^{22} + 269995 q^{23} + 97329 q^{25} + 336906 q^{26} - 901362 q^{28} + 543825 q^{29} - 633109 q^{31} + 837062 q^{32} - 529288 q^{34} + 287621 q^{35} - 867607 q^{37} + 1727169 q^{38} - 815662 q^{40} + 1428939 q^{41} - 477060 q^{43} + 1667926 q^{44} + 5305549 q^{46} + 1217849 q^{47} + 4350738 q^{49} - 4561369 q^{50} + 4175994 q^{52} + 3487068 q^{53} - 960484 q^{55} + 5363196 q^{56} - 3082906 q^{58} + 3491443 q^{59} + 998917 q^{61} + 5742614 q^{62} + 17531621 q^{64} + 6075816 q^{65} - 356026 q^{67} + 16149231 q^{68} - 548798 q^{70} + 12879428 q^{71} - 6176157 q^{73} + 5971906 q^{74} - 17624580 q^{76} - 239687 q^{77} - 18886490 q^{79} + 70463349 q^{80} - 19351611 q^{82} + 22824893 q^{83} - 7973079 q^{85} + 27502196 q^{86} - 62527651 q^{88} + 30609647 q^{89} - 36301521 q^{91} + 41388548 q^{92} + 1010176 q^{94} + 29303629 q^{95} - 26249806 q^{97} + 93110852 q^{98}+O(q^{100}) \) 17 * q + 32 * q^2 + 1166 * q^4 + 1072 * q^5 - 2407 * q^7 + 6645 * q^8 - 6391 * q^10 + 8888 * q^11 - 12702 * q^13 + 17555 * q^14 + 139226 * q^16 + 36167 * q^17 - 71037 * q^19 + 274883 * q^20 - 325182 * q^22 + 269995 * q^23 + 97329 * q^25 + 336906 * q^26 - 901362 * q^28 + 543825 * q^29 - 633109 * q^31 + 837062 * q^32 - 529288 * q^34 + 287621 * q^35 - 867607 * q^37 + 1727169 * q^38 - 815662 * q^40 + 1428939 * q^41 - 477060 * q^43 + 1667926 * q^44 + 5305549 * q^46 + 1217849 * q^47 + 4350738 * q^49 - 4561369 * q^50 + 4175994 * q^52 + 3487068 * q^53 - 960484 * q^55 + 5363196 * q^56 - 3082906 * q^58 + 3491443 * q^59 + 998917 * q^61 + 5742614 * q^62 + 17531621 * q^64 + 6075816 * q^65 - 356026 * q^67 + 16149231 * q^68 - 548798 * q^70 + 12879428 * q^71 - 6176157 * q^73 + 5971906 * q^74 - 17624580 * q^76 - 239687 * q^77 - 18886490 * q^79 + 70463349 * q^80 - 19351611 * q^82 + 22824893 * q^83 - 7973079 * q^85 + 27502196 * q^86 - 62527651 * q^88 + 30609647 * q^89 - 36301521 * q^91 + 41388548 * q^92 + 1010176 * q^94 + 29303629 * q^95 - 26249806 * q^97 + 93110852 * q^98

Introduction

L-functions

Modular forms

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