LMFDB - Embedded newform 75.14.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,14,Mod(1,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$80.4231967139$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 34660x^{3} - 255524x^{2} + 207252123x + 3641392989 $ x^5 - x^4 - 34660x^3 - 255524x^2 + 207252123*x + 3641392989
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-160.423$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-171.423 q^{2} -729.000 q^{3} +21194.0 q^{4} +124968. q^{6} +162072. q^{7} -2.22884e6 q^{8} +531441. q^{9} +6.40531e6 q^{11} -1.54504e7 q^{12} -2.02248e7 q^{13} -2.77829e7 q^{14} +2.08454e8 q^{16} +1.34243e8 q^{17} -9.11014e7 q^{18} +1.95770e8 q^{19} -1.18150e8 q^{21} -1.09802e9 q^{22} +1.34574e9 q^{23} +1.62482e9 q^{24} +3.46700e9 q^{26} -3.87420e8 q^{27} +3.43494e9 q^{28} +4.25977e9 q^{29} +5.99466e9 q^{31} -1.74752e10 q^{32} -4.66947e9 q^{33} -2.30124e10 q^{34} +1.12633e10 q^{36} -2.63847e10 q^{37} -3.35595e10 q^{38} +1.47439e10 q^{39} -2.26917e10 q^{41} +2.02537e10 q^{42} +1.75669e9 q^{43} +1.35754e11 q^{44} -2.30691e11 q^{46} +9.16446e10 q^{47} -1.51963e11 q^{48} -7.06217e10 q^{49} -9.78634e10 q^{51} -4.28643e11 q^{52} -1.75644e10 q^{53} +6.64129e10 q^{54} -3.61232e11 q^{56} -1.42716e11 q^{57} -7.30223e11 q^{58} +3.03625e11 q^{59} +1.25392e11 q^{61} -1.02762e12 q^{62} +8.61316e10 q^{63} +1.28801e12 q^{64} +8.00456e11 q^{66} +2.29310e11 q^{67} +2.84515e12 q^{68} -9.81042e11 q^{69} -1.77148e12 q^{71} -1.18450e12 q^{72} -6.66295e11 q^{73} +4.52296e12 q^{74} +4.14914e12 q^{76} +1.03812e12 q^{77} -2.52744e12 q^{78} +1.73192e12 q^{79} +2.82430e11 q^{81} +3.88988e12 q^{82} +1.27600e12 q^{83} -2.50407e12 q^{84} -3.01137e11 q^{86} -3.10537e12 q^{87} -1.42764e13 q^{88} -4.87003e12 q^{89} -3.27786e12 q^{91} +2.85215e13 q^{92} -4.37011e12 q^{93} -1.57100e13 q^{94} +1.27394e13 q^{96} +4.88961e12 q^{97} +1.21062e13 q^{98} +3.40404e12 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 54 q^{2} - 3645 q^{3} + 28944 q^{4} + 39366 q^{6} + 54809 q^{7} - 538596 q^{8} + 2657205 q^{9} + 4726458 q^{11} - 21100176 q^{12} + 9983301 q^{13} - 14754798 q^{14} + 204457656 q^{16} + 107620650 q^{17}+ \cdots + 2511833565978 q^{99}+O(q^{100}) $ 5 * q - 54 * q^2 - 3645 * q^3 + 28944 * q^4 + 39366 * q^6 + 54809 * q^7 - 538596 * q^8 + 2657205 * q^9 + 4726458 * q^11 - 21100176 * q^12 + 9983301 * q^13 - 14754798 * q^14 + 204457656 * q^16 + 107620650 * q^17 - 28697814 * q^18 + 103761183 * q^19 - 39955761 * q^21 - 1545351500 * q^22 + 446699502 * q^23 + 392636484 * q^24 + 6907508010 * q^26 - 1937102445 * q^27 - 512339152 * q^28 + 2075888466 * q^29 + 18575793161 * q^31 - 19667895336 * q^32 - 3445587882 * q^33 - 19327001132 * q^34 + 15382028304 * q^36 - 14764506566 * q^37 - 53806658562 * q^38 - 7277826429 * q^39 - 9049846824 * q^41 + 10756247742 * q^42 - 129538792611 * q^43 - 25241103360 * q^44 - 101769228324 * q^46 + 123016398138 * q^47 - 149049631224 * q^48 - 23480273638 * q^49 - 78455453850 * q^51 - 20058266480 * q^52 + 141387548652 * q^53 + 20920706406 * q^54 - 427567668660 * q^56 - 75641902407 * q^57 - 211776556444 * q^58 + 1188355601502 * q^59 + 117669052707 * q^61 + 647821896498 * q^62 + 29127749769 * q^63 + 1666031341760 * q^64 + 1126561243500 * q^66 + 418438807145 * q^67 + 3254390367552 * q^68 - 325643936958 * q^69 - 538646573160 * q^71 - 286231996836 * q^72 - 1032119974278 * q^73 + 8866479537972 * q^74 + 6314894350672 * q^76 + 2473561635138 * q^77 - 5035573339290 * q^78 + 4487823310160 * q^79 + 1412147682405 * q^81 + 6726410230448 * q^82 - 2792142569682 * q^83 + 373495241808 * q^84 - 59434309062 * q^86 - 1513322691714 * q^87 - 32640481328136 * q^88 + 8735447438208 * q^89 + 3335432909401 * q^91 + 40169368338624 * q^92 - 13541753214369 * q^93 + 14369118349876 * q^94 + 14337895699944 * q^96 + 13743433155685 * q^97 - 22777071296076 * q^98 + 2511833565978 * q^99

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 $

$2$	−171.423	−1.89398	−0.946989	−	0.321266i	$-0.895892\pi$
$2$	−171.423	−1.89398	−0.946989	+	0.321266i	$0.895892\pi$
$3$	−729.000	−0.577350
$3$	−729.000	−0.577350
$4$	21194.0	2.58715
$5$	0	0
$5$	0	0
$6$	124968.	1.09349
$7$	162072.	0.520679	0.260340	−	0.965517i	$-0.416165\pi$
$7$	162072.	0.520679	0.260340	+	0.965517i	$0.416165\pi$
$8$	−2.22884e6	−3.00603
$9$	531441.	0.333333
$10$	0	0
$11$	6.40531e6	1.09015	0.545077	−	0.838386i	$-0.316501\pi$
$11$	6.40531e6	1.09015	0.545077	+	0.838386i	$0.316501\pi$
$12$	−1.54504e7	−1.49369
$13$	−2.02248e7	−1.16212	−0.581061	−	0.813860i	$-0.697362\pi$
$13$	−2.02248e7	−1.16212	−0.581061	+	0.813860i	$0.697362\pi$
$14$	−2.77829e7	−0.986155
$15$	0	0
$16$	2.08454e8	3.10621
$17$	1.34243e8	1.34888	0.674442	−	0.738328i	$-0.264384\pi$
$17$	1.34243e8	1.34888	0.674442	+	0.738328i	$0.264384\pi$
$18$	−9.11014e7	−0.631326
$19$	1.95770e8	0.954657	0.477329	−	0.878725i	$-0.341605\pi$
$19$	1.95770e8	0.954657	0.477329	+	0.878725i	$0.341605\pi$
$20$	0	0
$21$	−1.18150e8	−0.300614
$22$	−1.09802e9	−2.06473
$23$	1.34574e9	1.89552	0.947761	−	0.318981i	$-0.103341\pi$
$23$	1.34574e9	1.89552	0.947761	+	0.318981i	$0.103341\pi$
$24$	1.62482e9	1.73553
$25$	0	0
$26$	3.46700e9	2.20103
$27$	−3.87420e8	−0.192450
$28$	3.43494e9	1.34708
$29$	4.25977e9	1.32984	0.664919	−	0.746915i	$-0.268466\pi$
$29$	4.25977e9	1.32984	0.664919	+	0.746915i	$0.268466\pi$
$30$	0	0
$31$	5.99466e9	1.21315	0.606574	−	0.795027i	$-0.292544\pi$
$31$	5.99466e9	1.21315	0.606574	+	0.795027i	$0.292544\pi$
$32$	−1.74752e10	−2.87705
$33$	−4.66947e9	−0.629400
$34$	−2.30124e10	−2.55476
$35$	0	0
$36$	1.12633e10	0.862384
$37$	−2.63847e10	−1.69060	−0.845301	−	0.534291i	$-0.820579\pi$
$37$	−2.63847e10	−1.69060	−0.845301	+	0.534291i	$0.820579\pi$
$38$	−3.35595e10	−1.80810
$39$	1.47439e10	0.670951
$40$	0	0
$41$	−2.26917e10	−0.746055	−0.373028	−	0.927820i	$-0.621680\pi$
$41$	−2.26917e10	−0.746055	−0.373028	+	0.927820i	$0.621680\pi$
$42$	2.02537e10	0.569357
$43$	1.75669e9	0.0423788	0.0211894	−	0.999775i	$-0.493255\pi$
$43$	1.75669e9	0.0423788	0.0211894	+	0.999775i	$0.493255\pi$
$44$	1.35754e11	2.82039
$45$	0	0
$46$	−2.30691e11	−3.59008
$47$	9.16446e10	1.24014	0.620070	−	0.784547i	$-0.287104\pi$
$47$	9.16446e10	1.24014	0.620070	+	0.784547i	$0.287104\pi$
$48$	−1.51963e11	−1.79337
$49$	−7.06217e10	−0.728893
$50$	0	0
$51$	−9.78634e10	−0.778779
$52$	−4.28643e11	−3.00659
$53$	−1.75644e10	−0.108853	−0.0544263	−	0.998518i	$-0.517333\pi$
$53$	−1.75644e10	−0.108853	−0.0544263	+	0.998518i	$0.517333\pi$
$54$	6.64129e10	0.364496
$55$	0	0
$56$	−3.61232e11	−1.56518
$57$	−1.42716e11	−0.551172
$58$	−7.30223e11	−2.51868
$59$	3.03625e11	0.937128	0.468564	−	0.883430i	$-0.344772\pi$
$59$	3.03625e11	0.937128	0.468564	+	0.883430i	$0.344772\pi$
$60$	0	0
$61$	1.25392e11	0.311621	0.155810	−	0.987787i	$-0.450201\pi$
$61$	1.25392e11	0.311621	0.155810	+	0.987787i	$0.450201\pi$
$62$	−1.02762e12	−2.29767
$63$	8.61316e10	0.173560
$64$	1.28801e12	2.34287
$65$	0	0
$66$	8.00456e11	1.19207
$67$	2.29310e11	0.309697	0.154848	−	0.987938i	$-0.450511\pi$
$67$	2.29310e11	0.309697	0.154848	+	0.987938i	$0.450511\pi$
$68$	2.84515e12	3.48977
$69$	−9.81042e11	−1.09438
$70$	0	0
$71$	−1.77148e12	−1.64118	−0.820592	−	0.571515i	$-0.806356\pi$
$71$	−1.77148e12	−1.64118	−0.820592	+	0.571515i	$0.806356\pi$
$72$	−1.18450e12	−1.00201
$73$	−6.66295e11	−0.515309	−0.257655	−	0.966237i	$-0.582950\pi$
$73$	−6.66295e11	−0.515309	−0.257655	+	0.966237i	$0.582950\pi$
$74$	4.52296e12	3.20196
$75$	0	0
$76$	4.14914e12	2.46984
$77$	1.03812e12	0.567620
$78$	−2.52744e12	−1.27077
$79$	1.73192e12	0.801587	0.400793	−	0.916168i	$-0.368734\pi$
$79$	1.73192e12	0.801587	0.400793	+	0.916168i	$0.368734\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	3.88988e12	1.41301
$83$	1.27600e12	0.428394	0.214197	−	0.976790i	$-0.431287\pi$
$83$	1.27600e12	0.428394	0.214197	+	0.976790i	$0.431287\pi$
$84$	−2.50407e12	−0.777735
$85$	0	0
$86$	−3.01137e11	−0.0802646
$87$	−3.10537e12	−0.767782
$88$	−1.42764e13	−3.27704
$89$	−4.87003e12	−1.03872	−0.519358	−	0.854557i	$-0.673829\pi$
$89$	−4.87003e12	−1.03872	−0.519358	+	0.854557i	$0.673829\pi$
$90$	0	0
$91$	−3.27786e12	−0.605093
$92$	2.85215e13	4.90401
$93$	−4.37011e12	−0.700411
$94$	−1.57100e13	−2.34880
$95$	0	0
$96$	1.27394e13	1.66107
$97$	4.88961e12	0.596016	0.298008	−	0.954563i	$-0.403678\pi$
$97$	4.88961e12	0.596016	0.298008	+	0.954563i	$0.403678\pi$
$98$	1.21062e13	1.38051
$99$	3.40404e12	0.363384
$100$	0	0
$101$	−3.69993e12	−0.346820	−0.173410	−	0.984850i	$-0.555479\pi$
$101$	−3.69993e12	−0.346820	−0.173410	+	0.984850i	$0.555479\pi$
$102$	1.67761e13	1.47499
$103$	1.06059e13	0.875196	0.437598	−	0.899171i	$-0.355829\pi$
$103$	1.06059e13	0.875196	0.437598	+	0.899171i	$0.355829\pi$
$104$	4.50777e13	3.49338
$105$	0	0
$106$	3.01094e12	0.206165
$107$	−8.97407e12	−0.578089	−0.289045	−	0.957316i	$-0.593338\pi$
$107$	−8.97407e12	−0.578089	−0.289045	+	0.957316i	$0.593338\pi$
$108$	−8.21097e12	−0.497898
$109$	−1.61009e13	−0.919557	−0.459779	−	0.888034i	$-0.652071\pi$
$109$	−1.61009e13	−0.919557	−0.459779	+	0.888034i	$0.652071\pi$
$110$	0	0
$111$	1.92345e13	0.976069
$112$	3.37845e13	1.61734
$113$	8.20355e12	0.370674	0.185337	−	0.982675i	$-0.440662\pi$
$113$	8.20355e12	0.370674	0.185337	+	0.982675i	$0.440662\pi$
$114$	2.44649e13	1.04391
$115$	0	0
$116$	9.02813e13	3.44049
$117$	−1.07483e13	−0.387374
$118$	−5.20483e13	−1.77490
$119$	2.17571e13	0.702336
$120$	0	0
$121$	6.50525e12	0.188434
$122$	−2.14952e13	−0.590203
$123$	1.65422e13	0.430735
$124$	1.27051e14	3.13860
$125$	0	0
$126$	−1.47650e13	−0.328718
$127$	−9.32098e12	−0.197123	−0.0985615	−	0.995131i	$-0.531424\pi$
$127$	−9.32098e12	−0.197123	−0.0985615	+	0.995131i	$0.531424\pi$
$128$	−7.76374e13	−1.56029
$129$	−1.28062e12	−0.0244674
$130$	0	0
$131$	−2.79642e13	−0.483436	−0.241718	−	0.970347i	$-0.577711\pi$
$131$	−2.79642e13	−0.483436	−0.241718	+	0.970347i	$0.577711\pi$
$132$	−9.89645e13	−1.62835
$133$	3.17288e13	0.497070
$134$	−3.93090e13	−0.586558
$135$	0	0
$136$	−2.99207e14	−4.05479
$137$	−6.65701e13	−0.860193	−0.430096	−	0.902783i	$-0.641520\pi$
$137$	−6.65701e13	−0.860193	−0.430096	+	0.902783i	$0.641520\pi$
$138$	1.68173e14	2.07273
$139$	9.02944e13	1.06185	0.530927	−	0.847418i	$-0.321844\pi$
$139$	9.02944e13	1.06185	0.530927	+	0.847418i	$0.321844\pi$
$140$	0	0
$141$	−6.68089e13	−0.715995
$142$	3.03673e14	3.10836
$143$	−1.29546e14	−1.26689
$144$	1.10781e14	1.03540
$145$	0	0
$146$	1.14219e14	0.975984
$147$	5.14832e13	0.420827
$148$	−5.59196e14	−4.37384
$149$	3.93096e13	0.294298	0.147149	−	0.989114i	$-0.452990\pi$
$149$	3.93096e13	0.294298	0.147149	+	0.989114i	$0.452990\pi$
$150$	0	0
$151$	−2.49867e14	−1.71538	−0.857688	−	0.514170i	$-0.828100\pi$
$151$	−2.49867e14	−1.71538	−0.857688	+	0.514170i	$0.828100\pi$
$152$	−4.36339e14	−2.86973
$153$	7.13424e13	0.449628
$154$	−1.77958e14	−1.07506
$155$	0	0
$156$	3.12481e14	1.73585
$157$	−1.41333e14	−0.753174	−0.376587	−	0.926381i	$-0.622902\pi$
$157$	−1.41333e14	−0.753174	−0.376587	+	0.926381i	$0.622902\pi$
$158$	−2.96891e14	−1.51819
$159$	1.28044e13	0.0628461
$160$	0	0
$161$	2.18106e14	0.986959
$162$	−4.84150e13	−0.210442
$163$	4.27510e14	1.78537	0.892683	−	0.450685i	$-0.148820\pi$
$163$	4.27510e14	1.78537	0.892683	+	0.450685i	$0.148820\pi$
$164$	−4.80926e14	−1.93016
$165$	0	0
$166$	−2.18736e14	−0.811369
$167$	2.90911e14	1.03777	0.518887	−	0.854843i	$-0.326346\pi$
$167$	2.90911e14	1.03777	0.518887	+	0.854843i	$0.326346\pi$
$168$	2.63338e14	0.903657
$169$	1.06166e14	0.350527
$170$	0	0
$171$	1.04040e14	0.318219
$172$	3.72311e13	0.109641
$173$	6.63985e14	1.88304	0.941518	−	0.336964i	$-0.109400\pi$
$173$	6.63985e14	1.88304	0.941518	+	0.336964i	$0.109400\pi$
$174$	5.32333e14	1.45416
$175$	0	0
$176$	1.33521e15	3.38624
$177$	−2.21342e14	−0.541051
$178$	8.34837e14	1.96730
$179$	5.30658e14	1.20579	0.602893	−	0.797822i	$-0.294015\pi$
$179$	5.30658e14	1.20579	0.602893	+	0.797822i	$0.294015\pi$
$180$	0	0
$181$	−6.50123e14	−1.37431	−0.687156	−	0.726510i	$-0.741141\pi$
$181$	−6.50123e14	−1.37431	−0.687156	+	0.726510i	$0.741141\pi$
$182$	5.61903e14	1.14603
$183$	−9.14109e13	−0.179914
$184$	−2.99943e15	−5.69800
$185$	0	0
$186$	7.49138e14	1.32656
$187$	8.59870e14	1.47049
$188$	1.94231e15	3.20843
$189$	−6.27900e13	−0.100205
$190$	0	0
$191$	−4.78551e14	−0.713200	−0.356600	−	0.934257i	$-0.616064\pi$
$191$	−4.78551e14	−0.713200	−0.356600	+	0.934257i	$0.616064\pi$
$192$	−9.38957e14	−1.35266
$193$	−1.31043e15	−1.82511	−0.912556	−	0.408951i	$-0.865895\pi$
$193$	−1.31043e15	−1.82511	−0.912556	+	0.408951i	$0.865895\pi$
$194$	−8.38194e14	−1.12884
$195$	0	0
$196$	−1.49675e15	−1.88576
$197$	−1.50108e14	−0.182967	−0.0914834	−	0.995807i	$-0.529161\pi$
$197$	−1.50108e14	−0.182967	−0.0914834	+	0.995807i	$0.529161\pi$
$198$	−5.83532e14	−0.688242
$199$	4.64828e13	0.0530576	0.0265288	−	0.999648i	$-0.491555\pi$
$199$	4.64828e13	0.0530576	0.0265288	+	0.999648i	$0.491555\pi$
$200$	0	0
$201$	−1.67167e14	−0.178803
$202$	6.34254e14	0.656869
$203$	6.90388e14	0.692419
$204$	−2.07411e15	−2.01482
$205$	0	0
$206$	−1.81810e15	−1.65760
$207$	7.15179e14	0.631841
$208$	−4.21593e15	−3.60979
$209$	1.25397e15	1.04072
$210$	0	0
$211$	6.63210e14	0.517386	0.258693	−	0.965960i	$-0.416708\pi$
$211$	6.63210e14	0.517386	0.258693	+	0.965960i	$0.416708\pi$
$212$	−3.72258e14	−0.281619
$213$	1.29141e15	0.947538
$214$	1.53836e15	1.09489
$215$	0	0
$216$	8.63498e14	0.578511
$217$	9.71566e14	0.631661
$218$	2.76008e15	1.74162
$219$	4.85729e14	0.297514
$220$	0	0
$221$	−2.71504e15	−1.56757
$222$	−3.29723e15	−1.84865
$223$	2.43567e15	1.32629	0.663143	−	0.748492i	$-0.269222\pi$
$223$	2.43567e15	1.32629	0.663143	+	0.748492i	$0.269222\pi$
$224$	−2.83224e15	−1.49802
$225$	0	0
$226$	−1.40628e15	−0.702048
$227$	2.90501e14	0.140922	0.0704611	−	0.997515i	$-0.477553\pi$
$227$	2.90501e14	0.140922	0.0704611	+	0.997515i	$0.477553\pi$
$228$	−3.02472e15	−1.42596
$229$	2.27636e15	1.04306	0.521531	−	0.853233i	$-0.325361\pi$
$229$	2.27636e15	1.04306	0.521531	+	0.853233i	$0.325361\pi$
$230$	0	0
$231$	−7.56790e14	−0.327716
$232$	−9.49433e15	−3.99754
$233$	3.97697e15	1.62831	0.814157	−	0.580644i	$-0.197199\pi$
$233$	3.97697e15	1.62831	0.814157	+	0.580644i	$0.197199\pi$
$234$	1.84250e15	0.733678
$235$	0	0
$236$	6.43501e15	2.42449
$237$	−1.26257e15	−0.462796
$238$	−3.72967e15	−1.33021
$239$	1.67198e15	0.580290	0.290145	−	0.956983i	$-0.406296\pi$
$239$	1.67198e15	0.580290	0.290145	+	0.956983i	$0.406296\pi$
$240$	0	0
$241$	1.34014e15	0.440594	0.220297	−	0.975433i	$-0.429297\pi$
$241$	1.34014e15	0.440594	0.220297	+	0.975433i	$0.429297\pi$
$242$	−1.11515e15	−0.356890
$243$	−2.05891e14	−0.0641500
$244$	2.65756e15	0.806211
$245$	0	0
$246$	−2.83572e15	−0.815803
$247$	−3.95940e15	−1.10943
$248$	−1.33611e16	−3.64676
$249$	−9.30205e14	−0.247334
$250$	0	0
$251$	3.02079e15	0.762501	0.381251	−	0.924472i	$-0.375493\pi$
$251$	3.02079e15	0.762501	0.381251	+	0.924472i	$0.375493\pi$
$252$	1.82547e15	0.449026
$253$	8.61985e15	2.06641
$254$	1.59783e15	0.373347
$255$	0	0
$256$	2.75750e15	0.612289
$257$	3.33335e15	0.721632	0.360816	−	0.932637i	$-0.382498\pi$
$257$	3.33335e15	0.721632	0.360816	+	0.932637i	$0.382498\pi$
$258$	2.19529e14	0.0463408
$259$	−4.27622e15	−0.880261
$260$	0	0
$261$	2.26381e15	0.443279
$262$	4.79371e15	0.915616
$263$	−3.46253e15	−0.645181	−0.322591	−	0.946539i	$-0.604554\pi$
$263$	−3.46253e15	−0.645181	−0.322591	+	0.946539i	$0.604554\pi$
$264$	1.04075e16	1.89200
$265$	0	0
$266$	−5.43905e15	−0.941440
$267$	3.55025e15	0.599703
$268$	4.85998e15	0.801232
$269$	6.87853e15	1.10689	0.553447	−	0.832884i	$-0.313312\pi$
$269$	6.87853e15	1.10689	0.553447	+	0.832884i	$0.313312\pi$
$270$	0	0
$271$	−2.62602e15	−0.402714	−0.201357	−	0.979518i	$-0.564535\pi$
$271$	−2.62602e15	−0.402714	−0.201357	+	0.979518i	$0.564535\pi$
$272$	2.79836e16	4.18991
$273$	2.38956e15	0.349351
$274$	1.14117e16	1.62919
$275$	0	0
$276$	−2.07921e16	−2.83133
$277$	9.79892e15	1.30335	0.651673	−	0.758500i	$-0.274067\pi$
$277$	9.79892e15	1.30335	0.651673	+	0.758500i	$0.274067\pi$
$278$	−1.54786e16	−2.01113
$279$	3.18581e15	0.404382
$280$	0	0
$281$	−4.26310e15	−0.516577	−0.258288	−	0.966068i	$-0.583158\pi$
$281$	−4.26310e15	−0.516577	−0.258288	+	0.966068i	$0.583158\pi$
$282$	1.14526e16	1.35608
$283$	1.39703e16	1.61657	0.808284	−	0.588793i	$-0.200397\pi$
$283$	1.39703e16	1.61657	0.808284	+	0.588793i	$0.200397\pi$
$284$	−3.75447e16	−4.24599
$285$	0	0
$286$	2.22072e16	2.39946
$287$	−3.67768e15	−0.388456
$288$	−9.28706e15	−0.959018
$289$	8.11668e15	0.819488
$290$	0	0
$291$	−3.56453e15	−0.344110
$292$	−1.41214e16	−1.33318
$293$	−6.22813e14	−0.0575067	−0.0287533	−	0.999587i	$-0.509154\pi$
$293$	−6.22813e14	−0.0575067	−0.0287533	+	0.999587i	$0.509154\pi$
$294$	−8.82543e15	−0.797036
$295$	0	0
$296$	5.88073e16	5.08200
$297$	−2.48155e15	−0.209800
$298$	−6.73858e15	−0.557395
$299$	−2.72172e16	−2.20283
$300$	0	0
$301$	2.84709e14	0.0220658
$302$	4.28331e16	3.24888
$303$	2.69725e15	0.200237
$304$	4.08090e16	2.96536
$305$	0	0
$306$	−1.22297e16	−0.851586
$307$	−4.30448e14	−0.0293441	−0.0146721	−	0.999892i	$-0.504670\pi$
$307$	−4.30448e14	−0.0293441	−0.0146721	+	0.999892i	$0.504670\pi$
$308$	2.20019e16	1.46852
$309$	−7.73169e15	−0.505294
$310$	0	0
$311$	−5.15074e15	−0.322795	−0.161398	−	0.986889i	$-0.551600\pi$
$311$	−5.15074e15	−0.322795	−0.161398	+	0.986889i	$0.551600\pi$
$312$	−3.28617e16	−2.01690
$313$	−2.72712e16	−1.63933	−0.819664	−	0.572845i	$-0.805840\pi$
$313$	−2.72712e16	−1.63933	−0.819664	+	0.572845i	$0.805840\pi$
$314$	2.42277e16	1.42650
$315$	0	0
$316$	3.67061e16	2.07383
$317$	−2.61056e16	−1.44494	−0.722469	−	0.691403i	$-0.756993\pi$
$317$	−2.61056e16	−1.44494	−0.722469	+	0.691403i	$0.756993\pi$
$318$	−2.19498e15	−0.119029
$319$	2.72851e16	1.44973
$320$	0	0
$321$	6.54209e15	0.333760
$322$	−3.73884e16	−1.86928
$323$	2.62808e16	1.28772
$324$	5.98580e15	0.287461
$325$	0	0
$326$	−7.32853e16	−3.38144
$327$	1.17376e16	0.530907
$328$	5.05760e16	2.24267
$329$	1.48530e16	0.645715
$330$	0	0
$331$	3.46176e16	1.44682	0.723412	−	0.690417i	$-0.242573\pi$
$331$	3.46176e16	1.44682	0.723412	+	0.690417i	$0.242573\pi$
$332$	2.70435e16	1.10832
$333$	−1.40219e16	−0.563534
$334$	−4.98689e16	−1.96552
$335$	0	0
$336$	−2.46289e16	−0.933770
$337$	−1.10574e16	−0.411207	−0.205603	−	0.978635i	$-0.565916\pi$
$337$	−1.10574e16	−0.411207	−0.205603	+	0.978635i	$0.565916\pi$
$338$	−1.81993e16	−0.663890
$339$	−5.98039e15	−0.214009
$340$	0	0
$341$	3.83976e16	1.32252
$342$	−1.78349e16	−0.602700
$343$	−2.71488e16	−0.900199
$344$	−3.91537e15	−0.127392
$345$	0	0
$346$	−1.13822e17	−3.56643
$347$	4.80581e15	0.147783	0.0738916	−	0.997266i	$-0.476458\pi$
$347$	4.80581e15	0.147783	0.0738916	+	0.997266i	$0.476458\pi$
$348$	−6.58151e16	−1.98637
$349$	−4.47644e16	−1.32607	−0.663036	−	0.748587i	$-0.730733\pi$
$349$	−4.47644e16	−1.32607	−0.663036	+	0.748587i	$0.730733\pi$
$350$	0	0
$351$	7.83549e15	0.223650
$352$	−1.11934e17	−3.13643
$353$	−2.63962e16	−0.726117	−0.363058	−	0.931766i	$-0.618267\pi$
$353$	−2.63962e16	−0.726117	−0.363058	+	0.931766i	$0.618267\pi$
$354$	3.79432e16	1.02474
$355$	0	0
$356$	−1.03215e17	−2.68732
$357$	−1.58609e16	−0.405494
$358$	−9.09672e16	−2.28373
$359$	1.93311e16	0.476588	0.238294	−	0.971193i	$-0.423412\pi$
$359$	1.93311e16	0.476588	0.238294	+	0.971193i	$0.423412\pi$
$360$	0	0
$361$	−3.72715e15	−0.0886299
$362$	1.11446e17	2.60292
$363$	−4.74233e15	−0.108792
$364$	−6.94709e16	−1.56547
$365$	0	0
$366$	1.56700e16	0.340754
$367$	−5.50709e16	−1.17650	−0.588251	−	0.808678i	$-0.700183\pi$
$367$	−5.50709e16	−1.17650	−0.588251	+	0.808678i	$0.700183\pi$
$368$	2.80524e17	5.88788
$369$	−1.20593e16	−0.248685
$370$	0	0
$371$	−2.84669e15	−0.0566773
$372$	−9.26198e16	−1.81207
$373$	4.45131e16	0.855817	0.427908	−	0.903822i	$-0.359251\pi$
$373$	4.45131e16	0.855817	0.427908	+	0.903822i	$0.359251\pi$
$374$	−1.47402e17	−2.78508
$375$	0	0
$376$	−2.04261e17	−3.72790
$377$	−8.61528e16	−1.54543
$378$	1.07637e16	0.189786
$379$	5.75094e16	0.996744	0.498372	−	0.866963i	$-0.333931\pi$
$379$	5.75094e16	0.996744	0.498372	+	0.866963i	$0.333931\pi$
$380$	0	0
$381$	6.79500e15	0.113809
$382$	8.20348e16	1.35079
$383$	−5.35837e16	−0.867443	−0.433722	−	0.901047i	$-0.642800\pi$
$383$	−5.35837e16	−0.867443	−0.433722	+	0.901047i	$0.642800\pi$
$384$	5.65977e16	0.900835
$385$	0	0
$386$	2.24637e17	3.45672
$387$	9.33574e14	0.0141263
$388$	1.03630e17	1.54199
$389$	7.60860e16	1.11335	0.556676	−	0.830730i	$-0.312077\pi$
$389$	7.60860e16	1.11335	0.556676	+	0.830730i	$0.312077\pi$
$390$	0	0
$391$	1.80656e17	2.55684
$392$	1.57404e17	2.19108
$393$	2.03859e16	0.279112
$394$	2.57320e16	0.346535
$395$	0	0
$396$	7.21451e16	0.940131
$397$	6.28610e16	0.805829	0.402915	−	0.915238i	$-0.367997\pi$
$397$	6.28610e16	0.805829	0.402915	+	0.915238i	$0.367997\pi$
$398$	−7.96824e15	−0.100490
$399$	−2.31303e16	−0.286984
$400$	0	0
$401$	6.07087e16	0.729143	0.364572	−	0.931175i	$-0.381215\pi$
$401$	6.07087e16	0.729143	0.364572	+	0.931175i	$0.381215\pi$
$402$	2.86563e16	0.338650
$403$	−1.21241e17	−1.40982
$404$	−7.84161e16	−0.897276
$405$	0	0
$406$	−1.18349e17	−1.31143
$407$	−1.69002e17	−1.84301
$408$	2.18122e17	2.34103
$409$	−1.10708e17	−1.16944	−0.584718	−	0.811236i	$-0.698795\pi$
$409$	−1.10708e17	−1.16944	−0.584718	+	0.811236i	$0.698795\pi$
$410$	0	0
$411$	4.85296e16	0.496633
$412$	2.24781e17	2.26426
$413$	4.92090e16	0.487943
$414$	−1.22598e17	−1.19669
$415$	0	0
$416$	3.53432e17	3.34349
$417$	−6.58247e16	−0.613062
$418$	−2.14959e17	−1.97111
$419$	−1.92248e17	−1.73568	−0.867840	−	0.496843i	$-0.834492\pi$
$419$	−1.92248e17	−1.73568	−0.867840	+	0.496843i	$0.834492\pi$
$420$	0	0
$421$	3.40528e16	0.298071	0.149035	−	0.988832i	$-0.452383\pi$
$421$	3.40528e16	0.298071	0.149035	+	0.988832i	$0.452383\pi$
$422$	−1.13690e17	−0.979918
$423$	4.87037e16	0.413380
$424$	3.91481e16	0.327215
$425$	0	0
$426$	−2.21378e17	−1.79462
$427$	2.03226e16	0.162255
$428$	−1.90196e17	−1.49560
$429$	9.44389e16	0.731440
$430$	0	0
$431$	1.29097e17	0.970098	0.485049	−	0.874487i	$-0.338802\pi$
$431$	1.29097e17	0.970098	0.485049	+	0.874487i	$0.338802\pi$
$432$	−8.07593e16	−0.597790
$433$	−2.87818e16	−0.209868	−0.104934	−	0.994479i	$-0.533463\pi$
$433$	−2.87818e16	−0.209868	−0.104934	+	0.994479i	$0.533463\pi$
$434$	−1.66549e17	−1.19635
$435$	0	0
$436$	−3.41242e17	−2.37904
$437$	2.63455e17	1.80957
$438$	−8.32653e16	−0.563485
$439$	1.71993e17	1.14681	0.573405	−	0.819272i	$-0.305622\pi$
$439$	1.71993e17	1.14681	0.573405	+	0.819272i	$0.305622\pi$
$440$	0	0
$441$	−3.75313e16	−0.242964
$442$	4.65421e17	2.96894
$443$	−2.30800e16	−0.145081	−0.0725406	−	0.997365i	$-0.523111\pi$
$443$	−2.30800e16	−0.145081	−0.0725406	+	0.997365i	$0.523111\pi$
$444$	4.07654e17	2.52524
$445$	0	0
$446$	−4.17531e17	−2.51196
$447$	−2.86567e16	−0.169913
$448$	2.08750e17	1.21989
$449$	−8.02803e16	−0.462389	−0.231195	−	0.972908i	$-0.574263\pi$
$449$	−8.02803e16	−0.462389	−0.231195	+	0.972908i	$0.574263\pi$
$450$	0	0
$451$	−1.45347e17	−0.813315
$452$	1.73866e17	0.958990
$453$	1.82153e17	0.990373
$454$	−4.97986e16	−0.266904
$455$	0	0
$456$	3.18091e17	1.65684
$457$	−1.74637e17	−0.896769	−0.448384	−	0.893841i	$-0.648000\pi$
$457$	−1.74637e17	−0.896769	−0.448384	+	0.893841i	$0.648000\pi$
$458$	−3.90221e17	−1.97554
$459$	−5.20086e16	−0.259593
$460$	0	0
$461$	1.27387e17	0.618113	0.309057	−	0.951044i	$-0.399987\pi$
$461$	1.27387e17	0.618113	0.309057	+	0.951044i	$0.399987\pi$
$462$	1.29731e17	0.620686
$463$	−1.45811e17	−0.687880	−0.343940	−	0.938992i	$-0.611762\pi$
$463$	−1.45811e17	−0.687880	−0.343940	+	0.938992i	$0.611762\pi$
$464$	8.87965e17	4.13075
$465$	0	0
$466$	−6.81745e17	−3.08399
$467$	−3.09767e17	−1.38190	−0.690949	−	0.722903i	$-0.742807\pi$
$467$	−3.09767e17	−1.38190	−0.690949	+	0.722903i	$0.742807\pi$
$468$	−2.27798e17	−1.00220
$469$	3.71647e16	0.161253
$470$	0	0
$471$	1.03032e17	0.434845
$472$	−6.76730e17	−2.81704
$473$	1.12521e16	0.0461994
$474$	2.16433e17	0.876526
$475$	0	0
$476$	4.61118e17	1.81705
$477$	−9.33442e15	−0.0362842
$478$	−2.86617e17	−1.09906
$479$	5.21405e16	0.197240	0.0986199	−	0.995125i	$-0.468557\pi$
$479$	5.21405e16	0.197240	0.0986199	+	0.995125i	$0.468557\pi$
$480$	0	0
$481$	5.33624e17	1.96468
$482$	−2.29731e17	−0.834476
$483$	−1.58999e17	−0.569821
$484$	1.37872e17	0.487507
$485$	0	0
$486$	3.52945e16	0.121499
$487$	1.16892e16	0.0397052	0.0198526	−	0.999803i	$-0.493680\pi$
$487$	1.16892e16	0.0397052	0.0198526	+	0.999803i	$0.493680\pi$
$488$	−2.79479e17	−0.936743
$489$	−3.11655e17	−1.03078
$490$	0	0
$491$	−1.09561e17	−0.352878	−0.176439	−	0.984312i	$-0.556458\pi$
$491$	−1.09561e17	−0.352878	−0.176439	+	0.984312i	$0.556458\pi$
$492$	3.50595e17	1.11438
$493$	5.71845e17	1.79380
$494$	6.78733e17	2.10123
$495$	0	0
$496$	1.24961e18	3.76829
$497$	−2.87107e17	−0.854530
$498$	1.59459e17	0.468444
$499$	3.19995e17	0.927876	0.463938	−	0.885868i	$-0.346436\pi$
$499$	3.19995e17	0.927876	0.463938	+	0.885868i	$0.346436\pi$
$500$	0	0
$501$	−2.12074e17	−0.599160
$502$	−5.17833e17	−1.44416
$503$	6.67305e17	1.83710	0.918549	−	0.395308i	$-0.129362\pi$
$503$	6.67305e17	1.83710	0.918549	+	0.395308i	$0.129362\pi$
$504$	−1.91974e17	−0.521726
$505$	0	0
$506$	−1.47764e18	−3.91373
$507$	−7.73949e16	−0.202377
$508$	−1.97548e17	−0.509987
$509$	3.20815e17	0.817690	0.408845	−	0.912604i	$-0.365932\pi$
$509$	3.20815e17	0.817690	0.408845	+	0.912604i	$0.365932\pi$
$510$	0	0
$511$	−1.07988e17	−0.268311
$512$	1.63305e17	0.400632
$513$	−7.58452e16	−0.183724
$514$	−5.71415e17	−1.36676
$515$	0	0
$516$	−2.71415e16	−0.0633010
$517$	5.87012e17	1.35194
$518$	7.33044e17	1.66720
$519$	−4.84045e17	−1.08717
$520$	0	0
$521$	−3.17985e17	−0.696565	−0.348283	−	0.937390i	$-0.613235\pi$
$521$	−3.17985e17	−0.696565	−0.348283	+	0.937390i	$0.613235\pi$
$522$	−3.88071e17	−0.839561
$523$	−4.44725e16	−0.0950234	−0.0475117	−	0.998871i	$-0.515129\pi$
$523$	−4.44725e16	−0.0950234	−0.0475117	+	0.998871i	$0.515129\pi$
$524$	−5.92672e17	−1.25072
$525$	0	0
$526$	5.93559e17	1.22196
$527$	8.04743e17	1.63639
$528$	−9.73369e17	−1.95505
$529$	1.30697e18	2.59301
$530$	0	0
$531$	1.61359e17	0.312376
$532$	6.72458e17	1.28600
$533$	4.58933e17	0.867007
$534$	−6.08596e17	−1.13582
$535$	0	0
$536$	−5.11094e17	−0.930958
$537$	−3.86850e17	−0.696161
$538$	−1.17914e18	−2.09643
$539$	−4.52354e17	−0.794605
$540$	0	0
$541$	2.56768e17	0.440310	0.220155	−	0.975465i	$-0.429344\pi$
$541$	2.56768e17	0.440310	0.220155	+	0.975465i	$0.429344\pi$
$542$	4.50160e17	0.762732
$543$	4.73940e17	0.793459
$544$	−2.34593e18	−3.88081
$545$	0	0
$546$	−4.09627e17	−0.661662
$547$	−1.95784e17	−0.312507	−0.156253	−	0.987717i	$-0.549942\pi$
$547$	−1.95784e17	−0.312507	−0.156253	+	0.987717i	$0.549942\pi$
$548$	−1.41088e18	−2.22545
$549$	6.66386e16	0.103874
$550$	0	0
$551$	8.33934e17	1.26954
$552$	2.18658e18	3.28974
$553$	2.80695e17	0.417370
$554$	−1.67976e18	−2.46851
$555$	0	0
$556$	1.91370e18	2.74718
$557$	−4.15036e17	−0.588881	−0.294440	−	0.955670i	$-0.595133\pi$
$557$	−4.15036e17	−0.588881	−0.294440	+	0.955670i	$0.595133\pi$
$558$	−5.46122e17	−0.765891
$559$	−3.55285e16	−0.0492494
$560$	0	0
$561$	−6.26845e17	−0.848988
$562$	7.30795e17	0.978385
$563$	8.31196e17	1.10002	0.550008	−	0.835160i	$-0.314625\pi$
$563$	8.31196e17	1.10002	0.550008	+	0.835160i	$0.314625\pi$
$564$	−1.41594e18	−1.85239
$565$	0	0
$566$	−2.39484e18	−3.06174
$567$	4.57739e16	0.0578533
$568$	3.94834e18	4.93345
$569$	8.66409e17	1.07027	0.535135	−	0.844767i	$-0.320261\pi$
$569$	8.66409e17	1.07027	0.535135	+	0.844767i	$0.320261\pi$
$570$	0	0
$571$	1.71702e17	0.207320	0.103660	−	0.994613i	$-0.466945\pi$
$571$	1.71702e17	0.207320	0.103660	+	0.994613i	$0.466945\pi$
$572$	−2.74559e18	−3.27764
$573$	3.48864e17	0.411766
$574$	6.30440e17	0.735727
$575$	0	0
$576$	6.84500e17	0.780957
$577$	1.30809e18	1.47569	0.737844	−	0.674972i	$-0.235844\pi$
$577$	1.30809e18	1.47569	0.737844	+	0.674972i	$0.235844\pi$
$578$	−1.39139e18	−1.55209
$579$	9.55300e17	1.05373
$580$	0	0
$581$	2.06804e17	0.223056
$582$	6.11043e17	0.651737
$583$	−1.12505e17	−0.118666
$584$	1.48506e18	1.54904
$585$	0	0
$586$	1.06765e17	0.108916
$587$	5.55292e16	0.0560239	0.0280120	−	0.999608i	$-0.491082\pi$
$587$	5.55292e16	0.0560239	0.0280120	+	0.999608i	$0.491082\pi$
$588$	1.09113e18	1.08874
$589$	1.17357e18	1.15814
$590$	0	0
$591$	1.09428e17	0.105636
$592$	−5.50000e18	−5.25136
$593$	8.89526e17	0.840046	0.420023	−	0.907513i	$-0.362022\pi$
$593$	8.89526e17	0.840046	0.420023	+	0.907513i	$0.362022\pi$
$594$	4.25395e17	0.397357
$595$	0	0
$596$	8.33126e17	0.761395
$597$	−3.38860e16	−0.0306328
$598$	4.66566e18	4.17211
$599$	−1.14465e17	−0.101251	−0.0506256	−	0.998718i	$-0.516122\pi$
$599$	−1.14465e17	−0.101251	−0.0506256	+	0.998718i	$0.516122\pi$
$600$	0	0
$601$	−1.39928e17	−0.121122	−0.0605609	−	0.998165i	$-0.519289\pi$
$601$	−1.39928e17	−0.121122	−0.0605609	+	0.998165i	$0.519289\pi$
$602$	−4.88058e16	−0.0417921
$603$	1.21865e17	0.103232
$604$	−5.29568e18	−4.43794
$605$	0	0
$606$	−4.62371e17	−0.379244
$607$	−1.78906e18	−1.45178	−0.725888	−	0.687813i	$-0.758571\pi$
$607$	−1.78906e18	−1.45178	−0.725888	+	0.687813i	$0.758571\pi$
$608$	−3.42112e18	−2.74660
$609$	−5.03293e17	−0.399768
$610$	0	0
$611$	−1.85349e18	−1.44119
$612$	1.51203e18	1.16326
$613$	−1.34925e18	−1.02707	−0.513534	−	0.858069i	$-0.671664\pi$
$613$	−1.34925e18	−1.02707	−0.513534	+	0.858069i	$0.671664\pi$
$614$	7.37888e16	0.0555771
$615$	0	0
$616$	−2.31380e18	−1.70629
$617$	−8.71819e17	−0.636169	−0.318085	−	0.948062i	$-0.603040\pi$
$617$	−8.71819e17	−0.636169	−0.318085	+	0.948062i	$0.603040\pi$
$618$	1.32539e18	0.957016
$619$	1.54715e18	1.10546	0.552728	−	0.833361i	$-0.313587\pi$
$619$	1.54715e18	1.10546	0.552728	+	0.833361i	$0.313587\pi$
$620$	0	0
$621$	−5.21366e17	−0.364793
$622$	8.82956e17	0.611367
$623$	−7.89295e17	−0.540838
$624$	3.07341e18	2.08411
$625$	0	0
$626$	4.67492e18	3.10485
$627$	−9.14141e17	−0.600861
$628$	−2.99540e18	−1.94858
$629$	−3.54197e18	−2.28042
$630$	0	0
$631$	1.22843e18	0.774746	0.387373	−	0.921923i	$-0.373383\pi$
$631$	1.22843e18	0.774746	0.387373	+	0.921923i	$0.373383\pi$
$632$	−3.86016e18	−2.40960
$633$	−4.83480e17	−0.298713
$634$	4.47512e18	2.73668
$635$	0	0
$636$	2.71376e17	0.162593
$637$	1.42831e18	0.847062
$638$	−4.67730e18	−2.74575
$639$	−9.41437e17	−0.547061
$640$	0	0
$641$	−2.84548e18	−1.62023	−0.810117	−	0.586268i	$-0.800596\pi$
$641$	−2.84548e18	−1.62023	−0.810117	+	0.586268i	$0.800596\pi$
$642$	−1.12147e18	−0.632134
$643$	−1.63945e18	−0.914800	−0.457400	−	0.889261i	$-0.651219\pi$
$643$	−1.63945e18	−0.914800	−0.457400	+	0.889261i	$0.651219\pi$
$644$	4.62253e18	2.55341
$645$	0	0
$646$	−4.50514e18	−2.43892
$647$	1.77607e17	0.0951879	0.0475940	−	0.998867i	$-0.484845\pi$
$647$	1.77607e17	0.0951879	0.0475940	+	0.998867i	$0.484845\pi$
$648$	−6.29490e17	−0.334004
$649$	1.94481e18	1.02161
$650$	0	0
$651$	−7.08271e17	−0.364690
$652$	9.06064e18	4.61901
$653$	7.71684e17	0.389497	0.194748	−	0.980853i	$-0.437611\pi$
$653$	7.71684e17	0.389497	0.194748	+	0.980853i	$0.437611\pi$
$654$	−2.01210e18	−1.00553
$655$	0	0
$656$	−4.73017e18	−2.31740
$657$	−3.54096e17	−0.171770
$658$	−2.54615e18	−1.22297
$659$	−1.20269e18	−0.572004	−0.286002	−	0.958229i	$-0.592326\pi$
$659$	−1.20269e18	−0.572004	−0.286002	+	0.958229i	$0.592326\pi$
$660$	0	0
$661$	−5.97449e17	−0.278607	−0.139303	−	0.990250i	$-0.544486\pi$
$661$	−5.97449e17	−0.278607	−0.139303	+	0.990250i	$0.544486\pi$
$662$	−5.93427e18	−2.74025
$663$	1.97926e18	0.905035
$664$	−2.84400e18	−1.28777
$665$	0	0
$666$	2.40368e18	1.06732
$667$	5.73252e18	2.52074
$668$	6.16555e18	2.68488
$669$	−1.77561e18	−0.765732
$670$	0	0
$671$	8.03176e17	0.339715
$672$	2.06471e18	0.864884
$673$	−1.39299e18	−0.577896	−0.288948	−	0.957345i	$-0.593305\pi$
$673$	−1.39299e18	−0.577896	−0.288948	+	0.957345i	$0.593305\pi$
$674$	1.89550e18	0.778817
$675$	0	0
$676$	2.25007e18	0.906866
$677$	−4.30164e17	−0.171715	−0.0858573	−	0.996307i	$-0.527363\pi$
$677$	−4.30164e17	−0.171715	−0.0858573	+	0.996307i	$0.527363\pi$
$678$	1.02518e18	0.405328
$679$	7.92469e17	0.310333
$680$	0	0
$681$	−2.11775e17	−0.0813615
$682$	−6.58225e18	−2.50482
$683$	−4.26023e18	−1.60583	−0.802913	−	0.596096i	$-0.796718\pi$
$683$	−4.26023e18	−1.60583	−0.802913	+	0.596096i	$0.796718\pi$
$684$	2.20502e18	0.823281
$685$	0	0
$686$	4.65393e18	1.70496
$687$	−1.65946e18	−0.602212
$688$	3.66188e17	0.131637
$689$	3.55235e17	0.126500
$690$	0	0
$691$	3.41734e17	0.119421	0.0597105	−	0.998216i	$-0.480982\pi$
$691$	3.41734e17	0.119421	0.0597105	+	0.998216i	$0.480982\pi$
$692$	1.40725e19	4.87170
$693$	5.51700e17	0.189207
$694$	−8.23828e17	−0.279898
$695$	0	0
$696$	6.92137e18	2.30798
$697$	−3.04620e18	−1.00634
$698$	7.67366e18	2.51155
$699$	−2.89921e18	−0.940108
$700$	0	0
$701$	2.89335e18	0.920945	0.460472	−	0.887674i	$-0.347680\pi$
$701$	2.89335e18	0.920945	0.460472	+	0.887674i	$0.347680\pi$
$702$	−1.34319e18	−0.423589
$703$	−5.16533e18	−1.61394
$704$	8.25008e18	2.55409
$705$	0	0
$706$	4.52493e18	1.37525
$707$	−5.99654e17	−0.180582
$708$	−4.69112e18	−1.39978
$709$	1.11626e18	0.330038	0.165019	−	0.986290i	$-0.447231\pi$
$709$	1.11626e18	0.330038	0.165019	+	0.986290i	$0.447231\pi$
$710$	0	0
$711$	9.20411e17	0.267196
$712$	1.08545e19	3.12241
$713$	8.06723e18	2.29955
$714$	2.71893e18	0.767997
$715$	0	0
$716$	1.12467e19	3.11955
$717$	−1.21887e18	−0.335031
$718$	−3.31380e18	−0.902646
$719$	−2.24400e18	−0.605738	−0.302869	−	0.953032i	$-0.597945\pi$
$719$	−2.24400e18	−0.605738	−0.302869	+	0.953032i	$0.597945\pi$
$720$	0	0
$721$	1.71892e18	0.455696
$722$	6.38921e17	0.167863
$723$	−9.76961e17	−0.254377
$724$	−1.37787e19	−3.55555
$725$	0	0
$726$	8.12946e17	0.206050
$727$	−2.69899e18	−0.677997	−0.338999	−	0.940787i	$-0.610088\pi$
$727$	−2.69899e18	−0.677997	−0.338999	+	0.940787i	$0.610088\pi$
$728$	7.30583e18	1.81893
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	2.35823e17	0.0571641
$732$	−1.93736e18	−0.465466
$733$	−6.95930e18	−1.65726	−0.828628	−	0.559800i	$-0.810878\pi$
$733$	−6.95930e18	−1.65726	−0.828628	+	0.559800i	$0.810878\pi$
$734$	9.44044e18	2.22827
$735$	0	0
$736$	−2.35170e19	−5.45352
$737$	1.46880e18	0.337617
$738$	2.06724e18	0.471004
$739$	−2.95895e18	−0.668266	−0.334133	−	0.942526i	$-0.608443\pi$
$739$	−2.95895e18	−0.668266	−0.334133	+	0.942526i	$0.608443\pi$
$740$	0	0
$741$	2.88640e18	0.640528
$742$	4.87989e17	0.107346
$743$	−1.75250e18	−0.382147	−0.191074	−	0.981576i	$-0.561197\pi$
$743$	−1.75250e18	−0.382147	−0.191074	+	0.981576i	$0.561197\pi$
$744$	9.74026e18	2.10546
$745$	0	0
$746$	−7.63059e18	−1.62090
$747$	6.78120e17	0.142798
$748$	1.82240e19	3.80438
$749$	−1.45444e18	−0.300999
$750$	0	0
$751$	3.37899e18	0.687269	0.343635	−	0.939103i	$-0.388342\pi$
$751$	3.37899e18	0.687269	0.343635	+	0.939103i	$0.388342\pi$
$752$	1.91037e19	3.85213
$753$	−2.20215e18	−0.440230
$754$	1.47686e19	2.92702
$755$	0	0
$756$	−1.33077e18	−0.259245
$757$	8.54187e18	1.64979	0.824897	−	0.565284i	$-0.191233\pi$
$757$	8.54187e18	1.64979	0.824897	+	0.565284i	$0.191233\pi$
$758$	−9.85845e18	−1.88781
$759$	−6.28387e18	−1.19304
$760$	0	0
$761$	−7.91490e18	−1.47722	−0.738610	−	0.674133i	$-0.764518\pi$
$761$	−7.91490e18	−1.47722	−0.738610	+	0.674133i	$0.764518\pi$
$762$	−1.16482e18	−0.215552
$763$	−2.60951e18	−0.478795
$764$	−1.01424e19	−1.84516
$765$	0	0
$766$	9.18550e18	1.64292
$767$	−6.14074e18	−1.08906
$768$	−2.01022e18	−0.353505
$769$	−4.76586e18	−0.831036	−0.415518	−	0.909585i	$-0.636400\pi$
$769$	−4.76586e18	−0.831036	−0.415518	+	0.909585i	$0.636400\pi$
$770$	0	0
$771$	−2.43001e18	−0.416635
$772$	−2.77731e19	−4.72185
$773$	−5.28988e18	−0.891824	−0.445912	−	0.895077i	$-0.647121\pi$
$773$	−5.28988e18	−0.891824	−0.445912	+	0.895077i	$0.647121\pi$
$774$	−1.60036e17	−0.0267549
$775$	0	0
$776$	−1.08982e19	−1.79164
$777$	3.11736e18	0.508219
$778$	−1.30429e19	−2.10866
$779$	−4.44234e18	−0.712227
$780$	0	0
$781$	−1.13469e19	−1.78914
$782$	−3.09687e19	−4.84260
$783$	−1.65032e18	−0.255927
$784$	−1.47214e19	−2.26409
$785$	0	0
$786$	−3.49462e18	−0.528631
$787$	1.03442e19	1.55189	0.775947	−	0.630798i	$-0.217272\pi$
$787$	1.03442e19	1.55189	0.775947	+	0.630798i	$0.217272\pi$
$788$	−3.18138e18	−0.473363
$789$	2.52419e18	0.372496
$790$	0	0
$791$	1.32957e18	0.193002
$792$	−7.58706e18	−1.09235
$793$	−2.53603e18	−0.362141
$794$	−1.07758e19	−1.52622
$795$	0	0
$796$	9.85155e17	0.137268
$797$	−1.34315e18	−0.185629	−0.0928147	−	0.995683i	$-0.529586\pi$
$797$	−1.34315e18	−0.185629	−0.0928147	+	0.995683i	$0.529586\pi$
$798$	3.96507e18	0.543541
$799$	1.23027e19	1.67280
$800$	0	0
$801$	−2.58813e18	−0.346238
$802$	−1.04069e19	−1.38098
$803$	−4.26782e18	−0.561766
$804$	−3.54293e18	−0.462592
$805$	0	0
$806$	2.07835e19	2.67018
$807$	−5.01445e18	−0.639065
$808$	8.24654e18	1.04255
$809$	−3.20015e18	−0.401333	−0.200667	−	0.979660i	$-0.564311\pi$
$809$	−3.20015e18	−0.401333	−0.200667	+	0.979660i	$0.564311\pi$
$810$	0	0
$811$	8.85553e18	1.09290	0.546448	−	0.837493i	$-0.315980\pi$
$811$	8.85553e18	1.09290	0.546448	+	0.837493i	$0.315980\pi$
$812$	1.46321e19	1.79139
$813$	1.91437e18	0.232507
$814$	2.89709e19	3.49063
$815$	0	0
$816$	−2.04000e19	−2.41905
$817$	3.43906e17	0.0404572
$818$	1.89779e19	2.21489
$819$	−1.74199e18	−0.201698
$820$	0	0
$821$	1.03933e19	1.18446	0.592232	−	0.805767i	$-0.298247\pi$
$821$	1.03933e19	1.18446	0.592232	+	0.805767i	$0.298247\pi$
$822$	−8.31911e18	−0.940611
$823$	4.86493e18	0.545730	0.272865	−	0.962052i	$-0.412029\pi$
$823$	4.86493e18	0.545730	0.272865	+	0.962052i	$0.412029\pi$
$824$	−2.36388e19	−2.63087
$825$	0	0
$826$	−8.43557e18	−0.924154
$827$	8.76105e18	0.952292	0.476146	−	0.879366i	$-0.342033\pi$
$827$	8.76105e18	0.952292	0.476146	+	0.879366i	$0.342033\pi$
$828$	1.51575e19	1.63467
$829$	9.26848e18	0.991754	0.495877	−	0.868393i	$-0.334847\pi$
$829$	9.26848e18	0.991754	0.495877	+	0.868393i	$0.334847\pi$
$830$	0	0
$831$	−7.14341e18	−0.752487
$832$	−2.60496e19	−2.72270
$833$	−9.48049e18	−0.983192
$834$	1.12839e19	1.16113
$835$	0	0
$836$	2.65765e19	2.69251
$837$	−2.32245e18	−0.233470
$838$	3.29557e19	3.28734
$839$	−3.96926e18	−0.392877	−0.196439	−	0.980516i	$-0.562938\pi$
$839$	−3.96926e18	−0.392877	−0.196439	+	0.980516i	$0.562938\pi$
$840$	0	0
$841$	7.88498e18	0.768469
$842$	−5.83744e18	−0.564539
$843$	3.10780e18	0.298246
$844$	1.40560e19	1.33856
$845$	0	0
$846$	−8.34895e18	−0.782933
$847$	1.05432e18	0.0981137
$848$	−3.66136e18	−0.338119
$849$	−1.01844e19	−0.933326
$850$	0	0
$851$	−3.55069e19	−3.20457
$852$	2.73701e19	2.45142
$853$	−1.77925e19	−1.58150	−0.790750	−	0.612140i	$-0.790309\pi$
$853$	−1.77925e19	−1.58150	−0.790750	+	0.612140i	$0.790309\pi$
$854$	−3.48376e18	−0.307307
$855$	0	0
$856$	2.00017e19	1.73775
$857$	3.27576e18	0.282446	0.141223	−	0.989978i	$-0.454896\pi$
$857$	3.27576e18	0.282446	0.141223	+	0.989978i	$0.454896\pi$
$858$	−1.61890e19	−1.38533
$859$	−1.16115e19	−0.986124	−0.493062	−	0.869994i	$-0.664122\pi$
$859$	−1.16115e19	−0.986124	−0.493062	+	0.869994i	$0.664122\pi$
$860$	0	0
$861$	2.68103e18	0.224275
$862$	−2.21303e19	−1.83734
$863$	3.08418e18	0.254138	0.127069	−	0.991894i	$-0.459443\pi$
$863$	3.08418e18	0.254138	0.127069	+	0.991894i	$0.459443\pi$
$864$	6.77026e18	0.553689
$865$	0	0
$866$	4.93387e18	0.397485
$867$	−5.91706e18	−0.473132
$868$	2.05913e19	1.63420
$869$	1.10935e19	0.873853
$870$	0	0
$871$	−4.63773e18	−0.359905
$872$	3.58864e19	2.76422
$873$	2.59854e18	0.198672
$874$	−4.51622e19	−3.42729
$875$	0	0
$876$	1.02945e19	0.769714
$877$	5.99187e18	0.444698	0.222349	−	0.974967i	$-0.428628\pi$
$877$	5.99187e18	0.444698	0.222349	+	0.974967i	$0.428628\pi$
$878$	−2.94836e19	−2.17203
$879$	4.54031e17	0.0332015
$880$	0	0
$881$	3.90778e18	0.281570	0.140785	−	0.990040i	$-0.455037\pi$
$881$	3.90778e18	0.281570	0.140785	+	0.990040i	$0.455037\pi$
$882$	6.43374e18	0.460169
$883$	1.78814e19	1.26957	0.634786	−	0.772688i	$-0.281088\pi$
$883$	1.78814e19	1.26957	0.634786	+	0.772688i	$0.281088\pi$
$884$	−5.75424e19	−4.05554
$885$	0	0
$886$	3.95645e18	0.274781
$887$	7.56032e18	0.521238	0.260619	−	0.965442i	$-0.416073\pi$
$887$	7.56032e18	0.521238	0.260619	+	0.965442i	$0.416073\pi$
$888$	−4.28705e19	−2.93410
$889$	−1.51067e18	−0.102638
$890$	0	0
$891$	1.80905e18	0.121128
$892$	5.16216e19	3.43131
$893$	1.79412e19	1.18391
$894$	4.91243e18	0.321812
$895$	0	0
$896$	−1.25828e19	−0.812412
$897$	1.98413e19	1.27180
$898$	1.37619e19	0.875755
$899$	2.55358e19	1.61329
$900$	0	0
$901$	−2.35790e18	−0.146830
$902$	2.49159e19	1.54040
$903$	−2.07553e17	−0.0127397
$904$	−1.82844e19	−1.11426
$905$	0	0
$906$	−3.12253e19	−1.87574
$907$	−1.88109e19	−1.12192	−0.560961	−	0.827842i	$-0.689568\pi$
$907$	−1.88109e19	−1.12192	−0.560961	+	0.827842i	$0.689568\pi$
$908$	6.15686e18	0.364587
$909$	−1.96629e18	−0.115607
$910$	0	0
$911$	−3.05113e19	−1.76844	−0.884221	−	0.467070i	$-0.845310\pi$
$911$	−3.05113e19	−1.76844	−0.884221	+	0.467070i	$0.845310\pi$
$912$	−2.97498e19	−1.71205
$913$	8.17318e18	0.467015
$914$	2.99368e19	1.69846
$915$	0	0
$916$	4.82450e19	2.69856
$917$	−4.53221e18	−0.251715
$918$	8.91549e18	0.491663
$919$	1.14631e19	0.627697	0.313849	−	0.949473i	$-0.398382\pi$
$919$	1.14631e19	0.627697	0.313849	+	0.949473i	$0.398382\pi$
$920$	0	0
$921$	3.13797e17	0.0169418
$922$	−2.18370e19	−1.17069
$923$	3.58278e19	1.90725
$924$	−1.60394e19	−0.847851
$925$	0	0
$926$	2.49953e19	1.30283
$927$	5.63640e18	0.291732
$928$	−7.44404e19	−3.82602
$929$	5.19474e18	0.265132	0.132566	−	0.991174i	$-0.457678\pi$
$929$	5.19474e18	0.265132	0.132566	+	0.991174i	$0.457678\pi$
$930$	0	0
$931$	−1.38256e19	−0.695843
$932$	8.42876e19	4.21270
$933$	3.75489e18	0.186366
$934$	5.31014e19	2.61729
$935$	0	0
$936$	2.39562e19	1.16446
$937$	2.77238e18	0.133828	0.0669138	−	0.997759i	$-0.478685\pi$
$937$	2.77238e18	0.133828	0.0669138	+	0.997759i	$0.478685\pi$
$938$	−6.37089e18	−0.305409
$939$	1.98807e19	0.946466
$940$	0	0
$941$	−7.87387e18	−0.369705	−0.184853	−	0.982766i	$-0.559181\pi$
$941$	−7.87387e18	−0.369705	−0.184853	+	0.982766i	$0.559181\pi$
$942$	−1.76620e19	−0.823587
$943$	−3.05370e19	−1.41416
$944$	6.32918e19	2.91091
$945$	0	0
$946$	−1.92887e18	−0.0875007
$947$	−8.23534e18	−0.371028	−0.185514	−	0.982642i	$-0.559395\pi$
$947$	−8.23534e18	−0.371028	−0.185514	+	0.982642i	$0.559395\pi$
$948$	−2.67588e19	−1.19733
$949$	1.34757e19	0.598852
$950$	0	0
$951$	1.90310e19	0.834235
$952$	−4.84930e19	−2.11125
$953$	2.33839e19	1.01114	0.505572	−	0.862784i	$-0.331281\pi$
$953$	2.33839e19	1.01114	0.505572	+	0.862784i	$0.331281\pi$
$954$	1.60014e18	0.0687215
$955$	0	0
$956$	3.54359e19	1.50130
$957$	−1.98908e19	−0.837000
$958$	−8.93810e18	−0.373568
$959$	−1.07891e19	−0.447885
$960$	0	0
$961$	1.15184e19	0.471726
$962$	−9.14757e19	−3.72107
$963$	−4.76919e18	−0.192696
$964$	2.84028e19	1.13988
$965$	0	0
$966$	2.72562e19	1.07923
$967$	2.07067e19	0.814401	0.407201	−	0.913339i	$-0.366505\pi$
$967$	2.07067e19	0.814401	0.407201	+	0.913339i	$0.366505\pi$
$968$	−1.44992e19	−0.566438
$969$	−1.91587e19	−0.743466
$970$	0	0
$971$	2.54963e19	0.976229	0.488115	−	0.872780i	$-0.337685\pi$
$971$	2.54963e19	0.976229	0.488115	+	0.872780i	$0.337685\pi$
$972$	−4.36365e18	−0.165966
$973$	1.46342e19	0.552886
$974$	−2.00381e18	−0.0752008
$975$	0	0
$976$	2.61385e19	0.967959
$977$	7.01123e18	0.257917	0.128958	−	0.991650i	$-0.458837\pi$
$977$	7.01123e18	0.257917	0.128958	+	0.991650i	$0.458837\pi$
$978$	5.34250e19	1.95228
$979$	−3.11940e19	−1.13236
$980$	0	0
$981$	−8.55670e18	−0.306519
$982$	1.87812e19	0.668343
$983$	2.52036e19	0.890973	0.445486	−	0.895289i	$-0.353031\pi$
$983$	2.52036e19	0.890973	0.445486	+	0.895289i	$0.353031\pi$
$984$	−3.68699e19	−1.29480
$985$	0	0
$986$	−9.80276e19	−3.39741
$987$	−1.08278e19	−0.372804
$988$	−8.39153e19	−2.87026
$989$	2.36403e18	0.0803300
$990$	0	0
$991$	8.95430e18	0.300298	0.150149	−	0.988663i	$-0.452025\pi$
$991$	8.95430e18	0.300298	0.150149	+	0.988663i	$0.452025\pi$
$992$	−1.04758e20	−3.49029
$993$	−2.52363e19	−0.835324
$994$	4.92169e19	1.61846
$995$	0	0
$996$	−1.97147e19	−0.639890
$997$	−3.47606e19	−1.12091	−0.560453	−	0.828186i	$-0.689373\pi$
$997$	−3.47606e19	−1.12091	−0.560453	+	0.828186i	$0.689373\pi$
$998$	−5.48546e19	−1.75738
$999$	1.02220e19	0.325356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.14.a.i.1.1	✓	5
5.2	odd	4		75.14.b.h.49.1		10
5.3	odd	4		75.14.b.h.49.10		10
5.4	even	2		75.14.a.j.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.14.a.i.1.1	✓	5	1.1	even	1	trivial
75.14.a.j.1.5	yes	5	5.4	even	2
75.14.b.h.49.1		10	5.2	odd	4
75.14.b.h.49.10		10	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-171.423 q^{2} -729.000 q^{3} +21194.0 q^{4} +124968. q^{6} +162072. q^{7} -2.22884e6 q^{8} +531441. q^{9} +6.40531e6 q^{11} -1.54504e7 q^{12} -2.02248e7 q^{13} -2.77829e7 q^{14} +2.08454e8 q^{16} +1.34243e8 q^{17} -9.11014e7 q^{18} +1.95770e8 q^{19} -1.18150e8 q^{21} -1.09802e9 q^{22} +1.34574e9 q^{23} +1.62482e9 q^{24} +3.46700e9 q^{26} -3.87420e8 q^{27} +3.43494e9 q^{28} +4.25977e9 q^{29} +5.99466e9 q^{31} -1.74752e10 q^{32} -4.66947e9 q^{33} -2.30124e10 q^{34} +1.12633e10 q^{36} -2.63847e10 q^{37} -3.35595e10 q^{38} +1.47439e10 q^{39} -2.26917e10 q^{41} +2.02537e10 q^{42} +1.75669e9 q^{43} +1.35754e11 q^{44} -2.30691e11 q^{46} +9.16446e10 q^{47} -1.51963e11 q^{48} -7.06217e10 q^{49} -9.78634e10 q^{51} -4.28643e11 q^{52} -1.75644e10 q^{53} +6.64129e10 q^{54} -3.61232e11 q^{56} -1.42716e11 q^{57} -7.30223e11 q^{58} +3.03625e11 q^{59} +1.25392e11 q^{61} -1.02762e12 q^{62} +8.61316e10 q^{63} +1.28801e12 q^{64} +8.00456e11 q^{66} +2.29310e11 q^{67} +2.84515e12 q^{68} -9.81042e11 q^{69} -1.77148e12 q^{71} -1.18450e12 q^{72} -6.66295e11 q^{73} +4.52296e12 q^{74} +4.14914e12 q^{76} +1.03812e12 q^{77} -2.52744e12 q^{78} +1.73192e12 q^{79} +2.82430e11 q^{81} +3.88988e12 q^{82} +1.27600e12 q^{83} -2.50407e12 q^{84} -3.01137e11 q^{86} -3.10537e12 q^{87} -1.42764e13 q^{88} -4.87003e12 q^{89} -3.27786e12 q^{91} +2.85215e13 q^{92} -4.37011e12 q^{93} -1.57100e13 q^{94} +1.27394e13 q^{96} +4.88961e12 q^{97} +1.21062e13 q^{98} +3.40404e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 54 q^{2} - 3645 q^{3} + 28944 q^{4} + 39366 q^{6} + 54809 q^{7} - 538596 q^{8} + 2657205 q^{9} + 4726458 q^{11} - 21100176 q^{12} + 9983301 q^{13} - 14754798 q^{14} + 204457656 q^{16} + 107620650 q^{17}+ \cdots + 2511833565978 q^{99}+O(q^{100}) \) 5 * q - 54 * q^2 - 3645 * q^3 + 28944 * q^4 + 39366 * q^6 + 54809 * q^7 - 538596 * q^8 + 2657205 * q^9 + 4726458 * q^11 - 21100176 * q^12 + 9983301 * q^13 - 14754798 * q^14 + 204457656 * q^16 + 107620650 * q^17 - 28697814 * q^18 + 103761183 * q^19 - 39955761 * q^21 - 1545351500 * q^22 + 446699502 * q^23 + 392636484 * q^24 + 6907508010 * q^26 - 1937102445 * q^27 - 512339152 * q^28 + 2075888466 * q^29 + 18575793161 * q^31 - 19667895336 * q^32 - 3445587882 * q^33 - 19327001132 * q^34 + 15382028304 * q^36 - 14764506566 * q^37 - 53806658562 * q^38 - 7277826429 * q^39 - 9049846824 * q^41 + 10756247742 * q^42 - 129538792611 * q^43 - 25241103360 * q^44 - 101769228324 * q^46 + 123016398138 * q^47 - 149049631224 * q^48 - 23480273638 * q^49 - 78455453850 * q^51 - 20058266480 * q^52 + 141387548652 * q^53 + 20920706406 * q^54 - 427567668660 * q^56 - 75641902407 * q^57 - 211776556444 * q^58 + 1188355601502 * q^59 + 117669052707 * q^61 + 647821896498 * q^62 + 29127749769 * q^63 + 1666031341760 * q^64 + 1126561243500 * q^66 + 418438807145 * q^67 + 3254390367552 * q^68 - 325643936958 * q^69 - 538646573160 * q^71 - 286231996836 * q^72 - 1032119974278 * q^73 + 8866479537972 * q^74 + 6314894350672 * q^76 + 2473561635138 * q^77 - 5035573339290 * q^78 + 4487823310160 * q^79 + 1412147682405 * q^81 + 6726410230448 * q^82 - 2792142569682 * q^83 + 373495241808 * q^84 - 59434309062 * q^86 - 1513322691714 * q^87 - 32640481328136 * q^88 + 8735447438208 * q^89 + 3335432909401 * q^91 + 40169368338624 * q^92 - 13541753214369 * q^93 + 14369118349876 * q^94 + 14337895699944 * q^96 + 13743433155685 * q^97 - 22777071296076 * q^98 + 2511833565978 * q^99

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