LMFDB - Embedded newform 45.12.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,12,Mod(1,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	45.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:	$34.5754431252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1801}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 450 $ x^2 - x - 450
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-20.7191$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.7191 q^{2} -1279.65 q^{4} +3125.00 q^{5} -72157.2 q^{7} -92239.5 q^{8} +O(q^{10})$	\(q+27.7191 q^{2} -1279.65 q^{4} +3125.00 q^{5} -72157.2 q^{7} -92239.5 q^{8} +86622.2 q^{10} +509499. q^{11} +1.85516e6 q^{13} -2.00013e6 q^{14} +63931.4 q^{16} -5.94676e6 q^{17} +6.02189e6 q^{19} -3.99891e6 q^{20} +1.41228e7 q^{22} +4.82627e7 q^{23} +9.76562e6 q^{25} +5.14234e7 q^{26} +9.23361e7 q^{28} +1.13550e7 q^{29} -1.72184e8 q^{31} +1.90679e8 q^{32} -1.64839e8 q^{34} -2.25491e8 q^{35} +6.25191e8 q^{37} +1.66921e8 q^{38} -2.88248e8 q^{40} +5.53971e8 q^{41} +1.52106e9 q^{43} -6.51981e8 q^{44} +1.33780e9 q^{46} -1.19456e9 q^{47} +3.22934e9 q^{49} +2.70694e8 q^{50} -2.37396e9 q^{52} -1.22320e9 q^{53} +1.59218e9 q^{55} +6.65575e9 q^{56} +3.14750e8 q^{58} +5.83637e9 q^{59} -6.61097e9 q^{61} -4.77277e9 q^{62} +5.15451e9 q^{64} +5.79738e9 q^{65} +1.66261e10 q^{67} +7.60978e9 q^{68} -6.25042e9 q^{70} -7.36200e9 q^{71} -6.35726e9 q^{73} +1.73297e10 q^{74} -7.70593e9 q^{76} -3.67640e10 q^{77} +2.47565e10 q^{79} +1.99786e8 q^{80} +1.53556e10 q^{82} -3.59416e10 q^{83} -1.85836e10 q^{85} +4.21625e10 q^{86} -4.69959e10 q^{88} -7.47690e10 q^{89} -1.33863e11 q^{91} -6.17594e10 q^{92} -3.31121e10 q^{94} +1.88184e10 q^{95} -1.66300e10 q^{97} +8.95144e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 13 q^{2} - 3111 q^{4} + 6250 q^{5} + 7784 q^{7} - 35139 q^{8}+O(q^{10}) $ 2 * q + 13 * q^2 - 3111 * q^4 + 6250 * q^5 + 7784 * q^7 - 35139 * q^8	\( 2 q + 13 q^{2} - 3111 q^{4} + 6250 q^{5} + 7784 q^{7} - 35139 q^{8} + 40625 q^{10} - 295568 q^{11} + 657492 q^{13} - 3176796 q^{14} + 2974065 q^{16} - 8579948 q^{17} + 17627976 q^{19} - 9721875 q^{20} + 25972696 q^{22} + 29841072 q^{23} + 19531250 q^{25} + 69052066 q^{26} - 54064108 q^{28} + 201881948 q^{29} - 71057008 q^{31} + 30902261 q^{32} - 126080702 q^{34} + 24325000 q^{35} + 705858484 q^{37} - 3909548 q^{38} - 109809375 q^{40} + 327655148 q^{41} + 3192552120 q^{43} + 822376568 q^{44} + 1608947832 q^{46} - 2053064720 q^{47} + 7642614770 q^{49} + 126953125 q^{50} - 180610022 q^{52} + 2304299452 q^{53} - 923650000 q^{55} + 11220431460 q^{56} - 2489635114 q^{58} + 1478770576 q^{59} - 8264891460 q^{61} - 6261263448 q^{62} + 1546316681 q^{64} + 2054662500 q^{65} + 24212177528 q^{67} + 12432065594 q^{68} - 9927487500 q^{70} + 20218888256 q^{71} + 25879583268 q^{73} + 16142374034 q^{74} - 28960704764 q^{76} - 101122045248 q^{77} + 22324995440 q^{79} + 9293953125 q^{80} + 18686740478 q^{82} - 48014508984 q^{83} - 26812337500 q^{85} + 17559698092 q^{86} - 92965609896 q^{88} - 79209683076 q^{89} - 229606712048 q^{91} - 28023036264 q^{92} - 20475612776 q^{94} + 55087425000 q^{95} - 37075227452 q^{97} + 24554976677 q^{98}+O(q^{100}) \) 2 * q + 13 * q^2 - 3111 * q^4 + 6250 * q^5 + 7784 * q^7 - 35139 * q^8 + 40625 * q^10 - 295568 * q^11 + 657492 * q^13 - 3176796 * q^14 + 2974065 * q^16 - 8579948 * q^17 + 17627976 * q^19 - 9721875 * q^20 + 25972696 * q^22 + 29841072 * q^23 + 19531250 * q^25 + 69052066 * q^26 - 54064108 * q^28 + 201881948 * q^29 - 71057008 * q^31 + 30902261 * q^32 - 126080702 * q^34 + 24325000 * q^35 + 705858484 * q^37 - 3909548 * q^38 - 109809375 * q^40 + 327655148 * q^41 + 3192552120 * q^43 + 822376568 * q^44 + 1608947832 * q^46 - 2053064720 * q^47 + 7642614770 * q^49 + 126953125 * q^50 - 180610022 * q^52 + 2304299452 * q^53 - 923650000 * q^55 + 11220431460 * q^56 - 2489635114 * q^58 + 1478770576 * q^59 - 8264891460 * q^61 - 6261263448 * q^62 + 1546316681 * q^64 + 2054662500 * q^65 + 24212177528 * q^67 + 12432065594 * q^68 - 9927487500 * q^70 + 20218888256 * q^71 + 25879583268 * q^73 + 16142374034 * q^74 - 28960704764 * q^76 - 101122045248 * q^77 + 22324995440 * q^79 + 9293953125 * q^80 + 18686740478 * q^82 - 48014508984 * q^83 - 26812337500 * q^85 + 17559698092 * q^86 - 92965609896 * q^88 - 79209683076 * q^89 - 229606712048 * q^91 - 28023036264 * q^92 - 20475612776 * q^94 + 55087425000 * q^95 - 37075227452 * q^97 + 24554976677 * 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 $

$2$	27.7191	0.612511	0.306256	−	0.951949i	$-0.400924\pi$
$2$	27.7191	0.612511	0.306256	+	0.951949i	$0.400924\pi$
$3$	0	0
$3$	0	0
$4$	−1279.65	−0.624830
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	0	0
$7$	−72157.2	−1.62271	−0.811355	−	0.584554i	$-0.801269\pi$
$7$	−72157.2	−1.62271	−0.811355	+	0.584554i	$0.801269\pi$
$8$	−92239.5	−0.995227
$9$	0	0
$10$	86622.2	0.273923
$11$	509499.	0.953857	0.476929	−	0.878942i	$-0.341750\pi$
$11$	509499.	0.953857	0.476929	+	0.878942i	$0.341750\pi$
$12$	0	0
$13$	1.85516e6	1.38578	0.692889	−	0.721044i	$-0.256338\pi$
$13$	1.85516e6	1.38578	0.692889	+	0.721044i	$0.256338\pi$
$14$	−2.00013e6	−0.993928
$15$	0	0
$16$	63931.4	0.0152424
$17$	−5.94676e6	−1.01581	−0.507904	−	0.861414i	$-0.669579\pi$
$17$	−5.94676e6	−1.01581	−0.507904	+	0.861414i	$0.669579\pi$
$18$	0	0
$19$	6.02189e6	0.557941	0.278971	−	0.960300i	$-0.410007\pi$
$19$	6.02189e6	0.557941	0.278971	+	0.960300i	$0.410007\pi$
$20$	−3.99891e6	−0.279432
$21$	0	0
$22$	1.41228e7	0.584248
$23$	4.82627e7	1.56354	0.781769	−	0.623568i	$-0.214318\pi$
$23$	4.82627e7	1.56354	0.781769	+	0.623568i	$0.214318\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	5.14234e7	0.848804
$27$	0	0
$28$	9.23361e7	1.01392
$29$	1.13550e7	0.102801	0.0514005	−	0.998678i	$-0.483632\pi$
$29$	1.13550e7	0.102801	0.0514005	+	0.998678i	$0.483632\pi$
$30$	0	0
$31$	−1.72184e8	−1.08020	−0.540098	−	0.841602i	$-0.681613\pi$
$31$	−1.72184e8	−1.08020	−0.540098	+	0.841602i	$0.681613\pi$
$32$	1.90679e8	1.00456
$33$	0	0
$34$	−1.64839e8	−0.622194
$35$	−2.25491e8	−0.725698
$36$	0	0
$37$	6.25191e8	1.48219	0.741094	−	0.671401i	$-0.234307\pi$
$37$	6.25191e8	1.48219	0.741094	+	0.671401i	$0.234307\pi$
$38$	1.66921e8	0.341745
$39$	0	0
$40$	−2.88248e8	−0.445079
$41$	5.53971e8	0.746751	0.373376	−	0.927680i	$-0.378200\pi$
$41$	5.53971e8	0.746751	0.373376	+	0.927680i	$0.378200\pi$
$42$	0	0
$43$	1.52106e9	1.57787	0.788934	−	0.614478i	$-0.210633\pi$
$43$	1.52106e9	1.57787	0.788934	+	0.614478i	$0.210633\pi$
$44$	−6.51981e8	−0.595998
$45$	0	0
$46$	1.33780e9	0.957685
$47$	−1.19456e9	−0.759747	−0.379873	−	0.925038i	$-0.624032\pi$
$47$	−1.19456e9	−0.759747	−0.379873	+	0.925038i	$0.624032\pi$
$48$	0	0
$49$	3.22934e9	1.63318
$50$	2.70694e8	0.122502
$51$	0	0
$52$	−2.37396e9	−0.865875
$53$	−1.22320e9	−0.401774	−0.200887	−	0.979614i	$-0.564382\pi$
$53$	−1.22320e9	−0.401774	−0.200887	+	0.979614i	$0.564382\pi$
$54$	0	0
$55$	1.59218e9	0.426578
$56$	6.65575e9	1.61496
$57$	0	0
$58$	3.14750e8	0.0629667
$59$	5.83637e9	1.06281	0.531407	−	0.847117i	$-0.321664\pi$
$59$	5.83637e9	1.06281	0.531407	+	0.847117i	$0.321664\pi$
$60$	0	0
$61$	−6.61097e9	−1.00219	−0.501096	−	0.865392i	$-0.667070\pi$
$61$	−6.61097e9	−1.00219	−0.501096	+	0.865392i	$0.667070\pi$
$62$	−4.77277e9	−0.661632
$63$	0	0
$64$	5.15451e9	0.600064
$65$	5.79738e9	0.619739
$66$	0	0
$67$	1.66261e10	1.50445	0.752226	−	0.658905i	$-0.228980\pi$
$67$	1.66261e10	1.50445	0.752226	+	0.658905i	$0.228980\pi$
$68$	7.60978e9	0.634707
$69$	0	0
$70$	−6.25042e9	−0.444498
$71$	−7.36200e9	−0.484256	−0.242128	−	0.970244i	$-0.577845\pi$
$71$	−7.36200e9	−0.484256	−0.242128	+	0.970244i	$0.577845\pi$
$72$	0	0
$73$	−6.35726e9	−0.358917	−0.179458	−	0.983766i	$-0.557435\pi$
$73$	−6.35726e9	−0.358917	−0.179458	+	0.983766i	$0.557435\pi$
$74$	1.73297e10	0.907857
$75$	0	0
$76$	−7.70593e9	−0.348618
$77$	−3.67640e10	−1.54783
$78$	0	0
$79$	2.47565e10	0.905191	0.452595	−	0.891716i	$-0.350498\pi$
$79$	2.47565e10	0.905191	0.452595	+	0.891716i	$0.350498\pi$
$80$	1.99786e8	0.00681663
$81$	0	0
$82$	1.53556e10	0.457393
$83$	−3.59416e10	−1.00154	−0.500769	−	0.865581i	$-0.666950\pi$
$83$	−3.59416e10	−1.00154	−0.500769	+	0.865581i	$0.666950\pi$
$84$	0	0
$85$	−1.85836e10	−0.454283
$86$	4.21625e10	0.966462
$87$	0	0
$88$	−4.69959e10	−0.949304
$89$	−7.47690e10	−1.41931	−0.709653	−	0.704551i	$-0.751148\pi$
$89$	−7.47690e10	−1.41931	−0.709653	+	0.704551i	$0.751148\pi$
$90$	0	0
$91$	−1.33863e11	−2.24871
$92$	−6.17594e10	−0.976945
$93$	0	0
$94$	−3.31121e10	−0.465354
$95$	1.88184e10	0.249519
$96$	0	0
$97$	−1.66300e10	−0.196629	−0.0983144	−	0.995155i	$-0.531345\pi$
$97$	−1.66300e10	−0.196629	−0.0983144	+	0.995155i	$0.531345\pi$
$98$	8.95144e10	1.00034
$99$	0	0
$100$	−1.24966e10	−0.124966
$101$	5.30247e10	0.502008	0.251004	−	0.967986i	$-0.419239\pi$
$101$	5.30247e10	0.502008	0.251004	+	0.967986i	$0.419239\pi$
$102$	0	0
$103$	1.00437e9	0.00853666	0.00426833	−	0.999991i	$-0.498641\pi$
$103$	1.00437e9	0.00853666	0.00426833	+	0.999991i	$0.498641\pi$
$104$	−1.71119e11	−1.37916
$105$	0	0
$106$	−3.39061e10	−0.246091
$107$	1.65238e11	1.13893	0.569466	−	0.822015i	$-0.307150\pi$
$107$	1.65238e11	1.13893	0.569466	+	0.822015i	$0.307150\pi$
$108$	0	0
$109$	7.04515e10	0.438576	0.219288	−	0.975660i	$-0.429627\pi$
$109$	7.04515e10	0.438576	0.219288	+	0.975660i	$0.429627\pi$
$110$	4.41339e10	0.261284
$111$	0	0
$112$	−4.61312e9	−0.0247341
$113$	−2.07466e11	−1.05929	−0.529646	−	0.848219i	$-0.677675\pi$
$113$	−2.07466e11	−1.05929	−0.529646	+	0.848219i	$0.677675\pi$
$114$	0	0
$115$	1.50821e11	0.699235
$116$	−1.45304e10	−0.0642331
$117$	0	0
$118$	1.61779e11	0.650985
$119$	4.29102e11	1.64836
$120$	0	0
$121$	−2.57228e10	−0.0901567
$122$	−1.83250e11	−0.613854
$123$	0	0
$124$	2.20335e11	0.674938
$125$	3.05176e10	0.0894427
$126$	0	0
$127$	1.17811e11	0.316422	0.158211	−	0.987405i	$-0.449427\pi$
$127$	1.17811e11	0.316422	0.158211	+	0.987405i	$0.449427\pi$
$128$	−2.47632e11	−0.637017
$129$	0	0
$130$	1.60698e11	0.379597
$131$	−4.60288e11	−1.04241	−0.521204	−	0.853432i	$-0.674517\pi$
$131$	−4.60288e11	−1.04241	−0.521204	+	0.853432i	$0.674517\pi$
$132$	0	0
$133$	−4.34523e11	−0.905376
$134$	4.60860e11	0.921494
$135$	0	0
$136$	5.48526e11	1.01096
$137$	5.80435e11	1.02752	0.513760	−	0.857934i	$-0.328252\pi$
$137$	5.80435e11	1.02752	0.513760	+	0.857934i	$0.328252\pi$
$138$	0	0
$139$	6.94847e11	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$139$	6.94847e11	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$140$	2.88550e11	0.453438
$141$	0	0
$142$	−2.04068e11	−0.296612
$143$	9.45203e11	1.32183
$144$	0	0
$145$	3.54843e10	0.0459740
$146$	−1.76217e11	−0.219841
$147$	0	0
$148$	−8.00027e11	−0.926115
$149$	1.12749e11	0.125774	0.0628869	−	0.998021i	$-0.479969\pi$
$149$	1.12749e11	0.125774	0.0628869	+	0.998021i	$0.479969\pi$
$150$	0	0
$151$	2.45742e11	0.254746	0.127373	−	0.991855i	$-0.459346\pi$
$151$	2.45742e11	0.254746	0.127373	+	0.991855i	$0.459346\pi$
$152$	−5.55457e11	−0.555278
$153$	0	0
$154$	−1.01907e12	−0.948065
$155$	−5.38074e11	−0.483078
$156$	0	0
$157$	−5.83317e11	−0.488042	−0.244021	−	0.969770i	$-0.578467\pi$
$157$	−5.83317e11	−0.488042	−0.244021	+	0.969770i	$0.578467\pi$
$158$	6.86227e11	0.554439
$159$	0	0
$160$	5.95871e11	0.449254
$161$	−3.48250e12	−2.53717
$162$	0	0
$163$	2.65801e12	1.80936	0.904681	−	0.426089i	$-0.140109\pi$
$163$	2.65801e12	1.80936	0.904681	+	0.426089i	$0.140109\pi$
$164$	−7.08890e11	−0.466592
$165$	0	0
$166$	−9.96269e11	−0.613454
$167$	1.68070e12	1.00126	0.500632	−	0.865660i	$-0.333101\pi$
$167$	1.68070e12	1.00126	0.500632	+	0.865660i	$0.333101\pi$
$168$	0	0
$169$	1.64947e12	0.920380
$170$	−5.15121e11	−0.278253
$171$	0	0
$172$	−1.94643e12	−0.985900
$173$	2.11009e12	1.03525	0.517627	−	0.855606i	$-0.326815\pi$
$173$	2.11009e12	1.03525	0.517627	+	0.855606i	$0.326815\pi$
$174$	0	0
$175$	−7.04661e11	−0.324542
$176$	3.25730e10	0.0145391
$177$	0	0
$178$	−2.07253e12	−0.869341
$179$	−8.73323e10	−0.0355208	−0.0177604	−	0.999842i	$-0.505654\pi$
$179$	−8.73323e10	−0.0355208	−0.0177604	+	0.999842i	$0.505654\pi$
$180$	0	0
$181$	2.60066e12	0.995066	0.497533	−	0.867445i	$-0.334239\pi$
$181$	2.60066e12	0.995066	0.497533	+	0.867445i	$0.334239\pi$
$182$	−3.71057e12	−1.37736
$183$	0	0
$184$	−4.45173e12	−1.55607
$185$	1.95372e12	0.662854
$186$	0	0
$187$	−3.02987e12	−0.968935
$188$	1.52862e12	0.474713
$189$	0	0
$190$	5.21630e11	0.152833
$191$	2.28556e12	0.650593	0.325296	−	0.945612i	$-0.394536\pi$
$191$	2.28556e12	0.650593	0.325296	+	0.945612i	$0.394536\pi$
$192$	0	0
$193$	3.11630e12	0.837671	0.418836	−	0.908062i	$-0.362438\pi$
$193$	3.11630e12	0.837671	0.418836	+	0.908062i	$0.362438\pi$
$194$	−4.60968e11	−0.120437
$195$	0	0
$196$	−4.13243e12	−1.02046
$197$	−3.68099e12	−0.883894	−0.441947	−	0.897041i	$-0.645712\pi$
$197$	−3.68099e12	−0.883894	−0.441947	+	0.897041i	$0.645712\pi$
$198$	0	0
$199$	−2.20305e12	−0.500417	−0.250208	−	0.968192i	$-0.580499\pi$
$199$	−2.20305e12	−0.500417	−0.250208	+	0.968192i	$0.580499\pi$
$200$	−9.00776e11	−0.199045
$201$	0	0
$202$	1.46980e12	0.307485
$203$	−8.19343e11	−0.166816
$204$	0	0
$205$	1.73116e12	0.333957
$206$	2.78402e10	0.00522880
$207$	0	0
$208$	1.18603e11	0.0211226
$209$	3.06815e12	0.532196
$210$	0	0
$211$	−1.08697e13	−1.78921	−0.894607	−	0.446853i	$-0.852545\pi$
$211$	−1.08697e13	−1.78921	−0.894607	+	0.446853i	$0.852545\pi$
$212$	1.56528e12	0.251040
$213$	0	0
$214$	4.58024e12	0.697609
$215$	4.75332e12	0.705644
$216$	0	0
$217$	1.24243e13	1.75284
$218$	1.95285e12	0.268633
$219$	0	0
$220$	−2.03744e12	−0.266539
$221$	−1.10322e13	−1.40768
$222$	0	0
$223$	9.89456e12	1.20149	0.600744	−	0.799441i	$-0.294871\pi$
$223$	9.89456e12	1.20149	0.600744	+	0.799441i	$0.294871\pi$
$224$	−1.37588e13	−1.63011
$225$	0	0
$226$	−5.75078e12	−0.648829
$227$	−5.42930e12	−0.597862	−0.298931	−	0.954275i	$-0.596630\pi$
$227$	−5.42930e12	−0.597862	−0.298931	+	0.954275i	$0.596630\pi$
$228$	0	0
$229$	2.17366e12	0.228085	0.114042	−	0.993476i	$-0.463620\pi$
$229$	2.17366e12	0.228085	0.114042	+	0.993476i	$0.463620\pi$
$230$	4.18062e12	0.428290
$231$	0	0
$232$	−1.04738e12	−0.102310
$233$	1.98130e13	1.89013	0.945067	−	0.326876i	$-0.105996\pi$
$233$	1.98130e13	1.89013	0.945067	+	0.326876i	$0.105996\pi$
$234$	0	0
$235$	−3.73299e12	−0.339769
$236$	−7.46853e12	−0.664078
$237$	0	0
$238$	1.18943e13	1.00964
$239$	−1.03457e12	−0.0858169	−0.0429085	−	0.999079i	$-0.513662\pi$
$239$	−1.03457e12	−0.0858169	−0.0429085	+	0.999079i	$0.513662\pi$
$240$	0	0
$241$	1.36081e13	1.07821	0.539106	−	0.842238i	$-0.318762\pi$
$241$	1.36081e13	1.07821	0.539106	+	0.842238i	$0.318762\pi$
$242$	−7.13011e11	−0.0552220
$243$	0	0
$244$	8.45973e12	0.626199
$245$	1.00917e13	0.730382
$246$	0	0
$247$	1.11716e13	0.773182
$248$	1.58821e13	1.07504
$249$	0	0
$250$	8.45920e11	0.0547847
$251$	2.28023e13	1.44469	0.722344	−	0.691534i	$-0.243065\pi$
$251$	2.28023e13	1.44469	0.722344	+	0.691534i	$0.243065\pi$
$252$	0	0
$253$	2.45898e13	1.49139
$254$	3.26562e12	0.193812
$255$	0	0
$256$	−1.74206e13	−0.990244
$257$	−6.13077e12	−0.341101	−0.170550	−	0.985349i	$-0.554555\pi$
$257$	−6.13077e12	−0.341101	−0.170550	+	0.985349i	$0.554555\pi$
$258$	0	0
$259$	−4.51121e13	−2.40516
$260$	−7.41863e12	−0.387231
$261$	0	0
$262$	−1.27588e13	−0.638486
$263$	3.25384e13	1.59456	0.797278	−	0.603613i	$-0.206273\pi$
$263$	3.25384e13	1.59456	0.797278	+	0.603613i	$0.206273\pi$
$264$	0	0
$265$	−3.82251e12	−0.179679
$266$	−1.20446e13	−0.554553
$267$	0	0
$268$	−2.12756e13	−0.940027
$269$	−3.66970e13	−1.58852	−0.794261	−	0.607577i	$-0.792142\pi$
$269$	−3.66970e13	−1.58852	−0.794261	+	0.607577i	$0.792142\pi$
$270$	0	0
$271$	1.87361e13	0.778659	0.389329	−	0.921099i	$-0.372707\pi$
$271$	1.87361e13	0.778659	0.389329	+	0.921099i	$0.372707\pi$
$272$	−3.80185e11	−0.0154834
$273$	0	0
$274$	1.60891e13	0.629368
$275$	4.97557e12	0.190771
$276$	0	0
$277$	−3.03044e13	−1.11652	−0.558260	−	0.829666i	$-0.688531\pi$
$277$	−3.03044e13	−1.11652	−0.558260	+	0.829666i	$0.688531\pi$
$278$	1.92605e13	0.695700
$279$	0	0
$280$	2.07992e13	0.722234
$281$	−4.93501e13	−1.68037	−0.840183	−	0.542304i	$-0.817552\pi$
$281$	−4.93501e13	−1.68037	−0.840183	+	0.542304i	$0.817552\pi$
$282$	0	0
$283$	−5.45122e12	−0.178512	−0.0892562	−	0.996009i	$-0.528449\pi$
$283$	−5.45122e12	−0.178512	−0.0892562	+	0.996009i	$0.528449\pi$
$284$	9.42079e12	0.302578
$285$	0	0
$286$	2.62002e13	0.809638
$287$	−3.99730e13	−1.21176
$288$	0	0
$289$	1.09207e12	0.0318649
$290$	9.83592e11	0.0281596
$291$	0	0
$292$	8.13507e12	0.224262
$293$	−4.85807e13	−1.31429	−0.657146	−	0.753763i	$-0.728237\pi$
$293$	−4.85807e13	−1.31429	−0.657146	+	0.753763i	$0.728237\pi$
$294$	0	0
$295$	1.82387e13	0.475305
$296$	−5.76673e13	−1.47511
$297$	0	0
$298$	3.12531e12	0.0770378
$299$	8.95352e13	2.16672
$300$	0	0
$301$	−1.09756e14	−2.56042
$302$	6.81176e12	0.156035
$303$	0	0
$304$	3.84988e11	0.00850439
$305$	−2.06593e13	−0.448194
$306$	0	0
$307$	−6.68455e13	−1.39898	−0.699489	−	0.714643i	$-0.746589\pi$
$307$	−6.68455e13	−1.39898	−0.699489	+	0.714643i	$0.746589\pi$
$308$	4.70451e13	0.967132
$309$	0	0
$310$	−1.49149e13	−0.295891
$311$	−7.26443e13	−1.41586	−0.707928	−	0.706284i	$-0.750370\pi$
$311$	−7.26443e13	−1.41586	−0.707928	+	0.706284i	$0.750370\pi$
$312$	0	0
$313$	6.27938e13	1.18147	0.590735	−	0.806866i	$-0.298838\pi$
$313$	6.27938e13	1.18147	0.590735	+	0.806866i	$0.298838\pi$
$314$	−1.61690e13	−0.298931
$315$	0	0
$316$	−3.16797e13	−0.565590
$317$	4.20135e13	0.737161	0.368581	−	0.929596i	$-0.379844\pi$
$317$	4.20135e13	0.737161	0.368581	+	0.929596i	$0.379844\pi$
$318$	0	0
$319$	5.78534e12	0.0980574
$320$	1.61078e13	0.268357
$321$	0	0
$322$	−9.65318e13	−1.55404
$323$	−3.58108e13	−0.566761
$324$	0	0
$325$	1.81168e13	0.277156
$326$	7.36778e13	1.10825
$327$	0	0
$328$	−5.10980e13	−0.743187
$329$	8.61960e13	1.23285
$330$	0	0
$331$	1.36334e14	1.88604	0.943019	−	0.332740i	$-0.107973\pi$
$331$	1.36334e14	1.88604	0.943019	+	0.332740i	$0.107973\pi$
$332$	4.59927e13	0.625792
$333$	0	0
$334$	4.65874e13	0.613286
$335$	5.19565e13	0.672811
$336$	0	0
$337$	9.94099e13	1.24585	0.622924	−	0.782282i	$-0.285944\pi$
$337$	9.94099e13	1.24585	0.622924	+	0.782282i	$0.285944\pi$
$338$	4.57218e13	0.563743
$339$	0	0
$340$	2.37806e13	0.283850
$341$	−8.77273e13	−1.03035
$342$	0	0
$343$	−9.03418e13	−1.02747
$344$	−1.40302e14	−1.57034
$345$	0	0
$346$	5.84897e13	0.634105
$347$	−7.96482e13	−0.849892	−0.424946	−	0.905219i	$-0.639707\pi$
$347$	−7.96482e13	−0.849892	−0.424946	+	0.905219i	$0.639707\pi$
$348$	0	0
$349$	−2.33239e13	−0.241135	−0.120568	−	0.992705i	$-0.538471\pi$
$349$	−2.33239e13	−0.241135	−0.120568	+	0.992705i	$0.538471\pi$
$350$	−1.95326e13	−0.198786
$351$	0	0
$352$	9.71505e13	0.958209
$353$	7.20065e13	0.699215	0.349608	−	0.936896i	$-0.386315\pi$
$353$	7.20065e13	0.699215	0.349608	+	0.936896i	$0.386315\pi$
$354$	0	0
$355$	−2.30062e13	−0.216566
$356$	9.56782e13	0.886825
$357$	0	0
$358$	−2.42077e12	−0.0217569
$359$	−1.02696e13	−0.0908941	−0.0454470	−	0.998967i	$-0.514471\pi$
$359$	−1.02696e13	−0.0908941	−0.0454470	+	0.998967i	$0.514471\pi$
$360$	0	0
$361$	−8.02270e13	−0.688702
$362$	7.20880e13	0.609489
$363$	0	0
$364$	1.71299e14	1.40506
$365$	−1.98664e13	−0.160513
$366$	0	0
$367$	1.03116e14	0.808470	0.404235	−	0.914655i	$-0.367538\pi$
$367$	1.03116e14	0.808470	0.404235	+	0.914655i	$0.367538\pi$
$368$	3.08550e12	0.0238321
$369$	0	0
$370$	5.41554e13	0.406006
$371$	8.82631e13	0.651962
$372$	0	0
$373$	1.30172e14	0.933509	0.466755	−	0.884387i	$-0.345423\pi$
$373$	1.30172e14	0.933509	0.466755	+	0.884387i	$0.345423\pi$
$374$	−8.39852e13	−0.593484
$375$	0	0
$376$	1.10185e14	0.756120
$377$	2.10653e13	0.142459
$378$	0	0
$379$	−6.26815e13	−0.411740	−0.205870	−	0.978579i	$-0.566002\pi$
$379$	−6.26815e13	−0.411740	−0.205870	+	0.978579i	$0.566002\pi$
$380$	−2.40810e13	−0.155907
$381$	0	0
$382$	6.33537e13	0.398495
$383$	−1.06310e14	−0.659145	−0.329572	−	0.944130i	$-0.606905\pi$
$383$	−1.06310e14	−0.659145	−0.329572	+	0.944130i	$0.606905\pi$
$384$	0	0
$385$	−1.14888e14	−0.692212
$386$	8.63809e13	0.513083
$387$	0	0
$388$	2.12806e13	0.122860
$389$	7.42336e13	0.422549	0.211275	−	0.977427i	$-0.432239\pi$
$389$	7.42336e13	0.422549	0.211275	+	0.977427i	$0.432239\pi$
$390$	0	0
$391$	−2.87007e14	−1.58825
$392$	−2.97873e14	−1.62539
$393$	0	0
$394$	−1.02034e14	−0.541395
$395$	7.73640e13	0.404814
$396$	0	0
$397$	−1.48552e14	−0.756018	−0.378009	−	0.925802i	$-0.623391\pi$
$397$	−1.48552e14	−0.756018	−0.378009	+	0.925802i	$0.623391\pi$
$398$	−6.10665e13	−0.306511
$399$	0	0
$400$	6.24331e11	0.00304849
$401$	9.20974e13	0.443561	0.221781	−	0.975097i	$-0.428813\pi$
$401$	9.20974e13	0.443561	0.221781	+	0.975097i	$0.428813\pi$
$402$	0	0
$403$	−3.19429e14	−1.49691
$404$	−6.78531e13	−0.313669
$405$	0	0
$406$	−2.27115e13	−0.102177
$407$	3.18534e14	1.41380
$408$	0	0
$409$	−3.23531e13	−0.139778	−0.0698889	−	0.997555i	$-0.522264\pi$
$409$	−3.23531e13	−0.139778	−0.0698889	+	0.997555i	$0.522264\pi$
$410$	4.79862e13	0.204553
$411$	0	0
$412$	−1.28524e12	−0.00533396
$413$	−4.21137e14	−1.72464
$414$	0	0
$415$	−1.12317e14	−0.447902
$416$	3.53740e14	1.39210
$417$	0	0
$418$	8.50463e13	0.325976
$419$	1.61755e14	0.611900	0.305950	−	0.952048i	$-0.401026\pi$
$419$	1.61755e14	0.611900	0.305950	+	0.952048i	$0.401026\pi$
$420$	0	0
$421$	−1.14462e14	−0.421803	−0.210902	−	0.977507i	$-0.567640\pi$
$421$	−1.14462e14	−0.421803	−0.210902	+	0.977507i	$0.567640\pi$
$422$	−3.01297e14	−1.09591
$423$	0	0
$424$	1.12828e14	0.399856
$425$	−5.80738e13	−0.203161
$426$	0	0
$427$	4.77029e14	1.62627
$428$	−2.11447e14	−0.711639
$429$	0	0
$430$	1.31758e14	0.432215
$431$	−1.01052e14	−0.327279	−0.163639	−	0.986520i	$-0.552323\pi$
$431$	−1.01052e14	−0.327279	−0.163639	+	0.986520i	$0.552323\pi$
$432$	0	0
$433$	1.21838e14	0.384680	0.192340	−	0.981328i	$-0.438392\pi$
$433$	1.21838e14	0.384680	0.192340	+	0.981328i	$0.438392\pi$
$434$	3.44390e14	1.07364
$435$	0	0
$436$	−9.01534e13	−0.274035
$437$	2.90633e14	0.872362
$438$	0	0
$439$	2.85736e14	0.836393	0.418197	−	0.908357i	$-0.362662\pi$
$439$	2.85736e14	0.836393	0.418197	+	0.908357i	$0.362662\pi$
$440$	−1.46862e14	−0.424542
$441$	0	0
$442$	−3.05803e14	−0.862222
$443$	−4.82498e14	−1.34362	−0.671808	−	0.740725i	$-0.734482\pi$
$443$	−4.82498e14	−1.34362	−0.671808	+	0.740725i	$0.734482\pi$
$444$	0	0
$445$	−2.33653e14	−0.634733
$446$	2.74268e14	0.735925
$447$	0	0
$448$	−3.71935e14	−0.973729
$449$	−6.86974e14	−1.77658	−0.888291	−	0.459281i	$-0.848107\pi$
$449$	−6.86974e14	−1.77658	−0.888291	+	0.459281i	$0.848107\pi$
$450$	0	0
$451$	2.82248e14	0.712294
$452$	2.65485e14	0.661878
$453$	0	0
$454$	−1.50495e14	−0.366197
$455$	−4.18323e14	−1.00566
$456$	0	0
$457$	−2.14744e14	−0.503944	−0.251972	−	0.967734i	$-0.581079\pi$
$457$	−2.14744e14	−0.503944	−0.251972	+	0.967734i	$0.581079\pi$
$458$	6.02519e13	0.139705
$459$	0	0
$460$	−1.92998e14	−0.436903
$461$	3.22127e14	0.720563	0.360281	−	0.932844i	$-0.382681\pi$
$461$	3.22127e14	0.720563	0.360281	+	0.932844i	$0.382681\pi$
$462$	0	0
$463$	8.81227e13	0.192483	0.0962415	−	0.995358i	$-0.469318\pi$
$463$	8.81227e13	0.192483	0.0962415	+	0.995358i	$0.469318\pi$
$464$	7.25940e11	0.00156694
$465$	0	0
$466$	5.49199e14	1.15773
$467$	−3.86746e14	−0.805718	−0.402859	−	0.915262i	$-0.631984\pi$
$467$	−3.86746e14	−0.805718	−0.402859	+	0.915262i	$0.631984\pi$
$468$	0	0
$469$	−1.19969e15	−2.44129
$470$	−1.03475e14	−0.208112
$471$	0	0
$472$	−5.38344e14	−1.05774
$473$	7.74980e14	1.50506
$474$	0	0
$475$	5.88076e13	0.111588
$476$	−5.49101e14	−1.02994
$477$	0	0
$478$	−2.86774e13	−0.0525638
$479$	5.88123e14	1.06567	0.532835	−	0.846219i	$-0.321127\pi$
$479$	5.88123e14	1.06567	0.532835	+	0.846219i	$0.321127\pi$
$480$	0	0
$481$	1.15983e15	2.05398
$482$	3.77205e14	0.660417
$483$	0	0
$484$	3.29162e13	0.0563326
$485$	−5.19687e13	−0.0879350
$486$	0	0
$487$	3.77139e14	0.623867	0.311933	−	0.950104i	$-0.399023\pi$
$487$	3.77139e14	0.623867	0.311933	+	0.950104i	$0.399023\pi$
$488$	6.09792e14	0.997408
$489$	0	0
$490$	2.79732e14	0.447367
$491$	3.71374e13	0.0587305	0.0293652	−	0.999569i	$-0.490651\pi$
$491$	3.71374e13	0.0587305	0.0293652	+	0.999569i	$0.490651\pi$
$492$	0	0
$493$	−6.75253e13	−0.104426
$494$	3.09667e14	0.473583
$495$	0	0
$496$	−1.10079e13	−0.0164648
$497$	5.31222e14	0.785806
$498$	0	0
$499$	2.72085e14	0.393688	0.196844	−	0.980435i	$-0.436931\pi$
$499$	2.72085e14	0.393688	0.196844	+	0.980435i	$0.436931\pi$
$500$	−3.90519e13	−0.0558865
$501$	0	0
$502$	6.32060e14	0.884888
$503$	−1.30904e15	−1.81272	−0.906358	−	0.422511i	$-0.861149\pi$
$503$	−1.30904e15	−1.81272	−0.906358	+	0.422511i	$0.861149\pi$
$504$	0	0
$505$	1.65702e14	0.224505
$506$	6.81606e14	0.913494
$507$	0	0
$508$	−1.50757e14	−0.197710
$509$	−1.41934e15	−1.84135	−0.920677	−	0.390326i	$-0.872362\pi$
$509$	−1.41934e15	−1.84135	−0.920677	+	0.390326i	$0.872362\pi$
$510$	0	0
$511$	4.58722e14	0.582418
$512$	2.42674e13	0.0304817
$513$	0	0
$514$	−1.69939e14	−0.208928
$515$	3.13865e12	0.00381771
$516$	0	0
$517$	−6.08626e14	−0.724690
$518$	−1.25047e15	−1.47319
$519$	0	0
$520$	−5.34748e14	−0.616780
$521$	5.15525e14	0.588359	0.294180	−	0.955750i	$-0.404954\pi$
$521$	5.15525e14	0.588359	0.294180	+	0.955750i	$0.404954\pi$
$522$	0	0
$523$	1.15496e15	1.29064	0.645322	−	0.763911i	$-0.276723\pi$
$523$	1.15496e15	1.29064	0.645322	+	0.763911i	$0.276723\pi$
$524$	5.89008e14	0.651328
$525$	0	0
$526$	9.01935e14	0.976683
$527$	1.02393e15	1.09727
$528$	0	0
$529$	1.37648e15	1.44465
$530$	−1.05957e14	−0.110055
$531$	0	0
$532$	5.56038e14	0.565706
$533$	1.02771e15	1.03483
$534$	0	0
$535$	5.16367e14	0.509346
$536$	−1.53358e15	−1.49727
$537$	0	0
$538$	−1.01721e15	−0.972988
$539$	1.64534e15	1.55782
$540$	0	0
$541$	−1.08911e15	−1.01038	−0.505190	−	0.863008i	$-0.668578\pi$
$541$	−1.08911e15	−1.01038	−0.505190	+	0.863008i	$0.668578\pi$
$542$	5.19347e14	0.476937
$543$	0	0
$544$	−1.13392e15	−1.02044
$545$	2.20161e14	0.196137
$546$	0	0
$547$	9.36391e14	0.817574	0.408787	−	0.912630i	$-0.365952\pi$
$547$	9.36391e14	0.817574	0.408787	+	0.912630i	$0.365952\pi$
$548$	−7.42754e14	−0.642025
$549$	0	0
$550$	1.37918e14	0.116850
$551$	6.83784e13	0.0573569
$552$	0	0
$553$	−1.78636e15	−1.46886
$554$	−8.40010e14	−0.683881
$555$	0	0
$556$	−8.89163e14	−0.709692
$557$	3.48191e14	0.275178	0.137589	−	0.990489i	$-0.456065\pi$
$557$	3.48191e14	0.275178	0.137589	+	0.990489i	$0.456065\pi$
$558$	0	0
$559$	2.82182e15	2.18658
$560$	−1.44160e13	−0.0110614
$561$	0	0
$562$	−1.36794e15	−1.02924
$563$	−8.78985e14	−0.654916	−0.327458	−	0.944866i	$-0.606192\pi$
$563$	−8.78985e14	−0.654916	−0.327458	+	0.944866i	$0.606192\pi$
$564$	0	0
$565$	−6.48332e14	−0.473730
$566$	−1.51103e14	−0.109341
$567$	0	0
$568$	6.79067e14	0.481944
$569$	1.45202e14	0.102060	0.0510299	−	0.998697i	$-0.483750\pi$
$569$	1.45202e14	0.102060	0.0510299	+	0.998697i	$0.483750\pi$
$570$	0	0
$571$	−1.99915e15	−1.37831	−0.689156	−	0.724613i	$-0.742019\pi$
$571$	−1.99915e15	−1.37831	−0.689156	+	0.724613i	$0.742019\pi$
$572$	−1.20953e15	−0.825921
$573$	0	0
$574$	−1.10802e15	−0.742216
$575$	4.71315e14	0.312708
$576$	0	0
$577$	−7.69862e13	−0.0501125	−0.0250562	−	0.999686i	$-0.507976\pi$
$577$	−7.69862e13	−0.0501125	−0.0250562	+	0.999686i	$0.507976\pi$
$578$	3.02712e13	0.0195176
$579$	0	0
$580$	−4.54075e13	−0.0287259
$581$	2.59345e15	1.62521
$582$	0	0
$583$	−6.23221e14	−0.383235
$584$	5.86390e14	0.357204
$585$	0	0
$586$	−1.34661e15	−0.805019
$587$	−8.86073e14	−0.524759	−0.262379	−	0.964965i	$-0.584507\pi$
$587$	−8.86073e14	−0.524759	−0.262379	+	0.964965i	$0.584507\pi$
$588$	0	0
$589$	−1.03687e15	−0.602685
$590$	5.05559e14	0.291129
$591$	0	0
$592$	3.99694e13	0.0225922
$593$	−9.32426e14	−0.522172	−0.261086	−	0.965316i	$-0.584081\pi$
$593$	−9.32426e14	−0.522172	−0.261086	+	0.965316i	$0.584081\pi$
$594$	0	0
$595$	1.34094e15	0.737169
$596$	−1.44280e14	−0.0785872
$597$	0	0
$598$	2.48183e15	1.32714
$599$	−1.16299e15	−0.616208	−0.308104	−	0.951353i	$-0.599695\pi$
$599$	−1.16299e15	−0.616208	−0.308104	+	0.951353i	$0.599695\pi$
$600$	0	0
$601$	−2.63081e15	−1.36861	−0.684306	−	0.729195i	$-0.739895\pi$
$601$	−2.63081e15	−1.36861	−0.684306	+	0.729195i	$0.739895\pi$
$602$	−3.04233e15	−1.56829
$603$	0	0
$604$	−3.14465e14	−0.159173
$605$	−8.03836e13	−0.0403193
$606$	0	0
$607$	1.56499e14	0.0770857	0.0385429	−	0.999257i	$-0.487728\pi$
$607$	1.56499e14	0.0770857	0.0385429	+	0.999257i	$0.487728\pi$
$608$	1.14825e15	0.560487
$609$	0	0
$610$	−5.72656e14	−0.274524
$611$	−2.21610e15	−1.05284
$612$	0	0
$613$	−1.63056e15	−0.760859	−0.380429	−	0.924810i	$-0.624224\pi$
$613$	−1.63056e15	−0.760859	−0.380429	+	0.924810i	$0.624224\pi$
$614$	−1.85290e15	−0.856890
$615$	0	0
$616$	3.39109e15	1.54044
$617$	−3.55060e15	−1.59858	−0.799289	−	0.600947i	$-0.794790\pi$
$617$	−3.55060e15	−1.59858	−0.799289	+	0.600947i	$0.794790\pi$
$618$	0	0
$619$	−7.99938e14	−0.353800	−0.176900	−	0.984229i	$-0.556607\pi$
$619$	−7.99938e14	−0.353800	−0.176900	+	0.984229i	$0.556607\pi$
$620$	6.88547e14	0.301842
$621$	0	0
$622$	−2.01363e15	−0.867228
$623$	5.39512e15	2.30312
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	1.74059e15	0.723664
$627$	0	0
$628$	7.46443e14	0.304943
$629$	−3.71786e15	−1.50562
$630$	0	0
$631$	3.38323e14	0.134639	0.0673194	−	0.997731i	$-0.478555\pi$
$631$	3.38323e14	0.134639	0.0673194	+	0.997731i	$0.478555\pi$
$632$	−2.28353e15	−0.900870
$633$	0	0
$634$	1.16457e15	0.451520
$635$	3.68160e14	0.141508
$636$	0	0
$637$	5.99095e15	2.26323
$638$	1.60364e14	0.0600613
$639$	0	0
$640$	−7.73849e14	−0.284883
$641$	−2.78171e15	−1.01529	−0.507647	−	0.861565i	$-0.669485\pi$
$641$	−2.78171e15	−1.01529	−0.507647	+	0.861565i	$0.669485\pi$
$642$	0	0
$643$	−1.37901e15	−0.494774	−0.247387	−	0.968917i	$-0.579572\pi$
$643$	−1.37901e15	−0.494774	−0.247387	+	0.968917i	$0.579572\pi$
$644$	4.45639e15	1.58530
$645$	0	0
$646$	−9.92642e14	−0.347147
$647$	4.51705e15	1.56632	0.783162	−	0.621818i	$-0.213606\pi$
$647$	4.51705e15	1.56632	0.783162	+	0.621818i	$0.213606\pi$
$648$	0	0
$649$	2.97362e15	1.01377
$650$	5.02182e14	0.169761
$651$	0	0
$652$	−3.40133e15	−1.13054
$653$	4.67633e15	1.54128	0.770642	−	0.637268i	$-0.219936\pi$
$653$	4.67633e15	1.54128	0.770642	+	0.637268i	$0.219936\pi$
$654$	0	0
$655$	−1.43840e15	−0.466179
$656$	3.54162e13	0.0113823
$657$	0	0
$658$	2.38927e15	0.755133
$659$	2.44697e15	0.766935	0.383468	−	0.923554i	$-0.374730\pi$
$659$	2.44697e15	0.766935	0.383468	+	0.923554i	$0.374730\pi$
$660$	0	0
$661$	−1.00863e15	−0.310902	−0.155451	−	0.987844i	$-0.549683\pi$
$661$	−1.00863e15	−0.310902	−0.155451	+	0.987844i	$0.549683\pi$
$662$	3.77906e15	1.15522
$663$	0	0
$664$	3.31523e15	0.996758
$665$	−1.35789e15	−0.404896
$666$	0	0
$667$	5.48022e14	0.160733
$668$	−2.15071e15	−0.625620
$669$	0	0
$670$	1.44019e15	0.412105
$671$	−3.36828e15	−0.955948
$672$	0	0
$673$	3.14110e15	0.877000	0.438500	−	0.898731i	$-0.355510\pi$
$673$	3.14110e15	0.877000	0.438500	+	0.898731i	$0.355510\pi$
$674$	2.75555e15	0.763096
$675$	0	0
$676$	−2.11075e15	−0.575081
$677$	1.77722e15	0.480291	0.240145	−	0.970737i	$-0.422805\pi$
$677$	1.77722e15	0.480291	0.240145	+	0.970737i	$0.422805\pi$
$678$	0	0
$679$	1.19997e15	0.319071
$680$	1.71414e15	0.452114
$681$	0	0
$682$	−2.43172e15	−0.631102
$683$	4.74129e15	1.22063	0.610313	−	0.792160i	$-0.291044\pi$
$683$	4.74129e15	1.22063	0.610313	+	0.792160i	$0.291044\pi$
$684$	0	0
$685$	1.81386e15	0.459521
$686$	−2.50419e15	−0.629340
$687$	0	0
$688$	9.72438e13	0.0240506
$689$	−2.26924e15	−0.556770
$690$	0	0
$691$	2.44203e15	0.589688	0.294844	−	0.955545i	$-0.404732\pi$
$691$	2.44203e15	0.589688	0.294844	+	0.955545i	$0.404732\pi$
$692$	−2.70018e15	−0.646858
$693$	0	0
$694$	−2.20777e15	−0.520568
$695$	2.17140e15	0.507952
$696$	0	0
$697$	−3.29433e15	−0.758555
$698$	−6.46517e14	−0.147698
$699$	0	0
$700$	9.01720e14	0.202783
$701$	−3.91146e15	−0.872749	−0.436375	−	0.899765i	$-0.643738\pi$
$701$	−3.91146e15	−0.872749	−0.436375	+	0.899765i	$0.643738\pi$
$702$	0	0
$703$	3.76483e15	0.826973
$704$	2.62621e15	0.572375
$705$	0	0
$706$	1.99596e15	0.428277
$707$	−3.82611e15	−0.814613
$708$	0	0
$709$	−7.52555e15	−1.57755	−0.788776	−	0.614681i	$-0.789285\pi$
$709$	−7.52555e15	−1.57755	−0.788776	+	0.614681i	$0.789285\pi$
$710$	−6.37712e14	−0.132649
$711$	0	0
$712$	6.89665e15	1.41253
$713$	−8.31004e15	−1.68893
$714$	0	0
$715$	2.95376e15	0.591142
$716$	1.11755e14	0.0221945
$717$	0	0
$718$	−2.84665e14	−0.0556737
$719$	−3.82053e15	−0.741507	−0.370753	−	0.928731i	$-0.620901\pi$
$719$	−3.82053e15	−0.741507	−0.370753	+	0.928731i	$0.620901\pi$
$720$	0	0
$721$	−7.24724e13	−0.0138525
$722$	−2.22382e15	−0.421838
$723$	0	0
$724$	−3.32794e15	−0.621747
$725$	1.10888e14	0.0205602
$726$	0	0
$727$	−7.15305e15	−1.30633	−0.653163	−	0.757217i	$-0.726558\pi$
$727$	−7.15305e15	−1.30633	−0.653163	+	0.757217i	$0.726558\pi$
$728$	1.23475e16	2.23798
$729$	0	0
$730$	−5.50679e14	−0.0983158
$731$	−9.04540e15	−1.60281
$732$	0	0
$733$	3.23209e15	0.564172	0.282086	−	0.959389i	$-0.408974\pi$
$733$	3.23209e15	0.564172	0.282086	+	0.959389i	$0.408974\pi$
$734$	2.85829e15	0.495197
$735$	0	0
$736$	9.20266e15	1.57067
$737$	8.47096e15	1.43503
$738$	0	0
$739$	−7.04037e15	−1.17504	−0.587518	−	0.809211i	$-0.699895\pi$
$739$	−7.04037e15	−1.17504	−0.587518	+	0.809211i	$0.699895\pi$
$740$	−2.50008e15	−0.414171
$741$	0	0
$742$	2.44657e15	0.399334
$743$	−1.76769e15	−0.286396	−0.143198	−	0.989694i	$-0.545739\pi$
$743$	−1.76769e15	−0.286396	−0.143198	+	0.989694i	$0.545739\pi$
$744$	0	0
$745$	3.52342e14	0.0562477
$746$	3.60825e15	0.571785
$747$	0	0
$748$	3.87717e15	0.605420
$749$	−1.19231e16	−1.84816
$750$	0	0
$751$	−6.97015e15	−1.06469	−0.532344	−	0.846528i	$-0.678689\pi$
$751$	−6.97015e15	−1.06469	−0.532344	+	0.846528i	$0.678689\pi$
$752$	−7.63698e13	−0.0115804
$753$	0	0
$754$	5.83912e14	0.0872579
$755$	7.67945e14	0.113926
$756$	0	0
$757$	−1.10779e16	−1.61968	−0.809841	−	0.586650i	$-0.800447\pi$
$757$	−1.10779e16	−1.61968	−0.809841	+	0.586650i	$0.800447\pi$
$758$	−1.73747e15	−0.252196
$759$	0	0
$760$	−1.73580e15	−0.248328
$761$	−1.03991e15	−0.147700	−0.0738500	−	0.997269i	$-0.523529\pi$
$761$	−1.03991e15	−0.147700	−0.0738500	+	0.997269i	$0.523529\pi$
$762$	0	0
$763$	−5.08359e15	−0.711681
$764$	−2.92472e15	−0.406510
$765$	0	0
$766$	−2.94682e15	−0.403734
$767$	1.08274e16	1.47282
$768$	0	0
$769$	−5.06484e15	−0.679158	−0.339579	−	0.940578i	$-0.610285\pi$
$769$	−5.06484e15	−0.679158	−0.339579	+	0.940578i	$0.610285\pi$
$770$	−3.18458e15	−0.423987
$771$	0	0
$772$	−3.98777e15	−0.523402
$773$	9.32127e15	1.21475	0.607376	−	0.794414i	$-0.292222\pi$
$773$	9.32127e15	1.21475	0.607376	+	0.794414i	$0.292222\pi$
$774$	0	0
$775$	−1.68148e15	−0.216039
$776$	1.53394e15	0.195690
$777$	0	0
$778$	2.05769e15	0.258816
$779$	3.33596e15	0.416643
$780$	0	0
$781$	−3.75093e15	−0.461911
$782$	−7.95557e15	−0.972823
$783$	0	0
$784$	2.06456e14	0.0248937
$785$	−1.82287e15	−0.218259
$786$	0	0
$787$	8.41800e15	0.993912	0.496956	−	0.867776i	$-0.334451\pi$
$787$	8.41800e15	0.993912	0.496956	+	0.867776i	$0.334451\pi$
$788$	4.71038e15	0.552283
$789$	0	0
$790$	2.14446e15	0.247953
$791$	1.49702e16	1.71892
$792$	0	0
$793$	−1.22644e16	−1.38882
$794$	−4.11774e15	−0.463069
$795$	0	0
$796$	2.81913e15	0.312676
$797$	−1.32426e16	−1.45866	−0.729328	−	0.684164i	$-0.760167\pi$
$797$	−1.32426e16	−1.45866	−0.729328	+	0.684164i	$0.760167\pi$
$798$	0	0
$799$	7.10375e15	0.771757
$800$	1.86210e15	0.200913
$801$	0	0
$802$	2.55286e15	0.271686
$803$	−3.23901e15	−0.342356
$804$	0	0
$805$	−1.08828e16	−1.13466
$806$	−8.85427e15	−0.916875
$807$	0	0
$808$	−4.89097e15	−0.499611
$809$	2.72166e15	0.276132	0.138066	−	0.990423i	$-0.455911\pi$
$809$	2.72166e15	0.276132	0.138066	+	0.990423i	$0.455911\pi$
$810$	0	0
$811$	1.27431e16	1.27544	0.637722	−	0.770267i	$-0.279877\pi$
$811$	1.27431e16	1.27544	0.637722	+	0.770267i	$0.279877\pi$
$812$	1.04847e15	0.104232
$813$	0	0
$814$	8.82947e15	0.865965
$815$	8.30630e15	0.809171
$816$	0	0
$817$	9.15968e15	0.880358
$818$	−8.96800e14	−0.0856155
$819$	0	0
$820$	−2.21528e15	−0.208666
$821$	−2.99243e15	−0.279986	−0.139993	−	0.990152i	$-0.544708\pi$
$821$	−2.99243e15	−0.279986	−0.139993	+	0.990152i	$0.544708\pi$
$822$	0	0
$823$	1.31983e16	1.21848	0.609239	−	0.792987i	$-0.291475\pi$
$823$	1.31983e16	1.21848	0.609239	+	0.792987i	$0.291475\pi$
$824$	−9.26423e13	−0.00849591
$825$	0	0
$826$	−1.16735e16	−1.05636
$827$	8.07855e15	0.726195	0.363097	−	0.931751i	$-0.381719\pi$
$827$	8.07855e15	0.726195	0.363097	+	0.931751i	$0.381719\pi$
$828$	0	0
$829$	−9.24683e15	−0.820244	−0.410122	−	0.912031i	$-0.634514\pi$
$829$	−9.24683e15	−0.820244	−0.410122	+	0.912031i	$0.634514\pi$
$830$	−3.11334e15	−0.274345
$831$	0	0
$832$	9.56245e15	0.831555
$833$	−1.92041e16	−1.65900
$834$	0	0
$835$	5.25218e15	0.447779
$836$	−3.92616e15	−0.332532
$837$	0	0
$838$	4.48370e15	0.374796
$839$	7.02444e14	0.0583339	0.0291669	−	0.999575i	$-0.490715\pi$
$839$	7.02444e14	0.0583339	0.0291669	+	0.999575i	$0.490715\pi$
$840$	0	0
$841$	−1.20716e16	−0.989432
$842$	−3.17278e15	−0.258359
$843$	0	0
$844$	1.39094e16	1.11795
$845$	5.15459e15	0.411606
$846$	0	0
$847$	1.85608e15	0.146298
$848$	−7.82012e13	−0.00612402
$849$	0	0
$850$	−1.60975e15	−0.124439
$851$	3.01734e16	2.31746
$852$	0	0
$853$	4.32915e15	0.328234	0.164117	−	0.986441i	$-0.447523\pi$
$853$	4.32915e15	0.328234	0.164117	+	0.986441i	$0.447523\pi$
$854$	1.32228e16	0.996106
$855$	0	0
$856$	−1.52414e16	−1.13350
$857$	−2.43435e16	−1.79882	−0.899412	−	0.437103i	$-0.856005\pi$
$857$	−2.43435e16	−1.79882	−0.899412	+	0.437103i	$0.856005\pi$
$858$	0	0
$859$	1.46780e16	1.07079	0.535394	−	0.844603i	$-0.320163\pi$
$859$	1.46780e16	1.07079	0.535394	+	0.844603i	$0.320163\pi$
$860$	−6.08260e15	−0.440908
$861$	0	0
$862$	−2.80106e15	−0.200462
$863$	−2.68085e16	−1.90640	−0.953199	−	0.302342i	$-0.902232\pi$
$863$	−2.68085e16	−1.90640	−0.953199	+	0.302342i	$0.902232\pi$
$864$	0	0
$865$	6.59403e15	0.462980
$866$	3.37724e15	0.235621
$867$	0	0
$868$	−1.58988e16	−1.09523
$869$	1.26134e16	0.863422
$870$	0	0
$871$	3.08441e16	2.08484
$872$	−6.49841e15	−0.436482
$873$	0	0
$874$	8.05608e15	0.534332
$875$	−2.20206e15	−0.145140
$876$	0	0
$877$	1.33110e16	0.866386	0.433193	−	0.901301i	$-0.357387\pi$
$877$	1.33110e16	0.866386	0.433193	+	0.901301i	$0.357387\pi$
$878$	7.92035e15	0.512300
$879$	0	0
$880$	1.01791e14	0.00650209
$881$	1.89533e16	1.20314	0.601570	−	0.798820i	$-0.294542\pi$
$881$	1.89533e16	1.20314	0.601570	+	0.798820i	$0.294542\pi$
$882$	0	0
$883$	−3.10076e15	−0.194395	−0.0971975	−	0.995265i	$-0.530988\pi$
$883$	−3.10076e15	−0.194395	−0.0971975	+	0.995265i	$0.530988\pi$
$884$	1.41174e16	0.879563
$885$	0	0
$886$	−1.33744e16	−0.822980
$887$	2.17069e16	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$887$	2.17069e16	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$888$	0	0
$889$	−8.50094e15	−0.513461
$890$	−6.47665e15	−0.388781
$891$	0	0
$892$	−1.26616e16	−0.750726
$893$	−7.19350e15	−0.423894
$894$	0	0
$895$	−2.72913e14	−0.0158854
$896$	1.78684e16	1.03369
$897$	0	0
$898$	−1.90423e16	−1.08818
$899$	−1.95514e15	−0.111045
$900$	0	0
$901$	7.27411e15	0.408125
$902$	7.82365e15	0.436288
$903$	0	0
$904$	1.91366e16	1.05424
$905$	8.12707e15	0.445007
$906$	0	0
$907$	−7.15520e15	−0.387063	−0.193532	−	0.981094i	$-0.561994\pi$
$907$	−7.15520e15	−0.387063	−0.193532	+	0.981094i	$0.561994\pi$
$908$	6.94761e15	0.373562
$909$	0	0
$910$	−1.15955e16	−0.615975
$911$	−2.19413e16	−1.15854	−0.579270	−	0.815135i	$-0.696662\pi$
$911$	−2.19413e16	−1.15854	−0.579270	+	0.815135i	$0.696662\pi$
$912$	0	0
$913$	−1.83122e16	−0.955325
$914$	−5.95252e15	−0.308672
$915$	0	0
$916$	−2.78153e15	−0.142514
$917$	3.32131e16	1.69152
$918$	0	0
$919$	−3.69170e15	−0.185777	−0.0928884	−	0.995677i	$-0.529610\pi$
$919$	−3.69170e15	−0.185777	−0.0928884	+	0.995677i	$0.529610\pi$
$920$	−1.39116e16	−0.695898
$921$	0	0
$922$	8.92906e15	0.441353
$923$	−1.36577e16	−0.671071
$924$	0	0
$925$	6.10538e15	0.296438
$926$	2.44268e15	0.117898
$927$	0	0
$928$	2.16515e15	0.103270
$929$	−2.85838e16	−1.35530	−0.677648	−	0.735387i	$-0.737000\pi$
$929$	−2.85838e16	−1.35530	−0.677648	+	0.735387i	$0.737000\pi$
$930$	0	0
$931$	1.94467e16	0.911221
$932$	−2.53537e16	−1.18101
$933$	0	0
$934$	−1.07203e16	−0.493512
$935$	−9.46833e15	−0.433321
$936$	0	0
$937$	1.92375e16	0.870125	0.435062	−	0.900400i	$-0.356726\pi$
$937$	1.92375e16	0.870125	0.435062	+	0.900400i	$0.356726\pi$
$938$	−3.32544e16	−1.49532
$939$	0	0
$940$	4.77693e15	0.212298
$941$	2.64255e16	1.16756	0.583781	−	0.811911i	$-0.301573\pi$
$941$	2.64255e16	1.16756	0.583781	+	0.811911i	$0.301573\pi$
$942$	0	0
$943$	2.67361e16	1.16757
$944$	3.73128e14	0.0161999
$945$	0	0
$946$	2.14817e16	0.921867
$947$	1.92888e16	0.822961	0.411481	−	0.911418i	$-0.365012\pi$
$947$	1.92888e16	0.822961	0.411481	+	0.911418i	$0.365012\pi$
$948$	0	0
$949$	−1.17937e16	−0.497379
$950$	1.63009e15	0.0683490
$951$	0	0
$952$	−3.95801e16	−1.64049
$953$	−1.47724e16	−0.608750	−0.304375	−	0.952552i	$-0.598448\pi$
$953$	−1.47724e16	−0.608750	−0.304375	+	0.952552i	$0.598448\pi$
$954$	0	0
$955$	7.14238e15	0.290954
$956$	1.32389e15	0.0536210
$957$	0	0
$958$	1.63022e16	0.652735
$959$	−4.18826e16	−1.66737
$960$	0	0
$961$	4.23870e15	0.166822
$962$	3.21495e16	1.25809
$963$	0	0
$964$	−1.74137e16	−0.673699
$965$	9.73843e15	0.374618
$966$	0	0
$967$	−6.09293e15	−0.231729	−0.115865	−	0.993265i	$-0.536964\pi$
$967$	−6.09293e15	−0.231729	−0.115865	+	0.993265i	$0.536964\pi$
$968$	2.37265e15	0.0897263
$969$	0	0
$970$	−1.44052e15	−0.0538612
$971$	−2.81975e16	−1.04835	−0.524174	−	0.851611i	$-0.675626\pi$
$971$	−2.81975e16	−1.04835	−0.524174	+	0.851611i	$0.675626\pi$
$972$	0	0
$973$	−5.01383e16	−1.84310
$974$	1.04539e16	0.382125
$975$	0	0
$976$	−4.22649e14	−0.0152759
$977$	1.84959e16	0.664745	0.332372	−	0.943148i	$-0.392151\pi$
$977$	1.84959e16	0.664745	0.332372	+	0.943148i	$0.392151\pi$
$978$	0	0
$979$	−3.80947e16	−1.35382
$980$	−1.29138e16	−0.456365
$981$	0	0
$982$	1.02942e15	0.0359731
$983$	2.21617e16	0.770121	0.385060	−	0.922891i	$-0.374181\pi$
$983$	2.21617e16	0.770121	0.385060	+	0.922891i	$0.374181\pi$
$984$	0	0
$985$	−1.15031e16	−0.395289
$986$	−1.87174e15	−0.0639621
$987$	0	0
$988$	−1.42958e16	−0.483108
$989$	7.34106e16	2.46706
$990$	0	0
$991$	7.87374e15	0.261683	0.130842	−	0.991403i	$-0.458232\pi$
$991$	7.87374e15	0.261683	0.130842	+	0.991403i	$0.458232\pi$
$992$	−3.28317e16	−1.08512
$993$	0	0
$994$	1.47250e16	0.481315
$995$	−6.88453e15	−0.223793
$996$	0	0
$997$	−6.16649e15	−0.198251	−0.0991254	−	0.995075i	$-0.531604\pi$
$997$	−6.16649e15	−0.198251	−0.0991254	+	0.995075i	$0.531604\pi$
$998$	7.54196e15	0.241138
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.12.a.c.1.2		2
3.2	odd	2		15.12.a.c.1.1	✓	2
5.2	odd	4		225.12.b.i.199.4		4
5.3	odd	4		225.12.b.i.199.1		4
5.4	even	2		225.12.a.i.1.1		2
12.11	even	2		240.12.a.m.1.2		2
15.2	even	4		75.12.b.d.49.1		4
15.8	even	4		75.12.b.d.49.4		4
15.14	odd	2		75.12.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.1	✓	2	3.2	odd	2
45.12.a.c.1.2		2	1.1	even	1	trivial
75.12.a.c.1.2		2	15.14	odd	2
75.12.b.d.49.1		4	15.2	even	4
75.12.b.d.49.4		4	15.8	even	4
225.12.a.i.1.1		2	5.4	even	2
225.12.b.i.199.1		4	5.3	odd	4
225.12.b.i.199.4		4	5.2	odd	4
240.12.a.m.1.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q+27.7191 q^{2} -1279.65 q^{4} +3125.00 q^{5} -72157.2 q^{7} -92239.5 q^{8} +O(q^{10})\)	\(q+27.7191 q^{2} -1279.65 q^{4} +3125.00 q^{5} -72157.2 q^{7} -92239.5 q^{8} +86622.2 q^{10} +509499. q^{11} +1.85516e6 q^{13} -2.00013e6 q^{14} +63931.4 q^{16} -5.94676e6 q^{17} +6.02189e6 q^{19} -3.99891e6 q^{20} +1.41228e7 q^{22} +4.82627e7 q^{23} +9.76562e6 q^{25} +5.14234e7 q^{26} +9.23361e7 q^{28} +1.13550e7 q^{29} -1.72184e8 q^{31} +1.90679e8 q^{32} -1.64839e8 q^{34} -2.25491e8 q^{35} +6.25191e8 q^{37} +1.66921e8 q^{38} -2.88248e8 q^{40} +5.53971e8 q^{41} +1.52106e9 q^{43} -6.51981e8 q^{44} +1.33780e9 q^{46} -1.19456e9 q^{47} +3.22934e9 q^{49} +2.70694e8 q^{50} -2.37396e9 q^{52} -1.22320e9 q^{53} +1.59218e9 q^{55} +6.65575e9 q^{56} +3.14750e8 q^{58} +5.83637e9 q^{59} -6.61097e9 q^{61} -4.77277e9 q^{62} +5.15451e9 q^{64} +5.79738e9 q^{65} +1.66261e10 q^{67} +7.60978e9 q^{68} -6.25042e9 q^{70} -7.36200e9 q^{71} -6.35726e9 q^{73} +1.73297e10 q^{74} -7.70593e9 q^{76} -3.67640e10 q^{77} +2.47565e10 q^{79} +1.99786e8 q^{80} +1.53556e10 q^{82} -3.59416e10 q^{83} -1.85836e10 q^{85} +4.21625e10 q^{86} -4.69959e10 q^{88} -7.47690e10 q^{89} -1.33863e11 q^{91} -6.17594e10 q^{92} -3.31121e10 q^{94} +1.88184e10 q^{95} -1.66300e10 q^{97} +8.95144e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 13 q^{2} - 3111 q^{4} + 6250 q^{5} + 7784 q^{7} - 35139 q^{8}+O(q^{10}) \) 2 * q + 13 * q^2 - 3111 * q^4 + 6250 * q^5 + 7784 * q^7 - 35139 * q^8	\( 2 q + 13 q^{2} - 3111 q^{4} + 6250 q^{5} + 7784 q^{7} - 35139 q^{8} + 40625 q^{10} - 295568 q^{11} + 657492 q^{13} - 3176796 q^{14} + 2974065 q^{16} - 8579948 q^{17} + 17627976 q^{19} - 9721875 q^{20} + 25972696 q^{22} + 29841072 q^{23} + 19531250 q^{25} + 69052066 q^{26} - 54064108 q^{28} + 201881948 q^{29} - 71057008 q^{31} + 30902261 q^{32} - 126080702 q^{34} + 24325000 q^{35} + 705858484 q^{37} - 3909548 q^{38} - 109809375 q^{40} + 327655148 q^{41} + 3192552120 q^{43} + 822376568 q^{44} + 1608947832 q^{46} - 2053064720 q^{47} + 7642614770 q^{49} + 126953125 q^{50} - 180610022 q^{52} + 2304299452 q^{53} - 923650000 q^{55} + 11220431460 q^{56} - 2489635114 q^{58} + 1478770576 q^{59} - 8264891460 q^{61} - 6261263448 q^{62} + 1546316681 q^{64} + 2054662500 q^{65} + 24212177528 q^{67} + 12432065594 q^{68} - 9927487500 q^{70} + 20218888256 q^{71} + 25879583268 q^{73} + 16142374034 q^{74} - 28960704764 q^{76} - 101122045248 q^{77} + 22324995440 q^{79} + 9293953125 q^{80} + 18686740478 q^{82} - 48014508984 q^{83} - 26812337500 q^{85} + 17559698092 q^{86} - 92965609896 q^{88} - 79209683076 q^{89} - 229606712048 q^{91} - 28023036264 q^{92} - 20475612776 q^{94} + 55087425000 q^{95} - 37075227452 q^{97} + 24554976677 q^{98}+O(q^{100}) \) 2 * q + 13 * q^2 - 3111 * q^4 + 6250 * q^5 + 7784 * q^7 - 35139 * q^8 + 40625 * q^10 - 295568 * q^11 + 657492 * q^13 - 3176796 * q^14 + 2974065 * q^16 - 8579948 * q^17 + 17627976 * q^19 - 9721875 * q^20 + 25972696 * q^22 + 29841072 * q^23 + 19531250 * q^25 + 69052066 * q^26 - 54064108 * q^28 + 201881948 * q^29 - 71057008 * q^31 + 30902261 * q^32 - 126080702 * q^34 + 24325000 * q^35 + 705858484 * q^37 - 3909548 * q^38 - 109809375 * q^40 + 327655148 * q^41 + 3192552120 * q^43 + 822376568 * q^44 + 1608947832 * q^46 - 2053064720 * q^47 + 7642614770 * q^49 + 126953125 * q^50 - 180610022 * q^52 + 2304299452 * q^53 - 923650000 * q^55 + 11220431460 * q^56 - 2489635114 * q^58 + 1478770576 * q^59 - 8264891460 * q^61 - 6261263448 * q^62 + 1546316681 * q^64 + 2054662500 * q^65 + 24212177528 * q^67 + 12432065594 * q^68 - 9927487500 * q^70 + 20218888256 * q^71 + 25879583268 * q^73 + 16142374034 * q^74 - 28960704764 * q^76 - 101122045248 * q^77 + 22324995440 * q^79 + 9293953125 * q^80 + 18686740478 * q^82 - 48014508984 * q^83 - 26812337500 * q^85 + 17559698092 * q^86 - 92965609896 * q^88 - 79209683076 * q^89 - 229606712048 * q^91 - 28023036264 * q^92 - 20475612776 * q^94 + 55087425000 * q^95 - 37075227452 * q^97 + 24554976677 * q^98

Introduction

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