LMFDB - Embedded newform 50.18.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [50,18,Mod(1,50)] 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("50.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	50.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,512,1308] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$91.6110436723$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{83281}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 20820 $ x^2 - x - 20820
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-143.792$ of defining polynomial
Character	$\chi$	$=$	50.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+256.000 q^{2} +9311.53 q^{3} +65536.0 q^{4} +2.38375e6 q^{6} +2.48655e7 q^{7} +1.67772e7 q^{8} -4.24355e7 q^{9} -1.90392e8 q^{11} +6.10241e8 q^{12} +1.57377e7 q^{13} +6.36558e9 q^{14} +4.29497e9 q^{16} +2.25502e10 q^{17} -1.08635e10 q^{18} +1.27984e11 q^{19} +2.31536e11 q^{21} -4.87405e10 q^{22} -2.59325e11 q^{23} +1.56222e11 q^{24} +4.02885e9 q^{26} -1.59763e12 q^{27} +1.62959e12 q^{28} -2.57953e12 q^{29} +7.91253e12 q^{31} +1.09951e12 q^{32} -1.77285e12 q^{33} +5.77284e12 q^{34} -2.78105e12 q^{36} +2.09888e13 q^{37} +3.27638e13 q^{38} +1.46542e11 q^{39} +1.98123e13 q^{41} +5.92733e13 q^{42} +4.60777e13 q^{43} -1.24776e13 q^{44} -6.63871e13 q^{46} -2.66827e14 q^{47} +3.99927e13 q^{48} +3.85664e14 q^{49} +2.09977e14 q^{51} +1.03139e12 q^{52} -4.00045e14 q^{53} -4.08994e14 q^{54} +4.17174e14 q^{56} +1.19172e15 q^{57} -6.60360e14 q^{58} +1.82611e15 q^{59} -1.48236e15 q^{61} +2.02561e15 q^{62} -1.05518e15 q^{63} +2.81475e14 q^{64} -4.53849e14 q^{66} +2.71410e15 q^{67} +1.47785e15 q^{68} -2.41471e15 q^{69} -3.87337e15 q^{71} -7.11949e14 q^{72} +2.85786e15 q^{73} +5.37314e15 q^{74} +8.38753e15 q^{76} -4.73421e15 q^{77} +3.75148e13 q^{78} +2.80988e15 q^{79} -9.39628e15 q^{81} +5.07194e15 q^{82} +7.04230e15 q^{83} +1.51740e16 q^{84} +1.17959e16 q^{86} -2.40194e16 q^{87} -3.19426e15 q^{88} +5.51966e16 q^{89} +3.91327e14 q^{91} -1.69951e16 q^{92} +7.36778e16 q^{93} -6.83076e16 q^{94} +1.02381e16 q^{96} -6.42662e16 q^{97} +9.87300e16 q^{98} +8.07940e15 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 512 q^{2} + 1308 q^{3} + 131072 q^{4} + 334848 q^{6} - 603844 q^{7} + 33554432 q^{8} - 107519094 q^{9} - 471481296 q^{11} + 85721088 q^{12} + 1541834228 q^{13} - 154584064 q^{14} + 8589934592 q^{16}+ \cdots + 26\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q + 512 * q^2 + 1308 * q^3 + 131072 * q^4 + 334848 * q^6 - 603844 * q^7 + 33554432 * q^8 - 107519094 * q^9 - 471481296 * q^11 + 85721088 * q^12 + 1541834228 * q^13 - 154584064 * q^14 + 8589934592 * q^16 - 32139900564 * q^17 - 27524888064 * q^18 + 128672529400 * q^19 + 435381246624 * q^21 - 120699211776 * q^22 - 650359859292 * q^23 + 21944598528 * q^24 + 394709562368 * q^26 - 43156034280 * q^27 - 39573520384 * q^28 + 2543054749980 * q^29 + 7839407998744 * q^31 + 2199023255552 * q^32 + 476857812816 * q^33 - 8227814544384 * q^34 - 7046371344384 * q^36 + 27805209097556 * q^37 + 32940167526400 * q^38 - 12067623537288 * q^39 - 37826364264156 * q^41 + 111457599135744 * q^42 - 29630453926852 * q^43 - 30898998214656 * q^44 - 166492123978752 * q^46 - 220791583022004 * q^47 + 5617817223168 * q^48 + 801722584677954 * q^49 + 647690491265544 * q^51 + 101045647966208 * q^52 - 1058306017600572 * q^53 - 11047944775680 * q^54 - 10130821218304 * q^56 + 1186209507102000 * q^57 + 651022015994880 * q^58 + 1527820717235160 * q^59 - 631738213122116 * q^61 + 2006888447678464 * q^62 + 602457597963468 * q^63 + 562949953421312 * q^64 + 122075600080896 * q^66 + 8843763493872596 * q^67 - 2106320523362304 * q^68 + 714953250788832 * q^69 - 3331901660912376 * q^71 - 1803871064162304 * q^72 + 10026350574292028 * q^73 + 7118133528974336 * q^74 + 8432682886758400 * q^76 + 2424946105073712 * q^77 - 3089311625545728 * q^78 + 8263606842055120 * q^79 - 13432683850659198 * q^81 - 9683549251623936 * q^82 + 10689526892336988 * q^83 + 28533145378750464 * q^84 - 7585396205274112 * q^86 - 65018171604068280 * q^87 - 7910143542951936 * q^88 + 55878412521904980 * q^89 - 38477396610603016 * q^91 - 42621983738560512 * q^92 + 74263001022197376 * q^93 - 56522645253633024 * q^94 + 1438161209131008 * q^96 + 56106580363859756 * q^97 + 205240981677556224 * q^98 + 26373671099096112 * 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 $

$2$	256.000	0.707107
$2$	256.000	0.707107
$3$	9311.53	0.819390	0.409695	−	0.912223i	$-0.365635\pi$
$3$	9311.53	0.819390	0.409695	+	0.912223i	$0.365635\pi$
$4$	65536.0	0.500000
$5$	0	0
$5$	0	0
$6$	2.38375e6	0.579396
$7$	2.48655e7	1.63029	0.815144	−	0.579258i	$-0.196658\pi$
$7$	2.48655e7	1.63029	0.815144	+	0.579258i	$0.196658\pi$
$8$	1.67772e7	0.353553
$9$	−4.24355e7	−0.328600
$10$	0	0
$11$	−1.90392e8	−0.267801	−0.133900	−	0.990995i	$-0.542750\pi$
$11$	−1.90392e8	−0.267801	−0.133900	+	0.990995i	$0.542750\pi$
$12$	6.10241e8	0.409695
$13$	1.57377e7	0.00535085	0.00267543	−	0.999996i	$-0.499148\pi$
$13$	1.57377e7	0.00535085	0.00267543	+	0.999996i	$0.499148\pi$
$14$	6.36558e9	1.15279
$15$	0	0
$16$	4.29497e9	0.250000
$17$	2.25502e10	0.784032	0.392016	−	0.919958i	$-0.371778\pi$
$17$	2.25502e10	0.784032	0.392016	+	0.919958i	$0.371778\pi$
$18$	−1.08635e10	−0.232355
$19$	1.27984e11	1.72882	0.864408	−	0.502791i	$-0.167694\pi$
$19$	1.27984e11	1.72882	0.864408	+	0.502791i	$0.167694\pi$
$20$	0	0
$21$	2.31536e11	1.33584
$22$	−4.87405e10	−0.189364
$23$	−2.59325e11	−0.690490	−0.345245	−	0.938513i	$-0.612204\pi$
$23$	−2.59325e11	−0.690490	−0.345245	+	0.938513i	$0.612204\pi$
$24$	1.56222e11	0.289698
$25$	0	0
$26$	4.02885e9	0.00378363
$27$	−1.59763e12	−1.08864
$28$	1.62959e12	0.815144
$29$	−2.57953e12	−0.957542	−0.478771	−	0.877940i	$-0.658918\pi$
$29$	−2.57953e12	−0.957542	−0.478771	+	0.877940i	$0.658918\pi$
$30$	0	0
$31$	7.91253e12	1.66625	0.833126	−	0.553083i	$-0.186549\pi$
$31$	7.91253e12	1.66625	0.833126	+	0.553083i	$0.186549\pi$
$32$	1.09951e12	0.176777
$33$	−1.77285e12	−0.219433
$34$	5.77284e12	0.554395
$35$	0	0
$36$	−2.78105e12	−0.164300
$37$	2.09888e13	0.982367	0.491184	−	0.871056i	$-0.336564\pi$
$37$	2.09888e13	0.982367	0.491184	+	0.871056i	$0.336564\pi$
$38$	3.27638e13	1.22246
$39$	1.46542e11	0.00438444
$40$	0	0
$41$	1.98123e13	0.387499	0.193750	−	0.981051i	$-0.437935\pi$
$41$	1.98123e13	0.387499	0.193750	+	0.981051i	$0.437935\pi$
$42$	5.92733e13	0.944583
$43$	4.60777e13	0.601186	0.300593	−	0.953753i	$-0.402815\pi$
$43$	4.60777e13	0.601186	0.300593	+	0.953753i	$0.402815\pi$
$44$	−1.24776e13	−0.133900
$45$	0	0
$46$	−6.63871e13	−0.488250
$47$	−2.66827e14	−1.63455	−0.817274	−	0.576250i	$-0.804516\pi$
$47$	−2.66827e14	−1.63455	−0.817274	+	0.576250i	$0.804516\pi$
$48$	3.99927e13	0.204847
$49$	3.85664e14	1.65784
$50$	0	0
$51$	2.09977e14	0.642428
$52$	1.03139e12	0.00267543
$53$	−4.00045e14	−0.882599	−0.441300	−	0.897360i	$-0.645482\pi$
$53$	−4.00045e14	−0.882599	−0.441300	+	0.897360i	$0.645482\pi$
$54$	−4.08994e14	−0.769786
$55$	0	0
$56$	4.17174e14	0.576394
$57$	1.19172e15	1.41657
$58$	−6.60360e14	−0.677084
$59$	1.82611e15	1.61914	0.809570	−	0.587024i	$-0.199701\pi$
$59$	1.82611e15	1.61914	0.809570	+	0.587024i	$0.199701\pi$
$60$	0	0
$61$	−1.48236e15	−0.990035	−0.495017	−	0.868883i	$-0.664838\pi$
$61$	−1.48236e15	−0.990035	−0.495017	+	0.868883i	$0.664838\pi$
$62$	2.02561e15	1.17822
$63$	−1.05518e15	−0.535713
$64$	2.81475e14	0.125000
$65$	0	0
$66$	−4.53849e14	−0.155163
$67$	2.71410e15	0.816563	0.408282	−	0.912856i	$-0.366128\pi$
$67$	2.71410e15	0.816563	0.408282	+	0.912856i	$0.366128\pi$
$68$	1.47785e15	0.392016
$69$	−2.41471e15	−0.565780
$70$	0	0
$71$	−3.87337e15	−0.711858	−0.355929	−	0.934513i	$-0.615836\pi$
$71$	−3.87337e15	−0.711858	−0.355929	+	0.934513i	$0.615836\pi$
$72$	−7.11949e14	−0.116178
$73$	2.85786e15	0.414759	0.207380	−	0.978261i	$-0.433506\pi$
$73$	2.85786e15	0.414759	0.207380	+	0.978261i	$0.433506\pi$
$74$	5.37314e15	0.694638
$75$	0	0
$76$	8.38753e15	0.864408
$77$	−4.73421e15	−0.436593
$78$	3.75148e13	0.00310026
$79$	2.80988e15	0.208381	0.104190	−	0.994557i	$-0.466775\pi$
$79$	2.80988e15	0.208381	0.104190	+	0.994557i	$0.466775\pi$
$80$	0	0
$81$	−9.39628e15	−0.563422
$82$	5.07194e15	0.274003
$83$	7.04230e15	0.343203	0.171601	−	0.985166i	$-0.445106\pi$
$83$	7.04230e15	0.343203	0.171601	+	0.985166i	$0.445106\pi$
$84$	1.51740e16	0.667921
$85$	0	0
$86$	1.17959e16	0.425103
$87$	−2.40194e16	−0.784600
$88$	−3.19426e15	−0.0946819
$89$	5.51966e16	1.48627	0.743135	−	0.669142i	$-0.233338\pi$
$89$	5.51966e16	1.48627	0.743135	+	0.669142i	$0.233338\pi$
$90$	0	0
$91$	3.91327e14	0.00872343
$92$	−1.69951e16	−0.345245
$93$	7.36778e16	1.36531
$94$	−6.83076e16	−1.15580
$95$	0	0
$96$	1.02381e16	0.144849
$97$	−6.42662e16	−0.832574	−0.416287	−	0.909233i	$-0.636669\pi$
$97$	−6.42662e16	−0.832574	−0.416287	+	0.909233i	$0.636669\pi$
$98$	9.87300e16	1.17227
$99$	8.07940e15	0.0879994
$100$	0	0
$101$	−1.56405e17	−1.43721	−0.718603	−	0.695421i	$-0.755218\pi$
$101$	−1.56405e17	−1.43721	−0.718603	+	0.695421i	$0.755218\pi$
$102$	5.37540e16	0.454265
$103$	2.33166e17	1.81363	0.906816	−	0.421527i	$-0.138506\pi$
$103$	2.33166e17	1.81363	0.906816	+	0.421527i	$0.138506\pi$
$104$	2.64035e14	0.00189181
$105$	0	0
$106$	−1.02411e17	−0.624092
$107$	1.05397e17	0.593017	0.296509	−	0.955030i	$-0.404178\pi$
$107$	1.05397e17	0.593017	0.296509	+	0.955030i	$0.404178\pi$
$108$	−1.04702e17	−0.544321
$109$	2.51895e17	1.21086	0.605431	−	0.795898i	$-0.293001\pi$
$109$	2.51895e17	1.21086	0.605431	+	0.795898i	$0.293001\pi$
$110$	0	0
$111$	1.95438e17	0.804942
$112$	1.06797e17	0.407572
$113$	−4.09395e17	−1.44869	−0.724346	−	0.689436i	$-0.757858\pi$
$113$	−4.09395e17	−1.44869	−0.724346	+	0.689436i	$0.757858\pi$
$114$	3.05081e17	1.00167
$115$	0	0
$116$	−1.69052e17	−0.478771
$117$	−6.67838e14	−0.00175829
$118$	4.67483e17	1.14490
$119$	5.60722e17	1.27820
$120$	0	0
$121$	−4.69198e17	−0.928283
$122$	−3.79484e17	−0.700060
$123$	1.84482e17	0.317513
$124$	5.18555e17	0.833126
$125$	0	0
$126$	−2.70126e17	−0.378806
$127$	6.42410e17	0.842327	0.421164	−	0.906985i	$-0.361622\pi$
$127$	6.42410e17	0.842327	0.421164	+	0.906985i	$0.361622\pi$
$128$	7.20576e16	0.0883883
$129$	4.29054e17	0.492606
$130$	0	0
$131$	3.57323e17	0.359961	0.179980	−	0.983670i	$-0.442397\pi$
$131$	3.57323e17	0.359961	0.179980	+	0.983670i	$0.442397\pi$
$132$	−1.16185e17	−0.109717
$133$	3.18238e18	2.81847
$134$	6.94809e17	0.577397
$135$	0	0
$136$	3.78329e17	0.277197
$137$	1.23821e18	0.852454	0.426227	−	0.904616i	$-0.359842\pi$
$137$	1.23821e18	0.852454	0.426227	+	0.904616i	$0.359842\pi$
$138$	−6.18166e17	−0.400067
$139$	1.95129e18	1.18767	0.593836	−	0.804586i	$-0.297613\pi$
$139$	1.95129e18	1.18767	0.593836	+	0.804586i	$0.297613\pi$
$140$	0	0
$141$	−2.48457e18	−1.33933
$142$	−9.91584e17	−0.503360
$143$	−2.99634e15	−0.00143296
$144$	−1.82259e17	−0.0821501
$145$	0	0
$146$	7.31611e17	0.293279
$147$	3.59112e18	1.35842
$148$	1.37552e18	0.491184
$149$	1.86793e18	0.629908	0.314954	−	0.949107i	$-0.398011\pi$
$149$	1.86793e18	0.629908	0.314954	+	0.949107i	$0.398011\pi$
$150$	0	0
$151$	5.18884e18	1.56231	0.781154	−	0.624339i	$-0.214632\pi$
$151$	5.18884e18	1.56231	0.781154	+	0.624339i	$0.214632\pi$
$152$	2.14721e18	0.611229
$153$	−9.56927e17	−0.257633
$154$	−1.21196e18	−0.308718
$155$	0	0
$156$	9.60379e15	0.00219222
$157$	−7.78987e18	−1.68416	−0.842080	−	0.539353i	$-0.818669\pi$
$157$	−7.78987e18	−1.68416	−0.842080	+	0.539353i	$0.818669\pi$
$158$	7.19330e17	0.147347
$159$	−3.72503e18	−0.723193
$160$	0	0
$161$	−6.44825e18	−1.12570
$162$	−2.40545e18	−0.398399
$163$	−6.16891e18	−0.969647	−0.484824	−	0.874612i	$-0.661116\pi$
$163$	−6.16891e18	−0.969647	−0.484824	+	0.874612i	$0.661116\pi$
$164$	1.29842e18	0.193750
$165$	0	0
$166$	1.80283e18	0.242681
$167$	−1.24726e19	−1.59539	−0.797693	−	0.603064i	$-0.793946\pi$
$167$	−1.24726e19	−1.59539	−0.797693	+	0.603064i	$0.793946\pi$
$168$	3.88453e18	0.472291
$169$	−8.65017e18	−0.999971
$170$	0	0
$171$	−5.43105e18	−0.568089
$172$	3.01975e18	0.300593
$173$	−1.64494e19	−1.55869	−0.779344	−	0.626596i	$-0.784448\pi$
$173$	−1.64494e19	−1.55869	−0.779344	+	0.626596i	$0.784448\pi$
$174$	−6.14896e18	−0.554796
$175$	0	0
$176$	−8.17729e17	−0.0669502
$177$	1.70039e19	1.32671
$178$	1.41303e19	1.05095
$179$	−1.53067e18	−0.108550	−0.0542751	−	0.998526i	$-0.517285\pi$
$179$	−1.53067e18	−0.108550	−0.0542751	+	0.998526i	$0.517285\pi$
$180$	0	0
$181$	−8.51444e18	−0.549399	−0.274700	−	0.961530i	$-0.588578\pi$
$181$	−8.51444e18	−0.549399	−0.274700	+	0.961530i	$0.588578\pi$
$182$	1.00180e17	0.00616840
$183$	−1.38031e19	−0.811224
$184$	−4.35075e18	−0.244125
$185$	0	0
$186$	1.88615e19	0.965420
$187$	−4.29338e18	−0.209965
$188$	−1.74868e19	−0.817274
$189$	−3.97260e19	−1.77480
$190$	0	0
$191$	1.05921e19	0.432710	0.216355	−	0.976315i	$-0.430583\pi$
$191$	1.05921e19	0.432710	0.216355	+	0.976315i	$0.430583\pi$
$192$	2.62096e18	0.102424
$193$	1.55768e19	0.582425	0.291213	−	0.956658i	$-0.405941\pi$
$193$	1.55768e19	0.582425	0.291213	+	0.956658i	$0.405941\pi$
$194$	−1.64521e19	−0.588719
$195$	0	0
$196$	2.52749e19	0.828920
$197$	−8.47320e18	−0.266124	−0.133062	−	0.991108i	$-0.542481\pi$
$197$	−8.47320e18	−0.266124	−0.133062	+	0.991108i	$0.542481\pi$
$198$	2.06833e18	0.0622250
$199$	9.48640e18	0.273433	0.136716	−	0.990610i	$-0.456345\pi$
$199$	9.48640e18	0.273433	0.136716	+	0.990610i	$0.456345\pi$
$200$	0	0
$201$	2.52724e19	0.669083
$202$	−4.00397e19	−1.01626
$203$	−6.41414e19	−1.56107
$204$	1.37610e19	0.321214
$205$	0	0
$206$	5.96905e19	1.28243
$207$	1.10046e19	0.226895
$208$	6.75930e16	0.00133771
$209$	−2.43671e19	−0.462978
$210$	0	0
$211$	−5.06102e19	−0.886824	−0.443412	−	0.896318i	$-0.646232\pi$
$211$	−5.06102e19	−0.886824	−0.443412	+	0.896318i	$0.646232\pi$
$212$	−2.62173e19	−0.441300
$213$	−3.60671e19	−0.583289
$214$	2.69817e19	0.419327
$215$	0	0
$216$	−2.68038e19	−0.384893
$217$	1.96749e20	2.71647
$218$	6.44851e19	0.856208
$219$	2.66110e19	0.339849
$220$	0	0
$221$	3.54888e17	0.00419524
$222$	5.00322e19	0.569180
$223$	−1.13017e20	−1.23752	−0.618759	−	0.785581i	$-0.712364\pi$
$223$	−1.13017e20	−1.23752	−0.618759	+	0.785581i	$0.712364\pi$
$224$	2.73399e19	0.288197
$225$	0	0
$226$	−1.04805e20	−1.02438
$227$	−1.34057e20	−1.26203	−0.631016	−	0.775770i	$-0.717362\pi$
$227$	−1.34057e20	−1.26203	−0.631016	+	0.775770i	$0.717362\pi$
$228$	7.81008e19	0.708287
$229$	−1.53092e20	−1.33768	−0.668840	−	0.743406i	$-0.733209\pi$
$229$	−1.53092e20	−1.33768	−0.668840	+	0.743406i	$0.733209\pi$
$230$	0	0
$231$	−4.40828e19	−0.357740
$232$	−4.32773e19	−0.338542
$233$	−1.54095e20	−1.16215	−0.581076	−	0.813849i	$-0.697368\pi$
$233$	−1.54095e20	−1.16215	−0.581076	+	0.813849i	$0.697368\pi$
$234$	−1.70966e17	−0.00124330
$235$	0	0
$236$	1.19676e20	0.809570
$237$	2.61643e19	0.170745
$238$	1.43545e20	0.903823
$239$	1.10626e20	0.672161	0.336081	−	0.941833i	$-0.390899\pi$
$239$	1.10626e20	0.672161	0.336081	+	0.941833i	$0.390899\pi$
$240$	0	0
$241$	−1.68590e20	−0.954303	−0.477152	−	0.878821i	$-0.658331\pi$
$241$	−1.68590e20	−0.954303	−0.477152	+	0.878821i	$0.658331\pi$
$242$	−1.20115e20	−0.656395
$243$	1.18825e20	0.626980
$244$	−9.71480e19	−0.495017
$245$	0	0
$246$	4.72275e19	0.224516
$247$	2.01417e18	0.00925064
$248$	1.32750e20	0.589109
$249$	6.55746e19	0.281217
$250$	0	0
$251$	−8.71684e19	−0.349247	−0.174623	−	0.984635i	$-0.555871\pi$
$251$	−8.71684e19	−0.349247	−0.174623	+	0.984635i	$0.555871\pi$
$252$	−6.91523e19	−0.267857
$253$	4.93735e19	0.184914
$254$	1.64457e20	0.595615
$255$	0	0
$256$	1.84467e19	0.0625000
$257$	−2.57989e20	−0.845610	−0.422805	−	0.906221i	$-0.638954\pi$
$257$	−2.57989e20	−0.845610	−0.422805	+	0.906221i	$0.638954\pi$
$258$	1.09838e20	0.348325
$259$	5.21899e20	1.60154
$260$	0	0
$261$	1.09464e20	0.314648
$262$	9.14747e19	0.254531
$263$	4.12838e20	1.11213	0.556065	−	0.831139i	$-0.312311\pi$
$263$	4.12838e20	1.11213	0.556065	+	0.831139i	$0.312311\pi$
$264$	−2.97434e19	−0.0775814
$265$	0	0
$266$	8.14689e20	1.99296
$267$	5.13965e20	1.21783
$268$	1.77871e20	0.408282
$269$	3.61622e20	0.804194	0.402097	−	0.915597i	$-0.368282\pi$
$269$	3.61622e20	0.804194	0.402097	+	0.915597i	$0.368282\pi$
$270$	0	0
$271$	−6.61685e20	−1.38170	−0.690848	−	0.723000i	$-0.742762\pi$
$271$	−6.61685e20	−1.38170	−0.690848	+	0.723000i	$0.742762\pi$
$272$	9.68522e19	0.196008
$273$	3.64385e18	0.00714789
$274$	3.16983e20	0.602776
$275$	0	0
$276$	−1.58250e20	−0.282890
$277$	−2.43700e20	−0.422452	−0.211226	−	0.977437i	$-0.567746\pi$
$277$	−2.43700e20	−0.422452	−0.211226	+	0.977437i	$0.567746\pi$
$278$	4.99531e20	0.839812
$279$	−3.35772e20	−0.547531
$280$	0	0
$281$	−9.47052e20	−1.45335	−0.726675	−	0.686981i	$-0.758935\pi$
$281$	−9.47052e20	−1.45335	−0.726675	+	0.686981i	$0.758935\pi$
$282$	−6.36049e20	−0.947050
$283$	3.67504e20	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$283$	3.67504e20	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$284$	−2.53845e20	−0.355929
$285$	0	0
$286$	−7.67064e17	−0.00101326
$287$	4.92642e20	0.631736
$288$	−4.66583e19	−0.0580889
$289$	−3.18730e20	−0.385293
$290$	0	0
$291$	−5.98417e20	−0.682203
$292$	1.87292e20	0.207380
$293$	−1.20904e21	−1.30037	−0.650183	−	0.759778i	$-0.725308\pi$
$293$	−1.20904e21	−1.30037	−0.650183	+	0.759778i	$0.725308\pi$
$294$	9.19328e20	0.960546
$295$	0	0
$296$	3.52134e20	0.347319
$297$	3.04177e20	0.291539
$298$	4.78190e20	0.445412
$299$	−4.08118e18	−0.00369471
$300$	0	0
$301$	1.14575e21	0.980106
$302$	1.32834e21	1.10472
$303$	−1.45637e21	−1.17763
$304$	5.49685e20	0.432204
$305$	0	0
$306$	−2.44973e20	−0.182174
$307$	4.76931e20	0.344969	0.172484	−	0.985012i	$-0.444821\pi$
$307$	4.76931e20	0.344969	0.172484	+	0.985012i	$0.444821\pi$
$308$	−3.10261e20	−0.218296
$309$	2.17114e21	1.48607
$310$	0	0
$311$	3.50418e20	0.227051	0.113525	−	0.993535i	$-0.463786\pi$
$311$	3.50418e20	0.227051	0.113525	+	0.993535i	$0.463786\pi$
$312$	2.45857e18	0.00155013
$313$	−2.04751e21	−1.25631	−0.628157	−	0.778087i	$-0.716190\pi$
$313$	−2.04751e21	−1.25631	−0.628157	+	0.778087i	$0.716190\pi$
$314$	−1.99421e21	−1.19088
$315$	0	0
$316$	1.84148e20	0.104190
$317$	−1.31663e21	−0.725205	−0.362602	−	0.931944i	$-0.618112\pi$
$317$	−1.31663e21	−0.725205	−0.362602	+	0.931944i	$0.618112\pi$
$318$	−9.53607e20	−0.511375
$319$	4.91123e20	0.256430
$320$	0	0
$321$	9.81411e20	0.485912
$322$	−1.65075e21	−0.795988
$323$	2.88605e21	1.35545
$324$	−6.15795e20	−0.281711
$325$	0	0
$326$	−1.57924e21	−0.685644
$327$	2.34553e21	0.992167
$328$	3.32394e20	0.137002
$329$	−6.63479e21	−2.66478
$330$	0	0
$331$	3.85862e21	1.47195	0.735977	−	0.677007i	$-0.236723\pi$
$331$	3.85862e21	1.47195	0.735977	+	0.677007i	$0.236723\pi$
$332$	4.61524e20	0.171601
$333$	−8.90672e20	−0.322806
$334$	−3.19298e21	−1.12811
$335$	0	0
$336$	9.94441e20	0.333960
$337$	−4.69149e21	−1.53623	−0.768115	−	0.640312i	$-0.778805\pi$
$337$	−4.69149e21	−1.53623	−0.768115	+	0.640312i	$0.778805\pi$
$338$	−2.21444e21	−0.707087
$339$	−3.81210e21	−1.18704
$340$	0	0
$341$	−1.50649e21	−0.446224
$342$	−1.39035e21	−0.401700
$343$	3.80526e21	1.07247
$344$	7.73055e20	0.212551
$345$	0	0
$346$	−4.21106e21	−1.10216
$347$	−2.51637e21	−0.642648	−0.321324	−	0.946969i	$-0.604128\pi$
$347$	−2.51637e21	−0.642648	−0.321324	+	0.946969i	$0.604128\pi$
$348$	−1.57413e21	−0.392300
$349$	−2.20222e21	−0.535606	−0.267803	−	0.963474i	$-0.586298\pi$
$349$	−2.20222e21	−0.535606	−0.267803	+	0.963474i	$0.586298\pi$
$350$	0	0
$351$	−2.51431e19	−0.00582516
$352$	−2.09339e20	−0.0473410
$353$	−1.30214e21	−0.287456	−0.143728	−	0.989617i	$-0.545909\pi$
$353$	−1.30214e21	−0.287456	−0.143728	+	0.989617i	$0.545909\pi$
$354$	4.35299e21	0.938123
$355$	0	0
$356$	3.61737e21	0.743135
$357$	5.22118e21	1.04734
$358$	−3.91851e20	−0.0767566
$359$	2.07162e21	0.396285	0.198143	−	0.980173i	$-0.436509\pi$
$359$	2.07162e21	0.396285	0.198143	+	0.980173i	$0.436509\pi$
$360$	0	0
$361$	1.08994e22	1.98880
$362$	−2.17970e21	−0.388484
$363$	−4.36895e21	−0.760625
$364$	2.56460e19	0.00436172
$365$	0	0
$366$	−3.53358e21	−0.573622
$367$	−4.69137e21	−0.744111	−0.372056	−	0.928210i	$-0.621347\pi$
$367$	−4.69137e21	−0.744111	−0.372056	+	0.928210i	$0.621347\pi$
$368$	−1.11379e21	−0.172622
$369$	−8.40743e20	−0.127332
$370$	0	0
$371$	−9.94732e21	−1.43889
$372$	4.82855e21	0.682655
$373$	7.77652e20	0.107463	0.0537317	−	0.998555i	$-0.482888\pi$
$373$	7.77652e20	0.107463	0.0537317	+	0.998555i	$0.482888\pi$
$374$	−1.09911e21	−0.148467
$375$	0	0
$376$	−4.47661e21	−0.577900
$377$	−4.05959e19	−0.00512367
$378$	−1.01699e22	−1.25497
$379$	3.50054e20	0.0422379	0.0211189	−	0.999777i	$-0.493277\pi$
$379$	3.50054e20	0.0422379	0.0211189	+	0.999777i	$0.493277\pi$
$380$	0	0
$381$	5.98183e21	0.690194
$382$	2.71157e21	0.305972
$383$	1.02315e22	1.12914	0.564570	−	0.825385i	$-0.309042\pi$
$383$	1.02315e22	1.12914	0.564570	+	0.825385i	$0.309042\pi$
$384$	6.70967e20	0.0724245
$385$	0	0
$386$	3.98765e21	0.411837
$387$	−1.95533e21	−0.197550
$388$	−4.21175e21	−0.416287
$389$	−4.13104e21	−0.399473	−0.199737	−	0.979850i	$-0.564009\pi$
$389$	−4.13104e21	−0.399473	−0.199737	+	0.979850i	$0.564009\pi$
$390$	0	0
$391$	−5.84781e21	−0.541366
$392$	6.47037e21	0.586135
$393$	3.32723e21	0.294948
$394$	−2.16914e21	−0.188178
$395$	0	0
$396$	5.29491e20	0.0439997
$397$	7.65334e21	0.622489	0.311244	−	0.950330i	$-0.399254\pi$
$397$	7.65334e21	0.622489	0.311244	+	0.950330i	$0.399254\pi$
$398$	2.42852e21	0.193346
$399$	2.96328e22	2.30942
$400$	0	0
$401$	−4.08273e20	−0.0304947	−0.0152473	−	0.999884i	$-0.504854\pi$
$401$	−4.08273e20	−0.0304947	−0.0152473	+	0.999884i	$0.504854\pi$
$402$	6.46974e21	0.473113
$403$	1.24525e20	0.00891588
$404$	−1.02502e22	−0.718603
$405$	0	0
$406$	−1.64202e22	−1.10384
$407$	−3.99612e21	−0.263079
$408$	3.52282e21	0.227133
$409$	2.00923e22	1.26877	0.634383	−	0.773019i	$-0.281254\pi$
$409$	2.00923e22	1.26877	0.634383	+	0.773019i	$0.281254\pi$
$410$	0	0
$411$	1.15297e22	0.698493
$412$	1.52808e22	0.906816
$413$	4.54071e22	2.63966
$414$	2.81717e21	0.160439
$415$	0	0
$416$	1.73038e19	0.000945906 0
$417$	1.81695e22	0.973167
$418$	−6.23798e21	−0.327375
$419$	3.31114e22	1.70278	0.851390	−	0.524533i	$-0.175760\pi$
$419$	3.31114e22	1.70278	0.851390	+	0.524533i	$0.175760\pi$
$420$	0	0
$421$	−2.31325e22	−1.14242	−0.571208	−	0.820805i	$-0.693525\pi$
$421$	−2.31325e22	−1.14242	−0.571208	+	0.820805i	$0.693525\pi$
$422$	−1.29562e22	−0.627079
$423$	1.13229e22	0.537113
$424$	−6.71163e21	−0.312046
$425$	0	0
$426$	−9.23317e21	−0.412448
$427$	−3.68597e22	−1.61404
$428$	6.90732e21	0.296509
$429$	−2.79005e19	−0.00117416
$430$	0	0
$431$	4.44596e22	1.79849	0.899247	−	0.437440i	$-0.144115\pi$
$431$	4.44596e22	1.79849	0.899247	+	0.437440i	$0.144115\pi$
$432$	−6.86178e21	−0.272160
$433$	−2.85304e22	−1.10959	−0.554793	−	0.831989i	$-0.687202\pi$
$433$	−2.85304e22	−1.10959	−0.554793	+	0.831989i	$0.687202\pi$
$434$	5.03678e22	1.92084
$435$	0	0
$436$	1.65082e22	0.605431
$437$	−3.31893e22	−1.19373
$438$	6.81242e21	0.240310
$439$	−4.28578e22	−1.48279	−0.741397	−	0.671066i	$-0.765837\pi$
$439$	−4.28578e22	−1.48279	−0.741397	+	0.671066i	$0.765837\pi$
$440$	0	0
$441$	−1.63658e22	−0.544767
$442$	9.08513e19	0.00296648
$443$	1.40464e22	0.449919	0.224959	−	0.974368i	$-0.427775\pi$
$443$	1.40464e22	0.449919	0.224959	+	0.974368i	$0.427775\pi$
$444$	1.28082e22	0.402471
$445$	0	0
$446$	−2.89323e22	−0.875057
$447$	1.73933e22	0.516140
$448$	6.99902e21	0.203786
$449$	−3.88401e22	−1.10965	−0.554826	−	0.831966i	$-0.687215\pi$
$449$	−3.88401e22	−1.10965	−0.554826	+	0.831966i	$0.687215\pi$
$450$	0	0
$451$	−3.77210e21	−0.103773
$452$	−2.68301e22	−0.724346
$453$	4.83161e22	1.28014
$454$	−3.43186e22	−0.892391
$455$	0	0
$456$	1.99938e22	0.500835
$457$	2.21160e22	0.543774	0.271887	−	0.962329i	$-0.412352\pi$
$457$	2.21160e22	0.543774	0.271887	+	0.962329i	$0.412352\pi$
$458$	−3.91917e22	−0.945883
$459$	−3.60269e22	−0.853530
$460$	0	0
$461$	1.69257e22	0.386446	0.193223	−	0.981155i	$-0.438106\pi$
$461$	1.69257e22	0.386446	0.193223	+	0.981155i	$0.438106\pi$
$462$	−1.12852e22	−0.252960
$463$	−5.00810e22	−1.10213	−0.551067	−	0.834461i	$-0.685779\pi$
$463$	−5.00810e22	−1.10213	−0.551067	+	0.834461i	$0.685779\pi$
$464$	−1.10790e22	−0.239385
$465$	0	0
$466$	−3.94483e22	−0.821765
$467$	6.87611e22	1.40653	0.703266	−	0.710927i	$-0.251724\pi$
$467$	6.87611e22	1.40653	0.703266	+	0.710927i	$0.251724\pi$
$468$	−4.37674e19	−0.000879146 0
$469$	6.74875e22	1.33123
$470$	0	0
$471$	−7.25357e22	−1.37998
$472$	3.06370e22	0.572452
$473$	−8.77284e21	−0.160998
$474$	6.69806e21	0.120735
$475$	0	0
$476$	3.67475e22	0.639099
$477$	1.69761e22	0.290022
$478$	2.83201e22	0.475290
$479$	−3.44291e22	−0.567641	−0.283821	−	0.958877i	$-0.591602\pi$
$479$	−3.44291e22	−0.567641	−0.283821	+	0.958877i	$0.591602\pi$
$480$	0	0
$481$	3.30316e20	0.00525650
$482$	−4.31590e22	−0.674794
$483$	−6.00431e22	−0.922385
$484$	−3.07493e22	−0.464141
$485$	0	0
$486$	3.04191e22	0.443342
$487$	−8.83844e20	−0.0126584	−0.00632921	−	0.999980i	$-0.502015\pi$
$487$	−8.83844e20	−0.0126584	−0.00632921	+	0.999980i	$0.502015\pi$
$488$	−2.48699e22	−0.350030
$489$	−5.74420e22	−0.794519
$490$	0	0
$491$	1.04515e23	1.39632	0.698158	−	0.715943i	$-0.254003\pi$
$491$	1.04515e23	1.39632	0.698158	+	0.715943i	$0.254003\pi$
$492$	1.20902e22	0.158757
$493$	−5.81688e22	−0.750744
$494$	5.15627e20	0.00654119
$495$	0	0
$496$	3.39841e22	0.416563
$497$	−9.63135e22	−1.16053
$498$	1.67871e22	0.198850
$499$	−5.12067e22	−0.596310	−0.298155	−	0.954518i	$-0.596371\pi$
$499$	−5.12067e22	−0.596310	−0.298155	+	0.954518i	$0.596371\pi$
$500$	0	0
$501$	−1.16139e23	−1.30724
$502$	−2.23151e22	−0.246955
$503$	−5.06660e22	−0.551301	−0.275650	−	0.961258i	$-0.588893\pi$
$503$	−5.06660e22	−0.551301	−0.275650	+	0.961258i	$0.588893\pi$
$504$	−1.77030e22	−0.189403
$505$	0	0
$506$	1.26396e22	0.130754
$507$	−8.05463e22	−0.819366
$508$	4.21010e22	0.421164
$509$	1.68111e23	1.65385	0.826923	−	0.562316i	$-0.190089\pi$
$509$	1.68111e23	1.65385	0.826923	+	0.562316i	$0.190089\pi$
$510$	0	0
$511$	7.10621e22	0.676177
$512$	4.72237e21	0.0441942
$513$	−2.04471e23	−1.88206
$514$	−6.60452e22	−0.597936
$515$	0	0
$516$	2.81185e22	0.246303
$517$	5.08018e22	0.437733
$518$	1.33606e23	1.13246
$519$	−1.53170e23	−1.27717
$520$	0	0
$521$	1.13516e23	0.916085	0.458042	−	0.888930i	$-0.348551\pi$
$521$	1.13516e23	0.916085	0.458042	+	0.888930i	$0.348551\pi$
$522$	2.80227e22	0.222490
$523$	−3.13139e22	−0.244609	−0.122305	−	0.992493i	$-0.539028\pi$
$523$	−3.13139e22	−0.244609	−0.122305	+	0.992493i	$0.539028\pi$
$524$	2.34175e22	0.179980
$525$	0	0
$526$	1.05686e23	0.786394
$527$	1.78429e23	1.30640
$528$	−7.61432e21	−0.0548583
$529$	−7.38008e22	−0.523224
$530$	0	0
$531$	−7.74918e22	−0.532050
$532$	2.08560e23	1.40923
$533$	3.11800e20	0.00207345
$534$	1.31575e23	0.861139
$535$	0	0
$536$	4.55350e22	0.288699
$537$	−1.42529e22	−0.0889449
$538$	9.25753e22	0.568651
$539$	−7.34275e22	−0.443971
$540$	0	0
$541$	−1.32903e23	−0.778680	−0.389340	−	0.921094i	$-0.627297\pi$
$541$	−1.32903e23	−0.778680	−0.389340	+	0.921094i	$0.627297\pi$
$542$	−1.69391e23	−0.977006
$543$	−7.92825e22	−0.450172
$544$	2.47942e22	0.138599
$545$	0	0
$546$	9.32826e20	0.00505432
$547$	−3.86923e22	−0.206411	−0.103205	−	0.994660i	$-0.532910\pi$
$547$	−3.86923e22	−0.206411	−0.103205	+	0.994660i	$0.532910\pi$
$548$	8.11477e22	0.426227
$549$	6.29047e22	0.325326
$550$	0	0
$551$	−3.30138e23	−1.65541
$552$	−4.05121e22	−0.200034
$553$	6.98692e22	0.339721
$554$	−6.23872e22	−0.298719
$555$	0	0
$556$	1.27880e23	0.593836
$557$	1.53384e23	0.701475	0.350737	−	0.936474i	$-0.385931\pi$
$557$	1.53384e23	0.701475	0.350737	+	0.936474i	$0.385931\pi$
$558$	−8.59576e22	−0.387163
$559$	7.25157e20	0.00321686
$560$	0	0
$561$	−3.99780e22	−0.172043
$562$	−2.42445e23	−1.02767
$563$	1.19252e23	0.497904	0.248952	−	0.968516i	$-0.419914\pi$
$563$	1.19252e23	0.497904	0.248952	+	0.968516i	$0.419914\pi$
$564$	−1.62829e23	−0.669666
$565$	0	0
$566$	9.40810e22	0.375459
$567$	−2.33644e23	−0.918540
$568$	−6.49844e22	−0.251680
$569$	−1.07363e23	−0.409638	−0.204819	−	0.978800i	$-0.565661\pi$
$569$	−1.07363e23	−0.409638	−0.204819	+	0.978800i	$0.565661\pi$
$570$	0	0
$571$	−2.25644e23	−0.835634	−0.417817	−	0.908531i	$-0.637205\pi$
$571$	−2.25644e23	−0.835634	−0.417817	+	0.908531i	$0.637205\pi$
$572$	−1.96368e20	−0.000716482 0
$573$	9.86284e22	0.354558
$574$	1.26116e23	0.446705
$575$	0	0
$576$	−1.19445e22	−0.0410750
$577$	1.29594e23	0.439127	0.219563	−	0.975598i	$-0.429537\pi$
$577$	1.29594e23	0.439127	0.219563	+	0.975598i	$0.429537\pi$
$578$	−8.15949e22	−0.272444
$579$	1.45044e23	0.477233
$580$	0	0
$581$	1.75111e23	0.559519
$582$	−1.53195e23	−0.482390
$583$	7.61655e22	0.236361
$584$	4.79469e22	0.146640
$585$	0	0
$586$	−3.09514e23	−0.919498
$587$	−3.21537e23	−0.941471	−0.470735	−	0.882274i	$-0.656011\pi$
$587$	−3.21537e23	−0.941471	−0.470735	+	0.882274i	$0.656011\pi$
$588$	2.35348e23	0.679208
$589$	1.01267e24	2.88064
$590$	0	0
$591$	−7.88985e22	−0.218060
$592$	9.01464e22	0.245592
$593$	8.23836e22	0.221246	0.110623	−	0.993862i	$-0.464715\pi$
$593$	8.23836e22	0.221246	0.110623	+	0.993862i	$0.464715\pi$
$594$	7.78694e22	0.206149
$595$	0	0
$596$	1.22417e23	0.314954
$597$	8.83330e22	0.224048
$598$	−1.04478e21	−0.00261255
$599$	−7.29015e22	−0.179725	−0.0898625	−	0.995954i	$-0.528643\pi$
$599$	−7.29015e22	−0.179725	−0.0898625	+	0.995954i	$0.528643\pi$
$600$	0	0
$601$	3.79880e23	0.910361	0.455180	−	0.890399i	$-0.349575\pi$
$601$	3.79880e23	0.910361	0.455180	+	0.890399i	$0.349575\pi$
$602$	2.93311e23	0.693040
$603$	−1.15174e23	−0.268323
$604$	3.40056e23	0.781154
$605$	0	0
$606$	−3.72831e23	−0.832711
$607$	3.76245e23	0.828641	0.414321	−	0.910131i	$-0.364019\pi$
$607$	3.76245e23	0.828641	0.414321	+	0.910131i	$0.364019\pi$
$608$	1.40719e23	0.305614
$609$	−5.97255e23	−1.27912
$610$	0	0
$611$	−4.19924e21	−0.00874623
$612$	−6.27132e22	−0.128817
$613$	4.60370e23	0.932595	0.466298	−	0.884628i	$-0.345588\pi$
$613$	4.60370e23	0.932595	0.466298	+	0.884628i	$0.345588\pi$
$614$	1.22094e23	0.243930
$615$	0	0
$616$	−7.94269e22	−0.154359
$617$	3.36863e23	0.645698	0.322849	−	0.946451i	$-0.395359\pi$
$617$	3.36863e23	0.645698	0.322849	+	0.946451i	$0.395359\pi$
$618$	5.55811e23	1.05081
$619$	1.49640e23	0.279047	0.139523	−	0.990219i	$-0.455443\pi$
$619$	1.49640e23	0.279047	0.139523	+	0.990219i	$0.455443\pi$
$620$	0	0
$621$	4.14306e23	0.751696
$622$	8.97070e22	0.160549
$623$	1.37249e24	2.42305
$624$	6.29394e20	0.00109611
$625$	0	0
$626$	−5.24161e23	−0.888348
$627$	−2.26895e23	−0.379360
$628$	−5.10517e23	−0.842080
$629$	4.73302e23	0.770208
$630$	0	0
$631$	−6.17252e21	−0.00977717	−0.00488858	−	0.999988i	$-0.501556\pi$
$631$	−6.17252e21	−0.00977717	−0.00488858	+	0.999988i	$0.501556\pi$
$632$	4.71420e22	0.0736737
$633$	−4.71259e23	−0.726654
$634$	−3.37058e23	−0.512797
$635$	0	0
$636$	−2.44124e23	−0.361596
$637$	6.06947e21	0.00887086
$638$	1.25728e23	0.181324
$639$	1.64369e23	0.233917
$640$	0	0
$641$	−1.59514e23	−0.221057	−0.110529	−	0.993873i	$-0.535254\pi$
$641$	−1.59514e23	−0.221057	−0.110529	+	0.993873i	$0.535254\pi$
$642$	2.51241e23	0.343592
$643$	−1.37854e24	−1.86049	−0.930244	−	0.366941i	$-0.880405\pi$
$643$	−1.37854e24	−1.86049	−0.930244	+	0.366941i	$0.880405\pi$
$644$	−4.22592e23	−0.562849
$645$	0	0
$646$	7.38829e23	0.958446
$647$	−3.15056e23	−0.403369	−0.201684	−	0.979451i	$-0.564641\pi$
$647$	−3.15056e23	−0.403369	−0.201684	+	0.979451i	$0.564641\pi$
$648$	−1.57643e23	−0.199200
$649$	−3.47677e23	−0.433607
$650$	0	0
$651$	1.83204e24	2.22585
$652$	−4.04286e23	−0.484824
$653$	2.27963e23	0.269838	0.134919	−	0.990857i	$-0.456923\pi$
$653$	2.27963e23	0.269838	0.134919	+	0.990857i	$0.456923\pi$
$654$	6.00455e23	0.701568
$655$	0	0
$656$	8.50930e22	0.0968749
$657$	−1.21275e23	−0.136290
$658$	−1.69851e24	−1.88429
$659$	1.18670e24	1.29961	0.649806	−	0.760100i	$-0.274850\pi$
$659$	1.18670e24	1.29961	0.649806	+	0.760100i	$0.274850\pi$
$660$	0	0
$661$	−1.50237e24	−1.60348	−0.801741	−	0.597672i	$-0.796092\pi$
$661$	−1.50237e24	−1.60348	−0.801741	+	0.597672i	$0.796092\pi$
$662$	9.87806e23	1.04083
$663$	3.30455e21	0.00343754
$664$	1.18150e23	0.121340
$665$	0	0
$666$	−2.28012e23	−0.228258
$667$	6.68936e23	0.661173
$668$	−8.17402e23	−0.797693
$669$	−1.05236e24	−1.01401
$670$	0	0
$671$	2.82230e23	0.265132
$672$	2.54577e23	0.236146
$673$	−2.53432e23	−0.232131	−0.116065	−	0.993242i	$-0.537028\pi$
$673$	−2.53432e23	−0.232131	−0.116065	+	0.993242i	$0.537028\pi$
$674$	−1.20102e24	−1.08628
$675$	0	0
$676$	−5.66897e23	−0.499986
$677$	−4.60323e23	−0.400921	−0.200460	−	0.979702i	$-0.564244\pi$
$677$	−4.60323e23	−0.400921	−0.200460	+	0.979702i	$0.564244\pi$
$678$	−9.75897e23	−0.839367
$679$	−1.59801e24	−1.35734
$680$	0	0
$681$	−1.24828e24	−1.03410
$682$	−3.85660e23	−0.315528
$683$	6.27345e23	0.506910	0.253455	−	0.967347i	$-0.418433\pi$
$683$	6.27345e23	0.506910	0.253455	+	0.967347i	$0.418433\pi$
$684$	−3.55929e23	−0.284045
$685$	0	0
$686$	9.74147e23	0.758349
$687$	−1.42553e24	−1.09608
$688$	1.97902e23	0.150296
$689$	−6.29579e21	−0.00472266
$690$	0	0
$691$	4.11645e23	0.301272	0.150636	−	0.988589i	$-0.451868\pi$
$691$	4.11645e23	0.301272	0.150636	+	0.988589i	$0.451868\pi$
$692$	−1.07803e24	−0.779344
$693$	2.00899e23	0.143464
$694$	−6.44190e23	−0.454421
$695$	0	0
$696$	−4.02978e23	−0.277398
$697$	4.46770e23	0.303812
$698$	−5.63770e23	−0.378731
$699$	−1.43486e24	−0.952255
$700$	0	0
$701$	2.88292e23	0.186737	0.0933683	−	0.995632i	$-0.470237\pi$
$701$	2.88292e23	0.186737	0.0933683	+	0.995632i	$0.470237\pi$
$702$	−6.43663e21	−0.00411901
$703$	2.68623e24	1.69833
$704$	−5.35907e22	−0.0334751
$705$	0	0
$706$	−3.33347e23	−0.203262
$707$	−3.88909e24	−2.34306
$708$	1.11436e24	0.663353
$709$	−2.85869e24	−1.68141	−0.840707	−	0.541491i	$-0.817860\pi$
$709$	−2.85869e24	−1.68141	−0.840707	+	0.541491i	$0.817860\pi$
$710$	0	0
$711$	−1.19239e23	−0.0684740
$712$	9.26046e23	0.525475
$713$	−2.05191e24	−1.15053
$714$	1.33662e24	0.740583
$715$	0	0
$716$	−1.00314e23	−0.0542751
$717$	1.03009e24	0.550762
$718$	5.30336e23	0.280216
$719$	−1.21983e24	−0.636946	−0.318473	−	0.947932i	$-0.603170\pi$
$719$	−1.21983e24	−0.636946	−0.318473	+	0.947932i	$0.603170\pi$
$720$	0	0
$721$	5.79780e24	2.95674
$722$	2.79025e24	1.40630
$723$	−1.56983e24	−0.781947
$724$	−5.58002e23	−0.274700
$725$	0	0
$726$	−1.11845e24	−0.537843
$727$	2.28060e24	1.08394	0.541972	−	0.840396i	$-0.317678\pi$
$727$	2.28060e24	1.08394	0.541972	+	0.840396i	$0.317678\pi$
$728$	6.56537e21	0.00308420
$729$	2.31988e24	1.07716
$730$	0	0
$731$	1.03906e24	0.471349
$732$	−9.04597e23	−0.405612
$733$	8.50533e23	0.376970	0.188485	−	0.982076i	$-0.439642\pi$
$733$	8.50533e23	0.376970	0.188485	+	0.982076i	$0.439642\pi$
$734$	−1.20099e24	−0.526166
$735$	0	0
$736$	−2.85130e23	−0.122062
$737$	−5.16744e23	−0.218676
$738$	−2.15230e23	−0.0900376
$739$	2.14385e24	0.886578	0.443289	−	0.896379i	$-0.353812\pi$
$739$	2.14385e24	0.886578	0.443289	+	0.896379i	$0.353812\pi$
$740$	0	0
$741$	1.87550e22	0.00757988
$742$	−2.54651e24	−1.01745
$743$	−2.23446e24	−0.882606	−0.441303	−	0.897358i	$-0.645484\pi$
$743$	−2.23446e24	−0.882606	−0.441303	+	0.897358i	$0.645484\pi$
$744$	1.23611e24	0.482710
$745$	0	0
$746$	1.99079e23	0.0759880
$747$	−2.98843e23	−0.112777
$748$	−2.81371e23	−0.104982
$749$	2.62076e24	0.966789
$750$	0	0
$751$	−1.71798e24	−0.619555	−0.309778	−	0.950809i	$-0.600255\pi$
$751$	−1.71798e24	−0.619555	−0.309778	+	0.950809i	$0.600255\pi$
$752$	−1.14601e24	−0.408637
$753$	−8.11671e23	−0.286169
$754$	−1.03926e22	−0.00362298
$755$	0	0
$756$	−2.60348e24	−0.887400
$757$	1.09729e24	0.369832	0.184916	−	0.982754i	$-0.440799\pi$
$757$	1.09729e24	0.369832	0.184916	+	0.982754i	$0.440799\pi$
$758$	8.96138e22	0.0298667
$759$	4.59743e23	0.151516
$760$	0	0
$761$	−1.70967e24	−0.550990	−0.275495	−	0.961302i	$-0.588842\pi$
$761$	−1.70967e24	−0.550990	−0.275495	+	0.961302i	$0.588842\pi$
$762$	1.53135e24	0.488041
$763$	6.26350e24	1.97405
$764$	6.94162e23	0.216355
$765$	0	0
$766$	2.61925e24	0.798423
$767$	2.87388e22	0.00866378
$768$	1.71767e23	0.0512119
$769$	1.22119e24	0.360090	0.180045	−	0.983658i	$-0.442376\pi$
$769$	1.22119e24	0.360090	0.180045	+	0.983658i	$0.442376\pi$
$770$	0	0
$771$	−2.40228e24	−0.692884
$772$	1.02084e24	0.291213
$773$	−2.59069e24	−0.730952	−0.365476	−	0.930821i	$-0.619094\pi$
$773$	−2.59069e24	−0.730952	−0.365476	+	0.930821i	$0.619094\pi$
$774$	−5.00564e23	−0.139689
$775$	0	0
$776$	−1.07821e24	−0.294359
$777$	4.85968e24	1.31229
$778$	−1.05755e24	−0.282470
$779$	2.53564e24	0.669915
$780$	0	0
$781$	7.37461e23	0.190636
$782$	−1.49704e24	−0.382804
$783$	4.12114e24	1.04242
$784$	1.65641e24	0.414460
$785$	0	0
$786$	8.51770e23	0.208560
$787$	−2.64413e24	−0.640467	−0.320234	−	0.947339i	$-0.603761\pi$
$787$	−2.64413e24	−0.640467	−0.320234	+	0.947339i	$0.603761\pi$
$788$	−5.55300e23	−0.133062
$789$	3.84415e24	0.911268
$790$	0	0
$791$	−1.01798e25	−2.36179
$792$	1.35550e23	0.0311125
$793$	−2.33290e22	−0.00529753
$794$	1.95925e24	0.440166
$795$	0	0
$796$	6.21701e23	0.136716
$797$	6.24909e24	1.35963	0.679815	−	0.733383i	$-0.262060\pi$
$797$	6.24909e24	1.35963	0.679815	+	0.733383i	$0.262060\pi$
$798$	7.58601e24	1.63301
$799$	−6.01699e24	−1.28154
$800$	0	0
$801$	−2.34230e24	−0.488388
$802$	−1.04518e23	−0.0215630
$803$	−5.44114e23	−0.111073
$804$	1.65625e24	0.334542
$805$	0	0
$806$	3.18784e22	0.00630448
$807$	3.36726e24	0.658948
$808$	−2.62404e24	−0.508129
$809$	5.76985e24	1.10561	0.552805	−	0.833311i	$-0.313557\pi$
$809$	5.76985e24	1.10561	0.552805	+	0.833311i	$0.313557\pi$
$810$	0	0
$811$	3.03433e24	0.569359	0.284679	−	0.958623i	$-0.408113\pi$
$811$	3.03433e24	0.569359	0.284679	+	0.958623i	$0.408113\pi$
$812$	−4.20357e24	−0.780534
$813$	−6.16130e24	−1.13215
$814$	−1.02301e24	−0.186025
$815$	0	0
$816$	9.01843e23	0.160607
$817$	5.89719e24	1.03934
$818$	5.14363e24	0.897153
$819$	−1.66061e22	−0.00286652
$820$	0	0
$821$	4.15299e24	0.702172	0.351086	−	0.936343i	$-0.385812\pi$
$821$	4.15299e24	0.702172	0.351086	+	0.936343i	$0.385812\pi$
$822$	2.95160e24	0.493909
$823$	−9.22848e23	−0.152838	−0.0764191	−	0.997076i	$-0.524349\pi$
$823$	−9.22848e23	−0.152838	−0.0764191	+	0.997076i	$0.524349\pi$
$824$	3.91188e24	0.641216
$825$	0	0
$826$	1.16242e25	1.86652
$827$	−7.67636e24	−1.22000	−0.609998	−	0.792403i	$-0.708830\pi$
$827$	−7.67636e24	−1.22000	−0.609998	+	0.792403i	$0.708830\pi$
$828$	7.21195e23	0.113448
$829$	8.23009e23	0.128142	0.0640709	−	0.997945i	$-0.479592\pi$
$829$	8.23009e23	0.128142	0.0640709	+	0.997945i	$0.479592\pi$
$830$	0	0
$831$	−2.26922e24	−0.346153
$832$	4.42977e21	0.000668857 0
$833$	8.69679e24	1.29980
$834$	4.65140e24	0.688133
$835$	0	0
$836$	−1.59692e24	−0.231489
$837$	−1.26413e25	−1.81395
$838$	8.47652e24	1.20405
$839$	−9.57083e24	−1.34578	−0.672888	−	0.739744i	$-0.734947\pi$
$839$	−9.57083e24	−1.34578	−0.672888	+	0.739744i	$0.734947\pi$
$840$	0	0
$841$	−6.03171e23	−0.0831140
$842$	−5.92192e24	−0.807811
$843$	−8.81851e24	−1.19086
$844$	−3.31679e24	−0.443412
$845$	0	0
$846$	2.89867e24	0.379796
$847$	−1.16669e25	−1.51337
$848$	−1.71818e24	−0.220650
$849$	3.42203e24	0.435079
$850$	0	0
$851$	−5.44292e24	−0.678314
$852$	−2.36369e24	−0.291645
$853$	−2.48597e24	−0.303690	−0.151845	−	0.988404i	$-0.548521\pi$
$853$	−2.48597e24	−0.303690	−0.151845	+	0.988404i	$0.548521\pi$
$854$	−9.43608e24	−1.14130
$855$	0	0
$856$	1.76827e24	0.209663
$857$	2.10495e24	0.247118	0.123559	−	0.992337i	$-0.460569\pi$
$857$	2.10495e24	0.247118	0.123559	+	0.992337i	$0.460569\pi$
$858$	−7.14254e21	−0.000830253 0
$859$	−7.58565e23	−0.0873074	−0.0436537	−	0.999047i	$-0.513900\pi$
$859$	−7.58565e23	−0.0873074	−0.0436537	+	0.999047i	$0.513900\pi$
$860$	0	0
$861$	4.58725e24	0.517638
$862$	1.13817e25	1.27173
$863$	5.83299e24	0.645356	0.322678	−	0.946509i	$-0.395417\pi$
$863$	5.83299e24	0.645356	0.322678	+	0.946509i	$0.395417\pi$
$864$	−1.75662e24	−0.192446
$865$	0	0
$866$	−7.30378e24	−0.784595
$867$	−2.96787e24	−0.315706
$868$	1.28942e25	1.35824
$869$	−5.34980e23	−0.0558046
$870$	0	0
$871$	4.27137e22	0.00436931
$872$	4.22610e24	0.428104
$873$	2.72717e24	0.273584
$874$	−8.49646e24	−0.844094
$875$	0	0
$876$	1.74398e24	0.169925
$877$	1.53835e25	1.48443	0.742214	−	0.670163i	$-0.233776\pi$
$877$	1.53835e25	1.48443	0.742214	+	0.670163i	$0.233776\pi$
$878$	−1.09716e25	−1.04849
$879$	−1.12580e25	−1.06551
$880$	0	0
$881$	−1.57727e25	−1.46424	−0.732118	−	0.681178i	$-0.761468\pi$
$881$	−1.57727e25	−1.46424	−0.732118	+	0.681178i	$0.761468\pi$
$882$	−4.18966e24	−0.385208
$883$	−1.75208e25	−1.59547	−0.797734	−	0.603009i	$-0.793968\pi$
$883$	−1.75208e25	−1.59547	−0.797734	+	0.603009i	$0.793968\pi$
$884$	2.32579e22	0.00209762
$885$	0	0
$886$	3.59588e24	0.318140
$887$	1.05988e25	0.928764	0.464382	−	0.885635i	$-0.346276\pi$
$887$	1.05988e25	0.928764	0.464382	+	0.885635i	$0.346276\pi$
$888$	3.27891e24	0.284590
$889$	1.59739e25	1.37324
$890$	0	0
$891$	1.78898e24	0.150885
$892$	−7.40667e24	−0.618759
$893$	−3.41494e25	−2.82583
$894$	4.45268e24	0.364966
$895$	0	0
$896$	1.79175e24	0.144098
$897$	−3.80020e22	−0.00302741
$898$	−9.94307e24	−0.784643
$899$	−2.04106e25	−1.59551
$900$	0	0
$901$	−9.02107e24	−0.691986
$902$	−9.65659e23	−0.0733784
$903$	1.06687e25	0.803089
$904$	−6.86852e24	−0.512190
$905$	0	0
$906$	1.23689e25	0.905195
$907$	−6.72762e24	−0.487753	−0.243876	−	0.969806i	$-0.578419\pi$
$907$	−6.72762e24	−0.487753	−0.243876	+	0.969806i	$0.578419\pi$
$908$	−8.78557e24	−0.631016
$909$	6.63712e24	0.472266
$910$	0	0
$911$	1.90734e25	1.33205	0.666027	−	0.745928i	$-0.267994\pi$
$911$	1.90734e25	1.33205	0.666027	+	0.745928i	$0.267994\pi$
$912$	5.11841e24	0.354143
$913$	−1.34080e24	−0.0919100
$914$	5.66169e24	0.384506
$915$	0	0
$916$	−1.00331e25	−0.668840
$917$	8.88503e24	0.586840
$918$	−9.22288e24	−0.603537
$919$	−1.73304e25	−1.12364	−0.561818	−	0.827261i	$-0.689898\pi$
$919$	−1.73304e25	−1.12364	−0.561818	+	0.827261i	$0.689898\pi$
$920$	0	0
$921$	4.44096e24	0.282664
$922$	4.33297e24	0.273258
$923$	−6.09580e22	−0.00380905
$924$	−2.88901e24	−0.178870
$925$	0	0
$926$	−1.28207e25	−0.779326
$927$	−9.89452e24	−0.595960
$928$	−2.83622e24	−0.169271
$929$	−6.38514e24	−0.377605	−0.188802	−	0.982015i	$-0.560461\pi$
$929$	−6.38514e24	−0.377605	−0.188802	+	0.982015i	$0.560461\pi$
$930$	0	0
$931$	4.93587e25	2.86610
$932$	−1.00988e25	−0.581076
$933$	3.26293e24	0.186043
$934$	1.76029e25	0.994568
$935$	0	0
$936$	−1.12045e22	−0.000621650 0
$937$	2.87864e25	1.58271	0.791354	−	0.611358i	$-0.209376\pi$
$937$	2.87864e25	1.58271	0.791354	+	0.611358i	$0.209376\pi$
$938$	1.72768e25	0.941324
$939$	−1.90654e25	−1.02941
$940$	0	0
$941$	−2.31234e25	−1.22614	−0.613071	−	0.790028i	$-0.710066\pi$
$941$	−2.31234e25	−1.22614	−0.613071	+	0.790028i	$0.710066\pi$
$942$	−1.85691e25	−0.975796
$943$	−5.13781e24	−0.267564
$944$	7.84307e24	0.404785
$945$	0	0
$946$	−2.24585e24	−0.113843
$947$	−1.60999e25	−0.808816	−0.404408	−	0.914579i	$-0.632522\pi$
$947$	−1.60999e25	−0.808816	−0.404408	+	0.914579i	$0.632522\pi$
$948$	1.71470e24	0.0853725
$949$	4.49761e22	0.00221932
$950$	0	0
$951$	−1.22599e25	−0.594225
$952$	9.40735e24	0.451911
$953$	1.12344e25	0.534883	0.267441	−	0.963574i	$-0.413822\pi$
$953$	1.12344e25	0.534883	0.267441	+	0.963574i	$0.413822\pi$
$954$	4.34588e24	0.205077
$955$	0	0
$956$	7.24996e24	0.336081
$957$	4.57311e24	0.210117
$958$	−8.81385e24	−0.401383
$959$	3.07889e25	1.38975
$960$	0	0
$961$	4.00580e25	1.77640
$962$	8.45610e22	0.00371691
$963$	−4.47259e24	−0.194866
$964$	−1.10487e25	−0.477152
$965$	0	0
$966$	−1.53710e25	−0.652225
$967$	−2.40854e24	−0.101305	−0.0506524	−	0.998716i	$-0.516130\pi$
$967$	−2.40854e24	−0.101305	−0.0506524	+	0.998716i	$0.516130\pi$
$968$	−7.87183e24	−0.328197
$969$	2.68736e25	1.11064
$970$	0	0
$971$	−2.55647e25	−1.03819	−0.519095	−	0.854716i	$-0.673731\pi$
$971$	−2.55647e25	−1.03819	−0.519095	+	0.854716i	$0.673731\pi$
$972$	7.78730e24	0.313490
$973$	4.85200e25	1.93625
$974$	−2.26264e23	−0.00895085
$975$	0	0
$976$	−6.36669e24	−0.247509
$977$	2.30301e25	0.887547	0.443774	−	0.896139i	$-0.353639\pi$
$977$	2.30301e25	0.887547	0.443774	+	0.896139i	$0.353639\pi$
$978$	−1.47052e25	−0.561810
$979$	−1.05090e25	−0.398024
$980$	0	0
$981$	−1.06893e25	−0.397889
$982$	2.67557e25	0.987345
$983$	2.04718e25	0.748949	0.374474	−	0.927237i	$-0.377823\pi$
$983$	2.04718e25	0.748949	0.374474	+	0.927237i	$0.377823\pi$
$984$	3.09510e24	0.112258
$985$	0	0
$986$	−1.48912e25	−0.530856
$987$	−6.17801e25	−2.18350
$988$	1.32001e23	0.00462532
$989$	−1.19491e25	−0.415113
$990$	0	0
$991$	1.20622e25	0.411907	0.205954	−	0.978562i	$-0.433970\pi$
$991$	1.20622e25	0.411907	0.205954	+	0.978562i	$0.433970\pi$
$992$	8.69992e24	0.294555
$993$	3.59297e25	1.20610
$994$	−2.46563e25	−0.820621
$995$	0	0
$996$	4.29750e24	0.140608
$997$	1.73789e23	0.00563786	0.00281893	−	0.999996i	$-0.499103\pi$
$997$	1.73789e23	0.00563786	0.00281893	+	0.999996i	$0.499103\pi$
$998$	−1.31089e25	−0.421655
$999$	−3.35325e25	−1.06945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.18.a.f.1.2		2
5.2	odd	4		50.18.b.d.49.3		4
5.3	odd	4		50.18.b.d.49.2		4
5.4	even	2		10.18.a.c.1.1	✓	2
15.14	odd	2		90.18.a.k.1.1		2
20.19	odd	2		80.18.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.c.1.1	✓	2	5.4	even	2
50.18.a.f.1.2		2	1.1	even	1	trivial
50.18.b.d.49.2		4	5.3	odd	4
50.18.b.d.49.3		4	5.2	odd	4
80.18.a.d.1.2		2	20.19	odd	2
90.18.a.k.1.1		2	15.14	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

\(f(q)\)	\(=\)	\(q+256.000 q^{2} +9311.53 q^{3} +65536.0 q^{4} +2.38375e6 q^{6} +2.48655e7 q^{7} +1.67772e7 q^{8} -4.24355e7 q^{9} -1.90392e8 q^{11} +6.10241e8 q^{12} +1.57377e7 q^{13} +6.36558e9 q^{14} +4.29497e9 q^{16} +2.25502e10 q^{17} -1.08635e10 q^{18} +1.27984e11 q^{19} +2.31536e11 q^{21} -4.87405e10 q^{22} -2.59325e11 q^{23} +1.56222e11 q^{24} +4.02885e9 q^{26} -1.59763e12 q^{27} +1.62959e12 q^{28} -2.57953e12 q^{29} +7.91253e12 q^{31} +1.09951e12 q^{32} -1.77285e12 q^{33} +5.77284e12 q^{34} -2.78105e12 q^{36} +2.09888e13 q^{37} +3.27638e13 q^{38} +1.46542e11 q^{39} +1.98123e13 q^{41} +5.92733e13 q^{42} +4.60777e13 q^{43} -1.24776e13 q^{44} -6.63871e13 q^{46} -2.66827e14 q^{47} +3.99927e13 q^{48} +3.85664e14 q^{49} +2.09977e14 q^{51} +1.03139e12 q^{52} -4.00045e14 q^{53} -4.08994e14 q^{54} +4.17174e14 q^{56} +1.19172e15 q^{57} -6.60360e14 q^{58} +1.82611e15 q^{59} -1.48236e15 q^{61} +2.02561e15 q^{62} -1.05518e15 q^{63} +2.81475e14 q^{64} -4.53849e14 q^{66} +2.71410e15 q^{67} +1.47785e15 q^{68} -2.41471e15 q^{69} -3.87337e15 q^{71} -7.11949e14 q^{72} +2.85786e15 q^{73} +5.37314e15 q^{74} +8.38753e15 q^{76} -4.73421e15 q^{77} +3.75148e13 q^{78} +2.80988e15 q^{79} -9.39628e15 q^{81} +5.07194e15 q^{82} +7.04230e15 q^{83} +1.51740e16 q^{84} +1.17959e16 q^{86} -2.40194e16 q^{87} -3.19426e15 q^{88} +5.51966e16 q^{89} +3.91327e14 q^{91} -1.69951e16 q^{92} +7.36778e16 q^{93} -6.83076e16 q^{94} +1.02381e16 q^{96} -6.42662e16 q^{97} +9.87300e16 q^{98} +8.07940e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 512 q^{2} + 1308 q^{3} + 131072 q^{4} + 334848 q^{6} - 603844 q^{7} + 33554432 q^{8} - 107519094 q^{9} - 471481296 q^{11} + 85721088 q^{12} + 1541834228 q^{13} - 154584064 q^{14} + 8589934592 q^{16}+ \cdots + 26\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q + 512 * q^2 + 1308 * q^3 + 131072 * q^4 + 334848 * q^6 - 603844 * q^7 + 33554432 * q^8 - 107519094 * q^9 - 471481296 * q^11 + 85721088 * q^12 + 1541834228 * q^13 - 154584064 * q^14 + 8589934592 * q^16 - 32139900564 * q^17 - 27524888064 * q^18 + 128672529400 * q^19 + 435381246624 * q^21 - 120699211776 * q^22 - 650359859292 * q^23 + 21944598528 * q^24 + 394709562368 * q^26 - 43156034280 * q^27 - 39573520384 * q^28 + 2543054749980 * q^29 + 7839407998744 * q^31 + 2199023255552 * q^32 + 476857812816 * q^33 - 8227814544384 * q^34 - 7046371344384 * q^36 + 27805209097556 * q^37 + 32940167526400 * q^38 - 12067623537288 * q^39 - 37826364264156 * q^41 + 111457599135744 * q^42 - 29630453926852 * q^43 - 30898998214656 * q^44 - 166492123978752 * q^46 - 220791583022004 * q^47 + 5617817223168 * q^48 + 801722584677954 * q^49 + 647690491265544 * q^51 + 101045647966208 * q^52 - 1058306017600572 * q^53 - 11047944775680 * q^54 - 10130821218304 * q^56 + 1186209507102000 * q^57 + 651022015994880 * q^58 + 1527820717235160 * q^59 - 631738213122116 * q^61 + 2006888447678464 * q^62 + 602457597963468 * q^63 + 562949953421312 * q^64 + 122075600080896 * q^66 + 8843763493872596 * q^67 - 2106320523362304 * q^68 + 714953250788832 * q^69 - 3331901660912376 * q^71 - 1803871064162304 * q^72 + 10026350574292028 * q^73 + 7118133528974336 * q^74 + 8432682886758400 * q^76 + 2424946105073712 * q^77 - 3089311625545728 * q^78 + 8263606842055120 * q^79 - 13432683850659198 * q^81 - 9683549251623936 * q^82 + 10689526892336988 * q^83 + 28533145378750464 * q^84 - 7585396205274112 * q^86 - 65018171604068280 * q^87 - 7910143542951936 * q^88 + 55878412521904980 * q^89 - 38477396610603016 * q^91 - 42621983738560512 * q^92 + 74263001022197376 * q^93 - 56522645253633024 * q^94 + 1438161209131008 * q^96 + 56106580363859756 * q^97 + 205240981677556224 * q^98 + 26373671099096112 * q^99

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