LMFDB - Embedded newform 25.20.a.f.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,20,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	25.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:	$57.2041741391$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 726881x^{6} + 160513523376x^{4} - 10607307647230976x^{2} + 32429098232548950016 $ x^8 - 726881x^6 + 160513523376x^4 - 10607307647230976*x^2 + 32429098232548950016
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{8}\cdot 5^{14} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$607.943$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1215.89 q^{2} +18754.1 q^{3} +954092. q^{4} +2.28029e7 q^{6} -1.79878e8 q^{7} +5.22593e8 q^{8} -8.10544e8 q^{9} +O(q^{10})$	\(q+1215.89 q^{2} +18754.1 q^{3} +954092. q^{4} +2.28029e7 q^{6} -1.79878e8 q^{7} +5.22593e8 q^{8} -8.10544e8 q^{9} -3.61960e9 q^{11} +1.78932e10 q^{12} -7.48125e9 q^{13} -2.18711e11 q^{14} +1.35195e11 q^{16} -2.32175e11 q^{17} -9.85529e11 q^{18} -2.12840e12 q^{19} -3.37345e12 q^{21} -4.40103e12 q^{22} +6.69298e12 q^{23} +9.80079e12 q^{24} -9.09635e12 q^{26} -3.69983e13 q^{27} -1.71620e14 q^{28} +7.26252e13 q^{29} +2.51555e14 q^{31} -1.09607e14 q^{32} -6.78825e13 q^{33} -2.82299e14 q^{34} -7.73334e14 q^{36} +2.00511e14 q^{37} -2.58790e15 q^{38} -1.40304e14 q^{39} -2.32187e15 q^{41} -4.10173e15 q^{42} -1.85667e15 q^{43} -3.45344e15 q^{44} +8.13791e15 q^{46} +9.40833e15 q^{47} +2.53547e15 q^{48} +2.09570e16 q^{49} -4.35425e15 q^{51} -7.13780e15 q^{52} +1.66277e16 q^{53} -4.49857e16 q^{54} -9.40028e16 q^{56} -3.99164e16 q^{57} +8.83040e16 q^{58} +1.59149e15 q^{59} -4.33362e16 q^{61} +3.05862e17 q^{62} +1.45799e17 q^{63} -2.04151e17 q^{64} -8.25375e16 q^{66} -2.54117e17 q^{67} -2.21517e17 q^{68} +1.25521e17 q^{69} -2.27564e17 q^{71} -4.23585e17 q^{72} -8.76951e17 q^{73} +2.43799e17 q^{74} -2.03069e18 q^{76} +6.51085e17 q^{77} -1.70594e17 q^{78} -5.76500e17 q^{79} +2.48193e17 q^{81} -2.82313e18 q^{82} -5.61483e17 q^{83} -3.21858e18 q^{84} -2.25751e18 q^{86} +1.36202e18 q^{87} -1.89158e18 q^{88} -2.36138e18 q^{89} +1.34571e18 q^{91} +6.38573e18 q^{92} +4.71769e18 q^{93} +1.14395e19 q^{94} -2.05559e18 q^{96} +1.90922e18 q^{97} +2.54814e19 q^{98} +2.93385e18 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1620744 q^{4} + 3365736 q^{6} - 345358584 q^{9}+O(q^{10}) $ 8 * q + 1620744 * q^4 + 3365736 * q^6 - 345358584 * q^9	$ 8 q + 1620744 q^{4} + 3365736 q^{6} - 345358584 q^{9} - 3379575264 q^{11} - 177250591032 q^{14} - 312730276832 q^{16} - 4547188380640 q^{19} - 2983154334624 q^{21} + 6176642779680 q^{24} + 15909228128496 q^{26} + 188222300345040 q^{29} + 72115006686976 q^{31} - 378440221985792 q^{34} - 964020253238712 q^{36} - 30\!\cdots\!28 q^{39}+ \cdots + 16\!\cdots\!72 q^{99}+O(q^{100}) $ 8 * q + 1620744 * q^4 + 3365736 * q^6 - 345358584 * q^9 - 3379575264 * q^11 - 177250591032 * q^14 - 312730276832 * q^16 - 4547188380640 * q^19 - 2983154334624 * q^21 + 6176642779680 * q^24 + 15909228128496 * q^26 + 188222300345040 * q^29 + 72115006686976 * q^31 - 378440221985792 * q^34 - 964020253238712 * q^36 - 3011653788049728 * q^39 - 3314269428857904 * q^41 - 489119304867552 * q^44 - 13716870314831944 * q^46 + 11129258011816744 * q^49 - 46867071411404544 * q^51 - 86068650834185040 * q^54 - 303831778832907360 * q^56 - 259648032778012320 * q^59 - 582893562753226064 * q^61 - 525415747276109696 * q^64 - 1871067587999976288 * q^66 - 302144302523815008 * q^69 - 2617007600602926144 * q^71 - 1847591013622215312 * q^74 - 7494494299858276320 * q^76 - 3935092091109861760 * q^79 - 8328177293660704152 * q^81 - 7683875738824129632 * q^84 - 14358563079146095224 * q^86 - 2120141101636921680 * q^89 - 3358638219973817664 * q^91 + 10251771654977709928 * q^94 + 15670161523483353216 * q^96 + 16256944625235082272 * q^99

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 $

$2$	1215.89	1.67922	0.839611	−	0.543188i	$-0.182783\pi$
$2$	1215.89	1.67922	0.839611	+	0.543188i	$0.182783\pi$
$3$	18754.1	0.550104	0.275052	−	0.961429i	$-0.411305\pi$
$3$	18754.1	0.550104	0.275052	+	0.961429i	$0.411305\pi$
$4$	954092.	1.81979
$5$	0	0
$5$	0	0
$6$	2.28029e7	0.923748
$7$	−1.79878e8	−1.68479	−0.842395	−	0.538861i	$-0.818855\pi$
$7$	−1.79878e8	−1.68479	−0.842395	+	0.538861i	$0.818855\pi$
$8$	5.22593e8	1.37660
$9$	−8.10544e8	−0.697385
$10$	0	0
$11$	−3.61960e9	−0.462840	−0.231420	−	0.972854i	$-0.574337\pi$
$11$	−3.61960e9	−0.462840	−0.231420	+	0.972854i	$0.574337\pi$
$12$	1.78932e10	1.00107
$13$	−7.48125e9	−0.195665	−0.0978323	−	0.995203i	$-0.531191\pi$
$13$	−7.48125e9	−0.195665	−0.0978323	+	0.995203i	$0.531191\pi$
$14$	−2.18711e11	−2.82914
$15$	0	0
$16$	1.35195e11	0.491837
$17$	−2.32175e11	−0.474844	−0.237422	−	0.971407i	$-0.576302\pi$
$17$	−2.32175e11	−0.474844	−0.237422	+	0.971407i	$0.576302\pi$
$18$	−9.85529e11	−1.17106
$19$	−2.12840e12	−1.51319	−0.756597	−	0.653881i	$-0.773140\pi$
$19$	−2.12840e12	−1.51319	−0.756597	+	0.653881i	$0.773140\pi$
$20$	0	0
$21$	−3.37345e12	−0.926810
$22$	−4.40103e12	−0.777210
$23$	6.69298e12	0.774828	0.387414	−	0.921906i	$-0.373368\pi$
$23$	6.69298e12	0.774828	0.387414	+	0.921906i	$0.373368\pi$
$24$	9.80079e12	0.757276
$25$	0	0
$26$	−9.09635e12	−0.328564
$27$	−3.69983e13	−0.933739
$28$	−1.71620e14	−3.06596
$29$	7.26252e13	0.929622	0.464811	−	0.885410i	$-0.346122\pi$
$29$	7.26252e13	0.929622	0.464811	+	0.885410i	$0.346122\pi$
$30$	0	0
$31$	2.51555e14	1.70882	0.854410	−	0.519600i	$-0.173919\pi$
$31$	2.51555e14	1.70882	0.854410	+	0.519600i	$0.173919\pi$
$32$	−1.09607e14	−0.550700
$33$	−6.78825e13	−0.254610
$34$	−2.82299e14	−0.797369
$35$	0	0
$36$	−7.73334e14	−1.26909
$37$	2.00511e14	0.253643	0.126821	−	0.991926i	$-0.459522\pi$
$37$	2.00511e14	0.253643	0.126821	+	0.991926i	$0.459522\pi$
$38$	−2.58790e15	−2.54099
$39$	−1.40304e14	−0.107636
$40$	0	0
$41$	−2.32187e15	−1.10762	−0.553811	−	0.832642i	$-0.686827\pi$
$41$	−2.32187e15	−1.10762	−0.553811	+	0.832642i	$0.686827\pi$
$42$	−4.10173e15	−1.55632
$43$	−1.85667e15	−0.563359	−0.281680	−	0.959508i	$-0.590892\pi$
$43$	−1.85667e15	−0.563359	−0.281680	+	0.959508i	$0.590892\pi$
$44$	−3.45344e15	−0.842269
$45$	0	0
$46$	8.13791e15	1.30111
$47$	9.40833e15	1.22626	0.613130	−	0.789982i	$-0.289910\pi$
$47$	9.40833e15	1.22626	0.613130	+	0.789982i	$0.289910\pi$
$48$	2.53547e15	0.270562
$49$	2.09570e16	1.83851
$50$	0	0
$51$	−4.35425e15	−0.261214
$52$	−7.13780e15	−0.356068
$53$	1.66277e16	0.692167	0.346084	−	0.938204i	$-0.387511\pi$
$53$	1.66277e16	0.692167	0.346084	+	0.938204i	$0.387511\pi$
$54$	−4.49857e16	−1.56796
$55$	0	0
$56$	−9.40028e16	−2.31929
$57$	−3.99164e16	−0.832415
$58$	8.83040e16	1.56104
$59$	1.59149e15	0.0239171	0.0119586	−	0.999928i	$-0.496193\pi$
$59$	1.59149e15	0.0239171	0.0119586	+	0.999928i	$0.496193\pi$
$60$	0	0
$61$	−4.33362e16	−0.474479	−0.237240	−	0.971451i	$-0.576243\pi$
$61$	−4.33362e16	−0.474479	−0.237240	+	0.971451i	$0.576243\pi$
$62$	3.05862e17	2.86949
$63$	1.45799e17	1.17495
$64$	−2.04151e17	−1.41659
$65$	0	0
$66$	−8.25375e16	−0.427547
$67$	−2.54117e17	−1.14110	−0.570550	−	0.821263i	$-0.693270\pi$
$67$	−2.54117e17	−1.14110	−0.570550	+	0.821263i	$0.693270\pi$
$68$	−2.21517e17	−0.864115
$69$	1.25521e17	0.426236
$70$	0	0
$71$	−2.27564e17	−0.589047	−0.294524	−	0.955644i	$-0.595161\pi$
$71$	−2.27564e17	−0.589047	−0.294524	+	0.955644i	$0.595161\pi$
$72$	−4.23585e17	−0.960023
$73$	−8.76951e17	−1.74344	−0.871722	−	0.490000i	$-0.836997\pi$
$73$	−8.76951e17	−1.74344	−0.871722	+	0.490000i	$0.836997\pi$
$74$	2.43799e17	0.425923
$75$	0	0
$76$	−2.03069e18	−2.75369
$77$	6.51085e17	0.779787
$78$	−1.70594e17	−0.180745
$79$	−5.76500e17	−0.541181	−0.270590	−	0.962695i	$-0.587219\pi$
$79$	−5.76500e17	−0.541181	−0.270590	+	0.962695i	$0.587219\pi$
$80$	0	0
$81$	2.48193e17	0.183731
$82$	−2.82313e18	−1.85994
$83$	−5.61483e17	−0.329682	−0.164841	−	0.986320i	$-0.552711\pi$
$83$	−5.61483e17	−0.329682	−0.164841	+	0.986320i	$0.552711\pi$
$84$	−3.21858e18	−1.68660
$85$	0	0
$86$	−2.25751e18	−0.946006
$87$	1.36202e18	0.511389
$88$	−1.89158e18	−0.637147
$89$	−2.36138e18	−0.714431	−0.357215	−	0.934022i	$-0.616274\pi$
$89$	−2.36138e18	−0.714431	−0.357215	+	0.934022i	$0.616274\pi$
$90$	0	0
$91$	1.34571e18	0.329654
$92$	6.38573e18	1.41002
$93$	4.71769e18	0.940029
$94$	1.14395e19	2.05916
$95$	0	0
$96$	−2.05559e18	−0.302943
$97$	1.90922e18	0.254991	0.127496	−	0.991839i	$-0.459306\pi$
$97$	1.90922e18	0.254991	0.127496	+	0.991839i	$0.459306\pi$
$98$	2.54814e19	3.08727
$99$	2.93385e18	0.322777
$100$	0	0
$101$	5.27814e18	0.480206	0.240103	−	0.970747i	$-0.422819\pi$
$101$	5.27814e18	0.480206	0.240103	+	0.970747i	$0.422819\pi$
$102$	−5.29427e18	−0.438636
$103$	−5.05461e18	−0.381710	−0.190855	−	0.981618i	$-0.561126\pi$
$103$	−5.05461e18	−0.381710	−0.190855	+	0.981618i	$0.561126\pi$
$104$	−3.90965e18	−0.269353
$105$	0	0
$106$	2.02174e19	1.16230
$107$	5.80087e18	0.305033	0.152517	−	0.988301i	$-0.451262\pi$
$107$	5.80087e18	0.305033	0.152517	+	0.988301i	$0.451262\pi$
$108$	−3.52998e19	−1.69921
$109$	4.20726e17	0.0185544	0.00927722	−	0.999957i	$-0.497047\pi$
$109$	4.20726e17	0.0185544	0.00927722	+	0.999957i	$0.497047\pi$
$110$	0	0
$111$	3.76042e18	0.139530
$112$	−2.43186e19	−0.828642
$113$	−1.52582e19	−0.477813	−0.238907	−	0.971043i	$-0.576789\pi$
$113$	−1.52582e19	−0.477813	−0.238907	+	0.971043i	$0.576789\pi$
$114$	−4.85338e19	−1.39781
$115$	0	0
$116$	6.92911e19	1.69171
$117$	6.06388e18	0.136454
$118$	1.93507e18	0.0401622
$119$	4.17631e19	0.800012
$120$	0	0
$121$	−4.80576e19	−0.785780
$122$	−5.26919e19	−0.796756
$123$	−4.35447e19	−0.609308
$124$	2.40006e20	3.10969
$125$	0	0
$126$	1.77275e20	1.97300
$127$	5.60376e19	0.578554	0.289277	−	0.957245i	$-0.406585\pi$
$127$	5.60376e19	0.578554	0.289277	+	0.957245i	$0.406585\pi$
$128$	−1.90759e20	−1.82806
$129$	−3.48203e19	−0.309907
$130$	0	0
$131$	−8.10990e19	−0.623646	−0.311823	−	0.950140i	$-0.600940\pi$
$131$	−8.10990e19	−0.623646	−0.311823	+	0.950140i	$0.600940\pi$
$132$	−6.47662e19	−0.463336
$133$	3.82852e20	2.54941
$134$	−3.08978e20	−1.91616
$135$	0	0
$136$	−1.21333e20	−0.653672
$137$	3.62932e20	1.82381	0.911905	−	0.410401i	$-0.134611\pi$
$137$	3.62932e20	1.82381	0.911905	+	0.410401i	$0.134611\pi$
$138$	1.52619e20	0.715746
$139$	−1.66383e20	−0.728564	−0.364282	−	0.931289i	$-0.618686\pi$
$139$	−1.66383e20	−0.728564	−0.364282	+	0.931289i	$0.618686\pi$
$140$	0	0
$141$	1.76445e20	0.674571
$142$	−2.76692e20	−0.989141
$143$	2.70792e19	0.0905613
$144$	−1.09582e20	−0.343000
$145$	0	0
$146$	−1.06627e21	−2.92763
$147$	3.93031e20	1.01137
$148$	1.91306e20	0.461576
$149$	1.41978e20	0.321330	0.160665	−	0.987009i	$-0.448636\pi$
$149$	1.41978e20	0.321330	0.160665	+	0.987009i	$0.448636\pi$
$150$	0	0
$151$	4.28309e20	0.854036	0.427018	−	0.904243i	$-0.359564\pi$
$151$	4.28309e20	0.854036	0.427018	+	0.904243i	$0.359564\pi$
$152$	−1.11229e21	−2.08307
$153$	1.88188e20	0.331149
$154$	7.91646e20	1.30944
$155$	0	0
$156$	−1.33863e20	−0.195875
$157$	7.80079e20	1.07422	0.537108	−	0.843513i	$-0.319517\pi$
$157$	7.80079e20	1.07422	0.537108	+	0.843513i	$0.319517\pi$
$158$	−7.00959e20	−0.908762
$159$	3.11838e20	0.380764
$160$	0	0
$161$	−1.20392e21	−1.30542
$162$	3.01775e20	0.308525
$163$	−1.09860e21	−1.05939	−0.529696	−	0.848188i	$-0.677694\pi$
$163$	−1.09860e21	−1.05939	−0.529696	+	0.848188i	$0.677694\pi$
$164$	−2.21528e21	−2.01564
$165$	0	0
$166$	−6.82699e20	−0.553608
$167$	−3.21947e19	−0.0246591	−0.0123296	−	0.999924i	$-0.503925\pi$
$167$	−3.21947e19	−0.0246591	−0.0123296	+	0.999924i	$0.503925\pi$
$168$	−1.76294e21	−1.27585
$169$	−1.40595e21	−0.961715
$170$	0	0
$171$	1.72516e21	1.05528
$172$	−1.77144e21	−1.02519
$173$	2.65247e21	1.45282	0.726410	−	0.687261i	$-0.241187\pi$
$173$	2.65247e21	1.45282	0.726410	+	0.687261i	$0.241187\pi$
$174$	1.65607e21	0.858736
$175$	0	0
$176$	−4.89353e20	−0.227642
$177$	2.98470e19	0.0131569
$178$	−2.87117e21	−1.19969
$179$	−1.15403e21	−0.457206	−0.228603	−	0.973520i	$-0.573416\pi$
$179$	−1.15403e21	−0.457206	−0.228603	+	0.973520i	$0.573416\pi$
$180$	0	0
$181$	−7.70360e20	−0.274629	−0.137315	−	0.990527i	$-0.543847\pi$
$181$	−7.70360e20	−0.274629	−0.137315	+	0.990527i	$0.543847\pi$
$182$	1.63623e21	0.553562
$183$	−8.12733e20	−0.261013
$184$	3.49771e21	1.06663
$185$	0	0
$186$	5.73618e21	1.57852
$187$	8.40382e20	0.219777
$188$	8.97641e21	2.23153
$189$	6.65516e21	1.57315
$190$	0	0
$191$	−3.70311e21	−0.792044	−0.396022	−	0.918241i	$-0.629610\pi$
$191$	−3.70311e21	−0.792044	−0.396022	+	0.918241i	$0.629610\pi$
$192$	−3.82868e21	−0.779270
$193$	5.44930e21	1.05571	0.527857	−	0.849333i	$-0.322996\pi$
$193$	5.44930e21	1.05571	0.527857	+	0.849333i	$0.322996\pi$
$194$	2.32140e21	0.428187
$195$	0	0
$196$	1.99949e22	3.34570
$197$	5.28276e21	0.842231	0.421116	−	0.907007i	$-0.361639\pi$
$197$	5.28276e21	0.842231	0.421116	+	0.907007i	$0.361639\pi$
$198$	3.56723e21	0.542015
$199$	−1.12121e22	−1.62398	−0.811991	−	0.583670i	$-0.801616\pi$
$199$	−1.12121e22	−1.62398	−0.811991	+	0.583670i	$0.801616\pi$
$200$	0	0
$201$	−4.76575e21	−0.627724
$202$	6.41762e21	0.806373
$203$	−1.30636e22	−1.56622
$204$	−4.15435e21	−0.475354
$205$	0	0
$206$	−6.14583e21	−0.640976
$207$	−5.42496e21	−0.540354
$208$	−1.01143e21	−0.0962351
$209$	7.70398e21	0.700366
$210$	0	0
$211$	1.02283e22	0.849416	0.424708	−	0.905330i	$-0.360377\pi$
$211$	1.02283e22	0.849416	0.424708	+	0.905330i	$0.360377\pi$
$212$	1.58644e22	1.25960
$213$	−4.26777e21	−0.324037
$214$	7.05320e21	0.512219
$215$	0	0
$216$	−1.93350e22	−1.28539
$217$	−4.52490e22	−2.87900
$218$	5.11556e20	0.0311570
$219$	−1.64465e22	−0.959077
$220$	0	0
$221$	1.73696e21	0.0929102
$222$	4.57224e21	0.234302
$223$	−9.30478e21	−0.456888	−0.228444	−	0.973557i	$-0.573364\pi$
$223$	−9.30478e21	−0.456888	−0.228444	+	0.973557i	$0.573364\pi$
$224$	1.97159e22	0.927814
$225$	0	0
$226$	−1.85523e22	−0.802355
$227$	7.54698e20	0.0312988	0.0156494	−	0.999878i	$-0.495018\pi$
$227$	7.54698e20	0.0312988	0.0156494	+	0.999878i	$0.495018\pi$
$228$	−3.80839e22	−1.51482
$229$	−6.69932e21	−0.255619	−0.127810	−	0.991799i	$-0.540795\pi$
$229$	−6.69932e21	−0.255619	−0.127810	+	0.991799i	$0.540795\pi$
$230$	0	0
$231$	1.22105e22	0.428964
$232$	3.79534e22	1.27972
$233$	−2.31221e22	−0.748421	−0.374211	−	0.927344i	$-0.622086\pi$
$233$	−2.31221e22	−0.748421	−0.374211	+	0.927344i	$0.622086\pi$
$234$	7.37299e21	0.229136
$235$	0	0
$236$	1.51843e21	0.0435241
$237$	−1.08118e22	−0.297706
$238$	5.07792e22	1.34340
$239$	−4.10044e22	−1.04244	−0.521220	−	0.853423i	$-0.674523\pi$
$239$	−4.10044e22	−1.04244	−0.521220	+	0.853423i	$0.674523\pi$
$240$	0	0
$241$	−5.88872e22	−1.38312	−0.691559	−	0.722320i	$-0.743076\pi$
$241$	−5.88872e22	−1.38312	−0.691559	+	0.722320i	$0.743076\pi$
$242$	−5.84325e22	−1.31950
$243$	4.76563e22	1.03481
$244$	−4.13467e22	−0.863451
$245$	0	0
$246$	−5.29454e22	−1.02316
$247$	1.59231e22	0.296079
$248$	1.31461e23	2.35237
$249$	−1.05301e22	−0.181359
$250$	0	0
$251$	2.44560e22	0.390378	0.195189	−	0.980766i	$-0.437468\pi$
$251$	2.44560e22	0.390378	0.195189	+	0.980766i	$0.437468\pi$
$252$	1.39105e23	2.13815
$253$	−2.42260e22	−0.358621
$254$	6.81354e22	0.971521
$255$	0	0
$256$	−1.24907e23	−1.65314
$257$	5.03201e22	0.641767	0.320883	−	0.947119i	$-0.396020\pi$
$257$	5.03201e22	0.641767	0.320883	+	0.947119i	$0.396020\pi$
$258$	−4.23376e22	−0.520402
$259$	−3.60675e22	−0.427335
$260$	0	0
$261$	−5.88659e22	−0.648304
$262$	−9.86072e22	−1.04724
$263$	6.26756e22	0.641976	0.320988	−	0.947083i	$-0.395985\pi$
$263$	6.26756e22	0.641976	0.320988	+	0.947083i	$0.395985\pi$
$264$	−3.54750e22	−0.350497
$265$	0	0
$266$	4.65505e23	4.28103
$267$	−4.42856e22	−0.393011
$268$	−2.42451e23	−2.07656
$269$	2.88988e22	0.238910	0.119455	−	0.992840i	$-0.461885\pi$
$269$	2.88988e22	0.238910	0.119455	+	0.992840i	$0.461885\pi$
$270$	0	0
$271$	2.97029e22	0.228871	0.114435	−	0.993431i	$-0.463494\pi$
$271$	2.97029e22	0.228871	0.114435	+	0.993431i	$0.463494\pi$
$272$	−3.13890e22	−0.233546
$273$	2.52376e22	0.181344
$274$	4.41284e23	3.06258
$275$	0	0
$276$	1.19759e23	0.775659
$277$	−3.03186e23	−1.89737	−0.948685	−	0.316223i	$-0.897585\pi$
$277$	−3.03186e23	−1.89737	−0.948685	+	0.316223i	$0.897585\pi$
$278$	−2.02302e23	−1.22342
$279$	−2.03896e23	−1.19171
$280$	0	0
$281$	−2.50158e23	−1.36617	−0.683084	−	0.730340i	$-0.739362\pi$
$281$	−2.50158e23	−1.36617	−0.683084	+	0.730340i	$0.739362\pi$
$282$	2.14537e23	1.13276
$283$	−1.87699e22	−0.0958276	−0.0479138	−	0.998851i	$-0.515257\pi$
$283$	−1.87699e22	−0.0958276	−0.0479138	+	0.998851i	$0.515257\pi$
$284$	−2.17117e23	−1.07194
$285$	0	0
$286$	3.29252e22	0.152073
$287$	4.17652e23	1.86611
$288$	8.88416e22	0.384050
$289$	−1.85167e23	−0.774523
$290$	0	0
$291$	3.58058e22	0.140272
$292$	−8.36692e23	−3.17270
$293$	−1.83821e23	−0.674764	−0.337382	−	0.941368i	$-0.609541\pi$
$293$	−1.83821e23	−0.674764	−0.337382	+	0.941368i	$0.609541\pi$
$294$	4.77881e23	1.69832
$295$	0	0
$296$	1.04786e23	0.349166
$297$	1.33919e23	0.432171
$298$	1.72629e23	0.539584
$299$	−5.00719e22	−0.151606
$300$	0	0
$301$	3.33974e23	0.949142
$302$	5.20775e23	1.43412
$303$	9.89869e22	0.264164
$304$	−2.87750e23	−0.744245
$305$	0	0
$306$	2.28815e23	0.556073
$307$	3.57199e23	0.841580	0.420790	−	0.907158i	$-0.361753\pi$
$307$	3.57199e23	0.841580	0.420790	+	0.907158i	$0.361753\pi$
$308$	6.21196e23	1.41905
$309$	−9.47948e22	−0.209981
$310$	0	0
$311$	−2.61658e22	−0.0545143	−0.0272571	−	0.999628i	$-0.508677\pi$
$311$	−2.61658e22	−0.0545143	−0.0272571	+	0.999628i	$0.508677\pi$
$312$	−7.33221e22	−0.148172
$313$	2.90894e23	0.570247	0.285124	−	0.958491i	$-0.407965\pi$
$313$	2.90894e23	0.570247	0.285124	+	0.958491i	$0.407965\pi$
$314$	9.48488e23	1.80385
$315$	0	0
$316$	−5.50035e23	−0.984833
$317$	1.00748e24	1.75055	0.875273	−	0.483628i	$-0.160681\pi$
$317$	1.00748e24	1.75055	0.875273	+	0.483628i	$0.160681\pi$
$318$	3.79160e23	0.639388
$319$	−2.62874e23	−0.430266
$320$	0	0
$321$	1.08790e23	0.167800
$322$	−1.46383e24	−2.19209
$323$	4.94162e23	0.718532
$324$	2.36799e23	0.334351
$325$	0	0
$326$	−1.33577e24	−1.77895
$327$	7.89036e21	0.0102069
$328$	−1.21339e24	−1.52476
$329$	−1.69235e24	−2.06599
$330$	0	0
$331$	−1.24050e24	−1.42965	−0.714827	−	0.699302i	$-0.753494\pi$
$331$	−1.24050e24	−1.42965	−0.714827	+	0.699302i	$0.753494\pi$
$332$	−5.35706e23	−0.599950
$333$	−1.62523e23	−0.176887
$334$	−3.91451e22	−0.0414082
$335$	0	0
$336$	−4.56074e23	−0.455840
$337$	−1.24721e24	−1.21187	−0.605937	−	0.795513i	$-0.707201\pi$
$337$	−1.24721e24	−1.21187	−0.605937	+	0.795513i	$0.707201\pi$
$338$	−1.70948e24	−1.61493
$339$	−2.86155e23	−0.262847
$340$	0	0
$341$	−9.10528e23	−0.790909
$342$	2.09760e24	1.77205
$343$	−1.71929e24	−1.41272
$344$	−9.70286e23	−0.775523
$345$	0	0
$346$	3.22510e24	2.43961
$347$	−8.82339e23	−0.649390	−0.324695	−	0.945819i	$-0.605262\pi$
$347$	−8.82339e23	−0.649390	−0.324695	+	0.945819i	$0.605262\pi$
$348$	1.29950e24	0.930619
$349$	6.79957e23	0.473849	0.236925	−	0.971528i	$-0.423861\pi$
$349$	6.79957e23	0.473849	0.236925	+	0.971528i	$0.423861\pi$
$350$	0	0
$351$	2.76793e23	0.182700
$352$	3.96735e23	0.254886
$353$	1.70063e24	1.06353	0.531765	−	0.846892i	$-0.321529\pi$
$353$	1.70063e24	1.06353	0.531765	+	0.846892i	$0.321529\pi$
$354$	3.62905e22	0.0220934
$355$	0	0
$356$	−2.25297e24	−1.30011
$357$	7.83231e23	0.440090
$358$	−1.40317e24	−0.767751
$359$	1.87542e24	0.999311	0.499656	−	0.866224i	$-0.333460\pi$
$359$	1.87542e24	0.999311	0.499656	+	0.866224i	$0.333460\pi$
$360$	0	0
$361$	2.55168e24	1.28976
$362$	−9.36670e23	−0.461164
$363$	−9.01278e23	−0.432261
$364$	1.28393e24	0.599899
$365$	0	0
$366$	−9.88191e23	−0.438299
$367$	−1.73965e24	−0.751856	−0.375928	−	0.926649i	$-0.622676\pi$
$367$	−1.73965e24	−0.751856	−0.375928	+	0.926649i	$0.622676\pi$
$368$	9.04859e23	0.381089
$369$	1.88198e24	0.772439
$370$	0	0
$371$	−2.99095e24	−1.16616
$372$	4.50111e24	1.71065
$373$	−1.95284e24	−0.723491	−0.361746	−	0.932277i	$-0.617819\pi$
$373$	−1.95284e24	−0.723491	−0.361746	+	0.932277i	$0.617819\pi$
$374$	1.02181e24	0.369054
$375$	0	0
$376$	4.91673e24	1.68808
$377$	−5.43327e23	−0.181894
$378$	8.09192e24	2.64167
$379$	−3.86854e24	−1.23161	−0.615806	−	0.787897i	$-0.711170\pi$
$379$	−3.86854e24	−1.23161	−0.615806	+	0.787897i	$0.711170\pi$
$380$	0	0
$381$	1.05094e24	0.318265
$382$	−4.50256e24	−1.33002
$383$	5.32239e24	1.53362	0.766810	−	0.641874i	$-0.221843\pi$
$383$	5.32239e24	1.53362	0.766810	+	0.641874i	$0.221843\pi$
$384$	−3.57752e24	−1.00562
$385$	0	0
$386$	6.62573e24	1.77278
$387$	1.50492e24	0.392878
$388$	1.82157e24	0.464030
$389$	−4.85730e24	−1.20746	−0.603731	−	0.797188i	$-0.706320\pi$
$389$	−4.85730e24	−1.20746	−0.603731	+	0.797188i	$0.706320\pi$
$390$	0	0
$391$	−1.55395e24	−0.367923
$392$	1.09520e25	2.53091
$393$	−1.52094e24	−0.343071
$394$	6.42323e24	1.41429
$395$	0	0
$396$	2.79916e24	0.587386
$397$	−5.78732e24	−1.18568	−0.592841	−	0.805320i	$-0.701994\pi$
$397$	−5.78732e24	−1.18568	−0.592841	+	0.805320i	$0.701994\pi$
$398$	−1.36326e25	−2.72703
$399$	7.18006e24	1.40244
$400$	0	0
$401$	6.31326e24	1.17593	0.587966	−	0.808886i	$-0.299929\pi$
$401$	6.31326e24	1.17593	0.587966	+	0.808886i	$0.299929\pi$
$402$	−5.79461e24	−1.05409
$403$	−1.88194e24	−0.334355
$404$	5.03583e24	0.873873
$405$	0	0
$406$	−1.58839e25	−2.63003
$407$	−7.25772e23	−0.117396
$408$	−2.27550e24	−0.359588
$409$	−5.68097e24	−0.877104	−0.438552	−	0.898706i	$-0.644509\pi$
$409$	−5.68097e24	−0.877104	−0.438552	+	0.898706i	$0.644509\pi$
$410$	0	0
$411$	6.80647e24	1.00329
$412$	−4.82256e24	−0.694631
$413$	−2.86273e23	−0.0402953
$414$	−6.59613e24	−0.907374
$415$	0	0
$416$	8.20001e23	0.107753
$417$	−3.12036e24	−0.400786
$418$	9.36716e24	1.17607
$419$	1.23198e25	1.51207	0.756033	−	0.654533i	$-0.227135\pi$
$419$	1.23198e25	1.51207	0.756033	+	0.654533i	$0.227135\pi$
$420$	0	0
$421$	−8.35914e24	−0.980577	−0.490289	−	0.871560i	$-0.663109\pi$
$421$	−8.35914e24	−0.980577	−0.490289	+	0.871560i	$0.663109\pi$
$422$	1.24365e25	1.42636
$423$	−7.62586e24	−0.855176
$424$	8.68952e24	0.952841
$425$	0	0
$426$	−5.18913e24	−0.544131
$427$	7.79520e24	0.799397
$428$	5.53457e24	0.555096
$429$	5.07846e23	0.0498182
$430$	0	0
$431$	−3.01638e23	−0.0283108	−0.0141554	−	0.999900i	$-0.504506\pi$
$431$	−3.01638e23	−0.0283108	−0.0141554	+	0.999900i	$0.504506\pi$
$432$	−5.00199e24	−0.459248
$433$	−1.90272e25	−1.70899	−0.854494	−	0.519461i	$-0.826133\pi$
$433$	−1.90272e25	−1.70899	−0.854494	+	0.519461i	$0.826133\pi$
$434$	−5.50177e25	−4.83448
$435$	0	0
$436$	4.01412e23	0.0337651
$437$	−1.42454e25	−1.17247
$438$	−1.99970e25	−1.61050
$439$	−5.53042e24	−0.435858	−0.217929	−	0.975965i	$-0.569930\pi$
$439$	−5.53042e24	−0.435858	−0.217929	+	0.975965i	$0.569930\pi$
$440$	0	0
$441$	−1.69866e25	−1.28215
$442$	2.11195e24	0.156017
$443$	2.30459e25	1.66632	0.833161	−	0.553031i	$-0.186529\pi$
$443$	2.30459e25	1.66632	0.833161	+	0.553031i	$0.186529\pi$
$444$	3.58779e24	0.253915
$445$	0	0
$446$	−1.13136e25	−0.767217
$447$	2.66267e24	0.176765
$448$	3.67223e25	2.38665
$449$	2.60186e25	1.65556	0.827778	−	0.561056i	$-0.189605\pi$
$449$	2.60186e25	1.65556	0.827778	+	0.561056i	$0.189605\pi$
$450$	0	0
$451$	8.40425e24	0.512651
$452$	−1.45577e25	−0.869519
$453$	8.03256e24	0.469809
$454$	9.17627e23	0.0525576
$455$	0	0
$456$	−2.08600e25	−1.14591
$457$	2.31329e25	1.24459	0.622295	−	0.782783i	$-0.286200\pi$
$457$	2.31329e25	1.24459	0.622295	+	0.782783i	$0.286200\pi$
$458$	−8.14561e24	−0.429241
$459$	8.59008e24	0.443380
$460$	0	0
$461$	2.86383e25	1.41837	0.709184	−	0.705023i	$-0.249063\pi$
$461$	2.86383e25	1.41837	0.709184	+	0.705023i	$0.249063\pi$
$462$	1.48466e25	0.720326
$463$	2.65604e25	1.26245	0.631227	−	0.775598i	$-0.282552\pi$
$463$	2.65604e25	1.26245	0.631227	+	0.775598i	$0.282552\pi$
$464$	9.81857e24	0.457223
$465$	0	0
$466$	−2.81139e25	−1.25677
$467$	1.59314e25	0.697819	0.348909	−	0.937156i	$-0.386552\pi$
$467$	1.59314e25	0.697819	0.348909	+	0.937156i	$0.386552\pi$
$468$	5.78550e24	0.248316
$469$	4.57100e25	1.92251
$470$	0	0
$471$	1.46297e25	0.590931
$472$	8.31701e23	0.0329244
$473$	6.72043e24	0.260745
$474$	−1.31459e25	−0.499914
$475$	0	0
$476$	3.98459e25	1.45585
$477$	−1.34775e25	−0.482707
$478$	−4.98567e25	−1.75049
$479$	−3.55269e25	−1.22284	−0.611420	−	0.791306i	$-0.709401\pi$
$479$	−3.55269e25	−1.22284	−0.611420	+	0.791306i	$0.709401\pi$
$480$	0	0
$481$	−1.50008e24	−0.0496289
$482$	−7.16002e25	−2.32256
$483$	−2.25784e25	−0.718118
$484$	−4.58514e25	−1.42995
$485$	0	0
$486$	5.79447e25	1.73768
$487$	5.62096e25	1.65305	0.826524	−	0.562902i	$-0.190315\pi$
$487$	5.62096e25	1.65305	0.826524	+	0.562902i	$0.190315\pi$
$488$	−2.26472e25	−0.653170
$489$	−2.06032e25	−0.582776
$490$	0	0
$491$	−5.81507e25	−1.58227	−0.791135	−	0.611642i	$-0.790509\pi$
$491$	−5.81507e25	−1.58227	−0.791135	+	0.611642i	$0.790509\pi$
$492$	−4.15457e25	−1.10881
$493$	−1.68618e25	−0.441425
$494$	1.93607e25	0.497182
$495$	0	0
$496$	3.40090e25	0.840461
$497$	4.09337e25	0.992420
$498$	−1.28034e25	−0.304542
$499$	−1.33885e25	−0.312447	−0.156224	−	0.987722i	$-0.549932\pi$
$499$	−1.33885e25	−0.312447	−0.156224	+	0.987722i	$0.549932\pi$
$500$	0	0
$501$	−6.03784e23	−0.0135651
$502$	2.97358e25	0.655532
$503$	−5.92007e25	−1.28065	−0.640326	−	0.768103i	$-0.721201\pi$
$503$	−5.92007e25	−1.28065	−0.640326	+	0.768103i	$0.721201\pi$
$504$	7.61934e25	1.61744
$505$	0	0
$506$	−2.94560e25	−0.602205
$507$	−2.63674e25	−0.529044
$508$	5.34650e25	1.05285
$509$	4.36448e25	0.843555	0.421777	−	0.906699i	$-0.361406\pi$
$509$	4.36448e25	0.843555	0.421777	+	0.906699i	$0.361406\pi$
$510$	0	0
$511$	1.57744e26	2.93734
$512$	−5.18605e25	−0.947920
$513$	7.87472e25	1.41293
$514$	6.11836e25	1.07767
$515$	0	0
$516$	−3.32218e25	−0.563964
$517$	−3.40544e25	−0.567562
$518$	−4.38540e25	−0.717590
$519$	4.97447e25	0.799203
$520$	0	0
$521$	−7.24724e25	−1.12257	−0.561286	−	0.827622i	$-0.689693\pi$
$521$	−7.24724e25	−1.12257	−0.561286	+	0.827622i	$0.689693\pi$
$522$	−7.15743e25	−1.08865
$523$	3.62351e25	0.541207	0.270604	−	0.962691i	$-0.412777\pi$
$523$	3.62351e25	0.541207	0.270604	+	0.962691i	$0.412777\pi$
$524$	−7.73760e25	−1.13490
$525$	0	0
$526$	7.62064e25	1.07802
$527$	−5.84047e25	−0.811423
$528$	−9.17739e24	−0.125227
$529$	−2.98194e25	−0.399641
$530$	0	0
$531$	−1.28997e24	−0.0166794
$532$	3.65276e26	4.63939
$533$	1.73705e25	0.216722
$534$	−5.38462e25	−0.659953
$535$	0	0
$536$	−1.32800e26	−1.57084
$537$	−2.16428e25	−0.251511
$538$	3.51377e25	0.401182
$539$	−7.58561e25	−0.850937
$540$	0	0
$541$	−4.47583e24	−0.0484730	−0.0242365	−	0.999706i	$-0.507715\pi$
$541$	−4.47583e24	−0.0484730	−0.0242365	+	0.999706i	$0.507715\pi$
$542$	3.61154e25	0.384325
$543$	−1.44474e25	−0.151075
$544$	2.54481e25	0.261497
$545$	0	0
$546$	3.06861e25	0.304517
$547$	−2.13519e25	−0.208237	−0.104118	−	0.994565i	$-0.533202\pi$
$547$	−2.13519e25	−0.208237	−0.104118	+	0.994565i	$0.533202\pi$
$548$	3.46271e26	3.31895
$549$	3.51259e25	0.330895
$550$	0	0
$551$	−1.54576e26	−1.40670
$552$	6.55965e25	0.586759
$553$	1.03699e26	0.911775
$554$	−3.68640e26	−3.18611
$555$	0	0
$556$	−1.58744e26	−1.32583
$557$	−7.41228e25	−0.608593	−0.304297	−	0.952577i	$-0.598421\pi$
$557$	−7.41228e25	−0.608593	−0.304297	+	0.952577i	$0.598421\pi$
$558$	−2.47914e26	−2.00114
$559$	1.38902e25	0.110229
$560$	0	0
$561$	1.57606e25	0.120900
$562$	−3.04164e26	−2.29410
$563$	−2.39164e26	−1.77364	−0.886820	−	0.462116i	$-0.847090\pi$
$563$	−2.39164e26	−1.77364	−0.886820	+	0.462116i	$0.847090\pi$
$564$	1.68345e26	1.22758
$565$	0	0
$566$	−2.28221e25	−0.160916
$567$	−4.46444e25	−0.309548
$568$	−1.18924e26	−0.810885
$569$	1.08135e26	0.725102	0.362551	−	0.931964i	$-0.381906\pi$
$569$	1.08135e26	0.725102	0.362551	+	0.931964i	$0.381906\pi$
$570$	0	0
$571$	−9.51568e25	−0.617158	−0.308579	−	0.951199i	$-0.599853\pi$
$571$	−9.51568e25	−0.617158	−0.308579	+	0.951199i	$0.599853\pi$
$572$	2.58360e25	0.164802
$573$	−6.94485e25	−0.435707
$574$	5.07818e26	3.13361
$575$	0	0
$576$	1.65474e26	0.987905
$577$	2.21789e26	1.30248	0.651238	−	0.758873i	$-0.274250\pi$
$577$	2.21789e26	1.30248	0.651238	+	0.758873i	$0.274250\pi$
$578$	−2.25142e26	−1.30060
$579$	1.02197e26	0.580753
$580$	0	0
$581$	1.00998e26	0.555444
$582$	4.35358e25	0.235547
$583$	−6.01856e25	−0.320362
$584$	−4.58289e26	−2.40003
$585$	0	0
$586$	−2.23505e26	−1.13308
$587$	−3.00717e26	−1.50002	−0.750008	−	0.661429i	$-0.769950\pi$
$587$	−3.00717e26	−1.50002	−0.750008	+	0.661429i	$0.769950\pi$
$588$	3.74988e26	1.84049
$589$	−5.35410e26	−2.58578
$590$	0	0
$591$	9.90735e25	0.463315
$592$	2.71082e25	0.124751
$593$	−2.27900e26	−1.03211	−0.516053	−	0.856557i	$-0.672599\pi$
$593$	−2.27900e26	−1.03211	−0.516053	+	0.856557i	$0.672599\pi$
$594$	1.62830e26	0.725712
$595$	0	0
$596$	1.35460e26	0.584751
$597$	−2.10273e26	−0.893360
$598$	−6.08817e25	−0.254581
$599$	−9.07164e25	−0.373363	−0.186681	−	0.982420i	$-0.559773\pi$
$599$	−9.07164e25	−0.373363	−0.186681	+	0.982420i	$0.559773\pi$
$600$	0	0
$601$	9.37859e25	0.373964	0.186982	−	0.982363i	$-0.440129\pi$
$601$	9.37859e25	0.373964	0.186982	+	0.982363i	$0.440129\pi$
$602$	4.06075e26	1.59382
$603$	2.05973e26	0.795786
$604$	4.08646e26	1.55416
$605$	0	0
$606$	1.20357e26	0.443589
$607$	−2.25098e26	−0.816732	−0.408366	−	0.912818i	$-0.633901\pi$
$607$	−2.25098e26	−0.816732	−0.408366	+	0.912818i	$0.633901\pi$
$608$	2.33289e26	0.833317
$609$	−2.44997e26	−0.861583
$610$	0	0
$611$	−7.03860e25	−0.239936
$612$	1.79549e26	0.602621
$613$	−8.30040e25	−0.274299	−0.137150	−	0.990550i	$-0.543794\pi$
$613$	−8.30040e25	−0.274299	−0.137150	+	0.990550i	$0.543794\pi$
$614$	4.34313e26	1.41320
$615$	0	0
$616$	3.40253e26	1.07346
$617$	4.41321e26	1.37103	0.685513	−	0.728061i	$-0.259578\pi$
$617$	4.41321e26	1.37103	0.685513	+	0.728061i	$0.259578\pi$
$618$	−1.15260e26	−0.352604
$619$	−2.41336e26	−0.727044	−0.363522	−	0.931586i	$-0.618426\pi$
$619$	−2.41336e26	−0.727044	−0.363522	+	0.931586i	$0.618426\pi$
$620$	0	0
$621$	−2.47629e26	−0.723487
$622$	−3.18146e25	−0.0915416
$623$	4.24758e26	1.20366
$624$	−1.89685e25	−0.0529394
$625$	0	0
$626$	3.53694e26	0.957572
$627$	1.44481e26	0.385275
$628$	7.44267e26	1.95485
$629$	−4.65538e25	−0.120441
$630$	0	0
$631$	−1.75895e26	−0.441545	−0.220773	−	0.975325i	$-0.570858\pi$
$631$	−1.75895e26	−0.441545	−0.220773	+	0.975325i	$0.570858\pi$
$632$	−3.01275e26	−0.744991
$633$	1.91823e26	0.467268
$634$	1.22498e27	2.93956
$635$	0	0
$636$	2.97522e26	0.692910
$637$	−1.56785e26	−0.359732
$638$	−3.19625e26	−0.722512
$639$	1.84451e26	0.410793
$640$	0	0
$641$	6.26722e26	1.35495	0.677475	−	0.735546i	$-0.263074\pi$
$641$	6.26722e26	1.35495	0.677475	+	0.735546i	$0.263074\pi$
$642$	1.32277e26	0.281774
$643$	4.06756e25	0.0853748	0.0426874	−	0.999088i	$-0.486408\pi$
$643$	4.06756e25	0.0853748	0.0426874	+	0.999088i	$0.486408\pi$
$644$	−1.14865e27	−2.37559
$645$	0	0
$646$	6.00846e26	1.20657
$647$	5.94304e26	1.17603	0.588014	−	0.808851i	$-0.299910\pi$
$647$	5.94304e26	1.17603	0.588014	+	0.808851i	$0.299910\pi$
$648$	1.29704e26	0.252925
$649$	−5.76055e24	−0.0110698
$650$	0	0
$651$	−8.48606e26	−1.58375
$652$	−1.04816e27	−1.92787
$653$	−4.38542e26	−0.794944	−0.397472	−	0.917614i	$-0.630112\pi$
$653$	−4.38542e26	−0.794944	−0.397472	+	0.917614i	$0.630112\pi$
$654$	9.59378e24	0.0171396
$655$	0	0
$656$	−3.13906e26	−0.544770
$657$	7.10807e26	1.21585
$658$	−2.05770e27	−3.46926
$659$	5.41264e26	0.899493	0.449747	−	0.893156i	$-0.351514\pi$
$659$	5.41264e26	0.899493	0.449747	+	0.893156i	$0.351514\pi$
$660$	0	0
$661$	7.96736e24	0.0128647	0.00643236	−	0.999979i	$-0.497953\pi$
$661$	7.96736e24	0.0128647	0.00643236	+	0.999979i	$0.497953\pi$
$662$	−1.50831e27	−2.40071
$663$	3.25752e25	0.0511103
$664$	−2.93427e26	−0.453841
$665$	0	0
$666$	−1.97610e26	−0.297032
$667$	4.86079e26	0.720297
$668$	−3.07167e25	−0.0448744
$669$	−1.74503e26	−0.251336
$670$	0	0
$671$	1.56860e26	0.219608
$672$	3.69755e26	0.510394
$673$	3.41589e26	0.464902	0.232451	−	0.972608i	$-0.425325\pi$
$673$	3.41589e26	0.464902	0.232451	+	0.972608i	$0.425325\pi$
$674$	−1.51647e27	−2.03500
$675$	0	0
$676$	−1.34141e27	−1.75012
$677$	−3.34901e26	−0.430849	−0.215424	−	0.976521i	$-0.569113\pi$
$677$	−3.34901e26	−0.430849	−0.215424	+	0.976521i	$0.569113\pi$
$678$	−3.47932e26	−0.441379
$679$	−3.43426e26	−0.429606
$680$	0	0
$681$	1.41537e25	0.0172176
$682$	−1.10710e27	−1.32811
$683$	1.09285e27	1.29290	0.646450	−	0.762956i	$-0.276253\pi$
$683$	1.09285e27	1.29290	0.646450	+	0.762956i	$0.276253\pi$
$684$	1.64597e27	1.92038
$685$	0	0
$686$	−2.09047e27	−2.37227
$687$	−1.25640e26	−0.140617
$688$	−2.51013e26	−0.277081
$689$	−1.24396e26	−0.135433
$690$	0	0
$691$	−2.01449e26	−0.213365	−0.106682	−	0.994293i	$-0.534023\pi$
$691$	−2.01449e26	−0.213365	−0.106682	+	0.994293i	$0.534023\pi$
$692$	2.53070e27	2.64382
$693$	−5.27733e26	−0.543812
$694$	−1.07282e27	−1.09047
$695$	0	0
$696$	7.11784e26	0.703980
$697$	5.39081e26	0.525948
$698$	8.26751e26	0.795698
$699$	−4.33635e26	−0.411710
$700$	0	0
$701$	−4.28823e26	−0.396239	−0.198120	−	0.980178i	$-0.563483\pi$
$701$	−4.28823e26	−0.396239	−0.198120	+	0.980178i	$0.563483\pi$
$702$	3.36549e26	0.306793
$703$	−4.26769e26	−0.383811
$704$	7.38947e26	0.655652
$705$	0	0
$706$	2.06777e27	1.78590
$707$	−9.49418e26	−0.809046
$708$	2.84768e25	0.0239428
$709$	1.91128e27	1.58557	0.792783	−	0.609504i	$-0.208631\pi$
$709$	1.91128e27	1.58557	0.792783	+	0.609504i	$0.208631\pi$
$710$	0	0
$711$	4.67279e26	0.377411
$712$	−1.23404e27	−0.983488
$713$	1.68365e27	1.32404
$714$	9.52320e26	0.739009
$715$	0	0
$716$	−1.10105e27	−0.832018
$717$	−7.69003e26	−0.573451
$718$	2.28030e27	1.67807
$719$	−2.62640e27	−1.90737	−0.953687	−	0.300799i	$-0.902747\pi$
$719$	−2.62640e27	−1.90737	−0.953687	+	0.300799i	$0.902747\pi$
$720$	0	0
$721$	9.09210e26	0.643101
$722$	3.10256e27	2.16579
$723$	−1.10438e27	−0.760859
$724$	−7.34994e26	−0.499767
$725$	0	0
$726$	−1.09585e27	−0.725862
$727$	7.24253e25	0.0473493	0.0236747	−	0.999720i	$-0.492463\pi$
$727$	7.24253e25	0.0473493	0.0236747	+	0.999720i	$0.492463\pi$
$728$	7.03258e26	0.453802
$729$	6.05287e26	0.385523
$730$	0	0
$731$	4.31074e26	0.267508
$732$	−7.75422e26	−0.474988
$733$	−1.12113e27	−0.677901	−0.338951	−	0.940804i	$-0.610072\pi$
$733$	−1.12113e27	−0.677901	−0.338951	+	0.940804i	$0.610072\pi$
$734$	−2.11522e27	−1.26253
$735$	0	0
$736$	−7.33601e26	−0.426698
$737$	9.19804e26	0.528146
$738$	2.28827e27	1.29710
$739$	2.25754e27	1.26332	0.631661	−	0.775245i	$-0.282373\pi$
$739$	2.25754e27	1.26332	0.631661	+	0.775245i	$0.282373\pi$
$740$	0	0
$741$	2.98624e26	0.162874
$742$	−3.63665e27	−1.95824
$743$	−2.84762e27	−1.51387	−0.756934	−	0.653491i	$-0.773304\pi$
$743$	−2.84762e27	−1.51387	−0.756934	+	0.653491i	$0.773304\pi$
$744$	2.46543e27	1.29405
$745$	0	0
$746$	−2.37443e27	−1.21490
$747$	4.55106e26	0.229915
$748$	8.01802e26	0.399947
$749$	−1.04345e27	−0.513917
$750$	0	0
$751$	1.75682e27	0.843623	0.421812	−	0.906684i	$-0.361394\pi$
$751$	1.75682e27	0.843623	0.421812	+	0.906684i	$0.361394\pi$
$752$	1.27196e27	0.603121
$753$	4.58652e26	0.214749
$754$	−6.60624e26	−0.305440
$755$	0	0
$756$	6.34963e27	2.86280
$757$	−3.07775e27	−1.37032	−0.685160	−	0.728392i	$-0.740268\pi$
$757$	−3.07775e27	−1.37032	−0.685160	+	0.728392i	$0.740268\pi$
$758$	−4.70370e27	−2.06815
$759$	−4.54337e26	−0.197279
$760$	0	0
$761$	−3.44812e27	−1.46025	−0.730127	−	0.683312i	$-0.760539\pi$
$761$	−3.44812e27	−1.46025	−0.730127	+	0.683312i	$0.760539\pi$
$762$	1.27782e27	0.534438
$763$	−7.56792e25	−0.0312603
$764$	−3.53310e27	−1.44135
$765$	0	0
$766$	6.47142e27	2.57529
$767$	−1.19063e25	−0.00467973
$768$	−2.34253e27	−0.909397
$769$	3.66284e27	1.40449	0.702243	−	0.711937i	$-0.252182\pi$
$769$	3.66284e27	1.40449	0.702243	+	0.711937i	$0.252182\pi$
$770$	0	0
$771$	9.43711e26	0.353039
$772$	5.19914e27	1.92117
$773$	1.93533e27	0.706399	0.353199	−	0.935548i	$-0.385094\pi$
$773$	1.93533e27	0.706399	0.353199	+	0.935548i	$0.385094\pi$
$774$	1.82981e27	0.659730
$775$	0	0
$776$	9.97747e26	0.351022
$777$	−6.76415e26	−0.235079
$778$	−5.90593e27	−2.02760
$779$	4.94188e27	1.67605
$780$	0	0
$781$	8.23693e26	0.272634
$782$	−1.88942e27	−0.617824
$783$	−2.68701e27	−0.868024
$784$	2.83329e27	0.904250
$785$	0	0
$786$	−1.84929e27	−0.576092
$787$	−4.47507e27	−1.37733	−0.688667	−	0.725077i	$-0.741804\pi$
$787$	−4.47507e27	−1.37733	−0.688667	+	0.725077i	$0.741804\pi$
$788$	5.04024e27	1.53268
$789$	1.17543e27	0.353154
$790$	0	0
$791$	2.74461e27	0.805015
$792$	1.53321e27	0.444337
$793$	3.24209e26	0.0928388
$794$	−7.03673e27	−1.99102
$795$	0	0
$796$	−1.06973e28	−2.95530
$797$	−2.99623e27	−0.817939	−0.408969	−	0.912548i	$-0.634112\pi$
$797$	−2.99623e27	−0.817939	−0.408969	+	0.912548i	$0.634112\pi$
$798$	8.73014e27	2.35501
$799$	−2.18438e27	−0.582283
$800$	0	0
$801$	1.91400e27	0.498233
$802$	7.67621e27	1.97465
$803$	3.17421e27	0.806935
$804$	−4.54697e27	−1.14232
$805$	0	0
$806$	−2.28823e27	−0.561457
$807$	5.41973e26	0.131425
$808$	2.75832e27	0.661054
$809$	8.97271e26	0.212526	0.106263	−	0.994338i	$-0.466111\pi$
$809$	8.97271e26	0.212526	0.106263	+	0.994338i	$0.466111\pi$
$810$	0	0
$811$	−7.80812e27	−1.80654	−0.903271	−	0.429070i	$-0.858841\pi$
$811$	−7.80812e27	−1.80654	−0.903271	+	0.429070i	$0.858841\pi$
$812$	−1.24639e28	−2.85018
$813$	5.57052e26	0.125903
$814$	−8.82456e26	−0.197134
$815$	0	0
$816$	−5.88673e26	−0.128475
$817$	3.95175e27	0.852472
$818$	−6.90742e27	−1.47285
$819$	−1.09076e27	−0.229895
$820$	0	0
$821$	5.65133e27	1.16383	0.581917	−	0.813248i	$-0.302303\pi$
$821$	5.65133e27	1.16383	0.581917	+	0.813248i	$0.302303\pi$
$822$	8.27590e27	1.68474
$823$	4.23337e27	0.851898	0.425949	−	0.904747i	$-0.359940\pi$
$823$	4.23337e27	0.851898	0.425949	+	0.904747i	$0.359940\pi$
$824$	−2.64150e27	−0.525464
$825$	0	0
$826$	−3.48075e26	−0.0676648
$827$	8.60057e27	1.65282	0.826408	−	0.563071i	$-0.190380\pi$
$827$	8.60057e27	1.65282	0.826408	+	0.563071i	$0.190380\pi$
$828$	−5.17591e27	−0.983328
$829$	2.40358e27	0.451431	0.225715	−	0.974193i	$-0.427528\pi$
$829$	2.40358e27	0.451431	0.225715	+	0.974193i	$0.427528\pi$
$830$	0	0
$831$	−5.68600e27	−1.04375
$832$	1.52731e27	0.277176
$833$	−4.86570e27	−0.873008
$834$	−3.79401e27	−0.673009
$835$	0	0
$836$	7.35031e27	1.27452
$837$	−9.30708e27	−1.59559
$838$	1.49795e28	2.53910
$839$	−2.28307e27	−0.382632	−0.191316	−	0.981529i	$-0.561275\pi$
$839$	−2.28307e27	−0.382632	−0.191316	+	0.981529i	$0.561275\pi$
$840$	0	0
$841$	−8.28845e26	−0.135804
$842$	−1.01638e28	−1.64661
$843$	−4.69150e27	−0.751535
$844$	9.75875e27	1.54576
$845$	0	0
$846$	−9.27218e27	−1.43603
$847$	8.64448e27	1.32387
$848$	2.24798e27	0.340434
$849$	−3.52013e26	−0.0527152
$850$	0	0
$851$	1.34202e27	0.196530
$852$	−4.07185e27	−0.589679
$853$	3.07333e27	0.440142	0.220071	−	0.975484i	$-0.429371\pi$
$853$	3.07333e27	0.440142	0.220071	+	0.975484i	$0.429371\pi$
$854$	9.47809e27	1.34237
$855$	0	0
$856$	3.03150e27	0.419910
$857$	4.56647e27	0.625551	0.312775	−	0.949827i	$-0.398741\pi$
$857$	4.56647e27	0.625551	0.312775	+	0.949827i	$0.398741\pi$
$858$	6.17483e26	0.0836558
$859$	−5.09546e27	−0.682728	−0.341364	−	0.939931i	$-0.610889\pi$
$859$	−5.09546e27	−0.682728	−0.341364	+	0.939931i	$0.610889\pi$
$860$	0	0
$861$	7.83271e27	1.02655
$862$	−3.66758e26	−0.0475402
$863$	4.49338e27	0.576064	0.288032	−	0.957621i	$-0.406999\pi$
$863$	4.49338e27	0.576064	0.288032	+	0.957621i	$0.406999\pi$
$864$	4.05528e27	0.514210
$865$	0	0
$866$	−2.31349e28	−2.86977
$867$	−3.47265e27	−0.426069
$868$	−4.31717e28	−5.23917
$869$	2.08670e27	0.250480
$870$	0	0
$871$	1.90111e27	0.223273
$872$	2.19869e26	0.0255421
$873$	−1.54751e27	−0.177827
$874$	−1.73208e28	−1.96883
$875$	0	0
$876$	−1.56914e28	−1.74532
$877$	1.20133e28	1.32180	0.660901	−	0.750473i	$-0.270174\pi$
$877$	1.20133e28	1.32180	0.660901	+	0.750473i	$0.270174\pi$
$878$	−6.72437e27	−0.731903
$879$	−3.44740e27	−0.371191
$880$	0	0
$881$	5.81954e27	0.613221	0.306611	−	0.951835i	$-0.400805\pi$
$881$	5.81954e27	0.613221	0.306611	+	0.951835i	$0.400805\pi$
$882$	−2.06538e28	−2.15302
$883$	1.68577e28	1.73849	0.869243	−	0.494386i	$-0.164607\pi$
$883$	1.68577e28	1.73849	0.869243	+	0.494386i	$0.164607\pi$
$884$	1.65722e27	0.169077
$885$	0	0
$886$	2.80212e28	2.79812
$887$	−8.09385e26	−0.0799614	−0.0399807	−	0.999200i	$-0.512730\pi$
$887$	−8.09385e26	−0.0799614	−0.0399807	+	0.999200i	$0.512730\pi$
$888$	1.96517e27	0.192078
$889$	−1.00799e28	−0.974742
$890$	0	0
$891$	−8.98362e26	−0.0850380
$892$	−8.87762e27	−0.831439
$893$	−2.00247e28	−1.85557
$894$	3.23750e27	0.296827
$895$	0	0
$896$	3.43133e28	3.07990
$897$	−9.39055e26	−0.0833994
$898$	3.16357e28	2.78004
$899$	1.82692e28	1.58856
$900$	0	0
$901$	−3.86054e27	−0.328672
$902$	1.02186e28	0.860855
$903$	6.26340e27	0.522127
$904$	−7.97384e27	−0.657760
$905$	0	0
$906$	9.76668e27	0.788913
$907$	2.84041e27	0.227045	0.113522	−	0.993535i	$-0.463787\pi$
$907$	2.84041e27	0.227045	0.113522	+	0.993535i	$0.463787\pi$
$908$	7.20051e26	0.0569571
$909$	−4.27816e27	−0.334889
$910$	0	0
$911$	−1.43331e28	−1.09879	−0.549397	−	0.835561i	$-0.685143\pi$
$911$	−1.43331e28	−1.09879	−0.549397	+	0.835561i	$0.685143\pi$
$912$	−5.39650e27	−0.409413
$913$	2.03235e27	0.152590
$914$	2.81270e28	2.08994
$915$	0	0
$916$	−6.39177e27	−0.465173
$917$	1.45879e28	1.05071
$918$	1.04446e28	0.744534
$919$	1.84550e27	0.130202	0.0651009	−	0.997879i	$-0.479263\pi$
$919$	1.84550e27	0.130202	0.0651009	+	0.997879i	$0.479263\pi$
$920$	0	0
$921$	6.69896e27	0.462957
$922$	3.48210e28	2.38176
$923$	1.70247e27	0.115256
$924$	1.16500e28	0.780624
$925$	0	0
$926$	3.22945e28	2.11994
$927$	4.09698e27	0.266199
$928$	−7.96026e27	−0.511943
$929$	−1.47827e28	−0.941033	−0.470516	−	0.882391i	$-0.655932\pi$
$929$	−1.47827e28	−0.941033	−0.470516	+	0.882391i	$0.655932\pi$
$930$	0	0
$931$	−4.46050e28	−2.78203
$932$	−2.20606e28	−1.36197
$933$	−4.90717e26	−0.0299886
$934$	1.93707e28	1.17179
$935$	0	0
$936$	3.16894e27	0.187843
$937$	−2.62877e28	−1.54250	−0.771251	−	0.636531i	$-0.780369\pi$
$937$	−2.62877e28	−1.54250	−0.771251	+	0.636531i	$0.780369\pi$
$938$	5.55782e28	3.22833
$939$	5.45546e27	0.313696
$940$	0	0
$941$	5.29025e27	0.298109	0.149054	−	0.988829i	$-0.452377\pi$
$941$	5.29025e27	0.298109	0.149054	+	0.988829i	$0.452377\pi$
$942$	1.77881e28	0.992305
$943$	−1.55403e28	−0.858217
$944$	2.15161e26	0.0117633
$945$	0	0
$946$	8.17128e27	0.437849
$947$	−2.77520e28	−1.47221	−0.736105	−	0.676868i	$-0.763337\pi$
$947$	−2.77520e28	−1.47221	−0.736105	+	0.676868i	$0.763337\pi$
$948$	−1.03154e28	−0.541761
$949$	6.56069e27	0.341130
$950$	0	0
$951$	1.88944e28	0.962984
$952$	2.18251e28	1.10130
$953$	4.86733e27	0.243169	0.121585	−	0.992581i	$-0.461202\pi$
$953$	4.86733e27	0.243169	0.121585	+	0.992581i	$0.461202\pi$
$954$	−1.63871e28	−0.810573
$955$	0	0
$956$	−3.91220e28	−1.89702
$957$	−4.92998e27	−0.236691
$958$	−4.31966e28	−2.05342
$959$	−6.52833e28	−3.07274
$960$	0	0
$961$	4.16091e28	1.92006
$962$	−1.82392e27	−0.0833380
$963$	−4.70186e27	−0.212726
$964$	−5.61838e28	−2.51698
$965$	0	0
$966$	−2.74528e28	−1.20588
$967$	1.90521e28	0.828689	0.414345	−	0.910120i	$-0.364011\pi$
$967$	1.90521e28	0.828689	0.414345	+	0.910120i	$0.364011\pi$
$968$	−2.51146e28	−1.08171
$969$	9.26759e27	0.395267
$970$	0	0
$971$	−4.36106e28	−1.82394	−0.911968	−	0.410261i	$-0.865438\pi$
$971$	−4.36106e28	−1.82394	−0.911968	+	0.410261i	$0.865438\pi$
$972$	4.54685e28	1.88313
$973$	2.99285e28	1.22748
$974$	6.83445e28	2.77583
$975$	0	0
$976$	−5.85884e27	−0.233367
$977$	−1.33368e28	−0.526081	−0.263041	−	0.964785i	$-0.584725\pi$
$977$	−1.33368e28	−0.526081	−0.263041	+	0.964785i	$0.584725\pi$
$978$	−2.50512e28	−0.978610
$979$	8.54724e27	0.330667
$980$	0	0
$981$	−3.41017e26	−0.0129396
$982$	−7.07046e28	−2.65698
$983$	−3.76958e28	−1.40293	−0.701463	−	0.712706i	$-0.747469\pi$
$983$	−3.76958e28	−1.40293	−0.701463	+	0.712706i	$0.747469\pi$
$984$	−2.27562e28	−0.838775
$985$	0	0
$986$	−2.05020e28	−0.741251
$987$	−3.17385e28	−1.13651
$988$	1.51921e28	0.538800
$989$	−1.24267e28	−0.436507
$990$	0	0
$991$	2.77322e28	0.955620	0.477810	−	0.878463i	$-0.341431\pi$
$991$	2.77322e28	0.955620	0.477810	+	0.878463i	$0.341431\pi$
$992$	−2.75723e28	−0.941047
$993$	−2.32645e28	−0.786459
$994$	4.97708e28	1.66649
$995$	0	0
$996$	−1.00467e28	−0.330035
$997$	4.18648e28	1.36221	0.681107	−	0.732184i	$-0.261499\pi$
$997$	4.18648e28	1.36221	0.681107	+	0.732184i	$0.261499\pi$
$998$	−1.62789e28	−0.524668
$999$	−7.41857e27	−0.236836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.20.a.f.1.8		8
5.2	odd	4		5.20.b.a.4.8	yes	8
5.3	odd	4		5.20.b.a.4.1	✓	8
5.4	even	2	inner	25.20.a.f.1.1		8
15.2	even	4		45.20.b.b.19.1		8
15.8	even	4		45.20.b.b.19.8		8
20.3	even	4		80.20.c.a.49.4		8
20.7	even	4		80.20.c.a.49.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.20.b.a.4.1	✓	8	5.3	odd	4
5.20.b.a.4.8	yes	8	5.2	odd	4
25.20.a.f.1.1		8	5.4	even	2	inner
25.20.a.f.1.8		8	1.1	even	1	trivial
45.20.b.b.19.1		8	15.2	even	4
45.20.b.b.19.8		8	15.8	even	4
80.20.c.a.49.4		8	20.3	even	4
80.20.c.a.49.5		8	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q+1215.89 q^{2} +18754.1 q^{3} +954092. q^{4} +2.28029e7 q^{6} -1.79878e8 q^{7} +5.22593e8 q^{8} -8.10544e8 q^{9} +O(q^{10})\)	\(q+1215.89 q^{2} +18754.1 q^{3} +954092. q^{4} +2.28029e7 q^{6} -1.79878e8 q^{7} +5.22593e8 q^{8} -8.10544e8 q^{9} -3.61960e9 q^{11} +1.78932e10 q^{12} -7.48125e9 q^{13} -2.18711e11 q^{14} +1.35195e11 q^{16} -2.32175e11 q^{17} -9.85529e11 q^{18} -2.12840e12 q^{19} -3.37345e12 q^{21} -4.40103e12 q^{22} +6.69298e12 q^{23} +9.80079e12 q^{24} -9.09635e12 q^{26} -3.69983e13 q^{27} -1.71620e14 q^{28} +7.26252e13 q^{29} +2.51555e14 q^{31} -1.09607e14 q^{32} -6.78825e13 q^{33} -2.82299e14 q^{34} -7.73334e14 q^{36} +2.00511e14 q^{37} -2.58790e15 q^{38} -1.40304e14 q^{39} -2.32187e15 q^{41} -4.10173e15 q^{42} -1.85667e15 q^{43} -3.45344e15 q^{44} +8.13791e15 q^{46} +9.40833e15 q^{47} +2.53547e15 q^{48} +2.09570e16 q^{49} -4.35425e15 q^{51} -7.13780e15 q^{52} +1.66277e16 q^{53} -4.49857e16 q^{54} -9.40028e16 q^{56} -3.99164e16 q^{57} +8.83040e16 q^{58} +1.59149e15 q^{59} -4.33362e16 q^{61} +3.05862e17 q^{62} +1.45799e17 q^{63} -2.04151e17 q^{64} -8.25375e16 q^{66} -2.54117e17 q^{67} -2.21517e17 q^{68} +1.25521e17 q^{69} -2.27564e17 q^{71} -4.23585e17 q^{72} -8.76951e17 q^{73} +2.43799e17 q^{74} -2.03069e18 q^{76} +6.51085e17 q^{77} -1.70594e17 q^{78} -5.76500e17 q^{79} +2.48193e17 q^{81} -2.82313e18 q^{82} -5.61483e17 q^{83} -3.21858e18 q^{84} -2.25751e18 q^{86} +1.36202e18 q^{87} -1.89158e18 q^{88} -2.36138e18 q^{89} +1.34571e18 q^{91} +6.38573e18 q^{92} +4.71769e18 q^{93} +1.14395e19 q^{94} -2.05559e18 q^{96} +1.90922e18 q^{97} +2.54814e19 q^{98} +2.93385e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1620744 q^{4} + 3365736 q^{6} - 345358584 q^{9}+O(q^{10}) \) 8 * q + 1620744 * q^4 + 3365736 * q^6 - 345358584 * q^9	\( 8 q + 1620744 q^{4} + 3365736 q^{6} - 345358584 q^{9} - 3379575264 q^{11} - 177250591032 q^{14} - 312730276832 q^{16} - 4547188380640 q^{19} - 2983154334624 q^{21} + 6176642779680 q^{24} + 15909228128496 q^{26} + 188222300345040 q^{29} + 72115006686976 q^{31} - 378440221985792 q^{34} - 964020253238712 q^{36} - 30\!\cdots\!28 q^{39}+ \cdots + 16\!\cdots\!72 q^{99}+O(q^{100}) \) 8 * q + 1620744 * q^4 + 3365736 * q^6 - 345358584 * q^9 - 3379575264 * q^11 - 177250591032 * q^14 - 312730276832 * q^16 - 4547188380640 * q^19 - 2983154334624 * q^21 + 6176642779680 * q^24 + 15909228128496 * q^26 + 188222300345040 * q^29 + 72115006686976 * q^31 - 378440221985792 * q^34 - 964020253238712 * q^36 - 3011653788049728 * q^39 - 3314269428857904 * q^41 - 489119304867552 * q^44 - 13716870314831944 * q^46 + 11129258011816744 * q^49 - 46867071411404544 * q^51 - 86068650834185040 * q^54 - 303831778832907360 * q^56 - 259648032778012320 * q^59 - 582893562753226064 * q^61 - 525415747276109696 * q^64 - 1871067587999976288 * q^66 - 302144302523815008 * q^69 - 2617007600602926144 * q^71 - 1847591013622215312 * q^74 - 7494494299858276320 * q^76 - 3935092091109861760 * q^79 - 8328177293660704152 * q^81 - 7683875738824129632 * q^84 - 14358563079146095224 * q^86 - 2120141101636921680 * q^89 - 3358638219973817664 * q^91 + 10251771654977709928 * q^94 + 15670161523483353216 * q^96 + 16256944625235082272 * q^99

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