LMFDB - Embedded newform 75.22.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	75.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:	$209.608008215$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 $ x^4 - x^3 - 3512166x^2 + 363520480x + 2321089280000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-849.272$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-625.272 q^{2} +59049.0 q^{3} -1.70619e6 q^{4} -3.69217e7 q^{6} -3.78633e8 q^{7} +2.37812e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-625.272 q^{2} +59049.0 q^{3} -1.70619e6 q^{4} -3.69217e7 q^{6} -3.78633e8 q^{7} +2.37812e9 q^{8} +3.48678e9 q^{9} -3.73130e10 q^{11} -1.00749e11 q^{12} +9.27472e11 q^{13} +2.36749e11 q^{14} +2.09116e12 q^{16} +1.40356e13 q^{17} -2.18019e12 q^{18} +3.31068e13 q^{19} -2.23579e13 q^{21} +2.33308e13 q^{22} -3.30762e14 q^{23} +1.40426e14 q^{24} -5.79922e14 q^{26} +2.05891e14 q^{27} +6.46019e14 q^{28} +2.35712e15 q^{29} +3.97094e15 q^{31} -6.29483e15 q^{32} -2.20330e15 q^{33} -8.77604e15 q^{34} -5.94911e15 q^{36} +4.11796e16 q^{37} -2.07007e16 q^{38} +5.47663e16 q^{39} -5.23046e16 q^{41} +1.39798e16 q^{42} -2.10603e17 q^{43} +6.36630e16 q^{44} +2.06816e17 q^{46} -5.77988e17 q^{47} +1.23481e17 q^{48} -4.15183e17 q^{49} +8.28786e17 q^{51} -1.58244e18 q^{52} -1.16230e18 q^{53} -1.28738e17 q^{54} -9.00436e17 q^{56} +1.95492e18 q^{57} -1.47384e18 q^{58} +4.17973e18 q^{59} +9.77562e18 q^{61} -2.48292e18 q^{62} -1.32021e18 q^{63} -4.49508e17 q^{64} +1.37766e18 q^{66} +1.53703e19 q^{67} -2.39473e19 q^{68} -1.95312e19 q^{69} -6.75134e18 q^{71} +8.29199e18 q^{72} +1.58050e19 q^{73} -2.57485e19 q^{74} -5.64864e19 q^{76} +1.41280e19 q^{77} -3.42438e19 q^{78} -1.48815e19 q^{79} +1.21577e19 q^{81} +3.27046e19 q^{82} +6.30804e19 q^{83} +3.81468e19 q^{84} +1.31684e20 q^{86} +1.39186e20 q^{87} -8.87349e19 q^{88} +1.22115e20 q^{89} -3.51172e20 q^{91} +5.64342e20 q^{92} +2.34480e20 q^{93} +3.61400e20 q^{94} -3.71703e20 q^{96} -5.41393e20 q^{97} +2.59602e20 q^{98} -1.30103e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9}+O(q^{10}) $ 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9	$ 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} + 3127645607697 q^{18} - 24270353300752 q^{19} + 13851567033696 q^{21} + 56303932793676 q^{22} - 10350924920928 q^{23} - 4538248036821 q^{24} + 474751622871378 q^{26} + 823564528378596 q^{27} + 18\!\cdots\!68 q^{28}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100}) $ 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 + 31491830256 * q^11 - 68681250027 * q^12 + 27017977768 * q^13 - 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 + 3127645607697 * q^18 - 24270353300752 * q^19 + 13851567033696 * q^21 + 56303932793676 * q^22 - 10350924920928 * q^23 - 4538248036821 * q^24 + 474751622871378 * q^26 + 823564528378596 * q^27 + 1868468795909968 * q^28 + 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 1859561084786544 * q^33 + 3862795623387294 * q^34 - 4055559132844323 * q^36 - 4813435696247096 * q^37 - 21307487199674508 * q^38 + 1595384569222632 * q^39 + 289731591445930344 * q^41 - 131874611571854928 * q^42 - 451091912658458000 * q^43 + 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 - 668711106779001975 * q^48 + 1545180753004755300 * q^49 + 173963965360807512 * q^51 - 884645329958480822 * q^52 - 697278335404085208 * q^53 + 184684345488900153 * q^54 + 5673275878853478000 * q^56 - 1433140092056104848 * q^57 - 12277858736996428626 * q^58 + 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 817921181772715104 * q^63 + 21525702776885072473 * q^64 + 3324690927533774124 * q^66 + 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 611211765655877472 * q^69 - 37390337803999713312 * q^71 - 267979008326243229 * q^72 + 37253672904265201432 * q^73 - 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 + 28033608578931999522 * q^78 - 109751291042259570400 * q^79 + 48630661836227715204 * q^81 + 210801413757052759338 * q^82 + 149023592088638482896 * q^83 + 110331213929687700432 * q^84 - 346200583098972939204 * q^86 + 279238273256843539704 * q^87 + 116461091469581805156 * q^88 + 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 - 64654161985536495264 * q^93 + 1096939878232058257512 * q^94 - 468455064452928935829 * q^96 + 236244923748673777528 * q^97 - 197058383612642903463 * q^98 + 109805222495560636656 * 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 $

$2$	−625.272	−0.431771	−0.215886	−	0.976419i	$-0.569264\pi$
$2$	−625.272	−0.431771	−0.215886	+	0.976419i	$0.569264\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	−1.70619e6	−0.813573
$5$	0	0
$5$	0	0
$6$	−3.69217e7	−0.249283
$7$	−3.78633e8	−0.506628	−0.253314	−	0.967384i	$-0.581521\pi$
$7$	−3.78633e8	−0.506628	−0.253314	+	0.967384i	$0.581521\pi$
$8$	2.37812e9	0.783049
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−3.73130e10	−0.433748	−0.216874	−	0.976200i	$-0.569586\pi$
$11$	−3.73130e10	−0.433748	−0.216874	+	0.976200i	$0.569586\pi$
$12$	−1.00749e11	−0.469717
$13$	9.27472e11	1.86593	0.932966	−	0.359965i	$-0.117211\pi$
$13$	9.27472e11	1.86593	0.932966	+	0.359965i	$0.117211\pi$
$14$	2.36749e11	0.218748
$15$	0	0
$16$	2.09116e12	0.475475
$17$	1.40356e13	1.68856	0.844279	−	0.535904i	$-0.180029\pi$
$17$	1.40356e13	1.68856	0.844279	+	0.535904i	$0.180029\pi$
$18$	−2.18019e12	−0.143924
$19$	3.31068e13	1.23881	0.619404	−	0.785072i	$-0.287374\pi$
$19$	3.31068e13	1.23881	0.619404	+	0.785072i	$0.287374\pi$
$20$	0	0
$21$	−2.23579e13	−0.292502
$22$	2.33308e13	0.187280
$23$	−3.30762e14	−1.66484	−0.832422	−	0.554143i	$-0.813046\pi$
$23$	−3.30762e14	−1.66484	−0.832422	+	0.554143i	$0.813046\pi$
$24$	1.40426e14	0.452094
$25$	0	0
$26$	−5.79922e14	−0.805656
$27$	2.05891e14	0.192450
$28$	6.46019e14	0.412179
$29$	2.35712e15	1.04041	0.520203	−	0.854043i	$-0.325856\pi$
$29$	2.35712e15	1.04041	0.520203	+	0.854043i	$0.325856\pi$
$30$	0	0
$31$	3.97094e15	0.870154	0.435077	−	0.900393i	$-0.356721\pi$
$31$	3.97094e15	0.870154	0.435077	+	0.900393i	$0.356721\pi$
$32$	−6.29483e15	−0.988346
$33$	−2.20330e15	−0.250425
$34$	−8.77604e15	−0.729071
$35$	0	0
$36$	−5.94911e15	−0.271191
$37$	4.11796e16	1.40788	0.703938	−	0.710262i	$-0.251423\pi$
$37$	4.11796e16	1.40788	0.703938	+	0.710262i	$0.251423\pi$
$38$	−2.07007e16	−0.534882
$39$	5.47663e16	1.07730
$40$	0	0
$41$	−5.23046e16	−0.608569	−0.304284	−	0.952581i	$-0.598417\pi$
$41$	−5.23046e16	−0.608569	−0.304284	+	0.952581i	$0.598417\pi$
$42$	1.39798e16	0.126294
$43$	−2.10603e17	−1.48609	−0.743046	−	0.669240i	$-0.766620\pi$
$43$	−2.10603e17	−1.48609	−0.743046	+	0.669240i	$0.766620\pi$
$44$	6.36630e16	0.352886
$45$	0	0
$46$	2.06816e17	0.718832
$47$	−5.77988e17	−1.60284	−0.801422	−	0.598099i	$-0.795923\pi$
$47$	−5.77988e17	−1.60284	−0.801422	+	0.598099i	$0.795923\pi$
$48$	1.23481e17	0.274516
$49$	−4.15183e17	−0.743328
$50$	0	0
$51$	8.28786e17	0.974890
$52$	−1.58244e18	−1.51807
$53$	−1.16230e18	−0.912894	−0.456447	−	0.889751i	$-0.650878\pi$
$53$	−1.16230e18	−0.912894	−0.456447	+	0.889751i	$0.650878\pi$
$54$	−1.28738e17	−0.0830944
$55$	0	0
$56$	−9.00436e17	−0.396715
$57$	1.95492e18	0.715226
$58$	−1.47384e18	−0.449217
$59$	4.17973e18	1.06464	0.532319	−	0.846544i	$-0.321321\pi$
$59$	4.17973e18	1.06464	0.532319	+	0.846544i	$0.321321\pi$
$60$	0	0
$61$	9.77562e18	1.75461	0.877306	−	0.479931i	$-0.159338\pi$
$61$	9.77562e18	1.75461	0.877306	+	0.479931i	$0.159338\pi$
$62$	−2.48292e18	−0.375707
$63$	−1.32021e18	−0.168876
$64$	−4.49508e17	−0.0487358
$65$	0	0
$66$	1.37766e18	0.108126
$67$	1.53703e19	1.03014	0.515071	−	0.857147i	$-0.327765\pi$
$67$	1.53703e19	1.03014	0.515071	+	0.857147i	$0.327765\pi$
$68$	−2.39473e19	−1.37377
$69$	−1.95312e19	−0.961198
$70$	0	0
$71$	−6.75134e18	−0.246137	−0.123069	−	0.992398i	$-0.539274\pi$
$71$	−6.75134e18	−0.246137	−0.123069	+	0.992398i	$0.539274\pi$
$72$	8.29199e18	0.261016
$73$	1.58050e19	0.430431	0.215216	−	0.976567i	$-0.430955\pi$
$73$	1.58050e19	0.430431	0.215216	+	0.976567i	$0.430955\pi$
$74$	−2.57485e19	−0.607880
$75$	0	0
$76$	−5.64864e19	−1.00786
$77$	1.41280e19	0.219749
$78$	−3.42438e19	−0.465146
$79$	−1.48815e19	−0.176833	−0.0884164	−	0.996084i	$-0.528181\pi$
$79$	−1.48815e19	−0.176833	−0.0884164	+	0.996084i	$0.528181\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	3.27046e19	0.262763
$83$	6.30804e19	0.446246	0.223123	−	0.974790i	$-0.428375\pi$
$83$	6.30804e19	0.446246	0.223123	+	0.974790i	$0.428375\pi$
$84$	3.81468e19	0.237972
$85$	0	0
$86$	1.31684e20	0.641652
$87$	1.39186e20	0.600679
$88$	−8.87349e19	−0.339646
$89$	1.22115e20	0.415119	0.207559	−	0.978222i	$-0.433448\pi$
$89$	1.22115e20	0.415119	0.207559	+	0.978222i	$0.433448\pi$
$90$	0	0
$91$	−3.51172e20	−0.945334
$92$	5.64342e20	1.35447
$93$	2.34480e20	0.502383
$94$	3.61400e20	0.692063
$95$	0	0
$96$	−3.71703e20	−0.570622
$97$	−5.41393e20	−0.745434	−0.372717	−	0.927945i	$-0.621574\pi$
$97$	−5.41393e20	−0.745434	−0.372717	+	0.927945i	$0.621574\pi$
$98$	2.59602e20	0.320948
$99$	−1.30103e20	−0.144583
$100$	0	0
$101$	4.54105e20	0.409055	0.204527	−	0.978861i	$-0.434434\pi$
$101$	4.54105e20	0.409055	0.204527	+	0.978861i	$0.434434\pi$
$102$	−5.18216e20	−0.420929
$103$	−1.33612e21	−0.979616	−0.489808	−	0.871830i	$-0.662933\pi$
$103$	−1.33612e21	−0.979616	−0.489808	+	0.871830i	$0.662933\pi$
$104$	2.20564e21	1.46112
$105$	0	0
$106$	7.26751e20	0.394161
$107$	1.60585e21	0.789179	0.394590	−	0.918858i	$-0.370887\pi$
$107$	1.60585e21	0.789179	0.394590	+	0.918858i	$0.370887\pi$
$108$	−3.51289e20	−0.156572
$109$	2.31707e21	0.937476	0.468738	−	0.883337i	$-0.344709\pi$
$109$	2.31707e21	0.937476	0.468738	+	0.883337i	$0.344709\pi$
$110$	0	0
$111$	2.43162e21	0.812837
$112$	−7.91784e20	−0.240889
$113$	−1.71334e21	−0.474810	−0.237405	−	0.971411i	$-0.576297\pi$
$113$	−1.71334e21	−0.474810	−0.237405	+	0.971411i	$0.576297\pi$
$114$	−1.22236e21	−0.308814
$115$	0	0
$116$	−4.02169e21	−0.846447
$117$	3.23390e21	0.621977
$118$	−2.61347e21	−0.459680
$119$	−5.31433e21	−0.855471
$120$	0	0
$121$	−6.00799e21	−0.811863
$122$	−6.11242e21	−0.757591
$123$	−3.08854e21	−0.351357
$124$	−6.77517e21	−0.707934
$125$	0	0
$126$	8.25492e20	0.0729159
$127$	−5.90533e21	−0.480071	−0.240035	−	0.970764i	$-0.577159\pi$
$127$	−5.90533e21	−0.480071	−0.240035	+	0.970764i	$0.577159\pi$
$128$	1.34823e22	1.00939
$129$	−1.24359e22	−0.857996
$130$	0	0
$131$	−2.01878e22	−1.18506	−0.592530	−	0.805548i	$-0.701871\pi$
$131$	−2.01878e22	−1.18506	−0.592530	+	0.805548i	$0.701871\pi$
$132$	3.75924e21	0.203739
$133$	−1.25353e22	−0.627615
$134$	−9.61063e21	−0.444786
$135$	0	0
$136$	3.33783e22	1.32222
$137$	2.23489e22	0.819768	0.409884	−	0.912138i	$-0.365569\pi$
$137$	2.23489e22	0.819768	0.409884	+	0.912138i	$0.365569\pi$
$138$	1.22123e22	0.415018
$139$	−7.44374e21	−0.234496	−0.117248	−	0.993103i	$-0.537407\pi$
$139$	−7.44374e21	−0.234496	−0.117248	+	0.993103i	$0.537407\pi$
$140$	0	0
$141$	−3.41296e22	−0.925403
$142$	4.22142e21	0.106275
$143$	−3.46068e22	−0.809344
$144$	7.29143e21	0.158492
$145$	0	0
$146$	−9.88241e21	−0.185848
$147$	−2.45161e22	−0.429160
$148$	−7.02601e22	−1.14541
$149$	−2.41970e21	−0.0367541	−0.0183770	−	0.999831i	$-0.505850\pi$
$149$	−2.41970e21	−0.0367541	−0.0183770	+	0.999831i	$0.505850\pi$
$150$	0	0
$151$	6.11367e22	0.807316	0.403658	−	0.914910i	$-0.367738\pi$
$151$	6.11367e22	0.807316	0.403658	+	0.914910i	$0.367738\pi$
$152$	7.87319e22	0.970048
$153$	4.89390e22	0.562853
$154$	−8.83382e21	−0.0948813
$155$	0	0
$156$	−9.34416e22	−0.876460
$157$	−1.98289e23	−1.73921	−0.869604	−	0.493750i	$-0.835626\pi$
$157$	−1.98289e23	−1.73921	−0.869604	+	0.493750i	$0.835626\pi$
$158$	9.30499e21	0.0763513
$159$	−6.86324e22	−0.527059
$160$	0	0
$161$	1.25238e23	0.843457
$162$	−7.60185e21	−0.0479746
$163$	8.73258e22	0.516622	0.258311	−	0.966062i	$-0.416834\pi$
$163$	8.73258e22	0.516622	0.258311	+	0.966062i	$0.416834\pi$
$164$	8.92415e22	0.495115
$165$	0	0
$166$	−3.94424e22	−0.192676
$167$	1.33825e23	0.613782	0.306891	−	0.951745i	$-0.400711\pi$
$167$	1.33825e23	0.613782	0.306891	+	0.951745i	$0.400711\pi$
$168$	−5.31698e22	−0.229043
$169$	6.13140e23	2.48170
$170$	0	0
$171$	1.15436e23	0.412936
$172$	3.59328e23	1.20905
$173$	−2.26996e23	−0.718677	−0.359338	−	0.933207i	$-0.616998\pi$
$173$	−2.26996e23	−0.718677	−0.359338	+	0.933207i	$0.616998\pi$
$174$	−8.70288e22	−0.259356
$175$	0	0
$176$	−7.80276e22	−0.206236
$177$	2.46809e23	0.614669
$178$	−7.63548e22	−0.179236
$179$	−5.38150e22	−0.119110	−0.0595548	−	0.998225i	$-0.518968\pi$
$179$	−5.38150e22	−0.119110	−0.0595548	+	0.998225i	$0.518968\pi$
$180$	0	0
$181$	5.67762e23	1.11826	0.559129	−	0.829081i	$-0.311136\pi$
$181$	5.67762e23	1.11826	0.559129	+	0.829081i	$0.311136\pi$
$182$	2.19578e23	0.408168
$183$	5.77241e23	1.01303
$184$	−7.86592e23	−1.30365
$185$	0	0
$186$	−1.46614e23	−0.216915
$187$	−5.23709e23	−0.732409
$188$	9.86156e23	1.30403
$189$	−7.79573e22	−0.0975007
$190$	0	0
$191$	−1.06663e22	−0.0119444	−0.00597219	−	0.999982i	$-0.501901\pi$
$191$	−1.06663e22	−0.0119444	−0.00597219	+	0.999982i	$0.501901\pi$
$192$	−2.65430e22	−0.0281376
$193$	−6.54963e23	−0.657453	−0.328727	−	0.944425i	$-0.606620\pi$
$193$	−6.54963e23	−0.657453	−0.328727	+	0.944425i	$0.606620\pi$
$194$	3.38518e23	0.321857
$195$	0	0
$196$	7.08379e23	0.604752
$197$	2.10493e24	1.70350	0.851752	−	0.523945i	$-0.175540\pi$
$197$	2.10493e24	1.70350	0.851752	+	0.523945i	$0.175540\pi$
$198$	8.13495e22	0.0624267
$199$	−7.42491e23	−0.540423	−0.270211	−	0.962801i	$-0.587094\pi$
$199$	−7.42491e23	−0.540423	−0.270211	+	0.962801i	$0.587094\pi$
$200$	0	0
$201$	9.07602e23	0.594753
$202$	−2.83939e23	−0.176618
$203$	−8.92484e23	−0.527099
$204$	−1.41406e24	−0.793144
$205$	0	0
$206$	8.35441e23	0.422970
$207$	−1.15330e24	−0.554948
$208$	1.93949e24	0.887204
$209$	−1.23531e24	−0.537331
$210$	0	0
$211$	−1.58957e23	−0.0625624	−0.0312812	−	0.999511i	$-0.509959\pi$
$211$	−1.58957e23	−0.0625624	−0.0312812	+	0.999511i	$0.509959\pi$
$212$	1.98310e24	0.742706
$213$	−3.98660e23	−0.142107
$214$	−1.00409e24	−0.340745
$215$	0	0
$216$	4.89634e23	0.150698
$217$	−1.50353e24	−0.440844
$218$	−1.44880e24	−0.404775
$219$	9.33268e23	0.248510
$220$	0	0
$221$	1.30176e25	3.15073
$222$	−1.52042e24	−0.350960
$223$	3.69828e24	0.814327	0.407164	−	0.913355i	$-0.366518\pi$
$223$	3.69828e24	0.814327	0.407164	+	0.913355i	$0.366518\pi$
$224$	2.38343e24	0.500724
$225$	0	0
$226$	1.07130e24	0.205009
$227$	4.74876e23	0.0867578	0.0433789	−	0.999059i	$-0.486188\pi$
$227$	4.74876e23	0.0867578	0.0433789	+	0.999059i	$0.486188\pi$
$228$	−3.33546e24	−0.581889
$229$	−5.00087e23	−0.0833245	−0.0416623	−	0.999132i	$-0.513265\pi$
$229$	−5.00087e23	−0.0833245	−0.0416623	+	0.999132i	$0.513265\pi$
$230$	0	0
$231$	8.34242e23	0.126872
$232$	5.60552e24	0.814689
$233$	1.84833e24	0.256769	0.128385	−	0.991724i	$-0.459021\pi$
$233$	1.84833e24	0.256769	0.128385	+	0.991724i	$0.459021\pi$
$234$	−2.02206e24	−0.268552
$235$	0	0
$236$	−7.13140e24	−0.866161
$237$	−8.78739e23	−0.102094
$238$	3.32290e24	0.369368
$239$	6.80041e24	0.723364	0.361682	−	0.932301i	$-0.382203\pi$
$239$	6.80041e24	0.723364	0.361682	+	0.932301i	$0.382203\pi$
$240$	0	0
$241$	1.15905e25	1.12960	0.564798	−	0.825229i	$-0.308954\pi$
$241$	1.15905e25	1.12960	0.564798	+	0.825229i	$0.308954\pi$
$242$	3.75662e24	0.350539
$243$	7.17898e23	0.0641500
$244$	−1.66790e25	−1.42751
$245$	0	0
$246$	1.93117e24	0.151706
$247$	3.07056e25	2.31153
$248$	9.44338e24	0.681373
$249$	3.72483e24	0.257640
$250$	0	0
$251$	7.75895e24	0.493434	0.246717	−	0.969088i	$-0.420648\pi$
$251$	7.75895e24	0.493434	0.246717	+	0.969088i	$0.420648\pi$
$252$	2.25253e24	0.137393
$253$	1.23417e25	0.722123
$254$	3.69244e24	0.207281
$255$	0	0
$256$	−7.48740e24	−0.387089
$257$	9.07103e24	0.450152	0.225076	−	0.974341i	$-0.427737\pi$
$257$	9.07103e24	0.450152	0.225076	+	0.974341i	$0.427737\pi$
$258$	7.77581e24	0.370458
$259$	−1.55920e25	−0.713270
$260$	0	0
$261$	8.21877e24	0.346802
$262$	1.26229e25	0.511675
$263$	−4.14573e25	−1.61460	−0.807301	−	0.590140i	$-0.799073\pi$
$263$	−4.14573e25	−1.61460	−0.807301	+	0.590140i	$0.799073\pi$
$264$	−5.23971e24	−0.196095
$265$	0	0
$266$	7.83799e24	0.270986
$267$	7.21074e24	0.239669
$268$	−2.62246e25	−0.838097
$269$	−4.62596e25	−1.42168	−0.710842	−	0.703352i	$-0.751686\pi$
$269$	−4.62596e25	−1.42168	−0.710842	+	0.703352i	$0.751686\pi$
$270$	0	0
$271$	3.86928e25	1.10015	0.550075	−	0.835115i	$-0.314599\pi$
$271$	3.86928e25	1.10015	0.550075	+	0.835115i	$0.314599\pi$
$272$	2.93506e25	0.802868
$273$	−2.07364e25	−0.545789
$274$	−1.39742e25	−0.353952
$275$	0	0
$276$	3.33238e25	0.782005
$277$	7.06699e25	1.59660	0.798302	−	0.602257i	$-0.205732\pi$
$277$	7.06699e25	1.59660	0.798302	+	0.602257i	$0.205732\pi$
$278$	4.65436e24	0.101249
$279$	1.38458e25	0.290051
$280$	0	0
$281$	7.68544e25	1.49366	0.746831	−	0.665014i	$-0.231575\pi$
$281$	7.68544e25	1.49366	0.746831	+	0.665014i	$0.231575\pi$
$282$	2.13403e25	0.399562
$283$	3.92878e24	0.0708761	0.0354381	−	0.999372i	$-0.488717\pi$
$283$	3.92878e24	0.0708761	0.0354381	+	0.999372i	$0.488717\pi$
$284$	1.15190e25	0.200251
$285$	0	0
$286$	2.16387e25	0.349452
$287$	1.98043e25	0.308318
$288$	−2.19487e25	−0.329449
$289$	1.27905e26	1.85123
$290$	0	0
$291$	−3.19687e25	−0.430377
$292$	−2.69663e25	−0.350188
$293$	3.51075e25	0.439835	0.219918	−	0.975518i	$-0.429421\pi$
$293$	3.51075e25	0.439835	0.219918	+	0.975518i	$0.429421\pi$
$294$	1.53292e25	0.185299
$295$	0	0
$296$	9.79301e25	1.10244
$297$	−7.68243e24	−0.0834748
$298$	1.51297e24	0.0158694
$299$	−3.06773e26	−3.10648
$300$	0	0
$301$	7.97413e25	0.752896
$302$	−3.82270e25	−0.348576
$303$	2.68144e25	0.236168
$304$	6.92316e25	0.589023
$305$	0	0
$306$	−3.06002e25	−0.243024
$307$	−1.03340e26	−0.793081	−0.396540	−	0.918017i	$-0.629789\pi$
$307$	−1.03340e26	−0.793081	−0.396540	+	0.918017i	$0.629789\pi$
$308$	−2.41050e25	−0.178782
$309$	−7.88968e25	−0.565582
$310$	0	0
$311$	−5.45153e25	−0.365203	−0.182602	−	0.983187i	$-0.558452\pi$
$311$	−5.45153e25	−0.365203	−0.182602	+	0.983187i	$0.558452\pi$
$312$	1.30241e26	0.843576
$313$	6.17771e25	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$313$	6.17771e25	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$314$	1.23984e26	0.750940
$315$	0	0
$316$	2.53907e25	0.143866
$317$	1.39085e26	0.762354	0.381177	−	0.924502i	$-0.375519\pi$
$317$	1.39085e26	0.762354	0.381177	+	0.924502i	$0.375519\pi$
$318$	4.29139e25	0.227569
$319$	−8.79513e25	−0.451274
$320$	0	0
$321$	9.48239e25	0.455633
$322$	−7.83075e25	−0.364181
$323$	4.64672e26	2.09180
$324$	−2.07433e25	−0.0903971
$325$	0	0
$326$	−5.46024e25	−0.223063
$327$	1.36820e26	0.541252
$328$	−1.24387e26	−0.476539
$329$	2.18846e26	0.812047
$330$	0	0
$331$	8.96648e25	0.312197	0.156098	−	0.987742i	$-0.450108\pi$
$331$	8.96648e25	0.312197	0.156098	+	0.987742i	$0.450108\pi$
$332$	−1.07627e26	−0.363054
$333$	1.43584e26	0.469292
$334$	−8.36769e25	−0.265013
$335$	0	0
$336$	−4.67540e25	−0.139077
$337$	4.20032e26	1.21107	0.605534	−	0.795820i	$-0.292960\pi$
$337$	4.20032e26	1.21107	0.605534	+	0.795820i	$0.292960\pi$
$338$	−3.83379e26	−1.07153
$339$	−1.01171e26	−0.274132
$340$	0	0
$341$	−1.48168e26	−0.377427
$342$	−7.21790e25	−0.178294
$343$	3.68686e26	0.883219
$344$	−5.00839e26	−1.16368
$345$	0	0
$346$	1.41934e26	0.310304
$347$	−5.85880e26	−1.24265	−0.621326	−	0.783552i	$-0.713406\pi$
$347$	−5.85880e26	−1.24265	−0.621326	+	0.783552i	$0.713406\pi$
$348$	−2.37477e26	−0.488696
$349$	5.60384e26	1.11897	0.559485	−	0.828840i	$-0.310999\pi$
$349$	5.60384e26	1.11897	0.559485	+	0.828840i	$0.310999\pi$
$350$	0	0
$351$	1.90958e26	0.359099
$352$	2.34879e26	0.428693
$353$	1.96463e26	0.348054	0.174027	−	0.984741i	$-0.444322\pi$
$353$	1.96463e26	0.348054	0.174027	+	0.984741i	$0.444322\pi$
$354$	−1.54323e26	−0.265397
$355$	0	0
$356$	−2.08350e26	−0.337729
$357$	−3.13806e26	−0.493907
$358$	3.36490e25	0.0514281
$359$	2.20403e26	0.327135	0.163567	−	0.986532i	$-0.447700\pi$
$359$	2.20403e26	0.327135	0.163567	+	0.986532i	$0.447700\pi$
$360$	0	0
$361$	3.81849e26	0.534646
$362$	−3.55006e26	−0.482831
$363$	−3.54766e26	−0.468729
$364$	5.99165e26	0.769098
$365$	0	0
$366$	−3.60932e26	−0.437396
$367$	−1.39438e27	−1.64205	−0.821025	−	0.570892i	$-0.806598\pi$
$367$	−1.39438e27	−1.64205	−0.821025	+	0.570892i	$0.806598\pi$
$368$	−6.91677e26	−0.791592
$369$	−1.82375e26	−0.202856
$370$	0	0
$371$	4.40084e26	0.462498
$372$	−4.00067e26	−0.408726
$373$	−8.14444e26	−0.808944	−0.404472	−	0.914550i	$-0.632545\pi$
$373$	−8.14444e26	−0.808944	−0.404472	+	0.914550i	$0.632545\pi$
$374$	3.27461e26	0.316233
$375$	0	0
$376$	−1.37453e27	−1.25511
$377$	2.18616e27	1.94133
$378$	4.87445e25	0.0420980
$379$	−2.19248e26	−0.184172	−0.0920862	−	0.995751i	$-0.529354\pi$
$379$	−2.19248e26	−0.184172	−0.0920862	+	0.995751i	$0.529354\pi$
$380$	0	0
$381$	−3.48704e26	−0.277169
$382$	6.66934e24	0.00515724
$383$	−1.46274e27	−1.10048	−0.550239	−	0.835007i	$-0.685463\pi$
$383$	−1.46274e27	−1.10048	−0.550239	+	0.835007i	$0.685463\pi$
$384$	7.96115e26	0.582771
$385$	0	0
$386$	4.09530e26	0.283870
$387$	−7.34327e26	−0.495364
$388$	9.23718e26	0.606465
$389$	1.72831e27	1.10446	0.552231	−	0.833691i	$-0.313777\pi$
$389$	1.72831e27	1.10446	0.552231	+	0.833691i	$0.313777\pi$
$390$	0	0
$391$	−4.64243e27	−2.81119
$392$	−9.87354e26	−0.582062
$393$	−1.19207e27	−0.684195
$394$	−1.31616e27	−0.735524
$395$	0	0
$396$	2.21979e26	0.117629
$397$	−7.87988e26	−0.406648	−0.203324	−	0.979111i	$-0.565175\pi$
$397$	−7.87988e26	−0.406648	−0.203324	+	0.979111i	$0.565175\pi$
$398$	4.64258e26	0.233339
$399$	−7.40199e26	−0.362354
$400$	0	0
$401$	−3.13221e27	−1.45491	−0.727453	−	0.686157i	$-0.759296\pi$
$401$	−3.13221e27	−1.45491	−0.727453	+	0.686157i	$0.759296\pi$
$402$	−5.67498e26	−0.256797
$403$	3.68294e27	1.62365
$404$	−7.74788e26	−0.332796
$405$	0	0
$406$	5.58045e26	0.227586
$407$	−1.53654e27	−0.610663
$408$	1.97095e27	0.763386
$409$	2.95830e27	1.11673	0.558363	−	0.829596i	$-0.311429\pi$
$409$	2.95830e27	1.11673	0.558363	+	0.829596i	$0.311429\pi$
$410$	0	0
$411$	1.31968e27	0.473293
$412$	2.27968e27	0.796990
$413$	−1.58259e27	−0.539376
$414$	7.21123e26	0.239611
$415$	0	0
$416$	−5.83828e27	−1.84419
$417$	−4.39545e26	−0.135386
$418$	7.72407e26	0.232004
$419$	1.05350e27	0.308595	0.154298	−	0.988024i	$-0.450689\pi$
$419$	1.05350e27	0.308595	0.154298	+	0.988024i	$0.450689\pi$
$420$	0	0
$421$	−3.19984e27	−0.891592	−0.445796	−	0.895135i	$-0.647079\pi$
$421$	−3.19984e27	−0.891592	−0.445796	+	0.895135i	$0.647079\pi$
$422$	9.93913e25	0.0270127
$423$	−2.01532e27	−0.534282
$424$	−2.76408e27	−0.714841
$425$	0	0
$426$	2.49271e26	0.0613579
$427$	−3.70138e27	−0.888936
$428$	−2.73988e27	−0.642055
$429$	−2.04350e27	−0.467275
$430$	0	0
$431$	6.57772e27	1.43240	0.716200	−	0.697895i	$-0.245880\pi$
$431$	6.57772e27	1.43240	0.716200	+	0.697895i	$0.245880\pi$
$432$	4.30552e26	0.0915053
$433$	9.38190e26	0.194611	0.0973056	−	0.995255i	$-0.468978\pi$
$433$	9.38190e26	0.194611	0.0973056	+	0.995255i	$0.468978\pi$
$434$	9.40116e26	0.190344
$435$	0	0
$436$	−3.95335e27	−0.762706
$437$	−1.09505e28	−2.06242
$438$	−5.83546e26	−0.107299
$439$	9.48686e27	1.70312	0.851559	−	0.524258i	$-0.175657\pi$
$439$	9.48686e27	1.70312	0.851559	+	0.524258i	$0.175657\pi$
$440$	0	0
$441$	−1.44765e27	−0.247776
$442$	−8.13953e27	−1.36040
$443$	−2.87822e27	−0.469770	−0.234885	−	0.972023i	$-0.575471\pi$
$443$	−2.87822e27	−0.469770	−0.234885	+	0.972023i	$0.575471\pi$
$444$	−4.14879e27	−0.661303
$445$	0	0
$446$	−2.31243e27	−0.351603
$447$	−1.42881e26	−0.0212200
$448$	1.70199e26	0.0246909
$449$	−7.14238e27	−1.01218	−0.506089	−	0.862482i	$-0.668909\pi$
$449$	−7.14238e27	−1.01218	−0.506089	+	0.862482i	$0.668909\pi$
$450$	0	0
$451$	1.95164e27	0.263965
$452$	2.92328e27	0.386293
$453$	3.61006e27	0.466104
$454$	−2.96926e26	−0.0374595
$455$	0	0
$456$	4.64904e27	0.560057
$457$	4.86346e26	0.0572566	0.0286283	−	0.999590i	$-0.490886\pi$
$457$	4.86346e26	0.0572566	0.0286283	+	0.999590i	$0.490886\pi$
$458$	3.12690e26	0.0359772
$459$	2.88980e27	0.324963
$460$	0	0
$461$	2.16847e26	0.0232966	0.0116483	−	0.999932i	$-0.496292\pi$
$461$	2.16847e26	0.0232966	0.0116483	+	0.999932i	$0.496292\pi$
$462$	−5.21628e26	−0.0547798
$463$	−1.29183e28	−1.32619	−0.663094	−	0.748536i	$-0.730757\pi$
$463$	−1.29183e28	−1.32619	−0.663094	+	0.748536i	$0.730757\pi$
$464$	4.92912e27	0.494687
$465$	0	0
$466$	−1.15571e27	−0.110866
$467$	2.06226e28	1.93426	0.967131	−	0.254277i	$-0.0818376\pi$
$467$	2.06226e28	1.93426	0.967131	+	0.254277i	$0.0818376\pi$
$468$	−5.51763e27	−0.506024
$469$	−5.81972e27	−0.521900
$470$	0	0
$471$	−1.17087e28	−1.00413
$472$	9.93990e27	0.833664
$473$	7.85824e27	0.644590
$474$	5.49450e26	0.0440815
$475$	0	0
$476$	9.06724e27	0.695989
$477$	−4.05268e27	−0.304298
$478$	−4.25210e27	−0.312328
$479$	3.66438e27	0.263317	0.131658	−	0.991295i	$-0.457970\pi$
$479$	3.66438e27	0.263317	0.131658	+	0.991295i	$0.457970\pi$
$480$	0	0
$481$	3.81930e28	2.62700
$482$	−7.24721e27	−0.487727
$483$	7.39515e27	0.486970
$484$	1.02507e28	0.660510
$485$	0	0
$486$	−4.48881e26	−0.0276981
$487$	6.79839e27	0.410537	0.205268	−	0.978706i	$-0.434193\pi$
$487$	6.79839e27	0.410537	0.205268	+	0.978706i	$0.434193\pi$
$488$	2.32476e28	1.37395
$489$	5.15650e27	0.298272
$490$	0	0
$491$	−1.83806e28	−1.01860	−0.509301	−	0.860588i	$-0.670096\pi$
$491$	−1.83806e28	−1.01860	−0.509301	+	0.860588i	$0.670096\pi$
$492$	5.26962e27	0.285855
$493$	3.30835e28	1.75679
$494$	−1.91994e28	−0.998053
$495$	0	0
$496$	8.30389e27	0.413736
$497$	2.55628e27	0.124700
$498$	−2.32903e27	−0.111242
$499$	7.03152e27	0.328847	0.164423	−	0.986390i	$-0.447424\pi$
$499$	7.03152e27	0.328847	0.164423	+	0.986390i	$0.447424\pi$
$500$	0	0
$501$	7.90222e27	0.354367
$502$	−4.85145e27	−0.213051
$503$	−1.71742e28	−0.738607	−0.369303	−	0.929309i	$-0.620404\pi$
$503$	−1.71742e28	−0.738607	−0.369303	+	0.929309i	$0.620404\pi$
$504$	−3.13963e27	−0.132238
$505$	0	0
$506$	−7.71694e27	−0.311792
$507$	3.62053e28	1.43281
$508$	1.00756e28	0.390573
$509$	−4.25735e28	−1.61660	−0.808300	−	0.588771i	$-0.799612\pi$
$509$	−4.25735e28	−1.61660	−0.808300	+	0.588771i	$0.799612\pi$
$510$	0	0
$511$	−5.98429e27	−0.218069
$512$	−2.35927e28	−0.842254
$513$	6.81639e27	0.238409
$514$	−5.67186e27	−0.194363
$515$	0	0
$516$	2.12180e28	0.698043
$517$	2.15665e28	0.695231
$518$	9.74922e27	0.307969
$519$	−1.34039e28	−0.414928
$520$	0	0
$521$	4.64258e27	0.138027	0.0690133	−	0.997616i	$-0.478015\pi$
$521$	4.64258e27	0.138027	0.0690133	+	0.997616i	$0.478015\pi$
$522$	−5.13897e27	−0.149739
$523$	−5.19658e28	−1.48406	−0.742028	−	0.670369i	$-0.766136\pi$
$523$	−5.19658e28	−1.48406	−0.742028	+	0.670369i	$0.766136\pi$
$524$	3.44442e28	0.964134
$525$	0	0
$526$	2.59221e28	0.697139
$527$	5.57344e28	1.46931
$528$	−4.60745e27	−0.119071
$529$	6.99320e28	1.77170
$530$	0	0
$531$	1.45738e28	0.354879
$532$	2.13876e28	0.510611
$533$	−4.85111e28	−1.13555
$534$	−4.50867e27	−0.103482
$535$	0	0
$536$	3.65525e28	0.806653
$537$	−3.17772e27	−0.0687680
$538$	2.89248e28	0.613843
$539$	1.54917e28	0.322417
$540$	0	0
$541$	3.03293e27	0.0607144	0.0303572	−	0.999539i	$-0.490336\pi$
$541$	3.03293e27	0.0607144	0.0303572	+	0.999539i	$0.490336\pi$
$542$	−2.41935e28	−0.475014
$543$	3.35258e28	0.645626
$544$	−8.83514e28	−1.66888
$545$	0	0
$546$	1.29659e28	0.235656
$547$	3.74354e27	0.0667445	0.0333722	−	0.999443i	$-0.489375\pi$
$547$	3.74354e27	0.0667445	0.0333722	+	0.999443i	$0.489375\pi$
$548$	−3.81315e28	−0.666942
$549$	3.40855e28	0.584871
$550$	0	0
$551$	7.80367e28	1.28886
$552$	−4.64475e28	−0.752665
$553$	5.63464e27	0.0895885
$554$	−4.41879e28	−0.689368
$555$	0	0
$556$	1.27004e28	0.190780
$557$	1.24990e28	0.184245	0.0921225	−	0.995748i	$-0.470635\pi$
$557$	1.24990e28	0.184245	0.0921225	+	0.995748i	$0.470635\pi$
$558$	−8.65740e27	−0.125236
$559$	−1.95328e29	−2.77295
$560$	0	0
$561$	−3.09245e28	−0.422856
$562$	−4.80549e28	−0.644920
$563$	1.37259e27	0.0180802	0.00904008	−	0.999959i	$-0.497122\pi$
$563$	1.37259e27	0.0180802	0.00904008	+	0.999959i	$0.497122\pi$
$564$	5.82315e28	0.752883
$565$	0	0
$566$	−2.45655e27	−0.0306023
$567$	−4.60330e27	−0.0562920
$568$	−1.60555e28	−0.192738
$569$	−4.80572e28	−0.566343	−0.283172	−	0.959069i	$-0.591387\pi$
$569$	−4.80572e28	−0.566343	−0.283172	+	0.959069i	$0.591387\pi$
$570$	0	0
$571$	−4.42408e28	−0.502508	−0.251254	−	0.967921i	$-0.580843\pi$
$571$	−4.42408e28	−0.502508	−0.251254	+	0.967921i	$0.580843\pi$
$572$	5.90457e28	0.658461
$573$	−6.29835e26	−0.00689609
$574$	−1.23831e28	−0.133123
$575$	0	0
$576$	−1.56734e27	−0.0162453
$577$	9.25254e28	0.941705	0.470852	−	0.882212i	$-0.343946\pi$
$577$	9.25254e28	0.941705	0.470852	+	0.882212i	$0.343946\pi$
$578$	−7.99754e28	−0.799308
$579$	−3.86749e28	−0.379581
$580$	0	0
$581$	−2.38843e28	−0.226081
$582$	1.99891e28	0.185824
$583$	4.33688e28	0.395966
$584$	3.75862e28	0.337049
$585$	0	0
$586$	−2.19517e28	−0.189908
$587$	1.94963e29	1.65673	0.828364	−	0.560190i	$-0.189272\pi$
$587$	1.94963e29	1.65673	0.828364	+	0.560190i	$0.189272\pi$
$588$	4.18291e28	0.349154
$589$	1.31465e29	1.07795
$590$	0	0
$591$	1.24294e29	0.983519
$592$	8.61133e28	0.669410
$593$	7.23704e28	0.552696	0.276348	−	0.961058i	$-0.410876\pi$
$593$	7.23704e28	0.552696	0.276348	+	0.961058i	$0.410876\pi$
$594$	4.80360e27	0.0360421
$595$	0	0
$596$	4.12846e27	0.0299022
$597$	−4.38433e28	−0.312013
$598$	1.91816e29	1.34129
$599$	1.37531e29	0.944970	0.472485	−	0.881339i	$-0.343357\pi$
$599$	1.37531e29	0.944970	0.472485	+	0.881339i	$0.343357\pi$
$600$	0	0
$601$	9.83934e28	0.652806	0.326403	−	0.945231i	$-0.394163\pi$
$601$	9.83934e28	0.652806	0.326403	+	0.945231i	$0.394163\pi$
$602$	−4.98600e28	−0.325079
$603$	5.35930e28	0.343381
$604$	−1.04311e29	−0.656811
$605$	0	0
$606$	−1.67663e28	−0.101971
$607$	−1.43248e29	−0.856262	−0.428131	−	0.903717i	$-0.640828\pi$
$607$	−1.43248e29	−0.856262	−0.428131	+	0.903717i	$0.640828\pi$
$608$	−2.08401e29	−1.22437
$609$	−5.27003e28	−0.304321
$610$	0	0
$611$	−5.36068e29	−2.99080
$612$	−8.34990e28	−0.457922
$613$	−2.28269e29	−1.23058	−0.615292	−	0.788299i	$-0.710962\pi$
$613$	−2.28269e29	−1.23058	−0.615292	+	0.788299i	$0.710962\pi$
$614$	6.46159e28	0.342430
$615$	0	0
$616$	3.35980e28	0.172074
$617$	−2.30467e28	−0.116042	−0.0580210	−	0.998315i	$-0.518479\pi$
$617$	−2.30467e28	−0.116042	−0.0580210	+	0.998315i	$0.518479\pi$
$618$	4.93319e28	0.244202
$619$	1.54693e29	0.752867	0.376433	−	0.926444i	$-0.377150\pi$
$619$	1.54693e29	0.752867	0.376433	+	0.926444i	$0.377150\pi$
$620$	0	0
$621$	−6.81010e28	−0.320399
$622$	3.40869e28	0.157684
$623$	−4.62366e28	−0.210311
$624$	1.14525e29	0.512228
$625$	0	0
$626$	−3.86275e28	−0.167057
$627$	−7.29441e28	−0.310228
$628$	3.38318e29	1.41497
$629$	5.77979e29	2.37728
$630$	0	0
$631$	3.94122e29	1.56792	0.783958	−	0.620814i	$-0.213198\pi$
$631$	3.94122e29	1.56792	0.783958	+	0.620814i	$0.213198\pi$
$632$	−3.53900e28	−0.138469
$633$	−9.38625e27	−0.0361204
$634$	−8.69657e28	−0.329163
$635$	0	0
$636$	1.17100e29	0.428802
$637$	−3.85070e29	−1.38700
$638$	5.49935e28	0.194847
$639$	−2.35405e28	−0.0820457
$640$	0	0
$641$	−1.02757e29	−0.346578	−0.173289	−	0.984871i	$-0.555439\pi$
$641$	−1.02757e29	−0.346578	−0.173289	+	0.984871i	$0.555439\pi$
$642$	−5.92907e28	−0.196729
$643$	−3.14849e29	−1.02775	−0.513875	−	0.857865i	$-0.671790\pi$
$643$	−3.14849e29	−1.02775	−0.513875	+	0.857865i	$0.671790\pi$
$644$	−2.13679e29	−0.686214
$645$	0	0
$646$	−2.90546e29	−0.903179
$647$	−1.46297e29	−0.447446	−0.223723	−	0.974653i	$-0.571821\pi$
$647$	−1.46297e29	−0.447446	−0.223723	+	0.974653i	$0.571821\pi$
$648$	2.89124e28	0.0870055
$649$	−1.55958e29	−0.461785
$650$	0	0
$651$	−8.87820e28	−0.254522
$652$	−1.48994e29	−0.420310
$653$	−7.78420e28	−0.216086	−0.108043	−	0.994146i	$-0.534458\pi$
$653$	−7.78420e28	−0.216086	−0.108043	+	0.994146i	$0.534458\pi$
$654$	−8.55500e28	−0.233697
$655$	0	0
$656$	−1.09377e29	−0.289359
$657$	5.51086e28	0.143477
$658$	−1.36838e29	−0.350618
$659$	−2.75206e29	−0.694003	−0.347001	−	0.937865i	$-0.612800\pi$
$659$	−2.75206e29	−0.694003	−0.347001	+	0.937865i	$0.612800\pi$
$660$	0	0
$661$	4.14436e29	1.01238	0.506188	−	0.862423i	$-0.331054\pi$
$661$	4.14436e29	1.01238	0.506188	+	0.862423i	$0.331054\pi$
$662$	−5.60649e28	−0.134798
$663$	7.68676e29	1.81908
$664$	1.50013e29	0.349433
$665$	0	0
$666$	−8.97793e28	−0.202627
$667$	−7.79646e29	−1.73211
$668$	−2.28330e29	−0.499356
$669$	2.18380e29	0.470152
$670$	0	0
$671$	−3.64758e29	−0.761060
$672$	1.40739e29	0.289093
$673$	4.45773e28	0.0901478	0.0450739	−	0.998984i	$-0.485648\pi$
$673$	4.45773e28	0.0901478	0.0450739	+	0.998984i	$0.485648\pi$
$674$	−2.62634e29	−0.522904
$675$	0	0
$676$	−1.04613e30	−2.01905
$677$	−4.10130e27	−0.00779365	−0.00389682	−	0.999992i	$-0.501240\pi$
$677$	−4.10130e27	−0.00779365	−0.00389682	+	0.999992i	$0.501240\pi$
$678$	6.32594e28	0.118362
$679$	2.04989e29	0.377658
$680$	0	0
$681$	2.80409e28	0.0500896
$682$	9.26453e28	0.162962
$683$	−2.97788e29	−0.515810	−0.257905	−	0.966170i	$-0.583032\pi$
$683$	−2.97788e29	−0.515810	−0.257905	+	0.966170i	$0.583032\pi$
$684$	−1.96956e29	−0.335954
$685$	0	0
$686$	−2.30529e29	−0.381349
$687$	−2.95296e28	−0.0481074
$688$	−4.40405e29	−0.706600
$689$	−1.07800e30	−1.70340
$690$	0	0
$691$	−7.07918e28	−0.108508	−0.0542542	−	0.998527i	$-0.517278\pi$
$691$	−7.07918e28	−0.108508	−0.0542542	+	0.998527i	$0.517278\pi$
$692$	3.87297e29	0.584696
$693$	4.92612e28	0.0732497
$694$	3.66334e29	0.536541
$695$	0	0
$696$	3.31000e29	0.470361
$697$	−7.34125e29	−1.02760
$698$	−3.50392e29	−0.483140
$699$	1.09142e29	0.148246
$700$	0	0
$701$	−1.23613e30	−1.62939	−0.814696	−	0.579889i	$-0.803096\pi$
$701$	−1.23613e30	−1.62939	−0.814696	+	0.579889i	$0.803096\pi$
$702$	−1.19401e29	−0.155049
$703$	1.36332e30	1.74409
$704$	1.67725e28	0.0211391
$705$	0	0
$706$	−1.22843e29	−0.150280
$707$	−1.71939e29	−0.207239
$708$	−4.21102e29	−0.500079
$709$	1.35420e30	1.58452	0.792260	−	0.610184i	$-0.208904\pi$
$709$	1.35420e30	1.58452	0.792260	+	0.610184i	$0.208904\pi$
$710$	0	0
$711$	−5.18886e28	−0.0589443
$712$	2.90403e29	0.325058
$713$	−1.31344e30	−1.44867
$714$	1.96214e29	0.213255
$715$	0	0
$716$	9.18185e28	0.0969044
$717$	4.01557e29	0.417635
$718$	−1.37812e29	−0.141247
$719$	6.73356e29	0.680129	0.340065	−	0.940402i	$-0.389551\pi$
$719$	6.73356e29	0.680129	0.340065	+	0.940402i	$0.389551\pi$
$720$	0	0
$721$	5.05901e29	0.496301
$722$	−2.38759e29	−0.230845
$723$	6.84408e29	0.652173
$724$	−9.68709e29	−0.909784
$725$	0	0
$726$	2.21825e29	0.202384
$727$	1.55564e30	1.39893	0.699467	−	0.714665i	$-0.253421\pi$
$727$	1.55564e30	1.39893	0.699467	+	0.714665i	$0.253421\pi$
$728$	−8.35129e29	−0.740243
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−2.95593e30	−2.50935
$732$	−9.84881e29	−0.824171
$733$	1.13297e30	0.934599	0.467299	−	0.884099i	$-0.345227\pi$
$733$	1.13297e30	0.934599	0.467299	+	0.884099i	$0.345227\pi$
$734$	8.71865e29	0.708990
$735$	0	0
$736$	2.08209e30	1.64544
$737$	−5.73513e29	−0.446822
$738$	1.14034e29	0.0875875
$739$	−5.13263e28	−0.0388663	−0.0194331	−	0.999811i	$-0.506186\pi$
$739$	−5.13263e28	−0.0388663	−0.0194331	+	0.999811i	$0.506186\pi$
$740$	0	0
$741$	1.81314e30	1.33456
$742$	−2.75172e29	−0.199693
$743$	−1.03039e30	−0.737255	−0.368627	−	0.929577i	$-0.620172\pi$
$743$	−1.03039e30	−0.737255	−0.368627	+	0.929577i	$0.620172\pi$
$744$	5.57622e29	0.393391
$745$	0	0
$746$	5.09249e29	0.349279
$747$	2.19948e29	0.148749
$748$	8.93546e29	0.595868
$749$	−6.08029e29	−0.399820
$750$	0	0
$751$	−6.58297e29	−0.420923	−0.210461	−	0.977602i	$-0.567497\pi$
$751$	−6.58297e29	−0.420923	−0.210461	+	0.977602i	$0.567497\pi$
$752$	−1.20867e30	−0.762113
$753$	4.58158e29	0.284884
$754$	−1.36695e30	−0.838209
$755$	0	0
$756$	1.33010e29	0.0793240
$757$	1.14028e30	0.670663	0.335332	−	0.942100i	$-0.391152\pi$
$757$	1.14028e30	0.670663	0.335332	+	0.942100i	$0.391152\pi$
$758$	1.37090e29	0.0795204
$759$	7.28767e29	0.416918
$760$	0	0
$761$	2.10124e30	1.16933	0.584664	−	0.811276i	$-0.301226\pi$
$761$	2.10124e30	1.16933	0.584664	+	0.811276i	$0.301226\pi$
$762$	2.18035e29	0.119674
$763$	−8.77319e29	−0.474952
$764$	1.81987e28	0.00971763
$765$	0	0
$766$	9.14612e29	0.475155
$767$	3.87658e30	1.98654
$768$	−4.42123e29	−0.223486
$769$	3.41637e30	1.70349	0.851743	−	0.523961i	$-0.175546\pi$
$769$	3.41637e30	1.70349	0.851743	+	0.523961i	$0.175546\pi$
$770$	0	0
$771$	5.35635e29	0.259895
$772$	1.11749e30	0.534887
$773$	−1.82343e30	−0.861001	−0.430500	−	0.902590i	$-0.641663\pi$
$773$	−1.82343e30	−0.861001	−0.430500	+	0.902590i	$0.641663\pi$
$774$	4.59154e29	0.213884
$775$	0	0
$776$	−1.28750e30	−0.583712
$777$	−9.20691e29	−0.411806
$778$	−1.08066e30	−0.476875
$779$	−1.73164e30	−0.753900
$780$	0	0
$781$	2.51913e29	0.106762
$782$	2.90278e30	1.21379
$783$	4.85310e29	0.200226
$784$	−8.68214e29	−0.353434
$785$	0	0
$786$	7.45368e29	0.295416
$787$	6.96799e29	0.272504	0.136252	−	0.990674i	$-0.456494\pi$
$787$	6.96799e29	0.272504	0.136252	+	0.990674i	$0.456494\pi$
$788$	−3.59141e30	−1.38593
$789$	−2.44801e30	−0.932191
$790$	0	0
$791$	6.48728e29	0.240552
$792$	−3.09400e29	−0.113215
$793$	9.06662e30	3.27399
$794$	4.92706e29	0.175579
$795$	0	0
$796$	1.26683e30	0.439674
$797$	1.48389e30	0.508263	0.254132	−	0.967170i	$-0.418210\pi$
$797$	1.48389e30	0.508263	0.254132	+	0.967170i	$0.418210\pi$
$798$	4.62825e29	0.156454
$799$	−8.11239e30	−2.70650
$800$	0	0
$801$	4.25787e29	0.138373
$802$	1.95848e30	0.628187
$803$	−5.89732e29	−0.186699
$804$	−1.54854e30	−0.483876
$805$	0	0
$806$	−2.30284e30	−0.701044
$807$	−2.73158e30	−0.820810
$808$	1.07992e30	0.320310
$809$	−3.52590e30	−1.03231	−0.516155	−	0.856495i	$-0.672637\pi$
$809$	−3.52590e30	−1.03231	−0.516155	+	0.856495i	$0.672637\pi$
$810$	0	0
$811$	3.08640e30	0.880508	0.440254	−	0.897873i	$-0.354888\pi$
$811$	3.08640e30	0.880508	0.440254	+	0.897873i	$0.354888\pi$
$812$	1.52275e30	0.428834
$813$	2.28477e30	0.635172
$814$	9.60753e29	0.263667
$815$	0	0
$816$	1.73313e30	0.463536
$817$	−6.97239e30	−1.84098
$818$	−1.84974e30	−0.482171
$819$	−1.22446e30	−0.315111
$820$	0	0
$821$	5.43721e30	1.36387	0.681934	−	0.731413i	$-0.261139\pi$
$821$	5.43721e30	1.36387	0.681934	+	0.731413i	$0.261139\pi$
$822$	−8.25160e29	−0.204355
$823$	3.97429e30	0.971766	0.485883	−	0.874024i	$-0.338498\pi$
$823$	3.97429e30	0.971766	0.485883	+	0.874024i	$0.338498\pi$
$824$	−3.17746e30	−0.767087
$825$	0	0
$826$	9.89546e29	0.232887
$827$	1.38664e30	0.322223	0.161111	−	0.986936i	$-0.448492\pi$
$827$	1.38664e30	0.322223	0.161111	+	0.986936i	$0.448492\pi$
$828$	1.96774e30	0.451491
$829$	7.15876e30	1.62187	0.810933	−	0.585139i	$-0.198960\pi$
$829$	7.15876e30	1.62187	0.810933	+	0.585139i	$0.198960\pi$
$830$	0	0
$831$	4.17299e30	0.921800
$832$	−4.16907e29	−0.0909377
$833$	−5.82732e30	−1.25515
$834$	2.74835e29	0.0584560
$835$	0	0
$836$	2.10768e30	0.437158
$837$	8.17582e29	0.167461
$838$	−6.58727e29	−0.133243
$839$	8.11621e30	1.62126	0.810630	−	0.585558i	$-0.199125\pi$
$839$	8.11621e30	1.62126	0.810630	+	0.585558i	$0.199125\pi$
$840$	0	0
$841$	4.23174e29	0.0824443
$842$	2.00077e30	0.384964
$843$	4.53817e30	0.862366
$844$	2.71210e29	0.0508991
$845$	0	0
$846$	1.26012e30	0.230688
$847$	2.27482e30	0.411313
$848$	−2.43055e30	−0.434058
$849$	2.31990e29	0.0409203
$850$	0	0
$851$	−1.36207e31	−2.34389
$852$	6.80188e29	0.115615
$853$	−5.85669e30	−0.983302	−0.491651	−	0.870792i	$-0.663606\pi$
$853$	−5.85669e30	−0.983302	−0.491651	+	0.870792i	$0.663606\pi$
$854$	2.31437e30	0.383817
$855$	0	0
$856$	3.81891e30	0.617966
$857$	2.04604e30	0.327051	0.163526	−	0.986539i	$-0.447713\pi$
$857$	2.04604e30	0.327051	0.163526	+	0.986539i	$0.447713\pi$
$858$	1.27774e30	0.201756
$859$	2.53649e30	0.395644	0.197822	−	0.980238i	$-0.436613\pi$
$859$	2.53649e30	0.395644	0.197822	+	0.980238i	$0.436613\pi$
$860$	0	0
$861$	1.16942e30	0.178008
$862$	−4.11286e30	−0.618469
$863$	5.67025e30	0.842343	0.421172	−	0.906981i	$-0.361619\pi$
$863$	5.67025e30	0.842343	0.421172	+	0.906981i	$0.361619\pi$
$864$	−1.29605e30	−0.190207
$865$	0	0
$866$	−5.86624e29	−0.0840275
$867$	7.55266e30	1.06881
$868$	2.56531e30	0.358659
$869$	5.55275e29	0.0767009
$870$	0	0
$871$	1.42555e31	1.92218
$872$	5.51026e30	0.734090
$873$	−1.88772e30	−0.248478
$874$	6.84702e30	0.890495
$875$	0	0
$876$	−1.59233e30	−0.202181
$877$	−5.52103e30	−0.692667	−0.346333	−	0.938112i	$-0.612573\pi$
$877$	−5.52103e30	−0.692667	−0.346333	+	0.938112i	$0.612573\pi$
$878$	−5.93186e30	−0.735358
$879$	2.07306e30	0.253939
$880$	0	0
$881$	−7.48461e30	−0.895204	−0.447602	−	0.894233i	$-0.647722\pi$
$881$	−7.48461e30	−0.895204	−0.447602	+	0.894233i	$0.647722\pi$
$882$	9.05176e29	0.106983
$883$	3.41506e30	0.398851	0.199426	−	0.979913i	$-0.436092\pi$
$883$	3.41506e30	0.398851	0.199426	+	0.979913i	$0.436092\pi$
$884$	−2.22104e31	−2.56335
$885$	0	0
$886$	1.79967e30	0.202833
$887$	3.07207e30	0.342164	0.171082	−	0.985257i	$-0.445274\pi$
$887$	3.07207e30	0.342164	0.171082	+	0.985257i	$0.445274\pi$
$888$	5.78267e30	0.636492
$889$	2.23596e30	0.243217
$890$	0	0
$891$	−4.53640e29	−0.0481942
$892$	−6.30996e30	−0.662515
$893$	−1.91353e31	−1.98562
$894$	8.93394e28	0.00916218
$895$	0	0
$896$	−5.10484e30	−0.511385
$897$	−1.81146e31	−1.79353
$898$	4.46593e30	0.437029
$899$	9.35999e30	0.905313
$900$	0	0
$901$	−1.63135e31	−1.54147
$902$	−1.22031e30	−0.113973
$903$	4.70865e30	0.434685
$904$	−4.07453e30	−0.371800
$905$	0	0
$906$	−2.25727e30	−0.201251
$907$	1.67239e31	1.47388	0.736940	−	0.675958i	$-0.236270\pi$
$907$	1.67239e31	1.47388	0.736940	+	0.675958i	$0.236270\pi$
$908$	−8.10227e29	−0.0705839
$909$	1.58337e30	0.136352
$910$	0	0
$911$	−2.31676e31	−1.94957	−0.974783	−	0.223153i	$-0.928365\pi$
$911$	−2.31676e31	−1.94957	−0.974783	+	0.223153i	$0.928365\pi$
$912$	4.08806e30	0.340072
$913$	−2.35372e30	−0.193558
$914$	−3.04099e29	−0.0247218
$915$	0	0
$916$	8.53242e29	0.0677906
$917$	7.64378e30	0.600385
$918$	−1.80691e30	−0.140310
$919$	9.52890e29	0.0731526	0.0365763	−	0.999331i	$-0.488355\pi$
$919$	9.52890e29	0.0731526	0.0365763	+	0.999331i	$0.488355\pi$
$920$	0	0
$921$	−6.10215e30	−0.457885
$922$	−1.35588e29	−0.0100588
$923$	−6.26168e30	−0.459275
$924$	−1.42337e30	−0.103220
$925$	0	0
$926$	8.07746e30	0.572610
$927$	−4.65878e30	−0.326539
$928$	−1.48377e31	−1.02828
$929$	2.43263e30	0.166691	0.0833453	−	0.996521i	$-0.473440\pi$
$929$	2.43263e30	0.166691	0.0833453	+	0.996521i	$0.473440\pi$
$930$	0	0
$931$	−1.37454e31	−0.920841
$932$	−3.15360e30	−0.208900
$933$	−3.21908e30	−0.210850
$934$	−1.28947e31	−0.835159
$935$	0	0
$936$	7.69059e30	0.487039
$937$	−1.14297e31	−0.715763	−0.357881	−	0.933767i	$-0.616501\pi$
$937$	−1.14297e31	−0.715763	−0.357881	+	0.933767i	$0.616501\pi$
$938$	3.63890e30	0.225341
$939$	3.64787e30	0.223384
$940$	0	0
$941$	7.76249e30	0.464847	0.232423	−	0.972615i	$-0.425334\pi$
$941$	7.76249e30	0.464847	0.232423	+	0.972615i	$0.425334\pi$
$942$	7.32115e30	0.433556
$943$	1.73004e31	1.01317
$944$	8.74049e30	0.506209
$945$	0	0
$946$	−4.91354e30	−0.278315
$947$	−3.25786e31	−1.82498	−0.912488	−	0.409103i	$-0.865842\pi$
$947$	−3.25786e31	−1.82498	−0.912488	+	0.409103i	$0.865842\pi$
$948$	1.49929e30	0.0830613
$949$	1.46587e31	0.803156
$950$	0	0
$951$	8.21280e30	0.440145
$952$	−1.26381e31	−0.669876
$953$	−3.62605e29	−0.0190090	−0.00950448	−	0.999955i	$-0.503025\pi$
$953$	−3.62605e29	−0.0190090	−0.00950448	+	0.999955i	$0.503025\pi$
$954$	2.53402e30	0.131387
$955$	0	0
$956$	−1.16028e31	−0.588510
$957$	−5.19344e30	−0.260543
$958$	−2.29124e30	−0.113693
$959$	−8.46205e30	−0.415318
$960$	0	0
$961$	−5.05711e30	−0.242833
$962$	−2.38810e31	−1.13426
$963$	5.59926e30	0.263060
$964$	−1.97756e31	−0.919010
$965$	0	0
$966$	−4.62398e30	−0.210260
$967$	−2.60517e30	−0.117181	−0.0585906	−	0.998282i	$-0.518661\pi$
$967$	−2.60517e30	−0.117181	−0.0585906	+	0.998282i	$0.518661\pi$
$968$	−1.42877e31	−0.635728
$969$	2.74384e31	1.20770
$970$	0	0
$971$	−3.70602e31	−1.59627	−0.798135	−	0.602479i	$-0.794180\pi$
$971$	−3.70602e31	−1.59627	−0.798135	+	0.602479i	$0.794180\pi$
$972$	−1.22487e30	−0.0521908
$973$	2.81845e30	0.118802
$974$	−4.25084e30	−0.177258
$975$	0	0
$976$	2.04424e31	0.834275
$977$	1.56254e30	0.0630866	0.0315433	−	0.999502i	$-0.489958\pi$
$977$	1.56254e30	0.0630866	0.0315433	+	0.999502i	$0.489958\pi$
$978$	−3.22422e30	−0.128785
$979$	−4.55646e30	−0.180057
$980$	0	0
$981$	8.07911e30	0.312492
$982$	1.14929e31	0.439804
$983$	1.61908e31	0.612996	0.306498	−	0.951871i	$-0.400843\pi$
$983$	1.61908e31	0.612996	0.306498	+	0.951871i	$0.400843\pi$
$984$	−7.34491e30	−0.275130
$985$	0	0
$986$	−2.06862e31	−0.758530
$987$	1.29226e31	0.468835
$988$	−5.23895e31	−1.88060
$989$	6.96595e31	2.47411
$990$	0	0
$991$	−1.36425e31	−0.474374	−0.237187	−	0.971464i	$-0.576225\pi$
$991$	−1.36425e31	−0.474374	−0.237187	+	0.971464i	$0.576225\pi$
$992$	−2.49964e31	−0.860013
$993$	5.29462e30	0.180247
$994$	−1.59837e30	−0.0538419
$995$	0	0
$996$	−6.35526e30	−0.209609
$997$	1.27367e31	0.415677	0.207839	−	0.978163i	$-0.433357\pi$
$997$	1.27367e31	0.415677	0.207839	+	0.978163i	$0.433357\pi$
$998$	−4.39661e30	−0.141987
$999$	8.47852e30	0.270946

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.h.1.2		4
5.2	odd	4		75.22.b.h.49.4		8
5.3	odd	4		75.22.b.h.49.5		8
5.4	even	2		15.22.a.e.1.3	✓	4
15.14	odd	2		45.22.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.e.1.3	✓	4	5.4	even	2
45.22.a.g.1.2		4	15.14	odd	2
75.22.a.h.1.2		4	1.1	even	1	trivial
75.22.b.h.49.4		8	5.2	odd	4
75.22.b.h.49.5		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-625.272 q^{2} +59049.0 q^{3} -1.70619e6 q^{4} -3.69217e7 q^{6} -3.78633e8 q^{7} +2.37812e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-625.272 q^{2} +59049.0 q^{3} -1.70619e6 q^{4} -3.69217e7 q^{6} -3.78633e8 q^{7} +2.37812e9 q^{8} +3.48678e9 q^{9} -3.73130e10 q^{11} -1.00749e11 q^{12} +9.27472e11 q^{13} +2.36749e11 q^{14} +2.09116e12 q^{16} +1.40356e13 q^{17} -2.18019e12 q^{18} +3.31068e13 q^{19} -2.23579e13 q^{21} +2.33308e13 q^{22} -3.30762e14 q^{23} +1.40426e14 q^{24} -5.79922e14 q^{26} +2.05891e14 q^{27} +6.46019e14 q^{28} +2.35712e15 q^{29} +3.97094e15 q^{31} -6.29483e15 q^{32} -2.20330e15 q^{33} -8.77604e15 q^{34} -5.94911e15 q^{36} +4.11796e16 q^{37} -2.07007e16 q^{38} +5.47663e16 q^{39} -5.23046e16 q^{41} +1.39798e16 q^{42} -2.10603e17 q^{43} +6.36630e16 q^{44} +2.06816e17 q^{46} -5.77988e17 q^{47} +1.23481e17 q^{48} -4.15183e17 q^{49} +8.28786e17 q^{51} -1.58244e18 q^{52} -1.16230e18 q^{53} -1.28738e17 q^{54} -9.00436e17 q^{56} +1.95492e18 q^{57} -1.47384e18 q^{58} +4.17973e18 q^{59} +9.77562e18 q^{61} -2.48292e18 q^{62} -1.32021e18 q^{63} -4.49508e17 q^{64} +1.37766e18 q^{66} +1.53703e19 q^{67} -2.39473e19 q^{68} -1.95312e19 q^{69} -6.75134e18 q^{71} +8.29199e18 q^{72} +1.58050e19 q^{73} -2.57485e19 q^{74} -5.64864e19 q^{76} +1.41280e19 q^{77} -3.42438e19 q^{78} -1.48815e19 q^{79} +1.21577e19 q^{81} +3.27046e19 q^{82} +6.30804e19 q^{83} +3.81468e19 q^{84} +1.31684e20 q^{86} +1.39186e20 q^{87} -8.87349e19 q^{88} +1.22115e20 q^{89} -3.51172e20 q^{91} +5.64342e20 q^{92} +2.34480e20 q^{93} +3.61400e20 q^{94} -3.71703e20 q^{96} -5.41393e20 q^{97} +2.59602e20 q^{98} -1.30103e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9}+O(q^{10}) \) 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9	\( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} + 3127645607697 q^{18} - 24270353300752 q^{19} + 13851567033696 q^{21} + 56303932793676 q^{22} - 10350924920928 q^{23} - 4538248036821 q^{24} + 474751622871378 q^{26} + 823564528378596 q^{27} + 18\!\cdots\!68 q^{28}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100}) \) 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 + 31491830256 * q^11 - 68681250027 * q^12 + 27017977768 * q^13 - 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 + 3127645607697 * q^18 - 24270353300752 * q^19 + 13851567033696 * q^21 + 56303932793676 * q^22 - 10350924920928 * q^23 - 4538248036821 * q^24 + 474751622871378 * q^26 + 823564528378596 * q^27 + 1868468795909968 * q^28 + 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 1859561084786544 * q^33 + 3862795623387294 * q^34 - 4055559132844323 * q^36 - 4813435696247096 * q^37 - 21307487199674508 * q^38 + 1595384569222632 * q^39 + 289731591445930344 * q^41 - 131874611571854928 * q^42 - 451091912658458000 * q^43 + 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 - 668711106779001975 * q^48 + 1545180753004755300 * q^49 + 173963965360807512 * q^51 - 884645329958480822 * q^52 - 697278335404085208 * q^53 + 184684345488900153 * q^54 + 5673275878853478000 * q^56 - 1433140092056104848 * q^57 - 12277858736996428626 * q^58 + 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 817921181772715104 * q^63 + 21525702776885072473 * q^64 + 3324690927533774124 * q^66 + 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 611211765655877472 * q^69 - 37390337803999713312 * q^71 - 267979008326243229 * q^72 + 37253672904265201432 * q^73 - 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 + 28033608578931999522 * q^78 - 109751291042259570400 * q^79 + 48630661836227715204 * q^81 + 210801413757052759338 * q^82 + 149023592088638482896 * q^83 + 110331213929687700432 * q^84 - 346200583098972939204 * q^86 + 279238273256843539704 * q^87 + 116461091469581805156 * q^88 + 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 - 64654161985536495264 * q^93 + 1096939878232058257512 * q^94 - 468455064452928935829 * q^96 + 236244923748673777528 * q^97 - 197058383612642903463 * q^98 + 109805222495560636656 * q^99

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