LMFDB - Embedded newform 75.20.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,20,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, 20, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 20 $
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:	$171.612522417$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{87481}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 21870 $ x^2 - x - 21870
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-147.386$ 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)

$f(q)$	$=$	$q+536.316 q^{2} +19683.0 q^{3} -236654. q^{4} +1.05563e7 q^{6} -1.13677e8 q^{7} -4.08105e8 q^{8} +3.87420e8 q^{9} +O(q^{10})$	\(q+536.316 q^{2} +19683.0 q^{3} -236654. q^{4} +1.05563e7 q^{6} -1.13677e8 q^{7} -4.08105e8 q^{8} +3.87420e8 q^{9} +6.43152e9 q^{11} -4.65805e9 q^{12} +5.75132e10 q^{13} -6.09670e10 q^{14} -9.47984e10 q^{16} -6.85083e11 q^{17} +2.07780e11 q^{18} +1.22275e12 q^{19} -2.23751e12 q^{21} +3.44932e12 q^{22} -5.02230e12 q^{23} -8.03273e12 q^{24} +3.08453e13 q^{26} +7.62560e12 q^{27} +2.69022e13 q^{28} +1.53748e14 q^{29} +3.92872e13 q^{31} +1.63123e14 q^{32} +1.26592e14 q^{33} -3.67421e14 q^{34} -9.16844e13 q^{36} -1.35709e15 q^{37} +6.55779e14 q^{38} +1.13203e15 q^{39} -5.75410e14 q^{41} -1.20001e15 q^{42} -3.36667e14 q^{43} -1.52204e15 q^{44} -2.69354e15 q^{46} +6.99998e15 q^{47} -1.86592e15 q^{48} +1.52367e15 q^{49} -1.34845e16 q^{51} -1.36107e16 q^{52} -1.69895e16 q^{53} +4.08973e15 q^{54} +4.63923e16 q^{56} +2.40674e16 q^{57} +8.24573e16 q^{58} -2.70689e16 q^{59} -5.11533e16 q^{61} +2.10704e16 q^{62} -4.40410e16 q^{63} +1.37187e17 q^{64} +6.78930e16 q^{66} -2.07437e17 q^{67} +1.62127e17 q^{68} -9.88540e16 q^{69} -2.35123e17 q^{71} -1.58108e17 q^{72} -7.43858e16 q^{73} -7.27828e17 q^{74} -2.89368e17 q^{76} -7.31118e17 q^{77} +6.07127e17 q^{78} -6.98697e17 q^{79} +1.50095e17 q^{81} -3.08601e17 q^{82} -3.25532e18 q^{83} +5.29515e17 q^{84} -1.80560e17 q^{86} +3.02622e18 q^{87} -2.62473e18 q^{88} -1.43575e18 q^{89} -6.53796e18 q^{91} +1.18855e18 q^{92} +7.73291e17 q^{93} +3.75420e18 q^{94} +3.21074e18 q^{96} -1.66911e18 q^{97} +8.17166e17 q^{98} +2.49170e18 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 702 q^{2} + 39366 q^{3} + 772484 q^{4} - 13817466 q^{6} - 113892064 q^{7} - 1008501624 q^{8} + 774840978 q^{9}+O(q^{10}) $ 2 * q - 702 * q^2 + 39366 * q^3 + 772484 * q^4 - 13817466 * q^6 - 113892064 * q^7 - 1008501624 * q^8 + 774840978 * q^9	$ 2 q - 702 q^{2} + 39366 q^{3} + 772484 q^{4} - 13817466 q^{6} - 113892064 q^{7} - 1008501624 q^{8} + 774840978 q^{9} - 6650071272 q^{11} + 15204802572 q^{12} + 44072356148 q^{13} - 60701219424 q^{14} + 119603620880 q^{16} - 336281471748 q^{17} - 271969183278 q^{18} + 602118925096 q^{19} - 2241737495712 q^{21} + 19648459326744 q^{22} - 2368252165968 q^{23} - 19850337465192 q^{24} + 47489309789364 q^{26} + 15251194969974 q^{27} + 26685589805888 q^{28} + 280977251970492 q^{29} + 41610149253712 q^{31} + 212406109003296 q^{32} - 130893352846776 q^{33} - 799347349171332 q^{34} + 299276129024676 q^{36} - 637994163989884 q^{37} + 14\!\cdots\!32 q^{38}+ \cdots - 25\!\cdots\!08 q^{99}+O(q^{100}) $ 2 * q - 702 * q^2 + 39366 * q^3 + 772484 * q^4 - 13817466 * q^6 - 113892064 * q^7 - 1008501624 * q^8 + 774840978 * q^9 - 6650071272 * q^11 + 15204802572 * q^12 + 44072356148 * q^13 - 60701219424 * q^14 + 119603620880 * q^16 - 336281471748 * q^17 - 271969183278 * q^18 + 602118925096 * q^19 - 2241737495712 * q^21 + 19648459326744 * q^22 - 2368252165968 * q^23 - 19850337465192 * q^24 + 47489309789364 * q^26 + 15251194969974 * q^27 + 26685589805888 * q^28 + 280977251970492 * q^29 + 41610149253712 * q^31 + 212406109003296 * q^32 - 130893352846776 * q^33 - 799347349171332 * q^34 + 299276129024676 * q^36 - 637994163989884 * q^37 + 1424314643816232 * q^38 + 867476186061084 * q^39 + 1150038430280532 * q^41 - 1194782101922592 * q^42 - 2820518953538200 * q^43 - 14723164414050768 * q^44 - 5980094010800208 * q^46 + 5603091319500480 * q^47 + 2354158069781040 * q^48 - 9875183230193070 * q^49 - 6619028208415884 * q^51 - 27174417188397208 * q^52 - 27537868857647628 * q^53 - 5353169434460874 * q^54 + 46521172561587840 * q^56 + 11851506802664568 * q^57 - 75092921304821316 * q^58 - 55153474421835816 * q^59 - 87192569066205620 * q^61 + 18193883108252112 * q^62 - 44124119128099296 * q^63 - 36250046218631104 * q^64 + 386740624928302152 * q^66 - 499299710877878248 * q^67 + 514116238109034744 * q^68 - 46614307382748144 * q^69 + 84915048436970544 * q^71 - 390714192327374136 * q^72 + 241433711161728332 * q^73 - 1618295867440077564 * q^74 - 915668569473570224 * q^76 - 728310910203957888 * q^77 + 934732084584051612 * q^78 + 1131591466033523440 * q^79 + 300189270593998242 * q^81 - 2445251486788059948 * q^82 - 1607743276003583784 * q^83 + 525252464149293504 * q^84 + 2895233411994836328 * q^86 + 5530475250535194036 * q^87 + 5229410429487705696 * q^88 + 4322797626369576276 * q^89 - 6535073681312344768 * q^91 + 3866849612692780896 * q^92 + 819012567760813296 * q^93 + 5483985680095257024 * q^94 + 4180789443511875168 * q^96 - 7902201235299458308 * q^97 + 14932538668319232402 * q^98 - 2576373864083092008 * 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 $

$2$	536.316	0.740688	0.370344	−	0.928895i	$-0.379240\pi$
$2$	536.316	0.740688	0.370344	+	0.928895i	$0.379240\pi$
$3$	19683.0	0.577350
$3$	19683.0	0.577350
$4$	−236654.	−0.451381
$5$	0	0
$5$	0	0
$6$	1.05563e7	0.427637
$7$	−1.13677e8	−1.06474	−0.532369	−	0.846512i	$-0.678698\pi$
$7$	−1.13677e8	−1.06474	−0.532369	+	0.846512i	$0.678698\pi$
$8$	−4.08105e8	−1.07502
$9$	3.87420e8	0.333333
$10$	0	0
$11$	6.43152e9	0.822400	0.411200	−	0.911545i	$-0.365110\pi$
$11$	6.43152e9	0.822400	0.411200	+	0.911545i	$0.365110\pi$
$12$	−4.65805e9	−0.260605
$13$	5.75132e10	1.50420	0.752101	−	0.659048i	$-0.229041\pi$
$13$	5.75132e10	1.50420	0.752101	+	0.659048i	$0.229041\pi$
$14$	−6.09670e10	−0.788639
$15$	0	0
$16$	−9.47984e10	−0.344874
$17$	−6.85083e11	−1.40113	−0.700565	−	0.713588i	$-0.747069\pi$
$17$	−6.85083e11	−1.40113	−0.700565	+	0.713588i	$0.747069\pi$
$18$	2.07780e11	0.246896
$19$	1.22275e12	0.869317	0.434658	−	0.900595i	$-0.356869\pi$
$19$	1.22275e12	0.869317	0.434658	+	0.900595i	$0.356869\pi$
$20$	0	0
$21$	−2.23751e12	−0.614727
$22$	3.44932e12	0.609142
$23$	−5.02230e12	−0.581418	−0.290709	−	0.956811i	$-0.593891\pi$
$23$	−5.02230e12	−0.581418	−0.290709	+	0.956811i	$0.593891\pi$
$24$	−8.03273e12	−0.620664
$25$	0	0
$26$	3.08453e13	1.11414
$27$	7.62560e12	0.192450
$28$	2.69022e13	0.480603
$29$	1.53748e14	1.96801	0.984006	−	0.178134i	$-0.0570060\pi$
$29$	1.53748e14	1.96801	0.984006	+	0.178134i	$0.0570060\pi$
$30$	0	0
$31$	3.92872e13	0.266880	0.133440	−	0.991057i	$-0.457398\pi$
$31$	3.92872e13	0.266880	0.133440	+	0.991057i	$0.457398\pi$
$32$	1.63123e14	0.819576
$33$	1.26592e14	0.474813
$34$	−3.67421e14	−1.03780
$35$	0	0
$36$	−9.16844e13	−0.150460
$37$	−1.35709e15	−1.71669	−0.858346	−	0.513072i	$-0.828507\pi$
$37$	−1.35709e15	−1.71669	−0.858346	+	0.513072i	$0.828507\pi$
$38$	6.55779e14	0.643893
$39$	1.13203e15	0.868451
$40$	0	0
$41$	−5.75410e14	−0.274493	−0.137246	−	0.990537i	$-0.543825\pi$
$41$	−5.75410e14	−0.274493	−0.137246	+	0.990537i	$0.543825\pi$
$42$	−1.20001e15	−0.455321
$43$	−3.36667e14	−0.102153	−0.0510763	−	0.998695i	$-0.516265\pi$
$43$	−3.36667e14	−0.102153	−0.0510763	+	0.998695i	$0.516265\pi$
$44$	−1.52204e15	−0.371215
$45$	0	0
$46$	−2.69354e15	−0.430650
$47$	6.99998e15	0.912362	0.456181	−	0.889887i	$-0.349217\pi$
$47$	6.99998e15	0.912362	0.456181	+	0.889887i	$0.349217\pi$
$48$	−1.86592e15	−0.199113
$49$	1.52367e15	0.133668
$50$	0	0
$51$	−1.34845e16	−0.808943
$52$	−1.36107e16	−0.678968
$53$	−1.69895e16	−0.707231	−0.353615	−	0.935391i	$-0.615048\pi$
$53$	−1.69895e16	−0.707231	−0.353615	+	0.935391i	$0.615048\pi$
$54$	4.08973e15	0.142546
$55$	0	0
$56$	4.63923e16	1.14462
$57$	2.40674e16	0.501900
$58$	8.24573e16	1.45768
$59$	−2.70689e16	−0.406796	−0.203398	−	0.979096i	$-0.565198\pi$
$59$	−2.70689e16	−0.406796	−0.203398	+	0.979096i	$0.565198\pi$
$60$	0	0
$61$	−5.11533e16	−0.560068	−0.280034	−	0.959990i	$-0.590346\pi$
$61$	−5.11533e16	−0.560068	−0.280034	+	0.959990i	$0.590346\pi$
$62$	2.10704e16	0.197675
$63$	−4.40410e16	−0.354913
$64$	1.37187e17	0.951925
$65$	0	0
$66$	6.78930e16	0.351688
$67$	−2.07437e17	−0.931487	−0.465743	−	0.884920i	$-0.654213\pi$
$67$	−2.07437e17	−0.931487	−0.465743	+	0.884920i	$0.654213\pi$
$68$	1.62127e17	0.632443
$69$	−9.88540e16	−0.335682
$70$	0	0
$71$	−2.35123e17	−0.608613	−0.304306	−	0.952574i	$-0.598425\pi$
$71$	−2.35123e17	−0.608613	−0.304306	+	0.952574i	$0.598425\pi$
$72$	−1.58108e17	−0.358340
$73$	−7.43858e16	−0.147885	−0.0739423	−	0.997263i	$-0.523558\pi$
$73$	−7.43858e16	−0.147885	−0.0739423	+	0.997263i	$0.523558\pi$
$74$	−7.27828e17	−1.27153
$75$	0	0
$76$	−2.89368e17	−0.392393
$77$	−7.31118e17	−0.875641
$78$	6.07127e17	0.643252
$79$	−6.98697e17	−0.655890	−0.327945	−	0.944697i	$-0.606356\pi$
$79$	−6.98697e17	−0.655890	−0.327945	+	0.944697i	$0.606356\pi$
$80$	0	0
$81$	1.50095e17	0.111111
$82$	−3.08601e17	−0.203314
$83$	−3.25532e18	−1.91140	−0.955701	−	0.294338i	$-0.904901\pi$
$83$	−3.25532e18	−1.91140	−0.955701	+	0.294338i	$0.904901\pi$
$84$	5.29515e17	0.277476
$85$	0	0
$86$	−1.80560e17	−0.0756633
$87$	3.02622e18	1.13623
$88$	−2.62473e18	−0.884097
$89$	−1.43575e18	−0.434385	−0.217193	−	0.976129i	$-0.569690\pi$
$89$	−1.43575e18	−0.434385	−0.217193	+	0.976129i	$0.569690\pi$
$90$	0	0
$91$	−6.53796e18	−1.60158
$92$	1.18855e18	0.262441
$93$	7.73291e17	0.154083
$94$	3.75420e18	0.675776
$95$	0	0
$96$	3.21074e18	0.473183
$97$	−1.66911e18	−0.222922	−0.111461	−	0.993769i	$-0.535553\pi$
$97$	−1.66911e18	−0.222922	−0.111461	+	0.993769i	$0.535553\pi$
$98$	8.17166e17	0.0990062
$99$	2.49170e18	0.274133
$100$	0	0
$101$	9.99068e18	0.908955	0.454477	−	0.890758i	$-0.349826\pi$
$101$	9.99068e18	0.908955	0.454477	+	0.890758i	$0.349826\pi$
$102$	−7.23194e18	−0.599175
$103$	−1.17501e18	−0.0887333	−0.0443666	−	0.999015i	$-0.514127\pi$
$103$	−1.17501e18	−0.0887333	−0.0443666	+	0.999015i	$0.514127\pi$
$104$	−2.34714e19	−1.61705
$105$	0	0
$106$	−9.11176e18	−0.523837
$107$	1.73164e19	0.910568	0.455284	−	0.890346i	$-0.349538\pi$
$107$	1.73164e19	0.910568	0.455284	+	0.890346i	$0.349538\pi$
$108$	−1.80462e18	−0.0868683
$109$	1.57748e19	0.695685	0.347842	−	0.937553i	$-0.386914\pi$
$109$	1.57748e19	0.695685	0.347842	+	0.937553i	$0.386914\pi$
$110$	0	0
$111$	−2.67116e19	−0.991132
$112$	1.07764e19	0.367201
$113$	−4.78817e19	−1.49942	−0.749712	−	0.661764i	$-0.769808\pi$
$113$	−4.78817e19	−1.49942	−0.749712	+	0.661764i	$0.769808\pi$
$114$	1.29077e19	0.371752
$115$	0	0
$116$	−3.63850e19	−0.888323
$117$	2.22818e19	0.501400
$118$	−1.45175e19	−0.301309
$119$	7.78785e19	1.49184
$120$	0	0
$121$	−1.97947e19	−0.323659
$122$	−2.74343e19	−0.414836
$123$	−1.13258e19	−0.158478
$124$	−9.29747e18	−0.120464
$125$	0	0
$126$	−2.36199e19	−0.262880
$127$	1.25232e20	1.29294	0.646472	−	0.762938i	$-0.276244\pi$
$127$	1.25232e20	1.29294	0.646472	+	0.762938i	$0.276244\pi$
$128$	−1.19478e19	−0.114497
$129$	−6.62661e18	−0.0589779
$130$	0	0
$131$	−2.09329e20	−1.60972	−0.804862	−	0.593462i	$-0.797761\pi$
$131$	−2.09329e20	−1.60972	−0.804862	+	0.593462i	$0.797761\pi$
$132$	−2.99583e19	−0.214321
$133$	−1.38999e20	−0.925595
$134$	−1.11252e20	−0.689941
$135$	0	0
$136$	2.79586e20	1.50624
$137$	−1.53205e20	−0.769888	−0.384944	−	0.922940i	$-0.625779\pi$
$137$	−1.53205e20	−0.769888	−0.384944	+	0.922940i	$0.625779\pi$
$138$	−5.30169e19	−0.248636
$139$	−2.54374e19	−0.111386	−0.0556932	−	0.998448i	$-0.517737\pi$
$139$	−2.54374e19	−0.111386	−0.0556932	+	0.998448i	$0.517737\pi$
$140$	0	0
$141$	1.37781e20	0.526752
$142$	−1.26100e20	−0.450792
$143$	3.69897e20	1.23705
$144$	−3.67268e19	−0.114958
$145$	0	0
$146$	−3.98942e19	−0.109536
$147$	2.99903e19	0.0771732
$148$	3.21160e20	0.774882
$149$	1.67659e20	0.379453	0.189727	−	0.981837i	$-0.439240\pi$
$149$	1.67659e20	0.379453	0.189727	+	0.981837i	$0.439240\pi$
$150$	0	0
$151$	−4.29868e20	−0.857144	−0.428572	−	0.903508i	$-0.640983\pi$
$151$	−4.29868e20	−0.857144	−0.428572	+	0.903508i	$0.640983\pi$
$152$	−4.99010e20	−0.934533
$153$	−2.65415e20	−0.467043
$154$	−3.92110e20	−0.648577
$155$	0	0
$156$	−2.67900e20	−0.392002
$157$	−9.16306e20	−1.26181	−0.630905	−	0.775860i	$-0.717316\pi$
$157$	−9.16306e20	−1.26181	−0.630905	+	0.775860i	$0.717316\pi$
$158$	−3.74722e20	−0.485810
$159$	−3.34405e20	−0.408320
$160$	0	0
$161$	5.70923e20	0.619058
$162$	8.04981e19	0.0822987
$163$	−4.50070e20	−0.434008	−0.217004	−	0.976171i	$-0.569628\pi$
$163$	−4.50070e20	−0.434008	−0.217004	+	0.976171i	$0.569628\pi$
$164$	1.36173e20	0.123901
$165$	0	0
$166$	−1.74588e21	−1.41575
$167$	1.62414e20	0.124399	0.0621993	−	0.998064i	$-0.480189\pi$
$167$	1.62414e20	0.124399	0.0621993	+	0.998064i	$0.480189\pi$
$168$	9.13140e20	0.660844
$169$	1.84585e21	1.26262
$170$	0	0
$171$	4.73718e20	0.289772
$172$	7.96734e19	0.0461098
$173$	3.29538e21	1.80496	0.902481	−	0.430731i	$-0.141744\pi$
$173$	3.29538e21	1.80496	0.902481	+	0.430731i	$0.141744\pi$
$174$	1.62301e21	0.841594
$175$	0	0
$176$	−6.09697e20	−0.283625
$177$	−5.32797e20	−0.234864
$178$	−7.70017e20	−0.321744
$179$	−9.81720e20	−0.388941	−0.194471	−	0.980908i	$-0.562299\pi$
$179$	−9.81720e20	−0.388941	−0.194471	+	0.980908i	$0.562299\pi$
$180$	0	0
$181$	1.51701e20	0.0540807	0.0270403	−	0.999634i	$-0.491392\pi$
$181$	1.51701e20	0.0540807	0.0270403	+	0.999634i	$0.491392\pi$
$182$	−3.50641e21	−1.18627
$183$	−1.00685e21	−0.323355
$184$	2.04963e21	0.625037
$185$	0	0
$186$	4.14728e20	0.114128
$187$	−4.40612e21	−1.15229
$188$	−1.65657e21	−0.411823
$189$	−8.66858e20	−0.204909
$190$	0	0
$191$	−5.13044e21	−1.09733	−0.548666	−	0.836042i	$-0.684864\pi$
$191$	−5.13044e21	−1.09733	−0.548666	+	0.836042i	$0.684864\pi$
$192$	2.70025e21	0.549594
$193$	9.72873e21	1.88478	0.942392	−	0.334510i	$-0.108571\pi$
$193$	9.72873e21	1.88478	0.942392	+	0.334510i	$0.108571\pi$
$194$	−8.95169e20	−0.165116
$195$	0	0
$196$	−3.60581e20	−0.0603351
$197$	−7.68506e21	−1.22523	−0.612616	−	0.790381i	$-0.709883\pi$
$197$	−7.68506e21	−1.22523	−0.612616	+	0.790381i	$0.709883\pi$
$198$	1.33634e21	0.203047
$199$	4.55786e21	0.660172	0.330086	−	0.943951i	$-0.392922\pi$
$199$	4.55786e21	0.660172	0.330086	+	0.943951i	$0.392922\pi$
$200$	0	0
$201$	−4.08299e21	−0.537794
$202$	5.35816e21	0.673252
$203$	−1.74777e22	−2.09542
$204$	3.19115e21	0.365141
$205$	0	0
$206$	−6.30174e20	−0.0657237
$207$	−1.94574e21	−0.193806
$208$	−5.45216e21	−0.518761
$209$	7.86413e21	0.714926
$210$	0	0
$211$	−1.71334e22	−1.42285	−0.711426	−	0.702761i	$-0.751950\pi$
$211$	−1.71334e22	−1.42285	−0.711426	+	0.702761i	$0.751950\pi$
$212$	4.02064e21	0.319230
$213$	−4.62793e21	−0.351383
$214$	9.28707e21	0.674447
$215$	0	0
$216$	−3.11204e21	−0.206888
$217$	−4.46607e21	−0.284157
$218$	8.46028e21	0.515286
$219$	−1.46414e21	−0.0853812
$220$	0	0
$221$	−3.94014e22	−2.10758
$222$	−1.43258e22	−0.734120
$223$	−3.17895e22	−1.56095	−0.780473	−	0.625189i	$-0.785022\pi$
$223$	−3.17895e22	−1.56095	−0.780473	+	0.625189i	$0.785022\pi$
$224$	−1.85434e22	−0.872634
$225$	0	0
$226$	−2.56797e22	−1.11061
$227$	−2.55774e22	−1.06074	−0.530372	−	0.847765i	$-0.677948\pi$
$227$	−2.55774e22	−1.06074	−0.530372	+	0.847765i	$0.677948\pi$
$228$	−5.69563e21	−0.226548
$229$	−2.23445e22	−0.852576	−0.426288	−	0.904588i	$-0.640179\pi$
$229$	−2.23445e22	−0.852576	−0.426288	+	0.904588i	$0.640179\pi$
$230$	0	0
$231$	−1.43906e22	−0.505551
$232$	−6.27452e22	−2.11565
$233$	2.52369e21	0.0816874	0.0408437	−	0.999166i	$-0.486995\pi$
$233$	2.52369e21	0.0816874	0.0408437	+	0.999166i	$0.486995\pi$
$234$	1.19501e22	0.371381
$235$	0	0
$236$	6.40595e21	0.183620
$237$	−1.37524e22	−0.378678
$238$	4.17675e22	1.10499
$239$	3.93020e22	0.999158	0.499579	−	0.866268i	$-0.333488\pi$
$239$	3.93020e22	0.999158	0.499579	+	0.866268i	$0.333488\pi$
$240$	0	0
$241$	−1.99943e22	−0.469618	−0.234809	−	0.972042i	$-0.575447\pi$
$241$	−1.99943e22	−0.469618	−0.234809	+	0.972042i	$0.575447\pi$
$242$	−1.06162e22	−0.239730
$243$	2.95431e21	0.0641500
$244$	1.21056e22	0.252804
$245$	0	0
$246$	−6.07420e21	−0.117383
$247$	7.03242e22	1.30763
$248$	−1.60333e22	−0.286901
$249$	−6.40745e22	−1.10355
$250$	0	0
$251$	−4.94152e22	−0.788788	−0.394394	−	0.918942i	$-0.629045\pi$
$251$	−4.94152e22	−0.788788	−0.394394	+	0.918942i	$0.629045\pi$
$252$	1.04225e22	0.160201
$253$	−3.23010e22	−0.478158
$254$	6.71638e22	0.957668
$255$	0	0
$256$	−7.83332e22	−1.03673
$257$	−1.79354e22	−0.228742	−0.114371	−	0.993438i	$-0.536485\pi$
$257$	−1.79354e22	−0.228742	−0.114371	+	0.993438i	$0.536485\pi$
$258$	−3.55395e21	−0.0436842
$259$	1.54271e23	1.82783
$260$	0	0
$261$	5.95650e22	0.656004
$262$	−1.12266e23	−1.19230
$263$	−2.64384e22	−0.270804	−0.135402	−	0.990791i	$-0.543233\pi$
$263$	−2.64384e22	−0.270804	−0.135402	+	0.990791i	$0.543233\pi$
$264$	−5.16626e22	−0.510434
$265$	0	0
$266$	−7.45473e22	−0.685577
$267$	−2.82599e22	−0.250792
$268$	4.90908e22	0.420455
$269$	−6.26512e22	−0.517944	−0.258972	−	0.965885i	$-0.583384\pi$
$269$	−6.26512e22	−0.517944	−0.258972	+	0.965885i	$0.583384\pi$
$270$	0	0
$271$	−8.23488e22	−0.634525	−0.317263	−	0.948338i	$-0.602764\pi$
$271$	−8.23488e22	−0.634525	−0.317263	+	0.948338i	$0.602764\pi$
$272$	6.49448e22	0.483214
$273$	−1.28687e23	−0.924673
$274$	−8.21662e22	−0.570247
$275$	0	0
$276$	2.33942e22	0.151520
$277$	−1.42704e23	−0.893057	−0.446528	−	0.894769i	$-0.647340\pi$
$277$	−1.42704e23	−0.893057	−0.446528	+	0.894769i	$0.647340\pi$
$278$	−1.36425e22	−0.0825026
$279$	1.52207e22	0.0889599
$280$	0	0
$281$	4.36986e22	0.238648	0.119324	−	0.992855i	$-0.461927\pi$
$281$	4.36986e22	0.238648	0.119324	+	0.992855i	$0.461927\pi$
$282$	7.38939e22	0.390159
$283$	2.42461e23	1.23786	0.618929	−	0.785447i	$-0.287567\pi$
$283$	2.42461e23	1.23786	0.618929	+	0.785447i	$0.287567\pi$
$284$	5.56427e22	0.274716
$285$	0	0
$286$	1.98382e23	0.916272
$287$	6.54112e22	0.292263
$288$	6.31970e22	0.273192
$289$	2.30266e23	0.963166
$290$	0	0
$291$	−3.28531e22	−0.128704
$292$	1.76037e22	0.0667522
$293$	−1.65262e22	−0.0606638	−0.0303319	−	0.999540i	$-0.509656\pi$
$293$	−1.65262e22	−0.0606638	−0.0303319	+	0.999540i	$0.509656\pi$
$294$	1.60843e22	0.0571613
$295$	0	0
$296$	5.53835e23	1.84548
$297$	4.90442e22	0.158271
$298$	8.99184e22	0.281057
$299$	−2.88849e23	−0.874570
$300$	0	0
$301$	3.82714e22	0.108766
$302$	−2.30545e23	−0.634876
$303$	1.96647e23	0.524785
$304$	−1.15915e23	−0.299805
$305$	0	0
$306$	−1.42346e23	−0.345934
$307$	7.14354e23	1.68306	0.841528	−	0.540213i	$-0.181656\pi$
$307$	7.14354e23	1.68306	0.841528	+	0.540213i	$0.181656\pi$
$308$	1.73022e23	0.395247
$309$	−2.31276e22	−0.0512302
$310$	0	0
$311$	3.74882e23	0.781037	0.390518	−	0.920595i	$-0.372296\pi$
$311$	3.74882e23	0.781037	0.390518	+	0.920595i	$0.372296\pi$
$312$	−4.61988e23	−0.933603
$313$	2.04132e23	0.400166	0.200083	−	0.979779i	$-0.435879\pi$
$313$	2.04132e23	0.400166	0.200083	+	0.979779i	$0.435879\pi$
$314$	−4.91429e23	−0.934607
$315$	0	0
$316$	1.65349e23	0.296056
$317$	2.74437e23	0.476848	0.238424	−	0.971161i	$-0.423369\pi$
$317$	2.74437e23	0.476848	0.238424	+	0.971161i	$0.423369\pi$
$318$	−1.79347e23	−0.302438
$319$	9.88832e23	1.61849
$320$	0	0
$321$	3.40839e23	0.525717
$322$	3.06195e23	0.458529
$323$	−8.37684e23	−1.21803
$324$	−3.55204e22	−0.0501534
$325$	0	0
$326$	−2.41379e23	−0.321464
$327$	3.10496e23	0.401654
$328$	2.34828e23	0.295085
$329$	−7.95740e23	−0.971427
$330$	0	0
$331$	−2.31592e23	−0.266906	−0.133453	−	0.991055i	$-0.542607\pi$
$331$	−2.31592e23	−0.266906	−0.133453	+	0.991055i	$0.542607\pi$
$332$	7.70384e23	0.862771
$333$	−5.25764e23	−0.572230
$334$	8.71049e22	0.0921406
$335$	0	0
$336$	2.12113e23	0.212004
$337$	1.11237e23	0.108085	0.0540425	−	0.998539i	$-0.482789\pi$
$337$	1.11237e23	0.108085	0.0540425	+	0.998539i	$0.482789\pi$
$338$	9.89960e23	0.935209
$339$	−9.42456e23	−0.865693
$340$	0	0
$341$	2.52677e23	0.219482
$342$	2.54062e23	0.214631
$343$	1.12259e24	0.922417
$344$	1.37395e23	0.109816
$345$	0	0
$346$	1.76737e24	1.33691
$347$	−2.05609e24	−1.51326	−0.756629	−	0.653844i	$-0.773155\pi$
$347$	−2.05609e24	−1.51326	−0.756629	+	0.653844i	$0.773155\pi$
$348$	−7.16165e23	−0.512874
$349$	−1.75492e24	−1.22297	−0.611487	−	0.791255i	$-0.709428\pi$
$349$	−1.75492e24	−1.22297	−0.611487	+	0.791255i	$0.709428\pi$
$350$	0	0
$351$	4.38573e23	0.289484
$352$	1.04913e24	0.674019
$353$	−8.21430e23	−0.513702	−0.256851	−	0.966451i	$-0.582685\pi$
$353$	−8.21430e23	−0.513702	−0.256851	+	0.966451i	$0.582685\pi$
$354$	−2.85747e23	−0.173961
$355$	0	0
$356$	3.39776e23	0.196073
$357$	1.53288e24	0.861313
$358$	−5.26512e23	−0.288084
$359$	5.29683e23	0.282240	0.141120	−	0.989992i	$-0.454930\pi$
$359$	5.29683e23	0.282240	0.141120	+	0.989992i	$0.454930\pi$
$360$	0	0
$361$	−4.83305e23	−0.244289
$362$	8.13596e22	0.0400569
$363$	−3.89618e23	−0.186864
$364$	1.54723e24	0.722923
$365$	0	0
$366$	−5.39990e23	−0.239505
$367$	2.30180e24	0.994809	0.497404	−	0.867519i	$-0.334287\pi$
$367$	2.30180e24	0.994809	0.497404	+	0.867519i	$0.334287\pi$
$368$	4.76106e23	0.200516
$369$	−2.22926e23	−0.0914976
$370$	0	0
$371$	1.93133e24	0.753016
$372$	−1.83002e23	−0.0695501
$373$	−1.64953e24	−0.611120	−0.305560	−	0.952173i	$-0.598844\pi$
$373$	−1.64953e24	−0.611120	−0.305560	+	0.952173i	$0.598844\pi$
$374$	−2.36307e24	−0.853487
$375$	0	0
$376$	−2.85673e24	−0.980808
$377$	8.84253e24	2.96029
$378$	−4.64910e23	−0.151774
$379$	3.36454e24	1.07116	0.535579	−	0.844485i	$-0.320094\pi$
$379$	3.36454e24	1.07116	0.535579	+	0.844485i	$0.320094\pi$
$380$	0	0
$381$	2.46494e24	0.746482
$382$	−2.75154e24	−0.812780
$383$	−1.55219e24	−0.447257	−0.223628	−	0.974674i	$-0.571790\pi$
$383$	−1.55219e24	−0.447257	−0.223628	+	0.974674i	$0.571790\pi$
$384$	−2.35168e23	−0.0661046
$385$	0	0
$386$	5.21767e24	1.39604
$387$	−1.30432e23	−0.0340509
$388$	3.95000e23	0.100623
$389$	−1.19996e23	−0.0298295	−0.0149148	−	0.999889i	$-0.504748\pi$
$389$	−1.19996e23	−0.0298295	−0.0149148	+	0.999889i	$0.504748\pi$
$390$	0	0
$391$	3.44070e24	0.814643
$392$	−6.21815e23	−0.143696
$393$	−4.12022e24	−0.929375
$394$	−4.12162e24	−0.907515
$395$	0	0
$396$	−5.89670e23	−0.123738
$397$	1.23812e24	0.253661	0.126831	−	0.991924i	$-0.459520\pi$
$397$	1.23812e24	0.253661	0.126831	+	0.991924i	$0.459520\pi$
$398$	2.44445e24	0.488981
$399$	−2.73592e24	−0.534392
$400$	0	0
$401$	−6.55287e24	−1.22056	−0.610281	−	0.792185i	$-0.708943\pi$
$401$	−6.55287e24	−1.22056	−0.610281	+	0.792185i	$0.708943\pi$
$402$	−2.18977e24	−0.398338
$403$	2.25954e24	0.401441
$404$	−2.36433e24	−0.410285
$405$	0	0
$406$	−9.37354e24	−1.55205
$407$	−8.72815e24	−1.41181
$408$	5.50309e24	0.869631
$409$	8.56848e24	1.32292	0.661458	−	0.749982i	$-0.269938\pi$
$409$	8.56848e24	1.32292	0.661458	+	0.749982i	$0.269938\pi$
$410$	0	0
$411$	−3.01553e24	−0.444495
$412$	2.78069e23	0.0400525
$413$	3.07712e24	0.433131
$414$	−1.04353e24	−0.143550
$415$	0	0
$416$	9.38171e24	1.23281
$417$	−5.00685e23	−0.0643090
$418$	4.21766e24	0.529537
$419$	8.09987e24	0.994134	0.497067	−	0.867712i	$-0.334410\pi$
$419$	8.09987e24	0.994134	0.497067	+	0.867712i	$0.334410\pi$
$420$	0	0
$421$	9.99748e24	1.17276	0.586382	−	0.810035i	$-0.300552\pi$
$421$	9.99748e24	1.17276	0.586382	+	0.810035i	$0.300552\pi$
$422$	−9.18890e24	−1.05389
$423$	2.71194e24	0.304121
$424$	6.93352e24	0.760288
$425$	0	0
$426$	−2.48203e24	−0.260265
$427$	5.81498e24	0.596326
$428$	−4.09799e24	−0.411013
$429$	7.28069e24	0.714214
$430$	0	0
$431$	−4.59733e24	−0.431491	−0.215745	−	0.976450i	$-0.569218\pi$
$431$	−4.59733e24	−0.431491	−0.215745	+	0.976450i	$0.569218\pi$
$432$	−7.22894e23	−0.0663711
$433$	7.51468e24	0.674956	0.337478	−	0.941333i	$-0.390426\pi$
$433$	7.51468e24	0.674956	0.337478	+	0.941333i	$0.390426\pi$
$434$	−2.39523e24	−0.210472
$435$	0	0
$436$	−3.73317e24	−0.314019
$437$	−6.14102e24	−0.505436
$438$	−7.85238e23	−0.0632408
$439$	−2.04197e25	−1.60930	−0.804649	−	0.593751i	$-0.797647\pi$
$439$	−2.04197e25	−1.60930	−0.804649	+	0.593751i	$0.797647\pi$
$440$	0	0
$441$	5.90299e23	0.0445560
$442$	−2.11316e25	−1.56106
$443$	1.84340e25	1.33286	0.666428	−	0.745569i	$-0.267822\pi$
$443$	1.84340e25	1.33286	0.666428	+	0.745569i	$0.267822\pi$
$444$	6.32140e24	0.447378
$445$	0	0
$446$	−1.70492e25	−1.15617
$447$	3.30004e24	0.219077
$448$	−1.55951e25	−1.01355
$449$	7.82616e24	0.497976	0.248988	−	0.968507i	$-0.419902\pi$
$449$	7.82616e24	0.497976	0.248988	+	0.968507i	$0.419902\pi$
$450$	0	0
$451$	−3.70076e24	−0.225743
$452$	1.13314e25	0.676811
$453$	−8.46108e24	−0.494872
$454$	−1.37176e25	−0.785681
$455$	0	0
$456$	−9.82201e24	−0.539553
$457$	−2.15612e25	−1.16003	−0.580015	−	0.814606i	$-0.696953\pi$
$457$	−2.15612e25	−1.16003	−0.580015	+	0.814606i	$0.696953\pi$
$458$	−1.19837e25	−0.631493
$459$	−5.22417e24	−0.269648
$460$	0	0
$461$	4.52754e24	0.224235	0.112118	−	0.993695i	$-0.464237\pi$
$461$	4.52754e24	0.224235	0.112118	+	0.993695i	$0.464237\pi$
$462$	−7.71791e24	−0.374456
$463$	−1.12552e25	−0.534974	−0.267487	−	0.963562i	$-0.586193\pi$
$463$	−1.12552e25	−0.534974	−0.267487	+	0.963562i	$0.586193\pi$
$464$	−1.45750e25	−0.678717
$465$	0	0
$466$	1.35350e24	0.0605049
$467$	−1.73792e25	−0.761237	−0.380618	−	0.924732i	$-0.624289\pi$
$467$	−1.73792e25	−0.761237	−0.380618	+	0.924732i	$0.624289\pi$
$468$	−5.27307e24	−0.226323
$469$	2.35810e25	0.991790
$470$	0	0
$471$	−1.80356e25	−0.728506
$472$	1.10469e25	0.437314
$473$	−2.16528e24	−0.0840104
$474$	−7.37565e24	−0.280483
$475$	0	0
$476$	−1.84302e25	−0.673387
$477$	−6.58210e24	−0.235744
$478$	2.10783e25	0.740065
$479$	4.26798e25	1.46904	0.734522	−	0.678585i	$-0.237407\pi$
$479$	4.26798e25	1.46904	0.734522	+	0.678585i	$0.237407\pi$
$480$	0	0
$481$	−7.80506e25	−2.58225
$482$	−1.07233e25	−0.347841
$483$	1.12375e25	0.357413
$484$	4.68448e24	0.146093
$485$	0	0
$486$	1.58444e24	0.0475152
$487$	6.57071e24	0.193236	0.0966178	−	0.995322i	$-0.469198\pi$
$487$	6.57071e24	0.193236	0.0966178	+	0.995322i	$0.469198\pi$
$488$	2.08759e25	0.602085
$489$	−8.85872e24	−0.250574
$490$	0	0
$491$	5.24168e25	1.42625	0.713126	−	0.701035i	$-0.247279\pi$
$491$	5.24168e25	1.42625	0.713126	+	0.701035i	$0.247279\pi$
$492$	2.68029e24	0.0715342
$493$	−1.05330e26	−2.75744
$494$	3.77160e25	0.968544
$495$	0	0
$496$	−3.72437e24	−0.0920400
$497$	2.67282e25	0.648013
$498$	−3.43642e25	−0.817386
$499$	−2.55146e25	−0.595434	−0.297717	−	0.954654i	$-0.596225\pi$
$499$	−2.55146e25	−0.595434	−0.297717	+	0.954654i	$0.596225\pi$
$500$	0	0
$501$	3.19679e24	0.0718216
$502$	−2.65021e25	−0.584246
$503$	5.69466e25	1.23189	0.615945	−	0.787789i	$-0.288775\pi$
$503$	5.69466e25	1.23189	0.615945	+	0.787789i	$0.288775\pi$
$504$	1.79733e25	0.381539
$505$	0	0
$506$	−1.73236e25	−0.354166
$507$	3.63319e25	0.728975
$508$	−2.96366e25	−0.583610
$509$	−3.26878e25	−0.631781	−0.315891	−	0.948796i	$-0.602303\pi$
$509$	−3.26878e25	−0.631781	−0.315891	+	0.948796i	$0.602303\pi$
$510$	0	0
$511$	8.45598e24	0.157458
$512$	−3.57472e25	−0.653398
$513$	9.32419e24	0.167300
$514$	−9.61903e24	−0.169427
$515$	0	0
$516$	1.56821e24	0.0266215
$517$	4.50205e25	0.750326
$518$	8.27377e25	1.35385
$519$	6.48630e25	1.04209
$520$	0	0
$521$	−9.30404e25	−1.44116	−0.720581	−	0.693370i	$-0.756125\pi$
$521$	−9.30404e25	−1.44116	−0.720581	+	0.693370i	$0.756125\pi$
$522$	3.19457e25	0.485895
$523$	−4.88002e25	−0.728879	−0.364440	−	0.931227i	$-0.618739\pi$
$523$	−4.88002e25	−0.728879	−0.364440	+	0.931227i	$0.618739\pi$
$524$	4.95384e25	0.726599
$525$	0	0
$526$	−1.41793e25	−0.200582
$527$	−2.69150e25	−0.373933
$528$	−1.20007e25	−0.163751
$529$	−4.93919e25	−0.661953
$530$	0	0
$531$	−1.04870e25	−0.135599
$532$	3.28946e25	0.417796
$533$	−3.30937e25	−0.412892
$534$	−1.51562e25	−0.185759
$535$	0	0
$536$	8.46562e25	1.00137
$537$	−1.93232e25	−0.224555
$538$	−3.36008e25	−0.383635
$539$	9.79948e24	0.109928
$540$	0	0
$541$	8.45858e25	0.916059	0.458030	−	0.888937i	$-0.348555\pi$
$541$	8.45858e25	0.916059	0.458030	+	0.888937i	$0.348555\pi$
$542$	−4.41649e25	−0.469985
$543$	2.98593e24	0.0312235
$544$	−1.11753e26	−1.14833
$545$	0	0
$546$	−6.90167e25	−0.684895
$547$	2.08591e25	0.203430	0.101715	−	0.994814i	$-0.467567\pi$
$547$	2.08591e25	0.203430	0.101715	+	0.994814i	$0.467567\pi$
$548$	3.62565e25	0.347513
$549$	−1.98179e25	−0.186689
$550$	0	0
$551$	1.87995e26	1.71083
$552$	4.03428e25	0.360865
$553$	7.94260e25	0.698352
$554$	−7.65345e25	−0.661477
$555$	0	0
$556$	6.01985e24	0.0502777
$557$	1.19573e25	0.0981768	0.0490884	−	0.998794i	$-0.484368\pi$
$557$	1.19573e25	0.0981768	0.0490884	+	0.998794i	$0.484368\pi$
$558$	8.16309e24	0.0658916
$559$	−1.93628e25	−0.153658
$560$	0	0
$561$	−8.67257e25	−0.665274
$562$	2.34362e25	0.176764
$563$	1.51109e26	1.12063	0.560313	−	0.828281i	$-0.310681\pi$
$563$	1.51109e26	1.12063	0.560313	+	0.828281i	$0.310681\pi$
$564$	−3.26063e25	−0.237766
$565$	0	0
$566$	1.30036e26	0.916866
$567$	−1.70624e25	−0.118304
$568$	9.59549e25	0.654271
$569$	−1.31930e26	−0.884661	−0.442331	−	0.896852i	$-0.645848\pi$
$569$	−1.31930e26	−0.884661	−0.442331	+	0.896852i	$0.645848\pi$
$570$	0	0
$571$	2.65239e26	1.72026	0.860130	−	0.510076i	$-0.170383\pi$
$571$	2.65239e26	1.72026	0.860130	+	0.510076i	$0.170383\pi$
$572$	−8.75375e25	−0.558383
$573$	−1.00982e26	−0.633544
$574$	3.50810e25	0.216476
$575$	0	0
$576$	5.31490e25	0.317308
$577$	9.99355e25	0.586880	0.293440	−	0.955977i	$-0.405200\pi$
$577$	9.99355e25	0.586880	0.293440	+	0.955977i	$0.405200\pi$
$578$	1.23496e26	0.713406
$579$	1.91491e26	1.08818
$580$	0	0
$581$	3.70057e26	2.03514
$582$	−1.76196e25	−0.0953296
$583$	−1.09269e26	−0.581626
$584$	3.03572e25	0.158979
$585$	0	0
$586$	−8.86324e24	−0.0449330
$587$	1.90494e26	0.950209	0.475104	−	0.879930i	$-0.342410\pi$
$587$	1.90494e26	0.950209	0.475104	+	0.879930i	$0.342410\pi$
$588$	−7.09732e24	−0.0348345
$589$	4.80384e25	0.232003
$590$	0	0
$591$	−1.51265e26	−0.707388
$592$	1.28650e26	0.592043
$593$	2.37669e26	1.07635	0.538173	−	0.842834i	$-0.319115\pi$
$593$	2.37669e26	1.07635	0.538173	+	0.842834i	$0.319115\pi$
$594$	2.63032e25	0.117229
$595$	0	0
$596$	−3.96772e25	−0.171278
$597$	8.97124e25	0.381150
$598$	−1.54914e26	−0.647784
$599$	−3.08975e26	−1.27165	−0.635826	−	0.771832i	$-0.719341\pi$
$599$	−3.08975e26	−1.27165	−0.635826	+	0.771832i	$0.719341\pi$
$600$	0	0
$601$	−2.61533e26	−1.04284	−0.521421	−	0.853299i	$-0.674598\pi$
$601$	−2.61533e26	−1.04284	−0.521421	+	0.853299i	$0.674598\pi$
$602$	2.05256e25	0.0805616
$603$	−8.03655e25	−0.310496
$604$	1.01730e26	0.386898
$605$	0	0
$606$	1.05465e26	0.388702
$607$	−5.37620e25	−0.195067	−0.0975334	−	0.995232i	$-0.531095\pi$
$607$	−5.37620e25	−0.195067	−0.0975334	+	0.995232i	$0.531095\pi$
$608$	1.99458e26	0.712471
$609$	−3.44013e26	−1.20979
$610$	0	0
$611$	4.02591e26	1.37238
$612$	6.28115e25	0.210814
$613$	−3.16087e26	−1.04456	−0.522278	−	0.852775i	$-0.674918\pi$
$613$	−3.16087e26	−1.04456	−0.522278	+	0.852775i	$0.674918\pi$
$614$	3.83119e26	1.24662
$615$	0	0
$616$	2.98373e26	0.941332
$617$	−1.98562e26	−0.616862	−0.308431	−	0.951247i	$-0.599804\pi$
$617$	−1.98562e26	−0.616862	−0.308431	+	0.951247i	$0.599804\pi$
$618$	−1.24037e25	−0.0379456
$619$	−4.82992e26	−1.45505	−0.727527	−	0.686080i	$-0.759330\pi$
$619$	−4.82992e26	−1.45505	−0.727527	+	0.686080i	$0.759330\pi$
$620$	0	0
$621$	−3.82981e25	−0.111894
$622$	2.01055e26	0.578505
$623$	1.63213e26	0.462507
$624$	−1.07315e26	−0.299507
$625$	0	0
$626$	1.09479e26	0.296398
$627$	1.54790e26	0.412763
$628$	2.16847e26	0.569556
$629$	9.29719e26	2.40531
$630$	0	0
$631$	−6.27693e26	−1.57568	−0.787841	−	0.615879i	$-0.788801\pi$
$631$	−6.27693e26	−1.57568	−0.787841	+	0.615879i	$0.788801\pi$
$632$	2.85141e26	0.705096
$633$	−3.37236e26	−0.821484
$634$	1.47185e26	0.353196
$635$	0	0
$636$	7.91382e25	0.184308
$637$	8.76310e25	0.201063
$638$	5.30326e26	1.19880
$639$	−9.10915e25	−0.202871
$640$	0	0
$641$	2.15915e26	0.466800	0.233400	−	0.972381i	$-0.425015\pi$
$641$	2.15915e26	0.466800	0.233400	+	0.972381i	$0.425015\pi$
$642$	1.82797e26	0.389392
$643$	−4.68779e26	−0.983929	−0.491964	−	0.870615i	$-0.663721\pi$
$643$	−4.68779e26	−0.983929	−0.491964	+	0.870615i	$0.663721\pi$
$644$	−1.35111e26	−0.279431
$645$	0	0
$646$	−4.49263e26	−0.902177
$647$	2.10829e26	0.417196	0.208598	−	0.978001i	$-0.433110\pi$
$647$	2.10829e26	0.417196	0.208598	+	0.978001i	$0.433110\pi$
$648$	−6.12543e25	−0.119447
$649$	−1.74094e26	−0.334549
$650$	0	0
$651$	−8.79057e25	−0.164058
$652$	1.06511e26	0.195903
$653$	−2.88194e26	−0.522408	−0.261204	−	0.965284i	$-0.584120\pi$
$653$	−2.88194e26	−0.522408	−0.261204	+	0.965284i	$0.584120\pi$
$654$	1.66524e26	0.297500
$655$	0	0
$656$	5.45480e25	0.0946656
$657$	−2.88186e25	−0.0492948
$658$	−4.26768e26	−0.719524
$659$	−7.36091e26	−1.22326	−0.611632	−	0.791143i	$-0.709486\pi$
$659$	−7.36091e26	−1.22326	−0.611632	+	0.791143i	$0.709486\pi$
$660$	0	0
$661$	3.05440e25	0.0493187	0.0246594	−	0.999696i	$-0.492150\pi$
$661$	3.05440e25	0.0493187	0.0246594	+	0.999696i	$0.492150\pi$
$662$	−1.24207e26	−0.197694
$663$	−7.75537e26	−1.21681
$664$	1.32851e27	2.05480
$665$	0	0
$666$	−2.81976e26	−0.423844
$667$	−7.72168e26	−1.14424
$668$	−3.84357e25	−0.0561512
$669$	−6.25713e26	−0.901213
$670$	0	0
$671$	−3.28994e26	−0.460600
$672$	−3.64989e26	−0.503816
$673$	9.62645e26	1.31016	0.655078	−	0.755561i	$-0.272636\pi$
$673$	9.62645e26	1.31016	0.655078	+	0.755561i	$0.272636\pi$
$674$	5.96582e25	0.0800573
$675$	0	0
$676$	−4.36828e26	−0.569923
$677$	−1.18974e27	−1.53059	−0.765295	−	0.643680i	$-0.777407\pi$
$677$	−1.18974e27	−1.53059	−0.765295	+	0.643680i	$0.777407\pi$
$678$	−5.05454e26	−0.641208
$679$	1.89740e26	0.237354
$680$	0	0
$681$	−5.03440e26	−0.612421
$682$	1.35514e26	0.162568
$683$	1.51240e27	1.78925	0.894624	−	0.446819i	$-0.147443\pi$
$683$	1.51240e27	1.78925	0.894624	+	0.446819i	$0.147443\pi$
$684$	−1.12107e26	−0.130798
$685$	0	0
$686$	6.02063e26	0.683224
$687$	−4.39806e26	−0.492235
$688$	3.19155e25	0.0352299
$689$	−9.77124e26	−1.06382
$690$	0	0
$691$	−1.40247e27	−1.48543	−0.742716	−	0.669606i	$-0.766463\pi$
$691$	−1.40247e27	−1.48543	−0.742716	+	0.669606i	$0.766463\pi$
$692$	−7.79864e26	−0.814725
$693$	−2.83250e26	−0.291880
$694$	−1.10272e27	−1.12085
$695$	0	0
$696$	−1.23501e27	−1.22147
$697$	3.94204e26	0.384600
$698$	−9.41194e26	−0.905842
$699$	4.96738e25	0.0471622
$700$	0	0
$701$	6.64167e26	0.613700	0.306850	−	0.951758i	$-0.400725\pi$
$701$	6.64167e26	0.613700	0.306850	+	0.951758i	$0.400725\pi$
$702$	2.35213e26	0.214417
$703$	−1.65938e27	−1.49235
$704$	8.82320e26	0.782863
$705$	0	0
$706$	−4.40546e26	−0.380493
$707$	−1.13572e27	−0.967799
$708$	1.26088e26	0.106013
$709$	1.03445e27	0.858163	0.429081	−	0.903266i	$-0.358837\pi$
$709$	1.03445e27	0.858163	0.429081	+	0.903266i	$0.358837\pi$
$710$	0	0
$711$	−2.70689e26	−0.218630
$712$	5.85938e26	0.466973
$713$	−1.97313e26	−0.155169
$714$	8.22109e26	0.637964
$715$	0	0
$716$	2.32328e26	0.175561
$717$	7.73581e26	0.576864
$718$	2.84077e26	0.209052
$719$	−1.59383e26	−0.115749	−0.0578744	−	0.998324i	$-0.518432\pi$
$719$	−1.59383e26	−0.115749	−0.0578744	+	0.998324i	$0.518432\pi$
$720$	0	0
$721$	1.33572e26	0.0944777
$722$	−2.59204e26	−0.180942
$723$	−3.93548e26	−0.271134
$724$	−3.59006e25	−0.0244110
$725$	0	0
$726$	−2.08958e26	−0.138408
$727$	−1.66059e27	−1.08564	−0.542819	−	0.839850i	$-0.682643\pi$
$727$	−1.66059e27	−1.08564	−0.542819	+	0.839850i	$0.682643\pi$
$728$	2.66817e27	1.72173
$729$	5.81497e25	0.0370370
$730$	0	0
$731$	2.30645e26	0.143129
$732$	2.38275e26	0.145956
$733$	1.36074e27	0.822786	0.411393	−	0.911458i	$-0.365042\pi$
$733$	1.36074e27	0.822786	0.411393	+	0.911458i	$0.365042\pi$
$734$	1.23449e27	0.736843
$735$	0	0
$736$	−8.19251e26	−0.476517
$737$	−1.33414e27	−0.766054
$738$	−1.19559e26	−0.0677712
$739$	9.59784e26	0.537095	0.268548	−	0.963266i	$-0.413456\pi$
$739$	9.59784e26	0.537095	0.268548	+	0.963266i	$0.413456\pi$
$740$	0	0
$741$	1.38419e27	0.754959
$742$	1.03580e27	0.557750
$743$	1.18287e27	0.628846	0.314423	−	0.949283i	$-0.398189\pi$
$743$	1.18287e27	0.628846	0.314423	+	0.949283i	$0.398189\pi$
$744$	−3.15584e26	−0.165643
$745$	0	0
$746$	−8.84670e26	−0.452650
$747$	−1.26118e27	−0.637134
$748$	1.04273e27	0.520121
$749$	−1.96849e27	−0.969516
$750$	0	0
$751$	−1.02382e27	−0.491637	−0.245819	−	0.969316i	$-0.579057\pi$
$751$	−1.02382e27	−0.491637	−0.245819	+	0.969316i	$0.579057\pi$
$752$	−6.63587e26	−0.314650
$753$	−9.72639e26	−0.455407
$754$	4.74239e27	2.19265
$755$	0	0
$756$	2.05145e26	0.0924920
$757$	−2.74415e25	−0.0122179	−0.00610897	−	0.999981i	$-0.501945\pi$
$757$	−2.74415e25	−0.0122179	−0.00610897	+	0.999981i	$0.501945\pi$
$758$	1.80446e27	0.793394
$759$	−6.35781e26	−0.276065
$760$	0	0
$761$	1.89240e27	0.801416	0.400708	−	0.916206i	$-0.368764\pi$
$761$	1.89240e27	0.801416	0.400708	+	0.916206i	$0.368764\pi$
$762$	1.32199e27	0.552910
$763$	−1.79324e27	−0.740722
$764$	1.21414e27	0.495314
$765$	0	0
$766$	−8.32465e26	−0.331278
$767$	−1.55682e27	−0.611902
$768$	−1.54183e27	−0.598557
$769$	−1.93865e27	−0.743359	−0.371679	−	0.928361i	$-0.621218\pi$
$769$	−1.93865e27	−0.743359	−0.371679	+	0.928361i	$0.621218\pi$
$770$	0	0
$771$	−3.53022e26	−0.132064
$772$	−2.30234e27	−0.850756
$773$	7.63732e24	0.00278764	0.00139382	−	0.999999i	$-0.499556\pi$
$773$	7.63732e24	0.00278764	0.00139382	+	0.999999i	$0.499556\pi$
$774$	−6.99525e25	−0.0252211
$775$	0	0
$776$	6.81171e26	0.239646
$777$	3.03651e27	1.05530
$778$	−6.43559e25	−0.0220944
$779$	−7.03582e26	−0.238621
$780$	0	0
$781$	−1.51220e27	−0.500523
$782$	1.84530e27	0.603396
$783$	1.17242e27	0.378744
$784$	−1.44441e26	−0.0460986
$785$	0	0
$786$	−2.20974e27	−0.688377
$787$	4.61341e27	1.41991	0.709956	−	0.704246i	$-0.248715\pi$
$787$	4.61341e27	1.41991	0.709956	+	0.704246i	$0.248715\pi$
$788$	1.81870e27	0.553046
$789$	−5.20387e26	−0.156349
$790$	0	0
$791$	5.44307e27	1.59649
$792$	−1.01688e27	−0.294699
$793$	−2.94199e27	−0.842455
$794$	6.64025e26	0.187884
$795$	0	0
$796$	−1.07863e27	−0.297989
$797$	−1.01219e27	−0.276316	−0.138158	−	0.990410i	$-0.544118\pi$
$797$	−1.01219e27	−0.276316	−0.138158	+	0.990410i	$0.544118\pi$
$798$	−1.46731e27	−0.395818
$799$	−4.79557e27	−1.27834
$800$	0	0
$801$	−5.56240e26	−0.144795
$802$	−3.51440e27	−0.904056
$803$	−4.78413e26	−0.121620
$804$	9.66255e26	0.242750
$805$	0	0
$806$	1.21183e27	0.297343
$807$	−1.23316e27	−0.299035
$808$	−4.07725e27	−0.977145
$809$	3.00853e27	0.712597	0.356298	−	0.934372i	$-0.384039\pi$
$809$	3.00853e27	0.712597	0.356298	+	0.934372i	$0.384039\pi$
$810$	0	0
$811$	3.54753e27	0.820782	0.410391	−	0.911910i	$-0.365392\pi$
$811$	3.54753e27	0.820782	0.410391	+	0.911910i	$0.365392\pi$
$812$	4.13615e27	0.945832
$813$	−1.62087e27	−0.366343
$814$	−4.68104e27	−1.04571
$815$	0	0
$816$	1.27831e27	0.278984
$817$	−4.11659e26	−0.0888030
$818$	4.59541e27	0.979869
$819$	−2.53294e27	−0.533860
$820$	0	0
$821$	−4.28797e27	−0.883063	−0.441532	−	0.897246i	$-0.645565\pi$
$821$	−4.28797e27	−0.883063	−0.441532	+	0.897246i	$0.645565\pi$
$822$	−1.61728e27	−0.329232
$823$	−3.20459e27	−0.644874	−0.322437	−	0.946591i	$-0.604502\pi$
$823$	−3.20459e27	−0.644874	−0.322437	+	0.946591i	$0.604502\pi$
$824$	4.79526e26	0.0953901
$825$	0	0
$826$	1.65031e27	0.320815
$827$	4.02192e27	0.772914	0.386457	−	0.922308i	$-0.373699\pi$
$827$	4.02192e27	0.772914	0.386457	+	0.922308i	$0.373699\pi$
$828$	4.60467e26	0.0874803
$829$	−1.58908e27	−0.298455	−0.149227	−	0.988803i	$-0.547679\pi$
$829$	−1.58908e27	−0.298455	−0.149227	+	0.988803i	$0.547679\pi$
$830$	0	0
$831$	−2.80885e27	−0.515607
$832$	7.89006e27	1.43189
$833$	−1.04384e27	−0.187286
$834$	−2.68525e26	−0.0476329
$835$	0	0
$836$	−1.86107e27	−0.322704
$837$	2.99589e26	0.0513610
$838$	4.34409e27	0.736343
$839$	−5.31813e27	−0.891293	−0.445646	−	0.895209i	$-0.647026\pi$
$839$	−5.31813e27	−0.891293	−0.445646	+	0.895209i	$0.647026\pi$
$840$	0	0
$841$	1.75351e28	2.87307
$842$	5.36180e27	0.868652
$843$	8.60120e26	0.137783
$844$	4.05467e27	0.642248
$845$	0	0
$846$	1.45445e27	0.225259
$847$	2.25021e27	0.344612
$848$	1.61058e27	0.243906
$849$	4.77236e27	0.714677
$850$	0	0
$851$	6.81572e27	0.998115
$852$	1.09522e27	0.158607
$853$	1.60722e27	0.230176	0.115088	−	0.993355i	$-0.463285\pi$
$853$	1.60722e27	0.230176	0.115088	+	0.993355i	$0.463285\pi$
$854$	3.11867e27	0.441691
$855$	0	0
$856$	−7.06692e27	−0.978879
$857$	−8.25782e27	−1.13122	−0.565611	−	0.824672i	$-0.691359\pi$
$857$	−8.25782e27	−1.13122	−0.565611	+	0.824672i	$0.691359\pi$
$858$	3.90475e27	0.529010
$859$	−9.31408e27	−1.24797	−0.623986	−	0.781436i	$-0.714488\pi$
$859$	−9.31408e27	−1.24797	−0.623986	+	0.781436i	$0.714488\pi$
$860$	0	0
$861$	1.28749e27	0.168738
$862$	−2.46562e27	−0.319600
$863$	−6.65457e27	−0.853134	−0.426567	−	0.904456i	$-0.640277\pi$
$863$	−6.65457e27	−0.853134	−0.426567	+	0.904456i	$0.640277\pi$
$864$	1.24391e27	0.157728
$865$	0	0
$866$	4.03024e27	0.499932
$867$	4.53234e27	0.556084
$868$	1.05691e27	0.128263
$869$	−4.49368e27	−0.539404
$870$	0	0
$871$	−1.19304e28	−1.40114
$872$	−6.43778e27	−0.747876
$873$	−6.46647e26	−0.0743074
$874$	−3.29352e27	−0.374371
$875$	0	0
$876$	3.46493e26	0.0385394
$877$	1.08073e27	0.118911	0.0594554	−	0.998231i	$-0.481064\pi$
$877$	1.08073e27	0.118911	0.0594554	+	0.998231i	$0.481064\pi$
$878$	−1.09514e28	−1.19199
$879$	−3.25285e26	−0.0350243
$880$	0	0
$881$	−1.38540e28	−1.45983	−0.729916	−	0.683536i	$-0.760441\pi$
$881$	−1.38540e28	−1.45983	−0.729916	+	0.683536i	$0.760441\pi$
$882$	3.16587e26	0.0330021
$883$	−6.76336e27	−0.697487	−0.348743	−	0.937218i	$-0.613392\pi$
$883$	−6.76336e27	−0.697487	−0.348743	+	0.937218i	$0.613392\pi$
$884$	9.32447e27	0.951322
$885$	0	0
$886$	9.88642e27	0.987231
$887$	1.60903e28	1.58960	0.794802	−	0.606869i	$-0.207575\pi$
$887$	1.60903e28	1.58960	0.794802	+	0.606869i	$0.207575\pi$
$888$	1.09011e28	1.06549
$889$	−1.42360e28	−1.37665
$890$	0	0
$891$	9.65336e26	0.0913777
$892$	7.52311e27	0.704581
$893$	8.55921e27	0.793131
$894$	1.76986e27	0.162268
$895$	0	0
$896$	1.35819e27	0.121909
$897$	−5.68541e27	−0.504933
$898$	4.19729e27	0.368845
$899$	6.04033e27	0.525223
$900$	0	0
$901$	1.16393e28	0.990922
$902$	−1.98478e27	−0.167205
$903$	7.53296e26	0.0627960
$904$	1.95408e28	1.61191
$905$	0	0
$906$	−4.53781e27	−0.366546
$907$	4.87731e26	0.0389862	0.0194931	−	0.999810i	$-0.493795\pi$
$907$	4.87731e26	0.0389862	0.0194931	+	0.999810i	$0.493795\pi$
$908$	6.05299e27	0.478800
$909$	3.87060e27	0.302985
$910$	0	0
$911$	2.04975e28	1.57136	0.785680	−	0.618634i	$-0.212313\pi$
$911$	2.04975e28	1.57136	0.785680	+	0.618634i	$0.212313\pi$
$912$	−2.28155e27	−0.173093
$913$	−2.09367e28	−1.57194
$914$	−1.15636e28	−0.859221
$915$	0	0
$916$	5.28790e27	0.384836
$917$	2.37960e28	1.71394
$918$	−2.80180e27	−0.199725
$919$	7.36209e27	0.519402	0.259701	−	0.965689i	$-0.416376\pi$
$919$	7.36209e27	0.519402	0.259701	+	0.965689i	$0.416376\pi$
$920$	0	0
$921$	1.40606e28	0.971713
$922$	2.42819e27	0.166088
$923$	−1.35227e28	−0.915476
$924$	3.40559e27	0.228196
$925$	0	0
$926$	−6.03632e27	−0.396249
$927$	−4.55222e26	−0.0295778
$928$	2.50797e28	1.61294
$929$	−9.18844e27	−0.584915	−0.292457	−	0.956279i	$-0.594473\pi$
$929$	−9.18844e27	−0.584915	−0.292457	+	0.956279i	$0.594473\pi$
$930$	0	0
$931$	1.86306e27	0.116200
$932$	−5.97241e26	−0.0368721
$933$	7.37881e27	0.450932
$934$	−9.32074e27	−0.563839
$935$	0	0
$936$	−9.09331e27	−0.539016
$937$	−1.55615e28	−0.913113	−0.456557	−	0.889694i	$-0.650917\pi$
$937$	−1.55615e28	−0.913113	−0.456557	+	0.889694i	$0.650917\pi$
$938$	1.26468e28	0.734607
$939$	4.01793e27	0.231036
$940$	0	0
$941$	−4.21264e27	−0.237385	−0.118693	−	0.992931i	$-0.537870\pi$
$941$	−4.21264e27	−0.237385	−0.118693	+	0.992931i	$0.537870\pi$
$942$	−9.67280e27	−0.539596
$943$	2.88989e27	0.159595
$944$	2.56609e27	0.140293
$945$	0	0
$946$	−1.16127e27	−0.0622255
$947$	−1.60906e27	−0.0853584	−0.0426792	−	0.999089i	$-0.513589\pi$
$947$	−1.60906e27	−0.0853584	−0.0426792	+	0.999089i	$0.513589\pi$
$948$	3.25456e27	0.170928
$949$	−4.27817e27	−0.222448
$950$	0	0
$951$	5.40175e27	0.275308
$952$	−3.17826e28	−1.60376
$953$	−2.00721e28	−1.00279	−0.501396	−	0.865218i	$-0.667180\pi$
$953$	−2.00721e28	−1.00279	−0.501396	+	0.865218i	$0.667180\pi$
$954$	−3.53008e27	−0.174612
$955$	0	0
$956$	−9.30095e27	−0.451001
$957$	1.94632e28	0.934437
$958$	2.28898e28	1.08810
$959$	1.74160e28	0.819729
$960$	0	0
$961$	−2.01272e28	−0.928775
$962$	−4.18598e28	−1.91264
$963$	6.70874e27	0.303523
$964$	4.73173e27	0.211977
$965$	0	0
$966$	6.02683e27	0.264732
$967$	−3.69606e28	−1.60764	−0.803818	−	0.594876i	$-0.797201\pi$
$967$	−3.69606e28	−1.60764	−0.803818	+	0.594876i	$0.797201\pi$
$968$	8.07830e27	0.347940
$969$	−1.64881e28	−0.703228
$970$	0	0
$971$	−2.54073e28	−1.06261	−0.531306	−	0.847180i	$-0.678299\pi$
$971$	−2.54073e28	−1.06261	−0.531306	+	0.847180i	$0.678299\pi$
$972$	−6.99149e26	−0.0289561
$973$	2.89166e27	0.118597
$974$	3.52397e27	0.143127
$975$	0	0
$976$	4.84925e27	0.193153
$977$	−8.52403e27	−0.336238	−0.168119	−	0.985767i	$-0.553769\pi$
$977$	−8.52403e27	−0.336238	−0.168119	+	0.985767i	$0.553769\pi$
$978$	−4.75107e27	−0.185598
$979$	−9.23408e27	−0.357238
$980$	0	0
$981$	6.11149e27	0.231895
$982$	2.81120e28	1.05641
$983$	−3.09592e28	−1.15221	−0.576105	−	0.817376i	$-0.695428\pi$
$983$	−3.09592e28	−1.15221	−0.576105	+	0.817376i	$0.695428\pi$
$984$	4.62211e27	0.170368
$985$	0	0
$986$	−5.64901e28	−2.04240
$987$	−1.56625e28	−0.560853
$988$	−1.66425e28	−0.590238
$989$	1.69084e27	0.0593934
$990$	0	0
$991$	−2.94856e28	−1.01604	−0.508020	−	0.861345i	$-0.669622\pi$
$991$	−2.94856e28	−1.01604	−0.508020	+	0.861345i	$0.669622\pi$
$992$	6.40864e27	0.218728
$993$	−4.55843e27	−0.154098
$994$	1.43347e28	0.479976
$995$	0	0
$996$	1.51635e28	0.498121
$997$	3.29038e28	1.07064	0.535318	−	0.844650i	$-0.320192\pi$
$997$	3.29038e28	1.07064	0.535318	+	0.844650i	$0.320192\pi$
$998$	−1.36839e28	−0.441031
$999$	−1.03486e28	−0.330377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.20.a.b.1.2		2
5.2	odd	4		75.20.b.b.49.3		4
5.3	odd	4		75.20.b.b.49.2		4
5.4	even	2		3.20.a.b.1.1	✓	2
15.14	odd	2		9.20.a.c.1.2		2
20.19	odd	2		48.20.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.20.a.b.1.1	✓	2	5.4	even	2
9.20.a.c.1.2		2	15.14	odd	2
48.20.a.j.1.1		2	20.19	odd	2
75.20.a.b.1.2		2	1.1	even	1	trivial
75.20.b.b.49.2		4	5.3	odd	4
75.20.b.b.49.3		4	5.2	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:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+536.316 q^{2} +19683.0 q^{3} -236654. q^{4} +1.05563e7 q^{6} -1.13677e8 q^{7} -4.08105e8 q^{8} +3.87420e8 q^{9} +O(q^{10})\)	\(q+536.316 q^{2} +19683.0 q^{3} -236654. q^{4} +1.05563e7 q^{6} -1.13677e8 q^{7} -4.08105e8 q^{8} +3.87420e8 q^{9} +6.43152e9 q^{11} -4.65805e9 q^{12} +5.75132e10 q^{13} -6.09670e10 q^{14} -9.47984e10 q^{16} -6.85083e11 q^{17} +2.07780e11 q^{18} +1.22275e12 q^{19} -2.23751e12 q^{21} +3.44932e12 q^{22} -5.02230e12 q^{23} -8.03273e12 q^{24} +3.08453e13 q^{26} +7.62560e12 q^{27} +2.69022e13 q^{28} +1.53748e14 q^{29} +3.92872e13 q^{31} +1.63123e14 q^{32} +1.26592e14 q^{33} -3.67421e14 q^{34} -9.16844e13 q^{36} -1.35709e15 q^{37} +6.55779e14 q^{38} +1.13203e15 q^{39} -5.75410e14 q^{41} -1.20001e15 q^{42} -3.36667e14 q^{43} -1.52204e15 q^{44} -2.69354e15 q^{46} +6.99998e15 q^{47} -1.86592e15 q^{48} +1.52367e15 q^{49} -1.34845e16 q^{51} -1.36107e16 q^{52} -1.69895e16 q^{53} +4.08973e15 q^{54} +4.63923e16 q^{56} +2.40674e16 q^{57} +8.24573e16 q^{58} -2.70689e16 q^{59} -5.11533e16 q^{61} +2.10704e16 q^{62} -4.40410e16 q^{63} +1.37187e17 q^{64} +6.78930e16 q^{66} -2.07437e17 q^{67} +1.62127e17 q^{68} -9.88540e16 q^{69} -2.35123e17 q^{71} -1.58108e17 q^{72} -7.43858e16 q^{73} -7.27828e17 q^{74} -2.89368e17 q^{76} -7.31118e17 q^{77} +6.07127e17 q^{78} -6.98697e17 q^{79} +1.50095e17 q^{81} -3.08601e17 q^{82} -3.25532e18 q^{83} +5.29515e17 q^{84} -1.80560e17 q^{86} +3.02622e18 q^{87} -2.62473e18 q^{88} -1.43575e18 q^{89} -6.53796e18 q^{91} +1.18855e18 q^{92} +7.73291e17 q^{93} +3.75420e18 q^{94} +3.21074e18 q^{96} -1.66911e18 q^{97} +8.17166e17 q^{98} +2.49170e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 702 q^{2} + 39366 q^{3} + 772484 q^{4} - 13817466 q^{6} - 113892064 q^{7} - 1008501624 q^{8} + 774840978 q^{9}+O(q^{10}) \) 2 * q - 702 * q^2 + 39366 * q^3 + 772484 * q^4 - 13817466 * q^6 - 113892064 * q^7 - 1008501624 * q^8 + 774840978 * q^9	\( 2 q - 702 q^{2} + 39366 q^{3} + 772484 q^{4} - 13817466 q^{6} - 113892064 q^{7} - 1008501624 q^{8} + 774840978 q^{9} - 6650071272 q^{11} + 15204802572 q^{12} + 44072356148 q^{13} - 60701219424 q^{14} + 119603620880 q^{16} - 336281471748 q^{17} - 271969183278 q^{18} + 602118925096 q^{19} - 2241737495712 q^{21} + 19648459326744 q^{22} - 2368252165968 q^{23} - 19850337465192 q^{24} + 47489309789364 q^{26} + 15251194969974 q^{27} + 26685589805888 q^{28} + 280977251970492 q^{29} + 41610149253712 q^{31} + 212406109003296 q^{32} - 130893352846776 q^{33} - 799347349171332 q^{34} + 299276129024676 q^{36} - 637994163989884 q^{37} + 14\!\cdots\!32 q^{38}+ \cdots - 25\!\cdots\!08 q^{99}+O(q^{100}) \) 2 * q - 702 * q^2 + 39366 * q^3 + 772484 * q^4 - 13817466 * q^6 - 113892064 * q^7 - 1008501624 * q^8 + 774840978 * q^9 - 6650071272 * q^11 + 15204802572 * q^12 + 44072356148 * q^13 - 60701219424 * q^14 + 119603620880 * q^16 - 336281471748 * q^17 - 271969183278 * q^18 + 602118925096 * q^19 - 2241737495712 * q^21 + 19648459326744 * q^22 - 2368252165968 * q^23 - 19850337465192 * q^24 + 47489309789364 * q^26 + 15251194969974 * q^27 + 26685589805888 * q^28 + 280977251970492 * q^29 + 41610149253712 * q^31 + 212406109003296 * q^32 - 130893352846776 * q^33 - 799347349171332 * q^34 + 299276129024676 * q^36 - 637994163989884 * q^37 + 1424314643816232 * q^38 + 867476186061084 * q^39 + 1150038430280532 * q^41 - 1194782101922592 * q^42 - 2820518953538200 * q^43 - 14723164414050768 * q^44 - 5980094010800208 * q^46 + 5603091319500480 * q^47 + 2354158069781040 * q^48 - 9875183230193070 * q^49 - 6619028208415884 * q^51 - 27174417188397208 * q^52 - 27537868857647628 * q^53 - 5353169434460874 * q^54 + 46521172561587840 * q^56 + 11851506802664568 * q^57 - 75092921304821316 * q^58 - 55153474421835816 * q^59 - 87192569066205620 * q^61 + 18193883108252112 * q^62 - 44124119128099296 * q^63 - 36250046218631104 * q^64 + 386740624928302152 * q^66 - 499299710877878248 * q^67 + 514116238109034744 * q^68 - 46614307382748144 * q^69 + 84915048436970544 * q^71 - 390714192327374136 * q^72 + 241433711161728332 * q^73 - 1618295867440077564 * q^74 - 915668569473570224 * q^76 - 728310910203957888 * q^77 + 934732084584051612 * q^78 + 1131591466033523440 * q^79 + 300189270593998242 * q^81 - 2445251486788059948 * q^82 - 1607743276003583784 * q^83 + 525252464149293504 * q^84 + 2895233411994836328 * q^86 + 5530475250535194036 * q^87 + 5229410429487705696 * q^88 + 4322797626369576276 * q^89 - 6535073681312344768 * q^91 + 3866849612692780896 * q^92 + 819012567760813296 * q^93 + 5483985680095257024 * q^94 + 4180789443511875168 * q^96 - 7902201235299458308 * q^97 + 14932538668319232402 * q^98 - 2576373864083092008 * q^99

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