LMFDB - Embedded newform 75.18.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,18,Mod(1,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,356] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$137.416565508$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{849}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 212 $ x^2 - x - 212
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3^{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			$-14.0688$ 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)

$f(q)$	$=$	\(q+702.477 q^{2} -6561.00 q^{3} +362402. q^{4} -4.60895e6 q^{6} +2.67481e6 q^{7} +1.62504e8 q^{8} +4.30467e7 q^{9} -9.86334e8 q^{11} -2.37772e9 q^{12} -2.81488e9 q^{13} +1.87899e9 q^{14} +6.66544e10 q^{16} -1.28501e10 q^{17} +3.02393e10 q^{18} -1.22560e11 q^{19} -1.75494e10 q^{21} -6.92877e11 q^{22} -1.14646e11 q^{23} -1.06619e12 q^{24} -1.97739e12 q^{26} -2.82430e11 q^{27} +9.69355e11 q^{28} -4.90027e12 q^{29} +7.64406e12 q^{31} +2.55235e13 q^{32} +6.47134e12 q^{33} -9.02688e12 q^{34} +1.56002e13 q^{36} -3.18983e12 q^{37} -8.60956e13 q^{38} +1.84684e13 q^{39} -5.53137e12 q^{41} -1.23281e13 q^{42} -2.42435e13 q^{43} -3.57449e14 q^{44} -8.05365e13 q^{46} +5.82161e13 q^{47} -4.37320e14 q^{48} -2.25476e14 q^{49} +8.43094e13 q^{51} -1.02012e15 q^{52} +6.45746e14 q^{53} -1.98400e14 q^{54} +4.34667e14 q^{56} +8.04117e14 q^{57} -3.44233e15 q^{58} -1.04433e12 q^{59} -1.56088e15 q^{61} +5.36978e15 q^{62} +1.15142e14 q^{63} +9.19314e15 q^{64} +4.54597e15 q^{66} -1.31151e15 q^{67} -4.65689e15 q^{68} +7.52195e14 q^{69} -7.09230e15 q^{71} +6.99526e15 q^{72} +1.17619e16 q^{73} -2.24078e15 q^{74} -4.44160e16 q^{76} -2.63826e15 q^{77} +1.29737e16 q^{78} -7.14674e15 q^{79} +1.85302e15 q^{81} -3.88566e15 q^{82} -9.04707e15 q^{83} -6.35994e15 q^{84} -1.70305e16 q^{86} +3.21507e16 q^{87} -1.60283e17 q^{88} -1.16786e16 q^{89} -7.52927e15 q^{91} -4.15481e16 q^{92} -5.01527e16 q^{93} +4.08955e16 q^{94} -1.67460e17 q^{96} +8.32538e16 q^{97} -1.58392e17 q^{98} -4.24585e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 356 q^{2} - 13122 q^{3} + 351376 q^{4} - 2335716 q^{6} + 20754552 q^{7} + 211737408 q^{8} + 86093442 q^{9} - 1131629912 q^{11} - 2305377936 q^{12} + 446672524 q^{13} - 4385222016 q^{14} + 51041317120 q^{16}+ \cdots - 48\!\cdots\!52 q^{99}+O(q^{100}) $ 2 * q + 356 * q^2 - 13122 * q^3 + 351376 * q^4 - 2335716 * q^6 + 20754552 * q^7 + 211737408 * q^8 + 86093442 * q^9 - 1131629912 * q^11 - 2305377936 * q^12 + 446672524 * q^13 - 4385222016 * q^14 + 51041317120 * q^16 - 20662021036 * q^17 + 15324632676 * q^18 - 194376127216 * q^19 - 136170615672 * q^21 - 642535396112 * q^22 - 281815042584 * q^23 - 1389209133888 * q^24 - 3107442504536 * q^26 - 564859072962 * q^27 + 770012242944 * q^28 - 4737062625892 * q^29 + 2256407074872 * q^31 + 24479953196032 * q^32 + 7424623852632 * q^33 - 6320225301928 * q^34 + 15125584638096 * q^36 + 33686114979452 * q^37 - 61213036677904 * q^38 - 2930618429964 * q^39 - 13051545859228 * q^41 + 28771441646976 * q^42 + 33149929851400 * q^43 - 355847272731712 * q^44 - 22616426690688 * q^46 - 84299585768440 * q^47 - 334882081624320 * q^48 - 131229302667470 * q^49 + 135563520017196 * q^51 - 1056079273719136 * q^52 - 155836696419964 * q^53 - 100544914987236 * q^54 + 1324797309480960 * q^56 + 1275301770664176 * q^57 - 3498878281127368 * q^58 + 963392423116456 * q^59 - 2858793214972660 * q^61 + 7236477385534944 * q^62 + 893415409423992 * q^63 + 11601155857485824 * q^64 + 4215674733890832 * q^66 - 6027859197776136 * q^67 - 4570758410249888 * q^68 + 1848988494393624 * q^69 + 2206536655060304 * q^71 + 9114601127439168 * q^72 + 18626002136017276 * q^73 - 15017443813698616 * q^74 - 43624167666234944 * q^76 - 5265163905377376 * q^77 + 20387930272260696 * q^78 + 4997891097934440 * q^79 + 3706040377703682 * q^81 - 1280087848985176 * q^82 + 23351553482511288 * q^83 - 5052050325955584 * q^84 - 36916031436093872 * q^86 + 31079867888477412 * q^87 - 167436503858533632 * q^88 - 60857257652216796 * q^89 + 51438794903148912 * q^91 - 39704913484919808 * q^92 - 14804286818235192 * q^93 + 90273897275494496 * q^94 - 160612972919165952 * q^96 - 42869862659782756 * q^97 - 191045888549711804 * q^98 - 48712957097118552 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	702.477	1.94034	0.970168	−	0.242432i	$-0.0779453\pi$
$2$	702.477	1.94034	0.970168	+	0.242432i	$0.0779453\pi$
$3$	−6561.00	−0.577350
$3$	−6561.00	−0.577350
$4$	362402.	2.76491
$5$	0	0
$5$	0	0
$6$	−4.60895e6	−1.12025
$7$	2.67481e6	0.175372	0.0876858	−	0.996148i	$-0.472053\pi$
$7$	2.67481e6	0.175372	0.0876858	+	0.996148i	$0.472053\pi$
$8$	1.62504e8	3.42451
$9$	4.30467e7	0.333333
$10$	0	0
$11$	−9.86334e8	−1.38735	−0.693675	−	0.720288i	$-0.744010\pi$
$11$	−9.86334e8	−1.38735	−0.693675	+	0.720288i	$0.744010\pi$
$12$	−2.37772e9	−1.59632
$13$	−2.81488e9	−0.957065	−0.478533	−	0.878070i	$-0.658831\pi$
$13$	−2.81488e9	−0.957065	−0.478533	+	0.878070i	$0.658831\pi$
$14$	1.87899e9	0.340280
$15$	0	0
$16$	6.66544e10	3.87980
$17$	−1.28501e10	−0.446776	−0.223388	−	0.974730i	$-0.571712\pi$
$17$	−1.28501e10	−0.446776	−0.223388	+	0.974730i	$0.571712\pi$
$18$	3.02393e10	0.646779
$19$	−1.22560e11	−1.65555	−0.827777	−	0.561057i	$-0.810395\pi$
$19$	−1.22560e11	−1.65555	−0.827777	+	0.561057i	$0.810395\pi$
$20$	0	0
$21$	−1.75494e10	−0.101251
$22$	−6.92877e11	−2.69193
$23$	−1.14646e11	−0.305263	−0.152631	−	0.988283i	$-0.548775\pi$
$23$	−1.14646e11	−0.305263	−0.152631	+	0.988283i	$0.548775\pi$
$24$	−1.06619e12	−1.97714
$25$	0	0
$26$	−1.97739e12	−1.85703
$27$	−2.82430e11	−0.192450
$28$	9.69355e11	0.484886
$29$	−4.90027e12	−1.81902	−0.909510	−	0.415682i	$-0.863543\pi$
$29$	−4.90027e12	−1.81902	−0.909510	+	0.415682i	$0.863543\pi$
$30$	0	0
$31$	7.64406e12	1.60972	0.804859	−	0.593466i	$-0.202241\pi$
$31$	7.64406e12	1.60972	0.804859	+	0.593466i	$0.202241\pi$
$32$	2.55235e13	4.10361
$33$	6.47134e12	0.800987
$34$	−9.02688e12	−0.866896
$35$	0	0
$36$	1.56002e13	0.921635
$37$	−3.18983e12	−0.149298	−0.0746488	−	0.997210i	$-0.523784\pi$
$37$	−3.18983e12	−0.149298	−0.0746488	+	0.997210i	$0.523784\pi$
$38$	−8.60956e13	−3.21233
$39$	1.84684e13	0.552562
$40$	0	0
$41$	−5.53137e12	−0.108186	−0.0540928	−	0.998536i	$-0.517227\pi$
$41$	−5.53137e12	−0.108186	−0.0540928	+	0.998536i	$0.517227\pi$
$42$	−1.23281e13	−0.196461
$43$	−2.42435e13	−0.316311	−0.158155	−	0.987414i	$-0.550555\pi$
$43$	−2.42435e13	−0.316311	−0.158155	+	0.987414i	$0.550555\pi$
$44$	−3.57449e14	−3.83589
$45$	0	0
$46$	−8.05365e13	−0.592313
$47$	5.82161e13	0.356625	0.178312	−	0.983974i	$-0.442936\pi$
$47$	5.82161e13	0.356625	0.178312	+	0.983974i	$0.442936\pi$
$48$	−4.37320e14	−2.24000
$49$	−2.25476e14	−0.969245
$50$	0	0
$51$	8.43094e13	0.257946
$52$	−1.02012e15	−2.64620
$53$	6.45746e14	1.42468	0.712339	−	0.701835i	$-0.247636\pi$
$53$	6.45746e14	1.42468	0.712339	+	0.701835i	$0.247636\pi$
$54$	−1.98400e14	−0.373418
$55$	0	0
$56$	4.34667e14	0.600562
$57$	8.04117e14	0.955835
$58$	−3.44233e15	−3.52951
$59$	−1.04433e12	−0.000925970 0	−0.000462985	−	1.00000i	$-0.500147\pi$
$59$	−1.04433e12	−0.000925970 0	−0.000462985		1.00000i	$0.500147\pi$
$60$	0	0
$61$	−1.56088e15	−1.04247	−0.521237	−	0.853412i	$-0.674529\pi$
$61$	−1.56088e15	−1.04247	−0.521237	+	0.853412i	$0.674529\pi$
$62$	5.36978e15	3.12340
$63$	1.15142e14	0.0584572
$64$	9.19314e15	4.08258
$65$	0	0
$66$	4.54597e15	1.55419
$67$	−1.31151e15	−0.394581	−0.197290	−	0.980345i	$-0.563214\pi$
$67$	−1.31151e15	−0.394581	−0.197290	+	0.980345i	$0.563214\pi$
$68$	−4.65689e15	−1.23529
$69$	7.52195e14	0.176244
$70$	0	0
$71$	−7.09230e15	−1.30344	−0.651720	−	0.758459i	$-0.725952\pi$
$71$	−7.09230e15	−1.30344	−0.651720	+	0.758459i	$0.725952\pi$
$72$	6.99526e15	1.14150
$73$	1.17619e16	1.70700	0.853501	−	0.521091i	$-0.174475\pi$
$73$	1.17619e16	1.70700	0.853501	+	0.521091i	$0.174475\pi$
$74$	−2.24078e15	−0.289688
$75$	0	0
$76$	−4.44160e16	−4.57745
$77$	−2.63826e15	−0.243302
$78$	1.29737e16	1.07216
$79$	−7.14674e15	−0.530003	−0.265001	−	0.964248i	$-0.585372\pi$
$79$	−7.14674e15	−0.530003	−0.265001	+	0.964248i	$0.585372\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	−3.88566e15	−0.209917
$83$	−9.04707e15	−0.440904	−0.220452	−	0.975398i	$-0.570753\pi$
$83$	−9.04707e15	−0.440904	−0.220452	+	0.975398i	$0.570753\pi$
$84$	−6.35994e15	−0.279949
$85$	0	0
$86$	−1.70305e16	−0.613749
$87$	3.21507e16	1.05021
$88$	−1.60283e17	−4.75100
$89$	−1.16786e16	−0.314467	−0.157233	−	0.987561i	$-0.550257\pi$
$89$	−1.16786e16	−0.314467	−0.157233	+	0.987561i	$0.550257\pi$
$90$	0	0
$91$	−7.52927e15	−0.167842
$92$	−4.15481e16	−0.844023
$93$	−5.01527e16	−0.929371
$94$	4.08955e16	0.691972
$95$	0	0
$96$	−1.67460e17	−2.36922
$97$	8.32538e16	1.07856	0.539280	−	0.842126i	$-0.318696\pi$
$97$	8.32538e16	1.07856	0.539280	+	0.842126i	$0.318696\pi$
$98$	−1.58392e17	−1.88066
$99$	−4.24585e16	−0.462450
$100$	0	0
$101$	−1.36210e16	−0.125164	−0.0625818	−	0.998040i	$-0.519933\pi$
$101$	−1.36210e16	−0.125164	−0.0625818	+	0.998040i	$0.519933\pi$
$102$	5.92254e16	0.500503
$103$	−1.19030e17	−0.925847	−0.462923	−	0.886398i	$-0.653199\pi$
$103$	−1.19030e17	−0.925847	−0.462923	+	0.886398i	$0.653199\pi$
$104$	−4.57429e17	−3.27748
$105$	0	0
$106$	4.53622e17	2.76436
$107$	−3.05375e17	−1.71819	−0.859096	−	0.511814i	$-0.828974\pi$
$107$	−3.05375e17	−1.71819	−0.859096	+	0.511814i	$0.828974\pi$
$108$	−1.02353e17	−0.532106
$109$	−2.94297e17	−1.41469	−0.707344	−	0.706869i	$-0.750107\pi$
$109$	−2.94297e17	−1.41469	−0.707344	+	0.706869i	$0.750107\pi$
$110$	0	0
$111$	2.09285e16	0.0861970
$112$	1.78288e17	0.680407
$113$	7.25034e16	0.256562	0.128281	−	0.991738i	$-0.459054\pi$
$113$	7.25034e16	0.256562	0.128281	+	0.991738i	$0.459054\pi$
$114$	5.64873e17	1.85464
$115$	0	0
$116$	−1.77587e18	−5.02942
$117$	−1.21171e17	−0.319022
$118$	−7.33620e14	−0.00179669
$119$	−3.43715e16	−0.0783519
$120$	0	0
$121$	4.67408e17	0.924742
$122$	−1.09648e18	−2.02275
$123$	3.62913e16	0.0624610
$124$	2.77022e18	4.45072
$125$	0	0
$126$	8.08844e16	0.113427
$127$	1.00107e18	1.31261	0.656304	−	0.754497i	$-0.272119\pi$
$127$	1.00107e18	1.31261	0.656304	+	0.754497i	$0.272119\pi$
$128$	3.11255e18	3.81797
$129$	1.59062e17	0.182622
$130$	0	0
$131$	6.36150e17	0.640846	0.320423	−	0.947275i	$-0.396175\pi$
$131$	6.36150e17	0.640846	0.320423	+	0.947275i	$0.396175\pi$
$132$	2.34522e18	2.21465
$133$	−3.27825e17	−0.290337
$134$	−9.21306e17	−0.765620
$135$	0	0
$136$	−2.08819e18	−1.52999
$137$	−2.75383e17	−0.189588	−0.0947942	−	0.995497i	$-0.530219\pi$
$137$	−2.75383e17	−0.189588	−0.0947942	+	0.995497i	$0.530219\pi$
$138$	5.28400e17	0.341972
$139$	3.61048e17	0.219755	0.109878	−	0.993945i	$-0.464954\pi$
$139$	3.61048e17	0.219755	0.109878	+	0.993945i	$0.464954\pi$
$140$	0	0
$141$	−3.81956e17	−0.205897
$142$	−4.98218e18	−2.52911
$143$	2.77641e18	1.32779
$144$	2.86926e18	1.29327
$145$	0	0
$146$	8.26248e18	3.31216
$147$	1.47935e18	0.559594
$148$	−1.15600e18	−0.412794
$149$	−4.02824e18	−1.35841	−0.679207	−	0.733947i	$-0.737676\pi$
$149$	−4.02824e18	−1.35841	−0.679207	+	0.733947i	$0.737676\pi$
$150$	0	0
$151$	−3.02464e18	−0.910689	−0.455345	−	0.890315i	$-0.650484\pi$
$151$	−3.02464e18	−0.910689	−0.455345	+	0.890315i	$0.650484\pi$
$152$	−1.99165e19	−5.66947
$153$	−5.53154e17	−0.148925
$154$	−1.85331e18	−0.472088
$155$	0	0
$156$	6.69300e18	1.52778
$157$	3.80909e18	0.823520	0.411760	−	0.911292i	$-0.364914\pi$
$157$	3.80909e18	0.823520	0.411760	+	0.911292i	$0.364914\pi$
$158$	−5.02042e18	−1.02838
$159$	−4.23674e18	−0.822539
$160$	0	0
$161$	−3.06657e17	−0.0535345
$162$	1.30170e18	0.215593
$163$	7.01276e18	1.10229	0.551143	−	0.834411i	$-0.314192\pi$
$163$	7.01276e18	1.10229	0.551143	+	0.834411i	$0.314192\pi$
$164$	−2.00458e18	−0.299123
$165$	0	0
$166$	−6.35535e18	−0.855502
$167$	5.20936e18	0.666337	0.333169	−	0.942867i	$-0.391882\pi$
$167$	5.20936e18	0.666337	0.333169	+	0.942867i	$0.391882\pi$
$168$	−2.85185e18	−0.346735
$169$	−7.26857e17	−0.0840256
$170$	0	0
$171$	−5.27581e18	−0.551851
$172$	−8.78590e18	−0.874570
$173$	−4.55714e18	−0.431818	−0.215909	−	0.976413i	$-0.569271\pi$
$173$	−4.55714e18	−0.431818	−0.215909	+	0.976413i	$0.569271\pi$
$174$	2.25851e19	2.03776
$175$	0	0
$176$	−6.57436e19	−5.38264
$177$	6.85187e15	0.000534609 0
$178$	−8.20393e18	−0.610171
$179$	1.27111e19	0.901428	0.450714	−	0.892668i	$-0.351169\pi$
$179$	1.27111e19	0.901428	0.450714	+	0.892668i	$0.351169\pi$
$180$	0	0
$181$	1.67839e19	1.08299	0.541494	−	0.840704i	$-0.317859\pi$
$181$	1.67839e19	1.08299	0.541494	+	0.840704i	$0.317859\pi$
$182$	−5.28914e18	−0.325670
$183$	1.02409e19	0.601872
$184$	−1.86305e19	−1.04538
$185$	0	0
$186$	−3.52311e19	−1.80329
$187$	1.26745e19	0.619835
$188$	2.10976e19	0.986034
$189$	−7.55445e17	−0.0337503
$190$	0	0
$191$	1.93607e19	0.790930	0.395465	−	0.918481i	$-0.370583\pi$
$191$	1.93607e19	0.790930	0.395465	+	0.918481i	$0.370583\pi$
$192$	−6.03162e19	−2.35708
$193$	−5.41103e18	−0.202322	−0.101161	−	0.994870i	$-0.532256\pi$
$193$	−5.41103e18	−0.202322	−0.101161	+	0.994870i	$0.532256\pi$
$194$	5.84839e19	2.09277
$195$	0	0
$196$	−8.17129e19	−2.67987
$197$	−1.69857e19	−0.533483	−0.266741	−	0.963768i	$-0.585947\pi$
$197$	−1.69857e19	−0.533483	−0.266741	+	0.963768i	$0.585947\pi$
$198$	−2.98261e19	−0.897309
$199$	−8.38703e18	−0.241745	−0.120872	−	0.992668i	$-0.538569\pi$
$199$	−8.38703e18	−0.241745	−0.120872	+	0.992668i	$0.538569\pi$
$200$	0	0
$201$	8.60482e18	0.227811
$202$	−9.56845e18	−0.242860
$203$	−1.31073e19	−0.319004
$204$	3.05539e19	0.713197
$205$	0	0
$206$	−8.36156e19	−1.79645
$207$	−4.93515e18	−0.101754
$208$	−1.87624e20	−3.71322
$209$	1.20885e20	2.29683
$210$	0	0
$211$	2.60946e19	0.457245	0.228623	−	0.973515i	$-0.426578\pi$
$211$	2.60946e19	0.457245	0.228623	+	0.973515i	$0.426578\pi$
$212$	2.34020e20	3.93910
$213$	4.65326e19	0.752542
$214$	−2.14519e20	−3.33387
$215$	0	0
$216$	−4.58959e19	−0.659048
$217$	2.04464e19	0.282299
$218$	−2.06737e20	−2.74497
$219$	−7.71700e19	−0.985539
$220$	0	0
$221$	3.61715e19	0.427594
$222$	1.47018e19	0.167251
$223$	−1.19753e19	−0.131128	−0.0655641	−	0.997848i	$-0.520885\pi$
$223$	−1.19753e19	−0.131128	−0.0655641	+	0.997848i	$0.520885\pi$
$224$	6.82705e19	0.719656
$225$	0	0
$226$	5.09320e19	0.497816
$227$	−1.24200e20	−1.16924	−0.584619	−	0.811308i	$-0.698756\pi$
$227$	−1.24200e20	−1.16924	−0.584619	+	0.811308i	$0.698756\pi$
$228$	2.91413e20	2.64279
$229$	1.03099e20	0.900853	0.450426	−	0.892814i	$-0.351272\pi$
$229$	1.03099e20	0.900853	0.450426	+	0.892814i	$0.351272\pi$
$230$	0	0
$231$	1.73096e19	0.140470
$232$	−7.96313e20	−6.22925
$233$	−1.13357e20	−0.854919	−0.427459	−	0.904035i	$-0.640591\pi$
$233$	−1.13357e20	−0.854919	−0.427459	+	0.904035i	$0.640591\pi$
$234$	−8.51201e19	−0.619010
$235$	0	0
$236$	−3.78468e17	−0.00256022
$237$	4.68898e19	0.305997
$238$	−2.41452e19	−0.152029
$239$	−8.29065e19	−0.503740	−0.251870	−	0.967761i	$-0.581046\pi$
$239$	−8.29065e19	−0.503740	−0.251870	+	0.967761i	$0.581046\pi$
$240$	0	0
$241$	−1.62688e20	−0.920896	−0.460448	−	0.887687i	$-0.652311\pi$
$241$	−1.62688e20	−0.920896	−0.460448	+	0.887687i	$0.652311\pi$
$242$	3.28343e20	1.79431
$243$	−1.21577e19	−0.0641500
$244$	−5.65664e20	−2.88234
$245$	0	0
$246$	2.54938e19	0.121195
$247$	3.44992e20	1.58447
$248$	1.24219e21	5.51250
$249$	5.93578e19	0.254556
$250$	0	0
$251$	4.10409e20	1.64433	0.822167	−	0.569247i	$-0.192765\pi$
$251$	4.10409e20	1.64433	0.822167	+	0.569247i	$0.192765\pi$
$252$	4.17276e19	0.161629
$253$	1.13080e20	0.423507
$254$	7.03232e20	2.54690
$255$	0	0
$256$	9.81533e20	3.32556
$257$	1.69582e20	0.555839	0.277920	−	0.960604i	$-0.410355\pi$
$257$	1.69582e20	0.555839	0.277920	+	0.960604i	$0.410355\pi$
$258$	1.11737e20	0.354348
$259$	−8.53218e18	−0.0261826
$260$	0	0
$261$	−2.10941e20	−0.606340
$262$	4.46881e20	1.24346
$263$	3.50624e20	0.944534	0.472267	−	0.881456i	$-0.343436\pi$
$263$	3.50624e20	0.944534	0.472267	+	0.881456i	$0.343436\pi$
$264$	1.05162e21	2.74299
$265$	0	0
$266$	−2.30289e20	−0.563352
$267$	7.66231e19	0.181557
$268$	−4.75294e20	−1.09098
$269$	2.92843e20	0.651240	0.325620	−	0.945501i	$-0.394427\pi$
$269$	2.92843e20	0.651240	0.325620	+	0.945501i	$0.394427\pi$
$270$	0	0
$271$	5.64043e20	1.17780	0.588902	−	0.808204i	$-0.299560\pi$
$271$	5.64043e20	1.17780	0.588902	+	0.808204i	$0.299560\pi$
$272$	−8.56515e20	−1.73340
$273$	4.93995e19	0.0969037
$274$	−1.93450e20	−0.367865
$275$	0	0
$276$	2.72597e20	0.487297
$277$	−7.92843e20	−1.37439	−0.687194	−	0.726474i	$-0.741158\pi$
$277$	−7.92843e20	−1.37439	−0.687194	+	0.726474i	$0.741158\pi$
$278$	2.53628e20	0.426399
$279$	3.29052e20	0.536573
$280$	0	0
$281$	−4.07531e20	−0.625399	−0.312700	−	0.949852i	$-0.601233\pi$
$281$	−4.07531e20	−0.625399	−0.312700	+	0.949852i	$0.601233\pi$
$282$	−2.68315e20	−0.399510
$283$	−9.05800e20	−1.30872	−0.654362	−	0.756182i	$-0.727063\pi$
$283$	−9.05800e20	−1.30872	−0.654362	+	0.756182i	$0.727063\pi$
$284$	−2.57026e21	−3.60389
$285$	0	0
$286$	1.95037e21	2.57635
$287$	−1.47953e19	−0.0189727
$288$	1.09870e21	1.36787
$289$	−6.62116e20	−0.800391
$290$	0	0
$291$	−5.46228e20	−0.622707
$292$	4.26254e21	4.71970
$293$	−1.22920e21	−1.32205	−0.661025	−	0.750364i	$-0.729878\pi$
$293$	−1.22920e21	−1.32205	−0.661025	+	0.750364i	$0.729878\pi$
$294$	1.03921e21	1.08580
$295$	0	0
$296$	−5.18359e20	−0.511271
$297$	2.78570e20	0.266996
$298$	−2.82974e21	−2.63578
$299$	3.22716e20	0.292157
$300$	0	0
$301$	−6.48468e19	−0.0554719
$302$	−2.12474e21	−1.76704
$303$	8.93675e19	0.0722632
$304$	−8.16918e21	−6.42322
$305$	0	0
$306$	−3.88578e20	−0.288965
$307$	1.33769e21	0.967565	0.483783	−	0.875188i	$-0.339262\pi$
$307$	1.33769e21	0.967565	0.483783	+	0.875188i	$0.339262\pi$
$308$	−9.56108e20	−0.672707
$309$	7.80954e20	0.534538
$310$	0	0
$311$	−8.26556e19	−0.0535561	−0.0267781	−	0.999641i	$-0.508525\pi$
$311$	−8.26556e19	−0.0535561	−0.0267781	+	0.999641i	$0.508525\pi$
$312$	3.00119e21	1.89226
$313$	2.88314e20	0.176905	0.0884523	−	0.996080i	$-0.471808\pi$
$313$	2.88314e20	0.176905	0.0884523	+	0.996080i	$0.471808\pi$
$314$	2.67580e21	1.59791
$315$	0	0
$316$	−2.58999e21	−1.46541
$317$	3.17162e21	1.74694	0.873468	−	0.486882i	$-0.161866\pi$
$317$	3.17162e21	1.74694	0.873468	+	0.486882i	$0.161866\pi$
$318$	−2.97621e21	−1.59600
$319$	4.83331e21	2.52362
$320$	0	0
$321$	2.00357e21	0.991999
$322$	−2.15420e20	−0.103875
$323$	1.57491e21	0.739662
$324$	6.71538e20	0.307212
$325$	0	0
$326$	4.92630e21	2.13881
$327$	1.93088e21	0.816771
$328$	−8.98868e20	−0.370483
$329$	1.55717e20	0.0625419
$330$	0	0
$331$	−1.58984e21	−0.606478	−0.303239	−	0.952915i	$-0.598068\pi$
$331$	−1.58984e21	−0.606478	−0.303239	+	0.952915i	$0.598068\pi$
$332$	−3.27867e21	−1.21906
$333$	−1.37312e20	−0.0497659
$334$	3.65945e21	1.29292
$335$	0	0
$336$	−1.16975e21	−0.392833
$337$	3.01551e20	0.0987432	0.0493716	−	0.998780i	$-0.484278\pi$
$337$	3.01551e20	0.0987432	0.0493716	+	0.998780i	$0.484278\pi$
$338$	−5.10600e20	−0.163038
$339$	−4.75695e20	−0.148126
$340$	0	0
$341$	−7.53960e21	−2.23324
$342$	−3.70613e21	−1.07078
$343$	−1.22535e21	−0.345350
$344$	−3.93967e21	−1.08321
$345$	0	0
$346$	−3.20129e21	−0.837873
$347$	3.81902e21	0.975331	0.487665	−	0.873031i	$-0.337849\pi$
$347$	3.81902e21	0.975331	0.487665	+	0.873031i	$0.337849\pi$
$348$	1.16515e22	2.90374
$349$	−3.13624e20	−0.0762769	−0.0381384	−	0.999272i	$-0.512143\pi$
$349$	−3.13624e20	−0.0762769	−0.0381384	+	0.999272i	$0.512143\pi$
$350$	0	0
$351$	7.95006e20	0.184187
$352$	−2.51747e22	−5.69314
$353$	4.65586e21	1.02781	0.513907	−	0.857846i	$-0.328198\pi$
$353$	4.65586e21	1.02781	0.513907	+	0.857846i	$0.328198\pi$
$354$	4.81328e18	0.00103732
$355$	0	0
$356$	−4.23234e21	−0.869471
$357$	2.25511e20	0.0452365
$358$	8.92923e21	1.74907
$359$	3.57780e21	0.684405	0.342203	−	0.939626i	$-0.388827\pi$
$359$	3.57780e21	0.684405	0.342203	+	0.939626i	$0.388827\pi$
$360$	0	0
$361$	9.54059e21	1.74086
$362$	1.17903e22	2.10136
$363$	−3.06667e21	−0.533900
$364$	−2.72862e21	−0.464068
$365$	0	0
$366$	7.19400e21	1.16783
$367$	5.72348e21	0.907818	0.453909	−	0.891048i	$-0.350029\pi$
$367$	5.72348e21	0.907818	0.453909	+	0.891048i	$0.350029\pi$
$368$	−7.64170e21	−1.18436
$369$	−2.38107e20	−0.0360619
$370$	0	0
$371$	1.72725e21	0.249848
$372$	−1.81754e22	−2.56962
$373$	−7.79326e21	−1.07695	−0.538473	−	0.842643i	$-0.680999\pi$
$373$	−7.79326e21	−1.07695	−0.538473	+	0.842643i	$0.680999\pi$
$374$	8.90352e21	1.20269
$375$	0	0
$376$	9.46034e21	1.22127
$377$	1.37937e22	1.74092
$378$	−5.30683e20	−0.0654869
$379$	4.41273e21	0.532444	0.266222	−	0.963912i	$-0.414225\pi$
$379$	4.41273e21	0.532444	0.266222	+	0.963912i	$0.414225\pi$
$380$	0	0
$381$	−6.56805e21	−0.757834
$382$	1.36005e22	1.53467
$383$	−7.42094e21	−0.818973	−0.409486	−	0.912316i	$-0.634292\pi$
$383$	−7.42094e21	−0.818973	−0.409486	+	0.912316i	$0.634292\pi$
$384$	−2.04215e22	−2.20430
$385$	0	0
$386$	−3.80112e21	−0.392572
$387$	−1.04360e21	−0.105437
$388$	3.01713e22	2.98212
$389$	4.44977e21	0.430295	0.215147	−	0.976582i	$-0.430977\pi$
$389$	4.44977e21	0.430295	0.215147	+	0.976582i	$0.430977\pi$
$390$	0	0
$391$	1.47322e21	0.136384
$392$	−3.66407e22	−3.31919
$393$	−4.17378e21	−0.369993
$394$	−1.19321e22	−1.03514
$395$	0	0
$396$	−1.53870e22	−1.27863
$397$	4.94222e20	0.0401979	0.0200989	−	0.999798i	$-0.493602\pi$
$397$	4.94222e20	0.0401979	0.0200989	+	0.999798i	$0.493602\pi$
$398$	−5.89170e21	−0.469066
$399$	2.15086e21	0.167626
$400$	0	0
$401$	2.49341e22	1.86237	0.931187	−	0.364542i	$-0.118775\pi$
$401$	2.49341e22	1.86237	0.931187	+	0.364542i	$0.118775\pi$
$402$	6.04469e21	0.442031
$403$	−2.15171e22	−1.54061
$404$	−4.93628e21	−0.346066
$405$	0	0
$406$	−9.20757e21	−0.618976
$407$	3.14624e21	0.207128
$408$	1.37006e22	0.883340
$409$	7.82942e21	0.494404	0.247202	−	0.968964i	$-0.420489\pi$
$409$	7.82942e21	0.494404	0.247202	+	0.968964i	$0.420489\pi$
$410$	0	0
$411$	1.80679e21	0.109459
$412$	−4.31366e22	−2.55988
$413$	−2.79339e18	−0.000162389 0
$414$	−3.46683e21	−0.197438
$415$	0	0
$416$	−7.18457e22	−3.92742
$417$	−2.36884e21	−0.126876
$418$	8.49191e22	4.45663
$419$	−1.37314e20	−0.00706150	−0.00353075	−	0.999994i	$-0.501124\pi$
$419$	−1.37314e20	−0.00706150	−0.00353075	+	0.999994i	$0.501124\pi$
$420$	0	0
$421$	−1.84158e21	−0.0909481	−0.0454740	−	0.998966i	$-0.514480\pi$
$421$	−1.84158e21	−0.0909481	−0.0454740	+	0.998966i	$0.514480\pi$
$422$	1.83308e22	0.887209
$423$	2.50601e21	0.118875
$424$	1.04936e23	4.87883
$425$	0	0
$426$	3.26881e22	1.46018
$427$	−4.17505e21	−0.182820
$428$	−1.10669e23	−4.75064
$429$	−1.82161e22	−0.766597
$430$	0	0
$431$	−3.30639e21	−0.133751	−0.0668756	−	0.997761i	$-0.521303\pi$
$431$	−3.30639e21	−0.133751	−0.0668756	+	0.997761i	$0.521303\pi$
$432$	−1.88252e22	−0.746668
$433$	−1.49902e22	−0.582990	−0.291495	−	0.956572i	$-0.594153\pi$
$433$	−1.49902e22	−0.582990	−0.291495	+	0.956572i	$0.594153\pi$
$434$	1.43631e22	0.547755
$435$	0	0
$436$	−1.06654e23	−3.91148
$437$	1.40511e22	0.505379
$438$	−5.42102e22	−1.91228
$439$	5.27352e22	1.82453	0.912267	−	0.409597i	$-0.134331\pi$
$439$	5.27352e22	1.82453	0.912267	+	0.409597i	$0.134331\pi$
$440$	0	0
$441$	−9.70600e21	−0.323082
$442$	2.54096e22	0.829676
$443$	−2.21332e22	−0.708944	−0.354472	−	0.935067i	$-0.615339\pi$
$443$	−2.21332e22	−0.708944	−0.354472	+	0.935067i	$0.615339\pi$
$444$	7.58452e21	0.238327
$445$	0	0
$446$	−8.41239e21	−0.254433
$447$	2.64293e22	0.784280
$448$	2.45899e22	0.715968
$449$	1.01171e22	0.289041	0.144521	−	0.989502i	$-0.453836\pi$
$449$	1.01171e22	0.289041	0.144521	+	0.989502i	$0.453836\pi$
$450$	0	0
$451$	5.45578e21	0.150091
$452$	2.62754e22	0.709369
$453$	1.98447e22	0.525787
$454$	−8.72479e22	−2.26872
$455$	0	0
$456$	1.30672e23	3.27327
$457$	7.09706e22	1.74498	0.872490	−	0.488632i	$-0.162504\pi$
$457$	7.09706e22	1.74498	0.872490	+	0.488632i	$0.162504\pi$
$458$	7.24248e22	1.74796
$459$	3.62924e21	0.0859821
$460$	0	0
$461$	1.09889e22	0.250899	0.125449	−	0.992100i	$-0.459963\pi$
$461$	1.09889e22	0.250899	0.125449	+	0.992100i	$0.459963\pi$
$462$	1.21596e22	0.272560
$463$	−1.78028e22	−0.391787	−0.195893	−	0.980625i	$-0.562761\pi$
$463$	−1.78028e22	−0.391787	−0.195893	+	0.980625i	$0.562761\pi$
$464$	−3.26625e23	−7.05743
$465$	0	0
$466$	−7.96310e22	−1.65883
$467$	−7.82704e22	−1.60105	−0.800524	−	0.599301i	$-0.795445\pi$
$467$	−7.82704e22	−1.60105	−0.800524	+	0.599301i	$0.795445\pi$
$468$	−4.39127e22	−0.882065
$469$	−3.50804e21	−0.0691983
$470$	0	0
$471$	−2.49914e22	−0.475459
$472$	−1.69708e20	−0.00317100
$473$	2.39122e22	0.438834
$474$	3.29390e22	0.593737
$475$	0	0
$476$	−1.24563e22	−0.216636
$477$	2.77973e22	0.474893
$478$	−5.82399e22	−0.977425
$479$	−1.17366e23	−1.93504	−0.967522	−	0.252785i	$-0.918653\pi$
$479$	−1.17366e23	−1.93504	−0.967522	+	0.252785i	$0.918653\pi$
$480$	0	0
$481$	8.97899e21	0.142888
$482$	−1.14285e23	−1.78685
$483$	2.01198e21	0.0309081
$484$	1.69390e23	2.55683
$485$	0	0
$486$	−8.54048e21	−0.124473
$487$	−4.40441e22	−0.630800	−0.315400	−	0.948959i	$-0.602139\pi$
$487$	−4.40441e22	−0.630800	−0.315400	+	0.948959i	$0.602139\pi$
$488$	−2.53648e23	−3.56996
$489$	−4.60107e22	−0.636405
$490$	0	0
$491$	−9.19512e21	−0.122847	−0.0614235	−	0.998112i	$-0.519564\pi$
$491$	−9.19512e21	−0.122847	−0.0614235	+	0.998112i	$0.519564\pi$
$492$	1.31520e22	0.172699
$493$	6.29689e22	0.812695
$494$	2.42349e23	3.07441
$495$	0	0
$496$	5.09511e23	6.24539
$497$	−1.89705e22	−0.228586
$498$	4.16975e22	0.493924
$499$	−9.12259e22	−1.06234	−0.531170	−	0.847265i	$-0.678247\pi$
$499$	−9.12259e22	−1.06234	−0.531170	+	0.847265i	$0.678247\pi$
$500$	0	0
$501$	−3.41786e22	−0.384710
$502$	2.88303e23	3.19056
$503$	4.20493e22	0.457542	0.228771	−	0.973480i	$-0.426529\pi$
$503$	4.20493e22	0.457542	0.228771	+	0.973480i	$0.426529\pi$
$504$	1.87110e22	0.200187
$505$	0	0
$506$	7.94359e22	0.821746
$507$	4.76891e21	0.0485122
$508$	3.62791e23	3.62924
$509$	3.12071e22	0.307010	0.153505	−	0.988148i	$-0.450944\pi$
$509$	3.12071e22	0.307010	0.153505	+	0.988148i	$0.450944\pi$
$510$	0	0
$511$	3.14609e22	0.299360
$512$	2.81536e23	2.63475
$513$	3.46146e22	0.318612
$514$	1.19128e23	1.07852
$515$	0	0
$516$	5.76443e22	0.504933
$517$	−5.74206e22	−0.494764
$518$	−5.99366e21	−0.0508030
$519$	2.98994e22	0.249310
$520$	0	0
$521$	−1.71736e23	−1.38593	−0.692965	−	0.720971i	$-0.743696\pi$
$521$	−1.71736e23	−1.38593	−0.692965	+	0.720971i	$0.743696\pi$
$522$	−1.48181e23	−1.17650
$523$	3.54202e22	0.276686	0.138343	−	0.990384i	$-0.455822\pi$
$523$	3.54202e22	0.276686	0.138343	+	0.990384i	$0.455822\pi$
$524$	2.30542e23	1.77188
$525$	0	0
$526$	2.46305e23	1.83271
$527$	−9.82268e22	−0.719184
$528$	4.31344e23	3.10767
$529$	−1.27906e23	−0.906815
$530$	0	0
$531$	−4.49551e19	−0.000308657 0
$532$	−1.18804e23	−0.802755
$533$	1.55701e22	0.103541
$534$	5.38260e22	0.352283
$535$	0	0
$536$	−2.13126e23	−1.35125
$537$	−8.33973e22	−0.520440
$538$	2.05716e23	1.26363
$539$	2.22395e23	1.34468
$540$	0	0
$541$	−1.03474e23	−0.606257	−0.303128	−	0.952950i	$-0.598031\pi$
$541$	−1.03474e23	−0.606257	−0.303128	+	0.952950i	$0.598031\pi$
$542$	3.96227e23	2.28534
$543$	−1.10119e23	−0.625264
$544$	−3.27979e23	−1.83339
$545$	0	0
$546$	3.47020e22	0.188026
$547$	−2.96616e23	−1.58235	−0.791176	−	0.611589i	$-0.790531\pi$
$547$	−2.96616e23	−1.58235	−0.791176	+	0.611589i	$0.790531\pi$
$548$	−9.97992e22	−0.524194
$549$	−6.71906e22	−0.347491
$550$	0	0
$551$	6.00578e23	3.01149
$552$	1.22235e23	0.603548
$553$	−1.91162e22	−0.0929474
$554$	−5.56954e23	−2.66678
$555$	0	0
$556$	1.30845e23	0.607603
$557$	5.46640e22	0.249996	0.124998	−	0.992157i	$-0.460108\pi$
$557$	5.46640e22	0.249996	0.124998	+	0.992157i	$0.460108\pi$
$558$	2.31151e23	1.04113
$559$	6.82427e22	0.302730
$560$	0	0
$561$	−8.31572e22	−0.357862
$562$	−2.86281e23	−1.21349
$563$	1.18516e23	0.494829	0.247414	−	0.968910i	$-0.420419\pi$
$563$	1.18516e23	0.494829	0.247414	+	0.968910i	$0.420419\pi$
$564$	−1.38422e23	−0.569287
$565$	0	0
$566$	−6.36304e23	−2.53936
$567$	4.95647e21	0.0194857
$568$	−1.15253e24	−4.46365
$569$	−4.39785e23	−1.67798	−0.838988	−	0.544149i	$-0.816853\pi$
$569$	−4.39785e23	−1.67798	−0.838988	+	0.544149i	$0.816853\pi$
$570$	0	0
$571$	−4.89721e22	−0.181360	−0.0906800	−	0.995880i	$-0.528904\pi$
$571$	−4.89721e22	−0.181360	−0.0906800	+	0.995880i	$0.528904\pi$
$572$	1.00618e24	3.67120
$573$	−1.27026e23	−0.456644
$574$	−1.03934e22	−0.0368134
$575$	0	0
$576$	3.95735e23	1.36086
$577$	−5.15333e23	−1.74620	−0.873099	−	0.487543i	$-0.837893\pi$
$577$	−5.15333e23	−1.74620	−0.873099	+	0.487543i	$0.837893\pi$
$578$	−4.65121e23	−1.55303
$579$	3.55018e22	0.116811
$580$	0	0
$581$	−2.41992e22	−0.0773220
$582$	−3.83713e23	−1.20826
$583$	−6.36922e23	−1.97653
$584$	1.91136e24	5.84565
$585$	0	0
$586$	−8.63485e23	−2.56522
$587$	3.59992e22	0.105407	0.0527035	−	0.998610i	$-0.483216\pi$
$587$	3.59992e22	0.105407	0.0527035	+	0.998610i	$0.483216\pi$
$588$	5.36118e23	1.54722
$589$	−9.36857e23	−2.66498
$590$	0	0
$591$	1.11443e23	0.308006
$592$	−2.12616e23	−0.579245
$593$	3.67652e23	0.987351	0.493676	−	0.869646i	$-0.335653\pi$
$593$	3.67652e23	0.987351	0.493676	+	0.869646i	$0.335653\pi$
$594$	1.95689e23	0.518062
$595$	0	0
$596$	−1.45984e24	−3.75588
$597$	5.50273e22	0.139571
$598$	2.26701e23	0.566882
$599$	−6.97132e23	−1.71865	−0.859324	−	0.511432i	$-0.829115\pi$
$599$	−6.97132e23	−1.71865	−0.859324	+	0.511432i	$0.829115\pi$
$600$	0	0
$601$	3.44330e23	0.825166	0.412583	−	0.910920i	$-0.364627\pi$
$601$	3.44330e23	0.825166	0.412583	+	0.910920i	$0.364627\pi$
$602$	−4.55534e22	−0.107634
$603$	−5.64562e22	−0.131527
$604$	−1.09614e24	−2.51797
$605$	0	0
$606$	6.27786e22	0.140215
$607$	3.08234e23	0.678855	0.339427	−	0.940632i	$-0.389767\pi$
$607$	3.08234e23	0.678855	0.339427	+	0.940632i	$0.389767\pi$
$608$	−3.12816e24	−6.79374
$609$	8.59970e22	0.184177
$610$	0	0
$611$	−1.63872e23	−0.341313
$612$	−2.00464e23	−0.411765
$613$	4.35176e23	0.881558	0.440779	−	0.897616i	$-0.354702\pi$
$613$	4.35176e23	0.881558	0.440779	+	0.897616i	$0.354702\pi$
$614$	9.39699e23	1.87740
$615$	0	0
$616$	−4.28727e23	−0.833190
$617$	−6.04231e23	−1.15819	−0.579094	−	0.815261i	$-0.696594\pi$
$617$	−6.04231e23	−1.15819	−0.579094	+	0.815261i	$0.696594\pi$
$618$	5.48602e23	1.03718
$619$	−3.33503e23	−0.621913	−0.310956	−	0.950424i	$-0.600649\pi$
$619$	−3.33503e23	−0.621913	−0.310956	+	0.950424i	$0.600649\pi$
$620$	0	0
$621$	3.23795e22	0.0587479
$622$	−5.80637e22	−0.103917
$623$	−3.12379e22	−0.0551485
$624$	1.23100e24	2.14383
$625$	0	0
$626$	2.02534e23	0.343255
$627$	−7.93128e23	−1.32608
$628$	1.38042e24	2.27695
$629$	4.09896e22	0.0667026
$630$	0	0
$631$	7.67831e23	1.21623	0.608116	−	0.793848i	$-0.291926\pi$
$631$	7.67831e23	1.21623	0.608116	+	0.793848i	$0.291926\pi$
$632$	−1.16137e24	−1.81500
$633$	−1.71206e23	−0.263991
$634$	2.22799e24	3.38964
$635$	0	0
$636$	−1.53540e24	−2.27424
$637$	6.34688e23	0.927631
$638$	3.39529e24	4.89667
$639$	−3.05300e23	−0.434480
$640$	0	0
$641$	−1.34739e24	−1.86724	−0.933620	−	0.358264i	$-0.883369\pi$
$641$	−1.34739e24	−1.86724	−0.933620	+	0.358264i	$0.883369\pi$
$642$	1.40746e24	1.92481
$643$	−1.07291e24	−1.44800	−0.723999	−	0.689801i	$-0.757698\pi$
$643$	−1.07291e24	−1.44800	−0.723999	+	0.689801i	$0.757698\pi$
$644$	−1.11133e23	−0.148018
$645$	0	0
$646$	1.10634e24	1.43519
$647$	−2.56767e23	−0.328740	−0.164370	−	0.986399i	$-0.552559\pi$
$647$	−2.56767e23	−0.328740	−0.164370	+	0.986399i	$0.552559\pi$
$648$	3.01123e23	0.380501
$649$	1.03006e21	0.00128465
$650$	0	0
$651$	−1.34149e23	−0.162985
$652$	2.54144e24	3.04772
$653$	−9.55573e23	−1.13110	−0.565551	−	0.824713i	$-0.691336\pi$
$653$	−9.55573e23	−1.13110	−0.565551	+	0.824713i	$0.691336\pi$
$654$	1.35640e24	1.58481
$655$	0	0
$656$	−3.68690e23	−0.419739
$657$	5.06312e23	0.569001
$658$	1.09388e23	0.121352
$659$	−4.48249e23	−0.490901	−0.245450	−	0.969409i	$-0.578936\pi$
$659$	−4.48249e23	−0.490901	−0.245450	+	0.969409i	$0.578936\pi$
$660$	0	0
$661$	3.31071e23	0.353353	0.176677	−	0.984269i	$-0.443465\pi$
$661$	3.31071e23	0.353353	0.176677	+	0.984269i	$0.443465\pi$
$662$	−1.11682e24	−1.17677
$663$	−2.37321e23	−0.246872
$664$	−1.47018e24	−1.50988
$665$	0	0
$666$	−9.64583e22	−0.0965625
$667$	5.61799e23	0.555279
$668$	1.88788e24	1.84236
$669$	7.85701e22	0.0757069
$670$	0	0
$671$	1.53955e24	1.44628
$672$	−4.47923e23	−0.415494
$673$	9.61434e23	0.880626	0.440313	−	0.897844i	$-0.354868\pi$
$673$	9.61434e23	0.880626	0.440313	+	0.897844i	$0.354868\pi$
$674$	2.11833e23	0.191595
$675$	0	0
$676$	−2.63414e23	−0.232323
$677$	−1.08092e24	−0.941432	−0.470716	−	0.882285i	$-0.656004\pi$
$677$	−1.08092e24	−0.941432	−0.470716	+	0.882285i	$0.656004\pi$
$678$	−3.34165e23	−0.287414
$679$	2.22688e23	0.189149
$680$	0	0
$681$	8.14879e23	0.675060
$682$	−5.29640e24	−4.33325
$683$	−4.56949e23	−0.369226	−0.184613	−	0.982811i	$-0.559103\pi$
$683$	−4.56949e23	−0.369226	−0.184613	+	0.982811i	$0.559103\pi$
$684$	−1.91196e24	−1.52582
$685$	0	0
$686$	−8.60778e23	−0.670095
$687$	−6.76434e23	−0.520107
$688$	−1.61594e24	−1.22722
$689$	−1.81770e24	−1.36351
$690$	0	0
$691$	−1.38418e24	−1.01305	−0.506523	−	0.862226i	$-0.669070\pi$
$691$	−1.38418e24	−1.01305	−0.506523	+	0.862226i	$0.669070\pi$
$692$	−1.65152e24	−1.19394
$693$	−1.13568e23	−0.0811007
$694$	2.68278e24	1.89247
$695$	0	0
$696$	5.22461e24	3.59646
$697$	7.10785e22	0.0483348
$698$	−2.20314e23	−0.148003
$699$	7.43739e23	0.493588
$700$	0	0
$701$	9.18978e22	0.0595254	0.0297627	−	0.999557i	$-0.490525\pi$
$701$	9.18978e22	0.0595254	0.0297627	+	0.999557i	$0.490525\pi$
$702$	5.58473e23	0.357385
$703$	3.90946e23	0.247170
$704$	−9.06751e24	−5.66396
$705$	0	0
$706$	3.27063e24	1.99431
$707$	−3.64336e22	−0.0219501
$708$	2.48313e21	0.00147814
$709$	−1.54021e24	−0.905913	−0.452956	−	0.891533i	$-0.649631\pi$
$709$	−1.54021e24	−0.905913	−0.452956	+	0.891533i	$0.649631\pi$
$710$	0	0
$711$	−3.07644e23	−0.176668
$712$	−1.89781e24	−1.07690
$713$	−8.76365e23	−0.491387
$714$	1.58417e23	0.0877740
$715$	0	0
$716$	4.60651e24	2.49236
$717$	5.43949e23	0.290834
$718$	2.51332e24	1.32798
$719$	3.23522e24	1.68931	0.844654	−	0.535313i	$-0.179806\pi$
$719$	3.23522e24	1.68931	0.844654	+	0.535313i	$0.179806\pi$
$720$	0	0
$721$	−3.18382e23	−0.162367
$722$	6.70204e24	3.37786
$723$	1.06740e24	0.531679
$724$	6.08250e24	2.99436
$725$	0	0
$726$	−2.15426e24	−1.03595
$727$	2.33512e24	1.10986	0.554929	−	0.831898i	$-0.312745\pi$
$727$	2.33512e24	1.10986	0.554929	+	0.831898i	$0.312745\pi$
$728$	−1.22354e24	−0.574777
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	3.11531e23	0.141320
$732$	3.71132e24	1.66412
$733$	−1.81465e24	−0.804282	−0.402141	−	0.915578i	$-0.631734\pi$
$733$	−1.81465e24	−0.804282	−0.402141	+	0.915578i	$0.631734\pi$
$734$	4.02061e24	1.76147
$735$	0	0
$736$	−2.92618e24	−1.25268
$737$	1.29359e24	0.547422
$738$	−1.67265e23	−0.0699722
$739$	−1.21901e24	−0.504114	−0.252057	−	0.967712i	$-0.581107\pi$
$739$	−1.21901e24	−0.504114	−0.252057	+	0.967712i	$0.581107\pi$
$740$	0	0
$741$	−2.26349e24	−0.914797
$742$	1.21335e24	0.484790
$743$	−4.70556e23	−0.185869	−0.0929344	−	0.995672i	$-0.529625\pi$
$743$	−4.70556e23	−0.185869	−0.0929344	+	0.995672i	$0.529625\pi$
$744$	−8.15001e24	−3.18264
$745$	0	0
$746$	−5.47458e24	−2.08964
$747$	−3.89447e23	−0.146968
$748$	4.59325e24	1.71379
$749$	−8.16820e23	−0.301322
$750$	0	0
$751$	−3.20027e24	−1.15411	−0.577056	−	0.816705i	$-0.695798\pi$
$751$	−3.20027e24	−1.15411	−0.577056	+	0.816705i	$0.695798\pi$
$752$	3.88036e24	1.38363
$753$	−2.69269e24	−0.949356
$754$	9.68975e24	3.37797
$755$	0	0
$756$	−2.73775e23	−0.0933164
$757$	5.33142e24	1.79692	0.898458	−	0.439060i	$-0.144688\pi$
$757$	5.33142e24	1.79692	0.898458	+	0.439060i	$0.144688\pi$
$758$	3.09984e24	1.03312
$759$	−7.41916e23	−0.244512
$760$	0	0
$761$	3.19775e24	1.03057	0.515283	−	0.857020i	$-0.327687\pi$
$761$	3.19775e24	1.03057	0.515283	+	0.857020i	$0.327687\pi$
$762$	−4.61390e24	−1.47045
$763$	−7.87188e23	−0.248096
$764$	7.01636e24	2.18685
$765$	0	0
$766$	−5.21304e24	−1.58908
$767$	2.93967e21	0.000886214 0
$768$	−6.43984e24	−1.92001
$769$	−3.24975e24	−0.958245	−0.479122	−	0.877748i	$-0.659045\pi$
$769$	−3.24975e24	−0.958245	−0.479122	+	0.877748i	$0.659045\pi$
$770$	0	0
$771$	−1.11263e24	−0.320914
$772$	−1.96097e24	−0.559401
$773$	−4.35391e24	−1.22844	−0.614220	−	0.789135i	$-0.710529\pi$
$773$	−4.35391e24	−1.22844	−0.614220	+	0.789135i	$0.710529\pi$
$774$	−7.33108e23	−0.204583
$775$	0	0
$776$	1.35291e25	3.69354
$777$	5.59797e22	0.0151165
$778$	3.12586e24	0.834917
$779$	6.77925e23	0.179107
$780$	0	0
$781$	6.99538e24	1.80833
$782$	1.03490e24	0.264631
$783$	1.38398e24	0.350070
$784$	−1.50290e25	−3.76048
$785$	0	0
$786$	−2.93199e24	−0.717910
$787$	−1.67282e24	−0.405194	−0.202597	−	0.979262i	$-0.564938\pi$
$787$	−1.67282e24	−0.405194	−0.202597	+	0.979262i	$0.564938\pi$
$788$	−6.15565e24	−1.47503
$789$	−2.30044e24	−0.545327
$790$	0	0
$791$	1.93933e23	0.0449936
$792$	−6.89966e24	−1.58367
$793$	4.39368e24	0.997715
$794$	3.47180e23	0.0779974
$795$	0	0
$796$	−3.03948e24	−0.668402
$797$	−9.07285e23	−0.197400	−0.0987002	−	0.995117i	$-0.531468\pi$
$797$	−9.07285e23	−0.197400	−0.0987002	+	0.995117i	$0.531468\pi$
$798$	1.51093e24	0.325251
$799$	−7.48082e23	−0.159331
$800$	0	0
$801$	−5.02724e23	−0.104822
$802$	1.75156e25	3.61363
$803$	−1.16012e25	−2.36821
$804$	3.11840e24	0.629877
$805$	0	0
$806$	−1.51153e25	−2.98929
$807$	−1.92135e24	−0.375994
$808$	−2.21347e24	−0.428624
$809$	−9.92795e23	−0.190238	−0.0951189	−	0.995466i	$-0.530323\pi$
$809$	−9.92795e23	−0.190238	−0.0951189	+	0.995466i	$0.530323\pi$
$810$	0	0
$811$	5.63576e24	1.05749	0.528744	−	0.848781i	$-0.322663\pi$
$811$	5.63576e24	1.05749	0.528744	+	0.848781i	$0.322663\pi$
$812$	−4.75011e24	−0.882017
$813$	−3.70068e24	−0.680006
$814$	2.21016e24	0.401898
$815$	0	0
$816$	5.61959e24	1.00078
$817$	2.97129e24	0.523670
$818$	5.49999e24	0.959310
$819$	−3.24110e23	−0.0559474
$820$	0	0
$821$	−9.18781e24	−1.55344	−0.776721	−	0.629845i	$-0.783118\pi$
$821$	−9.18781e24	−1.55344	−0.776721	+	0.629845i	$0.783118\pi$
$822$	1.26923e24	0.212387
$823$	−6.83078e24	−1.13128	−0.565642	−	0.824651i	$-0.691372\pi$
$823$	−6.83078e24	−1.13128	−0.565642	+	0.824651i	$0.691372\pi$
$824$	−1.93428e25	−3.17057
$825$	0	0
$826$	−1.96229e21	−0.000315089 0
$827$	9.93808e24	1.57945	0.789724	−	0.613462i	$-0.210224\pi$
$827$	9.93808e24	1.57945	0.789724	+	0.613462i	$0.210224\pi$
$828$	−1.78851e24	−0.281341
$829$	1.00372e25	1.56278	0.781391	−	0.624042i	$-0.214511\pi$
$829$	1.00372e25	1.56278	0.781391	+	0.624042i	$0.214511\pi$
$830$	0	0
$831$	5.20185e24	0.793503
$832$	−2.58776e25	−3.90729
$833$	2.89738e24	0.433035
$834$	−1.66405e24	−0.246182
$835$	0	0
$836$	4.38090e25	6.35053
$837$	−2.15891e24	−0.309790
$838$	−9.64602e22	−0.0137017
$839$	1.16269e25	1.63489	0.817446	−	0.576005i	$-0.195389\pi$
$839$	1.16269e25	1.63489	0.817446	+	0.576005i	$0.195389\pi$
$840$	0	0
$841$	1.67555e25	2.30883
$842$	−1.29367e24	−0.176470
$843$	2.67381e24	0.361074
$844$	9.45671e24	1.26424
$845$	0	0
$846$	1.76042e24	0.230657
$847$	1.25023e24	0.162174
$848$	4.30419e25	5.52747
$849$	5.94296e24	0.755592
$850$	0	0
$851$	3.65703e23	0.0455750
$852$	1.68635e25	2.08071
$853$	−1.09678e23	−0.0133984	−0.00669922	−	0.999978i	$-0.502132\pi$
$853$	−1.09678e23	−0.0133984	−0.00669922	+	0.999978i	$0.502132\pi$
$854$	−2.93287e24	−0.354733
$855$	0	0
$856$	−4.96246e25	−5.88397
$857$	−1.52818e25	−1.79406	−0.897030	−	0.441970i	$-0.854280\pi$
$857$	−1.52818e25	−1.79406	−0.897030	+	0.441970i	$0.854280\pi$
$858$	−1.27964e25	−1.48746
$859$	−1.62639e25	−1.87190	−0.935951	−	0.352130i	$-0.885457\pi$
$859$	−1.62639e25	−1.87190	−0.935951	+	0.352130i	$0.885457\pi$
$860$	0	0
$861$	9.70723e22	0.0109539
$862$	−2.32266e24	−0.259522
$863$	8.64833e24	0.956842	0.478421	−	0.878130i	$-0.341209\pi$
$863$	8.64833e24	0.956842	0.478421	+	0.878130i	$0.341209\pi$
$864$	−7.20859e24	−0.789739
$865$	0	0
$866$	−1.05303e25	−1.13120
$867$	4.34414e24	0.462106
$868$	7.40981e24	0.780530
$869$	7.04908e24	0.735299
$870$	0	0
$871$	3.69175e24	0.377640
$872$	−4.78244e25	−4.84462
$873$	3.58381e24	0.359520
$874$	9.87056e24	0.980606
$875$	0	0
$876$	−2.79665e25	−2.72492
$877$	−1.02051e25	−0.984740	−0.492370	−	0.870386i	$-0.663869\pi$
$877$	−1.02051e25	−0.984740	−0.492370	+	0.870386i	$0.663869\pi$
$878$	3.70452e25	3.54021
$879$	8.06479e24	0.763286
$880$	0	0
$881$	−9.18419e23	−0.0852601	−0.0426301	−	0.999091i	$-0.513574\pi$
$881$	−9.18419e23	−0.0852601	−0.0426301	+	0.999091i	$0.513574\pi$
$882$	−6.81824e24	−0.626887
$883$	−8.44090e24	−0.768640	−0.384320	−	0.923200i	$-0.625564\pi$
$883$	−8.44090e24	−0.768640	−0.384320	+	0.923200i	$0.625564\pi$
$884$	1.31086e25	1.18226
$885$	0	0
$886$	−1.55480e25	−1.37559
$887$	1.20407e25	1.05512	0.527558	−	0.849519i	$-0.323108\pi$
$887$	1.20407e25	1.05512	0.527558	+	0.849519i	$0.323108\pi$
$888$	3.40096e24	0.295183
$889$	2.67768e24	0.230194
$890$	0	0
$891$	−1.82770e24	−0.154150
$892$	−4.33988e24	−0.362557
$893$	−7.13498e24	−0.590412
$894$	1.85660e25	1.52177
$895$	0	0
$896$	8.32548e24	0.669563
$897$	−2.11734e24	−0.168677
$898$	7.10700e24	0.560838
$899$	−3.74580e25	−2.92811
$900$	0	0
$901$	−8.29789e24	−0.636512
$902$	3.83256e24	0.291228
$903$	4.25460e23	0.0320267
$904$	1.17821e25	0.878599
$905$	0	0
$906$	1.39404e25	1.02020
$907$	2.51467e25	1.82314	0.911568	−	0.411150i	$-0.134873\pi$
$907$	2.51467e25	1.82314	0.911568	+	0.411150i	$0.134873\pi$
$908$	−4.50104e25	−3.23284
$909$	−5.86340e23	−0.0417212
$910$	0	0
$911$	−8.57548e24	−0.598897	−0.299448	−	0.954112i	$-0.596803\pi$
$911$	−8.57548e24	−0.598897	−0.299448	+	0.954112i	$0.596803\pi$
$912$	5.35980e25	3.70845
$913$	8.92343e24	0.611688
$914$	4.98552e25	3.38585
$915$	0	0
$916$	3.73633e25	2.49077
$917$	1.70158e24	0.112386
$918$	2.54946e24	0.166834
$919$	1.26569e25	0.820626	0.410313	−	0.911945i	$-0.365419\pi$
$919$	1.26569e25	0.820626	0.410313	+	0.911945i	$0.365419\pi$
$920$	0	0
$921$	−8.77661e24	−0.558624
$922$	7.71948e24	0.486828
$923$	1.99640e25	1.24748
$924$	6.27303e24	0.388388
$925$	0	0
$926$	−1.25060e25	−0.760198
$927$	−5.12384e24	−0.308616
$928$	−1.25072e26	−7.46454
$929$	−1.61298e25	−0.953881	−0.476941	−	0.878936i	$-0.658254\pi$
$929$	−1.61298e25	−0.953881	−0.476941	+	0.878936i	$0.658254\pi$
$930$	0	0
$931$	2.76343e25	1.60464
$932$	−4.10810e25	−2.36377
$933$	5.42304e23	0.0309206
$934$	−5.49832e25	−3.10657
$935$	0	0
$936$	−1.96908e25	−1.09249
$937$	−2.27726e24	−0.125206	−0.0626032	−	0.998038i	$-0.519940\pi$
$937$	−2.27726e24	−0.125206	−0.0626032	+	0.998038i	$0.519940\pi$
$938$	−2.46432e24	−0.134268
$939$	−1.89163e24	−0.102136
$940$	0	0
$941$	−1.42919e25	−0.757839	−0.378919	−	0.925430i	$-0.623704\pi$
$941$	−1.42919e25	−0.757839	−0.378919	+	0.925430i	$0.623704\pi$
$942$	−1.75559e25	−0.922551
$943$	6.34151e23	0.0330251
$944$	−6.96095e22	−0.00359258
$945$	0	0
$946$	1.67978e25	0.851486
$947$	1.82863e24	0.0918653	0.0459326	−	0.998945i	$-0.485374\pi$
$947$	1.82863e24	0.0918653	0.0459326	+	0.998945i	$0.485374\pi$
$948$	1.69929e25	0.846053
$949$	−3.31084e25	−1.63371
$950$	0	0
$951$	−2.08090e25	−1.00859
$952$	−5.58550e24	−0.268317
$953$	−1.66599e25	−0.793201	−0.396601	−	0.917991i	$-0.629810\pi$
$953$	−1.66599e25	−0.793201	−0.396601	+	0.917991i	$0.629810\pi$
$954$	1.95269e25	0.921452
$955$	0	0
$956$	−3.00455e25	−1.39279
$957$	−3.17113e25	−1.45701
$958$	−8.24470e25	−3.75464
$959$	−7.36596e23	−0.0332484
$960$	0	0
$961$	3.58816e25	1.59119
$962$	6.30753e24	0.277250
$963$	−1.31454e25	−0.572731
$964$	−5.89584e25	−2.54619
$965$	0	0
$966$	1.41337e24	0.0599722
$967$	3.59794e25	1.51331	0.756657	−	0.653812i	$-0.226831\pi$
$967$	3.59794e25	1.51331	0.756657	+	0.653812i	$0.226831\pi$
$968$	7.59556e25	3.16679
$969$	−1.03330e25	−0.427044
$970$	0	0
$971$	−2.68946e25	−1.09220	−0.546099	−	0.837721i	$-0.683888\pi$
$971$	−2.68946e25	−1.09220	−0.546099	+	0.837721i	$0.683888\pi$
$972$	−4.40596e24	−0.177369
$973$	9.65735e23	0.0385389
$974$	−3.09400e25	−1.22396
$975$	0	0
$976$	−1.04039e26	−4.04459
$977$	−1.26185e25	−0.486301	−0.243151	−	0.969989i	$-0.578181\pi$
$977$	−1.26185e25	−0.486301	−0.243151	+	0.969989i	$0.578181\pi$
$978$	−3.23215e25	−1.23484
$979$	1.15190e25	0.436276
$980$	0	0
$981$	−1.26685e25	−0.471563
$982$	−6.45936e24	−0.238365
$983$	2.72481e25	0.996852	0.498426	−	0.866932i	$-0.333912\pi$
$983$	2.72481e25	0.996852	0.498426	+	0.866932i	$0.333912\pi$
$984$	5.89747e24	0.213898
$985$	0	0
$986$	4.42342e25	1.57690
$987$	−1.02166e24	−0.0361086
$988$	1.25026e26	4.38092
$989$	2.77944e24	0.0965580
$990$	0	0
$991$	−2.34443e25	−0.800593	−0.400296	−	0.916386i	$-0.631093\pi$
$991$	−2.34443e25	−0.800593	−0.400296	+	0.916386i	$0.631093\pi$
$992$	1.95103e26	6.60565
$993$	1.04309e25	0.350150
$994$	−1.33264e25	−0.443535
$995$	0	0
$996$	2.15114e25	0.703823
$997$	−5.52750e25	−1.79316	−0.896582	−	0.442877i	$-0.853958\pi$
$997$	−5.52750e25	−1.79316	−0.896582	+	0.442877i	$0.853958\pi$
$998$	−6.40841e25	−2.06130
$999$	9.00902e23	0.0287323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.18.a.c.1.2		2
5.2	odd	4		75.18.b.b.49.4		4
5.3	odd	4		75.18.b.b.49.1		4
5.4	even	2		15.18.a.a.1.1	✓	2
15.14	odd	2		45.18.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.a.a.1.1	✓	2	5.4	even	2
45.18.a.b.1.2		2	15.14	odd	2
75.18.a.c.1.2		2	1.1	even	1	trivial
75.18.b.b.49.1		4	5.3	odd	4
75.18.b.b.49.4		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+702.477 q^{2} -6561.00 q^{3} +362402. q^{4} -4.60895e6 q^{6} +2.67481e6 q^{7} +1.62504e8 q^{8} +4.30467e7 q^{9} -9.86334e8 q^{11} -2.37772e9 q^{12} -2.81488e9 q^{13} +1.87899e9 q^{14} +6.66544e10 q^{16} -1.28501e10 q^{17} +3.02393e10 q^{18} -1.22560e11 q^{19} -1.75494e10 q^{21} -6.92877e11 q^{22} -1.14646e11 q^{23} -1.06619e12 q^{24} -1.97739e12 q^{26} -2.82430e11 q^{27} +9.69355e11 q^{28} -4.90027e12 q^{29} +7.64406e12 q^{31} +2.55235e13 q^{32} +6.47134e12 q^{33} -9.02688e12 q^{34} +1.56002e13 q^{36} -3.18983e12 q^{37} -8.60956e13 q^{38} +1.84684e13 q^{39} -5.53137e12 q^{41} -1.23281e13 q^{42} -2.42435e13 q^{43} -3.57449e14 q^{44} -8.05365e13 q^{46} +5.82161e13 q^{47} -4.37320e14 q^{48} -2.25476e14 q^{49} +8.43094e13 q^{51} -1.02012e15 q^{52} +6.45746e14 q^{53} -1.98400e14 q^{54} +4.34667e14 q^{56} +8.04117e14 q^{57} -3.44233e15 q^{58} -1.04433e12 q^{59} -1.56088e15 q^{61} +5.36978e15 q^{62} +1.15142e14 q^{63} +9.19314e15 q^{64} +4.54597e15 q^{66} -1.31151e15 q^{67} -4.65689e15 q^{68} +7.52195e14 q^{69} -7.09230e15 q^{71} +6.99526e15 q^{72} +1.17619e16 q^{73} -2.24078e15 q^{74} -4.44160e16 q^{76} -2.63826e15 q^{77} +1.29737e16 q^{78} -7.14674e15 q^{79} +1.85302e15 q^{81} -3.88566e15 q^{82} -9.04707e15 q^{83} -6.35994e15 q^{84} -1.70305e16 q^{86} +3.21507e16 q^{87} -1.60283e17 q^{88} -1.16786e16 q^{89} -7.52927e15 q^{91} -4.15481e16 q^{92} -5.01527e16 q^{93} +4.08955e16 q^{94} -1.67460e17 q^{96} +8.32538e16 q^{97} -1.58392e17 q^{98} -4.24585e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 356 q^{2} - 13122 q^{3} + 351376 q^{4} - 2335716 q^{6} + 20754552 q^{7} + 211737408 q^{8} + 86093442 q^{9} - 1131629912 q^{11} - 2305377936 q^{12} + 446672524 q^{13} - 4385222016 q^{14} + 51041317120 q^{16}+ \cdots - 48\!\cdots\!52 q^{99}+O(q^{100}) \) 2 * q + 356 * q^2 - 13122 * q^3 + 351376 * q^4 - 2335716 * q^6 + 20754552 * q^7 + 211737408 * q^8 + 86093442 * q^9 - 1131629912 * q^11 - 2305377936 * q^12 + 446672524 * q^13 - 4385222016 * q^14 + 51041317120 * q^16 - 20662021036 * q^17 + 15324632676 * q^18 - 194376127216 * q^19 - 136170615672 * q^21 - 642535396112 * q^22 - 281815042584 * q^23 - 1389209133888 * q^24 - 3107442504536 * q^26 - 564859072962 * q^27 + 770012242944 * q^28 - 4737062625892 * q^29 + 2256407074872 * q^31 + 24479953196032 * q^32 + 7424623852632 * q^33 - 6320225301928 * q^34 + 15125584638096 * q^36 + 33686114979452 * q^37 - 61213036677904 * q^38 - 2930618429964 * q^39 - 13051545859228 * q^41 + 28771441646976 * q^42 + 33149929851400 * q^43 - 355847272731712 * q^44 - 22616426690688 * q^46 - 84299585768440 * q^47 - 334882081624320 * q^48 - 131229302667470 * q^49 + 135563520017196 * q^51 - 1056079273719136 * q^52 - 155836696419964 * q^53 - 100544914987236 * q^54 + 1324797309480960 * q^56 + 1275301770664176 * q^57 - 3498878281127368 * q^58 + 963392423116456 * q^59 - 2858793214972660 * q^61 + 7236477385534944 * q^62 + 893415409423992 * q^63 + 11601155857485824 * q^64 + 4215674733890832 * q^66 - 6027859197776136 * q^67 - 4570758410249888 * q^68 + 1848988494393624 * q^69 + 2206536655060304 * q^71 + 9114601127439168 * q^72 + 18626002136017276 * q^73 - 15017443813698616 * q^74 - 43624167666234944 * q^76 - 5265163905377376 * q^77 + 20387930272260696 * q^78 + 4997891097934440 * q^79 + 3706040377703682 * q^81 - 1280087848985176 * q^82 + 23351553482511288 * q^83 - 5052050325955584 * q^84 - 36916031436093872 * q^86 + 31079867888477412 * q^87 - 167436503858533632 * q^88 - 60857257652216796 * q^89 + 51438794903148912 * q^91 - 39704913484919808 * q^92 - 14804286818235192 * q^93 + 90273897275494496 * q^94 - 160612972919165952 * q^96 - 42869862659782756 * q^97 - 191045888549711804 * q^98 - 48712957097118552 * q^99

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