LMFDB - Embedded newform 45.18.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,18,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	45.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:	$82.4499393051$
Analytic rank:	$0$
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	$\chi$	$=$	45.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+702.477 q^{2} +362402. q^{4} -390625. q^{5} -2.67481e6 q^{7} +1.62504e8 q^{8} -2.74405e8 q^{10} +9.86334e8 q^{11} +2.81488e9 q^{13} -1.87899e9 q^{14} +6.66544e10 q^{16} -1.28501e10 q^{17} -1.22560e11 q^{19} -1.41563e11 q^{20} +6.92877e11 q^{22} -1.14646e11 q^{23} +1.52588e11 q^{25} +1.97739e12 q^{26} -9.69355e11 q^{28} +4.90027e12 q^{29} +7.64406e12 q^{31} +2.55235e13 q^{32} -9.02688e12 q^{34} +1.04485e12 q^{35} +3.18983e12 q^{37} -8.60956e13 q^{38} -6.34781e13 q^{40} +5.53137e12 q^{41} +2.42435e13 q^{43} +3.57449e14 q^{44} -8.05365e13 q^{46} +5.82161e13 q^{47} -2.25476e14 q^{49} +1.07189e14 q^{50} +1.02012e15 q^{52} +6.45746e14 q^{53} -3.85287e14 q^{55} -4.34667e14 q^{56} +3.44233e15 q^{58} +1.04433e12 q^{59} -1.56088e15 q^{61} +5.36978e15 q^{62} +9.19314e15 q^{64} -1.09956e15 q^{65} +1.31151e15 q^{67} -4.65689e15 q^{68} +7.33981e14 q^{70} +7.09230e15 q^{71} -1.17619e16 q^{73} +2.24078e15 q^{74} -4.44160e16 q^{76} -2.63826e15 q^{77} -7.14674e15 q^{79} -2.60369e16 q^{80} +3.88566e15 q^{82} -9.04707e15 q^{83} +5.01956e15 q^{85} +1.70305e16 q^{86} +1.60283e17 q^{88} +1.16786e16 q^{89} -7.52927e15 q^{91} -4.15481e16 q^{92} +4.08955e16 q^{94} +4.78750e16 q^{95} -8.32538e16 q^{97} -1.58392e17 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 356 q^{2} + 351376 q^{4} - 781250 q^{5} - 20754552 q^{7} + 211737408 q^{8} - 139062500 q^{10} + 1131629912 q^{11} - 446672524 q^{13} + 4385222016 q^{14} + 51041317120 q^{16} - 20662021036 q^{17} - 194376127216 q^{19}+ \cdots - 19\!\cdots\!04 q^{98}+O(q^{100}) $ 2 * q + 356 * q^2 + 351376 * q^4 - 781250 * q^5 - 20754552 * q^7 + 211737408 * q^8 - 139062500 * q^10 + 1131629912 * q^11 - 446672524 * q^13 + 4385222016 * q^14 + 51041317120 * q^16 - 20662021036 * q^17 - 194376127216 * q^19 - 137256250000 * q^20 + 642535396112 * q^22 - 281815042584 * q^23 + 305175781250 * q^25 + 3107442504536 * q^26 - 770012242944 * q^28 + 4737062625892 * q^29 + 2256407074872 * q^31 + 24479953196032 * q^32 - 6320225301928 * q^34 + 8107246875000 * q^35 - 33686114979452 * q^37 - 61213036677904 * q^38 - 82709925000000 * q^40 + 13051545859228 * q^41 - 33149929851400 * q^43 + 355847272731712 * q^44 - 22616426690688 * q^46 - 84299585768440 * q^47 - 131229302667470 * q^49 + 54321289062500 * q^50 + 1056079273719136 * q^52 - 155836696419964 * q^53 - 442042934375000 * q^55 - 1324797309480960 * q^56 + 3498878281127368 * q^58 - 963392423116456 * q^59 - 2858793214972660 * q^61 + 7236477385534944 * q^62 + 11601155857485824 * q^64 + 174481454687500 * q^65 + 6027859197776136 * q^67 - 4570758410249888 * q^68 - 1712977350000000 * q^70 - 2206536655060304 * q^71 - 18626002136017276 * q^73 + 15017443813698616 * q^74 - 43624167666234944 * q^76 - 5265163905377376 * q^77 + 4997891097934440 * q^79 - 19938014500000000 * q^80 + 1280087848985176 * q^82 + 23351553482511288 * q^83 + 8071101967187500 * q^85 + 36916031436093872 * q^86 + 167436503858533632 * q^88 + 60857257652216796 * q^89 + 51438794903148912 * q^91 - 39704913484919808 * q^92 + 90273897275494496 * q^94 + 75928174693750000 * q^95 + 42869862659782756 * q^97 - 191045888549711804 * q^98

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	$0.0779453\pi$
$3$	0	0
$3$	0	0
$4$	362402.	2.76491
$5$	−390625.	−0.447214
$5$	−390625.	−0.447214
$6$	0	0
$7$	−2.67481e6	−0.175372	−0.0876858	−	0.996148i	$-0.527947\pi$
$7$	−2.67481e6	−0.175372	−0.0876858	+	0.996148i	$0.527947\pi$
$8$	1.62504e8	3.42451
$9$	0	0
$10$	−2.74405e8	−0.867745
$11$	9.86334e8	1.38735	0.693675	−	0.720288i	$-0.255990\pi$
$11$	9.86334e8	1.38735	0.693675	+	0.720288i	$0.255990\pi$
$12$	0	0
$13$	2.81488e9	0.957065	0.478533	−	0.878070i	$-0.341169\pi$
$13$	2.81488e9	0.957065	0.478533	+	0.878070i	$0.341169\pi$
$14$	−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	$0.571712\pi$
$18$	0	0
$19$	−1.22560e11	−1.65555	−0.827777	−	0.561057i	$-0.810395\pi$
$19$	−1.22560e11	−1.65555	−0.827777	+	0.561057i	$0.810395\pi$
$20$	−1.41563e11	−1.23650
$21$	0	0
$22$	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	$0.548775\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	1.97739e12	1.85703
$27$	0	0
$28$	−9.69355e11	−0.484886
$29$	4.90027e12	1.81902	0.909510	−	0.415682i	$-0.136457\pi$
$29$	4.90027e12	1.81902	0.909510	+	0.415682i	$0.136457\pi$
$30$	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
$33$	0	0
$34$	−9.02688e12	−0.866896
$35$	1.04485e12	0.0784286
$36$	0	0
$37$	3.18983e12	0.149298	0.0746488	−	0.997210i	$-0.476216\pi$
$37$	3.18983e12	0.149298	0.0746488	+	0.997210i	$0.476216\pi$
$38$	−8.60956e13	−3.21233
$39$	0	0
$40$	−6.34781e13	−1.53149
$41$	5.53137e12	0.108186	0.0540928	−	0.998536i	$-0.482773\pi$
$41$	5.53137e12	0.108186	0.0540928	+	0.998536i	$0.482773\pi$
$42$	0	0
$43$	2.42435e13	0.316311	0.158155	−	0.987414i	$-0.449445\pi$
$43$	2.42435e13	0.316311	0.158155	+	0.987414i	$0.449445\pi$
$44$	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	$0.442936\pi$
$48$	0	0
$49$	−2.25476e14	−0.969245
$50$	1.07189e14	0.388067
$51$	0	0
$52$	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	$0.247636\pi$
$54$	0	0
$55$	−3.85287e14	−0.620442
$56$	−4.34667e14	−0.600562
$57$	0	0
$58$	3.44233e15	3.52951
$59$	1.04433e12	0.000925970 0	0.000462985	−	1.00000i	$-0.499853\pi$
$59$	1.04433e12	0.000925970 0	0.000462985		1.00000i	$0.499853\pi$
$60$	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
$63$	0	0
$64$	9.19314e15	4.08258
$65$	−1.09956e15	−0.428013
$66$	0	0
$67$	1.31151e15	0.394581	0.197290	−	0.980345i	$-0.436786\pi$
$67$	1.31151e15	0.394581	0.197290	+	0.980345i	$0.436786\pi$
$68$	−4.65689e15	−1.23529
$69$	0	0
$70$	7.33981e14	0.152178
$71$	7.09230e15	1.30344	0.651720	−	0.758459i	$-0.274048\pi$
$71$	7.09230e15	1.30344	0.651720	+	0.758459i	$0.274048\pi$
$72$	0	0
$73$	−1.17619e16	−1.70700	−0.853501	−	0.521091i	$-0.825525\pi$
$73$	−1.17619e16	−1.70700	−0.853501	+	0.521091i	$0.825525\pi$
$74$	2.24078e15	0.289688
$75$	0	0
$76$	−4.44160e16	−4.57745
$77$	−2.63826e15	−0.243302
$78$	0	0
$79$	−7.14674e15	−0.530003	−0.265001	−	0.964248i	$-0.585372\pi$
$79$	−7.14674e15	−0.530003	−0.265001	+	0.964248i	$0.585372\pi$
$80$	−2.60369e16	−1.73510
$81$	0	0
$82$	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	$0.570753\pi$
$84$	0	0
$85$	5.01956e15	0.199804
$86$	1.70305e16	0.613749
$87$	0	0
$88$	1.60283e17	4.75100
$89$	1.16786e16	0.314467	0.157233	−	0.987561i	$-0.449743\pi$
$89$	1.16786e16	0.314467	0.157233	+	0.987561i	$0.449743\pi$
$90$	0	0
$91$	−7.52927e15	−0.167842
$92$	−4.15481e16	−0.844023
$93$	0	0
$94$	4.08955e16	0.691972
$95$	4.78750e16	0.740386
$96$	0	0
$97$	−8.32538e16	−1.07856	−0.539280	−	0.842126i	$-0.681304\pi$
$97$	−8.32538e16	−1.07856	−0.539280	+	0.842126i	$0.681304\pi$
$98$	−1.58392e17	−1.88066
$99$	0	0
$100$	5.52981e16	0.552981
$101$	1.36210e16	0.125164	0.0625818	−	0.998040i	$-0.480067\pi$
$101$	1.36210e16	0.125164	0.0625818	+	0.998040i	$0.480067\pi$
$102$	0	0
$103$	1.19030e17	0.925847	0.462923	−	0.886398i	$-0.346801\pi$
$103$	1.19030e17	0.925847	0.462923	+	0.886398i	$0.346801\pi$
$104$	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	$0.828974\pi$
$108$	0	0
$109$	−2.94297e17	−1.41469	−0.707344	−	0.706869i	$-0.750107\pi$
$109$	−2.94297e17	−1.41469	−0.707344	+	0.706869i	$0.750107\pi$
$110$	−2.70655e17	−1.20387
$111$	0	0
$112$	−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	$0.459054\pi$
$114$	0	0
$115$	4.47838e16	0.136518
$116$	1.77587e18	5.02942
$117$	0	0
$118$	7.33620e14	0.00179669
$119$	3.43715e16	0.0783519
$120$	0	0
$121$	4.67408e17	0.924742
$122$	−1.09648e18	−2.02275
$123$	0	0
$124$	2.77022e18	4.45072
$125$	−5.96046e16	−0.0894427
$126$	0	0
$127$	−1.00107e18	−1.31261	−0.656304	−	0.754497i	$-0.727881\pi$
$127$	−1.00107e18	−1.31261	−0.656304	+	0.754497i	$0.727881\pi$
$128$	3.11255e18	3.81797
$129$	0	0
$130$	−7.72418e17	−0.830489
$131$	−6.36150e17	−0.640846	−0.320423	−	0.947275i	$-0.603825\pi$
$131$	−6.36150e17	−0.640846	−0.320423	+	0.947275i	$0.603825\pi$
$132$	0	0
$133$	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	$0.530219\pi$
$138$	0	0
$139$	3.61048e17	0.219755	0.109878	−	0.993945i	$-0.464954\pi$
$139$	3.61048e17	0.219755	0.109878	+	0.993945i	$0.464954\pi$
$140$	3.78654e17	0.216848
$141$	0	0
$142$	4.98218e18	2.52911
$143$	2.77641e18	1.32779
$144$	0	0
$145$	−1.91417e18	−0.813490
$146$	−8.26248e18	−3.31216
$147$	0	0
$148$	1.15600e18	0.412794
$149$	4.02824e18	1.35841	0.679207	−	0.733947i	$-0.262324\pi$
$149$	4.02824e18	1.35841	0.679207	+	0.733947i	$0.262324\pi$
$150$	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
$153$	0	0
$154$	−1.85331e18	−0.472088
$155$	−2.98596e18	−0.719888
$156$	0	0
$157$	−3.80909e18	−0.823520	−0.411760	−	0.911292i	$-0.635086\pi$
$157$	−3.80909e18	−0.823520	−0.411760	+	0.911292i	$0.635086\pi$
$158$	−5.02042e18	−1.02838
$159$	0	0
$160$	−9.97012e18	−1.83519
$161$	3.06657e17	0.0535345
$162$	0	0
$163$	−7.01276e18	−1.10229	−0.551143	−	0.834411i	$-0.685808\pi$
$163$	−7.01276e18	−1.10229	−0.551143	+	0.834411i	$0.685808\pi$
$164$	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	$0.391882\pi$
$168$	0	0
$169$	−7.26857e17	−0.0840256
$170$	3.52613e18	0.387688
$171$	0	0
$172$	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	$0.569271\pi$
$174$	0	0
$175$	−4.08143e17	−0.0350743
$176$	6.57436e19	5.38264
$177$	0	0
$178$	8.20393e18	0.610171
$179$	−1.27111e19	−0.901428	−0.450714	−	0.892668i	$-0.648831\pi$
$179$	−1.27111e19	−0.901428	−0.450714	+	0.892668i	$0.648831\pi$
$180$	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
$183$	0	0
$184$	−1.86305e19	−1.04538
$185$	−1.24603e18	−0.0667679
$186$	0	0
$187$	−1.26745e19	−0.619835
$188$	2.10976e19	0.986034
$189$	0	0
$190$	3.36311e19	1.43660
$191$	−1.93607e19	−0.790930	−0.395465	−	0.918481i	$-0.629417\pi$
$191$	−1.93607e19	−0.790930	−0.395465	+	0.918481i	$0.629417\pi$
$192$	0	0
$193$	5.41103e18	0.202322	0.101161	−	0.994870i	$-0.467744\pi$
$193$	5.41103e18	0.202322	0.101161	+	0.994870i	$0.467744\pi$
$194$	−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	$0.585947\pi$
$198$	0	0
$199$	−8.38703e18	−0.241745	−0.120872	−	0.992668i	$-0.538569\pi$
$199$	−8.38703e18	−0.241745	−0.120872	+	0.992668i	$0.538569\pi$
$200$	2.47961e19	0.684902
$201$	0	0
$202$	9.56845e18	0.242860
$203$	−1.31073e19	−0.319004
$204$	0	0
$205$	−2.16069e18	−0.0483821
$206$	8.36156e19	1.79645
$207$	0	0
$208$	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
$213$	0	0
$214$	−2.14519e20	−3.33387
$215$	−9.47013e18	−0.141459
$216$	0	0
$217$	−2.04464e19	−0.282299
$218$	−2.06737e20	−2.74497
$219$	0	0
$220$	−1.39629e20	−1.71546
$221$	−3.61715e19	−0.427594
$222$	0	0
$223$	1.19753e19	0.131128	0.0655641	−	0.997848i	$-0.479115\pi$
$223$	1.19753e19	0.131128	0.0655641	+	0.997848i	$0.479115\pi$
$224$	−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	$0.698756\pi$
$228$	0	0
$229$	1.03099e20	0.900853	0.450426	−	0.892814i	$-0.351272\pi$
$229$	1.03099e20	0.900853	0.450426	+	0.892814i	$0.351272\pi$
$230$	3.14596e19	0.264890
$231$	0	0
$232$	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	$0.640591\pi$
$234$	0	0
$235$	−2.27407e19	−0.159487
$236$	3.78468e17	0.00256022
$237$	0	0
$238$	2.41452e19	0.152029
$239$	8.29065e19	0.503740	0.251870	−	0.967761i	$-0.418954\pi$
$239$	8.29065e19	0.503740	0.251870	+	0.967761i	$0.418954\pi$
$240$	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
$243$	0	0
$244$	−5.65664e20	−2.88234
$245$	8.80765e19	0.433459
$246$	0	0
$247$	−3.44992e20	−1.58447
$248$	1.24219e21	5.51250
$249$	0	0
$250$	−4.18709e19	−0.173549
$251$	−4.10409e20	−1.64433	−0.822167	−	0.569247i	$-0.807235\pi$
$251$	−4.10409e20	−1.64433	−0.822167	+	0.569247i	$0.807235\pi$
$252$	0	0
$253$	−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	$0.410355\pi$
$258$	0	0
$259$	−8.53218e18	−0.0261826
$260$	−3.98484e20	−1.18341
$261$	0	0
$262$	−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	$0.343436\pi$
$264$	0	0
$265$	−2.52245e20	−0.637136
$266$	2.30289e20	0.563352
$267$	0	0
$268$	4.75294e20	1.09098
$269$	−2.92843e20	−0.651240	−0.325620	−	0.945501i	$-0.605573\pi$
$269$	−2.92843e20	−0.651240	−0.325620	+	0.945501i	$0.605573\pi$
$270$	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
$273$	0	0
$274$	−1.93450e20	−0.367865
$275$	1.50503e20	0.277470
$276$	0	0
$277$	7.92843e20	1.37439	0.687194	−	0.726474i	$-0.258842\pi$
$277$	7.92843e20	1.37439	0.687194	+	0.726474i	$0.258842\pi$
$278$	2.53628e20	0.426399
$279$	0	0
$280$	1.69792e20	0.268580
$281$	4.07531e20	0.625399	0.312700	−	0.949852i	$-0.398767\pi$
$281$	4.07531e20	0.625399	0.312700	+	0.949852i	$0.398767\pi$
$282$	0	0
$283$	9.05800e20	1.30872	0.654362	−	0.756182i	$-0.272937\pi$
$283$	9.05800e20	1.30872	0.654362	+	0.756182i	$0.272937\pi$
$284$	2.57026e21	3.60389
$285$	0	0
$286$	1.95037e21	2.57635
$287$	−1.47953e19	−0.0189727
$288$	0	0
$289$	−6.62116e20	−0.800391
$290$	−1.34466e21	−1.57845
$291$	0	0
$292$	−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	$0.729878\pi$
$294$	0	0
$295$	−4.07943e17	−0.000414106 0
$296$	5.18359e20	0.511271
$297$	0	0
$298$	2.82974e21	2.63578
$299$	−3.22716e20	−0.292157
$300$	0	0
$301$	−6.48468e19	−0.0554719
$302$	−2.12474e21	−1.76704
$303$	0	0
$304$	−8.16918e21	−6.42322
$305$	6.09717e20	0.466208
$306$	0	0
$307$	−1.33769e21	−0.967565	−0.483783	−	0.875188i	$-0.660738\pi$
$307$	−1.33769e21	−0.967565	−0.483783	+	0.875188i	$0.660738\pi$
$308$	−9.56108e20	−0.672707
$309$	0	0
$310$	−2.09757e21	−1.39683
$311$	8.26556e19	0.0535561	0.0267781	−	0.999641i	$-0.491475\pi$
$311$	8.26556e19	0.0535561	0.0267781	+	0.999641i	$0.491475\pi$
$312$	0	0
$313$	−2.88314e20	−0.176905	−0.0884523	−	0.996080i	$-0.528192\pi$
$313$	−2.88314e20	−0.176905	−0.0884523	+	0.996080i	$0.528192\pi$
$314$	−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	$0.161866\pi$
$318$	0	0
$319$	4.83331e21	2.52362
$320$	−3.59107e21	−1.82578
$321$	0	0
$322$	2.15420e20	0.103875
$323$	1.57491e21	0.739662
$324$	0	0
$325$	4.29517e20	0.191413
$326$	−4.92630e21	−2.13881
$327$	0	0
$328$	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
$333$	0	0
$334$	3.65945e21	1.29292
$335$	−5.12309e20	−0.176462
$336$	0	0
$337$	−3.01551e20	−0.0987432	−0.0493716	−	0.998780i	$-0.515722\pi$
$337$	−3.01551e20	−0.0987432	−0.0493716	+	0.998780i	$0.515722\pi$
$338$	−5.10600e20	−0.163038
$339$	0	0
$340$	1.81910e21	0.552440
$341$	7.53960e21	2.23324
$342$	0	0
$343$	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	$0.337849\pi$
$348$	0	0
$349$	−3.13624e20	−0.0762769	−0.0381384	−	0.999272i	$-0.512143\pi$
$349$	−3.13624e20	−0.0762769	−0.0381384	+	0.999272i	$0.512143\pi$
$350$	−2.86711e20	−0.0680560
$351$	0	0
$352$	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	$0.328198\pi$
$354$	0	0
$355$	−2.77043e21	−0.582916
$356$	4.23234e21	0.869471
$357$	0	0
$358$	−8.92923e21	−1.74907
$359$	−3.57780e21	−0.684405	−0.342203	−	0.939626i	$-0.611173\pi$
$359$	−3.57780e21	−0.684405	−0.342203	+	0.939626i	$0.611173\pi$
$360$	0	0
$361$	9.54059e21	1.74086
$362$	1.17903e22	2.10136
$363$	0	0
$364$	−2.72862e21	−0.464068
$365$	4.59450e21	0.763395
$366$	0	0
$367$	−5.72348e21	−0.907818	−0.453909	−	0.891048i	$-0.649971\pi$
$367$	−5.72348e21	−0.907818	−0.453909	+	0.891048i	$0.649971\pi$
$368$	−7.64170e21	−1.18436
$369$	0	0
$370$	−8.75305e20	−0.129552
$371$	−1.72725e21	−0.249848
$372$	0	0
$373$	7.79326e21	1.07695	0.538473	−	0.842643i	$-0.319001\pi$
$373$	7.79326e21	1.07695	0.538473	+	0.842643i	$0.319001\pi$
$374$	−8.90352e21	−1.20269
$375$	0	0
$376$	9.46034e21	1.22127
$377$	1.37937e22	1.74092
$378$	0	0
$379$	4.41273e21	0.532444	0.266222	−	0.963912i	$-0.414225\pi$
$379$	4.41273e21	0.532444	0.266222	+	0.963912i	$0.414225\pi$
$380$	1.73500e22	2.04710
$381$	0	0
$382$	−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	$0.634292\pi$
$384$	0	0
$385$	1.03057e21	0.108808
$386$	3.80112e21	0.392572
$387$	0	0
$388$	−3.01713e22	−2.98212
$389$	−4.44977e21	−0.430295	−0.215147	−	0.976582i	$-0.569023\pi$
$389$	−4.44977e21	−0.430295	−0.215147	+	0.976582i	$0.569023\pi$
$390$	0	0
$391$	1.47322e21	0.136384
$392$	−3.66407e22	−3.31919
$393$	0	0
$394$	−1.19321e22	−1.03514
$395$	2.79170e21	0.237024
$396$	0	0
$397$	−4.94222e20	−0.0401979	−0.0200989	−	0.999798i	$-0.506398\pi$
$397$	−4.94222e20	−0.0401979	−0.0200989	+	0.999798i	$0.506398\pi$
$398$	−5.89170e21	−0.469066
$399$	0	0
$400$	1.01707e22	0.775960
$401$	−2.49341e22	−1.86237	−0.931187	−	0.364542i	$-0.881225\pi$
$401$	−2.49341e22	−1.86237	−0.931187	+	0.364542i	$0.881225\pi$
$402$	0	0
$403$	2.15171e22	1.54061
$404$	4.93628e21	0.346066
$405$	0	0
$406$	−9.20757e21	−0.618976
$407$	3.14624e21	0.207128
$408$	0	0
$409$	7.82942e21	0.494404	0.247202	−	0.968964i	$-0.420489\pi$
$409$	7.82942e21	0.494404	0.247202	+	0.968964i	$0.420489\pi$
$410$	−1.51783e21	−0.0938775
$411$	0	0
$412$	4.31366e22	2.55988
$413$	−2.79339e18	−0.000162389 0
$414$	0	0
$415$	3.53401e21	0.197178
$416$	7.18457e22	3.92742
$417$	0	0
$418$	−8.49191e22	−4.45663
$419$	1.37314e20	0.00706150	0.00353075	−	0.999994i	$-0.498876\pi$
$419$	1.37314e20	0.00706150	0.00353075	+	0.999994i	$0.498876\pi$
$420$	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
$423$	0	0
$424$	1.04936e23	4.87883
$425$	−1.96077e21	−0.0893552
$426$	0	0
$427$	4.17505e21	0.182820
$428$	−1.10669e23	−4.75064
$429$	0	0
$430$	−6.65255e21	−0.274477
$431$	3.30639e21	0.133751	0.0668756	−	0.997761i	$-0.478697\pi$
$431$	3.30639e21	0.133751	0.0668756	+	0.997761i	$0.478697\pi$
$432$	0	0
$433$	1.49902e22	0.582990	0.291495	−	0.956572i	$-0.405847\pi$
$433$	1.49902e22	0.582990	0.291495	+	0.956572i	$0.405847\pi$
$434$	−1.43631e22	−0.547755
$435$	0	0
$436$	−1.06654e23	−3.91148
$437$	1.40511e22	0.505379
$438$	0	0
$439$	5.27352e22	1.82453	0.912267	−	0.409597i	$-0.134331\pi$
$439$	5.27352e22	1.82453	0.912267	+	0.409597i	$0.134331\pi$
$440$	−6.26106e22	−2.12471
$441$	0	0
$442$	−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	$0.615339\pi$
$444$	0	0
$445$	−4.56194e21	−0.140634
$446$	8.41239e21	0.254433
$447$	0	0
$448$	−2.45899e22	−0.715968
$449$	−1.01171e22	−0.289041	−0.144521	−	0.989502i	$-0.546164\pi$
$449$	−1.01171e22	−0.289041	−0.144521	+	0.989502i	$0.546164\pi$
$450$	0	0
$451$	5.45578e21	0.150091
$452$	2.62754e22	0.709369
$453$	0	0
$454$	−8.72479e22	−2.26872
$455$	2.94112e21	0.0750613
$456$	0	0
$457$	−7.09706e22	−1.74498	−0.872490	−	0.488632i	$-0.837496\pi$
$457$	−7.09706e22	−1.74498	−0.872490	+	0.488632i	$0.837496\pi$
$458$	7.24248e22	1.74796
$459$	0	0
$460$	1.62297e22	0.377459
$461$	−1.09889e22	−0.250899	−0.125449	−	0.992100i	$-0.540037\pi$
$461$	−1.09889e22	−0.250899	−0.125449	+	0.992100i	$0.540037\pi$
$462$	0	0
$463$	1.78028e22	0.391787	0.195893	−	0.980625i	$-0.437239\pi$
$463$	1.78028e22	0.391787	0.195893	+	0.980625i	$0.437239\pi$
$464$	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	$0.795445\pi$
$468$	0	0
$469$	−3.50804e21	−0.0691983
$470$	−1.59748e22	−0.309459
$471$	0	0
$472$	1.69708e20	0.00317100
$473$	2.39122e22	0.438834
$474$	0	0
$475$	−1.87012e22	−0.331111
$476$	1.24563e22	0.216636
$477$	0	0
$478$	5.82399e22	0.977425
$479$	1.17366e23	1.93504	0.967522	−	0.252785i	$-0.0813467\pi$
$479$	1.17366e23	1.93504	0.967522	+	0.252785i	$0.0813467\pi$
$480$	0	0
$481$	8.97899e21	0.142888
$482$	−1.14285e23	−1.78685
$483$	0	0
$484$	1.69390e23	2.55683
$485$	3.25210e22	0.482347
$486$	0	0
$487$	4.40441e22	0.630800	0.315400	−	0.948959i	$-0.397861\pi$
$487$	4.40441e22	0.630800	0.315400	+	0.948959i	$0.397861\pi$
$488$	−2.53648e23	−3.56996
$489$	0	0
$490$	6.18717e22	0.841057
$491$	9.19512e21	0.122847	0.0614235	−	0.998112i	$-0.480436\pi$
$491$	9.19512e21	0.122847	0.0614235	+	0.998112i	$0.480436\pi$
$492$	0	0
$493$	−6.29689e22	−0.812695
$494$	−2.42349e23	−3.07441
$495$	0	0
$496$	5.09511e23	6.24539
$497$	−1.89705e22	−0.228586
$498$	0	0
$499$	−9.12259e22	−1.06234	−0.531170	−	0.847265i	$-0.678247\pi$
$499$	−9.12259e22	−1.06234	−0.531170	+	0.847265i	$0.678247\pi$
$500$	−2.16008e22	−0.247301
$501$	0	0
$502$	−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	$0.426529\pi$
$504$	0	0
$505$	−5.32071e21	−0.0559749
$506$	−7.94359e22	−0.821746
$507$	0	0
$508$	−3.62791e23	−3.62924
$509$	−3.12071e22	−0.307010	−0.153505	−	0.988148i	$-0.549056\pi$
$509$	−3.12071e22	−0.307010	−0.153505	+	0.988148i	$0.549056\pi$
$510$	0	0
$511$	3.14609e22	0.299360
$512$	2.81536e23	2.63475
$513$	0	0
$514$	1.19128e23	1.07852
$515$	−4.64960e22	−0.414051
$516$	0	0
$517$	5.74206e22	0.494764
$518$	−5.99366e21	−0.0508030
$519$	0	0
$520$	−1.78683e23	−1.46573
$521$	1.71736e23	1.38593	0.692965	−	0.720971i	$-0.256304\pi$
$521$	1.71736e23	1.38593	0.692965	+	0.720971i	$0.256304\pi$
$522$	0	0
$523$	−3.54202e22	−0.276686	−0.138343	−	0.990384i	$-0.544178\pi$
$523$	−3.54202e22	−0.276686	−0.138343	+	0.990384i	$0.544178\pi$
$524$	−2.30542e23	−1.77188
$525$	0	0
$526$	2.46305e23	1.83271
$527$	−9.82268e22	−0.719184
$528$	0	0
$529$	−1.27906e23	−0.906815
$530$	−1.77196e23	−1.23626
$531$	0	0
$532$	1.18804e23	0.802755
$533$	1.55701e22	0.103541
$534$	0	0
$535$	1.19287e23	0.768399
$536$	2.13126e23	1.35125
$537$	0	0
$538$	−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
$543$	0	0
$544$	−3.27979e23	−1.83339
$545$	1.14960e23	0.632668
$546$	0	0
$547$	2.96616e23	1.58235	0.791176	−	0.611589i	$-0.209469\pi$
$547$	2.96616e23	1.58235	0.791176	+	0.611589i	$0.209469\pi$
$548$	−9.97992e22	−0.524194
$549$	0	0
$550$	1.05725e23	0.538385
$551$	−6.00578e23	−3.01149
$552$	0	0
$553$	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	$0.460108\pi$
$558$	0	0
$559$	6.82427e22	0.302730
$560$	6.96437e22	0.304287
$561$	0	0
$562$	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	$0.420419\pi$
$564$	0	0
$565$	−2.83217e22	−0.114738
$566$	6.36304e23	2.53936
$567$	0	0
$568$	1.15253e24	4.46365
$569$	4.39785e23	1.67798	0.838988	−	0.544149i	$-0.183147\pi$
$569$	4.39785e23	1.67798	0.838988	+	0.544149i	$0.183147\pi$
$570$	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
$573$	0	0
$574$	−1.03934e22	−0.0368134
$575$	−1.74937e22	−0.0610526
$576$	0	0
$577$	5.15333e23	1.74620	0.873099	−	0.487543i	$-0.162107\pi$
$577$	5.15333e23	1.74620	0.873099	+	0.487543i	$0.162107\pi$
$578$	−4.65121e23	−1.55303
$579$	0	0
$580$	−6.93698e23	−2.24922
$581$	2.41992e22	0.0773220
$582$	0	0
$583$	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	$0.483216\pi$
$588$	0	0
$589$	−9.36857e23	−2.66498
$590$	−2.86570e20	−0.000803506 0
$591$	0	0
$592$	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	$0.335653\pi$
$594$	0	0
$595$	−1.34264e22	−0.0350400
$596$	1.45984e24	3.75588
$597$	0	0
$598$	−2.26701e23	−0.566882
$599$	6.97132e23	1.71865	0.859324	−	0.511432i	$-0.170885\pi$
$599$	6.97132e23	1.71865	0.859324	+	0.511432i	$0.170885\pi$
$600$	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
$603$	0	0
$604$	−1.09614e24	−2.51797
$605$	−1.82581e23	−0.413557
$606$	0	0
$607$	−3.08234e23	−0.678855	−0.339427	−	0.940632i	$-0.610233\pi$
$607$	−3.08234e23	−0.678855	−0.339427	+	0.940632i	$0.610233\pi$
$608$	−3.12816e24	−6.79374
$609$	0	0
$610$	4.28312e23	0.904601
$611$	1.63872e23	0.341313
$612$	0	0
$613$	−4.35176e23	−0.881558	−0.440779	−	0.897616i	$-0.645298\pi$
$613$	−4.35176e23	−0.881558	−0.440779	+	0.897616i	$0.645298\pi$
$614$	−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	$0.696594\pi$
$618$	0	0
$619$	−3.33503e23	−0.621913	−0.310956	−	0.950424i	$-0.600649\pi$
$619$	−3.33503e23	−0.621913	−0.310956	+	0.950424i	$0.600649\pi$
$620$	−1.08212e24	−1.99042
$621$	0	0
$622$	5.80637e22	0.103917
$623$	−3.12379e22	−0.0551485
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	−2.02534e23	−0.343255
$627$	0	0
$628$	−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
$633$	0	0
$634$	2.22799e24	3.38964
$635$	3.91045e23	0.587016
$636$	0	0
$637$	−6.34688e23	−0.927631
$638$	3.39529e24	4.89667
$639$	0	0
$640$	−1.21584e24	−1.70745
$641$	1.34739e24	1.86724	0.933620	−	0.358264i	$-0.116631\pi$
$641$	1.34739e24	1.86724	0.933620	+	0.358264i	$0.116631\pi$
$642$	0	0
$643$	1.07291e24	1.44800	0.723999	−	0.689801i	$-0.242302\pi$
$643$	1.07291e24	1.44800	0.723999	+	0.689801i	$0.242302\pi$
$644$	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	$0.552559\pi$
$648$	0	0
$649$	1.03006e21	0.00128465
$650$	3.01726e23	0.371406
$651$	0	0
$652$	−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	$0.691336\pi$
$654$	0	0
$655$	2.48496e23	0.286595
$656$	3.68690e23	0.419739
$657$	0	0
$658$	−1.09388e23	−0.121352
$659$	4.48249e23	0.490901	0.245450	−	0.969409i	$-0.421064\pi$
$659$	4.48249e23	0.490901	0.245450	+	0.969409i	$0.421064\pi$
$660$	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
$663$	0	0
$664$	−1.47018e24	−1.50988
$665$	−1.28057e23	−0.129843
$666$	0	0
$667$	−5.61799e23	−0.555279
$668$	1.88788e24	1.84236
$669$	0	0
$670$	−3.59885e23	−0.342396
$671$	−1.53955e24	−1.44628
$672$	0	0
$673$	−9.61434e23	−0.880626	−0.440313	−	0.897844i	$-0.645132\pi$
$673$	−9.61434e23	−0.880626	−0.440313	+	0.897844i	$0.645132\pi$
$674$	−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	$0.656004\pi$
$678$	0	0
$679$	2.22688e23	0.189149
$680$	8.15698e23	0.684232
$681$	0	0
$682$	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	$0.559103\pi$
$684$	0	0
$685$	1.07571e23	0.0847865
$686$	8.60778e23	0.670095
$687$	0	0
$688$	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
$693$	0	0
$694$	2.68278e24	1.89247
$695$	−1.41035e23	−0.0982776
$696$	0	0
$697$	−7.10785e22	−0.0483348
$698$	−2.20314e23	−0.148003
$699$	0	0
$700$	−1.47912e23	−0.0969772
$701$	−9.18978e22	−0.0595254	−0.0297627	−	0.999557i	$-0.509475\pi$
$701$	−9.18978e22	−0.0595254	−0.0297627	+	0.999557i	$0.509475\pi$
$702$	0	0
$703$	−3.90946e23	−0.247170
$704$	9.06751e24	5.66396
$705$	0	0
$706$	3.27063e24	1.99431
$707$	−3.64336e22	−0.0219501
$708$	0	0
$709$	−1.54021e24	−0.905913	−0.452956	−	0.891533i	$-0.649631\pi$
$709$	−1.54021e24	−0.905913	−0.452956	+	0.891533i	$0.649631\pi$
$710$	−1.94616e24	−1.13105
$711$	0	0
$712$	1.89781e24	1.07690
$713$	−8.76365e23	−0.491387
$714$	0	0
$715$	−1.08454e24	−0.593804
$716$	−4.60651e24	−2.49236
$717$	0	0
$718$	−2.51332e24	−1.32798
$719$	−3.23522e24	−1.68931	−0.844654	−	0.535313i	$-0.820194\pi$
$719$	−3.23522e24	−1.68931	−0.844654	+	0.535313i	$0.820194\pi$
$720$	0	0
$721$	−3.18382e23	−0.162367
$722$	6.70204e24	3.37786
$723$	0	0
$724$	6.08250e24	2.99436
$725$	7.47722e23	0.363804
$726$	0	0
$727$	−2.33512e24	−1.10986	−0.554929	−	0.831898i	$-0.687255\pi$
$727$	−2.33512e24	−1.10986	−0.554929	+	0.831898i	$0.687255\pi$
$728$	−1.22354e24	−0.574777
$729$	0	0
$730$	3.22753e24	1.48124
$731$	−3.11531e23	−0.141320
$732$	0	0
$733$	1.81465e24	0.804282	0.402141	−	0.915578i	$-0.368266\pi$
$733$	1.81465e24	0.804282	0.402141	+	0.915578i	$0.368266\pi$
$734$	−4.02061e24	−1.76147
$735$	0	0
$736$	−2.92618e24	−1.25268
$737$	1.29359e24	0.547422
$738$	0	0
$739$	−1.21901e24	−0.504114	−0.252057	−	0.967712i	$-0.581107\pi$
$739$	−1.21901e24	−0.504114	−0.252057	+	0.967712i	$0.581107\pi$
$740$	−4.51562e23	−0.184607
$741$	0	0
$742$	−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	$0.529625\pi$
$744$	0	0
$745$	−1.57353e24	−0.607501
$746$	5.47458e24	2.08964
$747$	0	0
$748$	−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
$753$	0	0
$754$	9.68975e24	3.37797
$755$	1.18150e24	0.407273
$756$	0	0
$757$	−5.33142e24	−1.79692	−0.898458	−	0.439060i	$-0.855312\pi$
$757$	−5.33142e24	−1.79692	−0.898458	+	0.439060i	$0.855312\pi$
$758$	3.09984e24	1.03312
$759$	0	0
$760$	7.77988e24	2.53546
$761$	−3.19775e24	−1.03057	−0.515283	−	0.857020i	$-0.672313\pi$
$761$	−3.19775e24	−1.03057	−0.515283	+	0.857020i	$0.672313\pi$
$762$	0	0
$763$	7.87188e23	0.248096
$764$	−7.01636e24	−2.18685
$765$	0	0
$766$	−5.21304e24	−1.58908
$767$	2.93967e21	0.000886214 0
$768$	0	0
$769$	−3.24975e24	−0.958245	−0.479122	−	0.877748i	$-0.659045\pi$
$769$	−3.24975e24	−0.958245	−0.479122	+	0.877748i	$0.659045\pi$
$770$	7.23950e23	0.211124
$771$	0	0
$772$	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	$0.710529\pi$
$774$	0	0
$775$	1.16639e24	0.321944
$776$	−1.35291e25	−3.69354
$777$	0	0
$778$	−3.12586e24	−0.834917
$779$	−6.77925e23	−0.179107
$780$	0	0
$781$	6.99538e24	1.80833
$782$	1.03490e24	0.264631
$783$	0	0
$784$	−1.50290e25	−3.76048
$785$	1.48792e24	0.368289
$786$	0	0
$787$	1.67282e24	0.405194	0.202597	−	0.979262i	$-0.435062\pi$
$787$	1.67282e24	0.405194	0.202597	+	0.979262i	$0.435062\pi$
$788$	−6.15565e24	−1.47503
$789$	0	0
$790$	1.96110e24	0.459907
$791$	−1.93933e23	−0.0449936
$792$	0	0
$793$	−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	$0.531468\pi$
$798$	0	0
$799$	−7.48082e23	−0.159331
$800$	3.89458e24	0.820721
$801$	0	0
$802$	−1.75156e25	−3.61363
$803$	−1.16012e25	−2.36821
$804$	0	0
$805$	−1.19788e23	−0.0239413
$806$	1.51153e25	2.98929
$807$	0	0
$808$	2.21347e24	0.428624
$809$	9.92795e23	0.190238	0.0951189	−	0.995466i	$-0.469677\pi$
$809$	9.92795e23	0.190238	0.0951189	+	0.995466i	$0.469677\pi$
$810$	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
$813$	0	0
$814$	2.21016e24	0.401898
$815$	2.73936e24	0.492957
$816$	0	0
$817$	−2.97129e24	−0.523670
$818$	5.49999e24	0.959310
$819$	0	0
$820$	−7.83038e23	−0.133772
$821$	9.18781e24	1.55344	0.776721	−	0.629845i	$-0.216882\pi$
$821$	9.18781e24	1.55344	0.776721	+	0.629845i	$0.216882\pi$
$822$	0	0
$823$	6.83078e24	1.13128	0.565642	−	0.824651i	$-0.308628\pi$
$823$	6.83078e24	1.13128	0.565642	+	0.824651i	$0.308628\pi$
$824$	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	$0.210224\pi$
$828$	0	0
$829$	1.00372e25	1.56278	0.781391	−	0.624042i	$-0.214511\pi$
$829$	1.00372e25	1.56278	0.781391	+	0.624042i	$0.214511\pi$
$830$	2.48256e24	0.382592
$831$	0	0
$832$	2.58776e25	3.90729
$833$	2.89738e24	0.433035
$834$	0	0
$835$	−2.03491e24	−0.297995
$836$	−4.38090e25	−6.35053
$837$	0	0
$838$	9.64602e22	0.0137017
$839$	−1.16269e25	−1.63489	−0.817446	−	0.576005i	$-0.804611\pi$
$839$	−1.16269e25	−1.63489	−0.817446	+	0.576005i	$0.804611\pi$
$840$	0	0
$841$	1.67555e25	2.30883
$842$	−1.29367e24	−0.176470
$843$	0	0
$844$	9.45671e24	1.26424
$845$	2.83928e23	0.0375774
$846$	0	0
$847$	−1.25023e24	−0.162174
$848$	4.30419e25	5.52747
$849$	0	0
$850$	−1.37739e24	−0.173379
$851$	−3.65703e23	−0.0455750
$852$	0	0
$853$	1.09678e23	0.0133984	0.00669922	−	0.999978i	$-0.497868\pi$
$853$	1.09678e23	0.0133984	0.00669922	+	0.999978i	$0.497868\pi$
$854$	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	$0.854280\pi$
$858$	0	0
$859$	−1.62639e25	−1.87190	−0.935951	−	0.352130i	$-0.885457\pi$
$859$	−1.62639e25	−1.87190	−0.935951	+	0.352130i	$0.885457\pi$
$860$	−3.43199e24	−0.391119
$861$	0	0
$862$	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	$0.341209\pi$
$864$	0	0
$865$	1.78013e24	0.193115
$866$	1.05303e25	1.13120
$867$	0	0
$868$	−7.40981e24	−0.780530
$869$	−7.04908e24	−0.735299
$870$	0	0
$871$	3.69175e24	0.377640
$872$	−4.78244e25	−4.84462
$873$	0	0
$874$	9.87056e24	0.980606
$875$	1.59431e23	0.0156857
$876$	0	0
$877$	1.02051e25	0.984740	0.492370	−	0.870386i	$-0.336131\pi$
$877$	1.02051e25	0.984740	0.492370	+	0.870386i	$0.336131\pi$
$878$	3.70452e25	3.54021
$879$	0	0
$880$	−2.56811e25	−2.40719
$881$	9.18419e23	0.0852601	0.0426301	−	0.999091i	$-0.486426\pi$
$881$	9.18419e23	0.0852601	0.0426301	+	0.999091i	$0.486426\pi$
$882$	0	0
$883$	8.44090e24	0.768640	0.384320	−	0.923200i	$-0.374436\pi$
$883$	8.44090e24	0.768640	0.384320	+	0.923200i	$0.374436\pi$
$884$	−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	$0.323108\pi$
$888$	0	0
$889$	2.67768e24	0.230194
$890$	−3.20466e24	−0.272877
$891$	0	0
$892$	4.33988e24	0.362557
$893$	−7.13498e24	−0.590412
$894$	0	0
$895$	4.96526e24	0.403131
$896$	−8.32548e24	−0.669563
$897$	0	0
$898$	−7.10700e24	−0.560838
$899$	3.74580e25	2.92811
$900$	0	0
$901$	−8.29789e24	−0.636512
$902$	3.83256e24	0.291228
$903$	0	0
$904$	1.17821e25	0.878599
$905$	−6.55620e24	−0.484327
$906$	0	0
$907$	−2.51467e25	−1.82314	−0.911568	−	0.411150i	$-0.865127\pi$
$907$	−2.51467e25	−1.82314	−0.911568	+	0.411150i	$0.865127\pi$
$908$	−4.50104e25	−3.23284
$909$	0	0
$910$	2.06607e24	0.145644
$911$	8.57548e24	0.598897	0.299448	−	0.954112i	$-0.403197\pi$
$911$	8.57548e24	0.598897	0.299448	+	0.954112i	$0.403197\pi$
$912$	0	0
$913$	−8.92343e24	−0.611688
$914$	−4.98552e25	−3.38585
$915$	0	0
$916$	3.73633e25	2.49077
$917$	1.70158e24	0.112386
$918$	0	0
$919$	1.26569e25	0.820626	0.410313	−	0.911945i	$-0.365419\pi$
$919$	1.26569e25	0.820626	0.410313	+	0.911945i	$0.365419\pi$
$920$	7.27753e24	0.467507
$921$	0	0
$922$	−7.71948e24	−0.486828
$923$	1.99640e25	1.24748
$924$	0	0
$925$	4.86729e23	0.0298595
$926$	1.25060e25	0.760198
$927$	0	0
$928$	1.25072e26	7.46454
$929$	1.61298e25	0.953881	0.476941	−	0.878936i	$-0.341746\pi$
$929$	1.61298e25	0.953881	0.476941	+	0.878936i	$0.341746\pi$
$930$	0	0
$931$	2.76343e25	1.60464
$932$	−4.10810e25	−2.36377
$933$	0	0
$934$	−5.49832e25	−3.10657
$935$	4.95097e24	0.277199
$936$	0	0
$937$	2.27726e24	0.125206	0.0626032	−	0.998038i	$-0.480060\pi$
$937$	2.27726e24	0.125206	0.0626032	+	0.998038i	$0.480060\pi$
$938$	−2.46432e24	−0.134268
$939$	0	0
$940$	−8.24126e24	−0.440968
$941$	1.42919e25	0.757839	0.378919	−	0.925430i	$-0.376296\pi$
$941$	1.42919e25	0.757839	0.378919	+	0.925430i	$0.376296\pi$
$942$	0	0
$943$	−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	$0.485374\pi$
$948$	0	0
$949$	−3.31084e25	−1.63371
$950$	−1.31372e25	−0.642467
$951$	0	0
$952$	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	$0.629810\pi$
$954$	0	0
$955$	7.56278e24	0.353715
$956$	3.00455e25	1.39279
$957$	0	0
$958$	8.24470e25	3.75464
$959$	7.36596e23	0.0332484
$960$	0	0
$961$	3.58816e25	1.59119
$962$	6.30753e24	0.277250
$963$	0	0
$964$	−5.89584e25	−2.54619
$965$	−2.11368e24	−0.0904810
$966$	0	0
$967$	−3.59794e25	−1.51331	−0.756657	−	0.653812i	$-0.773169\pi$
$967$	−3.59794e25	−1.51331	−0.756657	+	0.653812i	$0.773169\pi$
$968$	7.59556e25	3.16679
$969$	0	0
$970$	2.28453e25	0.935915
$971$	2.68946e25	1.09220	0.546099	−	0.837721i	$-0.316112\pi$
$971$	2.68946e25	1.09220	0.546099	+	0.837721i	$0.316112\pi$
$972$	0	0
$973$	−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	$0.578181\pi$
$978$	0	0
$979$	1.15190e25	0.436276
$980$	3.19191e25	1.19847
$981$	0	0
$982$	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	$0.333912\pi$
$984$	0	0
$985$	6.63504e24	0.238581
$986$	−4.42342e25	−1.57690
$987$	0	0
$988$	−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
$993$	0	0
$994$	−1.33264e25	−0.443535
$995$	3.27619e24	0.108112
$996$	0	0
$997$	5.52750e25	1.79316	0.896582	−	0.442877i	$-0.146042\pi$
$997$	5.52750e25	1.79316	0.896582	+	0.442877i	$0.146042\pi$
$998$	−6.40841e25	−2.06130
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.a.a.1.1	✓	2	3.2	odd	2
45.18.a.b.1.2		2	1.1	even	1	trivial
75.18.a.c.1.2		2	15.14	odd	2
75.18.b.b.49.1		4	15.2	even	4
75.18.b.b.49.4		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q+702.477 q^{2} +362402. q^{4} -390625. q^{5} -2.67481e6 q^{7} +1.62504e8 q^{8} -2.74405e8 q^{10} +9.86334e8 q^{11} +2.81488e9 q^{13} -1.87899e9 q^{14} +6.66544e10 q^{16} -1.28501e10 q^{17} -1.22560e11 q^{19} -1.41563e11 q^{20} +6.92877e11 q^{22} -1.14646e11 q^{23} +1.52588e11 q^{25} +1.97739e12 q^{26} -9.69355e11 q^{28} +4.90027e12 q^{29} +7.64406e12 q^{31} +2.55235e13 q^{32} -9.02688e12 q^{34} +1.04485e12 q^{35} +3.18983e12 q^{37} -8.60956e13 q^{38} -6.34781e13 q^{40} +5.53137e12 q^{41} +2.42435e13 q^{43} +3.57449e14 q^{44} -8.05365e13 q^{46} +5.82161e13 q^{47} -2.25476e14 q^{49} +1.07189e14 q^{50} +1.02012e15 q^{52} +6.45746e14 q^{53} -3.85287e14 q^{55} -4.34667e14 q^{56} +3.44233e15 q^{58} +1.04433e12 q^{59} -1.56088e15 q^{61} +5.36978e15 q^{62} +9.19314e15 q^{64} -1.09956e15 q^{65} +1.31151e15 q^{67} -4.65689e15 q^{68} +7.33981e14 q^{70} +7.09230e15 q^{71} -1.17619e16 q^{73} +2.24078e15 q^{74} -4.44160e16 q^{76} -2.63826e15 q^{77} -7.14674e15 q^{79} -2.60369e16 q^{80} +3.88566e15 q^{82} -9.04707e15 q^{83} +5.01956e15 q^{85} +1.70305e16 q^{86} +1.60283e17 q^{88} +1.16786e16 q^{89} -7.52927e15 q^{91} -4.15481e16 q^{92} +4.08955e16 q^{94} +4.78750e16 q^{95} -8.32538e16 q^{97} -1.58392e17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 356 q^{2} + 351376 q^{4} - 781250 q^{5} - 20754552 q^{7} + 211737408 q^{8} - 139062500 q^{10} + 1131629912 q^{11} - 446672524 q^{13} + 4385222016 q^{14} + 51041317120 q^{16} - 20662021036 q^{17} - 194376127216 q^{19}+ \cdots - 19\!\cdots\!04 q^{98}+O(q^{100}) \) 2 * q + 356 * q^2 + 351376 * q^4 - 781250 * q^5 - 20754552 * q^7 + 211737408 * q^8 - 139062500 * q^10 + 1131629912 * q^11 - 446672524 * q^13 + 4385222016 * q^14 + 51041317120 * q^16 - 20662021036 * q^17 - 194376127216 * q^19 - 137256250000 * q^20 + 642535396112 * q^22 - 281815042584 * q^23 + 305175781250 * q^25 + 3107442504536 * q^26 - 770012242944 * q^28 + 4737062625892 * q^29 + 2256407074872 * q^31 + 24479953196032 * q^32 - 6320225301928 * q^34 + 8107246875000 * q^35 - 33686114979452 * q^37 - 61213036677904 * q^38 - 82709925000000 * q^40 + 13051545859228 * q^41 - 33149929851400 * q^43 + 355847272731712 * q^44 - 22616426690688 * q^46 - 84299585768440 * q^47 - 131229302667470 * q^49 + 54321289062500 * q^50 + 1056079273719136 * q^52 - 155836696419964 * q^53 - 442042934375000 * q^55 - 1324797309480960 * q^56 + 3498878281127368 * q^58 - 963392423116456 * q^59 - 2858793214972660 * q^61 + 7236477385534944 * q^62 + 11601155857485824 * q^64 + 174481454687500 * q^65 + 6027859197776136 * q^67 - 4570758410249888 * q^68 - 1712977350000000 * q^70 - 2206536655060304 * q^71 - 18626002136017276 * q^73 + 15017443813698616 * q^74 - 43624167666234944 * q^76 - 5265163905377376 * q^77 + 4997891097934440 * q^79 - 19938014500000000 * q^80 + 1280087848985176 * q^82 + 23351553482511288 * q^83 + 8071101967187500 * q^85 + 36916031436093872 * q^86 + 167436503858533632 * q^88 + 60857257652216796 * q^89 + 51438794903148912 * q^91 - 39704913484919808 * q^92 + 90273897275494496 * q^94 + 75928174693750000 * q^95 + 42869862659782756 * q^97 - 191045888549711804 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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