LMFDB - Embedded newform 90.16.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,16,Mod(1,90)] 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("90.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-256,0,32768,156250,0,-1761200] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$128.424154590$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5168130 $ x^2 - x - 5168130
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2272.85$ of defining polynomial
Character	$\chi$	$=$	90.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-128.000 q^{2} +16384.0 q^{4} +78125.0 q^{5} -307715. q^{7} -2.09715e6 q^{8} -1.00000e7 q^{10} +4.53233e7 q^{11} +2.10577e7 q^{13} +3.93876e7 q^{14} +2.68435e8 q^{16} -2.68450e9 q^{17} +2.06108e9 q^{19} +1.28000e9 q^{20} -5.80138e9 q^{22} +2.41078e10 q^{23} +6.10352e9 q^{25} -2.69539e9 q^{26} -5.04161e9 q^{28} -1.07547e11 q^{29} -1.24013e11 q^{31} -3.43597e10 q^{32} +3.43616e11 q^{34} -2.40403e10 q^{35} -9.96531e11 q^{37} -2.63819e11 q^{38} -1.63840e11 q^{40} -4.04851e11 q^{41} +2.24675e12 q^{43} +7.42577e11 q^{44} -3.08580e12 q^{46} +7.03758e11 q^{47} -4.65287e12 q^{49} -7.81250e11 q^{50} +3.45010e11 q^{52} +1.16581e13 q^{53} +3.54088e12 q^{55} +6.45326e11 q^{56} +1.37660e13 q^{58} -3.16978e12 q^{59} -2.64345e13 q^{61} +1.58737e13 q^{62} +4.39805e12 q^{64} +1.64513e12 q^{65} +5.89869e13 q^{67} -4.39828e13 q^{68} +3.07715e12 q^{70} +8.13879e13 q^{71} -7.13459e13 q^{73} +1.27556e14 q^{74} +3.37688e13 q^{76} -1.39467e13 q^{77} +5.66091e13 q^{79} +2.09715e13 q^{80} +5.18209e13 q^{82} +2.41022e14 q^{83} -2.09726e14 q^{85} -2.87584e14 q^{86} -9.50498e13 q^{88} -4.96924e13 q^{89} -6.47978e12 q^{91} +3.94983e14 q^{92} -9.00810e13 q^{94} +1.61022e14 q^{95} +1.22347e14 q^{97} +5.95568e14 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 256 q^{2} + 32768 q^{4} + 156250 q^{5} - 1761200 q^{7} - 4194304 q^{8} - 20000000 q^{10} - 11981640 q^{11} + 88928320 q^{13} + 225433600 q^{14} + 536870912 q^{16} - 234312756 q^{17} + 3086064208 q^{19}+ \cdots + 932840500876032 q^{98}+O(q^{100}) $ 2 * q - 256 * q^2 + 32768 * q^4 + 156250 * q^5 - 1761200 * q^7 - 4194304 * q^8 - 20000000 * q^10 - 11981640 * q^11 + 88928320 * q^13 + 225433600 * q^14 + 536870912 * q^16 - 234312756 * q^17 + 3086064208 * q^19 + 2560000000 * q^20 + 1533649920 * q^22 + 7357553280 * q^23 + 12207031250 * q^25 - 11382824960 * q^26 - 28855500800 * q^28 - 106516740300 * q^29 - 80812338008 * q^31 - 68719476736 * q^32 + 29992032768 * q^34 - 137593750000 * q^35 - 783787471400 * q^37 - 395016218624 * q^38 - 327680000000 * q^40 - 125811612720 * q^41 - 700693895840 * q^43 - 196307189760 * q^44 - 941766819840 * q^46 + 2028162668400 * q^47 - 7287816413094 * q^49 - 1562500000000 * q^50 + 1457001594880 * q^52 + 16011801636996 * q^53 - 936065625000 * q^55 + 3693504102400 * q^56 + 13634142758400 * q^58 + 20778903613320 * q^59 + 6823475639524 * q^61 + 10343979265024 * q^62 + 8796093022208 * q^64 + 6947525000000 * q^65 + 21554311388800 * q^67 - 3838980194304 * q^68 + 17612000000000 * q^70 + 48887974479120 * q^71 - 37196698499300 * q^73 + 100324796339200 * q^74 + 50562075983872 * q^76 + 69345170329512 * q^77 - 125355565580312 * q^79 + 41943040000000 * q^80 + 16103886428160 * q^82 + 365242719675648 * q^83 - 18305684062500 * q^85 + 89688818667520 * q^86 + 25127320289280 * q^88 + 329691731091240 * q^89 - 105128657395216 * q^91 + 120546152939520 * q^92 - 259604821555200 * q^94 + 241098766250000 * q^95 - 37249221079340 * q^97 + 932840500876032 * 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 $

$2$	−128.000	−0.707107
$2$	−128.000	−0.707107
$3$	0	0
$3$	0	0
$4$	16384.0	0.500000
$5$	78125.0	0.447214
$5$	78125.0	0.447214
$6$	0	0
$7$	−307715.	−0.141226	−0.0706128	−	0.997504i	$-0.522495\pi$
$7$	−307715.	−0.141226	−0.0706128	+	0.997504i	$0.522495\pi$
$8$	−2.09715e6	−0.353553
$9$	0	0
$10$	−1.00000e7	−0.316228
$11$	4.53233e7	0.701255	0.350628	−	0.936515i	$-0.385968\pi$
$11$	4.53233e7	0.701255	0.350628	+	0.936515i	$0.385968\pi$
$12$	0	0
$13$	2.10577e7	0.0930757	0.0465379	−	0.998917i	$-0.485181\pi$
$13$	2.10577e7	0.0930757	0.0465379	+	0.998917i	$0.485181\pi$
$14$	3.93876e7	0.0998616
$15$	0	0
$16$	2.68435e8	0.250000
$17$	−2.68450e9	−1.58671	−0.793353	−	0.608762i	$-0.791666\pi$
$17$	−2.68450e9	−1.58671	−0.793353	+	0.608762i	$0.791666\pi$
$18$	0	0
$19$	2.06108e9	0.528985	0.264493	−	0.964388i	$-0.414796\pi$
$19$	2.06108e9	0.528985	0.264493	+	0.964388i	$0.414796\pi$
$20$	1.28000e9	0.223607
$21$	0	0
$22$	−5.80138e9	−0.495862
$23$	2.41078e10	1.47639	0.738193	−	0.674590i	$-0.235680\pi$
$23$	2.41078e10	1.47639	0.738193	+	0.674590i	$0.235680\pi$
$24$	0	0
$25$	6.10352e9	0.200000
$26$	−2.69539e9	−0.0658145
$27$	0	0
$28$	−5.04161e9	−0.0706128
$29$	−1.07547e11	−1.15775	−0.578874	−	0.815417i	$-0.696508\pi$
$29$	−1.07547e11	−1.15775	−0.578874	+	0.815417i	$0.696508\pi$
$30$	0	0
$31$	−1.24013e11	−0.809570	−0.404785	−	0.914412i	$-0.632654\pi$
$31$	−1.24013e11	−0.809570	−0.404785	+	0.914412i	$0.632654\pi$
$32$	−3.43597e10	−0.176777
$33$	0	0
$34$	3.43616e11	1.12197
$35$	−2.40403e10	−0.0631580
$36$	0	0
$37$	−9.96531e11	−1.72575	−0.862875	−	0.505417i	$-0.831339\pi$
$37$	−9.96531e11	−1.72575	−0.862875	+	0.505417i	$0.831339\pi$
$38$	−2.63819e11	−0.374049
$39$	0	0
$40$	−1.63840e11	−0.158114
$41$	−4.04851e11	−0.324651	−0.162325	−	0.986737i	$-0.551899\pi$
$41$	−4.04851e11	−0.324651	−0.162325	+	0.986737i	$0.551899\pi$
$42$	0	0
$43$	2.24675e12	1.26050	0.630249	−	0.776393i	$-0.282953\pi$
$43$	2.24675e12	1.26050	0.630249	+	0.776393i	$0.282953\pi$
$44$	7.42577e11	0.350628
$45$	0	0
$46$	−3.08580e12	−1.04396
$47$	7.03758e11	0.202623	0.101312	−	0.994855i	$-0.467696\pi$
$47$	7.03758e11	0.202623	0.101312	+	0.994855i	$0.467696\pi$
$48$	0	0
$49$	−4.65287e12	−0.980055
$50$	−7.81250e11	−0.141421
$51$	0	0
$52$	3.45010e11	0.0465379
$53$	1.16581e13	1.36319	0.681596	−	0.731729i	$-0.261286\pi$
$53$	1.16581e13	1.36319	0.681596	+	0.731729i	$0.261286\pi$
$54$	0	0
$55$	3.54088e12	0.313611
$56$	6.45326e11	0.0499308
$57$	0	0
$58$	1.37660e13	0.818652
$59$	−3.16978e12	−0.165821	−0.0829104	−	0.996557i	$-0.526422\pi$
$59$	−3.16978e12	−0.165821	−0.0829104	+	0.996557i	$0.526422\pi$
$60$	0	0
$61$	−2.64345e13	−1.07695	−0.538477	−	0.842640i	$-0.681000\pi$
$61$	−2.64345e13	−1.07695	−0.538477	+	0.842640i	$0.681000\pi$
$62$	1.58737e13	0.572452
$63$	0	0
$64$	4.39805e12	0.125000
$65$	1.64513e12	0.0416247
$66$	0	0
$67$	5.89869e13	1.18904	0.594518	−	0.804083i	$-0.297343\pi$
$67$	5.89869e13	1.18904	0.594518	+	0.804083i	$0.297343\pi$
$68$	−4.39828e13	−0.793353
$69$	0	0
$70$	3.07715e12	0.0446595
$71$	8.13879e13	1.06199	0.530997	−	0.847374i	$-0.321818\pi$
$71$	8.13879e13	1.06199	0.530997	+	0.847374i	$0.321818\pi$
$72$	0	0
$73$	−7.13459e13	−0.755871	−0.377936	−	0.925832i	$-0.623366\pi$
$73$	−7.13459e13	−0.755871	−0.377936	+	0.925832i	$0.623366\pi$
$74$	1.27556e14	1.22029
$75$	0	0
$76$	3.37688e13	0.264493
$77$	−1.39467e13	−0.0990353
$78$	0	0
$79$	5.66091e13	0.331652	0.165826	−	0.986155i	$-0.446971\pi$
$79$	5.66091e13	0.331652	0.165826	+	0.986155i	$0.446971\pi$
$80$	2.09715e13	0.111803
$81$	0	0
$82$	5.18209e13	0.229563
$83$	2.41022e14	0.974923	0.487461	−	0.873145i	$-0.337923\pi$
$83$	2.41022e14	0.974923	0.487461	+	0.873145i	$0.337923\pi$
$84$	0	0
$85$	−2.09726e14	−0.709596
$86$	−2.87584e14	−0.891306
$87$	0	0
$88$	−9.50498e13	−0.247931
$89$	−4.96924e13	−0.119087	−0.0595435	−	0.998226i	$-0.518965\pi$
$89$	−4.96924e13	−0.119087	−0.0595435	+	0.998226i	$0.518965\pi$
$90$	0	0
$91$	−6.47978e12	−0.0131447
$92$	3.94983e14	0.738193
$93$	0	0
$94$	−9.00810e13	−0.143276
$95$	1.61022e14	0.236569
$96$	0	0
$97$	1.22347e14	0.153747	0.0768735	−	0.997041i	$-0.475506\pi$
$97$	1.22347e14	0.153747	0.0768735	+	0.997041i	$0.475506\pi$
$98$	5.95568e14	0.693004
$99$	0	0
$100$	1.00000e14	0.100000
$101$	1.74588e15	1.62034	0.810168	−	0.586197i	$-0.199376\pi$
$101$	1.74588e15	1.62034	0.810168	+	0.586197i	$0.199376\pi$
$102$	0	0
$103$	−7.70102e14	−0.616977	−0.308489	−	0.951228i	$-0.599823\pi$
$103$	−7.70102e14	−0.616977	−0.308489	+	0.951228i	$0.599823\pi$
$104$	−4.41613e13	−0.0329072
$105$	0	0
$106$	−1.49223e15	−0.963922
$107$	−6.55977e14	−0.394921	−0.197460	−	0.980311i	$-0.563269\pi$
$107$	−6.55977e14	−0.394921	−0.197460	+	0.980311i	$0.563269\pi$
$108$	0	0
$109$	−1.91917e15	−1.00558	−0.502789	−	0.864409i	$-0.667693\pi$
$109$	−1.91917e15	−1.00558	−0.502789	+	0.864409i	$0.667693\pi$
$110$	−4.53233e14	−0.221756
$111$	0	0
$112$	−8.26017e13	−0.0353064
$113$	−3.28069e14	−0.131183	−0.0655913	−	0.997847i	$-0.520893\pi$
$113$	−3.28069e14	−0.131183	−0.0655913	+	0.997847i	$0.520893\pi$
$114$	0	0
$115$	1.88343e15	0.660259
$116$	−1.76205e15	−0.578874
$117$	0	0
$118$	4.05732e14	0.117253
$119$	8.26061e14	0.224084
$120$	0	0
$121$	−2.12305e15	−0.508241
$122$	3.38361e15	0.761521
$123$	0	0
$124$	−2.03183e15	−0.404785
$125$	4.76837e14	0.0894427
$126$	0	0
$127$	−3.92241e15	−0.653168	−0.326584	−	0.945168i	$-0.605898\pi$
$127$	−3.92241e15	−0.653168	−0.326584	+	0.945168i	$0.605898\pi$
$128$	−5.62950e14	−0.0883883
$129$	0	0
$130$	−2.10577e14	−0.0294331
$131$	−7.79661e15	−1.02890	−0.514448	−	0.857522i	$-0.672003\pi$
$131$	−7.79661e15	−1.02890	−0.514448	+	0.857522i	$0.672003\pi$
$132$	0	0
$133$	−6.34227e14	−0.0747063
$134$	−7.55033e15	−0.840775
$135$	0	0
$136$	5.62980e15	0.560985
$137$	−1.16822e16	−1.10185	−0.550923	−	0.834556i	$-0.685724\pi$
$137$	−1.16822e16	−1.10185	−0.550923	+	0.834556i	$0.685724\pi$
$138$	0	0
$139$	−9.74490e15	−0.824454	−0.412227	−	0.911081i	$-0.635249\pi$
$139$	−9.74490e15	−0.824454	−0.412227	+	0.911081i	$0.635249\pi$
$140$	−3.93876e14	−0.0315790
$141$	0	0
$142$	−1.04176e16	−0.750943
$143$	9.54405e14	0.0652698
$144$	0	0
$145$	−8.40213e15	−0.517761
$146$	9.13228e15	0.534482
$147$	0	0
$148$	−1.63272e16	−0.862875
$149$	−4.23840e15	−0.212963	−0.106482	−	0.994315i	$-0.533959\pi$
$149$	−4.23840e15	−0.212963	−0.106482	+	0.994315i	$0.533959\pi$
$150$	0	0
$151$	−1.07118e16	−0.487006	−0.243503	−	0.969900i	$-0.578297\pi$
$151$	−1.07118e16	−0.487006	−0.243503	+	0.969900i	$0.578297\pi$
$152$	−4.32241e15	−0.187025
$153$	0	0
$154$	1.78517e15	0.0700285
$155$	−9.68851e15	−0.362051
$156$	0	0
$157$	−3.38094e16	−1.14760	−0.573799	−	0.818996i	$-0.694531\pi$
$157$	−3.38094e16	−1.14760	−0.573799	+	0.818996i	$0.694531\pi$
$158$	−7.24596e15	−0.234514
$159$	0	0
$160$	−2.68435e15	−0.0790569
$161$	−7.41835e15	−0.208504
$162$	0	0
$163$	−7.17509e16	−1.83832	−0.919158	−	0.393888i	$-0.871130\pi$
$163$	−7.17509e16	−1.83832	−0.919158	+	0.393888i	$0.871130\pi$
$164$	−6.63308e15	−0.162325
$165$	0	0
$166$	−3.08508e16	−0.689375
$167$	1.45005e16	0.309748	0.154874	−	0.987934i	$-0.450503\pi$
$167$	1.45005e16	0.309748	0.154874	+	0.987934i	$0.450503\pi$
$168$	0	0
$169$	−5.07425e16	−0.991337
$170$	2.68450e16	0.501760
$171$	0	0
$172$	3.68108e16	0.630249
$173$	5.47606e16	0.897681	0.448841	−	0.893612i	$-0.351837\pi$
$173$	5.47606e16	0.897681	0.448841	+	0.893612i	$0.351837\pi$
$174$	0	0
$175$	−1.87814e15	−0.0282451
$176$	1.21664e16	0.175314
$177$	0	0
$178$	6.36062e15	0.0842073
$179$	−5.63459e16	−0.715262	−0.357631	−	0.933863i	$-0.616415\pi$
$179$	−5.63459e16	−0.715262	−0.357631	+	0.933863i	$0.616415\pi$
$180$	0	0
$181$	−1.41337e17	−1.65069	−0.825345	−	0.564629i	$-0.809019\pi$
$181$	−1.41337e17	−1.65069	−0.825345	+	0.564629i	$0.809019\pi$
$182$	8.29412e14	0.00929469
$183$	0	0
$184$	−5.05578e16	−0.521981
$185$	−7.78540e16	−0.771779
$186$	0	0
$187$	−1.21670e17	−1.11269
$188$	1.15304e16	0.101312
$189$	0	0
$190$	−2.06108e16	−0.167280
$191$	8.48396e16	0.661985	0.330993	−	0.943633i	$-0.392616\pi$
$191$	8.48396e16	0.661985	0.330993	+	0.943633i	$0.392616\pi$
$192$	0	0
$193$	1.16613e17	0.841523	0.420761	−	0.907171i	$-0.361763\pi$
$193$	1.16613e17	0.841523	0.420761	+	0.907171i	$0.361763\pi$
$194$	−1.56605e16	−0.108716
$195$	0	0
$196$	−7.62327e16	−0.490028
$197$	6.60040e16	0.408388	0.204194	−	0.978930i	$-0.434543\pi$
$197$	6.60040e16	0.408388	0.204194	+	0.978930i	$0.434543\pi$
$198$	0	0
$199$	1.27765e16	0.0732847	0.0366424	−	0.999328i	$-0.488334\pi$
$199$	1.27765e16	0.0732847	0.0366424	+	0.999328i	$0.488334\pi$
$200$	−1.28000e16	−0.0707107
$201$	0	0
$202$	−2.23473e17	−1.14575
$203$	3.30939e16	0.163504
$204$	0	0
$205$	−3.16290e16	−0.145188
$206$	9.85731e16	0.436269
$207$	0	0
$208$	5.65264e15	0.0232689
$209$	9.34151e16	0.370954
$210$	0	0
$211$	−4.41097e16	−0.163086	−0.0815428	−	0.996670i	$-0.525985\pi$
$211$	−4.41097e16	−0.163086	−0.0815428	+	0.996670i	$0.525985\pi$
$212$	1.91006e17	0.681596
$213$	0	0
$214$	8.39650e16	0.279251
$215$	1.75528e17	0.563712
$216$	0	0
$217$	3.81607e16	0.114332
$218$	2.45654e17	0.711051
$219$	0	0
$220$	5.80138e16	0.156805
$221$	−5.65294e16	−0.147684
$222$	0	0
$223$	−6.09182e17	−1.48751	−0.743756	−	0.668451i	$-0.766958\pi$
$223$	−6.09182e17	−1.48751	−0.743756	+	0.668451i	$0.766958\pi$
$224$	1.05730e16	0.0249654
$225$	0	0
$226$	4.19928e16	0.0927601
$227$	−2.89704e17	−0.619100	−0.309550	−	0.950883i	$-0.600178\pi$
$227$	−2.89704e17	−0.619100	−0.309550	+	0.950883i	$0.600178\pi$
$228$	0	0
$229$	−3.51449e17	−0.703229	−0.351614	−	0.936145i	$-0.614367\pi$
$229$	−3.51449e17	−0.703229	−0.351614	+	0.936145i	$0.614367\pi$
$230$	−2.41078e17	−0.466874
$231$	0	0
$232$	2.25543e17	0.409326
$233$	1.34334e17	0.236056	0.118028	−	0.993010i	$-0.462343\pi$
$233$	1.34334e17	0.236056	0.118028	+	0.993010i	$0.462343\pi$
$234$	0	0
$235$	5.49811e16	0.0906159
$236$	−5.19337e16	−0.0829104
$237$	0	0
$238$	−1.05736e17	−0.158451
$239$	−1.18373e18	−1.71896	−0.859482	−	0.511166i	$-0.829214\pi$
$239$	−1.18373e18	−1.71896	−0.859482	+	0.511166i	$0.829214\pi$
$240$	0	0
$241$	7.96597e17	1.08670	0.543351	−	0.839505i	$-0.317155\pi$
$241$	7.96597e17	1.08670	0.543351	+	0.839505i	$0.317155\pi$
$242$	2.71750e17	0.359381
$243$	0	0
$244$	−4.33103e17	−0.538477
$245$	−3.63506e17	−0.438294
$246$	0	0
$247$	4.34017e16	0.0492357
$248$	2.60074e17	0.286226
$249$	0	0
$250$	−6.10352e16	−0.0632456
$251$	4.79038e17	0.481745	0.240872	−	0.970557i	$-0.422566\pi$
$251$	4.79038e17	0.481745	0.240872	+	0.970557i	$0.422566\pi$
$252$	0	0
$253$	1.09265e18	1.03532
$254$	5.02069e17	0.461860
$255$	0	0
$256$	7.20576e16	0.0625000
$257$	−5.68024e17	−0.478485	−0.239242	−	0.970960i	$-0.576899\pi$
$257$	−5.68024e17	−0.478485	−0.239242	+	0.970960i	$0.576899\pi$
$258$	0	0
$259$	3.06648e17	0.243720
$260$	2.69539e16	0.0208124
$261$	0	0
$262$	9.97966e17	0.727539
$263$	−1.56846e18	−1.11124	−0.555618	−	0.831438i	$-0.687518\pi$
$263$	−1.56846e18	−1.11124	−0.555618	+	0.831438i	$0.687518\pi$
$264$	0	0
$265$	9.10785e17	0.609638
$266$	8.11810e16	0.0528253
$267$	0	0
$268$	9.66442e17	0.594518
$269$	−1.70035e18	−1.01718	−0.508588	−	0.861010i	$-0.669832\pi$
$269$	−1.70035e18	−1.01718	−0.508588	+	0.861010i	$0.669832\pi$
$270$	0	0
$271$	−8.21525e17	−0.464891	−0.232445	−	0.972609i	$-0.574673\pi$
$271$	−8.21525e17	−0.464891	−0.232445	+	0.972609i	$0.574673\pi$
$272$	−7.20615e17	−0.396676
$273$	0	0
$274$	1.49532e18	0.779123
$275$	2.76631e17	0.140251
$276$	0	0
$277$	1.32628e17	0.0636849	0.0318425	−	0.999493i	$-0.489863\pi$
$277$	1.32628e17	0.0636849	0.0318425	+	0.999493i	$0.489863\pi$
$278$	1.24735e18	0.582977
$279$	0	0
$280$	5.04161e16	0.0223297
$281$	−2.95888e18	−1.27594	−0.637969	−	0.770062i	$-0.720225\pi$
$281$	−2.95888e18	−1.27594	−0.637969	+	0.770062i	$0.720225\pi$
$282$	0	0
$283$	4.78871e18	1.95803	0.979017	−	0.203778i	$-0.0653220\pi$
$283$	4.78871e18	1.95803	0.979017	+	0.203778i	$0.0653220\pi$
$284$	1.33346e18	0.530997
$285$	0	0
$286$	−1.22164e17	−0.0461527
$287$	1.24579e17	0.0458490
$288$	0	0
$289$	4.34411e18	1.51763
$290$	1.07547e18	0.366112
$291$	0	0
$292$	−1.16893e18	−0.377936
$293$	1.76126e18	0.555031	0.277515	−	0.960721i	$-0.410489\pi$
$293$	1.76126e18	0.555031	0.277515	+	0.960721i	$0.410489\pi$
$294$	0	0
$295$	−2.47639e17	−0.0741573
$296$	2.08988e18	0.610145
$297$	0	0
$298$	5.42515e17	0.150588
$299$	5.07656e17	0.137416
$300$	0	0
$301$	−6.91360e17	−0.178015
$302$	1.37111e18	0.344365
$303$	0	0
$304$	5.53268e17	0.132246
$305$	−2.06519e18	−0.481628
$306$	0	0
$307$	6.60778e18	1.46730	0.733648	−	0.679530i	$-0.237816\pi$
$307$	6.60778e18	1.46730	0.733648	+	0.679530i	$0.237816\pi$
$308$	−2.28502e17	−0.0495176
$309$	0	0
$310$	1.24013e18	0.256009
$311$	−2.80316e18	−0.564864	−0.282432	−	0.959287i	$-0.591141\pi$
$311$	−2.80316e18	−0.564864	−0.282432	+	0.959287i	$0.591141\pi$
$312$	0	0
$313$	9.39821e17	0.180494	0.0902470	−	0.995919i	$-0.471234\pi$
$313$	9.39821e17	0.180494	0.0902470	+	0.995919i	$0.471234\pi$
$314$	4.32760e18	0.811475
$315$	0	0
$316$	9.27483e17	0.165826
$317$	3.03729e18	0.530324	0.265162	−	0.964204i	$-0.414575\pi$
$317$	3.03729e18	0.530324	0.265162	+	0.964204i	$0.414575\pi$
$318$	0	0
$319$	−4.87439e18	−0.811877
$320$	3.43597e17	0.0559017
$321$	0	0
$322$	9.49549e17	0.147434
$323$	−5.53298e18	−0.839344
$324$	0	0
$325$	1.28526e17	0.0186151
$326$	9.18412e18	1.29989
$327$	0	0
$328$	8.49034e17	0.114781
$329$	−2.16557e17	−0.0286156
$330$	0	0
$331$	4.36555e18	0.551225	0.275612	−	0.961269i	$-0.411119\pi$
$331$	4.36555e18	0.551225	0.275612	+	0.961269i	$0.411119\pi$
$332$	3.94890e18	0.487461
$333$	0	0
$334$	−1.85606e18	−0.219025
$335$	4.60835e18	0.531753
$336$	0	0
$337$	−8.62580e18	−0.951864	−0.475932	−	0.879482i	$-0.657889\pi$
$337$	−8.62580e18	−0.951864	−0.475932	+	0.879482i	$0.657889\pi$
$338$	6.49504e18	0.700981
$339$	0	0
$340$	−3.43616e18	−0.354798
$341$	−5.62068e18	−0.567715
$342$	0	0
$343$	2.89266e18	0.279635
$344$	−4.71178e18	−0.445653
$345$	0	0
$346$	−7.00935e18	−0.634756
$347$	−1.87129e19	−1.65833	−0.829165	−	0.559004i	$-0.811184\pi$
$347$	−1.87129e19	−1.65833	−0.829165	+	0.559004i	$0.811184\pi$
$348$	0	0
$349$	1.00815e19	0.855722	0.427861	−	0.903845i	$-0.359267\pi$
$349$	1.00815e19	0.855722	0.427861	+	0.903845i	$0.359267\pi$
$350$	2.40403e17	0.0199723
$351$	0	0
$352$	−1.55730e18	−0.123966
$353$	−7.56297e18	−0.589362	−0.294681	−	0.955596i	$-0.595213\pi$
$353$	−7.56297e18	−0.589362	−0.294681	+	0.955596i	$0.595213\pi$
$354$	0	0
$355$	6.35843e18	0.474938
$356$	−8.14160e17	−0.0595435
$357$	0	0
$358$	7.21228e18	0.505766
$359$	1.93700e19	1.33021	0.665106	−	0.746749i	$-0.268386\pi$
$359$	1.93700e19	1.33021	0.665106	+	0.746749i	$0.268386\pi$
$360$	0	0
$361$	−1.09331e19	−0.720175
$362$	1.80911e19	1.16721
$363$	0	0
$364$	−1.06165e17	−0.00657234
$365$	−5.57390e18	−0.338036
$366$	0	0
$367$	−2.98418e19	−1.73712	−0.868561	−	0.495583i	$-0.834955\pi$
$367$	−2.98418e19	−1.73712	−0.868561	+	0.495583i	$0.834955\pi$
$368$	6.47140e18	0.369096
$369$	0	0
$370$	9.96531e18	0.545730
$371$	−3.58736e18	−0.192518
$372$	0	0
$373$	2.01374e19	1.03797	0.518987	−	0.854782i	$-0.326309\pi$
$373$	2.01374e19	1.03797	0.518987	+	0.854782i	$0.326309\pi$
$374$	1.55738e19	0.786788
$375$	0	0
$376$	−1.47589e18	−0.0716381
$377$	−2.26470e18	−0.107758
$378$	0	0
$379$	−2.46205e19	−1.12591	−0.562954	−	0.826488i	$-0.690335\pi$
$379$	−2.46205e19	−1.12591	−0.562954	+	0.826488i	$0.690335\pi$
$380$	2.63819e18	0.118285
$381$	0	0
$382$	−1.08595e19	−0.468094
$383$	−3.18081e19	−1.34446	−0.672229	−	0.740343i	$-0.734663\pi$
$383$	−3.18081e19	−1.34446	−0.672229	+	0.740343i	$0.734663\pi$
$384$	0	0
$385$	−1.08958e18	−0.0442899
$386$	−1.49264e19	−0.595047
$387$	0	0
$388$	2.00454e18	0.0768735
$389$	−4.33625e19	−1.63114	−0.815572	−	0.578655i	$-0.803578\pi$
$389$	−4.33625e19	−1.63114	−0.815572	+	0.578655i	$0.803578\pi$
$390$	0	0
$391$	−6.47175e19	−2.34259
$392$	9.75778e18	0.346502
$393$	0	0
$394$	−8.44851e18	−0.288774
$395$	4.42258e18	0.148319
$396$	0	0
$397$	8.12402e18	0.262326	0.131163	−	0.991361i	$-0.458129\pi$
$397$	8.12402e18	0.262326	0.131163	+	0.991361i	$0.458129\pi$
$398$	−1.63539e18	−0.0518201
$399$	0	0
$400$	1.63840e18	0.0500000
$401$	−3.51406e18	−0.105251	−0.0526255	−	0.998614i	$-0.516759\pi$
$401$	−3.51406e18	−0.105251	−0.0526255	+	0.998614i	$0.516759\pi$
$402$	0	0
$403$	−2.61143e18	−0.0753513
$404$	2.86046e19	0.810168
$405$	0	0
$406$	−4.23602e18	−0.115615
$407$	−4.51661e19	−1.21019
$408$	0	0
$409$	−6.44296e19	−1.66403	−0.832014	−	0.554755i	$-0.812812\pi$
$409$	−6.44296e19	−1.66403	−0.832014	+	0.554755i	$0.812812\pi$
$410$	4.04851e18	0.102664
$411$	0	0
$412$	−1.26174e19	−0.308489
$413$	9.75389e17	0.0234182
$414$	0	0
$415$	1.88298e19	0.435999
$416$	−7.23538e17	−0.0164536
$417$	0	0
$418$	−1.19571e19	−0.262304
$419$	1.12676e19	0.242788	0.121394	−	0.992604i	$-0.461263\pi$
$419$	1.12676e19	0.242788	0.121394	+	0.992604i	$0.461263\pi$
$420$	0	0
$421$	−4.80962e19	−0.999990	−0.499995	−	0.866028i	$-0.666665\pi$
$421$	−4.80962e19	−0.999990	−0.499995	+	0.866028i	$0.666665\pi$
$422$	5.64604e18	0.115319
$423$	0	0
$424$	−2.44487e19	−0.481961
$425$	−1.63849e19	−0.317341
$426$	0	0
$427$	8.13430e18	0.152094
$428$	−1.07475e19	−0.197460
$429$	0	0
$430$	−2.24675e19	−0.398604
$431$	−6.89397e19	−1.20196	−0.600980	−	0.799264i	$-0.705223\pi$
$431$	−6.89397e19	−1.20196	−0.600980	+	0.799264i	$0.705223\pi$
$432$	0	0
$433$	−1.39468e19	−0.234863	−0.117432	−	0.993081i	$-0.537466\pi$
$433$	−1.39468e19	−0.234863	−0.117432	+	0.993081i	$0.537466\pi$
$434$	−4.88457e18	−0.0808450
$435$	0	0
$436$	−3.14438e19	−0.502789
$437$	4.96883e19	0.780986
$438$	0	0
$439$	8.36739e19	1.27088	0.635442	−	0.772148i	$-0.280818\pi$
$439$	8.36739e19	1.27088	0.635442	+	0.772148i	$0.280818\pi$
$440$	−7.42577e18	−0.110878
$441$	0	0
$442$	7.23577e18	0.104428
$443$	2.44887e19	0.347486	0.173743	−	0.984791i	$-0.444414\pi$
$443$	2.44887e19	0.347486	0.173743	+	0.984791i	$0.444414\pi$
$444$	0	0
$445$	−3.88222e18	−0.0532573
$446$	7.79753e19	1.05183
$447$	0	0
$448$	−1.35335e18	−0.0176532
$449$	−5.82526e19	−0.747253	−0.373626	−	0.927579i	$-0.621886\pi$
$449$	−5.82526e19	−0.747253	−0.373626	+	0.927579i	$0.621886\pi$
$450$	0	0
$451$	−1.83492e19	−0.227663
$452$	−5.37507e18	−0.0655913
$453$	0	0
$454$	3.70821e19	0.437770
$455$	−5.06233e17	−0.00587848
$456$	0	0
$457$	−1.31249e20	−1.47477	−0.737385	−	0.675473i	$-0.763940\pi$
$457$	−1.31249e20	−1.47477	−0.737385	+	0.675473i	$0.763940\pi$
$458$	4.49855e19	0.497258
$459$	0	0
$460$	3.08580e19	0.330130
$461$	5.80642e19	0.611156	0.305578	−	0.952167i	$-0.401150\pi$
$461$	5.80642e19	0.611156	0.305578	+	0.952167i	$0.401150\pi$
$462$	0	0
$463$	1.59608e20	1.62629	0.813144	−	0.582062i	$-0.197754\pi$
$463$	1.59608e20	1.62629	0.813144	+	0.582062i	$0.197754\pi$
$464$	−2.88695e19	−0.289437
$465$	0	0
$466$	−1.71947e19	−0.166917
$467$	1.51070e20	1.44311	0.721556	−	0.692356i	$-0.243427\pi$
$467$	1.51070e20	1.44311	0.721556	+	0.692356i	$0.243427\pi$
$468$	0	0
$469$	−1.81512e19	−0.167922
$470$	−7.03758e18	−0.0640751
$471$	0	0
$472$	6.64751e18	0.0586265
$473$	1.01830e20	0.883931
$474$	0	0
$475$	1.25799e19	0.105797
$476$	1.35342e19	0.112042
$477$	0	0
$478$	1.51517e20	1.21549
$479$	1.63356e20	1.29009	0.645045	−	0.764145i	$-0.276839\pi$
$479$	1.63356e20	1.29009	0.645045	+	0.764145i	$0.276839\pi$
$480$	0	0
$481$	−2.09847e19	−0.160625
$482$	−1.01964e20	−0.768415
$483$	0	0
$484$	−3.47840e19	−0.254120
$485$	9.55839e18	0.0687577
$486$	0	0
$487$	9.23983e19	0.644461	0.322231	−	0.946661i	$-0.395567\pi$
$487$	9.23983e19	0.644461	0.322231	+	0.946661i	$0.395567\pi$
$488$	5.54371e19	0.380761
$489$	0	0
$490$	4.65287e19	0.309921
$491$	−1.96658e20	−1.29003	−0.645017	−	0.764168i	$-0.723150\pi$
$491$	−1.96658e20	−1.29003	−0.645017	+	0.764168i	$0.723150\pi$
$492$	0	0
$493$	2.88710e20	1.83701
$494$	−5.55542e18	−0.0348149
$495$	0	0
$496$	−3.32895e19	−0.202392
$497$	−2.50443e19	−0.149981
$498$	0	0
$499$	−4.51305e19	−0.262250	−0.131125	−	0.991366i	$-0.541859\pi$
$499$	−4.51305e19	−0.262250	−0.131125	+	0.991366i	$0.541859\pi$
$500$	7.81250e18	0.0447214
$501$	0	0
$502$	−6.13169e19	−0.340645
$503$	−3.11342e19	−0.170403	−0.0852017	−	0.996364i	$-0.527153\pi$
$503$	−3.11342e19	−0.170403	−0.0852017	+	0.996364i	$0.527153\pi$
$504$	0	0
$505$	1.36397e20	0.724637
$506$	−1.39859e20	−0.732084
$507$	0	0
$508$	−6.42648e19	−0.326584
$509$	−5.95101e19	−0.297994	−0.148997	−	0.988838i	$-0.547604\pi$
$509$	−5.95101e19	−0.297994	−0.148997	+	0.988838i	$0.547604\pi$
$510$	0	0
$511$	2.19542e19	0.106748
$512$	−9.22337e18	−0.0441942
$513$	0	0
$514$	7.27071e19	0.338340
$515$	−6.01642e19	−0.275921
$516$	0	0
$517$	3.18966e19	0.142091
$518$	−3.92509e19	−0.172336
$519$	0	0
$520$	−3.45010e18	−0.0147166
$521$	3.23934e20	1.36199	0.680994	−	0.732289i	$-0.261548\pi$
$521$	3.23934e20	1.36199	0.680994	+	0.732289i	$0.261548\pi$
$522$	0	0
$523$	−6.55328e19	−0.267729	−0.133865	−	0.991000i	$-0.542739\pi$
$523$	−6.55328e19	−0.267729	−0.133865	+	0.991000i	$0.542739\pi$
$524$	−1.27740e20	−0.514448
$525$	0	0
$526$	2.00763e20	0.785763
$527$	3.32913e20	1.28455
$528$	0	0
$529$	3.14553e20	1.17971
$530$	−1.16581e20	−0.431079
$531$	0	0
$532$	−1.03912e19	−0.0373532
$533$	−8.52524e18	−0.0302171
$534$	0	0
$535$	−5.12482e19	−0.176614
$536$	−1.23705e20	−0.420387
$537$	0	0
$538$	2.17645e20	0.719252
$539$	−2.10883e20	−0.687269
$540$	0	0
$541$	8.37496e19	0.265463	0.132731	−	0.991152i	$-0.457625\pi$
$541$	8.37496e19	0.265463	0.132731	+	0.991152i	$0.457625\pi$
$542$	1.05155e20	0.328728
$543$	0	0
$544$	9.22387e19	0.280493
$545$	−1.49936e20	−0.449708
$546$	0	0
$547$	−1.80363e19	−0.0526311	−0.0263156	−	0.999654i	$-0.508377\pi$
$547$	−1.80363e19	−0.0526311	−0.0263156	+	0.999654i	$0.508377\pi$
$548$	−1.91401e20	−0.550923
$549$	0	0
$550$	−3.54088e19	−0.0991725
$551$	−2.21664e20	−0.612432
$552$	0	0
$553$	−1.74195e19	−0.0468378
$554$	−1.69764e19	−0.0450320
$555$	0	0
$556$	−1.59660e20	−0.412227
$557$	−1.41919e20	−0.361515	−0.180757	−	0.983528i	$-0.557855\pi$
$557$	−1.41919e20	−0.361515	−0.180757	+	0.983528i	$0.557855\pi$
$558$	0	0
$559$	4.73115e19	0.117322
$560$	−6.45326e18	−0.0157895
$561$	0	0
$562$	3.78736e20	0.902225
$563$	6.76140e20	1.58937	0.794683	−	0.607025i	$-0.207637\pi$
$563$	6.76140e20	1.58937	0.794683	+	0.607025i	$0.207637\pi$
$564$	0	0
$565$	−2.56304e19	−0.0586666
$566$	−6.12955e20	−1.38454
$567$	0	0
$568$	−1.70683e20	−0.375471
$569$	3.21183e20	0.697286	0.348643	−	0.937256i	$-0.386643\pi$
$569$	3.21183e20	0.697286	0.348643	+	0.937256i	$0.386643\pi$
$570$	0	0
$571$	4.14110e20	0.875679	0.437840	−	0.899053i	$-0.355744\pi$
$571$	4.14110e20	0.875679	0.437840	+	0.899053i	$0.355744\pi$
$572$	1.56370e19	0.0326349
$573$	0	0
$574$	−1.59461e19	−0.0324201
$575$	1.47143e20	0.295277
$576$	0	0
$577$	3.54691e20	0.693476	0.346738	−	0.937962i	$-0.387289\pi$
$577$	3.54691e20	0.693476	0.346738	+	0.937962i	$0.387289\pi$
$578$	−5.56046e20	−1.07313
$579$	0	0
$580$	−1.37660e20	−0.258880
$581$	−7.41660e19	−0.137684
$582$	0	0
$583$	5.28381e20	0.955946
$584$	1.49623e20	0.267241
$585$	0	0
$586$	−2.25442e20	−0.392466
$587$	−7.23865e19	−0.124415	−0.0622074	−	0.998063i	$-0.519814\pi$
$587$	−7.23865e19	−0.124415	−0.0622074	+	0.998063i	$0.519814\pi$
$588$	0	0
$589$	−2.55601e20	−0.428251
$590$	3.16978e19	0.0524371
$591$	0	0
$592$	−2.67504e20	−0.431438
$593$	−1.85476e20	−0.295378	−0.147689	−	0.989034i	$-0.547183\pi$
$593$	−1.85476e20	−0.295378	−0.147689	+	0.989034i	$0.547183\pi$
$594$	0	0
$595$	6.45360e19	0.100213
$596$	−6.94419e19	−0.106482
$597$	0	0
$598$	−6.49800e19	−0.0971675
$599$	2.41908e20	0.357230	0.178615	−	0.983919i	$-0.442838\pi$
$599$	2.41908e20	0.357230	0.178615	+	0.983919i	$0.442838\pi$
$600$	0	0
$601$	−6.49372e20	−0.935266	−0.467633	−	0.883923i	$-0.654893\pi$
$601$	−6.49372e20	−0.935266	−0.467633	+	0.883923i	$0.654893\pi$
$602$	8.84941e19	0.125875
$603$	0	0
$604$	−1.75502e20	−0.243503
$605$	−1.65863e20	−0.227292
$606$	0	0
$607$	−1.78385e20	−0.238475	−0.119238	−	0.992866i	$-0.538045\pi$
$607$	−1.78385e20	−0.238475	−0.119238	+	0.992866i	$0.538045\pi$
$608$	−7.08183e19	−0.0935123
$609$	0	0
$610$	2.64345e20	0.340563
$611$	1.48195e19	0.0188593
$612$	0	0
$613$	−5.83650e20	−0.724768	−0.362384	−	0.932029i	$-0.618037\pi$
$613$	−5.83650e20	−0.724768	−0.362384	+	0.932029i	$0.618037\pi$
$614$	−8.45796e20	−1.03753
$615$	0	0
$616$	2.92483e19	0.0350143
$617$	1.59816e21	1.89008	0.945041	−	0.326953i	$-0.106022\pi$
$617$	1.59816e21	1.89008	0.945041	+	0.326953i	$0.106022\pi$
$618$	0	0
$619$	−5.07202e20	−0.585466	−0.292733	−	0.956194i	$-0.594565\pi$
$619$	−5.07202e20	−0.585466	−0.292733	+	0.956194i	$0.594565\pi$
$620$	−1.58737e20	−0.181025
$621$	0	0
$622$	3.58804e20	0.399419
$623$	1.52911e19	0.0168181
$624$	0	0
$625$	3.72529e19	0.0400000
$626$	−1.20297e20	−0.127628
$627$	0	0
$628$	−5.53933e20	−0.573799
$629$	2.67519e21	2.73826
$630$	0	0
$631$	−1.40119e21	−1.40048	−0.700241	−	0.713907i	$-0.746924\pi$
$631$	−1.40119e21	−1.40048	−0.700241	+	0.713907i	$0.746924\pi$
$632$	−1.18718e20	−0.117257
$633$	0	0
$634$	−3.88773e20	−0.374996
$635$	−3.06438e20	−0.292106
$636$	0	0
$637$	−9.79789e19	−0.0912193
$638$	6.23922e20	0.574084
$639$	0	0
$640$	−4.39805e19	−0.0395285
$641$	−2.06369e21	−1.83319	−0.916597	−	0.399812i	$-0.869075\pi$
$641$	−2.06369e21	−1.83319	−0.916597	+	0.399812i	$0.869075\pi$
$642$	0	0
$643$	−1.30785e21	−1.13494	−0.567472	−	0.823393i	$-0.692078\pi$
$643$	−1.30785e21	−1.13494	−0.567472	+	0.823393i	$0.692078\pi$
$644$	−1.21542e20	−0.104252
$645$	0	0
$646$	7.08221e20	0.593506
$647$	1.07597e21	0.891286	0.445643	−	0.895211i	$-0.352975\pi$
$647$	1.07597e21	0.891286	0.445643	+	0.895211i	$0.352975\pi$
$648$	0	0
$649$	−1.43665e20	−0.116283
$650$	−1.64513e19	−0.0131629
$651$	0	0
$652$	−1.17557e21	−0.919158
$653$	−8.55859e20	−0.661536	−0.330768	−	0.943712i	$-0.607308\pi$
$653$	−8.55859e20	−0.661536	−0.330768	+	0.943712i	$0.607308\pi$
$654$	0	0
$655$	−6.09110e20	−0.460136
$656$	−1.08676e20	−0.0811626
$657$	0	0
$658$	2.77193e19	0.0202343
$659$	5.37776e20	0.388115	0.194058	−	0.980990i	$-0.437835\pi$
$659$	5.37776e20	0.388115	0.194058	+	0.980990i	$0.437835\pi$
$660$	0	0
$661$	1.96289e21	1.38479	0.692396	−	0.721517i	$-0.256555\pi$
$661$	1.96289e21	1.38479	0.692396	+	0.721517i	$0.256555\pi$
$662$	−5.58790e20	−0.389775
$663$	0	0
$664$	−5.05459e20	−0.344687
$665$	−4.95490e19	−0.0334097
$666$	0	0
$667$	−2.59273e21	−1.70928
$668$	2.37576e20	0.154874
$669$	0	0
$670$	−5.89869e20	−0.376006
$671$	−1.19810e21	−0.755220
$672$	0	0
$673$	−2.82758e21	−1.74302	−0.871508	−	0.490382i	$-0.836857\pi$
$673$	−2.82758e21	−1.74302	−0.871508	+	0.490382i	$0.836857\pi$
$674$	1.10410e21	0.673069
$675$	0	0
$676$	−8.31365e20	−0.495668
$677$	2.22141e21	1.30983	0.654914	−	0.755704i	$-0.272705\pi$
$677$	2.22141e21	1.30983	0.654914	+	0.755704i	$0.272705\pi$
$678$	0	0
$679$	−3.76481e19	−0.0217130
$680$	4.39828e20	0.250880
$681$	0	0
$682$	7.19446e20	0.401435
$683$	2.75754e21	1.52183	0.760917	−	0.648849i	$-0.224749\pi$
$683$	2.75754e21	1.52183	0.760917	+	0.648849i	$0.224749\pi$
$684$	0	0
$685$	−9.12673e20	−0.492761
$686$	−3.70260e20	−0.197732
$687$	0	0
$688$	6.03108e20	0.315124
$689$	2.45492e20	0.126880
$690$	0	0
$691$	2.90151e21	1.46737	0.733684	−	0.679490i	$-0.237799\pi$
$691$	2.90151e21	1.46737	0.733684	+	0.679490i	$0.237799\pi$
$692$	8.97197e20	0.448841
$693$	0	0
$694$	2.39526e21	1.17262
$695$	−7.61320e20	−0.368707
$696$	0	0
$697$	1.08682e21	0.515125
$698$	−1.29043e21	−0.605087
$699$	0	0
$700$	−3.07715e19	−0.0141226
$701$	−1.75759e21	−0.798055	−0.399028	−	0.916939i	$-0.630652\pi$
$701$	−1.75759e21	−0.798055	−0.399028	+	0.916939i	$0.630652\pi$
$702$	0	0
$703$	−2.05393e21	−0.912897
$704$	1.99334e20	0.0876569
$705$	0	0
$706$	9.68060e20	0.416742
$707$	−5.37235e20	−0.228833
$708$	0	0
$709$	−1.58801e21	−0.662227	−0.331114	−	0.943591i	$-0.607424\pi$
$709$	−1.58801e21	−0.662227	−0.331114	+	0.943591i	$0.607424\pi$
$710$	−8.13879e20	−0.335832
$711$	0	0
$712$	1.04212e20	0.0421036
$713$	−2.98969e21	−1.19524
$714$	0	0
$715$	7.45629e19	0.0291896
$716$	−9.23172e20	−0.357631
$717$	0	0
$718$	−2.47936e21	−0.940602
$719$	−2.73391e21	−1.02640	−0.513201	−	0.858268i	$-0.671541\pi$
$719$	−2.73391e21	−1.02640	−0.513201	+	0.858268i	$0.671541\pi$
$720$	0	0
$721$	2.36972e20	0.0871331
$722$	1.39943e21	0.509240
$723$	0	0
$724$	−2.31566e21	−0.825345
$725$	−6.56416e20	−0.231550
$726$	0	0
$727$	4.66291e21	1.61120	0.805600	−	0.592460i	$-0.201843\pi$
$727$	4.66291e21	1.61120	0.805600	+	0.592460i	$0.201843\pi$
$728$	1.35891e19	0.00464735
$729$	0	0
$730$	7.13459e20	0.239027
$731$	−6.03141e21	−2.00004
$732$	0	0
$733$	3.24219e21	1.05331	0.526657	−	0.850078i	$-0.323445\pi$
$733$	3.24219e21	1.05331	0.526657	+	0.850078i	$0.323445\pi$
$734$	3.81976e21	1.22833
$735$	0	0
$736$	−8.28339e20	−0.260990
$737$	2.67348e21	0.833817
$738$	0	0
$739$	4.24307e21	1.29672	0.648362	−	0.761332i	$-0.275454\pi$
$739$	4.24307e21	1.29672	0.648362	+	0.761332i	$0.275454\pi$
$740$	−1.27556e21	−0.385890
$741$	0	0
$742$	4.59182e20	0.136131
$743$	3.88685e20	0.114073	0.0570363	−	0.998372i	$-0.481835\pi$
$743$	3.88685e20	0.114073	0.0570363	+	0.998372i	$0.481835\pi$
$744$	0	0
$745$	−3.31125e20	−0.0952401
$746$	−2.57758e21	−0.733958
$747$	0	0
$748$	−1.99345e21	−0.556343
$749$	2.01854e20	0.0557730
$750$	0	0
$751$	−3.84496e21	−1.04134	−0.520669	−	0.853759i	$-0.674317\pi$
$751$	−3.84496e21	−1.04134	−0.520669	+	0.853759i	$0.674317\pi$
$752$	1.88914e20	0.0506558
$753$	0	0
$754$	2.89882e20	0.0761966
$755$	−8.36857e20	−0.217796
$756$	0	0
$757$	−2.23187e21	−0.569442	−0.284721	−	0.958610i	$-0.591901\pi$
$757$	−2.23187e21	−0.569442	−0.284721	+	0.958610i	$0.591901\pi$
$758$	3.15143e21	0.796138
$759$	0	0
$760$	−3.37688e20	−0.0836399
$761$	−4.03178e21	−0.988808	−0.494404	−	0.869232i	$-0.664614\pi$
$761$	−4.03178e21	−0.988808	−0.494404	+	0.869232i	$0.664614\pi$
$762$	0	0
$763$	5.90559e20	0.142013
$764$	1.39001e21	0.330993
$765$	0	0
$766$	4.07144e21	0.950675
$767$	−6.67483e19	−0.0154339
$768$	0	0
$769$	2.90256e21	0.658163	0.329082	−	0.944301i	$-0.393261\pi$
$769$	2.90256e21	0.658163	0.329082	+	0.944301i	$0.393261\pi$
$770$	1.39467e20	0.0313177
$771$	0	0
$772$	1.91058e21	0.420761
$773$	−1.01296e21	−0.220925	−0.110463	−	0.993880i	$-0.535233\pi$
$773$	−1.01296e21	−0.220925	−0.110463	+	0.993880i	$0.535233\pi$
$774$	0	0
$775$	−7.56915e20	−0.161914
$776$	−2.56581e20	−0.0543578
$777$	0	0
$778$	5.55040e21	1.15339
$779$	−8.34432e20	−0.171735
$780$	0	0
$781$	3.68876e21	0.744729
$782$	8.28384e21	1.65646
$783$	0	0
$784$	−1.24900e21	−0.245014
$785$	−2.64136e21	−0.513222
$786$	0	0
$787$	5.06206e21	0.964978	0.482489	−	0.875902i	$-0.339733\pi$
$787$	5.06206e21	0.964978	0.482489	+	0.875902i	$0.339733\pi$
$788$	1.08141e21	0.204194
$789$	0	0
$790$	−5.66091e20	−0.104878
$791$	1.00952e20	0.0185264
$792$	0	0
$793$	−5.56650e20	−0.100238
$794$	−1.03987e21	−0.185493
$795$	0	0
$796$	2.09330e20	0.0366424
$797$	7.43121e21	1.28861	0.644306	−	0.764768i	$-0.277146\pi$
$797$	7.43121e21	1.28861	0.644306	+	0.764768i	$0.277146\pi$
$798$	0	0
$799$	−1.88924e21	−0.321503
$800$	−2.09715e20	−0.0353553
$801$	0	0
$802$	4.49799e20	0.0744237
$803$	−3.23363e21	−0.530059
$804$	0	0
$805$	−5.79559e20	−0.0932456
$806$	3.34263e20	0.0532814
$807$	0	0
$808$	−3.66139e21	−0.572876
$809$	3.66614e21	0.568323	0.284162	−	0.958776i	$-0.408285\pi$
$809$	3.66614e21	0.568323	0.284162	+	0.958776i	$0.408285\pi$
$810$	0	0
$811$	1.22030e22	1.85700	0.928500	−	0.371332i	$-0.121099\pi$
$811$	1.22030e22	1.85700	0.928500	+	0.371332i	$0.121099\pi$
$812$	5.42211e20	0.0817519
$813$	0	0
$814$	5.78126e21	0.855735
$815$	−5.60554e21	−0.822120
$816$	0	0
$817$	4.63075e21	0.666785
$818$	8.24699e21	1.17665
$819$	0	0
$820$	−5.18209e20	−0.0725941
$821$	−1.23404e22	−1.71299	−0.856495	−	0.516156i	$-0.827363\pi$
$821$	−1.23404e22	−1.71299	−0.856495	+	0.516156i	$0.827363\pi$
$822$	0	0
$823$	−1.61418e21	−0.220016	−0.110008	−	0.993931i	$-0.535088\pi$
$823$	−1.61418e21	−0.220016	−0.110008	+	0.993931i	$0.535088\pi$
$824$	1.61502e21	0.218134
$825$	0	0
$826$	−1.24850e20	−0.0165591
$827$	7.51350e21	0.987532	0.493766	−	0.869595i	$-0.335620\pi$
$827$	7.51350e21	0.987532	0.493766	+	0.869595i	$0.335620\pi$
$828$	0	0
$829$	8.41217e21	1.08580	0.542899	−	0.839798i	$-0.317327\pi$
$829$	8.41217e21	1.08580	0.542899	+	0.839798i	$0.317327\pi$
$830$	−2.41022e21	−0.308298
$831$	0	0
$832$	9.26129e19	0.0116345
$833$	1.24906e22	1.55506
$834$	0	0
$835$	1.13285e21	0.138523
$836$	1.53051e21	0.185477
$837$	0	0
$838$	−1.44226e21	−0.171677
$839$	4.19252e21	0.494607	0.247304	−	0.968938i	$-0.420456\pi$
$839$	4.19252e21	0.494607	0.247304	+	0.968938i	$0.420456\pi$
$840$	0	0
$841$	2.93722e21	0.340381
$842$	6.15632e21	0.707099
$843$	0	0
$844$	−7.22693e20	−0.0815428
$845$	−3.96426e21	−0.443339
$846$	0	0
$847$	6.53294e20	0.0717767
$848$	3.12944e21	0.340798
$849$	0	0
$850$	2.09726e21	0.224394
$851$	−2.40242e22	−2.54787
$852$	0	0
$853$	−9.01594e21	−0.939493	−0.469746	−	0.882801i	$-0.655655\pi$
$853$	−9.01594e21	−0.939493	−0.469746	+	0.882801i	$0.655655\pi$
$854$	−1.04119e21	−0.107546
$855$	0	0
$856$	1.37568e21	0.139626
$857$	3.51268e21	0.353413	0.176706	−	0.984264i	$-0.443456\pi$
$857$	3.51268e21	0.353413	0.176706	+	0.984264i	$0.443456\pi$
$858$	0	0
$859$	−1.12764e22	−1.11486	−0.557430	−	0.830224i	$-0.688212\pi$
$859$	−1.12764e22	−1.11486	−0.557430	+	0.830224i	$0.688212\pi$
$860$	2.87584e21	0.281856
$861$	0	0
$862$	8.82428e21	0.849914
$863$	−3.75344e21	−0.358384	−0.179192	−	0.983814i	$-0.557348\pi$
$863$	−3.75344e21	−0.358384	−0.179192	+	0.983814i	$0.557348\pi$
$864$	0	0
$865$	4.27817e21	0.401455
$866$	1.78519e21	0.166073
$867$	0	0
$868$	6.25225e20	0.0571660
$869$	2.56571e21	0.232573
$870$	0	0
$871$	1.24213e21	0.110670
$872$	4.02480e21	0.355526
$873$	0	0
$874$	−6.36010e21	−0.552241
$875$	−1.46730e20	−0.0126316
$876$	0	0
$877$	−8.42140e21	−0.712669	−0.356334	−	0.934359i	$-0.615974\pi$
$877$	−8.42140e21	−0.712669	−0.356334	+	0.934359i	$0.615974\pi$
$878$	−1.07103e22	−0.898651
$879$	0	0
$880$	9.50498e20	0.0784027
$881$	−2.60636e21	−0.213164	−0.106582	−	0.994304i	$-0.533991\pi$
$881$	−2.60636e21	−0.213164	−0.106582	+	0.994304i	$0.533991\pi$
$882$	0	0
$883$	1.49967e22	1.20584	0.602919	−	0.797802i	$-0.294004\pi$
$883$	1.49967e22	1.20584	0.602919	+	0.797802i	$0.294004\pi$
$884$	−9.26178e20	−0.0738419
$885$	0	0
$886$	−3.13455e21	−0.245710
$887$	1.91564e22	1.48897	0.744486	−	0.667639i	$-0.232695\pi$
$887$	1.91564e22	1.48897	0.744486	+	0.667639i	$0.232695\pi$
$888$	0	0
$889$	1.20699e21	0.0922442
$890$	4.96924e20	0.0376586
$891$	0	0
$892$	−9.98084e21	−0.743756
$893$	1.45050e21	0.107185
$894$	0	0
$895$	−4.40203e21	−0.319875
$896$	1.73228e20	0.0124827
$897$	0	0
$898$	7.45633e21	0.528388
$899$	1.33373e22	0.937278
$900$	0	0
$901$	−3.12960e22	−2.16298
$902$	2.34869e21	0.160982
$903$	0	0
$904$	6.88010e20	0.0463801
$905$	−1.10419e22	−0.738211
$906$	0	0
$907$	−2.49197e22	−1.63866	−0.819330	−	0.573323i	$-0.805654\pi$
$907$	−2.49197e22	−1.63866	−0.819330	+	0.573323i	$0.805654\pi$
$908$	−4.74651e21	−0.309550
$909$	0	0
$910$	6.47978e19	0.00415671
$911$	−2.98866e22	−1.90146	−0.950732	−	0.310014i	$-0.899666\pi$
$911$	−2.98866e22	−1.90146	−0.950732	+	0.310014i	$0.899666\pi$
$912$	0	0
$913$	1.09239e22	0.683670
$914$	1.67999e22	1.04282
$915$	0	0
$916$	−5.75815e21	−0.351614
$917$	2.39914e21	0.145306
$918$	0	0
$919$	3.97706e21	0.236971	0.118486	−	0.992956i	$-0.462196\pi$
$919$	3.97706e21	0.236971	0.118486	+	0.992956i	$0.462196\pi$
$920$	−3.94983e21	−0.233437
$921$	0	0
$922$	−7.43222e21	−0.432152
$923$	1.71384e21	0.0988458
$924$	0	0
$925$	−6.08234e21	−0.345150
$926$	−2.04298e22	−1.14996
$927$	0	0
$928$	3.69529e21	0.204663
$929$	−1.39808e22	−0.768092	−0.384046	−	0.923314i	$-0.625470\pi$
$929$	−1.39808e22	−0.768092	−0.384046	+	0.923314i	$0.625470\pi$
$930$	0	0
$931$	−9.58996e21	−0.518435
$932$	2.20092e21	0.118028
$933$	0	0
$934$	−1.93369e22	−1.02043
$935$	−9.50549e21	−0.497608
$936$	0	0
$937$	−6.98884e21	−0.360046	−0.180023	−	0.983662i	$-0.557617\pi$
$937$	−6.98884e21	−0.360046	−0.180023	+	0.983662i	$0.557617\pi$
$938$	2.32335e21	0.118739
$939$	0	0
$940$	9.00810e20	0.0453079
$941$	1.61505e21	0.0805868	0.0402934	−	0.999188i	$-0.487171\pi$
$941$	1.61505e21	0.0805868	0.0402934	+	0.999188i	$0.487171\pi$
$942$	0	0
$943$	−9.76008e21	−0.479309
$944$	−8.50881e20	−0.0414552
$945$	0	0
$946$	−1.30343e22	−0.625033
$947$	1.59416e22	0.758414	0.379207	−	0.925312i	$-0.376197\pi$
$947$	1.59416e22	0.758414	0.379207	+	0.925312i	$0.376197\pi$
$948$	0	0
$949$	−1.50238e21	−0.0703532
$950$	−1.61022e21	−0.0748098
$951$	0	0
$952$	−1.73238e21	−0.0792255
$953$	2.14420e22	0.972899	0.486449	−	0.873709i	$-0.338292\pi$
$953$	2.14420e22	0.972899	0.486449	+	0.873709i	$0.338292\pi$
$954$	0	0
$955$	6.62809e21	0.296049
$956$	−1.93942e22	−0.859482
$957$	0	0
$958$	−2.09096e22	−0.912231
$959$	3.59480e21	0.155609
$960$	0	0
$961$	−8.08605e21	−0.344596
$962$	2.68604e21	0.113579
$963$	0	0
$964$	1.30514e22	0.543351
$965$	9.11037e21	0.376340
$966$	0	0
$967$	3.74044e21	0.152133	0.0760667	−	0.997103i	$-0.475764\pi$
$967$	3.74044e21	0.152133	0.0760667	+	0.997103i	$0.475764\pi$
$968$	4.45235e21	0.179690
$969$	0	0
$970$	−1.22347e21	−0.0486191
$971$	−4.76468e22	−1.87884	−0.939419	−	0.342771i	$-0.888635\pi$
$971$	−4.76468e22	−1.87884	−0.939419	+	0.342771i	$0.888635\pi$
$972$	0	0
$973$	2.99865e21	0.116434
$974$	−1.18270e22	−0.455703
$975$	0	0
$976$	−7.09595e21	−0.269239
$977$	1.91084e22	0.719475	0.359737	−	0.933054i	$-0.382866\pi$
$977$	1.91084e22	0.719475	0.359737	+	0.933054i	$0.382866\pi$
$978$	0	0
$979$	−2.25222e21	−0.0835104
$980$	−5.95568e21	−0.219147
$981$	0	0
$982$	2.51723e22	0.912192
$983$	−5.33757e22	−1.91952	−0.959759	−	0.280826i	$-0.909392\pi$
$983$	−5.33757e22	−1.91952	−0.959759	+	0.280826i	$0.909392\pi$
$984$	0	0
$985$	5.15656e21	0.182637
$986$	−3.69549e22	−1.29896
$987$	0	0
$988$	7.11094e20	0.0246178
$989$	5.41644e22	1.86098
$990$	0	0
$991$	−2.36848e22	−0.801526	−0.400763	−	0.916182i	$-0.631255\pi$
$991$	−2.36848e22	−0.801526	−0.400763	+	0.916182i	$0.631255\pi$
$992$	4.26105e21	0.143113
$993$	0	0
$994$	3.20567e21	0.106052
$995$	9.98163e20	0.0327739
$996$	0	0
$997$	−2.47974e22	−0.802035	−0.401017	−	0.916070i	$-0.631343\pi$
$997$	−2.47974e22	−0.802035	−0.401017	+	0.916070i	$0.631343\pi$
$998$	5.77671e21	0.185439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.16.a.k.1.2	✓	2
3.2	odd	2		90.16.a.m.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.16.a.k.1.2	✓	2	1.1	even	1	trivial
90.16.a.m.1.2	yes	2	3.2	odd	2

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:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

\(f(q)\)	\(=\)	\(q-128.000 q^{2} +16384.0 q^{4} +78125.0 q^{5} -307715. q^{7} -2.09715e6 q^{8} -1.00000e7 q^{10} +4.53233e7 q^{11} +2.10577e7 q^{13} +3.93876e7 q^{14} +2.68435e8 q^{16} -2.68450e9 q^{17} +2.06108e9 q^{19} +1.28000e9 q^{20} -5.80138e9 q^{22} +2.41078e10 q^{23} +6.10352e9 q^{25} -2.69539e9 q^{26} -5.04161e9 q^{28} -1.07547e11 q^{29} -1.24013e11 q^{31} -3.43597e10 q^{32} +3.43616e11 q^{34} -2.40403e10 q^{35} -9.96531e11 q^{37} -2.63819e11 q^{38} -1.63840e11 q^{40} -4.04851e11 q^{41} +2.24675e12 q^{43} +7.42577e11 q^{44} -3.08580e12 q^{46} +7.03758e11 q^{47} -4.65287e12 q^{49} -7.81250e11 q^{50} +3.45010e11 q^{52} +1.16581e13 q^{53} +3.54088e12 q^{55} +6.45326e11 q^{56} +1.37660e13 q^{58} -3.16978e12 q^{59} -2.64345e13 q^{61} +1.58737e13 q^{62} +4.39805e12 q^{64} +1.64513e12 q^{65} +5.89869e13 q^{67} -4.39828e13 q^{68} +3.07715e12 q^{70} +8.13879e13 q^{71} -7.13459e13 q^{73} +1.27556e14 q^{74} +3.37688e13 q^{76} -1.39467e13 q^{77} +5.66091e13 q^{79} +2.09715e13 q^{80} +5.18209e13 q^{82} +2.41022e14 q^{83} -2.09726e14 q^{85} -2.87584e14 q^{86} -9.50498e13 q^{88} -4.96924e13 q^{89} -6.47978e12 q^{91} +3.94983e14 q^{92} -9.00810e13 q^{94} +1.61022e14 q^{95} +1.22347e14 q^{97} +5.95568e14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 256 q^{2} + 32768 q^{4} + 156250 q^{5} - 1761200 q^{7} - 4194304 q^{8} - 20000000 q^{10} - 11981640 q^{11} + 88928320 q^{13} + 225433600 q^{14} + 536870912 q^{16} - 234312756 q^{17} + 3086064208 q^{19}+ \cdots + 932840500876032 q^{98}+O(q^{100}) \) 2 * q - 256 * q^2 + 32768 * q^4 + 156250 * q^5 - 1761200 * q^7 - 4194304 * q^8 - 20000000 * q^10 - 11981640 * q^11 + 88928320 * q^13 + 225433600 * q^14 + 536870912 * q^16 - 234312756 * q^17 + 3086064208 * q^19 + 2560000000 * q^20 + 1533649920 * q^22 + 7357553280 * q^23 + 12207031250 * q^25 - 11382824960 * q^26 - 28855500800 * q^28 - 106516740300 * q^29 - 80812338008 * q^31 - 68719476736 * q^32 + 29992032768 * q^34 - 137593750000 * q^35 - 783787471400 * q^37 - 395016218624 * q^38 - 327680000000 * q^40 - 125811612720 * q^41 - 700693895840 * q^43 - 196307189760 * q^44 - 941766819840 * q^46 + 2028162668400 * q^47 - 7287816413094 * q^49 - 1562500000000 * q^50 + 1457001594880 * q^52 + 16011801636996 * q^53 - 936065625000 * q^55 + 3693504102400 * q^56 + 13634142758400 * q^58 + 20778903613320 * q^59 + 6823475639524 * q^61 + 10343979265024 * q^62 + 8796093022208 * q^64 + 6947525000000 * q^65 + 21554311388800 * q^67 - 3838980194304 * q^68 + 17612000000000 * q^70 + 48887974479120 * q^71 - 37196698499300 * q^73 + 100324796339200 * q^74 + 50562075983872 * q^76 + 69345170329512 * q^77 - 125355565580312 * q^79 + 41943040000000 * q^80 + 16103886428160 * q^82 + 365242719675648 * q^83 - 18305684062500 * q^85 + 89688818667520 * q^86 + 25127320289280 * q^88 + 329691731091240 * q^89 - 105128657395216 * q^91 + 120546152939520 * q^92 - 259604821555200 * q^94 + 241098766250000 * q^95 - 37249221079340 * q^97 + 932840500876032 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more