LMFDB - Embedded newform 64.18.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,18,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	64.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$117.262135901$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{9361}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2340 $ x^2 - x - 2340
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 4)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-47.8761$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15636.4 q^{3} +366730. q^{5} -2.37095e7 q^{7} +1.15358e8 q^{9} +O(q^{10})$	$q+15636.4 q^{3} +366730. q^{5} -2.37095e7 q^{7} +1.15358e8 q^{9} +1.33786e9 q^{11} +3.68345e9 q^{13} +5.73434e9 q^{15} -1.70006e10 q^{17} -2.59141e9 q^{19} -3.70732e11 q^{21} +1.96178e11 q^{23} -6.28449e11 q^{25} -2.15505e11 q^{27} +1.36209e12 q^{29} +2.25575e12 q^{31} +2.09194e13 q^{33} -8.69497e12 q^{35} -1.28643e12 q^{37} +5.75961e13 q^{39} +3.74268e13 q^{41} -2.61921e13 q^{43} +4.23052e13 q^{45} -1.94586e14 q^{47} +3.29509e14 q^{49} -2.65829e14 q^{51} +4.69769e14 q^{53} +4.90635e14 q^{55} -4.05205e13 q^{57} +1.58789e15 q^{59} +2.16562e15 q^{61} -2.73508e15 q^{63} +1.35083e15 q^{65} +1.54302e15 q^{67} +3.06753e15 q^{69} +4.51083e15 q^{71} -6.24410e15 q^{73} -9.82670e15 q^{75} -3.17201e16 q^{77} +7.95599e14 q^{79} -1.82671e16 q^{81} +2.62699e16 q^{83} -6.23462e15 q^{85} +2.12982e16 q^{87} +3.01291e16 q^{89} -8.73327e16 q^{91} +3.52719e16 q^{93} -9.50348e14 q^{95} +3.38333e16 q^{97} +1.54333e17 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5880 q^{3} - 604044 q^{5} - 25350160 q^{7} + 449174682 q^{9}+O(q^{10}) $ 2 * q - 5880 * q^3 - 604044 * q^5 - 25350160 * q^7 + 449174682 * q^9	$ 2 q - 5880 q^{3} - 604044 q^{5} - 25350160 q^{7} + 449174682 q^{9} + 1259648280 q^{11} + 1320052580 q^{13} + 26621930448 q^{15} - 27498226140 q^{17} + 101133633832 q^{19} - 335430207552 q^{21} - 134767491120 q^{23} - 448986850114 q^{25} - 4619416117680 q^{27} - 2337155582652 q^{29} - 278836113472 q^{31} + 22602380058720 q^{33} - 7102242382752 q^{35} - 20929802888140 q^{37} + 108447997173648 q^{39} + 4166592315732 q^{41} - 111143148534440 q^{43} - 281755357404444 q^{45} - 196772651157600 q^{47} + 99570326741874 q^{49} - 39956666918256 q^{51} + 487965122736660 q^{53} + 566565363246960 q^{55} - 22\!\cdots\!40 q^{57}+ \cdots + 12\!\cdots\!80 q^{99}+O(q^{100}) $ 2 * q - 5880 * q^3 - 604044 * q^5 - 25350160 * q^7 + 449174682 * q^9 + 1259648280 * q^11 + 1320052580 * q^13 + 26621930448 * q^15 - 27498226140 * q^17 + 101133633832 * q^19 - 335430207552 * q^21 - 134767491120 * q^23 - 448986850114 * q^25 - 4619416117680 * q^27 - 2337155582652 * q^29 - 278836113472 * q^31 + 22602380058720 * q^33 - 7102242382752 * q^35 - 20929802888140 * q^37 + 108447997173648 * q^39 + 4166592315732 * q^41 - 111143148534440 * q^43 - 281755357404444 * q^45 - 196772651157600 * q^47 + 99570326741874 * q^49 - 39956666918256 * q^51 + 487965122736660 * q^53 + 566565363246960 * q^55 - 2272313631999840 * q^57 + 2835904197813624 * q^59 + 67544034994436 * q^61 - 3282762121966800 * q^63 + 3645157343001768 * q^65 - 995171321546360 * q^67 + 10188302100105024 * q^69 - 3882245493215376 * q^71 - 12746425881769580 * q^73 - 13688080703624712 * q^75 - 31591755846002880 * q^77 + 14984271534065504 * q^79 + 33380230287529458 * q^81 + 43899417809893800 * q^83 + 3956216991540264 * q^85 + 100892756486768400 * q^87 + 51909007958846388 * q^89 - 83455169400462112 * q^91 + 89807116632334080 * q^93 - 101643889304423664 * q^95 - 49281007789848380 * q^97 + 128223271309133880 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	15636.4	1.37596	0.687982	−	0.725728i	$-0.258497\pi$
$3$	15636.4	1.37596	0.687982	+	0.725728i	$0.258497\pi$
$4$	0	0
$5$	366730.	0.419857	0.209928	−	0.977717i	$-0.432677\pi$
$5$	366730.	0.419857	0.209928	+	0.977717i	$0.432677\pi$
$6$	0	0
$7$	−2.37095e7	−1.55449	−0.777246	−	0.629196i	$-0.783384\pi$
$7$	−2.37095e7	−1.55449	−0.777246	+	0.629196i	$0.783384\pi$
$8$	0	0
$9$	1.15358e8	0.893277
$10$	0	0
$11$	1.33786e9	1.88180	0.940902	−	0.338679i	$-0.109980\pi$
$11$	1.33786e9	1.88180	0.940902	+	0.338679i	$0.109980\pi$
$12$	0	0
$13$	3.68345e9	1.25238	0.626191	−	0.779670i	$-0.284613\pi$
$13$	3.68345e9	1.25238	0.626191	+	0.779670i	$0.284613\pi$
$14$	0	0
$15$	5.73434e9	0.577708
$16$	0	0
$17$	−1.70006e10	−0.591082	−0.295541	−	0.955330i	$-0.595500\pi$
$17$	−1.70006e10	−0.591082	−0.295541	+	0.955330i	$0.595500\pi$
$18$	0	0
$19$	−2.59141e9	−0.0350051	−0.0175025	−	0.999847i	$-0.505572\pi$
$19$	−2.59141e9	−0.0350051	−0.0175025	+	0.999847i	$0.505572\pi$
$20$	0	0
$21$	−3.70732e11	−2.13893
$22$	0	0
$23$	1.96178e11	0.522353	0.261177	−	0.965291i	$-0.415889\pi$
$23$	1.96178e11	0.522353	0.261177	+	0.965291i	$0.415889\pi$
$24$	0	0
$25$	−6.28449e11	−0.823720
$26$	0	0
$27$	−2.15505e11	−0.146847
$28$	0	0
$29$	1.36209e12	0.505618	0.252809	−	0.967516i	$-0.418646\pi$
$29$	1.36209e12	0.505618	0.252809	+	0.967516i	$0.418646\pi$
$30$	0	0
$31$	2.25575e12	0.475025	0.237512	−	0.971384i	$-0.423668\pi$
$31$	2.25575e12	0.475025	0.237512	+	0.971384i	$0.423668\pi$
$32$	0	0
$33$	2.09194e13	2.58929
$34$	0	0
$35$	−8.69497e12	−0.652664
$36$	0	0
$37$	−1.28643e12	−0.0602105	−0.0301052	−	0.999547i	$-0.509584\pi$
$37$	−1.28643e12	−0.0602105	−0.0301052	+	0.999547i	$0.509584\pi$
$38$	0	0
$39$	5.75961e13	1.72323
$40$	0	0
$41$	3.74268e13	0.732014	0.366007	−	0.930612i	$-0.380725\pi$
$41$	3.74268e13	0.732014	0.366007	+	0.930612i	$0.380725\pi$
$42$	0	0
$43$	−2.61921e13	−0.341734	−0.170867	−	0.985294i	$-0.554657\pi$
$43$	−2.61921e13	−0.341734	−0.170867	+	0.985294i	$0.554657\pi$
$44$	0	0
$45$	4.23052e13	0.375048
$46$	0	0
$47$	−1.94586e14	−1.19201	−0.596006	−	0.802980i	$-0.703246\pi$
$47$	−1.94586e14	−1.19201	−0.596006	+	0.802980i	$0.703246\pi$
$48$	0	0
$49$	3.29509e14	1.41645
$50$	0	0
$51$	−2.65829e14	−0.813308
$52$	0	0
$53$	4.69769e14	1.03643	0.518214	−	0.855251i	$-0.326597\pi$
$53$	4.69769e14	1.03643	0.518214	+	0.855251i	$0.326597\pi$
$54$	0	0
$55$	4.90635e14	0.790088
$56$	0	0
$57$	−4.05205e13	−0.0481657
$58$	0	0
$59$	1.58789e15	1.40792	0.703962	−	0.710238i	$-0.251413\pi$
$59$	1.58789e15	1.40792	0.703962	+	0.710238i	$0.251413\pi$
$60$	0	0
$61$	2.16562e15	1.44637	0.723183	−	0.690656i	$-0.242678\pi$
$61$	2.16562e15	1.44637	0.723183	+	0.690656i	$0.242678\pi$
$62$	0	0
$63$	−2.73508e15	−1.38859
$64$	0	0
$65$	1.35083e15	0.525821
$66$	0	0
$67$	1.54302e15	0.464232	0.232116	−	0.972688i	$-0.425435\pi$
$67$	1.54302e15	0.464232	0.232116	+	0.972688i	$0.425435\pi$
$68$	0	0
$69$	3.06753e15	0.718739
$70$	0	0
$71$	4.51083e15	0.829011	0.414506	−	0.910047i	$-0.363954\pi$
$71$	4.51083e15	0.829011	0.414506	+	0.910047i	$0.363954\pi$
$72$	0	0
$73$	−6.24410e15	−0.906203	−0.453101	−	0.891459i	$-0.649682\pi$
$73$	−6.24410e15	−0.906203	−0.453101	+	0.891459i	$0.649682\pi$
$74$	0	0
$75$	−9.82670e15	−1.13341
$76$	0	0
$77$	−3.17201e16	−2.92525
$78$	0	0
$79$	7.95599e14	0.0590016	0.0295008	−	0.999565i	$-0.490608\pi$
$79$	7.95599e14	0.0590016	0.0295008	+	0.999565i	$0.490608\pi$
$80$	0	0
$81$	−1.82671e16	−1.09533
$82$	0	0
$83$	2.62699e16	1.28025	0.640125	−	0.768271i	$-0.278883\pi$
$83$	2.62699e16	1.28025	0.640125	+	0.768271i	$0.278883\pi$
$84$	0	0
$85$	−6.23462e15	−0.248170
$86$	0	0
$87$	2.12982e16	0.695712
$88$	0	0
$89$	3.01291e16	0.811280	0.405640	−	0.914033i	$-0.367049\pi$
$89$	3.01291e16	0.811280	0.405640	+	0.914033i	$0.367049\pi$
$90$	0	0
$91$	−8.73327e16	−1.94682
$92$	0	0
$93$	3.52719e16	0.653617
$94$	0	0
$95$	−9.50348e14	−0.0146971
$96$	0	0
$97$	3.38333e16	0.438314	0.219157	−	0.975690i	$-0.429669\pi$
$97$	3.38333e16	0.438314	0.219157	+	0.975690i	$0.429669\pi$
$98$	0	0
$99$	1.54333e17	1.68097
$100$	0	0
$101$	−5.50280e16	−0.505653	−0.252826	−	0.967512i	$-0.581360\pi$
$101$	−5.50280e16	−0.505653	−0.252826	+	0.967512i	$0.581360\pi$
$102$	0	0
$103$	2.99690e16	0.233107	0.116554	−	0.993184i	$-0.462815\pi$
$103$	2.99690e16	0.233107	0.116554	+	0.993184i	$0.462815\pi$
$104$	0	0
$105$	−1.35958e17	−0.898042
$106$	0	0
$107$	1.83477e17	1.03233	0.516166	−	0.856489i	$-0.327359\pi$
$107$	1.83477e17	1.03233	0.516166	+	0.856489i	$0.327359\pi$
$108$	0	0
$109$	2.65387e17	1.27572	0.637859	−	0.770153i	$-0.279820\pi$
$109$	2.65387e17	1.27572	0.637859	+	0.770153i	$0.279820\pi$
$110$	0	0
$111$	−2.01152e16	−0.0828475
$112$	0	0
$113$	−1.05318e17	−0.372681	−0.186341	−	0.982485i	$-0.559663\pi$
$113$	−1.05318e17	−0.372681	−0.186341	+	0.982485i	$0.559663\pi$
$114$	0	0
$115$	7.19444e16	0.219313
$116$	0	0
$117$	4.24915e17	1.11872
$118$	0	0
$119$	4.03075e17	0.918833
$120$	0	0
$121$	1.28444e18	2.54119
$122$	0	0
$123$	5.85221e17	1.00723
$124$	0	0
$125$	−5.10263e17	−0.765701
$126$	0	0
$127$	7.92218e17	1.03875	0.519377	−	0.854545i	$-0.326164\pi$
$127$	7.92218e17	1.03875	0.519377	+	0.854545i	$0.326164\pi$
$128$	0	0
$129$	−4.09551e17	−0.470213
$130$	0	0
$131$	−7.17088e17	−0.722381	−0.361191	−	0.932492i	$-0.617630\pi$
$131$	−7.17088e17	−0.722381	−0.361191	+	0.932492i	$0.617630\pi$
$132$	0	0
$133$	6.14411e16	0.0544152
$134$	0	0
$135$	−7.90322e16	−0.0616548
$136$	0	0
$137$	−1.94843e18	−1.34141	−0.670703	−	0.741726i	$-0.734008\pi$
$137$	−1.94843e18	−1.34141	−0.670703	+	0.741726i	$0.734008\pi$
$138$	0	0
$139$	1.23071e18	0.749085	0.374543	−	0.927210i	$-0.377800\pi$
$139$	1.23071e18	0.749085	0.374543	+	0.927210i	$0.377800\pi$
$140$	0	0
$141$	−3.04264e18	−1.64016
$142$	0	0
$143$	4.92796e18	2.35674
$144$	0	0
$145$	4.99518e17	0.212287
$146$	0	0
$147$	5.15235e18	1.94898
$148$	0	0
$149$	−2.33643e18	−0.787898	−0.393949	−	0.919132i	$-0.628891\pi$
$149$	−2.33643e18	−0.787898	−0.393949	+	0.919132i	$0.628891\pi$
$150$	0	0
$151$	−5.57631e17	−0.167897	−0.0839485	−	0.996470i	$-0.526753\pi$
$151$	−5.57631e17	−0.167897	−0.0839485	+	0.996470i	$0.526753\pi$
$152$	0	0
$153$	−1.96115e18	−0.528000
$154$	0	0
$155$	8.27250e17	0.199442
$156$	0	0
$157$	5.14946e18	1.11331	0.556653	−	0.830745i	$-0.312085\pi$
$157$	5.14946e18	1.11331	0.556653	+	0.830745i	$0.312085\pi$
$158$	0	0
$159$	7.34551e18	1.42609
$160$	0	0
$161$	−4.65129e18	−0.811994
$162$	0	0
$163$	1.04483e19	1.64230	0.821148	−	0.570715i	$-0.193334\pi$
$163$	1.04483e19	1.64230	0.821148	+	0.570715i	$0.193334\pi$
$164$	0	0
$165$	7.67178e18	1.08713
$166$	0	0
$167$	−1.10713e19	−1.41614	−0.708071	−	0.706141i	$-0.750435\pi$
$167$	−1.10713e19	−1.41614	−0.708071	+	0.706141i	$0.750435\pi$
$168$	0	0
$169$	4.91741e18	0.568459
$170$	0	0
$171$	−2.98940e17	−0.0312692
$172$	0	0
$173$	1.38307e18	0.131055	0.0655274	−	0.997851i	$-0.479127\pi$
$173$	1.38307e18	0.131055	0.0655274	+	0.997851i	$0.479127\pi$
$174$	0	0
$175$	1.49002e19	1.28047
$176$	0	0
$177$	2.48290e19	1.93725
$178$	0	0
$179$	1.50209e19	1.06524	0.532618	−	0.846356i	$-0.321208\pi$
$179$	1.50209e19	1.06524	0.532618	+	0.846356i	$0.321208\pi$
$180$	0	0
$181$	3.60783e17	0.0232798	0.0116399	−	0.999932i	$-0.496295\pi$
$181$	3.60783e17	0.0232798	0.0116399	+	0.999932i	$0.496295\pi$
$182$	0	0
$183$	3.38625e19	1.99015
$184$	0	0
$185$	−4.71773e17	−0.0252798
$186$	0	0
$187$	−2.27445e19	−1.11230
$188$	0	0
$189$	5.10952e18	0.228273
$190$	0	0
$191$	−8.97904e18	−0.366814	−0.183407	−	0.983037i	$-0.558713\pi$
$191$	−8.97904e18	−0.366814	−0.183407	+	0.983037i	$0.558713\pi$
$192$	0	0
$193$	−2.87785e19	−1.07605	−0.538023	−	0.842930i	$-0.680829\pi$
$193$	−2.87785e19	−1.07605	−0.538023	+	0.842930i	$0.680829\pi$
$194$	0	0
$195$	2.11222e19	0.723510
$196$	0	0
$197$	−5.29790e19	−1.66395	−0.831977	−	0.554811i	$-0.812791\pi$
$197$	−5.29790e19	−1.66395	−0.831977	+	0.554811i	$0.812791\pi$
$198$	0	0
$199$	−2.52925e19	−0.729021	−0.364510	−	0.931199i	$-0.618764\pi$
$199$	−2.52925e19	−0.729021	−0.364510	+	0.931199i	$0.618764\pi$
$200$	0	0
$201$	2.41273e19	0.638767
$202$	0	0
$203$	−3.22944e19	−0.785979
$204$	0	0
$205$	1.37255e19	0.307341
$206$	0	0
$207$	2.26307e19	0.466606
$208$	0	0
$209$	−3.46696e18	−0.0658727
$210$	0	0
$211$	6.87338e19	1.20440	0.602198	−	0.798346i	$-0.294292\pi$
$211$	6.87338e19	1.20440	0.602198	+	0.798346i	$0.294292\pi$
$212$	0	0
$213$	7.05333e19	1.14069
$214$	0	0
$215$	−9.60541e18	−0.143479
$216$	0	0
$217$	−5.34826e19	−0.738423
$218$	0	0
$219$	−9.76354e19	−1.24690
$220$	0	0
$221$	−6.26208e19	−0.740260
$222$	0	0
$223$	4.40564e18	0.0482412	0.0241206	−	0.999709i	$-0.492321\pi$
$223$	4.40564e18	0.0482412	0.0241206	+	0.999709i	$0.492321\pi$
$224$	0	0
$225$	−7.24965e19	−0.735810
$226$	0	0
$227$	3.27911e19	0.308700	0.154350	−	0.988016i	$-0.450672\pi$
$227$	3.27911e19	0.308700	0.154350	+	0.988016i	$0.450672\pi$
$228$	0	0
$229$	−9.54497e19	−0.834013	−0.417006	−	0.908904i	$-0.636921\pi$
$229$	−9.54497e19	−0.834013	−0.417006	+	0.908904i	$0.636921\pi$
$230$	0	0
$231$	−4.95989e20	−4.02504
$232$	0	0
$233$	−2.07634e20	−1.56593	−0.782965	−	0.622066i	$-0.786294\pi$
$233$	−2.07634e20	−1.56593	−0.782965	+	0.622066i	$0.786294\pi$
$234$	0	0
$235$	−7.13605e19	−0.500474
$236$	0	0
$237$	1.24403e19	0.0811841
$238$	0	0
$239$	2.76425e20	1.67956	0.839780	−	0.542927i	$-0.182684\pi$
$239$	2.76425e20	1.67956	0.839780	+	0.542927i	$0.182684\pi$
$240$	0	0
$241$	2.35403e20	1.33250	0.666249	−	0.745730i	$-0.267899\pi$
$241$	2.35403e20	1.33250	0.666249	+	0.745730i	$0.267899\pi$
$242$	0	0
$243$	−2.57802e20	−1.36029
$244$	0	0
$245$	1.20841e20	0.594705
$246$	0	0
$247$	−9.54535e18	−0.0438397
$248$	0	0
$249$	4.10768e20	1.76158
$250$	0	0
$251$	−3.91593e20	−1.56895	−0.784473	−	0.620163i	$-0.787067\pi$
$251$	−3.91593e20	−1.56895	−0.784473	+	0.620163i	$0.787067\pi$
$252$	0	0
$253$	2.62460e20	0.982967
$254$	0	0
$255$	−9.74872e19	−0.341473
$256$	0	0
$257$	4.14828e20	1.35968	0.679839	−	0.733361i	$-0.262049\pi$
$257$	4.14828e20	1.35968	0.679839	+	0.733361i	$0.262049\pi$
$258$	0	0
$259$	3.05006e19	0.0935968
$260$	0	0
$261$	1.57128e20	0.451657
$262$	0	0
$263$	−9.82407e19	−0.264647	−0.132324	−	0.991207i	$-0.542244\pi$
$263$	−9.82407e19	−0.264647	−0.132324	+	0.991207i	$0.542244\pi$
$264$	0	0
$265$	1.72278e20	0.435151
$266$	0	0
$267$	4.71112e20	1.11629
$268$	0	0
$269$	−5.22922e20	−1.16290	−0.581450	−	0.813582i	$-0.697514\pi$
$269$	−5.22922e20	−1.16290	−0.581450	+	0.813582i	$0.697514\pi$
$270$	0	0
$271$	−4.47142e20	−0.933698	−0.466849	−	0.884337i	$-0.654611\pi$
$271$	−4.47142e20	−0.933698	−0.466849	+	0.884337i	$0.654611\pi$
$272$	0	0
$273$	−1.36557e21	−2.67875
$274$	0	0
$275$	−8.40780e20	−1.55008
$276$	0	0
$277$	1.36272e20	0.236226	0.118113	−	0.993000i	$-0.462316\pi$
$277$	1.36272e20	0.236226	0.118113	+	0.993000i	$0.462316\pi$
$278$	0	0
$279$	2.60219e20	0.424329
$280$	0	0
$281$	1.26390e20	0.193958	0.0969790	−	0.995286i	$-0.469082\pi$
$281$	1.26390e20	0.193958	0.0969790	+	0.995286i	$0.469082\pi$
$282$	0	0
$283$	−9.81664e19	−0.141833	−0.0709167	−	0.997482i	$-0.522592\pi$
$283$	−9.81664e19	−0.141833	−0.0709167	+	0.997482i	$0.522592\pi$
$284$	0	0
$285$	−1.48601e19	−0.0202227
$286$	0	0
$287$	−8.87370e20	−1.13791
$288$	0	0
$289$	−5.38220e20	−0.650622
$290$	0	0
$291$	5.29033e20	0.603104
$292$	0	0
$293$	−1.01600e21	−1.09275	−0.546374	−	0.837541i	$-0.683992\pi$
$293$	−1.01600e21	−1.09275	−0.546374	+	0.837541i	$0.683992\pi$
$294$	0	0
$295$	5.82327e20	0.591126
$296$	0	0
$297$	−2.88317e20	−0.276338
$298$	0	0
$299$	7.22613e20	0.654185
$300$	0	0
$301$	6.21000e20	0.531223
$302$	0	0
$303$	−8.60442e20	−0.695760
$304$	0	0
$305$	7.94196e20	0.607266
$306$	0	0
$307$	1.04896e21	0.758720	0.379360	−	0.925249i	$-0.376144\pi$
$307$	1.04896e21	0.758720	0.379360	+	0.925249i	$0.376144\pi$
$308$	0	0
$309$	4.68609e20	0.320747
$310$	0	0
$311$	1.73873e21	1.12660	0.563298	−	0.826254i	$-0.309532\pi$
$311$	1.73873e21	1.12660	0.563298	+	0.826254i	$0.309532\pi$
$312$	0	0
$313$	−2.56401e21	−1.57323	−0.786616	−	0.617443i	$-0.788169\pi$
$313$	−2.56401e21	−1.57323	−0.786616	+	0.617443i	$0.788169\pi$
$314$	0	0
$315$	−1.00303e21	−0.583010
$316$	0	0
$317$	3.06492e20	0.168817	0.0844084	−	0.996431i	$-0.473100\pi$
$317$	3.06492e20	0.168817	0.0844084	+	0.996431i	$0.473100\pi$
$318$	0	0
$319$	1.82229e21	0.951474
$320$	0	0
$321$	2.86892e21	1.42045
$322$	0	0
$323$	4.40556e19	0.0206909
$324$	0	0
$325$	−2.31486e21	−1.03161
$326$	0	0
$327$	4.14971e21	1.75534
$328$	0	0
$329$	4.61354e21	1.85297
$330$	0	0
$331$	4.66802e21	1.78072	0.890359	−	0.455259i	$-0.150453\pi$
$331$	4.66802e21	1.78072	0.890359	+	0.455259i	$0.150453\pi$
$332$	0	0
$333$	−1.48400e20	−0.0537846
$334$	0	0
$335$	5.65871e20	0.194911
$336$	0	0
$337$	−1.64491e21	−0.538626	−0.269313	−	0.963053i	$-0.586797\pi$
$337$	−1.64491e21	−0.538626	−0.269313	+	0.963053i	$0.586797\pi$
$338$	0	0
$339$	−1.64681e21	−0.512796
$340$	0	0
$341$	3.01789e21	0.893904
$342$	0	0
$343$	−2.29694e21	−0.647365
$344$	0	0
$345$	1.12495e21	0.301767
$346$	0	0
$347$	−9.19804e20	−0.234906	−0.117453	−	0.993078i	$-0.537473\pi$
$347$	−9.19804e20	−0.234906	−0.117453	+	0.993078i	$0.537473\pi$
$348$	0	0
$349$	1.23910e21	0.301363	0.150681	−	0.988582i	$-0.451853\pi$
$349$	1.23910e21	0.301363	0.150681	+	0.988582i	$0.451853\pi$
$350$	0	0
$351$	−7.93804e20	−0.183909
$352$	0	0
$353$	−2.26089e21	−0.499109	−0.249554	−	0.968361i	$-0.580284\pi$
$353$	−2.26089e21	−0.499109	−0.249554	+	0.968361i	$0.580284\pi$
$354$	0	0
$355$	1.65426e21	0.348066
$356$	0	0
$357$	6.30266e21	1.26428
$358$	0	0
$359$	9.96795e21	1.90679	0.953395	−	0.301725i	$-0.0975624\pi$
$359$	9.96795e21	1.90679	0.953395	+	0.301725i	$0.0975624\pi$
$360$	0	0
$361$	−5.47367e21	−0.998775
$362$	0	0
$363$	2.00840e22	3.49658
$364$	0	0
$365$	−2.28990e21	−0.380475
$366$	0	0
$367$	−5.17139e21	−0.820248	−0.410124	−	0.912030i	$-0.634515\pi$
$367$	−5.17139e21	−0.820248	−0.410124	+	0.912030i	$0.634515\pi$
$368$	0	0
$369$	4.31748e21	0.653892
$370$	0	0
$371$	−1.11380e22	−1.61112
$372$	0	0
$373$	−1.06342e22	−1.46954	−0.734769	−	0.678317i	$-0.762710\pi$
$373$	−1.06342e22	−1.46954	−0.734769	+	0.678317i	$0.762710\pi$
$374$	0	0
$375$	−7.97870e21	−1.05358
$376$	0	0
$377$	5.01719e21	0.633226
$378$	0	0
$379$	−6.81105e21	−0.821827	−0.410914	−	0.911674i	$-0.634790\pi$
$379$	−6.81105e21	−0.821827	−0.410914	+	0.911674i	$0.634790\pi$
$380$	0	0
$381$	1.23875e22	1.42929
$382$	0	0
$383$	6.65462e21	0.734402	0.367201	−	0.930142i	$-0.380316\pi$
$383$	6.65462e21	0.734402	0.367201	+	0.930142i	$0.380316\pi$
$384$	0	0
$385$	−1.16327e22	−1.22819
$386$	0	0
$387$	−3.02146e21	−0.305263
$388$	0	0
$389$	−1.61442e22	−1.56115	−0.780574	−	0.625063i	$-0.785073\pi$
$389$	−1.61442e22	−1.56115	−0.780574	+	0.625063i	$0.785073\pi$
$390$	0	0
$391$	−3.33515e21	−0.308754
$392$	0	0
$393$	−1.12127e22	−0.993971
$394$	0	0
$395$	2.91770e20	0.0247722
$396$	0	0
$397$	−1.02258e21	−0.0831725	−0.0415862	−	0.999135i	$-0.513241\pi$
$397$	−1.02258e21	−0.0831725	−0.0415862	+	0.999135i	$0.513241\pi$
$398$	0	0
$399$	9.60719e20	0.0748733
$400$	0	0
$401$	−1.32825e22	−0.992095	−0.496047	−	0.868295i	$-0.665216\pi$
$401$	−1.32825e22	−0.992095	−0.496047	+	0.868295i	$0.665216\pi$
$402$	0	0
$403$	8.30895e21	0.594912
$404$	0	0
$405$	−6.69908e21	−0.459883
$406$	0	0
$407$	−1.72107e21	−0.113304
$408$	0	0
$409$	2.58540e22	1.63260	0.816300	−	0.577628i	$-0.196022\pi$
$409$	2.58540e22	1.63260	0.816300	+	0.577628i	$0.196022\pi$
$410$	0	0
$411$	−3.04665e22	−1.84573
$412$	0	0
$413$	−3.76481e22	−2.18861
$414$	0	0
$415$	9.63395e21	0.537521
$416$	0	0
$417$	1.92440e22	1.03071
$418$	0	0
$419$	1.13941e22	0.585951	0.292975	−	0.956120i	$-0.405355\pi$
$419$	1.13941e22	0.585951	0.292975	+	0.956120i	$0.405355\pi$
$420$	0	0
$421$	2.96045e22	1.46204	0.731021	−	0.682354i	$-0.239044\pi$
$421$	2.96045e22	1.46204	0.731021	+	0.682354i	$0.239044\pi$
$422$	0	0
$423$	−2.24471e22	−1.06480
$424$	0	0
$425$	1.06840e22	0.486887
$426$	0	0
$427$	−5.13457e22	−2.24837
$428$	0	0
$429$	7.70557e22	3.24278
$430$	0	0
$431$	−1.85742e22	−0.751369	−0.375684	−	0.926748i	$-0.622592\pi$
$431$	−1.85742e22	−0.751369	−0.375684	+	0.926748i	$0.622592\pi$
$432$	0	0
$433$	1.71171e21	0.0665707	0.0332854	−	0.999446i	$-0.489403\pi$
$433$	1.71171e21	0.0665707	0.0332854	+	0.999446i	$0.489403\pi$
$434$	0	0
$435$	7.81068e21	0.292099
$436$	0	0
$437$	−5.08379e20	−0.0182850
$438$	0	0
$439$	−3.04484e22	−1.05346	−0.526728	−	0.850034i	$-0.676581\pi$
$439$	−3.04484e22	−1.05346	−0.526728	+	0.850034i	$0.676581\pi$
$440$	0	0
$441$	3.80115e22	1.26528
$442$	0	0
$443$	−2.23180e22	−0.714864	−0.357432	−	0.933939i	$-0.616348\pi$
$443$	−2.23180e22	−0.714864	−0.357432	+	0.933939i	$0.616348\pi$
$444$	0	0
$445$	1.10492e22	0.340621
$446$	0	0
$447$	−3.65335e22	−1.08412
$448$	0	0
$449$	−5.77731e22	−1.65056	−0.825281	−	0.564723i	$-0.808983\pi$
$449$	−5.77731e22	−1.65056	−0.825281	+	0.564723i	$0.808983\pi$
$450$	0	0
$451$	5.00720e22	1.37751
$452$	0	0
$453$	−8.71936e21	−0.231020
$454$	0	0
$455$	−3.20275e22	−0.817384
$456$	0	0
$457$	−4.47877e22	−1.10121	−0.550606	−	0.834766i	$-0.685603\pi$
$457$	−4.47877e22	−1.10121	−0.550606	+	0.834766i	$0.685603\pi$
$458$	0	0
$459$	3.66372e21	0.0867989
$460$	0	0
$461$	−7.44258e22	−1.69928	−0.849641	−	0.527361i	$-0.823182\pi$
$461$	−7.44258e22	−1.69928	−0.849641	+	0.527361i	$0.823182\pi$
$462$	0	0
$463$	3.47065e22	0.763787	0.381893	−	0.924206i	$-0.375272\pi$
$463$	3.47065e22	0.763787	0.381893	+	0.924206i	$0.375272\pi$
$464$	0	0
$465$	1.29352e22	0.274425
$466$	0	0
$467$	−3.07656e22	−0.629321	−0.314661	−	0.949204i	$-0.601891\pi$
$467$	−3.07656e22	−0.629321	−0.314661	+	0.949204i	$0.601891\pi$
$468$	0	0
$469$	−3.65842e22	−0.721646
$470$	0	0
$471$	8.05192e22	1.53187
$472$	0	0
$473$	−3.50414e22	−0.643076
$474$	0	0
$475$	1.62857e21	0.0288344
$476$	0	0
$477$	5.41916e22	0.925817
$478$	0	0
$479$	2.50857e22	0.413595	0.206797	−	0.978384i	$-0.433696\pi$
$479$	2.50857e22	0.413595	0.206797	+	0.978384i	$0.433696\pi$
$480$	0	0
$481$	−4.73851e21	−0.0754065
$482$	0	0
$483$	−7.27295e22	−1.11728
$484$	0	0
$485$	1.24077e22	0.184029
$486$	0	0
$487$	−1.25508e23	−1.79752	−0.898761	−	0.438439i	$-0.855532\pi$
$487$	−1.25508e23	−1.79752	−0.898761	+	0.438439i	$0.855532\pi$
$488$	0	0
$489$	1.63374e23	2.25974
$490$	0	0
$491$	5.29654e22	0.707619	0.353809	−	0.935318i	$-0.384886\pi$
$491$	5.29654e22	0.707619	0.353809	+	0.935318i	$0.384886\pi$
$492$	0	0
$493$	−2.31563e22	−0.298862
$494$	0	0
$495$	5.65986e22	0.705767
$496$	0	0
$497$	−1.06949e23	−1.28869
$498$	0	0
$499$	2.81497e22	0.327807	0.163904	−	0.986476i	$-0.447591\pi$
$499$	2.81497e22	0.327807	0.163904	+	0.986476i	$0.447591\pi$
$500$	0	0
$501$	−1.73115e23	−1.94856
$502$	0	0
$503$	5.42160e22	0.589929	0.294964	−	0.955508i	$-0.404692\pi$
$503$	5.42160e22	0.589929	0.294964	+	0.955508i	$0.404692\pi$
$504$	0	0
$505$	−2.01804e22	−0.212302
$506$	0	0
$507$	7.68907e22	0.782179
$508$	0	0
$509$	1.78224e23	1.75333	0.876666	−	0.481099i	$-0.159762\pi$
$509$	1.78224e23	1.75333	0.876666	+	0.481099i	$0.159762\pi$
$510$	0	0
$511$	1.48044e23	1.40869
$512$	0	0
$513$	5.58463e20	0.00514040
$514$	0	0
$515$	1.09905e22	0.0978717
$516$	0	0
$517$	−2.60330e23	−2.24313
$518$	0	0
$519$	2.16263e22	0.180327
$520$	0	0
$521$	−1.94263e22	−0.156773	−0.0783863	−	0.996923i	$-0.524977\pi$
$521$	−1.94263e22	−0.156773	−0.0783863	+	0.996923i	$0.524977\pi$
$522$	0	0
$523$	−1.65768e23	−1.29490	−0.647451	−	0.762107i	$-0.724165\pi$
$523$	−1.65768e23	−1.29490	−0.647451	+	0.762107i	$0.724165\pi$
$524$	0	0
$525$	2.32986e23	1.76188
$526$	0	0
$527$	−3.83491e22	−0.280779
$528$	0	0
$529$	−1.02564e23	−0.727147
$530$	0	0
$531$	1.83176e23	1.25767
$532$	0	0
$533$	1.37860e23	0.916761
$534$	0	0
$535$	6.72864e22	0.433431
$536$	0	0
$537$	2.34874e23	1.46573
$538$	0	0
$539$	4.40839e23	2.66548
$540$	0	0
$541$	−2.94676e22	−0.172650	−0.0863251	−	0.996267i	$-0.527512\pi$
$541$	−2.94676e22	−0.172650	−0.0863251	+	0.996267i	$0.527512\pi$
$542$	0	0
$543$	5.64137e21	0.0320321
$544$	0	0
$545$	9.73253e22	0.535618
$546$	0	0
$547$	8.00474e22	0.427027	0.213513	−	0.976940i	$-0.431509\pi$
$547$	8.00474e22	0.427027	0.213513	+	0.976940i	$0.431509\pi$
$548$	0	0
$549$	2.49821e23	1.29201
$550$	0	0
$551$	−3.52974e21	−0.0176992
$552$	0	0
$553$	−1.88632e22	−0.0917176
$554$	0	0
$555$	−7.37684e21	−0.0347840
$556$	0	0
$557$	1.60669e23	0.734790	0.367395	−	0.930065i	$-0.380250\pi$
$557$	1.60669e23	0.734790	0.367395	+	0.930065i	$0.380250\pi$
$558$	0	0
$559$	−9.64772e22	−0.427981
$560$	0	0
$561$	−3.55643e23	−1.53049
$562$	0	0
$563$	−9.05044e22	−0.377875	−0.188937	−	0.981989i	$-0.560504\pi$
$563$	−9.05044e22	−0.377875	−0.188937	+	0.981989i	$0.560504\pi$
$564$	0	0
$565$	−3.86234e22	−0.156473
$566$	0	0
$567$	4.33103e23	1.70269
$568$	0	0
$569$	4.23350e23	1.61527	0.807635	−	0.589683i	$-0.200747\pi$
$569$	4.23350e23	1.61527	0.807635	+	0.589683i	$0.200747\pi$
$570$	0	0
$571$	1.34739e21	0.00498985	0.00249493	−	0.999997i	$-0.499206\pi$
$571$	1.34739e21	0.00498985	0.00249493	+	0.999997i	$0.499206\pi$
$572$	0	0
$573$	−1.40400e23	−0.504723
$574$	0	0
$575$	−1.23288e23	−0.430273
$576$	0	0
$577$	−4.74144e22	−0.160663	−0.0803315	−	0.996768i	$-0.525598\pi$
$577$	−4.74144e22	−0.160663	−0.0803315	+	0.996768i	$0.525598\pi$
$578$	0	0
$579$	−4.49993e23	−1.48060
$580$	0	0
$581$	−6.22846e23	−1.99014
$582$	0	0
$583$	6.28487e23	1.95036
$584$	0	0
$585$	1.55829e23	0.469703
$586$	0	0
$587$	1.06143e23	0.310790	0.155395	−	0.987852i	$-0.450335\pi$
$587$	1.06143e23	0.310790	0.155395	+	0.987852i	$0.450335\pi$
$588$	0	0
$589$	−5.84558e21	−0.0166283
$590$	0	0
$591$	−8.28403e23	−2.28954
$592$	0	0
$593$	−3.00230e23	−0.806285	−0.403142	−	0.915137i	$-0.632082\pi$
$593$	−3.00230e23	−0.806285	−0.403142	+	0.915137i	$0.632082\pi$
$594$	0	0
$595$	1.47820e23	0.385778
$596$	0	0
$597$	−3.95484e23	−1.00311
$598$	0	0
$599$	−1.55455e23	−0.383246	−0.191623	−	0.981469i	$-0.561375\pi$
$599$	−1.55455e23	−0.383246	−0.191623	+	0.981469i	$0.561375\pi$
$600$	0	0
$601$	−7.11005e22	−0.170388	−0.0851941	−	0.996364i	$-0.527151\pi$
$601$	−7.11005e22	−0.170388	−0.0851941	+	0.996364i	$0.527151\pi$
$602$	0	0
$603$	1.77999e23	0.414688
$604$	0	0
$605$	4.71041e23	1.06693
$606$	0	0
$607$	2.07763e23	0.457576	0.228788	−	0.973476i	$-0.426524\pi$
$607$	2.07763e23	0.457576	0.228788	+	0.973476i	$0.426524\pi$
$608$	0	0
$609$	−5.04970e23	−1.08148
$610$	0	0
$611$	−7.16749e23	−1.49285
$612$	0	0
$613$	−5.47051e23	−1.10819	−0.554095	−	0.832454i	$-0.686936\pi$
$613$	−5.47051e23	−1.10819	−0.554095	+	0.832454i	$0.686936\pi$
$614$	0	0
$615$	2.14618e23	0.422890
$616$	0	0
$617$	−5.79656e23	−1.11108	−0.555542	−	0.831489i	$-0.687489\pi$
$617$	−5.79656e23	−1.11108	−0.555542	+	0.831489i	$0.687489\pi$
$618$	0	0
$619$	−5.83066e23	−1.08729	−0.543647	−	0.839314i	$-0.682957\pi$
$619$	−5.83066e23	−1.08729	−0.543647	+	0.839314i	$0.682957\pi$
$620$	0	0
$621$	−4.22775e22	−0.0767062
$622$	0	0
$623$	−7.14345e23	−1.26113
$624$	0	0
$625$	2.92340e23	0.502236
$626$	0	0
$627$	−5.42109e22	−0.0906385
$628$	0	0
$629$	2.18701e22	0.0355894
$630$	0	0
$631$	−9.51360e23	−1.50694	−0.753469	−	0.657483i	$-0.771621\pi$
$631$	−9.51360e23	−1.50694	−0.753469	+	0.657483i	$0.771621\pi$
$632$	0	0
$633$	1.07475e24	1.65721
$634$	0	0
$635$	2.90530e23	0.436128
$636$	0	0
$637$	1.21373e24	1.77393
$638$	0	0
$639$	5.20360e23	0.740537
$640$	0	0
$641$	−9.77138e23	−1.35414	−0.677069	−	0.735920i	$-0.736750\pi$
$641$	−9.77138e23	−1.35414	−0.677069	+	0.735920i	$0.736750\pi$
$642$	0	0
$643$	5.56439e23	0.750973	0.375486	−	0.926828i	$-0.377476\pi$
$643$	5.56439e23	0.750973	0.375486	+	0.926828i	$0.377476\pi$
$644$	0	0
$645$	−1.50194e23	−0.197422
$646$	0	0
$647$	−1.15984e24	−1.48495	−0.742477	−	0.669871i	$-0.766349\pi$
$647$	−1.15984e24	−1.48495	−0.742477	+	0.669871i	$0.766349\pi$
$648$	0	0
$649$	2.12439e24	2.64944
$650$	0	0
$651$	−8.36278e23	−1.01604
$652$	0	0
$653$	9.87432e23	1.16881	0.584407	−	0.811461i	$-0.301327\pi$
$653$	9.87432e23	1.16881	0.584407	+	0.811461i	$0.301327\pi$
$654$	0	0
$655$	−2.62977e23	−0.303297
$656$	0	0
$657$	−7.20306e23	−0.809490
$658$	0	0
$659$	−4.97449e23	−0.544782	−0.272391	−	0.962187i	$-0.587814\pi$
$659$	−4.97449e23	−0.544782	−0.272391	+	0.962187i	$0.587814\pi$
$660$	0	0
$661$	1.16033e24	1.23843	0.619214	−	0.785222i	$-0.287451\pi$
$661$	1.16033e24	1.23843	0.619214	+	0.785222i	$0.287451\pi$
$662$	0	0
$663$	−9.79167e23	−1.01857
$664$	0	0
$665$	2.25323e22	0.0228466
$666$	0	0
$667$	2.67212e23	0.264111
$668$	0	0
$669$	6.88886e22	0.0663781
$670$	0	0
$671$	2.89730e24	2.72178
$672$	0	0
$673$	−1.13175e24	−1.03663	−0.518315	−	0.855190i	$-0.673441\pi$
$673$	−1.13175e24	−1.03663	−0.518315	+	0.855190i	$0.673441\pi$
$674$	0	0
$675$	1.35434e23	0.120961
$676$	0	0
$677$	−1.09418e24	−0.952983	−0.476492	−	0.879179i	$-0.658092\pi$
$677$	−1.09418e24	−0.952983	−0.476492	+	0.879179i	$0.658092\pi$
$678$	0	0
$679$	−8.02171e23	−0.681356
$680$	0	0
$681$	5.12736e23	0.424760
$682$	0	0
$683$	1.05400e22	0.00851660	0.00425830	−	0.999991i	$-0.498645\pi$
$683$	1.05400e22	0.00851660	0.00425830	+	0.999991i	$0.498645\pi$
$684$	0	0
$685$	−7.14548e23	−0.563198
$686$	0	0
$687$	−1.49249e24	−1.14757
$688$	0	0
$689$	1.73037e24	1.29800
$690$	0	0
$691$	−1.61636e24	−1.18297	−0.591485	−	0.806316i	$-0.701458\pi$
$691$	−1.61636e24	−1.18297	−0.591485	+	0.806316i	$0.701458\pi$
$692$	0	0
$693$	−3.65916e24	−2.61306
$694$	0	0
$695$	4.51339e23	0.314508
$696$	0	0
$697$	−6.36277e23	−0.432681
$698$	0	0
$699$	−3.24665e24	−2.15466
$700$	0	0
$701$	−1.86841e23	−0.121024	−0.0605118	−	0.998167i	$-0.519273\pi$
$701$	−1.86841e23	−0.121024	−0.0605118	+	0.998167i	$0.519273\pi$
$702$	0	0
$703$	3.33368e21	0.00210767
$704$	0	0
$705$	−1.11582e24	−0.688634
$706$	0	0
$707$	1.30469e24	0.786033
$708$	0	0
$709$	2.18773e23	0.128677	0.0643385	−	0.997928i	$-0.479506\pi$
$709$	2.18773e23	0.128677	0.0643385	+	0.997928i	$0.479506\pi$
$710$	0	0
$711$	9.17786e22	0.0527048
$712$	0	0
$713$	4.42529e23	0.248131
$714$	0	0
$715$	1.80723e24	0.989491
$716$	0	0
$717$	4.32230e24	2.31101
$718$	0	0
$719$	1.15548e24	0.603345	0.301673	−	0.953412i	$-0.402455\pi$
$719$	1.15548e24	0.603345	0.301673	+	0.953412i	$0.402455\pi$
$720$	0	0
$721$	−7.10550e23	−0.362364
$722$	0	0
$723$	3.68086e24	1.83347
$724$	0	0
$725$	−8.56003e23	−0.416488
$726$	0	0
$727$	3.57207e24	1.69777	0.848883	−	0.528581i	$-0.177276\pi$
$727$	3.57207e24	1.69777	0.848883	+	0.528581i	$0.177276\pi$
$728$	0	0
$729$	−1.67208e24	−0.776379
$730$	0	0
$731$	4.45280e23	0.201993
$732$	0	0
$733$	4.90391e23	0.217350	0.108675	−	0.994077i	$-0.465339\pi$
$733$	4.90391e23	0.217350	0.108675	+	0.994077i	$0.465339\pi$
$734$	0	0
$735$	1.88952e24	0.818293
$736$	0	0
$737$	2.06435e24	0.873594
$738$	0	0
$739$	2.17372e24	0.898928	0.449464	−	0.893298i	$-0.351615\pi$
$739$	2.17372e24	0.898928	0.449464	+	0.893298i	$0.351615\pi$
$740$	0	0
$741$	−1.49255e23	−0.0603219
$742$	0	0
$743$	−3.08963e24	−1.22040	−0.610199	−	0.792248i	$-0.708911\pi$
$743$	−3.08963e24	−1.22040	−0.610199	+	0.792248i	$0.708911\pi$
$744$	0	0
$745$	−8.56839e23	−0.330804
$746$	0	0
$747$	3.03044e24	1.14362
$748$	0	0
$749$	−4.35014e24	−1.60475
$750$	0	0
$751$	−4.58172e24	−1.65230	−0.826150	−	0.563451i	$-0.809474\pi$
$751$	−4.58172e24	−1.65230	−0.826150	+	0.563451i	$0.809474\pi$
$752$	0	0
$753$	−6.12312e24	−2.15881
$754$	0	0
$755$	−2.04500e23	−0.0704927
$756$	0	0
$757$	−3.43112e23	−0.115644	−0.0578218	−	0.998327i	$-0.518416\pi$
$757$	−3.43112e23	−0.115644	−0.0578218	+	0.998327i	$0.518416\pi$
$758$	0	0
$759$	4.10394e24	1.35253
$760$	0	0
$761$	3.01271e24	0.970930	0.485465	−	0.874256i	$-0.338650\pi$
$761$	3.01271e24	0.970930	0.485465	+	0.874256i	$0.338650\pi$
$762$	0	0
$763$	−6.29219e24	−1.98309
$764$	0	0
$765$	−7.19212e23	−0.221684
$766$	0	0
$767$	5.84893e24	1.76326
$768$	0	0
$769$	−1.62418e23	−0.0478918	−0.0239459	−	0.999713i	$-0.507623\pi$
$769$	−1.62418e23	−0.0478918	−0.0239459	+	0.999713i	$0.507623\pi$
$770$	0	0
$771$	6.48643e24	1.87087
$772$	0	0
$773$	4.43975e24	1.25266	0.626329	−	0.779559i	$-0.284557\pi$
$773$	4.43975e24	1.25266	0.626329	+	0.779559i	$0.284557\pi$
$774$	0	0
$775$	−1.41762e24	−0.391288
$776$	0	0
$777$	4.76921e23	0.128786
$778$	0	0
$779$	−9.69883e22	−0.0256242
$780$	0	0
$781$	6.03488e24	1.56004
$782$	0	0
$783$	−2.93537e23	−0.0742486
$784$	0	0
$785$	1.88846e24	0.467429
$786$	0	0
$787$	2.95546e24	0.715880	0.357940	−	0.933745i	$-0.383479\pi$
$787$	2.95546e24	0.715880	0.357940	+	0.933745i	$0.383479\pi$
$788$	0	0
$789$	−1.53613e24	−0.364145
$790$	0	0
$791$	2.49705e24	0.579331
$792$	0	0
$793$	7.97695e24	1.81140
$794$	0	0
$795$	2.69382e24	0.598752
$796$	0	0
$797$	6.01045e24	1.30771	0.653855	−	0.756620i	$-0.273151\pi$
$797$	6.01045e24	1.30771	0.653855	+	0.756620i	$0.273151\pi$
$798$	0	0
$799$	3.30808e24	0.704577
$800$	0	0
$801$	3.47563e24	0.724698
$802$	0	0
$803$	−8.35376e24	−1.70530
$804$	0	0
$805$	−1.70576e24	−0.340921
$806$	0	0
$807$	−8.17663e24	−1.60011
$808$	0	0
$809$	7.16704e24	1.37334	0.686668	−	0.726971i	$-0.259072\pi$
$809$	7.16704e24	1.37334	0.686668	+	0.726971i	$0.259072\pi$
$810$	0	0
$811$	5.73123e24	1.07540	0.537701	−	0.843136i	$-0.319293\pi$
$811$	5.73123e24	1.07540	0.537701	+	0.843136i	$0.319293\pi$
$812$	0	0
$813$	−6.99170e24	−1.28473
$814$	0	0
$815$	3.83171e24	0.689529
$816$	0	0
$817$	6.78745e22	0.0119624
$818$	0	0
$819$	−1.00745e25	−1.73905
$820$	0	0
$821$	−2.45653e24	−0.415341	−0.207670	−	0.978199i	$-0.566588\pi$
$821$	−2.45653e24	−0.415341	−0.207670	+	0.978199i	$0.566588\pi$
$822$	0	0
$823$	1.00636e25	1.66670	0.833348	−	0.552749i	$-0.186421\pi$
$823$	1.00636e25	1.66670	0.833348	+	0.552749i	$0.186421\pi$
$824$	0	0
$825$	−1.31468e25	−2.13286
$826$	0	0
$827$	2.31323e24	0.367639	0.183820	−	0.982960i	$-0.441154\pi$
$827$	2.31323e24	0.367639	0.183820	+	0.982960i	$0.441154\pi$
$828$	0	0
$829$	−5.49125e24	−0.854983	−0.427491	−	0.904019i	$-0.640603\pi$
$829$	−5.49125e24	−0.854983	−0.427491	+	0.904019i	$0.640603\pi$
$830$	0	0
$831$	2.13080e24	0.325038
$832$	0	0
$833$	−5.60185e24	−0.837237
$834$	0	0
$835$	−4.06016e24	−0.594577
$836$	0	0
$837$	−4.86126e23	−0.0697561
$838$	0	0
$839$	−2.39004e24	−0.336069	−0.168035	−	0.985781i	$-0.553742\pi$
$839$	−2.39004e24	−0.336069	−0.168035	+	0.985781i	$0.553742\pi$
$840$	0	0
$841$	−5.40186e24	−0.744351
$842$	0	0
$843$	1.97628e24	0.266879
$844$	0	0
$845$	1.80336e24	0.238671
$846$	0	0
$847$	−3.04533e25	−3.95026
$848$	0	0
$849$	−1.53497e24	−0.195158
$850$	0	0
$851$	−2.52370e23	−0.0314511
$852$	0	0
$853$	7.04660e24	0.860821	0.430410	−	0.902633i	$-0.358369\pi$
$853$	7.04660e24	0.860821	0.430410	+	0.902633i	$0.358369\pi$
$854$	0	0
$855$	−1.09630e23	−0.0131286
$856$	0	0
$857$	−6.71395e24	−0.788208	−0.394104	−	0.919066i	$-0.628945\pi$
$857$	−6.71395e24	−0.788208	−0.394104	+	0.919066i	$0.628945\pi$
$858$	0	0
$859$	8.58583e24	0.988189	0.494095	−	0.869408i	$-0.335500\pi$
$859$	8.58583e24	0.988189	0.494095	+	0.869408i	$0.335500\pi$
$860$	0	0
$861$	−1.38753e25	−1.56572
$862$	0	0
$863$	5.52514e24	0.611296	0.305648	−	0.952145i	$-0.401127\pi$
$863$	5.52514e24	0.611296	0.305648	+	0.952145i	$0.401127\pi$
$864$	0	0
$865$	5.07214e23	0.0550242
$866$	0	0
$867$	−8.41585e24	−0.895232
$868$	0	0
$869$	1.06440e24	0.111030
$870$	0	0
$871$	5.68364e24	0.581396
$872$	0	0
$873$	3.90294e24	0.391536
$874$	0	0
$875$	1.20981e25	1.19028
$876$	0	0
$877$	6.54824e24	0.631871	0.315935	−	0.948781i	$-0.397682\pi$
$877$	6.54824e24	0.631871	0.315935	+	0.948781i	$0.397682\pi$
$878$	0	0
$879$	−1.58867e25	−1.50358
$880$	0	0
$881$	5.78483e24	0.537026	0.268513	−	0.963276i	$-0.413468\pi$
$881$	5.78483e24	0.537026	0.268513	+	0.963276i	$0.413468\pi$
$882$	0	0
$883$	−1.75999e25	−1.60267	−0.801337	−	0.598213i	$-0.795878\pi$
$883$	−1.75999e25	−1.60267	−0.801337	+	0.598213i	$0.795878\pi$
$884$	0	0
$885$	9.10552e24	0.813368
$886$	0	0
$887$	−1.06620e25	−0.934300	−0.467150	−	0.884178i	$-0.654719\pi$
$887$	−1.06620e25	−0.934300	−0.467150	+	0.884178i	$0.654719\pi$
$888$	0	0
$889$	−1.87831e25	−1.61474
$890$	0	0
$891$	−2.44389e25	−2.06120
$892$	0	0
$893$	5.04254e23	0.0417265
$894$	0	0
$895$	5.50862e24	0.447246
$896$	0	0
$897$	1.12991e25	0.900136
$898$	0	0
$899$	3.07253e24	0.240181
$900$	0	0
$901$	−7.98634e24	−0.612615
$902$	0	0
$903$	9.71023e24	0.730943
$904$	0	0
$905$	1.32310e23	0.00977417
$906$	0	0
$907$	−1.63230e25	−1.18342	−0.591709	−	0.806152i	$-0.701547\pi$
$907$	−1.63230e25	−1.18342	−0.591709	+	0.806152i	$0.701547\pi$
$908$	0	0
$909$	−6.34792e24	−0.451688
$910$	0	0
$911$	−1.45971e25	−1.01944	−0.509719	−	0.860341i	$-0.670251\pi$
$911$	−1.45971e25	−1.01944	−0.509719	+	0.860341i	$0.670251\pi$
$912$	0	0
$913$	3.51456e25	2.40918
$914$	0	0
$915$	1.24184e25	0.835577
$916$	0	0
$917$	1.70018e25	1.12294
$918$	0	0
$919$	−1.83968e24	−0.119278	−0.0596390	−	0.998220i	$-0.518995\pi$
$919$	−1.83968e24	−0.119278	−0.0596390	+	0.998220i	$0.518995\pi$
$920$	0	0
$921$	1.64019e25	1.04397
$922$	0	0
$923$	1.66154e25	1.03824
$924$	0	0
$925$	8.08457e23	0.0495966
$926$	0	0
$927$	3.45716e24	0.208230
$928$	0	0
$929$	−7.16411e24	−0.423671	−0.211835	−	0.977305i	$-0.567944\pi$
$929$	−7.16411e24	−0.423671	−0.211835	+	0.977305i	$0.567944\pi$
$930$	0	0
$931$	−8.53894e23	−0.0495829
$932$	0	0
$933$	2.71875e25	1.55016
$934$	0	0
$935$	−8.34108e24	−0.467007
$936$	0	0
$937$	−1.68079e25	−0.924119	−0.462059	−	0.886849i	$-0.652889\pi$
$937$	−1.68079e25	−0.924119	−0.462059	+	0.886849i	$0.652889\pi$
$938$	0	0
$939$	−4.00920e25	−2.16471
$940$	0	0
$941$	−2.99139e25	−1.58621	−0.793105	−	0.609085i	$-0.791537\pi$
$941$	−2.99139e25	−1.58621	−0.793105	+	0.609085i	$0.791537\pi$
$942$	0	0
$943$	7.34232e24	0.382370
$944$	0	0
$945$	1.87381e24	0.0958420
$946$	0	0
$947$	1.00925e25	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$947$	1.00925e25	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$948$	0	0
$949$	−2.29998e25	−1.13491
$950$	0	0
$951$	4.79245e24	0.232286
$952$	0	0
$953$	9.95941e24	0.474181	0.237090	−	0.971488i	$-0.423806\pi$
$953$	9.95941e24	0.474181	0.237090	+	0.971488i	$0.423806\pi$
$954$	0	0
$955$	−3.29288e24	−0.154009
$956$	0	0
$957$	2.84941e25	1.30919
$958$	0	0
$959$	4.61963e25	2.08521
$960$	0	0
$961$	−1.74617e25	−0.774351
$962$	0	0
$963$	2.11655e25	0.922158
$964$	0	0
$965$	−1.05539e25	−0.451785
$966$	0	0
$967$	4.62055e24	0.194343	0.0971715	−	0.995268i	$-0.469020\pi$
$967$	4.62055e24	0.194343	0.0971715	+	0.995268i	$0.469020\pi$
$968$	0	0
$969$	6.88872e23	0.0284699
$970$	0	0
$971$	−1.50646e24	−0.0611780	−0.0305890	−	0.999532i	$-0.509738\pi$
$971$	−1.50646e24	−0.0611780	−0.0305890	+	0.999532i	$0.509738\pi$
$972$	0	0
$973$	−2.91796e25	−1.16445
$974$	0	0
$975$	−3.61962e25	−1.41946
$976$	0	0
$977$	−2.16292e25	−0.833560	−0.416780	−	0.909007i	$-0.636841\pi$
$977$	−2.16292e25	−0.833560	−0.416780	+	0.909007i	$0.636841\pi$
$978$	0	0
$979$	4.03087e25	1.52667
$980$	0	0
$981$	3.06145e25	1.13957
$982$	0	0
$983$	−1.09543e25	−0.400754	−0.200377	−	0.979719i	$-0.564217\pi$
$983$	−1.09543e25	−0.400754	−0.200377	+	0.979719i	$0.564217\pi$
$984$	0	0
$985$	−1.94290e25	−0.698622
$986$	0	0
$987$	7.21393e25	2.54962
$988$	0	0
$989$	−5.13831e24	−0.178506
$990$	0	0
$991$	2.10377e25	0.718410	0.359205	−	0.933259i	$-0.383048\pi$
$991$	2.10377e25	0.718410	0.359205	+	0.933259i	$0.383048\pi$
$992$	0	0
$993$	7.29913e25	2.45020
$994$	0	0
$995$	−9.27549e24	−0.306084
$996$	0	0
$997$	2.15374e25	0.698690	0.349345	−	0.936994i	$-0.386404\pi$
$997$	2.15374e25	0.698690	0.349345	+	0.936994i	$0.386404\pi$
$998$	0	0
$999$	2.77233e23	0.00884175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.18.a.g.1.2		2
4.3	odd	2		64.18.a.l.1.1		2
8.3	odd	2		4.18.a.a.1.2	✓	2
8.5	even	2		16.18.a.e.1.1		2
24.11	even	2		36.18.a.d.1.2		2
40.3	even	4		100.18.c.a.49.3		4
40.19	odd	2		100.18.a.b.1.1		2
40.27	even	4		100.18.c.a.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.18.a.a.1.2	✓	2	8.3	odd	2
16.18.a.e.1.1		2	8.5	even	2
36.18.a.d.1.2		2	24.11	even	2
64.18.a.g.1.2		2	1.1	even	1	trivial
64.18.a.l.1.1		2	4.3	odd	2
100.18.a.b.1.1		2	40.19	odd	2
100.18.c.a.49.2		4	40.27	even	4
100.18.c.a.49.3		4	40.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+15636.4 q^{3} +366730. q^{5} -2.37095e7 q^{7} +1.15358e8 q^{9} +O(q^{10})\)	\(q+15636.4 q^{3} +366730. q^{5} -2.37095e7 q^{7} +1.15358e8 q^{9} +1.33786e9 q^{11} +3.68345e9 q^{13} +5.73434e9 q^{15} -1.70006e10 q^{17} -2.59141e9 q^{19} -3.70732e11 q^{21} +1.96178e11 q^{23} -6.28449e11 q^{25} -2.15505e11 q^{27} +1.36209e12 q^{29} +2.25575e12 q^{31} +2.09194e13 q^{33} -8.69497e12 q^{35} -1.28643e12 q^{37} +5.75961e13 q^{39} +3.74268e13 q^{41} -2.61921e13 q^{43} +4.23052e13 q^{45} -1.94586e14 q^{47} +3.29509e14 q^{49} -2.65829e14 q^{51} +4.69769e14 q^{53} +4.90635e14 q^{55} -4.05205e13 q^{57} +1.58789e15 q^{59} +2.16562e15 q^{61} -2.73508e15 q^{63} +1.35083e15 q^{65} +1.54302e15 q^{67} +3.06753e15 q^{69} +4.51083e15 q^{71} -6.24410e15 q^{73} -9.82670e15 q^{75} -3.17201e16 q^{77} +7.95599e14 q^{79} -1.82671e16 q^{81} +2.62699e16 q^{83} -6.23462e15 q^{85} +2.12982e16 q^{87} +3.01291e16 q^{89} -8.73327e16 q^{91} +3.52719e16 q^{93} -9.50348e14 q^{95} +3.38333e16 q^{97} +1.54333e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5880 q^{3} - 604044 q^{5} - 25350160 q^{7} + 449174682 q^{9}+O(q^{10}) \) 2 * q - 5880 * q^3 - 604044 * q^5 - 25350160 * q^7 + 449174682 * q^9	\( 2 q - 5880 q^{3} - 604044 q^{5} - 25350160 q^{7} + 449174682 q^{9} + 1259648280 q^{11} + 1320052580 q^{13} + 26621930448 q^{15} - 27498226140 q^{17} + 101133633832 q^{19} - 335430207552 q^{21} - 134767491120 q^{23} - 448986850114 q^{25} - 4619416117680 q^{27} - 2337155582652 q^{29} - 278836113472 q^{31} + 22602380058720 q^{33} - 7102242382752 q^{35} - 20929802888140 q^{37} + 108447997173648 q^{39} + 4166592315732 q^{41} - 111143148534440 q^{43} - 281755357404444 q^{45} - 196772651157600 q^{47} + 99570326741874 q^{49} - 39956666918256 q^{51} + 487965122736660 q^{53} + 566565363246960 q^{55} - 22\!\cdots\!40 q^{57}+ \cdots + 12\!\cdots\!80 q^{99}+O(q^{100}) \) 2 * q - 5880 * q^3 - 604044 * q^5 - 25350160 * q^7 + 449174682 * q^9 + 1259648280 * q^11 + 1320052580 * q^13 + 26621930448 * q^15 - 27498226140 * q^17 + 101133633832 * q^19 - 335430207552 * q^21 - 134767491120 * q^23 - 448986850114 * q^25 - 4619416117680 * q^27 - 2337155582652 * q^29 - 278836113472 * q^31 + 22602380058720 * q^33 - 7102242382752 * q^35 - 20929802888140 * q^37 + 108447997173648 * q^39 + 4166592315732 * q^41 - 111143148534440 * q^43 - 281755357404444 * q^45 - 196772651157600 * q^47 + 99570326741874 * q^49 - 39956666918256 * q^51 + 487965122736660 * q^53 + 566565363246960 * q^55 - 2272313631999840 * q^57 + 2835904197813624 * q^59 + 67544034994436 * q^61 - 3282762121966800 * q^63 + 3645157343001768 * q^65 - 995171321546360 * q^67 + 10188302100105024 * q^69 - 3882245493215376 * q^71 - 12746425881769580 * q^73 - 13688080703624712 * q^75 - 31591755846002880 * q^77 + 14984271534065504 * q^79 + 33380230287529458 * q^81 + 43899417809893800 * q^83 + 3956216991540264 * q^85 + 100892756486768400 * q^87 + 51909007958846388 * q^89 - 83455169400462112 * q^91 + 89807116632334080 * q^93 - 101643889304423664 * q^95 - 49281007789848380 * q^97 + 128223271309133880 * q^99

Introduction

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