LMFDB - Embedded newform 20.10.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [20,10,Mod(1,20)] 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("20.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	20.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$10.3007167233$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{79}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.88819$ of defining polynomial
Character	$\chi$	$=$	20.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+12.2111 q^{3} -625.000 q^{5} +9622.57 q^{7} -19533.9 q^{9} +55626.3 q^{11} +169777. q^{13} -7631.94 q^{15} +207499. q^{17} +802445. q^{19} +117502. q^{21} -1.24189e6 q^{23} +390625. q^{25} -478882. q^{27} -4.28308e6 q^{29} -3.58713e6 q^{31} +679259. q^{33} -6.01410e6 q^{35} -2.89856e6 q^{37} +2.07317e6 q^{39} +2.51515e7 q^{41} -2.00204e7 q^{43} +1.22087e7 q^{45} +3.73010e7 q^{47} +5.22402e7 q^{49} +2.53380e6 q^{51} -2.55155e7 q^{53} -3.47665e7 q^{55} +9.79875e6 q^{57} -9.96495e7 q^{59} +2.00434e8 q^{61} -1.87966e8 q^{63} -1.06111e8 q^{65} -8.09951e7 q^{67} -1.51648e7 q^{69} -4.31522e7 q^{71} -3.40820e8 q^{73} +4.76997e6 q^{75} +5.35268e8 q^{77} +2.81089e8 q^{79} +3.78638e8 q^{81} -6.01017e8 q^{83} -1.29687e8 q^{85} -5.23011e7 q^{87} +5.39917e8 q^{89} +1.63369e9 q^{91} -4.38028e7 q^{93} -5.01528e8 q^{95} +4.23026e8 q^{97} -1.08660e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 260 q^{3} - 1250 q^{5} - 380 q^{7} + 34882 q^{9} + 102720 q^{11} + 179140 q^{13} + 162500 q^{15} + 316020 q^{17} + 137272 q^{19} + 2840312 q^{21} - 665460 q^{23} + 781250 q^{25} - 9933560 q^{27} - 6893748 q^{29}+ \cdots + 1476045120 q^{99}+O(q^{100}) $ 2 * q - 260 * q^3 - 1250 * q^5 - 380 * q^7 + 34882 * q^9 + 102720 * q^11 + 179140 * q^13 + 162500 * q^15 + 316020 * q^17 + 137272 * q^19 + 2840312 * q^21 - 665460 * q^23 + 781250 * q^25 - 9933560 * q^27 - 6893748 * q^29 + 291832 * q^31 - 12140160 * q^33 + 237500 * q^35 + 11261380 * q^37 - 475528 * q^39 + 29773452 * q^41 - 11708180 * q^43 - 21801250 * q^45 + 62493300 * q^47 + 111937914 * q^49 - 27006696 * q^51 + 9417780 * q^53 - 64200000 * q^55 + 190866320 * q^57 - 92930856 * q^59 + 195673924 * q^61 - 732264700 * q^63 - 111962500 * q^65 - 219767420 * q^67 - 172074264 * q^69 + 311207016 * q^71 - 99224060 * q^73 - 101562500 * q^75 + 64210560 * q^77 + 542261776 * q^79 + 1881238378 * q^81 - 1256915700 * q^83 - 197512500 * q^85 + 658353000 * q^87 - 462291852 * q^89 + 1540037768 * q^91 - 1099698160 * q^93 - 85795000 * q^95 + 1671716740 * q^97 + 1476045120 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	12.2111	0.0870381	0.0435191	−	0.999053i	$-0.486143\pi$
$3$	12.2111	0.0870381	0.0435191	+	0.999053i	$0.486143\pi$
$4$	0	0
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	9622.57	1.51478	0.757390	−	0.652962i	$-0.226474\pi$
$7$	9622.57	1.51478	0.757390	+	0.652962i	$0.226474\pi$
$8$	0	0
$9$	−19533.9	−0.992424
$10$	0	0
$11$	55626.3	1.14555	0.572774	−	0.819713i	$-0.305867\pi$
$11$	55626.3	1.14555	0.572774	+	0.819713i	$0.305867\pi$
$12$	0	0
$13$	169777.	1.64867	0.824335	−	0.566102i	$-0.191549\pi$
$13$	169777.	1.64867	0.824335	+	0.566102i	$0.191549\pi$
$14$	0	0
$15$	−7631.94	−0.0389246
$16$	0	0
$17$	207499.	0.602555	0.301278	−	0.953536i	$-0.402587\pi$
$17$	207499.	0.602555	0.301278	+	0.953536i	$0.402587\pi$
$18$	0	0
$19$	802445.	1.41262	0.706308	−	0.707904i	$-0.250359\pi$
$19$	802445.	1.41262	0.706308	+	0.707904i	$0.250359\pi$
$20$	0	0
$21$	117502.	0.131844
$22$	0	0
$23$	−1.24189e6	−0.925351	−0.462675	−	0.886528i	$-0.653110\pi$
$23$	−1.24189e6	−0.925351	−0.462675	+	0.886528i	$0.653110\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	−478882.	−0.173417
$28$	0	0
$29$	−4.28308e6	−1.12451	−0.562257	−	0.826963i	$-0.690067\pi$
$29$	−4.28308e6	−1.12451	−0.562257	+	0.826963i	$0.690067\pi$
$30$	0	0
$31$	−3.58713e6	−0.697620	−0.348810	−	0.937193i	$-0.613414\pi$
$31$	−3.58713e6	−0.697620	−0.348810	+	0.937193i	$0.613414\pi$
$32$	0	0
$33$	679259.	0.0997064
$34$	0	0
$35$	−6.01410e6	−0.677430
$36$	0	0
$37$	−2.89856e6	−0.254258	−0.127129	−	0.991886i	$-0.540576\pi$
$37$	−2.89856e6	−0.254258	−0.127129	+	0.991886i	$0.540576\pi$
$38$	0	0
$39$	2.07317e6	0.143497
$40$	0	0
$41$	2.51515e7	1.39007	0.695035	−	0.718975i	$-0.255389\pi$
$41$	2.51515e7	1.39007	0.695035	+	0.718975i	$0.255389\pi$
$42$	0	0
$43$	−2.00204e7	−0.893029	−0.446515	−	0.894776i	$-0.647335\pi$
$43$	−2.00204e7	−0.893029	−0.446515	+	0.894776i	$0.647335\pi$
$44$	0	0
$45$	1.22087e7	0.443826
$46$	0	0
$47$	3.73010e7	1.11501	0.557507	−	0.830172i	$-0.311758\pi$
$47$	3.73010e7	1.11501	0.557507	+	0.830172i	$0.311758\pi$
$48$	0	0
$49$	5.22402e7	1.29456
$50$	0	0
$51$	2.53380e6	0.0524453
$52$	0	0
$53$	−2.55155e7	−0.444184	−0.222092	−	0.975026i	$-0.571289\pi$
$53$	−2.55155e7	−0.444184	−0.222092	+	0.975026i	$0.571289\pi$
$54$	0	0
$55$	−3.47665e7	−0.512305
$56$	0	0
$57$	9.79875e6	0.122951
$58$	0	0
$59$	−9.96495e7	−1.07063	−0.535317	−	0.844651i	$-0.679808\pi$
$59$	−9.96495e7	−1.07063	−0.535317	+	0.844651i	$0.679808\pi$
$60$	0	0
$61$	2.00434e8	1.85347	0.926737	−	0.375710i	$-0.122601\pi$
$61$	2.00434e8	1.85347	0.926737	+	0.375710i	$0.122601\pi$
$62$	0	0
$63$	−1.87966e8	−1.50331
$64$	0	0
$65$	−1.06111e8	−0.737308
$66$	0	0
$67$	−8.09951e7	−0.491046	−0.245523	−	0.969391i	$-0.578960\pi$
$67$	−8.09951e7	−0.491046	−0.245523	+	0.969391i	$0.578960\pi$
$68$	0	0
$69$	−1.51648e7	−0.0805408
$70$	0	0
$71$	−4.31522e7	−0.201530	−0.100765	−	0.994910i	$-0.532129\pi$
$71$	−4.31522e7	−0.201530	−0.100765	+	0.994910i	$0.532129\pi$
$72$	0	0
$73$	−3.40820e8	−1.40466	−0.702332	−	0.711850i	$-0.747858\pi$
$73$	−3.40820e8	−1.40466	−0.702332	+	0.711850i	$0.747858\pi$
$74$	0	0
$75$	4.76997e6	0.0174076
$76$	0	0
$77$	5.35268e8	1.73525
$78$	0	0
$79$	2.81089e8	0.811935	0.405967	−	0.913888i	$-0.366935\pi$
$79$	2.81089e8	0.811935	0.405967	+	0.913888i	$0.366935\pi$
$80$	0	0
$81$	3.78638e8	0.977330
$82$	0	0
$83$	−6.01017e8	−1.39007	−0.695033	−	0.718978i	$-0.744610\pi$
$83$	−6.01017e8	−1.39007	−0.695033	+	0.718978i	$0.744610\pi$
$84$	0	0
$85$	−1.29687e8	−0.269471
$86$	0	0
$87$	−5.23011e7	−0.0978756
$88$	0	0
$89$	5.39917e8	0.912162	0.456081	−	0.889938i	$-0.349253\pi$
$89$	5.39917e8	0.912162	0.456081	+	0.889938i	$0.349253\pi$
$90$	0	0
$91$	1.63369e9	2.49737
$92$	0	0
$93$	−4.38028e7	−0.0607195
$94$	0	0
$95$	−5.01528e8	−0.631741
$96$	0	0
$97$	4.23026e8	0.485170	0.242585	−	0.970130i	$-0.422005\pi$
$97$	4.23026e8	0.485170	0.242585	+	0.970130i	$0.422005\pi$
$98$	0	0
$99$	−1.08660e9	−1.13687
$100$	0	0
$101$	−1.09286e9	−1.04501	−0.522503	−	0.852637i	$-0.675002\pi$
$101$	−1.09286e9	−1.04501	−0.522503	+	0.852637i	$0.675002\pi$
$102$	0	0
$103$	−8.50494e7	−0.0744567	−0.0372284	−	0.999307i	$-0.511853\pi$
$103$	−8.50494e7	−0.0744567	−0.0372284	+	0.999307i	$0.511853\pi$
$104$	0	0
$105$	−7.34389e7	−0.0589623
$106$	0	0
$107$	5.23881e8	0.386372	0.193186	−	0.981162i	$-0.438118\pi$
$107$	5.23881e8	0.386372	0.193186	+	0.981162i	$0.438118\pi$
$108$	0	0
$109$	−5.59417e8	−0.379591	−0.189796	−	0.981824i	$-0.560783\pi$
$109$	−5.59417e8	−0.379591	−0.189796	+	0.981824i	$0.560783\pi$
$110$	0	0
$111$	−3.53947e7	−0.0221302
$112$	0	0
$113$	−2.63801e9	−1.52203	−0.761016	−	0.648733i	$-0.775299\pi$
$113$	−2.63801e9	−1.52203	−0.761016	+	0.648733i	$0.775299\pi$
$114$	0	0
$115$	7.76179e8	0.413829
$116$	0	0
$117$	−3.31641e9	−1.63618
$118$	0	0
$119$	1.99668e9	0.912739
$120$	0	0
$121$	7.36341e8	0.312281
$122$	0	0
$123$	3.07128e8	0.120989
$124$	0	0
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	3.33268e9	1.13678	0.568391	−	0.822758i	$-0.307566\pi$
$127$	3.33268e9	1.13678	0.568391	+	0.822758i	$0.307566\pi$
$128$	0	0
$129$	−2.44472e8	−0.0777276
$130$	0	0
$131$	−3.78884e9	−1.12405	−0.562024	−	0.827121i	$-0.689977\pi$
$131$	−3.78884e9	−1.12405	−0.562024	+	0.827121i	$0.689977\pi$
$132$	0	0
$133$	7.72158e9	2.13980
$134$	0	0
$135$	2.99301e8	0.0775544
$136$	0	0
$137$	−2.15185e9	−0.521879	−0.260940	−	0.965355i	$-0.584032\pi$
$137$	−2.15185e9	−0.521879	−0.260940	+	0.965355i	$0.584032\pi$
$138$	0	0
$139$	2.75086e8	0.0625032	0.0312516	−	0.999512i	$-0.490051\pi$
$139$	2.75086e8	0.0625032	0.0312516	+	0.999512i	$0.490051\pi$
$140$	0	0
$141$	4.55487e8	0.0970487
$142$	0	0
$143$	9.44408e9	1.88863
$144$	0	0
$145$	2.67692e9	0.502898
$146$	0	0
$147$	6.37911e8	0.112676
$148$	0	0
$149$	−6.16582e9	−1.02483	−0.512416	−	0.858737i	$-0.671249\pi$
$149$	−6.16582e9	−1.02483	−0.512416	+	0.858737i	$0.671249\pi$
$150$	0	0
$151$	−3.28376e9	−0.514014	−0.257007	−	0.966410i	$-0.582736\pi$
$151$	−3.28376e9	−0.514014	−0.257007	+	0.966410i	$0.582736\pi$
$152$	0	0
$153$	−4.05327e9	−0.597990
$154$	0	0
$155$	2.24195e9	0.311985
$156$	0	0
$157$	−1.89255e9	−0.248599	−0.124299	−	0.992245i	$-0.539668\pi$
$157$	−1.89255e9	−0.248599	−0.124299	+	0.992245i	$0.539668\pi$
$158$	0	0
$159$	−3.11573e8	−0.0386610
$160$	0	0
$161$	−1.19501e10	−1.40170
$162$	0	0
$163$	4.14293e8	0.0459688	0.0229844	−	0.999736i	$-0.492683\pi$
$163$	4.14293e8	0.0459688	0.0229844	+	0.999736i	$0.492683\pi$
$164$	0	0
$165$	−4.24537e8	−0.0445900
$166$	0	0
$167$	−1.56225e10	−1.55427	−0.777136	−	0.629332i	$-0.783328\pi$
$167$	−1.56225e10	−1.55427	−0.777136	+	0.629332i	$0.783328\pi$
$168$	0	0
$169$	1.82198e10	1.71812
$170$	0	0
$171$	−1.56749e10	−1.40191
$172$	0	0
$173$	1.16058e10	0.985069	0.492535	−	0.870293i	$-0.336070\pi$
$173$	1.16058e10	0.985069	0.492535	+	0.870293i	$0.336070\pi$
$174$	0	0
$175$	3.75882e9	0.302956
$176$	0	0
$177$	−1.21683e9	−0.0931860
$178$	0	0
$179$	−2.45556e10	−1.78777	−0.893886	−	0.448294i	$-0.852032\pi$
$179$	−2.45556e10	−1.78777	−0.893886	+	0.448294i	$0.852032\pi$
$180$	0	0
$181$	3.21032e9	0.222328	0.111164	−	0.993802i	$-0.464542\pi$
$181$	3.21032e9	0.222328	0.111164	+	0.993802i	$0.464542\pi$
$182$	0	0
$183$	2.44752e9	0.161323
$184$	0	0
$185$	1.81160e9	0.113708
$186$	0	0
$187$	1.15424e10	0.690256
$188$	0	0
$189$	−4.60807e9	−0.262689
$190$	0	0
$191$	3.41548e10	1.85695	0.928477	−	0.371390i	$-0.121119\pi$
$191$	3.41548e10	1.85695	0.928477	+	0.371390i	$0.121119\pi$
$192$	0	0
$193$	2.50875e10	1.30152	0.650759	−	0.759284i	$-0.274451\pi$
$193$	2.50875e10	1.30152	0.650759	+	0.759284i	$0.274451\pi$
$194$	0	0
$195$	−1.29573e9	−0.0641739
$196$	0	0
$197$	9.64109e9	0.456067	0.228033	−	0.973653i	$-0.426771\pi$
$197$	9.64109e9	0.456067	0.228033	+	0.973653i	$0.426771\pi$
$198$	0	0
$199$	3.34318e9	0.151120	0.0755598	−	0.997141i	$-0.475926\pi$
$199$	3.34318e9	0.151120	0.0755598	+	0.997141i	$0.475926\pi$
$200$	0	0
$201$	−9.89040e8	−0.0427397
$202$	0	0
$203$	−4.12142e10	−1.70339
$204$	0	0
$205$	−1.57197e10	−0.621659
$206$	0	0
$207$	2.42589e10	0.918341
$208$	0	0
$209$	4.46371e10	1.61822
$210$	0	0
$211$	−1.76087e10	−0.611585	−0.305792	−	0.952098i	$-0.598921\pi$
$211$	−1.76087e10	−0.611585	−0.305792	+	0.952098i	$0.598921\pi$
$212$	0	0
$213$	−5.26936e8	−0.0175408
$214$	0	0
$215$	1.25128e10	0.399375
$216$	0	0
$217$	−3.45174e10	−1.05674
$218$	0	0
$219$	−4.16179e9	−0.122259
$220$	0	0
$221$	3.52287e10	0.993415
$222$	0	0
$223$	−5.00503e10	−1.35530	−0.677649	−	0.735386i	$-0.737001\pi$
$223$	−5.00503e10	−1.35530	−0.677649	+	0.735386i	$0.737001\pi$
$224$	0	0
$225$	−7.63043e9	−0.198485
$226$	0	0
$227$	−8.68117e9	−0.217001	−0.108501	−	0.994096i	$-0.534605\pi$
$227$	−8.68117e9	−0.217001	−0.108501	+	0.994096i	$0.534605\pi$
$228$	0	0
$229$	−2.96014e10	−0.711299	−0.355649	−	0.934619i	$-0.615740\pi$
$229$	−2.96014e10	−0.711299	−0.355649	+	0.934619i	$0.615740\pi$
$230$	0	0
$231$	6.53622e9	0.151033
$232$	0	0
$233$	−7.42617e10	−1.65068	−0.825341	−	0.564634i	$-0.809017\pi$
$233$	−7.42617e10	−1.65068	−0.825341	+	0.564634i	$0.809017\pi$
$234$	0	0
$235$	−2.33131e10	−0.498649
$236$	0	0
$237$	3.43240e9	0.0706693
$238$	0	0
$239$	−1.65577e10	−0.328253	−0.164126	−	0.986439i	$-0.552481\pi$
$239$	−1.65577e10	−0.328253	−0.164126	+	0.986439i	$0.552481\pi$
$240$	0	0
$241$	3.61072e10	0.689473	0.344737	−	0.938699i	$-0.387968\pi$
$241$	3.61072e10	0.689473	0.344737	+	0.938699i	$0.387968\pi$
$242$	0	0
$243$	1.40494e10	0.258482
$244$	0	0
$245$	−3.26501e10	−0.578945
$246$	0	0
$247$	1.36237e11	2.32894
$248$	0	0
$249$	−7.33909e9	−0.120989
$250$	0	0
$251$	2.32959e10	0.370465	0.185232	−	0.982695i	$-0.440696\pi$
$251$	2.32959e10	0.370465	0.185232	+	0.982695i	$0.440696\pi$
$252$	0	0
$253$	−6.90815e10	−1.06003
$254$	0	0
$255$	−1.58362e9	−0.0234542
$256$	0	0
$257$	−4.57624e10	−0.654350	−0.327175	−	0.944964i	$-0.606097\pi$
$257$	−4.57624e10	−0.654350	−0.327175	+	0.944964i	$0.606097\pi$
$258$	0	0
$259$	−2.78916e10	−0.385145
$260$	0	0
$261$	8.36651e10	1.11599
$262$	0	0
$263$	9.93911e10	1.28099	0.640496	−	0.767961i	$-0.278729\pi$
$263$	9.93911e10	1.28099	0.640496	+	0.767961i	$0.278729\pi$
$264$	0	0
$265$	1.59472e10	0.198645
$266$	0	0
$267$	6.59299e9	0.0793929
$268$	0	0
$269$	1.57475e11	1.83369	0.916845	−	0.399242i	$-0.130727\pi$
$269$	1.57475e11	1.83369	0.916845	+	0.399242i	$0.130727\pi$
$270$	0	0
$271$	2.79292e10	0.314555	0.157278	−	0.987554i	$-0.449728\pi$
$271$	2.79292e10	0.314555	0.157278	+	0.987554i	$0.449728\pi$
$272$	0	0
$273$	1.99492e10	0.217367
$274$	0	0
$275$	2.17290e10	0.229110
$276$	0	0
$277$	−1.68307e11	−1.71768	−0.858842	−	0.512240i	$-0.828816\pi$
$277$	−1.68307e11	−1.71768	−0.858842	+	0.512240i	$0.828816\pi$
$278$	0	0
$279$	7.00705e10	0.692335
$280$	0	0
$281$	−8.51877e10	−0.815076	−0.407538	−	0.913188i	$-0.633613\pi$
$281$	−8.51877e10	−0.815076	−0.407538	+	0.913188i	$0.633613\pi$
$282$	0	0
$283$	−2.02954e11	−1.88087	−0.940436	−	0.339970i	$-0.889583\pi$
$283$	−2.02954e11	−1.88087	−0.940436	+	0.339970i	$0.889583\pi$
$284$	0	0
$285$	−6.12422e9	−0.0549856
$286$	0	0
$287$	2.42022e11	2.10565
$288$	0	0
$289$	−7.55318e10	−0.636927
$290$	0	0
$291$	5.16561e9	0.0422283
$292$	0	0
$293$	1.31572e11	1.04294	0.521470	−	0.853270i	$-0.325384\pi$
$293$	1.31572e11	1.04294	0.521470	+	0.853270i	$0.325384\pi$
$294$	0	0
$295$	6.22810e10	0.478802
$296$	0	0
$297$	−2.66384e10	−0.198657
$298$	0	0
$299$	−2.10844e11	−1.52560
$300$	0	0
$301$	−1.92648e11	−1.35274
$302$	0	0
$303$	−1.33450e10	−0.0909554
$304$	0	0
$305$	−1.25271e11	−0.828899
$306$	0	0
$307$	−1.88785e11	−1.21296	−0.606479	−	0.795099i	$-0.707419\pi$
$307$	−1.88785e11	−1.21296	−0.606479	+	0.795099i	$0.707419\pi$
$308$	0	0
$309$	−1.03855e9	−0.00648057
$310$	0	0
$311$	−7.10454e10	−0.430640	−0.215320	−	0.976544i	$-0.569079\pi$
$311$	−7.10454e10	−0.430640	−0.215320	+	0.976544i	$0.569079\pi$
$312$	0	0
$313$	3.35032e11	1.97305	0.986523	−	0.163623i	$-0.0523182\pi$
$313$	3.35032e11	1.97305	0.986523	+	0.163623i	$0.0523182\pi$
$314$	0	0
$315$	1.17479e11	0.672299
$316$	0	0
$317$	−1.98163e11	−1.10219	−0.551094	−	0.834443i	$-0.685789\pi$
$317$	−1.98163e11	−1.10219	−0.551094	+	0.834443i	$0.685789\pi$
$318$	0	0
$319$	−2.38252e11	−1.28818
$320$	0	0
$321$	6.39717e9	0.0336291
$322$	0	0
$323$	1.66507e11	0.851179
$324$	0	0
$325$	6.63192e10	0.329734
$326$	0	0
$327$	−6.83110e9	−0.0330389
$328$	0	0
$329$	3.58931e11	1.68900
$330$	0	0
$331$	−1.60576e11	−0.735284	−0.367642	−	0.929967i	$-0.619835\pi$
$331$	−1.60576e11	−0.735284	−0.367642	+	0.929967i	$0.619835\pi$
$332$	0	0
$333$	5.66202e10	0.252332
$334$	0	0
$335$	5.06219e10	0.219602
$336$	0	0
$337$	8.77234e10	0.370494	0.185247	−	0.982692i	$-0.440692\pi$
$337$	8.77234e10	0.370494	0.185247	+	0.982692i	$0.440692\pi$
$338$	0	0
$339$	−3.22130e10	−0.132475
$340$	0	0
$341$	−1.99539e11	−0.799157
$342$	0	0
$343$	1.14379e11	0.446194
$344$	0	0
$345$	9.47800e9	0.0360189
$346$	0	0
$347$	1.89987e11	0.703464	0.351732	−	0.936101i	$-0.385593\pi$
$347$	1.89987e11	0.703464	0.351732	+	0.936101i	$0.385593\pi$
$348$	0	0
$349$	4.85678e11	1.75240	0.876201	−	0.481946i	$-0.160070\pi$
$349$	4.85678e11	1.75240	0.876201	+	0.481946i	$0.160070\pi$
$350$	0	0
$351$	−8.13031e10	−0.285907
$352$	0	0
$353$	5.15816e10	0.176811	0.0884054	−	0.996085i	$-0.471823\pi$
$353$	5.15816e10	0.176811	0.0884054	+	0.996085i	$0.471823\pi$
$354$	0	0
$355$	2.69701e10	0.0901270
$356$	0	0
$357$	2.43816e10	0.0794431
$358$	0	0
$359$	−1.79131e11	−0.569176	−0.284588	−	0.958650i	$-0.591857\pi$
$359$	−1.79131e11	−0.569176	−0.284588	+	0.958650i	$0.591857\pi$
$360$	0	0
$361$	3.21231e11	0.995485
$362$	0	0
$363$	8.99154e9	0.0271803
$364$	0	0
$365$	2.13012e11	0.628184
$366$	0	0
$367$	−2.76314e10	−0.0795071	−0.0397536	−	0.999210i	$-0.512657\pi$
$367$	−2.76314e10	−0.0795071	−0.0397536	+	0.999210i	$0.512657\pi$
$368$	0	0
$369$	−4.91307e11	−1.37954
$370$	0	0
$371$	−2.45525e11	−0.672842
$372$	0	0
$373$	2.35196e11	0.629129	0.314565	−	0.949236i	$-0.398142\pi$
$373$	2.35196e11	0.629129	0.314565	+	0.949236i	$0.398142\pi$
$374$	0	0
$375$	−2.98123e9	−0.00778493
$376$	0	0
$377$	−7.27168e11	−1.85395
$378$	0	0
$379$	−3.11965e11	−0.776658	−0.388329	−	0.921521i	$-0.626948\pi$
$379$	−3.11965e11	−0.776658	−0.388329	+	0.921521i	$0.626948\pi$
$380$	0	0
$381$	4.06958e10	0.0989434
$382$	0	0
$383$	−4.78535e10	−0.113637	−0.0568185	−	0.998385i	$-0.518096\pi$
$383$	−4.78535e10	−0.113637	−0.0568185	+	0.998385i	$0.518096\pi$
$384$	0	0
$385$	−3.34543e11	−0.776029
$386$	0	0
$387$	3.91077e11	0.886264
$388$	0	0
$389$	1.19528e11	0.264665	0.132332	−	0.991205i	$-0.457753\pi$
$389$	1.19528e11	0.264665	0.132332	+	0.991205i	$0.457753\pi$
$390$	0	0
$391$	−2.57691e11	−0.557575
$392$	0	0
$393$	−4.62659e10	−0.0978351
$394$	0	0
$395$	−1.75680e11	−0.363108
$396$	0	0
$397$	7.08892e10	0.143226	0.0716132	−	0.997432i	$-0.477185\pi$
$397$	7.08892e10	0.143226	0.0716132	+	0.997432i	$0.477185\pi$
$398$	0	0
$399$	9.42891e10	0.186245
$400$	0	0
$401$	5.62486e11	1.08633	0.543165	−	0.839626i	$-0.317226\pi$
$401$	5.62486e11	1.08633	0.543165	+	0.839626i	$0.317226\pi$
$402$	0	0
$403$	−6.09012e11	−1.15015
$404$	0	0
$405$	−2.36649e11	−0.437075
$406$	0	0
$407$	−1.61236e11	−0.291265
$408$	0	0
$409$	2.23341e11	0.394651	0.197325	−	0.980338i	$-0.436774\pi$
$409$	2.23341e11	0.394651	0.197325	+	0.980338i	$0.436774\pi$
$410$	0	0
$411$	−2.62765e10	−0.0454234
$412$	0	0
$413$	−9.58884e11	−1.62178
$414$	0	0
$415$	3.75636e11	0.621657
$416$	0	0
$417$	3.35911e9	0.00544016
$418$	0	0
$419$	3.54170e11	0.561370	0.280685	−	0.959800i	$-0.409438\pi$
$419$	3.54170e11	0.561370	0.280685	+	0.959800i	$0.409438\pi$
$420$	0	0
$421$	5.29294e11	0.821159	0.410580	−	0.911825i	$-0.365326\pi$
$421$	5.29294e11	0.821159	0.410580	+	0.911825i	$0.365326\pi$
$422$	0	0
$423$	−7.28634e11	−1.10657
$424$	0	0
$425$	8.10545e10	0.120511
$426$	0	0
$427$	1.92869e12	2.80761
$428$	0	0
$429$	1.15323e11	0.164383
$430$	0	0
$431$	1.26760e12	1.76944	0.884718	−	0.466127i	$-0.154351\pi$
$431$	1.26760e12	1.76944	0.884718	+	0.466127i	$0.154351\pi$
$432$	0	0
$433$	3.28861e11	0.449591	0.224795	−	0.974406i	$-0.427829\pi$
$433$	3.28861e11	0.449591	0.224795	+	0.974406i	$0.427829\pi$
$434$	0	0
$435$	3.26882e10	0.0437713
$436$	0	0
$437$	−9.96545e11	−1.30717
$438$	0	0
$439$	2.69396e11	0.346178	0.173089	−	0.984906i	$-0.444625\pi$
$439$	2.69396e11	0.346178	0.173089	+	0.984906i	$0.444625\pi$
$440$	0	0
$441$	−1.02045e12	−1.28475
$442$	0	0
$443$	4.20444e11	0.518671	0.259335	−	0.965787i	$-0.416497\pi$
$443$	4.20444e11	0.518671	0.259335	+	0.965787i	$0.416497\pi$
$444$	0	0
$445$	−3.37448e11	−0.407931
$446$	0	0
$447$	−7.52915e10	−0.0891995
$448$	0	0
$449$	1.19773e11	0.139075	0.0695376	−	0.997579i	$-0.477848\pi$
$449$	1.19773e11	0.139075	0.0695376	+	0.997579i	$0.477848\pi$
$450$	0	0
$451$	1.39909e12	1.59239
$452$	0	0
$453$	−4.00984e10	−0.0447388
$454$	0	0
$455$	−1.02106e12	−1.11686
$456$	0	0
$457$	1.34943e12	1.44719	0.723597	−	0.690223i	$-0.242488\pi$
$457$	1.34943e12	1.44719	0.723597	+	0.690223i	$0.242488\pi$
$458$	0	0
$459$	−9.93677e10	−0.104493
$460$	0	0
$461$	2.09834e11	0.216383	0.108191	−	0.994130i	$-0.465494\pi$
$461$	2.09834e11	0.216383	0.108191	+	0.994130i	$0.465494\pi$
$462$	0	0
$463$	−6.60348e11	−0.667819	−0.333909	−	0.942605i	$-0.608368\pi$
$463$	−6.60348e11	−0.667819	−0.333909	+	0.942605i	$0.608368\pi$
$464$	0	0
$465$	2.73767e10	0.0271546
$466$	0	0
$467$	−8.28966e11	−0.806512	−0.403256	−	0.915087i	$-0.632122\pi$
$467$	−8.28966e11	−0.806512	−0.403256	+	0.915087i	$0.632122\pi$
$468$	0	0
$469$	−7.79381e11	−0.743827
$470$	0	0
$471$	−2.31102e10	−0.0216376
$472$	0	0
$473$	−1.11366e12	−1.02301
$474$	0	0
$475$	3.13455e11	0.282523
$476$	0	0
$477$	4.98417e11	0.440819
$478$	0	0
$479$	−6.48710e11	−0.563042	−0.281521	−	0.959555i	$-0.590839\pi$
$479$	−6.48710e11	−0.563042	−0.281521	+	0.959555i	$0.590839\pi$
$480$	0	0
$481$	−4.92110e11	−0.419188
$482$	0	0
$483$	−1.45924e11	−0.122002
$484$	0	0
$485$	−2.64391e11	−0.216975
$486$	0	0
$487$	−1.75979e12	−1.41768	−0.708842	−	0.705367i	$-0.750782\pi$
$487$	−1.75979e12	−1.41768	−0.708842	+	0.705367i	$0.750782\pi$
$488$	0	0
$489$	5.05898e9	0.00400104
$490$	0	0
$491$	3.02135e11	0.234603	0.117302	−	0.993096i	$-0.462576\pi$
$491$	3.02135e11	0.234603	0.117302	+	0.993096i	$0.462576\pi$
$492$	0	0
$493$	−8.88736e11	−0.677582
$494$	0	0
$495$	6.79124e11	0.508424
$496$	0	0
$497$	−4.15235e11	−0.305274
$498$	0	0
$499$	5.11758e11	0.369498	0.184749	−	0.982786i	$-0.440853\pi$
$499$	5.11758e11	0.369498	0.184749	+	0.982786i	$0.440853\pi$
$500$	0	0
$501$	−1.90768e11	−0.135281
$502$	0	0
$503$	1.69213e12	1.17863	0.589314	−	0.807904i	$-0.299398\pi$
$503$	1.69213e12	1.17863	0.589314	+	0.807904i	$0.299398\pi$
$504$	0	0
$505$	6.83038e11	0.467341
$506$	0	0
$507$	2.22483e11	0.149542
$508$	0	0
$509$	7.10216e11	0.468986	0.234493	−	0.972118i	$-0.424657\pi$
$509$	7.10216e11	0.468986	0.234493	+	0.972118i	$0.424657\pi$
$510$	0	0
$511$	−3.27956e12	−2.12776
$512$	0	0
$513$	−3.84276e11	−0.244972
$514$	0	0
$515$	5.31559e10	0.0332981
$516$	0	0
$517$	2.07492e12	1.27730
$518$	0	0
$519$	1.41719e11	0.0857386
$520$	0	0
$521$	−8.88794e11	−0.528483	−0.264242	−	0.964456i	$-0.585122\pi$
$521$	−8.88794e11	−0.528483	−0.264242	+	0.964456i	$0.585122\pi$
$522$	0	0
$523$	1.01046e12	0.590555	0.295277	−	0.955412i	$-0.404588\pi$
$523$	1.01046e12	0.590555	0.295277	+	0.955412i	$0.404588\pi$
$524$	0	0
$525$	4.58993e10	0.0263687
$526$	0	0
$527$	−7.44327e11	−0.420355
$528$	0	0
$529$	−2.58873e11	−0.143726
$530$	0	0
$531$	1.94654e12	1.06252
$532$	0	0
$533$	4.27015e12	2.29177
$534$	0	0
$535$	−3.27426e11	−0.172791
$536$	0	0
$537$	−2.99851e11	−0.155604
$538$	0	0
$539$	2.90593e12	1.48298
$540$	0	0
$541$	3.39951e12	1.70619	0.853097	−	0.521753i	$-0.174722\pi$
$541$	3.39951e12	1.70619	0.853097	+	0.521753i	$0.174722\pi$
$542$	0	0
$543$	3.92015e10	0.0193510
$544$	0	0
$545$	3.49636e11	0.169758
$546$	0	0
$547$	−3.10308e12	−1.48201	−0.741004	−	0.671501i	$-0.765650\pi$
$547$	−3.10308e12	−1.48201	−0.741004	+	0.671501i	$0.765650\pi$
$548$	0	0
$549$	−3.91525e12	−1.83943
$550$	0	0
$551$	−3.43693e12	−1.58851
$552$	0	0
$553$	2.70479e12	1.22990
$554$	0	0
$555$	2.21217e10	0.00989691
$556$	0	0
$557$	−4.92558e11	−0.216825	−0.108412	−	0.994106i	$-0.534577\pi$
$557$	−4.92558e11	−0.216825	−0.108412	+	0.994106i	$0.534577\pi$
$558$	0	0
$559$	−3.39901e12	−1.47231
$560$	0	0
$561$	1.40946e11	0.0600786
$562$	0	0
$563$	1.67786e12	0.703832	0.351916	−	0.936032i	$-0.385530\pi$
$563$	1.67786e12	0.703832	0.351916	+	0.936032i	$0.385530\pi$
$564$	0	0
$565$	1.64876e12	0.680673
$566$	0	0
$567$	3.64347e12	1.48044
$568$	0	0
$569$	3.40646e12	1.36238	0.681190	−	0.732107i	$-0.261463\pi$
$569$	3.40646e12	1.36238	0.681190	+	0.732107i	$0.261463\pi$
$570$	0	0
$571$	−4.05234e12	−1.59530	−0.797651	−	0.603119i	$-0.793924\pi$
$571$	−4.05234e12	−1.59530	−0.797651	+	0.603119i	$0.793924\pi$
$572$	0	0
$573$	4.17068e11	0.161626
$574$	0	0
$575$	−4.85112e11	−0.185070
$576$	0	0
$577$	−4.55977e11	−0.171258	−0.0856291	−	0.996327i	$-0.527290\pi$
$577$	−4.55977e11	−0.171258	−0.0856291	+	0.996327i	$0.527290\pi$
$578$	0	0
$579$	3.06347e11	0.113282
$580$	0	0
$581$	−5.78333e12	−2.10565
$582$	0	0
$583$	−1.41933e12	−0.508834
$584$	0	0
$585$	2.07275e12	0.731722
$586$	0	0
$587$	−2.83858e12	−0.986801	−0.493401	−	0.869802i	$-0.664246\pi$
$587$	−2.83858e12	−0.986801	−0.493401	+	0.869802i	$0.664246\pi$
$588$	0	0
$589$	−2.87847e12	−0.985470
$590$	0	0
$591$	1.17728e11	0.0396952
$592$	0	0
$593$	−4.08687e12	−1.35720	−0.678601	−	0.734507i	$-0.737414\pi$
$593$	−4.08687e12	−1.35720	−0.678601	+	0.734507i	$0.737414\pi$
$594$	0	0
$595$	−1.24792e12	−0.408189
$596$	0	0
$597$	4.08240e10	0.0131532
$598$	0	0
$599$	4.47524e12	1.42035	0.710175	−	0.704025i	$-0.248616\pi$
$599$	4.47524e12	1.42035	0.710175	+	0.704025i	$0.248616\pi$
$600$	0	0
$601$	−4.83907e11	−0.151296	−0.0756478	−	0.997135i	$-0.524102\pi$
$601$	−4.83907e11	−0.151296	−0.0756478	+	0.997135i	$0.524102\pi$
$602$	0	0
$603$	1.58215e12	0.487326
$604$	0	0
$605$	−4.60213e11	−0.139656
$606$	0	0
$607$	−4.96609e12	−1.48479	−0.742395	−	0.669962i	$-0.766310\pi$
$607$	−4.96609e12	−1.48479	−0.742395	+	0.669962i	$0.766310\pi$
$608$	0	0
$609$	−5.03271e11	−0.148260
$610$	0	0
$611$	6.33285e12	1.83829
$612$	0	0
$613$	−2.43460e12	−0.696396	−0.348198	−	0.937421i	$-0.613206\pi$
$613$	−2.43460e12	−0.696396	−0.348198	+	0.937421i	$0.613206\pi$
$614$	0	0
$615$	−1.91955e11	−0.0541080
$616$	0	0
$617$	1.82614e12	0.507284	0.253642	−	0.967298i	$-0.418372\pi$
$617$	1.82614e12	0.507284	0.253642	+	0.967298i	$0.418372\pi$
$618$	0	0
$619$	−1.13058e12	−0.309523	−0.154761	−	0.987952i	$-0.549461\pi$
$619$	−1.13058e12	−0.309523	−0.154761	+	0.987952i	$0.549461\pi$
$620$	0	0
$621$	5.94716e11	0.160471
$622$	0	0
$623$	5.19539e12	1.38173
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	5.45068e11	0.140847
$628$	0	0
$629$	−6.01450e11	−0.153205
$630$	0	0
$631$	1.96900e12	0.494440	0.247220	−	0.968959i	$-0.420483\pi$
$631$	1.96900e12	0.494440	0.247220	+	0.968959i	$0.420483\pi$
$632$	0	0
$633$	−2.15022e11	−0.0532312
$634$	0	0
$635$	−2.08293e12	−0.508385
$636$	0	0
$637$	8.86918e12	2.13430
$638$	0	0
$639$	8.42930e11	0.200003
$640$	0	0
$641$	2.69141e12	0.629679	0.314840	−	0.949145i	$-0.398049\pi$
$641$	2.69141e12	0.629679	0.314840	+	0.949145i	$0.398049\pi$
$642$	0	0
$643$	−2.27990e12	−0.525977	−0.262989	−	0.964799i	$-0.584708\pi$
$643$	−2.27990e12	−0.525977	−0.262989	+	0.964799i	$0.584708\pi$
$644$	0	0
$645$	1.52795e11	0.0347608
$646$	0	0
$647$	−4.86871e12	−1.09231	−0.546153	−	0.837685i	$-0.683908\pi$
$647$	−4.86871e12	−1.09231	−0.546153	+	0.837685i	$0.683908\pi$
$648$	0	0
$649$	−5.54314e12	−1.22646
$650$	0	0
$651$	−4.21495e11	−0.0919768
$652$	0	0
$653$	2.30610e12	0.496329	0.248164	−	0.968718i	$-0.420173\pi$
$653$	2.30610e12	0.496329	0.248164	+	0.968718i	$0.420173\pi$
$654$	0	0
$655$	2.36802e12	0.502690
$656$	0	0
$657$	6.65754e12	1.39402
$658$	0	0
$659$	7.40069e12	1.52858	0.764289	−	0.644873i	$-0.223090\pi$
$659$	7.40069e12	1.52858	0.764289	+	0.644873i	$0.223090\pi$
$660$	0	0
$661$	−2.00760e12	−0.409045	−0.204522	−	0.978862i	$-0.565564\pi$
$661$	−2.00760e12	−0.409045	−0.204522	+	0.978862i	$0.565564\pi$
$662$	0	0
$663$	4.30181e11	0.0864650
$664$	0	0
$665$	−4.82599e12	−0.956949
$666$	0	0
$667$	5.31909e12	1.04057
$668$	0	0
$669$	−6.11170e11	−0.117963
$670$	0	0
$671$	1.11494e13	2.12324
$672$	0	0
$673$	−1.01154e13	−1.90071	−0.950355	−	0.311167i	$-0.899280\pi$
$673$	−1.01154e13	−1.90071	−0.950355	+	0.311167i	$0.899280\pi$
$674$	0	0
$675$	−1.87063e11	−0.0346834
$676$	0	0
$677$	−8.91166e12	−1.63046	−0.815229	−	0.579139i	$-0.803389\pi$
$677$	−8.91166e12	−1.63046	−0.815229	+	0.579139i	$0.803389\pi$
$678$	0	0
$679$	4.07059e12	0.734926
$680$	0	0
$681$	−1.06007e11	−0.0188874
$682$	0	0
$683$	−9.26863e12	−1.62976	−0.814878	−	0.579633i	$-0.803196\pi$
$683$	−9.26863e12	−1.62976	−0.814878	+	0.579633i	$0.803196\pi$
$684$	0	0
$685$	1.34491e12	0.233391
$686$	0	0
$687$	−3.61466e11	−0.0619101
$688$	0	0
$689$	−4.33195e12	−0.732313
$690$	0	0
$691$	1.03490e11	0.0172682	0.00863409	−	0.999963i	$-0.497252\pi$
$691$	1.03490e11	0.0172682	0.00863409	+	0.999963i	$0.497252\pi$
$692$	0	0
$693$	−1.04559e13	−1.72211
$694$	0	0
$695$	−1.71929e11	−0.0279523
$696$	0	0
$697$	5.21893e12	0.837594
$698$	0	0
$699$	−9.06818e11	−0.143672
$700$	0	0
$701$	4.77133e12	0.746291	0.373145	−	0.927773i	$-0.378279\pi$
$701$	4.77133e12	0.746291	0.373145	+	0.927773i	$0.378279\pi$
$702$	0	0
$703$	−2.32594e12	−0.359169
$704$	0	0
$705$	−2.84679e11	−0.0434015
$706$	0	0
$707$	−1.05161e13	−1.58295
$708$	0	0
$709$	7.58922e12	1.12795	0.563974	−	0.825793i	$-0.309272\pi$
$709$	7.58922e12	1.12795	0.563974	+	0.825793i	$0.309272\pi$
$710$	0	0
$711$	−5.49075e12	−0.805784
$712$	0	0
$713$	4.45480e12	0.645543
$714$	0	0
$715$	−5.90255e12	−0.844622
$716$	0	0
$717$	−2.02188e11	−0.0285705
$718$	0	0
$719$	1.50766e12	0.210389	0.105194	−	0.994452i	$-0.466454\pi$
$719$	1.50766e12	0.210389	0.105194	+	0.994452i	$0.466454\pi$
$720$	0	0
$721$	−8.18394e11	−0.112786
$722$	0	0
$723$	4.40909e11	0.0600105
$724$	0	0
$725$	−1.67308e12	−0.224903
$726$	0	0
$727$	1.12985e13	1.50009	0.750045	−	0.661387i	$-0.230032\pi$
$727$	1.12985e13	1.50009	0.750045	+	0.661387i	$0.230032\pi$
$728$	0	0
$729$	−7.28117e12	−0.954833
$730$	0	0
$731$	−4.15423e12	−0.538100
$732$	0	0
$733$	1.07062e12	0.136984	0.0684919	−	0.997652i	$-0.478181\pi$
$733$	1.07062e12	0.136984	0.0684919	+	0.997652i	$0.478181\pi$
$734$	0	0
$735$	−3.98694e11	−0.0503903
$736$	0	0
$737$	−4.50546e12	−0.562517
$738$	0	0
$739$	−1.44855e13	−1.78662	−0.893311	−	0.449439i	$-0.851624\pi$
$739$	−1.44855e13	−1.78662	−0.893311	+	0.449439i	$0.851624\pi$
$740$	0	0
$741$	1.66360e12	0.202707
$742$	0	0
$743$	1.24339e13	1.49678	0.748388	−	0.663261i	$-0.230828\pi$
$743$	1.24339e13	1.49678	0.748388	+	0.663261i	$0.230828\pi$
$744$	0	0
$745$	3.85364e12	0.458319
$746$	0	0
$747$	1.17402e13	1.37954
$748$	0	0
$749$	5.04108e12	0.585269
$750$	0	0
$751$	8.79097e12	1.00846	0.504228	−	0.863570i	$-0.331777\pi$
$751$	8.79097e12	1.00846	0.504228	+	0.863570i	$0.331777\pi$
$752$	0	0
$753$	2.84468e11	0.0322446
$754$	0	0
$755$	2.05235e12	0.229874
$756$	0	0
$757$	7.93877e12	0.878662	0.439331	−	0.898325i	$-0.355216\pi$
$757$	7.93877e12	0.878662	0.439331	+	0.898325i	$0.355216\pi$
$758$	0	0
$759$	−8.43562e11	−0.0922633
$760$	0	0
$761$	1.12763e13	1.21881	0.609405	−	0.792859i	$-0.291408\pi$
$761$	1.12763e13	1.21881	0.609405	+	0.792859i	$0.291408\pi$
$762$	0	0
$763$	−5.38303e12	−0.574998
$764$	0	0
$765$	2.53329e12	0.267429
$766$	0	0
$767$	−1.69182e13	−1.76512
$768$	0	0
$769$	3.80743e12	0.392612	0.196306	−	0.980543i	$-0.437105\pi$
$769$	3.80743e12	0.392612	0.196306	+	0.980543i	$0.437105\pi$
$770$	0	0
$771$	−5.58810e11	−0.0569534
$772$	0	0
$773$	−3.89254e12	−0.392126	−0.196063	−	0.980591i	$-0.562816\pi$
$773$	−3.89254e12	−0.392126	−0.196063	+	0.980591i	$0.562816\pi$
$774$	0	0
$775$	−1.40122e12	−0.139524
$776$	0	0
$777$	−3.40588e11	−0.0335223
$778$	0	0
$779$	2.01827e13	1.96364
$780$	0	0
$781$	−2.40040e12	−0.230862
$782$	0	0
$783$	2.05109e12	0.195010
$784$	0	0
$785$	1.18284e12	0.111177
$786$	0	0
$787$	−2.70256e12	−0.251124	−0.125562	−	0.992086i	$-0.540073\pi$
$787$	−2.70256e12	−0.251124	−0.125562	+	0.992086i	$0.540073\pi$
$788$	0	0
$789$	1.21368e12	0.111495
$790$	0	0
$791$	−2.53844e13	−2.30554
$792$	0	0
$793$	3.40291e13	3.05577
$794$	0	0
$795$	1.94733e11	0.0172897
$796$	0	0
$797$	−1.58748e12	−0.139363	−0.0696813	−	0.997569i	$-0.522198\pi$
$797$	−1.58748e12	−0.139363	−0.0696813	+	0.997569i	$0.522198\pi$
$798$	0	0
$799$	7.73994e12	0.671857
$800$	0	0
$801$	−1.05467e13	−0.905252
$802$	0	0
$803$	−1.89586e13	−1.60911
$804$	0	0
$805$	7.46883e12	0.626861
$806$	0	0
$807$	1.92294e12	0.159601
$808$	0	0
$809$	−4.55705e12	−0.374038	−0.187019	−	0.982356i	$-0.559883\pi$
$809$	−4.55705e12	−0.374038	−0.187019	+	0.982356i	$0.559883\pi$
$810$	0	0
$811$	−2.77014e12	−0.224858	−0.112429	−	0.993660i	$-0.535863\pi$
$811$	−2.77014e12	−0.224858	−0.112429	+	0.993660i	$0.535863\pi$
$812$	0	0
$813$	3.41047e11	0.0273783
$814$	0	0
$815$	−2.58933e11	−0.0205579
$816$	0	0
$817$	−1.60653e13	−1.26151
$818$	0	0
$819$	−3.19123e13	−2.47846
$820$	0	0
$821$	−1.21734e13	−0.935124	−0.467562	−	0.883960i	$-0.654868\pi$
$821$	−1.21734e13	−0.935124	−0.467562	+	0.883960i	$0.654868\pi$
$822$	0	0
$823$	1.75355e13	1.33235	0.666175	−	0.745796i	$-0.267931\pi$
$823$	1.75355e13	1.33235	0.666175	+	0.745796i	$0.267931\pi$
$824$	0	0
$825$	2.65336e11	0.0199413
$826$	0	0
$827$	−9.15138e12	−0.680318	−0.340159	−	0.940368i	$-0.610481\pi$
$827$	−9.15138e12	−0.680318	−0.340159	+	0.940368i	$0.610481\pi$
$828$	0	0
$829$	−5.76678e12	−0.424071	−0.212035	−	0.977262i	$-0.568009\pi$
$829$	−5.76678e12	−0.424071	−0.212035	+	0.977262i	$0.568009\pi$
$830$	0	0
$831$	−2.05522e12	−0.149504
$832$	0	0
$833$	1.08398e13	0.780044
$834$	0	0
$835$	9.76408e12	0.695092
$836$	0	0
$837$	1.71781e12	0.120979
$838$	0	0
$839$	8.69274e11	0.0605658	0.0302829	−	0.999541i	$-0.490359\pi$
$839$	8.69274e11	0.0605658	0.0302829	+	0.999541i	$0.490359\pi$
$840$	0	0
$841$	3.83759e12	0.264531
$842$	0	0
$843$	−1.04024e12	−0.0709427
$844$	0	0
$845$	−1.13873e13	−0.768365
$846$	0	0
$847$	7.08549e12	0.473037
$848$	0	0
$849$	−2.47830e12	−0.163708
$850$	0	0
$851$	3.59968e12	0.235278
$852$	0	0
$853$	−9.19943e12	−0.594963	−0.297482	−	0.954728i	$-0.596147\pi$
$853$	−9.19943e12	−0.594963	−0.297482	+	0.954728i	$0.596147\pi$
$854$	0	0
$855$	9.79680e12	0.626955
$856$	0	0
$857$	−6.57679e12	−0.416486	−0.208243	−	0.978077i	$-0.566775\pi$
$857$	−6.57679e12	−0.416486	−0.208243	+	0.978077i	$0.566775\pi$
$858$	0	0
$859$	−1.80030e13	−1.12817	−0.564085	−	0.825717i	$-0.690771\pi$
$859$	−1.80030e13	−1.12817	−0.564085	+	0.825717i	$0.690771\pi$
$860$	0	0
$861$	2.95536e12	0.183272
$862$	0	0
$863$	3.08737e13	1.89470	0.947350	−	0.320200i	$-0.103750\pi$
$863$	3.08737e13	1.89470	0.947350	+	0.320200i	$0.103750\pi$
$864$	0	0
$865$	−7.25361e12	−0.440536
$866$	0	0
$867$	−9.22328e11	−0.0554370
$868$	0	0
$869$	1.56359e13	0.930110
$870$	0	0
$871$	−1.37511e13	−0.809573
$872$	0	0
$873$	−8.26334e12	−0.481495
$874$	0	0
$875$	−2.34926e12	−0.135486
$876$	0	0
$877$	2.77535e12	0.158424	0.0792118	−	0.996858i	$-0.474760\pi$
$877$	2.77535e12	0.158424	0.0792118	+	0.996858i	$0.474760\pi$
$878$	0	0
$879$	1.60664e12	0.0907755
$880$	0	0
$881$	−1.64369e13	−0.919241	−0.459620	−	0.888116i	$-0.652014\pi$
$881$	−1.64369e13	−0.919241	−0.459620	+	0.888116i	$0.652014\pi$
$882$	0	0
$883$	−5.38106e12	−0.297882	−0.148941	−	0.988846i	$-0.547586\pi$
$883$	−5.38106e12	−0.297882	−0.148941	+	0.988846i	$0.547586\pi$
$884$	0	0
$885$	7.60520e11	0.0416741
$886$	0	0
$887$	1.61191e13	0.874347	0.437174	−	0.899377i	$-0.355980\pi$
$887$	1.61191e13	0.874347	0.437174	+	0.899377i	$0.355980\pi$
$888$	0	0
$889$	3.20690e13	1.72198
$890$	0	0
$891$	2.10622e13	1.11958
$892$	0	0
$893$	2.99320e13	1.57509
$894$	0	0
$895$	1.53473e13	0.799516
$896$	0	0
$897$	−2.57464e12	−0.132785
$898$	0	0
$899$	1.53639e13	0.784483
$900$	0	0
$901$	−5.29446e12	−0.267645
$902$	0	0
$903$	−2.35245e12	−0.117740
$904$	0	0
$905$	−2.00645e12	−0.0994281
$906$	0	0
$907$	1.50156e13	0.736734	0.368367	−	0.929681i	$-0.379917\pi$
$907$	1.50156e13	0.736734	0.368367	+	0.929681i	$0.379917\pi$
$908$	0	0
$909$	2.13478e13	1.03709
$910$	0	0
$911$	−2.12904e12	−0.102412	−0.0512061	−	0.998688i	$-0.516307\pi$
$911$	−2.12904e12	−0.102412	−0.0512061	+	0.998688i	$0.516307\pi$
$912$	0	0
$913$	−3.34324e13	−1.59239
$914$	0	0
$915$	−1.52970e12	−0.0721458
$916$	0	0
$917$	−3.64583e13	−1.70269
$918$	0	0
$919$	−2.82600e13	−1.30693	−0.653464	−	0.756957i	$-0.726685\pi$
$919$	−2.82600e13	−1.30693	−0.653464	+	0.756957i	$0.726685\pi$
$920$	0	0
$921$	−2.30528e12	−0.105574
$922$	0	0
$923$	−7.32625e12	−0.332257
$924$	0	0
$925$	−1.13225e12	−0.0508517
$926$	0	0
$927$	1.66135e12	0.0738927
$928$	0	0
$929$	−1.74749e11	−0.00769739	−0.00384869	−	0.999993i	$-0.501225\pi$
$929$	−1.74749e11	−0.00769739	−0.00384869	+	0.999993i	$0.501225\pi$
$930$	0	0
$931$	4.19199e13	1.82872
$932$	0	0
$933$	−8.67543e11	−0.0374821
$934$	0	0
$935$	−7.21402e12	−0.308692
$936$	0	0
$937$	6.04012e12	0.255987	0.127993	−	0.991775i	$-0.459146\pi$
$937$	6.04012e12	0.255987	0.127993	+	0.991775i	$0.459146\pi$
$938$	0	0
$939$	4.09112e12	0.171730
$940$	0	0
$941$	2.13848e13	0.889104	0.444552	−	0.895753i	$-0.353363\pi$
$941$	2.13848e13	0.889104	0.444552	+	0.895753i	$0.353363\pi$
$942$	0	0
$943$	−3.12353e13	−1.28630
$944$	0	0
$945$	2.88004e12	0.117478
$946$	0	0
$947$	4.58008e13	1.85054	0.925269	−	0.379312i	$-0.123839\pi$
$947$	4.58008e13	1.85054	0.925269	+	0.379312i	$0.123839\pi$
$948$	0	0
$949$	−5.78634e13	−2.31583
$950$	0	0
$951$	−2.41979e12	−0.0959323
$952$	0	0
$953$	−1.57835e13	−0.619847	−0.309924	−	0.950761i	$-0.600304\pi$
$953$	−1.57835e13	−0.619847	−0.309924	+	0.950761i	$0.600304\pi$
$954$	0	0
$955$	−2.13467e13	−0.830455
$956$	0	0
$957$	−2.90932e12	−0.112121
$958$	0	0
$959$	−2.07063e13	−0.790533
$960$	0	0
$961$	−1.35722e13	−0.513326
$962$	0	0
$963$	−1.02334e13	−0.383445
$964$	0	0
$965$	−1.56797e13	−0.582056
$966$	0	0
$967$	1.61550e13	0.594140	0.297070	−	0.954856i	$-0.403991\pi$
$967$	1.61550e13	0.594140	0.297070	+	0.954856i	$0.403991\pi$
$968$	0	0
$969$	2.03324e12	0.0740851
$970$	0	0
$971$	−5.05363e12	−0.182439	−0.0912194	−	0.995831i	$-0.529076\pi$
$971$	−5.05363e12	−0.182439	−0.0912194	+	0.995831i	$0.529076\pi$
$972$	0	0
$973$	2.64703e12	0.0946786
$974$	0	0
$975$	8.09831e11	0.0286994
$976$	0	0
$977$	−3.22726e13	−1.13321	−0.566603	−	0.823991i	$-0.691743\pi$
$977$	−3.22726e13	−1.13321	−0.566603	+	0.823991i	$0.691743\pi$
$978$	0	0
$979$	3.00336e13	1.04493
$980$	0	0
$981$	1.09276e13	0.376716
$982$	0	0
$983$	−2.11820e13	−0.723561	−0.361781	−	0.932263i	$-0.617831\pi$
$983$	−2.11820e13	−0.723561	−0.361781	+	0.932263i	$0.617831\pi$
$984$	0	0
$985$	−6.02568e12	−0.203959
$986$	0	0
$987$	4.38295e12	0.147007
$988$	0	0
$989$	2.48631e13	0.826365
$990$	0	0
$991$	−2.18946e13	−0.721116	−0.360558	−	0.932737i	$-0.617414\pi$
$991$	−2.18946e13	−0.721116	−0.360558	+	0.932737i	$0.617414\pi$
$992$	0	0
$993$	−1.96081e12	−0.0639977
$994$	0	0
$995$	−2.08949e12	−0.0675828
$996$	0	0
$997$	−4.15332e13	−1.33127	−0.665636	−	0.746277i	$-0.731840\pi$
$997$	−4.15332e13	−1.33127	−0.665636	+	0.746277i	$0.731840\pi$
$998$	0	0
$999$	1.38807e12	0.0440927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.a.b.1.2	✓	2
3.2	odd	2		180.10.a.e.1.2		2
4.3	odd	2		80.10.a.j.1.1		2
5.2	odd	4		100.10.c.c.49.2		4
5.3	odd	4		100.10.c.c.49.3		4
5.4	even	2		100.10.a.c.1.1		2
8.3	odd	2		320.10.a.l.1.2		2
8.5	even	2		320.10.a.t.1.1		2
20.3	even	4		400.10.c.l.49.2		4
20.7	even	4		400.10.c.l.49.3		4
20.19	odd	2		400.10.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.a.b.1.2	✓	2	1.1	even	1	trivial
80.10.a.j.1.1		2	4.3	odd	2
100.10.a.c.1.1		2	5.4	even	2
100.10.c.c.49.2		4	5.2	odd	4
100.10.c.c.49.3		4	5.3	odd	4
180.10.a.e.1.2		2	3.2	odd	2
320.10.a.l.1.2		2	8.3	odd	2
320.10.a.t.1.1		2	8.5	even	2
400.10.a.l.1.2		2	20.19	odd	2
400.10.c.l.49.2		4	20.3	even	4
400.10.c.l.49.3		4	20.7	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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	20.a (trivial)

\(f(q)\)	\(=\)	\(q+12.2111 q^{3} -625.000 q^{5} +9622.57 q^{7} -19533.9 q^{9} +55626.3 q^{11} +169777. q^{13} -7631.94 q^{15} +207499. q^{17} +802445. q^{19} +117502. q^{21} -1.24189e6 q^{23} +390625. q^{25} -478882. q^{27} -4.28308e6 q^{29} -3.58713e6 q^{31} +679259. q^{33} -6.01410e6 q^{35} -2.89856e6 q^{37} +2.07317e6 q^{39} +2.51515e7 q^{41} -2.00204e7 q^{43} +1.22087e7 q^{45} +3.73010e7 q^{47} +5.22402e7 q^{49} +2.53380e6 q^{51} -2.55155e7 q^{53} -3.47665e7 q^{55} +9.79875e6 q^{57} -9.96495e7 q^{59} +2.00434e8 q^{61} -1.87966e8 q^{63} -1.06111e8 q^{65} -8.09951e7 q^{67} -1.51648e7 q^{69} -4.31522e7 q^{71} -3.40820e8 q^{73} +4.76997e6 q^{75} +5.35268e8 q^{77} +2.81089e8 q^{79} +3.78638e8 q^{81} -6.01017e8 q^{83} -1.29687e8 q^{85} -5.23011e7 q^{87} +5.39917e8 q^{89} +1.63369e9 q^{91} -4.38028e7 q^{93} -5.01528e8 q^{95} +4.23026e8 q^{97} -1.08660e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 260 q^{3} - 1250 q^{5} - 380 q^{7} + 34882 q^{9} + 102720 q^{11} + 179140 q^{13} + 162500 q^{15} + 316020 q^{17} + 137272 q^{19} + 2840312 q^{21} - 665460 q^{23} + 781250 q^{25} - 9933560 q^{27} - 6893748 q^{29}+ \cdots + 1476045120 q^{99}+O(q^{100}) \) 2 * q - 260 * q^3 - 1250 * q^5 - 380 * q^7 + 34882 * q^9 + 102720 * q^11 + 179140 * q^13 + 162500 * q^15 + 316020 * q^17 + 137272 * q^19 + 2840312 * q^21 - 665460 * q^23 + 781250 * q^25 - 9933560 * q^27 - 6893748 * q^29 + 291832 * q^31 - 12140160 * q^33 + 237500 * q^35 + 11261380 * q^37 - 475528 * q^39 + 29773452 * q^41 - 11708180 * q^43 - 21801250 * q^45 + 62493300 * q^47 + 111937914 * q^49 - 27006696 * q^51 + 9417780 * q^53 - 64200000 * q^55 + 190866320 * q^57 - 92930856 * q^59 + 195673924 * q^61 - 732264700 * q^63 - 111962500 * q^65 - 219767420 * q^67 - 172074264 * q^69 + 311207016 * q^71 - 99224060 * q^73 - 101562500 * q^75 + 64210560 * q^77 + 542261776 * q^79 + 1881238378 * q^81 - 1256915700 * q^83 - 197512500 * q^85 + 658353000 * q^87 - 462291852 * q^89 + 1540037768 * q^91 - 1099698160 * q^93 - 85795000 * q^95 + 1671716740 * q^97 + 1476045120 * q^99

Introduction

L-functions

Modular forms

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Database

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