LMFDB - Embedded newform 180.10.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,10,Mod(1,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("180.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	180.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:	$92.7064505095$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.88819$ of defining polynomial
Character	$\chi$	$=$	180.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+625.000 q^{5} +9622.57 q^{7} +O(q^{10})$	$q+625.000 q^{5} +9622.57 q^{7} -55626.3 q^{11} +169777. q^{13} -207499. q^{17} +802445. q^{19} +1.24189e6 q^{23} +390625. q^{25} +4.28308e6 q^{29} -3.58713e6 q^{31} +6.01410e6 q^{35} -2.89856e6 q^{37} -2.51515e7 q^{41} -2.00204e7 q^{43} -3.73010e7 q^{47} +5.22402e7 q^{49} +2.55155e7 q^{53} -3.47665e7 q^{55} +9.96495e7 q^{59} +2.00434e8 q^{61} +1.06111e8 q^{65} -8.09951e7 q^{67} +4.31522e7 q^{71} -3.40820e8 q^{73} -5.35268e8 q^{77} +2.81089e8 q^{79} +6.01017e8 q^{83} -1.29687e8 q^{85} -5.39917e8 q^{89} +1.63369e9 q^{91} +5.01528e8 q^{95} +4.23026e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1250 q^{5} - 380 q^{7}+O(q^{10}) $ 2 * q + 1250 * q^5 - 380 * q^7	$ 2 q + 1250 q^{5} - 380 q^{7} - 102720 q^{11} + 179140 q^{13} - 316020 q^{17} + 137272 q^{19} + 665460 q^{23} + 781250 q^{25} + 6893748 q^{29} + 291832 q^{31} - 237500 q^{35} + 11261380 q^{37} - 29773452 q^{41} - 11708180 q^{43} - 62493300 q^{47} + 111937914 q^{49} - 9417780 q^{53} - 64200000 q^{55} + 92930856 q^{59} + 195673924 q^{61} + 111962500 q^{65} - 219767420 q^{67} - 311207016 q^{71} - 99224060 q^{73} - 64210560 q^{77} + 542261776 q^{79} + 1256915700 q^{83} - 197512500 q^{85} + 462291852 q^{89} + 1540037768 q^{91} + 85795000 q^{95} + 1671716740 q^{97}+O(q^{100}) $ 2 * q + 1250 * q^5 - 380 * q^7 - 102720 * q^11 + 179140 * q^13 - 316020 * q^17 + 137272 * q^19 + 665460 * q^23 + 781250 * q^25 + 6893748 * q^29 + 291832 * q^31 - 237500 * q^35 + 11261380 * q^37 - 29773452 * q^41 - 11708180 * q^43 - 62493300 * q^47 + 111937914 * q^49 - 9417780 * q^53 - 64200000 * q^55 + 92930856 * q^59 + 195673924 * q^61 + 111962500 * q^65 - 219767420 * q^67 - 311207016 * q^71 - 99224060 * q^73 - 64210560 * q^77 + 542261776 * q^79 + 1256915700 * q^83 - 197512500 * q^85 + 462291852 * q^89 + 1540037768 * q^91 + 85795000 * q^95 + 1671716740 * q^97

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
$3$	0	0
$3$	0	0
$4$	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
$9$	0	0
$10$	0	0
$11$	−55626.3	−1.14555	−0.572774	−	0.819713i	$-0.694133\pi$
$11$	−55626.3	−1.14555	−0.572774	+	0.819713i	$0.694133\pi$
$12$	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
$15$	0	0
$16$	0	0
$17$	−207499.	−0.602555	−0.301278	−	0.953536i	$-0.597413\pi$
$17$	−207499.	−0.602555	−0.301278	+	0.953536i	$0.597413\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	1.24189e6	0.925351	0.462675	−	0.886528i	$-0.346890\pi$
$23$	1.24189e6	0.925351	0.462675	+	0.886528i	$0.346890\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.28308e6	1.12451	0.562257	−	0.826963i	$-0.309933\pi$
$29$	4.28308e6	1.12451	0.562257	+	0.826963i	$0.309933\pi$
$30$	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
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	−2.51515e7	−1.39007	−0.695035	−	0.718975i	$-0.744611\pi$
$41$	−2.51515e7	−1.39007	−0.695035	+	0.718975i	$0.744611\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	−3.73010e7	−1.11501	−0.557507	−	0.830172i	$-0.688242\pi$
$47$	−3.73010e7	−1.11501	−0.557507	+	0.830172i	$0.688242\pi$
$48$	0	0
$49$	5.22402e7	1.29456
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.55155e7	0.444184	0.222092	−	0.975026i	$-0.428711\pi$
$53$	2.55155e7	0.444184	0.222092	+	0.975026i	$0.428711\pi$
$54$	0	0
$55$	−3.47665e7	−0.512305
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.96495e7	1.07063	0.535317	−	0.844651i	$-0.320192\pi$
$59$	9.96495e7	1.07063	0.535317	+	0.844651i	$0.320192\pi$
$60$	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
$63$	0	0
$64$	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
$69$	0	0
$70$	0	0
$71$	4.31522e7	0.201530	0.100765	−	0.994910i	$-0.467871\pi$
$71$	4.31522e7	0.201530	0.100765	+	0.994910i	$0.467871\pi$
$72$	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
$75$	0	0
$76$	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
$81$	0	0
$82$	0	0
$83$	6.01017e8	1.39007	0.695033	−	0.718978i	$-0.255390\pi$
$83$	6.01017e8	1.39007	0.695033	+	0.718978i	$0.255390\pi$
$84$	0	0
$85$	−1.29687e8	−0.269471
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.39917e8	−0.912162	−0.456081	−	0.889938i	$-0.650747\pi$
$89$	−5.39917e8	−0.912162	−0.456081	+	0.889938i	$0.650747\pi$
$90$	0	0
$91$	1.63369e9	2.49737
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	1.09286e9	1.04501	0.522503	−	0.852637i	$-0.324998\pi$
$101$	1.09286e9	1.04501	0.522503	+	0.852637i	$0.324998\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	−5.23881e8	−0.386372	−0.193186	−	0.981162i	$-0.561882\pi$
$107$	−5.23881e8	−0.386372	−0.193186	+	0.981162i	$0.561882\pi$
$108$	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
$111$	0	0
$112$	0	0
$113$	2.63801e9	1.52203	0.761016	−	0.648733i	$-0.224701\pi$
$113$	2.63801e9	1.52203	0.761016	+	0.648733i	$0.224701\pi$
$114$	0	0
$115$	7.76179e8	0.413829
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.99668e9	−0.912739
$120$	0	0
$121$	7.36341e8	0.312281
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	0	0
$131$	3.78884e9	1.12405	0.562024	−	0.827121i	$-0.310023\pi$
$131$	3.78884e9	1.12405	0.562024	+	0.827121i	$0.310023\pi$
$132$	0	0
$133$	7.72158e9	2.13980
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.15185e9	0.521879	0.260940	−	0.965355i	$-0.415968\pi$
$137$	2.15185e9	0.521879	0.260940	+	0.965355i	$0.415968\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	−9.44408e9	−1.88863
$144$	0	0
$145$	2.67692e9	0.502898
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.16582e9	1.02483	0.512416	−	0.858737i	$-0.328751\pi$
$149$	6.16582e9	1.02483	0.512416	+	0.858737i	$0.328751\pi$
$150$	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
$153$	0	0
$154$	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
$159$	0	0
$160$	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
$165$	0	0
$166$	0	0
$167$	1.56225e10	1.55427	0.777136	−	0.629332i	$-0.216672\pi$
$167$	1.56225e10	1.55427	0.777136	+	0.629332i	$0.216672\pi$
$168$	0	0
$169$	1.82198e10	1.71812
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.16058e10	−0.985069	−0.492535	−	0.870293i	$-0.663930\pi$
$173$	−1.16058e10	−0.985069	−0.492535	+	0.870293i	$0.663930\pi$
$174$	0	0
$175$	3.75882e9	0.302956
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.45556e10	1.78777	0.893886	−	0.448294i	$-0.147968\pi$
$179$	2.45556e10	1.78777	0.893886	+	0.448294i	$0.147968\pi$
$180$	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
$183$	0	0
$184$	0	0
$185$	−1.81160e9	−0.113708
$186$	0	0
$187$	1.15424e10	0.690256
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.41548e10	−1.85695	−0.928477	−	0.371390i	$-0.878881\pi$
$191$	−3.41548e10	−1.85695	−0.928477	+	0.371390i	$0.878881\pi$
$192$	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
$195$	0	0
$196$	0	0
$197$	−9.64109e9	−0.456067	−0.228033	−	0.973653i	$-0.573229\pi$
$197$	−9.64109e9	−0.456067	−0.228033	+	0.973653i	$0.573229\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	4.12142e10	1.70339
$204$	0	0
$205$	−1.57197e10	−0.621659
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	−1.25128e10	−0.399375
$216$	0	0
$217$	−3.45174e10	−1.05674
$218$	0	0
$219$	0	0
$220$	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
$225$	0	0
$226$	0	0
$227$	8.68117e9	0.217001	0.108501	−	0.994096i	$-0.465395\pi$
$227$	8.68117e9	0.217001	0.108501	+	0.994096i	$0.465395\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	7.42617e10	1.65068	0.825341	−	0.564634i	$-0.190983\pi$
$233$	7.42617e10	1.65068	0.825341	+	0.564634i	$0.190983\pi$
$234$	0	0
$235$	−2.33131e10	−0.498649
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.65577e10	0.328253	0.164126	−	0.986439i	$-0.447519\pi$
$239$	1.65577e10	0.328253	0.164126	+	0.986439i	$0.447519\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	3.26501e10	0.578945
$246$	0	0
$247$	1.36237e11	2.32894
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.32959e10	−0.370465	−0.185232	−	0.982695i	$-0.559304\pi$
$251$	−2.32959e10	−0.370465	−0.185232	+	0.982695i	$0.559304\pi$
$252$	0	0
$253$	−6.90815e10	−1.06003
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.57624e10	0.654350	0.327175	−	0.944964i	$-0.393903\pi$
$257$	4.57624e10	0.654350	0.327175	+	0.944964i	$0.393903\pi$
$258$	0	0
$259$	−2.78916e10	−0.385145
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.93911e10	−1.28099	−0.640496	−	0.767961i	$-0.721271\pi$
$263$	−9.93911e10	−1.28099	−0.640496	+	0.767961i	$0.721271\pi$
$264$	0	0
$265$	1.59472e10	0.198645
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.57475e11	−1.83369	−0.916845	−	0.399242i	$-0.869273\pi$
$269$	−1.57475e11	−1.83369	−0.916845	+	0.399242i	$0.869273\pi$
$270$	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
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	8.51877e10	0.815076	0.407538	−	0.913188i	$-0.366387\pi$
$281$	8.51877e10	0.815076	0.407538	+	0.913188i	$0.366387\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	−2.42022e11	−2.10565
$288$	0	0
$289$	−7.55318e10	−0.636927
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.31572e11	−1.04294	−0.521470	−	0.853270i	$-0.674616\pi$
$293$	−1.31572e11	−1.04294	−0.521470	+	0.853270i	$0.674616\pi$
$294$	0	0
$295$	6.22810e10	0.478802
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.10844e11	1.52560
$300$	0	0
$301$	−1.92648e11	−1.35274
$302$	0	0
$303$	0	0
$304$	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
$309$	0	0
$310$	0	0
$311$	7.10454e10	0.430640	0.215320	−	0.976544i	$-0.430921\pi$
$311$	7.10454e10	0.430640	0.215320	+	0.976544i	$0.430921\pi$
$312$	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
$315$	0	0
$316$	0	0
$317$	1.98163e11	1.10219	0.551094	−	0.834443i	$-0.314211\pi$
$317$	1.98163e11	1.10219	0.551094	+	0.834443i	$0.314211\pi$
$318$	0	0
$319$	−2.38252e11	−1.28818
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.66507e11	−0.851179
$324$	0	0
$325$	6.63192e10	0.329734
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	1.99539e11	0.799157
$342$	0	0
$343$	1.14379e11	0.446194
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.89987e11	−0.703464	−0.351732	−	0.936101i	$-0.614407\pi$
$347$	−1.89987e11	−0.703464	−0.351732	+	0.936101i	$0.614407\pi$
$348$	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
$351$	0	0
$352$	0	0
$353$	−5.15816e10	−0.176811	−0.0884054	−	0.996085i	$-0.528177\pi$
$353$	−5.15816e10	−0.176811	−0.0884054	+	0.996085i	$0.528177\pi$
$354$	0	0
$355$	2.69701e10	0.0901270
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.79131e11	0.569176	0.284588	−	0.958650i	$-0.408143\pi$
$359$	1.79131e11	0.569176	0.284588	+	0.958650i	$0.408143\pi$
$360$	0	0
$361$	3.21231e11	0.995485
$362$	0	0
$363$	0	0
$364$	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
$369$	0	0
$370$	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
$375$	0	0
$376$	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
$381$	0	0
$382$	0	0
$383$	4.78535e10	0.113637	0.0568185	−	0.998385i	$-0.481904\pi$
$383$	4.78535e10	0.113637	0.0568185	+	0.998385i	$0.481904\pi$
$384$	0	0
$385$	−3.34543e11	−0.776029
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.19528e11	−0.264665	−0.132332	−	0.991205i	$-0.542247\pi$
$389$	−1.19528e11	−0.264665	−0.132332	+	0.991205i	$0.542247\pi$
$390$	0	0
$391$	−2.57691e11	−0.557575
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	−5.62486e11	−1.08633	−0.543165	−	0.839626i	$-0.682774\pi$
$401$	−5.62486e11	−1.08633	−0.543165	+	0.839626i	$0.682774\pi$
$402$	0	0
$403$	−6.09012e11	−1.15015
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	9.58884e11	1.62178
$414$	0	0
$415$	3.75636e11	0.621657
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.54170e11	−0.561370	−0.280685	−	0.959800i	$-0.590562\pi$
$419$	−3.54170e11	−0.561370	−0.280685	+	0.959800i	$0.590562\pi$
$420$	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
$423$	0	0
$424$	0	0
$425$	−8.10545e10	−0.120511
$426$	0	0
$427$	1.92869e12	2.80761
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.26760e12	−1.76944	−0.884718	−	0.466127i	$-0.845649\pi$
$431$	−1.26760e12	−1.76944	−0.884718	+	0.466127i	$0.845649\pi$
$432$	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
$435$	0	0
$436$	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
$441$	0	0
$442$	0	0
$443$	−4.20444e11	−0.518671	−0.259335	−	0.965787i	$-0.583503\pi$
$443$	−4.20444e11	−0.518671	−0.259335	+	0.965787i	$0.583503\pi$
$444$	0	0
$445$	−3.37448e11	−0.407931
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.19773e11	−0.139075	−0.0695376	−	0.997579i	$-0.522152\pi$
$449$	−1.19773e11	−0.139075	−0.0695376	+	0.997579i	$0.522152\pi$
$450$	0	0
$451$	1.39909e12	1.59239
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	−2.09834e11	−0.216383	−0.108191	−	0.994130i	$-0.534506\pi$
$461$	−2.09834e11	−0.216383	−0.108191	+	0.994130i	$0.534506\pi$
$462$	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
$465$	0	0
$466$	0	0
$467$	8.28966e11	0.806512	0.403256	−	0.915087i	$-0.367878\pi$
$467$	8.28966e11	0.806512	0.403256	+	0.915087i	$0.367878\pi$
$468$	0	0
$469$	−7.79381e11	−0.743827
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.11366e12	1.02301
$474$	0	0
$475$	3.13455e11	0.282523
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.48710e11	0.563042	0.281521	−	0.959555i	$-0.409161\pi$
$479$	6.48710e11	0.563042	0.281521	+	0.959555i	$0.409161\pi$
$480$	0	0
$481$	−4.92110e11	−0.419188
$482$	0	0
$483$	0	0
$484$	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
$489$	0	0
$490$	0	0
$491$	−3.02135e11	−0.234603	−0.117302	−	0.993096i	$-0.537424\pi$
$491$	−3.02135e11	−0.234603	−0.117302	+	0.993096i	$0.537424\pi$
$492$	0	0
$493$	−8.88736e11	−0.677582
$494$	0	0
$495$	0	0
$496$	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
$501$	0	0
$502$	0	0
$503$	−1.69213e12	−1.17863	−0.589314	−	0.807904i	$-0.700602\pi$
$503$	−1.69213e12	−1.17863	−0.589314	+	0.807904i	$0.700602\pi$
$504$	0	0
$505$	6.83038e11	0.467341
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.10216e11	−0.468986	−0.234493	−	0.972118i	$-0.575343\pi$
$509$	−7.10216e11	−0.468986	−0.234493	+	0.972118i	$0.575343\pi$
$510$	0	0
$511$	−3.27956e12	−2.12776
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.31559e10	−0.0332981
$516$	0	0
$517$	2.07492e12	1.27730
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.88794e11	0.528483	0.264242	−	0.964456i	$-0.414878\pi$
$521$	8.88794e11	0.528483	0.264242	+	0.964456i	$0.414878\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	7.44327e11	0.420355
$528$	0	0
$529$	−2.58873e11	−0.143726
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.27015e12	−2.29177
$534$	0	0
$535$	−3.27426e11	−0.172791
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	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
$549$	0	0
$550$	0	0
$551$	3.43693e12	1.58851
$552$	0	0
$553$	2.70479e12	1.22990
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.92558e11	0.216825	0.108412	−	0.994106i	$-0.465423\pi$
$557$	4.92558e11	0.216825	0.108412	+	0.994106i	$0.465423\pi$
$558$	0	0
$559$	−3.39901e12	−1.47231
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.67786e12	−0.703832	−0.351916	−	0.936032i	$-0.614470\pi$
$563$	−1.67786e12	−0.703832	−0.351916	+	0.936032i	$0.614470\pi$
$564$	0	0
$565$	1.64876e12	0.680673
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.40646e12	−1.36238	−0.681190	−	0.732107i	$-0.738537\pi$
$569$	−3.40646e12	−1.36238	−0.681190	+	0.732107i	$0.738537\pi$
$570$	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
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	5.78333e12	2.10565
$582$	0	0
$583$	−1.41933e12	−0.508834
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.83858e12	0.986801	0.493401	−	0.869802i	$-0.335754\pi$
$587$	2.83858e12	0.986801	0.493401	+	0.869802i	$0.335754\pi$
$588$	0	0
$589$	−2.87847e12	−0.985470
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.08687e12	1.35720	0.678601	−	0.734507i	$-0.262586\pi$
$593$	4.08687e12	1.35720	0.678601	+	0.734507i	$0.262586\pi$
$594$	0	0
$595$	−1.24792e12	−0.408189
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.47524e12	−1.42035	−0.710175	−	0.704025i	$-0.751384\pi$
$599$	−4.47524e12	−1.42035	−0.710175	+	0.704025i	$0.751384\pi$
$600$	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
$603$	0	0
$604$	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
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	−1.82614e12	−0.507284	−0.253642	−	0.967298i	$-0.581628\pi$
$617$	−1.82614e12	−0.507284	−0.253642	+	0.967298i	$0.581628\pi$
$618$	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
$621$	0	0
$622$	0	0
$623$	−5.19539e12	−1.38173
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	2.08293e12	0.508385
$636$	0	0
$637$	8.86918e12	2.13430
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.69141e12	−0.629679	−0.314840	−	0.949145i	$-0.601951\pi$
$641$	−2.69141e12	−0.629679	−0.314840	+	0.949145i	$0.601951\pi$
$642$	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
$645$	0	0
$646$	0	0
$647$	4.86871e12	1.09231	0.546153	−	0.837685i	$-0.316092\pi$
$647$	4.86871e12	1.09231	0.546153	+	0.837685i	$0.316092\pi$
$648$	0	0
$649$	−5.54314e12	−1.22646
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.30610e12	−0.496329	−0.248164	−	0.968718i	$-0.579827\pi$
$653$	−2.30610e12	−0.496329	−0.248164	+	0.968718i	$0.579827\pi$
$654$	0	0
$655$	2.36802e12	0.502690
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.40069e12	−1.52858	−0.764289	−	0.644873i	$-0.776910\pi$
$659$	−7.40069e12	−1.52858	−0.764289	+	0.644873i	$0.776910\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	4.82599e12	0.956949
$666$	0	0
$667$	5.31909e12	1.04057
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	8.91166e12	1.63046	0.815229	−	0.579139i	$-0.196611\pi$
$677$	8.91166e12	1.63046	0.815229	+	0.579139i	$0.196611\pi$
$678$	0	0
$679$	4.07059e12	0.734926
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.26863e12	1.62976	0.814878	−	0.579633i	$-0.196804\pi$
$683$	9.26863e12	1.62976	0.814878	+	0.579633i	$0.196804\pi$
$684$	0	0
$685$	1.34491e12	0.233391
$686$	0	0
$687$	0	0
$688$	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
$693$	0	0
$694$	0	0
$695$	1.71929e11	0.0279523
$696$	0	0
$697$	5.21893e12	0.837594
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.77133e12	−0.746291	−0.373145	−	0.927773i	$-0.621721\pi$
$701$	−4.77133e12	−0.746291	−0.373145	+	0.927773i	$0.621721\pi$
$702$	0	0
$703$	−2.32594e12	−0.359169
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	−4.45480e12	−0.645543
$714$	0	0
$715$	−5.90255e12	−0.844622
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.50766e12	−0.210389	−0.105194	−	0.994452i	$-0.533546\pi$
$719$	−1.50766e12	−0.210389	−0.105194	+	0.994452i	$0.533546\pi$
$720$	0	0
$721$	−8.18394e11	−0.112786
$722$	0	0
$723$	0	0
$724$	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
$729$	0	0
$730$	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
$735$	0	0
$736$	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
$741$	0	0
$742$	0	0
$743$	−1.24339e13	−1.49678	−0.748388	−	0.663261i	$-0.769172\pi$
$743$	−1.24339e13	−1.49678	−0.748388	+	0.663261i	$0.769172\pi$
$744$	0	0
$745$	3.85364e12	0.458319
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	−1.12763e13	−1.21881	−0.609405	−	0.792859i	$-0.708592\pi$
$761$	−1.12763e13	−1.21881	−0.609405	+	0.792859i	$0.708592\pi$
$762$	0	0
$763$	−5.38303e12	−0.574998
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	3.89254e12	0.392126	0.196063	−	0.980591i	$-0.437184\pi$
$773$	3.89254e12	0.392126	0.196063	+	0.980591i	$0.437184\pi$
$774$	0	0
$775$	−1.40122e12	−0.139524
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.01827e13	−1.96364
$780$	0	0
$781$	−2.40040e12	−0.230862
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	2.53844e13	2.30554
$792$	0	0
$793$	3.40291e13	3.05577
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.58748e12	0.139363	0.0696813	−	0.997569i	$-0.477802\pi$
$797$	1.58748e12	0.139363	0.0696813	+	0.997569i	$0.477802\pi$
$798$	0	0
$799$	7.73994e12	0.671857
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.89586e13	1.60911
$804$	0	0
$805$	7.46883e12	0.626861
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.55705e12	0.374038	0.187019	−	0.982356i	$-0.440117\pi$
$809$	4.55705e12	0.374038	0.187019	+	0.982356i	$0.440117\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	2.58933e11	0.0205579
$816$	0	0
$817$	−1.60653e13	−1.26151
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.21734e13	0.935124	0.467562	−	0.883960i	$-0.345132\pi$
$821$	1.21734e13	0.935124	0.467562	+	0.883960i	$0.345132\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	9.15138e12	0.680318	0.340159	−	0.940368i	$-0.389519\pi$
$827$	9.15138e12	0.680318	0.340159	+	0.940368i	$0.389519\pi$
$828$	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
$831$	0	0
$832$	0	0
$833$	−1.08398e13	−0.780044
$834$	0	0
$835$	9.76408e12	0.695092
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.69274e11	−0.0605658	−0.0302829	−	0.999541i	$-0.509641\pi$
$839$	−8.69274e11	−0.0605658	−0.0302829	+	0.999541i	$0.509641\pi$
$840$	0	0
$841$	3.83759e12	0.264531
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.13873e13	0.768365
$846$	0	0
$847$	7.08549e12	0.473037
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	6.57679e12	0.416486	0.208243	−	0.978077i	$-0.433225\pi$
$857$	6.57679e12	0.416486	0.208243	+	0.978077i	$0.433225\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	−3.08737e13	−1.89470	−0.947350	−	0.320200i	$-0.896250\pi$
$863$	−3.08737e13	−1.89470	−0.947350	+	0.320200i	$0.896250\pi$
$864$	0	0
$865$	−7.25361e12	−0.440536
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.56359e13	−0.930110
$870$	0	0
$871$	−1.37511e13	−0.809573
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	1.64369e13	0.919241	0.459620	−	0.888116i	$-0.347986\pi$
$881$	1.64369e13	0.919241	0.459620	+	0.888116i	$0.347986\pi$
$882$	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
$885$	0	0
$886$	0	0
$887$	−1.61191e13	−0.874347	−0.437174	−	0.899377i	$-0.644020\pi$
$887$	−1.61191e13	−0.874347	−0.437174	+	0.899377i	$0.644020\pi$
$888$	0	0
$889$	3.20690e13	1.72198
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.99320e13	−1.57509
$894$	0	0
$895$	1.53473e13	0.799516
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.53639e13	−0.784483
$900$	0	0
$901$	−5.29446e12	−0.267645
$902$	0	0
$903$	0	0
$904$	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
$909$	0	0
$910$	0	0
$911$	2.12904e12	0.102412	0.0512061	−	0.998688i	$-0.483693\pi$
$911$	2.12904e12	0.102412	0.0512061	+	0.998688i	$0.483693\pi$
$912$	0	0
$913$	−3.34324e13	−1.59239
$914$	0	0
$915$	0	0
$916$	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
$921$	0	0
$922$	0	0
$923$	7.32625e12	0.332257
$924$	0	0
$925$	−1.13225e12	−0.0508517
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.74749e11	0.00769739	0.00384869	−	0.999993i	$-0.498775\pi$
$929$	1.74749e11	0.00769739	0.00384869	+	0.999993i	$0.498775\pi$
$930$	0	0
$931$	4.19199e13	1.82872
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	−2.13848e13	−0.889104	−0.444552	−	0.895753i	$-0.646637\pi$
$941$	−2.13848e13	−0.889104	−0.444552	+	0.895753i	$0.646637\pi$
$942$	0	0
$943$	−3.12353e13	−1.28630
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.58008e13	−1.85054	−0.925269	−	0.379312i	$-0.876161\pi$
$947$	−4.58008e13	−1.85054	−0.925269	+	0.379312i	$0.876161\pi$
$948$	0	0
$949$	−5.78634e13	−2.31583
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.57835e13	0.619847	0.309924	−	0.950761i	$-0.399696\pi$
$953$	1.57835e13	0.619847	0.309924	+	0.950761i	$0.399696\pi$
$954$	0	0
$955$	−2.13467e13	−0.830455
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.07063e13	0.790533
$960$	0	0
$961$	−1.35722e13	−0.513326
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	0	0
$971$	5.05363e12	0.182439	0.0912194	−	0.995831i	$-0.470924\pi$
$971$	5.05363e12	0.182439	0.0912194	+	0.995831i	$0.470924\pi$
$972$	0	0
$973$	2.64703e12	0.0946786
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.22726e13	1.13321	0.566603	−	0.823991i	$-0.308257\pi$
$977$	3.22726e13	1.13321	0.566603	+	0.823991i	$0.308257\pi$
$978$	0	0
$979$	3.00336e13	1.04493
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.11820e13	0.723561	0.361781	−	0.932263i	$-0.382169\pi$
$983$	2.11820e13	0.723561	0.361781	+	0.932263i	$0.382169\pi$
$984$	0	0
$985$	−6.02568e12	−0.203959
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.10.a.e.1.2		2
3.2	odd	2		20.10.a.b.1.2	✓	2
12.11	even	2		80.10.a.j.1.1		2
15.2	even	4		100.10.c.c.49.2		4
15.8	even	4		100.10.c.c.49.3		4
15.14	odd	2		100.10.a.c.1.1		2
24.5	odd	2		320.10.a.t.1.1		2
24.11	even	2		320.10.a.l.1.2		2
60.23	odd	4		400.10.c.l.49.2		4
60.47	odd	4		400.10.c.l.49.3		4
60.59	even	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	3.2	odd	2
80.10.a.j.1.1		2	12.11	even	2
100.10.a.c.1.1		2	15.14	odd	2
100.10.c.c.49.2		4	15.2	even	4
100.10.c.c.49.3		4	15.8	even	4
180.10.a.e.1.2		2	1.1	even	1	trivial
320.10.a.l.1.2		2	24.11	even	2
320.10.a.t.1.1		2	24.5	odd	2
400.10.a.l.1.2		2	60.59	even	2
400.10.c.l.49.2		4	60.23	odd	4
400.10.c.l.49.3		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	180.a (trivial)

\(f(q)\)	\(=\)	\(q+625.000 q^{5} +9622.57 q^{7} +O(q^{10})\)	\(q+625.000 q^{5} +9622.57 q^{7} -55626.3 q^{11} +169777. q^{13} -207499. q^{17} +802445. q^{19} +1.24189e6 q^{23} +390625. q^{25} +4.28308e6 q^{29} -3.58713e6 q^{31} +6.01410e6 q^{35} -2.89856e6 q^{37} -2.51515e7 q^{41} -2.00204e7 q^{43} -3.73010e7 q^{47} +5.22402e7 q^{49} +2.55155e7 q^{53} -3.47665e7 q^{55} +9.96495e7 q^{59} +2.00434e8 q^{61} +1.06111e8 q^{65} -8.09951e7 q^{67} +4.31522e7 q^{71} -3.40820e8 q^{73} -5.35268e8 q^{77} +2.81089e8 q^{79} +6.01017e8 q^{83} -1.29687e8 q^{85} -5.39917e8 q^{89} +1.63369e9 q^{91} +5.01528e8 q^{95} +4.23026e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1250 q^{5} - 380 q^{7}+O(q^{10}) \) 2 * q + 1250 * q^5 - 380 * q^7	\( 2 q + 1250 q^{5} - 380 q^{7} - 102720 q^{11} + 179140 q^{13} - 316020 q^{17} + 137272 q^{19} + 665460 q^{23} + 781250 q^{25} + 6893748 q^{29} + 291832 q^{31} - 237500 q^{35} + 11261380 q^{37} - 29773452 q^{41} - 11708180 q^{43} - 62493300 q^{47} + 111937914 q^{49} - 9417780 q^{53} - 64200000 q^{55} + 92930856 q^{59} + 195673924 q^{61} + 111962500 q^{65} - 219767420 q^{67} - 311207016 q^{71} - 99224060 q^{73} - 64210560 q^{77} + 542261776 q^{79} + 1256915700 q^{83} - 197512500 q^{85} + 462291852 q^{89} + 1540037768 q^{91} + 85795000 q^{95} + 1671716740 q^{97}+O(q^{100}) \) 2 * q + 1250 * q^5 - 380 * q^7 - 102720 * q^11 + 179140 * q^13 - 316020 * q^17 + 137272 * q^19 + 665460 * q^23 + 781250 * q^25 + 6893748 * q^29 + 291832 * q^31 - 237500 * q^35 + 11261380 * q^37 - 29773452 * q^41 - 11708180 * q^43 - 62493300 * q^47 + 111937914 * q^49 - 9417780 * q^53 - 64200000 * q^55 + 92930856 * q^59 + 195673924 * q^61 + 111962500 * q^65 - 219767420 * q^67 - 311207016 * q^71 - 99224060 * q^73 - 64210560 * q^77 + 542261776 * q^79 + 1256915700 * q^83 - 197512500 * q^85 + 462291852 * q^89 + 1540037768 * q^91 + 85795000 * q^95 + 1671716740 * q^97

Introduction

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