LMFDB - Embedded newform 80.20.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,20,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	80.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:	$183.053357245$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 214954323x^{2} - 341671644076x + 8077617181385444 $ x^4 - 2x^3 - 214954323x^2 - 341671644076*x + 8077617181385444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-9972.31$ of defining polynomial
Character	$\chi$	$=$	80.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+40661.2 q^{3} +1.95312e6 q^{5} +1.43021e8 q^{7} +4.91076e8 q^{9} +O(q^{10})$	$q+40661.2 q^{3} +1.95312e6 q^{5} +1.43021e8 q^{7} +4.91076e8 q^{9} +7.59820e9 q^{11} +5.72348e10 q^{13} +7.94165e10 q^{15} -8.94159e11 q^{17} +2.59184e12 q^{19} +5.81540e12 q^{21} +1.34802e13 q^{23} +3.81470e12 q^{25} -2.72913e13 q^{27} +3.06632e12 q^{29} -9.67405e13 q^{31} +3.08952e14 q^{33} +2.79337e14 q^{35} +8.83811e14 q^{37} +2.32724e15 q^{39} -4.21479e14 q^{41} -5.11756e15 q^{43} +9.59132e14 q^{45} +3.55281e15 q^{47} +9.05602e15 q^{49} -3.63576e16 q^{51} +2.49384e16 q^{53} +1.48402e16 q^{55} +1.05387e17 q^{57} -7.38757e16 q^{59} -1.08358e17 q^{61} +7.02340e16 q^{63} +1.11787e17 q^{65} +3.35811e16 q^{67} +5.48122e17 q^{69} -9.84049e16 q^{71} +3.43393e17 q^{73} +1.55110e17 q^{75} +1.08670e18 q^{77} +1.27181e18 q^{79} -1.68045e18 q^{81} -7.11817e16 q^{83} -1.74641e18 q^{85} +1.24680e17 q^{87} +4.93863e17 q^{89} +8.18576e18 q^{91} -3.93359e18 q^{93} +5.06218e18 q^{95} +4.61631e18 q^{97} +3.73129e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9}+O(q^{10}) $ 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9	$ 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9} + 2334973920 q^{11} + 57423224120 q^{13} + 6015625000 q^{15} - 465961763160 q^{17} + 1624818160624 q^{19} + 5400395639744 q^{21} + 10101669670680 q^{23} + 15258789062500 q^{25} - 56910507730480 q^{27} - 21802908180264 q^{29} + 69517593805936 q^{31} - 204641262341280 q^{33} - 289496171875000 q^{35} + 21\!\cdots\!60 q^{37}+ \cdots + 27\!\cdots\!80 q^{99}+O(q^{100}) $ 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9 + 2334973920 * q^11 + 57423224120 * q^13 + 6015625000 * q^15 - 465961763160 * q^17 + 1624818160624 * q^19 + 5400395639744 * q^21 + 10101669670680 * q^23 + 15258789062500 * q^25 - 56910507730480 * q^27 - 21802908180264 * q^29 + 69517593805936 * q^31 - 204641262341280 * q^33 - 289496171875000 * q^35 + 2156700568331960 * q^37 - 2385459176165456 * q^39 + 1367657291239944 * q^41 - 2895469359165560 * q^43 + 4359109601562500 * q^45 - 1740670005708600 * q^47 + 53534973320692932 * q^49 - 62580957868768368 * q^51 + 84253756978305240 * q^53 + 4560495937500000 * q^55 + 188957630874753440 * q^57 - 152838143056415952 * q^59 + 211425306591350168 * q^61 - 417796572994057400 * q^63 + 112154734609375000 * q^65 - 656919981274544360 * q^67 + 93524018475363648 * q^69 - 883122322400344752 * q^71 + 325226727237880520 * q^73 + 11749267578125000 * q^75 - 620955288586656480 * q^77 + 850256855296398112 * q^79 - 3380709519213288764 * q^81 + 3602925283549230600 * q^83 - 910081568671875000 * q^85 + 8179923434985534000 * q^87 + 418548299964421224 * q^89 + 10309921756152683504 * q^91 - 14102167255291233280 * q^93 + 3173472969968750000 * q^95 - 12135517460163032920 * q^97 + 27941442307774944480 * 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
$3$	40661.2	1.19269	0.596347	−	0.802727i	$-0.296618\pi$
$3$	40661.2	1.19269	0.596347	+	0.802727i	$0.296618\pi$
$4$	0	0
$5$	1.95312e6	0.447214
$5$	1.95312e6	0.447214
$6$	0	0
$7$	1.43021e8	1.33958	0.669788	−	0.742552i	$-0.266385\pi$
$7$	1.43021e8	1.33958	0.669788	+	0.742552i	$0.266385\pi$
$8$	0	0
$9$	4.91076e8	0.422517
$10$	0	0
$11$	7.59820e9	0.971583	0.485792	−	0.874075i	$-0.338531\pi$
$11$	7.59820e9	0.971583	0.485792	+	0.874075i	$0.338531\pi$
$12$	0	0
$13$	5.72348e10	1.49692	0.748460	−	0.663180i	$-0.230794\pi$
$13$	5.72348e10	1.49692	0.748460	+	0.663180i	$0.230794\pi$
$14$	0	0
$15$	7.94165e10	0.533389
$16$	0	0
$17$	−8.94159e11	−1.82873	−0.914366	−	0.404888i	$-0.867311\pi$
$17$	−8.94159e11	−1.82873	−0.914366	+	0.404888i	$0.867311\pi$
$18$	0	0
$19$	2.59184e12	1.84267	0.921337	−	0.388764i	$-0.127098\pi$
$19$	2.59184e12	1.84267	0.921337	+	0.388764i	$0.127098\pi$
$20$	0	0
$21$	5.81540e12	1.59770
$22$	0	0
$23$	1.34802e13	1.56056	0.780282	−	0.625427i	$-0.215075\pi$
$23$	1.34802e13	1.56056	0.780282	+	0.625427i	$0.215075\pi$
$24$	0	0
$25$	3.81470e12	0.200000
$26$	0	0
$27$	−2.72913e13	−0.688760
$28$	0	0
$29$	3.06632e12	0.0392497	0.0196249	−	0.999807i	$-0.493753\pi$
$29$	3.06632e12	0.0392497	0.0196249	+	0.999807i	$0.493753\pi$
$30$	0	0
$31$	−9.67405e13	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$31$	−9.67405e13	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$32$	0	0
$33$	3.08952e14	1.15880
$34$	0	0
$35$	2.79337e14	0.599077
$36$	0	0
$37$	8.83811e14	1.11800	0.559002	−	0.829167i	$-0.311185\pi$
$37$	8.83811e14	1.11800	0.559002	+	0.829167i	$0.311185\pi$
$38$	0	0
$39$	2.32724e15	1.78537
$40$	0	0
$41$	−4.21479e14	−0.201062	−0.100531	−	0.994934i	$-0.532054\pi$
$41$	−4.21479e14	−0.201062	−0.100531	+	0.994934i	$0.532054\pi$
$42$	0	0
$43$	−5.11756e15	−1.55279	−0.776394	−	0.630247i	$-0.782953\pi$
$43$	−5.11756e15	−1.55279	−0.776394	+	0.630247i	$0.782953\pi$
$44$	0	0
$45$	9.59132e14	0.188956
$46$	0	0
$47$	3.55281e15	0.463065	0.231533	−	0.972827i	$-0.425626\pi$
$47$	3.55281e15	0.463065	0.231533	+	0.972827i	$0.425626\pi$
$48$	0	0
$49$	9.05602e15	0.794464
$50$	0	0
$51$	−3.63576e16	−2.18112
$52$	0	0
$53$	2.49384e16	1.03812	0.519060	−	0.854738i	$-0.326282\pi$
$53$	2.49384e16	1.03812	0.519060	+	0.854738i	$0.326282\pi$
$54$	0	0
$55$	1.48402e16	0.434505
$56$	0	0
$57$	1.05387e17	2.19775
$58$	0	0
$59$	−7.38757e16	−1.11022	−0.555108	−	0.831778i	$-0.687323\pi$
$59$	−7.38757e16	−1.11022	−0.555108	+	0.831778i	$0.687323\pi$
$60$	0	0
$61$	−1.08358e17	−1.18639	−0.593194	−	0.805059i	$-0.702133\pi$
$61$	−1.08358e17	−1.18639	−0.593194	+	0.805059i	$0.702133\pi$
$62$	0	0
$63$	7.02340e16	0.565994
$64$	0	0
$65$	1.11787e17	0.669443
$66$	0	0
$67$	3.35811e16	0.150794	0.0753970	−	0.997154i	$-0.475978\pi$
$67$	3.35811e16	0.150794	0.0753970	+	0.997154i	$0.475978\pi$
$68$	0	0
$69$	5.48122e17	1.86127
$70$	0	0
$71$	−9.84049e16	−0.254720	−0.127360	−	0.991857i	$-0.540650\pi$
$71$	−9.84049e16	−0.254720	−0.127360	+	0.991857i	$0.540650\pi$
$72$	0	0
$73$	3.43393e17	0.682691	0.341346	−	0.939938i	$-0.389117\pi$
$73$	3.43393e17	0.682691	0.341346	+	0.939938i	$0.389117\pi$
$74$	0	0
$75$	1.55110e17	0.238539
$76$	0	0
$77$	1.08670e18	1.30151
$78$	0	0
$79$	1.27181e18	1.19389	0.596943	−	0.802283i	$-0.296382\pi$
$79$	1.27181e18	1.19389	0.596943	+	0.802283i	$0.296382\pi$
$80$	0	0
$81$	−1.68045e18	−1.24400
$82$	0	0
$83$	−7.11817e16	−0.0417952	−0.0208976	−	0.999782i	$-0.506652\pi$
$83$	−7.11817e16	−0.0417952	−0.0208976	+	0.999782i	$0.506652\pi$
$84$	0	0
$85$	−1.74641e18	−0.817834
$86$	0	0
$87$	1.24680e17	0.0468129
$88$	0	0
$89$	4.93863e17	0.149417	0.0747087	−	0.997205i	$-0.476197\pi$
$89$	4.93863e17	0.149417	0.0747087	+	0.997205i	$0.476197\pi$
$90$	0	0
$91$	8.18576e18	2.00524
$92$	0	0
$93$	−3.93359e18	−0.783793
$94$	0	0
$95$	5.06218e18	0.824069
$96$	0	0
$97$	4.61631e18	0.616544	0.308272	−	0.951298i	$-0.400249\pi$
$97$	4.61631e18	0.616544	0.308272	+	0.951298i	$0.400249\pi$
$98$	0	0
$99$	3.73129e18	0.410511
$100$	0	0
$101$	−8.21992e18	−0.747850	−0.373925	−	0.927459i	$-0.621988\pi$
$101$	−8.21992e18	−0.747850	−0.373925	+	0.927459i	$0.621988\pi$
$102$	0	0
$103$	−1.41223e19	−1.06648	−0.533240	−	0.845964i	$-0.679026\pi$
$103$	−1.41223e19	−1.06648	−0.533240	+	0.845964i	$0.679026\pi$
$104$	0	0
$105$	1.13582e19	0.714515
$106$	0	0
$107$	−1.78360e19	−0.937891	−0.468945	−	0.883227i	$-0.655366\pi$
$107$	−1.78360e19	−0.937891	−0.468945	+	0.883227i	$0.655366\pi$
$108$	0	0
$109$	1.75899e19	0.775731	0.387866	−	0.921716i	$-0.373212\pi$
$109$	1.75899e19	0.775731	0.387866	+	0.921716i	$0.373212\pi$
$110$	0	0
$111$	3.59369e19	1.33344
$112$	0	0
$113$	−3.67416e19	−1.15057	−0.575286	−	0.817953i	$-0.695109\pi$
$113$	−3.67416e19	−1.15057	−0.575286	+	0.817953i	$0.695109\pi$
$114$	0	0
$115$	2.63285e19	0.697906
$116$	0	0
$117$	2.81066e19	0.632475
$118$	0	0
$119$	−1.27883e20	−2.44973
$120$	0	0
$121$	−3.42649e18	−0.0560259
$122$	0	0
$123$	−1.71378e19	−0.239805
$124$	0	0
$125$	7.45058e18	0.0894427
$126$	0	0
$127$	−7.93443e19	−0.819182	−0.409591	−	0.912269i	$-0.634329\pi$
$127$	−7.93443e19	−0.819182	−0.409591	+	0.912269i	$0.634329\pi$
$128$	0	0
$129$	−2.08086e20	−1.85200
$130$	0	0
$131$	−2.34313e20	−1.80186	−0.900928	−	0.433969i	$-0.857113\pi$
$131$	−2.34313e20	−1.80186	−0.900928	+	0.433969i	$0.857113\pi$
$132$	0	0
$133$	3.70686e20	2.46840
$134$	0	0
$135$	−5.33032e19	−0.308023
$136$	0	0
$137$	−2.31382e20	−1.16274	−0.581372	−	0.813638i	$-0.697484\pi$
$137$	−2.31382e20	−1.16274	−0.581372	+	0.813638i	$0.697484\pi$
$138$	0	0
$139$	8.73285e19	0.382398	0.191199	−	0.981551i	$-0.438762\pi$
$139$	8.73285e19	0.382398	0.191199	+	0.981551i	$0.438762\pi$
$140$	0	0
$141$	1.44462e20	0.552295
$142$	0	0
$143$	4.34881e20	1.45438
$144$	0	0
$145$	5.98891e18	0.0175530
$146$	0	0
$147$	3.68229e20	0.947552
$148$	0	0
$149$	−5.94432e19	−0.134534	−0.0672671	−	0.997735i	$-0.521428\pi$
$149$	−5.94432e19	−0.134534	−0.0672671	+	0.997735i	$0.521428\pi$
$150$	0	0
$151$	7.44375e20	1.48426	0.742132	−	0.670254i	$-0.233815\pi$
$151$	7.44375e20	1.48426	0.742132	+	0.670254i	$0.233815\pi$
$152$	0	0
$153$	−4.39100e20	−0.772671
$154$	0	0
$155$	−1.88946e20	−0.293892
$156$	0	0
$157$	−5.76800e20	−0.794288	−0.397144	−	0.917756i	$-0.629999\pi$
$157$	−5.76800e20	−0.794288	−0.397144	+	0.917756i	$0.629999\pi$
$158$	0	0
$159$	1.01403e21	1.23816
$160$	0	0
$161$	1.92795e21	2.09050
$162$	0	0
$163$	2.49820e20	0.240904	0.120452	−	0.992719i	$-0.461566\pi$
$163$	2.49820e20	0.240904	0.120452	+	0.992719i	$0.461566\pi$
$164$	0	0
$165$	6.03422e20	0.518232
$166$	0	0
$167$	3.82581e20	0.293033	0.146517	−	0.989208i	$-0.453194\pi$
$167$	3.82581e20	0.293033	0.146517	+	0.989208i	$0.453194\pi$
$168$	0	0
$169$	1.81391e21	1.24077
$170$	0	0
$171$	1.27279e21	0.778562
$172$	0	0
$173$	−1.43264e20	−0.0784694	−0.0392347	−	0.999230i	$-0.512492\pi$
$173$	−1.43264e20	−0.0784694	−0.0392347	+	0.999230i	$0.512492\pi$
$174$	0	0
$175$	5.45581e20	0.267915
$176$	0	0
$177$	−3.00388e21	−1.32415
$178$	0	0
$179$	−2.24877e20	−0.0890925	−0.0445463	−	0.999007i	$-0.514184\pi$
$179$	−2.24877e20	−0.0890925	−0.0445463	+	0.999007i	$0.514184\pi$
$180$	0	0
$181$	1.14552e21	0.408372	0.204186	−	0.978932i	$-0.434545\pi$
$181$	1.14552e21	0.408372	0.204186	+	0.978932i	$0.434545\pi$
$182$	0	0
$183$	−4.40596e21	−1.41500
$184$	0	0
$185$	1.72619e21	0.499986
$186$	0	0
$187$	−6.79400e21	−1.77677
$188$	0	0
$189$	−3.90321e21	−0.922646
$190$	0	0
$191$	7.25070e21	1.55083	0.775413	−	0.631454i	$-0.217542\pi$
$191$	7.25070e21	1.55083	0.775413	+	0.631454i	$0.217542\pi$
$192$	0	0
$193$	8.34254e21	1.61623	0.808116	−	0.589023i	$-0.200487\pi$
$193$	8.34254e21	1.61623	0.808116	+	0.589023i	$0.200487\pi$
$194$	0	0
$195$	4.54539e21	0.798440
$196$	0	0
$197$	1.18539e22	1.88986	0.944932	−	0.327268i	$-0.106128\pi$
$197$	1.18539e22	1.88986	0.944932	+	0.327268i	$0.106128\pi$
$198$	0	0
$199$	−2.11398e21	−0.306194	−0.153097	−	0.988211i	$-0.548925\pi$
$199$	−2.11398e21	−0.306194	−0.153097	+	0.988211i	$0.548925\pi$
$200$	0	0
$201$	1.36545e21	0.179851
$202$	0	0
$203$	4.38547e20	0.0525780
$204$	0	0
$205$	−8.23201e20	−0.0899174
$206$	0	0
$207$	6.61980e21	0.659366
$208$	0	0
$209$	1.96933e22	1.79031
$210$	0	0
$211$	7.27374e21	0.604052	0.302026	−	0.953300i	$-0.402337\pi$
$211$	7.27374e21	0.604052	0.302026	+	0.953300i	$0.402337\pi$
$212$	0	0
$213$	−4.00127e21	−0.303802
$214$	0	0
$215$	−9.99523e21	−0.694428
$216$	0	0
$217$	−1.38359e22	−0.880318
$218$	0	0
$219$	1.39628e22	0.814241
$220$	0	0
$221$	−5.11771e22	−2.73747
$222$	0	0
$223$	−1.76340e21	−0.0865874	−0.0432937	−	0.999062i	$-0.513785\pi$
$223$	−1.76340e21	−0.0865874	−0.0432937	+	0.999062i	$0.513785\pi$
$224$	0	0
$225$	1.87331e21	0.0845035
$226$	0	0
$227$	2.19860e22	0.911801	0.455900	−	0.890031i	$-0.349317\pi$
$227$	2.19860e22	0.911801	0.455900	+	0.890031i	$0.349317\pi$
$228$	0	0
$229$	−1.21123e22	−0.462157	−0.231078	−	0.972935i	$-0.574225\pi$
$229$	−1.21123e22	−0.462157	−0.231078	+	0.972935i	$0.574225\pi$
$230$	0	0
$231$	4.41865e22	1.55230
$232$	0	0
$233$	3.16405e22	1.02415	0.512073	−	0.858942i	$-0.328878\pi$
$233$	3.16405e22	1.02415	0.512073	+	0.858942i	$0.328878\pi$
$234$	0	0
$235$	6.93908e21	0.207089
$236$	0	0
$237$	5.17132e22	1.42394
$238$	0	0
$239$	1.45831e22	0.370740	0.185370	−	0.982669i	$-0.440652\pi$
$239$	1.45831e22	0.370740	0.185370	+	0.982669i	$0.440652\pi$
$240$	0	0
$241$	6.91543e22	1.62427	0.812133	−	0.583472i	$-0.198306\pi$
$241$	6.91543e22	1.62427	0.812133	+	0.583472i	$0.198306\pi$
$242$	0	0
$243$	−3.66098e22	−0.794947
$244$	0	0
$245$	1.76875e22	0.355295
$246$	0	0
$247$	1.48343e23	2.75834
$248$	0	0
$249$	−2.89434e21	−0.0498489
$250$	0	0
$251$	−1.61464e22	−0.257736	−0.128868	−	0.991662i	$-0.541134\pi$
$251$	−1.61464e22	−0.257736	−0.128868	+	0.991662i	$0.541134\pi$
$252$	0	0
$253$	1.02425e23	1.51622
$254$	0	0
$255$	−7.10110e22	−0.975425
$256$	0	0
$257$	2.32362e22	0.296347	0.148174	−	0.988961i	$-0.452661\pi$
$257$	2.32362e22	0.296347	0.148174	+	0.988961i	$0.452661\pi$
$258$	0	0
$259$	1.26403e23	1.49765
$260$	0	0
$261$	1.50580e21	0.0165837
$262$	0	0
$263$	−1.43991e22	−0.147488	−0.0737439	−	0.997277i	$-0.523495\pi$
$263$	−1.43991e22	−0.147488	−0.0737439	+	0.997277i	$0.523495\pi$
$264$	0	0
$265$	4.87078e22	0.464262
$266$	0	0
$267$	2.00811e22	0.178209
$268$	0	0
$269$	7.18475e22	0.593971	0.296985	−	0.954882i	$-0.404019\pi$
$269$	7.18475e22	0.593971	0.296985	+	0.954882i	$0.404019\pi$
$270$	0	0
$271$	−1.09847e22	−0.0846406	−0.0423203	−	0.999104i	$-0.513475\pi$
$271$	−1.09847e22	−0.0846406	−0.0423203	+	0.999104i	$0.513475\pi$
$272$	0	0
$273$	3.32843e23	2.39163
$274$	0	0
$275$	2.89848e22	0.194317
$276$	0	0
$277$	−2.09588e23	−1.31162	−0.655812	−	0.754924i	$-0.727674\pi$
$277$	−2.09588e23	−1.31162	−0.655812	+	0.754924i	$0.727674\pi$
$278$	0	0
$279$	−4.75069e22	−0.277662
$280$	0	0
$281$	−3.34345e23	−1.82593	−0.912966	−	0.408035i	$-0.866214\pi$
$281$	−3.34345e23	−1.82593	−0.912966	+	0.408035i	$0.866214\pi$
$282$	0	0
$283$	−2.06812e23	−1.05586	−0.527928	−	0.849289i	$-0.677031\pi$
$283$	−2.06812e23	−1.05586	−0.527928	+	0.849289i	$0.677031\pi$
$284$	0	0
$285$	2.05835e23	0.982862
$286$	0	0
$287$	−6.02802e22	−0.269337
$288$	0	0
$289$	5.60449e23	2.34426
$290$	0	0
$291$	1.87705e23	0.735348
$292$	0	0
$293$	−1.67957e23	−0.616532	−0.308266	−	0.951300i	$-0.599749\pi$
$293$	−1.67957e23	−0.616532	−0.308266	+	0.951300i	$0.599749\pi$
$294$	0	0
$295$	−1.44289e23	−0.496504
$296$	0	0
$297$	−2.07364e23	−0.669187
$298$	0	0
$299$	7.71537e23	2.33604
$300$	0	0
$301$	−7.31916e23	−2.08008
$302$	0	0
$303$	−3.34232e23	−0.891956
$304$	0	0
$305$	−2.11636e23	−0.530569
$306$	0	0
$307$	−7.55761e22	−0.178062	−0.0890308	−	0.996029i	$-0.528377\pi$
$307$	−7.55761e22	−0.178062	−0.0890308	+	0.996029i	$0.528377\pi$
$308$	0	0
$309$	−5.74232e23	−1.27198
$310$	0	0
$311$	3.24572e23	0.676219	0.338110	−	0.941107i	$-0.390213\pi$
$311$	3.24572e23	0.676219	0.338110	+	0.941107i	$0.390213\pi$
$312$	0	0
$313$	1.11357e22	0.0218296	0.0109148	−	0.999940i	$-0.496526\pi$
$313$	1.11357e22	0.0218296	0.0109148	+	0.999940i	$0.496526\pi$
$314$	0	0
$315$	1.37176e23	0.253120
$316$	0	0
$317$	6.02050e22	0.104609	0.0523045	−	0.998631i	$-0.483343\pi$
$317$	6.02050e22	0.104609	0.0523045	+	0.998631i	$0.483343\pi$
$318$	0	0
$319$	2.32985e22	0.0381344
$320$	0	0
$321$	−7.25235e23	−1.11862
$322$	0	0
$323$	−2.31752e24	−3.36976
$324$	0	0
$325$	2.18334e23	0.299384
$326$	0	0
$327$	7.15227e23	0.925209
$328$	0	0
$329$	5.08125e23	0.620311
$330$	0	0
$331$	1.62352e24	1.87108	0.935539	−	0.353222i	$-0.114914\pi$
$331$	1.62352e24	1.87108	0.935539	+	0.353222i	$0.114914\pi$
$332$	0	0
$333$	4.34018e23	0.472376
$334$	0	0
$335$	6.55881e22	0.0674372
$336$	0	0
$337$	1.33817e24	1.30025	0.650127	−	0.759826i	$-0.274716\pi$
$337$	1.33817e24	1.30025	0.650127	+	0.759826i	$0.274716\pi$
$338$	0	0
$339$	−1.49396e24	−1.37228
$340$	0	0
$341$	−7.35054e23	−0.638488
$342$	0	0
$343$	−3.35080e23	−0.275331
$344$	0	0
$345$	1.07055e24	0.832387
$346$	0	0
$347$	−1.32060e24	−0.971941	−0.485970	−	0.873975i	$-0.661534\pi$
$347$	−1.32060e24	−0.971941	−0.485970	+	0.873975i	$0.661534\pi$
$348$	0	0
$349$	−7.12675e23	−0.496649	−0.248325	−	0.968677i	$-0.579880\pi$
$349$	−7.12675e23	−0.496649	−0.248325	+	0.968677i	$0.579880\pi$
$350$	0	0
$351$	−1.56201e24	−1.03102
$352$	0	0
$353$	−6.62780e23	−0.414486	−0.207243	−	0.978290i	$-0.566449\pi$
$353$	−6.62780e23	−0.414486	−0.207243	+	0.978290i	$0.566449\pi$
$354$	0	0
$355$	−1.92197e23	−0.113914
$356$	0	0
$357$	−5.19989e24	−2.92177
$358$	0	0
$359$	−3.98302e23	−0.212234	−0.106117	−	0.994354i	$-0.533842\pi$
$359$	−3.98302e23	−0.212234	−0.106117	+	0.994354i	$0.533842\pi$
$360$	0	0
$361$	4.73921e24	2.39545
$362$	0	0
$363$	−1.39325e23	−0.0668217
$364$	0	0
$365$	6.70689e23	0.305309
$366$	0	0
$367$	−7.27399e23	−0.314373	−0.157186	−	0.987569i	$-0.550242\pi$
$367$	−7.27399e23	−0.314373	−0.157186	+	0.987569i	$0.550242\pi$
$368$	0	0
$369$	−2.06978e23	−0.0849520
$370$	0	0
$371$	3.56671e24	1.39064
$372$	0	0
$373$	6.94139e23	0.257165	0.128583	−	0.991699i	$-0.458957\pi$
$373$	6.94139e23	0.257165	0.128583	+	0.991699i	$0.458957\pi$
$374$	0	0
$375$	3.02950e23	0.106678
$376$	0	0
$377$	1.75500e23	0.0587537
$378$	0	0
$379$	2.54877e24	0.811444	0.405722	−	0.913997i	$-0.367020\pi$
$379$	2.54877e24	0.811444	0.405722	+	0.913997i	$0.367020\pi$
$380$	0	0
$381$	−3.22624e24	−0.977033
$382$	0	0
$383$	−3.44993e24	−0.994082	−0.497041	−	0.867727i	$-0.665580\pi$
$383$	−3.44993e24	−0.994082	−0.497041	+	0.867727i	$0.665580\pi$
$384$	0	0
$385$	2.12246e24	0.582053
$386$	0	0
$387$	−2.51311e24	−0.656080
$388$	0	0
$389$	3.38722e24	0.842021	0.421010	−	0.907056i	$-0.361676\pi$
$389$	3.38722e24	0.842021	0.421010	+	0.907056i	$0.361676\pi$
$390$	0	0
$391$	−1.20534e25	−2.85386
$392$	0	0
$393$	−9.52748e24	−2.14906
$394$	0	0
$395$	2.48399e24	0.533922
$396$	0	0
$397$	−7.37327e24	−1.51060	−0.755302	−	0.655377i	$-0.772510\pi$
$397$	−7.37327e24	−1.51060	−0.755302	+	0.655377i	$0.772510\pi$
$398$	0	0
$399$	1.50726e25	2.94405
$400$	0	0
$401$	−8.34368e24	−1.55413	−0.777063	−	0.629423i	$-0.783291\pi$
$401$	−8.34368e24	−1.55413	−0.777063	+	0.629423i	$0.783291\pi$
$402$	0	0
$403$	−5.53693e24	−0.983719
$404$	0	0
$405$	−3.28214e24	−0.556332
$406$	0	0
$407$	6.71537e24	1.08623
$408$	0	0
$409$	1.25338e24	0.193514	0.0967568	−	0.995308i	$-0.469153\pi$
$409$	1.25338e24	0.193514	0.0967568	+	0.995308i	$0.469153\pi$
$410$	0	0
$411$	−9.40827e24	−1.38680
$412$	0	0
$413$	−1.05658e25	−1.48722
$414$	0	0
$415$	−1.39027e23	−0.0186914
$416$	0	0
$417$	3.55089e24	0.456083
$418$	0	0
$419$	−7.47362e24	−0.917271	−0.458636	−	0.888624i	$-0.651662\pi$
$419$	−7.47362e24	−0.917271	−0.458636	+	0.888624i	$0.651662\pi$
$420$	0	0
$421$	−3.56314e23	−0.0417978	−0.0208989	−	0.999782i	$-0.506653\pi$
$421$	−3.56314e23	−0.0417978	−0.0208989	+	0.999782i	$0.506653\pi$
$422$	0	0
$423$	1.74470e24	0.195653
$424$	0	0
$425$	−3.41095e24	−0.365747
$426$	0	0
$427$	−1.54974e25	−1.58926
$428$	0	0
$429$	1.76828e25	1.73463
$430$	0	0
$431$	3.32837e24	0.312390	0.156195	−	0.987726i	$-0.450077\pi$
$431$	3.32837e24	0.312390	0.156195	+	0.987726i	$0.450077\pi$
$432$	0	0
$433$	−1.56801e25	−1.40836	−0.704179	−	0.710023i	$-0.748685\pi$
$433$	−1.56801e25	−1.40836	−0.704179	+	0.710023i	$0.748685\pi$
$434$	0	0
$435$	2.43517e23	0.0209354
$436$	0	0
$437$	3.49385e25	2.87561
$438$	0	0
$439$	−1.09931e25	−0.866380	−0.433190	−	0.901303i	$-0.642612\pi$
$439$	−1.09931e25	−0.866380	−0.433190	+	0.901303i	$0.642612\pi$
$440$	0	0
$441$	4.44719e24	0.335675
$442$	0	0
$443$	2.52063e25	1.82253	0.911263	−	0.411824i	$-0.135108\pi$
$443$	2.52063e25	1.82253	0.911263	+	0.411824i	$0.135108\pi$
$444$	0	0
$445$	9.64575e23	0.0668215
$446$	0	0
$447$	−2.41703e24	−0.160458
$448$	0	0
$449$	1.29559e25	0.824381	0.412190	−	0.911098i	$-0.364764\pi$
$449$	1.29559e25	0.824381	0.412190	+	0.911098i	$0.364764\pi$
$450$	0	0
$451$	−3.20248e24	−0.195348
$452$	0	0
$453$	3.02672e25	1.77027
$454$	0	0
$455$	1.59878e25	0.896770
$456$	0	0
$457$	−1.20999e24	−0.0650994	−0.0325497	−	0.999470i	$-0.510363\pi$
$457$	−1.20999e24	−0.0650994	−0.0325497	+	0.999470i	$0.510363\pi$
$458$	0	0
$459$	2.44027e25	1.25956
$460$	0	0
$461$	1.83172e24	0.0907195	0.0453598	−	0.998971i	$-0.485557\pi$
$461$	1.83172e24	0.0907195	0.0453598	+	0.998971i	$0.485557\pi$
$462$	0	0
$463$	−1.28857e25	−0.612477	−0.306238	−	0.951955i	$-0.599070\pi$
$463$	−1.28857e25	−0.612477	−0.306238	+	0.951955i	$0.599070\pi$
$464$	0	0
$465$	−7.68279e24	−0.350523
$466$	0	0
$467$	1.24784e25	0.546574	0.273287	−	0.961933i	$-0.411889\pi$
$467$	1.24784e25	0.546574	0.273287	+	0.961933i	$0.411889\pi$
$468$	0	0
$469$	4.80279e24	0.202000
$470$	0	0
$471$	−2.34534e25	−0.947343
$472$	0	0
$473$	−3.88842e25	−1.50866
$474$	0	0
$475$	9.88708e24	0.368535
$476$	0	0
$477$	1.22466e25	0.438624
$478$	0	0
$479$	3.19995e25	1.10143	0.550714	−	0.834694i	$-0.314356\pi$
$479$	3.19995e25	1.10143	0.550714	+	0.834694i	$0.314356\pi$
$480$	0	0
$481$	5.05848e25	1.67356
$482$	0	0
$483$	7.83927e25	2.49332
$484$	0	0
$485$	9.01624e24	0.275727
$486$	0	0
$487$	−1.52720e25	−0.449128	−0.224564	−	0.974459i	$-0.572096\pi$
$487$	−1.52720e25	−0.449128	−0.224564	+	0.974459i	$0.572096\pi$
$488$	0	0
$489$	1.01580e25	0.287325
$490$	0	0
$491$	−1.74071e25	−0.473645	−0.236822	−	0.971553i	$-0.576106\pi$
$491$	−1.74071e25	−0.473645	−0.236822	+	0.971553i	$0.576106\pi$
$492$	0	0
$493$	−2.74178e24	−0.0717773
$494$	0	0
$495$	7.28768e24	0.183586
$496$	0	0
$497$	−1.40739e25	−0.341216
$498$	0	0
$499$	−2.83074e25	−0.660610	−0.330305	−	0.943874i	$-0.607152\pi$
$499$	−2.83074e25	−0.660610	−0.330305	+	0.943874i	$0.607152\pi$
$500$	0	0
$501$	1.55562e25	0.349499
$502$	0	0
$503$	−1.12443e25	−0.243241	−0.121621	−	0.992577i	$-0.538809\pi$
$503$	−1.12443e25	−0.243241	−0.121621	+	0.992577i	$0.538809\pi$
$504$	0	0
$505$	−1.60545e25	−0.334449
$506$	0	0
$507$	7.37556e25	1.47986
$508$	0	0
$509$	−8.09240e25	−1.56408	−0.782039	−	0.623229i	$-0.785820\pi$
$509$	−8.09240e25	−1.56408	−0.782039	+	0.623229i	$0.785820\pi$
$510$	0	0
$511$	4.91123e25	0.914517
$512$	0	0
$513$	−7.07345e25	−1.26916
$514$	0	0
$515$	−2.75827e25	−0.476945
$516$	0	0
$517$	2.69949e25	0.449906
$518$	0	0
$519$	−5.82531e24	−0.0935899
$520$	0	0
$521$	2.60762e25	0.403911	0.201956	−	0.979395i	$-0.435270\pi$
$521$	2.60762e25	0.403911	0.201956	+	0.979395i	$0.435270\pi$
$522$	0	0
$523$	−2.66904e25	−0.398648	−0.199324	−	0.979934i	$-0.563875\pi$
$523$	−2.66904e25	−0.398648	−0.199324	+	0.979934i	$0.563875\pi$
$524$	0	0
$525$	2.21840e25	0.319541
$526$	0	0
$527$	8.65015e25	1.20177
$528$	0	0
$529$	1.07100e26	1.43536
$530$	0	0
$531$	−3.62786e25	−0.469086
$532$	0	0
$533$	−2.41233e25	−0.300973
$534$	0	0
$535$	−3.48360e25	−0.419438
$536$	0	0
$537$	−9.14378e24	−0.106260
$538$	0	0
$539$	6.88094e25	0.771888
$540$	0	0
$541$	1.82562e26	1.97714	0.988570	−	0.150765i	$-0.0481737\pi$
$541$	1.82562e26	1.97714	0.988570	+	0.150765i	$0.0481737\pi$
$542$	0	0
$543$	4.65782e25	0.487062
$544$	0	0
$545$	3.43552e25	0.346918
$546$	0	0
$547$	−1.82236e26	−1.77727	−0.888635	−	0.458614i	$-0.848346\pi$
$547$	−1.82236e26	−1.77727	−0.888635	+	0.458614i	$0.848346\pi$
$548$	0	0
$549$	−5.32119e25	−0.501270
$550$	0	0
$551$	7.94741e24	0.0723245
$552$	0	0
$553$	1.81894e26	1.59930
$554$	0	0
$555$	7.01892e25	0.596330
$556$	0	0
$557$	−9.07761e25	−0.745327	−0.372664	−	0.927966i	$-0.621555\pi$
$557$	−9.07761e25	−0.745327	−0.372664	+	0.927966i	$0.621555\pi$
$558$	0	0
$559$	−2.92902e26	−2.32440
$560$	0	0
$561$	−2.76253e26	−2.11914
$562$	0	0
$563$	−2.46140e26	−1.82537	−0.912687	−	0.408660i	$-0.865996\pi$
$563$	−2.46140e26	−1.82537	−0.912687	+	0.408660i	$0.865996\pi$
$564$	0	0
$565$	−7.17610e25	−0.514551
$566$	0	0
$567$	−2.40340e26	−1.66643
$568$	0	0
$569$	−3.37478e25	−0.226297	−0.113149	−	0.993578i	$-0.536094\pi$
$569$	−3.37478e25	−0.226297	−0.113149	+	0.993578i	$0.536094\pi$
$570$	0	0
$571$	1.26807e26	0.822434	0.411217	−	0.911538i	$-0.365104\pi$
$571$	1.26807e26	0.822434	0.411217	+	0.911538i	$0.365104\pi$
$572$	0	0
$573$	2.94823e26	1.84966
$574$	0	0
$575$	5.14229e25	0.312113
$576$	0	0
$577$	−7.63131e25	−0.448155	−0.224078	−	0.974571i	$-0.571937\pi$
$577$	−7.63131e25	−0.448155	−0.224078	+	0.974571i	$0.571937\pi$
$578$	0	0
$579$	3.39218e26	1.92767
$580$	0	0
$581$	−1.01805e25	−0.0559879
$582$	0	0
$583$	1.89487e26	1.00862
$584$	0	0
$585$	5.48958e25	0.282851
$586$	0	0
$587$	1.02811e26	0.512834	0.256417	−	0.966566i	$-0.417458\pi$
$587$	1.02811e26	0.512834	0.256417	+	0.966566i	$0.417458\pi$
$588$	0	0
$589$	−2.50736e26	−1.21094
$590$	0	0
$591$	4.81992e26	2.25403
$592$	0	0
$593$	−1.16226e26	−0.526360	−0.263180	−	0.964747i	$-0.584771\pi$
$593$	−1.16226e26	−0.526360	−0.263180	+	0.964747i	$0.584771\pi$
$594$	0	0
$595$	−2.49772e26	−1.09555
$596$	0	0
$597$	−8.59570e25	−0.365195
$598$	0	0
$599$	−4.55677e26	−1.87544	−0.937718	−	0.347397i	$-0.887066\pi$
$599$	−4.55677e26	−1.87544	−0.937718	+	0.347397i	$0.887066\pi$
$600$	0	0
$601$	−2.69677e26	−1.07532	−0.537658	−	0.843163i	$-0.680691\pi$
$601$	−2.69677e26	−1.07532	−0.537658	+	0.843163i	$0.680691\pi$
$602$	0	0
$603$	1.64909e25	0.0637131
$604$	0	0
$605$	−6.69236e24	−0.0250555
$606$	0	0
$607$	7.90754e24	0.0286912	0.0143456	−	0.999897i	$-0.495433\pi$
$607$	7.90754e24	0.0286912	0.0143456	+	0.999897i	$0.495433\pi$
$608$	0	0
$609$	1.78319e25	0.0627094
$610$	0	0
$611$	2.03344e26	0.693171
$612$	0	0
$613$	5.57774e26	1.84325	0.921625	−	0.388082i	$-0.126862\pi$
$613$	5.57774e26	1.84325	0.921625	+	0.388082i	$0.126862\pi$
$614$	0	0
$615$	−3.34724e25	−0.107244
$616$	0	0
$617$	−5.18052e24	−0.0160940	−0.00804700	−	0.999968i	$-0.502561\pi$
$617$	−5.18052e24	−0.0160940	−0.00804700	+	0.999968i	$0.502561\pi$
$618$	0	0
$619$	−5.14356e26	−1.54954	−0.774770	−	0.632243i	$-0.782134\pi$
$619$	−5.14356e26	−1.54954	−0.774770	+	0.632243i	$0.782134\pi$
$620$	0	0
$621$	−3.67891e26	−1.07485
$622$	0	0
$623$	7.06326e25	0.200156
$624$	0	0
$625$	1.45519e25	0.0400000
$626$	0	0
$627$	8.00754e26	2.13529
$628$	0	0
$629$	−7.90268e26	−2.04453
$630$	0	0
$631$	−2.89800e26	−0.727476	−0.363738	−	0.931501i	$-0.618500\pi$
$631$	−2.89800e26	−0.727476	−0.363738	+	0.931501i	$0.618500\pi$
$632$	0	0
$633$	2.95759e26	0.720449
$634$	0	0
$635$	−1.54969e26	−0.366349
$636$	0	0
$637$	5.18320e26	1.18925
$638$	0	0
$639$	−4.83243e25	−0.107624
$640$	0	0
$641$	3.39194e24	0.00733325	0.00366663	−	0.999993i	$-0.498833\pi$
$641$	3.39194e24	0.00733325	0.00366663	+	0.999993i	$0.498833\pi$
$642$	0	0
$643$	−2.16164e26	−0.453710	−0.226855	−	0.973929i	$-0.572844\pi$
$643$	−2.16164e26	−0.453710	−0.226855	+	0.973929i	$0.572844\pi$
$644$	0	0
$645$	−4.06418e26	−0.828240
$646$	0	0
$647$	4.25070e26	0.841144	0.420572	−	0.907259i	$-0.361829\pi$
$647$	4.25070e26	0.841144	0.420572	+	0.907259i	$0.361829\pi$
$648$	0	0
$649$	−5.61322e26	−1.07867
$650$	0	0
$651$	−5.62585e26	−1.04995
$652$	0	0
$653$	−6.28862e26	−1.13993	−0.569967	−	0.821667i	$-0.693044\pi$
$653$	−6.28862e26	−1.13993	−0.569967	+	0.821667i	$0.693044\pi$
$654$	0	0
$655$	−4.57643e26	−0.805814
$656$	0	0
$657$	1.68632e26	0.288449
$658$	0	0
$659$	−3.92925e26	−0.652978	−0.326489	−	0.945201i	$-0.605866\pi$
$659$	−3.92925e26	−0.652978	−0.326489	+	0.945201i	$0.605866\pi$
$660$	0	0
$661$	−4.23909e26	−0.684477	−0.342238	−	0.939613i	$-0.611185\pi$
$661$	−4.23909e26	−0.684477	−0.342238	+	0.939613i	$0.611185\pi$
$662$	0	0
$663$	−2.08092e27	−3.26496
$664$	0	0
$665$	7.23997e26	1.10390
$666$	0	0
$667$	4.13346e25	0.0612517
$668$	0	0
$669$	−7.17021e25	−0.103272
$670$	0	0
$671$	−8.23324e26	−1.15268
$672$	0	0
$673$	4.51372e26	0.614316	0.307158	−	0.951659i	$-0.400622\pi$
$673$	4.51372e26	0.614316	0.307158	+	0.951659i	$0.400622\pi$
$674$	0	0
$675$	−1.04108e26	−0.137752
$676$	0	0
$677$	−1.32779e27	−1.70819	−0.854095	−	0.520117i	$-0.825888\pi$
$677$	−1.32779e27	−1.70819	−0.854095	+	0.520117i	$0.825888\pi$
$678$	0	0
$679$	6.60228e26	0.825907
$680$	0	0
$681$	8.93977e26	1.08750
$682$	0	0
$683$	4.24377e26	0.502059	0.251030	−	0.967979i	$-0.419231\pi$
$683$	4.24377e26	0.502059	0.251030	+	0.967979i	$0.419231\pi$
$684$	0	0
$685$	−4.51918e26	−0.519995
$686$	0	0
$687$	−4.92501e26	−0.551211
$688$	0	0
$689$	1.42735e27	1.55398
$690$	0	0
$691$	−1.01403e26	−0.107401	−0.0537006	−	0.998557i	$-0.517102\pi$
$691$	−1.01403e26	−0.107401	−0.0537006	+	0.998557i	$0.517102\pi$
$692$	0	0
$693$	5.33652e26	0.549911
$694$	0	0
$695$	1.70563e26	0.171013
$696$	0	0
$697$	3.76869e26	0.367688
$698$	0	0
$699$	1.28654e27	1.22149
$700$	0	0
$701$	−1.60977e26	−0.148745	−0.0743726	−	0.997231i	$-0.523695\pi$
$701$	−1.60977e26	−0.148745	−0.0743726	+	0.997231i	$0.523695\pi$
$702$	0	0
$703$	2.29070e27	2.06012
$704$	0	0
$705$	2.82152e26	0.246994
$706$	0	0
$707$	−1.17562e27	−1.00180
$708$	0	0
$709$	2.08980e26	0.173366	0.0866832	−	0.996236i	$-0.472373\pi$
$709$	2.08980e26	0.173366	0.0866832	+	0.996236i	$0.472373\pi$
$710$	0	0
$711$	6.24553e26	0.504438
$712$	0	0
$713$	−1.30408e27	−1.02554
$714$	0	0
$715$	8.49378e26	0.650420
$716$	0	0
$717$	5.92967e26	0.442180
$718$	0	0
$719$	−1.24431e27	−0.903658	−0.451829	−	0.892104i	$-0.649228\pi$
$719$	−1.24431e27	−0.903658	−0.451829	+	0.892104i	$0.649228\pi$
$720$	0	0
$721$	−2.01979e27	−1.42863
$722$	0	0
$723$	2.81190e27	1.93725
$724$	0	0
$725$	1.16971e25	0.00784995
$726$	0	0
$727$	1.54824e27	1.01219	0.506093	−	0.862479i	$-0.331089\pi$
$727$	1.54824e27	1.01219	0.506093	+	0.862479i	$0.331089\pi$
$728$	0	0
$729$	4.64527e26	0.295869
$730$	0	0
$731$	4.57591e27	2.83964
$732$	0	0
$733$	−5.27539e26	−0.318982	−0.159491	−	0.987199i	$-0.550985\pi$
$733$	−5.27539e26	−0.318982	−0.159491	+	0.987199i	$0.550985\pi$
$734$	0	0
$735$	7.19197e26	0.423758
$736$	0	0
$737$	2.55156e26	0.146509
$738$	0	0
$739$	−2.18433e27	−1.22235	−0.611175	−	0.791495i	$-0.709303\pi$
$739$	−2.18433e27	−1.22235	−0.611175	+	0.791495i	$0.709303\pi$
$740$	0	0
$741$	6.03183e27	3.28985
$742$	0	0
$743$	−3.19636e27	−1.69927	−0.849634	−	0.527373i	$-0.823177\pi$
$743$	−3.19636e27	−1.69927	−0.849634	+	0.527373i	$0.823177\pi$
$744$	0	0
$745$	−1.16100e26	−0.0601655
$746$	0	0
$747$	−3.49556e25	−0.0176592
$748$	0	0
$749$	−2.55092e27	−1.25638
$750$	0	0
$751$	−1.95546e27	−0.939008	−0.469504	−	0.882930i	$-0.655567\pi$
$751$	−1.95546e27	−0.939008	−0.469504	+	0.882930i	$0.655567\pi$
$752$	0	0
$753$	−6.56533e26	−0.307400
$754$	0	0
$755$	1.45386e27	0.663783
$756$	0	0
$757$	1.58893e27	0.707445	0.353723	−	0.935350i	$-0.384916\pi$
$757$	1.58893e27	0.707445	0.353723	+	0.935350i	$0.384916\pi$
$758$	0	0
$759$	4.16474e27	1.80838
$760$	0	0
$761$	9.93893e26	0.420906	0.210453	−	0.977604i	$-0.432506\pi$
$761$	9.93893e26	0.420906	0.210453	+	0.977604i	$0.432506\pi$
$762$	0	0
$763$	2.51572e27	1.03915
$764$	0	0
$765$	−8.57617e26	−0.345549
$766$	0	0
$767$	−4.22826e27	−1.66190
$768$	0	0
$769$	2.79733e27	1.07261	0.536307	−	0.844023i	$-0.319819\pi$
$769$	2.79733e27	1.07261	0.536307	+	0.844023i	$0.319819\pi$
$770$	0	0
$771$	9.44813e26	0.353451
$772$	0	0
$773$	5.07033e26	0.185068	0.0925339	−	0.995710i	$-0.470503\pi$
$773$	5.07033e26	0.185068	0.0925339	+	0.995710i	$0.470503\pi$
$774$	0	0
$775$	−3.69036e26	−0.131432
$776$	0	0
$777$	5.13971e27	1.78624
$778$	0	0
$779$	−1.09240e27	−0.370491
$780$	0	0
$781$	−7.47700e26	−0.247481
$782$	0	0
$783$	−8.36837e25	−0.0270336
$784$	0	0
$785$	−1.12656e27	−0.355217
$786$	0	0
$787$	4.37042e27	1.34513	0.672564	−	0.740039i	$-0.265193\pi$
$787$	4.37042e27	1.34513	0.672564	+	0.740039i	$0.265193\pi$
$788$	0	0
$789$	−5.85486e26	−0.175908
$790$	0	0
$791$	−5.25482e27	−1.54128
$792$	0	0
$793$	−6.20184e27	−1.77593
$794$	0	0
$795$	1.98052e27	0.553722
$796$	0	0
$797$	5.27098e27	1.43892	0.719461	−	0.694533i	$-0.244389\pi$
$797$	5.27098e27	1.43892	0.719461	+	0.694533i	$0.244389\pi$
$798$	0	0
$799$	−3.17678e27	−0.846822
$800$	0	0
$801$	2.42524e26	0.0631314
$802$	0	0
$803$	2.60917e27	0.663291
$804$	0	0
$805$	3.76552e27	0.934898
$806$	0	0
$807$	2.92141e27	0.708425
$808$	0	0
$809$	7.97624e27	1.88924	0.944621	−	0.328164i	$-0.106430\pi$
$809$	7.97624e27	1.88924	0.944621	+	0.328164i	$0.106430\pi$
$810$	0	0
$811$	−8.45165e27	−1.95544	−0.977718	−	0.209924i	$-0.932678\pi$
$811$	−8.45165e27	−1.95544	−0.977718	+	0.209924i	$0.932678\pi$
$812$	0	0
$813$	−4.46650e26	−0.100950
$814$	0	0
$815$	4.87929e26	0.107736
$816$	0	0
$817$	−1.32639e28	−2.86128
$818$	0	0
$819$	4.01983e27	0.847248
$820$	0	0
$821$	4.76435e27	0.981169	0.490585	−	0.871394i	$-0.336783\pi$
$821$	4.76435e27	0.981169	0.490585	+	0.871394i	$0.336783\pi$
$822$	0	0
$823$	1.13822e27	0.229049	0.114524	−	0.993420i	$-0.463466\pi$
$823$	1.13822e27	0.229049	0.114524	+	0.993420i	$0.463466\pi$
$824$	0	0
$825$	1.17856e27	0.231760
$826$	0	0
$827$	−4.53769e27	−0.872032	−0.436016	−	0.899939i	$-0.643611\pi$
$827$	−4.53769e27	−0.872032	−0.436016	+	0.899939i	$0.643611\pi$
$828$	0	0
$829$	−3.30719e27	−0.621142	−0.310571	−	0.950550i	$-0.600520\pi$
$829$	−3.30719e27	−0.621142	−0.310571	+	0.950550i	$0.600520\pi$
$830$	0	0
$831$	−8.52212e27	−1.56437
$832$	0	0
$833$	−8.09752e27	−1.45286
$834$	0	0
$835$	7.47228e26	0.131048
$836$	0	0
$837$	2.64017e27	0.452627
$838$	0	0
$839$	−3.21760e26	−0.0539254	−0.0269627	−	0.999636i	$-0.508584\pi$
$839$	−3.21760e26	−0.0539254	−0.0269627	+	0.999636i	$0.508584\pi$
$840$	0	0
$841$	−6.09386e27	−0.998459
$842$	0	0
$843$	−1.35949e28	−2.17778
$844$	0	0
$845$	3.54278e27	0.554889
$846$	0	0
$847$	−4.90059e26	−0.0750509
$848$	0	0
$849$	−8.40923e27	−1.25931
$850$	0	0
$851$	1.19139e28	1.74472
$852$	0	0
$853$	−7.67993e27	−1.09987	−0.549935	−	0.835207i	$-0.685348\pi$
$853$	−7.67993e27	−1.09987	−0.549935	+	0.835207i	$0.685348\pi$
$854$	0	0
$855$	2.48592e27	0.348184
$856$	0	0
$857$	−3.13490e27	−0.429444	−0.214722	−	0.976675i	$-0.568884\pi$
$857$	−3.13490e27	−0.429444	−0.214722	+	0.976675i	$0.568884\pi$
$858$	0	0
$859$	1.22945e28	1.64731	0.823653	−	0.567095i	$-0.191933\pi$
$859$	1.22945e28	1.64731	0.823653	+	0.567095i	$0.191933\pi$
$860$	0	0
$861$	−2.45107e27	−0.321237
$862$	0	0
$863$	5.91889e27	0.758818	0.379409	−	0.925229i	$-0.376127\pi$
$863$	5.91889e27	0.758818	0.379409	+	0.925229i	$0.376127\pi$
$864$	0	0
$865$	−2.79813e26	−0.0350926
$866$	0	0
$867$	2.27885e28	2.79599
$868$	0	0
$869$	9.66343e27	1.15996
$870$	0	0
$871$	1.92201e27	0.225727
$872$	0	0
$873$	2.26696e27	0.260500
$874$	0	0
$875$	1.06559e27	0.119815
$876$	0	0
$877$	9.36231e27	1.03012	0.515059	−	0.857155i	$-0.327770\pi$
$877$	9.36231e27	1.03012	0.515059	+	0.857155i	$0.327770\pi$
$878$	0	0
$879$	−6.82934e27	−0.735333
$880$	0	0
$881$	1.45150e27	0.152949	0.0764744	−	0.997072i	$-0.475634\pi$
$881$	1.45150e27	0.152949	0.0764744	+	0.997072i	$0.475634\pi$
$882$	0	0
$883$	−1.22192e27	−0.126013	−0.0630064	−	0.998013i	$-0.520069\pi$
$883$	−1.22192e27	−0.126013	−0.0630064	+	0.998013i	$0.520069\pi$
$884$	0	0
$885$	−5.86695e27	−0.592177
$886$	0	0
$887$	−9.73469e27	−0.961718	−0.480859	−	0.876798i	$-0.659675\pi$
$887$	−9.73469e27	−0.961718	−0.480859	+	0.876798i	$0.659675\pi$
$888$	0	0
$889$	−1.13479e28	−1.09736
$890$	0	0
$891$	−1.27684e28	−1.20865
$892$	0	0
$893$	9.20830e27	0.853279
$894$	0	0
$895$	−4.39213e26	−0.0398434
$896$	0	0
$897$	3.13716e28	2.78618
$898$	0	0
$899$	−2.96638e26	−0.0257934
$900$	0	0
$901$	−2.22989e28	−1.89845
$902$	0	0
$903$	−2.97606e28	−2.48090
$904$	0	0
$905$	2.23734e27	0.182629
$906$	0	0
$907$	4.95126e27	0.395773	0.197887	−	0.980225i	$-0.436592\pi$
$907$	4.95126e27	0.395773	0.197887	+	0.980225i	$0.436592\pi$
$908$	0	0
$909$	−4.03660e27	−0.315980
$910$	0	0
$911$	6.85325e27	0.525378	0.262689	−	0.964881i	$-0.415391\pi$
$911$	6.85325e27	0.525378	0.262689	+	0.964881i	$0.415391\pi$
$912$	0	0
$913$	−5.40853e26	−0.0406075
$914$	0	0
$915$	−8.60540e27	−0.632806
$916$	0	0
$917$	−3.35117e28	−2.41372
$918$	0	0
$919$	−2.40577e28	−1.69729	−0.848645	−	0.528962i	$-0.822581\pi$
$919$	−2.40577e28	−1.69729	−0.848645	+	0.528962i	$0.822581\pi$
$920$	0	0
$921$	−3.07302e27	−0.212373
$922$	0	0
$923$	−5.63219e27	−0.381295
$924$	0	0
$925$	3.37147e27	0.223601
$926$	0	0
$927$	−6.93514e27	−0.450607
$928$	0	0
$929$	1.69237e28	1.07732	0.538661	−	0.842523i	$-0.318930\pi$
$929$	1.69237e28	1.07732	0.538661	+	0.842523i	$0.318930\pi$
$930$	0	0
$931$	2.34717e28	1.46394
$932$	0	0
$933$	1.31975e28	0.806522
$934$	0	0
$935$	−1.32695e28	−0.794594
$936$	0	0
$937$	2.09214e28	1.22762	0.613811	−	0.789453i	$-0.289636\pi$
$937$	2.09214e28	1.22762	0.613811	+	0.789453i	$0.289636\pi$
$938$	0	0
$939$	4.52791e26	0.0260360
$940$	0	0
$941$	−3.14349e28	−1.77137	−0.885687	−	0.464282i	$-0.846312\pi$
$941$	−3.14349e28	−1.77137	−0.885687	+	0.464282i	$0.846312\pi$
$942$	0	0
$943$	−5.68161e27	−0.313769
$944$	0	0
$945$	−7.62346e27	−0.412620
$946$	0	0
$947$	1.40308e28	0.744318	0.372159	−	0.928169i	$-0.378618\pi$
$947$	1.40308e28	0.744318	0.372159	+	0.928169i	$0.378618\pi$
$948$	0	0
$949$	1.96540e28	1.02193
$950$	0	0
$951$	2.44801e27	0.124767
$952$	0	0
$953$	−5.95470e27	−0.297494	−0.148747	−	0.988875i	$-0.547524\pi$
$953$	−5.95470e27	−0.297494	−0.148747	+	0.988875i	$0.547524\pi$
$954$	0	0
$955$	1.41615e28	0.693551
$956$	0	0
$957$	9.47347e26	0.0454826
$958$	0	0
$959$	−3.30924e28	−1.55758
$960$	0	0
$961$	−1.23119e28	−0.568138
$962$	0	0
$963$	−8.75884e27	−0.396275
$964$	0	0
$965$	1.62940e28	0.722801
$966$	0	0
$967$	−1.33148e28	−0.579140	−0.289570	−	0.957157i	$-0.593512\pi$
$967$	−1.33148e28	−0.579140	−0.289570	+	0.957157i	$0.593512\pi$
$968$	0	0
$969$	−9.42331e28	−4.01909
$970$	0	0
$971$	−1.53320e28	−0.641235	−0.320618	−	0.947209i	$-0.603890\pi$
$971$	−1.53320e28	−0.641235	−0.320618	+	0.947209i	$0.603890\pi$
$972$	0	0
$973$	1.24898e28	0.512251
$974$	0	0
$975$	8.87771e27	0.357073
$976$	0	0
$977$	−1.00839e28	−0.397768	−0.198884	−	0.980023i	$-0.563732\pi$
$977$	−1.00839e28	−0.397768	−0.198884	+	0.980023i	$0.563732\pi$
$978$	0	0
$979$	3.75247e27	0.145171
$980$	0	0
$981$	8.63796e27	0.327760
$982$	0	0
$983$	−1.97851e28	−0.736343	−0.368171	−	0.929758i	$-0.620016\pi$
$983$	−1.97851e28	−0.736343	−0.368171	+	0.929758i	$0.620016\pi$
$984$	0	0
$985$	2.31521e28	0.845173
$986$	0	0
$987$	2.06610e28	0.739841
$988$	0	0
$989$	−6.89857e28	−2.42323
$990$	0	0
$991$	5.11948e27	0.176411	0.0882057	−	0.996102i	$-0.471887\pi$
$991$	5.11948e27	0.176411	0.0882057	+	0.996102i	$0.471887\pi$
$992$	0	0
$993$	6.60144e28	2.23162
$994$	0	0
$995$	−4.12887e27	−0.136934
$996$	0	0
$997$	−2.58463e28	−0.840997	−0.420498	−	0.907293i	$-0.638145\pi$
$997$	−2.58463e28	−0.840997	−0.420498	+	0.907293i	$0.638145\pi$
$998$	0	0
$999$	−2.41203e28	−0.770035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.a.h.1.4		4
4.3	odd	2		20.20.a.b.1.1	✓	4
20.3	even	4		100.20.c.c.49.2		8
20.7	even	4		100.20.c.c.49.7		8
20.19	odd	2		100.20.a.c.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.20.a.b.1.1	✓	4	4.3	odd	2
80.20.a.h.1.4		4	1.1	even	1	trivial
100.20.a.c.1.4		4	20.19	odd	2
100.20.c.c.49.2		8	20.3	even	4
100.20.c.c.49.7		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+40661.2 q^{3} +1.95312e6 q^{5} +1.43021e8 q^{7} +4.91076e8 q^{9} +O(q^{10})\)	\(q+40661.2 q^{3} +1.95312e6 q^{5} +1.43021e8 q^{7} +4.91076e8 q^{9} +7.59820e9 q^{11} +5.72348e10 q^{13} +7.94165e10 q^{15} -8.94159e11 q^{17} +2.59184e12 q^{19} +5.81540e12 q^{21} +1.34802e13 q^{23} +3.81470e12 q^{25} -2.72913e13 q^{27} +3.06632e12 q^{29} -9.67405e13 q^{31} +3.08952e14 q^{33} +2.79337e14 q^{35} +8.83811e14 q^{37} +2.32724e15 q^{39} -4.21479e14 q^{41} -5.11756e15 q^{43} +9.59132e14 q^{45} +3.55281e15 q^{47} +9.05602e15 q^{49} -3.63576e16 q^{51} +2.49384e16 q^{53} +1.48402e16 q^{55} +1.05387e17 q^{57} -7.38757e16 q^{59} -1.08358e17 q^{61} +7.02340e16 q^{63} +1.11787e17 q^{65} +3.35811e16 q^{67} +5.48122e17 q^{69} -9.84049e16 q^{71} +3.43393e17 q^{73} +1.55110e17 q^{75} +1.08670e18 q^{77} +1.27181e18 q^{79} -1.68045e18 q^{81} -7.11817e16 q^{83} -1.74641e18 q^{85} +1.24680e17 q^{87} +4.93863e17 q^{89} +8.18576e18 q^{91} -3.93359e18 q^{93} +5.06218e18 q^{95} +4.61631e18 q^{97} +3.73129e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9}+O(q^{10}) \) 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9	\( 4 q + 3080 q^{3} + 7812500 q^{5} - 148222040 q^{7} + 2231864116 q^{9} + 2334973920 q^{11} + 57423224120 q^{13} + 6015625000 q^{15} - 465961763160 q^{17} + 1624818160624 q^{19} + 5400395639744 q^{21} + 10101669670680 q^{23} + 15258789062500 q^{25} - 56910507730480 q^{27} - 21802908180264 q^{29} + 69517593805936 q^{31} - 204641262341280 q^{33} - 289496171875000 q^{35} + 21\!\cdots\!60 q^{37}+ \cdots + 27\!\cdots\!80 q^{99}+O(q^{100}) \) 4 * q + 3080 * q^3 + 7812500 * q^5 - 148222040 * q^7 + 2231864116 * q^9 + 2334973920 * q^11 + 57423224120 * q^13 + 6015625000 * q^15 - 465961763160 * q^17 + 1624818160624 * q^19 + 5400395639744 * q^21 + 10101669670680 * q^23 + 15258789062500 * q^25 - 56910507730480 * q^27 - 21802908180264 * q^29 + 69517593805936 * q^31 - 204641262341280 * q^33 - 289496171875000 * q^35 + 2156700568331960 * q^37 - 2385459176165456 * q^39 + 1367657291239944 * q^41 - 2895469359165560 * q^43 + 4359109601562500 * q^45 - 1740670005708600 * q^47 + 53534973320692932 * q^49 - 62580957868768368 * q^51 + 84253756978305240 * q^53 + 4560495937500000 * q^55 + 188957630874753440 * q^57 - 152838143056415952 * q^59 + 211425306591350168 * q^61 - 417796572994057400 * q^63 + 112154734609375000 * q^65 - 656919981274544360 * q^67 + 93524018475363648 * q^69 - 883122322400344752 * q^71 + 325226727237880520 * q^73 + 11749267578125000 * q^75 - 620955288586656480 * q^77 + 850256855296398112 * q^79 - 3380709519213288764 * q^81 + 3602925283549230600 * q^83 - 910081568671875000 * q^85 + 8179923434985534000 * q^87 + 418548299964421224 * q^89 + 10309921756152683504 * q^91 - 14102167255291233280 * q^93 + 3173472969968750000 * q^95 - 12135517460163032920 * q^97 + 27941442307774944480 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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