LMFDB - Embedded newform 80.20.a.e.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 1351720x + 139588750 $ x^3 - x^2 - 1351720*x + 139588750
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1210.70$ 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)

$f(q)$	$=$	$q+46767.8 q^{3} -1.95312e6 q^{5} +3.25847e7 q^{7} +1.02497e9 q^{9} +O(q^{10})$	$q+46767.8 q^{3} -1.95312e6 q^{5} +3.25847e7 q^{7} +1.02497e9 q^{9} -3.59337e9 q^{11} +7.05085e9 q^{13} -9.13434e10 q^{15} +4.80091e11 q^{17} -1.17333e12 q^{19} +1.52392e12 q^{21} -5.64342e12 q^{23} +3.81470e12 q^{25} -6.42101e12 q^{27} -1.21031e14 q^{29} +1.25351e14 q^{31} -1.68054e14 q^{33} -6.36420e13 q^{35} +5.31777e14 q^{37} +3.29753e14 q^{39} +8.87312e14 q^{41} -3.63662e14 q^{43} -2.00189e15 q^{45} +8.32293e15 q^{47} -1.03371e16 q^{49} +2.24528e16 q^{51} -2.34743e16 q^{53} +7.01829e15 q^{55} -5.48740e16 q^{57} -9.14089e14 q^{59} -9.45933e16 q^{61} +3.33982e16 q^{63} -1.37712e16 q^{65} -2.63147e17 q^{67} -2.63930e17 q^{69} +3.62022e17 q^{71} -5.68735e17 q^{73} +1.78405e17 q^{75} -1.17089e17 q^{77} -1.10844e18 q^{79} -1.49157e18 q^{81} -2.63110e18 q^{83} -9.37678e17 q^{85} -5.66036e18 q^{87} +4.21152e18 q^{89} +2.29750e17 q^{91} +5.86241e18 q^{93} +2.29166e18 q^{95} +8.83410e18 q^{97} -3.68308e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9}+O(q^{10}) $ 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9	$ 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9} - 6359213280 q^{11} - 33996539826 q^{13} + 66574218750 q^{15} - 60515110578 q^{17} + 1436218405908 q^{19} - 7119772762332 q^{21} - 1682066292342 q^{23} + 11444091796875 q^{25} - 91316335875732 q^{27} - 181580995192278 q^{29} + 3566464773252 q^{31} - 90839870844480 q^{33} - 224864402343750 q^{35} + 653714901466206 q^{37} + 43\!\cdots\!08 q^{39}+ \cdots - 37\!\cdots\!20 q^{99}+O(q^{100}) $ 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9 - 6359213280 * q^11 - 33996539826 * q^13 + 66574218750 * q^15 - 60515110578 * q^17 + 1436218405908 * q^19 - 7119772762332 * q^21 - 1682066292342 * q^23 + 11444091796875 * q^25 - 91316335875732 * q^27 - 181580995192278 * q^29 + 3566464773252 * q^31 - 90839870844480 * q^33 - 224864402343750 * q^35 + 653714901466206 * q^37 + 4339017394217508 * q^39 - 1257752912210622 * q^41 + 4361040178994610 * q^43 - 6111773490234375 * q^45 + 23174987171661798 * q^47 - 5454428919168921 * q^49 + 33371959424606244 * q^51 - 12180381712114362 * q^53 + 12420338437500000 * q^55 - 177419957801365416 * q^57 + 180023573560918116 * q^59 - 213491197168562514 * q^61 + 549307385783925246 * q^63 + 66399491847656250 * q^65 + 32494317103156134 * q^67 - 498099089220386004 * q^69 + 245957452394007276 * q^71 - 94437681636898506 * q^73 - 130027770996093750 * q^75 - 72972512123263680 * q^77 - 819623047726342536 * q^79 + 3159048884904329967 * q^81 - 2812828487653823982 * q^83 + 118193575347656250 * q^85 + 2510285506713629676 * q^87 + 4879439966436006078 * q^89 - 12213088562709029652 * q^91 + 16020238495381453176 * q^93 - 2805114074039062500 * q^95 + 8189756396617107846 * q^97 - 3795512975671396320 * 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
$3$	46767.8	1.37181	0.685907	−	0.727689i	$-0.259406\pi$
$3$	46767.8	1.37181	0.685907	+	0.727689i	$0.259406\pi$
$4$	0	0
$5$	−1.95312e6	−0.447214
$5$	−1.95312e6	−0.447214
$6$	0	0
$7$	3.25847e7	0.305199	0.152599	−	0.988288i	$-0.451236\pi$
$7$	3.25847e7	0.305199	0.152599	+	0.988288i	$0.451236\pi$
$8$	0	0
$9$	1.02497e9	0.881872
$10$	0	0
$11$	−3.59337e9	−0.459485	−0.229742	−	0.973252i	$-0.573788\pi$
$11$	−3.59337e9	−0.459485	−0.229742	+	0.973252i	$0.573788\pi$
$12$	0	0
$13$	7.05085e9	0.184408	0.0922039	−	0.995740i	$-0.470609\pi$
$13$	7.05085e9	0.184408	0.0922039	+	0.995740i	$0.470609\pi$
$14$	0	0
$15$	−9.13434e10	−0.613494
$16$	0	0
$17$	4.80091e11	0.981881	0.490941	−	0.871193i	$-0.336653\pi$
$17$	4.80091e11	0.981881	0.490941	+	0.871193i	$0.336653\pi$
$18$	0	0
$19$	−1.17333e12	−0.834182	−0.417091	−	0.908865i	$-0.636950\pi$
$19$	−1.17333e12	−0.834182	−0.417091	+	0.908865i	$0.636950\pi$
$20$	0	0
$21$	1.52392e12	0.418676
$22$	0	0
$23$	−5.64342e12	−0.653322	−0.326661	−	0.945142i	$-0.605924\pi$
$23$	−5.64342e12	−0.653322	−0.326661	+	0.945142i	$0.605924\pi$
$24$	0	0
$25$	3.81470e12	0.200000
$26$	0	0
$27$	−6.42101e12	−0.162049
$28$	0	0
$29$	−1.21031e14	−1.54923	−0.774615	−	0.632433i	$-0.782057\pi$
$29$	−1.21031e14	−1.54923	−0.774615	+	0.632433i	$0.782057\pi$
$30$	0	0
$31$	1.25351e14	0.851517	0.425758	−	0.904837i	$-0.360007\pi$
$31$	1.25351e14	0.851517	0.425758	+	0.904837i	$0.360007\pi$
$32$	0	0
$33$	−1.68054e14	−0.630327
$34$	0	0
$35$	−6.36420e13	−0.136489
$36$	0	0
$37$	5.31777e14	0.672688	0.336344	−	0.941739i	$-0.390810\pi$
$37$	5.31777e14	0.672688	0.336344	+	0.941739i	$0.390810\pi$
$38$	0	0
$39$	3.29753e14	0.252973
$40$	0	0
$41$	8.87312e14	0.423282	0.211641	−	0.977347i	$-0.432119\pi$
$41$	8.87312e14	0.423282	0.211641	+	0.977347i	$0.432119\pi$
$42$	0	0
$43$	−3.63662e14	−0.110344	−0.0551719	−	0.998477i	$-0.517571\pi$
$43$	−3.63662e14	−0.110344	−0.0551719	+	0.998477i	$0.517571\pi$
$44$	0	0
$45$	−2.00189e15	−0.394385
$46$	0	0
$47$	8.32293e15	1.08479	0.542396	−	0.840123i	$-0.317517\pi$
$47$	8.32293e15	1.08479	0.542396	+	0.840123i	$0.317517\pi$
$48$	0	0
$49$	−1.03371e16	−0.906854
$50$	0	0
$51$	2.24528e16	1.34696
$52$	0	0
$53$	−2.34743e16	−0.977174	−0.488587	−	0.872515i	$-0.662488\pi$
$53$	−2.34743e16	−0.977174	−0.488587	+	0.872515i	$0.662488\pi$
$54$	0	0
$55$	7.01829e15	0.205488
$56$	0	0
$57$	−5.48740e16	−1.14434
$58$	0	0
$59$	−9.14089e14	−0.0137371	−0.00686854	−	0.999976i	$-0.502186\pi$
$59$	−9.14089e14	−0.0137371	−0.00686854	+	0.999976i	$0.502186\pi$
$60$	0	0
$61$	−9.45933e16	−1.03568	−0.517841	−	0.855477i	$-0.673264\pi$
$61$	−9.45933e16	−1.03568	−0.517841	+	0.855477i	$0.673264\pi$
$62$	0	0
$63$	3.33982e16	0.269146
$64$	0	0
$65$	−1.37712e16	−0.0824697
$66$	0	0
$67$	−2.63147e17	−1.18165	−0.590824	−	0.806801i	$-0.701197\pi$
$67$	−2.63147e17	−1.18165	−0.590824	+	0.806801i	$0.701197\pi$
$68$	0	0
$69$	−2.63930e17	−0.896236
$70$	0	0
$71$	3.62022e17	0.937088	0.468544	−	0.883440i	$-0.344779\pi$
$71$	3.62022e17	0.937088	0.468544	+	0.883440i	$0.344779\pi$
$72$	0	0
$73$	−5.68735e17	−1.13069	−0.565344	−	0.824855i	$-0.691256\pi$
$73$	−5.68735e17	−1.13069	−0.565344	+	0.824855i	$0.691256\pi$
$74$	0	0
$75$	1.78405e17	0.274363
$76$	0	0
$77$	−1.17089e17	−0.140234
$78$	0	0
$79$	−1.10844e18	−1.04053	−0.520265	−	0.854005i	$-0.674167\pi$
$79$	−1.10844e18	−1.04053	−0.520265	+	0.854005i	$0.674167\pi$
$80$	0	0
$81$	−1.49157e18	−1.10417
$82$	0	0
$83$	−2.63110e18	−1.54488	−0.772441	−	0.635087i	$-0.780964\pi$
$83$	−2.63110e18	−1.54488	−0.772441	+	0.635087i	$0.780964\pi$
$84$	0	0
$85$	−9.37678e17	−0.439111
$86$	0	0
$87$	−5.66036e18	−2.12526
$88$	0	0
$89$	4.21152e18	1.27419	0.637094	−	0.770786i	$-0.280136\pi$
$89$	4.21152e18	1.27419	0.637094	+	0.770786i	$0.280136\pi$
$90$	0	0
$91$	2.29750e17	0.0562810
$92$	0	0
$93$	5.86241e18	1.16812
$94$	0	0
$95$	2.29166e18	0.373057
$96$	0	0
$97$	8.83410e18	1.17986	0.589930	−	0.807454i	$-0.299155\pi$
$97$	8.83410e18	1.17986	0.589930	+	0.807454i	$0.299155\pi$
$98$	0	0
$99$	−3.68308e18	−0.405207
$100$	0	0
$101$	−1.90143e19	−1.72992	−0.864961	−	0.501839i	$-0.832657\pi$
$101$	−1.90143e19	−1.72992	−0.864961	+	0.501839i	$0.832657\pi$
$102$	0	0
$103$	1.45262e19	1.09698	0.548491	−	0.836157i	$-0.315203\pi$
$103$	1.45262e19	1.09698	0.548491	+	0.836157i	$0.315203\pi$
$104$	0	0
$105$	−2.97640e18	−0.187237
$106$	0	0
$107$	−2.30227e19	−1.21063	−0.605314	−	0.795987i	$-0.706952\pi$
$107$	−2.30227e19	−1.21063	−0.605314	+	0.795987i	$0.706952\pi$
$108$	0	0
$109$	8.67782e18	0.382701	0.191350	−	0.981522i	$-0.438713\pi$
$109$	8.67782e18	0.382701	0.191350	+	0.981522i	$0.438713\pi$
$110$	0	0
$111$	2.48701e19	0.922802
$112$	0	0
$113$	2.06995e19	0.648208	0.324104	−	0.946021i	$-0.394937\pi$
$113$	2.06995e19	0.648208	0.324104	+	0.946021i	$0.394937\pi$
$114$	0	0
$115$	1.10223e19	0.292175
$116$	0	0
$117$	7.22688e18	0.162624
$118$	0	0
$119$	1.56436e19	0.299669
$120$	0	0
$121$	−4.82468e19	−0.788874
$122$	0	0
$123$	4.14976e19	0.580664
$124$	0	0
$125$	−7.45058e18	−0.0894427
$126$	0	0
$127$	1.24686e20	1.28731	0.643656	−	0.765315i	$-0.277417\pi$
$127$	1.24686e20	1.28731	0.643656	+	0.765315i	$0.277417\pi$
$128$	0	0
$129$	−1.70077e19	−0.151371
$130$	0	0
$131$	4.19201e19	0.322363	0.161181	−	0.986925i	$-0.448470\pi$
$131$	4.19201e19	0.322363	0.161181	+	0.986925i	$0.448470\pi$
$132$	0	0
$133$	−3.82326e19	−0.254591
$134$	0	0
$135$	1.25410e19	0.0724707
$136$	0	0
$137$	−1.79570e20	−0.902380	−0.451190	−	0.892428i	$-0.649000\pi$
$137$	−1.79570e20	−0.902380	−0.451190	+	0.892428i	$0.649000\pi$
$138$	0	0
$139$	1.39726e20	0.611837	0.305919	−	0.952058i	$-0.401036\pi$
$139$	1.39726e20	0.611837	0.305919	+	0.952058i	$0.401036\pi$
$140$	0	0
$141$	3.89245e20	1.48813
$142$	0	0
$143$	−2.53363e19	−0.0847326
$144$	0	0
$145$	2.36389e20	0.692837
$146$	0	0
$147$	−4.83445e20	−1.24403
$148$	0	0
$149$	−2.32818e20	−0.526922	−0.263461	−	0.964670i	$-0.584864\pi$
$149$	−2.32818e20	−0.526922	−0.263461	+	0.964670i	$0.584864\pi$
$150$	0	0
$151$	−9.16633e20	−1.82774	−0.913871	−	0.406005i	$-0.866921\pi$
$151$	−9.16633e20	−1.82774	−0.913871	+	0.406005i	$0.866921\pi$
$152$	0	0
$153$	4.92077e20	0.865894
$154$	0	0
$155$	−2.44827e20	−0.380810
$156$	0	0
$157$	−9.48000e20	−1.30545	−0.652727	−	0.757593i	$-0.726375\pi$
$157$	−9.48000e20	−1.30545	−0.652727	+	0.757593i	$0.726375\pi$
$158$	0	0
$159$	−1.09784e21	−1.34050
$160$	0	0
$161$	−1.83889e20	−0.199393
$162$	0	0
$163$	−1.35580e20	−0.130741	−0.0653707	−	0.997861i	$-0.520823\pi$
$163$	−1.35580e20	−0.130741	−0.0653707	+	0.997861i	$0.520823\pi$
$164$	0	0
$165$	3.28230e20	0.281891
$166$	0	0
$167$	1.14887e21	0.879962	0.439981	−	0.898007i	$-0.354985\pi$
$167$	1.14887e21	0.879962	0.439981	+	0.898007i	$0.354985\pi$
$168$	0	0
$169$	−1.41221e21	−0.965994
$170$	0	0
$171$	−1.20262e21	−0.735641
$172$	0	0
$173$	−1.20871e21	−0.662037	−0.331019	−	0.943624i	$-0.607392\pi$
$173$	−1.20871e21	−0.662037	−0.331019	+	0.943624i	$0.607392\pi$
$174$	0	0
$175$	1.24301e20	0.0610397
$176$	0	0
$177$	−4.27499e19	−0.0188447
$178$	0	0
$179$	−2.13691e20	−0.0846608	−0.0423304	−	0.999104i	$-0.513478\pi$
$179$	−2.13691e20	−0.0846608	−0.0423304	+	0.999104i	$0.513478\pi$
$180$	0	0
$181$	−1.55696e21	−0.555049	−0.277525	−	0.960718i	$-0.589514\pi$
$181$	−1.55696e21	−0.555049	−0.277525	+	0.960718i	$0.589514\pi$
$182$	0	0
$183$	−4.42392e21	−1.42076
$184$	0	0
$185$	−1.03863e21	−0.300835
$186$	0	0
$187$	−1.72514e21	−0.451159
$188$	0	0
$189$	−2.09227e20	−0.0494573
$190$	0	0
$191$	8.33121e21	1.78193	0.890966	−	0.454069i	$-0.150028\pi$
$191$	8.33121e21	1.78193	0.890966	+	0.454069i	$0.150028\pi$
$192$	0	0
$193$	−5.57024e21	−1.07914	−0.539572	−	0.841939i	$-0.681414\pi$
$193$	−5.57024e21	−1.07914	−0.539572	+	0.841939i	$0.681414\pi$
$194$	0	0
$195$	−6.44048e20	−0.113133
$196$	0	0
$197$	2.40017e20	0.0382660	0.0191330	−	0.999817i	$-0.493909\pi$
$197$	2.40017e20	0.0382660	0.0191330	+	0.999817i	$0.493909\pi$
$198$	0	0
$199$	5.67132e21	0.821448	0.410724	−	0.911760i	$-0.365276\pi$
$199$	5.67132e21	0.821448	0.410724	+	0.911760i	$0.365276\pi$
$200$	0	0
$201$	−1.23068e22	−1.62100
$202$	0	0
$203$	−3.94377e21	−0.472823
$204$	0	0
$205$	−1.73303e21	−0.189297
$206$	0	0
$207$	−5.78431e21	−0.576147
$208$	0	0
$209$	4.21620e21	0.383294
$210$	0	0
$211$	1.22489e22	1.01722	0.508608	−	0.860998i	$-0.330160\pi$
$211$	1.22489e22	1.01722	0.508608	+	0.860998i	$0.330160\pi$
$212$	0	0
$213$	1.69310e22	1.28551
$214$	0	0
$215$	7.10278e20	0.0493473
$216$	0	0
$217$	4.08454e21	0.259882
$218$	0	0
$219$	−2.65985e22	−1.55109
$220$	0	0
$221$	3.38505e21	0.181067
$222$	0	0
$223$	−3.14319e22	−1.54338	−0.771692	−	0.635996i	$-0.780589\pi$
$223$	−3.14319e22	−1.54338	−0.771692	+	0.635996i	$0.780589\pi$
$224$	0	0
$225$	3.90994e21	0.176374
$226$	0	0
$227$	−1.53381e22	−0.636100	−0.318050	−	0.948074i	$-0.603028\pi$
$227$	−1.53381e22	−0.636100	−0.318050	+	0.948074i	$0.603028\pi$
$228$	0	0
$229$	9.47620e21	0.361574	0.180787	−	0.983522i	$-0.442136\pi$
$229$	9.47620e21	0.361574	0.180787	+	0.983522i	$0.442136\pi$
$230$	0	0
$231$	−5.47599e21	−0.192375
$232$	0	0
$233$	−6.36950e21	−0.206169	−0.103085	−	0.994673i	$-0.532871\pi$
$233$	−6.36950e21	−0.206169	−0.103085	+	0.994673i	$0.532871\pi$
$234$	0	0
$235$	−1.62557e22	−0.485134
$236$	0	0
$237$	−5.18393e22	−1.42741
$238$	0	0
$239$	1.36502e22	0.347023	0.173511	−	0.984832i	$-0.444489\pi$
$239$	1.36502e22	0.347023	0.173511	+	0.984832i	$0.444489\pi$
$240$	0	0
$241$	−5.73885e22	−1.34792	−0.673959	−	0.738769i	$-0.735407\pi$
$241$	−5.73885e22	−1.34792	−0.673959	+	0.738769i	$0.735407\pi$
$242$	0	0
$243$	−6.22948e22	−1.35267
$244$	0	0
$245$	2.01897e22	0.405557
$246$	0	0
$247$	−8.27296e21	−0.153830
$248$	0	0
$249$	−1.23051e23	−2.11929
$250$	0	0
$251$	−6.93332e22	−1.10673	−0.553364	−	0.832939i	$-0.686656\pi$
$251$	−6.93332e22	−1.10673	−0.553364	+	0.832939i	$0.686656\pi$
$252$	0	0
$253$	2.02789e22	0.300192
$254$	0	0
$255$	−4.38532e22	−0.602378
$256$	0	0
$257$	−1.26982e23	−1.61949	−0.809743	−	0.586785i	$-0.800393\pi$
$257$	−1.26982e23	−1.61949	−0.809743	+	0.586785i	$0.800393\pi$
$258$	0	0
$259$	1.73278e22	0.205303
$260$	0	0
$261$	−1.24053e23	−1.36622
$262$	0	0
$263$	−1.21132e23	−1.24074	−0.620370	−	0.784309i	$-0.713018\pi$
$263$	−1.21132e23	−1.24074	−0.620370	+	0.784309i	$0.713018\pi$
$264$	0	0
$265$	4.58483e22	0.437006
$266$	0	0
$267$	1.96963e23	1.74795
$268$	0	0
$269$	6.14602e22	0.508098	0.254049	−	0.967191i	$-0.418238\pi$
$269$	6.14602e22	0.508098	0.254049	+	0.967191i	$0.418238\pi$
$270$	0	0
$271$	−8.72971e22	−0.672653	−0.336327	−	0.941745i	$-0.609185\pi$
$271$	−8.72971e22	−0.672653	−0.336327	+	0.941745i	$0.609185\pi$
$272$	0	0
$273$	1.07449e22	0.0772071
$274$	0	0
$275$	−1.37076e22	−0.0918969
$276$	0	0
$277$	−9.12791e22	−0.571233	−0.285617	−	0.958344i	$-0.592198\pi$
$277$	−9.12791e22	−0.571233	−0.285617	+	0.958344i	$0.592198\pi$
$278$	0	0
$279$	1.28481e23	0.750929
$280$	0	0
$281$	−2.94383e23	−1.60769	−0.803846	−	0.594837i	$-0.797217\pi$
$281$	−2.94383e23	−1.60769	−0.803846	+	0.594837i	$0.797217\pi$
$282$	0	0
$283$	−9.49890e22	−0.484956	−0.242478	−	0.970157i	$-0.577960\pi$
$283$	−9.49890e22	−0.484956	−0.242478	+	0.970157i	$0.577960\pi$
$284$	0	0
$285$	1.07176e23	0.511765
$286$	0	0
$287$	2.89128e22	0.129185
$288$	0	0
$289$	−8.58481e21	−0.0359088
$290$	0	0
$291$	4.13151e23	1.61855
$292$	0	0
$293$	4.28150e23	1.57164	0.785821	−	0.618454i	$-0.212241\pi$
$293$	4.28150e23	1.57164	0.785821	+	0.618454i	$0.212241\pi$
$294$	0	0
$295$	1.78533e21	0.00614341
$296$	0	0
$297$	2.30730e22	0.0744592
$298$	0	0
$299$	−3.97909e22	−0.120478
$300$	0	0
$301$	−1.18498e22	−0.0336768
$302$	0	0
$303$	−8.89256e23	−2.37313
$304$	0	0
$305$	1.84752e23	0.463172
$306$	0	0
$307$	−4.65406e23	−1.09652	−0.548261	−	0.836307i	$-0.684710\pi$
$307$	−4.65406e23	−1.09652	−0.548261	+	0.836307i	$0.684710\pi$
$308$	0	0
$309$	6.79360e23	1.50485
$310$	0	0
$311$	5.30167e23	1.10456	0.552280	−	0.833659i	$-0.313758\pi$
$311$	5.30167e23	1.10456	0.552280	+	0.833659i	$0.313758\pi$
$312$	0	0
$313$	2.65137e23	0.519755	0.259878	−	0.965642i	$-0.416318\pi$
$313$	2.65137e23	0.519755	0.259878	+	0.965642i	$0.416318\pi$
$314$	0	0
$315$	−6.52309e22	−0.120366
$316$	0	0
$317$	8.14658e23	1.41551	0.707754	−	0.706459i	$-0.249708\pi$
$317$	8.14658e23	1.41551	0.707754	+	0.706459i	$0.249708\pi$
$318$	0	0
$319$	4.34909e23	0.711848
$320$	0	0
$321$	−1.07672e24	−1.66076
$322$	0	0
$323$	−5.63305e23	−0.819067
$324$	0	0
$325$	2.68968e22	0.0368816
$326$	0	0
$327$	4.05843e23	0.524994
$328$	0	0
$329$	2.71200e23	0.331077
$330$	0	0
$331$	1.17921e24	1.35902	0.679511	−	0.733666i	$-0.262192\pi$
$331$	1.17921e24	1.35902	0.679511	+	0.733666i	$0.262192\pi$
$332$	0	0
$333$	5.45054e23	0.593224
$334$	0	0
$335$	5.13959e23	0.528449
$336$	0	0
$337$	−1.91941e24	−1.86502	−0.932508	−	0.361149i	$-0.882385\pi$
$337$	−1.91941e24	−1.86502	−0.932508	+	0.361149i	$0.882385\pi$
$338$	0	0
$339$	9.68070e23	0.889221
$340$	0	0
$341$	−4.50434e23	−0.391259
$342$	0	0
$343$	−7.08263e23	−0.581969
$344$	0	0
$345$	5.15489e23	0.400809
$346$	0	0
$347$	3.20014e23	0.235526	0.117763	−	0.993042i	$-0.462428\pi$
$347$	3.20014e23	0.235526	0.117763	+	0.993042i	$0.462428\pi$
$348$	0	0
$349$	1.07756e24	0.750928	0.375464	−	0.926837i	$-0.377483\pi$
$349$	1.07756e24	0.750928	0.375464	+	0.926837i	$0.377483\pi$
$350$	0	0
$351$	−4.52735e22	−0.0298832
$352$	0	0
$353$	−1.32729e24	−0.830051	−0.415025	−	0.909810i	$-0.636227\pi$
$353$	−1.32729e24	−0.830051	−0.415025	+	0.909810i	$0.636227\pi$
$354$	0	0
$355$	−7.07074e23	−0.419079
$356$	0	0
$357$	7.31619e23	0.411090
$358$	0	0
$359$	2.00966e24	1.07084	0.535422	−	0.844585i	$-0.320153\pi$
$359$	2.00966e24	1.07084	0.535422	+	0.844585i	$0.320153\pi$
$360$	0	0
$361$	−6.01719e23	−0.304141
$362$	0	0
$363$	−2.25640e24	−1.08219
$364$	0	0
$365$	1.11081e24	0.505659
$366$	0	0
$367$	3.11562e24	1.34653	0.673266	−	0.739400i	$-0.264891\pi$
$367$	3.11562e24	1.34653	0.673266	+	0.739400i	$0.264891\pi$
$368$	0	0
$369$	9.09465e23	0.373281
$370$	0	0
$371$	−7.64904e23	−0.298232
$372$	0	0
$373$	−2.67080e24	−0.989480	−0.494740	−	0.869041i	$-0.664737\pi$
$373$	−2.67080e24	−0.989480	−0.494740	+	0.869041i	$0.664737\pi$
$374$	0	0
$375$	−3.48447e23	−0.122699
$376$	0	0
$377$	−8.53372e23	−0.285690
$378$	0	0
$379$	6.23103e24	1.98375	0.991875	−	0.127215i	$-0.0406038\pi$
$379$	6.23103e24	1.98375	0.991875	+	0.127215i	$0.0406038\pi$
$380$	0	0
$381$	5.83131e24	1.76595
$382$	0	0
$383$	−4.18147e23	−0.120487	−0.0602435	−	0.998184i	$-0.519188\pi$
$383$	−4.18147e23	−0.120487	−0.0602435	+	0.998184i	$0.519188\pi$
$384$	0	0
$385$	2.28689e23	0.0627146
$386$	0	0
$387$	−3.72742e23	−0.0973091
$388$	0	0
$389$	4.55544e24	1.13242	0.566212	−	0.824260i	$-0.308408\pi$
$389$	4.55544e24	1.13242	0.566212	+	0.824260i	$0.308408\pi$
$390$	0	0
$391$	−2.70935e24	−0.641485
$392$	0	0
$393$	1.96051e24	0.442222
$394$	0	0
$395$	2.16492e24	0.465339
$396$	0	0
$397$	2.50919e23	0.0514072	0.0257036	−	0.999670i	$-0.491817\pi$
$397$	2.50919e23	0.0514072	0.0257036	+	0.999670i	$0.491817\pi$
$398$	0	0
$399$	−1.78805e24	−0.349252
$400$	0	0
$401$	3.50470e24	0.652799	0.326399	−	0.945232i	$-0.394165\pi$
$401$	3.50470e24	0.652799	0.326399	+	0.945232i	$0.394165\pi$
$402$	0	0
$403$	8.83834e23	0.157026
$404$	0	0
$405$	2.91323e24	0.493801
$406$	0	0
$407$	−1.91087e24	−0.309090
$408$	0	0
$409$	9.34948e24	1.44350	0.721749	−	0.692155i	$-0.243339\pi$
$409$	9.34948e24	1.44350	0.721749	+	0.692155i	$0.243339\pi$
$410$	0	0
$411$	−8.39811e24	−1.23790
$412$	0	0
$413$	−2.97853e22	−0.00419254
$414$	0	0
$415$	5.13886e24	0.690892
$416$	0	0
$417$	6.53467e24	0.839326
$418$	0	0
$419$	4.56745e24	0.560584	0.280292	−	0.959915i	$-0.409569\pi$
$419$	4.56745e24	0.560584	0.280292	+	0.959915i	$0.409569\pi$
$420$	0	0
$421$	−9.22031e24	−1.08160	−0.540799	−	0.841152i	$-0.681878\pi$
$421$	−9.22031e24	−1.08160	−0.540799	+	0.841152i	$0.681878\pi$
$422$	0	0
$423$	8.53072e24	0.956648
$424$	0	0
$425$	1.83140e24	0.196376
$426$	0	0
$427$	−3.08230e24	−0.316089
$428$	0	0
$429$	−1.18492e24	−0.116237
$430$	0	0
$431$	3.48157e24	0.326769	0.163385	−	0.986562i	$-0.447759\pi$
$431$	3.48157e24	0.326769	0.163385	+	0.986562i	$0.447759\pi$
$432$	0	0
$433$	−1.75714e25	−1.57824	−0.789118	−	0.614242i	$-0.789462\pi$
$433$	−1.75714e25	−1.57824	−0.789118	+	0.614242i	$0.789462\pi$
$434$	0	0
$435$	1.10554e25	0.950443
$436$	0	0
$437$	6.62158e24	0.544989
$438$	0	0
$439$	1.10377e25	0.869890	0.434945	−	0.900457i	$-0.356768\pi$
$439$	1.10377e25	0.869890	0.434945	+	0.900457i	$0.356768\pi$
$440$	0	0
$441$	−1.05952e25	−0.799729
$442$	0	0
$443$	3.18345e24	0.230177	0.115089	−	0.993355i	$-0.463285\pi$
$443$	3.18345e24	0.230177	0.115089	+	0.993355i	$0.463285\pi$
$444$	0	0
$445$	−8.22562e24	−0.569834
$446$	0	0
$447$	−1.08884e25	−0.722839
$448$	0	0
$449$	1.59901e25	1.01744	0.508722	−	0.860931i	$-0.330118\pi$
$449$	1.59901e25	1.01744	0.508722	+	0.860931i	$0.330118\pi$
$450$	0	0
$451$	−3.18844e24	−0.194492
$452$	0	0
$453$	−4.28689e25	−2.50732
$454$	0	0
$455$	−4.48730e23	−0.0251696
$456$	0	0
$457$	4.23958e24	0.228097	0.114048	−	0.993475i	$-0.463618\pi$
$457$	4.23958e24	0.228097	0.114048	+	0.993475i	$0.463618\pi$
$458$	0	0
$459$	−3.08267e24	−0.159113
$460$	0	0
$461$	1.96691e24	0.0974150	0.0487075	−	0.998813i	$-0.484490\pi$
$461$	1.96691e24	0.0974150	0.0487075	+	0.998813i	$0.484490\pi$
$462$	0	0
$463$	−5.01995e24	−0.238605	−0.119303	−	0.992858i	$-0.538066\pi$
$463$	−5.01995e24	−0.238605	−0.119303	+	0.992858i	$0.538066\pi$
$464$	0	0
$465$	−1.14500e25	−0.522400
$466$	0	0
$467$	−3.30902e25	−1.44940	−0.724702	−	0.689062i	$-0.758023\pi$
$467$	−3.30902e25	−1.44940	−0.724702	+	0.689062i	$0.758023\pi$
$468$	0	0
$469$	−8.57458e24	−0.360637
$470$	0	0
$471$	−4.43359e25	−1.79084
$472$	0	0
$473$	1.30677e24	0.0507013
$474$	0	0
$475$	−4.47589e24	−0.166836
$476$	0	0
$477$	−2.40604e25	−0.861743
$478$	0	0
$479$	−3.29895e25	−1.13550	−0.567751	−	0.823200i	$-0.692186\pi$
$479$	−3.29895e25	−1.13550	−0.567751	+	0.823200i	$0.692186\pi$
$480$	0	0
$481$	3.74948e24	0.124049
$482$	0	0
$483$	−8.60009e24	−0.273530
$484$	0	0
$485$	−1.72541e25	−0.527650
$486$	0	0
$487$	3.11382e25	0.915731	0.457865	−	0.889021i	$-0.348614\pi$
$487$	3.11382e25	0.915731	0.457865	+	0.889021i	$0.348614\pi$
$488$	0	0
$489$	−6.34077e24	−0.179353
$490$	0	0
$491$	−4.50248e25	−1.22512	−0.612558	−	0.790425i	$-0.709860\pi$
$491$	−4.50248e25	−1.22512	−0.612558	+	0.790425i	$0.709860\pi$
$492$	0	0
$493$	−5.81060e25	−1.52116
$494$	0	0
$495$	7.19351e24	0.181214
$496$	0	0
$497$	1.17964e25	0.285998
$498$	0	0
$499$	3.87532e25	0.904383	0.452191	−	0.891921i	$-0.350642\pi$
$499$	3.87532e25	0.904383	0.452191	+	0.891921i	$0.350642\pi$
$500$	0	0
$501$	5.37301e25	1.20714
$502$	0	0
$503$	−9.01002e24	−0.194908	−0.0974540	−	0.995240i	$-0.531070\pi$
$503$	−9.01002e24	−0.194908	−0.0974540	+	0.995240i	$0.531070\pi$
$504$	0	0
$505$	3.71372e25	0.773645
$506$	0	0
$507$	−6.60458e25	−1.32516
$508$	0	0
$509$	7.39109e24	0.142853	0.0714265	−	0.997446i	$-0.477245\pi$
$509$	7.39109e24	0.142853	0.0714265	+	0.997446i	$0.477245\pi$
$510$	0	0
$511$	−1.85321e25	−0.345084
$512$	0	0
$513$	7.53395e24	0.135179
$514$	0	0
$515$	−2.83716e25	−0.490585
$516$	0	0
$517$	−2.99073e25	−0.498445
$518$	0	0
$519$	−5.65285e25	−0.908192
$520$	0	0
$521$	1.11961e26	1.73423	0.867117	−	0.498104i	$-0.165970\pi$
$521$	1.11961e26	1.73423	0.867117	+	0.498104i	$0.165970\pi$
$522$	0	0
$523$	7.80297e25	1.16545	0.582726	−	0.812669i	$-0.301986\pi$
$523$	7.80297e25	1.16545	0.582726	+	0.812669i	$0.301986\pi$
$524$	0	0
$525$	5.81328e24	0.0837351
$526$	0	0
$527$	6.01801e25	0.836088
$528$	0	0
$529$	−4.27673e25	−0.573170
$530$	0	0
$531$	−9.36910e23	−0.0121143
$532$	0	0
$533$	6.25630e24	0.0780565
$534$	0	0
$535$	4.49662e25	0.541409
$536$	0	0
$537$	−9.99386e24	−0.116139
$538$	0	0
$539$	3.71451e25	0.416685
$540$	0	0
$541$	−1.01723e26	−1.10165	−0.550825	−	0.834621i	$-0.685687\pi$
$541$	−1.01723e26	−1.10165	−0.550825	+	0.834621i	$0.685687\pi$
$542$	0	0
$543$	−7.28157e25	−0.761424
$544$	0	0
$545$	−1.69489e25	−0.171149
$546$	0	0
$547$	1.25004e26	1.21912	0.609558	−	0.792741i	$-0.291347\pi$
$547$	1.25004e26	1.21912	0.609558	+	0.792741i	$0.291347\pi$
$548$	0	0
$549$	−9.69549e25	−0.913340
$550$	0	0
$551$	1.42009e26	1.29234
$552$	0	0
$553$	−3.61182e25	−0.317568
$554$	0	0
$555$	−4.85743e25	−0.412690
$556$	0	0
$557$	−4.11885e25	−0.338183	−0.169091	−	0.985600i	$-0.554083\pi$
$557$	−4.11885e25	−0.338183	−0.169091	+	0.985600i	$0.554083\pi$
$558$	0	0
$559$	−2.56413e24	−0.0203483
$560$	0	0
$561$	−8.06812e25	−0.618906
$562$	0	0
$563$	4.23839e25	0.314320	0.157160	−	0.987573i	$-0.449766\pi$
$563$	4.23839e25	0.314320	0.157160	+	0.987573i	$0.449766\pi$
$564$	0	0
$565$	−4.04287e25	−0.289888
$566$	0	0
$567$	−4.86026e25	−0.336992
$568$	0	0
$569$	2.56600e25	0.172064	0.0860320	−	0.996292i	$-0.472581\pi$
$569$	2.56600e25	0.172064	0.0860320	+	0.996292i	$0.472581\pi$
$570$	0	0
$571$	2.29740e26	1.49003	0.745013	−	0.667050i	$-0.232443\pi$
$571$	2.29740e26	1.49003	0.745013	+	0.667050i	$0.232443\pi$
$572$	0	0
$573$	3.89633e26	2.44448
$574$	0	0
$575$	−2.15279e25	−0.130664
$576$	0	0
$577$	1.01285e26	0.594806	0.297403	−	0.954752i	$-0.403880\pi$
$577$	1.01285e26	0.594806	0.297403	+	0.954752i	$0.403880\pi$
$578$	0	0
$579$	−2.60508e26	−1.48039
$580$	0	0
$581$	−8.57336e25	−0.471496
$582$	0	0
$583$	8.43518e25	0.448996
$584$	0	0
$585$	−1.41150e25	−0.0727277
$586$	0	0
$587$	−2.47017e26	−1.23216	−0.616078	−	0.787685i	$-0.711279\pi$
$587$	−2.47017e26	−1.23216	−0.616078	+	0.787685i	$0.711279\pi$
$588$	0	0
$589$	−1.47078e26	−0.710320
$590$	0	0
$591$	1.12251e25	0.0524939
$592$	0	0
$593$	2.36661e26	1.07178	0.535891	−	0.844287i	$-0.319976\pi$
$593$	2.36661e26	1.07178	0.535891	+	0.844287i	$0.319976\pi$
$594$	0	0
$595$	−3.05540e25	−0.134016
$596$	0	0
$597$	2.65235e26	1.12687
$598$	0	0
$599$	−1.37934e26	−0.567696	−0.283848	−	0.958869i	$-0.591611\pi$
$599$	−1.37934e26	−0.567696	−0.283848	+	0.958869i	$0.591611\pi$
$600$	0	0
$601$	−3.57697e26	−1.42629	−0.713145	−	0.701017i	$-0.752730\pi$
$601$	−3.57697e26	−1.42629	−0.713145	+	0.701017i	$0.752730\pi$
$602$	0	0
$603$	−2.69717e26	−1.04206
$604$	0	0
$605$	9.42321e25	0.352795
$606$	0	0
$607$	2.68308e26	0.973512	0.486756	−	0.873538i	$-0.338180\pi$
$607$	2.68308e26	0.973512	0.486756	+	0.873538i	$0.338180\pi$
$608$	0	0
$609$	−1.84441e26	−0.648625
$610$	0	0
$611$	5.86837e25	0.200044
$612$	0	0
$613$	3.31443e26	1.09530	0.547651	−	0.836707i	$-0.315522\pi$
$613$	3.31443e26	1.09530	0.547651	+	0.836707i	$0.315522\pi$
$614$	0	0
$615$	−8.10501e25	−0.259681
$616$	0	0
$617$	−1.25555e26	−0.390053	−0.195026	−	0.980798i	$-0.562479\pi$
$617$	−1.25555e26	−0.390053	−0.195026	+	0.980798i	$0.562479\pi$
$618$	0	0
$619$	−1.90269e26	−0.573202	−0.286601	−	0.958050i	$-0.592525\pi$
$619$	−1.90269e26	−0.573202	−0.286601	+	0.958050i	$0.592525\pi$
$620$	0	0
$621$	3.62364e25	0.105870
$622$	0	0
$623$	1.37231e26	0.388881
$624$	0	0
$625$	1.45519e25	0.0400000
$626$	0	0
$627$	1.97182e26	0.525807
$628$	0	0
$629$	2.55302e26	0.660499
$630$	0	0
$631$	5.76731e26	1.44775	0.723876	−	0.689930i	$-0.242359\pi$
$631$	5.76731e26	1.44775	0.723876	+	0.689930i	$0.242359\pi$
$632$	0	0
$633$	5.72854e26	1.39543
$634$	0	0
$635$	−2.43528e26	−0.575703
$636$	0	0
$637$	−7.28855e25	−0.167231
$638$	0	0
$639$	3.71060e26	0.826392
$640$	0	0
$641$	4.54638e26	0.982911	0.491456	−	0.870903i	$-0.336465\pi$
$641$	4.54638e26	0.982911	0.491456	+	0.870903i	$0.336465\pi$
$642$	0	0
$643$	1.28238e26	0.269161	0.134581	−	0.990903i	$-0.457031\pi$
$643$	1.28238e26	0.269161	0.134581	+	0.990903i	$0.457031\pi$
$644$	0	0
$645$	3.32181e25	0.0676952
$646$	0	0
$647$	6.05944e26	1.19906	0.599532	−	0.800351i	$-0.295354\pi$
$647$	6.05944e26	1.19906	0.599532	+	0.800351i	$0.295354\pi$
$648$	0	0
$649$	3.28466e24	0.00631198
$650$	0	0
$651$	1.91025e26	0.356509
$652$	0	0
$653$	7.23139e26	1.31083	0.655415	−	0.755269i	$-0.272494\pi$
$653$	7.23139e26	1.31083	0.655415	+	0.755269i	$0.272494\pi$
$654$	0	0
$655$	−8.18752e25	−0.144165
$656$	0	0
$657$	−5.82934e26	−0.997122
$658$	0	0
$659$	1.18676e26	0.197220	0.0986102	−	0.995126i	$-0.468560\pi$
$659$	1.18676e26	0.197220	0.0986102	+	0.995126i	$0.468560\pi$
$660$	0	0
$661$	9.85802e26	1.59175	0.795877	−	0.605459i	$-0.207010\pi$
$661$	9.85802e26	1.59175	0.795877	+	0.605459i	$0.207010\pi$
$662$	0	0
$663$	1.58311e26	0.248390
$664$	0	0
$665$	7.46731e25	0.113857
$666$	0	0
$667$	6.83029e26	1.01215
$668$	0	0
$669$	−1.47000e27	−2.11724
$670$	0	0
$671$	3.39908e26	0.475880
$672$	0	0
$673$	−8.20100e26	−1.11615	−0.558077	−	0.829789i	$-0.688461\pi$
$673$	−8.20100e26	−1.11615	−0.558077	+	0.829789i	$0.688461\pi$
$674$	0	0
$675$	−2.44942e25	−0.0324099
$676$	0	0
$677$	3.93371e26	0.506070	0.253035	−	0.967457i	$-0.418571\pi$
$677$	3.93371e26	0.506070	0.253035	+	0.967457i	$0.418571\pi$
$678$	0	0
$679$	2.87857e26	0.360092
$680$	0	0
$681$	−7.17329e26	−0.872611
$682$	0	0
$683$	5.45808e26	0.645719	0.322859	−	0.946447i	$-0.395356\pi$
$683$	5.45808e26	0.645719	0.322859	+	0.946447i	$0.395356\pi$
$684$	0	0
$685$	3.50723e26	0.403557
$686$	0	0
$687$	4.43181e26	0.496012
$688$	0	0
$689$	−1.65514e26	−0.180199
$690$	0	0
$691$	−4.58871e26	−0.486014	−0.243007	−	0.970024i	$-0.578134\pi$
$691$	−4.58871e26	−0.486014	−0.243007	+	0.970024i	$0.578134\pi$
$692$	0	0
$693$	−1.20012e26	−0.123669
$694$	0	0
$695$	−2.72902e26	−0.273622
$696$	0	0
$697$	4.25991e26	0.415613
$698$	0	0
$699$	−2.97887e26	−0.282826
$700$	0	0
$701$	−3.46098e26	−0.319800	−0.159900	−	0.987133i	$-0.551117\pi$
$701$	−3.46098e26	−0.319800	−0.159900	+	0.987133i	$0.551117\pi$
$702$	0	0
$703$	−6.23950e26	−0.561144
$704$	0	0
$705$	−7.60245e26	−0.665513
$706$	0	0
$707$	−6.19575e26	−0.527970
$708$	0	0
$709$	2.00209e27	1.66090	0.830450	−	0.557093i	$-0.188083\pi$
$709$	2.00209e27	1.66090	0.830450	+	0.557093i	$0.188083\pi$
$710$	0	0
$711$	−1.13611e27	−0.917614
$712$	0	0
$713$	−7.07410e26	−0.556315
$714$	0	0
$715$	4.94849e25	0.0378935
$716$	0	0
$717$	6.38388e26	0.476050
$718$	0	0
$719$	−2.29024e27	−1.66324	−0.831621	−	0.555343i	$-0.812587\pi$
$719$	−2.29024e27	−1.66324	−0.831621	+	0.555343i	$0.812587\pi$
$720$	0	0
$721$	4.73333e26	0.334797
$722$	0	0
$723$	−2.68394e27	−1.84909
$724$	0	0
$725$	−4.61697e26	−0.309846
$726$	0	0
$727$	−1.38455e27	−0.905171	−0.452585	−	0.891721i	$-0.649498\pi$
$727$	−1.38455e27	−0.905171	−0.452585	+	0.891721i	$0.649498\pi$
$728$	0	0
$729$	−1.17979e27	−0.751439
$730$	0	0
$731$	−1.74591e26	−0.108345
$732$	0	0
$733$	−9.23088e26	−0.558156	−0.279078	−	0.960268i	$-0.590029\pi$
$733$	−9.23088e26	−0.558156	−0.279078	+	0.960268i	$0.590029\pi$
$734$	0	0
$735$	9.44228e26	0.556349
$736$	0	0
$737$	9.45584e26	0.542949
$738$	0	0
$739$	−9.15481e26	−0.512303	−0.256152	−	0.966637i	$-0.582455\pi$
$739$	−9.15481e26	−0.512303	−0.256152	+	0.966637i	$0.582455\pi$
$740$	0	0
$741$	−3.86908e26	−0.211026
$742$	0	0
$743$	2.45243e27	1.30378	0.651888	−	0.758316i	$-0.273977\pi$
$743$	2.45243e27	1.30378	0.651888	+	0.758316i	$0.273977\pi$
$744$	0	0
$745$	4.54722e26	0.235647
$746$	0	0
$747$	−2.69679e27	−1.36239
$748$	0	0
$749$	−7.50189e26	−0.369482
$750$	0	0
$751$	7.87763e26	0.378282	0.189141	−	0.981950i	$-0.439430\pi$
$751$	7.87763e26	0.378282	0.189141	+	0.981950i	$0.439430\pi$
$752$	0	0
$753$	−3.24256e27	−1.51822
$754$	0	0
$755$	1.79030e27	0.817391
$756$	0	0
$757$	1.18712e26	0.0528547	0.0264274	−	0.999651i	$-0.491587\pi$
$757$	1.18712e26	0.0528547	0.0264274	+	0.999651i	$0.491587\pi$
$758$	0	0
$759$	9.48397e26	0.411807
$760$	0	0
$761$	−1.88106e27	−0.796617	−0.398308	−	0.917252i	$-0.630403\pi$
$761$	−1.88106e27	−0.796617	−0.398308	+	0.917252i	$0.630403\pi$
$762$	0	0
$763$	2.82765e26	0.116800
$764$	0	0
$765$	−9.61088e26	−0.387240
$766$	0	0
$767$	−6.44510e24	−0.00253323
$768$	0	0
$769$	2.49749e27	0.957642	0.478821	−	0.877913i	$-0.341064\pi$
$769$	2.49749e27	0.957642	0.478821	+	0.877913i	$0.341064\pi$
$770$	0	0
$771$	−5.93866e27	−2.22163
$772$	0	0
$773$	−2.93269e27	−1.07044	−0.535219	−	0.844713i	$-0.679771\pi$
$773$	−2.93269e27	−1.07044	−0.535219	+	0.844713i	$0.679771\pi$
$774$	0	0
$775$	4.78178e26	0.170303
$776$	0	0
$777$	8.10384e26	0.281638
$778$	0	0
$779$	−1.04111e27	−0.353094
$780$	0	0
$781$	−1.30088e27	−0.430578
$782$	0	0
$783$	7.77142e26	0.251052
$784$	0	0
$785$	1.85156e27	0.583817
$786$	0	0
$787$	−2.44798e27	−0.753440	−0.376720	−	0.926327i	$-0.622948\pi$
$787$	−2.44798e27	−0.753440	−0.376720	+	0.926327i	$0.622948\pi$
$788$	0	0
$789$	−5.66510e27	−1.70206
$790$	0	0
$791$	6.74487e26	0.197832
$792$	0	0
$793$	−6.66963e26	−0.190988
$794$	0	0
$795$	2.14422e27	0.599490
$796$	0	0
$797$	−3.77804e27	−1.03137	−0.515683	−	0.856780i	$-0.672462\pi$
$797$	−3.77804e27	−1.03137	−0.515683	+	0.856780i	$0.672462\pi$
$798$	0	0
$799$	3.99577e27	1.06514
$800$	0	0
$801$	4.31666e27	1.12367
$802$	0	0
$803$	2.04367e27	0.519533
$804$	0	0
$805$	3.59158e26	0.0891713
$806$	0	0
$807$	2.87436e27	0.697016
$808$	0	0
$809$	4.60493e27	1.09072	0.545358	−	0.838203i	$-0.316394\pi$
$809$	4.60493e27	1.09072	0.545358	+	0.838203i	$0.316394\pi$
$810$	0	0
$811$	−3.52743e27	−0.816133	−0.408066	−	0.912952i	$-0.633797\pi$
$811$	−3.52743e27	−0.816133	−0.408066	+	0.912952i	$0.633797\pi$
$812$	0	0
$813$	−4.08269e27	−0.922755
$814$	0	0
$815$	2.64805e26	0.0584693
$816$	0	0
$817$	4.26696e26	0.0920468
$818$	0	0
$819$	2.35486e26	0.0496327
$820$	0	0
$821$	1.49053e27	0.306958	0.153479	−	0.988152i	$-0.450952\pi$
$821$	1.49053e27	0.306958	0.153479	+	0.988152i	$0.450952\pi$
$822$	0	0
$823$	5.37479e27	1.08159	0.540796	−	0.841154i	$-0.318123\pi$
$823$	5.37479e27	1.08159	0.540796	+	0.841154i	$0.318123\pi$
$824$	0	0
$825$	−6.41074e26	−0.126065
$826$	0	0
$827$	8.84455e27	1.69970	0.849852	−	0.527021i	$-0.176691\pi$
$827$	8.84455e27	1.69970	0.849852	+	0.527021i	$0.176691\pi$
$828$	0	0
$829$	8.03394e27	1.50890	0.754450	−	0.656357i	$-0.227904\pi$
$829$	8.03394e27	1.50890	0.754450	+	0.656357i	$0.227904\pi$
$830$	0	0
$831$	−4.26892e27	−0.783626
$832$	0	0
$833$	−4.96277e27	−0.890423
$834$	0	0
$835$	−2.24388e27	−0.393531
$836$	0	0
$837$	−8.04882e26	−0.137988
$838$	0	0
$839$	8.98940e27	1.50658	0.753290	−	0.657688i	$-0.228466\pi$
$839$	8.98940e27	1.50658	0.753290	+	0.657688i	$0.228466\pi$
$840$	0	0
$841$	8.54527e27	1.40012
$842$	0	0
$843$	−1.37677e28	−2.20545
$844$	0	0
$845$	2.75821e27	0.432006
$846$	0	0
$847$	−1.57211e27	−0.240763
$848$	0	0
$849$	−4.44243e27	−0.665269
$850$	0	0
$851$	−3.00104e27	−0.439482
$852$	0	0
$853$	−4.12410e27	−0.590626	−0.295313	−	0.955400i	$-0.595424\pi$
$853$	−4.12410e27	−0.590626	−0.295313	+	0.955400i	$0.595424\pi$
$854$	0	0
$855$	2.34887e27	0.328989
$856$	0	0
$857$	1.22822e28	1.68251	0.841253	−	0.540641i	$-0.181818\pi$
$857$	1.22822e28	1.68251	0.841253	+	0.540641i	$0.181818\pi$
$858$	0	0
$859$	−1.29590e28	−1.73635	−0.868174	−	0.496261i	$-0.834706\pi$
$859$	−1.29590e28	−1.73635	−0.868174	+	0.496261i	$0.834706\pi$
$860$	0	0
$861$	1.35219e27	0.177218
$862$	0	0
$863$	−4.58089e27	−0.587282	−0.293641	−	0.955916i	$-0.594867\pi$
$863$	−4.58089e27	−0.587282	−0.293641	+	0.955916i	$0.594867\pi$
$864$	0	0
$865$	2.36075e27	0.296072
$866$	0	0
$867$	−4.01493e26	−0.0492602
$868$	0	0
$869$	3.98303e27	0.478107
$870$	0	0
$871$	−1.85541e27	−0.217905
$872$	0	0
$873$	9.05465e27	1.04049
$874$	0	0
$875$	−2.42775e26	−0.0272978
$876$	0	0
$877$	8.28844e27	0.911961	0.455981	−	0.889990i	$-0.349289\pi$
$877$	8.28844e27	0.911961	0.455981	+	0.889990i	$0.349289\pi$
$878$	0	0
$879$	2.00236e28	2.15600
$880$	0	0
$881$	−1.53869e28	−1.62136	−0.810682	−	0.585487i	$-0.800903\pi$
$881$	−1.53869e28	−1.62136	−0.810682	+	0.585487i	$0.800903\pi$
$882$	0	0
$883$	−8.67698e27	−0.894833	−0.447417	−	0.894326i	$-0.647656\pi$
$883$	−8.67698e27	−0.894833	−0.447417	+	0.894326i	$0.647656\pi$
$884$	0	0
$885$	8.34960e25	0.00842761
$886$	0	0
$887$	6.91409e27	0.683063	0.341531	−	0.939870i	$-0.389054\pi$
$887$	6.91409e27	0.683063	0.341531	+	0.939870i	$0.389054\pi$
$888$	0	0
$889$	4.06287e27	0.392886
$890$	0	0
$891$	5.35977e27	0.507351
$892$	0	0
$893$	−9.76554e27	−0.904914
$894$	0	0
$895$	4.17365e26	0.0378615
$896$	0	0
$897$	−1.86093e27	−0.165273
$898$	0	0
$899$	−1.51714e28	−1.31920
$900$	0	0
$901$	−1.12698e28	−0.959469
$902$	0	0
$903$	−5.54191e26	−0.0461983
$904$	0	0
$905$	3.04094e27	0.248226
$906$	0	0
$907$	9.64653e27	0.771084	0.385542	−	0.922690i	$-0.374014\pi$
$907$	9.64653e27	0.771084	0.385542	+	0.922690i	$0.374014\pi$
$908$	0	0
$909$	−1.94890e28	−1.52557
$910$	0	0
$911$	2.42290e28	1.85742	0.928711	−	0.370805i	$-0.120918\pi$
$911$	2.42290e28	1.85742	0.928711	+	0.370805i	$0.120918\pi$
$912$	0	0
$913$	9.45450e27	0.709849
$914$	0	0
$915$	8.64047e27	0.635385
$916$	0	0
$917$	1.36596e27	0.0983847
$918$	0	0
$919$	−1.71378e28	−1.20909	−0.604544	−	0.796572i	$-0.706645\pi$
$919$	−1.71378e28	−1.20909	−0.604544	+	0.796572i	$0.706645\pi$
$920$	0	0
$921$	−2.17660e28	−1.50422
$922$	0	0
$923$	2.55256e27	0.172806
$924$	0	0
$925$	2.02857e27	0.134538
$926$	0	0
$927$	1.48889e28	0.967398
$928$	0	0
$929$	6.82561e27	0.434502	0.217251	−	0.976116i	$-0.430291\pi$
$929$	6.82561e27	0.434502	0.217251	+	0.976116i	$0.430291\pi$
$930$	0	0
$931$	1.21289e28	0.756481
$932$	0	0
$933$	2.47947e28	1.51525
$934$	0	0
$935$	3.36942e27	0.201765
$936$	0	0
$937$	1.06271e28	0.623577	0.311789	−	0.950152i	$-0.399072\pi$
$937$	1.06271e28	0.623577	0.311789	+	0.950152i	$0.399072\pi$
$938$	0	0
$939$	1.23999e28	0.713008
$940$	0	0
$941$	1.99419e28	1.12374	0.561868	−	0.827227i	$-0.310083\pi$
$941$	1.99419e28	1.12374	0.561868	+	0.827227i	$0.310083\pi$
$942$	0	0
$943$	−5.00747e27	−0.276540
$944$	0	0
$945$	4.08646e26	0.0221180
$946$	0	0
$947$	−1.50706e28	−0.799479	−0.399739	−	0.916629i	$-0.630899\pi$
$947$	−1.50706e28	−0.799479	−0.399739	+	0.916629i	$0.630899\pi$
$948$	0	0
$949$	−4.01006e27	−0.208508
$950$	0	0
$951$	3.80998e28	1.94181
$952$	0	0
$953$	1.73573e27	0.0867162	0.0433581	−	0.999060i	$-0.486194\pi$
$953$	1.73573e27	0.0867162	0.0433581	+	0.999060i	$0.486194\pi$
$954$	0	0
$955$	−1.62719e28	−0.796905
$956$	0	0
$957$	2.03397e28	0.976522
$958$	0	0
$959$	−5.85125e27	−0.275405
$960$	0	0
$961$	−5.95768e27	−0.274919
$962$	0	0
$963$	−2.35975e28	−1.06762
$964$	0	0
$965$	1.08794e28	0.482608
$966$	0	0
$967$	4.20957e28	1.83099	0.915495	−	0.402329i	$-0.131799\pi$
$967$	4.20957e28	1.83099	0.915495	+	0.402329i	$0.131799\pi$
$968$	0	0
$969$	−2.63445e28	−1.12361
$970$	0	0
$971$	−3.71359e27	−0.155314	−0.0776571	−	0.996980i	$-0.524744\pi$
$971$	−3.71359e27	−0.155314	−0.0776571	+	0.996980i	$0.524744\pi$
$972$	0	0
$973$	4.55293e27	0.186732
$974$	0	0
$975$	1.25791e27	0.0505946
$976$	0	0
$977$	−1.36659e28	−0.539065	−0.269533	−	0.962991i	$-0.586869\pi$
$977$	−1.36659e28	−0.539065	−0.269533	+	0.962991i	$0.586869\pi$
$978$	0	0
$979$	−1.51335e28	−0.585470
$980$	0	0
$981$	8.89448e27	0.337493
$982$	0	0
$983$	1.40673e28	0.523544	0.261772	−	0.965130i	$-0.415693\pi$
$983$	1.40673e28	0.523544	0.261772	+	0.965130i	$0.415693\pi$
$984$	0	0
$985$	−4.68784e26	−0.0171131
$986$	0	0
$987$	1.26834e28	0.454176
$988$	0	0
$989$	2.05230e27	0.0720901
$990$	0	0
$991$	−3.22755e27	−0.111217	−0.0556087	−	0.998453i	$-0.517710\pi$
$991$	−3.22755e27	−0.111217	−0.0556087	+	0.998453i	$0.517710\pi$
$992$	0	0
$993$	5.51492e28	1.86432
$994$	0	0
$995$	−1.10768e28	−0.367363
$996$	0	0
$997$	1.09738e28	0.357070	0.178535	−	0.983934i	$-0.442864\pi$
$997$	1.09738e28	0.357070	0.178535	+	0.983934i	$0.442864\pi$
$998$	0	0
$999$	−3.41454e27	−0.109009

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.a.e.1.3		3
4.3	odd	2		20.20.a.a.1.1	✓	3
20.3	even	4		100.20.c.b.49.2		6
20.7	even	4		100.20.c.b.49.5		6
20.19	odd	2		100.20.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.20.a.a.1.1	✓	3	4.3	odd	2
80.20.a.e.1.3		3	1.1	even	1	trivial
100.20.a.b.1.3		3	20.19	odd	2
100.20.c.b.49.2		6	20.3	even	4
100.20.c.b.49.5		6	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)

\(f(q)\)	\(=\)	\(q+46767.8 q^{3} -1.95312e6 q^{5} +3.25847e7 q^{7} +1.02497e9 q^{9} +O(q^{10})\)	\(q+46767.8 q^{3} -1.95312e6 q^{5} +3.25847e7 q^{7} +1.02497e9 q^{9} -3.59337e9 q^{11} +7.05085e9 q^{13} -9.13434e10 q^{15} +4.80091e11 q^{17} -1.17333e12 q^{19} +1.52392e12 q^{21} -5.64342e12 q^{23} +3.81470e12 q^{25} -6.42101e12 q^{27} -1.21031e14 q^{29} +1.25351e14 q^{31} -1.68054e14 q^{33} -6.36420e13 q^{35} +5.31777e14 q^{37} +3.29753e14 q^{39} +8.87312e14 q^{41} -3.63662e14 q^{43} -2.00189e15 q^{45} +8.32293e15 q^{47} -1.03371e16 q^{49} +2.24528e16 q^{51} -2.34743e16 q^{53} +7.01829e15 q^{55} -5.48740e16 q^{57} -9.14089e14 q^{59} -9.45933e16 q^{61} +3.33982e16 q^{63} -1.37712e16 q^{65} -2.63147e17 q^{67} -2.63930e17 q^{69} +3.62022e17 q^{71} -5.68735e17 q^{73} +1.78405e17 q^{75} -1.17089e17 q^{77} -1.10844e18 q^{79} -1.49157e18 q^{81} -2.63110e18 q^{83} -9.37678e17 q^{85} -5.66036e18 q^{87} +4.21152e18 q^{89} +2.29750e17 q^{91} +5.86241e18 q^{93} +2.29166e18 q^{95} +8.83410e18 q^{97} -3.68308e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9}+O(q^{10}) \) 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9	\( 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9} - 6359213280 q^{11} - 33996539826 q^{13} + 66574218750 q^{15} - 60515110578 q^{17} + 1436218405908 q^{19} - 7119772762332 q^{21} - 1682066292342 q^{23} + 11444091796875 q^{25} - 91316335875732 q^{27} - 181580995192278 q^{29} + 3566464773252 q^{31} - 90839870844480 q^{33} - 224864402343750 q^{35} + 653714901466206 q^{37} + 43\!\cdots\!08 q^{39}+ \cdots - 37\!\cdots\!20 q^{99}+O(q^{100}) \) 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9 - 6359213280 * q^11 - 33996539826 * q^13 + 66574218750 * q^15 - 60515110578 * q^17 + 1436218405908 * q^19 - 7119772762332 * q^21 - 1682066292342 * q^23 + 11444091796875 * q^25 - 91316335875732 * q^27 - 181580995192278 * q^29 + 3566464773252 * q^31 - 90839870844480 * q^33 - 224864402343750 * q^35 + 653714901466206 * q^37 + 4339017394217508 * q^39 - 1257752912210622 * q^41 + 4361040178994610 * q^43 - 6111773490234375 * q^45 + 23174987171661798 * q^47 - 5454428919168921 * q^49 + 33371959424606244 * q^51 - 12180381712114362 * q^53 + 12420338437500000 * q^55 - 177419957801365416 * q^57 + 180023573560918116 * q^59 - 213491197168562514 * q^61 + 549307385783925246 * q^63 + 66399491847656250 * q^65 + 32494317103156134 * q^67 - 498099089220386004 * q^69 + 245957452394007276 * q^71 - 94437681636898506 * q^73 - 130027770996093750 * q^75 - 72972512123263680 * q^77 - 819623047726342536 * q^79 + 3159048884904329967 * q^81 - 2812828487653823982 * q^83 + 118193575347656250 * q^85 + 2510285506713629676 * q^87 + 4879439966436006078 * q^89 - 12213088562709029652 * q^91 + 16020238495381453176 * q^93 - 2805114074039062500 * q^95 + 8189756396617107846 * q^97 - 3795512975671396320 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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