LMFDB - Embedded newform 80.18.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,18,Mod(1,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1308] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$146.577669876$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{83281}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 20820 $ x^2 - x - 20820
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$144.792$ 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)

$f(q)$	$=$	$q-8003.53 q^{3} +390625. q^{5} -2.54694e7 q^{7} -6.50836e7 q^{9} +2.81089e8 q^{11} -1.52610e9 q^{13} -3.12638e9 q^{15} +5.46901e10 q^{17} -6.88947e8 q^{19} +2.03845e11 q^{21} -3.91035e11 q^{23} +1.52588e11 q^{25} +1.55448e12 q^{27} +5.12259e12 q^{29} +7.31204e10 q^{31} -2.24970e12 q^{33} -9.94897e12 q^{35} -6.81637e12 q^{37} +1.22142e13 q^{39} -5.76386e13 q^{41} -7.57081e13 q^{43} -2.54233e13 q^{45} +4.60351e13 q^{47} +4.16059e14 q^{49} -4.37714e14 q^{51} +6.58261e14 q^{53} +1.09800e14 q^{55} +5.51401e12 q^{57} +2.98287e14 q^{59} +8.50623e14 q^{61} +1.65764e15 q^{63} -5.96131e14 q^{65} +6.12967e15 q^{67} +3.12966e15 q^{69} -5.41472e14 q^{71} -7.16849e15 q^{73} -1.22124e15 q^{75} -7.15916e15 q^{77} -5.45373e15 q^{79} -4.03640e15 q^{81} +3.64723e15 q^{83} +2.13633e16 q^{85} -4.09988e16 q^{87} +6.81792e14 q^{89} +3.88687e16 q^{91} -5.85222e14 q^{93} -2.69120e14 q^{95} -1.20373e17 q^{97} -1.82943e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1308 q^{3} + 781250 q^{5} - 603844 q^{7} - 107519094 q^{9} + 471481296 q^{11} - 1541834228 q^{13} + 510937500 q^{15} + 32139900564 q^{17} - 128672529400 q^{19} + 435381246624 q^{21} - 650359859292 q^{23}+ \cdots - 26\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q + 1308 * q^3 + 781250 * q^5 - 603844 * q^7 - 107519094 * q^9 + 471481296 * q^11 - 1541834228 * q^13 + 510937500 * q^15 + 32139900564 * q^17 - 128672529400 * q^19 + 435381246624 * q^21 - 650359859292 * q^23 + 305175781250 * q^25 - 43156034280 * q^27 + 2543054749980 * q^29 - 7839407998744 * q^31 - 476857812816 * q^33 - 235876562500 * q^35 - 27805209097556 * q^37 + 12067623537288 * q^39 - 37826364264156 * q^41 - 29630453926852 * q^43 - 41999646093750 * q^45 - 220791583022004 * q^47 + 801722584677954 * q^49 - 647690491265544 * q^51 + 1058306017600572 * q^53 + 184172381250000 * q^55 - 1186209507102000 * q^57 - 1527820717235160 * q^59 - 631738213122116 * q^61 + 602457597963468 * q^63 - 602278995312500 * q^65 + 8843763493872596 * q^67 + 714953250788832 * q^69 + 3331901660912376 * q^71 - 10026350574292028 * q^73 + 199584960937500 * q^75 - 2424946105073712 * q^77 - 8263606842055120 * q^79 - 13432683850659198 * q^81 + 10689526892336988 * q^83 + 12554648657812500 * q^85 - 65018171604068280 * q^87 + 55878412521904980 * q^89 + 38477396610603016 * q^91 - 74263001022197376 * q^93 - 50262706796875000 * q^95 - 56106580363859756 * q^97 - 26373671099096112 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−8003.53	−0.704289	−0.352145	−	0.935946i	$-0.614548\pi$
$3$	−8003.53	−0.704289	−0.352145	+	0.935946i	$0.614548\pi$
$4$	0	0
$5$	390625.	0.447214
$5$	390625.	0.447214
$6$	0	0
$7$	−2.54694e7	−1.66988	−0.834939	−	0.550342i	$-0.814497\pi$
$7$	−2.54694e7	−1.66988	−0.834939	+	0.550342i	$0.814497\pi$
$8$	0	0
$9$	−6.50836e7	−0.503976
$10$	0	0
$11$	2.81089e8	0.395372	0.197686	−	0.980265i	$-0.436657\pi$
$11$	2.81089e8	0.395372	0.197686	+	0.980265i	$0.436657\pi$
$12$	0	0
$13$	−1.52610e9	−0.518876	−0.259438	−	0.965760i	$-0.583537\pi$
$13$	−1.52610e9	−0.518876	−0.259438	+	0.965760i	$0.583537\pi$
$14$	0	0
$15$	−3.12638e9	−0.314968
$16$	0	0
$17$	5.46901e10	1.90148	0.950742	−	0.309984i	$-0.100324\pi$
$17$	5.46901e10	1.90148	0.950742	+	0.309984i	$0.100324\pi$
$18$	0	0
$19$	−6.88947e8	−0.00930637	−0.00465318	−	0.999989i	$-0.501481\pi$
$19$	−6.88947e8	−0.00930637	−0.00465318	+	0.999989i	$0.501481\pi$
$20$	0	0
$21$	2.03845e11	1.17608
$22$	0	0
$23$	−3.91035e11	−1.04119	−0.520594	−	0.853804i	$-0.674290\pi$
$23$	−3.91035e11	−1.04119	−0.520594	+	0.853804i	$0.674290\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	0	0
$27$	1.55448e12	1.05923
$28$	0	0
$29$	5.12259e12	1.90154	0.950772	−	0.309892i	$-0.100293\pi$
$29$	5.12259e12	1.90154	0.950772	+	0.309892i	$0.100293\pi$
$30$	0	0
$31$	7.31204e10	0.0153980	0.00769900	−	0.999970i	$-0.497549\pi$
$31$	7.31204e10	0.0153980	0.00769900	+	0.999970i	$0.497549\pi$
$32$	0	0
$33$	−2.24970e12	−0.278456
$34$	0	0
$35$	−9.94897e12	−0.746792
$36$	0	0
$37$	−6.81637e12	−0.319035	−0.159517	−	0.987195i	$-0.550994\pi$
$37$	−6.81637e12	−0.319035	−0.159517	+	0.987195i	$0.550994\pi$
$38$	0	0
$39$	1.22142e13	0.365439
$40$	0	0
$41$	−5.76386e13	−1.12733	−0.563665	−	0.826004i	$-0.690609\pi$
$41$	−5.76386e13	−1.12733	−0.563665	+	0.826004i	$0.690609\pi$
$42$	0	0
$43$	−7.57081e13	−0.987781	−0.493890	−	0.869524i	$-0.664426\pi$
$43$	−7.57081e13	−0.987781	−0.493890	+	0.869524i	$0.664426\pi$
$44$	0	0
$45$	−2.54233e13	−0.225385
$46$	0	0
$47$	4.60351e13	0.282006	0.141003	−	0.990009i	$-0.454967\pi$
$47$	4.60351e13	0.282006	0.141003	+	0.990009i	$0.454967\pi$
$48$	0	0
$49$	4.16059e14	1.78849
$50$	0	0
$51$	−4.37714e14	−1.33919
$52$	0	0
$53$	6.58261e14	1.45229	0.726145	−	0.687541i	$-0.241310\pi$
$53$	6.58261e14	1.45229	0.726145	+	0.687541i	$0.241310\pi$
$54$	0	0
$55$	1.09800e14	0.176816
$56$	0	0
$57$	5.51401e12	0.00655437
$58$	0	0
$59$	2.98287e14	0.264479	0.132240	−	0.991218i	$-0.457783\pi$
$59$	2.98287e14	0.264479	0.132240	+	0.991218i	$0.457783\pi$
$60$	0	0
$61$	8.50623e14	0.568111	0.284056	−	0.958808i	$-0.408320\pi$
$61$	8.50623e14	0.568111	0.284056	+	0.958808i	$0.408320\pi$
$62$	0	0
$63$	1.65764e15	0.841580
$64$	0	0
$65$	−5.96131e14	−0.232048
$66$	0	0
$67$	6.12967e15	1.84417	0.922085	−	0.386987i	$-0.126484\pi$
$67$	6.12967e15	1.84417	0.922085	+	0.386987i	$0.126484\pi$
$68$	0	0
$69$	3.12966e15	0.733298
$70$	0	0
$71$	−5.41472e14	−0.0995131	−0.0497565	−	0.998761i	$-0.515845\pi$
$71$	−5.41472e14	−0.0995131	−0.0497565	+	0.998761i	$0.515845\pi$
$72$	0	0
$73$	−7.16849e15	−1.04036	−0.520180	−	0.854057i	$-0.674135\pi$
$73$	−7.16849e15	−1.04036	−0.520180	+	0.854057i	$0.674135\pi$
$74$	0	0
$75$	−1.22124e15	−0.140858
$76$	0	0
$77$	−7.15916e15	−0.660223
$78$	0	0
$79$	−5.45373e15	−0.404448	−0.202224	−	0.979339i	$-0.564817\pi$
$79$	−5.45373e15	−0.404448	−0.202224	+	0.979339i	$0.564817\pi$
$80$	0	0
$81$	−4.03640e15	−0.242031
$82$	0	0
$83$	3.64723e15	0.177746	0.0888728	−	0.996043i	$-0.471674\pi$
$83$	3.64723e15	0.177746	0.0888728	+	0.996043i	$0.471674\pi$
$84$	0	0
$85$	2.13633e16	0.850369
$86$	0	0
$87$	−4.09988e16	−1.33924
$88$	0	0
$89$	6.81792e14	0.0183585	0.00917925	−	0.999958i	$-0.497078\pi$
$89$	6.81792e14	0.0183585	0.00917925	+	0.999958i	$0.497078\pi$
$90$	0	0
$91$	3.88687e16	0.866460
$92$	0	0
$93$	−5.85222e14	−0.0108446
$94$	0	0
$95$	−2.69120e14	−0.00416193
$96$	0	0
$97$	−1.20373e17	−1.55944	−0.779720	−	0.626129i	$-0.784638\pi$
$97$	−1.20373e17	−1.55944	−0.779720	+	0.626129i	$0.784638\pi$
$98$	0	0
$99$	−1.82943e16	−0.199258
$100$	0	0
$101$	7.74092e16	0.711314	0.355657	−	0.934617i	$-0.384257\pi$
$101$	7.74092e16	0.711314	0.355657	+	0.934617i	$0.384257\pi$
$102$	0	0
$103$	1.39309e17	1.08359	0.541793	−	0.840512i	$-0.317746\pi$
$103$	1.39309e17	1.08359	0.541793	+	0.840512i	$0.317746\pi$
$104$	0	0
$105$	7.96270e16	0.525958
$106$	0	0
$107$	−2.75662e17	−1.55101	−0.775505	−	0.631341i	$-0.782505\pi$
$107$	−2.75662e17	−1.55101	−0.775505	+	0.631341i	$0.782505\pi$
$108$	0	0
$109$	1.37550e17	0.661202	0.330601	−	0.943771i	$-0.392749\pi$
$109$	1.37550e17	0.661202	0.330601	+	0.943771i	$0.392749\pi$
$110$	0	0
$111$	5.45550e16	0.224693
$112$	0	0
$113$	−3.90258e17	−1.38097	−0.690486	−	0.723345i	$-0.742604\pi$
$113$	−3.90258e17	−1.38097	−0.690486	+	0.723345i	$0.742604\pi$
$114$	0	0
$115$	−1.52748e17	−0.465634
$116$	0	0
$117$	9.93239e16	0.261501
$118$	0	0
$119$	−1.39292e18	−3.17525
$120$	0	0
$121$	−4.26436e17	−0.843681
$122$	0	0
$123$	4.61313e17	0.793966
$124$	0	0
$125$	5.96046e16	0.0894427
$126$	0	0
$127$	7.37824e17	0.967433	0.483717	−	0.875225i	$-0.339287\pi$
$127$	7.37824e17	0.967433	0.483717	+	0.875225i	$0.339287\pi$
$128$	0	0
$129$	6.05933e17	0.695684
$130$	0	0
$131$	1.45408e18	1.46482	0.732409	−	0.680865i	$-0.238396\pi$
$131$	1.45408e18	1.46482	0.732409	+	0.680865i	$0.238396\pi$
$132$	0	0
$133$	1.75470e16	0.0155405
$134$	0	0
$135$	6.07217e17	0.473704
$136$	0	0
$137$	−2.27534e18	−1.56647	−0.783236	−	0.621725i	$-0.786432\pi$
$137$	−2.27534e18	−1.56647	−0.783236	+	0.621725i	$0.786432\pi$
$138$	0	0
$139$	−1.41598e17	−0.0861849	−0.0430924	−	0.999071i	$-0.513721\pi$
$139$	−1.41598e17	−0.0861849	−0.0430924	+	0.999071i	$0.513721\pi$
$140$	0	0
$141$	−3.68444e17	−0.198614
$142$	0	0
$143$	−4.28969e17	−0.205149
$144$	0	0
$145$	2.00101e18	0.850396
$146$	0	0
$147$	−3.32994e18	−1.25962
$148$	0	0
$149$	−1.31089e18	−0.442063	−0.221032	−	0.975267i	$-0.570942\pi$
$149$	−1.31089e18	−0.442063	−0.221032	+	0.975267i	$0.570942\pi$
$150$	0	0
$151$	2.76548e18	0.832658	0.416329	−	0.909214i	$-0.363316\pi$
$151$	2.76548e18	0.832658	0.416329	+	0.909214i	$0.363316\pi$
$152$	0	0
$153$	−3.55943e18	−0.958303
$154$	0	0
$155$	2.85627e16	0.00688619
$156$	0	0
$157$	−3.81235e18	−0.824225	−0.412112	−	0.911133i	$-0.635209\pi$
$157$	−3.81235e18	−0.824225	−0.412112	+	0.911133i	$0.635209\pi$
$158$	0	0
$159$	−5.26842e18	−1.02283
$160$	0	0
$161$	9.95942e18	1.73866
$162$	0	0
$163$	−1.03773e19	−1.63114	−0.815570	−	0.578658i	$-0.803577\pi$
$163$	−1.03773e19	−1.63114	−0.815570	+	0.578658i	$0.803577\pi$
$164$	0	0
$165$	−8.78791e17	−0.124529
$166$	0	0
$167$	−1.21920e19	−1.55949	−0.779747	−	0.626095i	$-0.784652\pi$
$167$	−1.21920e19	−1.55949	−0.779747	+	0.626095i	$0.784652\pi$
$168$	0	0
$169$	−6.32145e18	−0.730768
$170$	0	0
$171$	4.48391e16	0.00469019
$172$	0	0
$173$	−3.88210e18	−0.367853	−0.183927	−	0.982940i	$-0.558881\pi$
$173$	−3.88210e18	−0.367853	−0.183927	+	0.982940i	$0.558881\pi$
$174$	0	0
$175$	−3.88632e18	−0.333976
$176$	0	0
$177$	−2.38735e18	−0.186270
$178$	0	0
$179$	1.62354e19	1.15136	0.575680	−	0.817675i	$-0.304737\pi$
$179$	1.62354e19	1.15136	0.575680	+	0.817675i	$0.304737\pi$
$180$	0	0
$181$	−2.32195e19	−1.49825	−0.749127	−	0.662427i	$-0.769527\pi$
$181$	−2.32195e19	−1.49825	−0.749127	+	0.662427i	$0.769527\pi$
$182$	0	0
$183$	−6.80799e18	−0.400115
$184$	0	0
$185$	−2.66264e18	−0.142677
$186$	0	0
$187$	1.53728e19	0.751793
$188$	0	0
$189$	−3.95915e19	−1.76879
$190$	0	0
$191$	−4.47928e19	−1.82989	−0.914944	−	0.403580i	$-0.867766\pi$
$191$	−4.47928e19	−1.82989	−0.914944	+	0.403580i	$0.867766\pi$
$192$	0	0
$193$	3.61154e19	1.35038	0.675189	−	0.737645i	$-0.264062\pi$
$193$	3.61154e19	1.35038	0.675189	+	0.737645i	$0.264062\pi$
$194$	0	0
$195$	4.77116e18	0.163429
$196$	0	0
$197$	−1.80503e19	−0.566920	−0.283460	−	0.958984i	$-0.591482\pi$
$197$	−1.80503e19	−0.566920	−0.283460	+	0.958984i	$0.591482\pi$
$198$	0	0
$199$	6.39636e18	0.184366	0.0921832	−	0.995742i	$-0.470615\pi$
$199$	6.39636e18	0.184366	0.0921832	+	0.995742i	$0.470615\pi$
$200$	0	0
$201$	−4.90590e19	−1.29883
$202$	0	0
$203$	−1.30469e20	−3.17535
$204$	0	0
$205$	−2.25151e19	−0.504157
$206$	0	0
$207$	2.54500e19	0.524734
$208$	0	0
$209$	−1.93655e17	−0.00367948
$210$	0	0
$211$	−7.72101e18	−0.135292	−0.0676462	−	0.997709i	$-0.521549\pi$
$211$	−7.72101e18	−0.135292	−0.0676462	+	0.997709i	$0.521549\pi$
$212$	0	0
$213$	4.33369e18	0.0700860
$214$	0	0
$215$	−2.95735e19	−0.441749
$216$	0	0
$217$	−1.86233e18	−0.0257128
$218$	0	0
$219$	5.73733e19	0.732715
$220$	0	0
$221$	−8.34623e19	−0.986634
$222$	0	0
$223$	1.08588e20	1.18902	0.594510	−	0.804088i	$-0.297346\pi$
$223$	1.08588e20	1.18902	0.594510	+	0.804088i	$0.297346\pi$
$224$	0	0
$225$	−9.93097e18	−0.100795
$226$	0	0
$227$	−1.95539e19	−0.184083	−0.0920415	−	0.995755i	$-0.529339\pi$
$227$	−1.95539e19	−0.184083	−0.0920415	+	0.995755i	$0.529339\pi$
$228$	0	0
$229$	1.20601e20	1.05378	0.526890	−	0.849933i	$-0.323358\pi$
$229$	1.20601e20	1.05378	0.526890	+	0.849933i	$0.323358\pi$
$230$	0	0
$231$	5.72986e19	0.464988
$232$	0	0
$233$	7.20388e19	0.543302	0.271651	−	0.962396i	$-0.412430\pi$
$233$	7.20388e19	0.543302	0.271651	+	0.962396i	$0.412430\pi$
$234$	0	0
$235$	1.79825e19	0.126117
$236$	0	0
$237$	4.36491e19	0.284849
$238$	0	0
$239$	−1.70425e20	−1.03550	−0.517750	−	0.855532i	$-0.673230\pi$
$239$	−1.70425e20	−1.03550	−0.517750	+	0.855532i	$0.673230\pi$
$240$	0	0
$241$	−1.71129e20	−0.968679	−0.484340	−	0.874880i	$-0.660940\pi$
$241$	−1.71129e20	−0.968679	−0.484340	+	0.874880i	$0.660940\pi$
$242$	0	0
$243$	−1.68440e20	−0.888775
$244$	0	0
$245$	1.62523e20	0.799839
$246$	0	0
$247$	1.05140e18	0.00482885
$248$	0	0
$249$	−2.91907e19	−0.125184
$250$	0	0
$251$	−1.85831e20	−0.744546	−0.372273	−	0.928123i	$-0.621422\pi$
$251$	−1.85831e20	−0.744546	−0.372273	+	0.928123i	$0.621422\pi$
$252$	0	0
$253$	−1.09916e20	−0.411657
$254$	0	0
$255$	−1.70982e20	−0.598906
$256$	0	0
$257$	1.68273e19	0.0551547	0.0275773	−	0.999620i	$-0.491221\pi$
$257$	1.68273e19	0.0551547	0.0275773	+	0.999620i	$0.491221\pi$
$258$	0	0
$259$	1.73609e20	0.532750
$260$	0	0
$261$	−3.33396e20	−0.958333
$262$	0	0
$263$	−2.77268e20	−0.746924	−0.373462	−	0.927645i	$-0.621829\pi$
$263$	−2.77268e20	−0.746924	−0.373462	+	0.927645i	$0.621829\pi$
$264$	0	0
$265$	2.57133e20	0.649484
$266$	0	0
$267$	−5.45675e18	−0.0129297
$268$	0	0
$269$	9.40799e19	0.209220	0.104610	−	0.994513i	$-0.466641\pi$
$269$	9.40799e19	0.209220	0.104610	+	0.994513i	$0.466641\pi$
$270$	0	0
$271$	6.17628e20	1.28970	0.644849	−	0.764310i	$-0.276920\pi$
$271$	6.17628e20	1.28970	0.644849	+	0.764310i	$0.276920\pi$
$272$	0	0
$273$	−3.11087e20	−0.610238
$274$	0	0
$275$	4.28907e19	0.0790744
$276$	0	0
$277$	−6.67865e20	−1.15774	−0.578869	−	0.815420i	$-0.696506\pi$
$277$	−6.67865e20	−1.15774	−0.578869	+	0.815420i	$0.696506\pi$
$278$	0	0
$279$	−4.75894e18	−0.00776023
$280$	0	0
$281$	−1.13640e20	−0.174393	−0.0871963	−	0.996191i	$-0.527791\pi$
$281$	−1.13640e20	−0.174393	−0.0871963	+	0.996191i	$0.527791\pi$
$282$	0	0
$283$	3.76957e20	0.544637	0.272318	−	0.962207i	$-0.412210\pi$
$283$	3.76957e20	0.544637	0.272318	+	0.962207i	$0.412210\pi$
$284$	0	0
$285$	2.15391e18	0.00293121
$286$	0	0
$287$	1.46802e21	1.88250
$288$	0	0
$289$	2.16376e21	2.61564
$290$	0	0
$291$	9.63408e20	1.09830
$292$	0	0
$293$	−3.57350e20	−0.384344	−0.192172	−	0.981361i	$-0.561553\pi$
$293$	−3.57350e20	−0.384344	−0.192172	+	0.981361i	$0.561553\pi$
$294$	0	0
$295$	1.16518e20	0.118279
$296$	0	0
$297$	4.36946e20	0.418792
$298$	0	0
$299$	5.96757e20	0.540247
$300$	0	0
$301$	1.92824e21	1.64947
$302$	0	0
$303$	−6.19547e20	−0.500971
$304$	0	0
$305$	3.32275e20	0.254067
$306$	0	0
$307$	−2.25658e21	−1.63220	−0.816102	−	0.577908i	$-0.803869\pi$
$307$	−2.25658e21	−1.63220	−0.816102	+	0.577908i	$0.803869\pi$
$308$	0	0
$309$	−1.11497e21	−0.763158
$310$	0	0
$311$	−5.76844e20	−0.373762	−0.186881	−	0.982383i	$-0.559838\pi$
$311$	−5.76844e20	−0.373762	−0.186881	+	0.982383i	$0.559838\pi$
$312$	0	0
$313$	2.63218e21	1.61506	0.807530	−	0.589827i	$-0.200804\pi$
$313$	2.63218e21	1.61506	0.807530	+	0.589827i	$0.200804\pi$
$314$	0	0
$315$	6.47515e20	0.376366
$316$	0	0
$317$	−9.95998e20	−0.548599	−0.274299	−	0.961644i	$-0.588446\pi$
$317$	−9.95998e20	−0.548599	−0.274299	+	0.961644i	$0.588446\pi$
$318$	0	0
$319$	1.43990e21	0.751817
$320$	0	0
$321$	2.20627e21	1.09236
$322$	0	0
$323$	−3.76786e19	−0.0176959
$324$	0	0
$325$	−2.32864e20	−0.103775
$326$	0	0
$327$	−1.10088e21	−0.465678
$328$	0	0
$329$	−1.17249e21	−0.470915
$330$	0	0
$331$	3.56520e21	1.36002	0.680011	−	0.733202i	$-0.261975\pi$
$331$	3.56520e21	1.36002	0.680011	+	0.733202i	$0.261975\pi$
$332$	0	0
$333$	4.43634e20	0.160786
$334$	0	0
$335$	2.39440e21	0.824738
$336$	0	0
$337$	3.65283e20	0.119612	0.0598060	−	0.998210i	$-0.480952\pi$
$337$	3.65283e20	0.119612	0.0598060	+	0.998210i	$0.480952\pi$
$338$	0	0
$339$	3.12344e21	0.972605
$340$	0	0
$341$	2.05533e19	0.00608793
$342$	0	0
$343$	−4.67180e21	−1.31669
$344$	0	0
$345$	1.22252e21	0.327941
$346$	0	0
$347$	2.06934e21	0.528483	0.264242	−	0.964457i	$-0.414878\pi$
$347$	2.06934e21	0.528483	0.264242	+	0.964457i	$0.414878\pi$
$348$	0	0
$349$	3.13803e20	0.0763204	0.0381602	−	0.999272i	$-0.487850\pi$
$349$	3.13803e20	0.0763204	0.0381602	+	0.999272i	$0.487850\pi$
$350$	0	0
$351$	−2.37228e21	−0.549611
$352$	0	0
$353$	−6.87901e21	−1.51859	−0.759296	−	0.650746i	$-0.774456\pi$
$353$	−6.87901e21	−1.51859	−0.759296	+	0.650746i	$0.774456\pi$
$354$	0	0
$355$	−2.11513e20	−0.0445036
$356$	0	0
$357$	1.11483e22	2.23629
$358$	0	0
$359$	9.50907e21	1.81901	0.909505	−	0.415693i	$-0.136461\pi$
$359$	9.50907e21	1.81901	0.909505	+	0.415693i	$0.136461\pi$
$360$	0	0
$361$	−5.47991e21	−0.999913
$362$	0	0
$363$	3.41300e21	0.594196
$364$	0	0
$365$	−2.80019e21	−0.465263
$366$	0	0
$367$	−1.88008e20	−0.0298205	−0.0149103	−	0.999889i	$-0.504746\pi$
$367$	−1.88008e20	−0.0298205	−0.0149103	+	0.999889i	$0.504746\pi$
$368$	0	0
$369$	3.75133e21	0.568147
$370$	0	0
$371$	−1.67655e22	−2.42515
$372$	0	0
$373$	9.66745e20	0.133594	0.0667970	−	0.997767i	$-0.478722\pi$
$373$	9.66745e20	0.133594	0.0667970	+	0.997767i	$0.478722\pi$
$374$	0	0
$375$	−4.77048e20	−0.0629936
$376$	0	0
$377$	−7.81756e21	−0.986665
$378$	0	0
$379$	−1.09478e22	−1.32097	−0.660484	−	0.750840i	$-0.729649\pi$
$379$	−1.09478e22	−1.32097	−0.660484	+	0.750840i	$0.729649\pi$
$380$	0	0
$381$	−5.90520e21	−0.681353
$382$	0	0
$383$	9.48135e21	1.04636	0.523179	−	0.852223i	$-0.324746\pi$
$383$	9.48135e21	1.04636	0.523179	+	0.852223i	$0.324746\pi$
$384$	0	0
$385$	−2.79655e21	−0.295261
$386$	0	0
$387$	4.92736e21	0.497818
$388$	0	0
$389$	8.68048e21	0.839406	0.419703	−	0.907661i	$-0.362134\pi$
$389$	8.68048e21	0.839406	0.419703	+	0.907661i	$0.362134\pi$
$390$	0	0
$391$	−2.13857e22	−1.97980
$392$	0	0
$393$	−1.16378e22	−1.03166
$394$	0	0
$395$	−2.13036e21	−0.180875
$396$	0	0
$397$	−5.61938e21	−0.457055	−0.228528	−	0.973537i	$-0.573391\pi$
$397$	−5.61938e21	−0.457055	−0.228528	+	0.973537i	$0.573391\pi$
$398$	0	0
$399$	−1.40438e20	−0.0109450
$400$	0	0
$401$	−4.97857e21	−0.371858	−0.185929	−	0.982563i	$-0.559530\pi$
$401$	−4.97857e21	−0.371858	−0.185929	+	0.982563i	$0.559530\pi$
$402$	0	0
$403$	−1.11589e20	−0.00798965
$404$	0	0
$405$	−1.57672e21	−0.108240
$406$	0	0
$407$	−1.91600e21	−0.126137
$408$	0	0
$409$	2.65693e20	0.0167777	0.00838883	−	0.999965i	$-0.497330\pi$
$409$	2.65693e20	0.0167777	0.00838883	+	0.999965i	$0.497330\pi$
$410$	0	0
$411$	1.82108e22	1.10325
$412$	0	0
$413$	−7.59717e21	−0.441648
$414$	0	0
$415$	1.42470e21	0.0794903
$416$	0	0
$417$	1.13328e21	0.0606991
$418$	0	0
$419$	−1.71242e22	−0.880626	−0.440313	−	0.897844i	$-0.645133\pi$
$419$	−1.71242e22	−0.880626	−0.440313	+	0.897844i	$0.645133\pi$
$420$	0	0
$421$	2.71839e22	1.34250	0.671249	−	0.741231i	$-0.265758\pi$
$421$	2.71839e22	1.34250	0.671249	+	0.741231i	$0.265758\pi$
$422$	0	0
$423$	−2.99613e21	−0.142124
$424$	0	0
$425$	8.34504e21	0.380297
$426$	0	0
$427$	−2.16648e22	−0.948677
$428$	0	0
$429$	3.43327e21	0.144484
$430$	0	0
$431$	2.34759e22	0.949656	0.474828	−	0.880079i	$-0.342510\pi$
$431$	2.34759e22	0.949656	0.474828	+	0.880079i	$0.342510\pi$
$432$	0	0
$433$	−1.88732e22	−0.734003	−0.367002	−	0.930220i	$-0.619616\pi$
$433$	−1.88732e22	−0.734003	−0.367002	+	0.930220i	$0.619616\pi$
$434$	0	0
$435$	−1.60152e22	−0.598925
$436$	0	0
$437$	2.69402e20	0.00968968
$438$	0	0
$439$	−5.51895e22	−1.90945	−0.954724	−	0.297494i	$-0.903849\pi$
$439$	−5.51895e22	−1.90945	−0.954724	+	0.297494i	$0.903849\pi$
$440$	0	0
$441$	−2.70786e22	−0.901359
$442$	0	0
$443$	1.20735e22	0.386723	0.193362	−	0.981128i	$-0.438061\pi$
$443$	1.20735e22	0.386723	0.193362	+	0.981128i	$0.438061\pi$
$444$	0	0
$445$	2.66325e20	0.00821017
$446$	0	0
$447$	1.04918e22	0.311341
$448$	0	0
$449$	2.57466e22	0.735573	0.367787	−	0.929910i	$-0.380116\pi$
$449$	2.57466e22	0.735573	0.367787	+	0.929910i	$0.380116\pi$
$450$	0	0
$451$	−1.62016e22	−0.445714
$452$	0	0
$453$	−2.21336e22	−0.586432
$454$	0	0
$455$	1.51831e22	0.387493
$456$	0	0
$457$	−1.01069e22	−0.248503	−0.124252	−	0.992251i	$-0.539653\pi$
$457$	−1.01069e22	−0.248503	−0.124252	+	0.992251i	$0.539653\pi$
$458$	0	0
$459$	8.50144e22	2.01412
$460$	0	0
$461$	1.68599e21	0.0384942	0.0192471	−	0.999815i	$-0.493873\pi$
$461$	1.68599e21	0.0384942	0.0192471	+	0.999815i	$0.493873\pi$
$462$	0	0
$463$	2.77945e22	0.611674	0.305837	−	0.952084i	$-0.401064\pi$
$463$	2.77945e22	0.611674	0.305837	+	0.952084i	$0.401064\pi$
$464$	0	0
$465$	−2.28602e20	−0.00484987
$466$	0	0
$467$	−8.26643e22	−1.69093	−0.845463	−	0.534034i	$-0.820676\pi$
$467$	−8.26643e22	−1.69093	−0.845463	+	0.534034i	$0.820676\pi$
$468$	0	0
$469$	−1.56119e23	−3.07954
$470$	0	0
$471$	3.05123e22	0.580493
$472$	0	0
$473$	−2.12807e22	−0.390541
$474$	0	0
$475$	−1.05125e20	−0.00186127
$476$	0	0
$477$	−4.28420e22	−0.731920
$478$	0	0
$479$	−2.59191e22	−0.427335	−0.213667	−	0.976906i	$-0.568541\pi$
$479$	−2.59191e22	−0.427335	−0.213667	+	0.976906i	$0.568541\pi$
$480$	0	0
$481$	1.04024e22	0.165540
$482$	0	0
$483$	−7.97106e22	−1.22452
$484$	0	0
$485$	−4.70206e22	−0.697403
$486$	0	0
$487$	4.14828e22	0.594116	0.297058	−	0.954859i	$-0.403994\pi$
$487$	4.14828e22	0.594116	0.297058	+	0.954859i	$0.403994\pi$
$488$	0	0
$489$	8.30554e22	1.14879
$490$	0	0
$491$	−3.08644e22	−0.412349	−0.206174	−	0.978515i	$-0.566101\pi$
$491$	−3.08644e22	−0.412349	−0.206174	+	0.978515i	$0.566101\pi$
$492$	0	0
$493$	2.80155e23	3.61575
$494$	0	0
$495$	−7.14620e21	−0.0891109
$496$	0	0
$497$	1.37910e22	0.166175
$498$	0	0
$499$	−3.24182e22	−0.377516	−0.188758	−	0.982024i	$-0.560446\pi$
$499$	−3.24182e22	−0.377516	−0.188758	+	0.982024i	$0.560446\pi$
$500$	0	0
$501$	9.75788e22	1.09833
$502$	0	0
$503$	−6.39324e22	−0.695654	−0.347827	−	0.937559i	$-0.613080\pi$
$503$	−6.39324e22	−0.695654	−0.347827	+	0.937559i	$0.613080\pi$
$504$	0	0
$505$	3.02380e22	0.318109
$506$	0	0
$507$	5.05939e22	0.514672
$508$	0	0
$509$	−9.33355e22	−0.918218	−0.459109	−	0.888380i	$-0.651831\pi$
$509$	−9.33355e22	−0.918218	−0.459109	+	0.888380i	$0.651831\pi$
$510$	0	0
$511$	1.82577e23	1.73728
$512$	0	0
$513$	−1.07095e21	−0.00985763
$514$	0	0
$515$	5.44176e22	0.484594
$516$	0	0
$517$	1.29400e22	0.111497
$518$	0	0
$519$	3.10705e22	0.259075
$520$	0	0
$521$	3.30201e22	0.266476	0.133238	−	0.991084i	$-0.457463\pi$
$521$	3.30201e22	0.266476	0.133238	+	0.991084i	$0.457463\pi$
$522$	0	0
$523$	−4.06002e22	−0.317149	−0.158575	−	0.987347i	$-0.550690\pi$
$523$	−4.06002e22	−0.317149	−0.158575	+	0.987347i	$0.550690\pi$
$524$	0	0
$525$	3.11043e22	0.235216
$526$	0	0
$527$	3.99896e21	0.0292790
$528$	0	0
$529$	1.18585e22	0.0840729
$530$	0	0
$531$	−1.94136e22	−0.133291
$532$	0	0
$533$	8.79621e22	0.584944
$534$	0	0
$535$	−1.07680e23	−0.693633
$536$	0	0
$537$	−1.29940e23	−0.810890
$538$	0	0
$539$	1.16949e23	0.707121
$540$	0	0
$541$	−4.20460e22	−0.246347	−0.123174	−	0.992385i	$-0.539307\pi$
$541$	−4.20460e22	−0.246347	−0.123174	+	0.992385i	$0.539307\pi$
$542$	0	0
$543$	1.85838e23	1.05520
$544$	0	0
$545$	5.37303e22	0.295699
$546$	0	0
$547$	2.01110e23	1.07285	0.536427	−	0.843947i	$-0.319774\pi$
$547$	2.01110e23	1.07285	0.536427	+	0.843947i	$0.319774\pi$
$548$	0	0
$549$	−5.53616e22	−0.286315
$550$	0	0
$551$	−3.52919e21	−0.0176965
$552$	0	0
$553$	1.38903e23	0.675380
$554$	0	0
$555$	2.13106e22	0.100486
$556$	0	0
$557$	1.46727e23	0.671030	0.335515	−	0.942035i	$-0.391090\pi$
$557$	1.46727e23	0.671030	0.335515	+	0.942035i	$0.391090\pi$
$558$	0	0
$559$	1.15538e23	0.512536
$560$	0	0
$561$	−1.23036e23	−0.529480
$562$	0	0
$563$	3.39706e23	1.41834	0.709172	−	0.705036i	$-0.249069\pi$
$563$	3.39706e23	1.41834	0.709172	+	0.705036i	$0.249069\pi$
$564$	0	0
$565$	−1.52445e23	−0.617590
$566$	0	0
$567$	1.02805e23	0.404163
$568$	0	0
$569$	2.72602e23	1.04010	0.520050	−	0.854136i	$-0.325913\pi$
$569$	2.72602e23	1.04010	0.520050	+	0.854136i	$0.325913\pi$
$570$	0	0
$571$	2.79882e22	0.103650	0.0518249	−	0.998656i	$-0.483496\pi$
$571$	2.79882e22	0.103650	0.0518249	+	0.998656i	$0.483496\pi$
$572$	0	0
$573$	3.58501e23	1.28877
$574$	0	0
$575$	−5.96672e22	−0.208238
$576$	0	0
$577$	3.86678e23	1.31025	0.655127	−	0.755519i	$-0.272615\pi$
$577$	3.86678e23	1.31025	0.655127	+	0.755519i	$0.272615\pi$
$578$	0	0
$579$	−2.89051e23	−0.951056
$580$	0	0
$581$	−9.28926e22	−0.296814
$582$	0	0
$583$	1.85030e23	0.574195
$584$	0	0
$585$	3.87984e22	0.116947
$586$	0	0
$587$	−3.64152e23	−1.06625	−0.533125	−	0.846036i	$-0.678983\pi$
$587$	−3.64152e23	−1.06625	−0.533125	+	0.846036i	$0.678983\pi$
$588$	0	0
$589$	−5.03761e19	−0.000143299 0
$590$	0	0
$591$	1.44466e23	0.399276
$592$	0	0
$593$	−6.86689e23	−1.84415	−0.922073	−	0.387017i	$-0.873506\pi$
$593$	−6.86689e23	−1.84415	−0.922073	+	0.387017i	$0.873506\pi$
$594$	0	0
$595$	−5.44110e23	−1.42001
$596$	0	0
$597$	−5.11935e22	−0.129847
$598$	0	0
$599$	3.82850e23	0.943846	0.471923	−	0.881640i	$-0.343560\pi$
$599$	3.82850e23	0.943846	0.471923	+	0.881640i	$0.343560\pi$
$600$	0	0
$601$	−2.66402e23	−0.638417	−0.319209	−	0.947684i	$-0.603417\pi$
$601$	−2.66402e23	−0.638417	−0.319209	+	0.947684i	$0.603417\pi$
$602$	0	0
$603$	−3.98941e23	−0.929418
$604$	0	0
$605$	−1.66577e23	−0.377306
$606$	0	0
$607$	1.11277e23	0.245076	0.122538	−	0.992464i	$-0.460897\pi$
$607$	1.11277e23	0.245076	0.122538	+	0.992464i	$0.460897\pi$
$608$	0	0
$609$	1.04421e24	2.23636
$610$	0	0
$611$	−7.02541e22	−0.146326
$612$	0	0
$613$	−3.91229e23	−0.792532	−0.396266	−	0.918136i	$-0.629694\pi$
$613$	−3.91229e23	−0.792532	−0.396266	+	0.918136i	$0.629694\pi$
$614$	0	0
$615$	1.80200e23	0.355072
$616$	0	0
$617$	−5.72626e23	−1.09761	−0.548804	−	0.835951i	$-0.684917\pi$
$617$	−5.72626e23	−1.09761	−0.548804	+	0.835951i	$0.684917\pi$
$618$	0	0
$619$	−7.86155e23	−1.46601	−0.733007	−	0.680222i	$-0.761884\pi$
$619$	−7.86155e23	−1.46601	−0.733007	+	0.680222i	$0.761884\pi$
$620$	0	0
$621$	−6.07855e23	−1.10286
$622$	0	0
$623$	−1.73648e22	−0.0306565
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	0	0
$627$	1.54993e21	0.00259142
$628$	0	0
$629$	−3.72788e23	−0.606640
$630$	0	0
$631$	−6.52712e23	−1.03389	−0.516943	−	0.856020i	$-0.672930\pi$
$631$	−6.52712e23	−1.03389	−0.516943	+	0.856020i	$0.672930\pi$
$632$	0	0
$633$	6.17954e22	0.0952850
$634$	0	0
$635$	2.88212e23	0.432649
$636$	0	0
$637$	−6.34945e23	−0.928007
$638$	0	0
$639$	3.52410e22	0.0501522
$640$	0	0
$641$	7.10734e23	0.984950	0.492475	−	0.870327i	$-0.336092\pi$
$641$	7.10734e23	0.984950	0.492475	+	0.870327i	$0.336092\pi$
$642$	0	0
$643$	−2.34278e23	−0.316183	−0.158092	−	0.987424i	$-0.550534\pi$
$643$	−2.34278e23	−0.316183	−0.158092	+	0.987424i	$0.550534\pi$
$644$	0	0
$645$	2.36692e23	0.311119
$646$	0	0
$647$	−1.50806e23	−0.193078	−0.0965391	−	0.995329i	$-0.530777\pi$
$647$	−1.50806e23	−0.193078	−0.0965391	+	0.995329i	$0.530777\pi$
$648$	0	0
$649$	8.38450e22	0.104568
$650$	0	0
$651$	1.49052e22	0.0181092
$652$	0	0
$653$	6.74375e22	0.0798251	0.0399126	−	0.999203i	$-0.487292\pi$
$653$	6.74375e22	0.0798251	0.0399126	+	0.999203i	$0.487292\pi$
$654$	0	0
$655$	5.68001e23	0.655086
$656$	0	0
$657$	4.66551e23	0.524317
$658$	0	0
$659$	−7.11301e22	−0.0778981	−0.0389491	−	0.999241i	$-0.512401\pi$
$659$	−7.11301e22	−0.0778981	−0.0389491	+	0.999241i	$0.512401\pi$
$660$	0	0
$661$	1.69805e24	1.81233	0.906164	−	0.422926i	$-0.138997\pi$
$661$	1.69805e24	1.81233	0.906164	+	0.422926i	$0.138997\pi$
$662$	0	0
$663$	6.67994e23	0.694876
$664$	0	0
$665$	6.85431e21	0.00694992
$666$	0	0
$667$	−2.00311e24	−1.97986
$668$	0	0
$669$	−8.69085e23	−0.837414
$670$	0	0
$671$	2.39101e23	0.224615
$672$	0	0
$673$	−1.39848e24	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$673$	−1.39848e24	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$674$	0	0
$675$	2.37194e23	0.211847
$676$	0	0
$677$	−9.38639e23	−0.817514	−0.408757	−	0.912643i	$-0.634038\pi$
$677$	−9.38639e23	−0.817514	−0.408757	+	0.912643i	$0.634038\pi$
$678$	0	0
$679$	3.06582e24	2.60408
$680$	0	0
$681$	1.56500e23	0.129648
$682$	0	0
$683$	−1.23254e24	−0.995922	−0.497961	−	0.867199i	$-0.665918\pi$
$683$	−1.23254e24	−0.995922	−0.497961	+	0.867199i	$0.665918\pi$
$684$	0	0
$685$	−8.88807e23	−0.700547
$686$	0	0
$687$	−9.65236e23	−0.742167
$688$	0	0
$689$	−1.00457e24	−0.753559
$690$	0	0
$691$	−3.83055e23	−0.280348	−0.140174	−	0.990127i	$-0.544766\pi$
$691$	−3.83055e23	−0.280348	−0.140174	+	0.990127i	$0.544766\pi$
$692$	0	0
$693$	4.65944e23	0.332737
$694$	0	0
$695$	−5.53117e22	−0.0385431
$696$	0	0
$697$	−3.15226e24	−2.14360
$698$	0	0
$699$	−5.76565e23	−0.382642
$700$	0	0
$701$	1.63235e24	1.05733	0.528665	−	0.848831i	$-0.322693\pi$
$701$	1.63235e24	1.05733	0.528665	+	0.848831i	$0.322693\pi$
$702$	0	0
$703$	4.69611e21	0.00296906
$704$	0	0
$705$	−1.43923e23	−0.0888227
$706$	0	0
$707$	−1.97156e24	−1.18781
$708$	0	0
$709$	−6.09431e23	−0.358453	−0.179226	−	0.983808i	$-0.557359\pi$
$709$	−6.09431e23	−0.358453	−0.179226	+	0.983808i	$0.557359\pi$
$710$	0	0
$711$	3.54948e23	0.203832
$712$	0	0
$713$	−2.85926e22	−0.0160322
$714$	0	0
$715$	−1.67566e23	−0.0917454
$716$	0	0
$717$	1.36400e24	0.729292
$718$	0	0
$719$	4.49148e23	0.234527	0.117264	−	0.993101i	$-0.462588\pi$
$719$	4.49148e23	0.234527	0.117264	+	0.993101i	$0.462588\pi$
$720$	0	0
$721$	−3.54812e24	−1.80946
$722$	0	0
$723$	1.36964e24	0.682230
$724$	0	0
$725$	7.81644e23	0.380309
$726$	0	0
$727$	5.00713e23	0.237983	0.118992	−	0.992895i	$-0.462034\pi$
$727$	5.00713e23	0.237983	0.118992	+	0.992895i	$0.462034\pi$
$728$	0	0
$729$	1.86938e24	0.867986
$730$	0	0
$731$	−4.14048e24	−1.87825
$732$	0	0
$733$	1.20329e24	0.533320	0.266660	−	0.963791i	$-0.414080\pi$
$733$	1.20329e24	0.533320	0.266660	+	0.963791i	$0.414080\pi$
$734$	0	0
$735$	−1.30076e24	−0.563318
$736$	0	0
$737$	1.72298e24	0.729133
$738$	0	0
$739$	2.19426e24	0.907426	0.453713	−	0.891148i	$-0.350099\pi$
$739$	2.19426e24	0.907426	0.453713	+	0.891148i	$0.350099\pi$
$740$	0	0
$741$	−8.41491e21	−0.00340091
$742$	0	0
$743$	9.89633e22	0.0390903	0.0195452	−	0.999809i	$-0.493778\pi$
$743$	9.89633e22	0.0390903	0.0195452	+	0.999809i	$0.493778\pi$
$744$	0	0
$745$	−5.12068e23	−0.197697
$746$	0	0
$747$	−2.37375e23	−0.0895796
$748$	0	0
$749$	7.02094e24	2.59000
$750$	0	0
$751$	1.01856e24	0.367322	0.183661	−	0.982990i	$-0.441205\pi$
$751$	1.01856e24	0.367322	0.183661	+	0.982990i	$0.441205\pi$
$752$	0	0
$753$	1.48731e24	0.524376
$754$	0	0
$755$	1.08027e24	0.372376
$756$	0	0
$757$	−1.43989e24	−0.485305	−0.242653	−	0.970113i	$-0.578018\pi$
$757$	−1.43989e24	−0.485305	−0.242653	+	0.970113i	$0.578018\pi$
$758$	0	0
$759$	8.79713e23	0.289925
$760$	0	0
$761$	−7.15424e23	−0.230565	−0.115283	−	0.993333i	$-0.536777\pi$
$761$	−7.15424e23	−0.230565	−0.115283	+	0.993333i	$0.536777\pi$
$762$	0	0
$763$	−3.50330e24	−1.10413
$764$	0	0
$765$	−1.39040e24	−0.428566
$766$	0	0
$767$	−4.55214e23	−0.137232
$768$	0	0
$769$	1.87177e24	0.551922	0.275961	−	0.961169i	$-0.411004\pi$
$769$	1.87177e24	0.551922	0.275961	+	0.961169i	$0.411004\pi$
$770$	0	0
$771$	−1.34678e23	−0.0388449
$772$	0	0
$773$	−1.86999e24	−0.527611	−0.263805	−	0.964576i	$-0.584978\pi$
$773$	−1.86999e24	−0.527611	−0.263805	+	0.964576i	$0.584978\pi$
$774$	0	0
$775$	1.11573e22	0.00307960
$776$	0	0
$777$	−1.38948e24	−0.375210
$778$	0	0
$779$	3.97099e22	0.0104913
$780$	0	0
$781$	−1.52202e23	−0.0393447
$782$	0	0
$783$	7.96294e24	2.01418
$784$	0	0
$785$	−1.48920e24	−0.368604
$786$	0	0
$787$	3.29304e24	0.797648	0.398824	−	0.917027i	$-0.369418\pi$
$787$	3.29304e24	0.797648	0.398824	+	0.917027i	$0.369418\pi$
$788$	0	0
$789$	2.21913e24	0.526051
$790$	0	0
$791$	9.93963e24	2.30606
$792$	0	0
$793$	−1.29813e24	−0.294779
$794$	0	0
$795$	−2.05798e24	−0.457425
$796$	0	0
$797$	−3.80089e24	−0.826969	−0.413485	−	0.910511i	$-0.635688\pi$
$797$	−3.80089e24	−0.826969	−0.413485	+	0.910511i	$0.635688\pi$
$798$	0	0
$799$	2.51767e24	0.536229
$800$	0	0
$801$	−4.43735e22	−0.00925225
$802$	0	0
$803$	−2.01498e24	−0.411329
$804$	0	0
$805$	3.89040e24	0.777552
$806$	0	0
$807$	−7.52972e23	−0.147351
$808$	0	0
$809$	1.38713e24	0.265799	0.132900	−	0.991129i	$-0.457571\pi$
$809$	1.38713e24	0.265799	0.132900	+	0.991129i	$0.457571\pi$
$810$	0	0
$811$	−2.43418e24	−0.456747	−0.228374	−	0.973574i	$-0.573341\pi$
$811$	−2.43418e24	−0.456747	−0.228374	+	0.973574i	$0.573341\pi$
$812$	0	0
$813$	−4.94321e24	−0.908321
$814$	0	0
$815$	−4.05365e24	−0.729468
$816$	0	0
$817$	5.21589e22	0.00919265
$818$	0	0
$819$	−2.52972e24	−0.436675
$820$	0	0
$821$	−7.07675e24	−1.19651	−0.598256	−	0.801305i	$-0.704140\pi$
$821$	−7.07675e24	−1.19651	−0.598256	+	0.801305i	$0.704140\pi$
$822$	0	0
$823$	5.31425e24	0.880124	0.440062	−	0.897968i	$-0.354957\pi$
$823$	5.31425e24	0.880124	0.440062	+	0.897968i	$0.354957\pi$
$824$	0	0
$825$	−3.43278e23	−0.0556912
$826$	0	0
$827$	−5.13487e24	−0.816080	−0.408040	−	0.912964i	$-0.633788\pi$
$827$	−5.13487e24	−0.816080	−0.408040	+	0.912964i	$0.633788\pi$
$828$	0	0
$829$	1.06738e25	1.66191	0.830954	−	0.556341i	$-0.187795\pi$
$829$	1.06738e25	1.66191	0.830954	+	0.556341i	$0.187795\pi$
$830$	0	0
$831$	5.34528e24	0.815383
$832$	0	0
$833$	2.27543e25	3.40079
$834$	0	0
$835$	−4.76248e24	−0.697427
$836$	0	0
$837$	1.13664e23	0.0163101
$838$	0	0
$839$	6.47302e24	0.910186	0.455093	−	0.890444i	$-0.349606\pi$
$839$	6.47302e24	0.910186	0.455093	+	0.890444i	$0.349606\pi$
$840$	0	0
$841$	1.89837e25	2.61587
$842$	0	0
$843$	9.09523e23	0.122823
$844$	0	0
$845$	−2.46931e24	−0.326809
$846$	0	0
$847$	1.08611e25	1.40885
$848$	0	0
$849$	−3.01699e24	−0.383582
$850$	0	0
$851$	2.66544e24	0.332175
$852$	0	0
$853$	2.69306e24	0.328988	0.164494	−	0.986378i	$-0.447401\pi$
$853$	2.69306e24	0.328988	0.164494	+	0.986378i	$0.447401\pi$
$854$	0	0
$855$	1.75153e22	0.00209752
$856$	0	0
$857$	−1.86785e24	−0.219283	−0.109641	−	0.993971i	$-0.534970\pi$
$857$	−1.86785e24	−0.219283	−0.109641	+	0.993971i	$0.534970\pi$
$858$	0	0
$859$	1.13124e25	1.30200	0.651000	−	0.759077i	$-0.274350\pi$
$859$	1.13124e25	1.30200	0.651000	+	0.759077i	$0.274350\pi$
$860$	0	0
$861$	−1.17493e25	−1.32583
$862$	0	0
$863$	2.37061e24	0.262282	0.131141	−	0.991364i	$-0.458136\pi$
$863$	2.37061e24	0.262282	0.131141	+	0.991364i	$0.458136\pi$
$864$	0	0
$865$	−1.51645e24	−0.164509
$866$	0	0
$867$	−1.73178e25	−1.84217
$868$	0	0
$869$	−1.53298e24	−0.159907
$870$	0	0
$871$	−9.35446e24	−0.956895
$872$	0	0
$873$	7.83429e24	0.785921
$874$	0	0
$875$	−1.51809e24	−0.149358
$876$	0	0
$877$	−6.79534e24	−0.655714	−0.327857	−	0.944727i	$-0.606326\pi$
$877$	−6.79534e24	−0.655714	−0.327857	+	0.944727i	$0.606326\pi$
$878$	0	0
$879$	2.86007e24	0.270689
$880$	0	0
$881$	−1.88402e25	−1.74900	−0.874501	−	0.485024i	$-0.838811\pi$
$881$	−1.88402e25	−1.74900	−0.874501	+	0.485024i	$0.838811\pi$
$882$	0	0
$883$	−3.85491e24	−0.351033	−0.175517	−	0.984476i	$-0.556160\pi$
$883$	−3.85491e24	−0.351033	−0.175517	+	0.984476i	$0.556160\pi$
$884$	0	0
$885$	−9.32557e23	−0.0833024
$886$	0	0
$887$	−8.32727e24	−0.729712	−0.364856	−	0.931064i	$-0.618882\pi$
$887$	−8.32727e24	−0.729712	−0.364856	+	0.931064i	$0.618882\pi$
$888$	0	0
$889$	−1.87919e25	−1.61550
$890$	0	0
$891$	−1.13459e24	−0.0956924
$892$	0	0
$893$	−3.17158e22	−0.00262445
$894$	0	0
$895$	6.34194e24	0.514904
$896$	0	0
$897$	−4.77617e24	−0.380491
$898$	0	0
$899$	3.74565e23	0.0292799
$900$	0	0
$901$	3.60004e25	2.76151
$902$	0	0
$903$	−1.54327e25	−1.16171
$904$	0	0
$905$	−9.07012e24	−0.670039
$906$	0	0
$907$	−1.92435e25	−1.39515	−0.697576	−	0.716511i	$-0.745738\pi$
$907$	−1.92435e25	−1.39515	−0.697576	+	0.716511i	$0.745738\pi$
$908$	0	0
$909$	−5.03807e24	−0.358485
$910$	0	0
$911$	1.57682e25	1.10123	0.550613	−	0.834761i	$-0.314394\pi$
$911$	1.57682e25	1.10123	0.550613	+	0.834761i	$0.314394\pi$
$912$	0	0
$913$	1.02519e24	0.0702756
$914$	0	0
$915$	−2.65937e24	−0.178937
$916$	0	0
$917$	−3.70346e25	−2.44607
$918$	0	0
$919$	−1.03058e25	−0.668189	−0.334094	−	0.942540i	$-0.608430\pi$
$919$	−1.03058e25	−0.668189	−0.334094	+	0.942540i	$0.608430\pi$
$920$	0	0
$921$	1.80606e25	1.14954
$922$	0	0
$923$	8.26339e23	0.0516349
$924$	0	0
$925$	−1.04010e24	−0.0638070
$926$	0	0
$927$	−9.06674e24	−0.546102
$928$	0	0
$929$	9.73688e24	0.575819	0.287910	−	0.957658i	$-0.407040\pi$
$929$	9.73688e24	0.575819	0.287910	+	0.957658i	$0.407040\pi$
$930$	0	0
$931$	−2.86642e23	−0.0166444
$932$	0	0
$933$	4.61679e24	0.263236
$934$	0	0
$935$	6.00499e24	0.336212
$936$	0	0
$937$	−3.22145e25	−1.77119	−0.885594	−	0.464459i	$-0.846249\pi$
$937$	−3.22145e25	−1.77119	−0.885594	+	0.464459i	$0.846249\pi$
$938$	0	0
$939$	−2.10667e25	−1.13747
$940$	0	0
$941$	−7.90374e24	−0.419103	−0.209552	−	0.977798i	$-0.567200\pi$
$941$	−7.90374e24	−0.419103	−0.209552	+	0.977798i	$0.567200\pi$
$942$	0	0
$943$	2.25387e25	1.17376
$944$	0	0
$945$	−1.54654e25	−0.791028
$946$	0	0
$947$	−2.36682e25	−1.18902	−0.594512	−	0.804086i	$-0.702655\pi$
$947$	−2.36682e25	−1.18902	−0.594512	+	0.804086i	$0.702655\pi$
$948$	0	0
$949$	1.09398e25	0.539818
$950$	0	0
$951$	7.97150e24	0.386372
$952$	0	0
$953$	9.09477e24	0.433014	0.216507	−	0.976281i	$-0.430534\pi$
$953$	9.09477e24	0.433014	0.216507	+	0.976281i	$0.430534\pi$
$954$	0	0
$955$	−1.74972e25	−0.818351
$956$	0	0
$957$	−1.15243e25	−0.529496
$958$	0	0
$959$	5.79516e25	2.61582
$960$	0	0
$961$	−2.25448e25	−0.999763
$962$	0	0
$963$	1.79411e25	0.781673
$964$	0	0
$965$	1.41076e25	0.603907
$966$	0	0
$967$	−3.62970e25	−1.52667	−0.763337	−	0.646001i	$-0.776440\pi$
$967$	−3.62970e25	−1.52667	−0.763337	+	0.646001i	$0.776440\pi$
$968$	0	0
$969$	3.01562e23	0.0124630
$970$	0	0
$971$	9.70396e24	0.394081	0.197040	−	0.980395i	$-0.436867\pi$
$971$	9.70396e24	0.394081	0.197040	+	0.980395i	$0.436867\pi$
$972$	0	0
$973$	3.60641e24	0.143918
$974$	0	0
$975$	1.86373e24	0.0730878
$976$	0	0
$977$	3.70141e25	1.42647	0.713236	−	0.700924i	$-0.247229\pi$
$977$	3.70141e25	1.42647	0.713236	+	0.700924i	$0.247229\pi$
$978$	0	0
$979$	1.91644e23	0.00725843
$980$	0	0
$981$	−8.95223e24	−0.333230
$982$	0	0
$983$	6.43184e24	0.235305	0.117652	−	0.993055i	$-0.462463\pi$
$983$	6.43184e24	0.235305	0.117652	+	0.993055i	$0.462463\pi$
$984$	0	0
$985$	−7.05090e24	−0.253534
$986$	0	0
$987$	9.38403e24	0.331661
$988$	0	0
$989$	2.96045e25	1.02847
$990$	0	0
$991$	−5.17070e25	−1.76573	−0.882863	−	0.469631i	$-0.844387\pi$
$991$	−5.17070e25	−1.76573	−0.882863	+	0.469631i	$0.844387\pi$
$992$	0	0
$993$	−2.85342e25	−0.957848
$994$	0	0
$995$	2.49858e24	0.0824512
$996$	0	0
$997$	−2.77457e25	−0.900093	−0.450047	−	0.893005i	$-0.648593\pi$
$997$	−2.77457e25	−0.900093	−0.450047	+	0.893005i	$0.648593\pi$
$998$	0	0
$999$	−1.05959e25	−0.337933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.18.a.d.1.1		2
4.3	odd	2		10.18.a.c.1.2	✓	2
12.11	even	2		90.18.a.k.1.2		2
20.3	even	4		50.18.b.d.49.4		4
20.7	even	4		50.18.b.d.49.1		4
20.19	odd	2		50.18.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.c.1.2	✓	2	4.3	odd	2
50.18.a.f.1.1		2	20.19	odd	2
50.18.b.d.49.1		4	20.7	even	4
50.18.b.d.49.4		4	20.3	even	4
80.18.a.d.1.1		2	1.1	even	1	trivial
90.18.a.k.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q-8003.53 q^{3} +390625. q^{5} -2.54694e7 q^{7} -6.50836e7 q^{9} +2.81089e8 q^{11} -1.52610e9 q^{13} -3.12638e9 q^{15} +5.46901e10 q^{17} -6.88947e8 q^{19} +2.03845e11 q^{21} -3.91035e11 q^{23} +1.52588e11 q^{25} +1.55448e12 q^{27} +5.12259e12 q^{29} +7.31204e10 q^{31} -2.24970e12 q^{33} -9.94897e12 q^{35} -6.81637e12 q^{37} +1.22142e13 q^{39} -5.76386e13 q^{41} -7.57081e13 q^{43} -2.54233e13 q^{45} +4.60351e13 q^{47} +4.16059e14 q^{49} -4.37714e14 q^{51} +6.58261e14 q^{53} +1.09800e14 q^{55} +5.51401e12 q^{57} +2.98287e14 q^{59} +8.50623e14 q^{61} +1.65764e15 q^{63} -5.96131e14 q^{65} +6.12967e15 q^{67} +3.12966e15 q^{69} -5.41472e14 q^{71} -7.16849e15 q^{73} -1.22124e15 q^{75} -7.15916e15 q^{77} -5.45373e15 q^{79} -4.03640e15 q^{81} +3.64723e15 q^{83} +2.13633e16 q^{85} -4.09988e16 q^{87} +6.81792e14 q^{89} +3.88687e16 q^{91} -5.85222e14 q^{93} -2.69120e14 q^{95} -1.20373e17 q^{97} -1.82943e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1308 q^{3} + 781250 q^{5} - 603844 q^{7} - 107519094 q^{9} + 471481296 q^{11} - 1541834228 q^{13} + 510937500 q^{15} + 32139900564 q^{17} - 128672529400 q^{19} + 435381246624 q^{21} - 650359859292 q^{23}+ \cdots - 26\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q + 1308 * q^3 + 781250 * q^5 - 603844 * q^7 - 107519094 * q^9 + 471481296 * q^11 - 1541834228 * q^13 + 510937500 * q^15 + 32139900564 * q^17 - 128672529400 * q^19 + 435381246624 * q^21 - 650359859292 * q^23 + 305175781250 * q^25 - 43156034280 * q^27 + 2543054749980 * q^29 - 7839407998744 * q^31 - 476857812816 * q^33 - 235876562500 * q^35 - 27805209097556 * q^37 + 12067623537288 * q^39 - 37826364264156 * q^41 - 29630453926852 * q^43 - 41999646093750 * q^45 - 220791583022004 * q^47 + 801722584677954 * q^49 - 647690491265544 * q^51 + 1058306017600572 * q^53 + 184172381250000 * q^55 - 1186209507102000 * q^57 - 1527820717235160 * q^59 - 631738213122116 * q^61 + 602457597963468 * q^63 - 602278995312500 * q^65 + 8843763493872596 * q^67 + 714953250788832 * q^69 + 3331901660912376 * q^71 - 10026350574292028 * q^73 + 199584960937500 * q^75 - 2424946105073712 * q^77 - 8263606842055120 * q^79 - 13432683850659198 * q^81 + 10689526892336988 * q^83 + 12554648657812500 * q^85 - 65018171604068280 * q^87 + 55878412521904980 * q^89 + 38477396610603016 * q^91 - 74263001022197376 * q^93 - 50262706796875000 * q^95 - 56106580363859756 * q^97 - 26373671099096112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more