LMFDB - Embedded newform 80.20.a.d.1.1

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:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2925852 $ x^2 - x - 2925852
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2}\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			$1711.01$ 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)

$f(q)$	$=$	$q-51072.2 q^{3} +1.95312e6 q^{5} -2.72650e7 q^{7} +1.44611e9 q^{9} -8.90778e9 q^{11} -3.88993e10 q^{13} -9.97505e10 q^{15} -4.92493e11 q^{17} -7.02926e11 q^{19} +1.39248e12 q^{21} +5.93310e12 q^{23} +3.81470e12 q^{25} -1.44969e13 q^{27} +1.42477e14 q^{29} -2.10392e14 q^{31} +4.54940e14 q^{33} -5.32519e13 q^{35} -1.00145e15 q^{37} +1.98667e15 q^{39} -4.08507e15 q^{41} -3.21023e15 q^{43} +2.82444e15 q^{45} +6.44781e15 q^{47} -1.06555e16 q^{49} +2.51527e16 q^{51} +1.88410e15 q^{53} -1.73980e16 q^{55} +3.59000e16 q^{57} +4.28479e15 q^{59} -1.19883e17 q^{61} -3.94282e16 q^{63} -7.59752e16 q^{65} -1.93776e17 q^{67} -3.03017e17 q^{69} -3.68964e17 q^{71} -9.09825e17 q^{73} -1.94825e17 q^{75} +2.42870e17 q^{77} -5.25508e17 q^{79} -9.40371e17 q^{81} +2.77280e18 q^{83} -9.61900e17 q^{85} -7.27660e18 q^{87} -3.88866e17 q^{89} +1.06059e18 q^{91} +1.07452e19 q^{93} -1.37290e18 q^{95} -9.73741e17 q^{97} -1.28817e19 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 33724 q^{3} + 3906250 q^{5} - 83061292 q^{7} + 584812954 q^{9} - 3549480144 q^{11} + 30250225564 q^{13} - 65867187500 q^{15} + 329655779412 q^{17} - 383076809800 q^{19} + 424515195104 q^{21} - 2082856096884 q^{23}+ \cdots - 17\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q - 33724 * q^3 + 3906250 * q^5 - 83061292 * q^7 + 584812954 * q^9 - 3549480144 * q^11 + 30250225564 * q^13 - 65867187500 * q^15 + 329655779412 * q^17 - 383076809800 * q^19 + 424515195104 * q^21 - 2082856096884 * q^23 + 7629394531250 * q^25 - 49602163340440 * q^27 + 166587684602220 * q^29 - 418151119038664 * q^31 + 547897533578928 * q^33 - 162229085937500 * q^35 - 1292219637694628 * q^37 + 3186297306918232 * q^39 - 3127192714527996 * q^41 + 1795992338357636 * q^43 + 1142212800781250 * q^45 + 6761653310165988 * q^47 - 18941182438871454 * q^49 + 39415550773661256 * q^51 - 9787568571066516 * q^53 - 6932578406250000 * q^55 + 41448804721422800 * q^57 + 25484316660294360 * q^59 - 77840503559636276 * q^61 + 8629133041886116 * q^63 + 59082471804687500 * q^65 - 439654835073204532 * q^67 - 442079200728708192 * q^69 + 29427662567526696 * q^71 - 1463308911724217756 * q^73 - 128646850585937500 * q^75 - 56103061801470576 * q^77 + 1235969155825112720 * q^79 - 548329353045127838 * q^81 + 877205635969635396 * q^83 + 643858944164062500 * q^85 - 6858312722579528040 * q^87 + 4172929077000032820 * q^89 - 2797700455846427144 * q^91 + 7140934358593491968 * q^93 - 748196894140625000 * q^95 + 11254578228347861332 * q^97 - 17496761012343831888 * 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
$3$	−51072.2	−1.49807	−0.749037	−	0.662529i	$-0.769483\pi$
$3$	−51072.2	−1.49807	−0.749037	+	0.662529i	$0.769483\pi$
$4$	0	0
$5$	1.95312e6	0.447214
$5$	1.95312e6	0.447214
$6$	0	0
$7$	−2.72650e7	−0.255372	−0.127686	−	0.991815i	$-0.540755\pi$
$7$	−2.72650e7	−0.255372	−0.127686	+	0.991815i	$0.540755\pi$
$8$	0	0
$9$	1.44611e9	1.24422
$10$	0	0
$11$	−8.90778e9	−1.13904	−0.569520	−	0.821977i	$-0.692871\pi$
$11$	−8.90778e9	−1.13904	−0.569520	+	0.821977i	$0.692871\pi$
$12$	0	0
$13$	−3.88993e10	−1.01737	−0.508686	−	0.860952i	$-0.669869\pi$
$13$	−3.88993e10	−1.01737	−0.508686	+	0.860952i	$0.669869\pi$
$14$	0	0
$15$	−9.97505e10	−0.669959
$16$	0	0
$17$	−4.92493e11	−1.00725	−0.503623	−	0.863924i	$-0.668000\pi$
$17$	−4.92493e11	−1.00725	−0.503623	+	0.863924i	$0.668000\pi$
$18$	0	0
$19$	−7.02926e11	−0.499747	−0.249873	−	0.968278i	$-0.580389\pi$
$19$	−7.02926e11	−0.499747	−0.249873	+	0.968278i	$0.580389\pi$
$20$	0	0
$21$	1.39248e12	0.382566
$22$	0	0
$23$	5.93310e12	0.686858	0.343429	−	0.939179i	$-0.388412\pi$
$23$	5.93310e12	0.686858	0.343429	+	0.939179i	$0.388412\pi$
$24$	0	0
$25$	3.81470e12	0.200000
$26$	0	0
$27$	−1.44969e13	−0.365864
$28$	0	0
$29$	1.42477e14	1.82374	0.911869	−	0.410481i	$-0.134639\pi$
$29$	1.42477e14	1.82374	0.911869	+	0.410481i	$0.134639\pi$
$30$	0	0
$31$	−2.10392e14	−1.42920	−0.714600	−	0.699533i	$-0.753391\pi$
$31$	−2.10392e14	−1.42920	−0.714600	+	0.699533i	$0.753391\pi$
$32$	0	0
$33$	4.54940e14	1.70637
$34$	0	0
$35$	−5.32519e13	−0.114206
$36$	0	0
$37$	−1.00145e15	−1.26681	−0.633404	−	0.773821i	$-0.718343\pi$
$37$	−1.00145e15	−1.26681	−0.633404	+	0.773821i	$0.718343\pi$
$38$	0	0
$39$	1.98667e15	1.52410
$40$	0	0
$41$	−4.08507e15	−1.94874	−0.974368	−	0.224961i	$-0.927775\pi$
$41$	−4.08507e15	−1.94874	−0.974368	+	0.224961i	$0.927775\pi$
$42$	0	0
$43$	−3.21023e15	−0.974060	−0.487030	−	0.873385i	$-0.661920\pi$
$43$	−3.21023e15	−0.974060	−0.487030	+	0.873385i	$0.661920\pi$
$44$	0	0
$45$	2.82444e15	0.556434
$46$	0	0
$47$	6.44781e15	0.840393	0.420196	−	0.907433i	$-0.361961\pi$
$47$	6.44781e15	0.840393	0.420196	+	0.907433i	$0.361961\pi$
$48$	0	0
$49$	−1.06555e16	−0.934785
$50$	0	0
$51$	2.51527e16	1.50893
$52$	0	0
$53$	1.88410e15	0.0784300	0.0392150	−	0.999231i	$-0.487514\pi$
$53$	1.88410e15	0.0784300	0.0392150	+	0.999231i	$0.487514\pi$
$54$	0	0
$55$	−1.73980e16	−0.509394
$56$	0	0
$57$	3.59000e16	0.748658
$58$	0	0
$59$	4.28479e15	0.0643924	0.0321962	−	0.999482i	$-0.489750\pi$
$59$	4.28479e15	0.0643924	0.0321962	+	0.999482i	$0.489750\pi$
$60$	0	0
$61$	−1.19883e17	−1.31257	−0.656287	−	0.754511i	$-0.727874\pi$
$61$	−1.19883e17	−1.31257	−0.656287	+	0.754511i	$0.727874\pi$
$62$	0	0
$63$	−3.94282e16	−0.317740
$64$	0	0
$65$	−7.59752e16	−0.454983
$66$	0	0
$67$	−1.93776e17	−0.870141	−0.435071	−	0.900396i	$-0.643277\pi$
$67$	−1.93776e17	−0.870141	−0.435071	+	0.900396i	$0.643277\pi$
$68$	0	0
$69$	−3.03017e17	−1.02896
$70$	0	0
$71$	−3.68964e17	−0.955058	−0.477529	−	0.878616i	$-0.658468\pi$
$71$	−3.68964e17	−0.955058	−0.477529	+	0.878616i	$0.658468\pi$
$72$	0	0
$73$	−9.09825e17	−1.80880	−0.904400	−	0.426685i	$-0.859681\pi$
$73$	−9.09825e17	−1.80880	−0.904400	+	0.426685i	$0.859681\pi$
$74$	0	0
$75$	−1.94825e17	−0.299615
$76$	0	0
$77$	2.42870e17	0.290879
$78$	0	0
$79$	−5.25508e17	−0.493313	−0.246656	−	0.969103i	$-0.579332\pi$
$79$	−5.25508e17	−0.493313	−0.246656	+	0.969103i	$0.579332\pi$
$80$	0	0
$81$	−9.40371e17	−0.696132
$82$	0	0
$83$	2.77280e18	1.62808	0.814042	−	0.580807i	$-0.197263\pi$
$83$	2.77280e18	1.62808	0.814042	+	0.580807i	$0.197263\pi$
$84$	0	0
$85$	−9.61900e17	−0.450454
$86$	0	0
$87$	−7.27660e18	−2.73209
$88$	0	0
$89$	−3.88866e17	−0.117651	−0.0588254	−	0.998268i	$-0.518736\pi$
$89$	−3.88866e17	−0.117651	−0.0588254	+	0.998268i	$0.518736\pi$
$90$	0	0
$91$	1.06059e18	0.259809
$92$	0	0
$93$	1.07452e19	2.14105
$94$	0	0
$95$	−1.37290e18	−0.223494
$96$	0	0
$97$	−9.73741e17	−0.130050	−0.0650252	−	0.997884i	$-0.520713\pi$
$97$	−9.73741e17	−0.130050	−0.0650252	+	0.997884i	$0.520713\pi$
$98$	0	0
$99$	−1.28817e19	−1.41722
$100$	0	0
$101$	−1.67461e19	−1.52356	−0.761781	−	0.647834i	$-0.775675\pi$
$101$	−1.67461e19	−1.52356	−0.761781	+	0.647834i	$0.775675\pi$
$102$	0	0
$103$	−4.40800e18	−0.332880	−0.166440	−	0.986052i	$-0.553227\pi$
$103$	−4.40800e18	−0.332880	−0.166440	+	0.986052i	$0.553227\pi$
$104$	0	0
$105$	2.71969e18	0.171089
$106$	0	0
$107$	8.71914e18	0.458488	0.229244	−	0.973369i	$-0.426375\pi$
$107$	8.71914e18	0.458488	0.229244	+	0.973369i	$0.426375\pi$
$108$	0	0
$109$	4.17114e19	1.83951	0.919757	−	0.392487i	$-0.128385\pi$
$109$	4.17114e19	1.83951	0.919757	+	0.392487i	$0.128385\pi$
$110$	0	0
$111$	5.11461e19	1.89777
$112$	0	0
$113$	−8.70592e18	−0.272627	−0.136314	−	0.990666i	$-0.543525\pi$
$113$	−8.70592e18	−0.272627	−0.136314	+	0.990666i	$0.543525\pi$
$114$	0	0
$115$	1.15881e19	0.307172
$116$	0	0
$117$	−5.62528e19	−1.26584
$118$	0	0
$119$	1.34278e19	0.257222
$120$	0	0
$121$	1.81895e19	0.297413
$122$	0	0
$123$	2.08634e20	2.91935
$124$	0	0
$125$	7.45058e18	0.0894427
$126$	0	0
$127$	−6.07257e19	−0.626956	−0.313478	−	0.949595i	$-0.601494\pi$
$127$	−6.07257e19	−0.626956	−0.313478	+	0.949595i	$0.601494\pi$
$128$	0	0
$129$	1.63954e20	1.45921
$130$	0	0
$131$	−1.94831e20	−1.49824	−0.749120	−	0.662434i	$-0.769523\pi$
$131$	−1.94831e20	−1.49824	−0.749120	+	0.662434i	$0.769523\pi$
$132$	0	0
$133$	1.91652e19	0.127622
$134$	0	0
$135$	−2.83143e19	−0.163620
$136$	0	0
$137$	1.21216e20	0.609135	0.304567	−	0.952491i	$-0.401488\pi$
$137$	1.21216e20	0.609135	0.304567	+	0.952491i	$0.401488\pi$
$138$	0	0
$139$	−2.22493e20	−0.974263	−0.487131	−	0.873329i	$-0.661957\pi$
$139$	−2.22493e20	−0.974263	−0.487131	+	0.873329i	$0.661957\pi$
$140$	0	0
$141$	−3.29304e20	−1.25897
$142$	0	0
$143$	3.46506e20	1.15883
$144$	0	0
$145$	2.78275e20	0.815600
$146$	0	0
$147$	5.44201e20	1.40038
$148$	0	0
$149$	3.46894e20	0.785104	0.392552	−	0.919730i	$-0.371592\pi$
$149$	3.46894e20	0.785104	0.392552	+	0.919730i	$0.371592\pi$
$150$	0	0
$151$	1.05205e19	0.0209776	0.0104888	−	0.999945i	$-0.496661\pi$
$151$	1.05205e19	0.0209776	0.0104888	+	0.999945i	$0.496661\pi$
$152$	0	0
$153$	−7.12200e20	−1.25324
$154$	0	0
$155$	−4.10922e20	−0.639158
$156$	0	0
$157$	−3.07714e19	−0.0423741	−0.0211871	−	0.999776i	$-0.506745\pi$
$157$	−3.07714e19	−0.0423741	−0.0211871	+	0.999776i	$0.506745\pi$
$158$	0	0
$159$	−9.62250e19	−0.117494
$160$	0	0
$161$	−1.61766e20	−0.175404
$162$	0	0
$163$	−5.76318e20	−0.555751	−0.277875	−	0.960617i	$-0.589630\pi$
$163$	−5.76318e20	−0.555751	−0.277875	+	0.960617i	$0.589630\pi$
$164$	0	0
$165$	8.88555e20	0.763110
$166$	0	0
$167$	−6.89550e20	−0.528153	−0.264076	−	0.964502i	$-0.585067\pi$
$167$	−6.89550e20	−0.528153	−0.264076	+	0.964502i	$0.585067\pi$
$168$	0	0
$169$	5.12350e19	0.0350464
$170$	0	0
$171$	−1.01651e21	−0.621797
$172$	0	0
$173$	−4.14244e19	−0.0226892	−0.0113446	−	0.999936i	$-0.503611\pi$
$173$	−4.14244e19	−0.0226892	−0.0113446	+	0.999936i	$0.503611\pi$
$174$	0	0
$175$	−1.04008e20	−0.0510744
$176$	0	0
$177$	−2.18834e20	−0.0964646
$178$	0	0
$179$	−3.65179e21	−1.44678	−0.723388	−	0.690441i	$-0.757416\pi$
$179$	−3.65179e21	−1.44678	−0.723388	+	0.690441i	$0.757416\pi$
$180$	0	0
$181$	1.05232e21	0.375145	0.187573	−	0.982251i	$-0.439938\pi$
$181$	1.05232e21	0.375145	0.187573	+	0.982251i	$0.439938\pi$
$182$	0	0
$183$	6.12269e21	1.96633
$184$	0	0
$185$	−1.95595e21	−0.566534
$186$	0	0
$187$	4.38702e21	1.14729
$188$	0	0
$189$	3.95258e20	0.0934316
$190$	0	0
$191$	6.63681e21	1.41952	0.709761	−	0.704442i	$-0.248803\pi$
$191$	6.63681e21	1.41952	0.709761	+	0.704442i	$0.248803\pi$
$192$	0	0
$193$	−2.65526e21	−0.514414	−0.257207	−	0.966356i	$-0.582802\pi$
$193$	−2.65526e21	−0.514414	−0.257207	+	0.966356i	$0.582802\pi$
$194$	0	0
$195$	3.88022e21	0.681597
$196$	0	0
$197$	1.02356e22	1.63186	0.815929	−	0.578153i	$-0.196226\pi$
$197$	1.02356e22	1.63186	0.815929	+	0.578153i	$0.196226\pi$
$198$	0	0
$199$	2.53082e21	0.366570	0.183285	−	0.983060i	$-0.441327\pi$
$199$	2.53082e21	0.366570	0.183285	+	0.983060i	$0.441327\pi$
$200$	0	0
$201$	9.89659e21	1.30354
$202$	0	0
$203$	−3.88462e21	−0.465732
$204$	0	0
$205$	−7.97865e21	−0.871501
$206$	0	0
$207$	8.57993e21	0.854605
$208$	0	0
$209$	6.26151e21	0.569232
$210$	0	0
$211$	1.43232e22	1.18948	0.594740	−	0.803918i	$-0.297255\pi$
$211$	1.43232e22	1.18948	0.594740	+	0.803918i	$0.297255\pi$
$212$	0	0
$213$	1.88438e22	1.43075
$214$	0	0
$215$	−6.26998e21	−0.435613
$216$	0	0
$217$	5.73633e21	0.364978
$218$	0	0
$219$	4.64668e22	2.70972
$220$	0	0
$221$	1.91576e22	1.02474
$222$	0	0
$223$	−1.40599e22	−0.690375	−0.345187	−	0.938534i	$-0.612185\pi$
$223$	−1.40599e22	−0.690375	−0.345187	+	0.938534i	$0.612185\pi$
$224$	0	0
$225$	5.51648e21	0.248845
$226$	0	0
$227$	−3.41441e22	−1.41602	−0.708011	−	0.706201i	$-0.750408\pi$
$227$	−3.41441e22	−1.41602	−0.708011	+	0.706201i	$0.750408\pi$
$228$	0	0
$229$	2.26978e22	0.866057	0.433028	−	0.901380i	$-0.357445\pi$
$229$	2.26978e22	0.866057	0.433028	+	0.901380i	$0.357445\pi$
$230$	0	0
$231$	−1.24039e22	−0.435758
$232$	0	0
$233$	3.14636e22	1.01842	0.509211	−	0.860642i	$-0.329937\pi$
$233$	3.14636e22	1.01842	0.509211	+	0.860642i	$0.329937\pi$
$234$	0	0
$235$	1.25934e22	0.375835
$236$	0	0
$237$	2.68389e22	0.739018
$238$	0	0
$239$	1.24777e22	0.317216	0.158608	−	0.987342i	$-0.449299\pi$
$239$	1.24777e22	0.317216	0.158608	+	0.987342i	$0.449299\pi$
$240$	0	0
$241$	−1.16264e21	−0.0273076	−0.0136538	−	0.999907i	$-0.504346\pi$
$241$	−1.16264e21	−0.0273076	−0.0136538	+	0.999907i	$0.504346\pi$
$242$	0	0
$243$	6.48761e22	1.40872
$244$	0	0
$245$	−2.08116e22	−0.418049
$246$	0	0
$247$	2.73433e22	0.508429
$248$	0	0
$249$	−1.41613e23	−2.43899
$250$	0	0
$251$	−1.09655e23	−1.75036	−0.875182	−	0.483793i	$-0.839259\pi$
$251$	−1.09655e23	−1.75036	−0.875182	+	0.483793i	$0.839259\pi$
$252$	0	0
$253$	−5.28507e22	−0.782359
$254$	0	0
$255$	4.91264e22	0.674813
$256$	0	0
$257$	−2.11998e22	−0.270375	−0.135187	−	0.990820i	$-0.543164\pi$
$257$	−2.11998e22	−0.270375	−0.135187	+	0.990820i	$0.543164\pi$
$258$	0	0
$259$	2.73044e22	0.323508
$260$	0	0
$261$	2.06037e23	2.26914
$262$	0	0
$263$	1.06654e23	1.09244	0.546222	−	0.837641i	$-0.316066\pi$
$263$	1.06654e23	1.09244	0.546222	+	0.837641i	$0.316066\pi$
$264$	0	0
$265$	3.67987e21	0.0350750
$266$	0	0
$267$	1.98603e22	0.176250
$268$	0	0
$269$	9.27742e22	0.766974	0.383487	−	0.923546i	$-0.374723\pi$
$269$	9.27742e22	0.766974	0.383487	+	0.923546i	$0.374723\pi$
$270$	0	0
$271$	9.46299e22	0.729156	0.364578	−	0.931173i	$-0.381213\pi$
$271$	9.46299e22	0.729156	0.364578	+	0.931173i	$0.381213\pi$
$272$	0	0
$273$	−5.41666e22	−0.389212
$274$	0	0
$275$	−3.39805e22	−0.227808
$276$	0	0
$277$	−7.04507e21	−0.0440887	−0.0220444	−	0.999757i	$-0.507018\pi$
$277$	−7.04507e21	−0.0440887	−0.0220444	+	0.999757i	$0.507018\pi$
$278$	0	0
$279$	−3.04251e23	−1.77824
$280$	0	0
$281$	−1.00259e22	−0.0547539	−0.0273770	−	0.999625i	$-0.508715\pi$
$281$	−1.00259e22	−0.0547539	−0.0273770	+	0.999625i	$0.508715\pi$
$282$	0	0
$283$	2.12867e23	1.08677	0.543383	−	0.839485i	$-0.317143\pi$
$283$	2.12867e23	1.08677	0.543383	+	0.839485i	$0.317143\pi$
$284$	0	0
$285$	7.01172e22	0.334810
$286$	0	0
$287$	1.11379e23	0.497653
$288$	0	0
$289$	3.47675e21	0.0145427
$290$	0	0
$291$	4.97311e22	0.194825
$292$	0	0
$293$	5.10289e23	1.87315	0.936577	−	0.350462i	$-0.113975\pi$
$293$	5.10289e23	1.87315	0.936577	+	0.350462i	$0.113975\pi$
$294$	0	0
$295$	8.36872e21	0.0287972
$296$	0	0
$297$	1.29135e23	0.416734
$298$	0	0
$299$	−2.30793e23	−0.698790
$300$	0	0
$301$	8.75268e22	0.248748
$302$	0	0
$303$	8.55260e23	2.28241
$304$	0	0
$305$	−2.34146e23	−0.587001
$306$	0	0
$307$	4.42100e23	1.04161	0.520806	−	0.853675i	$-0.325631\pi$
$307$	4.42100e23	1.04161	0.520806	+	0.853675i	$0.325631\pi$
$308$	0	0
$309$	2.25126e23	0.498678
$310$	0	0
$311$	3.04953e23	0.635344	0.317672	−	0.948201i	$-0.397099\pi$
$311$	3.04953e23	0.635344	0.317672	+	0.948201i	$0.397099\pi$
$312$	0	0
$313$	−6.44559e23	−1.26355	−0.631774	−	0.775153i	$-0.717673\pi$
$313$	−6.44559e23	−1.26355	−0.631774	+	0.775153i	$0.717673\pi$
$314$	0	0
$315$	−7.70083e22	−0.142098
$316$	0	0
$317$	1.02667e24	1.78389	0.891947	−	0.452140i	$-0.149339\pi$
$317$	1.02667e24	1.78389	0.891947	+	0.452140i	$0.149339\pi$
$318$	0	0
$319$	−1.26915e24	−2.07731
$320$	0	0
$321$	−4.45306e23	−0.686849
$322$	0	0
$323$	3.46186e23	0.503368
$324$	0	0
$325$	−1.48389e23	−0.203474
$326$	0	0
$327$	−2.13030e24	−2.75573
$328$	0	0
$329$	−1.75799e23	−0.214613
$330$	0	0
$331$	−8.39874e23	−0.967939	−0.483970	−	0.875085i	$-0.660806\pi$
$331$	−8.39874e23	−0.967939	−0.483970	+	0.875085i	$0.660806\pi$
$332$	0	0
$333$	−1.44820e24	−1.57619
$334$	0	0
$335$	−3.78469e23	−0.389139
$336$	0	0
$337$	−7.88161e23	−0.765827	−0.382913	−	0.923784i	$-0.625079\pi$
$337$	−7.88161e23	−0.765827	−0.382913	+	0.923784i	$0.625079\pi$
$338$	0	0
$339$	4.44631e23	0.408416
$340$	0	0
$341$	1.87413e24	1.62792
$342$	0	0
$343$	6.01313e23	0.494090
$344$	0	0
$345$	−5.91829e23	−0.460166
$346$	0	0
$347$	3.27984e23	0.241392	0.120696	−	0.992690i	$-0.461487\pi$
$347$	3.27984e23	0.241392	0.120696	+	0.992690i	$0.461487\pi$
$348$	0	0
$349$	−1.57618e24	−1.09841	−0.549204	−	0.835689i	$-0.685069\pi$
$349$	−1.57618e24	−1.09841	−0.549204	+	0.835689i	$0.685069\pi$
$350$	0	0
$351$	5.63920e23	0.372220
$352$	0	0
$353$	−1.44980e23	−0.0906667	−0.0453333	−	0.998972i	$-0.514435\pi$
$353$	−1.44980e23	−0.0906667	−0.0453333	+	0.998972i	$0.514435\pi$
$354$	0	0
$355$	−7.20633e23	−0.427115
$356$	0	0
$357$	−6.85788e23	−0.385338
$358$	0	0
$359$	−7.64624e23	−0.407428	−0.203714	−	0.979030i	$-0.565301\pi$
$359$	−7.64624e23	−0.407428	−0.203714	+	0.979030i	$0.565301\pi$
$360$	0	0
$361$	−1.48432e24	−0.750253
$362$	0	0
$363$	−9.28978e23	−0.445546
$364$	0	0
$365$	−1.77700e24	−0.808920
$366$	0	0
$367$	2.08854e24	0.902639	0.451320	−	0.892362i	$-0.350953\pi$
$367$	2.08854e24	0.902639	0.451320	+	0.892362i	$0.350953\pi$
$368$	0	0
$369$	−5.90747e24	−2.42466
$370$	0	0
$371$	−5.13698e22	−0.0200288
$372$	0	0
$373$	2.90964e24	1.07797	0.538984	−	0.842316i	$-0.318808\pi$
$373$	2.90964e24	1.07797	0.538984	+	0.842316i	$0.318808\pi$
$374$	0	0
$375$	−3.80518e23	−0.133992
$376$	0	0
$377$	−5.54224e24	−1.85542
$378$	0	0
$379$	−4.57048e24	−1.45509	−0.727544	−	0.686061i	$-0.759338\pi$
$379$	−4.57048e24	−1.45509	−0.727544	+	0.686061i	$0.759338\pi$
$380$	0	0
$381$	3.10140e24	0.939226
$382$	0	0
$383$	−4.74338e24	−1.36678	−0.683392	−	0.730052i	$-0.739496\pi$
$383$	−4.74338e24	−1.36678	−0.683392	+	0.730052i	$0.739496\pi$
$384$	0	0
$385$	4.74356e23	0.130085
$386$	0	0
$387$	−4.64235e24	−1.21195
$388$	0	0
$389$	−1.60295e24	−0.398472	−0.199236	−	0.979952i	$-0.563846\pi$
$389$	−1.60295e24	−0.398472	−0.199236	+	0.979952i	$0.563846\pi$
$390$	0	0
$391$	−2.92201e24	−0.691834
$392$	0	0
$393$	9.95047e24	2.24447
$394$	0	0
$395$	−1.02638e24	−0.220616
$396$	0	0
$397$	5.17477e24	1.06018	0.530092	−	0.847940i	$-0.322157\pi$
$397$	5.17477e24	1.06018	0.530092	+	0.847940i	$0.322157\pi$
$398$	0	0
$399$	−9.78812e23	−0.191186
$400$	0	0
$401$	1.04892e24	0.195377	0.0976883	−	0.995217i	$-0.468855\pi$
$401$	1.04892e24	0.195377	0.0976883	+	0.995217i	$0.468855\pi$
$402$	0	0
$403$	8.18410e24	1.45403
$404$	0	0
$405$	−1.83666e24	−0.311320
$406$	0	0
$407$	8.92066e24	1.44295
$408$	0	0
$409$	−9.84304e24	−1.51970	−0.759850	−	0.650098i	$-0.774728\pi$
$409$	−9.84304e24	−1.51970	−0.759850	+	0.650098i	$0.774728\pi$
$410$	0	0
$411$	−6.19076e24	−0.912529
$412$	0	0
$413$	−1.16825e23	−0.0164440
$414$	0	0
$415$	5.41562e24	0.728101
$416$	0	0
$417$	1.13632e25	1.45952
$418$	0	0
$419$	−1.39759e24	−0.171533	−0.0857664	−	0.996315i	$-0.527334\pi$
$419$	−1.39759e24	−0.171533	−0.0857664	+	0.996315i	$0.527334\pi$
$420$	0	0
$421$	2.96558e24	0.347880	0.173940	−	0.984756i	$-0.444350\pi$
$421$	2.96558e24	0.347880	0.173940	+	0.984756i	$0.444350\pi$
$422$	0	0
$423$	9.32425e24	1.04564
$424$	0	0
$425$	−1.87871e24	−0.201449
$426$	0	0
$427$	3.26861e24	0.335195
$428$	0	0
$429$	−1.76969e25	−1.73601
$430$	0	0
$431$	1.73742e25	1.63069	0.815343	−	0.578978i	$-0.196548\pi$
$431$	1.73742e25	1.63069	0.815343	+	0.578978i	$0.196548\pi$
$432$	0	0
$433$	1.97317e25	1.77227	0.886133	−	0.463432i	$-0.153382\pi$
$433$	1.97317e25	1.77227	0.886133	+	0.463432i	$0.153382\pi$
$434$	0	0
$435$	−1.42121e25	−1.22183
$436$	0	0
$437$	−4.17052e24	−0.343255
$438$	0	0
$439$	−2.37490e25	−1.87168	−0.935840	−	0.352425i	$-0.885357\pi$
$439$	−2.37490e25	−1.87168	−0.935840	+	0.352425i	$0.885357\pi$
$440$	0	0
$441$	−1.54091e25	−1.16308
$442$	0	0
$443$	1.08663e25	0.785679	0.392840	−	0.919607i	$-0.371493\pi$
$443$	1.08663e25	0.785679	0.392840	+	0.919607i	$0.371493\pi$
$444$	0	0
$445$	−7.59504e23	−0.0526150
$446$	0	0
$447$	−1.77166e25	−1.17614
$448$	0	0
$449$	1.64022e25	1.04367	0.521834	−	0.853047i	$-0.325248\pi$
$449$	1.64022e25	1.04367	0.521834	+	0.853047i	$0.325248\pi$
$450$	0	0
$451$	3.63889e25	2.21969
$452$	0	0
$453$	−5.37306e23	−0.0314260
$454$	0	0
$455$	2.07146e24	0.116190
$456$	0	0
$457$	−1.90127e25	−1.02291	−0.511457	−	0.859309i	$-0.670894\pi$
$457$	−1.90127e25	−1.02291	−0.511457	+	0.859309i	$0.670894\pi$
$458$	0	0
$459$	7.13963e24	0.368515
$460$	0	0
$461$	−2.73913e25	−1.35660	−0.678302	−	0.734783i	$-0.737284\pi$
$461$	−2.73913e25	−1.35660	−0.678302	+	0.734783i	$0.737284\pi$
$462$	0	0
$463$	−2.15999e25	−1.02667	−0.513337	−	0.858187i	$-0.671591\pi$
$463$	−2.15999e25	−1.02667	−0.513337	+	0.858187i	$0.671591\pi$
$464$	0	0
$465$	2.09867e25	0.957505
$466$	0	0
$467$	−6.57907e24	−0.288174	−0.144087	−	0.989565i	$-0.546024\pi$
$467$	−6.57907e24	−0.288174	−0.144087	+	0.989565i	$0.546024\pi$
$468$	0	0
$469$	5.28330e24	0.222210
$470$	0	0
$471$	1.57157e24	0.0634796
$472$	0	0
$473$	2.85960e25	1.10949
$474$	0	0
$475$	−2.68145e24	−0.0999494
$476$	0	0
$477$	2.72461e24	0.0975844
$478$	0	0
$479$	−1.84209e25	−0.634049	−0.317024	−	0.948417i	$-0.602684\pi$
$479$	−1.84209e25	−0.634049	−0.317024	+	0.948417i	$0.602684\pi$
$480$	0	0
$481$	3.89555e25	1.28882
$482$	0	0
$483$	8.26174e24	0.262769
$484$	0	0
$485$	−1.90184e24	−0.0581603
$486$	0	0
$487$	5.85949e25	1.72320	0.861598	−	0.507591i	$-0.169464\pi$
$487$	5.85949e25	1.72320	0.861598	+	0.507591i	$0.169464\pi$
$488$	0	0
$489$	2.94339e25	0.832555
$490$	0	0
$491$	−1.17567e24	−0.0319896	−0.0159948	−	0.999872i	$-0.505092\pi$
$491$	−1.17567e24	−0.0319896	−0.0159948	+	0.999872i	$0.505092\pi$
$492$	0	0
$493$	−7.01687e25	−1.83695
$494$	0	0
$495$	−2.51595e25	−0.633800
$496$	0	0
$497$	1.00598e25	0.243895
$498$	0	0
$499$	1.57255e25	0.366985	0.183492	−	0.983021i	$-0.441260\pi$
$499$	1.57255e25	0.366985	0.183492	+	0.983021i	$0.441260\pi$
$500$	0	0
$501$	3.52169e25	0.791211
$502$	0	0
$503$	2.64249e24	0.0571632	0.0285816	−	0.999591i	$-0.490901\pi$
$503$	2.64249e24	0.0571632	0.0285816	+	0.999591i	$0.490901\pi$
$504$	0	0
$505$	−3.27072e25	−0.681358
$506$	0	0
$507$	−2.61669e24	−0.0525020
$508$	0	0
$509$	7.94197e25	1.53500	0.767501	−	0.641048i	$-0.221500\pi$
$509$	7.94197e25	1.53500	0.767501	+	0.641048i	$0.221500\pi$
$510$	0	0
$511$	2.48064e25	0.461918
$512$	0	0
$513$	1.01903e25	0.182840
$514$	0	0
$515$	−8.60937e24	−0.148868
$516$	0	0
$517$	−5.74356e25	−0.957241
$518$	0	0
$519$	2.11564e24	0.0339900
$520$	0	0
$521$	4.66972e25	0.723324	0.361662	−	0.932309i	$-0.382209\pi$
$521$	4.66972e25	0.723324	0.361662	+	0.932309i	$0.382209\pi$
$522$	0	0
$523$	6.46849e25	0.966132	0.483066	−	0.875584i	$-0.339523\pi$
$523$	6.46849e25	0.966132	0.483066	+	0.875584i	$0.339523\pi$
$524$	0	0
$525$	5.31190e24	0.0765133
$526$	0	0
$527$	1.03617e26	1.43955
$528$	0	0
$529$	−3.94138e25	−0.528226
$530$	0	0
$531$	6.19628e24	0.0801186
$532$	0	0
$533$	1.58906e26	1.98259
$534$	0	0
$535$	1.70296e25	0.205042
$536$	0	0
$537$	1.86505e26	2.16738
$538$	0	0
$539$	9.49170e25	1.06476
$540$	0	0
$541$	1.28312e25	0.138961	0.0694805	−	0.997583i	$-0.477866\pi$
$541$	1.28312e25	0.138961	0.0694805	+	0.997583i	$0.477866\pi$
$542$	0	0
$543$	−5.37441e25	−0.561995
$544$	0	0
$545$	8.14676e25	0.822656
$546$	0	0
$547$	−5.22682e25	−0.509750	−0.254875	−	0.966974i	$-0.582034\pi$
$547$	−5.22682e25	−0.509750	−0.254875	+	0.966974i	$0.582034\pi$
$548$	0	0
$549$	−1.73364e26	−1.63314
$550$	0	0
$551$	−1.00150e26	−0.911408
$552$	0	0
$553$	1.43280e25	0.125978
$554$	0	0
$555$	9.98947e25	0.848710
$556$	0	0
$557$	5.10671e25	0.419292	0.209646	−	0.977777i	$-0.432769\pi$
$557$	5.10671e25	0.419292	0.209646	+	0.977777i	$0.432769\pi$
$558$	0	0
$559$	1.24876e26	0.990981
$560$	0	0
$561$	−2.24055e26	−1.71873
$562$	0	0
$563$	−1.95962e26	−1.45325	−0.726627	−	0.687032i	$-0.758913\pi$
$563$	−1.95962e26	−1.45325	−0.726627	+	0.687032i	$0.758913\pi$
$564$	0	0
$565$	−1.70037e25	−0.121923
$566$	0	0
$567$	2.56392e25	0.177773
$568$	0	0
$569$	−2.04702e25	−0.137264	−0.0686320	−	0.997642i	$-0.521863\pi$
$569$	−2.04702e25	−0.137264	−0.0686320	+	0.997642i	$0.521863\pi$
$570$	0	0
$571$	1.26019e26	0.817322	0.408661	−	0.912686i	$-0.365996\pi$
$571$	1.26019e26	0.817322	0.408661	+	0.912686i	$0.365996\pi$
$572$	0	0
$573$	−3.38957e26	−2.12655
$574$	0	0
$575$	2.26330e25	0.137372
$576$	0	0
$577$	−1.24271e26	−0.729792	−0.364896	−	0.931048i	$-0.618895\pi$
$577$	−1.24271e26	−0.729792	−0.364896	+	0.931048i	$0.618895\pi$
$578$	0	0
$579$	1.35610e26	0.770629
$580$	0	0
$581$	−7.56003e25	−0.415767
$582$	0	0
$583$	−1.67831e25	−0.0893349
$584$	0	0
$585$	−1.09869e26	−0.566100
$586$	0	0
$587$	−3.67268e26	−1.83198	−0.915991	−	0.401199i	$-0.868593\pi$
$587$	−3.67268e26	−1.83198	−0.915991	+	0.401199i	$0.868593\pi$
$588$	0	0
$589$	1.47890e26	0.714238
$590$	0	0
$591$	−5.22753e26	−2.44464
$592$	0	0
$593$	−1.40519e26	−0.636378	−0.318189	−	0.948027i	$-0.603075\pi$
$593$	−1.40519e26	−0.636378	−0.318189	+	0.948027i	$0.603075\pi$
$594$	0	0
$595$	2.62262e25	0.115033
$596$	0	0
$597$	−1.29255e26	−0.549149
$598$	0	0
$599$	−2.05613e26	−0.846244	−0.423122	−	0.906073i	$-0.639066\pi$
$599$	−2.05613e26	−0.846244	−0.423122	+	0.906073i	$0.639066\pi$
$600$	0	0
$601$	2.32345e26	0.926457	0.463228	−	0.886239i	$-0.346691\pi$
$601$	2.32345e26	0.926457	0.463228	+	0.886239i	$0.346691\pi$
$602$	0	0
$603$	−2.80222e26	−1.08265
$604$	0	0
$605$	3.55263e25	0.133007
$606$	0	0
$607$	−2.30237e26	−0.835376	−0.417688	−	0.908591i	$-0.637159\pi$
$607$	−2.30237e26	−0.835376	−0.417688	+	0.908591i	$0.637159\pi$
$608$	0	0
$609$	1.98396e26	0.697701
$610$	0	0
$611$	−2.50815e26	−0.854992
$612$	0	0
$613$	−7.21990e25	−0.238592	−0.119296	−	0.992859i	$-0.538064\pi$
$613$	−7.21990e25	−0.238592	−0.119296	+	0.992859i	$0.538064\pi$
$614$	0	0
$615$	4.07488e26	1.30557
$616$	0	0
$617$	−1.44998e26	−0.450458	−0.225229	−	0.974306i	$-0.572313\pi$
$617$	−1.44998e26	−0.450458	−0.225229	+	0.974306i	$0.572313\pi$
$618$	0	0
$619$	2.13817e26	0.644141	0.322071	−	0.946716i	$-0.395621\pi$
$619$	2.13817e26	0.644141	0.322071	+	0.946716i	$0.395621\pi$
$620$	0	0
$621$	−8.60116e25	−0.251297
$622$	0	0
$623$	1.06024e25	0.0300448
$624$	0	0
$625$	1.45519e25	0.0400000
$626$	0	0
$627$	−3.19789e26	−0.852751
$628$	0	0
$629$	4.93205e26	1.27599
$630$	0	0
$631$	−1.60890e26	−0.403877	−0.201939	−	0.979398i	$-0.564724\pi$
$631$	−1.60890e26	−0.403877	−0.201939	+	0.979398i	$0.564724\pi$
$632$	0	0
$633$	−7.31519e26	−1.78193
$634$	0	0
$635$	−1.18605e26	−0.280383
$636$	0	0
$637$	4.14492e26	0.951024
$638$	0	0
$639$	−5.33564e26	−1.18831
$640$	0	0
$641$	−5.97417e26	−1.29160	−0.645798	−	0.763509i	$-0.723475\pi$
$641$	−5.97417e26	−1.29160	−0.645798	+	0.763509i	$0.723475\pi$
$642$	0	0
$643$	8.08712e26	1.69742	0.848710	−	0.528858i	$-0.177380\pi$
$643$	8.08712e26	1.69742	0.848710	+	0.528858i	$0.177380\pi$
$644$	0	0
$645$	3.20222e26	0.652580
$646$	0	0
$647$	7.53176e26	1.49041	0.745205	−	0.666835i	$-0.232351\pi$
$647$	7.53176e26	1.49041	0.745205	+	0.666835i	$0.232351\pi$
$648$	0	0
$649$	−3.81679e25	−0.0733456
$650$	0	0
$651$	−2.92967e26	−0.546764
$652$	0	0
$653$	1.75162e25	0.0317515	0.0158758	−	0.999874i	$-0.494946\pi$
$653$	1.75162e25	0.0317515	0.0158758	+	0.999874i	$0.494946\pi$
$654$	0	0
$655$	−3.80530e26	−0.670033
$656$	0	0
$657$	−1.31571e27	−2.25055
$658$	0	0
$659$	1.10678e27	1.83929	0.919644	−	0.392754i	$-0.128477\pi$
$659$	1.10678e27	1.83929	0.919644	+	0.392754i	$0.128477\pi$
$660$	0	0
$661$	−1.90870e26	−0.308194	−0.154097	−	0.988056i	$-0.549247\pi$
$661$	−1.90870e26	−0.308194	−0.154097	+	0.988056i	$0.549247\pi$
$662$	0	0
$663$	−9.78423e26	−1.53514
$664$	0	0
$665$	3.74321e25	0.0570741
$666$	0	0
$667$	8.45327e26	1.25265
$668$	0	0
$669$	7.18069e26	1.03423
$670$	0	0
$671$	1.06789e27	1.49508
$672$	0	0
$673$	−1.19576e27	−1.62742	−0.813710	−	0.581270i	$-0.802556\pi$
$673$	−1.19576e27	−1.62742	−0.813710	+	0.581270i	$0.802556\pi$
$674$	0	0
$675$	−5.53014e25	−0.0731729
$676$	0	0
$677$	4.22916e26	0.544079	0.272040	−	0.962286i	$-0.412302\pi$
$677$	4.22916e26	0.544079	0.272040	+	0.962286i	$0.412302\pi$
$678$	0	0
$679$	2.65490e25	0.0332113
$680$	0	0
$681$	1.74382e27	2.12131
$682$	0	0
$683$	−7.22585e24	−0.00854854	−0.00427427	−	0.999991i	$-0.501361\pi$
$683$	−7.22585e24	−0.00854854	−0.00427427	+	0.999991i	$0.501361\pi$
$684$	0	0
$685$	2.36749e26	0.272413
$686$	0	0
$687$	−1.15923e27	−1.29742
$688$	0	0
$689$	−7.32900e25	−0.0797925
$690$	0	0
$691$	1.59913e26	0.169372	0.0846861	−	0.996408i	$-0.473011\pi$
$691$	1.59913e26	0.169372	0.0846861	+	0.996408i	$0.473011\pi$
$692$	0	0
$693$	3.51218e26	0.361919
$694$	0	0
$695$	−4.34557e26	−0.435703
$696$	0	0
$697$	2.01187e27	1.96285
$698$	0	0
$699$	−1.60692e27	−1.52567
$700$	0	0
$701$	−3.07030e26	−0.283700	−0.141850	−	0.989888i	$-0.545305\pi$
$701$	−3.07030e26	−0.283700	−0.141850	+	0.989888i	$0.545305\pi$
$702$	0	0
$703$	7.03942e26	0.633084
$704$	0	0
$705$	−6.43172e26	−0.563028
$706$	0	0
$707$	4.56582e26	0.389076
$708$	0	0
$709$	−2.50359e26	−0.207694	−0.103847	−	0.994593i	$-0.533115\pi$
$709$	−2.50359e26	−0.207694	−0.103847	+	0.994593i	$0.533115\pi$
$710$	0	0
$711$	−7.59944e26	−0.613791
$712$	0	0
$713$	−1.24828e27	−0.981657
$714$	0	0
$715$	6.76770e26	0.518244
$716$	0	0
$717$	−6.37264e26	−0.475212
$718$	0	0
$719$	4.99484e26	0.362741	0.181371	−	0.983415i	$-0.441947\pi$
$719$	4.99484e26	0.362741	0.181371	+	0.983415i	$0.441947\pi$
$720$	0	0
$721$	1.20184e26	0.0850083
$722$	0	0
$723$	5.93786e25	0.0409087
$724$	0	0
$725$	5.43505e26	0.364748
$726$	0	0
$727$	−1.17305e27	−0.766900	−0.383450	−	0.923562i	$-0.625264\pi$
$727$	−1.17305e27	−0.766900	−0.383450	+	0.923562i	$0.625264\pi$
$728$	0	0
$729$	−2.22041e27	−1.41423
$730$	0	0
$731$	1.58101e27	0.981117
$732$	0	0
$733$	7.29500e26	0.441101	0.220550	−	0.975376i	$-0.429215\pi$
$733$	7.29500e26	0.441101	0.220550	+	0.975376i	$0.429215\pi$
$734$	0	0
$735$	1.06289e27	0.626267
$736$	0	0
$737$	1.72612e27	0.991126
$738$	0	0
$739$	−2.41117e27	−1.34929	−0.674647	−	0.738141i	$-0.735704\pi$
$739$	−2.41117e27	−1.34929	−0.674647	+	0.738141i	$0.735704\pi$
$740$	0	0
$741$	−1.39648e27	−0.761663
$742$	0	0
$743$	−1.98675e27	−1.05621	−0.528105	−	0.849179i	$-0.677097\pi$
$743$	−1.98675e27	−1.05621	−0.528105	+	0.849179i	$0.677097\pi$
$744$	0	0
$745$	6.77527e26	0.351109
$746$	0	0
$747$	4.00978e27	2.02570
$748$	0	0
$749$	−2.37727e26	−0.117085
$750$	0	0
$751$	−4.77748e26	−0.229414	−0.114707	−	0.993399i	$-0.536593\pi$
$751$	−4.77748e26	−0.229414	−0.114707	+	0.993399i	$0.536593\pi$
$752$	0	0
$753$	5.60033e27	2.62217
$754$	0	0
$755$	2.05479e25	0.00938147
$756$	0	0
$757$	8.34040e25	0.0371344	0.0185672	−	0.999828i	$-0.494090\pi$
$757$	8.34040e25	0.0371344	0.0185672	+	0.999828i	$0.494090\pi$
$758$	0	0
$759$	2.69920e27	1.17203
$760$	0	0
$761$	−2.96552e27	−1.25587	−0.627937	−	0.778264i	$-0.716100\pi$
$761$	−2.96552e27	−1.25587	−0.627937	+	0.778264i	$0.716100\pi$
$762$	0	0
$763$	−1.13726e27	−0.469761
$764$	0	0
$765$	−1.39102e27	−0.560465
$766$	0	0
$767$	−1.66675e26	−0.0655111
$768$	0	0
$769$	9.68856e26	0.371500	0.185750	−	0.982597i	$-0.440528\pi$
$769$	9.68856e26	0.371500	0.185750	+	0.982597i	$0.440528\pi$
$770$	0	0
$771$	1.08272e27	0.405041
$772$	0	0
$773$	5.16072e27	1.88367	0.941836	−	0.336074i	$-0.109099\pi$
$773$	5.16072e27	1.88367	0.941836	+	0.336074i	$0.109099\pi$
$774$	0	0
$775$	−8.02582e26	−0.285840
$776$	0	0
$777$	−1.39450e27	−0.484638
$778$	0	0
$779$	2.87150e27	0.973875
$780$	0	0
$781$	3.28665e27	1.08785
$782$	0	0
$783$	−2.06547e27	−0.667241
$784$	0	0
$785$	−6.01004e25	−0.0189503
$786$	0	0
$787$	4.95523e27	1.52512	0.762559	−	0.646919i	$-0.223943\pi$
$787$	4.95523e27	1.52512	0.762559	+	0.646919i	$0.223943\pi$
$788$	0	0
$789$	−5.44708e27	−1.63656
$790$	0	0
$791$	2.37367e26	0.0696214
$792$	0	0
$793$	4.66336e27	1.33538
$794$	0	0
$795$	−1.87939e26	−0.0525449
$796$	0	0
$797$	9.51644e26	0.259789	0.129894	−	0.991528i	$-0.458536\pi$
$797$	9.51644e26	0.259789	0.129894	+	0.991528i	$0.458536\pi$
$798$	0	0
$799$	−3.17550e27	−0.846481
$800$	0	0
$801$	−5.62344e26	−0.146384
$802$	0	0
$803$	8.10452e27	2.06030
$804$	0	0
$805$	−3.15949e26	−0.0784433
$806$	0	0
$807$	−4.73819e27	−1.14898
$808$	0	0
$809$	6.64620e26	0.157421	0.0787105	−	0.996898i	$-0.474920\pi$
$809$	6.64620e26	0.157421	0.0787105	+	0.996898i	$0.474920\pi$
$810$	0	0
$811$	3.63702e27	0.841487	0.420743	−	0.907180i	$-0.361769\pi$
$811$	3.63702e27	0.841487	0.420743	+	0.907180i	$0.361769\pi$
$812$	0	0
$813$	−4.83296e27	−1.09233
$814$	0	0
$815$	−1.12562e27	−0.248539
$816$	0	0
$817$	2.25655e27	0.486783
$818$	0	0
$819$	1.53373e27	0.323260
$820$	0	0
$821$	−4.44788e26	−0.0915996	−0.0457998	−	0.998951i	$-0.514584\pi$
$821$	−4.44788e26	−0.0915996	−0.0457998	+	0.998951i	$0.514584\pi$
$822$	0	0
$823$	5.12412e27	1.03115	0.515574	−	0.856845i	$-0.327579\pi$
$823$	5.12412e27	1.03115	0.515574	+	0.856845i	$0.327579\pi$
$824$	0	0
$825$	1.73546e27	0.341273
$826$	0	0
$827$	5.00813e27	0.962439	0.481219	−	0.876600i	$-0.340194\pi$
$827$	5.00813e27	0.962439	0.481219	+	0.876600i	$0.340194\pi$
$828$	0	0
$829$	−3.98409e27	−0.748275	−0.374138	−	0.927373i	$-0.622061\pi$
$829$	−3.98409e27	−0.748275	−0.374138	+	0.927373i	$0.622061\pi$
$830$	0	0
$831$	3.59807e26	0.0660481
$832$	0	0
$833$	5.24777e27	0.941558
$834$	0	0
$835$	−1.34678e27	−0.236197
$836$	0	0
$837$	3.05004e27	0.522893
$838$	0	0
$839$	−9.17325e27	−1.53739	−0.768696	−	0.639615i	$-0.779094\pi$
$839$	−9.17325e27	−1.53739	−0.768696	+	0.639615i	$0.779094\pi$
$840$	0	0
$841$	1.41963e28	2.32602
$842$	0	0
$843$	5.12047e26	0.0820254
$844$	0	0
$845$	1.00068e26	0.0156732
$846$	0	0
$847$	−4.95936e26	−0.0759509
$848$	0	0
$849$	−1.08716e28	−1.62806
$850$	0	0
$851$	−5.94167e27	−0.870118
$852$	0	0
$853$	−7.79931e27	−1.11697	−0.558484	−	0.829516i	$-0.688617\pi$
$853$	−7.79931e27	−1.11697	−0.558484	+	0.829516i	$0.688617\pi$
$854$	0	0
$855$	−1.98537e27	−0.278076
$856$	0	0
$857$	5.48738e26	0.0751705	0.0375852	−	0.999293i	$-0.488033\pi$
$857$	5.48738e26	0.0751705	0.0375852	+	0.999293i	$0.488033\pi$
$858$	0	0
$859$	−2.88785e27	−0.386936	−0.193468	−	0.981107i	$-0.561974\pi$
$859$	−2.88785e27	−0.386936	−0.193468	+	0.981107i	$0.561974\pi$
$860$	0	0
$861$	−5.68839e27	−0.745521
$862$	0	0
$863$	1.27182e27	0.163051	0.0815257	−	0.996671i	$-0.474021\pi$
$863$	1.27182e27	0.163051	0.0815257	+	0.996671i	$0.474021\pi$
$864$	0	0
$865$	−8.09071e25	−0.0101469
$866$	0	0
$867$	−1.77566e26	−0.0217860
$868$	0	0
$869$	4.68111e27	0.561903
$870$	0	0
$871$	7.53776e27	0.885258
$872$	0	0
$873$	−1.40814e27	−0.161812
$874$	0	0
$875$	−2.03140e26	−0.0228412
$876$	0	0
$877$	−7.93254e27	−0.872803	−0.436401	−	0.899752i	$-0.643747\pi$
$877$	−7.93254e27	−0.872803	−0.436401	+	0.899752i	$0.643747\pi$
$878$	0	0
$879$	−2.60616e28	−2.80612
$880$	0	0
$881$	−4.65349e27	−0.490351	−0.245176	−	0.969479i	$-0.578846\pi$
$881$	−4.65349e27	−0.490351	−0.245176	+	0.969479i	$0.578846\pi$
$882$	0	0
$883$	−1.39322e28	−1.43678	−0.718392	−	0.695639i	$-0.755122\pi$
$883$	−1.39322e28	−1.43678	−0.718392	+	0.695639i	$0.755122\pi$
$884$	0	0
$885$	−4.27409e26	−0.0431403
$886$	0	0
$887$	2.98771e27	0.295165	0.147582	−	0.989050i	$-0.452851\pi$
$887$	2.98771e27	0.295165	0.147582	+	0.989050i	$0.452851\pi$
$888$	0	0
$889$	1.65569e27	0.160107
$890$	0	0
$891$	8.37662e27	0.792922
$892$	0	0
$893$	−4.53233e27	−0.419984
$894$	0	0
$895$	−7.13239e27	−0.647018
$896$	0	0
$897$	1.17871e28	1.04684
$898$	0	0
$899$	−2.99759e28	−2.60649
$900$	0	0
$901$	−9.27904e26	−0.0789982
$902$	0	0
$903$	−4.47019e27	−0.372642
$904$	0	0
$905$	2.05530e27	0.167770
$906$	0	0
$907$	1.01372e28	0.810308	0.405154	−	0.914248i	$-0.367218\pi$
$907$	1.01372e28	0.810308	0.405154	+	0.914248i	$0.367218\pi$
$908$	0	0
$909$	−2.42167e28	−1.89565
$910$	0	0
$911$	−1.01665e28	−0.779377	−0.389688	−	0.920947i	$-0.627417\pi$
$911$	−1.01665e28	−0.779377	−0.389688	+	0.920947i	$0.627417\pi$
$912$	0	0
$913$	−2.46995e28	−1.85445
$914$	0	0
$915$	1.19584e28	0.879371
$916$	0	0
$917$	5.31207e27	0.382609
$918$	0	0
$919$	8.89132e27	0.627291	0.313645	−	0.949540i	$-0.398450\pi$
$919$	8.89132e27	0.627291	0.313645	+	0.949540i	$0.398450\pi$
$920$	0	0
$921$	−2.25790e28	−1.56041
$922$	0	0
$923$	1.43524e28	0.971650
$924$	0	0
$925$	−3.82021e27	−0.253362
$926$	0	0
$927$	−6.37446e27	−0.414177
$928$	0	0
$929$	7.61744e26	0.0484909	0.0242454	−	0.999706i	$-0.492282\pi$
$929$	7.61744e26	0.0484909	0.0242454	+	0.999706i	$0.492282\pi$
$930$	0	0
$931$	7.49004e27	0.467156
$932$	0	0
$933$	−1.55746e28	−0.951791
$934$	0	0
$935$	8.56840e27	0.513085
$936$	0	0
$937$	−1.46040e28	−0.856931	−0.428465	−	0.903558i	$-0.640946\pi$
$937$	−1.46040e28	−0.856931	−0.428465	+	0.903558i	$0.640946\pi$
$938$	0	0
$939$	3.29191e28	1.89289
$940$	0	0
$941$	−1.23989e28	−0.698687	−0.349344	−	0.936995i	$-0.613595\pi$
$941$	−1.23989e28	−0.698687	−0.349344	+	0.936995i	$0.613595\pi$
$942$	0	0
$943$	−2.42371e28	−1.33850
$944$	0	0
$945$	7.71989e26	0.0417839
$946$	0	0
$947$	1.35611e28	0.719400	0.359700	−	0.933068i	$-0.382879\pi$
$947$	1.35611e28	0.719400	0.359700	+	0.933068i	$0.382879\pi$
$948$	0	0
$949$	3.53916e28	1.84022
$950$	0	0
$951$	−5.24345e28	−2.67240
$952$	0	0
$953$	−1.00835e28	−0.503764	−0.251882	−	0.967758i	$-0.581050\pi$
$953$	−1.00835e28	−0.503764	−0.251882	+	0.967758i	$0.581050\pi$
$954$	0	0
$955$	1.29625e28	0.634830
$956$	0	0
$957$	6.48183e28	3.11196
$958$	0	0
$959$	−3.30494e27	−0.155556
$960$	0	0
$961$	2.25941e28	1.04261
$962$	0	0
$963$	1.26089e28	0.570461
$964$	0	0
$965$	−5.18605e27	−0.230053
$966$	0	0
$967$	−1.24965e28	−0.543547	−0.271774	−	0.962361i	$-0.587610\pi$
$967$	−1.24965e28	−0.543547	−0.271774	+	0.962361i	$0.587610\pi$
$968$	0	0
$969$	−1.76805e28	−0.754082
$970$	0	0
$971$	9.87663e27	0.413072	0.206536	−	0.978439i	$-0.433781\pi$
$971$	9.87663e27	0.413072	0.206536	+	0.978439i	$0.433781\pi$
$972$	0	0
$973$	6.06627e27	0.248800
$974$	0	0
$975$	7.57856e27	0.304820
$976$	0	0
$977$	−2.84735e27	−0.112316	−0.0561581	−	0.998422i	$-0.517885\pi$
$977$	−2.84735e27	−0.112316	−0.0561581	+	0.998422i	$0.517885\pi$
$978$	0	0
$979$	3.46393e27	0.134009
$980$	0	0
$981$	6.03194e28	2.28877
$982$	0	0
$983$	2.10459e28	0.783265	0.391632	−	0.920122i	$-0.371910\pi$
$983$	2.10459e28	0.783265	0.391632	+	0.920122i	$0.371910\pi$
$984$	0	0
$985$	1.99913e28	0.729789
$986$	0	0
$987$	8.97846e27	0.321506
$988$	0	0
$989$	−1.90466e28	−0.669041
$990$	0	0
$991$	−5.03784e28	−1.73598	−0.867989	−	0.496583i	$-0.834588\pi$
$991$	−5.03784e28	−1.73598	−0.867989	+	0.496583i	$0.834588\pi$
$992$	0	0
$993$	4.28943e28	1.45004
$994$	0	0
$995$	4.94301e27	0.163935
$996$	0	0
$997$	−1.44919e28	−0.471544	−0.235772	−	0.971808i	$-0.575762\pi$
$997$	−1.44919e28	−0.471544	−0.235772	+	0.971808i	$0.575762\pi$
$998$	0	0
$999$	1.45179e28	0.463480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.a.d.1.1		2
4.3	odd	2		10.20.a.d.1.2	✓	2
12.11	even	2		90.20.a.g.1.1		2
20.3	even	4		50.20.b.f.49.2		4
20.7	even	4		50.20.b.f.49.3		4
20.19	odd	2		50.20.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.20.a.d.1.2	✓	2	4.3	odd	2
50.20.a.f.1.1		2	20.19	odd	2
50.20.b.f.49.2		4	20.3	even	4
50.20.b.f.49.3		4	20.7	even	4
80.20.a.d.1.1		2	1.1	even	1	trivial
90.20.a.g.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-51072.2 q^{3} +1.95312e6 q^{5} -2.72650e7 q^{7} +1.44611e9 q^{9} -8.90778e9 q^{11} -3.88993e10 q^{13} -9.97505e10 q^{15} -4.92493e11 q^{17} -7.02926e11 q^{19} +1.39248e12 q^{21} +5.93310e12 q^{23} +3.81470e12 q^{25} -1.44969e13 q^{27} +1.42477e14 q^{29} -2.10392e14 q^{31} +4.54940e14 q^{33} -5.32519e13 q^{35} -1.00145e15 q^{37} +1.98667e15 q^{39} -4.08507e15 q^{41} -3.21023e15 q^{43} +2.82444e15 q^{45} +6.44781e15 q^{47} -1.06555e16 q^{49} +2.51527e16 q^{51} +1.88410e15 q^{53} -1.73980e16 q^{55} +3.59000e16 q^{57} +4.28479e15 q^{59} -1.19883e17 q^{61} -3.94282e16 q^{63} -7.59752e16 q^{65} -1.93776e17 q^{67} -3.03017e17 q^{69} -3.68964e17 q^{71} -9.09825e17 q^{73} -1.94825e17 q^{75} +2.42870e17 q^{77} -5.25508e17 q^{79} -9.40371e17 q^{81} +2.77280e18 q^{83} -9.61900e17 q^{85} -7.27660e18 q^{87} -3.88866e17 q^{89} +1.06059e18 q^{91} +1.07452e19 q^{93} -1.37290e18 q^{95} -9.73741e17 q^{97} -1.28817e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 33724 q^{3} + 3906250 q^{5} - 83061292 q^{7} + 584812954 q^{9} - 3549480144 q^{11} + 30250225564 q^{13} - 65867187500 q^{15} + 329655779412 q^{17} - 383076809800 q^{19} + 424515195104 q^{21} - 2082856096884 q^{23}+ \cdots - 17\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q - 33724 * q^3 + 3906250 * q^5 - 83061292 * q^7 + 584812954 * q^9 - 3549480144 * q^11 + 30250225564 * q^13 - 65867187500 * q^15 + 329655779412 * q^17 - 383076809800 * q^19 + 424515195104 * q^21 - 2082856096884 * q^23 + 7629394531250 * q^25 - 49602163340440 * q^27 + 166587684602220 * q^29 - 418151119038664 * q^31 + 547897533578928 * q^33 - 162229085937500 * q^35 - 1292219637694628 * q^37 + 3186297306918232 * q^39 - 3127192714527996 * q^41 + 1795992338357636 * q^43 + 1142212800781250 * q^45 + 6761653310165988 * q^47 - 18941182438871454 * q^49 + 39415550773661256 * q^51 - 9787568571066516 * q^53 - 6932578406250000 * q^55 + 41448804721422800 * q^57 + 25484316660294360 * q^59 - 77840503559636276 * q^61 + 8629133041886116 * q^63 + 59082471804687500 * q^65 - 439654835073204532 * q^67 - 442079200728708192 * q^69 + 29427662567526696 * q^71 - 1463308911724217756 * q^73 - 128646850585937500 * q^75 - 56103061801470576 * q^77 + 1235969155825112720 * q^79 - 548329353045127838 * q^81 + 877205635969635396 * q^83 + 643858944164062500 * q^85 - 6858312722579528040 * q^87 + 4172929077000032820 * q^89 - 2797700455846427144 * q^91 + 7140934358593491968 * q^93 - 748196894140625000 * q^95 + 11254578228347861332 * q^97 - 17496761012343831888 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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