LMFDB - Embedded newform 80.18.a.d.1.2

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			1.2
Root			$-143.792$ 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)

$f(q)$	$=$	$q+9311.53 q^{3} +390625. q^{5} +2.48655e7 q^{7} -4.24355e7 q^{9} +1.90392e8 q^{11} -1.57377e7 q^{13} +3.63732e9 q^{15} -2.25502e10 q^{17} -1.27984e11 q^{19} +2.31536e11 q^{21} -2.59325e11 q^{23} +1.52588e11 q^{25} -1.59763e12 q^{27} -2.57953e12 q^{29} -7.91253e12 q^{31} +1.77285e12 q^{33} +9.71310e12 q^{35} -2.09888e13 q^{37} -1.46542e11 q^{39} +1.98123e13 q^{41} +4.60777e13 q^{43} -1.65764e13 q^{45} -2.66827e14 q^{47} +3.85664e14 q^{49} -2.09977e14 q^{51} +4.00045e14 q^{53} +7.43721e13 q^{55} -1.19172e15 q^{57} -1.82611e15 q^{59} -1.48236e15 q^{61} -1.05518e15 q^{63} -6.14754e12 q^{65} +2.71410e15 q^{67} -2.41471e15 q^{69} +3.87337e15 q^{71} -2.85786e15 q^{73} +1.42083e15 q^{75} +4.73421e15 q^{77} -2.80988e15 q^{79} -9.39628e15 q^{81} +7.04230e15 q^{83} -8.80866e15 q^{85} -2.40194e16 q^{87} +5.51966e16 q^{89} -3.91327e14 q^{91} -7.36778e16 q^{93} -4.99936e16 q^{95} +6.42662e16 q^{97} -8.07940e15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	9311.53	0.819390	0.409695	−	0.912223i	$-0.365635\pi$
$3$	9311.53	0.819390	0.409695	+	0.912223i	$0.365635\pi$
$4$	0	0
$5$	390625.	0.447214
$5$	390625.	0.447214
$6$	0	0
$7$	2.48655e7	1.63029	0.815144	−	0.579258i	$-0.196658\pi$
$7$	2.48655e7	1.63029	0.815144	+	0.579258i	$0.196658\pi$
$8$	0	0
$9$	−4.24355e7	−0.328600
$10$	0	0
$11$	1.90392e8	0.267801	0.133900	−	0.990995i	$-0.457250\pi$
$11$	1.90392e8	0.267801	0.133900	+	0.990995i	$0.457250\pi$
$12$	0	0
$13$	−1.57377e7	−0.00535085	−0.00267543	−	0.999996i	$-0.500852\pi$
$13$	−1.57377e7	−0.00535085	−0.00267543	+	0.999996i	$0.500852\pi$
$14$	0	0
$15$	3.63732e9	0.366442
$16$	0	0
$17$	−2.25502e10	−0.784032	−0.392016	−	0.919958i	$-0.628222\pi$
$17$	−2.25502e10	−0.784032	−0.392016	+	0.919958i	$0.628222\pi$
$18$	0	0
$19$	−1.27984e11	−1.72882	−0.864408	−	0.502791i	$-0.832306\pi$
$19$	−1.27984e11	−1.72882	−0.864408	+	0.502791i	$0.832306\pi$
$20$	0	0
$21$	2.31536e11	1.33584
$22$	0	0
$23$	−2.59325e11	−0.690490	−0.345245	−	0.938513i	$-0.612204\pi$
$23$	−2.59325e11	−0.690490	−0.345245	+	0.938513i	$0.612204\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	0	0
$27$	−1.59763e12	−1.08864
$28$	0	0
$29$	−2.57953e12	−0.957542	−0.478771	−	0.877940i	$-0.658918\pi$
$29$	−2.57953e12	−0.957542	−0.478771	+	0.877940i	$0.658918\pi$
$30$	0	0
$31$	−7.91253e12	−1.66625	−0.833126	−	0.553083i	$-0.813451\pi$
$31$	−7.91253e12	−1.66625	−0.833126	+	0.553083i	$0.813451\pi$
$32$	0	0
$33$	1.77285e12	0.219433
$34$	0	0
$35$	9.71310e12	0.729087
$36$	0	0
$37$	−2.09888e13	−0.982367	−0.491184	−	0.871056i	$-0.663436\pi$
$37$	−2.09888e13	−0.982367	−0.491184	+	0.871056i	$0.663436\pi$
$38$	0	0
$39$	−1.46542e11	−0.00438444
$40$	0	0
$41$	1.98123e13	0.387499	0.193750	−	0.981051i	$-0.437935\pi$
$41$	1.98123e13	0.387499	0.193750	+	0.981051i	$0.437935\pi$
$42$	0	0
$43$	4.60777e13	0.601186	0.300593	−	0.953753i	$-0.402815\pi$
$43$	4.60777e13	0.601186	0.300593	+	0.953753i	$0.402815\pi$
$44$	0	0
$45$	−1.65764e13	−0.146955
$46$	0	0
$47$	−2.66827e14	−1.63455	−0.817274	−	0.576250i	$-0.804516\pi$
$47$	−2.66827e14	−1.63455	−0.817274	+	0.576250i	$0.804516\pi$
$48$	0	0
$49$	3.85664e14	1.65784
$50$	0	0
$51$	−2.09977e14	−0.642428
$52$	0	0
$53$	4.00045e14	0.882599	0.441300	−	0.897360i	$-0.354518\pi$
$53$	4.00045e14	0.882599	0.441300	+	0.897360i	$0.354518\pi$
$54$	0	0
$55$	7.43721e13	0.119764
$56$	0	0
$57$	−1.19172e15	−1.41657
$58$	0	0
$59$	−1.82611e15	−1.61914	−0.809570	−	0.587024i	$-0.800299\pi$
$59$	−1.82611e15	−1.61914	−0.809570	+	0.587024i	$0.800299\pi$
$60$	0	0
$61$	−1.48236e15	−0.990035	−0.495017	−	0.868883i	$-0.664838\pi$
$61$	−1.48236e15	−0.990035	−0.495017	+	0.868883i	$0.664838\pi$
$62$	0	0
$63$	−1.05518e15	−0.535713
$64$	0	0
$65$	−6.14754e12	−0.00239297
$66$	0	0
$67$	2.71410e15	0.816563	0.408282	−	0.912856i	$-0.366128\pi$
$67$	2.71410e15	0.816563	0.408282	+	0.912856i	$0.366128\pi$
$68$	0	0
$69$	−2.41471e15	−0.565780
$70$	0	0
$71$	3.87337e15	0.711858	0.355929	−	0.934513i	$-0.384164\pi$
$71$	3.87337e15	0.711858	0.355929	+	0.934513i	$0.384164\pi$
$72$	0	0
$73$	−2.85786e15	−0.414759	−0.207380	−	0.978261i	$-0.566494\pi$
$73$	−2.85786e15	−0.414759	−0.207380	+	0.978261i	$0.566494\pi$
$74$	0	0
$75$	1.42083e15	0.163878
$76$	0	0
$77$	4.73421e15	0.436593
$78$	0	0
$79$	−2.80988e15	−0.208381	−0.104190	−	0.994557i	$-0.533225\pi$
$79$	−2.80988e15	−0.208381	−0.104190	+	0.994557i	$0.533225\pi$
$80$	0	0
$81$	−9.39628e15	−0.563422
$82$	0	0
$83$	7.04230e15	0.343203	0.171601	−	0.985166i	$-0.445106\pi$
$83$	7.04230e15	0.343203	0.171601	+	0.985166i	$0.445106\pi$
$84$	0	0
$85$	−8.80866e15	−0.350630
$86$	0	0
$87$	−2.40194e16	−0.784600
$88$	0	0
$89$	5.51966e16	1.48627	0.743135	−	0.669142i	$-0.233338\pi$
$89$	5.51966e16	1.48627	0.743135	+	0.669142i	$0.233338\pi$
$90$	0	0
$91$	−3.91327e14	−0.00872343
$92$	0	0
$93$	−7.36778e16	−1.36531
$94$	0	0
$95$	−4.99936e16	−0.773150
$96$	0	0
$97$	6.42662e16	0.832574	0.416287	−	0.909233i	$-0.363331\pi$
$97$	6.42662e16	0.832574	0.416287	+	0.909233i	$0.363331\pi$
$98$	0	0
$99$	−8.07940e15	−0.0879994
$100$	0	0
$101$	−1.56405e17	−1.43721	−0.718603	−	0.695421i	$-0.755218\pi$
$101$	−1.56405e17	−1.43721	−0.718603	+	0.695421i	$0.755218\pi$
$102$	0	0
$103$	2.33166e17	1.81363	0.906816	−	0.421527i	$-0.138506\pi$
$103$	2.33166e17	1.81363	0.906816	+	0.421527i	$0.138506\pi$
$104$	0	0
$105$	9.04438e16	0.597407
$106$	0	0
$107$	1.05397e17	0.593017	0.296509	−	0.955030i	$-0.404178\pi$
$107$	1.05397e17	0.593017	0.296509	+	0.955030i	$0.404178\pi$
$108$	0	0
$109$	2.51895e17	1.21086	0.605431	−	0.795898i	$-0.293001\pi$
$109$	2.51895e17	1.21086	0.605431	+	0.795898i	$0.293001\pi$
$110$	0	0
$111$	−1.95438e17	−0.804942
$112$	0	0
$113$	4.09395e17	1.44869	0.724346	−	0.689436i	$-0.242142\pi$
$113$	4.09395e17	1.44869	0.724346	+	0.689436i	$0.242142\pi$
$114$	0	0
$115$	−1.01299e17	−0.308796
$116$	0	0
$117$	6.67838e14	0.00175829
$118$	0	0
$119$	−5.60722e17	−1.27820
$120$	0	0
$121$	−4.69198e17	−0.928283
$122$	0	0
$123$	1.84482e17	0.317513
$124$	0	0
$125$	5.96046e16	0.0894427
$126$	0	0
$127$	6.42410e17	0.842327	0.421164	−	0.906985i	$-0.361622\pi$
$127$	6.42410e17	0.842327	0.421164	+	0.906985i	$0.361622\pi$
$128$	0	0
$129$	4.29054e17	0.492606
$130$	0	0
$131$	−3.57323e17	−0.359961	−0.179980	−	0.983670i	$-0.557603\pi$
$131$	−3.57323e17	−0.359961	−0.179980	+	0.983670i	$0.557603\pi$
$132$	0	0
$133$	−3.18238e18	−2.81847
$134$	0	0
$135$	−6.24075e17	−0.486855
$136$	0	0
$137$	−1.23821e18	−0.852454	−0.426227	−	0.904616i	$-0.640158\pi$
$137$	−1.23821e18	−0.852454	−0.426227	+	0.904616i	$0.640158\pi$
$138$	0	0
$139$	−1.95129e18	−1.18767	−0.593836	−	0.804586i	$-0.702387\pi$
$139$	−1.95129e18	−1.18767	−0.593836	+	0.804586i	$0.702387\pi$
$140$	0	0
$141$	−2.48457e18	−1.33933
$142$	0	0
$143$	−2.99634e15	−0.00143296
$144$	0	0
$145$	−1.00763e18	−0.428226
$146$	0	0
$147$	3.59112e18	1.35842
$148$	0	0
$149$	1.86793e18	0.629908	0.314954	−	0.949107i	$-0.398011\pi$
$149$	1.86793e18	0.629908	0.314954	+	0.949107i	$0.398011\pi$
$150$	0	0
$151$	−5.18884e18	−1.56231	−0.781154	−	0.624339i	$-0.785368\pi$
$151$	−5.18884e18	−1.56231	−0.781154	+	0.624339i	$0.785368\pi$
$152$	0	0
$153$	9.56927e17	0.257633
$154$	0	0
$155$	−3.09083e18	−0.745171
$156$	0	0
$157$	7.78987e18	1.68416	0.842080	−	0.539353i	$-0.181331\pi$
$157$	7.78987e18	1.68416	0.842080	+	0.539353i	$0.181331\pi$
$158$	0	0
$159$	3.72503e18	0.723193
$160$	0	0
$161$	−6.44825e18	−1.12570
$162$	0	0
$163$	−6.16891e18	−0.969647	−0.484824	−	0.874612i	$-0.661116\pi$
$163$	−6.16891e18	−0.969647	−0.484824	+	0.874612i	$0.661116\pi$
$164$	0	0
$165$	6.92518e17	0.0981336
$166$	0	0
$167$	−1.24726e19	−1.59539	−0.797693	−	0.603064i	$-0.793946\pi$
$167$	−1.24726e19	−1.59539	−0.797693	+	0.603064i	$0.793946\pi$
$168$	0	0
$169$	−8.65017e18	−0.999971
$170$	0	0
$171$	5.43105e18	0.568089
$172$	0	0
$173$	1.64494e19	1.55869	0.779344	−	0.626596i	$-0.215552\pi$
$173$	1.64494e19	1.55869	0.779344	+	0.626596i	$0.215552\pi$
$174$	0	0
$175$	3.79418e18	0.326058
$176$	0	0
$177$	−1.70039e19	−1.32671
$178$	0	0
$179$	1.53067e18	0.108550	0.0542751	−	0.998526i	$-0.482715\pi$
$179$	1.53067e18	0.108550	0.0542751	+	0.998526i	$0.482715\pi$
$180$	0	0
$181$	−8.51444e18	−0.549399	−0.274700	−	0.961530i	$-0.588578\pi$
$181$	−8.51444e18	−0.549399	−0.274700	+	0.961530i	$0.588578\pi$
$182$	0	0
$183$	−1.38031e19	−0.811224
$184$	0	0
$185$	−8.19877e18	−0.439328
$186$	0	0
$187$	−4.29338e18	−0.209965
$188$	0	0
$189$	−3.97260e19	−1.77480
$190$	0	0
$191$	−1.05921e19	−0.432710	−0.216355	−	0.976315i	$-0.569417\pi$
$191$	−1.05921e19	−0.432710	−0.216355	+	0.976315i	$0.569417\pi$
$192$	0	0
$193$	−1.55768e19	−0.582425	−0.291213	−	0.956658i	$-0.594059\pi$
$193$	−1.55768e19	−0.582425	−0.291213	+	0.956658i	$0.594059\pi$
$194$	0	0
$195$	−5.72431e16	−0.00196078
$196$	0	0
$197$	8.47320e18	0.266124	0.133062	−	0.991108i	$-0.457519\pi$
$197$	8.47320e18	0.266124	0.133062	+	0.991108i	$0.457519\pi$
$198$	0	0
$199$	−9.48640e18	−0.273433	−0.136716	−	0.990610i	$-0.543655\pi$
$199$	−9.48640e18	−0.273433	−0.136716	+	0.990610i	$0.543655\pi$
$200$	0	0
$201$	2.52724e19	0.669083
$202$	0	0
$203$	−6.41414e19	−1.56107
$204$	0	0
$205$	7.73916e18	0.173295
$206$	0	0
$207$	1.10046e19	0.226895
$208$	0	0
$209$	−2.43671e19	−0.462978
$210$	0	0
$211$	5.06102e19	0.886824	0.443412	−	0.896318i	$-0.353768\pi$
$211$	5.06102e19	0.886824	0.443412	+	0.896318i	$0.353768\pi$
$212$	0	0
$213$	3.60671e19	0.583289
$214$	0	0
$215$	1.79991e19	0.268858
$216$	0	0
$217$	−1.96749e20	−2.71647
$218$	0	0
$219$	−2.66110e19	−0.339849
$220$	0	0
$221$	3.54888e17	0.00419524
$222$	0	0
$223$	−1.13017e20	−1.23752	−0.618759	−	0.785581i	$-0.712364\pi$
$223$	−1.13017e20	−1.23752	−0.618759	+	0.785581i	$0.712364\pi$
$224$	0	0
$225$	−6.47514e18	−0.0657201
$226$	0	0
$227$	−1.34057e20	−1.26203	−0.631016	−	0.775770i	$-0.717362\pi$
$227$	−1.34057e20	−1.26203	−0.631016	+	0.775770i	$0.717362\pi$
$228$	0	0
$229$	−1.53092e20	−1.33768	−0.668840	−	0.743406i	$-0.733209\pi$
$229$	−1.53092e20	−1.33768	−0.668840	+	0.743406i	$0.733209\pi$
$230$	0	0
$231$	4.40828e19	0.357740
$232$	0	0
$233$	1.54095e20	1.16215	0.581076	−	0.813849i	$-0.302632\pi$
$233$	1.54095e20	1.16215	0.581076	+	0.813849i	$0.302632\pi$
$234$	0	0
$235$	−1.04229e20	−0.730992
$236$	0	0
$237$	−2.61643e19	−0.170745
$238$	0	0
$239$	−1.10626e20	−0.672161	−0.336081	−	0.941833i	$-0.609101\pi$
$239$	−1.10626e20	−0.672161	−0.336081	+	0.941833i	$0.609101\pi$
$240$	0	0
$241$	−1.68590e20	−0.954303	−0.477152	−	0.878821i	$-0.658331\pi$
$241$	−1.68590e20	−0.954303	−0.477152	+	0.878821i	$0.658331\pi$
$242$	0	0
$243$	1.18825e20	0.626980
$244$	0	0
$245$	1.50650e20	0.741408
$246$	0	0
$247$	2.01417e18	0.00925064
$248$	0	0
$249$	6.55746e19	0.281217
$250$	0	0
$251$	8.71684e19	0.349247	0.174623	−	0.984635i	$-0.444129\pi$
$251$	8.71684e19	0.349247	0.174623	+	0.984635i	$0.444129\pi$
$252$	0	0
$253$	−4.93735e19	−0.184914
$254$	0	0
$255$	−8.20221e19	−0.287303
$256$	0	0
$257$	2.57989e20	0.845610	0.422805	−	0.906221i	$-0.361046\pi$
$257$	2.57989e20	0.845610	0.422805	+	0.906221i	$0.361046\pi$
$258$	0	0
$259$	−5.21899e20	−1.60154
$260$	0	0
$261$	1.09464e20	0.314648
$262$	0	0
$263$	4.12838e20	1.11213	0.556065	−	0.831139i	$-0.312311\pi$
$263$	4.12838e20	1.11213	0.556065	+	0.831139i	$0.312311\pi$
$264$	0	0
$265$	1.56267e20	0.394710
$266$	0	0
$267$	5.13965e20	1.21783
$268$	0	0
$269$	3.61622e20	0.804194	0.402097	−	0.915597i	$-0.368282\pi$
$269$	3.61622e20	0.804194	0.402097	+	0.915597i	$0.368282\pi$
$270$	0	0
$271$	6.61685e20	1.38170	0.690848	−	0.723000i	$-0.257238\pi$
$271$	6.61685e20	1.38170	0.690848	+	0.723000i	$0.257238\pi$
$272$	0	0
$273$	−3.64385e18	−0.00714789
$274$	0	0
$275$	2.90516e19	0.0535602
$276$	0	0
$277$	2.43700e20	0.422452	0.211226	−	0.977437i	$-0.432254\pi$
$277$	2.43700e20	0.422452	0.211226	+	0.977437i	$0.432254\pi$
$278$	0	0
$279$	3.35772e20	0.547531
$280$	0	0
$281$	−9.47052e20	−1.45335	−0.726675	−	0.686981i	$-0.758935\pi$
$281$	−9.47052e20	−1.45335	−0.726675	+	0.686981i	$0.758935\pi$
$282$	0	0
$283$	3.67504e20	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$283$	3.67504e20	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$284$	0	0
$285$	−4.65517e20	−0.633511
$286$	0	0
$287$	4.92642e20	0.631736
$288$	0	0
$289$	−3.18730e20	−0.385293
$290$	0	0
$291$	5.98417e20	0.682203
$292$	0	0
$293$	1.20904e21	1.30037	0.650183	−	0.759778i	$-0.274692\pi$
$293$	1.20904e21	1.30037	0.650183	+	0.759778i	$0.274692\pi$
$294$	0	0
$295$	−7.13323e20	−0.724101
$296$	0	0
$297$	−3.04177e20	−0.291539
$298$	0	0
$299$	4.08118e18	0.00369471
$300$	0	0
$301$	1.14575e21	0.980106
$302$	0	0
$303$	−1.45637e21	−1.17763
$304$	0	0
$305$	−5.79047e20	−0.442757
$306$	0	0
$307$	4.76931e20	0.344969	0.172484	−	0.985012i	$-0.444821\pi$
$307$	4.76931e20	0.344969	0.172484	+	0.985012i	$0.444821\pi$
$308$	0	0
$309$	2.17114e21	1.48607
$310$	0	0
$311$	−3.50418e20	−0.227051	−0.113525	−	0.993535i	$-0.536214\pi$
$311$	−3.50418e20	−0.227051	−0.113525	+	0.993535i	$0.536214\pi$
$312$	0	0
$313$	2.04751e21	1.25631	0.628157	−	0.778087i	$-0.283810\pi$
$313$	2.04751e21	1.25631	0.628157	+	0.778087i	$0.283810\pi$
$314$	0	0
$315$	−4.12180e20	−0.239578
$316$	0	0
$317$	1.31663e21	0.725205	0.362602	−	0.931944i	$-0.381888\pi$
$317$	1.31663e21	0.725205	0.362602	+	0.931944i	$0.381888\pi$
$318$	0	0
$319$	−4.91123e20	−0.256430
$320$	0	0
$321$	9.81411e20	0.485912
$322$	0	0
$323$	2.88605e21	1.35545
$324$	0	0
$325$	−2.40138e18	−0.00107017
$326$	0	0
$327$	2.34553e21	0.992167
$328$	0	0
$329$	−6.63479e21	−2.66478
$330$	0	0
$331$	−3.85862e21	−1.47195	−0.735977	−	0.677007i	$-0.763277\pi$
$331$	−3.85862e21	−1.47195	−0.735977	+	0.677007i	$0.763277\pi$
$332$	0	0
$333$	8.90672e20	0.322806
$334$	0	0
$335$	1.06019e21	0.365178
$336$	0	0
$337$	4.69149e21	1.53623	0.768115	−	0.640312i	$-0.221195\pi$
$337$	4.69149e21	1.53623	0.768115	+	0.640312i	$0.221195\pi$
$338$	0	0
$339$	3.81210e21	1.18704
$340$	0	0
$341$	−1.50649e21	−0.446224
$342$	0	0
$343$	3.80526e21	1.07247
$344$	0	0
$345$	−9.43246e20	−0.253025
$346$	0	0
$347$	−2.51637e21	−0.642648	−0.321324	−	0.946969i	$-0.604128\pi$
$347$	−2.51637e21	−0.642648	−0.321324	+	0.946969i	$0.604128\pi$
$348$	0	0
$349$	−2.20222e21	−0.535606	−0.267803	−	0.963474i	$-0.586298\pi$
$349$	−2.20222e21	−0.535606	−0.267803	+	0.963474i	$0.586298\pi$
$350$	0	0
$351$	2.51431e19	0.00582516
$352$	0	0
$353$	1.30214e21	0.287456	0.143728	−	0.989617i	$-0.454091\pi$
$353$	1.30214e21	0.287456	0.143728	+	0.989617i	$0.454091\pi$
$354$	0	0
$355$	1.51304e21	0.318353
$356$	0	0
$357$	−5.22118e21	−1.04734
$358$	0	0
$359$	−2.07162e21	−0.396285	−0.198143	−	0.980173i	$-0.563491\pi$
$359$	−2.07162e21	−0.396285	−0.198143	+	0.980173i	$0.563491\pi$
$360$	0	0
$361$	1.08994e22	1.98880
$362$	0	0
$363$	−4.36895e21	−0.760625
$364$	0	0
$365$	−1.11635e21	−0.185486
$366$	0	0
$367$	−4.69137e21	−0.744111	−0.372056	−	0.928210i	$-0.621347\pi$
$367$	−4.69137e21	−0.744111	−0.372056	+	0.928210i	$0.621347\pi$
$368$	0	0
$369$	−8.40743e20	−0.127332
$370$	0	0
$371$	9.94732e21	1.43889
$372$	0	0
$373$	−7.77652e20	−0.107463	−0.0537317	−	0.998555i	$-0.517112\pi$
$373$	−7.77652e20	−0.107463	−0.0537317	+	0.998555i	$0.517112\pi$
$374$	0	0
$375$	5.55011e20	0.0732885
$376$	0	0
$377$	4.05959e19	0.00512367
$378$	0	0
$379$	−3.50054e20	−0.0422379	−0.0211189	−	0.999777i	$-0.506723\pi$
$379$	−3.50054e20	−0.0422379	−0.0211189	+	0.999777i	$0.506723\pi$
$380$	0	0
$381$	5.98183e21	0.690194
$382$	0	0
$383$	1.02315e22	1.12914	0.564570	−	0.825385i	$-0.309042\pi$
$383$	1.02315e22	1.12914	0.564570	+	0.825385i	$0.309042\pi$
$384$	0	0
$385$	1.84930e21	0.195250
$386$	0	0
$387$	−1.95533e21	−0.197550
$388$	0	0
$389$	−4.13104e21	−0.399473	−0.199737	−	0.979850i	$-0.564009\pi$
$389$	−4.13104e21	−0.399473	−0.199737	+	0.979850i	$0.564009\pi$
$390$	0	0
$391$	5.84781e21	0.541366
$392$	0	0
$393$	−3.32723e21	−0.294948
$394$	0	0
$395$	−1.09761e21	−0.0931907
$396$	0	0
$397$	−7.65334e21	−0.622489	−0.311244	−	0.950330i	$-0.600746\pi$
$397$	−7.65334e21	−0.622489	−0.311244	+	0.950330i	$0.600746\pi$
$398$	0	0
$399$	−2.96328e22	−2.30942
$400$	0	0
$401$	−4.08273e20	−0.0304947	−0.0152473	−	0.999884i	$-0.504854\pi$
$401$	−4.08273e20	−0.0304947	−0.0152473	+	0.999884i	$0.504854\pi$
$402$	0	0
$403$	1.24525e20	0.00891588
$404$	0	0
$405$	−3.67042e21	−0.251970
$406$	0	0
$407$	−3.99612e21	−0.263079
$408$	0	0
$409$	2.00923e22	1.26877	0.634383	−	0.773019i	$-0.281254\pi$
$409$	2.00923e22	1.26877	0.634383	+	0.773019i	$0.281254\pi$
$410$	0	0
$411$	−1.15297e22	−0.698493
$412$	0	0
$413$	−4.54071e22	−2.63966
$414$	0	0
$415$	2.75090e21	0.153485
$416$	0	0
$417$	−1.81695e22	−0.973167
$418$	0	0
$419$	−3.31114e22	−1.70278	−0.851390	−	0.524533i	$-0.824240\pi$
$419$	−3.31114e22	−1.70278	−0.851390	+	0.524533i	$0.824240\pi$
$420$	0	0
$421$	−2.31325e22	−1.14242	−0.571208	−	0.820805i	$-0.693525\pi$
$421$	−2.31325e22	−1.14242	−0.571208	+	0.820805i	$0.693525\pi$
$422$	0	0
$423$	1.13229e22	0.537113
$424$	0	0
$425$	−3.44088e21	−0.156806
$426$	0	0
$427$	−3.68597e22	−1.61404
$428$	0	0
$429$	−2.79005e19	−0.00117416
$430$	0	0
$431$	−4.44596e22	−1.79849	−0.899247	−	0.437440i	$-0.855885\pi$
$431$	−4.44596e22	−1.79849	−0.899247	+	0.437440i	$0.855885\pi$
$432$	0	0
$433$	2.85304e22	1.10959	0.554793	−	0.831989i	$-0.312798\pi$
$433$	2.85304e22	1.10959	0.554793	+	0.831989i	$0.312798\pi$
$434$	0	0
$435$	−9.38257e21	−0.350884
$436$	0	0
$437$	3.31893e22	1.19373
$438$	0	0
$439$	4.28578e22	1.48279	0.741397	−	0.671066i	$-0.234163\pi$
$439$	4.28578e22	1.48279	0.741397	+	0.671066i	$0.234163\pi$
$440$	0	0
$441$	−1.63658e22	−0.544767
$442$	0	0
$443$	1.40464e22	0.449919	0.224959	−	0.974368i	$-0.427775\pi$
$443$	1.40464e22	0.449919	0.224959	+	0.974368i	$0.427775\pi$
$444$	0	0
$445$	2.15612e22	0.664680
$446$	0	0
$447$	1.73933e22	0.516140
$448$	0	0
$449$	−3.88401e22	−1.10965	−0.554826	−	0.831966i	$-0.687215\pi$
$449$	−3.88401e22	−1.10965	−0.554826	+	0.831966i	$0.687215\pi$
$450$	0	0
$451$	3.77210e21	0.103773
$452$	0	0
$453$	−4.83161e22	−1.28014
$454$	0	0
$455$	−1.52862e20	−0.00390124
$456$	0	0
$457$	−2.21160e22	−0.543774	−0.271887	−	0.962329i	$-0.587648\pi$
$457$	−2.21160e22	−0.543774	−0.271887	+	0.962329i	$0.587648\pi$
$458$	0	0
$459$	3.60269e22	0.853530
$460$	0	0
$461$	1.69257e22	0.386446	0.193223	−	0.981155i	$-0.438106\pi$
$461$	1.69257e22	0.386446	0.193223	+	0.981155i	$0.438106\pi$
$462$	0	0
$463$	−5.00810e22	−1.10213	−0.551067	−	0.834461i	$-0.685779\pi$
$463$	−5.00810e22	−1.10213	−0.551067	+	0.834461i	$0.685779\pi$
$464$	0	0
$465$	−2.87804e22	−0.610585
$466$	0	0
$467$	6.87611e22	1.40653	0.703266	−	0.710927i	$-0.251724\pi$
$467$	6.87611e22	1.40653	0.703266	+	0.710927i	$0.251724\pi$
$468$	0	0
$469$	6.74875e22	1.33123
$470$	0	0
$471$	7.25357e22	1.37998
$472$	0	0
$473$	8.77284e21	0.160998
$474$	0	0
$475$	−1.95287e22	−0.345763
$476$	0	0
$477$	−1.69761e22	−0.290022
$478$	0	0
$479$	3.44291e22	0.567641	0.283821	−	0.958877i	$-0.408398\pi$
$479$	3.44291e22	0.567641	0.283821	+	0.958877i	$0.408398\pi$
$480$	0	0
$481$	3.30316e20	0.00525650
$482$	0	0
$483$	−6.00431e22	−0.922385
$484$	0	0
$485$	2.51040e22	0.372338
$486$	0	0
$487$	−8.83844e20	−0.0126584	−0.00632921	−	0.999980i	$-0.502015\pi$
$487$	−8.83844e20	−0.0126584	−0.00632921	+	0.999980i	$0.502015\pi$
$488$	0	0
$489$	−5.74420e22	−0.794519
$490$	0	0
$491$	−1.04515e23	−1.39632	−0.698158	−	0.715943i	$-0.745997\pi$
$491$	−1.04515e23	−1.39632	−0.698158	+	0.715943i	$0.745997\pi$
$492$	0	0
$493$	5.81688e22	0.750744
$494$	0	0
$495$	−3.15602e21	−0.0393545
$496$	0	0
$497$	9.63135e22	1.16053
$498$	0	0
$499$	5.12067e22	0.596310	0.298155	−	0.954518i	$-0.403629\pi$
$499$	5.12067e22	0.596310	0.298155	+	0.954518i	$0.403629\pi$
$500$	0	0
$501$	−1.16139e23	−1.30724
$502$	0	0
$503$	−5.06660e22	−0.551301	−0.275650	−	0.961258i	$-0.588893\pi$
$503$	−5.06660e22	−0.551301	−0.275650	+	0.961258i	$0.588893\pi$
$504$	0	0
$505$	−6.10957e22	−0.642738
$506$	0	0
$507$	−8.05463e22	−0.819366
$508$	0	0
$509$	1.68111e23	1.65385	0.826923	−	0.562316i	$-0.190089\pi$
$509$	1.68111e23	1.65385	0.826923	+	0.562316i	$0.190089\pi$
$510$	0	0
$511$	−7.10621e22	−0.676177
$512$	0	0
$513$	2.04471e23	1.88206
$514$	0	0
$515$	9.10805e22	0.811081
$516$	0	0
$517$	−5.08018e22	−0.437733
$518$	0	0
$519$	1.53170e23	1.27717
$520$	0	0
$521$	1.13516e23	0.916085	0.458042	−	0.888930i	$-0.348551\pi$
$521$	1.13516e23	0.916085	0.458042	+	0.888930i	$0.348551\pi$
$522$	0	0
$523$	−3.13139e22	−0.244609	−0.122305	−	0.992493i	$-0.539028\pi$
$523$	−3.13139e22	−0.244609	−0.122305	+	0.992493i	$0.539028\pi$
$524$	0	0
$525$	3.53296e22	0.267168
$526$	0	0
$527$	1.78429e23	1.30640
$528$	0	0
$529$	−7.38008e22	−0.523224
$530$	0	0
$531$	7.74918e22	0.532050
$532$	0	0
$533$	−3.11800e20	−0.00207345
$534$	0	0
$535$	4.11708e22	0.265205
$536$	0	0
$537$	1.42529e22	0.0889449
$538$	0	0
$539$	7.34275e22	0.443971
$540$	0	0
$541$	−1.32903e23	−0.778680	−0.389340	−	0.921094i	$-0.627297\pi$
$541$	−1.32903e23	−0.778680	−0.389340	+	0.921094i	$0.627297\pi$
$542$	0	0
$543$	−7.92825e22	−0.450172
$544$	0	0
$545$	9.83965e22	0.541514
$546$	0	0
$547$	−3.86923e22	−0.206411	−0.103205	−	0.994660i	$-0.532910\pi$
$547$	−3.86923e22	−0.206411	−0.103205	+	0.994660i	$0.532910\pi$
$548$	0	0
$549$	6.29047e22	0.325326
$550$	0	0
$551$	3.30138e23	1.65541
$552$	0	0
$553$	−6.98692e22	−0.339721
$554$	0	0
$555$	−7.63431e22	−0.359981
$556$	0	0
$557$	−1.53384e23	−0.701475	−0.350737	−	0.936474i	$-0.614069\pi$
$557$	−1.53384e23	−0.701475	−0.350737	+	0.936474i	$0.614069\pi$
$558$	0	0
$559$	−7.25157e20	−0.00321686
$560$	0	0
$561$	−3.99780e22	−0.172043
$562$	0	0
$563$	1.19252e23	0.497904	0.248952	−	0.968516i	$-0.419914\pi$
$563$	1.19252e23	0.497904	0.248952	+	0.968516i	$0.419914\pi$
$564$	0	0
$565$	1.59920e23	0.647875
$566$	0	0
$567$	−2.33644e23	−0.918540
$568$	0	0
$569$	−1.07363e23	−0.409638	−0.204819	−	0.978800i	$-0.565661\pi$
$569$	−1.07363e23	−0.409638	−0.204819	+	0.978800i	$0.565661\pi$
$570$	0	0
$571$	2.25644e23	0.835634	0.417817	−	0.908531i	$-0.362795\pi$
$571$	2.25644e23	0.835634	0.417817	+	0.908531i	$0.362795\pi$
$572$	0	0
$573$	−9.86284e22	−0.354558
$574$	0	0
$575$	−3.95698e22	−0.138098
$576$	0	0
$577$	−1.29594e23	−0.439127	−0.219563	−	0.975598i	$-0.570463\pi$
$577$	−1.29594e23	−0.439127	−0.219563	+	0.975598i	$0.570463\pi$
$578$	0	0
$579$	−1.45044e23	−0.477233
$580$	0	0
$581$	1.75111e23	0.559519
$582$	0	0
$583$	7.61655e22	0.236361
$584$	0	0
$585$	2.60874e20	0.000786332 0
$586$	0	0
$587$	−3.21537e23	−0.941471	−0.470735	−	0.882274i	$-0.656011\pi$
$587$	−3.21537e23	−0.941471	−0.470735	+	0.882274i	$0.656011\pi$
$588$	0	0
$589$	1.01267e24	2.88064
$590$	0	0
$591$	7.88985e22	0.218060
$592$	0	0
$593$	−8.23836e22	−0.221246	−0.110623	−	0.993862i	$-0.535285\pi$
$593$	−8.23836e22	−0.221246	−0.110623	+	0.993862i	$0.535285\pi$
$594$	0	0
$595$	−2.19032e23	−0.571628
$596$	0	0
$597$	−8.83330e22	−0.224048
$598$	0	0
$599$	7.29015e22	0.179725	0.0898625	−	0.995954i	$-0.471357\pi$
$599$	7.29015e22	0.179725	0.0898625	+	0.995954i	$0.471357\pi$
$600$	0	0
$601$	3.79880e23	0.910361	0.455180	−	0.890399i	$-0.349575\pi$
$601$	3.79880e23	0.910361	0.455180	+	0.890399i	$0.349575\pi$
$602$	0	0
$603$	−1.15174e23	−0.268323
$604$	0	0
$605$	−1.83280e23	−0.415141
$606$	0	0
$607$	3.76245e23	0.828641	0.414321	−	0.910131i	$-0.364019\pi$
$607$	3.76245e23	0.828641	0.414321	+	0.910131i	$0.364019\pi$
$608$	0	0
$609$	−5.97255e23	−1.27912
$610$	0	0
$611$	4.19924e21	0.00874623
$612$	0	0
$613$	−4.60370e23	−0.932595	−0.466298	−	0.884628i	$-0.654412\pi$
$613$	−4.60370e23	−0.932595	−0.466298	+	0.884628i	$0.654412\pi$
$614$	0	0
$615$	7.20635e22	0.141996
$616$	0	0
$617$	−3.36863e23	−0.645698	−0.322849	−	0.946451i	$-0.604641\pi$
$617$	−3.36863e23	−0.645698	−0.322849	+	0.946451i	$0.604641\pi$
$618$	0	0
$619$	−1.49640e23	−0.279047	−0.139523	−	0.990219i	$-0.544557\pi$
$619$	−1.49640e23	−0.279047	−0.139523	+	0.990219i	$0.544557\pi$
$620$	0	0
$621$	4.14306e23	0.751696
$622$	0	0
$623$	1.37249e24	2.42305
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	0	0
$627$	−2.26895e23	−0.379360
$628$	0	0
$629$	4.73302e23	0.770208
$630$	0	0
$631$	6.17252e21	0.00977717	0.00488858	−	0.999988i	$-0.498444\pi$
$631$	6.17252e21	0.00977717	0.00488858	+	0.999988i	$0.498444\pi$
$632$	0	0
$633$	4.71259e23	0.726654
$634$	0	0
$635$	2.50942e23	0.376700
$636$	0	0
$637$	−6.06947e21	−0.00887086
$638$	0	0
$639$	−1.64369e23	−0.233917
$640$	0	0
$641$	−1.59514e23	−0.221057	−0.110529	−	0.993873i	$-0.535254\pi$
$641$	−1.59514e23	−0.221057	−0.110529	+	0.993873i	$0.535254\pi$
$642$	0	0
$643$	−1.37854e24	−1.86049	−0.930244	−	0.366941i	$-0.880405\pi$
$643$	−1.37854e24	−1.86049	−0.930244	+	0.366941i	$0.880405\pi$
$644$	0	0
$645$	1.67599e23	0.220300
$646$	0	0
$647$	−3.15056e23	−0.403369	−0.201684	−	0.979451i	$-0.564641\pi$
$647$	−3.15056e23	−0.403369	−0.201684	+	0.979451i	$0.564641\pi$
$648$	0	0
$649$	−3.47677e23	−0.433607
$650$	0	0
$651$	−1.83204e24	−2.22585
$652$	0	0
$653$	−2.27963e23	−0.269838	−0.134919	−	0.990857i	$-0.543077\pi$
$653$	−2.27963e23	−0.269838	−0.134919	+	0.990857i	$0.543077\pi$
$654$	0	0
$655$	−1.39579e23	−0.160979
$656$	0	0
$657$	1.21275e23	0.136290
$658$	0	0
$659$	−1.18670e24	−1.29961	−0.649806	−	0.760100i	$-0.725150\pi$
$659$	−1.18670e24	−1.29961	−0.649806	+	0.760100i	$0.725150\pi$
$660$	0	0
$661$	−1.50237e24	−1.60348	−0.801741	−	0.597672i	$-0.796092\pi$
$661$	−1.50237e24	−1.60348	−0.801741	+	0.597672i	$0.796092\pi$
$662$	0	0
$663$	3.30455e21	0.00343754
$664$	0	0
$665$	−1.24312e24	−1.26046
$666$	0	0
$667$	6.68936e23	0.661173
$668$	0	0
$669$	−1.05236e24	−1.01401
$670$	0	0
$671$	−2.82230e23	−0.265132
$672$	0	0
$673$	2.53432e23	0.232131	0.116065	−	0.993242i	$-0.462972\pi$
$673$	2.53432e23	0.232131	0.116065	+	0.993242i	$0.462972\pi$
$674$	0	0
$675$	−2.43779e23	−0.217728
$676$	0	0
$677$	4.60323e23	0.400921	0.200460	−	0.979702i	$-0.435756\pi$
$677$	4.60323e23	0.400921	0.200460	+	0.979702i	$0.435756\pi$
$678$	0	0
$679$	1.59801e24	1.35734
$680$	0	0
$681$	−1.24828e24	−1.03410
$682$	0	0
$683$	6.27345e23	0.506910	0.253455	−	0.967347i	$-0.418433\pi$
$683$	6.27345e23	0.506910	0.253455	+	0.967347i	$0.418433\pi$
$684$	0	0
$685$	−4.83678e23	−0.381229
$686$	0	0
$687$	−1.42553e24	−1.09608
$688$	0	0
$689$	−6.29579e21	−0.00472266
$690$	0	0
$691$	−4.11645e23	−0.301272	−0.150636	−	0.988589i	$-0.548132\pi$
$691$	−4.11645e23	−0.301272	−0.150636	+	0.988589i	$0.548132\pi$
$692$	0	0
$693$	−2.00899e23	−0.143464
$694$	0	0
$695$	−7.62224e23	−0.531144
$696$	0	0
$697$	−4.46770e23	−0.303812
$698$	0	0
$699$	1.43486e24	0.952255
$700$	0	0
$701$	2.88292e23	0.186737	0.0933683	−	0.995632i	$-0.470237\pi$
$701$	2.88292e23	0.186737	0.0933683	+	0.995632i	$0.470237\pi$
$702$	0	0
$703$	2.68623e24	1.69833
$704$	0	0
$705$	−9.70534e23	−0.598967
$706$	0	0
$707$	−3.88909e24	−2.34306
$708$	0	0
$709$	−2.85869e24	−1.68141	−0.840707	−	0.541491i	$-0.817860\pi$
$709$	−2.85869e24	−1.68141	−0.840707	+	0.541491i	$0.817860\pi$
$710$	0	0
$711$	1.19239e23	0.0684740
$712$	0	0
$713$	2.05191e24	1.15053
$714$	0	0
$715$	−1.17045e21	−0.000640841 0
$716$	0	0
$717$	−1.03009e24	−0.550762
$718$	0	0
$719$	1.21983e24	0.636946	0.318473	−	0.947932i	$-0.396830\pi$
$719$	1.21983e24	0.636946	0.318473	+	0.947932i	$0.396830\pi$
$720$	0	0
$721$	5.79780e24	2.95674
$722$	0	0
$723$	−1.56983e24	−0.781947
$724$	0	0
$725$	−3.93605e23	−0.191508
$726$	0	0
$727$	2.28060e24	1.08394	0.541972	−	0.840396i	$-0.317678\pi$
$727$	2.28060e24	1.08394	0.541972	+	0.840396i	$0.317678\pi$
$728$	0	0
$729$	2.31988e24	1.07716
$730$	0	0
$731$	−1.03906e24	−0.471349
$732$	0	0
$733$	−8.50533e23	−0.376970	−0.188485	−	0.982076i	$-0.560358\pi$
$733$	−8.50533e23	−0.376970	−0.188485	+	0.982076i	$0.560358\pi$
$734$	0	0
$735$	1.40278e24	0.607503
$736$	0	0
$737$	5.16744e23	0.218676
$738$	0	0
$739$	−2.14385e24	−0.886578	−0.443289	−	0.896379i	$-0.646188\pi$
$739$	−2.14385e24	−0.886578	−0.443289	+	0.896379i	$0.646188\pi$
$740$	0	0
$741$	1.87550e22	0.00757988
$742$	0	0
$743$	−2.23446e24	−0.882606	−0.441303	−	0.897358i	$-0.645484\pi$
$743$	−2.23446e24	−0.882606	−0.441303	+	0.897358i	$0.645484\pi$
$744$	0	0
$745$	7.29660e23	0.281703
$746$	0	0
$747$	−2.98843e23	−0.112777
$748$	0	0
$749$	2.62076e24	0.966789
$750$	0	0
$751$	1.71798e24	0.619555	0.309778	−	0.950809i	$-0.399745\pi$
$751$	1.71798e24	0.619555	0.309778	+	0.950809i	$0.399745\pi$
$752$	0	0
$753$	8.11671e23	0.286169
$754$	0	0
$755$	−2.02689e24	−0.698685
$756$	0	0
$757$	−1.09729e24	−0.369832	−0.184916	−	0.982754i	$-0.559201\pi$
$757$	−1.09729e24	−0.369832	−0.184916	+	0.982754i	$0.559201\pi$
$758$	0	0
$759$	−4.59743e23	−0.151516
$760$	0	0
$761$	−1.70967e24	−0.550990	−0.275495	−	0.961302i	$-0.588842\pi$
$761$	−1.70967e24	−0.550990	−0.275495	+	0.961302i	$0.588842\pi$
$762$	0	0
$763$	6.26350e24	1.97405
$764$	0	0
$765$	3.73800e23	0.115217
$766$	0	0
$767$	2.87388e22	0.00866378
$768$	0	0
$769$	1.22119e24	0.360090	0.180045	−	0.983658i	$-0.442376\pi$
$769$	1.22119e24	0.360090	0.180045	+	0.983658i	$0.442376\pi$
$770$	0	0
$771$	2.40228e24	0.692884
$772$	0	0
$773$	2.59069e24	0.730952	0.365476	−	0.930821i	$-0.380906\pi$
$773$	2.59069e24	0.730952	0.365476	+	0.930821i	$0.380906\pi$
$774$	0	0
$775$	−1.20736e24	−0.333251
$776$	0	0
$777$	−4.85968e24	−1.31229
$778$	0	0
$779$	−2.53564e24	−0.669915
$780$	0	0
$781$	7.37461e23	0.190636
$782$	0	0
$783$	4.12114e24	1.04242
$784$	0	0
$785$	3.04292e24	0.753179
$786$	0	0
$787$	−2.64413e24	−0.640467	−0.320234	−	0.947339i	$-0.603761\pi$
$787$	−2.64413e24	−0.640467	−0.320234	+	0.947339i	$0.603761\pi$
$788$	0	0
$789$	3.84415e24	0.911268
$790$	0	0
$791$	1.01798e25	2.36179
$792$	0	0
$793$	2.33290e22	0.00529753
$794$	0	0
$795$	1.45509e24	0.323422
$796$	0	0
$797$	−6.24909e24	−1.35963	−0.679815	−	0.733383i	$-0.737940\pi$
$797$	−6.24909e24	−1.35963	−0.679815	+	0.733383i	$0.737940\pi$
$798$	0	0
$799$	6.01699e24	1.28154
$800$	0	0
$801$	−2.34230e24	−0.488388
$802$	0	0
$803$	−5.44114e23	−0.111073
$804$	0	0
$805$	−2.51885e24	−0.503427
$806$	0	0
$807$	3.36726e24	0.658948
$808$	0	0
$809$	5.76985e24	1.10561	0.552805	−	0.833311i	$-0.313557\pi$
$809$	5.76985e24	1.10561	0.552805	+	0.833311i	$0.313557\pi$
$810$	0	0
$811$	−3.03433e24	−0.569359	−0.284679	−	0.958623i	$-0.591887\pi$
$811$	−3.03433e24	−0.569359	−0.284679	+	0.958623i	$0.591887\pi$
$812$	0	0
$813$	6.16130e24	1.13215
$814$	0	0
$815$	−2.40973e24	−0.433639
$816$	0	0
$817$	−5.89719e24	−1.03934
$818$	0	0
$819$	1.66061e22	0.00286652
$820$	0	0
$821$	4.15299e24	0.702172	0.351086	−	0.936343i	$-0.385812\pi$
$821$	4.15299e24	0.702172	0.351086	+	0.936343i	$0.385812\pi$
$822$	0	0
$823$	−9.22848e23	−0.152838	−0.0764191	−	0.997076i	$-0.524349\pi$
$823$	−9.22848e23	−0.152838	−0.0764191	+	0.997076i	$0.524349\pi$
$824$	0	0
$825$	2.70515e23	0.0438867
$826$	0	0
$827$	−7.67636e24	−1.22000	−0.609998	−	0.792403i	$-0.708830\pi$
$827$	−7.67636e24	−1.22000	−0.609998	+	0.792403i	$0.708830\pi$
$828$	0	0
$829$	8.23009e23	0.128142	0.0640709	−	0.997945i	$-0.479592\pi$
$829$	8.23009e23	0.128142	0.0640709	+	0.997945i	$0.479592\pi$
$830$	0	0
$831$	2.26922e24	0.346153
$832$	0	0
$833$	−8.69679e24	−1.29980
$834$	0	0
$835$	−4.87209e24	−0.713478
$836$	0	0
$837$	1.26413e25	1.81395
$838$	0	0
$839$	9.57083e24	1.34578	0.672888	−	0.739744i	$-0.265053\pi$
$839$	9.57083e24	1.34578	0.672888	+	0.739744i	$0.265053\pi$
$840$	0	0
$841$	−6.03171e23	−0.0831140
$842$	0	0
$843$	−8.81851e24	−1.19086
$844$	0	0
$845$	−3.37897e24	−0.447201
$846$	0	0
$847$	−1.16669e25	−1.51337
$848$	0	0
$849$	3.42203e24	0.435079
$850$	0	0
$851$	5.44292e24	0.678314
$852$	0	0
$853$	2.48597e24	0.303690	0.151845	−	0.988404i	$-0.451479\pi$
$853$	2.48597e24	0.303690	0.151845	+	0.988404i	$0.451479\pi$
$854$	0	0
$855$	2.12150e24	0.254057
$856$	0	0
$857$	−2.10495e24	−0.247118	−0.123559	−	0.992337i	$-0.539431\pi$
$857$	−2.10495e24	−0.247118	−0.123559	+	0.992337i	$0.539431\pi$
$858$	0	0
$859$	7.58565e23	0.0873074	0.0436537	−	0.999047i	$-0.486100\pi$
$859$	7.58565e23	0.0873074	0.0436537	+	0.999047i	$0.486100\pi$
$860$	0	0
$861$	4.58725e24	0.517638
$862$	0	0
$863$	5.83299e24	0.645356	0.322678	−	0.946509i	$-0.395417\pi$
$863$	5.83299e24	0.645356	0.322678	+	0.946509i	$0.395417\pi$
$864$	0	0
$865$	6.42556e24	0.697067
$866$	0	0
$867$	−2.96787e24	−0.315706
$868$	0	0
$869$	−5.34980e23	−0.0558046
$870$	0	0
$871$	−4.27137e22	−0.00436931
$872$	0	0
$873$	−2.72717e24	−0.273584
$874$	0	0
$875$	1.48210e24	0.145817
$876$	0	0
$877$	−1.53835e25	−1.48443	−0.742214	−	0.670163i	$-0.766224\pi$
$877$	−1.53835e25	−1.48443	−0.742214	+	0.670163i	$0.766224\pi$
$878$	0	0
$879$	1.12580e25	1.06551
$880$	0	0
$881$	−1.57727e25	−1.46424	−0.732118	−	0.681178i	$-0.761468\pi$
$881$	−1.57727e25	−1.46424	−0.732118	+	0.681178i	$0.761468\pi$
$882$	0	0
$883$	−1.75208e25	−1.59547	−0.797734	−	0.603009i	$-0.793968\pi$
$883$	−1.75208e25	−1.59547	−0.797734	+	0.603009i	$0.793968\pi$
$884$	0	0
$885$	−6.64213e24	−0.593321
$886$	0	0
$887$	1.05988e25	0.928764	0.464382	−	0.885635i	$-0.346276\pi$
$887$	1.05988e25	0.928764	0.464382	+	0.885635i	$0.346276\pi$
$888$	0	0
$889$	1.59739e25	1.37324
$890$	0	0
$891$	−1.78898e24	−0.150885
$892$	0	0
$893$	3.41494e25	2.82583
$894$	0	0
$895$	5.97918e23	0.0485451
$896$	0	0
$897$	3.80020e22	0.00302741
$898$	0	0
$899$	2.04106e25	1.59551
$900$	0	0
$901$	−9.02107e24	−0.691986
$902$	0	0
$903$	1.06687e25	0.803089
$904$	0	0
$905$	−3.32595e24	−0.245699
$906$	0	0
$907$	−6.72762e24	−0.487753	−0.243876	−	0.969806i	$-0.578419\pi$
$907$	−6.72762e24	−0.487753	−0.243876	+	0.969806i	$0.578419\pi$
$908$	0	0
$909$	6.63712e24	0.472266
$910$	0	0
$911$	−1.90734e25	−1.33205	−0.666027	−	0.745928i	$-0.732006\pi$
$911$	−1.90734e25	−1.33205	−0.666027	+	0.745928i	$0.732006\pi$
$912$	0	0
$913$	1.34080e24	0.0919100
$914$	0	0
$915$	−5.39182e24	−0.362791
$916$	0	0
$917$	−8.88503e24	−0.586840
$918$	0	0
$919$	1.73304e25	1.12364	0.561818	−	0.827261i	$-0.310102\pi$
$919$	1.73304e25	1.12364	0.561818	+	0.827261i	$0.310102\pi$
$920$	0	0
$921$	4.44096e24	0.282664
$922$	0	0
$923$	−6.09580e22	−0.00380905
$924$	0	0
$925$	−3.20264e24	−0.196473
$926$	0	0
$927$	−9.89452e24	−0.595960
$928$	0	0
$929$	−6.38514e24	−0.377605	−0.188802	−	0.982015i	$-0.560461\pi$
$929$	−6.38514e24	−0.377605	−0.188802	+	0.982015i	$0.560461\pi$
$930$	0	0
$931$	−4.93587e25	−2.86610
$932$	0	0
$933$	−3.26293e24	−0.186043
$934$	0	0
$935$	−1.67710e24	−0.0938990
$936$	0	0
$937$	−2.87864e25	−1.58271	−0.791354	−	0.611358i	$-0.790624\pi$
$937$	−2.87864e25	−1.58271	−0.791354	+	0.611358i	$0.790624\pi$
$938$	0	0
$939$	1.90654e25	1.02941
$940$	0	0
$941$	−2.31234e25	−1.22614	−0.613071	−	0.790028i	$-0.710066\pi$
$941$	−2.31234e25	−1.22614	−0.613071	+	0.790028i	$0.710066\pi$
$942$	0	0
$943$	−5.13781e24	−0.267564
$944$	0	0
$945$	−1.55180e25	−0.793714
$946$	0	0
$947$	−1.60999e25	−0.808816	−0.404408	−	0.914579i	$-0.632522\pi$
$947$	−1.60999e25	−0.808816	−0.404408	+	0.914579i	$0.632522\pi$
$948$	0	0
$949$	4.49761e22	0.00221932
$950$	0	0
$951$	1.22599e25	0.594225
$952$	0	0
$953$	−1.12344e25	−0.534883	−0.267441	−	0.963574i	$-0.586178\pi$
$953$	−1.12344e25	−0.534883	−0.267441	+	0.963574i	$0.586178\pi$
$954$	0	0
$955$	−4.13753e24	−0.193514
$956$	0	0
$957$	−4.57311e24	−0.210117
$958$	0	0
$959$	−3.07889e25	−1.38975
$960$	0	0
$961$	4.00580e25	1.77640
$962$	0	0
$963$	−4.47259e24	−0.194866
$964$	0	0
$965$	−6.08468e24	−0.260469
$966$	0	0
$967$	−2.40854e24	−0.101305	−0.0506524	−	0.998716i	$-0.516130\pi$
$967$	−2.40854e24	−0.101305	−0.0506524	+	0.998716i	$0.516130\pi$
$968$	0	0
$969$	2.68736e25	1.11064
$970$	0	0
$971$	2.55647e25	1.03819	0.519095	−	0.854716i	$-0.326269\pi$
$971$	2.55647e25	1.03819	0.519095	+	0.854716i	$0.326269\pi$
$972$	0	0
$973$	−4.85200e25	−1.93625
$974$	0	0
$975$	−2.23606e22	−0.000876887 0
$976$	0	0
$977$	−2.30301e25	−0.887547	−0.443774	−	0.896139i	$-0.646361\pi$
$977$	−2.30301e25	−0.887547	−0.443774	+	0.896139i	$0.646361\pi$
$978$	0	0
$979$	1.05090e25	0.398024
$980$	0	0
$981$	−1.06893e25	−0.397889
$982$	0	0
$983$	2.04718e25	0.748949	0.374474	−	0.927237i	$-0.377823\pi$
$983$	2.04718e25	0.748949	0.374474	+	0.927237i	$0.377823\pi$
$984$	0	0
$985$	3.30984e24	0.119014
$986$	0	0
$987$	−6.17801e25	−2.18350
$988$	0	0
$989$	−1.19491e25	−0.415113
$990$	0	0
$991$	−1.20622e25	−0.411907	−0.205954	−	0.978562i	$-0.566030\pi$
$991$	−1.20622e25	−0.411907	−0.205954	+	0.978562i	$0.566030\pi$
$992$	0	0
$993$	−3.59297e25	−1.20610
$994$	0	0
$995$	−3.70563e24	−0.122283
$996$	0	0
$997$	−1.73789e23	−0.00563786	−0.00281893	−	0.999996i	$-0.500897\pi$
$997$	−1.73789e23	−0.00563786	−0.00281893	+	0.999996i	$0.500897\pi$
$998$	0	0
$999$	3.35325e25	1.06945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.18.a.d.1.2		2
4.3	odd	2		10.18.a.c.1.1	✓	2
12.11	even	2		90.18.a.k.1.1		2
20.3	even	4		50.18.b.d.49.3		4
20.7	even	4		50.18.b.d.49.2		4
20.19	odd	2		50.18.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.c.1.1	✓	2	4.3	odd	2
50.18.a.f.1.2		2	20.19	odd	2
50.18.b.d.49.2		4	20.7	even	4
50.18.b.d.49.3		4	20.3	even	4
80.18.a.d.1.2		2	1.1	even	1	trivial
90.18.a.k.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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+9311.53 q^{3} +390625. q^{5} +2.48655e7 q^{7} -4.24355e7 q^{9} +1.90392e8 q^{11} -1.57377e7 q^{13} +3.63732e9 q^{15} -2.25502e10 q^{17} -1.27984e11 q^{19} +2.31536e11 q^{21} -2.59325e11 q^{23} +1.52588e11 q^{25} -1.59763e12 q^{27} -2.57953e12 q^{29} -7.91253e12 q^{31} +1.77285e12 q^{33} +9.71310e12 q^{35} -2.09888e13 q^{37} -1.46542e11 q^{39} +1.98123e13 q^{41} +4.60777e13 q^{43} -1.65764e13 q^{45} -2.66827e14 q^{47} +3.85664e14 q^{49} -2.09977e14 q^{51} +4.00045e14 q^{53} +7.43721e13 q^{55} -1.19172e15 q^{57} -1.82611e15 q^{59} -1.48236e15 q^{61} -1.05518e15 q^{63} -6.14754e12 q^{65} +2.71410e15 q^{67} -2.41471e15 q^{69} +3.87337e15 q^{71} -2.85786e15 q^{73} +1.42083e15 q^{75} +4.73421e15 q^{77} -2.80988e15 q^{79} -9.39628e15 q^{81} +7.04230e15 q^{83} -8.80866e15 q^{85} -2.40194e16 q^{87} +5.51966e16 q^{89} -3.91327e14 q^{91} -7.36778e16 q^{93} -4.99936e16 q^{95} +6.42662e16 q^{97} -8.07940e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1308 q^{3} + 781250 q^{5} - 603844 q^{7} - 107519094 q^{9} + 471481296 q^{11} - 1541834228 q^{13} + 510937500 q^{15} + 32139900564 q^{17} - 128672529400 q^{19} + 435381246624 q^{21} - 650359859292 q^{23}+ \cdots - 26\!\cdots\!12 q^{99}+O(q^{100}) \) 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

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