LMFDB - Embedded newform 80.10.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,10,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, 10, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
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:	$41.2028668931$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1009}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 252 $ x^2 - x - 252
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$16.3824$ 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)

$f(q)$	$=$	$q-193.530 q^{3} +625.000 q^{5} -7647.66 q^{7} +17770.7 q^{9} +O(q^{10})$	$q-193.530 q^{3} +625.000 q^{5} -7647.66 q^{7} +17770.7 q^{9} +48361.0 q^{11} +100456. q^{13} -120956. q^{15} -201958. q^{17} +58048.1 q^{19} +1.48005e6 q^{21} +1.14078e6 q^{23} +390625. q^{25} +370091. q^{27} +1.56407e6 q^{29} +4.10744e6 q^{31} -9.35929e6 q^{33} -4.77979e6 q^{35} -1.70821e7 q^{37} -1.94412e7 q^{39} -8.52969e6 q^{41} -2.56254e7 q^{43} +1.11067e7 q^{45} -4.58297e7 q^{47} +1.81331e7 q^{49} +3.90848e7 q^{51} -5.56483e7 q^{53} +3.02257e7 q^{55} -1.12340e7 q^{57} -2.15699e7 q^{59} -1.15309e8 q^{61} -1.35904e8 q^{63} +6.27850e7 q^{65} +7.69254e7 q^{67} -2.20775e8 q^{69} -1.95870e8 q^{71} +3.40942e8 q^{73} -7.55975e7 q^{75} -3.69849e8 q^{77} +5.92965e8 q^{79} -4.21404e8 q^{81} -7.88477e8 q^{83} -1.26224e8 q^{85} -3.02693e8 q^{87} +8.40110e8 q^{89} -7.68253e8 q^{91} -7.94910e8 q^{93} +3.62801e7 q^{95} +2.35903e8 q^{97} +8.59408e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9}+O(q^{10}) $ 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9	$ 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9} - 23984 q^{11} + 115020 q^{13} - 162500 q^{15} + 412820 q^{17} + 296520 q^{19} + 1084704 q^{21} + 1049220 q^{23} + 781250 q^{25} + 2693080 q^{27} - 3666980 q^{29} - 1613144 q^{31} - 4550480 q^{33} - 1062500 q^{35} - 21121940 q^{37} - 20409272 q^{39} - 26957276 q^{41} - 52889700 q^{43} + 1566250 q^{45} - 58412180 q^{47} + 13154114 q^{49} - 1779784 q^{51} - 39035140 q^{53} - 14990000 q^{55} - 27085360 q^{57} + 54995560 q^{59} - 274579716 q^{61} - 226693140 q^{63} + 71887500 q^{65} + 318580 q^{67} - 214688928 q^{69} + 7130936 q^{71} + 120858180 q^{73} - 101562500 q^{75} - 800132400 q^{77} - 6877520 q^{79} - 275359358 q^{81} - 1402348740 q^{83} + 258012500 q^{85} + 45016840 q^{87} + 830088660 q^{89} - 681630904 q^{91} - 414660480 q^{93} + 185325000 q^{95} + 638394580 q^{97} + 1963732048 q^{99}+O(q^{100}) $ 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9 - 23984 * q^11 + 115020 * q^13 - 162500 * q^15 + 412820 * q^17 + 296520 * q^19 + 1084704 * q^21 + 1049220 * q^23 + 781250 * q^25 + 2693080 * q^27 - 3666980 * q^29 - 1613144 * q^31 - 4550480 * q^33 - 1062500 * q^35 - 21121940 * q^37 - 20409272 * q^39 - 26957276 * q^41 - 52889700 * q^43 + 1566250 * q^45 - 58412180 * q^47 + 13154114 * q^49 - 1779784 * q^51 - 39035140 * q^53 - 14990000 * q^55 - 27085360 * q^57 + 54995560 * q^59 - 274579716 * q^61 - 226693140 * q^63 + 71887500 * q^65 + 318580 * q^67 - 214688928 * q^69 + 7130936 * q^71 + 120858180 * q^73 - 101562500 * q^75 - 800132400 * q^77 - 6877520 * q^79 - 275359358 * q^81 - 1402348740 * q^83 + 258012500 * q^85 + 45016840 * q^87 + 830088660 * q^89 - 681630904 * q^91 - 414660480 * q^93 + 185325000 * q^95 + 638394580 * q^97 + 1963732048 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−193.530	−1.37944	−0.689718	−	0.724078i	$-0.742266\pi$
$3$	−193.530	−1.37944	−0.689718	+	0.724078i	$0.742266\pi$
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−7647.66	−1.20389	−0.601946	−	0.798537i	$-0.705608\pi$
$7$	−7647.66	−1.20389	−0.601946	+	0.798537i	$0.705608\pi$
$8$	0	0
$9$	17770.7	0.902844
$10$	0	0
$11$	48361.0	0.995929	0.497965	−	0.867197i	$-0.334081\pi$
$11$	48361.0	0.995929	0.497965	+	0.867197i	$0.334081\pi$
$12$	0	0
$13$	100456.	0.975507	0.487754	−	0.872981i	$-0.337816\pi$
$13$	100456.	0.975507	0.487754	+	0.872981i	$0.337816\pi$
$14$	0	0
$15$	−120956.	−0.616903
$16$	0	0
$17$	−201958.	−0.586463	−0.293231	−	0.956042i	$-0.594731\pi$
$17$	−201958.	−0.586463	−0.293231	+	0.956042i	$0.594731\pi$
$18$	0	0
$19$	58048.1	0.102187	0.0510936	−	0.998694i	$-0.483729\pi$
$19$	58048.1	0.102187	0.0510936	+	0.998694i	$0.483729\pi$
$20$	0	0
$21$	1.48005e6	1.66069
$22$	0	0
$23$	1.14078e6	0.850017	0.425009	−	0.905189i	$-0.360271\pi$
$23$	1.14078e6	0.850017	0.425009	+	0.905189i	$0.360271\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	370091.	0.134021
$28$	0	0
$29$	1.56407e6	0.410643	0.205322	−	0.978695i	$-0.434176\pi$
$29$	1.56407e6	0.410643	0.205322	+	0.978695i	$0.434176\pi$
$30$	0	0
$31$	4.10744e6	0.798810	0.399405	−	0.916775i	$-0.369217\pi$
$31$	4.10744e6	0.798810	0.399405	+	0.916775i	$0.369217\pi$
$32$	0	0
$33$	−9.35929e6	−1.37382
$34$	0	0
$35$	−4.77979e6	−0.538397
$36$	0	0
$37$	−1.70821e7	−1.49842	−0.749212	−	0.662330i	$-0.769568\pi$
$37$	−1.70821e7	−1.49842	−0.749212	+	0.662330i	$0.769568\pi$
$38$	0	0
$39$	−1.94412e7	−1.34565
$40$	0	0
$41$	−8.52969e6	−0.471418	−0.235709	−	0.971824i	$-0.575741\pi$
$41$	−8.52969e6	−0.471418	−0.235709	+	0.971824i	$0.575741\pi$
$42$	0	0
$43$	−2.56254e7	−1.14304	−0.571521	−	0.820587i	$-0.693646\pi$
$43$	−2.56254e7	−1.14304	−0.571521	+	0.820587i	$0.693646\pi$
$44$	0	0
$45$	1.11067e7	0.403764
$46$	0	0
$47$	−4.58297e7	−1.36996	−0.684978	−	0.728564i	$-0.740188\pi$
$47$	−4.58297e7	−1.36996	−0.684978	+	0.728564i	$0.740188\pi$
$48$	0	0
$49$	1.81331e7	0.449355
$50$	0	0
$51$	3.90848e7	0.808988
$52$	0	0
$53$	−5.56483e7	−0.968748	−0.484374	−	0.874861i	$-0.660953\pi$
$53$	−5.56483e7	−0.968748	−0.484374	+	0.874861i	$0.660953\pi$
$54$	0	0
$55$	3.02257e7	0.445393
$56$	0	0
$57$	−1.12340e7	−0.140961
$58$	0	0
$59$	−2.15699e7	−0.231747	−0.115873	−	0.993264i	$-0.536967\pi$
$59$	−2.15699e7	−0.231747	−0.115873	+	0.993264i	$0.536967\pi$
$60$	0	0
$61$	−1.15309e8	−1.06630	−0.533148	−	0.846022i	$-0.678991\pi$
$61$	−1.15309e8	−1.06630	−0.533148	+	0.846022i	$0.678991\pi$
$62$	0	0
$63$	−1.35904e8	−1.08693
$64$	0	0
$65$	6.27850e7	0.436260
$66$	0	0
$67$	7.69254e7	0.466373	0.233186	−	0.972432i	$-0.425085\pi$
$67$	7.69254e7	0.466373	0.233186	+	0.972432i	$0.425085\pi$
$68$	0	0
$69$	−2.20775e8	−1.17254
$70$	0	0
$71$	−1.95870e8	−0.914754	−0.457377	−	0.889273i	$-0.651211\pi$
$71$	−1.95870e8	−0.914754	−0.457377	+	0.889273i	$0.651211\pi$
$72$	0	0
$73$	3.40942e8	1.40517	0.702583	−	0.711601i	$-0.252030\pi$
$73$	3.40942e8	1.40517	0.702583	+	0.711601i	$0.252030\pi$
$74$	0	0
$75$	−7.55975e7	−0.275887
$76$	0	0
$77$	−3.69849e8	−1.19899
$78$	0	0
$79$	5.92965e8	1.71280	0.856401	−	0.516311i	$-0.172695\pi$
$79$	5.92965e8	1.71280	0.856401	+	0.516311i	$0.172695\pi$
$80$	0	0
$81$	−4.21404e8	−1.08772
$82$	0	0
$83$	−7.88477e8	−1.82363	−0.911817	−	0.410598i	$-0.865320\pi$
$83$	−7.88477e8	−1.82363	−0.911817	+	0.410598i	$0.865320\pi$
$84$	0	0
$85$	−1.26224e8	−0.262274
$86$	0	0
$87$	−3.02693e8	−0.566456
$88$	0	0
$89$	8.40110e8	1.41932	0.709661	−	0.704543i	$-0.248848\pi$
$89$	8.40110e8	1.41932	0.709661	+	0.704543i	$0.248848\pi$
$90$	0	0
$91$	−7.68253e8	−1.17441
$92$	0	0
$93$	−7.94910e8	−1.10191
$94$	0	0
$95$	3.62801e7	0.0456995
$96$	0	0
$97$	2.35903e8	0.270558	0.135279	−	0.990808i	$-0.456807\pi$
$97$	2.35903e8	0.270558	0.135279	+	0.990808i	$0.456807\pi$
$98$	0	0
$99$	8.59408e8	0.899169
$100$	0	0
$101$	−6.52957e8	−0.624365	−0.312182	−	0.950022i	$-0.601060\pi$
$101$	−6.52957e8	−0.624365	−0.312182	+	0.950022i	$0.601060\pi$
$102$	0	0
$103$	−9.53214e8	−0.834493	−0.417247	−	0.908793i	$-0.637005\pi$
$103$	−9.53214e8	−0.834493	−0.417247	+	0.908793i	$0.637005\pi$
$104$	0	0
$105$	9.25030e8	0.742684
$106$	0	0
$107$	1.33363e9	0.983573	0.491787	−	0.870716i	$-0.336344\pi$
$107$	1.33363e9	0.983573	0.491787	+	0.870716i	$0.336344\pi$
$108$	0	0
$109$	1.93368e9	1.31210	0.656049	−	0.754718i	$-0.272226\pi$
$109$	1.93368e9	1.31210	0.656049	+	0.754718i	$0.272226\pi$
$110$	0	0
$111$	3.30590e9	2.06698
$112$	0	0
$113$	−9.74289e8	−0.562128	−0.281064	−	0.959689i	$-0.590687\pi$
$113$	−9.74289e8	−0.562128	−0.281064	+	0.959689i	$0.590687\pi$
$114$	0	0
$115$	7.12989e8	0.380139
$116$	0	0
$117$	1.78517e9	0.880731
$118$	0	0
$119$	1.54450e9	0.706037
$120$	0	0
$121$	−1.91571e7	−0.00812446
$122$	0	0
$123$	1.65075e9	0.650290
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−1.25137e9	−0.426845	−0.213422	−	0.976960i	$-0.568461\pi$
$127$	−1.25137e9	−0.426845	−0.213422	+	0.976960i	$0.568461\pi$
$128$	0	0
$129$	4.95927e9	1.57675
$130$	0	0
$131$	−2.23060e9	−0.661760	−0.330880	−	0.943673i	$-0.607345\pi$
$131$	−2.23060e9	−0.661760	−0.330880	+	0.943673i	$0.607345\pi$
$132$	0	0
$133$	−4.43932e8	−0.123022
$134$	0	0
$135$	2.31307e8	0.0599359
$136$	0	0
$137$	−3.91202e9	−0.948764	−0.474382	−	0.880319i	$-0.657328\pi$
$137$	−3.91202e9	−0.948764	−0.474382	+	0.880319i	$0.657328\pi$
$138$	0	0
$139$	−1.64295e9	−0.373299	−0.186649	−	0.982427i	$-0.559763\pi$
$139$	−1.64295e9	−0.373299	−0.186649	+	0.982427i	$0.559763\pi$
$140$	0	0
$141$	8.86939e9	1.88977
$142$	0	0
$143$	4.85815e9	0.971537
$144$	0	0
$145$	9.77543e8	0.183645
$146$	0	0
$147$	−3.50929e9	−0.619856
$148$	0	0
$149$	−6.33624e9	−1.05316	−0.526579	−	0.850126i	$-0.676525\pi$
$149$	−6.33624e9	−1.05316	−0.526579	+	0.850126i	$0.676525\pi$
$150$	0	0
$151$	−5.67989e9	−0.889086	−0.444543	−	0.895757i	$-0.646634\pi$
$151$	−5.67989e9	−0.889086	−0.444543	+	0.895757i	$0.646634\pi$
$152$	0	0
$153$	−3.58893e9	−0.529484
$154$	0	0
$155$	2.56715e9	0.357238
$156$	0	0
$157$	−4.23621e9	−0.556454	−0.278227	−	0.960515i	$-0.589747\pi$
$157$	−4.23621e9	−0.556454	−0.278227	+	0.960515i	$0.589747\pi$
$158$	0	0
$159$	1.07696e10	1.33633
$160$	0	0
$161$	−8.72432e9	−1.02333
$162$	0	0
$163$	−8.56450e9	−0.950293	−0.475147	−	0.879907i	$-0.657605\pi$
$163$	−8.56450e9	−0.950293	−0.475147	+	0.879907i	$0.657605\pi$
$164$	0	0
$165$	−5.84956e9	−0.614391
$166$	0	0
$167$	−5.87750e9	−0.584748	−0.292374	−	0.956304i	$-0.594445\pi$
$167$	−5.87750e9	−0.584748	−0.292374	+	0.956304i	$0.594445\pi$
$168$	0	0
$169$	−5.13100e8	−0.0483851
$170$	0	0
$171$	1.03155e9	0.0922591
$172$	0	0
$173$	1.48147e10	1.25743	0.628715	−	0.777635i	$-0.283581\pi$
$173$	1.48147e10	1.25743	0.628715	+	0.777635i	$0.283581\pi$
$174$	0	0
$175$	−2.98737e9	−0.240778
$176$	0	0
$177$	4.17441e9	0.319680
$178$	0	0
$179$	−5.95512e9	−0.433563	−0.216781	−	0.976220i	$-0.569556\pi$
$179$	−5.95512e9	−0.433563	−0.216781	+	0.976220i	$0.569556\pi$
$180$	0	0
$181$	1.02707e10	0.711290	0.355645	−	0.934621i	$-0.384261\pi$
$181$	1.02707e10	0.711290	0.355645	+	0.934621i	$0.384261\pi$
$182$	0	0
$183$	2.23156e10	1.47089
$184$	0	0
$185$	−1.06763e10	−0.670116
$186$	0	0
$187$	−9.76689e9	−0.584075
$188$	0	0
$189$	−2.83033e9	−0.161346
$190$	0	0
$191$	−2.09899e9	−0.114120	−0.0570599	−	0.998371i	$-0.518173\pi$
$191$	−2.09899e9	−0.114120	−0.0570599	+	0.998371i	$0.518173\pi$
$192$	0	0
$193$	−1.95026e10	−1.01178	−0.505888	−	0.862599i	$-0.668835\pi$
$193$	−1.95026e10	−1.01178	−0.505888	+	0.862599i	$0.668835\pi$
$194$	0	0
$195$	−1.21507e10	−0.601793
$196$	0	0
$197$	6.38850e9	0.302204	0.151102	−	0.988518i	$-0.451718\pi$
$197$	6.38850e9	0.302204	0.151102	+	0.988518i	$0.451718\pi$
$198$	0	0
$199$	−1.32140e10	−0.597303	−0.298652	−	0.954362i	$-0.596537\pi$
$199$	−1.32140e10	−0.597303	−0.298652	+	0.954362i	$0.596537\pi$
$200$	0	0
$201$	−1.48873e10	−0.643331
$202$	0	0
$203$	−1.19615e10	−0.494370
$204$	0	0
$205$	−5.33106e9	−0.210824
$206$	0	0
$207$	2.02725e10	0.767433
$208$	0	0
$209$	2.80727e9	0.101771
$210$	0	0
$211$	−2.06410e9	−0.0716901	−0.0358450	−	0.999357i	$-0.511412\pi$
$211$	−2.06410e9	−0.0716901	−0.0358450	+	0.999357i	$0.511412\pi$
$212$	0	0
$213$	3.79065e10	1.26184
$214$	0	0
$215$	−1.60159e10	−0.511184
$216$	0	0
$217$	−3.14123e10	−0.961680
$218$	0	0
$219$	−6.59824e10	−1.93834
$220$	0	0
$221$	−2.02879e10	−0.572099
$222$	0	0
$223$	1.59589e10	0.432145	0.216073	−	0.976377i	$-0.430675\pi$
$223$	1.59589e10	0.432145	0.216073	+	0.976377i	$0.430675\pi$
$224$	0	0
$225$	6.94167e9	0.180569
$226$	0	0
$227$	−4.86308e10	−1.21561	−0.607807	−	0.794085i	$-0.707950\pi$
$227$	−4.86308e10	−1.21561	−0.607807	+	0.794085i	$0.707950\pi$
$228$	0	0
$229$	−6.34109e10	−1.52372	−0.761858	−	0.647744i	$-0.775713\pi$
$229$	−6.34109e10	−1.52372	−0.761858	+	0.647744i	$0.775713\pi$
$230$	0	0
$231$	7.15767e10	1.65393
$232$	0	0
$233$	2.45281e10	0.545209	0.272605	−	0.962126i	$-0.412115\pi$
$233$	2.45281e10	0.545209	0.272605	+	0.962126i	$0.412115\pi$
$234$	0	0
$235$	−2.86435e10	−0.612663
$236$	0	0
$237$	−1.14756e11	−2.36270
$238$	0	0
$239$	7.82086e10	1.55047	0.775236	−	0.631671i	$-0.217631\pi$
$239$	7.82086e10	1.55047	0.775236	+	0.631671i	$0.217631\pi$
$240$	0	0
$241$	6.59331e10	1.25900	0.629502	−	0.776999i	$-0.283259\pi$
$241$	6.59331e10	1.25900	0.629502	+	0.776999i	$0.283259\pi$
$242$	0	0
$243$	7.42696e10	1.36642
$244$	0	0
$245$	1.13332e10	0.200957
$246$	0	0
$247$	5.83128e9	0.0996844
$248$	0	0
$249$	1.52594e11	2.51559
$250$	0	0
$251$	−1.83321e10	−0.291527	−0.145764	−	0.989319i	$-0.546564\pi$
$251$	−1.83321e10	−0.291527	−0.145764	+	0.989319i	$0.546564\pi$
$252$	0	0
$253$	5.51694e10	0.846557
$254$	0	0
$255$	2.44280e10	0.361790
$256$	0	0
$257$	6.41469e10	0.917227	0.458613	−	0.888636i	$-0.348346\pi$
$257$	6.41469e10	0.917227	0.458613	+	0.888636i	$0.348346\pi$
$258$	0	0
$259$	1.30638e11	1.80394
$260$	0	0
$261$	2.77946e10	0.370747
$262$	0	0
$263$	−2.20901e10	−0.284706	−0.142353	−	0.989816i	$-0.545467\pi$
$263$	−2.20901e10	−0.284706	−0.142353	+	0.989816i	$0.545467\pi$
$264$	0	0
$265$	−3.47802e10	−0.433237
$266$	0	0
$267$	−1.62586e11	−1.95787
$268$	0	0
$269$	7.78901e10	0.906978	0.453489	−	0.891262i	$-0.350179\pi$
$269$	7.78901e10	0.906978	0.453489	+	0.891262i	$0.350179\pi$
$270$	0	0
$271$	9.36671e10	1.05493	0.527467	−	0.849576i	$-0.323142\pi$
$271$	9.36671e10	1.05493	0.527467	+	0.849576i	$0.323142\pi$
$272$	0	0
$273$	1.48680e11	1.62002
$274$	0	0
$275$	1.88910e10	0.199186
$276$	0	0
$277$	−7.45925e10	−0.761266	−0.380633	−	0.924726i	$-0.624294\pi$
$277$	−7.45925e10	−0.761266	−0.380633	+	0.924726i	$0.624294\pi$
$278$	0	0
$279$	7.29919e10	0.721200
$280$	0	0
$281$	−1.10771e11	−1.05986	−0.529930	−	0.848041i	$-0.677782\pi$
$281$	−1.10771e11	−1.05986	−0.529930	+	0.848041i	$0.677782\pi$
$282$	0	0
$283$	−1.44123e11	−1.33566	−0.667830	−	0.744314i	$-0.732777\pi$
$283$	−1.44123e11	−1.33566	−0.667830	+	0.744314i	$0.732777\pi$
$284$	0	0
$285$	−7.02126e9	−0.0630396
$286$	0	0
$287$	6.52321e10	0.567536
$288$	0	0
$289$	−7.78009e10	−0.656061
$290$	0	0
$291$	−4.56542e10	−0.373218
$292$	0	0
$293$	−1.39037e11	−1.10211	−0.551056	−	0.834469i	$-0.685775\pi$
$293$	−1.39037e11	−1.10211	−0.551056	+	0.834469i	$0.685775\pi$
$294$	0	0
$295$	−1.34812e10	−0.103640
$296$	0	0
$297$	1.78980e10	0.133475
$298$	0	0
$299$	1.14598e11	0.829198
$300$	0	0
$301$	1.95974e11	1.37610
$302$	0	0
$303$	1.26366e11	0.861271
$304$	0	0
$305$	−7.20679e10	−0.476862
$306$	0	0
$307$	2.71765e10	0.174611	0.0873053	−	0.996182i	$-0.472174\pi$
$307$	2.71765e10	0.174611	0.0873053	+	0.996182i	$0.472174\pi$
$308$	0	0
$309$	1.84475e11	1.15113
$310$	0	0
$311$	−9.82604e10	−0.595603	−0.297802	−	0.954628i	$-0.596253\pi$
$311$	−9.82604e10	−0.595603	−0.297802	+	0.954628i	$0.596253\pi$
$312$	0	0
$313$	−1.82397e11	−1.07416	−0.537078	−	0.843533i	$-0.680472\pi$
$313$	−1.82397e11	−1.07416	−0.537078	+	0.843533i	$0.680472\pi$
$314$	0	0
$315$	−8.49400e10	−0.486088
$316$	0	0
$317$	−1.52524e11	−0.848341	−0.424170	−	0.905582i	$-0.639434\pi$
$317$	−1.52524e11	−0.848341	−0.424170	+	0.905582i	$0.639434\pi$
$318$	0	0
$319$	7.56400e10	0.408972
$320$	0	0
$321$	−2.58096e11	−1.35678
$322$	0	0
$323$	−1.17233e10	−0.0599290
$324$	0	0
$325$	3.92406e10	0.195101
$326$	0	0
$327$	−3.74225e11	−1.80996
$328$	0	0
$329$	3.50490e11	1.64928
$330$	0	0
$331$	−4.08043e11	−1.86845	−0.934223	−	0.356690i	$-0.883905\pi$
$331$	−4.08043e11	−1.86845	−0.934223	+	0.356690i	$0.883905\pi$
$332$	0	0
$333$	−3.03561e11	−1.35284
$334$	0	0
$335$	4.80784e10	0.208568
$336$	0	0
$337$	−2.48513e11	−1.04958	−0.524789	−	0.851233i	$-0.675856\pi$
$337$	−2.48513e11	−1.04958	−0.524789	+	0.851233i	$0.675856\pi$
$338$	0	0
$339$	1.88554e11	0.775419
$340$	0	0
$341$	1.98640e11	0.795558
$342$	0	0
$343$	1.69935e11	0.662917
$344$	0	0
$345$	−1.37984e11	−0.524378
$346$	0	0
$347$	−3.44769e11	−1.27657	−0.638287	−	0.769799i	$-0.720356\pi$
$347$	−3.44769e11	−1.27657	−0.638287	+	0.769799i	$0.720356\pi$
$348$	0	0
$349$	4.24804e11	1.53276	0.766380	−	0.642388i	$-0.222056\pi$
$349$	4.24804e11	1.53276	0.766380	+	0.642388i	$0.222056\pi$
$350$	0	0
$351$	3.71779e10	0.130738
$352$	0	0
$353$	1.53549e11	0.526334	0.263167	−	0.964750i	$-0.415233\pi$
$353$	1.53549e11	0.526334	0.263167	+	0.964750i	$0.415233\pi$
$354$	0	0
$355$	−1.22418e11	−0.409091
$356$	0	0
$357$	−2.98907e11	−0.973933
$358$	0	0
$359$	3.08800e11	0.981189	0.490594	−	0.871388i	$-0.336780\pi$
$359$	3.08800e11	0.981189	0.490594	+	0.871388i	$0.336780\pi$
$360$	0	0
$361$	−3.19318e11	−0.989558
$362$	0	0
$363$	3.70745e9	0.0112072
$364$	0	0
$365$	2.13089e11	0.628410
$366$	0	0
$367$	1.26543e11	0.364116	0.182058	−	0.983288i	$-0.441724\pi$
$367$	1.26543e11	0.364116	0.182058	+	0.983288i	$0.441724\pi$
$368$	0	0
$369$	−1.51578e11	−0.425616
$370$	0	0
$371$	4.25579e11	1.16627
$372$	0	0
$373$	2.88008e10	0.0770397	0.0385199	−	0.999258i	$-0.487736\pi$
$373$	2.88008e10	0.0770397	0.0385199	+	0.999258i	$0.487736\pi$
$374$	0	0
$375$	−4.72484e10	−0.123381
$376$	0	0
$377$	1.57120e11	0.400586
$378$	0	0
$379$	−5.21974e11	−1.29949	−0.649744	−	0.760153i	$-0.725124\pi$
$379$	−5.21974e11	−1.29949	−0.649744	+	0.760153i	$0.725124\pi$
$380$	0	0
$381$	2.42178e11	0.588805
$382$	0	0
$383$	1.77381e10	0.0421223	0.0210611	−	0.999778i	$-0.493296\pi$
$383$	1.77381e10	0.0421223	0.0210611	+	0.999778i	$0.493296\pi$
$384$	0	0
$385$	−2.31155e11	−0.536205
$386$	0	0
$387$	−4.55380e11	−1.03199
$388$	0	0
$389$	−2.05874e10	−0.0455856	−0.0227928	−	0.999740i	$-0.507256\pi$
$389$	−2.05874e10	−0.0455856	−0.0227928	+	0.999740i	$0.507256\pi$
$390$	0	0
$391$	−2.30390e11	−0.498503
$392$	0	0
$393$	4.31686e11	0.912855
$394$	0	0
$395$	3.70603e11	0.765988
$396$	0	0
$397$	−8.47450e11	−1.71221	−0.856105	−	0.516803i	$-0.827122\pi$
$397$	−8.47450e11	−1.71221	−0.856105	+	0.516803i	$0.827122\pi$
$398$	0	0
$399$	8.59139e10	0.169701
$400$	0	0
$401$	−2.81253e11	−0.543185	−0.271592	−	0.962412i	$-0.587550\pi$
$401$	−2.81253e11	−0.543185	−0.271592	+	0.962412i	$0.587550\pi$
$402$	0	0
$403$	4.12616e11	0.779245
$404$	0	0
$405$	−2.63377e11	−0.486442
$406$	0	0
$407$	−8.26111e11	−1.49232
$408$	0	0
$409$	−6.89579e11	−1.21851	−0.609255	−	0.792975i	$-0.708531\pi$
$409$	−6.89579e11	−1.21851	−0.609255	+	0.792975i	$0.708531\pi$
$410$	0	0
$411$	7.57091e11	1.30876
$412$	0	0
$413$	1.64959e11	0.278998
$414$	0	0
$415$	−4.92798e11	−0.815553
$416$	0	0
$417$	3.17958e11	0.514942
$418$	0	0
$419$	4.76873e11	0.755857	0.377928	−	0.925835i	$-0.376637\pi$
$419$	4.76873e11	0.755857	0.377928	+	0.925835i	$0.376637\pi$
$420$	0	0
$421$	1.08180e12	1.67833	0.839165	−	0.543877i	$-0.183044\pi$
$421$	1.08180e12	1.67833	0.839165	+	0.543877i	$0.183044\pi$
$422$	0	0
$423$	−8.14424e11	−1.23686
$424$	0	0
$425$	−7.88897e10	−0.117293
$426$	0	0
$427$	8.81841e11	1.28370
$428$	0	0
$429$	−9.40196e11	−1.34017
$430$	0	0
$431$	3.36212e11	0.469316	0.234658	−	0.972078i	$-0.424603\pi$
$431$	3.36212e11	0.469316	0.234658	+	0.972078i	$0.424603\pi$
$432$	0	0
$433$	1.26577e12	1.73045	0.865225	−	0.501383i	$-0.167175\pi$
$433$	1.26577e12	1.73045	0.865225	+	0.501383i	$0.167175\pi$
$434$	0	0
$435$	−1.89183e11	−0.253327
$436$	0	0
$437$	6.62203e10	0.0868609
$438$	0	0
$439$	−2.16378e11	−0.278050	−0.139025	−	0.990289i	$-0.544397\pi$
$439$	−2.16378e11	−0.278050	−0.139025	+	0.990289i	$0.544397\pi$
$440$	0	0
$441$	3.22237e11	0.405697
$442$	0	0
$443$	3.36288e11	0.414853	0.207426	−	0.978251i	$-0.433491\pi$
$443$	3.36288e11	0.414853	0.207426	+	0.978251i	$0.433491\pi$
$444$	0	0
$445$	5.25069e11	0.634741
$446$	0	0
$447$	1.22625e12	1.45276
$448$	0	0
$449$	1.40309e12	1.62922	0.814608	−	0.580012i	$-0.196952\pi$
$449$	1.40309e12	1.62922	0.814608	+	0.580012i	$0.196952\pi$
$450$	0	0
$451$	−4.12505e11	−0.469499
$452$	0	0
$453$	1.09923e12	1.22644
$454$	0	0
$455$	−4.80158e11	−0.525210
$456$	0	0
$457$	1.26840e12	1.36030	0.680148	−	0.733075i	$-0.261916\pi$
$457$	1.26840e12	1.36030	0.680148	+	0.733075i	$0.261916\pi$
$458$	0	0
$459$	−7.47428e10	−0.0785981
$460$	0	0
$461$	−3.57771e11	−0.368936	−0.184468	−	0.982839i	$-0.559056\pi$
$461$	−3.57771e11	−0.368936	−0.184468	+	0.982839i	$0.559056\pi$
$462$	0	0
$463$	3.72773e11	0.376990	0.188495	−	0.982074i	$-0.439639\pi$
$463$	3.72773e11	0.376990	0.188495	+	0.982074i	$0.439639\pi$
$464$	0	0
$465$	−4.96819e11	−0.492788
$466$	0	0
$467$	1.30696e12	1.27155	0.635777	−	0.771873i	$-0.280680\pi$
$467$	1.30696e12	1.27155	0.635777	+	0.771873i	$0.280680\pi$
$468$	0	0
$469$	−5.88299e11	−0.561462
$470$	0	0
$471$	8.19832e11	0.767593
$472$	0	0
$473$	−1.23927e12	−1.13839
$474$	0	0
$475$	2.26750e10	0.0204374
$476$	0	0
$477$	−9.88908e11	−0.874628
$478$	0	0
$479$	−3.77216e10	−0.0327402	−0.0163701	−	0.999866i	$-0.505211\pi$
$479$	−3.77216e10	−0.0327402	−0.0163701	+	0.999866i	$0.505211\pi$
$480$	0	0
$481$	−1.71600e12	−1.46172
$482$	0	0
$483$	1.68841e12	1.41162
$484$	0	0
$485$	1.47440e11	0.120997
$486$	0	0
$487$	−1.89303e12	−1.52503	−0.762513	−	0.646973i	$-0.776035\pi$
$487$	−1.89303e12	−1.52503	−0.762513	+	0.646973i	$0.776035\pi$
$488$	0	0
$489$	1.65748e12	1.31087
$490$	0	0
$491$	7.57823e11	0.588438	0.294219	−	0.955738i	$-0.404940\pi$
$491$	7.57823e11	0.588438	0.294219	+	0.955738i	$0.404940\pi$
$492$	0	0
$493$	−3.15876e11	−0.240827
$494$	0	0
$495$	5.37130e11	0.402121
$496$	0	0
$497$	1.49794e12	1.10126
$498$	0	0
$499$	−3.90008e11	−0.281593	−0.140796	−	0.990039i	$-0.544966\pi$
$499$	−3.90008e11	−0.281593	−0.140796	+	0.990039i	$0.544966\pi$
$500$	0	0
$501$	1.13747e12	0.806623
$502$	0	0
$503$	3.44868e11	0.240213	0.120107	−	0.992761i	$-0.461676\pi$
$503$	3.44868e11	0.240213	0.120107	+	0.992761i	$0.461676\pi$
$504$	0	0
$505$	−4.08098e11	−0.279224
$506$	0	0
$507$	9.93001e10	0.0667442
$508$	0	0
$509$	−1.33209e12	−0.879636	−0.439818	−	0.898087i	$-0.644957\pi$
$509$	−1.33209e12	−0.879636	−0.439818	+	0.898087i	$0.644957\pi$
$510$	0	0
$511$	−2.60741e12	−1.69167
$512$	0	0
$513$	2.14831e10	0.0136952
$514$	0	0
$515$	−5.95759e11	−0.373197
$516$	0	0
$517$	−2.21637e12	−1.36438
$518$	0	0
$519$	−2.86707e12	−1.73455
$520$	0	0
$521$	−1.31345e12	−0.780985	−0.390493	−	0.920606i	$-0.627695\pi$
$521$	−1.31345e12	−0.780985	−0.390493	+	0.920606i	$0.627695\pi$
$522$	0	0
$523$	−1.60667e12	−0.939006	−0.469503	−	0.882931i	$-0.655567\pi$
$523$	−1.60667e12	−0.939006	−0.469503	+	0.882931i	$0.655567\pi$
$524$	0	0
$525$	5.78144e11	0.332138
$526$	0	0
$527$	−8.29529e11	−0.468472
$528$	0	0
$529$	−4.99767e11	−0.277471
$530$	0	0
$531$	−3.83311e11	−0.209231
$532$	0	0
$533$	−8.56858e11	−0.459871
$534$	0	0
$535$	8.33516e11	0.439867
$536$	0	0
$537$	1.15249e12	0.598072
$538$	0	0
$539$	8.76935e11	0.447525
$540$	0	0
$541$	−1.22163e12	−0.613132	−0.306566	−	0.951849i	$-0.599180\pi$
$541$	−1.22163e12	−0.613132	−0.306566	+	0.951849i	$0.599180\pi$
$542$	0	0
$543$	−1.98769e12	−0.981179
$544$	0	0
$545$	1.20855e12	0.586788
$546$	0	0
$547$	−1.65203e11	−0.0788996	−0.0394498	−	0.999222i	$-0.512561\pi$
$547$	−1.65203e11	−0.0788996	−0.0394498	+	0.999222i	$0.512561\pi$
$548$	0	0
$549$	−2.04911e12	−0.962698
$550$	0	0
$551$	9.07912e10	0.0419625
$552$	0	0
$553$	−4.53479e12	−2.06203
$554$	0	0
$555$	2.06619e12	0.924382
$556$	0	0
$557$	3.45024e12	1.51880	0.759401	−	0.650623i	$-0.225492\pi$
$557$	3.45024e12	1.51880	0.759401	+	0.650623i	$0.225492\pi$
$558$	0	0
$559$	−2.57422e12	−1.11505
$560$	0	0
$561$	1.89018e12	0.805695
$562$	0	0
$563$	−4.38944e11	−0.184129	−0.0920643	−	0.995753i	$-0.529347\pi$
$563$	−4.38944e11	−0.184129	−0.0920643	+	0.995753i	$0.529347\pi$
$564$	0	0
$565$	−6.08931e11	−0.251391
$566$	0	0
$567$	3.22275e12	1.30949
$568$	0	0
$569$	−2.03068e12	−0.812150	−0.406075	−	0.913840i	$-0.633103\pi$
$569$	−2.03068e12	−0.812150	−0.406075	+	0.913840i	$0.633103\pi$
$570$	0	0
$571$	−1.25554e12	−0.494276	−0.247138	−	0.968980i	$-0.579490\pi$
$571$	−1.25554e12	−0.494276	−0.247138	+	0.968980i	$0.579490\pi$
$572$	0	0
$573$	4.06217e11	0.157421
$574$	0	0
$575$	4.45618e11	0.170003
$576$	0	0
$577$	−1.45596e12	−0.546838	−0.273419	−	0.961895i	$-0.588155\pi$
$577$	−1.45596e12	−0.546838	−0.273419	+	0.961895i	$0.588155\pi$
$578$	0	0
$579$	3.77432e12	1.39568
$580$	0	0
$581$	6.03000e12	2.19546
$582$	0	0
$583$	−2.69121e12	−0.964805
$584$	0	0
$585$	1.11573e12	0.393875
$586$	0	0
$587$	7.51503e11	0.261252	0.130626	−	0.991432i	$-0.458301\pi$
$587$	7.51503e11	0.261252	0.130626	+	0.991432i	$0.458301\pi$
$588$	0	0
$589$	2.38429e11	0.0816281
$590$	0	0
$591$	−1.23636e12	−0.416872
$592$	0	0
$593$	−1.95284e12	−0.648514	−0.324257	−	0.945969i	$-0.605114\pi$
$593$	−1.95284e12	−0.648514	−0.324257	+	0.945969i	$0.605114\pi$
$594$	0	0
$595$	9.65315e11	0.315750
$596$	0	0
$597$	2.55730e12	0.823942
$598$	0	0
$599$	4.92766e12	1.56394	0.781969	−	0.623317i	$-0.214215\pi$
$599$	4.92766e12	1.56394	0.781969	+	0.623317i	$0.214215\pi$
$600$	0	0
$601$	−4.36498e12	−1.36473	−0.682366	−	0.731011i	$-0.739049\pi$
$601$	−4.36498e12	−1.36473	−0.682366	+	0.731011i	$0.739049\pi$
$602$	0	0
$603$	1.36702e12	0.421062
$604$	0	0
$605$	−1.19732e10	−0.00363337
$606$	0	0
$607$	1.49883e12	0.448128	0.224064	−	0.974574i	$-0.428068\pi$
$607$	1.49883e12	0.448128	0.224064	+	0.974574i	$0.428068\pi$
$608$	0	0
$609$	2.31490e12	0.681952
$610$	0	0
$611$	−4.60386e12	−1.33640
$612$	0	0
$613$	−3.19261e12	−0.913216	−0.456608	−	0.889668i	$-0.650936\pi$
$613$	−3.19261e12	−0.913216	−0.456608	+	0.889668i	$0.650936\pi$
$614$	0	0
$615$	1.03172e12	0.290819
$616$	0	0
$617$	−1.58167e12	−0.439371	−0.219685	−	0.975571i	$-0.570503\pi$
$617$	−1.58167e12	−0.439371	−0.219685	+	0.975571i	$0.570503\pi$
$618$	0	0
$619$	−2.53778e12	−0.694777	−0.347389	−	0.937721i	$-0.612932\pi$
$619$	−2.53778e12	−0.694777	−0.347389	+	0.937721i	$0.612932\pi$
$620$	0	0
$621$	4.22194e11	0.113920
$622$	0	0
$623$	−6.42488e12	−1.70871
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−5.43289e11	−0.140387
$628$	0	0
$629$	3.44987e12	0.878770
$630$	0	0
$631$	6.66150e12	1.67278	0.836392	−	0.548131i	$-0.184661\pi$
$631$	6.66150e12	1.67278	0.836392	+	0.548131i	$0.184661\pi$
$632$	0	0
$633$	3.99464e11	0.0988918
$634$	0	0
$635$	−7.82108e11	−0.190891
$636$	0	0
$637$	1.82158e12	0.438349
$638$	0	0
$639$	−3.48073e12	−0.825880
$640$	0	0
$641$	−6.22257e12	−1.45582	−0.727911	−	0.685671i	$-0.759509\pi$
$641$	−6.22257e12	−1.45582	−0.727911	+	0.685671i	$0.759509\pi$
$642$	0	0
$643$	4.55187e11	0.105012	0.0525061	−	0.998621i	$-0.483279\pi$
$643$	4.55187e11	0.105012	0.0525061	+	0.998621i	$0.483279\pi$
$644$	0	0
$645$	3.09954e12	0.705146
$646$	0	0
$647$	2.50192e12	0.561312	0.280656	−	0.959808i	$-0.409448\pi$
$647$	2.50192e12	0.561312	0.280656	+	0.959808i	$0.409448\pi$
$648$	0	0
$649$	−1.04314e12	−0.230803
$650$	0	0
$651$	6.07920e12	1.32658
$652$	0	0
$653$	3.54499e12	0.762966	0.381483	−	0.924376i	$-0.375413\pi$
$653$	3.54499e12	0.762966	0.381483	+	0.924376i	$0.375413\pi$
$654$	0	0
$655$	−1.39412e12	−0.295948
$656$	0	0
$657$	6.05877e12	1.26865
$658$	0	0
$659$	−5.93850e12	−1.22657	−0.613285	−	0.789862i	$-0.710152\pi$
$659$	−5.93850e12	−1.22657	−0.613285	+	0.789862i	$0.710152\pi$
$660$	0	0
$661$	−6.68673e12	−1.36241	−0.681204	−	0.732093i	$-0.738543\pi$
$661$	−6.68673e12	−1.36241	−0.681204	+	0.732093i	$0.738543\pi$
$662$	0	0
$663$	3.92630e12	0.789174
$664$	0	0
$665$	−2.77457e11	−0.0550173
$666$	0	0
$667$	1.78426e12	0.349054
$668$	0	0
$669$	−3.08851e12	−0.596117
$670$	0	0
$671$	−5.57645e12	−1.06196
$672$	0	0
$673$	8.43444e12	1.58485	0.792425	−	0.609969i	$-0.208818\pi$
$673$	8.43444e12	1.58485	0.792425	+	0.609969i	$0.208818\pi$
$674$	0	0
$675$	1.44567e11	0.0268041
$676$	0	0
$677$	9.48091e12	1.73461	0.867303	−	0.497781i	$-0.165852\pi$
$677$	9.48091e12	1.73461	0.867303	+	0.497781i	$0.165852\pi$
$678$	0	0
$679$	−1.80411e12	−0.325723
$680$	0	0
$681$	9.41150e12	1.67686
$682$	0	0
$683$	1.00045e13	1.75914	0.879571	−	0.475768i	$-0.157830\pi$
$683$	1.00045e13	1.75914	0.879571	+	0.475768i	$0.157830\pi$
$684$	0	0
$685$	−2.44501e12	−0.424300
$686$	0	0
$687$	1.22719e13	2.10187
$688$	0	0
$689$	−5.59021e12	−0.945021
$690$	0	0
$691$	−5.59146e12	−0.932983	−0.466491	−	0.884526i	$-0.654482\pi$
$691$	−5.59146e12	−0.932983	−0.466491	+	0.884526i	$0.654482\pi$
$692$	0	0
$693$	−6.57246e12	−1.08250
$694$	0	0
$695$	−1.02684e12	−0.166944
$696$	0	0
$697$	1.72264e12	0.276469
$698$	0	0
$699$	−4.74692e12	−0.752081
$700$	0	0
$701$	−6.41878e12	−1.00397	−0.501986	−	0.864876i	$-0.667397\pi$
$701$	−6.41878e12	−1.00397	−0.501986	+	0.864876i	$0.667397\pi$
$702$	0	0
$703$	−9.91586e11	−0.153120
$704$	0	0
$705$	5.54337e12	0.845129
$706$	0	0
$707$	4.99359e12	0.751667
$708$	0	0
$709$	1.98300e12	0.294724	0.147362	−	0.989083i	$-0.452922\pi$
$709$	1.98300e12	0.294724	0.147362	+	0.989083i	$0.452922\pi$
$710$	0	0
$711$	1.05374e13	1.54639
$712$	0	0
$713$	4.68569e12	0.679002
$714$	0	0
$715$	3.03635e12	0.434484
$716$	0	0
$717$	−1.51357e13	−2.13878
$718$	0	0
$719$	9.53823e12	1.33103	0.665515	−	0.746385i	$-0.268212\pi$
$719$	9.53823e12	1.33103	0.665515	+	0.746385i	$0.268212\pi$
$720$	0	0
$721$	7.28986e12	1.00464
$722$	0	0
$723$	−1.27600e13	−1.73671
$724$	0	0
$725$	6.10964e11	0.0821287
$726$	0	0
$727$	6.30135e11	0.0836621	0.0418310	−	0.999125i	$-0.486681\pi$
$727$	6.30135e11	0.0836621	0.0418310	+	0.999125i	$0.486681\pi$
$728$	0	0
$729$	−6.07886e12	−0.797165
$730$	0	0
$731$	5.17524e12	0.670352
$732$	0	0
$733$	2.79069e12	0.357061	0.178531	−	0.983934i	$-0.442866\pi$
$733$	2.79069e12	0.357061	0.178531	+	0.983934i	$0.442866\pi$
$734$	0	0
$735$	−2.19330e12	−0.277208
$736$	0	0
$737$	3.72019e12	0.464474
$738$	0	0
$739$	−7.26759e12	−0.896376	−0.448188	−	0.893939i	$-0.647931\pi$
$739$	−7.26759e12	−0.896376	−0.448188	+	0.893939i	$0.647931\pi$
$740$	0	0
$741$	−1.12852e12	−0.137508
$742$	0	0
$743$	9.19047e12	1.10634	0.553169	−	0.833069i	$-0.313418\pi$
$743$	9.19047e12	1.10634	0.553169	+	0.833069i	$0.313418\pi$
$744$	0	0
$745$	−3.96015e12	−0.470986
$746$	0	0
$747$	−1.40118e13	−1.64646
$748$	0	0
$749$	−1.01991e13	−1.18412
$750$	0	0
$751$	5.21242e11	0.0597943	0.0298972	−	0.999553i	$-0.490482\pi$
$751$	5.21242e11	0.0597943	0.0298972	+	0.999553i	$0.490482\pi$
$752$	0	0
$753$	3.54779e12	0.402143
$754$	0	0
$755$	−3.54993e12	−0.397612
$756$	0	0
$757$	1.15123e13	1.27417	0.637087	−	0.770792i	$-0.280139\pi$
$757$	1.15123e13	1.27417	0.637087	+	0.770792i	$0.280139\pi$
$758$	0	0
$759$	−1.06769e13	−1.16777
$760$	0	0
$761$	1.43517e13	1.55121	0.775607	−	0.631217i	$-0.217444\pi$
$761$	1.43517e13	1.55121	0.775607	+	0.631217i	$0.217444\pi$
$762$	0	0
$763$	−1.47882e13	−1.57962
$764$	0	0
$765$	−2.24308e12	−0.236793
$766$	0	0
$767$	−2.16682e12	−0.226071
$768$	0	0
$769$	−7.42318e12	−0.765458	−0.382729	−	0.923861i	$-0.625016\pi$
$769$	−7.42318e12	−0.765458	−0.382729	+	0.923861i	$0.625016\pi$
$770$	0	0
$771$	−1.24143e13	−1.26526
$772$	0	0
$773$	5.82774e12	0.587074	0.293537	−	0.955948i	$-0.405168\pi$
$773$	5.82774e12	0.587074	0.293537	+	0.955948i	$0.405168\pi$
$774$	0	0
$775$	1.60447e12	0.159762
$776$	0	0
$777$	−2.52824e13	−2.48842
$778$	0	0
$779$	−4.95132e11	−0.0481729
$780$	0	0
$781$	−9.47246e12	−0.911031
$782$	0	0
$783$	5.78848e11	0.0550347
$784$	0	0
$785$	−2.64763e12	−0.248854
$786$	0	0
$787$	−7.20033e11	−0.0669061	−0.0334531	−	0.999440i	$-0.510650\pi$
$787$	−7.20033e11	−0.0669061	−0.0334531	+	0.999440i	$0.510650\pi$
$788$	0	0
$789$	4.27508e12	0.392733
$790$	0	0
$791$	7.45103e12	0.676741
$792$	0	0
$793$	−1.15834e13	−1.04018
$794$	0	0
$795$	6.73100e12	0.597623
$796$	0	0
$797$	9.03491e12	0.793161	0.396580	−	0.918000i	$-0.370197\pi$
$797$	9.03491e12	0.793161	0.396580	+	0.918000i	$0.370197\pi$
$798$	0	0
$799$	9.25566e12	0.803428
$800$	0	0
$801$	1.49293e13	1.28143
$802$	0	0
$803$	1.64883e13	1.39945
$804$	0	0
$805$	−5.45270e12	−0.457646
$806$	0	0
$807$	−1.50740e13	−1.25112
$808$	0	0
$809$	−2.33217e13	−1.91422	−0.957111	−	0.289721i	$-0.906438\pi$
$809$	−2.33217e13	−1.91422	−0.957111	+	0.289721i	$0.906438\pi$
$810$	0	0
$811$	−1.16365e13	−0.944555	−0.472277	−	0.881450i	$-0.656568\pi$
$811$	−1.16365e13	−0.944555	−0.472277	+	0.881450i	$0.656568\pi$
$812$	0	0
$813$	−1.81273e13	−1.45521
$814$	0	0
$815$	−5.35281e12	−0.424984
$816$	0	0
$817$	−1.48750e12	−0.116804
$818$	0	0
$819$	−1.36524e13	−1.06030
$820$	0	0
$821$	−1.36931e13	−1.05186	−0.525931	−	0.850527i	$-0.676283\pi$
$821$	−1.36931e13	−1.05186	−0.525931	+	0.850527i	$0.676283\pi$
$822$	0	0
$823$	−8.13478e12	−0.618083	−0.309041	−	0.951049i	$-0.600008\pi$
$823$	−8.13478e12	−0.618083	−0.309041	+	0.951049i	$0.600008\pi$
$824$	0	0
$825$	−3.65597e12	−0.274764
$826$	0	0
$827$	−9.52838e12	−0.708344	−0.354172	−	0.935180i	$-0.615237\pi$
$827$	−9.52838e12	−0.708344	−0.354172	+	0.935180i	$0.615237\pi$
$828$	0	0
$829$	2.36561e13	1.73960	0.869798	−	0.493407i	$-0.164249\pi$
$829$	2.36561e13	1.73960	0.869798	+	0.493407i	$0.164249\pi$
$830$	0	0
$831$	1.44359e13	1.05012
$832$	0	0
$833$	−3.66212e12	−0.263530
$834$	0	0
$835$	−3.67344e12	−0.261507
$836$	0	0
$837$	1.52013e12	0.107057
$838$	0	0
$839$	1.43311e12	0.0998509	0.0499255	−	0.998753i	$-0.484102\pi$
$839$	1.43311e12	0.0998509	0.0499255	+	0.998753i	$0.484102\pi$
$840$	0	0
$841$	−1.20608e13	−0.831372
$842$	0	0
$843$	2.14375e13	1.46201
$844$	0	0
$845$	−3.20688e11	−0.0216385
$846$	0	0
$847$	1.46507e11	0.00978097
$848$	0	0
$849$	2.78921e13	1.84246
$850$	0	0
$851$	−1.94870e13	−1.27369
$852$	0	0
$853$	2.03077e13	1.31338	0.656690	−	0.754161i	$-0.271956\pi$
$853$	2.03077e13	1.31338	0.656690	+	0.754161i	$0.271956\pi$
$854$	0	0
$855$	6.44721e11	0.0412595
$856$	0	0
$857$	−1.23092e12	−0.0779498	−0.0389749	−	0.999240i	$-0.512409\pi$
$857$	−1.23092e12	−0.0779498	−0.0389749	+	0.999240i	$0.512409\pi$
$858$	0	0
$859$	1.69941e13	1.06495	0.532476	−	0.846445i	$-0.321262\pi$
$859$	1.69941e13	1.06495	0.532476	+	0.846445i	$0.321262\pi$
$860$	0	0
$861$	−1.26243e13	−0.782879
$862$	0	0
$863$	3.86270e12	0.237051	0.118526	−	0.992951i	$-0.462183\pi$
$863$	3.86270e12	0.237051	0.118526	+	0.992951i	$0.462183\pi$
$864$	0	0
$865$	9.25916e12	0.562340
$866$	0	0
$867$	1.50568e13	0.904995
$868$	0	0
$869$	2.86764e13	1.70583
$870$	0	0
$871$	7.72761e12	0.454950
$872$	0	0
$873$	4.19216e12	0.244272
$874$	0	0
$875$	−1.86710e12	−0.107679
$876$	0	0
$877$	−3.44305e12	−0.196538	−0.0982688	−	0.995160i	$-0.531330\pi$
$877$	−3.44305e12	−0.196538	−0.0982688	+	0.995160i	$0.531330\pi$
$878$	0	0
$879$	2.69077e13	1.52029
$880$	0	0
$881$	2.37092e13	1.32594	0.662971	−	0.748645i	$-0.269295\pi$
$881$	2.37092e13	1.32594	0.662971	+	0.748645i	$0.269295\pi$
$882$	0	0
$883$	3.07625e13	1.70294	0.851468	−	0.524407i	$-0.175713\pi$
$883$	3.07625e13	1.70294	0.851468	+	0.524407i	$0.175713\pi$
$884$	0	0
$885$	2.60901e12	0.142965
$886$	0	0
$887$	1.39503e13	0.756706	0.378353	−	0.925661i	$-0.376491\pi$
$887$	1.39503e13	0.756706	0.378353	+	0.925661i	$0.376491\pi$
$888$	0	0
$889$	9.57008e12	0.513875
$890$	0	0
$891$	−2.03795e13	−1.08329
$892$	0	0
$893$	−2.66032e12	−0.139992
$894$	0	0
$895$	−3.72195e12	−0.193895
$896$	0	0
$897$	−2.21782e13	−1.14383
$898$	0	0
$899$	6.42431e12	0.328026
$900$	0	0
$901$	1.12386e13	0.568134
$902$	0	0
$903$	−3.79268e13	−1.89824
$904$	0	0
$905$	6.41919e12	0.318099
$906$	0	0
$907$	−9.36080e12	−0.459283	−0.229641	−	0.973275i	$-0.573755\pi$
$907$	−9.36080e12	−0.459283	−0.229641	+	0.973275i	$0.573755\pi$
$908$	0	0
$909$	−1.16035e13	−0.563704
$910$	0	0
$911$	−6.17441e12	−0.297004	−0.148502	−	0.988912i	$-0.547445\pi$
$911$	−6.17441e12	−0.297004	−0.148502	+	0.988912i	$0.547445\pi$
$912$	0	0
$913$	−3.81316e13	−1.81621
$914$	0	0
$915$	1.39473e13	0.657800
$916$	0	0
$917$	1.70588e13	0.796687
$918$	0	0
$919$	−3.99328e12	−0.184676	−0.0923380	−	0.995728i	$-0.529434\pi$
$919$	−3.99328e12	−0.184676	−0.0923380	+	0.995728i	$0.529434\pi$
$920$	0	0
$921$	−5.25945e12	−0.240864
$922$	0	0
$923$	−1.96763e13	−0.892350
$924$	0	0
$925$	−6.67271e12	−0.299685
$926$	0	0
$927$	−1.69393e13	−0.753417
$928$	0	0
$929$	1.84456e13	0.812496	0.406248	−	0.913763i	$-0.366837\pi$
$929$	1.84456e13	0.812496	0.406248	+	0.913763i	$0.366837\pi$
$930$	0	0
$931$	1.05259e12	0.0459183
$932$	0	0
$933$	1.90163e13	0.821596
$934$	0	0
$935$	−6.10431e12	−0.261207
$936$	0	0
$937$	−8.90163e12	−0.377261	−0.188630	−	0.982048i	$-0.560405\pi$
$937$	−8.90163e12	−0.377261	−0.188630	+	0.982048i	$0.560405\pi$
$938$	0	0
$939$	3.52991e13	1.48173
$940$	0	0
$941$	−1.64341e13	−0.683270	−0.341635	−	0.939833i	$-0.610981\pi$
$941$	−1.64341e13	−0.683270	−0.341635	+	0.939833i	$0.610981\pi$
$942$	0	0
$943$	−9.73052e12	−0.400713
$944$	0	0
$945$	−1.76896e12	−0.0721563
$946$	0	0
$947$	−4.26042e13	−1.72138	−0.860691	−	0.509128i	$-0.829968\pi$
$947$	−4.26042e13	−1.72138	−0.860691	+	0.509128i	$0.829968\pi$
$948$	0	0
$949$	3.42497e13	1.37075
$950$	0	0
$951$	2.95178e13	1.17023
$952$	0	0
$953$	3.52200e12	0.138316	0.0691578	−	0.997606i	$-0.477969\pi$
$953$	3.52200e12	0.138316	0.0691578	+	0.997606i	$0.477969\pi$
$954$	0	0
$955$	−1.31187e12	−0.0510360
$956$	0	0
$957$	−1.46386e13	−0.564151
$958$	0	0
$959$	2.99178e13	1.14221
$960$	0	0
$961$	−9.56859e12	−0.361903
$962$	0	0
$963$	2.36994e13	0.888013
$964$	0	0
$965$	−1.21891e13	−0.452480
$966$	0	0
$967$	3.34889e13	1.23163	0.615816	−	0.787890i	$-0.288826\pi$
$967$	3.34889e13	1.23163	0.615816	+	0.787890i	$0.288826\pi$
$968$	0	0
$969$	2.26880e12	0.0826682
$970$	0	0
$971$	3.77122e13	1.36143	0.680715	−	0.732548i	$-0.261669\pi$
$971$	3.77122e13	1.36143	0.680715	+	0.732548i	$0.261669\pi$
$972$	0	0
$973$	1.25647e13	0.449411
$974$	0	0
$975$	−7.59422e12	−0.269130
$976$	0	0
$977$	−4.00215e13	−1.40530	−0.702648	−	0.711537i	$-0.747999\pi$
$977$	−4.00215e13	−1.40530	−0.702648	+	0.711537i	$0.747999\pi$
$978$	0	0
$979$	4.06286e13	1.41355
$980$	0	0
$981$	3.43629e13	1.18462
$982$	0	0
$983$	−3.31096e13	−1.13100	−0.565500	−	0.824748i	$-0.691317\pi$
$983$	−3.31096e13	−1.13100	−0.565500	+	0.824748i	$0.691317\pi$
$984$	0	0
$985$	3.99281e12	0.135150
$986$	0	0
$987$	−6.78301e13	−2.27507
$988$	0	0
$989$	−2.92330e13	−0.971606
$990$	0	0
$991$	−1.90619e12	−0.0627820	−0.0313910	−	0.999507i	$-0.509994\pi$
$991$	−1.90619e12	−0.0627820	−0.0313910	+	0.999507i	$0.509994\pi$
$992$	0	0
$993$	7.89684e13	2.57740
$994$	0	0
$995$	−8.25874e12	−0.267122
$996$	0	0
$997$	−2.13028e13	−0.682823	−0.341411	−	0.939914i	$-0.610905\pi$
$997$	−2.13028e13	−0.682823	−0.341411	+	0.939914i	$0.610905\pi$
$998$	0	0
$999$	−6.32195e12	−0.200820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.10.a.f.1.1		2
4.3	odd	2		5.10.a.b.1.1	✓	2
5.2	odd	4		400.10.c.p.49.4		4
5.3	odd	4		400.10.c.p.49.1		4
5.4	even	2		400.10.a.t.1.2		2
8.3	odd	2		320.10.a.k.1.1		2
8.5	even	2		320.10.a.s.1.2		2
12.11	even	2		45.10.a.f.1.2		2
20.3	even	4		25.10.b.b.24.4		4
20.7	even	4		25.10.b.b.24.1		4
20.19	odd	2		25.10.a.b.1.2		2
28.27	even	2		245.10.a.d.1.1		2
60.23	odd	4		225.10.b.h.199.1		4
60.47	odd	4		225.10.b.h.199.4		4
60.59	even	2		225.10.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.10.a.b.1.1	✓	2	4.3	odd	2
25.10.a.b.1.2		2	20.19	odd	2
25.10.b.b.24.1		4	20.7	even	4
25.10.b.b.24.4		4	20.3	even	4
45.10.a.f.1.2		2	12.11	even	2
80.10.a.f.1.1		2	1.1	even	1	trivial
225.10.a.h.1.1		2	60.59	even	2
225.10.b.h.199.1		4	60.23	odd	4
225.10.b.h.199.4		4	60.47	odd	4
245.10.a.d.1.1		2	28.27	even	2
320.10.a.k.1.1		2	8.3	odd	2
320.10.a.s.1.2		2	8.5	even	2
400.10.a.t.1.2		2	5.4	even	2
400.10.c.p.49.1		4	5.3	odd	4
400.10.c.p.49.4		4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q-193.530 q^{3} +625.000 q^{5} -7647.66 q^{7} +17770.7 q^{9} +O(q^{10})\)	\(q-193.530 q^{3} +625.000 q^{5} -7647.66 q^{7} +17770.7 q^{9} +48361.0 q^{11} +100456. q^{13} -120956. q^{15} -201958. q^{17} +58048.1 q^{19} +1.48005e6 q^{21} +1.14078e6 q^{23} +390625. q^{25} +370091. q^{27} +1.56407e6 q^{29} +4.10744e6 q^{31} -9.35929e6 q^{33} -4.77979e6 q^{35} -1.70821e7 q^{37} -1.94412e7 q^{39} -8.52969e6 q^{41} -2.56254e7 q^{43} +1.11067e7 q^{45} -4.58297e7 q^{47} +1.81331e7 q^{49} +3.90848e7 q^{51} -5.56483e7 q^{53} +3.02257e7 q^{55} -1.12340e7 q^{57} -2.15699e7 q^{59} -1.15309e8 q^{61} -1.35904e8 q^{63} +6.27850e7 q^{65} +7.69254e7 q^{67} -2.20775e8 q^{69} -1.95870e8 q^{71} +3.40942e8 q^{73} -7.55975e7 q^{75} -3.69849e8 q^{77} +5.92965e8 q^{79} -4.21404e8 q^{81} -7.88477e8 q^{83} -1.26224e8 q^{85} -3.02693e8 q^{87} +8.40110e8 q^{89} -7.68253e8 q^{91} -7.94910e8 q^{93} +3.62801e7 q^{95} +2.35903e8 q^{97} +8.59408e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9}+O(q^{10}) \) 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9	\( 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9} - 23984 q^{11} + 115020 q^{13} - 162500 q^{15} + 412820 q^{17} + 296520 q^{19} + 1084704 q^{21} + 1049220 q^{23} + 781250 q^{25} + 2693080 q^{27} - 3666980 q^{29} - 1613144 q^{31} - 4550480 q^{33} - 1062500 q^{35} - 21121940 q^{37} - 20409272 q^{39} - 26957276 q^{41} - 52889700 q^{43} + 1566250 q^{45} - 58412180 q^{47} + 13154114 q^{49} - 1779784 q^{51} - 39035140 q^{53} - 14990000 q^{55} - 27085360 q^{57} + 54995560 q^{59} - 274579716 q^{61} - 226693140 q^{63} + 71887500 q^{65} + 318580 q^{67} - 214688928 q^{69} + 7130936 q^{71} + 120858180 q^{73} - 101562500 q^{75} - 800132400 q^{77} - 6877520 q^{79} - 275359358 q^{81} - 1402348740 q^{83} + 258012500 q^{85} + 45016840 q^{87} + 830088660 q^{89} - 681630904 q^{91} - 414660480 q^{93} + 185325000 q^{95} + 638394580 q^{97} + 1963732048 q^{99}+O(q^{100}) \) 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9 - 23984 * q^11 + 115020 * q^13 - 162500 * q^15 + 412820 * q^17 + 296520 * q^19 + 1084704 * q^21 + 1049220 * q^23 + 781250 * q^25 + 2693080 * q^27 - 3666980 * q^29 - 1613144 * q^31 - 4550480 * q^33 - 1062500 * q^35 - 21121940 * q^37 - 20409272 * q^39 - 26957276 * q^41 - 52889700 * q^43 + 1566250 * q^45 - 58412180 * q^47 + 13154114 * q^49 - 1779784 * q^51 - 39035140 * q^53 - 14990000 * q^55 - 27085360 * q^57 + 54995560 * q^59 - 274579716 * q^61 - 226693140 * q^63 + 71887500 * q^65 + 318580 * q^67 - 214688928 * q^69 + 7130936 * q^71 + 120858180 * q^73 - 101562500 * q^75 - 800132400 * q^77 - 6877520 * q^79 - 275359358 * q^81 - 1402348740 * q^83 + 258012500 * q^85 + 45016840 * q^87 + 830088660 * q^89 - 681630904 * q^91 - 414660480 * q^93 + 185325000 * q^95 + 638394580 * q^97 + 1963732048 * q^99

Introduction

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