LMFDB - Embedded newform 80.14.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,14,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, 14); 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, 14, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-416] 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:	$85.7847431615$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4466x - 18720 $ x^3 - x^2 - 4466*x - 18720
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-64.1084$ 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)

$f(q)$	$=$	$q+573.185 q^{3} +15625.0 q^{5} +201493. q^{7} -1.26578e6 q^{9} +3.34359e6 q^{11} -7.80359e6 q^{13} +8.95602e6 q^{15} -8.71750e7 q^{17} +1.66766e7 q^{19} +1.15493e8 q^{21} -1.13518e9 q^{23} +2.44141e8 q^{25} -1.63937e9 q^{27} +2.60673e9 q^{29} -8.33139e8 q^{31} +1.91649e9 q^{33} +3.14833e9 q^{35} -1.05494e10 q^{37} -4.47290e9 q^{39} -4.33030e9 q^{41} -1.93854e9 q^{43} -1.97778e10 q^{45} -2.85468e10 q^{47} -5.62896e10 q^{49} -4.99674e10 q^{51} +1.23249e11 q^{53} +5.22435e10 q^{55} +9.55879e9 q^{57} +5.55404e11 q^{59} -4.10476e11 q^{61} -2.55046e11 q^{63} -1.21931e11 q^{65} -3.36861e11 q^{67} -6.50671e11 q^{69} -1.57323e12 q^{71} +2.05372e12 q^{73} +1.39938e11 q^{75} +6.73709e11 q^{77} +6.93000e11 q^{79} +1.07840e12 q^{81} -2.01116e12 q^{83} -1.36211e12 q^{85} +1.49414e12 q^{87} -8.51832e12 q^{89} -1.57237e12 q^{91} -4.77543e11 q^{93} +2.60572e11 q^{95} -7.99814e12 q^{97} -4.23225e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 416 q^{3} + 46875 q^{5} - 448292 q^{7} + 1286119 q^{9} + 6604004 q^{11} - 33501974 q^{13} - 6500000 q^{15} + 83129542 q^{17} - 97491100 q^{19} + 438200736 q^{21} - 316255836 q^{23} + 732421875 q^{25}+ \cdots - 126787366508 q^{99}+O(q^{100}) $ 3 * q - 416 * q^3 + 46875 * q^5 - 448292 * q^7 + 1286119 * q^9 + 6604004 * q^11 - 33501974 * q^13 - 6500000 * q^15 + 83129542 * q^17 - 97491100 * q^19 + 438200736 * q^21 - 316255836 * q^23 + 732421875 * q^25 - 6518951360 * q^27 + 2236171850 * q^29 - 7482994376 * q^31 + 359182912 * q^33 - 7004562500 * q^35 + 31447174242 * q^37 + 70188571072 * q^39 - 10752884434 * q^41 - 16930554856 * q^43 + 20095609375 * q^45 - 31934201692 * q^47 - 38956926629 * q^49 + 129369882944 * q^51 - 221149123934 * q^53 + 103187562500 * q^55 - 763110695680 * q^57 + 55436423900 * q^59 + 496161392746 * q^61 - 1085454385236 * q^63 - 523468343750 * q^65 - 459297824792 * q^67 - 333379292832 * q^69 - 521997878336 * q^71 + 2505025571086 * q^73 - 101562500000 * q^75 - 385562457456 * q^77 - 2990636883200 * q^79 + 320549052763 * q^81 - 5137135467696 * q^83 + 1298899093750 * q^85 - 6885420178880 * q^87 - 19423025958450 * q^89 + 6792522370184 * q^91 - 11095902136128 * q^93 - 1523298437500 * q^95 - 11088325396458 * q^97 - 126787366508 * 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
$3$	573.185	0.453949	0.226974	−	0.973901i	$-0.427117\pi$
$3$	573.185	0.453949	0.226974	+	0.973901i	$0.427117\pi$
$4$	0	0
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	201493.	0.647326	0.323663	−	0.946172i	$-0.395086\pi$
$7$	201493.	0.647326	0.323663	+	0.946172i	$0.395086\pi$
$8$	0	0
$9$	−1.26578e6	−0.793931
$10$	0	0
$11$	3.34359e6	0.569062	0.284531	−	0.958667i	$-0.408162\pi$
$11$	3.34359e6	0.569062	0.284531	+	0.958667i	$0.408162\pi$
$12$	0	0
$13$	−7.80359e6	−0.448397	−0.224198	−	0.974544i	$-0.571976\pi$
$13$	−7.80359e6	−0.448397	−0.224198	+	0.974544i	$0.571976\pi$
$14$	0	0
$15$	8.95602e6	0.203012
$16$	0	0
$17$	−8.71750e7	−0.875939	−0.437969	−	0.898990i	$-0.644302\pi$
$17$	−8.71750e7	−0.875939	−0.437969	+	0.898990i	$0.644302\pi$
$18$	0	0
$19$	1.66766e7	0.0813223	0.0406612	−	0.999173i	$-0.487054\pi$
$19$	1.66766e7	0.0813223	0.0406612	+	0.999173i	$0.487054\pi$
$20$	0	0
$21$	1.15493e8	0.293853
$22$	0	0
$23$	−1.13518e9	−1.59895	−0.799476	−	0.600698i	$-0.794889\pi$
$23$	−1.13518e9	−1.59895	−0.799476	+	0.600698i	$0.794889\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	0	0
$27$	−1.63937e9	−0.814352
$28$	0	0
$29$	2.60673e9	0.813783	0.406891	−	0.913477i	$-0.366613\pi$
$29$	2.60673e9	0.813783	0.406891	+	0.913477i	$0.366613\pi$
$30$	0	0
$31$	−8.33139e8	−0.168604	−0.0843018	−	0.996440i	$-0.526866\pi$
$31$	−8.33139e8	−0.168604	−0.0843018	+	0.996440i	$0.526866\pi$
$32$	0	0
$33$	1.91649e9	0.258325
$34$	0	0
$35$	3.14833e9	0.289493
$36$	0	0
$37$	−1.05494e10	−0.675953	−0.337976	−	0.941155i	$-0.609742\pi$
$37$	−1.05494e10	−0.675953	−0.337976	+	0.941155i	$0.609742\pi$
$38$	0	0
$39$	−4.47290e9	−0.203549
$40$	0	0
$41$	−4.33030e9	−0.142371	−0.0711857	−	0.997463i	$-0.522678\pi$
$41$	−4.33030e9	−0.142371	−0.0711857	+	0.997463i	$0.522678\pi$
$42$	0	0
$43$	−1.93854e9	−0.0467660	−0.0233830	−	0.999727i	$-0.507444\pi$
$43$	−1.93854e9	−0.0467660	−0.0233830	+	0.999727i	$0.507444\pi$
$44$	0	0
$45$	−1.97778e10	−0.355057
$46$	0	0
$47$	−2.85468e10	−0.386296	−0.193148	−	0.981170i	$-0.561870\pi$
$47$	−2.85468e10	−0.386296	−0.193148	+	0.981170i	$0.561870\pi$
$48$	0	0
$49$	−5.62896e10	−0.580969
$50$	0	0
$51$	−4.99674e10	−0.397631
$52$	0	0
$53$	1.23249e11	0.763819	0.381910	−	0.924200i	$-0.375267\pi$
$53$	1.23249e11	0.763819	0.381910	+	0.924200i	$0.375267\pi$
$54$	0	0
$55$	5.22435e10	0.254492
$56$	0	0
$57$	9.55879e9	0.0369162
$58$	0	0
$59$	5.55404e11	1.71424	0.857118	−	0.515120i	$-0.172253\pi$
$59$	5.55404e11	1.71424	0.857118	+	0.515120i	$0.172253\pi$
$60$	0	0
$61$	−4.10476e11	−1.02010	−0.510051	−	0.860144i	$-0.670373\pi$
$61$	−4.10476e11	−1.02010	−0.510051	+	0.860144i	$0.670373\pi$
$62$	0	0
$63$	−2.55046e11	−0.513932
$64$	0	0
$65$	−1.21931e11	−0.200529
$66$	0	0
$67$	−3.36861e11	−0.454951	−0.227475	−	0.973784i	$-0.573047\pi$
$67$	−3.36861e11	−0.454951	−0.227475	+	0.973784i	$0.573047\pi$
$68$	0	0
$69$	−6.50671e11	−0.725842
$70$	0	0
$71$	−1.57323e12	−1.45751	−0.728757	−	0.684772i	$-0.759902\pi$
$71$	−1.57323e12	−1.45751	−0.728757	+	0.684772i	$0.759902\pi$
$72$	0	0
$73$	2.05372e12	1.58834	0.794170	−	0.607695i	$-0.207906\pi$
$73$	2.05372e12	1.58834	0.794170	+	0.607695i	$0.207906\pi$
$74$	0	0
$75$	1.39938e11	0.0907897
$76$	0	0
$77$	6.73709e11	0.368369
$78$	0	0
$79$	6.93000e11	0.320743	0.160372	−	0.987057i	$-0.448731\pi$
$79$	6.93000e11	0.320743	0.160372	+	0.987057i	$0.448731\pi$
$80$	0	0
$81$	1.07840e12	0.424256
$82$	0	0
$83$	−2.01116e12	−0.675212	−0.337606	−	0.941288i	$-0.609617\pi$
$83$	−2.01116e12	−0.675212	−0.337606	+	0.941288i	$0.609617\pi$
$84$	0	0
$85$	−1.36211e12	−0.391732
$86$	0	0
$87$	1.49414e12	0.369416
$88$	0	0
$89$	−8.51832e12	−1.81685	−0.908424	−	0.418050i	$-0.862714\pi$
$89$	−8.51832e12	−1.81685	−0.908424	+	0.418050i	$0.862714\pi$
$90$	0	0
$91$	−1.57237e12	−0.290259
$92$	0	0
$93$	−4.77543e11	−0.0765374
$94$	0	0
$95$	2.60572e11	0.0363684
$96$	0	0
$97$	−7.99814e12	−0.974929	−0.487464	−	0.873143i	$-0.662078\pi$
$97$	−7.99814e12	−0.974929	−0.487464	+	0.873143i	$0.662078\pi$
$98$	0	0
$99$	−4.23225e12	−0.451796
$100$	0	0
$101$	−1.60387e13	−1.50342	−0.751710	−	0.659494i	$-0.770771\pi$
$101$	−1.60387e13	−1.50342	−0.751710	+	0.659494i	$0.770771\pi$
$102$	0	0
$103$	−1.72566e13	−1.42401	−0.712004	−	0.702176i	$-0.752212\pi$
$103$	−1.72566e13	−1.42401	−0.712004	+	0.702176i	$0.752212\pi$
$104$	0	0
$105$	1.80458e12	0.131415
$106$	0	0
$107$	3.98196e11	0.0256509	0.0128254	−	0.999918i	$-0.495917\pi$
$107$	3.98196e11	0.0256509	0.0128254	+	0.999918i	$0.495917\pi$
$108$	0	0
$109$	1.09886e13	0.627582	0.313791	−	0.949492i	$-0.398401\pi$
$109$	1.09886e13	0.627582	0.313791	+	0.949492i	$0.398401\pi$
$110$	0	0
$111$	−6.04676e12	−0.306848
$112$	0	0
$113$	−2.06154e13	−0.931497	−0.465748	−	0.884917i	$-0.654215\pi$
$113$	−2.06154e13	−0.931497	−0.465748	+	0.884917i	$0.654215\pi$
$114$	0	0
$115$	−1.77372e13	−0.715073
$116$	0	0
$117$	9.87764e12	0.355996
$118$	0	0
$119$	−1.75651e13	−0.567018
$120$	0	0
$121$	−2.33431e13	−0.676168
$122$	0	0
$123$	−2.48206e12	−0.0646293
$124$	0	0
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	−3.03181e13	−0.641177	−0.320588	−	0.947219i	$-0.603881\pi$
$127$	−3.03181e13	−0.641177	−0.320588	+	0.947219i	$0.603881\pi$
$128$	0	0
$129$	−1.11114e12	−0.0212294
$130$	0	0
$131$	7.79100e13	1.34688	0.673441	−	0.739241i	$-0.264815\pi$
$131$	7.79100e13	1.34688	0.673441	+	0.739241i	$0.264815\pi$
$132$	0	0
$133$	3.36022e12	0.0526420
$134$	0	0
$135$	−2.56152e13	−0.364190
$136$	0	0
$137$	−1.01051e14	−1.30574	−0.652871	−	0.757469i	$-0.726436\pi$
$137$	−1.01051e14	−1.30574	−0.652871	+	0.757469i	$0.726436\pi$
$138$	0	0
$139$	−1.42505e14	−1.67585	−0.837924	−	0.545787i	$-0.816231\pi$
$139$	−1.42505e14	−1.67585	−0.837924	+	0.545787i	$0.816231\pi$
$140$	0	0
$141$	−1.63626e13	−0.175359
$142$	0	0
$143$	−2.60920e13	−0.255166
$144$	0	0
$145$	4.07301e13	0.363935
$146$	0	0
$147$	−3.22643e13	−0.263730
$148$	0	0
$149$	−6.15555e13	−0.460846	−0.230423	−	0.973091i	$-0.574011\pi$
$149$	−6.15555e13	−0.460846	−0.230423	+	0.973091i	$0.574011\pi$
$150$	0	0
$151$	4.43226e13	0.304281	0.152141	−	0.988359i	$-0.451383\pi$
$151$	4.43226e13	0.304281	0.152141	+	0.988359i	$0.451383\pi$
$152$	0	0
$153$	1.10344e14	0.695435
$154$	0	0
$155$	−1.30178e13	−0.0754018
$156$	0	0
$157$	−1.81701e14	−0.968301	−0.484150	−	0.874985i	$-0.660871\pi$
$157$	−1.81701e14	−0.968301	−0.484150	+	0.874985i	$0.660871\pi$
$158$	0	0
$159$	7.06446e13	0.346735
$160$	0	0
$161$	−2.28732e14	−1.03504
$162$	0	0
$163$	−9.09336e13	−0.379756	−0.189878	−	0.981808i	$-0.560809\pi$
$163$	−9.09336e13	−0.379756	−0.189878	+	0.981808i	$0.560809\pi$
$164$	0	0
$165$	2.99452e13	0.115527
$166$	0	0
$167$	2.40022e14	0.856237	0.428118	−	0.903723i	$-0.359177\pi$
$167$	2.40022e14	0.856237	0.428118	+	0.903723i	$0.359177\pi$
$168$	0	0
$169$	−2.41979e14	−0.798940
$170$	0	0
$171$	−2.11090e13	−0.0645643
$172$	0	0
$173$	4.10274e14	1.16352	0.581760	−	0.813360i	$-0.302364\pi$
$173$	4.10274e14	1.16352	0.581760	+	0.813360i	$0.302364\pi$
$174$	0	0
$175$	4.91926e13	0.129465
$176$	0	0
$177$	3.18349e14	0.778175
$178$	0	0
$179$	2.37588e14	0.539858	0.269929	−	0.962880i	$-0.413000\pi$
$179$	2.37588e14	0.539858	0.269929	+	0.962880i	$0.413000\pi$
$180$	0	0
$181$	5.41620e14	1.14494	0.572472	−	0.819924i	$-0.305985\pi$
$181$	5.41620e14	1.14494	0.572472	+	0.819924i	$0.305985\pi$
$182$	0	0
$183$	−2.35279e14	−0.463074
$184$	0	0
$185$	−1.64834e14	−0.302295
$186$	0	0
$187$	−2.91477e14	−0.498464
$188$	0	0
$189$	−3.30322e14	−0.527151
$190$	0	0
$191$	2.56027e12	0.00381565	0.00190782	−	0.999998i	$-0.499393\pi$
$191$	2.56027e12	0.00381565	0.00190782	+	0.999998i	$0.499393\pi$
$192$	0	0
$193$	1.10496e15	1.53895	0.769475	−	0.638677i	$-0.220518\pi$
$193$	1.10496e15	1.53895	0.769475	+	0.638677i	$0.220518\pi$
$194$	0	0
$195$	−6.98891e13	−0.0910300
$196$	0	0
$197$	7.90145e14	0.963110	0.481555	−	0.876416i	$-0.340072\pi$
$197$	7.90145e14	0.963110	0.481555	+	0.876416i	$0.340072\pi$
$198$	0	0
$199$	2.97088e14	0.339109	0.169555	−	0.985521i	$-0.445767\pi$
$199$	2.97088e14	0.339109	0.169555	+	0.985521i	$0.445767\pi$
$200$	0	0
$201$	−1.93084e14	−0.206524
$202$	0	0
$203$	5.25237e14	0.526782
$204$	0	0
$205$	−6.76609e13	−0.0636704
$206$	0	0
$207$	1.43689e15	1.26946
$208$	0	0
$209$	5.57597e13	0.0462775
$210$	0	0
$211$	1.25602e15	0.979851	0.489926	−	0.871764i	$-0.337024\pi$
$211$	1.25602e15	0.979851	0.489926	+	0.871764i	$0.337024\pi$
$212$	0	0
$213$	−9.01752e14	−0.661637
$214$	0	0
$215$	−3.02897e13	−0.0209144
$216$	0	0
$217$	−1.67872e14	−0.109141
$218$	0	0
$219$	1.17716e15	0.721025
$220$	0	0
$221$	6.80278e14	0.392768
$222$	0	0
$223$	1.17841e15	0.641676	0.320838	−	0.947134i	$-0.396036\pi$
$223$	1.17841e15	0.641676	0.320838	+	0.947134i	$0.396036\pi$
$224$	0	0
$225$	−3.09029e14	−0.158786
$226$	0	0
$227$	−4.91498e14	−0.238426	−0.119213	−	0.992869i	$-0.538037\pi$
$227$	−4.91498e14	−0.238426	−0.119213	+	0.992869i	$0.538037\pi$
$228$	0	0
$229$	3.17657e15	1.45555	0.727777	−	0.685814i	$-0.240554\pi$
$229$	3.17657e15	1.45555	0.727777	+	0.685814i	$0.240554\pi$
$230$	0	0
$231$	3.86160e14	0.167221
$232$	0	0
$233$	−2.05201e14	−0.0840168	−0.0420084	−	0.999117i	$-0.513376\pi$
$233$	−2.05201e14	−0.0840168	−0.0420084	+	0.999117i	$0.513376\pi$
$234$	0	0
$235$	−4.46043e14	−0.172757
$236$	0	0
$237$	3.97218e14	0.145601
$238$	0	0
$239$	−4.78414e15	−1.66042	−0.830209	−	0.557453i	$-0.811779\pi$
$239$	−4.78414e15	−1.66042	−0.830209	+	0.557453i	$0.811779\pi$
$240$	0	0
$241$	2.83675e15	0.932633	0.466316	−	0.884618i	$-0.345581\pi$
$241$	2.83675e15	0.932633	0.466316	+	0.884618i	$0.345581\pi$
$242$	0	0
$243$	3.23181e15	1.00694
$244$	0	0
$245$	−8.79524e14	−0.259817
$246$	0	0
$247$	−1.30138e14	−0.0364647
$248$	0	0
$249$	−1.15277e15	−0.306512
$250$	0	0
$251$	−3.50075e15	−0.883654	−0.441827	−	0.897100i	$-0.645669\pi$
$251$	−3.50075e15	−0.883654	−0.441827	+	0.897100i	$0.645669\pi$
$252$	0	0
$253$	−3.79558e15	−0.909903
$254$	0	0
$255$	−7.80740e14	−0.177826
$256$	0	0
$257$	−7.14451e14	−0.154670	−0.0773352	−	0.997005i	$-0.524641\pi$
$257$	−7.14451e14	−0.154670	−0.0773352	+	0.997005i	$0.524641\pi$
$258$	0	0
$259$	−2.12563e15	−0.437562
$260$	0	0
$261$	−3.29955e15	−0.646087
$262$	0	0
$263$	6.41873e15	1.19601	0.598007	−	0.801491i	$-0.295959\pi$
$263$	6.41873e15	1.19601	0.598007	+	0.801491i	$0.295959\pi$
$264$	0	0
$265$	1.92577e15	0.341590
$266$	0	0
$267$	−4.88257e15	−0.824756
$268$	0	0
$269$	−1.09876e15	−0.176813	−0.0884063	−	0.996084i	$-0.528177\pi$
$269$	−1.09876e15	−0.176813	−0.0884063	+	0.996084i	$0.528177\pi$
$270$	0	0
$271$	−1.60206e15	−0.245684	−0.122842	−	0.992426i	$-0.539201\pi$
$271$	−1.60206e15	−0.245684	−0.122842	+	0.992426i	$0.539201\pi$
$272$	0	0
$273$	−9.01259e14	−0.131763
$274$	0	0
$275$	8.16305e14	0.113812
$276$	0	0
$277$	−6.91414e14	−0.0919644	−0.0459822	−	0.998942i	$-0.514642\pi$
$277$	−6.91414e14	−0.0919644	−0.0459822	+	0.998942i	$0.514642\pi$
$278$	0	0
$279$	1.05457e15	0.133860
$280$	0	0
$281$	2.20744e15	0.267484	0.133742	−	0.991016i	$-0.457301\pi$
$281$	2.20744e15	0.267484	0.133742	+	0.991016i	$0.457301\pi$
$282$	0	0
$283$	−8.67634e15	−1.00398	−0.501990	−	0.864873i	$-0.667399\pi$
$283$	−8.67634e15	−1.00398	−0.501990	+	0.864873i	$0.667399\pi$
$284$	0	0
$285$	1.49356e14	0.0165094
$286$	0	0
$287$	−8.72525e14	−0.0921606
$288$	0	0
$289$	−2.30511e15	−0.232731
$290$	0	0
$291$	−4.58442e15	−0.442568
$292$	0	0
$293$	−7.81345e15	−0.721445	−0.360723	−	0.932673i	$-0.617470\pi$
$293$	−7.81345e15	−0.721445	−0.360723	+	0.932673i	$0.617470\pi$
$294$	0	0
$295$	8.67818e15	0.766630
$296$	0	0
$297$	−5.48137e15	−0.463417
$298$	0	0
$299$	8.85851e15	0.716965
$300$	0	0
$301$	−3.90603e14	−0.0302728
$302$	0	0
$303$	−9.19315e15	−0.682475
$304$	0	0
$305$	−6.41368e15	−0.456203
$306$	0	0
$307$	−1.73583e16	−1.18334	−0.591668	−	0.806182i	$-0.701530\pi$
$307$	−1.73583e16	−1.18334	−0.591668	+	0.806182i	$0.701530\pi$
$308$	0	0
$309$	−9.89120e15	−0.646426
$310$	0	0
$311$	2.64097e15	0.165509	0.0827543	−	0.996570i	$-0.473628\pi$
$311$	2.64097e15	0.165509	0.0827543	+	0.996570i	$0.473628\pi$
$312$	0	0
$313$	1.04071e16	0.625594	0.312797	−	0.949820i	$-0.398734\pi$
$313$	1.04071e16	0.625594	0.312797	+	0.949820i	$0.398734\pi$
$314$	0	0
$315$	−3.98510e15	−0.229837
$316$	0	0
$317$	−1.64664e16	−0.911410	−0.455705	−	0.890131i	$-0.650613\pi$
$317$	−1.64664e16	−0.911410	−0.455705	+	0.890131i	$0.650613\pi$
$318$	0	0
$319$	8.71581e15	0.463093
$320$	0	0
$321$	2.28240e14	0.0116442
$322$	0	0
$323$	−1.45378e15	−0.0712334
$324$	0	0
$325$	−1.90517e15	−0.0896794
$326$	0	0
$327$	6.29851e15	0.284890
$328$	0	0
$329$	−5.75197e15	−0.250060
$330$	0	0
$331$	−1.40904e16	−0.588900	−0.294450	−	0.955667i	$-0.595137\pi$
$331$	−1.40904e16	−0.588900	−0.294450	+	0.955667i	$0.595137\pi$
$332$	0	0
$333$	1.33532e16	0.536660
$334$	0	0
$335$	−5.26345e15	−0.203460
$336$	0	0
$337$	1.51673e16	0.564044	0.282022	−	0.959408i	$-0.408995\pi$
$337$	1.51673e16	0.564044	0.282022	+	0.959408i	$0.408995\pi$
$338$	0	0
$339$	−1.18164e16	−0.422852
$340$	0	0
$341$	−2.78567e15	−0.0959460
$342$	0	0
$343$	−3.08644e16	−1.02340
$344$	0	0
$345$	−1.01667e16	−0.324606
$346$	0	0
$347$	−3.87222e15	−0.119074	−0.0595372	−	0.998226i	$-0.518962\pi$
$347$	−3.87222e15	−0.119074	−0.0595372	+	0.998226i	$0.518962\pi$
$348$	0	0
$349$	5.50673e16	1.63128	0.815640	−	0.578560i	$-0.196385\pi$
$349$	5.50673e16	1.63128	0.815640	+	0.578560i	$0.196385\pi$
$350$	0	0
$351$	1.27930e16	0.365153
$352$	0	0
$353$	−4.84424e15	−0.133257	−0.0666285	−	0.997778i	$-0.521224\pi$
$353$	−4.84424e15	−0.133257	−0.0666285	+	0.997778i	$0.521224\pi$
$354$	0	0
$355$	−2.45817e16	−0.651820
$356$	0	0
$357$	−1.00681e16	−0.257397
$358$	0	0
$359$	5.71730e16	1.40954	0.704770	−	0.709436i	$-0.251050\pi$
$359$	5.71730e16	1.40954	0.704770	+	0.709436i	$0.251050\pi$
$360$	0	0
$361$	−4.17749e16	−0.993387
$362$	0	0
$363$	−1.33799e16	−0.306946
$364$	0	0
$365$	3.20894e16	0.710328
$366$	0	0
$367$	3.48286e16	0.744057	0.372028	−	0.928221i	$-0.378662\pi$
$367$	3.48286e16	0.744057	0.372028	+	0.928221i	$0.378662\pi$
$368$	0	0
$369$	5.48121e15	0.113033
$370$	0	0
$371$	2.48338e16	0.494440
$372$	0	0
$373$	−2.14588e16	−0.412570	−0.206285	−	0.978492i	$-0.566137\pi$
$373$	−2.14588e16	−0.412570	−0.206285	+	0.978492i	$0.566137\pi$
$374$	0	0
$375$	2.18653e15	0.0406024
$376$	0	0
$377$	−2.03418e16	−0.364898
$378$	0	0
$379$	1.90361e16	0.329930	0.164965	−	0.986299i	$-0.447249\pi$
$379$	1.90361e16	0.329930	0.164965	+	0.986299i	$0.447249\pi$
$380$	0	0
$381$	−1.73779e16	−0.291061
$382$	0	0
$383$	1.03414e17	1.67412	0.837059	−	0.547112i	$-0.184273\pi$
$383$	1.03414e17	1.67412	0.837059	+	0.547112i	$0.184273\pi$
$384$	0	0
$385$	1.05267e16	0.164740
$386$	0	0
$387$	2.45377e15	0.0371290
$388$	0	0
$389$	9.11608e16	1.33394	0.666969	−	0.745085i	$-0.267591\pi$
$389$	9.11608e16	1.33394	0.666969	+	0.745085i	$0.267591\pi$
$390$	0	0
$391$	9.89596e16	1.40058
$392$	0	0
$393$	4.46568e16	0.611416
$394$	0	0
$395$	1.08281e16	0.143441
$396$	0	0
$397$	7.19989e15	0.0922970	0.0461485	−	0.998935i	$-0.485305\pi$
$397$	7.19989e15	0.0922970	0.0461485	+	0.998935i	$0.485305\pi$
$398$	0	0
$399$	1.92603e15	0.0238968
$400$	0	0
$401$	1.06602e17	1.28034	0.640172	−	0.768231i	$-0.278863\pi$
$401$	1.06602e17	1.28034	0.640172	+	0.768231i	$0.278863\pi$
$402$	0	0
$403$	6.50148e15	0.0756013
$404$	0	0
$405$	1.68500e16	0.189733
$406$	0	0
$407$	−3.52728e16	−0.384659
$408$	0	0
$409$	6.00154e16	0.633959	0.316980	−	0.948432i	$-0.397331\pi$
$409$	6.00154e16	0.633959	0.316980	+	0.948432i	$0.397331\pi$
$410$	0	0
$411$	−5.79210e16	−0.592740
$412$	0	0
$413$	1.11910e17	1.10967
$414$	0	0
$415$	−3.14244e16	−0.301964
$416$	0	0
$417$	−8.16819e16	−0.760749
$418$	0	0
$419$	−1.22532e17	−1.10626	−0.553130	−	0.833095i	$-0.686567\pi$
$419$	−1.22532e17	−1.10626	−0.553130	+	0.833095i	$0.686567\pi$
$420$	0	0
$421$	−1.19508e17	−1.04607	−0.523037	−	0.852310i	$-0.675201\pi$
$421$	−1.19508e17	−1.04607	−0.523037	+	0.852310i	$0.675201\pi$
$422$	0	0
$423$	3.61340e16	0.306693
$424$	0	0
$425$	−2.12829e16	−0.175188
$426$	0	0
$427$	−8.27080e16	−0.660338
$428$	0	0
$429$	−1.49555e16	−0.115832
$430$	0	0
$431$	3.18425e16	0.239279	0.119640	−	0.992817i	$-0.461826\pi$
$431$	3.18425e16	0.239279	0.119640	+	0.992817i	$0.461826\pi$
$432$	0	0
$433$	1.24850e16	0.0910366	0.0455183	−	0.998964i	$-0.485506\pi$
$433$	1.24850e16	0.0910366	0.0455183	+	0.998964i	$0.485506\pi$
$434$	0	0
$435$	2.33459e16	0.165208
$436$	0	0
$437$	−1.89310e16	−0.130030
$438$	0	0
$439$	2.58876e17	1.72613	0.863063	−	0.505095i	$-0.168543\pi$
$439$	2.58876e17	1.72613	0.863063	+	0.505095i	$0.168543\pi$
$440$	0	0
$441$	7.12503e16	0.461249
$442$	0	0
$443$	−1.53844e17	−0.967068	−0.483534	−	0.875326i	$-0.660647\pi$
$443$	−1.53844e17	−0.967068	−0.483534	+	0.875326i	$0.660647\pi$
$444$	0	0
$445$	−1.33099e17	−0.812519
$446$	0	0
$447$	−3.52827e16	−0.209201
$448$	0	0
$449$	−9.02887e16	−0.520034	−0.260017	−	0.965604i	$-0.583728\pi$
$449$	−9.02887e16	−0.520034	−0.260017	+	0.965604i	$0.583728\pi$
$450$	0	0
$451$	−1.44787e16	−0.0810182
$452$	0	0
$453$	2.54051e16	0.138128
$454$	0	0
$455$	−2.45683e16	−0.129808
$456$	0	0
$457$	−9.78643e16	−0.502538	−0.251269	−	0.967917i	$-0.580848\pi$
$457$	−9.78643e16	−0.502538	−0.251269	+	0.967917i	$0.580848\pi$
$458$	0	0
$459$	1.42912e17	0.713323
$460$	0	0
$461$	2.08140e17	1.00995	0.504976	−	0.863134i	$-0.331501\pi$
$461$	2.08140e17	1.00995	0.504976	+	0.863134i	$0.331501\pi$
$462$	0	0
$463$	2.44823e17	1.15498	0.577491	−	0.816397i	$-0.304032\pi$
$463$	2.44823e17	1.15498	0.577491	+	0.816397i	$0.304032\pi$
$464$	0	0
$465$	−7.46161e15	−0.0342286
$466$	0	0
$467$	−3.54401e17	−1.58101	−0.790506	−	0.612454i	$-0.790182\pi$
$467$	−3.54401e17	−1.58101	−0.790506	+	0.612454i	$0.790182\pi$
$468$	0	0
$469$	−6.78751e16	−0.294501
$470$	0	0
$471$	−1.04148e17	−0.439559
$472$	0	0
$473$	−6.48168e15	−0.0266128
$474$	0	0
$475$	4.07144e15	0.0162645
$476$	0	0
$477$	−1.56006e17	−0.606419
$478$	0	0
$479$	1.54040e17	0.582712	0.291356	−	0.956615i	$-0.405894\pi$
$479$	1.54040e17	0.582712	0.291356	+	0.956615i	$0.405894\pi$
$480$	0	0
$481$	8.23232e16	0.303095
$482$	0	0
$483$	−1.31106e17	−0.469856
$484$	0	0
$485$	−1.24971e17	−0.436001
$486$	0	0
$487$	−2.11302e17	−0.717736	−0.358868	−	0.933388i	$-0.616837\pi$
$487$	−2.11302e17	−0.717736	−0.358868	+	0.933388i	$0.616837\pi$
$488$	0	0
$489$	−5.21218e16	−0.172390
$490$	0	0
$491$	−2.61474e17	−0.842167	−0.421083	−	0.907022i	$-0.638350\pi$
$491$	−2.61474e17	−0.842167	−0.421083	+	0.907022i	$0.638350\pi$
$492$	0	0
$493$	−2.27241e17	−0.712824
$494$	0	0
$495$	−6.61289e16	−0.202049
$496$	0	0
$497$	−3.16995e17	−0.943487
$498$	0	0
$499$	−3.87524e17	−1.12369	−0.561843	−	0.827244i	$-0.689908\pi$
$499$	−3.87524e17	−1.12369	−0.561843	+	0.827244i	$0.689908\pi$
$500$	0	0
$501$	1.37577e17	0.388688
$502$	0	0
$503$	4.12680e17	1.13611	0.568056	−	0.822990i	$-0.307696\pi$
$503$	4.12680e17	1.13611	0.568056	+	0.822990i	$0.307696\pi$
$504$	0	0
$505$	−2.50605e17	−0.672350
$506$	0	0
$507$	−1.38699e17	−0.362678
$508$	0	0
$509$	8.09532e16	0.206333	0.103166	−	0.994664i	$-0.467103\pi$
$509$	8.09532e16	0.206333	0.103166	+	0.994664i	$0.467103\pi$
$510$	0	0
$511$	4.13811e17	1.02817
$512$	0	0
$513$	−2.73391e16	−0.0662250
$514$	0	0
$515$	−2.69634e17	−0.636836
$516$	0	0
$517$	−9.54485e16	−0.219827
$518$	0	0
$519$	2.35163e17	0.528178
$520$	0	0
$521$	−2.79201e17	−0.611607	−0.305803	−	0.952095i	$-0.598925\pi$
$521$	−2.79201e17	−0.611607	−0.305803	+	0.952095i	$0.598925\pi$
$522$	0	0
$523$	−9.97401e16	−0.213112	−0.106556	−	0.994307i	$-0.533982\pi$
$523$	−9.97401e16	−0.213112	−0.106556	+	0.994307i	$0.533982\pi$
$524$	0	0
$525$	2.81965e16	0.0587705
$526$	0	0
$527$	7.26289e16	0.147686
$528$	0	0
$529$	7.84606e17	1.55665
$530$	0	0
$531$	−7.03020e17	−1.36098
$532$	0	0
$533$	3.37919e16	0.0638389
$534$	0	0
$535$	6.22181e15	0.0114714
$536$	0	0
$537$	1.36182e17	0.245068
$538$	0	0
$539$	−1.88209e17	−0.330608
$540$	0	0
$541$	−2.15125e17	−0.368899	−0.184450	−	0.982842i	$-0.559050\pi$
$541$	−2.15125e17	−0.368899	−0.184450	+	0.982842i	$0.559050\pi$
$542$	0	0
$543$	3.10449e17	0.519746
$544$	0	0
$545$	1.71697e17	0.280663
$546$	0	0
$547$	−9.04541e17	−1.44381	−0.721906	−	0.691991i	$-0.756734\pi$
$547$	−9.04541e17	−1.44381	−0.721906	+	0.691991i	$0.756734\pi$
$548$	0	0
$549$	5.19573e17	0.809890
$550$	0	0
$551$	4.34714e16	0.0661787
$552$	0	0
$553$	1.39635e17	0.207625
$554$	0	0
$555$	−9.44806e16	−0.137227
$556$	0	0
$557$	−9.25130e17	−1.31263	−0.656317	−	0.754485i	$-0.727887\pi$
$557$	−9.25130e17	−1.31263	−0.656317	+	0.754485i	$0.727887\pi$
$558$	0	0
$559$	1.51276e16	0.0209697
$560$	0	0
$561$	−1.67070e17	−0.226277
$562$	0	0
$563$	7.77258e17	1.02863	0.514317	−	0.857600i	$-0.328046\pi$
$563$	7.77258e17	1.02863	0.514317	+	0.857600i	$0.328046\pi$
$564$	0	0
$565$	−3.22115e17	−0.416578
$566$	0	0
$567$	2.17291e17	0.274632
$568$	0	0
$569$	5.27222e17	0.651274	0.325637	−	0.945495i	$-0.394421\pi$
$569$	5.27222e17	0.651274	0.325637	+	0.945495i	$0.394421\pi$
$570$	0	0
$571$	8.57655e17	1.03557	0.517783	−	0.855512i	$-0.326757\pi$
$571$	8.57655e17	1.03557	0.517783	+	0.855512i	$0.326757\pi$
$572$	0	0
$573$	1.46751e15	0.00173211
$574$	0	0
$575$	−2.77144e17	−0.319790
$576$	0	0
$577$	−6.01001e17	−0.678004	−0.339002	−	0.940786i	$-0.610089\pi$
$577$	−6.01001e17	−0.678004	−0.339002	+	0.940786i	$0.610089\pi$
$578$	0	0
$579$	6.33347e17	0.698604
$580$	0	0
$581$	−4.05236e17	−0.437082
$582$	0	0
$583$	4.12094e17	0.434661
$584$	0	0
$585$	1.54338e17	0.159206
$586$	0	0
$587$	−2.25235e17	−0.227241	−0.113621	−	0.993524i	$-0.536245\pi$
$587$	−2.25235e17	−0.227241	−0.113621	+	0.993524i	$0.536245\pi$
$588$	0	0
$589$	−1.38940e16	−0.0137112
$590$	0	0
$591$	4.52899e17	0.437203
$592$	0	0
$593$	8.69033e17	0.820693	0.410347	−	0.911930i	$-0.365408\pi$
$593$	8.69033e17	0.820693	0.410347	+	0.911930i	$0.365408\pi$
$594$	0	0
$595$	−2.74455e17	−0.253578
$596$	0	0
$597$	1.70286e17	0.153938
$598$	0	0
$599$	1.94567e18	1.72105	0.860527	−	0.509405i	$-0.170134\pi$
$599$	1.94567e18	1.72105	0.860527	+	0.509405i	$0.170134\pi$
$600$	0	0
$601$	2.96725e17	0.256844	0.128422	−	0.991720i	$-0.459009\pi$
$601$	2.96725e17	0.256844	0.128422	+	0.991720i	$0.459009\pi$
$602$	0	0
$603$	4.26392e17	0.361199
$604$	0	0
$605$	−3.64737e17	−0.302391
$606$	0	0
$607$	−2.06953e18	−1.67937	−0.839683	−	0.543076i	$-0.817259\pi$
$607$	−2.06953e18	−1.67937	−0.839683	+	0.543076i	$0.817259\pi$
$608$	0	0
$609$	3.01058e17	0.239132
$610$	0	0
$611$	2.22767e17	0.173214
$612$	0	0
$613$	2.27988e18	1.73548	0.867738	−	0.497022i	$-0.165573\pi$
$613$	2.27988e18	1.73548	0.867738	+	0.497022i	$0.165573\pi$
$614$	0	0
$615$	−3.87822e16	−0.0289031
$616$	0	0
$617$	−2.03478e18	−1.48479	−0.742394	−	0.669964i	$-0.766310\pi$
$617$	−2.03478e18	−1.48479	−0.742394	+	0.669964i	$0.766310\pi$
$618$	0	0
$619$	7.79671e17	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$619$	7.79671e17	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$620$	0	0
$621$	1.86099e18	1.30211
$622$	0	0
$623$	−1.71638e18	−1.17609
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	0	0
$627$	3.19606e16	0.0210076
$628$	0	0
$629$	9.19643e17	0.592093
$630$	0	0
$631$	1.75763e18	1.10850	0.554252	−	0.832349i	$-0.313005\pi$
$631$	1.75763e18	1.10850	0.554252	+	0.832349i	$0.313005\pi$
$632$	0	0
$633$	7.19931e17	0.444802
$634$	0	0
$635$	−4.73720e17	−0.286743
$636$	0	0
$637$	4.39261e17	0.260505
$638$	0	0
$639$	1.99137e18	1.15717
$640$	0	0
$641$	2.32621e18	1.32456	0.662279	−	0.749257i	$-0.269589\pi$
$641$	2.32621e18	1.32456	0.662279	+	0.749257i	$0.269589\pi$
$642$	0	0
$643$	−2.60316e18	−1.45254	−0.726272	−	0.687408i	$-0.758749\pi$
$643$	−2.60316e18	−1.45254	−0.726272	+	0.687408i	$0.758749\pi$
$644$	0	0
$645$	−1.73616e16	−0.00949406
$646$	0	0
$647$	1.20524e18	0.645947	0.322974	−	0.946408i	$-0.395318\pi$
$647$	1.20524e18	0.645947	0.322974	+	0.946408i	$0.395318\pi$
$648$	0	0
$649$	1.85704e18	0.975507
$650$	0	0
$651$	−9.62216e16	−0.0495446
$652$	0	0
$653$	1.72181e18	0.869061	0.434531	−	0.900657i	$-0.356914\pi$
$653$	1.72181e18	0.869061	0.434531	+	0.900657i	$0.356914\pi$
$654$	0	0
$655$	1.21734e18	0.602344
$656$	0	0
$657$	−2.59957e18	−1.26103
$658$	0	0
$659$	3.66491e18	1.74304	0.871521	−	0.490358i	$-0.163134\pi$
$659$	3.66491e18	1.74304	0.871521	+	0.490358i	$0.163134\pi$
$660$	0	0
$661$	−3.66523e18	−1.70920	−0.854599	−	0.519289i	$-0.826197\pi$
$661$	−3.66523e18	−1.70920	−0.854599	+	0.519289i	$0.826197\pi$
$662$	0	0
$663$	3.89925e17	0.178297
$664$	0	0
$665$	5.25035e16	0.0235422
$666$	0	0
$667$	−2.95911e18	−1.30120
$668$	0	0
$669$	6.75448e17	0.291288
$670$	0	0
$671$	−1.37246e18	−0.580502
$672$	0	0
$673$	−3.60071e18	−1.49379	−0.746897	−	0.664940i	$-0.768457\pi$
$673$	−3.60071e18	−1.49379	−0.746897	+	0.664940i	$0.768457\pi$
$674$	0	0
$675$	−4.00237e17	−0.162870
$676$	0	0
$677$	1.46169e17	0.0583484	0.0291742	−	0.999574i	$-0.490712\pi$
$677$	1.46169e17	0.0583484	0.0291742	+	0.999574i	$0.490712\pi$
$678$	0	0
$679$	−1.61157e18	−0.631097
$680$	0	0
$681$	−2.81720e17	−0.108233
$682$	0	0
$683$	2.32270e18	0.875505	0.437753	−	0.899095i	$-0.355775\pi$
$683$	2.32270e18	0.875505	0.437753	+	0.899095i	$0.355775\pi$
$684$	0	0
$685$	−1.57892e18	−0.583946
$686$	0	0
$687$	1.82076e18	0.660747
$688$	0	0
$689$	−9.61786e17	−0.342494
$690$	0	0
$691$	6.92473e16	0.0241989	0.0120994	−	0.999927i	$-0.496149\pi$
$691$	6.92473e16	0.0241989	0.0120994	+	0.999927i	$0.496149\pi$
$692$	0	0
$693$	−8.52769e17	−0.292459
$694$	0	0
$695$	−2.22664e18	−0.749462
$696$	0	0
$697$	3.77493e17	0.124709
$698$	0	0
$699$	−1.17618e17	−0.0381393
$700$	0	0
$701$	−6.22383e16	−0.0198103	−0.00990514	−	0.999951i	$-0.503153\pi$
$701$	−6.22383e16	−0.0198103	−0.00990514	+	0.999951i	$0.503153\pi$
$702$	0	0
$703$	−1.75928e17	−0.0549700
$704$	0	0
$705$	−2.55665e17	−0.0784228
$706$	0	0
$707$	−3.23169e18	−0.973202
$708$	0	0
$709$	2.94114e18	0.869592	0.434796	−	0.900529i	$-0.356820\pi$
$709$	2.94114e18	0.869592	0.434796	+	0.900529i	$0.356820\pi$
$710$	0	0
$711$	−8.77187e17	−0.254648
$712$	0	0
$713$	9.45766e17	0.269589
$714$	0	0
$715$	−4.07687e17	−0.114114
$716$	0	0
$717$	−2.74220e18	−0.753744
$718$	0	0
$719$	−4.39694e18	−1.18690	−0.593448	−	0.804872i	$-0.702234\pi$
$719$	−4.39694e18	−1.18690	−0.593448	+	0.804872i	$0.702234\pi$
$720$	0	0
$721$	−3.47708e18	−0.921797
$722$	0	0
$723$	1.62598e18	0.423367
$724$	0	0
$725$	6.36408e17	0.162757
$726$	0	0
$727$	−6.21513e18	−1.56126	−0.780632	−	0.624991i	$-0.785103\pi$
$727$	−6.21513e18	−1.56126	−0.780632	+	0.624991i	$0.785103\pi$
$728$	0	0
$729$	1.33103e17	0.0328442
$730$	0	0
$731$	1.68992e17	0.0409641
$732$	0	0
$733$	−1.54714e18	−0.368429	−0.184215	−	0.982886i	$-0.558974\pi$
$733$	−1.54714e18	−0.368429	−0.184215	+	0.982886i	$0.558974\pi$
$734$	0	0
$735$	−5.04130e17	−0.117944
$736$	0	0
$737$	−1.12632e18	−0.258895
$738$	0	0
$739$	1.96669e18	0.444167	0.222084	−	0.975028i	$-0.428714\pi$
$739$	1.96669e18	0.444167	0.222084	+	0.975028i	$0.428714\pi$
$740$	0	0
$741$	−7.45929e16	−0.0165531
$742$	0	0
$743$	5.66492e18	1.23528	0.617641	−	0.786460i	$-0.288088\pi$
$743$	5.66492e18	1.23528	0.617641	+	0.786460i	$0.288088\pi$
$744$	0	0
$745$	−9.61805e17	−0.206097
$746$	0	0
$747$	2.54570e18	0.536071
$748$	0	0
$749$	8.02337e16	0.0166045
$750$	0	0
$751$	−4.65838e18	−0.947491	−0.473745	−	0.880662i	$-0.657098\pi$
$751$	−4.65838e18	−0.947491	−0.473745	+	0.880662i	$0.657098\pi$
$752$	0	0
$753$	−2.00658e18	−0.401133
$754$	0	0
$755$	6.92541e17	0.136079
$756$	0	0
$757$	−3.51348e18	−0.678602	−0.339301	−	0.940678i	$-0.610190\pi$
$757$	−3.51348e18	−0.678602	−0.339301	+	0.940678i	$0.610190\pi$
$758$	0	0
$759$	−2.17557e18	−0.413049
$760$	0	0
$761$	4.55489e18	0.850116	0.425058	−	0.905166i	$-0.360254\pi$
$761$	4.55489e18	0.850116	0.425058	+	0.905166i	$0.360254\pi$
$762$	0	0
$763$	2.21413e18	0.406250
$764$	0	0
$765$	1.72413e18	0.311008
$766$	0	0
$767$	−4.33414e18	−0.768658
$768$	0	0
$769$	5.81779e18	1.01446	0.507232	−	0.861810i	$-0.330669\pi$
$769$	5.81779e18	1.01446	0.507232	+	0.861810i	$0.330669\pi$
$770$	0	0
$771$	−4.09513e17	−0.0702124
$772$	0	0
$773$	−3.52257e18	−0.593871	−0.296936	−	0.954897i	$-0.595965\pi$
$773$	−3.52257e18	−0.593871	−0.296936	+	0.954897i	$0.595965\pi$
$774$	0	0
$775$	−2.03403e17	−0.0337207
$776$	0	0
$777$	−1.21838e18	−0.198631
$778$	0	0
$779$	−7.22147e16	−0.0115780
$780$	0	0
$781$	−5.26023e18	−0.829417
$782$	0	0
$783$	−4.27339e18	−0.662706
$784$	0	0
$785$	−2.83908e18	−0.433037
$786$	0	0
$787$	−1.48350e18	−0.222562	−0.111281	−	0.993789i	$-0.535495\pi$
$787$	−1.48350e18	−0.222562	−0.111281	+	0.993789i	$0.535495\pi$
$788$	0	0
$789$	3.67912e18	0.542929
$790$	0	0
$791$	−4.15386e18	−0.602982
$792$	0	0
$793$	3.20318e18	0.457411
$794$	0	0
$795$	1.10382e18	0.155065
$796$	0	0
$797$	3.72381e18	0.514645	0.257323	−	0.966326i	$-0.417160\pi$
$797$	3.72381e18	0.514645	0.257323	+	0.966326i	$0.417160\pi$
$798$	0	0
$799$	2.48856e18	0.338372
$800$	0	0
$801$	1.07823e19	1.44245
$802$	0	0
$803$	6.86680e18	0.903865
$804$	0	0
$805$	−3.57393e18	−0.462885
$806$	0	0
$807$	−6.29794e17	−0.0802639
$808$	0	0
$809$	−1.30799e19	−1.64036	−0.820181	−	0.572104i	$-0.806127\pi$
$809$	−1.30799e19	−1.64036	−0.820181	+	0.572104i	$0.806127\pi$
$810$	0	0
$811$	4.17968e18	0.515832	0.257916	−	0.966167i	$-0.416964\pi$
$811$	4.17968e18	0.515832	0.257916	+	0.966167i	$0.416964\pi$
$812$	0	0
$813$	−9.18274e17	−0.111528
$814$	0	0
$815$	−1.42084e18	−0.169832
$816$	0	0
$817$	−3.23283e16	−0.00380312
$818$	0	0
$819$	1.99028e18	0.230445
$820$	0	0
$821$	5.28864e18	0.602717	0.301358	−	0.953511i	$-0.402560\pi$
$821$	5.28864e18	0.602717	0.301358	+	0.953511i	$0.402560\pi$
$822$	0	0
$823$	−7.96616e18	−0.893614	−0.446807	−	0.894630i	$-0.647439\pi$
$823$	−7.96616e18	−0.893614	−0.446807	+	0.894630i	$0.647439\pi$
$824$	0	0
$825$	4.67894e17	0.0516650
$826$	0	0
$827$	1.49178e19	1.62150	0.810752	−	0.585389i	$-0.199058\pi$
$827$	1.49178e19	1.62150	0.810752	+	0.585389i	$0.199058\pi$
$828$	0	0
$829$	−8.50781e18	−0.910360	−0.455180	−	0.890399i	$-0.650425\pi$
$829$	−8.50781e18	−0.910360	−0.455180	+	0.890399i	$0.650425\pi$
$830$	0	0
$831$	−3.96308e17	−0.0417471
$832$	0	0
$833$	4.90704e18	0.508894
$834$	0	0
$835$	3.75034e18	0.382921
$836$	0	0
$837$	1.36582e18	0.137303
$838$	0	0
$839$	6.82788e18	0.675823	0.337912	−	0.941178i	$-0.390280\pi$
$839$	6.82788e18	0.675823	0.337912	+	0.941178i	$0.390280\pi$
$840$	0	0
$841$	−3.46561e18	−0.337758
$842$	0	0
$843$	1.26527e18	0.121424
$844$	0	0
$845$	−3.78092e18	−0.357297
$846$	0	0
$847$	−4.70348e18	−0.437701
$848$	0	0
$849$	−4.97315e18	−0.455755
$850$	0	0
$851$	1.19755e19	1.08082
$852$	0	0
$853$	6.17395e18	0.548775	0.274387	−	0.961619i	$-0.411525\pi$
$853$	6.17395e18	0.548775	0.274387	+	0.961619i	$0.411525\pi$
$854$	0	0
$855$	−3.29828e17	−0.0288740
$856$	0	0
$857$	1.69937e18	0.146525	0.0732625	−	0.997313i	$-0.476659\pi$
$857$	1.69937e18	0.146525	0.0732625	+	0.997313i	$0.476659\pi$
$858$	0	0
$859$	−3.92966e18	−0.333734	−0.166867	−	0.985979i	$-0.553365\pi$
$859$	−3.92966e18	−0.333734	−0.166867	+	0.985979i	$0.553365\pi$
$860$	0	0
$861$	−5.00118e17	−0.0418362
$862$	0	0
$863$	−6.77611e18	−0.558354	−0.279177	−	0.960240i	$-0.590062\pi$
$863$	−6.77611e18	−0.558354	−0.279177	+	0.960240i	$0.590062\pi$
$864$	0	0
$865$	6.41052e18	0.520342
$866$	0	0
$867$	−1.32125e18	−0.105648
$868$	0	0
$869$	2.31711e18	0.182523
$870$	0	0
$871$	2.62872e18	0.203998
$872$	0	0
$873$	1.01239e19	0.774026
$874$	0	0
$875$	7.68635e17	0.0578986
$876$	0	0
$877$	1.66517e19	1.23583	0.617917	−	0.786243i	$-0.287977\pi$
$877$	1.66517e19	1.23583	0.617917	+	0.786243i	$0.287977\pi$
$878$	0	0
$879$	−4.47855e18	−0.327499
$880$	0	0
$881$	1.72426e19	1.24239	0.621195	−	0.783656i	$-0.286647\pi$
$881$	1.72426e19	1.24239	0.621195	+	0.783656i	$0.286647\pi$
$882$	0	0
$883$	−3.03788e18	−0.215688	−0.107844	−	0.994168i	$-0.534395\pi$
$883$	−3.03788e18	−0.215688	−0.107844	+	0.994168i	$0.534395\pi$
$884$	0	0
$885$	4.97421e18	0.348011
$886$	0	0
$887$	−6.51405e18	−0.449105	−0.224552	−	0.974462i	$-0.572092\pi$
$887$	−6.51405e18	−0.449105	−0.224552	+	0.974462i	$0.572092\pi$
$888$	0	0
$889$	−6.10889e18	−0.415050
$890$	0	0
$891$	3.60573e18	0.241428
$892$	0	0
$893$	−4.76063e17	−0.0314145
$894$	0	0
$895$	3.71231e18	0.241432
$896$	0	0
$897$	5.07757e18	0.325465
$898$	0	0
$899$	−2.17177e18	−0.137207
$900$	0	0
$901$	−1.07442e19	−0.669059
$902$	0	0
$903$	−2.23888e17	−0.0137423
$904$	0	0
$905$	8.46282e18	0.512035
$906$	0	0
$907$	2.62324e19	1.56455	0.782277	−	0.622931i	$-0.214058\pi$
$907$	2.62324e19	1.56455	0.782277	+	0.622931i	$0.214058\pi$
$908$	0	0
$909$	2.03015e19	1.19361
$910$	0	0
$911$	9.93847e18	0.576037	0.288018	−	0.957625i	$-0.407004\pi$
$911$	9.93847e18	0.576037	0.288018	+	0.957625i	$0.407004\pi$
$912$	0	0
$913$	−6.72450e18	−0.384238
$914$	0	0
$915$	−3.67623e18	−0.207093
$916$	0	0
$917$	1.56983e19	0.871872
$918$	0	0
$919$	−1.61779e19	−0.885874	−0.442937	−	0.896553i	$-0.646063\pi$
$919$	−1.61779e19	−0.885874	−0.442937	+	0.896553i	$0.646063\pi$
$920$	0	0
$921$	−9.94953e18	−0.537174
$922$	0	0
$923$	1.22768e19	0.653545
$924$	0	0
$925$	−2.57554e18	−0.135191
$926$	0	0
$927$	2.18430e19	1.13056
$928$	0	0
$929$	2.14431e18	0.109442	0.0547211	−	0.998502i	$-0.482573\pi$
$929$	2.14431e18	0.109442	0.0547211	+	0.998502i	$0.482573\pi$
$930$	0	0
$931$	−9.38720e17	−0.0472458
$932$	0	0
$933$	1.51376e18	0.0751325
$934$	0	0
$935$	−4.55433e18	−0.222920
$936$	0	0
$937$	−3.39502e19	−1.63883	−0.819417	−	0.573199i	$-0.805702\pi$
$937$	−3.39502e19	−1.63883	−0.819417	+	0.573199i	$0.805702\pi$
$938$	0	0
$939$	5.96521e18	0.283988
$940$	0	0
$941$	−1.95970e19	−0.920149	−0.460074	−	0.887880i	$-0.652177\pi$
$941$	−1.95970e19	−0.920149	−0.460074	+	0.887880i	$0.652177\pi$
$942$	0	0
$943$	4.91568e18	0.227645
$944$	0	0
$945$	−5.16128e18	−0.235749
$946$	0	0
$947$	1.67818e19	0.756072	0.378036	−	0.925791i	$-0.376600\pi$
$947$	1.67818e19	0.756072	0.378036	+	0.925791i	$0.376600\pi$
$948$	0	0
$949$	−1.60264e19	−0.712207
$950$	0	0
$951$	−9.43831e18	−0.413734
$952$	0	0
$953$	9.68702e18	0.418877	0.209438	−	0.977822i	$-0.432836\pi$
$953$	9.68702e18	0.418877	0.209438	+	0.977822i	$0.432836\pi$
$954$	0	0
$955$	4.00042e16	0.00170641
$956$	0	0
$957$	4.99577e18	0.210221
$958$	0	0
$959$	−2.03611e19	−0.845241
$960$	0	0
$961$	−2.37234e19	−0.971573
$962$	0	0
$963$	−5.04029e17	−0.0203650
$964$	0	0
$965$	1.72650e19	0.688239
$966$	0	0
$967$	−2.34907e18	−0.0923899	−0.0461949	−	0.998932i	$-0.514710\pi$
$967$	−2.34907e18	−0.0923899	−0.0461949	+	0.998932i	$0.514710\pi$
$968$	0	0
$969$	−8.33287e17	−0.0323363
$970$	0	0
$971$	−2.00271e19	−0.766818	−0.383409	−	0.923579i	$-0.625250\pi$
$971$	−2.00271e19	−0.766818	−0.383409	+	0.923579i	$0.625250\pi$
$972$	0	0
$973$	−2.87138e19	−1.08482
$974$	0	0
$975$	−1.09202e18	−0.0407098
$976$	0	0
$977$	3.98189e19	1.46479	0.732394	−	0.680881i	$-0.238403\pi$
$977$	3.98189e19	1.46479	0.732394	+	0.680881i	$0.238403\pi$
$978$	0	0
$979$	−2.84817e19	−1.03390
$980$	0	0
$981$	−1.39092e19	−0.498257
$982$	0	0
$983$	−2.29985e19	−0.813022	−0.406511	−	0.913646i	$-0.633255\pi$
$983$	−2.29985e19	−0.813022	−0.406511	+	0.913646i	$0.633255\pi$
$984$	0	0
$985$	1.23460e19	0.430716
$986$	0	0
$987$	−3.29695e18	−0.113514
$988$	0	0
$989$	2.20060e18	0.0747765
$990$	0	0
$991$	−8.09114e18	−0.271351	−0.135675	−	0.990753i	$-0.543320\pi$
$991$	−8.09114e18	−0.271351	−0.135675	+	0.990753i	$0.543320\pi$
$992$	0	0
$993$	−8.07642e18	−0.267331
$994$	0	0
$995$	4.64200e18	0.151654
$996$	0	0
$997$	−2.26704e19	−0.731039	−0.365519	−	0.930804i	$-0.619109\pi$
$997$	−2.26704e19	−0.731039	−0.365519	+	0.930804i	$0.619109\pi$
$998$	0	0
$999$	1.72944e19	0.550464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.14.a.g.1.2		3
4.3	odd	2		5.14.a.b.1.3	✓	3
12.11	even	2		45.14.a.e.1.1		3
20.3	even	4		25.14.b.b.24.1		6
20.7	even	4		25.14.b.b.24.6		6
20.19	odd	2		25.14.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.14.a.b.1.3	✓	3	4.3	odd	2
25.14.a.b.1.1		3	20.19	odd	2
25.14.b.b.24.1		6	20.3	even	4
25.14.b.b.24.6		6	20.7	even	4
45.14.a.e.1.1		3	12.11	even	2
80.14.a.g.1.2		3	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+573.185 q^{3} +15625.0 q^{5} +201493. q^{7} -1.26578e6 q^{9} +3.34359e6 q^{11} -7.80359e6 q^{13} +8.95602e6 q^{15} -8.71750e7 q^{17} +1.66766e7 q^{19} +1.15493e8 q^{21} -1.13518e9 q^{23} +2.44141e8 q^{25} -1.63937e9 q^{27} +2.60673e9 q^{29} -8.33139e8 q^{31} +1.91649e9 q^{33} +3.14833e9 q^{35} -1.05494e10 q^{37} -4.47290e9 q^{39} -4.33030e9 q^{41} -1.93854e9 q^{43} -1.97778e10 q^{45} -2.85468e10 q^{47} -5.62896e10 q^{49} -4.99674e10 q^{51} +1.23249e11 q^{53} +5.22435e10 q^{55} +9.55879e9 q^{57} +5.55404e11 q^{59} -4.10476e11 q^{61} -2.55046e11 q^{63} -1.21931e11 q^{65} -3.36861e11 q^{67} -6.50671e11 q^{69} -1.57323e12 q^{71} +2.05372e12 q^{73} +1.39938e11 q^{75} +6.73709e11 q^{77} +6.93000e11 q^{79} +1.07840e12 q^{81} -2.01116e12 q^{83} -1.36211e12 q^{85} +1.49414e12 q^{87} -8.51832e12 q^{89} -1.57237e12 q^{91} -4.77543e11 q^{93} +2.60572e11 q^{95} -7.99814e12 q^{97} -4.23225e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 416 q^{3} + 46875 q^{5} - 448292 q^{7} + 1286119 q^{9} + 6604004 q^{11} - 33501974 q^{13} - 6500000 q^{15} + 83129542 q^{17} - 97491100 q^{19} + 438200736 q^{21} - 316255836 q^{23} + 732421875 q^{25}+ \cdots - 126787366508 q^{99}+O(q^{100}) \) 3 * q - 416 * q^3 + 46875 * q^5 - 448292 * q^7 + 1286119 * q^9 + 6604004 * q^11 - 33501974 * q^13 - 6500000 * q^15 + 83129542 * q^17 - 97491100 * q^19 + 438200736 * q^21 - 316255836 * q^23 + 732421875 * q^25 - 6518951360 * q^27 + 2236171850 * q^29 - 7482994376 * q^31 + 359182912 * q^33 - 7004562500 * q^35 + 31447174242 * q^37 + 70188571072 * q^39 - 10752884434 * q^41 - 16930554856 * q^43 + 20095609375 * q^45 - 31934201692 * q^47 - 38956926629 * q^49 + 129369882944 * q^51 - 221149123934 * q^53 + 103187562500 * q^55 - 763110695680 * q^57 + 55436423900 * q^59 + 496161392746 * q^61 - 1085454385236 * q^63 - 523468343750 * q^65 - 459297824792 * q^67 - 333379292832 * q^69 - 521997878336 * q^71 + 2505025571086 * q^73 - 101562500000 * q^75 - 385562457456 * q^77 - 2990636883200 * q^79 + 320549052763 * q^81 - 5137135467696 * q^83 + 1298899093750 * q^85 - 6885420178880 * q^87 - 19423025958450 * q^89 + 6792522370184 * q^91 - 11095902136128 * q^93 - 1523298437500 * q^95 - 11088325396458 * q^97 - 126787366508 * q^99

Introduction

L-functions

Modular forms

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Database

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