LMFDB - Embedded newform 80.14.a.g.1.3

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			1.3
Root			$69.3208$ 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)

$f(q)$	$=$	$q+1125.79 q^{3} +15625.0 q^{5} -324482. q^{7} -326912. q^{9} +1.64726e6 q^{11} +6.26700e6 q^{13} +1.75905e7 q^{15} +1.66481e8 q^{17} -3.12929e8 q^{19} -3.65300e8 q^{21} +6.32351e8 q^{23} +2.44141e8 q^{25} -2.16291e9 q^{27} -2.82750e9 q^{29} -7.61629e9 q^{31} +1.85448e9 q^{33} -5.07004e9 q^{35} +1.99161e10 q^{37} +7.05535e9 q^{39} -4.69877e10 q^{41} +7.85897e9 q^{43} -5.10799e9 q^{45} -8.31265e10 q^{47} +8.39984e9 q^{49} +1.87423e11 q^{51} -1.19285e11 q^{53} +2.57385e10 q^{55} -3.52294e11 q^{57} -4.20299e11 q^{59} +4.15504e11 q^{61} +1.06077e11 q^{63} +9.79220e10 q^{65} +1.02968e11 q^{67} +7.11897e11 q^{69} +4.00383e11 q^{71} +5.55011e11 q^{73} +2.74852e11 q^{75} -5.34508e11 q^{77} -1.60313e12 q^{79} -1.91379e12 q^{81} +2.64201e11 q^{83} +2.60127e12 q^{85} -3.18318e12 q^{87} -3.69637e12 q^{89} -2.03353e12 q^{91} -8.57437e12 q^{93} -4.88952e12 q^{95} -1.00920e13 q^{97} -5.38510e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1125.79	0.891601	0.445801	−	0.895132i	$-0.352919\pi$
$3$	1125.79	0.891601	0.445801	+	0.895132i	$0.352919\pi$
$4$	0	0
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	−324482.	−1.04245	−0.521223	−	0.853420i	$-0.674524\pi$
$7$	−324482.	−1.04245	−0.521223	+	0.853420i	$0.674524\pi$
$8$	0	0
$9$	−326912.	−0.205047
$10$	0	0
$11$	1.64726e6	0.280356	0.140178	−	0.990126i	$-0.455232\pi$
$11$	1.64726e6	0.280356	0.140178	+	0.990126i	$0.455232\pi$
$12$	0	0
$13$	6.26700e6	0.360104	0.180052	−	0.983657i	$-0.442373\pi$
$13$	6.26700e6	0.360104	0.180052	+	0.983657i	$0.442373\pi$
$14$	0	0
$15$	1.75905e7	0.398736
$16$	0	0
$17$	1.66481e8	1.67281	0.836406	−	0.548110i	$-0.184653\pi$
$17$	1.66481e8	1.67281	0.836406	+	0.548110i	$0.184653\pi$
$18$	0	0
$19$	−3.12929e8	−1.52598	−0.762988	−	0.646413i	$-0.776268\pi$
$19$	−3.12929e8	−1.52598	−0.762988	+	0.646413i	$0.776268\pi$
$20$	0	0
$21$	−3.65300e8	−0.929447
$22$	0	0
$23$	6.32351e8	0.890692	0.445346	−	0.895359i	$-0.353081\pi$
$23$	6.32351e8	0.890692	0.445346	+	0.895359i	$0.353081\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	0	0
$27$	−2.16291e9	−1.07442
$28$	0	0
$29$	−2.82750e9	−0.882704	−0.441352	−	0.897334i	$-0.645501\pi$
$29$	−2.82750e9	−0.882704	−0.441352	+	0.897334i	$0.645501\pi$
$30$	0	0
$31$	−7.61629e9	−1.54132	−0.770659	−	0.637247i	$-0.780073\pi$
$31$	−7.61629e9	−1.54132	−0.770659	+	0.637247i	$0.780073\pi$
$32$	0	0
$33$	1.85448e9	0.249966
$34$	0	0
$35$	−5.07004e9	−0.466196
$36$	0	0
$37$	1.99161e10	1.27613	0.638063	−	0.769984i	$-0.279736\pi$
$37$	1.99161e10	1.27613	0.638063	+	0.769984i	$0.279736\pi$
$38$	0	0
$39$	7.05535e9	0.321069
$40$	0	0
$41$	−4.69877e10	−1.54486	−0.772430	−	0.635099i	$-0.780959\pi$
$41$	−4.69877e10	−1.54486	−0.772430	+	0.635099i	$0.780959\pi$
$42$	0	0
$43$	7.85897e9	0.189592	0.0947961	−	0.995497i	$-0.469780\pi$
$43$	7.85897e9	0.189592	0.0947961	+	0.995497i	$0.469780\pi$
$44$	0	0
$45$	−5.10799e9	−0.0917000
$46$	0	0
$47$	−8.31265e10	−1.12487	−0.562437	−	0.826840i	$-0.690136\pi$
$47$	−8.31265e10	−1.12487	−0.562437	+	0.826840i	$0.690136\pi$
$48$	0	0
$49$	8.39984e9	0.0866955
$50$	0	0
$51$	1.87423e11	1.49148
$52$	0	0
$53$	−1.19285e11	−0.739255	−0.369628	−	0.929180i	$-0.620515\pi$
$53$	−1.19285e11	−0.739255	−0.369628	+	0.929180i	$0.620515\pi$
$54$	0	0
$55$	2.57385e10	0.125379
$56$	0	0
$57$	−3.52294e11	−1.36056
$58$	0	0
$59$	−4.20299e11	−1.29724	−0.648620	−	0.761112i	$-0.724654\pi$
$59$	−4.20299e11	−1.29724	−0.648620	+	0.761112i	$0.724654\pi$
$60$	0	0
$61$	4.15504e11	1.03260	0.516299	−	0.856409i	$-0.327309\pi$
$61$	4.15504e11	1.03260	0.516299	+	0.856409i	$0.327309\pi$
$62$	0	0
$63$	1.06077e11	0.213751
$64$	0	0
$65$	9.79220e10	0.161044
$66$	0	0
$67$	1.02968e11	0.139065	0.0695323	−	0.997580i	$-0.477849\pi$
$67$	1.02968e11	0.139065	0.0695323	+	0.997580i	$0.477849\pi$
$68$	0	0
$69$	7.11897e11	0.794142
$70$	0	0
$71$	4.00383e11	0.370933	0.185467	−	0.982651i	$-0.440620\pi$
$71$	4.00383e11	0.370933	0.185467	+	0.982651i	$0.440620\pi$
$72$	0	0
$73$	5.55011e11	0.429243	0.214621	−	0.976697i	$-0.431148\pi$
$73$	5.55011e11	0.429243	0.214621	+	0.976697i	$0.431148\pi$
$74$	0	0
$75$	2.74852e11	0.178320
$76$	0	0
$77$	−5.34508e11	−0.292257
$78$	0	0
$79$	−1.60313e12	−0.741980	−0.370990	−	0.928637i	$-0.620982\pi$
$79$	−1.60313e12	−0.741980	−0.370990	+	0.928637i	$0.620982\pi$
$80$	0	0
$81$	−1.91379e12	−0.752908
$82$	0	0
$83$	2.64201e11	0.0887005	0.0443503	−	0.999016i	$-0.485878\pi$
$83$	2.64201e11	0.0887005	0.0443503	+	0.999016i	$0.485878\pi$
$84$	0	0
$85$	2.60127e12	0.748104
$86$	0	0
$87$	−3.18318e12	−0.787020
$88$	0	0
$89$	−3.69637e12	−0.788388	−0.394194	−	0.919027i	$-0.628976\pi$
$89$	−3.69637e12	−0.788388	−0.394194	+	0.919027i	$0.628976\pi$
$90$	0	0
$91$	−2.03353e12	−0.375390
$92$	0	0
$93$	−8.57437e12	−1.37424
$94$	0	0
$95$	−4.88952e12	−0.682437
$96$	0	0
$97$	−1.00920e13	−1.23016	−0.615081	−	0.788464i	$-0.710877\pi$
$97$	−1.00920e13	−1.23016	−0.615081	+	0.788464i	$0.710877\pi$
$98$	0	0
$99$	−5.38510e11	−0.0574863
$100$	0	0
$101$	2.56737e12	0.240658	0.120329	−	0.992734i	$-0.461605\pi$
$101$	2.56737e12	0.240658	0.120329	+	0.992734i	$0.461605\pi$
$102$	0	0
$103$	−1.21037e13	−0.998791	−0.499395	−	0.866374i	$-0.666444\pi$
$103$	−1.21037e13	−0.998791	−0.499395	+	0.866374i	$0.666444\pi$
$104$	0	0
$105$	−5.70782e12	−0.415661
$106$	0	0
$107$	−9.37488e12	−0.603909	−0.301954	−	0.953322i	$-0.597639\pi$
$107$	−9.37488e12	−0.603909	−0.301954	+	0.953322i	$0.597639\pi$
$108$	0	0
$109$	−2.49908e13	−1.42727	−0.713637	−	0.700516i	$-0.752953\pi$
$109$	−2.49908e13	−1.42727	−0.713637	+	0.700516i	$0.752953\pi$
$110$	0	0
$111$	2.24214e13	1.13779
$112$	0	0
$113$	1.35600e13	0.612703	0.306352	−	0.951918i	$-0.400892\pi$
$113$	1.35600e13	0.612703	0.306352	+	0.951918i	$0.400892\pi$
$114$	0	0
$115$	9.88049e12	0.398330
$116$	0	0
$117$	−2.04876e12	−0.0738384
$118$	0	0
$119$	−5.40202e13	−1.74382
$120$	0	0
$121$	−3.18092e13	−0.921400
$122$	0	0
$123$	−5.28985e13	−1.37740
$124$	0	0
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	1.16836e13	0.247088	0.123544	−	0.992339i	$-0.460574\pi$
$127$	1.16836e13	0.247088	0.123544	+	0.992339i	$0.460574\pi$
$128$	0	0
$129$	8.84758e12	0.169041
$130$	0	0
$131$	−5.12734e13	−0.886398	−0.443199	−	0.896423i	$-0.646157\pi$
$131$	−5.12734e13	−0.886398	−0.443199	+	0.896423i	$0.646157\pi$
$132$	0	0
$133$	1.01540e14	1.59075
$134$	0	0
$135$	−3.37955e13	−0.480496
$136$	0	0
$137$	−7.15579e13	−0.924642	−0.462321	−	0.886713i	$-0.652983\pi$
$137$	−7.15579e13	−0.924642	−0.462321	+	0.886713i	$0.652983\pi$
$138$	0	0
$139$	1.16286e14	1.36751	0.683754	−	0.729712i	$-0.260346\pi$
$139$	1.16286e14	1.36751	0.683754	+	0.729712i	$0.260346\pi$
$140$	0	0
$141$	−9.35833e13	−1.00294
$142$	0	0
$143$	1.03234e13	0.100958
$144$	0	0
$145$	−4.41796e13	−0.394757
$146$	0	0
$147$	9.45649e12	0.0772978
$148$	0	0
$149$	−1.43657e14	−1.07552	−0.537758	−	0.843099i	$-0.680729\pi$
$149$	−1.43657e14	−1.07552	−0.537758	+	0.843099i	$0.680729\pi$
$150$	0	0
$151$	1.99719e13	0.137110	0.0685551	−	0.997647i	$-0.478161\pi$
$151$	1.99719e13	0.137110	0.0685551	+	0.997647i	$0.478161\pi$
$152$	0	0
$153$	−5.44246e13	−0.343006
$154$	0	0
$155$	−1.19005e14	−0.689299
$156$	0	0
$157$	2.46452e14	1.31336	0.656682	−	0.754168i	$-0.271959\pi$
$157$	2.46452e14	1.31336	0.656682	+	0.754168i	$0.271959\pi$
$158$	0	0
$159$	−1.34291e14	−0.659121
$160$	0	0
$161$	−2.05187e14	−0.928499
$162$	0	0
$163$	−1.33661e14	−0.558194	−0.279097	−	0.960263i	$-0.590035\pi$
$163$	−1.33661e14	−0.558194	−0.279097	+	0.960263i	$0.590035\pi$
$164$	0	0
$165$	2.89762e13	0.111788
$166$	0	0
$167$	1.57170e14	0.560676	0.280338	−	0.959901i	$-0.409553\pi$
$167$	1.57170e14	0.560676	0.280338	+	0.959901i	$0.409553\pi$
$168$	0	0
$169$	−2.63600e14	−0.870325
$170$	0	0
$171$	1.02300e14	0.312897
$172$	0	0
$173$	3.32272e14	0.942309	0.471155	−	0.882051i	$-0.343837\pi$
$173$	3.32272e14	0.942309	0.471155	+	0.882051i	$0.343837\pi$
$174$	0	0
$175$	−7.92193e13	−0.208489
$176$	0	0
$177$	−4.73170e14	−1.15662
$178$	0	0
$179$	−4.96701e14	−1.12863	−0.564313	−	0.825561i	$-0.690859\pi$
$179$	−4.96701e14	−1.12863	−0.564313	+	0.825561i	$0.690859\pi$
$180$	0	0
$181$	5.73895e14	1.21317	0.606585	−	0.795019i	$-0.292539\pi$
$181$	5.73895e14	1.21317	0.606585	+	0.795019i	$0.292539\pi$
$182$	0	0
$183$	4.67771e14	0.920665
$184$	0	0
$185$	3.11189e14	0.570701
$186$	0	0
$187$	2.74238e14	0.468984
$188$	0	0
$189$	7.01828e14	1.12003
$190$	0	0
$191$	9.21109e14	1.37276	0.686379	−	0.727244i	$-0.259199\pi$
$191$	9.21109e14	1.37276	0.686379	+	0.727244i	$0.259199\pi$
$192$	0	0
$193$	8.88779e14	1.23786	0.618929	−	0.785447i	$-0.287567\pi$
$193$	8.88779e14	1.23786	0.618929	+	0.785447i	$0.287567\pi$
$194$	0	0
$195$	1.10240e14	0.143587
$196$	0	0
$197$	−4.70943e13	−0.0574035	−0.0287017	−	0.999588i	$-0.509137\pi$
$197$	−4.70943e13	−0.0574035	−0.0287017	+	0.999588i	$0.509137\pi$
$198$	0	0
$199$	−4.94174e14	−0.564072	−0.282036	−	0.959404i	$-0.591010\pi$
$199$	−4.94174e14	−0.564072	−0.282036	+	0.959404i	$0.591010\pi$
$200$	0	0
$201$	1.15921e14	0.123990
$202$	0	0
$203$	9.17473e14	0.920172
$204$	0	0
$205$	−7.34183e14	−0.690883
$206$	0	0
$207$	−2.06723e14	−0.182634
$208$	0	0
$209$	−5.15477e14	−0.427817
$210$	0	0
$211$	1.22476e15	0.955466	0.477733	−	0.878505i	$-0.341459\pi$
$211$	1.22476e15	0.955466	0.477733	+	0.878505i	$0.341459\pi$
$212$	0	0
$213$	4.50748e14	0.330725
$214$	0	0
$215$	1.22796e14	0.0847882
$216$	0	0
$217$	2.47135e15	1.60674
$218$	0	0
$219$	6.24828e14	0.382713
$220$	0	0
$221$	1.04334e15	0.602387
$222$	0	0
$223$	1.23977e15	0.675087	0.337544	−	0.941310i	$-0.390404\pi$
$223$	1.23977e15	0.675087	0.337544	+	0.941310i	$0.390404\pi$
$224$	0	0
$225$	−7.98124e13	−0.0410095
$226$	0	0
$227$	3.31953e14	0.161031	0.0805154	−	0.996753i	$-0.474343\pi$
$227$	3.31953e14	0.161031	0.0805154	+	0.996753i	$0.474343\pi$
$228$	0	0
$229$	−1.92799e15	−0.883433	−0.441717	−	0.897155i	$-0.645630\pi$
$229$	−1.92799e15	−0.883433	−0.441717	+	0.897155i	$0.645630\pi$
$230$	0	0
$231$	−6.01746e14	−0.260576
$232$	0	0
$233$	−1.12721e15	−0.461522	−0.230761	−	0.973010i	$-0.574122\pi$
$233$	−1.12721e15	−0.461522	−0.230761	+	0.973010i	$0.574122\pi$
$234$	0	0
$235$	−1.29885e15	−0.503059
$236$	0	0
$237$	−1.80479e15	−0.661550
$238$	0	0
$239$	2.13044e15	0.739405	0.369703	−	0.929150i	$-0.379460\pi$
$239$	2.13044e15	0.739405	0.369703	+	0.929150i	$0.379460\pi$
$240$	0	0
$241$	3.70709e15	1.21877	0.609386	−	0.792874i	$-0.291416\pi$
$241$	3.70709e15	1.21877	0.609386	+	0.792874i	$0.291416\pi$
$242$	0	0
$243$	1.29385e15	0.403128
$244$	0	0
$245$	1.31248e14	0.0387714
$246$	0	0
$247$	−1.96113e15	−0.549510
$248$	0	0
$249$	2.97435e14	0.0790855
$250$	0	0
$251$	−1.69706e15	−0.428368	−0.214184	−	0.976793i	$-0.568709\pi$
$251$	−1.69706e15	−0.428368	−0.214184	+	0.976793i	$0.568709\pi$
$252$	0	0
$253$	1.04165e15	0.249711
$254$	0	0
$255$	2.92849e15	0.667011
$256$	0	0
$257$	−3.42768e14	−0.0742053	−0.0371027	−	0.999311i	$-0.511813\pi$
$257$	−3.42768e14	−0.0742053	−0.0371027	+	0.999311i	$0.511813\pi$
$258$	0	0
$259$	−6.46243e15	−1.33029
$260$	0	0
$261$	9.24342e14	0.180996
$262$	0	0
$263$	−4.63399e15	−0.863460	−0.431730	−	0.902003i	$-0.642097\pi$
$263$	−4.63399e15	−0.863460	−0.431730	+	0.902003i	$0.642097\pi$
$264$	0	0
$265$	−1.86384e15	−0.330605
$266$	0	0
$267$	−4.16134e15	−0.702927
$268$	0	0
$269$	−5.38103e15	−0.865915	−0.432957	−	0.901414i	$-0.642530\pi$
$269$	−5.38103e15	−0.865915	−0.432957	+	0.901414i	$0.642530\pi$
$270$	0	0
$271$	−8.17255e15	−1.25331	−0.626653	−	0.779298i	$-0.715576\pi$
$271$	−8.17255e15	−1.25331	−0.626653	+	0.779298i	$0.715576\pi$
$272$	0	0
$273$	−2.28934e15	−0.334698
$274$	0	0
$275$	4.02164e14	0.0560713
$276$	0	0
$277$	−7.61841e14	−0.101332	−0.0506659	−	0.998716i	$-0.516134\pi$
$277$	−7.61841e14	−0.101332	−0.0506659	+	0.998716i	$0.516134\pi$
$278$	0	0
$279$	2.48985e15	0.316043
$280$	0	0
$281$	−9.73851e15	−1.18005	−0.590027	−	0.807384i	$-0.700883\pi$
$281$	−9.73851e15	−1.18005	−0.590027	+	0.807384i	$0.700883\pi$
$282$	0	0
$283$	1.30296e16	1.50771	0.753857	−	0.657038i	$-0.228191\pi$
$283$	1.30296e16	1.50771	0.753857	+	0.657038i	$0.228191\pi$
$284$	0	0
$285$	−5.50459e15	−0.608461
$286$	0	0
$287$	1.52467e16	1.61044
$288$	0	0
$289$	1.78114e16	1.79830
$290$	0	0
$291$	−1.13615e16	−1.09681
$292$	0	0
$293$	−4.27472e15	−0.394701	−0.197351	−	0.980333i	$-0.563234\pi$
$293$	−4.27472e15	−0.394701	−0.197351	+	0.980333i	$0.563234\pi$
$294$	0	0
$295$	−6.56718e15	−0.580144
$296$	0	0
$297$	−3.56289e15	−0.301221
$298$	0	0
$299$	3.96295e15	0.320742
$300$	0	0
$301$	−2.55010e15	−0.197640
$302$	0	0
$303$	2.89033e15	0.214571
$304$	0	0
$305$	6.49224e15	0.461791
$306$	0	0
$307$	−1.84450e16	−1.25742	−0.628709	−	0.777640i	$-0.716417\pi$
$307$	−1.84450e16	−1.25742	−0.628709	+	0.777640i	$0.716417\pi$
$308$	0	0
$309$	−1.36262e16	−0.890523
$310$	0	0
$311$	2.21772e16	1.38984	0.694918	−	0.719089i	$-0.255441\pi$
$311$	2.21772e16	1.38984	0.694918	+	0.719089i	$0.255441\pi$
$312$	0	0
$313$	−2.02883e16	−1.21957	−0.609785	−	0.792567i	$-0.708744\pi$
$313$	−2.02883e16	−1.21957	−0.609785	+	0.792567i	$0.708744\pi$
$314$	0	0
$315$	1.65745e15	0.0955923
$316$	0	0
$317$	1.93347e16	1.07017	0.535085	−	0.844798i	$-0.320280\pi$
$317$	1.93347e16	1.07017	0.535085	+	0.844798i	$0.320280\pi$
$318$	0	0
$319$	−4.65763e15	−0.247472
$320$	0	0
$321$	−1.05542e16	−0.538446
$322$	0	0
$323$	−5.20968e16	−2.55267
$324$	0	0
$325$	1.53003e15	0.0720208
$326$	0	0
$327$	−2.81344e16	−1.27256
$328$	0	0
$329$	2.69731e16	1.17262
$330$	0	0
$331$	−4.61159e15	−0.192739	−0.0963693	−	0.995346i	$-0.530723\pi$
$331$	−4.61159e15	−0.192739	−0.0963693	+	0.995346i	$0.530723\pi$
$332$	0	0
$333$	−6.51081e15	−0.261666
$334$	0	0
$335$	1.60888e15	0.0621916
$336$	0	0
$337$	−3.91103e16	−1.45444	−0.727222	−	0.686402i	$-0.759189\pi$
$337$	−3.91103e16	−1.45444	−0.727222	+	0.686402i	$0.759189\pi$
$338$	0	0
$339$	1.52658e16	0.546287
$340$	0	0
$341$	−1.25460e16	−0.432119
$342$	0	0
$343$	2.87132e16	0.952071
$344$	0	0
$345$	1.11234e16	0.355151
$346$	0	0
$347$	−5.96375e15	−0.183391	−0.0916955	−	0.995787i	$-0.529229\pi$
$347$	−5.96375e15	−0.183391	−0.0916955	+	0.995787i	$0.529229\pi$
$348$	0	0
$349$	2.76636e15	0.0819490	0.0409745	−	0.999160i	$-0.486954\pi$
$349$	2.76636e15	0.0819490	0.0409745	+	0.999160i	$0.486954\pi$
$350$	0	0
$351$	−1.35550e16	−0.386904
$352$	0	0
$353$	1.98418e16	0.545816	0.272908	−	0.962040i	$-0.412015\pi$
$353$	1.98418e16	0.545816	0.272908	+	0.962040i	$0.412015\pi$
$354$	0	0
$355$	6.25598e15	0.165887
$356$	0	0
$357$	−6.08156e16	−1.55479
$358$	0	0
$359$	3.22798e16	0.795824	0.397912	−	0.917424i	$-0.369735\pi$
$359$	3.22798e16	0.795824	0.397912	+	0.917424i	$0.369735\pi$
$360$	0	0
$361$	5.58716e16	1.32860
$362$	0	0
$363$	−3.58106e16	−0.821522
$364$	0	0
$365$	8.67204e15	0.191963
$366$	0	0
$367$	4.42436e16	0.945194	0.472597	−	0.881279i	$-0.343317\pi$
$367$	4.42436e16	0.945194	0.472597	+	0.881279i	$0.343317\pi$
$368$	0	0
$369$	1.53608e16	0.316770
$370$	0	0
$371$	3.87060e16	0.770634
$372$	0	0
$373$	9.26267e16	1.78086	0.890428	−	0.455125i	$-0.150405\pi$
$373$	9.26267e16	1.78086	0.890428	+	0.455125i	$0.150405\pi$
$374$	0	0
$375$	4.29456e15	0.0797472
$376$	0	0
$377$	−1.77199e16	−0.317865
$378$	0	0
$379$	3.71415e16	0.643732	0.321866	−	0.946785i	$-0.395690\pi$
$379$	3.71415e16	0.643732	0.321866	+	0.946785i	$0.395690\pi$
$380$	0	0
$381$	1.31533e16	0.220304
$382$	0	0
$383$	1.22127e16	0.197706	0.0988532	−	0.995102i	$-0.468483\pi$
$383$	1.22127e16	0.197706	0.0988532	+	0.995102i	$0.468483\pi$
$384$	0	0
$385$	−8.35169e15	−0.130701
$386$	0	0
$387$	−2.56919e15	−0.0388754
$388$	0	0
$389$	7.53716e16	1.10290	0.551449	−	0.834209i	$-0.314075\pi$
$389$	7.53716e16	1.10290	0.551449	+	0.834209i	$0.314075\pi$
$390$	0	0
$391$	1.05275e17	1.48996
$392$	0	0
$393$	−5.77233e16	−0.790314
$394$	0	0
$395$	−2.50489e16	−0.331824
$396$	0	0
$397$	8.59809e16	1.10221	0.551104	−	0.834437i	$-0.314207\pi$
$397$	8.59809e16	1.10221	0.551104	+	0.834437i	$0.314207\pi$
$398$	0	0
$399$	1.14313e17	1.41831
$400$	0	0
$401$	−1.13598e17	−1.36437	−0.682186	−	0.731178i	$-0.738971\pi$
$401$	−1.13598e17	−1.36437	−0.682186	+	0.731178i	$0.738971\pi$
$402$	0	0
$403$	−4.77313e16	−0.555035
$404$	0	0
$405$	−2.99030e16	−0.336711
$406$	0	0
$407$	3.28071e16	0.357770
$408$	0	0
$409$	−3.50572e16	−0.370318	−0.185159	−	0.982709i	$-0.559280\pi$
$409$	−3.50572e16	−0.370318	−0.185159	+	0.982709i	$0.559280\pi$
$410$	0	0
$411$	−8.05594e16	−0.824412
$412$	0	0
$413$	1.36380e17	1.35230
$414$	0	0
$415$	4.12813e15	0.0396681
$416$	0	0
$417$	1.30914e17	1.21927
$418$	0	0
$419$	2.82535e15	0.0255083	0.0127541	−	0.999919i	$-0.495940\pi$
$419$	2.82535e15	0.0255083	0.0127541	+	0.999919i	$0.495940\pi$
$420$	0	0
$421$	8.17054e16	0.715183	0.357591	−	0.933878i	$-0.383598\pi$
$421$	8.17054e16	0.715183	0.357591	+	0.933878i	$0.383598\pi$
$422$	0	0
$423$	2.71750e16	0.230652
$424$	0	0
$425$	4.06448e16	0.334562
$426$	0	0
$427$	−1.34824e17	−1.07643
$428$	0	0
$429$	1.16220e16	0.0900139
$430$	0	0
$431$	−4.31353e16	−0.324139	−0.162069	−	0.986779i	$-0.551817\pi$
$431$	−4.31353e16	−0.324139	−0.162069	+	0.986779i	$0.551817\pi$
$432$	0	0
$433$	−6.25222e16	−0.455893	−0.227947	−	0.973674i	$-0.573201\pi$
$433$	−6.25222e16	−0.455893	−0.227947	+	0.973674i	$0.573201\pi$
$434$	0	0
$435$	−4.97371e16	−0.351966
$436$	0	0
$437$	−1.97881e17	−1.35917
$438$	0	0
$439$	7.25265e16	0.483590	0.241795	−	0.970327i	$-0.422264\pi$
$439$	7.25265e16	0.483590	0.241795	+	0.970327i	$0.422264\pi$
$440$	0	0
$441$	−2.74601e15	−0.0177767
$442$	0	0
$443$	2.76459e17	1.73783	0.868914	−	0.494962i	$-0.164818\pi$
$443$	2.76459e17	1.73783	0.868914	+	0.494962i	$0.164818\pi$
$444$	0	0
$445$	−5.77557e16	−0.352578
$446$	0	0
$447$	−1.61728e17	−0.958931
$448$	0	0
$449$	2.46120e17	1.41757	0.708786	−	0.705424i	$-0.249243\pi$
$449$	2.46120e17	1.41757	0.708786	+	0.705424i	$0.249243\pi$
$450$	0	0
$451$	−7.74012e16	−0.433112
$452$	0	0
$453$	2.24843e16	0.122248
$454$	0	0
$455$	−3.17740e16	−0.167879
$456$	0	0
$457$	−3.45983e17	−1.77664	−0.888319	−	0.459226i	$-0.848127\pi$
$457$	−3.45983e17	−1.77664	−0.888319	+	0.459226i	$0.848127\pi$
$458$	0	0
$459$	−3.60084e17	−1.79731
$460$	0	0
$461$	−1.04114e17	−0.505188	−0.252594	−	0.967572i	$-0.581284\pi$
$461$	−1.04114e17	−0.505188	−0.252594	+	0.967572i	$0.581284\pi$
$462$	0	0
$463$	1.70140e17	0.802656	0.401328	−	0.915934i	$-0.368549\pi$
$463$	1.70140e17	0.802656	0.401328	+	0.915934i	$0.368549\pi$
$464$	0	0
$465$	−1.33975e17	−0.614579
$466$	0	0
$467$	−1.84060e17	−0.821106	−0.410553	−	0.911837i	$-0.634664\pi$
$467$	−1.84060e17	−0.821106	−0.410553	+	0.911837i	$0.634664\pi$
$468$	0	0
$469$	−3.34114e16	−0.144968
$470$	0	0
$471$	2.77454e17	1.17100
$472$	0	0
$473$	1.29458e16	0.0531534
$474$	0	0
$475$	−7.63987e16	−0.305195
$476$	0	0
$477$	3.89958e16	0.151582
$478$	0	0
$479$	2.13215e17	0.806562	0.403281	−	0.915076i	$-0.367870\pi$
$479$	2.13215e17	0.806562	0.403281	+	0.915076i	$0.367870\pi$
$480$	0	0
$481$	1.24814e17	0.459538
$482$	0	0
$483$	−2.30998e17	−0.827851
$484$	0	0
$485$	−1.57688e17	−0.550145
$486$	0	0
$487$	3.74978e17	1.27370	0.636851	−	0.770987i	$-0.280237\pi$
$487$	3.74978e17	1.27370	0.636851	+	0.770987i	$0.280237\pi$
$488$	0	0
$489$	−1.50475e17	−0.497687
$490$	0	0
$491$	−4.11003e17	−1.32378	−0.661889	−	0.749602i	$-0.730245\pi$
$491$	−4.11003e17	−1.32378	−0.661889	+	0.749602i	$0.730245\pi$
$492$	0	0
$493$	−4.70725e17	−1.47660
$494$	0	0
$495$	−8.41421e15	−0.0257087
$496$	0	0
$497$	−1.29917e17	−0.386678
$498$	0	0
$499$	−1.73839e17	−0.504072	−0.252036	−	0.967718i	$-0.581100\pi$
$499$	−1.73839e17	−0.504072	−0.252036	+	0.967718i	$0.581100\pi$
$500$	0	0
$501$	1.76941e17	0.499899
$502$	0	0
$503$	3.28753e17	0.905062	0.452531	−	0.891749i	$-0.350521\pi$
$503$	3.28753e17	0.905062	0.452531	+	0.891749i	$0.350521\pi$
$504$	0	0
$505$	4.01152e16	0.107625
$506$	0	0
$507$	−2.96759e17	−0.775983
$508$	0	0
$509$	6.89978e16	0.175861	0.0879305	−	0.996127i	$-0.471975\pi$
$509$	6.89978e16	0.175861	0.0879305	+	0.996127i	$0.471975\pi$
$510$	0	0
$511$	−1.80091e17	−0.447463
$512$	0	0
$513$	6.76839e17	1.63954
$514$	0	0
$515$	−1.89120e17	−0.446673
$516$	0	0
$517$	−1.36931e17	−0.315365
$518$	0	0
$519$	3.74069e17	0.840164
$520$	0	0
$521$	8.38335e17	1.83642	0.918211	−	0.396092i	$-0.129634\pi$
$521$	8.38335e17	1.83642	0.918211	+	0.396092i	$0.129634\pi$
$522$	0	0
$523$	1.91687e17	0.409573	0.204786	−	0.978807i	$-0.434350\pi$
$523$	1.91687e17	0.409573	0.204786	+	0.978807i	$0.434350\pi$
$524$	0	0
$525$	−8.91846e16	−0.185889
$526$	0	0
$527$	−1.26797e18	−2.57834
$528$	0	0
$529$	−1.04168e17	−0.206668
$530$	0	0
$531$	1.37401e17	0.265996
$532$	0	0
$533$	−2.94472e17	−0.556311
$534$	0	0
$535$	−1.46483e17	−0.270076
$536$	0	0
$537$	−5.59183e17	−1.00628
$538$	0	0
$539$	1.38368e16	0.0243056
$540$	0	0
$541$	−8.65938e17	−1.48492	−0.742462	−	0.669888i	$-0.766342\pi$
$541$	−8.65938e17	−1.48492	−0.742462	+	0.669888i	$0.766342\pi$
$542$	0	0
$543$	6.46087e17	1.08166
$544$	0	0
$545$	−3.90481e17	−0.638296
$546$	0	0
$547$	−1.16237e18	−1.85536	−0.927678	−	0.373382i	$-0.878198\pi$
$547$	−1.16237e18	−1.85536	−0.927678	+	0.373382i	$0.878198\pi$
$548$	0	0
$549$	−1.35833e17	−0.211731
$550$	0	0
$551$	8.84806e17	1.34698
$552$	0	0
$553$	5.20187e17	0.773475
$554$	0	0
$555$	3.50335e17	0.508837
$556$	0	0
$557$	−1.02746e17	−0.145783	−0.0728917	−	0.997340i	$-0.523223\pi$
$557$	−1.02746e17	−0.145783	−0.0728917	+	0.997340i	$0.523223\pi$
$558$	0	0
$559$	4.92522e16	0.0682730
$560$	0	0
$561$	3.08736e17	0.418146
$562$	0	0
$563$	−7.17343e17	−0.949341	−0.474671	−	0.880164i	$-0.657433\pi$
$563$	−7.17343e17	−0.949341	−0.474671	+	0.880164i	$0.657433\pi$
$564$	0	0
$565$	2.11875e17	0.274009
$566$	0	0
$567$	6.20992e17	0.784867
$568$	0	0
$569$	−1.25353e17	−0.154848	−0.0774238	−	0.996998i	$-0.524669\pi$
$569$	−1.25353e17	−0.154848	−0.0774238	+	0.996998i	$0.524669\pi$
$570$	0	0
$571$	1.00269e17	0.121069	0.0605344	−	0.998166i	$-0.480720\pi$
$571$	1.00269e17	0.121069	0.0605344	+	0.998166i	$0.480720\pi$
$572$	0	0
$573$	1.03698e18	1.22395
$574$	0	0
$575$	1.54383e17	0.178138
$576$	0	0
$577$	−1.36098e18	−1.53535	−0.767675	−	0.640839i	$-0.778587\pi$
$577$	−1.36098e18	−1.53535	−0.767675	+	0.640839i	$0.778587\pi$
$578$	0	0
$579$	1.00058e18	1.10368
$580$	0	0
$581$	−8.57285e16	−0.0924656
$582$	0	0
$583$	−1.96495e17	−0.207255
$584$	0	0
$585$	−3.20118e16	−0.0330215
$586$	0	0
$587$	6.87785e16	0.0693913	0.0346956	−	0.999398i	$-0.488954\pi$
$587$	6.87785e16	0.0693913	0.0346956	+	0.999398i	$0.488954\pi$
$588$	0	0
$589$	2.38336e18	2.35201
$590$	0	0
$591$	−5.30185e16	−0.0511810
$592$	0	0
$593$	−3.46338e17	−0.327073	−0.163536	−	0.986537i	$-0.552290\pi$
$593$	−3.46338e17	−0.327073	−0.163536	+	0.986537i	$0.552290\pi$
$594$	0	0
$595$	−8.44066e17	−0.779859
$596$	0	0
$597$	−5.56338e17	−0.502928
$598$	0	0
$599$	1.08982e18	0.964004	0.482002	−	0.876170i	$-0.339910\pi$
$599$	1.08982e18	0.964004	0.482002	+	0.876170i	$0.339910\pi$
$600$	0	0
$601$	2.73058e17	0.236358	0.118179	−	0.992992i	$-0.462294\pi$
$601$	2.73058e17	0.236358	0.118179	+	0.992992i	$0.462294\pi$
$602$	0	0
$603$	−3.36615e16	−0.0285148
$604$	0	0
$605$	−4.97019e17	−0.412063
$606$	0	0
$607$	1.19160e18	0.966951	0.483476	−	0.875358i	$-0.339374\pi$
$607$	1.19160e18	0.966951	0.483476	+	0.875358i	$0.339374\pi$
$608$	0	0
$609$	1.03289e18	0.820426
$610$	0	0
$611$	−5.20954e17	−0.405072
$612$	0	0
$613$	3.63023e17	0.276338	0.138169	−	0.990409i	$-0.455878\pi$
$613$	3.63023e17	0.276338	0.138169	+	0.990409i	$0.455878\pi$
$614$	0	0
$615$	−8.26539e17	−0.615992
$616$	0	0
$617$	6.06720e17	0.442726	0.221363	−	0.975192i	$-0.428949\pi$
$617$	6.06720e17	0.442726	0.221363	+	0.975192i	$0.428949\pi$
$618$	0	0
$619$	2.29300e18	1.63838	0.819190	−	0.573523i	$-0.194424\pi$
$619$	2.29300e18	1.63838	0.819190	+	0.573523i	$0.194424\pi$
$620$	0	0
$621$	−1.36772e18	−0.956979
$622$	0	0
$623$	1.19941e18	0.821852
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	0	0
$627$	−5.80320e17	−0.381442
$628$	0	0
$629$	3.31566e18	2.13472
$630$	0	0
$631$	−3.33398e17	−0.210268	−0.105134	−	0.994458i	$-0.533527\pi$
$631$	−3.33398e17	−0.210268	−0.105134	+	0.994458i	$0.533527\pi$
$632$	0	0
$633$	1.37883e18	0.851895
$634$	0	0
$635$	1.82556e17	0.110501
$636$	0	0
$637$	5.26418e16	0.0312194
$638$	0	0
$639$	−1.30890e17	−0.0760589
$640$	0	0
$641$	−8.91188e17	−0.507449	−0.253724	−	0.967277i	$-0.581656\pi$
$641$	−8.91188e17	−0.507449	−0.253724	+	0.967277i	$0.581656\pi$
$642$	0	0
$643$	−5.58943e17	−0.311886	−0.155943	−	0.987766i	$-0.549842\pi$
$643$	−5.58943e17	−0.311886	−0.155943	+	0.987766i	$0.549842\pi$
$644$	0	0
$645$	1.38243e17	0.0755973
$646$	0	0
$647$	−1.62698e18	−0.871973	−0.435987	−	0.899953i	$-0.643601\pi$
$647$	−1.62698e18	−0.871973	−0.435987	+	0.899953i	$0.643601\pi$
$648$	0	0
$649$	−6.92344e17	−0.363690
$650$	0	0
$651$	2.78223e18	1.43257
$652$	0	0
$653$	−6.71206e17	−0.338782	−0.169391	−	0.985549i	$-0.554180\pi$
$653$	−6.71206e17	−0.338782	−0.169391	+	0.985549i	$0.554180\pi$
$654$	0	0
$655$	−8.01147e17	−0.396409
$656$	0	0
$657$	−1.81439e17	−0.0880150
$658$	0	0
$659$	1.08505e18	0.516053	0.258027	−	0.966138i	$-0.416928\pi$
$659$	1.08505e18	0.516053	0.258027	+	0.966138i	$0.416928\pi$
$660$	0	0
$661$	2.34214e18	1.09220	0.546100	−	0.837720i	$-0.316112\pi$
$661$	2.34214e18	1.09220	0.546100	+	0.837720i	$0.316112\pi$
$662$	0	0
$663$	1.17458e18	0.537089
$664$	0	0
$665$	1.58656e18	0.711404
$666$	0	0
$667$	−1.78797e18	−0.786217
$668$	0	0
$669$	1.39573e18	0.601909
$670$	0	0
$671$	6.84444e17	0.289495
$672$	0	0
$673$	7.92495e17	0.328775	0.164387	−	0.986396i	$-0.447435\pi$
$673$	7.92495e17	0.328775	0.164387	+	0.986396i	$0.447435\pi$
$674$	0	0
$675$	−5.28055e17	−0.214884
$676$	0	0
$677$	−2.35105e18	−0.938503	−0.469251	−	0.883065i	$-0.655476\pi$
$677$	−2.35105e18	−0.938503	−0.469251	+	0.883065i	$0.655476\pi$
$678$	0	0
$679$	3.27469e18	1.28238
$680$	0	0
$681$	3.73711e17	0.143575
$682$	0	0
$683$	3.22912e17	0.121717	0.0608583	−	0.998146i	$-0.480616\pi$
$683$	3.22912e17	0.121717	0.0608583	+	0.998146i	$0.480616\pi$
$684$	0	0
$685$	−1.11809e18	−0.413513
$686$	0	0
$687$	−2.17052e18	−0.787670
$688$	0	0
$689$	−7.47563e17	−0.266209
$690$	0	0
$691$	−3.36976e18	−1.17758	−0.588792	−	0.808284i	$-0.700396\pi$
$691$	−3.36976e18	−1.17758	−0.588792	+	0.808284i	$0.700396\pi$
$692$	0	0
$693$	1.74737e17	0.0599265
$694$	0	0
$695$	1.81696e18	0.611568
$696$	0	0
$697$	−7.82257e18	−2.58426
$698$	0	0
$699$	−1.26901e18	−0.411494
$700$	0	0
$701$	−5.38800e18	−1.71499	−0.857493	−	0.514495i	$-0.827979\pi$
$701$	−5.38800e18	−1.71499	−0.857493	+	0.514495i	$0.827979\pi$
$702$	0	0
$703$	−6.23233e18	−1.94734
$704$	0	0
$705$	−1.46224e18	−0.448528
$706$	0	0
$707$	−8.33067e17	−0.250873
$708$	0	0
$709$	−1.40376e18	−0.415044	−0.207522	−	0.978230i	$-0.566540\pi$
$709$	−1.40376e18	−0.415044	−0.207522	+	0.978230i	$0.566540\pi$
$710$	0	0
$711$	5.24082e17	0.152141
$712$	0	0
$713$	−4.81617e18	−1.37284
$714$	0	0
$715$	1.61303e17	0.0451496
$716$	0	0
$717$	2.39843e18	0.659254
$718$	0	0
$719$	1.38614e18	0.374170	0.187085	−	0.982344i	$-0.440096\pi$
$719$	1.38614e18	0.374170	0.187085	+	0.982344i	$0.440096\pi$
$720$	0	0
$721$	3.92742e18	1.04119
$722$	0	0
$723$	4.17342e18	1.08666
$724$	0	0
$725$	−6.90307e17	−0.176541
$726$	0	0
$727$	−4.20824e18	−1.05712	−0.528562	−	0.848894i	$-0.677269\pi$
$727$	−4.20824e18	−1.05712	−0.528562	+	0.848894i	$0.677269\pi$
$728$	0	0
$729$	4.50781e18	1.11234
$730$	0	0
$731$	1.30837e18	0.317152
$732$	0	0
$733$	7.61043e17	0.181231	0.0906156	−	0.995886i	$-0.471117\pi$
$733$	7.61043e17	0.181231	0.0906156	+	0.995886i	$0.471117\pi$
$734$	0	0
$735$	1.47758e17	0.0345686
$736$	0	0
$737$	1.69616e17	0.0389877
$738$	0	0
$739$	−6.25215e18	−1.41202	−0.706010	−	0.708202i	$-0.749507\pi$
$739$	−6.25215e18	−1.41202	−0.706010	+	0.708202i	$0.749507\pi$
$740$	0	0
$741$	−2.20783e18	−0.489944
$742$	0	0
$743$	−5.60504e18	−1.22223	−0.611113	−	0.791543i	$-0.709278\pi$
$743$	−5.60504e18	−1.22223	−0.611113	+	0.791543i	$0.709278\pi$
$744$	0	0
$745$	−2.24464e18	−0.480985
$746$	0	0
$747$	−8.63703e16	−0.0181878
$748$	0	0
$749$	3.04199e18	0.629543
$750$	0	0
$751$	−4.79388e18	−0.975053	−0.487526	−	0.873108i	$-0.662101\pi$
$751$	−4.79388e18	−0.975053	−0.487526	+	0.873108i	$0.662101\pi$
$752$	0	0
$753$	−1.91053e18	−0.381933
$754$	0	0
$755$	3.12061e17	0.0613175
$756$	0	0
$757$	2.47656e18	0.478328	0.239164	−	0.970979i	$-0.423127\pi$
$757$	2.47656e18	0.478328	0.239164	+	0.970979i	$0.423127\pi$
$758$	0	0
$759$	1.17268e18	0.222643
$760$	0	0
$761$	1.02209e19	1.90761	0.953804	−	0.300430i	$-0.0971301\pi$
$761$	1.02209e19	1.90761	0.953804	+	0.300430i	$0.0971301\pi$
$762$	0	0
$763$	8.10906e18	1.48786
$764$	0	0
$765$	−8.50385e17	−0.153397
$766$	0	0
$767$	−2.63402e18	−0.467142
$768$	0	0
$769$	−9.71765e18	−1.69449	−0.847247	−	0.531199i	$-0.821742\pi$
$769$	−9.71765e18	−1.69449	−0.847247	+	0.531199i	$0.821742\pi$
$770$	0	0
$771$	−3.85886e17	−0.0661616
$772$	0	0
$773$	6.33996e18	1.06886	0.534429	−	0.845213i	$-0.320527\pi$
$773$	6.33996e18	1.06886	0.534429	+	0.845213i	$0.320527\pi$
$774$	0	0
$775$	−1.85945e18	−0.308264
$776$	0	0
$777$	−7.27536e18	−1.18609
$778$	0	0
$779$	1.47038e19	2.35742
$780$	0	0
$781$	6.59536e17	0.103994
$782$	0	0
$783$	6.11563e18	0.948396
$784$	0	0
$785$	3.85082e18	0.587354
$786$	0	0
$787$	−6.11756e17	−0.0917789	−0.0458894	−	0.998947i	$-0.514612\pi$
$787$	−6.11756e17	−0.0917789	−0.0458894	+	0.998947i	$0.514612\pi$
$788$	0	0
$789$	−5.21691e18	−0.769862
$790$	0	0
$791$	−4.39999e18	−0.638711
$792$	0	0
$793$	2.60396e18	0.371843
$794$	0	0
$795$	−2.09829e18	−0.294768
$796$	0	0
$797$	1.19147e18	0.164665	0.0823327	−	0.996605i	$-0.473763\pi$
$797$	1.19147e18	0.164665	0.0823327	+	0.996605i	$0.473763\pi$
$798$	0	0
$799$	−1.38390e19	−1.88170
$800$	0	0
$801$	1.20839e18	0.161657
$802$	0	0
$803$	9.14249e17	0.120341
$804$	0	0
$805$	−3.20605e18	−0.415237
$806$	0	0
$807$	−6.05792e18	−0.772051
$808$	0	0
$809$	7.74613e18	0.971447	0.485724	−	0.874112i	$-0.338556\pi$
$809$	7.74613e18	0.971447	0.485724	+	0.874112i	$0.338556\pi$
$810$	0	0
$811$	4.31453e18	0.532474	0.266237	−	0.963908i	$-0.414220\pi$
$811$	4.31453e18	0.532474	0.266237	+	0.963908i	$0.414220\pi$
$812$	0	0
$813$	−9.20061e18	−1.11745
$814$	0	0
$815$	−2.08845e18	−0.249632
$816$	0	0
$817$	−2.45930e18	−0.289313
$818$	0	0
$819$	6.64786e17	0.0769726
$820$	0	0
$821$	−1.32454e19	−1.50950	−0.754752	−	0.656011i	$-0.772243\pi$
$821$	−1.32454e19	−1.50950	−0.754752	+	0.656011i	$0.772243\pi$
$822$	0	0
$823$	1.59982e19	1.79461	0.897307	−	0.441407i	$-0.145520\pi$
$823$	1.59982e19	1.79461	0.897307	+	0.441407i	$0.145520\pi$
$824$	0	0
$825$	4.52754e17	0.0499932
$826$	0	0
$827$	7.72194e18	0.839345	0.419672	−	0.907676i	$-0.362145\pi$
$827$	7.72194e18	0.839345	0.419672	+	0.907676i	$0.362145\pi$
$828$	0	0
$829$	−8.75341e18	−0.936640	−0.468320	−	0.883559i	$-0.655141\pi$
$829$	−8.75341e18	−0.936640	−0.468320	+	0.883559i	$0.655141\pi$
$830$	0	0
$831$	−8.57676e17	−0.0903476
$832$	0	0
$833$	1.39842e18	0.145025
$834$	0	0
$835$	2.45578e18	0.250742
$836$	0	0
$837$	1.64734e19	1.65603
$838$	0	0
$839$	−1.14368e19	−1.13201	−0.566007	−	0.824400i	$-0.691513\pi$
$839$	−1.14368e19	−1.13201	−0.566007	+	0.824400i	$0.691513\pi$
$840$	0	0
$841$	−2.26589e18	−0.220834
$842$	0	0
$843$	−1.09636e19	−1.05214
$844$	0	0
$845$	−4.11875e18	−0.389221
$846$	0	0
$847$	1.03215e19	0.960511
$848$	0	0
$849$	1.46686e19	1.34428
$850$	0	0
$851$	1.25940e19	1.13663
$852$	0	0
$853$	−9.31629e17	−0.0828084	−0.0414042	−	0.999142i	$-0.513183\pi$
$853$	−9.31629e17	−0.0828084	−0.0414042	+	0.999142i	$0.513183\pi$
$854$	0	0
$855$	1.59844e18	0.139932
$856$	0	0
$857$	9.56376e18	0.824619	0.412309	−	0.911044i	$-0.364722\pi$
$857$	9.56376e18	0.824619	0.412309	+	0.911044i	$0.364722\pi$
$858$	0	0
$859$	−7.17114e18	−0.609022	−0.304511	−	0.952509i	$-0.598493\pi$
$859$	−7.17114e18	−0.609022	−0.304511	+	0.952509i	$0.598493\pi$
$860$	0	0
$861$	1.71646e19	1.43587
$862$	0	0
$863$	−1.34010e19	−1.10425	−0.552125	−	0.833761i	$-0.686183\pi$
$863$	−1.34010e19	−1.10425	−0.552125	+	0.833761i	$0.686183\pi$
$864$	0	0
$865$	5.19174e18	0.421414
$866$	0	0
$867$	2.00520e19	1.60337
$868$	0	0
$869$	−2.64078e18	−0.208019
$870$	0	0
$871$	6.45302e17	0.0500778
$872$	0	0
$873$	3.29920e18	0.252241
$874$	0	0
$875$	−1.23780e18	−0.0932393
$876$	0	0
$877$	−8.23622e18	−0.611266	−0.305633	−	0.952149i	$-0.598868\pi$
$877$	−8.23622e18	−0.611266	−0.305633	+	0.952149i	$0.598868\pi$
$878$	0	0
$879$	−4.81246e18	−0.351916
$880$	0	0
$881$	7.58214e18	0.546322	0.273161	−	0.961968i	$-0.411931\pi$
$881$	7.58214e18	0.546322	0.273161	+	0.961968i	$0.411931\pi$
$882$	0	0
$883$	1.22388e19	0.868946	0.434473	−	0.900685i	$-0.356935\pi$
$883$	1.22388e19	0.868946	0.434473	+	0.900685i	$0.356935\pi$
$884$	0	0
$885$	−7.39328e18	−0.517257
$886$	0	0
$887$	−2.48370e18	−0.171236	−0.0856182	−	0.996328i	$-0.527287\pi$
$887$	−2.48370e18	−0.171236	−0.0856182	+	0.996328i	$0.527287\pi$
$888$	0	0
$889$	−3.79112e18	−0.257576
$890$	0	0
$891$	−3.15252e18	−0.211083
$892$	0	0
$893$	2.60127e19	1.71653
$894$	0	0
$895$	−7.76095e18	−0.504737
$896$	0	0
$897$	4.46146e18	0.285974
$898$	0	0
$899$	2.15350e19	1.36053
$900$	0	0
$901$	−1.98588e19	−1.23664
$902$	0	0
$903$	−2.87088e18	−0.176216
$904$	0	0
$905$	8.96710e18	0.542546
$906$	0	0
$907$	6.93837e18	0.413819	0.206909	−	0.978360i	$-0.433659\pi$
$907$	6.93837e18	0.413819	0.206909	+	0.978360i	$0.433659\pi$
$908$	0	0
$909$	−8.39303e17	−0.0493462
$910$	0	0
$911$	4.63486e18	0.268638	0.134319	−	0.990938i	$-0.457115\pi$
$911$	4.63486e18	0.268638	0.134319	+	0.990938i	$0.457115\pi$
$912$	0	0
$913$	4.35208e17	0.0248678
$914$	0	0
$915$	7.30893e18	0.411734
$916$	0	0
$917$	1.66373e19	0.924023
$918$	0	0
$919$	4.92534e18	0.269703	0.134851	−	0.990866i	$-0.456944\pi$
$919$	4.92534e18	0.269703	0.134851	+	0.990866i	$0.456944\pi$
$920$	0	0
$921$	−2.07653e19	−1.12112
$922$	0	0
$923$	2.50920e18	0.133575
$924$	0	0
$925$	4.86233e18	0.255225
$926$	0	0
$927$	3.95683e18	0.204799
$928$	0	0
$929$	−1.91203e19	−0.975870	−0.487935	−	0.872880i	$-0.662250\pi$
$929$	−1.91203e19	−0.975870	−0.487935	+	0.872880i	$0.662250\pi$
$930$	0	0
$931$	−2.62855e18	−0.132295
$932$	0	0
$933$	2.49669e19	1.23918
$934$	0	0
$935$	4.28497e18	0.209736
$936$	0	0
$937$	4.30002e18	0.207569	0.103785	−	0.994600i	$-0.466905\pi$
$937$	4.30002e18	0.207569	0.103785	+	0.994600i	$0.466905\pi$
$938$	0	0
$939$	−2.28404e19	−1.08737
$940$	0	0
$941$	5.18107e18	0.243269	0.121635	−	0.992575i	$-0.461186\pi$
$941$	5.18107e18	0.243269	0.121635	+	0.992575i	$0.461186\pi$
$942$	0	0
$943$	−2.97128e19	−1.37600
$944$	0	0
$945$	1.09661e19	0.500892
$946$	0	0
$947$	−1.94885e19	−0.878020	−0.439010	−	0.898482i	$-0.644671\pi$
$947$	−1.94885e19	−0.878020	−0.439010	+	0.898482i	$0.644671\pi$
$948$	0	0
$949$	3.47825e18	0.154572
$950$	0	0
$951$	2.17669e19	0.954165
$952$	0	0
$953$	−2.96510e19	−1.28214	−0.641070	−	0.767483i	$-0.721509\pi$
$953$	−2.96510e19	−1.28214	−0.641070	+	0.767483i	$0.721509\pi$
$954$	0	0
$955$	1.43923e19	0.613916
$956$	0	0
$957$	−5.24353e18	−0.220646
$958$	0	0
$959$	2.32193e19	0.963890
$960$	0	0
$961$	3.35903e19	1.37566
$962$	0	0
$963$	3.06476e18	0.123830
$964$	0	0
$965$	1.38872e19	0.553587
$966$	0	0
$967$	−1.25349e18	−0.0493004	−0.0246502	−	0.999696i	$-0.507847\pi$
$967$	−1.25349e18	−0.0493004	−0.0246502	+	0.999696i	$0.507847\pi$
$968$	0	0
$969$	−5.86502e19	−2.27596
$970$	0	0
$971$	2.17263e19	0.831879	0.415940	−	0.909392i	$-0.363453\pi$
$971$	2.17263e19	0.831879	0.415940	+	0.909392i	$0.363453\pi$
$972$	0	0
$973$	−3.77327e19	−1.42555
$974$	0	0
$975$	1.72250e18	0.0642139
$976$	0	0
$977$	−2.56555e19	−0.943768	−0.471884	−	0.881661i	$-0.656426\pi$
$977$	−2.56555e19	−0.943768	−0.471884	+	0.881661i	$0.656426\pi$
$978$	0	0
$979$	−6.08889e18	−0.221030
$980$	0	0
$981$	8.16977e18	0.292659
$982$	0	0
$983$	−1.02856e19	−0.363608	−0.181804	−	0.983335i	$-0.558194\pi$
$983$	−1.02856e19	−0.363608	−0.181804	+	0.983335i	$0.558194\pi$
$984$	0	0
$985$	−7.35849e17	−0.0256716
$986$	0	0
$987$	3.03661e19	1.04551
$988$	0	0
$989$	4.96963e18	0.168868
$990$	0	0
$991$	−3.91892e19	−1.31428	−0.657140	−	0.753769i	$-0.728234\pi$
$991$	−3.91892e19	−1.31428	−0.657140	+	0.753769i	$0.728234\pi$
$992$	0	0
$993$	−5.19170e18	−0.171846
$994$	0	0
$995$	−7.72147e18	−0.252261
$996$	0	0
$997$	1.60895e19	0.518829	0.259414	−	0.965766i	$-0.416470\pi$
$997$	1.60895e19	0.518829	0.259414	+	0.965766i	$0.416470\pi$
$998$	0	0
$999$	−4.30768e19	−1.37110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.14.a.g.1.3		3
4.3	odd	2		5.14.a.b.1.1	✓	3
12.11	even	2		45.14.a.e.1.3		3
20.3	even	4		25.14.b.b.24.5		6
20.7	even	4		25.14.b.b.24.2		6
20.19	odd	2		25.14.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.14.a.b.1.1	✓	3	4.3	odd	2
25.14.a.b.1.3		3	20.19	odd	2
25.14.b.b.24.2		6	20.7	even	4
25.14.b.b.24.5		6	20.3	even	4
45.14.a.e.1.3		3	12.11	even	2
80.14.a.g.1.3		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)

\(f(q)\)	\(=\)	\(q+1125.79 q^{3} +15625.0 q^{5} -324482. q^{7} -326912. q^{9} +1.64726e6 q^{11} +6.26700e6 q^{13} +1.75905e7 q^{15} +1.66481e8 q^{17} -3.12929e8 q^{19} -3.65300e8 q^{21} +6.32351e8 q^{23} +2.44141e8 q^{25} -2.16291e9 q^{27} -2.82750e9 q^{29} -7.61629e9 q^{31} +1.85448e9 q^{33} -5.07004e9 q^{35} +1.99161e10 q^{37} +7.05535e9 q^{39} -4.69877e10 q^{41} +7.85897e9 q^{43} -5.10799e9 q^{45} -8.31265e10 q^{47} +8.39984e9 q^{49} +1.87423e11 q^{51} -1.19285e11 q^{53} +2.57385e10 q^{55} -3.52294e11 q^{57} -4.20299e11 q^{59} +4.15504e11 q^{61} +1.06077e11 q^{63} +9.79220e10 q^{65} +1.02968e11 q^{67} +7.11897e11 q^{69} +4.00383e11 q^{71} +5.55011e11 q^{73} +2.74852e11 q^{75} -5.34508e11 q^{77} -1.60313e12 q^{79} -1.91379e12 q^{81} +2.64201e11 q^{83} +2.60127e12 q^{85} -3.18318e12 q^{87} -3.69637e12 q^{89} -2.03353e12 q^{91} -8.57437e12 q^{93} -4.88952e12 q^{95} -1.00920e13 q^{97} -5.38510e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 416 q^{3} + 46875 q^{5} - 448292 q^{7} + 1286119 q^{9} + 6604004 q^{11} - 33501974 q^{13} - 6500000 q^{15} + 83129542 q^{17} - 97491100 q^{19} + 438200736 q^{21} - 316255836 q^{23} + 732421875 q^{25}+ \cdots - 126787366508 q^{99}+O(q^{100}) \) 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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