LMFDB - Embedded newform 80.14.a.g.1.1

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			1.1
Root			$-4.21238$ 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)

$f(q)$	$=$	$q-2114.98 q^{3} +15625.0 q^{5} -325303. q^{7} +2.87881e6 q^{9} +1.61316e6 q^{11} -3.19654e7 q^{13} -3.30465e7 q^{15} +3.82330e6 q^{17} +1.98761e8 q^{19} +6.88008e8 q^{21} +1.86577e8 q^{23} +2.44141e8 q^{25} -2.71667e9 q^{27} +2.45694e9 q^{29} +9.66435e8 q^{31} -3.41179e9 q^{33} -5.08285e9 q^{35} +2.20805e10 q^{37} +6.76061e10 q^{39} +4.05651e10 q^{41} -2.28510e10 q^{43} +4.49814e10 q^{45} +7.97391e10 q^{47} +8.93279e9 q^{49} -8.08620e9 q^{51} -2.25113e11 q^{53} +2.52056e10 q^{55} -4.20376e11 q^{57} -7.96680e10 q^{59} +4.91133e11 q^{61} -9.36485e11 q^{63} -4.99459e11 q^{65} -2.25405e11 q^{67} -3.94606e11 q^{69} +6.50849e11 q^{71} -1.03710e11 q^{73} -5.16352e11 q^{75} -5.24764e11 q^{77} -2.08051e12 q^{79} +1.15594e12 q^{81} -3.39017e12 q^{83} +5.97390e10 q^{85} -5.19638e12 q^{87} -7.20835e12 q^{89} +1.03984e13 q^{91} -2.04399e12 q^{93} +3.10565e12 q^{95} +7.00185e12 q^{97} +4.64397e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2114.98	−1.67501	−0.837506	−	0.546428i	$-0.815987\pi$
$3$	−2114.98	−1.67501	−0.837506	+	0.546428i	$0.815987\pi$
$4$	0	0
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	−325303.	−1.04508	−0.522541	−	0.852614i	$-0.675016\pi$
$7$	−325303.	−1.04508	−0.522541	+	0.852614i	$0.675016\pi$
$8$	0	0
$9$	2.87881e6	1.80566
$10$	0	0
$11$	1.61316e6	0.274551	0.137276	−	0.990533i	$-0.456165\pi$
$11$	1.61316e6	0.274551	0.137276	+	0.990533i	$0.456165\pi$
$12$	0	0
$13$	−3.19654e7	−1.83674	−0.918371	−	0.395720i	$-0.870495\pi$
$13$	−3.19654e7	−1.83674	−0.918371	+	0.395720i	$0.870495\pi$
$14$	0	0
$15$	−3.30465e7	−0.749088
$16$	0	0
$17$	3.82330e6	0.0384167	0.0192084	−	0.999816i	$-0.493885\pi$
$17$	3.82330e6	0.0384167	0.0192084	+	0.999816i	$0.493885\pi$
$18$	0	0
$19$	1.98761e8	0.969245	0.484622	−	0.874723i	$-0.338957\pi$
$19$	1.98761e8	0.969245	0.484622	+	0.874723i	$0.338957\pi$
$20$	0	0
$21$	6.88008e8	1.75052
$22$	0	0
$23$	1.86577e8	0.262801	0.131400	−	0.991329i	$-0.458053\pi$
$23$	1.86577e8	0.262801	0.131400	+	0.991329i	$0.458053\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	0	0
$27$	−2.71667e9	−1.34950
$28$	0	0
$29$	2.45694e9	0.767022	0.383511	−	0.923536i	$-0.374715\pi$
$29$	2.45694e9	0.767022	0.383511	+	0.923536i	$0.374715\pi$
$30$	0	0
$31$	9.66435e8	0.195579	0.0977893	−	0.995207i	$-0.468823\pi$
$31$	9.66435e8	0.195579	0.0977893	+	0.995207i	$0.468823\pi$
$32$	0	0
$33$	−3.41179e9	−0.459877
$34$	0	0
$35$	−5.08285e9	−0.467375
$36$	0	0
$37$	2.20805e10	1.41481	0.707403	−	0.706810i	$-0.249867\pi$
$37$	2.20805e10	1.41481	0.707403	+	0.706810i	$0.249867\pi$
$38$	0	0
$39$	6.76061e10	3.07656
$40$	0	0
$41$	4.05651e10	1.33370	0.666850	−	0.745192i	$-0.267642\pi$
$41$	4.05651e10	1.33370	0.666850	+	0.745192i	$0.267642\pi$
$42$	0	0
$43$	−2.28510e10	−0.551264	−0.275632	−	0.961263i	$-0.588887\pi$
$43$	−2.28510e10	−0.551264	−0.275632	+	0.961263i	$0.588887\pi$
$44$	0	0
$45$	4.49814e10	0.807518
$46$	0	0
$47$	7.97391e10	1.07903	0.539517	−	0.841975i	$-0.318607\pi$
$47$	7.97391e10	1.07903	0.539517	+	0.841975i	$0.318607\pi$
$48$	0	0
$49$	8.93279e9	0.0921961
$50$	0	0
$51$	−8.08620e9	−0.0643485
$52$	0	0
$53$	−2.25113e11	−1.39511	−0.697553	−	0.716534i	$-0.745728\pi$
$53$	−2.25113e11	−1.39511	−0.697553	+	0.716534i	$0.745728\pi$
$54$	0	0
$55$	2.52056e10	0.122783
$56$	0	0
$57$	−4.20376e11	−1.62350
$58$	0	0
$59$	−7.96680e10	−0.245893	−0.122946	−	0.992413i	$-0.539234\pi$
$59$	−7.96680e10	−0.245893	−0.122946	+	0.992413i	$0.539234\pi$
$60$	0	0
$61$	4.91133e11	1.22055	0.610275	−	0.792190i	$-0.291059\pi$
$61$	4.91133e11	1.22055	0.610275	+	0.792190i	$0.291059\pi$
$62$	0	0
$63$	−9.36485e11	−1.88707
$64$	0	0
$65$	−4.99459e11	−0.821416
$66$	0	0
$67$	−2.25405e11	−0.304423	−0.152212	−	0.988348i	$-0.548640\pi$
$67$	−2.25405e11	−0.304423	−0.152212	+	0.988348i	$0.548640\pi$
$68$	0	0
$69$	−3.94606e11	−0.440194
$70$	0	0
$71$	6.50849e11	0.602978	0.301489	−	0.953470i	$-0.402516\pi$
$71$	6.50849e11	0.602978	0.301489	+	0.953470i	$0.402516\pi$
$72$	0	0
$73$	−1.03710e11	−0.0802087	−0.0401043	−	0.999195i	$-0.512769\pi$
$73$	−1.03710e11	−0.0802087	−0.0401043	+	0.999195i	$0.512769\pi$
$74$	0	0
$75$	−5.16352e11	−0.335002
$76$	0	0
$77$	−5.24764e11	−0.286929
$78$	0	0
$79$	−2.08051e12	−0.962927	−0.481464	−	0.876466i	$-0.659895\pi$
$79$	−2.08051e12	−0.962927	−0.481464	+	0.876466i	$0.659895\pi$
$80$	0	0
$81$	1.15594e12	0.454760
$82$	0	0
$83$	−3.39017e12	−1.13819	−0.569094	−	0.822272i	$-0.692706\pi$
$83$	−3.39017e12	−1.13819	−0.569094	+	0.822272i	$0.692706\pi$
$84$	0	0
$85$	5.97390e10	0.0171805
$86$	0	0
$87$	−5.19638e12	−1.28477
$88$	0	0
$89$	−7.20835e12	−1.53745	−0.768724	−	0.639580i	$-0.779108\pi$
$89$	−7.20835e12	−1.53745	−0.768724	+	0.639580i	$0.779108\pi$
$90$	0	0
$91$	1.03984e13	1.91955
$92$	0	0
$93$	−2.04399e12	−0.327597
$94$	0	0
$95$	3.10565e12	0.433459
$96$	0	0
$97$	7.00185e12	0.853486	0.426743	−	0.904373i	$-0.359661\pi$
$97$	7.00185e12	0.853486	0.426743	+	0.904373i	$0.359661\pi$
$98$	0	0
$99$	4.64397e12	0.495748
$100$	0	0
$101$	−1.36319e13	−1.27781	−0.638907	−	0.769284i	$-0.720613\pi$
$101$	−1.36319e13	−1.27781	−0.638907	+	0.769284i	$0.720613\pi$
$102$	0	0
$103$	6.72899e12	0.555275	0.277637	−	0.960686i	$-0.410449\pi$
$103$	6.72899e12	0.555275	0.277637	+	0.960686i	$0.410449\pi$
$104$	0	0
$105$	1.07501e13	0.782858
$106$	0	0
$107$	−5.49424e12	−0.353927	−0.176963	−	0.984217i	$-0.556627\pi$
$107$	−5.49424e12	−0.353927	−0.176963	+	0.984217i	$0.556627\pi$
$108$	0	0
$109$	1.05471e13	0.602369	0.301184	−	0.953566i	$-0.402618\pi$
$109$	1.05471e13	0.602369	0.301184	+	0.953566i	$0.402618\pi$
$110$	0	0
$111$	−4.66997e13	−2.36982
$112$	0	0
$113$	1.30490e13	0.589612	0.294806	−	0.955557i	$-0.404745\pi$
$113$	1.30490e13	0.589612	0.294806	+	0.955557i	$0.404745\pi$
$114$	0	0
$115$	2.91526e12	0.117528
$116$	0	0
$117$	−9.20224e13	−3.31654
$118$	0	0
$119$	−1.24373e12	−0.0401486
$120$	0	0
$121$	−3.19204e13	−0.924622
$122$	0	0
$123$	−8.57944e13	−2.23396
$124$	0	0
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	1.04514e13	0.221030	0.110515	−	0.993874i	$-0.464750\pi$
$127$	1.04514e13	0.221030	0.110515	+	0.993874i	$0.464750\pi$
$128$	0	0
$129$	4.83293e13	0.923374
$130$	0	0
$131$	−8.39814e12	−0.145184	−0.0725922	−	0.997362i	$-0.523127\pi$
$131$	−8.39814e12	−0.145184	−0.0725922	+	0.997362i	$0.523127\pi$
$132$	0	0
$133$	−6.46576e13	−1.01294
$134$	0	0
$135$	−4.24479e13	−0.603514
$136$	0	0
$137$	2.62849e13	0.339643	0.169822	−	0.985475i	$-0.445681\pi$
$137$	2.62849e13	0.339643	0.169822	+	0.985475i	$0.445681\pi$
$138$	0	0
$139$	−6.83006e13	−0.803209	−0.401604	−	0.915813i	$-0.631547\pi$
$139$	−6.83006e13	−0.803209	−0.401604	+	0.915813i	$0.631547\pi$
$140$	0	0
$141$	−1.68646e14	−1.80739
$142$	0	0
$143$	−5.15651e13	−0.504280
$144$	0	0
$145$	3.83897e13	0.343023
$146$	0	0
$147$	−1.88927e13	−0.154430
$148$	0	0
$149$	−1.70193e14	−1.27418	−0.637089	−	0.770790i	$-0.719862\pi$
$149$	−1.70193e14	−1.27418	−0.637089	+	0.770790i	$0.719862\pi$
$150$	0	0
$151$	−1.84944e13	−0.126967	−0.0634835	−	0.997983i	$-0.520221\pi$
$151$	−1.84944e13	−0.126967	−0.0634835	+	0.997983i	$0.520221\pi$
$152$	0	0
$153$	1.10066e13	0.0693677
$154$	0	0
$155$	1.51005e13	0.0874654
$156$	0	0
$157$	1.72832e14	0.921035	0.460517	−	0.887651i	$-0.347664\pi$
$157$	1.72832e14	0.921035	0.460517	+	0.887651i	$0.347664\pi$
$158$	0	0
$159$	4.76109e14	2.33682
$160$	0	0
$161$	−6.06939e13	−0.274648
$162$	0	0
$163$	−2.52994e14	−1.05655	−0.528275	−	0.849073i	$-0.677161\pi$
$163$	−2.52994e14	−1.05655	−0.528275	+	0.849073i	$0.677161\pi$
$164$	0	0
$165$	−5.33092e13	−0.205663
$166$	0	0
$167$	1.37321e14	0.489870	0.244935	−	0.969539i	$-0.421233\pi$
$167$	1.37321e14	0.489870	0.244935	+	0.969539i	$0.421233\pi$
$168$	0	0
$169$	7.18911e14	2.37362
$170$	0	0
$171$	5.72197e14	1.75013
$172$	0	0
$173$	−1.57684e14	−0.447186	−0.223593	−	0.974683i	$-0.571779\pi$
$173$	−1.57684e14	−0.447186	−0.223593	+	0.974683i	$0.571779\pi$
$174$	0	0
$175$	−7.94196e13	−0.209016
$176$	0	0
$177$	1.68496e14	0.411873
$178$	0	0
$179$	2.76902e14	0.629189	0.314595	−	0.949226i	$-0.398131\pi$
$179$	2.76902e14	0.629189	0.314595	+	0.949226i	$0.398131\pi$
$180$	0	0
$181$	−1.39415e14	−0.294712	−0.147356	−	0.989084i	$-0.547076\pi$
$181$	−1.39415e14	−0.294712	−0.147356	+	0.989084i	$0.547076\pi$
$182$	0	0
$183$	−1.03874e15	−2.04444
$184$	0	0
$185$	3.45007e14	0.632720
$186$	0	0
$187$	6.16758e12	0.0105474
$188$	0	0
$189$	8.83739e14	1.41034
$190$	0	0
$191$	−6.42718e14	−0.957864	−0.478932	−	0.877852i	$-0.658976\pi$
$191$	−6.42718e14	−0.957864	−0.478932	+	0.877852i	$0.658976\pi$
$192$	0	0
$193$	−1.22414e14	−0.170494	−0.0852470	−	0.996360i	$-0.527168\pi$
$193$	−1.22414e14	−0.170494	−0.0852470	+	0.996360i	$0.527168\pi$
$194$	0	0
$195$	1.05635e15	1.37588
$196$	0	0
$197$	−1.36916e15	−1.66887	−0.834435	−	0.551106i	$-0.814206\pi$
$197$	−1.36916e15	−1.66887	−0.834435	+	0.551106i	$0.814206\pi$
$198$	0	0
$199$	1.51380e15	1.72792	0.863961	−	0.503558i	$-0.167976\pi$
$199$	1.51380e15	1.72792	0.863961	+	0.503558i	$0.167976\pi$
$200$	0	0
$201$	4.76727e14	0.509912
$202$	0	0
$203$	−7.99250e14	−0.801601
$204$	0	0
$205$	6.33830e14	0.596448
$206$	0	0
$207$	5.37119e14	0.474530
$208$	0	0
$209$	3.20633e14	0.266108
$210$	0	0
$211$	−1.41808e15	−1.10628	−0.553141	−	0.833088i	$-0.686571\pi$
$211$	−1.41808e15	−1.10628	−0.553141	+	0.833088i	$0.686571\pi$
$212$	0	0
$213$	−1.37653e15	−1.00999
$214$	0	0
$215$	−3.57047e14	−0.246533
$216$	0	0
$217$	−3.14384e14	−0.204396
$218$	0	0
$219$	2.19344e14	0.134350
$220$	0	0
$221$	−1.22213e14	−0.0705616
$222$	0	0
$223$	3.48697e15	1.89874	0.949371	−	0.314157i	$-0.101722\pi$
$223$	3.48697e15	1.89874	0.949371	+	0.314157i	$0.101722\pi$
$224$	0	0
$225$	7.02835e14	0.361133
$226$	0	0
$227$	−1.43848e15	−0.697808	−0.348904	−	0.937158i	$-0.613446\pi$
$227$	−1.43848e15	−0.697808	−0.348904	+	0.937158i	$0.613446\pi$
$228$	0	0
$229$	1.00326e14	0.0459709	0.0229855	−	0.999736i	$-0.492683\pi$
$229$	1.00326e14	0.0459709	0.0229855	+	0.999736i	$0.492683\pi$
$230$	0	0
$231$	1.10986e15	0.480609
$232$	0	0
$233$	1.04926e15	0.429607	0.214803	−	0.976657i	$-0.431089\pi$
$233$	1.04926e15	0.429607	0.214803	+	0.976657i	$0.431089\pi$
$234$	0	0
$235$	1.24592e15	0.482559
$236$	0	0
$237$	4.40023e15	1.61291
$238$	0	0
$239$	4.68165e15	1.62485	0.812424	−	0.583067i	$-0.198147\pi$
$239$	4.68165e15	1.62485	0.812424	+	0.583067i	$0.198147\pi$
$240$	0	0
$241$	−1.22689e15	−0.403361	−0.201680	−	0.979451i	$-0.564640\pi$
$241$	−1.22689e15	−0.403361	−0.201680	+	0.979451i	$0.564640\pi$
$242$	0	0
$243$	1.88646e15	0.587770
$244$	0	0
$245$	1.39575e14	0.0412313
$246$	0	0
$247$	−6.35348e15	−1.78025
$248$	0	0
$249$	7.17014e15	1.90648
$250$	0	0
$251$	2.74250e15	0.692258	0.346129	−	0.938187i	$-0.387496\pi$
$251$	2.74250e15	0.692258	0.346129	+	0.938187i	$0.387496\pi$
$252$	0	0
$253$	3.00977e14	0.0721523
$254$	0	0
$255$	−1.26347e14	−0.0287775
$256$	0	0
$257$	−3.96785e15	−0.858994	−0.429497	−	0.903068i	$-0.641309\pi$
$257$	−3.96785e15	−0.858994	−0.429497	+	0.903068i	$0.641309\pi$
$258$	0	0
$259$	−7.18283e15	−1.47859
$260$	0	0
$261$	7.07307e15	1.38498
$262$	0	0
$263$	−4.19100e15	−0.780917	−0.390459	−	0.920621i	$-0.627684\pi$
$263$	−4.19100e15	−0.780917	−0.390459	+	0.920621i	$0.627684\pi$
$264$	0	0
$265$	−3.51739e15	−0.623910
$266$	0	0
$267$	1.52455e16	2.57524
$268$	0	0
$269$	1.20414e15	0.193771	0.0968854	−	0.995296i	$-0.469112\pi$
$269$	1.20414e15	0.193771	0.0968854	+	0.995296i	$0.469112\pi$
$270$	0	0
$271$	−8.17556e15	−1.25377	−0.626884	−	0.779113i	$-0.715670\pi$
$271$	−8.17556e15	−1.25377	−0.626884	+	0.779113i	$0.715670\pi$
$272$	0	0
$273$	−2.19924e16	−3.21526
$274$	0	0
$275$	3.93837e14	0.0549103
$276$	0	0
$277$	5.65506e15	0.752175	0.376088	−	0.926584i	$-0.377269\pi$
$277$	5.65506e15	0.752175	0.376088	+	0.926584i	$0.377269\pi$
$278$	0	0
$279$	2.78218e15	0.353149
$280$	0	0
$281$	−1.38960e16	−1.68383	−0.841917	−	0.539607i	$-0.818573\pi$
$281$	−1.38960e16	−1.68383	−0.841917	+	0.539607i	$0.818573\pi$
$282$	0	0
$283$	−1.68555e16	−1.95042	−0.975212	−	0.221271i	$-0.928979\pi$
$283$	−1.68555e16	−1.95042	−0.975212	+	0.221271i	$0.928979\pi$
$284$	0	0
$285$	−6.56837e15	−0.726050
$286$	0	0
$287$	−1.31959e16	−1.39383
$288$	0	0
$289$	−9.88996e15	−0.998524
$290$	0	0
$291$	−1.48088e16	−1.42960
$292$	0	0
$293$	8.49847e15	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$293$	8.49847e15	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$294$	0	0
$295$	−1.24481e15	−0.109967
$296$	0	0
$297$	−4.38241e15	−0.370506
$298$	0	0
$299$	−5.96400e15	−0.482697
$300$	0	0
$301$	7.43348e15	0.576116
$302$	0	0
$303$	2.88312e16	2.14035
$304$	0	0
$305$	7.67396e15	0.545846
$306$	0	0
$307$	2.68116e14	0.0182778	0.00913890	−	0.999958i	$-0.497091\pi$
$307$	2.68116e14	0.0182778	0.00913890	+	0.999958i	$0.497091\pi$
$308$	0	0
$309$	−1.42317e16	−0.930091
$310$	0	0
$311$	2.25419e16	1.41270	0.706348	−	0.707865i	$-0.250341\pi$
$311$	2.25419e16	1.41270	0.706348	+	0.707865i	$0.250341\pi$
$312$	0	0
$313$	2.91559e15	0.175262	0.0876312	−	0.996153i	$-0.472070\pi$
$313$	2.91559e15	0.175262	0.0876312	+	0.996153i	$0.472070\pi$
$314$	0	0
$315$	−1.46326e16	−0.843922
$316$	0	0
$317$	−2.90121e16	−1.60581	−0.802903	−	0.596109i	$-0.796713\pi$
$317$	−2.90121e16	−1.60581	−0.802903	+	0.596109i	$0.796713\pi$
$318$	0	0
$319$	3.96343e15	0.210587
$320$	0	0
$321$	1.16202e16	0.592831
$322$	0	0
$323$	7.59924e14	0.0372352
$324$	0	0
$325$	−7.80405e15	−0.367348
$326$	0	0
$327$	−2.23070e16	−1.00897
$328$	0	0
$329$	−2.59393e16	−1.12768
$330$	0	0
$331$	−3.67428e16	−1.53564	−0.767821	−	0.640665i	$-0.778659\pi$
$331$	−3.67428e16	−1.53564	−0.767821	+	0.640665i	$0.778659\pi$
$332$	0	0
$333$	6.35655e16	2.55466
$334$	0	0
$335$	−3.52196e15	−0.136142
$336$	0	0
$337$	2.62017e16	0.974395	0.487197	−	0.873292i	$-0.338019\pi$
$337$	2.62017e16	0.974395	0.487197	+	0.873292i	$0.338019\pi$
$338$	0	0
$339$	−2.75983e16	−0.987608
$340$	0	0
$341$	1.55901e15	0.0536964
$342$	0	0
$343$	2.86124e16	0.948729
$344$	0	0
$345$	−6.16572e15	−0.196861
$346$	0	0
$347$	−4.16211e15	−0.127989	−0.0639944	−	0.997950i	$-0.520384\pi$
$347$	−4.16211e15	−0.127989	−0.0639944	+	0.997950i	$0.520384\pi$
$348$	0	0
$349$	−3.79966e16	−1.12559	−0.562794	−	0.826597i	$-0.690274\pi$
$349$	−3.79966e16	−1.12559	−0.562794	+	0.826597i	$0.690274\pi$
$350$	0	0
$351$	8.68393e16	2.47868
$352$	0	0
$353$	6.24372e16	1.71754	0.858772	−	0.512358i	$-0.171228\pi$
$353$	6.24372e16	1.71754	0.858772	+	0.512358i	$0.171228\pi$
$354$	0	0
$355$	1.01695e16	0.269660
$356$	0	0
$357$	2.63046e15	0.0672494
$358$	0	0
$359$	−4.66231e16	−1.14944	−0.574721	−	0.818349i	$-0.694890\pi$
$359$	−4.66231e16	−1.14944	−0.574721	+	0.818349i	$0.694890\pi$
$360$	0	0
$361$	−2.54692e15	−0.0605646
$362$	0	0
$363$	6.75111e16	1.54875
$364$	0	0
$365$	−1.62047e15	−0.0358704
$366$	0	0
$367$	5.72445e16	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$367$	5.72445e16	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$368$	0	0
$369$	1.16779e17	2.40821
$370$	0	0
$371$	7.32298e16	1.45800
$372$	0	0
$373$	−2.68476e16	−0.516176	−0.258088	−	0.966121i	$-0.583092\pi$
$373$	−2.68476e16	−0.516176	−0.258088	+	0.966121i	$0.583092\pi$
$374$	0	0
$375$	−8.06800e15	−0.149818
$376$	0	0
$377$	−7.85371e16	−1.40882
$378$	0	0
$379$	−3.18194e16	−0.551489	−0.275744	−	0.961231i	$-0.588924\pi$
$379$	−3.18194e16	−0.551489	−0.275744	+	0.961231i	$0.588924\pi$
$380$	0	0
$381$	−2.21046e16	−0.370228
$382$	0	0
$383$	−1.08164e17	−1.75101	−0.875505	−	0.483208i	$-0.839471\pi$
$383$	−1.08164e17	−1.75101	−0.875505	+	0.483208i	$0.839471\pi$
$384$	0	0
$385$	−8.19943e15	−0.128318
$386$	0	0
$387$	−6.57837e16	−0.995398
$388$	0	0
$389$	−9.51462e16	−1.39226	−0.696128	−	0.717918i	$-0.745095\pi$
$389$	−9.51462e16	−1.39226	−0.696128	+	0.717918i	$0.745095\pi$
$390$	0	0
$391$	7.13339e14	0.0100959
$392$	0	0
$393$	1.77619e16	0.243185
$394$	0	0
$395$	−3.25079e16	−0.430634
$396$	0	0
$397$	−4.83241e16	−0.619477	−0.309739	−	0.950822i	$-0.600242\pi$
$397$	−4.83241e16	−0.619477	−0.309739	+	0.950822i	$0.600242\pi$
$398$	0	0
$399$	1.36749e17	1.69669
$400$	0	0
$401$	4.29285e16	0.515593	0.257797	−	0.966199i	$-0.417004\pi$
$401$	4.29285e16	0.515593	0.257797	+	0.966199i	$0.417004\pi$
$402$	0	0
$403$	−3.08925e16	−0.359228
$404$	0	0
$405$	1.80615e16	0.203375
$406$	0	0
$407$	3.56192e16	0.388437
$408$	0	0
$409$	8.36651e16	0.883777	0.441889	−	0.897070i	$-0.354309\pi$
$409$	8.36651e16	0.883777	0.441889	+	0.897070i	$0.354309\pi$
$410$	0	0
$411$	−5.55921e16	−0.568907
$412$	0	0
$413$	2.59162e16	0.256978
$414$	0	0
$415$	−5.29714e16	−0.509013
$416$	0	0
$417$	1.44454e17	1.34538
$418$	0	0
$419$	1.35543e17	1.22373	0.611865	−	0.790962i	$-0.290420\pi$
$419$	1.35543e17	1.22373	0.611865	+	0.790962i	$0.290420\pi$
$420$	0	0
$421$	5.25025e16	0.459565	0.229782	−	0.973242i	$-0.426199\pi$
$421$	5.25025e16	0.459565	0.229782	+	0.973242i	$0.426199\pi$
$422$	0	0
$423$	2.29554e17	1.94837
$424$	0	0
$425$	9.33423e14	0.00768334
$426$	0	0
$427$	−1.59767e17	−1.27557
$428$	0	0
$429$	1.09059e17	0.844675
$430$	0	0
$431$	−2.51192e15	−0.0188757	−0.00943785	−	0.999955i	$-0.503004\pi$
$431$	−2.51192e15	−0.0188757	−0.00943785	+	0.999955i	$0.503004\pi$
$432$	0	0
$433$	−7.31447e16	−0.533349	−0.266675	−	0.963787i	$-0.585925\pi$
$433$	−7.31447e16	−0.533349	−0.266675	+	0.963787i	$0.585925\pi$
$434$	0	0
$435$	−8.11934e16	−0.574567
$436$	0	0
$437$	3.70842e16	0.254718
$438$	0	0
$439$	7.23430e16	0.482367	0.241183	−	0.970480i	$-0.422464\pi$
$439$	7.23430e16	0.482367	0.241183	+	0.970480i	$0.422464\pi$
$440$	0	0
$441$	2.57158e16	0.166475
$442$	0	0
$443$	−2.25527e17	−1.41767	−0.708833	−	0.705377i	$-0.750778\pi$
$443$	−2.25527e17	−1.41767	−0.708833	+	0.705377i	$0.750778\pi$
$444$	0	0
$445$	−1.12630e17	−0.687568
$446$	0	0
$447$	3.59954e17	2.13426
$448$	0	0
$449$	5.33358e16	0.307197	0.153599	−	0.988133i	$-0.450914\pi$
$449$	5.33358e16	0.307197	0.153599	+	0.988133i	$0.450914\pi$
$450$	0	0
$451$	6.54379e16	0.366169
$452$	0	0
$453$	3.91153e16	0.212671
$454$	0	0
$455$	1.62475e17	0.858447
$456$	0	0
$457$	−9.83492e16	−0.505028	−0.252514	−	0.967593i	$-0.581257\pi$
$457$	−9.83492e16	−0.505028	−0.252514	+	0.967593i	$0.581257\pi$
$458$	0	0
$459$	−1.03866e16	−0.0518433
$460$	0	0
$461$	2.01882e17	0.979583	0.489791	−	0.871840i	$-0.337073\pi$
$461$	2.01882e17	0.979583	0.489791	+	0.871840i	$0.337073\pi$
$462$	0	0
$463$	1.21471e17	0.573057	0.286528	−	0.958072i	$-0.407499\pi$
$463$	1.21471e17	0.573057	0.286528	+	0.958072i	$0.407499\pi$
$464$	0	0
$465$	−3.19373e16	−0.146506
$466$	0	0
$467$	−4.32550e17	−1.92964	−0.964820	−	0.262910i	$-0.915318\pi$
$467$	−4.32550e17	−1.92964	−0.964820	+	0.262910i	$0.915318\pi$
$468$	0	0
$469$	7.33249e16	0.318147
$470$	0	0
$471$	−3.65536e17	−1.54274
$472$	0	0
$473$	−3.68622e16	−0.151350
$474$	0	0
$475$	4.85257e16	0.193849
$476$	0	0
$477$	−6.48057e17	−2.51909
$478$	0	0
$479$	1.80742e17	0.683719	0.341860	−	0.939751i	$-0.388943\pi$
$479$	1.80742e17	0.683719	0.341860	+	0.939751i	$0.388943\pi$
$480$	0	0
$481$	−7.05810e17	−2.59863
$482$	0	0
$483$	1.28366e17	0.460039
$484$	0	0
$485$	1.09404e17	0.381691
$486$	0	0
$487$	−1.79248e17	−0.608858	−0.304429	−	0.952535i	$-0.598466\pi$
$487$	−1.79248e17	−0.608858	−0.304429	+	0.952535i	$0.598466\pi$
$488$	0	0
$489$	5.35076e17	1.76973
$490$	0	0
$491$	−3.27397e17	−1.05450	−0.527248	−	0.849712i	$-0.676776\pi$
$491$	−3.27397e17	−1.05450	−0.527248	+	0.849712i	$0.676776\pi$
$492$	0	0
$493$	9.39362e15	0.0294665
$494$	0	0
$495$	7.25621e16	0.221705
$496$	0	0
$497$	−2.11723e17	−0.630161
$498$	0	0
$499$	4.14032e17	1.20055	0.600275	−	0.799794i	$-0.295058\pi$
$499$	4.14032e17	1.20055	0.600275	+	0.799794i	$0.295058\pi$
$500$	0	0
$501$	−2.90432e17	−0.820539
$502$	0	0
$503$	−1.04795e17	−0.288503	−0.144251	−	0.989541i	$-0.546077\pi$
$503$	−1.04795e17	−0.288503	−0.144251	+	0.989541i	$0.546077\pi$
$504$	0	0
$505$	−2.12999e17	−0.571456
$506$	0	0
$507$	−1.52048e18	−3.97584
$508$	0	0
$509$	4.12726e17	1.05195	0.525976	−	0.850499i	$-0.323700\pi$
$509$	4.12726e17	1.05195	0.525976	+	0.850499i	$0.323700\pi$
$510$	0	0
$511$	3.37371e16	0.0838246
$512$	0	0
$513$	−5.39968e17	−1.30799
$514$	0	0
$515$	1.05140e17	0.248326
$516$	0	0
$517$	1.28632e17	0.296250
$518$	0	0
$519$	3.33499e17	0.749042
$520$	0	0
$521$	2.21835e17	0.485942	0.242971	−	0.970034i	$-0.421878\pi$
$521$	2.21835e17	0.485942	0.242971	+	0.970034i	$0.421878\pi$
$522$	0	0
$523$	−7.40537e17	−1.58229	−0.791145	−	0.611629i	$-0.790514\pi$
$523$	−7.40537e17	−1.58229	−0.791145	+	0.611629i	$0.790514\pi$
$524$	0	0
$525$	1.67971e17	0.350105
$526$	0	0
$527$	3.69497e15	0.00751349
$528$	0	0
$529$	−4.69225e17	−0.930936
$530$	0	0
$531$	−2.29349e17	−0.444000
$532$	0	0
$533$	−1.29668e18	−2.44966
$534$	0	0
$535$	−8.58475e16	−0.158281
$536$	0	0
$537$	−5.85642e17	−1.05390
$538$	0	0
$539$	1.44100e16	0.0253126
$540$	0	0
$541$	1.10461e18	1.89420	0.947101	−	0.320936i	$-0.103997\pi$
$541$	1.10461e18	1.89420	0.947101	+	0.320936i	$0.103997\pi$
$542$	0	0
$543$	2.94859e17	0.493646
$544$	0	0
$545$	1.64799e17	0.269387
$546$	0	0
$547$	6.13521e17	0.979292	0.489646	−	0.871921i	$-0.337126\pi$
$547$	6.13521e17	0.979292	0.489646	+	0.871921i	$0.337126\pi$
$548$	0	0
$549$	1.41388e18	2.20390
$550$	0	0
$551$	4.88345e17	0.743432
$552$	0	0
$553$	6.76795e17	1.00634
$554$	0	0
$555$	−7.29683e17	−1.05981
$556$	0	0
$557$	−2.17270e17	−0.308276	−0.154138	−	0.988049i	$-0.549260\pi$
$557$	−2.17270e17	−0.308276	−0.154138	+	0.988049i	$0.549260\pi$
$558$	0	0
$559$	7.30440e17	1.01253
$560$	0	0
$561$	−1.30443e16	−0.0176670
$562$	0	0
$563$	1.16883e18	1.54684	0.773422	−	0.633892i	$-0.218543\pi$
$563$	1.16883e18	1.54684	0.773422	+	0.633892i	$0.218543\pi$
$564$	0	0
$565$	2.03890e17	0.263683
$566$	0	0
$567$	−3.76030e17	−0.475261
$568$	0	0
$569$	−4.86769e17	−0.601302	−0.300651	−	0.953734i	$-0.597204\pi$
$569$	−4.86769e17	−0.601302	−0.300651	+	0.953734i	$0.597204\pi$
$570$	0	0
$571$	3.75968e17	0.453958	0.226979	−	0.973900i	$-0.427115\pi$
$571$	3.75968e17	0.453958	0.226979	+	0.973900i	$0.427115\pi$
$572$	0	0
$573$	1.35934e18	1.60443
$574$	0	0
$575$	4.55510e16	0.0525601
$576$	0	0
$577$	2.49769e17	0.281771	0.140885	−	0.990026i	$-0.455005\pi$
$577$	2.49769e17	0.281771	0.140885	+	0.990026i	$0.455005\pi$
$578$	0	0
$579$	2.58903e17	0.285579
$580$	0	0
$581$	1.10283e18	1.18950
$582$	0	0
$583$	−3.63142e17	−0.383028
$584$	0	0
$585$	−1.43785e18	−1.48320
$586$	0	0
$587$	1.02718e18	1.03633	0.518165	−	0.855281i	$-0.326615\pi$
$587$	1.02718e18	1.03633	0.518165	+	0.855281i	$0.326615\pi$
$588$	0	0
$589$	1.92090e17	0.189564
$590$	0	0
$591$	2.89574e18	2.79538
$592$	0	0
$593$	1.25381e18	1.18406	0.592032	−	0.805914i	$-0.298326\pi$
$593$	1.25381e18	1.18406	0.592032	+	0.805914i	$0.298326\pi$
$594$	0	0
$595$	−1.94333e16	−0.0179550
$596$	0	0
$597$	−3.20166e18	−2.89429
$598$	0	0
$599$	5.76424e17	0.509879	0.254940	−	0.966957i	$-0.417944\pi$
$599$	5.76424e17	0.509879	0.254940	+	0.966957i	$0.417944\pi$
$600$	0	0
$601$	−1.23224e18	−1.06663	−0.533313	−	0.845918i	$-0.679053\pi$
$601$	−1.23224e18	−1.06663	−0.533313	+	0.845918i	$0.679053\pi$
$602$	0	0
$603$	−6.48899e17	−0.549686
$604$	0	0
$605$	−4.98757e17	−0.413503
$606$	0	0
$607$	−1.17936e18	−0.957019	−0.478510	−	0.878082i	$-0.658823\pi$
$607$	−1.17936e18	−0.957019	−0.478510	+	0.878082i	$0.658823\pi$
$608$	0	0
$609$	1.69040e18	1.34269
$610$	0	0
$611$	−2.54889e18	−1.98191
$612$	0	0
$613$	7.39177e17	0.562672	0.281336	−	0.959609i	$-0.409222\pi$
$613$	7.39177e17	0.562672	0.281336	+	0.959609i	$0.409222\pi$
$614$	0	0
$615$	−1.34054e18	−0.999058
$616$	0	0
$617$	2.13054e18	1.55467	0.777333	−	0.629090i	$-0.216572\pi$
$617$	2.13054e18	1.55467	0.777333	+	0.629090i	$0.216572\pi$
$618$	0	0
$619$	−1.66755e18	−1.19149	−0.595744	−	0.803175i	$-0.703143\pi$
$619$	−1.66755e18	−1.19149	−0.595744	+	0.803175i	$0.703143\pi$
$620$	0	0
$621$	−5.06867e17	−0.354649
$622$	0	0
$623$	2.34489e18	1.60676
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	0	0
$627$	−6.78132e17	−0.445733
$628$	0	0
$629$	8.44202e16	0.0543522
$630$	0	0
$631$	7.05851e16	0.0445166	0.0222583	−	0.999752i	$-0.492914\pi$
$631$	7.05851e16	0.0445166	0.0222583	+	0.999752i	$0.492914\pi$
$632$	0	0
$633$	2.99922e18	1.85304
$634$	0	0
$635$	1.63304e17	0.0988477
$636$	0	0
$637$	−2.85540e17	−0.169340
$638$	0	0
$639$	1.87367e18	1.08878
$640$	0	0
$641$	−2.27477e18	−1.29527	−0.647635	−	0.761951i	$-0.724242\pi$
$641$	−2.27477e18	−1.29527	−0.647635	+	0.761951i	$0.724242\pi$
$642$	0	0
$643$	1.40077e18	0.781621	0.390810	−	0.920471i	$-0.372195\pi$
$643$	1.40077e18	0.781621	0.390810	+	0.920471i	$0.372195\pi$
$644$	0	0
$645$	7.55146e17	0.412946
$646$	0	0
$647$	−3.49376e18	−1.87247	−0.936237	−	0.351369i	$-0.885716\pi$
$647$	−3.49376e18	−1.87247	−0.936237	+	0.351369i	$0.885716\pi$
$648$	0	0
$649$	−1.28517e17	−0.0675102
$650$	0	0
$651$	6.64915e17	0.342365
$652$	0	0
$653$	4.17303e17	0.210628	0.105314	−	0.994439i	$-0.466415\pi$
$653$	4.17303e17	0.210628	0.105314	+	0.994439i	$0.466415\pi$
$654$	0	0
$655$	−1.31221e17	−0.0649284
$656$	0	0
$657$	−2.98561e17	−0.144830
$658$	0	0
$659$	2.83630e18	1.34895	0.674477	−	0.738296i	$-0.264369\pi$
$659$	2.83630e18	1.34895	0.674477	+	0.738296i	$0.264369\pi$
$660$	0	0
$661$	8.05065e17	0.375423	0.187712	−	0.982224i	$-0.439893\pi$
$661$	8.05065e17	0.375423	0.187712	+	0.982224i	$0.439893\pi$
$662$	0	0
$663$	2.58478e17	0.118192
$664$	0	0
$665$	−1.01027e18	−0.453001
$666$	0	0
$667$	4.58408e17	0.201574
$668$	0	0
$669$	−7.37486e18	−3.18042
$670$	0	0
$671$	7.92274e17	0.335104
$672$	0	0
$673$	3.03151e18	1.25765	0.628827	−	0.777546i	$-0.283535\pi$
$673$	3.03151e18	1.25765	0.628827	+	0.777546i	$0.283535\pi$
$674$	0	0
$675$	−6.63249e17	−0.269900
$676$	0	0
$677$	−1.58711e18	−0.633551	−0.316775	−	0.948501i	$-0.602600\pi$
$677$	−1.58711e18	−0.633551	−0.316775	+	0.948501i	$0.602600\pi$
$678$	0	0
$679$	−2.27772e18	−0.891963
$680$	0	0
$681$	3.04236e18	1.16884
$682$	0	0
$683$	−9.37243e17	−0.353279	−0.176640	−	0.984276i	$-0.556523\pi$
$683$	−9.37243e17	−0.353279	−0.176640	+	0.984276i	$0.556523\pi$
$684$	0	0
$685$	4.10702e17	0.151893
$686$	0	0
$687$	−2.12187e17	−0.0770018
$688$	0	0
$689$	7.19582e18	2.56245
$690$	0	0
$691$	4.89005e18	1.70886	0.854428	−	0.519569i	$-0.173907\pi$
$691$	4.89005e18	1.70886	0.854428	+	0.519569i	$0.173907\pi$
$692$	0	0
$693$	−1.51070e18	−0.518097
$694$	0	0
$695$	−1.06720e18	−0.359206
$696$	0	0
$697$	1.55093e17	0.0512364
$698$	0	0
$699$	−2.21917e18	−0.719597
$700$	0	0
$701$	1.21554e18	0.386902	0.193451	−	0.981110i	$-0.438032\pi$
$701$	1.21554e18	0.386902	0.193451	+	0.981110i	$0.438032\pi$
$702$	0	0
$703$	4.38874e18	1.37129
$704$	0	0
$705$	−2.63510e18	−0.808292
$706$	0	0
$707$	4.43449e18	1.33542
$708$	0	0
$709$	2.62006e18	0.774658	0.387329	−	0.921942i	$-0.373398\pi$
$709$	2.62006e18	0.774658	0.387329	+	0.921942i	$0.373398\pi$
$710$	0	0
$711$	−5.98939e18	−1.73872
$712$	0	0
$713$	1.80314e17	0.0513982
$714$	0	0
$715$	−8.05705e17	−0.225521
$716$	0	0
$717$	−9.90160e18	−2.72164
$718$	0	0
$719$	2.75759e18	0.744375	0.372187	−	0.928158i	$-0.378608\pi$
$719$	2.75759e18	0.744375	0.372187	+	0.928158i	$0.378608\pi$
$720$	0	0
$721$	−2.18896e18	−0.580307
$722$	0	0
$723$	2.59484e18	0.675634
$724$	0	0
$725$	5.99839e17	0.153404
$726$	0	0
$727$	−5.02204e18	−1.26156	−0.630778	−	0.775963i	$-0.717264\pi$
$727$	−5.02204e18	−1.26156	−0.630778	+	0.775963i	$0.717264\pi$
$728$	0	0
$729$	−5.83276e18	−1.43928
$730$	0	0
$731$	−8.73661e16	−0.0211778
$732$	0	0
$733$	−4.44902e17	−0.105947	−0.0529734	−	0.998596i	$-0.516870\pi$
$733$	−4.44902e17	−0.105947	−0.0529734	+	0.998596i	$0.516870\pi$
$734$	0	0
$735$	−2.95198e17	−0.0690630
$736$	0	0
$737$	−3.63614e17	−0.0835798
$738$	0	0
$739$	−7.66043e18	−1.73007	−0.865037	−	0.501709i	$-0.832705\pi$
$739$	−7.66043e18	−1.73007	−0.865037	+	0.501709i	$0.832705\pi$
$740$	0	0
$741$	1.34375e19	2.98194
$742$	0	0
$743$	−2.91989e18	−0.636706	−0.318353	−	0.947972i	$-0.603130\pi$
$743$	−2.91989e18	−0.636706	−0.318353	+	0.947972i	$0.603130\pi$
$744$	0	0
$745$	−2.65926e18	−0.569830
$746$	0	0
$747$	−9.75967e18	−2.05519
$748$	0	0
$749$	1.78729e18	0.369882
$750$	0	0
$751$	−1.98445e18	−0.403628	−0.201814	−	0.979424i	$-0.564684\pi$
$751$	−1.98445e18	−0.403628	−0.201814	+	0.979424i	$0.564684\pi$
$752$	0	0
$753$	−5.80034e18	−1.15954
$754$	0	0
$755$	−2.88976e17	−0.0567814
$756$	0	0
$757$	−4.46722e18	−0.862807	−0.431404	−	0.902159i	$-0.641982\pi$
$757$	−4.46722e18	−0.862807	−0.431404	+	0.902159i	$0.641982\pi$
$758$	0	0
$759$	−6.36560e17	−0.120856
$760$	0	0
$761$	−7.81343e18	−1.45828	−0.729141	−	0.684363i	$-0.760080\pi$
$761$	−7.81343e18	−1.45828	−0.729141	+	0.684363i	$0.760080\pi$
$762$	0	0
$763$	−3.43101e18	−0.629525
$764$	0	0
$765$	1.71978e17	0.0310222
$766$	0	0
$767$	2.54662e18	0.451642
$768$	0	0
$769$	2.15334e17	0.0375483	0.0187742	−	0.999824i	$-0.494024\pi$
$769$	2.15334e17	0.0375483	0.0187742	+	0.999824i	$0.494024\pi$
$770$	0	0
$771$	8.39193e18	1.43883
$772$	0	0
$773$	−7.73551e18	−1.30413	−0.652067	−	0.758162i	$-0.726098\pi$
$773$	−7.73551e18	−1.30413	−0.652067	+	0.758162i	$0.726098\pi$
$774$	0	0
$775$	2.35946e17	0.0391157
$776$	0	0
$777$	1.51915e19	2.47665
$778$	0	0
$779$	8.06278e18	1.29268
$780$	0	0
$781$	1.04992e18	0.165548
$782$	0	0
$783$	−6.67469e18	−1.03509
$784$	0	0
$785$	2.70050e18	0.411899
$786$	0	0
$787$	−7.07633e18	−1.06163	−0.530814	−	0.847488i	$-0.678114\pi$
$787$	−7.07633e18	−1.06163	−0.530814	+	0.847488i	$0.678114\pi$
$788$	0	0
$789$	8.86387e18	1.30805
$790$	0	0
$791$	−4.24487e18	−0.616193
$792$	0	0
$793$	−1.56993e19	−2.24184
$794$	0	0
$795$	7.43920e18	1.04506
$796$	0	0
$797$	8.43425e17	0.116565	0.0582824	−	0.998300i	$-0.481438\pi$
$797$	8.43425e17	0.116565	0.0582824	+	0.998300i	$0.481438\pi$
$798$	0	0
$799$	3.04866e17	0.0414529
$800$	0	0
$801$	−2.07515e19	−2.77612
$802$	0	0
$803$	−1.67300e17	−0.0220214
$804$	0	0
$805$	−9.48342e17	−0.122826
$806$	0	0
$807$	−2.54674e18	−0.324568
$808$	0	0
$809$	5.80962e18	0.728588	0.364294	−	0.931284i	$-0.381310\pi$
$809$	5.80962e18	0.728588	0.364294	+	0.931284i	$0.381310\pi$
$810$	0	0
$811$	−1.15747e19	−1.42848	−0.714240	−	0.699901i	$-0.753227\pi$
$811$	−1.15747e19	−1.42848	−0.714240	+	0.699901i	$0.753227\pi$
$812$	0	0
$813$	1.72911e19	2.10008
$814$	0	0
$815$	−3.95302e18	−0.472503
$816$	0	0
$817$	−4.54189e18	−0.534310
$818$	0	0
$819$	2.99351e19	3.46606
$820$	0	0
$821$	4.73256e18	0.539343	0.269672	−	0.962952i	$-0.413085\pi$
$821$	4.73256e18	0.539343	0.269672	+	0.962952i	$0.413085\pi$
$822$	0	0
$823$	−5.39004e18	−0.604634	−0.302317	−	0.953207i	$-0.597760\pi$
$823$	−5.39004e18	−0.604634	−0.302317	+	0.953207i	$0.597760\pi$
$824$	0	0
$825$	−8.32956e17	−0.0919754
$826$	0	0
$827$	4.87314e18	0.529692	0.264846	−	0.964291i	$-0.414679\pi$
$827$	4.87314e18	0.529692	0.264846	+	0.964291i	$0.414679\pi$
$828$	0	0
$829$	−1.10498e19	−1.18236	−0.591182	−	0.806538i	$-0.701339\pi$
$829$	−1.10498e19	−1.18236	−0.591182	+	0.806538i	$0.701339\pi$
$830$	0	0
$831$	−1.19603e19	−1.25990
$832$	0	0
$833$	3.41527e16	0.00354187
$834$	0	0
$835$	2.14565e18	0.219077
$836$	0	0
$837$	−2.62548e18	−0.263933
$838$	0	0
$839$	−8.80967e17	−0.0871981	−0.0435990	−	0.999049i	$-0.513882\pi$
$839$	−8.80967e17	−0.0871981	−0.0435990	+	0.999049i	$0.513882\pi$
$840$	0	0
$841$	−4.22407e18	−0.411677
$842$	0	0
$843$	2.93898e19	2.82044
$844$	0	0
$845$	1.12330e19	1.06152
$846$	0	0
$847$	1.03838e19	0.966305
$848$	0	0
$849$	3.56490e19	3.26698
$850$	0	0
$851$	4.11970e18	0.371812
$852$	0	0
$853$	−5.43444e18	−0.483043	−0.241522	−	0.970395i	$-0.577646\pi$
$853$	−5.43444e18	−0.483043	−0.241522	+	0.970395i	$0.577646\pi$
$854$	0	0
$855$	8.94057e18	0.782682
$856$	0	0
$857$	3.79499e18	0.327217	0.163608	−	0.986525i	$-0.447687\pi$
$857$	3.79499e18	0.327217	0.163608	+	0.986525i	$0.447687\pi$
$858$	0	0
$859$	1.65792e17	0.0140802	0.00704010	−	0.999975i	$-0.497759\pi$
$859$	1.65792e17	0.0140802	0.00704010	+	0.999975i	$0.497759\pi$
$860$	0	0
$861$	2.79091e19	2.33467
$862$	0	0
$863$	−1.66111e19	−1.36877	−0.684383	−	0.729123i	$-0.739928\pi$
$863$	−1.66111e19	−1.36877	−0.684383	+	0.729123i	$0.739928\pi$
$864$	0	0
$865$	−2.46382e18	−0.199988
$866$	0	0
$867$	2.09171e19	1.67254
$868$	0	0
$869$	−3.35618e18	−0.264373
$870$	0	0
$871$	7.20517e18	0.559147
$872$	0	0
$873$	2.01570e19	1.54111
$874$	0	0
$875$	−1.24093e18	−0.0934750
$876$	0	0
$877$	−2.03717e18	−0.151192	−0.0755960	−	0.997139i	$-0.524086\pi$
$877$	−2.03717e18	−0.151192	−0.0755960	+	0.997139i	$0.524086\pi$
$878$	0	0
$879$	−1.79741e19	−1.31437
$880$	0	0
$881$	−1.97226e19	−1.42109	−0.710545	−	0.703652i	$-0.751551\pi$
$881$	−1.97226e19	−1.42109	−0.710545	+	0.703652i	$0.751551\pi$
$882$	0	0
$883$	−4.19713e18	−0.297994	−0.148997	−	0.988838i	$-0.547604\pi$
$883$	−4.19713e18	−0.297994	−0.148997	+	0.988838i	$0.547604\pi$
$884$	0	0
$885$	2.63275e18	0.184195
$886$	0	0
$887$	6.63464e18	0.457418	0.228709	−	0.973495i	$-0.426550\pi$
$887$	6.63464e18	0.457418	0.228709	+	0.973495i	$0.426550\pi$
$888$	0	0
$889$	−3.39988e18	−0.230995
$890$	0	0
$891$	1.86471e18	0.124855
$892$	0	0
$893$	1.58490e19	1.04585
$894$	0	0
$895$	4.32659e18	0.281382
$896$	0	0
$897$	1.26137e19	0.808523
$898$	0	0
$899$	2.37447e18	0.150013
$900$	0	0
$901$	−8.60673e17	−0.0535954
$902$	0	0
$903$	−1.57217e19	−0.965002
$904$	0	0
$905$	−2.17836e18	−0.131799
$906$	0	0
$907$	−5.59272e18	−0.333561	−0.166781	−	0.985994i	$-0.553337\pi$
$907$	−5.59272e18	−0.333561	−0.166781	+	0.985994i	$0.553337\pi$
$908$	0	0
$909$	−3.92437e19	−2.30730
$910$	0	0
$911$	−4.19586e18	−0.243193	−0.121597	−	0.992580i	$-0.538801\pi$
$911$	−4.19586e18	−0.243193	−0.121597	+	0.992580i	$0.538801\pi$
$912$	0	0
$913$	−5.46887e18	−0.312491
$914$	0	0
$915$	−1.62303e19	−0.914299
$916$	0	0
$917$	2.73194e18	0.151730
$918$	0	0
$919$	−6.83436e18	−0.374237	−0.187119	−	0.982337i	$-0.559915\pi$
$919$	−6.83436e18	−0.374237	−0.187119	+	0.982337i	$0.559915\pi$
$920$	0	0
$921$	−5.67061e17	−0.0306155
$922$	0	0
$923$	−2.08047e19	−1.10751
$924$	0	0
$925$	5.39074e18	0.282961
$926$	0	0
$927$	1.93715e19	1.00264
$928$	0	0
$929$	−7.68529e18	−0.392246	−0.196123	−	0.980579i	$-0.562835\pi$
$929$	−7.68529e18	−0.392246	−0.196123	+	0.980579i	$0.562835\pi$
$930$	0	0
$931$	1.77549e18	0.0893606
$932$	0	0
$933$	−4.76757e19	−2.36628
$934$	0	0
$935$	9.63684e16	0.00471693
$936$	0	0
$937$	1.82302e19	0.880004	0.440002	−	0.897997i	$-0.354978\pi$
$937$	1.82302e19	0.880004	0.440002	+	0.897997i	$0.354978\pi$
$938$	0	0
$939$	−6.16641e18	−0.293566
$940$	0	0
$941$	4.01498e18	0.188517	0.0942586	−	0.995548i	$-0.469952\pi$
$941$	4.01498e18	0.188517	0.0942586	+	0.995548i	$0.469952\pi$
$942$	0	0
$943$	7.56851e18	0.350497
$944$	0	0
$945$	1.38084e19	0.630721
$946$	0	0
$947$	−2.28347e19	−1.02877	−0.514387	−	0.857558i	$-0.671980\pi$
$947$	−2.28347e19	−1.02877	−0.514387	+	0.857558i	$0.671980\pi$
$948$	0	0
$949$	3.31513e18	0.147323
$950$	0	0
$951$	6.13599e19	2.68975
$952$	0	0
$953$	−3.72340e19	−1.61004	−0.805018	−	0.593250i	$-0.797845\pi$
$953$	−3.72340e19	−1.61004	−0.805018	+	0.593250i	$0.797845\pi$
$954$	0	0
$955$	−1.00425e19	−0.428370
$956$	0	0
$957$	−8.38257e18	−0.352736
$958$	0	0
$959$	−8.55056e18	−0.354955
$960$	0	0
$961$	−2.34836e19	−0.961749
$962$	0	0
$963$	−1.58169e19	−0.639073
$964$	0	0
$965$	−1.91272e18	−0.0762472
$966$	0	0
$967$	−1.86235e19	−0.732471	−0.366235	−	0.930522i	$-0.619353\pi$
$967$	−1.86235e19	−0.732471	−0.366235	+	0.930522i	$0.619353\pi$
$968$	0	0
$969$	−1.60722e18	−0.0623694
$970$	0	0
$971$	4.46491e19	1.70957	0.854787	−	0.518980i	$-0.173688\pi$
$971$	4.46491e19	1.70957	0.854787	+	0.518980i	$0.173688\pi$
$972$	0	0
$973$	2.22184e19	0.839419
$974$	0	0
$975$	1.65054e19	0.615313
$976$	0	0
$977$	3.67191e19	1.35076	0.675378	−	0.737471i	$-0.263980\pi$
$977$	3.67191e19	1.35076	0.675378	+	0.737471i	$0.263980\pi$
$978$	0	0
$979$	−1.16282e19	−0.422109
$980$	0	0
$981$	3.03632e19	1.08768
$982$	0	0
$983$	1.84031e19	0.650570	0.325285	−	0.945616i	$-0.394540\pi$
$983$	1.84031e19	0.650570	0.325285	+	0.945616i	$0.394540\pi$
$984$	0	0
$985$	−2.13931e19	−0.746341
$986$	0	0
$987$	5.48611e19	1.88888
$988$	0	0
$989$	−4.26346e18	−0.144873
$990$	0	0
$991$	−5.17113e19	−1.73423	−0.867115	−	0.498108i	$-0.834028\pi$
$991$	−5.17113e19	−1.73423	−0.867115	+	0.498108i	$0.834028\pi$
$992$	0	0
$993$	7.77102e19	2.57222
$994$	0	0
$995$	2.36532e19	0.772751
$996$	0	0
$997$	−4.39824e19	−1.41827	−0.709137	−	0.705071i	$-0.750915\pi$
$997$	−4.39824e19	−1.41827	−0.709137	+	0.705071i	$0.750915\pi$
$998$	0	0
$999$	−5.99853e19	−1.90928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.14.a.g.1.1		3
4.3	odd	2		5.14.a.b.1.2	✓	3
12.11	even	2		45.14.a.e.1.2		3
20.3	even	4		25.14.b.b.24.3		6
20.7	even	4		25.14.b.b.24.4		6
20.19	odd	2		25.14.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.14.a.b.1.2	✓	3	4.3	odd	2
25.14.a.b.1.2		3	20.19	odd	2
25.14.b.b.24.3		6	20.3	even	4
25.14.b.b.24.4		6	20.7	even	4
45.14.a.e.1.2		3	12.11	even	2
80.14.a.g.1.1		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)

\(f(q)\)	\(=\)	\(q-2114.98 q^{3} +15625.0 q^{5} -325303. q^{7} +2.87881e6 q^{9} +1.61316e6 q^{11} -3.19654e7 q^{13} -3.30465e7 q^{15} +3.82330e6 q^{17} +1.98761e8 q^{19} +6.88008e8 q^{21} +1.86577e8 q^{23} +2.44141e8 q^{25} -2.71667e9 q^{27} +2.45694e9 q^{29} +9.66435e8 q^{31} -3.41179e9 q^{33} -5.08285e9 q^{35} +2.20805e10 q^{37} +6.76061e10 q^{39} +4.05651e10 q^{41} -2.28510e10 q^{43} +4.49814e10 q^{45} +7.97391e10 q^{47} +8.93279e9 q^{49} -8.08620e9 q^{51} -2.25113e11 q^{53} +2.52056e10 q^{55} -4.20376e11 q^{57} -7.96680e10 q^{59} +4.91133e11 q^{61} -9.36485e11 q^{63} -4.99459e11 q^{65} -2.25405e11 q^{67} -3.94606e11 q^{69} +6.50849e11 q^{71} -1.03710e11 q^{73} -5.16352e11 q^{75} -5.24764e11 q^{77} -2.08051e12 q^{79} +1.15594e12 q^{81} -3.39017e12 q^{83} +5.97390e10 q^{85} -5.19638e12 q^{87} -7.20835e12 q^{89} +1.03984e13 q^{91} -2.04399e12 q^{93} +3.10565e12 q^{95} +7.00185e12 q^{97} +4.64397e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 416 q^{3} + 46875 q^{5} - 448292 q^{7} + 1286119 q^{9} + 6604004 q^{11} - 33501974 q^{13} - 6500000 q^{15} + 83129542 q^{17} - 97491100 q^{19} + 438200736 q^{21} - 316255836 q^{23} + 732421875 q^{25}+ \cdots - 126787366508 q^{99}+O(q^{100}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more