LMFDB - Embedded newform 80.20.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,20,Mod(1,80)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$183.053357245$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5694737x^{2} - 4823342760x - 829238069200 $ x^4 - 2x^3 - 5694737x^2 - 4823342760*x - 829238069200
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{4}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-698.569$ 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)

$f(q)$	$=$	$q+27215.3 q^{3} +1.95312e6 q^{5} +3.09073e6 q^{7} -4.21590e8 q^{9} +O(q^{10})$	$q+27215.3 q^{3} +1.95312e6 q^{5} +3.09073e6 q^{7} -4.21590e8 q^{9} -8.07566e9 q^{11} -4.39560e10 q^{13} +5.31549e10 q^{15} +7.50013e11 q^{17} +9.89618e11 q^{19} +8.41152e10 q^{21} +1.32170e13 q^{23} +3.81470e12 q^{25} -4.31050e13 q^{27} +1.43020e13 q^{29} -1.15261e14 q^{31} -2.19781e14 q^{33} +6.03659e12 q^{35} +1.12411e15 q^{37} -1.19628e15 q^{39} +4.88280e14 q^{41} -4.36116e14 q^{43} -8.23418e14 q^{45} -9.18620e15 q^{47} -1.13893e16 q^{49} +2.04118e16 q^{51} +1.23209e15 q^{53} -1.57728e16 q^{55} +2.69327e16 q^{57} -2.32458e16 q^{59} -1.60834e17 q^{61} -1.30302e15 q^{63} -8.58516e16 q^{65} +3.10649e17 q^{67} +3.59705e17 q^{69} -6.19865e17 q^{71} -7.55943e17 q^{73} +1.03818e17 q^{75} -2.49597e16 q^{77} +8.03273e17 q^{79} -6.83116e17 q^{81} -1.96399e18 q^{83} +1.46487e18 q^{85} +3.89232e17 q^{87} -3.13510e18 q^{89} -1.35856e17 q^{91} -3.13686e18 q^{93} +1.93285e18 q^{95} -4.89005e18 q^{97} +3.40462e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 46152 q^{3} + 7812500 q^{5} - 59073944 q^{7} + 25995060 q^{9}+O(q^{10}) $ 4 * q + 46152 * q^3 + 7812500 * q^5 - 59073944 * q^7 + 25995060 * q^9	$ 4 q + 46152 q^{3} + 7812500 q^{5} - 59073944 q^{7} + 25995060 q^{9} + 3675173728 q^{11} + 7175523640 q^{13} + 90140625000 q^{15} - 567763095896 q^{17} + 278184526832 q^{19} - 5545032556608 q^{21} + 3524328334936 q^{23} + 15258789062500 q^{25} + 39633120308304 q^{27} - 62219259151144 q^{29} - 247254182466832 q^{31} + 404147490823008 q^{33} - 115378796875000 q^{35} + 711533444521656 q^{37} - 29\!\cdots\!76 q^{39}+ \cdots + 21\!\cdots\!96 q^{99}+O(q^{100}) $ 4 * q + 46152 * q^3 + 7812500 * q^5 - 59073944 * q^7 + 25995060 * q^9 + 3675173728 * q^11 + 7175523640 * q^13 + 90140625000 * q^15 - 567763095896 * q^17 + 278184526832 * q^19 - 5545032556608 * q^21 + 3524328334936 * q^23 + 15258789062500 * q^25 + 39633120308304 * q^27 - 62219259151144 * q^29 - 247254182466832 * q^31 + 404147490823008 * q^33 - 115378796875000 * q^35 + 711533444521656 * q^37 - 2943955031648976 * q^39 - 868244518404600 * q^41 - 3716515028101432 * q^43 + 50771601562500 * q^45 - 8431486495352 * q^47 - 16013476405774140 * q^49 - 14635182514702320 * q^51 - 41519794873411368 * q^53 + 7178073687500000 * q^55 - 54294834644792928 * q^57 - 14480346320386000 * q^59 + 27840348176113048 * q^61 + 37085058206149896 * q^63 + 14014694609375000 * q^65 - 8174924905481000 * q^67 - 355071316348294848 * q^69 + 555093531720812496 * q^71 - 408571387318908216 * q^73 + 176055908203125000 * q^75 - 786239164619967200 * q^77 + 249424427517101088 * q^79 - 798177693248534844 * q^81 + 1696125608725987400 * q^83 - 1108912296671875000 * q^85 - 1094924005281685584 * q^87 - 3568552069941379992 * q^89 - 247232489018471312 * q^91 - 12281064825141550080 * q^93 + 543329153968750000 * q^95 - 22399415502269235032 * q^97 + 21952027783694553696 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	27215.3	0.798290	0.399145	−	0.916888i	$-0.369307\pi$
$3$	27215.3	0.798290	0.399145	+	0.916888i	$0.369307\pi$
$4$	0	0
$5$	1.95312e6	0.447214
$5$	1.95312e6	0.447214
$6$	0	0
$7$	3.09073e6	0.0289488	0.0144744	−	0.999895i	$-0.495392\pi$
$7$	3.09073e6	0.0289488	0.0144744	+	0.999895i	$0.495392\pi$
$8$	0	0
$9$	−4.21590e8	−0.362732
$10$	0	0
$11$	−8.07566e9	−1.03264	−0.516319	−	0.856397i	$-0.672698\pi$
$11$	−8.07566e9	−1.03264	−0.516319	+	0.856397i	$0.672698\pi$
$12$	0	0
$13$	−4.39560e10	−1.14963	−0.574813	−	0.818285i	$-0.694925\pi$
$13$	−4.39560e10	−1.14963	−0.574813	+	0.818285i	$0.694925\pi$
$14$	0	0
$15$	5.31549e10	0.357006
$16$	0	0
$17$	7.50013e11	1.53392	0.766962	−	0.641693i	$-0.221768\pi$
$17$	7.50013e11	1.53392	0.766962	+	0.641693i	$0.221768\pi$
$18$	0	0
$19$	9.89618e11	0.703572	0.351786	−	0.936080i	$-0.385575\pi$
$19$	9.89618e11	0.703572	0.351786	+	0.936080i	$0.385575\pi$
$20$	0	0
$21$	8.41152e10	0.0231095
$22$	0	0
$23$	1.32170e13	1.53010	0.765048	−	0.643973i	$-0.222715\pi$
$23$	1.32170e13	1.53010	0.765048	+	0.643973i	$0.222715\pi$
$24$	0	0
$25$	3.81470e12	0.200000
$26$	0	0
$27$	−4.31050e13	−1.08786
$28$	0	0
$29$	1.43020e13	0.183069	0.0915344	−	0.995802i	$-0.470823\pi$
$29$	1.43020e13	0.183069	0.0915344	+	0.995802i	$0.470823\pi$
$30$	0	0
$31$	−1.15261e14	−0.782973	−0.391487	−	0.920184i	$-0.628039\pi$
$31$	−1.15261e14	−0.782973	−0.391487	+	0.920184i	$0.628039\pi$
$32$	0	0
$33$	−2.19781e14	−0.824344
$34$	0	0
$35$	6.03659e12	0.0129463
$36$	0	0
$37$	1.12411e15	1.42198	0.710990	−	0.703203i	$-0.248247\pi$
$37$	1.12411e15	1.42198	0.710990	+	0.703203i	$0.248247\pi$
$38$	0	0
$39$	−1.19628e15	−0.917735
$40$	0	0
$41$	4.88280e14	0.232929	0.116464	−	0.993195i	$-0.462844\pi$
$41$	4.88280e14	0.232929	0.116464	+	0.993195i	$0.462844\pi$
$42$	0	0
$43$	−4.36116e14	−0.132328	−0.0661640	−	0.997809i	$-0.521076\pi$
$43$	−4.36116e14	−0.132328	−0.0661640	+	0.997809i	$0.521076\pi$
$44$	0	0
$45$	−8.23418e14	−0.162219
$46$	0	0
$47$	−9.18620e15	−1.19731	−0.598654	−	0.801008i	$-0.704298\pi$
$47$	−9.18620e15	−1.19731	−0.598654	+	0.801008i	$0.704298\pi$
$48$	0	0
$49$	−1.13893e16	−0.999162
$50$	0	0
$51$	2.04118e16	1.22452
$52$	0	0
$53$	1.23209e15	0.0512887	0.0256444	−	0.999671i	$-0.491836\pi$
$53$	1.23209e15	0.0512887	0.0256444	+	0.999671i	$0.491836\pi$
$54$	0	0
$55$	−1.57728e16	−0.461809
$56$	0	0
$57$	2.69327e16	0.561655
$58$	0	0
$59$	−2.32458e16	−0.349341	−0.174671	−	0.984627i	$-0.555886\pi$
$59$	−2.32458e16	−0.349341	−0.174671	+	0.984627i	$0.555886\pi$
$60$	0	0
$61$	−1.60834e17	−1.76094	−0.880468	−	0.474106i	$-0.842771\pi$
$61$	−1.60834e17	−1.76094	−0.880468	+	0.474106i	$0.842771\pi$
$62$	0	0
$63$	−1.30302e15	−0.0105007
$64$	0	0
$65$	−8.58516e16	−0.514128
$66$	0	0
$67$	3.10649e17	1.39495	0.697476	−	0.716608i	$-0.254306\pi$
$67$	3.10649e17	1.39495	0.697476	+	0.716608i	$0.254306\pi$
$68$	0	0
$69$	3.59705e17	1.22146
$70$	0	0
$71$	−6.19865e17	−1.60451	−0.802255	−	0.596981i	$-0.796367\pi$
$71$	−6.19865e17	−1.60451	−0.802255	+	0.596981i	$0.796367\pi$
$72$	0	0
$73$	−7.55943e17	−1.50287	−0.751436	−	0.659806i	$-0.770638\pi$
$73$	−7.55943e17	−1.50287	−0.751436	+	0.659806i	$0.770638\pi$
$74$	0	0
$75$	1.03818e17	0.159658
$76$	0	0
$77$	−2.49597e16	−0.0298936
$78$	0	0
$79$	8.03273e17	0.754060	0.377030	−	0.926201i	$-0.376945\pi$
$79$	8.03273e17	0.754060	0.377030	+	0.926201i	$0.376945\pi$
$80$	0	0
$81$	−6.83116e17	−0.505693
$82$	0	0
$83$	−1.96399e18	−1.15318	−0.576589	−	0.817034i	$-0.695617\pi$
$83$	−1.96399e18	−1.15318	−0.576589	+	0.817034i	$0.695617\pi$
$84$	0	0
$85$	1.46487e18	0.685992
$86$	0	0
$87$	3.89232e17	0.146142
$88$	0	0
$89$	−3.13510e18	−0.948518	−0.474259	−	0.880385i	$-0.657284\pi$
$89$	−3.13510e18	−0.948518	−0.474259	+	0.880385i	$0.657284\pi$
$90$	0	0
$91$	−1.35856e17	−0.0332802
$92$	0	0
$93$	−3.13686e18	−0.625040
$94$	0	0
$95$	1.93285e18	0.314647
$96$	0	0
$97$	−4.89005e18	−0.653103	−0.326552	−	0.945179i	$-0.605887\pi$
$97$	−4.89005e18	−0.653103	−0.326552	+	0.945179i	$0.605887\pi$
$98$	0	0
$99$	3.40462e18	0.374571
$100$	0	0
$101$	−1.12046e19	−1.01940	−0.509699	−	0.860353i	$-0.670243\pi$
$101$	−1.12046e19	−1.01940	−0.509699	+	0.860353i	$0.670243\pi$
$102$	0	0
$103$	1.38053e19	1.04254	0.521271	−	0.853391i	$-0.325458\pi$
$103$	1.38053e19	1.04254	0.521271	+	0.853391i	$0.325458\pi$
$104$	0	0
$105$	1.64287e17	0.0103349
$106$	0	0
$107$	6.96643e18	0.366323	0.183162	−	0.983083i	$-0.441367\pi$
$107$	6.96643e18	0.366323	0.183162	+	0.983083i	$0.441367\pi$
$108$	0	0
$109$	−2.71005e19	−1.19516	−0.597580	−	0.801809i	$-0.703871\pi$
$109$	−2.71005e19	−1.19516	−0.597580	+	0.801809i	$0.703871\pi$
$110$	0	0
$111$	3.05930e19	1.13515
$112$	0	0
$113$	−1.93633e19	−0.606366	−0.303183	−	0.952932i	$-0.598049\pi$
$113$	−1.93633e19	−0.606366	−0.303183	+	0.952932i	$0.598049\pi$
$114$	0	0
$115$	2.58145e19	0.684280
$116$	0	0
$117$	1.85314e19	0.417006
$118$	0	0
$119$	2.31809e18	0.0444052
$120$	0	0
$121$	4.05725e18	0.0663394
$122$	0	0
$123$	1.32887e19	0.185945
$124$	0	0
$125$	7.45058e18	0.0894427
$126$	0	0
$127$	1.21298e20	1.25233	0.626163	−	0.779693i	$-0.284625\pi$
$127$	1.21298e20	1.25233	0.626163	+	0.779693i	$0.284625\pi$
$128$	0	0
$129$	−1.18690e19	−0.105636
$130$	0	0
$131$	4.41655e19	0.339630	0.169815	−	0.985476i	$-0.445683\pi$
$131$	4.41655e19	0.339630	0.169815	+	0.985476i	$0.445683\pi$
$132$	0	0
$133$	3.05865e18	0.0203675
$134$	0	0
$135$	−8.41894e19	−0.486504
$136$	0	0
$137$	−7.38366e19	−0.371045	−0.185522	−	0.982640i	$-0.559398\pi$
$137$	−7.38366e19	−0.371045	−0.185522	+	0.982640i	$0.559398\pi$
$138$	0	0
$139$	1.33905e20	0.586351	0.293175	−	0.956059i	$-0.405288\pi$
$139$	1.33905e20	0.586351	0.293175	+	0.956059i	$0.405288\pi$
$140$	0	0
$141$	−2.50005e20	−0.955800
$142$	0	0
$143$	3.54974e20	1.18715
$144$	0	0
$145$	2.79335e19	0.0818709
$146$	0	0
$147$	−3.09964e20	−0.797621
$148$	0	0
$149$	−5.98446e20	−1.35443	−0.677213	−	0.735787i	$-0.736813\pi$
$149$	−5.98446e20	−1.35443	−0.677213	+	0.735787i	$0.736813\pi$
$150$	0	0
$151$	−4.29575e20	−0.856560	−0.428280	−	0.903646i	$-0.640880\pi$
$151$	−4.29575e20	−0.856560	−0.428280	+	0.903646i	$0.640880\pi$
$152$	0	0
$153$	−3.16198e20	−0.556404
$154$	0	0
$155$	−2.25119e20	−0.350156
$156$	0	0
$157$	3.32346e20	0.457661	0.228830	−	0.973466i	$-0.426510\pi$
$157$	3.32346e20	0.457661	0.228830	+	0.973466i	$0.426510\pi$
$158$	0	0
$159$	3.35317e19	0.0409433
$160$	0	0
$161$	4.08503e19	0.0442944
$162$	0	0
$163$	−1.72568e20	−0.166410	−0.0832049	−	0.996532i	$-0.526516\pi$
$163$	−1.72568e20	−0.166410	−0.0832049	+	0.996532i	$0.526516\pi$
$164$	0	0
$165$	−4.29261e20	−0.368658
$166$	0	0
$167$	−2.05209e21	−1.57178	−0.785888	−	0.618369i	$-0.787794\pi$
$167$	−2.05209e21	−1.57178	−0.785888	+	0.618369i	$0.787794\pi$
$168$	0	0
$169$	4.70211e20	0.321639
$170$	0	0
$171$	−4.17213e20	−0.255208
$172$	0	0
$173$	1.90009e21	1.04073	0.520363	−	0.853945i	$-0.325797\pi$
$173$	1.90009e21	1.04073	0.520363	+	0.853945i	$0.325797\pi$
$174$	0	0
$175$	1.17902e19	0.00578975
$176$	0	0
$177$	−6.32641e20	−0.278876
$178$	0	0
$179$	−1.97642e21	−0.783024	−0.391512	−	0.920173i	$-0.628048\pi$
$179$	−1.97642e21	−0.783024	−0.391512	+	0.920173i	$0.628048\pi$
$180$	0	0
$181$	−7.63756e20	−0.272275	−0.136138	−	0.990690i	$-0.543469\pi$
$181$	−7.63756e20	−0.272275	−0.136138	+	0.990690i	$0.543469\pi$
$182$	0	0
$183$	−4.37713e21	−1.40574
$184$	0	0
$185$	2.19553e21	0.635928
$186$	0	0
$187$	−6.05685e21	−1.58399
$188$	0	0
$189$	−1.33226e20	−0.0314921
$190$	0	0
$191$	−2.87324e19	−0.00614548	−0.00307274	−	0.999995i	$-0.500978\pi$
$191$	−2.87324e19	−0.00614548	−0.00307274	+	0.999995i	$0.500978\pi$
$192$	0	0
$193$	−7.08575e21	−1.37275	−0.686375	−	0.727248i	$-0.740799\pi$
$193$	−7.08575e21	−1.37275	−0.686375	+	0.727248i	$0.740799\pi$
$194$	0	0
$195$	−2.33648e21	−0.410424
$196$	0	0
$197$	7.05860e21	1.12536	0.562678	−	0.826676i	$-0.309771\pi$
$197$	7.05860e21	1.12536	0.562678	+	0.826676i	$0.309771\pi$
$198$	0	0
$199$	1.08983e22	1.57854	0.789271	−	0.614045i	$-0.210459\pi$
$199$	1.08983e22	1.57854	0.789271	+	0.614045i	$0.210459\pi$
$200$	0	0
$201$	8.45440e21	1.11358
$202$	0	0
$203$	4.42035e19	0.00529962
$204$	0	0
$205$	9.53673e20	0.104169
$206$	0	0
$207$	−5.57216e21	−0.555015
$208$	0	0
$209$	−7.99182e21	−0.726534
$210$	0	0
$211$	−1.00979e22	−0.838586	−0.419293	−	0.907851i	$-0.637722\pi$
$211$	−1.00979e22	−0.838586	−0.419293	+	0.907851i	$0.637722\pi$
$212$	0	0
$213$	−1.68698e22	−1.28087
$214$	0	0
$215$	−8.51789e20	−0.0591788
$216$	0	0
$217$	−3.56241e20	−0.0226661
$218$	0	0
$219$	−2.05732e22	−1.19973
$220$	0	0
$221$	−3.29676e22	−1.76344
$222$	0	0
$223$	−3.14720e21	−0.154535	−0.0772677	−	0.997010i	$-0.524620\pi$
$223$	−3.14720e21	−0.154535	−0.0772677	+	0.997010i	$0.524620\pi$
$224$	0	0
$225$	−1.60824e21	−0.0725465
$226$	0	0
$227$	1.77775e22	0.737267	0.368634	−	0.929575i	$-0.379826\pi$
$227$	1.77775e22	0.737267	0.368634	+	0.929575i	$0.379826\pi$
$228$	0	0
$229$	−4.44428e22	−1.69576	−0.847881	−	0.530187i	$-0.822122\pi$
$229$	−4.44428e22	−1.69576	−0.847881	+	0.530187i	$0.822122\pi$
$230$	0	0
$231$	−6.79286e20	−0.0238638
$232$	0	0
$233$	3.32328e22	1.07569	0.537844	−	0.843044i	$-0.319239\pi$
$233$	3.32328e22	1.07569	0.537844	+	0.843044i	$0.319239\pi$
$234$	0	0
$235$	−1.79418e22	−0.535453
$236$	0	0
$237$	2.18613e22	0.601959
$238$	0	0
$239$	−3.02208e22	−0.768292	−0.384146	−	0.923272i	$-0.625504\pi$
$239$	−3.02208e22	−0.768292	−0.384146	+	0.923272i	$0.625504\pi$
$240$	0	0
$241$	−3.13720e22	−0.736851	−0.368426	−	0.929657i	$-0.620103\pi$
$241$	−3.13720e22	−0.736851	−0.368426	+	0.929657i	$0.620103\pi$
$242$	0	0
$243$	3.15080e22	0.684166
$244$	0	0
$245$	−2.22448e22	−0.446839
$246$	0	0
$247$	−4.34997e22	−0.808844
$248$	0	0
$249$	−5.34505e22	−0.920572
$250$	0	0
$251$	2.19688e22	0.350675	0.175338	−	0.984508i	$-0.443898\pi$
$251$	2.19688e22	0.350675	0.175338	+	0.984508i	$0.443898\pi$
$252$	0	0
$253$	−1.06736e23	−1.58003
$254$	0	0
$255$	3.98668e22	0.547621
$256$	0	0
$257$	−1.52589e23	−1.94607	−0.973033	−	0.230664i	$-0.925910\pi$
$257$	−1.52589e23	−1.94607	−0.973033	+	0.230664i	$0.925910\pi$
$258$	0	0
$259$	3.47433e21	0.0411645
$260$	0	0
$261$	−6.02956e21	−0.0664050
$262$	0	0
$263$	−6.92386e22	−0.709199	−0.354600	−	0.935018i	$-0.615383\pi$
$263$	−6.92386e22	−0.709199	−0.354600	+	0.935018i	$0.615383\pi$
$264$	0	0
$265$	2.40643e21	0.0229370
$266$	0	0
$267$	−8.53225e22	−0.757193
$268$	0	0
$269$	−8.00310e22	−0.661625	−0.330812	−	0.943697i	$-0.607323\pi$
$269$	−8.00310e22	−0.661625	−0.330812	+	0.943697i	$0.607323\pi$
$270$	0	0
$271$	−2.83779e22	−0.218661	−0.109331	−	0.994005i	$-0.534871\pi$
$271$	−2.83779e22	−0.218661	−0.109331	+	0.994005i	$0.534871\pi$
$272$	0	0
$273$	−3.69737e21	−0.0265673
$274$	0	0
$275$	−3.08062e22	−0.206527
$276$	0	0
$277$	−1.79755e23	−1.12492	−0.562460	−	0.826824i	$-0.690145\pi$
$277$	−1.79755e23	−1.12492	−0.562460	+	0.826824i	$0.690145\pi$
$278$	0	0
$279$	4.85929e22	0.284010
$280$	0	0
$281$	2.94944e22	0.161076	0.0805378	−	0.996752i	$-0.474336\pi$
$281$	2.94944e22	0.161076	0.0805378	+	0.996752i	$0.474336\pi$
$282$	0	0
$283$	4.47235e22	0.228331	0.114166	−	0.993462i	$-0.463581\pi$
$283$	4.47235e22	0.228331	0.114166	+	0.993462i	$0.463581\pi$
$284$	0	0
$285$	5.26030e22	0.251180
$286$	0	0
$287$	1.50914e21	0.00674299
$288$	0	0
$289$	3.23446e23	1.35292
$290$	0	0
$291$	−1.33084e23	−0.521366
$292$	0	0
$293$	1.70855e23	0.627172	0.313586	−	0.949560i	$-0.398470\pi$
$293$	1.70855e23	0.627172	0.313586	+	0.949560i	$0.398470\pi$
$294$	0	0
$295$	−4.54019e22	−0.156230
$296$	0	0
$297$	3.48101e23	1.12336
$298$	0	0
$299$	−5.80967e23	−1.75904
$300$	0	0
$301$	−1.34792e21	−0.00383073
$302$	0	0
$303$	−3.04937e23	−0.813776
$304$	0	0
$305$	−3.14128e23	−0.787514
$306$	0	0
$307$	5.92916e23	1.39694	0.698471	−	0.715638i	$-0.253864\pi$
$307$	5.92916e23	1.39694	0.698471	+	0.715638i	$0.253864\pi$
$308$	0	0
$309$	3.75716e23	0.832252
$310$	0	0
$311$	−1.38272e22	−0.0288079	−0.0144040	−	0.999896i	$-0.504585\pi$
$311$	−1.38272e22	−0.0288079	−0.0144040	+	0.999896i	$0.504585\pi$
$312$	0	0
$313$	9.40325e23	1.84335	0.921673	−	0.387968i	$-0.126823\pi$
$313$	9.40325e23	1.84335	0.921673	+	0.387968i	$0.126823\pi$
$314$	0	0
$315$	−2.54496e21	−0.00469603
$316$	0	0
$317$	−2.99742e23	−0.520816	−0.260408	−	0.965499i	$-0.583857\pi$
$317$	−2.99742e23	−0.520816	−0.260408	+	0.965499i	$0.583857\pi$
$318$	0	0
$319$	−1.15498e23	−0.189044
$320$	0	0
$321$	1.89593e23	0.292432
$322$	0	0
$323$	7.42226e23	1.07923
$324$	0	0
$325$	−1.67679e23	−0.229925
$326$	0	0
$327$	−7.37549e23	−0.954085
$328$	0	0
$329$	−2.83921e22	−0.0346606
$330$	0	0
$331$	2.83775e23	0.327046	0.163523	−	0.986540i	$-0.447714\pi$
$331$	2.83775e23	0.327046	0.163523	+	0.986540i	$0.447714\pi$
$332$	0	0
$333$	−4.73914e23	−0.515798
$334$	0	0
$335$	6.06736e23	0.623841
$336$	0	0
$337$	1.26986e24	1.23387	0.616937	−	0.787013i	$-0.288373\pi$
$337$	1.26986e24	1.23387	0.616937	+	0.787013i	$0.288373\pi$
$338$	0	0
$339$	−5.26979e23	−0.484056
$340$	0	0
$341$	9.30810e23	0.808527
$342$	0	0
$343$	−7.04324e22	−0.0578733
$344$	0	0
$345$	7.02548e23	0.546254
$346$	0	0
$347$	1.28679e24	0.947063	0.473532	−	0.880777i	$-0.342979\pi$
$347$	1.28679e24	0.947063	0.473532	+	0.880777i	$0.342979\pi$
$348$	0	0
$349$	−2.17876e24	−1.51834	−0.759169	−	0.650894i	$-0.774394\pi$
$349$	−2.17876e24	−1.51834	−0.759169	+	0.650894i	$0.774394\pi$
$350$	0	0
$351$	1.89472e24	1.25063
$352$	0	0
$353$	1.05724e24	0.661173	0.330587	−	0.943776i	$-0.392753\pi$
$353$	1.05724e24	0.661173	0.330587	+	0.943776i	$0.392753\pi$
$354$	0	0
$355$	−1.21067e24	−0.717559
$356$	0	0
$357$	6.30874e22	0.0354482
$358$	0	0
$359$	−1.01501e24	−0.540845	−0.270422	−	0.962742i	$-0.587163\pi$
$359$	−1.01501e24	−0.540845	−0.270422	+	0.962742i	$0.587163\pi$
$360$	0	0
$361$	−9.99075e23	−0.504987
$362$	0	0
$363$	1.10419e23	0.0529581
$364$	0	0
$365$	−1.47645e24	−0.672105
$366$	0	0
$367$	1.42891e24	0.617559	0.308780	−	0.951134i	$-0.400079\pi$
$367$	1.42891e24	0.617559	0.308780	+	0.951134i	$0.400079\pi$
$368$	0	0
$369$	−2.05854e23	−0.0844907
$370$	0	0
$371$	3.80806e21	0.00148475
$372$	0	0
$373$	7.60962e23	0.281922	0.140961	−	0.990015i	$-0.454981\pi$
$373$	7.60962e23	0.281922	0.140961	+	0.990015i	$0.454981\pi$
$374$	0	0
$375$	2.02770e23	0.0714013
$376$	0	0
$377$	−6.28657e23	−0.210461
$378$	0	0
$379$	1.08981e24	0.346959	0.173480	−	0.984837i	$-0.444499\pi$
$379$	1.08981e24	0.346959	0.173480	+	0.984837i	$0.444499\pi$
$380$	0	0
$381$	3.30115e24	0.999719
$382$	0	0
$383$	6.34081e24	1.82708	0.913538	−	0.406754i	$-0.133339\pi$
$383$	6.34081e24	1.82708	0.913538	+	0.406754i	$0.133339\pi$
$384$	0	0
$385$	−4.87495e22	−0.0133688
$386$	0	0
$387$	1.83862e23	0.0479996
$388$	0	0
$389$	−8.53868e23	−0.212261	−0.106130	−	0.994352i	$-0.533846\pi$
$389$	−8.53868e23	−0.212261	−0.106130	+	0.994352i	$0.533846\pi$
$390$	0	0
$391$	9.91292e24	2.34705
$392$	0	0
$393$	1.20198e24	0.271123
$394$	0	0
$395$	1.56889e24	0.337226
$396$	0	0
$397$	6.19617e24	1.26944	0.634722	−	0.772740i	$-0.281115\pi$
$397$	6.19617e24	1.26944	0.634722	+	0.772740i	$0.281115\pi$
$398$	0	0
$399$	8.32419e22	0.0162592
$400$	0	0
$401$	−3.64139e24	−0.678260	−0.339130	−	0.940740i	$-0.610133\pi$
$401$	−3.64139e24	−0.678260	−0.339130	+	0.940740i	$0.610133\pi$
$402$	0	0
$403$	5.06642e24	0.900126
$404$	0	0
$405$	−1.33421e24	−0.226153
$406$	0	0
$407$	−9.07795e24	−1.46839
$408$	0	0
$409$	−7.80129e23	−0.120447	−0.0602234	−	0.998185i	$-0.519181\pi$
$409$	−7.80129e23	−0.120447	−0.0602234	+	0.998185i	$0.519181\pi$
$410$	0	0
$411$	−2.00948e24	−0.296202
$412$	0	0
$413$	−7.18465e22	−0.0101130
$414$	0	0
$415$	−3.83591e24	−0.515717
$416$	0	0
$417$	3.64427e24	0.468078
$418$	0	0
$419$	1.35681e25	1.66528	0.832640	−	0.553814i	$-0.186828\pi$
$419$	1.35681e25	1.66528	0.832640	+	0.553814i	$0.186828\pi$
$420$	0	0
$421$	−1.33328e25	−1.56402	−0.782010	−	0.623266i	$-0.785805\pi$
$421$	−1.33328e25	−1.56402	−0.782010	+	0.623266i	$0.785805\pi$
$422$	0	0
$423$	3.87281e24	0.434302
$424$	0	0
$425$	2.86107e24	0.306785
$426$	0	0
$427$	−4.97094e23	−0.0509769
$428$	0	0
$429$	9.66072e24	0.947687
$430$	0	0
$431$	6.98945e22	0.00656007	0.00328004	−	0.999995i	$-0.498956\pi$
$431$	6.98945e22	0.00656007	0.00328004	+	0.999995i	$0.498956\pi$
$432$	0	0
$433$	1.73903e25	1.56197	0.780985	−	0.624550i	$-0.214718\pi$
$433$	1.73903e25	1.56197	0.780985	+	0.624550i	$0.214718\pi$
$434$	0	0
$435$	7.60218e23	0.0653567
$436$	0	0
$437$	1.30798e25	1.07653
$438$	0	0
$439$	−1.96341e25	−1.54738	−0.773690	−	0.633564i	$-0.781591\pi$
$439$	−1.96341e25	−1.54738	−0.773690	+	0.633564i	$0.781591\pi$
$440$	0	0
$441$	4.80163e24	0.362428
$442$	0	0
$443$	−1.92714e25	−1.39341	−0.696704	−	0.717359i	$-0.745351\pi$
$443$	−1.92714e25	−1.39341	−0.696704	+	0.717359i	$0.745351\pi$
$444$	0	0
$445$	−6.12323e24	−0.424190
$446$	0	0
$447$	−1.62869e25	−1.08123
$448$	0	0
$449$	−2.56567e25	−1.63253	−0.816265	−	0.577678i	$-0.803959\pi$
$449$	−2.56567e25	−1.63253	−0.816265	+	0.577678i	$0.803959\pi$
$450$	0	0
$451$	−3.94319e24	−0.240531
$452$	0	0
$453$	−1.16910e25	−0.683784
$454$	0	0
$455$	−2.65344e23	−0.0148834
$456$	0	0
$457$	−4.04835e24	−0.217808	−0.108904	−	0.994052i	$-0.534734\pi$
$457$	−4.04835e24	−0.217808	−0.108904	+	0.994052i	$0.534734\pi$
$458$	0	0
$459$	−3.23293e25	−1.66869
$460$	0	0
$461$	2.56974e25	1.27271	0.636357	−	0.771395i	$-0.280440\pi$
$461$	2.56974e25	1.27271	0.636357	+	0.771395i	$0.280440\pi$
$462$	0	0
$463$	6.40726e24	0.304546	0.152273	−	0.988338i	$-0.451341\pi$
$463$	6.40726e24	0.304546	0.152273	+	0.988338i	$0.451341\pi$
$464$	0	0
$465$	−6.12669e24	−0.279526
$466$	0	0
$467$	−2.80604e25	−1.22909	−0.614545	−	0.788882i	$-0.710660\pi$
$467$	−2.80604e25	−1.22909	−0.614545	+	0.788882i	$0.710660\pi$
$468$	0	0
$469$	9.60133e23	0.0403821
$470$	0	0
$471$	9.04489e24	0.365346
$472$	0	0
$473$	3.52192e24	0.136647
$474$	0	0
$475$	3.77509e24	0.140714
$476$	0	0
$477$	−5.19437e23	−0.0186041
$478$	0	0
$479$	−2.91576e25	−1.00361	−0.501804	−	0.864981i	$-0.667330\pi$
$479$	−2.91576e25	−1.00361	−0.501804	+	0.864981i	$0.667330\pi$
$480$	0	0
$481$	−4.94115e25	−1.63474
$482$	0	0
$483$	1.11175e24	0.0353598
$484$	0	0
$485$	−9.55088e24	−0.292077
$486$	0	0
$487$	1.21464e25	0.357209	0.178605	−	0.983921i	$-0.442842\pi$
$487$	1.21464e25	0.357209	0.178605	+	0.983921i	$0.442842\pi$
$488$	0	0
$489$	−4.69650e24	−0.132843
$490$	0	0
$491$	6.51785e25	1.77350	0.886748	−	0.462253i	$-0.152959\pi$
$491$	6.51785e25	1.77350	0.886748	+	0.462253i	$0.152959\pi$
$492$	0	0
$493$	1.07266e25	0.280814
$494$	0	0
$495$	6.64964e24	0.167513
$496$	0	0
$497$	−1.91584e24	−0.0464486
$498$	0	0
$499$	−6.40335e25	−1.49435	−0.747175	−	0.664628i	$-0.768590\pi$
$499$	−6.40335e25	−1.49435	−0.747175	+	0.664628i	$0.768590\pi$
$500$	0	0
$501$	−5.58483e25	−1.25473
$502$	0	0
$503$	−5.29992e25	−1.14650	−0.573249	−	0.819381i	$-0.694317\pi$
$503$	−5.29992e25	−1.14650	−0.573249	+	0.819381i	$0.694317\pi$
$504$	0	0
$505$	−2.18840e25	−0.455889
$506$	0	0
$507$	1.27969e25	0.256761
$508$	0	0
$509$	−3.76060e24	−0.0726839	−0.0363419	−	0.999339i	$-0.511571\pi$
$509$	−3.76060e24	−0.0726839	−0.0363419	+	0.999339i	$0.511571\pi$
$510$	0	0
$511$	−2.33642e24	−0.0435063
$512$	0	0
$513$	−4.26575e25	−0.765385
$514$	0	0
$515$	2.69636e25	0.466239
$516$	0	0
$517$	7.41846e25	1.23639
$518$	0	0
$519$	5.17116e25	0.830802
$520$	0	0
$521$	−1.01672e26	−1.57486	−0.787431	−	0.616402i	$-0.788589\pi$
$521$	−1.01672e26	−1.57486	−0.787431	+	0.616402i	$0.788589\pi$
$522$	0	0
$523$	1.40906e25	0.210458	0.105229	−	0.994448i	$-0.466443\pi$
$523$	1.40906e25	0.210458	0.105229	+	0.994448i	$0.466443\pi$
$524$	0	0
$525$	3.20874e23	0.00462191
$526$	0	0
$527$	−8.64473e25	−1.20102
$528$	0	0
$529$	1.00074e26	1.34120
$530$	0	0
$531$	9.80019e24	0.126717
$532$	0	0
$533$	−2.14629e25	−0.267781
$534$	0	0
$535$	1.36063e25	0.163825
$536$	0	0
$537$	−5.37888e25	−0.625080
$538$	0	0
$539$	9.19765e25	1.03177
$540$	0	0
$541$	−1.47644e25	−0.159898	−0.0799490	−	0.996799i	$-0.525476\pi$
$541$	−1.47644e25	−0.159898	−0.0799490	+	0.996799i	$0.525476\pi$
$542$	0	0
$543$	−2.07858e25	−0.217355
$544$	0	0
$545$	−5.29307e25	−0.534492
$546$	0	0
$547$	2.81334e25	0.274374	0.137187	−	0.990545i	$-0.456194\pi$
$547$	2.81334e25	0.274374	0.137187	+	0.990545i	$0.456194\pi$
$548$	0	0
$549$	6.78058e25	0.638748
$550$	0	0
$551$	1.41535e25	0.128802
$552$	0	0
$553$	2.48270e24	0.0218291
$554$	0	0
$555$	5.97520e25	0.507656
$556$	0	0
$557$	−1.14539e26	−0.940436	−0.470218	−	0.882550i	$-0.655825\pi$
$557$	−1.14539e26	−0.940436	−0.470218	+	0.882550i	$0.655825\pi$
$558$	0	0
$559$	1.91699e25	0.152128
$560$	0	0
$561$	−1.64839e26	−1.26448
$562$	0	0
$563$	−7.02863e24	−0.0521244	−0.0260622	−	0.999660i	$-0.508297\pi$
$563$	−7.02863e24	−0.0521244	−0.0260622	+	0.999660i	$0.508297\pi$
$564$	0	0
$565$	−3.78190e25	−0.271175
$566$	0	0
$567$	−2.11133e24	−0.0146392
$568$	0	0
$569$	1.95768e26	1.31273	0.656364	−	0.754444i	$-0.272093\pi$
$569$	1.95768e26	1.31273	0.656364	+	0.754444i	$0.272093\pi$
$570$	0	0
$571$	2.58310e26	1.67532	0.837661	−	0.546191i	$-0.183923\pi$
$571$	2.58310e26	1.67532	0.837661	+	0.546191i	$0.183923\pi$
$572$	0	0
$573$	−7.81961e23	−0.00490587
$574$	0	0
$575$	5.04189e25	0.306019
$576$	0	0
$577$	1.85902e26	1.09173	0.545864	−	0.837874i	$-0.316201\pi$
$577$	1.85902e26	1.09173	0.545864	+	0.837874i	$0.316201\pi$
$578$	0	0
$579$	−1.92841e26	−1.09585
$580$	0	0
$581$	−6.07016e24	−0.0333831
$582$	0	0
$583$	−9.94995e24	−0.0529626
$584$	0	0
$585$	3.61941e25	0.186491
$586$	0	0
$587$	−3.62516e26	−1.80828	−0.904138	−	0.427240i	$-0.859486\pi$
$587$	−3.62516e26	−1.80828	−0.904138	+	0.427240i	$0.859486\pi$
$588$	0	0
$589$	−1.14065e26	−0.550878
$590$	0	0
$591$	1.92102e26	0.898361
$592$	0	0
$593$	1.26430e26	0.572572	0.286286	−	0.958144i	$-0.407579\pi$
$593$	1.26430e26	0.572572	0.286286	+	0.958144i	$0.407579\pi$
$594$	0	0
$595$	4.52752e24	0.0198586
$596$	0	0
$597$	2.96602e26	1.26014
$598$	0	0
$599$	1.79159e26	0.737367	0.368684	−	0.929555i	$-0.379809\pi$
$599$	1.79159e26	0.737367	0.368684	+	0.929555i	$0.379809\pi$
$600$	0	0
$601$	5.82667e25	0.232334	0.116167	−	0.993230i	$-0.462939\pi$
$601$	5.82667e25	0.232334	0.116167	+	0.993230i	$0.462939\pi$
$602$	0	0
$603$	−1.30966e26	−0.505994
$604$	0	0
$605$	7.92433e24	0.0296679
$606$	0	0
$607$	−4.86469e26	−1.76507	−0.882537	−	0.470243i	$-0.844166\pi$
$607$	−4.86469e26	−1.76507	−0.882537	+	0.470243i	$0.844166\pi$
$608$	0	0
$609$	1.20301e24	0.00423063
$610$	0	0
$611$	4.03789e26	1.37646
$612$	0	0
$613$	1.81522e26	0.599868	0.299934	−	0.953960i	$-0.403035\pi$
$613$	1.81522e26	0.599868	0.299934	+	0.953960i	$0.403035\pi$
$614$	0	0
$615$	2.59545e25	0.0831570
$616$	0	0
$617$	6.40323e26	1.98925	0.994627	−	0.103524i	$-0.0330119\pi$
$617$	6.40323e26	1.98925	0.994627	+	0.103524i	$0.0330119\pi$
$618$	0	0
$619$	−2.06913e26	−0.623341	−0.311670	−	0.950190i	$-0.600888\pi$
$619$	−2.06913e26	−0.623341	−0.311670	+	0.950190i	$0.600888\pi$
$620$	0	0
$621$	−5.69719e26	−1.66452
$622$	0	0
$623$	−9.68974e24	−0.0274584
$624$	0	0
$625$	1.45519e25	0.0400000
$626$	0	0
$627$	−2.17500e26	−0.579986
$628$	0	0
$629$	8.43098e26	2.18121
$630$	0	0
$631$	−4.35086e26	−1.09219	−0.546093	−	0.837725i	$-0.683885\pi$
$631$	−4.35086e26	−1.09219	−0.546093	+	0.837725i	$0.683885\pi$
$632$	0	0
$633$	−2.74817e26	−0.669436
$634$	0	0
$635$	2.36910e26	0.560057
$636$	0	0
$637$	5.00630e26	1.14866
$638$	0	0
$639$	2.61329e26	0.582008
$640$	0	0
$641$	6.37041e25	0.137726	0.0688630	−	0.997626i	$-0.478063\pi$
$641$	6.37041e25	0.137726	0.0688630	+	0.997626i	$0.478063\pi$
$642$	0	0
$643$	5.28854e26	1.11002	0.555011	−	0.831843i	$-0.312714\pi$
$643$	5.28854e26	1.11002	0.555011	+	0.831843i	$0.312714\pi$
$644$	0	0
$645$	−2.31817e25	−0.0472419
$646$	0	0
$647$	4.68543e26	0.927169	0.463585	−	0.886053i	$-0.346563\pi$
$647$	4.68543e26	0.927169	0.463585	+	0.886053i	$0.346563\pi$
$648$	0	0
$649$	1.87725e26	0.360743
$650$	0	0
$651$	−9.69521e24	−0.0180941
$652$	0	0
$653$	6.59110e25	0.119477	0.0597383	−	0.998214i	$-0.480973\pi$
$653$	6.59110e25	0.119477	0.0597383	+	0.998214i	$0.480973\pi$
$654$	0	0
$655$	8.62607e25	0.151887
$656$	0	0
$657$	3.18698e26	0.545140
$658$	0	0
$659$	7.65201e26	1.27164	0.635820	−	0.771838i	$-0.280662\pi$
$659$	7.65201e26	1.27164	0.635820	+	0.771838i	$0.280662\pi$
$660$	0	0
$661$	−8.13452e26	−1.31346	−0.656732	−	0.754124i	$-0.728062\pi$
$661$	−8.13452e26	−1.31346	−0.656732	+	0.754124i	$0.728062\pi$
$662$	0	0
$663$	−8.97221e26	−1.40774
$664$	0	0
$665$	5.97392e24	0.00910864
$666$	0	0
$667$	1.89029e26	0.280113
$668$	0	0
$669$	−8.56519e25	−0.123364
$670$	0	0
$671$	1.29884e27	1.81841
$672$	0	0
$673$	4.30859e25	0.0586398	0.0293199	−	0.999570i	$-0.490666\pi$
$673$	4.30859e25	0.0586398	0.0293199	+	0.999570i	$0.490666\pi$
$674$	0	0
$675$	−1.64432e26	−0.217571
$676$	0	0
$677$	−4.67369e26	−0.601267	−0.300633	−	0.953740i	$-0.597198\pi$
$677$	−4.67369e26	−0.601267	−0.300633	+	0.953740i	$0.597198\pi$
$678$	0	0
$679$	−1.51138e25	−0.0189065
$680$	0	0
$681$	4.83820e26	0.588553
$682$	0	0
$683$	−4.03068e25	−0.0476850	−0.0238425	−	0.999716i	$-0.507590\pi$
$683$	−4.03068e25	−0.0476850	−0.0238425	+	0.999716i	$0.507590\pi$
$684$	0	0
$685$	−1.44212e26	−0.165936
$686$	0	0
$687$	−1.20952e27	−1.35371
$688$	0	0
$689$	−5.41578e25	−0.0589628
$690$	0	0
$691$	1.31088e27	1.38842	0.694212	−	0.719771i	$-0.255753\pi$
$691$	1.31088e27	1.38842	0.694212	+	0.719771i	$0.255753\pi$
$692$	0	0
$693$	1.05228e25	0.0108434
$694$	0	0
$695$	2.61534e26	0.262224
$696$	0	0
$697$	3.66216e26	0.357295
$698$	0	0
$699$	9.04441e26	0.858711
$700$	0	0
$701$	−1.00287e27	−0.926668	−0.463334	−	0.886184i	$-0.653347\pi$
$701$	−1.00287e27	−0.926668	−0.463334	+	0.886184i	$0.653347\pi$
$702$	0	0
$703$	1.11244e27	1.00046
$704$	0	0
$705$	−4.88291e26	−0.427447
$706$	0	0
$707$	−3.46305e25	−0.0295103
$708$	0	0
$709$	−4.05934e26	−0.336756	−0.168378	−	0.985722i	$-0.553853\pi$
$709$	−4.05934e26	−0.336756	−0.168378	+	0.985722i	$0.553853\pi$
$710$	0	0
$711$	−3.38652e26	−0.273522
$712$	0	0
$713$	−1.52341e27	−1.19802
$714$	0	0
$715$	6.93309e26	0.530908
$716$	0	0
$717$	−8.22468e26	−0.613320
$718$	0	0
$719$	−2.40058e27	−1.74338	−0.871689	−	0.490059i	$-0.836975\pi$
$719$	−2.40058e27	−1.74338	−0.871689	+	0.490059i	$0.836975\pi$
$720$	0	0
$721$	4.26686e25	0.0301803
$722$	0	0
$723$	−8.53797e26	−0.588221
$724$	0	0
$725$	5.45576e25	0.0366138
$726$	0	0
$727$	−2.63808e27	−1.72469	−0.862345	−	0.506322i	$-0.831005\pi$
$727$	−2.63808e27	−1.72469	−0.862345	+	0.506322i	$0.831005\pi$
$728$	0	0
$729$	1.65146e27	1.05186
$730$	0	0
$731$	−3.27092e26	−0.202981
$732$	0	0
$733$	1.62181e26	0.0980647	0.0490324	−	0.998797i	$-0.484386\pi$
$733$	1.62181e26	0.0980647	0.0490324	+	0.998797i	$0.484386\pi$
$734$	0	0
$735$	−6.05399e26	−0.356707
$736$	0	0
$737$	−2.50870e27	−1.44048
$738$	0	0
$739$	9.83482e26	0.550357	0.275178	−	0.961393i	$-0.411263\pi$
$739$	9.83482e26	0.550357	0.275178	+	0.961393i	$0.411263\pi$
$740$	0	0
$741$	−1.18386e27	−0.645693
$742$	0	0
$743$	−5.57144e26	−0.296192	−0.148096	−	0.988973i	$-0.547315\pi$
$743$	−5.57144e26	−0.296192	−0.148096	+	0.988973i	$0.547315\pi$
$744$	0	0
$745$	−1.16884e27	−0.605718
$746$	0	0
$747$	8.27997e26	0.418295
$748$	0	0
$749$	2.15314e25	0.0106046
$750$	0	0
$751$	2.36925e26	0.113771	0.0568854	−	0.998381i	$-0.481883\pi$
$751$	2.36925e26	0.113771	0.0568854	+	0.998381i	$0.481883\pi$
$752$	0	0
$753$	5.97886e26	0.279941
$754$	0	0
$755$	−8.39013e26	−0.383065
$756$	0	0
$757$	−1.79972e27	−0.801297	−0.400648	−	0.916232i	$-0.631215\pi$
$757$	−1.79972e27	−0.801297	−0.400648	+	0.916232i	$0.631215\pi$
$758$	0	0
$759$	−2.90485e27	−1.26133
$760$	0	0
$761$	−2.93270e27	−1.24198	−0.620988	−	0.783820i	$-0.713268\pi$
$761$	−2.93270e27	−1.24198	−0.620988	+	0.783820i	$0.713268\pi$
$762$	0	0
$763$	−8.37605e25	−0.0345984
$764$	0	0
$765$	−6.17573e26	−0.248831
$766$	0	0
$767$	1.02179e27	0.401612
$768$	0	0
$769$	−1.17074e27	−0.448910	−0.224455	−	0.974484i	$-0.572060\pi$
$769$	−1.17074e27	−0.448910	−0.224455	+	0.974484i	$0.572060\pi$
$770$	0	0
$771$	−4.15275e27	−1.55353
$772$	0	0
$773$	1.75953e27	0.642231	0.321115	−	0.947040i	$-0.395942\pi$
$773$	1.75953e27	0.642231	0.321115	+	0.947040i	$0.395942\pi$
$774$	0	0
$775$	−4.39686e26	−0.156595
$776$	0	0
$777$	9.45549e25	0.0328613
$778$	0	0
$779$	4.83211e26	0.163882
$780$	0	0
$781$	5.00582e27	1.65688
$782$	0	0
$783$	−6.16485e26	−0.199153
$784$	0	0
$785$	6.49113e26	0.204672
$786$	0	0
$787$	3.17029e27	0.975751	0.487876	−	0.872913i	$-0.337772\pi$
$787$	3.17029e27	0.975751	0.487876	+	0.872913i	$0.337772\pi$
$788$	0	0
$789$	−1.88435e27	−0.566147
$790$	0	0
$791$	−5.98469e25	−0.0175536
$792$	0	0
$793$	7.06961e27	2.02442
$794$	0	0
$795$	6.54916e25	0.0183104
$796$	0	0
$797$	1.32138e26	0.0360723	0.0180361	−	0.999837i	$-0.494259\pi$
$797$	1.32138e26	0.0360723	0.0180361	+	0.999837i	$0.494259\pi$
$798$	0	0
$799$	−6.88976e27	−1.83658
$800$	0	0
$801$	1.32172e27	0.344058
$802$	0	0
$803$	6.10474e27	1.55192
$804$	0	0
$805$	7.97856e25	0.0198091
$806$	0	0
$807$	−2.17807e27	−0.528169
$808$	0	0
$809$	−1.55973e27	−0.369436	−0.184718	−	0.982792i	$-0.559137\pi$
$809$	−1.55973e27	−0.369436	−0.184718	+	0.982792i	$0.559137\pi$
$810$	0	0
$811$	−1.73006e27	−0.400279	−0.200140	−	0.979767i	$-0.564140\pi$
$811$	−1.73006e27	−0.400279	−0.200140	+	0.979767i	$0.564140\pi$
$812$	0	0
$813$	−7.72313e26	−0.174555
$814$	0	0
$815$	−3.37047e26	−0.0744207
$816$	0	0
$817$	−4.31588e26	−0.0931022
$818$	0	0
$819$	5.72756e25	0.0120718
$820$	0	0
$821$	9.32916e27	1.92124	0.960622	−	0.277859i	$-0.0896247\pi$
$821$	9.32916e27	1.92124	0.960622	+	0.277859i	$0.0896247\pi$
$822$	0	0
$823$	−4.58556e27	−0.922771	−0.461385	−	0.887200i	$-0.652647\pi$
$823$	−4.58556e27	−0.922771	−0.461385	+	0.887200i	$0.652647\pi$
$824$	0	0
$825$	−8.38400e26	−0.164869
$826$	0	0
$827$	−5.04516e27	−0.969555	−0.484777	−	0.874638i	$-0.661099\pi$
$827$	−5.04516e27	−0.969555	−0.484777	+	0.874638i	$0.661099\pi$
$828$	0	0
$829$	2.26868e27	0.426094	0.213047	−	0.977042i	$-0.431661\pi$
$829$	2.26868e27	0.426094	0.213047	+	0.977042i	$0.431661\pi$
$830$	0	0
$831$	−4.89207e27	−0.898014
$832$	0	0
$833$	−8.54215e27	−1.53264
$834$	0	0
$835$	−4.00799e27	−0.702920
$836$	0	0
$837$	4.96833e27	0.851762
$838$	0	0
$839$	9.66175e27	1.61926	0.809631	−	0.586939i	$-0.199667\pi$
$839$	9.66175e27	1.61926	0.809631	+	0.586939i	$0.199667\pi$
$840$	0	0
$841$	−5.89872e27	−0.966486
$842$	0	0
$843$	8.02699e26	0.128585
$844$	0	0
$845$	9.18380e26	0.143841
$846$	0	0
$847$	1.25399e25	0.00192044
$848$	0	0
$849$	1.21716e27	0.182274
$850$	0	0
$851$	1.48574e28	2.17577
$852$	0	0
$853$	4.37173e27	0.626092	0.313046	−	0.949738i	$-0.398651\pi$
$853$	4.37173e27	0.626092	0.313046	+	0.949738i	$0.398651\pi$
$854$	0	0
$855$	−8.14869e26	−0.114133
$856$	0	0
$857$	−7.84725e27	−1.07498	−0.537489	−	0.843271i	$-0.680627\pi$
$857$	−7.84725e27	−1.07498	−0.537489	+	0.843271i	$0.680627\pi$
$858$	0	0
$859$	1.05176e28	1.40923	0.704614	−	0.709590i	$-0.251120\pi$
$859$	1.05176e28	1.40923	0.704614	+	0.709590i	$0.251120\pi$
$860$	0	0
$861$	4.10718e25	0.00538287
$862$	0	0
$863$	1.75514e27	0.225013	0.112507	−	0.993651i	$-0.464112\pi$
$863$	1.75514e27	0.225013	0.112507	+	0.993651i	$0.464112\pi$
$864$	0	0
$865$	3.71112e27	0.465427
$866$	0	0
$867$	8.80268e27	1.08002
$868$	0	0
$869$	−6.48696e27	−0.778670
$870$	0	0
$871$	−1.36549e28	−1.60367
$872$	0	0
$873$	2.06160e27	0.236902
$874$	0	0
$875$	2.30278e25	0.00258926
$876$	0	0
$877$	−1.56391e28	−1.72075	−0.860373	−	0.509666i	$-0.829769\pi$
$877$	−1.56391e28	−1.72075	−0.860373	+	0.509666i	$0.829769\pi$
$878$	0	0
$879$	4.64988e27	0.500665
$880$	0	0
$881$	1.66747e28	1.75706	0.878532	−	0.477684i	$-0.158524\pi$
$881$	1.66747e28	1.75706	0.878532	+	0.477684i	$0.158524\pi$
$882$	0	0
$883$	1.50568e28	1.55277	0.776384	−	0.630261i	$-0.217052\pi$
$883$	1.50568e28	1.55277	0.776384	+	0.630261i	$0.217052\pi$
$884$	0	0
$885$	−1.23563e27	−0.124717
$886$	0	0
$887$	−1.27237e26	−0.0125701	−0.00628504	−	0.999980i	$-0.502001\pi$
$887$	−1.27237e26	−0.0125701	−0.00628504	+	0.999980i	$0.502001\pi$
$888$	0	0
$889$	3.74899e26	0.0362533
$890$	0	0
$891$	5.51662e27	0.522197
$892$	0	0
$893$	−9.09083e27	−0.842392
$894$	0	0
$895$	−3.86019e27	−0.350179
$896$	0	0
$897$	−1.58112e28	−1.40422
$898$	0	0
$899$	−1.64846e27	−0.143338
$900$	0	0
$901$	9.24083e26	0.0786730
$902$	0	0
$903$	−3.66840e25	−0.00305804
$904$	0	0
$905$	−1.49171e27	−0.121765
$906$	0	0
$907$	−2.25984e26	−0.0180637	−0.00903187	−	0.999959i	$-0.502875\pi$
$907$	−2.25984e26	−0.0180637	−0.00903187	+	0.999959i	$0.502875\pi$
$908$	0	0
$909$	4.72375e27	0.369769
$910$	0	0
$911$	−5.99523e27	−0.459601	−0.229801	−	0.973238i	$-0.573807\pi$
$911$	−5.99523e27	−0.459601	−0.229801	+	0.973238i	$0.573807\pi$
$912$	0	0
$913$	1.58605e28	1.19082
$914$	0	0
$915$	−8.54909e27	−0.628665
$916$	0	0
$917$	1.36504e26	0.00983186
$918$	0	0
$919$	1.91001e28	1.34753	0.673765	−	0.738946i	$-0.264676\pi$
$919$	1.91001e28	1.34753	0.673765	+	0.738946i	$0.264676\pi$
$920$	0	0
$921$	1.61364e28	1.11517
$922$	0	0
$923$	2.72468e28	1.84459
$924$	0	0
$925$	4.28815e27	0.284396
$926$	0	0
$927$	−5.82019e27	−0.378164
$928$	0	0
$929$	−2.36789e28	−1.50735	−0.753673	−	0.657249i	$-0.771720\pi$
$929$	−2.36789e28	−1.50735	−0.753673	+	0.657249i	$0.771720\pi$
$930$	0	0
$931$	−1.12711e28	−0.702982
$932$	0	0
$933$	−3.76313e26	−0.0229971
$934$	0	0
$935$	−1.18298e28	−0.708380
$936$	0	0
$937$	1.48823e28	0.873258	0.436629	−	0.899642i	$-0.356172\pi$
$937$	1.48823e28	0.873258	0.436629	+	0.899642i	$0.356172\pi$
$938$	0	0
$939$	2.55912e28	1.47153
$940$	0	0
$941$	1.83785e28	1.03564	0.517821	−	0.855489i	$-0.326743\pi$
$941$	1.83785e28	1.03564	0.517821	+	0.855489i	$0.326743\pi$
$942$	0	0
$943$	6.45361e27	0.356403
$944$	0	0
$945$	−2.60207e26	−0.0140837
$946$	0	0
$947$	1.80952e28	0.959926	0.479963	−	0.877289i	$-0.340650\pi$
$947$	1.80952e28	0.959926	0.479963	+	0.877289i	$0.340650\pi$
$948$	0	0
$949$	3.32282e28	1.72774
$950$	0	0
$951$	−8.15755e27	−0.415762
$952$	0	0
$953$	1.61100e28	0.804846	0.402423	−	0.915454i	$-0.368168\pi$
$953$	1.61100e28	0.804846	0.402423	+	0.915454i	$0.368168\pi$
$954$	0	0
$955$	−5.61180e25	−0.00274834
$956$	0	0
$957$	−3.14331e27	−0.150912
$958$	0	0
$959$	−2.28209e26	−0.0107413
$960$	0	0
$961$	−8.38553e27	−0.386953
$962$	0	0
$963$	−2.93698e27	−0.132877
$964$	0	0
$965$	−1.38394e28	−0.613912
$966$	0	0
$967$	−8.74663e27	−0.380443	−0.190221	−	0.981741i	$-0.560921\pi$
$967$	−8.74663e27	−0.380443	−0.190221	+	0.981741i	$0.560921\pi$
$968$	0	0
$969$	2.01999e28	0.861535
$970$	0	0
$971$	1.00606e28	0.420765	0.210382	−	0.977619i	$-0.432529\pi$
$971$	1.00606e28	0.420765	0.210382	+	0.977619i	$0.432529\pi$
$972$	0	0
$973$	4.13866e26	0.0169741
$974$	0	0
$975$	−4.56343e27	−0.183547
$976$	0	0
$977$	−4.62723e28	−1.82525	−0.912626	−	0.408796i	$-0.865949\pi$
$977$	−4.62723e28	−1.82525	−0.912626	+	0.408796i	$0.865949\pi$
$978$	0	0
$979$	2.53180e28	0.979475
$980$	0	0
$981$	1.14253e28	0.433523
$982$	0	0
$983$	−5.10887e28	−1.90137	−0.950685	−	0.310158i	$-0.899618\pi$
$983$	−5.10887e28	−1.90137	−0.950685	+	0.310158i	$0.899618\pi$
$984$	0	0
$985$	1.37863e28	0.503274
$986$	0	0
$987$	−7.72698e26	−0.0276692
$988$	0	0
$989$	−5.76415e27	−0.202474
$990$	0	0
$991$	−5.60178e28	−1.93031	−0.965153	−	0.261686i	$-0.915722\pi$
$991$	−5.60178e28	−1.93031	−0.965153	+	0.261686i	$0.915722\pi$
$992$	0	0
$993$	7.72302e27	0.261078
$994$	0	0
$995$	2.12858e28	0.705946
$996$	0	0
$997$	−3.60801e28	−1.17399	−0.586993	−	0.809592i	$-0.699689\pi$
$997$	−3.60801e28	−1.17399	−0.586993	+	0.809592i	$0.699689\pi$
$998$	0	0
$999$	−4.84548e28	−1.54691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.a.i.1.3		4
4.3	odd	2		40.20.a.a.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.20.a.a.1.2	✓	4	4.3	odd	2
80.20.a.i.1.3		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+27215.3 q^{3} +1.95312e6 q^{5} +3.09073e6 q^{7} -4.21590e8 q^{9} +O(q^{10})\)	\(q+27215.3 q^{3} +1.95312e6 q^{5} +3.09073e6 q^{7} -4.21590e8 q^{9} -8.07566e9 q^{11} -4.39560e10 q^{13} +5.31549e10 q^{15} +7.50013e11 q^{17} +9.89618e11 q^{19} +8.41152e10 q^{21} +1.32170e13 q^{23} +3.81470e12 q^{25} -4.31050e13 q^{27} +1.43020e13 q^{29} -1.15261e14 q^{31} -2.19781e14 q^{33} +6.03659e12 q^{35} +1.12411e15 q^{37} -1.19628e15 q^{39} +4.88280e14 q^{41} -4.36116e14 q^{43} -8.23418e14 q^{45} -9.18620e15 q^{47} -1.13893e16 q^{49} +2.04118e16 q^{51} +1.23209e15 q^{53} -1.57728e16 q^{55} +2.69327e16 q^{57} -2.32458e16 q^{59} -1.60834e17 q^{61} -1.30302e15 q^{63} -8.58516e16 q^{65} +3.10649e17 q^{67} +3.59705e17 q^{69} -6.19865e17 q^{71} -7.55943e17 q^{73} +1.03818e17 q^{75} -2.49597e16 q^{77} +8.03273e17 q^{79} -6.83116e17 q^{81} -1.96399e18 q^{83} +1.46487e18 q^{85} +3.89232e17 q^{87} -3.13510e18 q^{89} -1.35856e17 q^{91} -3.13686e18 q^{93} +1.93285e18 q^{95} -4.89005e18 q^{97} +3.40462e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 46152 q^{3} + 7812500 q^{5} - 59073944 q^{7} + 25995060 q^{9}+O(q^{10}) \) 4 * q + 46152 * q^3 + 7812500 * q^5 - 59073944 * q^7 + 25995060 * q^9	\( 4 q + 46152 q^{3} + 7812500 q^{5} - 59073944 q^{7} + 25995060 q^{9} + 3675173728 q^{11} + 7175523640 q^{13} + 90140625000 q^{15} - 567763095896 q^{17} + 278184526832 q^{19} - 5545032556608 q^{21} + 3524328334936 q^{23} + 15258789062500 q^{25} + 39633120308304 q^{27} - 62219259151144 q^{29} - 247254182466832 q^{31} + 404147490823008 q^{33} - 115378796875000 q^{35} + 711533444521656 q^{37} - 29\!\cdots\!76 q^{39}+ \cdots + 21\!\cdots\!96 q^{99}+O(q^{100}) \) 4 * q + 46152 * q^3 + 7812500 * q^5 - 59073944 * q^7 + 25995060 * q^9 + 3675173728 * q^11 + 7175523640 * q^13 + 90140625000 * q^15 - 567763095896 * q^17 + 278184526832 * q^19 - 5545032556608 * q^21 + 3524328334936 * q^23 + 15258789062500 * q^25 + 39633120308304 * q^27 - 62219259151144 * q^29 - 247254182466832 * q^31 + 404147490823008 * q^33 - 115378796875000 * q^35 + 711533444521656 * q^37 - 2943955031648976 * q^39 - 868244518404600 * q^41 - 3716515028101432 * q^43 + 50771601562500 * q^45 - 8431486495352 * q^47 - 16013476405774140 * q^49 - 14635182514702320 * q^51 - 41519794873411368 * q^53 + 7178073687500000 * q^55 - 54294834644792928 * q^57 - 14480346320386000 * q^59 + 27840348176113048 * q^61 + 37085058206149896 * q^63 + 14014694609375000 * q^65 - 8174924905481000 * q^67 - 355071316348294848 * q^69 + 555093531720812496 * q^71 - 408571387318908216 * q^73 + 176055908203125000 * q^75 - 786239164619967200 * q^77 + 249424427517101088 * q^79 - 798177693248534844 * q^81 + 1696125608725987400 * q^83 - 1108912296671875000 * q^85 - 1094924005281685584 * q^87 - 3568552069941379992 * q^89 - 247232489018471312 * q^91 - 12281064825141550080 * q^93 + 543329153968750000 * q^95 - 22399415502269235032 * q^97 + 21952027783694553696 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more