LMFDB - Embedded newform 80.10.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,10,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$41.2028668931$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 19x - 22 $ x^3 - x^2 - 19*x - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.92453$ 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)

$f(q)$	$=$	$q+192.296 q^{3} +625.000 q^{5} +10171.1 q^{7} +17294.6 q^{9} +O(q^{10})$	$q+192.296 q^{3} +625.000 q^{5} +10171.1 q^{7} +17294.6 q^{9} +57888.3 q^{11} -94906.1 q^{13} +120185. q^{15} +473946. q^{17} -951112. q^{19} +1.95585e6 q^{21} +1.31742e6 q^{23} +390625. q^{25} -459274. q^{27} +1.35218e6 q^{29} -4.00116e6 q^{31} +1.11317e7 q^{33} +6.35691e6 q^{35} -1.10039e7 q^{37} -1.82500e7 q^{39} +5.89469e6 q^{41} -3.32845e7 q^{43} +1.08091e7 q^{45} +2.55561e7 q^{47} +6.30967e7 q^{49} +9.11377e7 q^{51} +32490.4 q^{53} +3.61802e7 q^{55} -1.82895e8 q^{57} +1.03458e8 q^{59} +1.90488e8 q^{61} +1.75905e8 q^{63} -5.93163e7 q^{65} +2.56253e8 q^{67} +2.53335e8 q^{69} -1.46684e8 q^{71} -5.74145e7 q^{73} +7.51155e7 q^{75} +5.88785e8 q^{77} +3.38536e8 q^{79} -4.28726e8 q^{81} -7.02255e8 q^{83} +2.96216e8 q^{85} +2.60018e8 q^{87} -1.77075e8 q^{89} -9.65295e8 q^{91} -7.69405e8 q^{93} -5.94445e8 q^{95} -9.17589e7 q^{97} +1.00116e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 84 q^{3} + 1875 q^{5} + 5520 q^{7} + 47079 q^{9}+O(q^{10}) $ 3 * q - 84 * q^3 + 1875 * q^5 + 5520 * q^7 + 47079 * q^9	$ 3 q - 84 q^{3} + 1875 q^{5} + 5520 q^{7} + 47079 q^{9} - 5556 q^{11} - 83094 q^{13} - 52500 q^{15} + 367062 q^{17} - 1489116 q^{19} + 1573728 q^{21} + 499920 q^{23} + 1171875 q^{25} - 7695432 q^{27} + 5234682 q^{29} - 12708912 q^{31} + 30494064 q^{33} + 3450000 q^{35} + 21724434 q^{37} + 5806824 q^{39} + 27440478 q^{41} + 23218260 q^{43} + 29424375 q^{45} + 28701528 q^{47} + 27094923 q^{49} + 155348760 q^{51} - 45629982 q^{53} - 3472500 q^{55} + 5376144 q^{57} + 268721868 q^{59} + 155970138 q^{61} + 389747664 q^{63} - 51933750 q^{65} + 526916604 q^{67} - 172501728 q^{69} + 239894424 q^{71} + 198362430 q^{73} - 32812500 q^{75} + 385819008 q^{77} + 413839728 q^{79} + 1018787787 q^{81} - 371949828 q^{83} + 229413750 q^{85} + 503011752 q^{87} + 754926606 q^{89} - 1842389664 q^{91} + 989333760 q^{93} - 930697500 q^{95} - 903451002 q^{97} - 2871765540 q^{99}+O(q^{100}) $ 3 * q - 84 * q^3 + 1875 * q^5 + 5520 * q^7 + 47079 * q^9 - 5556 * q^11 - 83094 * q^13 - 52500 * q^15 + 367062 * q^17 - 1489116 * q^19 + 1573728 * q^21 + 499920 * q^23 + 1171875 * q^25 - 7695432 * q^27 + 5234682 * q^29 - 12708912 * q^31 + 30494064 * q^33 + 3450000 * q^35 + 21724434 * q^37 + 5806824 * q^39 + 27440478 * q^41 + 23218260 * q^43 + 29424375 * q^45 + 28701528 * q^47 + 27094923 * q^49 + 155348760 * q^51 - 45629982 * q^53 - 3472500 * q^55 + 5376144 * q^57 + 268721868 * q^59 + 155970138 * q^61 + 389747664 * q^63 - 51933750 * q^65 + 526916604 * q^67 - 172501728 * q^69 + 239894424 * q^71 + 198362430 * q^73 - 32812500 * q^75 + 385819008 * q^77 + 413839728 * q^79 + 1018787787 * q^81 - 371949828 * q^83 + 229413750 * q^85 + 503011752 * q^87 + 754926606 * q^89 - 1842389664 * q^91 + 989333760 * q^93 - 930697500 * q^95 - 903451002 * q^97 - 2871765540 * 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
$3$	192.296	1.37064	0.685321	−	0.728241i	$-0.259662\pi$
$3$	192.296	1.37064	0.685321	+	0.728241i	$0.259662\pi$
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	10171.1	1.60112	0.800561	−	0.599251i	$-0.204535\pi$
$7$	10171.1	1.60112	0.800561	+	0.599251i	$0.204535\pi$
$8$	0	0
$9$	17294.6	0.878658
$10$	0	0
$11$	57888.3	1.19213	0.596065	−	0.802936i	$-0.296730\pi$
$11$	57888.3	1.19213	0.596065	+	0.802936i	$0.296730\pi$
$12$	0	0
$13$	−94906.1	−0.921614	−0.460807	−	0.887500i	$-0.652440\pi$
$13$	−94906.1	−0.921614	−0.460807	+	0.887500i	$0.652440\pi$
$14$	0	0
$15$	120185.	0.612970
$16$	0	0
$17$	473946.	1.37629	0.688143	−	0.725575i	$-0.258426\pi$
$17$	473946.	1.37629	0.688143	+	0.725575i	$0.258426\pi$
$18$	0	0
$19$	−951112.	−1.67433	−0.837164	−	0.546953i	$-0.815788\pi$
$19$	−951112.	−1.67433	−0.837164	+	0.546953i	$0.815788\pi$
$20$	0	0
$21$	1.95585e6	2.19457
$22$	0	0
$23$	1.31742e6	0.981635	0.490817	−	0.871262i	$-0.336698\pi$
$23$	1.31742e6	0.981635	0.490817	+	0.871262i	$0.336698\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	−459274.	−0.166316
$28$	0	0
$29$	1.35218e6	0.355011	0.177506	−	0.984120i	$-0.443197\pi$
$29$	1.35218e6	0.355011	0.177506	+	0.984120i	$0.443197\pi$
$30$	0	0
$31$	−4.00116e6	−0.778140	−0.389070	−	0.921208i	$-0.627204\pi$
$31$	−4.00116e6	−0.778140	−0.389070	+	0.921208i	$0.627204\pi$
$32$	0	0
$33$	1.11317e7	1.63398
$34$	0	0
$35$	6.35691e6	0.716044
$36$	0	0
$37$	−1.10039e7	−0.965250	−0.482625	−	0.875827i	$-0.660317\pi$
$37$	−1.10039e7	−0.965250	−0.482625	+	0.875827i	$0.660317\pi$
$38$	0	0
$39$	−1.82500e7	−1.26320
$40$	0	0
$41$	5.89469e6	0.325787	0.162894	−	0.986644i	$-0.447917\pi$
$41$	5.89469e6	0.325787	0.162894	+	0.986644i	$0.447917\pi$
$42$	0	0
$43$	−3.32845e7	−1.48468	−0.742342	−	0.670021i	$-0.766285\pi$
$43$	−3.32845e7	−1.48468	−0.742342	+	0.670021i	$0.766285\pi$
$44$	0	0
$45$	1.08091e7	0.392948
$46$	0	0
$47$	2.55561e7	0.763933	0.381966	−	0.924176i	$-0.375247\pi$
$47$	2.55561e7	0.763933	0.381966	+	0.924176i	$0.375247\pi$
$48$	0	0
$49$	6.30967e7	1.56359
$50$	0	0
$51$	9.11377e7	1.88639
$52$	0	0
$53$	32490.4	0.000565605 0	0.000282802	−	1.00000i	$-0.499910\pi$
$53$	32490.4	0.000565605 0	0.000282802		1.00000i	$0.499910\pi$
$54$	0	0
$55$	3.61802e7	0.533137
$56$	0	0
$57$	−1.82895e8	−2.29490
$58$	0	0
$59$	1.03458e8	1.11155	0.555776	−	0.831332i	$-0.312421\pi$
$59$	1.03458e8	1.11155	0.555776	+	0.831332i	$0.312421\pi$
$60$	0	0
$61$	1.90488e8	1.76150	0.880749	−	0.473583i	$-0.157040\pi$
$61$	1.90488e8	1.76150	0.880749	+	0.473583i	$0.157040\pi$
$62$	0	0
$63$	1.75905e8	1.40684
$64$	0	0
$65$	−5.93163e7	−0.412158
$66$	0	0
$67$	2.56253e8	1.55357	0.776786	−	0.629764i	$-0.216848\pi$
$67$	2.56253e8	1.55357	0.776786	+	0.629764i	$0.216848\pi$
$68$	0	0
$69$	2.53335e8	1.34547
$70$	0	0
$71$	−1.46684e8	−0.685046	−0.342523	−	0.939509i	$-0.611282\pi$
$71$	−1.46684e8	−0.685046	−0.342523	+	0.939509i	$0.611282\pi$
$72$	0	0
$73$	−5.74145e7	−0.236630	−0.118315	−	0.992976i	$-0.537749\pi$
$73$	−5.74145e7	−0.236630	−0.118315	+	0.992976i	$0.537749\pi$
$74$	0	0
$75$	7.51155e7	0.274128
$76$	0	0
$77$	5.88785e8	1.90875
$78$	0	0
$79$	3.38536e8	0.977874	0.488937	−	0.872319i	$-0.337385\pi$
$79$	3.38536e8	0.977874	0.488937	+	0.872319i	$0.337385\pi$
$80$	0	0
$81$	−4.28726e8	−1.10662
$82$	0	0
$83$	−7.02255e8	−1.62422	−0.812108	−	0.583507i	$-0.801680\pi$
$83$	−7.02255e8	−1.62422	−0.812108	+	0.583507i	$0.801680\pi$
$84$	0	0
$85$	2.96216e8	0.615494
$86$	0	0
$87$	2.60018e8	0.486593
$88$	0	0
$89$	−1.77075e8	−0.299159	−0.149579	−	0.988750i	$-0.547792\pi$
$89$	−1.77075e8	−0.299159	−0.149579	+	0.988750i	$0.547792\pi$
$90$	0	0
$91$	−9.65295e8	−1.47562
$92$	0	0
$93$	−7.69405e8	−1.06655
$94$	0	0
$95$	−5.94445e8	−0.748782
$96$	0	0
$97$	−9.17589e7	−0.105239	−0.0526194	−	0.998615i	$-0.516757\pi$
$97$	−9.17589e7	−0.105239	−0.0526194	+	0.998615i	$0.516757\pi$
$98$	0	0
$99$	1.00116e9	1.04748
$100$	0	0
$101$	5.37650e7	0.0514107	0.0257053	−	0.999670i	$-0.491817\pi$
$101$	5.37650e7	0.0514107	0.0257053	+	0.999670i	$0.491817\pi$
$102$	0	0
$103$	−5.15056e8	−0.450907	−0.225454	−	0.974254i	$-0.572386\pi$
$103$	−5.15056e8	−0.450907	−0.225454	+	0.974254i	$0.572386\pi$
$104$	0	0
$105$	1.22241e9	0.981440
$106$	0	0
$107$	−1.68577e9	−1.24329	−0.621645	−	0.783299i	$-0.713535\pi$
$107$	−1.68577e9	−1.24329	−0.621645	+	0.783299i	$0.713535\pi$
$108$	0	0
$109$	1.68136e9	1.14088	0.570442	−	0.821338i	$-0.306772\pi$
$109$	1.68136e9	1.14088	0.570442	+	0.821338i	$0.306772\pi$
$110$	0	0
$111$	−2.11601e9	−1.32301
$112$	0	0
$113$	−2.04312e8	−0.117880	−0.0589401	−	0.998262i	$-0.518772\pi$
$113$	−2.04312e8	−0.117880	−0.0589401	+	0.998262i	$0.518772\pi$
$114$	0	0
$115$	8.23389e8	0.439000
$116$	0	0
$117$	−1.64137e9	−0.809784
$118$	0	0
$119$	4.82053e9	2.20360
$120$	0	0
$121$	9.93109e8	0.421175
$122$	0	0
$123$	1.13352e9	0.446537
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	6.15897e8	0.210083	0.105042	−	0.994468i	$-0.466502\pi$
$127$	6.15897e8	0.210083	0.105042	+	0.994468i	$0.466502\pi$
$128$	0	0
$129$	−6.40047e9	−2.03497
$130$	0	0
$131$	−2.67730e9	−0.794285	−0.397143	−	0.917757i	$-0.629998\pi$
$131$	−2.67730e9	−0.794285	−0.397143	+	0.917757i	$0.629998\pi$
$132$	0	0
$133$	−9.67381e9	−2.68080
$134$	0	0
$135$	−2.87046e8	−0.0743789
$136$	0	0
$137$	−6.28408e9	−1.52405	−0.762025	−	0.647547i	$-0.775795\pi$
$137$	−6.28408e9	−1.52405	−0.762025	+	0.647547i	$0.775795\pi$
$138$	0	0
$139$	−7.19913e9	−1.63574	−0.817868	−	0.575405i	$-0.804844\pi$
$139$	−7.19913e9	−1.63574	−0.817868	+	0.575405i	$0.804844\pi$
$140$	0	0
$141$	4.91434e9	1.04708
$142$	0	0
$143$	−5.49396e9	−1.09868
$144$	0	0
$145$	8.45110e8	0.158766
$146$	0	0
$147$	1.21332e10	2.14313
$148$	0	0
$149$	3.85695e9	0.641070	0.320535	−	0.947237i	$-0.396137\pi$
$149$	3.85695e9	0.641070	0.320535	+	0.947237i	$0.396137\pi$
$150$	0	0
$151$	3.54194e8	0.0554428	0.0277214	−	0.999616i	$-0.491175\pi$
$151$	3.54194e8	0.0554428	0.0277214	+	0.999616i	$0.491175\pi$
$152$	0	0
$153$	8.19672e9	1.20928
$154$	0	0
$155$	−2.50072e9	−0.347995
$156$	0	0
$157$	−1.47936e10	−1.94324	−0.971620	−	0.236549i	$-0.923984\pi$
$157$	−1.47936e10	−1.94324	−0.971620	+	0.236549i	$0.923984\pi$
$158$	0	0
$159$	6.24776e6	0.000775241 0
$160$	0	0
$161$	1.33996e10	1.57172
$162$	0	0
$163$	−1.17978e10	−1.30906	−0.654529	−	0.756037i	$-0.727133\pi$
$163$	−1.17978e10	−1.30906	−0.654529	+	0.756037i	$0.727133\pi$
$164$	0	0
$165$	6.95730e9	0.730740
$166$	0	0
$167$	1.11501e10	1.10931	0.554657	−	0.832079i	$-0.312850\pi$
$167$	1.11501e10	1.10931	0.554657	+	0.832079i	$0.312850\pi$
$168$	0	0
$169$	−1.59733e9	−0.150627
$170$	0	0
$171$	−1.64491e10	−1.47116
$172$	0	0
$173$	4.38553e9	0.372233	0.186116	−	0.982528i	$-0.440410\pi$
$173$	4.38553e9	0.372233	0.186116	+	0.982528i	$0.440410\pi$
$174$	0	0
$175$	3.97307e9	0.320225
$176$	0	0
$177$	1.98945e10	1.52354
$178$	0	0
$179$	−5.29799e9	−0.385720	−0.192860	−	0.981226i	$-0.561776\pi$
$179$	−5.29799e9	−0.385720	−0.192860	+	0.981226i	$0.561776\pi$
$180$	0	0
$181$	−5.24726e9	−0.363395	−0.181698	−	0.983354i	$-0.558159\pi$
$181$	−5.24726e9	−0.363395	−0.181698	+	0.983354i	$0.558159\pi$
$182$	0	0
$183$	3.66299e10	2.41438
$184$	0	0
$185$	−6.87746e9	−0.431673
$186$	0	0
$187$	2.74359e10	1.64071
$188$	0	0
$189$	−4.67130e9	−0.266293
$190$	0	0
$191$	−1.71321e10	−0.931454	−0.465727	−	0.884928i	$-0.654207\pi$
$191$	−1.71321e10	−0.931454	−0.465727	+	0.884928i	$0.654207\pi$
$192$	0	0
$193$	2.79758e10	1.45136	0.725680	−	0.688032i	$-0.241525\pi$
$193$	2.79758e10	1.45136	0.725680	+	0.688032i	$0.241525\pi$
$194$	0	0
$195$	−1.14063e10	−0.564921
$196$	0	0
$197$	3.34357e10	1.58166	0.790829	−	0.612037i	$-0.209650\pi$
$197$	3.34357e10	1.58166	0.790829	+	0.612037i	$0.209650\pi$
$198$	0	0
$199$	5.03726e9	0.227696	0.113848	−	0.993498i	$-0.463682\pi$
$199$	5.03726e9	0.227696	0.113848	+	0.993498i	$0.463682\pi$
$200$	0	0
$201$	4.92763e10	2.12939
$202$	0	0
$203$	1.37531e10	0.568417
$204$	0	0
$205$	3.68418e9	0.145696
$206$	0	0
$207$	2.27843e10	0.862521
$208$	0	0
$209$	−5.50583e10	−1.99602
$210$	0	0
$211$	−1.20805e10	−0.419578	−0.209789	−	0.977747i	$-0.567278\pi$
$211$	−1.20805e10	−0.419578	−0.209789	+	0.977747i	$0.567278\pi$
$212$	0	0
$213$	−2.82067e10	−0.938953
$214$	0	0
$215$	−2.08028e10	−0.663971
$216$	0	0
$217$	−4.06960e10	−1.24590
$218$	0	0
$219$	−1.10406e10	−0.324334
$220$	0	0
$221$	−4.49804e10	−1.26840
$222$	0	0
$223$	3.52614e10	0.954833	0.477417	−	0.878677i	$-0.341573\pi$
$223$	3.52614e10	0.954833	0.477417	+	0.878677i	$0.341573\pi$
$224$	0	0
$225$	6.75571e9	0.175732
$226$	0	0
$227$	−6.63314e10	−1.65807	−0.829035	−	0.559197i	$-0.811109\pi$
$227$	−6.63314e10	−1.65807	−0.829035	+	0.559197i	$0.811109\pi$
$228$	0	0
$229$	−4.84317e10	−1.16378	−0.581889	−	0.813269i	$-0.697686\pi$
$229$	−4.84317e10	−1.16378	−0.581889	+	0.813269i	$0.697686\pi$
$230$	0	0
$231$	1.13221e11	2.61621
$232$	0	0
$233$	−3.70184e10	−0.822842	−0.411421	−	0.911445i	$-0.634967\pi$
$233$	−3.70184e10	−0.822842	−0.411421	+	0.911445i	$0.634967\pi$
$234$	0	0
$235$	1.59726e10	0.341641
$236$	0	0
$237$	6.50990e10	1.34032
$238$	0	0
$239$	−4.26647e10	−0.845819	−0.422910	−	0.906172i	$-0.638991\pi$
$239$	−4.26647e10	−0.845819	−0.422910	+	0.906172i	$0.638991\pi$
$240$	0	0
$241$	−1.00238e10	−0.191406	−0.0957029	−	0.995410i	$-0.530510\pi$
$241$	−1.00238e10	−0.191406	−0.0957029	+	0.995410i	$0.530510\pi$
$242$	0	0
$243$	−7.34024e10	−1.35046
$244$	0	0
$245$	3.94354e10	0.699261
$246$	0	0
$247$	9.02663e10	1.54308
$248$	0	0
$249$	−1.35041e11	−2.22622
$250$	0	0
$251$	−4.05823e10	−0.645364	−0.322682	−	0.946507i	$-0.604584\pi$
$251$	−4.05823e10	−0.645364	−0.322682	+	0.946507i	$0.604584\pi$
$252$	0	0
$253$	7.62634e10	1.17024
$254$	0	0
$255$	5.69611e10	0.843621
$256$	0	0
$257$	8.66256e10	1.23865	0.619323	−	0.785137i	$-0.287407\pi$
$257$	8.66256e10	1.23865	0.619323	+	0.785137i	$0.287407\pi$
$258$	0	0
$259$	−1.11922e11	−1.54548
$260$	0	0
$261$	2.33854e10	0.311934
$262$	0	0
$263$	−1.46441e11	−1.88739	−0.943693	−	0.330821i	$-0.892674\pi$
$263$	−1.46441e11	−1.88739	−0.943693	+	0.330821i	$0.892674\pi$
$264$	0	0
$265$	2.03065e7	0.000252946 0
$266$	0	0
$267$	−3.40507e10	−0.410039
$268$	0	0
$269$	1.71634e11	1.99856	0.999282	−	0.0378975i	$-0.0120660\pi$
$269$	1.71634e11	1.99856	0.999282	+	0.0378975i	$0.0120660\pi$
$270$	0	0
$271$	−2.53661e10	−0.285688	−0.142844	−	0.989745i	$-0.545625\pi$
$271$	−2.53661e10	−0.285688	−0.142844	+	0.989745i	$0.545625\pi$
$272$	0	0
$273$	−1.85622e11	−2.02254
$274$	0	0
$275$	2.26126e10	0.238426
$276$	0	0
$277$	2.98573e10	0.304713	0.152357	−	0.988326i	$-0.451314\pi$
$277$	2.98573e10	0.304713	0.152357	+	0.988326i	$0.451314\pi$
$278$	0	0
$279$	−6.91985e10	−0.683719
$280$	0	0
$281$	−1.89029e9	−0.0180863	−0.00904316	−	0.999959i	$-0.502879\pi$
$281$	−1.89029e9	−0.0180863	−0.00904316	+	0.999959i	$0.502879\pi$
$282$	0	0
$283$	4.18336e10	0.387691	0.193846	−	0.981032i	$-0.437904\pi$
$283$	4.18336e10	0.387691	0.193846	+	0.981032i	$0.437904\pi$
$284$	0	0
$285$	−1.14309e11	−1.02631
$286$	0	0
$287$	5.99552e10	0.521625
$288$	0	0
$289$	1.06037e11	0.894163
$290$	0	0
$291$	−1.76448e10	−0.144245
$292$	0	0
$293$	−2.00579e10	−0.158994	−0.0794971	−	0.996835i	$-0.525331\pi$
$293$	−2.00579e10	−0.158994	−0.0794971	+	0.996835i	$0.525331\pi$
$294$	0	0
$295$	6.46612e10	0.497101
$296$	0	0
$297$	−2.65866e10	−0.198271
$298$	0	0
$299$	−1.25031e11	−0.904689
$300$	0	0
$301$	−3.38538e11	−2.37716
$302$	0	0
$303$	1.03388e10	0.0704656
$304$	0	0
$305$	1.19055e11	0.787766
$306$	0	0
$307$	−8.61702e10	−0.553649	−0.276824	−	0.960920i	$-0.589282\pi$
$307$	−8.61702e10	−0.553649	−0.276824	+	0.960920i	$0.589282\pi$
$308$	0	0
$309$	−9.90431e10	−0.618032
$310$	0	0
$311$	−3.34891e10	−0.202994	−0.101497	−	0.994836i	$-0.532363\pi$
$311$	−3.34891e10	−0.202994	−0.101497	+	0.994836i	$0.532363\pi$
$312$	0	0
$313$	1.55525e11	0.915907	0.457954	−	0.888976i	$-0.348583\pi$
$313$	1.55525e11	0.915907	0.457954	+	0.888976i	$0.348583\pi$
$314$	0	0
$315$	1.09940e11	0.629158
$316$	0	0
$317$	2.66820e11	1.48406	0.742031	−	0.670366i	$-0.233863\pi$
$317$	2.66820e11	1.48406	0.742031	+	0.670366i	$0.233863\pi$
$318$	0	0
$319$	7.82752e10	0.423220
$320$	0	0
$321$	−3.24167e11	−1.70411
$322$	0	0
$323$	−4.50776e11	−2.30435
$324$	0	0
$325$	−3.70727e10	−0.184323
$326$	0	0
$327$	3.23318e11	1.56374
$328$	0	0
$329$	2.59933e11	1.22315
$330$	0	0
$331$	3.33830e11	1.52862	0.764310	−	0.644849i	$-0.223080\pi$
$331$	3.33830e11	1.52862	0.764310	+	0.644849i	$0.223080\pi$
$332$	0	0
$333$	−1.90309e11	−0.848125
$334$	0	0
$335$	1.60158e11	0.694779
$336$	0	0
$337$	2.40469e11	1.01560	0.507802	−	0.861474i	$-0.330458\pi$
$337$	2.40469e11	1.01560	0.507802	+	0.861474i	$0.330458\pi$
$338$	0	0
$339$	−3.92883e10	−0.161571
$340$	0	0
$341$	−2.31620e11	−0.927645
$342$	0	0
$343$	2.31321e11	0.902384
$344$	0	0
$345$	1.58334e11	0.601712
$346$	0	0
$347$	3.85658e11	1.42797	0.713985	−	0.700161i	$-0.246888\pi$
$347$	3.85658e11	1.42797	0.713985	+	0.700161i	$0.246888\pi$
$348$	0	0
$349$	−4.50940e11	−1.62706	−0.813532	−	0.581521i	$-0.802458\pi$
$349$	−4.50940e11	−1.62706	−0.813532	+	0.581521i	$0.802458\pi$
$350$	0	0
$351$	4.35879e10	0.153279
$352$	0	0
$353$	−1.67282e11	−0.573407	−0.286703	−	0.958019i	$-0.592559\pi$
$353$	−1.67282e11	−0.573407	−0.286703	+	0.958019i	$0.592559\pi$
$354$	0	0
$355$	−9.16774e10	−0.306362
$356$	0	0
$357$	9.26967e11	3.02035
$358$	0	0
$359$	5.45859e11	1.73442	0.867212	−	0.497939i	$-0.165910\pi$
$359$	5.45859e11	1.73442	0.867212	+	0.497939i	$0.165910\pi$
$360$	0	0
$361$	5.81926e11	1.80337
$362$	0	0
$363$	1.90971e11	0.577280
$364$	0	0
$365$	−3.58841e10	−0.105824
$366$	0	0
$367$	−2.15503e11	−0.620093	−0.310046	−	0.950721i	$-0.600345\pi$
$367$	−2.15503e11	−0.620093	−0.310046	+	0.950721i	$0.600345\pi$
$368$	0	0
$369$	1.01947e11	0.286255
$370$	0	0
$371$	3.30461e8	0.000905603 0
$372$	0	0
$373$	−1.95969e11	−0.524201	−0.262100	−	0.965041i	$-0.584415\pi$
$373$	−1.95969e11	−0.524201	−0.262100	+	0.965041i	$0.584415\pi$
$374$	0	0
$375$	4.69472e10	0.122594
$376$	0	0
$377$	−1.28330e11	−0.327184
$378$	0	0
$379$	−5.15163e10	−0.128253	−0.0641266	−	0.997942i	$-0.520426\pi$
$379$	−5.15163e10	−0.128253	−0.0641266	+	0.997942i	$0.520426\pi$
$380$	0	0
$381$	1.18434e11	0.287949
$382$	0	0
$383$	−5.36021e11	−1.27288	−0.636440	−	0.771326i	$-0.719594\pi$
$383$	−5.36021e11	−1.27288	−0.636440	+	0.771326i	$0.719594\pi$
$384$	0	0
$385$	3.67991e11	0.853618
$386$	0	0
$387$	−5.75643e11	−1.30453
$388$	0	0
$389$	2.18865e10	0.0484621	0.0242311	−	0.999706i	$-0.492286\pi$
$389$	2.18865e10	0.0484621	0.0242311	+	0.999706i	$0.492286\pi$
$390$	0	0
$391$	6.24387e11	1.35101
$392$	0	0
$393$	−5.14834e11	−1.08868
$394$	0	0
$395$	2.11585e11	0.437319
$396$	0	0
$397$	6.65346e11	1.34428	0.672141	−	0.740423i	$-0.265375\pi$
$397$	6.65346e11	1.34428	0.672141	+	0.740423i	$0.265375\pi$
$398$	0	0
$399$	−1.86023e12	−3.67442
$400$	0	0
$401$	4.50006e11	0.869098	0.434549	−	0.900648i	$-0.356908\pi$
$401$	4.50006e11	0.869098	0.434549	+	0.900648i	$0.356908\pi$
$402$	0	0
$403$	3.79734e11	0.717145
$404$	0	0
$405$	−2.67954e11	−0.494895
$406$	0	0
$407$	−6.36999e11	−1.15070
$408$	0	0
$409$	−1.12038e12	−1.97975	−0.989877	−	0.141931i	$-0.954669\pi$
$409$	−1.12038e12	−1.97975	−0.989877	+	0.141931i	$0.954669\pi$
$410$	0	0
$411$	−1.20840e12	−2.08893
$412$	0	0
$413$	1.05228e12	1.77973
$414$	0	0
$415$	−4.38910e11	−0.726371
$416$	0	0
$417$	−1.38436e12	−2.24201
$418$	0	0
$419$	−5.43883e10	−0.0862069	−0.0431035	−	0.999071i	$-0.513725\pi$
$419$	−5.43883e10	−0.0862069	−0.0431035	+	0.999071i	$0.513725\pi$
$420$	0	0
$421$	−4.00163e11	−0.620823	−0.310411	−	0.950602i	$-0.600467\pi$
$421$	−4.00163e11	−0.620823	−0.310411	+	0.950602i	$0.600467\pi$
$422$	0	0
$423$	4.41984e11	0.671236
$424$	0	0
$425$	1.85135e11	0.275257
$426$	0	0
$427$	1.93746e12	2.82038
$428$	0	0
$429$	−1.05646e12	−1.50590
$430$	0	0
$431$	9.33544e10	0.130313	0.0651564	−	0.997875i	$-0.479245\pi$
$431$	9.33544e10	0.130313	0.0651564	+	0.997875i	$0.479245\pi$
$432$	0	0
$433$	8.43597e11	1.15329	0.576646	−	0.816994i	$-0.304361\pi$
$433$	8.43597e11	1.15329	0.576646	+	0.816994i	$0.304361\pi$
$434$	0	0
$435$	1.62511e11	0.217611
$436$	0	0
$437$	−1.25302e12	−1.64358
$438$	0	0
$439$	5.67234e11	0.728907	0.364454	−	0.931222i	$-0.381256\pi$
$439$	5.67234e11	0.728907	0.364454	+	0.931222i	$0.381256\pi$
$440$	0	0
$441$	1.09123e12	1.37386
$442$	0	0
$443$	−3.82986e11	−0.472461	−0.236231	−	0.971697i	$-0.575912\pi$
$443$	−3.82986e11	−0.472461	−0.236231	+	0.971697i	$0.575912\pi$
$444$	0	0
$445$	−1.10672e11	−0.133788
$446$	0	0
$447$	7.41675e11	0.878678
$448$	0	0
$449$	−2.85354e11	−0.331342	−0.165671	−	0.986181i	$-0.552979\pi$
$449$	−2.85354e11	−0.331342	−0.165671	+	0.986181i	$0.552979\pi$
$450$	0	0
$451$	3.41234e11	0.388381
$452$	0	0
$453$	6.81101e10	0.0759923
$454$	0	0
$455$	−6.03309e11	−0.659916
$456$	0	0
$457$	1.82219e11	0.195420	0.0977102	−	0.995215i	$-0.468848\pi$
$457$	1.82219e11	0.195420	0.0977102	+	0.995215i	$0.468848\pi$
$458$	0	0
$459$	−2.17671e11	−0.228899
$460$	0	0
$461$	−1.73033e12	−1.78433	−0.892164	−	0.451712i	$-0.850813\pi$
$461$	−1.73033e12	−1.78433	−0.892164	+	0.451712i	$0.850813\pi$
$462$	0	0
$463$	−8.13301e11	−0.822501	−0.411251	−	0.911522i	$-0.634908\pi$
$463$	−8.13301e11	−0.822501	−0.411251	+	0.911522i	$0.634908\pi$
$464$	0	0
$465$	−4.80878e11	−0.476976
$466$	0	0
$467$	1.08963e12	1.06011	0.530056	−	0.847963i	$-0.322171\pi$
$467$	1.08963e12	1.06011	0.530056	+	0.847963i	$0.322171\pi$
$468$	0	0
$469$	2.60636e12	2.48746
$470$	0	0
$471$	−2.84475e12	−2.66348
$472$	0	0
$473$	−1.92678e12	−1.76994
$474$	0	0
$475$	−3.71528e11	−0.334865
$476$	0	0
$477$	5.61909e8	0.000496973 0
$478$	0	0
$479$	−4.00913e11	−0.347969	−0.173984	−	0.984748i	$-0.555664\pi$
$479$	−4.00913e11	−0.347969	−0.173984	+	0.984748i	$0.555664\pi$
$480$	0	0
$481$	1.04434e12	0.889588
$482$	0	0
$483$	2.57668e12	2.15426
$484$	0	0
$485$	−5.73493e10	−0.0470642
$486$	0	0
$487$	1.16095e12	0.935262	0.467631	−	0.883924i	$-0.345108\pi$
$487$	1.16095e12	0.935262	0.467631	+	0.883924i	$0.345108\pi$
$488$	0	0
$489$	−2.26868e12	−1.79425
$490$	0	0
$491$	5.40746e10	0.0419881	0.0209941	−	0.999780i	$-0.493317\pi$
$491$	5.40746e10	0.0419881	0.0209941	+	0.999780i	$0.493317\pi$
$492$	0	0
$493$	6.40859e11	0.488597
$494$	0	0
$495$	6.25723e11	0.468445
$496$	0	0
$497$	−1.49193e12	−1.09684
$498$	0	0
$499$	−5.03600e11	−0.363608	−0.181804	−	0.983335i	$-0.558194\pi$
$499$	−5.03600e11	−0.363608	−0.181804	+	0.983335i	$0.558194\pi$
$500$	0	0
$501$	2.14411e12	1.52047
$502$	0	0
$503$	2.78019e10	0.0193650	0.00968252	−	0.999953i	$-0.496918\pi$
$503$	2.78019e10	0.0193650	0.00968252	+	0.999953i	$0.496918\pi$
$504$	0	0
$505$	3.36031e10	0.0229916
$506$	0	0
$507$	−3.07159e11	−0.206456
$508$	0	0
$509$	−3.00657e11	−0.198537	−0.0992685	−	0.995061i	$-0.531650\pi$
$509$	−3.00657e11	−0.198537	−0.0992685	+	0.995061i	$0.531650\pi$
$510$	0	0
$511$	−5.83966e11	−0.378873
$512$	0	0
$513$	4.36821e11	0.278468
$514$	0	0
$515$	−3.21910e11	−0.201652
$516$	0	0
$517$	1.47940e12	0.910707
$518$	0	0
$519$	8.43319e11	0.510198
$520$	0	0
$521$	1.01016e12	0.600648	0.300324	−	0.953837i	$-0.402905\pi$
$521$	1.01016e12	0.600648	0.300324	+	0.953837i	$0.402905\pi$
$522$	0	0
$523$	−1.23765e12	−0.723334	−0.361667	−	0.932307i	$-0.617792\pi$
$523$	−1.23765e12	−0.723334	−0.361667	+	0.932307i	$0.617792\pi$
$524$	0	0
$525$	7.64004e11	0.438913
$526$	0	0
$527$	−1.89633e12	−1.07094
$528$	0	0
$529$	−6.55495e10	−0.0363931
$530$	0	0
$531$	1.78927e12	0.976674
$532$	0	0
$533$	−5.59442e11	−0.300250
$534$	0	0
$535$	−1.05361e12	−0.556016
$536$	0	0
$537$	−1.01878e12	−0.528684
$538$	0	0
$539$	3.65256e12	1.86401
$540$	0	0
$541$	−6.45461e11	−0.323953	−0.161977	−	0.986795i	$-0.551787\pi$
$541$	−6.45461e11	−0.323953	−0.161977	+	0.986795i	$0.551787\pi$
$542$	0	0
$543$	−1.00903e12	−0.498085
$544$	0	0
$545$	1.05085e12	0.510218
$546$	0	0
$547$	−3.29872e12	−1.57544	−0.787722	−	0.616031i	$-0.788740\pi$
$547$	−3.29872e12	−1.57544	−0.787722	+	0.616031i	$0.788740\pi$
$548$	0	0
$549$	3.29441e12	1.54776
$550$	0	0
$551$	−1.28607e12	−0.594405
$552$	0	0
$553$	3.44327e12	1.56570
$554$	0	0
$555$	−1.32251e12	−0.591669
$556$	0	0
$557$	−9.52912e11	−0.419473	−0.209737	−	0.977758i	$-0.567261\pi$
$557$	−9.52912e11	−0.419473	−0.209737	+	0.977758i	$0.567261\pi$
$558$	0	0
$559$	3.15890e12	1.36831
$560$	0	0
$561$	5.27581e12	2.24883
$562$	0	0
$563$	8.63925e11	0.362400	0.181200	−	0.983446i	$-0.442002\pi$
$563$	8.63925e11	0.362400	0.181200	+	0.983446i	$0.442002\pi$
$564$	0	0
$565$	−1.27695e11	−0.0527176
$566$	0	0
$567$	−4.36060e12	−1.77183
$568$	0	0
$569$	2.09129e12	0.836391	0.418195	−	0.908357i	$-0.362663\pi$
$569$	2.09129e12	0.836391	0.418195	+	0.908357i	$0.362663\pi$
$570$	0	0
$571$	2.05490e12	0.808963	0.404481	−	0.914546i	$-0.367452\pi$
$571$	2.05490e12	0.808963	0.404481	+	0.914546i	$0.367452\pi$
$572$	0	0
$573$	−3.29444e12	−1.27669
$574$	0	0
$575$	5.14618e11	0.196327
$576$	0	0
$577$	−1.55913e12	−0.585588	−0.292794	−	0.956175i	$-0.594585\pi$
$577$	−1.55913e12	−0.585588	−0.292794	+	0.956175i	$0.594585\pi$
$578$	0	0
$579$	5.37963e12	1.98929
$580$	0	0
$581$	−7.14268e12	−2.60057
$582$	0	0
$583$	1.88081e9	0.000674275 0
$584$	0	0
$585$	−1.02585e12	−0.362146
$586$	0	0
$587$	7.89959e11	0.274621	0.137310	−	0.990528i	$-0.456154\pi$
$587$	7.89959e11	0.274621	0.137310	+	0.990528i	$0.456154\pi$
$588$	0	0
$589$	3.80555e12	1.30286
$590$	0	0
$591$	6.42955e12	2.16789
$592$	0	0
$593$	2.28530e12	0.758921	0.379460	−	0.925208i	$-0.376110\pi$
$593$	2.28530e12	0.758921	0.379460	+	0.925208i	$0.376110\pi$
$594$	0	0
$595$	3.01283e12	0.985481
$596$	0	0
$597$	9.68643e11	0.312089
$598$	0	0
$599$	−6.09900e11	−0.193570	−0.0967850	−	0.995305i	$-0.530856\pi$
$599$	−6.09900e11	−0.193570	−0.0967850	+	0.995305i	$0.530856\pi$
$600$	0	0
$601$	−1.71296e12	−0.535566	−0.267783	−	0.963479i	$-0.586291\pi$
$601$	−1.71296e12	−0.535566	−0.267783	+	0.963479i	$0.586291\pi$
$602$	0	0
$603$	4.43179e12	1.36506
$604$	0	0
$605$	6.20693e11	0.188355
$606$	0	0
$607$	3.69876e12	1.10588	0.552939	−	0.833222i	$-0.313506\pi$
$607$	3.69876e12	1.10588	0.552939	+	0.833222i	$0.313506\pi$
$608$	0	0
$609$	2.64465e12	0.779096
$610$	0	0
$611$	−2.42543e12	−0.704051
$612$	0	0
$613$	−4.20767e12	−1.20356	−0.601782	−	0.798660i	$-0.705542\pi$
$613$	−4.20767e12	−1.20356	−0.601782	+	0.798660i	$0.705542\pi$
$614$	0	0
$615$	7.08453e11	0.199698
$616$	0	0
$617$	−2.40944e11	−0.0669320	−0.0334660	−	0.999440i	$-0.510655\pi$
$617$	−2.40944e11	−0.0669320	−0.0334660	+	0.999440i	$0.510655\pi$
$618$	0	0
$619$	−2.59875e12	−0.711469	−0.355734	−	0.934587i	$-0.615769\pi$
$619$	−2.59875e12	−0.711469	−0.355734	+	0.934587i	$0.615769\pi$
$620$	0	0
$621$	−6.05058e11	−0.163262
$622$	0	0
$623$	−1.80104e12	−0.478990
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−1.05875e13	−2.73582
$628$	0	0
$629$	−5.21527e12	−1.32846
$630$	0	0
$631$	−4.03872e12	−1.01417	−0.507086	−	0.861895i	$-0.669278\pi$
$631$	−4.03872e12	−1.01417	−0.507086	+	0.861895i	$0.669278\pi$
$632$	0	0
$633$	−2.32302e12	−0.575091
$634$	0	0
$635$	3.84936e11	0.0939521
$636$	0	0
$637$	−5.98826e12	−1.44103
$638$	0	0
$639$	−2.53684e12	−0.601921
$640$	0	0
$641$	−7.22252e12	−1.68977	−0.844884	−	0.534949i	$-0.820331\pi$
$641$	−7.22252e12	−1.68977	−0.844884	+	0.534949i	$0.820331\pi$
$642$	0	0
$643$	−6.83682e11	−0.157727	−0.0788633	−	0.996885i	$-0.525129\pi$
$643$	−6.83682e11	−0.157727	−0.0788633	+	0.996885i	$0.525129\pi$
$644$	0	0
$645$	−4.00029e12	−0.910066
$646$	0	0
$647$	1.83462e12	0.411600	0.205800	−	0.978594i	$-0.434020\pi$
$647$	1.83462e12	0.411600	0.205800	+	0.978594i	$0.434020\pi$
$648$	0	0
$649$	5.98901e12	1.32512
$650$	0	0
$651$	−7.82566e12	−1.70768
$652$	0	0
$653$	1.05293e12	0.226615	0.113308	−	0.993560i	$-0.463855\pi$
$653$	1.05293e12	0.226615	0.113308	+	0.993560i	$0.463855\pi$
$654$	0	0
$655$	−1.67331e12	−0.355215
$656$	0	0
$657$	−9.92963e11	−0.207916
$658$	0	0
$659$	1.05822e12	0.218570	0.109285	−	0.994010i	$-0.465144\pi$
$659$	1.05822e12	0.218570	0.109285	+	0.994010i	$0.465144\pi$
$660$	0	0
$661$	6.24147e12	1.27169	0.635843	−	0.771818i	$-0.280653\pi$
$661$	6.24147e12	1.27169	0.635843	+	0.771818i	$0.280653\pi$
$662$	0	0
$663$	−8.64953e12	−1.73853
$664$	0	0
$665$	−6.04613e12	−1.19889
$666$	0	0
$667$	1.78139e12	0.348492
$668$	0	0
$669$	6.78061e12	1.30873
$670$	0	0
$671$	1.10270e13	2.09994
$672$	0	0
$673$	6.73998e12	1.26646	0.633229	−	0.773964i	$-0.281729\pi$
$673$	6.73998e12	1.26646	0.633229	+	0.773964i	$0.281729\pi$
$674$	0	0
$675$	−1.79404e11	−0.0332632
$676$	0	0
$677$	2.39301e12	0.437821	0.218910	−	0.975745i	$-0.429750\pi$
$677$	2.39301e12	0.437821	0.218910	+	0.975745i	$0.429750\pi$
$678$	0	0
$679$	−9.33285e11	−0.168500
$680$	0	0
$681$	−1.27552e13	−2.27262
$682$	0	0
$683$	−3.69120e12	−0.649044	−0.324522	−	0.945878i	$-0.605203\pi$
$683$	−3.69120e12	−0.649044	−0.324522	+	0.945878i	$0.605203\pi$
$684$	0	0
$685$	−3.92755e12	−0.681576
$686$	0	0
$687$	−9.31320e12	−1.59512
$688$	0	0
$689$	−3.08353e9	−0.000521269 0
$690$	0	0
$691$	9.81563e12	1.63782	0.818912	−	0.573920i	$-0.194578\pi$
$691$	9.81563e12	1.63782	0.818912	+	0.573920i	$0.194578\pi$
$692$	0	0
$693$	1.01828e13	1.67714
$694$	0	0
$695$	−4.49946e12	−0.731524
$696$	0	0
$697$	2.79377e12	0.448376
$698$	0	0
$699$	−7.11849e12	−1.12782
$700$	0	0
$701$	2.54103e12	0.397446	0.198723	−	0.980056i	$-0.436321\pi$
$701$	2.54103e12	0.397446	0.198723	+	0.980056i	$0.436321\pi$
$702$	0	0
$703$	1.04660e13	1.61615
$704$	0	0
$705$	3.07146e12	0.468267
$706$	0	0
$707$	5.46847e11	0.0823148
$708$	0	0
$709$	6.21876e12	0.924263	0.462131	−	0.886811i	$-0.347085\pi$
$709$	6.21876e12	0.924263	0.462131	+	0.886811i	$0.347085\pi$
$710$	0	0
$711$	5.85485e12	0.859217
$712$	0	0
$713$	−5.27122e12	−0.763850
$714$	0	0
$715$	−3.43372e12	−0.491347
$716$	0	0
$717$	−8.20423e12	−1.15932
$718$	0	0
$719$	−5.59469e11	−0.0780721	−0.0390361	−	0.999238i	$-0.512429\pi$
$719$	−5.59469e11	−0.0780721	−0.0390361	+	0.999238i	$0.512429\pi$
$720$	0	0
$721$	−5.23866e12	−0.721958
$722$	0	0
$723$	−1.92753e12	−0.262349
$724$	0	0
$725$	5.28194e11	0.0710023
$726$	0	0
$727$	−2.24660e11	−0.0298278	−0.0149139	−	0.999889i	$-0.504747\pi$
$727$	−2.24660e11	−0.0298278	−0.0149139	+	0.999889i	$0.504747\pi$
$728$	0	0
$729$	−5.67633e12	−0.744379
$730$	0	0
$731$	−1.57751e13	−2.04335
$732$	0	0
$733$	8.13738e12	1.04116	0.520579	−	0.853813i	$-0.325716\pi$
$733$	8.13738e12	1.04116	0.520579	+	0.853813i	$0.325716\pi$
$734$	0	0
$735$	7.58326e12	0.958436
$736$	0	0
$737$	1.48340e13	1.85206
$738$	0	0
$739$	−2.98426e12	−0.368075	−0.184038	−	0.982919i	$-0.558917\pi$
$739$	−2.98426e12	−0.368075	−0.184038	+	0.982919i	$0.558917\pi$
$740$	0	0
$741$	1.73578e13	2.11501
$742$	0	0
$743$	1.21518e13	1.46282	0.731409	−	0.681940i	$-0.238863\pi$
$743$	1.21518e13	1.46282	0.731409	+	0.681940i	$0.238863\pi$
$744$	0	0
$745$	2.41059e12	0.286695
$746$	0	0
$747$	−1.21452e13	−1.42713
$748$	0	0
$749$	−1.71461e13	−1.99066
$750$	0	0
$751$	8.91620e12	1.02282	0.511411	−	0.859336i	$-0.329123\pi$
$751$	8.91620e12	1.02282	0.511411	+	0.859336i	$0.329123\pi$
$752$	0	0
$753$	−7.80380e12	−0.884563
$754$	0	0
$755$	2.21372e11	0.0247948
$756$	0	0
$757$	6.20627e12	0.686909	0.343455	−	0.939169i	$-0.388403\pi$
$757$	6.20627e12	0.686909	0.343455	+	0.939169i	$0.388403\pi$
$758$	0	0
$759$	1.46651e13	1.60398
$760$	0	0
$761$	1.14448e13	1.23702	0.618510	−	0.785777i	$-0.287737\pi$
$761$	1.14448e13	1.23702	0.618510	+	0.785777i	$0.287737\pi$
$762$	0	0
$763$	1.71012e13	1.82669
$764$	0	0
$765$	5.12295e12	0.540809
$766$	0	0
$767$	−9.81879e12	−1.02442
$768$	0	0
$769$	1.59089e12	0.164048	0.0820242	−	0.996630i	$-0.473862\pi$
$769$	1.59089e12	0.164048	0.0820242	+	0.996630i	$0.473862\pi$
$770$	0	0
$771$	1.66577e13	1.69774
$772$	0	0
$773$	8.91182e12	0.897756	0.448878	−	0.893593i	$-0.351824\pi$
$773$	8.91182e12	0.897756	0.448878	+	0.893593i	$0.351824\pi$
$774$	0	0
$775$	−1.56295e12	−0.155628
$776$	0	0
$777$	−2.15220e13	−2.11831
$778$	0	0
$779$	−5.60651e12	−0.545474
$780$	0	0
$781$	−8.49128e12	−0.816665
$782$	0	0
$783$	−6.21019e11	−0.0590442
$784$	0	0
$785$	−9.24602e12	−0.869043
$786$	0	0
$787$	2.25771e12	0.209789	0.104894	−	0.994483i	$-0.466550\pi$
$787$	2.25771e12	0.209789	0.104894	+	0.994483i	$0.466550\pi$
$788$	0	0
$789$	−2.81599e13	−2.58693
$790$	0	0
$791$	−2.07807e12	−0.188741
$792$	0	0
$793$	−1.80784e13	−1.62342
$794$	0	0
$795$	3.90485e9	0.000346699 0
$796$	0	0
$797$	−5.89111e12	−0.517172	−0.258586	−	0.965988i	$-0.583256\pi$
$797$	−5.89111e12	−0.517172	−0.258586	+	0.965988i	$0.583256\pi$
$798$	0	0
$799$	1.21122e13	1.05139
$800$	0	0
$801$	−3.06244e12	−0.262858
$802$	0	0
$803$	−3.32363e12	−0.282093
$804$	0	0
$805$	8.37474e12	0.702894
$806$	0	0
$807$	3.30045e13	2.73931
$808$	0	0
$809$	1.86538e13	1.53108	0.765541	−	0.643387i	$-0.222471\pi$
$809$	1.86538e13	1.53108	0.765541	+	0.643387i	$0.222471\pi$
$810$	0	0
$811$	−1.60447e13	−1.30238	−0.651190	−	0.758915i	$-0.725730\pi$
$811$	−1.60447e13	−1.30238	−0.651190	+	0.758915i	$0.725730\pi$
$812$	0	0
$813$	−4.87780e12	−0.391576
$814$	0	0
$815$	−7.37365e12	−0.585428
$816$	0	0
$817$	3.16573e13	2.48585
$818$	0	0
$819$	−1.66944e13	−1.29656
$820$	0	0
$821$	1.64410e13	1.26294	0.631472	−	0.775398i	$-0.282451\pi$
$821$	1.64410e13	1.26294	0.631472	+	0.775398i	$0.282451\pi$
$822$	0	0
$823$	−1.11501e13	−0.847189	−0.423595	−	0.905852i	$-0.639232\pi$
$823$	−1.11501e13	−0.847189	−0.423595	+	0.905852i	$0.639232\pi$
$824$	0	0
$825$	4.34831e12	0.326797
$826$	0	0
$827$	6.39490e12	0.475400	0.237700	−	0.971339i	$-0.423607\pi$
$827$	6.39490e12	0.475400	0.237700	+	0.971339i	$0.423607\pi$
$828$	0	0
$829$	−9.81720e12	−0.721925	−0.360963	−	0.932580i	$-0.617552\pi$
$829$	−9.81720e12	−0.721925	−0.360963	+	0.932580i	$0.617552\pi$
$830$	0	0
$831$	5.74143e12	0.417653
$832$	0	0
$833$	2.99044e13	2.15195
$834$	0	0
$835$	6.96881e12	0.496100
$836$	0	0
$837$	1.83763e12	0.129417
$838$	0	0
$839$	−1.10783e12	−0.0771869	−0.0385934	−	0.999255i	$-0.512288\pi$
$839$	−1.10783e12	−0.0771869	−0.0385934	+	0.999255i	$0.512288\pi$
$840$	0	0
$841$	−1.26788e13	−0.873967
$842$	0	0
$843$	−3.63495e11	−0.0247899
$844$	0	0
$845$	−9.98330e11	−0.0673626
$846$	0	0
$847$	1.01010e13	0.674353
$848$	0	0
$849$	8.04442e12	0.531386
$850$	0	0
$851$	−1.44968e13	−0.947523
$852$	0	0
$853$	−2.98534e12	−0.193074	−0.0965368	−	0.995329i	$-0.530777\pi$
$853$	−2.98534e12	−0.193074	−0.0965368	+	0.995329i	$0.530777\pi$
$854$	0	0
$855$	−1.02807e13	−0.657923
$856$	0	0
$857$	5.91335e12	0.374472	0.187236	−	0.982315i	$-0.440047\pi$
$857$	5.91335e12	0.374472	0.187236	+	0.982315i	$0.440047\pi$
$858$	0	0
$859$	7.81309e12	0.489614	0.244807	−	0.969572i	$-0.421275\pi$
$859$	7.81309e12	0.489614	0.244807	+	0.969572i	$0.421275\pi$
$860$	0	0
$861$	1.15291e13	0.714961
$862$	0	0
$863$	−1.14191e13	−0.700780	−0.350390	−	0.936604i	$-0.613951\pi$
$863$	−1.14191e13	−0.700780	−0.350390	+	0.936604i	$0.613951\pi$
$864$	0	0
$865$	2.74096e12	0.166468
$866$	0	0
$867$	2.03904e13	1.22558
$868$	0	0
$869$	1.95973e13	1.16575
$870$	0	0
$871$	−2.43199e13	−1.43179
$872$	0	0
$873$	−1.58694e12	−0.0924689
$874$	0	0
$875$	2.48317e12	0.143209
$876$	0	0
$877$	−2.48087e13	−1.41614	−0.708070	−	0.706143i	$-0.750434\pi$
$877$	−2.48087e13	−1.41614	−0.708070	+	0.706143i	$0.750434\pi$
$878$	0	0
$879$	−3.85705e12	−0.217924
$880$	0	0
$881$	1.41256e13	0.789981	0.394990	−	0.918685i	$-0.370748\pi$
$881$	1.41256e13	0.789981	0.394990	+	0.918685i	$0.370748\pi$
$882$	0	0
$883$	6.79981e12	0.376421	0.188210	−	0.982129i	$-0.439731\pi$
$883$	6.79981e12	0.376421	0.188210	+	0.982129i	$0.439731\pi$
$884$	0	0
$885$	1.24341e13	0.681348
$886$	0	0
$887$	3.40051e13	1.84454	0.922269	−	0.386548i	$-0.126333\pi$
$887$	3.40051e13	1.84454	0.922269	+	0.386548i	$0.126333\pi$
$888$	0	0
$889$	6.26432e12	0.336369
$890$	0	0
$891$	−2.48183e13	−1.31923
$892$	0	0
$893$	−2.43068e13	−1.27907
$894$	0	0
$895$	−3.31124e12	−0.172499
$896$	0	0
$897$	−2.40430e13	−1.24000
$898$	0	0
$899$	−5.41027e12	−0.276249
$900$	0	0
$901$	1.53987e10	0.000778434 0
$902$	0	0
$903$	−6.50995e13	−3.25824
$904$	0	0
$905$	−3.27954e12	−0.162515
$906$	0	0
$907$	−6.14504e12	−0.301503	−0.150752	−	0.988572i	$-0.548169\pi$
$907$	−6.14504e12	−0.301503	−0.150752	+	0.988572i	$0.548169\pi$
$908$	0	0
$909$	9.29846e11	0.0451724
$910$	0	0
$911$	−9.58689e12	−0.461153	−0.230576	−	0.973054i	$-0.574061\pi$
$911$	−9.58689e12	−0.461153	−0.230576	+	0.973054i	$0.574061\pi$
$912$	0	0
$913$	−4.06524e13	−1.93628
$914$	0	0
$915$	2.28937e13	1.07974
$916$	0	0
$917$	−2.72310e13	−1.27175
$918$	0	0
$919$	4.02534e12	0.186158	0.0930791	−	0.995659i	$-0.470329\pi$
$919$	4.02534e12	0.186158	0.0930791	+	0.995659i	$0.470329\pi$
$920$	0	0
$921$	−1.65702e13	−0.758854
$922$	0	0
$923$	1.39212e13	0.631348
$924$	0	0
$925$	−4.29841e12	−0.193050
$926$	0	0
$927$	−8.90770e12	−0.396193
$928$	0	0
$929$	−8.51576e12	−0.375105	−0.187553	−	0.982255i	$-0.560055\pi$
$929$	−8.51576e12	−0.375105	−0.187553	+	0.982255i	$0.560055\pi$
$930$	0	0
$931$	−6.00120e13	−2.61797
$932$	0	0
$933$	−6.43982e12	−0.278231
$934$	0	0
$935$	1.71475e13	0.733749
$936$	0	0
$937$	3.36486e13	1.42606	0.713031	−	0.701132i	$-0.247322\pi$
$937$	3.36486e13	1.42606	0.713031	+	0.701132i	$0.247322\pi$
$938$	0	0
$939$	2.99068e13	1.25538
$940$	0	0
$941$	2.44396e12	0.101611	0.0508056	−	0.998709i	$-0.483821\pi$
$941$	2.44396e12	0.101611	0.0508056	+	0.998709i	$0.483821\pi$
$942$	0	0
$943$	7.76580e12	0.319804
$944$	0	0
$945$	−2.91956e12	−0.119090
$946$	0	0
$947$	−2.25484e13	−0.911047	−0.455524	−	0.890224i	$-0.650548\pi$
$947$	−2.25484e13	−0.911047	−0.455524	+	0.890224i	$0.650548\pi$
$948$	0	0
$949$	5.44899e12	0.218081
$950$	0	0
$951$	5.13083e13	2.03412
$952$	0	0
$953$	1.04485e13	0.410332	0.205166	−	0.978727i	$-0.434227\pi$
$953$	1.04485e13	0.410332	0.205166	+	0.978727i	$0.434227\pi$
$954$	0	0
$955$	−1.07076e13	−0.416559
$956$	0	0
$957$	1.50520e13	0.580083
$958$	0	0
$959$	−6.39157e13	−2.44019
$960$	0	0
$961$	−1.04304e13	−0.394497
$962$	0	0
$963$	−2.91548e13	−1.09243
$964$	0	0
$965$	1.74849e13	0.649068
$966$	0	0
$967$	3.19777e13	1.17606	0.588028	−	0.808841i	$-0.299905\pi$
$967$	3.19777e13	1.17606	0.588028	+	0.808841i	$0.299905\pi$
$968$	0	0
$969$	−8.66822e13	−3.15844
$970$	0	0
$971$	1.29931e13	0.469058	0.234529	−	0.972109i	$-0.424645\pi$
$971$	1.29931e13	0.469058	0.234529	+	0.972109i	$0.424645\pi$
$972$	0	0
$973$	−7.32227e13	−2.61902
$974$	0	0
$975$	−7.12892e12	−0.252641
$976$	0	0
$977$	4.35160e13	1.52800	0.764001	−	0.645215i	$-0.223232\pi$
$977$	4.35160e13	1.52800	0.764001	+	0.645215i	$0.223232\pi$
$978$	0	0
$979$	−1.02506e13	−0.356636
$980$	0	0
$981$	2.90785e13	1.00245
$982$	0	0
$983$	2.15549e13	0.736299	0.368150	−	0.929767i	$-0.379991\pi$
$983$	2.15549e13	0.736299	0.368150	+	0.929767i	$0.379991\pi$
$984$	0	0
$985$	2.08973e13	0.707339
$986$	0	0
$987$	4.99840e13	1.67650
$988$	0	0
$989$	−4.38498e13	−1.45742
$990$	0	0
$991$	1.42707e13	0.470017	0.235008	−	0.971993i	$-0.424488\pi$
$991$	1.42707e13	0.470017	0.235008	+	0.971993i	$0.424488\pi$
$992$	0	0
$993$	6.41941e13	2.09519
$994$	0	0
$995$	3.14829e12	0.101829
$996$	0	0
$997$	−2.66088e13	−0.852899	−0.426449	−	0.904511i	$-0.640236\pi$
$997$	−2.66088e13	−0.852899	−0.426449	+	0.904511i	$0.640236\pi$
$998$	0	0
$999$	5.05382e12	0.160537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.10.a.k.1.3		3
4.3	odd	2		40.10.a.d.1.1	✓	3
5.2	odd	4		400.10.c.r.49.2		6
5.3	odd	4		400.10.c.r.49.5		6
5.4	even	2		400.10.a.x.1.1		3
8.3	odd	2		320.10.a.u.1.3		3
8.5	even	2		320.10.a.v.1.1		3
12.11	even	2		360.10.a.j.1.1		3
20.3	even	4		200.10.c.f.49.2		6
20.7	even	4		200.10.c.f.49.5		6
20.19	odd	2		200.10.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.d.1.1	✓	3	4.3	odd	2
80.10.a.k.1.3		3	1.1	even	1	trivial
200.10.a.f.1.3		3	20.19	odd	2
200.10.c.f.49.2		6	20.3	even	4
200.10.c.f.49.5		6	20.7	even	4
320.10.a.u.1.3		3	8.3	odd	2
320.10.a.v.1.1		3	8.5	even	2
360.10.a.j.1.1		3	12.11	even	2
400.10.a.x.1.1		3	5.4	even	2
400.10.c.r.49.2		6	5.2	odd	4
400.10.c.r.49.5		6	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+192.296 q^{3} +625.000 q^{5} +10171.1 q^{7} +17294.6 q^{9} +O(q^{10})\)	\(q+192.296 q^{3} +625.000 q^{5} +10171.1 q^{7} +17294.6 q^{9} +57888.3 q^{11} -94906.1 q^{13} +120185. q^{15} +473946. q^{17} -951112. q^{19} +1.95585e6 q^{21} +1.31742e6 q^{23} +390625. q^{25} -459274. q^{27} +1.35218e6 q^{29} -4.00116e6 q^{31} +1.11317e7 q^{33} +6.35691e6 q^{35} -1.10039e7 q^{37} -1.82500e7 q^{39} +5.89469e6 q^{41} -3.32845e7 q^{43} +1.08091e7 q^{45} +2.55561e7 q^{47} +6.30967e7 q^{49} +9.11377e7 q^{51} +32490.4 q^{53} +3.61802e7 q^{55} -1.82895e8 q^{57} +1.03458e8 q^{59} +1.90488e8 q^{61} +1.75905e8 q^{63} -5.93163e7 q^{65} +2.56253e8 q^{67} +2.53335e8 q^{69} -1.46684e8 q^{71} -5.74145e7 q^{73} +7.51155e7 q^{75} +5.88785e8 q^{77} +3.38536e8 q^{79} -4.28726e8 q^{81} -7.02255e8 q^{83} +2.96216e8 q^{85} +2.60018e8 q^{87} -1.77075e8 q^{89} -9.65295e8 q^{91} -7.69405e8 q^{93} -5.94445e8 q^{95} -9.17589e7 q^{97} +1.00116e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 84 q^{3} + 1875 q^{5} + 5520 q^{7} + 47079 q^{9}+O(q^{10}) \) 3 * q - 84 * q^3 + 1875 * q^5 + 5520 * q^7 + 47079 * q^9	\( 3 q - 84 q^{3} + 1875 q^{5} + 5520 q^{7} + 47079 q^{9} - 5556 q^{11} - 83094 q^{13} - 52500 q^{15} + 367062 q^{17} - 1489116 q^{19} + 1573728 q^{21} + 499920 q^{23} + 1171875 q^{25} - 7695432 q^{27} + 5234682 q^{29} - 12708912 q^{31} + 30494064 q^{33} + 3450000 q^{35} + 21724434 q^{37} + 5806824 q^{39} + 27440478 q^{41} + 23218260 q^{43} + 29424375 q^{45} + 28701528 q^{47} + 27094923 q^{49} + 155348760 q^{51} - 45629982 q^{53} - 3472500 q^{55} + 5376144 q^{57} + 268721868 q^{59} + 155970138 q^{61} + 389747664 q^{63} - 51933750 q^{65} + 526916604 q^{67} - 172501728 q^{69} + 239894424 q^{71} + 198362430 q^{73} - 32812500 q^{75} + 385819008 q^{77} + 413839728 q^{79} + 1018787787 q^{81} - 371949828 q^{83} + 229413750 q^{85} + 503011752 q^{87} + 754926606 q^{89} - 1842389664 q^{91} + 989333760 q^{93} - 930697500 q^{95} - 903451002 q^{97} - 2871765540 q^{99}+O(q^{100}) \) 3 * q - 84 * q^3 + 1875 * q^5 + 5520 * q^7 + 47079 * q^9 - 5556 * q^11 - 83094 * q^13 - 52500 * q^15 + 367062 * q^17 - 1489116 * q^19 + 1573728 * q^21 + 499920 * q^23 + 1171875 * q^25 - 7695432 * q^27 + 5234682 * q^29 - 12708912 * q^31 + 30494064 * q^33 + 3450000 * q^35 + 21724434 * q^37 + 5806824 * q^39 + 27440478 * q^41 + 23218260 * q^43 + 29424375 * q^45 + 28701528 * q^47 + 27094923 * q^49 + 155348760 * q^51 - 45629982 * q^53 - 3472500 * q^55 + 5376144 * q^57 + 268721868 * q^59 + 155970138 * q^61 + 389747664 * q^63 - 51933750 * q^65 + 526916604 * q^67 - 172501728 * q^69 + 239894424 * q^71 + 198362430 * q^73 - 32812500 * q^75 + 385819008 * q^77 + 413839728 * q^79 + 1018787787 * q^81 - 371949828 * q^83 + 229413750 * q^85 + 503011752 * q^87 + 754926606 * q^89 - 1842389664 * q^91 + 989333760 * q^93 - 930697500 * q^95 - 903451002 * q^97 - 2871765540 * q^99

Introduction

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