LMFDB - Embedded newform 80.10.a.k.1.2

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			1.2
Root			$-1.41013$ 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)

$f(q)$	$=$	$q-13.6876 q^{3} +625.000 q^{5} -6441.91 q^{7} -19495.7 q^{9} +O(q^{10})$	$q-13.6876 q^{3} +625.000 q^{5} -6441.91 q^{7} -19495.7 q^{9} +10852.5 q^{11} +109106. q^{13} -8554.73 q^{15} +145197. q^{17} +188761. q^{19} +88174.1 q^{21} -2.57319e6 q^{23} +390625. q^{25} +536260. q^{27} +5.07219e6 q^{29} -2.12111e6 q^{31} -148544. q^{33} -4.02620e6 q^{35} +1.96791e7 q^{37} -1.49340e6 q^{39} +1.38412e7 q^{41} +1.50796e7 q^{43} -1.21848e7 q^{45} +2.89670e7 q^{47} +1.14466e6 q^{49} -1.98739e6 q^{51} +3.08138e7 q^{53} +6.78279e6 q^{55} -2.58368e6 q^{57} +1.41430e8 q^{59} -5.19485e7 q^{61} +1.25589e8 q^{63} +6.81915e7 q^{65} +1.43201e8 q^{67} +3.52207e7 q^{69} +1.73941e8 q^{71} +1.00728e8 q^{73} -5.34670e6 q^{75} -6.99106e7 q^{77} -1.08352e8 q^{79} +3.76393e8 q^{81} +3.13921e7 q^{83} +9.07479e7 q^{85} -6.94259e7 q^{87} +4.57018e7 q^{89} -7.02854e8 q^{91} +2.90328e7 q^{93} +1.17976e8 q^{95} -1.23012e8 q^{97} -2.11576e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−13.6876	−0.0975619	−0.0487810	−	0.998809i	$-0.515534\pi$
$3$	−13.6876	−0.0975619	−0.0487810	+	0.998809i	$0.515534\pi$
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−6441.91	−1.01408	−0.507042	−	0.861921i	$-0.669261\pi$
$7$	−6441.91	−1.01408	−0.507042	+	0.861921i	$0.669261\pi$
$8$	0	0
$9$	−19495.7	−0.990482
$10$	0	0
$11$	10852.5	0.223492	0.111746	−	0.993737i	$-0.464356\pi$
$11$	10852.5	0.223492	0.111746	+	0.993737i	$0.464356\pi$
$12$	0	0
$13$	109106.	1.05951	0.529755	−	0.848151i	$-0.322284\pi$
$13$	109106.	1.05951	0.529755	+	0.848151i	$0.322284\pi$
$14$	0	0
$15$	−8554.73	−0.0436310
$16$	0	0
$17$	145197.	0.421635	0.210817	−	0.977525i	$-0.432387\pi$
$17$	145197.	0.421635	0.210817	+	0.977525i	$0.432387\pi$
$18$	0	0
$19$	188761.	0.332293	0.166147	−	0.986101i	$-0.446867\pi$
$19$	188761.	0.332293	0.166147	+	0.986101i	$0.446867\pi$
$20$	0	0
$21$	88174.1	0.0989360
$22$	0	0
$23$	−2.57319e6	−1.91733	−0.958664	−	0.284542i	$-0.908159\pi$
$23$	−2.57319e6	−1.91733	−0.958664	+	0.284542i	$0.908159\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	536260.	0.194195
$28$	0	0
$29$	5.07219e6	1.33169	0.665847	−	0.746089i	$-0.268070\pi$
$29$	5.07219e6	1.33169	0.665847	+	0.746089i	$0.268070\pi$
$30$	0	0
$31$	−2.12111e6	−0.412511	−0.206256	−	0.978498i	$-0.566128\pi$
$31$	−2.12111e6	−0.412511	−0.206256	+	0.978498i	$0.566128\pi$
$32$	0	0
$33$	−148544.	−0.0218043
$34$	0	0
$35$	−4.02620e6	−0.453512
$36$	0	0
$37$	1.96791e7	1.72622	0.863111	−	0.505013i	$-0.168512\pi$
$37$	1.96791e7	1.72622	0.863111	+	0.505013i	$0.168512\pi$
$38$	0	0
$39$	−1.49340e6	−0.103368
$40$	0	0
$41$	1.38412e7	0.764975	0.382487	−	0.923961i	$-0.375068\pi$
$41$	1.38412e7	0.764975	0.382487	+	0.923961i	$0.375068\pi$
$42$	0	0
$43$	1.50796e7	0.672639	0.336319	−	0.941748i	$-0.390818\pi$
$43$	1.50796e7	0.672639	0.336319	+	0.941748i	$0.390818\pi$
$44$	0	0
$45$	−1.21848e7	−0.442957
$46$	0	0
$47$	2.89670e7	0.865891	0.432945	−	0.901420i	$-0.357474\pi$
$47$	2.89670e7	0.865891	0.432945	+	0.901420i	$0.357474\pi$
$48$	0	0
$49$	1.14466e6	0.0283657
$50$	0	0
$51$	−1.98739e6	−0.0411355
$52$	0	0
$53$	3.08138e7	0.536418	0.268209	−	0.963361i	$-0.413568\pi$
$53$	3.08138e7	0.536418	0.268209	+	0.963361i	$0.413568\pi$
$54$	0	0
$55$	6.78279e6	0.0999484
$56$	0	0
$57$	−2.58368e6	−0.0324192
$58$	0	0
$59$	1.41430e8	1.51952	0.759761	−	0.650202i	$-0.225316\pi$
$59$	1.41430e8	1.51952	0.759761	+	0.650202i	$0.225316\pi$
$60$	0	0
$61$	−5.19485e7	−0.480385	−0.240192	−	0.970725i	$-0.577211\pi$
$61$	−5.19485e7	−0.480385	−0.240192	+	0.970725i	$0.577211\pi$
$62$	0	0
$63$	1.25589e8	1.00443
$64$	0	0
$65$	6.81915e7	0.473827
$66$	0	0
$67$	1.43201e8	0.868181	0.434091	−	0.900869i	$-0.357070\pi$
$67$	1.43201e8	0.868181	0.434091	+	0.900869i	$0.357070\pi$
$68$	0	0
$69$	3.52207e7	0.187058
$70$	0	0
$71$	1.73941e8	0.812344	0.406172	−	0.913797i	$-0.366863\pi$
$71$	1.73941e8	0.812344	0.406172	+	0.913797i	$0.366863\pi$
$72$	0	0
$73$	1.00728e8	0.415144	0.207572	−	0.978220i	$-0.433444\pi$
$73$	1.00728e8	0.415144	0.207572	+	0.978220i	$0.433444\pi$
$74$	0	0
$75$	−5.34670e6	−0.0195124
$76$	0	0
$77$	−6.99106e7	−0.226639
$78$	0	0
$79$	−1.08352e8	−0.312979	−0.156490	−	0.987680i	$-0.550018\pi$
$79$	−1.08352e8	−0.312979	−0.156490	+	0.987680i	$0.550018\pi$
$80$	0	0
$81$	3.76393e8	0.971536
$82$	0	0
$83$	3.13921e7	0.0726053	0.0363027	−	0.999341i	$-0.488442\pi$
$83$	3.13921e7	0.0726053	0.0363027	+	0.999341i	$0.488442\pi$
$84$	0	0
$85$	9.07479e7	0.188561
$86$	0	0
$87$	−6.94259e7	−0.129923
$88$	0	0
$89$	4.57018e7	0.0772108	0.0386054	−	0.999255i	$-0.487708\pi$
$89$	4.57018e7	0.0772108	0.0386054	+	0.999255i	$0.487708\pi$
$90$	0	0
$91$	−7.02854e8	−1.07443
$92$	0	0
$93$	2.90328e7	0.0402454
$94$	0	0
$95$	1.17976e8	0.148606
$96$	0	0
$97$	−1.23012e8	−0.141083	−0.0705413	−	0.997509i	$-0.522473\pi$
$97$	−1.23012e8	−0.141083	−0.0705413	+	0.997509i	$0.522473\pi$
$98$	0	0
$99$	−2.11576e8	−0.221364
$100$	0	0
$101$	2.02496e9	1.93629	0.968145	−	0.250391i	$-0.0805593\pi$
$101$	2.02496e9	1.93629	0.968145	+	0.250391i	$0.0805593\pi$
$102$	0	0
$103$	−6.95510e8	−0.608886	−0.304443	−	0.952531i	$-0.598470\pi$
$103$	−6.95510e8	−0.608886	−0.304443	+	0.952531i	$0.598470\pi$
$104$	0	0
$105$	5.51088e7	0.0442455
$106$	0	0
$107$	−1.99939e7	−0.0147458	−0.00737292	−	0.999973i	$-0.502347\pi$
$107$	−1.99939e7	−0.0147458	−0.00737292	+	0.999973i	$0.502347\pi$
$108$	0	0
$109$	−1.72916e9	−1.17332	−0.586659	−	0.809834i	$-0.699557\pi$
$109$	−1.72916e9	−1.17332	−0.586659	+	0.809834i	$0.699557\pi$
$110$	0	0
$111$	−2.69359e8	−0.168414
$112$	0	0
$113$	−1.15724e9	−0.667680	−0.333840	−	0.942630i	$-0.608345\pi$
$113$	−1.15724e9	−0.667680	−0.333840	+	0.942630i	$0.608345\pi$
$114$	0	0
$115$	−1.60824e9	−0.857455
$116$	0	0
$117$	−2.12710e9	−1.04943
$118$	0	0
$119$	−9.35344e8	−0.427573
$120$	0	0
$121$	−2.24017e9	−0.950052
$122$	0	0
$123$	−1.89453e8	−0.0746324
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	5.43126e9	1.85261	0.926304	−	0.376777i	$-0.122968\pi$
$127$	5.43126e9	1.85261	0.926304	+	0.376777i	$0.122968\pi$
$128$	0	0
$129$	−2.06403e8	−0.0656240
$130$	0	0
$131$	−5.44539e9	−1.61551	−0.807753	−	0.589522i	$-0.799316\pi$
$131$	−5.44539e9	−1.61551	−0.807753	+	0.589522i	$0.799316\pi$
$132$	0	0
$133$	−1.21598e9	−0.336973
$134$	0	0
$135$	3.35163e8	0.0868468
$136$	0	0
$137$	−4.61719e9	−1.11979	−0.559893	−	0.828565i	$-0.689158\pi$
$137$	−4.61719e9	−1.11979	−0.559893	+	0.828565i	$0.689158\pi$
$138$	0	0
$139$	5.16328e9	1.17316	0.586582	−	0.809890i	$-0.300473\pi$
$139$	5.16328e9	1.17316	0.586582	+	0.809890i	$0.300473\pi$
$140$	0	0
$141$	−3.96488e8	−0.0844780
$142$	0	0
$143$	1.18407e9	0.236791
$144$	0	0
$145$	3.17012e9	0.595551
$146$	0	0
$147$	−1.56676e7	−0.00276742
$148$	0	0
$149$	−7.86663e9	−1.30753	−0.653763	−	0.756699i	$-0.726811\pi$
$149$	−7.86663e9	−1.30753	−0.653763	+	0.756699i	$0.726811\pi$
$150$	0	0
$151$	−1.00208e10	−1.56858	−0.784290	−	0.620395i	$-0.786972\pi$
$151$	−1.00208e10	−1.56858	−0.784290	+	0.620395i	$0.786972\pi$
$152$	0	0
$153$	−2.83070e9	−0.417622
$154$	0	0
$155$	−1.32569e9	−0.184481
$156$	0	0
$157$	2.95872e9	0.388647	0.194324	−	0.980937i	$-0.437749\pi$
$157$	2.95872e9	0.388647	0.194324	+	0.980937i	$0.437749\pi$
$158$	0	0
$159$	−4.21766e8	−0.0523340
$160$	0	0
$161$	1.65763e10	1.94433
$162$	0	0
$163$	3.24082e9	0.359592	0.179796	−	0.983704i	$-0.442456\pi$
$163$	3.24082e9	0.359592	0.179796	+	0.983704i	$0.442456\pi$
$164$	0	0
$165$	−9.28398e7	−0.00975116
$166$	0	0
$167$	1.51696e10	1.50921	0.754605	−	0.656179i	$-0.227828\pi$
$167$	1.51696e10	1.50921	0.754605	+	0.656179i	$0.227828\pi$
$168$	0	0
$169$	1.29970e9	0.122561
$170$	0	0
$171$	−3.68002e9	−0.329130
$172$	0	0
$173$	1.25199e10	1.06265	0.531327	−	0.847167i	$-0.321694\pi$
$173$	1.25199e10	1.06265	0.531327	+	0.847167i	$0.321694\pi$
$174$	0	0
$175$	−2.51637e9	−0.202817
$176$	0	0
$177$	−1.93583e9	−0.148248
$178$	0	0
$179$	−1.74627e10	−1.27137	−0.635686	−	0.771948i	$-0.719283\pi$
$179$	−1.74627e10	−1.27137	−0.635686	+	0.771948i	$0.719283\pi$
$180$	0	0
$181$	2.26092e10	1.56579	0.782893	−	0.622156i	$-0.213743\pi$
$181$	2.26092e10	1.56579	0.782893	+	0.622156i	$0.213743\pi$
$182$	0	0
$183$	7.11049e8	0.0468673
$184$	0	0
$185$	1.22994e10	0.771990
$186$	0	0
$187$	1.57574e9	0.0942318
$188$	0	0
$189$	−3.45454e9	−0.196930
$190$	0	0
$191$	8.50787e9	0.462563	0.231281	−	0.972887i	$-0.425708\pi$
$191$	8.50787e9	0.462563	0.231281	+	0.972887i	$0.425708\pi$
$192$	0	0
$193$	−1.81212e10	−0.940111	−0.470055	−	0.882637i	$-0.655766\pi$
$193$	−1.81212e10	−0.940111	−0.470055	+	0.882637i	$0.655766\pi$
$194$	0	0
$195$	−9.33375e8	−0.0462275
$196$	0	0
$197$	−1.86356e9	−0.0881547	−0.0440773	−	0.999028i	$-0.514035\pi$
$197$	−1.86356e9	−0.0881547	−0.0440773	+	0.999028i	$0.514035\pi$
$198$	0	0
$199$	3.57524e10	1.61609	0.808047	−	0.589118i	$-0.200525\pi$
$199$	3.57524e10	1.61609	0.808047	+	0.589118i	$0.200525\pi$
$200$	0	0
$201$	−1.96008e9	−0.0847014
$202$	0	0
$203$	−3.26746e10	−1.35045
$204$	0	0
$205$	8.65076e9	0.342107
$206$	0	0
$207$	5.01660e10	1.89908
$208$	0	0
$209$	2.04852e9	0.0742647
$210$	0	0
$211$	−7.16388e9	−0.248815	−0.124408	−	0.992231i	$-0.539703\pi$
$211$	−7.16388e9	−0.248815	−0.124408	+	0.992231i	$0.539703\pi$
$212$	0	0
$213$	−2.38083e9	−0.0792538
$214$	0	0
$215$	9.42476e9	0.300813
$216$	0	0
$217$	1.36640e10	0.418321
$218$	0	0
$219$	−1.37873e9	−0.0405023
$220$	0	0
$221$	1.58419e10	0.446726
$222$	0	0
$223$	4.74291e10	1.28432	0.642159	−	0.766571i	$-0.278039\pi$
$223$	4.74291e10	1.28432	0.642159	+	0.766571i	$0.278039\pi$
$224$	0	0
$225$	−7.61549e9	−0.198096
$226$	0	0
$227$	5.57370e9	0.139324	0.0696622	−	0.997571i	$-0.477808\pi$
$227$	5.57370e9	0.139324	0.0696622	+	0.997571i	$0.477808\pi$
$228$	0	0
$229$	−6.30694e10	−1.51551	−0.757756	−	0.652538i	$-0.773704\pi$
$229$	−6.30694e10	−1.51551	−0.757756	+	0.652538i	$0.773704\pi$
$230$	0	0
$231$	9.56906e8	0.0221114
$232$	0	0
$233$	−1.52414e10	−0.338785	−0.169392	−	0.985549i	$-0.554181\pi$
$233$	−1.52414e10	−0.338785	−0.169392	+	0.985549i	$0.554181\pi$
$234$	0	0
$235$	1.81044e10	0.387238
$236$	0	0
$237$	1.48308e9	0.0305349
$238$	0	0
$239$	−6.12794e10	−1.21485	−0.607427	−	0.794376i	$-0.707798\pi$
$239$	−6.12794e10	−1.21485	−0.607427	+	0.794376i	$0.707798\pi$
$240$	0	0
$241$	−6.42763e10	−1.22737	−0.613683	−	0.789553i	$-0.710313\pi$
$241$	−6.42763e10	−1.22737	−0.613683	+	0.789553i	$0.710313\pi$
$242$	0	0
$243$	−1.57071e10	−0.288980
$244$	0	0
$245$	7.15413e8	0.0126855
$246$	0	0
$247$	2.05951e10	0.352068
$248$	0	0
$249$	−4.29681e8	−0.00708351
$250$	0	0
$251$	−8.92346e10	−1.41906	−0.709531	−	0.704674i	$-0.751093\pi$
$251$	−8.92346e10	−1.41906	−0.709531	+	0.704674i	$0.751093\pi$
$252$	0	0
$253$	−2.79254e10	−0.428506
$254$	0	0
$255$	−1.24212e9	−0.0183964
$256$	0	0
$257$	1.08567e11	1.55239	0.776195	−	0.630493i	$-0.217147\pi$
$257$	1.08567e11	1.55239	0.776195	+	0.630493i	$0.217147\pi$
$258$	0	0
$259$	−1.26771e11	−1.75053
$260$	0	0
$261$	−9.88856e10	−1.31902
$262$	0	0
$263$	−4.15009e10	−0.534880	−0.267440	−	0.963575i	$-0.586178\pi$
$263$	−4.15009e10	−0.534880	−0.267440	+	0.963575i	$0.586178\pi$
$264$	0	0
$265$	1.92586e10	0.239894
$266$	0	0
$267$	−6.25546e8	−0.00753284
$268$	0	0
$269$	−9.06843e9	−0.105596	−0.0527979	−	0.998605i	$-0.516814\pi$
$269$	−9.06843e9	−0.105596	−0.0527979	+	0.998605i	$0.516814\pi$
$270$	0	0
$271$	8.86768e10	0.998730	0.499365	−	0.866392i	$-0.333567\pi$
$271$	8.86768e10	0.998730	0.499365	+	0.866392i	$0.333567\pi$
$272$	0	0
$273$	9.62036e9	0.104824
$274$	0	0
$275$	4.23924e9	0.0446983
$276$	0	0
$277$	1.42005e11	1.44925	0.724627	−	0.689141i	$-0.242012\pi$
$277$	1.42005e11	1.44925	0.724627	+	0.689141i	$0.242012\pi$
$278$	0	0
$279$	4.13524e10	0.408585
$280$	0	0
$281$	−3.47476e9	−0.0332466	−0.0166233	−	0.999862i	$-0.505292\pi$
$281$	−3.47476e9	−0.0332466	−0.0166233	+	0.999862i	$0.505292\pi$
$282$	0	0
$283$	−1.70869e11	−1.58352	−0.791759	−	0.610833i	$-0.790835\pi$
$283$	−1.70869e11	−1.58352	−0.791759	+	0.610833i	$0.790835\pi$
$284$	0	0
$285$	−1.61480e9	−0.0144983
$286$	0	0
$287$	−8.91640e10	−0.775748
$288$	0	0
$289$	−9.75058e10	−0.822224
$290$	0	0
$291$	1.68373e9	0.0137643
$292$	0	0
$293$	2.70019e10	0.214038	0.107019	−	0.994257i	$-0.465869\pi$
$293$	2.70019e10	0.214038	0.107019	+	0.994257i	$0.465869\pi$
$294$	0	0
$295$	8.83936e10	0.679551
$296$	0	0
$297$	5.81974e9	0.0434010
$298$	0	0
$299$	−2.80751e11	−2.03143
$300$	0	0
$301$	−9.71416e10	−0.682112
$302$	0	0
$303$	−2.77168e10	−0.188908
$304$	0	0
$305$	−3.24678e10	−0.214835
$306$	0	0
$307$	1.21361e11	0.779754	0.389877	−	0.920867i	$-0.372518\pi$
$307$	1.21361e11	0.779754	0.389877	+	0.920867i	$0.372518\pi$
$308$	0	0
$309$	9.51984e9	0.0594041
$310$	0	0
$311$	2.88321e11	1.74765	0.873825	−	0.486240i	$-0.161632\pi$
$311$	2.88321e11	1.74765	0.873825	+	0.486240i	$0.161632\pi$
$312$	0	0
$313$	−1.08067e11	−0.636420	−0.318210	−	0.948020i	$-0.603082\pi$
$313$	−1.08067e11	−0.636420	−0.318210	+	0.948020i	$0.603082\pi$
$314$	0	0
$315$	7.84933e10	0.449195
$316$	0	0
$317$	2.68432e11	1.49302	0.746512	−	0.665372i	$-0.231727\pi$
$317$	2.68432e11	1.49302	0.746512	+	0.665372i	$0.231727\pi$
$318$	0	0
$319$	5.50457e10	0.297622
$320$	0	0
$321$	2.73667e8	0.00143863
$322$	0	0
$323$	2.74075e10	0.140106
$324$	0	0
$325$	4.26197e10	0.211902
$326$	0	0
$327$	2.36680e10	0.114471
$328$	0	0
$329$	−1.86603e11	−0.878086
$330$	0	0
$331$	2.30150e11	1.05386	0.526932	−	0.849907i	$-0.323342\pi$
$331$	2.30150e11	1.05386	0.526932	+	0.849907i	$0.323342\pi$
$332$	0	0
$333$	−3.83656e11	−1.70979
$334$	0	0
$335$	8.95008e10	0.388262
$336$	0	0
$337$	3.04720e11	1.28696	0.643481	−	0.765462i	$-0.277489\pi$
$337$	3.04720e11	1.28696	0.643481	+	0.765462i	$0.277489\pi$
$338$	0	0
$339$	1.58397e10	0.0651402
$340$	0	0
$341$	−2.30193e10	−0.0921927
$342$	0	0
$343$	2.52581e11	0.985318
$344$	0	0
$345$	2.20129e10	0.0836550
$346$	0	0
$347$	4.45448e10	0.164936	0.0824679	−	0.996594i	$-0.473720\pi$
$347$	4.45448e10	0.164936	0.0824679	+	0.996594i	$0.473720\pi$
$348$	0	0
$349$	1.87710e11	0.677287	0.338644	−	0.940915i	$-0.390032\pi$
$349$	1.87710e11	0.677287	0.338644	+	0.940915i	$0.390032\pi$
$350$	0	0
$351$	5.85094e10	0.205752
$352$	0	0
$353$	4.16651e11	1.42819	0.714096	−	0.700048i	$-0.246838\pi$
$353$	4.16651e11	1.42819	0.714096	+	0.700048i	$0.246838\pi$
$354$	0	0
$355$	1.08713e11	0.363291
$356$	0	0
$357$	1.28026e10	0.0417148
$358$	0	0
$359$	−4.72035e11	−1.49985	−0.749927	−	0.661521i	$-0.769911\pi$
$359$	−4.72035e11	−1.49985	−0.749927	+	0.661521i	$0.769911\pi$
$360$	0	0
$361$	−2.87057e11	−0.889581
$362$	0	0
$363$	3.06625e10	0.0926889
$364$	0	0
$365$	6.29553e10	0.185658
$366$	0	0
$367$	6.85720e11	1.97310	0.986550	−	0.163457i	$-0.0522645\pi$
$367$	6.85720e11	1.97310	0.986550	+	0.163457i	$0.0522645\pi$
$368$	0	0
$369$	−2.69844e11	−0.757694
$370$	0	0
$371$	−1.98500e11	−0.543973
$372$	0	0
$373$	4.49370e11	1.20203	0.601014	−	0.799239i	$-0.294764\pi$
$373$	4.49370e11	1.20203	0.601014	+	0.799239i	$0.294764\pi$
$374$	0	0
$375$	−3.34169e9	−0.00872620
$376$	0	0
$377$	5.53408e11	1.41094
$378$	0	0
$379$	−2.25760e10	−0.0562045	−0.0281023	−	0.999605i	$-0.508946\pi$
$379$	−2.25760e10	−0.0562045	−0.0281023	+	0.999605i	$0.508946\pi$
$380$	0	0
$381$	−7.43406e10	−0.180744
$382$	0	0
$383$	−4.18750e11	−0.994399	−0.497199	−	0.867636i	$-0.665638\pi$
$383$	−4.18750e11	−0.994399	−0.497199	+	0.867636i	$0.665638\pi$
$384$	0	0
$385$	−4.36941e10	−0.101356
$386$	0	0
$387$	−2.93987e11	−0.666237
$388$	0	0
$389$	−2.62984e11	−0.582313	−0.291156	−	0.956676i	$-0.594040\pi$
$389$	−2.62984e11	−0.582313	−0.291156	+	0.956676i	$0.594040\pi$
$390$	0	0
$391$	−3.73618e11	−0.808412
$392$	0	0
$393$	7.45341e10	0.157612
$394$	0	0
$395$	−6.77201e10	−0.139969
$396$	0	0
$397$	1.06786e11	0.215753	0.107876	−	0.994164i	$-0.465595\pi$
$397$	1.06786e11	0.215753	0.107876	+	0.994164i	$0.465595\pi$
$398$	0	0
$399$	1.66439e10	0.0328758
$400$	0	0
$401$	1.17480e11	0.226889	0.113445	−	0.993544i	$-0.463812\pi$
$401$	1.17480e11	0.226889	0.113445	+	0.993544i	$0.463812\pi$
$402$	0	0
$403$	−2.31427e11	−0.437060
$404$	0	0
$405$	2.35245e11	0.434484
$406$	0	0
$407$	2.13566e11	0.385796
$408$	0	0
$409$	5.05173e11	0.892658	0.446329	−	0.894869i	$-0.352731\pi$
$409$	5.05173e11	0.892658	0.446329	+	0.894869i	$0.352731\pi$
$410$	0	0
$411$	6.31981e10	0.109249
$412$	0	0
$413$	−9.11079e11	−1.54092
$414$	0	0
$415$	1.96200e10	0.0324701
$416$	0	0
$417$	−7.06727e10	−0.114456
$418$	0	0
$419$	2.91162e11	0.461500	0.230750	−	0.973013i	$-0.425882\pi$
$419$	2.91162e11	0.461500	0.230750	+	0.973013i	$0.425882\pi$
$420$	0	0
$421$	5.62016e11	0.871926	0.435963	−	0.899965i	$-0.356408\pi$
$421$	5.62016e11	0.871926	0.435963	+	0.899965i	$0.356408\pi$
$422$	0	0
$423$	−5.64730e11	−0.857649
$424$	0	0
$425$	5.67174e10	0.0843270
$426$	0	0
$427$	3.34648e11	0.487150
$428$	0	0
$429$	−1.62071e10	−0.0231018
$430$	0	0
$431$	−1.16461e12	−1.62567	−0.812836	−	0.582492i	$-0.802078\pi$
$431$	−1.16461e12	−1.62567	−0.812836	+	0.582492i	$0.802078\pi$
$432$	0	0
$433$	−1.01628e12	−1.38938	−0.694688	−	0.719312i	$-0.744457\pi$
$433$	−1.01628e12	−1.38938	−0.694688	+	0.719312i	$0.744457\pi$
$434$	0	0
$435$	−4.33912e10	−0.0581031
$436$	0	0
$437$	−4.85718e11	−0.637115
$438$	0	0
$439$	−5.15097e11	−0.661910	−0.330955	−	0.943647i	$-0.607371\pi$
$439$	−5.15097e11	−0.661910	−0.330955	+	0.943647i	$0.607371\pi$
$440$	0	0
$441$	−2.23159e10	−0.0280957
$442$	0	0
$443$	1.00876e12	1.24443	0.622213	−	0.782848i	$-0.286234\pi$
$443$	1.00876e12	1.24443	0.622213	+	0.782848i	$0.286234\pi$
$444$	0	0
$445$	2.85636e10	0.0345297
$446$	0	0
$447$	1.07675e11	0.127565
$448$	0	0
$449$	−1.48908e12	−1.72905	−0.864527	−	0.502587i	$-0.832382\pi$
$449$	−1.48908e12	−1.72905	−0.864527	+	0.502587i	$0.832382\pi$
$450$	0	0
$451$	1.50211e11	0.170965
$452$	0	0
$453$	1.37160e11	0.153034
$454$	0	0
$455$	−4.39284e11	−0.480500
$456$	0	0
$457$	−1.37073e12	−1.47004	−0.735021	−	0.678044i	$-0.762828\pi$
$457$	−1.37073e12	−1.47004	−0.735021	+	0.678044i	$0.762828\pi$
$458$	0	0
$459$	7.78632e10	0.0818795
$460$	0	0
$461$	−6.87423e10	−0.0708875	−0.0354438	−	0.999372i	$-0.511284\pi$
$461$	−6.87423e10	−0.0708875	−0.0354438	+	0.999372i	$0.511284\pi$
$462$	0	0
$463$	−1.36281e11	−0.137822	−0.0689112	−	0.997623i	$-0.521953\pi$
$463$	−1.36281e11	−0.137822	−0.0689112	+	0.997623i	$0.521953\pi$
$464$	0	0
$465$	1.81455e10	0.0179983
$466$	0	0
$467$	1.01277e12	0.985337	0.492668	−	0.870217i	$-0.336022\pi$
$467$	1.01277e12	0.985337	0.492668	+	0.870217i	$0.336022\pi$
$468$	0	0
$469$	−9.22491e11	−0.880408
$470$	0	0
$471$	−4.04977e10	−0.0379172
$472$	0	0
$473$	1.63651e11	0.150329
$474$	0	0
$475$	7.37349e10	0.0664587
$476$	0	0
$477$	−6.00735e11	−0.531313
$478$	0	0
$479$	−1.04470e11	−0.0906740	−0.0453370	−	0.998972i	$-0.514436\pi$
$479$	−1.04470e11	−0.0906740	−0.0453370	+	0.998972i	$0.514436\pi$
$480$	0	0
$481$	2.14711e12	1.82895
$482$	0	0
$483$	−2.26889e11	−0.189693
$484$	0	0
$485$	−7.68823e10	−0.0630941
$486$	0	0
$487$	−7.77886e11	−0.626666	−0.313333	−	0.949643i	$-0.601446\pi$
$487$	−7.77886e11	−0.626666	−0.313333	+	0.949643i	$0.601446\pi$
$488$	0	0
$489$	−4.43589e10	−0.0350825
$490$	0	0
$491$	1.11203e10	0.00863479	0.00431739	−	0.999991i	$-0.498626\pi$
$491$	1.11203e10	0.00863479	0.00431739	+	0.999991i	$0.498626\pi$
$492$	0	0
$493$	7.36465e11	0.561488
$494$	0	0
$495$	−1.32235e11	−0.0989971
$496$	0	0
$497$	−1.12051e12	−0.823784
$498$	0	0
$499$	2.22906e12	1.60942	0.804711	−	0.593667i	$-0.202320\pi$
$499$	2.22906e12	1.60942	0.804711	+	0.593667i	$0.202320\pi$
$500$	0	0
$501$	−2.07635e11	−0.147241
$502$	0	0
$503$	−5.00423e11	−0.348563	−0.174282	−	0.984696i	$-0.555760\pi$
$503$	−5.00423e11	−0.348563	−0.174282	+	0.984696i	$0.555760\pi$
$504$	0	0
$505$	1.26560e12	0.865935
$506$	0	0
$507$	−1.77897e10	−0.0119573
$508$	0	0
$509$	−1.39812e11	−0.0923237	−0.0461619	−	0.998934i	$-0.514699\pi$
$509$	−1.39812e11	−0.0923237	−0.0461619	+	0.998934i	$0.514699\pi$
$510$	0	0
$511$	−6.48884e11	−0.420991
$512$	0	0
$513$	1.01225e11	0.0645298
$514$	0	0
$515$	−4.34694e11	−0.272302
$516$	0	0
$517$	3.14363e11	0.193519
$518$	0	0
$519$	−1.71366e11	−0.103675
$520$	0	0
$521$	−9.82729e11	−0.584338	−0.292169	−	0.956367i	$-0.594377\pi$
$521$	−9.82729e11	−0.584338	−0.292169	+	0.956367i	$0.594377\pi$
$522$	0	0
$523$	1.64699e12	0.962571	0.481286	−	0.876564i	$-0.340170\pi$
$523$	1.64699e12	0.962571	0.481286	+	0.876564i	$0.340170\pi$
$524$	0	0
$525$	3.44430e10	0.0197872
$526$	0	0
$527$	−3.07978e11	−0.173929
$528$	0	0
$529$	4.82014e12	2.67614
$530$	0	0
$531$	−2.75727e12	−1.50506
$532$	0	0
$533$	1.51017e12	0.810498
$534$	0	0
$535$	−1.24962e10	−0.00659454
$536$	0	0
$537$	2.39022e11	0.124037
$538$	0	0
$539$	1.24224e10	0.00633950
$540$	0	0
$541$	3.25662e12	1.63448	0.817240	−	0.576298i	$-0.195503\pi$
$541$	3.25662e12	1.63448	0.817240	+	0.576298i	$0.195503\pi$
$542$	0	0
$543$	−3.09465e11	−0.152761
$544$	0	0
$545$	−1.08072e12	−0.524723
$546$	0	0
$547$	1.35300e12	0.646181	0.323090	−	0.946368i	$-0.395278\pi$
$547$	1.35300e12	0.646181	0.323090	+	0.946368i	$0.395278\pi$
$548$	0	0
$549$	1.01277e12	0.475812
$550$	0	0
$551$	9.57432e11	0.442513
$552$	0	0
$553$	6.97995e11	0.317387
$554$	0	0
$555$	−1.68349e11	−0.0753169
$556$	0	0
$557$	−2.35526e12	−1.03679	−0.518394	−	0.855142i	$-0.673470\pi$
$557$	−2.35526e12	−1.03679	−0.518394	+	0.855142i	$0.673470\pi$
$558$	0	0
$559$	1.64528e12	0.712668
$560$	0	0
$561$	−2.15680e10	−0.00919344
$562$	0	0
$563$	3.17692e12	1.33266	0.666329	−	0.745658i	$-0.267865\pi$
$563$	3.17692e12	1.33266	0.666329	+	0.745658i	$0.267865\pi$
$564$	0	0
$565$	−7.23272e11	−0.298596
$566$	0	0
$567$	−2.42469e12	−0.985218
$568$	0	0
$569$	−3.46833e11	−0.138712	−0.0693562	−	0.997592i	$-0.522095\pi$
$569$	−3.46833e11	−0.138712	−0.0693562	+	0.997592i	$0.522095\pi$
$570$	0	0
$571$	2.20689e12	0.868796	0.434398	−	0.900721i	$-0.356961\pi$
$571$	2.20689e12	0.868796	0.434398	+	0.900721i	$0.356961\pi$
$572$	0	0
$573$	−1.16452e11	−0.0451285
$574$	0	0
$575$	−1.00515e12	−0.383465
$576$	0	0
$577$	3.69668e12	1.38842	0.694209	−	0.719773i	$-0.255754\pi$
$577$	3.69668e12	1.38842	0.694209	+	0.719773i	$0.255754\pi$
$578$	0	0
$579$	2.48035e11	0.0917190
$580$	0	0
$581$	−2.02225e11	−0.0736279
$582$	0	0
$583$	3.34405e11	0.119885
$584$	0	0
$585$	−1.32944e12	−0.469317
$586$	0	0
$587$	−9.28201e11	−0.322679	−0.161339	−	0.986899i	$-0.551581\pi$
$587$	−9.28201e11	−0.322679	−0.161339	+	0.986899i	$0.551581\pi$
$588$	0	0
$589$	−4.00383e11	−0.137075
$590$	0	0
$591$	2.55076e10	0.00860054
$592$	0	0
$593$	−1.46926e12	−0.487925	−0.243963	−	0.969785i	$-0.578447\pi$
$593$	−1.46926e12	−0.487925	−0.243963	+	0.969785i	$0.578447\pi$
$594$	0	0
$595$	−5.84590e11	−0.191216
$596$	0	0
$597$	−4.89363e11	−0.157669
$598$	0	0
$599$	3.83542e12	1.21728	0.608642	−	0.793445i	$-0.291715\pi$
$599$	3.83542e12	1.21728	0.608642	+	0.793445i	$0.291715\pi$
$600$	0	0
$601$	−4.56341e12	−1.42677	−0.713385	−	0.700772i	$-0.752839\pi$
$601$	−4.56341e12	−1.42677	−0.713385	+	0.700772i	$0.752839\pi$
$602$	0	0
$603$	−2.79180e12	−0.859918
$604$	0	0
$605$	−1.40011e12	−0.424876
$606$	0	0
$607$	−2.20405e12	−0.658981	−0.329490	−	0.944159i	$-0.606877\pi$
$607$	−2.20405e12	−0.658981	−0.329490	+	0.944159i	$0.606877\pi$
$608$	0	0
$609$	4.47235e11	0.131752
$610$	0	0
$611$	3.16048e12	0.917420
$612$	0	0
$613$	3.66833e12	1.04929	0.524646	−	0.851321i	$-0.324198\pi$
$613$	3.66833e12	1.04929	0.524646	+	0.851321i	$0.324198\pi$
$614$	0	0
$615$	−1.18408e11	−0.0333766
$616$	0	0
$617$	−6.55267e12	−1.82027	−0.910133	−	0.414316i	$-0.864021\pi$
$617$	−6.55267e12	−1.82027	−0.910133	+	0.414316i	$0.864021\pi$
$618$	0	0
$619$	1.01737e12	0.278529	0.139264	−	0.990255i	$-0.455526\pi$
$619$	1.01737e12	0.278529	0.139264	+	0.990255i	$0.455526\pi$
$620$	0	0
$621$	−1.37990e12	−0.372336
$622$	0	0
$623$	−2.94407e11	−0.0782982
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−2.80393e10	−0.00724541
$628$	0	0
$629$	2.85734e12	0.727836
$630$	0	0
$631$	6.83914e12	1.71739	0.858695	−	0.512486i	$-0.171276\pi$
$631$	6.83914e12	1.71739	0.858695	+	0.512486i	$0.171276\pi$
$632$	0	0
$633$	9.80561e10	0.0242749
$634$	0	0
$635$	3.39453e12	0.828511
$636$	0	0
$637$	1.24890e11	0.0300538
$638$	0	0
$639$	−3.39110e12	−0.804611
$640$	0	0
$641$	5.98404e11	0.140002	0.0700008	−	0.997547i	$-0.477700\pi$
$641$	5.98404e11	0.140002	0.0700008	+	0.997547i	$0.477700\pi$
$642$	0	0
$643$	2.69616e12	0.622009	0.311004	−	0.950408i	$-0.399335\pi$
$643$	2.69616e12	0.622009	0.311004	+	0.950408i	$0.399335\pi$
$644$	0	0
$645$	−1.29002e11	−0.0293479
$646$	0	0
$647$	−7.49457e12	−1.68143	−0.840713	−	0.541481i	$-0.817864\pi$
$647$	−7.49457e12	−1.68143	−0.840713	+	0.541481i	$0.817864\pi$
$648$	0	0
$649$	1.53486e12	0.339600
$650$	0	0
$651$	−1.87027e11	−0.0408122
$652$	0	0
$653$	−4.72232e12	−1.01636	−0.508178	−	0.861252i	$-0.669681\pi$
$653$	−4.72232e12	−1.01636	−0.508178	+	0.861252i	$0.669681\pi$
$654$	0	0
$655$	−3.40337e12	−0.722476
$656$	0	0
$657$	−1.96377e12	−0.411193
$658$	0	0
$659$	−3.87680e12	−0.800736	−0.400368	−	0.916354i	$-0.631118\pi$
$659$	−3.87680e12	−0.800736	−0.400368	+	0.916354i	$0.631118\pi$
$660$	0	0
$661$	−5.41631e12	−1.10356	−0.551781	−	0.833989i	$-0.686052\pi$
$661$	−5.41631e12	−1.10356	−0.551781	+	0.833989i	$0.686052\pi$
$662$	0	0
$663$	−2.16837e11	−0.0435835
$664$	0	0
$665$	−7.59990e11	−0.150699
$666$	0	0
$667$	−1.30517e13	−2.55329
$668$	0	0
$669$	−6.49188e11	−0.125301
$670$	0	0
$671$	−5.63769e11	−0.107362
$672$	0	0
$673$	3.19469e12	0.600289	0.300145	−	0.953894i	$-0.402965\pi$
$673$	3.19469e12	0.600289	0.300145	+	0.953894i	$0.402965\pi$
$674$	0	0
$675$	2.09477e11	0.0388390
$676$	0	0
$677$	1.51904e12	0.277920	0.138960	−	0.990298i	$-0.455624\pi$
$677$	1.51904e12	0.277920	0.138960	+	0.990298i	$0.455624\pi$
$678$	0	0
$679$	7.92431e11	0.143070
$680$	0	0
$681$	−7.62904e10	−0.0135928
$682$	0	0
$683$	−3.62219e12	−0.636910	−0.318455	−	0.947938i	$-0.603164\pi$
$683$	−3.62219e12	−0.636910	−0.318455	+	0.947938i	$0.603164\pi$
$684$	0	0
$685$	−2.88574e12	−0.500784
$686$	0	0
$687$	8.63267e11	0.147856
$688$	0	0
$689$	3.36198e12	0.568341
$690$	0	0
$691$	−5.71420e12	−0.953464	−0.476732	−	0.879049i	$-0.658179\pi$
$691$	−5.71420e12	−0.953464	−0.476732	+	0.879049i	$0.658179\pi$
$692$	0	0
$693$	1.36295e12	0.224482
$694$	0	0
$695$	3.22705e12	0.524655
$696$	0	0
$697$	2.00970e12	0.322540
$698$	0	0
$699$	2.08618e11	0.0330525
$700$	0	0
$701$	−4.71344e12	−0.737237	−0.368618	−	0.929581i	$-0.620169\pi$
$701$	−4.71344e12	−0.737237	−0.368618	+	0.929581i	$0.620169\pi$
$702$	0	0
$703$	3.71465e12	0.573612
$704$	0	0
$705$	−2.47805e11	−0.0377797
$706$	0	0
$707$	−1.30446e13	−1.96356
$708$	0	0
$709$	1.04827e13	1.55799	0.778996	−	0.627029i	$-0.215729\pi$
$709$	1.04827e13	1.55799	0.778996	+	0.627029i	$0.215729\pi$
$710$	0	0
$711$	2.11239e12	0.310000
$712$	0	0
$713$	5.45802e12	0.790919
$714$	0	0
$715$	7.40045e11	0.105896
$716$	0	0
$717$	8.38765e11	0.118523
$718$	0	0
$719$	7.75631e12	1.08237	0.541184	−	0.840904i	$-0.317976\pi$
$719$	7.75631e12	1.08237	0.541184	+	0.840904i	$0.317976\pi$
$720$	0	0
$721$	4.48042e12	0.617461
$722$	0	0
$723$	8.79785e11	0.119744
$724$	0	0
$725$	1.98132e12	0.266339
$726$	0	0
$727$	1.11988e13	1.48685	0.743425	−	0.668819i	$-0.233200\pi$
$727$	1.11988e13	1.48685	0.743425	+	0.668819i	$0.233200\pi$
$728$	0	0
$729$	−7.19355e12	−0.943342
$730$	0	0
$731$	2.18951e12	0.283608
$732$	0	0
$733$	−3.43273e12	−0.439210	−0.219605	−	0.975589i	$-0.570477\pi$
$733$	−3.43273e12	−0.439210	−0.219605	+	0.975589i	$0.570477\pi$
$734$	0	0
$735$	−9.79225e9	−0.00123763
$736$	0	0
$737$	1.55409e12	0.194031
$738$	0	0
$739$	−3.90905e12	−0.482137	−0.241069	−	0.970508i	$-0.577498\pi$
$739$	−3.90905e12	−0.482137	−0.241069	+	0.970508i	$0.577498\pi$
$740$	0	0
$741$	−2.81896e11	−0.0343484
$742$	0	0
$743$	−1.02378e13	−1.23241	−0.616207	−	0.787584i	$-0.711332\pi$
$743$	−1.02378e13	−1.23241	−0.616207	+	0.787584i	$0.711332\pi$
$744$	0	0
$745$	−4.91664e12	−0.584744
$746$	0	0
$747$	−6.12009e11	−0.0719142
$748$	0	0
$749$	1.28799e11	0.0149535
$750$	0	0
$751$	8.63831e12	0.990944	0.495472	−	0.868624i	$-0.334995\pi$
$751$	8.63831e12	0.990944	0.495472	+	0.868624i	$0.334995\pi$
$752$	0	0
$753$	1.22140e12	0.138447
$754$	0	0
$755$	−6.26301e12	−0.701490
$756$	0	0
$757$	−3.37911e12	−0.373999	−0.187000	−	0.982360i	$-0.559876\pi$
$757$	−3.37911e12	−0.373999	−0.187000	+	0.982360i	$0.559876\pi$
$758$	0	0
$759$	3.82231e11	0.0418059
$760$	0	0
$761$	−3.88319e12	−0.419718	−0.209859	−	0.977732i	$-0.567300\pi$
$761$	−3.88319e12	−0.419718	−0.209859	+	0.977732i	$0.567300\pi$
$762$	0	0
$763$	1.11391e13	1.18984
$764$	0	0
$765$	−1.76919e12	−0.186766
$766$	0	0
$767$	1.54309e13	1.60995
$768$	0	0
$769$	2.37701e12	0.245111	0.122556	−	0.992462i	$-0.460891\pi$
$769$	2.37701e12	0.245111	0.122556	+	0.992462i	$0.460891\pi$
$770$	0	0
$771$	−1.48602e12	−0.151454
$772$	0	0
$773$	1.55050e12	0.156194	0.0780971	−	0.996946i	$-0.475116\pi$
$773$	1.55050e12	0.156194	0.0780971	+	0.996946i	$0.475116\pi$
$774$	0	0
$775$	−8.28559e11	−0.0825022
$776$	0	0
$777$	1.73518e12	0.170786
$778$	0	0
$779$	2.61269e12	0.254196
$780$	0	0
$781$	1.88769e12	0.181552
$782$	0	0
$783$	2.72001e12	0.258609
$784$	0	0
$785$	1.84920e12	0.173808
$786$	0	0
$787$	1.11619e13	1.03717	0.518587	−	0.855025i	$-0.326458\pi$
$787$	1.11619e13	1.03717	0.518587	+	0.855025i	$0.326458\pi$
$788$	0	0
$789$	5.68046e11	0.0521839
$790$	0	0
$791$	7.45481e12	0.677084
$792$	0	0
$793$	−5.66792e12	−0.508972
$794$	0	0
$795$	−2.63604e11	−0.0234045
$796$	0	0
$797$	2.65374e12	0.232968	0.116484	−	0.993193i	$-0.462838\pi$
$797$	2.65374e12	0.232968	0.116484	+	0.993193i	$0.462838\pi$
$798$	0	0
$799$	4.20591e12	0.365090
$800$	0	0
$801$	−8.90986e11	−0.0764759
$802$	0	0
$803$	1.09315e12	0.0927812
$804$	0	0
$805$	1.03602e13	0.869531
$806$	0	0
$807$	1.24125e11	0.0103021
$808$	0	0
$809$	−5.65969e12	−0.464541	−0.232271	−	0.972651i	$-0.574616\pi$
$809$	−5.65969e12	−0.464541	−0.232271	+	0.972651i	$0.574616\pi$
$810$	0	0
$811$	−9.72531e12	−0.789422	−0.394711	−	0.918805i	$-0.629155\pi$
$811$	−9.72531e12	−0.789422	−0.394711	+	0.918805i	$0.629155\pi$
$812$	0	0
$813$	−1.21377e12	−0.0974380
$814$	0	0
$815$	2.02551e12	0.160815
$816$	0	0
$817$	2.84645e12	0.223513
$818$	0	0
$819$	1.37026e13	1.06420
$820$	0	0
$821$	1.57489e13	1.20978	0.604891	−	0.796309i	$-0.293217\pi$
$821$	1.57489e13	1.20978	0.604891	+	0.796309i	$0.293217\pi$
$822$	0	0
$823$	3.88455e12	0.295149	0.147575	−	0.989051i	$-0.452853\pi$
$823$	3.88455e12	0.295149	0.147575	+	0.989051i	$0.452853\pi$
$824$	0	0
$825$	−5.80249e10	−0.00436085
$826$	0	0
$827$	−1.86455e11	−0.0138611	−0.00693056	−	0.999976i	$-0.502206\pi$
$827$	−1.86455e11	−0.0138611	−0.00693056	+	0.999976i	$0.502206\pi$
$828$	0	0
$829$	−2.58795e12	−0.190310	−0.0951548	−	0.995462i	$-0.530335\pi$
$829$	−2.58795e12	−0.190310	−0.0951548	+	0.995462i	$0.530335\pi$
$830$	0	0
$831$	−1.94370e12	−0.141392
$832$	0	0
$833$	1.66201e11	0.0119600
$834$	0	0
$835$	9.48099e12	0.674939
$836$	0	0
$837$	−1.13747e12	−0.0801077
$838$	0	0
$839$	−2.53028e12	−0.176295	−0.0881475	−	0.996107i	$-0.528095\pi$
$839$	−2.53028e12	−0.176295	−0.0881475	+	0.996107i	$0.528095\pi$
$840$	0	0
$841$	1.12199e13	0.773407
$842$	0	0
$843$	4.75610e10	0.00324360
$844$	0	0
$845$	8.12313e11	0.0548110
$846$	0	0
$847$	1.44310e13	0.963432
$848$	0	0
$849$	2.33877e12	0.154491
$850$	0	0
$851$	−5.06380e13	−3.30973
$852$	0	0
$853$	−5.63946e12	−0.364726	−0.182363	−	0.983231i	$-0.558375\pi$
$853$	−5.63946e12	−0.364726	−0.182363	+	0.983231i	$0.558375\pi$
$854$	0	0
$855$	−2.30001e12	−0.147192
$856$	0	0
$857$	−1.49091e12	−0.0944143	−0.0472072	−	0.998885i	$-0.515032\pi$
$857$	−1.49091e12	−0.0944143	−0.0472072	+	0.998885i	$0.515032\pi$
$858$	0	0
$859$	−2.33948e13	−1.46605	−0.733027	−	0.680199i	$-0.761893\pi$
$859$	−2.33948e13	−1.46605	−0.733027	+	0.680199i	$0.761893\pi$
$860$	0	0
$861$	1.22044e12	0.0756835
$862$	0	0
$863$	−2.32336e13	−1.42583	−0.712915	−	0.701251i	$-0.752625\pi$
$863$	−2.32336e13	−1.42583	−0.712915	+	0.701251i	$0.752625\pi$
$864$	0	0
$865$	7.82491e12	0.475233
$866$	0	0
$867$	1.33462e12	0.0802178
$868$	0	0
$869$	−1.17589e12	−0.0699482
$870$	0	0
$871$	1.56242e13	0.919847
$872$	0	0
$873$	2.39819e12	0.139740
$874$	0	0
$875$	−1.57273e12	−0.0907024
$876$	0	0
$877$	−2.14368e13	−1.22366	−0.611830	−	0.790989i	$-0.709566\pi$
$877$	−2.14368e13	−1.22366	−0.611830	+	0.790989i	$0.709566\pi$
$878$	0	0
$879$	−3.69590e11	−0.0208819
$880$	0	0
$881$	2.26035e13	1.26411	0.632054	−	0.774925i	$-0.282212\pi$
$881$	2.26035e13	1.26411	0.632054	+	0.774925i	$0.282212\pi$
$882$	0	0
$883$	3.83289e12	0.212179	0.106090	−	0.994357i	$-0.466167\pi$
$883$	3.83289e12	0.212179	0.106090	+	0.994357i	$0.466167\pi$
$884$	0	0
$885$	−1.20989e12	−0.0662983
$886$	0	0
$887$	−4.25518e12	−0.230814	−0.115407	−	0.993318i	$-0.536817\pi$
$887$	−4.25518e12	−0.230814	−0.115407	+	0.993318i	$0.536817\pi$
$888$	0	0
$889$	−3.49877e13	−1.87870
$890$	0	0
$891$	4.08479e12	0.217130
$892$	0	0
$893$	5.46785e12	0.287730
$894$	0	0
$895$	−1.09142e13	−0.568575
$896$	0	0
$897$	3.84280e12	0.198190
$898$	0	0
$899$	−1.07587e13	−0.549338
$900$	0	0
$901$	4.47406e12	0.226173
$902$	0	0
$903$	1.32963e12	0.0665482
$904$	0	0
$905$	1.41308e13	0.700241
$906$	0	0
$907$	2.05253e12	0.100706	0.0503531	−	0.998731i	$-0.483965\pi$
$907$	2.05253e12	0.100706	0.0503531	+	0.998731i	$0.483965\pi$
$908$	0	0
$909$	−3.94779e13	−1.91786
$910$	0	0
$911$	−1.45629e13	−0.700509	−0.350255	−	0.936655i	$-0.613905\pi$
$911$	−1.45629e13	−0.700509	−0.350255	+	0.936655i	$0.613905\pi$
$912$	0	0
$913$	3.40681e11	0.0162267
$914$	0	0
$915$	4.44406e11	0.0209597
$916$	0	0
$917$	3.50788e13	1.63826
$918$	0	0
$919$	−3.74847e13	−1.73354	−0.866770	−	0.498708i	$-0.833808\pi$
$919$	−3.74847e13	−1.73354	−0.866770	+	0.498708i	$0.833808\pi$
$920$	0	0
$921$	−1.66114e12	−0.0760743
$922$	0	0
$923$	1.89781e13	0.860686
$924$	0	0
$925$	7.68714e12	0.345245
$926$	0	0
$927$	1.35594e13	0.603090
$928$	0	0
$929$	6.72075e12	0.296038	0.148019	−	0.988985i	$-0.452710\pi$
$929$	6.72075e12	0.296038	0.148019	+	0.988985i	$0.452710\pi$
$930$	0	0
$931$	2.16067e11	0.00942575
$932$	0	0
$933$	−3.94641e12	−0.170504
$934$	0	0
$935$	9.84838e11	0.0421417
$936$	0	0
$937$	−3.58761e13	−1.52047	−0.760233	−	0.649650i	$-0.774915\pi$
$937$	−3.58761e13	−1.52047	−0.760233	+	0.649650i	$0.774915\pi$
$938$	0	0
$939$	1.47917e12	0.0620904
$940$	0	0
$941$	−3.26759e13	−1.35855	−0.679273	−	0.733885i	$-0.737705\pi$
$941$	−3.26759e13	−1.35855	−0.679273	+	0.733885i	$0.737705\pi$
$942$	0	0
$943$	−3.56161e13	−1.46671
$944$	0	0
$945$	−2.15909e12	−0.0880699
$946$	0	0
$947$	4.16714e13	1.68369	0.841846	−	0.539718i	$-0.181469\pi$
$947$	4.16714e13	1.68369	0.841846	+	0.539718i	$0.181469\pi$
$948$	0	0
$949$	1.09901e13	0.439850
$950$	0	0
$951$	−3.67417e12	−0.145662
$952$	0	0
$953$	−3.65279e13	−1.43452	−0.717260	−	0.696806i	$-0.754604\pi$
$953$	−3.65279e13	−1.43452	−0.717260	+	0.696806i	$0.754604\pi$
$954$	0	0
$955$	5.31742e12	0.206864
$956$	0	0
$957$	−7.53441e11	−0.0290366
$958$	0	0
$959$	2.97435e13	1.13556
$960$	0	0
$961$	−2.19405e13	−0.829835
$962$	0	0
$963$	3.89793e11	0.0146055
$964$	0	0
$965$	−1.13257e13	−0.420430
$966$	0	0
$967$	2.77211e13	1.01951	0.509755	−	0.860320i	$-0.329736\pi$
$967$	2.77211e13	1.01951	0.509755	+	0.860320i	$0.329736\pi$
$968$	0	0
$969$	−3.75142e11	−0.0136691
$970$	0	0
$971$	−1.01665e13	−0.367016	−0.183508	−	0.983018i	$-0.558745\pi$
$971$	−1.01665e13	−0.367016	−0.183508	+	0.983018i	$0.558745\pi$
$972$	0	0
$973$	−3.32614e13	−1.18969
$974$	0	0
$975$	−5.83359e11	−0.0206736
$976$	0	0
$977$	2.67020e13	0.937603	0.468801	−	0.883304i	$-0.344686\pi$
$977$	2.67020e13	0.937603	0.468801	+	0.883304i	$0.344686\pi$
$978$	0	0
$979$	4.95977e11	0.0172560
$980$	0	0
$981$	3.37111e13	1.16215
$982$	0	0
$983$	−3.44738e13	−1.17760	−0.588801	−	0.808278i	$-0.700400\pi$
$983$	−3.44738e13	−1.17760	−0.588801	+	0.808278i	$0.700400\pi$
$984$	0	0
$985$	−1.16473e12	−0.0394240
$986$	0	0
$987$	2.55414e12	0.0856677
$988$	0	0
$989$	−3.88027e13	−1.28967
$990$	0	0
$991$	−3.39785e12	−0.111911	−0.0559555	−	0.998433i	$-0.517821\pi$
$991$	−3.39785e12	−0.111911	−0.0559555	+	0.998433i	$0.517821\pi$
$992$	0	0
$993$	−3.15019e12	−0.102817
$994$	0	0
$995$	2.23453e13	0.722739
$996$	0	0
$997$	4.27725e13	1.37100	0.685498	−	0.728074i	$-0.259584\pi$
$997$	4.27725e13	1.37100	0.685498	+	0.728074i	$0.259584\pi$
$998$	0	0
$999$	1.05531e13	0.335224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.10.a.k.1.2		3
4.3	odd	2		40.10.a.d.1.2	✓	3
5.2	odd	4		400.10.c.r.49.4		6
5.3	odd	4		400.10.c.r.49.3		6
5.4	even	2		400.10.a.x.1.2		3
8.3	odd	2		320.10.a.u.1.2		3
8.5	even	2		320.10.a.v.1.2		3
12.11	even	2		360.10.a.j.1.3		3
20.3	even	4		200.10.c.f.49.4		6
20.7	even	4		200.10.c.f.49.3		6
20.19	odd	2		200.10.a.f.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.d.1.2	✓	3	4.3	odd	2
80.10.a.k.1.2		3	1.1	even	1	trivial
200.10.a.f.1.2		3	20.19	odd	2
200.10.c.f.49.3		6	20.7	even	4
200.10.c.f.49.4		6	20.3	even	4
320.10.a.u.1.2		3	8.3	odd	2
320.10.a.v.1.2		3	8.5	even	2
360.10.a.j.1.3		3	12.11	even	2
400.10.a.x.1.2		3	5.4	even	2
400.10.c.r.49.3		6	5.3	odd	4
400.10.c.r.49.4		6	5.2	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)

\(f(q)\)	\(=\)	\(q-13.6876 q^{3} +625.000 q^{5} -6441.91 q^{7} -19495.7 q^{9} +O(q^{10})\)	\(q-13.6876 q^{3} +625.000 q^{5} -6441.91 q^{7} -19495.7 q^{9} +10852.5 q^{11} +109106. q^{13} -8554.73 q^{15} +145197. q^{17} +188761. q^{19} +88174.1 q^{21} -2.57319e6 q^{23} +390625. q^{25} +536260. q^{27} +5.07219e6 q^{29} -2.12111e6 q^{31} -148544. q^{33} -4.02620e6 q^{35} +1.96791e7 q^{37} -1.49340e6 q^{39} +1.38412e7 q^{41} +1.50796e7 q^{43} -1.21848e7 q^{45} +2.89670e7 q^{47} +1.14466e6 q^{49} -1.98739e6 q^{51} +3.08138e7 q^{53} +6.78279e6 q^{55} -2.58368e6 q^{57} +1.41430e8 q^{59} -5.19485e7 q^{61} +1.25589e8 q^{63} +6.81915e7 q^{65} +1.43201e8 q^{67} +3.52207e7 q^{69} +1.73941e8 q^{71} +1.00728e8 q^{73} -5.34670e6 q^{75} -6.99106e7 q^{77} -1.08352e8 q^{79} +3.76393e8 q^{81} +3.13921e7 q^{83} +9.07479e7 q^{85} -6.94259e7 q^{87} +4.57018e7 q^{89} -7.02854e8 q^{91} +2.90328e7 q^{93} +1.17976e8 q^{95} -1.23012e8 q^{97} -2.11576e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 84 q^{3} + 1875 q^{5} + 5520 q^{7} + 47079 q^{9}+O(q^{10}) \) 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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