LMFDB - Embedded newform 80.10.a.i.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{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			$4.69042$ 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)

$f(q)$	$=$	$q+245.617 q^{3} -625.000 q^{5} -12208.6 q^{7} +40644.5 q^{9} +O(q^{10})$	$q+245.617 q^{3} -625.000 q^{5} -12208.6 q^{7} +40644.5 q^{9} +30816.6 q^{11} +570.998 q^{13} -153510. q^{15} -486850. q^{17} -366058. q^{19} -2.99863e6 q^{21} -214861. q^{23} +390625. q^{25} +5.14850e6 q^{27} -3.29684e6 q^{29} -7.33158e6 q^{31} +7.56908e6 q^{33} +7.63036e6 q^{35} -1.25569e7 q^{37} +140247. q^{39} -2.37465e7 q^{41} +2.63974e7 q^{43} -2.54028e7 q^{45} +4.89681e7 q^{47} +1.08696e8 q^{49} -1.19579e8 q^{51} -4.11185e7 q^{53} -1.92604e7 q^{55} -8.99099e7 q^{57} -1.69195e8 q^{59} -6.50957e7 q^{61} -4.96212e8 q^{63} -356874. q^{65} +7.51098e7 q^{67} -5.27734e7 q^{69} +5.60303e7 q^{71} -1.46444e8 q^{73} +9.59440e7 q^{75} -3.76227e8 q^{77} +6.33189e8 q^{79} +4.64551e8 q^{81} -2.96211e7 q^{83} +3.04282e8 q^{85} -8.09759e8 q^{87} +3.79922e8 q^{89} -6.97107e6 q^{91} -1.80076e9 q^{93} +2.28786e8 q^{95} +4.10205e8 q^{97} +1.25253e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 116 q^{3} - 1250 q^{5} - 11284 q^{7} + 37762 q^{9}+O(q^{10}) $ 2 * q + 116 * q^3 - 1250 * q^5 - 11284 * q^7 + 37762 * q^9	$ 2 q + 116 q^{3} - 1250 q^{5} - 11284 q^{7} + 37762 q^{9} + 101408 q^{11} - 21372 q^{13} - 72500 q^{15} - 296780 q^{17} - 275832 q^{19} - 3118472 q^{21} + 585284 q^{23} + 781250 q^{25} + 8073368 q^{27} - 9928756 q^{29} - 5131480 q^{31} - 1580736 q^{33} + 7052500 q^{35} - 11007932 q^{37} + 2984424 q^{39} - 41835956 q^{41} + 23394052 q^{43} - 23601250 q^{45} - 11711748 q^{47} + 69197114 q^{49} - 144214840 q^{51} - 46384268 q^{53} - 63380000 q^{55} - 101604656 q^{57} - 178239576 q^{59} + 31825220 q^{61} - 498877204 q^{63} + 13357500 q^{65} - 89480628 q^{67} - 156485528 q^{69} - 112319176 q^{71} - 93294524 q^{73} + 45312500 q^{75} - 310959936 q^{77} + 191601328 q^{79} + 142176298 q^{81} + 17270436 q^{83} + 185487500 q^{85} + 49847352 q^{87} - 615067148 q^{89} - 27259176 q^{91} - 2085926640 q^{93} + 172395000 q^{95} - 996545468 q^{97} + 1049046048 q^{99}+O(q^{100}) $ 2 * q + 116 * q^3 - 1250 * q^5 - 11284 * q^7 + 37762 * q^9 + 101408 * q^11 - 21372 * q^13 - 72500 * q^15 - 296780 * q^17 - 275832 * q^19 - 3118472 * q^21 + 585284 * q^23 + 781250 * q^25 + 8073368 * q^27 - 9928756 * q^29 - 5131480 * q^31 - 1580736 * q^33 + 7052500 * q^35 - 11007932 * q^37 + 2984424 * q^39 - 41835956 * q^41 + 23394052 * q^43 - 23601250 * q^45 - 11711748 * q^47 + 69197114 * q^49 - 144214840 * q^51 - 46384268 * q^53 - 63380000 * q^55 - 101604656 * q^57 - 178239576 * q^59 + 31825220 * q^61 - 498877204 * q^63 + 13357500 * q^65 - 89480628 * q^67 - 156485528 * q^69 - 112319176 * q^71 - 93294524 * q^73 + 45312500 * q^75 - 310959936 * q^77 + 191601328 * q^79 + 142176298 * q^81 + 17270436 * q^83 + 185487500 * q^85 + 49847352 * q^87 - 615067148 * q^89 - 27259176 * q^91 - 2085926640 * q^93 + 172395000 * q^95 - 996545468 * q^97 + 1049046048 * 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
$3$	245.617	1.75070	0.875351	−	0.483488i	$-0.160630\pi$
$3$	245.617	1.75070	0.875351	+	0.483488i	$0.160630\pi$
$4$	0	0
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	−12208.6	−1.92187	−0.960935	−	0.276774i	$-0.910735\pi$
$7$	−12208.6	−1.92187	−0.960935	+	0.276774i	$0.910735\pi$
$8$	0	0
$9$	40644.5	2.06496
$10$	0	0
$11$	30816.6	0.634626	0.317313	−	0.948321i	$-0.397219\pi$
$11$	30816.6	0.634626	0.317313	+	0.948321i	$0.397219\pi$
$12$	0	0
$13$	570.998	0.00554484	0.00277242	−	0.999996i	$-0.499118\pi$
$13$	570.998	0.00554484	0.00277242	+	0.999996i	$0.499118\pi$
$14$	0	0
$15$	−153510.	−0.782938
$16$	0	0
$17$	−486850.	−1.41376	−0.706880	−	0.707334i	$-0.749898\pi$
$17$	−486850.	−1.41376	−0.706880	+	0.707334i	$0.749898\pi$
$18$	0	0
$19$	−366058.	−0.644404	−0.322202	−	0.946671i	$-0.604423\pi$
$19$	−366058.	−0.644404	−0.322202	+	0.946671i	$0.604423\pi$
$20$	0	0
$21$	−2.99863e6	−3.36462
$22$	0	0
$23$	−214861.	−0.160097	−0.0800483	−	0.996791i	$-0.525507\pi$
$23$	−214861.	−0.160097	−0.0800483	+	0.996791i	$0.525507\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	5.14850e6	1.86442
$28$	0	0
$29$	−3.29684e6	−0.865580	−0.432790	−	0.901495i	$-0.642471\pi$
$29$	−3.29684e6	−0.865580	−0.432790	+	0.901495i	$0.642471\pi$
$30$	0	0
$31$	−7.33158e6	−1.42584	−0.712918	−	0.701247i	$-0.752627\pi$
$31$	−7.33158e6	−1.42584	−0.712918	+	0.701247i	$0.752627\pi$
$32$	0	0
$33$	7.56908e6	1.11104
$34$	0	0
$35$	7.63036e6	0.859486
$36$	0	0
$37$	−1.25569e7	−1.10147	−0.550735	−	0.834680i	$-0.685653\pi$
$37$	−1.25569e7	−1.10147	−0.550735	+	0.834680i	$0.685653\pi$
$38$	0	0
$39$	140247.	0.00970737
$40$	0	0
$41$	−2.37465e7	−1.31242	−0.656209	−	0.754579i	$-0.727841\pi$
$41$	−2.37465e7	−1.31242	−0.656209	+	0.754579i	$0.727841\pi$
$42$	0	0
$43$	2.63974e7	1.17748	0.588738	−	0.808324i	$-0.299625\pi$
$43$	2.63974e7	1.17748	0.588738	+	0.808324i	$0.299625\pi$
$44$	0	0
$45$	−2.54028e7	−0.923476
$46$	0	0
$47$	4.89681e7	1.46377	0.731886	−	0.681427i	$-0.238640\pi$
$47$	4.89681e7	1.46377	0.731886	+	0.681427i	$0.238640\pi$
$48$	0	0
$49$	1.08696e8	2.69358
$50$	0	0
$51$	−1.19579e8	−2.47507
$52$	0	0
$53$	−4.11185e7	−0.715807	−0.357904	−	0.933759i	$-0.616508\pi$
$53$	−4.11185e7	−0.715807	−0.357904	+	0.933759i	$0.616508\pi$
$54$	0	0
$55$	−1.92604e7	−0.283814
$56$	0	0
$57$	−8.99099e7	−1.12816
$58$	0	0
$59$	−1.69195e8	−1.81783	−0.908917	−	0.416977i	$-0.863090\pi$
$59$	−1.69195e8	−1.81783	−0.908917	+	0.416977i	$0.863090\pi$
$60$	0	0
$61$	−6.50957e7	−0.601961	−0.300981	−	0.953630i	$-0.597314\pi$
$61$	−6.50957e7	−0.601961	−0.300981	+	0.953630i	$0.597314\pi$
$62$	0	0
$63$	−4.96212e8	−3.96858
$64$	0	0
$65$	−356874.	−0.00247973
$66$	0	0
$67$	7.51098e7	0.455365	0.227683	−	0.973735i	$-0.426885\pi$
$67$	7.51098e7	0.455365	0.227683	+	0.973735i	$0.426885\pi$
$68$	0	0
$69$	−5.27734e7	−0.280281
$70$	0	0
$71$	5.60303e7	0.261674	0.130837	−	0.991404i	$-0.458234\pi$
$71$	5.60303e7	0.261674	0.130837	+	0.991404i	$0.458234\pi$
$72$	0	0
$73$	−1.46444e8	−0.603556	−0.301778	−	0.953378i	$-0.597580\pi$
$73$	−1.46444e8	−0.603556	−0.301778	+	0.953378i	$0.597580\pi$
$74$	0	0
$75$	9.59440e7	0.350140
$76$	0	0
$77$	−3.76227e8	−1.21967
$78$	0	0
$79$	6.33189e8	1.82899	0.914496	−	0.404595i	$-0.132588\pi$
$79$	6.33189e8	1.82899	0.914496	+	0.404595i	$0.132588\pi$
$80$	0	0
$81$	4.64551e8	1.19909
$82$	0	0
$83$	−2.96211e7	−0.0685094	−0.0342547	−	0.999413i	$-0.510906\pi$
$83$	−2.96211e7	−0.0685094	−0.0342547	+	0.999413i	$0.510906\pi$
$84$	0	0
$85$	3.04282e8	0.632252
$86$	0	0
$87$	−8.09759e8	−1.51537
$88$	0	0
$89$	3.79922e8	0.641859	0.320930	−	0.947103i	$-0.396005\pi$
$89$	3.79922e8	0.641859	0.320930	+	0.947103i	$0.396005\pi$
$90$	0	0
$91$	−6.97107e6	−0.0106565
$92$	0	0
$93$	−1.80076e9	−2.49621
$94$	0	0
$95$	2.28786e8	0.288186
$96$	0	0
$97$	4.10205e8	0.470466	0.235233	−	0.971939i	$-0.424415\pi$
$97$	4.10205e8	0.470466	0.235233	+	0.971939i	$0.424415\pi$
$98$	0	0
$99$	1.25253e9	1.31048
$100$	0	0
$101$	−2.14867e8	−0.205458	−0.102729	−	0.994709i	$-0.532758\pi$
$101$	−2.14867e8	−0.205458	−0.102729	+	0.994709i	$0.532758\pi$
$102$	0	0
$103$	1.86010e9	1.62843	0.814214	−	0.580565i	$-0.197168\pi$
$103$	1.86010e9	1.62843	0.814214	+	0.580565i	$0.197168\pi$
$104$	0	0
$105$	1.87414e9	1.50470
$106$	0	0
$107$	1.09320e9	0.806253	0.403127	−	0.915144i	$-0.367923\pi$
$107$	1.09320e9	0.806253	0.403127	+	0.915144i	$0.367923\pi$
$108$	0	0
$109$	−1.39589e9	−0.947178	−0.473589	−	0.880746i	$-0.657042\pi$
$109$	−1.39589e9	−0.947178	−0.473589	+	0.880746i	$0.657042\pi$
$110$	0	0
$111$	−3.08417e9	−1.92835
$112$	0	0
$113$	2.21491e8	0.127792	0.0638959	−	0.997957i	$-0.479647\pi$
$113$	2.21491e8	0.127792	0.0638959	+	0.997957i	$0.479647\pi$
$114$	0	0
$115$	1.34288e8	0.0715974
$116$	0	0
$117$	2.32079e7	0.0114499
$118$	0	0
$119$	5.94375e9	2.71706
$120$	0	0
$121$	−1.40828e9	−0.597249
$122$	0	0
$123$	−5.83253e9	−2.29765
$124$	0	0
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	−2.05333e9	−0.700393	−0.350196	−	0.936676i	$-0.613885\pi$
$127$	−2.05333e9	−0.700393	−0.350196	+	0.936676i	$0.613885\pi$
$128$	0	0
$129$	6.48363e9	2.06141
$130$	0	0
$131$	−2.33558e9	−0.692905	−0.346452	−	0.938068i	$-0.612614\pi$
$131$	−2.33558e9	−0.692905	−0.346452	+	0.938068i	$0.612614\pi$
$132$	0	0
$133$	4.46905e9	1.23846
$134$	0	0
$135$	−3.21781e9	−0.833794
$136$	0	0
$137$	−1.97404e9	−0.478754	−0.239377	−	0.970927i	$-0.576943\pi$
$137$	−1.97404e9	−0.478754	−0.239377	+	0.970927i	$0.576943\pi$
$138$	0	0
$139$	−6.07919e7	−0.0138127	−0.00690636	−	0.999976i	$-0.502198\pi$
$139$	−6.07919e7	−0.0138127	−0.00690636	+	0.999976i	$0.502198\pi$
$140$	0	0
$141$	1.20274e10	2.56263
$142$	0	0
$143$	1.75962e7	0.00351890
$144$	0	0
$145$	2.06053e9	0.387099
$146$	0	0
$147$	2.66975e10	4.71566
$148$	0	0
$149$	2.74478e9	0.456215	0.228108	−	0.973636i	$-0.426746\pi$
$149$	2.74478e9	0.456215	0.228108	+	0.973636i	$0.426746\pi$
$150$	0	0
$151$	−5.89144e9	−0.922201	−0.461100	−	0.887348i	$-0.652545\pi$
$151$	−5.89144e9	−0.922201	−0.461100	+	0.887348i	$0.652545\pi$
$152$	0	0
$153$	−1.97878e10	−2.91935
$154$	0	0
$155$	4.58224e9	0.637654
$156$	0	0
$157$	3.28093e9	0.430971	0.215486	−	0.976507i	$-0.430867\pi$
$157$	3.28093e9	0.430971	0.215486	+	0.976507i	$0.430867\pi$
$158$	0	0
$159$	−1.00994e10	−1.25317
$160$	0	0
$161$	2.62315e9	0.307685
$162$	0	0
$163$	7.49559e9	0.831690	0.415845	−	0.909436i	$-0.363486\pi$
$163$	7.49559e9	0.831690	0.415845	+	0.909436i	$0.363486\pi$
$164$	0	0
$165$	−4.73067e9	−0.496873
$166$	0	0
$167$	5.76690e9	0.573744	0.286872	−	0.957969i	$-0.407385\pi$
$167$	5.76690e9	0.573744	0.286872	+	0.957969i	$0.407385\pi$
$168$	0	0
$169$	−1.06042e10	−0.999969
$170$	0	0
$171$	−1.48782e10	−1.33067
$172$	0	0
$173$	1.92662e10	1.63527	0.817634	−	0.575739i	$-0.195285\pi$
$173$	1.92662e10	1.63527	0.817634	+	0.575739i	$0.195285\pi$
$174$	0	0
$175$	−4.76898e9	−0.384374
$176$	0	0
$177$	−4.15572e10	−3.18249
$178$	0	0
$179$	−1.06626e10	−0.776289	−0.388145	−	0.921599i	$-0.626884\pi$
$179$	−1.06626e10	−0.776289	−0.388145	+	0.921599i	$0.626884\pi$
$180$	0	0
$181$	5.44288e9	0.376943	0.188471	−	0.982079i	$-0.439647\pi$
$181$	5.44288e9	0.376943	0.188471	+	0.982079i	$0.439647\pi$
$182$	0	0
$183$	−1.59886e10	−1.05385
$184$	0	0
$185$	7.84803e9	0.492593
$186$	0	0
$187$	−1.50031e10	−0.897209
$188$	0	0
$189$	−6.28559e10	−3.58317
$190$	0	0
$191$	8.86201e9	0.481817	0.240909	−	0.970548i	$-0.422555\pi$
$191$	8.86201e9	0.481817	0.240909	+	0.970548i	$0.422555\pi$
$192$	0	0
$193$	7.72843e9	0.400944	0.200472	−	0.979699i	$-0.435752\pi$
$193$	7.72843e9	0.400944	0.200472	+	0.979699i	$0.435752\pi$
$194$	0	0
$195$	−8.76541e7	−0.00434127
$196$	0	0
$197$	−1.63089e10	−0.771483	−0.385741	−	0.922607i	$-0.626054\pi$
$197$	−1.63089e10	−0.771483	−0.385741	+	0.922607i	$0.626054\pi$
$198$	0	0
$199$	−1.42219e10	−0.642863	−0.321432	−	0.946933i	$-0.604164\pi$
$199$	−1.42219e10	−0.642863	−0.321432	+	0.946933i	$0.604164\pi$
$200$	0	0
$201$	1.84482e10	0.797208
$202$	0	0
$203$	4.02498e10	1.66353
$204$	0	0
$205$	1.48416e10	0.586931
$206$	0	0
$207$	−8.73292e9	−0.330593
$208$	0	0
$209$	−1.12807e10	−0.408956
$210$	0	0
$211$	1.81268e10	0.629578	0.314789	−	0.949162i	$-0.398066\pi$
$211$	1.81268e10	0.629578	0.314789	+	0.949162i	$0.398066\pi$
$212$	0	0
$213$	1.37620e10	0.458113
$214$	0	0
$215$	−1.64983e10	−0.526584
$216$	0	0
$217$	8.95082e10	2.74027
$218$	0	0
$219$	−3.59690e10	−1.05665
$220$	0	0
$221$	−2.77991e8	−0.00783907
$222$	0	0
$223$	1.17231e10	0.317446	0.158723	−	0.987323i	$-0.449262\pi$
$223$	1.17231e10	0.317446	0.158723	+	0.987323i	$0.449262\pi$
$224$	0	0
$225$	1.58768e10	0.412991
$226$	0	0
$227$	5.29837e10	1.32442	0.662210	−	0.749318i	$-0.269618\pi$
$227$	5.29837e10	1.32442	0.662210	+	0.749318i	$0.269618\pi$
$228$	0	0
$229$	−3.97297e10	−0.954676	−0.477338	−	0.878720i	$-0.658398\pi$
$229$	−3.97297e10	−0.954676	−0.477338	+	0.878720i	$0.658398\pi$
$230$	0	0
$231$	−9.24077e10	−2.13528
$232$	0	0
$233$	−4.40536e10	−0.979218	−0.489609	−	0.871942i	$-0.662860\pi$
$233$	−4.40536e10	−0.979218	−0.489609	+	0.871942i	$0.662860\pi$
$234$	0	0
$235$	−3.06051e10	−0.654619
$236$	0	0
$237$	1.55522e11	3.20202
$238$	0	0
$239$	−3.45808e10	−0.685558	−0.342779	−	0.939416i	$-0.611368\pi$
$239$	−3.45808e10	−0.685558	−0.342779	+	0.939416i	$0.611368\pi$
$240$	0	0
$241$	4.75852e10	0.908646	0.454323	−	0.890837i	$-0.349881\pi$
$241$	4.75852e10	0.908646	0.454323	+	0.890837i	$0.349881\pi$
$242$	0	0
$243$	1.27635e10	0.234824
$244$	0	0
$245$	−6.79349e10	−1.20461
$246$	0	0
$247$	−2.09018e8	−0.00357312
$248$	0	0
$249$	−7.27544e9	−0.119940
$250$	0	0
$251$	1.21053e11	1.92506	0.962528	−	0.271183i	$-0.0874147\pi$
$251$	1.21053e11	1.92506	0.962528	+	0.271183i	$0.0874147\pi$
$252$	0	0
$253$	−6.62129e9	−0.101602
$254$	0	0
$255$	7.47366e10	1.10689
$256$	0	0
$257$	−3.14330e10	−0.449456	−0.224728	−	0.974422i	$-0.572149\pi$
$257$	−3.14330e10	−0.449456	−0.224728	+	0.974422i	$0.572149\pi$
$258$	0	0
$259$	1.53301e11	2.11688
$260$	0	0
$261$	−1.33999e11	−1.78738
$262$	0	0
$263$	−9.89276e10	−1.27502	−0.637509	−	0.770443i	$-0.720035\pi$
$263$	−9.89276e10	−1.27502	−0.637509	+	0.770443i	$0.720035\pi$
$264$	0	0
$265$	2.56991e10	0.320119
$266$	0	0
$267$	9.33152e10	1.12370
$268$	0	0
$269$	−2.37074e10	−0.276056	−0.138028	−	0.990428i	$-0.544076\pi$
$269$	−2.37074e10	−0.276056	−0.138028	+	0.990428i	$0.544076\pi$
$270$	0	0
$271$	1.03635e10	0.116720	0.0583600	−	0.998296i	$-0.481413\pi$
$271$	1.03635e10	0.116720	0.0583600	+	0.998296i	$0.481413\pi$
$272$	0	0
$273$	−1.71221e9	−0.0186563
$274$	0	0
$275$	1.20377e10	0.126925
$276$	0	0
$277$	−2.75040e10	−0.280696	−0.140348	−	0.990102i	$-0.544822\pi$
$277$	−2.75040e10	−0.280696	−0.140348	+	0.990102i	$0.544822\pi$
$278$	0	0
$279$	−2.97989e11	−2.94429
$280$	0	0
$281$	−3.01735e10	−0.288700	−0.144350	−	0.989527i	$-0.546109\pi$
$281$	−3.01735e10	−0.288700	−0.144350	+	0.989527i	$0.546109\pi$
$282$	0	0
$283$	−8.47861e10	−0.785753	−0.392876	−	0.919591i	$-0.628520\pi$
$283$	−8.47861e10	−0.785753	−0.392876	+	0.919591i	$0.628520\pi$
$284$	0	0
$285$	5.61937e10	0.504528
$286$	0	0
$287$	2.89911e11	2.52230
$288$	0	0
$289$	1.18435e11	0.998715
$290$	0	0
$291$	1.00753e11	0.823646
$292$	0	0
$293$	1.11476e10	0.0883644	0.0441822	−	0.999023i	$-0.485932\pi$
$293$	1.11476e10	0.0883644	0.0441822	+	0.999023i	$0.485932\pi$
$294$	0	0
$295$	1.05747e11	0.812960
$296$	0	0
$297$	1.58659e11	1.18321
$298$	0	0
$299$	−1.22685e8	−0.000887711 0
$300$	0	0
$301$	−3.22274e11	−2.26296
$302$	0	0
$303$	−5.27749e10	−0.359696
$304$	0	0
$305$	4.06848e10	0.269205
$306$	0	0
$307$	−1.81441e11	−1.16577	−0.582886	−	0.812554i	$-0.698076\pi$
$307$	−1.81441e11	−1.16577	−0.582886	+	0.812554i	$0.698076\pi$
$308$	0	0
$309$	4.56871e11	2.85089
$310$	0	0
$311$	1.34216e11	0.813548	0.406774	−	0.913529i	$-0.366654\pi$
$311$	1.34216e11	0.813548	0.406774	+	0.913529i	$0.366654\pi$
$312$	0	0
$313$	5.77513e10	0.340104	0.170052	−	0.985435i	$-0.445606\pi$
$313$	5.77513e10	0.340104	0.170052	+	0.985435i	$0.445606\pi$
$314$	0	0
$315$	3.10133e11	1.77480
$316$	0	0
$317$	1.44517e11	0.803805	0.401903	−	0.915682i	$-0.368349\pi$
$317$	1.44517e11	0.803805	0.401903	+	0.915682i	$0.368349\pi$
$318$	0	0
$319$	−1.01598e11	−0.549320
$320$	0	0
$321$	2.68507e11	1.41151
$322$	0	0
$323$	1.78215e11	0.911033
$324$	0	0
$325$	2.23046e8	0.00110897
$326$	0	0
$327$	−3.42853e11	−1.65823
$328$	0	0
$329$	−5.97832e11	−2.81318
$330$	0	0
$331$	−3.35086e11	−1.53437	−0.767185	−	0.641426i	$-0.778343\pi$
$331$	−3.35086e11	−1.53437	−0.767185	+	0.641426i	$0.778343\pi$
$332$	0	0
$333$	−5.10367e11	−2.27449
$334$	0	0
$335$	−4.69436e10	−0.203645
$336$	0	0
$337$	−3.69835e11	−1.56197	−0.780986	−	0.624549i	$-0.785283\pi$
$337$	−3.69835e11	−1.56197	−0.780986	+	0.624549i	$0.785283\pi$
$338$	0	0
$339$	5.44019e10	0.223725
$340$	0	0
$341$	−2.25935e11	−0.904874
$342$	0	0
$343$	−8.34362e11	−3.25485
$344$	0	0
$345$	3.29834e10	0.125346
$346$	0	0
$347$	−2.12865e11	−0.788172	−0.394086	−	0.919074i	$-0.628939\pi$
$347$	−2.12865e11	−0.788172	−0.394086	+	0.919074i	$0.628939\pi$
$348$	0	0
$349$	2.77173e11	1.00009	0.500043	−	0.866001i	$-0.333318\pi$
$349$	2.77173e11	1.00009	0.500043	+	0.866001i	$0.333318\pi$
$350$	0	0
$351$	2.93978e9	0.0103379
$352$	0	0
$353$	2.51725e10	0.0862861	0.0431430	−	0.999069i	$-0.486263\pi$
$353$	2.51725e10	0.0862861	0.0431430	+	0.999069i	$0.486263\pi$
$354$	0	0
$355$	−3.50189e10	−0.117024
$356$	0	0
$357$	1.45988e12	4.75676
$358$	0	0
$359$	−7.82440e10	−0.248614	−0.124307	−	0.992244i	$-0.539671\pi$
$359$	−7.82440e10	−0.248614	−0.124307	+	0.992244i	$0.539671\pi$
$360$	0	0
$361$	−1.88689e11	−0.584743
$362$	0	0
$363$	−3.45898e11	−1.04561
$364$	0	0
$365$	9.15272e10	0.269918
$366$	0	0
$367$	2.28170e11	0.656540	0.328270	−	0.944584i	$-0.393534\pi$
$367$	2.28170e11	0.656540	0.328270	+	0.944584i	$0.393534\pi$
$368$	0	0
$369$	−9.65165e11	−2.71008
$370$	0	0
$371$	5.01999e11	1.37569
$372$	0	0
$373$	−6.44643e11	−1.72437	−0.862183	−	0.506596i	$-0.830903\pi$
$373$	−6.44643e11	−1.72437	−0.862183	+	0.506596i	$0.830903\pi$
$374$	0	0
$375$	−5.99650e10	−0.156588
$376$	0	0
$377$	−1.88249e9	−0.00479950
$378$	0	0
$379$	−6.21976e11	−1.54845	−0.774225	−	0.632910i	$-0.781860\pi$
$379$	−6.21976e11	−1.54845	−0.774225	+	0.632910i	$0.781860\pi$
$380$	0	0
$381$	−5.04331e11	−1.22618
$382$	0	0
$383$	6.06377e11	1.43995	0.719976	−	0.693999i	$-0.244153\pi$
$383$	6.06377e11	1.43995	0.719976	+	0.693999i	$0.244153\pi$
$384$	0	0
$385$	2.35142e11	0.545453
$386$	0	0
$387$	1.07291e12	2.43144
$388$	0	0
$389$	−1.83635e11	−0.406613	−0.203306	−	0.979115i	$-0.565169\pi$
$389$	−1.83635e11	−0.406613	−0.203306	+	0.979115i	$0.565169\pi$
$390$	0	0
$391$	1.04605e11	0.226338
$392$	0	0
$393$	−5.73657e11	−1.21307
$394$	0	0
$395$	−3.95743e11	−0.817950
$396$	0	0
$397$	−1.65293e10	−0.0333962	−0.0166981	−	0.999861i	$-0.505315\pi$
$397$	−1.65293e10	−0.0333962	−0.0166981	+	0.999861i	$0.505315\pi$
$398$	0	0
$399$	1.09767e12	2.16818
$400$	0	0
$401$	7.71017e10	0.148907	0.0744534	−	0.997224i	$-0.476279\pi$
$401$	7.71017e10	0.148907	0.0744534	+	0.997224i	$0.476279\pi$
$402$	0	0
$403$	−4.18631e9	−0.00790604
$404$	0	0
$405$	−2.90344e11	−0.536248
$406$	0	0
$407$	−3.86960e11	−0.699022
$408$	0	0
$409$	3.04411e11	0.537905	0.268952	−	0.963153i	$-0.413323\pi$
$409$	3.04411e11	0.537905	0.268952	+	0.963153i	$0.413323\pi$
$410$	0	0
$411$	−4.84856e11	−0.838156
$412$	0	0
$413$	2.06563e12	3.49364
$414$	0	0
$415$	1.85132e10	0.0306383
$416$	0	0
$417$	−1.49315e10	−0.0241820
$418$	0	0
$419$	−4.05999e11	−0.643519	−0.321760	−	0.946821i	$-0.604274\pi$
$419$	−4.05999e11	−0.643519	−0.321760	+	0.946821i	$0.604274\pi$
$420$	0	0
$421$	−6.58349e11	−1.02138	−0.510689	−	0.859765i	$-0.670610\pi$
$421$	−6.58349e11	−1.02138	−0.510689	+	0.859765i	$0.670610\pi$
$422$	0	0
$423$	1.99029e12	3.02262
$424$	0	0
$425$	−1.90176e11	−0.282752
$426$	0	0
$427$	7.94727e11	1.15689
$428$	0	0
$429$	4.32193e9	0.00616055
$430$	0	0
$431$	−8.53467e11	−1.19135	−0.595674	−	0.803226i	$-0.703115\pi$
$431$	−8.53467e11	−1.19135	−0.595674	+	0.803226i	$0.703115\pi$
$432$	0	0
$433$	1.22911e12	1.68033	0.840167	−	0.542328i	$-0.182457\pi$
$433$	1.22911e12	1.68033	0.840167	+	0.542328i	$0.182457\pi$
$434$	0	0
$435$	5.06099e11	0.677695
$436$	0	0
$437$	7.86515e10	0.103167
$438$	0	0
$439$	−4.49370e11	−0.577449	−0.288725	−	0.957412i	$-0.593231\pi$
$439$	−4.49370e11	−0.577449	−0.288725	+	0.957412i	$0.593231\pi$
$440$	0	0
$441$	4.41789e12	5.56213
$442$	0	0
$443$	−3.14072e11	−0.387447	−0.193723	−	0.981056i	$-0.562056\pi$
$443$	−3.14072e11	−0.387447	−0.193723	+	0.981056i	$0.562056\pi$
$444$	0	0
$445$	−2.37451e11	−0.287048
$446$	0	0
$447$	6.74164e11	0.798697
$448$	0	0
$449$	−1.21107e12	−1.40625	−0.703124	−	0.711067i	$-0.748212\pi$
$449$	−1.21107e12	−1.40625	−0.703124	+	0.711067i	$0.748212\pi$
$450$	0	0
$451$	−7.31787e11	−0.832895
$452$	0	0
$453$	−1.44704e12	−1.61450
$454$	0	0
$455$	4.35692e9	0.00476572
$456$	0	0
$457$	−1.72885e12	−1.85410	−0.927052	−	0.374932i	$-0.877666\pi$
$457$	−1.72885e12	−1.85410	−0.927052	+	0.374932i	$0.877666\pi$
$458$	0	0
$459$	−2.50655e12	−2.63584
$460$	0	0
$461$	−7.05585e11	−0.727604	−0.363802	−	0.931476i	$-0.618522\pi$
$461$	−7.05585e11	−0.727604	−0.363802	+	0.931476i	$0.618522\pi$
$462$	0	0
$463$	−9.81254e11	−0.992355	−0.496177	−	0.868221i	$-0.665263\pi$
$463$	−9.81254e11	−0.992355	−0.496177	+	0.868221i	$0.665263\pi$
$464$	0	0
$465$	1.12547e12	1.11634
$466$	0	0
$467$	7.40988e11	0.720917	0.360459	−	0.932775i	$-0.382620\pi$
$467$	7.40988e11	0.720917	0.360459	+	0.932775i	$0.382620\pi$
$468$	0	0
$469$	−9.16984e11	−0.875153
$470$	0	0
$471$	8.05850e11	0.754502
$472$	0	0
$473$	8.13478e11	0.747258
$474$	0	0
$475$	−1.42991e11	−0.128881
$476$	0	0
$477$	−1.67124e12	−1.47811
$478$	0	0
$479$	−1.35876e12	−1.17933	−0.589664	−	0.807649i	$-0.700740\pi$
$479$	−1.35876e12	−1.17933	−0.589664	+	0.807649i	$0.700740\pi$
$480$	0	0
$481$	−7.16993e9	−0.00610748
$482$	0	0
$483$	6.44289e11	0.538665
$484$	0	0
$485$	−2.56378e11	−0.210399
$486$	0	0
$487$	8.22697e11	0.662765	0.331383	−	0.943496i	$-0.392485\pi$
$487$	8.22697e11	0.662765	0.331383	+	0.943496i	$0.392485\pi$
$488$	0	0
$489$	1.84104e12	1.45604
$490$	0	0
$491$	1.29179e12	1.00306	0.501529	−	0.865141i	$-0.332771\pi$
$491$	1.29179e12	1.00306	0.501529	+	0.865141i	$0.332771\pi$
$492$	0	0
$493$	1.60507e12	1.22372
$494$	0	0
$495$	−7.82830e11	−0.586063
$496$	0	0
$497$	−6.84050e11	−0.502903
$498$	0	0
$499$	1.63213e12	1.17843	0.589214	−	0.807977i	$-0.299438\pi$
$499$	1.63213e12	1.17843	0.589214	+	0.807977i	$0.299438\pi$
$500$	0	0
$501$	1.41645e12	1.00445
$502$	0	0
$503$	−5.81770e11	−0.405224	−0.202612	−	0.979259i	$-0.564943\pi$
$503$	−5.81770e11	−0.405224	−0.202612	+	0.979259i	$0.564943\pi$
$504$	0	0
$505$	1.34292e11	0.0918838
$506$	0	0
$507$	−2.60456e12	−1.75065
$508$	0	0
$509$	9.93884e11	0.656305	0.328152	−	0.944625i	$-0.393574\pi$
$509$	9.93884e11	0.656305	0.328152	+	0.944625i	$0.393574\pi$
$510$	0	0
$511$	1.78787e12	1.15996
$512$	0	0
$513$	−1.88465e12	−1.20144
$514$	0	0
$515$	−1.16256e12	−0.728255
$516$	0	0
$517$	1.50903e12	0.928948
$518$	0	0
$519$	4.73210e12	2.86287
$520$	0	0
$521$	1.77118e12	1.05316	0.526579	−	0.850126i	$-0.323475\pi$
$521$	1.77118e12	1.05316	0.526579	+	0.850126i	$0.323475\pi$
$522$	0	0
$523$	−6.02617e11	−0.352196	−0.176098	−	0.984373i	$-0.556348\pi$
$523$	−6.02617e11	−0.352196	−0.176098	+	0.984373i	$0.556348\pi$
$524$	0	0
$525$	−1.17134e12	−0.672924
$526$	0	0
$527$	3.56938e12	2.01579
$528$	0	0
$529$	−1.75499e12	−0.974369
$530$	0	0
$531$	−6.87686e12	−3.75375
$532$	0	0
$533$	−1.35592e10	−0.00727715
$534$	0	0
$535$	−6.83248e11	−0.360567
$536$	0	0
$537$	−2.61891e12	−1.35905
$538$	0	0
$539$	3.34964e12	1.70942
$540$	0	0
$541$	−2.95933e12	−1.48527	−0.742637	−	0.669695i	$-0.766425\pi$
$541$	−2.95933e12	−1.48527	−0.742637	+	0.669695i	$0.766425\pi$
$542$	0	0
$543$	1.33686e12	0.659914
$544$	0	0
$545$	8.72430e11	0.423591
$546$	0	0
$547$	−3.23821e12	−1.54654	−0.773270	−	0.634076i	$-0.781380\pi$
$547$	−3.23821e12	−1.54654	−0.773270	+	0.634076i	$0.781380\pi$
$548$	0	0
$549$	−2.64579e12	−1.24302
$550$	0	0
$551$	1.20683e12	0.557783
$552$	0	0
$553$	−7.73035e12	−3.51508
$554$	0	0
$555$	1.92761e12	0.862383
$556$	0	0
$557$	3.85748e12	1.69807	0.849035	−	0.528337i	$-0.177184\pi$
$557$	3.85748e12	1.69807	0.849035	+	0.528337i	$0.177184\pi$
$558$	0	0
$559$	1.50728e10	0.00652892
$560$	0	0
$561$	−3.68501e12	−1.57074
$562$	0	0
$563$	−1.80800e11	−0.0758423	−0.0379211	−	0.999281i	$-0.512074\pi$
$563$	−1.80800e11	−0.0758423	−0.0379211	+	0.999281i	$0.512074\pi$
$564$	0	0
$565$	−1.38432e11	−0.0571502
$566$	0	0
$567$	−5.67151e12	−2.30449
$568$	0	0
$569$	3.62005e12	1.44780	0.723902	−	0.689903i	$-0.242347\pi$
$569$	3.62005e12	1.44780	0.723902	+	0.689903i	$0.242347\pi$
$570$	0	0
$571$	1.55983e11	0.0614066	0.0307033	−	0.999529i	$-0.490225\pi$
$571$	1.55983e11	0.0614066	0.0307033	+	0.999529i	$0.490225\pi$
$572$	0	0
$573$	2.17666e12	0.843518
$574$	0	0
$575$	−8.39301e10	−0.0320193
$576$	0	0
$577$	−2.06129e12	−0.774191	−0.387095	−	0.922040i	$-0.626522\pi$
$577$	−2.06129e12	−0.774191	−0.387095	+	0.922040i	$0.626522\pi$
$578$	0	0
$579$	1.89823e12	0.701933
$580$	0	0
$581$	3.61632e11	0.131666
$582$	0	0
$583$	−1.26713e12	−0.454270
$584$	0	0
$585$	−1.45050e10	−0.00512053
$586$	0	0
$587$	−4.68577e12	−1.62896	−0.814478	−	0.580195i	$-0.802977\pi$
$587$	−4.68577e12	−1.62896	−0.814478	+	0.580195i	$0.802977\pi$
$588$	0	0
$589$	2.68378e12	0.918815
$590$	0	0
$591$	−4.00573e12	−1.35064
$592$	0	0
$593$	3.47697e12	1.15466	0.577331	−	0.816510i	$-0.304094\pi$
$593$	3.47697e12	1.15466	0.577331	+	0.816510i	$0.304094\pi$
$594$	0	0
$595$	−3.71485e12	−1.21511
$596$	0	0
$597$	−3.49313e12	−1.12546
$598$	0	0
$599$	−2.02512e12	−0.642732	−0.321366	−	0.946955i	$-0.604142\pi$
$599$	−2.02512e12	−0.642732	−0.321366	+	0.946955i	$0.604142\pi$
$600$	0	0
$601$	−4.54342e12	−1.42052	−0.710260	−	0.703939i	$-0.751423\pi$
$601$	−4.54342e12	−1.42052	−0.710260	+	0.703939i	$0.751423\pi$
$602$	0	0
$603$	3.05280e12	0.940309
$604$	0	0
$605$	8.80177e11	0.267098
$606$	0	0
$607$	5.77219e11	0.172581	0.0862903	−	0.996270i	$-0.472499\pi$
$607$	5.77219e11	0.172581	0.0862903	+	0.996270i	$0.472499\pi$
$608$	0	0
$609$	9.88601e12	2.91235
$610$	0	0
$611$	2.79607e10	0.00811639
$612$	0	0
$613$	8.72902e11	0.249686	0.124843	−	0.992177i	$-0.460157\pi$
$613$	8.72902e11	0.249686	0.124843	+	0.992177i	$0.460157\pi$
$614$	0	0
$615$	3.64533e12	1.02754
$616$	0	0
$617$	9.36327e11	0.260102	0.130051	−	0.991507i	$-0.458486\pi$
$617$	9.36327e11	0.260102	0.130051	+	0.991507i	$0.458486\pi$
$618$	0	0
$619$	3.44854e12	0.944120	0.472060	−	0.881566i	$-0.343510\pi$
$619$	3.44854e12	0.944120	0.472060	+	0.881566i	$0.343510\pi$
$620$	0	0
$621$	−1.10621e12	−0.298487
$622$	0	0
$623$	−4.63831e12	−1.23357
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−2.77072e12	−0.715960
$628$	0	0
$629$	6.11331e12	1.55721
$630$	0	0
$631$	2.14418e12	0.538430	0.269215	−	0.963080i	$-0.413236\pi$
$631$	2.14418e12	0.538430	0.269215	+	0.963080i	$0.413236\pi$
$632$	0	0
$633$	4.45224e12	1.10220
$634$	0	0
$635$	1.28333e12	0.313225
$636$	0	0
$637$	6.20651e10	0.0149355
$638$	0	0
$639$	2.27732e12	0.540345
$640$	0	0
$641$	−1.20747e12	−0.282498	−0.141249	−	0.989974i	$-0.545112\pi$
$641$	−1.20747e12	−0.282498	−0.141249	+	0.989974i	$0.545112\pi$
$642$	0	0
$643$	−3.02459e12	−0.697778	−0.348889	−	0.937164i	$-0.613441\pi$
$643$	−3.02459e12	−0.697778	−0.348889	+	0.937164i	$0.613441\pi$
$644$	0	0
$645$	−4.05227e12	−0.921891
$646$	0	0
$647$	5.19514e12	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$647$	5.19514e12	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$648$	0	0
$649$	−5.21403e12	−1.15365
$650$	0	0
$651$	2.19847e13	4.79740
$652$	0	0
$653$	4.53541e12	0.976129	0.488064	−	0.872808i	$-0.337703\pi$
$653$	4.53541e12	0.976129	0.488064	+	0.872808i	$0.337703\pi$
$654$	0	0
$655$	1.45974e12	0.309877
$656$	0	0
$657$	−5.95213e12	−1.24632
$658$	0	0
$659$	4.83190e10	0.00998007	0.00499004	−	0.999988i	$-0.498412\pi$
$659$	4.83190e10	0.00998007	0.00499004	+	0.999988i	$0.498412\pi$
$660$	0	0
$661$	7.81023e12	1.59132	0.795659	−	0.605745i	$-0.207125\pi$
$661$	7.81023e12	1.59132	0.795659	+	0.605745i	$0.207125\pi$
$662$	0	0
$663$	−6.82791e10	−0.0137239
$664$	0	0
$665$	−2.79315e12	−0.553857
$666$	0	0
$667$	7.08363e11	0.138576
$668$	0	0
$669$	2.87938e12	0.555753
$670$	0	0
$671$	−2.00603e12	−0.382020
$672$	0	0
$673$	2.11303e12	0.397043	0.198522	−	0.980096i	$-0.436386\pi$
$673$	2.11303e12	0.397043	0.198522	+	0.980096i	$0.436386\pi$
$674$	0	0
$675$	2.01113e12	0.372884
$676$	0	0
$677$	−3.37004e12	−0.616576	−0.308288	−	0.951293i	$-0.599756\pi$
$677$	−3.37004e12	−0.616576	−0.308288	+	0.951293i	$0.599756\pi$
$678$	0	0
$679$	−5.00802e12	−0.904175
$680$	0	0
$681$	1.30137e13	2.31867
$682$	0	0
$683$	5.33048e12	0.937288	0.468644	−	0.883387i	$-0.344743\pi$
$683$	5.33048e12	0.937288	0.468644	+	0.883387i	$0.344743\pi$
$684$	0	0
$685$	1.23377e12	0.214105
$686$	0	0
$687$	−9.75828e12	−1.67135
$688$	0	0
$689$	−2.34786e10	−0.00396904
$690$	0	0
$691$	−7.17813e12	−1.19773	−0.598866	−	0.800849i	$-0.704382\pi$
$691$	−7.17813e12	−1.19773	−0.598866	+	0.800849i	$0.704382\pi$
$692$	0	0
$693$	−1.52916e13	−2.51856
$694$	0	0
$695$	3.79950e10	0.00617724
$696$	0	0
$697$	1.15610e13	1.85544
$698$	0	0
$699$	−1.08203e13	−1.71432
$700$	0	0
$701$	−3.97694e12	−0.622039	−0.311019	−	0.950404i	$-0.600670\pi$
$701$	−3.97694e12	−0.622039	−0.311019	+	0.950404i	$0.600670\pi$
$702$	0	0
$703$	4.59653e12	0.709793
$704$	0	0
$705$	−7.51712e12	−1.14604
$706$	0	0
$707$	2.62322e12	0.394864
$708$	0	0
$709$	6.94724e12	1.03253	0.516267	−	0.856428i	$-0.327321\pi$
$709$	6.94724e12	1.03253	0.516267	+	0.856428i	$0.327321\pi$
$710$	0	0
$711$	2.57357e13	3.77679
$712$	0	0
$713$	1.57527e12	0.228272
$714$	0	0
$715$	−1.09976e10	−0.00157370
$716$	0	0
$717$	−8.49362e12	−1.20021
$718$	0	0
$719$	7.05915e12	0.985082	0.492541	−	0.870289i	$-0.336068\pi$
$719$	7.05915e12	0.985082	0.492541	+	0.870289i	$0.336068\pi$
$720$	0	0
$721$	−2.27092e13	−3.12963
$722$	0	0
$723$	1.16877e13	1.59077
$724$	0	0
$725$	−1.28783e12	−0.173116
$726$	0	0
$727$	−1.26483e13	−1.67929	−0.839645	−	0.543135i	$-0.817237\pi$
$727$	−1.26483e13	−1.67929	−0.839645	+	0.543135i	$0.817237\pi$
$728$	0	0
$729$	−6.00882e12	−0.787981
$730$	0	0
$731$	−1.28516e13	−1.66467
$732$	0	0
$733$	3.24806e12	0.415582	0.207791	−	0.978173i	$-0.433373\pi$
$733$	3.24806e12	0.415582	0.207791	+	0.978173i	$0.433373\pi$
$734$	0	0
$735$	−1.66859e13	−2.10891
$736$	0	0
$737$	2.31463e12	0.288987
$738$	0	0
$739$	1.71911e12	0.212033	0.106017	−	0.994364i	$-0.466190\pi$
$739$	1.71911e12	0.212033	0.106017	+	0.994364i	$0.466190\pi$
$740$	0	0
$741$	−5.13384e10	−0.00625547
$742$	0	0
$743$	−5.73458e12	−0.690322	−0.345161	−	0.938543i	$-0.612176\pi$
$743$	−5.73458e12	−0.690322	−0.345161	+	0.938543i	$0.612176\pi$
$744$	0	0
$745$	−1.71549e12	−0.204026
$746$	0	0
$747$	−1.20394e12	−0.141469
$748$	0	0
$749$	−1.33464e13	−1.54951
$750$	0	0
$751$	−1.33892e13	−1.53594	−0.767972	−	0.640484i	$-0.778734\pi$
$751$	−1.33892e13	−1.53594	−0.767972	+	0.640484i	$0.778734\pi$
$752$	0	0
$753$	2.97326e13	3.37020
$754$	0	0
$755$	3.68215e12	0.412421
$756$	0	0
$757$	−9.25082e12	−1.02388	−0.511940	−	0.859021i	$-0.671073\pi$
$757$	−9.25082e12	−1.02388	−0.511940	+	0.859021i	$0.671073\pi$
$758$	0	0
$759$	−1.62630e12	−0.177874
$760$	0	0
$761$	−4.22686e12	−0.456864	−0.228432	−	0.973560i	$-0.573360\pi$
$761$	−4.22686e12	−0.456864	−0.228432	+	0.973560i	$0.573360\pi$
$762$	0	0
$763$	1.70418e13	1.82035
$764$	0	0
$765$	1.23674e13	1.30557
$766$	0	0
$767$	−9.66102e10	−0.0100796
$768$	0	0
$769$	−6.92801e12	−0.714398	−0.357199	−	0.934028i	$-0.616268\pi$
$769$	−6.92801e12	−0.714398	−0.357199	+	0.934028i	$0.616268\pi$
$770$	0	0
$771$	−7.72047e12	−0.786863
$772$	0	0
$773$	9.81604e10	0.00988845	0.00494423	−	0.999988i	$-0.498426\pi$
$773$	9.81604e10	0.00988845	0.00494423	+	0.999988i	$0.498426\pi$
$774$	0	0
$775$	−2.86390e12	−0.285167
$776$	0	0
$777$	3.76534e13	3.70603
$778$	0	0
$779$	8.69259e12	0.845728
$780$	0	0
$781$	1.72666e12	0.166065
$782$	0	0
$783$	−1.69738e13	−1.61380
$784$	0	0
$785$	−2.05058e12	−0.192736
$786$	0	0
$787$	1.24632e13	1.15809	0.579047	−	0.815294i	$-0.303425\pi$
$787$	1.24632e13	1.15809	0.579047	+	0.815294i	$0.303425\pi$
$788$	0	0
$789$	−2.42983e13	−2.23218
$790$	0	0
$791$	−2.70409e12	−0.245599
$792$	0	0
$793$	−3.71695e10	−0.00333778
$794$	0	0
$795$	6.31212e12	0.560432
$796$	0	0
$797$	1.44045e13	1.26454	0.632272	−	0.774746i	$-0.282122\pi$
$797$	1.44045e13	1.26454	0.632272	+	0.774746i	$0.282122\pi$
$798$	0	0
$799$	−2.38402e13	−2.06942
$800$	0	0
$801$	1.54418e13	1.32541
$802$	0	0
$803$	−4.51290e12	−0.383032
$804$	0	0
$805$	−1.63947e12	−0.137601
$806$	0	0
$807$	−5.82292e12	−0.483292
$808$	0	0
$809$	9.86262e12	0.809513	0.404756	−	0.914425i	$-0.367356\pi$
$809$	9.86262e12	0.809513	0.404756	+	0.914425i	$0.367356\pi$
$810$	0	0
$811$	−2.85570e12	−0.231803	−0.115902	−	0.993261i	$-0.536976\pi$
$811$	−2.85570e12	−0.231803	−0.115902	+	0.993261i	$0.536976\pi$
$812$	0	0
$813$	2.54545e12	0.204342
$814$	0	0
$815$	−4.68474e12	−0.371943
$816$	0	0
$817$	−9.66296e12	−0.758771
$818$	0	0
$819$	−2.83336e11	−0.0220051
$820$	0	0
$821$	−7.69858e12	−0.591379	−0.295690	−	0.955284i	$-0.595549\pi$
$821$	−7.69858e12	−0.591379	−0.295690	+	0.955284i	$0.595549\pi$
$822$	0	0
$823$	5.64522e12	0.428925	0.214463	−	0.976732i	$-0.431200\pi$
$823$	5.64522e12	0.428925	0.214463	+	0.976732i	$0.431200\pi$
$824$	0	0
$825$	2.95667e12	0.222208
$826$	0	0
$827$	−4.67730e12	−0.347713	−0.173856	−	0.984771i	$-0.555623\pi$
$827$	−4.67730e12	−0.347713	−0.173856	+	0.984771i	$0.555623\pi$
$828$	0	0
$829$	1.03653e13	0.762227	0.381114	−	0.924528i	$-0.375541\pi$
$829$	1.03653e13	0.762227	0.381114	+	0.924528i	$0.375541\pi$
$830$	0	0
$831$	−6.75544e12	−0.491416
$832$	0	0
$833$	−5.29186e13	−3.80808
$834$	0	0
$835$	−3.60431e12	−0.256586
$836$	0	0
$837$	−3.77466e13	−2.65836
$838$	0	0
$839$	2.52107e13	1.75653	0.878264	−	0.478175i	$-0.158702\pi$
$839$	2.52107e13	1.75653	0.878264	+	0.478175i	$0.158702\pi$
$840$	0	0
$841$	−3.63798e12	−0.250772
$842$	0	0
$843$	−7.41111e12	−0.505428
$844$	0	0
$845$	6.62761e12	0.447200
$846$	0	0
$847$	1.71931e13	1.14784
$848$	0	0
$849$	−2.08249e13	−1.37562
$850$	0	0
$851$	2.69798e12	0.176342
$852$	0	0
$853$	−1.51992e13	−0.982992	−0.491496	−	0.870880i	$-0.663550\pi$
$853$	−1.51992e13	−0.982992	−0.491496	+	0.870880i	$0.663550\pi$
$854$	0	0
$855$	9.29890e12	0.595092
$856$	0	0
$857$	−3.04309e13	−1.92709	−0.963544	−	0.267551i	$-0.913785\pi$
$857$	−3.04309e13	−1.92709	−0.963544	+	0.267551i	$0.913785\pi$
$858$	0	0
$859$	−2.96056e13	−1.85526	−0.927629	−	0.373504i	$-0.878156\pi$
$859$	−2.96056e13	−1.85526	−0.927629	+	0.373504i	$0.878156\pi$
$860$	0	0
$861$	7.12069e13	4.41579
$862$	0	0
$863$	1.96521e13	1.20604	0.603018	−	0.797728i	$-0.293965\pi$
$863$	1.96521e13	1.20604	0.603018	+	0.797728i	$0.293965\pi$
$864$	0	0
$865$	−1.20414e13	−0.731314
$866$	0	0
$867$	2.90897e13	1.74845
$868$	0	0
$869$	1.95128e13	1.16073
$870$	0	0
$871$	4.28875e10	0.00252493
$872$	0	0
$873$	1.66726e13	0.971492
$874$	0	0
$875$	2.98061e12	0.171897
$876$	0	0
$877$	−8.33961e12	−0.476044	−0.238022	−	0.971260i	$-0.576499\pi$
$877$	−8.33961e12	−0.476044	−0.238022	+	0.971260i	$0.576499\pi$
$878$	0	0
$879$	2.73804e12	0.154700
$880$	0	0
$881$	−2.39222e13	−1.33786	−0.668929	−	0.743327i	$-0.733247\pi$
$881$	−2.39222e13	−1.33786	−0.668929	+	0.743327i	$0.733247\pi$
$882$	0	0
$883$	−1.18207e13	−0.654366	−0.327183	−	0.944961i	$-0.606099\pi$
$883$	−1.18207e13	−0.654366	−0.327183	+	0.944961i	$0.606099\pi$
$884$	0	0
$885$	2.59732e13	1.42325
$886$	0	0
$887$	1.38948e12	0.0753695	0.0376848	−	0.999290i	$-0.488002\pi$
$887$	1.38948e12	0.0753695	0.0376848	+	0.999290i	$0.488002\pi$
$888$	0	0
$889$	2.50682e13	1.34606
$890$	0	0
$891$	1.43159e13	0.760973
$892$	0	0
$893$	−1.79252e13	−0.943261
$894$	0	0
$895$	6.66411e12	0.347167
$896$	0	0
$897$	−3.01335e10	−0.00155412
$898$	0	0
$899$	2.41710e13	1.23418
$900$	0	0
$901$	2.00186e13	1.01198
$902$	0	0
$903$	−7.91559e13	−3.96176
$904$	0	0
$905$	−3.40180e12	−0.168574
$906$	0	0
$907$	2.26219e13	1.10993	0.554966	−	0.831873i	$-0.312731\pi$
$907$	2.26219e13	1.10993	0.554966	+	0.831873i	$0.312731\pi$
$908$	0	0
$909$	−8.73317e12	−0.424262
$910$	0	0
$911$	−2.00026e13	−0.962173	−0.481086	−	0.876673i	$-0.659758\pi$
$911$	−2.00026e13	−0.962173	−0.481086	+	0.876673i	$0.659758\pi$
$912$	0	0
$913$	−9.12824e11	−0.0434779
$914$	0	0
$915$	9.99287e12	0.471298
$916$	0	0
$917$	2.85141e13	1.33167
$918$	0	0
$919$	−7.89443e12	−0.365091	−0.182546	−	0.983197i	$-0.558434\pi$
$919$	−7.89443e12	−0.365091	−0.182546	+	0.983197i	$0.558434\pi$
$920$	0	0
$921$	−4.45650e13	−2.04092
$922$	0	0
$923$	3.19932e10	0.00145094
$924$	0	0
$925$	−4.90502e12	−0.220294
$926$	0	0
$927$	7.56028e13	3.36263
$928$	0	0
$929$	−2.04784e13	−0.902040	−0.451020	−	0.892514i	$-0.648940\pi$
$929$	−2.04784e13	−0.902040	−0.451020	+	0.892514i	$0.648940\pi$
$930$	0	0
$931$	−3.97890e13	−1.73576
$932$	0	0
$933$	3.29657e13	1.42428
$934$	0	0
$935$	9.37693e12	0.401244
$936$	0	0
$937$	3.76022e13	1.59362	0.796811	−	0.604228i	$-0.206518\pi$
$937$	3.76022e13	1.59362	0.796811	+	0.604228i	$0.206518\pi$
$938$	0	0
$939$	1.41847e13	0.595421
$940$	0	0
$941$	1.92548e13	0.800547	0.400273	−	0.916396i	$-0.368915\pi$
$941$	1.92548e13	0.800547	0.400273	+	0.916396i	$0.368915\pi$
$942$	0	0
$943$	5.10219e12	0.210114
$944$	0	0
$945$	3.92849e13	1.60244
$946$	0	0
$947$	1.79872e13	0.726755	0.363378	−	0.931642i	$-0.381623\pi$
$947$	1.79872e13	0.726755	0.363378	+	0.931642i	$0.381623\pi$
$948$	0	0
$949$	−8.36189e10	−0.00334662
$950$	0	0
$951$	3.54957e13	1.40722
$952$	0	0
$953$	2.38259e13	0.935688	0.467844	−	0.883811i	$-0.345031\pi$
$953$	2.38259e13	0.935688	0.467844	+	0.883811i	$0.345031\pi$
$954$	0	0
$955$	−5.53876e12	−0.215475
$956$	0	0
$957$	−2.49541e13	−0.961695
$958$	0	0
$959$	2.41002e13	0.920103
$960$	0	0
$961$	2.73124e13	1.03301
$962$	0	0
$963$	4.44325e13	1.66488
$964$	0	0
$965$	−4.83027e12	−0.179308
$966$	0	0
$967$	9.53191e12	0.350559	0.175279	−	0.984519i	$-0.443917\pi$
$967$	9.53191e12	0.350559	0.175279	+	0.984519i	$0.443917\pi$
$968$	0	0
$969$	4.37727e13	1.59495
$970$	0	0
$971$	−4.10968e13	−1.48362	−0.741808	−	0.670613i	$-0.766031\pi$
$971$	−4.10968e13	−1.48362	−0.741808	+	0.670613i	$0.766031\pi$
$972$	0	0
$973$	7.42183e11	0.0265463
$974$	0	0
$975$	5.47838e10	0.00194147
$976$	0	0
$977$	4.22249e13	1.48266	0.741332	−	0.671139i	$-0.234194\pi$
$977$	4.22249e13	1.48266	0.741332	+	0.671139i	$0.234194\pi$
$978$	0	0
$979$	1.17079e13	0.407341
$980$	0	0
$981$	−5.67352e13	−1.95588
$982$	0	0
$983$	−1.30018e13	−0.444134	−0.222067	−	0.975031i	$-0.571280\pi$
$983$	−1.30018e13	−0.444134	−0.222067	+	0.975031i	$0.571280\pi$
$984$	0	0
$985$	1.01931e13	0.345018
$986$	0	0
$987$	−1.46837e14	−4.92504
$988$	0	0
$989$	−5.67176e12	−0.188510
$990$	0	0
$991$	3.13811e13	1.03356	0.516781	−	0.856118i	$-0.327130\pi$
$991$	3.13811e13	1.03356	0.516781	+	0.856118i	$0.327130\pi$
$992$	0	0
$993$	−8.23026e13	−2.68622
$994$	0	0
$995$	8.88868e12	0.287497
$996$	0	0
$997$	4.24377e13	1.36027	0.680133	−	0.733089i	$-0.261922\pi$
$997$	4.24377e13	1.36027	0.680133	+	0.733089i	$0.261922\pi$
$998$	0	0
$999$	−6.46489e13	−2.05360

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.10.a.i.1.2		2
4.3	odd	2		40.10.a.a.1.1	✓	2
5.2	odd	4		400.10.c.k.49.1		4
5.3	odd	4		400.10.c.k.49.4		4
5.4	even	2		400.10.a.n.1.1		2
8.3	odd	2		320.10.a.r.1.2		2
8.5	even	2		320.10.a.m.1.1		2
12.11	even	2		360.10.a.i.1.2		2
20.3	even	4		200.10.c.c.49.1		4
20.7	even	4		200.10.c.c.49.4		4
20.19	odd	2		200.10.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.a.1.1	✓	2	4.3	odd	2
80.10.a.i.1.2		2	1.1	even	1	trivial
200.10.a.e.1.2		2	20.19	odd	2
200.10.c.c.49.1		4	20.3	even	4
200.10.c.c.49.4		4	20.7	even	4
320.10.a.m.1.1		2	8.5	even	2
320.10.a.r.1.2		2	8.3	odd	2
360.10.a.i.1.2		2	12.11	even	2
400.10.a.n.1.1		2	5.4	even	2
400.10.c.k.49.1		4	5.2	odd	4
400.10.c.k.49.4		4	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)

\(f(q)\)	\(=\)	\(q+245.617 q^{3} -625.000 q^{5} -12208.6 q^{7} +40644.5 q^{9} +O(q^{10})\)	\(q+245.617 q^{3} -625.000 q^{5} -12208.6 q^{7} +40644.5 q^{9} +30816.6 q^{11} +570.998 q^{13} -153510. q^{15} -486850. q^{17} -366058. q^{19} -2.99863e6 q^{21} -214861. q^{23} +390625. q^{25} +5.14850e6 q^{27} -3.29684e6 q^{29} -7.33158e6 q^{31} +7.56908e6 q^{33} +7.63036e6 q^{35} -1.25569e7 q^{37} +140247. q^{39} -2.37465e7 q^{41} +2.63974e7 q^{43} -2.54028e7 q^{45} +4.89681e7 q^{47} +1.08696e8 q^{49} -1.19579e8 q^{51} -4.11185e7 q^{53} -1.92604e7 q^{55} -8.99099e7 q^{57} -1.69195e8 q^{59} -6.50957e7 q^{61} -4.96212e8 q^{63} -356874. q^{65} +7.51098e7 q^{67} -5.27734e7 q^{69} +5.60303e7 q^{71} -1.46444e8 q^{73} +9.59440e7 q^{75} -3.76227e8 q^{77} +6.33189e8 q^{79} +4.64551e8 q^{81} -2.96211e7 q^{83} +3.04282e8 q^{85} -8.09759e8 q^{87} +3.79922e8 q^{89} -6.97107e6 q^{91} -1.80076e9 q^{93} +2.28786e8 q^{95} +4.10205e8 q^{97} +1.25253e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 116 q^{3} - 1250 q^{5} - 11284 q^{7} + 37762 q^{9}+O(q^{10}) \) 2 * q + 116 * q^3 - 1250 * q^5 - 11284 * q^7 + 37762 * q^9	\( 2 q + 116 q^{3} - 1250 q^{5} - 11284 q^{7} + 37762 q^{9} + 101408 q^{11} - 21372 q^{13} - 72500 q^{15} - 296780 q^{17} - 275832 q^{19} - 3118472 q^{21} + 585284 q^{23} + 781250 q^{25} + 8073368 q^{27} - 9928756 q^{29} - 5131480 q^{31} - 1580736 q^{33} + 7052500 q^{35} - 11007932 q^{37} + 2984424 q^{39} - 41835956 q^{41} + 23394052 q^{43} - 23601250 q^{45} - 11711748 q^{47} + 69197114 q^{49} - 144214840 q^{51} - 46384268 q^{53} - 63380000 q^{55} - 101604656 q^{57} - 178239576 q^{59} + 31825220 q^{61} - 498877204 q^{63} + 13357500 q^{65} - 89480628 q^{67} - 156485528 q^{69} - 112319176 q^{71} - 93294524 q^{73} + 45312500 q^{75} - 310959936 q^{77} + 191601328 q^{79} + 142176298 q^{81} + 17270436 q^{83} + 185487500 q^{85} + 49847352 q^{87} - 615067148 q^{89} - 27259176 q^{91} - 2085926640 q^{93} + 172395000 q^{95} - 996545468 q^{97} + 1049046048 q^{99}+O(q^{100}) \) 2 * q + 116 * q^3 - 1250 * q^5 - 11284 * q^7 + 37762 * q^9 + 101408 * q^11 - 21372 * q^13 - 72500 * q^15 - 296780 * q^17 - 275832 * q^19 - 3118472 * q^21 + 585284 * q^23 + 781250 * q^25 + 8073368 * q^27 - 9928756 * q^29 - 5131480 * q^31 - 1580736 * q^33 + 7052500 * q^35 - 11007932 * q^37 + 2984424 * q^39 - 41835956 * q^41 + 23394052 * q^43 - 23601250 * q^45 - 11711748 * q^47 + 69197114 * q^49 - 144214840 * q^51 - 46384268 * q^53 - 63380000 * q^55 - 101604656 * q^57 - 178239576 * q^59 + 31825220 * q^61 - 498877204 * q^63 + 13357500 * q^65 - 89480628 * q^67 - 156485528 * q^69 - 112319176 * q^71 - 93294524 * q^73 + 45312500 * q^75 - 310959936 * q^77 + 191601328 * q^79 + 142176298 * q^81 + 17270436 * q^83 + 185487500 * q^85 + 49847352 * q^87 - 615067148 * q^89 - 27259176 * q^91 - 2085926640 * q^93 + 172395000 * q^95 - 996545468 * q^97 + 1049046048 * q^99

Introduction

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