LMFDB - Embedded newform 80.10.a.f.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{1009}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 252 $ x^2 - x - 252
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-15.3824$ 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)

$f(q)$	$=$	$q-66.4705 q^{3} +625.000 q^{5} +5947.66 q^{7} -15264.7 q^{9} +O(q^{10})$	$q-66.4705 q^{3} +625.000 q^{5} +5947.66 q^{7} -15264.7 q^{9} -72345.0 q^{11} +14564.0 q^{13} -41544.0 q^{15} +614778. q^{17} +238472. q^{19} -395344. q^{21} -91562.8 q^{23} +390625. q^{25} +2.32299e6 q^{27} -5.23105e6 q^{29} -5.72058e6 q^{31} +4.80881e6 q^{33} +3.71729e6 q^{35} -4.03979e6 q^{37} -968079. q^{39} -1.84276e7 q^{41} -2.72643e7 q^{43} -9.54042e6 q^{45} -1.25825e7 q^{47} -4.97896e6 q^{49} -4.08646e7 q^{51} +1.66132e7 q^{53} -4.52157e7 q^{55} -1.58513e7 q^{57} +7.65654e7 q^{59} -1.59271e8 q^{61} -9.07891e7 q^{63} +9.10253e6 q^{65} -7.66068e7 q^{67} +6.08622e6 q^{69} +2.03001e8 q^{71} -2.20084e8 q^{73} -2.59650e7 q^{75} -4.30284e8 q^{77} -5.99842e8 q^{79} +1.46044e8 q^{81} -6.13872e8 q^{83} +3.84236e8 q^{85} +3.47710e8 q^{87} -1.00215e7 q^{89} +8.66220e7 q^{91} +3.80250e8 q^{93} +1.49045e8 q^{95} +4.02491e8 q^{97} +1.10432e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9}+O(q^{10}) $ 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9	$ 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9} - 23984 q^{11} + 115020 q^{13} - 162500 q^{15} + 412820 q^{17} + 296520 q^{19} + 1084704 q^{21} + 1049220 q^{23} + 781250 q^{25} + 2693080 q^{27} - 3666980 q^{29} - 1613144 q^{31} - 4550480 q^{33} - 1062500 q^{35} - 21121940 q^{37} - 20409272 q^{39} - 26957276 q^{41} - 52889700 q^{43} + 1566250 q^{45} - 58412180 q^{47} + 13154114 q^{49} - 1779784 q^{51} - 39035140 q^{53} - 14990000 q^{55} - 27085360 q^{57} + 54995560 q^{59} - 274579716 q^{61} - 226693140 q^{63} + 71887500 q^{65} + 318580 q^{67} - 214688928 q^{69} + 7130936 q^{71} + 120858180 q^{73} - 101562500 q^{75} - 800132400 q^{77} - 6877520 q^{79} - 275359358 q^{81} - 1402348740 q^{83} + 258012500 q^{85} + 45016840 q^{87} + 830088660 q^{89} - 681630904 q^{91} - 414660480 q^{93} + 185325000 q^{95} + 638394580 q^{97} + 1963732048 q^{99}+O(q^{100}) $ 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9 - 23984 * q^11 + 115020 * q^13 - 162500 * q^15 + 412820 * q^17 + 296520 * q^19 + 1084704 * q^21 + 1049220 * q^23 + 781250 * q^25 + 2693080 * q^27 - 3666980 * q^29 - 1613144 * q^31 - 4550480 * q^33 - 1062500 * q^35 - 21121940 * q^37 - 20409272 * q^39 - 26957276 * q^41 - 52889700 * q^43 + 1566250 * q^45 - 58412180 * q^47 + 13154114 * q^49 - 1779784 * q^51 - 39035140 * q^53 - 14990000 * q^55 - 27085360 * q^57 + 54995560 * q^59 - 274579716 * q^61 - 226693140 * q^63 + 71887500 * q^65 + 318580 * q^67 - 214688928 * q^69 + 7130936 * q^71 + 120858180 * q^73 - 101562500 * q^75 - 800132400 * q^77 - 6877520 * q^79 - 275359358 * q^81 - 1402348740 * q^83 + 258012500 * q^85 + 45016840 * q^87 + 830088660 * q^89 - 681630904 * q^91 - 414660480 * q^93 + 185325000 * q^95 + 638394580 * q^97 + 1963732048 * 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
$3$	−66.4705	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$3$	−66.4705	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	5947.66	0.936278	0.468139	−	0.883655i	$-0.344925\pi$
$7$	5947.66	0.936278	0.468139	+	0.883655i	$0.344925\pi$
$8$	0	0
$9$	−15264.7	−0.775526
$10$	0	0
$11$	−72345.0	−1.48985	−0.744924	−	0.667150i	$-0.767514\pi$
$11$	−72345.0	−1.48985	−0.744924	+	0.667150i	$0.767514\pi$
$12$	0	0
$13$	14564.0	0.141428	0.0707142	−	0.997497i	$-0.477472\pi$
$13$	14564.0	0.141428	0.0707142	+	0.997497i	$0.477472\pi$
$14$	0	0
$15$	−41544.0	−0.211884
$16$	0	0
$17$	614778.	1.78525	0.892623	−	0.450804i	$-0.148863\pi$
$17$	614778.	1.78525	0.892623	+	0.450804i	$0.148863\pi$
$18$	0	0
$19$	238472.	0.419803	0.209902	−	0.977722i	$-0.432686\pi$
$19$	238472.	0.419803	0.209902	+	0.977722i	$0.432686\pi$
$20$	0	0
$21$	−395344.	−0.443596
$22$	0	0
$23$	−91562.8	−0.0682251	−0.0341125	−	0.999418i	$-0.510860\pi$
$23$	−91562.8	−0.0682251	−0.0341125	+	0.999418i	$0.510860\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	2.32299e6	0.841221
$28$	0	0
$29$	−5.23105e6	−1.37340	−0.686701	−	0.726940i	$-0.740942\pi$
$29$	−5.23105e6	−1.37340	−0.686701	+	0.726940i	$0.740942\pi$
$30$	0	0
$31$	−5.72058e6	−1.11253	−0.556266	−	0.831004i	$-0.687766\pi$
$31$	−5.72058e6	−1.11253	−0.556266	+	0.831004i	$0.687766\pi$
$32$	0	0
$33$	4.80881e6	0.705870
$34$	0	0
$35$	3.71729e6	0.418716
$36$	0	0
$37$	−4.03979e6	−0.354365	−0.177183	−	0.984178i	$-0.556698\pi$
$37$	−4.03979e6	−0.354365	−0.177183	+	0.984178i	$0.556698\pi$
$38$	0	0
$39$	−968079.	−0.0670070
$40$	0	0
$41$	−1.84276e7	−1.01845	−0.509227	−	0.860632i	$-0.670069\pi$
$41$	−1.84276e7	−1.01845	−0.509227	+	0.860632i	$0.670069\pi$
$42$	0	0
$43$	−2.72643e7	−1.21615	−0.608074	−	0.793880i	$-0.708058\pi$
$43$	−2.72643e7	−1.21615	−0.608074	+	0.793880i	$0.708058\pi$
$44$	0	0
$45$	−9.54042e6	−0.346826
$46$	0	0
$47$	−1.25825e7	−0.376120	−0.188060	−	0.982157i	$-0.560220\pi$
$47$	−1.25825e7	−0.376120	−0.188060	+	0.982157i	$0.560220\pi$
$48$	0	0
$49$	−4.97896e6	−0.123383
$50$	0	0
$51$	−4.08646e7	−0.845826
$52$	0	0
$53$	1.66132e7	0.289209	0.144604	−	0.989490i	$-0.453809\pi$
$53$	1.66132e7	0.289209	0.144604	+	0.989490i	$0.453809\pi$
$54$	0	0
$55$	−4.52157e7	−0.666280
$56$	0	0
$57$	−1.58513e7	−0.198897
$58$	0	0
$59$	7.65654e7	0.822619	0.411309	−	0.911496i	$-0.365072\pi$
$59$	7.65654e7	0.822619	0.411309	+	0.911496i	$0.365072\pi$
$60$	0	0
$61$	−1.59271e8	−1.47283	−0.736415	−	0.676530i	$-0.763483\pi$
$61$	−1.59271e8	−1.47283	−0.736415	+	0.676530i	$0.763483\pi$
$62$	0	0
$63$	−9.07891e7	−0.726108
$64$	0	0
$65$	9.10253e6	0.0632487
$66$	0	0
$67$	−7.66068e7	−0.464441	−0.232221	−	0.972663i	$-0.574599\pi$
$67$	−7.66068e7	−0.464441	−0.232221	+	0.972663i	$0.574599\pi$
$68$	0	0
$69$	6.08622e6	0.0323241
$70$	0	0
$71$	2.03001e8	0.948057	0.474029	−	0.880509i	$-0.342799\pi$
$71$	2.03001e8	0.948057	0.474029	+	0.880509i	$0.342799\pi$
$72$	0	0
$73$	−2.20084e8	−0.907059	−0.453530	−	0.891241i	$-0.649835\pi$
$73$	−2.20084e8	−0.907059	−0.453530	+	0.891241i	$0.649835\pi$
$74$	0	0
$75$	−2.59650e7	−0.0947574
$76$	0	0
$77$	−4.30284e8	−1.39491
$78$	0	0
$79$	−5.99842e8	−1.73267	−0.866334	−	0.499465i	$-0.833530\pi$
$79$	−5.99842e8	−1.73267	−0.866334	+	0.499465i	$0.833530\pi$
$80$	0	0
$81$	1.46044e8	0.376966
$82$	0	0
$83$	−6.13872e8	−1.41980	−0.709899	−	0.704304i	$-0.751259\pi$
$83$	−6.13872e8	−1.41980	−0.709899	+	0.704304i	$0.751259\pi$
$84$	0	0
$85$	3.84236e8	0.798386
$86$	0	0
$87$	3.47710e8	0.650700
$88$	0	0
$89$	−1.00215e7	−0.0169309	−0.00846543	−	0.999964i	$-0.502695\pi$
$89$	−1.00215e7	−0.0169309	−0.00846543	+	0.999964i	$0.502695\pi$
$90$	0	0
$91$	8.66220e7	0.132416
$92$	0	0
$93$	3.80250e8	0.527103
$94$	0	0
$95$	1.49045e8	0.187742
$96$	0	0
$97$	4.02491e8	0.461619	0.230810	−	0.972999i	$-0.425863\pi$
$97$	4.02491e8	0.461619	0.230810	+	0.972999i	$0.425863\pi$
$98$	0	0
$99$	1.10432e9	1.15541
$100$	0	0
$101$	−1.30103e9	−1.24406	−0.622032	−	0.782992i	$-0.713693\pi$
$101$	−1.30103e9	−1.24406	−0.622032	+	0.782992i	$0.713693\pi$
$102$	0	0
$103$	2.45156e8	0.214622	0.107311	−	0.994225i	$-0.465776\pi$
$103$	2.45156e8	0.214622	0.107311	+	0.994225i	$0.465776\pi$
$104$	0	0
$105$	−2.47090e8	−0.198382
$106$	0	0
$107$	−1.00765e9	−0.743163	−0.371581	−	0.928400i	$-0.621184\pi$
$107$	−1.00765e9	−0.743163	−0.371581	+	0.928400i	$0.621184\pi$
$108$	0	0
$109$	7.68000e8	0.521125	0.260562	−	0.965457i	$-0.416092\pi$
$109$	7.68000e8	0.521125	0.260562	+	0.965457i	$0.416092\pi$
$110$	0	0
$111$	2.68527e8	0.167894
$112$	0	0
$113$	9.01969e8	0.520402	0.260201	−	0.965555i	$-0.416211\pi$
$113$	9.01969e8	0.520402	0.260201	+	0.965555i	$0.416211\pi$
$114$	0	0
$115$	−5.72268e7	−0.0305112
$116$	0	0
$117$	−2.22315e8	−0.109681
$118$	0	0
$119$	3.65649e9	1.67149
$120$	0	0
$121$	2.87586e9	1.21964
$122$	0	0
$123$	1.22489e9	0.482530
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	1.27265e9	0.434103	0.217052	−	0.976160i	$-0.430356\pi$
$127$	1.27265e9	0.434103	0.217052	+	0.976160i	$0.430356\pi$
$128$	0	0
$129$	1.81227e9	0.576195
$130$	0	0
$131$	−4.89056e8	−0.145090	−0.0725451	−	0.997365i	$-0.523112\pi$
$131$	−4.89056e8	−0.145090	−0.0725451	+	0.997365i	$0.523112\pi$
$132$	0	0
$133$	1.41835e9	0.393053
$134$	0	0
$135$	1.45187e9	0.376206
$136$	0	0
$137$	1.04222e9	0.252766	0.126383	−	0.991982i	$-0.459663\pi$
$137$	1.04222e9	0.252766	0.126383	+	0.991982i	$0.459663\pi$
$138$	0	0
$139$	−1.44082e9	−0.327372	−0.163686	−	0.986512i	$-0.552338\pi$
$139$	−1.44082e9	−0.327372	−0.163686	+	0.986512i	$0.552338\pi$
$140$	0	0
$141$	8.36366e8	0.178201
$142$	0	0
$143$	−1.05364e9	−0.210707
$144$	0	0
$145$	−3.26941e9	−0.614204
$146$	0	0
$147$	3.30954e8	0.0584574
$148$	0	0
$149$	5.29574e9	0.880214	0.440107	−	0.897945i	$-0.354941\pi$
$149$	5.29574e9	0.880214	0.440107	+	0.897945i	$0.354941\pi$
$150$	0	0
$151$	−3.60123e9	−0.563708	−0.281854	−	0.959457i	$-0.590949\pi$
$151$	−3.60123e9	−0.563708	−0.281854	+	0.959457i	$0.590949\pi$
$152$	0	0
$153$	−9.38438e9	−1.38450
$154$	0	0
$155$	−3.57536e9	−0.497539
$156$	0	0
$157$	3.01791e9	0.396423	0.198211	−	0.980159i	$-0.436487\pi$
$157$	3.01791e9	0.396423	0.198211	+	0.980159i	$0.436487\pi$
$158$	0	0
$159$	−1.10429e9	−0.137023
$160$	0	0
$161$	−5.44584e8	−0.0638776
$162$	0	0
$163$	9.02201e9	1.00106	0.500529	−	0.865720i	$-0.333139\pi$
$163$	9.02201e9	1.00106	0.500529	+	0.865720i	$0.333139\pi$
$164$	0	0
$165$	3.00551e9	0.315675
$166$	0	0
$167$	2.52141e9	0.250853	0.125426	−	0.992103i	$-0.459970\pi$
$167$	2.52141e9	0.250853	0.125426	+	0.992103i	$0.459970\pi$
$168$	0	0
$169$	−1.03924e10	−0.979998
$170$	0	0
$171$	−3.64020e9	−0.325568
$172$	0	0
$173$	−1.09608e10	−0.930329	−0.465164	−	0.885224i	$-0.654005\pi$
$173$	−1.09608e10	−0.930329	−0.465164	+	0.885224i	$0.654005\pi$
$174$	0	0
$175$	2.32330e9	0.187256
$176$	0	0
$177$	−5.08934e9	−0.389746
$178$	0	0
$179$	−6.46578e9	−0.470742	−0.235371	−	0.971906i	$-0.575630\pi$
$179$	−6.46578e9	−0.470742	−0.235371	+	0.971906i	$0.575630\pi$
$180$	0	0
$181$	−1.45339e8	−0.0100653	−0.00503267	−	0.999987i	$-0.501602\pi$
$181$	−1.45339e8	−0.0100653	−0.00503267	+	0.999987i	$0.501602\pi$
$182$	0	0
$183$	1.05868e10	0.697808
$184$	0	0
$185$	−2.52487e9	−0.158477
$186$	0	0
$187$	−4.44761e10	−2.65974
$188$	0	0
$189$	1.38163e10	0.787617
$190$	0	0
$191$	1.80577e10	0.981775	0.490888	−	0.871223i	$-0.336673\pi$
$191$	1.80577e10	0.981775	0.490888	+	0.871223i	$0.336673\pi$
$192$	0	0
$193$	3.63652e10	1.88659	0.943295	−	0.331956i	$-0.107708\pi$
$193$	3.63652e10	1.88659	0.943295	+	0.331956i	$0.107708\pi$
$194$	0	0
$195$	−6.05049e8	−0.0299664
$196$	0	0
$197$	−3.85969e10	−1.82580	−0.912902	−	0.408179i	$-0.866164\pi$
$197$	−3.85969e10	−1.82580	−0.912902	+	0.408179i	$0.866164\pi$
$198$	0	0
$199$	−2.78151e8	−0.0125731	−0.00628653	−	0.999980i	$-0.502001\pi$
$199$	−2.78151e8	−0.0125731	−0.00628653	+	0.999980i	$0.502001\pi$
$200$	0	0
$201$	5.09209e9	0.220046
$202$	0	0
$203$	−3.11125e10	−1.28589
$204$	0	0
$205$	−1.15172e10	−0.455466
$206$	0	0
$207$	1.39768e9	0.0529103
$208$	0	0
$209$	−1.72523e10	−0.625443
$210$	0	0
$211$	−2.36984e10	−0.823091	−0.411546	−	0.911389i	$-0.635011\pi$
$211$	−2.36984e10	−0.823091	−0.411546	+	0.911389i	$0.635011\pi$
$212$	0	0
$213$	−1.34935e10	−0.449177
$214$	0	0
$215$	−1.70402e10	−0.543878
$216$	0	0
$217$	−3.40241e10	−1.04164
$218$	0	0
$219$	1.46291e10	0.429753
$220$	0	0
$221$	8.95365e9	0.252485
$222$	0	0
$223$	3.57686e10	0.968567	0.484284	−	0.874911i	$-0.339080\pi$
$223$	3.57686e10	0.968567	0.484284	+	0.874911i	$0.339080\pi$
$224$	0	0
$225$	−5.96276e9	−0.155105
$226$	0	0
$227$	5.92891e9	0.148203	0.0741017	−	0.997251i	$-0.476391\pi$
$227$	5.92891e9	0.148203	0.0741017	+	0.997251i	$0.476391\pi$
$228$	0	0
$229$	4.15396e9	0.0998165	0.0499082	−	0.998754i	$-0.484107\pi$
$229$	4.15396e9	0.0998165	0.0499082	+	0.998754i	$0.484107\pi$
$230$	0	0
$231$	2.86012e10	0.660891
$232$	0	0
$233$	7.00301e10	1.55662	0.778310	−	0.627880i	$-0.216077\pi$
$233$	7.00301e10	1.55662	0.778310	+	0.627880i	$0.216077\pi$
$234$	0	0
$235$	−7.86407e9	−0.168206
$236$	0	0
$237$	3.98718e10	0.820916
$238$	0	0
$239$	8.12068e10	1.60991	0.804955	−	0.593336i	$-0.202189\pi$
$239$	8.12068e10	1.60991	0.804955	+	0.593336i	$0.202189\pi$
$240$	0	0
$241$	−7.73932e10	−1.47784	−0.738918	−	0.673795i	$-0.764663\pi$
$241$	−7.73932e10	−1.47784	−0.738918	+	0.673795i	$0.764663\pi$
$242$	0	0
$243$	−5.54310e10	−1.01982
$244$	0	0
$245$	−3.11185e9	−0.0551787
$246$	0	0
$247$	3.47312e9	0.0593722
$248$	0	0
$249$	4.08044e10	0.672682
$250$	0	0
$251$	6.34869e9	0.100961	0.0504804	−	0.998725i	$-0.483925\pi$
$251$	6.34869e9	0.100961	0.0504804	+	0.998725i	$0.483925\pi$
$252$	0	0
$253$	6.62412e9	0.101645
$254$	0	0
$255$	−2.55404e10	−0.378265
$256$	0	0
$257$	3.47169e10	0.496411	0.248206	−	0.968707i	$-0.420159\pi$
$257$	3.47169e10	0.496411	0.248206	+	0.968707i	$0.420159\pi$
$258$	0	0
$259$	−2.40273e10	−0.331784
$260$	0	0
$261$	7.98503e10	1.06511
$262$	0	0
$263$	7.47644e10	0.963593	0.481797	−	0.876283i	$-0.339984\pi$
$263$	7.47644e10	0.963593	0.481797	+	0.876283i	$0.339984\pi$
$264$	0	0
$265$	1.03832e10	0.129338
$266$	0	0
$267$	6.66136e8	0.00802162
$268$	0	0
$269$	7.94021e10	0.924585	0.462293	−	0.886727i	$-0.347027\pi$
$269$	7.94021e10	0.924585	0.462293	+	0.886727i	$0.347027\pi$
$270$	0	0
$271$	1.07816e11	1.21428	0.607142	−	0.794593i	$-0.292316\pi$
$271$	1.07816e11	1.21428	0.607142	+	0.794593i	$0.292316\pi$
$272$	0	0
$273$	−5.75780e9	−0.0627372
$274$	0	0
$275$	−2.82598e10	−0.297969
$276$	0	0
$277$	−1.47804e11	−1.50844	−0.754221	−	0.656621i	$-0.771985\pi$
$277$	−1.47804e11	−1.50844	−0.754221	+	0.656621i	$0.771985\pi$
$278$	0	0
$279$	8.73228e10	0.862797
$280$	0	0
$281$	7.71094e10	0.737783	0.368892	−	0.929472i	$-0.379737\pi$
$281$	7.71094e10	0.737783	0.368892	+	0.929472i	$0.379737\pi$
$282$	0	0
$283$	5.20411e10	0.482289	0.241145	−	0.970489i	$-0.422477\pi$
$283$	5.20411e10	0.482289	0.241145	+	0.970489i	$0.422477\pi$
$284$	0	0
$285$	−9.90709e9	−0.0889496
$286$	0	0
$287$	−1.09601e11	−0.953556
$288$	0	0
$289$	2.59364e11	2.18710
$290$	0	0
$291$	−2.67538e10	−0.218709
$292$	0	0
$293$	8.61823e10	0.683146	0.341573	−	0.939855i	$-0.389040\pi$
$293$	8.61823e10	0.683146	0.341573	+	0.939855i	$0.389040\pi$
$294$	0	0
$295$	4.78534e10	0.367886
$296$	0	0
$297$	−1.68057e11	−1.25329
$298$	0	0
$299$	−1.33352e9	−0.00964897
$300$	0	0
$301$	−1.62159e11	−1.13865
$302$	0	0
$303$	8.64804e10	0.589421
$304$	0	0
$305$	−9.95444e10	−0.658670
$306$	0	0
$307$	−2.85807e11	−1.83633	−0.918163	−	0.396203i	$-0.870328\pi$
$307$	−2.85807e11	−1.83633	−0.918163	+	0.396203i	$0.870328\pi$
$308$	0	0
$309$	−1.62956e10	−0.101685
$310$	0	0
$311$	−9.10066e10	−0.551634	−0.275817	−	0.961210i	$-0.588948\pi$
$311$	−9.10066e10	−0.551634	−0.275817	+	0.961210i	$0.588948\pi$
$312$	0	0
$313$	−1.10469e11	−0.650565	−0.325283	−	0.945617i	$-0.605459\pi$
$313$	−1.10469e11	−0.650565	−0.325283	+	0.945617i	$0.605459\pi$
$314$	0	0
$315$	−5.67432e10	−0.324725
$316$	0	0
$317$	−1.41331e11	−0.786087	−0.393043	−	0.919520i	$-0.628578\pi$
$317$	−1.41331e11	−0.786087	−0.393043	+	0.919520i	$0.628578\pi$
$318$	0	0
$319$	3.78440e11	2.04616
$320$	0	0
$321$	6.69792e10	0.352101
$322$	0	0
$323$	1.46607e11	0.749452
$324$	0	0
$325$	5.68908e9	0.0282857
$326$	0	0
$327$	−5.10493e10	−0.246902
$328$	0	0
$329$	−7.48365e10	−0.352153
$330$	0	0
$331$	−1.97402e11	−0.903911	−0.451956	−	0.892040i	$-0.649273\pi$
$331$	−1.97402e11	−0.903911	−0.451956	+	0.892040i	$0.649273\pi$
$332$	0	0
$333$	6.16661e10	0.274819
$334$	0	0
$335$	−4.78792e10	−0.207704
$336$	0	0
$337$	2.73223e10	0.115394	0.0576969	−	0.998334i	$-0.481624\pi$
$337$	2.73223e10	0.115394	0.0576969	+	0.998334i	$0.481624\pi$
$338$	0	0
$339$	−5.99543e10	−0.246560
$340$	0	0
$341$	4.13856e11	1.65750
$342$	0	0
$343$	−2.69623e11	−1.05180
$344$	0	0
$345$	3.80389e9	0.0144558
$346$	0	0
$347$	−3.10713e11	−1.15047	−0.575236	−	0.817987i	$-0.695090\pi$
$347$	−3.10713e11	−1.15047	−0.575236	+	0.817987i	$0.695090\pi$
$348$	0	0
$349$	−3.20017e10	−0.115467	−0.0577336	−	0.998332i	$-0.518387\pi$
$349$	−3.20017e10	−0.115467	−0.0577336	+	0.998332i	$0.518387\pi$
$350$	0	0
$351$	3.38321e10	0.118973
$352$	0	0
$353$	−3.87463e11	−1.32814	−0.664070	−	0.747671i	$-0.731172\pi$
$353$	−3.87463e11	−1.32814	−0.664070	+	0.747671i	$0.731172\pi$
$354$	0	0
$355$	1.26875e11	0.423984
$356$	0	0
$357$	−2.43049e11	−0.791929
$358$	0	0
$359$	−1.07798e11	−0.342520	−0.171260	−	0.985226i	$-0.554784\pi$
$359$	−1.07798e11	−0.342520	−0.171260	+	0.985226i	$0.554784\pi$
$360$	0	0
$361$	−2.65819e11	−0.823765
$362$	0	0
$363$	−1.91160e11	−0.577852
$364$	0	0
$365$	−1.37552e11	−0.405649
$366$	0	0
$367$	−3.11151e11	−0.895310	−0.447655	−	0.894206i	$-0.647741\pi$
$367$	−3.11151e11	−0.895310	−0.447655	+	0.894206i	$0.647741\pi$
$368$	0	0
$369$	2.81291e11	0.789837
$370$	0	0
$371$	9.88096e10	0.270780
$372$	0	0
$373$	2.69498e11	0.720884	0.360442	−	0.932782i	$-0.382626\pi$
$373$	2.69498e11	0.720884	0.360442	+	0.932782i	$0.382626\pi$
$374$	0	0
$375$	−1.62281e10	−0.0423768
$376$	0	0
$377$	−7.61852e10	−0.194238
$378$	0	0
$379$	2.99148e11	0.744749	0.372375	−	0.928083i	$-0.378544\pi$
$379$	2.99148e11	0.744749	0.372375	+	0.928083i	$0.378544\pi$
$380$	0	0
$381$	−8.45939e10	−0.205673
$382$	0	0
$383$	4.87824e11	1.15843	0.579214	−	0.815176i	$-0.303360\pi$
$383$	4.87824e11	1.15843	0.579214	+	0.815176i	$0.303360\pi$
$384$	0	0
$385$	−2.68927e11	−0.623823
$386$	0	0
$387$	4.16181e11	0.943155
$388$	0	0
$389$	−1.05812e10	−0.0234295	−0.0117147	−	0.999931i	$-0.503729\pi$
$389$	−1.05812e10	−0.0234295	−0.0117147	+	0.999931i	$0.503729\pi$
$390$	0	0
$391$	−5.62908e10	−0.121798
$392$	0	0
$393$	3.25078e10	0.0687418
$394$	0	0
$395$	−3.74902e11	−0.774873
$396$	0	0
$397$	−3.15739e11	−0.637928	−0.318964	−	0.947767i	$-0.603335\pi$
$397$	−3.15739e11	−0.637928	−0.318964	+	0.947767i	$0.603335\pi$
$398$	0	0
$399$	−9.42784e10	−0.186223
$400$	0	0
$401$	−2.96087e11	−0.571834	−0.285917	−	0.958254i	$-0.592298\pi$
$401$	−2.96087e11	−0.571834	−0.285917	+	0.958254i	$0.592298\pi$
$402$	0	0
$403$	−8.33148e10	−0.157344
$404$	0	0
$405$	9.12778e10	0.168584
$406$	0	0
$407$	2.92259e11	0.527950
$408$	0	0
$409$	7.70694e11	1.36184	0.680922	−	0.732356i	$-0.261579\pi$
$409$	7.70694e11	1.36184	0.680922	+	0.732356i	$0.261579\pi$
$410$	0	0
$411$	−6.92771e10	−0.119757
$412$	0	0
$413$	4.55385e11	0.770200
$414$	0	0
$415$	−3.83670e11	−0.634953
$416$	0	0
$417$	9.57717e10	0.155105
$418$	0	0
$419$	7.08818e11	1.12350	0.561748	−	0.827308i	$-0.310129\pi$
$419$	7.08818e11	1.12350	0.561748	+	0.827308i	$0.310129\pi$
$420$	0	0
$421$	−2.53189e11	−0.392803	−0.196401	−	0.980524i	$-0.562926\pi$
$421$	−2.53189e11	−0.392803	−0.196401	+	0.980524i	$0.562926\pi$
$422$	0	0
$423$	1.92068e11	0.291691
$424$	0	0
$425$	2.40148e11	0.357049
$426$	0	0
$427$	−9.47290e11	−1.37898
$428$	0	0
$429$	7.00357e10	0.0998302
$430$	0	0
$431$	3.58426e11	0.500325	0.250162	−	0.968204i	$-0.419516\pi$
$431$	3.58426e11	0.500325	0.250162	+	0.968204i	$0.419516\pi$
$432$	0	0
$433$	5.71936e11	0.781901	0.390951	−	0.920412i	$-0.372146\pi$
$433$	5.71936e11	0.781901	0.390951	+	0.920412i	$0.372146\pi$
$434$	0	0
$435$	2.17319e11	0.291002
$436$	0	0
$437$	−2.18352e10	−0.0286411
$438$	0	0
$439$	−1.16407e12	−1.49585	−0.747924	−	0.663784i	$-0.768949\pi$
$439$	−1.16407e12	−1.49585	−0.747924	+	0.663784i	$0.768949\pi$
$440$	0	0
$441$	7.60023e10	0.0956870
$442$	0	0
$443$	1.24504e12	1.53592	0.767958	−	0.640500i	$-0.221273\pi$
$443$	1.24504e12	1.53592	0.767958	+	0.640500i	$0.221273\pi$
$444$	0	0
$445$	−6.26346e9	−0.00757171
$446$	0	0
$447$	−3.52010e11	−0.417034
$448$	0	0
$449$	−3.98352e11	−0.462549	−0.231275	−	0.972888i	$-0.574290\pi$
$449$	−3.98352e11	−0.462549	−0.231275	+	0.972888i	$0.574290\pi$
$450$	0	0
$451$	1.33314e12	1.51734
$452$	0	0
$453$	2.39375e11	0.267078
$454$	0	0
$455$	5.41387e10	0.0592184
$456$	0	0
$457$	−2.64155e11	−0.283293	−0.141647	−	0.989917i	$-0.545240\pi$
$457$	−2.64155e11	−0.283293	−0.141647	+	0.989917i	$0.545240\pi$
$458$	0	0
$459$	1.42812e12	1.50179
$460$	0	0
$461$	1.54801e12	1.59632	0.798161	−	0.602444i	$-0.205806\pi$
$461$	1.54801e12	1.59632	0.798161	+	0.602444i	$0.205806\pi$
$462$	0	0
$463$	−1.39894e12	−1.41477	−0.707385	−	0.706829i	$-0.750125\pi$
$463$	−1.39894e12	−1.41477	−0.707385	+	0.706829i	$0.750125\pi$
$464$	0	0
$465$	2.37656e11	0.235728
$466$	0	0
$467$	1.13856e12	1.10772	0.553858	−	0.832611i	$-0.313155\pi$
$467$	1.13856e12	1.10772	0.553858	+	0.832611i	$0.313155\pi$
$468$	0	0
$469$	−4.55631e11	−0.434846
$470$	0	0
$471$	−2.00602e11	−0.187820
$472$	0	0
$473$	1.97244e12	1.81188
$474$	0	0
$475$	9.31531e10	0.0839607
$476$	0	0
$477$	−2.53595e11	−0.224289
$478$	0	0
$479$	1.31802e10	0.0114396	0.00571980	−	0.999984i	$-0.498179\pi$
$479$	1.31802e10	0.0114396	0.00571980	+	0.999984i	$0.498179\pi$
$480$	0	0
$481$	−5.88357e10	−0.0501174
$482$	0	0
$483$	3.61988e10	0.0302644
$484$	0	0
$485$	2.51557e11	0.206442
$486$	0	0
$487$	1.39614e12	1.12473	0.562367	−	0.826888i	$-0.309891\pi$
$487$	1.39614e12	1.12473	0.562367	+	0.826888i	$0.309891\pi$
$488$	0	0
$489$	−5.99697e11	−0.474288
$490$	0	0
$491$	4.13493e11	0.321071	0.160536	−	0.987030i	$-0.448678\pi$
$491$	4.13493e11	0.321071	0.160536	+	0.987030i	$0.448678\pi$
$492$	0	0
$493$	−3.21593e12	−2.45186
$494$	0	0
$495$	6.90202e11	0.516717
$496$	0	0
$497$	1.20738e12	0.887645
$498$	0	0
$499$	−2.65850e12	−1.91948	−0.959740	−	0.280890i	$-0.909370\pi$
$499$	−2.65850e12	−1.91948	−0.959740	+	0.280890i	$0.909370\pi$
$500$	0	0
$501$	−1.67599e11	−0.118851
$502$	0	0
$503$	−1.55866e12	−1.08567	−0.542834	−	0.839840i	$-0.682649\pi$
$503$	−1.55866e12	−1.08567	−0.542834	+	0.839840i	$0.682649\pi$
$504$	0	0
$505$	−8.13146e11	−0.556362
$506$	0	0
$507$	6.90787e11	0.464310
$508$	0	0
$509$	−4.51109e11	−0.297887	−0.148943	−	0.988846i	$-0.547587\pi$
$509$	−4.51109e11	−0.297887	−0.148943	+	0.988846i	$0.547587\pi$
$510$	0	0
$511$	−1.30898e12	−0.849260
$512$	0	0
$513$	5.53968e11	0.353148
$514$	0	0
$515$	1.53223e11	0.0959821
$516$	0	0
$517$	9.10282e11	0.560362
$518$	0	0
$519$	7.28573e11	0.440778
$520$	0	0
$521$	−2.47767e12	−1.47324	−0.736620	−	0.676307i	$-0.763579\pi$
$521$	−2.47767e12	−1.47324	−0.736620	+	0.676307i	$0.763579\pi$
$522$	0	0
$523$	6.07290e11	0.354926	0.177463	−	0.984127i	$-0.443211\pi$
$523$	6.07290e11	0.354926	0.177463	+	0.984127i	$0.443211\pi$
$524$	0	0
$525$	−1.54431e11	−0.0887193
$526$	0	0
$527$	−3.51689e12	−1.98614
$528$	0	0
$529$	−1.79277e12	−0.995345
$530$	0	0
$531$	−1.16875e12	−0.637962
$532$	0	0
$533$	−2.68380e11	−0.144038
$534$	0	0
$535$	−6.29783e11	−0.332352
$536$	0	0
$537$	4.29784e11	0.223031
$538$	0	0
$539$	3.60203e11	0.183822
$540$	0	0
$541$	1.12334e12	0.563798	0.281899	−	0.959444i	$-0.409036\pi$
$541$	1.12334e12	0.563798	0.281899	+	0.959444i	$0.409036\pi$
$542$	0	0
$543$	9.66075e9	0.00476883
$544$	0	0
$545$	4.80000e11	0.233054
$546$	0	0
$547$	1.03643e12	0.494991	0.247496	−	0.968889i	$-0.420392\pi$
$547$	1.03643e12	0.494991	0.247496	+	0.968889i	$0.420392\pi$
$548$	0	0
$549$	2.43122e12	1.14222
$550$	0	0
$551$	−1.24746e12	−0.576559
$552$	0	0
$553$	−3.56766e12	−1.62226
$554$	0	0
$555$	1.67829e11	0.0750843
$556$	0	0
$557$	6.38453e11	0.281048	0.140524	−	0.990077i	$-0.455121\pi$
$557$	6.38453e11	0.281048	0.140524	+	0.990077i	$0.455121\pi$
$558$	0	0
$559$	−3.97079e11	−0.171998
$560$	0	0
$561$	2.95635e12	1.26015
$562$	0	0
$563$	1.22208e12	0.512639	0.256319	−	0.966592i	$-0.417490\pi$
$563$	1.22208e12	0.512639	0.256319	+	0.966592i	$0.417490\pi$
$564$	0	0
$565$	5.63731e11	0.232731
$566$	0	0
$567$	8.68622e11	0.352945
$568$	0	0
$569$	−1.03020e12	−0.412018	−0.206009	−	0.978550i	$-0.566048\pi$
$569$	−1.03020e12	−0.412018	−0.206009	+	0.978550i	$0.566048\pi$
$570$	0	0
$571$	−2.35639e12	−0.927651	−0.463826	−	0.885927i	$-0.653524\pi$
$571$	−2.35639e12	−0.927651	−0.463826	+	0.885927i	$0.653524\pi$
$572$	0	0
$573$	−1.20030e12	−0.465152
$574$	0	0
$575$	−3.57667e10	−0.0136450
$576$	0	0
$577$	−6.23831e11	−0.234302	−0.117151	−	0.993114i	$-0.537376\pi$
$577$	−6.23831e11	−0.234302	−0.117151	+	0.993114i	$0.537376\pi$
$578$	0	0
$579$	−2.41721e12	−0.893842
$580$	0	0
$581$	−3.65110e12	−1.32933
$582$	0	0
$583$	−1.20188e12	−0.430877
$584$	0	0
$585$	−1.38947e11	−0.0490510
$586$	0	0
$587$	2.84599e12	0.989378	0.494689	−	0.869070i	$-0.335282\pi$
$587$	2.84599e12	0.989378	0.494689	+	0.869070i	$0.335282\pi$
$588$	0	0
$589$	−1.36420e12	−0.467045
$590$	0	0
$591$	2.56555e12	0.865042
$592$	0	0
$593$	−2.71615e12	−0.902002	−0.451001	−	0.892523i	$-0.648933\pi$
$593$	−2.71615e12	−0.902002	−0.451001	+	0.892523i	$0.648933\pi$
$594$	0	0
$595$	2.28531e12	0.747511
$596$	0	0
$597$	1.84888e10	0.00595696
$598$	0	0
$599$	−2.62092e12	−0.831827	−0.415914	−	0.909404i	$-0.636538\pi$
$599$	−2.62092e12	−0.831827	−0.415914	+	0.909404i	$0.636538\pi$
$600$	0	0
$601$	−3.04608e12	−0.952371	−0.476186	−	0.879345i	$-0.657981\pi$
$601$	−3.04608e12	−0.952371	−0.476186	+	0.879345i	$0.657981\pi$
$602$	0	0
$603$	1.16938e12	0.360186
$604$	0	0
$605$	1.79741e12	0.545442
$606$	0	0
$607$	2.30741e12	0.689883	0.344942	−	0.938624i	$-0.387899\pi$
$607$	2.30741e12	0.689883	0.344942	+	0.938624i	$0.387899\pi$
$608$	0	0
$609$	2.06806e12	0.609236
$610$	0	0
$611$	−1.83252e11	−0.0531941
$612$	0	0
$613$	3.11665e12	0.891488	0.445744	−	0.895160i	$-0.352939\pi$
$613$	3.11665e12	0.891488	0.445744	+	0.895160i	$0.352939\pi$
$614$	0	0
$615$	7.65557e11	0.215794
$616$	0	0
$617$	6.81198e12	1.89230	0.946151	−	0.323726i	$-0.104936\pi$
$617$	6.81198e12	1.89230	0.946151	+	0.323726i	$0.104936\pi$
$618$	0	0
$619$	4.84144e11	0.132546	0.0662730	−	0.997802i	$-0.478889\pi$
$619$	4.84144e11	0.132546	0.0662730	+	0.997802i	$0.478889\pi$
$620$	0	0
$621$	−2.12699e11	−0.0573924
$622$	0	0
$623$	−5.96046e10	−0.0158520
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	1.14677e12	0.296327
$628$	0	0
$629$	−2.48357e12	−0.632629
$630$	0	0
$631$	−4.81541e12	−1.20921	−0.604604	−	0.796526i	$-0.706669\pi$
$631$	−4.81541e12	−1.20921	−0.604604	+	0.796526i	$0.706669\pi$
$632$	0	0
$633$	1.57524e12	0.389970
$634$	0	0
$635$	7.95408e11	0.194137
$636$	0	0
$637$	−7.25138e10	−0.0174499
$638$	0	0
$639$	−3.09874e12	−0.735243
$640$	0	0
$641$	5.97949e12	1.39895	0.699476	−	0.714656i	$-0.253417\pi$
$641$	5.97949e12	1.39895	0.699476	+	0.714656i	$0.253417\pi$
$642$	0	0
$643$	−6.83223e12	−1.57621	−0.788104	−	0.615542i	$-0.788937\pi$
$643$	−6.83223e12	−1.57621	−0.788104	+	0.615542i	$0.788937\pi$
$644$	0	0
$645$	1.13267e12	0.257682
$646$	0	0
$647$	5.65905e12	1.26962	0.634811	−	0.772667i	$-0.281078\pi$
$647$	5.65905e12	1.26962	0.634811	+	0.772667i	$0.281078\pi$
$648$	0	0
$649$	−5.53913e12	−1.22558
$650$	0	0
$651$	2.26160e12	0.493515
$652$	0	0
$653$	−1.01707e12	−0.218898	−0.109449	−	0.993992i	$-0.534909\pi$
$653$	−1.01707e12	−0.218898	−0.109449	+	0.993992i	$0.534909\pi$
$654$	0	0
$655$	−3.05660e11	−0.0648863
$656$	0	0
$657$	3.35951e12	0.703448
$658$	0	0
$659$	3.72612e12	0.769614	0.384807	−	0.922997i	$-0.374268\pi$
$659$	3.72612e12	0.769614	0.384807	+	0.922997i	$0.374268\pi$
$660$	0	0
$661$	2.45746e11	0.0500704	0.0250352	−	0.999687i	$-0.492030\pi$
$661$	2.45746e11	0.0500704	0.0250352	+	0.999687i	$0.492030\pi$
$662$	0	0
$663$	−5.95153e11	−0.119624
$664$	0	0
$665$	8.86468e11	0.175779
$666$	0	0
$667$	4.78970e11	0.0937005
$668$	0	0
$669$	−2.37755e12	−0.458895
$670$	0	0
$671$	1.15225e13	2.19429
$672$	0	0
$673$	−2.74599e11	−0.0515979	−0.0257989	−	0.999667i	$-0.508213\pi$
$673$	−2.74599e11	−0.0515979	−0.0257989	+	0.999667i	$0.508213\pi$
$674$	0	0
$675$	9.07417e11	0.168244
$676$	0	0
$677$	9.25777e12	1.69378	0.846890	−	0.531767i	$-0.178472\pi$
$677$	9.25777e12	1.69378	0.846890	+	0.531767i	$0.178472\pi$
$678$	0	0
$679$	2.39388e12	0.432204
$680$	0	0
$681$	−3.94097e11	−0.0702169
$682$	0	0
$683$	−1.60300e12	−0.281865	−0.140932	−	0.990019i	$-0.545010\pi$
$683$	−1.60300e12	−0.281865	−0.140932	+	0.990019i	$0.545010\pi$
$684$	0	0
$685$	6.51390e11	0.113040
$686$	0	0
$687$	−2.76115e11	−0.0472918
$688$	0	0
$689$	2.41955e11	0.0409024
$690$	0	0
$691$	1.63810e12	0.273331	0.136665	−	0.990617i	$-0.456362\pi$
$691$	1.63810e12	0.273331	0.136665	+	0.990617i	$0.456362\pi$
$692$	0	0
$693$	6.56814e12	1.08179
$694$	0	0
$695$	−9.00509e11	−0.146405
$696$	0	0
$697$	−1.13289e13	−1.81819
$698$	0	0
$699$	−4.65493e12	−0.737507
$700$	0	0
$701$	−3.87066e11	−0.0605416	−0.0302708	−	0.999542i	$-0.509637\pi$
$701$	−3.87066e11	−0.0605416	−0.0302708	+	0.999542i	$0.509637\pi$
$702$	0	0
$703$	−9.63377e11	−0.148764
$704$	0	0
$705$	5.22728e11	0.0796939
$706$	0	0
$707$	−7.73811e12	−1.16479
$708$	0	0
$709$	6.64111e11	0.0987035	0.0493518	−	0.998781i	$-0.484284\pi$
$709$	6.64111e11	0.0987035	0.0493518	+	0.998781i	$0.484284\pi$
$710$	0	0
$711$	9.15640e12	1.34373
$712$	0	0
$713$	5.23792e11	0.0759026
$714$	0	0
$715$	−6.58523e11	−0.0942310
$716$	0	0
$717$	−5.39785e12	−0.762755
$718$	0	0
$719$	5.51431e12	0.769505	0.384752	−	0.923020i	$-0.374287\pi$
$719$	5.51431e12	0.769505	0.384752	+	0.923020i	$0.374287\pi$
$720$	0	0
$721$	1.45810e12	0.200946
$722$	0	0
$723$	5.14437e12	0.700180
$724$	0	0
$725$	−2.04338e12	−0.274680
$726$	0	0
$727$	1.24897e13	1.65824	0.829121	−	0.559070i	$-0.188842\pi$
$727$	1.24897e13	1.65824	0.829121	+	0.559070i	$0.188842\pi$
$728$	0	0
$729$	8.09935e11	0.106213
$730$	0	0
$731$	−1.67615e13	−2.17112
$732$	0	0
$733$	1.43648e12	0.183794	0.0918970	−	0.995769i	$-0.470707\pi$
$733$	1.43648e12	0.183794	0.0918970	+	0.995769i	$0.470707\pi$
$734$	0	0
$735$	2.06846e11	0.0261430
$736$	0	0
$737$	5.54212e12	0.691946
$738$	0	0
$739$	−1.36179e13	−1.67962	−0.839810	−	0.542881i	$-0.817333\pi$
$739$	−1.36179e13	−1.67962	−0.839810	+	0.542881i	$0.817333\pi$
$740$	0	0
$741$	−2.30860e11	−0.0281298
$742$	0	0
$743$	1.01703e10	0.00122429	0.000612145	−	1.00000i	$-0.499805\pi$
$743$	1.01703e10	0.00122429	0.000612145		1.00000i	$0.499805\pi$
$744$	0	0
$745$	3.30984e12	0.393644
$746$	0	0
$747$	9.37056e12	1.10109
$748$	0	0
$749$	−5.99317e12	−0.695807
$750$	0	0
$751$	−9.04893e12	−1.03805	−0.519024	−	0.854760i	$-0.673705\pi$
$751$	−9.04893e12	−1.03805	−0.519024	+	0.854760i	$0.673705\pi$
$752$	0	0
$753$	−4.22001e11	−0.0478339
$754$	0	0
$755$	−2.25077e12	−0.252098
$756$	0	0
$757$	3.89773e12	0.431400	0.215700	−	0.976460i	$-0.430797\pi$
$757$	3.89773e12	0.431400	0.215700	+	0.976460i	$0.430797\pi$
$758$	0	0
$759$	−4.40308e11	−0.0481580
$760$	0	0
$761$	−9.90923e12	−1.07105	−0.535524	−	0.844520i	$-0.679886\pi$
$761$	−9.90923e12	−1.07105	−0.535524	+	0.844520i	$0.679886\pi$
$762$	0	0
$763$	4.56780e12	0.487918
$764$	0	0
$765$	−5.86524e12	−0.619169
$766$	0	0
$767$	1.11510e12	0.116342
$768$	0	0
$769$	1.15295e13	1.18889	0.594444	−	0.804137i	$-0.297372\pi$
$769$	1.15295e13	1.18889	0.594444	+	0.804137i	$0.297372\pi$
$770$	0	0
$771$	−2.30765e12	−0.235193
$772$	0	0
$773$	−9.67555e12	−0.974693	−0.487347	−	0.873209i	$-0.662035\pi$
$773$	−9.67555e12	−0.974693	−0.487347	+	0.873209i	$0.662035\pi$
$774$	0	0
$775$	−2.23460e12	−0.222506
$776$	0	0
$777$	1.59711e12	0.157195
$778$	0	0
$779$	−4.39446e12	−0.427550
$780$	0	0
$781$	−1.46861e13	−1.41246
$782$	0	0
$783$	−1.21517e13	−1.15534
$784$	0	0
$785$	1.88620e12	0.177286
$786$	0	0
$787$	−2.10996e13	−1.96060	−0.980298	−	0.197525i	$-0.936710\pi$
$787$	−2.10996e13	−1.96060	−0.980298	+	0.197525i	$0.936710\pi$
$788$	0	0
$789$	−4.96962e12	−0.456538
$790$	0	0
$791$	5.36460e12	0.487241
$792$	0	0
$793$	−2.31963e12	−0.208300
$794$	0	0
$795$	−6.90179e11	−0.0612787
$796$	0	0
$797$	6.35783e12	0.558144	0.279072	−	0.960270i	$-0.409973\pi$
$797$	6.35783e12	0.558144	0.279072	+	0.960270i	$0.409973\pi$
$798$	0	0
$799$	−7.73545e12	−0.671467
$800$	0	0
$801$	1.52975e11	0.0131303
$802$	0	0
$803$	1.59220e13	1.35138
$804$	0	0
$805$	−3.40365e11	−0.0285669
$806$	0	0
$807$	−5.27790e12	−0.438056
$808$	0	0
$809$	−8.03990e12	−0.659906	−0.329953	−	0.943997i	$-0.607033\pi$
$809$	−8.03990e12	−0.659906	−0.329953	+	0.943997i	$0.607033\pi$
$810$	0	0
$811$	−8.17244e12	−0.663373	−0.331687	−	0.943390i	$-0.607618\pi$
$811$	−8.17244e12	−0.663373	−0.331687	+	0.943390i	$0.607618\pi$
$812$	0	0
$813$	−7.16656e12	−0.575312
$814$	0	0
$815$	5.63876e12	0.447687
$816$	0	0
$817$	−6.50177e12	−0.510543
$818$	0	0
$819$	−1.32226e12	−0.102692
$820$	0	0
$821$	2.18882e13	1.68138	0.840688	−	0.541519i	$-0.182151\pi$
$821$	2.18882e13	1.68138	0.840688	+	0.541519i	$0.182151\pi$
$822$	0	0
$823$	7.90628e12	0.600721	0.300360	−	0.953826i	$-0.402893\pi$
$823$	7.90628e12	0.600721	0.300360	+	0.953826i	$0.402893\pi$
$824$	0	0
$825$	1.87844e12	0.141174
$826$	0	0
$827$	−1.23397e13	−0.917340	−0.458670	−	0.888607i	$-0.651674\pi$
$827$	−1.23397e13	−0.917340	−0.458670	+	0.888607i	$0.651674\pi$
$828$	0	0
$829$	−1.05174e13	−0.773415	−0.386707	−	0.922203i	$-0.626388\pi$
$829$	−1.05174e13	−0.773415	−0.386707	+	0.922203i	$0.626388\pi$
$830$	0	0
$831$	9.82463e12	0.714680
$832$	0	0
$833$	−3.06096e12	−0.220270
$834$	0	0
$835$	1.57588e12	0.112185
$836$	0	0
$837$	−1.32888e13	−0.935885
$838$	0	0
$839$	2.42354e13	1.68858	0.844288	−	0.535889i	$-0.180024\pi$
$839$	2.42354e13	1.68858	0.844288	+	0.535889i	$0.180024\pi$
$840$	0	0
$841$	1.28567e13	0.886234
$842$	0	0
$843$	−5.12550e12	−0.349552
$844$	0	0
$845$	−6.49524e12	−0.438268
$846$	0	0
$847$	1.71046e13	1.14193
$848$	0	0
$849$	−3.45920e12	−0.228502
$850$	0	0
$851$	3.69895e11	0.0241766
$852$	0	0
$853$	1.18654e13	0.767381	0.383690	−	0.923462i	$-0.374653\pi$
$853$	1.18654e13	0.767381	0.383690	+	0.923462i	$0.374653\pi$
$854$	0	0
$855$	−2.27512e12	−0.145599
$856$	0	0
$857$	−2.22495e13	−1.40899	−0.704494	−	0.709710i	$-0.748826\pi$
$857$	−2.22495e13	−1.40899	−0.704494	+	0.709710i	$0.748826\pi$
$858$	0	0
$859$	1.23035e13	0.771006	0.385503	−	0.922707i	$-0.374028\pi$
$859$	1.23035e13	0.771006	0.385503	+	0.922707i	$0.374028\pi$
$860$	0	0
$861$	7.28523e12	0.451782
$862$	0	0
$863$	−1.67121e13	−1.02561	−0.512806	−	0.858505i	$-0.671394\pi$
$863$	−1.67121e13	−1.02561	−0.512806	+	0.858505i	$0.671394\pi$
$864$	0	0
$865$	−6.85053e12	−0.416056
$866$	0	0
$867$	−1.72400e13	−1.03622
$868$	0	0
$869$	4.33956e13	2.58141
$870$	0	0
$871$	−1.11570e12	−0.0656852
$872$	0	0
$873$	−6.14390e12	−0.357998
$874$	0	0
$875$	1.45207e12	0.0837433
$876$	0	0
$877$	1.11566e13	0.636845	0.318423	−	0.947949i	$-0.396847\pi$
$877$	1.11566e13	0.636845	0.318423	+	0.947949i	$0.396847\pi$
$878$	0	0
$879$	−5.72858e12	−0.323666
$880$	0	0
$881$	−6.45233e12	−0.360848	−0.180424	−	0.983589i	$-0.557747\pi$
$881$	−6.45233e12	−0.360848	−0.180424	+	0.983589i	$0.557747\pi$
$882$	0	0
$883$	1.06381e13	0.588898	0.294449	−	0.955667i	$-0.404864\pi$
$883$	1.06381e13	0.588898	0.294449	+	0.955667i	$0.404864\pi$
$884$	0	0
$885$	−3.18084e12	−0.174300
$886$	0	0
$887$	−2.10257e13	−1.14050	−0.570250	−	0.821471i	$-0.693154\pi$
$887$	−2.10257e13	−1.14050	−0.570250	+	0.821471i	$0.693154\pi$
$888$	0	0
$889$	7.56931e12	0.406442
$890$	0	0
$891$	−1.05656e13	−0.561622
$892$	0	0
$893$	−3.00058e12	−0.157897
$894$	0	0
$895$	−4.04111e12	−0.210522
$896$	0	0
$897$	8.86400e10	0.00457156
$898$	0	0
$899$	2.99246e13	1.52795
$900$	0	0
$901$	1.02134e13	0.516309
$902$	0	0
$903$	1.07788e13	0.539479
$904$	0	0
$905$	−9.08369e10	−0.00450136
$906$	0	0
$907$	8.98297e12	0.440745	0.220372	−	0.975416i	$-0.429273\pi$
$907$	8.98297e12	0.440745	0.220372	+	0.975416i	$0.429273\pi$
$908$	0	0
$909$	1.98599e13	0.964803
$910$	0	0
$911$	2.32715e13	1.11942	0.559709	−	0.828689i	$-0.310913\pi$
$911$	2.32715e13	1.11942	0.559709	+	0.828689i	$0.310913\pi$
$912$	0	0
$913$	4.44106e13	2.11528
$914$	0	0
$915$	6.61677e12	0.312069
$916$	0	0
$917$	−2.90874e12	−0.135845
$918$	0	0
$919$	1.25073e13	0.578420	0.289210	−	0.957266i	$-0.406607\pi$
$919$	1.25073e13	0.578420	0.289210	+	0.957266i	$0.406607\pi$
$920$	0	0
$921$	1.89977e13	0.870027
$922$	0	0
$923$	2.95651e12	0.134082
$924$	0	0
$925$	−1.57804e12	−0.0708731
$926$	0	0
$927$	−3.74223e12	−0.166445
$928$	0	0
$929$	7.76069e12	0.341845	0.170923	−	0.985284i	$-0.445325\pi$
$929$	7.76069e12	0.341845	0.170923	+	0.985284i	$0.445325\pi$
$930$	0	0
$931$	−1.18734e12	−0.0517968
$932$	0	0
$933$	6.04925e12	0.261357
$934$	0	0
$935$	−2.77976e13	−1.18947
$936$	0	0
$937$	−3.00977e13	−1.27557	−0.637786	−	0.770214i	$-0.720149\pi$
$937$	−3.00977e13	−1.27557	−0.637786	+	0.770214i	$0.720149\pi$
$938$	0	0
$939$	7.34293e12	0.308229
$940$	0	0
$941$	−4.73101e13	−1.96698	−0.983492	−	0.180950i	$-0.942083\pi$
$941$	−4.73101e13	−1.96698	−0.983492	+	0.180950i	$0.942083\pi$
$942$	0	0
$943$	1.68728e12	0.0694840
$944$	0	0
$945$	8.63522e12	0.352233
$946$	0	0
$947$	5.15925e12	0.208455	0.104227	−	0.994553i	$-0.466763\pi$
$947$	5.15925e12	0.208455	0.104227	+	0.994553i	$0.466763\pi$
$948$	0	0
$949$	−3.20531e12	−0.128284
$950$	0	0
$951$	9.39433e12	0.372438
$952$	0	0
$953$	−2.89086e13	−1.13530	−0.567648	−	0.823272i	$-0.692146\pi$
$953$	−2.89086e13	−1.13530	−0.567648	+	0.823272i	$0.692146\pi$
$954$	0	0
$955$	1.12861e13	0.439063
$956$	0	0
$957$	−2.51551e13	−0.969444
$958$	0	0
$959$	6.19879e12	0.236659
$960$	0	0
$961$	6.28542e12	0.237727
$962$	0	0
$963$	1.53815e13	0.576342
$964$	0	0
$965$	2.27282e13	0.843709
$966$	0	0
$967$	−2.34982e13	−0.864204	−0.432102	−	0.901825i	$-0.642228\pi$
$967$	−2.34982e13	−0.864204	−0.432102	+	0.901825i	$0.642228\pi$
$968$	0	0
$969$	−9.74505e12	−0.355081
$970$	0	0
$971$	−2.08721e13	−0.753493	−0.376747	−	0.926316i	$-0.622957\pi$
$971$	−2.08721e13	−0.753493	−0.376747	+	0.926316i	$0.622957\pi$
$972$	0	0
$973$	−8.56948e12	−0.306511
$974$	0	0
$975$	−3.78156e11	−0.0134014
$976$	0	0
$977$	3.84347e13	1.34958	0.674789	−	0.738011i	$-0.264235\pi$
$977$	3.84347e13	1.34958	0.674789	+	0.738011i	$0.264235\pi$
$978$	0	0
$979$	7.25008e11	0.0252244
$980$	0	0
$981$	−1.17233e13	−0.404146
$982$	0	0
$983$	−3.82774e13	−1.30753	−0.653764	−	0.756698i	$-0.726811\pi$
$983$	−3.82774e13	−1.30753	−0.653764	+	0.756698i	$0.726811\pi$
$984$	0	0
$985$	−2.41230e13	−0.816524
$986$	0	0
$987$	4.97442e12	0.166846
$988$	0	0
$989$	2.49640e12	0.0829718
$990$	0	0
$991$	5.18610e13	1.70808	0.854042	−	0.520204i	$-0.174144\pi$
$991$	5.18610e13	1.70808	0.854042	+	0.520204i	$0.174144\pi$
$992$	0	0
$993$	1.31214e13	0.428261
$994$	0	0
$995$	−1.73844e11	−0.00562285
$996$	0	0
$997$	2.46318e13	0.789529	0.394765	−	0.918782i	$-0.370826\pi$
$997$	2.46318e13	0.789529	0.394765	+	0.918782i	$0.370826\pi$
$998$	0	0
$999$	−9.38439e12	−0.298100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.10.a.f.1.2		2
4.3	odd	2		5.10.a.b.1.2	✓	2
5.2	odd	4		400.10.c.p.49.3		4
5.3	odd	4		400.10.c.p.49.2		4
5.4	even	2		400.10.a.t.1.1		2
8.3	odd	2		320.10.a.k.1.2		2
8.5	even	2		320.10.a.s.1.1		2
12.11	even	2		45.10.a.f.1.1		2
20.3	even	4		25.10.b.b.24.2		4
20.7	even	4		25.10.b.b.24.3		4
20.19	odd	2		25.10.a.b.1.1		2
28.27	even	2		245.10.a.d.1.2		2
60.23	odd	4		225.10.b.h.199.3		4
60.47	odd	4		225.10.b.h.199.2		4
60.59	even	2		225.10.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.10.a.b.1.2	✓	2	4.3	odd	2
25.10.a.b.1.1		2	20.19	odd	2
25.10.b.b.24.2		4	20.3	even	4
25.10.b.b.24.3		4	20.7	even	4
45.10.a.f.1.1		2	12.11	even	2
80.10.a.f.1.2		2	1.1	even	1	trivial
225.10.a.h.1.2		2	60.59	even	2
225.10.b.h.199.2		4	60.47	odd	4
225.10.b.h.199.3		4	60.23	odd	4
245.10.a.d.1.2		2	28.27	even	2
320.10.a.k.1.2		2	8.3	odd	2
320.10.a.s.1.1		2	8.5	even	2
400.10.a.t.1.1		2	5.4	even	2
400.10.c.p.49.2		4	5.3	odd	4
400.10.c.p.49.3		4	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)

\(f(q)\)	\(=\)	\(q-66.4705 q^{3} +625.000 q^{5} +5947.66 q^{7} -15264.7 q^{9} +O(q^{10})\)	\(q-66.4705 q^{3} +625.000 q^{5} +5947.66 q^{7} -15264.7 q^{9} -72345.0 q^{11} +14564.0 q^{13} -41544.0 q^{15} +614778. q^{17} +238472. q^{19} -395344. q^{21} -91562.8 q^{23} +390625. q^{25} +2.32299e6 q^{27} -5.23105e6 q^{29} -5.72058e6 q^{31} +4.80881e6 q^{33} +3.71729e6 q^{35} -4.03979e6 q^{37} -968079. q^{39} -1.84276e7 q^{41} -2.72643e7 q^{43} -9.54042e6 q^{45} -1.25825e7 q^{47} -4.97896e6 q^{49} -4.08646e7 q^{51} +1.66132e7 q^{53} -4.52157e7 q^{55} -1.58513e7 q^{57} +7.65654e7 q^{59} -1.59271e8 q^{61} -9.07891e7 q^{63} +9.10253e6 q^{65} -7.66068e7 q^{67} +6.08622e6 q^{69} +2.03001e8 q^{71} -2.20084e8 q^{73} -2.59650e7 q^{75} -4.30284e8 q^{77} -5.99842e8 q^{79} +1.46044e8 q^{81} -6.13872e8 q^{83} +3.84236e8 q^{85} +3.47710e8 q^{87} -1.00215e7 q^{89} +8.66220e7 q^{91} +3.80250e8 q^{93} +1.49045e8 q^{95} +4.02491e8 q^{97} +1.10432e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9}+O(q^{10}) \) 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9	\( 2 q - 260 q^{3} + 1250 q^{5} - 1700 q^{7} + 2506 q^{9} - 23984 q^{11} + 115020 q^{13} - 162500 q^{15} + 412820 q^{17} + 296520 q^{19} + 1084704 q^{21} + 1049220 q^{23} + 781250 q^{25} + 2693080 q^{27} - 3666980 q^{29} - 1613144 q^{31} - 4550480 q^{33} - 1062500 q^{35} - 21121940 q^{37} - 20409272 q^{39} - 26957276 q^{41} - 52889700 q^{43} + 1566250 q^{45} - 58412180 q^{47} + 13154114 q^{49} - 1779784 q^{51} - 39035140 q^{53} - 14990000 q^{55} - 27085360 q^{57} + 54995560 q^{59} - 274579716 q^{61} - 226693140 q^{63} + 71887500 q^{65} + 318580 q^{67} - 214688928 q^{69} + 7130936 q^{71} + 120858180 q^{73} - 101562500 q^{75} - 800132400 q^{77} - 6877520 q^{79} - 275359358 q^{81} - 1402348740 q^{83} + 258012500 q^{85} + 45016840 q^{87} + 830088660 q^{89} - 681630904 q^{91} - 414660480 q^{93} + 185325000 q^{95} + 638394580 q^{97} + 1963732048 q^{99}+O(q^{100}) \) 2 * q - 260 * q^3 + 1250 * q^5 - 1700 * q^7 + 2506 * q^9 - 23984 * q^11 + 115020 * q^13 - 162500 * q^15 + 412820 * q^17 + 296520 * q^19 + 1084704 * q^21 + 1049220 * q^23 + 781250 * q^25 + 2693080 * q^27 - 3666980 * q^29 - 1613144 * q^31 - 4550480 * q^33 - 1062500 * q^35 - 21121940 * q^37 - 20409272 * q^39 - 26957276 * q^41 - 52889700 * q^43 + 1566250 * q^45 - 58412180 * q^47 + 13154114 * q^49 - 1779784 * q^51 - 39035140 * q^53 - 14990000 * q^55 - 27085360 * q^57 + 54995560 * q^59 - 274579716 * q^61 - 226693140 * q^63 + 71887500 * q^65 + 318580 * q^67 - 214688928 * q^69 + 7130936 * q^71 + 120858180 * q^73 - 101562500 * q^75 - 800132400 * q^77 - 6877520 * q^79 - 275359358 * q^81 - 1402348740 * q^83 + 258012500 * q^85 + 45016840 * q^87 + 830088660 * q^89 - 681630904 * q^91 - 414660480 * q^93 + 185325000 * q^95 + 638394580 * q^97 + 1963732048 * q^99

Introduction

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