LMFDB - Embedded newform 1280.4.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(1,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.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:	$75.5224448073$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 38x^{3} + 93x^{2} - 108x - 60 $ x^6 - 2x^5 - 21x^4 + 38x^3 + 93x^2 - 108*x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.90720$ of defining polynomial
Character	$\chi$	$=$	1280.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.51777 q^{3} -5.00000 q^{5} +5.13620 q^{7} -24.6964 q^{9} -31.3403 q^{11} +4.75340 q^{13} +7.58886 q^{15} +108.154 q^{17} +89.8913 q^{19} -7.79558 q^{21} -68.5157 q^{23} +25.0000 q^{25} +78.4633 q^{27} +16.5719 q^{29} +300.523 q^{31} +47.5675 q^{33} -25.6810 q^{35} -327.879 q^{37} -7.21458 q^{39} +73.4968 q^{41} -0.836008 q^{43} +123.482 q^{45} -228.335 q^{47} -316.619 q^{49} -164.153 q^{51} +647.393 q^{53} +156.702 q^{55} -136.434 q^{57} -753.676 q^{59} -290.838 q^{61} -126.845 q^{63} -23.7670 q^{65} -801.801 q^{67} +103.991 q^{69} +767.674 q^{71} +48.3194 q^{73} -37.9443 q^{75} -160.970 q^{77} -451.701 q^{79} +547.712 q^{81} -976.099 q^{83} -540.771 q^{85} -25.1524 q^{87} +1204.25 q^{89} +24.4144 q^{91} -456.126 q^{93} -449.456 q^{95} -559.147 q^{97} +773.992 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 30 q^{5} + 14 q^{7} + 54 q^{9} - 44 q^{11} + 30 q^{15} - 152 q^{19} - 4 q^{21} + 302 q^{23} + 150 q^{25} - 216 q^{27} + 132 q^{31} - 116 q^{33} - 70 q^{35} + 68 q^{37} + 300 q^{39} - 20 q^{41}+ \cdots - 5516 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 - 30 * q^5 + 14 * q^7 + 54 * q^9 - 44 * q^11 + 30 * q^15 - 152 * q^19 - 4 * q^21 + 302 * q^23 + 150 * q^25 - 216 * q^27 + 132 * q^31 - 116 * q^33 - 70 * q^35 + 68 * q^37 + 300 * q^39 - 20 * q^41 - 602 * q^43 - 270 * q^45 + 470 * q^47 + 654 * q^49 - 612 * q^51 - 528 * q^53 + 220 * q^55 + 340 * q^57 - 472 * q^59 + 476 * q^61 + 650 * q^63 - 1206 * q^67 + 980 * q^69 - 796 * q^71 - 216 * q^73 - 150 * q^75 - 412 * q^77 - 1008 * q^79 + 1254 * q^81 - 1778 * q^83 - 984 * q^87 + 212 * q^89 - 3652 * q^91 + 1392 * q^93 + 760 * q^95 - 792 * q^97 - 5516 * q^99

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
$3$	−1.51777	−0.292095	−0.146048	−	0.989278i	$-0.546655\pi$
$3$	−1.51777	−0.292095	−0.146048	+	0.989278i	$0.546655\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	5.13620	0.277328	0.138664	−	0.990339i	$-0.455719\pi$
$7$	5.13620	0.277328	0.138664	+	0.990339i	$0.455719\pi$
$8$	0	0
$9$	−24.6964	−0.914680
$10$	0	0
$11$	−31.3403	−0.859042	−0.429521	−	0.903057i	$-0.641318\pi$
$11$	−31.3403	−0.859042	−0.429521	+	0.903057i	$0.641318\pi$
$12$	0	0
$13$	4.75340	0.101412	0.0507060	−	0.998714i	$-0.483853\pi$
$13$	4.75340	0.101412	0.0507060	+	0.998714i	$0.483853\pi$
$14$	0	0
$15$	7.58886	0.130629
$16$	0	0
$17$	108.154	1.54301	0.771507	−	0.636221i	$-0.219503\pi$
$17$	108.154	1.54301	0.771507	+	0.636221i	$0.219503\pi$
$18$	0	0
$19$	89.8913	1.08539	0.542697	−	0.839929i	$-0.317403\pi$
$19$	89.8913	1.08539	0.542697	+	0.839929i	$0.317403\pi$
$20$	0	0
$21$	−7.79558	−0.0810064
$22$	0	0
$23$	−68.5157	−0.621152	−0.310576	−	0.950548i	$-0.600522\pi$
$23$	−68.5157	−0.621152	−0.310576	+	0.950548i	$0.600522\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	78.4633	0.559269
$28$	0	0
$29$	16.5719	0.106115	0.0530573	−	0.998591i	$-0.483103\pi$
$29$	16.5719	0.106115	0.0530573	+	0.998591i	$0.483103\pi$
$30$	0	0
$31$	300.523	1.74115	0.870574	−	0.492037i	$-0.163748\pi$
$31$	300.523	1.74115	0.870574	+	0.492037i	$0.163748\pi$
$32$	0	0
$33$	47.5675	0.250922
$34$	0	0
$35$	−25.6810	−0.124025
$36$	0	0
$37$	−327.879	−1.45684	−0.728419	−	0.685132i	$-0.759744\pi$
$37$	−327.879	−1.45684	−0.728419	+	0.685132i	$0.759744\pi$
$38$	0	0
$39$	−7.21458	−0.0296220
$40$	0	0
$41$	73.4968	0.279958	0.139979	−	0.990154i	$-0.455297\pi$
$41$	73.4968	0.279958	0.139979	+	0.990154i	$0.455297\pi$
$42$	0	0
$43$	−0.836008	−0.00296489	−0.00148244	−	0.999999i	$-0.500472\pi$
$43$	−0.836008	−0.00296489	−0.00148244	+	0.999999i	$0.500472\pi$
$44$	0	0
$45$	123.482	0.409057
$46$	0	0
$47$	−228.335	−0.708639	−0.354319	−	0.935124i	$-0.615287\pi$
$47$	−228.335	−0.708639	−0.354319	+	0.935124i	$0.615287\pi$
$48$	0	0
$49$	−316.619	−0.923089
$50$	0	0
$51$	−164.153	−0.450707
$52$	0	0
$53$	647.393	1.67785	0.838927	−	0.544244i	$-0.183183\pi$
$53$	647.393	1.67785	0.838927	+	0.544244i	$0.183183\pi$
$54$	0	0
$55$	156.702	0.384175
$56$	0	0
$57$	−136.434	−0.317038
$58$	0	0
$59$	−753.676	−1.66305	−0.831527	−	0.555484i	$-0.812533\pi$
$59$	−753.676	−1.66305	−0.831527	+	0.555484i	$0.812533\pi$
$60$	0	0
$61$	−290.838	−0.610459	−0.305229	−	0.952279i	$-0.598733\pi$
$61$	−290.838	−0.610459	−0.305229	+	0.952279i	$0.598733\pi$
$62$	0	0
$63$	−126.845	−0.253667
$64$	0	0
$65$	−23.7670	−0.0453528
$66$	0	0
$67$	−801.801	−1.46202	−0.731012	−	0.682364i	$-0.760952\pi$
$67$	−801.801	−1.46202	−0.731012	+	0.682364i	$0.760952\pi$
$68$	0	0
$69$	103.991	0.181436
$70$	0	0
$71$	767.674	1.28318	0.641592	−	0.767046i	$-0.278274\pi$
$71$	767.674	1.28318	0.641592	+	0.767046i	$0.278274\pi$
$72$	0	0
$73$	48.3194	0.0774707	0.0387353	−	0.999250i	$-0.487667\pi$
$73$	48.3194	0.0774707	0.0387353	+	0.999250i	$0.487667\pi$
$74$	0	0
$75$	−37.9443	−0.0584191
$76$	0	0
$77$	−160.970	−0.238237
$78$	0	0
$79$	−451.701	−0.643296	−0.321648	−	0.946859i	$-0.604237\pi$
$79$	−451.701	−0.643296	−0.321648	+	0.946859i	$0.604237\pi$
$80$	0	0
$81$	547.712	0.751320
$82$	0	0
$83$	−976.099	−1.29085	−0.645427	−	0.763822i	$-0.723320\pi$
$83$	−976.099	−1.29085	−0.645427	+	0.763822i	$0.723320\pi$
$84$	0	0
$85$	−540.771	−0.690057
$86$	0	0
$87$	−25.1524	−0.0309956
$88$	0	0
$89$	1204.25	1.43428	0.717139	−	0.696930i	$-0.245451\pi$
$89$	1204.25	1.43428	0.717139	+	0.696930i	$0.245451\pi$
$90$	0	0
$91$	24.4144	0.0281244
$92$	0	0
$93$	−456.126	−0.508581
$94$	0	0
$95$	−449.456	−0.485403
$96$	0	0
$97$	−559.147	−0.585287	−0.292643	−	0.956222i	$-0.594535\pi$
$97$	−559.147	−0.585287	−0.292643	+	0.956222i	$0.594535\pi$
$98$	0	0
$99$	773.992	0.785749
$100$	0	0
$101$	542.070	0.534039	0.267020	−	0.963691i	$-0.413961\pi$
$101$	542.070	0.534039	0.267020	+	0.963691i	$0.413961\pi$
$102$	0	0
$103$	−1764.97	−1.68842	−0.844212	−	0.536009i	$-0.819931\pi$
$103$	−1764.97	−1.68842	−0.844212	+	0.536009i	$0.819931\pi$
$104$	0	0
$105$	38.9779	0.0362272
$106$	0	0
$107$	−442.840	−0.400103	−0.200051	−	0.979785i	$-0.564111\pi$
$107$	−442.840	−0.400103	−0.200051	+	0.979785i	$0.564111\pi$
$108$	0	0
$109$	1547.20	1.35959	0.679795	−	0.733403i	$-0.262069\pi$
$109$	1547.20	1.35959	0.679795	+	0.733403i	$0.262069\pi$
$110$	0	0
$111$	497.646	0.425536
$112$	0	0
$113$	788.524	0.656444	0.328222	−	0.944601i	$-0.393551\pi$
$113$	788.524	0.656444	0.328222	+	0.944601i	$0.393551\pi$
$114$	0	0
$115$	342.578	0.277788
$116$	0	0
$117$	−117.392	−0.0927596
$118$	0	0
$119$	555.501	0.427922
$120$	0	0
$121$	−348.785	−0.262047
$122$	0	0
$123$	−111.551	−0.0817744
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1543.92	1.07875	0.539374	−	0.842066i	$-0.318661\pi$
$127$	1543.92	1.07875	0.539374	+	0.842066i	$0.318661\pi$
$128$	0	0
$129$	1.26887	0.000866029 0
$130$	0	0
$131$	−1933.26	−1.28938	−0.644692	−	0.764442i	$-0.723014\pi$
$131$	−1933.26	−1.28938	−0.644692	+	0.764442i	$0.723014\pi$
$132$	0	0
$133$	461.699	0.301010
$134$	0	0
$135$	−392.317	−0.250113
$136$	0	0
$137$	−478.247	−0.298244	−0.149122	−	0.988819i	$-0.547645\pi$
$137$	−478.247	−0.298244	−0.149122	+	0.988819i	$0.547645\pi$
$138$	0	0
$139$	−2057.45	−1.25547	−0.627735	−	0.778427i	$-0.716018\pi$
$139$	−2057.45	−1.25547	−0.627735	+	0.778427i	$0.716018\pi$
$140$	0	0
$141$	346.560	0.206990
$142$	0	0
$143$	−148.973	−0.0871172
$144$	0	0
$145$	−82.8595	−0.0474559
$146$	0	0
$147$	480.556	0.269630
$148$	0	0
$149$	−2838.89	−1.56088	−0.780440	−	0.625231i	$-0.785005\pi$
$149$	−2838.89	−1.56088	−0.780440	+	0.625231i	$0.785005\pi$
$150$	0	0
$151$	−2187.09	−1.17869	−0.589346	−	0.807881i	$-0.700615\pi$
$151$	−2187.09	−1.17869	−0.589346	+	0.807881i	$0.700615\pi$
$152$	0	0
$153$	−2671.02	−1.41136
$154$	0	0
$155$	−1502.62	−0.778665
$156$	0	0
$157$	−936.231	−0.475920	−0.237960	−	0.971275i	$-0.576479\pi$
$157$	−936.231	−0.475920	−0.237960	+	0.971275i	$0.576479\pi$
$158$	0	0
$159$	−982.595	−0.490093
$160$	0	0
$161$	−351.910	−0.172263
$162$	0	0
$163$	−2329.68	−1.11948	−0.559738	−	0.828670i	$-0.689098\pi$
$163$	−2329.68	−1.11948	−0.559738	+	0.828670i	$0.689098\pi$
$164$	0	0
$165$	−237.837	−0.112216
$166$	0	0
$167$	2020.42	0.936195	0.468097	−	0.883677i	$-0.344940\pi$
$167$	2020.42	0.936195	0.468097	+	0.883677i	$0.344940\pi$
$168$	0	0
$169$	−2174.41	−0.989716
$170$	0	0
$171$	−2219.99	−0.992788
$172$	0	0
$173$	−912.153	−0.400866	−0.200433	−	0.979707i	$-0.564235\pi$
$173$	−912.153	−0.400866	−0.200433	+	0.979707i	$0.564235\pi$
$174$	0	0
$175$	128.405	0.0554657
$176$	0	0
$177$	1143.91	0.485771
$178$	0	0
$179$	−3226.95	−1.34745	−0.673726	−	0.738982i	$-0.735307\pi$
$179$	−3226.95	−1.34745	−0.673726	+	0.738982i	$0.735307\pi$
$180$	0	0
$181$	1003.43	0.412068	0.206034	−	0.978545i	$-0.433944\pi$
$181$	1003.43	0.412068	0.206034	+	0.978545i	$0.433944\pi$
$182$	0	0
$183$	441.426	0.178312
$184$	0	0
$185$	1639.40	0.651517
$186$	0	0
$187$	−3389.59	−1.32551
$188$	0	0
$189$	403.003	0.155101
$190$	0	0
$191$	−1272.08	−0.481907	−0.240953	−	0.970537i	$-0.577460\pi$
$191$	−1272.08	−0.481907	−0.240953	+	0.970537i	$0.577460\pi$
$192$	0	0
$193$	−730.914	−0.272603	−0.136301	−	0.990667i	$-0.543522\pi$
$193$	−730.914	−0.272603	−0.136301	+	0.990667i	$0.543522\pi$
$194$	0	0
$195$	36.0729	0.0132474
$196$	0	0
$197$	3582.02	1.29547	0.647737	−	0.761864i	$-0.275716\pi$
$197$	3582.02	1.29547	0.647737	+	0.761864i	$0.275716\pi$
$198$	0	0
$199$	2007.64	0.715166	0.357583	−	0.933881i	$-0.383601\pi$
$199$	2007.64	0.715166	0.357583	+	0.933881i	$0.383601\pi$
$200$	0	0
$201$	1216.95	0.427051
$202$	0	0
$203$	85.1165	0.0294286
$204$	0	0
$205$	−367.484	−0.125201
$206$	0	0
$207$	1692.09	0.568156
$208$	0	0
$209$	−2817.22	−0.932398
$210$	0	0
$211$	−1414.79	−0.461603	−0.230802	−	0.973001i	$-0.574135\pi$
$211$	−1414.79	−0.461603	−0.230802	+	0.973001i	$0.574135\pi$
$212$	0	0
$213$	−1165.15	−0.374812
$214$	0	0
$215$	4.18004	0.00132594
$216$	0	0
$217$	1543.55	0.482870
$218$	0	0
$219$	−73.3379	−0.0226288
$220$	0	0
$221$	514.100	0.156480
$222$	0	0
$223$	−1.69909	−0.000510221 0	−0.000255111	−	1.00000i	$-0.500081\pi$
$223$	−1.69909	−0.000510221 0	−0.000255111		1.00000i	$0.500081\pi$
$224$	0	0
$225$	−617.409	−0.182936
$226$	0	0
$227$	−3374.87	−0.986775	−0.493387	−	0.869810i	$-0.664241\pi$
$227$	−3374.87	−0.986775	−0.493387	+	0.869810i	$0.664241\pi$
$228$	0	0
$229$	−1330.69	−0.383994	−0.191997	−	0.981396i	$-0.561496\pi$
$229$	−1330.69	−0.383994	−0.191997	+	0.981396i	$0.561496\pi$
$230$	0	0
$231$	244.316	0.0695879
$232$	0	0
$233$	4373.02	1.22955	0.614776	−	0.788701i	$-0.289246\pi$
$233$	4373.02	1.22955	0.614776	+	0.788701i	$0.289246\pi$
$234$	0	0
$235$	1141.67	0.316913
$236$	0	0
$237$	685.580	0.187904
$238$	0	0
$239$	−794.613	−0.215060	−0.107530	−	0.994202i	$-0.534294\pi$
$239$	−794.613	−0.215060	−0.107530	+	0.994202i	$0.534294\pi$
$240$	0	0
$241$	617.471	0.165041	0.0825204	−	0.996589i	$-0.473703\pi$
$241$	617.471	0.165041	0.0825204	+	0.996589i	$0.473703\pi$
$242$	0	0
$243$	−2949.81	−0.778727
$244$	0	0
$245$	1583.10	0.412818
$246$	0	0
$247$	427.289	0.110072
$248$	0	0
$249$	1481.50	0.377052
$250$	0	0
$251$	907.026	0.228092	0.114046	−	0.993475i	$-0.463619\pi$
$251$	907.026	0.228092	0.114046	+	0.993475i	$0.463619\pi$
$252$	0	0
$253$	2147.30	0.533596
$254$	0	0
$255$	820.767	0.201563
$256$	0	0
$257$	−4350.70	−1.05599	−0.527994	−	0.849248i	$-0.677056\pi$
$257$	−4350.70	−1.05599	−0.527994	+	0.849248i	$0.677056\pi$
$258$	0	0
$259$	−1684.05	−0.404022
$260$	0	0
$261$	−409.266	−0.0970610
$262$	0	0
$263$	3606.18	0.845499	0.422750	−	0.906246i	$-0.361065\pi$
$263$	3606.18	0.845499	0.422750	+	0.906246i	$0.361065\pi$
$264$	0	0
$265$	−3236.96	−0.750359
$266$	0	0
$267$	−1827.78	−0.418946
$268$	0	0
$269$	−6501.85	−1.47370	−0.736849	−	0.676057i	$-0.763687\pi$
$269$	−6501.85	−1.47370	−0.736849	+	0.676057i	$0.763687\pi$
$270$	0	0
$271$	−1011.82	−0.226803	−0.113401	−	0.993549i	$-0.536175\pi$
$271$	−1011.82	−0.226803	−0.113401	+	0.993549i	$0.536175\pi$
$272$	0	0
$273$	−37.0555	−0.00821502
$274$	0	0
$275$	−783.508	−0.171808
$276$	0	0
$277$	−2618.50	−0.567979	−0.283990	−	0.958827i	$-0.591658\pi$
$277$	−2618.50	−0.567979	−0.283990	+	0.958827i	$0.591658\pi$
$278$	0	0
$279$	−7421.84	−1.59259
$280$	0	0
$281$	−2302.29	−0.488766	−0.244383	−	0.969679i	$-0.578585\pi$
$281$	−2302.29	−0.488766	−0.244383	+	0.969679i	$0.578585\pi$
$282$	0	0
$283$	−6591.42	−1.38452	−0.692260	−	0.721648i	$-0.743385\pi$
$283$	−6591.42	−1.38452	−0.692260	+	0.721648i	$0.743385\pi$
$284$	0	0
$285$	682.172	0.141784
$286$	0	0
$287$	377.494	0.0776403
$288$	0	0
$289$	6784.33	1.38089
$290$	0	0
$291$	848.658	0.170960
$292$	0	0
$293$	4765.54	0.950190	0.475095	−	0.879934i	$-0.342414\pi$
$293$	4765.54	0.950190	0.475095	+	0.879934i	$0.342414\pi$
$294$	0	0
$295$	3768.38	0.743741
$296$	0	0
$297$	−2459.06	−0.480436
$298$	0	0
$299$	−325.682	−0.0629923
$300$	0	0
$301$	−4.29390	−0.000822247 0
$302$	0	0
$303$	−822.739	−0.155990
$304$	0	0
$305$	1454.19	0.273005
$306$	0	0
$307$	117.958	0.0219290	0.0109645	−	0.999940i	$-0.496510\pi$
$307$	117.958	0.0219290	0.0109645	+	0.999940i	$0.496510\pi$
$308$	0	0
$309$	2678.82	0.493181
$310$	0	0
$311$	4085.06	0.744832	0.372416	−	0.928066i	$-0.378530\pi$
$311$	4085.06	0.744832	0.372416	+	0.928066i	$0.378530\pi$
$312$	0	0
$313$	−904.680	−0.163372	−0.0816861	−	0.996658i	$-0.526031\pi$
$313$	−904.680	−0.163372	−0.0816861	+	0.996658i	$0.526031\pi$
$314$	0	0
$315$	634.227	0.113443
$316$	0	0
$317$	5437.26	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$317$	5437.26	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$318$	0	0
$319$	−519.369	−0.0911569
$320$	0	0
$321$	672.131	0.116868
$322$	0	0
$323$	9722.12	1.67478
$324$	0	0
$325$	118.835	0.0202824
$326$	0	0
$327$	−2348.30	−0.397130
$328$	0	0
$329$	−1172.77	−0.196526
$330$	0	0
$331$	−5944.03	−0.987049	−0.493525	−	0.869732i	$-0.664292\pi$
$331$	−5944.03	−0.987049	−0.493525	+	0.869732i	$0.664292\pi$
$332$	0	0
$333$	8097.42	1.33254
$334$	0	0
$335$	4009.01	0.653837
$336$	0	0
$337$	5636.91	0.911164	0.455582	−	0.890194i	$-0.349431\pi$
$337$	5636.91	0.911164	0.455582	+	0.890194i	$0.349431\pi$
$338$	0	0
$339$	−1196.80	−0.191744
$340$	0	0
$341$	−9418.50	−1.49572
$342$	0	0
$343$	−3387.93	−0.533327
$344$	0	0
$345$	−519.956	−0.0811405
$346$	0	0
$347$	7761.12	1.20069	0.600344	−	0.799742i	$-0.295030\pi$
$347$	7761.12	1.20069	0.600344	+	0.799742i	$0.295030\pi$
$348$	0	0
$349$	−8709.62	−1.33586	−0.667930	−	0.744224i	$-0.732819\pi$
$349$	−8709.62	−1.33586	−0.667930	+	0.744224i	$0.732819\pi$
$350$	0	0
$351$	372.968	0.0567166
$352$	0	0
$353$	−986.323	−0.148716	−0.0743579	−	0.997232i	$-0.523691\pi$
$353$	−986.323	−0.148716	−0.0743579	+	0.997232i	$0.523691\pi$
$354$	0	0
$355$	−3838.37	−0.573858
$356$	0	0
$357$	−843.124	−0.124994
$358$	0	0
$359$	−11470.1	−1.68626	−0.843131	−	0.537708i	$-0.819290\pi$
$359$	−11470.1	−1.68626	−0.843131	+	0.537708i	$0.819290\pi$
$360$	0	0
$361$	1221.44	0.178078
$362$	0	0
$363$	529.376	0.0765428
$364$	0	0
$365$	−241.597	−0.0346459
$366$	0	0
$367$	−3330.42	−0.473697	−0.236848	−	0.971547i	$-0.576114\pi$
$367$	−3330.42	−0.473697	−0.236848	+	0.971547i	$0.576114\pi$
$368$	0	0
$369$	−1815.10	−0.256072
$370$	0	0
$371$	3325.14	0.465317
$372$	0	0
$373$	−9398.88	−1.30471	−0.652354	−	0.757915i	$-0.726218\pi$
$373$	−9398.88	−1.30471	−0.652354	+	0.757915i	$0.726218\pi$
$374$	0	0
$375$	189.722	0.0261258
$376$	0	0
$377$	78.7729	0.0107613
$378$	0	0
$379$	3840.58	0.520521	0.260260	−	0.965538i	$-0.416192\pi$
$379$	3840.58	0.520521	0.260260	+	0.965538i	$0.416192\pi$
$380$	0	0
$381$	−2343.32	−0.315097
$382$	0	0
$383$	2969.97	0.396236	0.198118	−	0.980178i	$-0.436517\pi$
$383$	2969.97	0.396236	0.198118	+	0.980178i	$0.436517\pi$
$384$	0	0
$385$	804.850	0.106543
$386$	0	0
$387$	20.6464	0.00271192
$388$	0	0
$389$	8349.10	1.08822	0.544108	−	0.839015i	$-0.316868\pi$
$389$	8349.10	1.08822	0.544108	+	0.839015i	$0.316868\pi$
$390$	0	0
$391$	−7410.26	−0.958447
$392$	0	0
$393$	2934.24	0.376623
$394$	0	0
$395$	2258.51	0.287691
$396$	0	0
$397$	4506.27	0.569680	0.284840	−	0.958575i	$-0.408060\pi$
$397$	4506.27	0.569680	0.284840	+	0.958575i	$0.408060\pi$
$398$	0	0
$399$	−700.754	−0.0879238
$400$	0	0
$401$	−7654.44	−0.953228	−0.476614	−	0.879113i	$-0.658136\pi$
$401$	−7654.44	−0.953228	−0.476614	+	0.879113i	$0.658136\pi$
$402$	0	0
$403$	1428.51	0.176573
$404$	0	0
$405$	−2738.56	−0.336001
$406$	0	0
$407$	10275.8	1.25148
$408$	0	0
$409$	−2603.53	−0.314758	−0.157379	−	0.987538i	$-0.550304\pi$
$409$	−2603.53	−0.314758	−0.157379	+	0.987538i	$0.550304\pi$
$410$	0	0
$411$	725.870	0.0871156
$412$	0	0
$413$	−3871.03	−0.461212
$414$	0	0
$415$	4880.50	0.577287
$416$	0	0
$417$	3122.74	0.366717
$418$	0	0
$419$	−525.993	−0.0613280	−0.0306640	−	0.999530i	$-0.509762\pi$
$419$	−525.993	−0.0613280	−0.0306640	+	0.999530i	$0.509762\pi$
$420$	0	0
$421$	15126.1	1.75107	0.875534	−	0.483157i	$-0.160510\pi$
$421$	15126.1	1.75107	0.875534	+	0.483157i	$0.160510\pi$
$422$	0	0
$423$	5639.03	0.648178
$424$	0	0
$425$	2703.86	0.308603
$426$	0	0
$427$	−1493.80	−0.169298
$428$	0	0
$429$	226.107	0.0254465
$430$	0	0
$431$	−4887.92	−0.546271	−0.273135	−	0.961976i	$-0.588061\pi$
$431$	−4887.92	−0.546271	−0.273135	+	0.961976i	$0.588061\pi$
$432$	0	0
$433$	−6944.15	−0.770704	−0.385352	−	0.922770i	$-0.625920\pi$
$433$	−6944.15	−0.770704	−0.385352	+	0.922770i	$0.625920\pi$
$434$	0	0
$435$	125.762	0.0138617
$436$	0	0
$437$	−6158.96	−0.674195
$438$	0	0
$439$	577.528	0.0627879	0.0313940	−	0.999507i	$-0.490005\pi$
$439$	577.528	0.0627879	0.0313940	+	0.999507i	$0.490005\pi$
$440$	0	0
$441$	7819.35	0.844331
$442$	0	0
$443$	−7039.42	−0.754973	−0.377486	−	0.926015i	$-0.623211\pi$
$443$	−7039.42	−0.754973	−0.377486	+	0.926015i	$0.623211\pi$
$444$	0	0
$445$	−6021.27	−0.641429
$446$	0	0
$447$	4308.79	0.455926
$448$	0	0
$449$	6396.99	0.672366	0.336183	−	0.941797i	$-0.390864\pi$
$449$	6396.99	0.672366	0.336183	+	0.941797i	$0.390864\pi$
$450$	0	0
$451$	−2303.41	−0.240496
$452$	0	0
$453$	3319.50	0.344291
$454$	0	0
$455$	−122.072	−0.0125776
$456$	0	0
$457$	7015.85	0.718135	0.359068	−	0.933312i	$-0.383095\pi$
$457$	7015.85	0.718135	0.359068	+	0.933312i	$0.383095\pi$
$458$	0	0
$459$	8486.14	0.862961
$460$	0	0
$461$	13374.1	1.35118	0.675588	−	0.737279i	$-0.263890\pi$
$461$	13374.1	1.35118	0.675588	+	0.737279i	$0.263890\pi$
$462$	0	0
$463$	−15414.3	−1.54722	−0.773611	−	0.633661i	$-0.781551\pi$
$463$	−15414.3	−1.54722	−0.773611	+	0.633661i	$0.781551\pi$
$464$	0	0
$465$	2280.63	0.227445
$466$	0	0
$467$	−1796.43	−0.178006	−0.0890032	−	0.996031i	$-0.528368\pi$
$467$	−1796.43	−0.178006	−0.0890032	+	0.996031i	$0.528368\pi$
$468$	0	0
$469$	−4118.21	−0.405461
$470$	0	0
$471$	1420.99	0.139014
$472$	0	0
$473$	26.2008	0.00254696
$474$	0	0
$475$	2247.28	0.217079
$476$	0	0
$477$	−15988.3	−1.53470
$478$	0	0
$479$	−11789.0	−1.12454	−0.562270	−	0.826954i	$-0.690072\pi$
$479$	−11789.0	−1.12454	−0.562270	+	0.826954i	$0.690072\pi$
$480$	0	0
$481$	−1558.54	−0.147741
$482$	0	0
$483$	534.119	0.0503173
$484$	0	0
$485$	2795.74	0.261748
$486$	0	0
$487$	18083.0	1.68258	0.841291	−	0.540582i	$-0.181796\pi$
$487$	18083.0	1.68258	0.841291	+	0.540582i	$0.181796\pi$
$488$	0	0
$489$	3535.92	0.326994
$490$	0	0
$491$	11259.2	1.03487	0.517435	−	0.855722i	$-0.326887\pi$
$491$	11259.2	1.03487	0.517435	+	0.855722i	$0.326887\pi$
$492$	0	0
$493$	1792.32	0.163736
$494$	0	0
$495$	−3869.96	−0.351397
$496$	0	0
$497$	3942.92	0.355864
$498$	0	0
$499$	4589.82	0.411761	0.205880	−	0.978577i	$-0.433994\pi$
$499$	4589.82	0.411761	0.205880	+	0.978577i	$0.433994\pi$
$500$	0	0
$501$	−3066.53	−0.273458
$502$	0	0
$503$	−5257.43	−0.466038	−0.233019	−	0.972472i	$-0.574860\pi$
$503$	−5257.43	−0.466038	−0.233019	+	0.972472i	$0.574860\pi$
$504$	0	0
$505$	−2710.35	−0.238830
$506$	0	0
$507$	3300.25	0.289091
$508$	0	0
$509$	2451.99	0.213522	0.106761	−	0.994285i	$-0.465952\pi$
$509$	2451.99	0.213522	0.106761	+	0.994285i	$0.465952\pi$
$510$	0	0
$511$	248.178	0.0214848
$512$	0	0
$513$	7053.17	0.607027
$514$	0	0
$515$	8824.85	0.755086
$516$	0	0
$517$	7156.08	0.608750
$518$	0	0
$519$	1384.44	0.117091
$520$	0	0
$521$	−10677.6	−0.897878	−0.448939	−	0.893562i	$-0.648198\pi$
$521$	−10677.6	−0.897878	−0.448939	+	0.893562i	$0.648198\pi$
$522$	0	0
$523$	19565.8	1.63585	0.817927	−	0.575323i	$-0.195124\pi$
$523$	19565.8	1.63585	0.817927	+	0.575323i	$0.195124\pi$
$524$	0	0
$525$	−194.889	−0.0162013
$526$	0	0
$527$	32502.9	2.68662
$528$	0	0
$529$	−7472.60	−0.614170
$530$	0	0
$531$	18613.1	1.52116
$532$	0	0
$533$	349.360	0.0283911
$534$	0	0
$535$	2214.20	0.178931
$536$	0	0
$537$	4897.78	0.393584
$538$	0	0
$539$	9922.95	0.792972
$540$	0	0
$541$	−3313.01	−0.263286	−0.131643	−	0.991297i	$-0.542025\pi$
$541$	−3313.01	−0.263286	−0.131643	+	0.991297i	$0.542025\pi$
$542$	0	0
$543$	−1522.98	−0.120363
$544$	0	0
$545$	−7736.02	−0.608027
$546$	0	0
$547$	25040.9	1.95735	0.978675	−	0.205414i	$-0.0658542\pi$
$547$	25040.9	1.95735	0.978675	+	0.205414i	$0.0658542\pi$
$548$	0	0
$549$	7182.64	0.558375
$550$	0	0
$551$	1489.67	0.115176
$552$	0	0
$553$	−2320.03	−0.178404
$554$	0	0
$555$	−2488.23	−0.190305
$556$	0	0
$557$	−2708.11	−0.206008	−0.103004	−	0.994681i	$-0.532845\pi$
$557$	−2708.11	−0.206008	−0.103004	+	0.994681i	$0.532845\pi$
$558$	0	0
$559$	−3.97388	−0.000300675 0
$560$	0	0
$561$	5144.62	0.387177
$562$	0	0
$563$	9374.67	0.701768	0.350884	−	0.936419i	$-0.385881\pi$
$563$	9374.67	0.701768	0.350884	+	0.936419i	$0.385881\pi$
$564$	0	0
$565$	−3942.62	−0.293571
$566$	0	0
$567$	2813.16	0.208363
$568$	0	0
$569$	−12092.5	−0.890938	−0.445469	−	0.895297i	$-0.646963\pi$
$569$	−12092.5	−0.890938	−0.445469	+	0.895297i	$0.646963\pi$
$570$	0	0
$571$	10847.4	0.795008	0.397504	−	0.917601i	$-0.369877\pi$
$571$	10847.4	0.795008	0.397504	+	0.917601i	$0.369877\pi$
$572$	0	0
$573$	1930.72	0.140763
$574$	0	0
$575$	−1712.89	−0.124230
$576$	0	0
$577$	−22325.8	−1.61080	−0.805402	−	0.592729i	$-0.798051\pi$
$577$	−22325.8	−1.61080	−0.805402	+	0.592729i	$0.798051\pi$
$578$	0	0
$579$	1109.36	0.0796261
$580$	0	0
$581$	−5013.44	−0.357990
$582$	0	0
$583$	−20289.5	−1.44135
$584$	0	0
$585$	586.959	0.0414833
$586$	0	0
$587$	−281.244	−0.0197754	−0.00988770	−	0.999951i	$-0.503147\pi$
$587$	−281.244	−0.0197754	−0.00988770	+	0.999951i	$0.503147\pi$
$588$	0	0
$589$	27014.4	1.88983
$590$	0	0
$591$	−5436.69	−0.378402
$592$	0	0
$593$	−934.658	−0.0647248	−0.0323624	−	0.999476i	$-0.510303\pi$
$593$	−934.658	−0.0647248	−0.0323624	+	0.999476i	$0.510303\pi$
$594$	0	0
$595$	−2777.51	−0.191372
$596$	0	0
$597$	−3047.14	−0.208897
$598$	0	0
$599$	−15278.2	−1.04215	−0.521077	−	0.853510i	$-0.674470\pi$
$599$	−15278.2	−1.04215	−0.521077	+	0.853510i	$0.674470\pi$
$600$	0	0
$601$	−10958.6	−0.743780	−0.371890	−	0.928277i	$-0.621290\pi$
$601$	−10958.6	−0.743780	−0.371890	+	0.928277i	$0.621290\pi$
$602$	0	0
$603$	19801.6	1.33728
$604$	0	0
$605$	1743.92	0.117191
$606$	0	0
$607$	24850.2	1.66167	0.830837	−	0.556515i	$-0.187862\pi$
$607$	24850.2	1.66167	0.830837	+	0.556515i	$0.187862\pi$
$608$	0	0
$609$	−129.187	−0.00859596
$610$	0	0
$611$	−1085.37	−0.0718645
$612$	0	0
$613$	3961.58	0.261022	0.130511	−	0.991447i	$-0.458338\pi$
$613$	3961.58	0.261022	0.130511	+	0.991447i	$0.458338\pi$
$614$	0	0
$615$	557.757	0.0365706
$616$	0	0
$617$	−20732.3	−1.35275	−0.676377	−	0.736555i	$-0.736451\pi$
$617$	−20732.3	−1.35275	−0.676377	+	0.736555i	$0.736451\pi$
$618$	0	0
$619$	−7801.96	−0.506603	−0.253301	−	0.967387i	$-0.581516\pi$
$619$	−7801.96	−0.506603	−0.253301	+	0.967387i	$0.581516\pi$
$620$	0	0
$621$	−5375.97	−0.347391
$622$	0	0
$623$	6185.29	0.397766
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4275.90	0.272349
$628$	0	0
$629$	−35461.5	−2.24792
$630$	0	0
$631$	−9359.85	−0.590507	−0.295253	−	0.955419i	$-0.595404\pi$
$631$	−9359.85	−0.590507	−0.295253	+	0.955419i	$0.595404\pi$
$632$	0	0
$633$	2147.33	0.134832
$634$	0	0
$635$	−7719.61	−0.482431
$636$	0	0
$637$	−1505.02	−0.0936123
$638$	0	0
$639$	−18958.8	−1.17370
$640$	0	0
$641$	19129.1	1.17871	0.589356	−	0.807873i	$-0.299381\pi$
$641$	19129.1	1.17871	0.589356	+	0.807873i	$0.299381\pi$
$642$	0	0
$643$	−11243.5	−0.689582	−0.344791	−	0.938680i	$-0.612050\pi$
$643$	−11243.5	−0.689582	−0.344791	+	0.938680i	$0.612050\pi$
$644$	0	0
$645$	−6.34435	−0.000387300 0
$646$	0	0
$647$	11887.6	0.722331	0.361166	−	0.932502i	$-0.382379\pi$
$647$	11887.6	0.722331	0.361166	+	0.932502i	$0.382379\pi$
$648$	0	0
$649$	23620.4	1.42863
$650$	0	0
$651$	−2342.75	−0.141044
$652$	0	0
$653$	−5327.46	−0.319264	−0.159632	−	0.987177i	$-0.551031\pi$
$653$	−5327.46	−0.319264	−0.159632	+	0.987177i	$0.551031\pi$
$654$	0	0
$655$	9666.28	0.576630
$656$	0	0
$657$	−1193.31	−0.0708609
$658$	0	0
$659$	26016.2	1.53786	0.768929	−	0.639334i	$-0.220790\pi$
$659$	26016.2	1.53786	0.768929	+	0.639334i	$0.220790\pi$
$660$	0	0
$661$	5961.64	0.350803	0.175401	−	0.984497i	$-0.443878\pi$
$661$	5961.64	0.350803	0.175401	+	0.984497i	$0.443878\pi$
$662$	0	0
$663$	−780.287	−0.0457071
$664$	0	0
$665$	−2308.50	−0.134616
$666$	0	0
$667$	−1135.43	−0.0659134
$668$	0	0
$669$	2.57883	0.000149033 0
$670$	0	0
$671$	9114.95	0.524410
$672$	0	0
$673$	17544.1	1.00487	0.502435	−	0.864615i	$-0.332438\pi$
$673$	17544.1	1.00487	0.502435	+	0.864615i	$0.332438\pi$
$674$	0	0
$675$	1961.58	0.111854
$676$	0	0
$677$	17000.0	0.965088	0.482544	−	0.875872i	$-0.339713\pi$
$677$	17000.0	0.965088	0.482544	+	0.875872i	$0.339713\pi$
$678$	0	0
$679$	−2871.89	−0.162317
$680$	0	0
$681$	5122.28	0.288232
$682$	0	0
$683$	−5386.68	−0.301780	−0.150890	−	0.988551i	$-0.548214\pi$
$683$	−5386.68	−0.301780	−0.150890	+	0.988551i	$0.548214\pi$
$684$	0	0
$685$	2391.23	0.133379
$686$	0	0
$687$	2019.69	0.112163
$688$	0	0
$689$	3077.32	0.170155
$690$	0	0
$691$	−18049.0	−0.993658	−0.496829	−	0.867849i	$-0.665502\pi$
$691$	−18049.0	−0.993658	−0.496829	+	0.867849i	$0.665502\pi$
$692$	0	0
$693$	3975.37	0.217910
$694$	0	0
$695$	10287.2	0.561464
$696$	0	0
$697$	7948.99	0.431979
$698$	0	0
$699$	−6637.24	−0.359147
$700$	0	0
$701$	−7991.13	−0.430558	−0.215279	−	0.976553i	$-0.569066\pi$
$701$	−7991.13	−0.430558	−0.215279	+	0.976553i	$0.569066\pi$
$702$	0	0
$703$	−29473.5	−1.58124
$704$	0	0
$705$	−1732.80	−0.0925688
$706$	0	0
$707$	2784.18	0.148104
$708$	0	0
$709$	11306.8	0.598920	0.299460	−	0.954109i	$-0.403194\pi$
$709$	11306.8	0.598920	0.299460	+	0.954109i	$0.403194\pi$
$710$	0	0
$711$	11155.4	0.588410
$712$	0	0
$713$	−20590.6	−1.08152
$714$	0	0
$715$	744.865	0.0389600
$716$	0	0
$717$	1206.04	0.0628180
$718$	0	0
$719$	−30968.9	−1.60632	−0.803160	−	0.595763i	$-0.796850\pi$
$719$	−30968.9	−1.60632	−0.803160	+	0.595763i	$0.796850\pi$
$720$	0	0
$721$	−9065.23	−0.468248
$722$	0	0
$723$	−937.181	−0.0482076
$724$	0	0
$725$	414.298	0.0212229
$726$	0	0
$727$	26520.2	1.35293	0.676466	−	0.736474i	$-0.263511\pi$
$727$	26520.2	1.35293	0.676466	+	0.736474i	$0.263511\pi$
$728$	0	0
$729$	−10311.1	−0.523858
$730$	0	0
$731$	−90.4178	−0.00457486
$732$	0	0
$733$	−15980.8	−0.805270	−0.402635	−	0.915361i	$-0.631906\pi$
$733$	−15980.8	−0.805270	−0.402635	+	0.915361i	$0.631906\pi$
$734$	0	0
$735$	−2402.78	−0.120582
$736$	0	0
$737$	25128.7	1.25594
$738$	0	0
$739$	10163.8	0.505927	0.252964	−	0.967476i	$-0.418595\pi$
$739$	10163.8	0.505927	0.252964	+	0.967476i	$0.418595\pi$
$740$	0	0
$741$	−648.528	−0.0321515
$742$	0	0
$743$	−37765.7	−1.86472	−0.932360	−	0.361530i	$-0.882254\pi$
$743$	−37765.7	−1.86472	−0.932360	+	0.361530i	$0.882254\pi$
$744$	0	0
$745$	14194.5	0.698047
$746$	0	0
$747$	24106.1	1.18072
$748$	0	0
$749$	−2274.51	−0.110960
$750$	0	0
$751$	20234.4	0.983175	0.491587	−	0.870828i	$-0.336417\pi$
$751$	20234.4	0.983175	0.491587	+	0.870828i	$0.336417\pi$
$752$	0	0
$753$	−1376.66	−0.0666245
$754$	0	0
$755$	10935.4	0.527127
$756$	0	0
$757$	−27066.9	−1.29956	−0.649778	−	0.760124i	$-0.725138\pi$
$757$	−27066.9	−1.29956	−0.649778	+	0.760124i	$0.725138\pi$
$758$	0	0
$759$	−3259.12	−0.155861
$760$	0	0
$761$	30797.4	1.46702	0.733512	−	0.679677i	$-0.237880\pi$
$761$	30797.4	1.46702	0.733512	+	0.679677i	$0.237880\pi$
$762$	0	0
$763$	7946.74	0.377053
$764$	0	0
$765$	13355.1	0.631182
$766$	0	0
$767$	−3582.52	−0.168654
$768$	0	0
$769$	−35490.2	−1.66425	−0.832125	−	0.554588i	$-0.812876\pi$
$769$	−35490.2	−1.66425	−0.832125	+	0.554588i	$0.812876\pi$
$770$	0	0
$771$	6603.37	0.308449
$772$	0	0
$773$	37364.0	1.73854	0.869269	−	0.494339i	$-0.164590\pi$
$773$	37364.0	1.73854	0.869269	+	0.494339i	$0.164590\pi$
$774$	0	0
$775$	7513.09	0.348230
$776$	0	0
$777$	2556.01	0.118013
$778$	0	0
$779$	6606.72	0.303864
$780$	0	0
$781$	−24059.1	−1.10231
$782$	0	0
$783$	1300.29	0.0593467
$784$	0	0
$785$	4681.16	0.212838
$786$	0	0
$787$	−28529.7	−1.29222	−0.646108	−	0.763246i	$-0.723605\pi$
$787$	−28529.7	−1.29222	−0.646108	+	0.763246i	$0.723605\pi$
$788$	0	0
$789$	−5473.35	−0.246967
$790$	0	0
$791$	4050.01	0.182051
$792$	0	0
$793$	−1382.47	−0.0619078
$794$	0	0
$795$	4912.98	0.219176
$796$	0	0
$797$	7130.44	0.316905	0.158452	−	0.987367i	$-0.449350\pi$
$797$	7130.44	0.316905	0.158452	+	0.987367i	$0.449350\pi$
$798$	0	0
$799$	−24695.3	−1.09344
$800$	0	0
$801$	−29740.7	−1.31191
$802$	0	0
$803$	−1514.35	−0.0665505
$804$	0	0
$805$	1759.55	0.0770385
$806$	0	0
$807$	9868.33	0.430461
$808$	0	0
$809$	11060.3	0.480665	0.240333	−	0.970691i	$-0.422743\pi$
$809$	11060.3	0.480665	0.240333	+	0.970691i	$0.422743\pi$
$810$	0	0
$811$	27381.5	1.18557	0.592784	−	0.805361i	$-0.298029\pi$
$811$	27381.5	1.18557	0.592784	+	0.805361i	$0.298029\pi$
$812$	0	0
$813$	1535.71	0.0662481
$814$	0	0
$815$	11648.4	0.500645
$816$	0	0
$817$	−75.1498	−0.00321807
$818$	0	0
$819$	−602.947	−0.0257249
$820$	0	0
$821$	−30033.9	−1.27672	−0.638362	−	0.769736i	$-0.720388\pi$
$821$	−30033.9	−1.27672	−0.638362	+	0.769736i	$0.720388\pi$
$822$	0	0
$823$	3186.93	0.134981	0.0674906	−	0.997720i	$-0.478501\pi$
$823$	3186.93	0.134981	0.0674906	+	0.997720i	$0.478501\pi$
$824$	0	0
$825$	1189.19	0.0501844
$826$	0	0
$827$	18326.0	0.770563	0.385282	−	0.922799i	$-0.374104\pi$
$827$	18326.0	0.770563	0.385282	+	0.922799i	$0.374104\pi$
$828$	0	0
$829$	−4370.27	−0.183095	−0.0915475	−	0.995801i	$-0.529181\pi$
$829$	−4370.27	−0.183095	−0.0915475	+	0.995801i	$0.529181\pi$
$830$	0	0
$831$	3974.28	0.165904
$832$	0	0
$833$	−34243.7	−1.42434
$834$	0	0
$835$	−10102.1	−0.418679
$836$	0	0
$837$	23580.1	0.973771
$838$	0	0
$839$	−29789.0	−1.22578	−0.612890	−	0.790168i	$-0.709993\pi$
$839$	−29789.0	−1.22578	−0.612890	+	0.790168i	$0.709993\pi$
$840$	0	0
$841$	−24114.4	−0.988740
$842$	0	0
$843$	3494.35	0.142766
$844$	0	0
$845$	10872.0	0.442614
$846$	0	0
$847$	−1791.43	−0.0726732
$848$	0	0
$849$	10004.3	0.404412
$850$	0	0
$851$	22464.9	0.904918
$852$	0	0
$853$	−39023.7	−1.56641	−0.783205	−	0.621764i	$-0.786416\pi$
$853$	−39023.7	−1.56641	−0.783205	+	0.621764i	$0.786416\pi$
$854$	0	0
$855$	11099.9	0.443988
$856$	0	0
$857$	2749.52	0.109594	0.0547969	−	0.998498i	$-0.482549\pi$
$857$	2749.52	0.109594	0.0547969	+	0.998498i	$0.482549\pi$
$858$	0	0
$859$	−48225.6	−1.91553	−0.957763	−	0.287559i	$-0.907156\pi$
$859$	−48225.6	−1.91553	−0.957763	+	0.287559i	$0.907156\pi$
$860$	0	0
$861$	−572.950	−0.0226784
$862$	0	0
$863$	12421.1	0.489942	0.244971	−	0.969530i	$-0.421222\pi$
$863$	12421.1	0.489942	0.244971	+	0.969530i	$0.421222\pi$
$864$	0	0
$865$	4560.77	0.179273
$866$	0	0
$867$	−10297.1	−0.403353
$868$	0	0
$869$	14156.5	0.552618
$870$	0	0
$871$	−3811.28	−0.148267
$872$	0	0
$873$	13808.9	0.535350
$874$	0	0
$875$	−642.024	−0.0248050
$876$	0	0
$877$	38301.8	1.47475	0.737377	−	0.675482i	$-0.236064\pi$
$877$	38301.8	1.47475	0.737377	+	0.675482i	$0.236064\pi$
$878$	0	0
$879$	−7233.00	−0.277546
$880$	0	0
$881$	14792.0	0.565671	0.282836	−	0.959168i	$-0.408725\pi$
$881$	14792.0	0.565671	0.282836	+	0.959168i	$0.408725\pi$
$882$	0	0
$883$	−2064.99	−0.0787002	−0.0393501	−	0.999225i	$-0.512529\pi$
$883$	−2064.99	−0.0787002	−0.0393501	+	0.999225i	$0.512529\pi$
$884$	0	0
$885$	−5719.54	−0.217243
$886$	0	0
$887$	19268.8	0.729406	0.364703	−	0.931124i	$-0.381171\pi$
$887$	19268.8	0.729406	0.364703	+	0.931124i	$0.381171\pi$
$888$	0	0
$889$	7929.89	0.299167
$890$	0	0
$891$	−17165.5	−0.645415
$892$	0	0
$893$	−20525.3	−0.769152
$894$	0	0
$895$	16134.8	0.602598
$896$	0	0
$897$	494.312	0.0183998
$898$	0	0
$899$	4980.24	0.184761
$900$	0	0
$901$	70018.3	2.58895
$902$	0	0
$903$	6.51717	0.000240175 0
$904$	0	0
$905$	−5017.14	−0.184282
$906$	0	0
$907$	−46355.6	−1.69704	−0.848519	−	0.529165i	$-0.822505\pi$
$907$	−46355.6	−1.69704	−0.848519	+	0.529165i	$0.822505\pi$
$908$	0	0
$909$	−13387.2	−0.488475
$910$	0	0
$911$	−23365.5	−0.849761	−0.424881	−	0.905249i	$-0.639684\pi$
$911$	−23365.5	−0.849761	−0.424881	+	0.905249i	$0.639684\pi$
$912$	0	0
$913$	30591.3	1.10890
$914$	0	0
$915$	−2207.13	−0.0797436
$916$	0	0
$917$	−9929.58	−0.357583
$918$	0	0
$919$	−36178.5	−1.29860	−0.649302	−	0.760530i	$-0.724939\pi$
$919$	−36178.5	−1.29860	−0.649302	+	0.760530i	$0.724939\pi$
$920$	0	0
$921$	−179.033	−0.00640535
$922$	0	0
$923$	3649.06	0.130130
$924$	0	0
$925$	−8196.98	−0.291367
$926$	0	0
$927$	43588.4	1.54437
$928$	0	0
$929$	17014.6	0.600895	0.300447	−	0.953798i	$-0.402864\pi$
$929$	17014.6	0.600895	0.300447	+	0.953798i	$0.402864\pi$
$930$	0	0
$931$	−28461.3	−1.00191
$932$	0	0
$933$	−6200.20	−0.217562
$934$	0	0
$935$	16947.9	0.592788
$936$	0	0
$937$	−16898.8	−0.589179	−0.294589	−	0.955624i	$-0.595183\pi$
$937$	−16898.8	−0.589179	−0.294589	+	0.955624i	$0.595183\pi$
$938$	0	0
$939$	1373.10	0.0477203
$940$	0	0
$941$	43995.9	1.52415	0.762074	−	0.647489i	$-0.224181\pi$
$941$	43995.9	1.52415	0.762074	+	0.647489i	$0.224181\pi$
$942$	0	0
$943$	−5035.68	−0.173897
$944$	0	0
$945$	−2015.01	−0.0693634
$946$	0	0
$947$	2588.13	0.0888099	0.0444050	−	0.999014i	$-0.485861\pi$
$947$	2588.13	0.0888099	0.0444050	+	0.999014i	$0.485861\pi$
$948$	0	0
$949$	229.681	0.00785646
$950$	0	0
$951$	−8252.52	−0.281395
$952$	0	0
$953$	−7309.24	−0.248447	−0.124223	−	0.992254i	$-0.539644\pi$
$953$	−7309.24	−0.248447	−0.124223	+	0.992254i	$0.539644\pi$
$954$	0	0
$955$	6360.38	0.215515
$956$	0	0
$957$	788.283	0.0266265
$958$	0	0
$959$	−2456.37	−0.0827115
$960$	0	0
$961$	60523.3	2.03160
$962$	0	0
$963$	10936.5	0.365966
$964$	0	0
$965$	3654.57	0.121912
$966$	0	0
$967$	27600.6	0.917866	0.458933	−	0.888471i	$-0.348232\pi$
$967$	27600.6	0.917866	0.458933	+	0.888471i	$0.348232\pi$
$968$	0	0
$969$	−14756.0	−0.489195
$970$	0	0
$971$	−22638.5	−0.748201	−0.374100	−	0.927388i	$-0.622048\pi$
$971$	−22638.5	−0.748201	−0.374100	+	0.927388i	$0.622048\pi$
$972$	0	0
$973$	−10567.5	−0.348178
$974$	0	0
$975$	−180.364	−0.00592440
$976$	0	0
$977$	−18390.3	−0.602208	−0.301104	−	0.953591i	$-0.597355\pi$
$977$	−18390.3	−0.602208	−0.301104	+	0.953591i	$0.597355\pi$
$978$	0	0
$979$	−37741.7	−1.23210
$980$	0	0
$981$	−38210.3	−1.24359
$982$	0	0
$983$	11139.1	0.361425	0.180712	−	0.983536i	$-0.442160\pi$
$983$	11139.1	0.361425	0.180712	+	0.983536i	$0.442160\pi$
$984$	0	0
$985$	−17910.1	−0.579353
$986$	0	0
$987$	1780.00	0.0574043
$988$	0	0
$989$	57.2797	0.00184165
$990$	0	0
$991$	−30232.3	−0.969083	−0.484542	−	0.874768i	$-0.661014\pi$
$991$	−30232.3	−0.969083	−0.484542	+	0.874768i	$0.661014\pi$
$992$	0	0
$993$	9021.68	0.288313
$994$	0	0
$995$	−10038.2	−0.319832
$996$	0	0
$997$	−9623.78	−0.305705	−0.152853	−	0.988249i	$-0.548846\pi$
$997$	−9623.78	−0.305705	−0.152853	+	0.988249i	$0.548846\pi$
$998$	0	0
$999$	−25726.5	−0.814764

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.a.ba.1.3		6
4.3	odd	2		1280.4.a.bc.1.4		6
8.3	odd	2		1280.4.a.bb.1.3		6
8.5	even	2		1280.4.a.bd.1.4		6
16.3	odd	4		40.4.d.a.21.8	yes	12
16.5	even	4		160.4.d.a.81.8		12
16.11	odd	4		40.4.d.a.21.7	✓	12
16.13	even	4		160.4.d.a.81.5		12
48.5	odd	4		1440.4.k.c.721.10		12
48.11	even	4		360.4.k.c.181.6		12
48.29	odd	4		1440.4.k.c.721.4		12
48.35	even	4		360.4.k.c.181.5		12
80.3	even	4		200.4.f.b.149.11		12
80.13	odd	4		800.4.f.c.49.8		12
80.19	odd	4		200.4.d.b.101.5		12
80.27	even	4		200.4.f.b.149.12		12
80.29	even	4		800.4.d.d.401.8		12
80.37	odd	4		800.4.f.c.49.7		12
80.43	even	4		200.4.f.c.149.1		12
80.53	odd	4		800.4.f.b.49.6		12
80.59	odd	4		200.4.d.b.101.6		12
80.67	even	4		200.4.f.c.149.2		12
80.69	even	4		800.4.d.d.401.5		12
80.77	odd	4		800.4.f.b.49.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.7	✓	12	16.11	odd	4
40.4.d.a.21.8	yes	12	16.3	odd	4
160.4.d.a.81.5		12	16.13	even	4
160.4.d.a.81.8		12	16.5	even	4
200.4.d.b.101.5		12	80.19	odd	4
200.4.d.b.101.6		12	80.59	odd	4
200.4.f.b.149.11		12	80.3	even	4
200.4.f.b.149.12		12	80.27	even	4
200.4.f.c.149.1		12	80.43	even	4
200.4.f.c.149.2		12	80.67	even	4
360.4.k.c.181.5		12	48.35	even	4
360.4.k.c.181.6		12	48.11	even	4
800.4.d.d.401.5		12	80.69	even	4
800.4.d.d.401.8		12	80.29	even	4
800.4.f.b.49.5		12	80.77	odd	4
800.4.f.b.49.6		12	80.53	odd	4
800.4.f.c.49.7		12	80.37	odd	4
800.4.f.c.49.8		12	80.13	odd	4
1280.4.a.ba.1.3		6	1.1	even	1	trivial
1280.4.a.bb.1.3		6	8.3	odd	2
1280.4.a.bc.1.4		6	4.3	odd	2
1280.4.a.bd.1.4		6	8.5	even	2
1440.4.k.c.721.4		12	48.29	odd	4
1440.4.k.c.721.10		12	48.5	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.51777 q^{3} -5.00000 q^{5} +5.13620 q^{7} -24.6964 q^{9} -31.3403 q^{11} +4.75340 q^{13} +7.58886 q^{15} +108.154 q^{17} +89.8913 q^{19} -7.79558 q^{21} -68.5157 q^{23} +25.0000 q^{25} +78.4633 q^{27} +16.5719 q^{29} +300.523 q^{31} +47.5675 q^{33} -25.6810 q^{35} -327.879 q^{37} -7.21458 q^{39} +73.4968 q^{41} -0.836008 q^{43} +123.482 q^{45} -228.335 q^{47} -316.619 q^{49} -164.153 q^{51} +647.393 q^{53} +156.702 q^{55} -136.434 q^{57} -753.676 q^{59} -290.838 q^{61} -126.845 q^{63} -23.7670 q^{65} -801.801 q^{67} +103.991 q^{69} +767.674 q^{71} +48.3194 q^{73} -37.9443 q^{75} -160.970 q^{77} -451.701 q^{79} +547.712 q^{81} -976.099 q^{83} -540.771 q^{85} -25.1524 q^{87} +1204.25 q^{89} +24.4144 q^{91} -456.126 q^{93} -449.456 q^{95} -559.147 q^{97} +773.992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 30 q^{5} + 14 q^{7} + 54 q^{9} - 44 q^{11} + 30 q^{15} - 152 q^{19} - 4 q^{21} + 302 q^{23} + 150 q^{25} - 216 q^{27} + 132 q^{31} - 116 q^{33} - 70 q^{35} + 68 q^{37} + 300 q^{39} - 20 q^{41}+ \cdots - 5516 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 - 30 * q^5 + 14 * q^7 + 54 * q^9 - 44 * q^11 + 30 * q^15 - 152 * q^19 - 4 * q^21 + 302 * q^23 + 150 * q^25 - 216 * q^27 + 132 * q^31 - 116 * q^33 - 70 * q^35 + 68 * q^37 + 300 * q^39 - 20 * q^41 - 602 * q^43 - 270 * q^45 + 470 * q^47 + 654 * q^49 - 612 * q^51 - 528 * q^53 + 220 * q^55 + 340 * q^57 - 472 * q^59 + 476 * q^61 + 650 * q^63 - 1206 * q^67 + 980 * q^69 - 796 * q^71 - 216 * q^73 - 150 * q^75 - 412 * q^77 - 1008 * q^79 + 1254 * q^81 - 1778 * q^83 - 984 * q^87 + 212 * q^89 - 3652 * q^91 + 1392 * q^93 + 760 * q^95 - 792 * q^97 - 5516 * q^99

Introduction

L-functions

Modular forms

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