LMFDB - Embedded newform 1280.4.a.bb.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)

$f(q)$	$=$	$q-1.51777 q^{3} +5.00000 q^{5} -5.13620 q^{7} -24.6964 q^{9} +O(q^{10})$	$q-1.51777 q^{3} +5.00000 q^{5} -5.13620 q^{7} -24.6964 q^{9} -31.3403 q^{11} -4.75340 q^{13} -7.58886 q^{15} +108.154 q^{17} +89.8913 q^{19} +7.79558 q^{21} +68.5157 q^{23} +25.0000 q^{25} +78.4633 q^{27} -16.5719 q^{29} -300.523 q^{31} +47.5675 q^{33} -25.6810 q^{35} +327.879 q^{37} +7.21458 q^{39} +73.4968 q^{41} -0.836008 q^{43} -123.482 q^{45} +228.335 q^{47} -316.619 q^{49} -164.153 q^{51} -647.393 q^{53} -156.702 q^{55} -136.434 q^{57} -753.676 q^{59} +290.838 q^{61} +126.845 q^{63} -23.7670 q^{65} -801.801 q^{67} -103.991 q^{69} -767.674 q^{71} +48.3194 q^{73} -37.9443 q^{75} +160.970 q^{77} +451.701 q^{79} +547.712 q^{81} -976.099 q^{83} +540.771 q^{85} +25.1524 q^{87} +1204.25 q^{89} +24.4144 q^{91} +456.126 q^{93} +449.456 q^{95} -559.147 q^{97} +773.992 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 30 q^{5} - 14 q^{7} + 54 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 30 * q^5 - 14 * q^7 + 54 * q^9	$ 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}+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

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
$3$	−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	$-0.544281\pi$
$7$	−5.13620	−0.277328	−0.138664	+	0.990339i	$0.544281\pi$
$8$	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	$-0.516147\pi$
$13$	−4.75340	−0.101412	−0.0507060	+	0.998714i	$0.516147\pi$
$14$	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	$-0.399478\pi$
$23$	68.5157	0.621152	0.310576	+	0.950548i	$0.399478\pi$
$24$	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	$-0.516897\pi$
$29$	−16.5719	−0.106115	−0.0530573	+	0.998591i	$0.516897\pi$
$30$	0	0
$31$	−300.523	−1.74115	−0.870574	−	0.492037i	$-0.836252\pi$
$31$	−300.523	−1.74115	−0.870574	+	0.492037i	$0.836252\pi$
$32$	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	$-0.240256\pi$
$37$	327.879	1.45684	0.728419	+	0.685132i	$0.240256\pi$
$38$	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	$-0.384713\pi$
$47$	228.335	0.708639	0.354319	+	0.935124i	$0.384713\pi$
$48$	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	$-0.816817\pi$
$53$	−647.393	−1.67785	−0.838927	+	0.544244i	$0.816817\pi$
$54$	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	$-0.401267\pi$
$61$	290.838	0.610459	0.305229	+	0.952279i	$0.401267\pi$
$62$	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	$-0.721726\pi$
$71$	−767.674	−1.28318	−0.641592	+	0.767046i	$0.721726\pi$
$72$	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	$-0.395763\pi$
$79$	451.701	0.643296	0.321648	+	0.946859i	$0.395763\pi$
$80$	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	$-0.586039\pi$
$101$	−542.070	−0.534039	−0.267020	+	0.963691i	$0.586039\pi$
$102$	0	0
$103$	1764.97	1.68842	0.844212	−	0.536009i	$-0.180069\pi$
$103$	1764.97	1.68842	0.844212	+	0.536009i	$0.180069\pi$
$104$	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	$-0.737931\pi$
$109$	−1547.20	−1.35959	−0.679795	+	0.733403i	$0.737931\pi$
$110$	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	$-0.681339\pi$
$127$	−1543.92	−1.07875	−0.539374	+	0.842066i	$0.681339\pi$
$128$	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	$-0.214995\pi$
$149$	2838.89	1.56088	0.780440	+	0.625231i	$0.214995\pi$
$150$	0	0
$151$	2187.09	1.17869	0.589346	−	0.807881i	$-0.299385\pi$
$151$	2187.09	1.17869	0.589346	+	0.807881i	$0.299385\pi$
$152$	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	$-0.423521\pi$
$157$	936.231	0.475920	0.237960	+	0.971275i	$0.423521\pi$
$158$	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	$-0.655060\pi$
$167$	−2020.42	−0.936195	−0.468097	+	0.883677i	$0.655060\pi$
$168$	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	$-0.435765\pi$
$173$	912.153	0.400866	0.200433	+	0.979707i	$0.435765\pi$
$174$	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	$-0.566056\pi$
$181$	−1003.43	−0.412068	−0.206034	+	0.978545i	$0.566056\pi$
$182$	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	$-0.422540\pi$
$191$	1272.08	0.481907	0.240953	+	0.970537i	$0.422540\pi$
$192$	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	$-0.724284\pi$
$197$	−3582.02	−1.29547	−0.647737	+	0.761864i	$0.724284\pi$
$198$	0	0
$199$	−2007.64	−0.715166	−0.357583	−	0.933881i	$-0.616399\pi$
$199$	−2007.64	−0.715166	−0.357583	+	0.933881i	$0.616399\pi$
$200$	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	$-0.499919\pi$
$223$	1.69909	0.000510221 0	0.000255111		1.00000i	$0.499919\pi$
$224$	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	$-0.438504\pi$
$229$	1330.69	0.383994	0.191997	+	0.981396i	$0.438504\pi$
$230$	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	$-0.465706\pi$
$239$	794.613	0.215060	0.107530	+	0.994202i	$0.465706\pi$
$240$	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	$-0.638935\pi$
$263$	−3606.18	−0.845499	−0.422750	+	0.906246i	$0.638935\pi$
$264$	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	$-0.236313\pi$
$269$	6501.85	1.47370	0.736849	+	0.676057i	$0.236313\pi$
$270$	0	0
$271$	1011.82	0.226803	0.113401	−	0.993549i	$-0.463825\pi$
$271$	1011.82	0.226803	0.113401	+	0.993549i	$0.463825\pi$
$272$	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	$-0.408342\pi$
$277$	2618.50	0.567979	0.283990	+	0.958827i	$0.408342\pi$
$278$	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	$-0.657586\pi$
$293$	−4765.54	−0.950190	−0.475095	+	0.879934i	$0.657586\pi$
$294$	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	$-0.621470\pi$
$311$	−4085.06	−0.744832	−0.372416	+	0.928066i	$0.621470\pi$
$312$	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	$-0.659974\pi$
$317$	−5437.26	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$318$	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	$-0.267181\pi$
$349$	8709.62	1.33586	0.667930	+	0.744224i	$0.267181\pi$
$350$	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	$-0.180710\pi$
$359$	11470.1	1.68626	0.843131	+	0.537708i	$0.180710\pi$
$360$	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	$-0.423886\pi$
$367$	3330.42	0.473697	0.236848	+	0.971547i	$0.423886\pi$
$368$	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	$-0.273782\pi$
$373$	9398.88	1.30471	0.652354	+	0.757915i	$0.273782\pi$
$374$	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	$-0.563483\pi$
$383$	−2969.97	−0.396236	−0.198118	+	0.980178i	$0.563483\pi$
$384$	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	$-0.683132\pi$
$389$	−8349.10	−1.08822	−0.544108	+	0.839015i	$0.683132\pi$
$390$	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	$-0.591940\pi$
$397$	−4506.27	−0.569680	−0.284840	+	0.958575i	$0.591940\pi$
$398$	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	$-0.839490\pi$
$421$	−15126.1	−1.75107	−0.875534	+	0.483157i	$0.839490\pi$
$422$	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	$-0.411939\pi$
$431$	4887.92	0.546271	0.273135	+	0.961976i	$0.411939\pi$
$432$	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	$-0.509995\pi$
$439$	−577.528	−0.0627879	−0.0313940	+	0.999507i	$0.509995\pi$
$440$	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	$-0.736110\pi$
$461$	−13374.1	−1.35118	−0.675588	+	0.737279i	$0.736110\pi$
$462$	0	0
$463$	15414.3	1.54722	0.773611	−	0.633661i	$-0.218449\pi$
$463$	15414.3	1.54722	0.773611	+	0.633661i	$0.218449\pi$
$464$	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	$-0.309928\pi$
$479$	11789.0	1.12454	0.562270	+	0.826954i	$0.309928\pi$
$480$	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	$-0.818204\pi$
$487$	−18083.0	−1.68258	−0.841291	+	0.540582i	$0.818204\pi$
$488$	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	$-0.425140\pi$
$503$	5257.43	0.466038	0.233019	+	0.972472i	$0.425140\pi$
$504$	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	$-0.534048\pi$
$509$	−2451.99	−0.213522	−0.106761	+	0.994285i	$0.534048\pi$
$510$	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	$-0.457975\pi$
$541$	3313.01	0.263286	0.131643	+	0.991297i	$0.457975\pi$
$542$	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	$-0.467155\pi$
$557$	2708.11	0.206008	0.103004	+	0.994681i	$0.467155\pi$
$558$	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	$-0.325530\pi$
$599$	15278.2	1.04215	0.521077	+	0.853510i	$0.325530\pi$
$600$	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	$-0.812138\pi$
$607$	−24850.2	−1.66167	−0.830837	+	0.556515i	$0.812138\pi$
$608$	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	$-0.541662\pi$
$613$	−3961.58	−0.261022	−0.130511	+	0.991447i	$0.541662\pi$
$614$	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	$-0.404596\pi$
$631$	9359.85	0.590507	0.295253	+	0.955419i	$0.404596\pi$
$632$	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	$-0.617621\pi$
$647$	−11887.6	−0.722331	−0.361166	+	0.932502i	$0.617621\pi$
$648$	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	$-0.448969\pi$
$653$	5327.46	0.319264	0.159632	+	0.987177i	$0.448969\pi$
$654$	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	$-0.556122\pi$
$661$	−5961.64	−0.350803	−0.175401	+	0.984497i	$0.556122\pi$
$662$	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	$-0.660287\pi$
$677$	−17000.0	−0.965088	−0.482544	+	0.875872i	$0.660287\pi$
$678$	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	$-0.430934\pi$
$701$	7991.13	0.430558	0.215279	+	0.976553i	$0.430934\pi$
$702$	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	$-0.596806\pi$
$709$	−11306.8	−0.598920	−0.299460	+	0.954109i	$0.596806\pi$
$710$	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	$-0.203150\pi$
$719$	30968.9	1.60632	0.803160	+	0.595763i	$0.203150\pi$
$720$	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	$-0.736489\pi$
$727$	−26520.2	−1.35293	−0.676466	+	0.736474i	$0.736489\pi$
$728$	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	$-0.368094\pi$
$733$	15980.8	0.805270	0.402635	+	0.915361i	$0.368094\pi$
$734$	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	$-0.117746\pi$
$743$	37765.7	1.86472	0.932360	+	0.361530i	$0.117746\pi$
$744$	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	$-0.663583\pi$
$751$	−20234.4	−0.983175	−0.491587	+	0.870828i	$0.663583\pi$
$752$	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	$-0.274862\pi$
$757$	27066.9	1.29956	0.649778	+	0.760124i	$0.274862\pi$
$758$	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	$-0.835410\pi$
$773$	−37364.0	−1.73854	−0.869269	+	0.494339i	$0.835410\pi$
$774$	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	$-0.550650\pi$
$797$	−7130.44	−0.316905	−0.158452	+	0.987367i	$0.550650\pi$
$798$	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	$-0.279612\pi$
$821$	30033.9	1.27672	0.638362	+	0.769736i	$0.279612\pi$
$822$	0	0
$823$	−3186.93	−0.134981	−0.0674906	−	0.997720i	$-0.521499\pi$
$823$	−3186.93	−0.134981	−0.0674906	+	0.997720i	$0.521499\pi$
$824$	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	$-0.470819\pi$
$829$	4370.27	0.183095	0.0915475	+	0.995801i	$0.470819\pi$
$830$	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	$-0.290007\pi$
$839$	29789.0	1.22578	0.612890	+	0.790168i	$0.290007\pi$
$840$	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	$-0.213584\pi$
$853$	39023.7	1.56641	0.783205	+	0.621764i	$0.213584\pi$
$854$	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	$-0.578778\pi$
$863$	−12421.1	−0.489942	−0.244971	+	0.969530i	$0.578778\pi$
$864$	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	$-0.763936\pi$
$877$	−38301.8	−1.47475	−0.737377	+	0.675482i	$0.763936\pi$
$878$	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	$-0.618829\pi$
$887$	−19268.8	−0.729406	−0.364703	+	0.931124i	$0.618829\pi$
$888$	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	$-0.360316\pi$
$911$	23365.5	0.849761	0.424881	+	0.905249i	$0.360316\pi$
$912$	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	$-0.275061\pi$
$919$	36178.5	1.29860	0.649302	+	0.760530i	$0.275061\pi$
$920$	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	$-0.775819\pi$
$941$	−43995.9	−1.52415	−0.762074	+	0.647489i	$0.775819\pi$
$942$	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	$-0.651768\pi$
$967$	−27600.6	−0.917866	−0.458933	+	0.888471i	$0.651768\pi$
$968$	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	$-0.557840\pi$
$983$	−11139.1	−0.361425	−0.180712	+	0.983536i	$0.557840\pi$
$984$	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	$-0.338986\pi$
$991$	30232.3	0.969083	0.484542	+	0.874768i	$0.338986\pi$
$992$	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	$-0.451154\pi$
$997$	9623.78	0.305705	0.152853	+	0.988249i	$0.451154\pi$
$998$	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.bb.1.3		6
4.3	odd	2		1280.4.a.bd.1.4		6
8.3	odd	2		1280.4.a.ba.1.3		6
8.5	even	2		1280.4.a.bc.1.4		6
16.3	odd	4		160.4.d.a.81.8		12
16.5	even	4		40.4.d.a.21.8	yes	12
16.11	odd	4		160.4.d.a.81.5		12
16.13	even	4		40.4.d.a.21.7	✓	12
48.5	odd	4		360.4.k.c.181.5		12
48.11	even	4		1440.4.k.c.721.4		12
48.29	odd	4		360.4.k.c.181.6		12
48.35	even	4		1440.4.k.c.721.10		12
80.3	even	4		800.4.f.b.49.6		12
80.13	odd	4		200.4.f.c.149.1		12
80.19	odd	4		800.4.d.d.401.5		12
80.27	even	4		800.4.f.b.49.5		12
80.29	even	4		200.4.d.b.101.6		12
80.37	odd	4		200.4.f.c.149.2		12
80.43	even	4		800.4.f.c.49.8		12
80.53	odd	4		200.4.f.b.149.11		12
80.59	odd	4		800.4.d.d.401.8		12
80.67	even	4		800.4.f.c.49.7		12
80.69	even	4		200.4.d.b.101.5		12
80.77	odd	4		200.4.f.b.149.12		12

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

\(f(q)\)	\(=\)	\(q-1.51777 q^{3} +5.00000 q^{5} -5.13620 q^{7} -24.6964 q^{9} +O(q^{10})\)	\(q-1.51777 q^{3} +5.00000 q^{5} -5.13620 q^{7} -24.6964 q^{9} -31.3403 q^{11} -4.75340 q^{13} -7.58886 q^{15} +108.154 q^{17} +89.8913 q^{19} +7.79558 q^{21} +68.5157 q^{23} +25.0000 q^{25} +78.4633 q^{27} -16.5719 q^{29} -300.523 q^{31} +47.5675 q^{33} -25.6810 q^{35} +327.879 q^{37} +7.21458 q^{39} +73.4968 q^{41} -0.836008 q^{43} -123.482 q^{45} +228.335 q^{47} -316.619 q^{49} -164.153 q^{51} -647.393 q^{53} -156.702 q^{55} -136.434 q^{57} -753.676 q^{59} +290.838 q^{61} +126.845 q^{63} -23.7670 q^{65} -801.801 q^{67} -103.991 q^{69} -767.674 q^{71} +48.3194 q^{73} -37.9443 q^{75} +160.970 q^{77} +451.701 q^{79} +547.712 q^{81} -976.099 q^{83} +540.771 q^{85} +25.1524 q^{87} +1204.25 q^{89} +24.4144 q^{91} +456.126 q^{93} +449.456 q^{95} -559.147 q^{97} +773.992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 30 q^{5} - 14 q^{7} + 54 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 30 * q^5 - 14 * q^7 + 54 * q^9	\( 6 q - 6 q^{3} + 30 q^{5} - 14 q^{7} + 54 q^{9} - 44 q^{11} - 30 q^{15} - 152 q^{19} + 4 q^{21} - 302 q^{23} + 150 q^{25} - 216 q^{27} - 132 q^{31} - 116 q^{33} - 70 q^{35} - 68 q^{37} - 300 q^{39} - 20 q^{41} - 602 q^{43} + 270 q^{45} - 470 q^{47} + 654 q^{49} - 612 q^{51} + 528 q^{53} - 220 q^{55} + 340 q^{57} - 472 q^{59} - 476 q^{61} - 650 q^{63} - 1206 q^{67} - 980 q^{69} + 796 q^{71} - 216 q^{73} - 150 q^{75} + 412 q^{77} + 1008 q^{79} + 1254 q^{81} - 1778 q^{83} + 984 q^{87} + 212 q^{89} - 3652 q^{91} - 1392 q^{93} - 760 q^{95} - 792 q^{97} - 5516 q^{99}+O(q^{100}) \) 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

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