LMFDB - Embedded newform 256.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,Mod(1,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("256.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	256.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:	$15.1044889615$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.29150 q^{3} +10.5830 q^{5} +8.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+5.29150 q^{3} +10.5830 q^{5} +8.00000 q^{7} +1.00000 q^{9} +15.8745 q^{11} +52.9150 q^{13} +56.0000 q^{15} -14.0000 q^{17} -37.0405 q^{19} +42.3320 q^{21} +152.000 q^{23} -13.0000 q^{25} -137.579 q^{27} -158.745 q^{29} +224.000 q^{31} +84.0000 q^{33} +84.6640 q^{35} -243.409 q^{37} +280.000 q^{39} +70.0000 q^{41} +439.195 q^{43} +10.5830 q^{45} +336.000 q^{47} -279.000 q^{49} -74.0810 q^{51} -31.7490 q^{53} +168.000 q^{55} -196.000 q^{57} -534.442 q^{59} +95.2470 q^{61} +8.00000 q^{63} +560.000 q^{65} +174.620 q^{67} +804.308 q^{69} +72.0000 q^{71} +294.000 q^{73} -68.7895 q^{75} +126.996 q^{77} -464.000 q^{79} -755.000 q^{81} -545.025 q^{83} -148.162 q^{85} -840.000 q^{87} -266.000 q^{89} +423.320 q^{91} +1185.30 q^{93} -392.000 q^{95} +994.000 q^{97} +15.8745 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 16 * q^7 + 2 * q^9	$ 2 q + 16 q^{7} + 2 q^{9} + 112 q^{15} - 28 q^{17} + 304 q^{23} - 26 q^{25} + 448 q^{31} + 168 q^{33} + 560 q^{39} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 392 q^{57} + 16 q^{63} + 1120 q^{65} + 144 q^{71} + 588 q^{73} - 928 q^{79} - 1510 q^{81} - 1680 q^{87} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100}) $ 2 * q + 16 * q^7 + 2 * q^9 + 112 * q^15 - 28 * q^17 + 304 * q^23 - 26 * q^25 + 448 * q^31 + 168 * q^33 + 560 * q^39 + 140 * q^41 + 672 * q^47 - 558 * q^49 + 336 * q^55 - 392 * q^57 + 16 * q^63 + 1120 * q^65 + 144 * q^71 + 588 * q^73 - 928 * q^79 - 1510 * q^81 - 1680 * q^87 - 532 * q^89 - 784 * q^95 + 1988 * q^97

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
$3$	5.29150	1.01835	0.509175	−	0.860663i	$-0.329951\pi$
$3$	5.29150	1.01835	0.509175	+	0.860663i	$0.329951\pi$
$4$	0	0
$5$	10.5830	0.946573	0.473286	−	0.880909i	$-0.343068\pi$
$5$	10.5830	0.946573	0.473286	+	0.880909i	$0.343068\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	1.00000	0.0370370
$10$	0	0
$11$	15.8745	0.435122	0.217561	−	0.976047i	$-0.430190\pi$
$11$	15.8745	0.435122	0.217561	+	0.976047i	$0.430190\pi$
$12$	0	0
$13$	52.9150	1.12892	0.564461	−	0.825460i	$-0.309084\pi$
$13$	52.9150	1.12892	0.564461	+	0.825460i	$0.309084\pi$
$14$	0	0
$15$	56.0000	0.963943
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	−37.0405	−0.447246	−0.223623	−	0.974676i	$-0.571788\pi$
$19$	−37.0405	−0.447246	−0.223623	+	0.974676i	$0.571788\pi$
$20$	0	0
$21$	42.3320	0.439886
$22$	0	0
$23$	152.000	1.37801	0.689004	−	0.724757i	$-0.258048\pi$
$23$	152.000	1.37801	0.689004	+	0.724757i	$0.258048\pi$
$24$	0	0
$25$	−13.0000	−0.104000
$26$	0	0
$27$	−137.579	−0.980633
$28$	0	0
$29$	−158.745	−1.01649	−0.508245	−	0.861212i	$-0.669706\pi$
$29$	−158.745	−1.01649	−0.508245	+	0.861212i	$0.669706\pi$
$30$	0	0
$31$	224.000	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$31$	224.000	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$32$	0	0
$33$	84.0000	0.443107
$34$	0	0
$35$	84.6640	0.408881
$36$	0	0
$37$	−243.409	−1.08152	−0.540760	−	0.841177i	$-0.681863\pi$
$37$	−243.409	−1.08152	−0.540760	+	0.841177i	$0.681863\pi$
$38$	0	0
$39$	280.000	1.14964
$40$	0	0
$41$	70.0000	0.266638	0.133319	−	0.991073i	$-0.457436\pi$
$41$	70.0000	0.266638	0.133319	+	0.991073i	$0.457436\pi$
$42$	0	0
$43$	439.195	1.55759	0.778797	−	0.627276i	$-0.215830\pi$
$43$	439.195	1.55759	0.778797	+	0.627276i	$0.215830\pi$
$44$	0	0
$45$	10.5830	0.0350583
$46$	0	0
$47$	336.000	1.04278	0.521390	−	0.853319i	$-0.325414\pi$
$47$	336.000	1.04278	0.521390	+	0.853319i	$0.325414\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	−74.0810	−0.203400
$52$	0	0
$53$	−31.7490	−0.0822842	−0.0411421	−	0.999153i	$-0.513100\pi$
$53$	−31.7490	−0.0822842	−0.0411421	+	0.999153i	$0.513100\pi$
$54$	0	0
$55$	168.000	0.411875
$56$	0	0
$57$	−196.000	−0.455453
$58$	0	0
$59$	−534.442	−1.17929	−0.589647	−	0.807661i	$-0.700733\pi$
$59$	−534.442	−1.17929	−0.589647	+	0.807661i	$0.700733\pi$
$60$	0	0
$61$	95.2470	0.199920	0.0999601	−	0.994991i	$-0.468128\pi$
$61$	95.2470	0.199920	0.0999601	+	0.994991i	$0.468128\pi$
$62$	0	0
$63$	8.00000	0.0159985
$64$	0	0
$65$	560.000	1.06861
$66$	0	0
$67$	174.620	0.318406	0.159203	−	0.987246i	$-0.449108\pi$
$67$	174.620	0.318406	0.159203	+	0.987246i	$0.449108\pi$
$68$	0	0
$69$	804.308	1.40329
$70$	0	0
$71$	72.0000	0.120350	0.0601748	−	0.998188i	$-0.480834\pi$
$71$	72.0000	0.120350	0.0601748	+	0.998188i	$0.480834\pi$
$72$	0	0
$73$	294.000	0.471371	0.235686	−	0.971829i	$-0.424266\pi$
$73$	294.000	0.471371	0.235686	+	0.971829i	$0.424266\pi$
$74$	0	0
$75$	−68.7895	−0.105908
$76$	0	0
$77$	126.996	0.187955
$78$	0	0
$79$	−464.000	−0.660811	−0.330406	−	0.943839i	$-0.607186\pi$
$79$	−464.000	−0.660811	−0.330406	+	0.943839i	$0.607186\pi$
$80$	0	0
$81$	−755.000	−1.03567
$82$	0	0
$83$	−545.025	−0.720774	−0.360387	−	0.932803i	$-0.617355\pi$
$83$	−545.025	−0.720774	−0.360387	+	0.932803i	$0.617355\pi$
$84$	0	0
$85$	−148.162	−0.189064
$86$	0	0
$87$	−840.000	−1.03514
$88$	0	0
$89$	−266.000	−0.316808	−0.158404	−	0.987374i	$-0.550635\pi$
$89$	−266.000	−0.316808	−0.158404	+	0.987374i	$0.550635\pi$
$90$	0	0
$91$	423.320	0.487649
$92$	0	0
$93$	1185.30	1.32161
$94$	0	0
$95$	−392.000	−0.423351
$96$	0	0
$97$	994.000	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$97$	994.000	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$98$	0	0
$99$	15.8745	0.0161156
$100$	0	0
$101$	−751.393	−0.740262	−0.370131	−	0.928980i	$-0.620687\pi$
$101$	−751.393	−0.740262	−0.370131	+	0.928980i	$0.620687\pi$
$102$	0	0
$103$	−1176.00	−1.12500	−0.562499	−	0.826798i	$-0.690160\pi$
$103$	−1176.00	−1.12500	−0.562499	+	0.826798i	$0.690160\pi$
$104$	0	0
$105$	448.000	0.416384
$106$	0	0
$107$	269.867	0.243822	0.121911	−	0.992541i	$-0.461098\pi$
$107$	269.867	0.243822	0.121911	+	0.992541i	$0.461098\pi$
$108$	0	0
$109$	−1894.36	−1.66465	−0.832324	−	0.554290i	$-0.812990\pi$
$109$	−1894.36	−1.66465	−0.832324	+	0.554290i	$0.812990\pi$
$110$	0	0
$111$	−1288.00	−1.10137
$112$	0	0
$113$	−1710.00	−1.42357	−0.711784	−	0.702398i	$-0.752113\pi$
$113$	−1710.00	−1.42357	−0.711784	+	0.702398i	$0.752113\pi$
$114$	0	0
$115$	1608.62	1.30439
$116$	0	0
$117$	52.9150	0.0418119
$118$	0	0
$119$	−112.000	−0.0862775
$120$	0	0
$121$	−1079.00	−0.810669
$122$	0	0
$123$	370.405	0.271531
$124$	0	0
$125$	−1460.45	−1.04502
$126$	0	0
$127$	−1664.00	−1.16265	−0.581323	−	0.813673i	$-0.697465\pi$
$127$	−1664.00	−1.16265	−0.581323	+	0.813673i	$0.697465\pi$
$128$	0	0
$129$	2324.00	1.58618
$130$	0	0
$131$	−672.021	−0.448204	−0.224102	−	0.974566i	$-0.571945\pi$
$131$	−672.021	−0.448204	−0.224102	+	0.974566i	$0.571945\pi$
$132$	0	0
$133$	−296.324	−0.193192
$134$	0	0
$135$	−1456.00	−0.928241
$136$	0	0
$137$	1062.00	0.662283	0.331142	−	0.943581i	$-0.392566\pi$
$137$	1062.00	0.662283	0.331142	+	0.943581i	$0.392566\pi$
$138$	0	0
$139$	−2693.37	−1.64352	−0.821759	−	0.569835i	$-0.807007\pi$
$139$	−2693.37	−1.64352	−0.821759	+	0.569835i	$0.807007\pi$
$140$	0	0
$141$	1777.94	1.06191
$142$	0	0
$143$	840.000	0.491219
$144$	0	0
$145$	−1680.00	−0.962182
$146$	0	0
$147$	−1476.33	−0.828337
$148$	0	0
$149$	−793.725	−0.436406	−0.218203	−	0.975903i	$-0.570020\pi$
$149$	−793.725	−0.436406	−0.218203	+	0.975903i	$0.570020\pi$
$150$	0	0
$151$	−744.000	−0.400966	−0.200483	−	0.979697i	$-0.564251\pi$
$151$	−744.000	−0.400966	−0.200483	+	0.979697i	$0.564251\pi$
$152$	0	0
$153$	−14.0000	−0.00739760
$154$	0	0
$155$	2370.59	1.22846
$156$	0	0
$157$	179.911	0.0914552	0.0457276	−	0.998954i	$-0.485439\pi$
$157$	179.911	0.0914552	0.0457276	+	0.998954i	$0.485439\pi$
$158$	0	0
$159$	−168.000	−0.0837941
$160$	0	0
$161$	1216.00	0.595244
$162$	0	0
$163$	−1772.65	−0.851809	−0.425905	−	0.904768i	$-0.640044\pi$
$163$	−1772.65	−0.851809	−0.425905	+	0.904768i	$0.640044\pi$
$164$	0	0
$165$	888.972	0.419433
$166$	0	0
$167$	1960.00	0.908200	0.454100	−	0.890951i	$-0.349961\pi$
$167$	1960.00	0.908200	0.454100	+	0.890951i	$0.349961\pi$
$168$	0	0
$169$	603.000	0.274465
$170$	0	0
$171$	−37.0405	−0.0165647
$172$	0	0
$173$	2000.19	0.879026	0.439513	−	0.898236i	$-0.355151\pi$
$173$	2000.19	0.879026	0.439513	+	0.898236i	$0.355151\pi$
$174$	0	0
$175$	−104.000	−0.0449238
$176$	0	0
$177$	−2828.00	−1.20094
$178$	0	0
$179$	3264.86	1.36328	0.681639	−	0.731688i	$-0.261267\pi$
$179$	3264.86	1.36328	0.681639	+	0.731688i	$0.261267\pi$
$180$	0	0
$181$	2338.84	0.960469	0.480235	−	0.877140i	$-0.340552\pi$
$181$	2338.84	0.960469	0.480235	+	0.877140i	$0.340552\pi$
$182$	0	0
$183$	504.000	0.203589
$184$	0	0
$185$	−2576.00	−1.02374
$186$	0	0
$187$	−222.243	−0.0869092
$188$	0	0
$189$	−1100.63	−0.423594
$190$	0	0
$191$	3904.00	1.47897	0.739486	−	0.673172i	$-0.235069\pi$
$191$	3904.00	1.47897	0.739486	+	0.673172i	$0.235069\pi$
$192$	0	0
$193$	3330.00	1.24196	0.620981	−	0.783826i	$-0.286734\pi$
$193$	3330.00	1.24196	0.620981	+	0.783826i	$0.286734\pi$
$194$	0	0
$195$	2963.24	1.08822
$196$	0	0
$197$	1195.88	0.432502	0.216251	−	0.976338i	$-0.430617\pi$
$197$	1195.88	0.432502	0.216251	+	0.976338i	$0.430617\pi$
$198$	0	0
$199$	1736.00	0.618401	0.309200	−	0.950997i	$-0.399939\pi$
$199$	1736.00	0.618401	0.309200	+	0.950997i	$0.399939\pi$
$200$	0	0
$201$	924.000	0.324248
$202$	0	0
$203$	−1269.96	−0.439083
$204$	0	0
$205$	740.810	0.252392
$206$	0	0
$207$	152.000	0.0510373
$208$	0	0
$209$	−588.000	−0.194607
$210$	0	0
$211$	−2915.62	−0.951277	−0.475638	−	0.879641i	$-0.657783\pi$
$211$	−2915.62	−0.951277	−0.475638	+	0.879641i	$0.657783\pi$
$212$	0	0
$213$	380.988	0.122558
$214$	0	0
$215$	4648.00	1.47438
$216$	0	0
$217$	1792.00	0.560594
$218$	0	0
$219$	1555.70	0.480021
$220$	0	0
$221$	−740.810	−0.225486
$222$	0	0
$223$	1568.00	0.470857	0.235428	−	0.971892i	$-0.424351\pi$
$223$	1568.00	0.470857	0.235428	+	0.971892i	$0.424351\pi$
$224$	0	0
$225$	−13.0000	−0.00385185
$226$	0	0
$227$	−1264.67	−0.369775	−0.184888	−	0.982760i	$-0.559192\pi$
$227$	−1264.67	−0.369775	−0.184888	+	0.982760i	$0.559192\pi$
$228$	0	0
$229$	−5153.92	−1.48725	−0.743626	−	0.668595i	$-0.766896\pi$
$229$	−5153.92	−1.48725	−0.743626	+	0.668595i	$0.766896\pi$
$230$	0	0
$231$	672.000	0.191404
$232$	0	0
$233$	838.000	0.235619	0.117809	−	0.993036i	$-0.462413\pi$
$233$	838.000	0.235619	0.117809	+	0.993036i	$0.462413\pi$
$234$	0	0
$235$	3555.89	0.987067
$236$	0	0
$237$	−2455.26	−0.672937
$238$	0	0
$239$	6288.00	1.70183	0.850914	−	0.525305i	$-0.176049\pi$
$239$	6288.00	1.70183	0.850914	+	0.525305i	$0.176049\pi$
$240$	0	0
$241$	−2926.00	−0.782076	−0.391038	−	0.920375i	$-0.627884\pi$
$241$	−2926.00	−0.782076	−0.391038	+	0.920375i	$0.627884\pi$
$242$	0	0
$243$	−280.450	−0.0740364
$244$	0	0
$245$	−2952.66	−0.769953
$246$	0	0
$247$	−1960.00	−0.504906
$248$	0	0
$249$	−2884.00	−0.734000
$250$	0	0
$251$	−5444.96	−1.36925	−0.684627	−	0.728894i	$-0.740035\pi$
$251$	−5444.96	−1.36925	−0.684627	+	0.728894i	$0.740035\pi$
$252$	0	0
$253$	2412.93	0.599602
$254$	0	0
$255$	−784.000	−0.192533
$256$	0	0
$257$	2562.00	0.621841	0.310921	−	0.950436i	$-0.399363\pi$
$257$	2562.00	0.621841	0.310921	+	0.950436i	$0.399363\pi$
$258$	0	0
$259$	−1947.27	−0.467172
$260$	0	0
$261$	−158.745	−0.0376478
$262$	0	0
$263$	5896.00	1.38237	0.691184	−	0.722679i	$-0.257089\pi$
$263$	5896.00	1.38237	0.691184	+	0.722679i	$0.257089\pi$
$264$	0	0
$265$	−336.000	−0.0778880
$266$	0	0
$267$	−1407.54	−0.322622
$268$	0	0
$269$	−5365.58	−1.21615	−0.608077	−	0.793878i	$-0.708059\pi$
$269$	−5365.58	−1.21615	−0.608077	+	0.793878i	$0.708059\pi$
$270$	0	0
$271$	−1680.00	−0.376578	−0.188289	−	0.982114i	$-0.560294\pi$
$271$	−1680.00	−0.376578	−0.188289	+	0.982114i	$0.560294\pi$
$272$	0	0
$273$	2240.00	0.496597
$274$	0	0
$275$	−206.369	−0.0452527
$276$	0	0
$277$	1576.87	0.342039	0.171019	−	0.985268i	$-0.445294\pi$
$277$	1576.87	0.342039	0.171019	+	0.985268i	$0.445294\pi$
$278$	0	0
$279$	224.000	0.0480664
$280$	0	0
$281$	2742.00	0.582114	0.291057	−	0.956706i	$-0.405993\pi$
$281$	2742.00	0.582114	0.291057	+	0.956706i	$0.405993\pi$
$282$	0	0
$283$	−2989.70	−0.627983	−0.313991	−	0.949426i	$-0.601666\pi$
$283$	−2989.70	−0.627983	−0.313991	+	0.949426i	$0.601666\pi$
$284$	0	0
$285$	−2074.27	−0.431120
$286$	0	0
$287$	560.000	0.115177
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	5259.75	1.05956
$292$	0	0
$293$	9238.96	1.84214	0.921068	−	0.389401i	$-0.127318\pi$
$293$	9238.96	1.84214	0.921068	+	0.389401i	$0.127318\pi$
$294$	0	0
$295$	−5656.00	−1.11629
$296$	0	0
$297$	−2184.00	−0.426695
$298$	0	0
$299$	8043.08	1.55566
$300$	0	0
$301$	3513.56	0.672818
$302$	0	0
$303$	−3976.00	−0.753846
$304$	0	0
$305$	1008.00	0.189239
$306$	0	0
$307$	2587.54	0.481039	0.240520	−	0.970644i	$-0.422682\pi$
$307$	2587.54	0.481039	0.240520	+	0.970644i	$0.422682\pi$
$308$	0	0
$309$	−6222.81	−1.14564
$310$	0	0
$311$	2744.00	0.500315	0.250157	−	0.968205i	$-0.419518\pi$
$311$	2744.00	0.500315	0.250157	+	0.968205i	$0.419518\pi$
$312$	0	0
$313$	−2282.00	−0.412097	−0.206048	−	0.978542i	$-0.566060\pi$
$313$	−2282.00	−0.412097	−0.206048	+	0.978542i	$0.566060\pi$
$314$	0	0
$315$	84.6640	0.0151437
$316$	0	0
$317$	9577.62	1.69695	0.848474	−	0.529237i	$-0.177522\pi$
$317$	9577.62	1.69695	0.848474	+	0.529237i	$0.177522\pi$
$318$	0	0
$319$	−2520.00	−0.442298
$320$	0	0
$321$	1428.00	0.248297
$322$	0	0
$323$	518.567	0.0893308
$324$	0	0
$325$	−687.895	−0.117408
$326$	0	0
$327$	−10024.0	−1.69519
$328$	0	0
$329$	2688.00	0.450438
$330$	0	0
$331$	4249.08	0.705590	0.352795	−	0.935701i	$-0.385231\pi$
$331$	4249.08	0.705590	0.352795	+	0.935701i	$0.385231\pi$
$332$	0	0
$333$	−243.409	−0.0400563
$334$	0	0
$335$	1848.00	0.301394
$336$	0	0
$337$	6130.00	0.990868	0.495434	−	0.868646i	$-0.335009\pi$
$337$	6130.00	0.990868	0.495434	+	0.868646i	$0.335009\pi$
$338$	0	0
$339$	−9048.47	−1.44969
$340$	0	0
$341$	3555.89	0.564699
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	8512.00	1.32832
$346$	0	0
$347$	−2481.71	−0.383935	−0.191967	−	0.981401i	$-0.561487\pi$
$347$	−2481.71	−0.383935	−0.191967	+	0.981401i	$0.561487\pi$
$348$	0	0
$349$	−328.073	−0.0503191	−0.0251595	−	0.999683i	$-0.508009\pi$
$349$	−328.073	−0.0503191	−0.0251595	+	0.999683i	$0.508009\pi$
$350$	0	0
$351$	−7280.00	−1.10706
$352$	0	0
$353$	−10206.0	−1.53884	−0.769420	−	0.638743i	$-0.779455\pi$
$353$	−10206.0	−1.53884	−0.769420	+	0.638743i	$0.779455\pi$
$354$	0	0
$355$	761.976	0.113920
$356$	0	0
$357$	−592.648	−0.0878607
$358$	0	0
$359$	3176.00	0.466916	0.233458	−	0.972367i	$-0.424996\pi$
$359$	3176.00	0.466916	0.233458	+	0.972367i	$0.424996\pi$
$360$	0	0
$361$	−5487.00	−0.799971
$362$	0	0
$363$	−5709.53	−0.825545
$364$	0	0
$365$	3111.40	0.446187
$366$	0	0
$367$	−11760.0	−1.67266	−0.836331	−	0.548225i	$-0.815304\pi$
$367$	−11760.0	−1.67266	−0.836331	+	0.548225i	$0.815304\pi$
$368$	0	0
$369$	70.0000	0.00987549
$370$	0	0
$371$	−253.992	−0.0355434
$372$	0	0
$373$	10974.6	1.52344	0.761719	−	0.647908i	$-0.224356\pi$
$373$	10974.6	1.52344	0.761719	+	0.647908i	$0.224356\pi$
$374$	0	0
$375$	−7728.00	−1.06419
$376$	0	0
$377$	−8400.00	−1.14754
$378$	0	0
$379$	−3074.36	−0.416674	−0.208337	−	0.978057i	$-0.566805\pi$
$379$	−3074.36	−0.416674	−0.208337	+	0.978057i	$0.566805\pi$
$380$	0	0
$381$	−8805.06	−1.18398
$382$	0	0
$383$	2688.00	0.358617	0.179309	−	0.983793i	$-0.442614\pi$
$383$	2688.00	0.358617	0.179309	+	0.983793i	$0.442614\pi$
$384$	0	0
$385$	1344.00	0.177913
$386$	0	0
$387$	439.195	0.0576887
$388$	0	0
$389$	−10487.8	−1.36697	−0.683484	−	0.729966i	$-0.739536\pi$
$389$	−10487.8	−1.36697	−0.683484	+	0.729966i	$0.739536\pi$
$390$	0	0
$391$	−2128.00	−0.275237
$392$	0	0
$393$	−3556.00	−0.456429
$394$	0	0
$395$	−4910.51	−0.625506
$396$	0	0
$397$	−5704.24	−0.721127	−0.360564	−	0.932735i	$-0.617416\pi$
$397$	−5704.24	−0.721127	−0.360564	+	0.932735i	$0.617416\pi$
$398$	0	0
$399$	−1568.00	−0.196737
$400$	0	0
$401$	12402.0	1.54445	0.772227	−	0.635346i	$-0.219143\pi$
$401$	12402.0	1.54445	0.772227	+	0.635346i	$0.219143\pi$
$402$	0	0
$403$	11853.0	1.46511
$404$	0	0
$405$	−7990.17	−0.980333
$406$	0	0
$407$	−3864.00	−0.470593
$408$	0	0
$409$	12278.0	1.48437	0.742186	−	0.670194i	$-0.233789\pi$
$409$	12278.0	1.48437	0.742186	+	0.670194i	$0.233789\pi$
$410$	0	0
$411$	5619.58	0.674436
$412$	0	0
$413$	−4275.53	−0.509407
$414$	0	0
$415$	−5768.00	−0.682265
$416$	0	0
$417$	−14252.0	−1.67368
$418$	0	0
$419$	−8207.12	−0.956907	−0.478454	−	0.878113i	$-0.658802\pi$
$419$	−8207.12	−0.956907	−0.478454	+	0.878113i	$0.658802\pi$
$420$	0	0
$421$	1449.87	0.167844	0.0839221	−	0.996472i	$-0.473255\pi$
$421$	1449.87	0.167844	0.0839221	+	0.996472i	$0.473255\pi$
$422$	0	0
$423$	336.000	0.0386215
$424$	0	0
$425$	182.000	0.0207725
$426$	0	0
$427$	761.976	0.0863574
$428$	0	0
$429$	4444.86	0.500233
$430$	0	0
$431$	7632.00	0.852948	0.426474	−	0.904500i	$-0.359756\pi$
$431$	7632.00	0.852948	0.426474	+	0.904500i	$0.359756\pi$
$432$	0	0
$433$	3794.00	0.421081	0.210540	−	0.977585i	$-0.432478\pi$
$433$	3794.00	0.421081	0.210540	+	0.977585i	$0.432478\pi$
$434$	0	0
$435$	−8889.72	−0.979838
$436$	0	0
$437$	−5630.16	−0.616309
$438$	0	0
$439$	1848.00	0.200912	0.100456	−	0.994942i	$-0.467970\pi$
$439$	1848.00	0.200912	0.100456	+	0.994942i	$0.467970\pi$
$440$	0	0
$441$	−279.000	−0.0301263
$442$	0	0
$443$	12334.5	1.32287	0.661433	−	0.750004i	$-0.269949\pi$
$443$	12334.5	1.32287	0.661433	+	0.750004i	$0.269949\pi$
$444$	0	0
$445$	−2815.08	−0.299882
$446$	0	0
$447$	−4200.00	−0.444414
$448$	0	0
$449$	−3582.00	−0.376492	−0.188246	−	0.982122i	$-0.560280\pi$
$449$	−3582.00	−0.376492	−0.188246	+	0.982122i	$0.560280\pi$
$450$	0	0
$451$	1111.22	0.116020
$452$	0	0
$453$	−3936.88	−0.408324
$454$	0	0
$455$	4480.00	0.461595
$456$	0	0
$457$	−2714.00	−0.277802	−0.138901	−	0.990306i	$-0.544357\pi$
$457$	−2714.00	−0.277802	−0.138901	+	0.990306i	$0.544357\pi$
$458$	0	0
$459$	1926.11	0.195867
$460$	0	0
$461$	8349.99	0.843596	0.421798	−	0.906690i	$-0.361399\pi$
$461$	8349.99	0.843596	0.421798	+	0.906690i	$0.361399\pi$
$462$	0	0
$463$	2224.00	0.223236	0.111618	−	0.993751i	$-0.464397\pi$
$463$	2224.00	0.223236	0.111618	+	0.993751i	$0.464397\pi$
$464$	0	0
$465$	12544.0	1.25100
$466$	0	0
$467$	10292.0	1.01982	0.509910	−	0.860228i	$-0.329679\pi$
$467$	10292.0	1.01982	0.509910	+	0.860228i	$0.329679\pi$
$468$	0	0
$469$	1396.96	0.137538
$470$	0	0
$471$	952.000	0.0931334
$472$	0	0
$473$	6972.00	0.677744
$474$	0	0
$475$	481.527	0.0465136
$476$	0	0
$477$	−31.7490	−0.00304756
$478$	0	0
$479$	17696.0	1.68800	0.843999	−	0.536345i	$-0.180195\pi$
$479$	17696.0	1.68800	0.843999	+	0.536345i	$0.180195\pi$
$480$	0	0
$481$	−12880.0	−1.22095
$482$	0	0
$483$	6434.47	0.606166
$484$	0	0
$485$	10519.5	0.984879
$486$	0	0
$487$	−1304.00	−0.121334	−0.0606672	−	0.998158i	$-0.519323\pi$
$487$	−1304.00	−0.121334	−0.0606672	+	0.998158i	$0.519323\pi$
$488$	0	0
$489$	−9380.00	−0.867440
$490$	0	0
$491$	−16662.9	−1.53154	−0.765772	−	0.643112i	$-0.777643\pi$
$491$	−16662.9	−1.53154	−0.765772	+	0.643112i	$0.777643\pi$
$492$	0	0
$493$	2222.43	0.203029
$494$	0	0
$495$	168.000	0.0152546
$496$	0	0
$497$	576.000	0.0519862
$498$	0	0
$499$	3095.53	0.277705	0.138853	−	0.990313i	$-0.455659\pi$
$499$	3095.53	0.277705	0.138853	+	0.990313i	$0.455659\pi$
$500$	0	0
$501$	10371.3	0.924865
$502$	0	0
$503$	19320.0	1.71260	0.856298	−	0.516481i	$-0.172758\pi$
$503$	19320.0	1.71260	0.856298	+	0.516481i	$0.172758\pi$
$504$	0	0
$505$	−7952.00	−0.700712
$506$	0	0
$507$	3190.78	0.279502
$508$	0	0
$509$	−4476.61	−0.389828	−0.194914	−	0.980820i	$-0.562443\pi$
$509$	−4476.61	−0.389828	−0.194914	+	0.980820i	$0.562443\pi$
$510$	0	0
$511$	2352.00	0.203613
$512$	0	0
$513$	5096.00	0.438585
$514$	0	0
$515$	−12445.6	−1.06489
$516$	0	0
$517$	5333.83	0.453737
$518$	0	0
$519$	10584.0	0.895156
$520$	0	0
$521$	2982.00	0.250756	0.125378	−	0.992109i	$-0.459986\pi$
$521$	2982.00	0.250756	0.125378	+	0.992109i	$0.459986\pi$
$522$	0	0
$523$	−2016.06	−0.168559	−0.0842794	−	0.996442i	$-0.526859\pi$
$523$	−2016.06	−0.168559	−0.0842794	+	0.996442i	$0.526859\pi$
$524$	0	0
$525$	−550.316	−0.0457481
$526$	0	0
$527$	−3136.00	−0.259215
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	−534.442	−0.0436776
$532$	0	0
$533$	3704.05	0.301014
$534$	0	0
$535$	2856.00	0.230796
$536$	0	0
$537$	17276.0	1.38830
$538$	0	0
$539$	−4428.99	−0.353933
$540$	0	0
$541$	15419.4	1.22539	0.612693	−	0.790321i	$-0.290086\pi$
$541$	15419.4	1.22539	0.612693	+	0.790321i	$0.290086\pi$
$542$	0	0
$543$	12376.0	0.978094
$544$	0	0
$545$	−20048.0	−1.57571
$546$	0	0
$547$	−12609.7	−0.985649	−0.492824	−	0.870129i	$-0.664035\pi$
$547$	−12609.7	−0.985649	−0.492824	+	0.870129i	$0.664035\pi$
$548$	0	0
$549$	95.2470	0.00740445
$550$	0	0
$551$	5880.00	0.454621
$552$	0	0
$553$	−3712.00	−0.285444
$554$	0	0
$555$	−13630.9	−1.04252
$556$	0	0
$557$	−7143.53	−0.543413	−0.271706	−	0.962380i	$-0.587588\pi$
$557$	−7143.53	−0.543413	−0.271706	+	0.962380i	$0.587588\pi$
$558$	0	0
$559$	23240.0	1.75840
$560$	0	0
$561$	−1176.00	−0.0885040
$562$	0	0
$563$	−7572.14	−0.566834	−0.283417	−	0.958997i	$-0.591468\pi$
$563$	−7572.14	−0.566834	−0.283417	+	0.958997i	$0.591468\pi$
$564$	0	0
$565$	−18096.9	−1.34751
$566$	0	0
$567$	−6040.00	−0.447365
$568$	0	0
$569$	−15594.0	−1.14892	−0.574459	−	0.818533i	$-0.694788\pi$
$569$	−15594.0	−1.14892	−0.574459	+	0.818533i	$0.694788\pi$
$570$	0	0
$571$	16737.0	1.22666	0.613330	−	0.789827i	$-0.289830\pi$
$571$	16737.0	1.22666	0.613330	+	0.789827i	$0.289830\pi$
$572$	0	0
$573$	20658.0	1.50611
$574$	0	0
$575$	−1976.00	−0.143313
$576$	0	0
$577$	6594.00	0.475757	0.237879	−	0.971295i	$-0.423548\pi$
$577$	6594.00	0.475757	0.237879	+	0.971295i	$0.423548\pi$
$578$	0	0
$579$	17620.7	1.26475
$580$	0	0
$581$	−4360.20	−0.311345
$582$	0	0
$583$	−504.000	−0.0358037
$584$	0	0
$585$	560.000	0.0395780
$586$	0	0
$587$	23213.8	1.63226	0.816130	−	0.577868i	$-0.196115\pi$
$587$	23213.8	1.63226	0.816130	+	0.577868i	$0.196115\pi$
$588$	0	0
$589$	−8297.08	−0.580433
$590$	0	0
$591$	6328.00	0.440438
$592$	0	0
$593$	14322.0	0.991794	0.495897	−	0.868381i	$-0.334839\pi$
$593$	14322.0	0.991794	0.495897	+	0.868381i	$0.334839\pi$
$594$	0	0
$595$	−1185.30	−0.0816679
$596$	0	0
$597$	9186.05	0.629749
$598$	0	0
$599$	16088.0	1.09739	0.548696	−	0.836022i	$-0.315124\pi$
$599$	16088.0	1.09739	0.548696	+	0.836022i	$0.315124\pi$
$600$	0	0
$601$	21238.0	1.44146	0.720729	−	0.693217i	$-0.243807\pi$
$601$	21238.0	1.44146	0.720729	+	0.693217i	$0.243807\pi$
$602$	0	0
$603$	174.620	0.0117928
$604$	0	0
$605$	−11419.1	−0.767357
$606$	0	0
$607$	−13664.0	−0.913681	−0.456841	−	0.889549i	$-0.651019\pi$
$607$	−13664.0	−0.913681	−0.456841	+	0.889549i	$0.651019\pi$
$608$	0	0
$609$	−6720.00	−0.447140
$610$	0	0
$611$	17779.4	1.17722
$612$	0	0
$613$	−20393.5	−1.34369	−0.671846	−	0.740690i	$-0.734499\pi$
$613$	−20393.5	−1.34369	−0.671846	+	0.740690i	$0.734499\pi$
$614$	0	0
$615$	3920.00	0.257024
$616$	0	0
$617$	3782.00	0.246771	0.123385	−	0.992359i	$-0.460625\pi$
$617$	3782.00	0.246771	0.123385	+	0.992359i	$0.460625\pi$
$618$	0	0
$619$	−5825.94	−0.378295	−0.189147	−	0.981949i	$-0.560572\pi$
$619$	−5825.94	−0.378295	−0.189147	+	0.981949i	$0.560572\pi$
$620$	0	0
$621$	−20912.0	−1.35132
$622$	0	0
$623$	−2128.00	−0.136848
$624$	0	0
$625$	−13831.0	−0.885184
$626$	0	0
$627$	−3111.40	−0.198178
$628$	0	0
$629$	3407.73	0.216017
$630$	0	0
$631$	−2056.00	−0.129712	−0.0648558	−	0.997895i	$-0.520659\pi$
$631$	−2056.00	−0.129712	−0.0648558	+	0.997895i	$0.520659\pi$
$632$	0	0
$633$	−15428.0	−0.968733
$634$	0	0
$635$	−17610.1	−1.10053
$636$	0	0
$637$	−14763.3	−0.918278
$638$	0	0
$639$	72.0000	0.00445740
$640$	0	0
$641$	11842.0	0.729689	0.364845	−	0.931068i	$-0.381122\pi$
$641$	11842.0	0.729689	0.364845	+	0.931068i	$0.381122\pi$
$642$	0	0
$643$	−16250.2	−0.996649	−0.498325	−	0.866991i	$-0.666051\pi$
$643$	−16250.2	−0.996649	−0.498325	+	0.866991i	$0.666051\pi$
$644$	0	0
$645$	24594.9	1.50143
$646$	0	0
$647$	−19320.0	−1.17395	−0.586976	−	0.809604i	$-0.699682\pi$
$647$	−19320.0	−1.17395	−0.586976	+	0.809604i	$0.699682\pi$
$648$	0	0
$649$	−8484.00	−0.513137
$650$	0	0
$651$	9482.37	0.570881
$652$	0	0
$653$	−2317.68	−0.138894	−0.0694470	−	0.997586i	$-0.522123\pi$
$653$	−2317.68	−0.138894	−0.0694470	+	0.997586i	$0.522123\pi$
$654$	0	0
$655$	−7112.00	−0.424258
$656$	0	0
$657$	294.000	0.0174582
$658$	0	0
$659$	27732.8	1.63932	0.819662	−	0.572847i	$-0.194161\pi$
$659$	27732.8	1.63932	0.819662	+	0.572847i	$0.194161\pi$
$660$	0	0
$661$	−22467.7	−1.32208	−0.661039	−	0.750352i	$-0.729884\pi$
$661$	−22467.7	−1.32208	−0.661039	+	0.750352i	$0.729884\pi$
$662$	0	0
$663$	−3920.00	−0.229623
$664$	0	0
$665$	−3136.00	−0.182870
$666$	0	0
$667$	−24129.3	−1.40073
$668$	0	0
$669$	8297.08	0.479497
$670$	0	0
$671$	1512.00	0.0869897
$672$	0	0
$673$	−10078.0	−0.577234	−0.288617	−	0.957445i	$-0.593195\pi$
$673$	−10078.0	−0.577234	−0.288617	+	0.957445i	$0.593195\pi$
$674$	0	0
$675$	1788.53	0.101986
$676$	0	0
$677$	−16160.2	−0.917413	−0.458707	−	0.888588i	$-0.651687\pi$
$677$	−16160.2	−0.917413	−0.458707	+	0.888588i	$0.651687\pi$
$678$	0	0
$679$	7952.00	0.449440
$680$	0	0
$681$	−6692.00	−0.376561
$682$	0	0
$683$	16356.0	0.916320	0.458160	−	0.888870i	$-0.348509\pi$
$683$	16356.0	0.916320	0.458160	+	0.888870i	$0.348509\pi$
$684$	0	0
$685$	11239.2	0.626899
$686$	0	0
$687$	−27272.0	−1.51454
$688$	0	0
$689$	−1680.00	−0.0928925
$690$	0	0
$691$	−29246.1	−1.61009	−0.805047	−	0.593211i	$-0.797860\pi$
$691$	−29246.1	−1.61009	−0.805047	+	0.593211i	$0.797860\pi$
$692$	0	0
$693$	126.996	0.00696130
$694$	0	0
$695$	−28504.0	−1.55571
$696$	0	0
$697$	−980.000	−0.0532570
$698$	0	0
$699$	4434.28	0.239943
$700$	0	0
$701$	2465.84	0.132858	0.0664290	−	0.997791i	$-0.478839\pi$
$701$	2465.84	0.132858	0.0664290	+	0.997791i	$0.478839\pi$
$702$	0	0
$703$	9016.00	0.483705
$704$	0	0
$705$	18816.0	1.00518
$706$	0	0
$707$	−6011.15	−0.319763
$708$	0	0
$709$	31674.9	1.67782	0.838912	−	0.544267i	$-0.183192\pi$
$709$	31674.9	1.67782	0.838912	+	0.544267i	$0.183192\pi$
$710$	0	0
$711$	−464.000	−0.0244745
$712$	0	0
$713$	34048.0	1.78837
$714$	0	0
$715$	8889.72	0.464975
$716$	0	0
$717$	33273.0	1.73306
$718$	0	0
$719$	−9296.00	−0.482173	−0.241086	−	0.970504i	$-0.577504\pi$
$719$	−9296.00	−0.482173	−0.241086	+	0.970504i	$0.577504\pi$
$720$	0	0
$721$	−9408.00	−0.485953
$722$	0	0
$723$	−15482.9	−0.796427
$724$	0	0
$725$	2063.69	0.105715
$726$	0	0
$727$	−21672.0	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$727$	−21672.0	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$728$	0	0
$729$	18901.0	0.960270
$730$	0	0
$731$	−6148.73	−0.311106
$732$	0	0
$733$	−9471.79	−0.477283	−0.238642	−	0.971108i	$-0.576702\pi$
$733$	−9471.79	−0.477283	−0.238642	+	0.971108i	$0.576702\pi$
$734$	0	0
$735$	−15624.0	−0.784082
$736$	0	0
$737$	2772.00	0.138545
$738$	0	0
$739$	6863.08	0.341627	0.170814	−	0.985303i	$-0.445360\pi$
$739$	6863.08	0.341627	0.170814	+	0.985303i	$0.445360\pi$
$740$	0	0
$741$	−10371.3	−0.514171
$742$	0	0
$743$	−17432.0	−0.860724	−0.430362	−	0.902656i	$-0.641614\pi$
$743$	−17432.0	−0.860724	−0.430362	+	0.902656i	$0.641614\pi$
$744$	0	0
$745$	−8400.00	−0.413090
$746$	0	0
$747$	−545.025	−0.0266953
$748$	0	0
$749$	2158.93	0.105321
$750$	0	0
$751$	−11632.0	−0.565190	−0.282595	−	0.959239i	$-0.591195\pi$
$751$	−11632.0	−0.565190	−0.282595	+	0.959239i	$0.591195\pi$
$752$	0	0
$753$	−28812.0	−1.39438
$754$	0	0
$755$	−7873.76	−0.379543
$756$	0	0
$757$	16731.7	0.803336	0.401668	−	0.915785i	$-0.368431\pi$
$757$	16731.7	0.803336	0.401668	+	0.915785i	$0.368431\pi$
$758$	0	0
$759$	12768.0	0.610605
$760$	0	0
$761$	−39466.0	−1.87995	−0.939975	−	0.341244i	$-0.889152\pi$
$761$	−39466.0	−1.87995	−0.939975	+	0.341244i	$0.889152\pi$
$762$	0	0
$763$	−15154.9	−0.719060
$764$	0	0
$765$	−148.162	−0.00700237
$766$	0	0
$767$	−28280.0	−1.33133
$768$	0	0
$769$	35266.0	1.65374	0.826869	−	0.562395i	$-0.190120\pi$
$769$	35266.0	1.65374	0.826869	+	0.562395i	$0.190120\pi$
$770$	0	0
$771$	13556.8	0.633252
$772$	0	0
$773$	−16244.9	−0.755872	−0.377936	−	0.925832i	$-0.623366\pi$
$773$	−16244.9	−0.755872	−0.377936	+	0.925832i	$0.623366\pi$
$774$	0	0
$775$	−2912.00	−0.134970
$776$	0	0
$777$	−10304.0	−0.475745
$778$	0	0
$779$	−2592.84	−0.119253
$780$	0	0
$781$	1142.96	0.0523668
$782$	0	0
$783$	21840.0	0.996805
$784$	0	0
$785$	1904.00	0.0865690
$786$	0	0
$787$	34844.5	1.57824	0.789119	−	0.614240i	$-0.210537\pi$
$787$	34844.5	1.57824	0.789119	+	0.614240i	$0.210537\pi$
$788$	0	0
$789$	31198.7	1.40774
$790$	0	0
$791$	−13680.0	−0.614924
$792$	0	0
$793$	5040.00	0.225694
$794$	0	0
$795$	−1777.94	−0.0793172
$796$	0	0
$797$	2550.50	0.113354	0.0566772	−	0.998393i	$-0.481949\pi$
$797$	2550.50	0.113354	0.0566772	+	0.998393i	$0.481949\pi$
$798$	0	0
$799$	−4704.00	−0.208280
$800$	0	0
$801$	−266.000	−0.0117336
$802$	0	0
$803$	4667.11	0.205104
$804$	0	0
$805$	12868.9	0.563441
$806$	0	0
$807$	−28392.0	−1.23847
$808$	0	0
$809$	24390.0	1.05996	0.529979	−	0.848010i	$-0.322200\pi$
$809$	24390.0	1.05996	0.529979	+	0.848010i	$0.322200\pi$
$810$	0	0
$811$	9582.91	0.414922	0.207461	−	0.978243i	$-0.433480\pi$
$811$	9582.91	0.414922	0.207461	+	0.978243i	$0.433480\pi$
$812$	0	0
$813$	−8889.72	−0.383489
$814$	0	0
$815$	−18760.0	−0.806300
$816$	0	0
$817$	−16268.0	−0.696628
$818$	0	0
$819$	423.320	0.0180611
$820$	0	0
$821$	8773.31	0.372948	0.186474	−	0.982460i	$-0.440294\pi$
$821$	8773.31	0.372948	0.186474	+	0.982460i	$0.440294\pi$
$822$	0	0
$823$	21688.0	0.918586	0.459293	−	0.888285i	$-0.348103\pi$
$823$	21688.0	0.918586	0.459293	+	0.888285i	$0.348103\pi$
$824$	0	0
$825$	−1092.00	−0.0460831
$826$	0	0
$827$	19446.3	0.817670	0.408835	−	0.912608i	$-0.365935\pi$
$827$	19446.3	0.817670	0.408835	+	0.912608i	$0.365935\pi$
$828$	0	0
$829$	−19546.8	−0.818925	−0.409462	−	0.912327i	$-0.634284\pi$
$829$	−19546.8	−0.818925	−0.409462	+	0.912327i	$0.634284\pi$
$830$	0	0
$831$	8344.00	0.348315
$832$	0	0
$833$	3906.00	0.162467
$834$	0	0
$835$	20742.7	0.859677
$836$	0	0
$837$	−30817.7	−1.27266
$838$	0	0
$839$	18760.0	0.771951	0.385976	−	0.922509i	$-0.373865\pi$
$839$	18760.0	0.771951	0.385976	+	0.922509i	$0.373865\pi$
$840$	0	0
$841$	811.000	0.0332527
$842$	0	0
$843$	14509.3	0.592796
$844$	0	0
$845$	6381.55	0.259801
$846$	0	0
$847$	−8632.00	−0.350176
$848$	0	0
$849$	−15820.0	−0.639506
$850$	0	0
$851$	−36998.2	−1.49034
$852$	0	0
$853$	−28732.9	−1.15333	−0.576667	−	0.816979i	$-0.695647\pi$
$853$	−28732.9	−1.15333	−0.576667	+	0.816979i	$0.695647\pi$
$854$	0	0
$855$	−392.000	−0.0156797
$856$	0	0
$857$	−8778.00	−0.349884	−0.174942	−	0.984579i	$-0.555974\pi$
$857$	−8778.00	−0.349884	−0.174942	+	0.984579i	$0.555974\pi$
$858$	0	0
$859$	5646.03	0.224261	0.112130	−	0.993693i	$-0.464233\pi$
$859$	5646.03	0.224261	0.112130	+	0.993693i	$0.464233\pi$
$860$	0	0
$861$	2963.24	0.117290
$862$	0	0
$863$	−9312.00	−0.367305	−0.183652	−	0.982991i	$-0.558792\pi$
$863$	−9312.00	−0.367305	−0.183652	+	0.982991i	$0.558792\pi$
$864$	0	0
$865$	21168.0	0.832062
$866$	0	0
$867$	−24960.0	−0.977724
$868$	0	0
$869$	−7365.77	−0.287534
$870$	0	0
$871$	9240.00	0.359455
$872$	0	0
$873$	994.000	0.0385359
$874$	0	0
$875$	−11683.6	−0.451405
$876$	0	0
$877$	137.579	0.00529728	0.00264864	−	0.999996i	$-0.499157\pi$
$877$	137.579	0.00529728	0.00264864	+	0.999996i	$0.499157\pi$
$878$	0	0
$879$	48888.0	1.87594
$880$	0	0
$881$	−31150.0	−1.19123	−0.595613	−	0.803272i	$-0.703091\pi$
$881$	−31150.0	−1.19123	−0.595613	+	0.803272i	$0.703091\pi$
$882$	0	0
$883$	12577.9	0.479366	0.239683	−	0.970851i	$-0.422957\pi$
$883$	12577.9	0.479366	0.239683	+	0.970851i	$0.422957\pi$
$884$	0	0
$885$	−29928.7	−1.13677
$886$	0	0
$887$	−37128.0	−1.40545	−0.702726	−	0.711460i	$-0.748034\pi$
$887$	−37128.0	−1.40545	−0.702726	+	0.711460i	$0.748034\pi$
$888$	0	0
$889$	−13312.0	−0.502216
$890$	0	0
$891$	−11985.3	−0.450641
$892$	0	0
$893$	−12445.6	−0.466379
$894$	0	0
$895$	34552.0	1.29044
$896$	0	0
$897$	42560.0	1.58421
$898$	0	0
$899$	−35558.9	−1.31919
$900$	0	0
$901$	444.486	0.0164351
$902$	0	0
$903$	18592.0	0.685164
$904$	0	0
$905$	24752.0	0.909154
$906$	0	0
$907$	−35204.4	−1.28880	−0.644400	−	0.764688i	$-0.722893\pi$
$907$	−35204.4	−1.28880	−0.644400	+	0.764688i	$0.722893\pi$
$908$	0	0
$909$	−751.393	−0.0274171
$910$	0	0
$911$	−10512.0	−0.382303	−0.191152	−	0.981561i	$-0.561222\pi$
$911$	−10512.0	−0.382303	−0.191152	+	0.981561i	$0.561222\pi$
$912$	0	0
$913$	−8652.00	−0.313625
$914$	0	0
$915$	5333.83	0.192712
$916$	0	0
$917$	−5376.17	−0.193606
$918$	0	0
$919$	46104.0	1.65488	0.827438	−	0.561557i	$-0.189798\pi$
$919$	46104.0	1.65488	0.827438	+	0.561557i	$0.189798\pi$
$920$	0	0
$921$	13692.0	0.489866
$922$	0	0
$923$	3809.88	0.135865
$924$	0	0
$925$	3164.32	0.112478
$926$	0	0
$927$	−1176.00	−0.0416666
$928$	0	0
$929$	−5726.00	−0.202222	−0.101111	−	0.994875i	$-0.532240\pi$
$929$	−5726.00	−0.202222	−0.101111	+	0.994875i	$0.532240\pi$
$930$	0	0
$931$	10334.3	0.363795
$932$	0	0
$933$	14519.9	0.509496
$934$	0	0
$935$	−2352.00	−0.0822659
$936$	0	0
$937$	−1274.00	−0.0444181	−0.0222091	−	0.999753i	$-0.507070\pi$
$937$	−1274.00	−0.0444181	−0.0222091	+	0.999753i	$0.507070\pi$
$938$	0	0
$939$	−12075.2	−0.419659
$940$	0	0
$941$	−26446.9	−0.916201	−0.458101	−	0.888900i	$-0.651470\pi$
$941$	−26446.9	−0.916201	−0.458101	+	0.888900i	$0.651470\pi$
$942$	0	0
$943$	10640.0	0.367430
$944$	0	0
$945$	−11648.0	−0.400962
$946$	0	0
$947$	23922.9	0.820897	0.410448	−	0.911884i	$-0.365372\pi$
$947$	23922.9	0.820897	0.410448	+	0.911884i	$0.365372\pi$
$948$	0	0
$949$	15557.0	0.532141
$950$	0	0
$951$	50680.0	1.72809
$952$	0	0
$953$	−38250.0	−1.30015	−0.650073	−	0.759872i	$-0.725262\pi$
$953$	−38250.0	−1.30015	−0.650073	+	0.759872i	$0.725262\pi$
$954$	0	0
$955$	41316.1	1.39995
$956$	0	0
$957$	−13334.6	−0.450414
$958$	0	0
$959$	8496.00	0.286079
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	269.867	0.00903046
$964$	0	0
$965$	35241.4	1.17561
$966$	0	0
$967$	−4664.00	−0.155103	−0.0775513	−	0.996988i	$-0.524710\pi$
$967$	−4664.00	−0.155103	−0.0775513	+	0.996988i	$0.524710\pi$
$968$	0	0
$969$	2744.00	0.0909701
$970$	0	0
$971$	−30971.2	−1.02360	−0.511798	−	0.859106i	$-0.671020\pi$
$971$	−30971.2	−1.02360	−0.511798	+	0.859106i	$0.671020\pi$
$972$	0	0
$973$	−21547.0	−0.709933
$974$	0	0
$975$	−3640.00	−0.119562
$976$	0	0
$977$	−4814.00	−0.157639	−0.0788196	−	0.996889i	$-0.525115\pi$
$977$	−4814.00	−0.157639	−0.0788196	+	0.996889i	$0.525115\pi$
$978$	0	0
$979$	−4222.62	−0.137850
$980$	0	0
$981$	−1894.36	−0.0616536
$982$	0	0
$983$	12376.0	0.401560	0.200780	−	0.979636i	$-0.435652\pi$
$983$	12376.0	0.401560	0.200780	+	0.979636i	$0.435652\pi$
$984$	0	0
$985$	12656.0	0.409395
$986$	0	0
$987$	14223.6	0.458704
$988$	0	0
$989$	66757.6	2.14638
$990$	0	0
$991$	45344.0	1.45348	0.726740	−	0.686912i	$-0.241034\pi$
$991$	45344.0	1.45348	0.726740	+	0.686912i	$0.241034\pi$
$992$	0	0
$993$	22484.0	0.718538
$994$	0	0
$995$	18372.1	0.585361
$996$	0	0
$997$	26002.4	0.825984	0.412992	−	0.910735i	$-0.364484\pi$
$997$	26002.4	0.825984	0.412992	+	0.910735i	$0.364484\pi$
$998$	0	0
$999$	33488.0	1.06057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.a.l.1.2		2
3.2	odd	2		2304.4.a.bn.1.1		2
4.3	odd	2		256.4.a.j.1.1		2
8.3	odd	2		256.4.a.j.1.2		2
8.5	even	2	inner	256.4.a.l.1.1		2
12.11	even	2		2304.4.a.v.1.1		2
16.3	odd	4		32.4.b.a.17.1		2
16.5	even	4		8.4.b.a.5.2	yes	2
16.11	odd	4		32.4.b.a.17.2		2
16.13	even	4		8.4.b.a.5.1	✓	2
24.5	odd	2		2304.4.a.bn.1.2		2
24.11	even	2		2304.4.a.v.1.2		2
48.5	odd	4		72.4.d.b.37.1		2
48.11	even	4		288.4.d.a.145.1		2
48.29	odd	4		72.4.d.b.37.2		2
48.35	even	4		288.4.d.a.145.2		2
80.3	even	4		800.4.f.a.49.3		4
80.13	odd	4		200.4.f.a.149.2		4
80.19	odd	4		800.4.d.a.401.2		2
80.27	even	4		800.4.f.a.49.4		4
80.29	even	4		200.4.d.a.101.2		2
80.37	odd	4		200.4.f.a.149.1		4
80.43	even	4		800.4.f.a.49.1		4
80.53	odd	4		200.4.f.a.149.4		4
80.59	odd	4		800.4.d.a.401.1		2
80.67	even	4		800.4.f.a.49.2		4
80.69	even	4		200.4.d.a.101.1		2
80.77	odd	4		200.4.f.a.149.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.b.a.5.1	✓	2	16.13	even	4
8.4.b.a.5.2	yes	2	16.5	even	4
32.4.b.a.17.1		2	16.3	odd	4
32.4.b.a.17.2		2	16.11	odd	4
72.4.d.b.37.1		2	48.5	odd	4
72.4.d.b.37.2		2	48.29	odd	4
200.4.d.a.101.1		2	80.69	even	4
200.4.d.a.101.2		2	80.29	even	4
200.4.f.a.149.1		4	80.37	odd	4
200.4.f.a.149.2		4	80.13	odd	4
200.4.f.a.149.3		4	80.77	odd	4
200.4.f.a.149.4		4	80.53	odd	4
256.4.a.j.1.1		2	4.3	odd	2
256.4.a.j.1.2		2	8.3	odd	2
256.4.a.l.1.1		2	8.5	even	2	inner
256.4.a.l.1.2		2	1.1	even	1	trivial
288.4.d.a.145.1		2	48.11	even	4
288.4.d.a.145.2		2	48.35	even	4
800.4.d.a.401.1		2	80.59	odd	4
800.4.d.a.401.2		2	80.19	odd	4
800.4.f.a.49.1		4	80.43	even	4
800.4.f.a.49.2		4	80.67	even	4
800.4.f.a.49.3		4	80.3	even	4
800.4.f.a.49.4		4	80.27	even	4
2304.4.a.v.1.1		2	12.11	even	2
2304.4.a.v.1.2		2	24.11	even	2
2304.4.a.bn.1.1		2	3.2	odd	2
2304.4.a.bn.1.2		2	24.5	odd	2

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q+5.29150 q^{3} +10.5830 q^{5} +8.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+5.29150 q^{3} +10.5830 q^{5} +8.00000 q^{7} +1.00000 q^{9} +15.8745 q^{11} +52.9150 q^{13} +56.0000 q^{15} -14.0000 q^{17} -37.0405 q^{19} +42.3320 q^{21} +152.000 q^{23} -13.0000 q^{25} -137.579 q^{27} -158.745 q^{29} +224.000 q^{31} +84.0000 q^{33} +84.6640 q^{35} -243.409 q^{37} +280.000 q^{39} +70.0000 q^{41} +439.195 q^{43} +10.5830 q^{45} +336.000 q^{47} -279.000 q^{49} -74.0810 q^{51} -31.7490 q^{53} +168.000 q^{55} -196.000 q^{57} -534.442 q^{59} +95.2470 q^{61} +8.00000 q^{63} +560.000 q^{65} +174.620 q^{67} +804.308 q^{69} +72.0000 q^{71} +294.000 q^{73} -68.7895 q^{75} +126.996 q^{77} -464.000 q^{79} -755.000 q^{81} -545.025 q^{83} -148.162 q^{85} -840.000 q^{87} -266.000 q^{89} +423.320 q^{91} +1185.30 q^{93} -392.000 q^{95} +994.000 q^{97} +15.8745 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 16 * q^7 + 2 * q^9	\( 2 q + 16 q^{7} + 2 q^{9} + 112 q^{15} - 28 q^{17} + 304 q^{23} - 26 q^{25} + 448 q^{31} + 168 q^{33} + 560 q^{39} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 392 q^{57} + 16 q^{63} + 1120 q^{65} + 144 q^{71} + 588 q^{73} - 928 q^{79} - 1510 q^{81} - 1680 q^{87} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100}) \) 2 * q + 16 * q^7 + 2 * q^9 + 112 * q^15 - 28 * q^17 + 304 * q^23 - 26 * q^25 + 448 * q^31 + 168 * q^33 + 560 * q^39 + 140 * q^41 + 672 * q^47 - 558 * q^49 + 336 * q^55 - 392 * q^57 + 16 * q^63 + 1120 * q^65 + 144 * q^71 + 588 * q^73 - 928 * q^79 - 1510 * q^81 - 1680 * q^87 - 532 * q^89 - 784 * q^95 + 1988 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more