LMFDB - Embedded newform 8048.2.a.v.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67525 q^{3} -2.92493 q^{5} +1.65914 q^{7} -0.193522 q^{9} +O(q^{10})$	$q-1.67525 q^{3} -2.92493 q^{5} +1.65914 q^{7} -0.193522 q^{9} -5.49209 q^{11} -4.79504 q^{13} +4.90001 q^{15} +5.49743 q^{17} +0.255062 q^{19} -2.77948 q^{21} -3.87181 q^{23} +3.55524 q^{25} +5.34996 q^{27} +2.98739 q^{29} +5.08489 q^{31} +9.20064 q^{33} -4.85288 q^{35} -0.611446 q^{37} +8.03291 q^{39} +3.44582 q^{41} +3.26488 q^{43} +0.566040 q^{45} +2.23291 q^{47} -4.24725 q^{49} -9.20959 q^{51} -1.60340 q^{53} +16.0640 q^{55} -0.427293 q^{57} +8.61014 q^{59} -2.48498 q^{61} -0.321081 q^{63} +14.0252 q^{65} +8.51103 q^{67} +6.48626 q^{69} +15.2777 q^{71} -0.465430 q^{73} -5.95594 q^{75} -9.11215 q^{77} +0.500221 q^{79} -8.38198 q^{81} -8.22195 q^{83} -16.0796 q^{85} -5.00465 q^{87} -0.0767541 q^{89} -7.95565 q^{91} -8.51849 q^{93} -0.746039 q^{95} -13.7565 q^{97} +1.06284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−1.67525	−0.967209	−0.483604	−	0.875287i	$-0.660673\pi$
$3$	−1.67525	−0.967209	−0.483604	+	0.875287i	$0.660673\pi$
$4$	0	0
$5$	−2.92493	−1.30807	−0.654035	−	0.756464i	$-0.726925\pi$
$5$	−2.92493	−1.30807	−0.654035	+	0.756464i	$0.726925\pi$
$6$	0	0
$7$	1.65914	0.627097	0.313548	−	0.949572i	$-0.398482\pi$
$7$	1.65914	0.627097	0.313548	+	0.949572i	$0.398482\pi$
$8$	0	0
$9$	−0.193522	−0.0645075
$10$	0	0
$11$	−5.49209	−1.65593	−0.827963	−	0.560782i	$-0.810500\pi$
$11$	−5.49209	−1.65593	−0.827963	+	0.560782i	$0.810500\pi$
$12$	0	0
$13$	−4.79504	−1.32990	−0.664952	−	0.746886i	$-0.731548\pi$
$13$	−4.79504	−1.32990	−0.664952	+	0.746886i	$0.731548\pi$
$14$	0	0
$15$	4.90001	1.26518
$16$	0	0
$17$	5.49743	1.33332	0.666661	−	0.745361i	$-0.267723\pi$
$17$	5.49743	1.33332	0.666661	+	0.745361i	$0.267723\pi$
$18$	0	0
$19$	0.255062	0.0585152	0.0292576	−	0.999572i	$-0.490686\pi$
$19$	0.255062	0.0585152	0.0292576	+	0.999572i	$0.490686\pi$
$20$	0	0
$21$	−2.77948	−0.606533
$22$	0	0
$23$	−3.87181	−0.807328	−0.403664	−	0.914907i	$-0.632263\pi$
$23$	−3.87181	−0.807328	−0.403664	+	0.914907i	$0.632263\pi$
$24$	0	0
$25$	3.55524	0.711049
$26$	0	0
$27$	5.34996	1.02960
$28$	0	0
$29$	2.98739	0.554745	0.277373	−	0.960762i	$-0.410536\pi$
$29$	2.98739	0.554745	0.277373	+	0.960762i	$0.410536\pi$
$30$	0	0
$31$	5.08489	0.913273	0.456637	−	0.889653i	$-0.349054\pi$
$31$	5.08489	0.913273	0.456637	+	0.889653i	$0.349054\pi$
$32$	0	0
$33$	9.20064	1.60163
$34$	0	0
$35$	−4.85288	−0.820287
$36$	0	0
$37$	−0.611446	−0.100521	−0.0502605	−	0.998736i	$-0.516005\pi$
$37$	−0.611446	−0.100521	−0.0502605	+	0.998736i	$0.516005\pi$
$38$	0	0
$39$	8.03291	1.28629
$40$	0	0
$41$	3.44582	0.538147	0.269073	−	0.963120i	$-0.413283\pi$
$41$	3.44582	0.538147	0.269073	+	0.963120i	$0.413283\pi$
$42$	0	0
$43$	3.26488	0.497890	0.248945	−	0.968518i	$-0.419916\pi$
$43$	3.26488	0.497890	0.248945	+	0.968518i	$0.419916\pi$
$44$	0	0
$45$	0.566040	0.0843803
$46$	0	0
$47$	2.23291	0.325703	0.162852	−	0.986651i	$-0.447931\pi$
$47$	2.23291	0.325703	0.162852	+	0.986651i	$0.447931\pi$
$48$	0	0
$49$	−4.24725	−0.606750
$50$	0	0
$51$	−9.20959	−1.28960
$52$	0	0
$53$	−1.60340	−0.220244	−0.110122	−	0.993918i	$-0.535124\pi$
$53$	−1.60340	−0.220244	−0.110122	+	0.993918i	$0.535124\pi$
$54$	0	0
$55$	16.0640	2.16607
$56$	0	0
$57$	−0.427293	−0.0565964
$58$	0	0
$59$	8.61014	1.12094	0.560472	−	0.828173i	$-0.310620\pi$
$59$	8.61014	1.12094	0.560472	+	0.828173i	$0.310620\pi$
$60$	0	0
$61$	−2.48498	−0.318169	−0.159084	−	0.987265i	$-0.550854\pi$
$61$	−2.48498	−0.318169	−0.159084	+	0.987265i	$0.550854\pi$
$62$	0	0
$63$	−0.321081	−0.0404524
$64$	0	0
$65$	14.0252	1.73961
$66$	0	0
$67$	8.51103	1.03979	0.519894	−	0.854231i	$-0.325972\pi$
$67$	8.51103	1.03979	0.519894	+	0.854231i	$0.325972\pi$
$68$	0	0
$69$	6.48626	0.780854
$70$	0	0
$71$	15.2777	1.81313	0.906563	−	0.422070i	$-0.138696\pi$
$71$	15.2777	1.81313	0.906563	+	0.422070i	$0.138696\pi$
$72$	0	0
$73$	−0.465430	−0.0544744	−0.0272372	−	0.999629i	$-0.508671\pi$
$73$	−0.465430	−0.0544744	−0.0272372	+	0.999629i	$0.508671\pi$
$74$	0	0
$75$	−5.95594	−0.687733
$76$	0	0
$77$	−9.11215	−1.03843
$78$	0	0
$79$	0.500221	0.0562793	0.0281396	−	0.999604i	$-0.491042\pi$
$79$	0.500221	0.0562793	0.0281396	+	0.999604i	$0.491042\pi$
$80$	0	0
$81$	−8.38198	−0.931331
$82$	0	0
$83$	−8.22195	−0.902477	−0.451238	−	0.892404i	$-0.649018\pi$
$83$	−8.22195	−0.902477	−0.451238	+	0.892404i	$0.649018\pi$
$84$	0	0
$85$	−16.0796	−1.74408
$86$	0	0
$87$	−5.00465	−0.536554
$88$	0	0
$89$	−0.0767541	−0.00813591	−0.00406796	−	0.999992i	$-0.501295\pi$
$89$	−0.0767541	−0.00813591	−0.00406796	+	0.999992i	$0.501295\pi$
$90$	0	0
$91$	−7.95565	−0.833978
$92$	0	0
$93$	−8.51849	−0.883326
$94$	0	0
$95$	−0.746039	−0.0765420
$96$	0	0
$97$	−13.7565	−1.39676	−0.698378	−	0.715729i	$-0.746095\pi$
$97$	−13.7565	−1.39676	−0.698378	+	0.715729i	$0.746095\pi$
$98$	0	0
$99$	1.06284	0.106820
$100$	0	0
$101$	−3.96776	−0.394807	−0.197404	−	0.980322i	$-0.563251\pi$
$101$	−3.96776	−0.394807	−0.197404	+	0.980322i	$0.563251\pi$
$102$	0	0
$103$	−4.48119	−0.441545	−0.220772	−	0.975325i	$-0.570858\pi$
$103$	−4.48119	−0.441545	−0.220772	+	0.975325i	$0.570858\pi$
$104$	0	0
$105$	8.12981	0.793388
$106$	0	0
$107$	12.9807	1.25489	0.627445	−	0.778661i	$-0.284101\pi$
$107$	12.9807	1.25489	0.627445	+	0.778661i	$0.284101\pi$
$108$	0	0
$109$	−10.9753	−1.05125	−0.525623	−	0.850718i	$-0.676168\pi$
$109$	−10.9753	−1.05125	−0.525623	+	0.850718i	$0.676168\pi$
$110$	0	0
$111$	1.02433	0.0972249
$112$	0	0
$113$	10.8044	1.01639	0.508195	−	0.861242i	$-0.330313\pi$
$113$	10.8044	1.01639	0.508195	+	0.861242i	$0.330313\pi$
$114$	0	0
$115$	11.3248	1.05604
$116$	0	0
$117$	0.927947	0.0857887
$118$	0	0
$119$	9.12101	0.836122
$120$	0	0
$121$	19.1630	1.74209
$122$	0	0
$123$	−5.77263	−0.520500
$124$	0	0
$125$	4.22582	0.377969
$126$	0	0
$127$	10.4196	0.924591	0.462295	−	0.886726i	$-0.347026\pi$
$127$	10.4196	0.924591	0.462295	+	0.886726i	$0.347026\pi$
$128$	0	0
$129$	−5.46951	−0.481564
$130$	0	0
$131$	7.12290	0.622330	0.311165	−	0.950356i	$-0.399281\pi$
$131$	7.12290	0.622330	0.311165	+	0.950356i	$0.399281\pi$
$132$	0	0
$133$	0.423184	0.0366947
$134$	0	0
$135$	−15.6483	−1.34679
$136$	0	0
$137$	−0.131855	−0.0112651	−0.00563257	−	0.999984i	$-0.501793\pi$
$137$	−0.131855	−0.0112651	−0.00563257	+	0.999984i	$0.501793\pi$
$138$	0	0
$139$	−13.6936	−1.16148	−0.580739	−	0.814090i	$-0.697236\pi$
$139$	−13.6936	−1.16148	−0.580739	+	0.814090i	$0.697236\pi$
$140$	0	0
$141$	−3.74069	−0.315023
$142$	0	0
$143$	26.3348	2.20222
$144$	0	0
$145$	−8.73793	−0.725646
$146$	0	0
$147$	7.11522	0.586854
$148$	0	0
$149$	−14.9476	−1.22456	−0.612278	−	0.790642i	$-0.709747\pi$
$149$	−14.9476	−1.22456	−0.612278	+	0.790642i	$0.709747\pi$
$150$	0	0
$151$	−12.2637	−0.998006	−0.499003	−	0.866600i	$-0.666300\pi$
$151$	−12.2637	−0.998006	−0.499003	+	0.866600i	$0.666300\pi$
$152$	0	0
$153$	−1.06388	−0.0860092
$154$	0	0
$155$	−14.8730	−1.19463
$156$	0	0
$157$	−15.1623	−1.21009	−0.605043	−	0.796193i	$-0.706844\pi$
$157$	−15.1623	−1.21009	−0.605043	+	0.796193i	$0.706844\pi$
$158$	0	0
$159$	2.68611	0.213022
$160$	0	0
$161$	−6.42388	−0.506272
$162$	0	0
$163$	20.8649	1.63426	0.817131	−	0.576453i	$-0.195563\pi$
$163$	20.8649	1.63426	0.817131	+	0.576453i	$0.195563\pi$
$164$	0	0
$165$	−26.9113	−2.09504
$166$	0	0
$167$	−5.55081	−0.429535	−0.214767	−	0.976665i	$-0.568899\pi$
$167$	−5.55081	−0.429535	−0.214767	+	0.976665i	$0.568899\pi$
$168$	0	0
$169$	9.99239	0.768645
$170$	0	0
$171$	−0.0493601	−0.00377466
$172$	0	0
$173$	15.6414	1.18919	0.594596	−	0.804024i	$-0.297312\pi$
$173$	15.6414	1.18919	0.594596	+	0.804024i	$0.297312\pi$
$174$	0	0
$175$	5.89865	0.445896
$176$	0	0
$177$	−14.4242	−1.08419
$178$	0	0
$179$	14.1615	1.05848	0.529239	−	0.848473i	$-0.322478\pi$
$179$	14.1615	1.05848	0.529239	+	0.848473i	$0.322478\pi$
$180$	0	0
$181$	−4.64577	−0.345318	−0.172659	−	0.984982i	$-0.555236\pi$
$181$	−4.64577	−0.345318	−0.172659	+	0.984982i	$0.555236\pi$
$182$	0	0
$183$	4.16297	0.307736
$184$	0	0
$185$	1.78844	0.131489
$186$	0	0
$187$	−30.1924	−2.20788
$188$	0	0
$189$	8.87635	0.645659
$190$	0	0
$191$	−0.862120	−0.0623808	−0.0311904	−	0.999513i	$-0.509930\pi$
$191$	−0.862120	−0.0623808	−0.0311904	+	0.999513i	$0.509930\pi$
$192$	0	0
$193$	−14.7217	−1.05969	−0.529845	−	0.848094i	$-0.677750\pi$
$193$	−14.7217	−1.05969	−0.529845	+	0.848094i	$0.677750\pi$
$194$	0	0
$195$	−23.4957	−1.68256
$196$	0	0
$197$	−4.83919	−0.344778	−0.172389	−	0.985029i	$-0.555149\pi$
$197$	−4.83919	−0.344778	−0.172389	+	0.985029i	$0.555149\pi$
$198$	0	0
$199$	−1.37690	−0.0976057	−0.0488028	−	0.998808i	$-0.515541\pi$
$199$	−1.37690	−0.0976057	−0.0488028	+	0.998808i	$0.515541\pi$
$200$	0	0
$201$	−14.2581	−1.00569
$202$	0	0
$203$	4.95651	0.347879
$204$	0	0
$205$	−10.0788	−0.703934
$206$	0	0
$207$	0.749281	0.0520786
$208$	0	0
$209$	−1.40082	−0.0968968
$210$	0	0
$211$	−6.30875	−0.434313	−0.217156	−	0.976137i	$-0.569678\pi$
$211$	−6.30875	−0.434313	−0.217156	+	0.976137i	$0.569678\pi$
$212$	0	0
$213$	−25.5940	−1.75367
$214$	0	0
$215$	−9.54958	−0.651276
$216$	0	0
$217$	8.43656	0.572711
$218$	0	0
$219$	0.779713	0.0526881
$220$	0	0
$221$	−26.3604	−1.77319
$222$	0	0
$223$	3.09863	0.207500	0.103750	−	0.994603i	$-0.466916\pi$
$223$	3.09863	0.207500	0.103750	+	0.994603i	$0.466916\pi$
$224$	0	0
$225$	−0.688019	−0.0458679
$226$	0	0
$227$	−11.8291	−0.785126	−0.392563	−	0.919725i	$-0.628411\pi$
$227$	−11.8291	−0.785126	−0.392563	+	0.919725i	$0.628411\pi$
$228$	0	0
$229$	2.41943	0.159880	0.0799402	−	0.996800i	$-0.474527\pi$
$229$	2.41943	0.159880	0.0799402	+	0.996800i	$0.474527\pi$
$230$	0	0
$231$	15.2652	1.00437
$232$	0	0
$233$	11.4383	0.749346	0.374673	−	0.927157i	$-0.377755\pi$
$233$	11.4383	0.749346	0.374673	+	0.927157i	$0.377755\pi$
$234$	0	0
$235$	−6.53111	−0.426043
$236$	0	0
$237$	−0.837998	−0.0544338
$238$	0	0
$239$	13.0091	0.841489	0.420745	−	0.907179i	$-0.361769\pi$
$239$	13.0091	0.841489	0.420745	+	0.907179i	$0.361769\pi$
$240$	0	0
$241$	−8.03111	−0.517329	−0.258665	−	0.965967i	$-0.583282\pi$
$241$	−8.03111	−0.517329	−0.258665	+	0.965967i	$0.583282\pi$
$242$	0	0
$243$	−2.00794	−0.128809
$244$	0	0
$245$	12.4229	0.793672
$246$	0	0
$247$	−1.22303	−0.0778196
$248$	0	0
$249$	13.7739	0.872883
$250$	0	0
$251$	1.45532	0.0918591	0.0459296	−	0.998945i	$-0.485375\pi$
$251$	1.45532	0.0918591	0.0459296	+	0.998945i	$0.485375\pi$
$252$	0	0
$253$	21.2643	1.33687
$254$	0	0
$255$	26.9375	1.68689
$256$	0	0
$257$	5.18126	0.323198	0.161599	−	0.986857i	$-0.448335\pi$
$257$	5.18126	0.323198	0.161599	+	0.986857i	$0.448335\pi$
$258$	0	0
$259$	−1.01448	−0.0630364
$260$	0	0
$261$	−0.578128	−0.0357852
$262$	0	0
$263$	22.3406	1.37758	0.688789	−	0.724962i	$-0.258143\pi$
$263$	22.3406	1.37758	0.688789	+	0.724962i	$0.258143\pi$
$264$	0	0
$265$	4.68985	0.288095
$266$	0	0
$267$	0.128583	0.00786913
$268$	0	0
$269$	3.64979	0.222532	0.111266	−	0.993791i	$-0.464510\pi$
$269$	3.64979	0.222532	0.111266	+	0.993791i	$0.464510\pi$
$270$	0	0
$271$	8.57494	0.520891	0.260445	−	0.965489i	$-0.416131\pi$
$271$	8.57494	0.520891	0.260445	+	0.965489i	$0.416131\pi$
$272$	0	0
$273$	13.3277	0.806631
$274$	0	0
$275$	−19.5257	−1.17744
$276$	0	0
$277$	−3.48305	−0.209276	−0.104638	−	0.994510i	$-0.533368\pi$
$277$	−3.48305	−0.209276	−0.104638	+	0.994510i	$0.533368\pi$
$278$	0	0
$279$	−0.984040	−0.0589129
$280$	0	0
$281$	−4.26109	−0.254196	−0.127098	−	0.991890i	$-0.540566\pi$
$281$	−4.26109	−0.254196	−0.127098	+	0.991890i	$0.540566\pi$
$282$	0	0
$283$	−5.98973	−0.356052	−0.178026	−	0.984026i	$-0.556971\pi$
$283$	−5.98973	−0.356052	−0.178026	+	0.984026i	$0.556971\pi$
$284$	0	0
$285$	1.24981	0.0740321
$286$	0	0
$287$	5.71710	0.337470
$288$	0	0
$289$	13.2217	0.777748
$290$	0	0
$291$	23.0456	1.35095
$292$	0	0
$293$	28.1934	1.64708	0.823539	−	0.567259i	$-0.191996\pi$
$293$	28.1934	1.64708	0.823539	+	0.567259i	$0.191996\pi$
$294$	0	0
$295$	−25.1841	−1.46627
$296$	0	0
$297$	−29.3825	−1.70494
$298$	0	0
$299$	18.5655	1.07367
$300$	0	0
$301$	5.41691	0.312225
$302$	0	0
$303$	6.64701	0.381861
$304$	0	0
$305$	7.26840	0.416187
$306$	0	0
$307$	−13.9513	−0.796243	−0.398121	−	0.917333i	$-0.630338\pi$
$307$	−13.9513	−0.796243	−0.398121	+	0.917333i	$0.630338\pi$
$308$	0	0
$309$	7.50713	0.427066
$310$	0	0
$311$	−22.7146	−1.28803	−0.644014	−	0.765014i	$-0.722732\pi$
$311$	−22.7146	−1.28803	−0.644014	+	0.765014i	$0.722732\pi$
$312$	0	0
$313$	−28.6935	−1.62185	−0.810927	−	0.585147i	$-0.801037\pi$
$313$	−28.6935	−1.62185	−0.810927	+	0.585147i	$0.801037\pi$
$314$	0	0
$315$	0.939141	0.0529146
$316$	0	0
$317$	−17.8556	−1.00287	−0.501434	−	0.865196i	$-0.667194\pi$
$317$	−17.8556	−1.00287	−0.501434	+	0.865196i	$0.667194\pi$
$318$	0	0
$319$	−16.4070	−0.918617
$320$	0	0
$321$	−21.7459	−1.21374
$322$	0	0
$323$	1.40218	0.0780196
$324$	0	0
$325$	−17.0475	−0.945627
$326$	0	0
$327$	18.3865	1.01677
$328$	0	0
$329$	3.70471	0.204247
$330$	0	0
$331$	−14.0693	−0.773320	−0.386660	−	0.922222i	$-0.626371\pi$
$331$	−14.0693	−0.773320	−0.386660	+	0.922222i	$0.626371\pi$
$332$	0	0
$333$	0.118328	0.00648436
$334$	0	0
$335$	−24.8942	−1.36012
$336$	0	0
$337$	20.5736	1.12072	0.560359	−	0.828250i	$-0.310663\pi$
$337$	20.5736	1.12072	0.560359	+	0.828250i	$0.310663\pi$
$338$	0	0
$339$	−18.1001	−0.983061
$340$	0	0
$341$	−27.9267	−1.51231
$342$	0	0
$343$	−18.6608	−1.00759
$344$	0	0
$345$	−18.9719	−1.02141
$346$	0	0
$347$	−34.1486	−1.83319	−0.916596	−	0.399815i	$-0.869074\pi$
$347$	−34.1486	−1.83319	−0.916596	+	0.399815i	$0.869074\pi$
$348$	0	0
$349$	30.4760	1.63134	0.815672	−	0.578515i	$-0.196367\pi$
$349$	30.4760	1.63134	0.815672	+	0.578515i	$0.196367\pi$
$350$	0	0
$351$	−25.6533	−1.36927
$352$	0	0
$353$	4.48599	0.238765	0.119383	−	0.992848i	$-0.461908\pi$
$353$	4.48599	0.238765	0.119383	+	0.992848i	$0.461908\pi$
$354$	0	0
$355$	−44.6862	−2.37170
$356$	0	0
$357$	−15.2800	−0.808704
$358$	0	0
$359$	16.7300	0.882976	0.441488	−	0.897267i	$-0.354451\pi$
$359$	16.7300	0.882976	0.441488	+	0.897267i	$0.354451\pi$
$360$	0	0
$361$	−18.9349	−0.996576
$362$	0	0
$363$	−32.1029	−1.68497
$364$	0	0
$365$	1.36135	0.0712564
$366$	0	0
$367$	19.8684	1.03712	0.518562	−	0.855040i	$-0.326467\pi$
$367$	19.8684	1.03712	0.518562	+	0.855040i	$0.326467\pi$
$368$	0	0
$369$	−0.666843	−0.0347145
$370$	0	0
$371$	−2.66027	−0.138114
$372$	0	0
$373$	−7.63562	−0.395357	−0.197679	−	0.980267i	$-0.563340\pi$
$373$	−7.63562	−0.395357	−0.197679	+	0.980267i	$0.563340\pi$
$374$	0	0
$375$	−7.07932	−0.365574
$376$	0	0
$377$	−14.3247	−0.737758
$378$	0	0
$379$	27.4902	1.41208	0.706039	−	0.708173i	$-0.250480\pi$
$379$	27.4902	1.41208	0.706039	+	0.708173i	$0.250480\pi$
$380$	0	0
$381$	−17.4555	−0.894272
$382$	0	0
$383$	−5.09662	−0.260425	−0.130213	−	0.991486i	$-0.541566\pi$
$383$	−5.09662	−0.260425	−0.130213	+	0.991486i	$0.541566\pi$
$384$	0	0
$385$	26.6524	1.35833
$386$	0	0
$387$	−0.631828	−0.0321176
$388$	0	0
$389$	−20.8271	−1.05597	−0.527987	−	0.849252i	$-0.677053\pi$
$389$	−20.8271	−1.05597	−0.527987	+	0.849252i	$0.677053\pi$
$390$	0	0
$391$	−21.2850	−1.07643
$392$	0	0
$393$	−11.9327	−0.601923
$394$	0	0
$395$	−1.46311	−0.0736173
$396$	0	0
$397$	34.1090	1.71188	0.855942	−	0.517073i	$-0.172978\pi$
$397$	34.1090	1.71188	0.855942	+	0.517073i	$0.172978\pi$
$398$	0	0
$399$	−0.708940	−0.0354914
$400$	0	0
$401$	23.5003	1.17355	0.586774	−	0.809751i	$-0.300398\pi$
$401$	23.5003	1.17355	0.586774	+	0.809751i	$0.300398\pi$
$402$	0	0
$403$	−24.3822	−1.21457
$404$	0	0
$405$	24.5168	1.21825
$406$	0	0
$407$	3.35811	0.166456
$408$	0	0
$409$	33.5861	1.66073	0.830363	−	0.557222i	$-0.188133\pi$
$409$	33.5861	1.66073	0.830363	+	0.557222i	$0.188133\pi$
$410$	0	0
$411$	0.220891	0.0108957
$412$	0	0
$413$	14.2854	0.702940
$414$	0	0
$415$	24.0487	1.18050
$416$	0	0
$417$	22.9403	1.12339
$418$	0	0
$419$	16.7632	0.818938	0.409469	−	0.912324i	$-0.365714\pi$
$419$	16.7632	0.818938	0.409469	+	0.912324i	$0.365714\pi$
$420$	0	0
$421$	−17.2099	−0.838760	−0.419380	−	0.907811i	$-0.637753\pi$
$421$	−17.2099	−0.838760	−0.419380	+	0.907811i	$0.637753\pi$
$422$	0	0
$423$	−0.432118	−0.0210103
$424$	0	0
$425$	19.5447	0.948057
$426$	0	0
$427$	−4.12293	−0.199523
$428$	0	0
$429$	−44.1174	−2.13001
$430$	0	0
$431$	−12.1118	−0.583404	−0.291702	−	0.956509i	$-0.594222\pi$
$431$	−12.1118	−0.583404	−0.291702	+	0.956509i	$0.594222\pi$
$432$	0	0
$433$	23.2582	1.11772	0.558860	−	0.829262i	$-0.311239\pi$
$433$	23.2582	1.11772	0.558860	+	0.829262i	$0.311239\pi$
$434$	0	0
$435$	14.6383	0.701851
$436$	0	0
$437$	−0.987550	−0.0472409
$438$	0	0
$439$	−33.9160	−1.61872	−0.809362	−	0.587310i	$-0.800187\pi$
$439$	−33.9160	−1.61872	−0.809362	+	0.587310i	$0.800187\pi$
$440$	0	0
$441$	0.821938	0.0391399
$442$	0	0
$443$	−15.8855	−0.754742	−0.377371	−	0.926062i	$-0.623172\pi$
$443$	−15.8855	−0.754742	−0.377371	+	0.926062i	$0.623172\pi$
$444$	0	0
$445$	0.224501	0.0106423
$446$	0	0
$447$	25.0411	1.18440
$448$	0	0
$449$	−28.1366	−1.32785	−0.663924	−	0.747800i	$-0.731110\pi$
$449$	−28.1366	−1.32785	−0.663924	+	0.747800i	$0.731110\pi$
$450$	0	0
$451$	−18.9247	−0.891131
$452$	0	0
$453$	20.5448	0.965280
$454$	0	0
$455$	23.2698	1.09090
$456$	0	0
$457$	−24.8513	−1.16249	−0.581247	−	0.813727i	$-0.697435\pi$
$457$	−24.8513	−1.16249	−0.581247	+	0.813727i	$0.697435\pi$
$458$	0	0
$459$	29.4110	1.37279
$460$	0	0
$461$	−23.0225	−1.07226	−0.536132	−	0.844134i	$-0.680115\pi$
$461$	−23.0225	−1.07226	−0.536132	+	0.844134i	$0.680115\pi$
$462$	0	0
$463$	−25.1625	−1.16940	−0.584700	−	0.811250i	$-0.698788\pi$
$463$	−25.1625	−1.16940	−0.584700	+	0.811250i	$0.698788\pi$
$464$	0	0
$465$	24.9160	1.15545
$466$	0	0
$467$	−17.8394	−0.825511	−0.412755	−	0.910842i	$-0.635434\pi$
$467$	−17.8394	−0.825511	−0.412755	+	0.910842i	$0.635434\pi$
$468$	0	0
$469$	14.1210	0.652047
$470$	0	0
$471$	25.4008	1.17041
$472$	0	0
$473$	−17.9310	−0.824470
$474$	0	0
$475$	0.906807	0.0416071
$476$	0	0
$477$	0.310294	0.0142074
$478$	0	0
$479$	−35.7755	−1.63463	−0.817313	−	0.576194i	$-0.804537\pi$
$479$	−35.7755	−1.63463	−0.817313	+	0.576194i	$0.804537\pi$
$480$	0	0
$481$	2.93191	0.133683
$482$	0	0
$483$	10.7616	0.489671
$484$	0	0
$485$	40.2367	1.82706
$486$	0	0
$487$	−17.8250	−0.807726	−0.403863	−	0.914819i	$-0.632333\pi$
$487$	−17.8250	−0.807726	−0.403863	+	0.914819i	$0.632333\pi$
$488$	0	0
$489$	−34.9539	−1.58067
$490$	0	0
$491$	26.0616	1.17614	0.588072	−	0.808808i	$-0.299887\pi$
$491$	26.0616	1.17614	0.588072	+	0.808808i	$0.299887\pi$
$492$	0	0
$493$	16.4230	0.739654
$494$	0	0
$495$	−3.10874	−0.139728
$496$	0	0
$497$	25.3478	1.13701
$498$	0	0
$499$	13.3714	0.598588	0.299294	−	0.954161i	$-0.403249\pi$
$499$	13.3714	0.598588	0.299294	+	0.954161i	$0.403249\pi$
$500$	0	0
$501$	9.29902	0.415450
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	11.6054	0.516436
$506$	0	0
$507$	−16.7398	−0.743440
$508$	0	0
$509$	1.40210	0.0621471	0.0310736	−	0.999517i	$-0.490107\pi$
$509$	1.40210	0.0621471	0.0310736	+	0.999517i	$0.490107\pi$
$510$	0	0
$511$	−0.772214	−0.0341607
$512$	0	0
$513$	1.36457	0.0602473
$514$	0	0
$515$	13.1072	0.577572
$516$	0	0
$517$	−12.2633	−0.539341
$518$	0	0
$519$	−26.2033	−1.15020
$520$	0	0
$521$	−26.2278	−1.14906	−0.574531	−	0.818483i	$-0.694816\pi$
$521$	−26.2278	−1.14906	−0.574531	+	0.818483i	$0.694816\pi$
$522$	0	0
$523$	−13.9265	−0.608965	−0.304482	−	0.952518i	$-0.598483\pi$
$523$	−13.9265	−0.608965	−0.304482	+	0.952518i	$0.598483\pi$
$524$	0	0
$525$	−9.88175	−0.431275
$526$	0	0
$527$	27.9538	1.21769
$528$	0	0
$529$	−8.00911	−0.348222
$530$	0	0
$531$	−1.66625	−0.0723093
$532$	0	0
$533$	−16.5228	−0.715684
$534$	0	0
$535$	−37.9676	−1.64148
$536$	0	0
$537$	−23.7241	−1.02377
$538$	0	0
$539$	23.3263	1.00473
$540$	0	0
$541$	−13.0315	−0.560266	−0.280133	−	0.959961i	$-0.590379\pi$
$541$	−13.0315	−0.560266	−0.280133	+	0.959961i	$0.590379\pi$
$542$	0	0
$543$	7.78285	0.333994
$544$	0	0
$545$	32.1021	1.37510
$546$	0	0
$547$	−11.4973	−0.491590	−0.245795	−	0.969322i	$-0.579049\pi$
$547$	−11.4973	−0.491590	−0.245795	+	0.969322i	$0.579049\pi$
$548$	0	0
$549$	0.480899	0.0205243
$550$	0	0
$551$	0.761970	0.0324610
$552$	0	0
$553$	0.829938	0.0352925
$554$	0	0
$555$	−2.99609	−0.127177
$556$	0	0
$557$	−16.4495	−0.696988	−0.348494	−	0.937311i	$-0.613307\pi$
$557$	−16.4495	−0.696988	−0.348494	+	0.937311i	$0.613307\pi$
$558$	0	0
$559$	−15.6552	−0.662146
$560$	0	0
$561$	50.5799	2.13548
$562$	0	0
$563$	23.3455	0.983898	0.491949	−	0.870624i	$-0.336285\pi$
$563$	23.3455	0.983898	0.491949	+	0.870624i	$0.336285\pi$
$564$	0	0
$565$	−31.6021	−1.32951
$566$	0	0
$567$	−13.9069	−0.584035
$568$	0	0
$569$	−33.4867	−1.40383	−0.701917	−	0.712258i	$-0.747672\pi$
$569$	−33.4867	−1.40383	−0.701917	+	0.712258i	$0.747672\pi$
$570$	0	0
$571$	−3.50020	−0.146479	−0.0732393	−	0.997314i	$-0.523334\pi$
$571$	−3.50020	−0.146479	−0.0732393	+	0.997314i	$0.523334\pi$
$572$	0	0
$573$	1.44427	0.0603353
$574$	0	0
$575$	−13.7652	−0.574049
$576$	0	0
$577$	−47.1850	−1.96434	−0.982169	−	0.188003i	$-0.939799\pi$
$577$	−47.1850	−1.96434	−0.982169	+	0.188003i	$0.939799\pi$
$578$	0	0
$579$	24.6626	1.02494
$580$	0	0
$581$	−13.6414	−0.565940
$582$	0	0
$583$	8.80602	0.364708
$584$	0	0
$585$	−2.71418	−0.112218
$586$	0	0
$587$	24.5496	1.01327	0.506636	−	0.862160i	$-0.330889\pi$
$587$	24.5496	1.01327	0.506636	+	0.862160i	$0.330889\pi$
$588$	0	0
$589$	1.29696	0.0534404
$590$	0	0
$591$	8.10688	0.333472
$592$	0	0
$593$	−13.6091	−0.558858	−0.279429	−	0.960166i	$-0.590145\pi$
$593$	−13.6091	−0.558858	−0.279429	+	0.960166i	$0.590145\pi$
$594$	0	0
$595$	−26.6784	−1.09371
$596$	0	0
$597$	2.30665	0.0944050
$598$	0	0
$599$	−6.59210	−0.269346	−0.134673	−	0.990890i	$-0.542998\pi$
$599$	−6.59210	−0.269346	−0.134673	+	0.990890i	$0.542998\pi$
$600$	0	0
$601$	16.8678	0.688054	0.344027	−	0.938960i	$-0.388209\pi$
$601$	16.8678	0.688054	0.344027	+	0.938960i	$0.388209\pi$
$602$	0	0
$603$	−1.64707	−0.0670740
$604$	0	0
$605$	−56.0506	−2.27878
$606$	0	0
$607$	23.4907	0.953460	0.476730	−	0.879050i	$-0.341822\pi$
$607$	23.4907	0.953460	0.476730	+	0.879050i	$0.341822\pi$
$608$	0	0
$609$	−8.30342	−0.336471
$610$	0	0
$611$	−10.7069	−0.433154
$612$	0	0
$613$	−39.7929	−1.60722	−0.803611	−	0.595155i	$-0.797091\pi$
$613$	−39.7929	−1.60722	−0.803611	+	0.595155i	$0.797091\pi$
$614$	0	0
$615$	16.8846	0.680851
$616$	0	0
$617$	5.90322	0.237655	0.118827	−	0.992915i	$-0.462086\pi$
$617$	5.90322	0.237655	0.118827	+	0.992915i	$0.462086\pi$
$618$	0	0
$619$	33.4915	1.34614	0.673069	−	0.739579i	$-0.264976\pi$
$619$	33.4915	1.34614	0.673069	+	0.739579i	$0.264976\pi$
$620$	0	0
$621$	−20.7140	−0.831225
$622$	0	0
$623$	−0.127346	−0.00510200
$624$	0	0
$625$	−30.1365	−1.20546
$626$	0	0
$627$	2.34673	0.0937194
$628$	0	0
$629$	−3.36138	−0.134027
$630$	0	0
$631$	−23.7313	−0.944729	−0.472364	−	0.881403i	$-0.656599\pi$
$631$	−23.7313	−0.944729	−0.472364	+	0.881403i	$0.656599\pi$
$632$	0	0
$633$	10.5688	0.420071
$634$	0	0
$635$	−30.4767	−1.20943
$636$	0	0
$637$	20.3657	0.806919
$638$	0	0
$639$	−2.95657	−0.116960
$640$	0	0
$641$	−16.7510	−0.661626	−0.330813	−	0.943696i	$-0.607323\pi$
$641$	−16.7510	−0.661626	−0.330813	+	0.943696i	$0.607323\pi$
$642$	0	0
$643$	−8.80580	−0.347267	−0.173633	−	0.984810i	$-0.555551\pi$
$643$	−8.80580	−0.347267	−0.173633	+	0.984810i	$0.555551\pi$
$644$	0	0
$645$	15.9980	0.629919
$646$	0	0
$647$	−28.3986	−1.11646	−0.558232	−	0.829685i	$-0.688520\pi$
$647$	−28.3986	−1.11646	−0.558232	+	0.829685i	$0.688520\pi$
$648$	0	0
$649$	−47.2876	−1.85620
$650$	0	0
$651$	−14.1334	−0.553931
$652$	0	0
$653$	−1.81491	−0.0710230	−0.0355115	−	0.999369i	$-0.511306\pi$
$653$	−1.81491	−0.0710230	−0.0355115	+	0.999369i	$0.511306\pi$
$654$	0	0
$655$	−20.8340	−0.814052
$656$	0	0
$657$	0.0900711	0.00351401
$658$	0	0
$659$	−3.09694	−0.120640	−0.0603199	−	0.998179i	$-0.519212\pi$
$659$	−3.09694	−0.120640	−0.0603199	+	0.998179i	$0.519212\pi$
$660$	0	0
$661$	−32.2197	−1.25320	−0.626601	−	0.779340i	$-0.715554\pi$
$661$	−32.2197	−1.25320	−0.626601	+	0.779340i	$0.715554\pi$
$662$	0	0
$663$	44.1603	1.71505
$664$	0	0
$665$	−1.23778	−0.0479992
$666$	0	0
$667$	−11.5666	−0.447861
$668$	0	0
$669$	−5.19099	−0.200695
$670$	0	0
$671$	13.6477	0.526864
$672$	0	0
$673$	29.9648	1.15506	0.577528	−	0.816371i	$-0.304017\pi$
$673$	29.9648	1.15506	0.577528	+	0.816371i	$0.304017\pi$
$674$	0	0
$675$	19.0204	0.732096
$676$	0	0
$677$	−29.5307	−1.13496	−0.567479	−	0.823388i	$-0.692081\pi$
$677$	−29.5307	−1.13496	−0.567479	+	0.823388i	$0.692081\pi$
$678$	0	0
$679$	−22.8239	−0.875901
$680$	0	0
$681$	19.8168	0.759380
$682$	0	0
$683$	35.7975	1.36975	0.684876	−	0.728659i	$-0.259856\pi$
$683$	35.7975	1.36975	0.684876	+	0.728659i	$0.259856\pi$
$684$	0	0
$685$	0.385668	0.0147356
$686$	0	0
$687$	−4.05316	−0.154638
$688$	0	0
$689$	7.68837	0.292904
$690$	0	0
$691$	4.39357	0.167139	0.0835695	−	0.996502i	$-0.473368\pi$
$691$	4.39357	0.167139	0.0835695	+	0.996502i	$0.473368\pi$
$692$	0	0
$693$	1.76340	0.0669862
$694$	0	0
$695$	40.0530	1.51930
$696$	0	0
$697$	18.9432	0.717523
$698$	0	0
$699$	−19.1620	−0.724774
$700$	0	0
$701$	−17.1846	−0.649052	−0.324526	−	0.945877i	$-0.605205\pi$
$701$	−17.1846	−0.649052	−0.324526	+	0.945877i	$0.605205\pi$
$702$	0	0
$703$	−0.155956	−0.00588201
$704$	0	0
$705$	10.9413	0.412072
$706$	0	0
$707$	−6.58308	−0.247582
$708$	0	0
$709$	9.98499	0.374994	0.187497	−	0.982265i	$-0.439962\pi$
$709$	9.98499	0.374994	0.187497	+	0.982265i	$0.439962\pi$
$710$	0	0
$711$	−0.0968040	−0.00363043
$712$	0	0
$713$	−19.6877	−0.737311
$714$	0	0
$715$	−77.0275	−2.88066
$716$	0	0
$717$	−21.7936	−0.813896
$718$	0	0
$719$	−30.3013	−1.13005	−0.565024	−	0.825075i	$-0.691133\pi$
$719$	−30.3013	−1.13005	−0.565024	+	0.825075i	$0.691133\pi$
$720$	0	0
$721$	−7.43493	−0.276891
$722$	0	0
$723$	13.4542	0.500365
$724$	0	0
$725$	10.6209	0.394451
$726$	0	0
$727$	−39.0522	−1.44837	−0.724183	−	0.689608i	$-0.757783\pi$
$727$	−39.0522	−1.44837	−0.724183	+	0.689608i	$0.757783\pi$
$728$	0	0
$729$	28.5097	1.05592
$730$	0	0
$731$	17.9485	0.663848
$732$	0	0
$733$	51.5101	1.90257	0.951285	−	0.308312i	$-0.0997640\pi$
$733$	51.5101	1.90257	0.951285	+	0.308312i	$0.0997640\pi$
$734$	0	0
$735$	−20.8116	−0.767646
$736$	0	0
$737$	−46.7433	−1.72181
$738$	0	0
$739$	22.6626	0.833659	0.416829	−	0.908985i	$-0.363141\pi$
$739$	22.6626	0.833659	0.416829	+	0.908985i	$0.363141\pi$
$740$	0	0
$741$	2.04889	0.0752678
$742$	0	0
$743$	13.2455	0.485929	0.242965	−	0.970035i	$-0.421880\pi$
$743$	13.2455	0.485929	0.242965	+	0.970035i	$0.421880\pi$
$744$	0	0
$745$	43.7208	1.60181
$746$	0	0
$747$	1.59113	0.0582165
$748$	0	0
$749$	21.5368	0.786937
$750$	0	0
$751$	−31.6069	−1.15335	−0.576677	−	0.816972i	$-0.695651\pi$
$751$	−31.6069	−1.15335	−0.576677	+	0.816972i	$0.695651\pi$
$752$	0	0
$753$	−2.43804	−0.0888470
$754$	0	0
$755$	35.8705	1.30546
$756$	0	0
$757$	−13.6152	−0.494852	−0.247426	−	0.968907i	$-0.579585\pi$
$757$	−13.6152	−0.494852	−0.247426	+	0.968907i	$0.579585\pi$
$758$	0	0
$759$	−35.6231	−1.29304
$760$	0	0
$761$	−8.24166	−0.298760	−0.149380	−	0.988780i	$-0.547728\pi$
$761$	−8.24166	−0.298760	−0.149380	+	0.988780i	$0.547728\pi$
$762$	0	0
$763$	−18.2096	−0.659233
$764$	0	0
$765$	3.11177	0.112506
$766$	0	0
$767$	−41.2859	−1.49075
$768$	0	0
$769$	11.2334	0.405088	0.202544	−	0.979273i	$-0.435079\pi$
$769$	11.2334	0.405088	0.202544	+	0.979273i	$0.435079\pi$
$770$	0	0
$771$	−8.67993	−0.312600
$772$	0	0
$773$	42.3532	1.52334	0.761669	−	0.647966i	$-0.224380\pi$
$773$	42.3532	1.52334	0.761669	+	0.647966i	$0.224380\pi$
$774$	0	0
$775$	18.0780	0.649382
$776$	0	0
$777$	1.69950	0.0609694
$778$	0	0
$779$	0.878897	0.0314898
$780$	0	0
$781$	−83.9063	−3.00240
$782$	0	0
$783$	15.9824	0.571166
$784$	0	0
$785$	44.3488	1.58288
$786$	0	0
$787$	32.3707	1.15389	0.576945	−	0.816783i	$-0.304245\pi$
$787$	32.3707	1.15389	0.576945	+	0.816783i	$0.304245\pi$
$788$	0	0
$789$	−37.4261	−1.33241
$790$	0	0
$791$	17.9260	0.637375
$792$	0	0
$793$	11.9156	0.423134
$794$	0	0
$795$	−7.85669	−0.278648
$796$	0	0
$797$	15.7618	0.558313	0.279157	−	0.960246i	$-0.409945\pi$
$797$	15.7618	0.558313	0.279157	+	0.960246i	$0.409945\pi$
$798$	0	0
$799$	12.2753	0.434267
$800$	0	0
$801$	0.0148536	0.000524827 0
$802$	0	0
$803$	2.55618	0.0902056
$804$	0	0
$805$	18.7894	0.662240
$806$	0	0
$807$	−6.11433	−0.215234
$808$	0	0
$809$	2.61862	0.0920658	0.0460329	−	0.998940i	$-0.485342\pi$
$809$	2.61862	0.0920658	0.0460329	+	0.998940i	$0.485342\pi$
$810$	0	0
$811$	−46.5376	−1.63416	−0.817078	−	0.576527i	$-0.804408\pi$
$811$	−46.5376	−1.63416	−0.817078	+	0.576527i	$0.804408\pi$
$812$	0	0
$813$	−14.3652	−0.503810
$814$	0	0
$815$	−61.0283	−2.13773
$816$	0	0
$817$	0.832747	0.0291341
$818$	0	0
$819$	1.53960	0.0537978
$820$	0	0
$821$	−12.8542	−0.448616	−0.224308	−	0.974518i	$-0.572012\pi$
$821$	−12.8542	−0.448616	−0.224308	+	0.974518i	$0.572012\pi$
$822$	0	0
$823$	−43.1243	−1.50322	−0.751609	−	0.659609i	$-0.770722\pi$
$823$	−43.1243	−1.50322	−0.751609	+	0.659609i	$0.770722\pi$
$824$	0	0
$825$	32.7105	1.13883
$826$	0	0
$827$	40.1984	1.39783	0.698917	−	0.715203i	$-0.253666\pi$
$827$	40.1984	1.39783	0.698917	+	0.715203i	$0.253666\pi$
$828$	0	0
$829$	17.0860	0.593422	0.296711	−	0.954967i	$-0.404110\pi$
$829$	17.0860	0.593422	0.296711	+	0.954967i	$0.404110\pi$
$830$	0	0
$831$	5.83500	0.202414
$832$	0	0
$833$	−23.3489	−0.808993
$834$	0	0
$835$	16.2358	0.561862
$836$	0	0
$837$	27.2040	0.940307
$838$	0	0
$839$	24.8422	0.857648	0.428824	−	0.903388i	$-0.358928\pi$
$839$	24.8422	0.857648	0.428824	+	0.903388i	$0.358928\pi$
$840$	0	0
$841$	−20.0755	−0.692258
$842$	0	0
$843$	7.13842	0.245860
$844$	0	0
$845$	−29.2271	−1.00544
$846$	0	0
$847$	31.7942	1.09246
$848$	0	0
$849$	10.0343	0.344377
$850$	0	0
$851$	2.36740	0.0811534
$852$	0	0
$853$	24.0326	0.822862	0.411431	−	0.911441i	$-0.365029\pi$
$853$	24.0326	0.822862	0.411431	+	0.911441i	$0.365029\pi$
$854$	0	0
$855$	0.144375	0.00493753
$856$	0	0
$857$	3.56986	0.121944	0.0609720	−	0.998139i	$-0.480580\pi$
$857$	3.56986	0.121944	0.0609720	+	0.998139i	$0.480580\pi$
$858$	0	0
$859$	20.5639	0.701630	0.350815	−	0.936445i	$-0.385905\pi$
$859$	20.5639	0.701630	0.350815	+	0.936445i	$0.385905\pi$
$860$	0	0
$861$	−9.57761	−0.326404
$862$	0	0
$863$	45.2368	1.53988	0.769939	−	0.638118i	$-0.220287\pi$
$863$	45.2368	1.53988	0.769939	+	0.638118i	$0.220287\pi$
$864$	0	0
$865$	−45.7500	−1.55555
$866$	0	0
$867$	−22.1497	−0.752245
$868$	0	0
$869$	−2.74726	−0.0931943
$870$	0	0
$871$	−40.8107	−1.38282
$872$	0	0
$873$	2.66218	0.0901012
$874$	0	0
$875$	7.01123	0.237023
$876$	0	0
$877$	−23.4083	−0.790443	−0.395222	−	0.918586i	$-0.629332\pi$
$877$	−23.4083	−0.790443	−0.395222	+	0.918586i	$0.629332\pi$
$878$	0	0
$879$	−47.2312	−1.59307
$880$	0	0
$881$	17.1411	0.577499	0.288749	−	0.957405i	$-0.406761\pi$
$881$	17.1411	0.577499	0.288749	+	0.957405i	$0.406761\pi$
$882$	0	0
$883$	−34.4628	−1.15977	−0.579883	−	0.814700i	$-0.696902\pi$
$883$	−34.4628	−1.15977	−0.579883	+	0.814700i	$0.696902\pi$
$884$	0	0
$885$	42.1898	1.41819
$886$	0	0
$887$	−0.775217	−0.0260293	−0.0130146	−	0.999915i	$-0.504143\pi$
$887$	−0.775217	−0.0260293	−0.0130146	+	0.999915i	$0.504143\pi$
$888$	0	0
$889$	17.2876	0.579808
$890$	0	0
$891$	46.0346	1.54222
$892$	0	0
$893$	0.569529	0.0190586
$894$	0	0
$895$	−41.4214	−1.38456
$896$	0	0
$897$	−31.1019	−1.03846
$898$	0	0
$899$	15.1906	0.506634
$900$	0	0
$901$	−8.81459	−0.293656
$902$	0	0
$903$	−9.07470	−0.301987
$904$	0	0
$905$	13.5886	0.451700
$906$	0	0
$907$	−29.1042	−0.966389	−0.483194	−	0.875513i	$-0.660524\pi$
$907$	−29.1042	−0.966389	−0.483194	+	0.875513i	$0.660524\pi$
$908$	0	0
$909$	0.767851	0.0254680
$910$	0	0
$911$	42.0173	1.39210	0.696048	−	0.717995i	$-0.254940\pi$
$911$	42.0173	1.39210	0.696048	+	0.717995i	$0.254940\pi$
$912$	0	0
$913$	45.1557	1.49443
$914$	0	0
$915$	−12.1764	−0.402540
$916$	0	0
$917$	11.8179	0.390261
$918$	0	0
$919$	4.95872	0.163573	0.0817865	−	0.996650i	$-0.473937\pi$
$919$	4.95872	0.163573	0.0817865	+	0.996650i	$0.473937\pi$
$920$	0	0
$921$	23.3720	0.770133
$922$	0	0
$923$	−73.2570	−2.41128
$924$	0	0
$925$	−2.17384	−0.0714754
$926$	0	0
$927$	0.867210	0.0284829
$928$	0	0
$929$	−28.2747	−0.927663	−0.463832	−	0.885923i	$-0.653526\pi$
$929$	−28.2747	−0.927663	−0.463832	+	0.885923i	$0.653526\pi$
$930$	0	0
$931$	−1.08331	−0.0355041
$932$	0	0
$933$	38.0528	1.24579
$934$	0	0
$935$	88.3107	2.88807
$936$	0	0
$937$	58.6513	1.91605	0.958027	−	0.286677i	$-0.0925505\pi$
$937$	58.6513	1.91605	0.958027	+	0.286677i	$0.0925505\pi$
$938$	0	0
$939$	48.0690	1.56867
$940$	0	0
$941$	35.5220	1.15798	0.578991	−	0.815334i	$-0.303446\pi$
$941$	35.5220	1.15798	0.578991	+	0.815334i	$0.303446\pi$
$942$	0	0
$943$	−13.3416	−0.434461
$944$	0	0
$945$	−25.9627	−0.844568
$946$	0	0
$947$	−7.64695	−0.248492	−0.124246	−	0.992251i	$-0.539651\pi$
$947$	−7.64695	−0.248492	−0.124246	+	0.992251i	$0.539651\pi$
$948$	0	0
$949$	2.23175	0.0724458
$950$	0	0
$951$	29.9126	0.969982
$952$	0	0
$953$	41.2516	1.33627	0.668135	−	0.744040i	$-0.267093\pi$
$953$	41.2516	1.33627	0.668135	+	0.744040i	$0.267093\pi$
$954$	0	0
$955$	2.52165	0.0815985
$956$	0	0
$957$	27.4859	0.888495
$958$	0	0
$959$	−0.218766	−0.00706433
$960$	0	0
$961$	−5.14388	−0.165932
$962$	0	0
$963$	−2.51205	−0.0809497
$964$	0	0
$965$	43.0600	1.38615
$966$	0	0
$967$	−44.2579	−1.42324	−0.711620	−	0.702565i	$-0.752038\pi$
$967$	−44.2579	−1.42324	−0.711620	+	0.702565i	$0.752038\pi$
$968$	0	0
$969$	−2.34901	−0.0754612
$970$	0	0
$971$	9.32573	0.299277	0.149638	−	0.988741i	$-0.452189\pi$
$971$	9.32573	0.299277	0.149638	+	0.988741i	$0.452189\pi$
$972$	0	0
$973$	−22.7197	−0.728359
$974$	0	0
$975$	28.5590	0.914618
$976$	0	0
$977$	27.6771	0.885470	0.442735	−	0.896653i	$-0.354008\pi$
$977$	27.6771	0.885470	0.442735	+	0.896653i	$0.354008\pi$
$978$	0	0
$979$	0.421540	0.0134725
$980$	0	0
$981$	2.12397	0.0678132
$982$	0	0
$983$	−58.9984	−1.88176	−0.940878	−	0.338744i	$-0.889998\pi$
$983$	−58.9984	−1.88176	−0.940878	+	0.338744i	$0.889998\pi$
$984$	0	0
$985$	14.1543	0.450994
$986$	0	0
$987$	−6.20633	−0.197550
$988$	0	0
$989$	−12.6410	−0.401960
$990$	0	0
$991$	24.7333	0.785679	0.392839	−	0.919607i	$-0.371493\pi$
$991$	24.7333	0.785679	0.392839	+	0.919607i	$0.371493\pi$
$992$	0	0
$993$	23.5697	0.747961
$994$	0	0
$995$	4.02733	0.127675
$996$	0	0
$997$	−47.7695	−1.51287	−0.756437	−	0.654066i	$-0.773062\pi$
$997$	−47.7695	−1.51287	−0.756437	+	0.654066i	$0.773062\pi$
$998$	0	0
$999$	−3.27121	−0.103497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.7		28
4.3	odd	2		4024.2.a.d.1.22	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.22	✓	28	4.3	odd	2
8048.2.a.v.1.7		28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67525 q^{3} -2.92493 q^{5} +1.65914 q^{7} -0.193522 q^{9} +O(q^{10})\)	\(q-1.67525 q^{3} -2.92493 q^{5} +1.65914 q^{7} -0.193522 q^{9} -5.49209 q^{11} -4.79504 q^{13} +4.90001 q^{15} +5.49743 q^{17} +0.255062 q^{19} -2.77948 q^{21} -3.87181 q^{23} +3.55524 q^{25} +5.34996 q^{27} +2.98739 q^{29} +5.08489 q^{31} +9.20064 q^{33} -4.85288 q^{35} -0.611446 q^{37} +8.03291 q^{39} +3.44582 q^{41} +3.26488 q^{43} +0.566040 q^{45} +2.23291 q^{47} -4.24725 q^{49} -9.20959 q^{51} -1.60340 q^{53} +16.0640 q^{55} -0.427293 q^{57} +8.61014 q^{59} -2.48498 q^{61} -0.321081 q^{63} +14.0252 q^{65} +8.51103 q^{67} +6.48626 q^{69} +15.2777 q^{71} -0.465430 q^{73} -5.95594 q^{75} -9.11215 q^{77} +0.500221 q^{79} -8.38198 q^{81} -8.22195 q^{83} -16.0796 q^{85} -5.00465 q^{87} -0.0767541 q^{89} -7.95565 q^{91} -8.51849 q^{93} -0.746039 q^{95} -13.7565 q^{97} +1.06284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more