LMFDB - Embedded newform 9984.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9984,2,Mod(1,9984)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9984, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9984.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9984 = 2^{8} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9984.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:	$79.7226413780$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4992)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	9984.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.82843 q^{5} +1.41421 q^{7} +1.00000 q^{9} -0.585786 q^{11} -1.00000 q^{13} -2.82843 q^{15} -2.82843 q^{17} -5.41421 q^{19} +1.41421 q^{21} +2.82843 q^{23} +3.00000 q^{25} +1.00000 q^{27} +3.65685 q^{29} +5.41421 q^{31} -0.585786 q^{33} -4.00000 q^{35} +2.00000 q^{37} -1.00000 q^{39} +6.82843 q^{41} +5.65685 q^{43} -2.82843 q^{45} +3.41421 q^{47} -5.00000 q^{49} -2.82843 q^{51} +1.17157 q^{53} +1.65685 q^{55} -5.41421 q^{57} +2.24264 q^{59} -9.65685 q^{61} +1.41421 q^{63} +2.82843 q^{65} -3.75736 q^{67} +2.82843 q^{69} +11.8995 q^{71} -7.65685 q^{73} +3.00000 q^{75} -0.828427 q^{77} +2.82843 q^{79} +1.00000 q^{81} -14.7279 q^{83} +8.00000 q^{85} +3.65685 q^{87} +2.82843 q^{89} -1.41421 q^{91} +5.41421 q^{93} +15.3137 q^{95} -11.6569 q^{97} -0.585786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 8 q^{19} + 6 q^{25} + 2 q^{27} - 4 q^{29} + 8 q^{31} - 4 q^{33} - 8 q^{35} + 4 q^{37} - 2 q^{39} + 8 q^{41} + 4 q^{47} - 10 q^{49} + 8 q^{53} - 8 q^{55} - 8 q^{57}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 8 * q^19 + 6 * q^25 + 2 * q^27 - 4 * q^29 + 8 * q^31 - 4 * q^33 - 8 * q^35 + 4 * q^37 - 2 * q^39 + 8 * q^41 + 4 * q^47 - 10 * q^49 + 8 * q^53 - 8 * q^55 - 8 * q^57 - 4 * q^59 - 8 * q^61 - 16 * q^67 + 4 * q^71 - 4 * q^73 + 6 * q^75 + 4 * q^77 + 2 * q^81 - 4 * q^83 + 16 * q^85 - 4 * q^87 + 8 * q^93 + 8 * q^95 - 12 * q^97 - 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.82843	−0.730297
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	5.41421	0.972421	0.486211	−	0.873842i	$-0.338379\pi$
$31$	5.41421	0.972421	0.486211	+	0.873842i	$0.338379\pi$
$32$	0	0
$33$	−0.585786	−0.101972
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.82843	1.06642	0.533211	−	0.845983i	$-0.320985\pi$
$41$	6.82843	1.06642	0.533211	+	0.845983i	$0.320985\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	3.41421	0.498014	0.249007	−	0.968502i	$-0.419896\pi$
$47$	3.41421	0.498014	0.249007	+	0.968502i	$0.419896\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	−2.82843	−0.396059
$52$	0	0
$53$	1.17157	0.160928	0.0804640	−	0.996758i	$-0.474360\pi$
$53$	1.17157	0.160928	0.0804640	+	0.996758i	$0.474360\pi$
$54$	0	0
$55$	1.65685	0.223410
$56$	0	0
$57$	−5.41421	−0.717130
$58$	0	0
$59$	2.24264	0.291967	0.145983	−	0.989287i	$-0.453365\pi$
$59$	2.24264	0.291967	0.145983	+	0.989287i	$0.453365\pi$
$60$	0	0
$61$	−9.65685	−1.23643	−0.618217	−	0.786008i	$-0.712145\pi$
$61$	−9.65685	−1.23643	−0.618217	+	0.786008i	$0.712145\pi$
$62$	0	0
$63$	1.41421	0.178174
$64$	0	0
$65$	2.82843	0.350823
$66$	0	0
$67$	−3.75736	−0.459034	−0.229517	−	0.973305i	$-0.573715\pi$
$67$	−3.75736	−0.459034	−0.229517	+	0.973305i	$0.573715\pi$
$68$	0	0
$69$	2.82843	0.340503
$70$	0	0
$71$	11.8995	1.41221	0.706105	−	0.708107i	$-0.250451\pi$
$71$	11.8995	1.41221	0.706105	+	0.708107i	$0.250451\pi$
$72$	0	0
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	−0.828427	−0.0944080
$78$	0	0
$79$	2.82843	0.318223	0.159111	−	0.987261i	$-0.449137\pi$
$79$	2.82843	0.318223	0.159111	+	0.987261i	$0.449137\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.7279	−1.61660	−0.808300	−	0.588771i	$-0.799612\pi$
$83$	−14.7279	−1.61660	−0.808300	+	0.588771i	$0.799612\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	3.65685	0.392056
$88$	0	0
$89$	2.82843	0.299813	0.149906	−	0.988700i	$-0.452103\pi$
$89$	2.82843	0.299813	0.149906	+	0.988700i	$0.452103\pi$
$90$	0	0
$91$	−1.41421	−0.148250
$92$	0	0
$93$	5.41421	0.561428
$94$	0	0
$95$	15.3137	1.57115
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	−0.585786	−0.0588738
$100$	0	0
$101$	−9.17157	−0.912606	−0.456303	−	0.889825i	$-0.650827\pi$
$101$	−9.17157	−0.912606	−0.456303	+	0.889825i	$0.650827\pi$
$102$	0	0
$103$	−9.17157	−0.903702	−0.451851	−	0.892093i	$-0.649236\pi$
$103$	−9.17157	−0.903702	−0.451851	+	0.892093i	$0.649236\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	1.17157	0.113260	0.0566301	−	0.998395i	$-0.481964\pi$
$107$	1.17157	0.113260	0.0566301	+	0.998395i	$0.481964\pi$
$108$	0	0
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−8.48528	−0.798228	−0.399114	−	0.916901i	$-0.630682\pi$
$113$	−8.48528	−0.798228	−0.399114	+	0.916901i	$0.630682\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.6569	−0.968805
$122$	0	0
$123$	6.82843	0.615699
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	13.6569	1.21185	0.605925	−	0.795522i	$-0.292803\pi$
$127$	13.6569	1.21185	0.605925	+	0.795522i	$0.292803\pi$
$128$	0	0
$129$	5.65685	0.498058
$130$	0	0
$131$	−9.17157	−0.801324	−0.400662	−	0.916226i	$-0.631220\pi$
$131$	−9.17157	−0.801324	−0.400662	+	0.916226i	$0.631220\pi$
$132$	0	0
$133$	−7.65685	−0.663933
$134$	0	0
$135$	−2.82843	−0.243432
$136$	0	0
$137$	0.485281	0.0414604	0.0207302	−	0.999785i	$-0.493401\pi$
$137$	0.485281	0.0414604	0.0207302	+	0.999785i	$0.493401\pi$
$138$	0	0
$139$	−14.1421	−1.19952	−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421	−1.19952	−0.599760	+	0.800180i	$0.704737\pi$
$140$	0	0
$141$	3.41421	0.287529
$142$	0	0
$143$	0.585786	0.0489859
$144$	0	0
$145$	−10.3431	−0.858952
$146$	0	0
$147$	−5.00000	−0.412393
$148$	0	0
$149$	9.17157	0.751365	0.375682	−	0.926749i	$-0.377408\pi$
$149$	9.17157	0.751365	0.375682	+	0.926749i	$0.377408\pi$
$150$	0	0
$151$	−0.242641	−0.0197458	−0.00987291	−	0.999951i	$-0.503143\pi$
$151$	−0.242641	−0.0197458	−0.00987291	+	0.999951i	$0.503143\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	−15.3137	−1.23003
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	0	0
$159$	1.17157	0.0929118
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	11.5563	0.905163	0.452582	−	0.891723i	$-0.350503\pi$
$163$	11.5563	0.905163	0.452582	+	0.891723i	$0.350503\pi$
$164$	0	0
$165$	1.65685	0.128986
$166$	0	0
$167$	−12.5858	−0.973917	−0.486959	−	0.873425i	$-0.661894\pi$
$167$	−12.5858	−0.973917	−0.486959	+	0.873425i	$0.661894\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.41421	−0.414035
$172$	0	0
$173$	12.4853	0.949238	0.474619	−	0.880191i	$-0.342586\pi$
$173$	12.4853	0.949238	0.474619	+	0.880191i	$0.342586\pi$
$174$	0	0
$175$	4.24264	0.320713
$176$	0	0
$177$	2.24264	0.168567
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−13.3137	−0.989600	−0.494800	−	0.869007i	$-0.664759\pi$
$181$	−13.3137	−0.989600	−0.494800	+	0.869007i	$0.664759\pi$
$182$	0	0
$183$	−9.65685	−0.713855
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	1.65685	0.121161
$188$	0	0
$189$	1.41421	0.102869
$190$	0	0
$191$	−1.65685	−0.119886	−0.0599429	−	0.998202i	$-0.519092\pi$
$191$	−1.65685	−0.119886	−0.0599429	+	0.998202i	$0.519092\pi$
$192$	0	0
$193$	1.31371	0.0945628	0.0472814	−	0.998882i	$-0.484944\pi$
$193$	1.31371	0.0945628	0.0472814	+	0.998882i	$0.484944\pi$
$194$	0	0
$195$	2.82843	0.202548
$196$	0	0
$197$	−18.8284	−1.34147	−0.670735	−	0.741697i	$-0.734021\pi$
$197$	−18.8284	−1.34147	−0.670735	+	0.741697i	$0.734021\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	−3.75736	−0.265024
$202$	0	0
$203$	5.17157	0.362973
$204$	0	0
$205$	−19.3137	−1.34893
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	3.17157	0.219382
$210$	0	0
$211$	−23.3137	−1.60498	−0.802491	−	0.596664i	$-0.796492\pi$
$211$	−23.3137	−1.60498	−0.802491	+	0.596664i	$0.796492\pi$
$212$	0	0
$213$	11.8995	0.815340
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	7.65685	0.519781
$218$	0	0
$219$	−7.65685	−0.517402
$220$	0	0
$221$	2.82843	0.190261
$222$	0	0
$223$	8.24264	0.551968	0.275984	−	0.961162i	$-0.410996\pi$
$223$	8.24264	0.551968	0.275984	+	0.961162i	$0.410996\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	2.72792	0.181059	0.0905293	−	0.995894i	$-0.471144\pi$
$227$	2.72792	0.181059	0.0905293	+	0.995894i	$0.471144\pi$
$228$	0	0
$229$	8.34315	0.551331	0.275665	−	0.961254i	$-0.411102\pi$
$229$	8.34315	0.551331	0.275665	+	0.961254i	$0.411102\pi$
$230$	0	0
$231$	−0.828427	−0.0545065
$232$	0	0
$233$	18.9706	1.24280	0.621401	−	0.783492i	$-0.286564\pi$
$233$	18.9706	1.24280	0.621401	+	0.783492i	$0.286564\pi$
$234$	0	0
$235$	−9.65685	−0.629944
$236$	0	0
$237$	2.82843	0.183726
$238$	0	0
$239$	21.0711	1.36297	0.681487	−	0.731830i	$-0.261334\pi$
$239$	21.0711	1.36297	0.681487	+	0.731830i	$0.261334\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	14.1421	0.903508
$246$	0	0
$247$	5.41421	0.344498
$248$	0	0
$249$	−14.7279	−0.933344
$250$	0	0
$251$	14.1421	0.892644	0.446322	−	0.894873i	$-0.352734\pi$
$251$	14.1421	0.892644	0.446322	+	0.894873i	$0.352734\pi$
$252$	0	0
$253$	−1.65685	−0.104166
$254$	0	0
$255$	8.00000	0.500979
$256$	0	0
$257$	−11.6569	−0.727135	−0.363567	−	0.931568i	$-0.618441\pi$
$257$	−11.6569	−0.727135	−0.363567	+	0.931568i	$0.618441\pi$
$258$	0	0
$259$	2.82843	0.175750
$260$	0	0
$261$	3.65685	0.226354
$262$	0	0
$263$	−4.48528	−0.276574	−0.138287	−	0.990392i	$-0.544160\pi$
$263$	−4.48528	−0.276574	−0.138287	+	0.990392i	$0.544160\pi$
$264$	0	0
$265$	−3.31371	−0.203559
$266$	0	0
$267$	2.82843	0.173097
$268$	0	0
$269$	−0.343146	−0.0209220	−0.0104610	−	0.999945i	$-0.503330\pi$
$269$	−0.343146	−0.0209220	−0.0104610	+	0.999945i	$0.503330\pi$
$270$	0	0
$271$	−12.7279	−0.773166	−0.386583	−	0.922255i	$-0.626345\pi$
$271$	−12.7279	−0.773166	−0.386583	+	0.922255i	$0.626345\pi$
$272$	0	0
$273$	−1.41421	−0.0855921
$274$	0	0
$275$	−1.75736	−0.105973
$276$	0	0
$277$	14.3431	0.861796	0.430898	−	0.902401i	$-0.358197\pi$
$277$	14.3431	0.861796	0.430898	+	0.902401i	$0.358197\pi$
$278$	0	0
$279$	5.41421	0.324140
$280$	0	0
$281$	−16.4853	−0.983429	−0.491715	−	0.870756i	$-0.663630\pi$
$281$	−16.4853	−0.983429	−0.491715	+	0.870756i	$0.663630\pi$
$282$	0	0
$283$	15.7990	0.939152	0.469576	−	0.882892i	$-0.344407\pi$
$283$	15.7990	0.939152	0.469576	+	0.882892i	$0.344407\pi$
$284$	0	0
$285$	15.3137	0.907106
$286$	0	0
$287$	9.65685	0.570026
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−11.6569	−0.683337
$292$	0	0
$293$	−2.82843	−0.165238	−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−2.82843	−0.165238	−0.0826192	+	0.996581i	$0.526329\pi$
$294$	0	0
$295$	−6.34315	−0.369312
$296$	0	0
$297$	−0.585786	−0.0339908
$298$	0	0
$299$	−2.82843	−0.163572
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	−9.17157	−0.526893
$304$	0	0
$305$	27.3137	1.56398
$306$	0	0
$307$	−11.7574	−0.671028	−0.335514	−	0.942035i	$-0.608910\pi$
$307$	−11.7574	−0.671028	−0.335514	+	0.942035i	$0.608910\pi$
$308$	0	0
$309$	−9.17157	−0.521753
$310$	0	0
$311$	−10.8284	−0.614024	−0.307012	−	0.951706i	$-0.599329\pi$
$311$	−10.8284	−0.614024	−0.307012	+	0.951706i	$0.599329\pi$
$312$	0	0
$313$	20.9706	1.18533	0.592663	−	0.805450i	$-0.298077\pi$
$313$	20.9706	1.18533	0.592663	+	0.805450i	$0.298077\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	18.1421	1.01896	0.509482	−	0.860481i	$-0.329837\pi$
$317$	18.1421	1.01896	0.509482	+	0.860481i	$0.329837\pi$
$318$	0	0
$319$	−2.14214	−0.119937
$320$	0	0
$321$	1.17157	0.0653908
$322$	0	0
$323$	15.3137	0.852078
$324$	0	0
$325$	−3.00000	−0.166410
$326$	0	0
$327$	−7.65685	−0.423425
$328$	0	0
$329$	4.82843	0.266200
$330$	0	0
$331$	−23.5563	−1.29477	−0.647387	−	0.762161i	$-0.724138\pi$
$331$	−23.5563	−1.29477	−0.647387	+	0.762161i	$0.724138\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	10.6274	0.580638
$336$	0	0
$337$	3.31371	0.180509	0.0902546	−	0.995919i	$-0.471232\pi$
$337$	3.31371	0.180509	0.0902546	+	0.995919i	$0.471232\pi$
$338$	0	0
$339$	−8.48528	−0.460857
$340$	0	0
$341$	−3.17157	−0.171750
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	−12.9706	−0.696296	−0.348148	−	0.937440i	$-0.613189\pi$
$347$	−12.9706	−0.696296	−0.348148	+	0.937440i	$0.613189\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−11.7990	−0.627997	−0.313998	−	0.949424i	$-0.601669\pi$
$353$	−11.7990	−0.627997	−0.313998	+	0.949424i	$0.601669\pi$
$354$	0	0
$355$	−33.6569	−1.78632
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	4.58579	0.242029	0.121014	−	0.992651i	$-0.461385\pi$
$359$	4.58579	0.242029	0.121014	+	0.992651i	$0.461385\pi$
$360$	0	0
$361$	10.3137	0.542827
$362$	0	0
$363$	−10.6569	−0.559340
$364$	0	0
$365$	21.6569	1.13357
$366$	0	0
$367$	−31.3137	−1.63456	−0.817281	−	0.576239i	$-0.804520\pi$
$367$	−31.3137	−1.63456	−0.817281	+	0.576239i	$0.804520\pi$
$368$	0	0
$369$	6.82843	0.355474
$370$	0	0
$371$	1.65685	0.0860196
$372$	0	0
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	−3.65685	−0.188338
$378$	0	0
$379$	30.3848	1.56076	0.780381	−	0.625305i	$-0.215025\pi$
$379$	30.3848	1.56076	0.780381	+	0.625305i	$0.215025\pi$
$380$	0	0
$381$	13.6569	0.699662
$382$	0	0
$383$	−11.2132	−0.572968	−0.286484	−	0.958085i	$-0.592487\pi$
$383$	−11.2132	−0.572968	−0.286484	+	0.958085i	$0.592487\pi$
$384$	0	0
$385$	2.34315	0.119418
$386$	0	0
$387$	5.65685	0.287554
$388$	0	0
$389$	12.6274	0.640235	0.320118	−	0.947378i	$-0.396278\pi$
$389$	12.6274	0.640235	0.320118	+	0.947378i	$0.396278\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	−9.17157	−0.462645
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−6.68629	−0.335575	−0.167788	−	0.985823i	$-0.553662\pi$
$397$	−6.68629	−0.335575	−0.167788	+	0.985823i	$0.553662\pi$
$398$	0	0
$399$	−7.65685	−0.383322
$400$	0	0
$401$	32.4853	1.62224	0.811119	−	0.584881i	$-0.198859\pi$
$401$	32.4853	1.62224	0.811119	+	0.584881i	$0.198859\pi$
$402$	0	0
$403$	−5.41421	−0.269701
$404$	0	0
$405$	−2.82843	−0.140546
$406$	0	0
$407$	−1.17157	−0.0580727
$408$	0	0
$409$	−5.31371	−0.262746	−0.131373	−	0.991333i	$-0.541939\pi$
$409$	−5.31371	−0.262746	−0.131373	+	0.991333i	$0.541939\pi$
$410$	0	0
$411$	0.485281	0.0239372
$412$	0	0
$413$	3.17157	0.156063
$414$	0	0
$415$	41.6569	2.04485
$416$	0	0
$417$	−14.1421	−0.692543
$418$	0	0
$419$	−28.4853	−1.39160	−0.695799	−	0.718237i	$-0.744949\pi$
$419$	−28.4853	−1.39160	−0.695799	+	0.718237i	$0.744949\pi$
$420$	0	0
$421$	−11.6569	−0.568120	−0.284060	−	0.958806i	$-0.591682\pi$
$421$	−11.6569	−0.568120	−0.284060	+	0.958806i	$0.591682\pi$
$422$	0	0
$423$	3.41421	0.166005
$424$	0	0
$425$	−8.48528	−0.411597
$426$	0	0
$427$	−13.6569	−0.660901
$428$	0	0
$429$	0.585786	0.0282820
$430$	0	0
$431$	−26.0416	−1.25438	−0.627191	−	0.778866i	$-0.715795\pi$
$431$	−26.0416	−1.25438	−0.627191	+	0.778866i	$0.715795\pi$
$432$	0	0
$433$	−20.6274	−0.991290	−0.495645	−	0.868525i	$-0.665068\pi$
$433$	−20.6274	−0.991290	−0.495645	+	0.868525i	$0.665068\pi$
$434$	0	0
$435$	−10.3431	−0.495916
$436$	0	0
$437$	−15.3137	−0.732554
$438$	0	0
$439$	−18.1421	−0.865877	−0.432938	−	0.901423i	$-0.642523\pi$
$439$	−18.1421	−0.865877	−0.432938	+	0.901423i	$0.642523\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	−3.51472	−0.166989	−0.0834947	−	0.996508i	$-0.526608\pi$
$443$	−3.51472	−0.166989	−0.0834947	+	0.996508i	$0.526608\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	9.17157	0.433801
$448$	0	0
$449$	24.4853	1.15553	0.577766	−	0.816203i	$-0.303925\pi$
$449$	24.4853	1.15553	0.577766	+	0.816203i	$0.303925\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	−0.242641	−0.0114003
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−5.31371	−0.248565	−0.124282	−	0.992247i	$-0.539663\pi$
$457$	−5.31371	−0.248565	−0.124282	+	0.992247i	$0.539663\pi$
$458$	0	0
$459$	−2.82843	−0.132020
$460$	0	0
$461$	26.8284	1.24952	0.624762	−	0.780815i	$-0.285196\pi$
$461$	26.8284	1.24952	0.624762	+	0.780815i	$0.285196\pi$
$462$	0	0
$463$	−18.3848	−0.854413	−0.427207	−	0.904154i	$-0.640502\pi$
$463$	−18.3848	−0.854413	−0.427207	+	0.904154i	$0.640502\pi$
$464$	0	0
$465$	−15.3137	−0.710156
$466$	0	0
$467$	−25.4558	−1.17796	−0.588978	−	0.808149i	$-0.700470\pi$
$467$	−25.4558	−1.17796	−0.588978	+	0.808149i	$0.700470\pi$
$468$	0	0
$469$	−5.31371	−0.245364
$470$	0	0
$471$	−17.3137	−0.797774
$472$	0	0
$473$	−3.31371	−0.152364
$474$	0	0
$475$	−16.2426	−0.745263
$476$	0	0
$477$	1.17157	0.0536426
$478$	0	0
$479$	−14.0416	−0.641578	−0.320789	−	0.947151i	$-0.603948\pi$
$479$	−14.0416	−0.641578	−0.320789	+	0.947151i	$0.603948\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	32.9706	1.49712
$486$	0	0
$487$	22.8701	1.03634	0.518171	−	0.855277i	$-0.326613\pi$
$487$	22.8701	1.03634	0.518171	+	0.855277i	$0.326613\pi$
$488$	0	0
$489$	11.5563	0.522596
$490$	0	0
$491$	36.7696	1.65939	0.829693	−	0.558219i	$-0.188515\pi$
$491$	36.7696	1.65939	0.829693	+	0.558219i	$0.188515\pi$
$492$	0	0
$493$	−10.3431	−0.465832
$494$	0	0
$495$	1.65685	0.0744701
$496$	0	0
$497$	16.8284	0.754858
$498$	0	0
$499$	−41.0122	−1.83596	−0.917979	−	0.396629i	$-0.870180\pi$
$499$	−41.0122	−1.83596	−0.917979	+	0.396629i	$0.870180\pi$
$500$	0	0
$501$	−12.5858	−0.562291
$502$	0	0
$503$	28.4853	1.27010	0.635048	−	0.772473i	$-0.280980\pi$
$503$	28.4853	1.27010	0.635048	+	0.772473i	$0.280980\pi$
$504$	0	0
$505$	25.9411	1.15436
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−19.7990	−0.877575	−0.438787	−	0.898591i	$-0.644592\pi$
$509$	−19.7990	−0.877575	−0.438787	+	0.898591i	$0.644592\pi$
$510$	0	0
$511$	−10.8284	−0.479021
$512$	0	0
$513$	−5.41421	−0.239043
$514$	0	0
$515$	25.9411	1.14310
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	0	0
$519$	12.4853	0.548043
$520$	0	0
$521$	−6.14214	−0.269092	−0.134546	−	0.990907i	$-0.542958\pi$
$521$	−6.14214	−0.269092	−0.134546	+	0.990907i	$0.542958\pi$
$522$	0	0
$523$	5.17157	0.226137	0.113069	−	0.993587i	$-0.463932\pi$
$523$	5.17157	0.226137	0.113069	+	0.993587i	$0.463932\pi$
$524$	0	0
$525$	4.24264	0.185164
$526$	0	0
$527$	−15.3137	−0.667076
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	2.24264	0.0973223
$532$	0	0
$533$	−6.82843	−0.295772
$534$	0	0
$535$	−3.31371	−0.143264
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	2.92893	0.126158
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−13.3137	−0.571346
$544$	0	0
$545$	21.6569	0.927678
$546$	0	0
$547$	−31.1127	−1.33028	−0.665141	−	0.746717i	$-0.731629\pi$
$547$	−31.1127	−1.33028	−0.665141	+	0.746717i	$0.731629\pi$
$548$	0	0
$549$	−9.65685	−0.412144
$550$	0	0
$551$	−19.7990	−0.843465
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	−5.65685	−0.240120
$556$	0	0
$557$	−25.4558	−1.07860	−0.539299	−	0.842114i	$-0.681311\pi$
$557$	−25.4558	−1.07860	−0.539299	+	0.842114i	$0.681311\pi$
$558$	0	0
$559$	−5.65685	−0.239259
$560$	0	0
$561$	1.65685	0.0699524
$562$	0	0
$563$	−28.2843	−1.19204	−0.596020	−	0.802970i	$-0.703252\pi$
$563$	−28.2843	−1.19204	−0.596020	+	0.802970i	$0.703252\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	1.41421	0.0593914
$568$	0	0
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	0	0
$571$	−38.8284	−1.62492	−0.812460	−	0.583018i	$-0.801872\pi$
$571$	−38.8284	−1.62492	−0.812460	+	0.583018i	$0.801872\pi$
$572$	0	0
$573$	−1.65685	−0.0692161
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	−45.3137	−1.88643	−0.943217	−	0.332177i	$-0.892217\pi$
$577$	−45.3137	−1.88643	−0.943217	+	0.332177i	$0.892217\pi$
$578$	0	0
$579$	1.31371	0.0545959
$580$	0	0
$581$	−20.8284	−0.864109
$582$	0	0
$583$	−0.686292	−0.0284233
$584$	0	0
$585$	2.82843	0.116941
$586$	0	0
$587$	−35.4142	−1.46170	−0.730851	−	0.682538i	$-0.760876\pi$
$587$	−35.4142	−1.46170	−0.730851	+	0.682538i	$0.760876\pi$
$588$	0	0
$589$	−29.3137	−1.20785
$590$	0	0
$591$	−18.8284	−0.774498
$592$	0	0
$593$	29.4558	1.20961	0.604803	−	0.796375i	$-0.293252\pi$
$593$	29.4558	1.20961	0.604803	+	0.796375i	$0.293252\pi$
$594$	0	0
$595$	11.3137	0.463817
$596$	0	0
$597$	−16.9706	−0.694559
$598$	0	0
$599$	−28.4853	−1.16388	−0.581939	−	0.813233i	$-0.697706\pi$
$599$	−28.4853	−1.16388	−0.581939	+	0.813233i	$0.697706\pi$
$600$	0	0
$601$	36.9706	1.50806	0.754030	−	0.656840i	$-0.228107\pi$
$601$	36.9706	1.50806	0.754030	+	0.656840i	$0.228107\pi$
$602$	0	0
$603$	−3.75736	−0.153011
$604$	0	0
$605$	30.1421	1.22545
$606$	0	0
$607$	−34.6274	−1.40548	−0.702742	−	0.711445i	$-0.748041\pi$
$607$	−34.6274	−1.40548	−0.702742	+	0.711445i	$0.748041\pi$
$608$	0	0
$609$	5.17157	0.209563
$610$	0	0
$611$	−3.41421	−0.138124
$612$	0	0
$613$	3.37258	0.136217	0.0681087	−	0.997678i	$-0.478304\pi$
$613$	3.37258	0.136217	0.0681087	+	0.997678i	$0.478304\pi$
$614$	0	0
$615$	−19.3137	−0.778804
$616$	0	0
$617$	−4.48528	−0.180571	−0.0902853	−	0.995916i	$-0.528778\pi$
$617$	−4.48528	−0.180571	−0.0902853	+	0.995916i	$0.528778\pi$
$618$	0	0
$619$	18.8701	0.758452	0.379226	−	0.925304i	$-0.376190\pi$
$619$	18.8701	0.758452	0.379226	+	0.925304i	$0.376190\pi$
$620$	0	0
$621$	2.82843	0.113501
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	3.17157	0.126660
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	36.0416	1.43479	0.717397	−	0.696664i	$-0.245333\pi$
$631$	36.0416	1.43479	0.717397	+	0.696664i	$0.245333\pi$
$632$	0	0
$633$	−23.3137	−0.926637
$634$	0	0
$635$	−38.6274	−1.53288
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	11.8995	0.470737
$640$	0	0
$641$	34.8284	1.37564	0.687820	−	0.725881i	$-0.258568\pi$
$641$	34.8284	1.37564	0.687820	+	0.725881i	$0.258568\pi$
$642$	0	0
$643$	16.0416	0.632620	0.316310	−	0.948656i	$-0.397556\pi$
$643$	16.0416	0.632620	0.316310	+	0.948656i	$0.397556\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−3.31371	−0.130275	−0.0651377	−	0.997876i	$-0.520749\pi$
$647$	−3.31371	−0.130275	−0.0651377	+	0.997876i	$0.520749\pi$
$648$	0	0
$649$	−1.31371	−0.0515676
$650$	0	0
$651$	7.65685	0.300096
$652$	0	0
$653$	−38.2843	−1.49818	−0.749090	−	0.662469i	$-0.769509\pi$
$653$	−38.2843	−1.49818	−0.749090	+	0.662469i	$0.769509\pi$
$654$	0	0
$655$	25.9411	1.01360
$656$	0	0
$657$	−7.65685	−0.298722
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	35.6569	1.38689	0.693445	−	0.720509i	$-0.256092\pi$
$661$	35.6569	1.38689	0.693445	+	0.720509i	$0.256092\pi$
$662$	0	0
$663$	2.82843	0.109847
$664$	0	0
$665$	21.6569	0.839817
$666$	0	0
$667$	10.3431	0.400488
$668$	0	0
$669$	8.24264	0.318679
$670$	0	0
$671$	5.65685	0.218380
$672$	0	0
$673$	10.6274	0.409657	0.204828	−	0.978798i	$-0.434336\pi$
$673$	10.6274	0.409657	0.204828	+	0.978798i	$0.434336\pi$
$674$	0	0
$675$	3.00000	0.115470
$676$	0	0
$677$	−6.82843	−0.262438	−0.131219	−	0.991353i	$-0.541889\pi$
$677$	−6.82843	−0.262438	−0.131219	+	0.991353i	$0.541889\pi$
$678$	0	0
$679$	−16.4853	−0.632647
$680$	0	0
$681$	2.72792	0.104534
$682$	0	0
$683$	−26.9289	−1.03041	−0.515203	−	0.857068i	$-0.672284\pi$
$683$	−26.9289	−1.03041	−0.515203	+	0.857068i	$0.672284\pi$
$684$	0	0
$685$	−1.37258	−0.0524437
$686$	0	0
$687$	8.34315	0.318311
$688$	0	0
$689$	−1.17157	−0.0446334
$690$	0	0
$691$	9.89949	0.376595	0.188297	−	0.982112i	$-0.439703\pi$
$691$	9.89949	0.376595	0.188297	+	0.982112i	$0.439703\pi$
$692$	0	0
$693$	−0.828427	−0.0314693
$694$	0	0
$695$	40.0000	1.51729
$696$	0	0
$697$	−19.3137	−0.731559
$698$	0	0
$699$	18.9706	0.717533
$700$	0	0
$701$	−24.6274	−0.930165	−0.465082	−	0.885267i	$-0.653975\pi$
$701$	−24.6274	−0.930165	−0.465082	+	0.885267i	$0.653975\pi$
$702$	0	0
$703$	−10.8284	−0.408402
$704$	0	0
$705$	−9.65685	−0.363698
$706$	0	0
$707$	−12.9706	−0.487808
$708$	0	0
$709$	18.9706	0.712454	0.356227	−	0.934399i	$-0.384063\pi$
$709$	18.9706	0.712454	0.356227	+	0.934399i	$0.384063\pi$
$710$	0	0
$711$	2.82843	0.106074
$712$	0	0
$713$	15.3137	0.573503
$714$	0	0
$715$	−1.65685	−0.0619628
$716$	0	0
$717$	21.0711	0.786913
$718$	0	0
$719$	−7.51472	−0.280252	−0.140126	−	0.990134i	$-0.544751\pi$
$719$	−7.51472	−0.280252	−0.140126	+	0.990134i	$0.544751\pi$
$720$	0	0
$721$	−12.9706	−0.483049
$722$	0	0
$723$	2.00000	0.0743808
$724$	0	0
$725$	10.9706	0.407436
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	−20.6274	−0.761891	−0.380946	−	0.924597i	$-0.624401\pi$
$733$	−20.6274	−0.761891	−0.380946	+	0.924597i	$0.624401\pi$
$734$	0	0
$735$	14.1421	0.521641
$736$	0	0
$737$	2.20101	0.0810753
$738$	0	0
$739$	6.10051	0.224411	0.112205	−	0.993685i	$-0.464209\pi$
$739$	6.10051	0.224411	0.112205	+	0.993685i	$0.464209\pi$
$740$	0	0
$741$	5.41421	0.198896
$742$	0	0
$743$	3.89949	0.143059	0.0715293	−	0.997438i	$-0.477212\pi$
$743$	3.89949	0.143059	0.0715293	+	0.997438i	$0.477212\pi$
$744$	0	0
$745$	−25.9411	−0.950409
$746$	0	0
$747$	−14.7279	−0.538866
$748$	0	0
$749$	1.65685	0.0605401
$750$	0	0
$751$	−0.201010	−0.00733496	−0.00366748	−	0.999993i	$-0.501167\pi$
$751$	−0.201010	−0.00733496	−0.00366748	+	0.999993i	$0.501167\pi$
$752$	0	0
$753$	14.1421	0.515368
$754$	0	0
$755$	0.686292	0.0249767
$756$	0	0
$757$	−23.3137	−0.847351	−0.423676	−	0.905814i	$-0.639260\pi$
$757$	−23.3137	−0.847351	−0.423676	+	0.905814i	$0.639260\pi$
$758$	0	0
$759$	−1.65685	−0.0601400
$760$	0	0
$761$	21.8579	0.792347	0.396173	−	0.918176i	$-0.370338\pi$
$761$	21.8579	0.792347	0.396173	+	0.918176i	$0.370338\pi$
$762$	0	0
$763$	−10.8284	−0.392015
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	−2.24264	−0.0809771
$768$	0	0
$769$	40.6274	1.46506	0.732531	−	0.680734i	$-0.238339\pi$
$769$	40.6274	1.46506	0.732531	+	0.680734i	$0.238339\pi$
$770$	0	0
$771$	−11.6569	−0.419811
$772$	0	0
$773$	−11.7990	−0.424380	−0.212190	−	0.977228i	$-0.568060\pi$
$773$	−11.7990	−0.424380	−0.212190	+	0.977228i	$0.568060\pi$
$774$	0	0
$775$	16.2426	0.583453
$776$	0	0
$777$	2.82843	0.101469
$778$	0	0
$779$	−36.9706	−1.32461
$780$	0	0
$781$	−6.97056	−0.249426
$782$	0	0
$783$	3.65685	0.130685
$784$	0	0
$785$	48.9706	1.74783
$786$	0	0
$787$	42.8701	1.52815	0.764076	−	0.645126i	$-0.223195\pi$
$787$	42.8701	1.52815	0.764076	+	0.645126i	$0.223195\pi$
$788$	0	0
$789$	−4.48528	−0.159680
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	9.65685	0.342925
$794$	0	0
$795$	−3.31371	−0.117525
$796$	0	0
$797$	−36.6274	−1.29741	−0.648705	−	0.761040i	$-0.724689\pi$
$797$	−36.6274	−1.29741	−0.648705	+	0.761040i	$0.724689\pi$
$798$	0	0
$799$	−9.65685	−0.341635
$800$	0	0
$801$	2.82843	0.0999376
$802$	0	0
$803$	4.48528	0.158282
$804$	0	0
$805$	−11.3137	−0.398756
$806$	0	0
$807$	−0.343146	−0.0120793
$808$	0	0
$809$	30.1421	1.05974	0.529871	−	0.848079i	$-0.322241\pi$
$809$	30.1421	1.05974	0.529871	+	0.848079i	$0.322241\pi$
$810$	0	0
$811$	−30.3848	−1.06695	−0.533477	−	0.845815i	$-0.679115\pi$
$811$	−30.3848	−1.06695	−0.533477	+	0.845815i	$0.679115\pi$
$812$	0	0
$813$	−12.7279	−0.446388
$814$	0	0
$815$	−32.6863	−1.14495
$816$	0	0
$817$	−30.6274	−1.07152
$818$	0	0
$819$	−1.41421	−0.0494166
$820$	0	0
$821$	29.4558	1.02802	0.514008	−	0.857785i	$-0.328160\pi$
$821$	29.4558	1.02802	0.514008	+	0.857785i	$0.328160\pi$
$822$	0	0
$823$	38.8284	1.35347	0.676737	−	0.736225i	$-0.263393\pi$
$823$	38.8284	1.35347	0.676737	+	0.736225i	$0.263393\pi$
$824$	0	0
$825$	−1.75736	−0.0611834
$826$	0	0
$827$	32.1838	1.11914	0.559570	−	0.828783i	$-0.310966\pi$
$827$	32.1838	1.11914	0.559570	+	0.828783i	$0.310966\pi$
$828$	0	0
$829$	33.9411	1.17882	0.589412	−	0.807833i	$-0.299359\pi$
$829$	33.9411	1.17882	0.589412	+	0.807833i	$0.299359\pi$
$830$	0	0
$831$	14.3431	0.497558
$832$	0	0
$833$	14.1421	0.489996
$834$	0	0
$835$	35.5980	1.23192
$836$	0	0
$837$	5.41421	0.187143
$838$	0	0
$839$	−39.4142	−1.36073	−0.680365	−	0.732874i	$-0.738179\pi$
$839$	−39.4142	−1.36073	−0.680365	+	0.732874i	$0.738179\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	−16.4853	−0.567783
$844$	0	0
$845$	−2.82843	−0.0973009
$846$	0	0
$847$	−15.0711	−0.517848
$848$	0	0
$849$	15.7990	0.542220
$850$	0	0
$851$	5.65685	0.193914
$852$	0	0
$853$	46.2843	1.58474	0.792372	−	0.610039i	$-0.208846\pi$
$853$	46.2843	1.58474	0.792372	+	0.610039i	$0.208846\pi$
$854$	0	0
$855$	15.3137	0.523718
$856$	0	0
$857$	−12.3431	−0.421634	−0.210817	−	0.977526i	$-0.567612\pi$
$857$	−12.3431	−0.421634	−0.210817	+	0.977526i	$0.567612\pi$
$858$	0	0
$859$	−32.2843	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$859$	−32.2843	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$860$	0	0
$861$	9.65685	0.329105
$862$	0	0
$863$	32.8701	1.11891	0.559455	−	0.828861i	$-0.311011\pi$
$863$	32.8701	1.11891	0.559455	+	0.828861i	$0.311011\pi$
$864$	0	0
$865$	−35.3137	−1.20070
$866$	0	0
$867$	−9.00000	−0.305656
$868$	0	0
$869$	−1.65685	−0.0562049
$870$	0	0
$871$	3.75736	0.127313
$872$	0	0
$873$	−11.6569	−0.394525
$874$	0	0
$875$	8.00000	0.270449
$876$	0	0
$877$	14.2843	0.482346	0.241173	−	0.970482i	$-0.422468\pi$
$877$	14.2843	0.482346	0.241173	+	0.970482i	$0.422468\pi$
$878$	0	0
$879$	−2.82843	−0.0954005
$880$	0	0
$881$	38.9706	1.31295	0.656476	−	0.754347i	$-0.272046\pi$
$881$	38.9706	1.31295	0.656476	+	0.754347i	$0.272046\pi$
$882$	0	0
$883$	16.2010	0.545207	0.272604	−	0.962126i	$-0.412115\pi$
$883$	16.2010	0.545207	0.272604	+	0.962126i	$0.412115\pi$
$884$	0	0
$885$	−6.34315	−0.213223
$886$	0	0
$887$	38.3431	1.28744	0.643718	−	0.765262i	$-0.277391\pi$
$887$	38.3431	1.28744	0.643718	+	0.765262i	$0.277391\pi$
$888$	0	0
$889$	19.3137	0.647761
$890$	0	0
$891$	−0.585786	−0.0196246
$892$	0	0
$893$	−18.4853	−0.618586
$894$	0	0
$895$	45.2548	1.51270
$896$	0	0
$897$	−2.82843	−0.0944384
$898$	0	0
$899$	19.7990	0.660333
$900$	0	0
$901$	−3.31371	−0.110396
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	37.6569	1.25176
$906$	0	0
$907$	27.1127	0.900262	0.450131	−	0.892962i	$-0.351377\pi$
$907$	27.1127	0.900262	0.450131	+	0.892962i	$0.351377\pi$
$908$	0	0
$909$	−9.17157	−0.304202
$910$	0	0
$911$	−57.6569	−1.91026	−0.955128	−	0.296192i	$-0.904283\pi$
$911$	−57.6569	−1.91026	−0.955128	+	0.296192i	$0.904283\pi$
$912$	0	0
$913$	8.62742	0.285526
$914$	0	0
$915$	27.3137	0.902963
$916$	0	0
$917$	−12.9706	−0.428326
$918$	0	0
$919$	23.1127	0.762418	0.381209	−	0.924489i	$-0.375508\pi$
$919$	23.1127	0.762418	0.381209	+	0.924489i	$0.375508\pi$
$920$	0	0
$921$	−11.7574	−0.387418
$922$	0	0
$923$	−11.8995	−0.391677
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−9.17157	−0.301234
$928$	0	0
$929$	−1.17157	−0.0384381	−0.0192190	−	0.999815i	$-0.506118\pi$
$929$	−1.17157	−0.0384381	−0.0192190	+	0.999815i	$0.506118\pi$
$930$	0	0
$931$	27.0711	0.887218
$932$	0	0
$933$	−10.8284	−0.354507
$934$	0	0
$935$	−4.68629	−0.153258
$936$	0	0
$937$	43.9411	1.43549	0.717747	−	0.696304i	$-0.245173\pi$
$937$	43.9411	1.43549	0.717747	+	0.696304i	$0.245173\pi$
$938$	0	0
$939$	20.9706	0.684348
$940$	0	0
$941$	32.7696	1.06826	0.534128	−	0.845403i	$-0.320640\pi$
$941$	32.7696	1.06826	0.534128	+	0.845403i	$0.320640\pi$
$942$	0	0
$943$	19.3137	0.628941
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	43.2132	1.40424	0.702120	−	0.712058i	$-0.252237\pi$
$947$	43.2132	1.40424	0.702120	+	0.712058i	$0.252237\pi$
$948$	0	0
$949$	7.65685	0.248552
$950$	0	0
$951$	18.1421	0.588299
$952$	0	0
$953$	2.28427	0.0739948	0.0369974	−	0.999315i	$-0.488221\pi$
$953$	2.28427	0.0739948	0.0369974	+	0.999315i	$0.488221\pi$
$954$	0	0
$955$	4.68629	0.151645
$956$	0	0
$957$	−2.14214	−0.0692454
$958$	0	0
$959$	0.686292	0.0221615
$960$	0	0
$961$	−1.68629	−0.0543965
$962$	0	0
$963$	1.17157	0.0377534
$964$	0	0
$965$	−3.71573	−0.119614
$966$	0	0
$967$	−2.58579	−0.0831533	−0.0415766	−	0.999135i	$-0.513238\pi$
$967$	−2.58579	−0.0831533	−0.0415766	+	0.999135i	$0.513238\pi$
$968$	0	0
$969$	15.3137	0.491947
$970$	0	0
$971$	−29.9411	−0.960856	−0.480428	−	0.877034i	$-0.659519\pi$
$971$	−29.9411	−0.960856	−0.480428	+	0.877034i	$0.659519\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	0	0
$975$	−3.00000	−0.0960769
$976$	0	0
$977$	−54.4264	−1.74126	−0.870628	−	0.491943i	$-0.836287\pi$
$977$	−54.4264	−1.74126	−0.870628	+	0.491943i	$0.836287\pi$
$978$	0	0
$979$	−1.65685	−0.0529533
$980$	0	0
$981$	−7.65685	−0.244465
$982$	0	0
$983$	25.7574	0.821532	0.410766	−	0.911741i	$-0.365261\pi$
$983$	25.7574	0.821532	0.410766	+	0.911741i	$0.365261\pi$
$984$	0	0
$985$	53.2548	1.69684
$986$	0	0
$987$	4.82843	0.153691
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	−11.0294	−0.350362	−0.175181	−	0.984536i	$-0.556051\pi$
$991$	−11.0294	−0.350362	−0.175181	+	0.984536i	$0.556051\pi$
$992$	0	0
$993$	−23.5563	−0.747538
$994$	0	0
$995$	48.0000	1.52170
$996$	0	0
$997$	11.3137	0.358309	0.179154	−	0.983821i	$-0.442664\pi$
$997$	11.3137	0.358309	0.179154	+	0.983821i	$0.442664\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9984.2.a.l.1.1		2
4.3	odd	2		9984.2.a.d.1.1		2
8.3	odd	2		9984.2.a.m.1.2		2
8.5	even	2		9984.2.a.e.1.2		2
16.3	odd	4		4992.2.g.b.2497.2	✓	4
16.5	even	4		4992.2.g.c.2497.1	yes	4
16.11	odd	4		4992.2.g.b.2497.3	yes	4
16.13	even	4		4992.2.g.c.2497.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4992.2.g.b.2497.2	✓	4	16.3	odd	4
4992.2.g.b.2497.3	yes	4	16.11	odd	4
4992.2.g.c.2497.1	yes	4	16.5	even	4
4992.2.g.c.2497.4	yes	4	16.13	even	4
9984.2.a.d.1.1		2	4.3	odd	2
9984.2.a.e.1.2		2	8.5	even	2
9984.2.a.l.1.1		2	1.1	even	1	trivial
9984.2.a.m.1.2		2	8.3	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:	\( N \)	\(=\)	\( 9984 = 2^{8} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9984.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.82843 q^{5} +1.41421 q^{7} +1.00000 q^{9} -0.585786 q^{11} -1.00000 q^{13} -2.82843 q^{15} -2.82843 q^{17} -5.41421 q^{19} +1.41421 q^{21} +2.82843 q^{23} +3.00000 q^{25} +1.00000 q^{27} +3.65685 q^{29} +5.41421 q^{31} -0.585786 q^{33} -4.00000 q^{35} +2.00000 q^{37} -1.00000 q^{39} +6.82843 q^{41} +5.65685 q^{43} -2.82843 q^{45} +3.41421 q^{47} -5.00000 q^{49} -2.82843 q^{51} +1.17157 q^{53} +1.65685 q^{55} -5.41421 q^{57} +2.24264 q^{59} -9.65685 q^{61} +1.41421 q^{63} +2.82843 q^{65} -3.75736 q^{67} +2.82843 q^{69} +11.8995 q^{71} -7.65685 q^{73} +3.00000 q^{75} -0.828427 q^{77} +2.82843 q^{79} +1.00000 q^{81} -14.7279 q^{83} +8.00000 q^{85} +3.65685 q^{87} +2.82843 q^{89} -1.41421 q^{91} +5.41421 q^{93} +15.3137 q^{95} -11.6569 q^{97} -0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 8 q^{19} + 6 q^{25} + 2 q^{27} - 4 q^{29} + 8 q^{31} - 4 q^{33} - 8 q^{35} + 4 q^{37} - 2 q^{39} + 8 q^{41} + 4 q^{47} - 10 q^{49} + 8 q^{53} - 8 q^{55} - 8 q^{57}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 8 * q^19 + 6 * q^25 + 2 * q^27 - 4 * q^29 + 8 * q^31 - 4 * q^33 - 8 * q^35 + 4 * q^37 - 2 * q^39 + 8 * q^41 + 4 * q^47 - 10 * q^49 + 8 * q^53 - 8 * q^55 - 8 * q^57 - 4 * q^59 - 8 * q^61 - 16 * q^67 + 4 * q^71 - 4 * q^73 + 6 * q^75 + 4 * q^77 + 2 * q^81 - 4 * q^83 + 16 * q^85 - 4 * q^87 + 8 * q^93 + 8 * q^95 - 12 * q^97 - 4 * q^99

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