LMFDB - Embedded newform 2000.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2000,2,Mod(1,2000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2000, 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("2000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2000 = 2^{4} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2000.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:	$15.9700804043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.54336$ of defining polynomial
Character	$\chi$	$=$	2000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.543362 q^{3} +2.11525 q^{7} -2.70476 q^{9} +O(q^{10})$	$q+0.543362 q^{3} +2.11525 q^{7} -2.70476 q^{9} +4.99442 q^{11} +6.32279 q^{13} +3.75836 q^{17} -1.90770 q^{19} +1.14934 q^{21} -9.35131 q^{23} -3.09975 q^{27} +7.22705 q^{29} -0.994424 q^{31} +2.71378 q^{33} -6.37984 q^{37} +3.43556 q^{39} +4.18247 q^{41} -1.45664 q^{43} -2.78501 q^{47} -2.52573 q^{49} +2.04215 q^{51} +1.52786 q^{53} -1.03657 q^{57} -1.47771 q^{59} +8.79148 q^{61} -5.72123 q^{63} +12.0811 q^{67} -5.08115 q^{69} +12.3798 q^{71} +9.09362 q^{73} +10.5644 q^{77} +8.91328 q^{79} +6.42999 q^{81} -12.1964 q^{83} +3.92690 q^{87} +3.08328 q^{89} +13.3743 q^{91} -0.540332 q^{93} +10.1382 q^{97} -13.5087 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9	$ 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9} + 4 q^{13} + 4 q^{17} + 8 q^{21} - 14 q^{23} - 22 q^{27} + 10 q^{29} + 16 q^{31} + 4 q^{33} + 24 q^{39} + 2 q^{41} - 12 q^{43} - 16 q^{47} + 2 q^{49} + 24 q^{53} + 24 q^{57} - 8 q^{59} + 6 q^{61} - 18 q^{63} + 16 q^{67} + 12 q^{69} + 24 q^{71} + 4 q^{73} + 32 q^{77} + 48 q^{79} + 16 q^{81} - 2 q^{83} + 22 q^{87} + 10 q^{89} + 8 q^{91} - 20 q^{93} + 4 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9 + 4 * q^13 + 4 * q^17 + 8 * q^21 - 14 * q^23 - 22 * q^27 + 10 * q^29 + 16 * q^31 + 4 * q^33 + 24 * q^39 + 2 * q^41 - 12 * q^43 - 16 * q^47 + 2 * q^49 + 24 * q^53 + 24 * q^57 - 8 * q^59 + 6 * q^61 - 18 * q^63 + 16 * q^67 + 12 * q^69 + 24 * q^71 + 4 * q^73 + 32 * q^77 + 48 * q^79 + 16 * q^81 - 2 * q^83 + 22 * q^87 + 10 * q^89 + 8 * q^91 - 20 * q^93 + 4 * q^97 - 8 * 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
$3$	0.543362	0.313710	0.156855	−	0.987622i	$-0.449864\pi$
$3$	0.543362	0.313710	0.156855	+	0.987622i	$0.449864\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.11525	0.799488	0.399744	−	0.916627i	$-0.369099\pi$
$7$	2.11525	0.799488	0.399744	+	0.916627i	$0.369099\pi$
$8$	0	0
$9$	−2.70476	−0.901586
$10$	0	0
$11$	4.99442	1.50588	0.752938	−	0.658092i	$-0.228636\pi$
$11$	4.99442	1.50588	0.752938	+	0.658092i	$0.228636\pi$
$12$	0	0
$13$	6.32279	1.75363	0.876813	−	0.480831i	$-0.159665\pi$
$13$	6.32279	1.75363	0.876813	+	0.480831i	$0.159665\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.75836	0.911535	0.455768	−	0.890099i	$-0.349365\pi$
$17$	3.75836	0.911535	0.455768	+	0.890099i	$0.349365\pi$
$18$	0	0
$19$	−1.90770	−0.437656	−0.218828	−	0.975763i	$-0.570223\pi$
$19$	−1.90770	−0.437656	−0.218828	+	0.975763i	$0.570223\pi$
$20$	0	0
$21$	1.14934	0.250807
$22$	0	0
$23$	−9.35131	−1.94988	−0.974942	−	0.222460i	$-0.928591\pi$
$23$	−9.35131	−1.94988	−0.974942	+	0.222460i	$0.928591\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.09975	−0.596547
$28$	0	0
$29$	7.22705	1.34203	0.671014	−	0.741444i	$-0.265859\pi$
$29$	7.22705	1.34203	0.671014	+	0.741444i	$0.265859\pi$
$30$	0	0
$31$	−0.994424	−0.178604	−0.0893019	−	0.996005i	$-0.528464\pi$
$31$	−0.994424	−0.178604	−0.0893019	+	0.996005i	$0.528464\pi$
$32$	0	0
$33$	2.71378	0.472408
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.37984	−1.04884	−0.524419	−	0.851460i	$-0.675718\pi$
$37$	−6.37984	−1.04884	−0.524419	+	0.851460i	$0.675718\pi$
$38$	0	0
$39$	3.43556	0.550131
$40$	0	0
$41$	4.18247	0.653192	0.326596	−	0.945164i	$-0.394098\pi$
$41$	4.18247	0.653192	0.326596	+	0.945164i	$0.394098\pi$
$42$	0	0
$43$	−1.45664	−0.222135	−0.111068	−	0.993813i	$-0.535427\pi$
$43$	−1.45664	−0.222135	−0.111068	+	0.993813i	$0.535427\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.78501	−0.406235	−0.203117	−	0.979154i	$-0.565107\pi$
$47$	−2.78501	−0.406235	−0.203117	+	0.979154i	$0.565107\pi$
$48$	0	0
$49$	−2.52573	−0.360819
$50$	0	0
$51$	2.04215	0.285958
$52$	0	0
$53$	1.52786	0.209868	0.104934	−	0.994479i	$-0.466537\pi$
$53$	1.52786	0.209868	0.104934	+	0.994479i	$0.466537\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.03657	−0.137297
$58$	0	0
$59$	−1.47771	−0.192382	−0.0961908	−	0.995363i	$-0.530666\pi$
$59$	−1.47771	−0.192382	−0.0961908	+	0.995363i	$0.530666\pi$
$60$	0	0
$61$	8.79148	1.12563	0.562817	−	0.826582i	$-0.309718\pi$
$61$	8.79148	1.12563	0.562817	+	0.826582i	$0.309718\pi$
$62$	0	0
$63$	−5.72123	−0.720807
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0811	1.47595	0.737974	−	0.674830i	$-0.235783\pi$
$67$	12.0811	1.47595	0.737974	+	0.674830i	$0.235783\pi$
$68$	0	0
$69$	−5.08115	−0.611698
$70$	0	0
$71$	12.3798	1.46922	0.734608	−	0.678492i	$-0.237366\pi$
$71$	12.3798	1.46922	0.734608	+	0.678492i	$0.237366\pi$
$72$	0	0
$73$	9.09362	1.06433	0.532164	−	0.846642i	$-0.321379\pi$
$73$	9.09362	1.06433	0.532164	+	0.846642i	$0.321379\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.5644	1.20393
$78$	0	0
$79$	8.91328	1.00282	0.501411	−	0.865209i	$-0.332814\pi$
$79$	8.91328	1.00282	0.501411	+	0.865209i	$0.332814\pi$
$80$	0	0
$81$	6.42999	0.714443
$82$	0	0
$83$	−12.1964	−1.33873	−0.669364	−	0.742935i	$-0.733433\pi$
$83$	−12.1964	−1.33873	−0.669364	+	0.742935i	$0.733433\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.92690	0.421008
$88$	0	0
$89$	3.08328	0.326827	0.163413	−	0.986558i	$-0.447750\pi$
$89$	3.08328	0.326827	0.163413	+	0.986558i	$0.447750\pi$
$90$	0	0
$91$	13.3743	1.40200
$92$	0	0
$93$	−0.540332	−0.0560298
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.1382	1.02938	0.514689	−	0.857377i	$-0.327907\pi$
$97$	10.1382	1.02938	0.514689	+	0.857377i	$0.327907\pi$
$98$	0	0
$99$	−13.5087	−1.35768
$100$	0	0
$101$	11.5202	1.14630	0.573149	−	0.819451i	$-0.305722\pi$
$101$	11.5202	1.14630	0.573149	+	0.819451i	$0.305722\pi$
$102$	0	0
$103$	6.20639	0.611534	0.305767	−	0.952106i	$-0.401087\pi$
$103$	6.20639	0.611534	0.305767	+	0.952106i	$0.401087\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.7168	−1.22938	−0.614690	−	0.788769i	$-0.710719\pi$
$107$	−12.7168	−1.22938	−0.614690	+	0.788769i	$0.710719\pi$
$108$	0	0
$109$	0.381966	0.0365857	0.0182929	−	0.999833i	$-0.494177\pi$
$109$	0.381966	0.0365857	0.0182929	+	0.999833i	$0.494177\pi$
$110$	0	0
$111$	−3.46656	−0.329031
$112$	0	0
$113$	−8.31164	−0.781893	−0.390947	−	0.920413i	$-0.627852\pi$
$113$	−8.31164	−0.781893	−0.390947	+	0.920413i	$0.627852\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−17.1016	−1.58105
$118$	0	0
$119$	7.94985	0.728761
$120$	0	0
$121$	13.9443	1.26766
$122$	0	0
$123$	2.27259	0.204913
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.3017	−0.914130	−0.457065	−	0.889433i	$-0.651099\pi$
$127$	−10.3017	−0.914130	−0.457065	+	0.889433i	$0.651099\pi$
$128$	0	0
$129$	−0.791482	−0.0696861
$130$	0	0
$131$	−3.08672	−0.269688	−0.134844	−	0.990867i	$-0.543053\pi$
$131$	−3.08672	−0.269688	−0.134844	+	0.990867i	$0.543053\pi$
$132$	0	0
$133$	−4.03526	−0.349901
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.1053	−1.37596	−0.687982	−	0.725728i	$-0.741503\pi$
$137$	−16.1053	−1.37596	−0.687982	+	0.725728i	$0.741503\pi$
$138$	0	0
$139$	−15.5477	−1.31874	−0.659370	−	0.751819i	$-0.729177\pi$
$139$	−15.5477	−1.31874	−0.659370	+	0.751819i	$0.729177\pi$
$140$	0	0
$141$	−1.51327	−0.127440
$142$	0	0
$143$	31.5787	2.64074
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.37239	−0.113193
$148$	0	0
$149$	−15.2636	−1.25044	−0.625222	−	0.780447i	$-0.714992\pi$
$149$	−15.2636	−1.25044	−0.625222	+	0.780447i	$0.714992\pi$
$150$	0	0
$151$	16.1313	1.31275	0.656373	−	0.754436i	$-0.272090\pi$
$151$	16.1313	1.31275	0.656373	+	0.754436i	$0.272090\pi$
$152$	0	0
$153$	−10.1654	−0.821827
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.241644	−0.0192853	−0.00964264	−	0.999954i	$-0.503069\pi$
$157$	−0.241644	−0.0192853	−0.00964264	+	0.999954i	$0.503069\pi$
$158$	0	0
$159$	0.830183	0.0658378
$160$	0	0
$161$	−19.7803	−1.55891
$162$	0	0
$163$	−0.314742	−0.0246525	−0.0123263	−	0.999924i	$-0.503924\pi$
$163$	−0.314742	−0.0246525	−0.0123263	+	0.999924i	$0.503924\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.71813	−0.597247	−0.298623	−	0.954371i	$-0.596527\pi$
$167$	−7.71813	−0.597247	−0.298623	+	0.954371i	$0.596527\pi$
$168$	0	0
$169$	26.9777	2.07521
$170$	0	0
$171$	5.15987	0.394585
$172$	0	0
$173$	13.7472	1.04518	0.522590	−	0.852584i	$-0.324966\pi$
$173$	13.7472	1.04518	0.522590	+	0.852584i	$0.324966\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.802932	−0.0603521
$178$	0	0
$179$	−1.29311	−0.0966518	−0.0483259	−	0.998832i	$-0.515389\pi$
$179$	−1.29311	−0.0966518	−0.0483259	+	0.998832i	$0.515389\pi$
$180$	0	0
$181$	−18.4589	−1.37204	−0.686018	−	0.727585i	$-0.740643\pi$
$181$	−18.4589	−1.37204	−0.686018	+	0.727585i	$0.740643\pi$
$182$	0	0
$183$	4.77696	0.353123
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.7708	1.37266
$188$	0	0
$189$	−6.55673	−0.476932
$190$	0	0
$191$	−5.90770	−0.427466	−0.213733	−	0.976892i	$-0.568562\pi$
$191$	−5.90770	−0.427466	−0.213733	+	0.976892i	$0.568562\pi$
$192$	0	0
$193$	−20.8520	−1.50096	−0.750479	−	0.660894i	$-0.770177\pi$
$193$	−20.8520	−1.50096	−0.750479	+	0.660894i	$0.770177\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.4735	1.38743	0.693713	−	0.720252i	$-0.255974\pi$
$197$	19.4735	1.38743	0.693713	+	0.720252i	$0.255974\pi$
$198$	0	0
$199$	23.1989	1.64452	0.822262	−	0.569109i	$-0.192712\pi$
$199$	23.1989	1.64452	0.822262	+	0.569109i	$0.192712\pi$
$200$	0	0
$201$	6.56444	0.463020
$202$	0	0
$203$	15.2870	1.07294
$204$	0	0
$205$	0	0
$206$	0	0
$207$	25.2930	1.75799
$208$	0	0
$209$	−9.52786	−0.659056
$210$	0	0
$211$	−23.7323	−1.63380	−0.816900	−	0.576780i	$-0.804309\pi$
$211$	−23.7323	−1.63380	−0.816900	+	0.576780i	$0.804309\pi$
$212$	0	0
$213$	6.72673	0.460908
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.10345	−0.142792
$218$	0	0
$219$	4.94112	0.333890
$220$	0	0
$221$	23.7633	1.59849
$222$	0	0
$223$	−10.6314	−0.711931	−0.355966	−	0.934499i	$-0.615848\pi$
$223$	−10.6314	−0.711931	−0.355966	+	0.934499i	$0.615848\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.0243	−1.66092	−0.830459	−	0.557079i	$-0.811922\pi$
$227$	−25.0243	−1.66092	−0.830459	+	0.557079i	$0.811922\pi$
$228$	0	0
$229$	11.8962	0.786126	0.393063	−	0.919511i	$-0.371415\pi$
$229$	11.8962	0.786126	0.393063	+	0.919511i	$0.371415\pi$
$230$	0	0
$231$	5.74031	0.377685
$232$	0	0
$233$	−16.0347	−1.05047	−0.525235	−	0.850957i	$-0.676023\pi$
$233$	−16.0347	−1.05047	−0.525235	+	0.850957i	$0.676023\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.84313	0.314595
$238$	0	0
$239$	11.0867	0.717141	0.358570	−	0.933503i	$-0.383264\pi$
$239$	11.0867	0.717141	0.358570	+	0.933503i	$0.383264\pi$
$240$	0	0
$241$	6.56473	0.422872	0.211436	−	0.977392i	$-0.432186\pi$
$241$	6.56473	0.422872	0.211436	+	0.977392i	$0.432186\pi$
$242$	0	0
$243$	12.7931	0.820675
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.0620	−0.767486
$248$	0	0
$249$	−6.62706	−0.419973
$250$	0	0
$251$	−1.22918	−0.0775849	−0.0387924	−	0.999247i	$-0.512351\pi$
$251$	−1.22918	−0.0775849	−0.0387924	+	0.999247i	$0.512351\pi$
$252$	0	0
$253$	−46.7044	−2.93628
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.52786	0.344819	0.172409	−	0.985025i	$-0.444845\pi$
$257$	5.52786	0.344819	0.172409	+	0.985025i	$0.444845\pi$
$258$	0	0
$259$	−13.4949	−0.838534
$260$	0	0
$261$	−19.5474	−1.20995
$262$	0	0
$263$	−6.91631	−0.426478	−0.213239	−	0.977000i	$-0.568401\pi$
$263$	−6.91631	−0.426478	−0.213239	+	0.977000i	$0.568401\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.67534	0.102529
$268$	0	0
$269$	2.94427	0.179515	0.0897577	−	0.995964i	$-0.471391\pi$
$269$	2.94427	0.179515	0.0897577	+	0.995964i	$0.471391\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	7.26706	0.439823
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.81408	0.169082	0.0845410	−	0.996420i	$-0.473058\pi$
$277$	2.81408	0.169082	0.0845410	+	0.996420i	$0.473058\pi$
$278$	0	0
$279$	2.68968	0.161027
$280$	0	0
$281$	−2.59393	−0.154741	−0.0773705	−	0.997002i	$-0.524652\pi$
$281$	−2.59393	−0.154741	−0.0773705	+	0.997002i	$0.524652\pi$
$282$	0	0
$283$	17.6090	1.04675	0.523374	−	0.852103i	$-0.324673\pi$
$283$	17.6090	1.04675	0.523374	+	0.852103i	$0.324673\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.84695	0.522219
$288$	0	0
$289$	−2.87476	−0.169103
$290$	0	0
$291$	5.50871	0.322926
$292$	0	0
$293$	−17.7392	−1.03634	−0.518168	−	0.855279i	$-0.673386\pi$
$293$	−17.7392	−1.03634	−0.518168	+	0.855279i	$0.673386\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.4815	−0.898325
$298$	0	0
$299$	−59.1264	−3.41937
$300$	0	0
$301$	−3.08115	−0.177594
$302$	0	0
$303$	6.25962	0.359606
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.4770	0.712102	0.356051	−	0.934466i	$-0.384123\pi$
$307$	12.4770	0.712102	0.356051	+	0.934466i	$0.384123\pi$
$308$	0	0
$309$	3.37232	0.191844
$310$	0	0
$311$	−20.2262	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$311$	−20.2262	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$312$	0	0
$313$	12.6456	0.714771	0.357385	−	0.933957i	$-0.383668\pi$
$313$	12.6456	0.714771	0.357385	+	0.933957i	$0.383668\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.67852	0.375103	0.187552	−	0.982255i	$-0.439945\pi$
$317$	6.67852	0.375103	0.187552	+	0.982255i	$0.439945\pi$
$318$	0	0
$319$	36.0949	2.02093
$320$	0	0
$321$	−6.90983	−0.385669
$322$	0	0
$323$	−7.16982	−0.398939
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.207546	0.0114773
$328$	0	0
$329$	−5.89097	−0.324780
$330$	0	0
$331$	−27.2628	−1.49850	−0.749250	−	0.662288i	$-0.769586\pi$
$331$	−27.2628	−1.49850	−0.749250	+	0.662288i	$0.769586\pi$
$332$	0	0
$333$	17.2559	0.945618
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.3835	−1.60062	−0.800310	−	0.599587i	$-0.795332\pi$
$337$	−29.3835	−1.60062	−0.800310	+	0.599587i	$0.795332\pi$
$338$	0	0
$339$	−4.51623	−0.245288
$340$	0	0
$341$	−4.96658	−0.268955
$342$	0	0
$343$	−20.1493	−1.08796
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.5007	−0.832119	−0.416059	−	0.909337i	$-0.636589\pi$
$347$	−15.5007	−0.832119	−0.416059	+	0.909337i	$0.636589\pi$
$348$	0	0
$349$	5.98540	0.320391	0.160196	−	0.987085i	$-0.448787\pi$
$349$	5.98540	0.320391	0.160196	+	0.987085i	$0.448787\pi$
$350$	0	0
$351$	−19.5991	−1.04612
$352$	0	0
$353$	22.8841	1.21800	0.608998	−	0.793171i	$-0.291572\pi$
$353$	22.8841	1.21800	0.608998	+	0.793171i	$0.291572\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.31964	0.228620
$358$	0	0
$359$	−11.3525	−0.599161	−0.299580	−	0.954071i	$-0.596847\pi$
$359$	−11.3525	−0.599161	−0.299580	+	0.954071i	$0.596847\pi$
$360$	0	0
$361$	−15.3607	−0.808457
$362$	0	0
$363$	7.57679	0.397678
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.8123	0.929794	0.464897	−	0.885365i	$-0.346091\pi$
$367$	17.8123	0.929794	0.464897	+	0.885365i	$0.346091\pi$
$368$	0	0
$369$	−11.3126	−0.588909
$370$	0	0
$371$	3.23181	0.167787
$372$	0	0
$373$	13.0366	0.675008	0.337504	−	0.941324i	$-0.390417\pi$
$373$	13.0366	0.675008	0.337504	+	0.941324i	$0.390417\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.6951	2.35342
$378$	0	0
$379$	31.2800	1.60675	0.803373	−	0.595476i	$-0.203036\pi$
$379$	31.2800	1.60675	0.803373	+	0.595476i	$0.203036\pi$
$380$	0	0
$381$	−5.59756	−0.286772
$382$	0	0
$383$	−13.3291	−0.681084	−0.340542	−	0.940229i	$-0.610611\pi$
$383$	−13.3291	−0.681084	−0.340542	+	0.940229i	$0.610611\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.93985	0.200274
$388$	0	0
$389$	2.66950	0.135349	0.0676746	−	0.997707i	$-0.478442\pi$
$389$	2.66950	0.135349	0.0676746	+	0.997707i	$0.478442\pi$
$390$	0	0
$391$	−35.1456	−1.77739
$392$	0	0
$393$	−1.67721	−0.0846040
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.9572	−1.25257	−0.626284	−	0.779595i	$-0.715425\pi$
$397$	−24.9572	−1.25257	−0.626284	+	0.779595i	$0.715425\pi$
$398$	0	0
$399$	−2.19260	−0.109768
$400$	0	0
$401$	−28.3039	−1.41343	−0.706716	−	0.707498i	$-0.749824\pi$
$401$	−28.3039	−1.41343	−0.706716	+	0.707498i	$0.749824\pi$
$402$	0	0
$403$	−6.28754	−0.313204
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−31.8636	−1.57942
$408$	0	0
$409$	−14.5647	−0.720180	−0.360090	−	0.932918i	$-0.617254\pi$
$409$	−14.5647	−0.720180	−0.360090	+	0.932918i	$0.617254\pi$
$410$	0	0
$411$	−8.75098	−0.431654
$412$	0	0
$413$	−3.12572	−0.153807
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.44803	−0.413702
$418$	0	0
$419$	−28.4610	−1.39041	−0.695205	−	0.718811i	$-0.744686\pi$
$419$	−28.4610	−1.39041	−0.695205	+	0.718811i	$0.744686\pi$
$420$	0	0
$421$	−20.5443	−1.00127	−0.500633	−	0.865660i	$-0.666899\pi$
$421$	−20.5443	−1.00127	−0.500633	+	0.865660i	$0.666899\pi$
$422$	0	0
$423$	7.53277	0.366256
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.5961	0.899931
$428$	0	0
$429$	17.1587	0.828428
$430$	0	0
$431$	−28.9753	−1.39569	−0.697845	−	0.716249i	$-0.745857\pi$
$431$	−28.9753	−1.39569	−0.697845	+	0.716249i	$0.745857\pi$
$432$	0	0
$433$	20.9002	1.00440	0.502199	−	0.864752i	$-0.332524\pi$
$433$	20.9002	1.00440	0.502199	+	0.864752i	$0.332524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.8395	0.853379
$438$	0	0
$439$	30.2546	1.44397	0.721987	−	0.691907i	$-0.243229\pi$
$439$	30.2546	1.44397	0.721987	+	0.691907i	$0.243229\pi$
$440$	0	0
$441$	6.83150	0.325309
$442$	0	0
$443$	−6.09346	−0.289509	−0.144754	−	0.989468i	$-0.546239\pi$
$443$	−6.09346	−0.289509	−0.144754	+	0.989468i	$0.546239\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.29367	−0.392277
$448$	0	0
$449$	−19.3135	−0.911459	−0.455730	−	0.890118i	$-0.650622\pi$
$449$	−19.3135	−0.911459	−0.455730	+	0.890118i	$0.650622\pi$
$450$	0	0
$451$	20.8890	0.983626
$452$	0	0
$453$	8.76513	0.411822
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.1289	0.614143	0.307071	−	0.951686i	$-0.400651\pi$
$457$	13.1289	0.614143	0.307071	+	0.951686i	$0.400651\pi$
$458$	0	0
$459$	−11.6500	−0.543773
$460$	0	0
$461$	−6.28409	−0.292679	−0.146340	−	0.989234i	$-0.546749\pi$
$461$	−6.28409	−0.292679	−0.146340	+	0.989234i	$0.546749\pi$
$462$	0	0
$463$	−0.705730	−0.0327981	−0.0163990	−	0.999866i	$-0.505220\pi$
$463$	−0.705730	−0.0327981	−0.0163990	+	0.999866i	$0.505220\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.1190	0.468251	0.234126	−	0.972206i	$-0.424777\pi$
$467$	10.1190	0.468251	0.234126	+	0.972206i	$0.424777\pi$
$468$	0	0
$469$	25.5546	1.18000
$470$	0	0
$471$	−0.131300	−0.00604999
$472$	0	0
$473$	−7.27507	−0.334508
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.13250	−0.189214
$478$	0	0
$479$	29.0533	1.32748	0.663739	−	0.747964i	$-0.268969\pi$
$479$	29.0533	1.32748	0.663739	+	0.747964i	$0.268969\pi$
$480$	0	0
$481$	−40.3384	−1.83927
$482$	0	0
$483$	−10.7479	−0.489045
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.59798	0.344297	0.172149	−	0.985071i	$-0.444929\pi$
$487$	7.59798	0.344297	0.172149	+	0.985071i	$0.444929\pi$
$488$	0	0
$489$	−0.171019	−0.00773375
$490$	0	0
$491$	11.6822	0.527208	0.263604	−	0.964631i	$-0.415089\pi$
$491$	11.6822	0.527208	0.263604	+	0.964631i	$0.415089\pi$
$492$	0	0
$493$	27.1618	1.22331
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.1864	1.17462
$498$	0	0
$499$	−4.21754	−0.188803	−0.0944015	−	0.995534i	$-0.530094\pi$
$499$	−4.21754	−0.188803	−0.0944015	+	0.995534i	$0.530094\pi$
$500$	0	0
$501$	−4.19374	−0.187362
$502$	0	0
$503$	−8.12640	−0.362338	−0.181169	−	0.983452i	$-0.557988\pi$
$503$	−8.12640	−0.362338	−0.181169	+	0.983452i	$0.557988\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.6587	0.651014
$508$	0	0
$509$	−20.8190	−0.922787	−0.461394	−	0.887196i	$-0.652650\pi$
$509$	−20.8190	−0.922787	−0.461394	+	0.887196i	$0.652650\pi$
$510$	0	0
$511$	19.2352	0.850917
$512$	0	0
$513$	5.91339	0.261083
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13.9095	−0.611739
$518$	0	0
$519$	7.46971	0.327884
$520$	0	0
$521$	15.7390	0.689539	0.344769	−	0.938687i	$-0.387957\pi$
$521$	15.7390	0.689539	0.344769	+	0.938687i	$0.387957\pi$
$522$	0	0
$523$	−39.6698	−1.73464	−0.867319	−	0.497753i	$-0.834159\pi$
$523$	−39.6698	−1.73464	−0.867319	+	0.497753i	$0.834159\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.73740	−0.162804
$528$	0	0
$529$	64.4471	2.80205
$530$	0	0
$531$	3.99685	0.173449
$532$	0	0
$533$	26.4449	1.14546
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.702628	−0.0303206
$538$	0	0
$539$	−12.6146	−0.543349
$540$	0	0
$541$	10.1162	0.434930	0.217465	−	0.976068i	$-0.430221\pi$
$541$	10.1162	0.434930	0.217465	+	0.976068i	$0.430221\pi$
$542$	0	0
$543$	−10.0298	−0.430422
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.91575	0.167425	0.0837127	−	0.996490i	$-0.473322\pi$
$547$	3.91575	0.167425	0.0837127	+	0.996490i	$0.473322\pi$
$548$	0	0
$549$	−23.7788	−1.01486
$550$	0	0
$551$	−13.7870	−0.587348
$552$	0	0
$553$	18.8538	0.801744
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.9095	−1.09782	−0.548910	−	0.835881i	$-0.684957\pi$
$557$	−25.9095	−1.09782	−0.548910	+	0.835881i	$0.684957\pi$
$558$	0	0
$559$	−9.21002	−0.389542
$560$	0	0
$561$	10.1994	0.430617
$562$	0	0
$563$	−30.7156	−1.29451	−0.647254	−	0.762275i	$-0.724083\pi$
$563$	−30.7156	−1.29451	−0.647254	+	0.762275i	$0.724083\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	13.6010	0.571189
$568$	0	0
$569$	−12.3448	−0.517519	−0.258760	−	0.965942i	$-0.583314\pi$
$569$	−12.3448	−0.517519	−0.258760	+	0.965942i	$0.583314\pi$
$570$	0	0
$571$	−33.9368	−1.42021	−0.710104	−	0.704096i	$-0.751352\pi$
$571$	−33.9368	−1.42021	−0.710104	+	0.704096i	$0.751352\pi$
$572$	0	0
$573$	−3.21002	−0.134100
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−45.2207	−1.88256	−0.941280	−	0.337626i	$-0.890376\pi$
$577$	−45.2207	−1.88256	−0.941280	+	0.337626i	$0.890376\pi$
$578$	0	0
$579$	−11.3302	−0.470866
$580$	0	0
$581$	−25.7984	−1.07030
$582$	0	0
$583$	7.63080	0.316035
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.6710	−1.55485	−0.777424	−	0.628976i	$-0.783474\pi$
$587$	−37.6710	−1.55485	−0.777424	+	0.628976i	$0.783474\pi$
$588$	0	0
$589$	1.89706	0.0781671
$590$	0	0
$591$	10.5811	0.435250
$592$	0	0
$593$	6.19524	0.254408	0.127204	−	0.991877i	$-0.459400\pi$
$593$	6.19524	0.254408	0.127204	+	0.991877i	$0.459400\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.6054	0.515904
$598$	0	0
$599$	4.03100	0.164702	0.0823510	−	0.996603i	$-0.473757\pi$
$599$	4.03100	0.164702	0.0823510	+	0.996603i	$0.473757\pi$
$600$	0	0
$601$	42.2631	1.72395	0.861974	−	0.506953i	$-0.169228\pi$
$601$	42.2631	1.72395	0.861974	+	0.506953i	$0.169228\pi$
$602$	0	0
$603$	−32.6766	−1.33069
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−30.9920	−1.25793	−0.628963	−	0.777436i	$-0.716520\pi$
$607$	−30.9920	−1.25793	−0.628963	+	0.777436i	$0.716520\pi$
$608$	0	0
$609$	8.30636	0.336591
$610$	0	0
$611$	−17.6090	−0.712384
$612$	0	0
$613$	−6.63443	−0.267962	−0.133981	−	0.990984i	$-0.542776\pi$
$613$	−6.63443	−0.267962	−0.133981	+	0.990984i	$0.542776\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.2862	0.856951	0.428475	−	0.903553i	$-0.359051\pi$
$617$	21.2862	0.856951	0.428475	+	0.903553i	$0.359051\pi$
$618$	0	0
$619$	−17.4244	−0.700346	−0.350173	−	0.936685i	$-0.613877\pi$
$619$	−17.4244	−0.700346	−0.350173	+	0.936685i	$0.613877\pi$
$620$	0	0
$621$	28.9867	1.16320
$622$	0	0
$623$	6.52189	0.261294
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.17708	−0.206753
$628$	0	0
$629$	−23.9777	−0.956053
$630$	0	0
$631$	−11.5148	−0.458396	−0.229198	−	0.973380i	$-0.573610\pi$
$631$	−11.5148	−0.458396	−0.229198	+	0.973380i	$0.573610\pi$
$632$	0	0
$633$	−12.8952	−0.512539
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−15.9697	−0.632742
$638$	0	0
$639$	−33.4845	−1.32462
$640$	0	0
$641$	−45.2710	−1.78810	−0.894048	−	0.447970i	$-0.852147\pi$
$641$	−45.2710	−1.78810	−0.894048	+	0.447970i	$0.852147\pi$
$642$	0	0
$643$	23.6449	0.932464	0.466232	−	0.884662i	$-0.345611\pi$
$643$	23.6449	0.932464	0.466232	+	0.884662i	$0.345611\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.6264	0.968165	0.484082	−	0.875022i	$-0.339154\pi$
$647$	24.6264	0.968165	0.484082	+	0.875022i	$0.339154\pi$
$648$	0	0
$649$	−7.38032	−0.289703
$650$	0	0
$651$	−1.14294	−0.0447952
$652$	0	0
$653$	7.93000	0.310325	0.155163	−	0.987889i	$-0.450410\pi$
$653$	7.93000	0.310325	0.155163	+	0.987889i	$0.450410\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−24.5960	−0.959582
$658$	0	0
$659$	20.3575	0.793017	0.396508	−	0.918031i	$-0.370222\pi$
$659$	20.3575	0.793017	0.396508	+	0.918031i	$0.370222\pi$
$660$	0	0
$661$	−39.5617	−1.53877	−0.769385	−	0.638785i	$-0.779437\pi$
$661$	−39.5617	−1.53877	−0.769385	+	0.638785i	$0.779437\pi$
$662$	0	0
$663$	12.9121	0.501463
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−67.5824	−2.61680
$668$	0	0
$669$	−5.77670	−0.223340
$670$	0	0
$671$	43.9084	1.69506
$672$	0	0
$673$	1.46282	0.0563874	0.0281937	−	0.999602i	$-0.491024\pi$
$673$	1.46282	0.0563874	0.0281937	+	0.999602i	$0.491024\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.0148	−0.500199	−0.250099	−	0.968220i	$-0.580463\pi$
$677$	−13.0148	−0.500199	−0.250099	+	0.968220i	$0.580463\pi$
$678$	0	0
$679$	21.4448	0.822975
$680$	0	0
$681$	−13.5972	−0.521047
$682$	0	0
$683$	−2.11337	−0.0808660	−0.0404330	−	0.999182i	$-0.512874\pi$
$683$	−2.11337	−0.0808660	−0.0404330	+	0.999182i	$0.512874\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.46397	0.246616
$688$	0	0
$689$	9.66037	0.368031
$690$	0	0
$691$	6.92443	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$691$	6.92443	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$692$	0	0
$693$	−28.5742	−1.08545
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.7192	0.595408
$698$	0	0
$699$	−8.71267	−0.329543
$700$	0	0
$701$	1.65310	0.0624369	0.0312185	−	0.999513i	$-0.490061\pi$
$701$	1.65310	0.0624369	0.0312185	+	0.999513i	$0.490061\pi$
$702$	0	0
$703$	12.1708	0.459031
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.3680	0.916452
$708$	0	0
$709$	−1.67982	−0.0630870	−0.0315435	−	0.999502i	$-0.510042\pi$
$709$	−1.67982	−0.0630870	−0.0315435	+	0.999502i	$0.510042\pi$
$710$	0	0
$711$	−24.1083	−0.904130
$712$	0	0
$713$	9.29917	0.348257
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.02410	0.224974
$718$	0	0
$719$	50.2441	1.87379	0.936895	−	0.349611i	$-0.113686\pi$
$719$	50.2441	1.87379	0.936895	+	0.349611i	$0.113686\pi$
$720$	0	0
$721$	13.1280	0.488914
$722$	0	0
$723$	3.56703	0.132659
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.2324	0.676203	0.338101	−	0.941110i	$-0.390215\pi$
$727$	18.2324	0.676203	0.338101	+	0.941110i	$0.390215\pi$
$728$	0	0
$729$	−12.3387	−0.456989
$730$	0	0
$731$	−5.47456	−0.202484
$732$	0	0
$733$	−51.5137	−1.90270	−0.951350	−	0.308112i	$-0.900303\pi$
$733$	−51.5137	−1.90270	−0.951350	+	0.308112i	$0.900303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.3384	2.22259
$738$	0	0
$739$	13.3261	0.490207	0.245103	−	0.969497i	$-0.421178\pi$
$739$	13.3261	0.490207	0.245103	+	0.969497i	$0.421178\pi$
$740$	0	0
$741$	−6.55403	−0.240768
$742$	0	0
$743$	−23.9448	−0.878448	−0.439224	−	0.898378i	$-0.644747\pi$
$743$	−23.9448	−0.878448	−0.439224	+	0.898378i	$0.644747\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.9883	1.20698
$748$	0	0
$749$	−26.8992	−0.982875
$750$	0	0
$751$	−0.0607894	−0.00221823	−0.00110912	−	0.999999i	$-0.500353\pi$
$751$	−0.0607894	−0.00221823	−0.00110912	+	0.999999i	$0.500353\pi$
$752$	0	0
$753$	−0.667887	−0.0243392
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.3402	1.06639	0.533194	−	0.845993i	$-0.320992\pi$
$757$	29.3402	1.06639	0.533194	+	0.845993i	$0.320992\pi$
$758$	0	0
$759$	−25.3774	−0.921142
$760$	0	0
$761$	33.1931	1.20325	0.601625	−	0.798779i	$-0.294520\pi$
$761$	33.1931	1.20325	0.601625	+	0.798779i	$0.294520\pi$
$762$	0	0
$763$	0.807952	0.0292498
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.34326	−0.337366
$768$	0	0
$769$	7.90557	0.285082	0.142541	−	0.989789i	$-0.454473\pi$
$769$	7.90557	0.285082	0.142541	+	0.989789i	$0.454473\pi$
$770$	0	0
$771$	3.00363	0.108173
$772$	0	0
$773$	−27.7151	−0.996843	−0.498421	−	0.866935i	$-0.666087\pi$
$773$	−27.7151	−0.996843	−0.498421	+	0.866935i	$0.666087\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.33263	−0.263057
$778$	0	0
$779$	−7.97890	−0.285874
$780$	0	0
$781$	61.8302	2.21246
$782$	0	0
$783$	−22.4020	−0.800583
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.1468	0.611217	0.305609	−	0.952157i	$-0.401140\pi$
$787$	17.1468	0.611217	0.305609	+	0.952157i	$0.401140\pi$
$788$	0	0
$789$	−3.75806	−0.133790
$790$	0	0
$791$	−17.5812	−0.625114
$792$	0	0
$793$	55.5867	1.97394
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.0575	−0.745896	−0.372948	−	0.927852i	$-0.621653\pi$
$797$	−21.0575	−0.745896	−0.372948	+	0.927852i	$0.621653\pi$
$798$	0	0
$799$	−10.4670	−0.370297
$800$	0	0
$801$	−8.33952	−0.294662
$802$	0	0
$803$	45.4174	1.60274
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.59981	0.0563158
$808$	0	0
$809$	38.8466	1.36577	0.682887	−	0.730524i	$-0.260724\pi$
$809$	38.8466	1.36577	0.682887	+	0.730524i	$0.260724\pi$
$810$	0	0
$811$	−31.2820	−1.09846	−0.549229	−	0.835672i	$-0.685079\pi$
$811$	−31.2820	−1.09846	−0.549229	+	0.835672i	$0.685079\pi$
$812$	0	0
$813$	8.69379	0.304905
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.77883	0.0972189
$818$	0	0
$819$	−36.1741	−1.26403
$820$	0	0
$821$	8.50658	0.296882	0.148441	−	0.988921i	$-0.452575\pi$
$821$	8.50658	0.296882	0.148441	+	0.988921i	$0.452575\pi$
$822$	0	0
$823$	44.6746	1.55726	0.778630	−	0.627483i	$-0.215915\pi$
$823$	44.6746	1.55726	0.778630	+	0.627483i	$0.215915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.6536	0.926836	0.463418	−	0.886140i	$-0.346623\pi$
$827$	26.6536	0.926836	0.463418	+	0.886140i	$0.346623\pi$
$828$	0	0
$829$	4.21967	0.146555	0.0732776	−	0.997312i	$-0.476654\pi$
$829$	4.21967	0.146555	0.0732776	+	0.997312i	$0.476654\pi$
$830$	0	0
$831$	1.52907	0.0530427
$832$	0	0
$833$	−9.49261	−0.328899
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.08246	0.106546
$838$	0	0
$839$	−17.1897	−0.593453	−0.296726	−	0.954963i	$-0.595895\pi$
$839$	−17.1897	−0.593453	−0.296726	+	0.954963i	$0.595895\pi$
$840$	0	0
$841$	23.2302	0.801041
$842$	0	0
$843$	−1.40944	−0.0485438
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29.4956	1.01348
$848$	0	0
$849$	9.56807	0.328375
$850$	0	0
$851$	59.6599	2.04511
$852$	0	0
$853$	28.2243	0.966381	0.483191	−	0.875515i	$-0.339478\pi$
$853$	28.2243	0.966381	0.483191	+	0.875515i	$0.339478\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.2310	−1.75002	−0.875008	−	0.484108i	$-0.839144\pi$
$857$	−51.2310	−1.75002	−0.875008	+	0.484108i	$0.839144\pi$
$858$	0	0
$859$	53.1686	1.81409	0.907044	−	0.421036i	$-0.138333\pi$
$859$	53.1686	1.81409	0.907044	+	0.421036i	$0.138333\pi$
$860$	0	0
$861$	4.80710	0.163825
$862$	0	0
$863$	26.8953	0.915526	0.457763	−	0.889074i	$-0.348651\pi$
$863$	26.8953	0.915526	0.457763	+	0.889074i	$0.348651\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.56203	−0.0530495
$868$	0	0
$869$	44.5167	1.51012
$870$	0	0
$871$	76.3866	2.58826
$872$	0	0
$873$	−27.4214	−0.928072
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.7008	0.530178	0.265089	−	0.964224i	$-0.414599\pi$
$877$	15.7008	0.530178	0.265089	+	0.964224i	$0.414599\pi$
$878$	0	0
$879$	−9.63881	−0.325109
$880$	0	0
$881$	2.32804	0.0784336	0.0392168	−	0.999231i	$-0.487514\pi$
$881$	2.32804	0.0784336	0.0392168	+	0.999231i	$0.487514\pi$
$882$	0	0
$883$	45.0863	1.51727	0.758637	−	0.651514i	$-0.225866\pi$
$883$	45.0863	1.51727	0.758637	+	0.651514i	$0.225866\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.9907	−1.54422	−0.772108	−	0.635491i	$-0.780798\pi$
$887$	−45.9907	−1.54422	−0.772108	+	0.635491i	$0.780798\pi$
$888$	0	0
$889$	−21.7907	−0.730836
$890$	0	0
$891$	32.1141	1.07586
$892$	0	0
$893$	5.31296	0.177791
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−32.1270	−1.07269
$898$	0	0
$899$	−7.18675	−0.239691
$900$	0	0
$901$	5.74226	0.191302
$902$	0	0
$903$	−1.67418	−0.0557132
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−41.9462	−1.39280	−0.696401	−	0.717653i	$-0.745216\pi$
$907$	−41.9462	−1.39280	−0.696401	+	0.717653i	$0.745216\pi$
$908$	0	0
$909$	−31.1592	−1.03349
$910$	0	0
$911$	38.6141	1.27934	0.639671	−	0.768649i	$-0.279071\pi$
$911$	38.6141	1.27934	0.639671	+	0.768649i	$0.279071\pi$
$912$	0	0
$913$	−60.9140	−2.01596
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.52918	−0.215613
$918$	0	0
$919$	46.7156	1.54100	0.770502	−	0.637437i	$-0.220005\pi$
$919$	46.7156	1.54100	0.770502	+	0.637437i	$0.220005\pi$
$920$	0	0
$921$	6.77955	0.223394
$922$	0	0
$923$	78.2751	2.57646
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.7868	−0.551350
$928$	0	0
$929$	−5.49036	−0.180133	−0.0900665	−	0.995936i	$-0.528708\pi$
$929$	−5.49036	−0.180133	−0.0900665	+	0.995936i	$0.528708\pi$
$930$	0	0
$931$	4.81834	0.157915
$932$	0	0
$933$	−10.9902	−0.359802
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.6902	0.577912	0.288956	−	0.957342i	$-0.406692\pi$
$937$	17.6902	0.577912	0.288956	+	0.957342i	$0.406692\pi$
$938$	0	0
$939$	6.87113	0.224231
$940$	0	0
$941$	41.0472	1.33810	0.669050	−	0.743217i	$-0.266701\pi$
$941$	41.0472	1.33810	0.669050	+	0.743217i	$0.266701\pi$
$942$	0	0
$943$	−39.1116	−1.27365
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.6027	−0.864472	−0.432236	−	0.901760i	$-0.642275\pi$
$947$	−26.6027	−0.864472	−0.432236	+	0.901760i	$0.642275\pi$
$948$	0	0
$949$	57.4970	1.86643
$950$	0	0
$951$	3.62886	0.117674
$952$	0	0
$953$	−22.1029	−0.715984	−0.357992	−	0.933725i	$-0.616539\pi$
$953$	−22.1029	−0.715984	−0.357992	+	0.933725i	$0.616539\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.6126	0.633986
$958$	0	0
$959$	−34.0666	−1.10007
$960$	0	0
$961$	−30.0111	−0.968101
$962$	0	0
$963$	34.3959	1.10839
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.05390	−0.0338912	−0.0169456	−	0.999856i	$-0.505394\pi$
$967$	−1.05390	−0.0338912	−0.0169456	+	0.999856i	$0.505394\pi$
$968$	0	0
$969$	−3.89581	−0.125151
$970$	0	0
$971$	1.47456	0.0473210	0.0236605	−	0.999720i	$-0.492468\pi$
$971$	1.47456	0.0473210	0.0236605	+	0.999720i	$0.492468\pi$
$972$	0	0
$973$	−32.8872	−1.05432
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.53670	−0.177135	−0.0885674	−	0.996070i	$-0.528229\pi$
$977$	−5.53670	−0.177135	−0.0885674	+	0.996070i	$0.528229\pi$
$978$	0	0
$979$	15.3992	0.492160
$980$	0	0
$981$	−1.03313	−0.0329852
$982$	0	0
$983$	4.18145	0.133368	0.0666838	−	0.997774i	$-0.478758\pi$
$983$	4.18145	0.133368	0.0666838	+	0.997774i	$0.478758\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.20093	−0.101887
$988$	0	0
$989$	13.6215	0.433138
$990$	0	0
$991$	21.5056	0.683147	0.341573	−	0.939855i	$-0.389040\pi$
$991$	21.5056	0.683147	0.341573	+	0.939855i	$0.389040\pi$
$992$	0	0
$993$	−14.8136	−0.470094
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.3446	−0.834341	−0.417171	−	0.908828i	$-0.636978\pi$
$997$	−26.3446	−0.834341	−0.417171	+	0.908828i	$0.636978\pi$
$998$	0	0
$999$	19.7759	0.625681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2000.2.a.m.1.3		4
4.3	odd	2		1000.2.a.h.1.2	yes	4
5.2	odd	4		2000.2.c.j.1249.4		8
5.3	odd	4		2000.2.c.j.1249.5		8
5.4	even	2		2000.2.a.r.1.2		4
8.3	odd	2		8000.2.a.ba.1.3		4
8.5	even	2		8000.2.a.bq.1.2		4
12.11	even	2		9000.2.a.ba.1.1		4
20.3	even	4		1000.2.c.d.249.4		8
20.7	even	4		1000.2.c.d.249.5		8
20.19	odd	2		1000.2.a.e.1.3	✓	4
40.19	odd	2		8000.2.a.br.1.2		4
40.29	even	2		8000.2.a.bb.1.3		4
60.59	even	2		9000.2.a.r.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.e.1.3	✓	4	20.19	odd	2
1000.2.a.h.1.2	yes	4	4.3	odd	2
1000.2.c.d.249.4		8	20.3	even	4
1000.2.c.d.249.5		8	20.7	even	4
2000.2.a.m.1.3		4	1.1	even	1	trivial
2000.2.a.r.1.2		4	5.4	even	2
2000.2.c.j.1249.4		8	5.2	odd	4
2000.2.c.j.1249.5		8	5.3	odd	4
8000.2.a.ba.1.3		4	8.3	odd	2
8000.2.a.bb.1.3		4	40.29	even	2
8000.2.a.bq.1.2		4	8.5	even	2
8000.2.a.br.1.2		4	40.19	odd	2
9000.2.a.r.1.4		4	60.59	even	2
9000.2.a.ba.1.1		4	12.11	even	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:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2000.a (trivial)

\(f(q)\)	\(=\)	\(q+0.543362 q^{3} +2.11525 q^{7} -2.70476 q^{9} +O(q^{10})\)	\(q+0.543362 q^{3} +2.11525 q^{7} -2.70476 q^{9} +4.99442 q^{11} +6.32279 q^{13} +3.75836 q^{17} -1.90770 q^{19} +1.14934 q^{21} -9.35131 q^{23} -3.09975 q^{27} +7.22705 q^{29} -0.994424 q^{31} +2.71378 q^{33} -6.37984 q^{37} +3.43556 q^{39} +4.18247 q^{41} -1.45664 q^{43} -2.78501 q^{47} -2.52573 q^{49} +2.04215 q^{51} +1.52786 q^{53} -1.03657 q^{57} -1.47771 q^{59} +8.79148 q^{61} -5.72123 q^{63} +12.0811 q^{67} -5.08115 q^{69} +12.3798 q^{71} +9.09362 q^{73} +10.5644 q^{77} +8.91328 q^{79} +6.42999 q^{81} -12.1964 q^{83} +3.92690 q^{87} +3.08328 q^{89} +13.3743 q^{91} -0.540332 q^{93} +10.1382 q^{97} -13.5087 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9	\( 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9} + 4 q^{13} + 4 q^{17} + 8 q^{21} - 14 q^{23} - 22 q^{27} + 10 q^{29} + 16 q^{31} + 4 q^{33} + 24 q^{39} + 2 q^{41} - 12 q^{43} - 16 q^{47} + 2 q^{49} + 24 q^{53} + 24 q^{57} - 8 q^{59} + 6 q^{61} - 18 q^{63} + 16 q^{67} + 12 q^{69} + 24 q^{71} + 4 q^{73} + 32 q^{77} + 48 q^{79} + 16 q^{81} - 2 q^{83} + 22 q^{87} + 10 q^{89} + 8 q^{91} - 20 q^{93} + 4 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9 + 4 * q^13 + 4 * q^17 + 8 * q^21 - 14 * q^23 - 22 * q^27 + 10 * q^29 + 16 * q^31 + 4 * q^33 + 24 * q^39 + 2 * q^41 - 12 * q^43 - 16 * q^47 + 2 * q^49 + 24 * q^53 + 24 * q^57 - 8 * q^59 + 6 * q^61 - 18 * q^63 + 16 * q^67 + 12 * q^69 + 24 * q^71 + 4 * q^73 + 32 * q^77 + 48 * q^79 + 16 * q^81 - 2 * q^83 + 22 * q^87 + 10 * q^89 + 8 * q^91 - 20 * q^93 + 4 * q^97 - 8 * q^99

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