LMFDB - Embedded newform 3536.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3536,2,Mod(1,3536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3536, 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("3536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3536 = 2^{4} \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3536.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:	$28.2351021547$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 221)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	3536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79129 q^{3} -1.00000 q^{5} +0.208712 q^{7} +0.208712 q^{9} +O(q^{10})$	$q+1.79129 q^{3} -1.00000 q^{5} +0.208712 q^{7} +0.208712 q^{9} +0.791288 q^{11} -1.00000 q^{13} -1.79129 q^{15} +1.00000 q^{17} -4.79129 q^{19} +0.373864 q^{21} -7.58258 q^{23} -4.00000 q^{25} -5.00000 q^{27} +9.00000 q^{29} +0.582576 q^{31} +1.41742 q^{33} -0.208712 q^{35} +0.582576 q^{37} -1.79129 q^{39} -9.00000 q^{43} -0.208712 q^{45} -3.58258 q^{47} -6.95644 q^{49} +1.79129 q^{51} -7.79129 q^{53} -0.791288 q^{55} -8.58258 q^{57} -8.58258 q^{59} +11.7913 q^{61} +0.0435608 q^{63} +1.00000 q^{65} +4.41742 q^{67} -13.5826 q^{69} -2.00000 q^{71} +8.58258 q^{73} -7.16515 q^{75} +0.165151 q^{77} +8.58258 q^{79} -9.58258 q^{81} -4.58258 q^{83} -1.00000 q^{85} +16.1216 q^{87} -15.9564 q^{89} -0.208712 q^{91} +1.04356 q^{93} +4.79129 q^{95} +14.9564 q^{97} +0.165151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9	$ 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9} - 3 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} - 5 q^{19} - 13 q^{21} - 6 q^{23} - 8 q^{25} - 10 q^{27} + 18 q^{29} - 8 q^{31} + 12 q^{33} - 5 q^{35} - 8 q^{37} + q^{39} - 18 q^{43} - 5 q^{45} + 2 q^{47} + 9 q^{49} - q^{51} - 11 q^{53} + 3 q^{55} - 8 q^{57} - 8 q^{59} + 19 q^{61} + 23 q^{63} + 2 q^{65} + 18 q^{67} - 18 q^{69} - 4 q^{71} + 8 q^{73} + 4 q^{75} - 18 q^{77} + 8 q^{79} - 10 q^{81} - 2 q^{85} - 9 q^{87} - 9 q^{89} - 5 q^{91} + 25 q^{93} + 5 q^{95} + 7 q^{97} - 18 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9 - 3 * q^11 - 2 * q^13 + q^15 + 2 * q^17 - 5 * q^19 - 13 * q^21 - 6 * q^23 - 8 * q^25 - 10 * q^27 + 18 * q^29 - 8 * q^31 + 12 * q^33 - 5 * q^35 - 8 * q^37 + q^39 - 18 * q^43 - 5 * q^45 + 2 * q^47 + 9 * q^49 - q^51 - 11 * q^53 + 3 * q^55 - 8 * q^57 - 8 * q^59 + 19 * q^61 + 23 * q^63 + 2 * q^65 + 18 * q^67 - 18 * q^69 - 4 * q^71 + 8 * q^73 + 4 * q^75 - 18 * q^77 + 8 * q^79 - 10 * q^81 - 2 * q^85 - 9 * q^87 - 9 * q^89 - 5 * q^91 + 25 * q^93 + 5 * q^95 + 7 * q^97 - 18 * 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
$3$	1.79129	1.03420	0.517100	−	0.855925i	$-0.327011\pi$
$3$	1.79129	1.03420	0.517100	+	0.855925i	$0.327011\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0.208712	0.0788858	0.0394429	−	0.999222i	$-0.487442\pi$
$7$	0.208712	0.0788858	0.0394429	+	0.999222i	$0.487442\pi$
$8$	0	0
$9$	0.208712	0.0695707
$10$	0	0
$11$	0.791288	0.238582	0.119291	−	0.992859i	$-0.461938\pi$
$11$	0.791288	0.238582	0.119291	+	0.992859i	$0.461938\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.79129	−0.462509
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.79129	−1.09920	−0.549598	−	0.835429i	$-0.685219\pi$
$19$	−4.79129	−1.09920	−0.549598	+	0.835429i	$0.685219\pi$
$20$	0	0
$21$	0.373864	0.0815837
$22$	0	0
$23$	−7.58258	−1.58108	−0.790538	−	0.612413i	$-0.790199\pi$
$23$	−7.58258	−1.58108	−0.790538	+	0.612413i	$0.790199\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	0.582576	0.104634	0.0523168	−	0.998631i	$-0.483339\pi$
$31$	0.582576	0.104634	0.0523168	+	0.998631i	$0.483339\pi$
$32$	0	0
$33$	1.41742	0.246742
$34$	0	0
$35$	−0.208712	−0.0352788
$36$	0	0
$37$	0.582576	0.0957749	0.0478874	−	0.998853i	$-0.484751\pi$
$37$	0.582576	0.0957749	0.0478874	+	0.998853i	$0.484751\pi$
$38$	0	0
$39$	−1.79129	−0.286836
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	−0.208712	−0.0311130
$46$	0	0
$47$	−3.58258	−0.522572	−0.261286	−	0.965261i	$-0.584147\pi$
$47$	−3.58258	−0.522572	−0.261286	+	0.965261i	$0.584147\pi$
$48$	0	0
$49$	−6.95644	−0.993777
$50$	0	0
$51$	1.79129	0.250830
$52$	0	0
$53$	−7.79129	−1.07022	−0.535108	−	0.844784i	$-0.679729\pi$
$53$	−7.79129	−1.07022	−0.535108	+	0.844784i	$0.679729\pi$
$54$	0	0
$55$	−0.791288	−0.106697
$56$	0	0
$57$	−8.58258	−1.13679
$58$	0	0
$59$	−8.58258	−1.11736	−0.558678	−	0.829385i	$-0.688691\pi$
$59$	−8.58258	−1.11736	−0.558678	+	0.829385i	$0.688691\pi$
$60$	0	0
$61$	11.7913	1.50972	0.754860	−	0.655886i	$-0.227705\pi$
$61$	11.7913	1.50972	0.754860	+	0.655886i	$0.227705\pi$
$62$	0	0
$63$	0.0435608	0.00548814
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	4.41742	0.539674	0.269837	−	0.962906i	$-0.413030\pi$
$67$	4.41742	0.539674	0.269837	+	0.962906i	$0.413030\pi$
$68$	0	0
$69$	−13.5826	−1.63515
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	8.58258	1.00451	0.502257	−	0.864718i	$-0.332503\pi$
$73$	8.58258	1.00451	0.502257	+	0.864718i	$0.332503\pi$
$74$	0	0
$75$	−7.16515	−0.827360
$76$	0	0
$77$	0.165151	0.0188207
$78$	0	0
$79$	8.58258	0.965615	0.482808	−	0.875726i	$-0.339617\pi$
$79$	8.58258	0.965615	0.482808	+	0.875726i	$0.339617\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	0	0
$83$	−4.58258	−0.503003	−0.251502	−	0.967857i	$-0.580924\pi$
$83$	−4.58258	−0.503003	−0.251502	+	0.967857i	$0.580924\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	16.1216	1.72842
$88$	0	0
$89$	−15.9564	−1.69138	−0.845690	−	0.533675i	$-0.820811\pi$
$89$	−15.9564	−1.69138	−0.845690	+	0.533675i	$0.820811\pi$
$90$	0	0
$91$	−0.208712	−0.0218790
$92$	0	0
$93$	1.04356	0.108212
$94$	0	0
$95$	4.79129	0.491576
$96$	0	0
$97$	14.9564	1.51860	0.759298	−	0.650743i	$-0.225542\pi$
$97$	14.9564	1.51860	0.759298	+	0.650743i	$0.225542\pi$
$98$	0	0
$99$	0.165151	0.0165983
$100$	0	0
$101$	6.16515	0.613455	0.306728	−	0.951797i	$-0.400766\pi$
$101$	6.16515	0.613455	0.306728	+	0.951797i	$0.400766\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	−0.373864	−0.0364853
$106$	0	0
$107$	−6.37386	−0.616185	−0.308092	−	0.951356i	$-0.599691\pi$
$107$	−6.37386	−0.616185	−0.308092	+	0.951356i	$0.599691\pi$
$108$	0	0
$109$	−16.9564	−1.62413	−0.812066	−	0.583565i	$-0.801657\pi$
$109$	−16.9564	−1.62413	−0.812066	+	0.583565i	$0.801657\pi$
$110$	0	0
$111$	1.04356	0.0990504
$112$	0	0
$113$	−3.79129	−0.356654	−0.178327	−	0.983971i	$-0.557069\pi$
$113$	−3.79129	−0.356654	−0.178327	+	0.983971i	$0.557069\pi$
$114$	0	0
$115$	7.58258	0.707079
$116$	0	0
$117$	−0.208712	−0.0192954
$118$	0	0
$119$	0.208712	0.0191326
$120$	0	0
$121$	−10.3739	−0.943079
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	5.95644	0.528549	0.264274	−	0.964448i	$-0.414868\pi$
$127$	5.95644	0.528549	0.264274	+	0.964448i	$0.414868\pi$
$128$	0	0
$129$	−16.1216	−1.41943
$130$	0	0
$131$	−10.5826	−0.924604	−0.462302	−	0.886723i	$-0.652976\pi$
$131$	−10.5826	−0.924604	−0.462302	+	0.886723i	$0.652976\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	−0.791288	−0.0676043	−0.0338021	−	0.999429i	$-0.510762\pi$
$137$	−0.791288	−0.0676043	−0.0338021	+	0.999429i	$0.510762\pi$
$138$	0	0
$139$	6.74773	0.572335	0.286167	−	0.958180i	$-0.407619\pi$
$139$	6.74773	0.572335	0.286167	+	0.958180i	$0.407619\pi$
$140$	0	0
$141$	−6.41742	−0.540445
$142$	0	0
$143$	−0.791288	−0.0661708
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	−12.4610	−1.02776
$148$	0	0
$149$	3.95644	0.324124	0.162062	−	0.986781i	$-0.448186\pi$
$149$	3.95644	0.324124	0.162062	+	0.986781i	$0.448186\pi$
$150$	0	0
$151$	−18.7477	−1.52567	−0.762834	−	0.646594i	$-0.776193\pi$
$151$	−18.7477	−1.52567	−0.762834	+	0.646594i	$0.776193\pi$
$152$	0	0
$153$	0.208712	0.0168734
$154$	0	0
$155$	−0.582576	−0.0467936
$156$	0	0
$157$	−5.41742	−0.432358	−0.216179	−	0.976354i	$-0.569359\pi$
$157$	−5.41742	−0.432358	−0.216179	+	0.976354i	$0.569359\pi$
$158$	0	0
$159$	−13.9564	−1.10682
$160$	0	0
$161$	−1.58258	−0.124724
$162$	0	0
$163$	−15.7913	−1.23687	−0.618435	−	0.785836i	$-0.712233\pi$
$163$	−15.7913	−1.23687	−0.618435	+	0.785836i	$0.712233\pi$
$164$	0	0
$165$	−1.41742	−0.110346
$166$	0	0
$167$	11.3303	0.876765	0.438383	−	0.898788i	$-0.355552\pi$
$167$	11.3303	0.876765	0.438383	+	0.898788i	$0.355552\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	8.16515	0.620785	0.310392	−	0.950608i	$-0.399540\pi$
$173$	8.16515	0.620785	0.310392	+	0.950608i	$0.399540\pi$
$174$	0	0
$175$	−0.834849	−0.0631086
$176$	0	0
$177$	−15.3739	−1.15557
$178$	0	0
$179$	−0.165151	−0.0123440	−0.00617200	−	0.999981i	$-0.501965\pi$
$179$	−0.165151	−0.0123440	−0.00617200	+	0.999981i	$0.501965\pi$
$180$	0	0
$181$	−19.7913	−1.47107	−0.735537	−	0.677484i	$-0.763070\pi$
$181$	−19.7913	−1.47107	−0.735537	+	0.677484i	$0.763070\pi$
$182$	0	0
$183$	21.1216	1.56135
$184$	0	0
$185$	−0.582576	−0.0428318
$186$	0	0
$187$	0.791288	0.0578647
$188$	0	0
$189$	−1.04356	−0.0759079
$190$	0	0
$191$	−9.74773	−0.705321	−0.352660	−	0.935751i	$-0.614723\pi$
$191$	−9.74773	−0.705321	−0.352660	+	0.935751i	$0.614723\pi$
$192$	0	0
$193$	0.417424	0.0300469	0.0150234	−	0.999887i	$-0.495218\pi$
$193$	0.417424	0.0300469	0.0150234	+	0.999887i	$0.495218\pi$
$194$	0	0
$195$	1.79129	0.128277
$196$	0	0
$197$	22.7477	1.62071	0.810354	−	0.585940i	$-0.199275\pi$
$197$	22.7477	1.62071	0.810354	+	0.585940i	$0.199275\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	7.91288	0.558131
$202$	0	0
$203$	1.87841	0.131838
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.58258	−0.109997
$208$	0	0
$209$	−3.79129	−0.262249
$210$	0	0
$211$	−11.4174	−0.786008	−0.393004	−	0.919537i	$-0.628564\pi$
$211$	−11.4174	−0.786008	−0.393004	+	0.919537i	$0.628564\pi$
$212$	0	0
$213$	−3.58258	−0.245474
$214$	0	0
$215$	9.00000	0.613795
$216$	0	0
$217$	0.121591	0.00825411
$218$	0	0
$219$	15.3739	1.03887
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−8.20871	−0.549696	−0.274848	−	0.961488i	$-0.588628\pi$
$223$	−8.20871	−0.549696	−0.274848	+	0.961488i	$0.588628\pi$
$224$	0	0
$225$	−0.834849	−0.0556566
$226$	0	0
$227$	7.58258	0.503273	0.251637	−	0.967822i	$-0.419031\pi$
$227$	7.58258	0.503273	0.251637	+	0.967822i	$0.419031\pi$
$228$	0	0
$229$	10.5826	0.699316	0.349658	−	0.936877i	$-0.386298\pi$
$229$	10.5826	0.699316	0.349658	+	0.936877i	$0.386298\pi$
$230$	0	0
$231$	0.295834	0.0194644
$232$	0	0
$233$	18.9129	1.23902	0.619512	−	0.784987i	$-0.287331\pi$
$233$	18.9129	1.23902	0.619512	+	0.784987i	$0.287331\pi$
$234$	0	0
$235$	3.58258	0.233701
$236$	0	0
$237$	15.3739	0.998640
$238$	0	0
$239$	25.7477	1.66548	0.832741	−	0.553663i	$-0.186770\pi$
$239$	25.7477	1.66548	0.832741	+	0.553663i	$0.186770\pi$
$240$	0	0
$241$	17.7477	1.14323	0.571616	−	0.820521i	$-0.306317\pi$
$241$	17.7477	1.14323	0.571616	+	0.820521i	$0.306317\pi$
$242$	0	0
$243$	−2.16515	−0.138895
$244$	0	0
$245$	6.95644	0.444431
$246$	0	0
$247$	4.79129	0.304862
$248$	0	0
$249$	−8.20871	−0.520206
$250$	0	0
$251$	−3.37386	−0.212956	−0.106478	−	0.994315i	$-0.533957\pi$
$251$	−3.37386	−0.212956	−0.106478	+	0.994315i	$0.533957\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	−1.79129	−0.112175
$256$	0	0
$257$	−10.1652	−0.634085	−0.317042	−	0.948411i	$-0.602690\pi$
$257$	−10.1652	−0.634085	−0.317042	+	0.948411i	$0.602690\pi$
$258$	0	0
$259$	0.121591	0.00755527
$260$	0	0
$261$	1.87841	0.116271
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	7.79129	0.478615
$266$	0	0
$267$	−28.5826	−1.74923
$268$	0	0
$269$	−20.3739	−1.24222	−0.621108	−	0.783725i	$-0.713317\pi$
$269$	−20.3739	−1.24222	−0.621108	+	0.783725i	$0.713317\pi$
$270$	0	0
$271$	−16.6261	−1.00997	−0.504983	−	0.863129i	$-0.668501\pi$
$271$	−16.6261	−1.00997	−0.504983	+	0.863129i	$0.668501\pi$
$272$	0	0
$273$	−0.373864	−0.0226273
$274$	0	0
$275$	−3.16515	−0.190866
$276$	0	0
$277$	−25.1652	−1.51203	−0.756014	−	0.654556i	$-0.772856\pi$
$277$	−25.1652	−1.51203	−0.756014	+	0.654556i	$0.772856\pi$
$278$	0	0
$279$	0.121591	0.00727944
$280$	0	0
$281$	12.1216	0.723113	0.361557	−	0.932350i	$-0.382245\pi$
$281$	12.1216	0.723113	0.361557	+	0.932350i	$0.382245\pi$
$282$	0	0
$283$	−9.79129	−0.582032	−0.291016	−	0.956718i	$-0.593993\pi$
$283$	−9.79129	−0.582032	−0.291016	+	0.956718i	$0.593993\pi$
$284$	0	0
$285$	8.58258	0.508388
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	26.7913	1.57053
$292$	0	0
$293$	31.3739	1.83288	0.916440	−	0.400171i	$-0.131049\pi$
$293$	31.3739	1.83288	0.916440	+	0.400171i	$0.131049\pi$
$294$	0	0
$295$	8.58258	0.499697
$296$	0	0
$297$	−3.95644	−0.229576
$298$	0	0
$299$	7.58258	0.438512
$300$	0	0
$301$	−1.87841	−0.108270
$302$	0	0
$303$	11.0436	0.634436
$304$	0	0
$305$	−11.7913	−0.675167
$306$	0	0
$307$	−15.2087	−0.868007	−0.434004	−	0.900911i	$-0.642899\pi$
$307$	−15.2087	−0.868007	−0.434004	+	0.900911i	$0.642899\pi$
$308$	0	0
$309$	1.79129	0.101903
$310$	0	0
$311$	−13.4174	−0.760832	−0.380416	−	0.924815i	$-0.624219\pi$
$311$	−13.4174	−0.760832	−0.380416	+	0.924815i	$0.624219\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	−0.0435608	−0.00245437
$316$	0	0
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	0	0
$319$	7.12159	0.398733
$320$	0	0
$321$	−11.4174	−0.637258
$322$	0	0
$323$	−4.79129	−0.266594
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−30.3739	−1.67968
$328$	0	0
$329$	−0.747727	−0.0412235
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	0	0
$333$	0.121591	0.00666313
$334$	0	0
$335$	−4.41742	−0.241350
$336$	0	0
$337$	−10.7913	−0.587839	−0.293919	−	0.955830i	$-0.594960\pi$
$337$	−10.7913	−0.587839	−0.293919	+	0.955830i	$0.594960\pi$
$338$	0	0
$339$	−6.79129	−0.368852
$340$	0	0
$341$	0.460985	0.0249637
$342$	0	0
$343$	−2.91288	−0.157281
$344$	0	0
$345$	13.5826	0.731261
$346$	0	0
$347$	15.7477	0.845382	0.422691	−	0.906274i	$-0.361086\pi$
$347$	15.7477	0.845382	0.422691	+	0.906274i	$0.361086\pi$
$348$	0	0
$349$	24.3739	1.30470	0.652352	−	0.757917i	$-0.273783\pi$
$349$	24.3739	1.30470	0.652352	+	0.757917i	$0.273783\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−14.3303	−0.762725	−0.381363	−	0.924426i	$-0.624545\pi$
$353$	−14.3303	−0.762725	−0.381363	+	0.924426i	$0.624545\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	0	0
$357$	0.373864	0.0197870
$358$	0	0
$359$	12.2087	0.644351	0.322176	−	0.946680i	$-0.395586\pi$
$359$	12.2087	0.644351	0.322176	+	0.946680i	$0.395586\pi$
$360$	0	0
$361$	3.95644	0.208234
$362$	0	0
$363$	−18.5826	−0.975332
$364$	0	0
$365$	−8.58258	−0.449233
$366$	0	0
$367$	24.9564	1.30272	0.651358	−	0.758771i	$-0.274200\pi$
$367$	24.9564	1.30272	0.651358	+	0.758771i	$0.274200\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.62614	−0.0844248
$372$	0	0
$373$	−14.1216	−0.731188	−0.365594	−	0.930774i	$-0.619134\pi$
$373$	−14.1216	−0.731188	−0.365594	+	0.930774i	$0.619134\pi$
$374$	0	0
$375$	16.1216	0.832515
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−8.74773	−0.449341	−0.224670	−	0.974435i	$-0.572131\pi$
$379$	−8.74773	−0.449341	−0.224670	+	0.974435i	$0.572131\pi$
$380$	0	0
$381$	10.6697	0.546625
$382$	0	0
$383$	16.9129	0.864208	0.432104	−	0.901824i	$-0.357771\pi$
$383$	16.9129	0.864208	0.432104	+	0.901824i	$0.357771\pi$
$384$	0	0
$385$	−0.165151	−0.00841689
$386$	0	0
$387$	−1.87841	−0.0954849
$388$	0	0
$389$	−23.9564	−1.21464	−0.607320	−	0.794457i	$-0.707755\pi$
$389$	−23.9564	−1.21464	−0.607320	+	0.794457i	$0.707755\pi$
$390$	0	0
$391$	−7.58258	−0.383467
$392$	0	0
$393$	−18.9564	−0.956226
$394$	0	0
$395$	−8.58258	−0.431836
$396$	0	0
$397$	1.16515	0.0584773	0.0292386	−	0.999572i	$-0.490692\pi$
$397$	1.16515	0.0584773	0.0292386	+	0.999572i	$0.490692\pi$
$398$	0	0
$399$	−1.79129	−0.0896766
$400$	0	0
$401$	3.95644	0.197575	0.0987876	−	0.995109i	$-0.468504\pi$
$401$	3.95644	0.197575	0.0987876	+	0.995109i	$0.468504\pi$
$402$	0	0
$403$	−0.582576	−0.0290202
$404$	0	0
$405$	9.58258	0.476162
$406$	0	0
$407$	0.460985	0.0228502
$408$	0	0
$409$	13.1652	0.650975	0.325487	−	0.945546i	$-0.394472\pi$
$409$	13.1652	0.650975	0.325487	+	0.945546i	$0.394472\pi$
$410$	0	0
$411$	−1.41742	−0.0699164
$412$	0	0
$413$	−1.79129	−0.0881435
$414$	0	0
$415$	4.58258	0.224950
$416$	0	0
$417$	12.0871	0.591909
$418$	0	0
$419$	24.3739	1.19074	0.595371	−	0.803451i	$-0.297005\pi$
$419$	24.3739	1.19074	0.595371	+	0.803451i	$0.297005\pi$
$420$	0	0
$421$	−10.7913	−0.525935	−0.262968	−	0.964805i	$-0.584701\pi$
$421$	−10.7913	−0.525935	−0.262968	+	0.964805i	$0.584701\pi$
$422$	0	0
$423$	−0.747727	−0.0363557
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	2.46099	0.119095
$428$	0	0
$429$	−1.41742	−0.0684339
$430$	0	0
$431$	39.1652	1.88652	0.943259	−	0.332057i	$-0.107742\pi$
$431$	39.1652	1.88652	0.943259	+	0.332057i	$0.107742\pi$
$432$	0	0
$433$	9.16515	0.440449	0.220225	−	0.975449i	$-0.429321\pi$
$433$	9.16515	0.440449	0.220225	+	0.975449i	$0.429321\pi$
$434$	0	0
$435$	−16.1216	−0.772971
$436$	0	0
$437$	36.3303	1.73791
$438$	0	0
$439$	4.62614	0.220793	0.110397	−	0.993888i	$-0.464788\pi$
$439$	4.62614	0.220793	0.110397	+	0.993888i	$0.464788\pi$
$440$	0	0
$441$	−1.45189	−0.0691378
$442$	0	0
$443$	8.58258	0.407770	0.203885	−	0.978995i	$-0.434643\pi$
$443$	8.58258	0.407770	0.203885	+	0.978995i	$0.434643\pi$
$444$	0	0
$445$	15.9564	0.756408
$446$	0	0
$447$	7.08712	0.335209
$448$	0	0
$449$	26.1216	1.23275	0.616377	−	0.787451i	$-0.288600\pi$
$449$	26.1216	1.23275	0.616377	+	0.787451i	$0.288600\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−33.5826	−1.57785
$454$	0	0
$455$	0.208712	0.00978458
$456$	0	0
$457$	−25.7042	−1.20239	−0.601195	−	0.799102i	$-0.705309\pi$
$457$	−25.7042	−1.20239	−0.601195	+	0.799102i	$0.705309\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	0.252273	0.0117495	0.00587476	−	0.999983i	$-0.498130\pi$
$461$	0.252273	0.0117495	0.00587476	+	0.999983i	$0.498130\pi$
$462$	0	0
$463$	11.7477	0.545963	0.272982	−	0.962019i	$-0.411990\pi$
$463$	11.7477	0.545963	0.272982	+	0.962019i	$0.411990\pi$
$464$	0	0
$465$	−1.04356	−0.0483940
$466$	0	0
$467$	−0.208712	−0.00965805	−0.00482902	−	0.999988i	$-0.501537\pi$
$467$	−0.208712	−0.00965805	−0.00482902	+	0.999988i	$0.501537\pi$
$468$	0	0
$469$	0.921970	0.0425726
$470$	0	0
$471$	−9.70417	−0.447144
$472$	0	0
$473$	−7.12159	−0.327451
$474$	0	0
$475$	19.1652	0.879357
$476$	0	0
$477$	−1.62614	−0.0744557
$478$	0	0
$479$	6.12159	0.279703	0.139851	−	0.990173i	$-0.455338\pi$
$479$	6.12159	0.279703	0.139851	+	0.990173i	$0.455338\pi$
$480$	0	0
$481$	−0.582576	−0.0265632
$482$	0	0
$483$	−2.83485	−0.128990
$484$	0	0
$485$	−14.9564	−0.679137
$486$	0	0
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	0	0
$489$	−28.2867	−1.27917
$490$	0	0
$491$	−9.95644	−0.449328	−0.224664	−	0.974436i	$-0.572128\pi$
$491$	−9.95644	−0.449328	−0.224664	+	0.974436i	$0.572128\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	−0.165151	−0.00742300
$496$	0	0
$497$	−0.417424	−0.0187240
$498$	0	0
$499$	−5.16515	−0.231224	−0.115612	−	0.993294i	$-0.536883\pi$
$499$	−5.16515	−0.231224	−0.115612	+	0.993294i	$0.536883\pi$
$500$	0	0
$501$	20.2958	0.906751
$502$	0	0
$503$	−17.6261	−0.785911	−0.392955	−	0.919558i	$-0.628547\pi$
$503$	−17.6261	−0.785911	−0.392955	+	0.919558i	$0.628547\pi$
$504$	0	0
$505$	−6.16515	−0.274346
$506$	0	0
$507$	1.79129	0.0795539
$508$	0	0
$509$	−26.5826	−1.17825	−0.589126	−	0.808041i	$-0.700528\pi$
$509$	−26.5826	−1.17825	−0.589126	+	0.808041i	$0.700528\pi$
$510$	0	0
$511$	1.79129	0.0792419
$512$	0	0
$513$	23.9564	1.05770
$514$	0	0
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	−2.83485	−0.124676
$518$	0	0
$519$	14.6261	0.642016
$520$	0	0
$521$	22.5826	0.989361	0.494680	−	0.869075i	$-0.335285\pi$
$521$	22.5826	0.989361	0.494680	+	0.869075i	$0.335285\pi$
$522$	0	0
$523$	−43.8693	−1.91827	−0.959136	−	0.282947i	$-0.908688\pi$
$523$	−43.8693	−1.91827	−0.959136	+	0.282947i	$0.908688\pi$
$524$	0	0
$525$	−1.49545	−0.0652670
$526$	0	0
$527$	0.582576	0.0253774
$528$	0	0
$529$	34.4955	1.49980
$530$	0	0
$531$	−1.79129	−0.0777353
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.37386	0.275566
$536$	0	0
$537$	−0.295834	−0.0127662
$538$	0	0
$539$	−5.50455	−0.237098
$540$	0	0
$541$	−2.79129	−0.120007	−0.0600034	−	0.998198i	$-0.519111\pi$
$541$	−2.79129	−0.120007	−0.0600034	+	0.998198i	$0.519111\pi$
$542$	0	0
$543$	−35.4519	−1.52139
$544$	0	0
$545$	16.9564	0.726334
$546$	0	0
$547$	−32.8693	−1.40539	−0.702695	−	0.711491i	$-0.748020\pi$
$547$	−32.8693	−1.40539	−0.702695	+	0.711491i	$0.748020\pi$
$548$	0	0
$549$	2.46099	0.105032
$550$	0	0
$551$	−43.1216	−1.83704
$552$	0	0
$553$	1.79129	0.0761733
$554$	0	0
$555$	−1.04356	−0.0442967
$556$	0	0
$557$	23.5826	0.999226	0.499613	−	0.866249i	$-0.333476\pi$
$557$	23.5826	0.999226	0.499613	+	0.866249i	$0.333476\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	1.41742	0.0598437
$562$	0	0
$563$	−17.0436	−0.718300	−0.359150	−	0.933280i	$-0.616933\pi$
$563$	−17.0436	−0.718300	−0.359150	+	0.933280i	$0.616933\pi$
$564$	0	0
$565$	3.79129	0.159501
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−7.62614	−0.319704	−0.159852	−	0.987141i	$-0.551102\pi$
$569$	−7.62614	−0.319704	−0.159852	+	0.987141i	$0.551102\pi$
$570$	0	0
$571$	−16.8348	−0.704516	−0.352258	−	0.935903i	$-0.614586\pi$
$571$	−16.8348	−0.704516	−0.352258	+	0.935903i	$0.614586\pi$
$572$	0	0
$573$	−17.4610	−0.729443
$574$	0	0
$575$	30.3303	1.26486
$576$	0	0
$577$	−29.2867	−1.21922	−0.609612	−	0.792700i	$-0.708675\pi$
$577$	−29.2867	−1.21922	−0.609612	+	0.792700i	$0.708675\pi$
$578$	0	0
$579$	0.747727	0.0310745
$580$	0	0
$581$	−0.956439	−0.0396798
$582$	0	0
$583$	−6.16515	−0.255334
$584$	0	0
$585$	0.208712	0.00862919
$586$	0	0
$587$	−0.956439	−0.0394765	−0.0197382	−	0.999805i	$-0.506283\pi$
$587$	−0.956439	−0.0394765	−0.0197382	+	0.999805i	$0.506283\pi$
$588$	0	0
$589$	−2.79129	−0.115013
$590$	0	0
$591$	40.7477	1.67614
$592$	0	0
$593$	−25.3739	−1.04198	−0.520990	−	0.853563i	$-0.674437\pi$
$593$	−25.3739	−1.04198	−0.520990	+	0.853563i	$0.674437\pi$
$594$	0	0
$595$	−0.208712	−0.00855636
$596$	0	0
$597$	−19.7042	−0.806438
$598$	0	0
$599$	−5.00000	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	42.4519	1.73165	0.865824	−	0.500348i	$-0.166795\pi$
$601$	42.4519	1.73165	0.865824	+	0.500348i	$0.166795\pi$
$602$	0	0
$603$	0.921970	0.0375455
$604$	0	0
$605$	10.3739	0.421758
$606$	0	0
$607$	−10.5826	−0.429533	−0.214767	−	0.976665i	$-0.568899\pi$
$607$	−10.5826	−0.429533	−0.214767	+	0.976665i	$0.568899\pi$
$608$	0	0
$609$	3.36477	0.136347
$610$	0	0
$611$	3.58258	0.144935
$612$	0	0
$613$	−16.2523	−0.656423	−0.328212	−	0.944604i	$-0.606446\pi$
$613$	−16.2523	−0.656423	−0.328212	+	0.944604i	$0.606446\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.4174	0.459648	0.229824	−	0.973232i	$-0.426185\pi$
$617$	11.4174	0.459648	0.229824	+	0.973232i	$0.426185\pi$
$618$	0	0
$619$	−48.1652	−1.93592	−0.967960	−	0.251103i	$-0.919207\pi$
$619$	−48.1652	−1.93592	−0.967960	+	0.251103i	$0.919207\pi$
$620$	0	0
$621$	37.9129	1.52139
$622$	0	0
$623$	−3.33030	−0.133426
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−6.79129	−0.271218
$628$	0	0
$629$	0.582576	0.0232288
$630$	0	0
$631$	−32.1652	−1.28047	−0.640237	−	0.768177i	$-0.721164\pi$
$631$	−32.1652	−1.28047	−0.640237	+	0.768177i	$0.721164\pi$
$632$	0	0
$633$	−20.4519	−0.812890
$634$	0	0
$635$	−5.95644	−0.236374
$636$	0	0
$637$	6.95644	0.275624
$638$	0	0
$639$	−0.417424	−0.0165131
$640$	0	0
$641$	−6.58258	−0.259996	−0.129998	−	0.991514i	$-0.541497\pi$
$641$	−6.58258	−0.259996	−0.129998	+	0.991514i	$0.541497\pi$
$642$	0	0
$643$	21.7042	0.855929	0.427964	−	0.903796i	$-0.359231\pi$
$643$	21.7042	0.855929	0.427964	+	0.903796i	$0.359231\pi$
$644$	0	0
$645$	16.1216	0.634787
$646$	0	0
$647$	29.7913	1.17122	0.585608	−	0.810594i	$-0.300856\pi$
$647$	29.7913	1.17122	0.585608	+	0.810594i	$0.300856\pi$
$648$	0	0
$649$	−6.79129	−0.266581
$650$	0	0
$651$	0.217804	0.00853640
$652$	0	0
$653$	−22.0780	−0.863980	−0.431990	−	0.901878i	$-0.642188\pi$
$653$	−22.0780	−0.863980	−0.431990	+	0.901878i	$0.642188\pi$
$654$	0	0
$655$	10.5826	0.413495
$656$	0	0
$657$	1.79129	0.0698848
$658$	0	0
$659$	34.9129	1.36001	0.680006	−	0.733206i	$-0.261977\pi$
$659$	34.9129	1.36001	0.680006	+	0.733206i	$0.261977\pi$
$660$	0	0
$661$	11.2523	0.437663	0.218831	−	0.975763i	$-0.429776\pi$
$661$	11.2523	0.437663	0.218831	+	0.975763i	$0.429776\pi$
$662$	0	0
$663$	−1.79129	−0.0695679
$664$	0	0
$665$	1.00000	0.0387783
$666$	0	0
$667$	−68.2432	−2.64239
$668$	0	0
$669$	−14.7042	−0.568496
$670$	0	0
$671$	9.33030	0.360192
$672$	0	0
$673$	−13.9129	−0.536302	−0.268151	−	0.963377i	$-0.586413\pi$
$673$	−13.9129	−0.536302	−0.268151	+	0.963377i	$0.586413\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	22.8693	0.878939	0.439470	−	0.898257i	$-0.355166\pi$
$677$	22.8693	0.878939	0.439470	+	0.898257i	$0.355166\pi$
$678$	0	0
$679$	3.12159	0.119796
$680$	0	0
$681$	13.5826	0.520485
$682$	0	0
$683$	31.9129	1.22111	0.610556	−	0.791973i	$-0.290946\pi$
$683$	31.9129	1.22111	0.610556	+	0.791973i	$0.290946\pi$
$684$	0	0
$685$	0.791288	0.0302336
$686$	0	0
$687$	18.9564	0.723233
$688$	0	0
$689$	7.79129	0.296824
$690$	0	0
$691$	−49.9129	−1.89878	−0.949388	−	0.314107i	$-0.898295\pi$
$691$	−49.9129	−1.89878	−0.949388	+	0.314107i	$0.898295\pi$
$692$	0	0
$693$	0.0344691	0.00130937
$694$	0	0
$695$	−6.74773	−0.255956
$696$	0	0
$697$	0	0
$698$	0	0
$699$	33.8784	1.28140
$700$	0	0
$701$	−4.04356	−0.152723	−0.0763616	−	0.997080i	$-0.524330\pi$
$701$	−4.04356	−0.152723	−0.0763616	+	0.997080i	$0.524330\pi$
$702$	0	0
$703$	−2.79129	−0.105275
$704$	0	0
$705$	6.41742	0.241694
$706$	0	0
$707$	1.28674	0.0483929
$708$	0	0
$709$	1.53901	0.0577989	0.0288995	−	0.999582i	$-0.490800\pi$
$709$	1.53901	0.0577989	0.0288995	+	0.999582i	$0.490800\pi$
$710$	0	0
$711$	1.79129	0.0671785
$712$	0	0
$713$	−4.41742	−0.165434
$714$	0	0
$715$	0.791288	0.0295925
$716$	0	0
$717$	46.1216	1.72244
$718$	0	0
$719$	−1.16515	−0.0434528	−0.0217264	−	0.999764i	$-0.506916\pi$
$719$	−1.16515	−0.0434528	−0.0217264	+	0.999764i	$0.506916\pi$
$720$	0	0
$721$	0.208712	0.00777285
$722$	0	0
$723$	31.7913	1.18233
$724$	0	0
$725$	−36.0000	−1.33701
$726$	0	0
$727$	4.33030	0.160602	0.0803010	−	0.996771i	$-0.474412\pi$
$727$	4.33030	0.160602	0.0803010	+	0.996771i	$0.474412\pi$
$728$	0	0
$729$	24.8693	0.921086
$730$	0	0
$731$	−9.00000	−0.332877
$732$	0	0
$733$	3.83485	0.141643	0.0708217	−	0.997489i	$-0.477438\pi$
$733$	3.83485	0.141643	0.0708217	+	0.997489i	$0.477438\pi$
$734$	0	0
$735$	12.4610	0.459630
$736$	0	0
$737$	3.49545	0.128757
$738$	0	0
$739$	37.7042	1.38697	0.693485	−	0.720471i	$-0.256074\pi$
$739$	37.7042	1.38697	0.693485	+	0.720471i	$0.256074\pi$
$740$	0	0
$741$	8.58258	0.315289
$742$	0	0
$743$	4.74773	0.174177	0.0870886	−	0.996201i	$-0.472244\pi$
$743$	4.74773	0.174177	0.0870886	+	0.996201i	$0.472244\pi$
$744$	0	0
$745$	−3.95644	−0.144953
$746$	0	0
$747$	−0.956439	−0.0349943
$748$	0	0
$749$	−1.33030	−0.0486082
$750$	0	0
$751$	22.3303	0.814844	0.407422	−	0.913240i	$-0.366428\pi$
$751$	22.3303	0.814844	0.407422	+	0.913240i	$0.366428\pi$
$752$	0	0
$753$	−6.04356	−0.220240
$754$	0	0
$755$	18.7477	0.682300
$756$	0	0
$757$	−8.37386	−0.304353	−0.152177	−	0.988353i	$-0.548628\pi$
$757$	−8.37386	−0.304353	−0.152177	+	0.988353i	$0.548628\pi$
$758$	0	0
$759$	−10.7477	−0.390118
$760$	0	0
$761$	42.9129	1.55559	0.777795	−	0.628518i	$-0.216338\pi$
$761$	42.9129	1.55559	0.777795	+	0.628518i	$0.216338\pi$
$762$	0	0
$763$	−3.53901	−0.128121
$764$	0	0
$765$	−0.208712	−0.00754600
$766$	0	0
$767$	8.58258	0.309899
$768$	0	0
$769$	−36.2087	−1.30572	−0.652860	−	0.757479i	$-0.726431\pi$
$769$	−36.2087	−1.30572	−0.652860	+	0.757479i	$0.726431\pi$
$770$	0	0
$771$	−18.2087	−0.655771
$772$	0	0
$773$	5.20871	0.187344	0.0936722	−	0.995603i	$-0.470139\pi$
$773$	5.20871	0.187344	0.0936722	+	0.995603i	$0.470139\pi$
$774$	0	0
$775$	−2.33030	−0.0837069
$776$	0	0
$777$	0.217804	0.00781367
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−1.58258	−0.0566290
$782$	0	0
$783$	−45.0000	−1.60817
$784$	0	0
$785$	5.41742	0.193356
$786$	0	0
$787$	51.2432	1.82662	0.913311	−	0.407263i	$-0.133517\pi$
$787$	51.2432	1.82662	0.913311	+	0.407263i	$0.133517\pi$
$788$	0	0
$789$	3.58258	0.127543
$790$	0	0
$791$	−0.791288	−0.0281350
$792$	0	0
$793$	−11.7913	−0.418721
$794$	0	0
$795$	13.9564	0.494984
$796$	0	0
$797$	−23.8693	−0.845495	−0.422747	−	0.906248i	$-0.638934\pi$
$797$	−23.8693	−0.845495	−0.422747	+	0.906248i	$0.638934\pi$
$798$	0	0
$799$	−3.58258	−0.126742
$800$	0	0
$801$	−3.33030	−0.117670
$802$	0	0
$803$	6.79129	0.239659
$804$	0	0
$805$	1.58258	0.0557785
$806$	0	0
$807$	−36.4955	−1.28470
$808$	0	0
$809$	2.33030	0.0819291	0.0409645	−	0.999161i	$-0.486957\pi$
$809$	2.33030	0.0819291	0.0409645	+	0.999161i	$0.486957\pi$
$810$	0	0
$811$	−11.8348	−0.415578	−0.207789	−	0.978174i	$-0.566627\pi$
$811$	−11.8348	−0.415578	−0.207789	+	0.978174i	$0.566627\pi$
$812$	0	0
$813$	−29.7822	−1.04451
$814$	0	0
$815$	15.7913	0.553145
$816$	0	0
$817$	43.1216	1.50863
$818$	0	0
$819$	−0.0435608	−0.00152214
$820$	0	0
$821$	−54.4083	−1.89886	−0.949432	−	0.313973i	$-0.898340\pi$
$821$	−54.4083	−1.89886	−0.949432	+	0.313973i	$0.898340\pi$
$822$	0	0
$823$	−16.5826	−0.578032	−0.289016	−	0.957324i	$-0.593328\pi$
$823$	−16.5826	−0.578032	−0.289016	+	0.957324i	$0.593328\pi$
$824$	0	0
$825$	−5.66970	−0.197394
$826$	0	0
$827$	40.4519	1.40665	0.703325	−	0.710868i	$-0.251698\pi$
$827$	40.4519	1.40665	0.703325	+	0.710868i	$0.251698\pi$
$828$	0	0
$829$	15.5826	0.541205	0.270603	−	0.962691i	$-0.412777\pi$
$829$	15.5826	0.541205	0.270603	+	0.962691i	$0.412777\pi$
$830$	0	0
$831$	−45.0780	−1.56374
$832$	0	0
$833$	−6.95644	−0.241026
$834$	0	0
$835$	−11.3303	−0.392101
$836$	0	0
$837$	−2.91288	−0.100684
$838$	0	0
$839$	12.7913	0.441604	0.220802	−	0.975319i	$-0.429132\pi$
$839$	12.7913	0.441604	0.220802	+	0.975319i	$0.429132\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	21.7133	0.747844
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−2.16515	−0.0743955
$848$	0	0
$849$	−17.5390	−0.601937
$850$	0	0
$851$	−4.41742	−0.151427
$852$	0	0
$853$	−25.8348	−0.884568	−0.442284	−	0.896875i	$-0.645832\pi$
$853$	−25.8348	−0.884568	−0.442284	+	0.896875i	$0.645832\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	36.0780	1.23240	0.616201	−	0.787589i	$-0.288671\pi$
$857$	36.0780	1.23240	0.616201	+	0.787589i	$0.288671\pi$
$858$	0	0
$859$	13.5390	0.461945	0.230973	−	0.972960i	$-0.425809\pi$
$859$	13.5390	0.461945	0.230973	+	0.972960i	$0.425809\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	0	0
$865$	−8.16515	−0.277623
$866$	0	0
$867$	1.79129	0.0608353
$868$	0	0
$869$	6.79129	0.230379
$870$	0	0
$871$	−4.41742	−0.149679
$872$	0	0
$873$	3.12159	0.105650
$874$	0	0
$875$	1.87841	0.0635018
$876$	0	0
$877$	13.6261	0.460122	0.230061	−	0.973176i	$-0.426107\pi$
$877$	13.6261	0.460122	0.230061	+	0.973176i	$0.426107\pi$
$878$	0	0
$879$	56.1996	1.89557
$880$	0	0
$881$	−33.5390	−1.12996	−0.564979	−	0.825105i	$-0.691116\pi$
$881$	−33.5390	−1.12996	−0.564979	+	0.825105i	$0.691116\pi$
$882$	0	0
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	0	0
$885$	15.3739	0.516787
$886$	0	0
$887$	28.0436	0.941611	0.470805	−	0.882237i	$-0.343963\pi$
$887$	28.0436	0.941611	0.470805	+	0.882237i	$0.343963\pi$
$888$	0	0
$889$	1.24318	0.0416950
$890$	0	0
$891$	−7.58258	−0.254026
$892$	0	0
$893$	17.1652	0.574410
$894$	0	0
$895$	0.165151	0.00552040
$896$	0	0
$897$	13.5826	0.453509
$898$	0	0
$899$	5.24318	0.174870
$900$	0	0
$901$	−7.79129	−0.259565
$902$	0	0
$903$	−3.36477	−0.111973
$904$	0	0
$905$	19.7913	0.657885
$906$	0	0
$907$	−36.3739	−1.20777	−0.603887	−	0.797070i	$-0.706382\pi$
$907$	−36.3739	−1.20777	−0.603887	+	0.797070i	$0.706382\pi$
$908$	0	0
$909$	1.28674	0.0426785
$910$	0	0
$911$	42.1652	1.39699	0.698497	−	0.715613i	$-0.253853\pi$
$911$	42.1652	1.39699	0.698497	+	0.715613i	$0.253853\pi$
$912$	0	0
$913$	−3.62614	−0.120008
$914$	0	0
$915$	−21.1216	−0.698258
$916$	0	0
$917$	−2.20871	−0.0729381
$918$	0	0
$919$	43.5826	1.43766	0.718828	−	0.695188i	$-0.244679\pi$
$919$	43.5826	1.43766	0.718828	+	0.695188i	$0.244679\pi$
$920$	0	0
$921$	−27.2432	−0.897693
$922$	0	0
$923$	2.00000	0.0658308
$924$	0	0
$925$	−2.33030	−0.0766199
$926$	0	0
$927$	0.208712	0.00685501
$928$	0	0
$929$	−18.3739	−0.602827	−0.301413	−	0.953494i	$-0.597458\pi$
$929$	−18.3739	−0.602827	−0.301413	+	0.953494i	$0.597458\pi$
$930$	0	0
$931$	33.3303	1.09236
$932$	0	0
$933$	−24.0345	−0.786853
$934$	0	0
$935$	−0.791288	−0.0258779
$936$	0	0
$937$	−27.7477	−0.906479	−0.453239	−	0.891389i	$-0.649732\pi$
$937$	−27.7477	−0.906479	−0.453239	+	0.891389i	$0.649732\pi$
$938$	0	0
$939$	−1.79129	−0.0584565
$940$	0	0
$941$	−25.2087	−0.821781	−0.410890	−	0.911685i	$-0.634782\pi$
$941$	−25.2087	−0.821781	−0.410890	+	0.911685i	$0.634782\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.04356	0.0339470
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−8.58258	−0.278602
$950$	0	0
$951$	30.4519	0.987470
$952$	0	0
$953$	18.4174	0.596599	0.298299	−	0.954472i	$-0.403581\pi$
$953$	18.4174	0.596599	0.298299	+	0.954472i	$0.403581\pi$
$954$	0	0
$955$	9.74773	0.315429
$956$	0	0
$957$	12.7568	0.412369
$958$	0	0
$959$	−0.165151	−0.00533302
$960$	0	0
$961$	−30.6606	−0.989052
$962$	0	0
$963$	−1.33030	−0.0428684
$964$	0	0
$965$	−0.417424	−0.0134374
$966$	0	0
$967$	35.4174	1.13895	0.569474	−	0.822009i	$-0.307147\pi$
$967$	35.4174	1.13895	0.569474	+	0.822009i	$0.307147\pi$
$968$	0	0
$969$	−8.58258	−0.275712
$970$	0	0
$971$	−48.9564	−1.57109	−0.785543	−	0.618807i	$-0.787616\pi$
$971$	−48.9564	−1.57109	−0.785543	+	0.618807i	$0.787616\pi$
$972$	0	0
$973$	1.40833	0.0451491
$974$	0	0
$975$	7.16515	0.229468
$976$	0	0
$977$	11.0436	0.353315	0.176657	−	0.984272i	$-0.443472\pi$
$977$	11.0436	0.353315	0.176657	+	0.984272i	$0.443472\pi$
$978$	0	0
$979$	−12.6261	−0.403533
$980$	0	0
$981$	−3.53901	−0.112992
$982$	0	0
$983$	−36.0780	−1.15071	−0.575355	−	0.817904i	$-0.695136\pi$
$983$	−36.0780	−1.15071	−0.575355	+	0.817904i	$0.695136\pi$
$984$	0	0
$985$	−22.7477	−0.724803
$986$	0	0
$987$	−1.33939	−0.0426334
$988$	0	0
$989$	68.2432	2.17001
$990$	0	0
$991$	3.20871	0.101928	0.0509641	−	0.998700i	$-0.483771\pi$
$991$	3.20871	0.101928	0.0509641	+	0.998700i	$0.483771\pi$
$992$	0	0
$993$	19.7042	0.625293
$994$	0	0
$995$	11.0000	0.348723
$996$	0	0
$997$	22.0780	0.699218	0.349609	−	0.936896i	$-0.386314\pi$
$997$	22.0780	0.699218	0.349609	+	0.936896i	$0.386314\pi$
$998$	0	0
$999$	−2.91288	−0.0921594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3536.2.a.r.1.2		2
4.3	odd	2		221.2.a.d.1.1	✓	2
12.11	even	2		1989.2.a.h.1.2		2
20.19	odd	2		5525.2.a.p.1.2		2
52.51	odd	2		2873.2.a.i.1.2		2
68.67	odd	2		3757.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.a.d.1.1	✓	2	4.3	odd	2
1989.2.a.h.1.2		2	12.11	even	2
2873.2.a.i.1.2		2	52.51	odd	2
3536.2.a.r.1.2		2	1.1	even	1	trivial
3757.2.a.g.1.1		2	68.67	odd	2
5525.2.a.p.1.2		2	20.19	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:	\( N \)	\(=\)	\( 3536 = 2^{4} \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3536.a (trivial)

\(f(q)\)	\(=\)	\(q+1.79129 q^{3} -1.00000 q^{5} +0.208712 q^{7} +0.208712 q^{9} +O(q^{10})\)	\(q+1.79129 q^{3} -1.00000 q^{5} +0.208712 q^{7} +0.208712 q^{9} +0.791288 q^{11} -1.00000 q^{13} -1.79129 q^{15} +1.00000 q^{17} -4.79129 q^{19} +0.373864 q^{21} -7.58258 q^{23} -4.00000 q^{25} -5.00000 q^{27} +9.00000 q^{29} +0.582576 q^{31} +1.41742 q^{33} -0.208712 q^{35} +0.582576 q^{37} -1.79129 q^{39} -9.00000 q^{43} -0.208712 q^{45} -3.58258 q^{47} -6.95644 q^{49} +1.79129 q^{51} -7.79129 q^{53} -0.791288 q^{55} -8.58258 q^{57} -8.58258 q^{59} +11.7913 q^{61} +0.0435608 q^{63} +1.00000 q^{65} +4.41742 q^{67} -13.5826 q^{69} -2.00000 q^{71} +8.58258 q^{73} -7.16515 q^{75} +0.165151 q^{77} +8.58258 q^{79} -9.58258 q^{81} -4.58258 q^{83} -1.00000 q^{85} +16.1216 q^{87} -15.9564 q^{89} -0.208712 q^{91} +1.04356 q^{93} +4.79129 q^{95} +14.9564 q^{97} +0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9	\( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 5 q^{9} - 3 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} - 5 q^{19} - 13 q^{21} - 6 q^{23} - 8 q^{25} - 10 q^{27} + 18 q^{29} - 8 q^{31} + 12 q^{33} - 5 q^{35} - 8 q^{37} + q^{39} - 18 q^{43} - 5 q^{45} + 2 q^{47} + 9 q^{49} - q^{51} - 11 q^{53} + 3 q^{55} - 8 q^{57} - 8 q^{59} + 19 q^{61} + 23 q^{63} + 2 q^{65} + 18 q^{67} - 18 q^{69} - 4 q^{71} + 8 q^{73} + 4 q^{75} - 18 q^{77} + 8 q^{79} - 10 q^{81} - 2 q^{85} - 9 q^{87} - 9 q^{89} - 5 q^{91} + 25 q^{93} + 5 q^{95} + 7 q^{97} - 18 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 5 * q^9 - 3 * q^11 - 2 * q^13 + q^15 + 2 * q^17 - 5 * q^19 - 13 * q^21 - 6 * q^23 - 8 * q^25 - 10 * q^27 + 18 * q^29 - 8 * q^31 + 12 * q^33 - 5 * q^35 - 8 * q^37 + q^39 - 18 * q^43 - 5 * q^45 + 2 * q^47 + 9 * q^49 - q^51 - 11 * q^53 + 3 * q^55 - 8 * q^57 - 8 * q^59 + 19 * q^61 + 23 * q^63 + 2 * q^65 + 18 * q^67 - 18 * q^69 - 4 * q^71 + 8 * q^73 + 4 * q^75 - 18 * q^77 + 8 * q^79 - 10 * q^81 - 2 * q^85 - 9 * q^87 - 9 * q^89 - 5 * q^91 + 25 * q^93 + 5 * q^95 + 7 * q^97 - 18 * q^99

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