LMFDB - Embedded newform 4864.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{5} -4.46410 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{5} -4.46410 q^{7} -2.00000 q^{9} -4.00000 q^{11} +4.46410 q^{13} -2.00000 q^{15} -7.92820 q^{17} +1.00000 q^{19} -4.46410 q^{21} +6.46410 q^{23} -1.00000 q^{25} -5.00000 q^{27} -2.46410 q^{29} -4.92820 q^{31} -4.00000 q^{33} +8.92820 q^{35} -2.00000 q^{37} +4.46410 q^{39} -6.92820 q^{41} +4.92820 q^{43} +4.00000 q^{45} +6.92820 q^{47} +12.9282 q^{49} -7.92820 q^{51} +12.4641 q^{53} +8.00000 q^{55} +1.00000 q^{57} -9.92820 q^{59} -8.92820 q^{61} +8.92820 q^{63} -8.92820 q^{65} -5.92820 q^{67} +6.46410 q^{69} +14.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} +17.8564 q^{77} -2.00000 q^{79} +1.00000 q^{81} +2.92820 q^{83} +15.8564 q^{85} -2.46410 q^{87} -0.928203 q^{89} -19.9282 q^{91} -4.92820 q^{93} -2.00000 q^{95} +12.9282 q^{97} +8.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9} - 8 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{21} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 2 q^{29} + 4 q^{31} - 8 q^{33} + 4 q^{35} - 4 q^{37} + 2 q^{39} - 4 q^{43} + 8 q^{45} + 12 q^{49} - 2 q^{51} + 18 q^{53} + 16 q^{55} + 2 q^{57} - 6 q^{59} - 4 q^{61} + 4 q^{63} - 4 q^{65} + 2 q^{67} + 6 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} + 8 q^{77} - 4 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 2 q^{87} + 12 q^{89} - 26 q^{91} + 4 q^{93} - 4 q^{95} + 12 q^{97} + 16 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9 - 8 * q^11 + 2 * q^13 - 4 * q^15 - 2 * q^17 + 2 * q^19 - 2 * q^21 + 6 * q^23 - 2 * q^25 - 10 * q^27 + 2 * q^29 + 4 * q^31 - 8 * q^33 + 4 * q^35 - 4 * q^37 + 2 * q^39 - 4 * q^43 + 8 * q^45 + 12 * q^49 - 2 * q^51 + 18 * q^53 + 16 * q^55 + 2 * q^57 - 6 * q^59 - 4 * q^61 + 4 * q^63 - 4 * q^65 + 2 * q^67 + 6 * q^69 + 28 * q^71 - 14 * q^73 - 2 * q^75 + 8 * q^77 - 4 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 2 * q^87 + 12 * q^89 - 26 * q^91 + 4 * q^93 - 4 * q^95 + 12 * q^97 + 16 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−4.46410	−1.68727	−0.843636	−	0.536916i	$-0.819589\pi$
$7$	−4.46410	−1.68727	−0.843636	+	0.536916i	$0.819589\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.46410	1.23812	0.619060	−	0.785344i	$-0.287514\pi$
$13$	4.46410	1.23812	0.619060	+	0.785344i	$0.287514\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−7.92820	−1.92287	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−7.92820	−1.92287	−0.961436	+	0.275029i	$0.911312\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.46410	−0.974147
$22$	0	0
$23$	6.46410	1.34786	0.673929	−	0.738796i	$-0.264605\pi$
$23$	6.46410	1.34786	0.673929	+	0.738796i	$0.264605\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−2.46410	−0.457572	−0.228786	−	0.973477i	$-0.573476\pi$
$29$	−2.46410	−0.457572	−0.228786	+	0.973477i	$0.573476\pi$
$30$	0	0
$31$	−4.92820	−0.885131	−0.442566	−	0.896736i	$-0.645932\pi$
$31$	−4.92820	−0.885131	−0.442566	+	0.896736i	$0.645932\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	8.92820	1.50914
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	4.46410	0.714828
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	12.9282	1.84689
$50$	0	0
$51$	−7.92820	−1.11017
$52$	0	0
$53$	12.4641	1.71208	0.856038	−	0.516913i	$-0.172919\pi$
$53$	12.4641	1.71208	0.856038	+	0.516913i	$0.172919\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−9.92820	−1.29254	−0.646271	−	0.763108i	$-0.723672\pi$
$59$	−9.92820	−1.29254	−0.646271	+	0.763108i	$0.723672\pi$
$60$	0	0
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	0	0
$63$	8.92820	1.12485
$64$	0	0
$65$	−8.92820	−1.10741
$66$	0	0
$67$	−5.92820	−0.724245	−0.362123	−	0.932130i	$-0.617948\pi$
$67$	−5.92820	−0.724245	−0.362123	+	0.932130i	$0.617948\pi$
$68$	0	0
$69$	6.46410	0.778186
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	17.8564	2.03493
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.92820	0.321412	0.160706	−	0.987002i	$-0.448623\pi$
$83$	2.92820	0.321412	0.160706	+	0.987002i	$0.448623\pi$
$84$	0	0
$85$	15.8564	1.71987
$86$	0	0
$87$	−2.46410	−0.264179
$88$	0	0
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	−19.9282	−2.08904
$92$	0	0
$93$	−4.92820	−0.511031
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	12.9282	1.31266	0.656330	−	0.754474i	$-0.272108\pi$
$97$	12.9282	1.31266	0.656330	+	0.754474i	$0.272108\pi$
$98$	0	0
$99$	8.00000	0.804030
$100$	0	0
$101$	−9.85641	−0.980749	−0.490375	−	0.871512i	$-0.663140\pi$
$101$	−9.85641	−0.980749	−0.490375	+	0.871512i	$0.663140\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	0	0
$105$	8.92820	0.871303
$106$	0	0
$107$	13.9282	1.34649	0.673245	−	0.739419i	$-0.264900\pi$
$107$	13.9282	1.34649	0.673245	+	0.739419i	$0.264900\pi$
$108$	0	0
$109$	−7.39230	−0.708054	−0.354027	−	0.935235i	$-0.615188\pi$
$109$	−7.39230	−0.708054	−0.354027	+	0.935235i	$0.615188\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	−12.9282	−1.20556
$116$	0	0
$117$	−8.92820	−0.825413
$118$	0	0
$119$	35.3923	3.24441
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−6.92820	−0.624695
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−0.928203	−0.0823647	−0.0411824	−	0.999152i	$-0.513112\pi$
$127$	−0.928203	−0.0823647	−0.0411824	+	0.999152i	$0.513112\pi$
$128$	0	0
$129$	4.92820	0.433904
$130$	0	0
$131$	7.07180	0.617866	0.308933	−	0.951084i	$-0.400028\pi$
$131$	7.07180	0.617866	0.308933	+	0.951084i	$0.400028\pi$
$132$	0	0
$133$	−4.46410	−0.387087
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	8.85641	0.756654	0.378327	−	0.925672i	$-0.376500\pi$
$137$	8.85641	0.756654	0.378327	+	0.925672i	$0.376500\pi$
$138$	0	0
$139$	−15.8564	−1.34492	−0.672461	−	0.740132i	$-0.734763\pi$
$139$	−15.8564	−1.34492	−0.672461	+	0.740132i	$0.734763\pi$
$140$	0	0
$141$	6.92820	0.583460
$142$	0	0
$143$	−17.8564	−1.49323
$144$	0	0
$145$	4.92820	0.409265
$146$	0	0
$147$	12.9282	1.06630
$148$	0	0
$149$	10.9282	0.895273	0.447637	−	0.894216i	$-0.352266\pi$
$149$	10.9282	0.895273	0.447637	+	0.894216i	$0.352266\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	15.8564	1.28191
$154$	0	0
$155$	9.85641	0.791686
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	12.4641	0.988468
$160$	0	0
$161$	−28.8564	−2.27420
$162$	0	0
$163$	23.8564	1.86858	0.934289	−	0.356517i	$-0.116036\pi$
$163$	23.8564	1.86858	0.934289	+	0.356517i	$0.116036\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	4.46410	0.337454
$176$	0	0
$177$	−9.92820	−0.746249
$178$	0	0
$179$	25.8564	1.93260	0.966299	−	0.257421i	$-0.0828728\pi$
$179$	25.8564	1.93260	0.966299	+	0.257421i	$0.0828728\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−8.92820	−0.659992
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	31.7128	2.31907
$188$	0	0
$189$	22.3205	1.62358
$190$	0	0
$191$	3.53590	0.255849	0.127924	−	0.991784i	$-0.459169\pi$
$191$	3.53590	0.255849	0.127924	+	0.991784i	$0.459169\pi$
$192$	0	0
$193$	9.85641	0.709480	0.354740	−	0.934965i	$-0.384569\pi$
$193$	9.85641	0.709480	0.354740	+	0.934965i	$0.384569\pi$
$194$	0	0
$195$	−8.92820	−0.639362
$196$	0	0
$197$	−6.92820	−0.493614	−0.246807	−	0.969065i	$-0.579381\pi$
$197$	−6.92820	−0.493614	−0.246807	+	0.969065i	$0.579381\pi$
$198$	0	0
$199$	−13.5359	−0.959534	−0.479767	−	0.877396i	$-0.659279\pi$
$199$	−13.5359	−0.959534	−0.479767	+	0.877396i	$0.659279\pi$
$200$	0	0
$201$	−5.92820	−0.418143
$202$	0	0
$203$	11.0000	0.772049
$204$	0	0
$205$	13.8564	0.967773
$206$	0	0
$207$	−12.9282	−0.898572
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−24.8564	−1.71119	−0.855593	−	0.517649i	$-0.826807\pi$
$211$	−24.8564	−1.71119	−0.855593	+	0.517649i	$0.826807\pi$
$212$	0	0
$213$	14.0000	0.959264
$214$	0	0
$215$	−9.85641	−0.672201
$216$	0	0
$217$	22.0000	1.49346
$218$	0	0
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−35.3923	−2.38074
$222$	0	0
$223$	26.9282	1.80325	0.901623	−	0.432523i	$-0.142377\pi$
$223$	26.9282	1.80325	0.901623	+	0.432523i	$0.142377\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	−1.92820	−0.127979	−0.0639897	−	0.997951i	$-0.520382\pi$
$227$	−1.92820	−0.127979	−0.0639897	+	0.997951i	$0.520382\pi$
$228$	0	0
$229$	−3.07180	−0.202990	−0.101495	−	0.994836i	$-0.532363\pi$
$229$	−3.07180	−0.202990	−0.101495	+	0.994836i	$0.532363\pi$
$230$	0	0
$231$	17.8564	1.17487
$232$	0	0
$233$	−15.8564	−1.03879	−0.519394	−	0.854535i	$-0.673842\pi$
$233$	−15.8564	−1.03879	−0.519394	+	0.854535i	$0.673842\pi$
$234$	0	0
$235$	−13.8564	−0.903892
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	16.3205	1.05569	0.527843	−	0.849342i	$-0.323001\pi$
$239$	16.3205	1.05569	0.527843	+	0.849342i	$0.323001\pi$
$240$	0	0
$241$	−24.7846	−1.59652	−0.798259	−	0.602315i	$-0.794245\pi$
$241$	−24.7846	−1.59652	−0.798259	+	0.602315i	$0.794245\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−25.8564	−1.65191
$246$	0	0
$247$	4.46410	0.284044
$248$	0	0
$249$	2.92820	0.185567
$250$	0	0
$251$	12.9282	0.816021	0.408010	−	0.912977i	$-0.366223\pi$
$251$	12.9282	0.816021	0.408010	+	0.912977i	$0.366223\pi$
$252$	0	0
$253$	−25.8564	−1.62558
$254$	0	0
$255$	15.8564	0.992967
$256$	0	0
$257$	−0.928203	−0.0578997	−0.0289499	−	0.999581i	$-0.509216\pi$
$257$	−0.928203	−0.0578997	−0.0289499	+	0.999581i	$0.509216\pi$
$258$	0	0
$259$	8.92820	0.554772
$260$	0	0
$261$	4.92820	0.305048
$262$	0	0
$263$	2.92820	0.180561	0.0902804	−	0.995916i	$-0.471224\pi$
$263$	2.92820	0.180561	0.0902804	+	0.995916i	$0.471224\pi$
$264$	0	0
$265$	−24.9282	−1.53133
$266$	0	0
$267$	−0.928203	−0.0568051
$268$	0	0
$269$	−7.85641	−0.479014	−0.239507	−	0.970895i	$-0.576986\pi$
$269$	−7.85641	−0.479014	−0.239507	+	0.970895i	$0.576986\pi$
$270$	0	0
$271$	4.60770	0.279898	0.139949	−	0.990159i	$-0.455306\pi$
$271$	4.60770	0.279898	0.139949	+	0.990159i	$0.455306\pi$
$272$	0	0
$273$	−19.9282	−1.20611
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−24.9282	−1.49779	−0.748895	−	0.662688i	$-0.769415\pi$
$277$	−24.9282	−1.49779	−0.748895	+	0.662688i	$0.769415\pi$
$278$	0	0
$279$	9.85641	0.590088
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	−18.9282	−1.12516	−0.562582	−	0.826741i	$-0.690192\pi$
$283$	−18.9282	−1.12516	−0.562582	+	0.826741i	$0.690192\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	30.9282	1.82563
$288$	0	0
$289$	45.8564	2.69744
$290$	0	0
$291$	12.9282	0.757865
$292$	0	0
$293$	8.32051	0.486089	0.243045	−	0.970015i	$-0.421854\pi$
$293$	8.32051	0.486089	0.243045	+	0.970015i	$0.421854\pi$
$294$	0	0
$295$	19.8564	1.15608
$296$	0	0
$297$	20.0000	1.16052
$298$	0	0
$299$	28.8564	1.66881
$300$	0	0
$301$	−22.0000	−1.26806
$302$	0	0
$303$	−9.85641	−0.566236
$304$	0	0
$305$	17.8564	1.02245
$306$	0	0
$307$	1.85641	0.105951	0.0529754	−	0.998596i	$-0.483130\pi$
$307$	1.85641	0.105951	0.0529754	+	0.998596i	$0.483130\pi$
$308$	0	0
$309$	−2.92820	−0.166580
$310$	0	0
$311$	−7.39230	−0.419179	−0.209590	−	0.977789i	$-0.567213\pi$
$311$	−7.39230	−0.419179	−0.209590	+	0.977789i	$0.567213\pi$
$312$	0	0
$313$	−16.8564	−0.952780	−0.476390	−	0.879234i	$-0.658055\pi$
$313$	−16.8564	−0.952780	−0.476390	+	0.879234i	$0.658055\pi$
$314$	0	0
$315$	−17.8564	−1.00609
$316$	0	0
$317$	14.4641	0.812385	0.406192	−	0.913788i	$-0.366856\pi$
$317$	14.4641	0.812385	0.406192	+	0.913788i	$0.366856\pi$
$318$	0	0
$319$	9.85641	0.551853
$320$	0	0
$321$	13.9282	0.777396
$322$	0	0
$323$	−7.92820	−0.441137
$324$	0	0
$325$	−4.46410	−0.247624
$326$	0	0
$327$	−7.39230	−0.408795
$328$	0	0
$329$	−30.9282	−1.70513
$330$	0	0
$331$	−9.78461	−0.537811	−0.268905	−	0.963167i	$-0.586662\pi$
$331$	−9.78461	−0.537811	−0.268905	+	0.963167i	$0.586662\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	11.8564	0.647785
$336$	0	0
$337$	−7.07180	−0.385225	−0.192613	−	0.981275i	$-0.561696\pi$
$337$	−7.07180	−0.385225	−0.192613	+	0.981275i	$0.561696\pi$
$338$	0	0
$339$	12.9282	0.702164
$340$	0	0
$341$	19.7128	1.06751
$342$	0	0
$343$	−26.4641	−1.42893
$344$	0	0
$345$	−12.9282	−0.696031
$346$	0	0
$347$	−27.8564	−1.49541	−0.747705	−	0.664031i	$-0.768844\pi$
$347$	−27.8564	−1.49541	−0.747705	+	0.664031i	$0.768844\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−22.3205	−1.19138
$352$	0	0
$353$	10.0718	0.536068	0.268034	−	0.963410i	$-0.413626\pi$
$353$	10.0718	0.536068	0.268034	+	0.963410i	$0.413626\pi$
$354$	0	0
$355$	−28.0000	−1.48609
$356$	0	0
$357$	35.3923	1.87316
$358$	0	0
$359$	24.4641	1.29117	0.645583	−	0.763690i	$-0.276614\pi$
$359$	24.4641	1.29117	0.645583	+	0.763690i	$0.276614\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	13.0718	0.682342	0.341171	−	0.940001i	$-0.389176\pi$
$367$	13.0718	0.682342	0.341171	+	0.940001i	$0.389176\pi$
$368$	0	0
$369$	13.8564	0.721336
$370$	0	0
$371$	−55.6410	−2.88874
$372$	0	0
$373$	−11.3923	−0.589871	−0.294936	−	0.955517i	$-0.595298\pi$
$373$	−11.3923	−0.589871	−0.294936	+	0.955517i	$0.595298\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−11.0000	−0.566529
$378$	0	0
$379$	6.85641	0.352190	0.176095	−	0.984373i	$-0.443653\pi$
$379$	6.85641	0.352190	0.176095	+	0.984373i	$0.443653\pi$
$380$	0	0
$381$	−0.928203	−0.0475533
$382$	0	0
$383$	−24.7846	−1.26643	−0.633217	−	0.773974i	$-0.718266\pi$
$383$	−24.7846	−1.26643	−0.633217	+	0.773974i	$0.718266\pi$
$384$	0	0
$385$	−35.7128	−1.82009
$386$	0	0
$387$	−9.85641	−0.501029
$388$	0	0
$389$	−25.8564	−1.31097	−0.655486	−	0.755207i	$-0.727536\pi$
$389$	−25.8564	−1.31097	−0.655486	+	0.755207i	$0.727536\pi$
$390$	0	0
$391$	−51.2487	−2.59176
$392$	0	0
$393$	7.07180	0.356725
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−8.92820	−0.448094	−0.224047	−	0.974578i	$-0.571927\pi$
$397$	−8.92820	−0.448094	−0.224047	+	0.974578i	$0.571927\pi$
$398$	0	0
$399$	−4.46410	−0.223485
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	−22.0000	−1.09590
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	8.85641	0.436854
$412$	0	0
$413$	44.3205	2.18087
$414$	0	0
$415$	−5.85641	−0.287480
$416$	0	0
$417$	−15.8564	−0.776492
$418$	0	0
$419$	−30.7846	−1.50393	−0.751963	−	0.659205i	$-0.770893\pi$
$419$	−30.7846	−1.50393	−0.751963	+	0.659205i	$0.770893\pi$
$420$	0	0
$421$	−11.5359	−0.562225	−0.281113	−	0.959675i	$-0.590703\pi$
$421$	−11.5359	−0.562225	−0.281113	+	0.959675i	$0.590703\pi$
$422$	0	0
$423$	−13.8564	−0.673722
$424$	0	0
$425$	7.92820	0.384574
$426$	0	0
$427$	39.8564	1.92879
$428$	0	0
$429$	−17.8564	−0.862115
$430$	0	0
$431$	12.9282	0.622730	0.311365	−	0.950290i	$-0.399214\pi$
$431$	12.9282	0.622730	0.311365	+	0.950290i	$0.399214\pi$
$432$	0	0
$433$	−24.7846	−1.19107	−0.595536	−	0.803328i	$-0.703060\pi$
$433$	−24.7846	−1.19107	−0.595536	+	0.803328i	$0.703060\pi$
$434$	0	0
$435$	4.92820	0.236289
$436$	0	0
$437$	6.46410	0.309220
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	−25.8564	−1.23126
$442$	0	0
$443$	−17.8564	−0.848383	−0.424192	−	0.905572i	$-0.639442\pi$
$443$	−17.8564	−0.848383	−0.424192	+	0.905572i	$0.639442\pi$
$444$	0	0
$445$	1.85641	0.0880021
$446$	0	0
$447$	10.9282	0.516886
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	27.7128	1.30495
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	39.8564	1.86850
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	39.6410	1.85028
$460$	0	0
$461$	13.8564	0.645357	0.322679	−	0.946509i	$-0.395417\pi$
$461$	13.8564	0.645357	0.322679	+	0.946509i	$0.395417\pi$
$462$	0	0
$463$	16.7846	0.780047	0.390023	−	0.920805i	$-0.372467\pi$
$463$	16.7846	0.780047	0.390023	+	0.920805i	$0.372467\pi$
$464$	0	0
$465$	9.85641	0.457080
$466$	0	0
$467$	2.78461	0.128856	0.0644282	−	0.997922i	$-0.479478\pi$
$467$	2.78461	0.128856	0.0644282	+	0.997922i	$0.479478\pi$
$468$	0	0
$469$	26.4641	1.22200
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−19.7128	−0.906396
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−24.9282	−1.14138
$478$	0	0
$479$	−16.7846	−0.766908	−0.383454	−	0.923560i	$-0.625266\pi$
$479$	−16.7846	−0.766908	−0.383454	+	0.923560i	$0.625266\pi$
$480$	0	0
$481$	−8.92820	−0.407091
$482$	0	0
$483$	−28.8564	−1.31301
$484$	0	0
$485$	−25.8564	−1.17408
$486$	0	0
$487$	34.6410	1.56973	0.784867	−	0.619664i	$-0.212731\pi$
$487$	34.6410	1.56973	0.784867	+	0.619664i	$0.212731\pi$
$488$	0	0
$489$	23.8564	1.07882
$490$	0	0
$491$	14.1436	0.638291	0.319146	−	0.947706i	$-0.396604\pi$
$491$	14.1436	0.638291	0.319146	+	0.947706i	$0.396604\pi$
$492$	0	0
$493$	19.5359	0.879853
$494$	0	0
$495$	−16.0000	−0.719147
$496$	0	0
$497$	−62.4974	−2.80339
$498$	0	0
$499$	−22.7846	−1.01998	−0.509990	−	0.860181i	$-0.670351\pi$
$499$	−22.7846	−1.01998	−0.509990	+	0.860181i	$0.670351\pi$
$500$	0	0
$501$	5.07180	0.226591
$502$	0	0
$503$	−22.3205	−0.995222	−0.497611	−	0.867400i	$-0.665789\pi$
$503$	−22.3205	−0.995222	−0.497611	+	0.867400i	$0.665789\pi$
$504$	0	0
$505$	19.7128	0.877209
$506$	0	0
$507$	6.92820	0.307692
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	31.2487	1.38236
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	5.85641	0.258064
$516$	0	0
$517$	−27.7128	−1.21881
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	13.0718	0.572686	0.286343	−	0.958127i	$-0.407560\pi$
$521$	13.0718	0.572686	0.286343	+	0.958127i	$0.407560\pi$
$522$	0	0
$523$	31.0000	1.35554	0.677768	−	0.735276i	$-0.262948\pi$
$523$	31.0000	1.35554	0.677768	+	0.735276i	$0.262948\pi$
$524$	0	0
$525$	4.46410	0.194829
$526$	0	0
$527$	39.0718	1.70199
$528$	0	0
$529$	18.7846	0.816722
$530$	0	0
$531$	19.8564	0.861695
$532$	0	0
$533$	−30.9282	−1.33965
$534$	0	0
$535$	−27.8564	−1.20434
$536$	0	0
$537$	25.8564	1.11579
$538$	0	0
$539$	−51.7128	−2.22743
$540$	0	0
$541$	12.7846	0.549653	0.274827	−	0.961494i	$-0.411380\pi$
$541$	12.7846	0.549653	0.274827	+	0.961494i	$0.411380\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	14.7846	0.633303
$546$	0	0
$547$	−6.14359	−0.262681	−0.131341	−	0.991337i	$-0.541928\pi$
$547$	−6.14359	−0.262681	−0.131341	+	0.991337i	$0.541928\pi$
$548$	0	0
$549$	17.8564	0.762093
$550$	0	0
$551$	−2.46410	−0.104974
$552$	0	0
$553$	8.92820	0.379666
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	0	0
$559$	22.0000	0.930501
$560$	0	0
$561$	31.7128	1.33892
$562$	0	0
$563$	31.7128	1.33654	0.668268	−	0.743921i	$-0.267036\pi$
$563$	31.7128	1.33654	0.668268	+	0.743921i	$0.267036\pi$
$564$	0	0
$565$	−25.8564	−1.08779
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	8.78461	0.368270	0.184135	−	0.982901i	$-0.441052\pi$
$569$	8.78461	0.368270	0.184135	+	0.982901i	$0.441052\pi$
$570$	0	0
$571$	−28.7846	−1.20460	−0.602299	−	0.798270i	$-0.705749\pi$
$571$	−28.7846	−1.20460	−0.602299	+	0.798270i	$0.705749\pi$
$572$	0	0
$573$	3.53590	0.147714
$574$	0	0
$575$	−6.46410	−0.269572
$576$	0	0
$577$	11.7846	0.490600	0.245300	−	0.969447i	$-0.421114\pi$
$577$	11.7846	0.490600	0.245300	+	0.969447i	$0.421114\pi$
$578$	0	0
$579$	9.85641	0.409618
$580$	0	0
$581$	−13.0718	−0.542310
$582$	0	0
$583$	−49.8564	−2.06484
$584$	0	0
$585$	17.8564	0.738272
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−4.92820	−0.203063
$590$	0	0
$591$	−6.92820	−0.284988
$592$	0	0
$593$	16.1436	0.662938	0.331469	−	0.943466i	$-0.392456\pi$
$593$	16.1436	0.662938	0.331469	+	0.943466i	$0.392456\pi$
$594$	0	0
$595$	−70.7846	−2.90189
$596$	0	0
$597$	−13.5359	−0.553987
$598$	0	0
$599$	9.85641	0.402722	0.201361	−	0.979517i	$-0.435464\pi$
$599$	9.85641	0.402722	0.201361	+	0.979517i	$0.435464\pi$
$600$	0	0
$601$	27.8564	1.13629	0.568143	−	0.822930i	$-0.307662\pi$
$601$	27.8564	1.13629	0.568143	+	0.822930i	$0.307662\pi$
$602$	0	0
$603$	11.8564	0.482830
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	39.8564	1.61772	0.808861	−	0.588000i	$-0.200085\pi$
$607$	39.8564	1.61772	0.808861	+	0.588000i	$0.200085\pi$
$608$	0	0
$609$	11.0000	0.445742
$610$	0	0
$611$	30.9282	1.25122
$612$	0	0
$613$	34.6410	1.39914	0.699569	−	0.714565i	$-0.253375\pi$
$613$	34.6410	1.39914	0.699569	+	0.714565i	$0.253375\pi$
$614$	0	0
$615$	13.8564	0.558744
$616$	0	0
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	0	0
$619$	27.8564	1.11964	0.559822	−	0.828613i	$-0.310870\pi$
$619$	27.8564	1.11964	0.559822	+	0.828613i	$0.310870\pi$
$620$	0	0
$621$	−32.3205	−1.29698
$622$	0	0
$623$	4.14359	0.166010
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	15.8564	0.632236
$630$	0	0
$631$	38.6410	1.53827	0.769137	−	0.639084i	$-0.220686\pi$
$631$	38.6410	1.53827	0.769137	+	0.639084i	$0.220686\pi$
$632$	0	0
$633$	−24.8564	−0.987953
$634$	0	0
$635$	1.85641	0.0736692
$636$	0	0
$637$	57.7128	2.28666
$638$	0	0
$639$	−28.0000	−1.10766
$640$	0	0
$641$	8.78461	0.346971	0.173486	−	0.984836i	$-0.444497\pi$
$641$	8.78461	0.346971	0.173486	+	0.984836i	$0.444497\pi$
$642$	0	0
$643$	−32.9282	−1.29856	−0.649281	−	0.760549i	$-0.724930\pi$
$643$	−32.9282	−1.29856	−0.649281	+	0.760549i	$0.724930\pi$
$644$	0	0
$645$	−9.85641	−0.388096
$646$	0	0
$647$	−13.5359	−0.532151	−0.266076	−	0.963952i	$-0.585727\pi$
$647$	−13.5359	−0.532151	−0.266076	+	0.963952i	$0.585727\pi$
$648$	0	0
$649$	39.7128	1.55886
$650$	0	0
$651$	22.0000	0.862248
$652$	0	0
$653$	−6.78461	−0.265502	−0.132751	−	0.991149i	$-0.542381\pi$
$653$	−6.78461	−0.265502	−0.132751	+	0.991149i	$0.542381\pi$
$654$	0	0
$655$	−14.1436	−0.552636
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	−2.85641	−0.111270	−0.0556349	−	0.998451i	$-0.517718\pi$
$659$	−2.85641	−0.111270	−0.0556349	+	0.998451i	$0.517718\pi$
$660$	0	0
$661$	30.4641	1.18492	0.592458	−	0.805601i	$-0.298158\pi$
$661$	30.4641	1.18492	0.592458	+	0.805601i	$0.298158\pi$
$662$	0	0
$663$	−35.3923	−1.37452
$664$	0	0
$665$	8.92820	0.346221
$666$	0	0
$667$	−15.9282	−0.616742
$668$	0	0
$669$	26.9282	1.04110
$670$	0	0
$671$	35.7128	1.37868
$672$	0	0
$673$	9.07180	0.349692	0.174846	−	0.984596i	$-0.444057\pi$
$673$	9.07180	0.349692	0.174846	+	0.984596i	$0.444057\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−7.39230	−0.284109	−0.142055	−	0.989859i	$-0.545371\pi$
$677$	−7.39230	−0.284109	−0.142055	+	0.989859i	$0.545371\pi$
$678$	0	0
$679$	−57.7128	−2.21481
$680$	0	0
$681$	−1.92820	−0.0738889
$682$	0	0
$683$	17.8564	0.683256	0.341628	−	0.939835i	$-0.389022\pi$
$683$	17.8564	0.683256	0.341628	+	0.939835i	$0.389022\pi$
$684$	0	0
$685$	−17.7128	−0.676772
$686$	0	0
$687$	−3.07180	−0.117196
$688$	0	0
$689$	55.6410	2.11975
$690$	0	0
$691$	−7.85641	−0.298872	−0.149436	−	0.988771i	$-0.547746\pi$
$691$	−7.85641	−0.298872	−0.149436	+	0.988771i	$0.547746\pi$
$692$	0	0
$693$	−35.7128	−1.35662
$694$	0	0
$695$	31.7128	1.20294
$696$	0	0
$697$	54.9282	2.08055
$698$	0	0
$699$	−15.8564	−0.599744
$700$	0	0
$701$	7.71281	0.291309	0.145654	−	0.989336i	$-0.453471\pi$
$701$	7.71281	0.291309	0.145654	+	0.989336i	$0.453471\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	−13.8564	−0.521862
$706$	0	0
$707$	44.0000	1.65479
$708$	0	0
$709$	10.7846	0.405025	0.202512	−	0.979280i	$-0.435089\pi$
$709$	10.7846	0.405025	0.202512	+	0.979280i	$0.435089\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−31.8564	−1.19303
$714$	0	0
$715$	35.7128	1.33558
$716$	0	0
$717$	16.3205	0.609501
$718$	0	0
$719$	13.3923	0.499449	0.249724	−	0.968317i	$-0.419660\pi$
$719$	13.3923	0.499449	0.249724	+	0.968317i	$0.419660\pi$
$720$	0	0
$721$	13.0718	0.486819
$722$	0	0
$723$	−24.7846	−0.921750
$724$	0	0
$725$	2.46410	0.0915144
$726$	0	0
$727$	0.320508	0.0118870	0.00594349	−	0.999982i	$-0.498108\pi$
$727$	0.320508	0.0118870	0.00594349	+	0.999982i	$0.498108\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−39.0718	−1.44512
$732$	0	0
$733$	43.8564	1.61987	0.809937	−	0.586517i	$-0.199501\pi$
$733$	43.8564	1.61987	0.809937	+	0.586517i	$0.199501\pi$
$734$	0	0
$735$	−25.8564	−0.953728
$736$	0	0
$737$	23.7128	0.873473
$738$	0	0
$739$	−9.85641	−0.362574	−0.181287	−	0.983430i	$-0.558026\pi$
$739$	−9.85641	−0.362574	−0.181287	+	0.983430i	$0.558026\pi$
$740$	0	0
$741$	4.46410	0.163993
$742$	0	0
$743$	−16.9282	−0.621036	−0.310518	−	0.950568i	$-0.600502\pi$
$743$	−16.9282	−0.621036	−0.310518	+	0.950568i	$0.600502\pi$
$744$	0	0
$745$	−21.8564	−0.800757
$746$	0	0
$747$	−5.85641	−0.214275
$748$	0	0
$749$	−62.1769	−2.27190
$750$	0	0
$751$	33.0718	1.20681	0.603404	−	0.797436i	$-0.293811\pi$
$751$	33.0718	1.20681	0.603404	+	0.797436i	$0.293811\pi$
$752$	0	0
$753$	12.9282	0.471130
$754$	0	0
$755$	−32.0000	−1.16460
$756$	0	0
$757$	1.21539	0.0441741	0.0220871	−	0.999756i	$-0.492969\pi$
$757$	1.21539	0.0441741	0.0220871	+	0.999756i	$0.492969\pi$
$758$	0	0
$759$	−25.8564	−0.938528
$760$	0	0
$761$	−48.7128	−1.76584	−0.882919	−	0.469525i	$-0.844425\pi$
$761$	−48.7128	−1.76584	−0.882919	+	0.469525i	$0.844425\pi$
$762$	0	0
$763$	33.0000	1.19468
$764$	0	0
$765$	−31.7128	−1.14658
$766$	0	0
$767$	−44.3205	−1.60032
$768$	0	0
$769$	−47.0000	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−47.0000	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$770$	0	0
$771$	−0.928203	−0.0334284
$772$	0	0
$773$	54.4641	1.95894	0.979469	−	0.201596i	$-0.0646128\pi$
$773$	54.4641	1.95894	0.979469	+	0.201596i	$0.0646128\pi$
$774$	0	0
$775$	4.92820	0.177026
$776$	0	0
$777$	8.92820	0.320298
$778$	0	0
$779$	−6.92820	−0.248229
$780$	0	0
$781$	−56.0000	−2.00384
$782$	0	0
$783$	12.3205	0.440299
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	−19.7846	−0.705245	−0.352623	−	0.935766i	$-0.614710\pi$
$787$	−19.7846	−0.705245	−0.352623	+	0.935766i	$0.614710\pi$
$788$	0	0
$789$	2.92820	0.104247
$790$	0	0
$791$	−57.7128	−2.05203
$792$	0	0
$793$	−39.8564	−1.41534
$794$	0	0
$795$	−24.9282	−0.884112
$796$	0	0
$797$	−30.1769	−1.06892	−0.534461	−	0.845193i	$-0.679485\pi$
$797$	−30.1769	−1.06892	−0.534461	+	0.845193i	$0.679485\pi$
$798$	0	0
$799$	−54.9282	−1.94322
$800$	0	0
$801$	1.85641	0.0655929
$802$	0	0
$803$	28.0000	0.988099
$804$	0	0
$805$	57.7128	2.03411
$806$	0	0
$807$	−7.85641	−0.276559
$808$	0	0
$809$	18.0718	0.635371	0.317685	−	0.948196i	$-0.397094\pi$
$809$	18.0718	0.635371	0.317685	+	0.948196i	$0.397094\pi$
$810$	0	0
$811$	1.14359	0.0401570	0.0200785	−	0.999798i	$-0.493608\pi$
$811$	1.14359	0.0401570	0.0200785	+	0.999798i	$0.493608\pi$
$812$	0	0
$813$	4.60770	0.161599
$814$	0	0
$815$	−47.7128	−1.67131
$816$	0	0
$817$	4.92820	0.172416
$818$	0	0
$819$	39.8564	1.39270
$820$	0	0
$821$	44.6410	1.55798	0.778991	−	0.627035i	$-0.215732\pi$
$821$	44.6410	1.55798	0.778991	+	0.627035i	$0.215732\pi$
$822$	0	0
$823$	−23.5359	−0.820410	−0.410205	−	0.911993i	$-0.634543\pi$
$823$	−23.5359	−0.820410	−0.410205	+	0.911993i	$0.634543\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−14.7128	−0.511615	−0.255807	−	0.966728i	$-0.582341\pi$
$827$	−14.7128	−0.511615	−0.255807	+	0.966728i	$0.582341\pi$
$828$	0	0
$829$	12.3205	0.427909	0.213954	−	0.976844i	$-0.431366\pi$
$829$	12.3205	0.427909	0.213954	+	0.976844i	$0.431366\pi$
$830$	0	0
$831$	−24.9282	−0.864750
$832$	0	0
$833$	−102.497	−3.55133
$834$	0	0
$835$	−10.1436	−0.351034
$836$	0	0
$837$	24.6410	0.851718
$838$	0	0
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	0	0
$841$	−22.9282	−0.790628
$842$	0	0
$843$	−4.00000	−0.137767
$844$	0	0
$845$	−13.8564	−0.476675
$846$	0	0
$847$	−22.3205	−0.766942
$848$	0	0
$849$	−18.9282	−0.649614
$850$	0	0
$851$	−12.9282	−0.443173
$852$	0	0
$853$	13.0718	0.447570	0.223785	−	0.974639i	$-0.428159\pi$
$853$	13.0718	0.447570	0.223785	+	0.974639i	$0.428159\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	49.8564	1.70306	0.851531	−	0.524304i	$-0.175674\pi$
$857$	49.8564	1.70306	0.851531	+	0.524304i	$0.175674\pi$
$858$	0	0
$859$	−21.0718	−0.718960	−0.359480	−	0.933153i	$-0.617046\pi$
$859$	−21.0718	−0.718960	−0.359480	+	0.933153i	$0.617046\pi$
$860$	0	0
$861$	30.9282	1.05403
$862$	0	0
$863$	−14.7846	−0.503274	−0.251637	−	0.967822i	$-0.580969\pi$
$863$	−14.7846	−0.503274	−0.251637	+	0.967822i	$0.580969\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	45.8564	1.55737
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−26.4641	−0.896702
$872$	0	0
$873$	−25.8564	−0.875107
$874$	0	0
$875$	−53.5692	−1.81097
$876$	0	0
$877$	31.2487	1.05519	0.527597	−	0.849495i	$-0.323093\pi$
$877$	31.2487	1.05519	0.527597	+	0.849495i	$0.323093\pi$
$878$	0	0
$879$	8.32051	0.280644
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	49.7128	1.67297	0.836485	−	0.547990i	$-0.184607\pi$
$883$	49.7128	1.67297	0.836485	+	0.547990i	$0.184607\pi$
$884$	0	0
$885$	19.8564	0.667466
$886$	0	0
$887$	12.9282	0.434087	0.217043	−	0.976162i	$-0.430359\pi$
$887$	12.9282	0.434087	0.217043	+	0.976162i	$0.430359\pi$
$888$	0	0
$889$	4.14359	0.138972
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	6.92820	0.231843
$894$	0	0
$895$	−51.7128	−1.72857
$896$	0	0
$897$	28.8564	0.963487
$898$	0	0
$899$	12.1436	0.405012
$900$	0	0
$901$	−98.8179	−3.29210
$902$	0	0
$903$	−22.0000	−0.732114
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	13.1436	0.436426	0.218213	−	0.975901i	$-0.429977\pi$
$907$	13.1436	0.436426	0.218213	+	0.975901i	$0.429977\pi$
$908$	0	0
$909$	19.7128	0.653833
$910$	0	0
$911$	7.85641	0.260294	0.130147	−	0.991495i	$-0.458455\pi$
$911$	7.85641	0.260294	0.130147	+	0.991495i	$0.458455\pi$
$912$	0	0
$913$	−11.7128	−0.387638
$914$	0	0
$915$	17.8564	0.590315
$916$	0	0
$917$	−31.5692	−1.04251
$918$	0	0
$919$	−5.39230	−0.177876	−0.0889379	−	0.996037i	$-0.528347\pi$
$919$	−5.39230	−0.177876	−0.0889379	+	0.996037i	$0.528347\pi$
$920$	0	0
$921$	1.85641	0.0611707
$922$	0	0
$923$	62.4974	2.05713
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	5.85641	0.192350
$928$	0	0
$929$	8.71281	0.285858	0.142929	−	0.989733i	$-0.454348\pi$
$929$	8.71281	0.285858	0.142929	+	0.989733i	$0.454348\pi$
$930$	0	0
$931$	12.9282	0.423705
$932$	0	0
$933$	−7.39230	−0.242013
$934$	0	0
$935$	−63.4256	−2.07424
$936$	0	0
$937$	−29.9282	−0.977712	−0.488856	−	0.872365i	$-0.662586\pi$
$937$	−29.9282	−0.977712	−0.488856	+	0.872365i	$0.662586\pi$
$938$	0	0
$939$	−16.8564	−0.550088
$940$	0	0
$941$	−41.1051	−1.33999	−0.669994	−	0.742366i	$-0.733703\pi$
$941$	−41.1051	−1.33999	−0.669994	+	0.742366i	$0.733703\pi$
$942$	0	0
$943$	−44.7846	−1.45839
$944$	0	0
$945$	−44.6410	−1.45217
$946$	0	0
$947$	16.9282	0.550093	0.275046	−	0.961431i	$-0.411307\pi$
$947$	16.9282	0.550093	0.275046	+	0.961431i	$0.411307\pi$
$948$	0	0
$949$	−31.2487	−1.01438
$950$	0	0
$951$	14.4641	0.469031
$952$	0	0
$953$	−7.21539	−0.233729	−0.116865	−	0.993148i	$-0.537284\pi$
$953$	−7.21539	−0.233729	−0.116865	+	0.993148i	$0.537284\pi$
$954$	0	0
$955$	−7.07180	−0.228838
$956$	0	0
$957$	9.85641	0.318612
$958$	0	0
$959$	−39.5359	−1.27668
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	−27.8564	−0.897660
$964$	0	0
$965$	−19.7128	−0.634578
$966$	0	0
$967$	24.7846	0.797019	0.398510	−	0.917164i	$-0.369528\pi$
$967$	24.7846	0.797019	0.398510	+	0.917164i	$0.369528\pi$
$968$	0	0
$969$	−7.92820	−0.254691
$970$	0	0
$971$	25.8564	0.829772	0.414886	−	0.909873i	$-0.363822\pi$
$971$	25.8564	0.829772	0.414886	+	0.909873i	$0.363822\pi$
$972$	0	0
$973$	70.7846	2.26925
$974$	0	0
$975$	−4.46410	−0.142966
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	3.71281	0.118662
$980$	0	0
$981$	14.7846	0.472036
$982$	0	0
$983$	20.6410	0.658346	0.329173	−	0.944270i	$-0.393230\pi$
$983$	20.6410	0.658346	0.329173	+	0.944270i	$0.393230\pi$
$984$	0	0
$985$	13.8564	0.441502
$986$	0	0
$987$	−30.9282	−0.984456
$988$	0	0
$989$	31.8564	1.01297
$990$	0	0
$991$	43.5692	1.38402	0.692011	−	0.721887i	$-0.256725\pi$
$991$	43.5692	1.38402	0.692011	+	0.721887i	$0.256725\pi$
$992$	0	0
$993$	−9.78461	−0.310505
$994$	0	0
$995$	27.0718	0.858234
$996$	0	0
$997$	42.7846	1.35500	0.677501	−	0.735522i	$-0.263063\pi$
$997$	42.7846	1.35500	0.677501	+	0.735522i	$0.263063\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.v.1.1		2
4.3	odd	2		4864.2.a.s.1.2		2
8.3	odd	2		4864.2.a.x.1.2		2
8.5	even	2		4864.2.a.u.1.1		2
16.3	odd	4		2432.2.c.c.1217.1	✓	4
16.5	even	4		2432.2.c.d.1217.2	yes	4
16.11	odd	4		2432.2.c.c.1217.3	yes	4
16.13	even	4		2432.2.c.d.1217.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.c.1217.1	✓	4	16.3	odd	4
2432.2.c.c.1217.3	yes	4	16.11	odd	4
2432.2.c.d.1217.2	yes	4	16.5	even	4
2432.2.c.d.1217.4	yes	4	16.13	even	4
4864.2.a.s.1.2		2	4.3	odd	2
4864.2.a.u.1.1		2	8.5	even	2
4864.2.a.v.1.1		2	1.1	even	1	trivial
4864.2.a.x.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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{5} -4.46410 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{5} -4.46410 q^{7} -2.00000 q^{9} -4.00000 q^{11} +4.46410 q^{13} -2.00000 q^{15} -7.92820 q^{17} +1.00000 q^{19} -4.46410 q^{21} +6.46410 q^{23} -1.00000 q^{25} -5.00000 q^{27} -2.46410 q^{29} -4.92820 q^{31} -4.00000 q^{33} +8.92820 q^{35} -2.00000 q^{37} +4.46410 q^{39} -6.92820 q^{41} +4.92820 q^{43} +4.00000 q^{45} +6.92820 q^{47} +12.9282 q^{49} -7.92820 q^{51} +12.4641 q^{53} +8.00000 q^{55} +1.00000 q^{57} -9.92820 q^{59} -8.92820 q^{61} +8.92820 q^{63} -8.92820 q^{65} -5.92820 q^{67} +6.46410 q^{69} +14.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} +17.8564 q^{77} -2.00000 q^{79} +1.00000 q^{81} +2.92820 q^{83} +15.8564 q^{85} -2.46410 q^{87} -0.928203 q^{89} -19.9282 q^{91} -4.92820 q^{93} -2.00000 q^{95} +12.9282 q^{97} +8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9} - 8 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{21} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 2 q^{29} + 4 q^{31} - 8 q^{33} + 4 q^{35} - 4 q^{37} + 2 q^{39} - 4 q^{43} + 8 q^{45} + 12 q^{49} - 2 q^{51} + 18 q^{53} + 16 q^{55} + 2 q^{57} - 6 q^{59} - 4 q^{61} + 4 q^{63} - 4 q^{65} + 2 q^{67} + 6 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} + 8 q^{77} - 4 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 2 q^{87} + 12 q^{89} - 26 q^{91} + 4 q^{93} - 4 q^{95} + 12 q^{97} + 16 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9 - 8 * q^11 + 2 * q^13 - 4 * q^15 - 2 * q^17 + 2 * q^19 - 2 * q^21 + 6 * q^23 - 2 * q^25 - 10 * q^27 + 2 * q^29 + 4 * q^31 - 8 * q^33 + 4 * q^35 - 4 * q^37 + 2 * q^39 - 4 * q^43 + 8 * q^45 + 12 * q^49 - 2 * q^51 + 18 * q^53 + 16 * q^55 + 2 * q^57 - 6 * q^59 - 4 * q^61 + 4 * q^63 - 4 * q^65 + 2 * q^67 + 6 * q^69 + 28 * q^71 - 14 * q^73 - 2 * q^75 + 8 * q^77 - 4 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 2 * q^87 + 12 * q^89 - 26 * q^91 + 4 * q^93 - 4 * q^95 + 12 * q^97 + 16 * q^99

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