LMFDB - Embedded newform 4864.2.a.v.1.2

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			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{5} +2.46410 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{5} +2.46410 q^{7} -2.00000 q^{9} -4.00000 q^{11} -2.46410 q^{13} -2.00000 q^{15} +5.92820 q^{17} +1.00000 q^{19} +2.46410 q^{21} -0.464102 q^{23} -1.00000 q^{25} -5.00000 q^{27} +4.46410 q^{29} +8.92820 q^{31} -4.00000 q^{33} -4.92820 q^{35} -2.00000 q^{37} -2.46410 q^{39} +6.92820 q^{41} -8.92820 q^{43} +4.00000 q^{45} -6.92820 q^{47} -0.928203 q^{49} +5.92820 q^{51} +5.53590 q^{53} +8.00000 q^{55} +1.00000 q^{57} +3.92820 q^{59} +4.92820 q^{61} -4.92820 q^{63} +4.92820 q^{65} +7.92820 q^{67} -0.464102 q^{69} +14.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} -9.85641 q^{77} -2.00000 q^{79} +1.00000 q^{81} -10.9282 q^{83} -11.8564 q^{85} +4.46410 q^{87} +12.9282 q^{89} -6.07180 q^{91} +8.92820 q^{93} -2.00000 q^{95} -0.928203 q^{97} +8.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	2.46410	0.931343	0.465671	−	0.884958i	$-0.345813\pi$
$7$	2.46410	0.931343	0.465671	+	0.884958i	$0.345813\pi$
$8$	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
$13$	−2.46410	−0.683419	−0.341709	−	0.939806i	$-0.611006\pi$
$13$	−2.46410	−0.683419	−0.341709	+	0.939806i	$0.611006\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	5.92820	1.43780	0.718900	−	0.695113i	$-0.244646\pi$
$17$	5.92820	1.43780	0.718900	+	0.695113i	$0.244646\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.46410	0.537711
$22$	0	0
$23$	−0.464102	−0.0967719	−0.0483859	−	0.998829i	$-0.515408\pi$
$23$	−0.464102	−0.0967719	−0.0483859	+	0.998829i	$0.515408\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	4.46410	0.828963	0.414481	−	0.910058i	$-0.363963\pi$
$29$	4.46410	0.828963	0.414481	+	0.910058i	$0.363963\pi$
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−4.92820	−0.833018
$36$	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
$39$	−2.46410	−0.394572
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	−0.928203	−0.132600
$50$	0	0
$51$	5.92820	0.830114
$52$	0	0
$53$	5.53590	0.760414	0.380207	−	0.924901i	$-0.375853\pi$
$53$	5.53590	0.760414	0.380207	+	0.924901i	$0.375853\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	3.92820	0.511409	0.255704	−	0.966755i	$-0.417693\pi$
$59$	3.92820	0.511409	0.255704	+	0.966755i	$0.417693\pi$
$60$	0	0
$61$	4.92820	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$61$	4.92820	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$62$	0	0
$63$	−4.92820	−0.620895
$64$	0	0
$65$	4.92820	0.611268
$66$	0	0
$67$	7.92820	0.968584	0.484292	−	0.874906i	$-0.339077\pi$
$67$	7.92820	0.968584	0.484292	+	0.874906i	$0.339077\pi$
$68$	0	0
$69$	−0.464102	−0.0558713
$70$	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
$77$	−9.85641	−1.12324
$78$	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
$83$	−10.9282	−1.19953	−0.599763	−	0.800178i	$-0.704739\pi$
$83$	−10.9282	−1.19953	−0.599763	+	0.800178i	$0.704739\pi$
$84$	0	0
$85$	−11.8564	−1.28601
$86$	0	0
$87$	4.46410	0.478602
$88$	0	0
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	−6.07180	−0.636497
$92$	0	0
$93$	8.92820	0.925812
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−0.928203	−0.0942448	−0.0471224	−	0.998889i	$-0.515005\pi$
$97$	−0.928203	−0.0942448	−0.0471224	+	0.998889i	$0.515005\pi$
$98$	0	0
$99$	8.00000	0.804030
$100$	0	0
$101$	17.8564	1.77678	0.888389	−	0.459091i	$-0.151825\pi$
$101$	17.8564	1.77678	0.888389	+	0.459091i	$0.151825\pi$
$102$	0	0
$103$	10.9282	1.07679	0.538394	−	0.842693i	$-0.319031\pi$
$103$	10.9282	1.07679	0.538394	+	0.842693i	$0.319031\pi$
$104$	0	0
$105$	−4.92820	−0.480943
$106$	0	0
$107$	0.0717968	0.00694086	0.00347043	−	0.999994i	$-0.498895\pi$
$107$	0.0717968	0.00694086	0.00347043	+	0.999994i	$0.498895\pi$
$108$	0	0
$109$	13.3923	1.28275	0.641375	−	0.767227i	$-0.278364\pi$
$109$	13.3923	1.28275	0.641375	+	0.767227i	$0.278364\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	0.928203	0.0865554
$116$	0	0
$117$	4.92820	0.455613
$118$	0	0
$119$	14.6077	1.33909
$120$	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
$127$	12.9282	1.14719	0.573596	−	0.819138i	$-0.305548\pi$
$127$	12.9282	1.14719	0.573596	+	0.819138i	$0.305548\pi$
$128$	0	0
$129$	−8.92820	−0.786084
$130$	0	0
$131$	20.9282	1.82851	0.914253	−	0.405144i	$-0.132779\pi$
$131$	20.9282	1.82851	0.914253	+	0.405144i	$0.132779\pi$
$132$	0	0
$133$	2.46410	0.213665
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	−18.8564	−1.61101	−0.805506	−	0.592588i	$-0.798106\pi$
$137$	−18.8564	−1.61101	−0.805506	+	0.592588i	$0.798106\pi$
$138$	0	0
$139$	11.8564	1.00565	0.502824	−	0.864389i	$-0.332295\pi$
$139$	11.8564	1.00565	0.502824	+	0.864389i	$0.332295\pi$
$140$	0	0
$141$	−6.92820	−0.583460
$142$	0	0
$143$	9.85641	0.824234
$144$	0	0
$145$	−8.92820	−0.741447
$146$	0	0
$147$	−0.928203	−0.0765569
$148$	0	0
$149$	−2.92820	−0.239888	−0.119944	−	0.992781i	$-0.538271\pi$
$149$	−2.92820	−0.239888	−0.119944	+	0.992781i	$0.538271\pi$
$150$	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
$153$	−11.8564	−0.958534
$154$	0	0
$155$	−17.8564	−1.43426
$156$	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
$159$	5.53590	0.439025
$160$	0	0
$161$	−1.14359	−0.0901278
$162$	0	0
$163$	−3.85641	−0.302057	−0.151029	−	0.988529i	$-0.548259\pi$
$163$	−3.85641	−0.302057	−0.151029	+	0.988529i	$0.548259\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	18.9282	1.46471	0.732354	−	0.680924i	$-0.238422\pi$
$167$	18.9282	1.46471	0.732354	+	0.680924i	$0.238422\pi$
$168$	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
$175$	−2.46410	−0.186269
$176$	0	0
$177$	3.92820	0.295262
$178$	0	0
$179$	−1.85641	−0.138754	−0.0693772	−	0.997591i	$-0.522101\pi$
$179$	−1.85641	−0.138754	−0.0693772	+	0.997591i	$0.522101\pi$
$180$	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
$183$	4.92820	0.364303
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−23.7128	−1.73405
$188$	0	0
$189$	−12.3205	−0.896185
$190$	0	0
$191$	10.4641	0.757156	0.378578	−	0.925569i	$-0.376413\pi$
$191$	10.4641	0.757156	0.378578	+	0.925569i	$0.376413\pi$
$192$	0	0
$193$	−17.8564	−1.28533	−0.642666	−	0.766146i	$-0.722172\pi$
$193$	−17.8564	−1.28533	−0.642666	+	0.766146i	$0.722172\pi$
$194$	0	0
$195$	4.92820	0.352916
$196$	0	0
$197$	6.92820	0.493614	0.246807	−	0.969065i	$-0.420619\pi$
$197$	6.92820	0.493614	0.246807	+	0.969065i	$0.420619\pi$
$198$	0	0
$199$	−20.4641	−1.45066	−0.725331	−	0.688400i	$-0.758313\pi$
$199$	−20.4641	−1.45066	−0.725331	+	0.688400i	$0.758313\pi$
$200$	0	0
$201$	7.92820	0.559212
$202$	0	0
$203$	11.0000	0.772049
$204$	0	0
$205$	−13.8564	−0.967773
$206$	0	0
$207$	0.928203	0.0645146
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	2.85641	0.196643	0.0983216	−	0.995155i	$-0.468653\pi$
$211$	2.85641	0.196643	0.0983216	+	0.995155i	$0.468653\pi$
$212$	0	0
$213$	14.0000	0.959264
$214$	0	0
$215$	17.8564	1.21780
$216$	0	0
$217$	22.0000	1.49346
$218$	0	0
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−14.6077	−0.982620
$222$	0	0
$223$	13.0718	0.875352	0.437676	−	0.899133i	$-0.355802\pi$
$223$	13.0718	0.875352	0.437676	+	0.899133i	$0.355802\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	11.9282	0.791703	0.395851	−	0.918315i	$-0.370450\pi$
$227$	11.9282	0.791703	0.395851	+	0.918315i	$0.370450\pi$
$228$	0	0
$229$	−16.9282	−1.11865	−0.559324	−	0.828949i	$-0.688939\pi$
$229$	−16.9282	−1.11865	−0.559324	+	0.828949i	$0.688939\pi$
$230$	0	0
$231$	−9.85641	−0.648504
$232$	0	0
$233$	11.8564	0.776739	0.388370	−	0.921504i	$-0.373038\pi$
$233$	11.8564	0.776739	0.388370	+	0.921504i	$0.373038\pi$
$234$	0	0
$235$	13.8564	0.903892
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	−18.3205	−1.18506	−0.592528	−	0.805550i	$-0.701870\pi$
$239$	−18.3205	−1.18506	−0.592528	+	0.805550i	$0.701870\pi$
$240$	0	0
$241$	16.7846	1.08119	0.540596	−	0.841282i	$-0.318199\pi$
$241$	16.7846	1.08119	0.540596	+	0.841282i	$0.318199\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	1.85641	0.118601
$246$	0	0
$247$	−2.46410	−0.156787
$248$	0	0
$249$	−10.9282	−0.692547
$250$	0	0
$251$	−0.928203	−0.0585877	−0.0292938	−	0.999571i	$-0.509326\pi$
$251$	−0.928203	−0.0585877	−0.0292938	+	0.999571i	$0.509326\pi$
$252$	0	0
$253$	1.85641	0.116711
$254$	0	0
$255$	−11.8564	−0.742477
$256$	0	0
$257$	12.9282	0.806439	0.403220	−	0.915103i	$-0.367891\pi$
$257$	12.9282	0.806439	0.403220	+	0.915103i	$0.367891\pi$
$258$	0	0
$259$	−4.92820	−0.306224
$260$	0	0
$261$	−8.92820	−0.552642
$262$	0	0
$263$	−10.9282	−0.673862	−0.336931	−	0.941529i	$-0.609389\pi$
$263$	−10.9282	−0.673862	−0.336931	+	0.941529i	$0.609389\pi$
$264$	0	0
$265$	−11.0718	−0.680135
$266$	0	0
$267$	12.9282	0.791193
$268$	0	0
$269$	19.8564	1.21067	0.605333	−	0.795972i	$-0.293040\pi$
$269$	19.8564	1.21067	0.605333	+	0.795972i	$0.293040\pi$
$270$	0	0
$271$	25.3923	1.54247	0.771236	−	0.636549i	$-0.219639\pi$
$271$	25.3923	1.54247	0.771236	+	0.636549i	$0.219639\pi$
$272$	0	0
$273$	−6.07180	−0.367482
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−11.0718	−0.665240	−0.332620	−	0.943061i	$-0.607933\pi$
$277$	−11.0718	−0.665240	−0.332620	+	0.943061i	$0.607933\pi$
$278$	0	0
$279$	−17.8564	−1.06904
$280$	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
$283$	−5.07180	−0.301487	−0.150744	−	0.988573i	$-0.548167\pi$
$283$	−5.07180	−0.301487	−0.150744	+	0.988573i	$0.548167\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	17.0718	1.00772
$288$	0	0
$289$	18.1436	1.06727
$290$	0	0
$291$	−0.928203	−0.0544122
$292$	0	0
$293$	−26.3205	−1.53766	−0.768830	−	0.639453i	$-0.779161\pi$
$293$	−26.3205	−1.53766	−0.768830	+	0.639453i	$0.779161\pi$
$294$	0	0
$295$	−7.85641	−0.457418
$296$	0	0
$297$	20.0000	1.16052
$298$	0	0
$299$	1.14359	0.0661357
$300$	0	0
$301$	−22.0000	−1.26806
$302$	0	0
$303$	17.8564	1.02582
$304$	0	0
$305$	−9.85641	−0.564376
$306$	0	0
$307$	−25.8564	−1.47570	−0.737852	−	0.674963i	$-0.764160\pi$
$307$	−25.8564	−1.47570	−0.737852	+	0.674963i	$0.764160\pi$
$308$	0	0
$309$	10.9282	0.621684
$310$	0	0
$311$	13.3923	0.759408	0.379704	−	0.925108i	$-0.376026\pi$
$311$	13.3923	0.759408	0.379704	+	0.925108i	$0.376026\pi$
$312$	0	0
$313$	10.8564	0.613640	0.306820	−	0.951767i	$-0.400735\pi$
$313$	10.8564	0.613640	0.306820	+	0.951767i	$0.400735\pi$
$314$	0	0
$315$	9.85641	0.555346
$316$	0	0
$317$	7.53590	0.423258	0.211629	−	0.977350i	$-0.432123\pi$
$317$	7.53590	0.423258	0.211629	+	0.977350i	$0.432123\pi$
$318$	0	0
$319$	−17.8564	−0.999767
$320$	0	0
$321$	0.0717968	0.00400730
$322$	0	0
$323$	5.92820	0.329854
$324$	0	0
$325$	2.46410	0.136684
$326$	0	0
$327$	13.3923	0.740596
$328$	0	0
$329$	−17.0718	−0.941199
$330$	0	0
$331$	31.7846	1.74704	0.873520	−	0.486788i	$-0.161832\pi$
$331$	31.7846	1.74704	0.873520	+	0.486788i	$0.161832\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	−15.8564	−0.866328
$336$	0	0
$337$	−20.9282	−1.14003	−0.570016	−	0.821634i	$-0.693063\pi$
$337$	−20.9282	−1.14003	−0.570016	+	0.821634i	$0.693063\pi$
$338$	0	0
$339$	−0.928203	−0.0504131
$340$	0	0
$341$	−35.7128	−1.93396
$342$	0	0
$343$	−19.5359	−1.05484
$344$	0	0
$345$	0.928203	0.0499728
$346$	0	0
$347$	−0.143594	−0.00770851	−0.00385425	−	0.999993i	$-0.501227\pi$
$347$	−0.143594	−0.00770851	−0.00385425	+	0.999993i	$0.501227\pi$
$348$	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
$351$	12.3205	0.657620
$352$	0	0
$353$	23.9282	1.27357	0.636785	−	0.771042i	$-0.280264\pi$
$353$	23.9282	1.27357	0.636785	+	0.771042i	$0.280264\pi$
$354$	0	0
$355$	−28.0000	−1.48609
$356$	0	0
$357$	14.6077	0.773121
$358$	0	0
$359$	17.5359	0.925509	0.462755	−	0.886486i	$-0.346861\pi$
$359$	17.5359	0.925509	0.462755	+	0.886486i	$0.346861\pi$
$360$	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
$367$	26.9282	1.40564	0.702820	−	0.711367i	$-0.251924\pi$
$367$	26.9282	1.40564	0.702820	+	0.711367i	$0.251924\pi$
$368$	0	0
$369$	−13.8564	−0.721336
$370$	0	0
$371$	13.6410	0.708206
$372$	0	0
$373$	9.39230	0.486315	0.243158	−	0.969987i	$-0.421817\pi$
$373$	9.39230	0.486315	0.243158	+	0.969987i	$0.421817\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−11.0000	−0.566529
$378$	0	0
$379$	−20.8564	−1.07132	−0.535661	−	0.844433i	$-0.679937\pi$
$379$	−20.8564	−1.07132	−0.535661	+	0.844433i	$0.679937\pi$
$380$	0	0
$381$	12.9282	0.662332
$382$	0	0
$383$	16.7846	0.857653	0.428827	−	0.903387i	$-0.358927\pi$
$383$	16.7846	0.857653	0.428827	+	0.903387i	$0.358927\pi$
$384$	0	0
$385$	19.7128	1.00466
$386$	0	0
$387$	17.8564	0.907692
$388$	0	0
$389$	1.85641	0.0941235	0.0470618	−	0.998892i	$-0.485014\pi$
$389$	1.85641	0.0941235	0.0470618	+	0.998892i	$0.485014\pi$
$390$	0	0
$391$	−2.75129	−0.139139
$392$	0	0
$393$	20.9282	1.05569
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	4.92820	0.247339	0.123670	−	0.992323i	$-0.460534\pi$
$397$	4.92820	0.247339	0.123670	+	0.992323i	$0.460534\pi$
$398$	0	0
$399$	2.46410	0.123359
$400$	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
$411$	−18.8564	−0.930118
$412$	0	0
$413$	9.67949	0.476297
$414$	0	0
$415$	21.8564	1.07289
$416$	0	0
$417$	11.8564	0.580611
$418$	0	0
$419$	10.7846	0.526863	0.263431	−	0.964678i	$-0.415146\pi$
$419$	10.7846	0.526863	0.263431	+	0.964678i	$0.415146\pi$
$420$	0	0
$421$	−18.4641	−0.899885	−0.449943	−	0.893057i	$-0.648556\pi$
$421$	−18.4641	−0.899885	−0.449943	+	0.893057i	$0.648556\pi$
$422$	0	0
$423$	13.8564	0.673722
$424$	0	0
$425$	−5.92820	−0.287560
$426$	0	0
$427$	12.1436	0.587670
$428$	0	0
$429$	9.85641	0.475872
$430$	0	0
$431$	−0.928203	−0.0447100	−0.0223550	−	0.999750i	$-0.507116\pi$
$431$	−0.928203	−0.0447100	−0.0223550	+	0.999750i	$0.507116\pi$
$432$	0	0
$433$	16.7846	0.806617	0.403308	−	0.915064i	$-0.367860\pi$
$433$	16.7846	0.806617	0.403308	+	0.915064i	$0.367860\pi$
$434$	0	0
$435$	−8.92820	−0.428075
$436$	0	0
$437$	−0.464102	−0.0222010
$438$	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
$441$	1.85641	0.0884003
$442$	0	0
$443$	9.85641	0.468292	0.234146	−	0.972201i	$-0.424771\pi$
$443$	9.85641	0.468292	0.234146	+	0.972201i	$0.424771\pi$
$444$	0	0
$445$	−25.8564	−1.22571
$446$	0	0
$447$	−2.92820	−0.138499
$448$	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
$455$	12.1436	0.569300
$456$	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
$459$	−29.6410	−1.38352
$460$	0	0
$461$	−13.8564	−0.645357	−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564	−0.645357	−0.322679	+	0.946509i	$0.604583\pi$
$462$	0	0
$463$	−24.7846	−1.15184	−0.575919	−	0.817507i	$-0.695356\pi$
$463$	−24.7846	−1.15184	−0.575919	+	0.817507i	$0.695356\pi$
$464$	0	0
$465$	−17.8564	−0.828071
$466$	0	0
$467$	−38.7846	−1.79474	−0.897369	−	0.441281i	$-0.854524\pi$
$467$	−38.7846	−1.79474	−0.897369	+	0.441281i	$0.854524\pi$
$468$	0	0
$469$	19.5359	0.902084
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	35.7128	1.64208
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−11.0718	−0.506943
$478$	0	0
$479$	24.7846	1.13244	0.566219	−	0.824255i	$-0.308406\pi$
$479$	24.7846	1.13244	0.566219	+	0.824255i	$0.308406\pi$
$480$	0	0
$481$	4.92820	0.224707
$482$	0	0
$483$	−1.14359	−0.0520353
$484$	0	0
$485$	1.85641	0.0842951
$486$	0	0
$487$	−34.6410	−1.56973	−0.784867	−	0.619664i	$-0.787269\pi$
$487$	−34.6410	−1.56973	−0.784867	+	0.619664i	$0.787269\pi$
$488$	0	0
$489$	−3.85641	−0.174393
$490$	0	0
$491$	41.8564	1.88895	0.944477	−	0.328579i	$-0.106570\pi$
$491$	41.8564	1.88895	0.944477	+	0.328579i	$0.106570\pi$
$492$	0	0
$493$	26.4641	1.19188
$494$	0	0
$495$	−16.0000	−0.719147
$496$	0	0
$497$	34.4974	1.54742
$498$	0	0
$499$	18.7846	0.840915	0.420457	−	0.907312i	$-0.361870\pi$
$499$	18.7846	0.840915	0.420457	+	0.907312i	$0.361870\pi$
$500$	0	0
$501$	18.9282	0.845650
$502$	0	0
$503$	12.3205	0.549344	0.274672	−	0.961538i	$-0.411431\pi$
$503$	12.3205	0.549344	0.274672	+	0.961538i	$0.411431\pi$
$504$	0	0
$505$	−35.7128	−1.58920
$506$	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
$511$	−17.2487	−0.763038
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	−21.8564	−0.963108
$516$	0	0
$517$	27.7128	1.21881
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	26.9282	1.17975	0.589873	−	0.807496i	$-0.299178\pi$
$521$	26.9282	1.17975	0.589873	+	0.807496i	$0.299178\pi$
$522$	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
$525$	−2.46410	−0.107542
$526$	0	0
$527$	52.9282	2.30559
$528$	0	0
$529$	−22.7846	−0.990635
$530$	0	0
$531$	−7.85641	−0.340939
$532$	0	0
$533$	−17.0718	−0.739462
$534$	0	0
$535$	−0.143594	−0.00620809
$536$	0	0
$537$	−1.85641	−0.0801099
$538$	0	0
$539$	3.71281	0.159922
$540$	0	0
$541$	−28.7846	−1.23755	−0.618774	−	0.785569i	$-0.712370\pi$
$541$	−28.7846	−1.23755	−0.618774	+	0.785569i	$0.712370\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−26.7846	−1.14733
$546$	0	0
$547$	−33.8564	−1.44760	−0.723798	−	0.690012i	$-0.757605\pi$
$547$	−33.8564	−1.44760	−0.723798	+	0.690012i	$0.757605\pi$
$548$	0	0
$549$	−9.85641	−0.420661
$550$	0	0
$551$	4.46410	0.190177
$552$	0	0
$553$	−4.92820	−0.209569
$554$	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
$561$	−23.7128	−1.00116
$562$	0	0
$563$	−23.7128	−0.999376	−0.499688	−	0.866205i	$-0.666552\pi$
$563$	−23.7128	−0.999376	−0.499688	+	0.866205i	$0.666552\pi$
$564$	0	0
$565$	1.85641	0.0780996
$566$	0	0
$567$	2.46410	0.103483
$568$	0	0
$569$	−32.7846	−1.37440	−0.687201	−	0.726467i	$-0.741161\pi$
$569$	−32.7846	−1.37440	−0.687201	+	0.726467i	$0.741161\pi$
$570$	0	0
$571$	12.7846	0.535019	0.267510	−	0.963555i	$-0.413799\pi$
$571$	12.7846	0.535019	0.267510	+	0.963555i	$0.413799\pi$
$572$	0	0
$573$	10.4641	0.437144
$574$	0	0
$575$	0.464102	0.0193544
$576$	0	0
$577$	−29.7846	−1.23995	−0.619975	−	0.784622i	$-0.712857\pi$
$577$	−29.7846	−1.23995	−0.619975	+	0.784622i	$0.712857\pi$
$578$	0	0
$579$	−17.8564	−0.742087
$580$	0	0
$581$	−26.9282	−1.11717
$582$	0	0
$583$	−22.1436	−0.917094
$584$	0	0
$585$	−9.85641	−0.407512
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	8.92820	0.367880
$590$	0	0
$591$	6.92820	0.284988
$592$	0	0
$593$	43.8564	1.80097	0.900483	−	0.434890i	$-0.143213\pi$
$593$	43.8564	1.80097	0.900483	+	0.434890i	$0.143213\pi$
$594$	0	0
$595$	−29.2154	−1.19771
$596$	0	0
$597$	−20.4641	−0.837540
$598$	0	0
$599$	−17.8564	−0.729593	−0.364796	−	0.931087i	$-0.618861\pi$
$599$	−17.8564	−0.729593	−0.364796	+	0.931087i	$0.618861\pi$
$600$	0	0
$601$	0.143594	0.00585730	0.00292865	−	0.999996i	$-0.499068\pi$
$601$	0.143594	0.00585730	0.00292865	+	0.999996i	$0.499068\pi$
$602$	0	0
$603$	−15.8564	−0.645723
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	12.1436	0.492893	0.246447	−	0.969156i	$-0.420737\pi$
$607$	12.1436	0.492893	0.246447	+	0.969156i	$0.420737\pi$
$608$	0	0
$609$	11.0000	0.445742
$610$	0	0
$611$	17.0718	0.690651
$612$	0	0
$613$	−34.6410	−1.39914	−0.699569	−	0.714565i	$-0.746625\pi$
$613$	−34.6410	−1.39914	−0.699569	+	0.714565i	$0.746625\pi$
$614$	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
$619$	0.143594	0.00577151	0.00288576	−	0.999996i	$-0.499081\pi$
$619$	0.143594	0.00577151	0.00288576	+	0.999996i	$0.499081\pi$
$620$	0	0
$621$	2.32051	0.0931188
$622$	0	0
$623$	31.8564	1.27630
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	−11.8564	−0.472746
$630$	0	0
$631$	−30.6410	−1.21980	−0.609900	−	0.792479i	$-0.708790\pi$
$631$	−30.6410	−1.21980	−0.609900	+	0.792479i	$0.708790\pi$
$632$	0	0
$633$	2.85641	0.113532
$634$	0	0
$635$	−25.8564	−1.02608
$636$	0	0
$637$	2.28719	0.0906217
$638$	0	0
$639$	−28.0000	−1.10766
$640$	0	0
$641$	−32.7846	−1.29491	−0.647457	−	0.762102i	$-0.724168\pi$
$641$	−32.7846	−1.29491	−0.647457	+	0.762102i	$0.724168\pi$
$642$	0	0
$643$	−19.0718	−0.752118	−0.376059	−	0.926596i	$-0.622721\pi$
$643$	−19.0718	−0.752118	−0.376059	+	0.926596i	$0.622721\pi$
$644$	0	0
$645$	17.8564	0.703095
$646$	0	0
$647$	−20.4641	−0.804527	−0.402263	−	0.915524i	$-0.631776\pi$
$647$	−20.4641	−0.804527	−0.402263	+	0.915524i	$0.631776\pi$
$648$	0	0
$649$	−15.7128	−0.616782
$650$	0	0
$651$	22.0000	0.862248
$652$	0	0
$653$	34.7846	1.36123	0.680613	−	0.732643i	$-0.261713\pi$
$653$	34.7846	1.36123	0.680613	+	0.732643i	$0.261713\pi$
$654$	0	0
$655$	−41.8564	−1.63547
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	24.8564	0.968268	0.484134	−	0.874994i	$-0.339135\pi$
$659$	24.8564	0.968268	0.484134	+	0.874994i	$0.339135\pi$
$660$	0	0
$661$	23.5359	0.915440	0.457720	−	0.889096i	$-0.348666\pi$
$661$	23.5359	0.915440	0.457720	+	0.889096i	$0.348666\pi$
$662$	0	0
$663$	−14.6077	−0.567316
$664$	0	0
$665$	−4.92820	−0.191108
$666$	0	0
$667$	−2.07180	−0.0802203
$668$	0	0
$669$	13.0718	0.505385
$670$	0	0
$671$	−19.7128	−0.761005
$672$	0	0
$673$	22.9282	0.883817	0.441909	−	0.897060i	$-0.354302\pi$
$673$	22.9282	0.883817	0.441909	+	0.897060i	$0.354302\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	13.3923	0.514708	0.257354	−	0.966317i	$-0.417149\pi$
$677$	13.3923	0.514708	0.257354	+	0.966317i	$0.417149\pi$
$678$	0	0
$679$	−2.28719	−0.0877742
$680$	0	0
$681$	11.9282	0.457090
$682$	0	0
$683$	−9.85641	−0.377145	−0.188572	−	0.982059i	$-0.560386\pi$
$683$	−9.85641	−0.377145	−0.188572	+	0.982059i	$0.560386\pi$
$684$	0	0
$685$	37.7128	1.44093
$686$	0	0
$687$	−16.9282	−0.645851
$688$	0	0
$689$	−13.6410	−0.519681
$690$	0	0
$691$	19.8564	0.755373	0.377687	−	0.925933i	$-0.376720\pi$
$691$	19.8564	0.755373	0.377687	+	0.925933i	$0.376720\pi$
$692$	0	0
$693$	19.7128	0.748828
$694$	0	0
$695$	−23.7128	−0.899478
$696$	0	0
$697$	41.0718	1.55571
$698$	0	0
$699$	11.8564	0.448450
$700$	0	0
$701$	−47.7128	−1.80209	−0.901044	−	0.433728i	$-0.857198\pi$
$701$	−47.7128	−1.80209	−0.901044	+	0.433728i	$0.857198\pi$
$702$	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
$709$	−30.7846	−1.15614	−0.578070	−	0.815987i	$-0.696194\pi$
$709$	−30.7846	−1.15614	−0.578070	+	0.815987i	$0.696194\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−4.14359	−0.155179
$714$	0	0
$715$	−19.7128	−0.737217
$716$	0	0
$717$	−18.3205	−0.684192
$718$	0	0
$719$	−7.39230	−0.275686	−0.137843	−	0.990454i	$-0.544017\pi$
$719$	−7.39230	−0.275686	−0.137843	+	0.990454i	$0.544017\pi$
$720$	0	0
$721$	26.9282	1.00286
$722$	0	0
$723$	16.7846	0.624226
$724$	0	0
$725$	−4.46410	−0.165793
$726$	0	0
$727$	−34.3205	−1.27288	−0.636439	−	0.771327i	$-0.719593\pi$
$727$	−34.3205	−1.27288	−0.636439	+	0.771327i	$0.719593\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−52.9282	−1.95762
$732$	0	0
$733$	16.1436	0.596277	0.298139	−	0.954523i	$-0.403634\pi$
$733$	16.1436	0.596277	0.298139	+	0.954523i	$0.403634\pi$
$734$	0	0
$735$	1.85641	0.0684746
$736$	0	0
$737$	−31.7128	−1.16816
$738$	0	0
$739$	17.8564	0.656859	0.328429	−	0.944529i	$-0.393481\pi$
$739$	17.8564	0.656859	0.328429	+	0.944529i	$0.393481\pi$
$740$	0	0
$741$	−2.46410	−0.0905210
$742$	0	0
$743$	−3.07180	−0.112693	−0.0563466	−	0.998411i	$-0.517945\pi$
$743$	−3.07180	−0.112693	−0.0563466	+	0.998411i	$0.517945\pi$
$744$	0	0
$745$	5.85641	0.214562
$746$	0	0
$747$	21.8564	0.799684
$748$	0	0
$749$	0.176915	0.00646432
$750$	0	0
$751$	46.9282	1.71243	0.856217	−	0.516616i	$-0.172808\pi$
$751$	46.9282	1.71243	0.856217	+	0.516616i	$0.172808\pi$
$752$	0	0
$753$	−0.928203	−0.0338256
$754$	0	0
$755$	−32.0000	−1.16460
$756$	0	0
$757$	42.7846	1.55503	0.777517	−	0.628862i	$-0.216479\pi$
$757$	42.7846	1.55503	0.777517	+	0.628862i	$0.216479\pi$
$758$	0	0
$759$	1.85641	0.0673833
$760$	0	0
$761$	6.71281	0.243339	0.121670	−	0.992571i	$-0.461175\pi$
$761$	6.71281	0.243339	0.121670	+	0.992571i	$0.461175\pi$
$762$	0	0
$763$	33.0000	1.19468
$764$	0	0
$765$	23.7128	0.857339
$766$	0	0
$767$	−9.67949	−0.349506
$768$	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
$771$	12.9282	0.465598
$772$	0	0
$773$	47.5359	1.70975	0.854874	−	0.518836i	$-0.173635\pi$
$773$	47.5359	1.70975	0.854874	+	0.518836i	$0.173635\pi$
$774$	0	0
$775$	−8.92820	−0.320711
$776$	0	0
$777$	−4.92820	−0.176798
$778$	0	0
$779$	6.92820	0.248229
$780$	0	0
$781$	−56.0000	−2.00384
$782$	0	0
$783$	−22.3205	−0.797670
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	21.7846	0.776537	0.388269	−	0.921546i	$-0.373073\pi$
$787$	21.7846	0.776537	0.388269	+	0.921546i	$0.373073\pi$
$788$	0	0
$789$	−10.9282	−0.389054
$790$	0	0
$791$	−2.28719	−0.0813230
$792$	0	0
$793$	−12.1436	−0.431232
$794$	0	0
$795$	−11.0718	−0.392676
$796$	0	0
$797$	32.1769	1.13976	0.569882	−	0.821726i	$-0.306989\pi$
$797$	32.1769	1.13976	0.569882	+	0.821726i	$0.306989\pi$
$798$	0	0
$799$	−41.0718	−1.45302
$800$	0	0
$801$	−25.8564	−0.913591
$802$	0	0
$803$	28.0000	0.988099
$804$	0	0
$805$	2.28719	0.0806128
$806$	0	0
$807$	19.8564	0.698979
$808$	0	0
$809$	31.9282	1.12254	0.561268	−	0.827634i	$-0.310314\pi$
$809$	31.9282	1.12254	0.561268	+	0.827634i	$0.310314\pi$
$810$	0	0
$811$	28.8564	1.01329	0.506643	−	0.862156i	$-0.330886\pi$
$811$	28.8564	1.01329	0.506643	+	0.862156i	$0.330886\pi$
$812$	0	0
$813$	25.3923	0.890547
$814$	0	0
$815$	7.71281	0.270168
$816$	0	0
$817$	−8.92820	−0.312358
$818$	0	0
$819$	12.1436	0.424331
$820$	0	0
$821$	−24.6410	−0.859977	−0.429989	−	0.902834i	$-0.641482\pi$
$821$	−24.6410	−0.859977	−0.429989	+	0.902834i	$0.641482\pi$
$822$	0	0
$823$	−30.4641	−1.06191	−0.530956	−	0.847399i	$-0.678167\pi$
$823$	−30.4641	−1.06191	−0.530956	+	0.847399i	$0.678167\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	40.7128	1.41572	0.707862	−	0.706351i	$-0.249660\pi$
$827$	40.7128	1.41572	0.707862	+	0.706351i	$0.249660\pi$
$828$	0	0
$829$	−22.3205	−0.775223	−0.387612	−	0.921823i	$-0.626700\pi$
$829$	−22.3205	−0.775223	−0.387612	+	0.921823i	$0.626700\pi$
$830$	0	0
$831$	−11.0718	−0.384076
$832$	0	0
$833$	−5.50258	−0.190653
$834$	0	0
$835$	−37.8564	−1.31007
$836$	0	0
$837$	−44.6410	−1.54302
$838$	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
$841$	−9.07180	−0.312821
$842$	0	0
$843$	−4.00000	−0.137767
$844$	0	0
$845$	13.8564	0.476675
$846$	0	0
$847$	12.3205	0.423338
$848$	0	0
$849$	−5.07180	−0.174064
$850$	0	0
$851$	0.928203	0.0318184
$852$	0	0
$853$	26.9282	0.922004	0.461002	−	0.887399i	$-0.347490\pi$
$853$	26.9282	0.922004	0.461002	+	0.887399i	$0.347490\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	22.1436	0.756411	0.378205	−	0.925722i	$-0.376541\pi$
$857$	22.1436	0.756411	0.378205	+	0.925722i	$0.376541\pi$
$858$	0	0
$859$	−34.9282	−1.19173	−0.595867	−	0.803083i	$-0.703192\pi$
$859$	−34.9282	−1.19173	−0.595867	+	0.803083i	$0.703192\pi$
$860$	0	0
$861$	17.0718	0.581805
$862$	0	0
$863$	26.7846	0.911759	0.455879	−	0.890042i	$-0.349325\pi$
$863$	26.7846	0.911759	0.455879	+	0.890042i	$0.349325\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	18.1436	0.616189
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−19.5359	−0.661949
$872$	0	0
$873$	1.85641	0.0628298
$874$	0	0
$875$	29.5692	0.999622
$876$	0	0
$877$	−17.2487	−0.582448	−0.291224	−	0.956655i	$-0.594062\pi$
$877$	−17.2487	−0.582448	−0.291224	+	0.956655i	$0.594062\pi$
$878$	0	0
$879$	−26.3205	−0.887769
$880$	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
$883$	−5.71281	−0.192251	−0.0961257	−	0.995369i	$-0.530645\pi$
$883$	−5.71281	−0.192251	−0.0961257	+	0.995369i	$0.530645\pi$
$884$	0	0
$885$	−7.85641	−0.264090
$886$	0	0
$887$	−0.928203	−0.0311660	−0.0155830	−	0.999879i	$-0.504960\pi$
$887$	−0.928203	−0.0311660	−0.0155830	+	0.999879i	$0.504960\pi$
$888$	0	0
$889$	31.8564	1.06843
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−6.92820	−0.231843
$894$	0	0
$895$	3.71281	0.124106
$896$	0	0
$897$	1.14359	0.0381835
$898$	0	0
$899$	39.8564	1.32929
$900$	0	0
$901$	32.8179	1.09332
$902$	0	0
$903$	−22.0000	−0.732114
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	40.8564	1.35661	0.678307	−	0.734778i	$-0.262714\pi$
$907$	40.8564	1.35661	0.678307	+	0.734778i	$0.262714\pi$
$908$	0	0
$909$	−35.7128	−1.18452
$910$	0	0
$911$	−19.8564	−0.657872	−0.328936	−	0.944352i	$-0.606690\pi$
$911$	−19.8564	−0.657872	−0.328936	+	0.944352i	$0.606690\pi$
$912$	0	0
$913$	43.7128	1.44668
$914$	0	0
$915$	−9.85641	−0.325843
$916$	0	0
$917$	51.5692	1.70297
$918$	0	0
$919$	15.3923	0.507745	0.253873	−	0.967238i	$-0.418296\pi$
$919$	15.3923	0.507745	0.253873	+	0.967238i	$0.418296\pi$
$920$	0	0
$921$	−25.8564	−0.851998
$922$	0	0
$923$	−34.4974	−1.13550
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	−21.8564	−0.717859
$928$	0	0
$929$	−46.7128	−1.53260	−0.766299	−	0.642484i	$-0.777904\pi$
$929$	−46.7128	−1.53260	−0.766299	+	0.642484i	$0.777904\pi$
$930$	0	0
$931$	−0.928203	−0.0304206
$932$	0	0
$933$	13.3923	0.438444
$934$	0	0
$935$	47.4256	1.55098
$936$	0	0
$937$	−16.0718	−0.525043	−0.262521	−	0.964926i	$-0.584554\pi$
$937$	−16.0718	−0.525043	−0.262521	+	0.964926i	$0.584554\pi$
$938$	0	0
$939$	10.8564	0.354285
$940$	0	0
$941$	35.1051	1.14439	0.572197	−	0.820116i	$-0.306091\pi$
$941$	35.1051	1.14439	0.572197	+	0.820116i	$0.306091\pi$
$942$	0	0
$943$	−3.21539	−0.104708
$944$	0	0
$945$	24.6410	0.801572
$946$	0	0
$947$	3.07180	0.0998200	0.0499100	−	0.998754i	$-0.484107\pi$
$947$	3.07180	0.0998200	0.0499100	+	0.998754i	$0.484107\pi$
$948$	0	0
$949$	17.2487	0.559917
$950$	0	0
$951$	7.53590	0.244368
$952$	0	0
$953$	−48.7846	−1.58029	−0.790144	−	0.612921i	$-0.789994\pi$
$953$	−48.7846	−1.58029	−0.790144	+	0.612921i	$0.789994\pi$
$954$	0	0
$955$	−20.9282	−0.677221
$956$	0	0
$957$	−17.8564	−0.577216
$958$	0	0
$959$	−46.4641	−1.50040
$960$	0	0
$961$	48.7128	1.57138
$962$	0	0
$963$	−0.143594	−0.00462724
$964$	0	0
$965$	35.7128	1.14964
$966$	0	0
$967$	−16.7846	−0.539757	−0.269878	−	0.962894i	$-0.586983\pi$
$967$	−16.7846	−0.539757	−0.269878	+	0.962894i	$0.586983\pi$
$968$	0	0
$969$	5.92820	0.190441
$970$	0	0
$971$	−1.85641	−0.0595749	−0.0297875	−	0.999556i	$-0.509483\pi$
$971$	−1.85641	−0.0595749	−0.0297875	+	0.999556i	$0.509483\pi$
$972$	0	0
$973$	29.2154	0.936602
$974$	0	0
$975$	2.46410	0.0789144
$976$	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
$979$	−51.7128	−1.65275
$980$	0	0
$981$	−26.7846	−0.855167
$982$	0	0
$983$	−48.6410	−1.55141	−0.775704	−	0.631097i	$-0.782605\pi$
$983$	−48.6410	−1.55141	−0.775704	+	0.631097i	$0.782605\pi$
$984$	0	0
$985$	−13.8564	−0.441502
$986$	0	0
$987$	−17.0718	−0.543401
$988$	0	0
$989$	4.14359	0.131759
$990$	0	0
$991$	−39.5692	−1.25696	−0.628479	−	0.777827i	$-0.716322\pi$
$991$	−39.5692	−1.25696	−0.628479	+	0.777827i	$0.716322\pi$
$992$	0	0
$993$	31.7846	1.00865
$994$	0	0
$995$	40.9282	1.29751
$996$	0	0
$997$	1.21539	0.0384918	0.0192459	−	0.999815i	$-0.493873\pi$
$997$	1.21539	0.0384918	0.0192459	+	0.999815i	$0.493873\pi$
$998$	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.1.2		2
4.3	odd	2		4864.2.a.s.1.1		2
8.3	odd	2		4864.2.a.x.1.1		2
8.5	even	2		4864.2.a.u.1.2		2
16.3	odd	4		2432.2.c.c.1217.2	✓	4
16.5	even	4		2432.2.c.d.1217.1	yes	4
16.11	odd	4		2432.2.c.c.1217.4	yes	4
16.13	even	4		2432.2.c.d.1217.3	yes	4

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{5} +2.46410 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{5} +2.46410 q^{7} -2.00000 q^{9} -4.00000 q^{11} -2.46410 q^{13} -2.00000 q^{15} +5.92820 q^{17} +1.00000 q^{19} +2.46410 q^{21} -0.464102 q^{23} -1.00000 q^{25} -5.00000 q^{27} +4.46410 q^{29} +8.92820 q^{31} -4.00000 q^{33} -4.92820 q^{35} -2.00000 q^{37} -2.46410 q^{39} +6.92820 q^{41} -8.92820 q^{43} +4.00000 q^{45} -6.92820 q^{47} -0.928203 q^{49} +5.92820 q^{51} +5.53590 q^{53} +8.00000 q^{55} +1.00000 q^{57} +3.92820 q^{59} +4.92820 q^{61} -4.92820 q^{63} +4.92820 q^{65} +7.92820 q^{67} -0.464102 q^{69} +14.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} -9.85641 q^{77} -2.00000 q^{79} +1.00000 q^{81} -10.9282 q^{83} -11.8564 q^{85} +4.46410 q^{87} +12.9282 q^{89} -6.07180 q^{91} +8.92820 q^{93} -2.00000 q^{95} -0.928203 q^{97} +8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 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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