LMFDB - Embedded newform 3584.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, 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("3584.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	3584.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765367 q^{3} +1.84776 q^{5} -1.00000 q^{7} -2.41421 q^{9} +O(q^{10})$	$q+0.765367 q^{3} +1.84776 q^{5} -1.00000 q^{7} -2.41421 q^{9} +3.69552 q^{11} +2.29610 q^{13} +1.41421 q^{15} +0.828427 q^{17} +3.37849 q^{19} -0.765367 q^{21} +0.585786 q^{23} -1.58579 q^{25} -4.14386 q^{27} +8.92177 q^{29} +1.17157 q^{31} +2.82843 q^{33} -1.84776 q^{35} -0.634051 q^{37} +1.75736 q^{39} -6.48528 q^{41} +1.53073 q^{43} -4.46088 q^{45} +4.00000 q^{47} +1.00000 q^{49} +0.634051 q^{51} -7.39104 q^{53} +6.82843 q^{55} +2.58579 q^{57} +15.0991 q^{59} -1.84776 q^{61} +2.41421 q^{63} +4.24264 q^{65} -6.49435 q^{67} +0.448342 q^{69} +4.82843 q^{71} -10.4853 q^{73} -1.21371 q^{75} -3.69552 q^{77} +8.82843 q^{79} +4.07107 q^{81} -2.29610 q^{83} +1.53073 q^{85} +6.82843 q^{87} +6.00000 q^{89} -2.29610 q^{91} +0.896683 q^{93} +6.24264 q^{95} -2.00000 q^{97} -8.92177 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 - 4 * q^9	$ 4 q - 4 q^{7} - 4 q^{9} - 8 q^{17} + 8 q^{23} - 12 q^{25} + 16 q^{31} + 24 q^{39} + 8 q^{41} + 16 q^{47} + 4 q^{49} + 16 q^{55} + 16 q^{57} + 4 q^{63} + 8 q^{71} - 8 q^{73} + 24 q^{79} - 12 q^{81} + 16 q^{87} + 24 q^{89} + 8 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^9 - 8 * q^17 + 8 * q^23 - 12 * q^25 + 16 * q^31 + 24 * q^39 + 8 * q^41 + 16 * q^47 + 4 * q^49 + 16 * q^55 + 16 * q^57 + 4 * q^63 + 8 * q^71 - 8 * q^73 + 24 * q^79 - 12 * q^81 + 16 * q^87 + 24 * q^89 + 8 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.765367	0.441885	0.220942	−	0.975287i	$-0.429087\pi$
$3$	0.765367	0.441885	0.220942	+	0.975287i	$0.429087\pi$
$4$	0	0
$5$	1.84776	0.826343	0.413171	−	0.910653i	$-0.364421\pi$
$5$	1.84776	0.826343	0.413171	+	0.910653i	$0.364421\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.41421	−0.804738
$10$	0	0
$11$	3.69552	1.11424	0.557120	−	0.830432i	$-0.311906\pi$
$11$	3.69552	1.11424	0.557120	+	0.830432i	$0.311906\pi$
$12$	0	0
$13$	2.29610	0.636824	0.318412	−	0.947952i	$-0.396851\pi$
$13$	2.29610	0.636824	0.318412	+	0.947952i	$0.396851\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	3.37849	0.775079	0.387540	−	0.921853i	$-0.373325\pi$
$19$	3.37849	0.775079	0.387540	+	0.921853i	$0.373325\pi$
$20$	0	0
$21$	−0.765367	−0.167017
$22$	0	0
$23$	0.585786	0.122145	0.0610725	−	0.998133i	$-0.480548\pi$
$23$	0.585786	0.122145	0.0610725	+	0.998133i	$0.480548\pi$
$24$	0	0
$25$	−1.58579	−0.317157
$26$	0	0
$27$	−4.14386	−0.797486
$28$	0	0
$29$	8.92177	1.65673	0.828366	−	0.560188i	$-0.189271\pi$
$29$	8.92177	1.65673	0.828366	+	0.560188i	$0.189271\pi$
$30$	0	0
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	−1.84776	−0.312328
$36$	0	0
$37$	−0.634051	−0.104237	−0.0521186	−	0.998641i	$-0.516597\pi$
$37$	−0.634051	−0.104237	−0.0521186	+	0.998641i	$0.516597\pi$
$38$	0	0
$39$	1.75736	0.281403
$40$	0	0
$41$	−6.48528	−1.01283	−0.506415	−	0.862290i	$-0.669030\pi$
$41$	−6.48528	−1.01283	−0.506415	+	0.862290i	$0.669030\pi$
$42$	0	0
$43$	1.53073	0.233435	0.116717	−	0.993165i	$-0.462763\pi$
$43$	1.53073	0.233435	0.116717	+	0.993165i	$0.462763\pi$
$44$	0	0
$45$	−4.46088	−0.664989
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.634051	0.0887849
$52$	0	0
$53$	−7.39104	−1.01524	−0.507618	−	0.861582i	$-0.669474\pi$
$53$	−7.39104	−1.01524	−0.507618	+	0.861582i	$0.669474\pi$
$54$	0	0
$55$	6.82843	0.920745
$56$	0	0
$57$	2.58579	0.342496
$58$	0	0
$59$	15.0991	1.96574	0.982868	−	0.184313i	$-0.0590061\pi$
$59$	15.0991	1.96574	0.982868	+	0.184313i	$0.0590061\pi$
$60$	0	0
$61$	−1.84776	−0.236581	−0.118291	−	0.992979i	$-0.537741\pi$
$61$	−1.84776	−0.236581	−0.118291	+	0.992979i	$0.537741\pi$
$62$	0	0
$63$	2.41421	0.304162
$64$	0	0
$65$	4.24264	0.526235
$66$	0	0
$67$	−6.49435	−0.793412	−0.396706	−	0.917946i	$-0.629847\pi$
$67$	−6.49435	−0.793412	−0.396706	+	0.917946i	$0.629847\pi$
$68$	0	0
$69$	0.448342	0.0539740
$70$	0	0
$71$	4.82843	0.573029	0.286514	−	0.958076i	$-0.407503\pi$
$71$	4.82843	0.573029	0.286514	+	0.958076i	$0.407503\pi$
$72$	0	0
$73$	−10.4853	−1.22721	−0.613605	−	0.789613i	$-0.710281\pi$
$73$	−10.4853	−1.22721	−0.613605	+	0.789613i	$0.710281\pi$
$74$	0	0
$75$	−1.21371	−0.140147
$76$	0	0
$77$	−3.69552	−0.421143
$78$	0	0
$79$	8.82843	0.993276	0.496638	−	0.867958i	$-0.334568\pi$
$79$	8.82843	0.993276	0.496638	+	0.867958i	$0.334568\pi$
$80$	0	0
$81$	4.07107	0.452341
$82$	0	0
$83$	−2.29610	−0.252030	−0.126015	−	0.992028i	$-0.540219\pi$
$83$	−2.29610	−0.252030	−0.126015	+	0.992028i	$0.540219\pi$
$84$	0	0
$85$	1.53073	0.166031
$86$	0	0
$87$	6.82843	0.732084
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.29610	−0.240697
$92$	0	0
$93$	0.896683	0.0929817
$94$	0	0
$95$	6.24264	0.640481
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−8.92177	−0.896672
$100$	0	0
$101$	14.0167	1.39471	0.697357	−	0.716724i	$-0.254359\pi$
$101$	14.0167	1.39471	0.697357	+	0.716724i	$0.254359\pi$
$102$	0	0
$103$	16.4853	1.62434	0.812172	−	0.583419i	$-0.198285\pi$
$103$	16.4853	1.62434	0.812172	+	0.583419i	$0.198285\pi$
$104$	0	0
$105$	−1.41421	−0.138013
$106$	0	0
$107$	−2.16478	−0.209278	−0.104639	−	0.994510i	$-0.533369\pi$
$107$	−2.16478	−0.209278	−0.104639	+	0.994510i	$0.533369\pi$
$108$	0	0
$109$	13.2513	1.26925	0.634624	−	0.772821i	$-0.281155\pi$
$109$	13.2513	1.26925	0.634624	+	0.772821i	$0.281155\pi$
$110$	0	0
$111$	−0.485281	−0.0460609
$112$	0	0
$113$	13.8995	1.30755	0.653777	−	0.756687i	$-0.273183\pi$
$113$	13.8995	1.30755	0.653777	+	0.756687i	$0.273183\pi$
$114$	0	0
$115$	1.08239	0.100934
$116$	0	0
$117$	−5.54328	−0.512476
$118$	0	0
$119$	−0.828427	−0.0759418
$120$	0	0
$121$	2.65685	0.241532
$122$	0	0
$123$	−4.96362	−0.447554
$124$	0	0
$125$	−12.1689	−1.08842
$126$	0	0
$127$	9.75736	0.865826	0.432913	−	0.901436i	$-0.357486\pi$
$127$	9.75736	0.865826	0.432913	+	0.901436i	$0.357486\pi$
$128$	0	0
$129$	1.17157	0.103151
$130$	0	0
$131$	−3.56420	−0.311406	−0.155703	−	0.987804i	$-0.549764\pi$
$131$	−3.56420	−0.311406	−0.155703	+	0.987804i	$0.549764\pi$
$132$	0	0
$133$	−3.37849	−0.292952
$134$	0	0
$135$	−7.65685	−0.658997
$136$	0	0
$137$	−20.9706	−1.79164	−0.895818	−	0.444421i	$-0.853409\pi$
$137$	−20.9706	−1.79164	−0.895818	+	0.444421i	$0.853409\pi$
$138$	0	0
$139$	−11.4036	−0.967239	−0.483620	−	0.875278i	$-0.660678\pi$
$139$	−11.4036	−0.967239	−0.483620	+	0.875278i	$0.660678\pi$
$140$	0	0
$141$	3.06147	0.257822
$142$	0	0
$143$	8.48528	0.709575
$144$	0	0
$145$	16.4853	1.36903
$146$	0	0
$147$	0.765367	0.0631264
$148$	0	0
$149$	15.6788	1.28445	0.642227	−	0.766515i	$-0.278011\pi$
$149$	15.6788	1.28445	0.642227	+	0.766515i	$0.278011\pi$
$150$	0	0
$151$	8.58579	0.698701	0.349351	−	0.936992i	$-0.386402\pi$
$151$	8.58579	0.698701	0.349351	+	0.936992i	$0.386402\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	2.16478	0.173880
$156$	0	0
$157$	−5.35757	−0.427580	−0.213790	−	0.976880i	$-0.568581\pi$
$157$	−5.35757	−0.427580	−0.213790	+	0.976880i	$0.568581\pi$
$158$	0	0
$159$	−5.65685	−0.448618
$160$	0	0
$161$	−0.585786	−0.0461664
$162$	0	0
$163$	22.4357	1.75730	0.878651	−	0.477464i	$-0.158444\pi$
$163$	22.4357	1.75730	0.878651	+	0.477464i	$0.158444\pi$
$164$	0	0
$165$	5.22625	0.406863
$166$	0	0
$167$	17.6569	1.36633	0.683164	−	0.730265i	$-0.260603\pi$
$167$	17.6569	1.36633	0.683164	+	0.730265i	$0.260603\pi$
$168$	0	0
$169$	−7.72792	−0.594456
$170$	0	0
$171$	−8.15640	−0.623736
$172$	0	0
$173$	−15.8101	−1.20202	−0.601009	−	0.799242i	$-0.705234\pi$
$173$	−15.8101	−1.20202	−0.601009	+	0.799242i	$0.705234\pi$
$174$	0	0
$175$	1.58579	0.119874
$176$	0	0
$177$	11.5563	0.868628
$178$	0	0
$179$	12.6173	0.943060	0.471530	−	0.881850i	$-0.343702\pi$
$179$	12.6173	0.943060	0.471530	+	0.881850i	$0.343702\pi$
$180$	0	0
$181$	−3.64113	−0.270643	−0.135321	−	0.990802i	$-0.543207\pi$
$181$	−3.64113	−0.270643	−0.135321	+	0.990802i	$0.543207\pi$
$182$	0	0
$183$	−1.41421	−0.104542
$184$	0	0
$185$	−1.17157	−0.0861358
$186$	0	0
$187$	3.06147	0.223877
$188$	0	0
$189$	4.14386	0.301421
$190$	0	0
$191$	13.6569	0.988175	0.494088	−	0.869412i	$-0.335502\pi$
$191$	13.6569	0.988175	0.494088	+	0.869412i	$0.335502\pi$
$192$	0	0
$193$	−10.5858	−0.761982	−0.380991	−	0.924579i	$-0.624417\pi$
$193$	−10.5858	−0.761982	−0.380991	+	0.924579i	$0.624417\pi$
$194$	0	0
$195$	3.24718	0.232535
$196$	0	0
$197$	10.8239	0.771173	0.385586	−	0.922672i	$-0.373999\pi$
$197$	10.8239	0.771173	0.385586	+	0.922672i	$0.373999\pi$
$198$	0	0
$199$	−6.14214	−0.435404	−0.217702	−	0.976015i	$-0.569856\pi$
$199$	−6.14214	−0.435404	−0.217702	+	0.976015i	$0.569856\pi$
$200$	0	0
$201$	−4.97056	−0.350596
$202$	0	0
$203$	−8.92177	−0.626185
$204$	0	0
$205$	−11.9832	−0.836946
$206$	0	0
$207$	−1.41421	−0.0982946
$208$	0	0
$209$	12.4853	0.863625
$210$	0	0
$211$	−23.9665	−1.64992	−0.824960	−	0.565191i	$-0.808802\pi$
$211$	−23.9665	−1.64992	−0.824960	+	0.565191i	$0.808802\pi$
$212$	0	0
$213$	3.69552	0.253213
$214$	0	0
$215$	2.82843	0.192897
$216$	0	0
$217$	−1.17157	−0.0795315
$218$	0	0
$219$	−8.02509	−0.542285
$220$	0	0
$221$	1.90215	0.127953
$222$	0	0
$223$	−1.17157	−0.0784543	−0.0392272	−	0.999230i	$-0.512490\pi$
$223$	−1.17157	−0.0784543	−0.0392272	+	0.999230i	$0.512490\pi$
$224$	0	0
$225$	3.82843	0.255228
$226$	0	0
$227$	−16.2584	−1.07911	−0.539554	−	0.841951i	$-0.681407\pi$
$227$	−16.2584	−1.07911	−0.539554	+	0.841951i	$0.681407\pi$
$228$	0	0
$229$	−25.7373	−1.70077	−0.850385	−	0.526161i	$-0.823631\pi$
$229$	−25.7373	−1.70077	−0.850385	+	0.526161i	$0.823631\pi$
$230$	0	0
$231$	−2.82843	−0.186097
$232$	0	0
$233$	−24.6274	−1.61340	−0.806698	−	0.590964i	$-0.798747\pi$
$233$	−24.6274	−1.61340	−0.806698	+	0.590964i	$0.798747\pi$
$234$	0	0
$235$	7.39104	0.482138
$236$	0	0
$237$	6.75699	0.438913
$238$	0	0
$239$	−3.89949	−0.252237	−0.126119	−	0.992015i	$-0.540252\pi$
$239$	−3.89949	−0.252237	−0.126119	+	0.992015i	$0.540252\pi$
$240$	0	0
$241$	−0.343146	−0.0221040	−0.0110520	−	0.999939i	$-0.503518\pi$
$241$	−0.343146	−0.0221040	−0.0110520	+	0.999939i	$0.503518\pi$
$242$	0	0
$243$	15.5474	0.997369
$244$	0	0
$245$	1.84776	0.118049
$246$	0	0
$247$	7.75736	0.493589
$248$	0	0
$249$	−1.75736	−0.111368
$250$	0	0
$251$	0.502734	0.0317323	0.0158662	−	0.999874i	$-0.494949\pi$
$251$	0.502734	0.0317323	0.0158662	+	0.999874i	$0.494949\pi$
$252$	0	0
$253$	2.16478	0.136099
$254$	0	0
$255$	1.17157	0.0733667
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	0.634051	0.0393980
$260$	0	0
$261$	−21.5391	−1.33323
$262$	0	0
$263$	−15.1716	−0.935519	−0.467760	−	0.883856i	$-0.654939\pi$
$263$	−15.1716	−0.935519	−0.467760	+	0.883856i	$0.654939\pi$
$264$	0	0
$265$	−13.6569	−0.838934
$266$	0	0
$267$	4.59220	0.281038
$268$	0	0
$269$	8.41904	0.513318	0.256659	−	0.966502i	$-0.417378\pi$
$269$	8.41904	0.513318	0.256659	+	0.966502i	$0.417378\pi$
$270$	0	0
$271$	−2.14214	−0.130125	−0.0650627	−	0.997881i	$-0.520725\pi$
$271$	−2.14214	−0.130125	−0.0650627	+	0.997881i	$0.520725\pi$
$272$	0	0
$273$	−1.75736	−0.106360
$274$	0	0
$275$	−5.86030	−0.353390
$276$	0	0
$277$	−3.43289	−0.206262	−0.103131	−	0.994668i	$-0.532886\pi$
$277$	−3.43289	−0.206262	−0.103131	+	0.994668i	$0.532886\pi$
$278$	0	0
$279$	−2.82843	−0.169334
$280$	0	0
$281$	12.9706	0.773759	0.386879	−	0.922130i	$-0.373553\pi$
$281$	12.9706	0.773759	0.386879	+	0.922130i	$0.373553\pi$
$282$	0	0
$283$	−20.1396	−1.19718	−0.598589	−	0.801057i	$-0.704272\pi$
$283$	−20.1396	−1.19718	−0.598589	+	0.801057i	$0.704272\pi$
$284$	0	0
$285$	4.77791	0.283019
$286$	0	0
$287$	6.48528	0.382814
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	−1.53073	−0.0897332
$292$	0	0
$293$	−9.68714	−0.565929	−0.282964	−	0.959130i	$-0.591318\pi$
$293$	−9.68714	−0.565929	−0.282964	+	0.959130i	$0.591318\pi$
$294$	0	0
$295$	27.8995	1.62437
$296$	0	0
$297$	−15.3137	−0.888591
$298$	0	0
$299$	1.34502	0.0777848
$300$	0	0
$301$	−1.53073	−0.0882300
$302$	0	0
$303$	10.7279	0.616303
$304$	0	0
$305$	−3.41421	−0.195497
$306$	0	0
$307$	−0.765367	−0.0436818	−0.0218409	−	0.999761i	$-0.506953\pi$
$307$	−0.765367	−0.0436818	−0.0218409	+	0.999761i	$0.506953\pi$
$308$	0	0
$309$	12.6173	0.717772
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	28.1421	1.59069	0.795344	−	0.606159i	$-0.207290\pi$
$313$	28.1421	1.59069	0.795344	+	0.606159i	$0.207290\pi$
$314$	0	0
$315$	4.46088	0.251342
$316$	0	0
$317$	5.59767	0.314396	0.157198	−	0.987567i	$-0.449754\pi$
$317$	5.59767	0.314396	0.157198	+	0.987567i	$0.449754\pi$
$318$	0	0
$319$	32.9706	1.84600
$320$	0	0
$321$	−1.65685	−0.0924766
$322$	0	0
$323$	2.79884	0.155731
$324$	0	0
$325$	−3.64113	−0.201973
$326$	0	0
$327$	10.1421	0.560861
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	23.7038	1.30288	0.651441	−	0.758700i	$-0.274165\pi$
$331$	23.7038	1.30288	0.651441	+	0.758700i	$0.274165\pi$
$332$	0	0
$333$	1.53073	0.0838837
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	6.58579	0.358751	0.179375	−	0.983781i	$-0.442592\pi$
$337$	6.58579	0.358751	0.179375	+	0.983781i	$0.442592\pi$
$338$	0	0
$339$	10.6382	0.577788
$340$	0	0
$341$	4.32957	0.234459
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.828427	0.0446010
$346$	0	0
$347$	19.0029	1.02013	0.510063	−	0.860137i	$-0.329622\pi$
$347$	19.0029	1.02013	0.510063	+	0.860137i	$0.329622\pi$
$348$	0	0
$349$	−30.7779	−1.64750	−0.823750	−	0.566953i	$-0.808122\pi$
$349$	−30.7779	−1.64750	−0.823750	+	0.566953i	$0.808122\pi$
$350$	0	0
$351$	−9.51472	−0.507858
$352$	0	0
$353$	−3.85786	−0.205333	−0.102667	−	0.994716i	$-0.532738\pi$
$353$	−3.85786	−0.205333	−0.102667	+	0.994716i	$0.532738\pi$
$354$	0	0
$355$	8.92177	0.473518
$356$	0	0
$357$	−0.634051	−0.0335575
$358$	0	0
$359$	−9.55635	−0.504365	−0.252182	−	0.967680i	$-0.581148\pi$
$359$	−9.55635	−0.504365	−0.252182	+	0.967680i	$0.581148\pi$
$360$	0	0
$361$	−7.58579	−0.399252
$362$	0	0
$363$	2.03347	0.106729
$364$	0	0
$365$	−19.3743	−1.01410
$366$	0	0
$367$	33.4558	1.74638	0.873190	−	0.487379i	$-0.162047\pi$
$367$	33.4558	1.74638	0.873190	+	0.487379i	$0.162047\pi$
$368$	0	0
$369$	15.6569	0.815063
$370$	0	0
$371$	7.39104	0.383723
$372$	0	0
$373$	−27.9246	−1.44588	−0.722941	−	0.690910i	$-0.757210\pi$
$373$	−27.9246	−1.44588	−0.722941	+	0.690910i	$0.757210\pi$
$374$	0	0
$375$	−9.31371	−0.480958
$376$	0	0
$377$	20.4853	1.05505
$378$	0	0
$379$	8.02509	0.412221	0.206111	−	0.978529i	$-0.433919\pi$
$379$	8.02509	0.412221	0.206111	+	0.978529i	$0.433919\pi$
$380$	0	0
$381$	7.46796	0.382595
$382$	0	0
$383$	−29.9411	−1.52992	−0.764960	−	0.644078i	$-0.777241\pi$
$383$	−29.9411	−1.52992	−0.764960	+	0.644078i	$0.777241\pi$
$384$	0	0
$385$	−6.82843	−0.348009
$386$	0	0
$387$	−3.69552	−0.187854
$388$	0	0
$389$	−20.6424	−1.04661	−0.523305	−	0.852145i	$-0.675301\pi$
$389$	−20.6424	−1.04661	−0.523305	+	0.852145i	$0.675301\pi$
$390$	0	0
$391$	0.485281	0.0245417
$392$	0	0
$393$	−2.72792	−0.137605
$394$	0	0
$395$	16.3128	0.820786
$396$	0	0
$397$	29.5098	1.48105	0.740526	−	0.672028i	$-0.234576\pi$
$397$	29.5098	1.48105	0.740526	+	0.672028i	$0.234576\pi$
$398$	0	0
$399$	−2.58579	−0.129451
$400$	0	0
$401$	−23.0711	−1.15211	−0.576057	−	0.817409i	$-0.695409\pi$
$401$	−23.0711	−1.15211	−0.576057	+	0.817409i	$0.695409\pi$
$402$	0	0
$403$	2.69005	0.134001
$404$	0	0
$405$	7.52235	0.373789
$406$	0	0
$407$	−2.34315	−0.116145
$408$	0	0
$409$	18.9706	0.938034	0.469017	−	0.883189i	$-0.344608\pi$
$409$	18.9706	0.938034	0.469017	+	0.883189i	$0.344608\pi$
$410$	0	0
$411$	−16.0502	−0.791697
$412$	0	0
$413$	−15.0991	−0.742978
$414$	0	0
$415$	−4.24264	−0.208263
$416$	0	0
$417$	−8.72792	−0.427408
$418$	0	0
$419$	29.0614	1.41974	0.709871	−	0.704331i	$-0.248753\pi$
$419$	29.0614	1.41974	0.709871	+	0.704331i	$0.248753\pi$
$420$	0	0
$421$	22.5445	1.09875	0.549377	−	0.835575i	$-0.314865\pi$
$421$	22.5445	1.09875	0.549377	+	0.835575i	$0.314865\pi$
$422$	0	0
$423$	−9.65685	−0.469532
$424$	0	0
$425$	−1.31371	−0.0637242
$426$	0	0
$427$	1.84776	0.0894193
$428$	0	0
$429$	6.49435	0.313550
$430$	0	0
$431$	7.41421	0.357130	0.178565	−	0.983928i	$-0.442855\pi$
$431$	7.41421	0.357130	0.178565	+	0.983928i	$0.442855\pi$
$432$	0	0
$433$	0.828427	0.0398117	0.0199058	−	0.999802i	$-0.493663\pi$
$433$	0.828427	0.0398117	0.0199058	+	0.999802i	$0.493663\pi$
$434$	0	0
$435$	12.6173	0.604953
$436$	0	0
$437$	1.97908	0.0946720
$438$	0	0
$439$	21.4558	1.02403	0.512016	−	0.858976i	$-0.328899\pi$
$439$	21.4558	1.02403	0.512016	+	0.858976i	$0.328899\pi$
$440$	0	0
$441$	−2.41421	−0.114963
$442$	0	0
$443$	−14.4107	−0.684671	−0.342335	−	0.939578i	$-0.611218\pi$
$443$	−14.4107	−0.684671	−0.342335	+	0.939578i	$0.611218\pi$
$444$	0	0
$445$	11.0866	0.525553
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−23.9665	−1.12854
$452$	0	0
$453$	6.57128	0.308746
$454$	0	0
$455$	−4.24264	−0.198898
$456$	0	0
$457$	−4.92893	−0.230566	−0.115283	−	0.993333i	$-0.536777\pi$
$457$	−4.92893	−0.230566	−0.115283	+	0.993333i	$0.536777\pi$
$458$	0	0
$459$	−3.43289	−0.160233
$460$	0	0
$461$	9.23880	0.430294	0.215147	−	0.976582i	$-0.430977\pi$
$461$	9.23880	0.430294	0.215147	+	0.976582i	$0.430977\pi$
$462$	0	0
$463$	−36.2843	−1.68627	−0.843137	−	0.537700i	$-0.819293\pi$
$463$	−36.2843	−1.68627	−0.843137	+	0.537700i	$0.819293\pi$
$464$	0	0
$465$	1.65685	0.0768348
$466$	0	0
$467$	23.6494	1.09437	0.547183	−	0.837013i	$-0.315700\pi$
$467$	23.6494	1.09437	0.547183	+	0.837013i	$0.315700\pi$
$468$	0	0
$469$	6.49435	0.299881
$470$	0	0
$471$	−4.10051	−0.188941
$472$	0	0
$473$	5.65685	0.260102
$474$	0	0
$475$	−5.35757	−0.245822
$476$	0	0
$477$	17.8435	0.817000
$478$	0	0
$479$	−35.7990	−1.63570	−0.817849	−	0.575433i	$-0.804833\pi$
$479$	−35.7990	−1.63570	−0.817849	+	0.575433i	$0.804833\pi$
$480$	0	0
$481$	−1.45584	−0.0663808
$482$	0	0
$483$	−0.448342	−0.0204002
$484$	0	0
$485$	−3.69552	−0.167805
$486$	0	0
$487$	−1.55635	−0.0705249	−0.0352625	−	0.999378i	$-0.511227\pi$
$487$	−1.55635	−0.0705249	−0.0352625	+	0.999378i	$0.511227\pi$
$488$	0	0
$489$	17.1716	0.776525
$490$	0	0
$491$	−7.91630	−0.357258	−0.178629	−	0.983917i	$-0.557166\pi$
$491$	−7.91630	−0.357258	−0.178629	+	0.983917i	$0.557166\pi$
$492$	0	0
$493$	7.39104	0.332876
$494$	0	0
$495$	−16.4853	−0.740958
$496$	0	0
$497$	−4.82843	−0.216585
$498$	0	0
$499$	−41.2848	−1.84816	−0.924080	−	0.382200i	$-0.875167\pi$
$499$	−41.2848	−1.84816	−0.924080	+	0.382200i	$0.875167\pi$
$500$	0	0
$501$	13.5140	0.603760
$502$	0	0
$503$	−16.4853	−0.735042	−0.367521	−	0.930015i	$-0.619793\pi$
$503$	−16.4853	−0.735042	−0.367521	+	0.930015i	$0.619793\pi$
$504$	0	0
$505$	25.8995	1.15251
$506$	0	0
$507$	−5.91470	−0.262681
$508$	0	0
$509$	7.97069	0.353295	0.176647	−	0.984274i	$-0.443475\pi$
$509$	7.97069	0.353295	0.176647	+	0.984274i	$0.443475\pi$
$510$	0	0
$511$	10.4853	0.463842
$512$	0	0
$513$	−14.0000	−0.618115
$514$	0	0
$515$	30.4608	1.34226
$516$	0	0
$517$	14.7821	0.650115
$518$	0	0
$519$	−12.1005	−0.531153
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	−13.8310	−0.604787	−0.302394	−	0.953183i	$-0.597786\pi$
$523$	−13.8310	−0.604787	−0.302394	+	0.953183i	$0.597786\pi$
$524$	0	0
$525$	1.21371	0.0529706
$526$	0	0
$527$	0.970563	0.0422784
$528$	0	0
$529$	−22.6569	−0.985081
$530$	0	0
$531$	−36.4524	−1.58190
$532$	0	0
$533$	−14.8909	−0.644995
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	9.65685	0.416724
$538$	0	0
$539$	3.69552	0.159177
$540$	0	0
$541$	−1.26810	−0.0545199	−0.0272600	−	0.999628i	$-0.508678\pi$
$541$	−1.26810	−0.0545199	−0.0272600	+	0.999628i	$0.508678\pi$
$542$	0	0
$543$	−2.78680	−0.119593
$544$	0	0
$545$	24.4853	1.04883
$546$	0	0
$547$	13.2513	0.566586	0.283293	−	0.959033i	$-0.408573\pi$
$547$	13.2513	0.566586	0.283293	+	0.959033i	$0.408573\pi$
$548$	0	0
$549$	4.46088	0.190386
$550$	0	0
$551$	30.1421	1.28410
$552$	0	0
$553$	−8.82843	−0.375423
$554$	0	0
$555$	−0.896683	−0.0380621
$556$	0	0
$557$	−18.7402	−0.794049	−0.397024	−	0.917808i	$-0.629957\pi$
$557$	−18.7402	−0.794049	−0.397024	+	0.917808i	$0.629957\pi$
$558$	0	0
$559$	3.51472	0.148657
$560$	0	0
$561$	2.34315	0.0989277
$562$	0	0
$563$	−23.7582	−1.00129	−0.500645	−	0.865653i	$-0.666904\pi$
$563$	−23.7582	−1.00129	−0.500645	+	0.865653i	$0.666904\pi$
$564$	0	0
$565$	25.6829	1.08049
$566$	0	0
$567$	−4.07107	−0.170969
$568$	0	0
$569$	−8.44365	−0.353976	−0.176988	−	0.984213i	$-0.556635\pi$
$569$	−8.44365	−0.353976	−0.176988	+	0.984213i	$0.556635\pi$
$570$	0	0
$571$	24.0753	1.00752	0.503759	−	0.863844i	$-0.331950\pi$
$571$	24.0753	1.00752	0.503759	+	0.863844i	$0.331950\pi$
$572$	0	0
$573$	10.4525	0.436660
$574$	0	0
$575$	−0.928932	−0.0387392
$576$	0	0
$577$	−33.1127	−1.37850	−0.689250	−	0.724524i	$-0.742060\pi$
$577$	−33.1127	−1.37850	−0.689250	+	0.724524i	$0.742060\pi$
$578$	0	0
$579$	−8.10201	−0.336708
$580$	0	0
$581$	2.29610	0.0952583
$582$	0	0
$583$	−27.3137	−1.13122
$584$	0	0
$585$	−10.2426	−0.423481
$586$	0	0
$587$	9.61021	0.396656	0.198328	−	0.980136i	$-0.436449\pi$
$587$	9.61021	0.396656	0.198328	+	0.980136i	$0.436449\pi$
$588$	0	0
$589$	3.95815	0.163093
$590$	0	0
$591$	8.28427	0.340769
$592$	0	0
$593$	18.6863	0.767354	0.383677	−	0.923467i	$-0.374658\pi$
$593$	18.6863	0.767354	0.383677	+	0.923467i	$0.374658\pi$
$594$	0	0
$595$	−1.53073	−0.0627540
$596$	0	0
$597$	−4.70099	−0.192399
$598$	0	0
$599$	−44.2843	−1.80941	−0.904703	−	0.426043i	$-0.859907\pi$
$599$	−44.2843	−1.80941	−0.904703	+	0.426043i	$0.859907\pi$
$600$	0	0
$601$	35.9411	1.46607	0.733035	−	0.680191i	$-0.238103\pi$
$601$	35.9411	1.46607	0.733035	+	0.680191i	$0.238103\pi$
$602$	0	0
$603$	15.6788	0.638488
$604$	0	0
$605$	4.90923	0.199588
$606$	0	0
$607$	−42.6274	−1.73019	−0.865097	−	0.501605i	$-0.832743\pi$
$607$	−42.6274	−1.73019	−0.865097	+	0.501605i	$0.832743\pi$
$608$	0	0
$609$	−6.82843	−0.276702
$610$	0	0
$611$	9.18440	0.371561
$612$	0	0
$613$	26.3939	1.06604	0.533020	−	0.846103i	$-0.321057\pi$
$613$	26.3939	1.06604	0.533020	+	0.846103i	$0.321057\pi$
$614$	0	0
$615$	−9.17157	−0.369834
$616$	0	0
$617$	−1.41421	−0.0569341	−0.0284670	−	0.999595i	$-0.509063\pi$
$617$	−1.41421	−0.0569341	−0.0284670	+	0.999595i	$0.509063\pi$
$618$	0	0
$619$	−39.5139	−1.58820	−0.794099	−	0.607788i	$-0.792057\pi$
$619$	−39.5139	−1.58820	−0.794099	+	0.607788i	$0.792057\pi$
$620$	0	0
$621$	−2.42742	−0.0974089
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−14.5563	−0.582254
$626$	0	0
$627$	9.55582	0.381623
$628$	0	0
$629$	−0.525265	−0.0209437
$630$	0	0
$631$	21.6569	0.862146	0.431073	−	0.902317i	$-0.358135\pi$
$631$	21.6569	0.862146	0.431073	+	0.902317i	$0.358135\pi$
$632$	0	0
$633$	−18.3431	−0.729075
$634$	0	0
$635$	18.0292	0.715469
$636$	0	0
$637$	2.29610	0.0909748
$638$	0	0
$639$	−11.6569	−0.461138
$640$	0	0
$641$	−11.2721	−0.445220	−0.222610	−	0.974908i	$-0.571458\pi$
$641$	−11.2721	−0.445220	−0.222610	+	0.974908i	$0.571458\pi$
$642$	0	0
$643$	−26.7109	−1.05338	−0.526688	−	0.850059i	$-0.676566\pi$
$643$	−26.7109	−1.05338	−0.526688	+	0.850059i	$0.676566\pi$
$644$	0	0
$645$	2.16478	0.0852383
$646$	0	0
$647$	−48.4853	−1.90615	−0.953077	−	0.302729i	$-0.902102\pi$
$647$	−48.4853	−1.90615	−0.953077	+	0.302729i	$0.902102\pi$
$648$	0	0
$649$	55.7990	2.19030
$650$	0	0
$651$	−0.896683	−0.0351438
$652$	0	0
$653$	3.69552	0.144617	0.0723084	−	0.997382i	$-0.476963\pi$
$653$	3.69552	0.144617	0.0723084	+	0.997382i	$0.476963\pi$
$654$	0	0
$655$	−6.58579	−0.257328
$656$	0	0
$657$	25.3137	0.987582
$658$	0	0
$659$	−21.9105	−0.853511	−0.426755	−	0.904367i	$-0.640344\pi$
$659$	−21.9105	−0.853511	−0.426755	+	0.904367i	$0.640344\pi$
$660$	0	0
$661$	−27.5307	−1.07082	−0.535410	−	0.844593i	$-0.679843\pi$
$661$	−27.5307	−1.07082	−0.535410	+	0.844593i	$0.679843\pi$
$662$	0	0
$663$	1.45584	0.0565403
$664$	0	0
$665$	−6.24264	−0.242079
$666$	0	0
$667$	5.22625	0.202361
$668$	0	0
$669$	−0.896683	−0.0346678
$670$	0	0
$671$	−6.82843	−0.263609
$672$	0	0
$673$	−16.2843	−0.627713	−0.313856	−	0.949471i	$-0.601621\pi$
$673$	−16.2843	−0.627713	−0.313856	+	0.949471i	$0.601621\pi$
$674$	0	0
$675$	6.57128	0.252929
$676$	0	0
$677$	−11.1409	−0.428181	−0.214091	−	0.976814i	$-0.568679\pi$
$677$	−11.1409	−0.428181	−0.214091	+	0.976814i	$0.568679\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−12.4437	−0.476842
$682$	0	0
$683$	12.2459	0.468575	0.234288	−	0.972167i	$-0.424724\pi$
$683$	12.2459	0.468575	0.234288	+	0.972167i	$0.424724\pi$
$684$	0	0
$685$	−38.7485	−1.48051
$686$	0	0
$687$	−19.6985	−0.751544
$688$	0	0
$689$	−16.9706	−0.646527
$690$	0	0
$691$	−37.4579	−1.42497	−0.712483	−	0.701689i	$-0.752430\pi$
$691$	−37.4579	−1.42497	−0.712483	+	0.701689i	$0.752430\pi$
$692$	0	0
$693$	8.92177	0.338910
$694$	0	0
$695$	−21.0711	−0.799271
$696$	0	0
$697$	−5.37258	−0.203501
$698$	0	0
$699$	−18.8490	−0.712935
$700$	0	0
$701$	−35.9497	−1.35780	−0.678901	−	0.734230i	$-0.737543\pi$
$701$	−35.9497	−1.35780	−0.678901	+	0.734230i	$0.737543\pi$
$702$	0	0
$703$	−2.14214	−0.0807922
$704$	0	0
$705$	5.65685	0.213049
$706$	0	0
$707$	−14.0167	−0.527152
$708$	0	0
$709$	35.0530	1.31644	0.658222	−	0.752824i	$-0.271309\pi$
$709$	35.0530	1.31644	0.658222	+	0.752824i	$0.271309\pi$
$710$	0	0
$711$	−21.3137	−0.799327
$712$	0	0
$713$	0.686292	0.0257018
$714$	0	0
$715$	15.6788	0.586352
$716$	0	0
$717$	−2.98454	−0.111460
$718$	0	0
$719$	−49.6569	−1.85189	−0.925944	−	0.377661i	$-0.876729\pi$
$719$	−49.6569	−1.85189	−0.925944	+	0.377661i	$0.876729\pi$
$720$	0	0
$721$	−16.4853	−0.613944
$722$	0	0
$723$	−0.262632	−0.00976740
$724$	0	0
$725$	−14.1480	−0.525444
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−0.313708	−0.0116188
$730$	0	0
$731$	1.26810	0.0469024
$732$	0	0
$733$	−30.7779	−1.13681	−0.568403	−	0.822750i	$-0.692439\pi$
$733$	−30.7779	−1.13681	−0.568403	+	0.822750i	$0.692439\pi$
$734$	0	0
$735$	1.41421	0.0521641
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−23.7038	−0.871960	−0.435980	−	0.899956i	$-0.643598\pi$
$739$	−23.7038	−0.871960	−0.435980	+	0.899956i	$0.643598\pi$
$740$	0	0
$741$	5.93723	0.218109
$742$	0	0
$743$	12.8701	0.472157	0.236078	−	0.971734i	$-0.424138\pi$
$743$	12.8701	0.472157	0.236078	+	0.971734i	$0.424138\pi$
$744$	0	0
$745$	28.9706	1.06140
$746$	0	0
$747$	5.54328	0.202818
$748$	0	0
$749$	2.16478	0.0790995
$750$	0	0
$751$	41.3553	1.50908	0.754539	−	0.656255i	$-0.227861\pi$
$751$	41.3553	1.50908	0.754539	+	0.656255i	$0.227861\pi$
$752$	0	0
$753$	0.384776	0.0140220
$754$	0	0
$755$	15.8645	0.577367
$756$	0	0
$757$	−40.6507	−1.47747	−0.738737	−	0.673993i	$-0.764578\pi$
$757$	−40.6507	−1.47747	−0.738737	+	0.673993i	$0.764578\pi$
$758$	0	0
$759$	1.65685	0.0601400
$760$	0	0
$761$	46.4853	1.68509	0.842545	−	0.538626i	$-0.181056\pi$
$761$	46.4853	1.68509	0.842545	+	0.538626i	$0.181056\pi$
$762$	0	0
$763$	−13.2513	−0.479731
$764$	0	0
$765$	−3.69552	−0.133612
$766$	0	0
$767$	34.6690	1.25183
$768$	0	0
$769$	−26.2843	−0.947835	−0.473918	−	0.880569i	$-0.657161\pi$
$769$	−26.2843	−0.947835	−0.473918	+	0.880569i	$0.657161\pi$
$770$	0	0
$771$	13.7766	0.496152
$772$	0	0
$773$	−12.3003	−0.442410	−0.221205	−	0.975227i	$-0.570999\pi$
$773$	−12.3003	−0.442410	−0.221205	+	0.975227i	$0.570999\pi$
$774$	0	0
$775$	−1.85786	−0.0667365
$776$	0	0
$777$	0.485281	0.0174094
$778$	0	0
$779$	−21.9105	−0.785024
$780$	0	0
$781$	17.8435	0.638492
$782$	0	0
$783$	−36.9706	−1.32122
$784$	0	0
$785$	−9.89949	−0.353328
$786$	0	0
$787$	31.1493	1.11035	0.555176	−	0.831733i	$-0.312651\pi$
$787$	31.1493	1.11035	0.555176	+	0.831733i	$0.312651\pi$
$788$	0	0
$789$	−11.6118	−0.413392
$790$	0	0
$791$	−13.8995	−0.494209
$792$	0	0
$793$	−4.24264	−0.150661
$794$	0	0
$795$	−10.4525	−0.370712
$796$	0	0
$797$	47.4302	1.68006	0.840032	−	0.542537i	$-0.182536\pi$
$797$	47.4302	1.68006	0.840032	+	0.542537i	$0.182536\pi$
$798$	0	0
$799$	3.31371	0.117231
$800$	0	0
$801$	−14.4853	−0.511812
$802$	0	0
$803$	−38.7485	−1.36741
$804$	0	0
$805$	−1.08239	−0.0381493
$806$	0	0
$807$	6.44365	0.226827
$808$	0	0
$809$	1.21320	0.0426540	0.0213270	−	0.999773i	$-0.493211\pi$
$809$	1.21320	0.0426540	0.0213270	+	0.999773i	$0.493211\pi$
$810$	0	0
$811$	21.6704	0.760950	0.380475	−	0.924791i	$-0.375761\pi$
$811$	21.6704	0.760950	0.380475	+	0.924791i	$0.375761\pi$
$812$	0	0
$813$	−1.63952	−0.0575005
$814$	0	0
$815$	41.4558	1.45213
$816$	0	0
$817$	5.17157	0.180930
$818$	0	0
$819$	5.54328	0.193698
$820$	0	0
$821$	20.3797	0.711258	0.355629	−	0.934627i	$-0.384267\pi$
$821$	20.3797	0.711258	0.355629	+	0.934627i	$0.384267\pi$
$822$	0	0
$823$	−33.7990	−1.17816	−0.589079	−	0.808075i	$-0.700509\pi$
$823$	−33.7990	−1.17816	−0.589079	+	0.808075i	$0.700509\pi$
$824$	0	0
$825$	−4.48528	−0.156157
$826$	0	0
$827$	−48.6758	−1.69262	−0.846311	−	0.532688i	$-0.821182\pi$
$827$	−48.6758	−1.69262	−0.846311	+	0.532688i	$0.821182\pi$
$828$	0	0
$829$	0.0543929	0.00188915	0.000944573	−	1.00000i	$-0.499699\pi$
$829$	0.0543929	0.00188915	0.000944573		1.00000i	$0.499699\pi$
$830$	0	0
$831$	−2.62742	−0.0911441
$832$	0	0
$833$	0.828427	0.0287033
$834$	0	0
$835$	32.6256	1.12906
$836$	0	0
$837$	−4.85483	−0.167808
$838$	0	0
$839$	−15.5147	−0.535628	−0.267814	−	0.963471i	$-0.586301\pi$
$839$	−15.5147	−0.535628	−0.267814	+	0.963471i	$0.586301\pi$
$840$	0	0
$841$	50.5980	1.74476
$842$	0	0
$843$	9.92724	0.341912
$844$	0	0
$845$	−14.2793	−0.491224
$846$	0	0
$847$	−2.65685	−0.0912906
$848$	0	0
$849$	−15.4142	−0.529014
$850$	0	0
$851$	−0.371418	−0.0127321
$852$	0	0
$853$	46.6423	1.59700	0.798501	−	0.601993i	$-0.205627\pi$
$853$	46.6423	1.59700	0.798501	+	0.601993i	$0.205627\pi$
$854$	0	0
$855$	−15.0711	−0.515420
$856$	0	0
$857$	−7.65685	−0.261553	−0.130777	−	0.991412i	$-0.541747\pi$
$857$	−7.65685	−0.261553	−0.130777	+	0.991412i	$0.541747\pi$
$858$	0	0
$859$	34.6591	1.18255	0.591276	−	0.806469i	$-0.298624\pi$
$859$	34.6591	1.18255	0.591276	+	0.806469i	$0.298624\pi$
$860$	0	0
$861$	4.96362	0.169160
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	−29.2132	−0.993279
$866$	0	0
$867$	−12.4860	−0.424046
$868$	0	0
$869$	32.6256	1.10675
$870$	0	0
$871$	−14.9117	−0.505263
$872$	0	0
$873$	4.82843	0.163417
$874$	0	0
$875$	12.1689	0.411385
$876$	0	0
$877$	19.8995	0.671960	0.335980	−	0.941869i	$-0.390933\pi$
$877$	19.8995	0.671960	0.335980	+	0.941869i	$0.390933\pi$
$878$	0	0
$879$	−7.41421	−0.250075
$880$	0	0
$881$	−36.4264	−1.22724	−0.613618	−	0.789603i	$-0.710287\pi$
$881$	−36.4264	−1.22724	−0.613618	+	0.789603i	$0.710287\pi$
$882$	0	0
$883$	42.1814	1.41952	0.709759	−	0.704444i	$-0.248804\pi$
$883$	42.1814	1.41952	0.709759	+	0.704444i	$0.248804\pi$
$884$	0	0
$885$	21.3533	0.717785
$886$	0	0
$887$	55.5980	1.86680	0.933399	−	0.358841i	$-0.116828\pi$
$887$	55.5980	1.86680	0.933399	+	0.358841i	$0.116828\pi$
$888$	0	0
$889$	−9.75736	−0.327251
$890$	0	0
$891$	15.0447	0.504017
$892$	0	0
$893$	13.5140	0.452228
$894$	0	0
$895$	23.3137	0.779291
$896$	0	0
$897$	1.02944	0.0343719
$898$	0	0
$899$	10.4525	0.348611
$900$	0	0
$901$	−6.12293	−0.203985
$902$	0	0
$903$	−1.17157	−0.0389875
$904$	0	0
$905$	−6.72792	−0.223644
$906$	0	0
$907$	−43.0781	−1.43039	−0.715193	−	0.698927i	$-0.753661\pi$
$907$	−43.0781	−1.43039	−0.715193	+	0.698927i	$0.753661\pi$
$908$	0	0
$909$	−33.8393	−1.12238
$910$	0	0
$911$	−48.1838	−1.59640	−0.798200	−	0.602393i	$-0.794214\pi$
$911$	−48.1838	−1.59640	−0.798200	+	0.602393i	$0.794214\pi$
$912$	0	0
$913$	−8.48528	−0.280822
$914$	0	0
$915$	−2.61313	−0.0863873
$916$	0	0
$917$	3.56420	0.117700
$918$	0	0
$919$	−13.1127	−0.432548	−0.216274	−	0.976333i	$-0.569390\pi$
$919$	−13.1127	−0.432548	−0.216274	+	0.976333i	$0.569390\pi$
$920$	0	0
$921$	−0.585786	−0.0193023
$922$	0	0
$923$	11.0866	0.364918
$924$	0	0
$925$	1.00547	0.0330596
$926$	0	0
$927$	−39.7990	−1.30717
$928$	0	0
$929$	33.7990	1.10891	0.554454	−	0.832214i	$-0.312927\pi$
$929$	33.7990	1.10891	0.554454	+	0.832214i	$0.312927\pi$
$930$	0	0
$931$	3.37849	0.110726
$932$	0	0
$933$	3.06147	0.100228
$934$	0	0
$935$	5.65685	0.184999
$936$	0	0
$937$	50.0833	1.63615	0.818074	−	0.575112i	$-0.195042\pi$
$937$	50.0833	1.63615	0.818074	+	0.575112i	$0.195042\pi$
$938$	0	0
$939$	21.5391	0.702901
$940$	0	0
$941$	−29.3240	−0.955936	−0.477968	−	0.878377i	$-0.658627\pi$
$941$	−29.3240	−0.955936	−0.477968	+	0.878377i	$0.658627\pi$
$942$	0	0
$943$	−3.79899	−0.123712
$944$	0	0
$945$	7.65685	0.249077
$946$	0	0
$947$	4.06694	0.132158	0.0660788	−	0.997814i	$-0.478951\pi$
$947$	4.06694	0.132158	0.0660788	+	0.997814i	$0.478951\pi$
$948$	0	0
$949$	−24.0753	−0.781516
$950$	0	0
$951$	4.28427	0.138927
$952$	0	0
$953$	12.3431	0.399834	0.199917	−	0.979813i	$-0.435933\pi$
$953$	12.3431	0.399834	0.199917	+	0.979813i	$0.435933\pi$
$954$	0	0
$955$	25.2346	0.816572
$956$	0	0
$957$	25.2346	0.815718
$958$	0	0
$959$	20.9706	0.677175
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	5.22625	0.168414
$964$	0	0
$965$	−19.5600	−0.629658
$966$	0	0
$967$	25.3553	0.815373	0.407686	−	0.913122i	$-0.366336\pi$
$967$	25.3553	0.815373	0.407686	+	0.913122i	$0.366336\pi$
$968$	0	0
$969$	2.14214	0.0688153
$970$	0	0
$971$	11.9288	0.382815	0.191407	−	0.981511i	$-0.438695\pi$
$971$	11.9288	0.382815	0.191407	+	0.981511i	$0.438695\pi$
$972$	0	0
$973$	11.4036	0.365582
$974$	0	0
$975$	−2.78680	−0.0892489
$976$	0	0
$977$	−29.3137	−0.937829	−0.468914	−	0.883244i	$-0.655355\pi$
$977$	−29.3137	−0.937829	−0.468914	+	0.883244i	$0.655355\pi$
$978$	0	0
$979$	22.1731	0.708656
$980$	0	0
$981$	−31.9916	−1.02141
$982$	0	0
$983$	−25.1716	−0.802848	−0.401424	−	0.915892i	$-0.631485\pi$
$983$	−25.1716	−0.802848	−0.401424	+	0.915892i	$0.631485\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	−3.06147	−0.0974476
$988$	0	0
$989$	0.896683	0.0285129
$990$	0	0
$991$	17.7990	0.565404	0.282702	−	0.959208i	$-0.408769\pi$
$991$	17.7990	0.565404	0.282702	+	0.959208i	$0.408769\pi$
$992$	0	0
$993$	18.1421	0.575723
$994$	0	0
$995$	−11.3492	−0.359793
$996$	0	0
$997$	17.2639	0.546753	0.273376	−	0.961907i	$-0.411860\pi$
$997$	17.2639	0.546753	0.273376	+	0.961907i	$0.411860\pi$
$998$	0	0
$999$	2.62742	0.0831278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.c.1.3	yes	4
4.3	odd	2		3584.2.a.d.1.2	yes	4
8.3	odd	2		3584.2.a.d.1.3	yes	4
8.5	even	2	inner	3584.2.a.c.1.2	✓	4
16.3	odd	4		3584.2.b.c.1793.2		4
16.5	even	4		3584.2.b.d.1793.2		4
16.11	odd	4		3584.2.b.c.1793.3		4
16.13	even	4		3584.2.b.d.1793.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.c.1.2	✓	4	8.5	even	2	inner
3584.2.a.c.1.3	yes	4	1.1	even	1	trivial
3584.2.a.d.1.2	yes	4	4.3	odd	2
3584.2.a.d.1.3	yes	4	8.3	odd	2
3584.2.b.c.1793.2		4	16.3	odd	4
3584.2.b.c.1793.3		4	16.11	odd	4
3584.2.b.d.1793.2		4	16.5	even	4
3584.2.b.d.1793.3		4	16.13	even	4

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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.a (trivial)

\(f(q)\)	\(=\)	\(q+0.765367 q^{3} +1.84776 q^{5} -1.00000 q^{7} -2.41421 q^{9} +O(q^{10})\)	\(q+0.765367 q^{3} +1.84776 q^{5} -1.00000 q^{7} -2.41421 q^{9} +3.69552 q^{11} +2.29610 q^{13} +1.41421 q^{15} +0.828427 q^{17} +3.37849 q^{19} -0.765367 q^{21} +0.585786 q^{23} -1.58579 q^{25} -4.14386 q^{27} +8.92177 q^{29} +1.17157 q^{31} +2.82843 q^{33} -1.84776 q^{35} -0.634051 q^{37} +1.75736 q^{39} -6.48528 q^{41} +1.53073 q^{43} -4.46088 q^{45} +4.00000 q^{47} +1.00000 q^{49} +0.634051 q^{51} -7.39104 q^{53} +6.82843 q^{55} +2.58579 q^{57} +15.0991 q^{59} -1.84776 q^{61} +2.41421 q^{63} +4.24264 q^{65} -6.49435 q^{67} +0.448342 q^{69} +4.82843 q^{71} -10.4853 q^{73} -1.21371 q^{75} -3.69552 q^{77} +8.82843 q^{79} +4.07107 q^{81} -2.29610 q^{83} +1.53073 q^{85} +6.82843 q^{87} +6.00000 q^{89} -2.29610 q^{91} +0.896683 q^{93} +6.24264 q^{95} -2.00000 q^{97} -8.92177 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 - 4 * q^9	\( 4 q - 4 q^{7} - 4 q^{9} - 8 q^{17} + 8 q^{23} - 12 q^{25} + 16 q^{31} + 24 q^{39} + 8 q^{41} + 16 q^{47} + 4 q^{49} + 16 q^{55} + 16 q^{57} + 4 q^{63} + 8 q^{71} - 8 q^{73} + 24 q^{79} - 12 q^{81} + 16 q^{87} + 24 q^{89} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^9 - 8 * q^17 + 8 * q^23 - 12 * q^25 + 16 * q^31 + 24 * q^39 + 8 * q^41 + 16 * q^47 + 4 * q^49 + 16 * q^55 + 16 * q^57 + 4 * q^63 + 8 * q^71 - 8 * q^73 + 24 * q^79 - 12 * q^81 + 16 * q^87 + 24 * q^89 + 8 * q^95 - 8 * q^97

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