LMFDB - Embedded newform 1536.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, 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("1536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.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:	$12.2650217505$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.41421 q^{5} -4.24264 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.41421 q^{5} -4.24264 q^{7} +1.00000 q^{9} -6.00000 q^{11} +5.65685 q^{13} +1.41421 q^{15} -6.00000 q^{17} -4.00000 q^{19} -4.24264 q^{21} +2.82843 q^{23} -3.00000 q^{25} +1.00000 q^{27} +1.41421 q^{29} -1.41421 q^{31} -6.00000 q^{33} -6.00000 q^{35} -8.48528 q^{37} +5.65685 q^{39} -2.00000 q^{41} +1.41421 q^{45} +2.82843 q^{47} +11.0000 q^{49} -6.00000 q^{51} -9.89949 q^{53} -8.48528 q^{55} -4.00000 q^{57} -4.00000 q^{59} -8.48528 q^{61} -4.24264 q^{63} +8.00000 q^{65} -8.00000 q^{67} +2.82843 q^{69} -2.82843 q^{71} +8.00000 q^{73} -3.00000 q^{75} +25.4558 q^{77} +12.7279 q^{79} +1.00000 q^{81} -2.00000 q^{83} -8.48528 q^{85} +1.41421 q^{87} +2.00000 q^{89} -24.0000 q^{91} -1.41421 q^{93} -5.65685 q^{95} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} - 12 q^{11} - 12 q^{17} - 8 q^{19} - 6 q^{25} + 2 q^{27} - 12 q^{33} - 12 q^{35} - 4 q^{41} + 22 q^{49} - 12 q^{51} - 8 q^{57} - 8 q^{59} + 16 q^{65} - 16 q^{67} + 16 q^{73} - 6 q^{75} + 2 q^{81} - 4 q^{83} + 4 q^{89} - 48 q^{91} + 4 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 12 * q^11 - 12 * q^17 - 8 * q^19 - 6 * q^25 + 2 * q^27 - 12 * q^33 - 12 * q^35 - 4 * q^41 + 22 * q^49 - 12 * q^51 - 8 * q^57 - 8 * q^59 + 16 * q^65 - 16 * q^67 + 16 * q^73 - 6 * q^75 + 2 * q^81 - 4 * q^83 + 4 * q^89 - 48 * q^91 + 4 * q^97 - 12 * 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
$3$	1.00000	0.577350
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−4.24264	−0.925820
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−6.00000	−1.01419
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	5.65685	0.905822
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.41421	0.210819
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−9.89949	−1.35980	−0.679900	−	0.733305i	$-0.737977\pi$
$53$	−9.89949	−1.35980	−0.679900	+	0.733305i	$0.737977\pi$
$54$	0	0
$55$	−8.48528	−1.14416
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	0	0
$63$	−4.24264	−0.534522
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	2.82843	0.340503
$70$	0	0
$71$	−2.82843	−0.335673	−0.167836	−	0.985815i	$-0.553678\pi$
$71$	−2.82843	−0.335673	−0.167836	+	0.985815i	$0.553678\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	25.4558	2.90096
$78$	0	0
$79$	12.7279	1.43200	0.716002	−	0.698099i	$-0.245970\pi$
$79$	12.7279	1.43200	0.716002	+	0.698099i	$0.245970\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−8.48528	−0.920358
$86$	0	0
$87$	1.41421	0.151620
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−24.0000	−2.51588
$92$	0	0
$93$	−1.41421	−0.146647
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−1.41421	−0.140720	−0.0703598	−	0.997522i	$-0.522415\pi$
$101$	−1.41421	−0.140720	−0.0703598	+	0.997522i	$0.522415\pi$
$102$	0	0
$103$	7.07107	0.696733	0.348367	−	0.937358i	$-0.386736\pi$
$103$	7.07107	0.696733	0.348367	+	0.937358i	$0.386736\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	16.9706	1.62549	0.812743	−	0.582623i	$-0.197974\pi$
$109$	16.9706	1.62549	0.812743	+	0.582623i	$0.197974\pi$
$110$	0	0
$111$	−8.48528	−0.805387
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	25.4558	2.33353
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	12.7279	1.12942	0.564710	−	0.825289i	$-0.308988\pi$
$127$	12.7279	1.12942	0.564710	+	0.825289i	$0.308988\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	16.9706	1.47153
$134$	0	0
$135$	1.41421	0.121716
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	2.82843	0.238197
$142$	0	0
$143$	−33.9411	−2.83830
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	11.0000	0.907265
$148$	0	0
$149$	−4.24264	−0.347571	−0.173785	−	0.984784i	$-0.555600\pi$
$149$	−4.24264	−0.347571	−0.173785	+	0.984784i	$0.555600\pi$
$150$	0	0
$151$	−12.7279	−1.03578	−0.517892	−	0.855446i	$-0.673283\pi$
$151$	−12.7279	−1.03578	−0.517892	+	0.855446i	$0.673283\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	2.82843	0.225733	0.112867	−	0.993610i	$-0.463997\pi$
$157$	2.82843	0.225733	0.112867	+	0.993610i	$0.463997\pi$
$158$	0	0
$159$	−9.89949	−0.785081
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	−8.48528	−0.660578
$166$	0	0
$167$	−22.6274	−1.75096	−0.875481	−	0.483252i	$-0.839455\pi$
$167$	−22.6274	−1.75096	−0.875481	+	0.483252i	$0.839455\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−7.07107	−0.537603	−0.268802	−	0.963196i	$-0.586628\pi$
$173$	−7.07107	−0.537603	−0.268802	+	0.963196i	$0.586628\pi$
$174$	0	0
$175$	12.7279	0.962140
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−16.9706	−1.26141	−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706	−1.26141	−0.630706	+	0.776022i	$0.717235\pi$
$182$	0	0
$183$	−8.48528	−0.627250
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	36.0000	2.63258
$188$	0	0
$189$	−4.24264	−0.308607
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	8.00000	0.572892
$196$	0	0
$197$	−4.24264	−0.302276	−0.151138	−	0.988513i	$-0.548294\pi$
$197$	−4.24264	−0.302276	−0.151138	+	0.988513i	$0.548294\pi$
$198$	0	0
$199$	−4.24264	−0.300753	−0.150376	−	0.988629i	$-0.548049\pi$
$199$	−4.24264	−0.300753	−0.150376	+	0.988629i	$0.548049\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−2.82843	−0.197546
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	−2.82843	−0.193801
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	−33.9411	−2.28313
$222$	0	0
$223$	15.5563	1.04173	0.520865	−	0.853639i	$-0.325609\pi$
$223$	15.5563	1.04173	0.520865	+	0.853639i	$0.325609\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	22.6274	1.49526	0.747631	−	0.664114i	$-0.231191\pi$
$229$	22.6274	1.49526	0.747631	+	0.664114i	$0.231191\pi$
$230$	0	0
$231$	25.4558	1.67487
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	12.7279	0.826767
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	15.5563	0.993859
$246$	0	0
$247$	−22.6274	−1.43975
$248$	0	0
$249$	−2.00000	−0.126745
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−16.9706	−1.06693
$254$	0	0
$255$	−8.48528	−0.531369
$256$	0	0
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	36.0000	2.23693
$260$	0	0
$261$	1.41421	0.0875376
$262$	0	0
$263$	16.9706	1.04645	0.523225	−	0.852195i	$-0.324729\pi$
$263$	16.9706	1.04645	0.523225	+	0.852195i	$0.324729\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	−12.7279	−0.776035	−0.388018	−	0.921652i	$-0.626840\pi$
$269$	−12.7279	−0.776035	−0.388018	+	0.921652i	$0.626840\pi$
$270$	0	0
$271$	4.24264	0.257722	0.128861	−	0.991663i	$-0.458868\pi$
$271$	4.24264	0.257722	0.128861	+	0.991663i	$0.458868\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	0	0
$275$	18.0000	1.08544
$276$	0	0
$277$	16.9706	1.01966	0.509831	−	0.860274i	$-0.329708\pi$
$277$	16.9706	1.01966	0.509831	+	0.860274i	$0.329708\pi$
$278$	0	0
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−5.65685	−0.335083
$286$	0	0
$287$	8.48528	0.500870
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	24.0416	1.40453	0.702264	−	0.711917i	$-0.252173\pi$
$293$	24.0416	1.40453	0.702264	+	0.711917i	$0.252173\pi$
$294$	0	0
$295$	−5.65685	−0.329355
$296$	0	0
$297$	−6.00000	−0.348155
$298$	0	0
$299$	16.0000	0.925304
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.41421	−0.0812444
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	7.07107	0.402259
$310$	0	0
$311$	22.6274	1.28308	0.641542	−	0.767088i	$-0.278295\pi$
$311$	22.6274	1.28308	0.641542	+	0.767088i	$0.278295\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	0	0
$315$	−6.00000	−0.338062
$316$	0	0
$317$	21.2132	1.19145	0.595726	−	0.803188i	$-0.296864\pi$
$317$	21.2132	1.19145	0.595726	+	0.803188i	$0.296864\pi$
$318$	0	0
$319$	−8.48528	−0.475085
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	−16.9706	−0.941357
$326$	0	0
$327$	16.9706	0.938474
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	0	0
$333$	−8.48528	−0.464991
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	8.48528	0.459504
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	−14.1421	−0.757011	−0.378506	−	0.925599i	$-0.623562\pi$
$349$	−14.1421	−0.757011	−0.378506	+	0.925599i	$0.623562\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	25.4558	1.34727
$358$	0	0
$359$	−31.1127	−1.64207	−0.821033	−	0.570881i	$-0.806602\pi$
$359$	−31.1127	−1.64207	−0.821033	+	0.570881i	$0.806602\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	25.0000	1.31216
$364$	0	0
$365$	11.3137	0.592187
$366$	0	0
$367$	−12.7279	−0.664392	−0.332196	−	0.943210i	$-0.607790\pi$
$367$	−12.7279	−0.664392	−0.332196	+	0.943210i	$0.607790\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	42.0000	2.18053
$372$	0	0
$373$	19.7990	1.02515	0.512576	−	0.858642i	$-0.328691\pi$
$373$	19.7990	1.02515	0.512576	+	0.858642i	$0.328691\pi$
$374$	0	0
$375$	−11.3137	−0.584237
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	12.7279	0.652071
$382$	0	0
$383$	5.65685	0.289052	0.144526	−	0.989501i	$-0.453834\pi$
$383$	5.65685	0.289052	0.144526	+	0.989501i	$0.453834\pi$
$384$	0	0
$385$	36.0000	1.83473
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.41421	−0.0717035	−0.0358517	−	0.999357i	$-0.511414\pi$
$389$	−1.41421	−0.0717035	−0.0358517	+	0.999357i	$0.511414\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	18.0000	0.905678
$396$	0	0
$397$	−31.1127	−1.56150	−0.780751	−	0.624843i	$-0.785163\pi$
$397$	−31.1127	−1.56150	−0.780751	+	0.624843i	$0.785163\pi$
$398$	0	0
$399$	16.9706	0.849591
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	1.41421	0.0702728
$406$	0	0
$407$	50.9117	2.52360
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	16.9706	0.835067
$414$	0	0
$415$	−2.82843	−0.138842
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.0000	−1.07477	−0.537385	−	0.843337i	$-0.680588\pi$
$419$	−22.0000	−1.07477	−0.537385	+	0.843337i	$0.680588\pi$
$420$	0	0
$421$	−11.3137	−0.551396	−0.275698	−	0.961244i	$-0.588909\pi$
$421$	−11.3137	−0.551396	−0.275698	+	0.961244i	$0.588909\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	18.0000	0.873128
$426$	0	0
$427$	36.0000	1.74216
$428$	0	0
$429$	−33.9411	−1.63869
$430$	0	0
$431$	−19.7990	−0.953684	−0.476842	−	0.878989i	$-0.658219\pi$
$431$	−19.7990	−0.953684	−0.476842	+	0.878989i	$0.658219\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	−11.3137	−0.541208
$438$	0	0
$439$	−9.89949	−0.472477	−0.236239	−	0.971695i	$-0.575915\pi$
$439$	−9.89949	−0.472477	−0.236239	+	0.971695i	$0.575915\pi$
$440$	0	0
$441$	11.0000	0.523810
$442$	0	0
$443$	22.0000	1.04525	0.522626	−	0.852562i	$-0.324953\pi$
$443$	22.0000	1.04525	0.522626	+	0.852562i	$0.324953\pi$
$444$	0	0
$445$	2.82843	0.134080
$446$	0	0
$447$	−4.24264	−0.200670
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	−12.7279	−0.598010
$454$	0	0
$455$	−33.9411	−1.59118
$456$	0	0
$457$	42.0000	1.96468	0.982339	−	0.187112i	$-0.0599128\pi$
$457$	42.0000	1.96468	0.982339	+	0.187112i	$0.0599128\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−35.3553	−1.64666	−0.823331	−	0.567561i	$-0.807887\pi$
$461$	−35.3553	−1.64666	−0.823331	+	0.567561i	$0.807887\pi$
$462$	0	0
$463$	29.6985	1.38021	0.690103	−	0.723711i	$-0.257565\pi$
$463$	29.6985	1.38021	0.690103	+	0.723711i	$0.257565\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	33.9411	1.56726
$470$	0	0
$471$	2.82843	0.130327
$472$	0	0
$473$	0	0
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	−9.89949	−0.453267
$478$	0	0
$479$	−8.48528	−0.387702	−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528	−0.387702	−0.193851	+	0.981031i	$0.562098\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	2.82843	0.128432
$486$	0	0
$487$	35.3553	1.60210	0.801052	−	0.598595i	$-0.204274\pi$
$487$	35.3553	1.60210	0.801052	+	0.598595i	$0.204274\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−8.48528	−0.382158
$494$	0	0
$495$	−8.48528	−0.381385
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	−22.6274	−1.01092
$502$	0	0
$503$	14.1421	0.630567	0.315283	−	0.948998i	$-0.397900\pi$
$503$	14.1421	0.630567	0.315283	+	0.948998i	$0.397900\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	19.0000	0.843820
$508$	0	0
$509$	−9.89949	−0.438787	−0.219394	−	0.975636i	$-0.570408\pi$
$509$	−9.89949	−0.438787	−0.219394	+	0.975636i	$0.570408\pi$
$510$	0	0
$511$	−33.9411	−1.50147
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	10.0000	0.440653
$516$	0	0
$517$	−16.9706	−0.746364
$518$	0	0
$519$	−7.07107	−0.310385
$520$	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
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	12.7279	0.555492
$526$	0	0
$527$	8.48528	0.369625
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−11.3137	−0.490051
$534$	0	0
$535$	5.65685	0.244567
$536$	0	0
$537$	−20.0000	−0.863064
$538$	0	0
$539$	−66.0000	−2.84282
$540$	0	0
$541$	22.6274	0.972829	0.486414	−	0.873728i	$-0.338305\pi$
$541$	22.6274	0.972829	0.486414	+	0.873728i	$0.338305\pi$
$542$	0	0
$543$	−16.9706	−0.728277
$544$	0	0
$545$	24.0000	1.02805
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	−8.48528	−0.362143
$550$	0	0
$551$	−5.65685	−0.240990
$552$	0	0
$553$	−54.0000	−2.29631
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	7.07107	0.299611	0.149805	−	0.988716i	$-0.452135\pi$
$557$	7.07107	0.299611	0.149805	+	0.988716i	$0.452135\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	36.0000	1.51992
$562$	0	0
$563$	−46.0000	−1.93867	−0.969334	−	0.245745i	$-0.920967\pi$
$563$	−46.0000	−1.93867	−0.969334	+	0.245745i	$0.920967\pi$
$564$	0	0
$565$	−14.1421	−0.594964
$566$	0	0
$567$	−4.24264	−0.178174
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	11.3137	0.472637
$574$	0	0
$575$	−8.48528	−0.353861
$576$	0	0
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	8.48528	0.352029
$582$	0	0
$583$	59.3970	2.45997
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	−44.0000	−1.81607	−0.908037	−	0.418890i	$-0.862419\pi$
$587$	−44.0000	−1.81607	−0.908037	+	0.418890i	$0.862419\pi$
$588$	0	0
$589$	5.65685	0.233087
$590$	0	0
$591$	−4.24264	−0.174519
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	36.0000	1.47586
$596$	0	0
$597$	−4.24264	−0.173640
$598$	0	0
$599$	−31.1127	−1.27123	−0.635615	−	0.772006i	$-0.719253\pi$
$599$	−31.1127	−1.27123	−0.635615	+	0.772006i	$0.719253\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	35.3553	1.43740
$606$	0	0
$607$	−9.89949	−0.401808	−0.200904	−	0.979611i	$-0.564388\pi$
$607$	−9.89949	−0.401808	−0.200904	+	0.979611i	$0.564388\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	2.82843	0.114239	0.0571195	−	0.998367i	$-0.481808\pi$
$613$	2.82843	0.114239	0.0571195	+	0.998367i	$0.481808\pi$
$614$	0	0
$615$	−2.82843	−0.114053
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	2.82843	0.113501
$622$	0	0
$623$	−8.48528	−0.339956
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	24.0000	0.958468
$628$	0	0
$629$	50.9117	2.02998
$630$	0	0
$631$	1.41421	0.0562990	0.0281495	−	0.999604i	$-0.491039\pi$
$631$	1.41421	0.0562990	0.0281495	+	0.999604i	$0.491039\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.0000	0.714308
$636$	0	0
$637$	62.2254	2.46546
$638$	0	0
$639$	−2.82843	−0.111891
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.1421	−0.555985	−0.277992	−	0.960583i	$-0.589669\pi$
$647$	−14.1421	−0.555985	−0.277992	+	0.960583i	$0.589669\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	6.00000	0.235159
$652$	0	0
$653$	−29.6985	−1.16219	−0.581096	−	0.813835i	$-0.697376\pi$
$653$	−29.6985	−1.16219	−0.581096	+	0.813835i	$0.697376\pi$
$654$	0	0
$655$	16.9706	0.663095
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−8.48528	−0.330039	−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528	−0.330039	−0.165020	+	0.986290i	$0.552769\pi$
$662$	0	0
$663$	−33.9411	−1.31816
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	4.00000	0.154881
$668$	0	0
$669$	15.5563	0.601443
$670$	0	0
$671$	50.9117	1.96542
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	−3.00000	−0.115470
$676$	0	0
$677$	−43.8406	−1.68493	−0.842466	−	0.538750i	$-0.818897\pi$
$677$	−43.8406	−1.68493	−0.842466	+	0.538750i	$0.818897\pi$
$678$	0	0
$679$	−8.48528	−0.325635
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	0	0
$685$	−25.4558	−0.972618
$686$	0	0
$687$	22.6274	0.863290
$688$	0	0
$689$	−56.0000	−2.13343
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	0	0
$693$	25.4558	0.966988
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	22.0000	0.832116
$700$	0	0
$701$	38.1838	1.44218	0.721090	−	0.692841i	$-0.243641\pi$
$701$	38.1838	1.44218	0.721090	+	0.692841i	$0.243641\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−16.9706	−0.637343	−0.318671	−	0.947865i	$-0.603237\pi$
$709$	−16.9706	−0.637343	−0.318671	+	0.947865i	$0.603237\pi$
$710$	0	0
$711$	12.7279	0.477334
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	−48.0000	−1.79510
$716$	0	0
$717$	11.3137	0.422518
$718$	0	0
$719$	31.1127	1.16031	0.580154	−	0.814507i	$-0.302992\pi$
$719$	31.1127	1.16031	0.580154	+	0.814507i	$0.302992\pi$
$720$	0	0
$721$	−30.0000	−1.11726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.24264	−0.157568
$726$	0	0
$727$	26.8701	0.996555	0.498278	−	0.867018i	$-0.333966\pi$
$727$	26.8701	0.996555	0.498278	+	0.867018i	$0.333966\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	28.2843	1.04470	0.522352	−	0.852730i	$-0.325055\pi$
$733$	28.2843	1.04470	0.522352	+	0.852730i	$0.325055\pi$
$734$	0	0
$735$	15.5563	0.573805
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−22.6274	−0.831239
$742$	0	0
$743$	−28.2843	−1.03765	−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843	−1.03765	−0.518825	+	0.854881i	$0.673630\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	−16.9706	−0.620091
$750$	0	0
$751$	7.07107	0.258027	0.129013	−	0.991643i	$-0.458819\pi$
$751$	7.07107	0.258027	0.129013	+	0.991643i	$0.458819\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−18.0000	−0.655087
$756$	0	0
$757$	−45.2548	−1.64481	−0.822407	−	0.568899i	$-0.807370\pi$
$757$	−45.2548	−1.64481	−0.822407	+	0.568899i	$0.807370\pi$
$758$	0	0
$759$	−16.9706	−0.615992
$760$	0	0
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	0	0
$763$	−72.0000	−2.60658
$764$	0	0
$765$	−8.48528	−0.306786
$766$	0	0
$767$	−22.6274	−0.817029
$768$	0	0
$769$	−28.0000	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	0	0
$773$	−18.3848	−0.661254	−0.330627	−	0.943761i	$-0.607260\pi$
$773$	−18.3848	−0.661254	−0.330627	+	0.943761i	$0.607260\pi$
$774$	0	0
$775$	4.24264	0.152400
$776$	0	0
$777$	36.0000	1.29149
$778$	0	0
$779$	8.00000	0.286630
$780$	0	0
$781$	16.9706	0.607254
$782$	0	0
$783$	1.41421	0.0505399
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	16.9706	0.604168
$790$	0	0
$791$	42.4264	1.50851
$792$	0	0
$793$	−48.0000	−1.70453
$794$	0	0
$795$	−14.0000	−0.496529
$796$	0	0
$797$	24.0416	0.851598	0.425799	−	0.904818i	$-0.359993\pi$
$797$	24.0416	0.851598	0.425799	+	0.904818i	$0.359993\pi$
$798$	0	0
$799$	−16.9706	−0.600375
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−48.0000	−1.69388
$804$	0	0
$805$	−16.9706	−0.598134
$806$	0	0
$807$	−12.7279	−0.448044
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	4.24264	0.148796
$814$	0	0
$815$	−11.3137	−0.396302
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−24.0000	−0.838628
$820$	0	0
$821$	−1.41421	−0.0493564	−0.0246782	−	0.999695i	$-0.507856\pi$
$821$	−1.41421	−0.0493564	−0.0246782	+	0.999695i	$0.507856\pi$
$822$	0	0
$823$	41.0122	1.42960	0.714798	−	0.699331i	$-0.246519\pi$
$823$	41.0122	1.42960	0.714798	+	0.699331i	$0.246519\pi$
$824$	0	0
$825$	18.0000	0.626680
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	−28.2843	−0.982353	−0.491177	−	0.871060i	$-0.663433\pi$
$829$	−28.2843	−0.982353	−0.491177	+	0.871060i	$0.663433\pi$
$830$	0	0
$831$	16.9706	0.588702
$832$	0	0
$833$	−66.0000	−2.28676
$834$	0	0
$835$	−32.0000	−1.10741
$836$	0	0
$837$	−1.41421	−0.0488824
$838$	0	0
$839$	−19.7990	−0.683537	−0.341769	−	0.939784i	$-0.611026\pi$
$839$	−19.7990	−0.683537	−0.341769	+	0.939784i	$0.611026\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	0	0
$843$	−18.0000	−0.619953
$844$	0	0
$845$	26.8701	0.924358
$846$	0	0
$847$	−106.066	−3.64447
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−42.4264	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$853$	−42.4264	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$854$	0	0
$855$	−5.65685	−0.193460
$856$	0	0
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	8.48528	0.289178
$862$	0	0
$863$	−11.3137	−0.385123	−0.192562	−	0.981285i	$-0.561680\pi$
$863$	−11.3137	−0.385123	−0.192562	+	0.981285i	$0.561680\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	19.0000	0.645274
$868$	0	0
$869$	−76.3675	−2.59059
$870$	0	0
$871$	−45.2548	−1.53340
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	−36.7696	−1.24162	−0.620810	−	0.783961i	$-0.713196\pi$
$877$	−36.7696	−1.24162	−0.620810	+	0.783961i	$0.713196\pi$
$878$	0	0
$879$	24.0416	0.810904
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	0	0
$885$	−5.65685	−0.190153
$886$	0	0
$887$	−39.5980	−1.32957	−0.664785	−	0.747035i	$-0.731477\pi$
$887$	−39.5980	−1.32957	−0.664785	+	0.747035i	$0.731477\pi$
$888$	0	0
$889$	−54.0000	−1.81110
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	−11.3137	−0.378599
$894$	0	0
$895$	−28.2843	−0.945439
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	−2.00000	−0.0667037
$900$	0	0
$901$	59.3970	1.97880
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	−1.41421	−0.0469065
$910$	0	0
$911$	−28.2843	−0.937100	−0.468550	−	0.883437i	$-0.655223\pi$
$911$	−28.2843	−0.937100	−0.468550	+	0.883437i	$0.655223\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	−12.0000	−0.396708
$916$	0	0
$917$	−50.9117	−1.68125
$918$	0	0
$919$	−26.8701	−0.886361	−0.443181	−	0.896432i	$-0.646150\pi$
$919$	−26.8701	−0.886361	−0.443181	+	0.896432i	$0.646150\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	25.4558	0.836983
$926$	0	0
$927$	7.07107	0.232244
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−44.0000	−1.44204
$932$	0	0
$933$	22.6274	0.740788
$934$	0	0
$935$	50.9117	1.66499
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	−1.41421	−0.0461020	−0.0230510	−	0.999734i	$-0.507338\pi$
$941$	−1.41421	−0.0461020	−0.0230510	+	0.999734i	$0.507338\pi$
$942$	0	0
$943$	−5.65685	−0.184213
$944$	0	0
$945$	−6.00000	−0.195180
$946$	0	0
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	45.2548	1.46903
$950$	0	0
$951$	21.2132	0.687885
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	−8.48528	−0.274290
$958$	0	0
$959$	76.3675	2.46604
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	16.9706	0.546302
$966$	0	0
$967$	21.2132	0.682171	0.341085	−	0.940032i	$-0.389205\pi$
$967$	21.2132	0.682171	0.341085	+	0.940032i	$0.389205\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−16.9706	−0.543493
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	16.9706	0.541828
$982$	0	0
$983$	−28.2843	−0.902128	−0.451064	−	0.892492i	$-0.648955\pi$
$983$	−28.2843	−0.902128	−0.451064	+	0.892492i	$0.648955\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−4.24264	−0.134772	−0.0673860	−	0.997727i	$-0.521466\pi$
$991$	−4.24264	−0.134772	−0.0673860	+	0.997727i	$0.521466\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	−6.00000	−0.190213
$996$	0	0
$997$	−31.1127	−0.985349	−0.492675	−	0.870214i	$-0.663981\pi$
$997$	−31.1127	−0.985349	−0.492675	+	0.870214i	$0.663981\pi$
$998$	0	0
$999$	−8.48528	−0.268462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.a.i.1.2	yes	2
3.2	odd	2		4608.2.a.l.1.1		2
4.3	odd	2		1536.2.a.d.1.2	yes	2
8.3	odd	2	inner	1536.2.a.i.1.1	yes	2
8.5	even	2		1536.2.a.d.1.1	✓	2
12.11	even	2		4608.2.a.g.1.1		2
16.3	odd	4		1536.2.d.c.769.1		4
16.5	even	4		1536.2.d.c.769.2		4
16.11	odd	4		1536.2.d.c.769.4		4
16.13	even	4		1536.2.d.c.769.3		4
24.5	odd	2		4608.2.a.g.1.2		2
24.11	even	2		4608.2.a.l.1.2		2
48.5	odd	4		4608.2.d.n.2305.2		4
48.11	even	4		4608.2.d.n.2305.1		4
48.29	odd	4		4608.2.d.n.2305.4		4
48.35	even	4		4608.2.d.n.2305.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.d.1.1	✓	2	8.5	even	2
1536.2.a.d.1.2	yes	2	4.3	odd	2
1536.2.a.i.1.1	yes	2	8.3	odd	2	inner
1536.2.a.i.1.2	yes	2	1.1	even	1	trivial
1536.2.d.c.769.1		4	16.3	odd	4
1536.2.d.c.769.2		4	16.5	even	4
1536.2.d.c.769.3		4	16.13	even	4
1536.2.d.c.769.4		4	16.11	odd	4
4608.2.a.g.1.1		2	12.11	even	2
4608.2.a.g.1.2		2	24.5	odd	2
4608.2.a.l.1.1		2	3.2	odd	2
4608.2.a.l.1.2		2	24.11	even	2
4608.2.d.n.2305.1		4	48.11	even	4
4608.2.d.n.2305.2		4	48.5	odd	4
4608.2.d.n.2305.3		4	48.35	even	4
4608.2.d.n.2305.4		4	48.29	odd	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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.41421 q^{5} -4.24264 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.41421 q^{5} -4.24264 q^{7} +1.00000 q^{9} -6.00000 q^{11} +5.65685 q^{13} +1.41421 q^{15} -6.00000 q^{17} -4.00000 q^{19} -4.24264 q^{21} +2.82843 q^{23} -3.00000 q^{25} +1.00000 q^{27} +1.41421 q^{29} -1.41421 q^{31} -6.00000 q^{33} -6.00000 q^{35} -8.48528 q^{37} +5.65685 q^{39} -2.00000 q^{41} +1.41421 q^{45} +2.82843 q^{47} +11.0000 q^{49} -6.00000 q^{51} -9.89949 q^{53} -8.48528 q^{55} -4.00000 q^{57} -4.00000 q^{59} -8.48528 q^{61} -4.24264 q^{63} +8.00000 q^{65} -8.00000 q^{67} +2.82843 q^{69} -2.82843 q^{71} +8.00000 q^{73} -3.00000 q^{75} +25.4558 q^{77} +12.7279 q^{79} +1.00000 q^{81} -2.00000 q^{83} -8.48528 q^{85} +1.41421 q^{87} +2.00000 q^{89} -24.0000 q^{91} -1.41421 q^{93} -5.65685 q^{95} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} - 12 q^{11} - 12 q^{17} - 8 q^{19} - 6 q^{25} + 2 q^{27} - 12 q^{33} - 12 q^{35} - 4 q^{41} + 22 q^{49} - 12 q^{51} - 8 q^{57} - 8 q^{59} + 16 q^{65} - 16 q^{67} + 16 q^{73} - 6 q^{75} + 2 q^{81} - 4 q^{83} + 4 q^{89} - 48 q^{91} + 4 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 12 * q^11 - 12 * q^17 - 8 * q^19 - 6 * q^25 + 2 * q^27 - 12 * q^33 - 12 * q^35 - 4 * q^41 + 22 * q^49 - 12 * q^51 - 8 * q^57 - 8 * q^59 + 16 * q^65 - 16 * q^67 + 16 * q^73 - 6 * q^75 + 2 * q^81 - 4 * q^83 + 4 * q^89 - 48 * q^91 + 4 * q^97 - 12 * q^99

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