LMFDB - Embedded newform 5625.2.a.y.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5625,2,Mod(1,5625)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5625, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,14,0,0,-10,0,0,0,0,0,-10,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.46980000000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 $ x^8 - 15x^6 + 80x^4 - 180*x^2 + 145
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.63148$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63148 q^{2} +0.661739 q^{4} +1.44512 q^{7} +2.18335 q^{8} +0.797518 q^{11} -3.02701 q^{13} -2.35770 q^{14} -4.88558 q^{16} -0.282098 q^{17} +2.88558 q^{19} -1.30114 q^{22} +3.47376 q^{23} +4.93852 q^{26} +0.956295 q^{28} -7.73271 q^{29} +2.38761 q^{31} +3.60404 q^{32} +0.460239 q^{34} -4.64195 q^{37} -4.70778 q^{38} +8.42719 q^{41} -6.07985 q^{43} +0.527748 q^{44} -5.66739 q^{46} +1.16013 q^{47} -4.91161 q^{49} -2.00309 q^{52} -11.7932 q^{53} +3.15522 q^{56} +12.6158 q^{58} +12.2420 q^{59} -0.868886 q^{61} -3.89535 q^{62} +3.89123 q^{64} -11.3892 q^{67} -0.186675 q^{68} +4.63389 q^{71} -10.8327 q^{73} +7.57327 q^{74} +1.90950 q^{76} +1.15251 q^{77} -4.73501 q^{79} -13.7488 q^{82} +16.6753 q^{83} +9.91918 q^{86} +1.74126 q^{88} -16.2163 q^{89} -4.37441 q^{91} +2.29872 q^{92} -1.89274 q^{94} -10.8409 q^{97} +8.01322 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} - 18 q^{16} + 2 q^{19} - 20 q^{22} - 10 q^{28} + 4 q^{31} + 50 q^{34} - 50 q^{43} - 30 q^{46} - 6 q^{49} - 30 q^{52} - 60 q^{58} + 46 q^{61} - 14 q^{64} - 40 q^{67}+ \cdots - 50 q^{97}+O(q^{100}) $ 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 - 18 * q^16 + 2 * q^19 - 20 * q^22 - 10 * q^28 + 4 * q^31 + 50 * q^34 - 50 * q^43 - 30 * q^46 - 6 * q^49 - 30 * q^52 - 60 * q^58 + 46 * q^61 - 14 * q^64 - 40 * q^67 - 50 * q^73 - 34 * q^76 - 12 * q^79 - 60 * q^82 - 70 * q^88 - 10 * q^91 - 20 * q^94 - 50 * q^97

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 $

$2$	−1.63148	−1.15363	−0.576817	−	0.816874i	$-0.695705\pi$
$2$	−1.63148	−1.15363	−0.576817	+	0.816874i	$0.695705\pi$
$3$	0	0
$3$	0	0
$4$	0.661739	0.330869
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.44512	0.546206	0.273103	−	0.961985i	$-0.411950\pi$
$7$	1.44512	0.546206	0.273103	+	0.961985i	$0.411950\pi$
$8$	2.18335	0.771931
$9$	0	0
$10$	0	0
$11$	0.797518	0.240461	0.120230	−	0.992746i	$-0.461637\pi$
$11$	0.797518	0.240461	0.120230	+	0.992746i	$0.461637\pi$
$12$	0	0
$13$	−3.02701	−0.839542	−0.419771	−	0.907630i	$-0.637890\pi$
$13$	−3.02701	−0.839542	−0.419771	+	0.907630i	$0.637890\pi$
$14$	−2.35770	−0.630121
$15$	0	0
$16$	−4.88558	−1.22139
$17$	−0.282098	−0.0684189	−0.0342094	−	0.999415i	$-0.510891\pi$
$17$	−0.282098	−0.0684189	−0.0342094	+	0.999415i	$0.510891\pi$
$18$	0	0
$19$	2.88558	0.661997	0.330999	−	0.943631i	$-0.392614\pi$
$19$	2.88558	0.661997	0.330999	+	0.943631i	$0.392614\pi$
$20$	0	0
$21$	0	0
$22$	−1.30114	−0.277403
$23$	3.47376	0.724329	0.362165	−	0.932114i	$-0.382038\pi$
$23$	3.47376	0.724329	0.362165	+	0.932114i	$0.382038\pi$
$24$	0	0
$25$	0	0
$26$	4.93852	0.968523
$27$	0	0
$28$	0.956295	0.180723
$29$	−7.73271	−1.43593	−0.717964	−	0.696080i	$-0.754926\pi$
$29$	−7.73271	−1.43593	−0.717964	+	0.696080i	$0.754926\pi$
$30$	0	0
$31$	2.38761	0.428828	0.214414	−	0.976743i	$-0.431216\pi$
$31$	2.38761	0.428828	0.214414	+	0.976743i	$0.431216\pi$
$32$	3.60404	0.637110
$33$	0	0
$34$	0.460239	0.0789303
$35$	0	0
$36$	0	0
$37$	−4.64195	−0.763133	−0.381566	−	0.924341i	$-0.624615\pi$
$37$	−4.64195	−0.763133	−0.381566	+	0.924341i	$0.624615\pi$
$38$	−4.70778	−0.763702
$39$	0	0
$40$	0	0
$41$	8.42719	1.31611	0.658053	−	0.752972i	$-0.271380\pi$
$41$	8.42719	1.31611	0.658053	+	0.752972i	$0.271380\pi$
$42$	0	0
$43$	−6.07985	−0.927169	−0.463584	−	0.886053i	$-0.653437\pi$
$43$	−6.07985	−0.927169	−0.463584	+	0.886053i	$0.653437\pi$
$44$	0.527748	0.0795611
$45$	0	0
$46$	−5.66739	−0.835610
$47$	1.16013	0.169223	0.0846114	−	0.996414i	$-0.473035\pi$
$47$	1.16013	0.169223	0.0846114	+	0.996414i	$0.473035\pi$
$48$	0	0
$49$	−4.91161	−0.701659
$50$	0	0
$51$	0	0
$52$	−2.00309	−0.277779
$53$	−11.7932	−1.61992	−0.809960	−	0.586485i	$-0.800511\pi$
$53$	−11.7932	−1.61992	−0.809960	+	0.586485i	$0.800511\pi$
$54$	0	0
$55$	0	0
$56$	3.15522	0.421633
$57$	0	0
$58$	12.6158	1.65653
$59$	12.2420	1.59378	0.796888	−	0.604127i	$-0.206478\pi$
$59$	12.2420	1.59378	0.796888	+	0.604127i	$0.206478\pi$
$60$	0	0
$61$	−0.868886	−0.111249	−0.0556247	−	0.998452i	$-0.517715\pi$
$61$	−0.868886	−0.111249	−0.0556247	+	0.998452i	$0.517715\pi$
$62$	−3.89535	−0.494710
$63$	0	0
$64$	3.89123	0.486403
$65$	0	0
$66$	0	0
$67$	−11.3892	−1.39141	−0.695706	−	0.718327i	$-0.744908\pi$
$67$	−11.3892	−1.39141	−0.695706	+	0.718327i	$0.744908\pi$
$68$	−0.186675	−0.0226377
$69$	0	0
$70$	0	0
$71$	4.63389	0.549942	0.274971	−	0.961453i	$-0.411332\pi$
$71$	4.63389	0.549942	0.274971	+	0.961453i	$0.411332\pi$
$72$	0	0
$73$	−10.8327	−1.26788	−0.633938	−	0.773384i	$-0.718563\pi$
$73$	−10.8327	−1.26788	−0.633938	+	0.773384i	$0.718563\pi$
$74$	7.57327	0.880375
$75$	0	0
$76$	1.90950	0.219035
$77$	1.15251	0.131341
$78$	0	0
$79$	−4.73501	−0.532730	−0.266365	−	0.963872i	$-0.585823\pi$
$79$	−4.73501	−0.532730	−0.266365	+	0.963872i	$0.585823\pi$
$80$	0	0
$81$	0	0
$82$	−13.7488	−1.51830
$83$	16.6753	1.83035	0.915177	−	0.403052i	$-0.132051\pi$
$83$	16.6753	1.83035	0.915177	+	0.403052i	$0.132051\pi$
$84$	0	0
$85$	0	0
$86$	9.91918	1.06961
$87$	0	0
$88$	1.74126	0.185619
$89$	−16.2163	−1.71892	−0.859462	−	0.511200i	$-0.829201\pi$
$89$	−16.2163	−1.71892	−0.859462	+	0.511200i	$0.829201\pi$
$90$	0	0
$91$	−4.37441	−0.458563
$92$	2.29872	0.239658
$93$	0	0
$94$	−1.89274	−0.195221
$95$	0	0
$96$	0	0
$97$	−10.8409	−1.10073	−0.550363	−	0.834925i	$-0.685511\pi$
$97$	−10.8409	−1.10073	−0.550363	+	0.834925i	$0.685511\pi$
$98$	8.01322	0.809457
$99$	0	0
$100$	0	0
$101$	5.97095	0.594132	0.297066	−	0.954857i	$-0.403992\pi$
$101$	5.97095	0.594132	0.297066	+	0.954857i	$0.403992\pi$
$102$	0	0
$103$	8.38631	0.826327	0.413164	−	0.910657i	$-0.364424\pi$
$103$	8.38631	0.826327	0.413164	+	0.910657i	$0.364424\pi$
$104$	−6.60903	−0.648069
$105$	0	0
$106$	19.2404	1.86879
$107$	−10.0314	−0.969775	−0.484888	−	0.874576i	$-0.661140\pi$
$107$	−10.0314	−0.969775	−0.484888	+	0.874576i	$0.661140\pi$
$108$	0	0
$109$	−3.57433	−0.342359	−0.171179	−	0.985240i	$-0.554758\pi$
$109$	−3.57433	−0.342359	−0.171179	+	0.985240i	$0.554758\pi$
$110$	0	0
$111$	0	0
$112$	−7.06027	−0.667133
$113$	−10.3012	−0.969056	−0.484528	−	0.874776i	$-0.661009\pi$
$113$	−10.3012	−0.969056	−0.484528	+	0.874776i	$0.661009\pi$
$114$	0	0
$115$	0	0
$116$	−5.11704	−0.475105
$117$	0	0
$118$	−19.9727	−1.83863
$119$	−0.407667	−0.0373708
$120$	0	0
$121$	−10.3640	−0.942179
$122$	1.41757	0.128341
$123$	0	0
$124$	1.57998	0.141886
$125$	0	0
$126$	0	0
$127$	−6.78181	−0.601788	−0.300894	−	0.953658i	$-0.597285\pi$
$127$	−6.78181	−0.601788	−0.300894	+	0.953658i	$0.597285\pi$
$128$	−13.5566	−1.19824
$129$	0	0
$130$	0	0
$131$	20.1690	1.76218	0.881088	−	0.472952i	$-0.156812\pi$
$131$	20.1690	1.76218	0.881088	+	0.472952i	$0.156812\pi$
$132$	0	0
$133$	4.17002	0.361587
$134$	18.5813	1.60518
$135$	0	0
$136$	−0.615920	−0.0528147
$137$	11.2562	0.961685	0.480843	−	0.876807i	$-0.340331\pi$
$137$	11.2562	0.961685	0.480843	+	0.876807i	$0.340331\pi$
$138$	0	0
$139$	−7.73034	−0.655679	−0.327839	−	0.944733i	$-0.606321\pi$
$139$	−7.73034	−0.655679	−0.327839	+	0.944733i	$0.606321\pi$
$140$	0	0
$141$	0	0
$142$	−7.56012	−0.634431
$143$	−2.41409	−0.201877
$144$	0	0
$145$	0	0
$146$	17.6734	1.46266
$147$	0	0
$148$	−3.07176	−0.252497
$149$	7.96312	0.652364	0.326182	−	0.945307i	$-0.394238\pi$
$149$	7.96312	0.652364	0.326182	+	0.945307i	$0.394238\pi$
$150$	0	0
$151$	20.0043	1.62793	0.813965	−	0.580914i	$-0.197305\pi$
$151$	20.0043	1.62793	0.813965	+	0.580914i	$0.197305\pi$
$152$	6.30023	0.511016
$153$	0	0
$154$	−1.88031	−0.151519
$155$	0	0
$156$	0	0
$157$	5.40707	0.431531	0.215765	−	0.976445i	$-0.430775\pi$
$157$	5.40707	0.431531	0.215765	+	0.976445i	$0.430775\pi$
$158$	7.72509	0.614575
$159$	0	0
$160$	0	0
$161$	5.02002	0.395633
$162$	0	0
$163$	−5.96446	−0.467172	−0.233586	−	0.972336i	$-0.575046\pi$
$163$	−5.96446	−0.467172	−0.233586	+	0.972336i	$0.575046\pi$
$164$	5.57660	0.435459
$165$	0	0
$166$	−27.2055	−2.11156
$167$	−8.52261	−0.659499	−0.329750	−	0.944068i	$-0.606964\pi$
$167$	−8.52261	−0.659499	−0.329750	+	0.944068i	$0.606964\pi$
$168$	0	0
$169$	−3.83720	−0.295170
$170$	0	0
$171$	0	0
$172$	−4.02327	−0.306772
$173$	7.42809	0.564747	0.282374	−	0.959305i	$-0.408878\pi$
$173$	7.42809	0.564747	0.282374	+	0.959305i	$0.408878\pi$
$174$	0	0
$175$	0	0
$176$	−3.89634	−0.293697
$177$	0	0
$178$	26.4566	1.98301
$179$	−20.7650	−1.55205	−0.776023	−	0.630704i	$-0.782766\pi$
$179$	−20.7650	−1.55205	−0.776023	+	0.630704i	$0.782766\pi$
$180$	0	0
$181$	19.7654	1.46915	0.734575	−	0.678528i	$-0.237382\pi$
$181$	19.7654	1.46915	0.734575	+	0.678528i	$0.237382\pi$
$182$	7.13678	0.529013
$183$	0	0
$184$	7.58444	0.559133
$185$	0	0
$186$	0	0
$187$	−0.224978	−0.0164521
$188$	0.767705	0.0559906
$189$	0	0
$190$	0	0
$191$	16.8195	1.21702	0.608509	−	0.793547i	$-0.291768\pi$
$191$	16.8195	1.21702	0.608509	+	0.793547i	$0.291768\pi$
$192$	0	0
$193$	−20.1885	−1.45320	−0.726602	−	0.687059i	$-0.758901\pi$
$193$	−20.1885	−1.45320	−0.726602	+	0.687059i	$0.758901\pi$
$194$	17.6867	1.26983
$195$	0	0
$196$	−3.25021	−0.232158
$197$	−12.3010	−0.876410	−0.438205	−	0.898875i	$-0.644386\pi$
$197$	−12.3010	−0.876410	−0.438205	+	0.898875i	$0.644386\pi$
$198$	0	0
$199$	19.6340	1.39182	0.695909	−	0.718130i	$-0.255002\pi$
$199$	19.6340	1.39182	0.695909	+	0.718130i	$0.255002\pi$
$200$	0	0
$201$	0	0
$202$	−9.74151	−0.685410
$203$	−11.1747	−0.784313
$204$	0	0
$205$	0	0
$206$	−13.6821	−0.953279
$207$	0	0
$208$	14.7887	1.02541
$209$	2.30130	0.159184
$210$	0	0
$211$	7.55552	0.520144	0.260072	−	0.965589i	$-0.416254\pi$
$211$	7.55552	0.520144	0.260072	+	0.965589i	$0.416254\pi$
$212$	−7.80402	−0.535982
$213$	0	0
$214$	16.3661	1.11877
$215$	0	0
$216$	0	0
$217$	3.45040	0.234228
$218$	5.83146	0.394956
$219$	0	0
$220$	0	0
$221$	0.853915	0.0574405
$222$	0	0
$223$	−1.60772	−0.107661	−0.0538303	−	0.998550i	$-0.517143\pi$
$223$	−1.60772	−0.107661	−0.0538303	+	0.998550i	$0.517143\pi$
$224$	5.20829	0.347993
$225$	0	0
$226$	16.8062	1.11793
$227$	5.53703	0.367506	0.183753	−	0.982972i	$-0.441175\pi$
$227$	5.53703	0.367506	0.183753	+	0.982972i	$0.441175\pi$
$228$	0	0
$229$	12.3562	0.816521	0.408260	−	0.912865i	$-0.366136\pi$
$229$	12.3562	0.816521	0.408260	+	0.912865i	$0.366136\pi$
$230$	0	0
$231$	0	0
$232$	−16.8832	−1.10844
$233$	6.66543	0.436667	0.218333	−	0.975874i	$-0.429938\pi$
$233$	6.66543	0.436667	0.218333	+	0.975874i	$0.429938\pi$
$234$	0	0
$235$	0	0
$236$	8.10102	0.527332
$237$	0	0
$238$	0.665103	0.0431122
$239$	−5.71449	−0.369640	−0.184820	−	0.982772i	$-0.559170\pi$
$239$	−5.71449	−0.369640	−0.184820	+	0.982772i	$0.559170\pi$
$240$	0	0
$241$	−21.1134	−1.36003	−0.680016	−	0.733198i	$-0.738027\pi$
$241$	−21.1134	−1.36003	−0.680016	+	0.733198i	$0.738027\pi$
$242$	16.9086	1.08693
$243$	0	0
$244$	−0.574975	−0.0368090
$245$	0	0
$246$	0	0
$247$	−8.73468	−0.555774
$248$	5.21300	0.331026
$249$	0	0
$250$	0	0
$251$	7.39926	0.467037	0.233519	−	0.972352i	$-0.424976\pi$
$251$	7.39926	0.467037	0.233519	+	0.972352i	$0.424976\pi$
$252$	0	0
$253$	2.77039	0.174173
$254$	11.0644	0.694243
$255$	0	0
$256$	14.3348	0.895928
$257$	−7.27627	−0.453881	−0.226941	−	0.973909i	$-0.572872\pi$
$257$	−7.27627	−0.453881	−0.226941	+	0.973909i	$0.572872\pi$
$258$	0	0
$259$	−6.70820	−0.416828
$260$	0	0
$261$	0	0
$262$	−32.9054	−2.03290
$263$	−8.62308	−0.531722	−0.265861	−	0.964011i	$-0.585656\pi$
$263$	−8.62308	−0.531722	−0.265861	+	0.964011i	$0.585656\pi$
$264$	0	0
$265$	0	0
$266$	−6.80332	−0.417139
$267$	0	0
$268$	−7.53667	−0.460375
$269$	−11.3763	−0.693626	−0.346813	−	0.937934i	$-0.612736\pi$
$269$	−11.3763	−0.693626	−0.346813	+	0.937934i	$0.612736\pi$
$270$	0	0
$271$	−13.6887	−0.831532	−0.415766	−	0.909472i	$-0.636487\pi$
$271$	−13.6887	−0.831532	−0.415766	+	0.909472i	$0.636487\pi$
$272$	1.37821	0.0835665
$273$	0	0
$274$	−18.3644	−1.10943
$275$	0	0
$276$	0	0
$277$	−26.4969	−1.59204	−0.796022	−	0.605267i	$-0.793066\pi$
$277$	−26.4969	−1.59204	−0.796022	+	0.605267i	$0.793066\pi$
$278$	12.6119	0.756413
$279$	0	0
$280$	0	0
$281$	11.1577	0.665612	0.332806	−	0.942995i	$-0.392005\pi$
$281$	11.1577	0.665612	0.332806	+	0.942995i	$0.392005\pi$
$282$	0	0
$283$	23.7714	1.41306	0.706532	−	0.707681i	$-0.250258\pi$
$283$	23.7714	1.41306	0.706532	+	0.707681i	$0.250258\pi$
$284$	3.06643	0.181959
$285$	0	0
$286$	3.93856	0.232892
$287$	12.1783	0.718865
$288$	0	0
$289$	−16.9204	−0.995319
$290$	0	0
$291$	0	0
$292$	−7.16844	−0.419501
$293$	0.0928455	0.00542409	0.00271204	−	0.999996i	$-0.499137\pi$
$293$	0.0928455	0.00542409	0.00271204	+	0.999996i	$0.499137\pi$
$294$	0	0
$295$	0	0
$296$	−10.1350	−0.589086
$297$	0	0
$298$	−12.9917	−0.752589
$299$	−10.5151	−0.608105
$300$	0	0
$301$	−8.78615	−0.506425
$302$	−32.6368	−1.87803
$303$	0	0
$304$	−14.0977	−0.808560
$305$	0	0
$306$	0	0
$307$	−12.5168	−0.714372	−0.357186	−	0.934033i	$-0.616264\pi$
$307$	−12.5168	−0.714372	−0.357186	+	0.934033i	$0.616264\pi$
$308$	0.762662	0.0434567
$309$	0	0
$310$	0	0
$311$	−30.5365	−1.73157	−0.865783	−	0.500420i	$-0.833179\pi$
$311$	−30.5365	−1.73157	−0.865783	+	0.500420i	$0.833179\pi$
$312$	0	0
$313$	13.4428	0.759833	0.379916	−	0.925021i	$-0.375953\pi$
$313$	13.4428	0.759833	0.379916	+	0.925021i	$0.375953\pi$
$314$	−8.82154	−0.497828
$315$	0	0
$316$	−3.13334	−0.176264
$317$	−12.4363	−0.698493	−0.349247	−	0.937031i	$-0.613562\pi$
$317$	−12.4363	−0.698493	−0.349247	+	0.937031i	$0.613562\pi$
$318$	0	0
$319$	−6.16697	−0.345284
$320$	0	0
$321$	0	0
$322$	−8.19008	−0.456415
$323$	−0.814017	−0.0452931
$324$	0	0
$325$	0	0
$326$	9.73091	0.538945
$327$	0	0
$328$	18.3995	1.01594
$329$	1.67654	0.0924304
$330$	0	0
$331$	1.86199	0.102344	0.0511721	−	0.998690i	$-0.483704\pi$
$331$	1.86199	0.102344	0.0511721	+	0.998690i	$0.483704\pi$
$332$	11.0347	0.605608
$333$	0	0
$334$	13.9045	0.760820
$335$	0	0
$336$	0	0
$337$	11.3165	0.616450	0.308225	−	0.951313i	$-0.400265\pi$
$337$	11.3165	0.616450	0.308225	+	0.951313i	$0.400265\pi$
$338$	6.26034	0.340517
$339$	0	0
$340$	0	0
$341$	1.90416	0.103116
$342$	0	0
$343$	−17.2138	−0.929456
$344$	−13.2745	−0.715711
$345$	0	0
$346$	−12.1188	−0.651511
$347$	25.4682	1.36721	0.683603	−	0.729854i	$-0.260412\pi$
$347$	25.4682	1.36721	0.683603	+	0.729854i	$0.260412\pi$
$348$	0	0
$349$	11.2236	0.600787	0.300394	−	0.953815i	$-0.402882\pi$
$349$	11.2236	0.600787	0.300394	+	0.953815i	$0.402882\pi$
$350$	0	0
$351$	0	0
$352$	2.87429	0.153200
$353$	13.0338	0.693722	0.346861	−	0.937917i	$-0.387248\pi$
$353$	13.0338	0.693722	0.346861	+	0.937917i	$0.387248\pi$
$354$	0	0
$355$	0	0
$356$	−10.7310	−0.568739
$357$	0	0
$358$	33.8777	1.79049
$359$	−29.5123	−1.55760	−0.778800	−	0.627273i	$-0.784171\pi$
$359$	−29.5123	−1.55760	−0.778800	+	0.627273i	$0.784171\pi$
$360$	0	0
$361$	−10.6734	−0.561760
$362$	−32.2469	−1.69486
$363$	0	0
$364$	−2.89472	−0.151724
$365$	0	0
$366$	0	0
$367$	−17.2070	−0.898199	−0.449100	−	0.893482i	$-0.648255\pi$
$367$	−17.2070	−0.898199	−0.449100	+	0.893482i	$0.648255\pi$
$368$	−16.9713	−0.884692
$369$	0	0
$370$	0	0
$371$	−17.0426	−0.884810
$372$	0	0
$373$	−16.2030	−0.838957	−0.419479	−	0.907765i	$-0.637787\pi$
$373$	−16.2030	−0.838957	−0.419479	+	0.907765i	$0.637787\pi$
$374$	0.367049	0.0189796
$375$	0	0
$376$	2.53298	0.130628
$377$	23.4070	1.20552
$378$	0	0
$379$	28.5047	1.46419	0.732095	−	0.681203i	$-0.238543\pi$
$379$	28.5047	1.46419	0.732095	+	0.681203i	$0.238543\pi$
$380$	0	0
$381$	0	0
$382$	−27.4408	−1.40399
$383$	−26.9679	−1.37800	−0.688998	−	0.724763i	$-0.741949\pi$
$383$	−26.9679	−1.37800	−0.688998	+	0.724763i	$0.741949\pi$
$384$	0	0
$385$	0	0
$386$	32.9373	1.67646
$387$	0	0
$388$	−7.17384	−0.364197
$389$	6.02735	0.305599	0.152799	−	0.988257i	$-0.451171\pi$
$389$	6.02735	0.305599	0.152799	+	0.988257i	$0.451171\pi$
$390$	0	0
$391$	−0.979942	−0.0495578
$392$	−10.7238	−0.541633
$393$	0	0
$394$	20.0689	1.01106
$395$	0	0
$396$	0	0
$397$	−18.4231	−0.924627	−0.462313	−	0.886717i	$-0.652980\pi$
$397$	−18.4231	−0.924627	−0.462313	+	0.886717i	$0.652980\pi$
$398$	−32.0325	−1.60565
$399$	0	0
$400$	0	0
$401$	−17.4430	−0.871063	−0.435532	−	0.900173i	$-0.643440\pi$
$401$	−17.4430	−0.871063	−0.435532	+	0.900173i	$0.643440\pi$
$402$	0	0
$403$	−7.22733	−0.360019
$404$	3.95121	0.196580
$405$	0	0
$406$	18.2314	0.904809
$407$	−3.70204	−0.183503
$408$	0	0
$409$	−21.9024	−1.08300	−0.541501	−	0.840700i	$-0.682144\pi$
$409$	−21.9024	−1.08300	−0.541501	+	0.840700i	$0.682144\pi$
$410$	0	0
$411$	0	0
$412$	5.54954	0.273406
$413$	17.6913	0.870530
$414$	0	0
$415$	0	0
$416$	−10.9095	−0.534881
$417$	0	0
$418$	−3.75453	−0.183640
$419$	−8.29532	−0.405253	−0.202626	−	0.979256i	$-0.564948\pi$
$419$	−8.29532	−0.405253	−0.202626	+	0.979256i	$0.564948\pi$
$420$	0	0
$421$	−11.0805	−0.540028	−0.270014	−	0.962856i	$-0.587028\pi$
$421$	−11.0805	−0.540028	−0.270014	+	0.962856i	$0.587028\pi$
$422$	−12.3267	−0.600055
$423$	0	0
$424$	−25.7487	−1.25047
$425$	0	0
$426$	0	0
$427$	−1.25565	−0.0607651
$428$	−6.63819	−0.320869
$429$	0	0
$430$	0	0
$431$	−25.2816	−1.21777	−0.608885	−	0.793259i	$-0.708383\pi$
$431$	−25.2816	−1.21777	−0.608885	+	0.793259i	$0.708383\pi$
$432$	0	0
$433$	−6.04358	−0.290436	−0.145218	−	0.989400i	$-0.546388\pi$
$433$	−6.04358	−0.290436	−0.145218	+	0.989400i	$0.546388\pi$
$434$	−5.62927	−0.270214
$435$	0	0
$436$	−2.36527	−0.113276
$437$	10.0238	0.479504
$438$	0	0
$439$	−18.0624	−0.862073	−0.431036	−	0.902335i	$-0.641852\pi$
$439$	−18.0624	−0.862073	−0.431036	+	0.902335i	$0.641852\pi$
$440$	0	0
$441$	0	0
$442$	−1.39315	−0.0662653
$443$	12.8138	0.608804	0.304402	−	0.952544i	$-0.401543\pi$
$443$	12.8138	0.608804	0.304402	+	0.952544i	$0.401543\pi$
$444$	0	0
$445$	0	0
$446$	2.62296	0.124201
$447$	0	0
$448$	5.62331	0.265676
$449$	−34.9139	−1.64769	−0.823845	−	0.566815i	$-0.808175\pi$
$449$	−34.9139	−1.64769	−0.823845	+	0.566815i	$0.808175\pi$
$450$	0	0
$451$	6.72083	0.316472
$452$	−6.81671	−0.320631
$453$	0	0
$454$	−9.03358	−0.423967
$455$	0	0
$456$	0	0
$457$	19.6673	0.919999	0.459999	−	0.887919i	$-0.347850\pi$
$457$	19.6673	0.919999	0.459999	+	0.887919i	$0.347850\pi$
$458$	−20.1589	−0.941966
$459$	0	0
$460$	0	0
$461$	−31.0991	−1.44843	−0.724215	−	0.689575i	$-0.757798\pi$
$461$	−31.0991	−1.44843	−0.724215	+	0.689575i	$0.757798\pi$
$462$	0	0
$463$	−38.9301	−1.80924	−0.904618	−	0.426222i	$-0.859844\pi$
$463$	−38.9301	−1.80924	−0.904618	+	0.426222i	$0.859844\pi$
$464$	37.7788	1.75384
$465$	0	0
$466$	−10.8745	−0.503753
$467$	11.3805	0.526626	0.263313	−	0.964710i	$-0.415185\pi$
$467$	11.3805	0.526626	0.263313	+	0.964710i	$0.415185\pi$
$468$	0	0
$469$	−16.4588	−0.759997
$470$	0	0
$471$	0	0
$472$	26.7286	1.23029
$473$	−4.84879	−0.222948
$474$	0	0
$475$	0	0
$476$	−0.269769	−0.0123649
$477$	0	0
$478$	9.32310	0.426429
$479$	24.6901	1.12812	0.564060	−	0.825734i	$-0.309238\pi$
$479$	24.6901	1.12812	0.564060	+	0.825734i	$0.309238\pi$
$480$	0	0
$481$	14.0512	0.640682
$482$	34.4461	1.56898
$483$	0	0
$484$	−6.85824	−0.311738
$485$	0	0
$486$	0	0
$487$	3.83392	0.173731	0.0868657	−	0.996220i	$-0.472315\pi$
$487$	3.83392	0.173731	0.0868657	+	0.996220i	$0.472315\pi$
$488$	−1.89708	−0.0858769
$489$	0	0
$490$	0	0
$491$	7.57778	0.341980	0.170990	−	0.985273i	$-0.445303\pi$
$491$	7.57778	0.341980	0.170990	+	0.985273i	$0.445303\pi$
$492$	0	0
$493$	2.18139	0.0982447
$494$	14.2505	0.641160
$495$	0	0
$496$	−11.6649	−0.523768
$497$	6.69656	0.300382
$498$	0	0
$499$	22.9554	1.02763	0.513813	−	0.857902i	$-0.328232\pi$
$499$	22.9554	1.02763	0.513813	+	0.857902i	$0.328232\pi$
$500$	0	0
$501$	0	0
$502$	−12.0718	−0.538789
$503$	31.2801	1.39471	0.697355	−	0.716726i	$-0.254360\pi$
$503$	31.2801	1.39471	0.697355	+	0.716726i	$0.254360\pi$
$504$	0	0
$505$	0	0
$506$	−4.51984	−0.200931
$507$	0	0
$508$	−4.48778	−0.199113
$509$	−23.1936	−1.02804	−0.514020	−	0.857778i	$-0.671844\pi$
$509$	−23.1936	−1.02804	−0.514020	+	0.857778i	$0.671844\pi$
$510$	0	0
$511$	−15.6547	−0.692521
$512$	3.72605	0.164670
$513$	0	0
$514$	11.8711	0.523612
$515$	0	0
$516$	0	0
$517$	0.925226	0.0406914
$518$	10.9443	0.480866
$519$	0	0
$520$	0	0
$521$	−31.0116	−1.35864	−0.679321	−	0.733841i	$-0.737726\pi$
$521$	−31.0116	−1.35864	−0.679321	+	0.733841i	$0.737726\pi$
$522$	0	0
$523$	11.3537	0.496462	0.248231	−	0.968701i	$-0.420151\pi$
$523$	11.3537	0.496462	0.248231	+	0.968701i	$0.420151\pi$
$524$	13.3466	0.583050
$525$	0	0
$526$	14.0684	0.613412
$527$	−0.673542	−0.0293399
$528$	0	0
$529$	−10.9330	−0.475347
$530$	0	0
$531$	0	0
$532$	2.75947	0.119638
$533$	−25.5092	−1.10493
$534$	0	0
$535$	0	0
$536$	−24.8666	−1.07407
$537$	0	0
$538$	18.5603	0.800190
$539$	−3.91710	−0.168721
$540$	0	0
$541$	−26.8120	−1.15274	−0.576368	−	0.817190i	$-0.695531\pi$
$541$	−26.8120	−1.15274	−0.576368	+	0.817190i	$0.695531\pi$
$542$	22.3330	0.959283
$543$	0	0
$544$	−1.01669	−0.0435904
$545$	0	0
$546$	0	0
$547$	−39.2250	−1.67714	−0.838570	−	0.544793i	$-0.816608\pi$
$547$	−39.2250	−1.67714	−0.838570	+	0.544793i	$0.816608\pi$
$548$	7.44869	0.318192
$549$	0	0
$550$	0	0
$551$	−22.3134	−0.950581
$552$	0	0
$553$	−6.84268	−0.290980
$554$	43.2293	1.83664
$555$	0	0
$556$	−5.11547	−0.216944
$557$	−44.3906	−1.88089	−0.940445	−	0.339946i	$-0.889591\pi$
$557$	−44.3906	−1.88089	−0.940445	+	0.339946i	$0.889591\pi$
$558$	0	0
$559$	18.4038	0.778397
$560$	0	0
$561$	0	0
$562$	−18.2036	−0.767873
$563$	33.4717	1.41066	0.705332	−	0.708877i	$-0.250798\pi$
$563$	33.4717	1.41066	0.705332	+	0.708877i	$0.250798\pi$
$564$	0	0
$565$	0	0
$566$	−38.7827	−1.63016
$567$	0	0
$568$	10.1174	0.424517
$569$	0.968748	0.0406120	0.0203060	−	0.999794i	$-0.493536\pi$
$569$	0.968748	0.0406120	0.0203060	+	0.999794i	$0.493536\pi$
$570$	0	0
$571$	25.5141	1.06773	0.533867	−	0.845569i	$-0.320738\pi$
$571$	25.5141	1.06773	0.533867	+	0.845569i	$0.320738\pi$
$572$	−1.59750	−0.0667948
$573$	0	0
$574$	−19.8688	−0.829306
$575$	0	0
$576$	0	0
$577$	34.3026	1.42804	0.714019	−	0.700127i	$-0.246873\pi$
$577$	34.3026	1.42804	0.714019	+	0.700127i	$0.246873\pi$
$578$	27.6054	1.14823
$579$	0	0
$580$	0	0
$581$	24.0979	0.999750
$582$	0	0
$583$	−9.40528	−0.389527
$584$	−23.6517	−0.978713
$585$	0	0
$586$	−0.151476	−0.00625741
$587$	25.3199	1.04507	0.522533	−	0.852619i	$-0.324987\pi$
$587$	25.3199	1.04507	0.522533	+	0.852619i	$0.324987\pi$
$588$	0	0
$589$	6.88965	0.283883
$590$	0	0
$591$	0	0
$592$	22.6786	0.932086
$593$	−0.909773	−0.0373599	−0.0186800	−	0.999826i	$-0.505946\pi$
$593$	−0.909773	−0.0373599	−0.0186800	+	0.999826i	$0.505946\pi$
$594$	0	0
$595$	0	0
$596$	5.26951	0.215847
$597$	0	0
$598$	17.1552	0.701530
$599$	2.86390	0.117016	0.0585080	−	0.998287i	$-0.481366\pi$
$599$	2.86390	0.117016	0.0585080	+	0.998287i	$0.481366\pi$
$600$	0	0
$601$	−3.46431	−0.141312	−0.0706560	−	0.997501i	$-0.522509\pi$
$601$	−3.46431	−0.141312	−0.0706560	+	0.997501i	$0.522509\pi$
$602$	14.3345	0.584229
$603$	0	0
$604$	13.2376	0.538632
$605$	0	0
$606$	0	0
$607$	13.9220	0.565075	0.282538	−	0.959256i	$-0.408824\pi$
$607$	13.9220	0.565075	0.282538	+	0.959256i	$0.408824\pi$
$608$	10.3997	0.421765
$609$	0	0
$610$	0	0
$611$	−3.51173	−0.142070
$612$	0	0
$613$	−4.86475	−0.196485	−0.0982427	−	0.995162i	$-0.531322\pi$
$613$	−4.86475	−0.196485	−0.0982427	+	0.995162i	$0.531322\pi$
$614$	20.4210	0.824124
$615$	0	0
$616$	2.51634	0.101386
$617$	−20.0261	−0.806220	−0.403110	−	0.915152i	$-0.632071\pi$
$617$	−20.0261	−0.806220	−0.403110	+	0.915152i	$0.632071\pi$
$618$	0	0
$619$	37.7672	1.51799	0.758996	−	0.651095i	$-0.225690\pi$
$619$	37.7672	1.51799	0.758996	+	0.651095i	$0.225690\pi$
$620$	0	0
$621$	0	0
$622$	49.8198	1.99759
$623$	−23.4346	−0.938886
$624$	0	0
$625$	0	0
$626$	−21.9317	−0.876568
$627$	0	0
$628$	3.57807	0.142780
$629$	1.30949	0.0522127
$630$	0	0
$631$	−25.0097	−0.995622	−0.497811	−	0.867286i	$-0.665863\pi$
$631$	−25.0097	−0.995622	−0.497811	+	0.867286i	$0.665863\pi$
$632$	−10.3382	−0.411231
$633$	0	0
$634$	20.2896	0.805805
$635$	0	0
$636$	0	0
$637$	14.8675	0.589072
$638$	10.0613	0.398331
$639$	0	0
$640$	0	0
$641$	−10.8546	−0.428731	−0.214365	−	0.976754i	$-0.568768\pi$
$641$	−10.8546	−0.428731	−0.214365	+	0.976754i	$0.568768\pi$
$642$	0	0
$643$	−31.0302	−1.22371	−0.611856	−	0.790969i	$-0.709577\pi$
$643$	−31.0302	−1.22371	−0.611856	+	0.790969i	$0.709577\pi$
$644$	3.32194	0.130903
$645$	0	0
$646$	1.32806	0.0522516
$647$	−49.2351	−1.93563	−0.967815	−	0.251662i	$-0.919023\pi$
$647$	−49.2351	−1.93563	−0.967815	+	0.251662i	$0.919023\pi$
$648$	0	0
$649$	9.76323	0.383240
$650$	0	0
$651$	0	0
$652$	−3.94691	−0.154573
$653$	42.5573	1.66540	0.832698	−	0.553727i	$-0.186795\pi$
$653$	42.5573	1.66540	0.832698	+	0.553727i	$0.186795\pi$
$654$	0	0
$655$	0	0
$656$	−41.1717	−1.60748
$657$	0	0
$658$	−2.73524	−0.106631
$659$	−32.0671	−1.24916	−0.624578	−	0.780963i	$-0.714729\pi$
$659$	−32.0671	−1.24916	−0.624578	+	0.780963i	$0.714729\pi$
$660$	0	0
$661$	38.0549	1.48016	0.740082	−	0.672517i	$-0.234787\pi$
$661$	38.0549	1.48016	0.740082	+	0.672517i	$0.234787\pi$
$662$	−3.03780	−0.118068
$663$	0	0
$664$	36.4081	1.41291
$665$	0	0
$666$	0	0
$667$	−26.8616	−1.04009
$668$	−5.63974	−0.218208
$669$	0	0
$670$	0	0
$671$	−0.692952	−0.0267511
$672$	0	0
$673$	−23.8508	−0.919381	−0.459691	−	0.888079i	$-0.652040\pi$
$673$	−23.8508	−0.919381	−0.459691	+	0.888079i	$0.652040\pi$
$674$	−18.4627	−0.711158
$675$	0	0
$676$	−2.53923	−0.0976626
$677$	−10.5516	−0.405530	−0.202765	−	0.979227i	$-0.564993\pi$
$677$	−10.5516	−0.405530	−0.202765	+	0.979227i	$0.564993\pi$
$678$	0	0
$679$	−15.6665	−0.601223
$680$	0	0
$681$	0	0
$682$	−3.10661	−0.118958
$683$	−19.9516	−0.763427	−0.381713	−	0.924281i	$-0.624666\pi$
$683$	−19.9516	−0.763427	−0.381713	+	0.924281i	$0.624666\pi$
$684$	0	0
$685$	0	0
$686$	28.0840	1.07225
$687$	0	0
$688$	29.7036	1.13244
$689$	35.6981	1.35999
$690$	0	0
$691$	32.7907	1.24742	0.623708	−	0.781657i	$-0.285625\pi$
$691$	32.7907	1.24742	0.623708	+	0.781657i	$0.285625\pi$
$692$	4.91545	0.186858
$693$	0	0
$694$	−41.5510	−1.57725
$695$	0	0
$696$	0	0
$697$	−2.37730	−0.0900465
$698$	−18.3112	−0.693088
$699$	0	0
$700$	0	0
$701$	−2.70090	−0.102012	−0.0510058	−	0.998698i	$-0.516243\pi$
$701$	−2.70090	−0.102012	−0.0510058	+	0.998698i	$0.516243\pi$
$702$	0	0
$703$	−13.3947	−0.505192
$704$	3.10332	0.116961
$705$	0	0
$706$	−21.2645	−0.800300
$707$	8.62877	0.324518
$708$	0	0
$709$	−30.2828	−1.13729	−0.568647	−	0.822582i	$-0.692533\pi$
$709$	−30.2828	−1.13729	−0.568647	+	0.822582i	$0.692533\pi$
$710$	0	0
$711$	0	0
$712$	−35.4059	−1.32689
$713$	8.29400	0.310613
$714$	0	0
$715$	0	0
$716$	−13.7410	−0.513525
$717$	0	0
$718$	48.1488	1.79690
$719$	−20.3313	−0.758228	−0.379114	−	0.925350i	$-0.623771\pi$
$719$	−20.3313	−0.758228	−0.379114	+	0.925350i	$0.623771\pi$
$720$	0	0
$721$	12.1193	0.451345
$722$	17.4135	0.648064
$723$	0	0
$724$	13.0795	0.486097
$725$	0	0
$726$	0	0
$727$	−21.6351	−0.802400	−0.401200	−	0.915991i	$-0.631407\pi$
$727$	−21.6351	−0.802400	−0.401200	+	0.915991i	$0.631407\pi$
$728$	−9.55087	−0.353979
$729$	0	0
$730$	0	0
$731$	1.71512	0.0634359
$732$	0	0
$733$	−42.5223	−1.57060	−0.785299	−	0.619116i	$-0.787491\pi$
$733$	−42.5223	−1.57060	−0.785299	+	0.619116i	$0.787491\pi$
$734$	28.0730	1.03619
$735$	0	0
$736$	12.5196	0.461478
$737$	−9.08308	−0.334580
$738$	0	0
$739$	−11.7281	−0.431425	−0.215713	−	0.976457i	$-0.569207\pi$
$739$	−11.7281	−0.431425	−0.215713	+	0.976457i	$0.569207\pi$
$740$	0	0
$741$	0	0
$742$	27.8048	1.02075
$743$	−27.4555	−1.00724	−0.503622	−	0.863924i	$-0.668000\pi$
$743$	−27.4555	−1.00724	−0.503622	+	0.863924i	$0.668000\pi$
$744$	0	0
$745$	0	0
$746$	26.4349	0.967849
$747$	0	0
$748$	−0.148877	−0.00544348
$749$	−14.4967	−0.529697
$750$	0	0
$751$	−25.8528	−0.943383	−0.471692	−	0.881764i	$-0.656356\pi$
$751$	−25.8528	−0.943383	−0.471692	+	0.881764i	$0.656356\pi$
$752$	−5.66792	−0.206688
$753$	0	0
$754$	−38.1881	−1.39073
$755$	0	0
$756$	0	0
$757$	28.8785	1.04961	0.524803	−	0.851224i	$-0.324139\pi$
$757$	28.8785	1.04961	0.524803	+	0.851224i	$0.324139\pi$
$758$	−46.5050	−1.68914
$759$	0	0
$760$	0	0
$761$	−40.6966	−1.47525	−0.737626	−	0.675210i	$-0.764053\pi$
$761$	−40.6966	−1.47525	−0.737626	+	0.675210i	$0.764053\pi$
$762$	0	0
$763$	−5.16535	−0.186998
$764$	11.1301	0.402674
$765$	0	0
$766$	43.9977	1.58970
$767$	−37.0567	−1.33804
$768$	0	0
$769$	12.8126	0.462033	0.231017	−	0.972950i	$-0.425795\pi$
$769$	12.8126	0.462033	0.231017	+	0.972950i	$0.425795\pi$
$770$	0	0
$771$	0	0
$772$	−13.3595	−0.480820
$773$	−29.1725	−1.04926	−0.524632	−	0.851329i	$-0.675797\pi$
$773$	−29.1725	−1.04926	−0.524632	+	0.851329i	$0.675797\pi$
$774$	0	0
$775$	0	0
$776$	−23.6695	−0.849685
$777$	0	0
$778$	−9.83352	−0.352549
$779$	24.3173	0.871258
$780$	0	0
$781$	3.69561	0.132239
$782$	1.59876	0.0571715
$783$	0	0
$784$	23.9961	0.857003
$785$	0	0
$786$	0	0
$787$	11.5839	0.412921	0.206460	−	0.978455i	$-0.433806\pi$
$787$	11.5839	0.412921	0.206460	+	0.978455i	$0.433806\pi$
$788$	−8.14005	−0.289977
$789$	0	0
$790$	0	0
$791$	−14.8865	−0.529304
$792$	0	0
$793$	2.63013	0.0933985
$794$	30.0569	1.06668
$795$	0	0
$796$	12.9926	0.460510
$797$	38.8479	1.37606	0.688031	−	0.725681i	$-0.258475\pi$
$797$	38.8479	1.37606	0.688031	+	0.725681i	$0.258475\pi$
$798$	0	0
$799$	−0.327271	−0.0115780
$800$	0	0
$801$	0	0
$802$	28.4580	1.00489
$803$	−8.63930	−0.304874
$804$	0	0
$805$	0	0
$806$	11.7913	0.415330
$807$	0	0
$808$	13.0367	0.458629
$809$	−32.9255	−1.15760	−0.578799	−	0.815470i	$-0.696479\pi$
$809$	−32.9255	−1.15760	−0.578799	+	0.815470i	$0.696479\pi$
$810$	0	0
$811$	−5.59127	−0.196336	−0.0981680	−	0.995170i	$-0.531298\pi$
$811$	−5.59127	−0.196336	−0.0981680	+	0.995170i	$0.531298\pi$
$812$	−7.39476	−0.259505
$813$	0	0
$814$	6.03982	0.211696
$815$	0	0
$816$	0	0
$817$	−17.5439	−0.613783
$818$	35.7333	1.24939
$819$	0	0
$820$	0	0
$821$	−13.8837	−0.484545	−0.242272	−	0.970208i	$-0.577893\pi$
$821$	−13.8837	−0.484545	−0.242272	+	0.970208i	$0.577893\pi$
$822$	0	0
$823$	−23.1074	−0.805473	−0.402736	−	0.915316i	$-0.631941\pi$
$823$	−23.1074	−0.805473	−0.402736	+	0.915316i	$0.631941\pi$
$824$	18.3103	0.637868
$825$	0	0
$826$	−28.8630	−1.00427
$827$	49.3260	1.71523	0.857616	−	0.514290i	$-0.171944\pi$
$827$	49.3260	1.71523	0.857616	+	0.514290i	$0.171944\pi$
$828$	0	0
$829$	41.0813	1.42681	0.713406	−	0.700751i	$-0.247152\pi$
$829$	41.0813	1.42681	0.713406	+	0.700751i	$0.247152\pi$
$830$	0	0
$831$	0	0
$832$	−11.7788	−0.408356
$833$	1.38556	0.0480067
$834$	0	0
$835$	0	0
$836$	1.52286	0.0526692
$837$	0	0
$838$	13.5337	0.467513
$839$	−6.97734	−0.240884	−0.120442	−	0.992720i	$-0.538431\pi$
$839$	−6.97734	−0.240884	−0.120442	+	0.992720i	$0.538431\pi$
$840$	0	0
$841$	30.7948	1.06189
$842$	18.0776	0.622995
$843$	0	0
$844$	4.99978	0.172100
$845$	0	0
$846$	0	0
$847$	−14.9772	−0.514624
$848$	57.6166	1.97856
$849$	0	0
$850$	0	0
$851$	−16.1250	−0.552759
$852$	0	0
$853$	−29.0382	−0.994249	−0.497125	−	0.867679i	$-0.665611\pi$
$853$	−29.0382	−0.994249	−0.497125	+	0.867679i	$0.665611\pi$
$854$	2.04857	0.0701006
$855$	0	0
$856$	−21.9021	−0.748600
$857$	4.63176	0.158218	0.0791090	−	0.996866i	$-0.474792\pi$
$857$	4.63176	0.158218	0.0791090	+	0.996866i	$0.474792\pi$
$858$	0	0
$859$	−10.8416	−0.369912	−0.184956	−	0.982747i	$-0.559214\pi$
$859$	−10.8416	−0.369912	−0.184956	+	0.982747i	$0.559214\pi$
$860$	0	0
$861$	0	0
$862$	41.2465	1.40486
$863$	−26.1400	−0.889815	−0.444908	−	0.895576i	$-0.646763\pi$
$863$	−26.1400	−0.889815	−0.444908	+	0.895576i	$0.646763\pi$
$864$	0	0
$865$	0	0
$866$	9.86000	0.335056
$867$	0	0
$868$	2.28326	0.0774990
$869$	−3.77625	−0.128101
$870$	0	0
$871$	34.4752	1.16815
$872$	−7.80402	−0.264277
$873$	0	0
$874$	−16.3537	−0.553172
$875$	0	0
$876$	0	0
$877$	40.1008	1.35411	0.677054	−	0.735933i	$-0.263256\pi$
$877$	40.1008	1.35411	0.677054	+	0.735933i	$0.263256\pi$
$878$	29.4686	0.994516
$879$	0	0
$880$	0	0
$881$	−34.4321	−1.16005	−0.580023	−	0.814600i	$-0.696956\pi$
$881$	−34.4321	−1.16005	−0.580023	+	0.814600i	$0.696956\pi$
$882$	0	0
$883$	−51.3034	−1.72650	−0.863249	−	0.504778i	$-0.831574\pi$
$883$	−51.3034	−1.72650	−0.863249	+	0.504778i	$0.831574\pi$
$884$	0.565069	0.0190053
$885$	0	0
$886$	−20.9056	−0.702336
$887$	11.5446	0.387631	0.193815	−	0.981038i	$-0.437914\pi$
$887$	11.5446	0.387631	0.193815	+	0.981038i	$0.437914\pi$
$888$	0	0
$889$	−9.80056	−0.328700
$890$	0	0
$891$	0	0
$892$	−1.06389	−0.0356216
$893$	3.34765	0.112025
$894$	0	0
$895$	0	0
$896$	−19.5909	−0.654486
$897$	0	0
$898$	56.9615	1.90083
$899$	−18.4627	−0.615766
$900$	0	0
$901$	3.32684	0.110833
$902$	−10.9649	−0.365092
$903$	0	0
$904$	−22.4911	−0.748044
$905$	0	0
$906$	0	0
$907$	29.4860	0.979067	0.489533	−	0.871985i	$-0.337167\pi$
$907$	29.4860	0.979067	0.489533	+	0.871985i	$0.337167\pi$
$908$	3.66407	0.121596
$909$	0	0
$910$	0	0
$911$	−6.96402	−0.230728	−0.115364	−	0.993323i	$-0.536804\pi$
$911$	−6.96402	−0.230728	−0.115364	+	0.993323i	$0.536804\pi$
$912$	0	0
$913$	13.2989	0.440128
$914$	−32.0869	−1.06134
$915$	0	0
$916$	8.17658	0.270162
$917$	29.1468	0.962511
$918$	0	0
$919$	−21.0012	−0.692767	−0.346384	−	0.938093i	$-0.612590\pi$
$919$	−21.0012	−0.692767	−0.346384	+	0.938093i	$0.612590\pi$
$920$	0	0
$921$	0	0
$922$	50.7377	1.67096
$923$	−14.0268	−0.461699
$924$	0	0
$925$	0	0
$926$	63.5139	2.08720
$927$	0	0
$928$	−27.8690	−0.914845
$929$	9.22471	0.302653	0.151326	−	0.988484i	$-0.451646\pi$
$929$	9.22471	0.302653	0.151326	+	0.988484i	$0.451646\pi$
$930$	0	0
$931$	−14.1729	−0.464496
$932$	4.41077	0.144480
$933$	0	0
$934$	−18.5671	−0.607533
$935$	0	0
$936$	0	0
$937$	35.5933	1.16278	0.581392	−	0.813624i	$-0.302508\pi$
$937$	35.5933	1.16278	0.581392	+	0.813624i	$0.302508\pi$
$938$	26.8523	0.876758
$939$	0	0
$940$	0	0
$941$	53.0875	1.73060	0.865302	−	0.501250i	$-0.167126\pi$
$941$	53.0875	1.73060	0.865302	+	0.501250i	$0.167126\pi$
$942$	0	0
$943$	29.2740	0.953294
$944$	−59.8094	−1.94663
$945$	0	0
$946$	7.91072	0.257200
$947$	−20.8873	−0.678746	−0.339373	−	0.940652i	$-0.610215\pi$
$947$	−20.8873	−0.678746	−0.339373	+	0.940652i	$0.610215\pi$
$948$	0	0
$949$	32.7908	1.06443
$950$	0	0
$951$	0	0
$952$	−0.890081	−0.0288477
$953$	−35.0040	−1.13389	−0.566945	−	0.823755i	$-0.691875\pi$
$953$	−35.0040	−1.13389	−0.566945	+	0.823755i	$0.691875\pi$
$954$	0	0
$955$	0	0
$956$	−3.78150	−0.122303
$957$	0	0
$958$	−40.2815	−1.30144
$959$	16.2667	0.525278
$960$	0	0
$961$	−25.2993	−0.816107
$962$	−22.9244	−0.739112
$963$	0	0
$964$	−13.9715	−0.449993
$965$	0	0
$966$	0	0
$967$	8.32372	0.267673	0.133836	−	0.991003i	$-0.457270\pi$
$967$	8.32372	0.267673	0.133836	+	0.991003i	$0.457270\pi$
$968$	−22.6282	−0.727297
$969$	0	0
$970$	0	0
$971$	−13.0277	−0.418080	−0.209040	−	0.977907i	$-0.567034\pi$
$971$	−13.0277	−0.418080	−0.209040	+	0.977907i	$0.567034\pi$
$972$	0	0
$973$	−11.1713	−0.358136
$974$	−6.25497	−0.200422
$975$	0	0
$976$	4.24501	0.135879
$977$	−48.0089	−1.53594	−0.767970	−	0.640486i	$-0.778733\pi$
$977$	−48.0089	−1.53594	−0.767970	+	0.640486i	$0.778733\pi$
$978$	0	0
$979$	−12.9328	−0.413334
$980$	0	0
$981$	0	0
$982$	−12.3630	−0.394520
$983$	30.3257	0.967240	0.483620	−	0.875278i	$-0.339322\pi$
$983$	30.3257	0.967240	0.483620	+	0.875278i	$0.339322\pi$
$984$	0	0
$985$	0	0
$986$	−3.55889	−0.113338
$987$	0	0
$988$	−5.78008	−0.183889
$989$	−21.1200	−0.671576
$990$	0	0
$991$	10.8408	0.344368	0.172184	−	0.985065i	$-0.444918\pi$
$991$	10.8408	0.344368	0.172184	+	0.985065i	$0.444918\pi$
$992$	8.60505	0.273211
$993$	0	0
$994$	−10.9253	−0.346530
$995$	0	0
$996$	0	0
$997$	−1.33544	−0.0422939	−0.0211470	−	0.999776i	$-0.506732\pi$
$997$	−1.33544	−0.0422939	−0.0211470	+	0.999776i	$0.506732\pi$
$998$	−37.4514	−1.18550
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.y.1.3	✓	8
3.2	odd	2	inner	5625.2.a.y.1.6	yes	8
5.4	even	2		5625.2.a.ba.1.6	yes	8
15.14	odd	2		5625.2.a.ba.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.y.1.3	✓	8	1.1	even	1	trivial
5625.2.a.y.1.6	yes	8	3.2	odd	2	inner
5625.2.a.ba.1.3	yes	8	15.14	odd	2
5625.2.a.ba.1.6	yes	8	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63148 q^{2} +0.661739 q^{4} +1.44512 q^{7} +2.18335 q^{8} +0.797518 q^{11} -3.02701 q^{13} -2.35770 q^{14} -4.88558 q^{16} -0.282098 q^{17} +2.88558 q^{19} -1.30114 q^{22} +3.47376 q^{23} +4.93852 q^{26} +0.956295 q^{28} -7.73271 q^{29} +2.38761 q^{31} +3.60404 q^{32} +0.460239 q^{34} -4.64195 q^{37} -4.70778 q^{38} +8.42719 q^{41} -6.07985 q^{43} +0.527748 q^{44} -5.66739 q^{46} +1.16013 q^{47} -4.91161 q^{49} -2.00309 q^{52} -11.7932 q^{53} +3.15522 q^{56} +12.6158 q^{58} +12.2420 q^{59} -0.868886 q^{61} -3.89535 q^{62} +3.89123 q^{64} -11.3892 q^{67} -0.186675 q^{68} +4.63389 q^{71} -10.8327 q^{73} +7.57327 q^{74} +1.90950 q^{76} +1.15251 q^{77} -4.73501 q^{79} -13.7488 q^{82} +16.6753 q^{83} +9.91918 q^{86} +1.74126 q^{88} -16.2163 q^{89} -4.37441 q^{91} +2.29872 q^{92} -1.89274 q^{94} -10.8409 q^{97} +8.01322 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} - 18 q^{16} + 2 q^{19} - 20 q^{22} - 10 q^{28} + 4 q^{31} + 50 q^{34} - 50 q^{43} - 30 q^{46} - 6 q^{49} - 30 q^{52} - 60 q^{58} + 46 q^{61} - 14 q^{64} - 40 q^{67}+ \cdots - 50 q^{97}+O(q^{100}) \) 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 - 18 * q^16 + 2 * q^19 - 20 * q^22 - 10 * q^28 + 4 * q^31 + 50 * q^34 - 50 * q^43 - 30 * q^46 - 6 * q^49 - 30 * q^52 - 60 * q^58 + 46 * q^61 - 14 * q^64 - 40 * q^67 - 50 * q^73 - 34 * q^76 - 12 * q^79 - 60 * q^82 - 70 * q^88 - 10 * q^91 - 20 * q^94 - 50 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more