LMFDB - Embedded newform 1800.4.a.bo.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1800,4,Mod(1,1800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1800.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,2,0,0,0,16,0,82,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$106.203438010$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 600)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.72015$ of defining polynomial
Character	$\chi$	$=$	1800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+21.8806 q^{7} -54.6418 q^{11} +82.7612 q^{13} +100.403 q^{17} -84.1194 q^{19} +0.880613 q^{23} +99.1194 q^{29} -78.9255 q^{31} +390.806 q^{37} -104.642 q^{41} +241.403 q^{43} -512.567 q^{47} +135.761 q^{49} -284.642 q^{53} +709.045 q^{59} +470.522 q^{61} -667.583 q^{67} +51.5378 q^{71} +371.284 q^{73} -1195.60 q^{77} -79.3428 q^{79} +682.180 q^{83} -628.478 q^{89} +1810.87 q^{91} +1519.09 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} + 16 q^{11} + 82 q^{13} - 8 q^{17} - 210 q^{19} - 40 q^{23} + 240 q^{29} + 218 q^{31} + 364 q^{37} - 84 q^{41} + 274 q^{43} - 524 q^{47} + 188 q^{49} - 444 q^{53} + 1084 q^{59} + 774 q^{61}+ \cdots + 2370 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 16 * q^11 + 82 * q^13 - 8 * q^17 - 210 * q^19 - 40 * q^23 + 240 * q^29 + 218 * q^31 + 364 * q^37 - 84 * q^41 + 274 * q^43 - 524 * q^47 + 188 * q^49 - 444 * q^53 + 1084 * q^59 + 774 * q^61 + 210 * q^67 - 1108 * q^71 + 492 * q^73 - 2600 * q^77 - 1328 * q^79 + 28 * q^83 - 1424 * q^89 + 1826 * q^91 + 2370 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	21.8806	1.18144	0.590721	−	0.806876i	$-0.298843\pi$
$7$	21.8806	1.18144	0.590721	+	0.806876i	$0.298843\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−54.6418	−1.49774	−0.748870	−	0.662717i	$-0.769403\pi$
$11$	−54.6418	−1.49774	−0.748870	+	0.662717i	$0.769403\pi$
$12$	0	0
$13$	82.7612	1.76568	0.882840	−	0.469674i	$-0.155629\pi$
$13$	82.7612	1.76568	0.882840	+	0.469674i	$0.155629\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	100.403	1.43243	0.716215	−	0.697879i	$-0.245873\pi$
$17$	100.403	1.43243	0.716215	+	0.697879i	$0.245873\pi$
$18$	0	0
$19$	−84.1194	−1.01570	−0.507850	−	0.861445i	$-0.669560\pi$
$19$	−84.1194	−1.01570	−0.507850	+	0.861445i	$0.669560\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.880613	0.00798350	0.00399175	−	0.999992i	$-0.498729\pi$
$23$	0.880613	0.00798350	0.00399175	+	0.999992i	$0.498729\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	99.1194	0.634690	0.317345	−	0.948310i	$-0.397209\pi$
$29$	99.1194	0.634690	0.317345	+	0.948310i	$0.397209\pi$
$30$	0	0
$31$	−78.9255	−0.457272	−0.228636	−	0.973512i	$-0.573427\pi$
$31$	−78.9255	−0.457272	−0.228636	+	0.973512i	$0.573427\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	390.806	1.73644	0.868218	−	0.496183i	$-0.165265\pi$
$37$	390.806	1.73644	0.868218	+	0.496183i	$0.165265\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−104.642	−0.398593	−0.199296	−	0.979939i	$-0.563866\pi$
$41$	−104.642	−0.398593	−0.199296	+	0.979939i	$0.563866\pi$
$42$	0	0
$43$	241.403	0.856131	0.428065	−	0.903748i	$-0.359195\pi$
$43$	241.403	0.856131	0.428065	+	0.903748i	$0.359195\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−512.567	−1.59076	−0.795379	−	0.606112i	$-0.792728\pi$
$47$	−512.567	−1.59076	−0.795379	+	0.606112i	$0.792728\pi$
$48$	0	0
$49$	135.761	0.395805
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−284.642	−0.737709	−0.368854	−	0.929487i	$-0.620250\pi$
$53$	−284.642	−0.737709	−0.368854	+	0.929487i	$0.620250\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	709.045	1.56457	0.782286	−	0.622919i	$-0.214053\pi$
$59$	709.045	1.56457	0.782286	+	0.622919i	$0.214053\pi$
$60$	0	0
$61$	470.522	0.987610	0.493805	−	0.869573i	$-0.335606\pi$
$61$	470.522	0.987610	0.493805	+	0.869573i	$0.335606\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−667.583	−1.21729	−0.608643	−	0.793444i	$-0.708286\pi$
$67$	−667.583	−1.21729	−0.608643	+	0.793444i	$0.708286\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	51.5378	0.0861466	0.0430733	−	0.999072i	$-0.486285\pi$
$71$	51.5378	0.0861466	0.0430733	+	0.999072i	$0.486285\pi$
$72$	0	0
$73$	371.284	0.595280	0.297640	−	0.954678i	$-0.403800\pi$
$73$	371.284	0.595280	0.297640	+	0.954678i	$0.403800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1195.60	−1.76949
$78$	0	0
$79$	−79.3428	−0.112997	−0.0564985	−	0.998403i	$-0.517994\pi$
$79$	−79.3428	−0.112997	−0.0564985	+	0.998403i	$0.517994\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	682.180	0.902156	0.451078	−	0.892485i	$-0.351040\pi$
$83$	682.180	0.902156	0.451078	+	0.892485i	$0.351040\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−628.478	−0.748522	−0.374261	−	0.927323i	$-0.622104\pi$
$89$	−628.478	−0.748522	−0.374261	+	0.927323i	$0.622104\pi$
$90$	0	0
$91$	1810.87	2.08605
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1519.09	1.59011	0.795053	−	0.606541i	$-0.207443\pi$
$97$	1519.09	1.59011	0.795053	+	0.606541i	$0.207443\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−977.777	−0.963291	−0.481646	−	0.876366i	$-0.659961\pi$
$101$	−977.777	−0.963291	−0.481646	+	0.876366i	$0.659961\pi$
$102$	0	0
$103$	759.194	0.726268	0.363134	−	0.931737i	$-0.381707\pi$
$103$	759.194	0.726268	0.363134	+	0.931737i	$0.381707\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−945.448	−0.854205	−0.427102	−	0.904203i	$-0.640466\pi$
$107$	−945.448	−0.854205	−0.427102	+	0.904203i	$0.640466\pi$
$108$	0	0
$109$	−688.314	−0.604849	−0.302425	−	0.953173i	$-0.597796\pi$
$109$	−688.314	−0.604849	−0.302425	+	0.953173i	$0.597796\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	350.865	0.292094	0.146047	−	0.989278i	$-0.453345\pi$
$113$	350.865	0.292094	0.146047	+	0.989278i	$0.453345\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2196.88	1.69233
$120$	0	0
$121$	1654.73	1.24322
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1588.21	1.10969	0.554846	−	0.831953i	$-0.312777\pi$
$127$	1588.21	1.10969	0.554846	+	0.831953i	$0.312777\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2156.02	1.43795	0.718977	−	0.695034i	$-0.244611\pi$
$131$	2156.02	1.43795	0.718977	+	0.695034i	$0.244611\pi$
$132$	0	0
$133$	−1840.58	−1.19999
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−316.716	−0.197510	−0.0987551	−	0.995112i	$-0.531486\pi$
$137$	−316.716	−0.197510	−0.0987551	+	0.995112i	$0.531486\pi$
$138$	0	0
$139$	−2624.87	−1.60171	−0.800857	−	0.598855i	$-0.795623\pi$
$139$	−2624.87	−1.60171	−0.800857	+	0.598855i	$0.795623\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4522.23	−2.64453
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2782.87	−1.53008	−0.765038	−	0.643985i	$-0.777280\pi$
$149$	−2782.87	−1.53008	−0.765038	+	0.643985i	$0.777280\pi$
$150$	0	0
$151$	1227.25	0.661407	0.330704	−	0.943735i	$-0.392714\pi$
$151$	1227.25	0.661407	0.330704	+	0.943735i	$0.392714\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3187.33	−1.62023	−0.810117	−	0.586268i	$-0.800596\pi$
$157$	−3187.33	−1.62023	−0.810117	+	0.586268i	$0.800596\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	19.2684	0.00943204
$162$	0	0
$163$	2481.64	1.19250	0.596249	−	0.802800i	$-0.296657\pi$
$163$	2481.64	1.19250	0.596249	+	0.802800i	$0.296657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1904.55	0.882508	0.441254	−	0.897382i	$-0.354534\pi$
$167$	1904.55	0.882508	0.441254	+	0.897382i	$0.354534\pi$
$168$	0	0
$169$	4652.42	2.11762
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3788.88	1.66511	0.832553	−	0.553946i	$-0.186879\pi$
$173$	3788.88	1.66511	0.832553	+	0.553946i	$0.186879\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1390.63	0.580672	0.290336	−	0.956925i	$-0.406233\pi$
$179$	1390.63	0.580672	0.290336	+	0.956925i	$0.406233\pi$
$180$	0	0
$181$	2512.55	1.03180	0.515902	−	0.856647i	$-0.327457\pi$
$181$	2512.55	1.03180	0.515902	+	0.856647i	$0.327457\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5486.21	−2.14541
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2872.82	−1.08833	−0.544163	−	0.838980i	$-0.683153\pi$
$191$	−2872.82	−1.08833	−0.544163	+	0.838980i	$0.683153\pi$
$192$	0	0
$193$	4130.79	1.54063	0.770314	−	0.637665i	$-0.220100\pi$
$193$	4130.79	1.54063	0.770314	+	0.637665i	$0.220100\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2243.52	−0.811393	−0.405697	−	0.914008i	$-0.632971\pi$
$197$	−2243.52	−0.811393	−0.405697	+	0.914008i	$0.632971\pi$
$198$	0	0
$199$	1111.28	0.395864	0.197932	−	0.980216i	$-0.436578\pi$
$199$	1111.28	0.395864	0.197932	+	0.980216i	$0.436578\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2168.79	0.749849
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4596.44	1.52125
$210$	0	0
$211$	−4546.63	−1.48343	−0.741713	−	0.670717i	$-0.765986\pi$
$211$	−4546.63	−1.48343	−0.741713	+	0.670717i	$0.765986\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1726.94	−0.540241
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8309.48	2.52921
$222$	0	0
$223$	3012.93	0.904755	0.452378	−	0.891826i	$-0.350576\pi$
$223$	3012.93	0.904755	0.452378	+	0.891826i	$0.350576\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1515.27	−0.443049	−0.221524	−	0.975155i	$-0.571103\pi$
$227$	−1515.27	−0.443049	−0.221524	+	0.975155i	$0.571103\pi$
$228$	0	0
$229$	2539.98	0.732956	0.366478	−	0.930427i	$-0.380564\pi$
$229$	2539.98	0.732956	0.366478	+	0.930427i	$0.380564\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4772.91	1.34199	0.670996	−	0.741461i	$-0.265867\pi$
$233$	4772.91	1.34199	0.670996	+	0.741461i	$0.265867\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−244.344	−0.0661309	−0.0330655	−	0.999453i	$-0.510527\pi$
$239$	−244.344	−0.0661309	−0.0330655	+	0.999453i	$0.510527\pi$
$240$	0	0
$241$	5573.18	1.48963	0.744813	−	0.667273i	$-0.232538\pi$
$241$	5573.18	1.48963	0.744813	+	0.667273i	$0.232538\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6961.82	−1.79340
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5786.78	1.45521	0.727606	−	0.685995i	$-0.240633\pi$
$251$	5786.78	1.45521	0.727606	+	0.685995i	$0.240633\pi$
$252$	0	0
$253$	−48.1183	−0.0119572
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7780.75	1.88852	0.944260	−	0.329200i	$-0.106779\pi$
$257$	7780.75	1.88852	0.944260	+	0.329200i	$0.106779\pi$
$258$	0	0
$259$	8551.08	2.05150
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5122.61	1.20104	0.600521	−	0.799609i	$-0.294960\pi$
$263$	5122.61	1.20104	0.600521	+	0.799609i	$0.294960\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7125.12	−1.61497	−0.807484	−	0.589890i	$-0.799171\pi$
$269$	−7125.12	−1.61497	−0.807484	+	0.589890i	$0.799171\pi$
$270$	0	0
$271$	1761.58	0.394865	0.197433	−	0.980316i	$-0.436740\pi$
$271$	1761.58	0.394865	0.197433	+	0.980316i	$0.436740\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5810.41	1.26034	0.630169	−	0.776458i	$-0.282986\pi$
$277$	5810.41	1.26034	0.630169	+	0.776458i	$0.282986\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4126.19	0.875972	0.437986	−	0.898982i	$-0.355692\pi$
$281$	4126.19	0.875972	0.437986	+	0.898982i	$0.355692\pi$
$282$	0	0
$283$	718.330	0.150884	0.0754422	−	0.997150i	$-0.475963\pi$
$283$	718.330	0.150884	0.0754422	+	0.997150i	$0.475963\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2289.63	−0.470914
$288$	0	0
$289$	5167.78	1.05186
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3756.21	0.748942	0.374471	−	0.927239i	$-0.377824\pi$
$293$	3756.21	0.748942	0.374471	+	0.927239i	$0.377824\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	72.8806	0.0140963
$300$	0	0
$301$	5282.05	1.01147
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7599.52	−1.41279	−0.706397	−	0.707816i	$-0.749681\pi$
$307$	−7599.52	−1.41279	−0.706397	+	0.707816i	$0.749681\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2850.03	−0.519648	−0.259824	−	0.965656i	$-0.583664\pi$
$311$	−2850.03	−0.519648	−0.259824	+	0.965656i	$0.583664\pi$
$312$	0	0
$313$	799.684	0.144411	0.0722057	−	0.997390i	$-0.476996\pi$
$313$	799.684	0.144411	0.0722057	+	0.997390i	$0.476996\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1003.79	0.177850	0.0889250	−	0.996038i	$-0.471657\pi$
$317$	1003.79	0.177850	0.0889250	+	0.996038i	$0.471657\pi$
$318$	0	0
$319$	−5416.07	−0.950600
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8445.84	−1.45492
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11215.3	−1.87939
$330$	0	0
$331$	8367.68	1.38951	0.694757	−	0.719245i	$-0.255512\pi$
$331$	8367.68	1.38951	0.694757	+	0.719245i	$0.255512\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5034.79	−0.813835	−0.406918	−	0.913465i	$-0.633396\pi$
$337$	−5034.79	−0.813835	−0.406918	+	0.913465i	$0.633396\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4312.64	0.684875
$342$	0	0
$343$	−4534.51	−0.713821
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2105.53	0.325737	0.162868	−	0.986648i	$-0.447925\pi$
$347$	2105.53	0.325737	0.162868	+	0.986648i	$0.447925\pi$
$348$	0	0
$349$	−8813.47	−1.35179	−0.675894	−	0.736999i	$-0.736242\pi$
$349$	−8813.47	−1.35179	−0.675894	+	0.736999i	$0.736242\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9348.98	−1.40962	−0.704810	−	0.709396i	$-0.748968\pi$
$353$	−9348.98	−1.40962	−0.704810	+	0.709396i	$0.748968\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9233.33	1.35743	0.678714	−	0.734403i	$-0.262538\pi$
$359$	9233.33	1.35743	0.678714	+	0.734403i	$0.262538\pi$
$360$	0	0
$361$	217.071	0.0316477
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6996.60	−0.995148	−0.497574	−	0.867421i	$-0.665776\pi$
$367$	−6996.60	−0.995148	−0.497574	+	0.867421i	$0.665776\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6228.14	−0.871560
$372$	0	0
$373$	2945.48	0.408877	0.204438	−	0.978879i	$-0.434463\pi$
$373$	2945.48	0.408877	0.204438	+	0.978879i	$0.434463\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8203.24	1.12066
$378$	0	0
$379$	8499.83	1.15200	0.575998	−	0.817451i	$-0.304613\pi$
$379$	8499.83	1.15200	0.575998	+	0.817451i	$0.304613\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6426.01	0.857320	0.428660	−	0.903466i	$-0.358986\pi$
$383$	6426.01	0.857320	0.428660	+	0.903466i	$0.358986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	675.805	0.0880840	0.0440420	−	0.999030i	$-0.485976\pi$
$389$	675.805	0.0880840	0.0440420	+	0.999030i	$0.485976\pi$
$390$	0	0
$391$	88.4162	0.0114358
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9331.43	−1.17967	−0.589837	−	0.807522i	$-0.700808\pi$
$397$	−9331.43	−1.17967	−0.589837	+	0.807522i	$0.700808\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	427.213	0.0532021	0.0266010	−	0.999646i	$-0.491532\pi$
$401$	427.213	0.0532021	0.0266010	+	0.999646i	$0.491532\pi$
$402$	0	0
$403$	−6531.97	−0.807396
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21354.4	−2.60073
$408$	0	0
$409$	1243.18	0.150297	0.0751484	−	0.997172i	$-0.476057\pi$
$409$	1243.18	0.150297	0.0751484	+	0.997172i	$0.476057\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15514.3	1.84845
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3802.17	0.443313	0.221656	−	0.975125i	$-0.428854\pi$
$419$	3802.17	0.443313	0.221656	+	0.975125i	$0.428854\pi$
$420$	0	0
$421$	−12785.6	−1.48012	−0.740059	−	0.672542i	$-0.765203\pi$
$421$	−12785.6	−1.48012	−0.740059	+	0.672542i	$0.765203\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10295.3	1.16680
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9588.11	1.07156	0.535781	−	0.844357i	$-0.320017\pi$
$431$	9588.11	1.07156	0.535781	+	0.844357i	$0.320017\pi$
$432$	0	0
$433$	8493.83	0.942697	0.471348	−	0.881947i	$-0.343767\pi$
$433$	8493.83	0.942697	0.471348	+	0.881947i	$0.343767\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−74.0766	−0.00810885
$438$	0	0
$439$	5167.40	0.561792	0.280896	−	0.959738i	$-0.409368\pi$
$439$	5167.40	0.561792	0.280896	+	0.959738i	$0.409368\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8192.86	−0.878679	−0.439339	−	0.898321i	$-0.644787\pi$
$443$	−8192.86	−0.878679	−0.439339	+	0.898321i	$0.644787\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18252.3	1.91844	0.959218	−	0.282666i	$-0.0912188\pi$
$449$	18252.3	1.91844	0.959218	+	0.282666i	$0.0912188\pi$
$450$	0	0
$451$	5717.82	0.596988
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−330.123	−0.0337910	−0.0168955	−	0.999857i	$-0.505378\pi$
$457$	−330.123	−0.0337910	−0.0168955	+	0.999857i	$0.505378\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4773.18	0.482232	0.241116	−	0.970496i	$-0.422487\pi$
$461$	4773.18	0.482232	0.241116	+	0.970496i	$0.422487\pi$
$462$	0	0
$463$	7860.46	0.788999	0.394499	−	0.918896i	$-0.370918\pi$
$463$	7860.46	0.788999	0.394499	+	0.918896i	$0.370918\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10520.8	−1.04250	−0.521249	−	0.853404i	$-0.674534\pi$
$467$	−10520.8	−1.04250	−0.521249	+	0.853404i	$0.674534\pi$
$468$	0	0
$469$	−14607.1	−1.43815
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−13190.7	−1.28226
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3013.64	0.287467	0.143734	−	0.989616i	$-0.454089\pi$
$479$	3013.64	0.287467	0.143734	+	0.989616i	$0.454089\pi$
$480$	0	0
$481$	32343.6	3.06599
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7048.81	0.655876	0.327938	−	0.944699i	$-0.393646\pi$
$487$	7048.81	0.655876	0.327938	+	0.944699i	$0.393646\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6338.68	−0.582608	−0.291304	−	0.956630i	$-0.594089\pi$
$491$	−6338.68	−0.582608	−0.291304	+	0.956630i	$0.594089\pi$
$492$	0	0
$493$	9951.89	0.909149
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1127.68	0.101777
$498$	0	0
$499$	−5402.66	−0.484682	−0.242341	−	0.970191i	$-0.577915\pi$
$499$	−5402.66	−0.484682	−0.242341	+	0.970191i	$0.577915\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7770.85	0.688837	0.344419	−	0.938816i	$-0.388076\pi$
$503$	7770.85	0.688837	0.344419	+	0.938816i	$0.388076\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−729.019	−0.0634837	−0.0317418	−	0.999496i	$-0.510105\pi$
$509$	−729.019	−0.0634837	−0.0317418	+	0.999496i	$0.510105\pi$
$510$	0	0
$511$	8123.91	0.703289
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	28007.6	2.38254
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14783.8	−1.24317	−0.621585	−	0.783346i	$-0.713511\pi$
$521$	−14783.8	−1.24317	−0.621585	+	0.783346i	$0.713511\pi$
$522$	0	0
$523$	395.434	0.0330614	0.0165307	−	0.999863i	$-0.494738\pi$
$523$	395.434	0.0330614	0.0165307	+	0.999863i	$0.494738\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7924.36	−0.655011
$528$	0	0
$529$	−12166.2	−0.999936
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8660.29	−0.703787
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7418.24	−0.592813
$540$	0	0
$541$	−16507.9	−1.31188	−0.655942	−	0.754811i	$-0.727728\pi$
$541$	−16507.9	−1.31188	−0.655942	+	0.754811i	$0.727728\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9242.35	0.722440	0.361220	−	0.932481i	$-0.382360\pi$
$547$	9242.35	0.722440	0.361220	+	0.932481i	$0.382360\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8337.86	−0.644655
$552$	0	0
$553$	−1736.07	−0.133499
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10449.5	0.794898	0.397449	−	0.917624i	$-0.369896\pi$
$557$	10449.5	0.794898	0.397449	+	0.917624i	$0.369896\pi$
$558$	0	0
$559$	19978.8	1.51165
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13670.1	1.02331	0.511656	−	0.859191i	$-0.329032\pi$
$563$	13670.1	1.02331	0.511656	+	0.859191i	$0.329032\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13616.7	−1.00323	−0.501617	−	0.865090i	$-0.667261\pi$
$569$	−13616.7	−1.00323	−0.501617	+	0.865090i	$0.667261\pi$
$570$	0	0
$571$	−11833.4	−0.867270	−0.433635	−	0.901089i	$-0.642769\pi$
$571$	−11833.4	−0.867270	−0.433635	+	0.901089i	$0.642769\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7131.30	0.514523	0.257262	−	0.966342i	$-0.417180\pi$
$577$	7131.30	0.514523	0.257262	+	0.966342i	$0.417180\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14926.5	1.06584
$582$	0	0
$583$	15553.4	1.10490
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25214.3	−1.77292	−0.886461	−	0.462804i	$-0.846843\pi$
$587$	−25214.3	−1.77292	−0.886461	+	0.462804i	$0.846843\pi$
$588$	0	0
$589$	6639.17	0.464452
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4218.04	−0.292098	−0.146049	−	0.989277i	$-0.546656\pi$
$593$	−4218.04	−0.292098	−0.146049	+	0.989277i	$0.546656\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5956.57	−0.406308	−0.203154	−	0.979147i	$-0.565119\pi$
$599$	−5956.57	−0.406308	−0.203154	+	0.979147i	$0.565119\pi$
$600$	0	0
$601$	−5182.29	−0.351731	−0.175865	−	0.984414i	$-0.556272\pi$
$601$	−5182.29	−0.351731	−0.175865	+	0.984414i	$0.556272\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10142.0	0.678171	0.339086	−	0.940755i	$-0.389882\pi$
$607$	10142.0	0.678171	0.339086	+	0.940755i	$0.389882\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−42420.7	−2.80877
$612$	0	0
$613$	−4022.95	−0.265066	−0.132533	−	0.991179i	$-0.542311\pi$
$613$	−4022.95	−0.265066	−0.132533	+	0.991179i	$0.542311\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27074.4	1.76657	0.883285	−	0.468837i	$-0.155327\pi$
$617$	27074.4	1.76657	0.883285	+	0.468837i	$0.155327\pi$
$618$	0	0
$619$	−17245.7	−1.11981	−0.559905	−	0.828557i	$-0.689162\pi$
$619$	−17245.7	−1.11981	−0.559905	+	0.828557i	$0.689162\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13751.5	−0.884336
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	39238.1	2.48732
$630$	0	0
$631$	−14254.4	−0.899302	−0.449651	−	0.893204i	$-0.648452\pi$
$631$	−14254.4	−0.899302	−0.449651	+	0.893204i	$0.648452\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11235.8	0.698865
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2611.61	−0.160924	−0.0804621	−	0.996758i	$-0.525640\pi$
$641$	−2611.61	−0.160924	−0.0804621	+	0.996758i	$0.525640\pi$
$642$	0	0
$643$	−27414.4	−1.68136	−0.840682	−	0.541529i	$-0.817846\pi$
$643$	−27414.4	−1.68136	−0.840682	+	0.541529i	$0.817846\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12820.8	−0.779041	−0.389521	−	0.921018i	$-0.627359\pi$
$647$	−12820.8	−0.779041	−0.389521	+	0.921018i	$0.627359\pi$
$648$	0	0
$649$	−38743.5	−2.34332
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15786.4	0.946049	0.473025	−	0.881049i	$-0.343162\pi$
$653$	15786.4	0.946049	0.473025	+	0.881049i	$0.343162\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4822.61	−0.285072	−0.142536	−	0.989790i	$-0.545526\pi$
$659$	−4822.61	−0.285072	−0.142536	+	0.989790i	$0.545526\pi$
$660$	0	0
$661$	−23624.9	−1.39017	−0.695084	−	0.718929i	$-0.744633\pi$
$661$	−23624.9	−1.39017	−0.695084	+	0.718929i	$0.744633\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	87.2858	0.00506705
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−25710.2	−1.47918
$672$	0	0
$673$	−8903.65	−0.509971	−0.254985	−	0.966945i	$-0.582071\pi$
$673$	−8903.65	−0.509971	−0.254985	+	0.966945i	$0.582071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9034.56	0.512890	0.256445	−	0.966559i	$-0.417449\pi$
$677$	9034.56	0.512890	0.256445	+	0.966559i	$0.417449\pi$
$678$	0	0
$679$	33238.6	1.87862
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28676.8	−1.60657	−0.803284	−	0.595596i	$-0.796916\pi$
$683$	−28676.8	−1.60657	−0.803284	+	0.595596i	$0.796916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23557.3	−1.30256
$690$	0	0
$691$	−20653.1	−1.13702	−0.568509	−	0.822677i	$-0.692480\pi$
$691$	−20653.1	−1.13702	−0.568509	+	0.822677i	$0.692480\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10506.4	−0.570957
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11000.1	0.592678	0.296339	−	0.955083i	$-0.404234\pi$
$701$	11000.1	0.592678	0.296339	+	0.955083i	$0.404234\pi$
$702$	0	0
$703$	−32874.4	−1.76370
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21394.4	−1.13807
$708$	0	0
$709$	−372.560	−0.0197345	−0.00986725	−	0.999951i	$-0.503141\pi$
$709$	−372.560	−0.0197345	−0.00986725	+	0.999951i	$0.503141\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−69.5028	−0.00365063
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7207.32	0.373835	0.186918	−	0.982376i	$-0.440150\pi$
$719$	7207.32	0.373835	0.186918	+	0.982376i	$0.440150\pi$
$720$	0	0
$721$	16611.6	0.858043
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23423.6	−1.19496	−0.597479	−	0.801885i	$-0.703831\pi$
$727$	−23423.6	−1.19496	−0.597479	+	0.801885i	$0.703831\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24237.6	1.22635
$732$	0	0
$733$	−20368.0	−1.02634	−0.513172	−	0.858286i	$-0.671530\pi$
$733$	−20368.0	−1.02634	−0.513172	+	0.858286i	$0.671530\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36477.9	1.82318
$738$	0	0
$739$	2603.64	0.129603	0.0648014	−	0.997898i	$-0.479359\pi$
$739$	2603.64	0.129603	0.0648014	+	0.997898i	$0.479359\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8627.83	−0.426009	−0.213004	−	0.977051i	$-0.568325\pi$
$743$	−8627.83	−0.426009	−0.213004	+	0.977051i	$0.568325\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20687.0	−1.00919
$750$	0	0
$751$	−18735.5	−0.910343	−0.455172	−	0.890404i	$-0.650422\pi$
$751$	−18735.5	−0.910343	−0.455172	+	0.890404i	$0.650422\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10787.3	0.517926	0.258963	−	0.965887i	$-0.416619\pi$
$757$	10787.3	0.517926	0.258963	+	0.965887i	$0.416619\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2175.95	0.103651	0.0518253	−	0.998656i	$-0.483496\pi$
$761$	2175.95	0.103651	0.0518253	+	0.998656i	$0.483496\pi$
$762$	0	0
$763$	−15060.7	−0.714594
$764$	0	0
$765$	0	0
$766$	0	0
$767$	58681.4	2.76253
$768$	0	0
$769$	18705.8	0.877176	0.438588	−	0.898688i	$-0.355479\pi$
$769$	18705.8	0.877176	0.438588	+	0.898688i	$0.355479\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8243.60	−0.383573	−0.191786	−	0.981437i	$-0.561428\pi$
$773$	−8243.60	−0.383573	−0.191786	+	0.981437i	$0.561428\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8802.41	0.404851
$780$	0	0
$781$	−2816.12	−0.129025
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−29442.1	−1.33354	−0.666770	−	0.745263i	$-0.732324\pi$
$787$	−29442.1	−1.33354	−0.666770	+	0.745263i	$0.732324\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7677.15	0.345092
$792$	0	0
$793$	38941.0	1.74380
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42045.4	−1.86866	−0.934332	−	0.356403i	$-0.884003\pi$
$797$	−42045.4	−1.86866	−0.934332	+	0.356403i	$0.884003\pi$
$798$	0	0
$799$	−51463.3	−2.27865
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20287.6	−0.891575
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−43757.6	−1.90165	−0.950825	−	0.309727i	$-0.899762\pi$
$809$	−43757.6	−1.90165	−0.950825	+	0.309727i	$0.899762\pi$
$810$	0	0
$811$	3273.00	0.141715	0.0708574	−	0.997486i	$-0.477426\pi$
$811$	3273.00	0.141715	0.0708574	+	0.997486i	$0.477426\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20306.7	−0.869572
$818$	0	0
$819$	0	0
$820$	0	0
$821$	40442.6	1.71919	0.859595	−	0.510976i	$-0.170716\pi$
$821$	40442.6	1.71919	0.859595	+	0.510976i	$0.170716\pi$
$822$	0	0
$823$	−27236.8	−1.15360	−0.576802	−	0.816884i	$-0.695700\pi$
$823$	−27236.8	−1.15360	−0.576802	+	0.816884i	$0.695700\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33344.5	1.40206	0.701028	−	0.713133i	$-0.252725\pi$
$827$	33344.5	1.40206	0.701028	+	0.713133i	$0.252725\pi$
$828$	0	0
$829$	−32085.0	−1.34422	−0.672110	−	0.740452i	$-0.734612\pi$
$829$	−32085.0	−1.34422	−0.672110	+	0.740452i	$0.734612\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13630.8	0.566964
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7072.24	0.291014	0.145507	−	0.989357i	$-0.453519\pi$
$839$	7072.24	0.291014	0.145507	+	0.989357i	$0.453519\pi$
$840$	0	0
$841$	−14564.3	−0.597169
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	36206.5	1.46880
$848$	0	0
$849$	0	0
$850$	0	0
$851$	344.149	0.0138628
$852$	0	0
$853$	−10225.0	−0.410431	−0.205215	−	0.978717i	$-0.565789\pi$
$853$	−10225.0	−0.410431	−0.205215	+	0.978717i	$0.565789\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9783.87	−0.389977	−0.194989	−	0.980805i	$-0.562467\pi$
$857$	−9783.87	−0.389977	−0.194989	+	0.980805i	$0.562467\pi$
$858$	0	0
$859$	−22931.8	−0.910853	−0.455427	−	0.890273i	$-0.650513\pi$
$859$	−22931.8	−0.910853	−0.455427	+	0.890273i	$0.650513\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23506.0	0.927175	0.463588	−	0.886051i	$-0.346562\pi$
$863$	23506.0	0.927175	0.463588	+	0.886051i	$0.346562\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4335.44	0.169240
$870$	0	0
$871$	−55250.0	−2.14934
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34672.0	1.33499	0.667497	−	0.744613i	$-0.267366\pi$
$877$	34672.0	1.33499	0.667497	+	0.744613i	$0.267366\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2920.92	0.111701	0.0558504	−	0.998439i	$-0.482213\pi$
$881$	2920.92	0.111701	0.0558504	+	0.998439i	$0.482213\pi$
$882$	0	0
$883$	7123.74	0.271498	0.135749	−	0.990743i	$-0.456656\pi$
$883$	7123.74	0.271498	0.135749	+	0.990743i	$0.456656\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7923.42	0.299935	0.149968	−	0.988691i	$-0.452083\pi$
$887$	7923.42	0.299935	0.149968	+	0.988691i	$0.452083\pi$
$888$	0	0
$889$	34751.0	1.31104
$890$	0	0
$891$	0	0
$892$	0	0
$893$	43116.9	1.61573
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7823.05	−0.290226
$900$	0	0
$901$	−28578.9	−1.05672
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19771.7	0.723825	0.361912	−	0.932212i	$-0.382124\pi$
$907$	19771.7	0.723825	0.361912	+	0.932212i	$0.382124\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38222.7	−1.39009	−0.695046	−	0.718966i	$-0.744616\pi$
$911$	−38222.7	−1.39009	−0.695046	+	0.718966i	$0.744616\pi$
$912$	0	0
$913$	−37275.5	−1.35119
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47174.9	1.69886
$918$	0	0
$919$	39237.3	1.40840	0.704199	−	0.710003i	$-0.251306\pi$
$919$	39237.3	1.40840	0.704199	+	0.710003i	$0.251306\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4265.33	0.152107
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20024.5	−0.707194	−0.353597	−	0.935398i	$-0.615042\pi$
$929$	−20024.5	−0.707194	−0.353597	+	0.935398i	$0.615042\pi$
$930$	0	0
$931$	−11420.2	−0.402020
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−8717.10	−0.303922	−0.151961	−	0.988386i	$-0.548559\pi$
$937$	−8717.10	−0.303922	−0.151961	+	0.988386i	$0.548559\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39276.2	−1.36065	−0.680323	−	0.732913i	$-0.738160\pi$
$941$	−39276.2	−1.36065	−0.680323	+	0.732913i	$0.738160\pi$
$942$	0	0
$943$	−92.1490	−0.00318217
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20080.8	0.689060	0.344530	−	0.938775i	$-0.388038\pi$
$947$	20080.8	0.689060	0.344530	+	0.938775i	$0.388038\pi$
$948$	0	0
$949$	30727.9	1.05107
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38549.9	−1.31034	−0.655170	−	0.755481i	$-0.727403\pi$
$953$	−38549.9	−1.31034	−0.655170	+	0.755481i	$0.727403\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6929.95	−0.233347
$960$	0	0
$961$	−23561.8	−0.790902
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−24108.2	−0.801724	−0.400862	−	0.916138i	$-0.631289\pi$
$967$	−24108.2	−0.801724	−0.400862	+	0.916138i	$0.631289\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41760.1	1.38017	0.690084	−	0.723729i	$-0.257573\pi$
$971$	41760.1	1.38017	0.690084	+	0.723729i	$0.257573\pi$
$972$	0	0
$973$	−57433.7	−1.89233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4532.52	−0.148422	−0.0742109	−	0.997243i	$-0.523644\pi$
$977$	−4532.52	−0.148422	−0.0742109	+	0.997243i	$0.523644\pi$
$978$	0	0
$979$	34341.2	1.12109
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48822.8	1.58414	0.792068	−	0.610432i	$-0.209004\pi$
$983$	48822.8	1.58414	0.792068	+	0.610432i	$0.209004\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	212.583	0.00683492
$990$	0	0
$991$	−26937.6	−0.863473	−0.431737	−	0.902000i	$-0.642099\pi$
$991$	−26937.6	−0.863473	−0.431737	+	0.902000i	$0.642099\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9191.03	−0.291959	−0.145979	−	0.989288i	$-0.546633\pi$
$997$	−9191.03	−0.291959	−0.145979	+	0.989288i	$0.546633\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.bo.1.2		2
3.2	odd	2		600.4.a.s.1.2	✓	2
5.2	odd	4		1800.4.f.z.649.4		4
5.3	odd	4		1800.4.f.z.649.1		4
5.4	even	2		1800.4.a.bm.1.1		2
12.11	even	2		1200.4.a.br.1.1		2
15.2	even	4		600.4.f.j.49.4		4
15.8	even	4		600.4.f.j.49.1		4
15.14	odd	2		600.4.a.u.1.1	yes	2
60.23	odd	4		1200.4.f.x.49.4		4
60.47	odd	4		1200.4.f.x.49.1		4
60.59	even	2		1200.4.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.s.1.2	✓	2	3.2	odd	2
600.4.a.u.1.1	yes	2	15.14	odd	2
600.4.f.j.49.1		4	15.8	even	4
600.4.f.j.49.4		4	15.2	even	4
1200.4.a.bp.1.2		2	60.59	even	2
1200.4.a.br.1.1		2	12.11	even	2
1200.4.f.x.49.1		4	60.47	odd	4
1200.4.f.x.49.4		4	60.23	odd	4
1800.4.a.bm.1.1		2	5.4	even	2
1800.4.a.bo.1.2		2	1.1	even	1	trivial
1800.4.f.z.649.1		4	5.3	odd	4
1800.4.f.z.649.4		4	5.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

\(f(q)\)	\(=\)	\(q+21.8806 q^{7} -54.6418 q^{11} +82.7612 q^{13} +100.403 q^{17} -84.1194 q^{19} +0.880613 q^{23} +99.1194 q^{29} -78.9255 q^{31} +390.806 q^{37} -104.642 q^{41} +241.403 q^{43} -512.567 q^{47} +135.761 q^{49} -284.642 q^{53} +709.045 q^{59} +470.522 q^{61} -667.583 q^{67} +51.5378 q^{71} +371.284 q^{73} -1195.60 q^{77} -79.3428 q^{79} +682.180 q^{83} -628.478 q^{89} +1810.87 q^{91} +1519.09 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} + 16 q^{11} + 82 q^{13} - 8 q^{17} - 210 q^{19} - 40 q^{23} + 240 q^{29} + 218 q^{31} + 364 q^{37} - 84 q^{41} + 274 q^{43} - 524 q^{47} + 188 q^{49} - 444 q^{53} + 1084 q^{59} + 774 q^{61}+ \cdots + 2370 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 16 * q^11 + 82 * q^13 - 8 * q^17 - 210 * q^19 - 40 * q^23 + 240 * q^29 + 218 * q^31 + 364 * q^37 - 84 * q^41 + 274 * q^43 - 524 * q^47 + 188 * q^49 - 444 * q^53 + 1084 * q^59 + 774 * q^61 + 210 * q^67 - 1108 * q^71 + 492 * q^73 - 2600 * q^77 - 1328 * q^79 + 28 * q^83 - 1424 * q^89 + 1826 * q^91 + 2370 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more