LMFDB - Embedded newform 900.6.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,6,Mod(1,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("900.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	900.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:	$144.345437832$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.535753.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 186x - 432 $ x^3 - x^2 - 186*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$15.1546$ of defining polynomial
Character	$\chi$	$=$	900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+121.510 q^{7} +O(q^{10})$	$q+121.510 q^{7} +699.623 q^{11} +436.743 q^{13} +1610.09 q^{17} +2778.55 q^{19} +1469.98 q^{23} -2233.05 q^{29} +4632.47 q^{31} +5109.62 q^{37} -7683.61 q^{41} -20417.9 q^{43} +21018.3 q^{47} -2042.21 q^{49} +6033.60 q^{53} +27762.7 q^{59} -48757.3 q^{61} -11289.5 q^{67} +15381.5 q^{71} -24443.1 q^{73} +85011.4 q^{77} +23718.9 q^{79} -18970.7 q^{83} -57532.2 q^{89} +53068.8 q^{91} -173854. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 88 q^{7}+O(q^{10}) $ 3 * q - 88 * q^7	$ 3 q - 88 q^{7} - 148 q^{11} + 868 q^{17} + 3000 q^{19} - 1052 q^{23} - 7962 q^{29} + 132 q^{31} - 11944 q^{37} - 8170 q^{41} - 26188 q^{43} + 37892 q^{47} + 13827 q^{49} + 11076 q^{53} + 46228 q^{59} + 3126 q^{61} - 126332 q^{67} + 80400 q^{71} - 42904 q^{73} + 234608 q^{77} - 64476 q^{79} + 126268 q^{83} + 38030 q^{89} - 49200 q^{91} - 331016 q^{97}+O(q^{100}) $ 3 * q - 88 * q^7 - 148 * q^11 + 868 * q^17 + 3000 * q^19 - 1052 * q^23 - 7962 * q^29 + 132 * q^31 - 11944 * q^37 - 8170 * q^41 - 26188 * q^43 + 37892 * q^47 + 13827 * q^49 + 11076 * q^53 + 46228 * q^59 + 3126 * q^61 - 126332 * q^67 + 80400 * q^71 - 42904 * q^73 + 234608 * q^77 - 64476 * q^79 + 126268 * q^83 + 38030 * q^89 - 49200 * q^91 - 331016 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	121.510	0.937278	0.468639	−	0.883390i	$-0.344744\pi$
$7$	121.510	0.937278	0.468639	+	0.883390i	$0.344744\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	699.623	1.74334	0.871670	−	0.490093i	$-0.163037\pi$
$11$	699.623	1.74334	0.871670	+	0.490093i	$0.163037\pi$
$12$	0	0
$13$	436.743	0.716749	0.358375	−	0.933578i	$-0.383331\pi$
$13$	436.743	0.716749	0.358375	+	0.933578i	$0.383331\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1610.09	1.35122	0.675611	−	0.737258i	$-0.263880\pi$
$17$	1610.09	1.35122	0.675611	+	0.737258i	$0.263880\pi$
$18$	0	0
$19$	2778.55	1.76577	0.882884	−	0.469590i	$-0.155598\pi$
$19$	2778.55	1.76577	0.882884	+	0.469590i	$0.155598\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1469.98	0.579420	0.289710	−	0.957115i	$-0.406441\pi$
$23$	1469.98	0.579420	0.289710	+	0.957115i	$0.406441\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2233.05	−0.493063	−0.246532	−	0.969135i	$-0.579291\pi$
$29$	−2233.05	−0.493063	−0.246532	+	0.969135i	$0.579291\pi$
$30$	0	0
$31$	4632.47	0.865781	0.432891	−	0.901446i	$-0.357494\pi$
$31$	4632.47	0.865781	0.432891	+	0.901446i	$0.357494\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5109.62	0.613598	0.306799	−	0.951774i	$-0.400742\pi$
$37$	5109.62	0.613598	0.306799	+	0.951774i	$0.400742\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7683.61	−0.713848	−0.356924	−	0.934133i	$-0.616174\pi$
$41$	−7683.61	−0.713848	−0.356924	+	0.934133i	$0.616174\pi$
$42$	0	0
$43$	−20417.9	−1.68399	−0.841996	−	0.539484i	$-0.818619\pi$
$43$	−20417.9	−1.68399	−0.841996	+	0.539484i	$0.818619\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	21018.3	1.38788	0.693942	−	0.720030i	$-0.255872\pi$
$47$	21018.3	1.38788	0.693942	+	0.720030i	$0.255872\pi$
$48$	0	0
$49$	−2042.21	−0.121510
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6033.60	0.295044	0.147522	−	0.989059i	$-0.452870\pi$
$53$	6033.60	0.295044	0.147522	+	0.989059i	$0.452870\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	27762.7	1.03832	0.519160	−	0.854677i	$-0.326245\pi$
$59$	27762.7	1.03832	0.519160	+	0.854677i	$0.326245\pi$
$60$	0	0
$61$	−48757.3	−1.67770	−0.838852	−	0.544359i	$-0.816773\pi$
$61$	−48757.3	−1.67770	−0.838852	+	0.544359i	$0.816773\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11289.5	−0.307246	−0.153623	−	0.988130i	$-0.549094\pi$
$67$	−11289.5	−0.307246	−0.153623	+	0.988130i	$0.549094\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15381.5	0.362121	0.181061	−	0.983472i	$-0.442047\pi$
$71$	15381.5	0.362121	0.181061	+	0.983472i	$0.442047\pi$
$72$	0	0
$73$	−24443.1	−0.536845	−0.268422	−	0.963301i	$-0.586502\pi$
$73$	−24443.1	−0.536845	−0.268422	+	0.963301i	$0.586502\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	85011.4	1.63399
$78$	0	0
$79$	23718.9	0.427589	0.213794	−	0.976879i	$-0.431418\pi$
$79$	23718.9	0.427589	0.213794	+	0.976879i	$0.431418\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−18970.7	−0.302266	−0.151133	−	0.988513i	$-0.548292\pi$
$83$	−18970.7	−0.302266	−0.151133	+	0.988513i	$0.548292\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−57532.2	−0.769902	−0.384951	−	0.922937i	$-0.625782\pi$
$89$	−57532.2	−0.769902	−0.384951	+	0.922937i	$0.625782\pi$
$90$	0	0
$91$	53068.8	0.671793
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−173854.	−1.87610	−0.938050	−	0.346499i	$-0.887371\pi$
$97$	−173854.	−1.87610	−0.938050	+	0.346499i	$0.887371\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	77773.0	0.758622	0.379311	−	0.925269i	$-0.376161\pi$
$101$	77773.0	0.758622	0.379311	+	0.925269i	$0.376161\pi$
$102$	0	0
$103$	−123836.	−1.15015	−0.575073	−	0.818102i	$-0.695026\pi$
$103$	−123836.	−1.15015	−0.575073	+	0.818102i	$0.695026\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	92949.4	0.784851	0.392426	−	0.919784i	$-0.371636\pi$
$107$	92949.4	0.784851	0.392426	+	0.919784i	$0.371636\pi$
$108$	0	0
$109$	−53772.7	−0.433506	−0.216753	−	0.976226i	$-0.569547\pi$
$109$	−53772.7	−0.433506	−0.216753	+	0.976226i	$0.569547\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	37540.9	0.276572	0.138286	−	0.990392i	$-0.455841\pi$
$113$	37540.9	0.276572	0.138286	+	0.990392i	$0.455841\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	195642.	1.26647
$120$	0	0
$121$	328421.	2.03923
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−154211.	−0.848409	−0.424204	−	0.905566i	$-0.639446\pi$
$127$	−154211.	−0.848409	−0.424204	+	0.905566i	$0.639446\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	136783.	0.696391	0.348196	−	0.937422i	$-0.386794\pi$
$131$	136783.	0.696391	0.348196	+	0.937422i	$0.386794\pi$
$132$	0	0
$133$	337623.	1.65502
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−303547.	−1.38173	−0.690867	−	0.722982i	$-0.742771\pi$
$137$	−303547.	−1.38173	−0.690867	+	0.722982i	$0.742771\pi$
$138$	0	0
$139$	139766.	0.613572	0.306786	−	0.951779i	$-0.400746\pi$
$139$	139766.	0.613572	0.306786	+	0.951779i	$0.400746\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	305555.	1.24954
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−160751.	−0.593184	−0.296592	−	0.955004i	$-0.595850\pi$
$149$	−160751.	−0.593184	−0.296592	+	0.955004i	$0.595850\pi$
$150$	0	0
$151$	−274592.	−0.980045	−0.490022	−	0.871710i	$-0.663011\pi$
$151$	−274592.	−0.980045	−0.490022	+	0.871710i	$0.663011\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−333596.	−1.08012	−0.540059	−	0.841627i	$-0.681598\pi$
$157$	−333596.	−1.08012	−0.540059	+	0.841627i	$0.681598\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	178618.	0.543077
$162$	0	0
$163$	159791.	0.471067	0.235533	−	0.971866i	$-0.424316\pi$
$163$	159791.	0.471067	0.235533	+	0.971866i	$0.424316\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	549868.	1.52569	0.762847	−	0.646579i	$-0.223801\pi$
$167$	549868.	1.52569	0.762847	+	0.646579i	$0.223801\pi$
$168$	0	0
$169$	−180549.	−0.486271
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−85278.5	−0.216633	−0.108316	−	0.994116i	$-0.534546\pi$
$173$	−85278.5	−0.216633	−0.108316	+	0.994116i	$0.534546\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−378124.	−0.882066	−0.441033	−	0.897491i	$-0.645388\pi$
$179$	−378124.	−0.882066	−0.441033	+	0.897491i	$0.645388\pi$
$180$	0	0
$181$	74902.5	0.169942	0.0849708	−	0.996383i	$-0.472920\pi$
$181$	74902.5	0.169942	0.0849708	+	0.996383i	$0.472920\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.12645e6	2.35564
$188$	0	0
$189$	0	0
$190$	0	0
$191$	810880.	1.60832	0.804161	−	0.594412i	$-0.202615\pi$
$191$	810880.	1.60832	0.804161	+	0.594412i	$0.202615\pi$
$192$	0	0
$193$	120513.	0.232884	0.116442	−	0.993197i	$-0.462851\pi$
$193$	120513.	0.232884	0.116442	+	0.993197i	$0.462851\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.04311e6	−1.91499	−0.957493	−	0.288456i	$-0.906858\pi$
$197$	−1.04311e6	−1.91499	−0.957493	+	0.288456i	$0.906858\pi$
$198$	0	0
$199$	87715.4	0.157016	0.0785079	−	0.996913i	$-0.474984\pi$
$199$	87715.4	0.157016	0.0785079	+	0.996913i	$0.474984\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−271338.	−0.462137
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.94393e6	3.07834
$210$	0	0
$211$	530246.	0.819919	0.409959	−	0.912104i	$-0.365543\pi$
$211$	530246.	0.819919	0.409959	+	0.912104i	$0.365543\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	562893.	0.811478
$218$	0	0
$219$	0	0
$220$	0	0
$221$	703193.	0.968488
$222$	0	0
$223$	−1.27008e6	−1.71028	−0.855140	−	0.518397i	$-0.826529\pi$
$223$	−1.27008e6	−1.71028	−0.855140	+	0.518397i	$0.826529\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−127005.	−0.163590	−0.0817948	−	0.996649i	$-0.526065\pi$
$227$	−127005.	−0.163590	−0.0817948	+	0.996649i	$0.526065\pi$
$228$	0	0
$229$	548288.	0.690908	0.345454	−	0.938436i	$-0.387725\pi$
$229$	548288.	0.690908	0.345454	+	0.938436i	$0.387725\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−794504.	−0.958752	−0.479376	−	0.877610i	$-0.659137\pi$
$233$	−794504.	−0.958752	−0.479376	+	0.877610i	$0.659137\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	584107.	0.661451	0.330725	−	0.943727i	$-0.392707\pi$
$239$	584107.	0.661451	0.330725	+	0.943727i	$0.392707\pi$
$240$	0	0
$241$	1.64168e6	1.82073	0.910365	−	0.413805i	$-0.135801\pi$
$241$	1.64168e6	1.82073	0.910365	+	0.413805i	$0.135801\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.21351e6	1.26561
$248$	0	0
$249$	0	0
$250$	0	0
$251$	569849.	0.570920	0.285460	−	0.958391i	$-0.407854\pi$
$251$	569849.	0.570920	0.285460	+	0.958391i	$0.407854\pi$
$252$	0	0
$253$	1.02843e6	1.01013
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−499302.	−0.471553	−0.235776	−	0.971807i	$-0.575763\pi$
$257$	−499302.	−0.471553	−0.235776	+	0.971807i	$0.575763\pi$
$258$	0	0
$259$	620872.	0.575112
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−716870.	−0.639074	−0.319537	−	0.947574i	$-0.603527\pi$
$263$	−716870.	−0.639074	−0.319537	+	0.947574i	$0.603527\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.19836e6	−1.85233	−0.926163	−	0.377124i	$-0.876913\pi$
$269$	−2.19836e6	−1.85233	−0.926163	+	0.377124i	$0.876913\pi$
$270$	0	0
$271$	−253384.	−0.209583	−0.104792	−	0.994494i	$-0.533418\pi$
$271$	−253384.	−0.209583	−0.104792	+	0.994494i	$0.533418\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.15596e6	0.905199	0.452600	−	0.891714i	$-0.350497\pi$
$277$	1.15596e6	0.905199	0.452600	+	0.891714i	$0.350497\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.37453e6	1.79396	0.896979	−	0.442074i	$-0.145757\pi$
$281$	2.37453e6	1.79396	0.896979	+	0.442074i	$0.145757\pi$
$282$	0	0
$283$	−1.56745e6	−1.16340	−0.581698	−	0.813405i	$-0.697611\pi$
$283$	−1.56745e6	−1.16340	−0.581698	+	0.813405i	$0.697611\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−933639.	−0.669074
$288$	0	0
$289$	1.17252e6	0.825803
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.74262e6	−1.18586	−0.592930	−	0.805254i	$-0.702029\pi$
$293$	−1.74262e6	−1.18586	−0.592930	+	0.805254i	$0.702029\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	642005.	0.415298
$300$	0	0
$301$	−2.48099e6	−1.57837
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3.02826e6	−1.83378	−0.916891	−	0.399137i	$-0.869310\pi$
$307$	−3.02826e6	−1.83378	−0.916891	+	0.399137i	$0.869310\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−139928.	−0.0820356	−0.0410178	−	0.999158i	$-0.513060\pi$
$311$	−139928.	−0.0820356	−0.0410178	+	0.999158i	$0.513060\pi$
$312$	0	0
$313$	2.76332e6	1.59430	0.797149	−	0.603782i	$-0.206340\pi$
$313$	2.76332e6	1.59430	0.797149	+	0.603782i	$0.206340\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.81991e6	1.01719	0.508595	−	0.861006i	$-0.330165\pi$
$317$	1.81991e6	1.01719	0.508595	+	0.861006i	$0.330165\pi$
$318$	0	0
$319$	−1.56229e6	−0.859577
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.47370e6	2.38595
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.55395e6	1.30083
$330$	0	0
$331$	2.72290e6	1.36603	0.683017	−	0.730403i	$-0.260668\pi$
$331$	2.72290e6	1.36603	0.683017	+	0.730403i	$0.260668\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.95014e6	−0.935385	−0.467693	−	0.883891i	$-0.654915\pi$
$337$	−1.95014e6	−0.935385	−0.467693	+	0.883891i	$0.654915\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.24098e6	1.50935
$342$	0	0
$343$	−2.29038e6	−1.05117
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.57423e6	0.701852	0.350926	−	0.936403i	$-0.385867\pi$
$347$	1.57423e6	0.701852	0.350926	+	0.936403i	$0.385867\pi$
$348$	0	0
$349$	−723241.	−0.317848	−0.158924	−	0.987291i	$-0.550803\pi$
$349$	−723241.	−0.317848	−0.158924	+	0.987291i	$0.550803\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.94949e6	0.832694	0.416347	−	0.909206i	$-0.363310\pi$
$353$	1.94949e6	0.832694	0.416347	+	0.909206i	$0.363310\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.57967e6	0.646891	0.323445	−	0.946247i	$-0.395159\pi$
$359$	1.57967e6	0.646891	0.323445	+	0.946247i	$0.395159\pi$
$360$	0	0
$361$	5.24423e6	2.11794
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−549351.	−0.212904	−0.106452	−	0.994318i	$-0.533949\pi$
$367$	−549351.	−0.212904	−0.106452	+	0.994318i	$0.533949\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	733145.	0.276538
$372$	0	0
$373$	4.11946e6	1.53309	0.766546	−	0.642189i	$-0.221974\pi$
$373$	4.11946e6	1.53309	0.766546	+	0.642189i	$0.221974\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−975266.	−0.353403
$378$	0	0
$379$	−481024.	−0.172016	−0.0860079	−	0.996294i	$-0.527411\pi$
$379$	−481024.	−0.172016	−0.0860079	+	0.996294i	$0.527411\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−74246.2	−0.0258629	−0.0129314	−	0.999916i	$-0.504116\pi$
$383$	−74246.2	−0.0258629	−0.0129314	+	0.999916i	$0.504116\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	373796.	0.125245	0.0626225	−	0.998037i	$-0.480054\pi$
$389$	373796.	0.125245	0.0626225	+	0.998037i	$0.480054\pi$
$390$	0	0
$391$	2.36680e6	0.782925
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.50136e6	−1.11496	−0.557482	−	0.830189i	$-0.688233\pi$
$397$	−3.50136e6	−1.11496	−0.557482	+	0.830189i	$0.688233\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.45598e6	0.452162	0.226081	−	0.974109i	$-0.427409\pi$
$401$	1.45598e6	0.452162	0.226081	+	0.974109i	$0.427409\pi$
$402$	0	0
$403$	2.02320e6	0.620548
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.57480e6	1.06971
$408$	0	0
$409$	55982.6	0.0165480	0.00827398	−	0.999966i	$-0.497366\pi$
$409$	55982.6	0.0165480	0.00827398	+	0.999966i	$0.497366\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.37345e6	0.973194
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.15797e6	−0.878765	−0.439382	−	0.898300i	$-0.644803\pi$
$419$	−3.15797e6	−0.878765	−0.439382	+	0.898300i	$0.644803\pi$
$420$	0	0
$421$	−4.18745e6	−1.15145	−0.575725	−	0.817644i	$-0.695280\pi$
$421$	−4.18745e6	−1.15145	−0.575725	+	0.817644i	$0.695280\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.92453e6	−1.57248
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.00857e6	1.29873	0.649367	−	0.760475i	$-0.275034\pi$
$431$	5.00857e6	1.29873	0.649367	+	0.760475i	$0.275034\pi$
$432$	0	0
$433$	−6.43666e6	−1.64984	−0.824918	−	0.565253i	$-0.808779\pi$
$433$	−6.43666e6	−1.64984	−0.824918	+	0.565253i	$0.808779\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.08442e6	1.02312
$438$	0	0
$439$	4.41638e6	1.09372	0.546859	−	0.837225i	$-0.315823\pi$
$439$	4.41638e6	1.09372	0.546859	+	0.837225i	$0.315823\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.58430e6	−0.383555	−0.191778	−	0.981438i	$-0.561425\pi$
$443$	−1.58430e6	−0.383555	−0.191778	+	0.981438i	$0.561425\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	654015.	0.153099	0.0765494	−	0.997066i	$-0.475610\pi$
$449$	654015.	0.153099	0.0765494	+	0.997066i	$0.475610\pi$
$450$	0	0
$451$	−5.37563e6	−1.24448
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.54972e6	1.01905	0.509523	−	0.860457i	$-0.329822\pi$
$457$	4.54972e6	1.01905	0.509523	+	0.860457i	$0.329822\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.13272e6	−1.34400	−0.672002	−	0.740549i	$-0.734566\pi$
$461$	−6.13272e6	−1.34400	−0.672002	+	0.740549i	$0.734566\pi$
$462$	0	0
$463$	−4.80795e6	−1.04233	−0.521167	−	0.853455i	$-0.674503\pi$
$463$	−4.80795e6	−1.04233	−0.521167	+	0.853455i	$0.674503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.52182e6	−1.17163	−0.585815	−	0.810445i	$-0.699225\pi$
$467$	−5.52182e6	−1.17163	−0.585815	+	0.810445i	$0.699225\pi$
$468$	0	0
$469$	−1.37179e6	−0.287975
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.42848e7	−2.93577
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.10086e6	−0.816650	−0.408325	−	0.912837i	$-0.633887\pi$
$479$	−4.10086e6	−0.816650	−0.408325	+	0.912837i	$0.633887\pi$
$480$	0	0
$481$	2.23159e6	0.439796
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.67945e6	0.320882	0.160441	−	0.987045i	$-0.448708\pi$
$487$	1.67945e6	0.320882	0.160441	+	0.987045i	$0.448708\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.85817e6	−0.722234	−0.361117	−	0.932521i	$-0.617604\pi$
$491$	−3.85817e6	−0.722234	−0.361117	+	0.932521i	$0.617604\pi$
$492$	0	0
$493$	−3.59540e6	−0.666238
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.86902e6	0.339408
$498$	0	0
$499$	3.53067e6	0.634755	0.317378	−	0.948299i	$-0.397198\pi$
$499$	3.53067e6	0.634755	0.317378	+	0.948299i	$0.397198\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.46451e6	1.49170	0.745851	−	0.666113i	$-0.232043\pi$
$503$	8.46451e6	1.49170	0.745851	+	0.666113i	$0.232043\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.12223e6	1.38957	0.694785	−	0.719217i	$-0.255499\pi$
$509$	8.12223e6	1.38957	0.694785	+	0.719217i	$0.255499\pi$
$510$	0	0
$511$	−2.97009e6	−0.503173
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.47049e7	2.41956
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.20656e6	−1.16314	−0.581572	−	0.813495i	$-0.697562\pi$
$521$	−7.20656e6	−1.16314	−0.581572	+	0.813495i	$0.697562\pi$
$522$	0	0
$523$	2.08544e6	0.333384	0.166692	−	0.986009i	$-0.446691\pi$
$523$	2.08544e6	0.333384	0.166692	+	0.986009i	$0.446691\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.45868e6	1.16986
$528$	0	0
$529$	−4.27549e6	−0.664273
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.35576e6	−0.511650
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.42878e6	−0.211833
$540$	0	0
$541$	−1.54682e6	−0.227220	−0.113610	−	0.993525i	$-0.536241\pi$
$541$	−1.54682e6	−0.227220	−0.113610	+	0.993525i	$0.536241\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	882763.	0.126147	0.0630734	−	0.998009i	$-0.479910\pi$
$547$	882763.	0.126147	0.0630734	+	0.998009i	$0.479910\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.20463e6	−0.870636
$552$	0	0
$553$	2.88209e6	0.400770
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.32281e7	−1.80659	−0.903294	−	0.429022i	$-0.858858\pi$
$557$	−1.32281e7	−1.80659	−0.903294	+	0.429022i	$0.858858\pi$
$558$	0	0
$559$	−8.91737e6	−1.20700
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.71554e6	0.228102	0.114051	−	0.993475i	$-0.463617\pi$
$563$	1.71554e6	0.228102	0.114051	+	0.993475i	$0.463617\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.06624e7	−1.38062	−0.690312	−	0.723512i	$-0.742527\pi$
$569$	−1.06624e7	−1.38062	−0.690312	+	0.723512i	$0.742527\pi$
$570$	0	0
$571$	2.13322e6	0.273807	0.136904	−	0.990584i	$-0.456285\pi$
$571$	2.13322e6	0.273807	0.136904	+	0.990584i	$0.456285\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.90217e6	0.362897	0.181448	−	0.983400i	$-0.441921\pi$
$577$	2.90217e6	0.362897	0.181448	+	0.983400i	$0.441921\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.30514e6	−0.283307
$582$	0	0
$583$	4.22124e6	0.514362
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.38744e7	−1.66195	−0.830975	−	0.556310i	$-0.812217\pi$
$587$	−1.38744e7	−1.66195	−0.830975	+	0.556310i	$0.812217\pi$
$588$	0	0
$589$	1.28715e7	1.52877
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.05925e6	0.123697	0.0618487	−	0.998086i	$-0.480300\pi$
$593$	1.05925e6	0.123697	0.0618487	+	0.998086i	$0.480300\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	821292.	0.0935256	0.0467628	−	0.998906i	$-0.485110\pi$
$599$	821292.	0.0935256	0.0467628	+	0.998906i	$0.485110\pi$
$600$	0	0
$601$	−8.26399e6	−0.933262	−0.466631	−	0.884452i	$-0.654532\pi$
$601$	−8.26399e6	−0.933262	−0.466631	+	0.884452i	$0.654532\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−252816.	−0.0278505	−0.0139252	−	0.999903i	$-0.504433\pi$
$607$	−252816.	−0.0278505	−0.0139252	+	0.999903i	$0.504433\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.17960e6	0.994765
$612$	0	0
$613$	5.25712e6	0.565062	0.282531	−	0.959258i	$-0.408826\pi$
$613$	5.25712e6	0.565062	0.282531	+	0.959258i	$0.408826\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.60527e6	0.698518	0.349259	−	0.937026i	$-0.386433\pi$
$617$	6.60527e6	0.698518	0.349259	+	0.937026i	$0.386433\pi$
$618$	0	0
$619$	−1.53364e7	−1.60878	−0.804392	−	0.594100i	$-0.797509\pi$
$619$	−1.53364e7	−1.60878	−0.804392	+	0.594100i	$0.797509\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.99076e6	−0.721613
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.22692e6	0.829108
$630$	0	0
$631$	−4.10137e6	−0.410068	−0.205034	−	0.978755i	$-0.565730\pi$
$631$	−4.10137e6	−0.410068	−0.205034	+	0.978755i	$0.565730\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−891922.	−0.0870920
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.80263e6	0.269414	0.134707	−	0.990885i	$-0.456991\pi$
$641$	2.80263e6	0.269414	0.134707	+	0.990885i	$0.456991\pi$
$642$	0	0
$643$	1.05243e7	1.00384	0.501920	−	0.864914i	$-0.332627\pi$
$643$	1.05243e7	1.00384	0.501920	+	0.864914i	$0.332627\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.43484e6	−0.792166	−0.396083	−	0.918215i	$-0.629631\pi$
$647$	−8.43484e6	−0.792166	−0.396083	+	0.918215i	$0.629631\pi$
$648$	0	0
$649$	1.94234e7	1.81014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.31348e6	0.487637	0.243818	−	0.969821i	$-0.421600\pi$
$653$	5.31348e6	0.487637	0.243818	+	0.969821i	$0.421600\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.80659e7	−1.62049	−0.810246	−	0.586091i	$-0.800666\pi$
$659$	−1.80659e7	−1.62049	−0.810246	+	0.586091i	$0.800666\pi$
$660$	0	0
$661$	7.01234e6	0.624251	0.312125	−	0.950041i	$-0.398959\pi$
$661$	7.01234e6	0.624251	0.312125	+	0.950041i	$0.398959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.28254e6	−0.285691
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.41117e7	−2.92481
$672$	0	0
$673$	−3.39367e6	−0.288823	−0.144412	−	0.989518i	$-0.546129\pi$
$673$	−3.39367e6	−0.288823	−0.144412	+	0.989518i	$0.546129\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.77774e6	−0.568347	−0.284173	−	0.958773i	$-0.591719\pi$
$677$	−6.77774e6	−0.568347	−0.284173	+	0.958773i	$0.591719\pi$
$678$	0	0
$679$	−2.11251e7	−1.75843
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.91859e6	0.813576	0.406788	−	0.913523i	$-0.366649\pi$
$683$	9.91859e6	0.813576	0.406788	+	0.913523i	$0.366649\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.63513e6	0.211473
$690$	0	0
$691$	46476.2	0.00370285	0.00185142	−	0.999998i	$-0.499411\pi$
$691$	46476.2	0.00370285	0.00185142	+	0.999998i	$0.499411\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.23713e7	−0.964567
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.23937e7	−0.952589	−0.476294	−	0.879286i	$-0.658020\pi$
$701$	−1.23937e7	−0.952589	−0.476294	+	0.879286i	$0.658020\pi$
$702$	0	0
$703$	1.41973e7	1.08347
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.45023e6	0.711040
$708$	0	0
$709$	−9.73930e6	−0.727632	−0.363816	−	0.931471i	$-0.618526\pi$
$709$	−9.73930e6	−0.727632	−0.363816	+	0.931471i	$0.618526\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.80966e6	0.501651
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.67144e7	−1.20578	−0.602891	−	0.797823i	$-0.705985\pi$
$719$	−1.67144e7	−1.20578	−0.602891	+	0.797823i	$0.705985\pi$
$720$	0	0
$721$	−1.50473e7	−1.07801
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.21859e7	0.855109	0.427555	−	0.903990i	$-0.359375\pi$
$727$	1.21859e7	0.855109	0.427555	+	0.903990i	$0.359375\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.28746e7	−2.27545
$732$	0	0
$733$	−1.90166e6	−0.130729	−0.0653647	−	0.997861i	$-0.520821\pi$
$733$	−1.90166e6	−0.130729	−0.0653647	+	0.997861i	$0.520821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.89837e6	−0.535635
$738$	0	0
$739$	−9.24528e6	−0.622743	−0.311372	−	0.950288i	$-0.600788\pi$
$739$	−9.24528e6	−0.622743	−0.311372	+	0.950288i	$0.600788\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.62223e6	−0.572991	−0.286495	−	0.958082i	$-0.592490\pi$
$743$	−8.62223e6	−0.572991	−0.286495	+	0.958082i	$0.592490\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.12943e7	0.735624
$750$	0	0
$751$	−2.53525e7	−1.64029	−0.820146	−	0.572154i	$-0.806108\pi$
$751$	−2.53525e7	−1.64029	−0.820146	+	0.572154i	$0.806108\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−156645.	−0.00993522	−0.00496761	−	0.999988i	$-0.501581\pi$
$757$	−156645.	−0.00993522	−0.00496761	+	0.999988i	$0.501581\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.56741e7	0.981118	0.490559	−	0.871408i	$-0.336793\pi$
$761$	1.56741e7	0.981118	0.490559	+	0.871408i	$0.336793\pi$
$762$	0	0
$763$	−6.53394e6	−0.406316
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.21251e7	0.744214
$768$	0	0
$769$	4.84095e6	0.295199	0.147600	−	0.989047i	$-0.452845\pi$
$769$	4.84095e6	0.295199	0.147600	+	0.989047i	$0.452845\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.85997e6	−0.111958	−0.0559792	−	0.998432i	$-0.517828\pi$
$773$	−1.85997e6	−0.111958	−0.0559792	+	0.998432i	$0.517828\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.13493e7	−1.26049
$780$	0	0
$781$	1.07613e7	0.631300
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.38288e7	−0.795881	−0.397940	−	0.917411i	$-0.630275\pi$
$787$	−1.38288e7	−0.795881	−0.397940	+	0.917411i	$0.630275\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.56161e6	0.259225
$792$	0	0
$793$	−2.12944e7	−1.20249
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.71374e6	0.541677	0.270839	−	0.962625i	$-0.412699\pi$
$797$	9.71374e6	0.541677	0.270839	+	0.962625i	$0.412699\pi$
$798$	0	0
$799$	3.38413e7	1.87534
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.71009e7	−0.935903
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.71666e6	−0.0922175	−0.0461088	−	0.998936i	$-0.514682\pi$
$809$	−1.71666e6	−0.0922175	−0.0461088	+	0.998936i	$0.514682\pi$
$810$	0	0
$811$	1.83595e7	0.980188	0.490094	−	0.871670i	$-0.336962\pi$
$811$	1.83595e7	0.980188	0.490094	+	0.871670i	$0.336962\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.67321e7	−2.97354
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.04696e7	0.542092	0.271046	−	0.962566i	$-0.412630\pi$
$821$	1.04696e7	0.542092	0.271046	+	0.962566i	$0.412630\pi$
$822$	0	0
$823$	−3.62514e7	−1.86563	−0.932815	−	0.360355i	$-0.882656\pi$
$823$	−3.62514e7	−1.86563	−0.932815	+	0.360355i	$0.882656\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.44656e7	1.24392	0.621959	−	0.783050i	$-0.286337\pi$
$827$	2.44656e7	1.24392	0.621959	+	0.783050i	$0.286337\pi$
$828$	0	0
$829$	−1.38073e7	−0.697789	−0.348894	−	0.937162i	$-0.613443\pi$
$829$	−1.38073e7	−0.697789	−0.348894	+	0.937162i	$0.613443\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.28814e6	−0.164187
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.90471e7	0.934164	0.467082	−	0.884214i	$-0.345305\pi$
$839$	1.90471e7	0.934164	0.467082	+	0.884214i	$0.345305\pi$
$840$	0	0
$841$	−1.55247e7	−0.756889
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.99066e7	1.91133
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.51106e6	0.355531
$852$	0	0
$853$	−9.14111e6	−0.430156	−0.215078	−	0.976597i	$-0.569001\pi$
$853$	−9.14111e6	−0.430156	−0.215078	+	0.976597i	$0.569001\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.39276e7	1.57798	0.788989	−	0.614407i	$-0.210605\pi$
$857$	3.39276e7	1.57798	0.788989	+	0.614407i	$0.210605\pi$
$858$	0	0
$859$	2.26635e7	1.04796	0.523979	−	0.851731i	$-0.324447\pi$
$859$	2.26635e7	1.04796	0.523979	+	0.851731i	$0.324447\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.01189e7	−0.919555	−0.459777	−	0.888034i	$-0.652071\pi$
$863$	−2.01189e7	−0.919555	−0.459777	+	0.888034i	$0.652071\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.65943e7	0.745433
$870$	0	0
$871$	−4.93059e6	−0.220218
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.99763e7	1.31607	0.658035	−	0.752988i	$-0.271388\pi$
$877$	2.99763e7	1.31607	0.658035	+	0.752988i	$0.271388\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.74388e7	−1.62511	−0.812554	−	0.582886i	$-0.801923\pi$
$881$	−3.74388e7	−1.62511	−0.812554	+	0.582886i	$0.801923\pi$
$882$	0	0
$883$	2.86458e7	1.23640	0.618200	−	0.786021i	$-0.287862\pi$
$883$	2.86458e7	1.23640	0.618200	+	0.786021i	$0.287862\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.81325e7	−1.20060	−0.600300	−	0.799775i	$-0.704952\pi$
$887$	−2.81325e7	−1.20060	−0.600300	+	0.799775i	$0.704952\pi$
$888$	0	0
$889$	−1.87382e7	−0.795195
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.84004e7	2.45068
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.03445e7	−0.426885
$900$	0	0
$901$	9.71462e6	0.398670
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.20623e7	−0.890498	−0.445249	−	0.895407i	$-0.646885\pi$
$907$	−2.20623e7	−0.890498	−0.445249	+	0.895407i	$0.646885\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.25081e7	0.499340	0.249670	−	0.968331i	$-0.419678\pi$
$911$	1.25081e7	0.499340	0.249670	+	0.968331i	$0.419678\pi$
$912$	0	0
$913$	−1.32724e7	−0.526952
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.66205e7	0.652712
$918$	0	0
$919$	−3.06659e6	−0.119775	−0.0598877	−	0.998205i	$-0.519074\pi$
$919$	−3.06659e6	−0.119775	−0.0598877	+	0.998205i	$0.519074\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.71777e6	0.259550
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.08272e7	1.55206	0.776032	−	0.630693i	$-0.217229\pi$
$929$	4.08272e7	1.55206	0.776032	+	0.630693i	$0.217229\pi$
$930$	0	0
$931$	−5.67439e6	−0.214558
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.07349e6	0.114362	0.0571812	−	0.998364i	$-0.481789\pi$
$937$	3.07349e6	0.114362	0.0571812	+	0.998364i	$0.481789\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.32976e6	−0.0857702	−0.0428851	−	0.999080i	$-0.513655\pi$
$941$	−2.32976e6	−0.0857702	−0.0428851	+	0.999080i	$0.513655\pi$
$942$	0	0
$943$	−1.12948e7	−0.413617
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.11096e7	0.764902	0.382451	−	0.923976i	$-0.375080\pi$
$947$	2.11096e7	0.764902	0.382451	+	0.923976i	$0.375080\pi$
$948$	0	0
$949$	−1.06753e7	−0.384783
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.66559e7	0.950738	0.475369	−	0.879787i	$-0.342315\pi$
$953$	2.66559e7	0.950738	0.475369	+	0.879787i	$0.342315\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.68841e7	−1.29507
$960$	0	0
$961$	−7.16939e6	−0.250423
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.22706e6	0.145369	0.0726846	−	0.997355i	$-0.476843\pi$
$967$	4.22706e6	0.145369	0.0726846	+	0.997355i	$0.476843\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−755839.	−0.0257265	−0.0128633	−	0.999917i	$-0.504095\pi$
$971$	−755839.	−0.0257265	−0.0128633	+	0.999917i	$0.504095\pi$
$972$	0	0
$973$	1.69831e7	0.575087
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−239167.	−0.00801613	−0.00400806	−	0.999992i	$-0.501276\pi$
$977$	−239167.	−0.00801613	−0.00400806	+	0.999992i	$0.501276\pi$
$978$	0	0
$979$	−4.02508e7	−1.34220
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.78483e7	−1.24929	−0.624644	−	0.780910i	$-0.714756\pi$
$983$	−3.78483e7	−1.24929	−0.624644	+	0.780910i	$0.714756\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.00140e7	−0.975738
$990$	0	0
$991$	−2.04454e7	−0.661321	−0.330661	−	0.943750i	$-0.607272\pi$
$991$	−2.04454e7	−0.661321	−0.330661	+	0.943750i	$0.607272\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.20062e6	0.0382532	0.0191266	−	0.999817i	$-0.493911\pi$
$997$	1.20062e6	0.0382532	0.0191266	+	0.999817i	$0.493911\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.a.w.1.3		3
3.2	odd	2		300.6.a.i.1.3		3
5.2	odd	4		180.6.d.d.109.4		6
5.3	odd	4		180.6.d.d.109.3		6
5.4	even	2		900.6.a.x.1.1		3
15.2	even	4		60.6.d.a.49.5	yes	6
15.8	even	4		60.6.d.a.49.2	✓	6
15.14	odd	2		300.6.a.j.1.1		3
20.3	even	4		720.6.f.m.289.3		6
20.7	even	4		720.6.f.m.289.4		6
60.23	odd	4		240.6.f.d.49.5		6
60.47	odd	4		240.6.f.d.49.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.d.a.49.2	✓	6	15.8	even	4
60.6.d.a.49.5	yes	6	15.2	even	4
180.6.d.d.109.3		6	5.3	odd	4
180.6.d.d.109.4		6	5.2	odd	4
240.6.f.d.49.2		6	60.47	odd	4
240.6.f.d.49.5		6	60.23	odd	4
300.6.a.i.1.3		3	3.2	odd	2
300.6.a.j.1.1		3	15.14	odd	2
720.6.f.m.289.3		6	20.3	even	4
720.6.f.m.289.4		6	20.7	even	4
900.6.a.w.1.3		3	1.1	even	1	trivial
900.6.a.x.1.1		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q+121.510 q^{7} +O(q^{10})\)	\(q+121.510 q^{7} +699.623 q^{11} +436.743 q^{13} +1610.09 q^{17} +2778.55 q^{19} +1469.98 q^{23} -2233.05 q^{29} +4632.47 q^{31} +5109.62 q^{37} -7683.61 q^{41} -20417.9 q^{43} +21018.3 q^{47} -2042.21 q^{49} +6033.60 q^{53} +27762.7 q^{59} -48757.3 q^{61} -11289.5 q^{67} +15381.5 q^{71} -24443.1 q^{73} +85011.4 q^{77} +23718.9 q^{79} -18970.7 q^{83} -57532.2 q^{89} +53068.8 q^{91} -173854. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 88 q^{7}+O(q^{10}) \) 3 * q - 88 * q^7	\( 3 q - 88 q^{7} - 148 q^{11} + 868 q^{17} + 3000 q^{19} - 1052 q^{23} - 7962 q^{29} + 132 q^{31} - 11944 q^{37} - 8170 q^{41} - 26188 q^{43} + 37892 q^{47} + 13827 q^{49} + 11076 q^{53} + 46228 q^{59} + 3126 q^{61} - 126332 q^{67} + 80400 q^{71} - 42904 q^{73} + 234608 q^{77} - 64476 q^{79} + 126268 q^{83} + 38030 q^{89} - 49200 q^{91} - 331016 q^{97}+O(q^{100}) \) 3 * q - 88 * q^7 - 148 * q^11 + 868 * q^17 + 3000 * q^19 - 1052 * q^23 - 7962 * q^29 + 132 * q^31 - 11944 * q^37 - 8170 * q^41 - 26188 * q^43 + 37892 * q^47 + 13827 * q^49 + 11076 * q^53 + 46228 * q^59 + 3126 * q^61 - 126332 * q^67 + 80400 * q^71 - 42904 * q^73 + 234608 * q^77 - 64476 * q^79 + 126268 * q^83 + 38030 * q^89 - 49200 * q^91 - 331016 * q^97

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