LMFDB - Embedded newform 1080.6.a.m.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,6,Mod(1,1080)] 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("1080.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1080.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-125,0,88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$173.214525398$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 100x^{3} - 199x^{2} + 871x + 1455 $ x^5 - 2x^4 - 100x^3 - 199x^2 + 871x + 1455
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.87411$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-25.0000 q^{5} +216.704 q^{7} -49.1051 q^{11} +460.124 q^{13} -547.924 q^{17} +1310.55 q^{19} -3491.31 q^{23} +625.000 q^{25} -8367.17 q^{29} -7545.52 q^{31} -5417.60 q^{35} -6504.07 q^{37} +9877.32 q^{41} +11141.4 q^{43} +8253.09 q^{47} +30153.7 q^{49} -23429.2 q^{53} +1227.63 q^{55} -39262.8 q^{59} +3174.48 q^{61} -11503.1 q^{65} -32494.2 q^{67} -21150.6 q^{71} +22041.7 q^{73} -10641.3 q^{77} -38586.9 q^{79} +49216.7 q^{83} +13698.1 q^{85} +93456.1 q^{89} +99710.8 q^{91} -32763.9 q^{95} +38920.1 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 125 q^{5} + 88 q^{7} - 134 q^{11} - 88 q^{13} - 211 q^{17} + 857 q^{19} - 461 q^{23} + 3125 q^{25} - 68 q^{29} + 1409 q^{31} - 2200 q^{35} + 5346 q^{37} - 750 q^{41} + 14486 q^{43} - 6704 q^{47} + 19005 q^{49}+ \cdots + 43184 q^{97}+O(q^{100}) $ 5 * q - 125 * q^5 + 88 * q^7 - 134 * q^11 - 88 * q^13 - 211 * q^17 + 857 * q^19 - 461 * q^23 + 3125 * q^25 - 68 * q^29 + 1409 * q^31 - 2200 * q^35 + 5346 * q^37 - 750 * q^41 + 14486 * q^43 - 6704 * q^47 + 19005 * q^49 - 8583 * q^53 + 3350 * q^55 - 25384 * q^59 + 38885 * q^61 + 2200 * q^65 + 23146 * q^67 - 62910 * q^71 + 54342 * q^73 - 5368 * q^77 + 24207 * q^79 - 20179 * q^83 + 5275 * q^85 + 10230 * q^89 + 41752 * q^91 - 21425 * q^95 + 43184 * 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
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	216.704	1.67156	0.835780	−	0.549064i	$-0.185016\pi$
$7$	216.704	1.67156	0.835780	+	0.549064i	$0.185016\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−49.1051	−0.122361	−0.0611807	−	0.998127i	$-0.519487\pi$
$11$	−49.1051	−0.122361	−0.0611807	+	0.998127i	$0.519487\pi$
$12$	0	0
$13$	460.124	0.755121	0.377561	−	0.925985i	$-0.376763\pi$
$13$	460.124	0.755121	0.377561	+	0.925985i	$0.376763\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−547.924	−0.459830	−0.229915	−	0.973211i	$-0.573845\pi$
$17$	−547.924	−0.459830	−0.229915	+	0.973211i	$0.573845\pi$
$18$	0	0
$19$	1310.55	0.832858	0.416429	−	0.909168i	$-0.363281\pi$
$19$	1310.55	0.832858	0.416429	+	0.909168i	$0.363281\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3491.31	−1.37616	−0.688080	−	0.725635i	$-0.741546\pi$
$23$	−3491.31	−1.37616	−0.688080	+	0.725635i	$0.741546\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8367.17	−1.84750	−0.923748	−	0.383002i	$-0.874890\pi$
$29$	−8367.17	−1.84750	−0.923748	+	0.383002i	$0.874890\pi$
$30$	0	0
$31$	−7545.52	−1.41021	−0.705107	−	0.709101i	$-0.749101\pi$
$31$	−7545.52	−1.41021	−0.705107	+	0.709101i	$0.749101\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5417.60	−0.747545
$36$	0	0
$37$	−6504.07	−0.781054	−0.390527	−	0.920591i	$-0.627707\pi$
$37$	−6504.07	−0.781054	−0.390527	+	0.920591i	$0.627707\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9877.32	0.917655	0.458828	−	0.888525i	$-0.348270\pi$
$41$	9877.32	0.917655	0.458828	+	0.888525i	$0.348270\pi$
$42$	0	0
$43$	11141.4	0.918900	0.459450	−	0.888204i	$-0.348047\pi$
$43$	11141.4	0.918900	0.459450	+	0.888204i	$0.348047\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8253.09	0.544969	0.272485	−	0.962160i	$-0.412155\pi$
$47$	8253.09	0.544969	0.272485	+	0.962160i	$0.412155\pi$
$48$	0	0
$49$	30153.7	1.79411
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−23429.2	−1.14569	−0.572845	−	0.819664i	$-0.694160\pi$
$53$	−23429.2	−1.14569	−0.572845	+	0.819664i	$0.694160\pi$
$54$	0	0
$55$	1227.63	0.0547217
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−39262.8	−1.46842	−0.734212	−	0.678920i	$-0.762448\pi$
$59$	−39262.8	−1.46842	−0.734212	+	0.678920i	$0.762448\pi$
$60$	0	0
$61$	3174.48	0.109232	0.0546158	−	0.998507i	$-0.482607\pi$
$61$	3174.48	0.109232	0.0546158	+	0.998507i	$0.482607\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11503.1	−0.337701
$66$	0	0
$67$	−32494.2	−0.884338	−0.442169	−	0.896932i	$-0.645791\pi$
$67$	−32494.2	−0.884338	−0.442169	+	0.896932i	$0.645791\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−21150.6	−0.497941	−0.248970	−	0.968511i	$-0.580092\pi$
$71$	−21150.6	−0.497941	−0.248970	+	0.968511i	$0.580092\pi$
$72$	0	0
$73$	22041.7	0.484104	0.242052	−	0.970263i	$-0.422180\pi$
$73$	22041.7	0.484104	0.242052	+	0.970263i	$0.422180\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10641.3	−0.204535
$78$	0	0
$79$	−38586.9	−0.695620	−0.347810	−	0.937565i	$-0.613075\pi$
$79$	−38586.9	−0.695620	−0.347810	+	0.937565i	$0.613075\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	49216.7	0.784182	0.392091	−	0.919926i	$-0.371752\pi$
$83$	49216.7	0.784182	0.392091	+	0.919926i	$0.371752\pi$
$84$	0	0
$85$	13698.1	0.205642
$86$	0	0
$87$	0	0
$88$	0	0
$89$	93456.1	1.25064	0.625321	−	0.780368i	$-0.284968\pi$
$89$	93456.1	1.25064	0.625321	+	0.780368i	$0.284968\pi$
$90$	0	0
$91$	99710.8	1.26223
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−32763.9	−0.372465
$96$	0	0
$97$	38920.1	0.419996	0.209998	−	0.977702i	$-0.432654\pi$
$97$	38920.1	0.419996	0.209998	+	0.977702i	$0.432654\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11259.7	−0.109831	−0.0549154	−	0.998491i	$-0.517489\pi$
$101$	−11259.7	−0.109831	−0.0549154	+	0.998491i	$0.517489\pi$
$102$	0	0
$103$	−149539.	−1.38887	−0.694436	−	0.719554i	$-0.744346\pi$
$103$	−149539.	−1.38887	−0.694436	+	0.719554i	$0.744346\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	45434.0	0.383638	0.191819	−	0.981430i	$-0.438561\pi$
$107$	45434.0	0.383638	0.191819	+	0.981430i	$0.438561\pi$
$108$	0	0
$109$	102271.	0.824489	0.412244	−	0.911073i	$-0.364745\pi$
$109$	102271.	0.824489	0.412244	+	0.911073i	$0.364745\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	92427.2	0.680932	0.340466	−	0.940257i	$-0.389415\pi$
$113$	92427.2	0.680932	0.340466	+	0.940257i	$0.389415\pi$
$114$	0	0
$115$	87282.8	0.615438
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−118737.	−0.768634
$120$	0	0
$121$	−158640.	−0.985028
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−240991.	−1.32584	−0.662922	−	0.748689i	$-0.730684\pi$
$127$	−240991.	−1.32584	−0.662922	+	0.748689i	$0.730684\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	129193.	0.657752	0.328876	−	0.944373i	$-0.393330\pi$
$131$	129193.	0.657752	0.328876	+	0.944373i	$0.393330\pi$
$132$	0	0
$133$	284003.	1.39217
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−23171.1	−0.105474	−0.0527370	−	0.998608i	$-0.516794\pi$
$137$	−23171.1	−0.105474	−0.0527370	+	0.998608i	$0.516794\pi$
$138$	0	0
$139$	−311404.	−1.36706	−0.683530	−	0.729922i	$-0.739556\pi$
$139$	−311404.	−1.36706	−0.683530	+	0.729922i	$0.739556\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−22594.4	−0.0923977
$144$	0	0
$145$	209179.	0.826225
$146$	0	0
$147$	0	0
$148$	0	0
$149$	159701.	0.589306	0.294653	−	0.955604i	$-0.404796\pi$
$149$	159701.	0.589306	0.294653	+	0.955604i	$0.404796\pi$
$150$	0	0
$151$	383268.	1.36792	0.683959	−	0.729521i	$-0.260257\pi$
$151$	383268.	1.36792	0.683959	+	0.729521i	$0.260257\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	188638.	0.630667
$156$	0	0
$157$	−177380.	−0.574321	−0.287161	−	0.957882i	$-0.592711\pi$
$157$	−177380.	−0.574321	−0.287161	+	0.957882i	$0.592711\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−756582.	−2.30034
$162$	0	0
$163$	−524956.	−1.54758	−0.773792	−	0.633440i	$-0.781642\pi$
$163$	−524956.	−1.54758	−0.773792	+	0.633440i	$0.781642\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−443301.	−1.23001	−0.615004	−	0.788524i	$-0.710845\pi$
$167$	−443301.	−1.23001	−0.615004	+	0.788524i	$0.710845\pi$
$168$	0	0
$169$	−159579.	−0.429792
$170$	0	0
$171$	0	0
$172$	0	0
$173$	169848.	0.431465	0.215732	−	0.976453i	$-0.430786\pi$
$173$	169848.	0.431465	0.215732	+	0.976453i	$0.430786\pi$
$174$	0	0
$175$	135440.	0.334312
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−320832.	−0.748421	−0.374210	−	0.927344i	$-0.622086\pi$
$179$	−320832.	−0.748421	−0.374210	+	0.927344i	$0.622086\pi$
$180$	0	0
$181$	145256.	0.329561	0.164781	−	0.986330i	$-0.447308\pi$
$181$	145256.	0.329561	0.164781	+	0.986330i	$0.447308\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	162602.	0.349298
$186$	0	0
$187$	26905.8	0.0562655
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−304719.	−0.604389	−0.302194	−	0.953246i	$-0.597719\pi$
$191$	−304719.	−0.604389	−0.302194	+	0.953246i	$0.597719\pi$
$192$	0	0
$193$	−865507.	−1.67254	−0.836271	−	0.548316i	$-0.815269\pi$
$193$	−865507.	−1.67254	−0.836271	+	0.548316i	$0.815269\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−254421.	−0.467076	−0.233538	−	0.972348i	$-0.575030\pi$
$197$	−254421.	−0.467076	−0.233538	+	0.972348i	$0.575030\pi$
$198$	0	0
$199$	1.01623e6	1.81910	0.909551	−	0.415592i	$-0.136425\pi$
$199$	1.01623e6	1.81910	0.909551	+	0.415592i	$0.136425\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.81320e6	−3.08820
$204$	0	0
$205$	−246933.	−0.410388
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−64354.9	−0.101910
$210$	0	0
$211$	1.09583e6	1.69448	0.847242	−	0.531208i	$-0.178262\pi$
$211$	1.09583e6	1.69448	0.847242	+	0.531208i	$0.178262\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−278535.	−0.410945
$216$	0	0
$217$	−1.63515e6	−2.35726
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−252113.	−0.347228
$222$	0	0
$223$	585305.	0.788170	0.394085	−	0.919074i	$-0.371062\pi$
$223$	585305.	0.788170	0.394085	+	0.919074i	$0.371062\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−456018.	−0.587377	−0.293689	−	0.955901i	$-0.594883\pi$
$227$	−456018.	−0.587377	−0.293689	+	0.955901i	$0.594883\pi$
$228$	0	0
$229$	−901326.	−1.13578	−0.567889	−	0.823105i	$-0.692240\pi$
$229$	−901326.	−1.13578	−0.567889	+	0.823105i	$0.692240\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.07629e6	−1.29879	−0.649394	−	0.760452i	$-0.724978\pi$
$233$	−1.07629e6	−1.29879	−0.649394	+	0.760452i	$0.724978\pi$
$234$	0	0
$235$	−206327.	−0.243718
$236$	0	0
$237$	0	0
$238$	0	0
$239$	62346.8	0.0706023	0.0353012	−	0.999377i	$-0.488761\pi$
$239$	62346.8	0.0706023	0.0353012	+	0.999377i	$0.488761\pi$
$240$	0	0
$241$	1.59833e6	1.77265	0.886324	−	0.463065i	$-0.153250\pi$
$241$	1.59833e6	1.77265	0.886324	+	0.463065i	$0.153250\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−753842.	−0.802352
$246$	0	0
$247$	603018.	0.628909
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.05801e6	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$251$	1.05801e6	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$252$	0	0
$253$	171441.	0.168389
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.52253e6	−1.43791	−0.718957	−	0.695055i	$-0.755380\pi$
$257$	−1.52253e6	−1.43791	−0.718957	+	0.695055i	$0.755380\pi$
$258$	0	0
$259$	−1.40946e6	−1.30558
$260$	0	0
$261$	0	0
$262$	0	0
$263$	66805.8	0.0595559	0.0297779	−	0.999557i	$-0.490520\pi$
$263$	66805.8	0.0595559	0.0297779	+	0.999557i	$0.490520\pi$
$264$	0	0
$265$	585729.	0.512368
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.10628e6	−1.77475	−0.887373	−	0.461051i	$-0.847472\pi$
$269$	−2.10628e6	−1.77475	−0.887373	+	0.461051i	$0.847472\pi$
$270$	0	0
$271$	1.00178e6	0.828606	0.414303	−	0.910139i	$-0.364025\pi$
$271$	1.00178e6	0.828606	0.414303	+	0.910139i	$0.364025\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−30690.7	−0.0244723
$276$	0	0
$277$	−1.72726e6	−1.35256	−0.676282	−	0.736643i	$-0.736410\pi$
$277$	−1.72726e6	−1.35256	−0.676282	+	0.736643i	$0.736410\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	468396.	0.353873	0.176936	−	0.984222i	$-0.443381\pi$
$281$	468396.	0.353873	0.176936	+	0.984222i	$0.443381\pi$
$282$	0	0
$283$	1.16704e6	0.866207	0.433103	−	0.901344i	$-0.357419\pi$
$283$	1.16704e6	0.866207	0.433103	+	0.901344i	$0.357419\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.14046e6	1.53392
$288$	0	0
$289$	−1.11964e6	−0.788556
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.52352e6	1.03676	0.518380	−	0.855151i	$-0.326535\pi$
$293$	1.52352e6	1.03676	0.518380	+	0.855151i	$0.326535\pi$
$294$	0	0
$295$	981571.	0.656700
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.60644e6	−1.03917
$300$	0	0
$301$	2.41439e6	1.53600
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−79362.1	−0.0488499
$306$	0	0
$307$	−604404.	−0.366000	−0.183000	−	0.983113i	$-0.558581\pi$
$307$	−604404.	−0.366000	−0.183000	+	0.983113i	$0.558581\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.50927e6	−0.884845	−0.442422	−	0.896807i	$-0.645881\pi$
$311$	−1.50927e6	−0.884845	−0.442422	+	0.896807i	$0.645881\pi$
$312$	0	0
$313$	−1.21293e6	−0.699801	−0.349900	−	0.936787i	$-0.613785\pi$
$313$	−1.21293e6	−0.699801	−0.349900	+	0.936787i	$0.613785\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.41548e6	−1.35007	−0.675034	−	0.737787i	$-0.735871\pi$
$317$	−2.41548e6	−1.35007	−0.675034	+	0.737787i	$0.735871\pi$
$318$	0	0
$319$	410870.	0.226062
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−718084.	−0.382973
$324$	0	0
$325$	287578.	0.151024
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.78848e6	0.910949
$330$	0	0
$331$	−2.11865e6	−1.06289	−0.531446	−	0.847092i	$-0.678351\pi$
$331$	−2.11865e6	−1.06289	−0.531446	+	0.847092i	$0.678351\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	812354.	0.395488
$336$	0	0
$337$	812616.	0.389772	0.194886	−	0.980826i	$-0.437566\pi$
$337$	812616.	0.389772	0.194886	+	0.980826i	$0.437566\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	370523.	0.172556
$342$	0	0
$343$	2.89228e6	1.32741
$344$	0	0
$345$	0	0
$346$	0	0
$347$	995907.	0.444012	0.222006	−	0.975045i	$-0.428739\pi$
$347$	995907.	0.444012	0.222006	+	0.975045i	$0.428739\pi$
$348$	0	0
$349$	−665885.	−0.292641	−0.146321	−	0.989237i	$-0.546743\pi$
$349$	−665885.	−0.292641	−0.146321	+	0.989237i	$0.546743\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.99177e6	1.27788	0.638941	−	0.769256i	$-0.279373\pi$
$353$	2.99177e6	1.27788	0.638941	+	0.769256i	$0.279373\pi$
$354$	0	0
$355$	528766.	0.222686
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.58779e6	−1.46924	−0.734618	−	0.678481i	$-0.762638\pi$
$359$	−3.58779e6	−1.46924	−0.734618	+	0.678481i	$0.762638\pi$
$360$	0	0
$361$	−758547.	−0.306347
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−551043.	−0.216498
$366$	0	0
$367$	−114041.	−0.0441973	−0.0220987	−	0.999756i	$-0.507035\pi$
$367$	−114041.	−0.0441973	−0.0220987	+	0.999756i	$0.507035\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.07720e6	−1.91509
$372$	0	0
$373$	34656.5	0.0128977	0.00644885	−	0.999979i	$-0.497947\pi$
$373$	34656.5	0.0128977	0.00644885	+	0.999979i	$0.497947\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.84994e6	−1.39508
$378$	0	0
$379$	−4.92297e6	−1.76047	−0.880236	−	0.474535i	$-0.842616\pi$
$379$	−4.92297e6	−1.76047	−0.880236	+	0.474535i	$0.842616\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.90994e6	−0.665309	−0.332655	−	0.943049i	$-0.607944\pi$
$383$	−1.90994e6	−0.665309	−0.332655	+	0.943049i	$0.607944\pi$
$384$	0	0
$385$	266032.	0.0914706
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.47691e6	−0.494856	−0.247428	−	0.968906i	$-0.579585\pi$
$389$	−1.47691e6	−0.494856	−0.247428	+	0.968906i	$0.579585\pi$
$390$	0	0
$391$	1.91297e6	0.632800
$392$	0	0
$393$	0	0
$394$	0	0
$395$	964672.	0.311091
$396$	0	0
$397$	2.77229e6	0.882799	0.441400	−	0.897311i	$-0.354482\pi$
$397$	2.77229e6	0.882799	0.441400	+	0.897311i	$0.354482\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.60313e6	−1.74008	−0.870042	−	0.492978i	$-0.835908\pi$
$401$	−5.60313e6	−1.74008	−0.870042	+	0.492978i	$0.835908\pi$
$402$	0	0
$403$	−3.47188e6	−1.06488
$404$	0	0
$405$	0	0
$406$	0	0
$407$	319383.	0.0955709
$408$	0	0
$409$	2.46880e6	0.729756	0.364878	−	0.931055i	$-0.381111\pi$
$409$	2.46880e6	0.729756	0.364878	+	0.931055i	$0.381111\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.50842e6	−2.45456
$414$	0	0
$415$	−1.23042e6	−0.350697
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.73295e6	−1.31703	−0.658517	−	0.752566i	$-0.728816\pi$
$419$	−4.73295e6	−1.31703	−0.658517	+	0.752566i	$0.728816\pi$
$420$	0	0
$421$	−6.01884e6	−1.65504	−0.827518	−	0.561439i	$-0.810248\pi$
$421$	−6.01884e6	−1.65504	−0.827518	+	0.561439i	$0.810248\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−342452.	−0.0919661
$426$	0	0
$427$	687923.	0.182587
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.10385e6	−0.286231	−0.143115	−	0.989706i	$-0.545712\pi$
$431$	−1.10385e6	−0.286231	−0.143115	+	0.989706i	$0.545712\pi$
$432$	0	0
$433$	−3.86946e6	−0.991814	−0.495907	−	0.868376i	$-0.665164\pi$
$433$	−3.86946e6	−0.991814	−0.495907	+	0.868376i	$0.665164\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.57555e6	−1.14615
$438$	0	0
$439$	4.87495e6	1.20728	0.603641	−	0.797256i	$-0.293716\pi$
$439$	4.87495e6	1.20728	0.603641	+	0.797256i	$0.293716\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.05403e6	−0.497277	−0.248638	−	0.968596i	$-0.579983\pi$
$443$	−2.05403e6	−0.497277	−0.248638	+	0.968596i	$0.579983\pi$
$444$	0	0
$445$	−2.33640e6	−0.559304
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.19976e6	1.45131	0.725653	−	0.688061i	$-0.241538\pi$
$449$	6.19976e6	1.45131	0.725653	+	0.688061i	$0.241538\pi$
$450$	0	0
$451$	−485026.	−0.112286
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.49277e6	−0.564487
$456$	0	0
$457$	1.72779e6	0.386990	0.193495	−	0.981101i	$-0.438018\pi$
$457$	1.72779e6	0.386990	0.193495	+	0.981101i	$0.438018\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.25908e6	−1.15254	−0.576272	−	0.817258i	$-0.695493\pi$
$461$	−5.25908e6	−1.15254	−0.576272	+	0.817258i	$0.695493\pi$
$462$	0	0
$463$	6.69549e6	1.45154	0.725771	−	0.687936i	$-0.241483\pi$
$463$	6.69549e6	1.45154	0.725771	+	0.687936i	$0.241483\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.11079e6	0.872234	0.436117	−	0.899890i	$-0.356353\pi$
$467$	4.11079e6	0.872234	0.436117	+	0.899890i	$0.356353\pi$
$468$	0	0
$469$	−7.04162e6	−1.47823
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−547099.	−0.112438
$474$	0	0
$475$	819096.	0.166572
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.54856e6	−0.308383	−0.154191	−	0.988041i	$-0.549277\pi$
$479$	−1.54856e6	−0.308383	−0.154191	+	0.988041i	$0.549277\pi$
$480$	0	0
$481$	−2.99268e6	−0.589791
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−973003.	−0.187828
$486$	0	0
$487$	−1.11511e6	−0.213057	−0.106528	−	0.994310i	$-0.533973\pi$
$487$	−1.11511e6	−0.213057	−0.106528	+	0.994310i	$0.533973\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.81193e6	1.46236	0.731180	−	0.682184i	$-0.238970\pi$
$491$	7.81193e6	1.46236	0.731180	+	0.682184i	$0.238970\pi$
$492$	0	0
$493$	4.58457e6	0.849535
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.58343e6	−0.832338
$498$	0	0
$499$	8.21463e6	1.47685	0.738426	−	0.674335i	$-0.235570\pi$
$499$	8.21463e6	1.47685	0.738426	+	0.674335i	$0.235570\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.70462e6	−0.829095	−0.414547	−	0.910028i	$-0.636060\pi$
$503$	−4.70462e6	−0.829095	−0.414547	+	0.910028i	$0.636060\pi$
$504$	0	0
$505$	281493.	0.0491178
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.48456e6	1.28048	0.640239	−	0.768176i	$-0.278835\pi$
$509$	7.48456e6	1.28048	0.640239	+	0.768176i	$0.278835\pi$
$510$	0	0
$511$	4.77653e6	0.809209
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.73848e6	0.621122
$516$	0	0
$517$	−405269.	−0.0666832
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.16350e7	1.87790	0.938949	−	0.344057i	$-0.111801\pi$
$521$	1.16350e7	1.87790	0.938949	+	0.344057i	$0.111801\pi$
$522$	0	0
$523$	−5.61332e6	−0.897357	−0.448679	−	0.893693i	$-0.648105\pi$
$523$	−5.61332e6	−0.897357	−0.448679	+	0.893693i	$0.648105\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.13437e6	0.648459
$528$	0	0
$529$	5.75291e6	0.893817
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.54479e6	0.692941
$534$	0	0
$535$	−1.13585e6	−0.171568
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.48070e6	−0.219530
$540$	0	0
$541$	−1.18616e7	−1.74241	−0.871204	−	0.490921i	$-0.836660\pi$
$541$	−1.18616e7	−1.74241	−0.871204	+	0.490921i	$0.836660\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.55677e6	−0.368723
$546$	0	0
$547$	1.03073e7	1.47291	0.736455	−	0.676486i	$-0.236498\pi$
$547$	1.03073e7	1.47291	0.736455	+	0.676486i	$0.236498\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.09656e7	−1.53870
$552$	0	0
$553$	−8.36193e6	−1.16277
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.90464e6	0.396693	0.198347	−	0.980132i	$-0.436443\pi$
$557$	2.90464e6	0.396693	0.198347	+	0.980132i	$0.436443\pi$
$558$	0	0
$559$	5.12642e6	0.693881
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.99840e6	−0.265712	−0.132856	−	0.991135i	$-0.542415\pi$
$563$	−1.99840e6	−0.265712	−0.132856	+	0.991135i	$0.542415\pi$
$564$	0	0
$565$	−2.31068e6	−0.304522
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−63428.7	−0.00821306	−0.00410653	−	0.999992i	$-0.501307\pi$
$569$	−63428.7	−0.00821306	−0.00410653	+	0.999992i	$0.501307\pi$
$570$	0	0
$571$	−9.55380e6	−1.22627	−0.613134	−	0.789979i	$-0.710092\pi$
$571$	−9.55380e6	−1.22627	−0.613134	+	0.789979i	$0.710092\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.18207e6	−0.275232
$576$	0	0
$577$	1.56175e6	0.195286	0.0976432	−	0.995221i	$-0.468870\pi$
$577$	1.56175e6	0.195286	0.0976432	+	0.995221i	$0.468870\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.06655e7	1.31081
$582$	0	0
$583$	1.15049e6	0.140188
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.11959e6	−0.373682	−0.186841	−	0.982390i	$-0.559825\pi$
$587$	−3.11959e6	−0.373682	−0.186841	+	0.982390i	$0.559825\pi$
$588$	0	0
$589$	−9.88881e6	−1.17451
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.77513e6	0.324076	0.162038	−	0.986785i	$-0.448193\pi$
$593$	2.77513e6	0.324076	0.162038	+	0.986785i	$0.448193\pi$
$594$	0	0
$595$	2.96843e6	0.343744
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.89325e6	−0.557225	−0.278612	−	0.960404i	$-0.589874\pi$
$599$	−4.89325e6	−0.557225	−0.278612	+	0.960404i	$0.589874\pi$
$600$	0	0
$601$	−8.48417e6	−0.958127	−0.479063	−	0.877780i	$-0.659024\pi$
$601$	−8.48417e6	−0.958127	−0.479063	+	0.877780i	$0.659024\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.96599e6	0.440518
$606$	0	0
$607$	1.78640e7	1.96792	0.983961	−	0.178382i	$-0.0570864\pi$
$607$	1.78640e7	1.96792	0.983961	+	0.178382i	$0.0570864\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.79745e6	0.411518
$612$	0	0
$613$	−1.03552e7	−1.11303	−0.556514	−	0.830838i	$-0.687861\pi$
$613$	−1.03552e7	−1.11303	−0.556514	+	0.830838i	$0.687861\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.80436e6	−0.190814	−0.0954068	−	0.995438i	$-0.530415\pi$
$617$	−1.80436e6	−0.190814	−0.0954068	+	0.995438i	$0.530415\pi$
$618$	0	0
$619$	6.41967e6	0.673420	0.336710	−	0.941608i	$-0.390686\pi$
$619$	6.41967e6	0.673420	0.336710	+	0.941608i	$0.390686\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.02523e7	2.09052
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.56373e6	0.359152
$630$	0	0
$631$	4.02548e6	0.402480	0.201240	−	0.979542i	$-0.435503\pi$
$631$	4.02548e6	0.402480	0.201240	+	0.979542i	$0.435503\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.02479e6	0.592935
$636$	0	0
$637$	1.38744e7	1.35477
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.83544e6	0.176439	0.0882197	−	0.996101i	$-0.471882\pi$
$641$	1.83544e6	0.176439	0.0882197	+	0.996101i	$0.471882\pi$
$642$	0	0
$643$	3.86958e6	0.369094	0.184547	−	0.982824i	$-0.440918\pi$
$643$	3.86958e6	0.369094	0.184547	+	0.982824i	$0.440918\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.75710e7	1.65020	0.825101	−	0.564986i	$-0.191118\pi$
$647$	1.75710e7	1.65020	0.825101	+	0.564986i	$0.191118\pi$
$648$	0	0
$649$	1.92800e6	0.179679
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.61584e6	−0.331838	−0.165919	−	0.986139i	$-0.553059\pi$
$653$	−3.61584e6	−0.331838	−0.165919	+	0.986139i	$0.553059\pi$
$654$	0	0
$655$	−3.22984e6	−0.294156
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.22385e6	0.647970	0.323985	−	0.946062i	$-0.394977\pi$
$659$	7.22385e6	0.647970	0.323985	+	0.946062i	$0.394977\pi$
$660$	0	0
$661$	−1.04789e6	−0.0932849	−0.0466424	−	0.998912i	$-0.514852\pi$
$661$	−1.04789e6	−0.0932849	−0.0466424	+	0.998912i	$0.514852\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.10006e6	−0.622599
$666$	0	0
$667$	2.92124e7	2.54245
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−155883.	−0.0133657
$672$	0	0
$673$	−5.45996e6	−0.464678	−0.232339	−	0.972635i	$-0.574638\pi$
$673$	−5.45996e6	−0.464678	−0.232339	+	0.972635i	$0.574638\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.33313e6	−0.195644	−0.0978222	−	0.995204i	$-0.531188\pi$
$677$	−2.33313e6	−0.195644	−0.0978222	+	0.995204i	$0.531188\pi$
$678$	0	0
$679$	8.43415e6	0.702048
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.08135e7	−0.886985	−0.443493	−	0.896278i	$-0.646261\pi$
$683$	−1.08135e7	−0.886985	−0.443493	+	0.896278i	$0.646261\pi$
$684$	0	0
$685$	579277.	0.0471694
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.07803e7	−0.865135
$690$	0	0
$691$	1.61286e7	1.28500	0.642498	−	0.766287i	$-0.277898\pi$
$691$	1.61286e7	1.28500	0.642498	+	0.766287i	$0.277898\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.78511e6	0.611368
$696$	0	0
$697$	−5.41202e6	−0.421966
$698$	0	0
$699$	0	0
$700$	0	0
$701$	595078.	0.0457381	0.0228691	−	0.999738i	$-0.492720\pi$
$701$	595078.	0.0457381	0.0228691	+	0.999738i	$0.492720\pi$
$702$	0	0
$703$	−8.52394e6	−0.650507
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.44003e6	−0.183589
$708$	0	0
$709$	−2.34844e7	−1.75455	−0.877273	−	0.479992i	$-0.840640\pi$
$709$	−2.34844e7	−1.75455	−0.877273	+	0.479992i	$0.840640\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.63438e7	1.94068
$714$	0	0
$715$	564861.	0.0413215
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.47052e7	−1.06084	−0.530418	−	0.847737i	$-0.677965\pi$
$719$	−1.47052e7	−1.06084	−0.530418	+	0.847737i	$0.677965\pi$
$720$	0	0
$721$	−3.24058e7	−2.32158
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.22948e6	−0.369499
$726$	0	0
$727$	−2.56423e6	−0.179937	−0.0899685	−	0.995945i	$-0.528677\pi$
$727$	−2.56423e6	−0.179937	−0.0899685	+	0.995945i	$0.528677\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.10463e6	−0.422538
$732$	0	0
$733$	1.23334e7	0.847858	0.423929	−	0.905695i	$-0.360651\pi$
$733$	1.23334e7	0.847858	0.423929	+	0.905695i	$0.360651\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.59563e6	0.108209
$738$	0	0
$739$	−2.54081e7	−1.71144	−0.855720	−	0.517440i	$-0.826885\pi$
$739$	−2.54081e7	−1.71144	−0.855720	+	0.517440i	$0.826885\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.62014e7	1.74121	0.870607	−	0.491979i	$-0.163726\pi$
$743$	2.62014e7	1.74121	0.870607	+	0.491979i	$0.163726\pi$
$744$	0	0
$745$	−3.99251e6	−0.263546
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.84574e6	0.641274
$750$	0	0
$751$	−9.37497e6	−0.606555	−0.303278	−	0.952902i	$-0.598081\pi$
$751$	−9.37497e6	−0.606555	−0.303278	+	0.952902i	$0.598081\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.58169e6	−0.611751
$756$	0	0
$757$	−2.95539e7	−1.87446	−0.937229	−	0.348714i	$-0.886619\pi$
$757$	−2.95539e7	−1.87446	−0.937229	+	0.348714i	$0.886619\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.92890e6	0.308524	0.154262	−	0.988030i	$-0.450700\pi$
$761$	4.92890e6	0.308524	0.154262	+	0.988030i	$0.450700\pi$
$762$	0	0
$763$	2.21625e7	1.37818
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.80658e7	−1.10884
$768$	0	0
$769$	766481.	0.0467396	0.0233698	−	0.999727i	$-0.492560\pi$
$769$	766481.	0.0467396	0.0233698	+	0.999727i	$0.492560\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.81694e6	−0.229756	−0.114878	−	0.993380i	$-0.536648\pi$
$773$	−3.81694e6	−0.229756	−0.114878	+	0.993380i	$0.536648\pi$
$774$	0	0
$775$	−4.71595e6	−0.282043
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.29448e7	0.764276
$780$	0	0
$781$	1.03860e6	0.0609288
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.43449e6	0.256844
$786$	0	0
$787$	2.63477e7	1.51637	0.758186	−	0.652038i	$-0.226086\pi$
$787$	2.63477e7	1.51637	0.758186	+	0.652038i	$0.226086\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.00294e7	1.13822
$792$	0	0
$793$	1.46066e6	0.0824831
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.43168e7	0.798365	0.399183	−	0.916871i	$-0.369294\pi$
$797$	1.43168e7	0.798365	0.399183	+	0.916871i	$0.369294\pi$
$798$	0	0
$799$	−4.52206e6	−0.250593
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.08236e6	−0.0592356
$804$	0	0
$805$	1.89145e7	1.02874
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.83586e7	1.52340	0.761699	−	0.647930i	$-0.224365\pi$
$809$	2.83586e7	1.52340	0.761699	+	0.647930i	$0.224365\pi$
$810$	0	0
$811$	−1.51736e7	−0.810096	−0.405048	−	0.914295i	$-0.632745\pi$
$811$	−1.51736e7	−0.810096	−0.405048	+	0.914295i	$0.632745\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.31239e7	0.692101
$816$	0	0
$817$	1.46014e7	0.765313
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.10718e7	−1.60883	−0.804413	−	0.594070i	$-0.797520\pi$
$821$	−3.10718e7	−1.60883	−0.804413	+	0.594070i	$0.797520\pi$
$822$	0	0
$823$	1.62540e7	0.836488	0.418244	−	0.908335i	$-0.362646\pi$
$823$	1.62540e7	0.836488	0.418244	+	0.908335i	$0.362646\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.47146e6	−0.125658	−0.0628290	−	0.998024i	$-0.520012\pi$
$827$	−2.47146e6	−0.125658	−0.0628290	+	0.998024i	$0.520012\pi$
$828$	0	0
$829$	−2.28848e6	−0.115654	−0.0578270	−	0.998327i	$-0.518417\pi$
$829$	−2.28848e6	−0.115654	−0.0578270	+	0.998327i	$0.518417\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.65219e7	−0.824988
$834$	0	0
$835$	1.10825e7	0.550076
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−616161.	−0.0302196	−0.0151098	−	0.999886i	$-0.504810\pi$
$839$	−616161.	−0.0302196	−0.0151098	+	0.999886i	$0.504810\pi$
$840$	0	0
$841$	4.94983e7	2.41324
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.98947e6	0.192209
$846$	0	0
$847$	−3.43779e7	−1.64653
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.27077e7	1.07486
$852$	0	0
$853$	−6.22473e6	−0.292919	−0.146460	−	0.989217i	$-0.546788\pi$
$853$	−6.22473e6	−0.292919	−0.146460	+	0.989217i	$0.546788\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.95218e7	0.907961	0.453980	−	0.891012i	$-0.350004\pi$
$857$	1.95218e7	0.907961	0.453980	+	0.891012i	$0.350004\pi$
$858$	0	0
$859$	4.91732e6	0.227377	0.113688	−	0.993516i	$-0.463733\pi$
$859$	4.91732e6	0.227377	0.113688	+	0.993516i	$0.463733\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.45672e7	−0.665809	−0.332905	−	0.942961i	$-0.608029\pi$
$863$	−1.45672e7	−0.665809	−0.332905	+	0.942961i	$0.608029\pi$
$864$	0	0
$865$	−4.24620e6	−0.192957
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.89481e6	0.0851170
$870$	0	0
$871$	−1.49514e7	−0.667783
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.38600e6	−0.149509
$876$	0	0
$877$	−2.79156e6	−0.122560	−0.0612799	−	0.998121i	$-0.519518\pi$
$877$	−2.79156e6	−0.122560	−0.0612799	+	0.998121i	$0.519518\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.75646e7	0.762427	0.381214	−	0.924487i	$-0.375506\pi$
$881$	1.75646e7	0.762427	0.381214	+	0.924487i	$0.375506\pi$
$882$	0	0
$883$	−7.02211e6	−0.303086	−0.151543	−	0.988451i	$-0.548424\pi$
$883$	−7.02211e6	−0.303086	−0.151543	+	0.988451i	$0.548424\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.32738e7	0.993248	0.496624	−	0.867966i	$-0.334573\pi$
$887$	2.32738e7	0.993248	0.496624	+	0.867966i	$0.334573\pi$
$888$	0	0
$889$	−5.22238e7	−2.21623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.08161e7	0.453882
$894$	0	0
$895$	8.02081e6	0.334704
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.31346e7	2.60536
$900$	0	0
$901$	1.28374e7	0.526823
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.63139e6	−0.147384
$906$	0	0
$907$	−1.10282e7	−0.445129	−0.222564	−	0.974918i	$-0.571443\pi$
$907$	−1.10282e7	−0.445129	−0.222564	+	0.974918i	$0.571443\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.95239e7	1.17863	0.589316	−	0.807903i	$-0.299397\pi$
$911$	2.95239e7	1.17863	0.589316	+	0.807903i	$0.299397\pi$
$912$	0	0
$913$	−2.41679e6	−0.0959537
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.79968e7	1.09947
$918$	0	0
$919$	−2.25710e7	−0.881582	−0.440791	−	0.897610i	$-0.645302\pi$
$919$	−2.25710e7	−0.881582	−0.440791	+	0.897610i	$0.645302\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.73193e6	−0.376006
$924$	0	0
$925$	−4.06505e6	−0.156211
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.02520e7	1.53020	0.765099	−	0.643913i	$-0.222690\pi$
$929$	4.02520e7	1.53020	0.765099	+	0.643913i	$0.222690\pi$
$930$	0	0
$931$	3.95180e7	1.49424
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−672646.	−0.0251627
$936$	0	0
$937$	−5.93146e6	−0.220705	−0.110353	−	0.993892i	$-0.535198\pi$
$937$	−5.93146e6	−0.220705	−0.110353	+	0.993892i	$0.535198\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.67266e7	0.615793	0.307896	−	0.951420i	$-0.400375\pi$
$941$	1.67266e7	0.615793	0.307896	+	0.951420i	$0.400375\pi$
$942$	0	0
$943$	−3.44848e7	−1.26284
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.32798e7	1.93058	0.965290	−	0.261181i	$-0.0841118\pi$
$947$	5.32798e7	1.93058	0.965290	+	0.261181i	$0.0841118\pi$
$948$	0	0
$949$	1.01419e7	0.365557
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.03774e7	−1.44015	−0.720073	−	0.693898i	$-0.755892\pi$
$953$	−4.03774e7	−1.44015	−0.720073	+	0.693898i	$0.755892\pi$
$954$	0	0
$955$	7.61798e6	0.270291
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.02127e6	−0.176306
$960$	0	0
$961$	2.83057e7	0.988703
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.16377e7	0.747984
$966$	0	0
$967$	4.90032e7	1.68523	0.842613	−	0.538520i	$-0.181016\pi$
$967$	4.90032e7	1.68523	0.842613	+	0.538520i	$0.181016\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.00304e7	0.681777	0.340889	−	0.940104i	$-0.389272\pi$
$971$	2.00304e7	0.681777	0.340889	+	0.940104i	$0.389272\pi$
$972$	0	0
$973$	−6.74826e7	−2.28512
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.10720e7	0.371099	0.185550	−	0.982635i	$-0.440593\pi$
$977$	1.10720e7	0.371099	0.185550	+	0.982635i	$0.440593\pi$
$978$	0	0
$979$	−4.58917e6	−0.153030
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.51692e7	1.49093	0.745467	−	0.666543i	$-0.232227\pi$
$983$	4.51692e7	1.49093	0.745467	+	0.666543i	$0.232227\pi$
$984$	0	0
$985$	6.36053e6	0.208883
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.88981e7	−1.26455
$990$	0	0
$991$	2.52900e7	0.818020	0.409010	−	0.912530i	$-0.365874\pi$
$991$	2.52900e7	0.818020	0.409010	+	0.912530i	$0.365874\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.54056e7	−0.813527
$996$	0	0
$997$	2.53395e7	0.807348	0.403674	−	0.914903i	$-0.367733\pi$
$997$	2.53395e7	0.807348	0.403674	+	0.914903i	$0.367733\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.6.a.m.1.5	✓	5
3.2	odd	2		1080.6.a.r.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.m.1.5	✓	5	1.1	even	1	trivial
1080.6.a.r.1.5	yes	5	3.2	odd	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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} +216.704 q^{7} -49.1051 q^{11} +460.124 q^{13} -547.924 q^{17} +1310.55 q^{19} -3491.31 q^{23} +625.000 q^{25} -8367.17 q^{29} -7545.52 q^{31} -5417.60 q^{35} -6504.07 q^{37} +9877.32 q^{41} +11141.4 q^{43} +8253.09 q^{47} +30153.7 q^{49} -23429.2 q^{53} +1227.63 q^{55} -39262.8 q^{59} +3174.48 q^{61} -11503.1 q^{65} -32494.2 q^{67} -21150.6 q^{71} +22041.7 q^{73} -10641.3 q^{77} -38586.9 q^{79} +49216.7 q^{83} +13698.1 q^{85} +93456.1 q^{89} +99710.8 q^{91} -32763.9 q^{95} +38920.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 125 q^{5} + 88 q^{7} - 134 q^{11} - 88 q^{13} - 211 q^{17} + 857 q^{19} - 461 q^{23} + 3125 q^{25} - 68 q^{29} + 1409 q^{31} - 2200 q^{35} + 5346 q^{37} - 750 q^{41} + 14486 q^{43} - 6704 q^{47} + 19005 q^{49}+ \cdots + 43184 q^{97}+O(q^{100}) \) 5 * q - 125 * q^5 + 88 * q^7 - 134 * q^11 - 88 * q^13 - 211 * q^17 + 857 * q^19 - 461 * q^23 + 3125 * q^25 - 68 * q^29 + 1409 * q^31 - 2200 * q^35 + 5346 * q^37 - 750 * q^41 + 14486 * q^43 - 6704 * q^47 + 19005 * q^49 - 8583 * q^53 + 3350 * q^55 - 25384 * q^59 + 38885 * q^61 + 2200 * q^65 + 23146 * q^67 - 62910 * q^71 + 54342 * q^73 - 5368 * q^77 + 24207 * q^79 - 20179 * q^83 + 5275 * q^85 + 10230 * q^89 + 41752 * q^91 - 21425 * q^95 + 43184 * q^97

Introduction

L-functions

Modular forms

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