LMFDB - Embedded newform 1728.4.a.bu.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(1,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-10,0,-19,0,0,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$101.955300490$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2708.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x - 7 $ x^3 - x^2 - 11*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.716463$ of defining polynomial
Character	$\chi$	$=$	1728.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+10.9734 q^{5} -22.3492 q^{7} +31.7250 q^{11} +68.2426 q^{13} -15.7250 q^{17} -52.8934 q^{19} -178.210 q^{23} -4.58534 q^{25} -294.687 q^{29} -78.2837 q^{31} -245.245 q^{35} +323.628 q^{37} +134.497 q^{41} -144.202 q^{43} +155.590 q^{47} +156.485 q^{49} +232.717 q^{53} +348.130 q^{55} -355.045 q^{59} -463.698 q^{61} +748.851 q^{65} +750.314 q^{67} -299.591 q^{71} +605.426 q^{73} -709.027 q^{77} -1004.10 q^{79} -381.791 q^{83} -172.556 q^{85} +1224.31 q^{89} -1525.17 q^{91} -580.419 q^{95} -895.299 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 10 q^{5} - 19 q^{7} + 42 q^{11} - 15 q^{13} + 6 q^{17} + 13 q^{19} - 42 q^{23} + 213 q^{25} - 328 q^{29} + 88 q^{31} + 90 q^{35} + 99 q^{37} + 424 q^{41} - 316 q^{43} + 534 q^{47} + 30 q^{49} - 692 q^{53}+ \cdots - 1197 q^{97}+O(q^{100}) $ 3 * q - 10 * q^5 - 19 * q^7 + 42 * q^11 - 15 * q^13 + 6 * q^17 + 13 * q^19 - 42 * q^23 + 213 * q^25 - 328 * q^29 + 88 * q^31 + 90 * q^35 + 99 * q^37 + 424 * q^41 - 316 * q^43 + 534 * q^47 + 30 * q^49 - 692 * q^53 - 748 * q^55 + 186 * q^59 + 147 * q^61 + 2242 * q^65 - 619 * q^67 + 540 * q^71 - 381 * q^73 - 2170 * q^77 - 1241 * q^79 - 132 * q^83 + 588 * q^85 + 2222 * q^89 - 737 * q^91 - 2262 * q^95 - 1197 * 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
$5$	10.9734	0.981487	0.490744	−	0.871304i	$-0.336725\pi$
$5$	10.9734	0.981487	0.490744	+	0.871304i	$0.336725\pi$
$6$	0	0
$7$	−22.3492	−1.20674	−0.603371	−	0.797461i	$-0.706176\pi$
$7$	−22.3492	−1.20674	−0.603371	+	0.797461i	$0.706176\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	31.7250	0.869585	0.434793	−	0.900531i	$-0.356822\pi$
$11$	31.7250	0.869585	0.434793	+	0.900531i	$0.356822\pi$
$12$	0	0
$13$	68.2426	1.45593	0.727965	−	0.685614i	$-0.240466\pi$
$13$	68.2426	1.45593	0.727965	+	0.685614i	$0.240466\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−15.7250	−0.224345	−0.112173	−	0.993689i	$-0.535781\pi$
$17$	−15.7250	−0.224345	−0.112173	+	0.993689i	$0.535781\pi$
$18$	0	0
$19$	−52.8934	−0.638663	−0.319331	−	0.947643i	$-0.603458\pi$
$19$	−52.8934	−0.638663	−0.319331	+	0.947643i	$0.603458\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−178.210	−1.61563	−0.807813	−	0.589439i	$-0.799349\pi$
$23$	−178.210	−1.61563	−0.807813	+	0.589439i	$0.799349\pi$
$24$	0	0
$25$	−4.58534	−0.0366827
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−294.687	−1.88696	−0.943482	−	0.331424i	$-0.892471\pi$
$29$	−294.687	−1.88696	−0.943482	+	0.331424i	$0.892471\pi$
$30$	0	0
$31$	−78.2837	−0.453554	−0.226777	−	0.973947i	$-0.572819\pi$
$31$	−78.2837	−0.453554	−0.226777	+	0.973947i	$0.572819\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−245.245	−1.18440
$36$	0	0
$37$	323.628	1.43795	0.718974	−	0.695037i	$-0.244612\pi$
$37$	323.628	1.43795	0.718974	+	0.695037i	$0.244612\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	134.497	0.512314	0.256157	−	0.966635i	$-0.417544\pi$
$41$	134.497	0.512314	0.256157	+	0.966635i	$0.417544\pi$
$42$	0	0
$43$	−144.202	−0.511408	−0.255704	−	0.966755i	$-0.582307\pi$
$43$	−144.202	−0.511408	−0.255704	+	0.966755i	$0.582307\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	155.590	0.482874	0.241437	−	0.970417i	$-0.422381\pi$
$47$	155.590	0.482874	0.241437	+	0.970417i	$0.422381\pi$
$48$	0	0
$49$	156.485	0.456225
$50$	0	0
$51$	0	0
$52$	0	0
$53$	232.717	0.603135	0.301567	−	0.953445i	$-0.402490\pi$
$53$	232.717	0.603135	0.301567	+	0.953445i	$0.402490\pi$
$54$	0	0
$55$	348.130	0.853487
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−355.045	−0.783440	−0.391720	−	0.920084i	$-0.628120\pi$
$59$	−355.045	−0.783440	−0.391720	+	0.920084i	$0.628120\pi$
$60$	0	0
$61$	−463.698	−0.973287	−0.486643	−	0.873601i	$-0.661779\pi$
$61$	−463.698	−0.973287	−0.486643	+	0.873601i	$0.661779\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	748.851	1.42898
$66$	0	0
$67$	750.314	1.36814	0.684071	−	0.729416i	$-0.260208\pi$
$67$	750.314	1.36814	0.684071	+	0.729416i	$0.260208\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−299.591	−0.500773	−0.250387	−	0.968146i	$-0.580558\pi$
$71$	−299.591	−0.500773	−0.250387	+	0.968146i	$0.580558\pi$
$72$	0	0
$73$	605.426	0.970682	0.485341	−	0.874325i	$-0.338696\pi$
$73$	605.426	0.970682	0.485341	+	0.874325i	$0.338696\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−709.027	−1.04936
$78$	0	0
$79$	−1004.10	−1.43000	−0.715001	−	0.699123i	$-0.753574\pi$
$79$	−1004.10	−1.43000	−0.715001	+	0.699123i	$0.753574\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−381.791	−0.504904	−0.252452	−	0.967609i	$-0.581237\pi$
$83$	−381.791	−0.504904	−0.252452	+	0.967609i	$0.581237\pi$
$84$	0	0
$85$	−172.556	−0.220192
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1224.31	1.45816	0.729082	−	0.684427i	$-0.239947\pi$
$89$	1224.31	1.45816	0.729082	+	0.684427i	$0.239947\pi$
$90$	0	0
$91$	−1525.17	−1.75693
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−580.419	−0.626839
$96$	0	0
$97$	−895.299	−0.937153	−0.468577	−	0.883423i	$-0.655233\pi$
$97$	−895.299	−0.937153	−0.468577	+	0.883423i	$0.655233\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1976.47	−1.94719	−0.973596	−	0.228279i	$-0.926690\pi$
$101$	−1976.47	−1.94719	−0.973596	+	0.228279i	$0.926690\pi$
$102$	0	0
$103$	−77.3991	−0.0740423	−0.0370212	−	0.999314i	$-0.511787\pi$
$103$	−77.3991	−0.0740423	−0.0370212	+	0.999314i	$0.511787\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1478.93	1.33620	0.668099	−	0.744072i	$-0.267108\pi$
$107$	1478.93	1.33620	0.668099	+	0.744072i	$0.267108\pi$
$108$	0	0
$109$	−1541.75	−1.35480	−0.677401	−	0.735614i	$-0.736894\pi$
$109$	−1541.75	−1.35480	−0.677401	+	0.735614i	$0.736894\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1333.67	1.11028	0.555138	−	0.831758i	$-0.312665\pi$
$113$	1333.67	1.11028	0.555138	+	0.831758i	$0.312665\pi$
$114$	0	0
$115$	−1955.56	−1.58572
$116$	0	0
$117$	0	0
$118$	0	0
$119$	351.440	0.270726
$120$	0	0
$121$	−324.526	−0.243821
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1421.99	−1.01749
$126$	0	0
$127$	−1747.86	−1.22124	−0.610620	−	0.791924i	$-0.709080\pi$
$127$	−1747.86	−1.22124	−0.610620	+	0.791924i	$0.709080\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1721.94	1.14845	0.574224	−	0.818698i	$-0.305304\pi$
$131$	1721.94	1.14845	0.574224	+	0.818698i	$0.305304\pi$
$132$	0	0
$133$	1182.12	0.770701
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1668.02	−1.04021	−0.520104	−	0.854103i	$-0.674107\pi$
$137$	−1668.02	−1.04021	−0.520104	+	0.854103i	$0.674107\pi$
$138$	0	0
$139$	−0.0513537	−3.13364e−5 0	−1.56682e−5	−	1.00000i	$-0.500005\pi$
$139$	−0.0513537	−3.13364e−5 0	−1.56682e−5		1.00000i	$0.500005\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2164.99	1.26606
$144$	0	0
$145$	−3233.70	−1.85203
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−750.148	−0.412447	−0.206223	−	0.978505i	$-0.566117\pi$
$149$	−750.148	−0.412447	−0.206223	+	0.978505i	$0.566117\pi$
$150$	0	0
$151$	−2096.18	−1.12970	−0.564849	−	0.825194i	$-0.691066\pi$
$151$	−2096.18	−1.12970	−0.564849	+	0.825194i	$0.691066\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−859.035	−0.445157
$156$	0	0
$157$	732.068	0.372136	0.186068	−	0.982537i	$-0.440426\pi$
$157$	732.068	0.372136	0.186068	+	0.982537i	$0.440426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3982.85	1.94964
$162$	0	0
$163$	−881.402	−0.423538	−0.211769	−	0.977320i	$-0.567922\pi$
$163$	−881.402	−0.423538	−0.211769	+	0.977320i	$0.567922\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1084.11	−0.502342	−0.251171	−	0.967943i	$-0.580816\pi$
$167$	−1084.11	−0.502342	−0.251171	+	0.967943i	$0.580816\pi$
$168$	0	0
$169$	2460.05	1.11973
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1367.53	−0.600992	−0.300496	−	0.953783i	$-0.597152\pi$
$173$	−1367.53	−0.600992	−0.300496	+	0.953783i	$0.597152\pi$
$174$	0	0
$175$	102.479	0.0442666
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4353.18	−1.81772	−0.908861	−	0.417100i	$-0.863047\pi$
$179$	−4353.18	−1.81772	−0.908861	+	0.417100i	$0.863047\pi$
$180$	0	0
$181$	1221.81	0.501749	0.250874	−	0.968020i	$-0.419282\pi$
$181$	1221.81	0.501749	0.250874	+	0.968020i	$0.419282\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3551.28	1.41133
$186$	0	0
$187$	−498.874	−0.195087
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4893.29	1.85375	0.926874	−	0.375374i	$-0.122486\pi$
$191$	4893.29	1.85375	0.926874	+	0.375374i	$0.122486\pi$
$192$	0	0
$193$	1313.07	0.489724	0.244862	−	0.969558i	$-0.421257\pi$
$193$	1313.07	0.489724	0.244862	+	0.969558i	$0.421257\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1433.46	−0.518424	−0.259212	−	0.965820i	$-0.583463\pi$
$197$	−1433.46	−0.518424	−0.259212	+	0.965820i	$0.583463\pi$
$198$	0	0
$199$	−2195.05	−0.781926	−0.390963	−	0.920406i	$-0.627858\pi$
$199$	−2195.05	−0.781926	−0.390963	+	0.920406i	$0.627858\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6586.00	2.27708
$204$	0	0
$205$	1475.88	0.502830
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1678.04	−0.555372
$210$	0	0
$211$	−5087.29	−1.65983	−0.829914	−	0.557891i	$-0.811611\pi$
$211$	−5087.29	−1.65983	−0.829914	+	0.557891i	$0.811611\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1582.38	−0.501940
$216$	0	0
$217$	1749.57	0.547322
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1073.11	−0.326631
$222$	0	0
$223$	−259.388	−0.0778918	−0.0389459	−	0.999241i	$-0.512400\pi$
$223$	−259.388	−0.0778918	−0.0389459	+	0.999241i	$0.512400\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2426.46	−0.709471	−0.354736	−	0.934967i	$-0.615429\pi$
$227$	−2426.46	−0.709471	−0.354736	+	0.934967i	$0.615429\pi$
$228$	0	0
$229$	1531.81	0.442029	0.221014	−	0.975271i	$-0.429063\pi$
$229$	1531.81	0.442029	0.221014	+	0.975271i	$0.429063\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	199.568	0.0561121	0.0280560	−	0.999606i	$-0.491068\pi$
$233$	199.568	0.0561121	0.0280560	+	0.999606i	$0.491068\pi$
$234$	0	0
$235$	1707.34	0.473935
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3220.94	−0.871737	−0.435869	−	0.900010i	$-0.643559\pi$
$239$	−3220.94	−0.871737	−0.435869	+	0.900010i	$0.643559\pi$
$240$	0	0
$241$	4118.46	1.10080	0.550401	−	0.834900i	$-0.314475\pi$
$241$	4118.46	1.10080	0.550401	+	0.834900i	$0.314475\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1717.17	0.447779
$246$	0	0
$247$	−3609.59	−0.929848
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2819.39	0.708998	0.354499	−	0.935056i	$-0.384651\pi$
$251$	2819.39	0.708998	0.354499	+	0.935056i	$0.384651\pi$
$252$	0	0
$253$	−5653.71	−1.40492
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6147.72	−1.49216	−0.746078	−	0.665859i	$-0.768065\pi$
$257$	−6147.72	−1.49216	−0.746078	+	0.665859i	$0.768065\pi$
$258$	0	0
$259$	−7232.81	−1.73523
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2733.39	−0.640866	−0.320433	−	0.947271i	$-0.603828\pi$
$263$	−2733.39	−0.640866	−0.320433	+	0.947271i	$0.603828\pi$
$264$	0	0
$265$	2553.69	0.591969
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8246.94	1.86924	0.934619	−	0.355651i	$-0.115741\pi$
$269$	8246.94	1.86924	0.934619	+	0.355651i	$0.115741\pi$
$270$	0	0
$271$	−7355.03	−1.64866	−0.824329	−	0.566111i	$-0.808447\pi$
$271$	−7355.03	−1.64866	−0.824329	+	0.566111i	$0.808447\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−145.470	−0.0318988
$276$	0	0
$277$	−2712.26	−0.588317	−0.294159	−	0.955757i	$-0.595039\pi$
$277$	−2712.26	−0.588317	−0.294159	+	0.955757i	$0.595039\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5128.42	1.08874	0.544370	−	0.838845i	$-0.316769\pi$
$281$	5128.42	1.08874	0.544370	+	0.838845i	$0.316769\pi$
$282$	0	0
$283$	−9016.16	−1.89383	−0.946917	−	0.321478i	$-0.895820\pi$
$283$	−9016.16	−1.89383	−0.946917	+	0.321478i	$0.895820\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3005.89	−0.618230
$288$	0	0
$289$	−4665.73	−0.949669
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2784.33	0.555161	0.277580	−	0.960702i	$-0.410467\pi$
$293$	2784.33	0.555161	0.277580	+	0.960702i	$0.410467\pi$
$294$	0	0
$295$	−3896.04	−0.768936
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12161.5	−2.35224
$300$	0	0
$301$	3222.78	0.617137
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5088.33	−0.955268
$306$	0	0
$307$	−2805.87	−0.521627	−0.260813	−	0.965389i	$-0.583991\pi$
$307$	−2805.87	−0.521627	−0.260813	+	0.965389i	$0.583991\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8723.91	−1.59063	−0.795317	−	0.606193i	$-0.792696\pi$
$311$	−8723.91	−1.59063	−0.795317	+	0.606193i	$0.792696\pi$
$312$	0	0
$313$	−9203.17	−1.66196	−0.830980	−	0.556302i	$-0.812220\pi$
$313$	−9203.17	−1.66196	−0.830980	+	0.556302i	$0.812220\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3298.56	−0.584434	−0.292217	−	0.956352i	$-0.594393\pi$
$317$	−3298.56	−0.584434	−0.292217	+	0.956352i	$0.594393\pi$
$318$	0	0
$319$	−9348.93	−1.64088
$320$	0	0
$321$	0	0
$322$	0	0
$323$	831.748	0.143281
$324$	0	0
$325$	−312.916	−0.0534075
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3477.30	−0.582704
$330$	0	0
$331$	10218.0	1.69677	0.848387	−	0.529376i	$-0.177574\pi$
$331$	10218.0	1.69677	0.848387	+	0.529376i	$0.177574\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8233.47	1.34281
$336$	0	0
$337$	5920.21	0.956957	0.478478	−	0.878099i	$-0.341189\pi$
$337$	5920.21	0.956957	0.478478	+	0.878099i	$0.341189\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2483.55	−0.394404
$342$	0	0
$343$	4168.45	0.656196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5887.39	0.910812	0.455406	−	0.890284i	$-0.349494\pi$
$347$	5887.39	0.910812	0.455406	+	0.890284i	$0.349494\pi$
$348$	0	0
$349$	1594.80	0.244606	0.122303	−	0.992493i	$-0.460972\pi$
$349$	1594.80	0.244606	0.122303	+	0.992493i	$0.460972\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6564.51	−0.989784	−0.494892	−	0.868955i	$-0.664792\pi$
$353$	−6564.51	−0.989784	−0.494892	+	0.868955i	$0.664792\pi$
$354$	0	0
$355$	−3287.52	−0.491503
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11386.2	1.67392	0.836961	−	0.547263i	$-0.184330\pi$
$359$	11386.2	1.67392	0.836961	+	0.547263i	$0.184330\pi$
$360$	0	0
$361$	−4061.28	−0.592110
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6643.56	0.952712
$366$	0	0
$367$	6432.28	0.914884	0.457442	−	0.889240i	$-0.348766\pi$
$367$	6432.28	0.914884	0.457442	+	0.889240i	$0.348766\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5201.03	−0.727827
$372$	0	0
$373$	35.4858	0.00492597	0.00246298	−	0.999997i	$-0.499216\pi$
$373$	35.4858	0.00492597	0.00246298	+	0.999997i	$0.499216\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−20110.2	−2.74729
$378$	0	0
$379$	−10109.0	−1.37009	−0.685046	−	0.728500i	$-0.740218\pi$
$379$	−10109.0	−1.37009	−0.685046	+	0.728500i	$0.740218\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9441.69	−1.25965	−0.629827	−	0.776735i	$-0.716874\pi$
$383$	−9441.69	−1.25965	−0.629827	+	0.776735i	$0.716874\pi$
$384$	0	0
$385$	−7780.41	−1.02994
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9435.27	1.22979	0.614893	−	0.788610i	$-0.289199\pi$
$389$	9435.27	1.22979	0.614893	+	0.788610i	$0.289199\pi$
$390$	0	0
$391$	2802.35	0.362458
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11018.4	−1.40353
$396$	0	0
$397$	−2098.74	−0.265321	−0.132661	−	0.991162i	$-0.542352\pi$
$397$	−2098.74	−0.265321	−0.132661	+	0.991162i	$0.542352\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	73.2567	0.00912285	0.00456143	−	0.999990i	$-0.498548\pi$
$401$	73.2567	0.00912285	0.00456143	+	0.999990i	$0.498548\pi$
$402$	0	0
$403$	−5342.28	−0.660343
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10267.1	1.25042
$408$	0	0
$409$	−8527.41	−1.03094	−0.515469	−	0.856908i	$-0.672382\pi$
$409$	−8527.41	−1.03094	−0.515469	+	0.856908i	$0.672382\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7934.97	0.945409
$414$	0	0
$415$	−4189.53	−0.495557
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11513.7	1.34243	0.671217	−	0.741261i	$-0.265772\pi$
$419$	11513.7	1.34243	0.671217	+	0.741261i	$0.265772\pi$
$420$	0	0
$421$	−4825.49	−0.558622	−0.279311	−	0.960201i	$-0.590106\pi$
$421$	−4825.49	−0.558622	−0.279311	+	0.960201i	$0.590106\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	72.1043	0.00822959
$426$	0	0
$427$	10363.3	1.17451
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3578.02	−0.399877	−0.199939	−	0.979808i	$-0.564074\pi$
$431$	−3578.02	−0.399877	−0.199939	+	0.979808i	$0.564074\pi$
$432$	0	0
$433$	1611.96	0.178905	0.0894523	−	0.995991i	$-0.471488\pi$
$433$	1611.96	0.178905	0.0894523	+	0.995991i	$0.471488\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9426.15	1.03184
$438$	0	0
$439$	−1062.02	−0.115462	−0.0577308	−	0.998332i	$-0.518387\pi$
$439$	−1062.02	−0.115462	−0.0577308	+	0.998332i	$0.518387\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8817.39	0.945659	0.472830	−	0.881154i	$-0.343233\pi$
$443$	8817.39	0.945659	0.472830	+	0.881154i	$0.343233\pi$
$444$	0	0
$445$	13434.8	1.43117
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8813.53	0.926362	0.463181	−	0.886264i	$-0.346708\pi$
$449$	8813.53	0.926362	0.463181	+	0.886264i	$0.346708\pi$
$450$	0	0
$451$	4266.91	0.445501
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16736.2	−1.72441
$456$	0	0
$457$	15573.0	1.59404	0.797020	−	0.603954i	$-0.206409\pi$
$457$	15573.0	1.59404	0.797020	+	0.603954i	$0.206409\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19330.8	1.95298	0.976490	−	0.215564i	$-0.0691591\pi$
$461$	19330.8	1.95298	0.976490	+	0.215564i	$0.0691591\pi$
$462$	0	0
$463$	−18338.0	−1.84069	−0.920343	−	0.391113i	$-0.872090\pi$
$463$	−18338.0	−1.84069	−0.920343	+	0.391113i	$0.872090\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2681.20	0.265677	0.132838	−	0.991138i	$-0.457591\pi$
$467$	2681.20	0.265677	0.132838	+	0.991138i	$0.457591\pi$
$468$	0	0
$469$	−16768.9	−1.65099
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4574.79	−0.444713
$474$	0	0
$475$	242.534	0.0234279
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12846.5	1.22541	0.612703	−	0.790313i	$-0.290082\pi$
$479$	12846.5	1.22541	0.612703	+	0.790313i	$0.290082\pi$
$480$	0	0
$481$	22085.2	2.09355
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9824.44	−0.919804
$486$	0	0
$487$	18380.1	1.71023	0.855114	−	0.518439i	$-0.173487\pi$
$487$	18380.1	1.71023	0.855114	+	0.518439i	$0.173487\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13692.0	1.25847	0.629237	−	0.777214i	$-0.283368\pi$
$491$	13692.0	1.25847	0.629237	+	0.777214i	$0.283368\pi$
$492$	0	0
$493$	4633.94	0.423331
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6695.61	0.604304
$498$	0	0
$499$	−2956.17	−0.265203	−0.132601	−	0.991169i	$-0.542333\pi$
$499$	−2956.17	−0.265203	−0.132601	+	0.991169i	$0.542333\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1778.68	0.157669	0.0788345	−	0.996888i	$-0.474880\pi$
$503$	1778.68	0.157669	0.0788345	+	0.996888i	$0.474880\pi$
$504$	0	0
$505$	−21688.5	−1.91114
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17954.3	−1.56347	−0.781737	−	0.623608i	$-0.785666\pi$
$509$	−17954.3	−1.56347	−0.781737	+	0.623608i	$0.785666\pi$
$510$	0	0
$511$	−13530.8	−1.17136
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−849.328	−0.0726716
$516$	0	0
$517$	4936.07	0.419900
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−374.174	−0.0314642	−0.0157321	−	0.999876i	$-0.505008\pi$
$521$	−374.174	−0.0314642	−0.0157321	+	0.999876i	$0.505008\pi$
$522$	0	0
$523$	−10002.9	−0.836322	−0.418161	−	0.908373i	$-0.637325\pi$
$523$	−10002.9	−0.836322	−0.418161	+	0.908373i	$0.637325\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1231.01	0.101753
$528$	0	0
$529$	19591.9	1.61025
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9178.41	0.745893
$534$	0	0
$535$	16228.8	1.31146
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4964.49	0.396727
$540$	0	0
$541$	−310.732	−0.0246939	−0.0123469	−	0.999924i	$-0.503930\pi$
$541$	−310.732	−0.0246939	−0.0123469	+	0.999924i	$0.503930\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16918.2	−1.32972
$546$	0	0
$547$	−2801.10	−0.218951	−0.109476	−	0.993989i	$-0.534917\pi$
$547$	−2801.10	−0.218951	−0.109476	+	0.993989i	$0.534917\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15587.0	1.20513
$552$	0	0
$553$	22440.8	1.72564
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13842.2	−1.05298	−0.526491	−	0.850181i	$-0.676493\pi$
$557$	−13842.2	−1.05298	−0.526491	+	0.850181i	$0.676493\pi$
$558$	0	0
$559$	−9840.69	−0.744574
$560$	0	0
$561$	0	0
$562$	0	0
$563$	729.262	0.0545910	0.0272955	−	0.999627i	$-0.491310\pi$
$563$	729.262	0.0545910	0.0272955	+	0.999627i	$0.491310\pi$
$564$	0	0
$565$	14634.9	1.08972
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10352.2	−0.762719	−0.381359	−	0.924427i	$-0.624544\pi$
$569$	−10352.2	−0.762719	−0.381359	+	0.924427i	$0.624544\pi$
$570$	0	0
$571$	14872.6	1.09002	0.545008	−	0.838431i	$-0.316527\pi$
$571$	14872.6	1.09002	0.545008	+	0.838431i	$0.316527\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	817.154	0.0592656
$576$	0	0
$577$	−19156.0	−1.38211	−0.691053	−	0.722804i	$-0.742853\pi$
$577$	−19156.0	−1.38211	−0.691053	+	0.722804i	$0.742853\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8532.72	0.609289
$582$	0	0
$583$	7382.94	0.524477
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25367.2	−1.78368	−0.891838	−	0.452354i	$-0.850584\pi$
$587$	−25367.2	−1.78368	−0.891838	+	0.452354i	$0.850584\pi$
$588$	0	0
$589$	4140.69	0.289668
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4024.03	−0.278663	−0.139331	−	0.990246i	$-0.544495\pi$
$593$	−4024.03	−0.278663	−0.139331	+	0.990246i	$0.544495\pi$
$594$	0	0
$595$	3856.48	0.265715
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16835.3	1.14837	0.574183	−	0.818727i	$-0.305320\pi$
$599$	16835.3	1.14837	0.574183	+	0.818727i	$0.305320\pi$
$600$	0	0
$601$	2514.04	0.170632	0.0853159	−	0.996354i	$-0.472810\pi$
$601$	2514.04	0.170632	0.0853159	+	0.996354i	$0.472810\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3561.14	−0.239308
$606$	0	0
$607$	18994.0	1.27009	0.635045	−	0.772475i	$-0.280982\pi$
$607$	18994.0	1.27009	0.635045	+	0.772475i	$0.280982\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10617.8	0.703031
$612$	0	0
$613$	−28340.0	−1.86728	−0.933639	−	0.358215i	$-0.883386\pi$
$613$	−28340.0	−1.86728	−0.933639	+	0.358215i	$0.883386\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27125.7	1.76992	0.884958	−	0.465671i	$-0.154187\pi$
$617$	27125.7	1.76992	0.884958	+	0.465671i	$0.154187\pi$
$618$	0	0
$619$	15468.5	1.00441	0.502207	−	0.864748i	$-0.332522\pi$
$619$	15468.5	1.00441	0.502207	+	0.864748i	$0.332522\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27362.3	−1.75963
$624$	0	0
$625$	−15030.8	−0.961972
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5089.04	−0.322596
$630$	0	0
$631$	18445.8	1.16374	0.581868	−	0.813283i	$-0.302322\pi$
$631$	18445.8	1.16374	0.581868	+	0.813283i	$0.302322\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19179.9	−1.19863
$636$	0	0
$637$	10679.0	0.664232
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11750.4	0.724045	0.362023	−	0.932169i	$-0.382086\pi$
$641$	11750.4	0.724045	0.362023	+	0.932169i	$0.382086\pi$
$642$	0	0
$643$	7999.11	0.490598	0.245299	−	0.969448i	$-0.421114\pi$
$643$	7999.11	0.490598	0.245299	+	0.969448i	$0.421114\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2713.51	0.164883	0.0824414	−	0.996596i	$-0.473728\pi$
$647$	2713.51	0.164883	0.0824414	+	0.996596i	$0.473728\pi$
$648$	0	0
$649$	−11263.8	−0.681268
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20082.2	−1.20349	−0.601745	−	0.798688i	$-0.705528\pi$
$653$	−20082.2	−1.20349	−0.601745	+	0.798688i	$0.705528\pi$
$654$	0	0
$655$	18895.5	1.12719
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11656.6	−0.689038	−0.344519	−	0.938779i	$-0.611958\pi$
$659$	−11656.6	−0.689038	−0.344519	+	0.938779i	$0.611958\pi$
$660$	0	0
$661$	32515.7	1.91334	0.956668	−	0.291181i	$-0.0940481\pi$
$661$	32515.7	1.91334	0.956668	+	0.291181i	$0.0940481\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12971.9	0.756433
$666$	0	0
$667$	52516.2	3.04863
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−14710.8	−0.846356
$672$	0	0
$673$	−25249.0	−1.44618	−0.723089	−	0.690755i	$-0.757278\pi$
$673$	−25249.0	−1.44618	−0.723089	+	0.690755i	$0.757278\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27184.3	−1.54324	−0.771622	−	0.636081i	$-0.780554\pi$
$677$	−27184.3	−1.54324	−0.771622	+	0.636081i	$0.780554\pi$
$678$	0	0
$679$	20009.2	1.13090
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10884.0	−0.609758	−0.304879	−	0.952391i	$-0.598616\pi$
$683$	−10884.0	−0.609758	−0.304879	+	0.952391i	$0.598616\pi$
$684$	0	0
$685$	−18303.8	−1.02095
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15881.2	0.878122
$690$	0	0
$691$	−24807.1	−1.36571	−0.682856	−	0.730553i	$-0.739262\pi$
$691$	−24807.1	−1.36571	−0.682856	+	0.730553i	$0.739262\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.563522	−3.07563e−5 0
$696$	0	0
$697$	−2114.96	−0.114935
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2863.18	0.154266	0.0771331	−	0.997021i	$-0.475423\pi$
$701$	2863.18	0.154266	0.0771331	+	0.997021i	$0.475423\pi$
$702$	0	0
$703$	−17117.8	−0.918363
$704$	0	0
$705$	0	0
$706$	0	0
$707$	44172.5	2.34976
$708$	0	0
$709$	−4449.63	−0.235697	−0.117849	−	0.993032i	$-0.537600\pi$
$709$	−4449.63	−0.235697	−0.117849	+	0.993032i	$0.537600\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13950.9	0.732773
$714$	0	0
$715$	23757.3	1.24262
$716$	0	0
$717$	0	0
$718$	0	0
$719$	676.876	0.0351088	0.0175544	−	0.999846i	$-0.494412\pi$
$719$	676.876	0.0351088	0.0175544	+	0.999846i	$0.494412\pi$
$720$	0	0
$721$	1729.81	0.0893500
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1351.24	0.0692190
$726$	0	0
$727$	−9843.01	−0.502142	−0.251071	−	0.967969i	$-0.580783\pi$
$727$	−9843.01	−0.502142	−0.251071	+	0.967969i	$0.580783\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2267.57	0.114732
$732$	0	0
$733$	−27638.2	−1.39269	−0.696344	−	0.717708i	$-0.745191\pi$
$733$	−27638.2	−1.39269	−0.696344	+	0.717708i	$0.745191\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23803.7	1.18972
$738$	0	0
$739$	−14257.1	−0.709684	−0.354842	−	0.934926i	$-0.615465\pi$
$739$	−14257.1	−0.709684	−0.354842	+	0.934926i	$0.615465\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1294.05	0.0638953	0.0319476	−	0.999490i	$-0.489829\pi$
$743$	1294.05	0.0638953	0.0319476	+	0.999490i	$0.489829\pi$
$744$	0	0
$745$	−8231.65	−0.404811
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−33052.8	−1.61245
$750$	0	0
$751$	24664.1	1.19841	0.599204	−	0.800596i	$-0.295484\pi$
$751$	24664.1	1.19841	0.599204	+	0.800596i	$0.295484\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−23002.1	−1.10878
$756$	0	0
$757$	−11814.1	−0.567226	−0.283613	−	0.958939i	$-0.591533\pi$
$757$	−11814.1	−0.567226	−0.283613	+	0.958939i	$0.591533\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4637.92	0.220926	0.110463	−	0.993880i	$-0.464767\pi$
$761$	4637.92	0.220926	0.110463	+	0.993880i	$0.464767\pi$
$762$	0	0
$763$	34456.9	1.63489
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24229.2	−1.14063
$768$	0	0
$769$	24182.7	1.13400	0.567002	−	0.823717i	$-0.308103\pi$
$769$	24182.7	1.13400	0.567002	+	0.823717i	$0.308103\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2881.48	0.134075	0.0670373	−	0.997750i	$-0.478645\pi$
$773$	2881.48	0.134075	0.0670373	+	0.997750i	$0.478645\pi$
$774$	0	0
$775$	358.957	0.0166376
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7114.00	−0.327196
$780$	0	0
$781$	−9504.52	−0.435465
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8033.24	0.365247
$786$	0	0
$787$	−14695.4	−0.665610	−0.332805	−	0.942996i	$-0.607995\pi$
$787$	−14695.4	−0.665610	−0.332805	+	0.942996i	$0.607995\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−29806.5	−1.33982
$792$	0	0
$793$	−31644.0	−1.41704
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9344.68	0.415315	0.207657	−	0.978202i	$-0.433416\pi$
$797$	9344.68	0.415315	0.207657	+	0.978202i	$0.433416\pi$
$798$	0	0
$799$	−2446.64	−0.108330
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19207.1	0.844091
$804$	0	0
$805$	43705.2	1.91355
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−15747.4	−0.684360	−0.342180	−	0.939634i	$-0.611165\pi$
$809$	−15747.4	−0.684360	−0.342180	+	0.939634i	$0.611165\pi$
$810$	0	0
$811$	−1830.55	−0.0792595	−0.0396297	−	0.999214i	$-0.512618\pi$
$811$	−1830.55	−0.0792595	−0.0396297	+	0.999214i	$0.512618\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9671.94	−0.415697
$816$	0	0
$817$	7627.32	0.326617
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23333.2	−0.991881	−0.495941	−	0.868356i	$-0.665177\pi$
$821$	−23333.2	−0.991881	−0.495941	+	0.868356i	$0.665177\pi$
$822$	0	0
$823$	35007.3	1.48272	0.741360	−	0.671107i	$-0.234181\pi$
$823$	35007.3	1.48272	0.741360	+	0.671107i	$0.234181\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7201.59	−0.302810	−0.151405	−	0.988472i	$-0.548380\pi$
$827$	−7201.59	−0.302810	−0.151405	+	0.988472i	$0.548380\pi$
$828$	0	0
$829$	−6531.90	−0.273658	−0.136829	−	0.990595i	$-0.543691\pi$
$829$	−6531.90	−0.273658	−0.136829	+	0.990595i	$0.543691\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2460.73	−0.102352
$834$	0	0
$835$	−11896.3	−0.493042
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4500.83	−0.185204	−0.0926019	−	0.995703i	$-0.529518\pi$
$839$	−4500.83	−0.185204	−0.0926019	+	0.995703i	$0.529518\pi$
$840$	0	0
$841$	62451.3	2.56063
$842$	0	0
$843$	0	0
$844$	0	0
$845$	26995.1	1.09900
$846$	0	0
$847$	7252.89	0.294229
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−57673.8	−2.32319
$852$	0	0
$853$	−25904.8	−1.03982	−0.519909	−	0.854222i	$-0.674034\pi$
$853$	−25904.8	−1.03982	−0.519909	+	0.854222i	$0.674034\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29705.0	1.18402	0.592009	−	0.805931i	$-0.298335\pi$
$857$	29705.0	1.18402	0.592009	+	0.805931i	$0.298335\pi$
$858$	0	0
$859$	16569.3	0.658135	0.329068	−	0.944306i	$-0.393266\pi$
$859$	16569.3	0.658135	0.329068	+	0.944306i	$0.393266\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28469.9	1.12297	0.561486	−	0.827486i	$-0.310230\pi$
$863$	28469.9	1.12297	0.561486	+	0.827486i	$0.310230\pi$
$864$	0	0
$865$	−15006.4	−0.589866
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−31855.1	−1.24351
$870$	0	0
$871$	51203.4	1.99192
$872$	0	0
$873$	0	0
$874$	0	0
$875$	31780.2	1.22785
$876$	0	0
$877$	−2050.79	−0.0789626	−0.0394813	−	0.999220i	$-0.512571\pi$
$877$	−2050.79	−0.0789626	−0.0394813	+	0.999220i	$0.512571\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28166.2	1.07712	0.538561	−	0.842587i	$-0.318968\pi$
$881$	28166.2	1.07712	0.538561	+	0.842587i	$0.318968\pi$
$882$	0	0
$883$	22424.3	0.854631	0.427316	−	0.904103i	$-0.359459\pi$
$883$	22424.3	0.854631	0.427316	+	0.904103i	$0.359459\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37071.4	1.40331	0.701654	−	0.712518i	$-0.252445\pi$
$887$	37071.4	1.40331	0.701654	+	0.712518i	$0.252445\pi$
$888$	0	0
$889$	39063.2	1.47372
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8229.67	−0.308393
$894$	0	0
$895$	−47769.0	−1.78407
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23069.2	0.855839
$900$	0	0
$901$	−3659.47	−0.135310
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13407.4	0.492460
$906$	0	0
$907$	19603.1	0.717651	0.358826	−	0.933405i	$-0.383177\pi$
$907$	19603.1	0.717651	0.358826	+	0.933405i	$0.383177\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43459.1	1.58053	0.790267	−	0.612763i	$-0.209942\pi$
$911$	43459.1	1.58053	0.790267	+	0.612763i	$0.209942\pi$
$912$	0	0
$913$	−12112.3	−0.439057
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−38483.9	−1.38588
$918$	0	0
$919$	12670.6	0.454804	0.227402	−	0.973801i	$-0.426977\pi$
$919$	12670.6	0.454804	0.227402	+	0.973801i	$0.426977\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−20444.9	−0.729091
$924$	0	0
$925$	−1483.94	−0.0527478
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20800.7	−0.734606	−0.367303	−	0.930101i	$-0.619719\pi$
$929$	−20800.7	−0.734606	−0.367303	+	0.930101i	$0.619719\pi$
$930$	0	0
$931$	−8277.04	−0.291374
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5474.33	−0.191476
$936$	0	0
$937$	15332.7	0.534574	0.267287	−	0.963617i	$-0.413873\pi$
$937$	15332.7	0.534574	0.267287	+	0.963617i	$0.413873\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3646.38	−0.126321	−0.0631607	−	0.998003i	$-0.520118\pi$
$941$	−3646.38	−0.126321	−0.0631607	+	0.998003i	$0.520118\pi$
$942$	0	0
$943$	−23968.7	−0.827708
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6052.46	0.207686	0.103843	−	0.994594i	$-0.466886\pi$
$947$	6052.46	0.207686	0.103843	+	0.994594i	$0.466886\pi$
$948$	0	0
$949$	41315.9	1.41325
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15214.3	0.517145	0.258573	−	0.965992i	$-0.416748\pi$
$953$	15214.3	0.517145	0.258573	+	0.965992i	$0.416748\pi$
$954$	0	0
$955$	53695.8	1.81943
$956$	0	0
$957$	0	0
$958$	0	0
$959$	37278.8	1.25526
$960$	0	0
$961$	−23662.7	−0.794289
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14408.8	0.480658
$966$	0	0
$967$	4933.04	0.164049	0.0820247	−	0.996630i	$-0.473861\pi$
$967$	4933.04	0.164049	0.0820247	+	0.996630i	$0.473861\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37446.1	1.23759	0.618796	−	0.785552i	$-0.287621\pi$
$971$	37446.1	1.23759	0.618796	+	0.785552i	$0.287621\pi$
$972$	0	0
$973$	1.14771	3.78150e−5 0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19752.7	−0.646822	−0.323411	−	0.946259i	$-0.604830\pi$
$977$	−19752.7	−0.646822	−0.323411	+	0.946259i	$0.604830\pi$
$978$	0	0
$979$	38841.2	1.26800
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16074.6	−0.521567	−0.260783	−	0.965397i	$-0.583981\pi$
$983$	−16074.6	−0.521567	−0.260783	+	0.965397i	$0.583981\pi$
$984$	0	0
$985$	−15729.8	−0.508826
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25698.2	0.826243
$990$	0	0
$991$	31583.2	1.01239	0.506193	−	0.862420i	$-0.331052\pi$
$991$	31583.2	1.01239	0.506193	+	0.862420i	$0.331052\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24087.1	−0.767451
$996$	0	0
$997$	31340.6	0.995554	0.497777	−	0.867305i	$-0.334150\pi$
$997$	31340.6	0.995554	0.497777	+	0.867305i	$0.334150\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.a.bu.1.3		3
3.2	odd	2		1728.4.a.ce.1.1		3
4.3	odd	2		1728.4.a.bv.1.3		3
8.3	odd	2		864.4.a.t.1.1	yes	3
8.5	even	2		864.4.a.s.1.1	yes	3
12.11	even	2		1728.4.a.cf.1.1		3
24.5	odd	2		864.4.a.i.1.3	✓	3
24.11	even	2		864.4.a.j.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.4.a.i.1.3	✓	3	24.5	odd	2
864.4.a.j.1.3	yes	3	24.11	even	2
864.4.a.s.1.1	yes	3	8.5	even	2
864.4.a.t.1.1	yes	3	8.3	odd	2
1728.4.a.bu.1.3		3	1.1	even	1	trivial
1728.4.a.bv.1.3		3	4.3	odd	2
1728.4.a.ce.1.1		3	3.2	odd	2
1728.4.a.cf.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

\(f(q)\)	\(=\)	\(q+10.9734 q^{5} -22.3492 q^{7} +31.7250 q^{11} +68.2426 q^{13} -15.7250 q^{17} -52.8934 q^{19} -178.210 q^{23} -4.58534 q^{25} -294.687 q^{29} -78.2837 q^{31} -245.245 q^{35} +323.628 q^{37} +134.497 q^{41} -144.202 q^{43} +155.590 q^{47} +156.485 q^{49} +232.717 q^{53} +348.130 q^{55} -355.045 q^{59} -463.698 q^{61} +748.851 q^{65} +750.314 q^{67} -299.591 q^{71} +605.426 q^{73} -709.027 q^{77} -1004.10 q^{79} -381.791 q^{83} -172.556 q^{85} +1224.31 q^{89} -1525.17 q^{91} -580.419 q^{95} -895.299 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 10 q^{5} - 19 q^{7} + 42 q^{11} - 15 q^{13} + 6 q^{17} + 13 q^{19} - 42 q^{23} + 213 q^{25} - 328 q^{29} + 88 q^{31} + 90 q^{35} + 99 q^{37} + 424 q^{41} - 316 q^{43} + 534 q^{47} + 30 q^{49} - 692 q^{53}+ \cdots - 1197 q^{97}+O(q^{100}) \) 3 * q - 10 * q^5 - 19 * q^7 + 42 * q^11 - 15 * q^13 + 6 * q^17 + 13 * q^19 - 42 * q^23 + 213 * q^25 - 328 * q^29 + 88 * q^31 + 90 * q^35 + 99 * q^37 + 424 * q^41 - 316 * q^43 + 534 * q^47 + 30 * q^49 - 692 * q^53 - 748 * q^55 + 186 * q^59 + 147 * q^61 + 2242 * q^65 - 619 * q^67 + 540 * q^71 - 381 * q^73 - 2170 * q^77 - 1241 * q^79 - 132 * q^83 + 588 * q^85 + 2222 * q^89 - 737 * q^91 - 2262 * q^95 - 1197 * q^97

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