LMFDB - Embedded newform 450.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,4,Mod(1,450)] 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("450.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,0,0,1,-8,0,0,-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:	$26.5508595026$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +1.00000 q^{7} -8.00000 q^{8} -42.0000 q^{11} +67.0000 q^{13} -2.00000 q^{14} +16.0000 q^{16} +54.0000 q^{17} -115.000 q^{19} +84.0000 q^{22} -162.000 q^{23} -134.000 q^{26} +4.00000 q^{28} +210.000 q^{29} -193.000 q^{31} -32.0000 q^{32} -108.000 q^{34} +286.000 q^{37} +230.000 q^{38} -12.0000 q^{41} -263.000 q^{43} -168.000 q^{44} +324.000 q^{46} +414.000 q^{47} -342.000 q^{49} +268.000 q^{52} -192.000 q^{53} -8.00000 q^{56} -420.000 q^{58} -690.000 q^{59} -733.000 q^{61} +386.000 q^{62} +64.0000 q^{64} -299.000 q^{67} +216.000 q^{68} +228.000 q^{71} -938.000 q^{73} -572.000 q^{74} -460.000 q^{76} -42.0000 q^{77} -160.000 q^{79} +24.0000 q^{82} -462.000 q^{83} +526.000 q^{86} +336.000 q^{88} +240.000 q^{89} +67.0000 q^{91} -648.000 q^{92} -828.000 q^{94} +511.000 q^{97} +684.000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.0539949	0.0269975	−	0.999636i	$-0.491405\pi$
$7$	1.00000	0.0539949	0.0269975	+	0.999636i	$0.491405\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−42.0000	−1.15123	−0.575613	−	0.817723i	$-0.695236\pi$
$11$	−42.0000	−1.15123	−0.575613	+	0.817723i	$0.695236\pi$
$12$	0	0
$13$	67.0000	1.42942	0.714710	−	0.699421i	$-0.246559\pi$
$13$	67.0000	1.42942	0.714710	+	0.699421i	$0.246559\pi$
$14$	−2.00000	−0.0381802
$15$	0	0
$16$	16.0000	0.250000
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	−115.000	−1.38857	−0.694284	−	0.719701i	$-0.744279\pi$
$19$	−115.000	−1.38857	−0.694284	+	0.719701i	$0.744279\pi$
$20$	0	0
$21$	0	0
$22$	84.0000	0.814039
$23$	−162.000	−1.46867	−0.734333	−	0.678789i	$-0.762505\pi$
$23$	−162.000	−1.46867	−0.734333	+	0.678789i	$0.762505\pi$
$24$	0	0
$25$	0	0
$26$	−134.000	−1.01075
$27$	0	0
$28$	4.00000	0.0269975
$29$	210.000	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$29$	210.000	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$30$	0	0
$31$	−193.000	−1.11819	−0.559094	−	0.829104i	$-0.688851\pi$
$31$	−193.000	−1.11819	−0.559094	+	0.829104i	$0.688851\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−108.000	−0.544760
$35$	0	0
$36$	0	0
$37$	286.000	1.27076	0.635380	−	0.772200i	$-0.280844\pi$
$37$	286.000	1.27076	0.635380	+	0.772200i	$0.280844\pi$
$38$	230.000	0.981866
$39$	0	0
$40$	0	0
$41$	−12.0000	−0.0457094	−0.0228547	−	0.999739i	$-0.507276\pi$
$41$	−12.0000	−0.0457094	−0.0228547	+	0.999739i	$0.507276\pi$
$42$	0	0
$43$	−263.000	−0.932724	−0.466362	−	0.884594i	$-0.654436\pi$
$43$	−263.000	−0.932724	−0.466362	+	0.884594i	$0.654436\pi$
$44$	−168.000	−0.575613
$45$	0	0
$46$	324.000	1.03850
$47$	414.000	1.28485	0.642427	−	0.766347i	$-0.277928\pi$
$47$	414.000	1.28485	0.642427	+	0.766347i	$0.277928\pi$
$48$	0	0
$49$	−342.000	−0.997085
$50$	0	0
$51$	0	0
$52$	268.000	0.714710
$53$	−192.000	−0.497608	−0.248804	−	0.968554i	$-0.580038\pi$
$53$	−192.000	−0.497608	−0.248804	+	0.968554i	$0.580038\pi$
$54$	0	0
$55$	0	0
$56$	−8.00000	−0.0190901
$57$	0	0
$58$	−420.000	−0.950840
$59$	−690.000	−1.52255	−0.761274	−	0.648430i	$-0.775426\pi$
$59$	−690.000	−1.52255	−0.761274	+	0.648430i	$0.775426\pi$
$60$	0	0
$61$	−733.000	−1.53854	−0.769271	−	0.638923i	$-0.779380\pi$
$61$	−733.000	−1.53854	−0.769271	+	0.638923i	$0.779380\pi$
$62$	386.000	0.790678
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−299.000	−0.545204	−0.272602	−	0.962127i	$-0.587884\pi$
$67$	−299.000	−0.545204	−0.272602	+	0.962127i	$0.587884\pi$
$68$	216.000	0.385204
$69$	0	0
$70$	0	0
$71$	228.000	0.381107	0.190554	−	0.981677i	$-0.438972\pi$
$71$	228.000	0.381107	0.190554	+	0.981677i	$0.438972\pi$
$72$	0	0
$73$	−938.000	−1.50390	−0.751949	−	0.659221i	$-0.770886\pi$
$73$	−938.000	−1.50390	−0.751949	+	0.659221i	$0.770886\pi$
$74$	−572.000	−0.898563
$75$	0	0
$76$	−460.000	−0.694284
$77$	−42.0000	−0.0621603
$78$	0	0
$79$	−160.000	−0.227866	−0.113933	−	0.993488i	$-0.536345\pi$
$79$	−160.000	−0.227866	−0.113933	+	0.993488i	$0.536345\pi$
$80$	0	0
$81$	0	0
$82$	24.0000	0.0323214
$83$	−462.000	−0.610977	−0.305488	−	0.952196i	$-0.598820\pi$
$83$	−462.000	−0.610977	−0.305488	+	0.952196i	$0.598820\pi$
$84$	0	0
$85$	0	0
$86$	526.000	0.659535
$87$	0	0
$88$	336.000	0.407020
$89$	240.000	0.285842	0.142921	−	0.989734i	$-0.454350\pi$
$89$	240.000	0.285842	0.142921	+	0.989734i	$0.454350\pi$
$90$	0	0
$91$	67.0000	0.0771814
$92$	−648.000	−0.734333
$93$	0	0
$94$	−828.000	−0.908529
$95$	0	0
$96$	0	0
$97$	511.000	0.534889	0.267444	−	0.963573i	$-0.413821\pi$
$97$	511.000	0.534889	0.267444	+	0.963573i	$0.413821\pi$
$98$	684.000	0.705045
$99$	0	0
$100$	0	0
$101$	−912.000	−0.898489	−0.449245	−	0.893409i	$-0.648307\pi$
$101$	−912.000	−0.898489	−0.449245	+	0.893409i	$0.648307\pi$
$102$	0	0
$103$	−668.000	−0.639029	−0.319515	−	0.947581i	$-0.603520\pi$
$103$	−668.000	−0.639029	−0.319515	+	0.947581i	$0.603520\pi$
$104$	−536.000	−0.505376
$105$	0	0
$106$	384.000	0.351862
$107$	−1296.00	−1.17093	−0.585463	−	0.810699i	$-0.699087\pi$
$107$	−1296.00	−1.17093	−0.585463	+	0.810699i	$0.699087\pi$
$108$	0	0
$109$	−1735.00	−1.52461	−0.762307	−	0.647216i	$-0.775933\pi$
$109$	−1735.00	−1.52461	−0.762307	+	0.647216i	$0.775933\pi$
$110$	0	0
$111$	0	0
$112$	16.0000	0.0134987
$113$	−1092.00	−0.909086	−0.454543	−	0.890725i	$-0.650197\pi$
$113$	−1092.00	−0.909086	−0.454543	+	0.890725i	$0.650197\pi$
$114$	0	0
$115$	0	0
$116$	840.000	0.672345
$117$	0	0
$118$	1380.00	1.07660
$119$	54.0000	0.0415981
$120$	0	0
$121$	433.000	0.325319
$122$	1466.00	1.08791
$123$	0	0
$124$	−772.000	−0.559094
$125$	0	0
$126$	0	0
$127$	16.0000	0.0111793	0.00558965	−	0.999984i	$-0.498221\pi$
$127$	16.0000	0.0111793	0.00558965	+	0.999984i	$0.498221\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−1992.00	−1.32856	−0.664282	−	0.747482i	$-0.731263\pi$
$131$	−1992.00	−1.32856	−0.664282	+	0.747482i	$0.731263\pi$
$132$	0	0
$133$	−115.000	−0.0749757
$134$	598.000	0.385517
$135$	0	0
$136$	−432.000	−0.272380
$137$	−2346.00	−1.46301	−0.731505	−	0.681836i	$-0.761182\pi$
$137$	−2346.00	−1.46301	−0.731505	+	0.681836i	$0.761182\pi$
$138$	0	0
$139$	2900.00	1.76960	0.884801	−	0.465968i	$-0.154294\pi$
$139$	2900.00	1.76960	0.884801	+	0.465968i	$0.154294\pi$
$140$	0	0
$141$	0	0
$142$	−456.000	−0.269484
$143$	−2814.00	−1.64558
$144$	0	0
$145$	0	0
$146$	1876.00	1.06342
$147$	0	0
$148$	1144.00	0.635380
$149$	2070.00	1.13813	0.569064	−	0.822293i	$-0.307306\pi$
$149$	2070.00	1.13813	0.569064	+	0.822293i	$0.307306\pi$
$150$	0	0
$151$	2237.00	1.20559	0.602796	−	0.797895i	$-0.294053\pi$
$151$	2237.00	1.20559	0.602796	+	0.797895i	$0.294053\pi$
$152$	920.000	0.490933
$153$	0	0
$154$	84.0000	0.0439540
$155$	0	0
$156$	0	0
$157$	241.000	0.122509	0.0612544	−	0.998122i	$-0.480490\pi$
$157$	241.000	0.122509	0.0612544	+	0.998122i	$0.480490\pi$
$158$	320.000	0.161126
$159$	0	0
$160$	0	0
$161$	−162.000	−0.0793006
$162$	0	0
$163$	3547.00	1.70443	0.852216	−	0.523190i	$-0.175258\pi$
$163$	3547.00	1.70443	0.852216	+	0.523190i	$0.175258\pi$
$164$	−48.0000	−0.0228547
$165$	0	0
$166$	924.000	0.432026
$167$	984.000	0.455953	0.227977	−	0.973667i	$-0.426789\pi$
$167$	984.000	0.455953	0.227977	+	0.973667i	$0.426789\pi$
$168$	0	0
$169$	2292.00	1.04324
$170$	0	0
$171$	0	0
$172$	−1052.00	−0.466362
$173$	3618.00	1.59001	0.795004	−	0.606604i	$-0.207469\pi$
$173$	3618.00	1.59001	0.795004	+	0.606604i	$0.207469\pi$
$174$	0	0
$175$	0	0
$176$	−672.000	−0.287806
$177$	0	0
$178$	−480.000	−0.202121
$179$	150.000	0.0626342	0.0313171	−	0.999509i	$-0.490030\pi$
$179$	150.000	0.0626342	0.0313171	+	0.999509i	$0.490030\pi$
$180$	0	0
$181$	197.000	0.0809000	0.0404500	−	0.999182i	$-0.487121\pi$
$181$	197.000	0.0809000	0.0404500	+	0.999182i	$0.487121\pi$
$182$	−134.000	−0.0545755
$183$	0	0
$184$	1296.00	0.519252
$185$	0	0
$186$	0	0
$187$	−2268.00	−0.886912
$188$	1656.00	0.642427
$189$	0	0
$190$	0	0
$191$	−1302.00	−0.493243	−0.246622	−	0.969112i	$-0.579320\pi$
$191$	−1302.00	−0.493243	−0.246622	+	0.969112i	$0.579320\pi$
$192$	0	0
$193$	−4163.00	−1.55264	−0.776319	−	0.630340i	$-0.782916\pi$
$193$	−4163.00	−1.55264	−0.776319	+	0.630340i	$0.782916\pi$
$194$	−1022.00	−0.378223
$195$	0	0
$196$	−1368.00	−0.498542
$197$	3054.00	1.10451	0.552255	−	0.833675i	$-0.313767\pi$
$197$	3054.00	1.10451	0.552255	+	0.833675i	$0.313767\pi$
$198$	0	0
$199$	3425.00	1.22006	0.610030	−	0.792379i	$-0.291158\pi$
$199$	3425.00	1.22006	0.610030	+	0.792379i	$0.291158\pi$
$200$	0	0
$201$	0	0
$202$	1824.00	0.635328
$203$	210.000	0.0726065
$204$	0	0
$205$	0	0
$206$	1336.00	0.451862
$207$	0	0
$208$	1072.00	0.357355
$209$	4830.00	1.59856
$210$	0	0
$211$	−2443.00	−0.797076	−0.398538	−	0.917152i	$-0.630482\pi$
$211$	−2443.00	−0.797076	−0.398538	+	0.917152i	$0.630482\pi$
$212$	−768.000	−0.248804
$213$	0	0
$214$	2592.00	0.827969
$215$	0	0
$216$	0	0
$217$	−193.000	−0.0603765
$218$	3470.00	1.07806
$219$	0	0
$220$	0	0
$221$	3618.00	1.10124
$222$	0	0
$223$	−23.0000	−0.00690670	−0.00345335	−	0.999994i	$-0.501099\pi$
$223$	−23.0000	−0.00690670	−0.00345335	+	0.999994i	$0.501099\pi$
$224$	−32.0000	−0.00954504
$225$	0	0
$226$	2184.00	0.642821
$227$	−1956.00	−0.571913	−0.285957	−	0.958243i	$-0.592311\pi$
$227$	−1956.00	−0.571913	−0.285957	+	0.958243i	$0.592311\pi$
$228$	0	0
$229$	1805.00	0.520864	0.260432	−	0.965492i	$-0.416135\pi$
$229$	1805.00	0.520864	0.260432	+	0.965492i	$0.416135\pi$
$230$	0	0
$231$	0	0
$232$	−1680.00	−0.475420
$233$	3468.00	0.975091	0.487546	−	0.873098i	$-0.337892\pi$
$233$	3468.00	0.975091	0.487546	+	0.873098i	$0.337892\pi$
$234$	0	0
$235$	0	0
$236$	−2760.00	−0.761274
$237$	0	0
$238$	−108.000	−0.0294143
$239$	−2640.00	−0.714508	−0.357254	−	0.934007i	$-0.616287\pi$
$239$	−2640.00	−0.714508	−0.357254	+	0.934007i	$0.616287\pi$
$240$	0	0
$241$	−5383.00	−1.43879	−0.719397	−	0.694599i	$-0.755582\pi$
$241$	−5383.00	−1.43879	−0.719397	+	0.694599i	$0.755582\pi$
$242$	−866.000	−0.230035
$243$	0	0
$244$	−2932.00	−0.769271
$245$	0	0
$246$	0	0
$247$	−7705.00	−1.98485
$248$	1544.00	0.395339
$249$	0	0
$250$	0	0
$251$	5028.00	1.26440	0.632200	−	0.774805i	$-0.282152\pi$
$251$	5028.00	1.26440	0.632200	+	0.774805i	$0.282152\pi$
$252$	0	0
$253$	6804.00	1.69077
$254$	−32.0000	−0.00790496
$255$	0	0
$256$	256.000	0.0625000
$257$	564.000	0.136892	0.0684462	−	0.997655i	$-0.478196\pi$
$257$	564.000	0.136892	0.0684462	+	0.997655i	$0.478196\pi$
$258$	0	0
$259$	286.000	0.0686146
$260$	0	0
$261$	0	0
$262$	3984.00	0.939436
$263$	−1812.00	−0.424839	−0.212420	−	0.977179i	$-0.568134\pi$
$263$	−1812.00	−0.424839	−0.212420	+	0.977179i	$0.568134\pi$
$264$	0	0
$265$	0	0
$266$	230.000	0.0530158
$267$	0	0
$268$	−1196.00	−0.272602
$269$	5190.00	1.17636	0.588178	−	0.808731i	$-0.299845\pi$
$269$	5190.00	1.17636	0.588178	+	0.808731i	$0.299845\pi$
$270$	0	0
$271$	4592.00	1.02931	0.514657	−	0.857396i	$-0.327919\pi$
$271$	4592.00	1.02931	0.514657	+	0.857396i	$0.327919\pi$
$272$	864.000	0.192602
$273$	0	0
$274$	4692.00	1.03450
$275$	0	0
$276$	0	0
$277$	2191.00	0.475251	0.237625	−	0.971357i	$-0.423631\pi$
$277$	2191.00	0.475251	0.237625	+	0.971357i	$0.423631\pi$
$278$	−5800.00	−1.25130
$279$	0	0
$280$	0	0
$281$	−7842.00	−1.66482	−0.832410	−	0.554160i	$-0.813040\pi$
$281$	−7842.00	−1.66482	−0.832410	+	0.554160i	$0.813040\pi$
$282$	0	0
$283$	247.000	0.0518821	0.0259410	−	0.999663i	$-0.491742\pi$
$283$	247.000	0.0518821	0.0259410	+	0.999663i	$0.491742\pi$
$284$	912.000	0.190554
$285$	0	0
$286$	5628.00	1.16360
$287$	−12.0000	−0.00246808
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	0	0
$292$	−3752.00	−0.751949
$293$	−5442.00	−1.08507	−0.542534	−	0.840034i	$-0.682535\pi$
$293$	−5442.00	−1.08507	−0.542534	+	0.840034i	$0.682535\pi$
$294$	0	0
$295$	0	0
$296$	−2288.00	−0.449281
$297$	0	0
$298$	−4140.00	−0.804778
$299$	−10854.0	−2.09934
$300$	0	0
$301$	−263.000	−0.0503624
$302$	−4474.00	−0.852483
$303$	0	0
$304$	−1840.00	−0.347142
$305$	0	0
$306$	0	0
$307$	3871.00	0.719641	0.359820	−	0.933022i	$-0.382838\pi$
$307$	3871.00	0.719641	0.359820	+	0.933022i	$0.382838\pi$
$308$	−168.000	−0.0310802
$309$	0	0
$310$	0	0
$311$	5718.00	1.04257	0.521283	−	0.853384i	$-0.325454\pi$
$311$	5718.00	1.04257	0.521283	+	0.853384i	$0.325454\pi$
$312$	0	0
$313$	3637.00	0.656790	0.328395	−	0.944540i	$-0.393492\pi$
$313$	3637.00	0.656790	0.328395	+	0.944540i	$0.393492\pi$
$314$	−482.000	−0.0866269
$315$	0	0
$316$	−640.000	−0.113933
$317$	−1296.00	−0.229623	−0.114812	−	0.993387i	$-0.536626\pi$
$317$	−1296.00	−0.229623	−0.114812	+	0.993387i	$0.536626\pi$
$318$	0	0
$319$	−8820.00	−1.54804
$320$	0	0
$321$	0	0
$322$	324.000	0.0560740
$323$	−6210.00	−1.06976
$324$	0	0
$325$	0	0
$326$	−7094.00	−1.20522
$327$	0	0
$328$	96.0000	0.0161607
$329$	414.000	0.0693756
$330$	0	0
$331$	5132.00	0.852206	0.426103	−	0.904675i	$-0.359886\pi$
$331$	5132.00	0.852206	0.426103	+	0.904675i	$0.359886\pi$
$332$	−1848.00	−0.305488
$333$	0	0
$334$	−1968.00	−0.322408
$335$	0	0
$336$	0	0
$337$	6751.00	1.09125	0.545624	−	0.838030i	$-0.316293\pi$
$337$	6751.00	1.09125	0.545624	+	0.838030i	$0.316293\pi$
$338$	−4584.00	−0.737683
$339$	0	0
$340$	0	0
$341$	8106.00	1.28729
$342$	0	0
$343$	−685.000	−0.107832
$344$	2104.00	0.329768
$345$	0	0
$346$	−7236.00	−1.12431
$347$	−5226.00	−0.808491	−0.404246	−	0.914651i	$-0.632466\pi$
$347$	−5226.00	−0.808491	−0.404246	+	0.914651i	$0.632466\pi$
$348$	0	0
$349$	−6190.00	−0.949407	−0.474704	−	0.880146i	$-0.657445\pi$
$349$	−6190.00	−0.949407	−0.474704	+	0.880146i	$0.657445\pi$
$350$	0	0
$351$	0	0
$352$	1344.00	0.203510
$353$	6618.00	0.997849	0.498924	−	0.866646i	$-0.333729\pi$
$353$	6618.00	0.997849	0.498924	+	0.866646i	$0.333729\pi$
$354$	0	0
$355$	0	0
$356$	960.000	0.142921
$357$	0	0
$358$	−300.000	−0.0442891
$359$	3420.00	0.502787	0.251394	−	0.967885i	$-0.419111\pi$
$359$	3420.00	0.502787	0.251394	+	0.967885i	$0.419111\pi$
$360$	0	0
$361$	6366.00	0.928124
$362$	−394.000	−0.0572049
$363$	0	0
$364$	268.000	0.0385907
$365$	0	0
$366$	0	0
$367$	871.000	0.123885	0.0619425	−	0.998080i	$-0.480270\pi$
$367$	871.000	0.123885	0.0619425	+	0.998080i	$0.480270\pi$
$368$	−2592.00	−0.367167
$369$	0	0
$370$	0	0
$371$	−192.000	−0.0268683
$372$	0	0
$373$	−6383.00	−0.886057	−0.443028	−	0.896508i	$-0.646096\pi$
$373$	−6383.00	−0.886057	−0.443028	+	0.896508i	$0.646096\pi$
$374$	4536.00	0.627142
$375$	0	0
$376$	−3312.00	−0.454264
$377$	14070.0	1.92213
$378$	0	0
$379$	−9865.00	−1.33702	−0.668511	−	0.743703i	$-0.733068\pi$
$379$	−9865.00	−1.33702	−0.668511	+	0.743703i	$0.733068\pi$
$380$	0	0
$381$	0	0
$382$	2604.00	0.348775
$383$	9828.00	1.31119	0.655597	−	0.755111i	$-0.272417\pi$
$383$	9828.00	1.31119	0.655597	+	0.755111i	$0.272417\pi$
$384$	0	0
$385$	0	0
$386$	8326.00	1.09788
$387$	0	0
$388$	2044.00	0.267444
$389$	−12540.0	−1.63446	−0.817228	−	0.576315i	$-0.804490\pi$
$389$	−12540.0	−1.63446	−0.817228	+	0.576315i	$0.804490\pi$
$390$	0	0
$391$	−8748.00	−1.13147
$392$	2736.00	0.352523
$393$	0	0
$394$	−6108.00	−0.781007
$395$	0	0
$396$	0	0
$397$	1381.00	0.174585	0.0872927	−	0.996183i	$-0.472178\pi$
$397$	1381.00	0.174585	0.0872927	+	0.996183i	$0.472178\pi$
$398$	−6850.00	−0.862712
$399$	0	0
$400$	0	0
$401$	−14232.0	−1.77235	−0.886175	−	0.463351i	$-0.846647\pi$
$401$	−14232.0	−1.77235	−0.886175	+	0.463351i	$0.846647\pi$
$402$	0	0
$403$	−12931.0	−1.59836
$404$	−3648.00	−0.449245
$405$	0	0
$406$	−420.000	−0.0513405
$407$	−12012.0	−1.46293
$408$	0	0
$409$	2645.00	0.319772	0.159886	−	0.987135i	$-0.448887\pi$
$409$	2645.00	0.319772	0.159886	+	0.987135i	$0.448887\pi$
$410$	0	0
$411$	0	0
$412$	−2672.00	−0.319515
$413$	−690.000	−0.0822099
$414$	0	0
$415$	0	0
$416$	−2144.00	−0.252688
$417$	0	0
$418$	−9660.00	−1.13035
$419$	−3000.00	−0.349784	−0.174892	−	0.984588i	$-0.555958\pi$
$419$	−3000.00	−0.349784	−0.174892	+	0.984588i	$0.555958\pi$
$420$	0	0
$421$	−11338.0	−1.31254	−0.656271	−	0.754525i	$-0.727867\pi$
$421$	−11338.0	−1.31254	−0.656271	+	0.754525i	$0.727867\pi$
$422$	4886.00	0.563618
$423$	0	0
$424$	1536.00	0.175931
$425$	0	0
$426$	0	0
$427$	−733.000	−0.0830734
$428$	−5184.00	−0.585463
$429$	0	0
$430$	0	0
$431$	3258.00	0.364112	0.182056	−	0.983288i	$-0.441725\pi$
$431$	3258.00	0.364112	0.182056	+	0.983288i	$0.441725\pi$
$432$	0	0
$433$	−1163.00	−0.129077	−0.0645384	−	0.997915i	$-0.520557\pi$
$433$	−1163.00	−0.129077	−0.0645384	+	0.997915i	$0.520557\pi$
$434$	386.000	0.0426926
$435$	0	0
$436$	−6940.00	−0.762307
$437$	18630.0	2.03934
$438$	0	0
$439$	6695.00	0.727870	0.363935	−	0.931424i	$-0.381433\pi$
$439$	6695.00	0.727870	0.363935	+	0.931424i	$0.381433\pi$
$440$	0	0
$441$	0	0
$442$	−7236.00	−0.778691
$443$	16368.0	1.75546	0.877728	−	0.479159i	$-0.159058\pi$
$443$	16368.0	1.75546	0.877728	+	0.479159i	$0.159058\pi$
$444$	0	0
$445$	0	0
$446$	46.0000	0.00488377
$447$	0	0
$448$	64.0000	0.00674937
$449$	−16380.0	−1.72165	−0.860824	−	0.508903i	$-0.830051\pi$
$449$	−16380.0	−1.72165	−0.860824	+	0.508903i	$0.830051\pi$
$450$	0	0
$451$	504.000	0.0526218
$452$	−4368.00	−0.454543
$453$	0	0
$454$	3912.00	0.404404
$455$	0	0
$456$	0	0
$457$	13786.0	1.41112	0.705560	−	0.708650i	$-0.250696\pi$
$457$	13786.0	1.41112	0.705560	+	0.708650i	$0.250696\pi$
$458$	−3610.00	−0.368306
$459$	0	0
$460$	0	0
$461$	−11832.0	−1.19538	−0.597691	−	0.801726i	$-0.703915\pi$
$461$	−11832.0	−1.19538	−0.597691	+	0.801726i	$0.703915\pi$
$462$	0	0
$463$	−3008.00	−0.301930	−0.150965	−	0.988539i	$-0.548238\pi$
$463$	−3008.00	−0.301930	−0.150965	+	0.988539i	$0.548238\pi$
$464$	3360.00	0.336173
$465$	0	0
$466$	−6936.00	−0.689494
$467$	4434.00	0.439360	0.219680	−	0.975572i	$-0.429499\pi$
$467$	4434.00	0.439360	0.219680	+	0.975572i	$0.429499\pi$
$468$	0	0
$469$	−299.000	−0.0294382
$470$	0	0
$471$	0	0
$472$	5520.00	0.538302
$473$	11046.0	1.07378
$474$	0	0
$475$	0	0
$476$	216.000	0.0207990
$477$	0	0
$478$	5280.00	0.505233
$479$	−7410.00	−0.706830	−0.353415	−	0.935467i	$-0.614980\pi$
$479$	−7410.00	−0.706830	−0.353415	+	0.935467i	$0.614980\pi$
$480$	0	0
$481$	19162.0	1.81645
$482$	10766.0	1.01738
$483$	0	0
$484$	1732.00	0.162660
$485$	0	0
$486$	0	0
$487$	8671.00	0.806818	0.403409	−	0.915020i	$-0.367825\pi$
$487$	8671.00	0.806818	0.403409	+	0.915020i	$0.367825\pi$
$488$	5864.00	0.543957
$489$	0	0
$490$	0	0
$491$	19368.0	1.78017	0.890087	−	0.455790i	$-0.150643\pi$
$491$	19368.0	1.78017	0.890087	+	0.455790i	$0.150643\pi$
$492$	0	0
$493$	11340.0	1.03596
$494$	15410.0	1.40350
$495$	0	0
$496$	−3088.00	−0.279547
$497$	228.000	0.0205779
$498$	0	0
$499$	−8875.00	−0.796192	−0.398096	−	0.917344i	$-0.630329\pi$
$499$	−8875.00	−0.796192	−0.398096	+	0.917344i	$0.630329\pi$
$500$	0	0
$501$	0	0
$502$	−10056.0	−0.894066
$503$	−10452.0	−0.926504	−0.463252	−	0.886227i	$-0.653318\pi$
$503$	−10452.0	−0.926504	−0.463252	+	0.886227i	$0.653318\pi$
$504$	0	0
$505$	0	0
$506$	−13608.0	−1.19555
$507$	0	0
$508$	64.0000	0.00558965
$509$	19770.0	1.72159	0.860796	−	0.508951i	$-0.169967\pi$
$509$	19770.0	1.72159	0.860796	+	0.508951i	$0.169967\pi$
$510$	0	0
$511$	−938.000	−0.0812029
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−1128.00	−0.0967976
$515$	0	0
$516$	0	0
$517$	−17388.0	−1.47916
$518$	−572.000	−0.0485178
$519$	0	0
$520$	0	0
$521$	11238.0	0.945001	0.472501	−	0.881330i	$-0.343351\pi$
$521$	11238.0	0.945001	0.472501	+	0.881330i	$0.343351\pi$
$522$	0	0
$523$	7447.00	0.622628	0.311314	−	0.950307i	$-0.399231\pi$
$523$	7447.00	0.622628	0.311314	+	0.950307i	$0.399231\pi$
$524$	−7968.00	−0.664282
$525$	0	0
$526$	3624.00	0.300407
$527$	−10422.0	−0.861460
$528$	0	0
$529$	14077.0	1.15698
$530$	0	0
$531$	0	0
$532$	−460.000	−0.0374878
$533$	−804.000	−0.0653379
$534$	0	0
$535$	0	0
$536$	2392.00	0.192759
$537$	0	0
$538$	−10380.0	−0.831810
$539$	14364.0	1.14787
$540$	0	0
$541$	−17623.0	−1.40050	−0.700251	−	0.713896i	$-0.746929\pi$
$541$	−17623.0	−1.40050	−0.700251	+	0.713896i	$0.746929\pi$
$542$	−9184.00	−0.727835
$543$	0	0
$544$	−1728.00	−0.136190
$545$	0	0
$546$	0	0
$547$	10096.0	0.789166	0.394583	−	0.918860i	$-0.370889\pi$
$547$	10096.0	0.789166	0.394583	+	0.918860i	$0.370889\pi$
$548$	−9384.00	−0.731505
$549$	0	0
$550$	0	0
$551$	−24150.0	−1.86720
$552$	0	0
$553$	−160.000	−0.0123036
$554$	−4382.00	−0.336053
$555$	0	0
$556$	11600.0	0.884801
$557$	14514.0	1.10409	0.552045	−	0.833814i	$-0.313848\pi$
$557$	14514.0	1.10409	0.552045	+	0.833814i	$0.313848\pi$
$558$	0	0
$559$	−17621.0	−1.33325
$560$	0	0
$561$	0	0
$562$	15684.0	1.17721
$563$	−10242.0	−0.766694	−0.383347	−	0.923604i	$-0.625229\pi$
$563$	−10242.0	−0.766694	−0.383347	+	0.923604i	$0.625229\pi$
$564$	0	0
$565$	0	0
$566$	−494.000	−0.0366862
$567$	0	0
$568$	−1824.00	−0.134742
$569$	6750.00	0.497319	0.248660	−	0.968591i	$-0.420010\pi$
$569$	6750.00	0.497319	0.248660	+	0.968591i	$0.420010\pi$
$570$	0	0
$571$	17117.0	1.25451	0.627254	−	0.778815i	$-0.284179\pi$
$571$	17117.0	1.25451	0.627254	+	0.778815i	$0.284179\pi$
$572$	−11256.0	−0.822792
$573$	0	0
$574$	24.0000	0.00174519
$575$	0	0
$576$	0	0
$577$	301.000	0.0217171	0.0108586	−	0.999941i	$-0.496544\pi$
$577$	301.000	0.0217171	0.0108586	+	0.999941i	$0.496544\pi$
$578$	3994.00	0.287420
$579$	0	0
$580$	0	0
$581$	−462.000	−0.0329897
$582$	0	0
$583$	8064.00	0.572859
$584$	7504.00	0.531708
$585$	0	0
$586$	10884.0	0.767259
$587$	−15456.0	−1.08678	−0.543388	−	0.839482i	$-0.682859\pi$
$587$	−15456.0	−1.08678	−0.543388	+	0.839482i	$0.682859\pi$
$588$	0	0
$589$	22195.0	1.55268
$590$	0	0
$591$	0	0
$592$	4576.00	0.317690
$593$	−9492.00	−0.657318	−0.328659	−	0.944449i	$-0.606597\pi$
$593$	−9492.00	−0.657318	−0.328659	+	0.944449i	$0.606597\pi$
$594$	0	0
$595$	0	0
$596$	8280.00	0.569064
$597$	0	0
$598$	21708.0	1.48446
$599$	−1500.00	−0.102318	−0.0511589	−	0.998691i	$-0.516291\pi$
$599$	−1500.00	−0.102318	−0.0511589	+	0.998691i	$0.516291\pi$
$600$	0	0
$601$	14627.0	0.992758	0.496379	−	0.868106i	$-0.334663\pi$
$601$	14627.0	0.992758	0.496379	+	0.868106i	$0.334663\pi$
$602$	526.000	0.0356116
$603$	0	0
$604$	8948.00	0.602796
$605$	0	0
$606$	0	0
$607$	−16184.0	−1.08219	−0.541094	−	0.840962i	$-0.681990\pi$
$607$	−16184.0	−1.08219	−0.541094	+	0.840962i	$0.681990\pi$
$608$	3680.00	0.245467
$609$	0	0
$610$	0	0
$611$	27738.0	1.83659
$612$	0	0
$613$	18502.0	1.21907	0.609534	−	0.792760i	$-0.291357\pi$
$613$	18502.0	1.21907	0.609534	+	0.792760i	$0.291357\pi$
$614$	−7742.00	−0.508863
$615$	0	0
$616$	336.000	0.0219770
$617$	−13896.0	−0.906697	−0.453348	−	0.891333i	$-0.649771\pi$
$617$	−13896.0	−0.906697	−0.453348	+	0.891333i	$0.649771\pi$
$618$	0	0
$619$	−9895.00	−0.642510	−0.321255	−	0.946993i	$-0.604105\pi$
$619$	−9895.00	−0.642510	−0.321255	+	0.946993i	$0.604105\pi$
$620$	0	0
$621$	0	0
$622$	−11436.0	−0.737206
$623$	240.000	0.0154340
$624$	0	0
$625$	0	0
$626$	−7274.00	−0.464421
$627$	0	0
$628$	964.000	0.0612544
$629$	15444.0	0.979003
$630$	0	0
$631$	467.000	0.0294627	0.0147314	−	0.999891i	$-0.495311\pi$
$631$	467.000	0.0294627	0.0147314	+	0.999891i	$0.495311\pi$
$632$	1280.00	0.0805628
$633$	0	0
$634$	2592.00	0.162368
$635$	0	0
$636$	0	0
$637$	−22914.0	−1.42525
$638$	17640.0	1.09463
$639$	0	0
$640$	0	0
$641$	−30612.0	−1.88627	−0.943137	−	0.332405i	$-0.892140\pi$
$641$	−30612.0	−1.88627	−0.943137	+	0.332405i	$0.892140\pi$
$642$	0	0
$643$	1852.00	0.113586	0.0567930	−	0.998386i	$-0.481913\pi$
$643$	1852.00	0.113586	0.0567930	+	0.998386i	$0.481913\pi$
$644$	−648.000	−0.0396503
$645$	0	0
$646$	12420.0	0.756437
$647$	−21156.0	−1.28551	−0.642757	−	0.766070i	$-0.722210\pi$
$647$	−21156.0	−1.28551	−0.642757	+	0.766070i	$0.722210\pi$
$648$	0	0
$649$	28980.0	1.75280
$650$	0	0
$651$	0	0
$652$	14188.0	0.852216
$653$	−9702.00	−0.581422	−0.290711	−	0.956811i	$-0.593892\pi$
$653$	−9702.00	−0.581422	−0.290711	+	0.956811i	$0.593892\pi$
$654$	0	0
$655$	0	0
$656$	−192.000	−0.0114273
$657$	0	0
$658$	−828.000	−0.0490559
$659$	−1980.00	−0.117041	−0.0585204	−	0.998286i	$-0.518638\pi$
$659$	−1980.00	−0.117041	−0.0585204	+	0.998286i	$0.518638\pi$
$660$	0	0
$661$	−20158.0	−1.18617	−0.593083	−	0.805142i	$-0.702089\pi$
$661$	−20158.0	−1.18617	−0.593083	+	0.805142i	$0.702089\pi$
$662$	−10264.0	−0.602601
$663$	0	0
$664$	3696.00	0.216013
$665$	0	0
$666$	0	0
$667$	−34020.0	−1.97490
$668$	3936.00	0.227977
$669$	0	0
$670$	0	0
$671$	30786.0	1.77121
$672$	0	0
$673$	16882.0	0.966944	0.483472	−	0.875360i	$-0.339376\pi$
$673$	16882.0	0.966944	0.483472	+	0.875360i	$0.339376\pi$
$674$	−13502.0	−0.771628
$675$	0	0
$676$	9168.00	0.521620
$677$	20934.0	1.18842	0.594209	−	0.804311i	$-0.297465\pi$
$677$	20934.0	1.18842	0.594209	+	0.804311i	$0.297465\pi$
$678$	0	0
$679$	511.000	0.0288813
$680$	0	0
$681$	0	0
$682$	−16212.0	−0.910249
$683$	−8712.00	−0.488075	−0.244038	−	0.969766i	$-0.578472\pi$
$683$	−8712.00	−0.488075	−0.244038	+	0.969766i	$0.578472\pi$
$684$	0	0
$685$	0	0
$686$	1370.00	0.0762490
$687$	0	0
$688$	−4208.00	−0.233181
$689$	−12864.0	−0.711291
$690$	0	0
$691$	−14128.0	−0.777792	−0.388896	−	0.921282i	$-0.627144\pi$
$691$	−14128.0	−0.777792	−0.388896	+	0.921282i	$0.627144\pi$
$692$	14472.0	0.795004
$693$	0	0
$694$	10452.0	0.571689
$695$	0	0
$696$	0	0
$697$	−648.000	−0.0352148
$698$	12380.0	0.671332
$699$	0	0
$700$	0	0
$701$	28278.0	1.52360	0.761801	−	0.647811i	$-0.224315\pi$
$701$	28278.0	1.52360	0.761801	+	0.647811i	$0.224315\pi$
$702$	0	0
$703$	−32890.0	−1.76454
$704$	−2688.00	−0.143903
$705$	0	0
$706$	−13236.0	−0.705586
$707$	−912.000	−0.0485138
$708$	0	0
$709$	8885.00	0.470639	0.235320	−	0.971918i	$-0.424386\pi$
$709$	8885.00	0.470639	0.235320	+	0.971918i	$0.424386\pi$
$710$	0	0
$711$	0	0
$712$	−1920.00	−0.101060
$713$	31266.0	1.64225
$714$	0	0
$715$	0	0
$716$	600.000	0.0313171
$717$	0	0
$718$	−6840.00	−0.355524
$719$	7530.00	0.390572	0.195286	−	0.980746i	$-0.437436\pi$
$719$	7530.00	0.390572	0.195286	+	0.980746i	$0.437436\pi$
$720$	0	0
$721$	−668.000	−0.0345043
$722$	−12732.0	−0.656283
$723$	0	0
$724$	788.000	0.0404500
$725$	0	0
$726$	0	0
$727$	1801.00	0.0918781	0.0459391	−	0.998944i	$-0.485372\pi$
$727$	1801.00	0.0918781	0.0459391	+	0.998944i	$0.485372\pi$
$728$	−536.000	−0.0272877
$729$	0	0
$730$	0	0
$731$	−14202.0	−0.718577
$732$	0	0
$733$	7882.00	0.397174	0.198587	−	0.980083i	$-0.436365\pi$
$733$	7882.00	0.397174	0.198587	+	0.980083i	$0.436365\pi$
$734$	−1742.00	−0.0876000
$735$	0	0
$736$	5184.00	0.259626
$737$	12558.0	0.627652
$738$	0	0
$739$	33860.0	1.68547	0.842734	−	0.538331i	$-0.180945\pi$
$739$	33860.0	1.68547	0.842734	+	0.538331i	$0.180945\pi$
$740$	0	0
$741$	0	0
$742$	384.000	0.0189988
$743$	−20652.0	−1.01972	−0.509858	−	0.860259i	$-0.670302\pi$
$743$	−20652.0	−1.01972	−0.509858	+	0.860259i	$0.670302\pi$
$744$	0	0
$745$	0	0
$746$	12766.0	0.626537
$747$	0	0
$748$	−9072.00	−0.443456
$749$	−1296.00	−0.0632240
$750$	0	0
$751$	7472.00	0.363059	0.181529	−	0.983386i	$-0.441895\pi$
$751$	7472.00	0.363059	0.181529	+	0.983386i	$0.441895\pi$
$752$	6624.00	0.321213
$753$	0	0
$754$	−28140.0	−1.35915
$755$	0	0
$756$	0	0
$757$	32251.0	1.54846	0.774229	−	0.632906i	$-0.218138\pi$
$757$	32251.0	1.54846	0.774229	+	0.632906i	$0.218138\pi$
$758$	19730.0	0.945417
$759$	0	0
$760$	0	0
$761$	−16812.0	−0.800834	−0.400417	−	0.916333i	$-0.631135\pi$
$761$	−16812.0	−0.800834	−0.400417	+	0.916333i	$0.631135\pi$
$762$	0	0
$763$	−1735.00	−0.0823214
$764$	−5208.00	−0.246622
$765$	0	0
$766$	−19656.0	−0.927154
$767$	−46230.0	−2.17636
$768$	0	0
$769$	−34645.0	−1.62462	−0.812309	−	0.583228i	$-0.801789\pi$
$769$	−34645.0	−1.62462	−0.812309	+	0.583228i	$0.801789\pi$
$770$	0	0
$771$	0	0
$772$	−16652.0	−0.776319
$773$	−8412.00	−0.391408	−0.195704	−	0.980663i	$-0.562699\pi$
$773$	−8412.00	−0.391408	−0.195704	+	0.980663i	$0.562699\pi$
$774$	0	0
$775$	0	0
$776$	−4088.00	−0.189112
$777$	0	0
$778$	25080.0	1.15573
$779$	1380.00	0.0634706
$780$	0	0
$781$	−9576.00	−0.438740
$782$	17496.0	0.800071
$783$	0	0
$784$	−5472.00	−0.249271
$785$	0	0
$786$	0	0
$787$	−18329.0	−0.830188	−0.415094	−	0.909778i	$-0.636251\pi$
$787$	−18329.0	−0.830188	−0.415094	+	0.909778i	$0.636251\pi$
$788$	12216.0	0.552255
$789$	0	0
$790$	0	0
$791$	−1092.00	−0.0490860
$792$	0	0
$793$	−49111.0	−2.19922
$794$	−2762.00	−0.123451
$795$	0	0
$796$	13700.0	0.610030
$797$	16044.0	0.713059	0.356529	−	0.934284i	$-0.383960\pi$
$797$	16044.0	0.713059	0.356529	+	0.934284i	$0.383960\pi$
$798$	0	0
$799$	22356.0	0.989860
$800$	0	0
$801$	0	0
$802$	28464.0	1.25324
$803$	39396.0	1.73133
$804$	0	0
$805$	0	0
$806$	25862.0	1.13021
$807$	0	0
$808$	7296.00	0.317664
$809$	24000.0	1.04301	0.521505	−	0.853248i	$-0.325371\pi$
$809$	24000.0	1.04301	0.521505	+	0.853248i	$0.325371\pi$
$810$	0	0
$811$	5117.00	0.221556	0.110778	−	0.993845i	$-0.464666\pi$
$811$	5117.00	0.221556	0.110778	+	0.993845i	$0.464666\pi$
$812$	840.000	0.0363032
$813$	0	0
$814$	24024.0	1.03445
$815$	0	0
$816$	0	0
$817$	30245.0	1.29515
$818$	−5290.00	−0.226113
$819$	0	0
$820$	0	0
$821$	−13542.0	−0.575663	−0.287831	−	0.957681i	$-0.592934\pi$
$821$	−13542.0	−0.575663	−0.287831	+	0.957681i	$0.592934\pi$
$822$	0	0
$823$	−1283.00	−0.0543409	−0.0271705	−	0.999631i	$-0.508650\pi$
$823$	−1283.00	−0.0543409	−0.0271705	+	0.999631i	$0.508650\pi$
$824$	5344.00	0.225931
$825$	0	0
$826$	1380.00	0.0581312
$827$	16344.0	0.687227	0.343613	−	0.939111i	$-0.388349\pi$
$827$	16344.0	0.687227	0.343613	+	0.939111i	$0.388349\pi$
$828$	0	0
$829$	−790.000	−0.0330975	−0.0165488	−	0.999863i	$-0.505268\pi$
$829$	−790.000	−0.0330975	−0.0165488	+	0.999863i	$0.505268\pi$
$830$	0	0
$831$	0	0
$832$	4288.00	0.178677
$833$	−18468.0	−0.768161
$834$	0	0
$835$	0	0
$836$	19320.0	0.799278
$837$	0	0
$838$	6000.00	0.247335
$839$	9990.00	0.411076	0.205538	−	0.978649i	$-0.434106\pi$
$839$	9990.00	0.411076	0.205538	+	0.978649i	$0.434106\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	22676.0	0.928108
$843$	0	0
$844$	−9772.00	−0.398538
$845$	0	0
$846$	0	0
$847$	433.000	0.0175656
$848$	−3072.00	−0.124402
$849$	0	0
$850$	0	0
$851$	−46332.0	−1.86632
$852$	0	0
$853$	−24743.0	−0.993182	−0.496591	−	0.867985i	$-0.665415\pi$
$853$	−24743.0	−0.993182	−0.496591	+	0.867985i	$0.665415\pi$
$854$	1466.00	0.0587418
$855$	0	0
$856$	10368.0	0.413985
$857$	−23556.0	−0.938924	−0.469462	−	0.882953i	$-0.655552\pi$
$857$	−23556.0	−0.938924	−0.469462	+	0.882953i	$0.655552\pi$
$858$	0	0
$859$	−34000.0	−1.35048	−0.675242	−	0.737597i	$-0.735961\pi$
$859$	−34000.0	−1.35048	−0.675242	+	0.737597i	$0.735961\pi$
$860$	0	0
$861$	0	0
$862$	−6516.00	−0.257466
$863$	−37032.0	−1.46070	−0.730350	−	0.683073i	$-0.760643\pi$
$863$	−37032.0	−1.46070	−0.730350	+	0.683073i	$0.760643\pi$
$864$	0	0
$865$	0	0
$866$	2326.00	0.0912710
$867$	0	0
$868$	−772.000	−0.0301882
$869$	6720.00	0.262325
$870$	0	0
$871$	−20033.0	−0.779325
$872$	13880.0	0.539032
$873$	0	0
$874$	−37260.0	−1.44203
$875$	0	0
$876$	0	0
$877$	−2519.00	−0.0969904	−0.0484952	−	0.998823i	$-0.515443\pi$
$877$	−2519.00	−0.0969904	−0.0484952	+	0.998823i	$0.515443\pi$
$878$	−13390.0	−0.514682
$879$	0	0
$880$	0	0
$881$	−43992.0	−1.68232	−0.841162	−	0.540783i	$-0.818128\pi$
$881$	−43992.0	−1.68232	−0.841162	+	0.540783i	$0.818128\pi$
$882$	0	0
$883$	19177.0	0.730869	0.365435	−	0.930837i	$-0.380920\pi$
$883$	19177.0	0.730869	0.365435	+	0.930837i	$0.380920\pi$
$884$	14472.0	0.550618
$885$	0	0
$886$	−32736.0	−1.24130
$887$	44994.0	1.70321	0.851607	−	0.524181i	$-0.175628\pi$
$887$	44994.0	1.70321	0.851607	+	0.524181i	$0.175628\pi$
$888$	0	0
$889$	16.0000	0.000603625 0
$890$	0	0
$891$	0	0
$892$	−92.0000	−0.00345335
$893$	−47610.0	−1.78411
$894$	0	0
$895$	0	0
$896$	−128.000	−0.00477252
$897$	0	0
$898$	32760.0	1.21739
$899$	−40530.0	−1.50362
$900$	0	0
$901$	−10368.0	−0.383361
$902$	−1008.00	−0.0372092
$903$	0	0
$904$	8736.00	0.321410
$905$	0	0
$906$	0	0
$907$	52396.0	1.91817	0.959085	−	0.283117i	$-0.0913686\pi$
$907$	52396.0	1.91817	0.959085	+	0.283117i	$0.0913686\pi$
$908$	−7824.00	−0.285957
$909$	0	0
$910$	0	0
$911$	−7242.00	−0.263379	−0.131689	−	0.991291i	$-0.542040\pi$
$911$	−7242.00	−0.263379	−0.131689	+	0.991291i	$0.542040\pi$
$912$	0	0
$913$	19404.0	0.703372
$914$	−27572.0	−0.997813
$915$	0	0
$916$	7220.00	0.260432
$917$	−1992.00	−0.0717357
$918$	0	0
$919$	4085.00	0.146629	0.0733143	−	0.997309i	$-0.476642\pi$
$919$	4085.00	0.146629	0.0733143	+	0.997309i	$0.476642\pi$
$920$	0	0
$921$	0	0
$922$	23664.0	0.845263
$923$	15276.0	0.544762
$924$	0	0
$925$	0	0
$926$	6016.00	0.213497
$927$	0	0
$928$	−6720.00	−0.237710
$929$	3030.00	0.107009	0.0535043	−	0.998568i	$-0.482961\pi$
$929$	3030.00	0.107009	0.0535043	+	0.998568i	$0.482961\pi$
$930$	0	0
$931$	39330.0	1.38452
$932$	13872.0	0.487546
$933$	0	0
$934$	−8868.00	−0.310674
$935$	0	0
$936$	0	0
$937$	−5759.00	−0.200788	−0.100394	−	0.994948i	$-0.532010\pi$
$937$	−5759.00	−0.200788	−0.100394	+	0.994948i	$0.532010\pi$
$938$	598.000	0.0208160
$939$	0	0
$940$	0	0
$941$	258.000	0.00893790	0.00446895	−	0.999990i	$-0.498577\pi$
$941$	258.000	0.00893790	0.00446895	+	0.999990i	$0.498577\pi$
$942$	0	0
$943$	1944.00	0.0671319
$944$	−11040.0	−0.380637
$945$	0	0
$946$	−22092.0	−0.759274
$947$	1374.00	0.0471478	0.0235739	−	0.999722i	$-0.492495\pi$
$947$	1374.00	0.0471478	0.0235739	+	0.999722i	$0.492495\pi$
$948$	0	0
$949$	−62846.0	−2.14970
$950$	0	0
$951$	0	0
$952$	−432.000	−0.0147071
$953$	9288.00	0.315706	0.157853	−	0.987463i	$-0.449543\pi$
$953$	9288.00	0.315706	0.157853	+	0.987463i	$0.449543\pi$
$954$	0	0
$955$	0	0
$956$	−10560.0	−0.357254
$957$	0	0
$958$	14820.0	0.499804
$959$	−2346.00	−0.0789951
$960$	0	0
$961$	7458.00	0.250344
$962$	−38324.0	−1.28442
$963$	0	0
$964$	−21532.0	−0.719397
$965$	0	0
$966$	0	0
$967$	21616.0	0.718846	0.359423	−	0.933175i	$-0.382974\pi$
$967$	21616.0	0.718846	0.359423	+	0.933175i	$0.382974\pi$
$968$	−3464.00	−0.115018
$969$	0	0
$970$	0	0
$971$	19098.0	0.631188	0.315594	−	0.948894i	$-0.397796\pi$
$971$	19098.0	0.631188	0.315594	+	0.948894i	$0.397796\pi$
$972$	0	0
$973$	2900.00	0.0955496
$974$	−17342.0	−0.570507
$975$	0	0
$976$	−11728.0	−0.384635
$977$	−18246.0	−0.597483	−0.298742	−	0.954334i	$-0.596567\pi$
$977$	−18246.0	−0.597483	−0.298742	+	0.954334i	$0.596567\pi$
$978$	0	0
$979$	−10080.0	−0.329069
$980$	0	0
$981$	0	0
$982$	−38736.0	−1.25877
$983$	−38772.0	−1.25802	−0.629011	−	0.777397i	$-0.716540\pi$
$983$	−38772.0	−1.25802	−0.629011	+	0.777397i	$0.716540\pi$
$984$	0	0
$985$	0	0
$986$	−22680.0	−0.732534
$987$	0	0
$988$	−30820.0	−0.992424
$989$	42606.0	1.36986
$990$	0	0
$991$	−23053.0	−0.738953	−0.369477	−	0.929240i	$-0.620463\pi$
$991$	−23053.0	−0.738953	−0.369477	+	0.929240i	$0.620463\pi$
$992$	6176.00	0.197670
$993$	0	0
$994$	−456.000	−0.0145507
$995$	0	0
$996$	0	0
$997$	10366.0	0.329282	0.164641	−	0.986354i	$-0.447353\pi$
$997$	10366.0	0.329282	0.164641	+	0.986354i	$0.447353\pi$
$998$	17750.0	0.562992
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.f.1.1		1
3.2	odd	2		150.4.a.h.1.1	yes	1
5.2	odd	4		450.4.c.a.199.1		2
5.3	odd	4		450.4.c.a.199.2		2
5.4	even	2		450.4.a.o.1.1		1
12.11	even	2		1200.4.a.i.1.1		1
15.2	even	4		150.4.c.e.49.2		2
15.8	even	4		150.4.c.e.49.1		2
15.14	odd	2		150.4.a.a.1.1	✓	1
60.23	odd	4		1200.4.f.c.49.1		2
60.47	odd	4		1200.4.f.c.49.2		2
60.59	even	2		1200.4.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.a.1.1	✓	1	15.14	odd	2
150.4.a.h.1.1	yes	1	3.2	odd	2
150.4.c.e.49.1		2	15.8	even	4
150.4.c.e.49.2		2	15.2	even	4
450.4.a.f.1.1		1	1.1	even	1	trivial
450.4.a.o.1.1		1	5.4	even	2
450.4.c.a.199.1		2	5.2	odd	4
450.4.c.a.199.2		2	5.3	odd	4
1200.4.a.i.1.1		1	12.11	even	2
1200.4.a.bb.1.1		1	60.59	even	2
1200.4.f.c.49.1		2	60.23	odd	4
1200.4.f.c.49.2		2	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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