LMFDB - Embedded newform 546.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,4,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("546.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.2150428631$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +12.0000 q^{5} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +12.0000 q^{5} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -24.0000 q^{10} -22.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} -14.0000 q^{14} -36.0000 q^{15} +16.0000 q^{16} -2.00000 q^{17} -18.0000 q^{18} -88.0000 q^{19} +48.0000 q^{20} -21.0000 q^{21} +44.0000 q^{22} -80.0000 q^{23} +24.0000 q^{24} +19.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} -22.0000 q^{29} +72.0000 q^{30} -92.0000 q^{31} -32.0000 q^{32} +66.0000 q^{33} +4.00000 q^{34} +84.0000 q^{35} +36.0000 q^{36} +118.000 q^{37} +176.000 q^{38} +39.0000 q^{39} -96.0000 q^{40} +324.000 q^{41} +42.0000 q^{42} +84.0000 q^{43} -88.0000 q^{44} +108.000 q^{45} +160.000 q^{46} -134.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -38.0000 q^{50} +6.00000 q^{51} -52.0000 q^{52} -194.000 q^{53} +54.0000 q^{54} -264.000 q^{55} -56.0000 q^{56} +264.000 q^{57} +44.0000 q^{58} +210.000 q^{59} -144.000 q^{60} -470.000 q^{61} +184.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -156.000 q^{65} -132.000 q^{66} -292.000 q^{67} -8.00000 q^{68} +240.000 q^{69} -168.000 q^{70} -66.0000 q^{71} -72.0000 q^{72} -506.000 q^{73} -236.000 q^{74} -57.0000 q^{75} -352.000 q^{76} -154.000 q^{77} -78.0000 q^{78} -776.000 q^{79} +192.000 q^{80} +81.0000 q^{81} -648.000 q^{82} -778.000 q^{83} -84.0000 q^{84} -24.0000 q^{85} -168.000 q^{86} +66.0000 q^{87} +176.000 q^{88} -920.000 q^{89} -216.000 q^{90} -91.0000 q^{91} -320.000 q^{92} +276.000 q^{93} +268.000 q^{94} -1056.00 q^{95} +96.0000 q^{96} -490.000 q^{97} -98.0000 q^{98} -198.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	6.00000	0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−24.0000	−0.758947
$11$	−22.0000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−22.0000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−14.0000	−0.267261
$15$	−36.0000	−0.619677
$16$	16.0000	0.250000
$17$	−2.00000	−0.0285336	−0.0142668	−	0.999898i	$-0.504541\pi$
$17$	−2.00000	−0.0285336	−0.0142668	+	0.999898i	$0.504541\pi$
$18$	−18.0000	−0.235702
$19$	−88.0000	−1.06256	−0.531279	−	0.847197i	$-0.678288\pi$
$19$	−88.0000	−1.06256	−0.531279	+	0.847197i	$0.678288\pi$
$20$	48.0000	0.536656
$21$	−21.0000	−0.218218
$22$	44.0000	0.426401
$23$	−80.0000	−0.725268	−0.362634	−	0.931932i	$-0.618122\pi$
$23$	−80.0000	−0.725268	−0.362634	+	0.931932i	$0.618122\pi$
$24$	24.0000	0.204124
$25$	19.0000	0.152000
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	−22.0000	−0.140872	−0.0704362	−	0.997516i	$-0.522439\pi$
$29$	−22.0000	−0.140872	−0.0704362	+	0.997516i	$0.522439\pi$
$30$	72.0000	0.438178
$31$	−92.0000	−0.533022	−0.266511	−	0.963832i	$-0.585871\pi$
$31$	−92.0000	−0.533022	−0.266511	+	0.963832i	$0.585871\pi$
$32$	−32.0000	−0.176777
$33$	66.0000	0.348155
$34$	4.00000	0.0201763
$35$	84.0000	0.405674
$36$	36.0000	0.166667
$37$	118.000	0.524299	0.262150	−	0.965027i	$-0.415569\pi$
$37$	118.000	0.524299	0.262150	+	0.965027i	$0.415569\pi$
$38$	176.000	0.751341
$39$	39.0000	0.160128
$40$	−96.0000	−0.379473
$41$	324.000	1.23415	0.617077	−	0.786903i	$-0.288317\pi$
$41$	324.000	1.23415	0.617077	+	0.786903i	$0.288317\pi$
$42$	42.0000	0.154303
$43$	84.0000	0.297904	0.148952	−	0.988844i	$-0.452410\pi$
$43$	84.0000	0.297904	0.148952	+	0.988844i	$0.452410\pi$
$44$	−88.0000	−0.301511
$45$	108.000	0.357771
$46$	160.000	0.512842
$47$	−134.000	−0.415870	−0.207935	−	0.978143i	$-0.566674\pi$
$47$	−134.000	−0.415870	−0.207935	+	0.978143i	$0.566674\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−38.0000	−0.107480
$51$	6.00000	0.0164739
$52$	−52.0000	−0.138675
$53$	−194.000	−0.502791	−0.251396	−	0.967884i	$-0.580890\pi$
$53$	−194.000	−0.502791	−0.251396	+	0.967884i	$0.580890\pi$
$54$	54.0000	0.136083
$55$	−264.000	−0.647232
$56$	−56.0000	−0.133631
$57$	264.000	0.613468
$58$	44.0000	0.0996118
$59$	210.000	0.463384	0.231692	−	0.972789i	$-0.425574\pi$
$59$	210.000	0.463384	0.231692	+	0.972789i	$0.425574\pi$
$60$	−144.000	−0.309839
$61$	−470.000	−0.986514	−0.493257	−	0.869884i	$-0.664194\pi$
$61$	−470.000	−0.986514	−0.493257	+	0.869884i	$0.664194\pi$
$62$	184.000	0.376904
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	−156.000	−0.297683
$66$	−132.000	−0.246183
$67$	−292.000	−0.532440	−0.266220	−	0.963912i	$-0.585775\pi$
$67$	−292.000	−0.532440	−0.266220	+	0.963912i	$0.585775\pi$
$68$	−8.00000	−0.0142668
$69$	240.000	0.418733
$70$	−168.000	−0.286855
$71$	−66.0000	−0.110321	−0.0551603	−	0.998478i	$-0.517567\pi$
$71$	−66.0000	−0.110321	−0.0551603	+	0.998478i	$0.517567\pi$
$72$	−72.0000	−0.117851
$73$	−506.000	−0.811272	−0.405636	−	0.914035i	$-0.632950\pi$
$73$	−506.000	−0.811272	−0.405636	+	0.914035i	$0.632950\pi$
$74$	−236.000	−0.370736
$75$	−57.0000	−0.0877572
$76$	−352.000	−0.531279
$77$	−154.000	−0.227921
$78$	−78.0000	−0.113228
$79$	−776.000	−1.10515	−0.552575	−	0.833463i	$-0.686355\pi$
$79$	−776.000	−1.10515	−0.552575	+	0.833463i	$0.686355\pi$
$80$	192.000	0.268328
$81$	81.0000	0.111111
$82$	−648.000	−0.872678
$83$	−778.000	−1.02887	−0.514437	−	0.857528i	$-0.671999\pi$
$83$	−778.000	−1.02887	−0.514437	+	0.857528i	$0.671999\pi$
$84$	−84.0000	−0.109109
$85$	−24.0000	−0.0306255
$86$	−168.000	−0.210650
$87$	66.0000	0.0813327
$88$	176.000	0.213201
$89$	−920.000	−1.09573	−0.547864	−	0.836567i	$-0.684559\pi$
$89$	−920.000	−1.09573	−0.547864	+	0.836567i	$0.684559\pi$
$90$	−216.000	−0.252982
$91$	−91.0000	−0.104828
$92$	−320.000	−0.362634
$93$	276.000	0.307741
$94$	268.000	0.294065
$95$	−1056.00	−1.14046
$96$	96.0000	0.102062
$97$	−490.000	−0.512907	−0.256453	−	0.966557i	$-0.582554\pi$
$97$	−490.000	−0.512907	−0.256453	+	0.966557i	$0.582554\pi$
$98$	−98.0000	−0.101015
$99$	−198.000	−0.201008
$100$	76.0000	0.0760000
$101$	966.000	0.951689	0.475845	−	0.879529i	$-0.342142\pi$
$101$	966.000	0.951689	0.475845	+	0.879529i	$0.342142\pi$
$102$	−12.0000	−0.0116488
$103$	−1344.00	−1.28571	−0.642856	−	0.765987i	$-0.722250\pi$
$103$	−1344.00	−1.28571	−0.642856	+	0.765987i	$0.722250\pi$
$104$	104.000	0.0980581
$105$	−252.000	−0.234216
$106$	388.000	0.355527
$107$	−52.0000	−0.0469816	−0.0234908	−	0.999724i	$-0.507478\pi$
$107$	−52.0000	−0.0469816	−0.0234908	+	0.999724i	$0.507478\pi$
$108$	−108.000	−0.0962250
$109$	−1542.00	−1.35502	−0.677508	−	0.735515i	$-0.736940\pi$
$109$	−1542.00	−1.35502	−0.677508	+	0.735515i	$0.736940\pi$
$110$	528.000	0.457662
$111$	−354.000	−0.302704
$112$	112.000	0.0944911
$113$	1766.00	1.47019	0.735094	−	0.677965i	$-0.237138\pi$
$113$	1766.00	1.47019	0.735094	+	0.677965i	$0.237138\pi$
$114$	−528.000	−0.433787
$115$	−960.000	−0.778439
$116$	−88.0000	−0.0704362
$117$	−117.000	−0.0924500
$118$	−420.000	−0.327662
$119$	−14.0000	−0.0107847
$120$	288.000	0.219089
$121$	−847.000	−0.636364
$122$	940.000	0.697571
$123$	−972.000	−0.712539
$124$	−368.000	−0.266511
$125$	−1272.00	−0.910169
$126$	−126.000	−0.0890871
$127$	696.000	0.486299	0.243150	−	0.969989i	$-0.421819\pi$
$127$	696.000	0.486299	0.243150	+	0.969989i	$0.421819\pi$
$128$	−128.000	−0.0883883
$129$	−252.000	−0.171995
$130$	312.000	0.210494
$131$	1504.00	1.00309	0.501546	−	0.865131i	$-0.332765\pi$
$131$	1504.00	1.00309	0.501546	+	0.865131i	$0.332765\pi$
$132$	264.000	0.174078
$133$	−616.000	−0.401609
$134$	584.000	0.376492
$135$	−324.000	−0.206559
$136$	16.0000	0.0100882
$137$	−648.000	−0.404105	−0.202052	−	0.979375i	$-0.564761\pi$
$137$	−648.000	−0.404105	−0.202052	+	0.979375i	$0.564761\pi$
$138$	−480.000	−0.296089
$139$	−1388.00	−0.846969	−0.423484	−	0.905903i	$-0.639193\pi$
$139$	−1388.00	−0.846969	−0.423484	+	0.905903i	$0.639193\pi$
$140$	336.000	0.202837
$141$	402.000	0.240103
$142$	132.000	0.0780084
$143$	286.000	0.167248
$144$	144.000	0.0833333
$145$	−264.000	−0.151200
$146$	1012.00	0.573656
$147$	−147.000	−0.0824786
$148$	472.000	0.262150
$149$	2420.00	1.33056	0.665282	−	0.746592i	$-0.268311\pi$
$149$	2420.00	1.33056	0.665282	+	0.746592i	$0.268311\pi$
$150$	114.000	0.0620537
$151$	−2268.00	−1.22230	−0.611150	−	0.791515i	$-0.709293\pi$
$151$	−2268.00	−1.22230	−0.611150	+	0.791515i	$0.709293\pi$
$152$	704.000	0.375671
$153$	−18.0000	−0.00951120
$154$	308.000	0.161165
$155$	−1104.00	−0.572099
$156$	156.000	0.0800641
$157$	−602.000	−0.306018	−0.153009	−	0.988225i	$-0.548896\pi$
$157$	−602.000	−0.306018	−0.153009	+	0.988225i	$0.548896\pi$
$158$	1552.00	0.781459
$159$	582.000	0.290287
$160$	−384.000	−0.189737
$161$	−560.000	−0.274125
$162$	−162.000	−0.0785674
$163$	1216.00	0.584322	0.292161	−	0.956369i	$-0.405626\pi$
$163$	1216.00	0.584322	0.292161	+	0.956369i	$0.405626\pi$
$164$	1296.00	0.617077
$165$	792.000	0.373679
$166$	1556.00	0.727524
$167$	−782.000	−0.362353	−0.181177	−	0.983451i	$-0.557991\pi$
$167$	−782.000	−0.362353	−0.181177	+	0.983451i	$0.557991\pi$
$168$	168.000	0.0771517
$169$	169.000	0.0769231
$170$	48.0000	0.0216555
$171$	−792.000	−0.354186
$172$	336.000	0.148952
$173$	3102.00	1.36324	0.681620	−	0.731706i	$-0.261276\pi$
$173$	3102.00	1.36324	0.681620	+	0.731706i	$0.261276\pi$
$174$	−132.000	−0.0575109
$175$	133.000	0.0574506
$176$	−352.000	−0.150756
$177$	−630.000	−0.267535
$178$	1840.00	0.774797
$179$	−2824.00	−1.17919	−0.589597	−	0.807698i	$-0.700713\pi$
$179$	−2824.00	−1.17919	−0.589597	+	0.807698i	$0.700713\pi$
$180$	432.000	0.178885
$181$	−842.000	−0.345776	−0.172888	−	0.984942i	$-0.555310\pi$
$181$	−842.000	−0.345776	−0.172888	+	0.984942i	$0.555310\pi$
$182$	182.000	0.0741249
$183$	1410.00	0.569564
$184$	640.000	0.256421
$185$	1416.00	0.562737
$186$	−552.000	−0.217605
$187$	44.0000	0.0172064
$188$	−536.000	−0.207935
$189$	−189.000	−0.0727393
$190$	2112.00	0.806424
$191$	2548.00	0.965271	0.482636	−	0.875821i	$-0.339680\pi$
$191$	2548.00	0.965271	0.482636	+	0.875821i	$0.339680\pi$
$192$	−192.000	−0.0721688
$193$	−1118.00	−0.416971	−0.208485	−	0.978025i	$-0.566853\pi$
$193$	−1118.00	−0.416971	−0.208485	+	0.978025i	$0.566853\pi$
$194$	980.000	0.362680
$195$	468.000	0.171868
$196$	196.000	0.0714286
$197$	1992.00	0.720427	0.360214	−	0.932870i	$-0.382704\pi$
$197$	1992.00	0.720427	0.360214	+	0.932870i	$0.382704\pi$
$198$	396.000	0.142134
$199$	−3536.00	−1.25960	−0.629800	−	0.776757i	$-0.716863\pi$
$199$	−3536.00	−1.25960	−0.629800	+	0.776757i	$0.716863\pi$
$200$	−152.000	−0.0537401
$201$	876.000	0.307404
$202$	−1932.00	−0.672946
$203$	−154.000	−0.0532447
$204$	24.0000	0.00823694
$205$	3888.00	1.32463
$206$	2688.00	0.909135
$207$	−720.000	−0.241756
$208$	−208.000	−0.0693375
$209$	1936.00	0.640746
$210$	504.000	0.165616
$211$	−2900.00	−0.946181	−0.473091	−	0.881014i	$-0.656862\pi$
$211$	−2900.00	−0.946181	−0.473091	+	0.881014i	$0.656862\pi$
$212$	−776.000	−0.251396
$213$	198.000	0.0636936
$214$	104.000	0.0332210
$215$	1008.00	0.319744
$216$	216.000	0.0680414
$217$	−644.000	−0.201463
$218$	3084.00	0.958141
$219$	1518.00	0.468388
$220$	−1056.00	−0.323616
$221$	26.0000	0.00791380
$222$	708.000	0.214044
$223$	4212.00	1.26483	0.632413	−	0.774631i	$-0.282064\pi$
$223$	4212.00	1.26483	0.632413	+	0.774631i	$0.282064\pi$
$224$	−224.000	−0.0668153
$225$	171.000	0.0506667
$226$	−3532.00	−1.03958
$227$	−2070.00	−0.605245	−0.302623	−	0.953110i	$-0.597862\pi$
$227$	−2070.00	−0.605245	−0.302623	+	0.953110i	$0.597862\pi$
$228$	1056.00	0.306734
$229$	−2198.00	−0.634270	−0.317135	−	0.948380i	$-0.602721\pi$
$229$	−2198.00	−0.634270	−0.317135	+	0.948380i	$0.602721\pi$
$230$	1920.00	0.550439
$231$	462.000	0.131590
$232$	176.000	0.0498059
$233$	−3378.00	−0.949786	−0.474893	−	0.880044i	$-0.657513\pi$
$233$	−3378.00	−0.949786	−0.474893	+	0.880044i	$0.657513\pi$
$234$	234.000	0.0653720
$235$	−1608.00	−0.446359
$236$	840.000	0.231692
$237$	2328.00	0.638058
$238$	28.0000	0.00762593
$239$	−2874.00	−0.777839	−0.388920	−	0.921272i	$-0.627152\pi$
$239$	−2874.00	−0.777839	−0.388920	+	0.921272i	$0.627152\pi$
$240$	−576.000	−0.154919
$241$	1926.00	0.514791	0.257395	−	0.966306i	$-0.417136\pi$
$241$	1926.00	0.514791	0.257395	+	0.966306i	$0.417136\pi$
$242$	1694.00	0.449977
$243$	−243.000	−0.0641500
$244$	−1880.00	−0.493257
$245$	588.000	0.153330
$246$	1944.00	0.503841
$247$	1144.00	0.294700
$248$	736.000	0.188452
$249$	2334.00	0.594021
$250$	2544.00	0.643587
$251$	−2024.00	−0.508979	−0.254490	−	0.967076i	$-0.581907\pi$
$251$	−2024.00	−0.508979	−0.254490	+	0.967076i	$0.581907\pi$
$252$	252.000	0.0629941
$253$	1760.00	0.437353
$254$	−1392.00	−0.343866
$255$	72.0000	0.0176816
$256$	256.000	0.0625000
$257$	5642.00	1.36941	0.684705	−	0.728820i	$-0.259931\pi$
$257$	5642.00	1.36941	0.684705	+	0.728820i	$0.259931\pi$
$258$	504.000	0.121619
$259$	826.000	0.198167
$260$	−624.000	−0.148842
$261$	−198.000	−0.0469574
$262$	−3008.00	−0.709293
$263$	−4680.00	−1.09727	−0.548633	−	0.836063i	$-0.684852\pi$
$263$	−4680.00	−1.09727	−0.548633	+	0.836063i	$0.684852\pi$
$264$	−528.000	−0.123091
$265$	−2328.00	−0.539652
$266$	1232.00	0.283980
$267$	2760.00	0.632619
$268$	−1168.00	−0.266220
$269$	3758.00	0.851782	0.425891	−	0.904775i	$-0.359961\pi$
$269$	3758.00	0.851782	0.425891	+	0.904775i	$0.359961\pi$
$270$	648.000	0.146059
$271$	128.000	0.0286917	0.0143458	−	0.999897i	$-0.495433\pi$
$271$	128.000	0.0286917	0.0143458	+	0.999897i	$0.495433\pi$
$272$	−32.0000	−0.00713340
$273$	273.000	0.0605228
$274$	1296.00	0.285745
$275$	−418.000	−0.0916594
$276$	960.000	0.209367
$277$	−826.000	−0.179168	−0.0895840	−	0.995979i	$-0.528554\pi$
$277$	−826.000	−0.179168	−0.0895840	+	0.995979i	$0.528554\pi$
$278$	2776.00	0.598897
$279$	−828.000	−0.177674
$280$	−672.000	−0.143427
$281$	−1448.00	−0.307404	−0.153702	−	0.988117i	$-0.549120\pi$
$281$	−1448.00	−0.307404	−0.153702	+	0.988117i	$0.549120\pi$
$282$	−804.000	−0.169778
$283$	6084.00	1.27794	0.638969	−	0.769233i	$-0.279361\pi$
$283$	6084.00	1.27794	0.638969	+	0.769233i	$0.279361\pi$
$284$	−264.000	−0.0551603
$285$	3168.00	0.658443
$286$	−572.000	−0.118262
$287$	2268.00	0.466466
$288$	−288.000	−0.0589256
$289$	−4909.00	−0.999186
$290$	528.000	0.106915
$291$	1470.00	0.296127
$292$	−2024.00	−0.405636
$293$	−648.000	−0.129203	−0.0646016	−	0.997911i	$-0.520578\pi$
$293$	−648.000	−0.129203	−0.0646016	+	0.997911i	$0.520578\pi$
$294$	294.000	0.0583212
$295$	2520.00	0.497356
$296$	−944.000	−0.185368
$297$	594.000	0.116052
$298$	−4840.00	−0.940851
$299$	1040.00	0.201153
$300$	−228.000	−0.0438786
$301$	588.000	0.112597
$302$	4536.00	0.864296
$303$	−2898.00	−0.549458
$304$	−1408.00	−0.265639
$305$	−5640.00	−1.05884
$306$	36.0000	0.00672543
$307$	5636.00	1.04776	0.523882	−	0.851791i	$-0.324483\pi$
$307$	5636.00	1.04776	0.523882	+	0.851791i	$0.324483\pi$
$308$	−616.000	−0.113961
$309$	4032.00	0.742306
$310$	2208.00	0.404535
$311$	−4964.00	−0.905089	−0.452544	−	0.891742i	$-0.649484\pi$
$311$	−4964.00	−0.905089	−0.452544	+	0.891742i	$0.649484\pi$
$312$	−312.000	−0.0566139
$313$	5934.00	1.07160	0.535798	−	0.844346i	$-0.320011\pi$
$313$	5934.00	1.07160	0.535798	+	0.844346i	$0.320011\pi$
$314$	1204.00	0.216387
$315$	756.000	0.135225
$316$	−3104.00	−0.552575
$317$	2560.00	0.453577	0.226789	−	0.973944i	$-0.427177\pi$
$317$	2560.00	0.453577	0.226789	+	0.973944i	$0.427177\pi$
$318$	−1164.00	−0.205264
$319$	484.000	0.0849492
$320$	768.000	0.134164
$321$	156.000	0.0271248
$322$	1120.00	0.193836
$323$	176.000	0.0303186
$324$	324.000	0.0555556
$325$	−247.000	−0.0421572
$326$	−2432.00	−0.413178
$327$	4626.00	0.782319
$328$	−2592.00	−0.436339
$329$	−938.000	−0.157184
$330$	−1584.00	−0.264231
$331$	4020.00	0.667550	0.333775	−	0.942653i	$-0.391677\pi$
$331$	4020.00	0.667550	0.333775	+	0.942653i	$0.391677\pi$
$332$	−3112.00	−0.514437
$333$	1062.00	0.174766
$334$	1564.00	0.256222
$335$	−3504.00	−0.571475
$336$	−336.000	−0.0545545
$337$	4734.00	0.765215	0.382607	−	0.923911i	$-0.375026\pi$
$337$	4734.00	0.765215	0.382607	+	0.923911i	$0.375026\pi$
$338$	−338.000	−0.0543928
$339$	−5298.00	−0.848814
$340$	−96.0000	−0.0153127
$341$	2024.00	0.321424
$342$	1584.00	0.250447
$343$	343.000	0.0539949
$344$	−672.000	−0.105325
$345$	2880.00	0.449432
$346$	−6204.00	−0.963957
$347$	−7864.00	−1.21660	−0.608302	−	0.793706i	$-0.708149\pi$
$347$	−7864.00	−1.21660	−0.608302	+	0.793706i	$0.708149\pi$
$348$	264.000	0.0406663
$349$	5870.00	0.900326	0.450163	−	0.892946i	$-0.351366\pi$
$349$	5870.00	0.900326	0.450163	+	0.892946i	$0.351366\pi$
$350$	−266.000	−0.0406237
$351$	351.000	0.0533761
$352$	704.000	0.106600
$353$	5532.00	0.834104	0.417052	−	0.908883i	$-0.363063\pi$
$353$	5532.00	0.834104	0.417052	+	0.908883i	$0.363063\pi$
$354$	1260.00	0.189176
$355$	−792.000	−0.118408
$356$	−3680.00	−0.547864
$357$	42.0000	0.00622654
$358$	5648.00	0.833816
$359$	5854.00	0.860619	0.430310	−	0.902681i	$-0.358404\pi$
$359$	5854.00	0.860619	0.430310	+	0.902681i	$0.358404\pi$
$360$	−864.000	−0.126491
$361$	885.000	0.129028
$362$	1684.00	0.244500
$363$	2541.00	0.367405
$364$	−364.000	−0.0524142
$365$	−6072.00	−0.870748
$366$	−2820.00	−0.402743
$367$	−7576.00	−1.07756	−0.538779	−	0.842447i	$-0.681114\pi$
$367$	−7576.00	−1.07756	−0.538779	+	0.842447i	$0.681114\pi$
$368$	−1280.00	−0.181317
$369$	2916.00	0.411385
$370$	−2832.00	−0.397915
$371$	−1358.00	−0.190037
$372$	1104.00	0.153870
$373$	−1502.00	−0.208500	−0.104250	−	0.994551i	$-0.533244\pi$
$373$	−1502.00	−0.208500	−0.104250	+	0.994551i	$0.533244\pi$
$374$	−88.0000	−0.0121668
$375$	3816.00	0.525486
$376$	1072.00	0.147032
$377$	286.000	0.0390710
$378$	378.000	0.0514344
$379$	−8584.00	−1.16340	−0.581702	−	0.813402i	$-0.697613\pi$
$379$	−8584.00	−1.16340	−0.581702	+	0.813402i	$0.697613\pi$
$380$	−4224.00	−0.570228
$381$	−2088.00	−0.280765
$382$	−5096.00	−0.682550
$383$	8598.00	1.14709	0.573547	−	0.819172i	$-0.305567\pi$
$383$	8598.00	1.14709	0.573547	+	0.819172i	$0.305567\pi$
$384$	384.000	0.0510310
$385$	−1848.00	−0.244631
$386$	2236.00	0.294843
$387$	756.000	0.0993014
$388$	−1960.00	−0.256453
$389$	−11666.0	−1.52054	−0.760270	−	0.649608i	$-0.774933\pi$
$389$	−11666.0	−1.52054	−0.760270	+	0.649608i	$0.774933\pi$
$390$	−936.000	−0.121529
$391$	160.000	0.0206945
$392$	−392.000	−0.0505076
$393$	−4512.00	−0.579136
$394$	−3984.00	−0.509419
$395$	−9312.00	−1.18617
$396$	−792.000	−0.100504
$397$	5334.00	0.674322	0.337161	−	0.941447i	$-0.390533\pi$
$397$	5334.00	0.674322	0.337161	+	0.941447i	$0.390533\pi$
$398$	7072.00	0.890672
$399$	1848.00	0.231869
$400$	304.000	0.0380000
$401$	1852.00	0.230635	0.115317	−	0.993329i	$-0.463212\pi$
$401$	1852.00	0.230635	0.115317	+	0.993329i	$0.463212\pi$
$402$	−1752.00	−0.217368
$403$	1196.00	0.147834
$404$	3864.00	0.475845
$405$	972.000	0.119257
$406$	308.000	0.0376497
$407$	−2596.00	−0.316164
$408$	−48.0000	−0.00582440
$409$	−13530.0	−1.63573	−0.817867	−	0.575407i	$-0.804844\pi$
$409$	−13530.0	−1.63573	−0.817867	+	0.575407i	$0.804844\pi$
$410$	−7776.00	−0.936657
$411$	1944.00	0.233310
$412$	−5376.00	−0.642856
$413$	1470.00	0.175143
$414$	1440.00	0.170947
$415$	−9336.00	−1.10430
$416$	416.000	0.0490290
$417$	4164.00	0.488997
$418$	−3872.00	−0.453076
$419$	1320.00	0.153905	0.0769525	−	0.997035i	$-0.475481\pi$
$419$	1320.00	0.153905	0.0769525	+	0.997035i	$0.475481\pi$
$420$	−1008.00	−0.117108
$421$	6322.00	0.731866	0.365933	−	0.930641i	$-0.380750\pi$
$421$	6322.00	0.731866	0.365933	+	0.930641i	$0.380750\pi$
$422$	5800.00	0.669051
$423$	−1206.00	−0.138623
$424$	1552.00	0.177764
$425$	−38.0000	−0.00433711
$426$	−396.000	−0.0450382
$427$	−3290.00	−0.372867
$428$	−208.000	−0.0234908
$429$	−858.000	−0.0965609
$430$	−2016.00	−0.226093
$431$	3498.00	0.390934	0.195467	−	0.980710i	$-0.437378\pi$
$431$	3498.00	0.390934	0.195467	+	0.980710i	$0.437378\pi$
$432$	−432.000	−0.0481125
$433$	−394.000	−0.0437285	−0.0218642	−	0.999761i	$-0.506960\pi$
$433$	−394.000	−0.0437285	−0.0218642	+	0.999761i	$0.506960\pi$
$434$	1288.00	0.142456
$435$	792.000	0.0872954
$436$	−6168.00	−0.677508
$437$	7040.00	0.770638
$438$	−3036.00	−0.331200
$439$	10528.0	1.14459	0.572294	−	0.820049i	$-0.306054\pi$
$439$	10528.0	1.14459	0.572294	+	0.820049i	$0.306054\pi$
$440$	2112.00	0.228831
$441$	441.000	0.0476190
$442$	−52.0000	−0.00559590
$443$	−5028.00	−0.539249	−0.269625	−	0.962965i	$-0.586900\pi$
$443$	−5028.00	−0.539249	−0.269625	+	0.962965i	$0.586900\pi$
$444$	−1416.00	−0.151352
$445$	−11040.0	−1.17606
$446$	−8424.00	−0.894368
$447$	−7260.00	−0.768202
$448$	448.000	0.0472456
$449$	−3012.00	−0.316581	−0.158291	−	0.987393i	$-0.550598\pi$
$449$	−3012.00	−0.316581	−0.158291	+	0.987393i	$0.550598\pi$
$450$	−342.000	−0.0358267
$451$	−7128.00	−0.744223
$452$	7064.00	0.735094
$453$	6804.00	0.705695
$454$	4140.00	0.427973
$455$	−1092.00	−0.112514
$456$	−2112.00	−0.216894
$457$	−5506.00	−0.563588	−0.281794	−	0.959475i	$-0.590930\pi$
$457$	−5506.00	−0.563588	−0.281794	+	0.959475i	$0.590930\pi$
$458$	4396.00	0.448497
$459$	54.0000	0.00549129
$460$	−3840.00	−0.389219
$461$	4764.00	0.481305	0.240652	−	0.970611i	$-0.422639\pi$
$461$	4764.00	0.481305	0.240652	+	0.970611i	$0.422639\pi$
$462$	−924.000	−0.0930484
$463$	5016.00	0.503484	0.251742	−	0.967794i	$-0.418996\pi$
$463$	5016.00	0.503484	0.251742	+	0.967794i	$0.418996\pi$
$464$	−352.000	−0.0352181
$465$	3312.00	0.330302
$466$	6756.00	0.671600
$467$	−8928.00	−0.884665	−0.442333	−	0.896851i	$-0.645849\pi$
$467$	−8928.00	−0.884665	−0.442333	+	0.896851i	$0.645849\pi$
$468$	−468.000	−0.0462250
$469$	−2044.00	−0.201243
$470$	3216.00	0.315623
$471$	1806.00	0.176680
$472$	−1680.00	−0.163831
$473$	−1848.00	−0.179643
$474$	−4656.00	−0.451175
$475$	−1672.00	−0.161509
$476$	−56.0000	−0.00539234
$477$	−1746.00	−0.167597
$478$	5748.00	0.550015
$479$	3166.00	0.302000	0.151000	−	0.988534i	$-0.451751\pi$
$479$	3166.00	0.302000	0.151000	+	0.988534i	$0.451751\pi$
$480$	1152.00	0.109545
$481$	−1534.00	−0.145415
$482$	−3852.00	−0.364012
$483$	1680.00	0.158266
$484$	−3388.00	−0.318182
$485$	−5880.00	−0.550509
$486$	486.000	0.0453609
$487$	−596.000	−0.0554565	−0.0277283	−	0.999615i	$-0.508827\pi$
$487$	−596.000	−0.0554565	−0.0277283	+	0.999615i	$0.508827\pi$
$488$	3760.00	0.348785
$489$	−3648.00	−0.337358
$490$	−1176.00	−0.108421
$491$	−6940.00	−0.637877	−0.318939	−	0.947775i	$-0.603326\pi$
$491$	−6940.00	−0.637877	−0.318939	+	0.947775i	$0.603326\pi$
$492$	−3888.00	−0.356269
$493$	44.0000	0.00401960
$494$	−2288.00	−0.208385
$495$	−2376.00	−0.215744
$496$	−1472.00	−0.133256
$497$	−462.000	−0.0416972
$498$	−4668.00	−0.420036
$499$	1156.00	0.103707	0.0518534	−	0.998655i	$-0.483487\pi$
$499$	1156.00	0.103707	0.0518534	+	0.998655i	$0.483487\pi$
$500$	−5088.00	−0.455085
$501$	2346.00	0.209205
$502$	4048.00	0.359903
$503$	2068.00	0.183315	0.0916576	−	0.995791i	$-0.470783\pi$
$503$	2068.00	0.183315	0.0916576	+	0.995791i	$0.470783\pi$
$504$	−504.000	−0.0445435
$505$	11592.0	1.02146
$506$	−3520.00	−0.309255
$507$	−507.000	−0.0444116
$508$	2784.00	0.243150
$509$	1140.00	0.0992723	0.0496362	−	0.998767i	$-0.484194\pi$
$509$	1140.00	0.0992723	0.0496362	+	0.998767i	$0.484194\pi$
$510$	−144.000	−0.0125028
$511$	−3542.00	−0.306632
$512$	−512.000	−0.0441942
$513$	2376.00	0.204489
$514$	−11284.0	−0.968319
$515$	−16128.0	−1.37997
$516$	−1008.00	−0.0859975
$517$	2948.00	0.250779
$518$	−1652.00	−0.140125
$519$	−9306.00	−0.787068
$520$	1248.00	0.105247
$521$	−13606.0	−1.14413	−0.572063	−	0.820210i	$-0.693857\pi$
$521$	−13606.0	−1.14413	−0.572063	+	0.820210i	$0.693857\pi$
$522$	396.000	0.0332039
$523$	−6868.00	−0.574219	−0.287110	−	0.957898i	$-0.592694\pi$
$523$	−6868.00	−0.574219	−0.287110	+	0.957898i	$0.592694\pi$
$524$	6016.00	0.501546
$525$	−399.000	−0.0331691
$526$	9360.00	0.775885
$527$	184.000	0.0152090
$528$	1056.00	0.0870388
$529$	−5767.00	−0.473987
$530$	4656.00	0.381592
$531$	1890.00	0.154461
$532$	−2464.00	−0.200804
$533$	−4212.00	−0.342293
$534$	−5520.00	−0.447329
$535$	−624.000	−0.0504259
$536$	2336.00	0.188246
$537$	8472.00	0.680808
$538$	−7516.00	−0.602301
$539$	−1078.00	−0.0861461
$540$	−1296.00	−0.103280
$541$	1170.00	0.0929801	0.0464900	−	0.998919i	$-0.485196\pi$
$541$	1170.00	0.0929801	0.0464900	+	0.998919i	$0.485196\pi$
$542$	−256.000	−0.0202881
$543$	2526.00	0.199634
$544$	64.0000	0.00504408
$545$	−18504.0	−1.45436
$546$	−546.000	−0.0427960
$547$	5524.00	0.431790	0.215895	−	0.976417i	$-0.430733\pi$
$547$	5524.00	0.431790	0.215895	+	0.976417i	$0.430733\pi$
$548$	−2592.00	−0.202052
$549$	−4230.00	−0.328838
$550$	836.000	0.0648130
$551$	1936.00	0.149685
$552$	−1920.00	−0.148045
$553$	−5432.00	−0.417707
$554$	1652.00	0.126691
$555$	−4248.00	−0.324897
$556$	−5552.00	−0.423484
$557$	6204.00	0.471942	0.235971	−	0.971760i	$-0.424173\pi$
$557$	6204.00	0.471942	0.235971	+	0.971760i	$0.424173\pi$
$558$	1656.00	0.125635
$559$	−1092.00	−0.0826238
$560$	1344.00	0.101419
$561$	−132.000	−0.00993413
$562$	2896.00	0.217367
$563$	23140.0	1.73221	0.866105	−	0.499861i	$-0.166616\pi$
$563$	23140.0	1.73221	0.866105	+	0.499861i	$0.166616\pi$
$564$	1608.00	0.120051
$565$	21192.0	1.57797
$566$	−12168.0	−0.903638
$567$	567.000	0.0419961
$568$	528.000	0.0390042
$569$	7086.00	0.522075	0.261037	−	0.965329i	$-0.415935\pi$
$569$	7086.00	0.522075	0.261037	+	0.965329i	$0.415935\pi$
$570$	−6336.00	−0.465589
$571$	5956.00	0.436516	0.218258	−	0.975891i	$-0.429963\pi$
$571$	5956.00	0.436516	0.218258	+	0.975891i	$0.429963\pi$
$572$	1144.00	0.0836242
$573$	−7644.00	−0.557300
$574$	−4536.00	−0.329841
$575$	−1520.00	−0.110241
$576$	576.000	0.0416667
$577$	8714.00	0.628715	0.314358	−	0.949305i	$-0.398211\pi$
$577$	8714.00	0.628715	0.314358	+	0.949305i	$0.398211\pi$
$578$	9818.00	0.706531
$579$	3354.00	0.240738
$580$	−1056.00	−0.0756000
$581$	−5446.00	−0.388878
$582$	−2940.00	−0.209393
$583$	4268.00	0.303195
$584$	4048.00	0.286828
$585$	−1404.00	−0.0992278
$586$	1296.00	0.0913605
$587$	17286.0	1.21545	0.607725	−	0.794147i	$-0.292082\pi$
$587$	17286.0	1.21545	0.607725	+	0.794147i	$0.292082\pi$
$588$	−588.000	−0.0412393
$589$	8096.00	0.566366
$590$	−5040.00	−0.351684
$591$	−5976.00	−0.415939
$592$	1888.00	0.131075
$593$	4936.00	0.341817	0.170908	−	0.985287i	$-0.445330\pi$
$593$	4936.00	0.341817	0.170908	+	0.985287i	$0.445330\pi$
$594$	−1188.00	−0.0820610
$595$	−168.000	−0.0115753
$596$	9680.00	0.665282
$597$	10608.0	0.727230
$598$	−2080.00	−0.142237
$599$	−18936.0	−1.29166	−0.645830	−	0.763481i	$-0.723488\pi$
$599$	−18936.0	−1.29166	−0.645830	+	0.763481i	$0.723488\pi$
$600$	456.000	0.0310269
$601$	62.0000	0.00420804	0.00210402	−	0.999998i	$-0.499330\pi$
$601$	62.0000	0.00420804	0.00210402	+	0.999998i	$0.499330\pi$
$602$	−1176.00	−0.0796182
$603$	−2628.00	−0.177480
$604$	−9072.00	−0.611150
$605$	−10164.0	−0.683017
$606$	5796.00	0.388525
$607$	22792.0	1.52405	0.762025	−	0.647547i	$-0.224205\pi$
$607$	22792.0	1.52405	0.762025	+	0.647547i	$0.224205\pi$
$608$	2816.00	0.187835
$609$	462.000	0.0307409
$610$	11280.0	0.748711
$611$	1742.00	0.115342
$612$	−72.0000	−0.00475560
$613$	30022.0	1.97810	0.989052	−	0.147571i	$-0.0471454\pi$
$613$	30022.0	1.97810	0.989052	+	0.147571i	$0.0471454\pi$
$614$	−11272.0	−0.740881
$615$	−11664.0	−0.764777
$616$	1232.00	0.0805823
$617$	26640.0	1.73823	0.869113	−	0.494613i	$-0.164690\pi$
$617$	26640.0	1.73823	0.869113	+	0.494613i	$0.164690\pi$
$618$	−8064.00	−0.524889
$619$	−180.000	−0.0116879	−0.00584395	−	0.999983i	$-0.501860\pi$
$619$	−180.000	−0.0116879	−0.00584395	+	0.999983i	$0.501860\pi$
$620$	−4416.00	−0.286050
$621$	2160.00	0.139578
$622$	9928.00	0.639994
$623$	−6440.00	−0.414146
$624$	624.000	0.0400320
$625$	−17639.0	−1.12890
$626$	−11868.0	−0.757733
$627$	−5808.00	−0.369935
$628$	−2408.00	−0.153009
$629$	−236.000	−0.0149602
$630$	−1512.00	−0.0956183
$631$	6652.00	0.419670	0.209835	−	0.977737i	$-0.432707\pi$
$631$	6652.00	0.419670	0.209835	+	0.977737i	$0.432707\pi$
$632$	6208.00	0.390729
$633$	8700.00	0.546278
$634$	−5120.00	−0.320727
$635$	8352.00	0.521951
$636$	2328.00	0.145143
$637$	−637.000	−0.0396214
$638$	−968.000	−0.0600682
$639$	−594.000	−0.0367735
$640$	−1536.00	−0.0948683
$641$	15258.0	0.940179	0.470090	−	0.882619i	$-0.344222\pi$
$641$	15258.0	0.940179	0.470090	+	0.882619i	$0.344222\pi$
$642$	−312.000	−0.0191802
$643$	−17124.0	−1.05024	−0.525120	−	0.851028i	$-0.675980\pi$
$643$	−17124.0	−1.05024	−0.525120	+	0.851028i	$0.675980\pi$
$644$	−2240.00	−0.137063
$645$	−3024.00	−0.184604
$646$	−352.000	−0.0214385
$647$	7032.00	0.427290	0.213645	−	0.976911i	$-0.431466\pi$
$647$	7032.00	0.427290	0.213645	+	0.976911i	$0.431466\pi$
$648$	−648.000	−0.0392837
$649$	−4620.00	−0.279431
$650$	494.000	0.0298097
$651$	1932.00	0.116315
$652$	4864.00	0.292161
$653$	−19254.0	−1.15385	−0.576927	−	0.816795i	$-0.695748\pi$
$653$	−19254.0	−1.15385	−0.576927	+	0.816795i	$0.695748\pi$
$654$	−9252.00	−0.553183
$655$	18048.0	1.07663
$656$	5184.00	0.308538
$657$	−4554.00	−0.270424
$658$	1876.00	0.111146
$659$	−1136.00	−0.0671506	−0.0335753	−	0.999436i	$-0.510689\pi$
$659$	−1136.00	−0.0671506	−0.0335753	+	0.999436i	$0.510689\pi$
$660$	3168.00	0.186840
$661$	20638.0	1.21441	0.607205	−	0.794545i	$-0.292291\pi$
$661$	20638.0	1.21441	0.607205	+	0.794545i	$0.292291\pi$
$662$	−8040.00	−0.472029
$663$	−78.0000	−0.00456903
$664$	6224.00	0.363762
$665$	−7392.00	−0.431052
$666$	−2124.00	−0.123579
$667$	1760.00	0.102170
$668$	−3128.00	−0.181177
$669$	−12636.0	−0.730248
$670$	7008.00	0.404094
$671$	10340.0	0.594890
$672$	672.000	0.0385758
$673$	14466.0	0.828564	0.414282	−	0.910149i	$-0.364033\pi$
$673$	14466.0	0.828564	0.414282	+	0.910149i	$0.364033\pi$
$674$	−9468.00	−0.541089
$675$	−513.000	−0.0292524
$676$	676.000	0.0384615
$677$	3162.00	0.179506	0.0897530	−	0.995964i	$-0.471392\pi$
$677$	3162.00	0.179506	0.0897530	+	0.995964i	$0.471392\pi$
$678$	10596.0	0.600202
$679$	−3430.00	−0.193861
$680$	192.000	0.0108277
$681$	6210.00	0.349439
$682$	−4048.00	−0.227281
$683$	−26334.0	−1.47532	−0.737659	−	0.675173i	$-0.764069\pi$
$683$	−26334.0	−1.47532	−0.737659	+	0.675173i	$0.764069\pi$
$684$	−3168.00	−0.177093
$685$	−7776.00	−0.433731
$686$	−686.000	−0.0381802
$687$	6594.00	0.366196
$688$	1344.00	0.0744760
$689$	2522.00	0.139449
$690$	−5760.00	−0.317796
$691$	11124.0	0.612412	0.306206	−	0.951965i	$-0.400940\pi$
$691$	11124.0	0.612412	0.306206	+	0.951965i	$0.400940\pi$
$692$	12408.0	0.681620
$693$	−1386.00	−0.0759737
$694$	15728.0	0.860269
$695$	−16656.0	−0.909062
$696$	−528.000	−0.0287554
$697$	−648.000	−0.0352148
$698$	−11740.0	−0.636627
$699$	10134.0	0.548359
$700$	532.000	0.0287253
$701$	6270.00	0.337824	0.168912	−	0.985631i	$-0.445975\pi$
$701$	6270.00	0.337824	0.168912	+	0.985631i	$0.445975\pi$
$702$	−702.000	−0.0377426
$703$	−10384.0	−0.557098
$704$	−1408.00	−0.0753778
$705$	4824.00	0.257705
$706$	−11064.0	−0.589800
$707$	6762.00	0.359705
$708$	−2520.00	−0.133768
$709$	17186.0	0.910344	0.455172	−	0.890404i	$-0.349578\pi$
$709$	17186.0	0.910344	0.455172	+	0.890404i	$0.349578\pi$
$710$	1584.00	0.0837274
$711$	−6984.00	−0.368383
$712$	7360.00	0.387398
$713$	7360.00	0.386584
$714$	−84.0000	−0.00440283
$715$	3432.00	0.179510
$716$	−11296.0	−0.589597
$717$	8622.00	0.449086
$718$	−11708.0	−0.608550
$719$	29244.0	1.51685	0.758426	−	0.651759i	$-0.225969\pi$
$719$	29244.0	1.51685	0.758426	+	0.651759i	$0.225969\pi$
$720$	1728.00	0.0894427
$721$	−9408.00	−0.485953
$722$	−1770.00	−0.0912363
$723$	−5778.00	−0.297215
$724$	−3368.00	−0.172888
$725$	−418.000	−0.0214126
$726$	−5082.00	−0.259794
$727$	22760.0	1.16110	0.580551	−	0.814224i	$-0.302837\pi$
$727$	22760.0	1.16110	0.580551	+	0.814224i	$0.302837\pi$
$728$	728.000	0.0370625
$729$	729.000	0.0370370
$730$	12144.0	0.615712
$731$	−168.000	−0.00850028
$732$	5640.00	0.284782
$733$	26546.0	1.33765	0.668826	−	0.743419i	$-0.266797\pi$
$733$	26546.0	1.33765	0.668826	+	0.743419i	$0.266797\pi$
$734$	15152.0	0.761949
$735$	−1764.00	−0.0885253
$736$	2560.00	0.128210
$737$	6424.00	0.321073
$738$	−5832.00	−0.290893
$739$	932.000	0.0463927	0.0231963	−	0.999731i	$-0.492616\pi$
$739$	932.000	0.0463927	0.0231963	+	0.999731i	$0.492616\pi$
$740$	5664.00	0.281369
$741$	−3432.00	−0.170145
$742$	2716.00	0.134377
$743$	−8418.00	−0.415648	−0.207824	−	0.978166i	$-0.566638\pi$
$743$	−8418.00	−0.415648	−0.207824	+	0.978166i	$0.566638\pi$
$744$	−2208.00	−0.108803
$745$	29040.0	1.42811
$746$	3004.00	0.147432
$747$	−7002.00	−0.342958
$748$	176.000	0.00860320
$749$	−364.000	−0.0177574
$750$	−7632.00	−0.371575
$751$	−15704.0	−0.763045	−0.381523	−	0.924359i	$-0.624600\pi$
$751$	−15704.0	−0.763045	−0.381523	+	0.924359i	$0.624600\pi$
$752$	−2144.00	−0.103968
$753$	6072.00	0.293859
$754$	−572.000	−0.0276273
$755$	−27216.0	−1.31191
$756$	−756.000	−0.0363696
$757$	−16734.0	−0.803445	−0.401722	−	0.915762i	$-0.631588\pi$
$757$	−16734.0	−0.803445	−0.401722	+	0.915762i	$0.631588\pi$
$758$	17168.0	0.822652
$759$	−5280.00	−0.252506
$760$	8448.00	0.403212
$761$	−11196.0	−0.533318	−0.266659	−	0.963791i	$-0.585920\pi$
$761$	−11196.0	−0.533318	−0.266659	+	0.963791i	$0.585920\pi$
$762$	4176.00	0.198531
$763$	−10794.0	−0.512148
$764$	10192.0	0.482636
$765$	−216.000	−0.0102085
$766$	−17196.0	−0.811118
$767$	−2730.00	−0.128520
$768$	−768.000	−0.0360844
$769$	18938.0	0.888065	0.444032	−	0.896011i	$-0.353548\pi$
$769$	18938.0	0.888065	0.444032	+	0.896011i	$0.353548\pi$
$770$	3696.00	0.172980
$771$	−16926.0	−0.790629
$772$	−4472.00	−0.208485
$773$	−13672.0	−0.636155	−0.318077	−	0.948065i	$-0.603037\pi$
$773$	−13672.0	−0.636155	−0.318077	+	0.948065i	$0.603037\pi$
$774$	−1512.00	−0.0702167
$775$	−1748.00	−0.0810194
$776$	3920.00	0.181340
$777$	−2478.00	−0.114412
$778$	23332.0	1.07518
$779$	−28512.0	−1.31136
$780$	1872.00	0.0859338
$781$	1452.00	0.0665258
$782$	−320.000	−0.0146332
$783$	594.000	0.0271109
$784$	784.000	0.0357143
$785$	−7224.00	−0.328453
$786$	9024.00	0.409511
$787$	16484.0	0.746622	0.373311	−	0.927706i	$-0.378223\pi$
$787$	16484.0	0.746622	0.373311	+	0.927706i	$0.378223\pi$
$788$	7968.00	0.360214
$789$	14040.0	0.633507
$790$	18624.0	0.838750
$791$	12362.0	0.555679
$792$	1584.00	0.0710669
$793$	6110.00	0.273610
$794$	−10668.0	−0.476818
$795$	6984.00	0.311568
$796$	−14144.0	−0.629800
$797$	17778.0	0.790124	0.395062	−	0.918654i	$-0.370723\pi$
$797$	17778.0	0.790124	0.395062	+	0.918654i	$0.370723\pi$
$798$	−3696.00	−0.163956
$799$	268.000	0.0118663
$800$	−608.000	−0.0268701
$801$	−8280.00	−0.365243
$802$	−3704.00	−0.163083
$803$	11132.0	0.489215
$804$	3504.00	0.153702
$805$	−6720.00	−0.294222
$806$	−2392.00	−0.104534
$807$	−11274.0	−0.491776
$808$	−7728.00	−0.336473
$809$	−33138.0	−1.44014	−0.720068	−	0.693903i	$-0.755889\pi$
$809$	−33138.0	−1.44014	−0.720068	+	0.693903i	$0.755889\pi$
$810$	−1944.00	−0.0843274
$811$	10888.0	0.471430	0.235715	−	0.971822i	$-0.424257\pi$
$811$	10888.0	0.471430	0.235715	+	0.971822i	$0.424257\pi$
$812$	−616.000	−0.0266224
$813$	−384.000	−0.0165652
$814$	5192.00	0.223562
$815$	14592.0	0.627160
$816$	96.0000	0.00411847
$817$	−7392.00	−0.316540
$818$	27060.0	1.15664
$819$	−819.000	−0.0349428
$820$	15552.0	0.662316
$821$	−13716.0	−0.583059	−0.291530	−	0.956562i	$-0.594164\pi$
$821$	−13716.0	−0.583059	−0.291530	+	0.956562i	$0.594164\pi$
$822$	−3888.00	−0.164975
$823$	−32016.0	−1.35602	−0.678012	−	0.735051i	$-0.737158\pi$
$823$	−32016.0	−1.35602	−0.678012	+	0.735051i	$0.737158\pi$
$824$	10752.0	0.454568
$825$	1254.00	0.0529196
$826$	−2940.00	−0.123845
$827$	40050.0	1.68401	0.842004	−	0.539471i	$-0.181376\pi$
$827$	40050.0	1.68401	0.842004	+	0.539471i	$0.181376\pi$
$828$	−2880.00	−0.120878
$829$	46602.0	1.95242	0.976209	−	0.216832i	$-0.0695724\pi$
$829$	46602.0	1.95242	0.976209	+	0.216832i	$0.0695724\pi$
$830$	18672.0	0.780861
$831$	2478.00	0.103443
$832$	−832.000	−0.0346688
$833$	−98.0000	−0.00407623
$834$	−8328.00	−0.345773
$835$	−9384.00	−0.388918
$836$	7744.00	0.320373
$837$	2484.00	0.102580
$838$	−2640.00	−0.108827
$839$	−18578.0	−0.764462	−0.382231	−	0.924067i	$-0.624844\pi$
$839$	−18578.0	−0.764462	−0.382231	+	0.924067i	$0.624844\pi$
$840$	2016.00	0.0828079
$841$	−23905.0	−0.980155
$842$	−12644.0	−0.517507
$843$	4344.00	0.177480
$844$	−11600.0	−0.473091
$845$	2028.00	0.0825625
$846$	2412.00	0.0980216
$847$	−5929.00	−0.240523
$848$	−3104.00	−0.125698
$849$	−18252.0	−0.737817
$850$	76.0000	0.00306680
$851$	−9440.00	−0.380257
$852$	792.000	0.0318468
$853$	21254.0	0.853134	0.426567	−	0.904456i	$-0.359723\pi$
$853$	21254.0	0.853134	0.426567	+	0.904456i	$0.359723\pi$
$854$	6580.00	0.263657
$855$	−9504.00	−0.380152
$856$	416.000	0.0166105
$857$	17818.0	0.710212	0.355106	−	0.934826i	$-0.384445\pi$
$857$	17818.0	0.710212	0.355106	+	0.934826i	$0.384445\pi$
$858$	1716.00	0.0682789
$859$	−29476.0	−1.17079	−0.585395	−	0.810748i	$-0.699060\pi$
$859$	−29476.0	−1.17079	−0.585395	+	0.810748i	$0.699060\pi$
$860$	4032.00	0.159872
$861$	−6804.00	−0.269314
$862$	−6996.00	−0.276432
$863$	21378.0	0.843239	0.421620	−	0.906773i	$-0.361462\pi$
$863$	21378.0	0.843239	0.421620	+	0.906773i	$0.361462\pi$
$864$	864.000	0.0340207
$865$	37224.0	1.46318
$866$	788.000	0.0309207
$867$	14727.0	0.576880
$868$	−2576.00	−0.100732
$869$	17072.0	0.666430
$870$	−1584.00	−0.0617272
$871$	3796.00	0.147672
$872$	12336.0	0.479071
$873$	−4410.00	−0.170969
$874$	−14080.0	−0.544923
$875$	−8904.00	−0.344012
$876$	6072.00	0.234194
$877$	−9218.00	−0.354926	−0.177463	−	0.984128i	$-0.556789\pi$
$877$	−9218.00	−0.354926	−0.177463	+	0.984128i	$0.556789\pi$
$878$	−21056.0	−0.809346
$879$	1944.00	0.0745956
$880$	−4224.00	−0.161808
$881$	36762.0	1.40584	0.702919	−	0.711270i	$-0.251880\pi$
$881$	36762.0	1.40584	0.702919	+	0.711270i	$0.251880\pi$
$882$	−882.000	−0.0336718
$883$	−40308.0	−1.53621	−0.768104	−	0.640325i	$-0.778800\pi$
$883$	−40308.0	−1.53621	−0.768104	+	0.640325i	$0.778800\pi$
$884$	104.000	0.00395690
$885$	−7560.00	−0.287149
$886$	10056.0	0.381307
$887$	−7512.00	−0.284361	−0.142181	−	0.989841i	$-0.545411\pi$
$887$	−7512.00	−0.284361	−0.142181	+	0.989841i	$0.545411\pi$
$888$	2832.00	0.107022
$889$	4872.00	0.183804
$890$	22080.0	0.831599
$891$	−1782.00	−0.0670025
$892$	16848.0	0.632413
$893$	11792.0	0.441886
$894$	14520.0	0.543201
$895$	−33888.0	−1.26564
$896$	−896.000	−0.0334077
$897$	−3120.00	−0.116136
$898$	6024.00	0.223857
$899$	2024.00	0.0750881
$900$	684.000	0.0253333
$901$	388.000	0.0143465
$902$	14256.0	0.526245
$903$	−1764.00	−0.0650080
$904$	−14128.0	−0.519790
$905$	−10104.0	−0.371125
$906$	−13608.0	−0.499002
$907$	−26164.0	−0.957841	−0.478920	−	0.877858i	$-0.658972\pi$
$907$	−26164.0	−0.957841	−0.478920	+	0.877858i	$0.658972\pi$
$908$	−8280.00	−0.302623
$909$	8694.00	0.317230
$910$	2184.00	0.0795592
$911$	−28300.0	−1.02922	−0.514611	−	0.857424i	$-0.672064\pi$
$911$	−28300.0	−1.02922	−0.514611	+	0.857424i	$0.672064\pi$
$912$	4224.00	0.153367
$913$	17116.0	0.620435
$914$	11012.0	0.398517
$915$	16920.0	0.611320
$916$	−8792.00	−0.317135
$917$	10528.0	0.379133
$918$	−108.000	−0.00388293
$919$	−15896.0	−0.570577	−0.285289	−	0.958442i	$-0.592089\pi$
$919$	−15896.0	−0.570577	−0.285289	+	0.958442i	$0.592089\pi$
$920$	7680.00	0.275220
$921$	−16908.0	−0.604927
$922$	−9528.00	−0.340334
$923$	858.000	0.0305974
$924$	1848.00	0.0657952
$925$	2242.00	0.0796935
$926$	−10032.0	−0.356017
$927$	−12096.0	−0.428570
$928$	704.000	0.0249029
$929$	−4108.00	−0.145080	−0.0725399	−	0.997366i	$-0.523110\pi$
$929$	−4108.00	−0.145080	−0.0725399	+	0.997366i	$0.523110\pi$
$930$	−6624.00	−0.233559
$931$	−4312.00	−0.151794
$932$	−13512.0	−0.474893
$933$	14892.0	0.522553
$934$	17856.0	0.625553
$935$	528.000	0.0184679
$936$	936.000	0.0326860
$937$	−43034.0	−1.50038	−0.750192	−	0.661220i	$-0.770039\pi$
$937$	−43034.0	−1.50038	−0.750192	+	0.661220i	$0.770039\pi$
$938$	4088.00	0.142301
$939$	−17802.0	−0.618686
$940$	−6432.00	−0.223179
$941$	−17568.0	−0.608608	−0.304304	−	0.952575i	$-0.598424\pi$
$941$	−17568.0	−0.608608	−0.304304	+	0.952575i	$0.598424\pi$
$942$	−3612.00	−0.124931
$943$	−25920.0	−0.895092
$944$	3360.00	0.115846
$945$	−2268.00	−0.0780720
$946$	3696.00	0.127027
$947$	−8426.00	−0.289132	−0.144566	−	0.989495i	$-0.546179\pi$
$947$	−8426.00	−0.289132	−0.144566	+	0.989495i	$0.546179\pi$
$948$	9312.00	0.319029
$949$	6578.00	0.225006
$950$	3344.00	0.114204
$951$	−7680.00	−0.261873
$952$	112.000	0.00381296
$953$	−25938.0	−0.881652	−0.440826	−	0.897593i	$-0.645314\pi$
$953$	−25938.0	−0.881652	−0.440826	+	0.897593i	$0.645314\pi$
$954$	3492.00	0.118509
$955$	30576.0	1.03604
$956$	−11496.0	−0.388920
$957$	−1452.00	−0.0490454
$958$	−6332.00	−0.213547
$959$	−4536.00	−0.152737
$960$	−2304.00	−0.0774597
$961$	−21327.0	−0.715887
$962$	3068.00	0.102824
$963$	−468.000	−0.0156605
$964$	7704.00	0.257395
$965$	−13416.0	−0.447540
$966$	−3360.00	−0.111911
$967$	13028.0	0.433249	0.216625	−	0.976255i	$-0.430495\pi$
$967$	13028.0	0.433249	0.216625	+	0.976255i	$0.430495\pi$
$968$	6776.00	0.224989
$969$	−528.000	−0.0175044
$970$	11760.0	0.389269
$971$	16524.0	0.546118	0.273059	−	0.961997i	$-0.411965\pi$
$971$	16524.0	0.546118	0.273059	+	0.961997i	$0.411965\pi$
$972$	−972.000	−0.0320750
$973$	−9716.00	−0.320124
$974$	1192.00	0.0392137
$975$	741.000	0.0243395
$976$	−7520.00	−0.246628
$977$	−18532.0	−0.606849	−0.303424	−	0.952856i	$-0.598130\pi$
$977$	−18532.0	−0.606849	−0.303424	+	0.952856i	$0.598130\pi$
$978$	7296.00	0.238548
$979$	20240.0	0.660749
$980$	2352.00	0.0766652
$981$	−13878.0	−0.451672
$982$	13880.0	0.451047
$983$	16646.0	0.540107	0.270053	−	0.962845i	$-0.412959\pi$
$983$	16646.0	0.540107	0.270053	+	0.962845i	$0.412959\pi$
$984$	7776.00	0.251921
$985$	23904.0	0.773243
$986$	−88.0000	−0.00284228
$987$	2814.00	0.0907504
$988$	4576.00	0.147350
$989$	−6720.00	−0.216060
$990$	4752.00	0.152554
$991$	−3016.00	−0.0966765	−0.0483382	−	0.998831i	$-0.515393\pi$
$991$	−3016.00	−0.0966765	−0.0483382	+	0.998831i	$0.515393\pi$
$992$	2944.00	0.0942259
$993$	−12060.0	−0.385410
$994$	924.000	0.0294844
$995$	−42432.0	−1.35194
$996$	9336.00	0.297011
$997$	33438.0	1.06218	0.531089	−	0.847316i	$-0.321783\pi$
$997$	33438.0	1.06218	0.531089	+	0.847316i	$0.321783\pi$
$998$	−2312.00	−0.0733317
$999$	−3186.00	−0.100901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.4.a.a.1.1	✓	1
3.2	odd	2		1638.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.a.1.1	✓	1	1.1	even	1	trivial
1638.4.a.i.1.1		1	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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