LMFDB - Embedded newform 546.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,4,Mod(1,546)] 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("546.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [1,2,-3,4,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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)

$f(q)$

$=$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -9.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -18.0000 q^{10} -18.0000 q^{11} -12.0000 q^{12} +13.0000 q^{13} +14.0000 q^{14} +27.0000 q^{15} +16.0000 q^{16} +60.0000 q^{17} +18.0000 q^{18} -43.0000 q^{19} -36.0000 q^{20} -21.0000 q^{21} -36.0000 q^{22} +9.00000 q^{23} -24.0000 q^{24} -44.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} -249.000 q^{29} +54.0000 q^{30} -79.0000 q^{31} +32.0000 q^{32} +54.0000 q^{33} +120.000 q^{34} -63.0000 q^{35} +36.0000 q^{36} -412.000 q^{37} -86.0000 q^{38} -39.0000 q^{39} -72.0000 q^{40} +222.000 q^{41} -42.0000 q^{42} -295.000 q^{43} -72.0000 q^{44} -81.0000 q^{45} +18.0000 q^{46} +411.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -88.0000 q^{50} -180.000 q^{51} +52.0000 q^{52} -237.000 q^{53} -54.0000 q^{54} +162.000 q^{55} +56.0000 q^{56} +129.000 q^{57} -498.000 q^{58} -384.000 q^{59} +108.000 q^{60} -466.000 q^{61} -158.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -117.000 q^{65} +108.000 q^{66} -1042.00 q^{67} +240.000 q^{68} -27.0000 q^{69} -126.000 q^{70} -288.000 q^{71} +72.0000 q^{72} -691.000 q^{73} -824.000 q^{74} +132.000 q^{75} -172.000 q^{76} -126.000 q^{77} -78.0000 q^{78} +1001.00 q^{79} -144.000 q^{80} +81.0000 q^{81} +444.000 q^{82} +39.0000 q^{83} -84.0000 q^{84} -540.000 q^{85} -590.000 q^{86} +747.000 q^{87} -144.000 q^{88} -339.000 q^{89} -162.000 q^{90} +91.0000 q^{91} +36.0000 q^{92} +237.000 q^{93} +822.000 q^{94} +387.000 q^{95} -96.0000 q^{96} +713.000 q^{97} +98.0000 q^{98} -162.000 q^{99} +O(q^{100})\)

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−9.00000	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−18.0000	−0.569210
$11$	−18.0000	−0.493382	−0.246691	−	0.969094i	$-0.579343\pi$
$11$	−18.0000	−0.493382	−0.246691	+	0.969094i	$0.579343\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	14.0000	0.267261
$15$	27.0000	0.464758
$16$	16.0000	0.250000
$17$	60.0000	0.856008	0.428004	−	0.903777i	$-0.359217\pi$
$17$	60.0000	0.856008	0.428004	+	0.903777i	$0.359217\pi$
$18$	18.0000	0.235702
$19$	−43.0000	−0.519204	−0.259602	−	0.965716i	$-0.583591\pi$
$19$	−43.0000	−0.519204	−0.259602	+	0.965716i	$0.583591\pi$
$20$	−36.0000	−0.402492
$21$	−21.0000	−0.218218
$22$	−36.0000	−0.348874
$23$	9.00000	0.0815926	0.0407963	−	0.999167i	$-0.487011\pi$
$23$	9.00000	0.0815926	0.0407963	+	0.999167i	$0.487011\pi$
$24$	−24.0000	−0.204124
$25$	−44.0000	−0.352000
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	−249.000	−1.59442	−0.797209	−	0.603703i	$-0.793691\pi$
$29$	−249.000	−1.59442	−0.797209	+	0.603703i	$0.793691\pi$
$30$	54.0000	0.328634
$31$	−79.0000	−0.457704	−0.228852	−	0.973461i	$-0.573497\pi$
$31$	−79.0000	−0.457704	−0.228852	+	0.973461i	$0.573497\pi$
$32$	32.0000	0.176777
$33$	54.0000	0.284854
$34$	120.000	0.605289
$35$	−63.0000	−0.304256
$36$	36.0000	0.166667
$37$	−412.000	−1.83060	−0.915302	−	0.402767i	$-0.868048\pi$
$37$	−412.000	−1.83060	−0.915302	+	0.402767i	$0.868048\pi$
$38$	−86.0000	−0.367133
$39$	−39.0000	−0.160128
$40$	−72.0000	−0.284605
$41$	222.000	0.845624	0.422812	−	0.906217i	$-0.361043\pi$
$41$	222.000	0.845624	0.422812	+	0.906217i	$0.361043\pi$
$42$	−42.0000	−0.154303
$43$	−295.000	−1.04621	−0.523106	−	0.852268i	$-0.675227\pi$
$43$	−295.000	−1.04621	−0.523106	+	0.852268i	$0.675227\pi$
$44$	−72.0000	−0.246691
$45$	−81.0000	−0.268328
$46$	18.0000	0.0576947
$47$	411.000	1.27554	0.637771	−	0.770226i	$-0.279856\pi$
$47$	411.000	1.27554	0.637771	+	0.770226i	$0.279856\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−88.0000	−0.248902
$51$	−180.000	−0.494217
$52$	52.0000	0.138675
$53$	−237.000	−0.614235	−0.307117	−	0.951672i	$-0.599364\pi$
$53$	−237.000	−0.614235	−0.307117	+	0.951672i	$0.599364\pi$
$54$	−54.0000	−0.136083
$55$	162.000	0.397165
$56$	56.0000	0.133631
$57$	129.000	0.299763
$58$	−498.000	−1.12742
$59$	−384.000	−0.847331	−0.423666	−	0.905819i	$-0.639257\pi$
$59$	−384.000	−0.847331	−0.423666	+	0.905819i	$0.639257\pi$
$60$	108.000	0.232379
$61$	−466.000	−0.978118	−0.489059	−	0.872251i	$-0.662660\pi$
$61$	−466.000	−0.978118	−0.489059	+	0.872251i	$0.662660\pi$
$62$	−158.000	−0.323645
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	−117.000	−0.223263
$66$	108.000	0.201422
$67$	−1042.00	−1.90001	−0.950004	−	0.312237i	$-0.898922\pi$
$67$	−1042.00	−1.90001	−0.950004	+	0.312237i	$0.898922\pi$
$68$	240.000	0.428004
$69$	−27.0000	−0.0471075
$70$	−126.000	−0.215141
$71$	−288.000	−0.481399	−0.240699	−	0.970600i	$-0.577377\pi$
$71$	−288.000	−0.481399	−0.240699	+	0.970600i	$0.577377\pi$
$72$	72.0000	0.117851
$73$	−691.000	−1.10788	−0.553941	−	0.832556i	$-0.686877\pi$
$73$	−691.000	−1.10788	−0.553941	+	0.832556i	$0.686877\pi$
$74$	−824.000	−1.29443
$75$	132.000	0.203227
$76$	−172.000	−0.259602
$77$	−126.000	−0.186481
$78$	−78.0000	−0.113228
$79$	1001.00	1.42559	0.712793	−	0.701374i	$-0.247430\pi$
$79$	1001.00	1.42559	0.712793	+	0.701374i	$0.247430\pi$
$80$	−144.000	−0.201246
$81$	81.0000	0.111111
$82$	444.000	0.597946
$83$	39.0000	0.0515760	0.0257880	−	0.999667i	$-0.491791\pi$
$83$	39.0000	0.0515760	0.0257880	+	0.999667i	$0.491791\pi$
$84$	−84.0000	−0.109109
$85$	−540.000	−0.689073
$86$	−590.000	−0.739783
$87$	747.000	0.920538
$88$	−144.000	−0.174437
$89$	−339.000	−0.403752	−0.201876	−	0.979411i	$-0.564704\pi$
$89$	−339.000	−0.403752	−0.201876	+	0.979411i	$0.564704\pi$
$90$	−162.000	−0.189737
$91$	91.0000	0.104828
$92$	36.0000	0.0407963
$93$	237.000	0.264255
$94$	822.000	0.901945
$95$	387.000	0.417951
$96$	−96.0000	−0.102062
$97$	713.000	0.746332	0.373166	−	0.927765i	$-0.378272\pi$
$97$	713.000	0.746332	0.373166	+	0.927765i	$0.378272\pi$
$98$	98.0000	0.101015
$99$	−162.000	−0.164461
$100$	−176.000	−0.176000
$101$	810.000	0.798000	0.399000	−	0.916951i	$-0.369357\pi$
$101$	810.000	0.798000	0.399000	+	0.916951i	$0.369357\pi$
$102$	−360.000	−0.349464
$103$	−1456.00	−1.39285	−0.696427	−	0.717628i	$-0.745228\pi$
$103$	−1456.00	−1.39285	−0.696427	+	0.717628i	$0.745228\pi$
$104$	104.000	0.0980581
$105$	189.000	0.175662
$106$	−474.000	−0.434330
$107$	−924.000	−0.834827	−0.417413	−	0.908717i	$-0.637063\pi$
$107$	−924.000	−0.834827	−0.417413	+	0.908717i	$0.637063\pi$
$108$	−108.000	−0.0962250
$109$	1154.00	1.01407	0.507033	−	0.861927i	$-0.330742\pi$
$109$	1154.00	1.01407	0.507033	+	0.861927i	$0.330742\pi$
$110$	324.000	0.280838
$111$	1236.00	1.05690
$112$	112.000	0.0944911
$113$	1725.00	1.43606	0.718028	−	0.696014i	$-0.245045\pi$
$113$	1725.00	1.43606	0.718028	+	0.696014i	$0.245045\pi$
$114$	258.000	0.211964
$115$	−81.0000	−0.0656808
$116$	−996.000	−0.797209
$117$	117.000	0.0924500
$118$	−768.000	−0.599154
$119$	420.000	0.323541
$120$	216.000	0.164317
$121$	−1007.00	−0.756574
$122$	−932.000	−0.691634
$123$	−666.000	−0.488221
$124$	−316.000	−0.228852
$125$	1521.00	1.08834
$126$	126.000	0.0890871
$127$	−1204.00	−0.841242	−0.420621	−	0.907236i	$-0.638188\pi$
$127$	−1204.00	−0.841242	−0.420621	+	0.907236i	$0.638188\pi$
$128$	128.000	0.0883883
$129$	885.000	0.604030
$130$	−234.000	−0.157870
$131$	828.000	0.552234	0.276117	−	0.961124i	$-0.410952\pi$
$131$	828.000	0.552234	0.276117	+	0.961124i	$0.410952\pi$
$132$	216.000	0.142427
$133$	−301.000	−0.196241
$134$	−2084.00	−1.34351
$135$	243.000	0.154919
$136$	480.000	0.302645
$137$	−2664.00	−1.66132	−0.830660	−	0.556780i	$-0.812037\pi$
$137$	−2664.00	−1.66132	−0.830660	+	0.556780i	$0.812037\pi$
$138$	−54.0000	−0.0333100
$139$	−322.000	−0.196487	−0.0982435	−	0.995162i	$-0.531322\pi$
$139$	−322.000	−0.196487	−0.0982435	+	0.995162i	$0.531322\pi$
$140$	−252.000	−0.152128
$141$	−1233.00	−0.736435
$142$	−576.000	−0.340400
$143$	−234.000	−0.136840
$144$	144.000	0.0833333
$145$	2241.00	1.28348
$146$	−1382.00	−0.783391
$147$	−147.000	−0.0824786
$148$	−1648.00	−0.915302
$149$	−1602.00	−0.880812	−0.440406	−	0.897799i	$-0.645165\pi$
$149$	−1602.00	−0.880812	−0.440406	+	0.897799i	$0.645165\pi$
$150$	264.000	0.143703
$151$	1928.00	1.03906	0.519531	−	0.854451i	$-0.326107\pi$
$151$	1928.00	1.03906	0.519531	+	0.854451i	$0.326107\pi$
$152$	−344.000	−0.183566
$153$	540.000	0.285336
$154$	−252.000	−0.131862
$155$	711.000	0.368444
$156$	−156.000	−0.0800641
$157$	1748.00	0.888571	0.444285	−	0.895885i	$-0.353458\pi$
$157$	1748.00	0.888571	0.444285	+	0.895885i	$0.353458\pi$
$158$	2002.00	1.00804
$159$	711.000	0.354629
$160$	−288.000	−0.142302
$161$	63.0000	0.0308391
$162$	162.000	0.0785674
$163$	3008.00	1.44543	0.722714	−	0.691147i	$-0.242894\pi$
$163$	3008.00	1.44543	0.722714	+	0.691147i	$0.242894\pi$
$164$	888.000	0.422812
$165$	−486.000	−0.229303
$166$	78.0000	0.0364697
$167$	−1767.00	−0.818770	−0.409385	−	0.912362i	$-0.634257\pi$
$167$	−1767.00	−0.818770	−0.409385	+	0.912362i	$0.634257\pi$
$168$	−168.000	−0.0771517
$169$	169.000	0.0769231
$170$	−1080.00	−0.487248
$171$	−387.000	−0.173068
$172$	−1180.00	−0.523106
$173$	96.0000	0.0421893	0.0210946	−	0.999777i	$-0.493285\pi$
$173$	96.0000	0.0421893	0.0210946	+	0.999777i	$0.493285\pi$
$174$	1494.00	0.650919
$175$	−308.000	−0.133043
$176$	−288.000	−0.123346
$177$	1152.00	0.489207
$178$	−678.000	−0.285496
$179$	3345.00	1.39674	0.698372	−	0.715735i	$-0.253908\pi$
$179$	3345.00	1.39674	0.698372	+	0.715735i	$0.253908\pi$
$180$	−324.000	−0.134164
$181$	2378.00	0.976549	0.488274	−	0.872690i	$-0.337627\pi$
$181$	2378.00	0.976549	0.488274	+	0.872690i	$0.337627\pi$
$182$	182.000	0.0741249
$183$	1398.00	0.564717
$184$	72.0000	0.0288473
$185$	3708.00	1.47361
$186$	474.000	0.186857
$187$	−1080.00	−0.422339
$188$	1644.00	0.637771
$189$	−189.000	−0.0727393
$190$	774.000	0.295536
$191$	−768.000	−0.290945	−0.145473	−	0.989362i	$-0.546470\pi$
$191$	−768.000	−0.290945	−0.145473	+	0.989362i	$0.546470\pi$
$192$	−192.000	−0.0721688
$193$	−3202.00	−1.19422	−0.597111	−	0.802158i	$-0.703685\pi$
$193$	−3202.00	−1.19422	−0.597111	+	0.802158i	$0.703685\pi$
$194$	1426.00	0.527736
$195$	351.000	0.128901
$196$	196.000	0.0714286
$197$	−774.000	−0.279925	−0.139962	−	0.990157i	$-0.544698\pi$
$197$	−774.000	−0.279925	−0.139962	+	0.990157i	$0.544698\pi$
$198$	−324.000	−0.116291
$199$	−1060.00	−0.377595	−0.188798	−	0.982016i	$-0.560459\pi$
$199$	−1060.00	−0.377595	−0.188798	+	0.982016i	$0.560459\pi$
$200$	−352.000	−0.124451
$201$	3126.00	1.09697
$202$	1620.00	0.564271
$203$	−1743.00	−0.602634
$204$	−720.000	−0.247108
$205$	−1998.00	−0.680714
$206$	−2912.00	−0.984896
$207$	81.0000	0.0271975
$208$	208.000	0.0693375
$209$	774.000	0.256166
$210$	378.000	0.124212
$211$	2189.00	0.714204	0.357102	−	0.934065i	$-0.383765\pi$
$211$	2189.00	0.714204	0.357102	+	0.934065i	$0.383765\pi$
$212$	−948.000	−0.307117
$213$	864.000	0.277936
$214$	−1848.00	−0.590312
$215$	2655.00	0.842184
$216$	−216.000	−0.0680414
$217$	−553.000	−0.172996
$218$	2308.00	0.717053
$219$	2073.00	0.639636
$220$	648.000	0.198583
$221$	780.000	0.237414
$222$	2472.00	0.747341
$223$	2891.00	0.868142	0.434071	−	0.900879i	$-0.357077\pi$
$223$	2891.00	0.868142	0.434071	+	0.900879i	$0.357077\pi$
$224$	224.000	0.0668153
$225$	−396.000	−0.117333
$226$	3450.00	1.01545
$227$	3132.00	0.915763	0.457881	−	0.889013i	$-0.348608\pi$
$227$	3132.00	0.915763	0.457881	+	0.889013i	$0.348608\pi$
$228$	516.000	0.149881
$229$	−1150.00	−0.331852	−0.165926	−	0.986138i	$-0.553061\pi$
$229$	−1150.00	−0.331852	−0.165926	+	0.986138i	$0.553061\pi$
$230$	−162.000	−0.0464433
$231$	378.000	0.107665
$232$	−1992.00	−0.563712
$233$	2847.00	0.800486	0.400243	−	0.916409i	$-0.368926\pi$
$233$	2847.00	0.800486	0.400243	+	0.916409i	$0.368926\pi$
$234$	234.000	0.0653720
$235$	−3699.00	−1.02679
$236$	−1536.00	−0.423666
$237$	−3003.00	−0.823062
$238$	840.000	0.228778
$239$	2220.00	0.600836	0.300418	−	0.953808i	$-0.402874\pi$
$239$	2220.00	0.600836	0.300418	+	0.953808i	$0.402874\pi$
$240$	432.000	0.116190
$241$	−6703.00	−1.79161	−0.895805	−	0.444447i	$-0.853400\pi$
$241$	−6703.00	−1.79161	−0.895805	+	0.444447i	$0.853400\pi$
$242$	−2014.00	−0.534979
$243$	−243.000	−0.0641500
$244$	−1864.00	−0.489059
$245$	−441.000	−0.114998
$246$	−1332.00	−0.345224
$247$	−559.000	−0.144001
$248$	−632.000	−0.161823
$249$	−117.000	−0.0297774
$250$	3042.00	0.769572
$251$	4770.00	1.19952	0.599760	−	0.800180i	$-0.295263\pi$
$251$	4770.00	1.19952	0.599760	+	0.800180i	$0.295263\pi$
$252$	252.000	0.0629941
$253$	−162.000	−0.0402563
$254$	−2408.00	−0.594848
$255$	1620.00	0.397837
$256$	256.000	0.0625000
$257$	−1062.00	−0.257766	−0.128883	−	0.991660i	$-0.541139\pi$
$257$	−1062.00	−0.257766	−0.128883	+	0.991660i	$0.541139\pi$
$258$	1770.00	0.427114
$259$	−2884.00	−0.691904
$260$	−468.000	−0.111631
$261$	−2241.00	−0.531473
$262$	1656.00	0.390489
$263$	5283.00	1.23865	0.619323	−	0.785137i	$-0.287407\pi$
$263$	5283.00	1.23865	0.619323	+	0.785137i	$0.287407\pi$
$264$	432.000	0.100711
$265$	2133.00	0.494450
$266$	−602.000	−0.138763
$267$	1017.00	0.233106
$268$	−4168.00	−0.950004
$269$	−6672.00	−1.51226	−0.756132	−	0.654419i	$-0.772913\pi$
$269$	−6672.00	−1.51226	−0.756132	+	0.654419i	$0.772913\pi$
$270$	486.000	0.109545
$271$	−5992.00	−1.34313	−0.671565	−	0.740946i	$-0.734377\pi$
$271$	−5992.00	−1.34313	−0.671565	+	0.740946i	$0.734377\pi$
$272$	960.000	0.214002
$273$	−273.000	−0.0605228
$274$	−5328.00	−1.17473
$275$	792.000	0.173671
$276$	−108.000	−0.0235538
$277$	−5731.00	−1.24311	−0.621557	−	0.783369i	$-0.713499\pi$
$277$	−5731.00	−1.24311	−0.621557	+	0.783369i	$0.713499\pi$
$278$	−644.000	−0.138937
$279$	−711.000	−0.152568
$280$	−504.000	−0.107571
$281$	6402.00	1.35911	0.679557	−	0.733622i	$-0.262172\pi$
$281$	6402.00	1.35911	0.679557	+	0.733622i	$0.262172\pi$
$282$	−2466.00	−0.520738
$283$	7796.00	1.63754	0.818770	−	0.574121i	$-0.194656\pi$
$283$	7796.00	1.63754	0.818770	+	0.574121i	$0.194656\pi$
$284$	−1152.00	−0.240699
$285$	−1161.00	−0.241304
$286$	−468.000	−0.0967602
$287$	1554.00	0.319616
$288$	288.000	0.0589256
$289$	−1313.00	−0.267250
$290$	4482.00	0.907559
$291$	−2139.00	−0.430895
$292$	−2764.00	−0.553941
$293$	6279.00	1.25196	0.625978	−	0.779841i	$-0.284700\pi$
$293$	6279.00	1.25196	0.625978	+	0.779841i	$0.284700\pi$
$294$	−294.000	−0.0583212
$295$	3456.00	0.682088
$296$	−3296.00	−0.647217
$297$	486.000	0.0949514
$298$	−3204.00	−0.622828
$299$	117.000	0.0226297
$300$	528.000	0.101614
$301$	−2065.00	−0.395431
$302$	3856.00	0.734728
$303$	−2430.00	−0.460726
$304$	−688.000	−0.129801
$305$	4194.00	0.787370
$306$	1080.00	0.201763
$307$	−1681.00	−0.312507	−0.156254	−	0.987717i	$-0.549942\pi$
$307$	−1681.00	−0.312507	−0.156254	+	0.987717i	$0.549942\pi$
$308$	−504.000	−0.0932405
$309$	4368.00	0.804165
$310$	1422.00	0.260530
$311$	−1770.00	−0.322725	−0.161363	−	0.986895i	$-0.551589\pi$
$311$	−1770.00	−0.322725	−0.161363	+	0.986895i	$0.551589\pi$
$312$	−312.000	−0.0566139
$313$	−4426.00	−0.799273	−0.399636	−	0.916674i	$-0.630864\pi$
$313$	−4426.00	−0.799273	−0.399636	+	0.916674i	$0.630864\pi$
$314$	3496.00	0.628314
$315$	−567.000	−0.101419
$316$	4004.00	0.712793
$317$	−5592.00	−0.990782	−0.495391	−	0.868670i	$-0.664975\pi$
$317$	−5592.00	−0.990782	−0.495391	+	0.868670i	$0.664975\pi$
$318$	1422.00	0.250760
$319$	4482.00	0.786658
$320$	−576.000	−0.100623
$321$	2772.00	0.481987
$322$	126.000	0.0218065
$323$	−2580.00	−0.444443
$324$	324.000	0.0555556
$325$	−572.000	−0.0976272
$326$	6016.00	1.02207
$327$	−3462.00	−0.585471
$328$	1776.00	0.298973
$329$	2877.00	0.482110
$330$	−972.000	−0.162142
$331$	−5938.00	−0.986048	−0.493024	−	0.870016i	$-0.664109\pi$
$331$	−5938.00	−0.986048	−0.493024	+	0.870016i	$0.664109\pi$
$332$	156.000	0.0257880
$333$	−3708.00	−0.610202
$334$	−3534.00	−0.578958
$335$	9378.00	1.52948
$336$	−336.000	−0.0545545
$337$	641.000	0.103613	0.0518064	−	0.998657i	$-0.483502\pi$
$337$	641.000	0.103613	0.0518064	+	0.998657i	$0.483502\pi$
$338$	338.000	0.0543928
$339$	−5175.00	−0.829107
$340$	−2160.00	−0.344537
$341$	1422.00	0.225823
$342$	−774.000	−0.122378
$343$	343.000	0.0539949
$344$	−2360.00	−0.369891
$345$	243.000	0.0379208
$346$	192.000	0.0298323
$347$	5736.00	0.887391	0.443695	−	0.896178i	$-0.353667\pi$
$347$	5736.00	0.887391	0.443695	+	0.896178i	$0.353667\pi$
$348$	2988.00	0.460269
$349$	−3535.00	−0.542190	−0.271095	−	0.962553i	$-0.587386\pi$
$349$	−3535.00	−0.542190	−0.271095	+	0.962553i	$0.587386\pi$
$350$	−616.000	−0.0940760
$351$	−351.000	−0.0533761
$352$	−576.000	−0.0872185
$353$	−3870.00	−0.583511	−0.291755	−	0.956493i	$-0.594239\pi$
$353$	−3870.00	−0.583511	−0.291755	+	0.956493i	$0.594239\pi$
$354$	2304.00	0.345922
$355$	2592.00	0.387519
$356$	−1356.00	−0.201876
$357$	−1260.00	−0.186796
$358$	6690.00	0.987647
$359$	7572.00	1.11319	0.556595	−	0.830784i	$-0.312108\pi$
$359$	7572.00	1.11319	0.556595	+	0.830784i	$0.312108\pi$
$360$	−648.000	−0.0948683
$361$	−5010.00	−0.730427
$362$	4756.00	0.690524
$363$	3021.00	0.436808
$364$	364.000	0.0524142
$365$	6219.00	0.891828
$366$	2796.00	0.399315
$367$	3206.00	0.456000	0.228000	−	0.973661i	$-0.426781\pi$
$367$	3206.00	0.456000	0.228000	+	0.973661i	$0.426781\pi$
$368$	144.000	0.0203981
$369$	1998.00	0.281875
$370$	7416.00	1.04200
$371$	−1659.00	−0.232159
$372$	948.000	0.132128
$373$	−9034.00	−1.25406	−0.627028	−	0.778997i	$-0.715729\pi$
$373$	−9034.00	−1.25406	−0.627028	+	0.778997i	$0.715729\pi$
$374$	−2160.00	−0.298639
$375$	−4563.00	−0.628353
$376$	3288.00	0.450972
$377$	−3237.00	−0.442212
$378$	−378.000	−0.0514344
$379$	8570.00	1.16151	0.580754	−	0.814079i	$-0.302758\pi$
$379$	8570.00	1.16151	0.580754	+	0.814079i	$0.302758\pi$
$380$	1548.00	0.208976
$381$	3612.00	0.485691
$382$	−1536.00	−0.205729
$383$	10836.0	1.44568	0.722838	−	0.691018i	$-0.242837\pi$
$383$	10836.0	1.44568	0.722838	+	0.691018i	$0.242837\pi$
$384$	−384.000	−0.0510310
$385$	1134.00	0.150114
$386$	−6404.00	−0.844443
$387$	−2655.00	−0.348737
$388$	2852.00	0.373166
$389$	−6282.00	−0.818792	−0.409396	−	0.912357i	$-0.634261\pi$
$389$	−6282.00	−0.818792	−0.409396	+	0.912357i	$0.634261\pi$
$390$	702.000	0.0911465
$391$	540.000	0.0698439
$392$	392.000	0.0505076
$393$	−2484.00	−0.318833
$394$	−1548.00	−0.197937
$395$	−9009.00	−1.14757
$396$	−648.000	−0.0822304
$397$	4745.00	0.599861	0.299930	−	0.953961i	$-0.403037\pi$
$397$	4745.00	0.599861	0.299930	+	0.953961i	$0.403037\pi$
$398$	−2120.00	−0.267000
$399$	903.000	0.113300
$400$	−704.000	−0.0880000
$401$	−5328.00	−0.663510	−0.331755	−	0.943366i	$-0.607641\pi$
$401$	−5328.00	−0.663510	−0.331755	+	0.943366i	$0.607641\pi$
$402$	6252.00	0.775675
$403$	−1027.00	−0.126944
$404$	3240.00	0.399000
$405$	−729.000	−0.0894427
$406$	−3486.00	−0.426126
$407$	7416.00	0.903188
$408$	−1440.00	−0.174732
$409$	−13795.0	−1.66777	−0.833886	−	0.551937i	$-0.813889\pi$
$409$	−13795.0	−1.66777	−0.833886	+	0.551937i	$0.813889\pi$
$410$	−3996.00	−0.481337
$411$	7992.00	0.959164
$412$	−5824.00	−0.696427
$413$	−2688.00	−0.320261
$414$	162.000	0.0192316
$415$	−351.000	−0.0415179
$416$	416.000	0.0490290
$417$	966.000	0.113442
$418$	1548.00	0.181137
$419$	4398.00	0.512784	0.256392	−	0.966573i	$-0.417466\pi$
$419$	4398.00	0.512784	0.256392	+	0.966573i	$0.417466\pi$
$420$	756.000	0.0878310
$421$	−484.000	−0.0560302	−0.0280151	−	0.999607i	$-0.508919\pi$
$421$	−484.000	−0.0560302	−0.0280151	+	0.999607i	$0.508919\pi$
$422$	4378.00	0.505018
$423$	3699.00	0.425181
$424$	−1896.00	−0.217165
$425$	−2640.00	−0.301315
$426$	1728.00	0.196530
$427$	−3262.00	−0.369694
$428$	−3696.00	−0.417413
$429$	702.000	0.0790044
$430$	5310.00	0.595514
$431$	−3414.00	−0.381547	−0.190773	−	0.981634i	$-0.561100\pi$
$431$	−3414.00	−0.381547	−0.190773	+	0.981634i	$0.561100\pi$
$432$	−432.000	−0.0481125
$433$	812.000	0.0901206	0.0450603	−	0.998984i	$-0.485652\pi$
$433$	812.000	0.0901206	0.0450603	+	0.998984i	$0.485652\pi$
$434$	−1106.00	−0.122326
$435$	−6723.00	−0.741019
$436$	4616.00	0.507033
$437$	−387.000	−0.0423632
$438$	4146.00	0.452291
$439$	7778.00	0.845612	0.422806	−	0.906220i	$-0.361045\pi$
$439$	7778.00	0.845612	0.422806	+	0.906220i	$0.361045\pi$
$440$	1296.00	0.140419
$441$	441.000	0.0476190
$442$	1560.00	0.167877
$443$	−6987.00	−0.749351	−0.374675	−	0.927156i	$-0.622246\pi$
$443$	−6987.00	−0.749351	−0.374675	+	0.927156i	$0.622246\pi$
$444$	4944.00	0.528450
$445$	3051.00	0.325014
$446$	5782.00	0.613869
$447$	4806.00	0.508537
$448$	448.000	0.0472456
$449$	16344.0	1.71786	0.858932	−	0.512089i	$-0.171128\pi$
$449$	16344.0	1.71786	0.858932	+	0.512089i	$0.171128\pi$
$450$	−792.000	−0.0829672
$451$	−3996.00	−0.417216
$452$	6900.00	0.718028
$453$	−5784.00	−0.599903
$454$	6264.00	0.647542
$455$	−819.000	−0.0843853
$456$	1032.00	0.105982
$457$	−17116.0	−1.75198	−0.875988	−	0.482334i	$-0.839789\pi$
$457$	−17116.0	−1.75198	−0.875988	+	0.482334i	$0.839789\pi$
$458$	−2300.00	−0.234655
$459$	−1620.00	−0.164739
$460$	−324.000	−0.0328404
$461$	−6882.00	−0.695286	−0.347643	−	0.937627i	$-0.613018\pi$
$461$	−6882.00	−0.695286	−0.347643	+	0.937627i	$0.613018\pi$
$462$	756.000	0.0761305
$463$	3854.00	0.386848	0.193424	−	0.981115i	$-0.438041\pi$
$463$	3854.00	0.386848	0.193424	+	0.981115i	$0.438041\pi$
$464$	−3984.00	−0.398605
$465$	−2133.00	−0.212722
$466$	5694.00	0.566029
$467$	1830.00	0.181333	0.0906663	−	0.995881i	$-0.471100\pi$
$467$	1830.00	0.181333	0.0906663	+	0.995881i	$0.471100\pi$
$468$	468.000	0.0462250
$469$	−7294.00	−0.718136
$470$	−7398.00	−0.726052
$471$	−5244.00	−0.513016
$472$	−3072.00	−0.299577
$473$	5310.00	0.516182
$474$	−6006.00	−0.581993
$475$	1892.00	0.182760
$476$	1680.00	0.161770
$477$	−2133.00	−0.204745
$478$	4440.00	0.424855
$479$	−12663.0	−1.20791	−0.603953	−	0.797020i	$-0.706409\pi$
$479$	−12663.0	−1.20791	−0.603953	+	0.797020i	$0.706409\pi$
$480$	864.000	0.0821584
$481$	−5356.00	−0.507718
$482$	−13406.0	−1.26686
$483$	−189.000	−0.0178050
$484$	−4028.00	−0.378287
$485$	−6417.00	−0.600785
$486$	−486.000	−0.0453609
$487$	−5578.00	−0.519021	−0.259511	−	0.965740i	$-0.583561\pi$
$487$	−5578.00	−0.519021	−0.259511	+	0.965740i	$0.583561\pi$
$488$	−3728.00	−0.345817
$489$	−9024.00	−0.834518
$490$	−882.000	−0.0813157
$491$	−17724.0	−1.62907	−0.814535	−	0.580115i	$-0.803008\pi$
$491$	−17724.0	−1.62907	−0.814535	+	0.580115i	$0.803008\pi$
$492$	−2664.00	−0.244111
$493$	−14940.0	−1.36484
$494$	−1118.00	−0.101824
$495$	1458.00	0.132388
$496$	−1264.00	−0.114426
$497$	−2016.00	−0.181952
$498$	−234.000	−0.0210558
$499$	−15820.0	−1.41924	−0.709620	−	0.704585i	$-0.751133\pi$
$499$	−15820.0	−1.41924	−0.709620	+	0.704585i	$0.751133\pi$
$500$	6084.00	0.544170
$501$	5301.00	0.472717
$502$	9540.00	0.848189
$503$	414.000	0.0366985	0.0183493	−	0.999832i	$-0.494159\pi$
$503$	414.000	0.0366985	0.0183493	+	0.999832i	$0.494159\pi$
$504$	504.000	0.0445435
$505$	−7290.00	−0.642378
$506$	−324.000	−0.0284655
$507$	−507.000	−0.0444116
$508$	−4816.00	−0.420621
$509$	−1521.00	−0.132450	−0.0662251	−	0.997805i	$-0.521096\pi$
$509$	−1521.00	−0.132450	−0.0662251	+	0.997805i	$0.521096\pi$
$510$	3240.00	0.281313
$511$	−4837.00	−0.418740
$512$	512.000	0.0441942
$513$	1161.00	0.0999209
$514$	−2124.00	−0.182268
$515$	13104.0	1.12123
$516$	3540.00	0.302015
$517$	−7398.00	−0.629330
$518$	−5768.00	−0.489250
$519$	−288.000	−0.0243580
$520$	−936.000	−0.0789352
$521$	2436.00	0.204843	0.102421	−	0.994741i	$-0.467341\pi$
$521$	2436.00	0.204843	0.102421	+	0.994741i	$0.467341\pi$
$522$	−4482.00	−0.375808
$523$	13178.0	1.10179	0.550893	−	0.834576i	$-0.314287\pi$
$523$	13178.0	1.10179	0.550893	+	0.834576i	$0.314287\pi$
$524$	3312.00	0.276117
$525$	924.000	0.0768127
$526$	10566.0	0.875855
$527$	−4740.00	−0.391798
$528$	864.000	0.0712136
$529$	−12086.0	−0.993343
$530$	4266.00	0.349629
$531$	−3456.00	−0.282444
$532$	−1204.00	−0.0981203
$533$	2886.00	0.234534
$534$	2034.00	0.164831
$535$	8316.00	0.672022
$536$	−8336.00	−0.671754
$537$	−10035.0	−0.806410
$538$	−13344.0	−1.06933
$539$	−882.000	−0.0704832
$540$	972.000	0.0774597
$541$	13700.0	1.08874	0.544371	−	0.838845i	$-0.316769\pi$
$541$	13700.0	1.08874	0.544371	+	0.838845i	$0.316769\pi$
$542$	−11984.0	−0.949736
$543$	−7134.00	−0.563811
$544$	1920.00	0.151322
$545$	−10386.0	−0.816307
$546$	−546.000	−0.0427960
$547$	−6649.00	−0.519727	−0.259864	−	0.965645i	$-0.583678\pi$
$547$	−6649.00	−0.519727	−0.259864	+	0.965645i	$0.583678\pi$
$548$	−10656.0	−0.830660
$549$	−4194.00	−0.326039
$550$	1584.00	0.122804
$551$	10707.0	0.827829
$552$	−216.000	−0.0166550
$553$	7007.00	0.538821
$554$	−11462.0	−0.879014
$555$	−11124.0	−0.850788
$556$	−1288.00	−0.0982435
$557$	−1272.00	−0.0967619	−0.0483809	−	0.998829i	$-0.515406\pi$
$557$	−1272.00	−0.0967619	−0.0483809	+	0.998829i	$0.515406\pi$
$558$	−1422.00	−0.107882
$559$	−3835.00	−0.290167
$560$	−1008.00	−0.0760639
$561$	3240.00	0.243838
$562$	12804.0	0.961039
$563$	−17052.0	−1.27648	−0.638238	−	0.769839i	$-0.720336\pi$
$563$	−17052.0	−1.27648	−0.638238	+	0.769839i	$0.720336\pi$
$564$	−4932.00	−0.368217
$565$	−15525.0	−1.15600
$566$	15592.0	1.15792
$567$	567.000	0.0419961
$568$	−2304.00	−0.170200
$569$	−19713.0	−1.45239	−0.726197	−	0.687487i	$-0.758714\pi$
$569$	−19713.0	−1.45239	−0.726197	+	0.687487i	$0.758714\pi$
$570$	−2322.00	−0.170628
$571$	16625.0	1.21845	0.609225	−	0.792998i	$-0.291481\pi$
$571$	16625.0	1.21845	0.609225	+	0.792998i	$0.291481\pi$
$572$	−936.000	−0.0684198
$573$	2304.00	0.167977
$574$	3108.00	0.226002
$575$	−396.000	−0.0287206
$576$	576.000	0.0416667
$577$	−7234.00	−0.521933	−0.260967	−	0.965348i	$-0.584041\pi$
$577$	−7234.00	−0.521933	−0.260967	+	0.965348i	$0.584041\pi$
$578$	−2626.00	−0.188974
$579$	9606.00	0.689485
$580$	8964.00	0.641741
$581$	273.000	0.0194939
$582$	−4278.00	−0.304689
$583$	4266.00	0.303053
$584$	−5528.00	−0.391696
$585$	−1053.00	−0.0744208
$586$	12558.0	0.885267
$587$	−7737.00	−0.544021	−0.272010	−	0.962294i	$-0.587689\pi$
$587$	−7737.00	−0.544021	−0.272010	+	0.962294i	$0.587689\pi$
$588$	−588.000	−0.0412393
$589$	3397.00	0.237642
$590$	6912.00	0.482309
$591$	2322.00	0.161615
$592$	−6592.00	−0.457651
$593$	18723.0	1.29656	0.648281	−	0.761401i	$-0.275488\pi$
$593$	18723.0	1.29656	0.648281	+	0.761401i	$0.275488\pi$
$594$	972.000	0.0671408
$595$	−3780.00	−0.260445
$596$	−6408.00	−0.440406
$597$	3180.00	0.218005
$598$	234.000	0.0160016
$599$	−24855.0	−1.69541	−0.847703	−	0.530472i	$-0.822015\pi$
$599$	−24855.0	−1.69541	−0.847703	+	0.530472i	$0.822015\pi$
$600$	1056.00	0.0718517
$601$	5114.00	0.347096	0.173548	−	0.984825i	$-0.444477\pi$
$601$	5114.00	0.347096	0.173548	+	0.984825i	$0.444477\pi$
$602$	−4130.00	−0.279612
$603$	−9378.00	−0.633336
$604$	7712.00	0.519531
$605$	9063.00	0.609030
$606$	−4860.00	−0.325782
$607$	16886.0	1.12913	0.564565	−	0.825389i	$-0.309044\pi$
$607$	16886.0	1.12913	0.564565	+	0.825389i	$0.309044\pi$
$608$	−1376.00	−0.0917832
$609$	5229.00	0.347931
$610$	8388.00	0.556754
$611$	5343.00	0.353772
$612$	2160.00	0.142668
$613$	812.000	0.0535014	0.0267507	−	0.999642i	$-0.491484\pi$
$613$	812.000	0.0535014	0.0267507	+	0.999642i	$0.491484\pi$
$614$	−3362.00	−0.220976
$615$	5994.00	0.393010
$616$	−1008.00	−0.0659310
$617$	11094.0	0.723870	0.361935	−	0.932203i	$-0.382116\pi$
$617$	11094.0	0.723870	0.361935	+	0.932203i	$0.382116\pi$
$618$	8736.00	0.568630
$619$	−21508.0	−1.39657	−0.698287	−	0.715818i	$-0.746054\pi$
$619$	−21508.0	−1.39657	−0.698287	+	0.715818i	$0.746054\pi$
$620$	2844.00	0.184222
$621$	−243.000	−0.0157025
$622$	−3540.00	−0.228201
$623$	−2373.00	−0.152604
$624$	−624.000	−0.0400320
$625$	−8189.00	−0.524096
$626$	−8852.00	−0.565171
$627$	−2322.00	−0.147898
$628$	6992.00	0.444285
$629$	−24720.0	−1.56701
$630$	−1134.00	−0.0717137
$631$	15014.0	0.947223	0.473612	−	0.880734i	$-0.342950\pi$
$631$	15014.0	0.947223	0.473612	+	0.880734i	$0.342950\pi$
$632$	8008.00	0.504021
$633$	−6567.00	−0.412346
$634$	−11184.0	−0.700589
$635$	10836.0	0.677187
$636$	2844.00	0.177314
$637$	637.000	0.0396214
$638$	8964.00	0.556251
$639$	−2592.00	−0.160466
$640$	−1152.00	−0.0711512
$641$	12321.0	0.759205	0.379602	−	0.925150i	$-0.376061\pi$
$641$	12321.0	0.759205	0.379602	+	0.925150i	$0.376061\pi$
$642$	5544.00	0.340817
$643$	13376.0	0.820370	0.410185	−	0.912002i	$-0.365464\pi$
$643$	13376.0	0.820370	0.410185	+	0.912002i	$0.365464\pi$
$644$	252.000	0.0154196
$645$	−7965.00	−0.486235
$646$	−5160.00	−0.314269
$647$	−15150.0	−0.920569	−0.460284	−	0.887772i	$-0.652253\pi$
$647$	−15150.0	−0.920569	−0.460284	+	0.887772i	$0.652253\pi$
$648$	648.000	0.0392837
$649$	6912.00	0.418058
$650$	−1144.00	−0.0690329
$651$	1659.00	0.0998792
$652$	12032.0	0.722714
$653$	−414.000	−0.0248102	−0.0124051	−	0.999923i	$-0.503949\pi$
$653$	−414.000	−0.0248102	−0.0124051	+	0.999923i	$0.503949\pi$
$654$	−6924.00	−0.413991
$655$	−7452.00	−0.444540
$656$	3552.00	0.211406
$657$	−6219.00	−0.369294
$658$	5754.00	0.340903
$659$	20631.0	1.21953	0.609765	−	0.792583i	$-0.291264\pi$
$659$	20631.0	1.21953	0.609765	+	0.792583i	$0.291264\pi$
$660$	−1944.00	−0.114652
$661$	25589.0	1.50574	0.752872	−	0.658167i	$-0.228668\pi$
$661$	25589.0	1.50574	0.752872	+	0.658167i	$0.228668\pi$
$662$	−11876.0	−0.697241
$663$	−2340.00	−0.137071
$664$	312.000	0.0182349
$665$	2709.00	0.157971
$666$	−7416.00	−0.431478
$667$	−2241.00	−0.130093
$668$	−7068.00	−0.409385
$669$	−8673.00	−0.501222
$670$	18756.0	1.08150
$671$	8388.00	0.482586
$672$	−672.000	−0.0385758
$673$	32105.0	1.83887	0.919433	−	0.393247i	$-0.128648\pi$
$673$	32105.0	1.83887	0.919433	+	0.393247i	$0.128648\pi$
$674$	1282.00	0.0732653
$675$	1188.00	0.0677424
$676$	676.000	0.0384615
$677$	29022.0	1.64757	0.823786	−	0.566901i	$-0.191858\pi$
$677$	29022.0	1.64757	0.823786	+	0.566901i	$0.191858\pi$
$678$	−10350.0	−0.586267
$679$	4991.00	0.282087
$680$	−4320.00	−0.243624
$681$	−9396.00	−0.528716
$682$	2844.00	0.159681
$683$	18960.0	1.06220	0.531101	−	0.847308i	$-0.321778\pi$
$683$	18960.0	1.06220	0.531101	+	0.847308i	$0.321778\pi$
$684$	−1548.00	−0.0865340
$685$	23976.0	1.33734
$686$	686.000	0.0381802
$687$	3450.00	0.191595
$688$	−4720.00	−0.261553
$689$	−3081.00	−0.170358
$690$	486.000	0.0268141
$691$	−20653.0	−1.13701	−0.568507	−	0.822678i	$-0.692479\pi$
$691$	−20653.0	−1.13701	−0.568507	+	0.822678i	$0.692479\pi$
$692$	384.000	0.0210946
$693$	−1134.00	−0.0621603
$694$	11472.0	0.627480
$695$	2898.00	0.158169
$696$	5976.00	0.325459
$697$	13320.0	0.723861
$698$	−7070.00	−0.383386
$699$	−8541.00	−0.462161
$700$	−1232.00	−0.0665217
$701$	−33543.0	−1.80728	−0.903639	−	0.428295i	$-0.859114\pi$
$701$	−33543.0	−1.80728	−0.903639	+	0.428295i	$0.859114\pi$
$702$	−702.000	−0.0377426
$703$	17716.0	0.950457
$704$	−1152.00	−0.0616728
$705$	11097.0	0.592819
$706$	−7740.00	−0.412604
$707$	5670.00	0.301616
$708$	4608.00	0.244603
$709$	2630.00	0.139311	0.0696557	−	0.997571i	$-0.477810\pi$
$709$	2630.00	0.139311	0.0696557	+	0.997571i	$0.477810\pi$
$710$	5184.00	0.274017
$711$	9009.00	0.475195
$712$	−2712.00	−0.142748
$713$	−711.000	−0.0373452
$714$	−2520.00	−0.132085
$715$	2106.00	0.110154
$716$	13380.0	0.698372
$717$	−6660.00	−0.346893
$718$	15144.0	0.787144
$719$	21414.0	1.11072	0.555360	−	0.831610i	$-0.312581\pi$
$719$	21414.0	1.11072	0.555360	+	0.831610i	$0.312581\pi$
$720$	−1296.00	−0.0670820
$721$	−10192.0	−0.526449
$722$	−10020.0	−0.516490
$723$	20109.0	1.03439
$724$	9512.00	0.488274
$725$	10956.0	0.561235
$726$	6042.00	0.308870
$727$	13466.0	0.686969	0.343484	−	0.939158i	$-0.388393\pi$
$727$	13466.0	0.686969	0.343484	+	0.939158i	$0.388393\pi$
$728$	728.000	0.0370625
$729$	729.000	0.0370370
$730$	12438.0	0.630618
$731$	−17700.0	−0.895565
$732$	5592.00	0.282358
$733$	−3895.00	−0.196269	−0.0981345	−	0.995173i	$-0.531288\pi$
$733$	−3895.00	−0.196269	−0.0981345	+	0.995173i	$0.531288\pi$
$734$	6412.00	0.322440
$735$	1323.00	0.0663940
$736$	288.000	0.0144237
$737$	18756.0	0.937430
$738$	3996.00	0.199315
$739$	−27250.0	−1.35644	−0.678219	−	0.734860i	$-0.737248\pi$
$739$	−27250.0	−1.35644	−0.678219	+	0.734860i	$0.737248\pi$
$740$	14832.0	0.736804
$741$	1677.00	0.0831392
$742$	−3318.00	−0.164161
$743$	−19632.0	−0.969352	−0.484676	−	0.874694i	$-0.661062\pi$
$743$	−19632.0	−0.969352	−0.484676	+	0.874694i	$0.661062\pi$
$744$	1896.00	0.0934284
$745$	14418.0	0.709040
$746$	−18068.0	−0.886751
$747$	351.000	0.0171920
$748$	−4320.00	−0.211170
$749$	−6468.00	−0.315535
$750$	−9126.00	−0.444313
$751$	−3499.00	−0.170014	−0.0850069	−	0.996380i	$-0.527091\pi$
$751$	−3499.00	−0.170014	−0.0850069	+	0.996380i	$0.527091\pi$
$752$	6576.00	0.318886
$753$	−14310.0	−0.692544
$754$	−6474.00	−0.312691
$755$	−17352.0	−0.836429
$756$	−756.000	−0.0363696
$757$	10469.0	0.502645	0.251323	−	0.967903i	$-0.419135\pi$
$757$	10469.0	0.502645	0.251323	+	0.967903i	$0.419135\pi$
$758$	17140.0	0.821310
$759$	486.000	0.0232420
$760$	3096.00	0.147768
$761$	5607.00	0.267088	0.133544	−	0.991043i	$-0.457364\pi$
$761$	5607.00	0.267088	0.133544	+	0.991043i	$0.457364\pi$
$762$	7224.00	0.343436
$763$	8078.00	0.383281
$764$	−3072.00	−0.145473
$765$	−4860.00	−0.229691
$766$	21672.0	1.02225
$767$	−4992.00	−0.235007
$768$	−768.000	−0.0360844
$769$	14843.0	0.696037	0.348018	−	0.937488i	$-0.386855\pi$
$769$	14843.0	0.696037	0.348018	+	0.937488i	$0.386855\pi$
$770$	2268.00	0.106147
$771$	3186.00	0.148821
$772$	−12808.0	−0.597111
$773$	−7518.00	−0.349811	−0.174905	−	0.984585i	$-0.555962\pi$
$773$	−7518.00	−0.349811	−0.174905	+	0.984585i	$0.555962\pi$
$774$	−5310.00	−0.246594
$775$	3476.00	0.161112
$776$	5704.00	0.263868
$777$	8652.00	0.399471
$778$	−12564.0	−0.578973
$779$	−9546.00	−0.439051
$780$	1404.00	0.0644503
$781$	5184.00	0.237514
$782$	1080.00	0.0493871
$783$	6723.00	0.306846
$784$	784.000	0.0357143
$785$	−15732.0	−0.715286
$786$	−4968.00	−0.225449
$787$	35813.0	1.62210	0.811052	−	0.584974i	$-0.198895\pi$
$787$	35813.0	1.62210	0.811052	+	0.584974i	$0.198895\pi$
$788$	−3096.00	−0.139962
$789$	−15849.0	−0.715132
$790$	−18018.0	−0.811458
$791$	12075.0	0.542778
$792$	−1296.00	−0.0581456
$793$	−6058.00	−0.271281
$794$	9490.00	0.424166
$795$	−6399.00	−0.285471
$796$	−4240.00	−0.188798
$797$	35718.0	1.58745	0.793724	−	0.608278i	$-0.208139\pi$
$797$	35718.0	1.58745	0.793724	+	0.608278i	$0.208139\pi$
$798$	1806.00	0.0801149
$799$	24660.0	1.09187
$800$	−1408.00	−0.0622254
$801$	−3051.00	−0.134584
$802$	−10656.0	−0.469173
$803$	12438.0	0.546610
$804$	12504.0	0.548485
$805$	−567.000	−0.0248250
$806$	−2054.00	−0.0897631
$807$	20016.0	0.873106
$808$	6480.00	0.282136
$809$	10401.0	0.452014	0.226007	−	0.974126i	$-0.427433\pi$
$809$	10401.0	0.452014	0.226007	+	0.974126i	$0.427433\pi$
$810$	−1458.00	−0.0632456
$811$	21548.0	0.932987	0.466494	−	0.884525i	$-0.345517\pi$
$811$	21548.0	0.932987	0.466494	+	0.884525i	$0.345517\pi$
$812$	−6972.00	−0.301317
$813$	17976.0	0.775456
$814$	14832.0	0.638650
$815$	−27072.0	−1.16355
$816$	−2880.00	−0.123554
$817$	12685.0	0.543197
$818$	−27590.0	−1.17929
$819$	819.000	0.0349428
$820$	−7992.00	−0.340357
$821$	−7554.00	−0.321116	−0.160558	−	0.987026i	$-0.551329\pi$
$821$	−7554.00	−0.321116	−0.160558	+	0.987026i	$0.551329\pi$
$822$	15984.0	0.678231
$823$	13448.0	0.569584	0.284792	−	0.958589i	$-0.408075\pi$
$823$	13448.0	0.569584	0.284792	+	0.958589i	$0.408075\pi$
$824$	−11648.0	−0.492448
$825$	−2376.00	−0.100269
$826$	−5376.00	−0.226459
$827$	−10728.0	−0.451087	−0.225544	−	0.974233i	$-0.572416\pi$
$827$	−10728.0	−0.451087	−0.225544	+	0.974233i	$0.572416\pi$
$828$	324.000	0.0135988
$829$	−42190.0	−1.76757	−0.883787	−	0.467889i	$-0.845015\pi$
$829$	−42190.0	−1.76757	−0.883787	+	0.467889i	$0.845015\pi$
$830$	−702.000	−0.0293576
$831$	17193.0	0.717712
$832$	832.000	0.0346688
$833$	2940.00	0.122287
$834$	1932.00	0.0802155
$835$	15903.0	0.659097
$836$	3096.00	0.128083
$837$	2133.00	0.0880851
$838$	8796.00	0.362593
$839$	−14856.0	−0.611306	−0.305653	−	0.952143i	$-0.598875\pi$
$839$	−14856.0	−0.611306	−0.305653	+	0.952143i	$0.598875\pi$
$840$	1512.00	0.0621059
$841$	37612.0	1.54217
$842$	−968.000	−0.0396193
$843$	−19206.0	−0.784685
$844$	8756.00	0.357102
$845$	−1521.00	−0.0619219
$846$	7398.00	0.300648
$847$	−7049.00	−0.285958
$848$	−3792.00	−0.153559
$849$	−23388.0	−0.945435
$850$	−5280.00	−0.213062
$851$	−3708.00	−0.149364
$852$	3456.00	0.138968
$853$	−29185.0	−1.17148	−0.585742	−	0.810498i	$-0.699197\pi$
$853$	−29185.0	−1.17148	−0.585742	+	0.810498i	$0.699197\pi$
$854$	−6524.00	−0.261413
$855$	3483.00	0.139317
$856$	−7392.00	−0.295156
$857$	−2178.00	−0.0868134	−0.0434067	−	0.999057i	$-0.513821\pi$
$857$	−2178.00	−0.0868134	−0.0434067	+	0.999057i	$0.513821\pi$
$858$	1404.00	0.0558645
$859$	11702.0	0.464805	0.232402	−	0.972620i	$-0.425341\pi$
$859$	11702.0	0.464805	0.232402	+	0.972620i	$0.425341\pi$
$860$	10620.0	0.421092
$861$	−4662.00	−0.184530
$862$	−6828.00	−0.269794
$863$	31044.0	1.22451	0.612254	−	0.790661i	$-0.290263\pi$
$863$	31044.0	1.22451	0.612254	+	0.790661i	$0.290263\pi$
$864$	−864.000	−0.0340207
$865$	−864.000	−0.0339617
$866$	1624.00	0.0637249
$867$	3939.00	0.154297
$868$	−2212.00	−0.0864979
$869$	−18018.0	−0.703359
$870$	−13446.0	−0.523979
$871$	−13546.0	−0.526968
$872$	9232.00	0.358526
$873$	6417.00	0.248777
$874$	−774.000	−0.0299553
$875$	10647.0	0.411353
$876$	8292.00	0.319818
$877$	13700.0	0.527498	0.263749	−	0.964591i	$-0.415041\pi$
$877$	13700.0	0.527498	0.263749	+	0.964591i	$0.415041\pi$
$878$	15556.0	0.597938
$879$	−18837.0	−0.722817
$880$	2592.00	0.0992913
$881$	−19710.0	−0.753742	−0.376871	−	0.926266i	$-0.623000\pi$
$881$	−19710.0	−0.753742	−0.376871	+	0.926266i	$0.623000\pi$
$882$	882.000	0.0336718
$883$	12476.0	0.475482	0.237741	−	0.971329i	$-0.423593\pi$
$883$	12476.0	0.475482	0.237741	+	0.971329i	$0.423593\pi$
$884$	3120.00	0.118707
$885$	−10368.0	−0.393804
$886$	−13974.0	−0.529871
$887$	14940.0	0.565542	0.282771	−	0.959187i	$-0.408746\pi$
$887$	14940.0	0.565542	0.282771	+	0.959187i	$0.408746\pi$
$888$	9888.00	0.373671
$889$	−8428.00	−0.317960
$890$	6102.00	0.229820
$891$	−1458.00	−0.0548202
$892$	11564.0	0.434071
$893$	−17673.0	−0.662267
$894$	9612.00	0.359590
$895$	−30105.0	−1.12436
$896$	896.000	0.0334077
$897$	−351.000	−0.0130653
$898$	32688.0	1.21471
$899$	19671.0	0.729772
$900$	−1584.00	−0.0586667
$901$	−14220.0	−0.525790
$902$	−7992.00	−0.295016
$903$	6195.00	0.228302
$904$	13800.0	0.507723
$905$	−21402.0	−0.786107
$906$	−11568.0	−0.424195
$907$	12953.0	0.474198	0.237099	−	0.971486i	$-0.423803\pi$
$907$	12953.0	0.474198	0.237099	+	0.971486i	$0.423803\pi$
$908$	12528.0	0.457881
$909$	7290.00	0.266000
$910$	−1638.00	−0.0596694
$911$	15411.0	0.560471	0.280236	−	0.959931i	$-0.409587\pi$
$911$	15411.0	0.560471	0.280236	+	0.959931i	$0.409587\pi$
$912$	2064.00	0.0749406
$913$	−702.000	−0.0254467
$914$	−34232.0	−1.23883
$915$	−12582.0	−0.454588
$916$	−4600.00	−0.165926
$917$	5796.00	0.208725
$918$	−3240.00	−0.116488
$919$	22520.0	0.808342	0.404171	−	0.914683i	$-0.367560\pi$
$919$	22520.0	0.808342	0.404171	+	0.914683i	$0.367560\pi$
$920$	−648.000	−0.0232217
$921$	5043.00	0.180426
$922$	−13764.0	−0.491641
$923$	−3744.00	−0.133516
$924$	1512.00	0.0538324
$925$	18128.0	0.644373
$926$	7708.00	0.273543
$927$	−13104.0	−0.464285
$928$	−7968.00	−0.281856
$929$	31803.0	1.12317	0.561584	−	0.827420i	$-0.310192\pi$
$929$	31803.0	1.12317	0.561584	+	0.827420i	$0.310192\pi$
$930$	−4266.00	−0.150417
$931$	−2107.00	−0.0741720
$932$	11388.0	0.400243
$933$	5310.00	0.186325
$934$	3660.00	0.128221
$935$	9720.00	0.339976
$936$	936.000	0.0326860
$937$	−30148.0	−1.05111	−0.525556	−	0.850759i	$-0.676143\pi$
$937$	−30148.0	−1.05111	−0.525556	+	0.850759i	$0.676143\pi$
$938$	−14588.0	−0.507799
$939$	13278.0	0.461460
$940$	−14796.0	−0.513396
$941$	5709.00	0.197777	0.0988885	−	0.995099i	$-0.468471\pi$
$941$	5709.00	0.197777	0.0988885	+	0.995099i	$0.468471\pi$
$942$	−10488.0	−0.362757
$943$	1998.00	0.0689966
$944$	−6144.00	−0.211833
$945$	1701.00	0.0585540
$946$	10620.0	0.364996
$947$	−42474.0	−1.45747	−0.728733	−	0.684798i	$-0.759890\pi$
$947$	−42474.0	−1.45747	−0.728733	+	0.684798i	$0.759890\pi$
$948$	−12012.0	−0.411531
$949$	−8983.00	−0.307271
$950$	3784.00	0.129231
$951$	16776.0	0.572028
$952$	3360.00	0.114389
$953$	10953.0	0.372301	0.186150	−	0.982521i	$-0.440399\pi$
$953$	10953.0	0.372301	0.186150	+	0.982521i	$0.440399\pi$
$954$	−4266.00	−0.144777
$955$	6912.00	0.234206
$956$	8880.00	0.300418
$957$	−13446.0	−0.454177
$958$	−25326.0	−0.854119
$959$	−18648.0	−0.627920
$960$	1728.00	0.0580948
$961$	−23550.0	−0.790507
$962$	−10712.0	−0.359011
$963$	−8316.00	−0.278276
$964$	−26812.0	−0.895805
$965$	28818.0	0.961331
$966$	−378.000	−0.0125900
$967$	−4498.00	−0.149582	−0.0747911	−	0.997199i	$-0.523829\pi$
$967$	−4498.00	−0.149582	−0.0747911	+	0.997199i	$0.523829\pi$
$968$	−8056.00	−0.267489
$969$	7740.00	0.256599
$970$	−12834.0	−0.424819
$971$	42978.0	1.42042	0.710211	−	0.703989i	$-0.248600\pi$
$971$	42978.0	1.42042	0.710211	+	0.703989i	$0.248600\pi$
$972$	−972.000	−0.0320750
$973$	−2254.00	−0.0742651
$974$	−11156.0	−0.367003
$975$	1716.00	0.0563651
$976$	−7456.00	−0.244529
$977$	−1554.00	−0.0508873	−0.0254436	−	0.999676i	$-0.508100\pi$
$977$	−1554.00	−0.0508873	−0.0254436	+	0.999676i	$0.508100\pi$
$978$	−18048.0	−0.590093
$979$	6102.00	0.199204
$980$	−1764.00	−0.0574989
$981$	10386.0	0.338022
$982$	−35448.0	−1.15193
$983$	15597.0	0.506070	0.253035	−	0.967457i	$-0.418571\pi$
$983$	15597.0	0.506070	0.253035	+	0.967457i	$0.418571\pi$
$984$	−5328.00	−0.172612
$985$	6966.00	0.225335
$986$	−29880.0	−0.965084
$987$	−8631.00	−0.278346
$988$	−2236.00	−0.0720006
$989$	−2655.00	−0.0853631
$990$	2916.00	0.0936127
$991$	52004.0	1.66696	0.833482	−	0.552546i	$-0.186344\pi$
$991$	52004.0	1.66696	0.833482	+	0.552546i	$0.186344\pi$
$992$	−2528.00	−0.0809114
$993$	17814.0	0.569295
$994$	−4032.00	−0.128659
$995$	9540.00	0.303958
$996$	−468.000	−0.0148887
$997$	16184.0	0.514095	0.257047	−	0.966399i	$-0.417250\pi$
$997$	16184.0	0.514095	0.257047	+	0.966399i	$0.417250\pi$
$998$	−31640.0	−1.00355
$999$	11124.0	0.352300

Twists

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

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

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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