LMFDB - Embedded newform 1638.4.a.v.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,4,Mod(1,1638)] 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("1638.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-6,0,12,-7,0,-21,-24,0,14,47] 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:	$96.6451285894$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.118088.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 50x - 32 $ x^3 - 50*x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.37163$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -6.57278 q^{5} -7.00000 q^{7} -8.00000 q^{8} +13.1456 q^{10} -18.0593 q^{11} +13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +42.3389 q^{17} +77.7068 q^{19} -26.2911 q^{20} +36.1186 q^{22} -43.8983 q^{23} -81.7985 q^{25} -26.0000 q^{26} -28.0000 q^{28} -121.075 q^{29} +248.214 q^{31} -32.0000 q^{32} -84.6779 q^{34} +46.0095 q^{35} -279.817 q^{37} -155.414 q^{38} +52.5823 q^{40} -425.980 q^{41} +179.185 q^{43} -72.2372 q^{44} +87.7966 q^{46} +216.321 q^{47} +49.0000 q^{49} +163.597 q^{50} +52.0000 q^{52} -713.003 q^{53} +118.700 q^{55} +56.0000 q^{56} +242.150 q^{58} -403.004 q^{59} +246.765 q^{61} -496.429 q^{62} +64.0000 q^{64} -85.4462 q^{65} -742.301 q^{67} +169.356 q^{68} -92.0190 q^{70} +65.6086 q^{71} +965.099 q^{73} +559.635 q^{74} +310.827 q^{76} +126.415 q^{77} +382.593 q^{79} -105.165 q^{80} +851.961 q^{82} +1347.88 q^{83} -278.285 q^{85} -358.371 q^{86} +144.474 q^{88} -830.091 q^{89} -91.0000 q^{91} -175.593 q^{92} -432.643 q^{94} -510.750 q^{95} +376.407 q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 12 q^{4} - 7 q^{5} - 21 q^{7} - 24 q^{8} + 14 q^{10} + 47 q^{11} + 39 q^{13} + 42 q^{14} + 48 q^{16} - 119 q^{17} + 101 q^{19} - 28 q^{20} - 94 q^{22} + 27 q^{23} + 266 q^{25} - 78 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) $ 3 * q - 6 * q^2 + 12 * q^4 - 7 * q^5 - 21 * q^7 - 24 * q^8 + 14 * q^10 + 47 * q^11 + 39 * q^13 + 42 * q^14 + 48 * q^16 - 119 * q^17 + 101 * q^19 - 28 * q^20 - 94 * q^22 + 27 * q^23 + 266 * q^25 - 78 * q^26 - 84 * q^28 - 345 * q^29 + 152 * q^31 - 96 * q^32 + 238 * q^34 + 49 * q^35 + 227 * q^37 - 202 * q^38 + 56 * q^40 - 452 * q^41 - 415 * q^43 + 188 * q^44 - 54 * q^46 - 240 * q^47 + 147 * q^49 - 532 * q^50 + 156 * q^52 - 874 * q^53 + 1123 * q^55 + 168 * q^56 + 690 * q^58 - 948 * q^59 + 951 * q^61 - 304 * q^62 + 192 * q^64 - 91 * q^65 + 742 * q^67 - 476 * q^68 - 98 * q^70 - 732 * q^71 - 585 * q^73 - 454 * q^74 + 404 * q^76 - 329 * q^77 - 618 * q^79 - 112 * q^80 + 904 * q^82 - 774 * q^83 + 1273 * q^85 + 830 * q^86 - 376 * q^88 - 730 * q^89 - 273 * q^91 + 108 * q^92 + 480 * q^94 + 2617 * q^95 + 2518 * q^97 - 294 * q^98

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
$5$	−6.57278	−0.587888	−0.293944	−	0.955823i	$-0.594968\pi$
$5$	−6.57278	−0.587888	−0.293944	+	0.955823i	$0.594968\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	13.1456	0.415699
$11$	−18.0593	−0.495008	−0.247504	−	0.968887i	$-0.579610\pi$
$11$	−18.0593	−0.495008	−0.247504	+	0.968887i	$0.579610\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	42.3389	0.604041	0.302021	−	0.953301i	$-0.402339\pi$
$17$	42.3389	0.604041	0.302021	+	0.953301i	$0.402339\pi$
$18$	0	0
$19$	77.7068	0.938272	0.469136	−	0.883126i	$-0.344565\pi$
$19$	77.7068	0.938272	0.469136	+	0.883126i	$0.344565\pi$
$20$	−26.2911	−0.293944
$21$	0	0
$22$	36.1186	0.350023
$23$	−43.8983	−0.397975	−0.198988	−	0.980002i	$-0.563765\pi$
$23$	−43.8983	−0.397975	−0.198988	+	0.980002i	$0.563765\pi$
$24$	0	0
$25$	−81.7985	−0.654388
$26$	−26.0000	−0.196116
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−121.075	−0.775276	−0.387638	−	0.921812i	$-0.626709\pi$
$29$	−121.075	−0.775276	−0.387638	+	0.921812i	$0.626709\pi$
$30$	0	0
$31$	248.214	1.43808	0.719042	−	0.694967i	$-0.244581\pi$
$31$	248.214	1.43808	0.719042	+	0.694967i	$0.244581\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−84.6779	−0.427122
$35$	46.0095	0.222201
$36$	0	0
$37$	−279.817	−1.24329	−0.621645	−	0.783299i	$-0.713535\pi$
$37$	−279.817	−1.24329	−0.621645	+	0.783299i	$0.713535\pi$
$38$	−155.414	−0.663459
$39$	0	0
$40$	52.5823	0.207850
$41$	−425.980	−1.62261	−0.811304	−	0.584624i	$-0.801242\pi$
$41$	−425.980	−1.62261	−0.811304	+	0.584624i	$0.801242\pi$
$42$	0	0
$43$	179.185	0.635477	0.317738	−	0.948178i	$-0.397077\pi$
$43$	179.185	0.635477	0.317738	+	0.948178i	$0.397077\pi$
$44$	−72.2372	−0.247504
$45$	0	0
$46$	87.7966	0.281411
$47$	216.321	0.671356	0.335678	−	0.941977i	$-0.391035\pi$
$47$	216.321	0.671356	0.335678	+	0.941977i	$0.391035\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	163.597	0.462722
$51$	0	0
$52$	52.0000	0.138675
$53$	−713.003	−1.84790	−0.923948	−	0.382518i	$-0.875057\pi$
$53$	−713.003	−1.84790	−0.923948	+	0.382518i	$0.875057\pi$
$54$	0	0
$55$	118.700	0.291009
$56$	56.0000	0.133631
$57$	0	0
$58$	242.150	0.548203
$59$	−403.004	−0.889264	−0.444632	−	0.895713i	$-0.646666\pi$
$59$	−403.004	−0.889264	−0.444632	+	0.895713i	$0.646666\pi$
$60$	0	0
$61$	246.765	0.517950	0.258975	−	0.965884i	$-0.416615\pi$
$61$	246.765	0.517950	0.258975	+	0.965884i	$0.416615\pi$
$62$	−496.429	−1.01688
$63$	0	0
$64$	64.0000	0.125000
$65$	−85.4462	−0.163051
$66$	0	0
$67$	−742.301	−1.35353	−0.676765	−	0.736199i	$-0.736619\pi$
$67$	−742.301	−1.35353	−0.676765	+	0.736199i	$0.736619\pi$
$68$	169.356	0.302021
$69$	0	0
$70$	−92.0190	−0.157120
$71$	65.6086	0.109666	0.0548331	−	0.998496i	$-0.482537\pi$
$71$	65.6086	0.109666	0.0548331	+	0.998496i	$0.482537\pi$
$72$	0	0
$73$	965.099	1.54735	0.773673	−	0.633585i	$-0.218417\pi$
$73$	965.099	1.54735	0.773673	+	0.633585i	$0.218417\pi$
$74$	559.635	0.879138
$75$	0	0
$76$	310.827	0.469136
$77$	126.415	0.187095
$78$	0	0
$79$	382.593	0.544874	0.272437	−	0.962174i	$-0.412170\pi$
$79$	382.593	0.544874	0.272437	+	0.962174i	$0.412170\pi$
$80$	−105.165	−0.146972
$81$	0	0
$82$	851.961	1.14736
$83$	1347.88	1.78252	0.891258	−	0.453496i	$-0.149823\pi$
$83$	1347.88	1.78252	0.891258	+	0.453496i	$0.149823\pi$
$84$	0	0
$85$	−278.285	−0.355108
$86$	−358.371	−0.449350
$87$	0	0
$88$	144.474	0.175012
$89$	−830.091	−0.988646	−0.494323	−	0.869278i	$-0.664584\pi$
$89$	−830.091	−0.988646	−0.494323	+	0.869278i	$0.664584\pi$
$90$	0	0
$91$	−91.0000	−0.104828
$92$	−175.593	−0.198988
$93$	0	0
$94$	−432.643	−0.474720
$95$	−510.750	−0.551599
$96$	0	0
$97$	376.407	0.394003	0.197002	−	0.980403i	$-0.436880\pi$
$97$	376.407	0.394003	0.197002	+	0.980403i	$0.436880\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	−327.194	−0.327194
$101$	−646.395	−0.636819	−0.318409	−	0.947953i	$-0.603149\pi$
$101$	−646.395	−0.636819	−0.318409	+	0.947953i	$0.603149\pi$
$102$	0	0
$103$	41.7528	0.0399420	0.0199710	−	0.999801i	$-0.493643\pi$
$103$	41.7528	0.0399420	0.0199710	+	0.999801i	$0.493643\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	1426.01	1.30666
$107$	536.924	0.485107	0.242554	−	0.970138i	$-0.422015\pi$
$107$	536.924	0.485107	0.242554	+	0.970138i	$0.422015\pi$
$108$	0	0
$109$	1959.48	1.72187	0.860935	−	0.508715i	$-0.169879\pi$
$109$	1959.48	1.72187	0.860935	+	0.508715i	$0.169879\pi$
$110$	−237.400	−0.205774
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−1503.52	−1.25168	−0.625838	−	0.779953i	$-0.715243\pi$
$113$	−1503.52	−1.25168	−0.625838	+	0.779953i	$0.715243\pi$
$114$	0	0
$115$	288.534	0.233965
$116$	−484.299	−0.387638
$117$	0	0
$118$	806.007	0.628805
$119$	−296.373	−0.228306
$120$	0	0
$121$	−1004.86	−0.754967
$122$	−493.529	−0.366246
$123$	0	0
$124$	992.857	0.719042
$125$	1359.24	0.972594
$126$	0	0
$127$	−506.342	−0.353784	−0.176892	−	0.984230i	$-0.556604\pi$
$127$	−506.342	−0.353784	−0.176892	+	0.984230i	$0.556604\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	170.892	0.115294
$131$	1140.76	0.760833	0.380416	−	0.924815i	$-0.375781\pi$
$131$	1140.76	0.760833	0.380416	+	0.924815i	$0.375781\pi$
$132$	0	0
$133$	−543.948	−0.354634
$134$	1484.60	0.957090
$135$	0	0
$136$	−338.712	−0.213561
$137$	2042.68	1.27386	0.636928	−	0.770923i	$-0.280205\pi$
$137$	2042.68	1.27386	0.636928	+	0.770923i	$0.280205\pi$
$138$	0	0
$139$	85.0793	0.0519161	0.0259580	−	0.999663i	$-0.491736\pi$
$139$	85.0793	0.0519161	0.0259580	+	0.999663i	$0.491736\pi$
$140$	184.038	0.111100
$141$	0	0
$142$	−131.217	−0.0775458
$143$	−234.771	−0.137291
$144$	0	0
$145$	795.798	0.455775
$146$	−1930.20	−1.09414
$147$	0	0
$148$	−1119.27	−0.621645
$149$	397.408	0.218503	0.109251	−	0.994014i	$-0.465155\pi$
$149$	397.408	0.218503	0.109251	+	0.994014i	$0.465155\pi$
$150$	0	0
$151$	−3099.79	−1.67058	−0.835290	−	0.549810i	$-0.814700\pi$
$151$	−3099.79	−1.67058	−0.835290	+	0.549810i	$0.814700\pi$
$152$	−621.655	−0.331729
$153$	0	0
$154$	−252.830	−0.132296
$155$	−1631.46	−0.845431
$156$	0	0
$157$	2827.44	1.43729	0.718645	−	0.695377i	$-0.244763\pi$
$157$	2827.44	1.43729	0.718645	+	0.695377i	$0.244763\pi$
$158$	−765.186	−0.385284
$159$	0	0
$160$	210.329	0.103925
$161$	307.288	0.150420
$162$	0	0
$163$	−3011.00	−1.44687	−0.723436	−	0.690392i	$-0.757438\pi$
$163$	−3011.00	−1.44687	−0.723436	+	0.690392i	$0.757438\pi$
$164$	−1703.92	−0.811304
$165$	0	0
$166$	−2695.76	−1.26043
$167$	−1853.19	−0.858708	−0.429354	−	0.903136i	$-0.641259\pi$
$167$	−1853.19	−0.858708	−0.429354	+	0.903136i	$0.641259\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	556.569	0.251100
$171$	0	0
$172$	716.741	0.317738
$173$	4400.17	1.93375	0.966874	−	0.255255i	$-0.0821593\pi$
$173$	4400.17	1.93375	0.966874	+	0.255255i	$0.0821593\pi$
$174$	0	0
$175$	572.590	0.247336
$176$	−288.949	−0.123752
$177$	0	0
$178$	1660.18	0.699078
$179$	3617.60	1.51057	0.755285	−	0.655397i	$-0.227498\pi$
$179$	3617.60	1.51057	0.755285	+	0.655397i	$0.227498\pi$
$180$	0	0
$181$	396.014	0.162627	0.0813135	−	0.996689i	$-0.474089\pi$
$181$	396.014	0.162627	0.0813135	+	0.996689i	$0.474089\pi$
$182$	182.000	0.0741249
$183$	0	0
$184$	351.186	0.140705
$185$	1839.18	0.730914
$186$	0	0
$187$	−764.612	−0.299005
$188$	865.286	0.335678
$189$	0	0
$190$	1021.50	0.390039
$191$	2672.76	1.01253	0.506267	−	0.862377i	$-0.331025\pi$
$191$	2672.76	1.01253	0.506267	+	0.862377i	$0.331025\pi$
$192$	0	0
$193$	1025.21	0.382363	0.191182	−	0.981555i	$-0.438768\pi$
$193$	1025.21	0.382363	0.191182	+	0.981555i	$0.438768\pi$
$194$	−752.814	−0.278603
$195$	0	0
$196$	196.000	0.0714286
$197$	3848.98	1.39202	0.696011	−	0.718032i	$-0.254957\pi$
$197$	3848.98	1.39202	0.696011	+	0.718032i	$0.254957\pi$
$198$	0	0
$199$	−4509.60	−1.60642	−0.803209	−	0.595697i	$-0.796876\pi$
$199$	−4509.60	−1.60642	−0.803209	+	0.595697i	$0.796876\pi$
$200$	654.388	0.231361
$201$	0	0
$202$	1292.79	0.450299
$203$	847.523	0.293027
$204$	0	0
$205$	2799.88	0.953912
$206$	−83.5057	−0.0282433
$207$	0	0
$208$	208.000	0.0693375
$209$	−1403.33	−0.464452
$210$	0	0
$211$	3323.88	1.08448	0.542240	−	0.840223i	$-0.317576\pi$
$211$	3323.88	1.08448	0.542240	+	0.840223i	$0.317576\pi$
$212$	−2852.01	−0.923948
$213$	0	0
$214$	−1073.85	−0.343022
$215$	−1177.75	−0.373589
$216$	0	0
$217$	−1737.50	−0.543545
$218$	−3918.95	−1.21755
$219$	0	0
$220$	474.800	0.145505
$221$	550.406	0.167531
$222$	0	0
$223$	1938.33	0.582062	0.291031	−	0.956714i	$-0.406002\pi$
$223$	1938.33	0.582062	0.291031	+	0.956714i	$0.406002\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	3007.04	0.885069
$227$	2557.44	0.747768	0.373884	−	0.927475i	$-0.378026\pi$
$227$	2557.44	0.747768	0.373884	+	0.927475i	$0.378026\pi$
$228$	0	0
$229$	−3873.48	−1.11776	−0.558879	−	0.829249i	$-0.688768\pi$
$229$	−3873.48	−1.11776	−0.558879	+	0.829249i	$0.688768\pi$
$230$	−577.068	−0.165438
$231$	0	0
$232$	968.598	0.274102
$233$	226.544	0.0636970	0.0318485	−	0.999493i	$-0.489861\pi$
$233$	226.544	0.0636970	0.0318485	+	0.999493i	$0.489861\pi$
$234$	0	0
$235$	−1421.83	−0.394682
$236$	−1612.01	−0.444632
$237$	0	0
$238$	592.745	0.161437
$239$	4141.26	1.12082	0.560410	−	0.828216i	$-0.310644\pi$
$239$	4141.26	1.12082	0.560410	+	0.828216i	$0.310644\pi$
$240$	0	0
$241$	2450.96	0.655104	0.327552	−	0.944833i	$-0.393776\pi$
$241$	2450.96	0.655104	0.327552	+	0.944833i	$0.393776\pi$
$242$	2009.72	0.533842
$243$	0	0
$244$	987.059	0.258975
$245$	−322.066	−0.0839839
$246$	0	0
$247$	1010.19	0.260230
$248$	−1985.71	−0.508439
$249$	0	0
$250$	−2718.48	−0.687728
$251$	−737.540	−0.185471	−0.0927353	−	0.995691i	$-0.529561\pi$
$251$	−737.540	−0.185471	−0.0927353	+	0.995691i	$0.529561\pi$
$252$	0	0
$253$	792.773	0.197001
$254$	1012.68	0.250163
$255$	0	0
$256$	256.000	0.0625000
$257$	−3381.82	−0.820825	−0.410412	−	0.911900i	$-0.634615\pi$
$257$	−3381.82	−0.820825	−0.410412	+	0.911900i	$0.634615\pi$
$258$	0	0
$259$	1958.72	0.469919
$260$	−341.785	−0.0815253
$261$	0	0
$262$	−2281.53	−0.537990
$263$	−3031.47	−0.710755	−0.355377	−	0.934723i	$-0.615648\pi$
$263$	−3031.47	−0.710755	−0.355377	+	0.934723i	$0.615648\pi$
$264$	0	0
$265$	4686.41	1.08635
$266$	1087.90	0.250764
$267$	0	0
$268$	−2969.20	−0.676765
$269$	−6048.97	−1.37105	−0.685524	−	0.728050i	$-0.740427\pi$
$269$	−6048.97	−1.37105	−0.685524	+	0.728050i	$0.740427\pi$
$270$	0	0
$271$	7457.62	1.67165	0.835827	−	0.548993i	$-0.184989\pi$
$271$	7457.62	1.67165	0.835827	+	0.548993i	$0.184989\pi$
$272$	677.423	0.151010
$273$	0	0
$274$	−4085.37	−0.900752
$275$	1477.23	0.323927
$276$	0	0
$277$	−3427.82	−0.743531	−0.371765	−	0.928327i	$-0.621247\pi$
$277$	−3427.82	−0.743531	−0.371765	+	0.928327i	$0.621247\pi$
$278$	−170.159	−0.0367102
$279$	0	0
$280$	−368.076	−0.0785598
$281$	3646.53	0.774142	0.387071	−	0.922050i	$-0.373487\pi$
$281$	3646.53	0.774142	0.387071	+	0.922050i	$0.373487\pi$
$282$	0	0
$283$	−3135.21	−0.658547	−0.329273	−	0.944235i	$-0.606804\pi$
$283$	−3135.21	−0.658547	−0.329273	+	0.944235i	$0.606804\pi$
$284$	262.434	0.0548331
$285$	0	0
$286$	469.542	0.0970790
$287$	2981.86	0.613289
$288$	0	0
$289$	−3120.41	−0.635134
$290$	−1591.60	−0.322282
$291$	0	0
$292$	3860.40	0.773673
$293$	6551.35	1.30626	0.653129	−	0.757246i	$-0.273456\pi$
$293$	6551.35	1.30626	0.653129	+	0.757246i	$0.273456\pi$
$294$	0	0
$295$	2648.85	0.522787
$296$	2238.54	0.439569
$297$	0	0
$298$	−794.816	−0.154505
$299$	−570.678	−0.110378
$300$	0	0
$301$	−1254.30	−0.240188
$302$	6199.58	1.18128
$303$	0	0
$304$	1243.31	0.234568
$305$	−1621.93	−0.304497
$306$	0	0
$307$	8630.62	1.60448	0.802240	−	0.597002i	$-0.203641\pi$
$307$	8630.62	1.60448	0.802240	+	0.597002i	$0.203641\pi$
$308$	505.661	0.0935477
$309$	0	0
$310$	3262.92	0.597810
$311$	5438.07	0.991526	0.495763	−	0.868458i	$-0.334888\pi$
$311$	5438.07	0.991526	0.495763	+	0.868458i	$0.334888\pi$
$312$	0	0
$313$	4103.38	0.741012	0.370506	−	0.928830i	$-0.379184\pi$
$313$	4103.38	0.741012	0.370506	+	0.928830i	$0.379184\pi$
$314$	−5654.89	−1.01632
$315$	0	0
$316$	1530.37	0.272437
$317$	4303.28	0.762449	0.381225	−	0.924482i	$-0.375502\pi$
$317$	4303.28	0.762449	0.381225	+	0.924482i	$0.375502\pi$
$318$	0	0
$319$	2186.53	0.383768
$320$	−420.658	−0.0734859
$321$	0	0
$322$	−614.576	−0.106363
$323$	3290.03	0.566755
$324$	0	0
$325$	−1063.38	−0.181495
$326$	6022.01	1.02309
$327$	0	0
$328$	3407.84	0.573679
$329$	−1514.25	−0.253749
$330$	0	0
$331$	−4518.12	−0.750266	−0.375133	−	0.926971i	$-0.622403\pi$
$331$	−4518.12	−0.750266	−0.375133	+	0.926971i	$0.622403\pi$
$332$	5391.51	0.891258
$333$	0	0
$334$	3706.38	0.607198
$335$	4878.98	0.795723
$336$	0	0
$337$	7466.07	1.20683	0.603416	−	0.797426i	$-0.293806\pi$
$337$	7466.07	1.20683	0.603416	+	0.797426i	$0.293806\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	−1113.14	−0.177554
$341$	−4482.58	−0.711863
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−1433.48	−0.224675
$345$	0	0
$346$	−8800.33	−1.36737
$347$	−504.632	−0.0780694	−0.0390347	−	0.999238i	$-0.512428\pi$
$347$	−504.632	−0.0780694	−0.0390347	+	0.999238i	$0.512428\pi$
$348$	0	0
$349$	8535.35	1.30913	0.654566	−	0.756005i	$-0.272851\pi$
$349$	8535.35	1.30913	0.654566	+	0.756005i	$0.272851\pi$
$350$	−1145.18	−0.174893
$351$	0	0
$352$	577.898	0.0875059
$353$	4330.89	0.653002	0.326501	−	0.945197i	$-0.394130\pi$
$353$	4330.89	0.653002	0.326501	+	0.945197i	$0.394130\pi$
$354$	0	0
$355$	−431.231	−0.0644714
$356$	−3320.36	−0.494323
$357$	0	0
$358$	−7235.20	−1.06813
$359$	6906.32	1.01532	0.507662	−	0.861556i	$-0.330510\pi$
$359$	6906.32	1.01532	0.507662	+	0.861556i	$0.330510\pi$
$360$	0	0
$361$	−820.646	−0.119645
$362$	−792.028	−0.114995
$363$	0	0
$364$	−364.000	−0.0524142
$365$	−6343.39	−0.909666
$366$	0	0
$367$	766.383	0.109005	0.0545025	−	0.998514i	$-0.482643\pi$
$367$	766.383	0.109005	0.0545025	+	0.998514i	$0.482643\pi$
$368$	−702.373	−0.0994938
$369$	0	0
$370$	−3678.36	−0.516834
$371$	4991.02	0.698439
$372$	0	0
$373$	5357.17	0.743657	0.371828	−	0.928302i	$-0.378731\pi$
$373$	5357.17	0.743657	0.371828	+	0.928302i	$0.378731\pi$
$374$	1529.22	0.211429
$375$	0	0
$376$	−1730.57	−0.237360
$377$	−1573.97	−0.215023
$378$	0	0
$379$	10300.5	1.39605	0.698024	−	0.716074i	$-0.254063\pi$
$379$	10300.5	1.39605	0.698024	+	0.716074i	$0.254063\pi$
$380$	−2043.00	−0.275799
$381$	0	0
$382$	−5345.51	−0.715969
$383$	3326.55	0.443809	0.221904	−	0.975068i	$-0.428773\pi$
$383$	3326.55	0.443809	0.221904	+	0.975068i	$0.428773\pi$
$384$	0	0
$385$	−830.899	−0.109991
$386$	−2050.42	−0.270372
$387$	0	0
$388$	1505.63	0.197002
$389$	7172.37	0.934843	0.467421	−	0.884035i	$-0.345183\pi$
$389$	7172.37	0.934843	0.467421	+	0.884035i	$0.345183\pi$
$390$	0	0
$391$	−1858.61	−0.240393
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−7697.95	−0.984308
$395$	−2514.70	−0.320325
$396$	0	0
$397$	−2251.67	−0.284655	−0.142328	−	0.989820i	$-0.545459\pi$
$397$	−2251.67	−0.284655	−0.142328	+	0.989820i	$0.545459\pi$
$398$	9019.21	1.13591
$399$	0	0
$400$	−1308.78	−0.163597
$401$	10811.6	1.34639	0.673197	−	0.739463i	$-0.264920\pi$
$401$	10811.6	1.34639	0.673197	+	0.739463i	$0.264920\pi$
$402$	0	0
$403$	3226.79	0.398853
$404$	−2585.58	−0.318409
$405$	0	0
$406$	−1695.05	−0.207201
$407$	5053.31	0.615438
$408$	0	0
$409$	−4439.26	−0.536693	−0.268347	−	0.963322i	$-0.586477\pi$
$409$	−4439.26	−0.536693	−0.268347	+	0.963322i	$0.586477\pi$
$410$	−5599.75	−0.674517
$411$	0	0
$412$	167.011	0.0199710
$413$	2821.02	0.336110
$414$	0	0
$415$	−8859.31	−1.04792
$416$	−416.000	−0.0490290
$417$	0	0
$418$	2806.66	0.328417
$419$	−4859.95	−0.566644	−0.283322	−	0.959025i	$-0.591437\pi$
$419$	−4859.95	−0.566644	−0.283322	+	0.959025i	$0.591437\pi$
$420$	0	0
$421$	12316.8	1.42585	0.712927	−	0.701238i	$-0.247369\pi$
$421$	12316.8	1.42585	0.712927	+	0.701238i	$0.247369\pi$
$422$	−6647.76	−0.766844
$423$	0	0
$424$	5704.02	0.653330
$425$	−3463.26	−0.395278
$426$	0	0
$427$	−1727.35	−0.195767
$428$	2147.70	0.242554
$429$	0	0
$430$	2355.49	0.264167
$431$	6239.42	0.697314	0.348657	−	0.937250i	$-0.386638\pi$
$431$	6239.42	0.697314	0.348657	+	0.937250i	$0.386638\pi$
$432$	0	0
$433$	5522.10	0.612876	0.306438	−	0.951891i	$-0.400863\pi$
$433$	5522.10	0.612876	0.306438	+	0.951891i	$0.400863\pi$
$434$	3475.00	0.384344
$435$	0	0
$436$	7837.91	0.860935
$437$	−3411.20	−0.373409
$438$	0	0
$439$	11311.7	1.22979	0.614896	−	0.788608i	$-0.289198\pi$
$439$	11311.7	1.22979	0.614896	+	0.788608i	$0.289198\pi$
$440$	−949.599	−0.102887
$441$	0	0
$442$	−1100.81	−0.118462
$443$	2739.63	0.293823	0.146912	−	0.989150i	$-0.453067\pi$
$443$	2739.63	0.293823	0.146912	+	0.989150i	$0.453067\pi$
$444$	0	0
$445$	5456.01	0.581212
$446$	−3876.65	−0.411580
$447$	0	0
$448$	−448.000	−0.0472456
$449$	8871.18	0.932421	0.466210	−	0.884674i	$-0.345619\pi$
$449$	8871.18	0.932421	0.466210	+	0.884674i	$0.345619\pi$
$450$	0	0
$451$	7692.91	0.803204
$452$	−6014.09	−0.625838
$453$	0	0
$454$	−5114.89	−0.528752
$455$	598.123	0.0616274
$456$	0	0
$457$	1435.24	0.146909	0.0734547	−	0.997299i	$-0.476598\pi$
$457$	1435.24	0.146909	0.0734547	+	0.997299i	$0.476598\pi$
$458$	7746.95	0.790374
$459$	0	0
$460$	1154.14	0.116982
$461$	−151.976	−0.0153541	−0.00767706	−	0.999971i	$-0.502444\pi$
$461$	−151.976	−0.0153541	−0.00767706	+	0.999971i	$0.502444\pi$
$462$	0	0
$463$	−1057.75	−0.106172	−0.0530860	−	0.998590i	$-0.516906\pi$
$463$	−1057.75	−0.106172	−0.0530860	+	0.998590i	$0.516906\pi$
$464$	−1937.20	−0.193819
$465$	0	0
$466$	−453.088	−0.0450406
$467$	−15537.7	−1.53961	−0.769806	−	0.638278i	$-0.779647\pi$
$467$	−15537.7	−1.53961	−0.769806	+	0.638278i	$0.779647\pi$
$468$	0	0
$469$	5196.11	0.511586
$470$	2843.67	0.279082
$471$	0	0
$472$	3224.03	0.314402
$473$	−3235.96	−0.314566
$474$	0	0
$475$	−6356.31	−0.613994
$476$	−1185.49	−0.114153
$477$	0	0
$478$	−8282.52	−0.792539
$479$	−2046.03	−0.195168	−0.0975842	−	0.995227i	$-0.531112\pi$
$479$	−2046.03	−0.195168	−0.0975842	+	0.995227i	$0.531112\pi$
$480$	0	0
$481$	−3637.63	−0.344826
$482$	−4901.92	−0.463228
$483$	0	0
$484$	−4019.44	−0.377484
$485$	−2474.04	−0.231630
$486$	0	0
$487$	10798.4	1.00477	0.502383	−	0.864645i	$-0.332457\pi$
$487$	10798.4	1.00477	0.502383	+	0.864645i	$0.332457\pi$
$488$	−1974.12	−0.183123
$489$	0	0
$490$	644.133	0.0593856
$491$	−15026.5	−1.38114	−0.690568	−	0.723267i	$-0.742640\pi$
$491$	−15026.5	−1.38114	−0.690568	+	0.723267i	$0.742640\pi$
$492$	0	0
$493$	−5126.18	−0.468299
$494$	−2020.38	−0.184010
$495$	0	0
$496$	3971.43	0.359521
$497$	−459.260	−0.0414499
$498$	0	0
$499$	2603.35	0.233551	0.116776	−	0.993158i	$-0.462744\pi$
$499$	2603.35	0.233551	0.116776	+	0.993158i	$0.462744\pi$
$500$	5436.97	0.486297
$501$	0	0
$502$	1475.08	0.131148
$503$	−4785.14	−0.424173	−0.212086	−	0.977251i	$-0.568026\pi$
$503$	−4785.14	−0.424173	−0.212086	+	0.977251i	$0.568026\pi$
$504$	0	0
$505$	4248.61	0.374378
$506$	−1585.55	−0.139301
$507$	0	0
$508$	−2025.37	−0.176892
$509$	−17105.9	−1.48960	−0.744799	−	0.667289i	$-0.767455\pi$
$509$	−17105.9	−1.48960	−0.744799	+	0.667289i	$0.767455\pi$
$510$	0	0
$511$	−6755.69	−0.584842
$512$	−512.000	−0.0441942
$513$	0	0
$514$	6763.64	0.580411
$515$	−274.432	−0.0234814
$516$	0	0
$517$	−3906.62	−0.332326
$518$	−3917.44	−0.332283
$519$	0	0
$520$	683.569	0.0576471
$521$	16650.7	1.40016	0.700079	−	0.714065i	$-0.253148\pi$
$521$	16650.7	1.40016	0.700079	+	0.714065i	$0.253148\pi$
$522$	0	0
$523$	−18616.3	−1.55647	−0.778234	−	0.627975i	$-0.783884\pi$
$523$	−18616.3	−1.55647	−0.778234	+	0.627975i	$0.783884\pi$
$524$	4563.06	0.380416
$525$	0	0
$526$	6062.94	0.502579
$527$	10509.1	0.868662
$528$	0	0
$529$	−10239.9	−0.841616
$530$	−9372.83	−0.768169
$531$	0	0
$532$	−2175.79	−0.177317
$533$	−5537.75	−0.450031
$534$	0	0
$535$	−3529.09	−0.285188
$536$	5938.41	0.478545
$537$	0	0
$538$	12097.9	0.969478
$539$	−884.906	−0.0707154
$540$	0	0
$541$	6924.25	0.550271	0.275136	−	0.961405i	$-0.411277\pi$
$541$	6924.25	0.550271	0.275136	+	0.961405i	$0.411277\pi$
$542$	−14915.2	−1.18204
$543$	0	0
$544$	−1354.85	−0.106780
$545$	−12879.2	−1.01227
$546$	0	0
$547$	11206.9	0.876000	0.438000	−	0.898975i	$-0.355687\pi$
$547$	11206.9	0.876000	0.438000	+	0.898975i	$0.355687\pi$
$548$	8170.74	0.636928
$549$	0	0
$550$	−2954.45	−0.229051
$551$	−9408.34	−0.727420
$552$	0	0
$553$	−2678.15	−0.205943
$554$	6855.65	0.525756
$555$	0	0
$556$	340.317	0.0259580
$557$	−24497.1	−1.86351	−0.931754	−	0.363091i	$-0.881721\pi$
$557$	−24497.1	−1.86351	−0.931754	+	0.363091i	$0.881721\pi$
$558$	0	0
$559$	2329.41	0.176250
$560$	736.152	0.0555501
$561$	0	0
$562$	−7293.06	−0.547401
$563$	12058.6	0.902680	0.451340	−	0.892352i	$-0.350946\pi$
$563$	12058.6	0.902680	0.451340	+	0.892352i	$0.350946\pi$
$564$	0	0
$565$	9882.32	0.735845
$566$	6270.42	0.465663
$567$	0	0
$568$	−524.869	−0.0387729
$569$	−4121.76	−0.303678	−0.151839	−	0.988405i	$-0.548520\pi$
$569$	−4121.76	−0.303678	−0.151839	+	0.988405i	$0.548520\pi$
$570$	0	0
$571$	−11260.1	−0.825257	−0.412629	−	0.910899i	$-0.635389\pi$
$571$	−11260.1	−0.825257	−0.412629	+	0.910899i	$0.635389\pi$
$572$	−939.084	−0.0686453
$573$	0	0
$574$	−5963.73	−0.433660
$575$	3590.82	0.260430
$576$	0	0
$577$	−5111.73	−0.368811	−0.184406	−	0.982850i	$-0.559036\pi$
$577$	−5111.73	−0.368811	−0.184406	+	0.982850i	$0.559036\pi$
$578$	6240.83	0.449108
$579$	0	0
$580$	3183.19	0.227888
$581$	−9435.15	−0.673728
$582$	0	0
$583$	12876.3	0.914723
$584$	−7720.79	−0.547070
$585$	0	0
$586$	−13102.7	−0.923664
$587$	12902.9	0.907253	0.453627	−	0.891192i	$-0.350130\pi$
$587$	12902.9	0.907253	0.453627	+	0.891192i	$0.350130\pi$
$588$	0	0
$589$	19287.9	1.34931
$590$	−5297.71	−0.369666
$591$	0	0
$592$	−4477.08	−0.310822
$593$	559.726	0.0387609	0.0193804	−	0.999812i	$-0.493831\pi$
$593$	559.726	0.0387609	0.0193804	+	0.999812i	$0.493831\pi$
$594$	0	0
$595$	1947.99	0.134218
$596$	1589.63	0.109251
$597$	0	0
$598$	1141.36	0.0780494
$599$	−23453.1	−1.59978	−0.799889	−	0.600148i	$-0.795108\pi$
$599$	−23453.1	−1.59978	−0.799889	+	0.600148i	$0.795108\pi$
$600$	0	0
$601$	−11687.3	−0.793239	−0.396619	−	0.917983i	$-0.629817\pi$
$601$	−11687.3	−0.793239	−0.396619	+	0.917983i	$0.629817\pi$
$602$	2508.59	0.169838
$603$	0	0
$604$	−12399.2	−0.835290
$605$	6604.73	0.443836
$606$	0	0
$607$	4531.12	0.302986	0.151493	−	0.988458i	$-0.451592\pi$
$607$	4531.12	0.302986	0.151493	+	0.988458i	$0.451592\pi$
$608$	−2486.62	−0.165865
$609$	0	0
$610$	3243.86	0.215312
$611$	2812.18	0.186201
$612$	0	0
$613$	−6577.62	−0.433389	−0.216695	−	0.976239i	$-0.569528\pi$
$613$	−6577.62	−0.433389	−0.216695	+	0.976239i	$0.569528\pi$
$614$	−17261.2	−1.13454
$615$	0	0
$616$	−1011.32	−0.0661482
$617$	−10214.3	−0.666473	−0.333236	−	0.942843i	$-0.608141\pi$
$617$	−10214.3	−0.666473	−0.333236	+	0.942843i	$0.608141\pi$
$618$	0	0
$619$	−17248.0	−1.11996	−0.559980	−	0.828506i	$-0.689191\pi$
$619$	−17248.0	−1.11996	−0.559980	+	0.828506i	$0.689191\pi$
$620$	−6525.83	−0.422716
$621$	0	0
$622$	−10876.1	−0.701115
$623$	5810.64	0.373673
$624$	0	0
$625$	1290.82	0.0826123
$626$	−8206.76	−0.523975
$627$	0	0
$628$	11309.8	0.718645
$629$	−11847.2	−0.750998
$630$	0	0
$631$	9107.60	0.574592	0.287296	−	0.957842i	$-0.407244\pi$
$631$	9107.60	0.574592	0.287296	+	0.957842i	$0.407244\pi$
$632$	−3060.74	−0.192642
$633$	0	0
$634$	−8606.57	−0.539133
$635$	3328.07	0.207985
$636$	0	0
$637$	637.000	0.0396214
$638$	−4373.05	−0.271365
$639$	0	0
$640$	841.316	0.0519624
$641$	−10871.5	−0.669887	−0.334944	−	0.942238i	$-0.608717\pi$
$641$	−10871.5	−0.669887	−0.334944	+	0.942238i	$0.608717\pi$
$642$	0	0
$643$	−8150.23	−0.499866	−0.249933	−	0.968263i	$-0.580409\pi$
$643$	−8150.23	−0.499866	−0.249933	+	0.968263i	$0.580409\pi$
$644$	1229.15	0.0752102
$645$	0	0
$646$	−6580.05	−0.400756
$647$	12505.3	0.759865	0.379933	−	0.925014i	$-0.375947\pi$
$647$	12505.3	0.759865	0.379933	+	0.925014i	$0.375947\pi$
$648$	0	0
$649$	7277.97	0.440193
$650$	2126.76	0.128336
$651$	0	0
$652$	−12044.0	−0.723436
$653$	11681.2	0.700033	0.350016	−	0.936744i	$-0.386176\pi$
$653$	11681.2	0.700033	0.350016	+	0.936744i	$0.386176\pi$
$654$	0	0
$655$	−7498.00	−0.447284
$656$	−6815.69	−0.405652
$657$	0	0
$658$	3028.50	0.179427
$659$	7375.64	0.435985	0.217992	−	0.975950i	$-0.430049\pi$
$659$	7375.64	0.435985	0.217992	+	0.975950i	$0.430049\pi$
$660$	0	0
$661$	13919.0	0.819044	0.409522	−	0.912300i	$-0.365696\pi$
$661$	13919.0	0.819044	0.409522	+	0.912300i	$0.365696\pi$
$662$	9036.23	0.530518
$663$	0	0
$664$	−10783.0	−0.630215
$665$	3575.25	0.208485
$666$	0	0
$667$	5314.98	0.308541
$668$	−7412.77	−0.429354
$669$	0	0
$670$	−9757.96	−0.562661
$671$	−4456.40	−0.256390
$672$	0	0
$673$	1972.29	0.112966	0.0564829	−	0.998404i	$-0.482011\pi$
$673$	1972.29	0.112966	0.0564829	+	0.998404i	$0.482011\pi$
$674$	−14932.1	−0.853360
$675$	0	0
$676$	676.000	0.0384615
$677$	−6306.27	−0.358005	−0.179003	−	0.983849i	$-0.557287\pi$
$677$	−6306.27	−0.358005	−0.179003	+	0.983849i	$0.557287\pi$
$678$	0	0
$679$	−2634.85	−0.148919
$680$	2226.28	0.125550
$681$	0	0
$682$	8965.16	0.503363
$683$	16430.3	0.920478	0.460239	−	0.887795i	$-0.347764\pi$
$683$	16430.3	0.920478	0.460239	+	0.887795i	$0.347764\pi$
$684$	0	0
$685$	−13426.1	−0.748884
$686$	686.000	0.0381802
$687$	0	0
$688$	2866.96	0.158869
$689$	−9269.04	−0.512514
$690$	0	0
$691$	−30739.4	−1.69230	−0.846151	−	0.532943i	$-0.821086\pi$
$691$	−30739.4	−1.69230	−0.846151	+	0.532943i	$0.821086\pi$
$692$	17600.7	0.966874
$693$	0	0
$694$	1009.26	0.0552034
$695$	−559.208	−0.0305208
$696$	0	0
$697$	−18035.6	−0.980123
$698$	−17070.7	−0.925696
$699$	0	0
$700$	2290.36	0.123668
$701$	28051.1	1.51138	0.755689	−	0.654930i	$-0.227302\pi$
$701$	28051.1	1.51138	0.755689	+	0.654930i	$0.227302\pi$
$702$	0	0
$703$	−21743.7	−1.16654
$704$	−1155.80	−0.0618760
$705$	0	0
$706$	−8661.77	−0.461742
$707$	4524.77	0.240695
$708$	0	0
$709$	−35964.1	−1.90502	−0.952510	−	0.304506i	$-0.901509\pi$
$709$	−35964.1	−1.90502	−0.952510	+	0.304506i	$0.901509\pi$
$710$	862.462	0.0455882
$711$	0	0
$712$	6640.73	0.349539
$713$	−10896.2	−0.572322
$714$	0	0
$715$	1543.10	0.0807114
$716$	14470.4	0.755285
$717$	0	0
$718$	−13812.6	−0.717943
$719$	−30052.3	−1.55878	−0.779389	−	0.626541i	$-0.784470\pi$
$719$	−30052.3	−1.55878	−0.779389	+	0.626541i	$0.784470\pi$
$720$	0	0
$721$	−292.270	−0.0150967
$722$	1641.29	0.0846019
$723$	0	0
$724$	1584.06	0.0813135
$725$	9903.74	0.507332
$726$	0	0
$727$	−20439.2	−1.04270	−0.521352	−	0.853342i	$-0.674572\pi$
$727$	−20439.2	−1.04270	−0.521352	+	0.853342i	$0.674572\pi$
$728$	728.000	0.0370625
$729$	0	0
$730$	12686.8	0.643231
$731$	7586.52	0.383854
$732$	0	0
$733$	25363.0	1.27804	0.639021	−	0.769189i	$-0.279340\pi$
$733$	25363.0	1.27804	0.639021	+	0.769189i	$0.279340\pi$
$734$	−1532.77	−0.0770782
$735$	0	0
$736$	1404.75	0.0703527
$737$	13405.4	0.670008
$738$	0	0
$739$	11265.6	0.560776	0.280388	−	0.959887i	$-0.409537\pi$
$739$	11265.6	0.560776	0.280388	+	0.959887i	$0.409537\pi$
$740$	7356.72	0.365457
$741$	0	0
$742$	−9982.04	−0.493871
$743$	−21110.7	−1.04236	−0.521182	−	0.853445i	$-0.674509\pi$
$743$	−21110.7	−1.04236	−0.521182	+	0.853445i	$0.674509\pi$
$744$	0	0
$745$	−2612.08	−0.128455
$746$	−10714.3	−0.525845
$747$	0	0
$748$	−3058.45	−0.149503
$749$	−3758.47	−0.183353
$750$	0	0
$751$	−33643.2	−1.63470	−0.817348	−	0.576145i	$-0.804557\pi$
$751$	−33643.2	−1.63470	−0.817348	+	0.576145i	$0.804557\pi$
$752$	3461.14	0.167839
$753$	0	0
$754$	3147.94	0.152044
$755$	20374.3	0.982113
$756$	0	0
$757$	−557.749	−0.0267791	−0.0133895	−	0.999910i	$-0.504262\pi$
$757$	−557.749	−0.0267791	−0.0133895	+	0.999910i	$0.504262\pi$
$758$	−20601.0	−0.987155
$759$	0	0
$760$	4086.00	0.195020
$761$	15083.4	0.718492	0.359246	−	0.933243i	$-0.383034\pi$
$761$	15083.4	0.718492	0.359246	+	0.933243i	$0.383034\pi$
$762$	0	0
$763$	−13716.3	−0.650806
$764$	10691.0	0.506267
$765$	0	0
$766$	−6653.10	−0.313820
$767$	−5239.05	−0.246638
$768$	0	0
$769$	−18531.2	−0.868987	−0.434494	−	0.900675i	$-0.643073\pi$
$769$	−18531.2	−0.868987	−0.434494	+	0.900675i	$0.643073\pi$
$770$	1661.80	0.0777754
$771$	0	0
$772$	4100.83	0.191182
$773$	18436.6	0.857852	0.428926	−	0.903340i	$-0.358892\pi$
$773$	18436.6	0.857852	0.428926	+	0.903340i	$0.358892\pi$
$774$	0	0
$775$	−20303.6	−0.941065
$776$	−3011.26	−0.139301
$777$	0	0
$778$	−14344.7	−0.661034
$779$	−33101.6	−1.52245
$780$	0	0
$781$	−1184.85	−0.0542857
$782$	3717.22	0.169984
$783$	0	0
$784$	784.000	0.0357143
$785$	−18584.2	−0.844965
$786$	0	0
$787$	−43153.4	−1.95458	−0.977288	−	0.211915i	$-0.932030\pi$
$787$	−43153.4	−1.95458	−0.977288	+	0.211915i	$0.932030\pi$
$788$	15395.9	0.696011
$789$	0	0
$790$	5029.40	0.226504
$791$	10524.7	0.473089
$792$	0	0
$793$	3207.94	0.143654
$794$	4503.34	0.201282
$795$	0	0
$796$	−18038.4	−0.803209
$797$	28095.4	1.24867	0.624336	−	0.781156i	$-0.285370\pi$
$797$	28095.4	1.24867	0.624336	+	0.781156i	$0.285370\pi$
$798$	0	0
$799$	9158.82	0.405527
$800$	2617.55	0.115681
$801$	0	0
$802$	−21623.2	−0.952045
$803$	−17429.0	−0.765949
$804$	0	0
$805$	−2019.74	−0.0884303
$806$	−6453.57	−0.282031
$807$	0	0
$808$	5171.16	0.225149
$809$	−73.9427	−0.00321346	−0.00160673	−	0.999999i	$-0.500511\pi$
$809$	−73.9427	−0.00321346	−0.00160673	+	0.999999i	$0.500511\pi$
$810$	0	0
$811$	18635.3	0.806872	0.403436	−	0.915008i	$-0.367816\pi$
$811$	18635.3	0.806872	0.403436	+	0.915008i	$0.367816\pi$
$812$	3390.09	0.146513
$813$	0	0
$814$	−10106.6	−0.435180
$815$	19790.7	0.850597
$816$	0	0
$817$	13923.9	0.596250
$818$	8878.53	0.379499
$819$	0	0
$820$	11199.5	0.476956
$821$	−34847.8	−1.48136	−0.740680	−	0.671858i	$-0.765496\pi$
$821$	−34847.8	−1.48136	−0.740680	+	0.671858i	$0.765496\pi$
$822$	0	0
$823$	−23482.4	−0.994587	−0.497293	−	0.867582i	$-0.665673\pi$
$823$	−23482.4	−0.994587	−0.497293	+	0.867582i	$0.665673\pi$
$824$	−334.023	−0.0141216
$825$	0	0
$826$	−5642.05	−0.237666
$827$	−26051.3	−1.09540	−0.547698	−	0.836676i	$-0.684496\pi$
$827$	−26051.3	−1.09540	−0.547698	+	0.836676i	$0.684496\pi$
$828$	0	0
$829$	42125.9	1.76489	0.882445	−	0.470416i	$-0.155896\pi$
$829$	42125.9	1.76489	0.882445	+	0.470416i	$0.155896\pi$
$830$	17718.6	0.740991
$831$	0	0
$832$	832.000	0.0346688
$833$	2074.61	0.0862916
$834$	0	0
$835$	12180.6	0.504824
$836$	−5613.33	−0.232226
$837$	0	0
$838$	9719.90	0.400678
$839$	44837.3	1.84500	0.922501	−	0.385995i	$-0.126142\pi$
$839$	44837.3	1.84500	0.922501	+	0.385995i	$0.126142\pi$
$840$	0	0
$841$	−9729.90	−0.398946
$842$	−24633.6	−1.00823
$843$	0	0
$844$	13295.5	0.542240
$845$	−1110.80	−0.0452221
$846$	0	0
$847$	7034.03	0.285351
$848$	−11408.0	−0.461974
$849$	0	0
$850$	6926.53	0.279503
$851$	12283.5	0.494798
$852$	0	0
$853$	6832.88	0.274271	0.137136	−	0.990552i	$-0.456210\pi$
$853$	6832.88	0.274271	0.137136	+	0.990552i	$0.456210\pi$
$854$	3454.71	0.138428
$855$	0	0
$856$	−4295.40	−0.171511
$857$	34793.5	1.38684	0.693421	−	0.720533i	$-0.256103\pi$
$857$	34793.5	1.38684	0.693421	+	0.720533i	$0.256103\pi$
$858$	0	0
$859$	12163.1	0.483118	0.241559	−	0.970386i	$-0.422341\pi$
$859$	12163.1	0.483118	0.241559	+	0.970386i	$0.422341\pi$
$860$	−4710.98	−0.186794
$861$	0	0
$862$	−12478.8	−0.493076
$863$	1008.87	0.0397941	0.0198971	−	0.999802i	$-0.493666\pi$
$863$	1008.87	0.0397941	0.0198971	+	0.999802i	$0.493666\pi$
$864$	0	0
$865$	−28921.3	−1.13683
$866$	−11044.2	−0.433369
$867$	0	0
$868$	−6950.00	−0.271772
$869$	−6909.36	−0.269717
$870$	0	0
$871$	−9649.91	−0.375401
$872$	−15675.8	−0.608773
$873$	0	0
$874$	6822.40	0.264040
$875$	−9514.69	−0.367606
$876$	0	0
$877$	−35771.9	−1.37734	−0.688672	−	0.725073i	$-0.741806\pi$
$877$	−35771.9	−1.37734	−0.688672	+	0.725073i	$0.741806\pi$
$878$	−22623.4	−0.869595
$879$	0	0
$880$	1899.20	0.0727523
$881$	−14557.1	−0.556689	−0.278344	−	0.960481i	$-0.589786\pi$
$881$	−14557.1	−0.556689	−0.278344	+	0.960481i	$0.589786\pi$
$882$	0	0
$883$	24368.1	0.928712	0.464356	−	0.885649i	$-0.346286\pi$
$883$	24368.1	0.928712	0.464356	+	0.885649i	$0.346286\pi$
$884$	2201.63	0.0837655
$885$	0	0
$886$	−5479.26	−0.207764
$887$	14649.3	0.554537	0.277268	−	0.960792i	$-0.410571\pi$
$887$	14649.3	0.554537	0.277268	+	0.960792i	$0.410571\pi$
$888$	0	0
$889$	3544.39	0.133718
$890$	−10912.0	−0.410979
$891$	0	0
$892$	7753.31	0.291031
$893$	16809.7	0.629914
$894$	0	0
$895$	−23777.7	−0.888045
$896$	896.000	0.0334077
$897$	0	0
$898$	−17742.4	−0.659321
$899$	−30052.5	−1.11491
$900$	0	0
$901$	−30187.8	−1.11621
$902$	−15385.8	−0.567951
$903$	0	0
$904$	12028.2	0.442534
$905$	−2602.91	−0.0956064
$906$	0	0
$907$	16643.1	0.609287	0.304644	−	0.952466i	$-0.401463\pi$
$907$	16643.1	0.609287	0.304644	+	0.952466i	$0.401463\pi$
$908$	10229.8	0.373884
$909$	0	0
$910$	−1196.25	−0.0435771
$911$	29484.9	1.07231	0.536157	−	0.844119i	$-0.319876\pi$
$911$	29484.9	1.07231	0.536157	+	0.844119i	$0.319876\pi$
$912$	0	0
$913$	−24341.8	−0.882360
$914$	−2870.48	−0.103881
$915$	0	0
$916$	−15493.9	−0.558879
$917$	−7985.35	−0.287568
$918$	0	0
$919$	−43199.5	−1.55062	−0.775310	−	0.631580i	$-0.782407\pi$
$919$	−43199.5	−1.55062	−0.775310	+	0.631580i	$0.782407\pi$
$920$	−2308.27	−0.0827190
$921$	0	0
$922$	303.953	0.0108570
$923$	852.911	0.0304159
$924$	0	0
$925$	22888.7	0.813594
$926$	2115.49	0.0750749
$927$	0	0
$928$	3874.39	0.137051
$929$	−14700.1	−0.519154	−0.259577	−	0.965722i	$-0.583583\pi$
$929$	−14700.1	−0.519154	−0.259577	+	0.965722i	$0.583583\pi$
$930$	0	0
$931$	3807.64	0.134039
$932$	906.176	0.0318485
$933$	0	0
$934$	31075.4	1.08867
$935$	5025.63	0.175781
$936$	0	0
$937$	319.330	0.0111335	0.00556673	−	0.999985i	$-0.498228\pi$
$937$	319.330	0.0111335	0.00556673	+	0.999985i	$0.498228\pi$
$938$	−10392.2	−0.361746
$939$	0	0
$940$	−5687.33	−0.197341
$941$	17188.4	0.595459	0.297729	−	0.954650i	$-0.403771\pi$
$941$	17188.4	0.595459	0.297729	+	0.954650i	$0.403771\pi$
$942$	0	0
$943$	18699.8	0.645758
$944$	−6448.06	−0.222316
$945$	0	0
$946$	6471.93	0.222432
$947$	9298.04	0.319056	0.159528	−	0.987193i	$-0.449003\pi$
$947$	9298.04	0.319056	0.159528	+	0.987193i	$0.449003\pi$
$948$	0	0
$949$	12546.3	0.429157
$950$	12712.6	0.434160
$951$	0	0
$952$	2370.98	0.0807184
$953$	38827.1	1.31976	0.659882	−	0.751369i	$-0.270606\pi$
$953$	38827.1	1.31976	0.659882	+	0.751369i	$0.270606\pi$
$954$	0	0
$955$	−17567.4	−0.595256
$956$	16565.0	0.560410
$957$	0	0
$958$	4092.07	0.138005
$959$	−14298.8	−0.481472
$960$	0	0
$961$	31819.3	1.06808
$962$	7275.25	0.243829
$963$	0	0
$964$	9803.83	0.327552
$965$	−6738.47	−0.224787
$966$	0	0
$967$	16033.4	0.533196	0.266598	−	0.963808i	$-0.414100\pi$
$967$	16033.4	0.533196	0.266598	+	0.963808i	$0.414100\pi$
$968$	8038.89	0.266921
$969$	0	0
$970$	4948.08	0.163787
$971$	−5182.02	−0.171266	−0.0856328	−	0.996327i	$-0.527291\pi$
$971$	−5182.02	−0.171266	−0.0856328	+	0.996327i	$0.527291\pi$
$972$	0	0
$973$	−595.555	−0.0196224
$974$	−21596.8	−0.710477
$975$	0	0
$976$	3948.23	0.129488
$977$	−30393.7	−0.995271	−0.497636	−	0.867386i	$-0.665798\pi$
$977$	−30393.7	−0.995271	−0.497636	+	0.867386i	$0.665798\pi$
$978$	0	0
$979$	14990.9	0.489387
$980$	−1288.27	−0.0419920
$981$	0	0
$982$	30053.1	0.976611
$983$	29485.7	0.956713	0.478356	−	0.878166i	$-0.341233\pi$
$983$	29485.7	0.956713	0.478356	+	0.878166i	$0.341233\pi$
$984$	0	0
$985$	−25298.5	−0.818352
$986$	10252.4	0.331137
$987$	0	0
$988$	4040.76	0.130115
$989$	−7865.93	−0.252904
$990$	0	0
$991$	−48854.3	−1.56600	−0.783000	−	0.622021i	$-0.786312\pi$
$991$	−48854.3	−1.56600	−0.783000	+	0.622021i	$0.786312\pi$
$992$	−7942.86	−0.254220
$993$	0	0
$994$	918.520	0.0293095
$995$	29640.6	0.944393
$996$	0	0
$997$	3620.03	0.114993	0.0574963	−	0.998346i	$-0.481688\pi$
$997$	3620.03	0.114993	0.0574963	+	0.998346i	$0.481688\pi$
$998$	−5206.71	−0.165146
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.v.1.2		3
3.2	odd	2		546.4.a.p.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.p.1.2	✓	3	3.2	odd	2
1638.4.a.v.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -6.57278 q^{5} -7.00000 q^{7} -8.00000 q^{8} +13.1456 q^{10} -18.0593 q^{11} +13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +42.3389 q^{17} +77.7068 q^{19} -26.2911 q^{20} +36.1186 q^{22} -43.8983 q^{23} -81.7985 q^{25} -26.0000 q^{26} -28.0000 q^{28} -121.075 q^{29} +248.214 q^{31} -32.0000 q^{32} -84.6779 q^{34} +46.0095 q^{35} -279.817 q^{37} -155.414 q^{38} +52.5823 q^{40} -425.980 q^{41} +179.185 q^{43} -72.2372 q^{44} +87.7966 q^{46} +216.321 q^{47} +49.0000 q^{49} +163.597 q^{50} +52.0000 q^{52} -713.003 q^{53} +118.700 q^{55} +56.0000 q^{56} +242.150 q^{58} -403.004 q^{59} +246.765 q^{61} -496.429 q^{62} +64.0000 q^{64} -85.4462 q^{65} -742.301 q^{67} +169.356 q^{68} -92.0190 q^{70} +65.6086 q^{71} +965.099 q^{73} +559.635 q^{74} +310.827 q^{76} +126.415 q^{77} +382.593 q^{79} -105.165 q^{80} +851.961 q^{82} +1347.88 q^{83} -278.285 q^{85} -358.371 q^{86} +144.474 q^{88} -830.091 q^{89} -91.0000 q^{91} -175.593 q^{92} -432.643 q^{94} -510.750 q^{95} +376.407 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 12 q^{4} - 7 q^{5} - 21 q^{7} - 24 q^{8} + 14 q^{10} + 47 q^{11} + 39 q^{13} + 42 q^{14} + 48 q^{16} - 119 q^{17} + 101 q^{19} - 28 q^{20} - 94 q^{22} + 27 q^{23} + 266 q^{25} - 78 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) \) 3 * q - 6 * q^2 + 12 * q^4 - 7 * q^5 - 21 * q^7 - 24 * q^8 + 14 * q^10 + 47 * q^11 + 39 * q^13 + 42 * q^14 + 48 * q^16 - 119 * q^17 + 101 * q^19 - 28 * q^20 - 94 * q^22 + 27 * q^23 + 266 * q^25 - 78 * q^26 - 84 * q^28 - 345 * q^29 + 152 * q^31 - 96 * q^32 + 238 * q^34 + 49 * q^35 + 227 * q^37 - 202 * q^38 + 56 * q^40 - 452 * q^41 - 415 * q^43 + 188 * q^44 - 54 * q^46 - 240 * q^47 + 147 * q^49 - 532 * q^50 + 156 * q^52 - 874 * q^53 + 1123 * q^55 + 168 * q^56 + 690 * q^58 - 948 * q^59 + 951 * q^61 - 304 * q^62 + 192 * q^64 - 91 * q^65 + 742 * q^67 - 476 * q^68 - 98 * q^70 - 732 * q^71 - 585 * q^73 - 454 * q^74 + 404 * q^76 - 329 * q^77 - 618 * q^79 - 112 * q^80 + 904 * q^82 - 774 * q^83 + 1273 * q^85 + 830 * q^86 - 376 * q^88 - 730 * q^89 - 273 * q^91 + 108 * q^92 + 480 * q^94 + 2617 * q^95 + 2518 * q^97 - 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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