LMFDB - Embedded newform 1638.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,4,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$96.6451285894$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} -18.0000 q^{10} +18.0000 q^{11} +13.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -60.0000 q^{17} -43.0000 q^{19} +36.0000 q^{20} -36.0000 q^{22} -9.00000 q^{23} -44.0000 q^{25} -26.0000 q^{26} +28.0000 q^{28} +249.000 q^{29} -79.0000 q^{31} -32.0000 q^{32} +120.000 q^{34} +63.0000 q^{35} -412.000 q^{37} +86.0000 q^{38} -72.0000 q^{40} -222.000 q^{41} -295.000 q^{43} +72.0000 q^{44} +18.0000 q^{46} -411.000 q^{47} +49.0000 q^{49} +88.0000 q^{50} +52.0000 q^{52} +237.000 q^{53} +162.000 q^{55} -56.0000 q^{56} -498.000 q^{58} +384.000 q^{59} -466.000 q^{61} +158.000 q^{62} +64.0000 q^{64} +117.000 q^{65} -1042.00 q^{67} -240.000 q^{68} -126.000 q^{70} +288.000 q^{71} -691.000 q^{73} +824.000 q^{74} -172.000 q^{76} +126.000 q^{77} +1001.00 q^{79} +144.000 q^{80} +444.000 q^{82} -39.0000 q^{83} -540.000 q^{85} +590.000 q^{86} -144.000 q^{88} +339.000 q^{89} +91.0000 q^{91} -36.0000 q^{92} +822.000 q^{94} -387.000 q^{95} +713.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−18.0000	−0.569210
$11$	18.0000	0.493382	0.246691	−	0.969094i	$-0.420657\pi$
$11$	18.0000	0.493382	0.246691	+	0.969094i	$0.420657\pi$
$12$	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
$17$	−60.0000	−0.856008	−0.428004	−	0.903777i	$-0.640783\pi$
$17$	−60.0000	−0.856008	−0.428004	+	0.903777i	$0.640783\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	−36.0000	−0.348874
$23$	−9.00000	−0.0815926	−0.0407963	−	0.999167i	$-0.512989\pi$
$23$	−9.00000	−0.0815926	−0.0407963	+	0.999167i	$0.512989\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	−26.0000	−0.196116
$27$	0	0
$28$	28.0000	0.188982
$29$	249.000	1.59442	0.797209	−	0.603703i	$-0.206309\pi$
$29$	249.000	1.59442	0.797209	+	0.603703i	$0.206309\pi$
$30$	0	0
$31$	−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
$33$	0	0
$34$	120.000	0.605289
$35$	63.0000	0.304256
$36$	0	0
$37$	−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
$39$	0	0
$40$	−72.0000	−0.284605
$41$	−222.000	−0.845624	−0.422812	−	0.906217i	$-0.638957\pi$
$41$	−222.000	−0.845624	−0.422812	+	0.906217i	$0.638957\pi$
$42$	0	0
$43$	−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
$45$	0	0
$46$	18.0000	0.0576947
$47$	−411.000	−1.27554	−0.637771	−	0.770226i	$-0.720144\pi$
$47$	−411.000	−1.27554	−0.637771	+	0.770226i	$0.720144\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	88.0000	0.248902
$51$	0	0
$52$	52.0000	0.138675
$53$	237.000	0.614235	0.307117	−	0.951672i	$-0.400636\pi$
$53$	237.000	0.614235	0.307117	+	0.951672i	$0.400636\pi$
$54$	0	0
$55$	162.000	0.397165
$56$	−56.0000	−0.133631
$57$	0	0
$58$	−498.000	−1.12742
$59$	384.000	0.847331	0.423666	−	0.905819i	$-0.360743\pi$
$59$	384.000	0.847331	0.423666	+	0.905819i	$0.360743\pi$
$60$	0	0
$61$	−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
$63$	0	0
$64$	64.0000	0.125000
$65$	117.000	0.223263
$66$	0	0
$67$	−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
$69$	0	0
$70$	−126.000	−0.215141
$71$	288.000	0.481399	0.240699	−	0.970600i	$-0.422623\pi$
$71$	288.000	0.481399	0.240699	+	0.970600i	$0.422623\pi$
$72$	0	0
$73$	−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
$75$	0	0
$76$	−172.000	−0.259602
$77$	126.000	0.186481
$78$	0	0
$79$	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
$81$	0	0
$82$	444.000	0.597946
$83$	−39.0000	−0.0515760	−0.0257880	−	0.999667i	$-0.508209\pi$
$83$	−39.0000	−0.0515760	−0.0257880	+	0.999667i	$0.508209\pi$
$84$	0	0
$85$	−540.000	−0.689073
$86$	590.000	0.739783
$87$	0	0
$88$	−144.000	−0.174437
$89$	339.000	0.403752	0.201876	−	0.979411i	$-0.435296\pi$
$89$	339.000	0.403752	0.201876	+	0.979411i	$0.435296\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	−36.0000	−0.0407963
$93$	0	0
$94$	822.000	0.901945
$95$	−387.000	−0.417951
$96$	0	0
$97$	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
$99$	0	0
$100$	−176.000	−0.176000
$101$	−810.000	−0.798000	−0.399000	−	0.916951i	$-0.630643\pi$
$101$	−810.000	−0.798000	−0.399000	+	0.916951i	$0.630643\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	−474.000	−0.434330
$107$	924.000	0.834827	0.417413	−	0.908717i	$-0.362937\pi$
$107$	924.000	0.834827	0.417413	+	0.908717i	$0.362937\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	112.000	0.0944911
$113$	−1725.00	−1.43606	−0.718028	−	0.696014i	$-0.754955\pi$
$113$	−1725.00	−1.43606	−0.718028	+	0.696014i	$0.754955\pi$
$114$	0	0
$115$	−81.0000	−0.0656808
$116$	996.000	0.797209
$117$	0	0
$118$	−768.000	−0.599154
$119$	−420.000	−0.323541
$120$	0	0
$121$	−1007.00	−0.756574
$122$	932.000	0.691634
$123$	0	0
$124$	−316.000	−0.228852
$125$	−1521.00	−1.08834
$126$	0	0
$127$	−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
$129$	0	0
$130$	−234.000	−0.157870
$131$	−828.000	−0.552234	−0.276117	−	0.961124i	$-0.589048\pi$
$131$	−828.000	−0.552234	−0.276117	+	0.961124i	$0.589048\pi$
$132$	0	0
$133$	−301.000	−0.196241
$134$	2084.00	1.34351
$135$	0	0
$136$	480.000	0.302645
$137$	2664.00	1.66132	0.830660	−	0.556780i	$-0.187963\pi$
$137$	2664.00	1.66132	0.830660	+	0.556780i	$0.187963\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	−576.000	−0.340400
$143$	234.000	0.136840
$144$	0	0
$145$	2241.00	1.28348
$146$	1382.00	0.783391
$147$	0	0
$148$	−1648.00	−0.915302
$149$	1602.00	0.880812	0.440406	−	0.897799i	$-0.354835\pi$
$149$	1602.00	0.880812	0.440406	+	0.897799i	$0.354835\pi$
$150$	0	0
$151$	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
$153$	0	0
$154$	−252.000	−0.131862
$155$	−711.000	−0.368444
$156$	0	0
$157$	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
$159$	0	0
$160$	−288.000	−0.142302
$161$	−63.0000	−0.0308391
$162$	0	0
$163$	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
$165$	0	0
$166$	78.0000	0.0364697
$167$	1767.00	0.818770	0.409385	−	0.912362i	$-0.365743\pi$
$167$	1767.00	0.818770	0.409385	+	0.912362i	$0.365743\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	1080.00	0.487248
$171$	0	0
$172$	−1180.00	−0.523106
$173$	−96.0000	−0.0421893	−0.0210946	−	0.999777i	$-0.506715\pi$
$173$	−96.0000	−0.0421893	−0.0210946	+	0.999777i	$0.506715\pi$
$174$	0	0
$175$	−308.000	−0.133043
$176$	288.000	0.123346
$177$	0	0
$178$	−678.000	−0.285496
$179$	−3345.00	−1.39674	−0.698372	−	0.715735i	$-0.746092\pi$
$179$	−3345.00	−1.39674	−0.698372	+	0.715735i	$0.746092\pi$
$180$	0	0
$181$	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
$183$	0	0
$184$	72.0000	0.0288473
$185$	−3708.00	−1.47361
$186$	0	0
$187$	−1080.00	−0.422339
$188$	−1644.00	−0.637771
$189$	0	0
$190$	774.000	0.295536
$191$	768.000	0.290945	0.145473	−	0.989362i	$-0.453530\pi$
$191$	768.000	0.290945	0.145473	+	0.989362i	$0.453530\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	196.000	0.0714286
$197$	774.000	0.279925	0.139962	−	0.990157i	$-0.455302\pi$
$197$	774.000	0.279925	0.139962	+	0.990157i	$0.455302\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	1620.00	0.564271
$203$	1743.00	0.602634
$204$	0	0
$205$	−1998.00	−0.680714
$206$	2912.00	0.984896
$207$	0	0
$208$	208.000	0.0693375
$209$	−774.000	−0.256166
$210$	0	0
$211$	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
$213$	0	0
$214$	−1848.00	−0.590312
$215$	−2655.00	−0.842184
$216$	0	0
$217$	−553.000	−0.172996
$218$	−2308.00	−0.717053
$219$	0	0
$220$	648.000	0.198583
$221$	−780.000	−0.237414
$222$	0	0
$223$	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
$225$	0	0
$226$	3450.00	1.01545
$227$	−3132.00	−0.915763	−0.457881	−	0.889013i	$-0.651392\pi$
$227$	−3132.00	−0.915763	−0.457881	+	0.889013i	$0.651392\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	−1992.00	−0.563712
$233$	−2847.00	−0.800486	−0.400243	−	0.916409i	$-0.631074\pi$
$233$	−2847.00	−0.800486	−0.400243	+	0.916409i	$0.631074\pi$
$234$	0	0
$235$	−3699.00	−1.02679
$236$	1536.00	0.423666
$237$	0	0
$238$	840.000	0.228778
$239$	−2220.00	−0.600836	−0.300418	−	0.953808i	$-0.597126\pi$
$239$	−2220.00	−0.600836	−0.300418	+	0.953808i	$0.597126\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	−1864.00	−0.489059
$245$	441.000	0.114998
$246$	0	0
$247$	−559.000	−0.144001
$248$	632.000	0.161823
$249$	0	0
$250$	3042.00	0.769572
$251$	−4770.00	−1.19952	−0.599760	−	0.800180i	$-0.704737\pi$
$251$	−4770.00	−1.19952	−0.599760	+	0.800180i	$0.704737\pi$
$252$	0	0
$253$	−162.000	−0.0402563
$254$	2408.00	0.594848
$255$	0	0
$256$	256.000	0.0625000
$257$	1062.00	0.257766	0.128883	−	0.991660i	$-0.458861\pi$
$257$	1062.00	0.257766	0.128883	+	0.991660i	$0.458861\pi$
$258$	0	0
$259$	−2884.00	−0.691904
$260$	468.000	0.111631
$261$	0	0
$262$	1656.00	0.390489
$263$	−5283.00	−1.23865	−0.619323	−	0.785137i	$-0.712593\pi$
$263$	−5283.00	−1.23865	−0.619323	+	0.785137i	$0.712593\pi$
$264$	0	0
$265$	2133.00	0.494450
$266$	602.000	0.138763
$267$	0	0
$268$	−4168.00	−0.950004
$269$	6672.00	1.51226	0.756132	−	0.654419i	$-0.227087\pi$
$269$	6672.00	1.51226	0.756132	+	0.654419i	$0.227087\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	−5328.00	−1.17473
$275$	−792.000	−0.173671
$276$	0	0
$277$	−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
$279$	0	0
$280$	−504.000	−0.107571
$281$	−6402.00	−1.35911	−0.679557	−	0.733622i	$-0.737828\pi$
$281$	−6402.00	−1.35911	−0.679557	+	0.733622i	$0.737828\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	−468.000	−0.0967602
$287$	−1554.00	−0.319616
$288$	0	0
$289$	−1313.00	−0.267250
$290$	−4482.00	−0.907559
$291$	0	0
$292$	−2764.00	−0.553941
$293$	−6279.00	−1.25196	−0.625978	−	0.779841i	$-0.715300\pi$
$293$	−6279.00	−1.25196	−0.625978	+	0.779841i	$0.715300\pi$
$294$	0	0
$295$	3456.00	0.682088
$296$	3296.00	0.647217
$297$	0	0
$298$	−3204.00	−0.622828
$299$	−117.000	−0.0226297
$300$	0	0
$301$	−2065.00	−0.395431
$302$	−3856.00	−0.734728
$303$	0	0
$304$	−688.000	−0.129801
$305$	−4194.00	−0.787370
$306$	0	0
$307$	−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
$309$	0	0
$310$	1422.00	0.260530
$311$	1770.00	0.322725	0.161363	−	0.986895i	$-0.448411\pi$
$311$	1770.00	0.322725	0.161363	+	0.986895i	$0.448411\pi$
$312$	0	0
$313$	−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
$315$	0	0
$316$	4004.00	0.712793
$317$	5592.00	0.990782	0.495391	−	0.868670i	$-0.335025\pi$
$317$	5592.00	0.990782	0.495391	+	0.868670i	$0.335025\pi$
$318$	0	0
$319$	4482.00	0.786658
$320$	576.000	0.100623
$321$	0	0
$322$	126.000	0.0218065
$323$	2580.00	0.444443
$324$	0	0
$325$	−572.000	−0.0976272
$326$	−6016.00	−1.02207
$327$	0	0
$328$	1776.00	0.298973
$329$	−2877.00	−0.482110
$330$	0	0
$331$	−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
$333$	0	0
$334$	−3534.00	−0.578958
$335$	−9378.00	−1.52948
$336$	0	0
$337$	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
$339$	0	0
$340$	−2160.00	−0.344537
$341$	−1422.00	−0.225823
$342$	0	0
$343$	343.000	0.0539949
$344$	2360.00	0.369891
$345$	0	0
$346$	192.000	0.0298323
$347$	−5736.00	−0.887391	−0.443695	−	0.896178i	$-0.646333\pi$
$347$	−5736.00	−0.887391	−0.443695	+	0.896178i	$0.646333\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	−576.000	−0.0872185
$353$	3870.00	0.583511	0.291755	−	0.956493i	$-0.405761\pi$
$353$	3870.00	0.583511	0.291755	+	0.956493i	$0.405761\pi$
$354$	0	0
$355$	2592.00	0.387519
$356$	1356.00	0.201876
$357$	0	0
$358$	6690.00	0.987647
$359$	−7572.00	−1.11319	−0.556595	−	0.830784i	$-0.687892\pi$
$359$	−7572.00	−1.11319	−0.556595	+	0.830784i	$0.687892\pi$
$360$	0	0
$361$	−5010.00	−0.730427
$362$	−4756.00	−0.690524
$363$	0	0
$364$	364.000	0.0524142
$365$	−6219.00	−0.891828
$366$	0	0
$367$	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
$369$	0	0
$370$	7416.00	1.04200
$371$	1659.00	0.232159
$372$	0	0
$373$	−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
$375$	0	0
$376$	3288.00	0.450972
$377$	3237.00	0.442212
$378$	0	0
$379$	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
$381$	0	0
$382$	−1536.00	−0.205729
$383$	−10836.0	−1.44568	−0.722838	−	0.691018i	$-0.757163\pi$
$383$	−10836.0	−1.44568	−0.722838	+	0.691018i	$0.757163\pi$
$384$	0	0
$385$	1134.00	0.150114
$386$	6404.00	0.844443
$387$	0	0
$388$	2852.00	0.373166
$389$	6282.00	0.818792	0.409396	−	0.912357i	$-0.365739\pi$
$389$	6282.00	0.818792	0.409396	+	0.912357i	$0.365739\pi$
$390$	0	0
$391$	540.000	0.0698439
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−1548.00	−0.197937
$395$	9009.00	1.14757
$396$	0	0
$397$	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
$399$	0	0
$400$	−704.000	−0.0880000
$401$	5328.00	0.663510	0.331755	−	0.943366i	$-0.392359\pi$
$401$	5328.00	0.663510	0.331755	+	0.943366i	$0.392359\pi$
$402$	0	0
$403$	−1027.00	−0.126944
$404$	−3240.00	−0.399000
$405$	0	0
$406$	−3486.00	−0.426126
$407$	−7416.00	−0.903188
$408$	0	0
$409$	−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
$411$	0	0
$412$	−5824.00	−0.696427
$413$	2688.00	0.320261
$414$	0	0
$415$	−351.000	−0.0415179
$416$	−416.000	−0.0490290
$417$	0	0
$418$	1548.00	0.181137
$419$	−4398.00	−0.512784	−0.256392	−	0.966573i	$-0.582534\pi$
$419$	−4398.00	−0.512784	−0.256392	+	0.966573i	$0.582534\pi$
$420$	0	0
$421$	−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
$423$	0	0
$424$	−1896.00	−0.217165
$425$	2640.00	0.301315
$426$	0	0
$427$	−3262.00	−0.369694
$428$	3696.00	0.417413
$429$	0	0
$430$	5310.00	0.595514
$431$	3414.00	0.381547	0.190773	−	0.981634i	$-0.438900\pi$
$431$	3414.00	0.381547	0.190773	+	0.981634i	$0.438900\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	4616.00	0.507033
$437$	387.000	0.0423632
$438$	0	0
$439$	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
$441$	0	0
$442$	1560.00	0.167877
$443$	6987.00	0.749351	0.374675	−	0.927156i	$-0.377754\pi$
$443$	6987.00	0.749351	0.374675	+	0.927156i	$0.377754\pi$
$444$	0	0
$445$	3051.00	0.325014
$446$	−5782.00	−0.613869
$447$	0	0
$448$	448.000	0.0472456
$449$	−16344.0	−1.71786	−0.858932	−	0.512089i	$-0.828872\pi$
$449$	−16344.0	−1.71786	−0.858932	+	0.512089i	$0.828872\pi$
$450$	0	0
$451$	−3996.00	−0.417216
$452$	−6900.00	−0.718028
$453$	0	0
$454$	6264.00	0.647542
$455$	819.000	0.0843853
$456$	0	0
$457$	−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
$459$	0	0
$460$	−324.000	−0.0328404
$461$	6882.00	0.695286	0.347643	−	0.937627i	$-0.386982\pi$
$461$	6882.00	0.695286	0.347643	+	0.937627i	$0.386982\pi$
$462$	0	0
$463$	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
$465$	0	0
$466$	5694.00	0.566029
$467$	−1830.00	−0.181333	−0.0906663	−	0.995881i	$-0.528900\pi$
$467$	−1830.00	−0.181333	−0.0906663	+	0.995881i	$0.528900\pi$
$468$	0	0
$469$	−7294.00	−0.718136
$470$	7398.00	0.726052
$471$	0	0
$472$	−3072.00	−0.299577
$473$	−5310.00	−0.516182
$474$	0	0
$475$	1892.00	0.182760
$476$	−1680.00	−0.161770
$477$	0	0
$478$	4440.00	0.424855
$479$	12663.0	1.20791	0.603953	−	0.797020i	$-0.293591\pi$
$479$	12663.0	1.20791	0.603953	+	0.797020i	$0.293591\pi$
$480$	0	0
$481$	−5356.00	−0.507718
$482$	13406.0	1.26686
$483$	0	0
$484$	−4028.00	−0.378287
$485$	6417.00	0.600785
$486$	0	0
$487$	−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
$489$	0	0
$490$	−882.000	−0.0813157
$491$	17724.0	1.62907	0.814535	−	0.580115i	$-0.196992\pi$
$491$	17724.0	1.62907	0.814535	+	0.580115i	$0.196992\pi$
$492$	0	0
$493$	−14940.0	−1.36484
$494$	1118.00	0.101824
$495$	0	0
$496$	−1264.00	−0.114426
$497$	2016.00	0.181952
$498$	0	0
$499$	−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
$501$	0	0
$502$	9540.00	0.848189
$503$	−414.000	−0.0366985	−0.0183493	−	0.999832i	$-0.505841\pi$
$503$	−414.000	−0.0366985	−0.0183493	+	0.999832i	$0.505841\pi$
$504$	0	0
$505$	−7290.00	−0.642378
$506$	324.000	0.0284655
$507$	0	0
$508$	−4816.00	−0.420621
$509$	1521.00	0.132450	0.0662251	−	0.997805i	$-0.478904\pi$
$509$	1521.00	0.132450	0.0662251	+	0.997805i	$0.478904\pi$
$510$	0	0
$511$	−4837.00	−0.418740
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−2124.00	−0.182268
$515$	−13104.0	−1.12123
$516$	0	0
$517$	−7398.00	−0.629330
$518$	5768.00	0.489250
$519$	0	0
$520$	−936.000	−0.0789352
$521$	−2436.00	−0.204843	−0.102421	−	0.994741i	$-0.532659\pi$
$521$	−2436.00	−0.204843	−0.102421	+	0.994741i	$0.532659\pi$
$522$	0	0
$523$	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
$525$	0	0
$526$	10566.0	0.875855
$527$	4740.00	0.391798
$528$	0	0
$529$	−12086.0	−0.993343
$530$	−4266.00	−0.349629
$531$	0	0
$532$	−1204.00	−0.0981203
$533$	−2886.00	−0.234534
$534$	0	0
$535$	8316.00	0.672022
$536$	8336.00	0.671754
$537$	0	0
$538$	−13344.0	−1.06933
$539$	882.000	0.0704832
$540$	0	0
$541$	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
$543$	0	0
$544$	1920.00	0.151322
$545$	10386.0	0.816307
$546$	0	0
$547$	−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
$549$	0	0
$550$	1584.00	0.122804
$551$	−10707.0	−0.827829
$552$	0	0
$553$	7007.00	0.538821
$554$	11462.0	0.879014
$555$	0	0
$556$	−1288.00	−0.0982435
$557$	1272.00	0.0967619	0.0483809	−	0.998829i	$-0.484594\pi$
$557$	1272.00	0.0967619	0.0483809	+	0.998829i	$0.484594\pi$
$558$	0	0
$559$	−3835.00	−0.290167
$560$	1008.00	0.0760639
$561$	0	0
$562$	12804.0	0.961039
$563$	17052.0	1.27648	0.638238	−	0.769839i	$-0.279664\pi$
$563$	17052.0	1.27648	0.638238	+	0.769839i	$0.279664\pi$
$564$	0	0
$565$	−15525.0	−1.15600
$566$	−15592.0	−1.15792
$567$	0	0
$568$	−2304.00	−0.170200
$569$	19713.0	1.45239	0.726197	−	0.687487i	$-0.241286\pi$
$569$	19713.0	1.45239	0.726197	+	0.687487i	$0.241286\pi$
$570$	0	0
$571$	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
$573$	0	0
$574$	3108.00	0.226002
$575$	396.000	0.0287206
$576$	0	0
$577$	−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
$579$	0	0
$580$	8964.00	0.641741
$581$	−273.000	−0.0194939
$582$	0	0
$583$	4266.00	0.303053
$584$	5528.00	0.391696
$585$	0	0
$586$	12558.0	0.885267
$587$	7737.00	0.544021	0.272010	−	0.962294i	$-0.412311\pi$
$587$	7737.00	0.544021	0.272010	+	0.962294i	$0.412311\pi$
$588$	0	0
$589$	3397.00	0.237642
$590$	−6912.00	−0.482309
$591$	0	0
$592$	−6592.00	−0.457651
$593$	−18723.0	−1.29656	−0.648281	−	0.761401i	$-0.724512\pi$
$593$	−18723.0	−1.29656	−0.648281	+	0.761401i	$0.724512\pi$
$594$	0	0
$595$	−3780.00	−0.260445
$596$	6408.00	0.440406
$597$	0	0
$598$	234.000	0.0160016
$599$	24855.0	1.69541	0.847703	−	0.530472i	$-0.177985\pi$
$599$	24855.0	1.69541	0.847703	+	0.530472i	$0.177985\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	7712.00	0.519531
$605$	−9063.00	−0.609030
$606$	0	0
$607$	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
$609$	0	0
$610$	8388.00	0.556754
$611$	−5343.00	−0.353772
$612$	0	0
$613$	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
$615$	0	0
$616$	−1008.00	−0.0659310
$617$	−11094.0	−0.723870	−0.361935	−	0.932203i	$-0.617884\pi$
$617$	−11094.0	−0.723870	−0.361935	+	0.932203i	$0.617884\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	−3540.00	−0.228201
$623$	2373.00	0.152604
$624$	0	0
$625$	−8189.00	−0.524096
$626$	8852.00	0.565171
$627$	0	0
$628$	6992.00	0.444285
$629$	24720.0	1.56701
$630$	0	0
$631$	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
$633$	0	0
$634$	−11184.0	−0.700589
$635$	−10836.0	−0.677187
$636$	0	0
$637$	637.000	0.0396214
$638$	−8964.00	−0.556251
$639$	0	0
$640$	−1152.00	−0.0711512
$641$	−12321.0	−0.759205	−0.379602	−	0.925150i	$-0.623939\pi$
$641$	−12321.0	−0.759205	−0.379602	+	0.925150i	$0.623939\pi$
$642$	0	0
$643$	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
$645$	0	0
$646$	−5160.00	−0.314269
$647$	15150.0	0.920569	0.460284	−	0.887772i	$-0.347747\pi$
$647$	15150.0	0.920569	0.460284	+	0.887772i	$0.347747\pi$
$648$	0	0
$649$	6912.00	0.418058
$650$	1144.00	0.0690329
$651$	0	0
$652$	12032.0	0.722714
$653$	414.000	0.0248102	0.0124051	−	0.999923i	$-0.496051\pi$
$653$	414.000	0.0248102	0.0124051	+	0.999923i	$0.496051\pi$
$654$	0	0
$655$	−7452.00	−0.444540
$656$	−3552.00	−0.211406
$657$	0	0
$658$	5754.00	0.340903
$659$	−20631.0	−1.21953	−0.609765	−	0.792583i	$-0.708736\pi$
$659$	−20631.0	−1.21953	−0.609765	+	0.792583i	$0.708736\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	312.000	0.0182349
$665$	−2709.00	−0.157971
$666$	0	0
$667$	−2241.00	−0.130093
$668$	7068.00	0.409385
$669$	0	0
$670$	18756.0	1.08150
$671$	−8388.00	−0.482586
$672$	0	0
$673$	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
$675$	0	0
$676$	676.000	0.0384615
$677$	−29022.0	−1.64757	−0.823786	−	0.566901i	$-0.808142\pi$
$677$	−29022.0	−1.64757	−0.823786	+	0.566901i	$0.808142\pi$
$678$	0	0
$679$	4991.00	0.282087
$680$	4320.00	0.243624
$681$	0	0
$682$	2844.00	0.159681
$683$	−18960.0	−1.06220	−0.531101	−	0.847308i	$-0.678222\pi$
$683$	−18960.0	−1.06220	−0.531101	+	0.847308i	$0.678222\pi$
$684$	0	0
$685$	23976.0	1.33734
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−4720.00	−0.261553
$689$	3081.00	0.170358
$690$	0	0
$691$	−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
$693$	0	0
$694$	11472.0	0.627480
$695$	−2898.00	−0.158169
$696$	0	0
$697$	13320.0	0.723861
$698$	7070.00	0.383386
$699$	0	0
$700$	−1232.00	−0.0665217
$701$	33543.0	1.80728	0.903639	−	0.428295i	$-0.140886\pi$
$701$	33543.0	1.80728	0.903639	+	0.428295i	$0.140886\pi$
$702$	0	0
$703$	17716.0	0.950457
$704$	1152.00	0.0616728
$705$	0	0
$706$	−7740.00	−0.412604
$707$	−5670.00	−0.301616
$708$	0	0
$709$	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
$711$	0	0
$712$	−2712.00	−0.142748
$713$	711.000	0.0373452
$714$	0	0
$715$	2106.00	0.110154
$716$	−13380.0	−0.698372
$717$	0	0
$718$	15144.0	0.787144
$719$	−21414.0	−1.11072	−0.555360	−	0.831610i	$-0.687419\pi$
$719$	−21414.0	−1.11072	−0.555360	+	0.831610i	$0.687419\pi$
$720$	0	0
$721$	−10192.0	−0.526449
$722$	10020.0	0.516490
$723$	0	0
$724$	9512.00	0.488274
$725$	−10956.0	−0.561235
$726$	0	0
$727$	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
$729$	0	0
$730$	12438.0	0.630618
$731$	17700.0	0.895565
$732$	0	0
$733$	−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
$735$	0	0
$736$	288.000	0.0144237
$737$	−18756.0	−0.937430
$738$	0	0
$739$	−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
$741$	0	0
$742$	−3318.00	−0.164161
$743$	19632.0	0.969352	0.484676	−	0.874694i	$-0.338938\pi$
$743$	19632.0	0.969352	0.484676	+	0.874694i	$0.338938\pi$
$744$	0	0
$745$	14418.0	0.709040
$746$	18068.0	0.886751
$747$	0	0
$748$	−4320.00	−0.211170
$749$	6468.00	0.315535
$750$	0	0
$751$	−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
$753$	0	0
$754$	−6474.00	−0.312691
$755$	17352.0	0.836429
$756$	0	0
$757$	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
$759$	0	0
$760$	3096.00	0.147768
$761$	−5607.00	−0.267088	−0.133544	−	0.991043i	$-0.542636\pi$
$761$	−5607.00	−0.267088	−0.133544	+	0.991043i	$0.542636\pi$
$762$	0	0
$763$	8078.00	0.383281
$764$	3072.00	0.145473
$765$	0	0
$766$	21672.0	1.02225
$767$	4992.00	0.235007
$768$	0	0
$769$	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
$771$	0	0
$772$	−12808.0	−0.597111
$773$	7518.00	0.349811	0.174905	−	0.984585i	$-0.444038\pi$
$773$	7518.00	0.349811	0.174905	+	0.984585i	$0.444038\pi$
$774$	0	0
$775$	3476.00	0.161112
$776$	−5704.00	−0.263868
$777$	0	0
$778$	−12564.0	−0.578973
$779$	9546.00	0.439051
$780$	0	0
$781$	5184.00	0.237514
$782$	−1080.00	−0.0493871
$783$	0	0
$784$	784.000	0.0357143
$785$	15732.0	0.715286
$786$	0	0
$787$	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
$789$	0	0
$790$	−18018.0	−0.811458
$791$	−12075.0	−0.542778
$792$	0	0
$793$	−6058.00	−0.271281
$794$	−9490.00	−0.424166
$795$	0	0
$796$	−4240.00	−0.188798
$797$	−35718.0	−1.58745	−0.793724	−	0.608278i	$-0.791861\pi$
$797$	−35718.0	−1.58745	−0.793724	+	0.608278i	$0.791861\pi$
$798$	0	0
$799$	24660.0	1.09187
$800$	1408.00	0.0622254
$801$	0	0
$802$	−10656.0	−0.469173
$803$	−12438.0	−0.546610
$804$	0	0
$805$	−567.000	−0.0248250
$806$	2054.00	0.0897631
$807$	0	0
$808$	6480.00	0.282136
$809$	−10401.0	−0.452014	−0.226007	−	0.974126i	$-0.572567\pi$
$809$	−10401.0	−0.452014	−0.226007	+	0.974126i	$0.572567\pi$
$810$	0	0
$811$	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
$813$	0	0
$814$	14832.0	0.638650
$815$	27072.0	1.16355
$816$	0	0
$817$	12685.0	0.543197
$818$	27590.0	1.17929
$819$	0	0
$820$	−7992.00	−0.340357
$821$	7554.00	0.321116	0.160558	−	0.987026i	$-0.448671\pi$
$821$	7554.00	0.321116	0.160558	+	0.987026i	$0.448671\pi$
$822$	0	0
$823$	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
$825$	0	0
$826$	−5376.00	−0.226459
$827$	10728.0	0.451087	0.225544	−	0.974233i	$-0.427584\pi$
$827$	10728.0	0.451087	0.225544	+	0.974233i	$0.427584\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	832.000	0.0346688
$833$	−2940.00	−0.122287
$834$	0	0
$835$	15903.0	0.659097
$836$	−3096.00	−0.128083
$837$	0	0
$838$	8796.00	0.362593
$839$	14856.0	0.611306	0.305653	−	0.952143i	$-0.401125\pi$
$839$	14856.0	0.611306	0.305653	+	0.952143i	$0.401125\pi$
$840$	0	0
$841$	37612.0	1.54217
$842$	968.000	0.0396193
$843$	0	0
$844$	8756.00	0.357102
$845$	1521.00	0.0619219
$846$	0	0
$847$	−7049.00	−0.285958
$848$	3792.00	0.153559
$849$	0	0
$850$	−5280.00	−0.213062
$851$	3708.00	0.149364
$852$	0	0
$853$	−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
$855$	0	0
$856$	−7392.00	−0.295156
$857$	2178.00	0.0868134	0.0434067	−	0.999057i	$-0.486179\pi$
$857$	2178.00	0.0868134	0.0434067	+	0.999057i	$0.486179\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	−6828.00	−0.269794
$863$	−31044.0	−1.22451	−0.612254	−	0.790661i	$-0.709737\pi$
$863$	−31044.0	−1.22451	−0.612254	+	0.790661i	$0.709737\pi$
$864$	0	0
$865$	−864.000	−0.0339617
$866$	−1624.00	−0.0637249
$867$	0	0
$868$	−2212.00	−0.0864979
$869$	18018.0	0.703359
$870$	0	0
$871$	−13546.0	−0.526968
$872$	−9232.00	−0.358526
$873$	0	0
$874$	−774.000	−0.0299553
$875$	−10647.0	−0.411353
$876$	0	0
$877$	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
$879$	0	0
$880$	2592.00	0.0992913
$881$	19710.0	0.753742	0.376871	−	0.926266i	$-0.377000\pi$
$881$	19710.0	0.753742	0.376871	+	0.926266i	$0.377000\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	−13974.0	−0.529871
$887$	−14940.0	−0.565542	−0.282771	−	0.959187i	$-0.591254\pi$
$887$	−14940.0	−0.565542	−0.282771	+	0.959187i	$0.591254\pi$
$888$	0	0
$889$	−8428.00	−0.317960
$890$	−6102.00	−0.229820
$891$	0	0
$892$	11564.0	0.434071
$893$	17673.0	0.662267
$894$	0	0
$895$	−30105.0	−1.12436
$896$	−896.000	−0.0334077
$897$	0	0
$898$	32688.0	1.21471
$899$	−19671.0	−0.729772
$900$	0	0
$901$	−14220.0	−0.525790
$902$	7992.00	0.295016
$903$	0	0
$904$	13800.0	0.507723
$905$	21402.0	0.786107
$906$	0	0
$907$	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
$909$	0	0
$910$	−1638.00	−0.0596694
$911$	−15411.0	−0.560471	−0.280236	−	0.959931i	$-0.590413\pi$
$911$	−15411.0	−0.560471	−0.280236	+	0.959931i	$0.590413\pi$
$912$	0	0
$913$	−702.000	−0.0254467
$914$	34232.0	1.23883
$915$	0	0
$916$	−4600.00	−0.165926
$917$	−5796.00	−0.208725
$918$	0	0
$919$	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
$921$	0	0
$922$	−13764.0	−0.491641
$923$	3744.00	0.133516
$924$	0	0
$925$	18128.0	0.644373
$926$	−7708.00	−0.273543
$927$	0	0
$928$	−7968.00	−0.281856
$929$	−31803.0	−1.12317	−0.561584	−	0.827420i	$-0.689808\pi$
$929$	−31803.0	−1.12317	−0.561584	+	0.827420i	$0.689808\pi$
$930$	0	0
$931$	−2107.00	−0.0741720
$932$	−11388.0	−0.400243
$933$	0	0
$934$	3660.00	0.128221
$935$	−9720.00	−0.339976
$936$	0	0
$937$	−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
$939$	0	0
$940$	−14796.0	−0.513396
$941$	−5709.00	−0.197777	−0.0988885	−	0.995099i	$-0.531529\pi$
$941$	−5709.00	−0.197777	−0.0988885	+	0.995099i	$0.531529\pi$
$942$	0	0
$943$	1998.00	0.0689966
$944$	6144.00	0.211833
$945$	0	0
$946$	10620.0	0.364996
$947$	42474.0	1.45747	0.728733	−	0.684798i	$-0.240110\pi$
$947$	42474.0	1.45747	0.728733	+	0.684798i	$0.240110\pi$
$948$	0	0
$949$	−8983.00	−0.307271
$950$	−3784.00	−0.129231
$951$	0	0
$952$	3360.00	0.114389
$953$	−10953.0	−0.372301	−0.186150	−	0.982521i	$-0.559601\pi$
$953$	−10953.0	−0.372301	−0.186150	+	0.982521i	$0.559601\pi$
$954$	0	0
$955$	6912.00	0.234206
$956$	−8880.00	−0.300418
$957$	0	0
$958$	−25326.0	−0.854119
$959$	18648.0	0.627920
$960$	0	0
$961$	−23550.0	−0.790507
$962$	10712.0	0.359011
$963$	0	0
$964$	−26812.0	−0.895805
$965$	−28818.0	−0.961331
$966$	0	0
$967$	−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
$969$	0	0
$970$	−12834.0	−0.424819
$971$	−42978.0	−1.42042	−0.710211	−	0.703989i	$-0.751400\pi$
$971$	−42978.0	−1.42042	−0.710211	+	0.703989i	$0.751400\pi$
$972$	0	0
$973$	−2254.00	−0.0742651
$974$	11156.0	0.367003
$975$	0	0
$976$	−7456.00	−0.244529
$977$	1554.00	0.0508873	0.0254436	−	0.999676i	$-0.491900\pi$
$977$	1554.00	0.0508873	0.0254436	+	0.999676i	$0.491900\pi$
$978$	0	0
$979$	6102.00	0.199204
$980$	1764.00	0.0574989
$981$	0	0
$982$	−35448.0	−1.15193
$983$	−15597.0	−0.506070	−0.253035	−	0.967457i	$-0.581429\pi$
$983$	−15597.0	−0.506070	−0.253035	+	0.967457i	$0.581429\pi$
$984$	0	0
$985$	6966.00	0.225335
$986$	29880.0	0.965084
$987$	0	0
$988$	−2236.00	−0.0720006
$989$	2655.00	0.0853631
$990$	0	0
$991$	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
$993$	0	0
$994$	−4032.00	−0.128659
$995$	−9540.00	−0.303958
$996$	0	0
$997$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.b.1.1	✓	1	3.2	odd	2
1638.4.a.f.1.1		1	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 $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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