LMFDB - Embedded newform 1638.4.a.g.1.1

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 = [1,-2,0,4,12,0,7,-8,0,-24,50] 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:	$1$
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)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +12.0000 q^{5} +7.00000 q^{7} -8.00000 q^{8} -24.0000 q^{10} +50.0000 q^{11} -13.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +58.0000 q^{17} -40.0000 q^{19} +48.0000 q^{20} -100.000 q^{22} +64.0000 q^{23} +19.0000 q^{25} +26.0000 q^{26} +28.0000 q^{28} +110.000 q^{29} +124.000 q^{31} -32.0000 q^{32} -116.000 q^{34} +84.0000 q^{35} -50.0000 q^{37} +80.0000 q^{38} -96.0000 q^{40} -84.0000 q^{41} -12.0000 q^{43} +200.000 q^{44} -128.000 q^{46} +82.0000 q^{47} +49.0000 q^{49} -38.0000 q^{50} -52.0000 q^{52} +442.000 q^{53} +600.000 q^{55} -56.0000 q^{56} -220.000 q^{58} +618.000 q^{59} -278.000 q^{61} -248.000 q^{62} +64.0000 q^{64} -156.000 q^{65} +20.0000 q^{67} +232.000 q^{68} -168.000 q^{70} +390.000 q^{71} -2.00000 q^{73} +100.000 q^{74} -160.000 q^{76} +350.000 q^{77} -680.000 q^{79} +192.000 q^{80} +168.000 q^{82} -322.000 q^{83} +696.000 q^{85} +24.0000 q^{86} -400.000 q^{88} -968.000 q^{89} -91.0000 q^{91} +256.000 q^{92} -164.000 q^{94} -480.000 q^{95} +1022.00 q^{97} -98.0000 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−24.0000	−0.758947
$11$	50.0000	1.37051	0.685253	−	0.728305i	$-0.259692\pi$
$11$	50.0000	1.37051	0.685253	+	0.728305i	$0.259692\pi$
$12$	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
$17$	58.0000	0.827474	0.413737	−	0.910396i	$-0.364223\pi$
$17$	58.0000	0.827474	0.413737	+	0.910396i	$0.364223\pi$
$18$	0	0
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	48.0000	0.536656
$21$	0	0
$22$	−100.000	−0.969094
$23$	64.0000	0.580214	0.290107	−	0.956994i	$-0.406309\pi$
$23$	64.0000	0.580214	0.290107	+	0.956994i	$0.406309\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	26.0000	0.196116
$27$	0	0
$28$	28.0000	0.188982
$29$	110.000	0.704362	0.352181	−	0.935932i	$-0.385440\pi$
$29$	110.000	0.704362	0.352181	+	0.935932i	$0.385440\pi$
$30$	0	0
$31$	124.000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	124.000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−116.000	−0.585113
$35$	84.0000	0.405674
$36$	0	0
$37$	−50.0000	−0.222161	−0.111080	−	0.993811i	$-0.535431\pi$
$37$	−50.0000	−0.222161	−0.111080	+	0.993811i	$0.535431\pi$
$38$	80.0000	0.341519
$39$	0	0
$40$	−96.0000	−0.379473
$41$	−84.0000	−0.319966	−0.159983	−	0.987120i	$-0.551144\pi$
$41$	−84.0000	−0.319966	−0.159983	+	0.987120i	$0.551144\pi$
$42$	0	0
$43$	−12.0000	−0.0425577	−0.0212789	−	0.999774i	$-0.506774\pi$
$43$	−12.0000	−0.0425577	−0.0212789	+	0.999774i	$0.506774\pi$
$44$	200.000	0.685253
$45$	0	0
$46$	−128.000	−0.410273
$47$	82.0000	0.254488	0.127244	−	0.991871i	$-0.459387\pi$
$47$	82.0000	0.254488	0.127244	+	0.991871i	$0.459387\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−38.0000	−0.107480
$51$	0	0
$52$	−52.0000	−0.138675
$53$	442.000	1.14554	0.572768	−	0.819718i	$-0.305870\pi$
$53$	442.000	1.14554	0.572768	+	0.819718i	$0.305870\pi$
$54$	0	0
$55$	600.000	1.47098
$56$	−56.0000	−0.133631
$57$	0	0
$58$	−220.000	−0.498059
$59$	618.000	1.36367	0.681837	−	0.731504i	$-0.261181\pi$
$59$	618.000	1.36367	0.681837	+	0.731504i	$0.261181\pi$
$60$	0	0
$61$	−278.000	−0.583512	−0.291756	−	0.956493i	$-0.594240\pi$
$61$	−278.000	−0.583512	−0.291756	+	0.956493i	$0.594240\pi$
$62$	−248.000	−0.508001
$63$	0	0
$64$	64.0000	0.125000
$65$	−156.000	−0.297683
$66$	0	0
$67$	20.0000	0.0364685	0.0182342	−	0.999834i	$-0.494196\pi$
$67$	20.0000	0.0364685	0.0182342	+	0.999834i	$0.494196\pi$
$68$	232.000	0.413737
$69$	0	0
$70$	−168.000	−0.286855
$71$	390.000	0.651894	0.325947	−	0.945388i	$-0.394317\pi$
$71$	390.000	0.651894	0.325947	+	0.945388i	$0.394317\pi$
$72$	0	0
$73$	−2.00000	−0.00320661	−0.00160330	−	0.999999i	$-0.500510\pi$
$73$	−2.00000	−0.00320661	−0.00160330	+	0.999999i	$0.500510\pi$
$74$	100.000	0.157091
$75$	0	0
$76$	−160.000	−0.241490
$77$	350.000	0.518003
$78$	0	0
$79$	−680.000	−0.968430	−0.484215	−	0.874949i	$-0.660895\pi$
$79$	−680.000	−0.968430	−0.484215	+	0.874949i	$0.660895\pi$
$80$	192.000	0.268328
$81$	0	0
$82$	168.000	0.226250
$83$	−322.000	−0.425832	−0.212916	−	0.977070i	$-0.568296\pi$
$83$	−322.000	−0.425832	−0.212916	+	0.977070i	$0.568296\pi$
$84$	0	0
$85$	696.000	0.888139
$86$	24.0000	0.0300929
$87$	0	0
$88$	−400.000	−0.484547
$89$	−968.000	−1.15290	−0.576448	−	0.817134i	$-0.695562\pi$
$89$	−968.000	−1.15290	−0.576448	+	0.817134i	$0.695562\pi$
$90$	0	0
$91$	−91.0000	−0.104828
$92$	256.000	0.290107
$93$	0	0
$94$	−164.000	−0.179950
$95$	−480.000	−0.518389
$96$	0	0
$97$	1022.00	1.06978	0.534889	−	0.844923i	$-0.320354\pi$
$97$	1022.00	1.06978	0.534889	+	0.844923i	$0.320354\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	76.0000	0.0760000
$101$	−54.0000	−0.0532000	−0.0266000	−	0.999646i	$-0.508468\pi$
$101$	−54.0000	−0.0532000	−0.0266000	+	0.999646i	$0.508468\pi$
$102$	0	0
$103$	1416.00	1.35459	0.677294	−	0.735712i	$-0.263152\pi$
$103$	1416.00	1.35459	0.677294	+	0.735712i	$0.263152\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	−884.000	−0.810016
$107$	−1252.00	−1.13117	−0.565586	−	0.824689i	$-0.691350\pi$
$107$	−1252.00	−1.13117	−0.565586	+	0.824689i	$0.691350\pi$
$108$	0	0
$109$	930.000	0.817228	0.408614	−	0.912707i	$-0.366012\pi$
$109$	930.000	0.817228	0.408614	+	0.912707i	$0.366012\pi$
$110$	−1200.00	−1.04014
$111$	0	0
$112$	112.000	0.0944911
$113$	1034.00	0.860801	0.430401	−	0.902638i	$-0.358372\pi$
$113$	1034.00	0.860801	0.430401	+	0.902638i	$0.358372\pi$
$114$	0	0
$115$	768.000	0.622751
$116$	440.000	0.352181
$117$	0	0
$118$	−1236.00	−0.964263
$119$	406.000	0.312756
$120$	0	0
$121$	1169.00	0.878287
$122$	556.000	0.412606
$123$	0	0
$124$	496.000	0.359211
$125$	−1272.00	−0.910169
$126$	0	0
$127$	−2280.00	−1.59305	−0.796525	−	0.604606i	$-0.793331\pi$
$127$	−2280.00	−1.59305	−0.796525	+	0.604606i	$0.793331\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	312.000	0.210494
$131$	1552.00	1.03511	0.517553	−	0.855651i	$-0.326843\pi$
$131$	1552.00	1.03511	0.517553	+	0.855651i	$0.326843\pi$
$132$	0	0
$133$	−280.000	−0.182549
$134$	−40.0000	−0.0257871
$135$	0	0
$136$	−464.000	−0.292556
$137$	−48.0000	−0.0299337	−0.0149668	−	0.999888i	$-0.504764\pi$
$137$	−48.0000	−0.0299337	−0.0149668	+	0.999888i	$0.504764\pi$
$138$	0	0
$139$	28.0000	0.0170858	0.00854291	−	0.999964i	$-0.497281\pi$
$139$	28.0000	0.0170858	0.00854291	+	0.999964i	$0.497281\pi$
$140$	336.000	0.202837
$141$	0	0
$142$	−780.000	−0.460959
$143$	−650.000	−0.380110
$144$	0	0
$145$	1320.00	0.756000
$146$	4.00000	0.00226741
$147$	0	0
$148$	−200.000	−0.111080
$149$	−2596.00	−1.42733	−0.713666	−	0.700486i	$-0.752967\pi$
$149$	−2596.00	−1.42733	−0.713666	+	0.700486i	$0.752967\pi$
$150$	0	0
$151$	−2052.00	−1.10589	−0.552945	−	0.833218i	$-0.686496\pi$
$151$	−2052.00	−1.10589	−0.552945	+	0.833218i	$0.686496\pi$
$152$	320.000	0.170759
$153$	0	0
$154$	−700.000	−0.366283
$155$	1488.00	0.771091
$156$	0	0
$157$	−2522.00	−1.28202	−0.641011	−	0.767532i	$-0.721485\pi$
$157$	−2522.00	−1.28202	−0.641011	+	0.767532i	$0.721485\pi$
$158$	1360.00	0.684783
$159$	0	0
$160$	−384.000	−0.189737
$161$	448.000	0.219300
$162$	0	0
$163$	−848.000	−0.407488	−0.203744	−	0.979024i	$-0.565311\pi$
$163$	−848.000	−0.407488	−0.203744	+	0.979024i	$0.565311\pi$
$164$	−336.000	−0.159983
$165$	0	0
$166$	644.000	0.301109
$167$	−854.000	−0.395716	−0.197858	−	0.980231i	$-0.563398\pi$
$167$	−854.000	−0.395716	−0.197858	+	0.980231i	$0.563398\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−1392.00	−0.628009
$171$	0	0
$172$	−48.0000	−0.0212789
$173$	−2046.00	−0.899159	−0.449579	−	0.893240i	$-0.648426\pi$
$173$	−2046.00	−0.899159	−0.449579	+	0.893240i	$0.648426\pi$
$174$	0	0
$175$	133.000	0.0574506
$176$	800.000	0.342627
$177$	0	0
$178$	1936.00	0.815221
$179$	1688.00	0.704844	0.352422	−	0.935841i	$-0.385358\pi$
$179$	1688.00	0.704844	0.352422	+	0.935841i	$0.385358\pi$
$180$	0	0
$181$	−266.000	−0.109235	−0.0546177	−	0.998507i	$-0.517394\pi$
$181$	−266.000	−0.109235	−0.0546177	+	0.998507i	$0.517394\pi$
$182$	182.000	0.0741249
$183$	0	0
$184$	−512.000	−0.205137
$185$	−600.000	−0.238448
$186$	0	0
$187$	2900.00	1.13406
$188$	328.000	0.127244
$189$	0	0
$190$	960.000	0.366556
$191$	−2876.00	−1.08953	−0.544765	−	0.838589i	$-0.683381\pi$
$191$	−2876.00	−1.08953	−0.544765	+	0.838589i	$0.683381\pi$
$192$	0	0
$193$	3058.00	1.14052	0.570258	−	0.821466i	$-0.306843\pi$
$193$	3058.00	1.14052	0.570258	+	0.821466i	$0.306843\pi$
$194$	−2044.00	−0.756447
$195$	0	0
$196$	196.000	0.0714286
$197$	2952.00	1.06762	0.533810	−	0.845604i	$-0.320760\pi$
$197$	2952.00	1.06762	0.533810	+	0.845604i	$0.320760\pi$
$198$	0	0
$199$	3064.00	1.09146	0.545732	−	0.837960i	$-0.316252\pi$
$199$	3064.00	1.09146	0.545732	+	0.837960i	$0.316252\pi$
$200$	−152.000	−0.0537401
$201$	0	0
$202$	108.000	0.0376181
$203$	770.000	0.266224
$204$	0	0
$205$	−1008.00	−0.343423
$206$	−2832.00	−0.957839
$207$	0	0
$208$	−208.000	−0.0693375
$209$	−2000.00	−0.661928
$210$	0	0
$211$	604.000	0.197067	0.0985334	−	0.995134i	$-0.468585\pi$
$211$	604.000	0.197067	0.0985334	+	0.995134i	$0.468585\pi$
$212$	1768.00	0.572768
$213$	0	0
$214$	2504.00	0.799859
$215$	−144.000	−0.0456778
$216$	0	0
$217$	868.000	0.271538
$218$	−1860.00	−0.577867
$219$	0	0
$220$	2400.00	0.735491
$221$	−754.000	−0.229500
$222$	0	0
$223$	5628.00	1.69004	0.845020	−	0.534735i	$-0.179589\pi$
$223$	5628.00	1.69004	0.845020	+	0.534735i	$0.179589\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−2068.00	−0.608678
$227$	1122.00	0.328061	0.164030	−	0.986455i	$-0.447551\pi$
$227$	1122.00	0.328061	0.164030	+	0.986455i	$0.447551\pi$
$228$	0	0
$229$	4426.00	1.27720	0.638599	−	0.769540i	$-0.279514\pi$
$229$	4426.00	1.27720	0.638599	+	0.769540i	$0.279514\pi$
$230$	−1536.00	−0.440351
$231$	0	0
$232$	−880.000	−0.249029
$233$	−2190.00	−0.615758	−0.307879	−	0.951425i	$-0.599619\pi$
$233$	−2190.00	−0.615758	−0.307879	+	0.951425i	$0.599619\pi$
$234$	0	0
$235$	984.000	0.273145
$236$	2472.00	0.681837
$237$	0	0
$238$	−812.000	−0.221152
$239$	1230.00	0.332896	0.166448	−	0.986050i	$-0.446770\pi$
$239$	1230.00	0.332896	0.166448	+	0.986050i	$0.446770\pi$
$240$	0	0
$241$	−786.000	−0.210086	−0.105043	−	0.994468i	$-0.533498\pi$
$241$	−786.000	−0.210086	−0.105043	+	0.994468i	$0.533498\pi$
$242$	−2338.00	−0.621043
$243$	0	0
$244$	−1112.00	−0.291756
$245$	588.000	0.153330
$246$	0	0
$247$	520.000	0.133955
$248$	−992.000	−0.254000
$249$	0	0
$250$	2544.00	0.643587
$251$	280.000	0.0704121	0.0352061	−	0.999380i	$-0.488791\pi$
$251$	280.000	0.0704121	0.0352061	+	0.999380i	$0.488791\pi$
$252$	0	0
$253$	3200.00	0.795187
$254$	4560.00	1.12646
$255$	0	0
$256$	256.000	0.0625000
$257$	3806.00	0.923781	0.461891	−	0.886937i	$-0.347171\pi$
$257$	3806.00	0.923781	0.461891	+	0.886937i	$0.347171\pi$
$258$	0	0
$259$	−350.000	−0.0839689
$260$	−624.000	−0.148842
$261$	0	0
$262$	−3104.00	−0.731930
$263$	3816.00	0.894694	0.447347	−	0.894360i	$-0.352369\pi$
$263$	3816.00	0.894694	0.447347	+	0.894360i	$0.352369\pi$
$264$	0	0
$265$	5304.00	1.22952
$266$	560.000	0.129082
$267$	0	0
$268$	80.0000	0.0182342
$269$	3026.00	0.685868	0.342934	−	0.939360i	$-0.388579\pi$
$269$	3026.00	0.685868	0.342934	+	0.939360i	$0.388579\pi$
$270$	0	0
$271$	2480.00	0.555901	0.277951	−	0.960595i	$-0.410345\pi$
$271$	2480.00	0.555901	0.277951	+	0.960595i	$0.410345\pi$
$272$	928.000	0.206869
$273$	0	0
$274$	96.0000	0.0211663
$275$	950.000	0.208317
$276$	0	0
$277$	3590.00	0.778708	0.389354	−	0.921088i	$-0.372698\pi$
$277$	3590.00	0.778708	0.389354	+	0.921088i	$0.372698\pi$
$278$	−56.0000	−0.0120815
$279$	0	0
$280$	−672.000	−0.143427
$281$	−4832.00	−1.02581	−0.512906	−	0.858445i	$-0.671431\pi$
$281$	−4832.00	−1.02581	−0.512906	+	0.858445i	$0.671431\pi$
$282$	0	0
$283$	−5700.00	−1.19728	−0.598639	−	0.801019i	$-0.704292\pi$
$283$	−5700.00	−1.19728	−0.598639	+	0.801019i	$0.704292\pi$
$284$	1560.00	0.325947
$285$	0	0
$286$	1300.00	0.268778
$287$	−588.000	−0.120936
$288$	0	0
$289$	−1549.00	−0.315286
$290$	−2640.00	−0.534573
$291$	0	0
$292$	−8.00000	−0.00160330
$293$	6384.00	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$293$	6384.00	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$294$	0	0
$295$	7416.00	1.46365
$296$	400.000	0.0785457
$297$	0	0
$298$	5192.00	1.00928
$299$	−832.000	−0.160922
$300$	0	0
$301$	−84.0000	−0.0160853
$302$	4104.00	0.781982
$303$	0	0
$304$	−640.000	−0.120745
$305$	−3336.00	−0.626291
$306$	0	0
$307$	3836.00	0.713134	0.356567	−	0.934270i	$-0.383947\pi$
$307$	3836.00	0.713134	0.356567	+	0.934270i	$0.383947\pi$
$308$	1400.00	0.259001
$309$	0	0
$310$	−2976.00	−0.545243
$311$	−6116.00	−1.11513	−0.557567	−	0.830132i	$-0.688265\pi$
$311$	−6116.00	−1.11513	−0.557567	+	0.830132i	$0.688265\pi$
$312$	0	0
$313$	−642.000	−0.115936	−0.0579680	−	0.998318i	$-0.518462\pi$
$313$	−642.000	−0.115936	−0.0579680	+	0.998318i	$0.518462\pi$
$314$	5044.00	0.906527
$315$	0	0
$316$	−2720.00	−0.484215
$317$	9136.00	1.61870	0.809352	−	0.587325i	$-0.199819\pi$
$317$	9136.00	1.61870	0.809352	+	0.587325i	$0.199819\pi$
$318$	0	0
$319$	5500.00	0.965332
$320$	768.000	0.134164
$321$	0	0
$322$	−896.000	−0.155069
$323$	−2320.00	−0.399654
$324$	0	0
$325$	−247.000	−0.0421572
$326$	1696.00	0.288137
$327$	0	0
$328$	672.000	0.113125
$329$	574.000	0.0961874
$330$	0	0
$331$	−3060.00	−0.508135	−0.254068	−	0.967186i	$-0.581769\pi$
$331$	−3060.00	−0.508135	−0.254068	+	0.967186i	$0.581769\pi$
$332$	−1288.00	−0.212916
$333$	0	0
$334$	1708.00	0.279813
$335$	240.000	0.0391421
$336$	0	0
$337$	4734.00	0.765215	0.382607	−	0.923911i	$-0.375026\pi$
$337$	4734.00	0.765215	0.382607	+	0.923911i	$0.375026\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	2784.00	0.444069
$341$	6200.00	0.984601
$342$	0	0
$343$	343.000	0.0539949
$344$	96.0000	0.0150464
$345$	0	0
$346$	4092.00	0.635801
$347$	11096.0	1.71661	0.858306	−	0.513138i	$-0.171517\pi$
$347$	11096.0	1.71661	0.858306	+	0.513138i	$0.171517\pi$
$348$	0	0
$349$	2366.00	0.362891	0.181446	−	0.983401i	$-0.441922\pi$
$349$	2366.00	0.362891	0.181446	+	0.983401i	$0.441922\pi$
$350$	−266.000	−0.0406237
$351$	0	0
$352$	−1600.00	−0.242274
$353$	11076.0	1.67002	0.835008	−	0.550237i	$-0.185463\pi$
$353$	11076.0	1.67002	0.835008	+	0.550237i	$0.185463\pi$
$354$	0	0
$355$	4680.00	0.699686
$356$	−3872.00	−0.576448
$357$	0	0
$358$	−3376.00	−0.498400
$359$	6406.00	0.941771	0.470885	−	0.882194i	$-0.343935\pi$
$359$	6406.00	0.941771	0.470885	+	0.882194i	$0.343935\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	532.000	0.0772412
$363$	0	0
$364$	−364.000	−0.0524142
$365$	−24.0000	−0.00344169
$366$	0	0
$367$	2000.00	0.284466	0.142233	−	0.989833i	$-0.454572\pi$
$367$	2000.00	0.284466	0.142233	+	0.989833i	$0.454572\pi$
$368$	1024.00	0.145054
$369$	0	0
$370$	1200.00	0.168608
$371$	3094.00	0.432972
$372$	0	0
$373$	−3662.00	−0.508341	−0.254170	−	0.967159i	$-0.581802\pi$
$373$	−3662.00	−0.508341	−0.254170	+	0.967159i	$0.581802\pi$
$374$	−5800.00	−0.801901
$375$	0	0
$376$	−656.000	−0.0899750
$377$	−1430.00	−0.195355
$378$	0	0
$379$	6008.00	0.814275	0.407138	−	0.913367i	$-0.366527\pi$
$379$	6008.00	0.814275	0.407138	+	0.913367i	$0.366527\pi$
$380$	−1920.00	−0.259195
$381$	0	0
$382$	5752.00	0.770413
$383$	−11202.0	−1.49451	−0.747253	−	0.664540i	$-0.768627\pi$
$383$	−11202.0	−1.49451	−0.747253	+	0.664540i	$0.768627\pi$
$384$	0	0
$385$	4200.00	0.555979
$386$	−6116.00	−0.806467
$387$	0	0
$388$	4088.00	0.534889
$389$	10858.0	1.41522	0.707612	−	0.706601i	$-0.249772\pi$
$389$	10858.0	1.41522	0.707612	+	0.706601i	$0.249772\pi$
$390$	0	0
$391$	3712.00	0.480112
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−5904.00	−0.754922
$395$	−8160.00	−1.03943
$396$	0	0
$397$	−5850.00	−0.739554	−0.369777	−	0.929120i	$-0.620566\pi$
$397$	−5850.00	−0.739554	−0.369777	+	0.929120i	$0.620566\pi$
$398$	−6128.00	−0.771781
$399$	0	0
$400$	304.000	0.0380000
$401$	11404.0	1.42017	0.710086	−	0.704115i	$-0.248656\pi$
$401$	11404.0	1.42017	0.710086	+	0.704115i	$0.248656\pi$
$402$	0	0
$403$	−1612.00	−0.199254
$404$	−216.000	−0.0266000
$405$	0	0
$406$	−1540.00	−0.188249
$407$	−2500.00	−0.304473
$408$	0	0
$409$	−10194.0	−1.23242	−0.616211	−	0.787581i	$-0.711333\pi$
$409$	−10194.0	−1.23242	−0.616211	+	0.787581i	$0.711333\pi$
$410$	2016.00	0.242837
$411$	0	0
$412$	5664.00	0.677294
$413$	4326.00	0.515420
$414$	0	0
$415$	−3864.00	−0.457051
$416$	416.000	0.0490290
$417$	0	0
$418$	4000.00	0.468054
$419$	−3192.00	−0.372170	−0.186085	−	0.982534i	$-0.559580\pi$
$419$	−3192.00	−0.372170	−0.186085	+	0.982534i	$0.559580\pi$
$420$	0	0
$421$	10474.0	1.21252	0.606261	−	0.795266i	$-0.292669\pi$
$421$	10474.0	1.21252	0.606261	+	0.795266i	$0.292669\pi$
$422$	−1208.00	−0.139347
$423$	0	0
$424$	−3536.00	−0.405008
$425$	1102.00	0.125776
$426$	0	0
$427$	−1946.00	−0.220547
$428$	−5008.00	−0.565586
$429$	0	0
$430$	288.000	0.0322991
$431$	−7902.00	−0.883123	−0.441561	−	0.897231i	$-0.645575\pi$
$431$	−7902.00	−0.883123	−0.441561	+	0.897231i	$0.645575\pi$
$432$	0	0
$433$	−16954.0	−1.88166	−0.940828	−	0.338884i	$-0.889951\pi$
$433$	−16954.0	−1.88166	−0.940828	+	0.338884i	$0.889951\pi$
$434$	−1736.00	−0.192006
$435$	0	0
$436$	3720.00	0.408614
$437$	−2560.00	−0.280232
$438$	0	0
$439$	13960.0	1.51771	0.758855	−	0.651260i	$-0.225759\pi$
$439$	13960.0	1.51771	0.758855	+	0.651260i	$0.225759\pi$
$440$	−4800.00	−0.520071
$441$	0	0
$442$	1508.00	0.162281
$443$	1884.00	0.202058	0.101029	−	0.994883i	$-0.467787\pi$
$443$	1884.00	0.202058	0.101029	+	0.994883i	$0.467787\pi$
$444$	0	0
$445$	−11616.0	−1.23742
$446$	−11256.0	−1.19504
$447$	0	0
$448$	448.000	0.0472456
$449$	11436.0	1.20200	0.601000	−	0.799249i	$-0.294769\pi$
$449$	11436.0	1.20200	0.601000	+	0.799249i	$0.294769\pi$
$450$	0	0
$451$	−4200.00	−0.438515
$452$	4136.00	0.430401
$453$	0	0
$454$	−2244.00	−0.231974
$455$	−1092.00	−0.112514
$456$	0	0
$457$	11486.0	1.17569	0.587847	−	0.808972i	$-0.299976\pi$
$457$	11486.0	1.17569	0.587847	+	0.808972i	$0.299976\pi$
$458$	−8852.00	−0.903115
$459$	0	0
$460$	3072.00	0.311376
$461$	−11172.0	−1.12870	−0.564351	−	0.825535i	$-0.690874\pi$
$461$	−11172.0	−1.12870	−0.564351	+	0.825535i	$0.690874\pi$
$462$	0	0
$463$	−11688.0	−1.17319	−0.586595	−	0.809880i	$-0.699532\pi$
$463$	−11688.0	−1.17319	−0.586595	+	0.809880i	$0.699532\pi$
$464$	1760.00	0.176090
$465$	0	0
$466$	4380.00	0.435407
$467$	2448.00	0.242569	0.121285	−	0.992618i	$-0.461299\pi$
$467$	2448.00	0.242569	0.121285	+	0.992618i	$0.461299\pi$
$468$	0	0
$469$	140.000	0.0137838
$470$	−1968.00	−0.193143
$471$	0	0
$472$	−4944.00	−0.482131
$473$	−600.000	−0.0583256
$474$	0	0
$475$	−760.000	−0.0734130
$476$	1624.00	0.156378
$477$	0	0
$478$	−2460.00	−0.235393
$479$	−698.000	−0.0665813	−0.0332906	−	0.999446i	$-0.510599\pi$
$479$	−698.000	−0.0665813	−0.0332906	+	0.999446i	$0.510599\pi$
$480$	0	0
$481$	650.000	0.0616163
$482$	1572.00	0.148553
$483$	0	0
$484$	4676.00	0.439144
$485$	12264.0	1.14821
$486$	0	0
$487$	−8156.00	−0.758899	−0.379449	−	0.925213i	$-0.623887\pi$
$487$	−8156.00	−0.758899	−0.379449	+	0.925213i	$0.623887\pi$
$488$	2224.00	0.206303
$489$	0	0
$490$	−1176.00	−0.108421
$491$	10484.0	0.963618	0.481809	−	0.876276i	$-0.339980\pi$
$491$	10484.0	0.963618	0.481809	+	0.876276i	$0.339980\pi$
$492$	0	0
$493$	6380.00	0.582841
$494$	−1040.00	−0.0947203
$495$	0	0
$496$	1984.00	0.179605
$497$	2730.00	0.246393
$498$	0	0
$499$	−6980.00	−0.626188	−0.313094	−	0.949722i	$-0.601365\pi$
$499$	−6980.00	−0.626188	−0.313094	+	0.949722i	$0.601365\pi$
$500$	−5088.00	−0.455085
$501$	0	0
$502$	−560.000	−0.0497889
$503$	−5180.00	−0.459175	−0.229587	−	0.973288i	$-0.573738\pi$
$503$	−5180.00	−0.459175	−0.229587	+	0.973288i	$0.573738\pi$
$504$	0	0
$505$	−648.000	−0.0571002
$506$	−6400.00	−0.562282
$507$	0	0
$508$	−9120.00	−0.796525
$509$	−4860.00	−0.423214	−0.211607	−	0.977355i	$-0.567870\pi$
$509$	−4860.00	−0.423214	−0.211607	+	0.977355i	$0.567870\pi$
$510$	0	0
$511$	−14.0000	−0.00121198
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−7612.00	−0.653212
$515$	16992.0	1.45390
$516$	0	0
$517$	4100.00	0.348777
$518$	700.000	0.0593750
$519$	0	0
$520$	1248.00	0.105247
$521$	17390.0	1.46232	0.731161	−	0.682205i	$-0.238979\pi$
$521$	17390.0	1.46232	0.731161	+	0.682205i	$0.238979\pi$
$522$	0	0
$523$	−12556.0	−1.04978	−0.524891	−	0.851170i	$-0.675894\pi$
$523$	−12556.0	−1.04978	−0.524891	+	0.851170i	$0.675894\pi$
$524$	6208.00	0.517553
$525$	0	0
$526$	−7632.00	−0.632645
$527$	7192.00	0.594475
$528$	0	0
$529$	−8071.00	−0.663352
$530$	−10608.0	−0.869400
$531$	0	0
$532$	−1120.00	−0.0912747
$533$	1092.00	0.0887425
$534$	0	0
$535$	−15024.0	−1.21410
$536$	−160.000	−0.0128936
$537$	0	0
$538$	−6052.00	−0.484982
$539$	2450.00	0.195787
$540$	0	0
$541$	−19398.0	−1.54156	−0.770781	−	0.637100i	$-0.780134\pi$
$541$	−19398.0	−1.54156	−0.770781	+	0.637100i	$0.780134\pi$
$542$	−4960.00	−0.393082
$543$	0	0
$544$	−1856.00	−0.146278
$545$	11160.0	0.877141
$546$	0	0
$547$	−9308.00	−0.727571	−0.363786	−	0.931483i	$-0.618516\pi$
$547$	−9308.00	−0.727571	−0.363786	+	0.931483i	$0.618516\pi$
$548$	−192.000	−0.0149668
$549$	0	0
$550$	−1900.00	−0.147302
$551$	−4400.00	−0.340193
$552$	0	0
$553$	−4760.00	−0.366032
$554$	−7180.00	−0.550630
$555$	0	0
$556$	112.000	0.00854291
$557$	19524.0	1.48520	0.742602	−	0.669733i	$-0.233592\pi$
$557$	19524.0	1.48520	0.742602	+	0.669733i	$0.233592\pi$
$558$	0	0
$559$	156.000	0.0118034
$560$	1344.00	0.101419
$561$	0	0
$562$	9664.00	0.725358
$563$	772.000	0.0577903	0.0288951	−	0.999582i	$-0.490801\pi$
$563$	772.000	0.0577903	0.0288951	+	0.999582i	$0.490801\pi$
$564$	0	0
$565$	12408.0	0.923909
$566$	11400.0	0.846604
$567$	0	0
$568$	−3120.00	−0.230479
$569$	8226.00	0.606067	0.303033	−	0.952980i	$-0.402001\pi$
$569$	8226.00	0.606067	0.303033	+	0.952980i	$0.402001\pi$
$570$	0	0
$571$	−11852.0	−0.868635	−0.434318	−	0.900760i	$-0.643010\pi$
$571$	−11852.0	−0.868635	−0.434318	+	0.900760i	$0.643010\pi$
$572$	−2600.00	−0.190055
$573$	0	0
$574$	1176.00	0.0855144
$575$	1216.00	0.0881925
$576$	0	0
$577$	11282.0	0.813996	0.406998	−	0.913429i	$-0.366576\pi$
$577$	11282.0	0.813996	0.406998	+	0.913429i	$0.366576\pi$
$578$	3098.00	0.222941
$579$	0	0
$580$	5280.00	0.378000
$581$	−2254.00	−0.160950
$582$	0	0
$583$	22100.0	1.56996
$584$	16.0000	0.00113371
$585$	0	0
$586$	−12768.0	−0.900070
$587$	−14994.0	−1.05429	−0.527145	−	0.849775i	$-0.676738\pi$
$587$	−14994.0	−1.05429	−0.527145	+	0.849775i	$0.676738\pi$
$588$	0	0
$589$	−4960.00	−0.346983
$590$	−14832.0	−1.03496
$591$	0	0
$592$	−800.000	−0.0555402
$593$	21544.0	1.49192	0.745958	−	0.665993i	$-0.231992\pi$
$593$	21544.0	1.49192	0.745958	+	0.665993i	$0.231992\pi$
$594$	0	0
$595$	4872.00	0.335685
$596$	−10384.0	−0.713666
$597$	0	0
$598$	1664.00	0.113789
$599$	4824.00	0.329054	0.164527	−	0.986373i	$-0.447390\pi$
$599$	4824.00	0.329054	0.164527	+	0.986373i	$0.447390\pi$
$600$	0	0
$601$	4718.00	0.320218	0.160109	−	0.987099i	$-0.448815\pi$
$601$	4718.00	0.320218	0.160109	+	0.987099i	$0.448815\pi$
$602$	168.000	0.0113740
$603$	0	0
$604$	−8208.00	−0.552945
$605$	14028.0	0.942677
$606$	0	0
$607$	4720.00	0.315616	0.157808	−	0.987470i	$-0.449557\pi$
$607$	4720.00	0.315616	0.157808	+	0.987470i	$0.449557\pi$
$608$	1280.00	0.0853797
$609$	0	0
$610$	6672.00	0.442855
$611$	−1066.00	−0.0705822
$612$	0	0
$613$	16078.0	1.05935	0.529677	−	0.848199i	$-0.322313\pi$
$613$	16078.0	1.05935	0.529677	+	0.848199i	$0.322313\pi$
$614$	−7672.00	−0.504262
$615$	0	0
$616$	−2800.00	−0.183142
$617$	−12840.0	−0.837794	−0.418897	−	0.908034i	$-0.637583\pi$
$617$	−12840.0	−0.837794	−0.418897	+	0.908034i	$0.637583\pi$
$618$	0	0
$619$	−22332.0	−1.45008	−0.725039	−	0.688707i	$-0.758179\pi$
$619$	−22332.0	−1.45008	−0.725039	+	0.688707i	$0.758179\pi$
$620$	5952.00	0.385545
$621$	0	0
$622$	12232.0	0.788519
$623$	−6776.00	−0.435754
$624$	0	0
$625$	−17639.0	−1.12890
$626$	1284.00	0.0819792
$627$	0	0
$628$	−10088.0	−0.641011
$629$	−2900.00	−0.183832
$630$	0	0
$631$	−23756.0	−1.49875	−0.749375	−	0.662146i	$-0.769646\pi$
$631$	−23756.0	−1.49875	−0.749375	+	0.662146i	$0.769646\pi$
$632$	5440.00	0.342392
$633$	0	0
$634$	−18272.0	−1.14460
$635$	−27360.0	−1.70984
$636$	0	0
$637$	−637.000	−0.0396214
$638$	−11000.0	−0.682593
$639$	0	0
$640$	−1536.00	−0.0948683
$641$	−10170.0	−0.626663	−0.313331	−	0.949644i	$-0.601445\pi$
$641$	−10170.0	−0.626663	−0.313331	+	0.949644i	$0.601445\pi$
$642$	0	0
$643$	−19404.0	−1.19008	−0.595038	−	0.803697i	$-0.702863\pi$
$643$	−19404.0	−1.19008	−0.595038	+	0.803697i	$0.702863\pi$
$644$	1792.00	0.109650
$645$	0	0
$646$	4640.00	0.282598
$647$	−13704.0	−0.832705	−0.416352	−	0.909203i	$-0.636692\pi$
$647$	−13704.0	−0.832705	−0.416352	+	0.909203i	$0.636692\pi$
$648$	0	0
$649$	30900.0	1.86892
$650$	494.000	0.0298097
$651$	0	0
$652$	−3392.00	−0.203744
$653$	19230.0	1.15242	0.576208	−	0.817303i	$-0.304532\pi$
$653$	19230.0	1.15242	0.576208	+	0.817303i	$0.304532\pi$
$654$	0	0
$655$	18624.0	1.11099
$656$	−1344.00	−0.0799914
$657$	0	0
$658$	−1148.00	−0.0680147
$659$	−18608.0	−1.09995	−0.549973	−	0.835182i	$-0.685362\pi$
$659$	−18608.0	−1.09995	−0.549973	+	0.835182i	$0.685362\pi$
$660$	0	0
$661$	−7394.00	−0.435088	−0.217544	−	0.976050i	$-0.569805\pi$
$661$	−7394.00	−0.435088	−0.217544	+	0.976050i	$0.569805\pi$
$662$	6120.00	0.359306
$663$	0	0
$664$	2576.00	0.150554
$665$	−3360.00	−0.195933
$666$	0	0
$667$	7040.00	0.408680
$668$	−3416.00	−0.197858
$669$	0	0
$670$	−480.000	−0.0276776
$671$	−13900.0	−0.799707
$672$	0	0
$673$	450.000	0.0257745	0.0128872	−	0.999917i	$-0.495898\pi$
$673$	450.000	0.0257745	0.0128872	+	0.999917i	$0.495898\pi$
$674$	−9468.00	−0.541089
$675$	0	0
$676$	676.000	0.0384615
$677$	−6666.00	−0.378427	−0.189214	−	0.981936i	$-0.560594\pi$
$677$	−6666.00	−0.378427	−0.189214	+	0.981936i	$0.560594\pi$
$678$	0	0
$679$	7154.00	0.404338
$680$	−5568.00	−0.314004
$681$	0	0
$682$	−12400.0	−0.696218
$683$	11082.0	0.620851	0.310425	−	0.950598i	$-0.399529\pi$
$683$	11082.0	0.620851	0.310425	+	0.950598i	$0.399529\pi$
$684$	0	0
$685$	−576.000	−0.0321282
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−192.000	−0.0106394
$689$	−5746.00	−0.317714
$690$	0	0
$691$	19308.0	1.06297	0.531484	−	0.847068i	$-0.321635\pi$
$691$	19308.0	1.06297	0.531484	+	0.847068i	$0.321635\pi$
$692$	−8184.00	−0.449579
$693$	0	0
$694$	−22192.0	−1.21383
$695$	336.000	0.0183384
$696$	0	0
$697$	−4872.00	−0.264763
$698$	−4732.00	−0.256603
$699$	0	0
$700$	532.000	0.0287253
$701$	−15318.0	−0.825325	−0.412663	−	0.910884i	$-0.635401\pi$
$701$	−15318.0	−0.825325	−0.412663	+	0.910884i	$0.635401\pi$
$702$	0	0
$703$	2000.00	0.107299
$704$	3200.00	0.171313
$705$	0	0
$706$	−22152.0	−1.18088
$707$	−378.000	−0.0201077
$708$	0	0
$709$	−16486.0	−0.873265	−0.436632	−	0.899640i	$-0.643829\pi$
$709$	−16486.0	−0.873265	−0.436632	+	0.899640i	$0.643829\pi$
$710$	−9360.00	−0.494753
$711$	0	0
$712$	7744.00	0.407610
$713$	7936.00	0.416838
$714$	0	0
$715$	−7800.00	−0.407977
$716$	6752.00	0.352422
$717$	0	0
$718$	−12812.0	−0.665933
$719$	−10740.0	−0.557072	−0.278536	−	0.960426i	$-0.589849\pi$
$719$	−10740.0	−0.557072	−0.278536	+	0.960426i	$0.589849\pi$
$720$	0	0
$721$	9912.00	0.511986
$722$	10518.0	0.542160
$723$	0	0
$724$	−1064.00	−0.0546177
$725$	2090.00	0.107063
$726$	0	0
$727$	22736.0	1.15988	0.579939	−	0.814660i	$-0.303076\pi$
$727$	22736.0	1.15988	0.579939	+	0.814660i	$0.303076\pi$
$728$	728.000	0.0370625
$729$	0	0
$730$	48.0000	0.00243364
$731$	−696.000	−0.0352154
$732$	0	0
$733$	10082.0	0.508032	0.254016	−	0.967200i	$-0.418248\pi$
$733$	10082.0	0.508032	0.254016	+	0.967200i	$0.418248\pi$
$734$	−4000.00	−0.201148
$735$	0	0
$736$	−2048.00	−0.102568
$737$	1000.00	0.0499803
$738$	0	0
$739$	−24676.0	−1.22831	−0.614155	−	0.789185i	$-0.710503\pi$
$739$	−24676.0	−1.22831	−0.614155	+	0.789185i	$0.710503\pi$
$740$	−2400.00	−0.119224
$741$	0	0
$742$	−6188.00	−0.306157
$743$	−29178.0	−1.44070	−0.720348	−	0.693613i	$-0.756018\pi$
$743$	−29178.0	−1.44070	−0.720348	+	0.693613i	$0.756018\pi$
$744$	0	0
$745$	−31152.0	−1.53197
$746$	7324.00	0.359451
$747$	0	0
$748$	11600.0	0.567029
$749$	−8764.00	−0.427543
$750$	0	0
$751$	13432.0	0.652651	0.326325	−	0.945258i	$-0.394190\pi$
$751$	13432.0	0.652651	0.326325	+	0.945258i	$0.394190\pi$
$752$	1312.00	0.0636220
$753$	0	0
$754$	2860.00	0.138137
$755$	−24624.0	−1.18697
$756$	0	0
$757$	114.000	0.00547345	0.00273672	−	0.999996i	$-0.499129\pi$
$757$	114.000	0.00547345	0.00273672	+	0.999996i	$0.499129\pi$
$758$	−12016.0	−0.575779
$759$	0	0
$760$	3840.00	0.183278
$761$	5052.00	0.240650	0.120325	−	0.992735i	$-0.461606\pi$
$761$	5052.00	0.240650	0.120325	+	0.992735i	$0.461606\pi$
$762$	0	0
$763$	6510.00	0.308883
$764$	−11504.0	−0.544765
$765$	0	0
$766$	22404.0	1.05677
$767$	−8034.00	−0.378215
$768$	0	0
$769$	−36862.0	−1.72858	−0.864290	−	0.502994i	$-0.832232\pi$
$769$	−36862.0	−1.72858	−0.864290	+	0.502994i	$0.832232\pi$
$770$	−8400.00	−0.393136
$771$	0	0
$772$	12232.0	0.570258
$773$	1472.00	0.0684918	0.0342459	−	0.999413i	$-0.489097\pi$
$773$	1472.00	0.0684918	0.0342459	+	0.999413i	$0.489097\pi$
$774$	0	0
$775$	2356.00	0.109200
$776$	−8176.00	−0.378223
$777$	0	0
$778$	−21716.0	−1.00072
$779$	3360.00	0.154537
$780$	0	0
$781$	19500.0	0.893425
$782$	−7424.00	−0.339491
$783$	0	0
$784$	784.000	0.0357143
$785$	−30264.0	−1.37601
$786$	0	0
$787$	−23620.0	−1.06984	−0.534919	−	0.844904i	$-0.679658\pi$
$787$	−23620.0	−1.06984	−0.534919	+	0.844904i	$0.679658\pi$
$788$	11808.0	0.533810
$789$	0	0
$790$	16320.0	0.734987
$791$	7238.00	0.325352
$792$	0	0
$793$	3614.00	0.161837
$794$	11700.0	0.522944
$795$	0	0
$796$	12256.0	0.545732
$797$	−13986.0	−0.621593	−0.310796	−	0.950476i	$-0.600596\pi$
$797$	−13986.0	−0.621593	−0.310796	+	0.950476i	$0.600596\pi$
$798$	0	0
$799$	4756.00	0.210582
$800$	−608.000	−0.0268701
$801$	0	0
$802$	−22808.0	−1.00421
$803$	−100.000	−0.00439467
$804$	0	0
$805$	5376.00	0.235378
$806$	3224.00	0.140894
$807$	0	0
$808$	432.000	0.0188090
$809$	30738.0	1.33584	0.667918	−	0.744235i	$-0.267186\pi$
$809$	30738.0	1.33584	0.667918	+	0.744235i	$0.267186\pi$
$810$	0	0
$811$	25480.0	1.10324	0.551618	−	0.834097i	$-0.314011\pi$
$811$	25480.0	1.10324	0.551618	+	0.834097i	$0.314011\pi$
$812$	3080.00	0.133112
$813$	0	0
$814$	5000.00	0.215295
$815$	−10176.0	−0.437362
$816$	0	0
$817$	480.000	0.0205546
$818$	20388.0	0.871454
$819$	0	0
$820$	−4032.00	−0.171712
$821$	−23484.0	−0.998291	−0.499146	−	0.866518i	$-0.666353\pi$
$821$	−23484.0	−0.998291	−0.499146	+	0.866518i	$0.666353\pi$
$822$	0	0
$823$	−1632.00	−0.0691227	−0.0345613	−	0.999403i	$-0.511003\pi$
$823$	−1632.00	−0.0691227	−0.0345613	+	0.999403i	$0.511003\pi$
$824$	−11328.0	−0.478919
$825$	0	0
$826$	−8652.00	−0.364457
$827$	−25734.0	−1.08205	−0.541027	−	0.841005i	$-0.681964\pi$
$827$	−25734.0	−1.08205	−0.541027	+	0.841005i	$0.681964\pi$
$828$	0	0
$829$	22314.0	0.934858	0.467429	−	0.884031i	$-0.345180\pi$
$829$	22314.0	0.934858	0.467429	+	0.884031i	$0.345180\pi$
$830$	7728.00	0.323184
$831$	0	0
$832$	−832.000	−0.0346688
$833$	2842.00	0.118211
$834$	0	0
$835$	−10248.0	−0.424727
$836$	−8000.00	−0.330964
$837$	0	0
$838$	6384.00	0.263164
$839$	−8330.00	−0.342769	−0.171385	−	0.985204i	$-0.554824\pi$
$839$	−8330.00	−0.342769	−0.171385	+	0.985204i	$0.554824\pi$
$840$	0	0
$841$	−12289.0	−0.503875
$842$	−20948.0	−0.857382
$843$	0	0
$844$	2416.00	0.0985334
$845$	2028.00	0.0825625
$846$	0	0
$847$	8183.00	0.331961
$848$	7072.00	0.286384
$849$	0	0
$850$	−2204.00	−0.0889371
$851$	−3200.00	−0.128901
$852$	0	0
$853$	−4522.00	−0.181513	−0.0907563	−	0.995873i	$-0.528928\pi$
$853$	−4522.00	−0.181513	−0.0907563	+	0.995873i	$0.528928\pi$
$854$	3892.00	0.155950
$855$	0	0
$856$	10016.0	0.399930
$857$	40126.0	1.59939	0.799695	−	0.600406i	$-0.204994\pi$
$857$	40126.0	1.59939	0.799695	+	0.600406i	$0.204994\pi$
$858$	0	0
$859$	34484.0	1.36971	0.684854	−	0.728680i	$-0.259866\pi$
$859$	34484.0	1.36971	0.684854	+	0.728680i	$0.259866\pi$
$860$	−576.000	−0.0228389
$861$	0	0
$862$	15804.0	0.624462
$863$	6714.00	0.264829	0.132414	−	0.991194i	$-0.457727\pi$
$863$	6714.00	0.264829	0.132414	+	0.991194i	$0.457727\pi$
$864$	0	0
$865$	−24552.0	−0.965079
$866$	33908.0	1.33053
$867$	0	0
$868$	3472.00	0.135769
$869$	−34000.0	−1.32724
$870$	0	0
$871$	−260.000	−0.0101145
$872$	−7440.00	−0.288934
$873$	0	0
$874$	5120.00	0.198154
$875$	−8904.00	−0.344012
$876$	0	0
$877$	−18362.0	−0.707002	−0.353501	−	0.935434i	$-0.615009\pi$
$877$	−18362.0	−0.707002	−0.353501	+	0.935434i	$0.615009\pi$
$878$	−27920.0	−1.07318
$879$	0	0
$880$	9600.00	0.367745
$881$	−18018.0	−0.689037	−0.344519	−	0.938779i	$-0.611958\pi$
$881$	−18018.0	−0.689037	−0.344519	+	0.938779i	$0.611958\pi$
$882$	0	0
$883$	−5892.00	−0.224554	−0.112277	−	0.993677i	$-0.535814\pi$
$883$	−5892.00	−0.224554	−0.112277	+	0.993677i	$0.535814\pi$
$884$	−3016.00	−0.114750
$885$	0	0
$886$	−3768.00	−0.142876
$887$	35544.0	1.34549	0.672746	−	0.739874i	$-0.265115\pi$
$887$	35544.0	1.34549	0.672746	+	0.739874i	$0.265115\pi$
$888$	0	0
$889$	−15960.0	−0.602116
$890$	23232.0	0.874987
$891$	0	0
$892$	22512.0	0.845020
$893$	−3280.00	−0.122913
$894$	0	0
$895$	20256.0	0.756518
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−22872.0	−0.849943
$899$	13640.0	0.506028
$900$	0	0
$901$	25636.0	0.947901
$902$	8400.00	0.310077
$903$	0	0
$904$	−8272.00	−0.304339
$905$	−3192.00	−0.117244
$906$	0	0
$907$	21020.0	0.769523	0.384762	−	0.923016i	$-0.374284\pi$
$907$	21020.0	0.769523	0.384762	+	0.923016i	$0.374284\pi$
$908$	4488.00	0.164030
$909$	0	0
$910$	2184.00	0.0795592
$911$	−30172.0	−1.09730	−0.548651	−	0.836051i	$-0.684859\pi$
$911$	−30172.0	−1.09730	−0.548651	+	0.836051i	$0.684859\pi$
$912$	0	0
$913$	−16100.0	−0.583606
$914$	−22972.0	−0.831342
$915$	0	0
$916$	17704.0	0.638599
$917$	10864.0	0.391233
$918$	0	0
$919$	25528.0	0.916312	0.458156	−	0.888872i	$-0.348510\pi$
$919$	25528.0	0.916312	0.458156	+	0.888872i	$0.348510\pi$
$920$	−6144.00	−0.220176
$921$	0	0
$922$	22344.0	0.798113
$923$	−5070.00	−0.180803
$924$	0	0
$925$	−950.000	−0.0337684
$926$	23376.0	0.829571
$927$	0	0
$928$	−3520.00	−0.124515
$929$	28460.0	1.00510	0.502552	−	0.864547i	$-0.332394\pi$
$929$	28460.0	1.00510	0.502552	+	0.864547i	$0.332394\pi$
$930$	0	0
$931$	−1960.00	−0.0689972
$932$	−8760.00	−0.307879
$933$	0	0
$934$	−4896.00	−0.171522
$935$	34800.0	1.21720
$936$	0	0
$937$	−10682.0	−0.372429	−0.186214	−	0.982509i	$-0.559622\pi$
$937$	−10682.0	−0.372429	−0.186214	+	0.982509i	$0.559622\pi$
$938$	−280.000	−0.00974661
$939$	0	0
$940$	3936.00	0.136573
$941$	−49992.0	−1.73187	−0.865937	−	0.500154i	$-0.833277\pi$
$941$	−49992.0	−1.73187	−0.865937	+	0.500154i	$0.833277\pi$
$942$	0	0
$943$	−5376.00	−0.185649
$944$	9888.00	0.340918
$945$	0	0
$946$	1200.00	0.0412425
$947$	−24338.0	−0.835141	−0.417571	−	0.908644i	$-0.637118\pi$
$947$	−24338.0	−0.835141	−0.417571	+	0.908644i	$0.637118\pi$
$948$	0	0
$949$	26.0000	0.000889353 0
$950$	1520.00	0.0519109
$951$	0	0
$952$	−3248.00	−0.110576
$953$	−24558.0	−0.834745	−0.417372	−	0.908736i	$-0.637049\pi$
$953$	−24558.0	−0.834745	−0.417372	+	0.908736i	$0.637049\pi$
$954$	0	0
$955$	−34512.0	−1.16941
$956$	4920.00	0.166448
$957$	0	0
$958$	1396.00	0.0470801
$959$	−336.000	−0.0113139
$960$	0	0
$961$	−14415.0	−0.483871
$962$	−1300.00	−0.0435693
$963$	0	0
$964$	−3144.00	−0.105043
$965$	36696.0	1.22413
$966$	0	0
$967$	−17764.0	−0.590746	−0.295373	−	0.955382i	$-0.595444\pi$
$967$	−17764.0	−0.590746	−0.295373	+	0.955382i	$0.595444\pi$
$968$	−9352.00	−0.310521
$969$	0	0
$970$	−24528.0	−0.811904
$971$	−39300.0	−1.29886	−0.649432	−	0.760420i	$-0.724993\pi$
$971$	−39300.0	−1.29886	−0.649432	+	0.760420i	$0.724993\pi$
$972$	0	0
$973$	196.000	0.00645783
$974$	16312.0	0.536622
$975$	0	0
$976$	−4448.00	−0.145878
$977$	−3940.00	−0.129019	−0.0645096	−	0.997917i	$-0.520548\pi$
$977$	−3940.00	−0.129019	−0.0645096	+	0.997917i	$0.520548\pi$
$978$	0	0
$979$	−48400.0	−1.58005
$980$	2352.00	0.0766652
$981$	0	0
$982$	−20968.0	−0.681381
$983$	−33682.0	−1.09287	−0.546434	−	0.837502i	$-0.684015\pi$
$983$	−33682.0	−1.09287	−0.546434	+	0.837502i	$0.684015\pi$
$984$	0	0
$985$	35424.0	1.14589
$986$	−12760.0	−0.412131
$987$	0	0
$988$	2080.00	0.0669773
$989$	−768.000	−0.0246926
$990$	0	0
$991$	−15352.0	−0.492101	−0.246051	−	0.969257i	$-0.579133\pi$
$991$	−15352.0	−0.492101	−0.246051	+	0.969257i	$0.579133\pi$
$992$	−3968.00	−0.127000
$993$	0	0
$994$	−5460.00	−0.174226
$995$	36768.0	1.17148
$996$	0	0
$997$	6942.00	0.220517	0.110258	−	0.993903i	$-0.464832\pi$
$997$	6942.00	0.220517	0.110258	+	0.993903i	$0.464832\pi$
$998$	13960.0	0.442782
$999$	0	0

Twists

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

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

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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