LMFDB - Embedded newform 546.4.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,4,Mod(1,546)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("546.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,2,3,4,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

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

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -12.0000 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -24.0000 q^{10} -50.0000 q^{11} +12.0000 q^{12} -13.0000 q^{13} +14.0000 q^{14} -36.0000 q^{15} +16.0000 q^{16} -58.0000 q^{17} +18.0000 q^{18} -40.0000 q^{19} -48.0000 q^{20} +21.0000 q^{21} -100.000 q^{22} -64.0000 q^{23} +24.0000 q^{24} +19.0000 q^{25} -26.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -110.000 q^{29} -72.0000 q^{30} +124.000 q^{31} +32.0000 q^{32} -150.000 q^{33} -116.000 q^{34} -84.0000 q^{35} +36.0000 q^{36} -50.0000 q^{37} -80.0000 q^{38} -39.0000 q^{39} -96.0000 q^{40} +84.0000 q^{41} +42.0000 q^{42} -12.0000 q^{43} -200.000 q^{44} -108.000 q^{45} -128.000 q^{46} -82.0000 q^{47} +48.0000 q^{48} +49.0000 q^{49} +38.0000 q^{50} -174.000 q^{51} -52.0000 q^{52} -442.000 q^{53} +54.0000 q^{54} +600.000 q^{55} +56.0000 q^{56} -120.000 q^{57} -220.000 q^{58} -618.000 q^{59} -144.000 q^{60} -278.000 q^{61} +248.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} +156.000 q^{65} -300.000 q^{66} +20.0000 q^{67} -232.000 q^{68} -192.000 q^{69} -168.000 q^{70} -390.000 q^{71} +72.0000 q^{72} -2.00000 q^{73} -100.000 q^{74} +57.0000 q^{75} -160.000 q^{76} -350.000 q^{77} -78.0000 q^{78} -680.000 q^{79} -192.000 q^{80} +81.0000 q^{81} +168.000 q^{82} +322.000 q^{83} +84.0000 q^{84} +696.000 q^{85} -24.0000 q^{86} -330.000 q^{87} -400.000 q^{88} +968.000 q^{89} -216.000 q^{90} -91.0000 q^{91} -256.000 q^{92} +372.000 q^{93} -164.000 q^{94} +480.000 q^{95} +96.0000 q^{96} +1022.00 q^{97} +98.0000 q^{98} -450.000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	6.00000	0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−24.0000	−0.758947
$11$	−50.0000	−1.37051	−0.685253	−	0.728305i	$-0.740308\pi$
$11$	−50.0000	−1.37051	−0.685253	+	0.728305i	$0.740308\pi$
$12$	12.0000	0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	14.0000	0.267261
$15$	−36.0000	−0.619677
$16$	16.0000	0.250000
$17$	−58.0000	−0.827474	−0.413737	−	0.910396i	$-0.635777\pi$
$17$	−58.0000	−0.827474	−0.413737	+	0.910396i	$0.635777\pi$
$18$	18.0000	0.235702
$19$	−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
$21$	21.0000	0.218218
$22$	−100.000	−0.969094
$23$	−64.0000	−0.580214	−0.290107	−	0.956994i	$-0.593691\pi$
$23$	−64.0000	−0.580214	−0.290107	+	0.956994i	$0.593691\pi$
$24$	24.0000	0.204124
$25$	19.0000	0.152000
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	−110.000	−0.704362	−0.352181	−	0.935932i	$-0.614560\pi$
$29$	−110.000	−0.704362	−0.352181	+	0.935932i	$0.614560\pi$
$30$	−72.0000	−0.438178
$31$	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
$33$	−150.000	−0.791262
$34$	−116.000	−0.585113
$35$	−84.0000	−0.405674
$36$	36.0000	0.166667
$37$	−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
$39$	−39.0000	−0.160128
$40$	−96.0000	−0.379473
$41$	84.0000	0.319966	0.159983	−	0.987120i	$-0.448856\pi$
$41$	84.0000	0.319966	0.159983	+	0.987120i	$0.448856\pi$
$42$	42.0000	0.154303
$43$	−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
$45$	−108.000	−0.357771
$46$	−128.000	−0.410273
$47$	−82.0000	−0.254488	−0.127244	−	0.991871i	$-0.540613\pi$
$47$	−82.0000	−0.254488	−0.127244	+	0.991871i	$0.540613\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	38.0000	0.107480
$51$	−174.000	−0.477743
$52$	−52.0000	−0.138675
$53$	−442.000	−1.14554	−0.572768	−	0.819718i	$-0.694130\pi$
$53$	−442.000	−1.14554	−0.572768	+	0.819718i	$0.694130\pi$
$54$	54.0000	0.136083
$55$	600.000	1.47098
$56$	56.0000	0.133631
$57$	−120.000	−0.278849
$58$	−220.000	−0.498059
$59$	−618.000	−1.36367	−0.681837	−	0.731504i	$-0.738819\pi$
$59$	−618.000	−1.36367	−0.681837	+	0.731504i	$0.738819\pi$
$60$	−144.000	−0.309839
$61$	−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
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	156.000	0.297683
$66$	−300.000	−0.559507
$67$	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
$69$	−192.000	−0.334987
$70$	−168.000	−0.286855
$71$	−390.000	−0.651894	−0.325947	−	0.945388i	$-0.605683\pi$
$71$	−390.000	−0.651894	−0.325947	+	0.945388i	$0.605683\pi$
$72$	72.0000	0.117851
$73$	−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
$75$	57.0000	0.0877572
$76$	−160.000	−0.241490
$77$	−350.000	−0.518003
$78$	−78.0000	−0.113228
$79$	−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
$81$	81.0000	0.111111
$82$	168.000	0.226250
$83$	322.000	0.425832	0.212916	−	0.977070i	$-0.431704\pi$
$83$	322.000	0.425832	0.212916	+	0.977070i	$0.431704\pi$
$84$	84.0000	0.109109
$85$	696.000	0.888139
$86$	−24.0000	−0.0300929
$87$	−330.000	−0.406663
$88$	−400.000	−0.484547
$89$	968.000	1.15290	0.576448	−	0.817134i	$-0.304438\pi$
$89$	968.000	1.15290	0.576448	+	0.817134i	$0.304438\pi$
$90$	−216.000	−0.252982
$91$	−91.0000	−0.104828
$92$	−256.000	−0.290107
$93$	372.000	0.414781
$94$	−164.000	−0.179950
$95$	480.000	0.518389
$96$	96.0000	0.102062
$97$	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
$99$	−450.000	−0.456835
$100$	76.0000	0.0760000
$101$	54.0000	0.0532000	0.0266000	−	0.999646i	$-0.491532\pi$
$101$	54.0000	0.0532000	0.0266000	+	0.999646i	$0.491532\pi$
$102$	−348.000	−0.337815
$103$	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
$105$	−252.000	−0.234216
$106$	−884.000	−0.810016
$107$	1252.00	1.13117	0.565586	−	0.824689i	$-0.308650\pi$
$107$	1252.00	1.13117	0.565586	+	0.824689i	$0.308650\pi$
$108$	108.000	0.0962250
$109$	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
$111$	−150.000	−0.128265
$112$	112.000	0.0944911
$113$	−1034.00	−0.860801	−0.430401	−	0.902638i	$-0.641628\pi$
$113$	−1034.00	−0.860801	−0.430401	+	0.902638i	$0.641628\pi$
$114$	−240.000	−0.197176
$115$	768.000	0.622751
$116$	−440.000	−0.352181
$117$	−117.000	−0.0924500
$118$	−1236.00	−0.964263
$119$	−406.000	−0.312756
$120$	−288.000	−0.219089
$121$	1169.00	0.878287
$122$	−556.000	−0.412606
$123$	252.000	0.184732
$124$	496.000	0.359211
$125$	1272.00	0.910169
$126$	126.000	0.0890871
$127$	−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
$129$	−36.0000	−0.0245707
$130$	312.000	0.210494
$131$	−1552.00	−1.03511	−0.517553	−	0.855651i	$-0.673157\pi$
$131$	−1552.00	−1.03511	−0.517553	+	0.855651i	$0.673157\pi$
$132$	−600.000	−0.395631
$133$	−280.000	−0.182549
$134$	40.0000	0.0257871
$135$	−324.000	−0.206559
$136$	−464.000	−0.292556
$137$	48.0000	0.0299337	0.0149668	−	0.999888i	$-0.495236\pi$
$137$	48.0000	0.0299337	0.0149668	+	0.999888i	$0.495236\pi$
$138$	−384.000	−0.236871
$139$	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
$141$	−246.000	−0.146929
$142$	−780.000	−0.460959
$143$	650.000	0.380110
$144$	144.000	0.0833333
$145$	1320.00	0.756000
$146$	−4.00000	−0.00226741
$147$	147.000	0.0824786
$148$	−200.000	−0.111080
$149$	2596.00	1.42733	0.713666	−	0.700486i	$-0.247033\pi$
$149$	2596.00	1.42733	0.713666	+	0.700486i	$0.247033\pi$
$150$	114.000	0.0620537
$151$	−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
$153$	−522.000	−0.275825
$154$	−700.000	−0.366283
$155$	−1488.00	−0.771091
$156$	−156.000	−0.0800641
$157$	−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
$159$	−1326.00	−0.661375
$160$	−384.000	−0.189737
$161$	−448.000	−0.219300
$162$	162.000	0.0785674
$163$	−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
$165$	1800.00	0.849272
$166$	644.000	0.301109
$167$	854.000	0.395716	0.197858	−	0.980231i	$-0.436602\pi$
$167$	854.000	0.395716	0.197858	+	0.980231i	$0.436602\pi$
$168$	168.000	0.0771517
$169$	169.000	0.0769231
$170$	1392.00	0.628009
$171$	−360.000	−0.160993
$172$	−48.0000	−0.0212789
$173$	2046.00	0.899159	0.449579	−	0.893240i	$-0.351574\pi$
$173$	2046.00	0.899159	0.449579	+	0.893240i	$0.351574\pi$
$174$	−660.000	−0.287554
$175$	133.000	0.0574506
$176$	−800.000	−0.342627
$177$	−1854.00	−0.787317
$178$	1936.00	0.815221
$179$	−1688.00	−0.704844	−0.352422	−	0.935841i	$-0.614642\pi$
$179$	−1688.00	−0.704844	−0.352422	+	0.935841i	$0.614642\pi$
$180$	−432.000	−0.178885
$181$	−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
$183$	−834.000	−0.336891
$184$	−512.000	−0.205137
$185$	600.000	0.238448
$186$	744.000	0.293294
$187$	2900.00	1.13406
$188$	−328.000	−0.127244
$189$	189.000	0.0727393
$190$	960.000	0.366556
$191$	2876.00	1.08953	0.544765	−	0.838589i	$-0.316619\pi$
$191$	2876.00	1.08953	0.544765	+	0.838589i	$0.316619\pi$
$192$	192.000	0.0721688
$193$	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
$195$	468.000	0.171868
$196$	196.000	0.0714286
$197$	−2952.00	−1.06762	−0.533810	−	0.845604i	$-0.679240\pi$
$197$	−2952.00	−1.06762	−0.533810	+	0.845604i	$0.679240\pi$
$198$	−900.000	−0.323031
$199$	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
$201$	60.0000	0.0210551
$202$	108.000	0.0376181
$203$	−770.000	−0.266224
$204$	−696.000	−0.238871
$205$	−1008.00	−0.343423
$206$	2832.00	0.957839
$207$	−576.000	−0.193405
$208$	−208.000	−0.0693375
$209$	2000.00	0.661928
$210$	−504.000	−0.165616
$211$	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
$213$	−1170.00	−0.376371
$214$	2504.00	0.799859
$215$	144.000	0.0456778
$216$	216.000	0.0680414
$217$	868.000	0.271538
$218$	1860.00	0.577867
$219$	−6.00000	−0.00185134
$220$	2400.00	0.735491
$221$	754.000	0.229500
$222$	−300.000	−0.0906968
$223$	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
$225$	171.000	0.0506667
$226$	−2068.00	−0.608678
$227$	−1122.00	−0.328061	−0.164030	−	0.986455i	$-0.552449\pi$
$227$	−1122.00	−0.328061	−0.164030	+	0.986455i	$0.552449\pi$
$228$	−480.000	−0.139424
$229$	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
$231$	−1050.00	−0.299069
$232$	−880.000	−0.249029
$233$	2190.00	0.615758	0.307879	−	0.951425i	$-0.400381\pi$
$233$	2190.00	0.615758	0.307879	+	0.951425i	$0.400381\pi$
$234$	−234.000	−0.0653720
$235$	984.000	0.273145
$236$	−2472.00	−0.681837
$237$	−2040.00	−0.559123
$238$	−812.000	−0.221152
$239$	−1230.00	−0.332896	−0.166448	−	0.986050i	$-0.553230\pi$
$239$	−1230.00	−0.332896	−0.166448	+	0.986050i	$0.553230\pi$
$240$	−576.000	−0.154919
$241$	−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
$243$	243.000	0.0641500
$244$	−1112.00	−0.291756
$245$	−588.000	−0.153330
$246$	504.000	0.130625
$247$	520.000	0.133955
$248$	992.000	0.254000
$249$	966.000	0.245854
$250$	2544.00	0.643587
$251$	−280.000	−0.0704121	−0.0352061	−	0.999380i	$-0.511209\pi$
$251$	−280.000	−0.0704121	−0.0352061	+	0.999380i	$0.511209\pi$
$252$	252.000	0.0629941
$253$	3200.00	0.795187
$254$	−4560.00	−1.12646
$255$	2088.00	0.512767
$256$	256.000	0.0625000
$257$	−3806.00	−0.923781	−0.461891	−	0.886937i	$-0.652829\pi$
$257$	−3806.00	−0.923781	−0.461891	+	0.886937i	$0.652829\pi$
$258$	−72.0000	−0.0173741
$259$	−350.000	−0.0839689
$260$	624.000	0.148842
$261$	−990.000	−0.234787
$262$	−3104.00	−0.731930
$263$	−3816.00	−0.894694	−0.447347	−	0.894360i	$-0.647631\pi$
$263$	−3816.00	−0.894694	−0.447347	+	0.894360i	$0.647631\pi$
$264$	−1200.00	−0.279753
$265$	5304.00	1.22952
$266$	−560.000	−0.129082
$267$	2904.00	0.665625
$268$	80.0000	0.0182342
$269$	−3026.00	−0.685868	−0.342934	−	0.939360i	$-0.611421\pi$
$269$	−3026.00	−0.685868	−0.342934	+	0.939360i	$0.611421\pi$
$270$	−648.000	−0.146059
$271$	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
$273$	−273.000	−0.0605228
$274$	96.0000	0.0211663
$275$	−950.000	−0.208317
$276$	−768.000	−0.167493
$277$	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
$279$	1116.00	0.239474
$280$	−672.000	−0.143427
$281$	4832.00	1.02581	0.512906	−	0.858445i	$-0.328569\pi$
$281$	4832.00	1.02581	0.512906	+	0.858445i	$0.328569\pi$
$282$	−492.000	−0.103894
$283$	−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
$285$	1440.00	0.299292
$286$	1300.00	0.268778
$287$	588.000	0.120936
$288$	288.000	0.0589256
$289$	−1549.00	−0.315286
$290$	2640.00	0.534573
$291$	3066.00	0.617636
$292$	−8.00000	−0.00160330
$293$	−6384.00	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$293$	−6384.00	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$294$	294.000	0.0583212
$295$	7416.00	1.46365
$296$	−400.000	−0.0785457
$297$	−1350.00	−0.263754
$298$	5192.00	1.00928
$299$	832.000	0.160922
$300$	228.000	0.0438786
$301$	−84.0000	−0.0160853
$302$	−4104.00	−0.781982
$303$	162.000	0.0307150
$304$	−640.000	−0.120745
$305$	3336.00	0.626291
$306$	−1044.00	−0.195038
$307$	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
$309$	4248.00	0.782072
$310$	−2976.00	−0.545243
$311$	6116.00	1.11513	0.557567	−	0.830132i	$-0.311735\pi$
$311$	6116.00	1.11513	0.557567	+	0.830132i	$0.311735\pi$
$312$	−312.000	−0.0566139
$313$	−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
$315$	−756.000	−0.135225
$316$	−2720.00	−0.484215
$317$	−9136.00	−1.61870	−0.809352	−	0.587325i	$-0.800181\pi$
$317$	−9136.00	−1.61870	−0.809352	+	0.587325i	$0.800181\pi$
$318$	−2652.00	−0.467663
$319$	5500.00	0.965332
$320$	−768.000	−0.134164
$321$	3756.00	0.653082
$322$	−896.000	−0.155069
$323$	2320.00	0.399654
$324$	324.000	0.0555556
$325$	−247.000	−0.0421572
$326$	−1696.00	−0.288137
$327$	2790.00	0.471827
$328$	672.000	0.113125
$329$	−574.000	−0.0961874
$330$	3600.00	0.600526
$331$	−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
$333$	−450.000	−0.0740536
$334$	1708.00	0.279813
$335$	−240.000	−0.0391421
$336$	336.000	0.0545545
$337$	4734.00	0.765215	0.382607	−	0.923911i	$-0.375026\pi$
$337$	4734.00	0.765215	0.382607	+	0.923911i	$0.375026\pi$
$338$	338.000	0.0543928
$339$	−3102.00	−0.496984
$340$	2784.00	0.444069
$341$	−6200.00	−0.984601
$342$	−720.000	−0.113840
$343$	343.000	0.0539949
$344$	−96.0000	−0.0150464
$345$	2304.00	0.359545
$346$	4092.00	0.635801
$347$	−11096.0	−1.71661	−0.858306	−	0.513138i	$-0.828483\pi$
$347$	−11096.0	−1.71661	−0.858306	+	0.513138i	$0.828483\pi$
$348$	−1320.00	−0.203332
$349$	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
$351$	−351.000	−0.0533761
$352$	−1600.00	−0.242274
$353$	−11076.0	−1.67002	−0.835008	−	0.550237i	$-0.814537\pi$
$353$	−11076.0	−1.67002	−0.835008	+	0.550237i	$0.814537\pi$
$354$	−3708.00	−0.556717
$355$	4680.00	0.699686
$356$	3872.00	0.576448
$357$	−1218.00	−0.180570
$358$	−3376.00	−0.498400
$359$	−6406.00	−0.941771	−0.470885	−	0.882194i	$-0.656065\pi$
$359$	−6406.00	−0.941771	−0.470885	+	0.882194i	$0.656065\pi$
$360$	−864.000	−0.126491
$361$	−5259.00	−0.766730
$362$	−532.000	−0.0772412
$363$	3507.00	0.507079
$364$	−364.000	−0.0524142
$365$	24.0000	0.00344169
$366$	−1668.00	−0.238218
$367$	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
$369$	756.000	0.106655
$370$	1200.00	0.168608
$371$	−3094.00	−0.432972
$372$	1488.00	0.207390
$373$	−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
$375$	3816.00	0.525486
$376$	−656.000	−0.0899750
$377$	1430.00	0.195355
$378$	378.000	0.0514344
$379$	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
$381$	−6840.00	−0.919748
$382$	5752.00	0.770413
$383$	11202.0	1.49451	0.747253	−	0.664540i	$-0.231373\pi$
$383$	11202.0	1.49451	0.747253	+	0.664540i	$0.231373\pi$
$384$	384.000	0.0510310
$385$	4200.00	0.555979
$386$	6116.00	0.806467
$387$	−108.000	−0.0141859
$388$	4088.00	0.534889
$389$	−10858.0	−1.41522	−0.707612	−	0.706601i	$-0.750228\pi$
$389$	−10858.0	−1.41522	−0.707612	+	0.706601i	$0.750228\pi$
$390$	936.000	0.121529
$391$	3712.00	0.480112
$392$	392.000	0.0505076
$393$	−4656.00	−0.597619
$394$	−5904.00	−0.754922
$395$	8160.00	1.03943
$396$	−1800.00	−0.228418
$397$	−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
$399$	−840.000	−0.105395
$400$	304.000	0.0380000
$401$	−11404.0	−1.42017	−0.710086	−	0.704115i	$-0.751344\pi$
$401$	−11404.0	−1.42017	−0.710086	+	0.704115i	$0.751344\pi$
$402$	120.000	0.0148882
$403$	−1612.00	−0.199254
$404$	216.000	0.0266000
$405$	−972.000	−0.119257
$406$	−1540.00	−0.188249
$407$	2500.00	0.304473
$408$	−1392.00	−0.168908
$409$	−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
$411$	144.000	0.0172822
$412$	5664.00	0.677294
$413$	−4326.00	−0.515420
$414$	−1152.00	−0.136758
$415$	−3864.00	−0.457051
$416$	−416.000	−0.0490290
$417$	84.0000	0.00986450
$418$	4000.00	0.468054
$419$	3192.00	0.372170	0.186085	−	0.982534i	$-0.440420\pi$
$419$	3192.00	0.372170	0.186085	+	0.982534i	$0.440420\pi$
$420$	−1008.00	−0.117108
$421$	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
$423$	−738.000	−0.0848293
$424$	−3536.00	−0.405008
$425$	−1102.00	−0.125776
$426$	−2340.00	−0.266135
$427$	−1946.00	−0.220547
$428$	5008.00	0.565586
$429$	1950.00	0.219457
$430$	288.000	0.0322991
$431$	7902.00	0.883123	0.441561	−	0.897231i	$-0.354425\pi$
$431$	7902.00	0.883123	0.441561	+	0.897231i	$0.354425\pi$
$432$	432.000	0.0481125
$433$	−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
$435$	3960.00	0.436477
$436$	3720.00	0.408614
$437$	2560.00	0.280232
$438$	−12.0000	−0.00130909
$439$	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
$441$	441.000	0.0476190
$442$	1508.00	0.162281
$443$	−1884.00	−0.202058	−0.101029	−	0.994883i	$-0.532213\pi$
$443$	−1884.00	−0.202058	−0.101029	+	0.994883i	$0.532213\pi$
$444$	−600.000	−0.0641323
$445$	−11616.0	−1.23742
$446$	11256.0	1.19504
$447$	7788.00	0.824071
$448$	448.000	0.0472456
$449$	−11436.0	−1.20200	−0.601000	−	0.799249i	$-0.705231\pi$
$449$	−11436.0	−1.20200	−0.601000	+	0.799249i	$0.705231\pi$
$450$	342.000	0.0358267
$451$	−4200.00	−0.438515
$452$	−4136.00	−0.430401
$453$	−6156.00	−0.638486
$454$	−2244.00	−0.231974
$455$	1092.00	0.112514
$456$	−960.000	−0.0985880
$457$	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
$459$	−1566.00	−0.159248
$460$	3072.00	0.311376
$461$	11172.0	1.12870	0.564351	−	0.825535i	$-0.309126\pi$
$461$	11172.0	1.12870	0.564351	+	0.825535i	$0.309126\pi$
$462$	−2100.00	−0.211474
$463$	−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
$465$	−4464.00	−0.445189
$466$	4380.00	0.435407
$467$	−2448.00	−0.242569	−0.121285	−	0.992618i	$-0.538701\pi$
$467$	−2448.00	−0.242569	−0.121285	+	0.992618i	$0.538701\pi$
$468$	−468.000	−0.0462250
$469$	140.000	0.0137838
$470$	1968.00	0.193143
$471$	−7566.00	−0.740176
$472$	−4944.00	−0.482131
$473$	600.000	0.0583256
$474$	−4080.00	−0.395360
$475$	−760.000	−0.0734130
$476$	−1624.00	−0.156378
$477$	−3978.00	−0.381845
$478$	−2460.00	−0.235393
$479$	698.000	0.0665813	0.0332906	−	0.999446i	$-0.489401\pi$
$479$	698.000	0.0665813	0.0332906	+	0.999446i	$0.489401\pi$
$480$	−1152.00	−0.109545
$481$	650.000	0.0616163
$482$	−1572.00	−0.148553
$483$	−1344.00	−0.126613
$484$	4676.00	0.439144
$485$	−12264.0	−1.14821
$486$	486.000	0.0453609
$487$	−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
$489$	−2544.00	−0.235263
$490$	−1176.00	−0.108421
$491$	−10484.0	−0.963618	−0.481809	−	0.876276i	$-0.660020\pi$
$491$	−10484.0	−0.963618	−0.481809	+	0.876276i	$0.660020\pi$
$492$	1008.00	0.0923662
$493$	6380.00	0.582841
$494$	1040.00	0.0947203
$495$	5400.00	0.490327
$496$	1984.00	0.179605
$497$	−2730.00	−0.246393
$498$	1932.00	0.173845
$499$	−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
$501$	2562.00	0.228467
$502$	−560.000	−0.0497889
$503$	5180.00	0.459175	0.229587	−	0.973288i	$-0.426262\pi$
$503$	5180.00	0.459175	0.229587	+	0.973288i	$0.426262\pi$
$504$	504.000	0.0445435
$505$	−648.000	−0.0571002
$506$	6400.00	0.562282
$507$	507.000	0.0444116
$508$	−9120.00	−0.796525
$509$	4860.00	0.423214	0.211607	−	0.977355i	$-0.432130\pi$
$509$	4860.00	0.423214	0.211607	+	0.977355i	$0.432130\pi$
$510$	4176.00	0.362581
$511$	−14.0000	−0.00121198
$512$	512.000	0.0441942
$513$	−1080.00	−0.0929496
$514$	−7612.00	−0.653212
$515$	−16992.0	−1.45390
$516$	−144.000	−0.0122854
$517$	4100.00	0.348777
$518$	−700.000	−0.0593750
$519$	6138.00	0.519130
$520$	1248.00	0.105247
$521$	−17390.0	−1.46232	−0.731161	−	0.682205i	$-0.761021\pi$
$521$	−17390.0	−1.46232	−0.731161	+	0.682205i	$0.761021\pi$
$522$	−1980.00	−0.166020
$523$	−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
$525$	399.000	0.0331691
$526$	−7632.00	−0.632645
$527$	−7192.00	−0.594475
$528$	−2400.00	−0.197816
$529$	−8071.00	−0.663352
$530$	10608.0	0.869400
$531$	−5562.00	−0.454558
$532$	−1120.00	−0.0912747
$533$	−1092.00	−0.0887425
$534$	5808.00	0.470668
$535$	−15024.0	−1.21410
$536$	160.000	0.0128936
$537$	−5064.00	−0.406942
$538$	−6052.00	−0.484982
$539$	−2450.00	−0.195787
$540$	−1296.00	−0.103280
$541$	−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
$543$	−798.000	−0.0630671
$544$	−1856.00	−0.146278
$545$	−11160.0	−0.877141
$546$	−546.000	−0.0427960
$547$	−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
$549$	−2502.00	−0.194504
$550$	−1900.00	−0.147302
$551$	4400.00	0.340193
$552$	−1536.00	−0.118436
$553$	−4760.00	−0.366032
$554$	7180.00	0.550630
$555$	1800.00	0.137668
$556$	112.000	0.00854291
$557$	−19524.0	−1.48520	−0.742602	−	0.669733i	$-0.766408\pi$
$557$	−19524.0	−1.48520	−0.742602	+	0.669733i	$0.766408\pi$
$558$	2232.00	0.169334
$559$	156.000	0.0118034
$560$	−1344.00	−0.101419
$561$	8700.00	0.654749
$562$	9664.00	0.725358
$563$	−772.000	−0.0577903	−0.0288951	−	0.999582i	$-0.509199\pi$
$563$	−772.000	−0.0577903	−0.0288951	+	0.999582i	$0.509199\pi$
$564$	−984.000	−0.0734643
$565$	12408.0	0.923909
$566$	−11400.0	−0.846604
$567$	567.000	0.0419961
$568$	−3120.00	−0.230479
$569$	−8226.00	−0.606067	−0.303033	−	0.952980i	$-0.597999\pi$
$569$	−8226.00	−0.606067	−0.303033	+	0.952980i	$0.597999\pi$
$570$	2880.00	0.211631
$571$	−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
$573$	8628.00	0.629040
$574$	1176.00	0.0855144
$575$	−1216.00	−0.0881925
$576$	576.000	0.0416667
$577$	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
$579$	9174.00	0.658477
$580$	5280.00	0.378000
$581$	2254.00	0.160950
$582$	6132.00	0.436735
$583$	22100.0	1.56996
$584$	−16.0000	−0.00113371
$585$	1404.00	0.0992278
$586$	−12768.0	−0.900070
$587$	14994.0	1.05429	0.527145	−	0.849775i	$-0.323262\pi$
$587$	14994.0	1.05429	0.527145	+	0.849775i	$0.323262\pi$
$588$	588.000	0.0412393
$589$	−4960.00	−0.346983
$590$	14832.0	1.03496
$591$	−8856.00	−0.616391
$592$	−800.000	−0.0555402
$593$	−21544.0	−1.49192	−0.745958	−	0.665993i	$-0.768008\pi$
$593$	−21544.0	−1.49192	−0.745958	+	0.665993i	$0.768008\pi$
$594$	−2700.00	−0.186502
$595$	4872.00	0.335685
$596$	10384.0	0.713666
$597$	9192.00	0.630157
$598$	1664.00	0.113789
$599$	−4824.00	−0.329054	−0.164527	−	0.986373i	$-0.552610\pi$
$599$	−4824.00	−0.329054	−0.164527	+	0.986373i	$0.552610\pi$
$600$	456.000	0.0310269
$601$	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
$603$	180.000	0.0121562
$604$	−8208.00	−0.552945
$605$	−14028.0	−0.942677
$606$	324.000	0.0217188
$607$	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
$609$	−2310.00	−0.153704
$610$	6672.00	0.442855
$611$	1066.00	0.0705822
$612$	−2088.00	−0.137912
$613$	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
$615$	−3024.00	−0.198276
$616$	−2800.00	−0.183142
$617$	12840.0	0.837794	0.418897	−	0.908034i	$-0.362417\pi$
$617$	12840.0	0.837794	0.418897	+	0.908034i	$0.362417\pi$
$618$	8496.00	0.553008
$619$	−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
$621$	−1728.00	−0.111662
$622$	12232.0	0.788519
$623$	6776.00	0.435754
$624$	−624.000	−0.0400320
$625$	−17639.0	−1.12890
$626$	−1284.00	−0.0819792
$627$	6000.00	0.382164
$628$	−10088.0	−0.641011
$629$	2900.00	0.183832
$630$	−1512.00	−0.0956183
$631$	−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
$633$	1812.00	0.113777
$634$	−18272.0	−1.14460
$635$	27360.0	1.70984
$636$	−5304.00	−0.330688
$637$	−637.000	−0.0396214
$638$	11000.0	0.682593
$639$	−3510.00	−0.217298
$640$	−1536.00	−0.0948683
$641$	10170.0	0.626663	0.313331	−	0.949644i	$-0.398555\pi$
$641$	10170.0	0.626663	0.313331	+	0.949644i	$0.398555\pi$
$642$	7512.00	0.461799
$643$	−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
$645$	432.000	0.0263721
$646$	4640.00	0.282598
$647$	13704.0	0.832705	0.416352	−	0.909203i	$-0.363308\pi$
$647$	13704.0	0.832705	0.416352	+	0.909203i	$0.363308\pi$
$648$	648.000	0.0392837
$649$	30900.0	1.86892
$650$	−494.000	−0.0298097
$651$	2604.00	0.156772
$652$	−3392.00	−0.203744
$653$	−19230.0	−1.15242	−0.576208	−	0.817303i	$-0.695468\pi$
$653$	−19230.0	−1.15242	−0.576208	+	0.817303i	$0.695468\pi$
$654$	5580.00	0.333632
$655$	18624.0	1.11099
$656$	1344.00	0.0799914
$657$	−18.0000	−0.00106887
$658$	−1148.00	−0.0680147
$659$	18608.0	1.09995	0.549973	−	0.835182i	$-0.314638\pi$
$659$	18608.0	1.09995	0.549973	+	0.835182i	$0.314638\pi$
$660$	7200.00	0.424636
$661$	−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
$663$	2262.00	0.132502
$664$	2576.00	0.150554
$665$	3360.00	0.195933
$666$	−900.000	−0.0523638
$667$	7040.00	0.408680
$668$	3416.00	0.197858
$669$	16884.0	0.975745
$670$	−480.000	−0.0276776
$671$	13900.0	0.799707
$672$	672.000	0.0385758
$673$	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
$675$	513.000	0.0292524
$676$	676.000	0.0384615
$677$	6666.00	0.378427	0.189214	−	0.981936i	$-0.439406\pi$
$677$	6666.00	0.378427	0.189214	+	0.981936i	$0.439406\pi$
$678$	−6204.00	−0.351421
$679$	7154.00	0.404338
$680$	5568.00	0.314004
$681$	−3366.00	−0.189406
$682$	−12400.0	−0.696218
$683$	−11082.0	−0.620851	−0.310425	−	0.950598i	$-0.600471\pi$
$683$	−11082.0	−0.620851	−0.310425	+	0.950598i	$0.600471\pi$
$684$	−1440.00	−0.0804967
$685$	−576.000	−0.0321282
$686$	686.000	0.0381802
$687$	13278.0	0.737391
$688$	−192.000	−0.0106394
$689$	5746.00	0.317714
$690$	4608.00	0.254237
$691$	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
$693$	−3150.00	−0.172668
$694$	−22192.0	−1.21383
$695$	−336.000	−0.0183384
$696$	−2640.00	−0.143777
$697$	−4872.00	−0.264763
$698$	4732.00	0.256603
$699$	6570.00	0.355508
$700$	532.000	0.0287253
$701$	15318.0	0.825325	0.412663	−	0.910884i	$-0.364599\pi$
$701$	15318.0	0.825325	0.412663	+	0.910884i	$0.364599\pi$
$702$	−702.000	−0.0377426
$703$	2000.00	0.107299
$704$	−3200.00	−0.171313
$705$	2952.00	0.157700
$706$	−22152.0	−1.18088
$707$	378.000	0.0201077
$708$	−7416.00	−0.393659
$709$	−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
$711$	−6120.00	−0.322810
$712$	7744.00	0.407610
$713$	−7936.00	−0.416838
$714$	−2436.00	−0.127682
$715$	−7800.00	−0.407977
$716$	−6752.00	−0.352422
$717$	−3690.00	−0.192197
$718$	−12812.0	−0.665933
$719$	10740.0	0.557072	0.278536	−	0.960426i	$-0.410151\pi$
$719$	10740.0	0.557072	0.278536	+	0.960426i	$0.410151\pi$
$720$	−1728.00	−0.0894427
$721$	9912.00	0.511986
$722$	−10518.0	−0.542160
$723$	−2358.00	−0.121293
$724$	−1064.00	−0.0546177
$725$	−2090.00	−0.107063
$726$	7014.00	0.358559
$727$	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
$729$	729.000	0.0370370
$730$	48.0000	0.00243364
$731$	696.000	0.0352154
$732$	−3336.00	−0.168446
$733$	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
$735$	−1764.00	−0.0885253
$736$	−2048.00	−0.102568
$737$	−1000.00	−0.0499803
$738$	1512.00	0.0754167
$739$	−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
$741$	1560.00	0.0773388
$742$	−6188.00	−0.306157
$743$	29178.0	1.44070	0.720348	−	0.693613i	$-0.243982\pi$
$743$	29178.0	1.44070	0.720348	+	0.693613i	$0.243982\pi$
$744$	2976.00	0.146647
$745$	−31152.0	−1.53197
$746$	−7324.00	−0.359451
$747$	2898.00	0.141944
$748$	11600.0	0.567029
$749$	8764.00	0.427543
$750$	7632.00	0.371575
$751$	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
$753$	−840.000	−0.0406525
$754$	2860.00	0.138137
$755$	24624.0	1.18697
$756$	756.000	0.0363696
$757$	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
$759$	9600.00	0.459101
$760$	3840.00	0.183278
$761$	−5052.00	−0.240650	−0.120325	−	0.992735i	$-0.538394\pi$
$761$	−5052.00	−0.240650	−0.120325	+	0.992735i	$0.538394\pi$
$762$	−13680.0	−0.650360
$763$	6510.00	0.308883
$764$	11504.0	0.544765
$765$	6264.00	0.296046
$766$	22404.0	1.05677
$767$	8034.00	0.378215
$768$	768.000	0.0360844
$769$	−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
$771$	−11418.0	−0.533345
$772$	12232.0	0.570258
$773$	−1472.00	−0.0684918	−0.0342459	−	0.999413i	$-0.510903\pi$
$773$	−1472.00	−0.0684918	−0.0342459	+	0.999413i	$0.510903\pi$
$774$	−216.000	−0.0100310
$775$	2356.00	0.109200
$776$	8176.00	0.378223
$777$	−1050.00	−0.0484795
$778$	−21716.0	−1.00072
$779$	−3360.00	−0.154537
$780$	1872.00	0.0859338
$781$	19500.0	0.893425
$782$	7424.00	0.339491
$783$	−2970.00	−0.135554
$784$	784.000	0.0357143
$785$	30264.0	1.37601
$786$	−9312.00	−0.422580
$787$	−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
$789$	−11448.0	−0.516552
$790$	16320.0	0.734987
$791$	−7238.00	−0.325352
$792$	−3600.00	−0.161516
$793$	3614.00	0.161837
$794$	−11700.0	−0.522944
$795$	15912.0	0.709862
$796$	12256.0	0.545732
$797$	13986.0	0.621593	0.310796	−	0.950476i	$-0.399404\pi$
$797$	13986.0	0.621593	0.310796	+	0.950476i	$0.399404\pi$
$798$	−1680.00	−0.0745255
$799$	4756.00	0.210582
$800$	608.000	0.0268701
$801$	8712.00	0.384299
$802$	−22808.0	−1.00421
$803$	100.000	0.00439467
$804$	240.000	0.0105275
$805$	5376.00	0.235378
$806$	−3224.00	−0.140894
$807$	−9078.00	−0.395986
$808$	432.000	0.0188090
$809$	−30738.0	−1.33584	−0.667918	−	0.744235i	$-0.732814\pi$
$809$	−30738.0	−1.33584	−0.667918	+	0.744235i	$0.732814\pi$
$810$	−1944.00	−0.0843274
$811$	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
$813$	7440.00	0.320950
$814$	5000.00	0.215295
$815$	10176.0	0.437362
$816$	−2784.00	−0.119436
$817$	480.000	0.0205546
$818$	−20388.0	−0.871454
$819$	−819.000	−0.0349428
$820$	−4032.00	−0.171712
$821$	23484.0	0.998291	0.499146	−	0.866518i	$-0.333647\pi$
$821$	23484.0	0.998291	0.499146	+	0.866518i	$0.333647\pi$
$822$	288.000	0.0122204
$823$	−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
$825$	−2850.00	−0.120272
$826$	−8652.00	−0.364457
$827$	25734.0	1.08205	0.541027	−	0.841005i	$-0.318036\pi$
$827$	25734.0	1.08205	0.541027	+	0.841005i	$0.318036\pi$
$828$	−2304.00	−0.0967023
$829$	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
$831$	10770.0	0.449587
$832$	−832.000	−0.0346688
$833$	−2842.00	−0.118211
$834$	168.000	0.00697526
$835$	−10248.0	−0.424727
$836$	8000.00	0.330964
$837$	3348.00	0.138260
$838$	6384.00	0.263164
$839$	8330.00	0.342769	0.171385	−	0.985204i	$-0.445176\pi$
$839$	8330.00	0.342769	0.171385	+	0.985204i	$0.445176\pi$
$840$	−2016.00	−0.0828079
$841$	−12289.0	−0.503875
$842$	20948.0	0.857382
$843$	14496.0	0.592252
$844$	2416.00	0.0985334
$845$	−2028.00	−0.0825625
$846$	−1476.00	−0.0599834
$847$	8183.00	0.331961
$848$	−7072.00	−0.286384
$849$	−17100.0	−0.691249
$850$	−2204.00	−0.0889371
$851$	3200.00	0.128901
$852$	−4680.00	−0.188186
$853$	−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
$855$	4320.00	0.172796
$856$	10016.0	0.399930
$857$	−40126.0	−1.59939	−0.799695	−	0.600406i	$-0.795006\pi$
$857$	−40126.0	−1.59939	−0.799695	+	0.600406i	$0.795006\pi$
$858$	3900.00	0.155179
$859$	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
$861$	1764.00	0.0698223
$862$	15804.0	0.624462
$863$	−6714.00	−0.264829	−0.132414	−	0.991194i	$-0.542273\pi$
$863$	−6714.00	−0.264829	−0.132414	+	0.991194i	$0.542273\pi$
$864$	864.000	0.0340207
$865$	−24552.0	−0.965079
$866$	−33908.0	−1.33053
$867$	−4647.00	−0.182030
$868$	3472.00	0.135769
$869$	34000.0	1.32724
$870$	7920.00	0.308636
$871$	−260.000	−0.0101145
$872$	7440.00	0.288934
$873$	9198.00	0.356592
$874$	5120.00	0.198154
$875$	8904.00	0.344012
$876$	−24.0000	−0.000925668 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
$879$	−19152.0	−0.734904
$880$	9600.00	0.367745
$881$	18018.0	0.689037	0.344519	−	0.938779i	$-0.388042\pi$
$881$	18018.0	0.689037	0.344519	+	0.938779i	$0.388042\pi$
$882$	882.000	0.0336718
$883$	−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
$885$	22248.0	0.845038
$886$	−3768.00	−0.142876
$887$	−35544.0	−1.34549	−0.672746	−	0.739874i	$-0.734885\pi$
$887$	−35544.0	−1.34549	−0.672746	+	0.739874i	$0.734885\pi$
$888$	−1200.00	−0.0453484
$889$	−15960.0	−0.602116
$890$	−23232.0	−0.874987
$891$	−4050.00	−0.152278
$892$	22512.0	0.845020
$893$	3280.00	0.122913
$894$	15576.0	0.582706
$895$	20256.0	0.756518
$896$	896.000	0.0334077
$897$	2496.00	0.0929086
$898$	−22872.0	−0.849943
$899$	−13640.0	−0.506028
$900$	684.000	0.0253333
$901$	25636.0	0.947901
$902$	−8400.00	−0.310077
$903$	−252.000	−0.00928686
$904$	−8272.00	−0.304339
$905$	3192.00	0.117244
$906$	−12312.0	−0.451478
$907$	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
$909$	486.000	0.0177333
$910$	2184.00	0.0795592
$911$	30172.0	1.09730	0.548651	−	0.836051i	$-0.315141\pi$
$911$	30172.0	1.09730	0.548651	+	0.836051i	$0.315141\pi$
$912$	−1920.00	−0.0697122
$913$	−16100.0	−0.583606
$914$	22972.0	0.831342
$915$	10008.0	0.361589
$916$	17704.0	0.638599
$917$	−10864.0	−0.391233
$918$	−3132.00	−0.112605
$919$	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
$921$	11508.0	0.411728
$922$	22344.0	0.798113
$923$	5070.00	0.180803
$924$	−4200.00	−0.149534
$925$	−950.000	−0.0337684
$926$	−23376.0	−0.829571
$927$	12744.0	0.451530
$928$	−3520.00	−0.124515
$929$	−28460.0	−1.00510	−0.502552	−	0.864547i	$-0.667606\pi$
$929$	−28460.0	−1.00510	−0.502552	+	0.864547i	$0.667606\pi$
$930$	−8928.00	−0.314796
$931$	−1960.00	−0.0689972
$932$	8760.00	0.307879
$933$	18348.0	0.643823
$934$	−4896.00	−0.171522
$935$	−34800.0	−1.21720
$936$	−936.000	−0.0326860
$937$	−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
$939$	−1926.00	−0.0669357
$940$	3936.00	0.136573
$941$	49992.0	1.73187	0.865937	−	0.500154i	$-0.166723\pi$
$941$	49992.0	1.73187	0.865937	+	0.500154i	$0.166723\pi$
$942$	−15132.0	−0.523383
$943$	−5376.00	−0.185649
$944$	−9888.00	−0.340918
$945$	−2268.00	−0.0780720
$946$	1200.00	0.0412425
$947$	24338.0	0.835141	0.417571	−	0.908644i	$-0.362882\pi$
$947$	24338.0	0.835141	0.417571	+	0.908644i	$0.362882\pi$
$948$	−8160.00	−0.279562
$949$	26.0000	0.000889353 0
$950$	−1520.00	−0.0519109
$951$	−27408.0	−0.934559
$952$	−3248.00	−0.110576
$953$	24558.0	0.834745	0.417372	−	0.908736i	$-0.362951\pi$
$953$	24558.0	0.834745	0.417372	+	0.908736i	$0.362951\pi$
$954$	−7956.00	−0.270005
$955$	−34512.0	−1.16941
$956$	−4920.00	−0.166448
$957$	16500.0	0.557335
$958$	1396.00	0.0470801
$959$	336.000	0.0113139
$960$	−2304.00	−0.0774597
$961$	−14415.0	−0.483871
$962$	1300.00	0.0435693
$963$	11268.0	0.377057
$964$	−3144.00	−0.105043
$965$	−36696.0	−1.22413
$966$	−2688.00	−0.0895290
$967$	−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
$969$	6960.00	0.230740
$970$	−24528.0	−0.811904
$971$	39300.0	1.29886	0.649432	−	0.760420i	$-0.275007\pi$
$971$	39300.0	1.29886	0.649432	+	0.760420i	$0.275007\pi$
$972$	972.000	0.0320750
$973$	196.000	0.00645783
$974$	−16312.0	−0.536622
$975$	−741.000	−0.0243395
$976$	−4448.00	−0.145878
$977$	3940.00	0.129019	0.0645096	−	0.997917i	$-0.479452\pi$
$977$	3940.00	0.129019	0.0645096	+	0.997917i	$0.479452\pi$
$978$	−5088.00	−0.166356
$979$	−48400.0	−1.58005
$980$	−2352.00	−0.0766652
$981$	8370.00	0.272409
$982$	−20968.0	−0.681381
$983$	33682.0	1.09287	0.546434	−	0.837502i	$-0.315985\pi$
$983$	33682.0	1.09287	0.546434	+	0.837502i	$0.315985\pi$
$984$	2016.00	0.0653127
$985$	35424.0	1.14589
$986$	12760.0	0.412131
$987$	−1722.00	−0.0555338
$988$	2080.00	0.0669773
$989$	768.000	0.0246926
$990$	10800.0	0.346714
$991$	−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
$993$	−9180.00	−0.293372
$994$	−5460.00	−0.174226
$995$	−36768.0	−1.17148
$996$	3864.00	0.122927
$997$	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
$999$	−1350.00	−0.0427549

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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