LMFDB - Embedded newform 57.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,6,Mod(1,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("57.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	57.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:	$9.14187772934$
Analytic rank:	$0$
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	$\chi$	$=$	57.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +9.00000 q^{3} -28.0000 q^{4} -98.0000 q^{5} -18.0000 q^{6} +240.000 q^{7} +120.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +9.00000 q^{3} -28.0000 q^{4} -98.0000 q^{5} -18.0000 q^{6} +240.000 q^{7} +120.000 q^{8} +81.0000 q^{9} +196.000 q^{10} +336.000 q^{11} -252.000 q^{12} +342.000 q^{13} -480.000 q^{14} -882.000 q^{15} +656.000 q^{16} -6.00000 q^{17} -162.000 q^{18} -361.000 q^{19} +2744.00 q^{20} +2160.00 q^{21} -672.000 q^{22} +2836.00 q^{23} +1080.00 q^{24} +6479.00 q^{25} -684.000 q^{26} +729.000 q^{27} -6720.00 q^{28} -5902.00 q^{29} +1764.00 q^{30} +2744.00 q^{31} -5152.00 q^{32} +3024.00 q^{33} +12.0000 q^{34} -23520.0 q^{35} -2268.00 q^{36} +13670.0 q^{37} +722.000 q^{38} +3078.00 q^{39} -11760.0 q^{40} +10990.0 q^{41} -4320.00 q^{42} -4996.00 q^{43} -9408.00 q^{44} -7938.00 q^{45} -5672.00 q^{46} -17124.0 q^{47} +5904.00 q^{48} +40793.0 q^{49} -12958.0 q^{50} -54.0000 q^{51} -9576.00 q^{52} -4470.00 q^{53} -1458.00 q^{54} -32928.0 q^{55} +28800.0 q^{56} -3249.00 q^{57} +11804.0 q^{58} +26292.0 q^{59} +24696.0 q^{60} +29134.0 q^{61} -5488.00 q^{62} +19440.0 q^{63} -10688.0 q^{64} -33516.0 q^{65} -6048.00 q^{66} -42052.0 q^{67} +168.000 q^{68} +25524.0 q^{69} +47040.0 q^{70} -26112.0 q^{71} +9720.00 q^{72} -49046.0 q^{73} -27340.0 q^{74} +58311.0 q^{75} +10108.0 q^{76} +80640.0 q^{77} -6156.00 q^{78} +79056.0 q^{79} -64288.0 q^{80} +6561.00 q^{81} -21980.0 q^{82} +9472.00 q^{83} -60480.0 q^{84} +588.000 q^{85} +9992.00 q^{86} -53118.0 q^{87} +40320.0 q^{88} +82894.0 q^{89} +15876.0 q^{90} +82080.0 q^{91} -79408.0 q^{92} +24696.0 q^{93} +34248.0 q^{94} +35378.0 q^{95} -46368.0 q^{96} +39850.0 q^{97} -81586.0 q^{98} +27216.0 q^{99} +O(q^{100})\)

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	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−2.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−28.0000	−0.875000
$5$	−98.0000	−1.75308	−0.876539	−	0.481331i	$-0.840153\pi$
$5$	−98.0000	−1.75308	−0.876539	+	0.481331i	$0.840153\pi$
$6$	−18.0000	−0.204124
$7$	240.000	1.85125	0.925627	−	0.378436i	$-0.123538\pi$
$7$	240.000	1.85125	0.925627	+	0.378436i	$0.123538\pi$
$8$	120.000	0.662913
$9$	81.0000	0.333333
$10$	196.000	0.619806
$11$	336.000	0.837255	0.418627	−	0.908158i	$-0.362511\pi$
$11$	336.000	0.837255	0.418627	+	0.908158i	$0.362511\pi$
$12$	−252.000	−0.505181
$13$	342.000	0.561265	0.280632	−	0.959815i	$-0.409456\pi$
$13$	342.000	0.561265	0.280632	+	0.959815i	$0.409456\pi$
$14$	−480.000	−0.654517
$15$	−882.000	−1.01214
$16$	656.000	0.640625
$17$	−6.00000	−0.00503534	−0.00251767	−	0.999997i	$-0.500801\pi$
$17$	−6.00000	−0.00503534	−0.00251767	+	0.999997i	$0.500801\pi$
$18$	−162.000	−0.117851
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	2744.00	1.53394
$21$	2160.00	1.06882
$22$	−672.000	−0.296014
$23$	2836.00	1.11786	0.558929	−	0.829216i	$-0.311212\pi$
$23$	2836.00	1.11786	0.558929	+	0.829216i	$0.311212\pi$
$24$	1080.00	0.382733
$25$	6479.00	2.07328
$26$	−684.000	−0.198437
$27$	729.000	0.192450
$28$	−6720.00	−1.61985
$29$	−5902.00	−1.30318	−0.651590	−	0.758572i	$-0.725898\pi$
$29$	−5902.00	−1.30318	−0.651590	+	0.758572i	$0.725898\pi$
$30$	1764.00	0.357845
$31$	2744.00	0.512838	0.256419	−	0.966566i	$-0.417457\pi$
$31$	2744.00	0.512838	0.256419	+	0.966566i	$0.417457\pi$
$32$	−5152.00	−0.889408
$33$	3024.00	0.483389
$34$	12.0000	0.00178026
$35$	−23520.0	−3.24539
$36$	−2268.00	−0.291667
$37$	13670.0	1.64159	0.820794	−	0.571224i	$-0.193531\pi$
$37$	13670.0	1.64159	0.820794	+	0.571224i	$0.193531\pi$
$38$	722.000	0.0811107
$39$	3078.00	0.324046
$40$	−11760.0	−1.16214
$41$	10990.0	1.02103	0.510514	−	0.859869i	$-0.329455\pi$
$41$	10990.0	1.02103	0.510514	+	0.859869i	$0.329455\pi$
$42$	−4320.00	−0.377886
$43$	−4996.00	−0.412051	−0.206026	−	0.978547i	$-0.566053\pi$
$43$	−4996.00	−0.412051	−0.206026	+	0.978547i	$0.566053\pi$
$44$	−9408.00	−0.732598
$45$	−7938.00	−0.584359
$46$	−5672.00	−0.395222
$47$	−17124.0	−1.13073	−0.565367	−	0.824839i	$-0.691266\pi$
$47$	−17124.0	−1.13073	−0.565367	+	0.824839i	$0.691266\pi$
$48$	5904.00	0.369865
$49$	40793.0	2.42714
$50$	−12958.0	−0.733015
$51$	−54.0000	−0.00290716
$52$	−9576.00	−0.491107
$53$	−4470.00	−0.218584	−0.109292	−	0.994010i	$-0.534858\pi$
$53$	−4470.00	−0.218584	−0.109292	+	0.994010i	$0.534858\pi$
$54$	−1458.00	−0.0680414
$55$	−32928.0	−1.46777
$56$	28800.0	1.22722
$57$	−3249.00	−0.132453
$58$	11804.0	0.460743
$59$	26292.0	0.983317	0.491659	−	0.870788i	$-0.336391\pi$
$59$	26292.0	0.983317	0.491659	+	0.870788i	$0.336391\pi$
$60$	24696.0	0.885622
$61$	29134.0	1.00248	0.501240	−	0.865308i	$-0.332877\pi$
$61$	29134.0	1.00248	0.501240	+	0.865308i	$0.332877\pi$
$62$	−5488.00	−0.181315
$63$	19440.0	0.617085
$64$	−10688.0	−0.326172
$65$	−33516.0	−0.983940
$66$	−6048.00	−0.170904
$67$	−42052.0	−1.14446	−0.572229	−	0.820094i	$-0.693921\pi$
$67$	−42052.0	−1.14446	−0.572229	+	0.820094i	$0.693921\pi$
$68$	168.000	0.00440592
$69$	25524.0	0.645396
$70$	47040.0	1.14742
$71$	−26112.0	−0.614744	−0.307372	−	0.951589i	$-0.599450\pi$
$71$	−26112.0	−0.614744	−0.307372	+	0.951589i	$0.599450\pi$
$72$	9720.00	0.220971
$73$	−49046.0	−1.07720	−0.538600	−	0.842562i	$-0.681047\pi$
$73$	−49046.0	−1.07720	−0.538600	+	0.842562i	$0.681047\pi$
$74$	−27340.0	−0.580389
$75$	58311.0	1.19701
$76$	10108.0	0.200739
$77$	80640.0	1.54997
$78$	−6156.00	−0.114568
$79$	79056.0	1.42517	0.712586	−	0.701585i	$-0.247524\pi$
$79$	79056.0	1.42517	0.712586	+	0.701585i	$0.247524\pi$
$80$	−64288.0	−1.12307
$81$	6561.00	0.111111
$82$	−21980.0	−0.360988
$83$	9472.00	0.150920	0.0754599	−	0.997149i	$-0.475958\pi$
$83$	9472.00	0.150920	0.0754599	+	0.997149i	$0.475958\pi$
$84$	−60480.0	−0.935220
$85$	588.000	0.00882734
$86$	9992.00	0.145682
$87$	−53118.0	−0.752391
$88$	40320.0	0.555027
$89$	82894.0	1.10930	0.554649	−	0.832085i	$-0.312853\pi$
$89$	82894.0	1.10930	0.554649	+	0.832085i	$0.312853\pi$
$90$	15876.0	0.206602
$91$	82080.0	1.03904
$92$	−79408.0	−0.978126
$93$	24696.0	0.296087
$94$	34248.0	0.399775
$95$	35378.0	0.402184
$96$	−46368.0	−0.513500
$97$	39850.0	0.430030	0.215015	−	0.976611i	$-0.431020\pi$
$97$	39850.0	0.430030	0.215015	+	0.976611i	$0.431020\pi$
$98$	−81586.0	−0.858125
$99$	27216.0	0.279085
$100$	−181412.	−1.81412
$101$	99542.0	0.970964	0.485482	−	0.874247i	$-0.338644\pi$
$101$	99542.0	0.970964	0.485482	+	0.874247i	$0.338644\pi$
$102$	108.000	0.00102783
$103$	69368.0	0.644267	0.322134	−	0.946694i	$-0.395600\pi$
$103$	69368.0	0.644267	0.322134	+	0.946694i	$0.395600\pi$
$104$	41040.0	0.372069
$105$	−211680.	−1.87373
$106$	8940.00	0.0772810
$107$	−110204.	−0.930546	−0.465273	−	0.885167i	$-0.654044\pi$
$107$	−110204.	−0.930546	−0.465273	+	0.885167i	$0.654044\pi$
$108$	−20412.0	−0.168394
$109$	−217786.	−1.75575	−0.877877	−	0.478886i	$-0.841041\pi$
$109$	−217786.	−1.75575	−0.877877	+	0.478886i	$0.841041\pi$
$110$	65856.0	0.518936
$111$	123030.	0.947771
$112$	157440.	1.18596
$113$	−84618.0	−0.623400	−0.311700	−	0.950181i	$-0.600898\pi$
$113$	−84618.0	−0.623400	−0.311700	+	0.950181i	$0.600898\pi$
$114$	6498.00	0.0468293
$115$	−277928.	−1.95969
$116$	165256.	1.14028
$117$	27702.0	0.187088
$118$	−52584.0	−0.347655
$119$	−1440.00	−0.00932170
$120$	−105840.	−0.670960
$121$	−48155.0	−0.299005
$122$	−58268.0	−0.354430
$123$	98910.0	0.589491
$124$	−76832.0	−0.448733
$125$	−328692.	−1.88154
$126$	−38880.0	−0.218172
$127$	89608.0	0.492989	0.246495	−	0.969144i	$-0.420721\pi$
$127$	89608.0	0.492989	0.246495	+	0.969144i	$0.420721\pi$
$128$	186240.	1.00473
$129$	−44964.0	−0.237898
$130$	67032.0	0.347875
$131$	255992.	1.30331	0.651656	−	0.758515i	$-0.274075\pi$
$131$	255992.	1.30331	0.651656	+	0.758515i	$0.274075\pi$
$132$	−84672.0	−0.422966
$133$	−86640.0	−0.424707
$134$	84104.0	0.404627
$135$	−71442.0	−0.337380
$136$	−720.000	−0.00333799
$137$	50226.0	0.228627	0.114313	−	0.993445i	$-0.463533\pi$
$137$	50226.0	0.228627	0.114313	+	0.993445i	$0.463533\pi$
$138$	−51048.0	−0.228182
$139$	−242108.	−1.06285	−0.531425	−	0.847105i	$-0.678343\pi$
$139$	−242108.	−1.06285	−0.531425	+	0.847105i	$0.678343\pi$
$140$	658560.	2.83972
$141$	−154116.	−0.652830
$142$	52224.0	0.217345
$143$	114912.	0.469921
$144$	53136.0	0.213542
$145$	578396.	2.28457
$146$	98092.0	0.380848
$147$	367137.	1.40131
$148$	−382760.	−1.43639
$149$	−352506.	−1.30077	−0.650385	−	0.759604i	$-0.725393\pi$
$149$	−352506.	−1.30077	−0.650385	+	0.759604i	$0.725393\pi$
$150$	−116622.	−0.423207
$151$	−272120.	−0.971221	−0.485611	−	0.874175i	$-0.661403\pi$
$151$	−272120.	−0.971221	−0.485611	+	0.874175i	$0.661403\pi$
$152$	−43320.0	−0.152083
$153$	−486.000	−0.00167845
$154$	−161280.	−0.547998
$155$	−268912.	−0.899044
$156$	−86184.0	−0.283541
$157$	124942.	0.404538	0.202269	−	0.979330i	$-0.435168\pi$
$157$	124942.	0.404538	0.202269	+	0.979330i	$0.435168\pi$
$158$	−158112.	−0.503874
$159$	−40230.0	−0.126199
$160$	504896.	1.55920
$161$	680640.	2.06944
$162$	−13122.0	−0.0392837
$163$	−28900.0	−0.0851979	−0.0425989	−	0.999092i	$-0.513564\pi$
$163$	−28900.0	−0.0851979	−0.0425989	+	0.999092i	$0.513564\pi$
$164$	−307720.	−0.893400
$165$	−296352.	−0.847419
$166$	−18944.0	−0.0533582
$167$	370200.	1.02718	0.513588	−	0.858037i	$-0.328316\pi$
$167$	370200.	1.02718	0.513588	+	0.858037i	$0.328316\pi$
$168$	259200.	0.708536
$169$	−254329.	−0.684982
$170$	−1176.00	−0.00312094
$171$	−29241.0	−0.0764719
$172$	139888.	0.360545
$173$	−277414.	−0.704714	−0.352357	−	0.935866i	$-0.614620\pi$
$173$	−277414.	−0.704714	−0.352357	+	0.935866i	$0.614620\pi$
$174$	106236.	0.266010
$175$	1.55496e6	3.83817
$176$	220416.	0.536366
$177$	236628.	0.567718
$178$	−165788.	−0.392196
$179$	−439108.	−1.02433	−0.512164	−	0.858888i	$-0.671156\pi$
$179$	−439108.	−1.02433	−0.512164	+	0.858888i	$0.671156\pi$
$180$	222264.	0.511314
$181$	25406.0	0.0576421	0.0288211	−	0.999585i	$-0.490825\pi$
$181$	25406.0	0.0576421	0.0288211	+	0.999585i	$0.490825\pi$
$182$	−164160.	−0.367357
$183$	262206.	0.578782
$184$	340320.	0.741042
$185$	−1.33966e6	−2.87783
$186$	−49392.0	−0.104683
$187$	−2016.00	−0.00421586
$188$	479472.	0.989393
$189$	174960.	0.356274
$190$	−70756.0	−0.142193
$191$	640644.	1.27067	0.635336	−	0.772236i	$-0.280862\pi$
$191$	640644.	1.27067	0.635336	+	0.772236i	$0.280862\pi$
$192$	−96192.0	−0.188315
$193$	−224846.	−0.434502	−0.217251	−	0.976116i	$-0.569709\pi$
$193$	−224846.	−0.434502	−0.217251	+	0.976116i	$0.569709\pi$
$194$	−79700.0	−0.152039
$195$	−301644.	−0.568078
$196$	−1.14220e6	−2.12375
$197$	325438.	0.597452	0.298726	−	0.954339i	$-0.403438\pi$
$197$	325438.	0.597452	0.298726	+	0.954339i	$0.403438\pi$
$198$	−54432.0	−0.0986714
$199$	−983576.	−1.76066	−0.880329	−	0.474363i	$-0.842679\pi$
$199$	−983576.	−1.76066	−0.880329	+	0.474363i	$0.842679\pi$
$200$	777480.	1.37440
$201$	−378468.	−0.660753
$202$	−199084.	−0.343287
$203$	−1.41648e6	−2.41252
$204$	1512.00	0.00254376
$205$	−1.07702e6	−1.78994
$206$	−138736.	−0.227783
$207$	229716.	0.372619
$208$	224352.	0.359560
$209$	−121296.	−0.192079
$210$	423360.	0.662463
$211$	−866308.	−1.33957	−0.669786	−	0.742554i	$-0.733614\pi$
$211$	−866308.	−1.33957	−0.669786	+	0.742554i	$0.733614\pi$
$212$	125160.	0.191261
$213$	−235008.	−0.354923
$214$	220408.	0.328998
$215$	489608.	0.722358
$216$	87480.0	0.127578
$217$	658560.	0.949393
$218$	435572.	0.620753
$219$	−441414.	−0.621922
$220$	921984.	1.28430
$221$	−2052.00	−0.00282616
$222$	−246060.	−0.335088
$223$	−76928.0	−0.103591	−0.0517955	−	0.998658i	$-0.516494\pi$
$223$	−76928.0	−0.103591	−0.0517955	+	0.998658i	$0.516494\pi$
$224$	−1.23648e6	−1.64652
$225$	524799.	0.691093
$226$	169236.	0.220405
$227$	−969612.	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$227$	−969612.	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$228$	90972.0	0.115897
$229$	840150.	1.05869	0.529344	−	0.848407i	$-0.322438\pi$
$229$	840150.	1.05869	0.529344	+	0.848407i	$0.322438\pi$
$230$	555856.	0.692856
$231$	725760.	0.894876
$232$	−708240.	−0.863894
$233$	575914.	0.694973	0.347486	−	0.937685i	$-0.387035\pi$
$233$	575914.	0.694973	0.347486	+	0.937685i	$0.387035\pi$
$234$	−55404.0	−0.0661457
$235$	1.67815e6	1.98226
$236$	−736176.	−0.860402
$237$	711504.	0.822823
$238$	2880.00	0.00329572
$239$	−352188.	−0.398823	−0.199411	−	0.979916i	$-0.563903\pi$
$239$	−352188.	−0.398823	−0.199411	+	0.979916i	$0.563903\pi$
$240$	−578592.	−0.648402
$241$	451290.	0.500510	0.250255	−	0.968180i	$-0.419485\pi$
$241$	451290.	0.500510	0.250255	+	0.968180i	$0.419485\pi$
$242$	96310.0	0.105714
$243$	59049.0	0.0641500
$244$	−815752.	−0.877170
$245$	−3.99771e6	−4.25497
$246$	−197820.	−0.208417
$247$	−123462.	−0.128763
$248$	329280.	0.339967
$249$	85248.0	0.0871336
$250$	657384.	0.665226
$251$	145752.	0.146026	0.0730130	−	0.997331i	$-0.476739\pi$
$251$	145752.	0.146026	0.0730130	+	0.997331i	$0.476739\pi$
$252$	−544320.	−0.539949
$253$	952896.	0.935932
$254$	−179216.	−0.174298
$255$	5292.00	0.00509647
$256$	−30464.0	−0.0290527
$257$	−1.87818e6	−1.77380	−0.886899	−	0.461964i	$-0.847145\pi$
$257$	−1.87818e6	−1.77380	−0.886899	+	0.461964i	$0.847145\pi$
$258$	89928.0	0.0841096
$259$	3.28080e6	3.03900
$260$	938448.	0.860948
$261$	−478062.	−0.434393
$262$	−511984.	−0.460790
$263$	895068.	0.797933	0.398967	−	0.916965i	$-0.369369\pi$
$263$	895068.	0.797933	0.398967	+	0.916965i	$0.369369\pi$
$264$	362880.	0.320445
$265$	438060.	0.383194
$266$	173280.	0.150157
$267$	746046.	0.640453
$268$	1.17746e6	1.00140
$269$	1.27291e6	1.07255	0.536273	−	0.844045i	$-0.319832\pi$
$269$	1.27291e6	1.07255	0.536273	+	0.844045i	$0.319832\pi$
$270$	142884.	0.119282
$271$	306672.	0.253659	0.126830	−	0.991925i	$-0.459520\pi$
$271$	306672.	0.253659	0.126830	+	0.991925i	$0.459520\pi$
$272$	−3936.00	−0.00322577
$273$	738720.	0.599892
$274$	−100452.	−0.0808318
$275$	2.17694e6	1.73586
$276$	−714672.	−0.564721
$277$	−192026.	−0.150370	−0.0751849	−	0.997170i	$-0.523955\pi$
$277$	−192026.	−0.150370	−0.0751849	+	0.997170i	$0.523955\pi$
$278$	484216.	0.375774
$279$	222264.	0.170946
$280$	−2.82240e6	−2.15141
$281$	1.00055e6	0.755915	0.377958	−	0.925823i	$-0.376627\pi$
$281$	1.00055e6	0.755915	0.377958	+	0.925823i	$0.376627\pi$
$282$	308232.	0.230810
$283$	−1.26847e6	−0.941485	−0.470743	−	0.882271i	$-0.656014\pi$
$283$	−1.26847e6	−0.941485	−0.470743	+	0.882271i	$0.656014\pi$
$284$	731136.	0.537901
$285$	318402.	0.232201
$286$	−229824.	−0.166142
$287$	2.63760e6	1.89018
$288$	−417312.	−0.296469
$289$	−1.41982e6	−0.999975
$290$	−1.15679e6	−0.807719
$291$	358650.	0.248278
$292$	1.37329e6	0.942550
$293$	−1.52560e6	−1.03818	−0.519088	−	0.854721i	$-0.673728\pi$
$293$	−1.52560e6	−1.03818	−0.519088	+	0.854721i	$0.673728\pi$
$294$	−734274.	−0.495439
$295$	−2.57662e6	−1.72383
$296$	1.64040e6	1.08823
$297$	244944.	0.161130
$298$	705012.	0.459892
$299$	969912.	0.627414
$300$	−1.63271e6	−1.04738
$301$	−1.19904e6	−0.762812
$302$	544240.	0.343379
$303$	895878.	0.560586
$304$	−236816.	−0.146969
$305$	−2.85513e6	−1.75742
$306$	972.000	0.000593421 0
$307$	−1.19665e6	−0.724639	−0.362320	−	0.932054i	$-0.618015\pi$
$307$	−1.19665e6	−0.724639	−0.362320	+	0.932054i	$0.618015\pi$
$308$	−2.25792e6	−1.35623
$309$	624312.	0.371968
$310$	537824.	0.317860
$311$	2.37144e6	1.39031	0.695155	−	0.718859i	$-0.255336\pi$
$311$	2.37144e6	1.39031	0.695155	+	0.718859i	$0.255336\pi$
$312$	369360.	0.214814
$313$	353738.	0.204090	0.102045	−	0.994780i	$-0.467461\pi$
$313$	353738.	0.204090	0.102045	+	0.994780i	$0.467461\pi$
$314$	−249884.	−0.143026
$315$	−1.90512e6	−1.08180
$316$	−2.21357e6	−1.24703
$317$	2.70427e6	1.51148	0.755738	−	0.654874i	$-0.227278\pi$
$317$	2.70427e6	1.51148	0.755738	+	0.654874i	$0.227278\pi$
$318$	80460.0	0.0446182
$319$	−1.98307e6	−1.09109
$320$	1.04742e6	0.571805
$321$	−991836.	−0.537251
$322$	−1.36128e6	−0.731657
$323$	2166.00	0.00115519
$324$	−183708.	−0.0972222
$325$	2.21582e6	1.16366
$326$	57800.0	0.0301220
$327$	−1.96007e6	−1.01369
$328$	1.31880e6	0.676853
$329$	−4.10976e6	−2.09328
$330$	592704.	0.299608
$331$	−327444.	−0.164273	−0.0821367	−	0.996621i	$-0.526174\pi$
$331$	−327444.	−0.164273	−0.0821367	+	0.996621i	$0.526174\pi$
$332$	−265216.	−0.132055
$333$	1.10727e6	0.547196
$334$	−740400.	−0.363162
$335$	4.12110e6	2.00632
$336$	1.41696e6	0.684714
$337$	367946.	0.176486	0.0882428	−	0.996099i	$-0.471875\pi$
$337$	367946.	0.176486	0.0882428	+	0.996099i	$0.471875\pi$
$338$	508658.	0.242178
$339$	−761562.	−0.359920
$340$	−16464.0	−0.00772393
$341$	921984.	0.429376
$342$	58482.0	0.0270369
$343$	5.75664e6	2.64201
$344$	−599520.	−0.273154
$345$	−2.50135e6	−1.13143
$346$	554828.	0.249154
$347$	566160.	0.252415	0.126208	−	0.992004i	$-0.459719\pi$
$347$	566160.	0.252415	0.126208	+	0.992004i	$0.459719\pi$
$348$	1.48730e6	0.658342
$349$	−4.50687e6	−1.98067	−0.990333	−	0.138713i	$-0.955703\pi$
$349$	−4.50687e6	−1.98067	−0.990333	+	0.138713i	$0.955703\pi$
$350$	−3.10992e6	−1.35700
$351$	249318.	0.108015
$352$	−1.73107e6	−0.744661
$353$	−1.09778e6	−0.468899	−0.234450	−	0.972128i	$-0.575329\pi$
$353$	−1.09778e6	−0.468899	−0.234450	+	0.972128i	$0.575329\pi$
$354$	−473256.	−0.200719
$355$	2.55898e6	1.07769
$356$	−2.32103e6	−0.970635
$357$	−12960.0	−0.00538189
$358$	878216.	0.362154
$359$	−1.95223e6	−0.799456	−0.399728	−	0.916634i	$-0.630895\pi$
$359$	−1.95223e6	−0.799456	−0.399728	+	0.916634i	$0.630895\pi$
$360$	−952560.	−0.387379
$361$	130321.	0.0526316
$362$	−50812.0	−0.0203796
$363$	−433395.	−0.172630
$364$	−2.29824e6	−0.909163
$365$	4.80651e6	1.88842
$366$	−524412.	−0.204630
$367$	1.49163e6	0.578091	0.289046	−	0.957315i	$-0.406662\pi$
$367$	1.49163e6	0.578091	0.289046	+	0.957315i	$0.406662\pi$
$368$	1.86042e6	0.716128
$369$	890190.	0.340343
$370$	2.67932e6	1.01747
$371$	−1.07280e6	−0.404654
$372$	−691488.	−0.259076
$373$	1.48305e6	0.551928	0.275964	−	0.961168i	$-0.411003\pi$
$373$	1.48305e6	0.551928	0.275964	+	0.961168i	$0.411003\pi$
$374$	4032.00	0.00149053
$375$	−2.95823e6	−1.08631
$376$	−2.05488e6	−0.749578
$377$	−2.01848e6	−0.731429
$378$	−349920.	−0.125962
$379$	−2.79436e6	−0.999272	−0.499636	−	0.866235i	$-0.666533\pi$
$379$	−2.79436e6	−0.999272	−0.499636	+	0.866235i	$0.666533\pi$
$380$	−990584.	−0.351911
$381$	806472.	0.284627
$382$	−1.28129e6	−0.449250
$383$	2.06910e6	0.720748	0.360374	−	0.932808i	$-0.382649\pi$
$383$	2.06910e6	0.720748	0.360374	+	0.932808i	$0.382649\pi$
$384$	1.67616e6	0.580079
$385$	−7.90272e6	−2.71722
$386$	449692.	0.153620
$387$	−404676.	−0.137350
$388$	−1.11580e6	−0.376276
$389$	4.61696e6	1.54697	0.773485	−	0.633815i	$-0.218512\pi$
$389$	4.61696e6	1.54697	0.773485	+	0.633815i	$0.218512\pi$
$390$	603288.	0.200846
$391$	−17016.0	−0.00562880
$392$	4.89516e6	1.60898
$393$	2.30393e6	0.752467
$394$	−650876.	−0.211231
$395$	−7.74749e6	−2.49844
$396$	−762048.	−0.244199
$397$	875870.	0.278910	0.139455	−	0.990228i	$-0.455465\pi$
$397$	875870.	0.278910	0.139455	+	0.990228i	$0.455465\pi$
$398$	1.96715e6	0.622487
$399$	−779760.	−0.245205
$400$	4.25022e6	1.32820
$401$	−3.36615e6	−1.04538	−0.522689	−	0.852524i	$-0.675071\pi$
$401$	−3.36615e6	−1.04538	−0.522689	+	0.852524i	$0.675071\pi$
$402$	756936.	0.233611
$403$	938448.	0.287838
$404$	−2.78718e6	−0.849593
$405$	−642978.	−0.194786
$406$	2.83296e6	0.852954
$407$	4.59312e6	1.37443
$408$	−6480.00	−0.00192719
$409$	6.58655e6	1.94693	0.973463	−	0.228844i	$-0.0734947\pi$
$409$	6.58655e6	1.94693	0.973463	+	0.228844i	$0.0734947\pi$
$410$	2.15404e6	0.632840
$411$	452034.	0.131998
$412$	−1.94230e6	−0.563734
$413$	6.31008e6	1.82037
$414$	−459432.	−0.131741
$415$	−928256.	−0.264574
$416$	−1.76198e6	−0.499193
$417$	−2.17897e6	−0.613637
$418$	242592.	0.0679103
$419$	−1.06775e6	−0.297122	−0.148561	−	0.988903i	$-0.547464\pi$
$419$	−1.06775e6	−0.297122	−0.148561	+	0.988903i	$0.547464\pi$
$420$	5.92704e6	1.63951
$421$	3.60621e6	0.991620	0.495810	−	0.868431i	$-0.334871\pi$
$421$	3.60621e6	0.991620	0.495810	+	0.868431i	$0.334871\pi$
$422$	1.73262e6	0.473610
$423$	−1.38704e6	−0.376911
$424$	−536400.	−0.144902
$425$	−38874.0	−0.0104397
$426$	470016.	0.125484
$427$	6.99216e6	1.85584
$428$	3.08571e6	0.814228
$429$	1.03421e6	0.271309
$430$	−979216.	−0.255392
$431$	1.05310e6	0.273071	0.136535	−	0.990635i	$-0.456403\pi$
$431$	1.05310e6	0.273071	0.136535	+	0.990635i	$0.456403\pi$
$432$	478224.	0.123288
$433$	−3.45697e6	−0.886087	−0.443044	−	0.896500i	$-0.646101\pi$
$433$	−3.45697e6	−0.886087	−0.443044	+	0.896500i	$0.646101\pi$
$434$	−1.31712e6	−0.335661
$435$	5.20556e6	1.31900
$436$	6.09801e6	1.53628
$437$	−1.02380e6	−0.256454
$438$	882828.	0.219883
$439$	−5.88610e6	−1.45769	−0.728847	−	0.684676i	$-0.759944\pi$
$439$	−5.88610e6	−1.45769	−0.728847	+	0.684676i	$0.759944\pi$
$440$	−3.95136e6	−0.973005
$441$	3.30423e6	0.809048
$442$	4104.00	0.000999198 0
$443$	−4.80216e6	−1.16259	−0.581296	−	0.813692i	$-0.697454\pi$
$443$	−4.80216e6	−1.16259	−0.581296	+	0.813692i	$0.697454\pi$
$444$	−3.44484e6	−0.829300
$445$	−8.12361e6	−1.94468
$446$	153856.	0.0366250
$447$	−3.17255e6	−0.751000
$448$	−2.56512e6	−0.603827
$449$	−3.17967e6	−0.744330	−0.372165	−	0.928167i	$-0.621384\pi$
$449$	−3.17967e6	−0.744330	−0.372165	+	0.928167i	$0.621384\pi$
$450$	−1.04960e6	−0.244338
$451$	3.69264e6	0.854861
$452$	2.36930e6	0.545475
$453$	−2.44908e6	−0.560735
$454$	1.93922e6	0.441559
$455$	−8.04384e6	−1.82152
$456$	−389880.	−0.0878049
$457$	483546.	0.108305	0.0541523	−	0.998533i	$-0.482754\pi$
$457$	483546.	0.108305	0.0541523	+	0.998533i	$0.482754\pi$
$458$	−1.68030e6	−0.374303
$459$	−4374.00	−0.000969052 0
$460$	7.78198e6	1.71473
$461$	707982.	0.155156	0.0775782	−	0.996986i	$-0.475281\pi$
$461$	707982.	0.155156	0.0775782	+	0.996986i	$0.475281\pi$
$462$	−1.45152e6	−0.316387
$463$	8.24091e6	1.78658	0.893291	−	0.449479i	$-0.148390\pi$
$463$	8.24091e6	1.78658	0.893291	+	0.449479i	$0.148390\pi$
$464$	−3.87171e6	−0.834849
$465$	−2.42021e6	−0.519063
$466$	−1.15183e6	−0.245710
$467$	2.13640e6	0.453305	0.226652	−	0.973976i	$-0.427222\pi$
$467$	2.13640e6	0.453305	0.226652	+	0.973976i	$0.427222\pi$
$468$	−775656.	−0.163702
$469$	−1.00925e7	−2.11868
$470$	−3.35630e6	−0.700836
$471$	1.12448e6	0.233560
$472$	3.15504e6	0.651853
$473$	−1.67866e6	−0.344992
$474$	−1.42301e6	−0.290912
$475$	−2.33892e6	−0.475643
$476$	40320.0	0.00815649
$477$	−362070.	−0.0728612
$478$	704376.	0.141005
$479$	1.20736e6	0.240436	0.120218	−	0.992748i	$-0.461641\pi$
$479$	1.20736e6	0.240436	0.120218	+	0.992748i	$0.461641\pi$
$480$	4.54406e6	0.900205
$481$	4.67514e6	0.921365
$482$	−902580.	−0.176957
$483$	6.12576e6	1.19479
$484$	1.34834e6	0.261629
$485$	−3.90530e6	−0.753876
$486$	−118098.	−0.0226805
$487$	−8.77970e6	−1.67748	−0.838739	−	0.544533i	$-0.816707\pi$
$487$	−8.77970e6	−1.67748	−0.838739	+	0.544533i	$0.816707\pi$
$488$	3.49608e6	0.664556
$489$	−260100.	−0.0491890
$490$	7.99543e6	1.50436
$491$	4.23400e6	0.792587	0.396294	−	0.918124i	$-0.370296\pi$
$491$	4.23400e6	0.792587	0.396294	+	0.918124i	$0.370296\pi$
$492$	−2.76948e6	−0.515805
$493$	35412.0	0.00656195
$494$	246924.	0.0455246
$495$	−2.66717e6	−0.489257
$496$	1.80006e6	0.328537
$497$	−6.26688e6	−1.13805
$498$	−170496.	−0.0308064
$499$	−1.73450e6	−0.311834	−0.155917	−	0.987770i	$-0.549833\pi$
$499$	−1.73450e6	−0.311834	−0.155917	+	0.987770i	$0.549833\pi$
$500$	9.20338e6	1.64635
$501$	3.33180e6	0.593041
$502$	−291504.	−0.0516280
$503$	−4.72352e6	−0.832427	−0.416214	−	0.909267i	$-0.636643\pi$
$503$	−4.72352e6	−0.832427	−0.416214	+	0.909267i	$0.636643\pi$
$504$	2.33280e6	0.409073
$505$	−9.75512e6	−1.70217
$506$	−1.90579e6	−0.330902
$507$	−2.28896e6	−0.395475
$508$	−2.50902e6	−0.431366
$509$	−2.49741e6	−0.427264	−0.213632	−	0.976914i	$-0.568529\pi$
$509$	−2.49741e6	−0.427264	−0.213632	+	0.976914i	$0.568529\pi$
$510$	−10584.0	−0.00180187
$511$	−1.17710e7	−1.99417
$512$	−5.89875e6	−0.994455
$513$	−263169.	−0.0441511
$514$	3.75636e6	0.627132
$515$	−6.79806e6	−1.12945
$516$	1.25899e6	0.208161
$517$	−5.75366e6	−0.946713
$518$	−6.56160e6	−1.07445
$519$	−2.49673e6	−0.406867
$520$	−4.02192e6	−0.652266
$521$	2.86413e6	0.462272	0.231136	−	0.972921i	$-0.425756\pi$
$521$	2.86413e6	0.462272	0.231136	+	0.972921i	$0.425756\pi$
$522$	956124.	0.153581
$523$	2.46228e6	0.393625	0.196812	−	0.980441i	$-0.436941\pi$
$523$	2.46228e6	0.393625	0.196812	+	0.980441i	$0.436941\pi$
$524$	−7.16778e6	−1.14040
$525$	1.39946e7	2.21597
$526$	−1.79014e6	−0.282112
$527$	−16464.0	−0.00258231
$528$	1.98374e6	0.309671
$529$	1.60655e6	0.249606
$530$	−876120.	−0.135480
$531$	2.12965e6	0.327772
$532$	2.42592e6	0.371619
$533$	3.75858e6	0.573067
$534$	−1.49209e6	−0.226434
$535$	1.08000e7	1.63132
$536$	−5.04624e6	−0.758675
$537$	−3.95197e6	−0.591396
$538$	−2.54581e6	−0.379202
$539$	1.37064e7	2.03214
$540$	2.00038e6	0.295207
$541$	6.88192e6	1.01092	0.505459	−	0.862850i	$-0.331323\pi$
$541$	6.88192e6	1.01092	0.505459	+	0.862850i	$0.331323\pi$
$542$	−613344.	−0.0896821
$543$	228654.	0.0332797
$544$	30912.0	0.00447847
$545$	2.13430e7	3.07797
$546$	−1.47744e6	−0.212094
$547$	−4.99680e6	−0.714041	−0.357021	−	0.934097i	$-0.616207\pi$
$547$	−4.99680e6	−0.714041	−0.357021	+	0.934097i	$0.616207\pi$
$548$	−1.40633e6	−0.200048
$549$	2.35985e6	0.334160
$550$	−4.35389e6	−0.613720
$551$	2.13062e6	0.298970
$552$	3.06288e6	0.427841
$553$	1.89734e7	2.63836
$554$	384052.	0.0531638
$555$	−1.20569e7	−1.66152
$556$	6.77902e6	0.929994
$557$	−8.42887e6	−1.15115	−0.575574	−	0.817750i	$-0.695221\pi$
$557$	−8.42887e6	−1.15115	−0.575574	+	0.817750i	$0.695221\pi$
$558$	−444528.	−0.0604385
$559$	−1.70863e6	−0.231270
$560$	−1.54291e7	−2.07908
$561$	−18144.0	−0.00243403
$562$	−2.00110e6	−0.267256
$563$	−1.01781e7	−1.35330	−0.676652	−	0.736303i	$-0.736570\pi$
$563$	−1.01781e7	−1.35330	−0.676652	+	0.736303i	$0.736570\pi$
$564$	4.31525e6	0.571226
$565$	8.29256e6	1.09287
$566$	2.53694e6	0.332865
$567$	1.57464e6	0.205695
$568$	−3.13344e6	−0.407522
$569$	1.13792e7	1.47344	0.736719	−	0.676199i	$-0.236374\pi$
$569$	1.13792e7	1.47344	0.736719	+	0.676199i	$0.236374\pi$
$570$	−636804.	−0.0820954
$571$	−7.57426e6	−0.972187	−0.486094	−	0.873907i	$-0.661579\pi$
$571$	−7.57426e6	−0.972187	−0.486094	+	0.873907i	$0.661579\pi$
$572$	−3.21754e6	−0.411181
$573$	5.76580e6	0.733623
$574$	−5.27520e6	−0.668281
$575$	1.83744e7	2.31763
$576$	−865728.	−0.108724
$577$	−6.06488e6	−0.758372	−0.379186	−	0.925320i	$-0.623796\pi$
$577$	−6.06488e6	−0.758372	−0.379186	+	0.925320i	$0.623796\pi$
$578$	2.83964e6	0.353544
$579$	−2.02361e6	−0.250860
$580$	−1.61951e7	−1.99900
$581$	2.27328e6	0.279391
$582$	−717300.	−0.0877796
$583$	−1.50192e6	−0.183010
$584$	−5.88552e6	−0.714090
$585$	−2.71480e6	−0.327980
$586$	3.05120e6	0.367051
$587$	1.62011e7	1.94066	0.970330	−	0.241783i	$-0.0777323\pi$
$587$	1.62011e7	1.94066	0.970330	+	0.241783i	$0.0777323\pi$
$588$	−1.02798e7	−1.22615
$589$	−990584.	−0.117653
$590$	5.15323e6	0.609466
$591$	2.92894e6	0.344939
$592$	8.96752e6	1.05164
$593$	9.92291e6	1.15878	0.579392	−	0.815049i	$-0.303290\pi$
$593$	9.92291e6	1.15878	0.579392	+	0.815049i	$0.303290\pi$
$594$	−489888.	−0.0569680
$595$	141120.	0.0163417
$596$	9.87017e6	1.13817
$597$	−8.85218e6	−1.01652
$598$	−1.93982e6	−0.221824
$599$	−9.55020e6	−1.08754	−0.543770	−	0.839234i	$-0.683004\pi$
$599$	−9.55020e6	−1.08754	−0.543770	+	0.839234i	$0.683004\pi$
$600$	6.99732e6	0.793512
$601$	1.57009e6	0.177312	0.0886560	−	0.996062i	$-0.471743\pi$
$601$	1.57009e6	0.177312	0.0886560	+	0.996062i	$0.471743\pi$
$602$	2.39808e6	0.269695
$603$	−3.40621e6	−0.381486
$604$	7.61936e6	0.849818
$605$	4.71919e6	0.524178
$606$	−1.79176e6	−0.198197
$607$	8.83870e6	0.973680	0.486840	−	0.873491i	$-0.338149\pi$
$607$	8.83870e6	0.973680	0.486840	+	0.873491i	$0.338149\pi$
$608$	1.85987e6	0.204044
$609$	−1.27483e7	−1.39287
$610$	5.71026e6	0.621343
$611$	−5.85641e6	−0.634641
$612$	13608.0	0.00146864
$613$	−4.94068e6	−0.531050	−0.265525	−	0.964104i	$-0.585545\pi$
$613$	−4.94068e6	−0.531050	−0.265525	+	0.964104i	$0.585545\pi$
$614$	2.39330e6	0.256199
$615$	−9.69318e6	−1.03342
$616$	9.67680e6	1.02750
$617$	−5.44608e6	−0.575932	−0.287966	−	0.957641i	$-0.592979\pi$
$617$	−5.44608e6	−0.575932	−0.287966	+	0.957641i	$0.592979\pi$
$618$	−1.24862e6	−0.131511
$619$	9.25648e6	0.971000	0.485500	−	0.874237i	$-0.338637\pi$
$619$	9.25648e6	0.971000	0.485500	+	0.874237i	$0.338637\pi$
$620$	7.52954e6	0.786663
$621$	2.06744e6	0.215132
$622$	−4.74289e6	−0.491549
$623$	1.98946e7	2.05359
$624$	2.01917e6	0.207592
$625$	1.19649e7	1.22521
$626$	−707476.	−0.0721566
$627$	−1.09166e6	−0.110897
$628$	−3.49838e6	−0.353971
$629$	−82020.0	−0.00826596
$630$	3.81024e6	0.382473
$631$	−1.84220e7	−1.84189	−0.920944	−	0.389694i	$-0.872581\pi$
$631$	−1.84220e7	−1.84189	−0.920944	+	0.389694i	$0.872581\pi$
$632$	9.48672e6	0.944764
$633$	−7.79677e6	−0.773402
$634$	−5.40853e6	−0.534387
$635$	−8.78158e6	−0.864248
$636$	1.12644e6	0.110424
$637$	1.39512e7	1.36227
$638$	3.96614e6	0.385760
$639$	−2.11507e6	−0.204915
$640$	−1.82515e7	−1.76136
$641$	−1.83640e7	−1.76532	−0.882660	−	0.470013i	$-0.844249\pi$
$641$	−1.83640e7	−1.76532	−0.882660	+	0.470013i	$0.844249\pi$
$642$	1.98367e6	0.189947
$643$	5.30824e6	0.506318	0.253159	−	0.967425i	$-0.418530\pi$
$643$	5.30824e6	0.506318	0.253159	+	0.967425i	$0.418530\pi$
$644$	−1.90579e7	−1.81076
$645$	4.40647e6	0.417053
$646$	−4332.00	−0.000408420 0
$647$	−3.32276e6	−0.312060	−0.156030	−	0.987752i	$-0.549870\pi$
$647$	−3.32276e6	−0.312060	−0.156030	+	0.987752i	$0.549870\pi$
$648$	787320.	0.0736570
$649$	8.83411e6	0.823287
$650$	−4.43164e6	−0.411416
$651$	5.92704e6	0.548132
$652$	809200.	0.0745482
$653$	−5.81096e6	−0.533292	−0.266646	−	0.963795i	$-0.585915\pi$
$653$	−5.81096e6	−0.533292	−0.266646	+	0.963795i	$0.585915\pi$
$654$	3.92015e6	0.358392
$655$	−2.50872e7	−2.28481
$656$	7.20944e6	0.654097
$657$	−3.97273e6	−0.359067
$658$	8.21952e6	0.740085
$659$	−1.60066e7	−1.43577	−0.717886	−	0.696160i	$-0.754890\pi$
$659$	−1.60066e7	−1.43577	−0.717886	+	0.696160i	$0.754890\pi$
$660$	8.29786e6	0.741491
$661$	−1.27998e7	−1.13946	−0.569731	−	0.821831i	$-0.692953\pi$
$661$	−1.27998e7	−1.13946	−0.569731	+	0.821831i	$0.692953\pi$
$662$	654888.	0.0580794
$663$	−18468.0	−0.00163168
$664$	1.13664e6	0.100047
$665$	8.49072e6	0.744544
$666$	−2.21454e6	−0.193463
$667$	−1.67381e7	−1.45677
$668$	−1.03656e7	−0.898780
$669$	−692352.	−0.0598083
$670$	−8.24219e6	−0.709342
$671$	9.78902e6	0.839331
$672$	−1.11283e7	−0.950619
$673$	9.83823e6	0.837296	0.418648	−	0.908149i	$-0.362504\pi$
$673$	9.83823e6	0.837296	0.418648	+	0.908149i	$0.362504\pi$
$674$	−735892.	−0.0623971
$675$	4.72319e6	0.399003
$676$	7.12121e6	0.599359
$677$	−8.54417e6	−0.716470	−0.358235	−	0.933631i	$-0.616621\pi$
$677$	−8.54417e6	−0.716470	−0.358235	+	0.933631i	$0.616621\pi$
$678$	1.52312e6	0.127251
$679$	9.56400e6	0.796095
$680$	70560.0	0.00585176
$681$	−8.72651e6	−0.721062
$682$	−1.84397e6	−0.151807
$683$	−1.00816e7	−0.826947	−0.413473	−	0.910516i	$-0.635685\pi$
$683$	−1.00816e7	−0.826947	−0.413473	+	0.910516i	$0.635685\pi$
$684$	818748.	0.0669129
$685$	−4.92215e6	−0.400800
$686$	−1.15133e7	−0.934090
$687$	7.56135e6	0.611234
$688$	−3.27738e6	−0.263970
$689$	−1.52874e6	−0.122683
$690$	5.00270e6	0.400020
$691$	−7.71320e6	−0.614525	−0.307262	−	0.951625i	$-0.599413\pi$
$691$	−7.71320e6	−0.614525	−0.307262	+	0.951625i	$0.599413\pi$
$692$	7.76759e6	0.616625
$693$	6.53184e6	0.516657
$694$	−1.13232e6	−0.0892422
$695$	2.37266e7	1.86326
$696$	−6.37416e6	−0.498769
$697$	−65940.0	−0.00514123
$698$	9.01373e6	0.700271
$699$	5.18323e6	0.401243
$700$	−4.35389e7	−3.35840
$701$	4.92440e6	0.378493	0.189247	−	0.981930i	$-0.439395\pi$
$701$	4.92440e6	0.378493	0.189247	+	0.981930i	$0.439395\pi$
$702$	−498636.	−0.0381892
$703$	−4.93487e6	−0.376606
$704$	−3.59117e6	−0.273089
$705$	1.51034e7	1.14446
$706$	2.19556e6	0.165781
$707$	2.38901e7	1.79750
$708$	−6.62558e6	−0.496754
$709$	5.35468e6	0.400053	0.200027	−	0.979790i	$-0.435897\pi$
$709$	5.35468e6	0.400053	0.200027	+	0.979790i	$0.435897\pi$
$710$	−5.11795e6	−0.381022
$711$	6.40354e6	0.475057
$712$	9.94728e6	0.735367
$713$	7.78198e6	0.573280
$714$	25920.0	0.00190278
$715$	−1.12614e7	−0.823809
$716$	1.22950e7	0.896286
$717$	−3.16969e6	−0.230260
$718$	3.90446e6	0.282650
$719$	9.39507e6	0.677763	0.338881	−	0.940829i	$-0.389951\pi$
$719$	9.39507e6	0.677763	0.338881	+	0.940829i	$0.389951\pi$
$720$	−5.20733e6	−0.374355
$721$	1.66483e7	1.19270
$722$	−260642.	−0.0186081
$723$	4.06161e6	0.288970
$724$	−711368.	−0.0504368
$725$	−3.82391e7	−2.70186
$726$	866790.	0.0610341
$727$	6.83055e6	0.479314	0.239657	−	0.970858i	$-0.422965\pi$
$727$	6.83055e6	0.479314	0.239657	+	0.970858i	$0.422965\pi$
$728$	9.84960e6	0.688795
$729$	531441.	0.0370370
$730$	−9.61302e6	−0.667656
$731$	29976.0	0.00207482
$732$	−7.34177e6	−0.506434
$733$	1.29536e7	0.890491	0.445245	−	0.895409i	$-0.353116\pi$
$733$	1.29536e7	0.890491	0.445245	+	0.895409i	$0.353116\pi$
$734$	−2.98326e6	−0.204386
$735$	−3.59794e7	−2.45661
$736$	−1.46111e7	−0.994232
$737$	−1.41295e7	−0.958202
$738$	−1.78038e6	−0.120329
$739$	1.15292e7	0.776582	0.388291	−	0.921537i	$-0.373066\pi$
$739$	1.15292e7	0.776582	0.388291	+	0.921537i	$0.373066\pi$
$740$	3.75105e7	2.51810
$741$	−1.11116e6	−0.0743413
$742$	2.14560e6	0.143067
$743$	3.20131e6	0.212743	0.106372	−	0.994326i	$-0.466077\pi$
$743$	3.20131e6	0.212743	0.106372	+	0.994326i	$0.466077\pi$
$744$	2.96352e6	0.196280
$745$	3.45456e7	2.28035
$746$	−2.96609e6	−0.195136
$747$	767232.	0.0503066
$748$	56448.0	0.00368888
$749$	−2.64490e7	−1.72268
$750$	5.91646e6	0.384068
$751$	5.52822e6	0.357673	0.178836	−	0.983879i	$-0.442767\pi$
$751$	5.52822e6	0.357673	0.178836	+	0.983879i	$0.442767\pi$
$752$	−1.12333e7	−0.724377
$753$	1.31177e6	0.0843082
$754$	4.03697e6	0.258599
$755$	2.66678e7	1.70263
$756$	−4.89888e6	−0.311740
$757$	1.62708e7	1.03198	0.515989	−	0.856595i	$-0.327425\pi$
$757$	1.62708e7	1.03198	0.515989	+	0.856595i	$0.327425\pi$
$758$	5.58871e6	0.353296
$759$	8.57606e6	0.540360
$760$	4.24536e6	0.266613
$761$	−8.40043e6	−0.525823	−0.262912	−	0.964820i	$-0.584683\pi$
$761$	−8.40043e6	−0.525823	−0.262912	+	0.964820i	$0.584683\pi$
$762$	−1.61294e6	−0.100631
$763$	−5.22686e7	−3.25035
$764$	−1.79380e7	−1.11184
$765$	47628.0	0.00294245
$766$	−4.13819e6	−0.254823
$767$	8.99186e6	0.551901
$768$	−274176.	−0.0167736
$769$	7.66543e6	0.467434	0.233717	−	0.972305i	$-0.424911\pi$
$769$	7.66543e6	0.467434	0.233717	+	0.972305i	$0.424911\pi$
$770$	1.58054e7	0.960682
$771$	−1.69036e7	−1.02410
$772$	6.29569e6	0.380189
$773$	−2.59829e7	−1.56401	−0.782004	−	0.623273i	$-0.785802\pi$
$773$	−2.59829e7	−1.56401	−0.782004	+	0.623273i	$0.785802\pi$
$774$	809352.	0.0485607
$775$	1.77784e7	1.06326
$776$	4.78200e6	0.285072
$777$	2.95272e7	1.75457
$778$	−9.23392e6	−0.546937
$779$	−3.96739e6	−0.234240
$780$	8.44603e6	0.497068
$781$	−8.77363e6	−0.514697
$782$	34032.0	0.00199008
$783$	−4.30256e6	−0.250797
$784$	2.67602e7	1.55489
$785$	−1.22443e7	−0.709186
$786$	−4.60786e6	−0.266037
$787$	2.59845e7	1.49547	0.747734	−	0.663999i	$-0.231142\pi$
$787$	2.59845e7	1.49547	0.747734	+	0.663999i	$0.231142\pi$
$788$	−9.11226e6	−0.522770
$789$	8.05561e6	0.460687
$790$	1.54950e7	0.883330
$791$	−2.03083e7	−1.15407
$792$	3.26592e6	0.185009
$793$	9.96383e6	0.562656
$794$	−1.75174e6	−0.0986094
$795$	3.94254e6	0.221237
$796$	2.75401e7	1.54058
$797$	2.12011e7	1.18226	0.591129	−	0.806577i	$-0.298683\pi$
$797$	2.12011e7	1.18226	0.591129	+	0.806577i	$0.298683\pi$
$798$	1.55952e6	0.0866929
$799$	102744.	0.00569363
$800$	−3.33798e7	−1.84399
$801$	6.71441e6	0.369766
$802$	6.73231e6	0.369597
$803$	−1.64795e7	−0.901891
$804$	1.05971e7	0.578159
$805$	−6.67027e7	−3.62789
$806$	−1.87690e6	−0.101766
$807$	1.14562e7	0.619234
$808$	1.19450e7	0.643664
$809$	−1.06374e7	−0.571429	−0.285714	−	0.958315i	$-0.592231\pi$
$809$	−1.06374e7	−0.571429	−0.285714	+	0.958315i	$0.592231\pi$
$810$	1.28596e6	0.0688674
$811$	−1.19205e7	−0.636416	−0.318208	−	0.948021i	$-0.603081\pi$
$811$	−1.19205e7	−0.636416	−0.318208	+	0.948021i	$0.603081\pi$
$812$	3.96614e7	2.11095
$813$	2.76005e6	0.146450
$814$	−9.18624e6	−0.485933
$815$	2.83220e6	0.149358
$816$	−35424.0	−0.00186240
$817$	1.80356e6	0.0945311
$818$	−1.31731e7	−0.688342
$819$	6.64848e6	0.346348
$820$	3.01566e7	1.56620
$821$	8.14431e6	0.421693	0.210847	−	0.977519i	$-0.432378\pi$
$821$	8.14431e6	0.421693	0.210847	+	0.977519i	$0.432378\pi$
$822$	−904068.	−0.0466683
$823$	−9.57675e6	−0.492854	−0.246427	−	0.969161i	$-0.579257\pi$
$823$	−9.57675e6	−0.492854	−0.246427	+	0.969161i	$0.579257\pi$
$824$	8.32416e6	0.427093
$825$	1.95925e7	1.00220
$826$	−1.26202e7	−0.643598
$827$	−2.95107e6	−0.150043	−0.0750214	−	0.997182i	$-0.523903\pi$
$827$	−2.95107e6	−0.150043	−0.0750214	+	0.997182i	$0.523903\pi$
$828$	−6.43205e6	−0.326042
$829$	8.79546e6	0.444501	0.222250	−	0.974990i	$-0.428660\pi$
$829$	8.79546e6	0.444501	0.222250	+	0.974990i	$0.428660\pi$
$830$	1.85651e6	0.0935411
$831$	−1.72823e6	−0.0868160
$832$	−3.65530e6	−0.183069
$833$	−244758.	−0.0122215
$834$	4.35794e6	0.216953
$835$	−3.62796e7	−1.80072
$836$	3.39629e6	0.168069
$837$	2.00038e6	0.0986956
$838$	2.13550e6	0.105049
$839$	−1.46842e6	−0.0720185	−0.0360093	−	0.999351i	$-0.511465\pi$
$839$	−1.46842e6	−0.0720185	−0.0360093	+	0.999351i	$0.511465\pi$
$840$	−2.54016e7	−1.24212
$841$	1.43225e7	0.698277
$842$	−7.21241e6	−0.350591
$843$	9.00495e6	0.436428
$844$	2.42566e7	1.17213
$845$	2.49242e7	1.20083
$846$	2.77409e6	0.133258
$847$	−1.15572e7	−0.553534
$848$	−2.93232e6	−0.140030
$849$	−1.14162e7	−0.543567
$850$	77748.0	0.00369098
$851$	3.87681e7	1.83506
$852$	6.58022e6	0.310557
$853$	2.37789e7	1.11897	0.559486	−	0.828840i	$-0.310999\pi$
$853$	2.37789e7	1.11897	0.559486	+	0.828840i	$0.310999\pi$
$854$	−1.39843e7	−0.656140
$855$	2.86562e6	0.134061
$856$	−1.32245e7	−0.616871
$857$	−4.09013e7	−1.90233	−0.951163	−	0.308688i	$-0.900110\pi$
$857$	−4.09013e7	−1.90233	−0.951163	+	0.308688i	$0.900110\pi$
$858$	−2.06842e6	−0.0959223
$859$	−7.99731e6	−0.369795	−0.184897	−	0.982758i	$-0.559195\pi$
$859$	−7.99731e6	−0.369795	−0.184897	+	0.982758i	$0.559195\pi$
$860$	−1.37090e7	−0.632063
$861$	2.37384e7	1.09130
$862$	−2.10619e6	−0.0965450
$863$	−4.40634e6	−0.201396	−0.100698	−	0.994917i	$-0.532108\pi$
$863$	−4.40634e6	−0.201396	−0.100698	+	0.994917i	$0.532108\pi$
$864$	−3.75581e6	−0.171167
$865$	2.71866e7	1.23542
$866$	6.91395e6	0.313279
$867$	−1.27784e7	−0.577336
$868$	−1.84397e7	−0.830719
$869$	2.65628e7	1.19323
$870$	−1.04111e7	−0.466337
$871$	−1.43818e7	−0.642344
$872$	−2.61343e7	−1.16391
$873$	3.22785e6	0.143343
$874$	2.04759e6	0.0906703
$875$	−7.88861e7	−3.48321
$876$	1.23596e7	0.544182
$877$	−2.55057e7	−1.11980	−0.559898	−	0.828561i	$-0.689160\pi$
$877$	−2.55057e7	−1.11980	−0.559898	+	0.828561i	$0.689160\pi$
$878$	1.17722e7	0.515373
$879$	−1.37304e7	−0.599391
$880$	−2.16008e7	−0.940292
$881$	−3.50822e7	−1.52281	−0.761407	−	0.648274i	$-0.775491\pi$
$881$	−3.50822e7	−1.52281	−0.761407	+	0.648274i	$0.775491\pi$
$882$	−6.60847e6	−0.286042
$883$	−2.47814e7	−1.06961	−0.534803	−	0.844977i	$-0.679614\pi$
$883$	−2.47814e7	−1.06961	−0.534803	+	0.844977i	$0.679614\pi$
$884$	57456.0	0.00247289
$885$	−2.31895e7	−0.995254
$886$	9.60432e6	0.411038
$887$	−6.85463e6	−0.292533	−0.146267	−	0.989245i	$-0.546726\pi$
$887$	−6.85463e6	−0.292533	−0.146267	+	0.989245i	$0.546726\pi$
$888$	1.47636e7	0.628290
$889$	2.15059e7	0.912649
$890$	1.62472e7	0.687550
$891$	2.20450e6	0.0930283
$892$	2.15398e6	0.0906422
$893$	6.18176e6	0.259408
$894$	6.34511e6	0.265519
$895$	4.30326e7	1.79573
$896$	4.46976e7	1.86001
$897$	8.72921e6	0.362238
$898$	6.35933e6	0.263160
$899$	−1.61951e7	−0.668319
$900$	−1.46944e7	−0.604707
$901$	26820.0	0.00110064
$902$	−7.38528e6	−0.302239
$903$	−1.07914e7	−0.440410
$904$	−1.01542e7	−0.413260
$905$	−2.48979e6	−0.101051
$906$	4.89816e6	0.198250
$907$	−3.64392e7	−1.47079	−0.735395	−	0.677639i	$-0.763003\pi$
$907$	−3.64392e7	−1.47079	−0.735395	+	0.677639i	$0.763003\pi$
$908$	2.71491e7	1.09280
$909$	8.06290e6	0.323655
$910$	1.60877e7	0.644006
$911$	−2.83123e7	−1.13026	−0.565132	−	0.825001i	$-0.691175\pi$
$911$	−2.83123e7	−1.13026	−0.565132	+	0.825001i	$0.691175\pi$
$912$	−2.13134e6	−0.0848529
$913$	3.18259e6	0.126358
$914$	−967092.	−0.0382915
$915$	−2.56962e7	−1.01465
$916$	−2.35242e7	−0.926352
$917$	6.14381e7	2.41276
$918$	8748.00	0.000342612 0
$919$	−3.75587e6	−0.146697	−0.0733486	−	0.997306i	$-0.523369\pi$
$919$	−3.75587e6	−0.146697	−0.0733486	+	0.997306i	$0.523369\pi$
$920$	−3.33514e7	−1.29910
$921$	−1.07699e7	−0.418371
$922$	−1.41596e6	−0.0548561
$923$	−8.93030e6	−0.345034
$924$	−2.03213e7	−0.783017
$925$	8.85679e7	3.40347
$926$	−1.64818e7	−0.631652
$927$	5.61881e6	0.214756
$928$	3.04071e7	1.15906
$929$	−5.55366e6	−0.211125	−0.105563	−	0.994413i	$-0.533664\pi$
$929$	−5.55366e6	−0.211125	−0.105563	+	0.994413i	$0.533664\pi$
$930$	4.84042e6	0.183517
$931$	−1.47263e7	−0.556825
$932$	−1.61256e7	−0.608101
$933$	2.13430e7	0.802696
$934$	−4.27280e6	−0.160267
$935$	197568.	0.00739073
$936$	3.32424e6	0.124023
$937$	2.46166e7	0.915965	0.457982	−	0.888961i	$-0.348572\pi$
$937$	2.46166e7	0.915965	0.457982	+	0.888961i	$0.348572\pi$
$938$	2.01850e7	0.749067
$939$	3.18364e6	0.117831
$940$	−4.69883e7	−1.73448
$941$	−3.89956e7	−1.43563	−0.717813	−	0.696236i	$-0.754857\pi$
$941$	−3.89956e7	−1.43563	−0.717813	+	0.696236i	$0.754857\pi$
$942$	−2.24896e6	−0.0825760
$943$	3.11676e7	1.14137
$944$	1.72476e7	0.629938
$945$	−1.71461e7	−0.624576
$946$	3.35731e6	0.121973
$947$	2.16559e7	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$947$	2.16559e7	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$948$	−1.99221e7	−0.719970
$949$	−1.67737e7	−0.604594
$950$	4.67784e6	0.168165
$951$	2.43384e7	0.872651
$952$	−172800.	−0.00617947
$953$	−1.94047e7	−0.692109	−0.346054	−	0.938214i	$-0.612479\pi$
$953$	−1.94047e7	−0.692109	−0.346054	+	0.938214i	$0.612479\pi$
$954$	724140.	0.0257603
$955$	−6.27831e7	−2.22759
$956$	9.86126e6	0.348970
$957$	−1.78476e7	−0.629943
$958$	−2.41473e6	−0.0850070
$959$	1.20542e7	0.423246
$960$	9.42682e6	0.330131
$961$	−2.10996e7	−0.736998
$962$	−9.35028e6	−0.325752
$963$	−8.92652e6	−0.310182
$964$	−1.26361e7	−0.437947
$965$	2.20349e7	0.761716
$966$	−1.22515e7	−0.422423
$967$	7.26789e6	0.249944	0.124972	−	0.992160i	$-0.460116\pi$
$967$	7.26789e6	0.249944	0.124972	+	0.992160i	$0.460116\pi$
$968$	−5.77860e6	−0.198214
$969$	19494.0	0.000666947 0
$970$	7.81060e6	0.266536
$971$	−2.43391e7	−0.828431	−0.414216	−	0.910179i	$-0.635944\pi$
$971$	−2.43391e7	−0.828431	−0.414216	+	0.910179i	$0.635944\pi$
$972$	−1.65337e6	−0.0561313
$973$	−5.81059e7	−1.96761
$974$	1.75594e7	0.593078
$975$	1.99424e7	0.671839
$976$	1.91119e7	0.642213
$977$	−3.09415e7	−1.03706	−0.518531	−	0.855059i	$-0.673521\pi$
$977$	−3.09415e7	−1.03706	−0.518531	+	0.855059i	$0.673521\pi$
$978$	520200.	0.0173909
$979$	2.78524e7	0.928765
$980$	1.11936e8	3.72310
$981$	−1.76407e7	−0.585251
$982$	−8.46800e6	−0.280222
$983$	3.53169e7	1.16573	0.582865	−	0.812569i	$-0.301931\pi$
$983$	3.53169e7	1.16573	0.582865	+	0.812569i	$0.301931\pi$
$984$	1.18692e7	0.390781
$985$	−3.18929e7	−1.04738
$986$	−70824.0	−0.00232000
$987$	−3.69878e7	−1.20855
$988$	3.45694e6	0.112668
$989$	−1.41687e7	−0.460615
$990$	5.33434e6	0.172979
$991$	4.96000e7	1.60434	0.802172	−	0.597092i	$-0.203677\pi$
$991$	4.96000e7	1.60434	0.802172	+	0.597092i	$0.203677\pi$
$992$	−1.41371e7	−0.456122
$993$	−2.94700e6	−0.0948432
$994$	1.25338e7	0.402361
$995$	9.63904e7	3.08657
$996$	−2.38694e6	−0.0762419
$997$	−5.33551e7	−1.69996	−0.849978	−	0.526818i	$-0.823385\pi$
$997$	−5.33551e7	−1.69996	−0.849978	+	0.526818i	$0.823385\pi$
$998$	3.46900e6	0.110250
$999$	9.96543e6	0.315924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.6.a.a.1.1	✓	1
3.2	odd	2		171.6.a.c.1.1		1
4.3	odd	2		912.6.a.a.1.1		1
19.18	odd	2		1083.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.6.a.a.1.1	✓	1	1.1	even	1	trivial
171.6.a.c.1.1		1	3.2	odd	2
912.6.a.a.1.1		1	4.3	odd	2
1083.6.a.b.1.1		1	19.18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	57.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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