LMFDB - Embedded newform 222.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [222,6,Mod(1,222)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(222, 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("222.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 222 = 2 \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	222.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:	$35.6052079985$
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	$\chi$	$=$	222.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -94.0000 q^{5} +36.0000 q^{6} +188.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -94.0000 q^{5} +36.0000 q^{6} +188.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -376.000 q^{10} -740.000 q^{11} +144.000 q^{12} -462.000 q^{13} +752.000 q^{14} -846.000 q^{15} +256.000 q^{16} +138.000 q^{17} +324.000 q^{18} -722.000 q^{19} -1504.00 q^{20} +1692.00 q^{21} -2960.00 q^{22} -1466.00 q^{23} +576.000 q^{24} +5711.00 q^{25} -1848.00 q^{26} +729.000 q^{27} +3008.00 q^{28} -6190.00 q^{29} -3384.00 q^{30} +2618.00 q^{31} +1024.00 q^{32} -6660.00 q^{33} +552.000 q^{34} -17672.0 q^{35} +1296.00 q^{36} -1369.00 q^{37} -2888.00 q^{38} -4158.00 q^{39} -6016.00 q^{40} -15954.0 q^{41} +6768.00 q^{42} -20906.0 q^{43} -11840.0 q^{44} -7614.00 q^{45} -5864.00 q^{46} +4420.00 q^{47} +2304.00 q^{48} +18537.0 q^{49} +22844.0 q^{50} +1242.00 q^{51} -7392.00 q^{52} +1658.00 q^{53} +2916.00 q^{54} +69560.0 q^{55} +12032.0 q^{56} -6498.00 q^{57} -24760.0 q^{58} -32274.0 q^{59} -13536.0 q^{60} +2098.00 q^{61} +10472.0 q^{62} +15228.0 q^{63} +4096.00 q^{64} +43428.0 q^{65} -26640.0 q^{66} +46556.0 q^{67} +2208.00 q^{68} -13194.0 q^{69} -70688.0 q^{70} +33228.0 q^{71} +5184.00 q^{72} -55442.0 q^{73} -5476.00 q^{74} +51399.0 q^{75} -11552.0 q^{76} -139120. q^{77} -16632.0 q^{78} -25546.0 q^{79} -24064.0 q^{80} +6561.00 q^{81} -63816.0 q^{82} +76920.0 q^{83} +27072.0 q^{84} -12972.0 q^{85} -83624.0 q^{86} -55710.0 q^{87} -47360.0 q^{88} -15134.0 q^{89} -30456.0 q^{90} -86856.0 q^{91} -23456.0 q^{92} +23562.0 q^{93} +17680.0 q^{94} +67868.0 q^{95} +9216.00 q^{96} -80654.0 q^{97} +74148.0 q^{98} -59940.0 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	−94.0000	−1.68152	−0.840762	−	0.541406i	$-0.817892\pi$
$5$	−94.0000	−1.68152	−0.840762	+	0.541406i	$0.817892\pi$
$6$	36.0000	0.408248
$7$	188.000	1.45015	0.725075	−	0.688670i	$-0.241805\pi$
$7$	188.000	1.45015	0.725075	+	0.688670i	$0.241805\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−376.000	−1.18902
$11$	−740.000	−1.84395	−0.921977	−	0.387245i	$-0.873427\pi$
$11$	−740.000	−1.84395	−0.921977	+	0.387245i	$0.873427\pi$
$12$	144.000	0.288675
$13$	−462.000	−0.758200	−0.379100	−	0.925356i	$-0.623766\pi$
$13$	−462.000	−0.758200	−0.379100	+	0.925356i	$0.623766\pi$
$14$	752.000	1.02541
$15$	−846.000	−0.970828
$16$	256.000	0.250000
$17$	138.000	0.115813	0.0579064	−	0.998322i	$-0.481557\pi$
$17$	138.000	0.115813	0.0579064	+	0.998322i	$0.481557\pi$
$18$	324.000	0.235702
$19$	−722.000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−722.000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1504.00	−0.840762
$21$	1692.00	0.837244
$22$	−2960.00	−1.30387
$23$	−1466.00	−0.577849	−0.288925	−	0.957352i	$-0.593298\pi$
$23$	−1466.00	−0.577849	−0.288925	+	0.957352i	$0.593298\pi$
$24$	576.000	0.204124
$25$	5711.00	1.82752
$26$	−1848.00	−0.536128
$27$	729.000	0.192450
$28$	3008.00	0.725075
$29$	−6190.00	−1.36677	−0.683385	−	0.730058i	$-0.739493\pi$
$29$	−6190.00	−1.36677	−0.683385	+	0.730058i	$0.739493\pi$
$30$	−3384.00	−0.686479
$31$	2618.00	0.489289	0.244644	−	0.969613i	$-0.421329\pi$
$31$	2618.00	0.489289	0.244644	+	0.969613i	$0.421329\pi$
$32$	1024.00	0.176777
$33$	−6660.00	−1.06461
$34$	552.000	0.0818921
$35$	−17672.0	−2.43846
$36$	1296.00	0.166667
$37$	−1369.00	−0.164399
$37$	−1369.00	−0.164399
$38$	−2888.00	−0.324443
$39$	−4158.00	−0.437747
$40$	−6016.00	−0.594508
$41$	−15954.0	−1.48221	−0.741105	−	0.671389i	$-0.765698\pi$
$41$	−15954.0	−1.48221	−0.741105	+	0.671389i	$0.765698\pi$
$42$	6768.00	0.592021
$43$	−20906.0	−1.72425	−0.862124	−	0.506697i	$-0.830866\pi$
$43$	−20906.0	−1.72425	−0.862124	+	0.506697i	$0.830866\pi$
$44$	−11840.0	−0.921977
$45$	−7614.00	−0.560508
$46$	−5864.00	−0.408601
$47$	4420.00	0.291862	0.145931	−	0.989295i	$-0.453382\pi$
$47$	4420.00	0.291862	0.145931	+	0.989295i	$0.453382\pi$
$48$	2304.00	0.144338
$49$	18537.0	1.10293
$50$	22844.0	1.29225
$51$	1242.00	0.0668646
$52$	−7392.00	−0.379100
$53$	1658.00	0.0810765	0.0405382	−	0.999178i	$-0.487093\pi$
$53$	1658.00	0.0810765	0.0405382	+	0.999178i	$0.487093\pi$
$54$	2916.00	0.136083
$55$	69560.0	3.10065
$56$	12032.0	0.512705
$57$	−6498.00	−0.264906
$58$	−24760.0	−0.966453
$59$	−32274.0	−1.20704	−0.603522	−	0.797347i	$-0.706236\pi$
$59$	−32274.0	−1.20704	−0.603522	+	0.797347i	$0.706236\pi$
$60$	−13536.0	−0.485414
$61$	2098.00	0.0721906	0.0360953	−	0.999348i	$-0.488508\pi$
$61$	2098.00	0.0721906	0.0360953	+	0.999348i	$0.488508\pi$
$62$	10472.0	0.345980
$63$	15228.0	0.483383
$64$	4096.00	0.125000
$65$	43428.0	1.27493
$66$	−26640.0	−0.752791
$67$	46556.0	1.26704	0.633518	−	0.773728i	$-0.281610\pi$
$67$	46556.0	1.26704	0.633518	+	0.773728i	$0.281610\pi$
$68$	2208.00	0.0579064
$69$	−13194.0	−0.333621
$70$	−70688.0	−1.72425
$71$	33228.0	0.782273	0.391136	−	0.920333i	$-0.372082\pi$
$71$	33228.0	0.782273	0.391136	+	0.920333i	$0.372082\pi$
$72$	5184.00	0.117851
$73$	−55442.0	−1.21768	−0.608838	−	0.793295i	$-0.708364\pi$
$73$	−55442.0	−1.21768	−0.608838	+	0.793295i	$0.708364\pi$
$74$	−5476.00	−0.116248
$75$	51399.0	1.05512
$76$	−11552.0	−0.229416
$77$	−139120.	−2.67401
$78$	−16632.0	−0.309534
$79$	−25546.0	−0.460527	−0.230264	−	0.973128i	$-0.573959\pi$
$79$	−25546.0	−0.460527	−0.230264	+	0.973128i	$0.573959\pi$
$80$	−24064.0	−0.420381
$81$	6561.00	0.111111
$82$	−63816.0	−1.04808
$83$	76920.0	1.22559	0.612793	−	0.790243i	$-0.290046\pi$
$83$	76920.0	1.22559	0.612793	+	0.790243i	$0.290046\pi$
$84$	27072.0	0.418622
$85$	−12972.0	−0.194742
$86$	−83624.0	−1.21923
$87$	−55710.0	−0.789105
$88$	−47360.0	−0.651936
$89$	−15134.0	−0.202525	−0.101263	−	0.994860i	$-0.532288\pi$
$89$	−15134.0	−0.202525	−0.101263	+	0.994860i	$0.532288\pi$
$90$	−30456.0	−0.396339
$91$	−86856.0	−1.09950
$92$	−23456.0	−0.288925
$93$	23562.0	0.282491
$94$	17680.0	0.206378
$95$	67868.0	0.771536
$96$	9216.00	0.102062
$97$	−80654.0	−0.870355	−0.435178	−	0.900345i	$-0.643314\pi$
$97$	−80654.0	−0.870355	−0.435178	+	0.900345i	$0.643314\pi$
$98$	74148.0	0.779892
$99$	−59940.0	−0.614651
$100$	91376.0	0.913760
$101$	47370.0	0.462062	0.231031	−	0.972946i	$-0.425790\pi$
$101$	47370.0	0.462062	0.231031	+	0.972946i	$0.425790\pi$
$102$	4968.00	0.0472804
$103$	70378.0	0.653648	0.326824	−	0.945085i	$-0.394022\pi$
$103$	70378.0	0.653648	0.326824	+	0.945085i	$0.394022\pi$
$104$	−29568.0	−0.268064
$105$	−159048.	−1.40785
$106$	6632.00	0.0573297
$107$	208584.	1.76125	0.880626	−	0.473812i	$-0.157122\pi$
$107$	208584.	1.76125	0.880626	+	0.473812i	$0.157122\pi$
$108$	11664.0	0.0962250
$109$	−44982.0	−0.362637	−0.181319	−	0.983424i	$-0.558037\pi$
$109$	−44982.0	−0.362637	−0.181319	+	0.983424i	$0.558037\pi$
$110$	278240.	2.19249
$111$	−12321.0	−0.0949158
$112$	48128.0	0.362537
$113$	−78470.0	−0.578106	−0.289053	−	0.957313i	$-0.593340\pi$
$113$	−78470.0	−0.578106	−0.289053	+	0.957313i	$0.593340\pi$
$114$	−25992.0	−0.187317
$115$	137804.	0.971666
$116$	−99040.0	−0.683385
$117$	−37422.0	−0.252733
$118$	−129096.	−0.853508
$119$	25944.0	0.167946
$120$	−54144.0	−0.343239
$121$	386549.	2.40017
$122$	8392.00	0.0510465
$123$	−143586.	−0.855755
$124$	41888.0	0.244644
$125$	−243084.	−1.39149
$126$	60912.0	0.341803
$127$	−88044.0	−0.484385	−0.242192	−	0.970228i	$-0.577867\pi$
$127$	−88044.0	−0.484385	−0.242192	+	0.970228i	$0.577867\pi$
$128$	16384.0	0.0883883
$129$	−188154.	−0.995495
$130$	173712.	0.901512
$131$	332234.	1.69148	0.845738	−	0.533598i	$-0.179161\pi$
$131$	332234.	1.69148	0.845738	+	0.533598i	$0.179161\pi$
$132$	−106560.	−0.532304
$133$	−135736.	−0.665374
$134$	186224.	0.895929
$135$	−68526.0	−0.323609
$136$	8832.00	0.0409460
$137$	323158.	1.47100	0.735501	−	0.677523i	$-0.236947\pi$
$137$	323158.	1.47100	0.735501	+	0.677523i	$0.236947\pi$
$138$	−52776.0	−0.235906
$139$	−7268.00	−0.0319064	−0.0159532	−	0.999873i	$-0.505078\pi$
$139$	−7268.00	−0.0319064	−0.0159532	+	0.999873i	$0.505078\pi$
$140$	−282752.	−1.21923
$141$	39780.0	0.168507
$142$	132912.	0.553151
$143$	341880.	1.39809
$144$	20736.0	0.0833333
$145$	581860.	2.29826
$146$	−221768.	−0.861027
$147$	166833.	0.636779
$148$	−21904.0	−0.0821995
$149$	−502346.	−1.85369	−0.926845	−	0.375443i	$-0.877490\pi$
$149$	−502346.	−1.85369	−0.926845	+	0.375443i	$0.877490\pi$
$150$	205596.	0.746082
$151$	60080.0	0.214431	0.107215	−	0.994236i	$-0.465807\pi$
$151$	60080.0	0.214431	0.107215	+	0.994236i	$0.465807\pi$
$152$	−46208.0	−0.162221
$153$	11178.0	0.0386043
$154$	−556480.	−1.89081
$155$	−246092.	−0.822751
$156$	−66528.0	−0.218873
$157$	−301746.	−0.976995	−0.488498	−	0.872565i	$-0.662455\pi$
$157$	−301746.	−0.976995	−0.488498	+	0.872565i	$0.662455\pi$
$158$	−102184.	−0.325642
$159$	14922.0	0.0468095
$160$	−96256.0	−0.297254
$161$	−275608.	−0.837967
$162$	26244.0	0.0785674
$163$	553770.	1.63253	0.816264	−	0.577680i	$-0.196042\pi$
$163$	553770.	1.63253	0.816264	+	0.577680i	$0.196042\pi$
$164$	−255264.	−0.741105
$165$	626040.	1.79016
$166$	307680.	0.866621
$167$	454310.	1.26055	0.630277	−	0.776371i	$-0.282942\pi$
$167$	454310.	1.26055	0.630277	+	0.776371i	$0.282942\pi$
$168$	108288.	0.296011
$169$	−157849.	−0.425133
$170$	−51888.0	−0.137703
$171$	−58482.0	−0.152944
$172$	−334496.	−0.862124
$173$	−269066.	−0.683508	−0.341754	−	0.939789i	$-0.611021\pi$
$173$	−269066.	−0.683508	−0.341754	+	0.939789i	$0.611021\pi$
$174$	−222840.	−0.557982
$175$	1.07367e6	2.65018
$176$	−189440.	−0.460988
$177$	−290466.	−0.696887
$178$	−60536.0	−0.143207
$179$	−814294.	−1.89954	−0.949770	−	0.312947i	$-0.898684\pi$
$179$	−814294.	−1.89954	−0.949770	+	0.312947i	$0.898684\pi$
$180$	−121824.	−0.280254
$181$	545594.	1.23786	0.618932	−	0.785444i	$-0.287566\pi$
$181$	545594.	1.23786	0.618932	+	0.785444i	$0.287566\pi$
$182$	−347424.	−0.777466
$183$	18882.0	0.0416793
$184$	−93824.0	−0.204300
$185$	128686.	0.276441
$186$	94248.0	0.199751
$187$	−102120.	−0.213554
$188$	70720.0	0.145931
$189$	137052.	0.279081
$190$	271472.	0.545558
$191$	110478.	0.219125	0.109563	−	0.993980i	$-0.465055\pi$
$191$	110478.	0.219125	0.109563	+	0.993980i	$0.465055\pi$
$192$	36864.0	0.0721688
$193$	475914.	0.919677	0.459838	−	0.888003i	$-0.347907\pi$
$193$	475914.	0.919677	0.459838	+	0.888003i	$0.347907\pi$
$194$	−322616.	−0.615434
$195$	390852.	0.736081
$196$	296592.	0.551467
$197$	994310.	1.82539	0.912696	−	0.408639i	$-0.133996\pi$
$197$	994310.	1.82539	0.912696	+	0.408639i	$0.133996\pi$
$198$	−239760.	−0.434624
$199$	−416606.	−0.745749	−0.372875	−	0.927882i	$-0.621628\pi$
$199$	−416606.	−0.745749	−0.372875	+	0.927882i	$0.621628\pi$
$200$	365504.	0.646126
$201$	419004.	0.731523
$202$	189480.	0.326727
$203$	−1.16372e6	−1.98202
$204$	19872.0	0.0334323
$205$	1.49968e6	2.49237
$206$	281512.	0.462199
$207$	−118746.	−0.192616
$208$	−118272.	−0.189550
$209$	534280.	0.846064
$210$	−636192.	−0.995497
$211$	−838608.	−1.29674	−0.648370	−	0.761325i	$-0.724549\pi$
$211$	−838608.	−1.29674	−0.648370	+	0.761325i	$0.724549\pi$
$212$	26528.0	0.0405382
$213$	299052.	0.451646
$214$	834336.	1.24539
$215$	1.96516e6	2.89936
$216$	46656.0	0.0680414
$217$	492184.	0.709542
$218$	−179928.	−0.256423
$219$	−498978.	−0.703026
$220$	1.11296e6	1.55033
$221$	−63756.0	−0.0878093
$222$	−49284.0	−0.0671156
$223$	−244716.	−0.329534	−0.164767	−	0.986333i	$-0.552687\pi$
$223$	−244716.	−0.329534	−0.164767	+	0.986333i	$0.552687\pi$
$224$	192512.	0.256353
$225$	462591.	0.609173
$226$	−313880.	−0.408783
$227$	13838.0	0.0178241	0.00891207	−	0.999960i	$-0.497163\pi$
$227$	13838.0	0.0178241	0.00891207	+	0.999960i	$0.497163\pi$
$228$	−103968.	−0.132453
$229$	−1.28806e6	−1.62311	−0.811553	−	0.584279i	$-0.801377\pi$
$229$	−1.28806e6	−1.62311	−0.811553	+	0.584279i	$0.801377\pi$
$230$	551216.	0.687072
$231$	−1.25208e6	−1.54384
$232$	−396160.	−0.483226
$233$	−937810.	−1.13168	−0.565842	−	0.824514i	$-0.691449\pi$
$233$	−937810.	−1.13168	−0.565842	+	0.824514i	$0.691449\pi$
$234$	−149688.	−0.178709
$235$	−415480.	−0.490773
$236$	−516384.	−0.603522
$237$	−229914.	−0.265885
$238$	103776.	0.118756
$239$	−1.00922e6	−1.14285	−0.571426	−	0.820653i	$-0.693610\pi$
$239$	−1.00922e6	−1.14285	−0.571426	+	0.820653i	$0.693610\pi$
$240$	−216576.	−0.242707
$241$	1.29498e6	1.43622	0.718108	−	0.695932i	$-0.245008\pi$
$241$	1.29498e6	1.43622	0.718108	+	0.695932i	$0.245008\pi$
$242$	1.54620e6	1.69717
$243$	59049.0	0.0641500
$244$	33568.0	0.0360953
$245$	−1.74248e6	−1.85461
$246$	−574344.	−0.605110
$247$	333564.	0.347886
$248$	167552.	0.172990
$249$	692280.	0.707593
$250$	−972336.	−0.983935
$251$	−1.25151e6	−1.25386	−0.626931	−	0.779074i	$-0.715689\pi$
$251$	−1.25151e6	−1.25386	−0.626931	+	0.779074i	$0.715689\pi$
$252$	243648.	0.241692
$253$	1.08484e6	1.06553
$254$	−352176.	−0.342512
$255$	−116748.	−0.112434
$256$	65536.0	0.0625000
$257$	−192606.	−0.181902	−0.0909509	−	0.995855i	$-0.528991\pi$
$257$	−192606.	−0.181902	−0.0909509	+	0.995855i	$0.528991\pi$
$258$	−752616.	−0.703921
$259$	−257372.	−0.238403
$260$	694848.	0.637465
$261$	−501390.	−0.455590
$262$	1.32894e6	1.19605
$263$	465576.	0.415051	0.207525	−	0.978230i	$-0.433459\pi$
$263$	465576.	0.415051	0.207525	+	0.978230i	$0.433459\pi$
$264$	−426240.	−0.376395
$265$	−155852.	−0.136332
$266$	−542944.	−0.470491
$267$	−136206.	−0.116928
$268$	744896.	0.633518
$269$	−1.63715e6	−1.37946	−0.689730	−	0.724067i	$-0.742271\pi$
$269$	−1.63715e6	−1.37946	−0.689730	+	0.724067i	$0.742271\pi$
$270$	−274104.	−0.228826
$271$	−1.89574e6	−1.56803	−0.784017	−	0.620739i	$-0.786833\pi$
$271$	−1.89574e6	−1.56803	−0.784017	+	0.620739i	$0.786833\pi$
$272$	35328.0	0.0289532
$273$	−781704.	−0.634798
$274$	1.29263e6	1.04016
$275$	−4.22614e6	−3.36986
$276$	−211104.	−0.166811
$277$	94546.0	0.0740361	0.0370181	−	0.999315i	$-0.488214\pi$
$277$	94546.0	0.0740361	0.0370181	+	0.999315i	$0.488214\pi$
$278$	−29072.0	−0.0225612
$279$	212058.	0.163096
$280$	−1.13101e6	−0.862126
$281$	1.46025e6	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$281$	1.46025e6	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$282$	159120.	0.119152
$283$	2.52122e6	1.87130	0.935652	−	0.352924i	$-0.114813\pi$
$283$	2.52122e6	1.87130	0.935652	+	0.352924i	$0.114813\pi$
$284$	531648.	0.391136
$285$	610812.	0.445446
$286$	1.36752e6	0.988595
$287$	−2.99935e6	−2.14943
$288$	82944.0	0.0589256
$289$	−1.40081e6	−0.986587
$290$	2.32744e6	1.62511
$291$	−725886.	−0.502500
$292$	−887072.	−0.608838
$293$	−16086.0	−0.0109466	−0.00547330	−	0.999985i	$-0.501742\pi$
$293$	−16086.0	−0.0109466	−0.00547330	+	0.999985i	$0.501742\pi$
$294$	667332.	0.450271
$295$	3.03376e6	2.02967
$296$	−87616.0	−0.0581238
$297$	−539460.	−0.354869
$298$	−2.00938e6	−1.31076
$299$	677292.	0.438125
$300$	822384.	0.527560
$301$	−3.93033e6	−2.50042
$302$	240320.	0.151626
$303$	426330.	0.266771
$304$	−184832.	−0.114708
$305$	−197212.	−0.121390
$306$	44712.0	0.0272974
$307$	−663364.	−0.401704	−0.200852	−	0.979622i	$-0.564371\pi$
$307$	−663364.	−0.401704	−0.200852	+	0.979622i	$0.564371\pi$
$308$	−2.22592e6	−1.33700
$309$	633402.	0.377384
$310$	−984368.	−0.581773
$311$	965866.	0.566260	0.283130	−	0.959082i	$-0.408627\pi$
$311$	965866.	0.566260	0.283130	+	0.959082i	$0.408627\pi$
$312$	−266112.	−0.154767
$313$	1.34329e6	0.775013	0.387506	−	0.921867i	$-0.373336\pi$
$313$	1.34329e6	0.775013	0.387506	+	0.921867i	$0.373336\pi$
$314$	−1.20698e6	−0.690840
$315$	−1.43143e6	−0.812820
$316$	−408736.	−0.230264
$317$	1.20845e6	0.675433	0.337716	−	0.941248i	$-0.390346\pi$
$317$	1.20845e6	0.675433	0.337716	+	0.941248i	$0.390346\pi$
$318$	59688.0	0.0330993
$319$	4.58060e6	2.52026
$320$	−385024.	−0.210190
$321$	1.87726e6	1.01686
$322$	−1.10243e6	−0.592532
$323$	−99636.0	−0.0531386
$324$	104976.	0.0555556
$325$	−2.63848e6	−1.38563
$326$	2.21508e6	1.15437
$327$	−404838.	−0.209369
$328$	−1.02106e6	−0.524041
$329$	830960.	0.423244
$330$	2.50416e6	1.26584
$331$	−487370.	−0.244506	−0.122253	−	0.992499i	$-0.539012\pi$
$331$	−487370.	−0.244506	−0.122253	+	0.992499i	$0.539012\pi$
$332$	1.23072e6	0.612793
$333$	−110889.	−0.0547997
$334$	1.81724e6	0.891346
$335$	−4.37626e6	−2.13055
$336$	433152.	0.209311
$337$	−2.96924e6	−1.42420	−0.712100	−	0.702078i	$-0.752256\pi$
$337$	−2.96924e6	−1.42420	−0.712100	+	0.702078i	$0.752256\pi$
$338$	−631396.	−0.300615
$339$	−706230.	−0.333770
$340$	−207552.	−0.0973710
$341$	−1.93732e6	−0.902226
$342$	−233928.	−0.108148
$343$	325240.	0.149269
$344$	−1.33798e6	−0.609614
$345$	1.24024e6	0.560992
$346$	−1.07626e6	−0.483313
$347$	−4.34782e6	−1.93842	−0.969210	−	0.246234i	$-0.920807\pi$
$347$	−4.34782e6	−1.93842	−0.969210	+	0.246234i	$0.920807\pi$
$348$	−891360.	−0.394553
$349$	181554.	0.0797889	0.0398944	−	0.999204i	$-0.487298\pi$
$349$	181554.	0.0797889	0.0398944	+	0.999204i	$0.487298\pi$
$350$	4.29467e6	1.87396
$351$	−336798.	−0.145916
$352$	−757760.	−0.325968
$353$	1.50420e6	0.642494	0.321247	−	0.946995i	$-0.395898\pi$
$353$	1.50420e6	0.642494	0.321247	+	0.946995i	$0.395898\pi$
$354$	−1.16186e6	−0.492773
$355$	−3.12343e6	−1.31541
$356$	−242144.	−0.101263
$357$	233496.	0.0969636
$358$	−3.25718e6	−1.34318
$359$	1.84040e6	0.753663	0.376831	−	0.926282i	$-0.377014\pi$
$359$	1.84040e6	0.753663	0.376831	+	0.926282i	$0.377014\pi$
$360$	−487296.	−0.198169
$361$	−1.95482e6	−0.789474
$362$	2.18238e6	0.875302
$363$	3.47894e6	1.38574
$364$	−1.38970e6	−0.549751
$365$	5.21155e6	2.04755
$366$	75528.0	0.0294717
$367$	4.44121e6	1.72122	0.860610	−	0.509265i	$-0.170083\pi$
$367$	4.44121e6	1.72122	0.860610	+	0.509265i	$0.170083\pi$
$368$	−375296.	−0.144462
$369$	−1.29227e6	−0.494070
$370$	514744.	0.195473
$371$	311704.	0.117573
$372$	376992.	0.141246
$373$	−3.54930e6	−1.32090	−0.660451	−	0.750869i	$-0.729635\pi$
$373$	−3.54930e6	−1.32090	−0.660451	+	0.750869i	$0.729635\pi$
$374$	−408480.	−0.151005
$375$	−2.18776e6	−0.803379
$376$	282880.	0.103189
$377$	2.85978e6	1.03629
$378$	548208.	0.197340
$379$	−1.65670e6	−0.592443	−0.296222	−	0.955119i	$-0.595727\pi$
$379$	−1.65670e6	−0.592443	−0.296222	+	0.955119i	$0.595727\pi$
$380$	1.08589e6	0.385768
$381$	−792396.	−0.279660
$382$	441912.	0.154945
$383$	−4.14211e6	−1.44286	−0.721430	−	0.692487i	$-0.756515\pi$
$383$	−4.14211e6	−1.44286	−0.721430	+	0.692487i	$0.756515\pi$
$384$	147456.	0.0510310
$385$	1.30773e7	4.49641
$386$	1.90366e6	0.650310
$387$	−1.69339e6	−0.574749
$388$	−1.29046e6	−0.435178
$389$	−2.85961e6	−0.958147	−0.479074	−	0.877775i	$-0.659027\pi$
$389$	−2.85961e6	−0.958147	−0.479074	+	0.877775i	$0.659027\pi$
$390$	1.56341e6	0.520488
$391$	−202308.	−0.0669223
$392$	1.18637e6	0.389946
$393$	2.99011e6	0.976574
$394$	3.97724e6	1.29075
$395$	2.40132e6	0.774387
$396$	−959040.	−0.307326
$397$	−1.31689e6	−0.419348	−0.209674	−	0.977771i	$-0.567240\pi$
$397$	−1.31689e6	−0.419348	−0.209674	+	0.977771i	$0.567240\pi$
$398$	−1.66642e6	−0.527324
$399$	−1.22162e6	−0.384154
$400$	1.46202e6	0.456880
$401$	−4.67400e6	−1.45154	−0.725768	−	0.687940i	$-0.758515\pi$
$401$	−4.67400e6	−1.45154	−0.725768	+	0.687940i	$0.758515\pi$
$402$	1.67602e6	0.517265
$403$	−1.20952e6	−0.370979
$404$	757920.	0.231031
$405$	−616734.	−0.186836
$406$	−4.65488e6	−1.40150
$407$	1.01306e6	0.303144
$408$	79488.0	0.0236402
$409$	−3.24561e6	−0.959373	−0.479687	−	0.877440i	$-0.659250\pi$
$409$	−3.24561e6	−0.959373	−0.479687	+	0.877440i	$0.659250\pi$
$410$	5.99870e6	1.76237
$411$	2.90842e6	0.849284
$412$	1.12605e6	0.326824
$413$	−6.06751e6	−1.75039
$414$	−474984.	−0.136200
$415$	−7.23048e6	−2.06085
$416$	−473088.	−0.134032
$417$	−65412.0	−0.0184212
$418$	2.13712e6	0.598258
$419$	1.22440e6	0.340713	0.170356	−	0.985383i	$-0.445508\pi$
$419$	1.22440e6	0.340713	0.170356	+	0.985383i	$0.445508\pi$
$420$	−2.54477e6	−0.703923
$421$	−2.59503e6	−0.713571	−0.356785	−	0.934186i	$-0.616127\pi$
$421$	−2.59503e6	−0.713571	−0.356785	+	0.934186i	$0.616127\pi$
$422$	−3.35443e6	−0.916934
$423$	358020.	0.0972873
$424$	106112.	0.0286649
$425$	788118.	0.211650
$426$	1.19621e6	0.319362
$427$	394424.	0.104687
$428$	3.33734e6	0.880626
$429$	3.07692e6	0.807185
$430$	7.86066e6	2.05016
$431$	−1.12313e6	−0.291232	−0.145616	−	0.989341i	$-0.546516\pi$
$431$	−1.12313e6	−0.291232	−0.145616	+	0.989341i	$0.546516\pi$
$432$	186624.	0.0481125
$433$	2.68657e6	0.688619	0.344310	−	0.938856i	$-0.388113\pi$
$433$	2.68657e6	0.688619	0.344310	+	0.938856i	$0.388113\pi$
$434$	1.96874e6	0.501722
$435$	5.23674e6	1.32690
$436$	−719712.	−0.181319
$437$	1.05845e6	0.265135
$438$	−1.99591e6	−0.497114
$439$	7.30208e6	1.80836	0.904181	−	0.427149i	$-0.140482\pi$
$439$	7.30208e6	1.80836	0.904181	+	0.427149i	$0.140482\pi$
$440$	4.45184e6	1.09625
$441$	1.50150e6	0.367644
$442$	−255024.	−0.0620905
$443$	735252.	0.178003	0.0890014	−	0.996031i	$-0.471632\pi$
$443$	735252.	0.178003	0.0890014	+	0.996031i	$0.471632\pi$
$444$	−197136.	−0.0474579
$445$	1.42260e6	0.340551
$446$	−978864.	−0.233016
$447$	−4.52111e6	−1.07023
$448$	770048.	0.181269
$449$	802618.	0.187885	0.0939427	−	0.995578i	$-0.470053\pi$
$449$	802618.	0.187885	0.0939427	+	0.995578i	$0.470053\pi$
$450$	1.85036e6	0.430751
$451$	1.18060e7	2.73313
$452$	−1.25552e6	−0.289053
$453$	540720.	0.123802
$454$	55352.0	0.0126036
$455$	8.16446e6	1.84884
$456$	−415872.	−0.0936586
$457$	−1.32721e6	−0.297270	−0.148635	−	0.988892i	$-0.547488\pi$
$457$	−1.32721e6	−0.297270	−0.148635	+	0.988892i	$0.547488\pi$
$458$	−5.15223e6	−1.14771
$459$	100602.	0.0222882
$460$	2.20486e6	0.485833
$461$	5.45153e6	1.19472	0.597360	−	0.801973i	$-0.296216\pi$
$461$	5.45153e6	1.19472	0.597360	+	0.801973i	$0.296216\pi$
$462$	−5.00832e6	−1.09166
$463$	−3.45611e6	−0.749263	−0.374632	−	0.927174i	$-0.622231\pi$
$463$	−3.45611e6	−0.749263	−0.374632	+	0.927174i	$0.622231\pi$
$464$	−1.58464e6	−0.341693
$465$	−2.21483e6	−0.475015
$466$	−3.75124e6	−0.800221
$467$	8.49264e6	1.80198	0.900991	−	0.433838i	$-0.142841\pi$
$467$	8.49264e6	1.80198	0.900991	+	0.433838i	$0.142841\pi$
$468$	−598752.	−0.126367
$469$	8.75253e6	1.83739
$470$	−1.66192e6	−0.347029
$471$	−2.71571e6	−0.564068
$472$	−2.06554e6	−0.426754
$473$	1.54704e7	3.17943
$474$	−919656.	−0.188009
$475$	−4.12334e6	−0.838524
$476$	415104.	0.0839730
$477$	134298.	0.0270255
$478$	−4.03687e6	−0.808119
$479$	−7.47323e6	−1.48823	−0.744115	−	0.668052i	$-0.767128\pi$
$479$	−7.47323e6	−1.48823	−0.744115	+	0.668052i	$0.767128\pi$
$480$	−866304.	−0.171620
$481$	632478.	0.124647
$482$	5.17991e6	1.01556
$483$	−2.48047e6	−0.483801
$484$	6.18478e6	1.20008
$485$	7.58148e6	1.46352
$486$	236196.	0.0453609
$487$	−3.04349e6	−0.581500	−0.290750	−	0.956799i	$-0.593905\pi$
$487$	−3.04349e6	−0.581500	−0.290750	+	0.956799i	$0.593905\pi$
$488$	134272.	0.0255232
$489$	4.98393e6	0.942540
$490$	−6.96991e6	−1.31141
$491$	789608.	0.147811	0.0739057	−	0.997265i	$-0.476454\pi$
$491$	789608.	0.147811	0.0739057	+	0.997265i	$0.476454\pi$
$492$	−2.29738e6	−0.427877
$493$	−854220.	−0.158290
$494$	1.33426e6	0.245992
$495$	5.63436e6	1.03355
$496$	670208.	0.122322
$497$	6.24686e6	1.13441
$498$	2.76912e6	0.500344
$499$	−8.37433e6	−1.50556	−0.752781	−	0.658271i	$-0.771288\pi$
$499$	−8.37433e6	−1.50556	−0.752781	+	0.658271i	$0.771288\pi$
$500$	−3.88934e6	−0.695747
$501$	4.08879e6	0.727781
$502$	−5.00604e6	−0.886615
$503$	−5.42707e6	−0.956412	−0.478206	−	0.878248i	$-0.658713\pi$
$503$	−5.42707e6	−0.956412	−0.478206	+	0.878248i	$0.658713\pi$
$504$	974592.	0.170902
$505$	−4.45278e6	−0.776967
$506$	4.33936e6	0.753441
$507$	−1.42064e6	−0.245451
$508$	−1.40870e6	−0.242192
$509$	−1.62849e6	−0.278606	−0.139303	−	0.990250i	$-0.544486\pi$
$509$	−1.62849e6	−0.278606	−0.139303	+	0.990250i	$0.544486\pi$
$510$	−466992.	−0.0795031
$511$	−1.04231e7	−1.76581
$512$	262144.	0.0441942
$513$	−526338.	−0.0883022
$514$	−770424.	−0.128624
$515$	−6.61553e6	−1.09912
$516$	−3.01046e6	−0.497748
$517$	−3.27080e6	−0.538180
$518$	−1.02949e6	−0.168576
$519$	−2.42159e6	−0.394623
$520$	2.77939e6	0.450756
$521$	−2.28000e6	−0.367994	−0.183997	−	0.982927i	$-0.558904\pi$
$521$	−2.28000e6	−0.367994	−0.183997	+	0.982927i	$0.558904\pi$
$522$	−2.00556e6	−0.322151
$523$	2.78085e6	0.444552	0.222276	−	0.974984i	$-0.428651\pi$
$523$	2.78085e6	0.444552	0.222276	+	0.974984i	$0.428651\pi$
$524$	5.31574e6	0.845738
$525$	9.66301e6	1.53008
$526$	1.86230e6	0.293485
$527$	361284.	0.0566660
$528$	−1.70496e6	−0.266152
$529$	−4.28719e6	−0.666091
$530$	−623408.	−0.0964013
$531$	−2.61419e6	−0.402348
$532$	−2.17178e6	−0.332687
$533$	7.37075e6	1.12381
$534$	−544824.	−0.0826805
$535$	−1.96069e7	−2.96159
$536$	2.97958e6	0.447965
$537$	−7.32865e6	−1.09670
$538$	−6.54862e6	−0.975425
$539$	−1.37174e7	−2.03376
$540$	−1.09642e6	−0.161805
$541$	−1.25932e7	−1.84988	−0.924939	−	0.380117i	$-0.875884\pi$
$541$	−1.25932e7	−1.84988	−0.924939	+	0.380117i	$0.875884\pi$
$542$	−7.58296e6	−1.10877
$543$	4.91035e6	0.714681
$544$	141312.	0.0204730
$545$	4.22831e6	0.609783
$546$	−3.12682e6	−0.448870
$547$	4.31470e6	0.616570	0.308285	−	0.951294i	$-0.400245\pi$
$547$	4.31470e6	0.616570	0.308285	+	0.951294i	$0.400245\pi$
$548$	5.17053e6	0.735501
$549$	169938.	0.0240635
$550$	−1.69046e7	−2.38285
$551$	4.46918e6	0.627117
$552$	−844416.	−0.117953
$553$	−4.80265e6	−0.667833
$554$	378184.	0.0523515
$555$	1.15817e6	0.159603
$556$	−116288.	−0.0159532
$557$	−3.91112e6	−0.534150	−0.267075	−	0.963676i	$-0.586057\pi$
$557$	−3.91112e6	−0.534150	−0.267075	+	0.963676i	$0.586057\pi$
$558$	848232.	0.115327
$559$	9.65857e6	1.30732
$560$	−4.52403e6	−0.609615
$561$	−919080.	−0.123295
$562$	5.84100e6	0.780093
$563$	−1.01933e7	−1.35533	−0.677663	−	0.735373i	$-0.737007\pi$
$563$	−1.01933e7	−1.35533	−0.677663	+	0.735373i	$0.737007\pi$
$564$	636480.	0.0842533
$565$	7.37618e6	0.972099
$566$	1.00849e7	1.32321
$567$	1.23347e6	0.161128
$568$	2.12659e6	0.276575
$569$	−6.27879e6	−0.813009	−0.406504	−	0.913649i	$-0.633252\pi$
$569$	−6.27879e6	−0.813009	−0.406504	+	0.913649i	$0.633252\pi$
$570$	2.44325e6	0.314978
$571$	−1.19061e6	−0.152819	−0.0764097	−	0.997077i	$-0.524346\pi$
$571$	−1.19061e6	−0.152819	−0.0764097	+	0.997077i	$0.524346\pi$
$572$	5.47008e6	0.699043
$573$	994302.	0.126512
$574$	−1.19974e7	−1.51987
$575$	−8.37233e6	−1.05603
$576$	331776.	0.0416667
$577$	4.06805e6	0.508682	0.254341	−	0.967115i	$-0.418141\pi$
$577$	4.06805e6	0.508682	0.254341	+	0.967115i	$0.418141\pi$
$578$	−5.60325e6	−0.697623
$579$	4.28323e6	0.530976
$580$	9.30976e6	1.14913
$581$	1.44610e7	1.77728
$582$	−2.90354e6	−0.355321
$583$	−1.22692e6	−0.149501
$584$	−3.54829e6	−0.430513
$585$	3.51767e6	0.424977
$586$	−64344.0	−0.00774041
$587$	4.89307e6	0.586120	0.293060	−	0.956094i	$-0.405326\pi$
$587$	4.89307e6	0.586120	0.293060	+	0.956094i	$0.405326\pi$
$588$	2.66933e6	0.318389
$589$	−1.89020e6	−0.224501
$590$	1.21350e7	1.43519
$591$	8.94879e6	1.05389
$592$	−350464.	−0.0410997
$593$	7.16991e6	0.837292	0.418646	−	0.908150i	$-0.362505\pi$
$593$	7.16991e6	0.837292	0.418646	+	0.908150i	$0.362505\pi$
$594$	−2.15784e6	−0.250930
$595$	−2.43874e6	−0.282405
$596$	−8.03754e6	−0.926845
$597$	−3.74945e6	−0.430559
$598$	2.70917e6	0.309801
$599$	−1.70565e7	−1.94233	−0.971166	−	0.238404i	$-0.923376\pi$
$599$	−1.70565e7	−1.94233	−0.971166	+	0.238404i	$0.923376\pi$
$600$	3.28954e6	0.373041
$601$	511338.	0.0577460	0.0288730	−	0.999583i	$-0.490808\pi$
$601$	511338.	0.0577460	0.0288730	+	0.999583i	$0.490808\pi$
$602$	−1.57213e7	−1.76806
$603$	3.77104e6	0.422345
$604$	961280.	0.107215
$605$	−3.63356e7	−4.03593
$606$	1.70532e6	0.188636
$607$	1.23271e7	1.35797	0.678986	−	0.734151i	$-0.262420\pi$
$607$	1.23271e7	1.35797	0.678986	+	0.734151i	$0.262420\pi$
$608$	−739328.	−0.0811107
$609$	−1.04735e7	−1.14432
$610$	−788848.	−0.0858359
$611$	−2.04204e6	−0.221290
$612$	178848.	0.0193021
$613$	9.03550e6	0.971183	0.485591	−	0.874186i	$-0.338604\pi$
$613$	9.03550e6	0.971183	0.485591	+	0.874186i	$0.338604\pi$
$614$	−2.65346e6	−0.284047
$615$	1.34971e7	1.43897
$616$	−8.90368e6	−0.945405
$617$	4.58459e6	0.484827	0.242414	−	0.970173i	$-0.422061\pi$
$617$	4.58459e6	0.484827	0.242414	+	0.970173i	$0.422061\pi$
$618$	2.53361e6	0.266851
$619$	6.21266e6	0.651705	0.325852	−	0.945421i	$-0.394349\pi$
$619$	6.21266e6	0.651705	0.325852	+	0.945421i	$0.394349\pi$
$620$	−3.93747e6	−0.411375
$621$	−1.06871e6	−0.111207
$622$	3.86346e6	0.400406
$623$	−2.84519e6	−0.293692
$624$	−1.06445e6	−0.109437
$625$	5.00302e6	0.512309
$626$	5.37316e6	0.548017
$627$	4.80852e6	0.488475
$628$	−4.82794e6	−0.488498
$629$	−188922.	−0.0190395
$630$	−5.72573e6	−0.574750
$631$	−1.82294e6	−0.182263	−0.0911317	−	0.995839i	$-0.529048\pi$
$631$	−1.82294e6	−0.182263	−0.0911317	+	0.995839i	$0.529048\pi$
$632$	−1.63494e6	−0.162821
$633$	−7.54747e6	−0.748673
$634$	4.83382e6	0.477603
$635$	8.27614e6	0.814504
$636$	238752.	0.0234048
$637$	−8.56409e6	−0.836244
$638$	1.83224e7	1.78209
$639$	2.69147e6	0.260758
$640$	−1.54010e6	−0.148627
$641$	1.61309e7	1.55065	0.775327	−	0.631560i	$-0.217585\pi$
$641$	1.61309e7	1.55065	0.775327	+	0.631560i	$0.217585\pi$
$642$	7.50902e6	0.719028
$643$	1.04288e7	0.994734	0.497367	−	0.867540i	$-0.334300\pi$
$643$	1.04288e7	0.994734	0.497367	+	0.867540i	$0.334300\pi$
$644$	−4.40973e6	−0.418984
$645$	1.76865e7	1.67395
$646$	−398544.	−0.0375747
$647$	−3.53863e6	−0.332334	−0.166167	−	0.986098i	$-0.553139\pi$
$647$	−3.53863e6	−0.332334	−0.166167	+	0.986098i	$0.553139\pi$
$648$	419904.	0.0392837
$649$	2.38828e7	2.22573
$650$	−1.05539e7	−0.979785
$651$	4.42966e6	0.409654
$652$	8.86032e6	0.816264
$653$	−1.17876e7	−1.08179	−0.540893	−	0.841091i	$-0.681914\pi$
$653$	−1.17876e7	−1.08179	−0.540893	+	0.841091i	$0.681914\pi$
$654$	−1.61935e6	−0.148046
$655$	−3.12300e7	−2.84426
$656$	−4.08422e6	−0.370553
$657$	−4.49080e6	−0.405892
$658$	3.32384e6	0.299278
$659$	−6.49666e6	−0.582742	−0.291371	−	0.956610i	$-0.594111\pi$
$659$	−6.49666e6	−0.582742	−0.291371	+	0.956610i	$0.594111\pi$
$660$	1.00166e7	0.895081
$661$	9.33195e6	0.830747	0.415373	−	0.909651i	$-0.363651\pi$
$661$	9.33195e6	0.830747	0.415373	+	0.909651i	$0.363651\pi$
$662$	−1.94948e6	−0.172892
$663$	−573804.	−0.0506967
$664$	4.92288e6	0.433310
$665$	1.27592e7	1.11884
$666$	−443556.	−0.0387492
$667$	9.07454e6	0.789787
$668$	7.26896e6	0.630277
$669$	−2.20244e6	−0.190257
$670$	−1.75051e7	−1.50653
$671$	−1.55252e6	−0.133116
$672$	1.73261e6	0.148005
$673$	1.63467e7	1.39121	0.695606	−	0.718424i	$-0.255136\pi$
$673$	1.63467e7	1.39121	0.695606	+	0.718424i	$0.255136\pi$
$674$	−1.18770e7	−1.00706
$675$	4.16332e6	0.351706
$676$	−2.52558e6	−0.212567
$677$	1.37588e7	1.15374	0.576871	−	0.816835i	$-0.304273\pi$
$677$	1.37588e7	1.15374	0.576871	+	0.816835i	$0.304273\pi$
$678$	−2.82492e6	−0.236011
$679$	−1.51630e7	−1.26215
$680$	−830208.	−0.0688517
$681$	124542.	0.0102908
$682$	−7.74928e6	−0.637970
$683$	1.59236e7	1.30614	0.653071	−	0.757297i	$-0.273480\pi$
$683$	1.59236e7	1.30614	0.653071	+	0.757297i	$0.273480\pi$
$684$	−935712.	−0.0764719
$685$	−3.03769e7	−2.47353
$686$	1.30096e6	0.105549
$687$	−1.15925e7	−0.937100
$688$	−5.35194e6	−0.431062
$689$	−765996.	−0.0614722
$690$	4.96094e6	0.396681
$691$	478964.	0.0381599	0.0190800	−	0.999818i	$-0.493926\pi$
$691$	478964.	0.0381599	0.0190800	+	0.999818i	$0.493926\pi$
$692$	−4.30506e6	−0.341754
$693$	−1.12687e7	−0.891336
$694$	−1.73913e7	−1.37067
$695$	683192.	0.0536514
$696$	−3.56544e6	−0.278991
$697$	−2.20165e6	−0.171659
$698$	726216.	0.0564192
$699$	−8.44029e6	−0.653378
$700$	1.71787e7	1.32509
$701$	−1.15565e7	−0.888241	−0.444121	−	0.895967i	$-0.646484\pi$
$701$	−1.15565e7	−0.888241	−0.444121	+	0.895967i	$0.646484\pi$
$702$	−1.34719e6	−0.103178
$703$	988418.	0.0754314
$704$	−3.03104e6	−0.230494
$705$	−3.73932e6	−0.283348
$706$	6.01681e6	0.454312
$707$	8.90556e6	0.670058
$708$	−4.64746e6	−0.348443
$709$	1.19797e7	0.895015	0.447507	−	0.894280i	$-0.352312\pi$
$709$	1.19797e7	0.895015	0.447507	+	0.894280i	$0.352312\pi$
$710$	−1.24937e7	−0.930135
$711$	−2.06923e6	−0.153509
$712$	−968576.	−0.0716034
$713$	−3.83799e6	−0.282735
$714$	933984.	0.0685636
$715$	−3.21367e7	−2.35091
$716$	−1.30287e7	−0.949770
$717$	−9.08296e6	−0.659826
$718$	7.36162e6	0.532920
$719$	9.00832e6	0.649863	0.324931	−	0.945738i	$-0.394659\pi$
$719$	9.00832e6	0.649863	0.324931	+	0.945738i	$0.394659\pi$
$720$	−1.94918e6	−0.140127
$721$	1.32311e7	0.947887
$722$	−7.81926e6	−0.558242
$723$	1.16548e7	0.829200
$724$	8.72950e6	0.618932
$725$	−3.53511e7	−2.49780
$726$	1.39158e7	0.979863
$727$	−8.25072e6	−0.578970	−0.289485	−	0.957183i	$-0.593484\pi$
$727$	−8.25072e6	−0.578970	−0.289485	+	0.957183i	$0.593484\pi$
$728$	−5.55878e6	−0.388733
$729$	531441.	0.0370370
$730$	2.08462e7	1.44784
$731$	−2.88503e6	−0.199690
$732$	302112.	0.0208396
$733$	1.59454e7	1.09616	0.548081	−	0.836426i	$-0.315359\pi$
$733$	1.59454e7	1.09616	0.548081	+	0.836426i	$0.315359\pi$
$734$	1.77648e7	1.21709
$735$	−1.56823e7	−1.07076
$736$	−1.50118e6	−0.102150
$737$	−3.44514e7	−2.33635
$738$	−5.16910e6	−0.349360
$739$	−5.06241e6	−0.340994	−0.170497	−	0.985358i	$-0.554537\pi$
$739$	−5.06241e6	−0.340994	−0.170497	+	0.985358i	$0.554537\pi$
$740$	2.05898e6	0.138220
$741$	3.00208e6	0.200852
$742$	1.24682e6	0.0831367
$743$	−2.30579e7	−1.53231	−0.766157	−	0.642654i	$-0.777833\pi$
$743$	−2.30579e7	−1.53231	−0.766157	+	0.642654i	$0.777833\pi$
$744$	1.50797e6	0.0998757
$745$	4.72205e7	3.11702
$746$	−1.41972e7	−0.934018
$747$	6.23052e6	0.408529
$748$	−1.63392e6	−0.106777
$749$	3.92138e7	2.55408
$750$	−8.75102e6	−0.568075
$751$	1.30144e7	0.842026	0.421013	−	0.907055i	$-0.361675\pi$
$751$	1.30144e7	0.842026	0.421013	+	0.907055i	$0.361675\pi$
$752$	1.13152e6	0.0729655
$753$	−1.12636e7	−0.723918
$754$	1.14391e7	0.732764
$755$	−5.64752e6	−0.360571
$756$	2.19283e6	0.139541
$757$	8.02216e6	0.508805	0.254403	−	0.967098i	$-0.418121\pi$
$757$	8.02216e6	0.508805	0.254403	+	0.967098i	$0.418121\pi$
$758$	−6.62682e6	−0.418921
$759$	9.76356e6	0.615182
$760$	4.34355e6	0.272779
$761$	7.63406e6	0.477852	0.238926	−	0.971038i	$-0.423205\pi$
$761$	7.63406e6	0.477852	0.238926	+	0.971038i	$0.423205\pi$
$762$	−3.16958e6	−0.197749
$763$	−8.45662e6	−0.525878
$764$	1.76765e6	0.109563
$765$	−1.05073e6	−0.0649140
$766$	−1.65684e7	−1.02026
$767$	1.49106e7	0.915180
$768$	589824.	0.0360844
$769$	−1.57980e6	−0.0963354	−0.0481677	−	0.998839i	$-0.515338\pi$
$769$	−1.57980e6	−0.0963354	−0.0481677	+	0.998839i	$0.515338\pi$
$770$	5.23091e7	3.17944
$771$	−1.73345e6	−0.105021
$772$	7.61462e6	0.459838
$773$	1.36785e7	0.823361	0.411681	−	0.911328i	$-0.364942\pi$
$773$	1.36785e7	0.823361	0.411681	+	0.911328i	$0.364942\pi$
$774$	−6.77354e6	−0.406409
$775$	1.49514e7	0.894185
$776$	−5.16186e6	−0.307717
$777$	−2.31635e6	−0.137642
$778$	−1.14384e7	−0.677512
$779$	1.15188e7	0.680085
$780$	6.25363e6	0.368041
$781$	−2.45887e7	−1.44248
$782$	−809232.	−0.0473212
$783$	−4.51251e6	−0.263035
$784$	4.74547e6	0.275733
$785$	2.83641e7	1.64284
$786$	1.19604e7	0.690542
$787$	1.42563e7	0.820484	0.410242	−	0.911977i	$-0.365444\pi$
$787$	1.42563e7	0.820484	0.410242	+	0.911977i	$0.365444\pi$
$788$	1.59090e7	0.912696
$789$	4.19018e6	0.239630
$790$	9.60530e6	0.547574
$791$	−1.47524e7	−0.838340
$792$	−3.83616e6	−0.217312
$793$	−969276.	−0.0547349
$794$	−5.26758e6	−0.296524
$795$	−1.40267e6	−0.0787113
$796$	−6.66570e6	−0.372875
$797$	6.14004e6	0.342394	0.171197	−	0.985237i	$-0.445237\pi$
$797$	6.14004e6	0.342394	0.171197	+	0.985237i	$0.445237\pi$
$798$	−4.88650e6	−0.271638
$799$	609960.	0.0338014
$800$	5.84806e6	0.323063
$801$	−1.22585e6	−0.0675083
$802$	−1.86960e7	−1.02639
$803$	4.10271e7	2.24534
$804$	6.70406e6	0.365762
$805$	2.59072e7	1.40906
$806$	−4.83806e6	−0.262322
$807$	−1.47344e7	−0.796431
$808$	3.03168e6	0.163363
$809$	1.14816e7	0.616783	0.308391	−	0.951260i	$-0.400209\pi$
$809$	1.14816e7	0.616783	0.308391	+	0.951260i	$0.400209\pi$
$810$	−2.46694e6	−0.132113
$811$	1.41562e7	0.755779	0.377889	−	0.925851i	$-0.376650\pi$
$811$	1.41562e7	0.755779	0.377889	+	0.925851i	$0.376650\pi$
$812$	−1.86195e7	−0.991011
$813$	−1.70617e7	−0.905305
$814$	4.05224e6	0.214355
$815$	−5.20544e7	−2.74513
$816$	317952.	0.0167161
$817$	1.50941e7	0.791139
$818$	−1.29824e7	−0.678379
$819$	−7.03534e6	−0.366501
$820$	2.39948e7	1.24619
$821$	−8.12257e6	−0.420567	−0.210284	−	0.977640i	$-0.567439\pi$
$821$	−8.12257e6	−0.420567	−0.210284	+	0.977640i	$0.567439\pi$
$822$	1.16337e7	0.600534
$823$	−2.34067e7	−1.20460	−0.602298	−	0.798272i	$-0.705748\pi$
$823$	−2.34067e7	−1.20460	−0.602298	+	0.798272i	$0.705748\pi$
$824$	4.50419e6	0.231099
$825$	−3.80353e7	−1.94559
$826$	−2.42700e7	−1.23771
$827$	2.94496e7	1.49733	0.748663	−	0.662951i	$-0.230696\pi$
$827$	2.94496e7	1.49733	0.748663	+	0.662951i	$0.230696\pi$
$828$	−1.89994e6	−0.0963082
$829$	−3.37105e7	−1.70364	−0.851822	−	0.523832i	$-0.824502\pi$
$829$	−3.37105e7	−1.70364	−0.851822	+	0.523832i	$0.824502\pi$
$830$	−2.89219e7	−1.45724
$831$	850914.	0.0427448
$832$	−1.89235e6	−0.0947750
$833$	2.55811e6	0.127734
$834$	−261648.	−0.0130257
$835$	−4.27051e7	−2.11965
$836$	8.54848e6	0.423032
$837$	1.90852e6	0.0941637
$838$	4.89760e6	0.240920
$839$	−2.09190e7	−1.02597	−0.512987	−	0.858396i	$-0.671461\pi$
$839$	−2.09190e7	−1.02597	−0.512987	+	0.858396i	$0.671461\pi$
$840$	−1.01791e7	−0.497749
$841$	1.78050e7	0.868062
$842$	−1.03801e7	−0.504571
$843$	1.31422e7	0.636943
$844$	−1.34177e7	−0.648370
$845$	1.48378e7	0.714871
$846$	1.43208e6	0.0687925
$847$	7.26712e7	3.48060
$848$	424448.	0.0202691
$849$	2.26910e7	1.08040
$850$	3.15247e6	0.149659
$851$	2.00695e6	0.0949978
$852$	4.78483e6	0.225823
$853$	−1.86368e7	−0.876999	−0.438499	−	0.898731i	$-0.644490\pi$
$853$	−1.86368e7	−0.876999	−0.438499	+	0.898731i	$0.644490\pi$
$854$	1.57770e6	0.0740250
$855$	5.49731e6	0.257179
$856$	1.33494e7	0.622697
$857$	−2.51977e7	−1.17195	−0.585975	−	0.810329i	$-0.699288\pi$
$857$	−2.51977e7	−1.17195	−0.585975	+	0.810329i	$0.699288\pi$
$858$	1.23077e7	0.570766
$859$	−2.33184e7	−1.07824	−0.539121	−	0.842228i	$-0.681243\pi$
$859$	−2.33184e7	−1.07824	−0.539121	+	0.842228i	$0.681243\pi$
$860$	3.14426e7	1.44968
$861$	−2.69942e7	−1.24097
$862$	−4.49254e6	−0.205932
$863$	1.29389e7	0.591384	0.295692	−	0.955283i	$-0.404450\pi$
$863$	1.29389e7	0.591384	0.295692	+	0.955283i	$0.404450\pi$
$864$	746496.	0.0340207
$865$	2.52922e7	1.14933
$866$	1.07463e7	0.486927
$867$	−1.26073e7	−0.569606
$868$	7.87494e6	0.354771
$869$	1.89040e7	0.849191
$870$	2.09470e7	0.938259
$871$	−2.15089e7	−0.960666
$872$	−2.87885e6	−0.128212
$873$	−6.53297e6	−0.290118
$874$	4.23381e6	0.187479
$875$	−4.56998e7	−2.01787
$876$	−7.98365e6	−0.351513
$877$	1.20623e7	0.529580	0.264790	−	0.964306i	$-0.414697\pi$
$877$	1.20623e7	0.529580	0.264790	+	0.964306i	$0.414697\pi$
$878$	2.92083e7	1.27871
$879$	−144774.	−0.00632002
$880$	1.78074e7	0.775163
$881$	−1.99491e6	−0.0865931	−0.0432966	−	0.999062i	$-0.513786\pi$
$881$	−1.99491e6	−0.0865931	−0.0432966	+	0.999062i	$0.513786\pi$
$882$	6.00599e6	0.259964
$883$	1.31081e7	0.565766	0.282883	−	0.959154i	$-0.408709\pi$
$883$	1.31081e7	0.565766	0.282883	+	0.959154i	$0.408709\pi$
$884$	−1.02010e6	−0.0439046
$885$	2.73038e7	1.17183
$886$	2.94101e6	0.125867
$887$	3.89434e7	1.66197	0.830987	−	0.556291i	$-0.187776\pi$
$887$	3.89434e7	1.66197	0.830987	+	0.556291i	$0.187776\pi$
$888$	−788544.	−0.0335578
$889$	−1.65523e7	−0.702430
$890$	5.69038e6	0.240806
$891$	−4.85514e6	−0.204884
$892$	−3.91546e6	−0.164767
$893$	−3.19124e6	−0.133915
$894$	−1.80845e7	−0.756766
$895$	7.65436e7	3.19412
$896$	3.08019e6	0.128176
$897$	6.09563e6	0.252952
$898$	3.21047e6	0.132855
$899$	−1.62054e7	−0.668746
$900$	7.40146e6	0.304587
$901$	228804.	0.00938970
$902$	4.72238e7	1.93261
$903$	−3.53730e7	−1.44362
$904$	−5.02208e6	−0.204391
$905$	−5.12858e7	−2.08150
$906$	2.16288e6	0.0875411
$907$	1.91123e7	0.771425	0.385712	−	0.922619i	$-0.373956\pi$
$907$	1.91123e7	0.771425	0.385712	+	0.922619i	$0.373956\pi$
$908$	221408.	0.00891207
$909$	3.83697e6	0.154021
$910$	3.26579e7	1.30733
$911$	−4.90044e7	−1.95632	−0.978158	−	0.207860i	$-0.933350\pi$
$911$	−4.90044e7	−1.95632	−0.978158	+	0.207860i	$0.933350\pi$
$912$	−1.66349e6	−0.0662266
$913$	−5.69208e7	−2.25993
$914$	−5.30886e6	−0.210201
$915$	−1.77491e6	−0.0700847
$916$	−2.06089e7	−0.811553
$917$	6.24600e7	2.45289
$918$	402408.	0.0157601
$919$	−3.40843e7	−1.33127	−0.665634	−	0.746278i	$-0.731839\pi$
$919$	−3.40843e7	−1.33127	−0.665634	+	0.746278i	$0.731839\pi$
$920$	8.81946e6	0.343536
$921$	−5.97028e6	−0.231924
$922$	2.18061e7	0.844795
$923$	−1.53513e7	−0.593119
$924$	−2.00333e7	−0.771920
$925$	−7.81836e6	−0.300442
$926$	−1.38244e7	−0.529809
$927$	5.70062e6	0.217883
$928$	−6.33856e6	−0.241613
$929$	9.12293e6	0.346813	0.173406	−	0.984850i	$-0.444523\pi$
$929$	9.12293e6	0.346813	0.173406	+	0.984850i	$0.444523\pi$
$930$	−8.85931e6	−0.335887
$931$	−1.33837e7	−0.506061
$932$	−1.50050e7	−0.565842
$933$	8.69279e6	0.326930
$934$	3.39706e7	1.27419
$935$	9.59928e6	0.359095
$936$	−2.39501e6	−0.0893547
$937$	−3.17882e7	−1.18282	−0.591408	−	0.806372i	$-0.701428\pi$
$937$	−3.17882e7	−1.18282	−0.591408	+	0.806372i	$0.701428\pi$
$938$	3.50101e7	1.29923
$939$	1.20896e7	0.447454
$940$	−6.64768e6	−0.245386
$941$	−3.17451e7	−1.16870	−0.584349	−	0.811503i	$-0.698650\pi$
$941$	−3.17451e7	−1.16870	−0.584349	+	0.811503i	$0.698650\pi$
$942$	−1.08629e7	−0.398857
$943$	2.33886e7	0.856494
$944$	−8.26214e6	−0.301761
$945$	−1.28829e7	−0.469282
$946$	6.18818e7	2.24820
$947$	315374.	0.0114275	0.00571375	−	0.999984i	$-0.498181\pi$
$947$	315374.	0.0114275	0.00571375	+	0.999984i	$0.498181\pi$
$948$	−3.67862e6	−0.132943
$949$	2.56142e7	0.923242
$950$	−1.64934e7	−0.592926
$951$	1.08761e7	0.389961
$952$	1.66042e6	0.0593779
$953$	4.90143e7	1.74820	0.874100	−	0.485747i	$-0.161452\pi$
$953$	4.90143e7	1.74820	0.874100	+	0.485747i	$0.161452\pi$
$954$	537192.	0.0191099
$955$	−1.03849e7	−0.368464
$956$	−1.61475e7	−0.571426
$957$	4.12254e7	1.45507
$958$	−2.98929e7	−1.05234
$959$	6.07537e7	2.13317
$960$	−3.46522e6	−0.121353
$961$	−2.17752e7	−0.760596
$962$	2.52991e6	0.0881389
$963$	1.68953e7	0.587084
$964$	2.07196e7	0.718108
$965$	−4.47359e7	−1.54646
$966$	−9.92189e6	−0.342099
$967$	5.06161e6	0.174069	0.0870346	−	0.996205i	$-0.472261\pi$
$967$	5.06161e6	0.174069	0.0870346	+	0.996205i	$0.472261\pi$
$968$	2.47391e7	0.848587
$969$	−896724.	−0.0306796
$970$	3.03259e7	1.03487
$971$	−2.25994e7	−0.769218	−0.384609	−	0.923080i	$-0.625664\pi$
$971$	−2.25994e7	−0.769218	−0.384609	+	0.923080i	$0.625664\pi$
$972$	944784.	0.0320750
$973$	−1.36638e6	−0.0462691
$974$	−1.21740e7	−0.411182
$975$	−2.37463e7	−0.799991
$976$	537088.	0.0180477
$977$	−3.29721e7	−1.10512	−0.552560	−	0.833473i	$-0.686349\pi$
$977$	−3.29721e7	−1.10512	−0.552560	+	0.833473i	$0.686349\pi$
$978$	1.99357e7	0.666476
$979$	1.11992e7	0.373447
$980$	−2.78796e7	−0.927304
$981$	−3.64354e6	−0.120879
$982$	3.15843e6	0.104518
$983$	−5.16350e7	−1.70435	−0.852177	−	0.523253i	$-0.824718\pi$
$983$	−5.16350e7	−1.70435	−0.852177	+	0.523253i	$0.824718\pi$
$984$	−9.18950e6	−0.302555
$985$	−9.34651e7	−3.06944
$986$	−3.41688e6	−0.111928
$987$	7.47864e6	0.244360
$988$	5.33702e6	0.173943
$989$	3.06482e7	0.996355
$990$	2.25374e7	0.730830
$991$	−4.20225e7	−1.35924	−0.679622	−	0.733562i	$-0.737856\pi$
$991$	−4.20225e7	−1.35924	−0.679622	+	0.733562i	$0.737856\pi$
$992$	2.68083e6	0.0864949
$993$	−4.38633e6	−0.141165
$994$	2.49875e7	0.802151
$995$	3.91610e7	1.25399
$996$	1.10765e7	0.353796
$997$	2.22651e7	0.709393	0.354696	−	0.934982i	$-0.384584\pi$
$997$	2.22651e7	0.709393	0.354696	+	0.934982i	$0.384584\pi$
$998$	−3.34973e7	−1.06459
$999$	−998001.	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	222.6.a.b.1.1	✓	1
3.2	odd	2		666.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.6.a.b.1.1	✓	1	1.1	even	1	trivial
666.6.a.c.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	222.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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