LMFDB - Embedded newform 21.6.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [21,6,Mod(1,21)] 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("21.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	21.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$3.36806021607$
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	$\chi$	$=$	21.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} -34.0000 q^{5} -9.00000 q^{6} -49.0000 q^{7} -63.0000 q^{8} +81.0000 q^{9} -34.0000 q^{10} -340.000 q^{11} +279.000 q^{12} +454.000 q^{13} -49.0000 q^{14} +306.000 q^{15} +929.000 q^{16} -798.000 q^{17} +81.0000 q^{18} +892.000 q^{19} +1054.00 q^{20} +441.000 q^{21} -340.000 q^{22} -3192.00 q^{23} +567.000 q^{24} -1969.00 q^{25} +454.000 q^{26} -729.000 q^{27} +1519.00 q^{28} -8242.00 q^{29} +306.000 q^{30} -2496.00 q^{31} +2945.00 q^{32} +3060.00 q^{33} -798.000 q^{34} +1666.00 q^{35} -2511.00 q^{36} +9798.00 q^{37} +892.000 q^{38} -4086.00 q^{39} +2142.00 q^{40} +19834.0 q^{41} +441.000 q^{42} -17236.0 q^{43} +10540.0 q^{44} -2754.00 q^{45} -3192.00 q^{46} +8928.00 q^{47} -8361.00 q^{48} +2401.00 q^{49} -1969.00 q^{50} +7182.00 q^{51} -14074.0 q^{52} +150.000 q^{53} -729.000 q^{54} +11560.0 q^{55} +3087.00 q^{56} -8028.00 q^{57} -8242.00 q^{58} -42396.0 q^{59} -9486.00 q^{60} +14758.0 q^{61} -2496.00 q^{62} -3969.00 q^{63} -26783.0 q^{64} -15436.0 q^{65} +3060.00 q^{66} -1676.00 q^{67} +24738.0 q^{68} +28728.0 q^{69} +1666.00 q^{70} +14568.0 q^{71} -5103.00 q^{72} +78378.0 q^{73} +9798.00 q^{74} +17721.0 q^{75} -27652.0 q^{76} +16660.0 q^{77} -4086.00 q^{78} -2272.00 q^{79} -31586.0 q^{80} +6561.00 q^{81} +19834.0 q^{82} -37764.0 q^{83} -13671.0 q^{84} +27132.0 q^{85} -17236.0 q^{86} +74178.0 q^{87} +21420.0 q^{88} -117286. q^{89} -2754.00 q^{90} -22246.0 q^{91} +98952.0 q^{92} +22464.0 q^{93} +8928.00 q^{94} -30328.0 q^{95} -26505.0 q^{96} +10002.0 q^{97} +2401.00 q^{98} -27540.0 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.176777	0.0883883	−	0.996086i	$-0.471828\pi$
$2$	1.00000	0.176777	0.0883883	+	0.996086i	$0.471828\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−31.0000	−0.968750
$5$	−34.0000	−0.608210	−0.304105	−	0.952638i	$-0.598357\pi$
$5$	−34.0000	−0.608210	−0.304105	+	0.952638i	$0.598357\pi$
$6$	−9.00000	−0.102062
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−63.0000	−0.348029
$9$	81.0000	0.333333
$10$	−34.0000	−0.107517
$11$	−340.000	−0.847222	−0.423611	−	0.905844i	$-0.639238\pi$
$11$	−340.000	−0.847222	−0.423611	+	0.905844i	$0.639238\pi$
$12$	279.000	0.559308
$13$	454.000	0.745071	0.372535	−	0.928018i	$-0.378489\pi$
$13$	454.000	0.745071	0.372535	+	0.928018i	$0.378489\pi$
$14$	−49.0000	−0.0668153
$15$	306.000	0.351150
$16$	929.000	0.907227
$17$	−798.000	−0.669700	−0.334850	−	0.942271i	$-0.608686\pi$
$17$	−798.000	−0.669700	−0.334850	+	0.942271i	$0.608686\pi$
$18$	81.0000	0.0589256
$19$	892.000	0.566867	0.283433	−	0.958992i	$-0.408527\pi$
$19$	892.000	0.566867	0.283433	+	0.958992i	$0.408527\pi$
$20$	1054.00	0.589204
$21$	441.000	0.218218
$22$	−340.000	−0.149769
$23$	−3192.00	−1.25818	−0.629091	−	0.777332i	$-0.716573\pi$
$23$	−3192.00	−1.25818	−0.629091	+	0.777332i	$0.716573\pi$
$24$	567.000	0.200935
$25$	−1969.00	−0.630080
$26$	454.000	0.131711
$27$	−729.000	−0.192450
$28$	1519.00	0.366153
$29$	−8242.00	−1.81986	−0.909929	−	0.414764i	$-0.863864\pi$
$29$	−8242.00	−1.81986	−0.909929	+	0.414764i	$0.863864\pi$
$30$	306.000	0.0620752
$31$	−2496.00	−0.466488	−0.233244	−	0.972418i	$-0.574934\pi$
$31$	−2496.00	−0.466488	−0.233244	+	0.972418i	$0.574934\pi$
$32$	2945.00	0.508406
$33$	3060.00	0.489144
$34$	−798.000	−0.118387
$35$	1666.00	0.229882
$36$	−2511.00	−0.322917
$37$	9798.00	1.17661	0.588306	−	0.808639i	$-0.299795\pi$
$37$	9798.00	1.17661	0.588306	+	0.808639i	$0.299795\pi$
$38$	892.000	0.100209
$39$	−4086.00	−0.430167
$40$	2142.00	0.211675
$41$	19834.0	1.84268	0.921342	−	0.388754i	$-0.127094\pi$
$41$	19834.0	1.84268	0.921342	+	0.388754i	$0.127094\pi$
$42$	441.000	0.0385758
$43$	−17236.0	−1.42156	−0.710780	−	0.703414i	$-0.751658\pi$
$43$	−17236.0	−1.42156	−0.710780	+	0.703414i	$0.751658\pi$
$44$	10540.0	0.820746
$45$	−2754.00	−0.202737
$46$	−3192.00	−0.222417
$47$	8928.00	0.589535	0.294767	−	0.955569i	$-0.404758\pi$
$47$	8928.00	0.589535	0.294767	+	0.955569i	$0.404758\pi$
$48$	−8361.00	−0.523788
$49$	2401.00	0.142857
$50$	−1969.00	−0.111383
$51$	7182.00	0.386652
$52$	−14074.0	−0.721787
$53$	150.000	0.00733502	0.00366751	−	0.999993i	$-0.498833\pi$
$53$	150.000	0.00733502	0.00366751	+	0.999993i	$0.498833\pi$
$54$	−729.000	−0.0340207
$55$	11560.0	0.515289
$56$	3087.00	0.131543
$57$	−8028.00	−0.327281
$58$	−8242.00	−0.321709
$59$	−42396.0	−1.58560	−0.792802	−	0.609479i	$-0.791379\pi$
$59$	−42396.0	−1.58560	−0.792802	+	0.609479i	$0.791379\pi$
$60$	−9486.00	−0.340177
$61$	14758.0	0.507812	0.253906	−	0.967229i	$-0.418285\pi$
$61$	14758.0	0.507812	0.253906	+	0.967229i	$0.418285\pi$
$62$	−2496.00	−0.0824642
$63$	−3969.00	−0.125988
$64$	−26783.0	−0.817352
$65$	−15436.0	−0.453160
$66$	3060.00	0.0864692
$67$	−1676.00	−0.0456128	−0.0228064	−	0.999740i	$-0.507260\pi$
$67$	−1676.00	−0.0456128	−0.0228064	+	0.999740i	$0.507260\pi$
$68$	24738.0	0.648772
$69$	28728.0	0.726411
$70$	1666.00	0.0406378
$71$	14568.0	0.342968	0.171484	−	0.985187i	$-0.445144\pi$
$71$	14568.0	0.342968	0.171484	+	0.985187i	$0.445144\pi$
$72$	−5103.00	−0.116010
$73$	78378.0	1.72142	0.860710	−	0.509095i	$-0.170020\pi$
$73$	78378.0	1.72142	0.860710	+	0.509095i	$0.170020\pi$
$74$	9798.00	0.207998
$75$	17721.0	0.363777
$76$	−27652.0	−0.549152
$77$	16660.0	0.320220
$78$	−4086.00	−0.0760435
$79$	−2272.00	−0.0409582	−0.0204791	−	0.999790i	$-0.506519\pi$
$79$	−2272.00	−0.0409582	−0.0204791	+	0.999790i	$0.506519\pi$
$80$	−31586.0	−0.551785
$81$	6561.00	0.111111
$82$	19834.0	0.325743
$83$	−37764.0	−0.601704	−0.300852	−	0.953671i	$-0.597271\pi$
$83$	−37764.0	−0.601704	−0.300852	+	0.953671i	$0.597271\pi$
$84$	−13671.0	−0.211399
$85$	27132.0	0.407319
$86$	−17236.0	−0.251299
$87$	74178.0	1.05070
$88$	21420.0	0.294858
$89$	−117286.	−1.56954	−0.784768	−	0.619790i	$-0.787218\pi$
$89$	−117286.	−1.56954	−0.784768	+	0.619790i	$0.787218\pi$
$90$	−2754.00	−0.0358391
$91$	−22246.0	−0.281610
$92$	98952.0	1.21886
$93$	22464.0	0.269327
$94$	8928.00	0.104216
$95$	−30328.0	−0.344774
$96$	−26505.0	−0.293528
$97$	10002.0	0.107934	0.0539669	−	0.998543i	$-0.482813\pi$
$97$	10002.0	0.107934	0.0539669	+	0.998543i	$0.482813\pi$
$98$	2401.00	0.0252538
$99$	−27540.0	−0.282407
$100$	61039.0	0.610390
$101$	−108770.	−1.06098	−0.530488	−	0.847692i	$-0.677991\pi$
$101$	−108770.	−1.06098	−0.530488	+	0.847692i	$0.677991\pi$
$102$	7182.00	0.0683510
$103$	−199192.	−1.85003	−0.925015	−	0.379930i	$-0.875948\pi$
$103$	−199192.	−1.85003	−0.925015	+	0.379930i	$0.875948\pi$
$104$	−28602.0	−0.259306
$105$	−14994.0	−0.132722
$106$	150.000	0.00129666
$107$	−79972.0	−0.675272	−0.337636	−	0.941277i	$-0.609627\pi$
$107$	−79972.0	−0.675272	−0.337636	+	0.941277i	$0.609627\pi$
$108$	22599.0	0.186436
$109$	−46098.0	−0.371634	−0.185817	−	0.982584i	$-0.559493\pi$
$109$	−46098.0	−0.371634	−0.185817	+	0.982584i	$0.559493\pi$
$110$	11560.0	0.0910911
$111$	−88182.0	−0.679317
$112$	−45521.0	−0.342899
$113$	262706.	1.93541	0.967707	−	0.252078i	$-0.0811138\pi$
$113$	262706.	1.93541	0.967707	+	0.252078i	$0.0811138\pi$
$114$	−8028.00	−0.0578556
$115$	108528.	0.765239
$116$	255502.	1.76299
$117$	36774.0	0.248357
$118$	−42396.0	−0.280298
$119$	39102.0	0.253123
$120$	−19278.0	−0.122211
$121$	−45451.0	−0.282215
$122$	14758.0	0.0897693
$123$	−178506.	−1.06387
$124$	77376.0	0.451910
$125$	173196.	0.991432
$126$	−3969.00	−0.0222718
$127$	196608.	1.08166	0.540831	−	0.841131i	$-0.318110\pi$
$127$	196608.	1.08166	0.540831	+	0.841131i	$0.318110\pi$
$128$	−121023.	−0.652894
$129$	155124.	0.820738
$130$	−15436.0	−0.0801081
$131$	−77140.0	−0.392737	−0.196368	−	0.980530i	$-0.562915\pi$
$131$	−77140.0	−0.392737	−0.196368	+	0.980530i	$0.562915\pi$
$132$	−94860.0	−0.473858
$133$	−43708.0	−0.214255
$134$	−1676.00	−0.00806329
$135$	24786.0	0.117050
$136$	50274.0	0.233075
$137$	208170.	0.947582	0.473791	−	0.880637i	$-0.342885\pi$
$137$	208170.	0.947582	0.473791	+	0.880637i	$0.342885\pi$
$138$	28728.0	0.128413
$139$	−275580.	−1.20979	−0.604896	−	0.796304i	$-0.706785\pi$
$139$	−275580.	−1.20979	−0.604896	+	0.796304i	$0.706785\pi$
$140$	−51646.0	−0.222698
$141$	−80352.0	−0.340368
$142$	14568.0	0.0606288
$143$	−154360.	−0.631240
$144$	75249.0	0.302409
$145$	280228.	1.10686
$146$	78378.0	0.304307
$147$	−21609.0	−0.0824786
$148$	−303738.	−1.13984
$149$	−296106.	−1.09265	−0.546326	−	0.837573i	$-0.683974\pi$
$149$	−296106.	−1.09265	−0.546326	+	0.837573i	$0.683974\pi$
$150$	17721.0	0.0643073
$151$	−426472.	−1.52212	−0.761059	−	0.648683i	$-0.775320\pi$
$151$	−426472.	−1.52212	−0.761059	+	0.648683i	$0.775320\pi$
$152$	−56196.0	−0.197286
$153$	−64638.0	−0.223233
$154$	16660.0	0.0566074
$155$	84864.0	0.283723
$156$	126666.	0.416724
$157$	178486.	0.577903	0.288952	−	0.957344i	$-0.406693\pi$
$157$	178486.	0.577903	0.288952	+	0.957344i	$0.406693\pi$
$158$	−2272.00	−0.00724045
$159$	−1350.00	−0.00423488
$160$	−100130.	−0.309218
$161$	156408.	0.475548
$162$	6561.00	0.0196419
$163$	252772.	0.745178	0.372589	−	0.927996i	$-0.378470\pi$
$163$	252772.	0.745178	0.372589	+	0.927996i	$0.378470\pi$
$164$	−614854.	−1.78510
$165$	−104040.	−0.297502
$166$	−37764.0	−0.106367
$167$	508088.	1.40977	0.704884	−	0.709322i	$-0.250999\pi$
$167$	508088.	1.40977	0.704884	+	0.709322i	$0.250999\pi$
$168$	−27783.0	−0.0759462
$169$	−165177.	−0.444870
$170$	27132.0	0.0720045
$171$	72252.0	0.188956
$172$	534316.	1.37714
$173$	−221834.	−0.563525	−0.281762	−	0.959484i	$-0.590919\pi$
$173$	−221834.	−0.563525	−0.281762	+	0.959484i	$0.590919\pi$
$174$	74178.0	0.185739
$175$	96481.0	0.238148
$176$	−315860.	−0.768622
$177$	381564.	0.915449
$178$	−117286.	−0.277457
$179$	−113564.	−0.264916	−0.132458	−	0.991189i	$-0.542287\pi$
$179$	−113564.	−0.264916	−0.132458	+	0.991189i	$0.542287\pi$
$180$	85374.0	0.196401
$181$	663118.	1.50451	0.752254	−	0.658873i	$-0.228967\pi$
$181$	663118.	1.50451	0.752254	+	0.658873i	$0.228967\pi$
$182$	−22246.0	−0.0497821
$183$	−132822.	−0.293185
$184$	201096.	0.437884
$185$	−333132.	−0.715628
$186$	22464.0	0.0476107
$187$	271320.	0.567385
$188$	−276768.	−0.571112
$189$	35721.0	0.0727393
$190$	−30328.0	−0.0609480
$191$	505664.	1.00295	0.501474	−	0.865173i	$-0.332791\pi$
$191$	505664.	1.00295	0.501474	+	0.865173i	$0.332791\pi$
$192$	241047.	0.471899
$193$	−432382.	−0.835554	−0.417777	−	0.908550i	$-0.637191\pi$
$193$	−432382.	−0.835554	−0.417777	+	0.908550i	$0.637191\pi$
$194$	10002.0	0.0190802
$195$	138924.	0.261632
$196$	−74431.0	−0.138393
$197$	−131962.	−0.242261	−0.121130	−	0.992637i	$-0.538652\pi$
$197$	−131962.	−0.242261	−0.121130	+	0.992637i	$0.538652\pi$
$198$	−27540.0	−0.0499230
$199$	298536.	0.534397	0.267199	−	0.963642i	$-0.413902\pi$
$199$	298536.	0.534397	0.267199	+	0.963642i	$0.413902\pi$
$200$	124047.	0.219286
$201$	15084.0	0.0263346
$202$	−108770.	−0.187556
$203$	403858.	0.687842
$204$	−222642.	−0.374569
$205$	−674356.	−1.12074
$206$	−199192.	−0.327042
$207$	−258552.	−0.419394
$208$	421766.	0.675948
$209$	−303280.	−0.480262
$210$	−14994.0	−0.0234622
$211$	−1.17062e6	−1.81013	−0.905065	−	0.425273i	$-0.860178\pi$
$211$	−1.17062e6	−1.81013	−0.905065	+	0.425273i	$0.860178\pi$
$212$	−4650.00	−0.00710581
$213$	−131112.	−0.198013
$214$	−79972.0	−0.119372
$215$	586024.	0.864608
$216$	45927.0	0.0669782
$217$	122304.	0.176316
$218$	−46098.0	−0.0656963
$219$	−705402.	−0.993863
$220$	−358360.	−0.499186
$221$	−362292.	−0.498974
$222$	−88182.0	−0.120087
$223$	399376.	0.537799	0.268899	−	0.963168i	$-0.413340\pi$
$223$	399376.	0.537799	0.268899	+	0.963168i	$0.413340\pi$
$224$	−144305.	−0.192159
$225$	−159489.	−0.210027
$226$	262706.	0.342136
$227$	707916.	0.911837	0.455918	−	0.890022i	$-0.349311\pi$
$227$	707916.	0.911837	0.455918	+	0.890022i	$0.349311\pi$
$228$	248868.	0.317053
$229$	−735778.	−0.927167	−0.463584	−	0.886053i	$-0.653437\pi$
$229$	−735778.	−0.927167	−0.463584	+	0.886053i	$0.653437\pi$
$230$	108528.	0.135276
$231$	−149940.	−0.184879
$232$	519246.	0.633364
$233$	−208758.	−0.251915	−0.125957	−	0.992036i	$-0.540200\pi$
$233$	−208758.	−0.251915	−0.125957	+	0.992036i	$0.540200\pi$
$234$	36774.0	0.0439037
$235$	−303552.	−0.358561
$236$	1.31428e6	1.53605
$237$	20448.0	0.0236472
$238$	39102.0	0.0447462
$239$	713376.	0.807837	0.403919	−	0.914795i	$-0.367648\pi$
$239$	713376.	0.807837	0.403919	+	0.914795i	$0.367648\pi$
$240$	284274.	0.318573
$241$	−505246.	−0.560351	−0.280176	−	0.959949i	$-0.590393\pi$
$241$	−505246.	−0.560351	−0.280176	+	0.959949i	$0.590393\pi$
$242$	−45451.0	−0.0498890
$243$	−59049.0	−0.0641500
$244$	−457498.	−0.491943
$245$	−81634.0	−0.0868872
$246$	−178506.	−0.188068
$247$	404968.	0.422356
$248$	157248.	0.162351
$249$	339876.	0.347394
$250$	173196.	0.175262
$251$	317108.	0.317704	0.158852	−	0.987302i	$-0.449221\pi$
$251$	317108.	0.317704	0.158852	+	0.987302i	$0.449221\pi$
$252$	123039.	0.122051
$253$	1.08528e6	1.06596
$254$	196608.	0.191213
$255$	−244188.	−0.235166
$256$	736033.	0.701936
$257$	−1.44285e6	−1.36266	−0.681329	−	0.731977i	$-0.738598\pi$
$257$	−1.44285e6	−1.36266	−0.681329	+	0.731977i	$0.738598\pi$
$258$	155124.	0.145087
$259$	−480102.	−0.444717
$260$	478516.	0.438999
$261$	−667602.	−0.606619
$262$	−77140.0	−0.0694267
$263$	271496.	0.242033	0.121016	−	0.992651i	$-0.461385\pi$
$263$	271496.	0.242033	0.121016	+	0.992651i	$0.461385\pi$
$264$	−192780.	−0.170236
$265$	−5100.00	−0.00446124
$266$	−43708.0	−0.0378754
$267$	1.05557e6	0.906172
$268$	51956.0	0.0441874
$269$	850614.	0.716724	0.358362	−	0.933583i	$-0.383335\pi$
$269$	850614.	0.716724	0.358362	+	0.933583i	$0.383335\pi$
$270$	24786.0	0.0206917
$271$	−540128.	−0.446759	−0.223380	−	0.974732i	$-0.571709\pi$
$271$	−540128.	−0.446759	−0.223380	+	0.974732i	$0.571709\pi$
$272$	−741342.	−0.607570
$273$	200214.	0.162588
$274$	208170.	0.167510
$275$	669460.	0.533818
$276$	−890568.	−0.703711
$277$	513574.	0.402164	0.201082	−	0.979574i	$-0.435554\pi$
$277$	513574.	0.402164	0.201082	+	0.979574i	$0.435554\pi$
$278$	−275580.	−0.213863
$279$	−202176.	−0.155496
$280$	−104958.	−0.0800056
$281$	−1.35642e6	−1.02478	−0.512388	−	0.858754i	$-0.671239\pi$
$281$	−1.35642e6	−1.02478	−0.512388	+	0.858754i	$0.671239\pi$
$282$	−80352.0	−0.0601692
$283$	286756.	0.212837	0.106418	−	0.994321i	$-0.466062\pi$
$283$	286756.	0.212837	0.106418	+	0.994321i	$0.466062\pi$
$284$	−451608.	−0.332251
$285$	272952.	0.199055
$286$	−154360.	−0.111589
$287$	−971866.	−0.696469
$288$	238545.	0.169469
$289$	−783053.	−0.551501
$290$	280228.	0.195667
$291$	−90018.0	−0.0623156
$292$	−2.42972e6	−1.66763
$293$	−1.70727e6	−1.16180	−0.580901	−	0.813974i	$-0.697300\pi$
$293$	−1.70727e6	−1.16180	−0.580901	+	0.813974i	$0.697300\pi$
$294$	−21609.0	−0.0145803
$295$	1.44146e6	0.964381
$296$	−617274.	−0.409495
$297$	247860.	0.163048
$298$	−296106.	−0.193155
$299$	−1.44917e6	−0.937434
$300$	−549351.	−0.352409
$301$	844564.	0.537299
$302$	−426472.	−0.269075
$303$	978930.	0.612555
$304$	828668.	0.514276
$305$	−501772.	−0.308857
$306$	−64638.0	−0.0394625
$307$	−546788.	−0.331111	−0.165555	−	0.986201i	$-0.552942\pi$
$307$	−546788.	−0.331111	−0.165555	+	0.986201i	$0.552942\pi$
$308$	−516460.	−0.310213
$309$	1.79273e6	1.06812
$310$	84864.0	0.0501556
$311$	3.23426e6	1.89616	0.948079	−	0.318035i	$-0.103023\pi$
$311$	3.23426e6	1.89616	0.948079	+	0.318035i	$0.103023\pi$
$312$	257418.	0.149711
$313$	1.81313e6	1.04609	0.523044	−	0.852306i	$-0.324796\pi$
$313$	1.81313e6	1.04609	0.523044	+	0.852306i	$0.324796\pi$
$314$	178486.	0.102160
$315$	134946.	0.0766273
$316$	70432.0	0.0396782
$317$	−1.27658e6	−0.713509	−0.356754	−	0.934198i	$-0.616117\pi$
$317$	−1.27658e6	−0.713509	−0.356754	+	0.934198i	$0.616117\pi$
$318$	−1350.00	−0.000748628 0
$319$	2.80228e6	1.54182
$320$	910622.	0.497122
$321$	719748.	0.389868
$322$	156408.	0.0840658
$323$	−711816.	−0.379631
$324$	−203391.	−0.107639
$325$	−893926.	−0.469454
$326$	252772.	0.131730
$327$	414882.	0.214563
$328$	−1.24954e6	−0.641307
$329$	−437472.	−0.222823
$330$	−104040.	−0.0525915
$331$	−1.73621e6	−0.871029	−0.435515	−	0.900182i	$-0.643434\pi$
$331$	−1.73621e6	−0.871029	−0.435515	+	0.900182i	$0.643434\pi$
$332$	1.17068e6	0.582901
$333$	793638.	0.392204
$334$	508088.	0.249214
$335$	56984.0	0.0277422
$336$	409689.	0.197973
$337$	2.07215e6	0.993907	0.496953	−	0.867777i	$-0.334452\pi$
$337$	2.07215e6	0.993907	0.496953	+	0.867777i	$0.334452\pi$
$338$	−165177.	−0.0786426
$339$	−2.36435e6	−1.11741
$340$	−841092.	−0.394590
$341$	848640.	0.395219
$342$	72252.0	0.0334029
$343$	−117649.	−0.0539949
$344$	1.08587e6	0.494744
$345$	−976752.	−0.441811
$346$	−221834.	−0.0996180
$347$	−1.65146e6	−0.736282	−0.368141	−	0.929770i	$-0.620006\pi$
$347$	−1.65146e6	−0.736282	−0.368141	+	0.929770i	$0.620006\pi$
$348$	−2.29952e6	−1.01786
$349$	1.26645e6	0.556578	0.278289	−	0.960497i	$-0.410233\pi$
$349$	1.26645e6	0.556578	0.278289	+	0.960497i	$0.410233\pi$
$350$	96481.0	0.0420990
$351$	−330966.	−0.143389
$352$	−1.00130e6	−0.430732
$353$	573218.	0.244840	0.122420	−	0.992478i	$-0.460934\pi$
$353$	573218.	0.244840	0.122420	+	0.992478i	$0.460934\pi$
$354$	381564.	0.161830
$355$	−495312.	−0.208597
$356$	3.63587e6	1.52049
$357$	−351918.	−0.146141
$358$	−113564.	−0.0468310
$359$	4.46322e6	1.82773	0.913866	−	0.406016i	$-0.133082\pi$
$359$	4.46322e6	1.82773	0.913866	+	0.406016i	$0.133082\pi$
$360$	173502.	0.0705583
$361$	−1.68044e6	−0.678662
$362$	663118.	0.265962
$363$	409059.	0.162937
$364$	689626.	0.272810
$365$	−2.66485e6	−1.04699
$366$	−132822.	−0.0518283
$367$	−4.50797e6	−1.74709	−0.873546	−	0.486742i	$-0.838185\pi$
$367$	−4.50797e6	−1.74709	−0.873546	+	0.486742i	$0.838185\pi$
$368$	−2.96537e6	−1.14146
$369$	1.60655e6	0.614228
$370$	−333132.	−0.126506
$371$	−7350.00	−0.00277238
$372$	−696384.	−0.260910
$373$	1.66535e6	0.619774	0.309887	−	0.950773i	$-0.399709\pi$
$373$	1.66535e6	0.619774	0.309887	+	0.950773i	$0.399709\pi$
$374$	271320.	0.100300
$375$	−1.55876e6	−0.572403
$376$	−562464.	−0.205175
$377$	−3.74187e6	−1.35592
$378$	35721.0	0.0128586
$379$	−2.53232e6	−0.905568	−0.452784	−	0.891620i	$-0.649569\pi$
$379$	−2.53232e6	−0.905568	−0.452784	+	0.891620i	$0.649569\pi$
$380$	940168.	0.334000
$381$	−1.76947e6	−0.624498
$382$	505664.	0.177298
$383$	796368.	0.277407	0.138703	−	0.990334i	$-0.455707\pi$
$383$	796368.	0.277407	0.138703	+	0.990334i	$0.455707\pi$
$384$	1.08921e6	0.376949
$385$	−566440.	−0.194761
$386$	−432382.	−0.147706
$387$	−1.39612e6	−0.473853
$388$	−310062.	−0.104561
$389$	1.94799e6	0.652699	0.326349	−	0.945249i	$-0.394181\pi$
$389$	1.94799e6	0.652699	0.326349	+	0.945249i	$0.394181\pi$
$390$	138924.	0.0462504
$391$	2.54722e6	0.842605
$392$	−151263.	−0.0497184
$393$	694260.	0.226747
$394$	−131962.	−0.0428261
$395$	77248.0	0.0249112
$396$	853740.	0.273582
$397$	1.08116e6	0.344281	0.172140	−	0.985072i	$-0.444932\pi$
$397$	1.08116e6	0.344281	0.172140	+	0.985072i	$0.444932\pi$
$398$	298536.	0.0944689
$399$	393372.	0.123700
$400$	−1.82920e6	−0.571625
$401$	2.76770e6	0.859524	0.429762	−	0.902942i	$-0.358598\pi$
$401$	2.76770e6	0.859524	0.429762	+	0.902942i	$0.358598\pi$
$402$	15084.0	0.00465534
$403$	−1.13318e6	−0.347566
$404$	3.37187e6	1.02782
$405$	−223074.	−0.0675789
$406$	403858.	0.121594
$407$	−3.33132e6	−0.996851
$408$	−452466.	−0.134566
$409$	2.36350e6	0.698630	0.349315	−	0.937005i	$-0.386414\pi$
$409$	2.36350e6	0.698630	0.349315	+	0.937005i	$0.386414\pi$
$410$	−674356.	−0.198121
$411$	−1.87353e6	−0.547087
$412$	6.17495e6	1.79222
$413$	2.07740e6	0.599302
$414$	−258552.	−0.0741391
$415$	1.28398e6	0.365963
$416$	1.33703e6	0.378798
$417$	2.48022e6	0.698474
$418$	−303280.	−0.0848991
$419$	−2.98669e6	−0.831104	−0.415552	−	0.909569i	$-0.636412\pi$
$419$	−2.98669e6	−0.831104	−0.415552	+	0.909569i	$0.636412\pi$
$420$	464814.	0.128575
$421$	−3.46331e6	−0.952326	−0.476163	−	0.879357i	$-0.657973\pi$
$421$	−3.46331e6	−0.952326	−0.476163	+	0.879357i	$0.657973\pi$
$422$	−1.17062e6	−0.319989
$423$	723168.	0.196512
$424$	−9450.00	−0.00255280
$425$	1.57126e6	0.421965
$426$	−131112.	−0.0350041
$427$	−723142.	−0.191935
$428$	2.47913e6	0.654169
$429$	1.38924e6	0.364447
$430$	586024.	0.152843
$431$	2.33693e6	0.605971	0.302986	−	0.952995i	$-0.402017\pi$
$431$	2.33693e6	0.605971	0.302986	+	0.952995i	$0.402017\pi$
$432$	−677241.	−0.174596
$433$	−3.50838e6	−0.899264	−0.449632	−	0.893214i	$-0.648445\pi$
$433$	−3.50838e6	−0.899264	−0.449632	+	0.893214i	$0.648445\pi$
$434$	122304.	0.0311685
$435$	−2.52205e6	−0.639044
$436$	1.42904e6	0.360021
$437$	−2.84726e6	−0.713221
$438$	−705402.	−0.175692
$439$	3.54833e6	0.878744	0.439372	−	0.898305i	$-0.355201\pi$
$439$	3.54833e6	0.878744	0.439372	+	0.898305i	$0.355201\pi$
$440$	−728280.	−0.179336
$441$	194481.	0.0476190
$442$	−362292.	−0.0882070
$443$	1.76833e6	0.428109	0.214055	−	0.976822i	$-0.431333\pi$
$443$	1.76833e6	0.428109	0.214055	+	0.976822i	$0.431333\pi$
$444$	2.73364e6	0.658088
$445$	3.98772e6	0.954608
$446$	399376.	0.0950703
$447$	2.66495e6	0.630842
$448$	1.31237e6	0.308930
$449$	−5.52579e6	−1.29354	−0.646768	−	0.762687i	$-0.723880\pi$
$449$	−5.52579e6	−1.29354	−0.646768	+	0.762687i	$0.723880\pi$
$450$	−159489.	−0.0371278
$451$	−6.74356e6	−1.56116
$452$	−8.14389e6	−1.87493
$453$	3.83825e6	0.878795
$454$	707916.	0.161191
$455$	756364.	0.171278
$456$	505764.	0.113903
$457$	−2.96226e6	−0.663488	−0.331744	−	0.943369i	$-0.607637\pi$
$457$	−2.96226e6	−0.663488	−0.331744	+	0.943369i	$0.607637\pi$
$458$	−735778.	−0.163902
$459$	581742.	0.128884
$460$	−3.36437e6	−0.741325
$461$	2.11884e6	0.464350	0.232175	−	0.972674i	$-0.425416\pi$
$461$	2.11884e6	0.464350	0.232175	+	0.972674i	$0.425416\pi$
$462$	−149940.	−0.0326823
$463$	3.19226e6	0.692062	0.346031	−	0.938223i	$-0.387529\pi$
$463$	3.19226e6	0.692062	0.346031	+	0.938223i	$0.387529\pi$
$464$	−7.65682e6	−1.65102
$465$	−763776.	−0.163807
$466$	−208758.	−0.0445326
$467$	−7.42621e6	−1.57571	−0.787853	−	0.615863i	$-0.788807\pi$
$467$	−7.42621e6	−1.57571	−0.787853	+	0.615863i	$0.788807\pi$
$468$	−1.13999e6	−0.240596
$469$	82124.0	0.0172400
$470$	−303552.	−0.0633853
$471$	−1.60637e6	−0.333653
$472$	2.67095e6	0.551837
$473$	5.86024e6	1.20438
$474$	20448.0	0.00418028
$475$	−1.75635e6	−0.357171
$476$	−1.21216e6	−0.245213
$477$	12150.0	0.00244501
$478$	713376.	0.142807
$479$	−3.39685e6	−0.676453	−0.338226	−	0.941065i	$-0.609827\pi$
$479$	−3.39685e6	−0.676453	−0.338226	+	0.941065i	$0.609827\pi$
$480$	901170.	0.178527
$481$	4.44829e6	0.876659
$482$	−505246.	−0.0990570
$483$	−1.40767e6	−0.274558
$484$	1.40898e6	0.273396
$485$	−340068.	−0.0656465
$486$	−59049.0	−0.0113402
$487$	−3.71382e6	−0.709574	−0.354787	−	0.934947i	$-0.615447\pi$
$487$	−3.71382e6	−0.709574	−0.354787	+	0.934947i	$0.615447\pi$
$488$	−929754.	−0.176733
$489$	−2.27495e6	−0.430229
$490$	−81634.0	−0.0153596
$491$	5.57494e6	1.04361	0.521803	−	0.853066i	$-0.325260\pi$
$491$	5.57494e6	1.04361	0.521803	+	0.853066i	$0.325260\pi$
$492$	5.53369e6	1.03063
$493$	6.57712e6	1.21876
$494$	404968.	0.0746626
$495$	936360.	0.171763
$496$	−2.31878e6	−0.423210
$497$	−713832.	−0.129630
$498$	339876.	0.0614111
$499$	3.92698e6	0.706004	0.353002	−	0.935623i	$-0.385161\pi$
$499$	3.92698e6	0.706004	0.353002	+	0.935623i	$0.385161\pi$
$500$	−5.36908e6	−0.960450
$501$	−4.57279e6	−0.813930
$502$	317108.	0.0561627
$503$	6.42079e6	1.13154	0.565768	−	0.824564i	$-0.308580\pi$
$503$	6.42079e6	1.13154	0.565768	+	0.824564i	$0.308580\pi$
$504$	250047.	0.0438475
$505$	3.69818e6	0.645297
$506$	1.08528e6	0.188437
$507$	1.48659e6	0.256846
$508$	−6.09485e6	−1.04786
$509$	146278.	0.0250256	0.0125128	−	0.999922i	$-0.496017\pi$
$509$	146278.	0.0250256	0.0125128	+	0.999922i	$0.496017\pi$
$510$	−244188.	−0.0415718
$511$	−3.84052e6	−0.650636
$512$	4.60877e6	0.776980
$513$	−650268.	−0.109094
$514$	−1.44285e6	−0.240886
$515$	6.77253e6	1.12521
$516$	−4.80884e6	−0.795090
$517$	−3.03552e6	−0.499467
$518$	−480102.	−0.0786157
$519$	1.99651e6	0.325351
$520$	972468.	0.157713
$521$	7.70937e6	1.24430	0.622149	−	0.782899i	$-0.286260\pi$
$521$	7.70937e6	1.24430	0.622149	+	0.782899i	$0.286260\pi$
$522$	−667602.	−0.107236
$523$	−569420.	−0.0910287	−0.0455144	−	0.998964i	$-0.514493\pi$
$523$	−569420.	−0.0910287	−0.0455144	+	0.998964i	$0.514493\pi$
$524$	2.39134e6	0.380464
$525$	−868329.	−0.137495
$526$	271496.	0.0427857
$527$	1.99181e6	0.312407
$528$	2.84274e6	0.443764
$529$	3.75252e6	0.583021
$530$	−5100.00	−0.000788643 0
$531$	−3.43408e6	−0.528535
$532$	1.35495e6	0.207560
$533$	9.00464e6	1.37293
$534$	1.05557e6	0.160190
$535$	2.71905e6	0.410707
$536$	105588.	0.0158746
$537$	1.02208e6	0.152949
$538$	850614.	0.126700
$539$	−816340.	−0.121032
$540$	−768366.	−0.113392
$541$	−9.44802e6	−1.38787	−0.693933	−	0.720040i	$-0.744124\pi$
$541$	−9.44802e6	−1.38787	−0.693933	+	0.720040i	$0.744124\pi$
$542$	−540128.	−0.0789766
$543$	−5.96806e6	−0.868628
$544$	−2.35011e6	−0.340479
$545$	1.56733e6	0.226032
$546$	200214.	0.0287417
$547$	−1.35321e6	−0.193374	−0.0966869	−	0.995315i	$-0.530825\pi$
$547$	−1.35321e6	−0.193374	−0.0966869	+	0.995315i	$0.530825\pi$
$548$	−6.45327e6	−0.917970
$549$	1.19540e6	0.169271
$550$	669460.	0.0943665
$551$	−7.35186e6	−1.03162
$552$	−1.80986e6	−0.252812
$553$	111328.	0.0154807
$554$	513574.	0.0710933
$555$	2.99819e6	0.413168
$556$	8.54298e6	1.17199
$557$	8.19390e6	1.11906	0.559529	−	0.828811i	$-0.310982\pi$
$557$	8.19390e6	1.11906	0.559529	+	0.828811i	$0.310982\pi$
$558$	−202176.	−0.0274881
$559$	−7.82514e6	−1.05916
$560$	1.54771e6	0.208555
$561$	−2.44188e6	−0.327580
$562$	−1.35642e6	−0.181157
$563$	−1.05796e7	−1.40669	−0.703347	−	0.710847i	$-0.748312\pi$
$563$	−1.05796e7	−1.40669	−0.703347	+	0.710847i	$0.748312\pi$
$564$	2.49091e6	0.329732
$565$	−8.93200e6	−1.17714
$566$	286756.	0.0376246
$567$	−321489.	−0.0419961
$568$	−917784.	−0.119363
$569$	−1.20205e7	−1.55648	−0.778238	−	0.627969i	$-0.783886\pi$
$569$	−1.20205e7	−1.55648	−0.778238	+	0.627969i	$0.783886\pi$
$570$	272952.	0.0351884
$571$	−2.48948e6	−0.319534	−0.159767	−	0.987155i	$-0.551074\pi$
$571$	−2.48948e6	−0.319534	−0.159767	+	0.987155i	$0.551074\pi$
$572$	4.78516e6	0.611514
$573$	−4.55098e6	−0.579053
$574$	−971866.	−0.123119
$575$	6.28505e6	0.792755
$576$	−2.16942e6	−0.272451
$577$	8.21322e6	1.02701	0.513504	−	0.858087i	$-0.328347\pi$
$577$	8.21322e6	1.02701	0.513504	+	0.858087i	$0.328347\pi$
$578$	−783053.	−0.0974926
$579$	3.89144e6	0.482407
$580$	−8.68707e6	−1.07227
$581$	1.85044e6	0.227423
$582$	−90018.0	−0.0110159
$583$	−51000.0	−0.00621439
$584$	−4.93781e6	−0.599105
$585$	−1.25032e6	−0.151053
$586$	−1.70727e6	−0.205380
$587$	−1.21827e6	−0.145931	−0.0729655	−	0.997334i	$-0.523246\pi$
$587$	−1.21827e6	−0.145931	−0.0729655	+	0.997334i	$0.523246\pi$
$588$	669879.	0.0799012
$589$	−2.22643e6	−0.264436
$590$	1.44146e6	0.170480
$591$	1.18766e6	0.139869
$592$	9.10234e6	1.06745
$593$	−8.42379e6	−0.983718	−0.491859	−	0.870675i	$-0.663683\pi$
$593$	−8.42379e6	−0.983718	−0.491859	+	0.870675i	$0.663683\pi$
$594$	247860.	0.0288231
$595$	−1.32947e6	−0.153952
$596$	9.17929e6	1.05851
$597$	−2.68682e6	−0.308534
$598$	−1.44917e6	−0.165717
$599$	8.21254e6	0.935212	0.467606	−	0.883937i	$-0.345117\pi$
$599$	8.21254e6	0.935212	0.467606	+	0.883937i	$0.345117\pi$
$600$	−1.11642e6	−0.126605
$601$	3.25478e6	0.367566	0.183783	−	0.982967i	$-0.441166\pi$
$601$	3.25478e6	0.367566	0.183783	+	0.982967i	$0.441166\pi$
$602$	844564.	0.0949820
$603$	−135756.	−0.0152043
$604$	1.32206e7	1.47455
$605$	1.54533e6	0.171646
$606$	978930.	0.108285
$607$	7.82101e6	0.861571	0.430785	−	0.902454i	$-0.358237\pi$
$607$	7.82101e6	0.861571	0.430785	+	0.902454i	$0.358237\pi$
$608$	2.62694e6	0.288198
$609$	−3.63472e6	−0.397126
$610$	−501772.	−0.0545986
$611$	4.05331e6	0.439245
$612$	2.00378e6	0.216257
$613$	−9.51670e6	−1.02290	−0.511452	−	0.859312i	$-0.670892\pi$
$613$	−9.51670e6	−1.02290	−0.511452	+	0.859312i	$0.670892\pi$
$614$	−546788.	−0.0585326
$615$	6.06920e6	0.647059
$616$	−1.04958e6	−0.111446
$617$	−7.04895e6	−0.745438	−0.372719	−	0.927944i	$-0.621574\pi$
$617$	−7.04895e6	−0.745438	−0.372719	+	0.927944i	$0.621574\pi$
$618$	1.79273e6	0.188818
$619$	−6.32174e6	−0.663147	−0.331574	−	0.943429i	$-0.607580\pi$
$619$	−6.32174e6	−0.663147	−0.331574	+	0.943429i	$0.607580\pi$
$620$	−2.63078e6	−0.274856
$621$	2.32697e6	0.242137
$622$	3.23426e6	0.335197
$623$	5.74701e6	0.593229
$624$	−3.79589e6	−0.390259
$625$	264461.	0.0270808
$626$	1.81313e6	0.184924
$627$	2.72952e6	0.277279
$628$	−5.53307e6	−0.559844
$629$	−7.81880e6	−0.787977
$630$	134946.	0.0135459
$631$	8.61236e6	0.861090	0.430545	−	0.902569i	$-0.358321\pi$
$631$	8.61236e6	0.861090	0.430545	+	0.902569i	$0.358321\pi$
$632$	143136.	0.0142546
$633$	1.05356e7	1.04508
$634$	−1.27658e6	−0.126132
$635$	−6.68467e6	−0.657879
$636$	41850.0	0.00410254
$637$	1.09005e6	0.106439
$638$	2.80228e6	0.272559
$639$	1.18001e6	0.114323
$640$	4.11478e6	0.397097
$641$	−5.22829e6	−0.502590	−0.251295	−	0.967910i	$-0.580857\pi$
$641$	−5.22829e6	−0.502590	−0.251295	+	0.967910i	$0.580857\pi$
$642$	719748.	0.0689196
$643$	1.61373e7	1.53923	0.769615	−	0.638508i	$-0.220448\pi$
$643$	1.61373e7	1.53923	0.769615	+	0.638508i	$0.220448\pi$
$644$	−4.84865e6	−0.460687
$645$	−5.27422e6	−0.499182
$646$	−711816.	−0.0671099
$647$	−1.58749e7	−1.49090	−0.745451	−	0.666560i	$-0.767766\pi$
$647$	−1.58749e7	−1.49090	−0.745451	+	0.666560i	$0.767766\pi$
$648$	−413343.	−0.0386699
$649$	1.44146e7	1.34336
$650$	−893926.	−0.0829886
$651$	−1.10074e6	−0.101796
$652$	−7.83593e6	−0.721891
$653$	−5.94112e6	−0.545237	−0.272619	−	0.962122i	$-0.587890\pi$
$653$	−5.94112e6	−0.545237	−0.272619	+	0.962122i	$0.587890\pi$
$654$	414882.	0.0379298
$655$	2.62276e6	0.238867
$656$	1.84258e7	1.67173
$657$	6.34862e6	0.573807
$658$	−437472.	−0.0393900
$659$	−7.64430e6	−0.685684	−0.342842	−	0.939393i	$-0.611390\pi$
$659$	−7.64430e6	−0.685684	−0.342842	+	0.939393i	$0.611390\pi$
$660$	3.22524e6	0.288205
$661$	−7.58688e6	−0.675398	−0.337699	−	0.941254i	$-0.609649\pi$
$661$	−7.58688e6	−0.675398	−0.337699	+	0.941254i	$0.609649\pi$
$662$	−1.73621e6	−0.153978
$663$	3.26063e6	0.288083
$664$	2.37913e6	0.209410
$665$	1.48607e6	0.130312
$666$	793638.	0.0693325
$667$	2.63085e7	2.28971
$668$	−1.57507e7	−1.36571
$669$	−3.59438e6	−0.310498
$670$	56984.0	0.00490417
$671$	−5.01772e6	−0.430229
$672$	1.29874e6	0.110943
$673$	−2.06681e7	−1.75899	−0.879494	−	0.475910i	$-0.842119\pi$
$673$	−2.06681e7	−1.75899	−0.879494	+	0.475910i	$0.842119\pi$
$674$	2.07215e6	0.175700
$675$	1.43540e6	0.121259
$676$	5.12049e6	0.430968
$677$	7.89541e6	0.662068	0.331034	−	0.943619i	$-0.392602\pi$
$677$	7.89541e6	0.662068	0.331034	+	0.943619i	$0.392602\pi$
$678$	−2.36435e6	−0.197532
$679$	−490098.	−0.0407951
$680$	−1.70932e6	−0.141759
$681$	−6.37124e6	−0.526449
$682$	848640.	0.0698655
$683$	−1.96015e7	−1.60782	−0.803911	−	0.594750i	$-0.797251\pi$
$683$	−1.96015e7	−1.60782	−0.803911	+	0.594750i	$0.797251\pi$
$684$	−2.23981e6	−0.183051
$685$	−7.07778e6	−0.576329
$686$	−117649.	−0.00954504
$687$	6.62200e6	0.535300
$688$	−1.60122e7	−1.28968
$689$	68100.0	0.00546511
$690$	−976752.	−0.0781019
$691$	−1.72710e7	−1.37601	−0.688005	−	0.725706i	$-0.741513\pi$
$691$	−1.72710e7	−1.37601	−0.688005	+	0.725706i	$0.741513\pi$
$692$	6.87685e6	0.545914
$693$	1.34946e6	0.106740
$694$	−1.65146e6	−0.130158
$695$	9.36972e6	0.735808
$696$	−4.67321e6	−0.365673
$697$	−1.58275e7	−1.23405
$698$	1.26645e6	0.0983900
$699$	1.87882e6	0.145443
$700$	−2.99091e6	−0.230706
$701$	−5.36344e6	−0.412238	−0.206119	−	0.978527i	$-0.566083\pi$
$701$	−5.36344e6	−0.412238	−0.206119	+	0.978527i	$0.566083\pi$
$702$	−330966.	−0.0253478
$703$	8.73982e6	0.666982
$704$	9.10622e6	0.692479
$705$	2.73197e6	0.207015
$706$	573218.	0.0432821
$707$	5.32973e6	0.401011
$708$	−1.18285e7	−0.886841
$709$	−1.73733e7	−1.29798	−0.648988	−	0.760798i	$-0.724808\pi$
$709$	−1.73733e7	−1.29798	−0.648988	+	0.760798i	$0.724808\pi$
$710$	−495312.	−0.0368751
$711$	−184032.	−0.0136527
$712$	7.38902e6	0.546244
$713$	7.96723e6	0.586926
$714$	−351918.	−0.0258343
$715$	5.24824e6	0.383927
$716$	3.52048e6	0.256637
$717$	−6.42038e6	−0.466405
$718$	4.46322e6	0.323100
$719$	424608.	0.0306313	0.0153157	−	0.999883i	$-0.495125\pi$
$719$	424608.	0.0306313	0.0153157	+	0.999883i	$0.495125\pi$
$720$	−2.55847e6	−0.183928
$721$	9.76041e6	0.699246
$722$	−1.68044e6	−0.119972
$723$	4.54721e6	0.323519
$724$	−2.05567e7	−1.45749
$725$	1.62285e7	1.14666
$726$	409059.	0.0288034
$727$	2.18290e7	1.53179	0.765893	−	0.642968i	$-0.222297\pi$
$727$	2.18290e7	1.53179	0.765893	+	0.642968i	$0.222297\pi$
$728$	1.40150e6	0.0980086
$729$	531441.	0.0370370
$730$	−2.66485e6	−0.185083
$731$	1.37543e7	0.952020
$732$	4.11748e6	0.284023
$733$	2.17675e7	1.49640	0.748202	−	0.663470i	$-0.230917\pi$
$733$	2.17675e7	1.49640	0.748202	+	0.663470i	$0.230917\pi$
$734$	−4.50797e6	−0.308845
$735$	734706.	0.0501644
$736$	−9.40044e6	−0.639667
$737$	569840.	0.0386442
$738$	1.60655e6	0.108581
$739$	6.21786e6	0.418822	0.209411	−	0.977828i	$-0.432845\pi$
$739$	6.21786e6	0.418822	0.209411	+	0.977828i	$0.432845\pi$
$740$	1.03271e7	0.693264
$741$	−3.64471e6	−0.243847
$742$	−7350.00	−0.000490092 0
$743$	3.77647e6	0.250966	0.125483	−	0.992096i	$-0.459952\pi$
$743$	3.77647e6	0.250966	0.125483	+	0.992096i	$0.459952\pi$
$744$	−1.41523e6	−0.0937336
$745$	1.00676e7	0.664562
$746$	1.66535e6	0.109562
$747$	−3.05888e6	−0.200568
$748$	−8.41092e6	−0.549654
$749$	3.91863e6	0.255229
$750$	−1.55876e6	−0.101188
$751$	−2.88795e6	−0.186849	−0.0934244	−	0.995626i	$-0.529781\pi$
$751$	−2.88795e6	−0.186849	−0.0934244	+	0.995626i	$0.529781\pi$
$752$	8.29411e6	0.534842
$753$	−2.85397e6	−0.183427
$754$	−3.74187e6	−0.239696
$755$	1.45000e7	0.925768
$756$	−1.10735e6	−0.0704662
$757$	1.25519e6	0.0796104	0.0398052	−	0.999207i	$-0.487326\pi$
$757$	1.25519e6	0.0796104	0.0398052	+	0.999207i	$0.487326\pi$
$758$	−2.53232e6	−0.160083
$759$	−9.76752e6	−0.615432
$760$	1.91066e6	0.119991
$761$	−1.42623e7	−0.892746	−0.446373	−	0.894847i	$-0.647284\pi$
$761$	−1.42623e7	−0.892746	−0.446373	+	0.894847i	$0.647284\pi$
$762$	−1.76947e6	−0.110397
$763$	2.25880e6	0.140465
$764$	−1.56756e7	−0.971606
$765$	2.19769e6	0.135773
$766$	796368.	0.0490390
$767$	−1.92478e7	−1.18139
$768$	−6.62430e6	−0.405263
$769$	−2.02261e7	−1.23338	−0.616689	−	0.787207i	$-0.711526\pi$
$769$	−2.02261e7	−1.23338	−0.616689	+	0.787207i	$0.711526\pi$
$770$	−566440.	−0.0344292
$771$	1.29856e7	0.786732
$772$	1.34038e7	0.809443
$773$	2.62288e7	1.57881	0.789406	−	0.613872i	$-0.210389\pi$
$773$	2.62288e7	1.57881	0.789406	+	0.613872i	$0.210389\pi$
$774$	−1.39612e6	−0.0837663
$775$	4.91462e6	0.293925
$776$	−630126.	−0.0375641
$777$	4.32092e6	0.256758
$778$	1.94799e6	0.115382
$779$	1.76919e7	1.04456
$780$	−4.30664e6	−0.253456
$781$	−4.95312e6	−0.290570
$782$	2.54722e6	0.148953
$783$	6.00842e6	0.350232
$784$	2.23053e6	0.129604
$785$	−6.06852e6	−0.351487
$786$	694260.	0.0400835
$787$	−9.92829e6	−0.571397	−0.285698	−	0.958320i	$-0.592226\pi$
$787$	−9.92829e6	−0.571397	−0.285698	+	0.958320i	$0.592226\pi$
$788$	4.09082e6	0.234690
$789$	−2.44346e6	−0.139738
$790$	77248.0	0.00440372
$791$	−1.28726e7	−0.731518
$792$	1.73502e6	0.0982860
$793$	6.70013e6	0.378356
$794$	1.08116e6	0.0608608
$795$	45900.0	0.00257570
$796$	−9.25462e6	−0.517697
$797$	1.09033e7	0.608014	0.304007	−	0.952670i	$-0.401675\pi$
$797$	1.09033e7	0.608014	0.304007	+	0.952670i	$0.401675\pi$
$798$	393372.	0.0218674
$799$	−7.12454e6	−0.394812
$800$	−5.79871e6	−0.320336
$801$	−9.50017e6	−0.523179
$802$	2.76770e6	0.151944
$803$	−2.66485e7	−1.45843
$804$	−467604.	−0.0255116
$805$	−5.31787e6	−0.289233
$806$	−1.13318e6	−0.0614416
$807$	−7.65553e6	−0.413801
$808$	6.85251e6	0.369251
$809$	6.06398e6	0.325751	0.162876	−	0.986647i	$-0.447923\pi$
$809$	6.06398e6	0.325751	0.162876	+	0.986647i	$0.447923\pi$
$810$	−223074.	−0.0119464
$811$	−8.59438e6	−0.458841	−0.229421	−	0.973327i	$-0.573683\pi$
$811$	−8.59438e6	−0.458841	−0.229421	+	0.973327i	$0.573683\pi$
$812$	−1.25196e7	−0.666347
$813$	4.86115e6	0.257937
$814$	−3.33132e6	−0.176220
$815$	−8.59425e6	−0.453225
$816$	6.67208e6	0.350781
$817$	−1.53745e7	−0.805835
$818$	2.36350e6	0.123501
$819$	−1.80193e6	−0.0938701
$820$	2.09050e7	1.08572
$821$	−2.01396e6	−0.104278	−0.0521391	−	0.998640i	$-0.516604\pi$
$821$	−2.01396e6	−0.104278	−0.0521391	+	0.998640i	$0.516604\pi$
$822$	−1.87353e6	−0.0967122
$823$	−2.64679e7	−1.36213	−0.681067	−	0.732221i	$-0.738484\pi$
$823$	−2.64679e7	−1.36213	−0.681067	+	0.732221i	$0.738484\pi$
$824$	1.25491e7	0.643864
$825$	−6.02514e6	−0.308200
$826$	2.07740e6	0.105943
$827$	−3.90229e6	−0.198407	−0.0992033	−	0.995067i	$-0.531629\pi$
$827$	−3.90229e6	−0.198407	−0.0992033	+	0.995067i	$0.531629\pi$
$828$	8.01511e6	0.406288
$829$	−1.95595e7	−0.988487	−0.494244	−	0.869323i	$-0.664555\pi$
$829$	−1.95595e7	−0.988487	−0.494244	+	0.869323i	$0.664555\pi$
$830$	1.28398e6	0.0646937
$831$	−4.62217e6	−0.232190
$832$	−1.21595e7	−0.608985
$833$	−1.91600e6	−0.0956715
$834$	2.48022e6	0.123474
$835$	−1.72750e7	−0.857436
$836$	9.40168e6	0.465254
$837$	1.81958e6	0.0897756
$838$	−2.98669e6	−0.146920
$839$	−2.45448e7	−1.20380	−0.601901	−	0.798570i	$-0.705590\pi$
$839$	−2.45448e7	−1.20380	−0.601901	+	0.798570i	$0.705590\pi$
$840$	944622.	0.0461913
$841$	4.74194e7	2.31188
$842$	−3.46331e6	−0.168349
$843$	1.22078e7	0.591655
$844$	3.62892e7	1.75356
$845$	5.61602e6	0.270574
$846$	723168.	0.0347387
$847$	2.22710e6	0.106667
$848$	139350.	0.00665453
$849$	−2.58080e6	−0.122881
$850$	1.57126e6	0.0745936
$851$	−3.12752e7	−1.48039
$852$	4.06447e6	0.191825
$853$	3.38305e7	1.59197	0.795987	−	0.605314i	$-0.206952\pi$
$853$	3.38305e7	1.59197	0.795987	+	0.605314i	$0.206952\pi$
$854$	−723142.	−0.0339296
$855$	−2.45657e6	−0.114925
$856$	5.03824e6	0.235014
$857$	3.18009e7	1.47907	0.739534	−	0.673120i	$-0.235046\pi$
$857$	3.18009e7	1.47907	0.739534	+	0.673120i	$0.235046\pi$
$858$	1.38924e6	0.0644257
$859$	638420.	0.0295205	0.0147602	−	0.999891i	$-0.495301\pi$
$859$	638420.	0.0295205	0.0147602	+	0.999891i	$0.495301\pi$
$860$	−1.81667e7	−0.837589
$861$	8.74679e6	0.402106
$862$	2.33693e6	0.107122
$863$	−4.22256e6	−0.192996	−0.0964981	−	0.995333i	$-0.530764\pi$
$863$	−4.22256e6	−0.192996	−0.0964981	+	0.995333i	$0.530764\pi$
$864$	−2.14690e6	−0.0978427
$865$	7.54236e6	0.342742
$866$	−3.50838e6	−0.158969
$867$	7.04748e6	0.318409
$868$	−3.79142e6	−0.170806
$869$	772480.	0.0347007
$870$	−2.52205e6	−0.112968
$871$	−760904.	−0.0339848
$872$	2.90417e6	0.129340
$873$	810162.	0.0359779
$874$	−2.84726e6	−0.126081
$875$	−8.48660e6	−0.374726
$876$	2.18675e7	0.962805
$877$	−2.45043e7	−1.07583	−0.537915	−	0.842999i	$-0.680788\pi$
$877$	−2.45043e7	−1.07583	−0.537915	+	0.842999i	$0.680788\pi$
$878$	3.54833e6	0.155341
$879$	1.53654e7	0.670767
$880$	1.07392e7	0.467484
$881$	−2.77630e7	−1.20511	−0.602555	−	0.798078i	$-0.705850\pi$
$881$	−2.77630e7	−1.20511	−0.602555	+	0.798078i	$0.705850\pi$
$882$	194481.	0.00841794
$883$	3.30170e7	1.42507	0.712534	−	0.701638i	$-0.247548\pi$
$883$	3.30170e7	1.42507	0.712534	+	0.701638i	$0.247548\pi$
$884$	1.12311e7	0.483381
$885$	−1.29732e7	−0.556786
$886$	1.76833e6	0.0756797
$887$	4.34462e6	0.185414	0.0927070	−	0.995693i	$-0.470448\pi$
$887$	4.34462e6	0.185414	0.0927070	+	0.995693i	$0.470448\pi$
$888$	5.55547e6	0.236422
$889$	−9.63379e6	−0.408830
$890$	3.98772e6	0.168752
$891$	−2.23074e6	−0.0941358
$892$	−1.23807e7	−0.520993
$893$	7.96378e6	0.334188
$894$	2.66495e6	0.111518
$895$	3.86118e6	0.161125
$896$	5.93013e6	0.246771
$897$	1.30425e7	0.541228
$898$	−5.52579e6	−0.228667
$899$	2.05720e7	0.848942
$900$	4.94416e6	0.203463
$901$	−119700.	−0.00491227
$902$	−6.74356e6	−0.275977
$903$	−7.60108e6	−0.310210
$904$	−1.65505e7	−0.673580
$905$	−2.25460e7	−0.915057
$906$	3.83825e6	0.155350
$907$	1.96499e7	0.793128	0.396564	−	0.918007i	$-0.370203\pi$
$907$	1.96499e7	0.793128	0.396564	+	0.918007i	$0.370203\pi$
$908$	−2.19454e7	−0.883342
$909$	−8.81037e6	−0.353659
$910$	756364.	0.0302780
$911$	−7.26518e6	−0.290035	−0.145018	−	0.989429i	$-0.546324\pi$
$911$	−7.26518e6	−0.290035	−0.145018	+	0.989429i	$0.546324\pi$
$912$	−7.45801e6	−0.296918
$913$	1.28398e7	0.509777
$914$	−2.96226e6	−0.117289
$915$	4.51595e6	0.178318
$916$	2.28091e7	0.898193
$917$	3.77986e6	0.148440
$918$	581742.	0.0227837
$919$	9.82532e6	0.383758	0.191879	−	0.981419i	$-0.438542\pi$
$919$	9.82532e6	0.383758	0.191879	+	0.981419i	$0.438542\pi$
$920$	−6.83726e6	−0.266326
$921$	4.92109e6	0.191167
$922$	2.11884e6	0.0820863
$923$	6.61387e6	0.255536
$924$	4.64814e6	0.179102
$925$	−1.92923e7	−0.741359
$926$	3.19226e6	0.122340
$927$	−1.61346e7	−0.616677
$928$	−2.42727e7	−0.925226
$929$	2.71152e7	1.03080	0.515399	−	0.856951i	$-0.327644\pi$
$929$	2.71152e7	1.03080	0.515399	+	0.856951i	$0.327644\pi$
$930$	−763776.	−0.0289573
$931$	2.14169e6	0.0809809
$932$	6.47150e6	0.244042
$933$	−2.91084e7	−1.09475
$934$	−7.42621e6	−0.278548
$935$	−9.22488e6	−0.345089
$936$	−2.31676e6	−0.0864354
$937$	−4.53522e7	−1.68752	−0.843761	−	0.536720i	$-0.819663\pi$
$937$	−4.53522e7	−1.68752	−0.843761	+	0.536720i	$0.819663\pi$
$938$	82124.0	0.00304764
$939$	−1.63182e7	−0.603959
$940$	9.41011e6	0.347356
$941$	4.65780e7	1.71477	0.857387	−	0.514672i	$-0.172086\pi$
$941$	4.65780e7	1.71477	0.857387	+	0.514672i	$0.172086\pi$
$942$	−1.60637e6	−0.0589820
$943$	−6.33101e7	−2.31843
$944$	−3.93859e7	−1.43850
$945$	−1.21451e6	−0.0442408
$946$	5.86024e6	0.212906
$947$	2.53799e7	0.919632	0.459816	−	0.888014i	$-0.347915\pi$
$947$	2.53799e7	0.919632	0.459816	+	0.888014i	$0.347915\pi$
$948$	−633888.	−0.0229082
$949$	3.55836e7	1.28258
$950$	−1.75635e6	−0.0631396
$951$	1.14892e7	0.411944
$952$	−2.46343e6	−0.0880942
$953$	1.52948e7	0.545520	0.272760	−	0.962082i	$-0.412063\pi$
$953$	1.52948e7	0.545520	0.272760	+	0.962082i	$0.412063\pi$
$954$	12150.0	0.000432220 0
$955$	−1.71926e7	−0.610004
$956$	−2.21147e7	−0.782592
$957$	−2.52205e7	−0.890173
$958$	−3.39685e6	−0.119581
$959$	−1.02003e7	−0.358152
$960$	−8.19560e6	−0.287014
$961$	−2.23991e7	−0.782389
$962$	4.44829e6	0.154973
$963$	−6.47773e6	−0.225091
$964$	1.56626e7	0.542840
$965$	1.47010e7	0.508192
$966$	−1.40767e6	−0.0485354
$967$	−5.71465e6	−0.196527	−0.0982637	−	0.995160i	$-0.531329\pi$
$967$	−5.71465e6	−0.196527	−0.0982637	+	0.995160i	$0.531329\pi$
$968$	2.86341e6	0.0982190
$969$	6.40634e6	0.219180
$970$	−340068.	−0.0116048
$971$	1.30250e7	0.443332	0.221666	−	0.975123i	$-0.428851\pi$
$971$	1.30250e7	0.443332	0.221666	+	0.975123i	$0.428851\pi$
$972$	1.83052e6	0.0621453
$973$	1.35034e7	0.457258
$974$	−3.71382e6	−0.125436
$975$	8.04533e6	0.271039
$976$	1.37102e7	0.460700
$977$	−1.70360e7	−0.570992	−0.285496	−	0.958380i	$-0.592158\pi$
$977$	−1.70360e7	−0.570992	−0.285496	+	0.958380i	$0.592158\pi$
$978$	−2.27495e6	−0.0760544
$979$	3.98772e7	1.32974
$980$	2.53065e6	0.0841720
$981$	−3.73394e6	−0.123878
$982$	5.57494e6	0.184485
$983$	−1.36985e7	−0.452156	−0.226078	−	0.974109i	$-0.572590\pi$
$983$	−1.36985e7	−0.452156	−0.226078	+	0.974109i	$0.572590\pi$
$984$	1.12459e7	0.370259
$985$	4.48671e6	0.147346
$986$	6.57712e6	0.215448
$987$	3.93725e6	0.128647
$988$	−1.25540e7	−0.409157
$989$	5.50173e7	1.78858
$990$	936360.	0.0303637
$991$	−3.49088e7	−1.12915	−0.564574	−	0.825383i	$-0.690959\pi$
$991$	−3.49088e7	−1.12915	−0.564574	+	0.825383i	$0.690959\pi$
$992$	−7.35072e6	−0.237165
$993$	1.56259e7	0.502889
$994$	−713832.	−0.0229155
$995$	−1.01502e7	−0.325026
$996$	−1.05362e7	−0.336538
$997$	875662.	0.0278996	0.0139498	−	0.999903i	$-0.495559\pi$
$997$	875662.	0.0278996	0.0139498	+	0.999903i	$0.495559\pi$
$998$	3.92698e6	0.124805
$999$	−7.14274e6	−0.226439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.6.a.b.1.1	✓	1
3.2	odd	2		63.6.a.c.1.1		1
4.3	odd	2		336.6.a.l.1.1		1
5.2	odd	4		525.6.d.d.274.2		2
5.3	odd	4		525.6.d.d.274.1		2
5.4	even	2		525.6.a.c.1.1		1
7.2	even	3		147.6.e.f.67.1		2
7.3	odd	6		147.6.e.e.79.1		2
7.4	even	3		147.6.e.f.79.1		2
7.5	odd	6		147.6.e.e.67.1		2
7.6	odd	2		147.6.a.e.1.1		1
12.11	even	2		1008.6.a.t.1.1		1
21.20	even	2		441.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.b.1.1	✓	1	1.1	even	1	trivial
63.6.a.c.1.1		1	3.2	odd	2
147.6.a.e.1.1		1	7.6	odd	2
147.6.e.e.67.1		2	7.5	odd	6
147.6.e.e.79.1		2	7.3	odd	6
147.6.e.f.67.1		2	7.2	even	3
147.6.e.f.79.1		2	7.4	even	3
336.6.a.l.1.1		1	4.3	odd	2
441.6.a.d.1.1		1	21.20	even	2
525.6.a.c.1.1		1	5.4	even	2
525.6.d.d.274.1		2	5.3	odd	4
525.6.d.d.274.2		2	5.2	odd	4
1008.6.a.t.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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