LMFDB - Embedded newform 63.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, 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("63.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.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:	$10.1041806482$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -31.0000 q^{4} +34.0000 q^{5} -49.0000 q^{7} +63.0000 q^{8} +O(q^{10})$

\(q-1.00000 q^{2} -31.0000 q^{4} +34.0000 q^{5} -49.0000 q^{7} +63.0000 q^{8} -34.0000 q^{10} +340.000 q^{11} +454.000 q^{13} +49.0000 q^{14} +929.000 q^{16} +798.000 q^{17} +892.000 q^{19} -1054.00 q^{20} -340.000 q^{22} +3192.00 q^{23} -1969.00 q^{25} -454.000 q^{26} +1519.00 q^{28} +8242.00 q^{29} -2496.00 q^{31} -2945.00 q^{32} -798.000 q^{34} -1666.00 q^{35} +9798.00 q^{37} -892.000 q^{38} +2142.00 q^{40} -19834.0 q^{41} -17236.0 q^{43} -10540.0 q^{44} -3192.00 q^{46} -8928.00 q^{47} +2401.00 q^{49} +1969.00 q^{50} -14074.0 q^{52} -150.000 q^{53} +11560.0 q^{55} -3087.00 q^{56} -8242.00 q^{58} +42396.0 q^{59} +14758.0 q^{61} +2496.00 q^{62} -26783.0 q^{64} +15436.0 q^{65} -1676.00 q^{67} -24738.0 q^{68} +1666.00 q^{70} -14568.0 q^{71} +78378.0 q^{73} -9798.00 q^{74} -27652.0 q^{76} -16660.0 q^{77} -2272.00 q^{79} +31586.0 q^{80} +19834.0 q^{82} +37764.0 q^{83} +27132.0 q^{85} +17236.0 q^{86} +21420.0 q^{88} +117286. q^{89} -22246.0 q^{91} -98952.0 q^{92} +8928.00 q^{94} +30328.0 q^{95} +10002.0 q^{97} -2401.00 q^{98} +O(q^{100})\)

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 $

$2$	−1.00000	−0.176777	−0.0883883	−	0.996086i	$-0.528172\pi$
$2$	−1.00000	−0.176777	−0.0883883	+	0.996086i	$0.528172\pi$
$3$	0	0
$3$	0	0
$4$	−31.0000	−0.968750
$5$	34.0000	0.608210	0.304105	−	0.952638i	$-0.401643\pi$
$5$	34.0000	0.608210	0.304105	+	0.952638i	$0.401643\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	63.0000	0.348029
$9$	0	0
$10$	−34.0000	−0.107517
$11$	340.000	0.847222	0.423611	−	0.905844i	$-0.360762\pi$
$11$	340.000	0.847222	0.423611	+	0.905844i	$0.360762\pi$
$12$	0	0
$13$	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
$15$	0	0
$16$	929.000	0.907227
$17$	798.000	0.669700	0.334850	−	0.942271i	$-0.391314\pi$
$17$	798.000	0.669700	0.334850	+	0.942271i	$0.391314\pi$
$18$	0	0
$19$	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
$21$	0	0
$22$	−340.000	−0.149769
$23$	3192.00	1.25818	0.629091	−	0.777332i	$-0.283427\pi$
$23$	3192.00	1.25818	0.629091	+	0.777332i	$0.283427\pi$
$24$	0	0
$25$	−1969.00	−0.630080
$26$	−454.000	−0.131711
$27$	0	0
$28$	1519.00	0.366153
$29$	8242.00	1.81986	0.909929	−	0.414764i	$-0.136136\pi$
$29$	8242.00	1.81986	0.909929	+	0.414764i	$0.136136\pi$
$30$	0	0
$31$	−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
$33$	0	0
$34$	−798.000	−0.118387
$35$	−1666.00	−0.229882
$36$	0	0
$37$	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
$39$	0	0
$40$	2142.00	0.211675
$41$	−19834.0	−1.84268	−0.921342	−	0.388754i	$-0.872906\pi$
$41$	−19834.0	−1.84268	−0.921342	+	0.388754i	$0.872906\pi$
$42$	0	0
$43$	−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
$45$	0	0
$46$	−3192.00	−0.222417
$47$	−8928.00	−0.589535	−0.294767	−	0.955569i	$-0.595242\pi$
$47$	−8928.00	−0.589535	−0.294767	+	0.955569i	$0.595242\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	1969.00	0.111383
$51$	0	0
$52$	−14074.0	−0.721787
$53$	−150.000	−0.00733502	−0.00366751	−	0.999993i	$-0.501167\pi$
$53$	−150.000	−0.00733502	−0.00366751	+	0.999993i	$0.501167\pi$
$54$	0	0
$55$	11560.0	0.515289
$56$	−3087.00	−0.131543
$57$	0	0
$58$	−8242.00	−0.321709
$59$	42396.0	1.58560	0.792802	−	0.609479i	$-0.208621\pi$
$59$	42396.0	1.58560	0.792802	+	0.609479i	$0.208621\pi$
$60$	0	0
$61$	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
$63$	0	0
$64$	−26783.0	−0.817352
$65$	15436.0	0.453160
$66$	0	0
$67$	−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
$69$	0	0
$70$	1666.00	0.0406378
$71$	−14568.0	−0.342968	−0.171484	−	0.985187i	$-0.554856\pi$
$71$	−14568.0	−0.342968	−0.171484	+	0.985187i	$0.554856\pi$
$72$	0	0
$73$	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
$75$	0	0
$76$	−27652.0	−0.549152
$77$	−16660.0	−0.320220
$78$	0	0
$79$	−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
$81$	0	0
$82$	19834.0	0.325743
$83$	37764.0	0.601704	0.300852	−	0.953671i	$-0.402729\pi$
$83$	37764.0	0.601704	0.300852	+	0.953671i	$0.402729\pi$
$84$	0	0
$85$	27132.0	0.407319
$86$	17236.0	0.251299
$87$	0	0
$88$	21420.0	0.294858
$89$	117286.	1.56954	0.784768	−	0.619790i	$-0.212782\pi$
$89$	117286.	1.56954	0.784768	+	0.619790i	$0.212782\pi$
$90$	0	0
$91$	−22246.0	−0.281610
$92$	−98952.0	−1.21886
$93$	0	0
$94$	8928.00	0.104216
$95$	30328.0	0.344774
$96$	0	0
$97$	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
$99$	0	0
$100$	61039.0	0.610390
$101$	108770.	1.06098	0.530488	−	0.847692i	$-0.322009\pi$
$101$	108770.	1.06098	0.530488	+	0.847692i	$0.322009\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	150.000	0.00129666
$107$	79972.0	0.675272	0.337636	−	0.941277i	$-0.390373\pi$
$107$	79972.0	0.675272	0.337636	+	0.941277i	$0.390373\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	−45521.0	−0.342899
$113$	−262706.	−1.93541	−0.967707	−	0.252078i	$-0.918886\pi$
$113$	−262706.	−1.93541	−0.967707	+	0.252078i	$0.918886\pi$
$114$	0	0
$115$	108528.	0.765239
$116$	−255502.	−1.76299
$117$	0	0
$118$	−42396.0	−0.280298
$119$	−39102.0	−0.253123
$120$	0	0
$121$	−45451.0	−0.282215
$122$	−14758.0	−0.0897693
$123$	0	0
$124$	77376.0	0.451910
$125$	−173196.	−0.991432
$126$	0	0
$127$	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
$129$	0	0
$130$	−15436.0	−0.0801081
$131$	77140.0	0.392737	0.196368	−	0.980530i	$-0.437085\pi$
$131$	77140.0	0.392737	0.196368	+	0.980530i	$0.437085\pi$
$132$	0	0
$133$	−43708.0	−0.214255
$134$	1676.00	0.00806329
$135$	0	0
$136$	50274.0	0.233075
$137$	−208170.	−0.947582	−0.473791	−	0.880637i	$-0.657115\pi$
$137$	−208170.	−0.947582	−0.473791	+	0.880637i	$0.657115\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	14568.0	0.0606288
$143$	154360.	0.631240
$144$	0	0
$145$	280228.	1.10686
$146$	−78378.0	−0.304307
$147$	0	0
$148$	−303738.	−1.13984
$149$	296106.	1.09265	0.546326	−	0.837573i	$-0.316026\pi$
$149$	296106.	1.09265	0.546326	+	0.837573i	$0.316026\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	16660.0	0.0566074
$155$	−84864.0	−0.283723
$156$	0	0
$157$	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
$159$	0	0
$160$	−100130.	−0.309218
$161$	−156408.	−0.475548
$162$	0	0
$163$	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
$165$	0	0
$166$	−37764.0	−0.106367
$167$	−508088.	−1.40977	−0.704884	−	0.709322i	$-0.749001\pi$
$167$	−508088.	−1.40977	−0.704884	+	0.709322i	$0.749001\pi$
$168$	0	0
$169$	−165177.	−0.444870
$170$	−27132.0	−0.0720045
$171$	0	0
$172$	534316.	1.37714
$173$	221834.	0.563525	0.281762	−	0.959484i	$-0.409081\pi$
$173$	221834.	0.563525	0.281762	+	0.959484i	$0.409081\pi$
$174$	0	0
$175$	96481.0	0.238148
$176$	315860.	0.768622
$177$	0	0
$178$	−117286.	−0.277457
$179$	113564.	0.264916	0.132458	−	0.991189i	$-0.457713\pi$
$179$	113564.	0.264916	0.132458	+	0.991189i	$0.457713\pi$
$180$	0	0
$181$	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
$183$	0	0
$184$	201096.	0.437884
$185$	333132.	0.715628
$186$	0	0
$187$	271320.	0.567385
$188$	276768.	0.571112
$189$	0	0
$190$	−30328.0	−0.0609480
$191$	−505664.	−1.00295	−0.501474	−	0.865173i	$-0.667209\pi$
$191$	−505664.	−1.00295	−0.501474	+	0.865173i	$0.667209\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	−74431.0	−0.138393
$197$	131962.	0.242261	0.121130	−	0.992637i	$-0.461348\pi$
$197$	131962.	0.242261	0.121130	+	0.992637i	$0.461348\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	−108770.	−0.187556
$203$	−403858.	−0.687842
$204$	0	0
$205$	−674356.	−1.12074
$206$	199192.	0.327042
$207$	0	0
$208$	421766.	0.675948
$209$	303280.	0.480262
$210$	0	0
$211$	−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
$213$	0	0
$214$	−79972.0	−0.119372
$215$	−586024.	−0.864608
$216$	0	0
$217$	122304.	0.176316
$218$	46098.0	0.0656963
$219$	0	0
$220$	−358360.	−0.499186
$221$	362292.	0.498974
$222$	0	0
$223$	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
$225$	0	0
$226$	262706.	0.342136
$227$	−707916.	−0.911837	−0.455918	−	0.890022i	$-0.650689\pi$
$227$	−707916.	−0.911837	−0.455918	+	0.890022i	$0.650689\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	519246.	0.633364
$233$	208758.	0.251915	0.125957	−	0.992036i	$-0.459800\pi$
$233$	208758.	0.251915	0.125957	+	0.992036i	$0.459800\pi$
$234$	0	0
$235$	−303552.	−0.358561
$236$	−1.31428e6	−1.53605
$237$	0	0
$238$	39102.0	0.0447462
$239$	−713376.	−0.807837	−0.403919	−	0.914795i	$-0.632352\pi$
$239$	−713376.	−0.807837	−0.403919	+	0.914795i	$0.632352\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	−457498.	−0.491943
$245$	81634.0	0.0868872
$246$	0	0
$247$	404968.	0.422356
$248$	−157248.	−0.162351
$249$	0	0
$250$	173196.	0.175262
$251$	−317108.	−0.317704	−0.158852	−	0.987302i	$-0.550779\pi$
$251$	−317108.	−0.317704	−0.158852	+	0.987302i	$0.550779\pi$
$252$	0	0
$253$	1.08528e6	1.06596
$254$	−196608.	−0.191213
$255$	0	0
$256$	736033.	0.701936
$257$	1.44285e6	1.36266	0.681329	−	0.731977i	$-0.261402\pi$
$257$	1.44285e6	1.36266	0.681329	+	0.731977i	$0.261402\pi$
$258$	0	0
$259$	−480102.	−0.444717
$260$	−478516.	−0.438999
$261$	0	0
$262$	−77140.0	−0.0694267
$263$	−271496.	−0.242033	−0.121016	−	0.992651i	$-0.538615\pi$
$263$	−271496.	−0.242033	−0.121016	+	0.992651i	$0.538615\pi$
$264$	0	0
$265$	−5100.00	−0.00446124
$266$	43708.0	0.0378754
$267$	0	0
$268$	51956.0	0.0441874
$269$	−850614.	−0.716724	−0.358362	−	0.933583i	$-0.616665\pi$
$269$	−850614.	−0.716724	−0.358362	+	0.933583i	$0.616665\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	208170.	0.167510
$275$	−669460.	−0.533818
$276$	0	0
$277$	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
$279$	0	0
$280$	−104958.	−0.0800056
$281$	1.35642e6	1.02478	0.512388	−	0.858754i	$-0.328761\pi$
$281$	1.35642e6	1.02478	0.512388	+	0.858754i	$0.328761\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	−154360.	−0.111589
$287$	971866.	0.696469
$288$	0	0
$289$	−783053.	−0.551501
$290$	−280228.	−0.195667
$291$	0	0
$292$	−2.42972e6	−1.66763
$293$	1.70727e6	1.16180	0.580901	−	0.813974i	$-0.302700\pi$
$293$	1.70727e6	1.16180	0.580901	+	0.813974i	$0.302700\pi$
$294$	0	0
$295$	1.44146e6	0.964381
$296$	617274.	0.409495
$297$	0	0
$298$	−296106.	−0.193155
$299$	1.44917e6	0.937434
$300$	0	0
$301$	844564.	0.537299
$302$	426472.	0.269075
$303$	0	0
$304$	828668.	0.514276
$305$	501772.	0.308857
$306$	0	0
$307$	−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
$309$	0	0
$310$	84864.0	0.0501556
$311$	−3.23426e6	−1.89616	−0.948079	−	0.318035i	$-0.896977\pi$
$311$	−3.23426e6	−1.89616	−0.948079	+	0.318035i	$0.896977\pi$
$312$	0	0
$313$	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
$315$	0	0
$316$	70432.0	0.0396782
$317$	1.27658e6	0.713509	0.356754	−	0.934198i	$-0.383883\pi$
$317$	1.27658e6	0.713509	0.356754	+	0.934198i	$0.383883\pi$
$318$	0	0
$319$	2.80228e6	1.54182
$320$	−910622.	−0.497122
$321$	0	0
$322$	156408.	0.0840658
$323$	711816.	0.379631
$324$	0	0
$325$	−893926.	−0.469454
$326$	−252772.	−0.131730
$327$	0	0
$328$	−1.24954e6	−0.641307
$329$	437472.	0.222823
$330$	0	0
$331$	−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
$333$	0	0
$334$	508088.	0.249214
$335$	−56984.0	−0.0277422
$336$	0	0
$337$	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
$339$	0	0
$340$	−841092.	−0.394590
$341$	−848640.	−0.395219
$342$	0	0
$343$	−117649.	−0.0539949
$344$	−1.08587e6	−0.494744
$345$	0	0
$346$	−221834.	−0.0996180
$347$	1.65146e6	0.736282	0.368141	−	0.929770i	$-0.379994\pi$
$347$	1.65146e6	0.736282	0.368141	+	0.929770i	$0.379994\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	−1.00130e6	−0.430732
$353$	−573218.	−0.244840	−0.122420	−	0.992478i	$-0.539066\pi$
$353$	−573218.	−0.244840	−0.122420	+	0.992478i	$0.539066\pi$
$354$	0	0
$355$	−495312.	−0.208597
$356$	−3.63587e6	−1.52049
$357$	0	0
$358$	−113564.	−0.0468310
$359$	−4.46322e6	−1.82773	−0.913866	−	0.406016i	$-0.866918\pi$
$359$	−4.46322e6	−1.82773	−0.913866	+	0.406016i	$0.866918\pi$
$360$	0	0
$361$	−1.68044e6	−0.678662
$362$	−663118.	−0.265962
$363$	0	0
$364$	689626.	0.272810
$365$	2.66485e6	1.04699
$366$	0	0
$367$	−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
$369$	0	0
$370$	−333132.	−0.126506
$371$	7350.00	0.00277238
$372$	0	0
$373$	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
$375$	0	0
$376$	−562464.	−0.205175
$377$	3.74187e6	1.35592
$378$	0	0
$379$	−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
$381$	0	0
$382$	505664.	0.177298
$383$	−796368.	−0.277407	−0.138703	−	0.990334i	$-0.544293\pi$
$383$	−796368.	−0.277407	−0.138703	+	0.990334i	$0.544293\pi$
$384$	0	0
$385$	−566440.	−0.194761
$386$	432382.	0.147706
$387$	0	0
$388$	−310062.	−0.104561
$389$	−1.94799e6	−0.652699	−0.326349	−	0.945249i	$-0.605819\pi$
$389$	−1.94799e6	−0.652699	−0.326349	+	0.945249i	$0.605819\pi$
$390$	0	0
$391$	2.54722e6	0.842605
$392$	151263.	0.0497184
$393$	0	0
$394$	−131962.	−0.0428261
$395$	−77248.0	−0.0249112
$396$	0	0
$397$	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
$399$	0	0
$400$	−1.82920e6	−0.571625
$401$	−2.76770e6	−0.859524	−0.429762	−	0.902942i	$-0.641402\pi$
$401$	−2.76770e6	−0.859524	−0.429762	+	0.902942i	$0.641402\pi$
$402$	0	0
$403$	−1.13318e6	−0.347566
$404$	−3.37187e6	−1.02782
$405$	0	0
$406$	403858.	0.121594
$407$	3.33132e6	0.996851
$408$	0	0
$409$	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
$411$	0	0
$412$	6.17495e6	1.79222
$413$	−2.07740e6	−0.599302
$414$	0	0
$415$	1.28398e6	0.365963
$416$	−1.33703e6	−0.378798
$417$	0	0
$418$	−303280.	−0.0848991
$419$	2.98669e6	0.831104	0.415552	−	0.909569i	$-0.363588\pi$
$419$	2.98669e6	0.831104	0.415552	+	0.909569i	$0.363588\pi$
$420$	0	0
$421$	−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
$423$	0	0
$424$	−9450.00	−0.00255280
$425$	−1.57126e6	−0.421965
$426$	0	0
$427$	−723142.	−0.191935
$428$	−2.47913e6	−0.654169
$429$	0	0
$430$	586024.	0.152843
$431$	−2.33693e6	−0.605971	−0.302986	−	0.952995i	$-0.597983\pi$
$431$	−2.33693e6	−0.605971	−0.302986	+	0.952995i	$0.597983\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	1.42904e6	0.360021
$437$	2.84726e6	0.713221
$438$	0	0
$439$	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
$441$	0	0
$442$	−362292.	−0.0882070
$443$	−1.76833e6	−0.428109	−0.214055	−	0.976822i	$-0.568667\pi$
$443$	−1.76833e6	−0.428109	−0.214055	+	0.976822i	$0.568667\pi$
$444$	0	0
$445$	3.98772e6	0.954608
$446$	−399376.	−0.0950703
$447$	0	0
$448$	1.31237e6	0.308930
$449$	5.52579e6	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$449$	5.52579e6	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$450$	0	0
$451$	−6.74356e6	−1.56116
$452$	8.14389e6	1.87493
$453$	0	0
$454$	707916.	0.161191
$455$	−756364.	−0.171278
$456$	0	0
$457$	−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
$459$	0	0
$460$	−3.36437e6	−0.741325
$461$	−2.11884e6	−0.464350	−0.232175	−	0.972674i	$-0.574584\pi$
$461$	−2.11884e6	−0.464350	−0.232175	+	0.972674i	$0.574584\pi$
$462$	0	0
$463$	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
$465$	0	0
$466$	−208758.	−0.0445326
$467$	7.42621e6	1.57571	0.787853	−	0.615863i	$-0.211193\pi$
$467$	7.42621e6	1.57571	0.787853	+	0.615863i	$0.211193\pi$
$468$	0	0
$469$	82124.0	0.0172400
$470$	303552.	0.0633853
$471$	0	0
$472$	2.67095e6	0.551837
$473$	−5.86024e6	−1.20438
$474$	0	0
$475$	−1.75635e6	−0.357171
$476$	1.21216e6	0.245213
$477$	0	0
$478$	713376.	0.142807
$479$	3.39685e6	0.676453	0.338226	−	0.941065i	$-0.390173\pi$
$479$	3.39685e6	0.676453	0.338226	+	0.941065i	$0.390173\pi$
$480$	0	0
$481$	4.44829e6	0.876659
$482$	505246.	0.0990570
$483$	0	0
$484$	1.40898e6	0.273396
$485$	340068.	0.0656465
$486$	0	0
$487$	−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
$489$	0	0
$490$	−81634.0	−0.0153596
$491$	−5.57494e6	−1.04361	−0.521803	−	0.853066i	$-0.674740\pi$
$491$	−5.57494e6	−1.04361	−0.521803	+	0.853066i	$0.674740\pi$
$492$	0	0
$493$	6.57712e6	1.21876
$494$	−404968.	−0.0746626
$495$	0	0
$496$	−2.31878e6	−0.423210
$497$	713832.	0.129630
$498$	0	0
$499$	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
$501$	0	0
$502$	317108.	0.0561627
$503$	−6.42079e6	−1.13154	−0.565768	−	0.824564i	$-0.691420\pi$
$503$	−6.42079e6	−1.13154	−0.565768	+	0.824564i	$0.691420\pi$
$504$	0	0
$505$	3.69818e6	0.645297
$506$	−1.08528e6	−0.188437
$507$	0	0
$508$	−6.09485e6	−1.04786
$509$	−146278.	−0.0250256	−0.0125128	−	0.999922i	$-0.503983\pi$
$509$	−146278.	−0.0250256	−0.0125128	+	0.999922i	$0.503983\pi$
$510$	0	0
$511$	−3.84052e6	−0.650636
$512$	−4.60877e6	−0.776980
$513$	0	0
$514$	−1.44285e6	−0.240886
$515$	−6.77253e6	−1.12521
$516$	0	0
$517$	−3.03552e6	−0.499467
$518$	480102.	0.0786157
$519$	0	0
$520$	972468.	0.157713
$521$	−7.70937e6	−1.24430	−0.622149	−	0.782899i	$-0.713740\pi$
$521$	−7.70937e6	−1.24430	−0.622149	+	0.782899i	$0.713740\pi$
$522$	0	0
$523$	−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
$525$	0	0
$526$	271496.	0.0427857
$527$	−1.99181e6	−0.312407
$528$	0	0
$529$	3.75252e6	0.583021
$530$	5100.00	0.000788643 0
$531$	0	0
$532$	1.35495e6	0.207560
$533$	−9.00464e6	−1.37293
$534$	0	0
$535$	2.71905e6	0.410707
$536$	−105588.	−0.0158746
$537$	0	0
$538$	850614.	0.126700
$539$	816340.	0.121032
$540$	0	0
$541$	−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
$543$	0	0
$544$	−2.35011e6	−0.340479
$545$	−1.56733e6	−0.226032
$546$	0	0
$547$	−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
$549$	0	0
$550$	669460.	0.0943665
$551$	7.35186e6	1.03162
$552$	0	0
$553$	111328.	0.0154807
$554$	−513574.	−0.0710933
$555$	0	0
$556$	8.54298e6	1.17199
$557$	−8.19390e6	−1.11906	−0.559529	−	0.828811i	$-0.689018\pi$
$557$	−8.19390e6	−1.11906	−0.559529	+	0.828811i	$0.689018\pi$
$558$	0	0
$559$	−7.82514e6	−1.05916
$560$	−1.54771e6	−0.208555
$561$	0	0
$562$	−1.35642e6	−0.181157
$563$	1.05796e7	1.40669	0.703347	−	0.710847i	$-0.251688\pi$
$563$	1.05796e7	1.40669	0.703347	+	0.710847i	$0.251688\pi$
$564$	0	0
$565$	−8.93200e6	−1.17714
$566$	−286756.	−0.0376246
$567$	0	0
$568$	−917784.	−0.119363
$569$	1.20205e7	1.55648	0.778238	−	0.627969i	$-0.216114\pi$
$569$	1.20205e7	1.55648	0.778238	+	0.627969i	$0.216114\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	−971866.	−0.123119
$575$	−6.28505e6	−0.792755
$576$	0	0
$577$	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
$579$	0	0
$580$	−8.68707e6	−1.07227
$581$	−1.85044e6	−0.227423
$582$	0	0
$583$	−51000.0	−0.00621439
$584$	4.93781e6	0.599105
$585$	0	0
$586$	−1.70727e6	−0.205380
$587$	1.21827e6	0.145931	0.0729655	−	0.997334i	$-0.476754\pi$
$587$	1.21827e6	0.145931	0.0729655	+	0.997334i	$0.476754\pi$
$588$	0	0
$589$	−2.22643e6	−0.264436
$590$	−1.44146e6	−0.170480
$591$	0	0
$592$	9.10234e6	1.06745
$593$	8.42379e6	0.983718	0.491859	−	0.870675i	$-0.336317\pi$
$593$	8.42379e6	0.983718	0.491859	+	0.870675i	$0.336317\pi$
$594$	0	0
$595$	−1.32947e6	−0.153952
$596$	−9.17929e6	−1.05851
$597$	0	0
$598$	−1.44917e6	−0.165717
$599$	−8.21254e6	−0.935212	−0.467606	−	0.883937i	$-0.654883\pi$
$599$	−8.21254e6	−0.935212	−0.467606	+	0.883937i	$0.654883\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	1.32206e7	1.47455
$605$	−1.54533e6	−0.171646
$606$	0	0
$607$	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
$609$	0	0
$610$	−501772.	−0.0545986
$611$	−4.05331e6	−0.439245
$612$	0	0
$613$	−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
$615$	0	0
$616$	−1.04958e6	−0.111446
$617$	7.04895e6	0.745438	0.372719	−	0.927944i	$-0.378426\pi$
$617$	7.04895e6	0.745438	0.372719	+	0.927944i	$0.378426\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	3.23426e6	0.335197
$623$	−5.74701e6	−0.593229
$624$	0	0
$625$	264461.	0.0270808
$626$	−1.81313e6	−0.184924
$627$	0	0
$628$	−5.53307e6	−0.559844
$629$	7.81880e6	0.787977
$630$	0	0
$631$	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
$633$	0	0
$634$	−1.27658e6	−0.126132
$635$	6.68467e6	0.657879
$636$	0	0
$637$	1.09005e6	0.106439
$638$	−2.80228e6	−0.272559
$639$	0	0
$640$	4.11478e6	0.397097
$641$	5.22829e6	0.502590	0.251295	−	0.967910i	$-0.419143\pi$
$641$	5.22829e6	0.502590	0.251295	+	0.967910i	$0.419143\pi$
$642$	0	0
$643$	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
$645$	0	0
$646$	−711816.	−0.0671099
$647$	1.58749e7	1.49090	0.745451	−	0.666560i	$-0.232234\pi$
$647$	1.58749e7	1.49090	0.745451	+	0.666560i	$0.232234\pi$
$648$	0	0
$649$	1.44146e7	1.34336
$650$	893926.	0.0829886
$651$	0	0
$652$	−7.83593e6	−0.721891
$653$	5.94112e6	0.545237	0.272619	−	0.962122i	$-0.412110\pi$
$653$	5.94112e6	0.545237	0.272619	+	0.962122i	$0.412110\pi$
$654$	0	0
$655$	2.62276e6	0.238867
$656$	−1.84258e7	−1.67173
$657$	0	0
$658$	−437472.	−0.0393900
$659$	7.64430e6	0.685684	0.342842	−	0.939393i	$-0.388610\pi$
$659$	7.64430e6	0.685684	0.342842	+	0.939393i	$0.388610\pi$
$660$	0	0
$661$	−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
$663$	0	0
$664$	2.37913e6	0.209410
$665$	−1.48607e6	−0.130312
$666$	0	0
$667$	2.63085e7	2.28971
$668$	1.57507e7	1.36571
$669$	0	0
$670$	56984.0	0.00490417
$671$	5.01772e6	0.430229
$672$	0	0
$673$	−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
$675$	0	0
$676$	5.12049e6	0.430968
$677$	−7.89541e6	−0.662068	−0.331034	−	0.943619i	$-0.607398\pi$
$677$	−7.89541e6	−0.662068	−0.331034	+	0.943619i	$0.607398\pi$
$678$	0	0
$679$	−490098.	−0.0407951
$680$	1.70932e6	0.141759
$681$	0	0
$682$	848640.	0.0698655
$683$	1.96015e7	1.60782	0.803911	−	0.594750i	$-0.202749\pi$
$683$	1.96015e7	1.60782	0.803911	+	0.594750i	$0.202749\pi$
$684$	0	0
$685$	−7.07778e6	−0.576329
$686$	117649.	0.00954504
$687$	0	0
$688$	−1.60122e7	−1.28968
$689$	−68100.0	−0.00546511
$690$	0	0
$691$	−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
$693$	0	0
$694$	−1.65146e6	−0.130158
$695$	−9.36972e6	−0.735808
$696$	0	0
$697$	−1.58275e7	−1.23405
$698$	−1.26645e6	−0.0983900
$699$	0	0
$700$	−2.99091e6	−0.230706
$701$	5.36344e6	0.412238	0.206119	−	0.978527i	$-0.433917\pi$
$701$	5.36344e6	0.412238	0.206119	+	0.978527i	$0.433917\pi$
$702$	0	0
$703$	8.73982e6	0.666982
$704$	−9.10622e6	−0.692479
$705$	0	0
$706$	573218.	0.0432821
$707$	−5.32973e6	−0.401011
$708$	0	0
$709$	−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
$711$	0	0
$712$	7.38902e6	0.546244
$713$	−7.96723e6	−0.586926
$714$	0	0
$715$	5.24824e6	0.383927
$716$	−3.52048e6	−0.256637
$717$	0	0
$718$	4.46322e6	0.323100
$719$	−424608.	−0.0306313	−0.0153157	−	0.999883i	$-0.504875\pi$
$719$	−424608.	−0.0306313	−0.0153157	+	0.999883i	$0.504875\pi$
$720$	0	0
$721$	9.76041e6	0.699246
$722$	1.68044e6	0.119972
$723$	0	0
$724$	−2.05567e7	−1.45749
$725$	−1.62285e7	−1.14666
$726$	0	0
$727$	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
$729$	0	0
$730$	−2.66485e6	−0.185083
$731$	−1.37543e7	−0.952020
$732$	0	0
$733$	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
$735$	0	0
$736$	−9.40044e6	−0.639667
$737$	−569840.	−0.0386442
$738$	0	0
$739$	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
$741$	0	0
$742$	−7350.00	−0.000490092 0
$743$	−3.77647e6	−0.250966	−0.125483	−	0.992096i	$-0.540048\pi$
$743$	−3.77647e6	−0.250966	−0.125483	+	0.992096i	$0.540048\pi$
$744$	0	0
$745$	1.00676e7	0.664562
$746$	−1.66535e6	−0.109562
$747$	0	0
$748$	−8.41092e6	−0.549654
$749$	−3.91863e6	−0.255229
$750$	0	0
$751$	−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
$753$	0	0
$754$	−3.74187e6	−0.239696
$755$	−1.45000e7	−0.925768
$756$	0	0
$757$	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
$759$	0	0
$760$	1.91066e6	0.119991
$761$	1.42623e7	0.892746	0.446373	−	0.894847i	$-0.352716\pi$
$761$	1.42623e7	0.892746	0.446373	+	0.894847i	$0.352716\pi$
$762$	0	0
$763$	2.25880e6	0.140465
$764$	1.56756e7	0.971606
$765$	0	0
$766$	796368.	0.0490390
$767$	1.92478e7	1.18139
$768$	0	0
$769$	−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
$771$	0	0
$772$	1.34038e7	0.809443
$773$	−2.62288e7	−1.57881	−0.789406	−	0.613872i	$-0.789611\pi$
$773$	−2.62288e7	−1.57881	−0.789406	+	0.613872i	$0.789611\pi$
$774$	0	0
$775$	4.91462e6	0.293925
$776$	630126.	0.0375641
$777$	0	0
$778$	1.94799e6	0.115382
$779$	−1.76919e7	−1.04456
$780$	0	0
$781$	−4.95312e6	−0.290570
$782$	−2.54722e6	−0.148953
$783$	0	0
$784$	2.23053e6	0.129604
$785$	6.06852e6	0.351487
$786$	0	0
$787$	−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
$789$	0	0
$790$	77248.0	0.00440372
$791$	1.28726e7	0.731518
$792$	0	0
$793$	6.70013e6	0.378356
$794$	−1.08116e6	−0.0608608
$795$	0	0
$796$	−9.25462e6	−0.517697
$797$	−1.09033e7	−0.608014	−0.304007	−	0.952670i	$-0.598325\pi$
$797$	−1.09033e7	−0.608014	−0.304007	+	0.952670i	$0.598325\pi$
$798$	0	0
$799$	−7.12454e6	−0.394812
$800$	5.79871e6	0.320336
$801$	0	0
$802$	2.76770e6	0.151944
$803$	2.66485e7	1.45843
$804$	0	0
$805$	−5.31787e6	−0.289233
$806$	1.13318e6	0.0614416
$807$	0	0
$808$	6.85251e6	0.369251
$809$	−6.06398e6	−0.325751	−0.162876	−	0.986647i	$-0.552077\pi$
$809$	−6.06398e6	−0.325751	−0.162876	+	0.986647i	$0.552077\pi$
$810$	0	0
$811$	−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
$813$	0	0
$814$	−3.33132e6	−0.176220
$815$	8.59425e6	0.453225
$816$	0	0
$817$	−1.53745e7	−0.805835
$818$	−2.36350e6	−0.123501
$819$	0	0
$820$	2.09050e7	1.08572
$821$	2.01396e6	0.104278	0.0521391	−	0.998640i	$-0.483396\pi$
$821$	2.01396e6	0.104278	0.0521391	+	0.998640i	$0.483396\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	2.07740e6	0.105943
$827$	3.90229e6	0.198407	0.0992033	−	0.995067i	$-0.468371\pi$
$827$	3.90229e6	0.198407	0.0992033	+	0.995067i	$0.468371\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	−1.21595e7	−0.608985
$833$	1.91600e6	0.0956715
$834$	0	0
$835$	−1.72750e7	−0.857436
$836$	−9.40168e6	−0.465254
$837$	0	0
$838$	−2.98669e6	−0.146920
$839$	2.45448e7	1.20380	0.601901	−	0.798570i	$-0.294410\pi$
$839$	2.45448e7	1.20380	0.601901	+	0.798570i	$0.294410\pi$
$840$	0	0
$841$	4.74194e7	2.31188
$842$	3.46331e6	0.168349
$843$	0	0
$844$	3.62892e7	1.75356
$845$	−5.61602e6	−0.270574
$846$	0	0
$847$	2.22710e6	0.106667
$848$	−139350.	−0.00665453
$849$	0	0
$850$	1.57126e6	0.0745936
$851$	3.12752e7	1.48039
$852$	0	0
$853$	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
$855$	0	0
$856$	5.03824e6	0.235014
$857$	−3.18009e7	−1.47907	−0.739534	−	0.673120i	$-0.764954\pi$
$857$	−3.18009e7	−1.47907	−0.739534	+	0.673120i	$0.764954\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	2.33693e6	0.107122
$863$	4.22256e6	0.192996	0.0964981	−	0.995333i	$-0.469236\pi$
$863$	4.22256e6	0.192996	0.0964981	+	0.995333i	$0.469236\pi$
$864$	0	0
$865$	7.54236e6	0.342742
$866$	3.50838e6	0.158969
$867$	0	0
$868$	−3.79142e6	−0.170806
$869$	−772480.	−0.0347007
$870$	0	0
$871$	−760904.	−0.0339848
$872$	−2.90417e6	−0.129340
$873$	0	0
$874$	−2.84726e6	−0.126081
$875$	8.48660e6	0.374726
$876$	0	0
$877$	−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
$879$	0	0
$880$	1.07392e7	0.467484
$881$	2.77630e7	1.20511	0.602555	−	0.798078i	$-0.294150\pi$
$881$	2.77630e7	1.20511	0.602555	+	0.798078i	$0.294150\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	1.76833e6	0.0756797
$887$	−4.34462e6	−0.185414	−0.0927070	−	0.995693i	$-0.529552\pi$
$887$	−4.34462e6	−0.185414	−0.0927070	+	0.995693i	$0.529552\pi$
$888$	0	0
$889$	−9.63379e6	−0.408830
$890$	−3.98772e6	−0.168752
$891$	0	0
$892$	−1.23807e7	−0.520993
$893$	−7.96378e6	−0.334188
$894$	0	0
$895$	3.86118e6	0.161125
$896$	−5.93013e6	−0.246771
$897$	0	0
$898$	−5.52579e6	−0.228667
$899$	−2.05720e7	−0.848942
$900$	0	0
$901$	−119700.	−0.00491227
$902$	6.74356e6	0.275977
$903$	0	0
$904$	−1.65505e7	−0.673580
$905$	2.25460e7	0.915057
$906$	0	0
$907$	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
$909$	0	0
$910$	756364.	0.0302780
$911$	7.26518e6	0.290035	0.145018	−	0.989429i	$-0.453676\pi$
$911$	7.26518e6	0.290035	0.145018	+	0.989429i	$0.453676\pi$
$912$	0	0
$913$	1.28398e7	0.509777
$914$	2.96226e6	0.117289
$915$	0	0
$916$	2.28091e7	0.898193
$917$	−3.77986e6	−0.148440
$918$	0	0
$919$	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
$921$	0	0
$922$	2.11884e6	0.0820863
$923$	−6.61387e6	−0.255536
$924$	0	0
$925$	−1.92923e7	−0.741359
$926$	−3.19226e6	−0.122340
$927$	0	0
$928$	−2.42727e7	−0.925226
$929$	−2.71152e7	−1.03080	−0.515399	−	0.856951i	$-0.672356\pi$
$929$	−2.71152e7	−1.03080	−0.515399	+	0.856951i	$0.672356\pi$
$930$	0	0
$931$	2.14169e6	0.0809809
$932$	−6.47150e6	−0.244042
$933$	0	0
$934$	−7.42621e6	−0.278548
$935$	9.22488e6	0.345089
$936$	0	0
$937$	−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
$939$	0	0
$940$	9.41011e6	0.347356
$941$	−4.65780e7	−1.71477	−0.857387	−	0.514672i	$-0.827914\pi$
$941$	−4.65780e7	−1.71477	−0.857387	+	0.514672i	$0.827914\pi$
$942$	0	0
$943$	−6.33101e7	−2.31843
$944$	3.93859e7	1.43850
$945$	0	0
$946$	5.86024e6	0.212906
$947$	−2.53799e7	−0.919632	−0.459816	−	0.888014i	$-0.652085\pi$
$947$	−2.53799e7	−0.919632	−0.459816	+	0.888014i	$0.652085\pi$
$948$	0	0
$949$	3.55836e7	1.28258
$950$	1.75635e6	0.0631396
$951$	0	0
$952$	−2.46343e6	−0.0880942
$953$	−1.52948e7	−0.545520	−0.272760	−	0.962082i	$-0.587937\pi$
$953$	−1.52948e7	−0.545520	−0.272760	+	0.962082i	$0.587937\pi$
$954$	0	0
$955$	−1.71926e7	−0.610004
$956$	2.21147e7	0.782592
$957$	0	0
$958$	−3.39685e6	−0.119581
$959$	1.02003e7	0.358152
$960$	0	0
$961$	−2.23991e7	−0.782389
$962$	−4.44829e6	−0.154973
$963$	0	0
$964$	1.56626e7	0.542840
$965$	−1.47010e7	−0.508192
$966$	0	0
$967$	−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
$969$	0	0
$970$	−340068.	−0.0116048
$971$	−1.30250e7	−0.443332	−0.221666	−	0.975123i	$-0.571149\pi$
$971$	−1.30250e7	−0.443332	−0.221666	+	0.975123i	$0.571149\pi$
$972$	0	0
$973$	1.35034e7	0.457258
$974$	3.71382e6	0.125436
$975$	0	0
$976$	1.37102e7	0.460700
$977$	1.70360e7	0.570992	0.285496	−	0.958380i	$-0.407842\pi$
$977$	1.70360e7	0.570992	0.285496	+	0.958380i	$0.407842\pi$
$978$	0	0
$979$	3.98772e7	1.32974
$980$	−2.53065e6	−0.0841720
$981$	0	0
$982$	5.57494e6	0.184485
$983$	1.36985e7	0.452156	0.226078	−	0.974109i	$-0.427410\pi$
$983$	1.36985e7	0.452156	0.226078	+	0.974109i	$0.427410\pi$
$984$	0	0
$985$	4.48671e6	0.147346
$986$	−6.57712e6	−0.215448
$987$	0	0
$988$	−1.25540e7	−0.409157
$989$	−5.50173e7	−1.78858
$990$	0	0
$991$	−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
$993$	0	0
$994$	−713832.	−0.0229155
$995$	1.01502e7	0.325026
$996$	0	0
$997$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.6.a.c.1.1		1
3.2	odd	2		21.6.a.b.1.1	✓	1
4.3	odd	2		1008.6.a.t.1.1		1
7.6	odd	2		441.6.a.d.1.1		1
12.11	even	2		336.6.a.l.1.1		1
15.2	even	4		525.6.d.d.274.2		2
15.8	even	4		525.6.d.d.274.1		2
15.14	odd	2		525.6.a.c.1.1		1
21.2	odd	6		147.6.e.f.67.1		2
21.5	even	6		147.6.e.e.67.1		2
21.11	odd	6		147.6.e.f.79.1		2
21.17	even	6		147.6.e.e.79.1		2
21.20	even	2		147.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.b.1.1	✓	1	3.2	odd	2
63.6.a.c.1.1		1	1.1	even	1	trivial
147.6.a.e.1.1		1	21.20	even	2
147.6.e.e.67.1		2	21.5	even	6
147.6.e.e.79.1		2	21.17	even	6
147.6.e.f.67.1		2	21.2	odd	6
147.6.e.f.79.1		2	21.11	odd	6
336.6.a.l.1.1		1	12.11	even	2
441.6.a.d.1.1		1	7.6	odd	2
525.6.a.c.1.1		1	15.14	odd	2
525.6.d.d.274.1		2	15.8	even	4
525.6.d.d.274.2		2	15.2	even	4
1008.6.a.t.1.1		1	4.3	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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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