LMFDB - Embedded newform 63.6.a.b.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)

$f(q)$

$=$

$q-5.00000 q^{2} -7.00000 q^{4} -94.0000 q^{5} -49.0000 q^{7} +195.000 q^{8} +O(q^{10})$

\(q-5.00000 q^{2} -7.00000 q^{4} -94.0000 q^{5} -49.0000 q^{7} +195.000 q^{8} +470.000 q^{10} -52.0000 q^{11} -770.000 q^{13} +245.000 q^{14} -751.000 q^{16} +2022.00 q^{17} +1732.00 q^{19} +658.000 q^{20} +260.000 q^{22} +576.000 q^{23} +5711.00 q^{25} +3850.00 q^{26} +343.000 q^{28} -5518.00 q^{29} +6336.00 q^{31} -2485.00 q^{32} -10110.0 q^{34} +4606.00 q^{35} -7338.00 q^{37} -8660.00 q^{38} -18330.0 q^{40} +3262.00 q^{41} +5420.00 q^{43} +364.000 q^{44} -2880.00 q^{46} -864.000 q^{47} +2401.00 q^{49} -28555.0 q^{50} +5390.00 q^{52} -4182.00 q^{53} +4888.00 q^{55} -9555.00 q^{56} +27590.0 q^{58} +11220.0 q^{59} -45602.0 q^{61} -31680.0 q^{62} +36457.0 q^{64} +72380.0 q^{65} +1396.00 q^{67} -14154.0 q^{68} -23030.0 q^{70} -18720.0 q^{71} +46362.0 q^{73} +36690.0 q^{74} -12124.0 q^{76} +2548.00 q^{77} +97424.0 q^{79} +70594.0 q^{80} -16310.0 q^{82} +81228.0 q^{83} -190068. q^{85} -27100.0 q^{86} -10140.0 q^{88} +3182.00 q^{89} +37730.0 q^{91} -4032.00 q^{92} +4320.00 q^{94} -162808. q^{95} +4914.00 q^{97} -12005.0 q^{98} +O(q^{100})\)

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 $

$2$	−5.00000	−0.883883	−0.441942	−	0.897044i	$-0.645710\pi$
$2$	−5.00000	−0.883883	−0.441942	+	0.897044i	$0.645710\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.218750
$5$	−94.0000	−1.68152	−0.840762	−	0.541406i	$-0.817892\pi$
$5$	−94.0000	−1.68152	−0.840762	+	0.541406i	$0.817892\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	195.000	1.07723
$9$	0	0
$10$	470.000	1.48627
$11$	−52.0000	−0.129575	−0.0647876	−	0.997899i	$-0.520637\pi$
$11$	−52.0000	−0.129575	−0.0647876	+	0.997899i	$0.520637\pi$
$12$	0	0
$13$	−770.000	−1.26367	−0.631833	−	0.775104i	$-0.717697\pi$
$13$	−770.000	−1.26367	−0.631833	+	0.775104i	$0.717697\pi$
$14$	245.000	0.334077
$15$	0	0
$16$	−751.000	−0.733398
$17$	2022.00	1.69691	0.848455	−	0.529267i	$-0.177533\pi$
$17$	2022.00	1.69691	0.848455	+	0.529267i	$0.177533\pi$
$18$	0	0
$19$	1732.00	1.10069	0.550344	−	0.834938i	$-0.314497\pi$
$19$	1732.00	1.10069	0.550344	+	0.834938i	$0.314497\pi$
$20$	658.000	0.367833
$21$	0	0
$22$	260.000	0.114529
$23$	576.000	0.227040	0.113520	−	0.993536i	$-0.463787\pi$
$23$	576.000	0.227040	0.113520	+	0.993536i	$0.463787\pi$
$24$	0	0
$25$	5711.00	1.82752
$26$	3850.00	1.11693
$27$	0	0
$28$	343.000	0.0826797
$29$	−5518.00	−1.21839	−0.609196	−	0.793020i	$-0.708508\pi$
$29$	−5518.00	−1.21839	−0.609196	+	0.793020i	$0.708508\pi$
$30$	0	0
$31$	6336.00	1.18416	0.592081	−	0.805879i	$-0.298307\pi$
$31$	6336.00	1.18416	0.592081	+	0.805879i	$0.298307\pi$
$32$	−2485.00	−0.428994
$33$	0	0
$34$	−10110.0	−1.49987
$35$	4606.00	0.635556
$36$	0	0
$37$	−7338.00	−0.881198	−0.440599	−	0.897704i	$-0.645234\pi$
$37$	−7338.00	−0.881198	−0.440599	+	0.897704i	$0.645234\pi$
$38$	−8660.00	−0.972879
$39$	0	0
$40$	−18330.0	−1.81139
$41$	3262.00	0.303057	0.151528	−	0.988453i	$-0.451580\pi$
$41$	3262.00	0.303057	0.151528	+	0.988453i	$0.451580\pi$
$42$	0	0
$43$	5420.00	0.447021	0.223511	−	0.974701i	$-0.428248\pi$
$43$	5420.00	0.447021	0.223511	+	0.974701i	$0.428248\pi$
$44$	364.000	0.0283446
$45$	0	0
$46$	−2880.00	−0.200677
$47$	−864.000	−0.0570518	−0.0285259	−	0.999593i	$-0.509081\pi$
$47$	−864.000	−0.0570518	−0.0285259	+	0.999593i	$0.509081\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−28555.0	−1.61531
$51$	0	0
$52$	5390.00	0.276427
$53$	−4182.00	−0.204500	−0.102250	−	0.994759i	$-0.532604\pi$
$53$	−4182.00	−0.204500	−0.102250	+	0.994759i	$0.532604\pi$
$54$	0	0
$55$	4888.00	0.217884
$56$	−9555.00	−0.407156
$57$	0	0
$58$	27590.0	1.07692
$59$	11220.0	0.419626	0.209813	−	0.977741i	$-0.432714\pi$
$59$	11220.0	0.419626	0.209813	+	0.977741i	$0.432714\pi$
$60$	0	0
$61$	−45602.0	−1.56913	−0.784566	−	0.620046i	$-0.787114\pi$
$61$	−45602.0	−1.56913	−0.784566	+	0.620046i	$0.787114\pi$
$62$	−31680.0	−1.04666
$63$	0	0
$64$	36457.0	1.11258
$65$	72380.0	2.12488
$66$	0	0
$67$	1396.00	0.0379925	0.0189963	−	0.999820i	$-0.493953\pi$
$67$	1396.00	0.0379925	0.0189963	+	0.999820i	$0.493953\pi$
$68$	−14154.0	−0.371199
$69$	0	0
$70$	−23030.0	−0.561757
$71$	−18720.0	−0.440717	−0.220359	−	0.975419i	$-0.570723\pi$
$71$	−18720.0	−0.440717	−0.220359	+	0.975419i	$0.570723\pi$
$72$	0	0
$73$	46362.0	1.01825	0.509126	−	0.860692i	$-0.329969\pi$
$73$	46362.0	1.01825	0.509126	+	0.860692i	$0.329969\pi$
$74$	36690.0	0.778876
$75$	0	0
$76$	−12124.0	−0.240775
$77$	2548.00	0.0489748
$78$	0	0
$79$	97424.0	1.75630	0.878149	−	0.478387i	$-0.158778\pi$
$79$	97424.0	1.75630	0.878149	+	0.478387i	$0.158778\pi$
$80$	70594.0	1.23323
$81$	0	0
$82$	−16310.0	−0.267867
$83$	81228.0	1.29423	0.647114	−	0.762394i	$-0.275976\pi$
$83$	81228.0	1.29423	0.647114	+	0.762394i	$0.275976\pi$
$84$	0	0
$85$	−190068.	−2.85339
$86$	−27100.0	−0.395115
$87$	0	0
$88$	−10140.0	−0.139583
$89$	3182.00	0.0425819	0.0212910	−	0.999773i	$-0.493222\pi$
$89$	3182.00	0.0425819	0.0212910	+	0.999773i	$0.493222\pi$
$90$	0	0
$91$	37730.0	0.477621
$92$	−4032.00	−0.0496651
$93$	0	0
$94$	4320.00	0.0504271
$95$	−162808.	−1.85083
$96$	0	0
$97$	4914.00	0.0530281	0.0265140	−	0.999648i	$-0.491559\pi$
$97$	4914.00	0.0530281	0.0265140	+	0.999648i	$0.491559\pi$
$98$	−12005.0	−0.126269
$99$	0	0
$100$	−39977.0	−0.399770
$101$	166354.	1.62267	0.811334	−	0.584583i	$-0.198742\pi$
$101$	166354.	1.62267	0.811334	+	0.584583i	$0.198742\pi$
$102$	0	0
$103$	157160.	1.45965	0.729825	−	0.683634i	$-0.239601\pi$
$103$	157160.	1.45965	0.729825	+	0.683634i	$0.239601\pi$
$104$	−150150.	−1.36126
$105$	0	0
$106$	20910.0	0.180755
$107$	6764.00	0.0571142	0.0285571	−	0.999592i	$-0.490909\pi$
$107$	6764.00	0.0571142	0.0285571	+	0.999592i	$0.490909\pi$
$108$	0	0
$109$	178398.	1.43821	0.719107	−	0.694899i	$-0.244551\pi$
$109$	178398.	1.43821	0.719107	+	0.694899i	$0.244551\pi$
$110$	−24440.0	−0.192584
$111$	0	0
$112$	36799.0	0.277199
$113$	45134.0	0.332512	0.166256	−	0.986083i	$-0.446832\pi$
$113$	45134.0	0.332512	0.166256	+	0.986083i	$0.446832\pi$
$114$	0	0
$115$	−54144.0	−0.381773
$116$	38626.0	0.266523
$117$	0	0
$118$	−56100.0	−0.370901
$119$	−99078.0	−0.641372
$120$	0	0
$121$	−158347.	−0.983210
$122$	228010.	1.38693
$123$	0	0
$124$	−44352.0	−0.259035
$125$	−243084.	−1.39149
$126$	0	0
$127$	−205056.	−1.12814	−0.564070	−	0.825727i	$-0.690765\pi$
$127$	−205056.	−1.12814	−0.564070	+	0.825727i	$0.690765\pi$
$128$	−102765.	−0.554396
$129$	0	0
$130$	−361900.	−1.87815
$131$	−72964.0	−0.371476	−0.185738	−	0.982599i	$-0.559468\pi$
$131$	−72964.0	−0.371476	−0.185738	+	0.982599i	$0.559468\pi$
$132$	0	0
$133$	−84868.0	−0.416021
$134$	−6980.00	−0.0335810
$135$	0	0
$136$	394290.	1.82797
$137$	94182.0	0.428713	0.214356	−	0.976756i	$-0.431235\pi$
$137$	94182.0	0.428713	0.214356	+	0.976756i	$0.431235\pi$
$138$	0	0
$139$	−47796.0	−0.209824	−0.104912	−	0.994482i	$-0.533456\pi$
$139$	−47796.0	−0.209824	−0.104912	+	0.994482i	$0.533456\pi$
$140$	−32242.0	−0.139028
$141$	0	0
$142$	93600.0	0.389543
$143$	40040.0	0.163740
$144$	0	0
$145$	518692.	2.04875
$146$	−231810.	−0.900016
$147$	0	0
$148$	51366.0	0.192762
$149$	124266.	0.458550	0.229275	−	0.973362i	$-0.426364\pi$
$149$	124266.	0.458550	0.229275	+	0.973362i	$0.426364\pi$
$150$	0	0
$151$	−446296.	−1.59287	−0.796436	−	0.604723i	$-0.793284\pi$
$151$	−446296.	−1.59287	−0.796436	+	0.604723i	$0.793284\pi$
$152$	337740.	1.18570
$153$	0	0
$154$	−12740.0	−0.0432880
$155$	−595584.	−1.99119
$156$	0	0
$157$	−159746.	−0.517227	−0.258613	−	0.965981i	$-0.583266\pi$
$157$	−159746.	−0.517227	−0.258613	+	0.965981i	$0.583266\pi$
$158$	−487120.	−1.55236
$159$	0	0
$160$	233590.	0.721364
$161$	−28224.0	−0.0858132
$162$	0	0
$163$	247252.	0.728905	0.364452	−	0.931222i	$-0.381256\pi$
$163$	247252.	0.728905	0.364452	+	0.931222i	$0.381256\pi$
$164$	−22834.0	−0.0662937
$165$	0	0
$166$	−406140.	−1.14395
$167$	684488.	1.89922	0.949609	−	0.313438i	$-0.101481\pi$
$167$	684488.	1.89922	0.949609	+	0.313438i	$0.101481\pi$
$168$	0	0
$169$	221607.	0.596852
$170$	950340.	2.52207
$171$	0	0
$172$	−37940.0	−0.0977859
$173$	610474.	1.55079	0.775393	−	0.631479i	$-0.217552\pi$
$173$	610474.	1.55079	0.775393	+	0.631479i	$0.217552\pi$
$174$	0	0
$175$	−279839.	−0.690738
$176$	39052.0	0.0950302
$177$	0	0
$178$	−15910.0	−0.0376374
$179$	−662252.	−1.54487	−0.772433	−	0.635097i	$-0.780960\pi$
$179$	−662252.	−1.54487	−0.772433	+	0.635097i	$0.780960\pi$
$180$	0	0
$181$	154630.	0.350830	0.175415	−	0.984495i	$-0.443873\pi$
$181$	154630.	0.350830	0.175415	+	0.984495i	$0.443873\pi$
$182$	−188650.	−0.422161
$183$	0	0
$184$	112320.	0.244575
$185$	689772.	1.48175
$186$	0	0
$187$	−105144.	−0.219877
$188$	6048.00	0.0124801
$189$	0	0
$190$	814040.	1.63592
$191$	−486904.	−0.965739	−0.482870	−	0.875692i	$-0.660406\pi$
$191$	−486904.	−0.965739	−0.482870	+	0.875692i	$0.660406\pi$
$192$	0	0
$193$	620546.	1.19917	0.599585	−	0.800311i	$-0.295332\pi$
$193$	620546.	1.19917	0.599585	+	0.800311i	$0.295332\pi$
$194$	−24570.0	−0.0468706
$195$	0	0
$196$	−16807.0	−0.0312500
$197$	236570.	0.434304	0.217152	−	0.976138i	$-0.430323\pi$
$197$	236570.	0.434304	0.217152	+	0.976138i	$0.430323\pi$
$198$	0	0
$199$	82104.0	0.146971	0.0734855	−	0.997296i	$-0.476588\pi$
$199$	82104.0	0.146971	0.0734855	+	0.997296i	$0.476588\pi$
$200$	1.11364e6	1.96866
$201$	0	0
$202$	−831770.	−1.43425
$203$	270382.	0.460509
$204$	0	0
$205$	−306628.	−0.509597
$206$	−785800.	−1.29016
$207$	0	0
$208$	578270.	0.926771
$209$	−90064.0	−0.142622
$210$	0	0
$211$	99892.0	0.154463	0.0772315	−	0.997013i	$-0.475392\pi$
$211$	99892.0	0.154463	0.0772315	+	0.997013i	$0.475392\pi$
$212$	29274.0	0.0447345
$213$	0	0
$214$	−33820.0	−0.0504823
$215$	−509480.	−0.751677
$216$	0	0
$217$	−310464.	−0.447571
$218$	−891990.	−1.27121
$219$	0	0
$220$	−34216.0	−0.0476620
$221$	−1.55694e6	−2.14433
$222$	0	0
$223$	−186704.	−0.251415	−0.125708	−	0.992067i	$-0.540120\pi$
$223$	−186704.	−0.251415	−0.125708	+	0.992067i	$0.540120\pi$
$224$	121765.	0.162145
$225$	0	0
$226$	−225670.	−0.293902
$227$	−336372.	−0.433267	−0.216633	−	0.976253i	$-0.569508\pi$
$227$	−336372.	−0.433267	−0.216633	+	0.976253i	$0.569508\pi$
$228$	0	0
$229$	−926314.	−1.16727	−0.583633	−	0.812018i	$-0.698369\pi$
$229$	−926314.	−1.16727	−0.583633	+	0.812018i	$0.698369\pi$
$230$	270720.	0.337443
$231$	0	0
$232$	−1.07601e6	−1.31249
$233$	−1.25711e6	−1.51700	−0.758499	−	0.651675i	$-0.774067\pi$
$233$	−1.25711e6	−1.51700	−0.758499	+	0.651675i	$0.774067\pi$
$234$	0	0
$235$	81216.0	0.0959339
$236$	−78540.0	−0.0917933
$237$	0	0
$238$	495390.	0.566898
$239$	347016.	0.392966	0.196483	−	0.980507i	$-0.437048\pi$
$239$	347016.	0.392966	0.196483	+	0.980507i	$0.437048\pi$
$240$	0	0
$241$	99170.0	0.109986	0.0549930	−	0.998487i	$-0.482486\pi$
$241$	99170.0	0.109986	0.0549930	+	0.998487i	$0.482486\pi$
$242$	791735.	0.869043
$243$	0	0
$244$	319214.	0.343247
$245$	−225694.	−0.240218
$246$	0	0
$247$	−1.33364e6	−1.39090
$248$	1.23552e6	1.27562
$249$	0	0
$250$	1.21542e6	1.22992
$251$	−344428.	−0.345076	−0.172538	−	0.985003i	$-0.555197\pi$
$251$	−344428.	−0.345076	−0.172538	+	0.985003i	$0.555197\pi$
$252$	0	0
$253$	−29952.0	−0.0294188
$254$	1.02528e6	0.997145
$255$	0	0
$256$	−652799.	−0.622558
$257$	−295130.	−0.278728	−0.139364	−	0.990241i	$-0.544506\pi$
$257$	−295130.	−0.278728	−0.139364	+	0.990241i	$0.544506\pi$
$258$	0	0
$259$	359562.	0.333061
$260$	−506660.	−0.464818
$261$	0	0
$262$	364820.	0.328341
$263$	1.27246e6	1.13437	0.567187	−	0.823589i	$-0.308032\pi$
$263$	1.27246e6	1.13437	0.567187	+	0.823589i	$0.308032\pi$
$264$	0	0
$265$	393108.	0.343872
$266$	424340.	0.367714
$267$	0	0
$268$	−9772.00	−0.00831087
$269$	−276774.	−0.233209	−0.116604	−	0.993178i	$-0.537201\pi$
$269$	−276774.	−0.233209	−0.116604	+	0.993178i	$0.537201\pi$
$270$	0	0
$271$	−1.28994e6	−1.06695	−0.533476	−	0.845815i	$-0.679115\pi$
$271$	−1.28994e6	−1.06695	−0.533476	+	0.845815i	$0.679115\pi$
$272$	−1.51852e6	−1.24451
$273$	0	0
$274$	−470910.	−0.378932
$275$	−296972.	−0.236801
$276$	0	0
$277$	1.71655e6	1.34418	0.672089	−	0.740470i	$-0.265397\pi$
$277$	1.71655e6	1.34418	0.672089	+	0.740470i	$0.265397\pi$
$278$	238980.	0.185460
$279$	0	0
$280$	898170.	0.684642
$281$	1.47218e6	1.11223	0.556116	−	0.831104i	$-0.312291\pi$
$281$	1.47218e6	1.11223	0.556116	+	0.831104i	$0.312291\pi$
$282$	0	0
$283$	1.02881e6	0.763607	0.381804	−	0.924244i	$-0.375303\pi$
$283$	1.02881e6	0.763607	0.381804	+	0.924244i	$0.375303\pi$
$284$	131040.	0.0964069
$285$	0	0
$286$	−200200.	−0.144727
$287$	−159838.	−0.114545
$288$	0	0
$289$	2.66863e6	1.87950
$290$	−2.59346e6	−1.81086
$291$	0	0
$292$	−324534.	−0.222742
$293$	1.18607e6	0.807123	0.403562	−	0.914952i	$-0.367772\pi$
$293$	1.18607e6	0.807123	0.403562	+	0.914952i	$0.367772\pi$
$294$	0	0
$295$	−1.05468e6	−0.705612
$296$	−1.43091e6	−0.949255
$297$	0	0
$298$	−621330.	−0.405305
$299$	−443520.	−0.286903
$300$	0	0
$301$	−265580.	−0.168958
$302$	2.23148e6	1.40791
$303$	0	0
$304$	−1.30073e6	−0.807242
$305$	4.28659e6	2.63853
$306$	0	0
$307$	1.51892e6	0.919788	0.459894	−	0.887974i	$-0.347887\pi$
$307$	1.51892e6	0.919788	0.459894	+	0.887974i	$0.347887\pi$
$308$	−17836.0	−0.0107132
$309$	0	0
$310$	2.97792e6	1.75998
$311$	−212808.	−0.124763	−0.0623817	−	0.998052i	$-0.519870\pi$
$311$	−212808.	−0.124763	−0.0623817	+	0.998052i	$0.519870\pi$
$312$	0	0
$313$	−1894.00	−0.00109275	−0.000546373	−	1.00000i	$-0.500174\pi$
$313$	−1894.00	−0.00109275	−0.000546373		1.00000i	$0.500174\pi$
$314$	798730.	0.457168
$315$	0	0
$316$	−681968.	−0.384190
$317$	1.57898e6	0.882527	0.441263	−	0.897378i	$-0.354530\pi$
$317$	1.57898e6	0.882527	0.441263	+	0.897378i	$0.354530\pi$
$318$	0	0
$319$	286936.	0.157873
$320$	−3.42696e6	−1.87083
$321$	0	0
$322$	141120.	0.0758488
$323$	3.50210e6	1.86777
$324$	0	0
$325$	−4.39747e6	−2.30938
$326$	−1.23626e6	−0.644267
$327$	0	0
$328$	636090.	0.326463
$329$	42336.0	0.0215635
$330$	0	0
$331$	−3.39471e6	−1.70307	−0.851535	−	0.524298i	$-0.824328\pi$
$331$	−3.39471e6	−1.70307	−0.851535	+	0.524298i	$0.824328\pi$
$332$	−568596.	−0.283112
$333$	0	0
$334$	−3.42244e6	−1.67869
$335$	−131224.	−0.0638853
$336$	0	0
$337$	2.02731e6	0.972403	0.486201	−	0.873847i	$-0.338382\pi$
$337$	2.02731e6	0.972403	0.486201	+	0.873847i	$0.338382\pi$
$338$	−1.10804e6	−0.527548
$339$	0	0
$340$	1.33048e6	0.624180
$341$	−329472.	−0.153438
$342$	0	0
$343$	−117649.	−0.0539949
$344$	1.05690e6	0.481546
$345$	0	0
$346$	−3.05237e6	−1.37071
$347$	−3.48885e6	−1.55546	−0.777730	−	0.628598i	$-0.783629\pi$
$347$	−3.48885e6	−1.55546	−0.777730	+	0.628598i	$0.783629\pi$
$348$	0	0
$349$	965566.	0.424344	0.212172	−	0.977232i	$-0.431946\pi$
$349$	965566.	0.424344	0.212172	+	0.977232i	$0.431946\pi$
$350$	1.39919e6	0.610532
$351$	0	0
$352$	129220.	0.0555870
$353$	−1.15393e6	−0.492882	−0.246441	−	0.969158i	$-0.579261\pi$
$353$	−1.15393e6	−0.492882	−0.246441	+	0.969158i	$0.579261\pi$
$354$	0	0
$355$	1.75968e6	0.741076
$356$	−22274.0	−0.00931479
$357$	0	0
$358$	3.31126e6	1.36548
$359$	−1.61110e6	−0.659762	−0.329881	−	0.944022i	$-0.607009\pi$
$359$	−1.61110e6	−0.659762	−0.329881	+	0.944022i	$0.607009\pi$
$360$	0	0
$361$	523725.	0.211512
$362$	−773150.	−0.310093
$363$	0	0
$364$	−264110.	−0.104480
$365$	−4.35803e6	−1.71221
$366$	0	0
$367$	3.67747e6	1.42523	0.712614	−	0.701557i	$-0.247511\pi$
$367$	3.67747e6	1.42523	0.712614	+	0.701557i	$0.247511\pi$
$368$	−432576.	−0.166511
$369$	0	0
$370$	−3.44886e6	−1.30970
$371$	204918.	0.0772939
$372$	0	0
$373$	649766.	0.241816	0.120908	−	0.992664i	$-0.461419\pi$
$373$	649766.	0.241816	0.120908	+	0.992664i	$0.461419\pi$
$374$	525720.	0.194346
$375$	0	0
$376$	−168480.	−0.0614580
$377$	4.24886e6	1.53964
$378$	0	0
$379$	320700.	0.114683	0.0573417	−	0.998355i	$-0.481738\pi$
$379$	320700.	0.114683	0.0573417	+	0.998355i	$0.481738\pi$
$380$	1.13966e6	0.404869
$381$	0	0
$382$	2.43452e6	0.853601
$383$	2.36189e6	0.822740	0.411370	−	0.911469i	$-0.365050\pi$
$383$	2.36189e6	0.822740	0.411370	+	0.911469i	$0.365050\pi$
$384$	0	0
$385$	−239512.	−0.0823522
$386$	−3.10273e6	−1.05993
$387$	0	0
$388$	−34398.0	−0.0115999
$389$	3.53390e6	1.18408	0.592039	−	0.805910i	$-0.298323\pi$
$389$	3.53390e6	1.18408	0.592039	+	0.805910i	$0.298323\pi$
$390$	0	0
$391$	1.16467e6	0.385267
$392$	468195.	0.153890
$393$	0	0
$394$	−1.18285e6	−0.383874
$395$	−9.15786e6	−2.95326
$396$	0	0
$397$	4.04811e6	1.28907	0.644534	−	0.764575i	$-0.277051\pi$
$397$	4.04811e6	1.28907	0.644534	+	0.764575i	$0.277051\pi$
$398$	−410520.	−0.129905
$399$	0	0
$400$	−4.28896e6	−1.34030
$401$	−2.07645e6	−0.644853	−0.322426	−	0.946595i	$-0.604498\pi$
$401$	−2.07645e6	−0.644853	−0.322426	+	0.946595i	$0.604498\pi$
$402$	0	0
$403$	−4.87872e6	−1.49638
$404$	−1.16448e6	−0.354959
$405$	0	0
$406$	−1.35191e6	−0.407036
$407$	381576.	0.114181
$408$	0	0
$409$	2.57431e6	0.760945	0.380472	−	0.924792i	$-0.375761\pi$
$409$	2.57431e6	0.760945	0.380472	+	0.924792i	$0.375761\pi$
$410$	1.53314e6	0.450425
$411$	0	0
$412$	−1.10012e6	−0.319299
$413$	−549780.	−0.158604
$414$	0	0
$415$	−7.63543e6	−2.17627
$416$	1.91345e6	0.542105
$417$	0	0
$418$	450320.	0.126061
$419$	−848148.	−0.236013	−0.118007	−	0.993013i	$-0.537650\pi$
$419$	−848148.	−0.236013	−0.118007	+	0.993013i	$0.537650\pi$
$420$	0	0
$421$	1.43682e6	0.395092	0.197546	−	0.980294i	$-0.436703\pi$
$421$	1.43682e6	0.395092	0.197546	+	0.980294i	$0.436703\pi$
$422$	−499460.	−0.136527
$423$	0	0
$424$	−815490.	−0.220295
$425$	1.15476e7	3.10114
$426$	0	0
$427$	2.23450e6	0.593076
$428$	−47348.0	−0.0124937
$429$	0	0
$430$	2.54740e6	0.664394
$431$	−2.35438e6	−0.610496	−0.305248	−	0.952273i	$-0.598739\pi$
$431$	−2.35438e6	−0.610496	−0.305248	+	0.952273i	$0.598739\pi$
$432$	0	0
$433$	−3.78808e6	−0.970955	−0.485478	−	0.874249i	$-0.661354\pi$
$433$	−3.78808e6	−0.970955	−0.485478	+	0.874249i	$0.661354\pi$
$434$	1.55232e6	0.395601
$435$	0	0
$436$	−1.24879e6	−0.314609
$437$	997632.	0.249900
$438$	0	0
$439$	−3.64322e6	−0.902245	−0.451123	−	0.892462i	$-0.648976\pi$
$439$	−3.64322e6	−0.902245	−0.451123	+	0.892462i	$0.648976\pi$
$440$	953160.	0.234711
$441$	0	0
$442$	7.78470e6	1.89534
$443$	−2.48389e6	−0.601345	−0.300672	−	0.953728i	$-0.597211\pi$
$443$	−2.48389e6	−0.601345	−0.300672	+	0.953728i	$0.597211\pi$
$444$	0	0
$445$	−299108.	−0.0716025
$446$	933520.	0.222222
$447$	0	0
$448$	−1.78639e6	−0.420515
$449$	2.63177e6	0.616074	0.308037	−	0.951374i	$-0.400328\pi$
$449$	2.63177e6	0.616074	0.308037	+	0.951374i	$0.400328\pi$
$450$	0	0
$451$	−169624.	−0.0392686
$452$	−315938.	−0.0727371
$453$	0	0
$454$	1.68186e6	0.382957
$455$	−3.54662e6	−0.803131
$456$	0	0
$457$	−1.16130e6	−0.260109	−0.130054	−	0.991507i	$-0.541515\pi$
$457$	−1.16130e6	−0.260109	−0.130054	+	0.991507i	$0.541515\pi$
$458$	4.63157e6	1.03173
$459$	0	0
$460$	379008.	0.0835129
$461$	−2.81385e6	−0.616663	−0.308332	−	0.951279i	$-0.599771\pi$
$461$	−2.81385e6	−0.616663	−0.308332	+	0.951279i	$0.599771\pi$
$462$	0	0
$463$	6.84299e6	1.48352	0.741760	−	0.670665i	$-0.233991\pi$
$463$	6.84299e6	1.48352	0.741760	+	0.670665i	$0.233991\pi$
$464$	4.14402e6	0.893566
$465$	0	0
$466$	6.28557e6	1.34085
$467$	−3.34314e6	−0.709353	−0.354676	−	0.934989i	$-0.615409\pi$
$467$	−3.34314e6	−0.709353	−0.354676	+	0.934989i	$0.615409\pi$
$468$	0	0
$469$	−68404.0	−0.0143598
$470$	−406080.	−0.0847944
$471$	0	0
$472$	2.18790e6	0.452035
$473$	−281840.	−0.0579228
$474$	0	0
$475$	9.89145e6	2.01153
$476$	693546.	0.140300
$477$	0	0
$478$	−1.73508e6	−0.347336
$479$	4.28248e6	0.852818	0.426409	−	0.904530i	$-0.359778\pi$
$479$	4.28248e6	0.852818	0.426409	+	0.904530i	$0.359778\pi$
$480$	0	0
$481$	5.65026e6	1.11354
$482$	−495850.	−0.0972149
$483$	0	0
$484$	1.10843e6	0.215077
$485$	−461916.	−0.0891679
$486$	0	0
$487$	−8.93175e6	−1.70653	−0.853266	−	0.521477i	$-0.825381\pi$
$487$	−8.93175e6	−1.70653	−0.853266	+	0.521477i	$0.825381\pi$
$488$	−8.89239e6	−1.69032
$489$	0	0
$490$	1.12847e6	0.212324
$491$	−2.75306e6	−0.515361	−0.257681	−	0.966230i	$-0.582958\pi$
$491$	−2.75306e6	−0.515361	−0.257681	+	0.966230i	$0.582958\pi$
$492$	0	0
$493$	−1.11574e7	−2.06750
$494$	6.66820e6	1.22939
$495$	0	0
$496$	−4.75834e6	−0.868462
$497$	917280.	0.166575
$498$	0	0
$499$	4.80408e6	0.863693	0.431846	−	0.901947i	$-0.357862\pi$
$499$	4.80408e6	0.863693	0.431846	+	0.901947i	$0.357862\pi$
$500$	1.70159e6	0.304389
$501$	0	0
$502$	1.72214e6	0.305007
$503$	6.02465e6	1.06172	0.530862	−	0.847458i	$-0.321868\pi$
$503$	6.02465e6	1.06172	0.530862	+	0.847458i	$0.321868\pi$
$504$	0	0
$505$	−1.56373e7	−2.72855
$506$	149760.	0.0260028
$507$	0	0
$508$	1.43539e6	0.246781
$509$	8.42987e6	1.44220	0.721101	−	0.692830i	$-0.243636\pi$
$509$	8.42987e6	1.44220	0.721101	+	0.692830i	$0.243636\pi$
$510$	0	0
$511$	−2.27174e6	−0.384863
$512$	6.55248e6	1.10466
$513$	0	0
$514$	1.47565e6	0.246363
$515$	−1.47730e7	−2.45444
$516$	0	0
$517$	44928.0	0.00739249
$518$	−1.79781e6	−0.294388
$519$	0	0
$520$	1.41141e7	2.28899
$521$	−9.25058e6	−1.49305	−0.746525	−	0.665357i	$-0.768279\pi$
$521$	−9.25058e6	−1.49305	−0.746525	+	0.665357i	$0.768279\pi$
$522$	0	0
$523$	5.84494e6	0.934385	0.467192	−	0.884156i	$-0.345266\pi$
$523$	5.84494e6	0.934385	0.467192	+	0.884156i	$0.345266\pi$
$524$	510748.	0.0812603
$525$	0	0
$526$	−6.36232e6	−1.00265
$527$	1.28114e7	2.00942
$528$	0	0
$529$	−6.10457e6	−0.948453
$530$	−1.96554e6	−0.303943
$531$	0	0
$532$	594076.	0.0910045
$533$	−2.51174e6	−0.382963
$534$	0	0
$535$	−635816.	−0.0960389
$536$	272220.	0.0409268
$537$	0	0
$538$	1.38387e6	0.206129
$539$	−124852.	−0.0185107
$540$	0	0
$541$	9.22533e6	1.35515	0.677577	−	0.735452i	$-0.263030\pi$
$541$	9.22533e6	1.35515	0.677577	+	0.735452i	$0.263030\pi$
$542$	6.44968e6	0.943061
$543$	0	0
$544$	−5.02467e6	−0.727965
$545$	−1.67694e7	−2.41839
$546$	0	0
$547$	−6.44337e6	−0.920757	−0.460378	−	0.887723i	$-0.652286\pi$
$547$	−6.44337e6	−0.920757	−0.460378	+	0.887723i	$0.652286\pi$
$548$	−659274.	−0.0937809
$549$	0	0
$550$	1.48486e6	0.209305
$551$	−9.55718e6	−1.34107
$552$	0	0
$553$	−4.77378e6	−0.663818
$554$	−8.58275e6	−1.18810
$555$	0	0
$556$	334572.	0.0458989
$557$	−3.74213e6	−0.511070	−0.255535	−	0.966800i	$-0.582252\pi$
$557$	−3.74213e6	−0.511070	−0.255535	+	0.966800i	$0.582252\pi$
$558$	0	0
$559$	−4.17340e6	−0.564886
$560$	−3.45911e6	−0.466116
$561$	0	0
$562$	−7.36091e6	−0.983084
$563$	1.46384e7	1.94635	0.973176	−	0.230060i	$-0.0738923\pi$
$563$	1.46384e7	1.94635	0.973176	+	0.230060i	$0.0738923\pi$
$564$	0	0
$565$	−4.24260e6	−0.559127
$566$	−5.14406e6	−0.674940
$567$	0	0
$568$	−3.65040e6	−0.474755
$569$	−1.41805e7	−1.83616	−0.918078	−	0.396400i	$-0.870259\pi$
$569$	−1.41805e7	−1.83616	−0.918078	+	0.396400i	$0.870259\pi$
$570$	0	0
$571$	−1.25160e6	−0.160648	−0.0803242	−	0.996769i	$-0.525596\pi$
$571$	−1.25160e6	−0.160648	−0.0803242	+	0.996769i	$0.525596\pi$
$572$	−280280.	−0.0358181
$573$	0	0
$574$	799190.	0.101244
$575$	3.28954e6	0.414921
$576$	0	0
$577$	5.94378e6	0.743230	0.371615	−	0.928387i	$-0.378804\pi$
$577$	5.94378e6	0.743230	0.371615	+	0.928387i	$0.378804\pi$
$578$	−1.33431e7	−1.66126
$579$	0	0
$580$	−3.63084e6	−0.448165
$581$	−3.98017e6	−0.489172
$582$	0	0
$583$	217464.	0.0264982
$584$	9.04059e6	1.09689
$585$	0	0
$586$	−5.93033e6	−0.713403
$587$	6.46192e6	0.774046	0.387023	−	0.922070i	$-0.373503\pi$
$587$	6.46192e6	0.774046	0.387023	+	0.922070i	$0.373503\pi$
$588$	0	0
$589$	1.09740e7	1.30339
$590$	5.27340e6	0.623678
$591$	0	0
$592$	5.51084e6	0.646269
$593$	2.34605e6	0.273969	0.136984	−	0.990573i	$-0.456259\pi$
$593$	2.34605e6	0.273969	0.136984	+	0.990573i	$0.456259\pi$
$594$	0	0
$595$	9.31333e6	1.07848
$596$	−869862.	−0.100308
$597$	0	0
$598$	2.21760e6	0.253589
$599$	1.34959e7	1.53686	0.768432	−	0.639931i	$-0.221037\pi$
$599$	1.34959e7	1.53686	0.768432	+	0.639931i	$0.221037\pi$
$600$	0	0
$601$	3.87849e6	0.438002	0.219001	−	0.975725i	$-0.429720\pi$
$601$	3.87849e6	0.438002	0.219001	+	0.975725i	$0.429720\pi$
$602$	1.32790e6	0.149339
$603$	0	0
$604$	3.12407e6	0.348441
$605$	1.48846e7	1.65329
$606$	0	0
$607$	−533488.	−0.0587696	−0.0293848	−	0.999568i	$-0.509355\pi$
$607$	−533488.	−0.0587696	−0.0293848	+	0.999568i	$0.509355\pi$
$608$	−4.30402e6	−0.472188
$609$	0	0
$610$	−2.14329e7	−2.33215
$611$	665280.	0.0720944
$612$	0	0
$613$	5.14610e6	0.553130	0.276565	−	0.960995i	$-0.410804\pi$
$613$	5.14610e6	0.553130	0.276565	+	0.960995i	$0.410804\pi$
$614$	−7.59458e6	−0.812986
$615$	0	0
$616$	496860.	0.0527573
$617$	2.37860e6	0.251541	0.125770	−	0.992059i	$-0.459860\pi$
$617$	2.37860e6	0.251541	0.125770	+	0.992059i	$0.459860\pi$
$618$	0	0
$619$	1.60023e7	1.67863	0.839317	−	0.543642i	$-0.182955\pi$
$619$	1.60023e7	1.67863	0.839317	+	0.543642i	$0.182955\pi$
$620$	4.16909e6	0.435574
$621$	0	0
$622$	1.06404e6	0.110276
$623$	−155918.	−0.0160944
$624$	0	0
$625$	5.00302e6	0.512309
$626$	9470.00	0.000965860 0
$627$	0	0
$628$	1.11822e6	0.113143
$629$	−1.48374e7	−1.49531
$630$	0	0
$631$	1.23459e7	1.23439	0.617193	−	0.786812i	$-0.288270\pi$
$631$	1.23459e7	1.23439	0.617193	+	0.786812i	$0.288270\pi$
$632$	1.89977e7	1.89194
$633$	0	0
$634$	−7.89489e6	−0.780051
$635$	1.92753e7	1.89699
$636$	0	0
$637$	−1.84877e6	−0.180524
$638$	−1.43468e6	−0.139541
$639$	0	0
$640$	9.65991e6	0.932230
$641$	3.43755e6	0.330449	0.165224	−	0.986256i	$-0.447165\pi$
$641$	3.43755e6	0.330449	0.165224	+	0.986256i	$0.447165\pi$
$642$	0	0
$643$	−1.62191e7	−1.54703	−0.773515	−	0.633778i	$-0.781503\pi$
$643$	−1.62191e7	−1.54703	−0.773515	+	0.633778i	$0.781503\pi$
$644$	197568.	0.0187716
$645$	0	0
$646$	−1.75105e7	−1.65089
$647$	−1.19929e7	−1.12632	−0.563160	−	0.826348i	$-0.690415\pi$
$647$	−1.19929e7	−1.12632	−0.563160	+	0.826348i	$0.690415\pi$
$648$	0	0
$649$	−583440.	−0.0543731
$650$	2.19873e7	2.04122
$651$	0	0
$652$	−1.73076e6	−0.159448
$653$	−1.58009e6	−0.145011	−0.0725053	−	0.997368i	$-0.523099\pi$
$653$	−1.58009e6	−0.145011	−0.0725053	+	0.997368i	$0.523099\pi$
$654$	0	0
$655$	6.85862e6	0.624645
$656$	−2.44976e6	−0.222262
$657$	0	0
$658$	−211680.	−0.0190597
$659$	−6.98358e6	−0.626419	−0.313209	−	0.949684i	$-0.601404\pi$
$659$	−6.98358e6	−0.626419	−0.313209	+	0.949684i	$0.601404\pi$
$660$	0	0
$661$	3.69602e6	0.329027	0.164513	−	0.986375i	$-0.447395\pi$
$661$	3.69602e6	0.329027	0.164513	+	0.986375i	$0.447395\pi$
$662$	1.69735e7	1.50532
$663$	0	0
$664$	1.58395e7	1.39418
$665$	7.97759e6	0.699548
$666$	0	0
$667$	−3.17837e6	−0.276624
$668$	−4.79142e6	−0.415454
$669$	0	0
$670$	656120.	0.0564672
$671$	2.37130e6	0.203320
$672$	0	0
$673$	1.84688e6	0.157182	0.0785908	−	0.996907i	$-0.474958\pi$
$673$	1.84688e6	0.157182	0.0785908	+	0.996907i	$0.474958\pi$
$674$	−1.01366e7	−0.859491
$675$	0	0
$676$	−1.55125e6	−0.130561
$677$	7.68501e6	0.644426	0.322213	−	0.946667i	$-0.395573\pi$
$677$	7.68501e6	0.644426	0.322213	+	0.946667i	$0.395573\pi$
$678$	0	0
$679$	−240786.	−0.0200427
$680$	−3.70633e7	−3.07377
$681$	0	0
$682$	1.64736e6	0.135621
$683$	−7.12180e6	−0.584168	−0.292084	−	0.956393i	$-0.594349\pi$
$683$	−7.12180e6	−0.584168	−0.292084	+	0.956393i	$0.594349\pi$
$684$	0	0
$685$	−8.85311e6	−0.720891
$686$	588245.	0.0477252
$687$	0	0
$688$	−4.07042e6	−0.327845
$689$	3.22014e6	0.258420
$690$	0	0
$691$	−3.23787e6	−0.257967	−0.128983	−	0.991647i	$-0.541171\pi$
$691$	−3.23787e6	−0.257967	−0.128983	+	0.991647i	$0.541171\pi$
$692$	−4.27332e6	−0.339234
$693$	0	0
$694$	1.74443e7	1.37485
$695$	4.49282e6	0.352823
$696$	0	0
$697$	6.59576e6	0.514260
$698$	−4.82783e6	−0.375071
$699$	0	0
$700$	1.95887e6	0.151099
$701$	7.39163e6	0.568127	0.284063	−	0.958805i	$-0.408317\pi$
$701$	7.39163e6	0.568127	0.284063	+	0.958805i	$0.408317\pi$
$702$	0	0
$703$	−1.27094e7	−0.969923
$704$	−1.89576e6	−0.144163
$705$	0	0
$706$	5.76965e6	0.435650
$707$	−8.15135e6	−0.613311
$708$	0	0
$709$	−5.33361e6	−0.398479	−0.199240	−	0.979951i	$-0.563847\pi$
$709$	−5.33361e6	−0.398479	−0.199240	+	0.979951i	$0.563847\pi$
$710$	−8.79840e6	−0.655025
$711$	0	0
$712$	620490.	0.0458706
$713$	3.64954e6	0.268852
$714$	0	0
$715$	−3.76376e6	−0.275332
$716$	4.63576e6	0.337939
$717$	0	0
$718$	8.05552e6	0.583153
$719$	1.14564e7	0.826468	0.413234	−	0.910625i	$-0.364399\pi$
$719$	1.14564e7	0.826468	0.413234	+	0.910625i	$0.364399\pi$
$720$	0	0
$721$	−7.70084e6	−0.551696
$722$	−2.61862e6	−0.186952
$723$	0	0
$724$	−1.08241e6	−0.0767442
$725$	−3.15133e7	−2.22663
$726$	0	0
$727$	−2.49540e7	−1.75107	−0.875536	−	0.483153i	$-0.839492\pi$
$727$	−2.49540e7	−1.75107	−0.875536	+	0.483153i	$0.839492\pi$
$728$	7.35735e6	0.514509
$729$	0	0
$730$	2.17901e7	1.51340
$731$	1.09592e7	0.758555
$732$	0	0
$733$	−1.43398e7	−0.985789	−0.492894	−	0.870089i	$-0.664061\pi$
$733$	−1.43398e7	−0.985789	−0.492894	+	0.870089i	$0.664061\pi$
$734$	−1.83874e7	−1.25974
$735$	0	0
$736$	−1.43136e6	−0.0973990
$737$	−72592.0	−0.00492289
$738$	0	0
$739$	922932.	0.0621668	0.0310834	−	0.999517i	$-0.490104\pi$
$739$	922932.	0.0621668	0.0310834	+	0.999517i	$0.490104\pi$
$740$	−4.82840e6	−0.324134
$741$	0	0
$742$	−1.02459e6	−0.0683188
$743$	9.38995e6	0.624010	0.312005	−	0.950081i	$-0.398999\pi$
$743$	9.38995e6	0.624010	0.312005	+	0.950081i	$0.398999\pi$
$744$	0	0
$745$	−1.16810e7	−0.771062
$746$	−3.24883e6	−0.213737
$747$	0	0
$748$	736008.	0.0480982
$749$	−331436.	−0.0215871
$750$	0	0
$751$	−408032.	−0.0263994	−0.0131997	−	0.999913i	$-0.504202\pi$
$751$	−408032.	−0.0263994	−0.0131997	+	0.999913i	$0.504202\pi$
$752$	648864.	0.0418417
$753$	0	0
$754$	−2.12443e7	−1.36086
$755$	4.19518e7	2.67845
$756$	0	0
$757$	2.59605e7	1.64654	0.823271	−	0.567649i	$-0.192147\pi$
$757$	2.59605e7	1.64654	0.823271	+	0.567649i	$0.192147\pi$
$758$	−1.60350e6	−0.101367
$759$	0	0
$760$	−3.17476e7	−1.99378
$761$	−1.83554e7	−1.14895	−0.574477	−	0.818521i	$-0.694794\pi$
$761$	−1.83554e7	−1.14895	−0.574477	+	0.818521i	$0.694794\pi$
$762$	0	0
$763$	−8.74150e6	−0.543594
$764$	3.40833e6	0.211255
$765$	0	0
$766$	−1.18094e7	−0.727206
$767$	−8.63940e6	−0.530268
$768$	0	0
$769$	−747166.	−0.0455618	−0.0227809	−	0.999740i	$-0.507252\pi$
$769$	−747166.	−0.0455618	−0.0227809	+	0.999740i	$0.507252\pi$
$770$	1.19756e6	0.0727898
$771$	0	0
$772$	−4.34382e6	−0.262318
$773$	−2.02692e7	−1.22008	−0.610038	−	0.792372i	$-0.708846\pi$
$773$	−2.02692e7	−1.22008	−0.610038	+	0.792372i	$0.708846\pi$
$774$	0	0
$775$	3.61849e7	2.16408
$776$	958230.	0.0571236
$777$	0	0
$778$	−1.76695e7	−1.04659
$779$	5.64978e6	0.333571
$780$	0	0
$781$	973440.	0.0571060
$782$	−5.82336e6	−0.340531
$783$	0	0
$784$	−1.80315e6	−0.104771
$785$	1.50161e7	0.869729
$786$	0	0
$787$	−4.69982e6	−0.270486	−0.135243	−	0.990812i	$-0.543181\pi$
$787$	−4.69982e6	−0.270486	−0.135243	+	0.990812i	$0.543181\pi$
$788$	−1.65599e6	−0.0950041
$789$	0	0
$790$	4.57893e7	2.61033
$791$	−2.21157e6	−0.125678
$792$	0	0
$793$	3.51135e7	1.98286
$794$	−2.02406e7	−1.13939
$795$	0	0
$796$	−574728.	−0.0321499
$797$	−584710.	−0.0326058	−0.0163029	−	0.999867i	$-0.505190\pi$
$797$	−584710.	−0.0326058	−0.0163029	+	0.999867i	$0.505190\pi$
$798$	0	0
$799$	−1.74701e6	−0.0968117
$800$	−1.41918e7	−0.783996
$801$	0	0
$802$	1.03822e7	0.569975
$803$	−2.41082e6	−0.131940
$804$	0	0
$805$	2.65306e6	0.144297
$806$	2.43936e7	1.32263
$807$	0	0
$808$	3.24390e7	1.74799
$809$	1.64013e7	0.881061	0.440531	−	0.897738i	$-0.354790\pi$
$809$	1.64013e7	0.881061	0.440531	+	0.897738i	$0.354790\pi$
$810$	0	0
$811$	−304948.	−0.0162807	−0.00814036	−	0.999967i	$-0.502591\pi$
$811$	−304948.	−0.0162807	−0.00814036	+	0.999967i	$0.502591\pi$
$812$	−1.89267e6	−0.100736
$813$	0	0
$814$	−1.90788e6	−0.100923
$815$	−2.32417e7	−1.22567
$816$	0	0
$817$	9.38744e6	0.492031
$818$	−1.28716e7	−0.672587
$819$	0	0
$820$	2.14640e6	0.111474
$821$	−3.43428e7	−1.77819	−0.889095	−	0.457722i	$-0.848665\pi$
$821$	−3.43428e7	−1.77819	−0.889095	+	0.457722i	$0.848665\pi$
$822$	0	0
$823$	1.56684e7	0.806351	0.403176	−	0.915123i	$-0.367906\pi$
$823$	1.56684e7	0.806351	0.403176	+	0.915123i	$0.367906\pi$
$824$	3.06462e7	1.57238
$825$	0	0
$826$	2.74890e6	0.140187
$827$	2.96886e7	1.50948	0.754738	−	0.656026i	$-0.227764\pi$
$827$	2.96886e7	1.50948	0.754738	+	0.656026i	$0.227764\pi$
$828$	0	0
$829$	−2.30708e7	−1.16594	−0.582970	−	0.812494i	$-0.698110\pi$
$829$	−2.30708e7	−1.16594	−0.582970	+	0.812494i	$0.698110\pi$
$830$	3.81772e7	1.92357
$831$	0	0
$832$	−2.80719e7	−1.40593
$833$	4.85482e6	0.242416
$834$	0	0
$835$	−6.43419e7	−3.19358
$836$	630448.	0.0311985
$837$	0	0
$838$	4.24074e6	0.208608
$839$	−2.32642e7	−1.14100	−0.570498	−	0.821299i	$-0.693250\pi$
$839$	−2.32642e7	−1.14100	−0.570498	+	0.821299i	$0.693250\pi$
$840$	0	0
$841$	9.93718e6	0.484477
$842$	−7.18411e6	−0.349215
$843$	0	0
$844$	−699244.	−0.0337888
$845$	−2.08311e7	−1.00362
$846$	0	0
$847$	7.75900e6	0.371619
$848$	3.14068e6	0.149980
$849$	0	0
$850$	−5.77382e7	−2.74104
$851$	−4.22669e6	−0.200067
$852$	0	0
$853$	1.91515e7	0.901219	0.450610	−	0.892721i	$-0.351207\pi$
$853$	1.91515e7	0.901219	0.450610	+	0.892721i	$0.351207\pi$
$854$	−1.11725e7	−0.524210
$855$	0	0
$856$	1.31898e6	0.0615253
$857$	5.34683e6	0.248682	0.124341	−	0.992240i	$-0.460318\pi$
$857$	5.34683e6	0.248682	0.124341	+	0.992240i	$0.460318\pi$
$858$	0	0
$859$	3.95858e7	1.83045	0.915223	−	0.402948i	$-0.132014\pi$
$859$	3.95858e7	1.83045	0.915223	+	0.402948i	$0.132014\pi$
$860$	3.56636e6	0.164429
$861$	0	0
$862$	1.17719e7	0.539607
$863$	2.50284e7	1.14395	0.571973	−	0.820272i	$-0.306178\pi$
$863$	2.50284e7	1.14395	0.571973	+	0.820272i	$0.306178\pi$
$864$	0	0
$865$	−5.73846e7	−2.60768
$866$	1.89404e7	0.858211
$867$	0	0
$868$	2.17325e6	0.0979062
$869$	−5.06605e6	−0.227573
$870$	0	0
$871$	−1.07492e6	−0.0480099
$872$	3.47876e7	1.54929
$873$	0	0
$874$	−4.98816e6	−0.220883
$875$	1.19111e7	0.525935
$876$	0	0
$877$	−5.02589e6	−0.220655	−0.110328	−	0.993895i	$-0.535190\pi$
$877$	−5.02589e6	−0.220655	−0.110328	+	0.993895i	$0.535190\pi$
$878$	1.82161e7	0.797480
$879$	0	0
$880$	−3.67089e6	−0.159795
$881$	2.60490e7	1.13071	0.565356	−	0.824847i	$-0.308739\pi$
$881$	2.60490e7	1.13071	0.565356	+	0.824847i	$0.308739\pi$
$882$	0	0
$883$	−6.82462e6	−0.294562	−0.147281	−	0.989095i	$-0.547052\pi$
$883$	−6.82462e6	−0.294562	−0.147281	+	0.989095i	$0.547052\pi$
$884$	1.08986e7	0.469072
$885$	0	0
$886$	1.24195e7	0.531519
$887$	−2.33835e7	−0.997931	−0.498965	−	0.866622i	$-0.666287\pi$
$887$	−2.33835e7	−0.997931	−0.498965	+	0.866622i	$0.666287\pi$
$888$	0	0
$889$	1.00477e7	0.426397
$890$	1.49554e6	0.0632882
$891$	0	0
$892$	1.30693e6	0.0549971
$893$	−1.49645e6	−0.0627961
$894$	0	0
$895$	6.22517e7	2.59773
$896$	5.03548e6	0.209542
$897$	0	0
$898$	−1.31589e7	−0.544537
$899$	−3.49620e7	−1.44277
$900$	0	0
$901$	−8.45600e6	−0.347019
$902$	848120.	0.0347089
$903$	0	0
$904$	8.80113e6	0.358193
$905$	−1.45352e7	−0.589930
$906$	0	0
$907$	3.95959e7	1.59820	0.799102	−	0.601196i	$-0.205309\pi$
$907$	3.95959e7	1.59820	0.799102	+	0.601196i	$0.205309\pi$
$908$	2.35460e6	0.0947771
$909$	0	0
$910$	1.77331e7	0.709874
$911$	4.67570e6	0.186660	0.0933300	−	0.995635i	$-0.470249\pi$
$911$	4.67570e6	0.186660	0.0933300	+	0.995635i	$0.470249\pi$
$912$	0	0
$913$	−4.22386e6	−0.167700
$914$	5.80651e6	0.229906
$915$	0	0
$916$	6.48420e6	0.255339
$917$	3.57524e6	0.140405
$918$	0	0
$919$	−4.92594e6	−0.192398	−0.0961990	−	0.995362i	$-0.530669\pi$
$919$	−4.92594e6	−0.192398	−0.0961990	+	0.995362i	$0.530669\pi$
$920$	−1.05581e7	−0.411259
$921$	0	0
$922$	1.40692e7	0.545058
$923$	1.44144e7	0.556919
$924$	0	0
$925$	−4.19073e7	−1.61041
$926$	−3.42150e7	−1.31126
$927$	0	0
$928$	1.37122e7	0.522683
$929$	3.23688e7	1.23052	0.615258	−	0.788326i	$-0.289052\pi$
$929$	3.23688e7	1.23052	0.615258	+	0.788326i	$0.289052\pi$
$930$	0	0
$931$	4.15853e6	0.157241
$932$	8.79980e6	0.331843
$933$	0	0
$934$	1.67157e7	0.626985
$935$	9.88354e6	0.369729
$936$	0	0
$937$	−3.32337e7	−1.23660	−0.618301	−	0.785941i	$-0.712179\pi$
$937$	−3.32337e7	−1.23660	−0.618301	+	0.785941i	$0.712179\pi$
$938$	342020.	0.0126924
$939$	0	0
$940$	−568512.	−0.0209855
$941$	2.66426e7	0.980852	0.490426	−	0.871483i	$-0.336841\pi$
$941$	2.66426e7	0.980852	0.490426	+	0.871483i	$0.336841\pi$
$942$	0	0
$943$	1.87891e6	0.0688061
$944$	−8.42622e6	−0.307753
$945$	0	0
$946$	1.40920e6	0.0511970
$947$	−3.14663e7	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$947$	−3.14663e7	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$948$	0	0
$949$	−3.56987e7	−1.28673
$950$	−4.94573e7	−1.77796
$951$	0	0
$952$	−1.93202e7	−0.690907
$953$	1.34516e7	0.479779	0.239890	−	0.970800i	$-0.422889\pi$
$953$	1.34516e7	0.479779	0.239890	+	0.970800i	$0.422889\pi$
$954$	0	0
$955$	4.57690e7	1.62391
$956$	−2.42911e6	−0.0859613
$957$	0	0
$958$	−2.14124e7	−0.753792
$959$	−4.61492e6	−0.162038
$960$	0	0
$961$	1.15157e7	0.402238
$962$	−2.82513e7	−0.984239
$963$	0	0
$964$	−694190.	−0.0240595
$965$	−5.83313e7	−2.01643
$966$	0	0
$967$	−2.84963e7	−0.979992	−0.489996	−	0.871725i	$-0.663002\pi$
$967$	−2.84963e7	−0.979992	−0.489996	+	0.871725i	$0.663002\pi$
$968$	−3.08777e7	−1.05915
$969$	0	0
$970$	2.30958e6	0.0788141
$971$	−1.81858e7	−0.618990	−0.309495	−	0.950901i	$-0.600160\pi$
$971$	−1.81858e7	−0.618990	−0.309495	+	0.950901i	$0.600160\pi$
$972$	0	0
$973$	2.34200e6	0.0793059
$974$	4.46588e7	1.50837
$975$	0	0
$976$	3.42471e7	1.15080
$977$	−3.20941e7	−1.07569	−0.537847	−	0.843042i	$-0.680762\pi$
$977$	−3.20941e7	−1.07569	−0.537847	+	0.843042i	$0.680762\pi$
$978$	0	0
$979$	−165464.	−0.00551756
$980$	1.57986e6	0.0525476
$981$	0	0
$982$	1.37653e7	0.455519
$983$	−1.56154e7	−0.515429	−0.257715	−	0.966221i	$-0.582969\pi$
$983$	−1.56154e7	−0.515429	−0.257715	+	0.966221i	$0.582969\pi$
$984$	0	0
$985$	−2.22376e7	−0.730293
$986$	5.57870e7	1.82743
$987$	0	0
$988$	9.33548e6	0.304260
$989$	3.12192e6	0.101492
$990$	0	0
$991$	4.84499e7	1.56714	0.783572	−	0.621301i	$-0.213395\pi$
$991$	4.84499e7	1.56714	0.783572	+	0.621301i	$0.213395\pi$
$992$	−1.57450e7	−0.507998
$993$	0	0
$994$	−4.58640e6	−0.147233
$995$	−7.71778e6	−0.247135
$996$	0	0
$997$	−4.54336e7	−1.44757	−0.723784	−	0.690027i	$-0.757599\pi$
$997$	−4.54336e7	−1.44757	−0.723784	+	0.690027i	$0.757599\pi$
$998$	−2.40204e7	−0.763404
$999$	0	0

Twists

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

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