LMFDB - Embedded newform 63.6.a.e.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 7)
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)

$f(q)$

$=$

$q+10.0000 q^{2} +68.0000 q^{4} +56.0000 q^{5} -49.0000 q^{7} +360.000 q^{8} +O(q^{10})$

\(q+10.0000 q^{2} +68.0000 q^{4} +56.0000 q^{5} -49.0000 q^{7} +360.000 q^{8} +560.000 q^{10} -232.000 q^{11} -140.000 q^{13} -490.000 q^{14} +1424.00 q^{16} +1722.00 q^{17} -98.0000 q^{19} +3808.00 q^{20} -2320.00 q^{22} -1824.00 q^{23} +11.0000 q^{25} -1400.00 q^{26} -3332.00 q^{28} -3418.00 q^{29} -7644.00 q^{31} +2720.00 q^{32} +17220.0 q^{34} -2744.00 q^{35} -10398.0 q^{37} -980.000 q^{38} +20160.0 q^{40} +17962.0 q^{41} +10880.0 q^{43} -15776.0 q^{44} -18240.0 q^{46} -9324.00 q^{47} +2401.00 q^{49} +110.000 q^{50} -9520.00 q^{52} -2262.00 q^{53} -12992.0 q^{55} -17640.0 q^{56} -34180.0 q^{58} +2730.00 q^{59} +25648.0 q^{61} -76440.0 q^{62} -18368.0 q^{64} -7840.00 q^{65} -48404.0 q^{67} +117096. q^{68} -27440.0 q^{70} +58560.0 q^{71} +68082.0 q^{73} -103980. q^{74} -6664.00 q^{76} +11368.0 q^{77} +31784.0 q^{79} +79744.0 q^{80} +179620. q^{82} +20538.0 q^{83} +96432.0 q^{85} +108800. q^{86} -83520.0 q^{88} +50582.0 q^{89} +6860.00 q^{91} -124032. q^{92} -93240.0 q^{94} -5488.00 q^{95} -58506.0 q^{97} +24010.0 q^{98} +O(q^{100})\)

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 $

$2$	10.0000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	10.0000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	0	0
$3$	0	0
$4$	68.0000	2.12500
$5$	56.0000	1.00176	0.500879	−	0.865517i	$-0.333010\pi$
$5$	56.0000	1.00176	0.500879	+	0.865517i	$0.333010\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	360.000	1.98874
$9$	0	0
$10$	560.000	1.77088
$11$	−232.000	−0.578104	−0.289052	−	0.957313i	$-0.593340\pi$
$11$	−232.000	−0.578104	−0.289052	+	0.957313i	$0.593340\pi$
$12$	0	0
$13$	−140.000	−0.229757	−0.114879	−	0.993380i	$-0.536648\pi$
$13$	−140.000	−0.229757	−0.114879	+	0.993380i	$0.536648\pi$
$14$	−490.000	−0.668153
$15$	0	0
$16$	1424.00	1.39062
$17$	1722.00	1.44514	0.722572	−	0.691296i	$-0.242960\pi$
$17$	1722.00	1.44514	0.722572	+	0.691296i	$0.242960\pi$
$18$	0	0
$19$	−98.0000	−0.0622791	−0.0311395	−	0.999515i	$-0.509914\pi$
$19$	−98.0000	−0.0622791	−0.0311395	+	0.999515i	$0.509914\pi$
$20$	3808.00	2.12874
$21$	0	0
$22$	−2320.00	−1.02195
$23$	−1824.00	−0.718961	−0.359480	−	0.933153i	$-0.617046\pi$
$23$	−1824.00	−0.718961	−0.359480	+	0.933153i	$0.617046\pi$
$24$	0	0
$25$	11.0000	0.00352000
$26$	−1400.00	−0.406158
$27$	0	0
$28$	−3332.00	−0.803175
$29$	−3418.00	−0.754705	−0.377352	−	0.926070i	$-0.623165\pi$
$29$	−3418.00	−0.754705	−0.377352	+	0.926070i	$0.623165\pi$
$30$	0	0
$31$	−7644.00	−1.42862	−0.714310	−	0.699830i	$-0.753259\pi$
$31$	−7644.00	−1.42862	−0.714310	+	0.699830i	$0.753259\pi$
$32$	2720.00	0.469563
$33$	0	0
$34$	17220.0	2.55468
$35$	−2744.00	−0.378629
$36$	0	0
$37$	−10398.0	−1.24866	−0.624332	−	0.781159i	$-0.714629\pi$
$37$	−10398.0	−1.24866	−0.624332	+	0.781159i	$0.714629\pi$
$38$	−980.000	−0.110095
$39$	0	0
$40$	20160.0	1.99223
$41$	17962.0	1.66876	0.834382	−	0.551186i	$-0.185825\pi$
$41$	17962.0	1.66876	0.834382	+	0.551186i	$0.185825\pi$
$42$	0	0
$43$	10880.0	0.897342	0.448671	−	0.893697i	$-0.351898\pi$
$43$	10880.0	0.897342	0.448671	+	0.893697i	$0.351898\pi$
$44$	−15776.0	−1.22847
$45$	0	0
$46$	−18240.0	−1.27096
$47$	−9324.00	−0.615684	−0.307842	−	0.951438i	$-0.599607\pi$
$47$	−9324.00	−0.615684	−0.307842	+	0.951438i	$0.599607\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	110.000	0.00622254
$51$	0	0
$52$	−9520.00	−0.488235
$53$	−2262.00	−0.110612	−0.0553061	−	0.998469i	$-0.517613\pi$
$53$	−2262.00	−0.110612	−0.0553061	+	0.998469i	$0.517613\pi$
$54$	0	0
$55$	−12992.0	−0.579121
$56$	−17640.0	−0.751672
$57$	0	0
$58$	−34180.0	−1.33414
$59$	2730.00	0.102102	0.0510508	−	0.998696i	$-0.483743\pi$
$59$	2730.00	0.102102	0.0510508	+	0.998696i	$0.483743\pi$
$60$	0	0
$61$	25648.0	0.882529	0.441264	−	0.897377i	$-0.354530\pi$
$61$	25648.0	0.882529	0.441264	+	0.897377i	$0.354530\pi$
$62$	−76440.0	−2.52547
$63$	0	0
$64$	−18368.0	−0.560547
$65$	−7840.00	−0.230161
$66$	0	0
$67$	−48404.0	−1.31733	−0.658664	−	0.752437i	$-0.728878\pi$
$67$	−48404.0	−1.31733	−0.658664	+	0.752437i	$0.728878\pi$
$68$	117096.	3.07093
$69$	0	0
$70$	−27440.0	−0.669328
$71$	58560.0	1.37865	0.689327	−	0.724450i	$-0.257906\pi$
$71$	58560.0	1.37865	0.689327	+	0.724450i	$0.257906\pi$
$72$	0	0
$73$	68082.0	1.49529	0.747645	−	0.664099i	$-0.231185\pi$
$73$	68082.0	1.49529	0.747645	+	0.664099i	$0.231185\pi$
$74$	−103980.	−2.20735
$75$	0	0
$76$	−6664.00	−0.132343
$77$	11368.0	0.218503
$78$	0	0
$79$	31784.0	0.572982	0.286491	−	0.958083i	$-0.407511\pi$
$79$	31784.0	0.572982	0.286491	+	0.958083i	$0.407511\pi$
$80$	79744.0	1.39307
$81$	0	0
$82$	179620.	2.94999
$83$	20538.0	0.327237	0.163619	−	0.986524i	$-0.447683\pi$
$83$	20538.0	0.327237	0.163619	+	0.986524i	$0.447683\pi$
$84$	0	0
$85$	96432.0	1.44768
$86$	108800.	1.58629
$87$	0	0
$88$	−83520.0	−1.14970
$89$	50582.0	0.676894	0.338447	−	0.940985i	$-0.390098\pi$
$89$	50582.0	0.676894	0.338447	+	0.940985i	$0.390098\pi$
$90$	0	0
$91$	6860.00	0.0868402
$92$	−124032.	−1.52779
$93$	0	0
$94$	−93240.0	−1.08839
$95$	−5488.00	−0.0623886
$96$	0	0
$97$	−58506.0	−0.631351	−0.315676	−	0.948867i	$-0.602231\pi$
$97$	−58506.0	−0.631351	−0.315676	+	0.948867i	$0.602231\pi$
$98$	24010.0	0.252538
$99$	0	0
$100$	748.000	0.00748000
$101$	−38696.0	−0.377453	−0.188726	−	0.982030i	$-0.560436\pi$
$101$	−38696.0	−0.377453	−0.188726	+	0.982030i	$0.560436\pi$
$102$	0	0
$103$	53060.0	0.492804	0.246402	−	0.969168i	$-0.420752\pi$
$103$	53060.0	0.492804	0.246402	+	0.969168i	$0.420752\pi$
$104$	−50400.0	−0.456927
$105$	0	0
$106$	−22620.0	−0.195537
$107$	146324.	1.23554	0.617769	−	0.786360i	$-0.288037\pi$
$107$	146324.	1.23554	0.617769	+	0.786360i	$0.288037\pi$
$108$	0	0
$109$	92898.0	0.748928	0.374464	−	0.927241i	$-0.377827\pi$
$109$	92898.0	0.748928	0.374464	+	0.927241i	$0.377827\pi$
$110$	−129920.	−1.02375
$111$	0	0
$112$	−69776.0	−0.525607
$113$	83354.0	0.614088	0.307044	−	0.951695i	$-0.400660\pi$
$113$	83354.0	0.614088	0.307044	+	0.951695i	$0.400660\pi$
$114$	0	0
$115$	−102144.	−0.720225
$116$	−232424.	−1.60375
$117$	0	0
$118$	27300.0	0.180492
$119$	−84378.0	−0.546213
$120$	0	0
$121$	−107227.	−0.665795
$122$	256480.	1.56011
$123$	0	0
$124$	−519792.	−3.03582
$125$	−174384.	−0.998232
$126$	0	0
$127$	60384.0	0.332210	0.166105	−	0.986108i	$-0.446881\pi$
$127$	60384.0	0.332210	0.166105	+	0.986108i	$0.446881\pi$
$128$	−270720.	−1.46048
$129$	0	0
$130$	−78400.0	−0.406872
$131$	61586.0	0.313548	0.156774	−	0.987635i	$-0.449891\pi$
$131$	61586.0	0.313548	0.156774	+	0.987635i	$0.449891\pi$
$132$	0	0
$133$	4802.00	0.0235393
$134$	−484040.	−2.32873
$135$	0	0
$136$	619920.	2.87401
$137$	204462.	0.930703	0.465352	−	0.885126i	$-0.345928\pi$
$137$	204462.	0.930703	0.465352	+	0.885126i	$0.345928\pi$
$138$	0	0
$139$	−35406.0	−0.155432	−0.0777159	−	0.996976i	$-0.524763\pi$
$139$	−35406.0	−0.155432	−0.0777159	+	0.996976i	$0.524763\pi$
$140$	−186592.	−0.804587
$141$	0	0
$142$	585600.	2.43714
$143$	32480.0	0.132824
$144$	0	0
$145$	−191408.	−0.756032
$146$	680820.	2.64332
$147$	0	0
$148$	−707064.	−2.65341
$149$	20226.0	0.0746353	0.0373177	−	0.999303i	$-0.488119\pi$
$149$	20226.0	0.0746353	0.0373177	+	0.999303i	$0.488119\pi$
$150$	0	0
$151$	70904.0	0.253063	0.126531	−	0.991963i	$-0.459616\pi$
$151$	70904.0	0.253063	0.126531	+	0.991963i	$0.459616\pi$
$152$	−35280.0	−0.123857
$153$	0	0
$154$	113680.	0.386262
$155$	−428064.	−1.43113
$156$	0	0
$157$	293524.	0.950374	0.475187	−	0.879885i	$-0.342380\pi$
$157$	293524.	0.950374	0.475187	+	0.879885i	$0.342380\pi$
$158$	317840.	1.01290
$159$	0	0
$160$	152320.	0.470389
$161$	89376.0	0.271742
$162$	0	0
$163$	13192.0	0.0388903	0.0194452	−	0.999811i	$-0.493810\pi$
$163$	13192.0	0.0388903	0.0194452	+	0.999811i	$0.493810\pi$
$164$	1.22142e6	3.54612
$165$	0	0
$166$	205380.	0.578479
$167$	−493612.	−1.36960	−0.684801	−	0.728730i	$-0.740111\pi$
$167$	−493612.	−1.36960	−0.684801	+	0.728730i	$0.740111\pi$
$168$	0	0
$169$	−351693.	−0.947212
$170$	964320.	2.55917
$171$	0	0
$172$	739840.	1.90685
$173$	−240716.	−0.611490	−0.305745	−	0.952113i	$-0.598906\pi$
$173$	−240716.	−0.611490	−0.305745	+	0.952113i	$0.598906\pi$
$174$	0	0
$175$	−539.000	−0.00133043
$176$	−330368.	−0.803926
$177$	0	0
$178$	505820.	1.19659
$179$	−294932.	−0.688001	−0.344001	−	0.938969i	$-0.611782\pi$
$179$	−294932.	−0.688001	−0.344001	+	0.938969i	$0.611782\pi$
$180$	0	0
$181$	−336980.	−0.764553	−0.382277	−	0.924048i	$-0.624860\pi$
$181$	−336980.	−0.764553	−0.382277	+	0.924048i	$0.624860\pi$
$182$	68600.0	0.153513
$183$	0	0
$184$	−656640.	−1.42982
$185$	−582288.	−1.25086
$186$	0	0
$187$	−399504.	−0.835444
$188$	−634032.	−1.30833
$189$	0	0
$190$	−54880.0	−0.110288
$191$	−358264.	−0.710591	−0.355296	−	0.934754i	$-0.615620\pi$
$191$	−358264.	−0.710591	−0.355296	+	0.934754i	$0.615620\pi$
$192$	0	0
$193$	−989554.	−1.91226	−0.956128	−	0.292948i	$-0.905364\pi$
$193$	−989554.	−1.91226	−0.956128	+	0.292948i	$0.905364\pi$
$194$	−585060.	−1.11608
$195$	0	0
$196$	163268.	0.303571
$197$	990050.	1.81757	0.908786	−	0.417263i	$-0.137011\pi$
$197$	990050.	1.81757	0.908786	+	0.417263i	$0.137011\pi$
$198$	0	0
$199$	−840756.	−1.50500	−0.752501	−	0.658591i	$-0.771153\pi$
$199$	−840756.	−1.50500	−0.752501	+	0.658591i	$0.771153\pi$
$200$	3960.00	0.00700036
$201$	0	0
$202$	−386960.	−0.667249
$203$	167482.	0.285252
$204$	0	0
$205$	1.00587e6	1.67170
$206$	530600.	0.871163
$207$	0	0
$208$	−199360.	−0.319506
$209$	22736.0	0.0360038
$210$	0	0
$211$	1.15073e6	1.77938	0.889689	−	0.456568i	$-0.150921\pi$
$211$	1.15073e6	1.77938	0.889689	+	0.456568i	$0.150921\pi$
$212$	−153816.	−0.235051
$213$	0	0
$214$	1.46324e6	2.18414
$215$	609280.	0.898919
$216$	0	0
$217$	374556.	0.539967
$218$	928980.	1.32393
$219$	0	0
$220$	−883456.	−1.23063
$221$	−241080.	−0.332032
$222$	0	0
$223$	−824264.	−1.10995	−0.554976	−	0.831866i	$-0.687273\pi$
$223$	−824264.	−1.10995	−0.554976	+	0.831866i	$0.687273\pi$
$224$	−133280.	−0.177478
$225$	0	0
$226$	833540.	1.08556
$227$	−74382.0	−0.0958083	−0.0479042	−	0.998852i	$-0.515254\pi$
$227$	−74382.0	−0.0958083	−0.0479042	+	0.998852i	$0.515254\pi$
$228$	0	0
$229$	1.13196e6	1.42640	0.713199	−	0.700961i	$-0.247245\pi$
$229$	1.13196e6	1.42640	0.713199	+	0.700961i	$0.247245\pi$
$230$	−1.02144e6	−1.27319
$231$	0	0
$232$	−1.23048e6	−1.50091
$233$	198726.	0.239809	0.119904	−	0.992785i	$-0.461741\pi$
$233$	198726.	0.239809	0.119904	+	0.992785i	$0.461741\pi$
$234$	0	0
$235$	−522144.	−0.616766
$236$	185640.	0.216966
$237$	0	0
$238$	−843780.	−0.965577
$239$	−482904.	−0.546847	−0.273424	−	0.961894i	$-0.588156\pi$
$239$	−482904.	−0.546847	−0.273424	+	0.961894i	$0.588156\pi$
$240$	0	0
$241$	805910.	0.893807	0.446904	−	0.894582i	$-0.352527\pi$
$241$	805910.	0.893807	0.446904	+	0.894582i	$0.352527\pi$
$242$	−1.07227e6	−1.17697
$243$	0	0
$244$	1.74406e6	1.87537
$245$	134456.	0.143108
$246$	0	0
$247$	13720.0	0.0143091
$248$	−2.75184e6	−2.84115
$249$	0	0
$250$	−1.74384e6	−1.76464
$251$	−430738.	−0.431548	−0.215774	−	0.976443i	$-0.569227\pi$
$251$	−430738.	−0.431548	−0.215774	+	0.976443i	$0.569227\pi$
$252$	0	0
$253$	423168.	0.415634
$254$	603840.	0.587270
$255$	0	0
$256$	−2.11942e6	−2.02124
$257$	1.17691e6	1.11150	0.555751	−	0.831349i	$-0.312431\pi$
$257$	1.17691e6	1.11150	0.555751	+	0.831349i	$0.312431\pi$
$258$	0	0
$259$	509502.	0.471951
$260$	−533120.	−0.489093
$261$	0	0
$262$	615860.	0.554279
$263$	−1.29098e6	−1.15088	−0.575438	−	0.817845i	$-0.695169\pi$
$263$	−1.29098e6	−1.15088	−0.575438	+	0.817845i	$0.695169\pi$
$264$	0	0
$265$	−126672.	−0.110807
$266$	48020.0	0.0416119
$267$	0	0
$268$	−3.29147e6	−2.79932
$269$	1.27756e6	1.07646	0.538232	−	0.842797i	$-0.319093\pi$
$269$	1.27756e6	1.07646	0.538232	+	0.842797i	$0.319093\pi$
$270$	0	0
$271$	1.65054e6	1.36522	0.682612	−	0.730781i	$-0.260844\pi$
$271$	1.65054e6	1.36522	0.682612	+	0.730781i	$0.260844\pi$
$272$	2.45213e6	2.00965
$273$	0	0
$274$	2.04462e6	1.64527
$275$	−2552.00	−0.00203493
$276$	0	0
$277$	−1.06409e6	−0.833257	−0.416628	−	0.909077i	$-0.636788\pi$
$277$	−1.06409e6	−0.833257	−0.416628	+	0.909077i	$0.636788\pi$
$278$	−354060.	−0.274767
$279$	0	0
$280$	−987840.	−0.752994
$281$	22342.0	0.0168794	0.00843969	−	0.999964i	$-0.497314\pi$
$281$	22342.0	0.0168794	0.00843969	+	0.999964i	$0.497314\pi$
$282$	0	0
$283$	−2.49574e6	−1.85239	−0.926196	−	0.377042i	$-0.876941\pi$
$283$	−2.49574e6	−1.85239	−0.926196	+	0.377042i	$0.876941\pi$
$284$	3.98208e6	2.92964
$285$	0	0
$286$	324800.	0.234802
$287$	−880138.	−0.630734
$288$	0	0
$289$	1.54543e6	1.08844
$290$	−1.91408e6	−1.33649
$291$	0	0
$292$	4.62958e6	3.17749
$293$	1.93178e6	1.31458	0.657291	−	0.753637i	$-0.271702\pi$
$293$	1.93178e6	1.31458	0.657291	+	0.753637i	$0.271702\pi$
$294$	0	0
$295$	152880.	0.102281
$296$	−3.74328e6	−2.48326
$297$	0	0
$298$	202260.	0.131938
$299$	255360.	0.165187
$300$	0	0
$301$	−533120.	−0.339163
$302$	709040.	0.447356
$303$	0	0
$304$	−139552.	−0.0866068
$305$	1.43629e6	0.884081
$306$	0	0
$307$	−459074.	−0.277995	−0.138997	−	0.990293i	$-0.544388\pi$
$307$	−459074.	−0.277995	−0.138997	+	0.990293i	$0.544388\pi$
$308$	773024.	0.464319
$309$	0	0
$310$	−4.28064e6	−2.52991
$311$	−667128.	−0.391118	−0.195559	−	0.980692i	$-0.562652\pi$
$311$	−667128.	−0.391118	−0.195559	+	0.980692i	$0.562652\pi$
$312$	0	0
$313$	−111034.	−0.0640612	−0.0320306	−	0.999487i	$-0.510197\pi$
$313$	−111034.	−0.0640612	−0.0320306	+	0.999487i	$0.510197\pi$
$314$	2.93524e6	1.68004
$315$	0	0
$316$	2.16131e6	1.21759
$317$	68778.0	0.0384416	0.0192208	−	0.999815i	$-0.493881\pi$
$317$	68778.0	0.0384416	0.0192208	+	0.999815i	$0.493881\pi$
$318$	0	0
$319$	792976.	0.436298
$320$	−1.02861e6	−0.561533
$321$	0	0
$322$	893760.	0.480376
$323$	−168756.	−0.0900022
$324$	0	0
$325$	−1540.00	−0.000808746 0
$326$	131920.	0.0687490
$327$	0	0
$328$	6.46632e6	3.31874
$329$	456876.	0.232707
$330$	0	0
$331$	−564448.	−0.283174	−0.141587	−	0.989926i	$-0.545221\pi$
$331$	−564448.	−0.283174	−0.141587	+	0.989926i	$0.545221\pi$
$332$	1.39658e6	0.695379
$333$	0	0
$334$	−4.93612e6	−2.42114
$335$	−2.71062e6	−1.31965
$336$	0	0
$337$	2.07729e6	0.996376	0.498188	−	0.867069i	$-0.333999\pi$
$337$	2.07729e6	0.996376	0.498188	+	0.867069i	$0.333999\pi$
$338$	−3.51693e6	−1.67445
$339$	0	0
$340$	6.55738e6	3.07633
$341$	1.77341e6	0.825891
$342$	0	0
$343$	−117649.	−0.0539949
$344$	3.91680e6	1.78458
$345$	0	0
$346$	−2.40716e6	−1.08097
$347$	53248.0	0.0237399	0.0118700	−	0.999930i	$-0.496222\pi$
$347$	53248.0	0.0237399	0.0118700	+	0.999930i	$0.496222\pi$
$348$	0	0
$349$	−2.27200e6	−0.998494	−0.499247	−	0.866460i	$-0.666390\pi$
$349$	−2.27200e6	−0.998494	−0.499247	+	0.866460i	$0.666390\pi$
$350$	−5390.00	−0.00235190
$351$	0	0
$352$	−631040.	−0.271456
$353$	−4.00645e6	−1.71129	−0.855644	−	0.517565i	$-0.826838\pi$
$353$	−4.00645e6	−1.71129	−0.855644	+	0.517565i	$0.826838\pi$
$354$	0	0
$355$	3.27936e6	1.38108
$356$	3.43958e6	1.43840
$357$	0	0
$358$	−2.94932e6	−1.21623
$359$	−73784.0	−0.0302152	−0.0151076	−	0.999886i	$-0.504809\pi$
$359$	−73784.0	−0.0302152	−0.0151076	+	0.999886i	$0.504809\pi$
$360$	0	0
$361$	−2.46650e6	−0.996121
$362$	−3.36980e6	−1.35155
$363$	0	0
$364$	466480.	0.184535
$365$	3.81259e6	1.49792
$366$	0	0
$367$	1.40431e6	0.544250	0.272125	−	0.962262i	$-0.412274\pi$
$367$	1.40431e6	0.544250	0.272125	+	0.962262i	$0.412274\pi$
$368$	−2.59738e6	−0.999805
$369$	0	0
$370$	−5.82288e6	−2.21123
$371$	110838.	0.0418075
$372$	0	0
$373$	−1.60323e6	−0.596657	−0.298329	−	0.954463i	$-0.596429\pi$
$373$	−1.60323e6	−0.596657	−0.298329	+	0.954463i	$0.596429\pi$
$374$	−3.99504e6	−1.47687
$375$	0	0
$376$	−3.35664e6	−1.22443
$377$	478520.	0.173399
$378$	0	0
$379$	−4.77012e6	−1.70581	−0.852906	−	0.522064i	$-0.825162\pi$
$379$	−4.77012e6	−1.70581	−0.852906	+	0.522064i	$0.825162\pi$
$380$	−373184.	−0.132576
$381$	0	0
$382$	−3.58264e6	−1.25616
$383$	2.23079e6	0.777072	0.388536	−	0.921434i	$-0.372981\pi$
$383$	2.23079e6	0.777072	0.388536	+	0.921434i	$0.372981\pi$
$384$	0	0
$385$	636608.	0.218887
$386$	−9.89554e6	−3.38042
$387$	0	0
$388$	−3.97841e6	−1.34162
$389$	−4.84024e6	−1.62178	−0.810892	−	0.585196i	$-0.801018\pi$
$389$	−4.84024e6	−1.62178	−0.810892	+	0.585196i	$0.801018\pi$
$390$	0	0
$391$	−3.14093e6	−1.03900
$392$	864360.	0.284105
$393$	0	0
$394$	9.90050e6	3.21304
$395$	1.77990e6	0.573989
$396$	0	0
$397$	995820.	0.317106	0.158553	−	0.987350i	$-0.449317\pi$
$397$	995820.	0.317106	0.158553	+	0.987350i	$0.449317\pi$
$398$	−8.40756e6	−2.66049
$399$	0	0
$400$	15664.0	0.00489500
$401$	3.31605e6	1.02982	0.514909	−	0.857245i	$-0.327826\pi$
$401$	3.31605e6	1.02982	0.514909	+	0.857245i	$0.327826\pi$
$402$	0	0
$403$	1.07016e6	0.328236
$404$	−2.63133e6	−0.802087
$405$	0	0
$406$	1.67482e6	0.504258
$407$	2.41234e6	0.721858
$408$	0	0
$409$	3.07273e6	0.908274	0.454137	−	0.890932i	$-0.349948\pi$
$409$	3.07273e6	0.908274	0.454137	+	0.890932i	$0.349948\pi$
$410$	1.00587e7	2.95517
$411$	0	0
$412$	3.60808e6	1.04721
$413$	−133770.	−0.0385908
$414$	0	0
$415$	1.15013e6	0.327813
$416$	−380800.	−0.107886
$417$	0	0
$418$	227360.	0.0636463
$419$	−2.81438e6	−0.783154	−0.391577	−	0.920145i	$-0.628070\pi$
$419$	−2.81438e6	−0.783154	−0.391577	+	0.920145i	$0.628070\pi$
$420$	0	0
$421$	3.05802e6	0.840883	0.420441	−	0.907320i	$-0.361875\pi$
$421$	3.05802e6	0.840883	0.420441	+	0.907320i	$0.361875\pi$
$422$	1.15073e7	3.14552
$423$	0	0
$424$	−814320.	−0.219979
$425$	18942.0	0.00508690
$426$	0	0
$427$	−1.25675e6	−0.333565
$428$	9.95003e6	2.62552
$429$	0	0
$430$	6.09280e6	1.58908
$431$	−1.93750e6	−0.502398	−0.251199	−	0.967936i	$-0.580825\pi$
$431$	−1.93750e6	−0.502398	−0.251199	+	0.967936i	$0.580825\pi$
$432$	0	0
$433$	3.94790e6	1.01192	0.505961	−	0.862557i	$-0.331138\pi$
$433$	3.94790e6	1.01192	0.505961	+	0.862557i	$0.331138\pi$
$434$	3.74556e6	0.954536
$435$	0	0
$436$	6.31706e6	1.59147
$437$	178752.	0.0447762
$438$	0	0
$439$	−7.41770e6	−1.83700	−0.918498	−	0.395426i	$-0.870597\pi$
$439$	−7.41770e6	−1.83700	−0.918498	+	0.395426i	$0.870597\pi$
$440$	−4.67712e6	−1.15172
$441$	0	0
$442$	−2.41080e6	−0.586956
$443$	−1.40269e6	−0.339589	−0.169794	−	0.985480i	$-0.554310\pi$
$443$	−1.40269e6	−0.339589	−0.169794	+	0.985480i	$0.554310\pi$
$444$	0	0
$445$	2.83259e6	0.678085
$446$	−8.24264e6	−1.96214
$447$	0	0
$448$	900032.	0.211867
$449$	590574.	0.138248	0.0691239	−	0.997608i	$-0.477980\pi$
$449$	590574.	0.138248	0.0691239	+	0.997608i	$0.477980\pi$
$450$	0	0
$451$	−4.16718e6	−0.964720
$452$	5.66807e6	1.30494
$453$	0	0
$454$	−743820.	−0.169367
$455$	384160.	0.0869929
$456$	0	0
$457$	−2.90484e6	−0.650627	−0.325313	−	0.945606i	$-0.605470\pi$
$457$	−2.90484e6	−0.650627	−0.325313	+	0.945606i	$0.605470\pi$
$458$	1.13196e7	2.52154
$459$	0	0
$460$	−6.94579e6	−1.53048
$461$	922684.	0.202209	0.101105	−	0.994876i	$-0.467762\pi$
$461$	922684.	0.202209	0.101105	+	0.994876i	$0.467762\pi$
$462$	0	0
$463$	7.18235e6	1.55709	0.778546	−	0.627588i	$-0.215958\pi$
$463$	7.18235e6	1.55709	0.778546	+	0.627588i	$0.215958\pi$
$464$	−4.86723e6	−1.04951
$465$	0	0
$466$	1.98726e6	0.423926
$467$	612570.	0.129976	0.0649881	−	0.997886i	$-0.479299\pi$
$467$	612570.	0.129976	0.0649881	+	0.997886i	$0.479299\pi$
$468$	0	0
$469$	2.37180e6	0.497904
$470$	−5.22144e6	−1.09030
$471$	0	0
$472$	982800.	0.203053
$473$	−2.52416e6	−0.518757
$474$	0	0
$475$	−1078.00	−0.000219222 0
$476$	−5.73770e6	−1.16070
$477$	0	0
$478$	−4.82904e6	−0.966699
$479$	−2.60330e6	−0.518424	−0.259212	−	0.965820i	$-0.583463\pi$
$479$	−2.60330e6	−0.518424	−0.259212	+	0.965820i	$0.583463\pi$
$480$	0	0
$481$	1.45572e6	0.286890
$482$	8.05910e6	1.58004
$483$	0	0
$484$	−7.29144e6	−1.41482
$485$	−3.27634e6	−0.632461
$486$	0	0
$487$	5.46309e6	1.04380	0.521898	−	0.853008i	$-0.325224\pi$
$487$	5.46309e6	1.04380	0.521898	+	0.853008i	$0.325224\pi$
$488$	9.23328e6	1.75512
$489$	0	0
$490$	1.34456e6	0.252982
$491$	−1.64090e6	−0.307170	−0.153585	−	0.988135i	$-0.549082\pi$
$491$	−1.64090e6	−0.307170	−0.153585	+	0.988135i	$0.549082\pi$
$492$	0	0
$493$	−5.88580e6	−1.09066
$494$	137200.	0.0252951
$495$	0	0
$496$	−1.08851e7	−1.98667
$497$	−2.86944e6	−0.521082
$498$	0	0
$499$	2.99796e6	0.538983	0.269491	−	0.963003i	$-0.413144\pi$
$499$	2.99796e6	0.538983	0.269491	+	0.963003i	$0.413144\pi$
$500$	−1.18581e7	−2.12124
$501$	0	0
$502$	−4.30738e6	−0.762876
$503$	6.89405e6	1.21494	0.607469	−	0.794343i	$-0.292185\pi$
$503$	6.89405e6	1.21494	0.607469	+	0.794343i	$0.292185\pi$
$504$	0	0
$505$	−2.16698e6	−0.378117
$506$	4.23168e6	0.734745
$507$	0	0
$508$	4.10611e6	0.705946
$509$	−2.30476e6	−0.394305	−0.197152	−	0.980373i	$-0.563169\pi$
$509$	−2.30476e6	−0.394305	−0.197152	+	0.980373i	$0.563169\pi$
$510$	0	0
$511$	−3.33602e6	−0.565166
$512$	−1.25312e7	−2.11260
$513$	0	0
$514$	1.17691e7	1.96488
$515$	2.97136e6	0.493671
$516$	0	0
$517$	2.16317e6	0.355929
$518$	5.09502e6	0.834299
$519$	0	0
$520$	−2.82240e6	−0.457731
$521$	1.20960e7	1.95231	0.976155	−	0.217073i	$-0.0696509\pi$
$521$	1.20960e7	1.95231	0.976155	+	0.217073i	$0.0696509\pi$
$522$	0	0
$523$	5.48443e6	0.876753	0.438377	−	0.898791i	$-0.355554\pi$
$523$	5.48443e6	0.876753	0.438377	+	0.898791i	$0.355554\pi$
$524$	4.18785e6	0.666289
$525$	0	0
$526$	−1.29098e7	−2.03448
$527$	−1.31630e7	−2.06456
$528$	0	0
$529$	−3.10937e6	−0.483095
$530$	−1.26672e6	−0.195880
$531$	0	0
$532$	326536.	0.0500210
$533$	−2.51468e6	−0.383411
$534$	0	0
$535$	8.19414e6	1.23771
$536$	−1.74254e7	−2.61982
$537$	0	0
$538$	1.27756e7	1.90294
$539$	−557032.	−0.0825863
$540$	0	0
$541$	−6.71799e6	−0.986839	−0.493420	−	0.869791i	$-0.664253\pi$
$541$	−6.71799e6	−0.986839	−0.493420	+	0.869791i	$0.664253\pi$
$542$	1.65054e7	2.41340
$543$	0	0
$544$	4.68384e6	0.678586
$545$	5.20229e6	0.750245
$546$	0	0
$547$	−5.00235e6	−0.714835	−0.357418	−	0.933945i	$-0.616343\pi$
$547$	−5.00235e6	−0.714835	−0.357418	+	0.933945i	$0.616343\pi$
$548$	1.39034e7	1.97774
$549$	0	0
$550$	−25520.0	−0.00359728
$551$	334964.	0.0470023
$552$	0	0
$553$	−1.55742e6	−0.216567
$554$	−1.06409e7	−1.47300
$555$	0	0
$556$	−2.40761e6	−0.330293
$557$	−9.01961e6	−1.23183	−0.615913	−	0.787814i	$-0.711213\pi$
$557$	−9.01961e6	−1.23183	−0.615913	+	0.787814i	$0.711213\pi$
$558$	0	0
$559$	−1.52320e6	−0.206171
$560$	−3.90746e6	−0.526531
$561$	0	0
$562$	223420.	0.0298388
$563$	−1.24051e7	−1.64941	−0.824707	−	0.565561i	$-0.808660\pi$
$563$	−1.24051e7	−1.64941	−0.824707	+	0.565561i	$0.808660\pi$
$564$	0	0
$565$	4.66782e6	0.615167
$566$	−2.49574e7	−3.27460
$567$	0	0
$568$	2.10816e7	2.74178
$569$	−6.48804e6	−0.840103	−0.420052	−	0.907500i	$-0.637988\pi$
$569$	−6.48804e6	−0.840103	−0.420052	+	0.907500i	$0.637988\pi$
$570$	0	0
$571$	−1.02285e7	−1.31287	−0.656435	−	0.754382i	$-0.727936\pi$
$571$	−1.02285e7	−1.31287	−0.656435	+	0.754382i	$0.727936\pi$
$572$	2.20864e6	0.282251
$573$	0	0
$574$	−8.80138e6	−1.11499
$575$	−20064.0	−0.00253074
$576$	0	0
$577$	2.65338e6	0.331787	0.165894	−	0.986144i	$-0.446949\pi$
$577$	2.65338e6	0.331787	0.165894	+	0.986144i	$0.446949\pi$
$578$	1.54543e7	1.92411
$579$	0	0
$580$	−1.30157e7	−1.60657
$581$	−1.00636e6	−0.123684
$582$	0	0
$583$	524784.	0.0639454
$584$	2.45095e7	2.97374
$585$	0	0
$586$	1.93178e7	2.32387
$587$	1.43044e7	1.71346	0.856729	−	0.515766i	$-0.172493\pi$
$587$	1.43044e7	1.71346	0.856729	+	0.515766i	$0.172493\pi$
$588$	0	0
$589$	749112.	0.0889731
$590$	1.52880e6	0.180809
$591$	0	0
$592$	−1.48068e7	−1.73642
$593$	1.00265e7	1.17088	0.585442	−	0.810714i	$-0.300921\pi$
$593$	1.00265e7	1.17088	0.585442	+	0.810714i	$0.300921\pi$
$594$	0	0
$595$	−4.72517e6	−0.547173
$596$	1.37537e6	0.158600
$597$	0	0
$598$	2.55360e6	0.292011
$599$	7.52292e6	0.856681	0.428341	−	0.903617i	$-0.359098\pi$
$599$	7.52292e6	0.856681	0.428341	+	0.903617i	$0.359098\pi$
$600$	0	0
$601$	3.38625e6	0.382413	0.191207	−	0.981550i	$-0.438760\pi$
$601$	3.38625e6	0.382413	0.191207	+	0.981550i	$0.438760\pi$
$602$	−5.33120e6	−0.599562
$603$	0	0
$604$	4.82147e6	0.537759
$605$	−6.00471e6	−0.666966
$606$	0	0
$607$	−6.90861e6	−0.761060	−0.380530	−	0.924769i	$-0.624258\pi$
$607$	−6.90861e6	−0.761060	−0.380530	+	0.924769i	$0.624258\pi$
$608$	−266560.	−0.0292439
$609$	0	0
$610$	1.43629e7	1.56285
$611$	1.30536e6	0.141458
$612$	0	0
$613$	−9.68896e6	−1.04142	−0.520710	−	0.853734i	$-0.674333\pi$
$613$	−9.68896e6	−1.04142	−0.520710	+	0.853734i	$0.674333\pi$
$614$	−4.59074e6	−0.491430
$615$	0	0
$616$	4.09248e6	0.434545
$617$	7.84742e6	0.829877	0.414939	−	0.909849i	$-0.363803\pi$
$617$	7.84742e6	0.829877	0.414939	+	0.909849i	$0.363803\pi$
$618$	0	0
$619$	−1.01972e7	−1.06968	−0.534840	−	0.844953i	$-0.679628\pi$
$619$	−1.01972e7	−1.06968	−0.534840	+	0.844953i	$0.679628\pi$
$620$	−2.91084e7	−3.04115
$621$	0	0
$622$	−6.67128e6	−0.691406
$623$	−2.47852e6	−0.255842
$624$	0	0
$625$	−9.79988e6	−1.00351
$626$	−1.11034e6	−0.113245
$627$	0	0
$628$	1.99596e7	2.01954
$629$	−1.79054e7	−1.80450
$630$	0	0
$631$	−8.36258e6	−0.836116	−0.418058	−	0.908420i	$-0.637289\pi$
$631$	−8.36258e6	−0.836116	−0.418058	+	0.908420i	$0.637289\pi$
$632$	1.14422e7	1.13951
$633$	0	0
$634$	687780.	0.0679558
$635$	3.38150e6	0.332794
$636$	0	0
$637$	−336140.	−0.0328225
$638$	7.92976e6	0.771273
$639$	0	0
$640$	−1.51603e7	−1.46305
$641$	−1.10283e6	−0.106014	−0.0530070	−	0.998594i	$-0.516881\pi$
$641$	−1.10283e6	−0.106014	−0.0530070	+	0.998594i	$0.516881\pi$
$642$	0	0
$643$	1.71354e7	1.63443	0.817217	−	0.576330i	$-0.195516\pi$
$643$	1.71354e7	1.63443	0.817217	+	0.576330i	$0.195516\pi$
$644$	6.07757e6	0.577451
$645$	0	0
$646$	−1.68756e6	−0.159103
$647$	54964.0	0.00516200	0.00258100	−	0.999997i	$-0.499178\pi$
$647$	54964.0	0.00516200	0.00258100	+	0.999997i	$0.499178\pi$
$648$	0	0
$649$	−633360.	−0.0590254
$650$	−15400.0	−0.00142968
$651$	0	0
$652$	897056.	0.0826420
$653$	485166.	0.0445254	0.0222627	−	0.999752i	$-0.492913\pi$
$653$	485166.	0.0445254	0.0222627	+	0.999752i	$0.492913\pi$
$654$	0	0
$655$	3.44882e6	0.314099
$656$	2.55779e7	2.32063
$657$	0	0
$658$	4.56876e6	0.411371
$659$	2.72136e6	0.244103	0.122051	−	0.992524i	$-0.461053\pi$
$659$	2.72136e6	0.244103	0.122051	+	0.992524i	$0.461053\pi$
$660$	0	0
$661$	−2.14525e6	−0.190974	−0.0954869	−	0.995431i	$-0.530441\pi$
$661$	−2.14525e6	−0.190974	−0.0954869	+	0.995431i	$0.530441\pi$
$662$	−5.64448e6	−0.500586
$663$	0	0
$664$	7.39368e6	0.650789
$665$	268912.	0.0235807
$666$	0	0
$667$	6.23443e6	0.542603
$668$	−3.35656e7	−2.91041
$669$	0	0
$670$	−2.71062e7	−2.33283
$671$	−5.95034e6	−0.510194
$672$	0	0
$673$	2.92796e6	0.249188	0.124594	−	0.992208i	$-0.460237\pi$
$673$	2.92796e6	0.249188	0.124594	+	0.992208i	$0.460237\pi$
$674$	2.07729e7	1.76136
$675$	0	0
$676$	−2.39151e7	−2.01282
$677$	1.34992e7	1.13198	0.565988	−	0.824414i	$-0.308495\pi$
$677$	1.34992e7	1.13198	0.565988	+	0.824414i	$0.308495\pi$
$678$	0	0
$679$	2.86679e6	0.238628
$680$	3.47155e7	2.87906
$681$	0	0
$682$	1.77341e7	1.45998
$683$	5.42972e6	0.445375	0.222688	−	0.974890i	$-0.428517\pi$
$683$	5.42972e6	0.445375	0.222688	+	0.974890i	$0.428517\pi$
$684$	0	0
$685$	1.14499e7	0.932340
$686$	−1.17649e6	−0.0954504
$687$	0	0
$688$	1.54931e7	1.24787
$689$	316680.	0.0254140
$690$	0	0
$691$	2.08280e7	1.65940	0.829702	−	0.558207i	$-0.188510\pi$
$691$	2.08280e7	1.65940	0.829702	+	0.558207i	$0.188510\pi$
$692$	−1.63687e7	−1.29942
$693$	0	0
$694$	532480.	0.0419667
$695$	−1.98274e6	−0.155705
$696$	0	0
$697$	3.09306e7	2.41160
$698$	−2.27200e7	−1.76510
$699$	0	0
$700$	−36652.0	−0.00282717
$701$	−2.35141e7	−1.80731	−0.903655	−	0.428261i	$-0.859126\pi$
$701$	−2.35141e7	−1.80731	−0.903655	+	0.428261i	$0.859126\pi$
$702$	0	0
$703$	1.01900e6	0.0777656
$704$	4.26138e6	0.324055
$705$	0	0
$706$	−4.00645e7	−3.02516
$707$	1.89610e6	0.142664
$708$	0	0
$709$	−1.95747e7	−1.46244	−0.731221	−	0.682140i	$-0.761049\pi$
$709$	−1.95747e7	−1.46244	−0.731221	+	0.682140i	$0.761049\pi$
$710$	3.27936e7	2.44142
$711$	0	0
$712$	1.82095e7	1.34617
$713$	1.39427e7	1.02712
$714$	0	0
$715$	1.81888e6	0.133057
$716$	−2.00554e7	−1.46200
$717$	0	0
$718$	−737840.	−0.0534135
$719$	2.61152e7	1.88396	0.941978	−	0.335674i	$-0.108964\pi$
$719$	2.61152e7	1.88396	0.941978	+	0.335674i	$0.108964\pi$
$720$	0	0
$721$	−2.59994e6	−0.186262
$722$	−2.46650e7	−1.76091
$723$	0	0
$724$	−2.29146e7	−1.62468
$725$	−37598.0	−0.00265656
$726$	0	0
$727$	1.54126e7	1.08154	0.540768	−	0.841172i	$-0.318134\pi$
$727$	1.54126e7	1.08154	0.540768	+	0.841172i	$0.318134\pi$
$728$	2.46960e6	0.172702
$729$	0	0
$730$	3.81259e7	2.64797
$731$	1.87354e7	1.29679
$732$	0	0
$733$	−1.69868e7	−1.16776	−0.583878	−	0.811841i	$-0.698465\pi$
$733$	−1.69868e7	−1.16776	−0.583878	+	0.811841i	$0.698465\pi$
$734$	1.40431e7	0.962107
$735$	0	0
$736$	−4.96128e6	−0.337597
$737$	1.12297e7	0.761554
$738$	0	0
$739$	2.01511e6	0.135734	0.0678669	−	0.997694i	$-0.478381\pi$
$739$	2.01511e6	0.135734	0.0678669	+	0.997694i	$0.478381\pi$
$740$	−3.95956e7	−2.65808
$741$	0	0
$742$	1.10838e6	0.0739059
$743$	1.51381e7	1.00600	0.503001	−	0.864286i	$-0.332229\pi$
$743$	1.51381e7	1.00600	0.503001	+	0.864286i	$0.332229\pi$
$744$	0	0
$745$	1.13266e6	0.0747666
$746$	−1.60323e7	−1.05475
$747$	0	0
$748$	−2.71663e7	−1.77532
$749$	−7.16988e6	−0.466989
$750$	0	0
$751$	7.21401e6	0.466742	0.233371	−	0.972388i	$-0.425024\pi$
$751$	7.21401e6	0.466742	0.233371	+	0.972388i	$0.425024\pi$
$752$	−1.32774e7	−0.856185
$753$	0	0
$754$	4.78520e6	0.306529
$755$	3.97062e6	0.253508
$756$	0	0
$757$	−1.09697e7	−0.695755	−0.347877	−	0.937540i	$-0.613097\pi$
$757$	−1.09697e7	−0.695755	−0.347877	+	0.937540i	$0.613097\pi$
$758$	−4.77012e7	−3.01548
$759$	0	0
$760$	−1.97568e6	−0.124075
$761$	−1.92442e7	−1.20459	−0.602293	−	0.798275i	$-0.705746\pi$
$761$	−1.92442e7	−1.20459	−0.602293	+	0.798275i	$0.705746\pi$
$762$	0	0
$763$	−4.55200e6	−0.283068
$764$	−2.43620e7	−1.51001
$765$	0	0
$766$	2.23079e7	1.37368
$767$	−382200.	−0.0234586
$768$	0	0
$769$	8.21185e6	0.500755	0.250378	−	0.968148i	$-0.419445\pi$
$769$	8.21185e6	0.500755	0.250378	+	0.968148i	$0.419445\pi$
$770$	6.36608e6	0.386941
$771$	0	0
$772$	−6.72897e7	−4.06355
$773$	−1.86187e7	−1.12073	−0.560363	−	0.828247i	$-0.689338\pi$
$773$	−1.86187e7	−1.12073	−0.560363	+	0.828247i	$0.689338\pi$
$774$	0	0
$775$	−84084.0	−0.00502874
$776$	−2.10622e7	−1.25559
$777$	0	0
$778$	−4.84024e7	−2.86694
$779$	−1.76028e6	−0.103929
$780$	0	0
$781$	−1.35859e7	−0.797006
$782$	−3.14093e7	−1.83671
$783$	0	0
$784$	3.41902e6	0.198661
$785$	1.64373e7	0.952045
$786$	0	0
$787$	2.62501e7	1.51075	0.755377	−	0.655291i	$-0.227454\pi$
$787$	2.62501e7	1.51075	0.755377	+	0.655291i	$0.227454\pi$
$788$	6.73234e7	3.86234
$789$	0	0
$790$	1.77990e7	1.01468
$791$	−4.08435e6	−0.232103
$792$	0	0
$793$	−3.59072e6	−0.202768
$794$	9.95820e6	0.560570
$795$	0	0
$796$	−5.71714e7	−3.19813
$797$	1.00373e7	0.559720	0.279860	−	0.960041i	$-0.409712\pi$
$797$	1.00373e7	0.559720	0.279860	+	0.960041i	$0.409712\pi$
$798$	0	0
$799$	−1.60559e7	−0.889751
$800$	29920.0	0.00165286
$801$	0	0
$802$	3.31605e7	1.82048
$803$	−1.57950e7	−0.864433
$804$	0	0
$805$	5.00506e6	0.272220
$806$	1.07016e7	0.580245
$807$	0	0
$808$	−1.39306e7	−0.750655
$809$	−1.40884e7	−0.756816	−0.378408	−	0.925639i	$-0.623528\pi$
$809$	−1.40884e7	−0.756816	−0.378408	+	0.925639i	$0.623528\pi$
$810$	0	0
$811$	1.81433e7	0.968646	0.484323	−	0.874889i	$-0.339066\pi$
$811$	1.81433e7	0.968646	0.484323	+	0.874889i	$0.339066\pi$
$812$	1.13888e7	0.606160
$813$	0	0
$814$	2.41234e7	1.27608
$815$	738752.	0.0389587
$816$	0	0
$817$	−1.06624e6	−0.0558856
$818$	3.07273e7	1.60562
$819$	0	0
$820$	6.83993e7	3.55236
$821$	2.13669e7	1.10633	0.553164	−	0.833072i	$-0.313420\pi$
$821$	2.13669e7	1.10633	0.553164	+	0.833072i	$0.313420\pi$
$822$	0	0
$823$	1.78017e7	0.916142	0.458071	−	0.888916i	$-0.348541\pi$
$823$	1.78017e7	0.916142	0.458071	+	0.888916i	$0.348541\pi$
$824$	1.91016e7	0.980058
$825$	0	0
$826$	−1.33770e6	−0.0682195
$827$	−1.62921e7	−0.828350	−0.414175	−	0.910197i	$-0.635930\pi$
$827$	−1.62921e7	−0.828350	−0.414175	+	0.910197i	$0.635930\pi$
$828$	0	0
$829$	−2.08499e6	−0.105370	−0.0526851	−	0.998611i	$-0.516778\pi$
$829$	−2.08499e6	−0.105370	−0.0526851	+	0.998611i	$0.516778\pi$
$830$	1.15013e7	0.579497
$831$	0	0
$832$	2.57152e6	0.128790
$833$	4.13452e6	0.206449
$834$	0	0
$835$	−2.76423e7	−1.37201
$836$	1.54605e6	0.0765081
$837$	0	0
$838$	−2.81438e7	−1.38443
$839$	2.27850e7	1.11749	0.558745	−	0.829340i	$-0.311283\pi$
$839$	2.27850e7	1.11749	0.558745	+	0.829340i	$0.311283\pi$
$840$	0	0
$841$	−8.82842e6	−0.430421
$842$	3.05802e7	1.48648
$843$	0	0
$844$	7.82498e7	3.78118
$845$	−1.96948e7	−0.948877
$846$	0	0
$847$	5.25412e6	0.251647
$848$	−3.22109e6	−0.153820
$849$	0	0
$850$	189420.	0.00899246
$851$	1.89660e7	0.897740
$852$	0	0
$853$	−2.26975e7	−1.06808	−0.534042	−	0.845458i	$-0.679328\pi$
$853$	−2.26975e7	−1.06808	−0.534042	+	0.845458i	$0.679328\pi$
$854$	−1.25675e7	−0.589664
$855$	0	0
$856$	5.26766e7	2.45716
$857$	−2.52900e7	−1.17624	−0.588120	−	0.808774i	$-0.700132\pi$
$857$	−2.52900e7	−1.17624	−0.588120	+	0.808774i	$0.700132\pi$
$858$	0	0
$859$	−1.03947e7	−0.480652	−0.240326	−	0.970692i	$-0.577254\pi$
$859$	−1.03947e7	−0.480652	−0.240326	+	0.970692i	$0.577254\pi$
$860$	4.14310e7	1.91020
$861$	0	0
$862$	−1.93750e7	−0.888122
$863$	−4.33399e7	−1.98089	−0.990447	−	0.137892i	$-0.955967\pi$
$863$	−4.33399e7	−1.98089	−0.990447	+	0.137892i	$0.955967\pi$
$864$	0	0
$865$	−1.34801e7	−0.612566
$866$	3.94790e7	1.78884
$867$	0	0
$868$	2.54698e7	1.14743
$869$	−7.37389e6	−0.331243
$870$	0	0
$871$	6.77656e6	0.302666
$872$	3.34433e7	1.48942
$873$	0	0
$874$	1.78752e6	0.0791539
$875$	8.54482e6	0.377296
$876$	0	0
$877$	3.71659e7	1.63172	0.815861	−	0.578248i	$-0.196264\pi$
$877$	3.71659e7	1.63172	0.815861	+	0.578248i	$0.196264\pi$
$878$	−7.41770e7	−3.24738
$879$	0	0
$880$	−1.85006e7	−0.805340
$881$	−9.04785e6	−0.392740	−0.196370	−	0.980530i	$-0.562915\pi$
$881$	−9.04785e6	−0.392740	−0.196370	+	0.980530i	$0.562915\pi$
$882$	0	0
$883$	3.29679e7	1.42295	0.711474	−	0.702712i	$-0.248028\pi$
$883$	3.29679e7	1.42295	0.711474	+	0.702712i	$0.248028\pi$
$884$	−1.63934e7	−0.705569
$885$	0	0
$886$	−1.40269e7	−0.600313
$887$	1.61099e7	0.687517	0.343758	−	0.939058i	$-0.388300\pi$
$887$	1.61099e7	0.687517	0.343758	+	0.939058i	$0.388300\pi$
$888$	0	0
$889$	−2.95882e6	−0.125564
$890$	2.83259e7	1.19870
$891$	0	0
$892$	−5.60500e7	−2.35865
$893$	913752.	0.0383442
$894$	0	0
$895$	−1.65162e7	−0.689211
$896$	1.32653e7	0.552009
$897$	0	0
$898$	5.90574e6	0.244390
$899$	2.61272e7	1.07819
$900$	0	0
$901$	−3.89516e6	−0.159850
$902$	−4.16718e7	−1.70540
$903$	0	0
$904$	3.00074e7	1.22126
$905$	−1.88709e7	−0.765898
$906$	0	0
$907$	−4.47286e7	−1.80537	−0.902686	−	0.430300i	$-0.858408\pi$
$907$	−4.47286e7	−1.80537	−0.902686	+	0.430300i	$0.858408\pi$
$908$	−5.05798e6	−0.203593
$909$	0	0
$910$	3.84160e6	0.153783
$911$	6.60518e6	0.263687	0.131844	−	0.991271i	$-0.457910\pi$
$911$	6.60518e6	0.263687	0.131844	+	0.991271i	$0.457910\pi$
$912$	0	0
$913$	−4.76482e6	−0.189177
$914$	−2.90484e7	−1.15016
$915$	0	0
$916$	7.69730e7	3.03110
$917$	−3.01771e6	−0.118510
$918$	0	0
$919$	−3.08930e7	−1.20662	−0.603311	−	0.797506i	$-0.706152\pi$
$919$	−3.08930e7	−1.20662	−0.603311	+	0.797506i	$0.706152\pi$
$920$	−3.67718e7	−1.43234
$921$	0	0
$922$	9.22684e6	0.357459
$923$	−8.19840e6	−0.316756
$924$	0	0
$925$	−114378.	−0.00439530
$926$	7.18235e7	2.75258
$927$	0	0
$928$	−9.29696e6	−0.354381
$929$	4.87215e6	0.185217	0.0926087	−	0.995703i	$-0.470479\pi$
$929$	4.87215e6	0.185217	0.0926087	+	0.995703i	$0.470479\pi$
$930$	0	0
$931$	−235298.	−0.00889701
$932$	1.35134e7	0.509593
$933$	0	0
$934$	6.12570e6	0.229767
$935$	−2.23722e7	−0.836913
$936$	0	0
$937$	−3.25004e7	−1.20932	−0.604658	−	0.796485i	$-0.706690\pi$
$937$	−3.25004e7	−1.20932	−0.604658	+	0.796485i	$0.706690\pi$
$938$	2.37180e7	0.880177
$939$	0	0
$940$	−3.55058e7	−1.31063
$941$	2.64040e6	0.0972066	0.0486033	−	0.998818i	$-0.484523\pi$
$941$	2.64040e6	0.0972066	0.0486033	+	0.998818i	$0.484523\pi$
$942$	0	0
$943$	−3.27627e7	−1.19978
$944$	3.88752e6	0.141985
$945$	0	0
$946$	−2.52416e7	−0.917042
$947$	4.08179e7	1.47903	0.739513	−	0.673142i	$-0.235056\pi$
$947$	4.08179e7	1.47903	0.739513	+	0.673142i	$0.235056\pi$
$948$	0	0
$949$	−9.53148e6	−0.343554
$950$	−10780.0	−0.000387534 0
$951$	0	0
$952$	−3.03761e7	−1.08627
$953$	6.71983e6	0.239677	0.119838	−	0.992793i	$-0.461762\pi$
$953$	6.71983e6	0.239677	0.119838	+	0.992793i	$0.461762\pi$
$954$	0	0
$955$	−2.00628e7	−0.711841
$956$	−3.28375e7	−1.16205
$957$	0	0
$958$	−2.60330e7	−0.916454
$959$	−1.00186e7	−0.351773
$960$	0	0
$961$	2.98016e7	1.04095
$962$	1.45572e7	0.507154
$963$	0	0
$964$	5.48019e7	1.89934
$965$	−5.54150e7	−1.91562
$966$	0	0
$967$	−2.78979e6	−0.0959413	−0.0479707	−	0.998849i	$-0.515275\pi$
$967$	−2.78979e6	−0.0959413	−0.0479707	+	0.998849i	$0.515275\pi$
$968$	−3.86017e7	−1.32409
$969$	0	0
$970$	−3.27634e7	−1.11804
$971$	−3.33594e7	−1.13545	−0.567727	−	0.823217i	$-0.692177\pi$
$971$	−3.33594e7	−1.13545	−0.567727	+	0.823217i	$0.692177\pi$
$972$	0	0
$973$	1.73489e6	0.0587477
$974$	5.46309e7	1.84519
$975$	0	0
$976$	3.65228e7	1.22727
$977$	7.60033e6	0.254739	0.127370	−	0.991855i	$-0.459347\pi$
$977$	7.60033e6	0.254739	0.127370	+	0.991855i	$0.459347\pi$
$978$	0	0
$979$	−1.17350e7	−0.391316
$980$	9.14301e6	0.304105
$981$	0	0
$982$	−1.64090e7	−0.543004
$983$	5.79760e6	0.191366	0.0956829	−	0.995412i	$-0.469497\pi$
$983$	5.79760e6	0.191366	0.0956829	+	0.995412i	$0.469497\pi$
$984$	0	0
$985$	5.54428e7	1.82077
$986$	−5.88580e7	−1.92803
$987$	0	0
$988$	932960.	0.0304068
$989$	−1.98451e7	−0.645153
$990$	0	0
$991$	1.26825e7	0.410224	0.205112	−	0.978739i	$-0.434244\pi$
$991$	1.26825e7	0.410224	0.205112	+	0.978739i	$0.434244\pi$
$992$	−2.07917e7	−0.670827
$993$	0	0
$994$	−2.86944e7	−0.921152
$995$	−4.70823e7	−1.50765
$996$	0	0
$997$	1.44400e7	0.460077	0.230039	−	0.973182i	$-0.426115\pi$
$997$	1.44400e7	0.460077	0.230039	+	0.973182i	$0.426115\pi$
$998$	2.99796e7	0.952796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.6.a.e.1.1		1
3.2	odd	2		7.6.a.a.1.1	✓	1
4.3	odd	2		1008.6.a.y.1.1		1
7.6	odd	2		441.6.a.k.1.1		1
12.11	even	2		112.6.a.g.1.1		1
15.2	even	4		175.6.b.a.99.1		2
15.8	even	4		175.6.b.a.99.2		2
15.14	odd	2		175.6.a.b.1.1		1
21.2	odd	6		49.6.c.c.18.1		2
21.5	even	6		49.6.c.b.18.1		2
21.11	odd	6		49.6.c.c.30.1		2
21.17	even	6		49.6.c.b.30.1		2
21.20	even	2		49.6.a.a.1.1		1
24.5	odd	2		448.6.a.m.1.1		1
24.11	even	2		448.6.a.c.1.1		1
33.32	even	2		847.6.a.b.1.1		1
84.83	odd	2		784.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.a.1.1	✓	1	3.2	odd	2
49.6.a.a.1.1		1	21.20	even	2
49.6.c.b.18.1		2	21.5	even	6
49.6.c.b.30.1		2	21.17	even	6
49.6.c.c.18.1		2	21.2	odd	6
49.6.c.c.30.1		2	21.11	odd	6
63.6.a.e.1.1		1	1.1	even	1	trivial
112.6.a.g.1.1		1	12.11	even	2
175.6.a.b.1.1		1	15.14	odd	2
175.6.b.a.99.1		2	15.2	even	4
175.6.b.a.99.2		2	15.8	even	4
441.6.a.k.1.1		1	7.6	odd	2
448.6.a.c.1.1		1	24.11	even	2
448.6.a.m.1.1		1	24.5	odd	2
784.6.a.c.1.1		1	84.83	odd	2
847.6.a.b.1.1		1	33.32	even	2
1008.6.a.y.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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