LMFDB - Embedded newform 63.6.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [63,6,Mod(1,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [1,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.1041806482$
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)

$f(q)$

$=$

\(q-10.0000 q^{2} +68.0000 q^{4} +106.000 q^{5} -49.0000 q^{7} -360.000 q^{8} -1060.00 q^{10} -92.0000 q^{11} +670.000 q^{13} +490.000 q^{14} +1424.00 q^{16} +222.000 q^{17} -908.000 q^{19} +7208.00 q^{20} +920.000 q^{22} +1176.00 q^{23} +8111.00 q^{25} -6700.00 q^{26} -3332.00 q^{28} -1118.00 q^{29} +3696.00 q^{31} -2720.00 q^{32} -2220.00 q^{34} -5194.00 q^{35} +4182.00 q^{37} +9080.00 q^{38} -38160.0 q^{40} +6662.00 q^{41} -3700.00 q^{43} -6256.00 q^{44} -11760.0 q^{46} +7056.00 q^{47} +2401.00 q^{49} -81110.0 q^{50} +45560.0 q^{52} +37578.0 q^{53} -9752.00 q^{55} +17640.0 q^{56} +11180.0 q^{58} -32700.0 q^{59} -10802.0 q^{61} -36960.0 q^{62} -18368.0 q^{64} +71020.0 q^{65} +64996.0 q^{67} +15096.0 q^{68} +51940.0 q^{70} +61320.0 q^{71} +38922.0 q^{73} -41820.0 q^{74} -61744.0 q^{76} +4508.00 q^{77} -88096.0 q^{79} +150944. q^{80} -66620.0 q^{82} -71892.0 q^{83} +23532.0 q^{85} +37000.0 q^{86} +33120.0 q^{88} -111818. q^{89} -32830.0 q^{91} +79968.0 q^{92} -70560.0 q^{94} -96248.0 q^{95} -150846. q^{97} -24010.0 q^{98} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−10.0000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−10.0000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	0	0
$3$	0	0
$4$	68.0000	2.12500
$5$	106.000	1.89619	0.948093	−	0.317994i	$-0.103009\pi$
$5$	106.000	1.89619	0.948093	+	0.317994i	$0.103009\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−360.000	−1.98874
$9$	0	0
$10$	−1060.00	−3.35201
$11$	−92.0000	−0.229248	−0.114624	−	0.993409i	$-0.536566\pi$
$11$	−92.0000	−0.229248	−0.114624	+	0.993409i	$0.536566\pi$
$12$	0	0
$13$	670.000	1.09955	0.549777	−	0.835312i	$-0.314713\pi$
$13$	670.000	1.09955	0.549777	+	0.835312i	$0.314713\pi$
$14$	490.000	0.668153
$15$	0	0
$16$	1424.00	1.39062
$17$	222.000	0.186308	0.0931538	−	0.995652i	$-0.470305\pi$
$17$	222.000	0.186308	0.0931538	+	0.995652i	$0.470305\pi$
$18$	0	0
$19$	−908.000	−0.577035	−0.288517	−	0.957475i	$-0.593162\pi$
$19$	−908.000	−0.577035	−0.288517	+	0.957475i	$0.593162\pi$
$20$	7208.00	4.02939
$21$	0	0
$22$	920.000	0.405258
$23$	1176.00	0.463541	0.231770	−	0.972771i	$-0.425548\pi$
$23$	1176.00	0.463541	0.231770	+	0.972771i	$0.425548\pi$
$24$	0	0
$25$	8111.00	2.59552
$26$	−6700.00	−1.94375
$27$	0	0
$28$	−3332.00	−0.803175
$29$	−1118.00	−0.246858	−0.123429	−	0.992353i	$-0.539389\pi$
$29$	−1118.00	−0.246858	−0.123429	+	0.992353i	$0.539389\pi$
$30$	0	0
$31$	3696.00	0.690761	0.345380	−	0.938463i	$-0.387750\pi$
$31$	3696.00	0.690761	0.345380	+	0.938463i	$0.387750\pi$
$32$	−2720.00	−0.469563
$33$	0	0
$34$	−2220.00	−0.329348
$35$	−5194.00	−0.716691
$36$	0	0
$37$	4182.00	0.502203	0.251102	−	0.967961i	$-0.419207\pi$
$37$	4182.00	0.502203	0.251102	+	0.967961i	$0.419207\pi$
$38$	9080.00	1.02006
$39$	0	0
$40$	−38160.0	−3.77102
$41$	6662.00	0.618935	0.309467	−	0.950910i	$-0.399849\pi$
$41$	6662.00	0.618935	0.309467	+	0.950910i	$0.399849\pi$
$42$	0	0
$43$	−3700.00	−0.305162	−0.152581	−	0.988291i	$-0.548759\pi$
$43$	−3700.00	−0.305162	−0.152581	+	0.988291i	$0.548759\pi$
$44$	−6256.00	−0.487153
$45$	0	0
$46$	−11760.0	−0.819432
$47$	7056.00	0.465923	0.232961	−	0.972486i	$-0.425158\pi$
$47$	7056.00	0.465923	0.232961	+	0.972486i	$0.425158\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−81110.0	−4.58827
$51$	0	0
$52$	45560.0	2.33655
$53$	37578.0	1.83757	0.918785	−	0.394758i	$-0.129172\pi$
$53$	37578.0	1.83757	0.918785	+	0.394758i	$0.129172\pi$
$54$	0	0
$55$	−9752.00	−0.434697
$56$	17640.0	0.751672
$57$	0	0
$58$	11180.0	0.436387
$59$	−32700.0	−1.22298	−0.611488	−	0.791254i	$-0.709429\pi$
$59$	−32700.0	−1.22298	−0.611488	+	0.791254i	$0.709429\pi$
$60$	0	0
$61$	−10802.0	−0.371689	−0.185844	−	0.982579i	$-0.559502\pi$
$61$	−10802.0	−0.371689	−0.185844	+	0.982579i	$0.559502\pi$
$62$	−36960.0	−1.22110
$63$	0	0
$64$	−18368.0	−0.560547
$65$	71020.0	2.08496
$66$	0	0
$67$	64996.0	1.76889	0.884443	−	0.466649i	$-0.154539\pi$
$67$	64996.0	1.76889	0.884443	+	0.466649i	$0.154539\pi$
$68$	15096.0	0.395904
$69$	0	0
$70$	51940.0	1.26694
$71$	61320.0	1.44363	0.721816	−	0.692085i	$-0.243308\pi$
$71$	61320.0	1.44363	0.721816	+	0.692085i	$0.243308\pi$
$72$	0	0
$73$	38922.0	0.854846	0.427423	−	0.904052i	$-0.359421\pi$
$73$	38922.0	0.854846	0.427423	+	0.904052i	$0.359421\pi$
$74$	−41820.0	−0.887779
$75$	0	0
$76$	−61744.0	−1.22620
$77$	4508.00	0.0866477
$78$	0	0
$79$	−88096.0	−1.58814	−0.794069	−	0.607827i	$-0.792041\pi$
$79$	−88096.0	−1.58814	−0.794069	+	0.607827i	$0.792041\pi$
$80$	150944.	2.63688
$81$	0	0
$82$	−66620.0	−1.09413
$83$	−71892.0	−1.14547	−0.572737	−	0.819739i	$-0.694118\pi$
$83$	−71892.0	−1.14547	−0.572737	+	0.819739i	$0.694118\pi$
$84$	0	0
$85$	23532.0	0.353274
$86$	37000.0	0.539455
$87$	0	0
$88$	33120.0	0.455915
$89$	−111818.	−1.49636	−0.748181	−	0.663495i	$-0.769073\pi$
$89$	−111818.	−1.49636	−0.748181	+	0.663495i	$0.769073\pi$
$90$	0	0
$91$	−32830.0	−0.415592
$92$	79968.0	0.985024
$93$	0	0
$94$	−70560.0	−0.823643
$95$	−96248.0	−1.09416
$96$	0	0
$97$	−150846.	−1.62781	−0.813906	−	0.580996i	$-0.802663\pi$
$97$	−150846.	−1.62781	−0.813906	+	0.580996i	$0.802663\pi$
$98$	−24010.0	−0.252538
$99$	0	0
$100$	551548.	5.51548
$101$	137354.	1.33979	0.669897	−	0.742454i	$-0.266338\pi$
$101$	137354.	1.33979	0.669897	+	0.742454i	$0.266338\pi$
$102$	0	0
$103$	28760.0	0.267113	0.133557	−	0.991041i	$-0.457360\pi$
$103$	28760.0	0.267113	0.133557	+	0.991041i	$0.457360\pi$
$104$	−241200.	−2.18672
$105$	0	0
$106$	−375780.	−3.24840
$107$	−22556.0	−0.190460	−0.0952298	−	0.995455i	$-0.530359\pi$
$107$	−22556.0	−0.190460	−0.0952298	+	0.995455i	$0.530359\pi$
$108$	0	0
$109$	19998.0	0.161221	0.0806103	−	0.996746i	$-0.474313\pi$
$109$	19998.0	0.161221	0.0806103	+	0.996746i	$0.474313\pi$
$110$	97520.0	0.768444
$111$	0	0
$112$	−69776.0	−0.525607
$113$	−17906.0	−0.131918	−0.0659588	−	0.997822i	$-0.521011\pi$
$113$	−17906.0	−0.131918	−0.0659588	+	0.997822i	$0.521011\pi$
$114$	0	0
$115$	124656.	0.878959
$116$	−76024.0	−0.524573
$117$	0	0
$118$	327000.	2.16194
$119$	−10878.0	−0.0704177
$120$	0	0
$121$	−152587.	−0.947445
$122$	108020.	0.657059
$123$	0	0
$124$	251328.	1.46787
$125$	528516.	3.02540
$126$	0	0
$127$	66864.0	0.367860	0.183930	−	0.982939i	$-0.441118\pi$
$127$	66864.0	0.367860	0.183930	+	0.982939i	$0.441118\pi$
$128$	270720.	1.46048
$129$	0	0
$130$	−710200.	−3.68572
$131$	−153764.	−0.782846	−0.391423	−	0.920211i	$-0.628017\pi$
$131$	−153764.	−0.782846	−0.391423	+	0.920211i	$0.628017\pi$
$132$	0	0
$133$	44492.0	0.218099
$134$	−649960.	−3.12698
$135$	0	0
$136$	−79920.0	−0.370517
$137$	−255978.	−1.16520	−0.582601	−	0.812758i	$-0.697965\pi$
$137$	−255978.	−1.16520	−0.582601	+	0.812758i	$0.697965\pi$
$138$	0	0
$139$	282924.	1.24203	0.621016	−	0.783798i	$-0.286720\pi$
$139$	282924.	1.24203	0.621016	+	0.783798i	$0.286720\pi$
$140$	−353192.	−1.52297
$141$	0	0
$142$	−613200.	−2.55200
$143$	−61640.0	−0.252071
$144$	0	0
$145$	−118508.	−0.468088
$146$	−389220.	−1.51117
$147$	0	0
$148$	284376.	1.06718
$149$	−408054.	−1.50575	−0.752873	−	0.658165i	$-0.771333\pi$
$149$	−408054.	−1.50575	−0.752873	+	0.658165i	$0.771333\pi$
$150$	0	0
$151$	362504.	1.29381	0.646905	−	0.762571i	$-0.276063\pi$
$151$	362504.	1.29381	0.646905	+	0.762571i	$0.276063\pi$
$152$	326880.	1.14757
$153$	0	0
$154$	−45080.0	−0.153173
$155$	391776.	1.30981
$156$	0	0
$157$	−152786.	−0.494691	−0.247346	−	0.968927i	$-0.579558\pi$
$157$	−152786.	−0.494691	−0.247346	+	0.968927i	$0.579558\pi$
$158$	880960.	2.80746
$159$	0	0
$160$	−288320.	−0.890379
$161$	−57624.0	−0.175202
$162$	0	0
$163$	−150428.	−0.443465	−0.221733	−	0.975107i	$-0.571171\pi$
$163$	−150428.	−0.443465	−0.221733	+	0.975107i	$0.571171\pi$
$164$	453016.	1.31524
$165$	0	0
$166$	718920.	2.02493
$167$	7288.00	0.0202217	0.0101108	−	0.999949i	$-0.496782\pi$
$167$	7288.00	0.0202217	0.0101108	+	0.999949i	$0.496782\pi$
$168$	0	0
$169$	77607.0	0.209018
$170$	−235320.	−0.624506
$171$	0	0
$172$	−251600.	−0.648469
$173$	289154.	0.734537	0.367269	−	0.930115i	$-0.380293\pi$
$173$	289154.	0.734537	0.367269	+	0.930115i	$0.380293\pi$
$174$	0	0
$175$	−397439.	−0.981014
$176$	−131008.	−0.318798
$177$	0	0
$178$	1.11818e6	2.64522
$179$	−199492.	−0.465364	−0.232682	−	0.972553i	$-0.574750\pi$
$179$	−199492.	−0.465364	−0.232682	+	0.972553i	$0.574750\pi$
$180$	0	0
$181$	240550.	0.545769	0.272885	−	0.962047i	$-0.412022\pi$
$181$	240550.	0.545769	0.272885	+	0.962047i	$0.412022\pi$
$182$	328300.	0.734670
$183$	0	0
$184$	−423360.	−0.921861
$185$	443292.	0.952271
$186$	0	0
$187$	−20424.0	−0.0427107
$188$	479808.	0.990086
$189$	0	0
$190$	962480.	1.93423
$191$	−290384.	−0.575956	−0.287978	−	0.957637i	$-0.592983\pi$
$191$	−290384.	−0.575956	−0.287978	+	0.957637i	$0.592983\pi$
$192$	0	0
$193$	−171454.	−0.331325	−0.165663	−	0.986182i	$-0.552976\pi$
$193$	−171454.	−0.331325	−0.165663	+	0.986182i	$0.552976\pi$
$194$	1.50846e6	2.87759
$195$	0	0
$196$	163268.	0.303571
$197$	−401990.	−0.737989	−0.368994	−	0.929432i	$-0.620298\pi$
$197$	−401990.	−0.737989	−0.368994	+	0.929432i	$0.620298\pi$
$198$	0	0
$199$	−259176.	−0.463940	−0.231970	−	0.972723i	$-0.574517\pi$
$199$	−259176.	−0.463940	−0.231970	+	0.972723i	$0.574517\pi$
$200$	−2.91996e6	−5.16181
$201$	0	0
$202$	−1.37354e6	−2.36844
$203$	54782.0	0.0933035
$204$	0	0
$205$	706172.	1.17362
$206$	−287600.	−0.472194
$207$	0	0
$208$	954080.	1.52907
$209$	83536.0	0.132284
$210$	0	0
$211$	−1.19179e6	−1.84286	−0.921431	−	0.388542i	$-0.872979\pi$
$211$	−1.19179e6	−1.84286	−0.921431	+	0.388542i	$0.872979\pi$
$212$	2.55530e6	3.90484
$213$	0	0
$214$	225560.	0.336688
$215$	−392200.	−0.578644
$216$	0	0
$217$	−181104.	−0.261083
$218$	−199980.	−0.285000
$219$	0	0
$220$	−663136.	−0.923732
$221$	148740.	0.204855
$222$	0	0
$223$	−218384.	−0.294075	−0.147038	−	0.989131i	$-0.546974\pi$
$223$	−218384.	−0.294075	−0.147038	+	0.989131i	$0.546974\pi$
$224$	133280.	0.177478
$225$	0	0
$226$	179060.	0.233199
$227$	−582852.	−0.750747	−0.375374	−	0.926874i	$-0.622486\pi$
$227$	−582852.	−0.750747	−0.375374	+	0.926874i	$0.622486\pi$
$228$	0	0
$229$	961046.	1.21103	0.605516	−	0.795833i	$-0.292967\pi$
$229$	961046.	1.21103	0.605516	+	0.795833i	$0.292967\pi$
$230$	−1.24656e6	−1.55379
$231$	0	0
$232$	402480.	0.490935
$233$	−605994.	−0.731271	−0.365636	−	0.930758i	$-0.619148\pi$
$233$	−605994.	−0.731271	−0.365636	+	0.930758i	$0.619148\pi$
$234$	0	0
$235$	747936.	0.883476
$236$	−2.22360e6	−2.59882
$237$	0	0
$238$	108780.	0.124482
$239$	−1.17014e6	−1.32509	−0.662544	−	0.749023i	$-0.730523\pi$
$239$	−1.17014e6	−1.32509	−0.662544	+	0.749023i	$0.730523\pi$
$240$	0	0
$241$	−1.23691e6	−1.37181	−0.685907	−	0.727689i	$-0.740595\pi$
$241$	−1.23691e6	−1.37181	−0.685907	+	0.727689i	$0.740595\pi$
$242$	1.52587e6	1.67486
$243$	0	0
$244$	−734536.	−0.789839
$245$	254506.	0.270884
$246$	0	0
$247$	−608360.	−0.634480
$248$	−1.33056e6	−1.37374
$249$	0	0
$250$	−5.28516e6	−5.34821
$251$	−959708.	−0.961512	−0.480756	−	0.876854i	$-0.659638\pi$
$251$	−959708.	−0.961512	−0.480756	+	0.876854i	$0.659638\pi$
$252$	0	0
$253$	−108192.	−0.106266
$254$	−668640.	−0.650291
$255$	0	0
$256$	−2.11942e6	−2.02124
$257$	1.21259e6	1.14520	0.572600	−	0.819835i	$-0.305935\pi$
$257$	1.21259e6	1.14520	0.572600	+	0.819835i	$0.305935\pi$
$258$	0	0
$259$	−204918.	−0.189815
$260$	4.82936e6	4.43054
$261$	0	0
$262$	1.53764e6	1.38389
$263$	1.25274e6	1.11679	0.558397	−	0.829574i	$-0.311417\pi$
$263$	1.25274e6	1.11679	0.558397	+	0.829574i	$0.311417\pi$
$264$	0	0
$265$	3.98327e6	3.48437
$266$	−444920.	−0.385547
$267$	0	0
$268$	4.41973e6	3.75888
$269$	136866.	0.115323	0.0576614	−	0.998336i	$-0.481636\pi$
$269$	136866.	0.115323	0.0576614	+	0.998336i	$0.481636\pi$
$270$	0	0
$271$	−960896.	−0.794791	−0.397396	−	0.917647i	$-0.630086\pi$
$271$	−960896.	−0.794791	−0.397396	+	0.917647i	$0.630086\pi$
$272$	316128.	0.259084
$273$	0	0
$274$	2.55978e6	2.05981
$275$	−746212.	−0.595019
$276$	0	0
$277$	905830.	0.709328	0.354664	−	0.934994i	$-0.384595\pi$
$277$	905830.	0.709328	0.354664	+	0.934994i	$0.384595\pi$
$278$	−2.82924e6	−2.19562
$279$	0	0
$280$	1.86984e6	1.42531
$281$	33062.0	0.0249783	0.0124892	−	0.999922i	$-0.496024\pi$
$281$	33062.0	0.0249783	0.0124892	+	0.999922i	$0.496024\pi$
$282$	0	0
$283$	−863588.	−0.640974	−0.320487	−	0.947253i	$-0.603847\pi$
$283$	−863588.	−0.640974	−0.320487	+	0.947253i	$0.603847\pi$
$284$	4.16976e6	3.06772
$285$	0	0
$286$	616400.	0.445602
$287$	−326438.	−0.233935
$288$	0	0
$289$	−1.37057e6	−0.965289
$290$	1.18508e6	0.827471
$291$	0	0
$292$	2.64670e6	1.81655
$293$	1.33755e6	0.910206	0.455103	−	0.890439i	$-0.349602\pi$
$293$	1.33755e6	0.910206	0.455103	+	0.890439i	$0.349602\pi$
$294$	0	0
$295$	−3.46620e6	−2.31899
$296$	−1.50552e6	−0.998751
$297$	0	0
$298$	4.08054e6	2.66181
$299$	787920.	0.509688
$300$	0	0
$301$	181300.	0.115340
$302$	−3.62504e6	−2.28715
$303$	0	0
$304$	−1.29299e6	−0.802439
$305$	−1.14501e6	−0.704791
$306$	0	0
$307$	−1.32820e6	−0.804301	−0.402151	−	0.915573i	$-0.631737\pi$
$307$	−1.32820e6	−0.804301	−0.402151	+	0.915573i	$0.631737\pi$
$308$	306544.	0.184126
$309$	0	0
$310$	−3.91776e6	−2.31544
$311$	665832.	0.390359	0.195179	−	0.980768i	$-0.437471\pi$
$311$	665832.	0.390359	0.195179	+	0.980768i	$0.437471\pi$
$312$	0	0
$313$	−3.09021e6	−1.78290	−0.891451	−	0.453116i	$-0.850312\pi$
$313$	−3.09021e6	−1.78290	−0.891451	+	0.453116i	$0.850312\pi$
$314$	1.52786e6	0.874499
$315$	0	0
$316$	−5.99053e6	−3.37479
$317$	974178.	0.544490	0.272245	−	0.962228i	$-0.412234\pi$
$317$	974178.	0.544490	0.272245	+	0.962228i	$0.412234\pi$
$318$	0	0
$319$	102856.	0.0565917
$320$	−1.94701e6	−1.06290
$321$	0	0
$322$	576240.	0.309716
$323$	−201576.	−0.107506
$324$	0	0
$325$	5.43437e6	2.85391
$326$	1.50428e6	0.783943
$327$	0	0
$328$	−2.39832e6	−1.23090
$329$	−345744.	−0.176102
$330$	0	0
$331$	781772.	0.392202	0.196101	−	0.980584i	$-0.437172\pi$
$331$	781772.	0.392202	0.196101	+	0.980584i	$0.437172\pi$
$332$	−4.88866e6	−2.43413
$333$	0	0
$334$	−72880.0	−0.0357472
$335$	6.88958e6	3.35413
$336$	0	0
$337$	348754.	0.167280	0.0836401	−	0.996496i	$-0.473345\pi$
$337$	348754.	0.167280	0.0836401	+	0.996496i	$0.473345\pi$
$338$	−776070.	−0.369495
$339$	0	0
$340$	1.60018e6	0.750707
$341$	−340032.	−0.158356
$342$	0	0
$343$	−117649.	−0.0539949
$344$	1.33200e6	0.606887
$345$	0	0
$346$	−2.89154e6	−1.29849
$347$	−2.50625e6	−1.11738	−0.558690	−	0.829376i	$-0.688696\pi$
$347$	−2.50625e6	−1.11738	−0.558690	+	0.829376i	$0.688696\pi$
$348$	0	0
$349$	3.05861e6	1.34419	0.672094	−	0.740466i	$-0.265395\pi$
$349$	3.05861e6	1.34419	0.672094	+	0.740466i	$0.265395\pi$
$350$	3.97439e6	1.73420
$351$	0	0
$352$	250240.	0.107647
$353$	3.49291e6	1.49194	0.745969	−	0.665981i	$-0.231987\pi$
$353$	3.49291e6	1.49194	0.745969	+	0.665981i	$0.231987\pi$
$354$	0	0
$355$	6.49992e6	2.73739
$356$	−7.60362e6	−3.17977
$357$	0	0
$358$	1.99492e6	0.822655
$359$	−2.12034e6	−0.868301	−0.434150	−	0.900840i	$-0.642951\pi$
$359$	−2.12034e6	−0.868301	−0.434150	+	0.900840i	$0.642951\pi$
$360$	0	0
$361$	−1.65163e6	−0.667031
$362$	−2.40550e6	−0.964793
$363$	0	0
$364$	−2.23244e6	−0.883133
$365$	4.12573e6	1.62095
$366$	0	0
$367$	746592.	0.289346	0.144673	−	0.989479i	$-0.453787\pi$
$367$	746592.	0.289346	0.144673	+	0.989479i	$0.453787\pi$
$368$	1.67462e6	0.644611
$369$	0	0
$370$	−4.43292e6	−1.68339
$371$	−1.84132e6	−0.694536
$372$	0	0
$373$	−939034.	−0.349469	−0.174735	−	0.984616i	$-0.555907\pi$
$373$	−939034.	−0.349469	−0.174735	+	0.984616i	$0.555907\pi$
$374$	204240.	0.0755026
$375$	0	0
$376$	−2.54016e6	−0.926598
$377$	−749060.	−0.271433
$378$	0	0
$379$	5.16534e6	1.84714	0.923572	−	0.383424i	$-0.125255\pi$
$379$	5.16534e6	1.84714	0.923572	+	0.383424i	$0.125255\pi$
$380$	−6.54486e6	−2.32510
$381$	0	0
$382$	2.90384e6	1.01816
$383$	−400512.	−0.139514	−0.0697571	−	0.997564i	$-0.522222\pi$
$383$	−400512.	−0.139514	−0.0697571	+	0.997564i	$0.522222\pi$
$384$	0	0
$385$	477848.	0.164300
$386$	1.71454e6	0.585706
$387$	0	0
$388$	−1.02575e7	−3.45910
$389$	−306822.	−0.102805	−0.0514023	−	0.998678i	$-0.516369\pi$
$389$	−306822.	−0.102805	−0.0514023	+	0.998678i	$0.516369\pi$
$390$	0	0
$391$	261072.	0.0863611
$392$	−864360.	−0.284105
$393$	0	0
$394$	4.01990e6	1.30459
$395$	−9.33818e6	−3.01141
$396$	0	0
$397$	−3.83421e6	−1.22095	−0.610477	−	0.792034i	$-0.709022\pi$
$397$	−3.83421e6	−1.22095	−0.610477	+	0.792034i	$0.709022\pi$
$398$	2.59176e6	0.820138
$399$	0	0
$400$	1.15501e7	3.60940
$401$	3.29355e6	1.02283	0.511415	−	0.859334i	$-0.329122\pi$
$401$	3.29355e6	1.02283	0.511415	+	0.859334i	$0.329122\pi$
$402$	0	0
$403$	2.47632e6	0.759529
$404$	9.34007e6	2.84706
$405$	0	0
$406$	−547820.	−0.164939
$407$	−384744.	−0.115129
$408$	0	0
$409$	−1.35473e6	−0.400445	−0.200223	−	0.979750i	$-0.564167\pi$
$409$	−1.35473e6	−0.400445	−0.200223	+	0.979750i	$0.564167\pi$
$410$	−7.06172e6	−2.07468
$411$	0	0
$412$	1.95568e6	0.567616
$413$	1.60230e6	0.462241
$414$	0	0
$415$	−7.62055e6	−2.17203
$416$	−1.82240e6	−0.516310
$417$	0	0
$418$	−835360.	−0.233848
$419$	−5.08199e6	−1.41416	−0.707080	−	0.707134i	$-0.749988\pi$
$419$	−5.08199e6	−1.41416	−0.707080	+	0.707134i	$0.749988\pi$
$420$	0	0
$421$	628022.	0.172691	0.0863455	−	0.996265i	$-0.472481\pi$
$421$	628022.	0.172691	0.0863455	+	0.996265i	$0.472481\pi$
$422$	1.19179e7	3.25775
$423$	0	0
$424$	−1.35281e7	−3.65445
$425$	1.80064e6	0.483565
$426$	0	0
$427$	529298.	0.140485
$428$	−1.53381e6	−0.404726
$429$	0	0
$430$	3.92200e6	1.02291
$431$	3.00086e6	0.778132	0.389066	−	0.921210i	$-0.372798\pi$
$431$	3.00086e6	0.778132	0.389066	+	0.921210i	$0.372798\pi$
$432$	0	0
$433$	1.21496e6	0.311417	0.155709	−	0.987803i	$-0.450234\pi$
$433$	1.21496e6	0.311417	0.155709	+	0.987803i	$0.450234\pi$
$434$	1.81104e6	0.461534
$435$	0	0
$436$	1.35986e6	0.342594
$437$	−1.06781e6	−0.267479
$438$	0	0
$439$	4.00654e6	0.992219	0.496110	−	0.868260i	$-0.334761\pi$
$439$	4.00654e6	0.992219	0.496110	+	0.868260i	$0.334761\pi$
$440$	3.51072e6	0.864499
$441$	0	0
$442$	−1.48740e6	−0.362136
$443$	5.44751e6	1.31883	0.659415	−	0.751779i	$-0.270804\pi$
$443$	5.44751e6	1.31883	0.659415	+	0.751779i	$0.270804\pi$
$444$	0	0
$445$	−1.18527e7	−2.83738
$446$	2.18384e6	0.519857
$447$	0	0
$448$	900032.	0.211867
$449$	1.81577e6	0.425056	0.212528	−	0.977155i	$-0.431830\pi$
$449$	1.81577e6	0.425056	0.212528	+	0.977155i	$0.431830\pi$
$450$	0	0
$451$	−612904.	−0.141890
$452$	−1.21761e6	−0.280325
$453$	0	0
$454$	5.82852e6	1.32715
$455$	−3.47998e6	−0.788040
$456$	0	0
$457$	−5.30082e6	−1.18728	−0.593639	−	0.804731i	$-0.702309\pi$
$457$	−5.30082e6	−1.18728	−0.593639	+	0.804731i	$0.702309\pi$
$458$	−9.61046e6	−2.14082
$459$	0	0
$460$	8.47661e6	1.86779
$461$	−3.20381e6	−0.702124	−0.351062	−	0.936352i	$-0.614179\pi$
$461$	−3.20381e6	−0.702124	−0.351062	+	0.936352i	$0.614179\pi$
$462$	0	0
$463$	−1.11853e6	−0.242490	−0.121245	−	0.992623i	$-0.538689\pi$
$463$	−1.11853e6	−0.242490	−0.121245	+	0.992623i	$0.538689\pi$
$464$	−1.59203e6	−0.343287
$465$	0	0
$466$	6.05994e6	1.29272
$467$	3.85134e6	0.817184	0.408592	−	0.912717i	$-0.366020\pi$
$467$	3.85134e6	0.817184	0.408592	+	0.912717i	$0.366020\pi$
$468$	0	0
$469$	−3.18480e6	−0.668576
$470$	−7.47936e6	−1.56178
$471$	0	0
$472$	1.17720e7	2.43218
$473$	340400.	0.0699579
$474$	0	0
$475$	−7.36479e6	−1.49770
$476$	−739704.	−0.149638
$477$	0	0
$478$	1.17014e7	2.34245
$479$	−1.43536e6	−0.285839	−0.142920	−	0.989734i	$-0.545649\pi$
$479$	−1.43536e6	−0.285839	−0.142920	+	0.989734i	$0.545649\pi$
$480$	0	0
$481$	2.80194e6	0.552200
$482$	1.23691e7	2.42505
$483$	0	0
$484$	−1.03759e7	−2.01332
$485$	−1.59897e7	−3.08664
$486$	0	0
$487$	4.61097e6	0.880987	0.440494	−	0.897756i	$-0.354803\pi$
$487$	4.61097e6	0.880987	0.440494	+	0.897756i	$0.354803\pi$
$488$	3.88872e6	0.739192
$489$	0	0
$490$	−2.54506e6	−0.478859
$491$	−7.40518e6	−1.38622	−0.693110	−	0.720832i	$-0.743760\pi$
$491$	−7.40518e6	−1.38622	−0.693110	+	0.720832i	$0.743760\pi$
$492$	0	0
$493$	−248196.	−0.0459915
$494$	6.08360e6	1.12161
$495$	0	0
$496$	5.26310e6	0.960589
$497$	−3.00468e6	−0.545641
$498$	0	0
$499$	3.93432e6	0.707325	0.353662	−	0.935373i	$-0.384936\pi$
$499$	3.93432e6	0.707325	0.353662	+	0.935373i	$0.384936\pi$
$500$	3.59391e7	6.42898
$501$	0	0
$502$	9.59708e6	1.69973
$503$	−3.40975e6	−0.600901	−0.300450	−	0.953797i	$-0.597137\pi$
$503$	−3.40975e6	−0.600901	−0.300450	+	0.953797i	$0.597137\pi$
$504$	0	0
$505$	1.45595e7	2.54050
$506$	1.08192e6	0.187853
$507$	0	0
$508$	4.54675e6	0.781703
$509$	7.72383e6	1.32141	0.660706	−	0.750645i	$-0.270257\pi$
$509$	7.72383e6	1.32141	0.660706	+	0.750645i	$0.270257\pi$
$510$	0	0
$511$	−1.90718e6	−0.323102
$512$	1.25312e7	2.11260
$513$	0	0
$514$	−1.21259e7	−2.02445
$515$	3.04856e6	0.506497
$516$	0	0
$517$	−649152.	−0.106812
$518$	2.04918e6	0.335549
$519$	0	0
$520$	−2.55672e7	−4.14643
$521$	4.77658e6	0.770944	0.385472	−	0.922719i	$-0.374039\pi$
$521$	4.77658e6	0.770944	0.385472	+	0.922719i	$0.374039\pi$
$522$	0	0
$523$	−9.28754e6	−1.48473	−0.742363	−	0.669998i	$-0.766295\pi$
$523$	−9.28754e6	−1.48473	−0.742363	+	0.669998i	$0.766295\pi$
$524$	−1.04560e7	−1.66355
$525$	0	0
$526$	−1.25274e7	−1.97423
$527$	820512.	0.128694
$528$	0	0
$529$	−5.05337e6	−0.785130
$530$	−3.98327e7	−6.15956
$531$	0	0
$532$	3.02546e6	0.463459
$533$	4.46354e6	0.680552
$534$	0	0
$535$	−2.39094e6	−0.361147
$536$	−2.33986e7	−3.51785
$537$	0	0
$538$	−1.36866e6	−0.203864
$539$	−220892.	−0.0327498
$540$	0	0
$541$	7.72917e6	1.13538	0.567688	−	0.823244i	$-0.307838\pi$
$541$	7.72917e6	1.13538	0.567688	+	0.823244i	$0.307838\pi$
$542$	9.60896e6	1.40501
$543$	0	0
$544$	−603840.	−0.0874832
$545$	2.11979e6	0.305704
$546$	0	0
$547$	−8.60361e6	−1.22945	−0.614727	−	0.788740i	$-0.710734\pi$
$547$	−8.60361e6	−1.22945	−0.614727	+	0.788740i	$0.710734\pi$
$548$	−1.74065e7	−2.47605
$549$	0	0
$550$	7.46212e6	1.05185
$551$	1.01514e6	0.142445
$552$	0	0
$553$	4.31670e6	0.600260
$554$	−9.05830e6	−1.25393
$555$	0	0
$556$	1.92388e7	2.63932
$557$	1.77723e6	0.242721	0.121360	−	0.992609i	$-0.461274\pi$
$557$	1.77723e6	0.242721	0.121360	+	0.992609i	$0.461274\pi$
$558$	0	0
$559$	−2.47900e6	−0.335542
$560$	−7.39626e6	−0.996648
$561$	0	0
$562$	−330620.	−0.0441559
$563$	−2.68860e6	−0.357482	−0.178741	−	0.983896i	$-0.557202\pi$
$563$	−2.68860e6	−0.357482	−0.178741	+	0.983896i	$0.557202\pi$
$564$	0	0
$565$	−1.89804e6	−0.250140
$566$	8.63588e6	1.13309
$567$	0	0
$568$	−2.20752e7	−2.87100
$569$	−5.32630e6	−0.689675	−0.344838	−	0.938662i	$-0.612066\pi$
$569$	−5.32630e6	−0.689675	−0.344838	+	0.938662i	$0.612066\pi$
$570$	0	0
$571$	1.33992e7	1.71984	0.859921	−	0.510427i	$-0.170513\pi$
$571$	1.33992e7	1.71984	0.859921	+	0.510427i	$0.170513\pi$
$572$	−4.19152e6	−0.535650
$573$	0	0
$574$	3.26438e6	0.413543
$575$	9.53854e6	1.20313
$576$	0	0
$577$	−1.10502e6	−0.138176	−0.0690878	−	0.997611i	$-0.522009\pi$
$577$	−1.10502e6	−0.138176	−0.0690878	+	0.997611i	$0.522009\pi$
$578$	1.37057e7	1.70641
$579$	0	0
$580$	−8.05854e6	−0.994687
$581$	3.52271e6	0.432949
$582$	0	0
$583$	−3.45718e6	−0.421260
$584$	−1.40119e7	−1.70007
$585$	0	0
$586$	−1.33755e7	−1.60903
$587$	−5.97288e6	−0.715465	−0.357732	−	0.933824i	$-0.616450\pi$
$587$	−5.97288e6	−0.715465	−0.357732	+	0.933824i	$0.616450\pi$
$588$	0	0
$589$	−3.35597e6	−0.398593
$590$	3.46620e7	4.09943
$591$	0	0
$592$	5.95517e6	0.698377
$593$	1.11945e7	1.30728	0.653639	−	0.756807i	$-0.273242\pi$
$593$	1.11945e7	1.30728	0.653639	+	0.756807i	$0.273242\pi$
$594$	0	0
$595$	−1.15307e6	−0.133525
$596$	−2.77477e7	−3.19971
$597$	0	0
$598$	−7.87920e6	−0.901009
$599$	−1.09055e7	−1.24187	−0.620937	−	0.783860i	$-0.713248\pi$
$599$	−1.09055e7	−1.24187	−0.620937	+	0.783860i	$0.713248\pi$
$600$	0	0
$601$	7.39737e6	0.835394	0.417697	−	0.908586i	$-0.362837\pi$
$601$	7.39737e6	0.835394	0.417697	+	0.908586i	$0.362837\pi$
$602$	−1.81300e6	−0.203895
$603$	0	0
$604$	2.46503e7	2.74935
$605$	−1.61742e7	−1.79653
$606$	0	0
$607$	7.13355e6	0.785840	0.392920	−	0.919573i	$-0.371465\pi$
$607$	7.13355e6	0.785840	0.392920	+	0.919573i	$0.371465\pi$
$608$	2.46976e6	0.270954
$609$	0	0
$610$	1.14501e7	1.24591
$611$	4.72752e6	0.512307
$612$	0	0
$613$	−1.71264e7	−1.84083	−0.920416	−	0.390939i	$-0.872150\pi$
$613$	−1.71264e7	−1.84083	−0.920416	+	0.390939i	$0.872150\pi$
$614$	1.32820e7	1.42182
$615$	0	0
$616$	−1.62288e6	−0.172320
$617$	−2.29924e6	−0.243149	−0.121574	−	0.992582i	$-0.538794\pi$
$617$	−2.29924e6	−0.243149	−0.121574	+	0.992582i	$0.538794\pi$
$618$	0	0
$619$	−1.85176e6	−0.194249	−0.0971245	−	0.995272i	$-0.530965\pi$
$619$	−1.85176e6	−0.194249	−0.0971245	+	0.995272i	$0.530965\pi$
$620$	2.66408e7	2.78335
$621$	0	0
$622$	−6.65832e6	−0.690063
$623$	5.47908e6	0.565572
$624$	0	0
$625$	3.06758e7	3.14120
$626$	3.09021e7	3.15176
$627$	0	0
$628$	−1.03894e7	−1.05122
$629$	928404.	0.0935643
$630$	0	0
$631$	9.25978e6	0.925822	0.462911	−	0.886405i	$-0.346805\pi$
$631$	9.25978e6	0.925822	0.462911	+	0.886405i	$0.346805\pi$
$632$	3.17146e7	3.15839
$633$	0	0
$634$	−9.74178e6	−0.962532
$635$	7.08758e6	0.697532
$636$	0	0
$637$	1.60867e6	0.157079
$638$	−1.02856e6	−0.100041
$639$	0	0
$640$	2.86963e7	2.76934
$641$	−1.79419e7	−1.72474	−0.862369	−	0.506280i	$-0.831020\pi$
$641$	−1.79419e7	−1.72474	−0.862369	+	0.506280i	$0.831020\pi$
$642$	0	0
$643$	6.70020e6	0.639087	0.319544	−	0.947572i	$-0.396470\pi$
$643$	6.70020e6	0.639087	0.319544	+	0.947572i	$0.396470\pi$
$644$	−3.91843e6	−0.372304
$645$	0	0
$646$	2.01576e6	0.190045
$647$	1.12549e7	1.05701	0.528507	−	0.848929i	$-0.322752\pi$
$647$	1.12549e7	1.05701	0.528507	+	0.848929i	$0.322752\pi$
$648$	0	0
$649$	3.00840e6	0.280365
$650$	−5.43437e7	−5.04505
$651$	0	0
$652$	−1.02291e7	−0.942364
$653$	−1.31704e7	−1.20869	−0.604347	−	0.796721i	$-0.706566\pi$
$653$	−1.31704e7	−1.20869	−0.604347	+	0.796721i	$0.706566\pi$
$654$	0	0
$655$	−1.62990e7	−1.48442
$656$	9.48669e6	0.860706
$657$	0	0
$658$	3.45744e6	0.311308
$659$	1.43453e7	1.28676	0.643380	−	0.765547i	$-0.277532\pi$
$659$	1.43453e7	1.28676	0.643380	+	0.765547i	$0.277532\pi$
$660$	0	0
$661$	−14138.0	−0.00125859	−0.000629295	−	1.00000i	$-0.500200\pi$
$661$	−14138.0	−0.00125859	−0.000629295		1.00000i	$0.500200\pi$
$662$	−7.81772e6	−0.693322
$663$	0	0
$664$	2.58811e7	2.27805
$665$	4.71615e6	0.413555
$666$	0	0
$667$	−1.31477e6	−0.114429
$668$	495584.	0.0429711
$669$	0	0
$670$	−6.88958e7	−5.92933
$671$	993784.	0.0852090
$672$	0	0
$673$	1.37787e7	1.17266	0.586329	−	0.810073i	$-0.300573\pi$
$673$	1.37787e7	1.17266	0.586329	+	0.810073i	$0.300573\pi$
$674$	−3.48754e6	−0.295712
$675$	0	0
$676$	5.27728e6	0.444164
$677$	−1.26155e7	−1.05787	−0.528936	−	0.848662i	$-0.677409\pi$
$677$	−1.26155e7	−1.05787	−0.528936	+	0.848662i	$0.677409\pi$
$678$	0	0
$679$	7.39145e6	0.615255
$680$	−8.47152e6	−0.702569
$681$	0	0
$682$	3.40032e6	0.279936
$683$	1.08656e6	0.0891258	0.0445629	−	0.999007i	$-0.485810\pi$
$683$	1.08656e6	0.0891258	0.0445629	+	0.999007i	$0.485810\pi$
$684$	0	0
$685$	−2.71337e7	−2.20944
$686$	1.17649e6	0.0954504
$687$	0	0
$688$	−5.26880e6	−0.424366
$689$	2.51773e7	2.02051
$690$	0	0
$691$	1.91229e7	1.52356	0.761780	−	0.647836i	$-0.224326\pi$
$691$	1.91229e7	1.52356	0.761780	+	0.647836i	$0.224326\pi$
$692$	1.96625e7	1.56089
$693$	0	0
$694$	2.50625e7	1.97527
$695$	2.99899e7	2.35512
$696$	0	0
$697$	1.47896e6	0.115312
$698$	−3.05861e7	−2.37621
$699$	0	0
$700$	−2.70259e7	−2.08466
$701$	−1.15000e7	−0.883897	−0.441948	−	0.897040i	$-0.645713\pi$
$701$	−1.15000e7	−0.883897	−0.441948	+	0.897040i	$0.645713\pi$
$702$	0	0
$703$	−3.79726e6	−0.289789
$704$	1.68986e6	0.128504
$705$	0	0
$706$	−3.49291e7	−2.63740
$707$	−6.73035e6	−0.506394
$708$	0	0
$709$	6.97551e6	0.521147	0.260574	−	0.965454i	$-0.416088\pi$
$709$	6.97551e6	0.521147	0.260574	+	0.965454i	$0.416088\pi$
$710$	−6.49992e7	−4.83907
$711$	0	0
$712$	4.02545e7	2.97587
$713$	4.34650e6	0.320196
$714$	0	0
$715$	−6.53384e6	−0.477973
$716$	−1.35655e7	−0.988899
$717$	0	0
$718$	2.12034e7	1.53495
$719$	1.44264e6	0.104072	0.0520362	−	0.998645i	$-0.483429\pi$
$719$	1.44264e6	0.104072	0.0520362	+	0.998645i	$0.483429\pi$
$720$	0	0
$721$	−1.40924e6	−0.100959
$722$	1.65164e7	1.17916
$723$	0	0
$724$	1.63574e7	1.15976
$725$	−9.06810e6	−0.640724
$726$	0	0
$727$	−1.90334e7	−1.33561	−0.667807	−	0.744334i	$-0.732767\pi$
$727$	−1.90334e7	−1.33561	−0.667807	+	0.744334i	$0.732767\pi$
$728$	1.18188e7	0.826504
$729$	0	0
$730$	−4.12573e7	−2.86546
$731$	−821400.	−0.0568540
$732$	0	0
$733$	5.45585e6	0.375062	0.187531	−	0.982259i	$-0.439952\pi$
$733$	5.45585e6	0.375062	0.187531	+	0.982259i	$0.439952\pi$
$734$	−7.46592e6	−0.511497
$735$	0	0
$736$	−3.19872e6	−0.217662
$737$	−5.97963e6	−0.405514
$738$	0	0
$739$	7.85197e6	0.528893	0.264446	−	0.964400i	$-0.414811\pi$
$739$	7.85197e6	0.528893	0.264446	+	0.964400i	$0.414811\pi$
$740$	3.01439e7	2.02358
$741$	0	0
$742$	1.84132e7	1.22778
$743$	1.48695e7	0.988154	0.494077	−	0.869418i	$-0.335506\pi$
$743$	1.48695e7	0.988154	0.494077	+	0.869418i	$0.335506\pi$
$744$	0	0
$745$	−4.32537e7	−2.85518
$746$	9.39034e6	0.617781
$747$	0	0
$748$	−1.38883e6	−0.0907603
$749$	1.10524e6	0.0719869
$750$	0	0
$751$	−2.51309e7	−1.62595	−0.812977	−	0.582295i	$-0.802155\pi$
$751$	−2.51309e7	−1.62595	−0.812977	+	0.582295i	$0.802155\pi$
$752$	1.00477e7	0.647924
$753$	0	0
$754$	7.49060e6	0.479831
$755$	3.84254e7	2.45330
$756$	0	0
$757$	−1.97874e7	−1.25501	−0.627507	−	0.778611i	$-0.715925\pi$
$757$	−1.97874e7	−1.25501	−0.627507	+	0.778611i	$0.715925\pi$
$758$	−5.16534e7	−3.26532
$759$	0	0
$760$	3.46493e7	2.17601
$761$	1.49642e7	0.936678	0.468339	−	0.883549i	$-0.344853\pi$
$761$	1.49642e7	0.936678	0.468339	+	0.883549i	$0.344853\pi$
$762$	0	0
$763$	−979902.	−0.0609356
$764$	−1.97461e7	−1.22391
$765$	0	0
$766$	4.00512e6	0.246629
$767$	−2.19090e7	−1.34473
$768$	0	0
$769$	5.06419e6	0.308812	0.154406	−	0.988007i	$-0.450654\pi$
$769$	5.06419e6	0.308812	0.154406	+	0.988007i	$0.450654\pi$
$770$	−4.77848e6	−0.290444
$771$	0	0
$772$	−1.16589e7	−0.704066
$773$	−2.63025e7	−1.58324	−0.791622	−	0.611011i	$-0.790763\pi$
$773$	−2.63025e7	−1.58324	−0.791622	+	0.611011i	$0.790763\pi$
$774$	0	0
$775$	2.99783e7	1.79288
$776$	5.43046e7	3.23729
$777$	0	0
$778$	3.06822e6	0.181735
$779$	−6.04910e6	−0.357147
$780$	0	0
$781$	−5.64144e6	−0.330950
$782$	−2.61072e6	−0.152666
$783$	0	0
$784$	3.41902e6	0.198661
$785$	−1.61953e7	−0.938027
$786$	0	0
$787$	1.00525e7	0.578545	0.289273	−	0.957247i	$-0.406587\pi$
$787$	1.00525e7	0.578545	0.289273	+	0.957247i	$0.406587\pi$
$788$	−2.73353e7	−1.56823
$789$	0	0
$790$	9.33818e7	5.32346
$791$	877394.	0.0498601
$792$	0	0
$793$	−7.23734e6	−0.408692
$794$	3.83421e7	2.15836
$795$	0	0
$796$	−1.76240e7	−0.985873
$797$	−2.78516e7	−1.55312	−0.776560	−	0.630044i	$-0.783037\pi$
$797$	−2.78516e7	−1.55312	−0.776560	+	0.630044i	$0.783037\pi$
$798$	0	0
$799$	1.56643e6	0.0868050
$800$	−2.20619e7	−1.21876
$801$	0	0
$802$	−3.29355e7	−1.80812
$803$	−3.58082e6	−0.195972
$804$	0	0
$805$	−6.10814e6	−0.332215
$806$	−2.47632e7	−1.34267
$807$	0	0
$808$	−4.94474e7	−2.66450
$809$	1.96936e7	1.05792	0.528961	−	0.848646i	$-0.322582\pi$
$809$	1.96936e7	1.05792	0.528961	+	0.848646i	$0.322582\pi$
$810$	0	0
$811$	5.05617e6	0.269942	0.134971	−	0.990850i	$-0.456906\pi$
$811$	5.05617e6	0.269942	0.134971	+	0.990850i	$0.456906\pi$
$812$	3.72518e6	0.198270
$813$	0	0
$814$	3.84744e6	0.203522
$815$	−1.59454e7	−0.840893
$816$	0	0
$817$	3.35960e6	0.176089
$818$	1.35473e7	0.707894
$819$	0	0
$820$	4.80197e7	2.49393
$821$	2.82324e6	0.146181	0.0730904	−	0.997325i	$-0.476714\pi$
$821$	2.82324e6	0.146181	0.0730904	+	0.997325i	$0.476714\pi$
$822$	0	0
$823$	2.54741e7	1.31099	0.655494	−	0.755201i	$-0.272461\pi$
$823$	2.54741e7	1.31099	0.655494	+	0.755201i	$0.272461\pi$
$824$	−1.03536e7	−0.531219
$825$	0	0
$826$	−1.60230e7	−0.817135
$827$	1.75616e7	0.892893	0.446446	−	0.894810i	$-0.352689\pi$
$827$	1.75616e7	0.892893	0.446446	+	0.894810i	$0.352689\pi$
$828$	0	0
$829$	9.14728e6	0.462280	0.231140	−	0.972920i	$-0.425754\pi$
$829$	9.14728e6	0.462280	0.231140	+	0.972920i	$0.425754\pi$
$830$	7.62055e7	3.83965
$831$	0	0
$832$	−1.23066e7	−0.616351
$833$	533022.	0.0266154
$834$	0	0
$835$	772528.	0.0383441
$836$	5.68045e6	0.281104
$837$	0	0
$838$	5.08199e7	2.49991
$839$	1.09891e7	0.538961	0.269481	−	0.963006i	$-0.413148\pi$
$839$	1.09891e7	0.538961	0.269481	+	0.963006i	$0.413148\pi$
$840$	0	0
$841$	−1.92612e7	−0.939061
$842$	−6.28022e6	−0.305277
$843$	0	0
$844$	−8.10416e7	−3.91608
$845$	8.22634e6	0.396337
$846$	0	0
$847$	7.47676e6	0.358101
$848$	5.35111e7	2.55537
$849$	0	0
$850$	−1.80064e7	−0.854831
$851$	4.91803e6	0.232792
$852$	0	0
$853$	−1.34854e7	−0.634586	−0.317293	−	0.948328i	$-0.602774\pi$
$853$	−1.34854e7	−0.634586	−0.317293	+	0.948328i	$0.602774\pi$
$854$	−5.29298e6	−0.248345
$855$	0	0
$856$	8.12016e6	0.378774
$857$	−1.07032e7	−0.497808	−0.248904	−	0.968528i	$-0.580070\pi$
$857$	−1.07032e7	−0.497808	−0.248904	+	0.968528i	$0.580070\pi$
$858$	0	0
$859$	−1.33747e7	−0.618446	−0.309223	−	0.950990i	$-0.600069\pi$
$859$	−1.33747e7	−0.618446	−0.309223	+	0.950990i	$0.600069\pi$
$860$	−2.66696e7	−1.22962
$861$	0	0
$862$	−3.00086e7	−1.37556
$863$	−1.99768e7	−0.913058	−0.456529	−	0.889708i	$-0.650908\pi$
$863$	−1.99768e7	−0.913058	−0.456529	+	0.889708i	$0.650908\pi$
$864$	0	0
$865$	3.06503e7	1.39282
$866$	−1.21496e7	−0.550513
$867$	0	0
$868$	−1.23151e7	−0.554802
$869$	8.10483e6	0.364078
$870$	0	0
$871$	4.35473e7	1.94498
$872$	−7.19928e6	−0.320625
$873$	0	0
$874$	1.06781e7	0.472840
$875$	−2.58973e7	−1.14349
$876$	0	0
$877$	8.81107e6	0.386838	0.193419	−	0.981116i	$-0.438042\pi$
$877$	8.81107e6	0.386838	0.193419	+	0.981116i	$0.438042\pi$
$878$	−4.00654e7	−1.75401
$879$	0	0
$880$	−1.38868e7	−0.604501
$881$	−4.01078e7	−1.74096	−0.870481	−	0.492202i	$-0.836192\pi$
$881$	−4.01078e7	−1.74096	−0.870481	+	0.492202i	$0.836192\pi$
$882$	0	0
$883$	1.49664e7	0.645976	0.322988	−	0.946403i	$-0.395313\pi$
$883$	1.49664e7	0.645976	0.322988	+	0.946403i	$0.395313\pi$
$884$	1.01143e7	0.435317
$885$	0	0
$886$	−5.44751e7	−2.33138
$887$	575144.	0.0245453	0.0122726	−	0.999925i	$-0.496093\pi$
$887$	575144.	0.0245453	0.0122726	+	0.999925i	$0.496093\pi$
$888$	0	0
$889$	−3.27634e6	−0.139038
$890$	1.18527e8	5.01583
$891$	0	0
$892$	−1.48501e7	−0.624910
$893$	−6.40685e6	−0.268854
$894$	0	0
$895$	−2.11462e7	−0.882417
$896$	−1.32653e7	−0.552009
$897$	0	0
$898$	−1.81577e7	−0.751400
$899$	−4.13213e6	−0.170520
$900$	0	0
$901$	8.34232e6	0.342353
$902$	6.12904e6	0.250828
$903$	0	0
$904$	6.44616e6	0.262349
$905$	2.54983e7	1.03488
$906$	0	0
$907$	−3.33367e7	−1.34557	−0.672783	−	0.739840i	$-0.734901\pi$
$907$	−3.33367e7	−1.34557	−0.672783	+	0.739840i	$0.734901\pi$
$908$	−3.96339e7	−1.59534
$909$	0	0
$910$	3.47998e7	1.39307
$911$	−2.17451e7	−0.868090	−0.434045	−	0.900891i	$-0.642914\pi$
$911$	−2.17451e7	−0.868090	−0.434045	+	0.900891i	$0.642914\pi$
$912$	0	0
$913$	6.61406e6	0.262598
$914$	5.30082e7	2.09883
$915$	0	0
$916$	6.53511e7	2.57344
$917$	7.53444e6	0.295888
$918$	0	0
$919$	−4.03986e6	−0.157789	−0.0788947	−	0.996883i	$-0.525139\pi$
$919$	−4.03986e6	−0.157789	−0.0788947	+	0.996883i	$0.525139\pi$
$920$	−4.48762e7	−1.74802
$921$	0	0
$922$	3.20381e7	1.24119
$923$	4.10844e7	1.58735
$924$	0	0
$925$	3.39202e7	1.30348
$926$	1.11853e7	0.428666
$927$	0	0
$928$	3.04096e6	0.115915
$929$	−7.69679e6	−0.292597	−0.146299	−	0.989240i	$-0.546736\pi$
$929$	−7.69679e6	−0.292597	−0.146299	+	0.989240i	$0.546736\pi$
$930$	0	0
$931$	−2.18011e6	−0.0824335
$932$	−4.12076e7	−1.55395
$933$	0	0
$934$	−3.85134e7	−1.44459
$935$	−2.16494e6	−0.0809874
$936$	0	0
$937$	−453558.	−0.0168766	−0.00843828	−	0.999964i	$-0.502686\pi$
$937$	−453558.	−0.0168766	−0.00843828	+	0.999964i	$0.502686\pi$
$938$	3.18480e7	1.18189
$939$	0	0
$940$	5.08596e7	1.87739
$941$	2.43852e7	0.897745	0.448873	−	0.893596i	$-0.351826\pi$
$941$	2.43852e7	0.897745	0.448873	+	0.893596i	$0.351826\pi$
$942$	0	0
$943$	7.83451e6	0.286901
$944$	−4.65648e7	−1.70070
$945$	0	0
$946$	−3.40400e6	−0.123669
$947$	2.18745e7	0.792615	0.396308	−	0.918118i	$-0.370291\pi$
$947$	2.18745e7	0.792615	0.396308	+	0.918118i	$0.370291\pi$
$948$	0	0
$949$	2.60777e7	0.939949
$950$	7.36479e7	2.64759
$951$	0	0
$952$	3.91608e6	0.140042
$953$	3.93319e7	1.40286	0.701428	−	0.712741i	$-0.252546\pi$
$953$	3.93319e7	1.40286	0.701428	+	0.712741i	$0.252546\pi$
$954$	0	0
$955$	−3.07807e7	−1.09212
$956$	−7.95698e7	−2.81581
$957$	0	0
$958$	1.43536e7	0.505297
$959$	1.25429e7	0.440405
$960$	0	0
$961$	−1.49687e7	−0.522849
$962$	−2.80194e7	−0.976160
$963$	0	0
$964$	−8.41099e7	−2.91511
$965$	−1.81741e7	−0.628254
$966$	0	0
$967$	3.85234e7	1.32483	0.662413	−	0.749139i	$-0.269532\pi$
$967$	3.85234e7	1.32483	0.662413	+	0.749139i	$0.269532\pi$
$968$	5.49313e7	1.88422
$969$	0	0
$970$	1.59897e8	5.45645
$971$	−2.43543e7	−0.828947	−0.414473	−	0.910061i	$-0.636034\pi$
$971$	−2.43543e7	−0.828947	−0.414473	+	0.910061i	$0.636034\pi$
$972$	0	0
$973$	−1.38633e7	−0.469444
$974$	−4.61097e7	−1.55738
$975$	0	0
$976$	−1.53820e7	−0.516880
$977$	−1.62534e7	−0.544763	−0.272382	−	0.962189i	$-0.587811\pi$
$977$	−1.62534e7	−0.544763	−0.272382	+	0.962189i	$0.587811\pi$
$978$	0	0
$979$	1.02873e7	0.343038
$980$	1.73064e7	0.575628
$981$	0	0
$982$	7.40518e7	2.45051
$983$	252456.	0.00833301	0.00416650	−	0.999991i	$-0.498674\pi$
$983$	252456.	0.00833301	0.00416650	+	0.999991i	$0.498674\pi$
$984$	0	0
$985$	−4.26109e7	−1.39936
$986$	2.48196e6	0.0813022
$987$	0	0
$988$	−4.13685e7	−1.34827
$989$	−4.35120e6	−0.141455
$990$	0	0
$991$	−2.49728e7	−0.807761	−0.403880	−	0.914812i	$-0.632339\pi$
$991$	−2.49728e7	−0.807761	−0.403880	+	0.914812i	$0.632339\pi$
$992$	−1.00531e7	−0.324356
$993$	0	0
$994$	3.00468e7	0.964567
$995$	−2.74727e7	−0.879717
$996$	0	0
$997$	−3.60983e7	−1.15013	−0.575067	−	0.818106i	$-0.695024\pi$
$997$	−3.60983e7	−1.15013	−0.575067	+	0.818106i	$0.695024\pi$
$998$	−3.93432e7	−1.25039
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.d.1.1	✓	1	3.2	odd	2
63.6.a.a.1.1		1	1.1	even	1	trivial
147.6.a.g.1.1		1	21.20	even	2
147.6.e.a.67.1		2	21.2	odd	6
147.6.e.a.79.1		2	21.11	odd	6
147.6.e.b.67.1		2	21.5	even	6
147.6.e.b.79.1		2	21.17	even	6
336.6.a.a.1.1		1	12.11	even	2
441.6.a.b.1.1		1	7.6	odd	2
525.6.a.a.1.1		1	15.14	odd	2
525.6.d.a.274.1		2	15.8	even	4
525.6.d.a.274.2		2	15.2	even	4
1008.6.a.bc.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 $\F_{q}$
Belyi maps

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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