LMFDB - Embedded newform 450.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} +1.00000 q^{7} -64.0000 q^{8} +O(q^{10})$

$q-4.00000 q^{2} +16.0000 q^{4} +1.00000 q^{7} -64.0000 q^{8} +210.000 q^{11} +667.000 q^{13} -4.00000 q^{14} +256.000 q^{16} +114.000 q^{17} +581.000 q^{19} -840.000 q^{22} -4350.00 q^{23} -2668.00 q^{26} +16.0000 q^{28} +126.000 q^{29} +7583.00 q^{31} -1024.00 q^{32} -456.000 q^{34} +3742.00 q^{37} -2324.00 q^{38} +2856.00 q^{41} +18241.0 q^{43} +3360.00 q^{44} +17400.0 q^{46} -23370.0 q^{47} -16806.0 q^{49} +10672.0 q^{52} -21684.0 q^{53} -64.0000 q^{56} -504.000 q^{58} +32310.0 q^{59} -7165.00 q^{61} -30332.0 q^{62} +4096.00 q^{64} -59579.0 q^{67} +1824.00 q^{68} +43080.0 q^{71} +28942.0 q^{73} -14968.0 q^{74} +9296.00 q^{76} +210.000 q^{77} +27608.0 q^{79} -11424.0 q^{82} -1782.00 q^{83} -72964.0 q^{86} -13440.0 q^{88} -50208.0 q^{89} +667.000 q^{91} -69600.0 q^{92} +93480.0 q^{94} -142793. q^{97} +67224.0 q^{98} +O(q^{100})$

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.00771356	0.00385678	−	0.999993i	$-0.498772\pi$
$7$	1.00000	0.00771356	0.00385678	+	0.999993i	$0.498772\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	0	0
$11$	210.000	0.523284	0.261642	−	0.965165i	$-0.415736\pi$
$11$	210.000	0.523284	0.261642	+	0.965165i	$0.415736\pi$
$12$	0	0
$13$	667.000	1.09463	0.547315	−	0.836927i	$-0.315650\pi$
$13$	667.000	1.09463	0.547315	+	0.836927i	$0.315650\pi$
$14$	−4.00000	−0.00545431
$15$	0	0
$16$	256.000	0.250000
$17$	114.000	0.0956715	0.0478357	−	0.998855i	$-0.484768\pi$
$17$	114.000	0.0956715	0.0478357	+	0.998855i	$0.484768\pi$
$18$	0	0
$19$	581.000	0.369226	0.184613	−	0.982811i	$-0.440897\pi$
$19$	581.000	0.369226	0.184613	+	0.982811i	$0.440897\pi$
$20$	0	0
$21$	0	0
$22$	−840.000	−0.370018
$23$	−4350.00	−1.71463	−0.857314	−	0.514795i	$-0.827868\pi$
$23$	−4350.00	−1.71463	−0.857314	+	0.514795i	$0.827868\pi$
$24$	0	0
$25$	0	0
$26$	−2668.00	−0.774020
$27$	0	0
$28$	16.0000	0.00385678
$29$	126.000	0.0278212	0.0139106	−	0.999903i	$-0.495572\pi$
$29$	126.000	0.0278212	0.0139106	+	0.999903i	$0.495572\pi$
$30$	0	0
$31$	7583.00	1.41722	0.708609	−	0.705601i	$-0.249323\pi$
$31$	7583.00	1.41722	0.708609	+	0.705601i	$0.249323\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−456.000	−0.0676500
$35$	0	0
$36$	0	0
$37$	3742.00	0.449365	0.224683	−	0.974432i	$-0.427865\pi$
$37$	3742.00	0.449365	0.224683	+	0.974432i	$0.427865\pi$
$38$	−2324.00	−0.261082
$39$	0	0
$40$	0	0
$41$	2856.00	0.265337	0.132669	−	0.991160i	$-0.457645\pi$
$41$	2856.00	0.265337	0.132669	+	0.991160i	$0.457645\pi$
$42$	0	0
$43$	18241.0	1.50445	0.752225	−	0.658907i	$-0.228981\pi$
$43$	18241.0	1.50445	0.752225	+	0.658907i	$0.228981\pi$
$44$	3360.00	0.261642
$45$	0	0
$46$	17400.0	1.21242
$47$	−23370.0	−1.54317	−0.771586	−	0.636126i	$-0.780536\pi$
$47$	−23370.0	−1.54317	−0.771586	+	0.636126i	$0.780536\pi$
$48$	0	0
$49$	−16806.0	−0.999941
$50$	0	0
$51$	0	0
$52$	10672.0	0.547315
$53$	−21684.0	−1.06035	−0.530176	−	0.847888i	$-0.677874\pi$
$53$	−21684.0	−1.06035	−0.530176	+	0.847888i	$0.677874\pi$
$54$	0	0
$55$	0	0
$56$	−64.0000	−0.00272716
$57$	0	0
$58$	−504.000	−0.0196725
$59$	32310.0	1.20839	0.604195	−	0.796837i	$-0.293495\pi$
$59$	32310.0	1.20839	0.604195	+	0.796837i	$0.293495\pi$
$60$	0	0
$61$	−7165.00	−0.246542	−0.123271	−	0.992373i	$-0.539339\pi$
$61$	−7165.00	−0.246542	−0.123271	+	0.992373i	$0.539339\pi$
$62$	−30332.0	−1.00212
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−59579.0	−1.62146	−0.810730	−	0.585420i	$-0.800929\pi$
$67$	−59579.0	−1.62146	−0.810730	+	0.585420i	$0.800929\pi$
$68$	1824.00	0.0478357
$69$	0	0
$70$	0	0
$71$	43080.0	1.01421	0.507107	−	0.861883i	$-0.330715\pi$
$71$	43080.0	1.01421	0.507107	+	0.861883i	$0.330715\pi$
$72$	0	0
$73$	28942.0	0.635655	0.317827	−	0.948149i	$-0.397047\pi$
$73$	28942.0	0.635655	0.317827	+	0.948149i	$0.397047\pi$
$74$	−14968.0	−0.317749
$75$	0	0
$76$	9296.00	0.184613
$77$	210.000	0.00403638
$78$	0	0
$79$	27608.0	0.497700	0.248850	−	0.968542i	$-0.419947\pi$
$79$	27608.0	0.497700	0.248850	+	0.968542i	$0.419947\pi$
$80$	0	0
$81$	0	0
$82$	−11424.0	−0.187622
$83$	−1782.00	−0.0283931	−0.0141965	−	0.999899i	$-0.504519\pi$
$83$	−1782.00	−0.0283931	−0.0141965	+	0.999899i	$0.504519\pi$
$84$	0	0
$85$	0	0
$86$	−72964.0	−1.06381
$87$	0	0
$88$	−13440.0	−0.185009
$89$	−50208.0	−0.671890	−0.335945	−	0.941882i	$-0.609056\pi$
$89$	−50208.0	−0.671890	−0.335945	+	0.941882i	$0.609056\pi$
$90$	0	0
$91$	667.000	0.00844350
$92$	−69600.0	−0.857314
$93$	0	0
$94$	93480.0	1.09119
$95$	0	0
$96$	0	0
$97$	−142793.	−1.54091	−0.770456	−	0.637494i	$-0.779971\pi$
$97$	−142793.	−1.54091	−0.770456	+	0.637494i	$0.779971\pi$
$98$	67224.0	0.707065
$99$	0	0
$100$	0	0
$101$	164052.	1.60021	0.800107	−	0.599857i	$-0.204776\pi$
$101$	164052.	1.60021	0.800107	+	0.599857i	$0.204776\pi$
$102$	0	0
$103$	107236.	0.995973	0.497986	−	0.867185i	$-0.334073\pi$
$103$	107236.	0.995973	0.497986	+	0.867185i	$0.334073\pi$
$104$	−42688.0	−0.387010
$105$	0	0
$106$	86736.0	0.749782
$107$	21012.0	0.177422	0.0887111	−	0.996057i	$-0.471725\pi$
$107$	21012.0	0.177422	0.0887111	+	0.996057i	$0.471725\pi$
$108$	0	0
$109$	237545.	1.91505	0.957524	−	0.288354i	$-0.0931079\pi$
$109$	237545.	1.91505	0.957524	+	0.288354i	$0.0931079\pi$
$110$	0	0
$111$	0	0
$112$	256.000	0.00192839
$113$	233388.	1.71942	0.859711	−	0.510781i	$-0.170644\pi$
$113$	233388.	1.71942	0.859711	+	0.510781i	$0.170644\pi$
$114$	0	0
$115$	0	0
$116$	2016.00	0.0139106
$117$	0	0
$118$	−129240.	−0.854460
$119$	114.000	0.000737968 0
$120$	0	0
$121$	−116951.	−0.726174
$122$	28660.0	0.174332
$123$	0	0
$124$	121328.	0.708609
$125$	0	0
$126$	0	0
$127$	−138800.	−0.763625	−0.381813	−	0.924240i	$-0.624700\pi$
$127$	−138800.	−0.763625	−0.381813	+	0.924240i	$0.624700\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	0	0
$131$	161340.	0.821417	0.410709	−	0.911767i	$-0.365281\pi$
$131$	161340.	0.821417	0.410709	+	0.911767i	$0.365281\pi$
$132$	0	0
$133$	581.000	0.00284805
$134$	238316.	1.14655
$135$	0	0
$136$	−7296.00	−0.0338250
$137$	69054.0	0.314331	0.157166	−	0.987572i	$-0.449764\pi$
$137$	69054.0	0.314331	0.157166	+	0.987572i	$0.449764\pi$
$138$	0	0
$139$	224396.	0.985095	0.492547	−	0.870286i	$-0.336066\pi$
$139$	224396.	0.985095	0.492547	+	0.870286i	$0.336066\pi$
$140$	0	0
$141$	0	0
$142$	−172320.	−0.717158
$143$	140070.	0.572803
$144$	0	0
$145$	0	0
$146$	−115768.	−0.449476
$147$	0	0
$148$	59872.0	0.224683
$149$	407646.	1.50424	0.752121	−	0.659025i	$-0.229031\pi$
$149$	407646.	1.50424	0.752121	+	0.659025i	$0.229031\pi$
$150$	0	0
$151$	216053.	0.771113	0.385556	−	0.922684i	$-0.374010\pi$
$151$	216053.	0.771113	0.385556	+	0.922684i	$0.374010\pi$
$152$	−37184.0	−0.130541
$153$	0	0
$154$	−840.000	−0.00285415
$155$	0	0
$156$	0	0
$157$	188617.	0.610705	0.305353	−	0.952239i	$-0.401226\pi$
$157$	188617.	0.610705	0.305353	+	0.952239i	$0.401226\pi$
$158$	−110432.	−0.351927
$159$	0	0
$160$	0	0
$161$	−4350.00	−0.0132259
$162$	0	0
$163$	422251.	1.24481	0.622403	−	0.782697i	$-0.286157\pi$
$163$	422251.	1.24481	0.622403	+	0.782697i	$0.286157\pi$
$164$	45696.0	0.132669
$165$	0	0
$166$	7128.00	0.0200769
$167$	41700.0	0.115703	0.0578515	−	0.998325i	$-0.481575\pi$
$167$	41700.0	0.115703	0.0578515	+	0.998325i	$0.481575\pi$
$168$	0	0
$169$	73596.0	0.198215
$170$	0	0
$171$	0	0
$172$	291856.	0.752225
$173$	266142.	0.676080	0.338040	−	0.941132i	$-0.390236\pi$
$173$	266142.	0.676080	0.338040	+	0.941132i	$0.390236\pi$
$174$	0	0
$175$	0	0
$176$	53760.0	0.130821
$177$	0	0
$178$	200832.	0.475098
$179$	51846.0	0.120944	0.0604718	−	0.998170i	$-0.480739\pi$
$179$	51846.0	0.120944	0.0604718	+	0.998170i	$0.480739\pi$
$180$	0	0
$181$	499349.	1.13294	0.566471	−	0.824082i	$-0.308308\pi$
$181$	499349.	1.13294	0.566471	+	0.824082i	$0.308308\pi$
$182$	−2668.00	−0.00597045
$183$	0	0
$184$	278400.	0.606212
$185$	0	0
$186$	0	0
$187$	23940.0	0.0500634
$188$	−373920.	−0.771586
$189$	0	0
$190$	0	0
$191$	−705906.	−1.40011	−0.700057	−	0.714087i	$-0.746842\pi$
$191$	−705906.	−1.40011	−0.700057	+	0.714087i	$0.746842\pi$
$192$	0	0
$193$	519973.	1.00482	0.502409	−	0.864630i	$-0.332447\pi$
$193$	519973.	1.00482	0.502409	+	0.864630i	$0.332447\pi$
$194$	571172.	1.08959
$195$	0	0
$196$	−268896.	−0.499970
$197$	−200394.	−0.367891	−0.183945	−	0.982936i	$-0.558887\pi$
$197$	−200394.	−0.367891	−0.183945	+	0.982936i	$0.558887\pi$
$198$	0	0
$199$	553673.	0.991107	0.495554	−	0.868577i	$-0.334965\pi$
$199$	553673.	0.991107	0.495554	+	0.868577i	$0.334965\pi$
$200$	0	0
$201$	0	0
$202$	−656208.	−1.13152
$203$	126.000	0.000214600 0
$204$	0	0
$205$	0	0
$206$	−428944.	−0.704259
$207$	0	0
$208$	170752.	0.273658
$209$	122010.	0.193210
$210$	0	0
$211$	777365.	1.20204	0.601020	−	0.799234i	$-0.294761\pi$
$211$	777365.	1.20204	0.601020	+	0.799234i	$0.294761\pi$
$212$	−346944.	−0.530176
$213$	0	0
$214$	−84048.0	−0.125456
$215$	0	0
$216$	0	0
$217$	7583.00	0.0109318
$218$	−950180.	−1.35414
$219$	0	0
$220$	0	0
$221$	76038.0	0.104725
$222$	0	0
$223$	776185.	1.04521	0.522604	−	0.852575i	$-0.324960\pi$
$223$	776185.	1.04521	0.522604	+	0.852575i	$0.324960\pi$
$224$	−1024.00	−0.00136358
$225$	0	0
$226$	−933552.	−1.21581
$227$	919992.	1.18500	0.592501	−	0.805569i	$-0.298140\pi$
$227$	919992.	1.18500	0.592501	+	0.805569i	$0.298140\pi$
$228$	0	0
$229$	−1.13220e6	−1.42670	−0.713350	−	0.700808i	$-0.752823\pi$
$229$	−1.13220e6	−1.42670	−0.713350	+	0.700808i	$0.752823\pi$
$230$	0	0
$231$	0	0
$232$	−8064.00	−0.00983627
$233$	823128.	0.993293	0.496647	−	0.867953i	$-0.334564\pi$
$233$	823128.	0.993293	0.496647	+	0.867953i	$0.334564\pi$
$234$	0	0
$235$	0	0
$236$	516960.	0.604195
$237$	0	0
$238$	−456.000	−0.000521822 0
$239$	−1.32836e6	−1.50426	−0.752129	−	0.659016i	$-0.770973\pi$
$239$	−1.32836e6	−1.50426	−0.752129	+	0.659016i	$0.770973\pi$
$240$	0	0
$241$	94937.0	0.105291	0.0526457	−	0.998613i	$-0.483235\pi$
$241$	94937.0	0.105291	0.0526457	+	0.998613i	$0.483235\pi$
$242$	467804.	0.513482
$243$	0	0
$244$	−114640.	−0.123271
$245$	0	0
$246$	0	0
$247$	387527.	0.404166
$248$	−485312.	−0.501062
$249$	0	0
$250$	0	0
$251$	128124.	0.128365	0.0641824	−	0.997938i	$-0.479556\pi$
$251$	128124.	0.128365	0.0641824	+	0.997938i	$0.479556\pi$
$252$	0	0
$253$	−913500.	−0.897237
$254$	555200.	0.539964
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.72285e6	−1.62710	−0.813552	−	0.581493i	$-0.802469\pi$
$257$	−1.72285e6	−1.62710	−0.813552	+	0.581493i	$0.802469\pi$
$258$	0	0
$259$	3742.00	0.00346621
$260$	0	0
$261$	0	0
$262$	−645360.	−0.580830
$263$	692160.	0.617045	0.308523	−	0.951217i	$-0.400165\pi$
$263$	692160.	0.617045	0.308523	+	0.951217i	$0.400165\pi$
$264$	0	0
$265$	0	0
$266$	−2324.00	−0.00201387
$267$	0	0
$268$	−953264.	−0.810730
$269$	−690894.	−0.582144	−0.291072	−	0.956701i	$-0.594012\pi$
$269$	−690894.	−0.582144	−0.291072	+	0.956701i	$0.594012\pi$
$270$	0	0
$271$	−1.04690e6	−0.865925	−0.432963	−	0.901412i	$-0.642532\pi$
$271$	−1.04690e6	−0.865925	−0.432963	+	0.901412i	$0.642532\pi$
$272$	29184.0	0.0239179
$273$	0	0
$274$	−276216.	−0.222266
$275$	0	0
$276$	0	0
$277$	1.07137e6	0.838955	0.419478	−	0.907766i	$-0.362213\pi$
$277$	1.07137e6	0.838955	0.419478	+	0.907766i	$0.362213\pi$
$278$	−897584.	−0.696567
$279$	0	0
$280$	0	0
$281$	1.40416e6	1.06084	0.530422	−	0.847734i	$-0.322034\pi$
$281$	1.40416e6	1.06084	0.530422	+	0.847734i	$0.322034\pi$
$282$	0	0
$283$	−1.68203e6	−1.24844	−0.624221	−	0.781248i	$-0.714584\pi$
$283$	−1.68203e6	−1.24844	−0.624221	+	0.781248i	$0.714584\pi$
$284$	689280.	0.507107
$285$	0	0
$286$	−560280.	−0.405033
$287$	2856.00	0.00204670
$288$	0	0
$289$	−1.40686e6	−0.990847
$290$	0	0
$291$	0	0
$292$	463072.	0.317827
$293$	1.76293e6	1.19968	0.599839	−	0.800120i	$-0.295231\pi$
$293$	1.76293e6	1.19968	0.599839	+	0.800120i	$0.295231\pi$
$294$	0	0
$295$	0	0
$296$	−239488.	−0.158875
$297$	0	0
$298$	−1.63058e6	−1.06366
$299$	−2.90145e6	−1.87688
$300$	0	0
$301$	18241.0	0.0116047
$302$	−864212.	−0.545259
$303$	0	0
$304$	148736.	0.0923065
$305$	0	0
$306$	0	0
$307$	−471905.	−0.285765	−0.142882	−	0.989740i	$-0.545637\pi$
$307$	−471905.	−0.285765	−0.142882	+	0.989740i	$0.545637\pi$
$308$	3360.00	0.00201819
$309$	0	0
$310$	0	0
$311$	1.64439e6	0.964060	0.482030	−	0.876155i	$-0.339900\pi$
$311$	1.64439e6	0.964060	0.482030	+	0.876155i	$0.339900\pi$
$312$	0	0
$313$	−2.37908e6	−1.37262	−0.686308	−	0.727311i	$-0.740770\pi$
$313$	−2.37908e6	−1.37262	−0.686308	+	0.727311i	$0.740770\pi$
$314$	−754468.	−0.431834
$315$	0	0
$316$	441728.	0.248850
$317$	1.32679e6	0.741574	0.370787	−	0.928718i	$-0.379088\pi$
$317$	1.32679e6	0.741574	0.370787	+	0.928718i	$0.379088\pi$
$318$	0	0
$319$	26460.0	0.0145584
$320$	0	0
$321$	0	0
$322$	17400.0	0.00935211
$323$	66234.0	0.0353244
$324$	0	0
$325$	0	0
$326$	−1.68900e6	−0.880211
$327$	0	0
$328$	−182784.	−0.0938110
$329$	−23370.0	−0.0119033
$330$	0	0
$331$	661172.	0.331699	0.165850	−	0.986151i	$-0.446963\pi$
$331$	661172.	0.331699	0.165850	+	0.986151i	$0.446963\pi$
$332$	−28512.0	−0.0141965
$333$	0	0
$334$	−166800.	−0.0818144
$335$	0	0
$336$	0	0
$337$	1.53675e6	0.737104	0.368552	−	0.929607i	$-0.379854\pi$
$337$	1.53675e6	0.737104	0.368552	+	0.929607i	$0.379854\pi$
$338$	−294384.	−0.140159
$339$	0	0
$340$	0	0
$341$	1.59243e6	0.741608
$342$	0	0
$343$	−33613.0	−0.0154267
$344$	−1.16742e6	−0.531903
$345$	0	0
$346$	−1.06457e6	−0.478061
$347$	3.86576e6	1.72350	0.861749	−	0.507334i	$-0.169369\pi$
$347$	3.86576e6	1.72350	0.861749	+	0.507334i	$0.169369\pi$
$348$	0	0
$349$	−4.23391e6	−1.86071	−0.930354	−	0.366663i	$-0.880500\pi$
$349$	−4.23391e6	−1.86071	−0.930354	+	0.366663i	$0.880500\pi$
$350$	0	0
$351$	0	0
$352$	−215040.	−0.0925044
$353$	−2.41459e6	−1.03135	−0.515675	−	0.856784i	$-0.672459\pi$
$353$	−2.41459e6	−1.03135	−0.515675	+	0.856784i	$0.672459\pi$
$354$	0	0
$355$	0	0
$356$	−803328.	−0.335945
$357$	0	0
$358$	−207384.	−0.0855200
$359$	−2.94802e6	−1.20724	−0.603620	−	0.797272i	$-0.706276\pi$
$359$	−2.94802e6	−1.20724	−0.603620	+	0.797272i	$0.706276\pi$
$360$	0	0
$361$	−2.13854e6	−0.863672
$362$	−1.99740e6	−0.801111
$363$	0	0
$364$	10672.0	0.00422175
$365$	0	0
$366$	0	0
$367$	−679793.	−0.263458	−0.131729	−	0.991286i	$-0.542053\pi$
$367$	−679793.	−0.263458	−0.131729	+	0.991286i	$0.542053\pi$
$368$	−1.11360e6	−0.428657
$369$	0	0
$370$	0	0
$371$	−21684.0	−0.00817908
$372$	0	0
$373$	3.60888e6	1.34308	0.671538	−	0.740970i	$-0.265634\pi$
$373$	3.60888e6	1.34308	0.671538	+	0.740970i	$0.265634\pi$
$374$	−95760.0	−0.0354002
$375$	0	0
$376$	1.49568e6	0.545593
$377$	84042.0	0.0304539
$378$	0	0
$379$	−386809.	−0.138324	−0.0691622	−	0.997605i	$-0.522033\pi$
$379$	−386809.	−0.138324	−0.0691622	+	0.997605i	$0.522033\pi$
$380$	0	0
$381$	0	0
$382$	2.82362e6	0.990030
$383$	−645060.	−0.224700	−0.112350	−	0.993669i	$-0.535838\pi$
$383$	−645060.	−0.224700	−0.112350	+	0.993669i	$0.535838\pi$
$384$	0	0
$385$	0	0
$386$	−2.07989e6	−0.710514
$387$	0	0
$388$	−2.28469e6	−0.770456
$389$	812976.	0.272398	0.136199	−	0.990682i	$-0.456511\pi$
$389$	812976.	0.272398	0.136199	+	0.990682i	$0.456511\pi$
$390$	0	0
$391$	−495900.	−0.164041
$392$	1.07558e6	0.353532
$393$	0	0
$394$	801576.	0.260138
$395$	0	0
$396$	0	0
$397$	988813.	0.314875	0.157437	−	0.987529i	$-0.449677\pi$
$397$	988813.	0.314875	0.157437	+	0.987529i	$0.449677\pi$
$398$	−2.21469e6	−0.700819
$399$	0	0
$400$	0	0
$401$	−2.14580e6	−0.666391	−0.333195	−	0.942858i	$-0.608127\pi$
$401$	−2.14580e6	−0.666391	−0.333195	+	0.942858i	$0.608127\pi$
$402$	0	0
$403$	5.05786e6	1.55133
$404$	2.62483e6	0.800107
$405$	0	0
$406$	−504.000	−0.000151745 0
$407$	785820.	0.235146
$408$	0	0
$409$	−5.77658e6	−1.70751	−0.853753	−	0.520678i	$-0.825679\pi$
$409$	−5.77658e6	−1.70751	−0.853753	+	0.520678i	$0.825679\pi$
$410$	0	0
$411$	0	0
$412$	1.71578e6	0.497986
$413$	32310.0	0.00932099
$414$	0	0
$415$	0	0
$416$	−683008.	−0.193505
$417$	0	0
$418$	−488040.	−0.136620
$419$	5.07318e6	1.41171	0.705855	−	0.708357i	$-0.250563\pi$
$419$	5.07318e6	1.41171	0.705855	+	0.708357i	$0.250563\pi$
$420$	0	0
$421$	−3.01297e6	−0.828494	−0.414247	−	0.910164i	$-0.635955\pi$
$421$	−3.01297e6	−0.828494	−0.414247	+	0.910164i	$0.635955\pi$
$422$	−3.10946e6	−0.849970
$423$	0	0
$424$	1.38778e6	0.374891
$425$	0	0
$426$	0	0
$427$	−7165.00	−0.00190172
$428$	336192.	0.0887111
$429$	0	0
$430$	0	0
$431$	−5.59449e6	−1.45067	−0.725333	−	0.688398i	$-0.758314\pi$
$431$	−5.59449e6	−1.45067	−0.725333	+	0.688398i	$0.758314\pi$
$432$	0	0
$433$	−5.63454e6	−1.44424	−0.722119	−	0.691769i	$-0.756832\pi$
$433$	−5.63454e6	−1.44424	−0.722119	+	0.691769i	$0.756832\pi$
$434$	−30332.0	−0.00772995
$435$	0	0
$436$	3.80072e6	0.957524
$437$	−2.52735e6	−0.633085
$438$	0	0
$439$	860663.	0.213143	0.106572	−	0.994305i	$-0.466013\pi$
$439$	860663.	0.213143	0.106572	+	0.994305i	$0.466013\pi$
$440$	0	0
$441$	0	0
$442$	−304152.	−0.0740517
$443$	−2.75335e6	−0.666580	−0.333290	−	0.942824i	$-0.608159\pi$
$443$	−2.75335e6	−0.666580	−0.333290	+	0.942824i	$0.608159\pi$
$444$	0	0
$445$	0	0
$446$	−3.10474e6	−0.739074
$447$	0	0
$448$	4096.00	0.000964195 0
$449$	2.83168e6	0.662869	0.331434	−	0.943478i	$-0.392467\pi$
$449$	2.83168e6	0.662869	0.331434	+	0.943478i	$0.392467\pi$
$450$	0	0
$451$	599760.	0.138847
$452$	3.73421e6	0.859711
$453$	0	0
$454$	−3.67997e6	−0.837924
$455$	0	0
$456$	0	0
$457$	4.66235e6	1.04427	0.522136	−	0.852862i	$-0.325135\pi$
$457$	4.66235e6	1.04427	0.522136	+	0.852862i	$0.325135\pi$
$458$	4.52878e6	1.00883
$459$	0	0
$460$	0	0
$461$	5.27154e6	1.15527	0.577637	−	0.816294i	$-0.303975\pi$
$461$	5.27154e6	1.15527	0.577637	+	0.816294i	$0.303975\pi$
$462$	0	0
$463$	7.16954e6	1.55431	0.777157	−	0.629307i	$-0.216661\pi$
$463$	7.16954e6	1.55431	0.777157	+	0.629307i	$0.216661\pi$
$464$	32256.0	0.00695530
$465$	0	0
$466$	−3.29251e6	−0.702365
$467$	4.30441e6	0.913316	0.456658	−	0.889642i	$-0.349046\pi$
$467$	4.30441e6	0.913316	0.456658	+	0.889642i	$0.349046\pi$
$468$	0	0
$469$	−59579.0	−0.0125072
$470$	0	0
$471$	0	0
$472$	−2.06784e6	−0.427230
$473$	3.83061e6	0.787254
$474$	0	0
$475$	0	0
$476$	1824.00	0.000368984 0
$477$	0	0
$478$	5.31346e6	1.06367
$479$	−190506.	−0.0379376	−0.0189688	−	0.999820i	$-0.506038\pi$
$479$	−190506.	−0.0379376	−0.0189688	+	0.999820i	$0.506038\pi$
$480$	0	0
$481$	2.49591e6	0.491889
$482$	−379748.	−0.0744523
$483$	0	0
$484$	−1.87122e6	−0.363087
$485$	0	0
$486$	0	0
$487$	128887.	0.0246256	0.0123128	−	0.999924i	$-0.496081\pi$
$487$	128887.	0.0246256	0.0123128	+	0.999924i	$0.496081\pi$
$488$	458560.	0.0871659
$489$	0	0
$490$	0	0
$491$	304296.	0.0569630	0.0284815	−	0.999594i	$-0.490933\pi$
$491$	304296.	0.0569630	0.0284815	+	0.999594i	$0.490933\pi$
$492$	0	0
$493$	14364.0	0.00266169
$494$	−1.55011e6	−0.285788
$495$	0	0
$496$	1.94125e6	0.354305
$497$	43080.0	0.00782321
$498$	0	0
$499$	6.41974e6	1.15416	0.577080	−	0.816688i	$-0.304192\pi$
$499$	6.41974e6	1.15416	0.577080	+	0.816688i	$0.304192\pi$
$500$	0	0
$501$	0	0
$502$	−512496.	−0.0907677
$503$	−1.55534e6	−0.274098	−0.137049	−	0.990564i	$-0.543762\pi$
$503$	−1.55534e6	−0.274098	−0.137049	+	0.990564i	$0.543762\pi$
$504$	0	0
$505$	0	0
$506$	3.65400e6	0.634442
$507$	0	0
$508$	−2.22080e6	−0.381813
$509$	541506.	0.0926422	0.0463211	−	0.998927i	$-0.485250\pi$
$509$	541506.	0.0926422	0.0463211	+	0.998927i	$0.485250\pi$
$510$	0	0
$511$	28942.0	0.00490316
$512$	−262144.	−0.0441942
$513$	0	0
$514$	6.89141e6	1.15054
$515$	0	0
$516$	0	0
$517$	−4.90770e6	−0.807517
$518$	−14968.0	−0.00245098
$519$	0	0
$520$	0	0
$521$	888762.	0.143447	0.0717235	−	0.997425i	$-0.477150\pi$
$521$	888762.	0.143447	0.0717235	+	0.997425i	$0.477150\pi$
$522$	0	0
$523$	−7.69950e6	−1.23086	−0.615430	−	0.788192i	$-0.711018\pi$
$523$	−7.69950e6	−1.23086	−0.615430	+	0.788192i	$0.711018\pi$
$524$	2.58144e6	0.410709
$525$	0	0
$526$	−2.76864e6	−0.436317
$527$	864462.	0.135587
$528$	0	0
$529$	1.24862e7	1.93995
$530$	0	0
$531$	0	0
$532$	9296.00	0.00142402
$533$	1.90495e6	0.290446
$534$	0	0
$535$	0	0
$536$	3.81306e6	0.573273
$537$	0	0
$538$	2.76358e6	0.411638
$539$	−3.52926e6	−0.523253
$540$	0	0
$541$	−3.01878e6	−0.443443	−0.221721	−	0.975110i	$-0.571168\pi$
$541$	−3.01878e6	−0.443443	−0.221721	+	0.975110i	$0.571168\pi$
$542$	4.18758e6	0.612301
$543$	0	0
$544$	−116736.	−0.0169125
$545$	0	0
$546$	0	0
$547$	−7.23481e6	−1.03385	−0.516926	−	0.856030i	$-0.672924\pi$
$547$	−7.23481e6	−1.03385	−0.516926	+	0.856030i	$0.672924\pi$
$548$	1.10486e6	0.157166
$549$	0	0
$550$	0	0
$551$	73206.0	0.0102723
$552$	0	0
$553$	27608.0	0.00383904
$554$	−4.28547e6	−0.593231
$555$	0	0
$556$	3.59034e6	0.492547
$557$	−9.88712e6	−1.35030	−0.675152	−	0.737678i	$-0.735922\pi$
$557$	−9.88712e6	−1.35030	−0.675152	+	0.737678i	$0.735922\pi$
$558$	0	0
$559$	1.21667e7	1.64682
$560$	0	0
$561$	0	0
$562$	−5.61665e6	−0.750130
$563$	946362.	0.125831	0.0629153	−	0.998019i	$-0.479960\pi$
$563$	946362.	0.125831	0.0629153	+	0.998019i	$0.479960\pi$
$564$	0	0
$565$	0	0
$566$	6.72813e6	0.882782
$567$	0	0
$568$	−2.75712e6	−0.358579
$569$	4.51552e6	0.584692	0.292346	−	0.956313i	$-0.405564\pi$
$569$	4.51552e6	0.584692	0.292346	+	0.956313i	$0.405564\pi$
$570$	0	0
$571$	7.57824e6	0.972699	0.486349	−	0.873764i	$-0.338328\pi$
$571$	7.57824e6	0.972699	0.486349	+	0.873764i	$0.338328\pi$
$572$	2.24112e6	0.286401
$573$	0	0
$574$	−11424.0	−0.00144723
$575$	0	0
$576$	0	0
$577$	−1.11688e7	−1.39659	−0.698295	−	0.715811i	$-0.746057\pi$
$577$	−1.11688e7	−1.39659	−0.698295	+	0.715811i	$0.746057\pi$
$578$	5.62744e6	0.700635
$579$	0	0
$580$	0	0
$581$	−1782.00	−0.000219012 0
$582$	0	0
$583$	−4.55364e6	−0.554865
$584$	−1.85229e6	−0.224738
$585$	0	0
$586$	−7.05170e6	−0.848301
$587$	1.01053e6	0.121047	0.0605236	−	0.998167i	$-0.480723\pi$
$587$	1.01053e6	0.121047	0.0605236	+	0.998167i	$0.480723\pi$
$588$	0	0
$589$	4.40572e6	0.523274
$590$	0	0
$591$	0	0
$592$	957952.	0.112341
$593$	1.13674e7	1.32746	0.663732	−	0.747970i	$-0.268971\pi$
$593$	1.13674e7	1.32746	0.663732	+	0.747970i	$0.268971\pi$
$594$	0	0
$595$	0	0
$596$	6.52234e6	0.752121
$597$	0	0
$598$	1.16058e7	1.32716
$599$	−2.56732e6	−0.292356	−0.146178	−	0.989258i	$-0.546697\pi$
$599$	−2.56732e6	−0.292356	−0.146178	+	0.989258i	$0.546697\pi$
$600$	0	0
$601$	1.59837e7	1.80506	0.902530	−	0.430626i	$-0.141707\pi$
$601$	1.59837e7	1.80506	0.902530	+	0.430626i	$0.141707\pi$
$602$	−72964.0	−0.00820573
$603$	0	0
$604$	3.45685e6	0.385556
$605$	0	0
$606$	0	0
$607$	1.01626e7	1.11952	0.559760	−	0.828655i	$-0.310893\pi$
$607$	1.01626e7	1.11952	0.559760	+	0.828655i	$0.310893\pi$
$608$	−594944.	−0.0652705
$609$	0	0
$610$	0	0
$611$	−1.55878e7	−1.68920
$612$	0	0
$613$	−5.83042e6	−0.626684	−0.313342	−	0.949640i	$-0.601449\pi$
$613$	−5.83042e6	−0.626684	−0.313342	+	0.949640i	$0.601449\pi$
$614$	1.88762e6	0.202066
$615$	0	0
$616$	−13440.0	−0.00142708
$617$	−1.51029e7	−1.59716	−0.798578	−	0.601891i	$-0.794414\pi$
$617$	−1.51029e7	−1.59716	−0.798578	+	0.601891i	$0.794414\pi$
$618$	0	0
$619$	−5.38515e6	−0.564900	−0.282450	−	0.959282i	$-0.591147\pi$
$619$	−5.38515e6	−0.564900	−0.282450	+	0.959282i	$0.591147\pi$
$620$	0	0
$621$	0	0
$622$	−6.57756e6	−0.681693
$623$	−50208.0	−0.00518266
$624$	0	0
$625$	0	0
$626$	9.51633e6	0.970585
$627$	0	0
$628$	3.01787e6	0.305353
$629$	426588.	0.0429914
$630$	0	0
$631$	−5.59345e6	−0.559251	−0.279625	−	0.960109i	$-0.590210\pi$
$631$	−5.59345e6	−0.559251	−0.279625	+	0.960109i	$0.590210\pi$
$632$	−1.76691e6	−0.175963
$633$	0	0
$634$	−5.30717e6	−0.524372
$635$	0	0
$636$	0	0
$637$	−1.12096e7	−1.09457
$638$	−105840.	−0.0102943
$639$	0	0
$640$	0	0
$641$	1.00072e7	0.961985	0.480993	−	0.876725i	$-0.340276\pi$
$641$	1.00072e7	0.961985	0.480993	+	0.876725i	$0.340276\pi$
$642$	0	0
$643$	9.95036e6	0.949098	0.474549	−	0.880229i	$-0.342611\pi$
$643$	9.95036e6	0.949098	0.474549	+	0.880229i	$0.342611\pi$
$644$	−69600.0	−0.00661294
$645$	0	0
$646$	−264936.	−0.0249781
$647$	−9.16904e6	−0.861119	−0.430560	−	0.902562i	$-0.641684\pi$
$647$	−9.16904e6	−0.861119	−0.430560	+	0.902562i	$0.641684\pi$
$648$	0	0
$649$	6.78510e6	0.632331
$650$	0	0
$651$	0	0
$652$	6.75602e6	0.622403
$653$	−1.77127e7	−1.62555	−0.812777	−	0.582575i	$-0.802045\pi$
$653$	−1.77127e7	−1.62555	−0.812777	+	0.582575i	$0.802045\pi$
$654$	0	0
$655$	0	0
$656$	731136.	0.0663344
$657$	0	0
$658$	93480.0	0.00841693
$659$	2.07949e7	1.86527	0.932637	−	0.360816i	$-0.117502\pi$
$659$	2.07949e7	1.86527	0.932637	+	0.360816i	$0.117502\pi$
$660$	0	0
$661$	−1.51745e6	−0.135086	−0.0675429	−	0.997716i	$-0.521516\pi$
$661$	−1.51745e6	−0.135086	−0.0675429	+	0.997716i	$0.521516\pi$
$662$	−2.64469e6	−0.234547
$663$	0	0
$664$	114048.	0.0100385
$665$	0	0
$666$	0	0
$667$	−548100.	−0.0477029
$668$	667200.	0.0578515
$669$	0	0
$670$	0	0
$671$	−1.50465e6	−0.129012
$672$	0	0
$673$	2.90483e6	0.247219	0.123610	−	0.992331i	$-0.460553\pi$
$673$	2.90483e6	0.247219	0.123610	+	0.992331i	$0.460553\pi$
$674$	−6.14700e6	−0.521211
$675$	0	0
$676$	1.17754e6	0.0991077
$677$	−7.06049e6	−0.592057	−0.296028	−	0.955179i	$-0.595662\pi$
$677$	−7.06049e6	−0.592057	−0.296028	+	0.955179i	$0.595662\pi$
$678$	0	0
$679$	−142793.	−0.0118859
$680$	0	0
$681$	0	0
$682$	−6.36972e6	−0.524396
$683$	3.81415e6	0.312857	0.156429	−	0.987689i	$-0.450002\pi$
$683$	3.81415e6	0.312857	0.156429	+	0.987689i	$0.450002\pi$
$684$	0	0
$685$	0	0
$686$	134452.	0.0109083
$687$	0	0
$688$	4.66970e6	0.376112
$689$	−1.44632e7	−1.16069
$690$	0	0
$691$	−8.95166e6	−0.713195	−0.356597	−	0.934258i	$-0.616063\pi$
$691$	−8.95166e6	−0.713195	−0.356597	+	0.934258i	$0.616063\pi$
$692$	4.25827e6	0.338040
$693$	0	0
$694$	−1.54630e7	−1.21870
$695$	0	0
$696$	0	0
$697$	325584.	0.0253852
$698$	1.69356e7	1.31572
$699$	0	0
$700$	0	0
$701$	−8.45297e6	−0.649702	−0.324851	−	0.945765i	$-0.605314\pi$
$701$	−8.45297e6	−0.649702	−0.324851	+	0.945765i	$0.605314\pi$
$702$	0	0
$703$	2.17410e6	0.165917
$704$	860160.	0.0654105
$705$	0	0
$706$	9.65834e6	0.729274
$707$	164052.	0.0123433
$708$	0	0
$709$	−1.18033e7	−0.881834	−0.440917	−	0.897548i	$-0.645347\pi$
$709$	−1.18033e7	−0.881834	−0.440917	+	0.897548i	$0.645347\pi$
$710$	0	0
$711$	0	0
$712$	3.21331e6	0.237549
$713$	−3.29861e7	−2.43000
$714$	0	0
$715$	0	0
$716$	829536.	0.0604718
$717$	0	0
$718$	1.17921e7	0.853648
$719$	2.68089e7	1.93400	0.967000	−	0.254778i	$-0.0820024\pi$
$719$	2.68089e7	1.93400	0.967000	+	0.254778i	$0.0820024\pi$
$720$	0	0
$721$	107236.	0.00768250
$722$	8.55415e6	0.610709
$723$	0	0
$724$	7.98958e6	0.566471
$725$	0	0
$726$	0	0
$727$	2.00638e6	0.140792	0.0703958	−	0.997519i	$-0.477574\pi$
$727$	2.00638e6	0.140792	0.0703958	+	0.997519i	$0.477574\pi$
$728$	−42688.0	−0.00298523
$729$	0	0
$730$	0	0
$731$	2.07947e6	0.143933
$732$	0	0
$733$	9.16554e6	0.630083	0.315042	−	0.949078i	$-0.397981\pi$
$733$	9.16554e6	0.630083	0.315042	+	0.949078i	$0.397981\pi$
$734$	2.71917e6	0.186293
$735$	0	0
$736$	4.45440e6	0.303106
$737$	−1.25116e7	−0.848484
$738$	0	0
$739$	1.55897e7	1.05009	0.525047	−	0.851073i	$-0.324048\pi$
$739$	1.55897e7	1.05009	0.525047	+	0.851073i	$0.324048\pi$
$740$	0	0
$741$	0	0
$742$	86736.0	0.00578349
$743$	−3.78304e6	−0.251402	−0.125701	−	0.992068i	$-0.540118\pi$
$743$	−3.78304e6	−0.251402	−0.125701	+	0.992068i	$0.540118\pi$
$744$	0	0
$745$	0	0
$746$	−1.44355e7	−0.949698
$747$	0	0
$748$	383040.	0.0250317
$749$	21012.0	0.00136856
$750$	0	0
$751$	1.24726e7	0.806972	0.403486	−	0.914986i	$-0.367798\pi$
$751$	1.24726e7	0.806972	0.403486	+	0.914986i	$0.367798\pi$
$752$	−5.98272e6	−0.385793
$753$	0	0
$754$	−336168.	−0.0215342
$755$	0	0
$756$	0	0
$757$	−1.87121e6	−0.118682	−0.0593408	−	0.998238i	$-0.518900\pi$
$757$	−1.87121e6	−0.118682	−0.0593408	+	0.998238i	$0.518900\pi$
$758$	1.54724e6	0.0978101
$759$	0	0
$760$	0	0
$761$	−2.53638e7	−1.58764	−0.793821	−	0.608152i	$-0.791911\pi$
$761$	−2.53638e7	−1.58764	−0.793821	+	0.608152i	$0.791911\pi$
$762$	0	0
$763$	237545.	0.0147718
$764$	−1.12945e7	−0.700057
$765$	0	0
$766$	2.58024e6	0.158887
$767$	2.15508e7	1.32274
$768$	0	0
$769$	−1.43729e7	−0.876452	−0.438226	−	0.898865i	$-0.644393\pi$
$769$	−1.43729e7	−0.876452	−0.438226	+	0.898865i	$0.644393\pi$
$770$	0	0
$771$	0	0
$772$	8.31957e6	0.502409
$773$	−1.64185e7	−0.988289	−0.494145	−	0.869380i	$-0.664519\pi$
$773$	−1.64185e7	−0.988289	−0.494145	+	0.869380i	$0.664519\pi$
$774$	0	0
$775$	0	0
$776$	9.13875e6	0.544794
$777$	0	0
$778$	−3.25190e6	−0.192614
$779$	1.65934e6	0.0979695
$780$	0	0
$781$	9.04680e6	0.530722
$782$	1.98360e6	0.115994
$783$	0	0
$784$	−4.30234e6	−0.249985
$785$	0	0
$786$	0	0
$787$	1.46320e7	0.842106	0.421053	−	0.907036i	$-0.361661\pi$
$787$	1.46320e7	0.842106	0.421053	+	0.907036i	$0.361661\pi$
$788$	−3.20630e6	−0.183945
$789$	0	0
$790$	0	0
$791$	233388.	0.0132629
$792$	0	0
$793$	−4.77906e6	−0.269873
$794$	−3.95525e6	−0.222650
$795$	0	0
$796$	8.85877e6	0.495554
$797$	2.61796e7	1.45988	0.729941	−	0.683511i	$-0.239548\pi$
$797$	2.61796e7	1.45988	0.729941	+	0.683511i	$0.239548\pi$
$798$	0	0
$799$	−2.66418e6	−0.147637
$800$	0	0
$801$	0	0
$802$	8.58322e6	0.471210
$803$	6.07782e6	0.332628
$804$	0	0
$805$	0	0
$806$	−2.02314e7	−1.09696
$807$	0	0
$808$	−1.04993e7	−0.565761
$809$	−2.99051e7	−1.60648	−0.803238	−	0.595658i	$-0.796891\pi$
$809$	−2.99051e7	−1.60648	−0.803238	+	0.595658i	$0.796891\pi$
$810$	0	0
$811$	−3.12873e7	−1.67038	−0.835190	−	0.549961i	$-0.814642\pi$
$811$	−3.12873e7	−1.67038	−0.835190	+	0.549961i	$0.814642\pi$
$812$	2016.00	0.000107300 0
$813$	0	0
$814$	−3.14328e6	−0.166273
$815$	0	0
$816$	0	0
$817$	1.05980e7	0.555482
$818$	2.31063e7	1.20739
$819$	0	0
$820$	0	0
$821$	3.24291e7	1.67910	0.839550	−	0.543282i	$-0.182818\pi$
$821$	3.24291e7	1.67910	0.839550	+	0.543282i	$0.182818\pi$
$822$	0	0
$823$	−1.91700e7	−0.986560	−0.493280	−	0.869871i	$-0.664202\pi$
$823$	−1.91700e7	−0.986560	−0.493280	+	0.869871i	$0.664202\pi$
$824$	−6.86310e6	−0.352130
$825$	0	0
$826$	−129240.	−0.00659093
$827$	−98052.0	−0.00498532	−0.00249266	−	0.999997i	$-0.500793\pi$
$827$	−98052.0	−0.00498532	−0.00249266	+	0.999997i	$0.500793\pi$
$828$	0	0
$829$	5.77426e6	0.291817	0.145908	−	0.989298i	$-0.453390\pi$
$829$	5.77426e6	0.291817	0.145908	+	0.989298i	$0.453390\pi$
$830$	0	0
$831$	0	0
$832$	2.73203e6	0.136829
$833$	−1.91588e6	−0.0956658
$834$	0	0
$835$	0	0
$836$	1.95216e6	0.0966050
$837$	0	0
$838$	−2.02927e7	−0.998229
$839$	1.69354e7	0.830599	0.415299	−	0.909685i	$-0.363677\pi$
$839$	1.69354e7	0.830599	0.415299	+	0.909685i	$0.363677\pi$
$840$	0	0
$841$	−2.04953e7	−0.999226
$842$	1.20519e7	0.585834
$843$	0	0
$844$	1.24378e7	0.601020
$845$	0	0
$846$	0	0
$847$	−116951.	−0.00560138
$848$	−5.55110e6	−0.265088
$849$	0	0
$850$	0	0
$851$	−1.62777e7	−0.770494
$852$	0	0
$853$	−5.04391e6	−0.237353	−0.118677	−	0.992933i	$-0.537865\pi$
$853$	−5.04391e6	−0.237353	−0.118677	+	0.992933i	$0.537865\pi$
$854$	28660.0	0.00134472
$855$	0	0
$856$	−1.34477e6	−0.0627282
$857$	1.28829e7	0.599188	0.299594	−	0.954067i	$-0.403149\pi$
$857$	1.28829e7	0.599188	0.299594	+	0.954067i	$0.403149\pi$
$858$	0	0
$859$	−3.59522e7	−1.66243	−0.831213	−	0.555954i	$-0.812353\pi$
$859$	−3.59522e7	−1.66243	−0.831213	+	0.555954i	$0.812353\pi$
$860$	0	0
$861$	0	0
$862$	2.23780e7	1.02578
$863$	2.20345e7	1.00711	0.503554	−	0.863964i	$-0.332025\pi$
$863$	2.20345e7	1.00711	0.503554	+	0.863964i	$0.332025\pi$
$864$	0	0
$865$	0	0
$866$	2.25382e7	1.02123
$867$	0	0
$868$	121328.	0.00546590
$869$	5.79768e6	0.260438
$870$	0	0
$871$	−3.97392e7	−1.77490
$872$	−1.52029e7	−0.677072
$873$	0	0
$874$	1.01094e7	0.447658
$875$	0	0
$876$	0	0
$877$	−1.34531e7	−0.590641	−0.295320	−	0.955398i	$-0.595426\pi$
$877$	−1.34531e7	−0.590641	−0.295320	+	0.955398i	$0.595426\pi$
$878$	−3.44265e6	−0.150715
$879$	0	0
$880$	0	0
$881$	−2.36331e7	−1.02584	−0.512922	−	0.858435i	$-0.671437\pi$
$881$	−2.36331e7	−1.02584	−0.512922	+	0.858435i	$0.671437\pi$
$882$	0	0
$883$	−3.49588e7	−1.50888	−0.754439	−	0.656370i	$-0.772091\pi$
$883$	−3.49588e7	−1.50888	−0.754439	+	0.656370i	$0.772091\pi$
$884$	1.21661e6	0.0523625
$885$	0	0
$886$	1.10134e7	0.471343
$887$	3.11168e7	1.32796	0.663982	−	0.747749i	$-0.268865\pi$
$887$	3.11168e7	1.32796	0.663982	+	0.747749i	$0.268865\pi$
$888$	0	0
$889$	−138800.	−0.00589027
$890$	0	0
$891$	0	0
$892$	1.24190e7	0.522604
$893$	−1.35780e7	−0.569779
$894$	0	0
$895$	0	0
$896$	−16384.0	−0.000681789 0
$897$	0	0
$898$	−1.13267e7	−0.468719
$899$	955458.	0.0394287
$900$	0	0
$901$	−2.47198e6	−0.101445
$902$	−2.39904e6	−0.0981796
$903$	0	0
$904$	−1.49368e7	−0.607907
$905$	0	0
$906$	0	0
$907$	1.97320e7	0.796438	0.398219	−	0.917290i	$-0.369628\pi$
$907$	1.97320e7	0.796438	0.398219	+	0.917290i	$0.369628\pi$
$908$	1.47199e7	0.592501
$909$	0	0
$910$	0	0
$911$	4.40720e7	1.75941	0.879704	−	0.475521i	$-0.157740\pi$
$911$	4.40720e7	1.75941	0.879704	+	0.475521i	$0.157740\pi$
$912$	0	0
$913$	−374220.	−0.0148576
$914$	−1.86494e7	−0.738412
$915$	0	0
$916$	−1.81151e7	−0.713350
$917$	161340.	0.00633605
$918$	0	0
$919$	−4.33621e7	−1.69364	−0.846821	−	0.531879i	$-0.821486\pi$
$919$	−4.33621e7	−1.69364	−0.846821	+	0.531879i	$0.821486\pi$
$920$	0	0
$921$	0	0
$922$	−2.10862e7	−0.816902
$923$	2.87344e7	1.11019
$924$	0	0
$925$	0	0
$926$	−2.86781e7	−1.09907
$927$	0	0
$928$	−129024.	−0.00491814
$929$	−1.65235e7	−0.628147	−0.314074	−	0.949399i	$-0.601694\pi$
$929$	−1.65235e7	−0.628147	−0.314074	+	0.949399i	$0.601694\pi$
$930$	0	0
$931$	−9.76429e6	−0.369204
$932$	1.31700e7	0.496647
$933$	0	0
$934$	−1.72176e7	−0.645812
$935$	0	0
$936$	0	0
$937$	−2.03546e7	−0.757381	−0.378691	−	0.925523i	$-0.623626\pi$
$937$	−2.03546e7	−0.757381	−0.378691	+	0.925523i	$0.623626\pi$
$938$	238316.	0.00884395
$939$	0	0
$940$	0	0
$941$	5.12406e7	1.88643	0.943215	−	0.332184i	$-0.107786\pi$
$941$	5.12406e7	1.88643	0.943215	+	0.332184i	$0.107786\pi$
$942$	0	0
$943$	−1.24236e7	−0.454955
$944$	8.27136e6	0.302097
$945$	0	0
$946$	−1.53224e7	−0.556673
$947$	2.79736e7	1.01361	0.506807	−	0.862059i	$-0.330826\pi$
$947$	2.79736e7	1.01361	0.506807	+	0.862059i	$0.330826\pi$
$948$	0	0
$949$	1.93043e7	0.695807
$950$	0	0
$951$	0	0
$952$	−7296.00	−0.000260911 0
$953$	2.68337e7	0.957080	0.478540	−	0.878066i	$-0.341166\pi$
$953$	2.68337e7	0.957080	0.478540	+	0.878066i	$0.341166\pi$
$954$	0	0
$955$	0	0
$956$	−2.12538e7	−0.752129
$957$	0	0
$958$	762024.	0.0268259
$959$	69054.0	0.00242461
$960$	0	0
$961$	2.88727e7	1.00851
$962$	−9.98366e6	−0.347818
$963$	0	0
$964$	1.51899e6	0.0526457
$965$	0	0
$966$	0	0
$967$	5.24831e7	1.80490	0.902450	−	0.430794i	$-0.141766\pi$
$967$	5.24831e7	1.80490	0.902450	+	0.430794i	$0.141766\pi$
$968$	7.48486e6	0.256741
$969$	0	0
$970$	0	0
$971$	−5.01029e7	−1.70536	−0.852678	−	0.522437i	$-0.825023\pi$
$971$	−5.01029e7	−1.70536	−0.852678	+	0.522437i	$0.825023\pi$
$972$	0	0
$973$	224396.	0.00759859
$974$	−515548.	−0.0174129
$975$	0	0
$976$	−1.83424e6	−0.0616356
$977$	−4.50181e7	−1.50887	−0.754433	−	0.656377i	$-0.772088\pi$
$977$	−4.50181e7	−1.50887	−0.754433	+	0.656377i	$0.772088\pi$
$978$	0	0
$979$	−1.05437e7	−0.351589
$980$	0	0
$981$	0	0
$982$	−1.21718e6	−0.0402789
$983$	−3.26907e7	−1.07905	−0.539523	−	0.841971i	$-0.681395\pi$
$983$	−3.26907e7	−1.07905	−0.539523	+	0.841971i	$0.681395\pi$
$984$	0	0
$985$	0	0
$986$	−57456.0	−0.00188210
$987$	0	0
$988$	6.20043e6	0.202083
$989$	−7.93484e7	−2.57957
$990$	0	0
$991$	6.24606e6	0.202033	0.101016	−	0.994885i	$-0.467791\pi$
$991$	6.24606e6	0.202033	0.101016	+	0.994885i	$0.467791\pi$
$992$	−7.76499e6	−0.250531
$993$	0	0
$994$	−172320.	−0.00553184
$995$	0	0
$996$	0	0
$997$	−2.85010e7	−0.908077	−0.454039	−	0.890982i	$-0.650017\pi$
$997$	−2.85010e7	−0.908077	−0.454039	+	0.890982i	$0.650017\pi$
$998$	−2.56790e7	−0.816115
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.g.1.1		1
3.2	odd	2		150.6.a.k.1.1	yes	1
5.2	odd	4		450.6.c.k.199.1		2
5.3	odd	4		450.6.c.k.199.2		2
5.4	even	2		450.6.a.r.1.1		1
15.2	even	4		150.6.c.a.49.2		2
15.8	even	4		150.6.c.a.49.1		2
15.14	odd	2		150.6.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.e.1.1	✓	1	15.14	odd	2
150.6.a.k.1.1	yes	1	3.2	odd	2
150.6.c.a.49.1		2	15.8	even	4
150.6.c.a.49.2		2	15.2	even	4
450.6.a.g.1.1		1	1.1	even	1	trivial
450.6.a.r.1.1		1	5.4	even	2
450.6.c.k.199.1		2	5.2	odd	4
450.6.c.k.199.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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