LMFDB - Embedded newform 150.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,6,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	150.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:	$24.0575729719$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -1.00000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -1.00000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -210.000 q^{11} +144.000 q^{12} -667.000 q^{13} +4.00000 q^{14} +256.000 q^{16} +114.000 q^{17} -324.000 q^{18} +581.000 q^{19} -9.00000 q^{21} +840.000 q^{22} -4350.00 q^{23} -576.000 q^{24} +2668.00 q^{26} +729.000 q^{27} -16.0000 q^{28} -126.000 q^{29} +7583.00 q^{31} -1024.00 q^{32} -1890.00 q^{33} -456.000 q^{34} +1296.00 q^{36} -3742.00 q^{37} -2324.00 q^{38} -6003.00 q^{39} -2856.00 q^{41} +36.0000 q^{42} -18241.0 q^{43} -3360.00 q^{44} +17400.0 q^{46} -23370.0 q^{47} +2304.00 q^{48} -16806.0 q^{49} +1026.00 q^{51} -10672.0 q^{52} -21684.0 q^{53} -2916.00 q^{54} +64.0000 q^{56} +5229.00 q^{57} +504.000 q^{58} -32310.0 q^{59} -7165.00 q^{61} -30332.0 q^{62} -81.0000 q^{63} +4096.00 q^{64} +7560.00 q^{66} +59579.0 q^{67} +1824.00 q^{68} -39150.0 q^{69} -43080.0 q^{71} -5184.00 q^{72} -28942.0 q^{73} +14968.0 q^{74} +9296.00 q^{76} +210.000 q^{77} +24012.0 q^{78} +27608.0 q^{79} +6561.00 q^{81} +11424.0 q^{82} -1782.00 q^{83} -144.000 q^{84} +72964.0 q^{86} -1134.00 q^{87} +13440.0 q^{88} +50208.0 q^{89} +667.000 q^{91} -69600.0 q^{92} +68247.0 q^{93} +93480.0 q^{94} -9216.00 q^{96} +142793. q^{97} +67224.0 q^{98} -17010.0 q^{99} +O(q^{100})\)

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−36.0000	−0.408248
$7$	−1.00000	−0.00771356	−0.00385678	−	0.999993i	$-0.501228\pi$
$7$	−1.00000	−0.00771356	−0.00385678	+	0.999993i	$0.501228\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	0	0
$11$	−210.000	−0.523284	−0.261642	−	0.965165i	$-0.584264\pi$
$11$	−210.000	−0.523284	−0.261642	+	0.965165i	$0.584264\pi$
$12$	144.000	0.288675
$13$	−667.000	−1.09463	−0.547315	−	0.836927i	$-0.684350\pi$
$13$	−667.000	−1.09463	−0.547315	+	0.836927i	$0.684350\pi$
$14$	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	$0.484768\pi$
$18$	−324.000	−0.235702
$19$	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
$21$	−9.00000	−0.00445343
$22$	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	$0.827868\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	2668.00	0.774020
$27$	729.000	0.192450
$28$	−16.0000	−0.00385678
$29$	−126.000	−0.0278212	−0.0139106	−	0.999903i	$-0.504428\pi$
$29$	−126.000	−0.0278212	−0.0139106	+	0.999903i	$0.504428\pi$
$30$	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
$33$	−1890.00	−0.302118
$34$	−456.000	−0.0676500
$35$	0	0
$36$	1296.00	0.166667
$37$	−3742.00	−0.449365	−0.224683	−	0.974432i	$-0.572135\pi$
$37$	−3742.00	−0.449365	−0.224683	+	0.974432i	$0.572135\pi$
$38$	−2324.00	−0.261082
$39$	−6003.00	−0.631985
$40$	0	0
$41$	−2856.00	−0.265337	−0.132669	−	0.991160i	$-0.542355\pi$
$41$	−2856.00	−0.265337	−0.132669	+	0.991160i	$0.542355\pi$
$42$	36.0000	0.00314905
$43$	−18241.0	−1.50445	−0.752225	−	0.658907i	$-0.771019\pi$
$43$	−18241.0	−1.50445	−0.752225	+	0.658907i	$0.771019\pi$
$44$	−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	$0.780536\pi$
$48$	2304.00	0.144338
$49$	−16806.0	−0.999941
$50$	0	0
$51$	1026.00	0.0552360
$52$	−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	$0.677874\pi$
$54$	−2916.00	−0.136083
$55$	0	0
$56$	64.0000	0.00272716
$57$	5229.00	0.213173
$58$	504.000	0.0196725
$59$	−32310.0	−1.20839	−0.604195	−	0.796837i	$-0.706505\pi$
$59$	−32310.0	−1.20839	−0.604195	+	0.796837i	$0.706505\pi$
$60$	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
$63$	−81.0000	−0.00257119
$64$	4096.00	0.125000
$65$	0	0
$66$	7560.00	0.213630
$67$	59579.0	1.62146	0.810730	−	0.585420i	$-0.199071\pi$
$67$	59579.0	1.62146	0.810730	+	0.585420i	$0.199071\pi$
$68$	1824.00	0.0478357
$69$	−39150.0	−0.989940
$70$	0	0
$71$	−43080.0	−1.01421	−0.507107	−	0.861883i	$-0.669285\pi$
$71$	−43080.0	−1.01421	−0.507107	+	0.861883i	$0.669285\pi$
$72$	−5184.00	−0.117851
$73$	−28942.0	−0.635655	−0.317827	−	0.948149i	$-0.602953\pi$
$73$	−28942.0	−0.635655	−0.317827	+	0.948149i	$0.602953\pi$
$74$	14968.0	0.317749
$75$	0	0
$76$	9296.00	0.184613
$77$	210.000	0.00403638
$78$	24012.0	0.446881
$79$	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
$81$	6561.00	0.111111
$82$	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	$0.504519\pi$
$84$	−144.000	−0.00222671
$85$	0	0
$86$	72964.0	1.06381
$87$	−1134.00	−0.0160626
$88$	13440.0	0.185009
$89$	50208.0	0.671890	0.335945	−	0.941882i	$-0.390944\pi$
$89$	50208.0	0.671890	0.335945	+	0.941882i	$0.390944\pi$
$90$	0	0
$91$	667.000	0.00844350
$92$	−69600.0	−0.857314
$93$	68247.0	0.818231
$94$	93480.0	1.09119
$95$	0	0
$96$	−9216.00	−0.102062
$97$	142793.	1.54091	0.770456	−	0.637494i	$-0.220029\pi$
$97$	142793.	1.54091	0.770456	+	0.637494i	$0.220029\pi$
$98$	67224.0	0.707065
$99$	−17010.0	−0.174428
$100$	0	0
$101$	−164052.	−1.60021	−0.800107	−	0.599857i	$-0.795224\pi$
$101$	−164052.	−1.60021	−0.800107	+	0.599857i	$0.795224\pi$
$102$	−4104.00	−0.0390577
$103$	−107236.	−0.995973	−0.497986	−	0.867185i	$-0.665927\pi$
$103$	−107236.	−0.995973	−0.497986	+	0.867185i	$0.665927\pi$
$104$	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	$0.471725\pi$
$108$	11664.0	0.0962250
$109$	237545.	1.91505	0.957524	−	0.288354i	$-0.0931079\pi$
$109$	237545.	1.91505	0.957524	+	0.288354i	$0.0931079\pi$
$110$	0	0
$111$	−33678.0	−0.259441
$112$	−256.000	−0.00192839
$113$	233388.	1.71942	0.859711	−	0.510781i	$-0.170644\pi$
$113$	233388.	1.71942	0.859711	+	0.510781i	$0.170644\pi$
$114$	−20916.0	−0.150736
$115$	0	0
$116$	−2016.00	−0.0139106
$117$	−54027.0	−0.364877
$118$	129240.	0.854460
$119$	−114.000	−0.000737968 0
$120$	0	0
$121$	−116951.	−0.726174
$122$	28660.0	0.174332
$123$	−25704.0	−0.153193
$124$	121328.	0.708609
$125$	0	0
$126$	324.000	0.00181810
$127$	138800.	0.763625	0.381813	−	0.924240i	$-0.375300\pi$
$127$	138800.	0.763625	0.381813	+	0.924240i	$0.375300\pi$
$128$	−16384.0	−0.0883883
$129$	−164169.	−0.868594
$130$	0	0
$131$	−161340.	−0.821417	−0.410709	−	0.911767i	$-0.634719\pi$
$131$	−161340.	−0.821417	−0.410709	+	0.911767i	$0.634719\pi$
$132$	−30240.0	−0.151059
$133$	−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	$0.449764\pi$
$138$	156600.	0.699994
$139$	224396.	0.985095	0.492547	−	0.870286i	$-0.336066\pi$
$139$	224396.	0.985095	0.492547	+	0.870286i	$0.336066\pi$
$140$	0	0
$141$	−210330.	−0.890950
$142$	172320.	0.717158
$143$	140070.	0.572803
$144$	20736.0	0.0833333
$145$	0	0
$146$	115768.	0.449476
$147$	−151254.	−0.577316
$148$	−59872.0	−0.224683
$149$	−407646.	−1.50424	−0.752121	−	0.659025i	$-0.770969\pi$
$149$	−407646.	−1.50424	−0.752121	+	0.659025i	$0.770969\pi$
$150$	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
$153$	9234.00	0.0318905
$154$	−840.000	−0.00285415
$155$	0	0
$156$	−96048.0	−0.315993
$157$	−188617.	−0.610705	−0.305353	−	0.952239i	$-0.598774\pi$
$157$	−188617.	−0.610705	−0.305353	+	0.952239i	$0.598774\pi$
$158$	−110432.	−0.351927
$159$	−195156.	−0.612194
$160$	0	0
$161$	4350.00	0.0132259
$162$	−26244.0	−0.0785674
$163$	−422251.	−1.24481	−0.622403	−	0.782697i	$-0.713843\pi$
$163$	−422251.	−1.24481	−0.622403	+	0.782697i	$0.713843\pi$
$164$	−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	$0.481575\pi$
$168$	576.000	0.00157452
$169$	73596.0	0.198215
$170$	0	0
$171$	47061.0	0.123075
$172$	−291856.	−0.752225
$173$	266142.	0.676080	0.338040	−	0.941132i	$-0.390236\pi$
$173$	266142.	0.676080	0.338040	+	0.941132i	$0.390236\pi$
$174$	4536.00	0.0113579
$175$	0	0
$176$	−53760.0	−0.130821
$177$	−290790.	−0.697664
$178$	−200832.	−0.475098
$179$	−51846.0	−0.120944	−0.0604718	−	0.998170i	$-0.519261\pi$
$179$	−51846.0	−0.120944	−0.0604718	+	0.998170i	$0.519261\pi$
$180$	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
$183$	−64485.0	−0.142341
$184$	278400.	0.606212
$185$	0	0
$186$	−272988.	−0.578577
$187$	−23940.0	−0.0500634
$188$	−373920.	−0.771586
$189$	−729.000	−0.00148448
$190$	0	0
$191$	705906.	1.40011	0.700057	−	0.714087i	$-0.253158\pi$
$191$	705906.	1.40011	0.700057	+	0.714087i	$0.253158\pi$
$192$	36864.0	0.0721688
$193$	−519973.	−1.00482	−0.502409	−	0.864630i	$-0.667553\pi$
$193$	−519973.	−1.00482	−0.502409	+	0.864630i	$0.667553\pi$
$194$	−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	$0.558887\pi$
$198$	68040.0	0.123339
$199$	553673.	0.991107	0.495554	−	0.868577i	$-0.334965\pi$
$199$	553673.	0.991107	0.495554	+	0.868577i	$0.334965\pi$
$200$	0	0
$201$	536211.	0.936150
$202$	656208.	1.13152
$203$	126.000	0.000214600 0
$204$	16416.0	0.0276180
$205$	0	0
$206$	428944.	0.704259
$207$	−352350.	−0.571542
$208$	−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
$213$	−387720.	−0.585557
$214$	−84048.0	−0.125456
$215$	0	0
$216$	−46656.0	−0.0680414
$217$	−7583.00	−0.0109318
$218$	−950180.	−1.35414
$219$	−260478.	−0.366996
$220$	0	0
$221$	−76038.0	−0.104725
$222$	134712.	0.183453
$223$	−776185.	−1.04521	−0.522604	−	0.852575i	$-0.675040\pi$
$223$	−776185.	−1.04521	−0.522604	+	0.852575i	$0.675040\pi$
$224$	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	$0.298140\pi$
$228$	83664.0	0.106586
$229$	−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
$231$	1890.00	0.00233041
$232$	8064.00	0.00983627
$233$	823128.	0.993293	0.496647	−	0.867953i	$-0.334564\pi$
$233$	823128.	0.993293	0.496647	+	0.867953i	$0.334564\pi$
$234$	216108.	0.258007
$235$	0	0
$236$	−516960.	−0.604195
$237$	248472.	0.287347
$238$	456.000	0.000521822 0
$239$	1.32836e6	1.50426	0.752129	−	0.659016i	$-0.229027\pi$
$239$	1.32836e6	1.50426	0.752129	+	0.659016i	$0.229027\pi$
$240$	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
$243$	59049.0	0.0641500
$244$	−114640.	−0.123271
$245$	0	0
$246$	102816.	0.108324
$247$	−387527.	−0.404166
$248$	−485312.	−0.501062
$249$	−16038.0	−0.0163928
$250$	0	0
$251$	−128124.	−0.128365	−0.0641824	−	0.997938i	$-0.520444\pi$
$251$	−128124.	−0.128365	−0.0641824	+	0.997938i	$0.520444\pi$
$252$	−1296.00	−0.00128559
$253$	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	$0.802469\pi$
$258$	656676.	0.614189
$259$	3742.00	0.00346621
$260$	0	0
$261$	−10206.0	−0.00927373
$262$	645360.	0.580830
$263$	692160.	0.617045	0.308523	−	0.951217i	$-0.400165\pi$
$263$	692160.	0.617045	0.308523	+	0.951217i	$0.400165\pi$
$264$	120960.	0.106815
$265$	0	0
$266$	2324.00	0.00201387
$267$	451872.	0.387916
$268$	953264.	0.810730
$269$	690894.	0.582144	0.291072	−	0.956701i	$-0.405988\pi$
$269$	690894.	0.582144	0.291072	+	0.956701i	$0.405988\pi$
$270$	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
$273$	6003.00	0.00487486
$274$	−276216.	−0.222266
$275$	0	0
$276$	−626400.	−0.494970
$277$	−1.07137e6	−0.838955	−0.419478	−	0.907766i	$-0.637787\pi$
$277$	−1.07137e6	−0.838955	−0.419478	+	0.907766i	$0.637787\pi$
$278$	−897584.	−0.696567
$279$	614223.	0.472406
$280$	0	0
$281$	−1.40416e6	−1.06084	−0.530422	−	0.847734i	$-0.677966\pi$
$281$	−1.40416e6	−1.06084	−0.530422	+	0.847734i	$0.677966\pi$
$282$	841320.	0.629997
$283$	1.68203e6	1.24844	0.624221	−	0.781248i	$-0.285416\pi$
$283$	1.68203e6	1.24844	0.624221	+	0.781248i	$0.285416\pi$
$284$	−689280.	−0.507107
$285$	0	0
$286$	−560280.	−0.405033
$287$	2856.00	0.00204670
$288$	−82944.0	−0.0589256
$289$	−1.40686e6	−0.990847
$290$	0	0
$291$	1.28514e6	0.889645
$292$	−463072.	−0.317827
$293$	1.76293e6	1.19968	0.599839	−	0.800120i	$-0.295231\pi$
$293$	1.76293e6	1.19968	0.599839	+	0.800120i	$0.295231\pi$
$294$	605016.	0.408224
$295$	0	0
$296$	239488.	0.158875
$297$	−153090.	−0.100706
$298$	1.63058e6	1.06366
$299$	2.90145e6	1.87688
$300$	0	0
$301$	18241.0	0.0116047
$302$	−864212.	−0.545259
$303$	−1.47647e6	−0.923884
$304$	148736.	0.0923065
$305$	0	0
$306$	−36936.0	−0.0225500
$307$	471905.	0.285765	0.142882	−	0.989740i	$-0.454363\pi$
$307$	471905.	0.285765	0.142882	+	0.989740i	$0.454363\pi$
$308$	3360.00	0.00201819
$309$	−965124.	−0.575025
$310$	0	0
$311$	−1.64439e6	−0.964060	−0.482030	−	0.876155i	$-0.660100\pi$
$311$	−1.64439e6	−0.964060	−0.482030	+	0.876155i	$0.660100\pi$
$312$	384192.	0.223440
$313$	2.37908e6	1.37262	0.686308	−	0.727311i	$-0.259230\pi$
$313$	2.37908e6	1.37262	0.686308	+	0.727311i	$0.259230\pi$
$314$	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	$0.379088\pi$
$318$	780624.	0.432887
$319$	26460.0	0.0145584
$320$	0	0
$321$	189108.	0.102435
$322$	−17400.0	−0.00935211
$323$	66234.0	0.0353244
$324$	104976.	0.0555556
$325$	0	0
$326$	1.68900e6	0.880211
$327$	2.13790e6	1.10565
$328$	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
$333$	−303102.	−0.149788
$334$	−166800.	−0.0818144
$335$	0	0
$336$	−2304.00	−0.00111336
$337$	−1.53675e6	−0.737104	−0.368552	−	0.929607i	$-0.620146\pi$
$337$	−1.53675e6	−0.737104	−0.368552	+	0.929607i	$0.620146\pi$
$338$	−294384.	−0.140159
$339$	2.10049e6	0.992709
$340$	0	0
$341$	−1.59243e6	−0.741608
$342$	−188244.	−0.0870274
$343$	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	$0.169369\pi$
$348$	−18144.0	−0.00803128
$349$	−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
$351$	−486243.	−0.210662
$352$	215040.	0.0925044
$353$	−2.41459e6	−1.03135	−0.515675	−	0.856784i	$-0.672459\pi$
$353$	−2.41459e6	−1.03135	−0.515675	+	0.856784i	$0.672459\pi$
$354$	1.16316e6	0.493323
$355$	0	0
$356$	803328.	0.335945
$357$	−1026.00	−0.000426066 0
$358$	207384.	0.0855200
$359$	2.94802e6	1.20724	0.603620	−	0.797272i	$-0.293724\pi$
$359$	2.94802e6	1.20724	0.603620	+	0.797272i	$0.293724\pi$
$360$	0	0
$361$	−2.13854e6	−0.863672
$362$	−1.99740e6	−0.801111
$363$	−1.05256e6	−0.419257
$364$	10672.0	0.00422175
$365$	0	0
$366$	257940.	0.100651
$367$	679793.	0.263458	0.131729	−	0.991286i	$-0.457947\pi$
$367$	679793.	0.263458	0.131729	+	0.991286i	$0.457947\pi$
$368$	−1.11360e6	−0.428657
$369$	−231336.	−0.0884458
$370$	0	0
$371$	21684.0	0.00817908
$372$	1.09195e6	0.409116
$373$	−3.60888e6	−1.34308	−0.671538	−	0.740970i	$-0.734366\pi$
$373$	−3.60888e6	−1.34308	−0.671538	+	0.740970i	$0.734366\pi$
$374$	95760.0	0.0354002
$375$	0	0
$376$	1.49568e6	0.545593
$377$	84042.0	0.0304539
$378$	2916.00	0.00104968
$379$	−386809.	−0.138324	−0.0691622	−	0.997605i	$-0.522033\pi$
$379$	−386809.	−0.138324	−0.0691622	+	0.997605i	$0.522033\pi$
$380$	0	0
$381$	1.24920e6	0.440879
$382$	−2.82362e6	−0.990030
$383$	−645060.	−0.224700	−0.112350	−	0.993669i	$-0.535838\pi$
$383$	−645060.	−0.224700	−0.112350	+	0.993669i	$0.535838\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	2.07989e6	0.710514
$387$	−1.47752e6	−0.501483
$388$	2.28469e6	0.770456
$389$	−812976.	−0.272398	−0.136199	−	0.990682i	$-0.543489\pi$
$389$	−812976.	−0.272398	−0.136199	+	0.990682i	$0.543489\pi$
$390$	0	0
$391$	−495900.	−0.164041
$392$	1.07558e6	0.353532
$393$	−1.45206e6	−0.474245
$394$	801576.	0.260138
$395$	0	0
$396$	−272160.	−0.0872140
$397$	−988813.	−0.314875	−0.157437	−	0.987529i	$-0.550323\pi$
$397$	−988813.	−0.314875	−0.157437	+	0.987529i	$0.550323\pi$
$398$	−2.21469e6	−0.700819
$399$	−5229.00	−0.00164432
$400$	0	0
$401$	2.14580e6	0.666391	0.333195	−	0.942858i	$-0.391873\pi$
$401$	2.14580e6	0.666391	0.333195	+	0.942858i	$0.391873\pi$
$402$	−2.14484e6	−0.661958
$403$	−5.05786e6	−1.55133
$404$	−2.62483e6	−0.800107
$405$	0	0
$406$	−504.000	−0.000151745 0
$407$	785820.	0.235146
$408$	−65664.0	−0.0195289
$409$	−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
$411$	621486.	0.181479
$412$	−1.71578e6	−0.497986
$413$	32310.0	0.00932099
$414$	1.40940e6	0.404141
$415$	0	0
$416$	683008.	0.193505
$417$	2.01956e6	0.568745
$418$	488040.	0.136620
$419$	−5.07318e6	−1.41171	−0.705855	−	0.708357i	$-0.749437\pi$
$419$	−5.07318e6	−1.41171	−0.705855	+	0.708357i	$0.749437\pi$
$420$	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
$423$	−1.89297e6	−0.514390
$424$	1.38778e6	0.374891
$425$	0	0
$426$	1.55088e6	0.414051
$427$	7165.00	0.00190172
$428$	336192.	0.0887111
$429$	1.26063e6	0.330708
$430$	0	0
$431$	5.59449e6	1.45067	0.725333	−	0.688398i	$-0.241686\pi$
$431$	5.59449e6	1.45067	0.725333	+	0.688398i	$0.241686\pi$
$432$	186624.	0.0481125
$433$	5.63454e6	1.44424	0.722119	−	0.691769i	$-0.243168\pi$
$433$	5.63454e6	1.44424	0.722119	+	0.691769i	$0.243168\pi$
$434$	30332.0	0.00772995
$435$	0	0
$436$	3.80072e6	0.957524
$437$	−2.52735e6	−0.633085
$438$	1.04191e6	0.259505
$439$	860663.	0.213143	0.106572	−	0.994305i	$-0.466013\pi$
$439$	860663.	0.213143	0.106572	+	0.994305i	$0.466013\pi$
$440$	0	0
$441$	−1.36129e6	−0.333314
$442$	304152.	0.0740517
$443$	−2.75335e6	−0.666580	−0.333290	−	0.942824i	$-0.608159\pi$
$443$	−2.75335e6	−0.666580	−0.333290	+	0.942824i	$0.608159\pi$
$444$	−538848.	−0.129721
$445$	0	0
$446$	3.10474e6	0.739074
$447$	−3.66881e6	−0.868474
$448$	−4096.00	−0.000964195 0
$449$	−2.83168e6	−0.662869	−0.331434	−	0.943478i	$-0.607533\pi$
$449$	−2.83168e6	−0.662869	−0.331434	+	0.943478i	$0.607533\pi$
$450$	0	0
$451$	599760.	0.138847
$452$	3.73421e6	0.859711
$453$	1.94448e6	0.445202
$454$	−3.67997e6	−0.837924
$455$	0	0
$456$	−334656.	−0.0753679
$457$	−4.66235e6	−1.04427	−0.522136	−	0.852862i	$-0.674865\pi$
$457$	−4.66235e6	−1.04427	−0.522136	+	0.852862i	$0.674865\pi$
$458$	4.52878e6	1.00883
$459$	83106.0	0.0184120
$460$	0	0
$461$	−5.27154e6	−1.15527	−0.577637	−	0.816294i	$-0.696025\pi$
$461$	−5.27154e6	−1.15527	−0.577637	+	0.816294i	$0.696025\pi$
$462$	−7560.00	−0.00164785
$463$	−7.16954e6	−1.55431	−0.777157	−	0.629307i	$-0.783339\pi$
$463$	−7.16954e6	−1.55431	−0.777157	+	0.629307i	$0.783339\pi$
$464$	−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	$0.349046\pi$
$468$	−864432.	−0.182438
$469$	−59579.0	−0.0125072
$470$	0	0
$471$	−1.69755e6	−0.352591
$472$	2.06784e6	0.427230
$473$	3.83061e6	0.787254
$474$	−993888.	−0.203185
$475$	0	0
$476$	−1824.00	−0.000368984 0
$477$	−1.75640e6	−0.353450
$478$	−5.31346e6	−1.06367
$479$	190506.	0.0379376	0.0189688	−	0.999820i	$-0.493962\pi$
$479$	190506.	0.0379376	0.0189688	+	0.999820i	$0.493962\pi$
$480$	0	0
$481$	2.49591e6	0.491889
$482$	−379748.	−0.0744523
$483$	39150.0	0.00763597
$484$	−1.87122e6	−0.363087
$485$	0	0
$486$	−236196.	−0.0453609
$487$	−128887.	−0.0246256	−0.0123128	−	0.999924i	$-0.503919\pi$
$487$	−128887.	−0.0246256	−0.0123128	+	0.999924i	$0.503919\pi$
$488$	458560.	0.0871659
$489$	−3.80026e6	−0.718689
$490$	0	0
$491$	−304296.	−0.0569630	−0.0284815	−	0.999594i	$-0.509067\pi$
$491$	−304296.	−0.0569630	−0.0284815	+	0.999594i	$0.509067\pi$
$492$	−411264.	−0.0765963
$493$	−14364.0	−0.00266169
$494$	1.55011e6	0.285788
$495$	0	0
$496$	1.94125e6	0.354305
$497$	43080.0	0.00782321
$498$	64152.0	0.0115914
$499$	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
$501$	375300.	0.0668012
$502$	512496.	0.0907677
$503$	−1.55534e6	−0.274098	−0.137049	−	0.990564i	$-0.543762\pi$
$503$	−1.55534e6	−0.274098	−0.137049	+	0.990564i	$0.543762\pi$
$504$	5184.00	0.000909052 0
$505$	0	0
$506$	−3.65400e6	−0.634442
$507$	662364.	0.114440
$508$	2.22080e6	0.381813
$509$	−541506.	−0.0926422	−0.0463211	−	0.998927i	$-0.514750\pi$
$509$	−541506.	−0.0926422	−0.0463211	+	0.998927i	$0.514750\pi$
$510$	0	0
$511$	28942.0	0.00490316
$512$	−262144.	−0.0441942
$513$	423549.	0.0710576
$514$	6.89141e6	1.15054
$515$	0	0
$516$	−2.62670e6	−0.434297
$517$	4.90770e6	0.807517
$518$	−14968.0	−0.00245098
$519$	2.39528e6	0.390335
$520$	0	0
$521$	−888762.	−0.143447	−0.0717235	−	0.997425i	$-0.522850\pi$
$521$	−888762.	−0.143447	−0.0717235	+	0.997425i	$0.522850\pi$
$522$	40824.0	0.00655752
$523$	7.69950e6	1.23086	0.615430	−	0.788192i	$-0.288982\pi$
$523$	7.69950e6	1.23086	0.615430	+	0.788192i	$0.288982\pi$
$524$	−2.58144e6	−0.410709
$525$	0	0
$526$	−2.76864e6	−0.436317
$527$	864462.	0.135587
$528$	−483840.	−0.0755296
$529$	1.24862e7	1.93995
$530$	0	0
$531$	−2.61711e6	−0.402796
$532$	−9296.00	−0.00142402
$533$	1.90495e6	0.290446
$534$	−1.80749e6	−0.274298
$535$	0	0
$536$	−3.81306e6	−0.573273
$537$	−466614.	−0.0698268
$538$	−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
$543$	4.49414e6	0.654104
$544$	−116736.	−0.0169125
$545$	0	0
$546$	−24012.0	−0.00344704
$547$	7.23481e6	1.03385	0.516926	−	0.856030i	$-0.327076\pi$
$547$	7.23481e6	1.03385	0.516926	+	0.856030i	$0.327076\pi$
$548$	1.10486e6	0.157166
$549$	−580365.	−0.0821808
$550$	0	0
$551$	−73206.0	−0.0102723
$552$	2.50560e6	0.349997
$553$	−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	$0.735922\pi$
$558$	−2.45689e6	−0.334042
$559$	1.21667e7	1.64682
$560$	0	0
$561$	−215460.	−0.0289041
$562$	5.61665e6	0.750130
$563$	946362.	0.125831	0.0629153	−	0.998019i	$-0.479960\pi$
$563$	946362.	0.125831	0.0629153	+	0.998019i	$0.479960\pi$
$564$	−3.36528e6	−0.445475
$565$	0	0
$566$	−6.72813e6	−0.882782
$567$	−6561.00	−0.000857062 0
$568$	2.75712e6	0.358579
$569$	−4.51552e6	−0.584692	−0.292346	−	0.956313i	$-0.594436\pi$
$569$	−4.51552e6	−0.584692	−0.292346	+	0.956313i	$0.594436\pi$
$570$	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
$573$	6.35315e6	0.808356
$574$	−11424.0	−0.00144723
$575$	0	0
$576$	331776.	0.0416667
$577$	1.11688e7	1.39659	0.698295	−	0.715811i	$-0.253943\pi$
$577$	1.11688e7	1.39659	0.698295	+	0.715811i	$0.253943\pi$
$578$	5.62744e6	0.700635
$579$	−4.67976e6	−0.580132
$580$	0	0
$581$	1782.00	0.000219012 0
$582$	−5.14055e6	−0.629074
$583$	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	$0.480723\pi$
$588$	−2.42006e6	−0.288658
$589$	4.40572e6	0.523274
$590$	0	0
$591$	−1.80355e6	−0.212402
$592$	−957952.	−0.112341
$593$	1.13674e7	1.32746	0.663732	−	0.747970i	$-0.268971\pi$
$593$	1.13674e7	1.32746	0.663732	+	0.747970i	$0.268971\pi$
$594$	612360.	0.0712100
$595$	0	0
$596$	−6.52234e6	−0.752121
$597$	4.98306e6	0.572216
$598$	−1.16058e7	−1.32716
$599$	2.56732e6	0.292356	0.146178	−	0.989258i	$-0.453303\pi$
$599$	2.56732e6	0.292356	0.146178	+	0.989258i	$0.453303\pi$
$600$	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
$603$	4.82590e6	0.540487
$604$	3.45685e6	0.385556
$605$	0	0
$606$	5.90587e6	0.653285
$607$	−1.01626e7	−1.11952	−0.559760	−	0.828655i	$-0.689107\pi$
$607$	−1.01626e7	−1.11952	−0.559760	+	0.828655i	$0.689107\pi$
$608$	−594944.	−0.0652705
$609$	1134.00	0.000123900 0
$610$	0	0
$611$	1.55878e7	1.68920
$612$	147744.	0.0159452
$613$	5.83042e6	0.626684	0.313342	−	0.949640i	$-0.398551\pi$
$613$	5.83042e6	0.626684	0.313342	+	0.949640i	$0.398551\pi$
$614$	−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	$0.794414\pi$
$618$	3.86050e6	0.406604
$619$	−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
$621$	−3.17115e6	−0.329980
$622$	6.57756e6	0.681693
$623$	−50208.0	−0.00518266
$624$	−1.53677e6	−0.157996
$625$	0	0
$626$	−9.51633e6	−0.970585
$627$	−1.09809e6	−0.111550
$628$	−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
$633$	6.99628e6	0.693998
$634$	−5.30717e6	−0.524372
$635$	0	0
$636$	−3.12250e6	−0.306097
$637$	1.12096e7	1.09457
$638$	−105840.	−0.0102943
$639$	−3.48948e6	−0.338072
$640$	0	0
$641$	−1.00072e7	−0.961985	−0.480993	−	0.876725i	$-0.659724\pi$
$641$	−1.00072e7	−0.961985	−0.480993	+	0.876725i	$0.659724\pi$
$642$	−756432.	−0.0724323
$643$	−9.95036e6	−0.949098	−0.474549	−	0.880229i	$-0.657389\pi$
$643$	−9.95036e6	−0.949098	−0.474549	+	0.880229i	$0.657389\pi$
$644$	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	$0.641684\pi$
$648$	−419904.	−0.0392837
$649$	6.78510e6	0.632331
$650$	0	0
$651$	−68247.0	−0.00631148
$652$	−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	$0.802045\pi$
$654$	−8.55162e6	−0.781815
$655$	0	0
$656$	−731136.	−0.0663344
$657$	−2.34430e6	−0.211885
$658$	−93480.0	−0.00841693
$659$	−2.07949e7	−1.86527	−0.932637	−	0.360816i	$-0.882498\pi$
$659$	−2.07949e7	−1.86527	−0.932637	+	0.360816i	$0.882498\pi$
$660$	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
$663$	−684342.	−0.0604630
$664$	114048.	0.0100385
$665$	0	0
$666$	1.21241e6	0.105916
$667$	548100.	0.0477029
$668$	667200.	0.0578515
$669$	−6.98566e6	−0.603452
$670$	0	0
$671$	1.50465e6	0.129012
$672$	9216.00	0.000787262 0
$673$	−2.90483e6	−0.247219	−0.123610	−	0.992331i	$-0.539447\pi$
$673$	−2.90483e6	−0.247219	−0.123610	+	0.992331i	$0.539447\pi$
$674$	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	$0.595662\pi$
$678$	−8.40197e6	−0.701951
$679$	−142793.	−0.0118859
$680$	0	0
$681$	8.27993e6	0.684162
$682$	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	$0.450002\pi$
$684$	752976.	0.0615376
$685$	0	0
$686$	−134452.	−0.0109083
$687$	−1.01898e7	−0.823705
$688$	−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
$693$	17010.0	0.00134546
$694$	−1.54630e7	−1.21870
$695$	0	0
$696$	72576.0	0.00567897
$697$	−325584.	−0.0253852
$698$	1.69356e7	1.31572
$699$	7.40815e6	0.573478
$700$	0	0
$701$	8.45297e6	0.649702	0.324851	−	0.945765i	$-0.394686\pi$
$701$	8.45297e6	0.649702	0.324851	+	0.945765i	$0.394686\pi$
$702$	1.94497e6	0.148960
$703$	−2.17410e6	−0.165917
$704$	−860160.	−0.0654105
$705$	0	0
$706$	9.65834e6	0.729274
$707$	164052.	0.0123433
$708$	−4.65264e6	−0.348832
$709$	−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
$711$	2.23625e6	0.165900
$712$	−3.21331e6	−0.237549
$713$	−3.29861e7	−2.43000
$714$	4104.00	0.000301274 0
$715$	0	0
$716$	−829536.	−0.0604718
$717$	1.19553e7	0.868484
$718$	−1.17921e7	−0.853648
$719$	−2.68089e7	−1.93400	−0.967000	−	0.254778i	$-0.917998\pi$
$719$	−2.68089e7	−1.93400	−0.967000	+	0.254778i	$0.917998\pi$
$720$	0	0
$721$	107236.	0.00768250
$722$	8.55415e6	0.610709
$723$	854433.	0.0607900
$724$	7.98958e6	0.566471
$725$	0	0
$726$	4.21024e6	0.296459
$727$	−2.00638e6	−0.140792	−0.0703958	−	0.997519i	$-0.522426\pi$
$727$	−2.00638e6	−0.140792	−0.0703958	+	0.997519i	$0.522426\pi$
$728$	−42688.0	−0.00298523
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.07947e6	−0.143933
$732$	−1.03176e6	−0.0711707
$733$	−9.16554e6	−0.630083	−0.315042	−	0.949078i	$-0.602019\pi$
$733$	−9.16554e6	−0.630083	−0.315042	+	0.949078i	$0.602019\pi$
$734$	−2.71917e6	−0.186293
$735$	0	0
$736$	4.45440e6	0.303106
$737$	−1.25116e7	−0.848484
$738$	925344.	0.0625406
$739$	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
$741$	−3.48774e6	−0.233345
$742$	−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	$0.540118\pi$
$744$	−4.36781e6	−0.289289
$745$	0	0
$746$	1.44355e7	0.949698
$747$	−144342.	−0.00946436
$748$	−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
$753$	−1.15312e6	−0.0741115
$754$	−336168.	−0.0215342
$755$	0	0
$756$	−11664.0	−0.000742238 0
$757$	1.87121e6	0.118682	0.0593408	−	0.998238i	$-0.481100\pi$
$757$	1.87121e6	0.118682	0.0593408	+	0.998238i	$0.481100\pi$
$758$	1.54724e6	0.0978101
$759$	8.22150e6	0.518020
$760$	0	0
$761$	2.53638e7	1.58764	0.793821	−	0.608152i	$-0.208089\pi$
$761$	2.53638e7	1.58764	0.793821	+	0.608152i	$0.208089\pi$
$762$	−4.99680e6	−0.311749
$763$	−237545.	−0.0147718
$764$	1.12945e7	0.700057
$765$	0	0
$766$	2.58024e6	0.158887
$767$	2.15508e7	1.32274
$768$	589824.	0.0360844
$769$	−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
$771$	−1.55057e7	−0.939409
$772$	−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	$0.664519\pi$
$774$	5.91008e6	0.354602
$775$	0	0
$776$	−9.13875e6	−0.544794
$777$	33678.0	0.00200121
$778$	3.25190e6	0.192614
$779$	−1.65934e6	−0.0979695
$780$	0	0
$781$	9.04680e6	0.530722
$782$	1.98360e6	0.115994
$783$	−91854.0	−0.00535419
$784$	−4.30234e6	−0.249985
$785$	0	0
$786$	5.80824e6	0.335342
$787$	−1.46320e7	−0.842106	−0.421053	−	0.907036i	$-0.638339\pi$
$787$	−1.46320e7	−0.842106	−0.421053	+	0.907036i	$0.638339\pi$
$788$	−3.20630e6	−0.183945
$789$	6.22944e6	0.356251
$790$	0	0
$791$	−233388.	−0.0132629
$792$	1.08864e6	0.0616696
$793$	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	$0.239548\pi$
$798$	20916.0	0.00116271
$799$	−2.66418e6	−0.147637
$800$	0	0
$801$	4.06685e6	0.223963
$802$	−8.58322e6	−0.471210
$803$	6.07782e6	0.332628
$804$	8.57938e6	0.468075
$805$	0	0
$806$	2.02314e7	1.09696
$807$	6.21805e6	0.336101
$808$	1.04993e7	0.565761
$809$	2.99051e7	1.60648	0.803238	−	0.595658i	$-0.203109\pi$
$809$	2.99051e7	1.60648	0.803238	+	0.595658i	$0.203109\pi$
$810$	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
$813$	−9.42206e6	−0.499942
$814$	−3.14328e6	−0.166273
$815$	0	0
$816$	262656.	0.0138090
$817$	−1.05980e7	−0.555482
$818$	2.31063e7	1.20739
$819$	54027.0	0.00281450
$820$	0	0
$821$	−3.24291e7	−1.67910	−0.839550	−	0.543282i	$-0.817182\pi$
$821$	−3.24291e7	−1.67910	−0.839550	+	0.543282i	$0.817182\pi$
$822$	−2.48594e6	−0.128325
$823$	1.91700e7	0.986560	0.493280	−	0.869871i	$-0.335798\pi$
$823$	1.91700e7	0.986560	0.493280	+	0.869871i	$0.335798\pi$
$824$	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	$0.500793\pi$
$828$	−5.63760e6	−0.285771
$829$	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
$831$	−9.64230e6	−0.484371
$832$	−2.73203e6	−0.136829
$833$	−1.91588e6	−0.0956658
$834$	−8.07826e6	−0.402163
$835$	0	0
$836$	−1.95216e6	−0.0966050
$837$	5.52801e6	0.272744
$838$	2.02927e7	0.998229
$839$	−1.69354e7	−0.830599	−0.415299	−	0.909685i	$-0.636323\pi$
$839$	−1.69354e7	−0.830599	−0.415299	+	0.909685i	$0.636323\pi$
$840$	0	0
$841$	−2.04953e7	−0.999226
$842$	1.20519e7	0.585834
$843$	−1.26375e7	−0.612478
$844$	1.24378e7	0.601020
$845$	0	0
$846$	7.57188e6	0.363729
$847$	116951.	0.00560138
$848$	−5.55110e6	−0.265088
$849$	1.51383e7	0.720788
$850$	0	0
$851$	1.62777e7	0.770494
$852$	−6.20352e6	−0.292779
$853$	5.04391e6	0.237353	0.118677	−	0.992933i	$-0.462135\pi$
$853$	5.04391e6	0.237353	0.118677	+	0.992933i	$0.462135\pi$
$854$	−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	$0.403149\pi$
$858$	−5.04252e6	−0.233846
$859$	−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
$861$	25704.0	0.00118166
$862$	−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	$0.332025\pi$
$864$	−746496.	−0.0340207
$865$	0	0
$866$	−2.25382e7	−1.02123
$867$	−1.26617e7	−0.572066
$868$	−121328.	−0.00546590
$869$	−5.79768e6	−0.260438
$870$	0	0
$871$	−3.97392e7	−1.77490
$872$	−1.52029e7	−0.677072
$873$	1.15662e7	0.513637
$874$	1.01094e7	0.447658
$875$	0	0
$876$	−4.16765e6	−0.183498
$877$	1.34531e7	0.590641	0.295320	−	0.955398i	$-0.404574\pi$
$877$	1.34531e7	0.590641	0.295320	+	0.955398i	$0.404574\pi$
$878$	−3.44265e6	−0.150715
$879$	1.58663e7	0.692635
$880$	0	0
$881$	2.36331e7	1.02584	0.512922	−	0.858435i	$-0.328563\pi$
$881$	2.36331e7	1.02584	0.512922	+	0.858435i	$0.328563\pi$
$882$	5.44514e6	0.235688
$883$	3.49588e7	1.50888	0.754439	−	0.656370i	$-0.227909\pi$
$883$	3.49588e7	1.50888	0.754439	+	0.656370i	$0.227909\pi$
$884$	−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	$0.268865\pi$
$888$	2.15539e6	0.0917263
$889$	−138800.	−0.00589027
$890$	0	0
$891$	−1.37781e6	−0.0581427
$892$	−1.24190e7	−0.522604
$893$	−1.35780e7	−0.569779
$894$	1.46753e7	0.614104
$895$	0	0
$896$	16384.0	0.000681789 0
$897$	2.61130e7	1.08362
$898$	1.13267e7	0.468719
$899$	−955458.	−0.0394287
$900$	0	0
$901$	−2.47198e6	−0.101445
$902$	−2.39904e6	−0.0981796
$903$	164169.	0.00669995
$904$	−1.49368e7	−0.607907
$905$	0	0
$906$	−7.77791e6	−0.314805
$907$	−1.97320e7	−0.796438	−0.398219	−	0.917290i	$-0.630372\pi$
$907$	−1.97320e7	−0.796438	−0.398219	+	0.917290i	$0.630372\pi$
$908$	1.47199e7	0.592501
$909$	−1.32882e7	−0.533405
$910$	0	0
$911$	−4.40720e7	−1.75941	−0.879704	−	0.475521i	$-0.842260\pi$
$911$	−4.40720e7	−1.75941	−0.879704	+	0.475521i	$0.842260\pi$
$912$	1.33862e6	0.0532932
$913$	374220.	0.0148576
$914$	1.86494e7	0.738412
$915$	0	0
$916$	−1.81151e7	−0.713350
$917$	161340.	0.00633605
$918$	−332424.	−0.0130192
$919$	−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
$921$	4.24714e6	0.164986
$922$	2.10862e7	0.816902
$923$	2.87344e7	1.11019
$924$	30240.0	0.00116520
$925$	0	0
$926$	2.86781e7	1.09907
$927$	−8.68612e6	−0.331991
$928$	129024.	0.00491814
$929$	1.65235e7	0.628147	0.314074	−	0.949399i	$-0.398306\pi$
$929$	1.65235e7	0.628147	0.314074	+	0.949399i	$0.398306\pi$
$930$	0	0
$931$	−9.76429e6	−0.369204
$932$	1.31700e7	0.496647
$933$	−1.47995e7	−0.556600
$934$	−1.72176e7	−0.645812
$935$	0	0
$936$	3.45773e6	0.129003
$937$	2.03546e7	0.757381	0.378691	−	0.925523i	$-0.376374\pi$
$937$	2.03546e7	0.757381	0.378691	+	0.925523i	$0.376374\pi$
$938$	238316.	0.00884395
$939$	2.14117e7	0.792480
$940$	0	0
$941$	−5.12406e7	−1.88643	−0.943215	−	0.332184i	$-0.892214\pi$
$941$	−5.12406e7	−1.88643	−0.943215	+	0.332184i	$0.892214\pi$
$942$	6.79021e6	0.249319
$943$	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	$0.330826\pi$
$948$	3.97555e6	0.143673
$949$	1.93043e7	0.695807
$950$	0	0
$951$	1.19411e7	0.428148
$952$	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	$0.341166\pi$
$954$	7.02562e6	0.249927
$955$	0	0
$956$	2.12538e7	0.752129
$957$	238140.	0.00840529
$958$	−762024.	−0.0268259
$959$	−69054.0	−0.00242461
$960$	0	0
$961$	2.88727e7	1.00851
$962$	−9.98366e6	−0.347818
$963$	1.70197e6	0.0591407
$964$	1.51899e6	0.0526457
$965$	0	0
$966$	−156600.	−0.00539944
$967$	−5.24831e7	−1.80490	−0.902450	−	0.430794i	$-0.858234\pi$
$967$	−5.24831e7	−1.80490	−0.902450	+	0.430794i	$0.858234\pi$
$968$	7.48486e6	0.256741
$969$	596106.	0.0203945
$970$	0	0
$971$	5.01029e7	1.70536	0.852678	−	0.522437i	$-0.174977\pi$
$971$	5.01029e7	1.70536	0.852678	+	0.522437i	$0.174977\pi$
$972$	944784.	0.0320750
$973$	−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	$0.772088\pi$
$978$	1.52010e7	0.508190
$979$	−1.05437e7	−0.351589
$980$	0	0
$981$	1.92411e7	0.638349
$982$	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	$0.681395\pi$
$984$	1.64506e6	0.0541618
$985$	0	0
$986$	57456.0	0.00188210
$987$	210330.	0.00687240
$988$	−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
$993$	5.95055e6	0.191507
$994$	−172320.	−0.00553184
$995$	0	0
$996$	−256608.	−0.00819638
$997$	2.85010e7	0.908077	0.454039	−	0.890982i	$-0.349983\pi$
$997$	2.85010e7	0.908077	0.454039	+	0.890982i	$0.349983\pi$
$998$	−2.56790e7	−0.816115
$999$	−2.72792e6	−0.0864804

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.e.1.1	✓	1	1.1	even	1	trivial
150.6.a.k.1.1	yes	1	5.4	even	2
150.6.c.a.49.1		2	5.2	odd	4
150.6.c.a.49.2		2	5.3	odd	4
450.6.a.g.1.1		1	15.14	odd	2
450.6.a.r.1.1		1	3.2	odd	2
450.6.c.k.199.1		2	15.8	even	4
450.6.c.k.199.2		2	15.2	even	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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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