LMFDB - Embedded newform 60.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,6,Mod(1,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("60.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	60.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:	$9.62302918878$
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	$\chi$	$=$	60.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} +25.0000 q^{5} -16.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} +25.0000 q^{5} -16.0000 q^{7} +81.0000 q^{9} -564.000 q^{11} -370.000 q^{13} -225.000 q^{15} -1086.00 q^{17} -2860.00 q^{19} +144.000 q^{21} +1584.00 q^{23} +625.000 q^{25} -729.000 q^{27} +1134.00 q^{29} -6016.00 q^{31} +5076.00 q^{33} -400.000 q^{35} -538.000 q^{37} +3330.00 q^{39} +11370.0 q^{41} +5444.00 q^{43} +2025.00 q^{45} +10296.0 q^{47} -16551.0 q^{49} +9774.00 q^{51} +34758.0 q^{53} -14100.0 q^{55} +25740.0 q^{57} -26196.0 q^{59} +9422.00 q^{61} -1296.00 q^{63} -9250.00 q^{65} -51124.0 q^{67} -14256.0 q^{69} +14520.0 q^{71} -22678.0 q^{73} -5625.00 q^{75} +9024.00 q^{77} -97312.0 q^{79} +6561.00 q^{81} -7956.00 q^{83} -27150.0 q^{85} -10206.0 q^{87} -47910.0 q^{89} +5920.00 q^{91} +54144.0 q^{93} -71500.0 q^{95} +140738. q^{97} -45684.0 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−16.0000	−0.123417	−0.0617085	−	0.998094i	$-0.519655\pi$
$7$	−16.0000	−0.123417	−0.0617085	+	0.998094i	$0.519655\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−564.000	−1.40539	−0.702696	−	0.711490i	$-0.748021\pi$
$11$	−564.000	−1.40539	−0.702696	+	0.711490i	$0.748021\pi$
$12$	0	0
$13$	−370.000	−0.607216	−0.303608	−	0.952797i	$-0.598191\pi$
$13$	−370.000	−0.607216	−0.303608	+	0.952797i	$0.598191\pi$
$14$	0	0
$15$	−225.000	−0.258199
$16$	0	0
$17$	−1086.00	−0.911397	−0.455698	−	0.890134i	$-0.650610\pi$
$17$	−1086.00	−0.911397	−0.455698	+	0.890134i	$0.650610\pi$
$18$	0	0
$19$	−2860.00	−1.81753	−0.908766	−	0.417306i	$-0.862974\pi$
$19$	−2860.00	−1.81753	−0.908766	+	0.417306i	$0.862974\pi$
$20$	0	0
$21$	144.000	0.0712548
$22$	0	0
$23$	1584.00	0.624361	0.312180	−	0.950023i	$-0.398941\pi$
$23$	1584.00	0.624361	0.312180	+	0.950023i	$0.398941\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	1134.00	0.250391	0.125195	−	0.992132i	$-0.460044\pi$
$29$	1134.00	0.250391	0.125195	+	0.992132i	$0.460044\pi$
$30$	0	0
$31$	−6016.00	−1.12436	−0.562178	−	0.827016i	$-0.690036\pi$
$31$	−6016.00	−1.12436	−0.562178	+	0.827016i	$0.690036\pi$
$32$	0	0
$33$	5076.00	0.811403
$34$	0	0
$35$	−400.000	−0.0551937
$36$	0	0
$37$	−538.000	−0.0646068	−0.0323034	−	0.999478i	$-0.510284\pi$
$37$	−538.000	−0.0646068	−0.0323034	+	0.999478i	$0.510284\pi$
$38$	0	0
$39$	3330.00	0.350576
$40$	0	0
$41$	11370.0	1.05633	0.528166	−	0.849141i	$-0.322880\pi$
$41$	11370.0	1.05633	0.528166	+	0.849141i	$0.322880\pi$
$42$	0	0
$43$	5444.00	0.449001	0.224500	−	0.974474i	$-0.427925\pi$
$43$	5444.00	0.449001	0.224500	+	0.974474i	$0.427925\pi$
$44$	0	0
$45$	2025.00	0.149071
$46$	0	0
$47$	10296.0	0.679867	0.339933	−	0.940449i	$-0.389595\pi$
$47$	10296.0	0.679867	0.339933	+	0.940449i	$0.389595\pi$
$48$	0	0
$49$	−16551.0	−0.984768
$50$	0	0
$51$	9774.00	0.526195
$52$	0	0
$53$	34758.0	1.69967	0.849836	−	0.527047i	$-0.176701\pi$
$53$	34758.0	1.69967	0.849836	+	0.527047i	$0.176701\pi$
$54$	0	0
$55$	−14100.0	−0.628510
$56$	0	0
$57$	25740.0	1.04935
$58$	0	0
$59$	−26196.0	−0.979727	−0.489863	−	0.871799i	$-0.662953\pi$
$59$	−26196.0	−0.979727	−0.489863	+	0.871799i	$0.662953\pi$
$60$	0	0
$61$	9422.00	0.324204	0.162102	−	0.986774i	$-0.448173\pi$
$61$	9422.00	0.324204	0.162102	+	0.986774i	$0.448173\pi$
$62$	0	0
$63$	−1296.00	−0.0411390
$64$	0	0
$65$	−9250.00	−0.271555
$66$	0	0
$67$	−51124.0	−1.39135	−0.695677	−	0.718354i	$-0.744896\pi$
$67$	−51124.0	−1.39135	−0.695677	+	0.718354i	$0.744896\pi$
$68$	0	0
$69$	−14256.0	−0.360475
$70$	0	0
$71$	14520.0	0.341838	0.170919	−	0.985285i	$-0.445326\pi$
$71$	14520.0	0.341838	0.170919	+	0.985285i	$0.445326\pi$
$72$	0	0
$73$	−22678.0	−0.498078	−0.249039	−	0.968493i	$-0.580115\pi$
$73$	−22678.0	−0.498078	−0.249039	+	0.968493i	$0.580115\pi$
$74$	0	0
$75$	−5625.00	−0.115470
$76$	0	0
$77$	9024.00	0.173449
$78$	0	0
$79$	−97312.0	−1.75428	−0.877140	−	0.480236i	$-0.840551\pi$
$79$	−97312.0	−1.75428	−0.877140	+	0.480236i	$0.840551\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−7956.00	−0.126765	−0.0633825	−	0.997989i	$-0.520189\pi$
$83$	−7956.00	−0.126765	−0.0633825	+	0.997989i	$0.520189\pi$
$84$	0	0
$85$	−27150.0	−0.407589
$86$	0	0
$87$	−10206.0	−0.144563
$88$	0	0
$89$	−47910.0	−0.641137	−0.320569	−	0.947225i	$-0.603874\pi$
$89$	−47910.0	−0.641137	−0.320569	+	0.947225i	$0.603874\pi$
$90$	0	0
$91$	5920.00	0.0749408
$92$	0	0
$93$	54144.0	0.649147
$94$	0	0
$95$	−71500.0	−0.812825
$96$	0	0
$97$	140738.	1.51874	0.759368	−	0.650662i	$-0.225508\pi$
$97$	140738.	1.51874	0.759368	+	0.650662i	$0.225508\pi$
$98$	0	0
$99$	−45684.0	−0.468464
$100$	0	0
$101$	85398.0	0.832999	0.416499	−	0.909136i	$-0.363257\pi$
$101$	85398.0	0.832999	0.416499	+	0.909136i	$0.363257\pi$
$102$	0	0
$103$	198656.	1.84505	0.922526	−	0.385935i	$-0.126121\pi$
$103$	198656.	1.84505	0.922526	+	0.385935i	$0.126121\pi$
$104$	0	0
$105$	3600.00	0.0318661
$106$	0	0
$107$	−116268.	−0.981750	−0.490875	−	0.871230i	$-0.663323\pi$
$107$	−116268.	−0.981750	−0.490875	+	0.871230i	$0.663323\pi$
$108$	0	0
$109$	−146722.	−1.18285	−0.591424	−	0.806361i	$-0.701434\pi$
$109$	−146722.	−1.18285	−0.591424	+	0.806361i	$0.701434\pi$
$110$	0	0
$111$	4842.00	0.0373007
$112$	0	0
$113$	210882.	1.55361	0.776807	−	0.629738i	$-0.216838\pi$
$113$	210882.	1.55361	0.776807	+	0.629738i	$0.216838\pi$
$114$	0	0
$115$	39600.0	0.279223
$116$	0	0
$117$	−29970.0	−0.202405
$118$	0	0
$119$	17376.0	0.112482
$120$	0	0
$121$	157045.	0.975126
$122$	0	0
$123$	−102330.	−0.609874
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	244424.	1.34473	0.672364	−	0.740221i	$-0.265279\pi$
$127$	244424.	1.34473	0.672364	+	0.740221i	$0.265279\pi$
$128$	0	0
$129$	−48996.0	−0.259231
$130$	0	0
$131$	−145308.	−0.739795	−0.369897	−	0.929073i	$-0.620607\pi$
$131$	−145308.	−0.739795	−0.369897	+	0.929073i	$0.620607\pi$
$132$	0	0
$133$	45760.0	0.224314
$134$	0	0
$135$	−18225.0	−0.0860663
$136$	0	0
$137$	−125478.	−0.571171	−0.285586	−	0.958353i	$-0.592188\pi$
$137$	−125478.	−0.571171	−0.285586	+	0.958353i	$0.592188\pi$
$138$	0	0
$139$	251756.	1.10520	0.552602	−	0.833445i	$-0.313635\pi$
$139$	251756.	1.10520	0.552602	+	0.833445i	$0.313635\pi$
$140$	0	0
$141$	−92664.0	−0.392521
$142$	0	0
$143$	208680.	0.853377
$144$	0	0
$145$	28350.0	0.111978
$146$	0	0
$147$	148959.	0.568556
$148$	0	0
$149$	−167322.	−0.617430	−0.308715	−	0.951155i	$-0.599899\pi$
$149$	−167322.	−0.617430	−0.308715	+	0.951155i	$0.599899\pi$
$150$	0	0
$151$	68120.0	0.243126	0.121563	−	0.992584i	$-0.461209\pi$
$151$	68120.0	0.243126	0.121563	+	0.992584i	$0.461209\pi$
$152$	0	0
$153$	−87966.0	−0.303799
$154$	0	0
$155$	−150400.	−0.502827
$156$	0	0
$157$	−361762.	−1.17132	−0.585658	−	0.810559i	$-0.699164\pi$
$157$	−361762.	−1.17132	−0.585658	+	0.810559i	$0.699164\pi$
$158$	0	0
$159$	−312822.	−0.981306
$160$	0	0
$161$	−25344.0	−0.0770567
$162$	0	0
$163$	121820.	0.359128	0.179564	−	0.983746i	$-0.442531\pi$
$163$	121820.	0.359128	0.179564	+	0.983746i	$0.442531\pi$
$164$	0	0
$165$	126900.	0.362871
$166$	0	0
$167$	−409008.	−1.13486	−0.567428	−	0.823423i	$-0.692061\pi$
$167$	−409008.	−1.13486	−0.567428	+	0.823423i	$0.692061\pi$
$168$	0	0
$169$	−234393.	−0.631288
$170$	0	0
$171$	−231660.	−0.605844
$172$	0	0
$173$	−729954.	−1.85430	−0.927151	−	0.374689i	$-0.877749\pi$
$173$	−729954.	−1.85430	−0.927151	+	0.374689i	$0.877749\pi$
$174$	0	0
$175$	−10000.0	−0.0246834
$176$	0	0
$177$	235764.	0.565645
$178$	0	0
$179$	704484.	1.64338	0.821691	−	0.569933i	$-0.193031\pi$
$179$	704484.	1.64338	0.821691	+	0.569933i	$0.193031\pi$
$180$	0	0
$181$	−405850.	−0.920808	−0.460404	−	0.887709i	$-0.652295\pi$
$181$	−405850.	−0.920808	−0.460404	+	0.887709i	$0.652295\pi$
$182$	0	0
$183$	−84798.0	−0.187179
$184$	0	0
$185$	−13450.0	−0.0288930
$186$	0	0
$187$	612504.	1.28087
$188$	0	0
$189$	11664.0	0.0237516
$190$	0	0
$191$	183024.	0.363015	0.181508	−	0.983390i	$-0.441902\pi$
$191$	183024.	0.363015	0.181508	+	0.983390i	$0.441902\pi$
$192$	0	0
$193$	−853054.	−1.64848	−0.824239	−	0.566242i	$-0.808397\pi$
$193$	−853054.	−1.64848	−0.824239	+	0.566242i	$0.808397\pi$
$194$	0	0
$195$	83250.0	0.156783
$196$	0	0
$197$	−476394.	−0.874582	−0.437291	−	0.899320i	$-0.644062\pi$
$197$	−476394.	−0.874582	−0.437291	+	0.899320i	$0.644062\pi$
$198$	0	0
$199$	470648.	0.842488	0.421244	−	0.906947i	$-0.361594\pi$
$199$	470648.	0.842488	0.421244	+	0.906947i	$0.361594\pi$
$200$	0	0
$201$	460116.	0.803299
$202$	0	0
$203$	−18144.0	−0.0309025
$204$	0	0
$205$	284250.	0.472406
$206$	0	0
$207$	128304.	0.208120
$208$	0	0
$209$	1.61304e6	2.55434
$210$	0	0
$211$	−14140.0	−0.0218647	−0.0109323	−	0.999940i	$-0.503480\pi$
$211$	−14140.0	−0.0218647	−0.0109323	+	0.999940i	$0.503480\pi$
$212$	0	0
$213$	−130680.	−0.197360
$214$	0	0
$215$	136100.	0.200799
$216$	0	0
$217$	96256.0	0.138765
$218$	0	0
$219$	204102.	0.287566
$220$	0	0
$221$	401820.	0.553415
$222$	0	0
$223$	−1.08052e6	−1.45503	−0.727513	−	0.686094i	$-0.759324\pi$
$223$	−1.08052e6	−1.45503	−0.727513	+	0.686094i	$0.759324\pi$
$224$	0	0
$225$	50625.0	0.0666667
$226$	0	0
$227$	−340500.	−0.438584	−0.219292	−	0.975659i	$-0.570375\pi$
$227$	−340500.	−0.438584	−0.219292	+	0.975659i	$0.570375\pi$
$228$	0	0
$229$	−787594.	−0.992462	−0.496231	−	0.868191i	$-0.665283\pi$
$229$	−787594.	−0.992462	−0.496231	+	0.868191i	$0.665283\pi$
$230$	0	0
$231$	−81216.0	−0.100141
$232$	0	0
$233$	1.51881e6	1.83279	0.916397	−	0.400271i	$-0.131084\pi$
$233$	1.51881e6	1.83279	0.916397	+	0.400271i	$0.131084\pi$
$234$	0	0
$235$	257400.	0.304046
$236$	0	0
$237$	875808.	1.01283
$238$	0	0
$239$	1.18181e6	1.33830	0.669148	−	0.743129i	$-0.266659\pi$
$239$	1.18181e6	1.33830	0.669148	+	0.743129i	$0.266659\pi$
$240$	0	0
$241$	−523342.	−0.580421	−0.290210	−	0.956963i	$-0.593725\pi$
$241$	−523342.	−0.580421	−0.290210	+	0.956963i	$0.593725\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−413775.	−0.440402
$246$	0	0
$247$	1.05820e6	1.10363
$248$	0	0
$249$	71604.0	0.0731878
$250$	0	0
$251$	−1.27741e6	−1.27981	−0.639907	−	0.768453i	$-0.721027\pi$
$251$	−1.27741e6	−1.27981	−0.639907	+	0.768453i	$0.721027\pi$
$252$	0	0
$253$	−893376.	−0.877471
$254$	0	0
$255$	244350.	0.235322
$256$	0	0
$257$	−1.93112e6	−1.82379	−0.911897	−	0.410418i	$-0.865383\pi$
$257$	−1.93112e6	−1.82379	−0.911897	+	0.410418i	$0.865383\pi$
$258$	0	0
$259$	8608.00	0.00797357
$260$	0	0
$261$	91854.0	0.0834635
$262$	0	0
$263$	−822336.	−0.733094	−0.366547	−	0.930399i	$-0.619460\pi$
$263$	−822336.	−0.733094	−0.366547	+	0.930399i	$0.619460\pi$
$264$	0	0
$265$	868950.	0.760116
$266$	0	0
$267$	431190.	0.370161
$268$	0	0
$269$	1.59181e6	1.34125	0.670625	−	0.741797i	$-0.266026\pi$
$269$	1.59181e6	1.34125	0.670625	+	0.741797i	$0.266026\pi$
$270$	0	0
$271$	−1.21106e6	−1.00171	−0.500854	−	0.865532i	$-0.666981\pi$
$271$	−1.21106e6	−1.00171	−0.500854	+	0.865532i	$0.666981\pi$
$272$	0	0
$273$	−53280.0	−0.0432671
$274$	0	0
$275$	−352500.	−0.281078
$276$	0	0
$277$	−2.34356e6	−1.83517	−0.917587	−	0.397536i	$-0.869865\pi$
$277$	−2.34356e6	−1.83517	−0.917587	+	0.397536i	$0.869865\pi$
$278$	0	0
$279$	−487296.	−0.374785
$280$	0	0
$281$	1.86211e6	1.40682	0.703410	−	0.710784i	$-0.251660\pi$
$281$	1.86211e6	1.40682	0.703410	+	0.710784i	$0.251660\pi$
$282$	0	0
$283$	108212.	0.0803173	0.0401587	−	0.999193i	$-0.487214\pi$
$283$	108212.	0.0803173	0.0401587	+	0.999193i	$0.487214\pi$
$284$	0	0
$285$	643500.	0.469285
$286$	0	0
$287$	−181920.	−0.130369
$288$	0	0
$289$	−240461.	−0.169356
$290$	0	0
$291$	−1.26664e6	−0.876842
$292$	0	0
$293$	−959658.	−0.653052	−0.326526	−	0.945188i	$-0.605878\pi$
$293$	−959658.	−0.653052	−0.326526	+	0.945188i	$0.605878\pi$
$294$	0	0
$295$	−654900.	−0.438147
$296$	0	0
$297$	411156.	0.270468
$298$	0	0
$299$	−586080.	−0.379122
$300$	0	0
$301$	−87104.0	−0.0554143
$302$	0	0
$303$	−768582.	−0.480932
$304$	0	0
$305$	235550.	0.144988
$306$	0	0
$307$	1.62729e6	0.985416	0.492708	−	0.870195i	$-0.336007\pi$
$307$	1.62729e6	0.985416	0.492708	+	0.870195i	$0.336007\pi$
$308$	0	0
$309$	−1.78790e6	−1.06524
$310$	0	0
$311$	−2.84086e6	−1.66551	−0.832757	−	0.553639i	$-0.813239\pi$
$311$	−2.84086e6	−1.66551	−0.832757	+	0.553639i	$0.813239\pi$
$312$	0	0
$313$	1.61715e6	0.933014	0.466507	−	0.884517i	$-0.345512\pi$
$313$	1.61715e6	0.933014	0.466507	+	0.884517i	$0.345512\pi$
$314$	0	0
$315$	−32400.0	−0.0183979
$316$	0	0
$317$	−2.04367e6	−1.14225	−0.571126	−	0.820863i	$-0.693493\pi$
$317$	−2.04367e6	−1.14225	−0.571126	+	0.820863i	$0.693493\pi$
$318$	0	0
$319$	−639576.	−0.351897
$320$	0	0
$321$	1.04641e6	0.566813
$322$	0	0
$323$	3.10596e6	1.65649
$324$	0	0
$325$	−231250.	−0.121443
$326$	0	0
$327$	1.32050e6	0.682918
$328$	0	0
$329$	−164736.	−0.0839071
$330$	0	0
$331$	1.00425e6	0.503817	0.251908	−	0.967751i	$-0.418942\pi$
$331$	1.00425e6	0.503817	0.251908	+	0.967751i	$0.418942\pi$
$332$	0	0
$333$	−43578.0	−0.0215356
$334$	0	0
$335$	−1.27810e6	−0.622233
$336$	0	0
$337$	2.91960e6	1.40039	0.700195	−	0.713952i	$-0.253096\pi$
$337$	2.91960e6	1.40039	0.700195	+	0.713952i	$0.253096\pi$
$338$	0	0
$339$	−1.89794e6	−0.896980
$340$	0	0
$341$	3.39302e6	1.58016
$342$	0	0
$343$	533728.	0.244954
$344$	0	0
$345$	−356400.	−0.161209
$346$	0	0
$347$	−2.21075e6	−0.985634	−0.492817	−	0.870133i	$-0.664033\pi$
$347$	−2.21075e6	−0.985634	−0.492817	+	0.870133i	$0.664033\pi$
$348$	0	0
$349$	375566.	0.165053	0.0825264	−	0.996589i	$-0.473701\pi$
$349$	375566.	0.165053	0.0825264	+	0.996589i	$0.473701\pi$
$350$	0	0
$351$	269730.	0.116859
$352$	0	0
$353$	1.49992e6	0.640666	0.320333	−	0.947305i	$-0.396205\pi$
$353$	1.49992e6	0.640666	0.320333	+	0.947305i	$0.396205\pi$
$354$	0	0
$355$	363000.	0.152875
$356$	0	0
$357$	−156384.	−0.0649414
$358$	0	0
$359$	2.49626e6	1.02224	0.511122	−	0.859508i	$-0.329230\pi$
$359$	2.49626e6	1.02224	0.511122	+	0.859508i	$0.329230\pi$
$360$	0	0
$361$	5.70350e6	2.30342
$362$	0	0
$363$	−1.41340e6	−0.562989
$364$	0	0
$365$	−566950.	−0.222747
$366$	0	0
$367$	4.09714e6	1.58787	0.793937	−	0.608000i	$-0.208028\pi$
$367$	4.09714e6	1.58787	0.793937	+	0.608000i	$0.208028\pi$
$368$	0	0
$369$	920970.	0.352111
$370$	0	0
$371$	−556128.	−0.209768
$372$	0	0
$373$	1.24213e6	0.462271	0.231135	−	0.972922i	$-0.425756\pi$
$373$	1.24213e6	0.462271	0.231135	+	0.972922i	$0.425756\pi$
$374$	0	0
$375$	−140625.	−0.0516398
$376$	0	0
$377$	−419580.	−0.152041
$378$	0	0
$379$	−2.28413e6	−0.816814	−0.408407	−	0.912800i	$-0.633916\pi$
$379$	−2.28413e6	−0.816814	−0.408407	+	0.912800i	$0.633916\pi$
$380$	0	0
$381$	−2.19982e6	−0.776379
$382$	0	0
$383$	−565128.	−0.196857	−0.0984283	−	0.995144i	$-0.531382\pi$
$383$	−565128.	−0.196857	−0.0984283	+	0.995144i	$0.531382\pi$
$384$	0	0
$385$	225600.	0.0775688
$386$	0	0
$387$	440964.	0.149667
$388$	0	0
$389$	−3.37367e6	−1.13039	−0.565196	−	0.824957i	$-0.691199\pi$
$389$	−3.37367e6	−1.13039	−0.565196	+	0.824957i	$0.691199\pi$
$390$	0	0
$391$	−1.72022e6	−0.569040
$392$	0	0
$393$	1.30777e6	0.427121
$394$	0	0
$395$	−2.43280e6	−0.784537
$396$	0	0
$397$	−745138.	−0.237280	−0.118640	−	0.992937i	$-0.537853\pi$
$397$	−745138.	−0.237280	−0.118640	+	0.992937i	$0.537853\pi$
$398$	0	0
$399$	−411840.	−0.129508
$400$	0	0
$401$	864786.	0.268564	0.134282	−	0.990943i	$-0.457127\pi$
$401$	864786.	0.268564	0.134282	+	0.990943i	$0.457127\pi$
$402$	0	0
$403$	2.22592e6	0.682727
$404$	0	0
$405$	164025.	0.0496904
$406$	0	0
$407$	303432.	0.0907978
$408$	0	0
$409$	−5.70189e6	−1.68543	−0.842715	−	0.538359i	$-0.819044\pi$
$409$	−5.70189e6	−1.68543	−0.842715	+	0.538359i	$0.819044\pi$
$410$	0	0
$411$	1.12930e6	0.329766
$412$	0	0
$413$	419136.	0.120915
$414$	0	0
$415$	−198900.	−0.0566911
$416$	0	0
$417$	−2.26580e6	−0.638090
$418$	0	0
$419$	−2.91947e6	−0.812398	−0.406199	−	0.913785i	$-0.633146\pi$
$419$	−2.91947e6	−0.812398	−0.406199	+	0.913785i	$0.633146\pi$
$420$	0	0
$421$	1.08815e6	0.299215	0.149608	−	0.988745i	$-0.452199\pi$
$421$	1.08815e6	0.299215	0.149608	+	0.988745i	$0.452199\pi$
$422$	0	0
$423$	833976.	0.226622
$424$	0	0
$425$	−678750.	−0.182279
$426$	0	0
$427$	−150752.	−0.0400123
$428$	0	0
$429$	−1.87812e6	−0.492697
$430$	0	0
$431$	4.64573e6	1.20465	0.602325	−	0.798251i	$-0.294241\pi$
$431$	4.64573e6	1.20465	0.602325	+	0.798251i	$0.294241\pi$
$432$	0	0
$433$	−702094.	−0.179960	−0.0899799	−	0.995944i	$-0.528680\pi$
$433$	−702094.	−0.179960	−0.0899799	+	0.995944i	$0.528680\pi$
$434$	0	0
$435$	−255150.	−0.0646506
$436$	0	0
$437$	−4.53024e6	−1.13480
$438$	0	0
$439$	−1.63343e6	−0.404520	−0.202260	−	0.979332i	$-0.564829\pi$
$439$	−1.63343e6	−0.404520	−0.202260	+	0.979332i	$0.564829\pi$
$440$	0	0
$441$	−1.34063e6	−0.328256
$442$	0	0
$443$	5.39500e6	1.30612	0.653058	−	0.757308i	$-0.273486\pi$
$443$	5.39500e6	1.30612	0.653058	+	0.757308i	$0.273486\pi$
$444$	0	0
$445$	−1.19775e6	−0.286725
$446$	0	0
$447$	1.50590e6	0.356473
$448$	0	0
$449$	−918462.	−0.215003	−0.107502	−	0.994205i	$-0.534285\pi$
$449$	−918462.	−0.215003	−0.107502	+	0.994205i	$0.534285\pi$
$450$	0	0
$451$	−6.41268e6	−1.48456
$452$	0	0
$453$	−613080.	−0.140369
$454$	0	0
$455$	148000.	0.0335145
$456$	0	0
$457$	−1.55695e6	−0.348726	−0.174363	−	0.984681i	$-0.555787\pi$
$457$	−1.55695e6	−0.348726	−0.174363	+	0.984681i	$0.555787\pi$
$458$	0	0
$459$	791694.	0.175398
$460$	0	0
$461$	−5.19299e6	−1.13806	−0.569030	−	0.822316i	$-0.692681\pi$
$461$	−5.19299e6	−1.13806	−0.569030	+	0.822316i	$0.692681\pi$
$462$	0	0
$463$	−665848.	−0.144352	−0.0721760	−	0.997392i	$-0.522994\pi$
$463$	−665848.	−0.144352	−0.0721760	+	0.997392i	$0.522994\pi$
$464$	0	0
$465$	1.35360e6	0.290307
$466$	0	0
$467$	6.84073e6	1.45148	0.725739	−	0.687970i	$-0.241498\pi$
$467$	6.84073e6	1.45148	0.725739	+	0.687970i	$0.241498\pi$
$468$	0	0
$469$	817984.	0.171717
$470$	0	0
$471$	3.25586e6	0.676259
$472$	0	0
$473$	−3.07042e6	−0.631022
$474$	0	0
$475$	−1.78750e6	−0.363506
$476$	0	0
$477$	2.81540e6	0.566557
$478$	0	0
$479$	2.26214e6	0.450486	0.225243	−	0.974303i	$-0.427682\pi$
$479$	2.26214e6	0.450486	0.225243	+	0.974303i	$0.427682\pi$
$480$	0	0
$481$	199060.	0.0392303
$482$	0	0
$483$	228096.	0.0444887
$484$	0	0
$485$	3.51845e6	0.679199
$486$	0	0
$487$	−6.78443e6	−1.29626	−0.648128	−	0.761531i	$-0.724448\pi$
$487$	−6.78443e6	−1.29626	−0.648128	+	0.761531i	$0.724448\pi$
$488$	0	0
$489$	−1.09638e6	−0.207343
$490$	0	0
$491$	−6.75964e6	−1.26538	−0.632688	−	0.774407i	$-0.718048\pi$
$491$	−6.75964e6	−1.26538	−0.632688	+	0.774407i	$0.718048\pi$
$492$	0	0
$493$	−1.23152e6	−0.228205
$494$	0	0
$495$	−1.14210e6	−0.209503
$496$	0	0
$497$	−232320.	−0.0421886
$498$	0	0
$499$	8.73256e6	1.56997	0.784983	−	0.619517i	$-0.212671\pi$
$499$	8.73256e6	1.56997	0.784983	+	0.619517i	$0.212671\pi$
$500$	0	0
$501$	3.68107e6	0.655209
$502$	0	0
$503$	4.75426e6	0.837843	0.418921	−	0.908022i	$-0.362408\pi$
$503$	4.75426e6	0.837843	0.418921	+	0.908022i	$0.362408\pi$
$504$	0	0
$505$	2.13495e6	0.372528
$506$	0	0
$507$	2.10954e6	0.364475
$508$	0	0
$509$	−2.13074e6	−0.364532	−0.182266	−	0.983249i	$-0.558343\pi$
$509$	−2.13074e6	−0.364532	−0.182266	+	0.983249i	$0.558343\pi$
$510$	0	0
$511$	362848.	0.0614713
$512$	0	0
$513$	2.08494e6	0.349784
$514$	0	0
$515$	4.96640e6	0.825132
$516$	0	0
$517$	−5.80694e6	−0.955479
$518$	0	0
$519$	6.56959e6	1.07058
$520$	0	0
$521$	1.10967e7	1.79101	0.895507	−	0.445048i	$-0.146813\pi$
$521$	1.10967e7	1.79101	0.895507	+	0.445048i	$0.146813\pi$
$522$	0	0
$523$	941252.	0.150471	0.0752353	−	0.997166i	$-0.476029\pi$
$523$	941252.	0.150471	0.0752353	+	0.997166i	$0.476029\pi$
$524$	0	0
$525$	90000.0	0.0142510
$526$	0	0
$527$	6.53338e6	1.02473
$528$	0	0
$529$	−3.92729e6	−0.610174
$530$	0	0
$531$	−2.12188e6	−0.326576
$532$	0	0
$533$	−4.20690e6	−0.641422
$534$	0	0
$535$	−2.90670e6	−0.439052
$536$	0	0
$537$	−6.34036e6	−0.948807
$538$	0	0
$539$	9.33476e6	1.38399
$540$	0	0
$541$	2.39896e6	0.352395	0.176197	−	0.984355i	$-0.443620\pi$
$541$	2.39896e6	0.352395	0.176197	+	0.984355i	$0.443620\pi$
$542$	0	0
$543$	3.65265e6	0.531629
$544$	0	0
$545$	−3.66805e6	−0.528986
$546$	0	0
$547$	1.55851e6	0.222711	0.111355	−	0.993781i	$-0.464481\pi$
$547$	1.55851e6	0.222711	0.111355	+	0.993781i	$0.464481\pi$
$548$	0	0
$549$	763182.	0.108068
$550$	0	0
$551$	−3.24324e6	−0.455093
$552$	0	0
$553$	1.55699e6	0.216508
$554$	0	0
$555$	121050.	0.0166814
$556$	0	0
$557$	1.19451e7	1.63137	0.815685	−	0.578496i	$-0.196360\pi$
$557$	1.19451e7	1.63137	0.815685	+	0.578496i	$0.196360\pi$
$558$	0	0
$559$	−2.01428e6	−0.272640
$560$	0	0
$561$	−5.51254e6	−0.739510
$562$	0	0
$563$	−9.63906e6	−1.28163	−0.640817	−	0.767694i	$-0.721404\pi$
$563$	−9.63906e6	−1.28163	−0.640817	+	0.767694i	$0.721404\pi$
$564$	0	0
$565$	5.27205e6	0.694798
$566$	0	0
$567$	−104976.	−0.0137130
$568$	0	0
$569$	830106.	0.107486	0.0537431	−	0.998555i	$-0.482885\pi$
$569$	830106.	0.107486	0.0537431	+	0.998555i	$0.482885\pi$
$570$	0	0
$571$	−3.32914e6	−0.427309	−0.213654	−	0.976909i	$-0.568537\pi$
$571$	−3.32914e6	−0.427309	−0.213654	+	0.976909i	$0.568537\pi$
$572$	0	0
$573$	−1.64722e6	−0.209587
$574$	0	0
$575$	990000.	0.124872
$576$	0	0
$577$	4.40026e6	0.550223	0.275111	−	0.961412i	$-0.411285\pi$
$577$	4.40026e6	0.550223	0.275111	+	0.961412i	$0.411285\pi$
$578$	0	0
$579$	7.67749e6	0.951749
$580$	0	0
$581$	127296.	0.0156450
$582$	0	0
$583$	−1.96035e7	−2.38870
$584$	0	0
$585$	−749250.	−0.0905184
$586$	0	0
$587$	−7.25848e6	−0.869461	−0.434731	−	0.900561i	$-0.643156\pi$
$587$	−7.25848e6	−0.869461	−0.434731	+	0.900561i	$0.643156\pi$
$588$	0	0
$589$	1.72058e7	2.04355
$590$	0	0
$591$	4.28755e6	0.504940
$592$	0	0
$593$	−1.55171e7	−1.81206	−0.906032	−	0.423210i	$-0.860903\pi$
$593$	−1.55171e7	−1.81206	−0.906032	+	0.423210i	$0.860903\pi$
$594$	0	0
$595$	434400.	0.0503034
$596$	0	0
$597$	−4.23583e6	−0.486410
$598$	0	0
$599$	−1.02670e7	−1.16917	−0.584583	−	0.811334i	$-0.698742\pi$
$599$	−1.02670e7	−1.16917	−0.584583	+	0.811334i	$0.698742\pi$
$600$	0	0
$601$	−9.42362e6	−1.06422	−0.532110	−	0.846675i	$-0.678601\pi$
$601$	−9.42362e6	−1.06422	−0.532110	+	0.846675i	$0.678601\pi$
$602$	0	0
$603$	−4.14104e6	−0.463785
$604$	0	0
$605$	3.92612e6	0.436090
$606$	0	0
$607$	−7.67289e6	−0.845254	−0.422627	−	0.906304i	$-0.638892\pi$
$607$	−7.67289e6	−0.845254	−0.422627	+	0.906304i	$0.638892\pi$
$608$	0	0
$609$	163296.	0.0178415
$610$	0	0
$611$	−3.80952e6	−0.412826
$612$	0	0
$613$	−2.30598e6	−0.247859	−0.123929	−	0.992291i	$-0.539550\pi$
$613$	−2.30598e6	−0.247859	−0.123929	+	0.992291i	$0.539550\pi$
$614$	0	0
$615$	−2.55825e6	−0.272744
$616$	0	0
$617$	−505254.	−0.0534314	−0.0267157	−	0.999643i	$-0.508505\pi$
$617$	−505254.	−0.0534314	−0.0267157	+	0.999643i	$0.508505\pi$
$618$	0	0
$619$	4.61380e6	0.483986	0.241993	−	0.970278i	$-0.422199\pi$
$619$	4.61380e6	0.483986	0.241993	+	0.970278i	$0.422199\pi$
$620$	0	0
$621$	−1.15474e6	−0.120158
$622$	0	0
$623$	766560.	0.0791272
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.45174e7	−1.47475
$628$	0	0
$629$	584268.	0.0588824
$630$	0	0
$631$	4.77327e6	0.477247	0.238623	−	0.971112i	$-0.423304\pi$
$631$	4.77327e6	0.477247	0.238623	+	0.971112i	$0.423304\pi$
$632$	0	0
$633$	127260.	0.0126236
$634$	0	0
$635$	6.11060e6	0.601381
$636$	0	0
$637$	6.12387e6	0.597967
$638$	0	0
$639$	1.17612e6	0.113946
$640$	0	0
$641$	−1.09254e7	−1.05025	−0.525125	−	0.851025i	$-0.675981\pi$
$641$	−1.09254e7	−1.05025	−0.525125	+	0.851025i	$0.675981\pi$
$642$	0	0
$643$	1.13952e6	0.108691	0.0543454	−	0.998522i	$-0.482693\pi$
$643$	1.13952e6	0.108691	0.0543454	+	0.998522i	$0.482693\pi$
$644$	0	0
$645$	−1.22490e6	−0.115931
$646$	0	0
$647$	5.10955e6	0.479868	0.239934	−	0.970789i	$-0.422874\pi$
$647$	5.10955e6	0.479868	0.239934	+	0.970789i	$0.422874\pi$
$648$	0	0
$649$	1.47745e7	1.37690
$650$	0	0
$651$	−866304.	−0.0801157
$652$	0	0
$653$	−7.81591e6	−0.717293	−0.358646	−	0.933474i	$-0.616762\pi$
$653$	−7.81591e6	−0.717293	−0.358646	+	0.933474i	$0.616762\pi$
$654$	0	0
$655$	−3.63270e6	−0.330846
$656$	0	0
$657$	−1.83692e6	−0.166026
$658$	0	0
$659$	−1.05905e7	−0.949954	−0.474977	−	0.879998i	$-0.657544\pi$
$659$	−1.05905e7	−0.949954	−0.474977	+	0.879998i	$0.657544\pi$
$660$	0	0
$661$	4.67092e6	0.415814	0.207907	−	0.978149i	$-0.433335\pi$
$661$	4.67092e6	0.415814	0.207907	+	0.978149i	$0.433335\pi$
$662$	0	0
$663$	−3.61638e6	−0.319514
$664$	0	0
$665$	1.14400e6	0.100316
$666$	0	0
$667$	1.79626e6	0.156334
$668$	0	0
$669$	9.72468e6	0.840059
$670$	0	0
$671$	−5.31401e6	−0.455634
$672$	0	0
$673$	2.07728e7	1.76790	0.883948	−	0.467585i	$-0.154876\pi$
$673$	2.07728e7	1.76790	0.883948	+	0.467585i	$0.154876\pi$
$674$	0	0
$675$	−455625.	−0.0384900
$676$	0	0
$677$	1.67497e7	1.40454	0.702270	−	0.711911i	$-0.252170\pi$
$677$	1.67497e7	1.40454	0.702270	+	0.711911i	$0.252170\pi$
$678$	0	0
$679$	−2.25181e6	−0.187438
$680$	0	0
$681$	3.06450e6	0.253216
$682$	0	0
$683$	4.41700e6	0.362306	0.181153	−	0.983455i	$-0.442017\pi$
$683$	4.41700e6	0.362306	0.181153	+	0.983455i	$0.442017\pi$
$684$	0	0
$685$	−3.13695e6	−0.255435
$686$	0	0
$687$	7.08835e6	0.572998
$688$	0	0
$689$	−1.28605e7	−1.03207
$690$	0	0
$691$	1.86481e7	1.48573	0.742863	−	0.669443i	$-0.233467\pi$
$691$	1.86481e7	1.48573	0.742863	+	0.669443i	$0.233467\pi$
$692$	0	0
$693$	730944.	0.0578164
$694$	0	0
$695$	6.29390e6	0.494263
$696$	0	0
$697$	−1.23478e7	−0.962739
$698$	0	0
$699$	−1.36693e7	−1.05816
$700$	0	0
$701$	7.28389e6	0.559845	0.279923	−	0.960023i	$-0.409691\pi$
$701$	7.28389e6	0.559845	0.279923	+	0.960023i	$0.409691\pi$
$702$	0	0
$703$	1.53868e6	0.117425
$704$	0	0
$705$	−2.31660e6	−0.175541
$706$	0	0
$707$	−1.36637e6	−0.102806
$708$	0	0
$709$	1.82063e7	1.36021	0.680105	−	0.733115i	$-0.261934\pi$
$709$	1.82063e7	1.36021	0.680105	+	0.733115i	$0.261934\pi$
$710$	0	0
$711$	−7.88227e6	−0.584760
$712$	0	0
$713$	−9.52934e6	−0.702003
$714$	0	0
$715$	5.21700e6	0.381642
$716$	0	0
$717$	−1.06363e7	−0.772666
$718$	0	0
$719$	6.46901e6	0.466676	0.233338	−	0.972396i	$-0.425035\pi$
$719$	6.46901e6	0.466676	0.233338	+	0.972396i	$0.425035\pi$
$720$	0	0
$721$	−3.17850e6	−0.227711
$722$	0	0
$723$	4.71008e6	0.335106
$724$	0	0
$725$	708750.	0.0500781
$726$	0	0
$727$	−1.39991e7	−0.982343	−0.491172	−	0.871063i	$-0.663431\pi$
$727$	−1.39991e7	−0.982343	−0.491172	+	0.871063i	$0.663431\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.91218e6	−0.409218
$732$	0	0
$733$	2.51019e7	1.72562	0.862811	−	0.505526i	$-0.168702\pi$
$733$	2.51019e7	1.72562	0.862811	+	0.505526i	$0.168702\pi$
$734$	0	0
$735$	3.72398e6	0.254266
$736$	0	0
$737$	2.88339e7	1.95540
$738$	0	0
$739$	9.73514e6	0.655739	0.327870	−	0.944723i	$-0.393669\pi$
$739$	9.73514e6	0.655739	0.327870	+	0.944723i	$0.393669\pi$
$740$	0	0
$741$	−9.52380e6	−0.637184
$742$	0	0
$743$	3.03370e6	0.201604	0.100802	−	0.994906i	$-0.467859\pi$
$743$	3.03370e6	0.201604	0.100802	+	0.994906i	$0.467859\pi$
$744$	0	0
$745$	−4.18305e6	−0.276123
$746$	0	0
$747$	−644436.	−0.0422550
$748$	0	0
$749$	1.86029e6	0.121165
$750$	0	0
$751$	−6.07557e6	−0.393086	−0.196543	−	0.980495i	$-0.562971\pi$
$751$	−6.07557e6	−0.393086	−0.196543	+	0.980495i	$0.562971\pi$
$752$	0	0
$753$	1.14967e7	0.738901
$754$	0	0
$755$	1.70300e6	0.108729
$756$	0	0
$757$	−4.82103e6	−0.305774	−0.152887	−	0.988244i	$-0.548857\pi$
$757$	−4.82103e6	−0.305774	−0.152887	+	0.988244i	$0.548857\pi$
$758$	0	0
$759$	8.04038e6	0.506608
$760$	0	0
$761$	5.82748e6	0.364770	0.182385	−	0.983227i	$-0.441618\pi$
$761$	5.82748e6	0.364770	0.182385	+	0.983227i	$0.441618\pi$
$762$	0	0
$763$	2.34755e6	0.145984
$764$	0	0
$765$	−2.19915e6	−0.135863
$766$	0	0
$767$	9.69252e6	0.594906
$768$	0	0
$769$	−2.71546e7	−1.65587	−0.827936	−	0.560822i	$-0.810485\pi$
$769$	−2.71546e7	−1.65587	−0.827936	+	0.560822i	$0.810485\pi$
$770$	0	0
$771$	1.73801e7	1.05297
$772$	0	0
$773$	6678.00	0.000401974 0	0.000200987	−	1.00000i	$-0.499936\pi$
$773$	6678.00	0.000401974 0	0.000200987		1.00000i	$0.499936\pi$
$774$	0	0
$775$	−3.76000e6	−0.224871
$776$	0	0
$777$	−77472.0	−0.00460354
$778$	0	0
$779$	−3.25182e7	−1.91992
$780$	0	0
$781$	−8.18928e6	−0.480417
$782$	0	0
$783$	−826686.	−0.0481877
$784$	0	0
$785$	−9.04405e6	−0.523828
$786$	0	0
$787$	1.44352e7	0.830781	0.415391	−	0.909643i	$-0.363645\pi$
$787$	1.44352e7	0.830781	0.415391	+	0.909643i	$0.363645\pi$
$788$	0	0
$789$	7.40102e6	0.423252
$790$	0	0
$791$	−3.37411e6	−0.191742
$792$	0	0
$793$	−3.48614e6	−0.196862
$794$	0	0
$795$	−7.82055e6	−0.438853
$796$	0	0
$797$	−1.90792e6	−0.106393	−0.0531967	−	0.998584i	$-0.516941\pi$
$797$	−1.90792e6	−0.106393	−0.0531967	+	0.998584i	$0.516941\pi$
$798$	0	0
$799$	−1.11815e7	−0.619629
$800$	0	0
$801$	−3.88071e6	−0.213712
$802$	0	0
$803$	1.27904e7	0.699995
$804$	0	0
$805$	−633600.	−0.0344608
$806$	0	0
$807$	−1.43263e7	−0.774371
$808$	0	0
$809$	−3.48543e7	−1.87234	−0.936171	−	0.351546i	$-0.885656\pi$
$809$	−3.48543e7	−1.87234	−0.936171	+	0.351546i	$0.885656\pi$
$810$	0	0
$811$	1.41263e7	0.754182	0.377091	−	0.926176i	$-0.376924\pi$
$811$	1.41263e7	0.754182	0.377091	+	0.926176i	$0.376924\pi$
$812$	0	0
$813$	1.08995e7	0.578336
$814$	0	0
$815$	3.04550e6	0.160607
$816$	0	0
$817$	−1.55698e7	−0.816073
$818$	0	0
$819$	479520.	0.0249803
$820$	0	0
$821$	1.78858e7	0.926082	0.463041	−	0.886337i	$-0.346758\pi$
$821$	1.78858e7	0.926082	0.463041	+	0.886337i	$0.346758\pi$
$822$	0	0
$823$	−1.73695e7	−0.893897	−0.446948	−	0.894560i	$-0.647489\pi$
$823$	−1.73695e7	−0.893897	−0.446948	+	0.894560i	$0.647489\pi$
$824$	0	0
$825$	3.17250e6	0.162281
$826$	0	0
$827$	−1.84745e7	−0.939310	−0.469655	−	0.882850i	$-0.655622\pi$
$827$	−1.84745e7	−0.939310	−0.469655	+	0.882850i	$0.655622\pi$
$828$	0	0
$829$	−3.02647e7	−1.52950	−0.764750	−	0.644327i	$-0.777138\pi$
$829$	−3.02647e7	−1.52950	−0.764750	+	0.644327i	$0.777138\pi$
$830$	0	0
$831$	2.10921e7	1.05954
$832$	0	0
$833$	1.79744e7	0.897515
$834$	0	0
$835$	−1.02252e7	−0.507523
$836$	0	0
$837$	4.38566e6	0.216382
$838$	0	0
$839$	−2.10237e7	−1.03111	−0.515555	−	0.856856i	$-0.672414\pi$
$839$	−2.10237e7	−1.03111	−0.515555	+	0.856856i	$0.672414\pi$
$840$	0	0
$841$	−1.92252e7	−0.937305
$842$	0	0
$843$	−1.67590e7	−0.812228
$844$	0	0
$845$	−5.85982e6	−0.282321
$846$	0	0
$847$	−2.51272e6	−0.120347
$848$	0	0
$849$	−973908.	−0.0463712
$850$	0	0
$851$	−852192.	−0.0403379
$852$	0	0
$853$	−2.60690e7	−1.22674	−0.613369	−	0.789796i	$-0.710186\pi$
$853$	−2.60690e7	−1.22674	−0.613369	+	0.789796i	$0.710186\pi$
$854$	0	0
$855$	−5.79150e6	−0.270942
$856$	0	0
$857$	489738.	0.0227778	0.0113889	−	0.999935i	$-0.496375\pi$
$857$	489738.	0.0227778	0.0113889	+	0.999935i	$0.496375\pi$
$858$	0	0
$859$	−2.17067e7	−1.00371	−0.501857	−	0.864951i	$-0.667350\pi$
$859$	−2.17067e7	−1.00371	−0.501857	+	0.864951i	$0.667350\pi$
$860$	0	0
$861$	1.63728e6	0.0752688
$862$	0	0
$863$	2.91038e7	1.33022	0.665108	−	0.746747i	$-0.268385\pi$
$863$	2.91038e7	1.33022	0.665108	+	0.746747i	$0.268385\pi$
$864$	0	0
$865$	−1.82488e7	−0.829269
$866$	0	0
$867$	2.16415e6	0.0977776
$868$	0	0
$869$	5.48840e7	2.46545
$870$	0	0
$871$	1.89159e7	0.844853
$872$	0	0
$873$	1.13998e7	0.506245
$874$	0	0
$875$	−250000.	−0.0110387
$876$	0	0
$877$	2.11648e7	0.929215	0.464607	−	0.885517i	$-0.346195\pi$
$877$	2.11648e7	0.929215	0.464607	+	0.885517i	$0.346195\pi$
$878$	0	0
$879$	8.63692e6	0.377039
$880$	0	0
$881$	−9.60651e6	−0.416990	−0.208495	−	0.978023i	$-0.566857\pi$
$881$	−9.60651e6	−0.416990	−0.208495	+	0.978023i	$0.566857\pi$
$882$	0	0
$883$	2.58825e7	1.11713	0.558566	−	0.829460i	$-0.311352\pi$
$883$	2.58825e7	1.11713	0.558566	+	0.829460i	$0.311352\pi$
$884$	0	0
$885$	5.89410e6	0.252964
$886$	0	0
$887$	2.43249e7	1.03811	0.519053	−	0.854742i	$-0.326285\pi$
$887$	2.43249e7	1.03811	0.519053	+	0.854742i	$0.326285\pi$
$888$	0	0
$889$	−3.91078e6	−0.165962
$890$	0	0
$891$	−3.70040e6	−0.156155
$892$	0	0
$893$	−2.94466e7	−1.23568
$894$	0	0
$895$	1.76121e7	0.734943
$896$	0	0
$897$	5.27472e6	0.218886
$898$	0	0
$899$	−6.82214e6	−0.281528
$900$	0	0
$901$	−3.77472e7	−1.54908
$902$	0	0
$903$	783936.	0.0319935
$904$	0	0
$905$	−1.01462e7	−0.411798
$906$	0	0
$907$	−7.87371e6	−0.317805	−0.158903	−	0.987294i	$-0.550796\pi$
$907$	−7.87371e6	−0.317805	−0.158903	+	0.987294i	$0.550796\pi$
$908$	0	0
$909$	6.91724e6	0.277666
$910$	0	0
$911$	4.74478e7	1.89418	0.947088	−	0.320975i	$-0.104010\pi$
$911$	4.74478e7	1.89418	0.947088	+	0.320975i	$0.104010\pi$
$912$	0	0
$913$	4.48718e6	0.178155
$914$	0	0
$915$	−2.11995e6	−0.0837091
$916$	0	0
$917$	2.32493e6	0.0913032
$918$	0	0
$919$	605288.	0.0236414	0.0118207	−	0.999930i	$-0.496237\pi$
$919$	605288.	0.0236414	0.0118207	+	0.999930i	$0.496237\pi$
$920$	0	0
$921$	−1.46456e7	−0.568930
$922$	0	0
$923$	−5.37240e6	−0.207570
$924$	0	0
$925$	−336250.	−0.0129214
$926$	0	0
$927$	1.60911e7	0.615017
$928$	0	0
$929$	−9.52438e6	−0.362074	−0.181037	−	0.983476i	$-0.557945\pi$
$929$	−9.52438e6	−0.362074	−0.181037	+	0.983476i	$0.557945\pi$
$930$	0	0
$931$	4.73359e7	1.78985
$932$	0	0
$933$	2.55677e7	0.961585
$934$	0	0
$935$	1.53126e7	0.572822
$936$	0	0
$937$	−9.15708e6	−0.340728	−0.170364	−	0.985381i	$-0.554494\pi$
$937$	−9.15708e6	−0.340728	−0.170364	+	0.985381i	$0.554494\pi$
$938$	0	0
$939$	−1.45543e7	−0.538676
$940$	0	0
$941$	2.46291e6	0.0906723	0.0453361	−	0.998972i	$-0.485564\pi$
$941$	2.46291e6	0.0906723	0.0453361	+	0.998972i	$0.485564\pi$
$942$	0	0
$943$	1.80101e7	0.659533
$944$	0	0
$945$	291600.	0.0106220
$946$	0	0
$947$	8.13961e6	0.294937	0.147468	−	0.989067i	$-0.452888\pi$
$947$	8.13961e6	0.294937	0.147468	+	0.989067i	$0.452888\pi$
$948$	0	0
$949$	8.39086e6	0.302441
$950$	0	0
$951$	1.83930e7	0.659479
$952$	0	0
$953$	1.65014e7	0.588557	0.294278	−	0.955720i	$-0.404921\pi$
$953$	1.65014e7	0.588557	0.294278	+	0.955720i	$0.404921\pi$
$954$	0	0
$955$	4.57560e6	0.162345
$956$	0	0
$957$	5.75618e6	0.203168
$958$	0	0
$959$	2.00765e6	0.0704922
$960$	0	0
$961$	7.56310e6	0.264175
$962$	0	0
$963$	−9.41771e6	−0.327250
$964$	0	0
$965$	−2.13264e7	−0.737222
$966$	0	0
$967$	−2.72505e7	−0.937148	−0.468574	−	0.883424i	$-0.655232\pi$
$967$	−2.72505e7	−0.937148	−0.468574	+	0.883424i	$0.655232\pi$
$968$	0	0
$969$	−2.79536e7	−0.956377
$970$	0	0
$971$	−1.77172e7	−0.603040	−0.301520	−	0.953460i	$-0.597494\pi$
$971$	−1.77172e7	−0.603040	−0.301520	+	0.953460i	$0.597494\pi$
$972$	0	0
$973$	−4.02810e6	−0.136401
$974$	0	0
$975$	2.08125e6	0.0701153
$976$	0	0
$977$	2.57062e7	0.861591	0.430795	−	0.902450i	$-0.358233\pi$
$977$	2.57062e7	0.861591	0.430795	+	0.902450i	$0.358233\pi$
$978$	0	0
$979$	2.70212e7	0.901049
$980$	0	0
$981$	−1.18845e7	−0.394283
$982$	0	0
$983$	4.05067e6	0.133704	0.0668518	−	0.997763i	$-0.478705\pi$
$983$	4.05067e6	0.133704	0.0668518	+	0.997763i	$0.478705\pi$
$984$	0	0
$985$	−1.19098e7	−0.391125
$986$	0	0
$987$	1.48262e6	0.0484438
$988$	0	0
$989$	8.62330e6	0.280338
$990$	0	0
$991$	−1.71299e6	−0.0554078	−0.0277039	−	0.999616i	$-0.508820\pi$
$991$	−1.71299e6	−0.0554078	−0.0277039	+	0.999616i	$0.508820\pi$
$992$	0	0
$993$	−9.03827e6	−0.290879
$994$	0	0
$995$	1.17662e7	0.376772
$996$	0	0
$997$	2.09874e7	0.668683	0.334341	−	0.942452i	$-0.391486\pi$
$997$	2.09874e7	0.668683	0.334341	+	0.942452i	$0.391486\pi$
$998$	0	0
$999$	392202.	0.0124336

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.a.b.1.1	✓	1
3.2	odd	2		180.6.a.a.1.1		1
4.3	odd	2		240.6.a.m.1.1		1
5.2	odd	4		300.6.d.a.49.2		2
5.3	odd	4		300.6.d.a.49.1		2
5.4	even	2		300.6.a.e.1.1		1
8.3	odd	2		960.6.a.c.1.1		1
8.5	even	2		960.6.a.r.1.1		1
12.11	even	2		720.6.a.f.1.1		1
15.2	even	4		900.6.d.i.649.1		2
15.8	even	4		900.6.d.i.649.2		2
15.14	odd	2		900.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.a.b.1.1	✓	1	1.1	even	1	trivial
180.6.a.a.1.1		1	3.2	odd	2
240.6.a.m.1.1		1	4.3	odd	2
300.6.a.e.1.1		1	5.4	even	2
300.6.d.a.49.1		2	5.3	odd	4
300.6.d.a.49.2		2	5.2	odd	4
720.6.a.f.1.1		1	12.11	even	2
900.6.a.g.1.1		1	15.14	odd	2
900.6.d.i.649.1		2	15.2	even	4
900.6.d.i.649.2		2	15.8	even	4
960.6.a.c.1.1		1	8.3	odd	2
960.6.a.r.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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