LMFDB - Embedded newform 50.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,6,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	50.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:	$8.01919099065$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +14.0000 q^{3} +16.0000 q^{4} -56.0000 q^{6} +158.000 q^{7} -64.0000 q^{8} -47.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +14.0000 q^{3} +16.0000 q^{4} -56.0000 q^{6} +158.000 q^{7} -64.0000 q^{8} -47.0000 q^{9} -148.000 q^{11} +224.000 q^{12} +684.000 q^{13} -632.000 q^{14} +256.000 q^{16} +2048.00 q^{17} +188.000 q^{18} +2220.00 q^{19} +2212.00 q^{21} +592.000 q^{22} -1246.00 q^{23} -896.000 q^{24} -2736.00 q^{26} -4060.00 q^{27} +2528.00 q^{28} -270.000 q^{29} -2048.00 q^{31} -1024.00 q^{32} -2072.00 q^{33} -8192.00 q^{34} -752.000 q^{36} -4372.00 q^{37} -8880.00 q^{38} +9576.00 q^{39} -2398.00 q^{41} -8848.00 q^{42} +2294.00 q^{43} -2368.00 q^{44} +4984.00 q^{46} -10682.0 q^{47} +3584.00 q^{48} +8157.00 q^{49} +28672.0 q^{51} +10944.0 q^{52} +2964.00 q^{53} +16240.0 q^{54} -10112.0 q^{56} +31080.0 q^{57} +1080.00 q^{58} -39740.0 q^{59} -42298.0 q^{61} +8192.00 q^{62} -7426.00 q^{63} +4096.00 q^{64} +8288.00 q^{66} +32098.0 q^{67} +32768.0 q^{68} -17444.0 q^{69} -4248.00 q^{71} +3008.00 q^{72} +30104.0 q^{73} +17488.0 q^{74} +35520.0 q^{76} -23384.0 q^{77} -38304.0 q^{78} +35280.0 q^{79} -45419.0 q^{81} +9592.00 q^{82} -27826.0 q^{83} +35392.0 q^{84} -9176.00 q^{86} -3780.00 q^{87} +9472.00 q^{88} -85210.0 q^{89} +108072. q^{91} -19936.0 q^{92} -28672.0 q^{93} +42728.0 q^{94} -14336.0 q^{96} -97232.0 q^{97} -32628.0 q^{98} +6956.00 q^{99} +O(q^{100})\)

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
$3$	14.0000	0.898100	0.449050	−	0.893507i	$-0.351762\pi$
$3$	14.0000	0.898100	0.449050	+	0.893507i	$0.351762\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−56.0000	−0.635053
$7$	158.000	1.21874	0.609371	−	0.792885i	$-0.291422\pi$
$7$	158.000	1.21874	0.609371	+	0.792885i	$0.291422\pi$
$8$	−64.0000	−0.353553
$9$	−47.0000	−0.193416
$10$	0	0
$11$	−148.000	−0.368791	−0.184395	−	0.982852i	$-0.559033\pi$
$11$	−148.000	−0.368791	−0.184395	+	0.982852i	$0.559033\pi$
$12$	224.000	0.449050
$13$	684.000	1.12253	0.561265	−	0.827636i	$-0.310315\pi$
$13$	684.000	1.12253	0.561265	+	0.827636i	$0.310315\pi$
$14$	−632.000	−0.861781
$15$	0	0
$16$	256.000	0.250000
$17$	2048.00	1.71873	0.859365	−	0.511363i	$-0.170859\pi$
$17$	2048.00	1.71873	0.859365	+	0.511363i	$0.170859\pi$
$18$	188.000	0.136766
$19$	2220.00	1.41081	0.705406	−	0.708804i	$-0.250765\pi$
$19$	2220.00	1.41081	0.705406	+	0.708804i	$0.250765\pi$
$20$	0	0
$21$	2212.00	1.09455
$22$	592.000	0.260774
$23$	−1246.00	−0.491132	−0.245566	−	0.969380i	$-0.578974\pi$
$23$	−1246.00	−0.491132	−0.245566	+	0.969380i	$0.578974\pi$
$24$	−896.000	−0.317526
$25$	0	0
$26$	−2736.00	−0.793748
$27$	−4060.00	−1.07181
$28$	2528.00	0.609371
$29$	−270.000	−0.0596168	−0.0298084	−	0.999556i	$-0.509490\pi$
$29$	−270.000	−0.0596168	−0.0298084	+	0.999556i	$0.509490\pi$
$30$	0	0
$31$	−2048.00	−0.382759	−0.191380	−	0.981516i	$-0.561296\pi$
$31$	−2048.00	−0.382759	−0.191380	+	0.981516i	$0.561296\pi$
$32$	−1024.00	−0.176777
$33$	−2072.00	−0.331211
$34$	−8192.00	−1.21533
$35$	0	0
$36$	−752.000	−0.0967078
$37$	−4372.00	−0.525020	−0.262510	−	0.964929i	$-0.584550\pi$
$37$	−4372.00	−0.525020	−0.262510	+	0.964929i	$0.584550\pi$
$38$	−8880.00	−0.997594
$39$	9576.00	1.00814
$40$	0	0
$41$	−2398.00	−0.222787	−0.111393	−	0.993776i	$-0.535531\pi$
$41$	−2398.00	−0.222787	−0.111393	+	0.993776i	$0.535531\pi$
$42$	−8848.00	−0.773966
$43$	2294.00	0.189200	0.0946002	−	0.995515i	$-0.469843\pi$
$43$	2294.00	0.189200	0.0946002	+	0.995515i	$0.469843\pi$
$44$	−2368.00	−0.184395
$45$	0	0
$46$	4984.00	0.347283
$47$	−10682.0	−0.705355	−0.352678	−	0.935745i	$-0.614729\pi$
$47$	−10682.0	−0.705355	−0.352678	+	0.935745i	$0.614729\pi$
$48$	3584.00	0.224525
$49$	8157.00	0.485333
$50$	0	0
$51$	28672.0	1.54359
$52$	10944.0	0.561265
$53$	2964.00	0.144940	0.0724700	−	0.997371i	$-0.476912\pi$
$53$	2964.00	0.144940	0.0724700	+	0.997371i	$0.476912\pi$
$54$	16240.0	0.757882
$55$	0	0
$56$	−10112.0	−0.430891
$57$	31080.0	1.26705
$58$	1080.00	0.0421555
$59$	−39740.0	−1.48627	−0.743135	−	0.669141i	$-0.766662\pi$
$59$	−39740.0	−1.48627	−0.743135	+	0.669141i	$0.766662\pi$
$60$	0	0
$61$	−42298.0	−1.45544	−0.727722	−	0.685873i	$-0.759421\pi$
$61$	−42298.0	−1.45544	−0.727722	+	0.685873i	$0.759421\pi$
$62$	8192.00	0.270652
$63$	−7426.00	−0.235724
$64$	4096.00	0.125000
$65$	0	0
$66$	8288.00	0.234202
$67$	32098.0	0.873556	0.436778	−	0.899569i	$-0.356119\pi$
$67$	32098.0	0.873556	0.436778	+	0.899569i	$0.356119\pi$
$68$	32768.0	0.859365
$69$	−17444.0	−0.441086
$70$	0	0
$71$	−4248.00	−0.100009	−0.0500044	−	0.998749i	$-0.515924\pi$
$71$	−4248.00	−0.100009	−0.0500044	+	0.998749i	$0.515924\pi$
$72$	3008.00	0.0683828
$73$	30104.0	0.661176	0.330588	−	0.943775i	$-0.392753\pi$
$73$	30104.0	0.661176	0.330588	+	0.943775i	$0.392753\pi$
$74$	17488.0	0.371245
$75$	0	0
$76$	35520.0	0.705406
$77$	−23384.0	−0.449461
$78$	−38304.0	−0.712866
$79$	35280.0	0.636005	0.318003	−	0.948090i	$-0.396988\pi$
$79$	35280.0	0.636005	0.318003	+	0.948090i	$0.396988\pi$
$80$	0	0
$81$	−45419.0	−0.769175
$82$	9592.00	0.157534
$83$	−27826.0	−0.443359	−0.221680	−	0.975120i	$-0.571154\pi$
$83$	−27826.0	−0.443359	−0.221680	+	0.975120i	$0.571154\pi$
$84$	35392.0	0.547277
$85$	0	0
$86$	−9176.00	−0.133785
$87$	−3780.00	−0.0535419
$88$	9472.00	0.130387
$89$	−85210.0	−1.14029	−0.570145	−	0.821544i	$-0.693113\pi$
$89$	−85210.0	−1.14029	−0.570145	+	0.821544i	$0.693113\pi$
$90$	0	0
$91$	108072.	1.36807
$92$	−19936.0	−0.245566
$93$	−28672.0	−0.343756
$94$	42728.0	0.498762
$95$	0	0
$96$	−14336.0	−0.158763
$97$	−97232.0	−1.04925	−0.524626	−	0.851333i	$-0.675795\pi$
$97$	−97232.0	−1.04925	−0.524626	+	0.851333i	$0.675795\pi$
$98$	−32628.0	−0.343183
$99$	6956.00	0.0713299
$100$	0	0
$101$	−4298.00	−0.0419240	−0.0209620	−	0.999780i	$-0.506673\pi$
$101$	−4298.00	−0.0419240	−0.0209620	+	0.999780i	$0.506673\pi$
$102$	−114688.	−1.09148
$103$	124114.	1.15273	0.576365	−	0.817192i	$-0.304471\pi$
$103$	124114.	1.15273	0.576365	+	0.817192i	$0.304471\pi$
$104$	−43776.0	−0.396874
$105$	0	0
$106$	−11856.0	−0.102488
$107$	−42342.0	−0.357530	−0.178765	−	0.983892i	$-0.557210\pi$
$107$	−42342.0	−0.357530	−0.178765	+	0.983892i	$0.557210\pi$
$108$	−64960.0	−0.535904
$109$	−35990.0	−0.290145	−0.145073	−	0.989421i	$-0.546342\pi$
$109$	−35990.0	−0.290145	−0.145073	+	0.989421i	$0.546342\pi$
$110$	0	0
$111$	−61208.0	−0.471521
$112$	40448.0	0.304686
$113$	−228816.	−1.68574	−0.842869	−	0.538118i	$-0.819135\pi$
$113$	−228816.	−1.68574	−0.842869	+	0.538118i	$0.819135\pi$
$114$	−124320.	−0.895940
$115$	0	0
$116$	−4320.00	−0.0298084
$117$	−32148.0	−0.217115
$118$	158960.	1.05095
$119$	323584.	2.09469
$120$	0	0
$121$	−139147.	−0.863993
$122$	169192.	1.02915
$123$	−33572.0	−0.200085
$124$	−32768.0	−0.191380
$125$	0	0
$126$	29704.0	0.166682
$127$	175238.	0.964093	0.482047	−	0.876146i	$-0.339894\pi$
$127$	175238.	0.964093	0.482047	+	0.876146i	$0.339894\pi$
$128$	−16384.0	−0.0883883
$129$	32116.0	0.169921
$130$	0	0
$131$	299652.	1.52559	0.762797	−	0.646638i	$-0.223826\pi$
$131$	299652.	1.52559	0.762797	+	0.646638i	$0.223826\pi$
$132$	−33152.0	−0.165606
$133$	350760.	1.71942
$134$	−128392.	−0.617698
$135$	0	0
$136$	−131072.	−0.607663
$137$	107928.	0.491284	0.245642	−	0.969361i	$-0.421001\pi$
$137$	107928.	0.491284	0.245642	+	0.969361i	$0.421001\pi$
$138$	69776.0	0.311895
$139$	−196460.	−0.862456	−0.431228	−	0.902243i	$-0.641920\pi$
$139$	−196460.	−0.862456	−0.431228	+	0.902243i	$0.641920\pi$
$140$	0	0
$141$	−149548.	−0.633480
$142$	16992.0	0.0707170
$143$	−101232.	−0.413978
$144$	−12032.0	−0.0483539
$145$	0	0
$146$	−120416.	−0.467522
$147$	114198.	0.435878
$148$	−69952.0	−0.262510
$149$	138850.	0.512366	0.256183	−	0.966628i	$-0.417535\pi$
$149$	138850.	0.512366	0.256183	+	0.966628i	$0.417535\pi$
$150$	0	0
$151$	416152.	1.48528	0.742642	−	0.669688i	$-0.233572\pi$
$151$	416152.	1.48528	0.742642	+	0.669688i	$0.233572\pi$
$152$	−142080.	−0.498797
$153$	−96256.0	−0.332429
$154$	93536.0	0.317817
$155$	0	0
$156$	153216.	0.504072
$157$	433108.	1.40232	0.701160	−	0.713004i	$-0.252666\pi$
$157$	433108.	1.40232	0.701160	+	0.713004i	$0.252666\pi$
$158$	−141120.	−0.449724
$159$	41496.0	0.130171
$160$	0	0
$161$	−196868.	−0.598564
$162$	181676.	0.543889
$163$	149134.	0.439651	0.219825	−	0.975539i	$-0.429451\pi$
$163$	149134.	0.439651	0.219825	+	0.975539i	$0.429451\pi$
$164$	−38368.0	−0.111393
$165$	0	0
$166$	111304.	0.313502
$167$	−559602.	−1.55270	−0.776351	−	0.630301i	$-0.782932\pi$
$167$	−559602.	−1.55270	−0.776351	+	0.630301i	$0.782932\pi$
$168$	−141568.	−0.386983
$169$	96563.0	0.260072
$170$	0	0
$171$	−104340.	−0.272873
$172$	36704.0	0.0946002
$173$	343804.	0.873365	0.436682	−	0.899616i	$-0.356153\pi$
$173$	343804.	0.873365	0.436682	+	0.899616i	$0.356153\pi$
$174$	15120.0	0.0378598
$175$	0	0
$176$	−37888.0	−0.0921977
$177$	−556360.	−1.33482
$178$	340840.	0.806307
$179$	23980.0	0.0559392	0.0279696	−	0.999609i	$-0.491096\pi$
$179$	23980.0	0.0559392	0.0279696	+	0.999609i	$0.491096\pi$
$180$	0	0
$181$	−651898.	−1.47905	−0.739526	−	0.673128i	$-0.764950\pi$
$181$	−651898.	−1.47905	−0.739526	+	0.673128i	$0.764950\pi$
$182$	−432288.	−0.967375
$183$	−592172.	−1.30713
$184$	79744.0	0.173641
$185$	0	0
$186$	114688.	0.243072
$187$	−303104.	−0.633852
$188$	−170912.	−0.352678
$189$	−641480.	−1.30626
$190$	0	0
$191$	202752.	0.402144	0.201072	−	0.979576i	$-0.435557\pi$
$191$	202752.	0.402144	0.201072	+	0.979576i	$0.435557\pi$
$192$	57344.0	0.112263
$193$	−452656.	−0.874732	−0.437366	−	0.899284i	$-0.644089\pi$
$193$	−452656.	−0.874732	−0.437366	+	0.899284i	$0.644089\pi$
$194$	388928.	0.741933
$195$	0	0
$196$	130512.	0.242667
$197$	337468.	0.619537	0.309768	−	0.950812i	$-0.399748\pi$
$197$	337468.	0.619537	0.309768	+	0.950812i	$0.399748\pi$
$198$	−27824.0	−0.0504379
$199$	−561000.	−1.00422	−0.502112	−	0.864803i	$-0.667443\pi$
$199$	−561000.	−1.00422	−0.502112	+	0.864803i	$0.667443\pi$
$200$	0	0
$201$	449372.	0.784541
$202$	17192.0	0.0296448
$203$	−42660.0	−0.0726576
$204$	458752.	0.771796
$205$	0	0
$206$	−496456.	−0.815103
$207$	58562.0	0.0949927
$208$	175104.	0.280632
$209$	−328560.	−0.520294
$210$	0	0
$211$	−805548.	−1.24562	−0.622810	−	0.782373i	$-0.714009\pi$
$211$	−805548.	−1.24562	−0.622810	+	0.782373i	$0.714009\pi$
$212$	47424.0	0.0724700
$213$	−59472.0	−0.0898180
$214$	169368.	0.252812
$215$	0	0
$216$	259840.	0.378941
$217$	−323584.	−0.466485
$218$	143960.	0.205164
$219$	421456.	0.593802
$220$	0	0
$221$	1.40083e6	1.92932
$222$	244832.	0.333415
$223$	1.21855e6	1.64090	0.820451	−	0.571717i	$-0.193722\pi$
$223$	1.21855e6	1.64090	0.820451	+	0.571717i	$0.193722\pi$
$224$	−161792.	−0.215445
$225$	0	0
$226$	915264.	1.19200
$227$	564338.	0.726900	0.363450	−	0.931614i	$-0.381599\pi$
$227$	564338.	0.726900	0.363450	+	0.931614i	$0.381599\pi$
$228$	497280.	0.633525
$229$	560330.	0.706082	0.353041	−	0.935608i	$-0.385148\pi$
$229$	560330.	0.706082	0.353041	+	0.935608i	$0.385148\pi$
$230$	0	0
$231$	−327376.	−0.403661
$232$	17280.0	0.0210777
$233$	−293576.	−0.354267	−0.177134	−	0.984187i	$-0.556682\pi$
$233$	−293576.	−0.354267	−0.177134	+	0.984187i	$0.556682\pi$
$234$	128592.	0.153523
$235$	0	0
$236$	−635840.	−0.743135
$237$	493920.	0.571197
$238$	−1.29434e6	−1.48117
$239$	584240.	0.661602	0.330801	−	0.943701i	$-0.392681\pi$
$239$	584240.	0.661602	0.330801	+	0.943701i	$0.392681\pi$
$240$	0	0
$241$	−563798.	−0.625289	−0.312645	−	0.949870i	$-0.601215\pi$
$241$	−563798.	−0.625289	−0.312645	+	0.949870i	$0.601215\pi$
$242$	556588.	0.610936
$243$	350714.	0.381011
$244$	−676768.	−0.727722
$245$	0	0
$246$	134288.	0.141481
$247$	1.51848e6	1.58368
$248$	131072.	0.135326
$249$	−389564.	−0.398181
$250$	0	0
$251$	−1.01975e6	−1.02167	−0.510833	−	0.859680i	$-0.670663\pi$
$251$	−1.01975e6	−1.02167	−0.510833	+	0.859680i	$0.670663\pi$
$252$	−118816.	−0.117862
$253$	184408.	0.181125
$254$	−700952.	−0.681717
$255$	0	0
$256$	65536.0	0.0625000
$257$	657408.	0.620872	0.310436	−	0.950594i	$-0.399525\pi$
$257$	657408.	0.620872	0.310436	+	0.950594i	$0.399525\pi$
$258$	−128464.	−0.120152
$259$	−690776.	−0.639864
$260$	0	0
$261$	12690.0	0.0115308
$262$	−1.19861e6	−1.07876
$263$	−562366.	−0.501337	−0.250668	−	0.968073i	$-0.580650\pi$
$263$	−562366.	−0.501337	−0.250668	+	0.968073i	$0.580650\pi$
$264$	132608.	0.117101
$265$	0	0
$266$	−1.40304e6	−1.21581
$267$	−1.19294e6	−1.02410
$268$	513568.	0.436778
$269$	366570.	0.308870	0.154435	−	0.988003i	$-0.450644\pi$
$269$	366570.	0.308870	0.154435	+	0.988003i	$0.450644\pi$
$270$	0	0
$271$	1.16075e6	0.960099	0.480050	−	0.877241i	$-0.340619\pi$
$271$	1.16075e6	0.960099	0.480050	+	0.877241i	$0.340619\pi$
$272$	524288.	0.429682
$273$	1.51301e6	1.22867
$274$	−431712.	−0.347390
$275$	0	0
$276$	−279104.	−0.220543
$277$	−2.51501e6	−1.96943	−0.984715	−	0.174172i	$-0.944275\pi$
$277$	−2.51501e6	−1.96943	−0.984715	+	0.174172i	$0.944275\pi$
$278$	785840.	0.609849
$279$	96256.0	0.0740316
$280$	0	0
$281$	2.08600e6	1.57597	0.787987	−	0.615692i	$-0.211124\pi$
$281$	2.08600e6	1.57597	0.787987	+	0.615692i	$0.211124\pi$
$282$	598192.	0.447938
$283$	−2.23803e6	−1.66111	−0.830556	−	0.556935i	$-0.811977\pi$
$283$	−2.23803e6	−1.66111	−0.830556	+	0.556935i	$0.811977\pi$
$284$	−67968.0	−0.0500044
$285$	0	0
$286$	404928.	0.292727
$287$	−378884.	−0.271520
$288$	48128.0	0.0341914
$289$	2.77445e6	1.95403
$290$	0	0
$291$	−1.36125e6	−0.942334
$292$	481664.	0.330588
$293$	−975756.	−0.664006	−0.332003	−	0.943278i	$-0.607724\pi$
$293$	−975756.	−0.664006	−0.332003	+	0.943278i	$0.607724\pi$
$294$	−456792.	−0.308212
$295$	0	0
$296$	279808.	0.185623
$297$	600880.	0.395273
$298$	−555400.	−0.362297
$299$	−852264.	−0.551310
$300$	0	0
$301$	362452.	0.230587
$302$	−1.66461e6	−1.05025
$303$	−60172.0	−0.0376520
$304$	568320.	0.352703
$305$	0	0
$306$	385024.	0.235063
$307$	87858.0	0.0532029	0.0266015	−	0.999646i	$-0.491531\pi$
$307$	87858.0	0.0532029	0.0266015	+	0.999646i	$0.491531\pi$
$308$	−374144.	−0.224730
$309$	1.73760e6	1.03527
$310$	0	0
$311$	599352.	0.351383	0.175692	−	0.984445i	$-0.443784\pi$
$311$	599352.	0.351383	0.175692	+	0.984445i	$0.443784\pi$
$312$	−612864.	−0.356433
$313$	−2.09342e6	−1.20780	−0.603900	−	0.797060i	$-0.706387\pi$
$313$	−2.09342e6	−1.20780	−0.603900	+	0.797060i	$0.706387\pi$
$314$	−1.73243e6	−0.991590
$315$	0	0
$316$	564480.	0.318003
$317$	−2.41625e6	−1.35050	−0.675249	−	0.737590i	$-0.735964\pi$
$317$	−2.41625e6	−1.35050	−0.675249	+	0.737590i	$0.735964\pi$
$318$	−165984.	−0.0920446
$319$	39960.0	0.0219861
$320$	0	0
$321$	−592788.	−0.321097
$322$	787472.	0.423249
$323$	4.54656e6	2.42480
$324$	−726704.	−0.384587
$325$	0	0
$326$	−596536.	−0.310880
$327$	−503860.	−0.260580
$328$	153472.	0.0787670
$329$	−1.68776e6	−0.859647
$330$	0	0
$331$	−1.64095e6	−0.823237	−0.411618	−	0.911356i	$-0.635036\pi$
$331$	−1.64095e6	−0.823237	−0.411618	+	0.911356i	$0.635036\pi$
$332$	−445216.	−0.221680
$333$	205484.	0.101547
$334$	2.23841e6	1.09793
$335$	0	0
$336$	566272.	0.273638
$337$	2.18773e6	1.04935	0.524673	−	0.851304i	$-0.324188\pi$
$337$	2.18773e6	1.04935	0.524673	+	0.851304i	$0.324188\pi$
$338$	−386252.	−0.183899
$339$	−3.20342e6	−1.51396
$340$	0	0
$341$	303104.	0.141158
$342$	417360.	0.192950
$343$	−1.36670e6	−0.627246
$344$	−146816.	−0.0668925
$345$	0	0
$346$	−1.37522e6	−0.617562
$347$	2.74502e6	1.22383	0.611916	−	0.790923i	$-0.290399\pi$
$347$	2.74502e6	1.22383	0.611916	+	0.790923i	$0.290399\pi$
$348$	−60480.0	−0.0267709
$349$	−2.65115e6	−1.16512	−0.582560	−	0.812788i	$-0.697949\pi$
$349$	−2.65115e6	−1.16512	−0.582560	+	0.812788i	$0.697949\pi$
$350$	0	0
$351$	−2.77704e6	−1.20313
$352$	151552.	0.0651936
$353$	3.05766e6	1.30603	0.653015	−	0.757345i	$-0.273504\pi$
$353$	3.05766e6	1.30603	0.653015	+	0.757345i	$0.273504\pi$
$354$	2.22544e6	0.943860
$355$	0	0
$356$	−1.36336e6	−0.570145
$357$	4.53018e6	1.88124
$358$	−95920.0	−0.0395550
$359$	3.79356e6	1.55350	0.776749	−	0.629810i	$-0.216867\pi$
$359$	3.79356e6	1.55350	0.776749	+	0.629810i	$0.216867\pi$
$360$	0	0
$361$	2.45230e6	0.990389
$362$	2.60759e6	1.04585
$363$	−1.94806e6	−0.775953
$364$	1.72915e6	0.684037
$365$	0	0
$366$	2.36869e6	0.924283
$367$	−3.11060e6	−1.20553	−0.602767	−	0.797917i	$-0.705935\pi$
$367$	−3.11060e6	−1.20553	−0.602767	+	0.797917i	$0.705935\pi$
$368$	−318976.	−0.122783
$369$	112706.	0.0430905
$370$	0	0
$371$	468312.	0.176645
$372$	−458752.	−0.171878
$373$	−1.41520e6	−0.526677	−0.263339	−	0.964703i	$-0.584824\pi$
$373$	−1.41520e6	−0.526677	−0.263339	+	0.964703i	$0.584824\pi$
$374$	1.21242e6	0.448201
$375$	0	0
$376$	683648.	0.249381
$377$	−184680.	−0.0669216
$378$	2.56592e6	0.923663
$379$	−3.90262e6	−1.39559	−0.697796	−	0.716297i	$-0.745836\pi$
$379$	−3.90262e6	−1.39559	−0.697796	+	0.716297i	$0.745836\pi$
$380$	0	0
$381$	2.45333e6	0.865852
$382$	−811008.	−0.284359
$383$	695674.	0.242331	0.121165	−	0.992632i	$-0.461337\pi$
$383$	695674.	0.242331	0.121165	+	0.992632i	$0.461337\pi$
$384$	−229376.	−0.0793816
$385$	0	0
$386$	1.81062e6	0.618529
$387$	−107818.	−0.0365943
$388$	−1.55571e6	−0.524626
$389$	498290.	0.166958	0.0834792	−	0.996510i	$-0.473397\pi$
$389$	498290.	0.166958	0.0834792	+	0.996510i	$0.473397\pi$
$390$	0	0
$391$	−2.55181e6	−0.844124
$392$	−522048.	−0.171591
$393$	4.19513e6	1.37014
$394$	−1.34987e6	−0.438079
$395$	0	0
$396$	111296.	0.0356649
$397$	1.09567e6	0.348901	0.174451	−	0.984666i	$-0.444185\pi$
$397$	1.09567e6	0.348901	0.174451	+	0.984666i	$0.444185\pi$
$398$	2.24400e6	0.710093
$399$	4.91064e6	1.54421
$400$	0	0
$401$	−2.49160e6	−0.773779	−0.386890	−	0.922126i	$-0.626451\pi$
$401$	−2.49160e6	−0.773779	−0.386890	+	0.922126i	$0.626451\pi$
$402$	−1.79749e6	−0.554755
$403$	−1.40083e6	−0.429659
$404$	−68768.0	−0.0209620
$405$	0	0
$406$	170640.	0.0513766
$407$	647056.	0.193623
$408$	−1.83501e6	−0.545742
$409$	−3.63349e6	−1.07403	−0.537014	−	0.843573i	$-0.680448\pi$
$409$	−3.63349e6	−1.07403	−0.537014	+	0.843573i	$0.680448\pi$
$410$	0	0
$411$	1.51099e6	0.441222
$412$	1.98582e6	0.576365
$413$	−6.27892e6	−1.81138
$414$	−234248.	−0.0671700
$415$	0	0
$416$	−700416.	−0.198437
$417$	−2.75044e6	−0.774572
$418$	1.31424e6	0.367904
$419$	−3.64378e6	−1.01395	−0.506976	−	0.861960i	$-0.669237\pi$
$419$	−3.64378e6	−1.01395	−0.506976	+	0.861960i	$0.669237\pi$
$420$	0	0
$421$	−1.82530e6	−0.501913	−0.250957	−	0.967998i	$-0.580745\pi$
$421$	−1.82530e6	−0.501913	−0.250957	+	0.967998i	$0.580745\pi$
$422$	3.22219e6	0.880786
$423$	502054.	0.136427
$424$	−189696.	−0.0512441
$425$	0	0
$426$	237888.	0.0635109
$427$	−6.68308e6	−1.77381
$428$	−677472.	−0.178765
$429$	−1.41725e6	−0.371794
$430$	0	0
$431$	2.85435e6	0.740141	0.370070	−	0.929004i	$-0.379334\pi$
$431$	2.85435e6	0.740141	0.370070	+	0.929004i	$0.379334\pi$
$432$	−1.03936e6	−0.267952
$433$	−587776.	−0.150658	−0.0753290	−	0.997159i	$-0.524001\pi$
$433$	−587776.	−0.150658	−0.0753290	+	0.997159i	$0.524001\pi$
$434$	1.29434e6	0.329855
$435$	0	0
$436$	−575840.	−0.145073
$437$	−2.76612e6	−0.692895
$438$	−1.68582e6	−0.419882
$439$	6.11604e6	1.51464	0.757319	−	0.653045i	$-0.226509\pi$
$439$	6.11604e6	1.51464	0.757319	+	0.653045i	$0.226509\pi$
$440$	0	0
$441$	−383379.	−0.0938711
$442$	−5.60333e6	−1.36424
$443$	−2.35771e6	−0.570795	−0.285398	−	0.958409i	$-0.592126\pi$
$443$	−2.35771e6	−0.570795	−0.285398	+	0.958409i	$0.592126\pi$
$444$	−979328.	−0.235760
$445$	0	0
$446$	−4.87422e6	−1.16029
$447$	1.94390e6	0.460156
$448$	647168.	0.152343
$449$	5.49735e6	1.28688	0.643439	−	0.765497i	$-0.277507\pi$
$449$	5.49735e6	1.28688	0.643439	+	0.765497i	$0.277507\pi$
$450$	0	0
$451$	354904.	0.0821617
$452$	−3.66106e6	−0.842869
$453$	5.82613e6	1.33393
$454$	−2.25735e6	−0.513996
$455$	0	0
$456$	−1.98912e6	−0.447970
$457$	−1.16039e6	−0.259905	−0.129952	−	0.991520i	$-0.541482\pi$
$457$	−1.16039e6	−0.259905	−0.129952	+	0.991520i	$0.541482\pi$
$458$	−2.24132e6	−0.499275
$459$	−8.31488e6	−1.84215
$460$	0	0
$461$	−2.30330e6	−0.504775	−0.252387	−	0.967626i	$-0.581216\pi$
$461$	−2.30330e6	−0.504775	−0.252387	+	0.967626i	$0.581216\pi$
$462$	1.30950e6	0.285431
$463$	2.71343e6	0.588257	0.294128	−	0.955766i	$-0.404971\pi$
$463$	2.71343e6	0.588257	0.294128	+	0.955766i	$0.404971\pi$
$464$	−69120.0	−0.0149042
$465$	0	0
$466$	1.17430e6	0.250505
$467$	4.05050e6	0.859441	0.429721	−	0.902962i	$-0.358612\pi$
$467$	4.05050e6	0.859441	0.429721	+	0.902962i	$0.358612\pi$
$468$	−514368.	−0.108557
$469$	5.07148e6	1.06464
$470$	0	0
$471$	6.06351e6	1.25942
$472$	2.54336e6	0.525476
$473$	−339512.	−0.0697754
$474$	−1.97568e6	−0.403897
$475$	0	0
$476$	5.17734e6	1.04734
$477$	−139308.	−0.0280337
$478$	−2.33696e6	−0.467823
$479$	5.60528e6	1.11624	0.558121	−	0.829759i	$-0.311522\pi$
$479$	5.60528e6	1.11624	0.558121	+	0.829759i	$0.311522\pi$
$480$	0	0
$481$	−2.99045e6	−0.589350
$482$	2.25519e6	0.442146
$483$	−2.75615e6	−0.537570
$484$	−2.22635e6	−0.431997
$485$	0	0
$486$	−1.40286e6	−0.269415
$487$	7.13168e6	1.36260	0.681301	−	0.732003i	$-0.261414\pi$
$487$	7.13168e6	1.36260	0.681301	+	0.732003i	$0.261414\pi$
$488$	2.70707e6	0.514577
$489$	2.08788e6	0.394850
$490$	0	0
$491$	5.88145e6	1.10098	0.550492	−	0.834841i	$-0.314440\pi$
$491$	5.88145e6	1.10098	0.550492	+	0.834841i	$0.314440\pi$
$492$	−537152.	−0.100042
$493$	−552960.	−0.102465
$494$	−6.07392e6	−1.11983
$495$	0	0
$496$	−524288.	−0.0956898
$497$	−671184.	−0.121885
$498$	1.55826e6	0.281556
$499$	1.75710e6	0.315897	0.157948	−	0.987447i	$-0.449512\pi$
$499$	1.75710e6	0.315897	0.157948	+	0.987447i	$0.449512\pi$
$500$	0	0
$501$	−7.83443e6	−1.39448
$502$	4.07899e6	0.722426
$503$	4.91411e6	0.866015	0.433007	−	0.901390i	$-0.357452\pi$
$503$	4.91411e6	0.866015	0.433007	+	0.901390i	$0.357452\pi$
$504$	475264.	0.0833410
$505$	0	0
$506$	−737632.	−0.128075
$507$	1.35188e6	0.233571
$508$	2.80381e6	0.482047
$509$	−5.75499e6	−0.984578	−0.492289	−	0.870432i	$-0.663840\pi$
$509$	−5.75499e6	−0.984578	−0.492289	+	0.870432i	$0.663840\pi$
$510$	0	0
$511$	4.75643e6	0.805803
$512$	−262144.	−0.0441942
$513$	−9.01320e6	−1.51212
$514$	−2.62963e6	−0.439023
$515$	0	0
$516$	513856.	0.0849605
$517$	1.58094e6	0.260128
$518$	2.76310e6	0.452452
$519$	4.81326e6	0.784369
$520$	0	0
$521$	−1.61980e6	−0.261437	−0.130718	−	0.991420i	$-0.541728\pi$
$521$	−1.61980e6	−0.261437	−0.130718	+	0.991420i	$0.541728\pi$
$522$	−50760.0	−0.00815352
$523$	1.19117e7	1.90422	0.952112	−	0.305751i	$-0.0989074\pi$
$523$	1.19117e7	1.90422	0.952112	+	0.305751i	$0.0989074\pi$
$524$	4.79443e6	0.762797
$525$	0	0
$526$	2.24946e6	0.354499
$527$	−4.19430e6	−0.657860
$528$	−530432.	−0.0828028
$529$	−4.88383e6	−0.758789
$530$	0	0
$531$	1.86778e6	0.287468
$532$	5.61216e6	0.859708
$533$	−1.64023e6	−0.250085
$534$	4.77176e6	0.724145
$535$	0	0
$536$	−2.05427e6	−0.308849
$537$	335720.	0.0502391
$538$	−1.46628e6	−0.218404
$539$	−1.20724e6	−0.178986
$540$	0	0
$541$	4.07630e6	0.598788	0.299394	−	0.954130i	$-0.403215\pi$
$541$	4.07630e6	0.598788	0.299394	+	0.954130i	$0.403215\pi$
$542$	−4.64301e6	−0.678893
$543$	−9.12657e6	−1.32834
$544$	−2.09715e6	−0.303831
$545$	0	0
$546$	−6.05203e6	−0.868800
$547$	1.23680e7	1.76739	0.883694	−	0.468065i	$-0.155049\pi$
$547$	1.23680e7	1.76739	0.883694	+	0.468065i	$0.155049\pi$
$548$	1.72685e6	0.245642
$549$	1.98801e6	0.281505
$550$	0	0
$551$	−599400.	−0.0841081
$552$	1.11642e6	0.155947
$553$	5.57424e6	0.775127
$554$	1.00600e7	1.39260
$555$	0	0
$556$	−3.14336e6	−0.431228
$557$	130308.	0.0177964	0.00889822	−	0.999960i	$-0.497168\pi$
$557$	130308.	0.0177964	0.00889822	+	0.999960i	$0.497168\pi$
$558$	−385024.	−0.0523483
$559$	1.56910e6	0.212383
$560$	0	0
$561$	−4.24346e6	−0.569262
$562$	−8.34401e6	−1.11438
$563$	−5.91687e6	−0.786721	−0.393361	−	0.919384i	$-0.628688\pi$
$563$	−5.91687e6	−0.786721	−0.393361	+	0.919384i	$0.628688\pi$
$564$	−2.39277e6	−0.316740
$565$	0	0
$566$	8.95210e6	1.17458
$567$	−7.17620e6	−0.937426
$568$	271872.	0.0353585
$569$	−9.03013e6	−1.16927	−0.584633	−	0.811298i	$-0.698761\pi$
$569$	−9.03013e6	−1.16927	−0.584633	+	0.811298i	$0.698761\pi$
$570$	0	0
$571$	−1.07093e7	−1.37459	−0.687294	−	0.726379i	$-0.741202\pi$
$571$	−1.07093e7	−1.37459	−0.687294	+	0.726379i	$0.741202\pi$
$572$	−1.61971e6	−0.206989
$573$	2.83853e6	0.361166
$574$	1.51554e6	0.191994
$575$	0	0
$576$	−192512.	−0.0241770
$577$	−1.22051e6	−0.152617	−0.0763084	−	0.997084i	$-0.524313\pi$
$577$	−1.22051e6	−0.152617	−0.0763084	+	0.997084i	$0.524313\pi$
$578$	−1.10978e7	−1.38171
$579$	−6.33718e6	−0.785597
$580$	0	0
$581$	−4.39651e6	−0.540341
$582$	5.44499e6	0.666331
$583$	−438672.	−0.0534526
$584$	−1.92666e6	−0.233761
$585$	0	0
$586$	3.90302e6	0.469523
$587$	−1.47104e7	−1.76210	−0.881049	−	0.473026i	$-0.843162\pi$
$587$	−1.47104e7	−1.76210	−0.881049	+	0.473026i	$0.843162\pi$
$588$	1.82717e6	0.217939
$589$	−4.54656e6	−0.540001
$590$	0	0
$591$	4.72455e6	0.556406
$592$	−1.11923e6	−0.131255
$593$	8.52014e6	0.994970	0.497485	−	0.867472i	$-0.334257\pi$
$593$	8.52014e6	0.994970	0.497485	+	0.867472i	$0.334257\pi$
$594$	−2.40352e6	−0.279500
$595$	0	0
$596$	2.22160e6	0.256183
$597$	−7.85400e6	−0.901893
$598$	3.40906e6	0.389835
$599$	2.90100e6	0.330355	0.165177	−	0.986264i	$-0.447180\pi$
$599$	2.90100e6	0.330355	0.165177	+	0.986264i	$0.447180\pi$
$600$	0	0
$601$	5.72760e6	0.646825	0.323412	−	0.946258i	$-0.395170\pi$
$601$	5.72760e6	0.646825	0.323412	+	0.946258i	$0.395170\pi$
$602$	−1.44981e6	−0.163049
$603$	−1.50861e6	−0.168959
$604$	6.65843e6	0.742642
$605$	0	0
$606$	240688.	0.0266240
$607$	−8.79924e6	−0.969334	−0.484667	−	0.874699i	$-0.661059\pi$
$607$	−8.79924e6	−0.969334	−0.484667	+	0.874699i	$0.661059\pi$
$608$	−2.27328e6	−0.249399
$609$	−597240.	−0.0652538
$610$	0	0
$611$	−7.30649e6	−0.791782
$612$	−1.54010e6	−0.166215
$613$	1.03408e6	0.111149	0.0555744	−	0.998455i	$-0.482301\pi$
$613$	1.03408e6	0.111149	0.0555744	+	0.998455i	$0.482301\pi$
$614$	−351432.	−0.0376201
$615$	0	0
$616$	1.49658e6	0.158908
$617$	1.29854e7	1.37323	0.686616	−	0.727020i	$-0.259095\pi$
$617$	1.29854e7	1.37323	0.686616	+	0.727020i	$0.259095\pi$
$618$	−6.95038e6	−0.732045
$619$	7.92002e6	0.830806	0.415403	−	0.909637i	$-0.363641\pi$
$619$	7.92002e6	0.830806	0.415403	+	0.909637i	$0.363641\pi$
$620$	0	0
$621$	5.05876e6	0.526399
$622$	−2.39741e6	−0.248465
$623$	−1.34632e7	−1.38972
$624$	2.45146e6	0.252036
$625$	0	0
$626$	8.37366e6	0.854043
$627$	−4.59984e6	−0.467276
$628$	6.92973e6	0.701160
$629$	−8.95386e6	−0.902368
$630$	0	0
$631$	1.68218e7	1.68189	0.840945	−	0.541120i	$-0.181999\pi$
$631$	1.68218e7	1.68189	0.840945	+	0.541120i	$0.181999\pi$
$632$	−2.25792e6	−0.224862
$633$	−1.12777e7	−1.11869
$634$	9.66501e6	0.954947
$635$	0	0
$636$	663936.	0.0650854
$637$	5.57939e6	0.544801
$638$	−159840.	−0.0155465
$639$	199656.	0.0193433
$640$	0	0
$641$	−1.55154e7	−1.49148	−0.745741	−	0.666236i	$-0.767904\pi$
$641$	−1.55154e7	−1.49148	−0.745741	+	0.666236i	$0.767904\pi$
$642$	2.37115e6	0.227050
$643$	1.05801e7	1.00916	0.504582	−	0.863364i	$-0.331646\pi$
$643$	1.05801e7	1.00916	0.504582	+	0.863364i	$0.331646\pi$
$644$	−3.14989e6	−0.299282
$645$	0	0
$646$	−1.81862e7	−1.71460
$647$	1.37883e7	1.29494	0.647471	−	0.762090i	$-0.275827\pi$
$647$	1.37883e7	1.29494	0.647471	+	0.762090i	$0.275827\pi$
$648$	2.90682e6	0.271944
$649$	5.88152e6	0.548123
$650$	0	0
$651$	−4.53018e6	−0.418950
$652$	2.38614e6	0.219825
$653$	−1.58924e6	−0.145850	−0.0729248	−	0.997337i	$-0.523233\pi$
$653$	−1.58924e6	−0.145850	−0.0729248	+	0.997337i	$0.523233\pi$
$654$	2.01544e6	0.184258
$655$	0	0
$656$	−613888.	−0.0556967
$657$	−1.41489e6	−0.127882
$658$	6.75102e6	0.607862
$659$	−9.12434e6	−0.818442	−0.409221	−	0.912435i	$-0.634199\pi$
$659$	−9.12434e6	−0.818442	−0.409221	+	0.912435i	$0.634199\pi$
$660$	0	0
$661$	6.50310e6	0.578918	0.289459	−	0.957190i	$-0.406525\pi$
$661$	6.50310e6	0.578918	0.289459	+	0.957190i	$0.406525\pi$
$662$	6.56379e6	0.582116
$663$	1.96116e7	1.73273
$664$	1.78086e6	0.156751
$665$	0	0
$666$	−821936.	−0.0718046
$667$	336420.	0.0292797
$668$	−8.95363e6	−0.776351
$669$	1.70598e7	1.47369
$670$	0	0
$671$	6.26010e6	0.536754
$672$	−2.26509e6	−0.193492
$673$	−2.17810e6	−0.185370	−0.0926850	−	0.995695i	$-0.529545\pi$
$673$	−2.17810e6	−0.185370	−0.0926850	+	0.995695i	$0.529545\pi$
$674$	−8.75091e6	−0.741999
$675$	0	0
$676$	1.54501e6	0.130036
$677$	3.98419e6	0.334094	0.167047	−	0.985949i	$-0.446577\pi$
$677$	3.98419e6	0.334094	0.167047	+	0.985949i	$0.446577\pi$
$678$	1.28137e7	1.07053
$679$	−1.53627e7	−1.27877
$680$	0	0
$681$	7.90073e6	0.652829
$682$	−1.21242e6	−0.0998138
$683$	−5.91563e6	−0.485231	−0.242616	−	0.970122i	$-0.578005\pi$
$683$	−5.91563e6	−0.485231	−0.242616	+	0.970122i	$0.578005\pi$
$684$	−1.66944e6	−0.136436
$685$	0	0
$686$	5.46680e6	0.443530
$687$	7.84462e6	0.634133
$688$	587264.	0.0473001
$689$	2.02738e6	0.162700
$690$	0	0
$691$	−1.55471e7	−1.23867	−0.619335	−	0.785127i	$-0.712598\pi$
$691$	−1.55471e7	−1.23867	−0.619335	+	0.785127i	$0.712598\pi$
$692$	5.50086e6	0.436682
$693$	1.09905e6	0.0869328
$694$	−1.09801e7	−0.865379
$695$	0	0
$696$	241920.	0.0189299
$697$	−4.91110e6	−0.382910
$698$	1.06046e7	0.823864
$699$	−4.11006e6	−0.318167
$700$	0	0
$701$	−2.27103e7	−1.74553	−0.872766	−	0.488139i	$-0.837676\pi$
$701$	−2.27103e7	−1.74553	−0.872766	+	0.488139i	$0.837676\pi$
$702$	1.11082e7	0.850745
$703$	−9.70584e6	−0.740704
$704$	−606208.	−0.0460988
$705$	0	0
$706$	−1.22307e7	−0.923502
$707$	−679084.	−0.0510946
$708$	−8.90176e6	−0.667410
$709$	6.29841e6	0.470560	0.235280	−	0.971928i	$-0.424399\pi$
$709$	6.29841e6	0.470560	0.235280	+	0.971928i	$0.424399\pi$
$710$	0	0
$711$	−1.65816e6	−0.123013
$712$	5.45344e6	0.403154
$713$	2.55181e6	0.187985
$714$	−1.81207e7	−1.33024
$715$	0	0
$716$	383680.	0.0279696
$717$	8.17936e6	0.594185
$718$	−1.51742e7	−1.09849
$719$	2.11911e7	1.52873	0.764367	−	0.644782i	$-0.223052\pi$
$719$	2.11911e7	1.52873	0.764367	+	0.644782i	$0.223052\pi$
$720$	0	0
$721$	1.96100e7	1.40488
$722$	−9.80920e6	−0.700311
$723$	−7.89317e6	−0.561572
$724$	−1.04304e7	−0.739526
$725$	0	0
$726$	7.79223e6	0.548682
$727$	1.35610e7	0.951605	0.475803	−	0.879552i	$-0.342158\pi$
$727$	1.35610e7	0.951605	0.475803	+	0.879552i	$0.342158\pi$
$728$	−6.91661e6	−0.483687
$729$	1.59468e7	1.11136
$730$	0	0
$731$	4.69811e6	0.325185
$732$	−9.47475e6	−0.653567
$733$	2.69413e7	1.85208	0.926038	−	0.377429i	$-0.123192\pi$
$733$	2.69413e7	1.85208	0.926038	+	0.377429i	$0.123192\pi$
$734$	1.24424e7	0.852441
$735$	0	0
$736$	1.27590e6	0.0868207
$737$	−4.75050e6	−0.322160
$738$	−450824.	−0.0304696
$739$	2.77414e6	0.186860	0.0934302	−	0.995626i	$-0.470217\pi$
$739$	2.77414e6	0.186860	0.0934302	+	0.995626i	$0.470217\pi$
$740$	0	0
$741$	2.12587e7	1.42230
$742$	−1.87325e6	−0.124907
$743$	1.85538e7	1.23299	0.616497	−	0.787358i	$-0.288551\pi$
$743$	1.85538e7	1.23299	0.616497	+	0.787358i	$0.288551\pi$
$744$	1.83501e6	0.121536
$745$	0	0
$746$	5.66078e6	0.372417
$747$	1.30782e6	0.0857526
$748$	−4.84966e6	−0.316926
$749$	−6.69004e6	−0.435736
$750$	0	0
$751$	−2.19285e6	−0.141876	−0.0709380	−	0.997481i	$-0.522599\pi$
$751$	−2.19285e6	−0.141876	−0.0709380	+	0.997481i	$0.522599\pi$
$752$	−2.73459e6	−0.176339
$753$	−1.42765e7	−0.917558
$754$	738720.	0.0473207
$755$	0	0
$756$	−1.02637e7	−0.653128
$757$	−9.48749e6	−0.601744	−0.300872	−	0.953665i	$-0.597278\pi$
$757$	−9.48749e6	−0.601744	−0.300872	+	0.953665i	$0.597278\pi$
$758$	1.56105e7	0.986832
$759$	2.58171e6	0.162668
$760$	0	0
$761$	9.69580e6	0.606907	0.303453	−	0.952846i	$-0.401860\pi$
$761$	9.69580e6	0.606907	0.303453	+	0.952846i	$0.401860\pi$
$762$	−9.81333e6	−0.612250
$763$	−5.68642e6	−0.353612
$764$	3.24403e6	0.201072
$765$	0	0
$766$	−2.78270e6	−0.171354
$767$	−2.71822e7	−1.66838
$768$	917504.	0.0561313
$769$	9.32787e6	0.568809	0.284405	−	0.958704i	$-0.408204\pi$
$769$	9.32787e6	0.568809	0.284405	+	0.958704i	$0.408204\pi$
$770$	0	0
$771$	9.20371e6	0.557606
$772$	−7.24250e6	−0.437366
$773$	−9.68080e6	−0.582723	−0.291362	−	0.956613i	$-0.594108\pi$
$773$	−9.68080e6	−0.582723	−0.291362	+	0.956613i	$0.594108\pi$
$774$	431272.	0.0258761
$775$	0	0
$776$	6.22285e6	0.370967
$777$	−9.67086e6	−0.574662
$778$	−1.99316e6	−0.118057
$779$	−5.32356e6	−0.314310
$780$	0	0
$781$	628704.	0.0368824
$782$	1.02072e7	0.596886
$783$	1.09620e6	0.0638977
$784$	2.08819e6	0.121333
$785$	0	0
$786$	−1.67805e7	−0.968833
$787$	−5.52302e6	−0.317863	−0.158931	−	0.987290i	$-0.550805\pi$
$787$	−5.52302e6	−0.317863	−0.158931	+	0.987290i	$0.550805\pi$
$788$	5.39949e6	0.309768
$789$	−7.87312e6	−0.450251
$790$	0	0
$791$	−3.61529e7	−2.05448
$792$	−445184.	−0.0252189
$793$	−2.89318e7	−1.63378
$794$	−4.38267e6	−0.246711
$795$	0	0
$796$	−8.97600e6	−0.502112
$797$	−1.71119e7	−0.954230	−0.477115	−	0.878841i	$-0.658318\pi$
$797$	−1.71119e7	−0.954230	−0.477115	+	0.878841i	$0.658318\pi$
$798$	−1.96426e7	−1.09192
$799$	−2.18767e7	−1.21232
$800$	0	0
$801$	4.00487e6	0.220550
$802$	9.96639e6	0.547145
$803$	−4.45539e6	−0.243836
$804$	7.18995e6	0.392271
$805$	0	0
$806$	5.60333e6	0.303814
$807$	5.13198e6	0.277397
$808$	275072.	0.0148224
$809$	−1.45309e7	−0.780586	−0.390293	−	0.920691i	$-0.627626\pi$
$809$	−1.45309e7	−0.780586	−0.390293	+	0.920691i	$0.627626\pi$
$810$	0	0
$811$	−2.13545e7	−1.14009	−0.570044	−	0.821614i	$-0.693074\pi$
$811$	−2.13545e7	−1.14009	−0.570044	+	0.821614i	$0.693074\pi$
$812$	−682560.	−0.0363288
$813$	1.62505e7	0.862266
$814$	−2.58822e6	−0.136912
$815$	0	0
$816$	7.34003e6	0.385898
$817$	5.09268e6	0.266926
$818$	1.45340e7	0.759453
$819$	−5.07938e6	−0.264607
$820$	0	0
$821$	3.67967e7	1.90525	0.952623	−	0.304154i	$-0.0983737\pi$
$821$	3.67967e7	1.90525	0.952623	+	0.304154i	$0.0983737\pi$
$822$	−6.04397e6	−0.311991
$823$	−3.30668e7	−1.70174	−0.850870	−	0.525376i	$-0.823925\pi$
$823$	−3.30668e7	−1.70174	−0.850870	+	0.525376i	$0.823925\pi$
$824$	−7.94330e6	−0.407552
$825$	0	0
$826$	2.51157e7	1.28084
$827$	1.77309e7	0.901505	0.450752	−	0.892649i	$-0.351156\pi$
$827$	1.77309e7	0.901505	0.450752	+	0.892649i	$0.351156\pi$
$828$	936992.	0.0474963
$829$	1.29375e7	0.653830	0.326915	−	0.945054i	$-0.393991\pi$
$829$	1.29375e7	0.653830	0.326915	+	0.945054i	$0.393991\pi$
$830$	0	0
$831$	−3.52102e7	−1.76875
$832$	2.80166e6	0.140316
$833$	1.67055e7	0.834157
$834$	1.10018e7	0.547705
$835$	0	0
$836$	−5.25696e6	−0.260147
$837$	8.31488e6	0.410244
$838$	1.45751e7	0.716972
$839$	3.31812e7	1.62738	0.813688	−	0.581302i	$-0.197457\pi$
$839$	3.31812e7	1.62738	0.813688	+	0.581302i	$0.197457\pi$
$840$	0	0
$841$	−2.04382e7	−0.996446
$842$	7.30119e6	0.354906
$843$	2.92040e7	1.41538
$844$	−1.28888e7	−0.622810
$845$	0	0
$846$	−2.00822e6	−0.0964683
$847$	−2.19852e7	−1.05299
$848$	758784.	0.0362350
$849$	−3.13324e7	−1.49185
$850$	0	0
$851$	5.44751e6	0.257854
$852$	−951552.	−0.0449090
$853$	−5.17224e6	−0.243392	−0.121696	−	0.992567i	$-0.538833\pi$
$853$	−5.17224e6	−0.243392	−0.121696	+	0.992567i	$0.538833\pi$
$854$	2.67323e7	1.25427
$855$	0	0
$856$	2.70989e6	0.126406
$857$	−1.05320e7	−0.489845	−0.244922	−	0.969543i	$-0.578762\pi$
$857$	−1.05320e7	−0.489845	−0.244922	+	0.969543i	$0.578762\pi$
$858$	5.66899e6	0.262898
$859$	−1.14741e7	−0.530563	−0.265282	−	0.964171i	$-0.585465\pi$
$859$	−1.14741e7	−0.530563	−0.265282	+	0.964171i	$0.585465\pi$
$860$	0	0
$861$	−5.30438e6	−0.243852
$862$	−1.14174e7	−0.523359
$863$	1.92722e7	0.880856	0.440428	−	0.897788i	$-0.354827\pi$
$863$	1.92722e7	0.880856	0.440428	+	0.897788i	$0.354827\pi$
$864$	4.15744e6	0.189471
$865$	0	0
$866$	2.35110e6	0.106531
$867$	3.88423e7	1.75492
$868$	−5.17734e6	−0.233243
$869$	−5.22144e6	−0.234553
$870$	0	0
$871$	2.19550e7	0.980593
$872$	2.30336e6	0.102582
$873$	4.56990e6	0.202942
$874$	1.10645e7	0.489951
$875$	0	0
$876$	6.74330e6	0.296901
$877$	2.30524e7	1.01208	0.506042	−	0.862509i	$-0.331108\pi$
$877$	2.30524e7	1.01208	0.506042	+	0.862509i	$0.331108\pi$
$878$	−2.44642e7	−1.07101
$879$	−1.36606e7	−0.596344
$880$	0	0
$881$	2.26690e7	0.983994	0.491997	−	0.870597i	$-0.336267\pi$
$881$	2.26690e7	0.983994	0.491997	+	0.870597i	$0.336267\pi$
$882$	1.53352e6	0.0663769
$883$	3.67337e6	0.158549	0.0792745	−	0.996853i	$-0.474740\pi$
$883$	3.67337e6	0.158549	0.0792745	+	0.996853i	$0.474740\pi$
$884$	2.24133e7	0.964662
$885$	0	0
$886$	9.43082e6	0.403613
$887$	3.39649e7	1.44951	0.724755	−	0.689007i	$-0.241953\pi$
$887$	3.39649e7	1.44951	0.724755	+	0.689007i	$0.241953\pi$
$888$	3.91731e6	0.166708
$889$	2.76876e7	1.17498
$890$	0	0
$891$	6.72201e6	0.283665
$892$	1.94969e7	0.820451
$893$	−2.37140e7	−0.995123
$894$	−7.77560e6	−0.325379
$895$	0	0
$896$	−2.58867e6	−0.107723
$897$	−1.19317e7	−0.495132
$898$	−2.19894e7	−0.909960
$899$	552960.	0.0228189
$900$	0	0
$901$	6.07027e6	0.249113
$902$	−1.41962e6	−0.0580971
$903$	5.07433e6	0.207090
$904$	1.46442e7	0.595999
$905$	0	0
$906$	−2.33045e7	−0.943234
$907$	−2.13327e7	−0.861050	−0.430525	−	0.902579i	$-0.641672\pi$
$907$	−2.13327e7	−0.861050	−0.430525	+	0.902579i	$0.641672\pi$
$908$	9.02941e6	0.363450
$909$	202006.	0.00810876
$910$	0	0
$911$	−1.03512e7	−0.413235	−0.206617	−	0.978422i	$-0.566246\pi$
$911$	−1.03512e7	−0.413235	−0.206617	+	0.978422i	$0.566246\pi$
$912$	7.95648e6	0.316763
$913$	4.11825e6	0.163507
$914$	4.64157e6	0.183780
$915$	0	0
$916$	8.96528e6	0.353041
$917$	4.73450e7	1.85931
$918$	3.32595e7	1.30259
$919$	−2.59019e7	−1.01168	−0.505839	−	0.862628i	$-0.668817\pi$
$919$	−2.59019e7	−1.01168	−0.505839	+	0.862628i	$0.668817\pi$
$920$	0	0
$921$	1.23001e6	0.0477816
$922$	9.21319e6	0.356930
$923$	−2.90563e6	−0.112263
$924$	−5.23802e6	−0.201831
$925$	0	0
$926$	−1.08537e7	−0.415960
$927$	−5.83336e6	−0.222956
$928$	276480.	0.0105389
$929$	−3.13230e7	−1.19076	−0.595379	−	0.803445i	$-0.702998\pi$
$929$	−3.13230e7	−1.19076	−0.595379	+	0.803445i	$0.702998\pi$
$930$	0	0
$931$	1.81085e7	0.684714
$932$	−4.69722e6	−0.177134
$933$	8.39093e6	0.315577
$934$	−1.62020e7	−0.607717
$935$	0	0
$936$	2.05747e6	0.0767617
$937$	−2.08461e7	−0.775667	−0.387833	−	0.921729i	$-0.626776\pi$
$937$	−2.08461e7	−0.775667	−0.387833	+	0.921729i	$0.626776\pi$
$938$	−2.02859e7	−0.752814
$939$	−2.93078e7	−1.08472
$940$	0	0
$941$	−3.82929e7	−1.40976	−0.704878	−	0.709328i	$-0.748998\pi$
$941$	−3.82929e7	−1.40976	−0.704878	+	0.709328i	$0.748998\pi$
$942$	−2.42540e7	−0.890547
$943$	2.98791e6	0.109418
$944$	−1.01734e7	−0.371568
$945$	0	0
$946$	1.35805e6	0.0493387
$947$	−4.25088e7	−1.54029	−0.770147	−	0.637866i	$-0.779817\pi$
$947$	−4.25088e7	−1.54029	−0.770147	+	0.637866i	$0.779817\pi$
$948$	7.90272e6	0.285598
$949$	2.05911e7	0.742189
$950$	0	0
$951$	−3.38275e7	−1.21288
$952$	−2.07094e7	−0.740585
$953$	3.91855e7	1.39763	0.698816	−	0.715302i	$-0.253711\pi$
$953$	3.91855e7	1.39763	0.698816	+	0.715302i	$0.253711\pi$
$954$	557232.	0.0198228
$955$	0	0
$956$	9.34784e6	0.330801
$957$	559440.	0.0197458
$958$	−2.24211e7	−0.789303
$959$	1.70526e7	0.598749
$960$	0	0
$961$	−2.44348e7	−0.853495
$962$	1.19618e7	0.416734
$963$	1.99007e6	0.0691518
$964$	−9.02077e6	−0.312645
$965$	0	0
$966$	1.10246e7	0.380120
$967$	−1.84836e7	−0.635653	−0.317827	−	0.948149i	$-0.602953\pi$
$967$	−1.84836e7	−0.635653	−0.317827	+	0.948149i	$0.602953\pi$
$968$	8.90541e6	0.305468
$969$	6.36518e7	2.17772
$970$	0	0
$971$	3.95031e7	1.34457	0.672284	−	0.740294i	$-0.265314\pi$
$971$	3.95031e7	1.34457	0.672284	+	0.740294i	$0.265314\pi$
$972$	5.61142e6	0.190505
$973$	−3.10407e7	−1.05111
$974$	−2.85267e7	−0.963506
$975$	0	0
$976$	−1.08283e7	−0.363861
$977$	3.29043e7	1.10285	0.551425	−	0.834225i	$-0.314084\pi$
$977$	3.29043e7	1.10285	0.551425	+	0.834225i	$0.314084\pi$
$978$	−8.35150e6	−0.279201
$979$	1.26111e7	0.420529
$980$	0	0
$981$	1.69153e6	0.0561186
$982$	−2.35258e7	−0.778513
$983$	−2.65797e7	−0.877338	−0.438669	−	0.898649i	$-0.644550\pi$
$983$	−2.65797e7	−0.877338	−0.438669	+	0.898649i	$0.644550\pi$
$984$	2.14861e6	0.0707407
$985$	0	0
$986$	2.21184e6	0.0724538
$987$	−2.36286e7	−0.772049
$988$	2.42957e7	0.791839
$989$	−2.85832e6	−0.0929225
$990$	0	0
$991$	1.92964e7	0.624153	0.312077	−	0.950057i	$-0.398975\pi$
$991$	1.92964e7	0.624153	0.312077	+	0.950057i	$0.398975\pi$
$992$	2.09715e6	0.0676629
$993$	−2.29733e7	−0.739349
$994$	2.68474e6	0.0861858
$995$	0	0
$996$	−6.23302e6	−0.199090
$997$	−5.12017e7	−1.63135	−0.815674	−	0.578511i	$-0.803634\pi$
$997$	−5.12017e7	−1.63135	−0.815674	+	0.578511i	$0.803634\pi$
$998$	−7.02840e6	−0.223373
$999$	1.77503e7	0.562720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.a.c.1.1		1
3.2	odd	2		450.6.a.w.1.1		1
4.3	odd	2		400.6.a.c.1.1		1
5.2	odd	4		10.6.b.a.9.1	✓	2
5.3	odd	4		10.6.b.a.9.2	yes	2
5.4	even	2		50.6.a.e.1.1		1
15.2	even	4		90.6.c.a.19.2		2
15.8	even	4		90.6.c.a.19.1		2
15.14	odd	2		450.6.a.c.1.1		1
20.3	even	4		80.6.c.c.49.1		2
20.7	even	4		80.6.c.c.49.2		2
20.19	odd	2		400.6.a.k.1.1		1
40.3	even	4		320.6.c.a.129.2		2
40.13	odd	4		320.6.c.b.129.1		2
40.27	even	4		320.6.c.a.129.1		2
40.37	odd	4		320.6.c.b.129.2		2
60.23	odd	4		720.6.f.a.289.1		2
60.47	odd	4		720.6.f.a.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.b.a.9.1	✓	2	5.2	odd	4
10.6.b.a.9.2	yes	2	5.3	odd	4
50.6.a.c.1.1		1	1.1	even	1	trivial
50.6.a.e.1.1		1	5.4	even	2
80.6.c.c.49.1		2	20.3	even	4
80.6.c.c.49.2		2	20.7	even	4
90.6.c.a.19.1		2	15.8	even	4
90.6.c.a.19.2		2	15.2	even	4
320.6.c.a.129.1		2	40.27	even	4
320.6.c.a.129.2		2	40.3	even	4
320.6.c.b.129.1		2	40.13	odd	4
320.6.c.b.129.2		2	40.37	odd	4
400.6.a.c.1.1		1	4.3	odd	2
400.6.a.k.1.1		1	20.19	odd	2
450.6.a.c.1.1		1	15.14	odd	2
450.6.a.w.1.1		1	3.2	odd	2
720.6.f.a.289.1		2	60.23	odd	4
720.6.f.a.289.2		2	60.47	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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