LMFDB - Embedded newform 450.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.1727189158$
Analytic rank:	$1$
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	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} -158.000 q^{7} -64.0000 q^{8} +O(q^{10})$

$q-4.00000 q^{2} +16.0000 q^{4} -158.000 q^{7} -64.0000 q^{8} +148.000 q^{11} -684.000 q^{13} +632.000 q^{14} +256.000 q^{16} +2048.00 q^{17} +2220.00 q^{19} -592.000 q^{22} -1246.00 q^{23} +2736.00 q^{26} -2528.00 q^{28} +270.000 q^{29} -2048.00 q^{31} -1024.00 q^{32} -8192.00 q^{34} +4372.00 q^{37} -8880.00 q^{38} +2398.00 q^{41} -2294.00 q^{43} +2368.00 q^{44} +4984.00 q^{46} -10682.0 q^{47} +8157.00 q^{49} -10944.0 q^{52} +2964.00 q^{53} +10112.0 q^{56} -1080.00 q^{58} +39740.0 q^{59} -42298.0 q^{61} +8192.00 q^{62} +4096.00 q^{64} -32098.0 q^{67} +32768.0 q^{68} +4248.00 q^{71} -30104.0 q^{73} -17488.0 q^{74} +35520.0 q^{76} -23384.0 q^{77} +35280.0 q^{79} -9592.00 q^{82} -27826.0 q^{83} +9176.00 q^{86} -9472.00 q^{88} +85210.0 q^{89} +108072. q^{91} -19936.0 q^{92} +42728.0 q^{94} +97232.0 q^{97} -32628.0 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−158.000	−1.21874	−0.609371	−	0.792885i	$-0.708578\pi$
$7$	−158.000	−1.21874	−0.609371	+	0.792885i	$0.708578\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	0	0
$11$	148.000	0.368791	0.184395	−	0.982852i	$-0.440967\pi$
$11$	148.000	0.368791	0.184395	+	0.982852i	$0.440967\pi$
$12$	0	0
$13$	−684.000	−1.12253	−0.561265	−	0.827636i	$-0.689685\pi$
$13$	−684.000	−1.12253	−0.561265	+	0.827636i	$0.689685\pi$
$14$	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	$0.170859\pi$
$18$	0	0
$19$	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
$21$	0	0
$22$	−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	$0.578974\pi$
$24$	0	0
$25$	0	0
$26$	2736.00	0.793748
$27$	0	0
$28$	−2528.00	−0.609371
$29$	270.000	0.0596168	0.0298084	−	0.999556i	$-0.490510\pi$
$29$	270.000	0.0596168	0.0298084	+	0.999556i	$0.490510\pi$
$30$	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
$33$	0	0
$34$	−8192.00	−1.21533
$35$	0	0
$36$	0	0
$37$	4372.00	0.525020	0.262510	−	0.964929i	$-0.415450\pi$
$37$	4372.00	0.525020	0.262510	+	0.964929i	$0.415450\pi$
$38$	−8880.00	−0.997594
$39$	0	0
$40$	0	0
$41$	2398.00	0.222787	0.111393	−	0.993776i	$-0.464469\pi$
$41$	2398.00	0.222787	0.111393	+	0.993776i	$0.464469\pi$
$42$	0	0
$43$	−2294.00	−0.189200	−0.0946002	−	0.995515i	$-0.530157\pi$
$43$	−2294.00	−0.189200	−0.0946002	+	0.995515i	$0.530157\pi$
$44$	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	$0.614729\pi$
$48$	0	0
$49$	8157.00	0.485333
$50$	0	0
$51$	0	0
$52$	−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	$0.476912\pi$
$54$	0	0
$55$	0	0
$56$	10112.0	0.430891
$57$	0	0
$58$	−1080.00	−0.0421555
$59$	39740.0	1.48627	0.743135	−	0.669141i	$-0.233338\pi$
$59$	39740.0	1.48627	0.743135	+	0.669141i	$0.233338\pi$
$60$	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
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−32098.0	−0.873556	−0.436778	−	0.899569i	$-0.643881\pi$
$67$	−32098.0	−0.873556	−0.436778	+	0.899569i	$0.643881\pi$
$68$	32768.0	0.859365
$69$	0	0
$70$	0	0
$71$	4248.00	0.100009	0.0500044	−	0.998749i	$-0.484076\pi$
$71$	4248.00	0.100009	0.0500044	+	0.998749i	$0.484076\pi$
$72$	0	0
$73$	−30104.0	−0.661176	−0.330588	−	0.943775i	$-0.607247\pi$
$73$	−30104.0	−0.661176	−0.330588	+	0.943775i	$0.607247\pi$
$74$	−17488.0	−0.371245
$75$	0	0
$76$	35520.0	0.705406
$77$	−23384.0	−0.449461
$78$	0	0
$79$	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
$81$	0	0
$82$	−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	$0.571154\pi$
$84$	0	0
$85$	0	0
$86$	9176.00	0.133785
$87$	0	0
$88$	−9472.00	−0.130387
$89$	85210.0	1.14029	0.570145	−	0.821544i	$-0.306887\pi$
$89$	85210.0	1.14029	0.570145	+	0.821544i	$0.306887\pi$
$90$	0	0
$91$	108072.	1.36807
$92$	−19936.0	−0.245566
$93$	0	0
$94$	42728.0	0.498762
$95$	0	0
$96$	0	0
$97$	97232.0	1.04925	0.524626	−	0.851333i	$-0.324205\pi$
$97$	97232.0	1.04925	0.524626	+	0.851333i	$0.324205\pi$
$98$	−32628.0	−0.343183
$99$	0	0
$100$	0	0
$101$	4298.00	0.0419240	0.0209620	−	0.999780i	$-0.493327\pi$
$101$	4298.00	0.0419240	0.0209620	+	0.999780i	$0.493327\pi$
$102$	0	0
$103$	−124114.	−1.15273	−0.576365	−	0.817192i	$-0.695529\pi$
$103$	−124114.	−1.15273	−0.576365	+	0.817192i	$0.695529\pi$
$104$	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	$0.557210\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	−40448.0	−0.304686
$113$	−228816.	−1.68574	−0.842869	−	0.538118i	$-0.819135\pi$
$113$	−228816.	−1.68574	−0.842869	+	0.538118i	$0.819135\pi$
$114$	0	0
$115$	0	0
$116$	4320.00	0.0298084
$117$	0	0
$118$	−158960.	−1.05095
$119$	−323584.	−2.09469
$120$	0	0
$121$	−139147.	−0.863993
$122$	169192.	1.02915
$123$	0	0
$124$	−32768.0	−0.191380
$125$	0	0
$126$	0	0
$127$	−175238.	−0.964093	−0.482047	−	0.876146i	$-0.660106\pi$
$127$	−175238.	−0.964093	−0.482047	+	0.876146i	$0.660106\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	0	0
$131$	−299652.	−1.52559	−0.762797	−	0.646638i	$-0.776174\pi$
$131$	−299652.	−1.52559	−0.762797	+	0.646638i	$0.776174\pi$
$132$	0	0
$133$	−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	$0.421001\pi$
$138$	0	0
$139$	−196460.	−0.862456	−0.431228	−	0.902243i	$-0.641920\pi$
$139$	−196460.	−0.862456	−0.431228	+	0.902243i	$0.641920\pi$
$140$	0	0
$141$	0	0
$142$	−16992.0	−0.0707170
$143$	−101232.	−0.413978
$144$	0	0
$145$	0	0
$146$	120416.	0.467522
$147$	0	0
$148$	69952.0	0.262510
$149$	−138850.	−0.512366	−0.256183	−	0.966628i	$-0.582465\pi$
$149$	−138850.	−0.512366	−0.256183	+	0.966628i	$0.582465\pi$
$150$	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
$153$	0	0
$154$	93536.0	0.317817
$155$	0	0
$156$	0	0
$157$	−433108.	−1.40232	−0.701160	−	0.713004i	$-0.747334\pi$
$157$	−433108.	−1.40232	−0.701160	+	0.713004i	$0.747334\pi$
$158$	−141120.	−0.449724
$159$	0	0
$160$	0	0
$161$	196868.	0.598564
$162$	0	0
$163$	−149134.	−0.439651	−0.219825	−	0.975539i	$-0.570549\pi$
$163$	−149134.	−0.439651	−0.219825	+	0.975539i	$0.570549\pi$
$164$	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	$0.782932\pi$
$168$	0	0
$169$	96563.0	0.260072
$170$	0	0
$171$	0	0
$172$	−36704.0	−0.0946002
$173$	343804.	0.873365	0.436682	−	0.899616i	$-0.356153\pi$
$173$	343804.	0.873365	0.436682	+	0.899616i	$0.356153\pi$
$174$	0	0
$175$	0	0
$176$	37888.0	0.0921977
$177$	0	0
$178$	−340840.	−0.806307
$179$	−23980.0	−0.0559392	−0.0279696	−	0.999609i	$-0.508904\pi$
$179$	−23980.0	−0.0559392	−0.0279696	+	0.999609i	$0.508904\pi$
$180$	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
$183$	0	0
$184$	79744.0	0.173641
$185$	0	0
$186$	0	0
$187$	303104.	0.633852
$188$	−170912.	−0.352678
$189$	0	0
$190$	0	0
$191$	−202752.	−0.402144	−0.201072	−	0.979576i	$-0.564443\pi$
$191$	−202752.	−0.402144	−0.201072	+	0.979576i	$0.564443\pi$
$192$	0	0
$193$	452656.	0.874732	0.437366	−	0.899284i	$-0.355911\pi$
$193$	452656.	0.874732	0.437366	+	0.899284i	$0.355911\pi$
$194$	−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	$0.399748\pi$
$198$	0	0
$199$	−561000.	−1.00422	−0.502112	−	0.864803i	$-0.667443\pi$
$199$	−561000.	−1.00422	−0.502112	+	0.864803i	$0.667443\pi$
$200$	0	0
$201$	0	0
$202$	−17192.0	−0.0296448
$203$	−42660.0	−0.0726576
$204$	0	0
$205$	0	0
$206$	496456.	0.815103
$207$	0	0
$208$	−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
$213$	0	0
$214$	169368.	0.252812
$215$	0	0
$216$	0	0
$217$	323584.	0.466485
$218$	143960.	0.205164
$219$	0	0
$220$	0	0
$221$	−1.40083e6	−1.92932
$222$	0	0
$223$	−1.21855e6	−1.64090	−0.820451	−	0.571717i	$-0.806278\pi$
$223$	−1.21855e6	−1.64090	−0.820451	+	0.571717i	$0.806278\pi$
$224$	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	$0.381599\pi$
$228$	0	0
$229$	560330.	0.706082	0.353041	−	0.935608i	$-0.385148\pi$
$229$	560330.	0.706082	0.353041	+	0.935608i	$0.385148\pi$
$230$	0	0
$231$	0	0
$232$	−17280.0	−0.0210777
$233$	−293576.	−0.354267	−0.177134	−	0.984187i	$-0.556682\pi$
$233$	−293576.	−0.354267	−0.177134	+	0.984187i	$0.556682\pi$
$234$	0	0
$235$	0	0
$236$	635840.	0.743135
$237$	0	0
$238$	1.29434e6	1.48117
$239$	−584240.	−0.661602	−0.330801	−	0.943701i	$-0.607319\pi$
$239$	−584240.	−0.661602	−0.330801	+	0.943701i	$0.607319\pi$
$240$	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
$243$	0	0
$244$	−676768.	−0.727722
$245$	0	0
$246$	0	0
$247$	−1.51848e6	−1.58368
$248$	131072.	0.135326
$249$	0	0
$250$	0	0
$251$	1.01975e6	1.02167	0.510833	−	0.859680i	$-0.329337\pi$
$251$	1.01975e6	1.02167	0.510833	+	0.859680i	$0.329337\pi$
$252$	0	0
$253$	−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	$0.399525\pi$
$258$	0	0
$259$	−690776.	−0.639864
$260$	0	0
$261$	0	0
$262$	1.19861e6	1.07876
$263$	−562366.	−0.501337	−0.250668	−	0.968073i	$-0.580650\pi$
$263$	−562366.	−0.501337	−0.250668	+	0.968073i	$0.580650\pi$
$264$	0	0
$265$	0	0
$266$	1.40304e6	1.21581
$267$	0	0
$268$	−513568.	−0.436778
$269$	−366570.	−0.308870	−0.154435	−	0.988003i	$-0.549356\pi$
$269$	−366570.	−0.308870	−0.154435	+	0.988003i	$0.549356\pi$
$270$	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
$273$	0	0
$274$	−431712.	−0.347390
$275$	0	0
$276$	0	0
$277$	2.51501e6	1.96943	0.984715	−	0.174172i	$-0.0557250\pi$
$277$	2.51501e6	1.96943	0.984715	+	0.174172i	$0.0557250\pi$
$278$	785840.	0.609849
$279$	0	0
$280$	0	0
$281$	−2.08600e6	−1.57597	−0.787987	−	0.615692i	$-0.788876\pi$
$281$	−2.08600e6	−1.57597	−0.787987	+	0.615692i	$0.788876\pi$
$282$	0	0
$283$	2.23803e6	1.66111	0.830556	−	0.556935i	$-0.188023\pi$
$283$	2.23803e6	1.66111	0.830556	+	0.556935i	$0.188023\pi$
$284$	67968.0	0.0500044
$285$	0	0
$286$	404928.	0.292727
$287$	−378884.	−0.271520
$288$	0	0
$289$	2.77445e6	1.95403
$290$	0	0
$291$	0	0
$292$	−481664.	−0.330588
$293$	−975756.	−0.664006	−0.332003	−	0.943278i	$-0.607724\pi$
$293$	−975756.	−0.664006	−0.332003	+	0.943278i	$0.607724\pi$
$294$	0	0
$295$	0	0
$296$	−279808.	−0.185623
$297$	0	0
$298$	555400.	0.362297
$299$	852264.	0.551310
$300$	0	0
$301$	362452.	0.230587
$302$	−1.66461e6	−1.05025
$303$	0	0
$304$	568320.	0.352703
$305$	0	0
$306$	0	0
$307$	−87858.0	−0.0532029	−0.0266015	−	0.999646i	$-0.508469\pi$
$307$	−87858.0	−0.0532029	−0.0266015	+	0.999646i	$0.508469\pi$
$308$	−374144.	−0.224730
$309$	0	0
$310$	0	0
$311$	−599352.	−0.351383	−0.175692	−	0.984445i	$-0.556216\pi$
$311$	−599352.	−0.351383	−0.175692	+	0.984445i	$0.556216\pi$
$312$	0	0
$313$	2.09342e6	1.20780	0.603900	−	0.797060i	$-0.293613\pi$
$313$	2.09342e6	1.20780	0.603900	+	0.797060i	$0.293613\pi$
$314$	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	$0.735964\pi$
$318$	0	0
$319$	39960.0	0.0219861
$320$	0	0
$321$	0	0
$322$	−787472.	−0.423249
$323$	4.54656e6	2.42480
$324$	0	0
$325$	0	0
$326$	596536.	0.310880
$327$	0	0
$328$	−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
$333$	0	0
$334$	2.23841e6	1.09793
$335$	0	0
$336$	0	0
$337$	−2.18773e6	−1.04935	−0.524673	−	0.851304i	$-0.675812\pi$
$337$	−2.18773e6	−1.04935	−0.524673	+	0.851304i	$0.675812\pi$
$338$	−386252.	−0.183899
$339$	0	0
$340$	0	0
$341$	−303104.	−0.141158
$342$	0	0
$343$	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	$0.290399\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	−151552.	−0.0651936
$353$	3.05766e6	1.30603	0.653015	−	0.757345i	$-0.273504\pi$
$353$	3.05766e6	1.30603	0.653015	+	0.757345i	$0.273504\pi$
$354$	0	0
$355$	0	0
$356$	1.36336e6	0.570145
$357$	0	0
$358$	95920.0	0.0395550
$359$	−3.79356e6	−1.55350	−0.776749	−	0.629810i	$-0.783133\pi$
$359$	−3.79356e6	−1.55350	−0.776749	+	0.629810i	$0.783133\pi$
$360$	0	0
$361$	2.45230e6	0.990389
$362$	2.60759e6	1.04585
$363$	0	0
$364$	1.72915e6	0.684037
$365$	0	0
$366$	0	0
$367$	3.11060e6	1.20553	0.602767	−	0.797917i	$-0.294065\pi$
$367$	3.11060e6	1.20553	0.602767	+	0.797917i	$0.294065\pi$
$368$	−318976.	−0.122783
$369$	0	0
$370$	0	0
$371$	−468312.	−0.176645
$372$	0	0
$373$	1.41520e6	0.526677	0.263339	−	0.964703i	$-0.415176\pi$
$373$	1.41520e6	0.526677	0.263339	+	0.964703i	$0.415176\pi$
$374$	−1.21242e6	−0.448201
$375$	0	0
$376$	683648.	0.249381
$377$	−184680.	−0.0669216
$378$	0	0
$379$	−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
$381$	0	0
$382$	811008.	0.284359
$383$	695674.	0.242331	0.121165	−	0.992632i	$-0.461337\pi$
$383$	695674.	0.242331	0.121165	+	0.992632i	$0.461337\pi$
$384$	0	0
$385$	0	0
$386$	−1.81062e6	−0.618529
$387$	0	0
$388$	1.55571e6	0.524626
$389$	−498290.	−0.166958	−0.0834792	−	0.996510i	$-0.526603\pi$
$389$	−498290.	−0.166958	−0.0834792	+	0.996510i	$0.526603\pi$
$390$	0	0
$391$	−2.55181e6	−0.844124
$392$	−522048.	−0.171591
$393$	0	0
$394$	−1.34987e6	−0.438079
$395$	0	0
$396$	0	0
$397$	−1.09567e6	−0.348901	−0.174451	−	0.984666i	$-0.555815\pi$
$397$	−1.09567e6	−0.348901	−0.174451	+	0.984666i	$0.555815\pi$
$398$	2.24400e6	0.710093
$399$	0	0
$400$	0	0
$401$	2.49160e6	0.773779	0.386890	−	0.922126i	$-0.373549\pi$
$401$	2.49160e6	0.773779	0.386890	+	0.922126i	$0.373549\pi$
$402$	0	0
$403$	1.40083e6	0.429659
$404$	68768.0	0.0209620
$405$	0	0
$406$	170640.	0.0513766
$407$	647056.	0.193623
$408$	0	0
$409$	−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
$411$	0	0
$412$	−1.98582e6	−0.576365
$413$	−6.27892e6	−1.81138
$414$	0	0
$415$	0	0
$416$	700416.	0.198437
$417$	0	0
$418$	−1.31424e6	−0.367904
$419$	3.64378e6	1.01395	0.506976	−	0.861960i	$-0.330763\pi$
$419$	3.64378e6	1.01395	0.506976	+	0.861960i	$0.330763\pi$
$420$	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
$423$	0	0
$424$	−189696.	−0.0512441
$425$	0	0
$426$	0	0
$427$	6.68308e6	1.77381
$428$	−677472.	−0.178765
$429$	0	0
$430$	0	0
$431$	−2.85435e6	−0.740141	−0.370070	−	0.929004i	$-0.620666\pi$
$431$	−2.85435e6	−0.740141	−0.370070	+	0.929004i	$0.620666\pi$
$432$	0	0
$433$	587776.	0.150658	0.0753290	−	0.997159i	$-0.475999\pi$
$433$	587776.	0.150658	0.0753290	+	0.997159i	$0.475999\pi$
$434$	−1.29434e6	−0.329855
$435$	0	0
$436$	−575840.	−0.145073
$437$	−2.76612e6	−0.692895
$438$	0	0
$439$	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
$441$	0	0
$442$	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	$0.592126\pi$
$444$	0	0
$445$	0	0
$446$	4.87422e6	1.16029
$447$	0	0
$448$	−647168.	−0.152343
$449$	−5.49735e6	−1.28688	−0.643439	−	0.765497i	$-0.722493\pi$
$449$	−5.49735e6	−1.28688	−0.643439	+	0.765497i	$0.722493\pi$
$450$	0	0
$451$	354904.	0.0821617
$452$	−3.66106e6	−0.842869
$453$	0	0
$454$	−2.25735e6	−0.513996
$455$	0	0
$456$	0	0
$457$	1.16039e6	0.259905	0.129952	−	0.991520i	$-0.458518\pi$
$457$	1.16039e6	0.259905	0.129952	+	0.991520i	$0.458518\pi$
$458$	−2.24132e6	−0.499275
$459$	0	0
$460$	0	0
$461$	2.30330e6	0.504775	0.252387	−	0.967626i	$-0.418784\pi$
$461$	2.30330e6	0.504775	0.252387	+	0.967626i	$0.418784\pi$
$462$	0	0
$463$	−2.71343e6	−0.588257	−0.294128	−	0.955766i	$-0.595029\pi$
$463$	−2.71343e6	−0.588257	−0.294128	+	0.955766i	$0.595029\pi$
$464$	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	$0.358612\pi$
$468$	0	0
$469$	5.07148e6	1.06464
$470$	0	0
$471$	0	0
$472$	−2.54336e6	−0.525476
$473$	−339512.	−0.0697754
$474$	0	0
$475$	0	0
$476$	−5.17734e6	−1.04734
$477$	0	0
$478$	2.33696e6	0.467823
$479$	−5.60528e6	−1.11624	−0.558121	−	0.829759i	$-0.688478\pi$
$479$	−5.60528e6	−1.11624	−0.558121	+	0.829759i	$0.688478\pi$
$480$	0	0
$481$	−2.99045e6	−0.589350
$482$	2.25519e6	0.442146
$483$	0	0
$484$	−2.22635e6	−0.431997
$485$	0	0
$486$	0	0
$487$	−7.13168e6	−1.36260	−0.681301	−	0.732003i	$-0.738586\pi$
$487$	−7.13168e6	−1.36260	−0.681301	+	0.732003i	$0.738586\pi$
$488$	2.70707e6	0.514577
$489$	0	0
$490$	0	0
$491$	−5.88145e6	−1.10098	−0.550492	−	0.834841i	$-0.685560\pi$
$491$	−5.88145e6	−1.10098	−0.550492	+	0.834841i	$0.685560\pi$
$492$	0	0
$493$	552960.	0.102465
$494$	6.07392e6	1.11983
$495$	0	0
$496$	−524288.	−0.0956898
$497$	−671184.	−0.121885
$498$	0	0
$499$	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
$501$	0	0
$502$	−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	$0.357452\pi$
$504$	0	0
$505$	0	0
$506$	737632.	0.128075
$507$	0	0
$508$	−2.80381e6	−0.482047
$509$	5.75499e6	0.984578	0.492289	−	0.870432i	$-0.336160\pi$
$509$	5.75499e6	0.984578	0.492289	+	0.870432i	$0.336160\pi$
$510$	0	0
$511$	4.75643e6	0.805803
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−2.62963e6	−0.439023
$515$	0	0
$516$	0	0
$517$	−1.58094e6	−0.260128
$518$	2.76310e6	0.452452
$519$	0	0
$520$	0	0
$521$	1.61980e6	0.261437	0.130718	−	0.991420i	$-0.458272\pi$
$521$	1.61980e6	0.261437	0.130718	+	0.991420i	$0.458272\pi$
$522$	0	0
$523$	−1.19117e7	−1.90422	−0.952112	−	0.305751i	$-0.901093\pi$
$523$	−1.19117e7	−1.90422	−0.952112	+	0.305751i	$0.901093\pi$
$524$	−4.79443e6	−0.762797
$525$	0	0
$526$	2.24946e6	0.354499
$527$	−4.19430e6	−0.657860
$528$	0	0
$529$	−4.88383e6	−0.758789
$530$	0	0
$531$	0	0
$532$	−5.61216e6	−0.859708
$533$	−1.64023e6	−0.250085
$534$	0	0
$535$	0	0
$536$	2.05427e6	0.308849
$537$	0	0
$538$	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
$543$	0	0
$544$	−2.09715e6	−0.303831
$545$	0	0
$546$	0	0
$547$	−1.23680e7	−1.76739	−0.883694	−	0.468065i	$-0.844951\pi$
$547$	−1.23680e7	−1.76739	−0.883694	+	0.468065i	$0.844951\pi$
$548$	1.72685e6	0.245642
$549$	0	0
$550$	0	0
$551$	599400.	0.0841081
$552$	0	0
$553$	−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	$0.497168\pi$
$558$	0	0
$559$	1.56910e6	0.212383
$560$	0	0
$561$	0	0
$562$	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	$0.628688\pi$
$564$	0	0
$565$	0	0
$566$	−8.95210e6	−1.17458
$567$	0	0
$568$	−271872.	−0.0353585
$569$	9.03013e6	1.16927	0.584633	−	0.811298i	$-0.301239\pi$
$569$	9.03013e6	1.16927	0.584633	+	0.811298i	$0.301239\pi$
$570$	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
$573$	0	0
$574$	1.51554e6	0.191994
$575$	0	0
$576$	0	0
$577$	1.22051e6	0.152617	0.0763084	−	0.997084i	$-0.475687\pi$
$577$	1.22051e6	0.152617	0.0763084	+	0.997084i	$0.475687\pi$
$578$	−1.10978e7	−1.38171
$579$	0	0
$580$	0	0
$581$	4.39651e6	0.540341
$582$	0	0
$583$	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	$0.843162\pi$
$588$	0	0
$589$	−4.54656e6	−0.540001
$590$	0	0
$591$	0	0
$592$	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	$0.334257\pi$
$594$	0	0
$595$	0	0
$596$	−2.22160e6	−0.256183
$597$	0	0
$598$	−3.40906e6	−0.389835
$599$	−2.90100e6	−0.330355	−0.165177	−	0.986264i	$-0.552820\pi$
$599$	−2.90100e6	−0.330355	−0.165177	+	0.986264i	$0.552820\pi$
$600$	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
$603$	0	0
$604$	6.65843e6	0.742642
$605$	0	0
$606$	0	0
$607$	8.79924e6	0.969334	0.484667	−	0.874699i	$-0.338941\pi$
$607$	8.79924e6	0.969334	0.484667	+	0.874699i	$0.338941\pi$
$608$	−2.27328e6	−0.249399
$609$	0	0
$610$	0	0
$611$	7.30649e6	0.791782
$612$	0	0
$613$	−1.03408e6	−0.111149	−0.0555744	−	0.998455i	$-0.517699\pi$
$613$	−1.03408e6	−0.111149	−0.0555744	+	0.998455i	$0.517699\pi$
$614$	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	$0.259095\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	2.39741e6	0.248465
$623$	−1.34632e7	−1.38972
$624$	0	0
$625$	0	0
$626$	−8.37366e6	−0.854043
$627$	0	0
$628$	−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
$633$	0	0
$634$	9.66501e6	0.954947
$635$	0	0
$636$	0	0
$637$	−5.57939e6	−0.544801
$638$	−159840.	−0.0155465
$639$	0	0
$640$	0	0
$641$	1.55154e7	1.49148	0.745741	−	0.666236i	$-0.232096\pi$
$641$	1.55154e7	1.49148	0.745741	+	0.666236i	$0.232096\pi$
$642$	0	0
$643$	−1.05801e7	−1.00916	−0.504582	−	0.863364i	$-0.668354\pi$
$643$	−1.05801e7	−1.00916	−0.504582	+	0.863364i	$0.668354\pi$
$644$	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	$0.275827\pi$
$648$	0	0
$649$	5.88152e6	0.548123
$650$	0	0
$651$	0	0
$652$	−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	$0.523233\pi$
$654$	0	0
$655$	0	0
$656$	613888.	0.0556967
$657$	0	0
$658$	−6.75102e6	−0.607862
$659$	9.12434e6	0.818442	0.409221	−	0.912435i	$-0.365801\pi$
$659$	9.12434e6	0.818442	0.409221	+	0.912435i	$0.365801\pi$
$660$	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
$663$	0	0
$664$	1.78086e6	0.156751
$665$	0	0
$666$	0	0
$667$	−336420.	−0.0292797
$668$	−8.95363e6	−0.776351
$669$	0	0
$670$	0	0
$671$	−6.26010e6	−0.536754
$672$	0	0
$673$	2.17810e6	0.185370	0.0926850	−	0.995695i	$-0.470455\pi$
$673$	2.17810e6	0.185370	0.0926850	+	0.995695i	$0.470455\pi$
$674$	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	$0.446577\pi$
$678$	0	0
$679$	−1.53627e7	−1.27877
$680$	0	0
$681$	0	0
$682$	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	$0.578005\pi$
$684$	0	0
$685$	0	0
$686$	−5.46680e6	−0.443530
$687$	0	0
$688$	−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
$693$	0	0
$694$	−1.09801e7	−0.865379
$695$	0	0
$696$	0	0
$697$	4.91110e6	0.382910
$698$	1.06046e7	0.823864
$699$	0	0
$700$	0	0
$701$	2.27103e7	1.74553	0.872766	−	0.488139i	$-0.162324\pi$
$701$	2.27103e7	1.74553	0.872766	+	0.488139i	$0.162324\pi$
$702$	0	0
$703$	9.70584e6	0.740704
$704$	606208.	0.0460988
$705$	0	0
$706$	−1.22307e7	−0.923502
$707$	−679084.	−0.0510946
$708$	0	0
$709$	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
$711$	0	0
$712$	−5.45344e6	−0.403154
$713$	2.55181e6	0.187985
$714$	0	0
$715$	0	0
$716$	−383680.	−0.0279696
$717$	0	0
$718$	1.51742e7	1.09849
$719$	−2.11911e7	−1.52873	−0.764367	−	0.644782i	$-0.776948\pi$
$719$	−2.11911e7	−1.52873	−0.764367	+	0.644782i	$0.776948\pi$
$720$	0	0
$721$	1.96100e7	1.40488
$722$	−9.80920e6	−0.700311
$723$	0	0
$724$	−1.04304e7	−0.739526
$725$	0	0
$726$	0	0
$727$	−1.35610e7	−0.951605	−0.475803	−	0.879552i	$-0.657842\pi$
$727$	−1.35610e7	−0.951605	−0.475803	+	0.879552i	$0.657842\pi$
$728$	−6.91661e6	−0.483687
$729$	0	0
$730$	0	0
$731$	−4.69811e6	−0.325185
$732$	0	0
$733$	−2.69413e7	−1.85208	−0.926038	−	0.377429i	$-0.876808\pi$
$733$	−2.69413e7	−1.85208	−0.926038	+	0.377429i	$0.876808\pi$
$734$	−1.24424e7	−0.852441
$735$	0	0
$736$	1.27590e6	0.0868207
$737$	−4.75050e6	−0.322160
$738$	0	0
$739$	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
$741$	0	0
$742$	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	$0.288551\pi$
$744$	0	0
$745$	0	0
$746$	−5.66078e6	−0.372417
$747$	0	0
$748$	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
$753$	0	0
$754$	738720.	0.0473207
$755$	0	0
$756$	0	0
$757$	9.48749e6	0.601744	0.300872	−	0.953665i	$-0.402722\pi$
$757$	9.48749e6	0.601744	0.300872	+	0.953665i	$0.402722\pi$
$758$	1.56105e7	0.986832
$759$	0	0
$760$	0	0
$761$	−9.69580e6	−0.606907	−0.303453	−	0.952846i	$-0.598140\pi$
$761$	−9.69580e6	−0.606907	−0.303453	+	0.952846i	$0.598140\pi$
$762$	0	0
$763$	5.68642e6	0.353612
$764$	−3.24403e6	−0.201072
$765$	0	0
$766$	−2.78270e6	−0.171354
$767$	−2.71822e7	−1.66838
$768$	0	0
$769$	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
$771$	0	0
$772$	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	$0.594108\pi$
$774$	0	0
$775$	0	0
$776$	−6.22285e6	−0.370967
$777$	0	0
$778$	1.99316e6	0.118057
$779$	5.32356e6	0.314310
$780$	0	0
$781$	628704.	0.0368824
$782$	1.02072e7	0.596886
$783$	0	0
$784$	2.08819e6	0.121333
$785$	0	0
$786$	0	0
$787$	5.52302e6	0.317863	0.158931	−	0.987290i	$-0.449195\pi$
$787$	5.52302e6	0.317863	0.158931	+	0.987290i	$0.449195\pi$
$788$	5.39949e6	0.309768
$789$	0	0
$790$	0	0
$791$	3.61529e7	2.05448
$792$	0	0
$793$	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	$0.658318\pi$
$798$	0	0
$799$	−2.18767e7	−1.21232
$800$	0	0
$801$	0	0
$802$	−9.96639e6	−0.547145
$803$	−4.45539e6	−0.243836
$804$	0	0
$805$	0	0
$806$	−5.60333e6	−0.303814
$807$	0	0
$808$	−275072.	−0.0148224
$809$	1.45309e7	0.780586	0.390293	−	0.920691i	$-0.372374\pi$
$809$	1.45309e7	0.780586	0.390293	+	0.920691i	$0.372374\pi$
$810$	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
$813$	0	0
$814$	−2.58822e6	−0.136912
$815$	0	0
$816$	0	0
$817$	−5.09268e6	−0.266926
$818$	1.45340e7	0.759453
$819$	0	0
$820$	0	0
$821$	−3.67967e7	−1.90525	−0.952623	−	0.304154i	$-0.901626\pi$
$821$	−3.67967e7	−1.90525	−0.952623	+	0.304154i	$0.901626\pi$
$822$	0	0
$823$	3.30668e7	1.70174	0.850870	−	0.525376i	$-0.176075\pi$
$823$	3.30668e7	1.70174	0.850870	+	0.525376i	$0.176075\pi$
$824$	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	$0.351156\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	−2.80166e6	−0.140316
$833$	1.67055e7	0.834157
$834$	0	0
$835$	0	0
$836$	5.25696e6	0.260147
$837$	0	0
$838$	−1.45751e7	−0.716972
$839$	−3.31812e7	−1.62738	−0.813688	−	0.581302i	$-0.802543\pi$
$839$	−3.31812e7	−1.62738	−0.813688	+	0.581302i	$0.802543\pi$
$840$	0	0
$841$	−2.04382e7	−0.996446
$842$	7.30119e6	0.354906
$843$	0	0
$844$	−1.28888e7	−0.622810
$845$	0	0
$846$	0	0
$847$	2.19852e7	1.05299
$848$	758784.	0.0362350
$849$	0	0
$850$	0	0
$851$	−5.44751e6	−0.257854
$852$	0	0
$853$	5.17224e6	0.243392	0.121696	−	0.992567i	$-0.461167\pi$
$853$	5.17224e6	0.243392	0.121696	+	0.992567i	$0.461167\pi$
$854$	−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	$0.578762\pi$
$858$	0	0
$859$	−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
$861$	0	0
$862$	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	$0.354827\pi$
$864$	0	0
$865$	0	0
$866$	−2.35110e6	−0.106531
$867$	0	0
$868$	5.17734e6	0.233243
$869$	5.22144e6	0.234553
$870$	0	0
$871$	2.19550e7	0.980593
$872$	2.30336e6	0.102582
$873$	0	0
$874$	1.10645e7	0.489951
$875$	0	0
$876$	0	0
$877$	−2.30524e7	−1.01208	−0.506042	−	0.862509i	$-0.668892\pi$
$877$	−2.30524e7	−1.01208	−0.506042	+	0.862509i	$0.668892\pi$
$878$	−2.44642e7	−1.07101
$879$	0	0
$880$	0	0
$881$	−2.26690e7	−0.983994	−0.491997	−	0.870597i	$-0.663733\pi$
$881$	−2.26690e7	−0.983994	−0.491997	+	0.870597i	$0.663733\pi$
$882$	0	0
$883$	−3.67337e6	−0.158549	−0.0792745	−	0.996853i	$-0.525260\pi$
$883$	−3.67337e6	−0.158549	−0.0792745	+	0.996853i	$0.525260\pi$
$884$	−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	$0.241953\pi$
$888$	0	0
$889$	2.76876e7	1.17498
$890$	0	0
$891$	0	0
$892$	−1.94969e7	−0.820451
$893$	−2.37140e7	−0.995123
$894$	0	0
$895$	0	0
$896$	2.58867e6	0.107723
$897$	0	0
$898$	2.19894e7	0.909960
$899$	−552960.	−0.0228189
$900$	0	0
$901$	6.07027e6	0.249113
$902$	−1.41962e6	−0.0580971
$903$	0	0
$904$	1.46442e7	0.595999
$905$	0	0
$906$	0	0
$907$	2.13327e7	0.861050	0.430525	−	0.902579i	$-0.358328\pi$
$907$	2.13327e7	0.861050	0.430525	+	0.902579i	$0.358328\pi$
$908$	9.02941e6	0.363450
$909$	0	0
$910$	0	0
$911$	1.03512e7	0.413235	0.206617	−	0.978422i	$-0.433754\pi$
$911$	1.03512e7	0.413235	0.206617	+	0.978422i	$0.433754\pi$
$912$	0	0
$913$	−4.11825e6	−0.163507
$914$	−4.64157e6	−0.183780
$915$	0	0
$916$	8.96528e6	0.353041
$917$	4.73450e7	1.85931
$918$	0	0
$919$	−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
$921$	0	0
$922$	−9.21319e6	−0.356930
$923$	−2.90563e6	−0.112263
$924$	0	0
$925$	0	0
$926$	1.08537e7	0.415960
$927$	0	0
$928$	−276480.	−0.0105389
$929$	3.13230e7	1.19076	0.595379	−	0.803445i	$-0.297002\pi$
$929$	3.13230e7	1.19076	0.595379	+	0.803445i	$0.297002\pi$
$930$	0	0
$931$	1.81085e7	0.684714
$932$	−4.69722e6	−0.177134
$933$	0	0
$934$	−1.62020e7	−0.607717
$935$	0	0
$936$	0	0
$937$	2.08461e7	0.775667	0.387833	−	0.921729i	$-0.373224\pi$
$937$	2.08461e7	0.775667	0.387833	+	0.921729i	$0.373224\pi$
$938$	−2.02859e7	−0.752814
$939$	0	0
$940$	0	0
$941$	3.82929e7	1.40976	0.704878	−	0.709328i	$-0.251002\pi$
$941$	3.82929e7	1.40976	0.704878	+	0.709328i	$0.251002\pi$
$942$	0	0
$943$	−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	$0.779817\pi$
$948$	0	0
$949$	2.05911e7	0.742189
$950$	0	0
$951$	0	0
$952$	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	$0.253711\pi$
$954$	0	0
$955$	0	0
$956$	−9.34784e6	−0.330801
$957$	0	0
$958$	2.24211e7	0.789303
$959$	−1.70526e7	−0.598749
$960$	0	0
$961$	−2.44348e7	−0.853495
$962$	1.19618e7	0.416734
$963$	0	0
$964$	−9.02077e6	−0.312645
$965$	0	0
$966$	0	0
$967$	1.84836e7	0.635653	0.317827	−	0.948149i	$-0.397047\pi$
$967$	1.84836e7	0.635653	0.317827	+	0.948149i	$0.397047\pi$
$968$	8.90541e6	0.305468
$969$	0	0
$970$	0	0
$971$	−3.95031e7	−1.34457	−0.672284	−	0.740294i	$-0.734686\pi$
$971$	−3.95031e7	−1.34457	−0.672284	+	0.740294i	$0.734686\pi$
$972$	0	0
$973$	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	$0.314084\pi$
$978$	0	0
$979$	1.26111e7	0.420529
$980$	0	0
$981$	0	0
$982$	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	$0.644550\pi$
$984$	0	0
$985$	0	0
$986$	−2.21184e6	−0.0724538
$987$	0	0
$988$	−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
$993$	0	0
$994$	2.68474e6	0.0861858
$995$	0	0
$996$	0	0
$997$	5.12017e7	1.63135	0.815674	−	0.578511i	$-0.196366\pi$
$997$	5.12017e7	1.63135	0.815674	+	0.578511i	$0.196366\pi$
$998$	−7.02840e6	−0.223373
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.b.a.9.1	✓	2	15.8	even	4
10.6.b.a.9.2	yes	2	15.2	even	4
50.6.a.c.1.1		1	15.14	odd	2
50.6.a.e.1.1		1	3.2	odd	2
80.6.c.c.49.1		2	60.47	odd	4
80.6.c.c.49.2		2	60.23	odd	4
90.6.c.a.19.1		2	5.2	odd	4
90.6.c.a.19.2		2	5.3	odd	4
320.6.c.a.129.1		2	120.83	odd	4
320.6.c.a.129.2		2	120.107	odd	4
320.6.c.b.129.1		2	120.77	even	4
320.6.c.b.129.2		2	120.53	even	4
400.6.a.c.1.1		1	60.59	even	2
400.6.a.k.1.1		1	12.11	even	2
450.6.a.c.1.1		1	1.1	even	1	trivial
450.6.a.w.1.1		1	5.4	even	2
720.6.f.a.289.1		2	20.7	even	4
720.6.f.a.289.2		2	20.3	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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