LMFDB - Embedded newform 1089.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,6,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1089.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:	$174.657979776$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{2} +49.0000 q^{4} -24.0000 q^{5} -72.0000 q^{7} -153.000 q^{8} +O(q^{10})$

$q-9.00000 q^{2} +49.0000 q^{4} -24.0000 q^{5} -72.0000 q^{7} -153.000 q^{8} +216.000 q^{10} -306.000 q^{13} +648.000 q^{14} -191.000 q^{16} +1206.00 q^{17} +774.000 q^{19} -1176.00 q^{20} +4626.00 q^{23} -2549.00 q^{25} +2754.00 q^{26} -3528.00 q^{28} -7686.00 q^{29} +5428.00 q^{31} +6615.00 q^{32} -10854.0 q^{34} +1728.00 q^{35} +3454.00 q^{37} -6966.00 q^{38} +3672.00 q^{40} +7866.00 q^{41} -15786.0 q^{43} -41634.0 q^{46} +6402.00 q^{47} -11623.0 q^{49} +22941.0 q^{50} -14994.0 q^{52} +21684.0 q^{53} +11016.0 q^{56} +69174.0 q^{58} +27420.0 q^{59} -52866.0 q^{61} -48852.0 q^{62} -53423.0 q^{64} +7344.00 q^{65} +25012.0 q^{67} +59094.0 q^{68} -15552.0 q^{70} -65058.0 q^{71} +26676.0 q^{73} -31086.0 q^{74} +37926.0 q^{76} +18612.0 q^{79} +4584.00 q^{80} -70794.0 q^{82} -28944.0 q^{85} +142074. q^{86} +41670.0 q^{89} +22032.0 q^{91} +226674. q^{92} -57618.0 q^{94} -18576.0 q^{95} +40694.0 q^{97} +104607. q^{98} +O(q^{100})$

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 $

$2$	−9.00000	−1.59099	−0.795495	−	0.605960i	$-0.792789\pi$
$2$	−9.00000	−1.59099	−0.795495	+	0.605960i	$0.792789\pi$
$3$	0	0
$3$	0	0
$4$	49.0000	1.53125
$5$	−24.0000	−0.429325	−0.214663	−	0.976688i	$-0.568865\pi$
$5$	−24.0000	−0.429325	−0.214663	+	0.976688i	$0.568865\pi$
$6$	0	0
$7$	−72.0000	−0.555376	−0.277688	−	0.960671i	$-0.589568\pi$
$7$	−72.0000	−0.555376	−0.277688	+	0.960671i	$0.589568\pi$
$8$	−153.000	−0.845214
$9$	0	0
$10$	216.000	0.683052
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−306.000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$13$	−306.000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$14$	648.000	0.883598
$15$	0	0
$16$	−191.000	−0.186523
$17$	1206.00	1.01210	0.506052	−	0.862503i	$-0.331104\pi$
$17$	1206.00	1.01210	0.506052	+	0.862503i	$0.331104\pi$
$18$	0	0
$19$	774.000	0.491878	0.245939	−	0.969285i	$-0.420904\pi$
$19$	774.000	0.491878	0.245939	+	0.969285i	$0.420904\pi$
$20$	−1176.00	−0.657404
$21$	0	0
$22$	0	0
$23$	4626.00	1.82342	0.911709	−	0.410838i	$-0.134764\pi$
$23$	4626.00	1.82342	0.911709	+	0.410838i	$0.134764\pi$
$24$	0	0
$25$	−2549.00	−0.815680
$26$	2754.00	0.798970
$27$	0	0
$28$	−3528.00	−0.850420
$29$	−7686.00	−1.69709	−0.848546	−	0.529122i	$-0.822522\pi$
$29$	−7686.00	−1.69709	−0.848546	+	0.529122i	$0.822522\pi$
$30$	0	0
$31$	5428.00	1.01446	0.507231	−	0.861810i	$-0.330669\pi$
$31$	5428.00	1.01446	0.507231	+	0.861810i	$0.330669\pi$
$32$	6615.00	1.14197
$33$	0	0
$34$	−10854.0	−1.61025
$35$	1728.00	0.238437
$36$	0	0
$37$	3454.00	0.414780	0.207390	−	0.978258i	$-0.433503\pi$
$37$	3454.00	0.414780	0.207390	+	0.978258i	$0.433503\pi$
$38$	−6966.00	−0.782572
$39$	0	0
$40$	3672.00	0.362871
$41$	7866.00	0.730793	0.365396	−	0.930852i	$-0.380933\pi$
$41$	7866.00	0.730793	0.365396	+	0.930852i	$0.380933\pi$
$42$	0	0
$43$	−15786.0	−1.30197	−0.650985	−	0.759091i	$-0.725644\pi$
$43$	−15786.0	−1.30197	−0.650985	+	0.759091i	$0.725644\pi$
$44$	0	0
$45$	0	0
$46$	−41634.0	−2.90104
$47$	6402.00	0.422738	0.211369	−	0.977406i	$-0.432208\pi$
$47$	6402.00	0.422738	0.211369	+	0.977406i	$0.432208\pi$
$48$	0	0
$49$	−11623.0	−0.691557
$50$	22941.0	1.29774
$51$	0	0
$52$	−14994.0	−0.768970
$53$	21684.0	1.06035	0.530176	−	0.847888i	$-0.322126\pi$
$53$	21684.0	1.06035	0.530176	+	0.847888i	$0.322126\pi$
$54$	0	0
$55$	0	0
$56$	11016.0	0.469412
$57$	0	0
$58$	69174.0	2.70006
$59$	27420.0	1.02550	0.512752	−	0.858537i	$-0.328626\pi$
$59$	27420.0	1.02550	0.512752	+	0.858537i	$0.328626\pi$
$60$	0	0
$61$	−52866.0	−1.81908	−0.909540	−	0.415616i	$-0.863566\pi$
$61$	−52866.0	−1.81908	−0.909540	+	0.415616i	$0.863566\pi$
$62$	−48852.0	−1.61400
$63$	0	0
$64$	−53423.0	−1.63034
$65$	7344.00	0.215600
$66$	0	0
$67$	25012.0	0.680709	0.340354	−	0.940297i	$-0.389453\pi$
$67$	25012.0	0.680709	0.340354	+	0.940297i	$0.389453\pi$
$68$	59094.0	1.54978
$69$	0	0
$70$	−15552.0	−0.379351
$71$	−65058.0	−1.53163	−0.765817	−	0.643059i	$-0.777665\pi$
$71$	−65058.0	−1.53163	−0.765817	+	0.643059i	$0.777665\pi$
$72$	0	0
$73$	26676.0	0.585887	0.292943	−	0.956130i	$-0.405365\pi$
$73$	26676.0	0.585887	0.292943	+	0.956130i	$0.405365\pi$
$74$	−31086.0	−0.659911
$75$	0	0
$76$	37926.0	0.753187
$77$	0	0
$78$	0	0
$79$	18612.0	0.335525	0.167763	−	0.985827i	$-0.446346\pi$
$79$	18612.0	0.335525	0.167763	+	0.985827i	$0.446346\pi$
$80$	4584.00	0.0800792
$81$	0	0
$82$	−70794.0	−1.16268
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−28944.0	−0.434521
$86$	142074.	2.07142
$87$	0	0
$88$	0	0
$89$	41670.0	0.557633	0.278817	−	0.960344i	$-0.410058\pi$
$89$	41670.0	0.557633	0.278817	+	0.960344i	$0.410058\pi$
$90$	0	0
$91$	22032.0	0.278901
$92$	226674.	2.79211
$93$	0	0
$94$	−57618.0	−0.672572
$95$	−18576.0	−0.211175
$96$	0	0
$97$	40694.0	0.439138	0.219569	−	0.975597i	$-0.429535\pi$
$97$	40694.0	0.439138	0.219569	+	0.975597i	$0.429535\pi$
$98$	104607.	1.10026
$99$	0	0
$100$	−124901.	−1.24901
$101$	−76554.0	−0.746731	−0.373366	−	0.927684i	$-0.621796\pi$
$101$	−76554.0	−0.746731	−0.373366	+	0.927684i	$0.621796\pi$
$102$	0	0
$103$	−81748.0	−0.759249	−0.379624	−	0.925141i	$-0.623947\pi$
$103$	−81748.0	−0.759249	−0.379624	+	0.925141i	$0.623947\pi$
$104$	46818.0	0.424453
$105$	0	0
$106$	−195156.	−1.68701
$107$	−53748.0	−0.453840	−0.226920	−	0.973913i	$-0.572866\pi$
$107$	−53748.0	−0.453840	−0.226920	+	0.973913i	$0.572866\pi$
$108$	0	0
$109$	213390.	1.72031	0.860157	−	0.510029i	$-0.170365\pi$
$109$	213390.	1.72031	0.860157	+	0.510029i	$0.170365\pi$
$110$	0	0
$111$	0	0
$112$	13752.0	0.103591
$113$	141594.	1.04315	0.521577	−	0.853204i	$-0.325344\pi$
$113$	141594.	1.04315	0.521577	+	0.853204i	$0.325344\pi$
$114$	0	0
$115$	−111024.	−0.782839
$116$	−376614.	−2.59867
$117$	0	0
$118$	−246780.	−1.63157
$119$	−86832.0	−0.562098
$120$	0	0
$121$	0	0
$122$	475794.	2.89414
$123$	0	0
$124$	265972.	1.55339
$125$	136176.	0.779517
$126$	0	0
$127$	−196812.	−1.08279	−0.541393	−	0.840770i	$-0.682103\pi$
$127$	−196812.	−1.08279	−0.541393	+	0.840770i	$0.682103\pi$
$128$	269127.	1.45189
$129$	0	0
$130$	−66096.0	−0.343018
$131$	−68436.0	−0.348423	−0.174211	−	0.984708i	$-0.555738\pi$
$131$	−68436.0	−0.348423	−0.174211	+	0.984708i	$0.555738\pi$
$132$	0	0
$133$	−55728.0	−0.273177
$134$	−225108.	−1.08300
$135$	0	0
$136$	−184518.	−0.855444
$137$	106806.	0.486177	0.243088	−	0.970004i	$-0.421839\pi$
$137$	106806.	0.486177	0.243088	+	0.970004i	$0.421839\pi$
$138$	0	0
$139$	−362322.	−1.59059	−0.795294	−	0.606224i	$-0.792684\pi$
$139$	−362322.	−1.59059	−0.795294	+	0.606224i	$0.792684\pi$
$140$	84672.0	0.365107
$141$	0	0
$142$	585522.	2.43681
$143$	0	0
$144$	0	0
$145$	184464.	0.728604
$146$	−240084.	−0.932140
$147$	0	0
$148$	169246.	0.635132
$149$	−149814.	−0.552824	−0.276412	−	0.961039i	$-0.589145\pi$
$149$	−149814.	−0.552824	−0.276412	+	0.961039i	$0.589145\pi$
$150$	0	0
$151$	335772.	1.19840	0.599200	−	0.800599i	$-0.295485\pi$
$151$	335772.	1.19840	0.599200	+	0.800599i	$0.295485\pi$
$152$	−118422.	−0.415742
$153$	0	0
$154$	0	0
$155$	−130272.	−0.435534
$156$	0	0
$157$	155458.	0.503343	0.251671	−	0.967813i	$-0.419020\pi$
$157$	155458.	0.503343	0.251671	+	0.967813i	$0.419020\pi$
$158$	−167508.	−0.533818
$159$	0	0
$160$	−158760.	−0.490277
$161$	−333072.	−1.01268
$162$	0	0
$163$	−213964.	−0.630771	−0.315385	−	0.948964i	$-0.602134\pi$
$163$	−213964.	−0.630771	−0.315385	+	0.948964i	$0.602134\pi$
$164$	385434.	1.11903
$165$	0	0
$166$	0	0
$167$	−425880.	−1.18167	−0.590835	−	0.806793i	$-0.701202\pi$
$167$	−425880.	−1.18167	−0.590835	+	0.806793i	$0.701202\pi$
$168$	0	0
$169$	−277657.	−0.747811
$170$	260496.	0.691319
$171$	0	0
$172$	−773514.	−1.99364
$173$	−723438.	−1.83775	−0.918874	−	0.394551i	$-0.870900\pi$
$173$	−723438.	−1.83775	−0.918874	+	0.394551i	$0.870900\pi$
$174$	0	0
$175$	183528.	0.453009
$176$	0	0
$177$	0	0
$178$	−375030.	−0.887189
$179$	−620112.	−1.44656	−0.723282	−	0.690553i	$-0.757367\pi$
$179$	−620112.	−1.44656	−0.723282	+	0.690553i	$0.757367\pi$
$180$	0	0
$181$	584354.	1.32580	0.662902	−	0.748706i	$-0.269324\pi$
$181$	584354.	1.32580	0.662902	+	0.748706i	$0.269324\pi$
$182$	−198288.	−0.443729
$183$	0	0
$184$	−707778.	−1.54118
$185$	−82896.0	−0.178076
$186$	0	0
$187$	0	0
$188$	313698.	0.647317
$189$	0	0
$190$	167184.	0.335978
$191$	−206310.	−0.409201	−0.204601	−	0.978846i	$-0.565590\pi$
$191$	−206310.	−0.409201	−0.204601	+	0.978846i	$0.565590\pi$
$192$	0	0
$193$	−915300.	−1.76877	−0.884383	−	0.466763i	$-0.845420\pi$
$193$	−915300.	−1.76877	−0.884383	+	0.466763i	$0.845420\pi$
$194$	−366246.	−0.698664
$195$	0	0
$196$	−569527.	−1.05895
$197$	−219978.	−0.403844	−0.201922	−	0.979402i	$-0.564719\pi$
$197$	−219978.	−0.403844	−0.201922	+	0.979402i	$0.564719\pi$
$198$	0	0
$199$	563056.	1.00790	0.503952	−	0.863732i	$-0.331879\pi$
$199$	563056.	1.00790	0.503952	+	0.863732i	$0.331879\pi$
$200$	389997.	0.689424
$201$	0	0
$202$	688986.	1.18804
$203$	553392.	0.942525
$204$	0	0
$205$	−188784.	−0.313748
$206$	735732.	1.20796
$207$	0	0
$208$	58446.0	0.0936691
$209$	0	0
$210$	0	0
$211$	−220662.	−0.341210	−0.170605	−	0.985340i	$-0.554572\pi$
$211$	−220662.	−0.341210	−0.170605	+	0.985340i	$0.554572\pi$
$212$	1.06252e6	1.62366
$213$	0	0
$214$	483732.	0.722055
$215$	378864.	0.558968
$216$	0	0
$217$	−390816.	−0.563408
$218$	−1.92051e6	−2.73700
$219$	0	0
$220$	0	0
$221$	−369036.	−0.508262
$222$	0	0
$223$	−1.28150e6	−1.72566	−0.862830	−	0.505495i	$-0.831310\pi$
$223$	−1.28150e6	−1.72566	−0.862830	+	0.505495i	$0.831310\pi$
$224$	−476280.	−0.634223
$225$	0	0
$226$	−1.27435e6	−1.65965
$227$	−1.45480e6	−1.87386	−0.936931	−	0.349515i	$-0.886346\pi$
$227$	−1.45480e6	−1.87386	−0.936931	+	0.349515i	$0.886346\pi$
$228$	0	0
$229$	−4382.00	−0.00552184	−0.00276092	−	0.999996i	$-0.500879\pi$
$229$	−4382.00	−0.00552184	−0.00276092	+	0.999996i	$0.500879\pi$
$230$	999216.	1.24549
$231$	0	0
$232$	1.17596e6	1.43441
$233$	1.07231e6	1.29399	0.646997	−	0.762493i	$-0.276025\pi$
$233$	1.07231e6	1.29399	0.646997	+	0.762493i	$0.276025\pi$
$234$	0	0
$235$	−153648.	−0.181492
$236$	1.34358e6	1.57030
$237$	0	0
$238$	781488.	0.894293
$239$	−828648.	−0.938373	−0.469186	−	0.883099i	$-0.655453\pi$
$239$	−828648.	−0.938373	−0.469186	+	0.883099i	$0.655453\pi$
$240$	0	0
$241$	601200.	0.666770	0.333385	−	0.942791i	$-0.391809\pi$
$241$	601200.	0.666770	0.333385	+	0.942791i	$0.391809\pi$
$242$	0	0
$243$	0	0
$244$	−2.59043e6	−2.78547
$245$	278952.	0.296903
$246$	0	0
$247$	−236844.	−0.247013
$248$	−830484.	−0.857437
$249$	0	0
$250$	−1.22558e6	−1.24020
$251$	−1.13940e6	−1.14154	−0.570771	−	0.821109i	$-0.693356\pi$
$251$	−1.13940e6	−1.14154	−0.570771	+	0.821109i	$0.693356\pi$
$252$	0	0
$253$	0	0
$254$	1.77131e6	1.72270
$255$	0	0
$256$	−712607.	−0.679595
$257$	466542.	0.440614	0.220307	−	0.975431i	$-0.429294\pi$
$257$	466542.	0.440614	0.220307	+	0.975431i	$0.429294\pi$
$258$	0	0
$259$	−248688.	−0.230359
$260$	359856.	0.330138
$261$	0	0
$262$	615924.	0.554337
$263$	720504.	0.642313	0.321157	−	0.947026i	$-0.395928\pi$
$263$	720504.	0.642313	0.321157	+	0.947026i	$0.395928\pi$
$264$	0	0
$265$	−520416.	−0.455235
$266$	501552.	0.434622
$267$	0	0
$268$	1.22559e6	1.04234
$269$	1.22492e6	1.03212	0.516058	−	0.856554i	$-0.327399\pi$
$269$	1.22492e6	1.03212	0.516058	+	0.856554i	$0.327399\pi$
$270$	0	0
$271$	1.37840e6	1.14013	0.570064	−	0.821601i	$-0.306919\pi$
$271$	1.37840e6	1.14013	0.570064	+	0.821601i	$0.306919\pi$
$272$	−230346.	−0.188781
$273$	0	0
$274$	−961254.	−0.773503
$275$	0	0
$276$	0	0
$277$	2.33627e6	1.82947	0.914733	−	0.404059i	$-0.132401\pi$
$277$	2.33627e6	1.82947	0.914733	+	0.404059i	$0.132401\pi$
$278$	3.26090e6	2.53061
$279$	0	0
$280$	−264384.	−0.201530
$281$	2.37652e6	1.79546	0.897731	−	0.440545i	$-0.145215\pi$
$281$	2.37652e6	1.79546	0.897731	+	0.440545i	$0.145215\pi$
$282$	0	0
$283$	1.11433e6	0.827077	0.413539	−	0.910487i	$-0.364293\pi$
$283$	1.11433e6	0.827077	0.413539	+	0.910487i	$0.364293\pi$
$284$	−3.18784e6	−2.34531
$285$	0	0
$286$	0	0
$287$	−566352.	−0.405865
$288$	0	0
$289$	34579.0	0.0243539
$290$	−1.66018e6	−1.15920
$291$	0	0
$292$	1.30712e6	0.897139
$293$	558558.	0.380101	0.190051	−	0.981774i	$-0.439135\pi$
$293$	558558.	0.380101	0.190051	+	0.981774i	$0.439135\pi$
$294$	0	0
$295$	−658080.	−0.440275
$296$	−528462.	−0.350578
$297$	0	0
$298$	1.34833e6	0.879537
$299$	−1.41556e6	−0.915691
$300$	0	0
$301$	1.13659e6	0.723083
$302$	−3.02195e6	−1.90664
$303$	0	0
$304$	−147834.	−0.0917467
$305$	1.26878e6	0.780977
$306$	0	0
$307$	1.22369e6	0.741015	0.370507	−	0.928830i	$-0.379184\pi$
$307$	1.22369e6	0.741015	0.370507	+	0.928830i	$0.379184\pi$
$308$	0	0
$309$	0	0
$310$	1.17245e6	0.692930
$311$	1.91858e6	1.12481	0.562404	−	0.826863i	$-0.309877\pi$
$311$	1.91858e6	1.12481	0.562404	+	0.826863i	$0.309877\pi$
$312$	0	0
$313$	−801626.	−0.462499	−0.231250	−	0.972894i	$-0.574281\pi$
$313$	−801626.	−0.462499	−0.231250	+	0.972894i	$0.574281\pi$
$314$	−1.39912e6	−0.800814
$315$	0	0
$316$	911988.	0.513773
$317$	2.34522e6	1.31080	0.655398	−	0.755283i	$-0.272501\pi$
$317$	2.34522e6	1.31080	0.655398	+	0.755283i	$0.272501\pi$
$318$	0	0
$319$	0	0
$320$	1.28215e6	0.699946
$321$	0	0
$322$	2.99765e6	1.61117
$323$	933444.	0.497831
$324$	0	0
$325$	779994.	0.409622
$326$	1.92568e6	1.00355
$327$	0	0
$328$	−1.20350e6	−0.617676
$329$	−460944.	−0.234779
$330$	0	0
$331$	3.31005e6	1.66060	0.830300	−	0.557317i	$-0.188169\pi$
$331$	3.31005e6	1.66060	0.830300	+	0.557317i	$0.188169\pi$
$332$	0	0
$333$	0	0
$334$	3.83292e6	1.88002
$335$	−600288.	−0.292245
$336$	0	0
$337$	−1.44803e6	−0.694548	−0.347274	−	0.937764i	$-0.612893\pi$
$337$	−1.44803e6	−0.694548	−0.347274	+	0.937764i	$0.612893\pi$
$338$	2.49891e6	1.18976
$339$	0	0
$340$	−1.41826e6	−0.665361
$341$	0	0
$342$	0	0
$343$	2.04696e6	0.939451
$344$	2.41526e6	1.10044
$345$	0	0
$346$	6.51094e6	2.92384
$347$	−391752.	−0.174658	−0.0873288	−	0.996180i	$-0.527833\pi$
$347$	−391752.	−0.174658	−0.0873288	+	0.996180i	$0.527833\pi$
$348$	0	0
$349$	2.65603e6	1.16726	0.583632	−	0.812019i	$-0.301631\pi$
$349$	2.65603e6	1.16726	0.583632	+	0.812019i	$0.301631\pi$
$350$	−1.65175e6	−0.720734
$351$	0	0
$352$	0	0
$353$	1.40614e6	0.600610	0.300305	−	0.953843i	$-0.402912\pi$
$353$	1.40614e6	0.600610	0.300305	+	0.953843i	$0.402912\pi$
$354$	0	0
$355$	1.56139e6	0.657569
$356$	2.04183e6	0.853876
$357$	0	0
$358$	5.58101e6	2.30147
$359$	3.14755e6	1.28895	0.644476	−	0.764624i	$-0.277076\pi$
$359$	3.14755e6	1.28895	0.644476	+	0.764624i	$0.277076\pi$
$360$	0	0
$361$	−1.87702e6	−0.758057
$362$	−5.25919e6	−2.10934
$363$	0	0
$364$	1.07957e6	0.427068
$365$	−640224.	−0.251536
$366$	0	0
$367$	−1.74806e6	−0.677470	−0.338735	−	0.940882i	$-0.609999\pi$
$367$	−1.74806e6	−0.677470	−0.338735	+	0.940882i	$0.609999\pi$
$368$	−883566.	−0.340110
$369$	0	0
$370$	746064.	0.283316
$371$	−1.56125e6	−0.588894
$372$	0	0
$373$	−3.30925e6	−1.23156	−0.615782	−	0.787917i	$-0.711160\pi$
$373$	−3.30925e6	−1.23156	−0.615782	+	0.787917i	$0.711160\pi$
$374$	0	0
$375$	0	0
$376$	−979506.	−0.357304
$377$	2.35192e6	0.852253
$378$	0	0
$379$	−4.10254e6	−1.46708	−0.733542	−	0.679644i	$-0.762134\pi$
$379$	−4.10254e6	−1.46708	−0.733542	+	0.679644i	$0.762134\pi$
$380$	−910224.	−0.323362
$381$	0	0
$382$	1.85679e6	0.651035
$383$	1.12151e6	0.390668	0.195334	−	0.980737i	$-0.437421\pi$
$383$	1.12151e6	0.390668	0.195334	+	0.980737i	$0.437421\pi$
$384$	0	0
$385$	0	0
$386$	8.23770e6	2.81409
$387$	0	0
$388$	1.99401e6	0.672430
$389$	3.08818e6	1.03473	0.517366	−	0.855764i	$-0.326913\pi$
$389$	3.08818e6	1.03473	0.517366	+	0.855764i	$0.326913\pi$
$390$	0	0
$391$	5.57896e6	1.84549
$392$	1.77832e6	0.584513
$393$	0	0
$394$	1.97980e6	0.642512
$395$	−446688.	−0.144049
$396$	0	0
$397$	506594.	0.161318	0.0806592	−	0.996742i	$-0.474297\pi$
$397$	506594.	0.161318	0.0806592	+	0.996742i	$0.474297\pi$
$398$	−5.06750e6	−1.60356
$399$	0	0
$400$	486859.	0.152143
$401$	2.44671e6	0.759839	0.379919	−	0.925020i	$-0.375952\pi$
$401$	2.44671e6	0.759839	0.379919	+	0.925020i	$0.375952\pi$
$402$	0	0
$403$	−1.66097e6	−0.509447
$404$	−3.75115e6	−1.14343
$405$	0	0
$406$	−4.98053e6	−1.49955
$407$	0	0
$408$	0	0
$409$	2.48818e6	0.735483	0.367742	−	0.929928i	$-0.380131\pi$
$409$	2.48818e6	0.735483	0.367742	+	0.929928i	$0.380131\pi$
$410$	1.69906e6	0.499170
$411$	0	0
$412$	−4.00565e6	−1.16260
$413$	−1.97424e6	−0.569541
$414$	0	0
$415$	0	0
$416$	−2.02419e6	−0.573480
$417$	0	0
$418$	0	0
$419$	−405672.	−0.112886	−0.0564430	−	0.998406i	$-0.517976\pi$
$419$	−405672.	−0.112886	−0.0564430	+	0.998406i	$0.517976\pi$
$420$	0	0
$421$	−4.36275e6	−1.19965	−0.599825	−	0.800131i	$-0.704763\pi$
$421$	−4.36275e6	−1.19965	−0.599825	+	0.800131i	$0.704763\pi$
$422$	1.98596e6	0.542861
$423$	0	0
$424$	−3.31765e6	−0.896223
$425$	−3.07409e6	−0.825553
$426$	0	0
$427$	3.80635e6	1.01027
$428$	−2.63365e6	−0.694943
$429$	0	0
$430$	−3.40978e6	−0.889313
$431$	5.71615e6	1.48221	0.741106	−	0.671388i	$-0.234301\pi$
$431$	5.71615e6	1.48221	0.741106	+	0.671388i	$0.234301\pi$
$432$	0	0
$433$	−5.84150e6	−1.49729	−0.748643	−	0.662973i	$-0.769294\pi$
$433$	−5.84150e6	−1.49729	−0.748643	+	0.662973i	$0.769294\pi$
$434$	3.51734e6	0.896377
$435$	0	0
$436$	1.04561e7	2.63423
$437$	3.58052e6	0.896898
$438$	0	0
$439$	−393336.	−0.0974097	−0.0487049	−	0.998813i	$-0.515509\pi$
$439$	−393336.	−0.0974097	−0.0487049	+	0.998813i	$0.515509\pi$
$440$	0	0
$441$	0	0
$442$	3.32132e6	0.808641
$443$	1.65260e6	0.400092	0.200046	−	0.979787i	$-0.435891\pi$
$443$	1.65260e6	0.400092	0.200046	+	0.979787i	$0.435891\pi$
$444$	0	0
$445$	−1.00008e6	−0.239406
$446$	1.15335e7	2.74551
$447$	0	0
$448$	3.84646e6	0.905453
$449$	−1.73199e6	−0.405443	−0.202721	−	0.979236i	$-0.564979\pi$
$449$	−1.73199e6	−0.405443	−0.202721	+	0.979236i	$0.564979\pi$
$450$	0	0
$451$	0	0
$452$	6.93811e6	1.59733
$453$	0	0
$454$	1.30932e7	2.98130
$455$	−528768.	−0.119739
$456$	0	0
$457$	5.66489e6	1.26882	0.634411	−	0.772996i	$-0.281243\pi$
$457$	5.66489e6	1.26882	0.634411	+	0.772996i	$0.281243\pi$
$458$	39438.0	0.00878519
$459$	0	0
$460$	−5.44018e6	−1.19872
$461$	827982.	0.181455	0.0907274	−	0.995876i	$-0.471081\pi$
$461$	827982.	0.181455	0.0907274	+	0.995876i	$0.471081\pi$
$462$	0	0
$463$	−3.24255e6	−0.702965	−0.351483	−	0.936194i	$-0.614322\pi$
$463$	−3.24255e6	−0.702965	−0.351483	+	0.936194i	$0.614322\pi$
$464$	1.46803e6	0.316547
$465$	0	0
$466$	−9.65083e6	−2.05873
$467$	6.29414e6	1.33550	0.667751	−	0.744385i	$-0.267257\pi$
$467$	6.29414e6	1.33550	0.667751	+	0.744385i	$0.267257\pi$
$468$	0	0
$469$	−1.80086e6	−0.378050
$470$	1.38283e6	0.288752
$471$	0	0
$472$	−4.19526e6	−0.866770
$473$	0	0
$474$	0	0
$475$	−1.97293e6	−0.401215
$476$	−4.25477e6	−0.860713
$477$	0	0
$478$	7.45783e6	1.49294
$479$	7.28863e6	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$479$	7.28863e6	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$480$	0	0
$481$	−1.05692e6	−0.208296
$482$	−5.41080e6	−1.06083
$483$	0	0
$484$	0	0
$485$	−976656.	−0.188533
$486$	0	0
$487$	−2.01570e6	−0.385126	−0.192563	−	0.981285i	$-0.561680\pi$
$487$	−2.01570e6	−0.385126	−0.192563	+	0.981285i	$0.561680\pi$
$488$	8.08850e6	1.53751
$489$	0	0
$490$	−2.51057e6	−0.472369
$491$	538596.	0.100823	0.0504115	−	0.998729i	$-0.483947\pi$
$491$	538596.	0.100823	0.0504115	+	0.998729i	$0.483947\pi$
$492$	0	0
$493$	−9.26932e6	−1.71763
$494$	2.13160e6	0.392995
$495$	0	0
$496$	−1.03675e6	−0.189221
$497$	4.68418e6	0.850633
$498$	0	0
$499$	5.15588e6	0.926939	0.463469	−	0.886113i	$-0.346604\pi$
$499$	5.15588e6	0.926939	0.463469	+	0.886113i	$0.346604\pi$
$500$	6.67262e6	1.19364
$501$	0	0
$502$	1.02546e7	1.81618
$503$	4.64962e6	0.819402	0.409701	−	0.912220i	$-0.365633\pi$
$503$	4.64962e6	0.819402	0.409701	+	0.912220i	$0.365633\pi$
$504$	0	0
$505$	1.83730e6	0.320591
$506$	0	0
$507$	0	0
$508$	−9.64379e6	−1.65801
$509$	−6.73988e6	−1.15308	−0.576538	−	0.817070i	$-0.695597\pi$
$509$	−6.73988e6	−1.15308	−0.576538	+	0.817070i	$0.695597\pi$
$510$	0	0
$511$	−1.92067e6	−0.325388
$512$	−2.19860e6	−0.370656
$513$	0	0
$514$	−4.19888e6	−0.701012
$515$	1.96195e6	0.325965
$516$	0	0
$517$	0	0
$518$	2.23819e6	0.366499
$519$	0	0
$520$	−1.12363e6	−0.182228
$521$	2.62743e6	0.424069	0.212035	−	0.977262i	$-0.431991\pi$
$521$	2.62743e6	0.424069	0.212035	+	0.977262i	$0.431991\pi$
$522$	0	0
$523$	8.18041e6	1.30774	0.653869	−	0.756608i	$-0.273145\pi$
$523$	8.18041e6	1.30774	0.653869	+	0.756608i	$0.273145\pi$
$524$	−3.35336e6	−0.533522
$525$	0	0
$526$	−6.48454e6	−1.02191
$527$	6.54617e6	1.02674
$528$	0	0
$529$	1.49635e7	2.32485
$530$	4.68374e6	0.724275
$531$	0	0
$532$	−2.73067e6	−0.418302
$533$	−2.40700e6	−0.366993
$534$	0	0
$535$	1.28995e6	0.194845
$536$	−3.82684e6	−0.575344
$537$	0	0
$538$	−1.10243e7	−1.64209
$539$	0	0
$540$	0	0
$541$	−6.44510e6	−0.946752	−0.473376	−	0.880860i	$-0.656965\pi$
$541$	−6.44510e6	−0.946752	−0.473376	+	0.880860i	$0.656965\pi$
$542$	−1.24056e7	−1.81393
$543$	0	0
$544$	7.97769e6	1.15579
$545$	−5.12136e6	−0.738574
$546$	0	0
$547$	−6.72241e6	−0.960631	−0.480315	−	0.877096i	$-0.659478\pi$
$547$	−6.72241e6	−0.960631	−0.480315	+	0.877096i	$0.659478\pi$
$548$	5.23349e6	0.744458
$549$	0	0
$550$	0	0
$551$	−5.94896e6	−0.834761
$552$	0	0
$553$	−1.34006e6	−0.186343
$554$	−2.10265e7	−2.91066
$555$	0	0
$556$	−1.77538e7	−2.43559
$557$	−2.71874e6	−0.371304	−0.185652	−	0.982616i	$-0.559440\pi$
$557$	−2.71874e6	−0.371304	−0.185652	+	0.982616i	$0.559440\pi$
$558$	0	0
$559$	4.83052e6	0.653829
$560$	−330048.	−0.0444741
$561$	0	0
$562$	−2.13887e7	−2.85656
$563$	817848.	0.108743	0.0543715	−	0.998521i	$-0.482684\pi$
$563$	817848.	0.108743	0.0543715	+	0.998521i	$0.482684\pi$
$564$	0	0
$565$	−3.39826e6	−0.447852
$566$	−1.00289e7	−1.31587
$567$	0	0
$568$	9.95387e6	1.29456
$569$	8.33672e6	1.07948	0.539740	−	0.841832i	$-0.318523\pi$
$569$	8.33672e6	1.07948	0.539740	+	0.841832i	$0.318523\pi$
$570$	0	0
$571$	8.20715e6	1.05342	0.526711	−	0.850044i	$-0.323425\pi$
$571$	8.20715e6	1.05342	0.526711	+	0.850044i	$0.323425\pi$
$572$	0	0
$573$	0	0
$574$	5.09717e6	0.645727
$575$	−1.17917e7	−1.48732
$576$	0	0
$577$	−605126.	−0.0756670	−0.0378335	−	0.999284i	$-0.512046\pi$
$577$	−605126.	−0.0756670	−0.0378335	+	0.999284i	$0.512046\pi$
$578$	−311211.	−0.0387468
$579$	0	0
$580$	9.03874e6	1.11568
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−4.08143e6	−0.495199
$585$	0	0
$586$	−5.02702e6	−0.604737
$587$	1.00479e7	1.20359	0.601797	−	0.798649i	$-0.294452\pi$
$587$	1.00479e7	1.20359	0.601797	+	0.798649i	$0.294452\pi$
$588$	0	0
$589$	4.20127e6	0.498991
$590$	5.92272e6	0.700473
$591$	0	0
$592$	−659714.	−0.0773662
$593$	−2.95609e6	−0.345208	−0.172604	−	0.984991i	$-0.555218\pi$
$593$	−2.95609e6	−0.345208	−0.172604	+	0.984991i	$0.555218\pi$
$594$	0	0
$595$	2.08397e6	0.241323
$596$	−7.34089e6	−0.846511
$597$	0	0
$598$	1.27400e7	1.45686
$599$	1.27169e7	1.44815	0.724075	−	0.689722i	$-0.242267\pi$
$599$	1.27169e7	1.44815	0.724075	+	0.689722i	$0.242267\pi$
$600$	0	0
$601$	−1.62535e7	−1.83553	−0.917763	−	0.397128i	$-0.870007\pi$
$601$	−1.62535e7	−1.83553	−0.917763	+	0.397128i	$0.870007\pi$
$602$	−1.02293e7	−1.15042
$603$	0	0
$604$	1.64528e7	1.83505
$605$	0	0
$606$	0	0
$607$	1.20900e7	1.33185	0.665923	−	0.746020i	$-0.268038\pi$
$607$	1.20900e7	1.33185	0.665923	+	0.746020i	$0.268038\pi$
$608$	5.12001e6	0.561710
$609$	0	0
$610$	−1.14191e7	−1.24253
$611$	−1.95901e6	−0.212292
$612$	0	0
$613$	2.83160e6	0.304355	0.152177	−	0.988353i	$-0.451371\pi$
$613$	2.83160e6	0.304355	0.152177	+	0.988353i	$0.451371\pi$
$614$	−1.10132e7	−1.17895
$615$	0	0
$616$	0	0
$617$	1.62285e6	0.171619	0.0858095	−	0.996312i	$-0.472652\pi$
$617$	1.62285e6	0.171619	0.0858095	+	0.996312i	$0.472652\pi$
$618$	0	0
$619$	1.25560e6	0.131712	0.0658561	−	0.997829i	$-0.479022\pi$
$619$	1.25560e6	0.131712	0.0658561	+	0.997829i	$0.479022\pi$
$620$	−6.38333e6	−0.666911
$621$	0	0
$622$	−1.72672e7	−1.78956
$623$	−3.00024e6	−0.309696
$624$	0	0
$625$	4.69740e6	0.481014
$626$	7.21463e6	0.735832
$627$	0	0
$628$	7.61744e6	0.770744
$629$	4.16552e6	0.419801
$630$	0	0
$631$	−579044.	−0.0578946	−0.0289473	−	0.999581i	$-0.509216\pi$
$631$	−579044.	−0.0578946	−0.0289473	+	0.999581i	$0.509216\pi$
$632$	−2.84764e6	−0.283591
$633$	0	0
$634$	−2.11070e7	−2.08547
$635$	4.72349e6	0.464867
$636$	0	0
$637$	3.55664e6	0.347289
$638$	0	0
$639$	0	0
$640$	−6.45905e6	−0.623331
$641$	1.12839e7	1.08471	0.542354	−	0.840150i	$-0.317533\pi$
$641$	1.12839e7	1.08471	0.542354	+	0.840150i	$0.317533\pi$
$642$	0	0
$643$	1.60866e7	1.53439	0.767196	−	0.641413i	$-0.221652\pi$
$643$	1.60866e7	1.53439	0.767196	+	0.641413i	$0.221652\pi$
$644$	−1.63205e7	−1.55067
$645$	0	0
$646$	−8.40100e6	−0.792044
$647$	−7.80003e6	−0.732547	−0.366274	−	0.930507i	$-0.619367\pi$
$647$	−7.80003e6	−0.732547	−0.366274	+	0.930507i	$0.619367\pi$
$648$	0	0
$649$	0	0
$650$	−7.01995e6	−0.651704
$651$	0	0
$652$	−1.04842e7	−0.965868
$653$	3.81000e6	0.349657	0.174828	−	0.984599i	$-0.444063\pi$
$653$	3.81000e6	0.349657	0.174828	+	0.984599i	$0.444063\pi$
$654$	0	0
$655$	1.64246e6	0.149587
$656$	−1.50241e6	−0.136310
$657$	0	0
$658$	4.14850e6	0.373530
$659$	2.82564e6	0.253456	0.126728	−	0.991937i	$-0.459552\pi$
$659$	2.82564e6	0.253456	0.126728	+	0.991937i	$0.459552\pi$
$660$	0	0
$661$	−1.52399e7	−1.35668	−0.678340	−	0.734748i	$-0.737300\pi$
$661$	−1.52399e7	−1.35668	−0.678340	+	0.734748i	$0.737300\pi$
$662$	−2.97905e7	−2.64200
$663$	0	0
$664$	0	0
$665$	1.33747e6	0.117282
$666$	0	0
$667$	−3.55554e7	−3.09451
$668$	−2.08681e7	−1.80943
$669$	0	0
$670$	5.40259e6	0.464960
$671$	0	0
$672$	0	0
$673$	1.33261e7	1.13414	0.567068	−	0.823671i	$-0.308078\pi$
$673$	1.33261e7	1.13414	0.567068	+	0.823671i	$0.308078\pi$
$674$	1.30323e7	1.10502
$675$	0	0
$676$	−1.36052e7	−1.14509
$677$	−9.75076e6	−0.817649	−0.408824	−	0.912613i	$-0.634061\pi$
$677$	−9.75076e6	−0.817649	−0.408824	+	0.912613i	$0.634061\pi$
$678$	0	0
$679$	−2.92997e6	−0.243887
$680$	4.42843e6	0.367263
$681$	0	0
$682$	0	0
$683$	−3.79889e6	−0.311605	−0.155803	−	0.987788i	$-0.549796\pi$
$683$	−3.79889e6	−0.311605	−0.155803	+	0.987788i	$0.549796\pi$
$684$	0	0
$685$	−2.56334e6	−0.208728
$686$	−1.84226e7	−1.49466
$687$	0	0
$688$	3.01513e6	0.242848
$689$	−6.63530e6	−0.532492
$690$	0	0
$691$	1.04595e7	0.833328	0.416664	−	0.909060i	$-0.363199\pi$
$691$	1.04595e7	0.833328	0.416664	+	0.909060i	$0.363199\pi$
$692$	−3.54485e7	−2.81405
$693$	0	0
$694$	3.52577e6	0.277879
$695$	8.69573e6	0.682879
$696$	0	0
$697$	9.48640e6	0.739638
$698$	−2.39042e7	−1.85710
$699$	0	0
$700$	8.99287e6	0.693671
$701$	−2.79859e6	−0.215102	−0.107551	−	0.994200i	$-0.534301\pi$
$701$	−2.79859e6	−0.215102	−0.107551	+	0.994200i	$0.534301\pi$
$702$	0	0
$703$	2.67340e6	0.204021
$704$	0	0
$705$	0	0
$706$	−1.26553e7	−0.955564
$707$	5.51189e6	0.414717
$708$	0	0
$709$	−2.45413e7	−1.83350	−0.916752	−	0.399457i	$-0.869199\pi$
$709$	−2.45413e7	−1.83350	−0.916752	+	0.399457i	$0.869199\pi$
$710$	−1.40525e7	−1.04619
$711$	0	0
$712$	−6.37551e6	−0.471319
$713$	2.51099e7	1.84979
$714$	0	0
$715$	0	0
$716$	−3.03855e7	−2.21505
$717$	0	0
$718$	−2.83280e7	−2.05071
$719$	4.81287e6	0.347202	0.173601	−	0.984816i	$-0.444460\pi$
$719$	4.81287e6	0.347202	0.173601	+	0.984816i	$0.444460\pi$
$720$	0	0
$721$	5.88586e6	0.421669
$722$	1.68932e7	1.20606
$723$	0	0
$724$	2.86333e7	2.03014
$725$	1.95916e7	1.38428
$726$	0	0
$727$	1.22938e7	0.862677	0.431339	−	0.902190i	$-0.358041\pi$
$727$	1.22938e7	0.862677	0.431339	+	0.902190i	$0.358041\pi$
$728$	−3.37090e6	−0.235731
$729$	0	0
$730$	5.76202e6	0.400191
$731$	−1.90379e7	−1.31773
$732$	0	0
$733$	−2.60411e6	−0.179019	−0.0895097	−	0.995986i	$-0.528530\pi$
$733$	−2.60411e6	−0.179019	−0.0895097	+	0.995986i	$0.528530\pi$
$734$	1.57325e7	1.07785
$735$	0	0
$736$	3.06010e7	2.08229
$737$	0	0
$738$	0	0
$739$	1.85178e7	1.24732	0.623661	−	0.781695i	$-0.285645\pi$
$739$	1.85178e7	1.24732	0.623661	+	0.781695i	$0.285645\pi$
$740$	−4.06190e6	−0.272678
$741$	0	0
$742$	1.40512e7	0.936925
$743$	3.01104e6	0.200099	0.100049	−	0.994982i	$-0.468100\pi$
$743$	3.01104e6	0.200099	0.100049	+	0.994982i	$0.468100\pi$
$744$	0	0
$745$	3.59554e6	0.237341
$746$	2.97832e7	1.95941
$747$	0	0
$748$	0	0
$749$	3.86986e6	0.252052
$750$	0	0
$751$	1.41805e7	0.917470	0.458735	−	0.888573i	$-0.348303\pi$
$751$	1.41805e7	0.917470	0.458735	+	0.888573i	$0.348303\pi$
$752$	−1.22278e6	−0.0788505
$753$	0	0
$754$	−2.11672e7	−1.35593
$755$	−8.05853e6	−0.514504
$756$	0	0
$757$	−7.79645e6	−0.494490	−0.247245	−	0.968953i	$-0.579525\pi$
$757$	−7.79645e6	−0.494490	−0.247245	+	0.968953i	$0.579525\pi$
$758$	3.69229e7	2.33412
$759$	0	0
$760$	2.84213e6	0.178488
$761$	−1.09794e7	−0.687252	−0.343626	−	0.939107i	$-0.611655\pi$
$761$	−1.09794e7	−0.687252	−0.343626	+	0.939107i	$0.611655\pi$
$762$	0	0
$763$	−1.53641e7	−0.955422
$764$	−1.01092e7	−0.626589
$765$	0	0
$766$	−1.00936e7	−0.621549
$767$	−8.39052e6	−0.514992
$768$	0	0
$769$	1.88030e7	1.14660	0.573299	−	0.819347i	$-0.305664\pi$
$769$	1.88030e7	1.14660	0.573299	+	0.819347i	$0.305664\pi$
$770$	0	0
$771$	0	0
$772$	−4.48497e7	−2.70842
$773$	1.21929e7	0.733935	0.366967	−	0.930234i	$-0.380396\pi$
$773$	1.21929e7	0.733935	0.366967	+	0.930234i	$0.380396\pi$
$774$	0	0
$775$	−1.38360e7	−0.827476
$776$	−6.22618e6	−0.371165
$777$	0	0
$778$	−2.77936e7	−1.64625
$779$	6.08828e6	0.359461
$780$	0	0
$781$	0	0
$782$	−5.02106e7	−2.93615
$783$	0	0
$784$	2.21999e6	0.128992
$785$	−3.73099e6	−0.216098
$786$	0	0
$787$	−2.75304e7	−1.58444	−0.792221	−	0.610234i	$-0.791075\pi$
$787$	−2.75304e7	−1.58444	−0.792221	+	0.610234i	$0.791075\pi$
$788$	−1.07789e7	−0.618386
$789$	0	0
$790$	4.02019e6	0.229181
$791$	−1.01948e7	−0.579343
$792$	0	0
$793$	1.61770e7	0.913513
$794$	−4.55935e6	−0.256656
$795$	0	0
$796$	2.75897e7	1.54335
$797$	−2.46936e7	−1.37701	−0.688507	−	0.725230i	$-0.741733\pi$
$797$	−2.46936e7	−1.37701	−0.688507	+	0.725230i	$0.741733\pi$
$798$	0	0
$799$	7.72081e6	0.427854
$800$	−1.68616e7	−0.931483
$801$	0	0
$802$	−2.20204e7	−1.20890
$803$	0	0
$804$	0	0
$805$	7.99373e6	0.434770
$806$	1.49487e7	0.810525
$807$	0	0
$808$	1.17128e7	0.631148
$809$	2.40376e7	1.29128	0.645639	−	0.763642i	$-0.276591\pi$
$809$	2.40376e7	1.29128	0.645639	+	0.763642i	$0.276591\pi$
$810$	0	0
$811$	−3.38751e6	−0.180854	−0.0904271	−	0.995903i	$-0.528823\pi$
$811$	−3.38751e6	−0.180854	−0.0904271	+	0.995903i	$0.528823\pi$
$812$	2.71162e7	1.44324
$813$	0	0
$814$	0	0
$815$	5.13514e6	0.270806
$816$	0	0
$817$	−1.22184e7	−0.640410
$818$	−2.23936e7	−1.17015
$819$	0	0
$820$	−9.25042e6	−0.480426
$821$	2.68551e6	0.139049	0.0695247	−	0.997580i	$-0.477852\pi$
$821$	2.68551e6	0.139049	0.0695247	+	0.997580i	$0.477852\pi$
$822$	0	0
$823$	1.52561e7	0.785134	0.392567	−	0.919723i	$-0.371587\pi$
$823$	1.52561e7	0.785134	0.392567	+	0.919723i	$0.371587\pi$
$824$	1.25074e7	0.641727
$825$	0	0
$826$	1.77682e7	0.906134
$827$	2.51270e7	1.27755	0.638773	−	0.769395i	$-0.279442\pi$
$827$	2.51270e7	1.27755	0.638773	+	0.769395i	$0.279442\pi$
$828$	0	0
$829$	3.00676e7	1.51954	0.759770	−	0.650191i	$-0.225311\pi$
$829$	3.00676e7	1.51954	0.759770	+	0.650191i	$0.225311\pi$
$830$	0	0
$831$	0	0
$832$	1.63474e7	0.818731
$833$	−1.40173e7	−0.699927
$834$	0	0
$835$	1.02211e7	0.507320
$836$	0	0
$837$	0	0
$838$	3.65105e6	0.179600
$839$	−3.77681e7	−1.85234	−0.926170	−	0.377107i	$-0.876919\pi$
$839$	−3.77681e7	−1.85234	−0.926170	+	0.377107i	$0.876919\pi$
$840$	0	0
$841$	3.85634e7	1.88012
$842$	3.92647e7	1.90863
$843$	0	0
$844$	−1.08124e7	−0.522477
$845$	6.66377e6	0.321054
$846$	0	0
$847$	0	0
$848$	−4.14164e6	−0.197780
$849$	0	0
$850$	2.76668e7	1.31345
$851$	1.59782e7	0.756317
$852$	0	0
$853$	254934.	0.0119965	0.00599826	−	0.999982i	$-0.498091\pi$
$853$	254934.	0.0119965	0.00599826	+	0.999982i	$0.498091\pi$
$854$	−3.42572e7	−1.60734
$855$	0	0
$856$	8.22344e6	0.383592
$857$	4.24942e6	0.197641	0.0988207	−	0.995105i	$-0.468493\pi$
$857$	4.24942e6	0.197641	0.0988207	+	0.995105i	$0.468493\pi$
$858$	0	0
$859$	−2.24416e7	−1.03770	−0.518850	−	0.854865i	$-0.673640\pi$
$859$	−2.24416e7	−1.03770	−0.518850	+	0.854865i	$0.673640\pi$
$860$	1.85643e7	0.855920
$861$	0	0
$862$	−5.14454e7	−2.35819
$863$	−2.01011e6	−0.0918742	−0.0459371	−	0.998944i	$-0.514627\pi$
$863$	−2.01011e6	−0.0918742	−0.0459371	+	0.998944i	$0.514627\pi$
$864$	0	0
$865$	1.73625e7	0.788991
$866$	5.25735e7	2.38217
$867$	0	0
$868$	−1.91500e7	−0.862718
$869$	0	0
$870$	0	0
$871$	−7.65367e6	−0.341841
$872$	−3.26487e7	−1.45403
$873$	0	0
$874$	−3.22247e7	−1.42696
$875$	−9.80467e6	−0.432925
$876$	0	0
$877$	7.26086e6	0.318778	0.159389	−	0.987216i	$-0.449048\pi$
$877$	7.26086e6	0.318778	0.159389	+	0.987216i	$0.449048\pi$
$878$	3.54002e6	0.154978
$879$	0	0
$880$	0	0
$881$	4.04800e7	1.75712	0.878558	−	0.477636i	$-0.158506\pi$
$881$	4.04800e7	1.75712	0.878558	+	0.477636i	$0.158506\pi$
$882$	0	0
$883$	−2.00636e7	−0.865979	−0.432990	−	0.901399i	$-0.642541\pi$
$883$	−2.00636e7	−0.865979	−0.432990	+	0.901399i	$0.642541\pi$
$884$	−1.80828e7	−0.778277
$885$	0	0
$886$	−1.48734e7	−0.636542
$887$	−1.42543e7	−0.608325	−0.304163	−	0.952620i	$-0.598377\pi$
$887$	−1.42543e7	−0.608325	−0.304163	+	0.952620i	$0.598377\pi$
$888$	0	0
$889$	1.41705e7	0.601353
$890$	9.00072e6	0.380892
$891$	0	0
$892$	−6.27933e7	−2.64242
$893$	4.95515e6	0.207935
$894$	0	0
$895$	1.48827e7	0.621046
$896$	−1.93771e7	−0.806343
$897$	0	0
$898$	1.55879e7	0.645055
$899$	−4.17196e7	−1.72163
$900$	0	0
$901$	2.61509e7	1.07319
$902$	0	0
$903$	0	0
$904$	−2.16639e7	−0.881689
$905$	−1.40245e7	−0.569201
$906$	0	0
$907$	−3.25752e7	−1.31483	−0.657413	−	0.753530i	$-0.728349\pi$
$907$	−3.25752e7	−1.31483	−0.657413	+	0.753530i	$0.728349\pi$
$908$	−7.12850e7	−2.86935
$909$	0	0
$910$	4.75891e6	0.190504
$911$	−7.00624e6	−0.279698	−0.139849	−	0.990173i	$-0.544662\pi$
$911$	−7.00624e6	−0.279698	−0.139849	+	0.990173i	$0.544662\pi$
$912$	0	0
$913$	0	0
$914$	−5.09840e7	−2.01868
$915$	0	0
$916$	−214718.	−0.00845531
$917$	4.92739e6	0.193506
$918$	0	0
$919$	2.17803e7	0.850695	0.425348	−	0.905030i	$-0.360152\pi$
$919$	2.17803e7	0.850695	0.425348	+	0.905030i	$0.360152\pi$
$920$	1.69867e7	0.661666
$921$	0	0
$922$	−7.45184e6	−0.288693
$923$	1.99077e7	0.769162
$924$	0	0
$925$	−8.80425e6	−0.338328
$926$	2.91829e7	1.11841
$927$	0	0
$928$	−5.08429e7	−1.93803
$929$	4.88614e7	1.85749	0.928745	−	0.370720i	$-0.120889\pi$
$929$	4.88614e7	1.85749	0.928745	+	0.370720i	$0.120889\pi$
$930$	0	0
$931$	−8.99620e6	−0.340161
$932$	5.25434e7	1.98143
$933$	0	0
$934$	−5.66473e7	−2.12477
$935$	0	0
$936$	0	0
$937$	1.16064e6	0.0431866	0.0215933	−	0.999767i	$-0.493126\pi$
$937$	1.16064e6	0.0431866	0.0215933	+	0.999767i	$0.493126\pi$
$938$	1.62078e7	0.601473
$939$	0	0
$940$	−7.52875e6	−0.277909
$941$	4.50576e7	1.65880	0.829400	−	0.558656i	$-0.188683\pi$
$941$	4.50576e7	1.65880	0.829400	+	0.558656i	$0.188683\pi$
$942$	0	0
$943$	3.63881e7	1.33254
$944$	−5.23722e6	−0.191281
$945$	0	0
$946$	0	0
$947$	1.06091e7	0.384418	0.192209	−	0.981354i	$-0.438435\pi$
$947$	1.06091e7	0.384418	0.192209	+	0.981354i	$0.438435\pi$
$948$	0	0
$949$	−8.16286e6	−0.294223
$950$	1.77563e7	0.638329
$951$	0	0
$952$	1.32853e7	0.475093
$953$	−3.63059e7	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$953$	−3.63059e7	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$954$	0	0
$955$	4.95144e6	0.175680
$956$	−4.06038e7	−1.43688
$957$	0	0
$958$	−6.55977e7	−2.30927
$959$	−7.69003e6	−0.270011
$960$	0	0
$961$	834033.	0.0291323
$962$	9.51232e6	0.331397
$963$	0	0
$964$	2.94588e7	1.02099
$965$	2.19672e7	0.759375
$966$	0	0
$967$	−2.10387e7	−0.723524	−0.361762	−	0.932270i	$-0.617825\pi$
$967$	−2.10387e7	−0.723524	−0.361762	+	0.932270i	$0.617825\pi$
$968$	0	0
$969$	0	0
$970$	8.78990e6	0.299954
$971$	−3.58180e7	−1.21914	−0.609570	−	0.792732i	$-0.708658\pi$
$971$	−3.58180e7	−1.21914	−0.609570	+	0.792732i	$0.708658\pi$
$972$	0	0
$973$	2.60872e7	0.883375
$974$	1.81413e7	0.612731
$975$	0	0
$976$	1.00974e7	0.339301
$977$	1.22823e6	0.0411664	0.0205832	−	0.999788i	$-0.493448\pi$
$977$	1.22823e6	0.0411664	0.0205832	+	0.999788i	$0.493448\pi$
$978$	0	0
$979$	0	0
$980$	1.36686e7	0.454632
$981$	0	0
$982$	−4.84736e6	−0.160408
$983$	−3.68929e6	−0.121775	−0.0608876	−	0.998145i	$-0.519393\pi$
$983$	−3.68929e6	−0.121775	−0.0608876	+	0.998145i	$0.519393\pi$
$984$	0	0
$985$	5.27947e6	0.173380
$986$	8.34238e7	2.73274
$987$	0	0
$988$	−1.16054e7	−0.378239
$989$	−7.30260e7	−2.37403
$990$	0	0
$991$	1.24253e7	0.401905	0.200952	−	0.979601i	$-0.435596\pi$
$991$	1.24253e7	0.401905	0.200952	+	0.979601i	$0.435596\pi$
$992$	3.59062e7	1.15849
$993$	0	0
$994$	−4.21576e7	−1.35335
$995$	−1.35133e7	−0.432718
$996$	0	0
$997$	−2.87328e7	−0.915462	−0.457731	−	0.889091i	$-0.651338\pi$
$997$	−2.87328e7	−0.915462	−0.457731	+	0.889091i	$0.651338\pi$
$998$	−4.64029e7	−1.47475
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.a.1.1		1
3.2	odd	2		363.6.a.e.1.1	yes	1
11.10	odd	2		1089.6.a.i.1.1		1
33.32	even	2		363.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.6.a.a.1.1	✓	1	33.32	even	2
363.6.a.e.1.1	yes	1	3.2	odd	2
1089.6.a.a.1.1		1	1.1	even	1	trivial
1089.6.a.i.1.1		1	11.10	odd	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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