LMFDB - Embedded newform 882.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.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:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +8.00000 q^{8} +O(q^{10})$

$q+2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +8.00000 q^{8} -12.0000 q^{10} +30.0000 q^{11} -2.00000 q^{13} +16.0000 q^{16} -66.0000 q^{17} +52.0000 q^{19} -24.0000 q^{20} +60.0000 q^{22} +114.000 q^{23} -89.0000 q^{25} -4.00000 q^{26} +72.0000 q^{29} +196.000 q^{31} +32.0000 q^{32} -132.000 q^{34} -286.000 q^{37} +104.000 q^{38} -48.0000 q^{40} +378.000 q^{41} +164.000 q^{43} +120.000 q^{44} +228.000 q^{46} +228.000 q^{47} -178.000 q^{50} -8.00000 q^{52} -348.000 q^{53} -180.000 q^{55} +144.000 q^{58} +348.000 q^{59} +106.000 q^{61} +392.000 q^{62} +64.0000 q^{64} +12.0000 q^{65} +596.000 q^{67} -264.000 q^{68} +630.000 q^{71} +1042.00 q^{73} -572.000 q^{74} +208.000 q^{76} -88.0000 q^{79} -96.0000 q^{80} +756.000 q^{82} +1440.00 q^{83} +396.000 q^{85} +328.000 q^{86} +240.000 q^{88} -1374.00 q^{89} +456.000 q^{92} +456.000 q^{94} -312.000 q^{95} +34.0000 q^{97} +O(q^{100})$

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	0	0
$10$	−12.0000	−0.379473
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	−2.00000	−0.0426692	−0.0213346	−	0.999772i	$-0.506792\pi$
$13$	−2.00000	−0.0426692	−0.0213346	+	0.999772i	$0.506792\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	52.0000	0.627875	0.313937	−	0.949444i	$-0.398352\pi$
$19$	52.0000	0.627875	0.313937	+	0.949444i	$0.398352\pi$
$20$	−24.0000	−0.268328
$21$	0	0
$22$	60.0000	0.581456
$23$	114.000	1.03351	0.516753	−	0.856134i	$-0.327141\pi$
$23$	114.000	1.03351	0.516753	+	0.856134i	$0.327141\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	−4.00000	−0.0301717
$27$	0	0
$28$	0	0
$29$	72.0000	0.461037	0.230518	−	0.973068i	$-0.425958\pi$
$29$	72.0000	0.461037	0.230518	+	0.973068i	$0.425958\pi$
$30$	0	0
$31$	196.000	1.13557	0.567785	−	0.823177i	$-0.307801\pi$
$31$	196.000	1.13557	0.567785	+	0.823177i	$0.307801\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−132.000	−0.665818
$35$	0	0
$36$	0	0
$37$	−286.000	−1.27076	−0.635380	−	0.772200i	$-0.719156\pi$
$37$	−286.000	−1.27076	−0.635380	+	0.772200i	$0.719156\pi$
$38$	104.000	0.443974
$39$	0	0
$40$	−48.0000	−0.189737
$41$	378.000	1.43985	0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000	1.43985	0.719923	+	0.694054i	$0.244177\pi$
$42$	0	0
$43$	164.000	0.581622	0.290811	−	0.956780i	$-0.406075\pi$
$43$	164.000	0.581622	0.290811	+	0.956780i	$0.406075\pi$
$44$	120.000	0.411152
$45$	0	0
$46$	228.000	0.730799
$47$	228.000	0.707600	0.353800	−	0.935321i	$-0.384889\pi$
$47$	228.000	0.707600	0.353800	+	0.935321i	$0.384889\pi$
$48$	0	0
$49$	0	0
$50$	−178.000	−0.503460
$51$	0	0
$52$	−8.00000	−0.0213346
$53$	−348.000	−0.901915	−0.450957	−	0.892546i	$-0.648917\pi$
$53$	−348.000	−0.901915	−0.450957	+	0.892546i	$0.648917\pi$
$54$	0	0
$55$	−180.000	−0.441294
$56$	0	0
$57$	0	0
$58$	144.000	0.326002
$59$	348.000	0.767894	0.383947	−	0.923355i	$-0.374565\pi$
$59$	348.000	0.767894	0.383947	+	0.923355i	$0.374565\pi$
$60$	0	0
$61$	106.000	0.222490	0.111245	−	0.993793i	$-0.464516\pi$
$61$	106.000	0.222490	0.111245	+	0.993793i	$0.464516\pi$
$62$	392.000	0.802969
$63$	0	0
$64$	64.0000	0.125000
$65$	12.0000	0.0228987
$66$	0	0
$67$	596.000	1.08676	0.543381	−	0.839487i	$-0.317144\pi$
$67$	596.000	1.08676	0.543381	+	0.839487i	$0.317144\pi$
$68$	−264.000	−0.470804
$69$	0	0
$70$	0	0
$71$	630.000	1.05306	0.526530	−	0.850157i	$-0.323493\pi$
$71$	630.000	1.05306	0.526530	+	0.850157i	$0.323493\pi$
$72$	0	0
$73$	1042.00	1.67064	0.835321	−	0.549762i	$-0.185282\pi$
$73$	1042.00	1.67064	0.835321	+	0.549762i	$0.185282\pi$
$74$	−572.000	−0.898563
$75$	0	0
$76$	208.000	0.313937
$77$	0	0
$78$	0	0
$79$	−88.0000	−0.125326	−0.0626631	−	0.998035i	$-0.519959\pi$
$79$	−88.0000	−0.125326	−0.0626631	+	0.998035i	$0.519959\pi$
$80$	−96.0000	−0.134164
$81$	0	0
$82$	756.000	1.01812
$83$	1440.00	1.90434	0.952172	−	0.305563i	$-0.0988446\pi$
$83$	1440.00	1.90434	0.952172	+	0.305563i	$0.0988446\pi$
$84$	0	0
$85$	396.000	0.505320
$86$	328.000	0.411269
$87$	0	0
$88$	240.000	0.290728
$89$	−1374.00	−1.63645	−0.818223	−	0.574901i	$-0.805041\pi$
$89$	−1374.00	−1.63645	−0.818223	+	0.574901i	$0.805041\pi$
$90$	0	0
$91$	0	0
$92$	456.000	0.516753
$93$	0	0
$94$	456.000	0.500349
$95$	−312.000	−0.336953
$96$	0	0
$97$	34.0000	0.0355895	0.0177947	−	0.999842i	$-0.494335\pi$
$97$	34.0000	0.0355895	0.0177947	+	0.999842i	$0.494335\pi$
$98$	0	0
$99$	0	0
$100$	−356.000	−0.356000
$101$	438.000	0.431511	0.215756	−	0.976447i	$-0.430779\pi$
$101$	438.000	0.431511	0.215756	+	0.976447i	$0.430779\pi$
$102$	0	0
$103$	−1676.00	−1.60331	−0.801656	−	0.597785i	$-0.796048\pi$
$103$	−1676.00	−1.60331	−0.801656	+	0.597785i	$0.796048\pi$
$104$	−16.0000	−0.0150859
$105$	0	0
$106$	−696.000	−0.637750
$107$	2022.00	1.82686	0.913430	−	0.406995i	$-0.133423\pi$
$107$	2022.00	1.82686	0.913430	+	0.406995i	$0.133423\pi$
$108$	0	0
$109$	−502.000	−0.441127	−0.220564	−	0.975373i	$-0.570790\pi$
$109$	−502.000	−0.441127	−0.220564	+	0.975373i	$0.570790\pi$
$110$	−360.000	−0.312042
$111$	0	0
$112$	0	0
$113$	2016.00	1.67831	0.839156	−	0.543890i	$-0.183049\pi$
$113$	2016.00	1.67831	0.839156	+	0.543890i	$0.183049\pi$
$114$	0	0
$115$	−684.000	−0.554638
$116$	288.000	0.230518
$117$	0	0
$118$	696.000	0.542983
$119$	0	0
$120$	0	0
$121$	−431.000	−0.323817
$122$	212.000	0.157324
$123$	0	0
$124$	784.000	0.567785
$125$	1284.00	0.918756
$126$	0	0
$127$	1784.00	1.24649	0.623246	−	0.782026i	$-0.285814\pi$
$127$	1784.00	1.24649	0.623246	+	0.782026i	$0.285814\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	24.0000	0.0161918
$131$	−1608.00	−1.07246	−0.536228	−	0.844073i	$-0.680151\pi$
$131$	−1608.00	−1.07246	−0.536228	+	0.844073i	$0.680151\pi$
$132$	0	0
$133$	0	0
$134$	1192.00	0.768456
$135$	0	0
$136$	−528.000	−0.332909
$137$	−2580.00	−1.60894	−0.804468	−	0.593996i	$-0.797550\pi$
$137$	−2580.00	−1.60894	−0.804468	+	0.593996i	$0.797550\pi$
$138$	0	0
$139$	−2144.00	−1.30829	−0.654143	−	0.756371i	$-0.726970\pi$
$139$	−2144.00	−1.30829	−0.654143	+	0.756371i	$0.726970\pi$
$140$	0	0
$141$	0	0
$142$	1260.00	0.744626
$143$	−60.0000	−0.0350871
$144$	0	0
$145$	−432.000	−0.247418
$146$	2084.00	1.18132
$147$	0	0
$148$	−1144.00	−0.635380
$149$	1500.00	0.824730	0.412365	−	0.911019i	$-0.364703\pi$
$149$	1500.00	0.824730	0.412365	+	0.911019i	$0.364703\pi$
$150$	0	0
$151$	−1240.00	−0.668277	−0.334138	−	0.942524i	$-0.608445\pi$
$151$	−1240.00	−0.668277	−0.334138	+	0.942524i	$0.608445\pi$
$152$	416.000	0.221987
$153$	0	0
$154$	0	0
$155$	−1176.00	−0.609410
$156$	0	0
$157$	−614.000	−0.312118	−0.156059	−	0.987748i	$-0.549879\pi$
$157$	−614.000	−0.312118	−0.156059	+	0.987748i	$0.549879\pi$
$158$	−176.000	−0.0886190
$159$	0	0
$160$	−192.000	−0.0948683
$161$	0	0
$162$	0	0
$163$	92.0000	0.0442086	0.0221043	−	0.999756i	$-0.492963\pi$
$163$	92.0000	0.0442086	0.0221043	+	0.999756i	$0.492963\pi$
$164$	1512.00	0.719923
$165$	0	0
$166$	2880.00	1.34657
$167$	3924.00	1.81825	0.909126	−	0.416520i	$-0.136750\pi$
$167$	3924.00	1.81825	0.909126	+	0.416520i	$0.136750\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	792.000	0.357315
$171$	0	0
$172$	656.000	0.290811
$173$	1902.00	0.835875	0.417938	−	0.908476i	$-0.362753\pi$
$173$	1902.00	0.835875	0.417938	+	0.908476i	$0.362753\pi$
$174$	0	0
$175$	0	0
$176$	480.000	0.205576
$177$	0	0
$178$	−2748.00	−1.15714
$179$	−6.00000	−0.00250537	−0.00125268	−	0.999999i	$-0.500399\pi$
$179$	−6.00000	−0.00250537	−0.00125268	+	0.999999i	$0.500399\pi$
$180$	0	0
$181$	2878.00	1.18188	0.590939	−	0.806716i	$-0.298757\pi$
$181$	2878.00	1.18188	0.590939	+	0.806716i	$0.298757\pi$
$182$	0	0
$183$	0	0
$184$	912.000	0.365400
$185$	1716.00	0.681961
$186$	0	0
$187$	−1980.00	−0.774288
$188$	912.000	0.353800
$189$	0	0
$190$	−624.000	−0.238262
$191$	−354.000	−0.134108	−0.0670538	−	0.997749i	$-0.521360\pi$
$191$	−354.000	−0.134108	−0.0670538	+	0.997749i	$0.521360\pi$
$192$	0	0
$193$	−4858.00	−1.81185	−0.905924	−	0.423441i	$-0.860822\pi$
$193$	−4858.00	−1.81185	−0.905924	+	0.423441i	$0.860822\pi$
$194$	68.0000	0.0251655
$195$	0	0
$196$	0	0
$197$	−396.000	−0.143217	−0.0716087	−	0.997433i	$-0.522813\pi$
$197$	−396.000	−0.143217	−0.0716087	+	0.997433i	$0.522813\pi$
$198$	0	0
$199$	−1712.00	−0.609852	−0.304926	−	0.952376i	$-0.598632\pi$
$199$	−1712.00	−0.609852	−0.304926	+	0.952376i	$0.598632\pi$
$200$	−712.000	−0.251730
$201$	0	0
$202$	876.000	0.305124
$203$	0	0
$204$	0	0
$205$	−2268.00	−0.772702
$206$	−3352.00	−1.13371
$207$	0	0
$208$	−32.0000	−0.0106673
$209$	1560.00	0.516304
$210$	0	0
$211$	−772.000	−0.251880	−0.125940	−	0.992038i	$-0.540195\pi$
$211$	−772.000	−0.251880	−0.125940	+	0.992038i	$0.540195\pi$
$212$	−1392.00	−0.450957
$213$	0	0
$214$	4044.00	1.29179
$215$	−984.000	−0.312131
$216$	0	0
$217$	0	0
$218$	−1004.00	−0.311924
$219$	0	0
$220$	−720.000	−0.220647
$221$	132.000	0.0401777
$222$	0	0
$223$	−776.000	−0.233026	−0.116513	−	0.993189i	$-0.537172\pi$
$223$	−776.000	−0.233026	−0.116513	+	0.993189i	$0.537172\pi$
$224$	0	0
$225$	0	0
$226$	4032.00	1.18675
$227$	1788.00	0.522792	0.261396	−	0.965232i	$-0.415817\pi$
$227$	1788.00	0.522792	0.261396	+	0.965232i	$0.415817\pi$
$228$	0	0
$229$	−5402.00	−1.55884	−0.779420	−	0.626502i	$-0.784486\pi$
$229$	−5402.00	−1.55884	−0.779420	+	0.626502i	$0.784486\pi$
$230$	−1368.00	−0.392188
$231$	0	0
$232$	576.000	0.163001
$233$	3012.00	0.846878	0.423439	−	0.905925i	$-0.360823\pi$
$233$	3012.00	0.846878	0.423439	+	0.905925i	$0.360823\pi$
$234$	0	0
$235$	−1368.00	−0.379738
$236$	1392.00	0.383947
$237$	0	0
$238$	0	0
$239$	3546.00	0.959714	0.479857	−	0.877347i	$-0.340689\pi$
$239$	3546.00	0.959714	0.479857	+	0.877347i	$0.340689\pi$
$240$	0	0
$241$	3562.00	0.952069	0.476034	−	0.879427i	$-0.342074\pi$
$241$	3562.00	0.952069	0.476034	+	0.879427i	$0.342074\pi$
$242$	−862.000	−0.228973
$243$	0	0
$244$	424.000	0.111245
$245$	0	0
$246$	0	0
$247$	−104.000	−0.0267909
$248$	1568.00	0.401484
$249$	0	0
$250$	2568.00	0.649658
$251$	−3348.00	−0.841928	−0.420964	−	0.907077i	$-0.638308\pi$
$251$	−3348.00	−0.841928	−0.420964	+	0.907077i	$0.638308\pi$
$252$	0	0
$253$	3420.00	0.849856
$254$	3568.00	0.881402
$255$	0	0
$256$	256.000	0.0625000
$257$	−366.000	−0.0888344	−0.0444172	−	0.999013i	$-0.514143\pi$
$257$	−366.000	−0.0888344	−0.0444172	+	0.999013i	$0.514143\pi$
$258$	0	0
$259$	0	0
$260$	48.0000	0.0114494
$261$	0	0
$262$	−3216.00	−0.758340
$263$	4170.00	0.977693	0.488846	−	0.872370i	$-0.337418\pi$
$263$	4170.00	0.977693	0.488846	+	0.872370i	$0.337418\pi$
$264$	0	0
$265$	2088.00	0.484018
$266$	0	0
$267$	0	0
$268$	2384.00	0.543381
$269$	−6078.00	−1.37763	−0.688814	−	0.724938i	$-0.741869\pi$
$269$	−6078.00	−1.37763	−0.688814	+	0.724938i	$0.741869\pi$
$270$	0	0
$271$	−2468.00	−0.553212	−0.276606	−	0.960983i	$-0.589210\pi$
$271$	−2468.00	−0.553212	−0.276606	+	0.960983i	$0.589210\pi$
$272$	−1056.00	−0.235402
$273$	0	0
$274$	−5160.00	−1.13769
$275$	−2670.00	−0.585480
$276$	0	0
$277$	−394.000	−0.0854627	−0.0427313	−	0.999087i	$-0.513606\pi$
$277$	−394.000	−0.0854627	−0.0427313	+	0.999087i	$0.513606\pi$
$278$	−4288.00	−0.925098
$279$	0	0
$280$	0	0
$281$	−396.000	−0.0840690	−0.0420345	−	0.999116i	$-0.513384\pi$
$281$	−396.000	−0.0840690	−0.0420345	+	0.999116i	$0.513384\pi$
$282$	0	0
$283$	1348.00	0.283146	0.141573	−	0.989928i	$-0.454784\pi$
$283$	1348.00	0.283146	0.141573	+	0.989928i	$0.454784\pi$
$284$	2520.00	0.526530
$285$	0	0
$286$	−120.000	−0.0248103
$287$	0	0
$288$	0	0
$289$	−557.000	−0.113373
$290$	−864.000	−0.174951
$291$	0	0
$292$	4168.00	0.835321
$293$	−7506.00	−1.49660	−0.748302	−	0.663358i	$-0.769131\pi$
$293$	−7506.00	−1.49660	−0.748302	+	0.663358i	$0.769131\pi$
$294$	0	0
$295$	−2088.00	−0.412095
$296$	−2288.00	−0.449281
$297$	0	0
$298$	3000.00	0.583172
$299$	−228.000	−0.0440989
$300$	0	0
$301$	0	0
$302$	−2480.00	−0.472543
$303$	0	0
$304$	832.000	0.156969
$305$	−636.000	−0.119401
$306$	0	0
$307$	−1748.00	−0.324963	−0.162481	−	0.986712i	$-0.551950\pi$
$307$	−1748.00	−0.324963	−0.162481	+	0.986712i	$0.551950\pi$
$308$	0	0
$309$	0	0
$310$	−2352.00	−0.430918
$311$	1140.00	0.207857	0.103928	−	0.994585i	$-0.466859\pi$
$311$	1140.00	0.207857	0.103928	+	0.994585i	$0.466859\pi$
$312$	0	0
$313$	−146.000	−0.0263655	−0.0131828	−	0.999913i	$-0.504196\pi$
$313$	−146.000	−0.0263655	−0.0131828	+	0.999913i	$0.504196\pi$
$314$	−1228.00	−0.220701
$315$	0	0
$316$	−352.000	−0.0626631
$317$	−8148.00	−1.44365	−0.721825	−	0.692075i	$-0.756697\pi$
$317$	−8148.00	−1.44365	−0.721825	+	0.692075i	$0.756697\pi$
$318$	0	0
$319$	2160.00	0.379112
$320$	−384.000	−0.0670820
$321$	0	0
$322$	0	0
$323$	−3432.00	−0.591212
$324$	0	0
$325$	178.000	0.0303805
$326$	184.000	0.0312602
$327$	0	0
$328$	3024.00	0.509062
$329$	0	0
$330$	0	0
$331$	−9700.00	−1.61076	−0.805378	−	0.592762i	$-0.798038\pi$
$331$	−9700.00	−1.61076	−0.805378	+	0.592762i	$0.798038\pi$
$332$	5760.00	0.952172
$333$	0	0
$334$	7848.00	1.28570
$335$	−3576.00	−0.583217
$336$	0	0
$337$	8174.00	1.32126	0.660632	−	0.750710i	$-0.270288\pi$
$337$	8174.00	1.32126	0.660632	+	0.750710i	$0.270288\pi$
$338$	−4386.00	−0.705819
$339$	0	0
$340$	1584.00	0.252660
$341$	5880.00	0.933783
$342$	0	0
$343$	0	0
$344$	1312.00	0.205635
$345$	0	0
$346$	3804.00	0.591053
$347$	−4038.00	−0.624701	−0.312350	−	0.949967i	$-0.601116\pi$
$347$	−4038.00	−0.624701	−0.312350	+	0.949967i	$0.601116\pi$
$348$	0	0
$349$	−10766.0	−1.65126	−0.825631	−	0.564210i	$-0.809181\pi$
$349$	−10766.0	−1.65126	−0.825631	+	0.564210i	$0.809181\pi$
$350$	0	0
$351$	0	0
$352$	960.000	0.145364
$353$	−3666.00	−0.552752	−0.276376	−	0.961050i	$-0.589134\pi$
$353$	−3666.00	−0.552752	−0.276376	+	0.961050i	$0.589134\pi$
$354$	0	0
$355$	−3780.00	−0.565131
$356$	−5496.00	−0.818223
$357$	0	0
$358$	−12.0000	−0.00177156
$359$	−5106.00	−0.750653	−0.375326	−	0.926893i	$-0.622469\pi$
$359$	−5106.00	−0.750653	−0.375326	+	0.926893i	$0.622469\pi$
$360$	0	0
$361$	−4155.00	−0.605773
$362$	5756.00	0.835714
$363$	0	0
$364$	0	0
$365$	−6252.00	−0.896561
$366$	0	0
$367$	5776.00	0.821539	0.410769	−	0.911739i	$-0.365260\pi$
$367$	5776.00	0.821539	0.410769	+	0.911739i	$0.365260\pi$
$368$	1824.00	0.258377
$369$	0	0
$370$	3432.00	0.482219
$371$	0	0
$372$	0	0
$373$	8462.00	1.17465	0.587327	−	0.809350i	$-0.300180\pi$
$373$	8462.00	1.17465	0.587327	+	0.809350i	$0.300180\pi$
$374$	−3960.00	−0.547505
$375$	0	0
$376$	1824.00	0.250175
$377$	−144.000	−0.0196721
$378$	0	0
$379$	6860.00	0.929748	0.464874	−	0.885377i	$-0.346100\pi$
$379$	6860.00	0.929748	0.464874	+	0.885377i	$0.346100\pi$
$380$	−1248.00	−0.168476
$381$	0	0
$382$	−708.000	−0.0948284
$383$	696.000	0.0928562	0.0464281	−	0.998922i	$-0.485216\pi$
$383$	696.000	0.0928562	0.0464281	+	0.998922i	$0.485216\pi$
$384$	0	0
$385$	0	0
$386$	−9716.00	−1.28117
$387$	0	0
$388$	136.000	0.0177947
$389$	11136.0	1.45146	0.725730	−	0.687980i	$-0.241502\pi$
$389$	11136.0	1.45146	0.725730	+	0.687980i	$0.241502\pi$
$390$	0	0
$391$	−7524.00	−0.973159
$392$	0	0
$393$	0	0
$394$	−792.000	−0.101270
$395$	528.000	0.0672571
$396$	0	0
$397$	−10838.0	−1.37014	−0.685068	−	0.728480i	$-0.740227\pi$
$397$	−10838.0	−1.37014	−0.685068	+	0.728480i	$0.740227\pi$
$398$	−3424.00	−0.431230
$399$	0	0
$400$	−1424.00	−0.178000
$401$	−8364.00	−1.04159	−0.520796	−	0.853681i	$-0.674365\pi$
$401$	−8364.00	−1.04159	−0.520796	+	0.853681i	$0.674365\pi$
$402$	0	0
$403$	−392.000	−0.0484539
$404$	1752.00	0.215756
$405$	0	0
$406$	0	0
$407$	−8580.00	−1.04495
$408$	0	0
$409$	1762.00	0.213020	0.106510	−	0.994312i	$-0.466032\pi$
$409$	1762.00	0.213020	0.106510	+	0.994312i	$0.466032\pi$
$410$	−4536.00	−0.546383
$411$	0	0
$412$	−6704.00	−0.801656
$413$	0	0
$414$	0	0
$415$	−8640.00	−1.02198
$416$	−64.0000	−0.00754293
$417$	0	0
$418$	3120.00	0.365082
$419$	−14580.0	−1.69995	−0.849976	−	0.526822i	$-0.823383\pi$
$419$	−14580.0	−1.69995	−0.849976	+	0.526822i	$0.823383\pi$
$420$	0	0
$421$	8534.00	0.987938	0.493969	−	0.869480i	$-0.335546\pi$
$421$	8534.00	0.987938	0.493969	+	0.869480i	$0.335546\pi$
$422$	−1544.00	−0.178106
$423$	0	0
$424$	−2784.00	−0.318875
$425$	5874.00	0.670426
$426$	0	0
$427$	0	0
$428$	8088.00	0.913430
$429$	0	0
$430$	−1968.00	−0.220710
$431$	5934.00	0.663180	0.331590	−	0.943424i	$-0.392415\pi$
$431$	5934.00	0.663180	0.331590	+	0.943424i	$0.392415\pi$
$432$	0	0
$433$	14758.0	1.63793	0.818966	−	0.573843i	$-0.194548\pi$
$433$	14758.0	1.63793	0.818966	+	0.573843i	$0.194548\pi$
$434$	0	0
$435$	0	0
$436$	−2008.00	−0.220564
$437$	5928.00	0.648912
$438$	0	0
$439$	11392.0	1.23852	0.619260	−	0.785186i	$-0.287433\pi$
$439$	11392.0	1.23852	0.619260	+	0.785186i	$0.287433\pi$
$440$	−1440.00	−0.156021
$441$	0	0
$442$	264.000	0.0284100
$443$	7026.00	0.753533	0.376767	−	0.926308i	$-0.377036\pi$
$443$	7026.00	0.753533	0.376767	+	0.926308i	$0.377036\pi$
$444$	0	0
$445$	8244.00	0.878209
$446$	−1552.00	−0.164774
$447$	0	0
$448$	0	0
$449$	−3384.00	−0.355681	−0.177841	−	0.984059i	$-0.556911\pi$
$449$	−3384.00	−0.355681	−0.177841	+	0.984059i	$0.556911\pi$
$450$	0	0
$451$	11340.0	1.18399
$452$	8064.00	0.839156
$453$	0	0
$454$	3576.00	0.369670
$455$	0	0
$456$	0	0
$457$	−4282.00	−0.438301	−0.219150	−	0.975691i	$-0.570329\pi$
$457$	−4282.00	−0.438301	−0.219150	+	0.975691i	$0.570329\pi$
$458$	−10804.0	−1.10227
$459$	0	0
$460$	−2736.00	−0.277319
$461$	−16650.0	−1.68214	−0.841071	−	0.540924i	$-0.818075\pi$
$461$	−16650.0	−1.68214	−0.841071	+	0.540924i	$0.818075\pi$
$462$	0	0
$463$	−9664.00	−0.970031	−0.485015	−	0.874506i	$-0.661186\pi$
$463$	−9664.00	−0.970031	−0.485015	+	0.874506i	$0.661186\pi$
$464$	1152.00	0.115259
$465$	0	0
$466$	6024.00	0.598834
$467$	12324.0	1.22117	0.610585	−	0.791950i	$-0.290934\pi$
$467$	12324.0	1.22117	0.610585	+	0.791950i	$0.290934\pi$
$468$	0	0
$469$	0	0
$470$	−2736.00	−0.268515
$471$	0	0
$472$	2784.00	0.271491
$473$	4920.00	0.478270
$474$	0	0
$475$	−4628.00	−0.447047
$476$	0	0
$477$	0	0
$478$	7092.00	0.678620
$479$	−18660.0	−1.77995	−0.889976	−	0.456007i	$-0.849279\pi$
$479$	−18660.0	−1.77995	−0.889976	+	0.456007i	$0.849279\pi$
$480$	0	0
$481$	572.000	0.0542224
$482$	7124.00	0.673214
$483$	0	0
$484$	−1724.00	−0.161908
$485$	−204.000	−0.0190993
$486$	0	0
$487$	−3400.00	−0.316363	−0.158181	−	0.987410i	$-0.550563\pi$
$487$	−3400.00	−0.316363	−0.158181	+	0.987410i	$0.550563\pi$
$488$	848.000	0.0786622
$489$	0	0
$490$	0	0
$491$	2970.00	0.272982	0.136491	−	0.990641i	$-0.456418\pi$
$491$	2970.00	0.272982	0.136491	+	0.990641i	$0.456418\pi$
$492$	0	0
$493$	−4752.00	−0.434116
$494$	−208.000	−0.0189441
$495$	0	0
$496$	3136.00	0.283892
$497$	0	0
$498$	0	0
$499$	−988.000	−0.0886352	−0.0443176	−	0.999017i	$-0.514111\pi$
$499$	−988.000	−0.0886352	−0.0443176	+	0.999017i	$0.514111\pi$
$500$	5136.00	0.459378
$501$	0	0
$502$	−6696.00	−0.595333
$503$	−5184.00	−0.459529	−0.229765	−	0.973246i	$-0.573796\pi$
$503$	−5184.00	−0.459529	−0.229765	+	0.973246i	$0.573796\pi$
$504$	0	0
$505$	−2628.00	−0.231573
$506$	6840.00	0.600939
$507$	0	0
$508$	7136.00	0.623246
$509$	−16854.0	−1.46766	−0.733831	−	0.679332i	$-0.762270\pi$
$509$	−16854.0	−1.46766	−0.733831	+	0.679332i	$0.762270\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	0	0
$514$	−732.000	−0.0628154
$515$	10056.0	0.860428
$516$	0	0
$517$	6840.00	0.581862
$518$	0	0
$519$	0	0
$520$	96.0000	0.00809592
$521$	4398.00	0.369827	0.184914	−	0.982755i	$-0.440800\pi$
$521$	4398.00	0.369827	0.184914	+	0.982755i	$0.440800\pi$
$522$	0	0
$523$	10672.0	0.892264	0.446132	−	0.894967i	$-0.352801\pi$
$523$	10672.0	0.892264	0.446132	+	0.894967i	$0.352801\pi$
$524$	−6432.00	−0.536228
$525$	0	0
$526$	8340.00	0.691333
$527$	−12936.0	−1.06926
$528$	0	0
$529$	829.000	0.0681351
$530$	4176.00	0.342253
$531$	0	0
$532$	0	0
$533$	−756.000	−0.0614371
$534$	0	0
$535$	−12132.0	−0.980396
$536$	4768.00	0.384228
$537$	0	0
$538$	−12156.0	−0.974131
$539$	0	0
$540$	0	0
$541$	20702.0	1.64519	0.822596	−	0.568627i	$-0.192525\pi$
$541$	20702.0	1.64519	0.822596	+	0.568627i	$0.192525\pi$
$542$	−4936.00	−0.391180
$543$	0	0
$544$	−2112.00	−0.166455
$545$	3012.00	0.236734
$546$	0	0
$547$	−22876.0	−1.78813	−0.894065	−	0.447937i	$-0.852159\pi$
$547$	−22876.0	−1.78813	−0.894065	+	0.447937i	$0.852159\pi$
$548$	−10320.0	−0.804468
$549$	0	0
$550$	−5340.00	−0.413997
$551$	3744.00	0.289473
$552$	0	0
$553$	0	0
$554$	−788.000	−0.0604312
$555$	0	0
$556$	−8576.00	−0.654143
$557$	−12876.0	−0.979486	−0.489743	−	0.871867i	$-0.662909\pi$
$557$	−12876.0	−0.979486	−0.489743	+	0.871867i	$0.662909\pi$
$558$	0	0
$559$	−328.000	−0.0248174
$560$	0	0
$561$	0	0
$562$	−792.000	−0.0594457
$563$	6900.00	0.516519	0.258260	−	0.966076i	$-0.416851\pi$
$563$	6900.00	0.516519	0.258260	+	0.966076i	$0.416851\pi$
$564$	0	0
$565$	−12096.0	−0.900677
$566$	2696.00	0.200214
$567$	0	0
$568$	5040.00	0.372313
$569$	14676.0	1.08128	0.540641	−	0.841253i	$-0.318182\pi$
$569$	14676.0	1.08128	0.540641	+	0.841253i	$0.318182\pi$
$570$	0	0
$571$	380.000	0.0278503	0.0139251	−	0.999903i	$-0.495567\pi$
$571$	380.000	0.0278503	0.0139251	+	0.999903i	$0.495567\pi$
$572$	−240.000	−0.0175435
$573$	0	0
$574$	0	0
$575$	−10146.0	−0.735856
$576$	0	0
$577$	11806.0	0.851803	0.425901	−	0.904770i	$-0.359957\pi$
$577$	11806.0	0.851803	0.425901	+	0.904770i	$0.359957\pi$
$578$	−1114.00	−0.0801666
$579$	0	0
$580$	−1728.00	−0.123709
$581$	0	0
$582$	0	0
$583$	−10440.0	−0.741648
$584$	8336.00	0.590661
$585$	0	0
$586$	−15012.0	−1.05826
$587$	19188.0	1.34919	0.674594	−	0.738189i	$-0.264319\pi$
$587$	19188.0	1.34919	0.674594	+	0.738189i	$0.264319\pi$
$588$	0	0
$589$	10192.0	0.712995
$590$	−4176.00	−0.291395
$591$	0	0
$592$	−4576.00	−0.317690
$593$	−690.000	−0.0477823	−0.0238912	−	0.999715i	$-0.507606\pi$
$593$	−690.000	−0.0477823	−0.0238912	+	0.999715i	$0.507606\pi$
$594$	0	0
$595$	0	0
$596$	6000.00	0.412365
$597$	0	0
$598$	−456.000	−0.0311827
$599$	−20490.0	−1.39766	−0.698830	−	0.715287i	$-0.746296\pi$
$599$	−20490.0	−1.39766	−0.698830	+	0.715287i	$0.746296\pi$
$600$	0	0
$601$	11590.0	0.786632	0.393316	−	0.919403i	$-0.371328\pi$
$601$	11590.0	0.786632	0.393316	+	0.919403i	$0.371328\pi$
$602$	0	0
$603$	0	0
$604$	−4960.00	−0.334138
$605$	2586.00	0.173778
$606$	0	0
$607$	6424.00	0.429559	0.214779	−	0.976663i	$-0.431097\pi$
$607$	6424.00	0.429559	0.214779	+	0.976663i	$0.431097\pi$
$608$	1664.00	0.110994
$609$	0	0
$610$	−1272.00	−0.0844291
$611$	−456.000	−0.0301928
$612$	0	0
$613$	−9682.00	−0.637932	−0.318966	−	0.947766i	$-0.603336\pi$
$613$	−9682.00	−0.637932	−0.318966	+	0.947766i	$0.603336\pi$
$614$	−3496.00	−0.229784
$615$	0	0
$616$	0	0
$617$	5076.00	0.331203	0.165601	−	0.986193i	$-0.447044\pi$
$617$	5076.00	0.331203	0.165601	+	0.986193i	$0.447044\pi$
$618$	0	0
$619$	−22664.0	−1.47164	−0.735818	−	0.677179i	$-0.763202\pi$
$619$	−22664.0	−1.47164	−0.735818	+	0.677179i	$0.763202\pi$
$620$	−4704.00	−0.304705
$621$	0	0
$622$	2280.00	0.146977
$623$	0	0
$624$	0	0
$625$	3421.00	0.218944
$626$	−292.000	−0.0186432
$627$	0	0
$628$	−2456.00	−0.156059
$629$	18876.0	1.19656
$630$	0	0
$631$	−8584.00	−0.541559	−0.270779	−	0.962641i	$-0.587281\pi$
$631$	−8584.00	−0.541559	−0.270779	+	0.962641i	$0.587281\pi$
$632$	−704.000	−0.0443095
$633$	0	0
$634$	−16296.0	−1.02082
$635$	−10704.0	−0.668937
$636$	0	0
$637$	0	0
$638$	4320.00	0.268073
$639$	0	0
$640$	−768.000	−0.0474342
$641$	372.000	0.0229222	0.0114611	−	0.999934i	$-0.496352\pi$
$641$	372.000	0.0229222	0.0114611	+	0.999934i	$0.496352\pi$
$642$	0	0
$643$	−3188.00	−0.195525	−0.0977624	−	0.995210i	$-0.531169\pi$
$643$	−3188.00	−0.195525	−0.0977624	+	0.995210i	$0.531169\pi$
$644$	0	0
$645$	0	0
$646$	−6864.00	−0.418050
$647$	12732.0	0.773642	0.386821	−	0.922155i	$-0.373573\pi$
$647$	12732.0	0.773642	0.386821	+	0.922155i	$0.373573\pi$
$648$	0	0
$649$	10440.0	0.631442
$650$	356.000	0.0214823
$651$	0	0
$652$	368.000	0.0221043
$653$	−3576.00	−0.214303	−0.107151	−	0.994243i	$-0.534173\pi$
$653$	−3576.00	−0.214303	−0.107151	+	0.994243i	$0.534173\pi$
$654$	0	0
$655$	9648.00	0.575540
$656$	6048.00	0.359961
$657$	0	0
$658$	0	0
$659$	11430.0	0.675644	0.337822	−	0.941210i	$-0.390310\pi$
$659$	11430.0	0.675644	0.337822	+	0.941210i	$0.390310\pi$
$660$	0	0
$661$	−22646.0	−1.33257	−0.666284	−	0.745698i	$-0.732116\pi$
$661$	−22646.0	−1.33257	−0.666284	+	0.745698i	$0.732116\pi$
$662$	−19400.0	−1.13898
$663$	0	0
$664$	11520.0	0.673287
$665$	0	0
$666$	0	0
$667$	8208.00	0.476484
$668$	15696.0	0.909126
$669$	0	0
$670$	−7152.00	−0.412397
$671$	3180.00	0.182955
$672$	0	0
$673$	−13570.0	−0.777244	−0.388622	−	0.921397i	$-0.627049\pi$
$673$	−13570.0	−0.777244	−0.388622	+	0.921397i	$0.627049\pi$
$674$	16348.0	0.934275
$675$	0	0
$676$	−8772.00	−0.499090
$677$	2838.00	0.161113	0.0805563	−	0.996750i	$-0.474330\pi$
$677$	2838.00	0.161113	0.0805563	+	0.996750i	$0.474330\pi$
$678$	0	0
$679$	0	0
$680$	3168.00	0.178658
$681$	0	0
$682$	11760.0	0.660284
$683$	−6558.00	−0.367401	−0.183701	−	0.982982i	$-0.558808\pi$
$683$	−6558.00	−0.367401	−0.183701	+	0.982982i	$0.558808\pi$
$684$	0	0
$685$	15480.0	0.863446
$686$	0	0
$687$	0	0
$688$	2624.00	0.145406
$689$	696.000	0.0384840
$690$	0	0
$691$	21832.0	1.20192	0.600961	−	0.799278i	$-0.294785\pi$
$691$	21832.0	1.20192	0.600961	+	0.799278i	$0.294785\pi$
$692$	7608.00	0.417938
$693$	0	0
$694$	−8076.00	−0.441730
$695$	12864.0	0.702100
$696$	0	0
$697$	−24948.0	−1.35577
$698$	−21532.0	−1.16762
$699$	0	0
$700$	0	0
$701$	16200.0	0.872847	0.436423	−	0.899741i	$-0.356245\pi$
$701$	16200.0	0.872847	0.436423	+	0.899741i	$0.356245\pi$
$702$	0	0
$703$	−14872.0	−0.797878
$704$	1920.00	0.102788
$705$	0	0
$706$	−7332.00	−0.390855
$707$	0	0
$708$	0	0
$709$	36722.0	1.94517	0.972584	−	0.232553i	$-0.0747080\pi$
$709$	36722.0	1.94517	0.972584	+	0.232553i	$0.0747080\pi$
$710$	−7560.00	−0.399608
$711$	0	0
$712$	−10992.0	−0.578571
$713$	22344.0	1.17362
$714$	0	0
$715$	360.000	0.0188297
$716$	−24.0000	−0.00125268
$717$	0	0
$718$	−10212.0	−0.530792
$719$	−13776.0	−0.714545	−0.357273	−	0.934000i	$-0.616293\pi$
$719$	−13776.0	−0.714545	−0.357273	+	0.934000i	$0.616293\pi$
$720$	0	0
$721$	0	0
$722$	−8310.00	−0.428347
$723$	0	0
$724$	11512.0	0.590939
$725$	−6408.00	−0.328258
$726$	0	0
$727$	−34220.0	−1.74574	−0.872868	−	0.487957i	$-0.837742\pi$
$727$	−34220.0	−1.74574	−0.872868	+	0.487957i	$0.837742\pi$
$728$	0	0
$729$	0	0
$730$	−12504.0	−0.633964
$731$	−10824.0	−0.547661
$732$	0	0
$733$	13750.0	0.692862	0.346431	−	0.938075i	$-0.387393\pi$
$733$	13750.0	0.692862	0.346431	+	0.938075i	$0.387393\pi$
$734$	11552.0	0.580916
$735$	0	0
$736$	3648.00	0.182700
$737$	17880.0	0.893648
$738$	0	0
$739$	39836.0	1.98294	0.991469	−	0.130344i	$-0.0416081\pi$
$739$	39836.0	1.98294	0.991469	+	0.130344i	$0.0416081\pi$
$740$	6864.00	0.340981
$741$	0	0
$742$	0	0
$743$	34470.0	1.70199	0.850997	−	0.525170i	$-0.175998\pi$
$743$	34470.0	1.70199	0.850997	+	0.525170i	$0.175998\pi$
$744$	0	0
$745$	−9000.00	−0.442597
$746$	16924.0	0.830605
$747$	0	0
$748$	−7920.00	−0.387144
$749$	0	0
$750$	0	0
$751$	5240.00	0.254608	0.127304	−	0.991864i	$-0.459368\pi$
$751$	5240.00	0.254608	0.127304	+	0.991864i	$0.459368\pi$
$752$	3648.00	0.176900
$753$	0	0
$754$	−288.000	−0.0139103
$755$	7440.00	0.358635
$756$	0	0
$757$	18578.0	0.891980	0.445990	−	0.895038i	$-0.352852\pi$
$757$	18578.0	0.891980	0.445990	+	0.895038i	$0.352852\pi$
$758$	13720.0	0.657431
$759$	0	0
$760$	−2496.00	−0.119131
$761$	−30534.0	−1.45448	−0.727238	−	0.686385i	$-0.759196\pi$
$761$	−30534.0	−1.45448	−0.727238	+	0.686385i	$0.759196\pi$
$762$	0	0
$763$	0	0
$764$	−1416.00	−0.0670538
$765$	0	0
$766$	1392.00	0.0656593
$767$	−696.000	−0.0327655
$768$	0	0
$769$	39958.0	1.87376	0.936881	−	0.349650i	$-0.113699\pi$
$769$	39958.0	1.87376	0.936881	+	0.349650i	$0.113699\pi$
$770$	0	0
$771$	0	0
$772$	−19432.0	−0.905924
$773$	3966.00	0.184537	0.0922685	−	0.995734i	$-0.470588\pi$
$773$	3966.00	0.184537	0.0922685	+	0.995734i	$0.470588\pi$
$774$	0	0
$775$	−17444.0	−0.808525
$776$	272.000	0.0125828
$777$	0	0
$778$	22272.0	1.02634
$779$	19656.0	0.904043
$780$	0	0
$781$	18900.0	0.865935
$782$	−15048.0	−0.688127
$783$	0	0
$784$	0	0
$785$	3684.00	0.167500
$786$	0	0
$787$	3760.00	0.170304	0.0851522	−	0.996368i	$-0.472862\pi$
$787$	3760.00	0.170304	0.0851522	+	0.996368i	$0.472862\pi$
$788$	−1584.00	−0.0716087
$789$	0	0
$790$	1056.00	0.0475580
$791$	0	0
$792$	0	0
$793$	−212.000	−0.00949349
$794$	−21676.0	−0.968832
$795$	0	0
$796$	−6848.00	−0.304926
$797$	24102.0	1.07119	0.535594	−	0.844476i	$-0.320088\pi$
$797$	24102.0	1.07119	0.535594	+	0.844476i	$0.320088\pi$
$798$	0	0
$799$	−15048.0	−0.666283
$800$	−2848.00	−0.125865
$801$	0	0
$802$	−16728.0	−0.736516
$803$	31260.0	1.37378
$804$	0	0
$805$	0	0
$806$	−784.000	−0.0342621
$807$	0	0
$808$	3504.00	0.152562
$809$	11712.0	0.508989	0.254494	−	0.967074i	$-0.418091\pi$
$809$	11712.0	0.508989	0.254494	+	0.967074i	$0.418091\pi$
$810$	0	0
$811$	−37424.0	−1.62039	−0.810194	−	0.586162i	$-0.800638\pi$
$811$	−37424.0	−1.62039	−0.810194	+	0.586162i	$0.800638\pi$
$812$	0	0
$813$	0	0
$814$	−17160.0	−0.738892
$815$	−552.000	−0.0237248
$816$	0	0
$817$	8528.00	0.365186
$818$	3524.00	0.150628
$819$	0	0
$820$	−9072.00	−0.386351
$821$	13452.0	0.571837	0.285918	−	0.958254i	$-0.407701\pi$
$821$	13452.0	0.571837	0.285918	+	0.958254i	$0.407701\pi$
$822$	0	0
$823$	20432.0	0.865389	0.432694	−	0.901541i	$-0.357563\pi$
$823$	20432.0	0.865389	0.432694	+	0.901541i	$0.357563\pi$
$824$	−13408.0	−0.566857
$825$	0	0
$826$	0	0
$827$	24390.0	1.02554	0.512771	−	0.858525i	$-0.328619\pi$
$827$	24390.0	1.02554	0.512771	+	0.858525i	$0.328619\pi$
$828$	0	0
$829$	28510.0	1.19444	0.597221	−	0.802076i	$-0.296271\pi$
$829$	28510.0	1.19444	0.597221	+	0.802076i	$0.296271\pi$
$830$	−17280.0	−0.722648
$831$	0	0
$832$	−128.000	−0.00533366
$833$	0	0
$834$	0	0
$835$	−23544.0	−0.975777
$836$	6240.00	0.258152
$837$	0	0
$838$	−29160.0	−1.20205
$839$	36972.0	1.52135	0.760677	−	0.649131i	$-0.224867\pi$
$839$	36972.0	1.52135	0.760677	+	0.649131i	$0.224867\pi$
$840$	0	0
$841$	−19205.0	−0.787445
$842$	17068.0	0.698577
$843$	0	0
$844$	−3088.00	−0.125940
$845$	13158.0	0.535679
$846$	0	0
$847$	0	0
$848$	−5568.00	−0.225479
$849$	0	0
$850$	11748.0	0.474062
$851$	−32604.0	−1.31334
$852$	0	0
$853$	14074.0	0.564929	0.282465	−	0.959278i	$-0.408848\pi$
$853$	14074.0	0.564929	0.282465	+	0.959278i	$0.408848\pi$
$854$	0	0
$855$	0	0
$856$	16176.0	0.645893
$857$	8826.00	0.351797	0.175899	−	0.984408i	$-0.443717\pi$
$857$	8826.00	0.351797	0.175899	+	0.984408i	$0.443717\pi$
$858$	0	0
$859$	20500.0	0.814262	0.407131	−	0.913370i	$-0.366529\pi$
$859$	20500.0	0.814262	0.407131	+	0.913370i	$0.366529\pi$
$860$	−3936.00	−0.156066
$861$	0	0
$862$	11868.0	0.468939
$863$	7674.00	0.302695	0.151348	−	0.988481i	$-0.451639\pi$
$863$	7674.00	0.302695	0.151348	+	0.988481i	$0.451639\pi$
$864$	0	0
$865$	−11412.0	−0.448578
$866$	29516.0	1.15819
$867$	0	0
$868$	0	0
$869$	−2640.00	−0.103056
$870$	0	0
$871$	−1192.00	−0.0463713
$872$	−4016.00	−0.155962
$873$	0	0
$874$	11856.0	0.458850
$875$	0	0
$876$	0	0
$877$	−8890.00	−0.342296	−0.171148	−	0.985245i	$-0.554748\pi$
$877$	−8890.00	−0.342296	−0.171148	+	0.985245i	$0.554748\pi$
$878$	22784.0	0.875766
$879$	0	0
$880$	−2880.00	−0.110324
$881$	−738.000	−0.0282223	−0.0141112	−	0.999900i	$-0.504492\pi$
$881$	−738.000	−0.0282223	−0.0141112	+	0.999900i	$0.504492\pi$
$882$	0	0
$883$	20.0000	0.000762235 0	0.000381118	−	1.00000i	$-0.499879\pi$
$883$	20.0000	0.000762235 0	0.000381118		1.00000i	$0.499879\pi$
$884$	528.000	0.0200889
$885$	0	0
$886$	14052.0	0.532829
$887$	−39804.0	−1.50675	−0.753375	−	0.657591i	$-0.771576\pi$
$887$	−39804.0	−1.50675	−0.753375	+	0.657591i	$0.771576\pi$
$888$	0	0
$889$	0	0
$890$	16488.0	0.620988
$891$	0	0
$892$	−3104.00	−0.116513
$893$	11856.0	0.444284
$894$	0	0
$895$	36.0000	0.00134452
$896$	0	0
$897$	0	0
$898$	−6768.00	−0.251505
$899$	14112.0	0.523539
$900$	0	0
$901$	22968.0	0.849251
$902$	22680.0	0.837208
$903$	0	0
$904$	16128.0	0.593373
$905$	−17268.0	−0.634263
$906$	0	0
$907$	29180.0	1.06825	0.534127	−	0.845404i	$-0.320640\pi$
$907$	29180.0	1.06825	0.534127	+	0.845404i	$0.320640\pi$
$908$	7152.00	0.261396
$909$	0	0
$910$	0	0
$911$	−48258.0	−1.75506	−0.877530	−	0.479523i	$-0.840810\pi$
$911$	−48258.0	−1.75506	−0.877530	+	0.479523i	$0.840810\pi$
$912$	0	0
$913$	43200.0	1.56595
$914$	−8564.00	−0.309926
$915$	0	0
$916$	−21608.0	−0.779420
$917$	0	0
$918$	0	0
$919$	25760.0	0.924640	0.462320	−	0.886713i	$-0.347017\pi$
$919$	25760.0	0.924640	0.462320	+	0.886713i	$0.347017\pi$
$920$	−5472.00	−0.196094
$921$	0	0
$922$	−33300.0	−1.18945
$923$	−1260.00	−0.0449333
$924$	0	0
$925$	25454.0	0.904781
$926$	−19328.0	−0.685915
$927$	0	0
$928$	2304.00	0.0815005
$929$	44778.0	1.58140	0.790699	−	0.612205i	$-0.209717\pi$
$929$	44778.0	1.58140	0.790699	+	0.612205i	$0.209717\pi$
$930$	0	0
$931$	0	0
$932$	12048.0	0.423439
$933$	0	0
$934$	24648.0	0.863498
$935$	11880.0	0.415527
$936$	0	0
$937$	44494.0	1.55129	0.775643	−	0.631171i	$-0.217426\pi$
$937$	44494.0	1.55129	0.775643	+	0.631171i	$0.217426\pi$
$938$	0	0
$939$	0	0
$940$	−5472.00	−0.189869
$941$	7458.00	0.258368	0.129184	−	0.991621i	$-0.458764\pi$
$941$	7458.00	0.258368	0.129184	+	0.991621i	$0.458764\pi$
$942$	0	0
$943$	43092.0	1.48809
$944$	5568.00	0.191973
$945$	0	0
$946$	9840.00	0.338188
$947$	17790.0	0.610451	0.305226	−	0.952280i	$-0.401268\pi$
$947$	17790.0	0.610451	0.305226	+	0.952280i	$0.401268\pi$
$948$	0	0
$949$	−2084.00	−0.0712850
$950$	−9256.00	−0.316110
$951$	0	0
$952$	0	0
$953$	−5832.00	−0.198234	−0.0991170	−	0.995076i	$-0.531602\pi$
$953$	−5832.00	−0.198234	−0.0991170	+	0.995076i	$0.531602\pi$
$954$	0	0
$955$	2124.00	0.0719697
$956$	14184.0	0.479857
$957$	0	0
$958$	−37320.0	−1.25862
$959$	0	0
$960$	0	0
$961$	8625.00	0.289517
$962$	1144.00	0.0383410
$963$	0	0
$964$	14248.0	0.476034
$965$	29148.0	0.972339
$966$	0	0
$967$	−13264.0	−0.441098	−0.220549	−	0.975376i	$-0.570785\pi$
$967$	−13264.0	−0.441098	−0.220549	+	0.975376i	$0.570785\pi$
$968$	−3448.00	−0.114486
$969$	0	0
$970$	−408.000	−0.0135052
$971$	−3984.00	−0.131671	−0.0658356	−	0.997830i	$-0.520971\pi$
$971$	−3984.00	−0.131671	−0.0658356	+	0.997830i	$0.520971\pi$
$972$	0	0
$973$	0	0
$974$	−6800.00	−0.223702
$975$	0	0
$976$	1696.00	0.0556226
$977$	−38940.0	−1.27513	−0.637564	−	0.770397i	$-0.720058\pi$
$977$	−38940.0	−1.27513	−0.637564	+	0.770397i	$0.720058\pi$
$978$	0	0
$979$	−41220.0	−1.34566
$980$	0	0
$981$	0	0
$982$	5940.00	0.193028
$983$	8808.00	0.285790	0.142895	−	0.989738i	$-0.454359\pi$
$983$	8808.00	0.285790	0.142895	+	0.989738i	$0.454359\pi$
$984$	0	0
$985$	2376.00	0.0768585
$986$	−9504.00	−0.306967
$987$	0	0
$988$	−416.000	−0.0133955
$989$	18696.0	0.601110
$990$	0	0
$991$	18488.0	0.592624	0.296312	−	0.955091i	$-0.404243\pi$
$991$	18488.0	0.592624	0.296312	+	0.955091i	$0.404243\pi$
$992$	6272.00	0.200742
$993$	0	0
$994$	0	0
$995$	10272.0	0.327281
$996$	0	0
$997$	5002.00	0.158892	0.0794458	−	0.996839i	$-0.474685\pi$
$997$	5002.00	0.158892	0.0794458	+	0.996839i	$0.474685\pi$
$998$	−1976.00	−0.0626745
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.m.1.1		1
3.2	odd	2		882.4.a.e.1.1		1
7.2	even	3		882.4.g.h.361.1		2
7.3	odd	6		882.4.g.e.667.1		2
7.4	even	3		882.4.g.h.667.1		2
7.5	odd	6		882.4.g.e.361.1		2
7.6	odd	2		126.4.a.g.1.1	yes	1
21.2	odd	6		882.4.g.q.361.1		2
21.5	even	6		882.4.g.t.361.1		2
21.11	odd	6		882.4.g.q.667.1		2
21.17	even	6		882.4.g.t.667.1		2
21.20	even	2		126.4.a.b.1.1	✓	1
28.27	even	2		1008.4.a.n.1.1		1
84.83	odd	2		1008.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.a.b.1.1	✓	1	21.20	even	2
126.4.a.g.1.1	yes	1	7.6	odd	2
882.4.a.e.1.1		1	3.2	odd	2
882.4.a.m.1.1		1	1.1	even	1	trivial
882.4.g.e.361.1		2	7.5	odd	6
882.4.g.e.667.1		2	7.3	odd	6
882.4.g.h.361.1		2	7.2	even	3
882.4.g.h.667.1		2	7.4	even	3
882.4.g.q.361.1		2	21.2	odd	6
882.4.g.q.667.1		2	21.11	odd	6
882.4.g.t.361.1		2	21.5	even	6
882.4.g.t.667.1		2	21.17	even	6
1008.4.a.g.1.1		1	84.83	odd	2
1008.4.a.n.1.1		1	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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